Symmetries of the Standard Model without and with a Right-Handed Neutrino

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Geometry of the Standard Model

a r X i v :0803.2668v 1 [m a t h .D G ] 18 M a r 2008Presented at Conference on Geometry in Be ¸dlewo,Poland,September 2002Geometry of the Standard ModelAndrzej Derdzinski0.Introduction.The Standard Model of particles and interactions is the currently-accepted theory of elementary particles.It can be naturally divided into the classical part ,a description of which is possible in the language of vector bundles over the spacetime and operations on them,and a ﬁeld quantization procedure that transforms the classical part into a reasonable model of physical reality.This note covers only the classical part of the Standard Model.Similar but more detailed expositions of this topic can be found in the following texts:A.Derdzinski,Geometry of the Standard Model of Elementary Particles ,Texts and Monographsin Physics,Springer-Verlag,Berlin-Heidelberg-New York,1992.A.Derdzinski,Geometry of elementary particles ,Proceedings of Symposia in Pure Mathematics54(1993),(edited by R.E.Greene and S.-T.Yau),Part 2,157–171.1.Interactions.Aside from gravity,the known kinds of particle interactions,ordered by decreasing strength,are the strong,electromagnetic,and weak forces.The latter two may be combined into the electroweak interaction (§5).The strength of an interaction amounts to the probability of its occurrence in the given circumstances.2.Taxonomy of particle species.(See also §3,§5.)known particlesz{interaction carriers:matter particles8species of gluons,z {γ(the photon),leptonshadronsW −,W +,Z 0z{z{leptons proper,3generations:e ,νe µ,νµτ,ντantileptons,3generations:e +,νµτ+,}|2ANDRZEJ DERDZINSKI4.A physics-geometry dictionary.Table4.1.Particles and bundlesphysicsa PARTICLE species(M,g);the particle is represented by(or lives in)ζSECTIONSψof the bundleζthe particleEVOLUTION of the statesa MATTER particlegeometry:operations onmatter particlesthe GENERALIZATION(all n species involved live here).a natural surjective MORPHISMζ1...ζn→ζa COMPOSITE systemsymmetric/skewsymmetric in any group of iden-ANTIPARTICLE formationgeometry:Yang-Millsﬁeldsa NATURAL VECTOR BUNDLEη→M ofa FREE matter particlecies);naturality amounts to direct observabilitya NON-NATURAL vector bundleδ→M(the interac-of some given kindCARRIERS oftions are the compatible connections inδthe INTERACTING-PARTICLE BUNDLEα=α(δ,η), ING matterthe free-particle bundleη(basic example:α=δη)Usually,neitherαnor C(δ)is natural.This contradicts the obvious requirement that carriers of interactions and interacting matter should be directly observable. One resolves this problem by“restoring”naturality of the bundles in question using bound states,or symmetry breaking,as described below.(Notations:N is theﬁbre dimension of theﬁxed interaction bundleδ;an integer k>0represents the product vector bundle M×GEOMETRY OF THE STANDARD MODEL 3Table 4.4.Bound states and symmetry breakingphysics BOUND STATESobtained by naturally “canceling”the δ-related factorsselection of an ADDITIONAL STRUCTURE(the interacting particle comes in N separate versions),(a thought experiment)dim G species of matter particles living in T ∗=T ∗Mbreaking (in nature,for in-5.The standard model.Table 5.1.Geometry of interactionsinter-ELECTROWEAKMAGNETICWeyl,Gell-Mann,Zweig,1961–1967The model describes one gen-a uniﬁed descrip-Hadrons appearat a time.Choose,e.g.,e ,νe :netism for all par-of quarks and com-trino.Their free-particle bun-the fact that the(abbreviation:where σdenotes a ﬁxed Diracquantized ,i.e.,oc-in several ﬂa-ed orientable,and σ=σL +σRof a ﬁxed amount.σL .G,δG =U(2),δ=ιwhat δis a complexline bundle 3-space bundle a Hermitian, and a unit sec-of δﬁbre metric , free-par-a ﬁxed Dirac spinor any ηantiquarksα=λk ηif parti-α=ρσacting or,if neutrinos are massive,of electron chargeα=σbundleλk )(antiquarks)4ANDRZEJ DERDZINSKITable5.2.Bound states and symmetrybreaking in the standard modelinter-ELECTROWEAKMAGNETIConly if cancelρfactors states:ρ→1,α1...αntral systems,ρ3→1,getting ζasλλk1...λk n=ons proper),¯q tri-are natural)C(λ)=T∗.Mat-3ter:same as es in3colors. breakingone species,carriers appear asChoice of a sectionφofιduces U(2)to U(1).Callλinteraction bundle,soι=spontaneous Thus,α=σL+λσdescrib-too strong much breaking charges,and the summandsrepresent the carriers:thematterlike weak-interac-ed),and Z(neutral).。

Yet Another Extension of the Standard Model Oases in the Desert

Stanford Linear Accelerator Center Stanford University, Stanford, CA 94309 and University of Hawaii, Honolulu, HI 96822
Submitted to Physical Review D.
SLAC-PUB-9212 UH-511-1000-02 May 2002
arXiv:hep-ph/0206116v1 12 Jun 2002
Yet Another Extension of the Standard Model: Oases in the Desert? ∗
J. D. Bjorken, S. Pakvasa and S. F. Tuan
Work supported by the Department of Energy, contract DE-AC03-76SF00515 and grant DE-FG-0394ER40833.
∗
ABSTRACT We have searched for conceptually simple extensions of the standard model, and describe here a candidate model which we ﬁnd attractive. Our starting point is the assumption that oﬀ-diagonal CKM mixing matrix elements are directly related by lowest order perturbation theory to the quark mass matrices. This appears to be most easily and naturally implemented by assuming that all oﬀ-diagonal elements reside in the down-quark mass matrix. This assumption is in turn naturally realized by introducing three generations of heavy, electroweak-singlet down quarks which couple to the Higgs sector diagonally in ﬂavor, while mass-mixing oﬀ-diagonally with the light down-quarks. Anomaly cancellation then naturally leads to inclusion of electroweak vector-doublet leptons. It is then only a short step to completing the extension to three generations of fundamental 27 representations of E (6). Consequences of this picture include (1) the hypothesis of “Stech texture” for the down-quark mass matrix (imaginary oﬀ-diagonal elements) leads to an approximate right unitarity triangle (γ ≈ π/2), and a value of sin 2β between 0.64 and 0.80; (2) Assuming only that the third generation B couples to the Higgs sector at least as strongly as does the top quark, the mass of the B is roughly estimated to lie between 1.7 TeV and 10 TeV, with lower-generation quarks no heavier. The corresponding guess for the new leptons is a factor two lower, 0.8 TeV to 5 TeV; (3) Within the validity of the model, ﬂavor and CP violation are “infrared” in nature, induced by semi-soft mass mixing terms, not Yukawa couplings; (4) The “Mexican hat” structure of the Higgs potential may be radiatively induced by the new heavy down-quark one-loop contributions to the potential; (5) A subset of the precision electroweak experiments are sensitive to the physics induced by the heavy quarks and/or leptons; (6) If the Higgs couplings of the new quarks are ﬂavor symmetric, then there necessarily must be at least one “oasis” in the desert, induced by new radiative corrections to the top quark and Higgs coupling constants, and roughly at 103 TeV; (7) If the Higgs couplings of the new down-quarks are hierarchical and equal to the usual Higgs couplings to right-handed up quarks, then, in the limit in which gauge coupling constants are set to zero, parity violation is also “infrared”, induced by semi-soft mass terms.

Signals of Doubly-Charged Higgsinos at the CERN Large Hadron Collider

DESY-08-060, IZTECH-P-08-03, CUMQ-HEP-149, HIP-2008-15/TH
Signals of Doubly-Charged Higgsinos at the CERN Large Hadron Collider
Durmu¸ s A. Demir1,2 , Mariana Frank3 , Katri Huitu4 , Santosh Kumar Rai4 , and Ismail Turan3
Department of Physics, Izmir Institute of Technology, IZTECH, TR35430 Izmir, Turkey. 2 Deutsches Elektronen - Synchrotron, DESY, D-22603 Hamburg, Germany. 3 Department of Physics, Concordia University, 7141 Sherbrooke Street West, Montreal, Quebec, CANADA H4B 1R6. and 4 Department of Physics, University of Helsinki and Helsinki Institute of Physics, P.O. Box 64, FIN-00014 University of Helsinki, Finland. Several supersymmetric models with extended gauge structures, motivated by either grand uniﬁcation or by neutrino mass generation, predict light doubly-charged Higgsinos. In this work we study productions and decays of doubly-charged Higgsinos present in left-right supersymmetric models, and show that they invariably lead to novel collider signals not found in the minimal supersymmetric model (MSSM) or in any of its extensions motivated by the µ problem or even in extra dimensional theories. We investigate their distinctive signatures at the Large Hadron Collider (LHC) in both pair– and single–production modes, and show that they are powerful tools in determining the underlying model via the measurements at the LHC experiments.

Handbook of economic finance(MIT Stephen A.Ross)

Arbitrage,State Prices and Portfolio Theory Handbook of the Economics of FinancePhilip Dybvig Washington University in Saint Louis Stephen A.RossMITFirst draft:September,2001 This draft:September19,2002AbstractNeoclassicalﬁnancial models provide the foundation for our understanding of ﬁnance.This chapter introduces the main ideas of neoclassicalﬁnance in a single-period context that avoids the technical difﬁculties of continuous-time models,but preserves the principal intuitions of the subject.The starting point of the analysis is the formulation of standard portfolio choice problems.A central conceptual result is the Fundamental Theorem of Asset Pricing,which asserts the equivalence of absence of arbitrage,the existence of a positive linear pricing rule,and the existence of an optimum for some agent who prefers more to less.A related conceptual result is the Pricing Rule Representation Theorem, which asserts that a positive linear pricing rule can be represented as using state prices,risk-neutral expectations,or a state-price density.Different equivalent rep-resentations are useful in different contexts.Many applied results can be derived from theﬁrst-order conditions of the portfolio choice problem.Theﬁrst-order conditions say that marginal utility in each state is proportional to a consistent state-price density,where the constant of proportion-ality is determined by the budget constaint.If markets are complete,the implicit state-price density is uniquely determined by investment opportunities and must be the same as viewed by all agents,thus simplifying the choice problem.Solv-ingﬁrst-order conditions for quantities gives us optimal portfolio choice,solving them for prices gives us asset pricing models,solving them for utilities gives us preferences,and solving them for for probabilities gives us beliefs.We look at two popular asset pricing models,the CAPM and the APT,as well as complete-markets pricing.In the case of the CAPM,theﬁrst-order conditions link nicely to the traditional measures of portfolio performance.Further conceptual results include aggregation and mutual fund separation theory, both of which are useful for understanding equilibrium and asset pricing.The modern quantitative approach toﬁnance has its original roots in neoclassicaleconomics.Neoclassical economics studies an idealized world in which marketswork smoothly without impediments such as transaction costs,taxes,asymme-try of information,or indivisibilities.This chapter considers what we learn fromsingle-period neoclassical models inﬁnance.While dynamic models are becom-ing more and more common,single-period models contain a surprisingly largeamount of the intuition and intellectual content of modernﬁnance,and are alsocommonly used by investment practitioners for the construction of optimal port-folios and communication of investment results.Focusing on a single period isalso consistent with an important theme.While general equilibrium theory seeksgreat generality and abstraction,ﬁnance has work to be done and seeks speciﬁcmodels with strong assumptions and deﬁnite implications that can be tested andimplemented in practice.1Portfolio ProblemsIn our analysis,there are two points of time,0and1,with an interval of time inbetween during which nothing happens.At time zero,our champion(the agent)is making decisions that will affect the allocation of consumption between non-random consumption,c0,at time0,and random consumption cωacross statesω12Ωrevealed at time1.At time0and in each state at time1,there is a single consumption good,and therefore consumption at time0or in a state attime1is a real number.This abstraction of a single good is obviously not“true”in any literal sense,but this is not a problem,and indeed any useful theoreticalmodel is much simpler than reality.The abstraction does,however face us withthe question of how to interpret our simple model(in this case with a single good)in a practical context that is more complex(has multiple goods).In using a single-good model,there are two usual practices:either use nominal values and measureconsumption in dollars,or use real values and measure consumption in inﬂation-adjusted dollars.Depending on the context,one or the other can make the mostsense.In this article,we will normally think of the consumption units as beingthe numeraire,so that“cashﬂows”or“claims to consumption”have the samemeaning.Following the usual practice from general equilibrium theory of thinking of units2of consumption at various times and in different states of nature as different goods, a typical consumption vector is C c0c1cΩ,where the real number c0de-notes consumption of the single good at time zero,and the vector c c1cΩof real numbers c1cΩdenotes random consumption of the single good in each state1Ωat time1.If this were a typical exercise in general equilibrium theory,we would have a price vector for consumption across goods.For example,we might have the following choice problem,which is named after two great pioneers of general equilibrium theory,Kenneth Arrow and Gerard Debreu:Problem1Arrow-Debreu ProblemChoose consumptions C c0c1cΩtomaximize utility of consumption U C subject tothe budget constraintc0Ω∑ω1pωcωW(1)Here,U is the utility function that represents preferences,p is the price vector, and W is wealth,which might be replaced by the market value of an endowment. We are taking consumption at time0to be the numeraire,and pωis the price of the Arrow-Debreu security which is a claim to one unit of consumption at time1 in stateω.Theﬁrst-order condition for Problem1is the existence of a positive Lagrangian multiplierλ(the marginal utility of wealth)such that U0c0λ,and for allω1Ω,UωcωλpωThis is the usual result from neoclassical economics that the gradient of the util-ity function is proportional to prices.Specializing to the leading case inﬁnance of time-separable von Neumann-Morgenstern preferences,named after John von Neumann and Oscar Morgenstern,two great pioneers of utility theory,we have that U C v c0∑Ωω1πωu cω.We will take v and u to be differentiable, strictly increasing(more is preferred to less),and strictly concave(risk averse).3Here,πωis the probability of stateω.In this case,theﬁrst-order condition is the existence ofλsuch that(2)v c0λand for allω12n,(3)πωu cωλpωor equivalently(4)u cωλρωwhereρωpωπωis the state-price density(also called the stochastic discount factor or pricing kernel),which is a measure of priced relative scarcity in state of natureω.Therefore,the marginal utility of consumption in a state is pro-portional to the relative scarcity.There is a solution if the problem is feasible, prices and probabilities are positive,the von Neumann-Morgenstern utility func-tion is increasing and strictly concave,and there is satisﬁed the Inada condition lim c∞u c0.1There are different motivations of von Neumann-Morgenstern preferences in the literature and the probabilities may be objective or subjective. What is important for us that the von Neumann-Morgenstern utility function rep-resents preferences in the sense that expected utility is higher for more preferred consumption patterns.2Using von Neumann-Morganstern preferences has been popular in part because of axiomatic derivations of the theory(see,for example,Herstein and Milnor [1953]or Luce and Raiffa[1957],chapter2).There is also a large literature on alternatives and extensions to von Neumann-Morgenstern preferences.Forsingle-period models,see Knight[1921],Bewley[1988],Machina[1982],Blume, Brandenburger,and Dekel[1991],and Fishburn[1988].There is an even richer set of models in multiple periods,for example,time-separable von Neumann-Morgenstern(the traditional standard),habit formation(e.g.Duesenberry[1949], Pollak[1970],Abel[1990],Constantinides[1991],and Dybvig[1995]),local substitutability over time(Hindy and Huang[1992]),interpersonal dependence (Duesenberry[1949]and Abel[1990]),preference for resolution of uncertainty (Kreps and Porteus[1978]),time preference dependent on consumption(Bergman [1985]),and general recursive utility(Epstein and Zin[1989]).Recently,there have also been some attempts to revive the age-old idea of study-ingﬁnancial situations using psychological theories(like prospect theory,Kahne-man and Tversky[1979]).Unfortunately,these models do not translate well to ﬁnancial markets.For example,in prospect theory framing matters,that is,the observed phenomenon of an agent making different decisions when facing identi-cal decision problems described differently.However,this is an alien concept for ﬁnancial economists and when they proxy for it in models they substitute some-thing more familiar(for example,some history dependence as in Barberis,Huang, and Santos[2001]).Another problem with the psychological theories is that they tend to be isolated stories rather than a general speciﬁcation,and they are often hard to generalize.For example,prospect theory says that agents put extra weight on very unlikely outcomes,but it is not at all clear what this means in a model with a continuum of states.This literature also has problems with using ex post explanations(positive correlations of returns are underreaction and negative cor-relations are overreactions)and a lack of clarity of how much is going on that cannot be explained by traditional models(and much of it can).In actualﬁnancial markets,Arrow-Debreu securities do not trade directly,even if they can be constructed indirectly using a portfolio of securities.A security is characterized by its cashﬂows.This description would not be adequate for analysis of taxes,since different sources of cashﬂow might have very different tax treatment,but we are looking at models without taxes.For an asset like a common stock or a bond,the cashﬂow might be negative at time0,from payment of the price,and positive or zero in each state at time1,the positive amount coming from any repayment of principal,dividends,coupons,or proceeds from sale of the asset.For a futures contract,the cashﬂow would be0at time0,and the cashﬂow in different states at time1could be positive,negative,or zero, depending on news about the value of the underlying commodity.In general,we5think of the negative of the initial cashﬂow as the price of a security.We denote by P P1P N the vector of prices of the N securities1N,and we denote by X the payoff matrix.We have that P n is the price we pay for one unit of security n and Xωn is the payoff per unit of security n at time1in the single state of nature ω.With the choice of a portfolio of assets,our choice problem might becomeProblem2First Portfolio Choice ProblemChoose portfolio holdingsΘΘ1Θn and consumptions C c0cΩto maximize utility of consumption U C subject toportfolio payoffs c c1cωXΘandbudget constraint c0PΘW.Here,Θis the vector of portfolio weights.Time0consumption is the numeraire, and wealth W is now chosen in time0consumption units and the entire endow-ment is received at time0.In the budget constraint,the term PΘis the cost of the portfolio holding,which is the sum across securities n of the price P n times the number of shares or other unitΘn.The matrix product XΘsays that the consump-tion in stateωis cω∑n XωnΘn,i.e.the sum across securities n of the payoff Xωn of security n in stateω,times the number of shares or other unitsΘn of security n our champion is holding.Theﬁrst-order condition for Problem2is the existence of a vector of shadow prices p and a Lagrangian multiplierλsuch that(5)πωu cωλpωwhere(6)P pXTheﬁrst equation is the same as in the Arrow-Debreu model,with an implicit shadow price vector in place of the given Arrow-Debreu prices.The second equa-tion is a pricing equation that says the prices of all assets must be consistent with6the shadow prices of the states.For the Arrow-Debreu model itself,the state-space tableau X is I,the identity matrix,and the price vector P is p,the vector of Arrow-Debreu state prices.For the Arrow-Debreu model,the pricing equation determines the shadow prices as equal to the state prices.Even if the assets are not the Arrow-Debreu securities,Problem2may be essen-tially equivalent to the Arrow-Debreu model in Problem1.In economic terms, the important feature of the Arrow-Debreu problem is that all payoff patterns are spanned,i.e.,each potential payoff pattern can be generated at some price by some portolio of assets.Linear algebra tells us that all payoff patterns can be generated if the payoff matrix X has full row rank.If X has full row rank,p is determined(or over-determined)by(6).If p is uniquely determined by the pricing equation(and therefore also all Arrow-Debreu assets can be purchased as portfolios of assets in the economy),we say that markets are complete,and for all practical purposes we are in an Arrow-Debreu world.For the choice problem to have a solution for any agent who prefers more to less, we also need for the price of each payoff pattern to be unique(the“law of one price”)and positive,or else there would be arbitrage(i.e.,a“money pump”or a “free lunch”).If there is no arbitrage,then there is at least one vector of positive state prices p solving the pricing equation(6).There is an arbitrage if the vector of state prices is overdetermined or if all consistent vectors of state prices assign a negative or zero price to some state.The notion of absence of arbitrage is a central concept inﬁnance,and we develop its implications more fully in the section on preference-free results.So far,we have been stating portfolio problems in prices and quantities,as we would in general equilibrium theory.However,it is also common to describe assets in terms of rates of return,which are relative price changes(often expressed as percentages).The return to security n,which is the relative change in total value (including any dividends,splits,warrant issues,coupons,stock issues,and the like as well as change in the price).There is not an absolute standard of what is meant by return,in different contexts this can be the rate of rate of return,one plus the rate of return,or the difference between two rates of return.It is necessary to ﬁgure which is intended by asking or from ing the notation above,the rate of return in stateωis rωn=Xωn P n P n.3Often,consumption at the outsetis suppressed,and we specialize to von Neumann-Morgenstern expected utility. In this case,we have the following common form of portfolio problem. Problem3Portfolio Problem using ReturnsChoose portfolio proportionsθθ1θn and consumptions c c1cΩtomaximize expected utility of consumption∑Ωω1πθu cωsubject tothe consumption equation c Wθ1r and the budget constraintθ11.Here,ππ1πΩis a vector of state probabilities,u is the von Neumann-Morgenstern utility function,and1is a vector of1’s.The dimensionality of1is determined implicitly from the context,here the dimensionality is the number of assets.Theﬁrst-order condition for an optimum is the existence of shadow state price density vectorρand shadow marginal utility of wealthλsuch that(7)u cωλρωand(8)1E1rρThese equations say that the state-price density is consistent with the marginal valuation by the agent and with pricing in the market.As ourﬁnal typical problem,let us consider a mean-variance optimization.This optimization is predicated on the assumption that investors care only about mean and variance(typically preferring more mean and less variance),so we have a utility function V m v in mean m and variance v.For this problem,suppose there is a risk-free asset paying a return r(although the market-level implications of mean-variance analysis can also be derived in a general model without a riskyasset).In this case,portfolio proportions in the risky assets are unconstrained (need not sum to1)because the slack can be taken up by the risk-free asset.We denote byµthe vector of mean risky asset returns and byσthe covariance matrix of risky returns.Then our champion solves the following choice problem. Problem4Mean-variance optimizationChoose portfolio proportionsθθ1θn tomaximize the mean-variance utility function V rµr1θθΣθ.Theﬁrst-order condition for the problem is(9)µr1λΣθwhereλis twice the marginal rate of substitution V v m v V m m v,evaluated at m rµr1θand vθΣθ,whereθis the optimal choice of portfolio pro-portions.Theﬁrst-order condition(9)says that mean excess return for each asset is proportional to the marginal contribution of volatility to the agent’s optimal portfolio.We have seen a few of the typical types of portfolio problem.There are a lot of variations.The problem might be stated in terms of excess returns(rate of return less a risk-free rate)or total return(one plus the rate of return).Or,we might constrain portfolio holdings to be positive(no short sales)or we might require consumption to be nonnegative(limited liability).Many other variations adapt the basic portfolio problem to handle institutional features not present in a neoclassical formulation,such as transaction costs,bid-ask spreads,or taxes. These extensions are very interesting,but beyond the scope of what we are doing here,which is to explore the neoclassical foundations.2Absence of Arbitrage and Preference-free ResultsBefore considering speciﬁc solutions and applications,let us consider some gen-eral results that are useful for thinking about portfolio choice.These results are9preference-free in the sense that they do not depend on any speciﬁc assumptions about preferences but only depend on an assumption that agents prefer more to less.Central to this section is the notion of an arbitrage,which is a“money pump”or a“free lunch”.If there is arbitrage,linearity of the neoclassical problem im-plies that any candidate optimum can be dominated by adding the arbitrage.As a result,no agent who prefers more to less would have an optimum if there ex-ists arbitrage.Furthermore,this seemingly weak assumption is enough to obtain two useful theorems.The Fundamental Theorem of Asset Pricing says that the following are equivalent:absence of arbitrage,existence of a consistent positive linear pricing rule,and existence of an optimum for some hypothetical agent who prefers more to less.The Pricing Rule Representation Theorem gives different equivalent forms for the consistent positive linear pricing rule,using state prices, risk-neutral probabilites(martingale valuation),state-price density(or stochastic discount factor or pricing kernel),or an abstract positive linear operator.The re-sults in this section are from Cox and Ross[1975],Ross[1976c,1978b],and Dybvig and Ross[1987].The results have been formalized in continuous time by Harrison and Kreps[1979]and Harrison and Pliska[1981].Occasionally,the theorems in this section can be applied directly to obtain an in-teresting result.For example,linearity of the pricing rule is enough to derive put-call parity without constructing the arbitrage.More often,the results in this sec-tion help to answer conceptual questions.For example,an option pricing formula that is derived using absence of arbitrage is always consistent with equilibrium, as can be seen from the Fundamental Theorem.By the Fundamental Theorem, absence of arbitrage implies there is an optimum for some hypothetical agent who prefers more to less;we can therefore construct an equilibrium in the single-agent pure exchange economy in which this agent is endowed with the optimal holding. By construction the equilibrium in this economy will have the desired pricing,and therefore any no-arbitrage pricing result is consistent with some equilibrium.In this section,we will work in the context of Problem2.An arbitrage is a change in the portfolio that makes all agents who prefer more to less better off.We make all such agents better off if we increase consumption sometime,and in some state of nature,and we never decrease consumption.By combining the two constraints in Problem2,we can write the consumption C associated with any portfolio choice10Θusing the stacked matrix equationC WPXΘTheﬁrst row,W PΘ,is consumption at time0,which is wealth W less the cost of our portfolio.The remaining rows,XΘ,give the random consumption across states at time1.Now,when we move from the portfolio choiceΘto the portfolio choiceΘη, the initial wealth term cancels and the change in consumption can now be writtenas∆CPXηThis will be an arbitrage it∆C is never negative and is positive in at least one component,which we will write as4∆C0orPXη0Some authors describe taxonomies of different types of arbitrage,having perhaps a negative price today and zero payoff tomorrow,a zero price today and a non-negative but not identically zero payoff tomorrow,or a negative price today and a positive payoff tomorrow.These are all examples of arbitrages that are sub-sumed by our general formula.The important thing is that there is an increase in consumption in some state of nature at some point of time and there is never any decrease in consumption.Fundamental Theorem of Asset PricingTheorem1Fundamental Theorem of Asset Pricing The following conditions on prices P and payoffs X are equivalent:i Absence of arbitrage:ηP Xη0.ii Existence of a consistent positive linear pricing rule(positive state prices): p0P p X.iii Some agent with strictly increasing preferences U has an optimum in Prob-lem2.P ROOF We prove the equivalence by showing i ii,ii iii,and iii i.i ii:This is the most subtle part,and it follows from a separation theorem or the duality theorem from linear programming.From the deﬁnition of absence of arbitrage,we have that the setsS1PXηηℜnandS2xℜΩ1x0must be disjoint.Therefore,there is a separating hyperplane z such that z x0for all x S1and z x0for all x S2.(See...,theorem...)Normalizing so that the ﬁrst component(the shadow price of time zero consumption)is1,we will see that p deﬁned by1p z z0is the consistent linear pricing rule we seek.Constancy of zx for x S1implies that1p P X0,which is to say that P p X,i.e. p is a consistent linear pricing rule.Furthermore,z x positive for x S2implies z0and consequently p0,and p is indeed the desired consistent positive linear pricing rule.ii iii:This part is proven by construction.Let U C1p C,thenΘ0 solves Problem2.To see this,note that the objective function U C is constant and equal to W for allΘ:U C1p C1p WPXΘW P p XΘW12(The motivation this construction is observation that the existence of the consistent linear pricing rule with state prices p implies that all feasible consumptions satisfy 1p C W.)iii i:This part is obvious,since any candidate optimum is dominated by adding the arbitrage,and therefore there can be no arbitrage if there is an optimum. More formally,adding an arbitrage implies the change of consumption∆C0, which implies an increase in U C.Pricing Rule Representation TheoremDepending on the context,there are different useful ways of representing the pric-ing rule.For some abstract applications(like proving put-call parity),it is easiestto use a general abstract representation as a linear operator L c such that c0 L c0.For asset pricing applications,it is often useful to use either the thestate-price representation we used in the Fundamental Theorem,L c∑ωpωcω,or risk-neutral probabilities,L c1r1E cω1r1∑ωπωcω.The intuition behind the risk-neutral representation(or martingale representation5)is that the price is the expected discounted value computed using a shadow risk-free rate(equal to the actual risk-free rate if there is one)and artiﬁcial risk-neutral probabilitiesπthat assign positive probability to the same states as do the true probabilities.Risk-neutral pricing says that all investments are fair gambles once we have adjusted for time preference by discounting and for risk preference by adjusting the probabilities.Theﬁnal representation using the state-price density (or stochastic discount factor)ρto write L c Eρωcω∑ωπωρωcω.The state price density simpliﬁesﬁrst-order conditions of portfolio choice problems because the state-price density measures priced scarcity of consumption.The state-price density is also handy for continuous-state models in which individual states have zero state probabilities and state prices but there exists a well-deﬁned positive ratio of the two.Theorem2Pricing Rule Representation Theorem The consistent positive lin-ear pricing rule can be represented equivalently usingi an abstract linear functional L c that is positive:c0L c0ii positive state prices p0:L c∑Ωω1pωcωiii positive risk-neutral probabilitiesπ0summing to1with associated shadow risk-free rate r:L c1r1E cω1r1∑ωπωcωiv positive state-price densitiesρ0:L c Eρc∑ωπωρωcω.P ROOF i ii:This is the known form of a linear operator inℜΩ.ii iii:Noteﬁrst that the shadow risk-free rate must price the riskless asset c1:Ω∑ω1pω11r1E1which implies(since E11)that r1p1 1.Then,matching coefﬁcients inΩ∑ω1pωcω1r1∑ωπωcωwe have thatπp1p,which sums to1as required and inherits positivity from p.iii iv:Simply letρω1r1πω(which is the same as pωπω).iv i:immediate.time s of receiving subsequent cashﬂows c s1,c s2...c t is given by t∑τs1E sρτ1r s1r s11rτ(11)Note that unless the riskfree rate is nonrandom,we cannot take the discount fac-tors out of the expectation.6This is because of the way that the law of iterated expectations works.For example,consider the value V0at time0of the cashﬂow in time2.V01r11E0V1(12)1r11E01r21E1c21r11E01r21c2Now,1r11is outside the expectation(as could be1r s11in(11)),but 1r21cannot come outside the expectation unless it is nonrandom.7So,it is best to remember that when interest rates are stochastic,discounting for risk-neutral valuation should use the rolled-over spot rate,within the expectation.3Various Analyses:Arrow-Debreu WorldThe portfolio problem is the starting point of a lot of types of analysis inﬁnance. Here are some implications that can be drawn from portfolio problems(usually through theﬁrst-order conditions):optimal portfolio choice(asset allocation or stock selection)portfolio efﬁciencyaggregation and market-level implicationsasset pricing and performance measurementpayoff distribution pricingrecovery or estimation of preferencesinference of expectationsWe can think of many of these distinctions as a question of what we are solving for when we look at theﬁrst-order conditions.In optimal portfolio choice and its aggregation,we are solving for the portfolio choice given the preferences and be-liefs about returns.In asset pricing,we are computing the prices(or restrictions on expected returns)given preferences,beliefs about payoffs,and the optimal choice (which is itself often derived using an aggregation result).In recovery,we derive preferences from beliefs and idealized observations about portfolio choice,e.g.at all wealth levels.Estimation of preferences is similar,but works with noisy obser-vations of demand at aﬁnite set of data points and uses a restriction in the func-tional form or smoothing in the statistical procedure to identify preferences.And, inference of expectations derives probability beliefs from preferences,prices,and the(observed)optimal demand.In this section,we illustrate the various analyses in the case of an Arrow-Debreu world.Analysis of the complete-markets model has been developed by many people over a period of time.Some of the more important works include some of the origi-nal work on competitive equilibrium such as Arrow and Debreu[1954],Debreu17。

