Constraints on the rare tau decays from mu -- e gamma in the supersymmetric see-saw model
Decay Constants $f_{D_s^}$ and $f_{D_s}$ from ${bar{B}}^0to D^+ l^- {bar{nu}}$ and ${bar{B}
form factor.
PACS index : 12.15.-y, 13.20.-v, 13.25.Hw, 14.40.Nd, 14.65.Fy Keywards : Factorization, Non-leptonic Decays, Decay Constant, Penguin Effects
∗ experimentally from leptonic B and Ds decays. For instance, determine fB , fBs fDs and fDs
+ the decay rate for Ds is given by [1]
+ Γ(Ds
m2 G2 2 2 l 1 − m M → ℓ ν ) = F fD D s 2 8π s ℓ MD s
1/2
(4)
.
(5)
In the zero lepton-mass limit, 0 ≤ q 2 ≤ (mB − mD )2 .
2
For the q 2 dependence of the form factors, Wirbel et al. [8] assumed a simple pole formula for both F1 (q 2 ) and F0 (q 2 ) (we designate this scenario ’pole/pole’): q2 F1 (q ) = F1 (0) /(1 − 2 ), mF1
∗ amount to about 11 % for B → DDs and 5 % for B → DDs , which have been mentioned in
时序约束英语
时序约束英语Time Constraint in EnglishThe concept of time constraint has become increasingly relevant in the modern world, where efficiency and productivity are highly valued. In the realm of language learning, the ability to communicate effectively under time pressure is a crucial skill that can have a significant impact on an individual's academic and professional success. This essay will delve into the importance of time constraint in the context of English language proficiency, exploring its implications and strategies for effective management.One of the primary challenges posed by time constraint in English language learning is the need to formulate and express thoughts and ideas in a concise and coherent manner. In real-world situations, such as job interviews, presentations, or academic examinations, individuals are often required to respond to questions or prompts within a limited timeframe. This can be particularly daunting for non-native English speakers, who may struggle to organize their thoughts and articulate their responses fluently.To address this challenge, learners must develop strategies toenhance their ability to think and communicate quickly in English. This may involve practicing timed exercises, such as writing short essays or engaging in mock interviews, to improve their ability to generate and express ideas under time pressure. Additionally, learners can focus on developing a strong command of English grammar, vocabulary, and idioms, which can help them formulate more precise and natural-sounding responses.Another aspect of time constraint in English language learning is the ability to comprehend and interpret information rapidly. In academic or professional settings, individuals may be required to read and understand lengthy texts, listen to presentations, or engage in discussions within a limited timeframe. Effective time management in these situations can be the difference between success and failure.To enhance their ability to comprehend and interpret information quickly, learners can engage in activities that challenge their reading and listening skills. This may include reading articles or books within a specified time limit, listening to audio recordings and answering comprehension questions, or participating in group discussions where they must process and respond to information in real-time.Moreover, the ability to manage time effectively can also have a significant impact on language learners' performance in high-stakes assessments, such as standardized tests or job-related examinations.In these scenarios, learners must not only demonstrate their language proficiency but also their ability to allocate their time wisely and complete the required tasks within the allotted timeframe.To prepare for such assessments, learners can engage in practice tests and simulations, familiarizing themselves with the format and timing of the exam. They can also develop strategies for managing their time, such as prioritizing tasks, allocating appropriate time to each section, and practicing efficient time-keeping techniques.In addition to the practical benefits of time constraint in English language learning, there are also cognitive and psychological aspects to consider. The ability to think and communicate quickly under pressure can foster cognitive flexibility, problem-solving skills, and resilience – all of which are highly valued in academic and professional settings.Furthermore, the experience of successfully navigating time-constrained situations can boost an individual's confidence and self-efficacy, further enhancing their language proficiency and overall performance. As learners gain experience in managing time constraints, they may develop a greater sense of control over their learning process and a deeper appreciation for the importance of time management in language acquisition.In conclusion, time constraint is a crucial aspect of English language learning that can have a significant impact on an individual's academic and professional success. By developing strategies to enhance their ability to think, communicate, and comprehend information quickly, learners can not only improve their language proficiency but also cultivate valuable cognitive and psychological skills that can serve them well in a wide range of contexts. As the demands of the modern world continue to evolve, the ability to navigate time constraints in English language learning will remain a vital skill for individuals seeking to thrive in an increasingly competitive global landscape.。
Isospin constraints on the $tauto Kbar Knpinu$ decay mode
a r X i v :h e p -p h /9708473v 2 7 O c t 1997Isospin constraints on the τ→K ¯Knπνdecay modeAndr´e Roug´e ∗LPNHE Ecole Polytechnique-IN2P3/CNRSF-91128Palaiseau CedexAugust 1997AbstractThe construction of the complete isospin relations and inequalities between thepossible charge configurations of a τ→K ¯Knπνdecay mode is presented.Detailedapplications to the cases of two and three pions are given.X-LPNHE 97/081IntroductionThe isospin constraints on the τhadronic decay modes are known for the nπand Knπmodes[1,2].For the K ¯Knπfinal states,only the simplest decay mode K ¯Kπhas been ing the formalism of symmetry classes introduced by Pais[3],we generalize the relations for an arbitrary value of n and give a geometrical representation of the constraints.2The general methodThe K ¯Knπsystem produced by a τdecay has isospin 1;the possible values of the K ¯Kisospin I K ¯K are 0and 1and the isospin I nπof the n pion system is 1for I K ¯K =0and 0,1or 2for I K ¯K =1.Since there is no second-class current in the Standard Model,interferences between amplitudes with I K ¯K =0and I K ¯K =1vanish in the partial widths [2].Therefore we have the relationΓK 0¯K 0(nπ)−=ΓK +K −(nπ)−,(1)which is true for each charge configuration of the nπsystem,and,since ΓK S K S (nπ)−=ΓK L K L (nπ)−,ΓK +K −(nπ)−=ΓK S K L (nπ)−+2ΓK S K S (nπ)−,(2)using the most easily observable states.The amplitudes are classified by the values of I K¯K and I nπ.To complete the classifi-cation,we use the isospin symmetry class[3]of the nπsystem i.e.the representation of the permutation group S n to which belongs the state.It is characterized by the lengths of the three rows of its Young diagram(n1n2n3).Due to the Pauli principle,the momentum and isospin states have the same symmetry.Thus integration over the momenta kills the interferences between amplitudes in different classes and there is no contribution from them in the partial widths.Since I nπ=0and I nπ=1amplitudes belong to different symmetry classes[3],their in-terferences vanish.The presence of I nπ=2amplitudes makes the problem more intricate since they share symmetry properties with some I nπ=1or I nπ=0amplitudes[4,5].For instance,in the case n=2the symmetry class(200)is shared by I nπ=0and I nπ=2; in the case n=3,the symmetry class(210)is shared by I nπ=1and I nπ=2.Therefore the allowed domains in the space of the charge configuration fractions(f cc=Γcc/ΓK¯Knπ) must be determined separately for each symmetry class and I K¯K,taking