Field Theory and Standard Model

Abstract This is a short introduction to the Standard Model and the underlying concepts of quantum ﬁeld theory.
Lectures given at the European School of High-Energy Physics, August 2005, Kitzb¨ uhel, Austria
Contents
1 Introduction 1.1 Theoretical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Phenomenological Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Quantisation of Fields 2.1 Why Fields? . . . . . . . . . . . . . . . . . . . . 2.1.1 Quantisation in Quantum Mechanics . . 2.1.2 Special Relativity Requires Antiparticles 2.2 Multiparticle States and Fies, Creation and Annihilation . . . . 2.2.2 Charge and Momentum . . . . . . . . . 2.2.3 Field Operator . . . . . . . . . . . . . . 2.2.4 Propagator . . . . . . . . . . . . . . . . 2.3 Canonical Quantisation . . . . . . . . . . . . . . 2.4 Fermions . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Canonical Quantisation of Fermions . . . 2.5 Interactions . . . . . . . . . . . . . . . . . . . . 2.5.1 φ4 Theory . . . . . . . . . . . . . . . . . 2.5.2 Fermions . . . . . . . . . . . . . . . . . . 3 Gauge Theories 3.1 Global Symmetries v Gauge Symmetries 3.2 Abelian Gauge Theories . . . . . . . . . 3.3 Non-Abelian Gauge Theories . . . . . . . 3.4 Quantisation . . . . . . . . . . . . . . . . 4 Quantum Corrections 4.1 Anomalous Magnetic Moment . . . 4.2 Divergences . . . . . . . . . . . . . 4.2.1 Dimensional Regularisation 4.2.2 Renormalisation . . . . . . . 4.2.3 Running Coupling in QED . 4.2.4 Running Coupling in QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 5 6 6 6 8 9 9 10 11 12 12 14 15 17 18 20 22 22 24 27 29 32 32 35 36 38 40 41

standard model标准模型

CPT violation and the standard modelDon Colladay and V.Alan Kostelecky´Department of Physics,Indiana University,Bloomington,Indiana47405͑Received22January1997͒Spontaneous CPT breaking arising in string theory has been suggested as a possible observable experimen-tal signature in neutral-meson systems.We provide a theoretical framework for the treatment of low-energy effects of spontaneous CPT violation and the attendant partial Lorentz breaking.The analysis is within the context of conventional relativistic quantum mechanics and quantumﬁeld theory in four dimensions.We use the framework to develop a CPT-violating extension to the minimal standard model that could serve as a basis for establishing quantitative CPT bounds.͓S0556-2821͑97͒05211-9͔PACS number͑s͒:11.30.Er,11.25.Ϫw,12.60.ϪiI.INTRODUCTIONAmong the symmetries of the minimal standard model is invariance under CPT.Indeed,CPT invariance holds under mild technical assumptions for any local relativistic point-particleﬁeld theory͓1–5͔.Numerous experiments have con-ﬁrmed this result͓6͔,including in particular high-precision tests using neutral-kaon interferometry͓7,8͔.The simulta-neous existence of a general theoretical proof of CPT invari-ance in particle physics and accurate experimental tests makes CPT violation an attractive candidate signature for nonparticle physics such as string theory͓9,10͔.The assumptions needed to prove the CPT theorem are invalid for strings,which are extended objects.Moreover, since the critical string dimensionality is larger than four,it is plausible that higher-dimensional Lorentz breaking would be incorporated in a realistic model.In fact,a mechanism is known in string theory that can cause spontaneous CPT vio-lation͓9͔with accompanying partial Lorentz-symmetry breaking͓11͔.The effect can be traced to string interactions that are absent in conventional four-dimensional renormaliz-able gauge theory.Under suitable circumstances,these inter-actions can cause instabilities in Lorentz-tensor potentials, thereby inducing spontaneous CPT and Lorentz breaking.If in a realistic theory the spontaneous CPT and partial Lorentz violation extend to the four-dimensional spacetime,detect-able effects might occur in interferometric experiments with neutral kaons͓9,10͔,neutral B d or B s mesons͓10,12͔,or neutral D mesons͓10,13͔.For example,the quantities pa-rametrizing indirect CPT violation in these systems could be nonzero.There may also be implications for baryogenesis ͓14͔.In the present paper,our goal is to develop within an effective-theory approach a plausible CPT-violating exten-sion of the minimal standard model that provides a theoreti-cal basis for establishing quantitative bounds on CPT invari-ance.The idea is to incorporate notions of spontaneous CPT and Lorentz breaking while maintaining the usual gauge structure and properties like renormalizability.To achieve this,weﬁrst establish a conceptual framework and a procedure for treating spontaneous CPT and Lorentz viola-tion in the context of conventional quantum theory.We seek a general methodology that is compatible with desirable fea-tures like microscopic causality while being sufﬁciently de-tailed to permit explicit calculations.We suppose that underlying the effective four-dimensional action is a complete fundamental theory that is based on conventional quantum physics͓15͔and is dynami-cally CPT and Poincare´invariant.The fundamental theory is assumed to undergo spontaneous CPT and Lorentz breaking. In a Poincare´-observer frame in the low-energy effective ac-tion,this process is taken toﬁx the form of any CPT-and Lorentz-violating terms.Since interferometric tests of CPT violation are so sensi-tive,we focus speciﬁcally on CPT violation and the associ-ated Lorentz-breaking issues in a low-energy effective theory without gravity͓21͔.For the most part,effects from deriva-tive couplings and possible CPT-preserving but Lorentz-breaking terms in the action are disregarded,and any CPT-violating terms are taken to be small enough to avoid issues with standard experimental tests of Lorentz symmetry.A partial justiﬁcation for the latter assumption is that the absence of signals for CPT violation in the neutral-kaonsystem provides one of the best bounds on Lorentz invari-ance.Our focus on the low-energy effective model bypassesvarious important theoretical issues regarding the structure ofthe underlying fundamental theory and its behavior at scalesabove electroweak uniﬁcation,including the origin and ͑renormalization-group͒stability of the suppression of CPT breaking and the issue of modeﬂuctuations around Lorentz-tensor expectation values.Since these topics involve the Lor-entz structure of the fundamental theory,they are likely to berelated to the difﬁcult hierarchy problems associated withcompactiﬁcation and the cosmological constant.The ideas underlying our theoretical framework are de-scribed in Sec.II.A simple model is used to illustrate con-cepts associated with CPT and Lorentz breaking,including the possibility of eliminating some CPT-violating effects throughﬁeld redeﬁnitions.The associated relativistic quan-tum mechanics is discussed in Sec.III.Section IV contains a treatment of some issues in quantumﬁeld theory.A CPT-violating extension of the minimal standard model is provided in Sec.V,and the physically observable subset ofPHYSICAL REVIEW D1JUNE1997VOLUME55,NUMBER11550556-2821/97/55͑11͒/6760͑15͒/$10.006760©1997The American Physical SocietyCPT-breaking terms is established.We summarize in Sec. VI.Some of the more technical results are presented in the appendixes.II.BASICSA.Effective model for spontaneous CPT violationWe begin our considerations with a simple model within which many of the basic features of spontaneous CPT vio-lation can be examined.The model involves a single massive Diracﬁeld␺(x)in four dimensions with Lagrangian densityLϭL0ϪLЈ,͑1͒where L0is the usual free-ﬁeld Dirac Lagrangian for a fer-mion␺of mass m,and where LЈcontains extra CPT-violating terms to be described below.For the present discussion,we follow an approach in which the C,P,T and Lorentz properties of␺are assumed to be conventionally determined by the free-ﬁeld theory L0and are used to estab-lish the corresponding properties of LЈ͓22͔.This method is intrinsically perturbative,which is particularly appropriate here since any CPT-violating effects must be small.In Sec. II C,we consider the possibility of alternative deﬁnitions of C,P,T and Lorentz properties that could encompass the full structure of L.We are interested in possible forms of LЈthat could arise as effective contributions from spontaneous CPT violation in a more complete theory.To our knowledge,string theory forms the only class of͑gauge͒theories in four or more dimensions that are quantum consistent,dynamically Poin-care´invariant,and known to admit an explicit mechanism ͓9͔for spontaneous CPT violation triggered by interactions in the Lagrangian.However,to keep the treatment as general as possible we assume only that the spontaneous CPT vio-lation arises from nonzero expectation values acquired by one or more Lorentz tensors T,so LЈis taken to be an effective four-dimensional Lagrangian obtained from an un-derlying theory involving Poincare´-invariant interactions of ␺with T.The discussion that follows is independent of any speciﬁcs of string theory and should therefore be relevant to a nonstring model with spontaneous CPT violation,if such a model is eventually formulated.Even applying the stringent requirement of dynamical Poincare´invariance,an unbroken realistic theory can in prin-ciple include terms with derivatives,powers of tensorﬁelds, and powers of various terms quadratic in fermionﬁelds. However,any CPT-breaking term that is to be part of a four-dimensional effective theory must have mass dimension four.In the effective Lagrangian,each combination ofﬁelds and derivatives of dimension greater than four therefore must have a corresponding weighting factor of a negative power Ϫk of at least one mass scale M that is large compared to the scale m of the effective theory.In a realistic theory with the string scenario,M might be the Planck mass or perhaps a smaller mass scale associated with compactiﬁcation and uni-ﬁcation.Moreover,since the expectations͗T͘of the tensors T are assumed to be Lorentz and possibly CPT violating, any terms that survive in LЈafter the spontaneous symmetry breaking must on physical grounds be suppressed,presum-ably by at least one power of m/M relative to the scale of the effective theory.A hierarchy of possible terms in LЈthus emerges,labeled by kϭ0,1,2,....Omitting Lorentz indices for simplicity,the leading terms with kр2have the schematic formLЈʛ␭M k͗T͘•␺¯⌫͑iץ͒k␺ϩH.c.͑2͒In this expression,the parameter␭is a dimensionless cou-pling constant,(iץ)k represents k four-derivatives acting in some combination on the fermionﬁelds,and⌫represents some gamma-matrix structure.Terms with kу3and with more quadratic fermion factors also appear,but these are further suppressed.Note that contributions of the form͑2͒arise in string theory͓10͔.Note also that naive power count-ing indicates the dominant terms with kр1are renormaliz-able.For kϭ0,the above considerations indicate that the domi-nant terms of the form͑2͒must have expectations͗T͘ϳm2/M.In the present work,we focus primarily on this relatively simple case.Most of the general features arising from CPT and Lorentz violation together with some of our more speciﬁc results remain valid when terms with other values of k are considered,but it remains an open issue to investigate the detailed properties of terms with kϭ1and expectations͗T͘ϳm or those with kϭ2and expectations ͗T͘ϳM.Both these could in principle contribute leading effects in the low-energy effective action.Each contribution to LЈfrom an expression of the form ͑2͒is a fermion bilinear involving a4ϫ4spinor matrix⌫. Regardless of the complexity and number of the tensors T inducing the breaking,⌫can be decomposed as a linear com-bination of the usual16basis elements of the gamma-matrix algebra.Only the subset of these that produce CPT-violating bilinears are of interest for our present pur-poses,and they permit us to provide explicit and relatively simple expressions for the possible CPT-violating contribu-tions to LЈ.For the case kϭ0of interest here,weﬁnd two possible types of CPT-violating term:L aЈϵa␮␺¯␥␮␺,L bЈϵb␮␺¯␥5␥␮␺.͑3͒For completeness,we provide here also the terms appearing for the case kϭ1,where weﬁnd three types of relevant contribution:L cЈϵ12ic␣␺¯ץJ␣␺,L dЈϵ12d␣␺¯␥5ץJ␣␺,L eЈϵ12ie␮␯␣␺¯␴␮␯ץJ␣␺,͑4͒where AץJ␮BϵAץ␮BϪ(ץ␮A)B.In all these expressions,the quantities a␮,b␮,c␣,d␣,and e␮␯␣must be real as conse-quences of their origins in spontaneous symmetry breaking and of the presumed hermiticity of the underlying theory. They are combinations of coupling constants,tensor expec-tations,mass parameters,and coefﬁcients arising from the decomposition of⌫.In keeping with their interpretation as effective coupling constants arising from a scenario with spontaneous symme-try breaking,a␮,b␮,c␣,d␣,and e␮␯␣are invariant under CPT transformations.Together with the standard556761CPT VIOLATION AND THE STANDARD MODELCPT-transformation properties ascribed to␺,this invariance causes the terms in Eqs.͑3͒and͑4͒to break CPT͓23͔.As discussed above,in the remainder of this work we restrict ourselves largely to the expressions in Eq.͑3͒.Allowing both kinds of term in Eq.͑3͒to appear in LЈproduces a model Lagrangian of the formLϭ12i␺¯␥␮ץJ␮␺Ϫa␮␺¯␥␮␺Ϫb␮␺¯␥5␥␮␺Ϫm␺¯␺.͑5͒The variational procedure generates a modiﬁed Dirac equa-tion:͑i␥␮ץ␮Ϫa␮␥␮Ϫb␮␥5␥␮Ϫm͒␺ϭ0.͑6͒Associated with this Dirac-type equation is a modiﬁed Klein-Gordon equation.Proceeding with the usual squaring procedure,in which the Dirac-equation operator with oppo-site mass sign is applied to the Dirac equation from the left, leads to the Klein-Gordon-type expression͓͑iץϪa͒2Ϫb2Ϫm2ϩ2i␥5␴␮␯b␮͑iץ␯Ϫa␯͔͒␺͑x͒ϭ0.͑7͒This equation is second order in derivatives,but unlike the usual Klein-Gordon case it contains off-diagonal terms in the spinor space.These may be eliminated by repeating the squaring procedure,this time applying the operator in Eq.͑7͒with opposite sign for the off-diagonal piece.The result is a fourth-order equation satisﬁed by each spinor component of any solution to the modiﬁed Dirac equation:͕͓͑iץϪa͒2Ϫb2Ϫm2͔2ϩ4b2͑iץϪa͒2Ϫ4͓b␮͑iץ␮Ϫa␮͔͒2͖␺͑x͒ϭ0.͑8͒B.Continuous symmetriesConsider next the continuous symmetries of the model with Lagrangian͑5͒.For deﬁniteness,we begin with an analysis in a given oriented inertial frame in which values of the quantities a␮and b␮are assumed to have been speciﬁed. The effects of rotations and boosts are considered later.The CPT-violating terms in Eq.͑5͒leave unaffected the usual global U͑1͒gauge invariance,which has conserved current j␮ϭ␺¯␥␮␺.Charge is therefore conserved in the model.These terms also leave unaffected the usual breaking of the chiral U͑1͒current j5␮ϭ␺¯␥5␥␮␺due to the mass term. In what follows,we denote the volume integrals of the cur-rent densities j␮and j5␮by J␮and J5␮,respectively.The model is also invariant under translations provided the tensor expectations are assumed constant,i.e.,provided the possibility of CPT-breaking soliton-type solutions in the underlying theory is disregarded.This leads to a conserved canonical energy-momentum tensor⌰␮␯given by⌰␮␯ϭ12i␺¯␥␮ץJ␯␺,ץ␮⌰␮␯ϭ0,͑9͒and a corresponding conserved four-momentum P␮.These expressions have the same form as in the free theory.Note, however,that constancy of the energy and momentum does not necessarily imply conventional behavior under boosts or rotations.Note also that the presence of the CPT-violating terms in the Dirac equation destroys the usual symmetriz-ability property of⌰␮␯.The antisymmetric part⌰͓␮␯͔is⌰͓␮␯͔ϵ⌰␮␯Ϫ⌰␯␮ϭϪ14ץ␣͓␺¯͕␥␣,␴␮␯͖␺͔Ϫa[␮j␯]Ϫb[␮j5␯],͑10͒which is no longer a total divergence.The conventional con-struction of a symmetric energy-momentum tensor,involv-ing a subtraction of this antisymmetric part from the canoni-cal energy-momentum tensor,would affect the conserved energy and momentum and is therefore presumably inappli-cable in the present case.The implications of this for a more complete low-energy effective theory that includes gravity remain to be explored.Next,consider the effect of Lorentz transformations,i.e., rotations and boosts.Conventional Lorentz transformations in special relativity relate observations made in two inertial frames with differing orientations and velocities.These transformations can be implemented as coordinate changes, and we call them observer Lorentz transformations.It is also possible to consider transformations that relate the properties of two particles with differing spin orientation or momentum within a speciﬁc oriented inertial frame.We call these par-ticle Lorentz transformations.For free particles under usual circumstances,the two kinds of transformation are͑in-versely͒related.However,this equivalence fails for particles under the action of a backgroundﬁeld.The reader is warned to avoid confusing observer Lorentz transformations͑which involve coordinate changes͒or par-ticle Lorentz transformations͑which involve boosts on par-ticles or localizedﬁelds but not on backgroundﬁelds͒with a third type of Lorentz transformation that within a speciﬁed inertial frame boosts all particles andﬁelds simultaneously, including background ones.The latter are sometimes called ͑inverse͒active Lorentz transformations.For the case of con-ventional free particles,they coincide with particle Lorentz transformations.We have chosen to avoid applying the terms active and passive here because they are insufﬁcient to dis-tinguish the three kinds of transformation and because in any case their interpretation varies in the literature.The distinction between observer and particle transforma-tions is relevant for the present model,where the CPT-violating terms can be regarded as arising from con-stant backgroundﬁelds a␮and b␮.The point is that these eight quantities transform as two four-vectors under observer Lorentz transformations and as eight scalars under particle Lorentz transformations,whereas they are coupled to cur-rents that transform as four-vectors under both types of trans-formation.This means that observer Lorentz symmetry is still an invariance of the model,but the particle Lorentz group is͑partly͒broken.Physical situations with features like this can readily be identiﬁed.For example,an electron with momentum perpen-dicular to a uniform background magneticﬁeld moves in a circle.Suppose in the same observer frame we instanta-neously increase the magnitude of the electron momentum without changing its direction,causing the electron to move in a circle of larger radius.This͑instantaneous͒particle boost676255DON COLLADAY AND V.ALAN KOSTELECKY´leaves the backgroundﬁeld unaffected.However,if insteadan observer boost perpendicular to the magneticﬁeld is ap-plied,the electron no longer moves in a circle.This is viewed in the new inertial frame as an EϫB drift caused bythe presence of an electricﬁeld.In this example,the back-ground magneticﬁeld transforms into a different electromag-neticﬁeld under observer boosts but͑by deﬁnition͒is un-changed by particle boosts,in analogy to the transformation of a␮and b␮in the CPT-violating model.From the viewpoint of this example,the unconventional aspect of the CPT-violating model is merely that the con-stantﬁelds a␮and b␮are a global feature of the model.Theycannot be regarded as arising from localized experimentalconditions,which would cause them to transform under par-ticle Lorentz transformations as four-vectors rather than as scalars.The behavior of a␮and b␮as backgroundﬁelds andhence as scalars under particle Lorentz transformations is aconsequence of their origin as nonzero expectation values ofLorentz tensors in the underlying theory.These Lorentz-tensor expectations break those parts of the particle Lorentzgroup that cannot be implemented as unitary transformationson the vacuum.This is in parallel with other situations in-volving spontaneous symmetry breaking,such as ones com-monly encountered in the treatment of internal symmetries.The preservation of observer Lorentz symmetry is an im-portant feature of the model.It is a consequence of observerLorentz invariance of the underlying fundamental theory.This symmetry is unaffected by the appearance of tensorexpectation values by virtue of its implementation via coor-dinate transformations.As an illustration of its use in theeffective model,we show that it permits a further classiﬁca-tion of types of CPT-violating term according to the ob-server Lorentz properties of a␮and b␮.Thus,for example, if b␮is future timelike in one inertial frame,it must be futuretimelike in all frames.This implies that a class of inertial frames can be found in which b␮ϭb(1,0,0,0),where calcu-lations are potentially simpliﬁed.A similar argument for the lightlike or spacelike cases shows that the CPT-violating physics of the four components of b␮can in each case bereduced to knowledge of its Lorentz type and a single num-ber specifying its magnitude.Inertial frames within this ideal class are determined by the little group of b␮,which can in turn be used to simplify͑partially͒the form of a␮.The reader is cautioned that the class of inertial framesselected in this way may be distinct from experimentallyrelevant inertial frames such as,for example,those deﬁnedusing the microwave background radiation and interpretingthe dipole component in terms of the motion of the Earth.The point is that,given an inertial frame,the process ofspontaneous Lorentz violation in the underlying theory is assumed to produce some values of a␮and b␮.In this spe-ciﬁc inertial frame,there is no reason a priori why these values should take the ideal form described above.One is merely assured of the existence of some frame in which the ideal form can be attained.The current J␭␮␯for particle Lorentz transformationstakes the usual form when expressed in terms of the energy-momentum tensor:J␭␮␯ϭx[␮⌰␭␯]ϩ14␺¯͕␥␭,␴␮␯͖␺.͑11͒This current is conserved at the level of the underlying theory with spontaneous symmetry breaking,but in the ef-fective low-energy theory where the spontaneous breaking appears as an explicit symmetry violation the conservation property is destroyed.In the latter case,the corresponding Lorentz charges M␮␯obeydM␮␯dtϭϪa[␮J␯]Ϫb[␮J5␯].͑12͒Given explicit values of a␮and b␮in some inertial frame, Eq.͑12͒can be used directly to determine which Lorentz symmetries are violated.Note that if either a␮or b␮van-ishes,the Lorentz group is broken to the little group of the nonzero four-vector.This means that the largest Lorentz-symmetry subgroup that can remain as an invariance of the model Lagrangian͑5͒is SO͑3͒,E͑2͒,or SO͑2,1͒.Since a␮and b␮represent two four-vectors in four-dimensional space-time,they can deﬁne a two-dimensional plane.Transforma-tions involving the two orthogonal dimensions have no effect on this plane.This means that the smallest Lorentz-symmetry subgroup that can remain is a compact or noncom-pact U͑1͒.In a realistic low-energy effective theory,CPT-violating terms would break the particle Lorentz group in a manner related to the breaking given by Eq.͑12͒.Since no zeroth-order CPT violation has been observed in experiments, CPT-violating effects in the string scenario are expected to be suppressed by at least one power of the Planck mass rela-tive to the scale of the effective theory.However,the inter-esting and involved issue of exactly how small the magni-tudes of a␮and b␮͑or their equivalents in a realistic model͒must be to satisfy current experimental constraints lies be-yond the scope of the present work.We conﬁne our remarks here to noting that the partial breaking of particle Lorentz invariance discussed above generates an effective boost de-pendence in the CPT-breaking parameters.This could pro-vide a deﬁnite experimental signature for our framework if CPT violation were detected at some future date.C.Field redeﬁnitionsFor the discussions in the previous subsections,we adopted a practical approach to the deﬁnition of CPT and Lorentz transformations.It involves treating C,P,T and Lorentz properties of␺as being deﬁned via the free-ﬁeld theory L0and subsequently using them to establish the sym-metry properties of LЈ.This approach requires caution,how-ever,because in principle alternative deﬁnitions of the sym-metry transformations could exist that would leave the full theory L invariant.Considerﬁrst an apparently CPT-and Lorentz-violating model formed with a␮only,deﬁned in a given inertial frame by the LagrangianL͓␺͔ϭL0͓␺͔ϪL aЈ͓␺͔.͑13͒Introducing in this frame aﬁeld redeﬁnition of␺by a spacetime-dependent phase,␹ϭexp͑ia•x͒␺,͑14͒556763CPT VIOLATION AND THE STANDARD MODELthe Lagrangian expressed in terms of the newﬁeld is L͓␺ϭexp(Ϫia•x)␹͔ϵL0͓␹͔.This shows that the model is equivalent to a conventional free Dirac theory,in which there is no CPT or Lorentz breaking,and thereby provides an example of redeﬁning symmetry transformations to main-tain invariance͓24͔.The connection between the Poincare´generators in the two forms of the theory can be found explicitly by substitut-ing␺ϭ␺͓␹͔in the Poincare´generators for L͓␺͔and ex-tracting the combinations needed to reproduce the usual Poincare´generators for L0͓␹͔.Weﬁnd that the charge and chiral currents j␮and j5␮take the same functional forms in both theories but that the form of the canonical energy-momentum tensor changes,⌰␮␯ϭ12i␹¯␥␮ץJ␯␹ϩa␯␹¯␥␮␹,͑15͒producing a corresponding change in the Lorentz current J␭␮␯.This means that in the original theory L͓␺͔we could introduce modiﬁed Poincare´currents⌰˜␮␯and J˜␭␮␯that have corresponding conserved charges generating an unbro-ken Poincare´algebra.These currents are given as functionals of␺by⌰˜␮␯ϭ⌰␮␯Ϫa␯j␮,J˜␭␮␯ϭJ␭␮␯Ϫx[␮a␯]j␭.͑16͒The existence of this connection between the two theories depends critically on the existence of the conserved current j␮.In the model͑5͒with both a␮and b␮terms,the compo-nent L aЈcan be eliminated by aﬁeld redeﬁnition as before but there is no similar transformation removing L bЈbecauseconservation of the chiral current j5␮is violated by the mass. In the massless limit of this model the chiral current is con-served,and we can eliminate both a␮and b␮via theﬁeld redeﬁnition␹ϭexp͑ia•xϪib•xy5͒␺.͑17͒For the situation with m 0,however,this redeﬁnition would introduce spacetime-dependent mass parameters.The term L aЈin Eq.͑3͒is reminiscent of a local U͑1͒coupling,although there is no local U͑1͒invariance in the theory͑5͒.It is natural and relevant to our later consider-ations of the standard model to ask how the above discussion ofﬁeld redeﬁnitions is affected if the U͑1͒invariance of the original theory is gauged.Then,the term L aЈhas the same form as a coupling to a constant background electromagnetic potential.At the classical level,this would be expected to have no effect since it is pure gauge.However,a conven-tional quantum-ﬁeld gauge transformation involving both␺and the electromagnetic potential A␮cannot eliminate a␮, since the theory is invariant under such transformations.In-stead,the electromagneticﬁeld can be taken as the sum of a classical c-number backgroundﬁeld A␮and a quantumﬁeld A␮,whereupon a␮can be regarded as contributing to an effective A␮.Conventional classical gauge transformations can be performed on the c-number potential A␮,while leav-ing the quantumﬁelds␺and A␮unaffected.This changes the Lagrangian but should not change the physics.In fact, the resulting gauge-transformed Lagrangian is unitarily equivalent to the original one under aﬁeld redeﬁnition on␺of the form discussed above for the ungauged model.To summarize,in the gauged theory the CPT-breaking term L aЈcan be interpreted as a background gauge choice and eliminated via aﬁeld redeﬁnition as in the ungauged case. We note in passing that related issues arise for certain non-linear gauge choices͓25͔and in the context of efforts to interpret the photon as a Nambu-Goldstone boson arising from͑unphysical͒spontaneous Lorentz breaking͓26–32͔.In typical models of the latter type,a four-vector bilinear con-densate͗␺¯␥␮␺͘plays a role having some similarities to that of a␮.The model͑5͒involves only a single fermionﬁeld.All CPT-violating effects can also be removed from certain theories describing more than one fermionﬁeld in which each fermion has a term of the form L aЈ.For example,this is possible if there is no fermion mixing and each such CPT-violating term involves the same value of a␮,or if the fermions have no interactions or mixings that acquire spacetime-dependence upon performing theﬁeld redeﬁni-tions.However,in generic multifermion theories with CPT violation involving fermion-bilinear terms,it is impossible to eliminate all CPT-breaking effects throughﬁeld redeﬁni-tions.Nonetheless,since Lagrangian terms that spontane-ously break CPT necessarily involve paired fermionﬁelds, at least one of the quantities a␮can be removed.This means that only differences between values of a␮are observable. Examples appear in the context of the CPT-violating exten-sion of the standard model discussed in Sec.V.III.RELATIVISTIC QUANTUM MECHANICS In this section,we discuss some aspects of relativistic quantum mechanics based on Eq.͑6͒,with␺regarded as a four-component wave function.The results obtained provide further insight into the nature of the CPT-violating terms and are precursors to the quantumﬁeld theory.The analo-gous treatment in the context of the standard model involves several fermionﬁelds,for which CPT-violating terms of the form L aЈcannot be altogether eliminated.We therefore ex-plicitly include the quantity a␮in the following analysis, even though it could be eliminated by aﬁeld redeﬁnition for the simple one-fermion case.In fact,the reinterpretation of negative-energy solutions causes the explicit effects of a␮to be more involved than might otherwise be expected.The modiﬁed Dirac equation͑6͒can be solved by assum-ing the usual plane-wave dependence,␺͑x͒ϭeϪi␭␮x␮w͑␭ជ͒.͑18͒In this equation,w(␭ជ)is a four-component spinor satisfying ͑␭␮␥␮Ϫa␮␥␮Ϫb␮␥5␥␮Ϫm͒w͑␭ជ͒ϭ0.͑19͒For a nontrivial solution to exist,the determinant of the ma-trix acting on w(␭ជ)in this equation must vanish.This means that␭␮ϵ(␭0,␭ជ),where͓33͔␭0ϭ␭0(␭ជ),must satisfy the requirement͓͑␭Ϫa͒2Ϫb2Ϫm2͔2ϩ4b2͑␭Ϫa͒2Ϫ4͓b␮͑␭␮Ϫa␮͔͒2ϭ0.͑20͒676455DON COLLADAY AND V.ALAN KOSTELECKY´。

my model的英语作文

When writing an essay in English about My Model,its important to consider the context in which the term model is being used.Here are a few different approaches you might take,depending on the specific meaning of model in your essay:1.A Role Model:Begin by introducing who your role model is and why they are important to you. Discuss the qualities and achievements of your role model that you admire. Explain how their actions or life story has influenced your own life or goals.Example Paragraph:My role model is Malala Yousafzai,a Pakistani activist for female education and the youngest Nobel Prize laureate.Her courage and determination to fight for girls education rights in the face of adversity have deeply inspired me.Malalas story has taught me the importance of standing up for what I believe in,even when it is difficult.2.A Fashion Model:Describe the physical attributes and style of the model.Discuss the impact they have had on the fashion industry or their unique contributions to it.Explain why you find their work or presence in the industry notable.Example Paragraph:Kendall Jenner is a fashion model who has made a significant impact on the industry with her unique style and presence.Her tall and slender physique,combined with her ability to carry off diverse looks,has made her a favorite among designers and fashion enthusiasts alike.I admire her for her versatility and the way she uses her platform to promote body positivity.3.A Model in Science or Technology:Introduce the model as a theoretical framework or a practical tool used in a specific field.Explain the principles behind the model and how it is applied.Discuss the benefits or limitations of the model and its implications in the real world.Example Paragraph:The Standard Model in physics is a theoretical framework that describes three of the four known fundamental forces excluding gravity and classifies all known elementary particles.It has been instrumental in understanding the behavior of subatomic particles and predicting the existence of new particles,such as the Higgs boson.However,the models inability to incorporate gravity or dark matter has led to ongoing research for amore comprehensive theory.4.A Model in Business or Economics:Introduce the business or economic model and its purpose.Explain how the model works and the strategies it employs.Discuss the success or challenges associated with the model and its potential for future growth.Example Paragraph:The subscriptionbased business model has become increasingly popular in recent years, particularly in the software panies like Adobe have transitioned from selling packaged software to offering services on a subscription basis,allowing for continuous revenue streams and a more predictable income.This model has been successful in fostering customer loyalty and providing a steady income,although it requires ongoing innovation to maintain customer interest.5.A Model in Art or Design:Describe the aesthetic or functional qualities of the model.Discuss the creative process or design principles that inform the model.Explain the cultural or historical significance of the model and its influence on contemporary art or design.Example Paragraph:The Eames Lounge Chair,designed by Charles and Ray Eames,is a model of modern furniture that has become an icon of midcentury design.Its elegant form,made from molded plywood and leather,exemplifies the designers commitment to blending comfort with aesthetics.The chairs timeless appeal has made it a staple in both residential and commercial settings,influencing countless furniture designs that followed. Remember to structure your essay with a clear introduction,body paragraphs that develop your points,and a conclusion that summarizes your main e specific examples and evidence to support your claims,and ensure your writing is clear,concise, and engaging.。

科学仪器服务公司SIMION

The Industry Standard in 3D Ion and Electron Optics Simulations Scientific Instrument Services, Inc.1027 Old York Rd, Ringoes, NJ 08551Phone: (908) 788-5550Scientific Instument Services, Inc™ SIMION ™Version 8.1SIMION 8.1S IMION 8.1 is a software package used primarily to calculate electric fields, when given a configuration of electrodes withvoltages, and calculate trajectories of charged particles in those fields, when given particle initial conditions, including optional RF, magnetic field, and collisional effects are supported. In this, SIMION provides extensive supporting functionality in defin-ing your system geometry and conditions, recording and visualizing results, and extending the simulation capabilities with user pro-gramming. It is an affordable but versatile platform, widely used for over 35 years to simulate lens, mass spec, and other types of particle optics systems.Typical usage of SIMION is illustrated below for a simple three-element Einzel lens. The geometry consisting of three ring elec-trodes with given voltages is defined (top), and the fields and particle trajectories are calculated and displayed.Electrostatic field solving:SIMION solves fields in 2D and 3D arrays of up to nearbillions of points, with optimizations for systems with symmetry and mirroring, accord-ing to the finite difference method with much optimized linear-time solving. Smallarrays solve in under a minute; very large arrays may take roughly an hour depending onconditions. A “workbench” strategy allows you to position, size, and orient instances(3D images) of different grid densities and symmetries to permit the simulation of muchlarger systems that don't easily fit into a single array. Some magnetic field solving capa-bilities are also available (see following page).Particle trajectory solving: Particle trajectories are calculated given the previouslycalculated or defined fields. The method is Runge-Kutta with relativistic corrections andvariable-length dynamically adjusting and controllable time steps. Particle mass, charge,and other parameters can be defined individually or according to some pattern or distrib-ution. User programming can modify the system during particle flight to inject noveleffects (such ion-gas scattering). Particle tracing is fast _millions of particles can behandled—and they display in real-time. Basic charge repulsion effects, including a pois-son solver can help estimate the onset of space-charge.Viewing of the system is highly interactive, allowing adjustment of parametersand viewing of the system even during particle flight (trajectory calculation). SIMIONsupports cutting away volumes to see trajectories inside, zooming, viewing potentialenergy surfaces, contour lines, and trajectories, and reflying particles as dots for movieeffects.S IMION is suitable for a wide variety of systems: from ion flight through simple electrostatic and magnetic lenses to particle guns to highly complex instruments, including time-of-flight, hemispherical analyzers, ion traps, quadrupoles, ICR cells, and other MS, ion source and detector optics.Time-dependent or RF (low frequency) voltages:Electrode voltages may be controlled in a general way during particle flight via simple user programs _ e.g. to step or oscillate electrode voltages in some manner. Quadrupole mass filter, multipole, and ion trap simulations (above) in the megahertz range are regularly performed. SIMION applies the quasistatic approximation with superposition, which gives fast calculations (assuming the absence of induced magnetic field or radiation effects as would occur in “high frequency” systems having the wavelength below the length of your system).Magnetic fields: SIMION will import magnetic fields, define them analytically or solve them in restricted cases (e.g. Biot-Savart wire currents - left), optionally superimposed on an electrostatic field (e.g.penning trap or ICR cell - right) for the pur-pose of particle flying.ApplicationsRF Quad Mass Filter RF Ion Trap RF Ion Trap (Potential Energy Display)Ion Confinement in Air SolenoidICR CellIon-neutral collisions: SIMION can handle the effects of particles colliding against a background gas, such as for the buffer gas of the ion trap (top), the back-ground gas in an RF ion-funnel (right), or in ion mobility. Multiple collision models are included: Stokes' law, hard-sphere, and a mobility model optimized for high pres-sure “atmospheric” conditions. The parti-cles will diffuse and randomly scatter away from their normal trajectories.RF Ion Funnel Atmospheric Pressure ExampleDefine Your SimulationComplex CAD Modelimported from STL file(left) to a SIMION arrayGeometry (GEM) defi-nition file exampleGeometry definition: A system geometry can be defined by whichever method is most convenient for you: an interactive 3D paint-like program(called “Modify”), CAD import from STL format (supported by most CAD packages), a solid geometry defined mathe-matically via a text file(“GEM files”), and programmatic manipulation of arrays from such languages as Lua, Perl, Python, and C++.Particle initial conditions can be defined in various ways. The“FLY2” format in SIMION allows quick definition of many types ofparticles random distributions and sequences. Particles may also beexhaustively enumerated (optionally imported from a text file).Analysis and Programming SIMION has a number of capabilities for collecting data.•Package contents: a 450-page printed manual, installation CD with software license key number (for receiving softwareupdates), and quick start notes. The installation CD installs the software, examples, and additional documentation.•Documentation:SIMION comes with a 450-page printed manual. Additional documentation and course notes are available electronically, in the examples, or on the web site. See the web site for the user group, software updates, latest SIMION tips, articles, and links to some of the hundreds of scholarly papers that use SIMION.•Updates:Free updates to 8.1.x versions of 8.1 are provided as free downloads from .•Support:Free basic support via email, phone, and forum •Supported systems: Formally tested on Windows 10/8.1/8/Vista/XP, as well as Wine/Linux (and Crossover/Mac). Latest system compatibility information is on .In the example above, trajectories are calcu-lated while phase space data is interactively plotted in Excel via the Lua COM interfaceSIMION can optimize voltages and geometry with simplex optimizer and batch mode capabilities. At left is a SIMION generated surface plot of beam size as a func-tion of two lens voltages. At right is one of the many user programming examples (scattering at surface).Programming in Lua Surface Plot in ExcelScattering Effects at Surface User programming allows the simulation to be extended in many novel ways. During ion flight, you may control electrode voltages (example at right), define or modify fields, scatter or deflect ions (e.g.ion-gas collision models), tune (optimize) lens voltages, compute results, export data to programs like Excel via COM or command-line interfaces, and do many other things. The Lua scripting language is directly embedded in SIMION, and Lua may also call C/C++ or COM routines. Programming may also be used to operate SIMION in batch mode , such as for geometry optimization or to read/manipu-late potential array files.Contents Data recording:The simulation parameters you are interested in (e.g. ion position, velocity, KE, and voltage) can be recorded at various stages in particle flight (e.g. when hitting an electrode and crossing a plane). Data can be recording to the screen or to delim-ited text file for subsequent analysis of fields and trajectories (right). Analysis can be done via SIMION user programming, in a program or language of your choice like Excel, and MATLAB ®.Features in SIMION 8.1 (and 8.2EA/beta)Poisson solver (Refine), fully Dielectric materials (Refine)Supplemental Documentation Integration with Lua/C, Excel, gnuplot, Origin,Large 64-bit array sizes up to 20billion points / 190 GB Improved curved surface handling (“surface enhancement”) gives order of magnitude field accuracy improvement Multicore Refines (8.1)Oblong, non-square grid cells.More AccurateMore Versatile CompatibilityNested refining techniquesSome permeability and mag-High quality 3D (OpenGL)graphics on View screen More examples and documentation New GUI dialog library New programming API’s:。