interferences into account when necessary.The complete allowed domain is the convex hull of the sub-domains corresponding to the different I K¯K and symmetry classes and its projections are the convex hulls of their projections.For n≤5,which is always true in aτdecay,the isospin values and the symmetry class characterize unambiguously the amplitude properties[4,5],therefore there is,at most, one interference term per class.The sub-domain,for such a symmetry class associated with two different I nπvalues,is then a two-dimensional one since the partial widthsΓcc are linear functions of three quantities:the sums of squared amplitudes for the two values of I nπand the interference term.Its boundary is determined by the Schwarz’s inequality [6].This boundary is an ellipse;it can be parametrized by writing the sums of squared amplitudes for the two values of I nπasρ[1±cosθ]/2and the largest interference term allowed by the Schwarz’s inequality as kρsinθ,where the coefficient k depends on the coupling coefficients.The most general domain is hence the convex hull of a set of points corresponding to the symmetry classes without I nπ=2and a set of ellipses.The cases n=2and n=3are presented in detail in the following sections.They both have the property that only one symmetry class is associated with two isospin values. Higher values of n are not expected,for some time,to be of experimental interest.3The decayτ→K¯K2πνThe possible states that can be observed for aτ→νK¯Kππdecay are the following:K S K Sπ0π−K S K Lπ0π−K L K Lπ0π−K+K−π0π−K+¯K0π−π−K0K−π+π−K0K−π0π0.As mentioned before,not all the corresponding partial widths are independent and we can use the four fractions:2f K+K−π0π−=f K+K−π0π−+f K0¯K0π0π−,f K+¯K0π−π−,f K0K−π+π−and f K0K−π0π0,whose sum is equal to1,to describe the possible charge configurations in a three-dimensional space.The ratio K S K S/K S K L is a free parameter independent of the charge configuration fractions.The partial widths for all the charge configurations can be expressed as functions ofthe positive quantities S[IK¯K ,Iππ]which are the sums of the squared absolute values ofthe amplitudes with the given values of the isospins and the interference term I of the Iππ=0and Iππ=2amplitudes.With only two pions the coefficients are readily obtained from a Clebsch-Gordan table and we getΓK+K−π0π−+ΓK0¯K0π0π−=2ΓK+K−π0π−=S[0,1]+110S[1,2]ΓK+¯K0π−π−=63S[1,0]+130S[1,2]+IΓK0K−π0π0=130S[1,2]−I.The partial widthΓK¯Kππis the sum over the charge configurations:ΓK¯Kππ= ccΓcc= sc S sc.(4) The Schwarz’s inequality bounding the interference term I is|I|≤254(f K0K−π+π−−2f K0K−π0π0+16f K+¯K0π−π−).As explained before,the complete domain is the convex hull of this ellipse and the two points[0,1]and[1,1].Calling I the point of the ellipse for which f K0K−π0π0=0,the domain is made of the tetrahedron having the points I,[0,1],[1,1]and[1,0]for vertices and of the two half-cones whose bases are the halves of the ellipse delimited by the points I and[1,0]and whose vertices are the points[1,1]and[0,1]respectively.0.512f(K -K +π-π0)f (K 0K -π0π0)+f (K 0K -π+π-)0,1]0.512f(K -K +π-π0)f (K 0K +π-π-)+f (K 0K -π+π-)0,1]Figure 1:Projections of the allowed domain on the planes x/y ,x =2f K +K −π0π−,y =f K 0K −π+π−+f K 0K −π0π0and x =2f K +K −π0π−,y =f K +¯K 0π−π−+f K 0K −π+π−.The classes of amplitudes are labelled by the isospin values,[I K ¯K ,I 2π].For practical purposes it is useful to draw the projections of the domain on two-dimensional planes.The method is very simple:we first draw the projection of the ellipse on the plane and then the tangents to the projected ellipse from the projections of the points [0,1]and [1,1].A first simple example is the projection on the plane x/y ,with x =2f K +K −π0π−and y =f K 0K −π+π−+f K 0K −π0π0.Here the ellipse projection is a mere segment and the projected domain is the polygon whose vertices are the (projected)points [0,1],[1,1],[1,0]and [1,2].More interesting is the projection x =2f K +K −π0π−,y =f K +¯K 0π−π−+f K 0K −π+π−,since the two final states K +¯K0π−π−and K 0K −π+π−have the same topology:one K 0and three charged hadrons.The complement 1−x −y is the fraction f K 0K −π0π0of decays with two π0’s.The projected ellipse has vertical tangents at the points [1,0]and [1,2].It is also tangent to the line x +y =1at the point I (x =1/4),for which W [1,0]=W [1,2]/5since f K 0K −π0π0can be 0,because of the interference,only when the two contributions have the same modulus.The second tangent from [0,1]touch the ellipse at the point of coordinates x =2/23and y =12/23.The allowed domain is shown on Fig.1.The main constraint is the inequalityf K 0K −π0π0≤3implies the dominance of I K¯K=1and a small value the dominance of I K¯K=0.With one dominant isospin for the K¯K system,the ratio K S K S/K S K L measures the proportions of the two G-parities i.e.the contributions of axial and vector currents.4The decayτ→K¯K3πνThefinal states for the decayτ→K¯Kπππνare:K0¯K0π+π−π−K+K−π+π−π−K0¯K0π0π0π−K+K−π0π0π−K0K−π+π−π0K0K−π0π0π0K+¯K0π−π−π0.The relations between the K0¯K0and K+K−final states are the same as in the τ→K¯Kππνdecay.Thus the charge configurations are described in a four-dimensional space by thefive fractions:2f K+K−π+π−π−=f K+K−π+π−π−+f K0¯K0π+π−π−,2f K+K−π0π0π−= f K+K−π0π0π−+f K0¯K0π0π0π−,f K0K−π+π−π0,f K0K−π0π0π0and f K+¯K0π−π−π0.We shall label the amplitudes by the two isospin values and the symmetry class: [I K¯K,(n1n2n3)I3π].The relations between the partial widths for the charge configurations and the amplitudes use both Clebsch-Gordan coefficients and the similar coefficients for the symmetry classes[4,5].With the notations defined in the previous section,they can be written2ΓK+K−π−π0π0=ΓK0¯K0π−π0π0+ΓK+K−π−π0π0=12S[0,(210)1]+14S[1,(210)1]+35S[0,(300)1]+15S[1,(300)1]+120S[1,(210)2]−IΓK0K−π+π−π0=S[1,(111)0]+12S[1,(210)1]+1√10S[1,(300)1](8)ΓK+¯K0π−π−π0=3√25S[1,(210)1]S[1,(210)2].(11)2f(K +K -π-π-π+)f (K 0K +π-π-π0)+f (K 0K -π+π-π0)0Figure 2:Projection of the allowed domain on the plane x/y ,x =2f K +K −π+π−π−,y =f K 0K −π+π−π0+f K +¯K 0π−π−π0.The classes of amplitudes are labelledby the isospin values and symmetry classes,[I K ¯K ,(n 1n 2n 3)I3π].The sub-domains are points for the classes [0,(300)1],[0,(210)1],[1,(300)1]and [1,(111)0].For the interfering classes [1,(210)1]and [1,(210)2],the plane of the two-dimensional sub-domain is determined by the two relations:ΓK 0K −π0π0π0=0and 2(1+13)ΓK +K −π−π0π0+2(1−13)ΓK +K −π+π−π−−ΓK 0K −π+π−π0−1To distinguish twoπ0from threeπ0decays,we can use a third coordinate z= f K0K−π0π0π0.The three-dimensional domain is the cone having for basis the above de-scribed contour in the x/y plane and,for vertex,the point[1,(300)1].5SummaryWe have presented the complete isospin constraints on theτ→K¯Knπνdecay modes in the space of charge configurations with some details in the cases n=2and n=3.The geometrical method adopted allows to draw very easily any wanted projection of the multi-dimensional domain and hence obtain the most restrictive inequalities for a given set of measurements.References[1]F.J.Gilman and S.H.Rhie,Phys.Rev.D31(1985)1066[2]A.Roug´e,Z.Phys.C70(1996)109[3]A.Pais,Ann.Phys.9(1960)548[4]A.Pais,Ann.Phys.22(1963)274[5]H.Pilkuhn,Nucl.Phys.22(1961)168[6]L.Michel,Nuovo Cim.22(1961)203。
瓶颈管理英文版
❖ 1986年后半年，Drs. Goldratt and Robert E. Fox共 同除了Goldratt Research Institute。OPT的理念经 10年的发展逐步演进为众所周知的TOC。
The opportunities to make more money by increasing Throughput, on the other hand, are unlimited.
7
瓶颈管理的学习框架
8
什么是瓶颈管理？
TOC是一套管理理念与管理工具的集 合，它把企业在实现其目标过程中现 存的或潜伏的制约因素称为“瓶颈” 或“约束”，通过逐个识别和消除这 些“瓶颈”，使得企业的改进方向与 改进策略明确化，从而达到帮助企业 更有效地实现其目标的目的。
❖T＝S－TVE
21
瓶颈管理的核心理念是什么？
TOC作业指标体系中的三个概念 产销率(Throughput, T)—有效产出
❖ 必须把产成品与卖出品区别开来。 ❖在制品和未销售出去的产成品只能是库存，不能 带来T的增加。 ❖只有那些销售出去的产品才能真正给企业带来利 润。 ❖产销率衡量了企业在单位时间内能够生产、销售 产品而最终获利多少的能力。
❖ Increase Throughput
❖ Reduce Inventory
❖ Reduce Operating Expense
As Dr. Goldratt notes, the opportunities to make more money through reductions in inventory and operating expense are limited by zero.