手性与对称性

Chapter5Chiral Dynamics5.1What is spontaneous symmetry breaking?Symmetries and their breakings are important part of modern physics.Spacetime symmetry and its supersymmetric extensions are the basis for building quantumﬁeld theories.Internal symmetries, such as isospin(proton and neutron,up and down quark symmetry),ﬂavor,color etc.,form the fundamental structure of the standard model.On the other hand,studying symmetry breakings is as interesting as studying symmetries themselves.As far as we know,there are three ways to break a symmetry:explicit breaking,spontaneous breaking,andﬁnally anomalous breaking.In this part of the lectures we will concern ourselves with theﬁrst two types of breakings of the so-called chiral symmetry,the exact meaning of which will become clear later.We will come to the anomalous symmetry breaking towards the end of the course.In quantum mechnics,a symmetry of a hamiltonian is usually reﬂected in its energy spectrum. For instance,the rotational symmetry of a three-dimensional system often leads to a2ℓ+1-fold degeneracy of the spectrum.This standard realization of a symmetry is called Wigner-Weyl mode. On the other hand,in the late50’s Nambu and Goldstone discovered a new way through which a symmetry of a system can manifest itself:spontaneous breaking of the symmetry.This realization of a symmetry is called Nambu-Goldstone mode.To understand the Nambu-Goldstone realization of a symmetry,let us recall a related problem in statistical mechanics:second-order phase transitions.We have many examples of the second-order phase transitions in which a continuous change of order parameters happens.Consider a piece of magnetic material.Its hamiltonian is certainly rotationally symmetric and therefore normally one would expect its ground state wave function is also rotationally symmetric.This apparently is not the case below a certain critical temperature at which a spontaneous magnetization occurs. The magnitization vector points to a certain direction in space,and hence the rotational symmetry is lost.We say in this case that the rotational symmetry is spontaneously broken.Likewise,for a conductor below a certain temperature,the electromagnetic U(1)symmetry is spontaneously broken and the wavefunction of the Cooper pairs developes certain classical value.A useful mathematical formulation of the SSB is the concept of the eﬀective action.Let us introduce thisﬁrst.Consider a scalarﬁeld theory with lagrangian density L(φ).We deﬁne the green’s function8182CHAPTER5.CHIRAL DYNAMICS functional or generating functional Z(j)asZ(j)=∞i=0i n[Dφ]e i d4x L(x).(5.2)We deﬁne the connected green’s function G(n)c throughW(j)=∞i=1i nδj(x),(5.4)from which one can solve j(x)as a functional ofφ(x).Perform now the Legendre transformation,Γ(φ)= W− d4xj(x)φ(x) |j=j(φ)(5.5) ThenΓ(φ)is the generating functional for the one-particle irreducible Green’s functionsΓ(n)(x1,···,x n),Γ(φ)= n=11δφ(x).(5.7)Eﬀective action can be computed through the shift ofﬁeld in the lagrangianφ→φ+φc,and calculating the1PI contribution to the eﬀective W.There are two popular usage of the eﬀective action formalism:First,the eﬀective action containsall the1PI which are the target for renormalization study.The renormalization condition can5.1.WHAT IS SPONTANEOUS SYMMETRY BREAKING?83 easily expressed in terms of1PI,like the mass of the particles and coupling constants.Moreover, the symmetry of these1PI can be expressed in terms of the Ward-Takahashi identities which can be summarized in terms of a simple equation for the eﬀective action.This equation can be used to prove the Goldstone theorem.Second,the eﬀective action can be used as a thermodynamic function with natural variableφc which diagnoses the phase structure of the system.For instance, according to Colemann-Weinberg,the natural phase of the massless scalar electrodynamics is the Higgs phase in which the vector and scalar particles aquire mass through radiative corrections. Another use of the eﬀective action is in cosmology.The spontanous symmetry breaking happens only if there is a degeneracy in the vacuum.This degeneracy can arise from certain symmetry of the original lagrangian.Consider a symmetry transformation ofﬁelds,φi(x)→φ′i(x)= j L ijφj(x),(5.8)here we have assumed multipleﬁelds with i=1,...,n.If the action and measure are both invariant, then the eﬀective action is invariant under a similar transformation of the classicalﬁeldsΓ[φ]=Γ[Lφ].(5.9) As we mentioned before,the vaccum state is a solution¯φof−Γ[φ]at its minimum.If the solution is invariant L¯φ=¯φ,i.e.the vacuum is invariant under the symmetry transformation,the vacuum is unique.On the other hand,if L¯φ=¯φ,the solution is not.Then we have many degenerate vacua which are all physically equivalent.By choosing a particular barφas the true vaccum,we have a spontaneous symmetry breaking.According to the above discussion,the key condition for SSB is there are multiple,equivalent vacua.Although it is easy toﬁnd ground state degeneracies in the classical systems,in quantum systems it is diﬃcult to have multiple vacuum.For instance,in a potential with a double well,the ground state is a non-degenerate symmetrical state.In other words,the real vacuum is a linear combination of the various classical vacua.The same thing happens for a rotationally symmetric system in which the ground state has J=0,i.e.,allθangles are equally probable.There are special cases in quantum mechanics in which the ground state may be degenerate. For instace,in an atom with a ground state J=0,the state can be prepared in the eigenstates of J2and J z.However,there is no SSB because the states of diﬀerent J z are not equivalent vacua in the sense that they blong to the same Hilbert space and are easily connected through a transitions operators.Therefore,the spontaneous symmetry breaking happens only if the volume of the system is approaching inﬁnity and the transition rate between the degenerate states goes to zero.In this case,it turns out that the vacuum states are not representations of the symmetry generators. Rather they are eigenstates of the conjugating coordinate operators and are superposition of states with symmetry quantum numbers.Any perturbation which causes the transition between diﬀerent vacua have exponentially small matrix elements.On the other hand,the diagonal matrix elements of the perturbation is much larger than the oﬀ-diagonal matrix elements.In other words,the vacuum states are those with deﬁnite¯φ,or in the rotationally symmetric system,deﬁniteθ.So in the limit of inﬁnit volume,the states with deﬁnite¯φbecome the exact vacua.It can be shows that with local hamiltonian and operators,diﬀerent vacua obey the super-selection rule.Assume the degenerate vacua are|v i andv i|v j =δij(5.10)84CHAPTER5.CHIRAL DYNAMICS By considering the matrix element of v i|A( x)B(0)|v j in the limit of x→∞,it can be shownu i|A(0)|u j =δij a i.(5.11) Therefore the local operators have noﬁnite matrix elements between diﬀerent vacuum states. 5.1.1SSB and Space(-time)DimensionsIn aﬁnite quantum mechanical system,there is no SSB.For discrete symmetry,such as Z2symmetry (σi→−σi)in the Ising model,it cannot be broken in one-dimensional(0+1)system.This is known in1938to Peierls.But,it can be broken in two-dimensional(1+1)system.For example, the Onsagar solution contains a spontaneous magnetization for a two-dimensional Ising model.For continuous symmetry,it cannot be spontaneously broken in two-dimensional system.This is called the Mermin-Wagner-Coleman theorem.For example,the classical Heisenberg model consists of interactions of spins living on a n-dimensional sphere.The system has O(n)symmetry.This model has spontaneous symmetry breaking only in3D.To see the MWC theorem,let’s assume there is a SSB in2D.Then we have massless Goldstone bosons.The correlation of these massless Goldstone bosons reads0|φ(x)φ(0)|0 = d2k2πk1cos(k1x1)e ik1x0(5.12)which is hopelessly infrared divergent.This strongﬂuctation will destroy any long-range order. In a two-dimensional classical Heisenberg model,an disordered phase has as much weight as an ordered one.5.2SSB of the continuous symmetry and Goldstone TheoremIn the case of the spontaneous breaking of a continuous symmetry,a theorem can be proved.The theorem says that the spectrum of physical particles must contain one particle of zero mass and spin for each broken symmetry generator.Those particles are called Goldstone bosons.Consider an inﬁnitesimal transformationφi→φi+iǫa(t aφ)i.(5.13) The same transformation leaves the eﬀective action invariantijd4xδΓ∂φit a ijφj=0,(5.15)This relation is true independent ofφ.Diﬀerentiate the above equation with respect toφk and take φ=¯φin a vacuum,∂2V(φ)5.2.SSB OF THE CONTINUOUS SYMMETRY AND GOLDSTONE THEOREM85According to the deﬁnition of the eﬀective potential,we have∂2V (φ)2∂µφi ∂µφi −14(φi φi )2(5.19)In the tree approximation Γ=V 3L ,we haveV =14(φi φi )2(5.20)If M 2is negative,we have¯φi ¯φi =−M 2/g(5.21)We can choose a solution as ¯φi =(0, 0∂φi ∂φj |φ=¯φ=2g ¯φi ¯φj =(0,...,0,2|M 2|)(5.22)Thus the last particle now has mass √86CHAPTER5.CHIRAL DYNAMICS we have the eﬀective classical hamiltonianH eﬀ= N1∂φi 1···φi N (5.27)From equations derived earlier,it is easy to see that the amplitude for a zero-momentum Goldstone boson disappearing into the vacuum is zero.The amplitude for a zero-momentum goldstone boson to make transition to another boson is zero.Finally,the amplitude for three massless Goldstone bosons to make transtion is zero.This is in fact true to all orders.Let us consider now the interactions of Goldstone bosons with other massive particles.The following approach assumes exact symmetry.To calculate the process ofα→β+B a,we start from the matrix element with the corresponding conserved currentβ|Jµa|α .(5.28) The current supports a momentum transfer q=pα−pβ.Clearly the most important contributionto matrix element comes from the Goldstone boson pole which has the following structureiF qµMβB,αFqµNµβ+J,α.(5.30)This is a form of Ward identity.If Nµhas no pole,then the process of emitting a Goldstone boson vanishes as q→0.This is called the Adler zero.The most important contribution in the regular term comes from the Feynman diagrams in which J acting on the external line.In this case,there is a heavy-particle pole which enhance the contribution.The pole contribution can often be calculated or extracted from experimental data,from example,the nucleon pole contribution is related to the neutron beta decay constant g A.Knowing g A,we can calculate the meson-nucleon interaction as we shall do in the next section.The above result can also be derived from a theory with explicit breaking of the symmetry.This approach is called PCAC.In this case,the masses of the Goldstone bosons are not exactly zero, butﬁnite.They are called pseudo-Goldstone bosons.Let us consider the SSB of an approximate symmetry.In this case,the vacuum is no longer degenerate,and strictly speaking,there is no spontaneous symmetry breaking.This is very much like a magnet in an external magneticﬁeld(ﬁrst order phase transition).In the following we would like toﬁnd the constraint on the vacuum from the symmetry breaking eﬀects;we also want to derive the masses of the pseudo-Goldstone bosons.5.3.PION AS GOLDSTONE BOSON,PCAC87Now the eﬀective potential has two terms V(φ)=V0(φ)+V1(φ).The real solution isφ=φ0+φ1 which is no longer degenerate.The condition onφ0andφ1is contained in the expanded version of ∂V(φ)/∂φi|φ=¯φ=0∂2V0=0(5.31)∂φiUsing the equation we found early,we have∂V1(φ0)(t aφ0)i(5.33)∂φi∂φjwhich vanish to the zeroth order.To the frist order,weﬁndM2ab=− cd F−1ac F−1bd 0|[T a,[T b,H1]]|0 (5.34) where T a is the quantum generator of the symmetry group.5.3pion as goldstone boson,PCACOne of the most interesting examples of SSB is exhibited by fundamental strong interactions:quan-tum chromodynamics.Consider the QCD lagrangian.The only parameters with mass dimension are quark masses.For ordinary matter,we just consider up and down quarkﬂavors.The QCD scale ΛQCD is about200MeV,which is much larger than the up and down quark masses(5to9MeV). Therefore,to a good approximation,we can negelect the quark masses in the QCD lagrangian. Then the QCD lagrangian has the U(2)×U(2)chiral symmetry.Recall the chiral projection operators P L=(1−γ5)/2and P R=(1+γ5)/2,whereγ5is diag (-1,1),which project out the left-handed and right-handed quarkﬁelds,ψL,R=P L,Rψ.(5.35) Then the QCD lagrangian we can be written in terms ofL=ψR(i D)ψR−188CHAPTER5.CHIRAL DYNAMICS where U L,R are unitary matrices in the two-dimensionalﬂavor space.Since U(2)=U(1)×SU(2),we have two U(1)symmtries.From now on,we focus on the two SU(2)symmetries,leaving the U(1) symmetries to later discussion.According to Noether’s theorem,the SU L(2)×SU R(2)chiral symmetry leads to the the following conserved currents,jµL,R=¯ψL,R t aγµψL,R,(5.38) where t a=τa/2andτa is the usual Pauli matrices.We have the vector and axial vector currents from the linear combinations,j aµV=¯ψt aγµψ=jµL+jµRj bµA=¯ψt aγµγ5ψ=jµR−jµL.(5.39) From the above currents,we can deﬁne the charges Q a and Q a5in the usual way.And it is easy to see that the charges obey the following algebra:[Q a,Q b]=iǫabc Q c;[Q5a,Q b]=iǫabc Q5c;[Q5a,Q5b]=iǫabc Q c.(5.40)From the above,weﬁnd that Q a forms a subgroup of the chiral symmetry group and is called the isospin group.From the experimental hadron spectrum,weﬁnd that the isospin subgroup is realized in Wigner-Weyl mode.For instance,the pion comes in with three charge states and near degenerate mass.The proton and neutron also have nearly degenerate mass.However,the spectrum does not show the full chiral symmetry.For instance,the three pion states do not form an irreducible reps of the chiral group.Together a scalar particleσ,they form(1/2,1/2)reps. Therefore,if the chiral symmetry is realized fully in Wigner-Weyl mode,there must be a scalar particle with the same mass as the pion.We do not see such a particle in Nature.Thus,the chiral group SU L(2)×SU(2)R must break spontaneously to the isospin subgroup SU(2).Thus the QCD vacuum|0 satisﬁesQ a|0 =0,Q5a|0 =0.(5.41) According to Goldstone’s theorem,there are three massless spin-0pseudo-scalar bosons.They are pseudoscalars because Q5a changes sign under parity transformation.Of course,in the real world,we don’t have massless pseudoscalars.We have pions.The pion masses are indeed much smaller than a typical hadron mass.For instance,the rho meson has mass 770MeV.The nucleon mass is940MeV.And the pion mass is140MeV.The pions are called pseudo-Goldstone bosons because the chiral symmetry is not exact.It is broken by theﬁnite up and down quark masses.H1=m u¯u u+m d¯dd.(5.42) If we write u in terms of left and right-handedﬁelds,we haveH1=m u(¯u L u R+¯u R u L)+m d(¯d L d R+¯d R d L).(5.43)Therefore the left and right-handedﬁelds are now coupled through the mass terms.The mass operator transforms as the components of(1/2,1/2)representations of the chiral group.Using the relation we found earlier,we can calculate the pion mass,m2π=−(m u+m d) 0|¯u u+¯dd|0 /f2π(5.44)5.3.PION AS GOLDSTONE BOSON,PCAC89 where 0|¯u u+¯dd|0 is the chiral condensate.Since¯u u is a part of the representation(1/2,1/2),it vacuum expectation value vanishes ordinarily because of the chiral symmetry.However,it has a vacuum expectation value because of the vacuum is no longer chirally symmetric(chiral singlet). In fact,the vacuum contains all(k,k)type of representations because the vacuum has zero isospin. Any chiral tensor of type(k,k)has non-zero vacuum expectation value.The pion decay constant fπis deﬁned from0|jµa(x)|πb =ipµδab fπe−ip·x.(5.45) It can be measured from the semi-leptonic weak decayπ+→µ+νµrateG2F m2µf2π(m2π−m2µ)2Γ=U(p′)t a[g A(q2)γµγ5+g p(q2)qµγ5]U(p),(5.48) where q=p−p′and U’s are the on-shell Dirac spinors of the nucleon states.Multiplying qµto both sides of the equation and using current conservation and Direc equation(p−M)U(p)=0, we have−2Mg A(q2)+q2g P(q2)=0.(5.49) g A(q2)in the limit of q2→0is just the neutron decay constant(the axial current is part of the weak interaction current)and has been measured accuratelyg A(0)=1.257.(5.50) Thus according to the above equation g P(q)must have a pole in1/q2.This pole corresponds to the intermediate massless pion contribution to the interaction between the the axial current and the nucleon.If we introduce the pion-nucleon interaction vertex gπNN¯Niγ5τa Nπa,the contribution to the axial current matrix element is(ifπqµ).(5.51)i2gπNNq2In the limit of q2→0,weﬁnd the following celebrated Goldberger-Treiman relationg A(0)M=gπNN fπ.(5.52) Using gπNN from experimental data(g2πNN/4π=14.6),weﬁnd that the above relation is obeyed at better than10%level.90CHAPTER5.CHIRAL DYNAMICS According to the recipe derived from the previous section,we calculate the interactions between the soft pion and the nucleon system as follows.First use a vertex iqµ/fπconnecting the Goldstone boson to the axial current.Then the non-singular part of the axial current interaction with the nu-cleon is approximated through the g Aγµγ5vertex.This yields the eﬀective pion-nucleon interaction vertex i qγ5/fπ.This is a peudo-vector interaction.Another way to study the interactions among the pions and with other particles is through what is called the PCAC(partially-conserved axial-vector current),in which we assume there is a small explicit symmetry breaking through nonvanishing quark masses.Applying the derivative operator to the current matrix between the vacuum and the pion,we have0|∂µjµa|πb =m2πδab fπ.(5.53) The right-hand side is proportional to the pion mass squared.This motivates the assumption that∂µj aµ=m2πfππa,(5.54)whereπa is a pion interpolatingﬁeld.Of course,the above relation is in some sense empty because any pseudo-scalar operator can be used as an interpolatingﬁeld for pion.The content of the PCAC is that axial current at zero momentum transfer(this is the place where we know how to calculate the matrix element)is dominated by the pion contribution at q2=m2π.In other words, the variation of the matrix elements of the axial current from q2=0to m2πis smooth.In fact,we can derive the Goldberger-Treiman relation using PCAC andﬁnd now one has to use g A(0)instead of g A(m2π).The content of PCAC is that the variation of this small.Therefore,when the pion energy is small,we can calculate using PCAC.PCAC can be used to study the multi-pion interactions.For instance,consider the amplitudeT abµν= d4xe iqx H(p2)|T A aµ(x)A bν(0)|H(p1) (5.55) Applying diﬀerentical operators to the above quantity,we derive a Ward ing PCAC, one can calculate the pion-nucleon scattering amplitude at low-energy.However,it turns out that it is much easier to get the predictions using the low-energy eﬀective theory.5.4the linearσmodelMany of the essential physics exhibited in spontaneous breaking of the chiral symmetry can be illustrated by a simple phenomenological model.This is very similar to the Ginsburg-Landau theory for second-order phase transitions.This model isﬁrst introduced by Gell-Mann and Levy, and is called the linearσmodel.The lagrangian is,L=L S+cσ,L S=¯ψ[i∂+g(σ+i π·τγ5)]ψ+12(σ2+ π2)−λ5.5.EFFECTIVE FIELD THEORY:CHIRAL PERTURBATION THEORY WITH PIONS91 term cσ,the lagrangian is clearly symmetric under the chiral SU L(2)×SU R(2),and the correspond-ing vector and axial vector current isτajµa=¯ψγµψ+(σ∂µπa−πa∂µσ)(5.57)2After introducing the symmetry breaking term,the axial vector current is no longer conserved.We have instead∂µA aµ=−cπa(5.58) according to the equation of motion.The above has the form of PCAC.Whenµ2<0,the spontaneous symmetry breaking happens.The potential has its minimum not atπa=σ=0but atπ2+σ2=v2,where v2=−µ2/λ.Thus,the shape of the potential is a Mexican hat.There are inﬁnite many degenerate minima.We need to choose a particular direction as our vacuum state.If we want to keep the isospin group intact,we takeσ =v.(5.59) The pion excitation corresponds to the motion along the minima and therefore has zero energy unless the wavelength isﬁnite.Theσmass corresponds to the curvature in theσdirection and is 2λv2.The nucleon also get its mass from spontaneous symmetry breaking and is−gv.From the PCAC,weﬁnd that fπ=−v.When the symmetry breaking term is introduced,the Mexican hat is tilted.In this case,the minimum of the potential is unique and the pion excitations do have mass.5.5eﬀectiveﬁeld theory:Chiral Perturbation theory with pionsCurrent algebra and Ward identity approach were popular in the60’s for calculating Goldstone boson interactions.However,they are tedious.In1967,Weinberg used the nonlinearly-transformed eﬀective lagrangian to study the Goldstone boson interactions.This is the precursor of eﬀective ﬁeld theory approach which is popular today.The key observation is that when the Goldstone boson energy is small,the coupling is weak. Therefore their interactions must be calculable in perturbation theory.However,in the strong interactions,we also have the usual QCD or hadron(rho meson or nucleon)mass scale.The physics at these two diﬀerent scales have to be separated before one can apply chiral perturbation theory.The physics at QCD or hadron mass scale can be parametrized in terms of various low-enegy constants which can be determined from experimental data.Through a particular model,we demonstrate the separation of physics through nonlinear trans-formations.Weﬁrst perform a symmetry transformation at every point of the spacetime to get rid of the Goldstone boson degrees of freedom.We then re-introduce them through the spacetime-dependent symmetry transformation.When the Goldstone-bosonﬁelds are constant,the transfor-mation is the usual chiral tranformation;and the Goldstone bosonﬁelds disappear.Therefore,in the new lagrangian,the Goldstone boson interaction must have derivative-type interactions.Consider the linear sigma model.Let us introduce a(1/2,1/2)2×2matrixU=σ+i π·τ(5.60)92CHAPTER5.CHIRAL DYNAMICSUnder the chiral transformation,we haveU→U L UU†R(5.61) We can write the linear sigma model asL=14Tr[UU†]−λπ2+σ2.We reintroduce back the goldstone boson by parametrizing the U including the axial transformation parameters,U=σe i πa(x)·τa/fπ(5.64) whereπa=fπθa A is now the Goldstone bosonﬁeld.For the convenience,we call the exponential factorΣ.Now substituting U=σΣinto the original lagrangian,we get,L=14σ2Tr[∂µΣ∂µΣ†]−14σ4.(5.65) Now the Goldstone bosonﬁelds contain derivatives and therefore the above lagrangian will pro-duce appropriate Goldstone boson interactions.Since theσparticle has a typical hadronic mass, its eﬀects can be integrated out completely and theσis then replaced by its expectation value. Therefore,the eﬀective intereaction lagrangian for pion is justL(2)ππ=f25.5.EFFECTIVE FIELD THEORY:CHIRAL PERTURBATION THEORY WITH PIONS93 Using L=I− i V i+1,we haveν= i V i(d i−2)+2L+2.(5.69)Therefore the lowest power of Q in any pion process is2.We can use the above leading order lagrangian to calculate the interactions between the pions. Expand in1/fπto to the second order,we have[(∂µ π· π)2− π2(∂µ π)2]+...(5.70) L(2)ππ=16f2πThe second term can be used to calculate the S-matrix element between pion scattering.Assume the incoming pions with momenta p A and p B and isospin indices a and b and the outgoing pions with momenta p C and p D and isospin indices c and d.We have the following leading-order invariant amplitude(S=1−iM),M=−f−2π(δabδcd s+δacδbd t+δadδbc u)(5.71) where s=(p A+p b)2,t=(p A−p C)2and u=(p A−p D)2are called Mandelstam variables.5.5.1Scalar and Pseudoscalar SourcesWe can include the quark mass eﬀects at this order.The quark mass term transforms like(1/2,1/2) under chiral transformations.In general,let us introduce s and p source in the QCD lagrangianL sp=−¯ψs(x)ψ+¯ψiγ5p(x)ψ=−¯ψR(s+ip)ψL−¯ψL(s−ip)ψR(5.72) Call s−ip=χand s+ip=χ†.Then the interaction is invariant ifχ→LχR;χ†→Rχ†L†(5.73) Without the p source,χ∼χ†∼s∼m q,which counts as second order in momentum.The eﬀective lagrangian then containχas a O(p2)external source.The lowest order isL(2m)ππ=B Tr(Σχ†+Σ†χ).(5.74) When expanded to the leading order,the above gives the pion mass contribution if B=f2π/4and χ=m2π.The next-order contribution ism2π94CHAPTER5.CHIRAL DYNAMICSAt the threshold where s=4m2π,t=u=0,we haveM=−m2πf−2π[3δabδcd−δacδbd−δadδbc].(5.77) The scattering amplitude f=−M/8π√16f4π −1µ2−1µ2−1µ2 −14c2(t2+u2) +crossing(5.78)where c1and c2are constants which must be determined from experimental data.In fact,there are also pion mass contribution at this order which we will not go into.The p4-order mass term include the followingL4Tr(DµΣ†DµΣ)Tr(χ†Σ+χΣ†)+L5Tr(DµΣ†DµΣ)(χ†Σ+χΣ†)+L6(Tr(χ†Σ+χΣ†))2+L7(Tr(χ†Σ−χΣ†))2+L8Tr(χ†Σχ†Σ+χΣ†χΣ†)+H2Tr(χ†χ)(5.79) where H2is pointless because there is no meson depedence.5.5.2Electromagnetic and Axial InteractionsWhen there are electromagnetic and weak interactions with the Goldstone boson system,we need to construct a gauge theory in which the eﬀective theory is gauge invariant under gauge transfor-mations.Introduce the the following coupling the QCD lagrangianL=¯ψ(γµvµ(x)+γµγ5aµ(x))ψ=¯ψLγµ(vµ−aµ)ψL+¯ψRγµ(vµ+aµ)ψR(5.80) If vµand aµare gaugeﬁelds,under gauge transformation,they must transform in the following way,vµ−aµ→L(vµ−aµ)L†+iL∂µL†vµ+aµ→R(vµ−aµ)R†+iR∂µR†(5.81) The above equation means that these gaugeﬁelds have to appear together withΣin the following formDµΣ=∂µΣ−i(v−a)µΣ+iΣ(v+a)µ(5.82)5.6.BANKS-CASHER FORMULA AND VAFA-WITTEN THEOREM95 Then all the partial derivatives will be replaced by the above covariant derivatives.For example,consider the electromagnetic interaction of the pions.In this case,we replace vµ=−ie(τ3/2+1/6)Aµwhere e is the charge of a proton andτ3is the isospin and1/6is the hypercharge.Then the partial derivative becomes,DµΣ=∂µΣ+ieAµ[τ3∂x1+∂A2∂x3+∂A496CHAPTER5.CHIRAL DYNAMICS The electricﬁeld E in the Euclidean space is the imaginary of that in the Minkowski space and so E2→−E2,and FµνFµν=−2(E2−B2)→2(B2+E2)=FµνFµν.We also deﬁne the Euclidean version of theγmatrix withγE4=γ0andγE i=−iγi and the commutators now become{γEµ,γEν}=2δµν(5.91) The newγmatrices are hermitian.The QCD lagrangian is nowL QCD=−4FµνFµν (5.92)Notice thatγµDµis now an antihermitian operator.We can deﬁne the Euclidean L to absorb the minus sign.Consider now the exponential factor exp(iS)in the path integral.After rotation,the integral d4x becomes−i d4x.The−i here cancels the i in front of the iS and deﬁne the Eulidean action asS E=− d4x L(5.93) Therefore the integration meansure becomes exp(−S E)Let us see how the spontaneous symmetry breaking takes place in QCD.To this goal,we need to introduce an explicit breaking of the symmetry.For example,we give a small mass to quarks. Consider the expectation value of ¯u u .We writeV4d4x u(x)¯u(x)=− [DA]e−S Y M Det(D+M)1D+m u].(5.94)where Tr is over spatial,color,and spin indices.Now consider the eigenstates of D.Because it is an anti-hermitian operator,we haveD|λ =iλ|λ ,(5.95) whereλis real.The diﬀerent|λ are orthogonal and therefore we haveTr[1iλi+m u.(5.96)On the other hand,we have Tr(D+M)=Tr(−D+M)because(γ5)2=1.We get thenTr 1iλi+m u+1m2u+λ2i(5.97)Intrduce now aδ(λ−λi)and integration overλ.We have then¯u u =−dλρ(λ)m uZV4 [DA]exp(−S Y M)Det(D+M)i2δ(λ−λi)(5.99)。

On a Factorization of Symmetric Matrices and Antilinear Symmetries

2
3
Motivation: Consequences of antilinear symmetries
Consider a diagonalizable linear operator H : H → H acting in a ﬁnite-dimensional complex inner-product space H with inner-product ( , ). Let n label the eigenvalues En of H , µn be the multiplicity of En , and ψn,a be the eigenvectors corresponding to the eigenvalue En where a ∈ {1, 2, · · · , µn } is the degeneracy label. Then it is well-known [3] that the adjoint ˜n of H ∗ are complex conjugate of those of H , i.e., H ∗ of H is diagonalizable; the eigenvalues E ˜n = E ¯n ; the multiplicity of E ˜n is equal to µn ; and one can choose the eigenvectors φn,a of H ∗ E in such a way that for all spectral labels m, n and degeneracy labels a, b, (φn,a, ψn,a ) = δn,m δa,b . Clearly, both sets of eigenvectors ψn,a of H and eigenvectors φa,n of H ∗ form bases of H; {ψn,a , φn,a } is a complete biorthonormal system. Recently [6], we have shown that if the eigenvalues of H are real, then H has an antilinear symmetry. More generally, we proved the following theorem. Theorem 1: The presence of an antilinear symmetry of H is a necessary and suﬃcient condition for the eigenvalues of H to either be real or come in complex conjugate pairs. The proof of Theorem 1 uses the following lemma. Lemma 1: Every diagonalizable linear operator H : H → H acting in a ﬁnitedimensional complex inner-product space H is T -Hermitian, H ∗ = T H T −1 , for some Hermitian, invertible, antilinear operator T : H → H. It turns out [6] that any such T may be expressed in terms of the eigenvectors φn,a of H ∗ according to