Early-universe constraints on a Primordial Scaling Field
a r X i v :a s t r o -p h /0104162v 2 28 M a y 2001Early-universe constraints on Dark EnergyRachel Bean ♯,Steen H.Hansen ♭and Alessandro Melchiorri ♭♯Theoretical Physics,The Blackett Laboratory,Imperial College,Prince Consort Road,London,U.K.♭NAPL,University of Oxford,Keble road,OX13RH,Oxford,UK In the past years ’quintessence’models have been considered which can produce the accel-erated expansion in the universe suggested by recent astronomical observations.One of the key differences between quintessence and a cosmological constant is that the energy density in quintessence,Ωφ,could be a significant fraction of the overall energy even in the early uni-verse,while the cosmological constant will be dynamically relevant only at late times.We use standard Big Bang Nucleosynthesis and the observed abundances of primordial nuclides to put constraints on Ωφat temperatures near T ∼1MeV .We point out that current ex-perimental data does not support the presence of such a field,providing the strong constraint Ωφ(MeV)<0.045at 2σC.L.and strengthening previous results.We also consider the effect a scaling field has on CMB anisotropies using the recent data from Boomerang and DASI,providing the CMB constraint Ωφ≤0.39at 2σduring the radiation dominated epoch.Introduction.Recent astronomical observations [1]suggest that the energy density of the universe is dominated by a dark energy component with nega-tive pressure which causes the expansion rate of the universe to accelerate.One of the main goals for cos-mology,and for fundamental physics,is ascertaining the nature of the dark energy [2].In the past years scaling fields have been con-sidered which can produce an accelerated expansion in the present epoch.The scaling field is known as “quintessence”and a vast category of “tracker”quintessence models have been created (see for exam-ple [3,4]and references therein),in which the field approaches an attractor solution at early times,with its energy density scaling as a fraction of the domi-nant component.The desired late time accelerated expansion behaviour is then set up independently of initial conditions,with the quintessential field domi-nating the energy content.Let us remind the reader of the two key differ-ences between the general quintessential model and a cosmological constant:firstly,for quintessence,the equation-of-state parameter w φ=p/ρvaries in time,usually approaching a present value w 0<−1/3,whilst for the cosmological constant remains fixed at w Λ=−1.Secondly,during the attractor regime the energy density in quintessence Ωφis,in general,a sig-nificant fraction of the dominant component whilst ΩΛis only comparable to it at late times.Future supernovae luminosity distance data,as might be obtained by the proposed SNAP satel-lite,will probably have the potential to discriminate between different dark energy theories [5].These datasets will only be able to probe the late time be-haviour of the dark energy component at red-shift z <2,however.Furthermore,since the luminosity distance depends on w through a multiple integral re-lation,it will be difficult to infer a precise measure-ment of w (z )from these datasets alone [6].In this Letter we take a different approach to the problem,focusing our attention on the early time be-haviour of the quintessence field,when the trackingregime is maintained in a wide class of models,and Ωφis a significant (≥0.01,say)fraction of the overall density.In particular,we will use standard big bang nucle-osynthesis and the observed abundances of primordial nuclides to put constraints on the amplitude of Ωφat temperatures near T ∼1MeV.The inclusion of a scaling field increases the expansion rate of the uni-verse,and changes the ratio of neutrons to protons at freeze-out and hence the predicted abundances of light elements.The presence of this field in the radiation domi-nated regime also has important effects on the shape of the spectrum of the cosmic microwave background anisotropies.We use the recent anisotropy power spectrum data obtained by the Boomerang [7]and DASI [8]experiments to obtain further,independent constraints on Ωφduring the radiation dominated epoch.There are a wide variety of quintessential models;we limit our analysis to the most general ones,with attractor solutions established well before nucleosyn-thesis.More specifically,we study a tracker model based on the exponential potential V =V 0e −λφ[9].If the dominant component scales as ρn =ρ0(a 0λ2.However,the pure exponential potential,since it simply mimics the scaling of the dominant matter in the attractor regime,cannot produce an ac-celerated expanding universe in the matter dominated epoch.Therefore,we focus our attention on a recently pro-posed model by Albrecht and Skordis (referred to as the AS model from herein)[10],motivated by physics in the low-energy limit of M-theory,which includes a factor in front of the exponential,so that it takes the form V =V 0 (φ0−φ)2+A e−λφ.The prefactor introduces a small minimum in the potential.When the potential gets trapped in this minimum its kinetic energy disappears,triggering a period of accelerated-1.0-0.8-0.6-0.4-0.20.0-1.0-0.8-0.6-0.4-0.20.00.2-1.0-0.8-0.6-0.4-0.20.00.2log 10(a)w t o tlog 10(a)ΩQFIG.1.Top panel:Time behaviour of the fractional energy density Ωφfor the Albrecht and Skordis model to-gether with the constraints presented in the paper.The parameters of the models are (assuming h =0.65and Ωφ=0.65)λ=3,φ0=87.089,A =0.01and λ=8,φ0=25.797,A =0.01.Bottom panel:Time behaviour for the overall equation of state parameter w tot for the two models.Luminosity distance data will not be useful in differentiating the two models.expansion,which never ends if Aλ2<1[11].In Fig.1we introduce and summarise the main re-sults of the paper.In the figure,the BBN constraints obtained in section 2,and the CMB constrains ob-tained in section 3are shown together with two dif-ferent versions of the AS model which both satisfy the condition Ωφ=0.65today.Constraints from BBN.In the last few years impor-tant experimental progress has been made in the mea-surement of light element primordial abundances.For the 4He mass fraction,Y He ,two marginally compati-ble measurements have been obtained from regression against zero metallicity in blue compact galaxies.A low value Y He =0.234±0.003[12]and a high one Y He =0.244±0.002[13]give realistic bounds.We use the high value in our analysis;if one instead con-sidered the low value,the bounds obtained would be even stronger.Observations in different quasar absorption line sys-tems give a relative abundance for deuterium,critical in fixing the baryon fraction,of D/H =(3.0±0.4)·10−5[14].Recently a new measurement of deuterium in the damped Lyman-αsystem was presented [15],leading to the weighted abundance D/H =(2.2±0.2)·10−5.We use the value from [14]in our analysis;the use of [15]leads to even stronger bound.In the standard BBN scenario,the primordial abun-dances are a function of the baryon density η∼Ωb h 2only.To constrain the energy density of a primor-FIG. 2.1,2and 3σlikelihood contours in the (Ωb h 2,Ωφ(1MeV))parameter space derived from 4He and D abundances.dial field a T ∼MeV,we modified the standard BBN code [16],including the quintessence energy compo-nent Ωφ.We then performed a likelihood analysis in the parameter space (Ωb h 2,ΩBBN φ)using the observed abundances Y He and D/H .In Fig.2we plot the 1,2and 3σlikelihood contours in the (Ωb h 2,ΩBBN φ)plane.Our main result is that the experimental data for 4He and D does not favour the presence of a dark energy component,providing the strong constraint Ωφ(MeV)<0.045at 2σ(corresponding to λ>9for the exponential potential scenario),strengthening sig-nificantly the previous limit of [17],Ωφ(MeV)<0.2.The reason for the difference is due to the improve-ment in the measurements of the observed abun-dances,especially for the deuterium,which now corre-sponds to approximately ∆N eff<0.2−0.3additional effective neutrinos (see,e.g.[18]),whereas Ref.[17]used the conservative value ∆N eff<1.5.One could worry about the effect of any underesti-mated systematic errors,and we therefore multiplied the error-bars of the observed abundances by a factor of 2.Even taking this into account,there is still a strong constraint Ωφ(MeV)<0.09(λ>6.5)at 2σ.Constraints from CMB.The effects of a scaling field on the angular power spectrum of the CMB anisotropies are several [10].Firstly,if the energy den-sity in the scaling quintessence is significant during the radiation epoch,this would change the equality redshift and modify the structure of the peaks in the CMB spectrum (see e.g.