Remodeling the Pentagon After the Events of 22306

a r X i v :h e p -p h /0606313v 1 29 J u n 2006Preprint typeset in JHEP style.-PAPER VERSIONT.Banks Department of Physics University of California,Santa Cruz,CA 95064E-mail:banks@ and NHETC,Rutgers University Piscataway,NJ 08540Abstract:The meta-stable SUSY breaking mechanism of Intriligator Seiberg and Shih can be used to simplify the Pentagon model of TeV scale physics.The simpliﬁed model has only a single scalar ﬁeld and no troublesome low energy axion.One signif-icant signature is l +l −X plus missing energy,where X might be the two photons of gauge mediated models,but is likely to be diﬀerent.There is a new strongly interacting sector with a scale around 1.5TeV.The penta-hadrons of this sector have masses of order 6TeV or more.Dark matter is probably the pseudo-goldstone boson of sponta-neously broken penta-baryon number.This can be a viable dark matter candidate if an appropriate asymmetry in penta-baryon number is generated in the early universe.The pseudo-Goldstone particle has a mass of ∼1eV and is produced predominantly in ﬂavor changing charged current decays of ordinary particles.The model solves the ﬂavor problems of SUSY,but has two low energy CP violating phases,whose value is strongly constrained by experiment.Keywords:Cosmological SUSY Breaking.1.IntroductionIn a recent paper[1],I proposed an explicit model,which implemented the idea of Cos-mological SUSY Breaking(CSB).Apart from theﬁelds of the supersymmetric standard model(SSM),the model consisted of the Pentagon-a new strongly interacting SU(5) super-QCD with5ﬂavors of penta-quark,and three singletﬁelds,S,G,T with a variety of Yukawa couplings to the Pentagon and the SSM.The intricate pattern of singlets was required to motivate the possibility of a meta-stable SUSY breaking vacuum state of theﬂat space quantumﬁeld theory.It also caused a potential phenomenological problem-a low scale QCD axion.One could invoke higher dimension operators,which raised the axion mass and made it barely compatible with observation.This of course1removed the model’s solution of the strong CP problem.It also introduced aﬁne tun-ing of the electroweak scale,of order1%.This was required to raise the axion decay constant above the laboratory bounds.The Pentagon Paper was written before the world changed on02/23/06.The Neo-conservative1revolution in SUSY breaking,heralded by the paper of Intriligator, Seiberg,and Shih(ISS)[3]now provides us with a plethora of calculable SUSY breaking models,where SUSY is broken in a meta-stable state-exactly what is needed to solve the problems at the Pentagon.ISS prove that when a mass term m ISS tr P˜P is added to SUSY QCD with N C+1≤N F≤3N C4m P1This term is motivated by the fact,pointed out to the author by N.Seiberg,that meta-stable SUSY breaking was advocated in unpublished work by M.Dine,which was done in the run-up to the Aﬄeck Dine Seiberg[2]discovery of models of dynamical SUSY breaking.2rapid deployment to solve all2of the problems of the supersymmetric standard model. It involves a single scalarﬁeld S with discrete R charge2.The only marginal/relevant couplings of S are encoded in a superpotential:W S=g S SP A i P j A Y i j+gµSH u H d+g T S3.Y is the unique traceless SU(1,2,3)invariant matrix in the fundamental representation of SU(5).Note that the SU(1,2,3)standard model gauge group is embedded in the obvious fashion into the vector SU(5)ﬂavor group of the Pentagon.In fact,we could consider a more general version of the model in which we require only that the linear combination of penta-quark bilinears to which S couples,is linearly independent of the combination to which m ISS couples.The form we have written may have a group theoretic justiﬁcation at the uniﬁcation scale,which we will discuss below.2.Known knownsThe full low energy Lagrangian of our model isL=d4θ[P∗e V P+Q∗e V Q+L∗e V L+(¯U)∗e V¯U+(¯D)∗e V¯D+(¯E)∗e V¯E]+[ d2θ(( τi W iα)2+P A i˜P j A(m ISSδi j+g S SY i j)+gµSH u H d+g T S31+H u Q m(λu)m n¯U n+H d Q m(λd)m n¯D n+H d L m(λL)m n¯E n+2As is conventional in communications from the Pentagon,we are here indulging in a bit of hyper-bole.3The parameter m ISS is assumed to be induced by Cosmological SUSY Breaking ,CSB,as in[4].The c.c.,λis a tunable parameter3,and m ISS scales likeλ1/4as λ→0.In this limit,the low energy Lagrangian has an U(1)R symmetry,which has no SU P(5)anomaly.P and˜P have R charge zero.A discrete Z4subgroup of this U(1) is assumed to be an exact symmetry of the S-matrix whenλ=0.S has R charge2. The R transformation properties of the SSM chiral multiplets can be chosen so that the only perturbative baryon or lepton number violating interaction of dimension≤5 is the neutrino see-saw term,d2θ(H u L m)(H u L n)S mn3Actually it is discretely tunable.π(RM P)2is the logarithm of an integer number of states.R is the dS radius.In a more ambitious model based on holographic cosmology,one gets a distribution of asymptotically dS universes with diﬀerentλ,andλcan be anthropically selected.However,this model has no other testable consequences once the value ofλis chosen,so there is no point in discussing it here.4were not under analytic control.That state had a non-vanishing expectation value for the penta-baryon number violating operators<ǫA1...A N C P i1A1...P i N CA NC>=ΛN C NCǫi1...i N C,<ǫA1...A N C ˜P A1i1...˜P A N Ci NC>=ΛN C NCǫi1...i N C.By contrast,the meson operators P A i˜P j A have vanishing expectation value in this state.This model is merged with the hypothesis of CSB by making two assumptions. First the parameter m is determined so that the gravitino mass at the meta-stable SUSY violating vacuum is given by the CSB formulam3/2=γλ1/4(M p/m P),where M P=√Λ5.Note that this is a term in the Lagrangian at a scale≫Λ5where the Pentagon interac-tions are weak.From the point of view of standard eﬀectiveﬁeld theory,it is extremely peculiar to have the IR scaleΛ5appear in the denominator of this parameter.However, the diagrams contributing to the argument for the CSB formula that I presented in[4], combine infrared propagation in a single horizon volume,with UV dynamics in the vicinity of the horizon.It is plausible that they contain such inverse IR scales.The second input from CSB is that we add a constant W0to the superpotential to tune the c.c.at the meta-stable minimum toλ.Again,there is no reason to do this in eﬀectiveﬁeld theory,though in this case it would be the strategy of any eﬀectiveﬁeld theorist who wanted the meta-stable vacuum to be of phenomenological relevance.The logic for adding W0in CSB is diﬀerent.λis viewed as a high energy input parameter determined by the number of quantum states in the asymptotic de Sitter space to which the system is converging.It cannot be renormalized,and the constants in eﬀectiveﬁeld theory must be tuned to reproduce its input value.It should be emphasized that from the point of view of low energy eﬀectiveﬁeld theory,the model is deﬁned without reference to CSB.Thus,the eﬀectiveﬁeld theorist can simply take the mass term m ISS to be a parameter of unknown provenance,or imagine that it arises as a consequence of the dynamics of another gauge group,not included in the Pentagon model.Our insistence on the origin of m ISS in CSB is a strong constraint,because it bounds the scale of SUSY breaking in the model.We will see that this makes it more diﬃcult toﬁnd a working version of the model.5One other place where CSB plays a role in our considerations is our decision to tune the c.c.(almost)to zero near a particular SUSY violating minimum of the eﬀective potential of theﬂat spaceﬁeld theory.It would not be consistent with the hypothesis of CSB to perform this tuning at the SUSic minimum.We will in fact have to choose between two SUSY violating minima of the potential on phenomenological grounds.It is important to note that once we have done this tuning,the supersymmetric states no longer have anything to do with the theory.Our meta-stable SUSY violating world can tunnel to the negative c.c.region of the potential,but the resulting bubble is a low entropy,highly non-supersymmetric,Big Crunch geometry.These tunneling amplitudes are too small to be of any interest.Before sailing for murkier waters,in which we will have to swim in order to get interesting phenomenological consequences of this model,I will list the approximate symmetries of the low energy Lagrangian.The pure N F=N C=5model has a U(5)×U(5)symmetry.The axial U(1)is an R symmetry under which P and˜P have R charge0.This is also a symmetry of the Yukawa couplings of the Sﬁeld,if S has R charge2and the up and down Higgsﬁelds have opposite R charge.On the other hand the mass term m ISS breaks U(1)R and is required to be fairly large for phenomenological reasons.This term also breaks SU(5)×SU(5)to its diagonal subgroup.The Yukawa coupling g S and the standard model gauge couplings break the diagonal SU(5)down to SU(1,2,3)of the standard model.The meta-stable ISS vacuum spontaneously breaks the remaining U(1)Penta-baryon symmetry.The Goldstone boson is theﬁeld deﬁned by<B>=Λ5e ib/Λ5=<˜B>∗,in the ISS vacuum.We call this the pentonﬁeld,though a catchier name might be found,if one were willing to put in the eﬀort to think about it.In addition to these continuous low energy symmetries,the Pentagon has an anomaly free Z5subgroup of the axial symmetry which gives P and˜P charge one. The mass term m ISS breaks this symmetry as well as the U(1)R but preserves a diago-nal Z R5subgroup.This is inconsistent with the tenets of CSB,according to which the c.c.breaks all R symmetries.It would also prevent us from generating gaugino masses. The couplings g S and g T violate this Z R5if they are both non-zero.These couplings must therefore be large at low energy.The usual chiral Lagrangian predictions for a Goldstone boson relate the emission amplitude for a single Goldstone boson in a transition between twoﬁnal states to the change in the spontaneously broken quantum number in the transition.This would predict zero coupling to ordinary quarks and leptons since they do not carry penta-6baryon number.However,there are dimension5operatorsc BΛ5∂µbJµL,which are allowed by the symmetries of the low energy Lagrangian.These are in fact generated by the diagrams of Figure1,involving standard model couplings at the scale Λ5.The nominal estimates for the coeﬃcients arec B∼α23(Λ5),c L∼α22(Λ5),where these are running gauge couplings,at the indicatedscale.Figure1:Diagrams leading to penton interaction with standard model currents.Gauge bosons are any charge and color neutral pair in SU(1,2,3).The RHS loop could contain leptons.These terms in the action are total derivatives if we neglect the violation of baryon and lepton number by electroweak instantons,but they will be important in the early universe if there are temperatures as high as the electroweak sphaleron mass.Under laboratory conditions,the dominant dimension5coupling of the penton to normal matter comes from similar dimensionﬁve couplings to non-conserved neutral hadronic currents like beauty,charm,and strangeness.They are suppressed by two powers ofα3(Λ5)and are proportional to quark mass diﬀerences in units of the QCD scale(for light quarks).They are of the formα23m FΛ5JµF,where JµF is aﬂavor current.Note that analogous couplings to axial currents are suppressed by further powers of electroweak gauge couplings,because the SU(5)×SU(3)gauge theory is invariant under parity.Thus pentons will be predominantly emitted in charged current weak decays,since the amplitudes are proportional to the divergence of the correspondingﬂavor current.As we will see in the section on dark7matter,pentons are very light,.1−1eV,and will escape from the detector as missing energy.Models with very low energy SUSY breaking cannot contain the usual SUSic can-didates for dark matter.Even if R parity is conserved,the LSP is the gravitino,which is relativistic at the usual scale of matter domination.It is also relatively strongly coupled and so the NLSP is not cosmologically meta-stable.In previous attempts to construct a phenomenology based on CSB,I suggested that baryons of the new strong interactions(penta-baryons in the Pentagon model)would be the dark matter.This is not possible if penta-baryon number is spontaneously broken.Instead I will suggest that the dark matter is a condensate of pentons.I will show that this is reasonable if the early universe produces a suﬃciently large penta-baryon asymmetry.3.Known unknownsWe will accept the hypothesis of ISS that the Pentagon model has a meta-stable SUSY violating state withﬂat space vacuum energy of order m2ISSΛ25.ISS characterize this as a state which has vanishing expectation value for penta-meson operators.However,in the presence of m ISS and the couplings g S and g T there is no symmetry which prevents the combinations P A i˜P j A(δi j,Y i j)from getting non-zero VEVs.Thus,we expect these bilinears to have VEVs of order KΛ25at the SUSY violating minimum,where K involves powers of g S,T if these are small.We will tune the parameter W0in the SUGRA formula for the eﬀective potential,so that the c.c.at this SUSY violating minimum is of order the observed c.c.,λ.Actually we must resolve one further ambiguity in choosing the SUSY violating vacuum.When m ISS=0there are two solutions of the F and D term constraints for the Higgsﬁelds and the singlet S:h u,d=0,g S p A i˜p j A Y i j=−3g T s2,ands=0,g S p A i˜p j A Y i j=−gµh u h d,where lower case letter represent the scalar components of chiral superﬁelds.The Higgs VEVs in the second equation are oriented so that electromagnetism is unbroken and tanβ,the ratio of Higgs VEVs is one.In the presence of the SUSY breaking parameter m ISS there will be a similar ambiguity in the choice of VEVS at the SUSY violating vacuum.However,s=0is no longer a stationary point of the eﬀective action since there are no unbroken symmetries which preserve it.On the other hand,theﬁrst minimum will still have h u,d=0.We8will choose to tune the c.c.toλat the stationary point where SU(2)×U(1)is broken .The other SUSY violating stationary point of theﬂat space potential may then have either positive or negative c.c.,while the erstwhile SUSic states will have negative c.c. .None of the other stationary points of the potential represent long lived states of the universe once gravity is taken into account.The Pentagon model thus has a stationary point of its eﬀective potential with spontaneously broken SUSY(F∼m ISSΛ5),and SU(2)×U(1)→U(1)EM with |h u|∼|h d|∼Λ5,which we will take to represent the real world.SUSY breaking is communicated to the standard model by two distinct mecha-nisms.Since s,h u,h d are all non-zero,the Higgs superﬁelds have F terms which will contribute tree level masses to squarks and sleptons.In addition there are more or less conventional gauge mediated contributions.The latter are the dominant contributions for gaugino masses as well as the masses of squarks and sleptons,apart from the top squark.Gaugino masses are estimated from one loop standard model diagrams with pentaquark superﬁelds in the loop,and arbitrary numbers of penta-gauge bosons.If we compare to conventional gauge mediated scenarios,these diagrams are enhanced by a factor of5 Y2∼16.7(where the sum is over the weak hypercharges in the5 representation of SU(5)GUT).As we will see,this means that the gaugino to squark or slepton mass ratios are larger by a factor∼4than they are in conventional gauge mediated models.The corresponding two loop contribution to e.g.the right handed charged slepton squared mass is enhanced by the same factor.Thusm˜eR∼4α1Λ5.In the standard modelα1250so we need F/Λ5>6.25TeV in order to satisfy the experimental bounds on this mass.The CSB prediction for F is of order10TeV2so this impliesΛ5∼1.6TeV.The ISS mass parameter then satisﬁesm ISS4A proof of this fact has not yet been found.9The phenomenologically preferred value for m ISS/Λ5is not in the perturbative regime.For example,a corresponding ratio in QCD would correspond to quark masses of order600MeV.Thus,it is not implausible that the meta-stable state exists in the phenomenologically required region.Note also that the additional hypercharged states in the Pentagon model might make the couplingα1,which appears in the es-timate for the slepton mass,slightly larger,and consequently loosen the bound on F/Λ5.Nonetheless,these considerations suggest that the lightest charged sleptons in the Pentagon model of SUSY breaking cannot be signiﬁcantly heavier than the current experimental lower limits.Thus,we expect to see these sleptons produced at LHC. Since the LSP is the light gravitino,with couplings of order1/F,slepton pair produc-tion will result in spectacularﬁnal states with two hard leptons,other hard particles, and missing energy,a classic signal for low energy SUSY breaking[5].It is however likely that the NLSP in this model is not a gaugino,because of the extra factor of4 in gaugino masses.The other hard particles in theﬁnal state depend on the nature of the NLSP,which we cannot determine at this time.It is worth pointing out that these estimates of the scale of SUSY breaking give a gravitino mass of order5×10−3eV.Such gravitinos are perfectly consistent with Big Bang Nucleosynthesis,in sharp contrast to conventional gauge mediated models.They are light enough,and their longitudinal components strongly coupled enough,that one might imagineﬁnding them in experiments probing for short distance modiﬁcations of gravity.It has also been suggested that they might be found at the LHC[6].Another diﬀerence between the Pentagon model and conventional gauge mediated models is that the SU(2)×U(1)violating top squark mass generated by the F term of H u is comparable to or larger than the gauge mediated mass.This is because our model eﬀectively generates a sizable eﬀectiveµterm,from the VEV of S.Unfortunately,the strong coupling physics at the scaleΛ5prevents us from making very precise statements about the spectrum of superpartner masses.In particular,there are three potentially worrisome tuning problems that I have not had the calculational skill to address.First,our estimates of gaugino and right handed slepton masses depended on the assumption that the couplings g S and g T were strong enough that there is no further loop suppression of gaugino masses(recall that these couplings broke the discrete R symmetry left over by m ISS).In particular one may worry that g T is not asymptotically free(it would appear to be renormalized only by wave function renormalization of the gauge invariant Sﬁeld)so that a large value at low energy may lead to a Landau pole well below the uniﬁcation scale.The second potential tuning is the ratio of the electroweak scale,250GeV toΛ5. For g S of order1,this is of order1/6.It scales like g S for small g S,since the VEV of the bilinear P Y˜P is of order g S,but small g S would alter our estimates for gaugino10and charged slepton masses in an unpleasant fashion.Finally,one may worry aboutthe“little hierarchy problem”.Precision electroweak measurements seem to prefer aHiggs mass below200GeV,and this may also be a little tuned in the current model.It is hard to tell whether one should take factors of6−10seriously in a model whereit is so hard to make precise calculations.One thing that appears safe is direct interference of the Pentagon degrees of freedomwith precision electroweak measurements.These would primarily aﬀect the Peskin-Takeuchi S-parameter,but with our estimates ofΛ5and m ISS the eﬀects seem to besmall5.These same estimates suggest that the expected rich“penta-hadron”spectrummay be beyond the discovery reach of the LHC.Scaling up from QCD we might expectpenta-mesons in the6−10TeV range.Penta-baryons will probably be unstable todecays into penta-mesons and pentons,with life-times of orderΛ−15.The penton itselfis the only light remnant of the penta-hadron spectrum.As discussed above,it shouldbe produced in association with ordinary charge changing weak decays and can besearched for in low energy experiments,rather than the LHC.4.Baryon number,lepton number,andﬂavor4.1B and L violating operators of dimension4and5A central element in CSB is the discrete R symmetry which guarantees Poincare in-variance in the the limiting model.This can be put to other uses.In[7]I showed thatit can eliminate all unwanted dimension4and5baryon and lepton number violatingoperators in the supersymmetric standard model.This is suﬃcient to account for ex-perimental bounds on baryon and lepton number violating processes.The interaction d2θH2u L2,should not be forbidden by R.I will adopt the philosophy of a previous paper and insist that the texture of quark and lepton Yukawa couplings,as well asneutrino masses,are determined by physics at the uniﬁcation scale.We will choose the R charge of SSMﬁelds to be independent of quark and leptonﬂavor,and denote it by the name of the correspondingﬁeld.All R charges are tobe understood modulo N,where Z N is the R symmetry group.In the remodeledPentagon model,we must choose N=4in order to accommodate the g T S3term inthe superpotential.We will also impose anomaly freedom for the discrete R symmetry.That is,the’t Hooft interactions generated by all instantons should be invariant.Thisleads to the three constraints(all equations in this section are equalities mod4):5(P+˜P)=0(4.1)6Q+3¯U+3¯D+5(P+˜P)=0(4.2)H u+H d+9Q+3L+5(P+˜P)=0(4.3) In writing these equations,we have taken into account the gaugino charges,and dropped terms that are explicitly zero ing theﬁrst anomaly equation and dropping more terms which vanish mod4,the second two equations can be simpliﬁed to:6Q+3(¯U+¯D)=0,(4.4)3Q+L=0.(4.5) The condition that the standard Yukawa couplings are allowed by R symmetry isL+H d+¯E=Q+H d+¯D=Q+H u+¯U=2.(4.6) The coupling SH u H d requiresH u+H d=0.(4.7) Note that these conditions forbid the standardµterm d2θH u H d,.We will also impose2L+2H u=2to allow the dimension5F term which can generate neutrino masses.The renormalizable dynamics of the Pentagon gauge theory preserves allﬂavor symmetries of the standard model.This forbids the generation of the neutrino seesaw term with coeﬃcient1We can solve for all of the R charges in terms of L and H d:Q=−3L¯E=2−L−Hd¯D=2+3L−Hd¯U=2+3L+H.dIn addition we have the relation2L=2H d+2.The inequalities which forbid dangerous operators are all satisﬁed if and only if L is odd and H d is even.Any choice satisﬁes the last constraint,so we have four solutions L=±1,H d=0,2.4.2Flavor and CPThe Pentagon shares with generic gauge mediated models the property that the only terms in the low energy Lagrangian that are not invariant under the SU(3)Q×SU(3)¯U×SU(3)¯Dﬂavor group of the standard model,are the quark and lepton Yukawa couplings, and the neutrino seesaw term.As a consequence it has a GIM mechanism,andﬂavor changing neutral currents are suppressed below experimental upper bounds.Similarly, leptonﬂavor changing processes likeµ→e+γare within experimental limits.Quark and leptonﬂavor changing processes,in addition to those induced by the neutrino seesaw term,will come from dimension6operators.We can assume that they are scaled by the same operator M U∼1015GeV as the neutrino seesaw.Note that the restriction to dimension6operators is non-trivial and depends on the fact that dimension5superpotential terms like Q2¯U¯D(with variousﬂavor combinations)are forbidden by the Z4R symmetry,although they conserve baryon and lepton number. Flavor violation comes predominantly through SSM loop graphs.The remodeled Pentagon thus solves most of the problems of generic SUSic models. However,it does have CP violating phases in addition to the usual CKM parameter. To see this,perform the following sequence of transformations.•A U A(1)transformation on quarks,to eliminate the QCDθangle,θ3.•A similar U A P(5)transformation on penta-quarks to eliminateθ5.•An anomaly free U R(1)transformation,to eliminate arg m ISS.13•A common phase rotation on H u,d to eliminate argdetλuλd.•A phase rotation of S to eliminate arg gµ.We are left with the phases of g S,T(as well as phases in the neutrino see-sawterm and CKM matrix)as physical CP violating parameters.When we integrate outscales≫ΛQCD,these phases will infect the determinant of the renormalized quarkmass matrix and are likely to give rise to an electric dipole moment for the neutronwhich is incompatible with experimental bounds.The upside of this result is that thepotentially troublesome axion,which roamed the halls of the old Pentagon,no longerexists.Counting one of these two phases as a stand in forθ3we see that the Pentagonmodel has six new physical parameters,the absolute values of m ISS,Λ5,g S,T,µand onecombination of the phases of g S,T in addition to the parameters in the standard modelLagrangian.Of these,|m ISS|is roughly determined by the rules of CSB.The three Yukawa couplings of the Sﬁeld are required to be reasonably large.We will discussthe consequences of this assumption in the section on uniﬁcation of couplings.5.Dark matterCSB,like any model with a maximum SUSY breaking scale of order<100TeV,doesnot have a cosmologically stable massive LSP.Even if R-parity is preserved,the LSP isthe gravitino,and its longitudinal components are so strongly coupled that the NLSPwill decay to it rapidly,probably in typical particle detectors,and certainly with non-cosmological lifetimes.In previous discussions of CSB,I have suggested baryons of thenew strong gauge group as dark matter candidates and in[8]we showed that this wasa viable option if a suﬃciently large penta-baryon asymmetry is generated in the earlyuniverse.The meta-stable ISS SUSY breaking state also breaks penta-baryon numberspontaneously,so penta-baryons are no longer cosmologically stable.Instead,I want to show that with a suﬃciently large penta-baryon number asym-metry,the penton can be the dark matter.Assume a penta-baryon asymmetry toentropy ratioǫis generated in the early universe at or after inﬂationary reheating,andthat there is no signiﬁcant entropy production thereafter.The universe is radiationdominated and the penta-baryon density at temperature T will ben P B=ǫGT3,where G counts the eﬀective number of massless degrees of freedom in the plasma.Once T drops belowΛ5the penta-baryon density can be written in terms of the penton14ﬁeld bn P B=Λ5˙b,so we get the equation of motion˙b=ǫG T3M2U d2θ(aP5+b˜P5).We will take the mass scale in these interactions to be the same order of magnitude,∼1015GeV,as that which enters thedimension5operator that leads to neutrino masses and mixings.Below the scaleΛ5these interactions will give rise to a potential for b of orderΛ65M U∼2eV.The potential will begin to aﬀect the cosmological evolution of the penton whenthe kinetic energy generated by the asymmetry is of order the potential.This happensat a temperature T∗given byǫ2G2(T∗)6=Λ85M U∼10−12,this temperature is belowΛ5and so our description of the eﬀects of the explicit symmetry breaking is valid.We will see thatǫhas to be quite large if we want the penton to be dark matter.Indeed,below the scale T∗the penton density will grow relative to the radiationenergy density by a factor T∗ργ=ǫ2G(T∗T.Note that(T∗ǫGM U,so thatρpM UΛ5。

String Inspired Singlet Extensions of the Minimal Supersymmetric Standard Model

a r X i v :h e p -p h /0612359v 3 28 A u g 2007MIFP-06-26String Inspired Singlet Extensions of the Minimal Supersymmetric Standard ModelTianjun Li 1,21George P.and Cynthia W.Mitchell Institute for Fundamental Physics,Texas A &M University,College Station,TX 77843,USA2Institute of Theoretical Physics,Chinese Academy of Sciences,Beijing 100080,ChinaThe only allowed Higgs superpotential term at stringy tree level in the string derived Singlet Extensions of the Minimal Supersymmetric Standard Model (SEMSSM)is hSH d H u ,which leads to an additional global U (1)symmetry in the Higgs potential.We propose the string inspired SEMSSM where the global U (1)symmetry is broken by the additional superpotential terms or supersymmetry breaking soft terms that can be obtained naturally due to the instanton eﬀects or anomalous U (1)A gauge symmetry.In these models,we can solve the µproblem and the ﬁne-tuning problem for the lightest CP-even Higgs boson mass in the MSSM,generate the baryon asymmetry via electroweak baryogenesis,and predict the new Higgs physics which can be tested at the LHC and ILC.PACS numbers:11.25.Mj,12.10.Kt,12.60.FrIntroduction –The Minimal Supersymmetric Stan-dard Model (MSSM)can solve the gauge hierarchy prob-lem elegantly due to supersymmetry,has neutralino as cold dark matter candidate,and accommodates the gauge coupling uniﬁcation [1,2].So,it is the most nat-ural extension of the Standard Model (SM).However,there are a few problems within the MSSM.The bilinear supersymmetric Higgs mass term µH d H u in the superpo-tential,where H d and H u are one pair of Higgs doublets,does not violate supersymmetry and gauge symmetry.Then the natural scale for µis about Planck scale but not the TeV scale,which leads to the µproblem.More-over,in order to have the lightest CP-even Higgs boson mass larger than the low bound 114GeV from the LEP experiment,there exists a few percent ﬁne-tuning [3].To solve the µproblem,the Next to the Minimal Su-persymmetric Standard Model (NMSSM)was proposed in which a SM singlet S and a Z 3discrete symmetry are introduced [4].The µH d H u term is forbidden by the Z 3symmetry,and the superpotential in the NMSSM isW =hSH d H u +κ2the ﬁne-tuning problem for the lightest CP-even Higgs boson mass in the MSSM.We also calculate the Higgs boson masses,chargino masses and neutralino masses at tree level,and predict the new Higgs physics which can be tested at the LHC and ILC.A more detail discussions will be presented elsewhere [15].Model Building –Let us consider the most general SEMSSM.The generic superpotential is W =hSH d H u +µH d H u +m 2S +µ′3!S 3,(3)where µ,m 2,and µ′are mass parameters.The corre-sponding F -term scalar potential isV F =|hH d H u +m 2+µ′S +κ8|H u |2−|H d |2 2,(5)where G 2=g 2Y +g 22;g Y and g 2are respectively the coupling constants for U (1)Y and SU (2)L .Moreover,weintroduce the supersymmetry breaking soft terms V Isoftand V IIsoft as follows V I soft =m 2H d |H d |2+m 2H u |H u |2+m 2S |S |2,(6)V IIsoft =− A h hSH d H u +BµB H d H u +A X m 2X S+13!A κκX S 3+H .C .,(7)where m 2H d ,m 2H u ,and m 2S are supersymmetry breakingsoft masses,A h ,B ,µB ,A X ,m 2X ,B ′,µ′B ,and A κare supersymmetry breaking soft mass parameters,and κX is the coupling constant.In addition,if µ=0,m 2=0,µ′=0,or κ=0,we assume µB =µ,m 2X =m 2,µ′B =µ′,or κX =κ,respectively.However,even if µ=0,m 2=0,µ′=0,or κ=0,we can show that µB ,m 2X ,µ′B ,or κX might not be zero in general,so the global U (1)symmetry in the Higgs potential can be broken by the supersymmetry breaking soft terms [15].In the string derived models,the terms µH d H u ,µ′S 2/2!,and κS 3/3!in superpotential in Eq.(3),which are forbidden at stringy tree level,might be generated due to the instanton eﬀects.And the eﬀective µis about M string e −A where M string is the string scale around 1017GeV.So,the µproblem in the MSSM is solved if A ∼33[9].Similar result holds for µ′.However,the κS 3/3!term generated from instanton eﬀects might be highly suppressed.To construct the string inspired SEMSSM,we con-sider the models with an anomalous U (1)A gauge sym-metry [14].In string model building,there generically exists one anomalous U (1)gauge symmetry in the het-erotic string model building [14]or up to four in theType II orientifold model building [16].The correspond-ing anomalies are cancelled by the (generalized)Green-Schwarz mechanism [17].We introduce a SM singlet ﬁeld φwith U (1)A charge −1.To cancel the Fayet-Iliopoulos term of U (1)A ,we assume that φobtains a VEV so that the U (1)A D-ﬂatness and supersymmetry can be preserved.Interestingly, φ /M Pl is about 0.22,where M Pl is the Planck scale [14].Moreover,to break the su-persymmetry,we introduce a hidden sector superﬁeld Z whose F component acquires a VEV around 1021GeV 2.We assume that the U (1)A charges for S and Z are n +p/q and m +p ′/q ′,respectively,where m and n are integers,(p ,q )and (p ′and q ′)are relatively prime positive integers,or p/q or p ′/q ′is zero.To have the hSH d H u term in superpotential,we also assume that the total U (1)A charges for H d and H u are −n −p/q ,but we do not give the explicit charges for H d and H u which are irrelevant to our discussions.Moreover,if m +p ′/q ′is non-zero,the gaugino masses can not be generated via F-terms ZW αW α/M Pl .To have the gaug-ino masses,we can introduce another U (1)A -uncharged hidden-sector superﬁeld Z ′whose F component acquires a VEV.In fact,in the string model building,both dilaton and moduli ﬁelds can break the supersymmetry due to their F-component VEVs.In addition,the supersymme-try breaking soft mass terms in V Isoft can be generated via D-term operatorsd 4xd 2θd 2ZZM PlhSH d H u +H .C ..(9)Model A –We choose the following U (1)A charges for S and Zm +n =47,p/q =1/5,p ′/q ′=4/5.(10)Then the U (1)A allowed renormalizable superpotential isW =hSH d H u .(11)The additional supersymmetry breaking soft term V IIsoft can be generated via the following operator (the other operators are forbidden by U (1)A or negligible)d 4xd 2θM string ZSφ3where A h ∼A X ∼102GeV,and m 2X ∼104−6GeV 2.Interestingly,the global U (1)symmetry in the Higgs po-tential is indeed broken by the supersymmetry breaking soft term A X m 2X S .Model B –We choose the following U (1)A charges for S and Zn =−22,p/q =0,m =0,p ′/q ′=0.(14)The additional relevant F-term and D-term operators ared 4xd 2θ(M string H d H u +ZH d H u )φθZZSφM Pl23+H .C ..(21)So,the superpotential isW =hSH d H u +µ′H 0dS H 01H 03A 0211921367210251123188452062141231616617121412016767181225where µ′∼102GeV.And the supersymmetry breakingsoft terms in V IIsoft areV IIsoft=−A h hSH d H u +1H 0d 2+ H 0u2≃174.1GeV.For Model A,we choose:h =0.7,m 2H d =−0.1,m 2H u =−0.2,m 2S =0.1,A h =1.0,A X =0.68,m 2X =0.6.And the VEVs for the Higgs ﬁelds at the minimum are H 0d =0.7031, H 0u =0.75,and S =1.2563.For Model B,we choose:h =0.7,µ=−0.2,m 2=−0.3,m 2H d =−0.1,m 2H u =−0.1,m 2S =0.1,A h =0.6,B =−0.1,µB =−0.2,A X =−1.9,m 2X =−0.3.And the VEVs for the Higgs ﬁelds are H 0d =0.8625, H 0u =0.8625,and S =1.3156.For Model C,we choose:h =0.7,µ=−0.1,m 2H d =−0.1,m 2H u =−0.1,m 2S =−0.6,A h =2.0,B =−0.6,µB =−0.1.And the VEVs for the Higgs ﬁelds are H 0d =1.5875, H 0u =1.5875,and S =2.075.For Model D,we choose:h =0.7,µ′=−0.3,m 2H d =−0.1,m 2H u =−0.4,m 2S =−0.68,A h =2.0,B ′=−0.6,µ′B =−0.3.And the VEVs for the Higgs ﬁelds are H 0d =1.6375, H 0u =1.7203,and S =2.275.We present the Higgs VEVs,the charged Higgs bo-son (H ±)mass,CP-even Higgs boson (H 01,H 02,and H 03)masses,and CP-odd Higgs boson (A 01and A 02)masses4 TABLE II:The chargino and neutralino masses in GeV.Model˜χ±1˜χ01˜χ03˜χ05>033488217A16368169314>032881184B11881156316>032980185C12080156316>033068200D132********in Table I.Interestingly,the couplings of the CP-even Higgs boson H01and H02with Z0gauge boson almost vanish,and only the heaviest CP-even Higgs boson H03 can couple to Z0[15].Thus,theﬁne-tuning problem for the lightest CP-even Higgs boson mass in the MSSM from the LEP constraints can be completely relaxed,and we have new Higgs physics at the LHC and ILC because H03has mass around190GeV.Moreover,to calculate the chargino and neutralino masses,we choose the positive and negative gaugino masses M1and M2for U(1)Y and SU(2)L:(1)M1=150GeV,and M2=300GeV;(2) M1=−150GeV,and M2=−300GeV.The masses for charginos(˜χ±1and˜χ±2)and neutralinos(˜χ0i with i=1,2,...,5)are given in Table II.Conclusions–In the string derived SEMSSM, hSH d H u is the only allowed superpotential term at stringy tree level.Then there exist an additional global U(1)symmetry in the Higgs potential,and the axion problem.We propose the string inspired SEMSSM in which the global U(1)symmetry is broken by the addi-tional superpotential terms or supersymmetry breaking soft terms.The extra superpotential terms can be ob-tained via instanton eﬀects in the string derived models. With anomalous U(1)A gauge symmetry,we present four simple and concrete SEMSSM.In these models,we can naturally solve theµproblem and theﬁne-tuning prob-lem for the lightest CP-even Higgs boson mass in the MSSM,and generated the observed baryon asymmetry via electroweak baryogenesis.In addition,we calculate the Higgs boson masses,chargino masses and neutralino masses at tree level,and predict the new Higgs physics which can be tested at the LHC and ILC.Conﬁrmation one of these models at the future colliders might give us indirect implication of string theory. Acknowledgments–This research was supported in part by the Cambridge-Mitchell Collaboration in Theo-retical Cosmology.[1]S.Dimopoulos and H.Georgi,Nucl.Phys.B193,150(1981);S.Dimopoulos,S.Raby and F.Wilczek,Phys.Rev.D24,1681(1981);N.Sakai,Z.Phys.C11,153 (1981);L.E.Ibanez and G.G.Ross,Phys.Lett.B105, 439(1981);M.B.Einhorn and D.R.T.Jones,Nucl.Phys.B196,475(1982);W.J.Marciano and G.Sen-janovic,Phys.Rev.D25,3092(1982).[2]U.Amaldi,W.de Boer and H.Furstenau,Phys.Lett.B260,447(1991);J.R.Ellis,S.Kelley andD.V.Nanopoulos,Phys.Lett.B249,441(1990);n-gacker and M.X.Luo,Phys.Rev.D44,817(1991);C.Giunti,C.W.Kim and U.W.Lee,Mod.Phys.Lett.A6(1991)1745.[3]Y.Okada,M.Yamaguchi and T.Yanagida,Prog.Theor.Phys.85,1(1991);J.R.Ellis,G.RidolﬁandF.Zwirner,Phys.Lett.B257,83(1991);H.E.Haberand R.Hempﬂing,Phys.Rev.Lett.66,1815(1991);M.Carena,J.R.Espinosa,M.Quiros and C.E.M.Wag-ner,Phys.Lett.B355,209(1995).[4]P.Fayet,Nucl.Phys.B90,104(1975);H.P.Nilles,M.Srednicki and D.Wyler,Phys.Lett.B120,346 (1983);J.M.Frere, D.R.T.Jones and S.Raby, Nucl.Phys.B222,11(1983);J.P.Derendinger andC.A.Savoy,Nucl.Phys.B237,307(1984).[5]R.Dermisek and J.F.Gunion,Phys.Rev.Lett.95,041801(2005).[6]M.Pietroni,Nucl.Phys.B402(1993)27.[7]C.Panagiotakopoulos and K.Tamvakis,Phys.Lett.B469,145(1999);C.Panagiotakopoulos and A.Pilaftsis, Phys.Rev.D63,055003(2001);A.Dedes,C.Hugonie, S.Moretti and K.Tamvakis,Phys.Rev.D63,055009 (2001).[8]M.Cvetic,D.A.Demir,J.R.Espinosa,L.L.Everett andngacker,Phys.Rev.D56,2861(1997)[Erratum-ibid.D58,119905(1998)];J.Erler,ngacker and T.Li,Phys.Rev.D66,015002(2002);V.Barger,n-gacker,H.S.Lee and G.Shaughnessy,Phys.Rev.D73, 115010(2006).[9]R.Blumenhagen,M.Cvetic and T.Weigand,Nucl.Phys.B771,113(2007);L.E.Ibanez and A.M.Uranga,JHEP 0703,052(2007).[10]V.Braun,Y.H.He,B.A.Ovrut and T.Pantev,Phys.Lett.B618,252(2005);V.Bouchard and R.Donagi, Phys.Lett.B633,783(2006);V.Bouchard,M.Cvetic and R.Donagi,Nucl.Phys.B745,62(2006).[11]C.M.Chen,T.Li and D.V.Nanopoulos,Nucl.Phys.B751,260(2006).[12]E.Witten,Phys.Lett.B155,151(1985);T.Li,J.L.Lopez and D.V.Nanopoulos,Phys.Rev.D56, 2602(1997).[13]D.Cremades,L.E.Ibanez and F.Marchesano,JHEP0307,038(2003);M.Cvetic and I.Papadimitriou,Phys.Rev.D68,046001(2003)[Erratum-ibid.D70,029903 (2004)];S.A.Abel and A.W.Owen,Nucl.Phys.B663, 197(2003).[14]H.K.Dreiner,H.Murayama and M.Thormeier,Nucl.Phys.B729,278(2005),and references therein.[15]T.Li,in preparation.[16]L.E.Ibanez,F.Marchesano and R.Rabadan,JHEP0111,002(2001).[17]M.B.Green and J.H.Schwarz,Phys.Lett.B149,1175 (1984).。

The Higgs Particle希格斯粒子

1961 – 1968 Glashow, Weinberg and Salam (GWS) developed a theory that unifies the electromagnetic and weak forces into one electroweak force.
Electromagnetic Force – mediator: photon (mass = 0) felt by electrically charged particles
August 22, 2002
UCI Quarknet
Higgs Mechanism
Goldstone’s Theorem - The spontaneous breaking of a continuous global symmetry is always accompanied by the appearance of massless scalar particles called Goldstone bosons.
W’s
August 22, 2002
UCI Quarknet
W+
W-
Z0
The Higgs Field and Higgs Boson
The neutral Higgs field permeates space and all particles acquire mass via their interactions with this field.
August 22, 2002
UCI Quarknet
Spontaneous Symmetry Breaking
This is a phenomenon that can occur when the symmetries of the equations of motion of a system do not hold for the ground state of the system.

The Standard Model and Fermion Masses

τ3 =
ψ L and isotopic matrices ψ R
I 0 0 I , τ1 = affecting the four upper and four lower components of 0 −I I 0
operator Ф1 . Now, equations (2) can be written as
Consider the density of Hamiltonian of a Dirac particle with mass m f interacting with an arbitrary boson field B µ ℋD = ψ † (α p + β m f + qα µ B µ )ψ = ψ † ( PL + PR ) (α p + β m f + qα µ B µ ) ( PL + PR )ψ =
GG † GG † † =ψ L p + qα µ B µ )ψ R + ψ L β m fψ R +ψ R β m fψ L (α p + qα µ B µ )ψ L +ψ R† (α
.
GG
GG
(1)
In (1) q − is the coupling constant; PL =
1− γ 5 1+ γ 5 , PR = − are the left and 2 2
∂ . Linear forms of SU (2) − invariant ∂t
equations for fermion fields relative to p0 can be obtained using the FoldyWouthuysen transformation [6] in the specially introduced isotopic space. Introduce the eight-component field operator Ф1 =

公共课英语二模拟题2020年(31)_真题-无答案

公共课英语二模拟题2020年(31)(总分100,考试时间180分钟)阅读理解In the first year or so of Web business, most of the action has revolved around efforts to tap the consumer market. More recently, as the Web proved to be more than a fashion, companies have started to buy and sell products and services with one another. Such business-to-business sales make sense because business people typically know what product they're looking for.Nonetheless, **panies still hesitate to use the Web because of doubts about its reliability. "Businesses need to feel they can trust the pathway between them and the supplier," says senior analyst Blane Erwin of Forrester Research. **panies are limiting the risk by conducting online transactions only with established business partners who are given access to **pany's private intranet.Another major shift in the model for **merce concerns the technology available for marketing. Until recently, Internet marketing activities have focused on strategies to "pull" customers into sites. In the past year, however, **panies have developed tools that **panies to "push" information directly out to consumers, transmitting marketing messages directly to targeted customers. Most notably, the PointCast Network uses a screen saver to deliver a continually updated stream of news and advertisements to subscribers' computer monitors. Subscribers can customize the information they want to receive and proceed directly to a company's Web site. Companies such as Virtual Vineyards are already starting to use similar technologies to push messages to customers about special sales, product offering, or other events. But push technology has earned the contempt of many Web users. Online culture thinks highly of the notion that the information flowing onto the **es there by specific request. **mercial promotion begins to fill the screen uninvited, the distinction between the Web and television fades. That's a prospect that horrifies Net purists.But it is hardly inevitable **panies on the Web will need to resort to push strategies to make money. The examples of Virtual Vineyards, , and other pioneers show that a Web site selling the right kind of products with the right mix of interactivity, hospitality, and security will attract online customers. And the cost of computing power continues to free fall, which is a good sign for any enterprise setting up shop in silicon. People looking back 5 or 10 years from now may well wonder why so **panies took the online plunge.1. 1.According to Paragraphs 1 and 2, what do we learn about the present web business?A. Web business is no longer in fashion.B. Business-to-business sales are the trend.C. Web business is prosperous in the consumer market.D. **panies still lack confidence in web business.2. 2.Established business partners are preferred in web business because _____.A. they are more creditable than othersB. they specify the products they wantC. they have access to **pany's private intranetD. they are capable of conducting online transactions3. 3.PointCast Network in Paragraph 3 is most probably _____.A. a company that develops the latest push softwareB. a tool that promotes a company's online marketingC. the **pany that used an online push softwareD. the most popular software that helps a company push4. purists are most worried that _____.A. only the requested **es to the screenB. the Net is filled **mercial promotionC. the difference between the Web and TV will fadeD. push technology will dominate the screen5. 5.What can be inferred from the last paragraph about ?A. Its success is attributed to push strategies.B. It is prosperous without push strategies.C. It is highly concerned about the cost of computing power.D. It is a good example of the flourishing online business.On August 18th US News & World Report released its 2007 rankings of America's top colleges. The survey began in 1983 as an unofficial opinion poll, when the magazine asked 662 college presidents to identify the country's best places of learning. It has since changed into an annually frightening experience for reputable universities. A strong showing in the rankings spurs student interest and alumni giving while a slip has grave consequences for public relations.University administrators deeply dislike the survey. Many reject the idea that schools can be stacked up against one another in any meaningful way. And the survey's methodology is suspect. The rankings are still based partly on peer evaluations. **pare rates of alumni giving, which has little to do with the transmission of knowledge. Besides, the magazine's data are supplied by the schools and unproved.But whether the rankings are fair is beside the point, because they are wildly influential. In the 1983 survey barely half of the presidents approached bothered to respond. Today, only a handful dare refuse.Most, in fact, do more than simply fill out the survey. Competition between colleges for top students is increasing, partly because of the very popularity of rankings. Colin Diver, the president of Reed College in Oregon, considers that "rankings create powerful incentives to manipulate data and distort institutional behaviour." A school may game the system by luring applications from students who stand no chance of admission, or by leaning on alumni to arrange jobs for graduates. Reed is one of the few prominent colleges that dares to despise taking part in the US News survey.In some ways, the scramble to attract applicants has helped students. Universities such as Duke in North Carolina and Rice in Houston are devoting more money to scholarships. That seems a reasonable response to the challenge of the rankings, as the National Centre for Education Statistics reckons that roughly two-thirds of undergraduates rely on financial aid.Other colleges, though, are trying to drum up excitement by offering privileges that wouldhave been unheard of a generation ago. Students at the University of California, Los Angeles (UCLA) now appreciate weekly maid service in the dorms. "The elevators", replied an enthusiastic respondent to an online survey, "smell lemon fresh." Students at Pennsylvania State University enjoy free access to Napster, the music-sharing service. Multi-million dollar gyms have become so common that they are unremarkable.University officials, defending this strategy, often imply that they are only responding to student demand. Discouraging words for those who believe that a college's job is to educate, not indulge.6. 6.According to Paragraph 1, the top universities take the annual rankings as _____.A. a severe testB. a routine scheduleC. a chance to distinguish themselvesD. an official public-opinion poll7. 7.The universities administrators most probably agree that the rankings should _____.A. **pletely on peer evaluationsB. count out the rates of alumni givingC. be done by a more convincing magazineD. be based on date supplied by the presidents8. 8.According to Colin Diver, why does a school lean on alumni to arrange jobs for graduates?A. To attract more top applicants.B. To make the rankings reliable.C. To take advantage of its public relations.D. To become more popular in the rankings.9. 9.The author tends to think that the weekly maid service of the UCLA is meant to be _____.A. beneficialB. innovativeC. appealingD. indulgent10. 10.The text is intended to tell most clearly about _____.A. the result of **petition for top studentsB. the defects in the American educational systemC. the influence of the rankings of top universitiesD. the ways to increase the number of top applicantsLatino youths need better education for Arizona to take full advantage of the possibilities their exploding population offers. Arizona's fast-growing Latino population offers the state tremendous promise and a challenge. Even more than the aging of the baby boomers, the Latino boom is fundamentally reorienting the state's economic and social structure.Immigration and natural increase have added 600,000 young Latino residents to the state's population in the past decade. Half of the population younger than 18 in both Phoenix and Tucson is now Latino. Within 20 years, Latinos will make up half of the homegrown entry-level labor pool in the state's two largest labor markets.What is more, Hispanics are becoming key economic players. Most people don't notice it, but Latinos born in Arizona make up much of their immigrant parents' economic and educational deficits. For example, second-generation Mexican-Americans secure an average of 12 grades ofschooling where their parents obtained less than nine. That means they erase 70 percent of their parents' lag behind third-generation non-Hispanic Whites in a single generation.All of this hands the state a golden opportunity. At a time when many states will straggle with labor shortages because of modest population growth, Arizona has a priceless chance to build a populous, hardworking and skilled workforce on which to base future prosperity. The problem is that Arizona and its Latino residents may not be able to seize this opportunity. Far too many of Arizona's Latinos drop out of high school or fail to obtain the basic education needed for more advanced study. As a result, educational deficits are holding back many Latinos—and the state as well. To be sure, construction and low-end service jobs continue to absorb tens of thousands of Latino immigrants with little formal education. But over the long term, most of Arizona's Latino citizens remain ill-prepared to prosper in an increasingly demanding knowledge economy.For the reason, the educational uplift of Arizona's huge Latino population must move to the center of the state's agenda. After all, the education deficits of Arizona's Latino population will severely cramp the fortunes of hardworking people if they go unaddressed and could well undercut the state's ability to compete in the new economy. At the entry level, slower growth rates may create **petition for low-skill jobs, displacing Latinos from a significant means of support. At the higher end, shortages of Latinos educationally ready to move up will make it that much harder for knowledge-**panies to staff high-skill positions.11. 11.The Latino population is changing Arizona's _____.A. aging problemB. educational systemC. economic structureD. financial deficits12. 12.What can be inferred from Paragraph 3?A. The Latino population in Arizona is made up of Hispanics and Mexican-Americans.B. The first-generation Latinos are immigrants instead of being born in America.C. 70 percent of the first-generation Latinos had less schooling than nine years.D. The educational system used to be in favor of the non-Hispanic Whites.13. 13."educational deficits" (Para. 4) most probably means that _____.A. the state did not put much money into educationB. many Latinos are too poor to obtain educationC. education is not a profitable enterpriseD. many Latinos are not well-educated14. 14.According to the author, Arizona should give highest priority to _____.A. controlling the Latino populationB. enhancing the educational level of the Latino populationC. improving the knowledge-based economyD. building the Latino population into hardworking and skilled workforce15. 15.It is implied that, in the long run, most Latinos in Arizona will _____.A. be joblessB. be badly-paidC. do low-skill jobsD. do high-skill jobsThe destruction of our natural resources and contamination of our food supply continue tooccur, largely because of the extreme difficulty in affixing legal responsibility on those who continue to treat our environment with reckless abandon. Attempts to prevent pollution by legislation, economic incentives and friendly persuasion have been met by lawsuits, personal and industrial denial and long delays—not only in accepting responsibility, but more importantly, in doing something about it.It seems that only when government decides it can afford tax incentives or production sacrifices is there any initiative for change. Where is industry's and our recognition that protecting mankind's great treasure is the single most important responsibility? If ever there will be time for environmental health professionals to come to the frontlines and provide leadership to solve environmental problems, that time is now.We are being asked, and, in fact, the public is demanding that we take positive action. It is our responsibility as professionals in environmental health to make the difference. Yes, the ecologists, the environmental activists and the conservationists serve to communicate, stimulate thinking and promote behavioral change. However, it is those of us who are paid to make the decisions to develop, improve and enforce environmental standards, I submit, who must lead the charge.We must recognize that environmental health issues do not stop at city limits, county lines, state or even federal boundaries. We can no longer afford to be tunnel-visioned in our approach. We must visualize issues from every perspective to make the objective decisions. We must express our views clearly to prevent media distortion and public confusion.I believe we have a three-part mission for the present. First, we must continue to press for improvements in the quality of life that people can make for themselves. Second, we must investigate and understand the link between environment and health. Third, we must be able to communicate technical information in a form that citizens can understand. If we can accomplish these three goals in this decade, maybe we can finally stop environmental degradation, and not merely hold it back. We will then be able to spend pollution dollars truly on prevention rather than on bandages.16. 16.We can infer from the first two paragraphs that the industrialists disregard environmental protection chiefly because _____.A. they are unaware of the consequences of what they are doingB. they are reluctant to sacrifice their own economic interestsC. time has not **e for them to put due emphasis on itD. it is difficult for them to take effective measures17. 17.The main task now facing ecologists, environmental activists and conservationists is _____.A. to prevent pollution by legislation, economic incentives and persuasionB. to arouse public awareness of the importance of environmental protectionC. to take radical measures to control environmental pollutionD. to improve the quality of life by enforcing environmental standards18. 18.The word "tunnel-visioned" (Para. 4) most probably means "_____".A. narrow-mindedB. blind to the factsC. short-sightedD. able to see only one aspect19. 19.Which of the following, according to the author, should play the leading role in the solution of environmental problems?A. Legislation and government intervention.B. The industry's understanding and support.C. The efforts of environmental health professionals.D. The cooperation of ecologists, environmental activists and conservationists.20. 20.Which of the following is true according to the last paragraph?A. Efforts should be exerted on pollution prevention instead of on remedial measures.B. More money should be spent in order to stop pollution.C. Ordinary citizens have no access to technical information on pollution.D. Environmental degradation will be stopped by the end of this decade.。