[19]).Secondly,since the inclusion of a scaling field changes the overall content in matter and energy,the angular diameter distance of the acoustic horizon size at recombination will change.This would result in a shift of the peak positions on the angular spectrum.It is important to note that this effect does not quali-tatively add any new features additional to those pro-01002003004005006007008009001000110020406080100FIG.3.CMB anisotropy power spectra for the Al-brecht-Skordis models with λ=3,φ0=87.22and A =0.009(long dash),λ=8,φ0=32.329and A =0.01(short dash)(both with Ωφ=0.65and h =0.65),and a cosmological constant with ΩΛ=0.65,N ν=3.04(full line)and N ν=7.8(dash-dot).duced by the presence of a cosmological constant [20].Third,the time-varying Newtonian potential after decoupling will produce anisotropies at large angu-lar scales through the Integrated Sachs-Wolfe (ISW)effect.Again,this effect will be difficult to disentan-gle from the same effect generated by a cosmologi-cal constant,especially in view of the affect of cos-mic variance and/or gravity waves on the large scale anisotropies.Finally,the perturbations in the scaling field about the homogeneous solution will also slightly affect the baryon-photon fluid modifying the structure of the spectral peaks.However,this effect is generally neg-ligible.From these considerations,supported also by re-cent CMB analysis [21,22],we can conclude that the CMB anisotropies alone cannot give significant con-straints on w φat late times.If,however,Ωφis signif-icant during the radiation dominated epoch it would leave a characteristic imprint on the CMB spectrum.The CMB anisotropies can then provide a useful cross check on the bounds obtained from BBN.To obtain an upper bound on Ωφat last scatter-ing,we perform a likelihood analysis on the recent Boomerang [7]and DASI [8]data.The anisotropy power spectrum from BOOMERang and DASI was estimated in 19bins between ℓ=75and ℓ=1025and in 9bins,from ℓ=100to ℓ=864respec-tively.Our database of models is sampled as in [23],we include the effect of the beam uncertainties for the BOOMERanG data and we use the public available covariance matrix and window functions for the DASI experiment.There are naturally degeneracies between Ωm and ΩΛwhich are broken by the inclusion of SN1a data [1].0.010.111001000FIG.4.Matter power spectra for 3models in fig. 3.The predictions support those in the CMB spectra,the quintessence model in agreement with BBN λ=8(short dash)mimics the ΩΛ=0.65spectrum with N ν=3.0(full line).The model with λ=3(long dash)is in clear dis-agreement with observations.By finding the remaining “nuisance”parameters which maximise the likelihood,we obtain Ωφ<0.39at 2σlevel during the radiation dominated epoch.Therefore,while there is no evidence from the CMB anisotropies for a presence of a scaling field in the ra-diation dominated regime,the bounds obtained are actually larger than those from BBN.In Fig.3we plot the CMB power spectra for 4alter-native scenarios.The CMB spectrum for the model which satisfies the BBN constraints is practically in-distinguishable from the spectrum obtained with a cosmological constant.Nonetheless,if the dark energy component during radiation is significant,the change in the redshift of equality leaves a characteristic im-print in the CMB spectrum,breaking the geometri-cal degeneracy.This is also found when considering non-minimally coupled scalar fields [24],even when the scalar is a small fraction of the energy density at last scattering.In the minimally coupled models con-sidered here,this is equivalent to an increase in the neutrino effective number,i.e.altering the number of relativistic degrees of freedom at last scattering.In Fig.4we have plotted the corresponding mat-ter power spectra together with the decorrelated data points of Ref.[25].As one can see,the model with λ=3is in disagreement with the data,producing less power than the model with λ=8,with this last one still mimicking a cosmological constant.The less power can be still explained by the increment in the radiation energy component which shift the equality at late time and the position of the turn-around in the matter spectrum towards larger scales.A bias fac-tor could in principle solve the discrepancy between the λ=3model and the 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Complete reducibility of systems of equations with respect to
1
1.1
Introduction
Framework
Since the 1960’s, the theory of finite semigroups has drawn a strong motivation from applications in computer science, namely as a natural algebraic framework for classifying combinatorial phenomena described through finite automata, rational languages, or various kinds of logical formalisms [30, 31, 40, 41, 51]. A central question in problems arising from such applications is to determine effectively whether a given finite semigroup belongs to a fixed pseudovariety (that is a class closed under taking homomorphic images, subsemigroups, and finite direct products) or to show that there is no algorithm that solves the membership problem for the pseudovariety. Pseudovarieties are often defined by describing a set of generators, which are usually constructed by applying some natural algebraic construction to members of given pseudovarieties. Thus, for two given pseudovarieties U and V, the central problem translates into solving the membership problem for their join U ∨ V, their semidirect or wreath product U ∗ V, their bilateral semidirect or block product U V, their Mal’cev product U m V, or the power P V. While there is no hope for a universal solution of this type of problem, as it has been shown that none of these operators preserves decidability of the membership problem [1, 46, 47, 23], in many particular cases the problem can be solved by exploring special structural features of the semigroups involved.
The partition function of 2d string theory
a rXiv:h ep-th/92831v111A ug1992IASSNS-HEP-92/48YCTP-P22-92hepth@xxx.9208031The Partition Function of 2D String Theory Robbert Dijkgraaf 1School of Natural Sciences Institute for Advanced Study Princeton,NJ 08540Gregory Moore and Ronen Plesser Department of Physics Yale University New Haven,CT 06511-8167We derive a compact and explicit expression for the generating functional of all correla-tion functions of tachyon operators in 2D string theory.This expression makes manifest relations of the c =1system to KP flow and W 1+∞constraints.Moreover we derive aKontsevich-Penner integral representation of this generating functional.August 11,19921.IntroductionOne of the beautiful aspects of the matrix-model formulation of c<1string theory is that it gives a natural and mathematically precise formulation of the partition function of strings moving in different backgrounds.This result began with Kazakov’s fundamental discovery of the appearance of matterfields in the one-matrix model[1]and culminated in the discovery of the generalized KdVflow equations and the associated W N constraints in the c<1matrix models coupled to gravity[2–6].Recently these results have been further deepened through the use of a Kontsevich matrix model representation for the tau functions relevant to theseflows[7],see also[8,9].Analogous results in the c=1model have been strangely absent,and this paper is afirst step in an attempt to change that situation. Using recently developed techniques for calculating tachyon correlators in the c=1model we derive a simple and compact expression(equation(3.10))for the generating functional of tachyon correlators,or equivalently the string partition function in an arbitrary tachyon background,valid to all orders in string perturbation theory.In Euclidean space this quantity can be interpreted as the partition function of a nonlinear sigma model as a function of an infinite set of coupling constants t k,¯t k for a set of marginal operators. Upon appropriate analytic continuation to Minkowski space the partition function may be interpreted as the string S-matrix in a coherent state basis.One immediate consequence of our result(3.10)is that the partition function is natu-rally represented as a tau function of the Toda hierarchy.From this result we obtain W∞flow equations(equation4.10)when the c=1coordinate X is compactified at the self-dual radius.Moreover,this expression can be used to derive a Kontsevich-Penner representa-tion of the partition function as a matrix integral,as described in sectionfive below.In section six we discuss how time-independent changes in the matrix model backgroundfit into our formalism,and in section seven we discuss some open problems and the relation of this work to other recent papers on c=1and W∞.2.Defining correlation functionsIn a particular background,string propagation in a two-dimensional spacetime is described on the string worldsheet by the conformalfield theory of a massless scalar X coupled to a c=25Liouville theoryφwith worldsheet action(excluding ghosts)A= 12R(2)φ+µe√Via its dual interpretation as the conformal gauge action for the coupling of X to two-dimensional gravity,(2.1)isexpressible asthecontinuumlimitofa sumover discretized surfaces.The discrete sum,as is by now well known,is generated by a matrix integral.In the double scaling limit which leads to the continuum theory this is in turn equivalent to a theory of free nonrelativistic fermions with actionS = ∞−∞dxdλˆψ† i ddλ2−V (λ) ˆψ.(2.2)The potential V (λ)in (2.2)is required to approach −1Γ(−|q |) Σe iqX/√2(1−12φso that the bounding circles C have lengths ℓ= C e φ/√sinπ|p |µ−|p |/2I |p |(2√r 2−p 2µ−r/2I r (2√2The behavior of V (λ)for negative λis irrelevant to all orders of perturbation theory in 1/µ.Indeed,the results of this paper should be interpreted in this perturbative sense.Many results are true in the nonperturbative context and we will indicate this in the appropriate places.Where we mention nonperturbative results we will refer to potentials which grow sufficiently rapidly for large negative λ.In [10]these were termed “type I”models.whereˆB r,p are redundant operators for p/∈Z Z.We may thus extract tachyon correlators from macroscopic loop amplitudes asni=1W(ℓi,q i) =n i=1Γ(−|q i|)ℓ|q i|i n i=1T q i +O(ℓ2i) +analytic inℓi.(2.5)The matrix model formulation of the theory leads to a simple computation of the appro-priate limits of loop amplitudes.In the matrix model the macroscopic loop is related by a Laplace transform to the eigenvalue densityˆρ(λ,x)=ˆψ†ˆψ(λ,x):W(ℓ,x)= ∞0e−ℓλˆρ(λ,x)dλW(ℓ,q)= ∞−∞e iqx W(ℓ,x)dx.(2.6) DefiningˆW(z,x)= ∞0e−zℓW(ℓ,x)dℓ(2.7) we recoverˆρ(λ,x)=−i2)/βwith m∈Z Z.The eigenvalue correlator can be written as:ˆρ(λi ,q i ) =∞ m =−∞ σ∈Σn n k =1I (Q σk ,λσ(k ),λσ(k +1))(3.1)where Σn is the set of permutations of n objects,Q σk ≡p m +q σ(1)+···+q σ(k ).The λdependence of the correlator is determined by the function I (q,λ1,λ2)=(I (−q,λ1,λ2))∗=λ1|1√4|λ21−λ22|R [q,λ1,λ2]=R q exp i (µ+iq )log (λ1λ2)−i 4λ2with an infinite wallat λ=0it is given byR q =i 1−ie −π(µ+iq ) 2−iµ+q )2+iµ−q ).(3.3)Inserting(3.2)in the expression (3.1)leads to a sum of terms.The calculation of tachyon correlators requires the extraction of those terms in the sum with the correct asymptotic dependence on λi .For each permutation σ,at most a finite number of terms in the sum over the loop momentum p m contribute to the result.A graphical procedure for performing this extraction was developed in [10]and used to derive an explicit expression for arbitrary tachyon correlators.We divide the tachyon insertions into “incoming”(q <0)and ‘outgoing”(q >0)particles.As in a Feynman diagram there is a vertex in the (x,λ)half-space corresponding to each operator ˆρ(x,λ).While the final result will of course be independent of the order in which the λi are increased to infinity,in intermediate steps we will choose some order and locate the vertices accordingly.Points are connected by line segments,representing the integral I ,to form a one-loop graph.Since the expression for I in (3.2)has two terms we have both direct and reflected propagators as in fig.1.Each line segment carries a momentum and an arrow.Note that in fig.1the reflected propagator,which we call simply a “bounce,”is composedof two segments with opposite arrows and momenta.These line segments are joined to form a one-loop graph according to the following rules:RH1.Lines with positive(negative)momenta slope upwards to the right(left).RH2.At any vertex arrows are conserved and momentum is conserved as time flows upwards.In particular momentum q i is inserted at the vertex infig.2.RH3.Outgoing vertices at(x out,λout)all have later times than incoming vertices (x in,λin):x out>x in.Diagrams drawn according to these rules correspond to possible physical processes in real time and were hence termed“real histories”.The connected tachyon correlation function is found by summing the terms in(3.1)corresponding to all real histories,and reads schematicallyni=1T q i =(−i)n RH± m bounces R Q(−R Q)∗.(3.4)The graphical rules allow one to convert(3.4)into an explicit formula for the amplitude [10].In the next subsection we will show that this result may be written quite simply in terms of free fermionicfields,representing a fermionized version of the free relativistic bosonicfield which describes the asymptotic behavior of the tachyon.3.2.Free Energy in terms of free oscillatorsOne of the central results of[10]is that the graphical rules described above are equiv-alent to the composition of three transformations on the scattering states:fermionization, free fermion scattering,and bosonization:i f→b◦S ff◦i b→f as infig.3.The various real his-tories correspond to the possible contractions among the incoming and outgoing fermions, and the fermion scattering matrix describes a simple one-body process,given essentially by the phase shift in the nonrelativistic problem.It should be noted that this does not imply the(false)statement that bosonization is exact for the nonrelativistic fermion prob-lem.Rather,it is a statement about the asymptotics of certain correlators in the theory for a particular class of potentials.Here we will rewrite the tachyon amplitude using this formulation as a matrix element of a certain operator in the conformalfield theory of a free Weyl fermion.It is convenient to define rescaled tachyon vertex operators V q =µ1−|q |/2T q and two free scalar fields ∂φin/out = n αin/out n z −n −1,such thatn i =1V n i /βm j =1V −n ′j /β =−(iµ)n2with expansionsψ(z )=m ∈Z Z ψm +12z −m −1{ψr ,¯ψs }=δr +s,0.(3.6)Now the result of [10]states that (3.5)is equivalent toψin−(m +12)¯ψin −(m +12).(3.7)Unitarity of the tachyon S -matrix is equivalent to the identityR q R ∗−q =1(3.8)on the reflection factors.3Using this,we can rewrite (3.7)as a unitary transformationψin (z )=Sψout (z )S −1¯ψin (z )=S ¯ψout (z )S −1S =:exp m ∈Z Z log R p m ψout−(m +12 :.(3.9)Thus we may write the full generating functional for connected Green’s functions in terms of a single free boson with modes αn :µ2F ≡ e n ≥1t n V n/β+ n ≥1¯t n V −n/β c=−1µ2F 1+···.This formula is anenormous simplification over previous expressions for c =1amplitudes.The generating function for all amplitudes isZ =e µ2F .(3.11)4.W 1+∞constraintsIn correlation functions of tachyons with integer (Euclidean)momentum,the bounce factors R q of(3.3)simplify due to the following identityR ∗ξR n −ξ=(−iµ)−n1µ∂Zw n dzz )m 0|eiµ n ≥1t n αn ψ(z )¯ψ(w )Se iµ n ≥1¯t n α−n |0(4.2)At the self-dual radius β=1,where all tachyon momenta are integral,we may simplify the sum on m using (4.1)(−iµ)−n (−iµ+z ∂z )m (4.3)the latter sum acting like a delta function.Now integrate by parts and use the identity(−iµ−z ∂∂z)n z iµ.(4.4)It is convenient to bosonizeψ(z)=eφ(z),¯ψ(z)=e−φ(z)and shift the zero mode:˜φ(z)=φ(z)+iµlog z.(4.5)Taking the operator product of the two exponentials inφ,and using the delta function and charge conservation wefind the operator:dw(iµ)−n1∂¯t n=Z−1 dw(iµ)−(n+1)w+ n>0nt n w n−1−1∂t n w−n−1.(4.8) The genus zero result of[18]is easily obtained from this as the leading term at largeµ. (Note that this was obtained atβ=∞but genus zero correlators are independent ofβ[17].)The operators P(n)(z)=:e−˜φ(z)∂n e˜φ(z):and their derivatives generate the algebra W1+∞[19].The standard generators are related to these byW(n)(z)=n−1l=0(−1)l(2n−2)l∂l P(n−l)(z).(4.9)The rescaling of the scalarfield required to obtain(4.8)is simply a change of basis effected by the operator:e log(iµ)πφ˜φ:whereπφis the momentum conjugate to˜φ.Inserting this we can rewrite(4.7)as∂Z5.Tau-functions and the Kontsevich-Penner matrix integralIn this section we will point out that the above reformulation of the generating func-tional of the c=1string represents mathematically aτ-function of the Toda Lattice hierarchy.The Toda Lattice naturally contains the KP and KdV hierarchies,and thus the c=1results are closely related to the expressions obtained for c<1.We will also show how to rewrite the partition function(at the self-dual radius)as a matrix integral, generalizing expressions previously considered by Kontsevich[7]and Penner[20].5.1.Grassmannians and tau-functionsLet usfirst briefly explain the notion of a tau-function and its relation with the universal Grassmannian.For more details see e.g.[21]and[22].We will focus here on the relation with conformalfield theory instead of the Lax pair formulation.Consider a two-dimensional free chiral scalarfieldϕ(z),with the usual mode expansion∂ϕ(z)= nαn z−n−1.