Drs. Stefan Bold

Strong limits and Inaccessibility with non-wellorderablepowersets.MSc Thesis(Afstudeerscriptie)written byIoanna M.Dimitriou(born November27th,1978in Thessaloniki,Greece)under the supervision of Dr.Benedikt L¨o we,and submitted to the Board of Examiners in partial fulﬁllment of the requirements for the degree ofMSc in Logicat the Universiteit van Amsterdam.Date of the public defense:Members of the Thesis Committee: January30,2006Prof.Dr.Peter van Emde BoasDrs.Stefan BoldDr.Benedikt L¨o weDrs.Brian SemmesDr.Yde VenemaContents0Introduction.2 1Notation and preliminaries.31.1The reals (5)1.2New cardinal notions (6)2Part I.Axiomatic approach92.1Fragments of AC (9)2.2Determinacy (10)2.3Figura’s generalisations (12)2.4Inaccessibilities (14)3Part II.Approach by construction163.1The Blass model (16)3.2Symmetric submodels (19)3.3The Feferman-L´e vy model (21)3.4A new model (33)3.5The two-step version (35)4Conclusion and open question4010Introduction.This thesis is about set theory without the axiom of choice.The theory of or-dinals and their powersets without the axiom of choice is not a popular subject in set theoretic practise;in this thesis,we will shed a little light on some basic questions in this area.Our basic theory is ZF,unless otherwise stated.In this thesis we will be interested in the concept of a cardinal being a strong limit.This concept is one of the basic properties of the ordinary theory of cardinals and their powersets. It is well studied in the ZFC context and is typically deﬁned as follows:⇐⇒∀λ<κ(2λ<κ)κis a strong limit defwhere we read“2λ<κ”as“some(any)ordinal in bijection with the powerset ofλis smaller thanκ”.In the ZFC context,this ordinal always exists,if2λis not wellorderable,it may not.As it turns out,this deﬁnition is equivalent in ZFC to four other deﬁnitions(where<is replaced by relations<s,<i,<¯s,<¯ı) that are more appropriate for an investigation without the axiom of choice.We look at this subject from two diﬀerent points of view,thus this thesis includes two parts.Theﬁrst part is looking at the problem from an axiomatic point of view,i.e.,we see what diﬀerent answers we can have when we assume diﬀerent axioms.It starts with the axiom of choice,the axiom of determinacy, weaker forms of them which are involved in this study and some generalisations of statements incompatible with the more famous axioms above.We end this part with a discussion on several notions of being an inaccessible cardinal,i.e.,a regular strong limit cardinal.These are deﬁned using the alternative deﬁnitions of strong-limitedness we mentioned above.We also deﬁne the notion of being aβ-inaccessible cardinal that uses the set of ultraﬁlters on a cardinal and it is connected with the axiom of determinacy.The notion of inaccessibility is connected with a metamathematical point of view in set theory.It is known that the existence of inaccessible cardinals is equivalent to ZFC having a set model(see[1,Ch.IV,Lemma6.3]).By G¨o del’s Incompleteness and Completeness Theorems this is actually a metamathemat-ical proof that these cardinals’existence cannot be proven in ZFC.From this metamathematical point of view,theorems that talk about the consistency of a theory motivate us to deﬁne a consistency strength hierarchy between theories that contain ZF.This is because we conventionally accept ZF as consistent.We will not go into the details of this hierarchy but we will just state that a theory1T has stronger consistency strength than a theory T if T can prove the consistency of the same or more theories than T can.Therefore the theory ZFC+“there is an inaccessible cardinal”is stronger than ZFC.All this creates the natural question of what happens in non-AC environments.This leads us at the second part of the thesis,where we look at the problem by constructing generic models by forcing.First we take a brief look at a model 1By theory we mean a recursively axiomatisable theory that contains ZF.2by Blass where all ultraﬁlters are principal and where the notion of being aβ-strong limit becomes trivial.Afterwards we will describe the method of taking symmetric submodels of generic extensions and then we will study in depth the Feferman-L´e vy model,a symmetric submodel.This model will answer most of our questions in this part and this will lead us to attempt a generalisation of it,hoping this will solve our last question.This attempt will fail but this failure will make the problem clearer and might help to lead us in a new way to approach this in the future.1Notation and preliminaries.We assume that the reader is familiar with the basic ZF theory,which at large is covered by Jech’s“Set Theory”([2]).Nevertheless,below we deﬁne notions essential for this thesis and we state some more or less standard notation.Deﬁnition1(Relations).A relation R is deﬁned to be any set of ordered pairs.The domain of the relation is the set of all elements that occur as the ﬁrst coordinate of a pair in the relation and it’s denoted by dom(R).The range is the set of all elements that occur as the second coordinate of a pair in the relation and it’s denoted by rng(R).Deﬁnition2(Functions).Let A and B be sets.If there is a surjection from A onto B,we write A →B.If there is an injection from A into B we write A →B and if there is a bijection from A onto B then we write A →B.For A,B sets,the set A B is the set of all functions from A to B.For two functions f and g,if rng(f)⊆dom(g)then we can deﬁne their composition to be the following set.f◦g def={(x,y);∃z∈rng(f)∩dom(g)[(x,z)∈f and(z,y)∈g]}.For two sets A,B with relations R A and R B respectively,we write A,R A ∼= B,R B if there is an isomorphism between the structures A,R A and B,R B .Deﬁnition3(Sequences).Aﬁnite sequence of elements of a set X is an element of the set<ωX.For such a sequence s we denote by lh(s)the length of s,i.e.,the smallest n∈ωsuch that s∈n X.We denote the last element of s as last(s),i.e.,last(s)def=s(lh(s)−1).A countable sequence of elements of a set X is an element of the setωX.The ordinals are deﬁned as usual,we write Ord for the class of all ordinals. The relation<between ordinals is deﬁned to be equal to∈ Ord×Ord.For a set x,its transitive closure is deﬁned as trcl(x)def=Sn∈ωx n,wherex0def=x and x n+1def=Sx n.We will now describe the constructible universe,L and the constructible uni-verse relative to a set a,L[a].For a detailed introduction in these constructions, see[9,Ch.II].3Deﬁnition4(L and L[a]).For a set X,Def[X]denotes the set of all subsets of X which are deﬁnable in the structure X,∈ from a formulaϕof set theory that has only one free variable.L is deﬁned recursively as follows.L0def=∅Lα+1def=Def(Lα)Lλdef=[β∈λLβforλlimit,and L def=[α∈OrdLαIt is well known that this is the smallest inner model of ZFC and that it satisﬁes the generalised continuum hypothesis(GCH).For a,X sets,Def a[X]denotes the set of all subsets of X which are deﬁnable in the structure X,∈,a∩X from a formulaϕ∈Fmas(L X(˚a))that has only one free variable.The unary predicate symbol˚a(x)is intended to be interpreted as x∈a.Similarly to L,L[a]is deﬁned recursively as followsL0[a]def=∅Lα+1[a]def=Def a(Lα[a])Lλ[a]def=[β∈λLβ[a]forλlimit,and L def=[α∈OrdLα[a]It is known that L[a]is a model of ZFC and it’s clear that L⊆L[a],for every set a.While working without the axiom of choice,we have to be very careful with the notion of cardinality.This notion is meant to describe the size of a set.When all sets are wellorderable,then we can use some kind of numbering,namely the ordinals,to measure the size of any set.But when choice fails there are non wellorderable sets and thus ordinals cannot be used to describe the size of any set.This is because the original deﬁnition says that the cardinality of a set X is theﬁrst ordinal that is in bijection with X.Since ordinals are wellordered by∈,they cannot be used for this deﬁnition in a non-choice environment.We deﬁne cardinality as follows.Deﬁnition5(Cardinality).For every set X,the cardinality of X is deﬁned to be||X||def={Y;Y →X}.In ZF this is a proper class if X is non-empty,which is strange since cardinal-ity deﬁned under AC is always a set,i.e.,a cardinal.This will not be a problem for us but we mention that there is a set-deﬁnition of cardinality without choice due to Scott.He deﬁned cardinality as||X||∩Vαwhereαis minimal such that the deﬁned set is non-empty and in some papers this set is called a cardinal.In this thesis we deﬁne cardinals as follows.Deﬁnition6(Cardinals).An ordinalκis said to be a cardinal if it is an initial ordinal,i.e.,if\(Ord∩||κ||)=κ.Denote the class of all cardinals by Card.Note that this is equivalent to the traditional deﬁnition;κ∈Card⇐⇒κ=min{α∈Ord;α∈||κ||}.4If A is a wellorderable set then there is a unique cardinalκthat is the minimal ordinal in||A||.We write|A|forκ.For wellorderable sets we often say“the cardinality of A”when we mean“the cardinal of A”,i.e.,|A|.Conversely, when for a set A we say“the cardinality of A isκ”then we imply that A is wellorderable.Since cardinals are ordinals,we order them by<c which is deﬁned to be equal to< Card×Card.For ease of notation we will denote this relation also as<.By the context it will be clear which one of the two relations we are using. Deﬁnition7(Successor and limit cardinal).For a cardinalκ,we writeκ+ for theﬁrst ordinalαfor which there is no surjection fromκontoα.Clearly, as an initial ordinal,κ+is a cardinal.The cardinalκ+is called a successor cardinal.A cardinalκis said to be a limit cardinal if it is not a successor. For every wellorderable set A,we denote by2|A|the cardinality of℘(A)andfor a non-wellorderable set B we denote the cardinality of its powerset by2||B||.Since we deﬁned only wellorderable cardinals,we can still make use of car-dinal arithmetic up to the point that it does not involve any exponentiation of cardinals.Because then we are not guaranteed that the result of the exponen-tiation is wellorderable.But we can still do addition and multiplication at this point.If we had allowed non wellorderable cardinals to exist,then nor addition nor multiplication would be deﬁnable.This is because these deﬁnitions use transﬁ-nite recursion and so apply to wellorderable sets only.1.1The realsIn this thesis,we deﬁne the reals as the powerset of the natural numbers,i.e.,R def=℘(ω),which has cardinality the continuum.By“the continuum”we mean the cardi-nality of the reals when they are deﬁned the standard way,i.e.,as the unique orderedﬁeld in which every non-empty bounded set has a supremum.Naturally,the continuum is not always a cardinal.In fact,the statement “the continuum is a cardinal”is equivalent to the axiom of choice for subsets of the reals(see[3,Forms79,79A]).The statement“for every set X,X is wellorderable”(and therefore X has a cardinal)is equivalent to the axiom of choice itself(see[3,Forms1,1E]).We will discuss the axiom of choice and axiomatic fragments of it in§2.1.We are interested mostly in cardinalities that include only non-wellorderable sets and in particular we are interested mostly in the cardinality of the reals. Therefore any set with cardinality2ωcould be used instead of the actual reals. Thus we are justiﬁed in our deﬁnition of the reals for the purposes of this the-sis.Only in one proof,the proof of Theorem3.21,we will study a topological5property of the reals.There we are going to use the Cantor set,i.e.,ω2which is homeomorphic to a compact subset of R \Q ,where by Q we mean here the rationals.Note that the Cantor set consists of countable sequences of zeroes and ones,which actually are characteristic functions for subsets of ω.According to our deﬁnition for R ,we will refer to a subset of ωas a real number.In our topological discussion we are going to need some basics from the projective hierarchy.We refer the reader to [7,1E]for more details on this hierarchy and to [7,1B]for more details on the Borel hierarchy.We will deﬁne only the sets we will need for our proof of Theorem 3.21.Deﬁnition 8(Topology of the reals and the set Πe 11).If we assume thereals to be the Cantor set,then we can deﬁne a metric on 2ωas follows.Forx,y ∈2ω,deﬁned (x,y )def=§0if x =y 2−n if n is least such that x (n +1)=y (n +1)The set Σe 01is deﬁned to be the set of all open subsets of the reals,according tothe metric d .The set Πe 01is the set of complements of sets in Σe 01,i.e.,as the set of all closed subsets of R according to the metric d .For A a subset of 2ω×2ωwe deﬁne its projection ∃R A as follows:∃R A def={y ∈2ω;∃x ∈2ω(x,y )∈A }.The set Σe11is deﬁned as follows:Σe11def ={∃R A ;A ∈Πe 01},i.e.,as the set of all projections of closed sets of reals according to the metric d .Finally,the set Πe 11is the set of all complements of sets in Σe11.1.2New cardinal notionsIf we assume AC ,then a cardinal is called a strong limit if for all λ<κwehave 2λ<κ.Without the axiom of choice,2λis not necessarily an ordinal,so this deﬁnition of strong limit does not always make sense.If we are to work in non-choice environments we need another notion of strong limit.The purpose of the next deﬁnition is to state possible other ways of talking about strong limits and later about diﬀerent notions of inaccessibility,without involving the axiom of choice.Deﬁnition 9(s ,i ,¯s ,¯ı-strong,strong limit).For sets X ,Y and for ordinals α,β,we deﬁne the following relations.X <s αdef⇐⇒there is some β<αand a surjection from βonto X .X <i αdef ⇐⇒there is some β<αand an injection from X into β.X <¯s Y def ⇐⇒there is no surjection from X onto Y .X <¯ıY def ⇐⇒there is no injection from Y into X .6Note that for theﬁrst two cases we need the right hand side to be an ordinal, for the others we don’t.Now,for x∈{s,i,¯s,¯ı},a cardinalκis called x-strong if for all cardinalsλwithλ<xκwe have2λ<xκ.If the cardinal is also a limit cardinal,then we call it an x-strong limit.First we’ll prove the following easy lemma.Lemma1.1.For every X,Y sets and for everyα∈Card,the following hold:(a)X →Y implies that Y →X.(b)X<sαis equivalent to X<iα.(c)X<iαimplies that X<¯sα.(d)X<¯s Y implies that X<¯ıY.(e)Ifλ<α,then2λ<sαimplies that2λis well orderable.Proof.Let X,Y be arbitrary sets andαan arbitrary cardinal.(a)Without loss of generality assume that X is not empty and let x0be anelement of X.Let f be the injection from X into Y.Deﬁne g:Y→X asfollows.g(y)def=§f−1(y)if y∈f[X]x0otherwise.(b)To show from left to right,assume that there is aβinαand a functionf:β →X.Fixβ,f and deﬁne g:X→βfor every x in X to beg(x)def=min f−1(x).Since f−1(x)is a subset ofβ,g is well deﬁned and clearly an injection.To show now the direction from right to left,assume that there is aβin αsuch that X →β.By(a)this implies thatβ →X and thus X<sαholds.(c)Assume there is aβinαsuch that X →β.By(a)we have that there is afunction g:β →X.Assume for a contradiction that there is a surjectionf from X ontoα.Then g◦f would be a surjection fromβontoα.Butβ<α,contradiction.(d)By contraposition of(a),X →αimpliesα →X.(e)Letλ<αsuch that2λ<iα,i.e.,such that there exists aγinαand afunction g:2β →α.Deﬁne a relation<∗on2β:x<∗y def⇐⇒g(x)<g(y).Clearly,this is a wellorder.qed7Schematically,the above lemma says the following.?>=<89:;<s ?>=<89:;<i?>=<89:;<¯s?>=<89:;<¯ıThe next lemma with the next corollary show that these newly deﬁned relationsare all equivalent under the axiom of choice.Lemma 1.2.Assume AC .Then for two sets A and B it holds that if A →B then B →A .Proof.Let A ,B be arbitrary non-empty sets.If one of them was empty,then the lemma would trivially hold.Let there be a function f :A →B .Look at the set {f (a );a ∈A }.This is a set of non-empty sets because f is a total function.By AC there is a function g such that for every a in A ,g (f (a ))∈f (a ).Deﬁnea function h :B →A for everyb in B to be h (b )def=g (f (a )).Since for every b in B there is an a in A such that f (a )=b ,this is a total function.And since f is a function,h is an injection.qed Corollary 1.3.Assume AC .Then <=<s =<i =<¯s =<¯ı.Proof.Work with the notation in Lemma 1.1.By [3,Forms 1,1E],there are ordinals α,βsuch that α →X and Y →β.Therefore we work with the ordinals α,βinstead of the sets.By Lemma 1.1it suﬃces to show that α<βimplies that α<¯ıβ,that α<¯ıβimplies that α<¯s βand that the latter implies that α<i β.But these relations are obviously equal for ordinals.qed The result of Lemma 1.1is a general result about these relations.We are interested in the notions of x -strong cardinals and that splits in two studies,one of x -strong successor cardinals and one for x -strong limit cardinals.From ZF we have a basic lemma that involves powersets and surjections and since it’s a useful one we prove it below.Lemma 1.4.For every inﬁnite cardinal λ,℘(λ) →λ+.Proof.Fix a cardinal λ.If α<λ+then by deﬁnition there is a cardinal κ≤λand a bijection f :κ →α.For every bijection f from a cardinal κbelow λ+to an ordinal αbelow λ+,deﬁne a relation R f ⊆λ×λbyγR f βdef⇐⇒f (γ)∈f (β).Clearly, λ,R f ∼= α,∈ for every such f .Now we can deﬁne F :℘(λ×λ)→λ+as follows.F (R )def=§0if λ,R is not isomorphic to an ordinal,and αif αis the unique ordinal such that λ,R ∼= α,∈ holds.This is a surjection from ℘(λ×λ)onto λ+and since λ×λ →λholds,we havethat ℘(λ)surjects onto λ+.qed This lemma is essential,as we’ll see in the next lemma,since it makes ¯s -strongness impossible for successor cardinals.8Lemma 1.5(Successor case).Successor cardinals can never be s ,i ,¯s -strong.Proof.By Lemma 1.1we know that it’s enough to show that no successor car-dinal can be ¯s -strong.Assume for a contradiction that for κ∈Card,κ+is ¯s -strong,i.e.,that for every λ<κ+,2λ →κ+.But κ<κ+and thus we have that there is no surjection from 2κonto κ+,which is a contradiction to Lemma 1.4.qed What about ¯ı-strongness?In §3.3we’ll see that the existence of ¯ı-strong successor cardinals is consistent with ZF .For the limit case we will see that in the diagram below,the arrow from <s and <i to <¯s cannot be reversed because it is consistent with ZF that there is an ¯s -strong limit cardinal which is not an s -strong limit cardinal.The question whether ¯ıimplies or not ¯s is an open question for this thesis,but one would expect that it does not.At the moment we have the following picture:Successor case:Limit case:?>=<89:;<s C C C C C C C C C C C C C C ?>=<89:;<i {{{{{{{{{{{{{{?>=<89:;<¯s<¯ı?>=<89:;<s ?>=<89:;<i ?>=<89:;<¯s <¯ıNote that according to Corollary 1.3,if we assume AC then the diagrams abovecollapse.2Part I.Axiomatic approachIn this part we are going to examine diﬀerent axiomatic systems that might give answers or insights on our questions.We are going to see ZFC which is ZF +AC and ZF +AD .Also we are going to investigate weaker systems,some systems in between ZF and ZFC and some systems in between ZF and ZF +AD .We will see that even though AC is incompatible with AD ,there are systems which are weaker than both and stronger than ZF and with them we will look at diﬀerent notions of inaccessibility.2.1Fragments of ACWe are going to present some axiomatic fragments of AC .We do that because as we will see later,they are helpful to our questions.These axiomatic fragments are listed in the lemma below.After the lemma we give a diagram that shows precisely how the implications between these fragments are.The role of the lemma is actually more of a deﬁnition since the proofs are not given.Next to each statement in the lemma there is a statement like [Form n ]where n is a natural number.This refers to [3]where these statements have these form numbers.If there is no other abbreviation and one is needed,we use the form9number of a statement.The book [3]is part of the project “Consequences of AC ”by P.Howard and J.E.Rubin.This is one of the most useful books for this kind of research since it includes a large amount of known implications between statements that follow from the axiom of choice and of course the references.Lemma 2.1.The following are implied by AC ,the axiom of choice [Form 1].•DC ;the axiom of dependent choices [Form 43].•AC (R );the axiom of choice for families of sets of reals [Form 79A].•AC ω(R );the axiom of choice for countable families of sets of reals [Form 94].•CUC (R );the statement “Countable unions of countable sets of reals are countable”[Form 6].•“ω1is regular”[Form 34].•“ω1 →R ”[Form 170].•“R is not a countable union of countable sets”[Form 38].In particular,the implications are as follows.AC F F F F F F F F F AC (R )J J J J J J J J J J AC ω(R )N N N N N N N N N N N CUC (R )Form 38DC ω1 →R h h h h h h h h h h h h h h h h h h hh h h ω1is regularp p p p p p p p p p p In this diagram,an arrow means proper implication.Most of the proofs of the implications above are rather simple and well known.In this thesis we have included the proof of “ω1 →R ⇒Form 38”and of “ω1is regular ⇒Form 38”,see Theorem 2.7(b)for κ=ω.Next we are going to see the axiom of determinacy,an axiom inconsistent with AC (R )and therefore AC itself.2.2DeterminacyHere we assume that the reader is a bit familiar with the theory of determinacy.For details and information on this theory we refer the reader to Kanamori’s “The Higher Inﬁnite”([8,Chapter 6]).The axiom of determinacy or AD states that for every subset A ⊆ωω,the ω-game on A is determined.The axiom of determinacy is broadly used in descriptive set theory and has some very nice consequences.As seen in [8,Corollary 27.4]AD is incompatible with AC (R ).Under the axiom of determinacy ω1is not injectable into the reals and the reals are not injectable in ω1either.This is also a motivation for this study,since this is a axiomatic system that has answers for our questions.The following theorem states important consequences of AD .Theorem 2.2.The following hold:10(a)AD implies the perfect set property of R (PSP ),i.e.,that every subset ofR is either countable or contains a subset that is homeomorphic to ω2(a perfect set).(b)PSP implies that ω1≤2ω.(c)AD implies AC ω(R ),i.e.,that every countable set of subsets of R has achoice function.Proof.For the proof of (a)see [8,Proposition 27.5]and for (c)see [8,Proposi-tion 27.10].We will prove (b).Let PSP hold.Assume for a contradiction that ω1≤R holds,and call fthe injection from ω1into R .Then X def =f [ω1]is an uncountable subset ofR and by PSP this means that X is homeomorphic to the reals.But X is in bijection with ω1via f .This makes R wellordered and thus makes AC (R )hold.Contradiction.qed Schematically,this is as follows,where an arrow means proper implication.AD r r r r r r r r r r r PSP AC ω(R )NN N N N N N N N N N ω1 →R CUC (R )ω1is regularp p p p p p p p p p p Form 38In the next lemma we see that the axiom of determinacy does give an answer to our questions.In particular,it separates the notion of ¯s from ¯ı-strong cardinal.Lemma 2.3.Assume AD .Then ω1is an ¯ı-strong cardinal and not an ¯s -strong cardinal.Proof.This is clear since ω1being a ¯ı-strong cardinal is equivalent to ω1not being injectable into ℘(R ),i.e.,the reals.The latter is true under AD by Theorem 2.2(b).Also,by Lemma 1.5we have that ω1as a successor can never be ¯s -strong.qed This is a nice result,since now we know that ¯ı-strongness for successor car-dinals can be a meaningful notion.So we see that AD does indeed have answers for our questions and these answers are well known.One might think here that there is no reason for re-searching further our questions,but the reason comes from a theorem of Woodin which showed that AD has the same consistency strength with a strong large cardinal axiom 2(see [8,Theorem 32.16]).These strong interconnections of AD2Inparticular AD is equiconsistent with the statement “There exist inﬁnitely many Woodin cardinals”.11with large cardinal axioms lead us to consider our questions still open and to continue our reasearch without involving AD.As seen in the proof of Lemma2.3,AD gave us a result only because it implies thatω1≤R.Next we are going to look at some interesting statements that implyω1 →2ωand generalisations of this that do not require strong statements like AD.2.3Figura’s generalisationsThe statement“countable unions of countable sets of reals are not always count-able”or¬CUC,is inconsistent with both the axiom of determinacy and the axiom of choice.We will see that it can be a fruitful statement.Lemma2.4.If X is a countable union of countable sets,then there is no surjection from X ontoω1ω.Proof.Let X=Sn∈ωX n for some countable sets X n.Let f be any functionfrom X toω1ω.Since f is a function,for every n∈ωthe set f[X n]is countable and thereforeω1\{f(x)(n);x∈X n}is not empty.Letαn be the minimal element of this set.Deﬁne the function Z:ω→ω1as Z(n)def=αn.We claim that Z is not in the range of f.Assume for a contradiction that it is,then there must be some z∈X such that f(z)=Z.Let n be such that z∈X n.By construction Z(n)=αn=f(z)(n),so Z=f(z).Therefore,f cannot be such a surjection.qed Theorem2.5.If R is a countable union of countable sets,every wellorderable subset of the reals has cardinality≤ω.Proof.We prove this by contraposition.Assume that there is an injection from ω1into2ω.Then there is an injection fromω1ωinto(2ω)ω=(2ω×ω)=2ω. But clearly there is an injection from2ωintoω1ω.By the Schr¨o der-Bernstein theorem,we get that2ω=ω1ω.If R was a countable union of countable sets, we would have a contradiction to Lemma2.4.qed Therefore,if we manage to have a model where R is a countable union of countable sets(a statement that contradicts both AC and AD),then that model will be a witness of the diﬀerence between¯s and¯ı-strong cardinals.Such a model was constructed by Feferman and L´e vy in1963.For the abstract see[5] and for a more detailed version see[2,Example15.57].We will study this model in Part II,§3.3.The statements that are studied above are going to be useful in our study for successor cardinals,so we take the time to look at some generalisations of them hoping to reach a separation result for limit cardinals.To do that we’ll use the following generalisations that are due to Figura(see[6]).12Deﬁnition10.Letκbe a cardinal.Then,CP(κ)def⇐⇒“κ+is a singular cardinal.”WOP(κ)def⇐⇒“every wellorderable subset of℘(κ)has cardinality≤κ.”DP(κ)def⇐⇒“℘(κ)is a union ofκsets of cardinality≤κ.”These notions are generalisations of the ones forκ=ωwhich are more well known.In particular,WOP(ω)means that“every wellorderable subset of R has cardinality≤ω”.This is clearly equivalent to saying thatω1 →R,i.e.,¬Form170.DP(ω)is“R is a union ofωsets of cardinality≤ω,i.e.,¬Form38 and CP(ω)is“ω1is singular”,i.e.,¬Form34.It is known that Form34implies Form38,so with contraposition it is true that DP(ω)implies CP(ω).The next theorem generalizes this result as well as the result of Theorem2.5.But before we start we need a lemma similar to Lemma2.4.Lemma2.6.If X is a union ofκ-many sets of cardinality≤κ,then there cannot be a surjection from2κontoκ+κ.Proof.The proof is similar to the proof of Lemma2.4.Let X=Sα∈κXαfor some sets Xαof cardinality≤κ.Let f be any func-tion from X toκ+κ.Since f is a function,for everyα∈κthe set f[Xα]has cardinality≤κand therefore for everyα∈κ,the setκ+\{f(x)(α);x∈Xα} is not empty.Letβαbe the minimal element of this set.Deﬁne the function Z:κ→κ+as Z(α)def=βα.We claim that Z is not in the range of f.Assume for a contradiction that it is,then there must be some z∈X such that f(z)=Z.Letαbe such that z∈Xα.By construction Z(α)=βα=f(z)(α),so Z=f(z).Therefore,f cannot be such a surjection.qed Theorem2.7(Figura).Letκ∈Card.(a)DP(κ)implies WOP(κ).(b)DP(κ)implies CP(κ).Proof.Letκbe a cardinal.(a)The proof is similar to the one of Theorem2.5.It is a proof by contrapo-sition,i.e.,we’ll prove that if there is an injection fromκ+into2κ,then 2κcannot be aκ-union of sets of cardinality≤κ.So by our assumption, there is an injection fromκ+into2κ.Then there is an injection fromκ+κinto(2κ)κ=(2κ×κ)=2κ.But clearly there is an injection from2κinto κ+κ.By the Schr¨o der-Bernstein theorem,we get that2κ=κ+κ.If2κwas aκ-union of sets with cardinality≤κ,we would have a contradiction to Lemma2.6.(b)Assume that DP(κ)holds,i.e.,that℘(κ)is a union ofκsets of cardinality≤κ.Then,look atcof(κ+)=inf{α∈Ord;∃ βi;i∈α [∀i∈α,βi<κ+andκ+=[i∈αβi]}13。

after the establishment of the model

after the establishment of themodelAfter the Establishment of the ModelAfter the establishment of the model, a new era of possibility and opportunity unfolded. The model, a complex and meticulously crafted framework, represented a significant milestone in our journey of exploration and understanding. It was not just a set of equations or algorithms; it was a window to a world of unknown potential.The initial stages of the model's creation were filled with challenges and uncertainties. We faced numerous questions: Would the model be accurate? Would it be able to predict and explain real-world phenomena? Would it hold up under scrutiny and rigorous testing?As we built the model, we drew upon a vast array of data, theories, and insights from experts across various fields. The process was iterative and collaborative, with team members constantly refining and iterating based on feedback and results. It was a testament to the power of collective intelligence and the value of interdisciplinary collaboration.Once the model was finally established, the sense of accomplishment was immense. But more than that, it sparked a sense of curiosity and excitement. We were now equipped with a tool that had the potential to revolutionize our understanding of the world.The applications of the model were vast and diverse. It could be used to predict market trends, optimize complex systems, and even aid in scientific discoveries. Its impact was felt across various industries and sectors, sparking innovation and driving progress.However, with the establishment of the model also came a new set of challenges. There were ethical considerations to address, such as how to ensure fairness and transparency in its applications. There were also practical challenges, such as howto scale and implement the model in real-world scenarios.Despite these challenges, the establishment of the model marked a significant step forward in our quest to understand and shape the world. It was a reminder that with every new tool and technology, there comes a new set of opportunities and responsibilities. As we moved forward with the model, it was important to remember that its power and potential were limitless, but only if we used it wisely and responsibly.。