(5.1)The reader is encouraged to think about this scalarfield as the target space tachyonfield at spatial infinity with a periodic Euclidean time coordinate.We have a Hilbert space H built on the vacuum|0 ,and as in the case of a harmonic oscillator one can consider coherent states,∞ n=1it nα−n(5.2)|t =expand their Hermitian conjugates∞ n=1−it nαn(5.3)t|= 0|exp(The parameters t n are considered to be real here.)Now to any state|W in the Hilbert space H we can associate a coherent state wavefunctionτW(t)by considering the inner productτW(t)= t|W .(5.4)This function is a tau-function of the KP hierarchy if and only if the state|W lies in the so-called Grassmannian.To explain the concept of the Grassmannian we have to turn to the alternative de-scription of this chiral conformalfield theory in terms of chiral Weyl fermionsψ(z),¯ψ(z) by means of the well-known bosonization formulas4i∂ϕ=¯ψψ,ψ=e iϕ,¯ψ=e−iϕ.(5.5)Loosely speaking,the Grassmannian can be defined as the collection of all fermionic Bogo-liobov transforms of the vacuum|0 .That is,the state|W belongs to the Grassmannian if it is annihilated by particular linear combinations of the fermionic creation and annihilation operators.)|W =0,n≥0,(5.6) (ψn+12or equivalently,|W =S·|0 ,S=exp n,m A nm¯ψ−n−12.(5.7)Note that the operator S can be considered as an element of the infinite-dimensional linear group,S∈GL(∞,4Since we do not wish toflaunt tradition we change conventions for bosonization in this section relative to the previous sections.with|¯t and t|the coherent states(5.2)and(5.3)and S a general GL(∞,z−n i.(5.12)nWith this choice of parameterization,and after taking into account a normal ordering contribution,the tau-function can be written asdet v j−1(z i)τ(t)=The techniques of the double-scaled matrix models leads to two important results.First, the partition functionτ(t)is a tau-function of the KP hierarchy,that is,it can be written asτ(t)= t|W = t|S|0 ,(5.16) for some state|W and matrix S∈GL(∞,ipz p−1z p+1/λ· ∞−∞dy·y n·e i(z p y−y p+1p+1)/λ.(5.19)p+1Here Y and Z are both N×N Hermitian matrices,and the parameterization of the KP times t k in terms of the matrix Z isλt k=∆(V′(z))2·det V′′(Z)·5.3.The Kontsevich-Penner integralWe have seen that the c=1partition function can be succinctly written as a tau-function of the Toda Lattice hierarchyτ(t,¯t)= t|S|¯t .(5.23)Forfixed¯t k we recover a tau-function of the KP hierarchy,which we can study with the techniques of the previous subsection.Indeed the operators O n of the minimal models should now be compared to the outgoing tachyons of the c=1model.We want to determine in more detail the element W(¯t)in the Grassmannian that parametrizes this particular orbit of the KPflows.To this end we have to consider the state|W(¯t) =S·U(¯t)·|0 ,U(¯t)=exp∞ n=1iµ¯t nα−n.(5.24)We will describe|W(¯t) by giving a basis v k(z;¯t),k≥0,of one-particle wave-functions. First we observe that the operator U(¯t)acts on the wave-functions z n by simple multipli-cationU(¯t):z n→exp iµ¯t k z−k ·z n.(5.25) Similarly we have for the action of S a multiplicationS:z n→R p n·z n.(5.26)We have already seen that the reflection factors R pncontain all the relevant information of the c=1matrix model.At radiusβthey can be chosen to beR pn =(−iµ)−n+1βΓ(12Γ(1with a normalization constant c k such that v k(z;0)=z k.(This corresponds to the normal ordering of the S-matrix in(3.9).)Since the reflection factor is basically a gamma function, the result can be expressed as a Laplace transformv k(z;¯t)=c′(z)· ∞0dy·y k·y−iµβ+(β−1)/2e iµ(y/z)βexp iµ¯t k y−k (5.29) Here the constant c′(z)is given byc′(z)=β(−iµ/zβ)1√2−iµ).(5.30)These integral representations are of Kontsevich type if and only ifβ=1,that is,only at the self-dual radius.Indeed in that case we havev k(z;¯t)=c′(z)· ∞0dy·y k·exp iµ y/z−log y+ ¯t k y−k (5.31) Therefore,following the procedure in[8,9],we can write the following matrix integral representation for the generating functional.Define the integralσ(Z,¯t)= dY e iµT r[Y Z−1+V(Y)],(5.32) whereV(Y)=−log Y+ ¯t k Y−k,(5.33) and we integrate over positive definite matrices Y.Then we haveτ(t,¯t)=σ(Z,¯t)nT rZ−n.(5.35) Note that with this normalizationτ(t,0)=1,which is appropriate since we consider normalized correlation functions.In order to write down the result(5.34)we had to treat the incoming and outgoing tachyons very differently,parametrizing the outgoing states through(5.35),whereas the coupling coefficients to the incoming states enter the matrix integral in a much more straightforward fashion.Equation(5.34)should be considered as an asymptotic expansion inµ−1,but,for small enough t k,¯t k the expansion in these variables will be convergent.In some cases,(e.g.the sine-Gordon case considered in[26]) the expansion has afinite radius of convergence,and as we increase|t k|beyond the radius of convergence we can have phase transitions.5.4.The partition functionMatrix integrals of the above type have appeared in the work of the mathematicians Harer and Zagier[27]and Penner[20]in their investigations of the Euler characteristic of the moduli space M g,s of Riemann surfaces with g handles and s punctures.(See[28]for more details on these wonderful calculations.)The double scaling limit of this so-called Penner integral was considered by Distler and Vafa[29]who also speculated on the relation with c=1string theory.Their work has been followed by a number of papers concerned with double scaling limits and multi-critical behaviour of matrix models with logarithmic potentials[25].All these papers considered essentially the case Z=1and¯t n=0,in the notation of(5.32).Distler and Vafa noticed that—after a double scaling limit and an analytic contin-uation—the Penner matrix integral could reproduce the c=1partition function at the self-dual radiusβ=1.Recall that the free energy at that radius is given by[30]∂2Fxe−iµx x/22µ2logµ−12g(2g−2)µ2−2g.(5.37)(Up to analytic terms inµ.)This makes one wonder whether our result(5.34)can be sharpened to give the unnormalized correlation functions.To this end let us put the incoming coupling constants¯t k to zero(and thereby also t k=0)and take a closer look at the integralσ(Z)= dY e iµT r[Y Z−1−log Y].(5.38) First of all it has a trivial Z-dependenceσ(Z)=(det Z)N−iµ·σ(1).(5.39) Actually,it is convenient to work with the quantity F defined bye F=(πi/µ)−N2As an asymptotic expansion in 1/µit has the representatione F = dY ·e iµ ∞k =21dY·eiµ1µ)−N/2 (−iµ)iµΓ(−iµ) N N−1p =1(1−p/iµ)N −p(5.44)from which one may obtain the formulae:F g,s =(−1)s B 2g2g (2g −2)1−(1−x )2−2g ,g ≥2,F 1(x )=−12(1−x )2log(1−x )+32x .(5.47)The double-scaling limit considered by Distler and Vafa in[29]keeps N−iµfixed,while sending N,µ→∞(and x→1in(5.47).).This is clearly only possible for imaginary µ,which is precisely the case they study.However,here we want to consider a simpler limit in whichµis keptfixed,but N tends to infinity.We already mentioned that the parameterization(5.35)only makes sense in this limit.Indeed,the absence of a double scaling limit is very much in the spirit of Kontsevich integrals.The contribution for genus 2or higher have a smooth limit,as is evident from(5.47).(Recall,we send x→∞.) However,we have to worry about the genus zero and one pieces,which have to be corrected by hand.(This is by the way also true for the double scaling limit.)Combining all ingredients we obtain the followingfinal result for the unnormalized generating functional for the c=1string theoryτ(t,¯t)=c(Z)· dY exp iµT r Y Z−1−log Y+ ¯t k Y−k .(5.48) where the normalization constant is given byc(Z)=e−iµN(2πi/µ)N2/2(det Z)iµ−N(5.49)(1+iN/µ)112µ112e32N/µ.The expression(5.48)has a smooth large N limit.6.Other BackgroundsThe results of the previous sections comprise in principle a calculation of the partition function in arbitrary tachyon backgrounds(subject to the equations of motion).The full space of classical backgrounds in the theory includes in addition to these excitations of the“discrete states”corresponding to global modes like the radius of the1D universe and generalizations thereof.Of these,the ones best understood in terms of the matrix model are the zero-momentum excitations which are thought to be represented by variations in the double-scaled potential.In this section we study the dependence of the amplitudes on these extra parameters.We note that in principle the formulation of section three applies in arbitrary potentials.What we add here is a study of the variation of the reflection factor R q,hence of the partition function,under variations of the potential.6.1.Dependence on βThe most obvious parameter is β,the radius at which we compactify the scalar field X .The formulas of section four are valid for arbitrary β,however as pointed out in [17],correlation functions at different radii are related.The relation is most simply written in terms of rescaled couplings t n .DefiningˆF [t n ,¯t n ;β;µ]≡µ2F [µn 2β−1¯t n ;β;µ](6.1)so that derivatives of ˆFyield correlation functions of T q ,we have ˆF [t n ,¯t n ;β;µ]=1∂µ2β∂2β−1t (n/β),µn∂t n ∂¯t n =µnn −1 m =0R ∗p mR n −p m =i n n −1 m =0(−iµ−m )n .(6.3)Inverting the operator in (6.2)assin(∂∂2−iµ−x )n (6.5)in agreement with the result of [10].6.2.Other zero-momentum modesThe matrix model naturally suggests candidate representatives of the special states at zero X momentum.Operators with the appropriate quantum numbers may be introduced as generating variations in the double-scaled potential V (λ).Their correlators may thus be studied by analysis of the variation of the partition function Z computed above under these changes in V .From the definition of I (q,λ1,λ2)we can obtain directly constraints on thevariation of R q.Essentially these follow upon integration by parts from the linear Gelfand-Dikii equation satisfied by a product of Sturm-Liouville eigenfunctions[31].Explicitly,we haveL q,k R q=0k≥−1L q,k=−k(k2−1)∂∂s+2 p≥0s p(2k+p+2)∂k5In[33]proposals for c=1flow equations were made by taking the N→∞limit of the W N constraints of the c<1models.It should be noted that,although our equations have some similarities to the proposals of[33],they are not equivalent.From the relation of these results to a Kontsevich-type matrix model it appears that we have taken a step closer to a unified description of all the c≤1models along the lines proposed by[8,9].Moreover,the description(5.48)of the partition function is a strong hint that the c=1correlators have a description in terms of a topologicalfield theory.If this is so then the present results provide a direct bridge between a topologicalfield theory at the self-dual radius and the local physics of the c=1tachyon in the uncompactified theory.There have been many discussions of W∞symmetry in the c=1system.Our con-straints are related to the results of[15,34–36].The other modes of the W∞currents appearing in equation(4.10)define a set of operatorsσn(T q)whose correlation functions are determined by the subleading terms proportional toλ−|q|−2n in the largeλasymp-totics of the eigenvalue correlators.6These“operators”exist at any radius for X and have free fermion representations as fermion bilinears.Their correlators are also given by a Toda tau function generalizing that in(3.10).Note that these operators appear at any momentum q and are related to fractional powers ofℓ(or,equivalently,ofλ).Therefore, at generic q they cannot be the special state operators but rather are related to contact terms associated to singular geometries created by intersecting macroscopic loops[15].At integer q the distinction between special states and theσn(T q)is less clear.We hope to return to the subtleties of these contact terms in a future publication.The W∞symmetry we have discussed might also be related to the W∞Ward identities of[37–43].In these references the Liouvillefield is treated as a freefield,in other words, one works atµ=0.One should be cautious about identifying these W∞symmetries with those of the matrix model.As we have emphasized,the W∞modes of the matrix model σn(T q)are constructed from the tachyon degrees of freedom in distinction to the W∞currents of[37–43].Moreover,our Ward identities are highly nonlinear when expressed in terms of the correlation functions7in contrast to the quadratic identities of[39–43]. Finally the ghost sector of the theory is crucial in[39–43],leading to many more“special state operators”at given X,φmomenta than are considered in[15,34–36].Clearly there is a certain amount of tension between these two approaches and further work is needed to see if these differences are superficial or essential.We must emphasize that at c =1the W ∞-constraints are actually somewhat sec-ondary,since we have an explicit solution of the appropriate Toda tau function given by (3.10).Analogous representations for thec <1tau functions (at nontopological points)replace the simple operator S by complicated and uncomputable objects like the “star operators”of [44].This is why the Virasoro constraints at c <1are essential to the actual computation of amplitudes.It would be interesting to investigate further the physical properties of these different time-dependent backgrounds.In [18,26]some results along these lines were discussed.Our result (3.10)should allow a much more complete analysis of the space of time-dependent backgrounds in 2D string theory and the various phase transitions occurring as one in-creases the coordinates t k .What is needed for further progress is a more effective way to compute the tau function (perhaps from the Kontsevich representation)or a deeper understanding of the infinite dimensional geometry of the associated Grassmannian.AcknowledgementsWe would like to thank T.Banks,E.Martinec,N.Seiberg,C.Vafa,and H.Verlinde for discussions.This work is supported by DOE grant DE-AC02-76ER03075and by a Presidential Young Investigator Award (G.M.,R.P.)and by the W.M.Keck foundation (R.D.).G.M would like to thank the Isaac Newton Institute for Mathematical Sciences for hospitality.Appendix A.Dependence on the potentialIn this appendix we will derive constraints on the dependence of the free energy upon the double-scaled matrix model potential V (λ).We restrict attention to variations of the potential which preserve the asymptotics V (λ)∼−1H −µ−iq |λ2 =−∞−∞dλδV (λ)I (q,λ1,λ)I (q,λ,λ2).(A.1)We now recall the calculation of I from[10](see appendix A of this work for a detailed calculation for particular potentials).For simplicity let q>0.We will make use of the eigenfunctions of H=d22∓iz e±iλ2.(A.2) In terms of these we can write the resolvent quite easily by imposing the boundary condi-tions and the defining property(H−z)I(q,λ1,λ2)=δ(λ1−λ2)as in[10]I(q,λ1,λ2)=−iθ(λ1−λ2) χ−(z,λ1)χ+(z,λ2)+R qχ−(z,λ1)χ−(z,λ2) +(λ1↔λ2).(A.3)The reflection factor R q contains all the effects of the potential,and for the standard V is given by(3.3).Inserting this into(A.1)and neglecting terms of orderδV(λ1,2)for large λi,wefind that a variation of V yieldsδR q=−i ∞−∞dλδV(λ)ψ(z,λ)2(A.4) whereψ=χ++R qχ−is the solution satisfying the boundary conditions at smallλ.The integrand F(z,λ)=ψ(z,λ)2in(A.4)satisfies a differential equation[31]following from that satisfied byψF′′′−4(V(λ)+z)F′−2V′F=0(A.5) where primes denoteλdifferentiation.Let us choose as a convenient set of variations of the potentialδV(λ)=ǫe−ℓλ.Inserting this in(A.4)and integrating by parts wefind8[ℓ3−4zℓ−4ℓV(−d dℓ)]δR q=0.(A.6) The integration by parts is justified by the limiting conditions we have imposed uponψandδV.Formally expanding V= n≥0s nλn the bounce factor becomes a function of the s j: R q=R q[s1,s2,...].RewritingδV as a motion in s j and inserting the resulting expression forδR q in(A.6)we obtain(after shifting s0)L q,k R q=0k≥−1+2 p≥0s p(2k+p+2)∂L q,k=−k(k2−1)∂∂sk8The similarity of this to the WdW equation of[16]is no coincidence;setting z=µand λ1=λ2wefind that(A.1)is essentially the WdW wavefunction of the cosmological constant.。
Constraints on the Parameters of the Unseen Pulsar in the PWN G0.9+0.1 from Radio, X-Ray, a
1. Introduction The H.E.S detection of the composite supernova remnant SNR G0.9+0.1 in very high energy (VHE) gammarays [Aharonian et al. 2005] with a significance of ≈ 13σ , after 50 hours of observations of the Galactic Center (GC) region during March-September 2004 with the full telescope system (see also [Aharonian et al. 2006]). The unresolved gamma-ray radiation, which appears to be associated with the plerionic remnant core, corresponds to a photon flux above 200 GeV of (5.7 ± 0.7stat ± 1.2sys ) × 10−12 /cm2 /s and luminosity of ∼ 2 × 1034 ergs/s, making this one of the faintest VHE gamma-ray sources. For a radially symmetric Gaussian emission region, a 95% confidence limit of 1.3′ ∼ 3.2 pc on the source radius is obtained, and similarly for emission from a uniform thin shell, with extension < 2.2′ ∼ 5.4 pc. This is the first time that SNR G0.9+0.1 has been detected at gamma-ray energies [1], subsequent to the earlier unconstraining upper limit given by HEGRA [Aharonian et al. 2002], and H.E.S.S. observations in 2003 (resulting in a 4σ detection) [Aharonian et al. 2004]. MAGIC also detected
constraint
sions which do not involve default negated literals, or are incomplete 15]. The novelty of our procedure is the simultaneous treatment of positive and negative recursions, together with general constraint handling features used to implement constructive negation, enabling us to address in this way the oundering problem in its full generality. For lack of space we only provide a high-level description of our procedure focusing on its major features.