A straw drift chamber spectrometer for studies of rare kaon decays

a rX iv:physics /03847v1[physics.ins -det]12Aug2003A straw drift chamber spectrometer for studies of rare kaon decays ng a ,1,D.Ambrose a ,C.Arroyo b ,M.Bachman d ,D.Connor c ,M.Eckhause d ,K.M.Ecklund b ,S.Graessle a ,M.Hamela a ,S.Hamilton a ,A.D.Hancock d ,K.Hartman b ,M.Hebert b ,C.H.Hoﬀd ,G.W.Hoﬀmann a ,G.M.Irwin b ,J.R.Kane d ,N.Kanematsu c ,Y.Kuang d ,R.Lee c ,M.Marcin a ,R.D.Martin d ,J.McDonough a ,der a ,W.R.Molzon c ,D.Ouimette b ,M.Pommot-Maia b ,M.Proga a ,P.J.Riley a ,J.L.Ritchie a ,P.D.Rubin e ,V.I.Vassilakopoulos a , B.Ware a ,R.E.Welsh d ,S.G.Wojcicki b ,S.Worm a (BNL E871Collaboration)a University of Texas,Austin,Texas,78712b Stanford University,Stanford,California,94305c University of California,Irvine,California,92697d College of William and Mary,Williamsburg,Virginia,23187e University of Richmond,Richmond,Virginia,231731IntroductionWe present a comprehensive description of a two-arm tracking system consist-ing of eight planar drift chambers based on5mm diameter straws designed, constructed,and operated in Alternating Gradient Synchrotron experiment 871at Brookhaven National Laboratory.Following a brief motivation for the experiment,we describe mechanical and electronics design considerations,im-portant construction details,and three years experience in operating the sys-tem.1.1Physics motivationDespite abundant experimental conﬁrmations of the Standard Model of ele-mentary particles and interactions,the theory is generally perceived as incom-plete,or a low energy realization of a more general theory with a full symmetry at higher energy.There are indeed many basic questions to which answers will have to come from outside the current model.This is the main motivation for theoretical and experimental explorations beyond the Standard Model.In particular,leptonﬂavor violation occurs naturally in many extensions of the Standard Model;the process K L→µ±e∓is a sensitive probe of such viola-tion.In general,rare kaon decay experiments provided a promising avenue for discovering phenomena outside the Standard Model.The primary goal of experiment E871was to search for separate lepton number violation in the decay K L→µ±e∓.The experiment achieved a90%C.L.limit of4.7×10−12[1],thus probing new interactions in nature in the200-TeV mass scale.To achieve this sensitivity,the experiment ran with the very high beam intensities made possible by the Booster at the AGS.The experiment also made theﬁrst observation of the decay K L→e+e−,mea-suring a branching fraction of8.7+5.7−4.1×10−12based on four observed events[2]. Over6,000thousand K L→µ+µ−events were also observed,for a branching fraction of(7.19±0.17)×10−9,reducing the uncertainty for this decay mode by a factor of three compared to previous attempts[3].1.2Experimental setupTwo important experimental elements relevant to this work were the AGS high-intensity neutral beam and the E871two-arm spectrometer.A novel beam stop[4,5]was located in the upstream spectrometer magnet to improve downstream acceptance,tracking,and particle identiﬁcation.Fig.1.BNL E871apparatus.The essential requirements on the spectrometer were good kinematic recon-struction and reliable,redundant particle identiﬁcation,both to be accom-plished in a high rate environment.The spectrometer,shown in Figure1,was built in the B5beam line of the AGS.The neutral beam was produced by an intense primary24.1GeV/c beam of up to1.7×1013protons delivered in a1.6s spill onto a1.5-interaction-length platinum target.A system of sweep-ing magnets and collimators angled at3.75◦with respect to the proton beam direction formed a neutral beam with mostly neutrons and kaons at an ap-proximate solid angle of4×16mrad2.Particles emerging from the11m long evacuated decay volume were tracked and identiﬁed in the E871spectrometer. The main features of the apparatus were:•two magnets for redundant momentum measurement,tuned to form a“par-allel trigger”for two-body K L→µ±e∓decays(i.e.,charged particles trav-eling downstream of the second analyzing magnets triggered the apparatus only if they traveled nearly parallel to the neutral beam direction);•a beam-stop placed in theﬁrst magnet to absorb the neutral beam •redundantﬁnely-segmented fast straw drift chambers,followed by conven-tional drift chambers in regions of lower rate•redundant particle identiﬁcation of muons and electrons•multi-level trigger with fast on-line reconstruction•fast custom-designed massively-parallel data acquisition system.The neutral beam was absorbed in a beam stop specially designed and tested for this conﬁguration.The eﬀective strengths of the magneticﬁelds(+418and −216MeV/c)were set such that trajectories of two-body kaon decays emerged nearly parallel to the neutral beam direction downstream of the second mag-net[5].Such an arrangement simpliﬁed triggering and provided theﬁrst level rejection of the three-body decays.The intense primary proton beam produced roughly2×108K L decays per AGS spill,resulting in high hit rates in the upstream straw drift chambers. Additional rates in chambers resulted from leakage of low energy particles (mainly charged,but also neutrons and photons)from the beam-stop.The beam stop shielded the downstream part of the spectrometer,where rates were substantially reduced,thus minimizing the probability of pattern-recognition or particle-identiﬁcation errors.Rejection of background due to common kaon decays depended crucially on precise,redundant and low-occupancy tracking.1.3Tracking system requirementsSince tracking resolution at low momentum is dominated by multiple Coulomb scattering,maximal background suppression required that the tracking sys-tem be low mass,eﬃcient,redundant,fast(i.e.,possess low occupancy),and provide good position measurement.When designing experiment871the fol-lowing requirements for the tracking were thus imposed:(1)minimize multiple Coulomb scattering by limiting the amount of material;(2)provide tracking redundancy with low rate of wire failures and low celloccupancy;(3)provide high hit eﬃciency for minimum ionizing particles;(4)sustain good position resolution in high rate environment;(5)assure good vertex reconstruction of two-body decays.The corollary from the above list are general hardware features desired of the system:•high segmentation(i.e.,small cell size)•fast drift velocity gas•fast-timing electronics with short pulse tails•minimal cross-talk•mechanical robustness.We discuss the design and material choices made to accomplish the above requirements in the following sections.Beam Side View Off Beam Side ViewTop ViewFront View Fig.2.Front,top,and side views of an E871straw drift chamber.The ampliﬁers are shown in their nominal locations on the top and beam-out sides.2Mechanical designAs Figure 1illustrates,there were two straw stations between the decay tank and ﬁrst spectrometer magnet,and two more between the magnets,followed by two conventional drift chamber stations downstream of the second magnet.An assembly drawing of a complete chamber module with x and y views is shown in Figure 2.2.1Choice of materialsThe technical possibility of making thin-wall tubes suggested the replacement of traditional drift chambers —with the electric ﬁeld shaped by wires —by chambers based on a continuous cylindrical cathode surface,or “straw”.The name derives from the manufacturing process,which is similar to that of paper drinking straws.Thin,small straws with wall thicknesses of approximately 30µm and 5-mm diameter,were originally used in the vertex detectors of several high energy physics experiments at e +e −colliders [6],as well as in balloon ﬂight experiments [7].They will be part of the ATLAS detector at the CERN Large Hadron Collider [8].Table 1summarizes the amounts of material in an individual tracking cham-ber,expressed in units of radiation length.The traditional 20µm gold-plated tungsten wire was used for the anode.As shown in the table,the total radia-tion thickness of a typical E871straw chamber was 0.21%of radiation length.Table1Radiation thickness of a straw drift chamber with3x and2y layers(in units of 10−4of X0).The straw intra-wall adhesive layer is accounted for by multiplying the number of straw layers(5)by120%.mean radiationthickness(cm)layers X0(cm)thickness(%X0)The thickness of conventional drift chambers was very similar.The materials most often used for straw walls are polyester(e.g.Mylar[10]) and polyimide(such as Kapton[11]).In our experience Kapton has the better dimensional stability of the two.Mylar straws installed for the1995run of E879showed signs of shrinkage over the course of the running period,making the individual straws appear twisted.This motivated our move to Kapton for rebuilt1996chambers;these chambers displayed no such eﬀect.We attribute this diﬀerence in stability to the higher water content of Mylar relative to Kapton;Mylar became more brittle or“crinkly”with age—a sign of dehy-dration.Two metals are typically considered for cathode lamination:aluminum and copper.Another option is a carbon-doped substrate material with controlled resistivity.At the time of our construction,doped Polycarbonate sheets were available and we explored their use[12].However,the thinnest sheets available were15µm,and since Polycarbonate is mechanically weaker than Mylar,we did not pursue this option.The use of resistive substrate and copper cathodes was motivated by eﬀects related to the aging/oxidizing of the aluminum cath-ode,as it had been suggested that a(semi-)conductive base material would remedy some of these problems.Similar comments can be made about Kapton. The thickness of Mylar and Kapton can be controlled with high precision;thin foils can be as little as2µm thick.Metalizing Mylar or Kapton up to1000˚A is easy and bining Mylar or Kapton with copper provided a strong straw base and led to less severe anode aging.Copper lamination was chosen because of its much better electrical aging properties than aluminum.Copper oxide is a semiconductor,which should lessen the possibility of cathodeﬁeld-emission eﬀects,as opposed to aluminum dioxide,which is an insulator.Copper cathodes also have a relatively high work function,which provides some soft-photon absorption at the straw walls.2.2Physical design considerationsWe studied several possible planar cell patterns,trying to optimize eﬃ-ciency and redundancy while minimizing radiation thickness and construction complexity.The conﬁgurations fall into three categories:close-packed(Fig-ure3(A)),rotated close-packed(Figure3(B)),and staggered(Figure3(C)). Close-pack arrangements are bonded together along the straw,forming self-supporting structures,while staggered designs require individually tensioned straws.The self-supporting nature of close-pack designs oﬀers simplicity of design, since the array needs no external tension.Staggered straws oﬀer greaterﬂex-ibility,with aﬂatter distribution of mean path lengths across the active area of the chamber,but require tension to keep from sagging.We studied the relative hit multiplicities of the various geometries via Monte Carlo simulation for the acceptance for tracks with angle ranges with respect to the neutral beam direction of−250to250,20to250,and20to150mr.To maintain high redundancy,we wanted to minimize the number of tracks for which two or fewer hits were observed per single-view tracking station.The simulation used the straw’s inner diameter of0.5cm as the size of the active area of each straw.Simulated E871tracks were projected onto the xz plane and any track crossing a straw’s active area was considered a hit.The studied geometries are shown in Figure3and summarized in Ta-ble 2.2.In theﬁrst design,a conventional3-view close-pack,theﬁnite straw-wall thickness allowed some2-hit tracks.The mean straw density was 1.97straws/cm/view.The second design,a“rotated”close pack,allowed no 2-hit tracks,while some tracks could have as many as6hits.The straw den-sity here was higher:2.27straws/cm/view.Both designs required straws to be glued together for mechanical stability.The third design was a staggered-straw geometry.This design presented a smoother gas path-length distribution and might have allowed replacement of individual straws.Moreover the position of the oﬀset straw could be placed oﬀ-center in order to optimize the chamber for a desired angular acceptance.All geometries are shown for three measurement views.Table2.2compares the close-pack and rotated close-pack geometries to stag-(A) (B)Fig.3.(A)Close-pack,(B)rotated close-pack,and(C)staggered geometries.Tracksare vertical in all cases.gered straws at the same straw density.All conﬁgurations with2.27straws/cm/view appear to be hit-equivalent.The view densities used here should be consid-ered the two extremes.Only a skewed-stagger design,which provides plentyof straw-to-straw separation while still limiting the2-hit probability and view density,allows a continuous density adjustment.In summary,we found that the rotated close-pack and staggered designs with2.27straws/cm/view had similar hit characteristics.The mean numbers of hits were3.45and3.46respectively and neither had appreciable2-hit probability. Due to the unavoidable dead space between adjacent straws,the conventional close-pack design allowed about3%2-hit tracks.The tensioning required of staggered conﬁgurations would make chambers susceptible to“creep”,or relaxation of tension,which is expected from any plastic stretched beyond its glass transition point.In laboratory testing we observed this eﬀect in our Mylar straws.While tensioning or gluing of thex(vertical-straw)view would have been unnecessary;y views,with horizon-tal straws,required either tension or structural support to maintain shape. Prototype chambers of both close pack and individually-tensioned,staggered straws were built and tested in cosmic-ray telescopes and in the AGS neutral beam,but the technical challenges associated with the individually-tensioned conﬁguration led us to choose the conventional close pack design for the E871 spectrometer.Comparison of the hit-straw multiplicity,¯n,and distributions of the number of hit straws,n,close-pack(A)or rotated close-pack(B)geometry to staggered geometries of acceptances(C)-250–250mr,(D)20–250mr and(E)20–150mr.N.N.is the nearest-neighbor distance in cm.1.97straws/cmPercent n-hitsN.N.¯n n=234562.27straws/cmPercent n-hitsN.N.¯n n=23456SDC (y)DCH (y)Fig.4.Straw and conventional drift cell conﬁgurations for x and y views.Example tracks with(perfect)distances of closest approach(DOCAs)are shown.2.3Mechanical assemblyClose-pack cell conﬁgurations for the E871straw chambers are shown in com-parison to those of our conventional drift chambers in Figure4.For redun-Straw drift chamber characteristicsdancy in the momentum-measuring horizontal(xz)plane,the x-measuring views each had a redundant,third plane of drift cells.Each drift cell in the straw drift chambers consisted of a5-mm diameter cylindrical copper-plated Mylar or Kapton cathode[13,14]with a20-micron gold-plated tungsten sense wire[15]along its axis.The cylindrical cathodes were wound on precision man-drels,where they were built up from two25.4-µm layers of material bonded with a thin layer of4-8µm of adhesive.The inner layer had the0.1-µm cop-per layer vapor-deposited[16]on one side which faced the inside of the straw tube.The original straws in the experiment were made of Mylar.However,all but one of the chambers(called SDC4LY)were rebuilt with Kapton for the 1996run period.Cylindrical brass sleeves provided the mechanical support and electrical contact between the straws and the end-plates.Feedthroughs pass through the end-plates and into the sleeves.Individual straws were bonded into closely-packed arrays whose cross section is shown in Figure4.Sense wires were precisely positioned within the straws by molded Ultem[17]inserts,which also acted as gas feedthroughs.Ground contact between the copper cathodes and the aluminum chamber frames was made through gold-plated brass sleeves,which were press-ﬁt onto the Ultem feedthroughs and silver-epoxied into the straws.More silver epoxy formed the electrical contact between the sleeves and the chamber frames,while also providing a gas seal.Figures5,6,7show a cross-section assembly drawing of the inserts,sleeves,and end-plate gas manifold.A more detailed end-plate assembly for one of the chambers,including ampliﬁer-card mounting and gas connections,is shown in Figure8.The straw drift chamber characteristics are summarized in Table3.Straw tubeCollarWireholder / Ultem (Feedthru) V -grooveSense wireGas hole Brass pinEndplate ConductiveEpoxy Au -platedbrass sleeve Fig.5.Straw end assembly drawing.Fig.6.Standard and truncated ULTEM inserts,with plugsThe feedthrough assembly served a number of functions,including positioning,tensioning,and electrically isolating the anode wire from the end-plate.It also supported the electrical contact of the anode wire to the outside signal and high voltage,and provided gas ﬂow into or out of the straw.A sketch of the end-plate and feedthrough assembly is shown in Figure 5.The wire was positioned at the center of the straw at the feedthrough with a precision of about 25µm using a V-shaped groove molded into the feedthrough near one end (Figure 7).Tension was maintained with a brass pin inserted and clamped into the slotted Ultem end opposite the V-groove.The pin diameter,about 50µm smaller than the inside diameter of the feedthrough,allowed the anode wire to extend along the pin.The wire was compressed between the brass pin and the (slotted)feedthrough wall with an Ultem ring,which slid over the outside of the feedthrough.The pin end of the feedthrough,the pin,and the compression ring extended outside the chamber end-plate.The Ultem material had a resistance of at least 100MΩthrough its wall thick-ness of 0.10inches.Gas was supplied to the straw tubes in parallel through a gas manifold milled into the end-plates.From the manifold,the gas entered the interior of the Ultem feedthrough via a pair of holes in the feedthrough walls.Fig.7.Wire-positioning V-grooveTable4Adhesives and encapsulants used in the E871straw chambersapplication name pot life cure time Since the straw assembly was essentially irreversible,we did a test assembly of the complete chamber frame,including end-plates and corner blocks.This was then dismantled for installation of straws.All straw-end pieces were de-burred and cleaned in an ultrasonic bath.Prior to assembly,holes with critical clearance,such as the Ultem clearance holes in the end-plates,were tested and cleaned by inserting the requisite gauge pin.Most gasﬁttings were installed in the frame pieces prior to assembly,since the assembled frame had insuﬃcient clearance for installation.With the exception of theﬁrst chamber built(called SDC2R),which had a stainless steel frame,all frame pieces were aluminum. The straws,of Mylar or Kapton,were shipped with Tyvek[18]liners,in ei-ther PVC tubes or wooden crates.Thus theﬁrst step in the straw preparation was unpacking,de-lining,and inspection.Straws with visible inner surface irregularities(dents or lumps of glue)were rejected.Following a previous ex-periment[19],we constructed a simple straw cutting method consisting of a precision bored Al block,with a hole diameter about25µm larger than the outer diameter of straws,with a transverse slit cut through the bore.Cutting a straw only required insertion into the bore and sliding a razor into the slit. Straws were installed in each x and y view of each module separately.Be-fore the two views were combined,their straws were laminated,potted,and structurally glued to the view sub-frame.Straws were held together within an array through spot application of Epo-Tek301-2,a very liquid epoxy with an8-hour pot life and two day cure[23].After securing the straws preciselyin place with specially-made brackets mounted on the chamber frames,epoxy was applied in small(∼0.5mm-diameter)drops to4–6points along each tan-gent between straws.A gas-pressure dispenser was used,with tubular steel tips which allowed inner straws to be reached.At this point the two viewsﬁt loosely in their frames.Bolting and pinning the x and y views together completed and squared the module frame,making it possible to bond the straw arrays to the frames by applying Tra-Bond2143D epoxy[24]around each insert outside the chamber end-plates,but still inside the gas manifold.Electrical contact between the straw inner surface(via the gold-plated brass‘sleeve’)was made byﬁlling the gap between straw/sleeve end and inner end-plate surface with EPO-TEK410E silver-ﬁlled epoxy[23]. Although before installation individual straw assemblies were tested for leaks, the silver epoxy should form an additional gas seal;the2143D epoxy forms the manifold-straw seal.The frame assembly was completed by gluing down of the manifold cover with a bead of silver epoxy joining the aluminum cover to the end-plate andﬁlling the2143D epoxy around each ultem insert. Wires were installed after chamber assembly.We used20-µm,gold-plated tungsten wire.The wire was threaded through a blunted sewing needle,fed through the top insert,and lowered down the straw and out the bottom insert. To provide working clearance and hold the wire against the interior v-notch of the insert,the wire was deﬂected by a horizontal bar suspended below the chamber.A gold-plated brass pin was inserted into the top insert and clamped in place with an Ultem collar.Then tension was applied to the wire by clamping a70-g mass below the bottom insert and deﬂecting bar(some tension was lost to friction at the bar and at the Ultem notch).Wire tension was locked in by installing another brass pin in the bottom insert,again with a clamping Ultem collar.Because of the awkward position and close-pack clearances of this step,we designed a special tool for pin installation.Once we were satisﬁed with the wire installation,a drop of epoxy(301-2)at each end ensured the pins and wire ends stayed in place.We checked wire tensions by measuring the resonant frequency of a driven wire in a constant magneticﬁeld.A custom device[28]performed the frequency measurement and provided the driving frequency,while a permanent magnet held near the wire center supplied theﬁeld.Any wires not satisfying minimum tension or standoﬀlimits were removed and replaced.3Readout electronicsSignals from the straw sense wires were ampliﬁed and discriminated on6-channel boards directly mounted on the straw end pins.The ampliﬁer circuitFig.8.SDC chamber assembly.(Figure9)is a four-transistor design,decoupled from the high-power discrimi-nation stage by a1-to-2transformer[29].The gain was about20mV/µA with less than1.5%cross talk,and a typical operating threshold of1.5µA.The digitized signal was converted to a30-ns pulse and sent over a333-ft long, 500-ns delay line by a96-channel driver board,to be recorded by a6-bit, 1.75-ns least-count TDC.Since particle positions were determined from the drift times of ionization electrons,position resolution was directly aﬀected by the precision of the drift-time measurement.The high particleﬂux expected in experiment871required a small cell size to minimize occupancy,with a high drift velocity gas for reduced dead time.The small cell size of the straw chambers called for tight packing of front-end channels,increasing the possibility of cross talk.Our plan to use a CF4drift gas,meanwhile,implied small pulse heights,requiring a high gain front-end ampliﬁer.In addition,the front-end electronics were required to dovetail with existing equipment:32-channel gray code TDCs with1.875ns least count[33] coupled to the front-end electronics through97-conductor(32ECL channel) 500ns delay lines.Front-end components designed and built speciﬁcally for E871included6-channel ampliﬁer/discriminator boards,which mounted di-rectly onto the chamber frames,and96-channel cable driver boards.A blockR2539+5.0VETEST R5512K R302.4KC14.1uF R1710K C17.1uFQ2BFT25R18680C222.2pF Q3BFT25C20.1uF R44680C12.1uF Xform-Q4BFT25R3222R1910R354.3K Q1BFT92R38330C3220pFD8BAV99CH0 INC18390pF R34100R20200R271K Xform+LOGIC 0R515.1+5VCVTH Xform+3144 13T1B PE8309C25.1uF R4343K C26.1uF12 138********1517U2B LOGIC 1CH0 OUTCH0 OUT C24.1uFR39B 220Xform-R39A220R525.1-5VCFig.9.Ampliﬁer schematic for the straw drift chambers.A single ampliﬁer and discriminator circuit is shown.500 ns DelayCablesTDCs32 channelCable Driver Boards 16 x 6 channels in 3 x 32 channels out on twist-n-flat cable Differential signalsAmp/Disc BoardsStraws ~8000 total6 straws/board Fig.10.E871straw chamber readoutdiagram of the straw chamber electronics setup is shown in Figure 10.Initial designs of the front end ampliﬁer based on an integrated-circuit am-pliﬁer proved costly (more than 12dollars per channel),while an ASIC-based design missed our time window.We therefore pursued a much less expensive option (less than ﬁve dollars per channel),based on a four-transistor discrete ampliﬁer design.Our initial design,with high-voltage distribution and the analog ampliﬁer circuit on the chamber-mounted board and the discriminator on a separate driver board,taught us that twist-n-ﬂat cable made an excellentantenna.The discriminator stage was subsequently moved onto the chamber-mounted board.Theﬁnal6-channel ampliﬁer/discriminator layout consisted of high-voltage, analog,and digital sections coexisting on one four-layer board.Copious grounding and surface-mounted components eased the tight space constraints, while SIP sockets facilitated mounting the cards directly onto the straw end pins.The bias high voltage and front-end ampliﬁer low-voltage power,along with odd-and even-channel pulser signals for rudimentary electronics diagno-sis,were delivered by chamber-mounted busses.The digital half of the ampli-ﬁer boards were supplied by a separate cable bus.3.1Early testingEarly bench tests with simulated straw pulses reported gains of(22.±1.2)mV/µA.Stable operation was achieved at thresholds as low as0.9µA.We found that DIP-package multi-channel transformers in our original ampliﬁer design caused too much cross talk.After replacing these with single-channel packages,cross talk between adjacent channels was reduced to1.5%.The front-end ampliﬁers were also tested on prototype straw chambers.During cosmic ray tests the ampliﬁers again operated at thresholds below1µA,with overall straw eﬃciencies measured above97%.Single-wire position resolution for the cosmic ray tests was120µm for the Ar-ethane(50-50)gas-ﬁll.3.2Full-system performanceDuring actual data taking for E871,the complete6500-channel straw cham-ber readout system performed nearly as well as in tests.Even in the noisy environment of a high energy physics experiment,all chambers operated at or below a1.3-µA threshold with a mean eﬃciency of96%.We experienced a signiﬁcant board failure rate due to cold solder joints on the ampliﬁer cards;a test stand was built to diagnose and repair these faults. The entire mechanical design and the front-end electronics system has sub-sequently been copied for use in the Fermilab FOCUS experiment(FNAL 831)[27].。

ModelTestAnswer

ModelTestAnswerModel Test 1快速阅读1.答案为Y；题目意思是：汽车尽管在生活中必不可少，但给城市生活的各个方面带来了负面影响。

根据全文，尤其是小标题，不难得出答案为Y。

2. 答案为N；题目意思是：汽车产生的社会隔离问题在郊区并不存在，因为汽车在郊区不普及。

而原文第二段第二句表明这一问题不仅在郊区存在，而且更为突出，所以题目错误。

3. 答案为Y；题目意思是：人们步行、骑车或是乘坐公交车辆时，会更关注环境。

与原文第三段第四句中“feel a sense of responsibility for the environments”的意思相同，因此本句答案为Y。

4. 答案为Y；题目意思是：汽车交通引起的噪音、危险和污染是街区生活和社会交往减少的原因。

原文第四段开头两句话就指出，街区生活随着车辆在街道上的增加而消失，并指出直接原因是噪音、危险和污染，因此题目正确。

5. 答案为Y；题目意思是：欧洲将对汽车制定更严格的标准以减少与交通相关的污染。

根据原文第十段最后一句的内容判断本句答案为Y。

6. 答案为N；题目意思是：由于更高的汽车技术，在美国，汽车所消耗的国民收入在减少。

而文章第十四段指出，每年随着驾车人数和行驶距离的增加，汽车所消耗的国民收入随之增加，所以题目错误。

7. 答案为NG；题目意思是：在南亚，将近一半的市区土地被用来建造汽车公路。

而原文第十五段谈论汽车公路的占地问题是以美国为例，没有提到亚洲的情况。

8. 答案为the car；题目意思是：造成美国年轻人死亡最主要的原因是什么?根据原文第八段第三句话指出汽车是造成美国年轻人死亡最主要的原因。

9. 答案为community；题目意思是：消除城市车辆将有助于社区感的恢复。

根据第七段第一句话，可直接找到答案为community。

10. 答案为family life；题目意思是：由于交通拥挤，父母没有时间与孩子相处，对家庭生活造成不良影响。

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This means that the mixed gauge-gravitational anomaly 1 is absent. The number of SU(2) fermion doublets is even. Hence the SU(2) global anomaly 2 is also absent. Because left-handed and right-handed fermions transform diﬀerently, there are potentially several axial-vector triangle anomalies 3 in the standard model. However, they are all canceled as follows: [SU (3)]2 U (1) : 2(1/6) − (2/3) − (−1/3) = 0, 3(1/6) + (−1/2) = 0, (4) (5) (6)
1
Cancellation of Anomalies
In the minimal standard model, under the gauge group SU (3)C × SU (2)L × U (1)Y , the quarks and leptons transform as: u d ∼ (3, 2, 1/6),
Hence B − L itself cannot be gauged without the addition of one νR per family. It is already well-known that if there is one νR per family, then the standard model may be extended to become a left-right symmetric model with 1 Q = T3L + Y = T3L + T3R + (B − L). 2 (12)
If there is only one νR ∼ (1, 1, 0), then it can couple only to one linear combination of νiL . Assuming that its Majorana mass is MR , the 4 × 4 mass matrix c spanning νiL and νR is given by 0 0 0 m1 0 0 0 m2 . M= (17) 0 0 0 m3 m1 m2 m3 M R If m1,2,3 << MR , one linear combination of νiL gets a seesaw mass 8 mν0 ∼
α ∂α JB +L =
g2 ˜ iµν , Nf Wiµν W 16π 2
(7)
˜ its dual, g is the corresponding where Wiµν is the SU (2)L ﬁeld tensor and W gauge coupling, and Nf is the number of families of quarks and leptons. Nonperturbative pseudo-particle contributions to the above result in the selection rule ∆(B + L) = 2Nf ν, (8) where ν= Note that
2 2
1 − 1 = 0,
(13) (14)
2
[2(−1/2)2 − (−1)2 ](1 − 1) = 0,
2 2
2(−1/2)(1) − (−1)(1) + 2(−1/2)(−1) − (−1)(−1) = 0, (15) 2
(Li − Lj3; 2(−1)3 − (−1)3 = 0.
Given the particle content of the standard model without and with a right-handed neutrino, the requirement that all anomalies cancel singles out a set of possible global symmetries which can be gauged. I review this topic and propose a new gauge symmetry B − 3Lτ in the context of the minimal standard model consisting of the usual three families of quarks and leptons plus just one νR . The many interesting phenomenological consequences of this hypothesis are brieﬂy discussed.
[SU (2)]2 U (1) : [U (1)]3 :
6(1/6)3 + 2(−1/2)3 − 3(2/3)3 − 3(−1/3)3 − (−1)3 = 0. 1
2
Global and Gaugeable Symmetries
There are four global symmetries in the minimal standard model, corresponding to baryon number B and the three lepton numbers Le , Lµ , and Lτ . Each α qiL γ α qiL + q ¯iR γ α qiR ), etc.] is conserved at the classical current [i.e. JB = i (¯ (tree) level. However, at the quantum level [i.e. when loops are included], each current by itself is not conserved. In fact, it was shown long ago 4 that
L
uR ∼ (3, 1, 2/3),
dR ∼ (3, 1, −1/3);
(1)
ν l
∼ (1, 2, −1/2),
L
lR ∼ (1, 1, −1).
(2)
The sum of the U(1) axial charges is 6(1/6) + 2(−1/2) − 3(2/3) − 3(−1/3) − (−1) = 0. (3)
(16)
The Z ′ boson associated with this U(1) may show up in e+ e− or µ+ µ− collisions as a resonance decaying into charged-lepton pairs. Without νR , all neutrinos are massless, protected respectively by the additive lepton numbers Le , Lµ , Lτ , unless a Higgs triplet is added 6 . In general, there may be a certain number of νR ’s, not necessarily equal to the number of νL ’s, as discussed for example in Ref. 7. 3 Standard Model + one νR
SYMMETRIES OF THE STANDARD MODEL WITHOUT AND WITH A RIGHT-HANDED NEUTRINO
arXiv:hep-ph/9803208v1 28 Feb 1998
ERNEST MA Department of Physics, University of California Riverside, CA 92521, USA
As mentioned already, B cannot be gauged by itself without adding to the particle content of the standard model. Nevertheless, the phenomenology of such a possibility has been discussed 12 . On the other hand, if the one νR added to the minimal standard model is required to be ντ R , then B − 3Lτ can be gauged 13 . The various axial-vector triangle anomalies are canceled as follows: [SU (2)]2 U (1)X : 3 3(3)(1/3) − 3 = 0, (19)
α ∂α JB −L ∝ 3(3)(1/3) − 1 − 1 − 1 = 0;
g2 32π 2
˜ iµν = ±1, ±2, ... d4 xWiµν W
(9) (10)
hence the electroweak phase transition conserves B − L. However, the chiral B − L anomaly is nonzero: [U (1)B −L ]3 : 6(1/3)3 + 2(−1)3 − 6(1/3)3 − (−1)3 = 0; (11)
2 2 m2 1 + m2 + m3 . MR