2
1 Introduction
The applications of logic programming are wide-ranging, namely for knowledge representation. Nevertheless, a mechanism for explicitly declaring the falsity of propositions is not available in the restricted language of normal logic programs. The importance of extending logic programming with an explicit form of non-classical negation, \:", alongside the usual default or implicit one, has been stressed for use in deductive databases, knowledge representation, and non-monotonic reasoning. This has been pointed out by several authors, and an enhanced language and semantics have been proposed. For a recent study of this explicit form of negation consult 2]. However, the introduction of explicit negation permits and requires being able to reason with, or at least detect, contradictory information. Indeed, information is not only normally incomplete but contradictory as well. The idea of introducing paraconsistent reasoning in logic programming is fairly recent. Constraint logic programming adds to knowledge espressivity by merging two declarative programming paradigms: constraint solving and logic programming. A constraint logic program is a logic program parametrized with a constraint system de ning the meaning of several primitive predicates (the basic constraints). The advantage of such frameworks is the smooth integration of a domain associated with constraints, often an arithmetic domain, into a logic programming language with clear semantical properties, namely the equivalence of operational, model theoretic, and xpoint semantics. The starting point of this paper is our paraconsistent well-founded semantics with explicit negation (WFSX p ) 1, 8], to which we add general constraints. In order to have an usable working system several issues must be addressed: soundness, completeness, termination and e ciency. We achieve this by relying on a proof procedure which integrates tabulation and constraint handling features and, in particular, constructive negation. The tabulation mechanisms provide a solution to the problems of termination of the proof procedure, namely negative and positive recursions. The integration of constraint technology in tabulation procedures is not new, see for instance 6]. However, the systems present in the literature only take care of positive recursions, i.e. recur-
约束的英文作文优美
约束的英文作文优美英文：Constraints are a necessary part of life. They exist in various forms, such as rules, laws, social norms, and personal beliefs. While some might view constraints as limiting, they can also serve as a guide to help us make better decisions and achieve our goals.For example, traffic laws are a type of constraint that keeps us safe on the road. Without these laws, chaos would ensue, and accidents would be more frequent. Similarly, social norms and etiquette provide a framework for us to interact with others in a respectful and appropriate manner. By following these constraints, we can build stronger relationships and avoid offending others.However, constraints can also be self-imposed, such as personal beliefs or goals. While these constraints maylimit our choices in the short term, they can ultimatelylead to greater success and fulfillment. For instance, an athlete who sets a goal to run a marathon may have to sacrifice leisure time and unhealthy habits to achievetheir goal. But in the end, the sense of accomplishment and personal growth they experience can be incredibly rewarding.In conclusion, constraints are an essential aspect oflife that can serve as a guide to help us make better decisions and achieve our goals. While they may seemlimiting at times, they ultimately contribute to our personal growth and success.中文：约束是生活中必不可少的一部分。
约束判断的英语
约束判断的英语Judgment and ConstraintThe concept of judgment and constraint is a complex and multifaceted topic that has been extensively explored in various fields, including philosophy, psychology, and cognitive science. At its core, judgment and constraint refer to the processes by which individuals make decisions, form opinions, and navigate the complexities of the world around them.One of the fundamental aspects of judgment and constraint is the role of cognitive biases. Cognitive biases are systematic patterns of deviation from rationality in judgment and decision-making, which can lead to systematic errors in perception, memory, and reasoning. These biases can have a significant impact on how individuals perceive and interpret information, ultimately shaping their judgments and decisions.For example, the confirmation bias, which is the tendency to seek out and interpret information in a way that confirms one's existing beliefs and preconceptions, can lead individuals to disregard or dismiss evidence that contradicts their established views. Similarly,the availability heuristic, which is the tendency to rely on information that is readily available or easily recalled, can cause individuals to make judgments based on incomplete or biased information.Another important aspect of judgment and constraint is the role of social and cultural factors. Individuals' judgments and decisions are often heavily influenced by the norms, values, and beliefs of the communities and societies in which they are embedded. This can lead to the formation of in-group biases, where individuals tend to favor and trust information and perspectives that align with their own social and cultural affiliations.Furthermore, the concept of constraint in judgment and decision-making is closely tied to the notion of cognitive resources. Individuals have a finite capacity for processing and analyzing information, and when faced with complex or ambiguous situations, they often resort to heuristics and shortcuts to simplify the decision-making process. This can result in suboptimal or biased judgments, particularly when the constraints on cognitive resources are significant.In recent years, there has been a growing interest in the development of techniques and strategies to mitigate the negative effects of cognitive biases and constraints on judgment and decision-making. One such approach is the use of debiasingtechniques, which aim to raise awareness of cognitive biases and provide individuals with tools and strategies to overcome them.Another approach is the development of decision support systems and algorithms that can assist individuals in making more informed and rational judgments. These systems can leverage vast amounts of data, sophisticated analytical techniques, and advanced computational power to provide individuals with more accurate and unbiased information, ultimately helping them make better-informed decisions.In conclusion, the concept of judgment and constraint is a complex and multifaceted topic that has significant implications for how individuals perceive, interpret, and make decisions about the world around them. By understanding the underlying cognitive and social factors that shape judgment and decision-making, researchers and practitioners can work towards developing more effective strategies and tools to help individuals navigate the complexities of the modern world.。
约束的英文作文模板初中
约束的英文作文模板初中英文：When it comes to constraints, I believe that they can be both positive and negative. On one hand, constraints can provide structure and guidance, helping us to focus and achieve our goals. For example, a deadline can be a constraint that motivates us to work efficiently and complete a project on time. On the other hand, constraints can also limit our creativity and hinder our progress. For instance, if we are too focused on following a set of rules or guidelines, we may overlook innovative solutions or alternative approaches.In my experience, constraints are most effective when they are flexible and adaptable. This means that while they provide a framework for our work, they also allow for some degree of creativity and experimentation. For example, if I am given a specific topic to write about, I may choose to approach it from a unique angle or use a different writingstyle than what is expected. This allows me to meet the requirements of the constraint while also expressing my individuality and creativity.Overall, I believe that constraints can be a valuable tool for achieving success, but only if they are used thoughtfully and with flexibility. By embracing constraints and finding ways to work within them, we can challenge ourselves to think creatively and achieve our goals.中文：谈到约束，我认为它们既可以是正面的，也可以是负面的。
约束的英文作文范文
约束的英文作文范文英文：Constraints are an inevitable part of life. We all face them at some point in our lives, whether it's in our personal relationships, academic pursuits, or professional careers. Constraints can be frustrating, but they can also be an opportunity for growth and creativity.One way to approach constraints is to view them as a challenge. Instead of seeing them as roadblocks, we can see them as opportunities to think outside the box and come up with innovative solutions. For example, if we have alimited budget for a project, we can think of ways to maximize our resources and still achieve our goals.Another approach is to prioritize our goals and focus on what's most important. Sometimes we have to make tough choices and sacrifice certain things in order to achieve our desired outcome. For instance, if we want to excel inour career, we may have to put in extra hours of work and sacrifice some of our leisure time.Ultimately, constraints can be a source of motivation and inspiration. They can push us to be more creative, resourceful, and determined. As the saying goes, "necessity is the mother of invention."中文：约束是生活中不可避免的一部分。
约束的作文800字
约束的作文800字"英文回答，"Constraints are a part of life, whether we like it or not. They can come in many forms, such as time constraints, financial constraints, or even personal constraints. For example, when I was in college, I had a lot of time constraints because I was juggling a part-time job and a full course load. This meant that I had to be very organized and efficient with my time in order to meet all of my deadlines.Constraints can be frustrating, but they can also be a source of creativity and innovation. When we are forced to work within certain limitations, it can push us to think outside the box and come up with new solutions. For instance, when I was planning a party on a tight budget, I had to get creative with decorations and food choices. In the end, the constraints actually made the party more unique and memorable."中文回答，"约束是生活中不可避免的一部分，不管我们喜欢与否。
A Plea Must Be Specific
A Plea Must Be SpecificAnother constraint upon the federal judiciary is that judges will hear no case on the merits unless the petition-er is first able to cite a specific part of the Constitution as the basis of the plea.For example,the First Amend-ment forbids government from mak-ing a law respecting an establishmen of religion.In 1989 the state of New York created a special school district solely for the benefit ofthe Satmar Hasids,a group ofHasidic Jews with East European roots that strongly re-sists assimilation into modern society.Most ofthe children attended parochial schools in the Village of Kiryas Joel,but these private schools werent able to accommodate retardeand disabled students,and the Satmars claimed that such children within their community would be trauma-tized ifforced to attend a public school.Responding to this situation,the state legislature created a special district encompassing a single school that served only handicapped children from the Hasidic Jewish community.This arrangement was challenged by the association representing New York states school boards.In June 1994 the U.S.Supreme Court ruled that the cre-ation ofthe one-school district effec-tively delegated political power to the orthodox Jewish group and therefore violated the First Amendments ban on governmental establishment of religion. Whether or not everyone agrees that the New York law was con-stitutional,few,ifany,would doubt that the school board association metthe specific criteria for securing judi-cial review:The Constitution clearly forbids the government from delegat-ing political power to a specific reli-gious entity.The government here readily acknowledged that it had passed a law for the unique benefit of a singular religious community.However,ifone went into court and contended that a particular law or official action violated the spirit of the Bill of Rights or offended the values ofthe Founders, a judge surel would dismiss the proceeding.For if judges were free to give concrete,sub-stantive meaning to vague generalities such as these,there would be little check on what they could do.In the real world this principle is not as sim-ple and clear-cut as it sounds,becausethe Constitution contains many claus-es that are open to a wide variety of in-terpretations,giving federal judges sufficient room to maneuver and make policy.。
约束英文作文素材
约束英文作文素材1. Constraints can be frustrating, but they also force us to be creative and think outside the box. When we are faced with limitations, we are forced to come up with innovative solutions that we might not have thought of otherwise. Constraints can also help us focus on what is truly important and prioritize our goals. In the end, constraints can be a blessing in disguise, pushing us to achieve greatness.2. However, constraints can also be limiting and stifling. When we are restricted in our choices, we may feel trapped and unable to express ourselves fully. This can lead to frustration and a lack of motivation. It is important to find a balance between embracing constraints as a challenge and recognizing when they are hindering our progress.3. Constraints can also be cultural or societal, such as gender or racial stereotypes. These constraints can bedeeply ingrained and difficult to overcome, but it is important to recognize them and work towards breaking them down. By challenging these constraints, we can create a more inclusive and diverse society where everyone has equal opportunities.4. In the creative industries, constraints are often seen as a necessary part of the creative process. Whether it is a limited budget or a tight deadline, these constraints can push artists and designers to create their best work. However, it is important to remember that constraints should not be used as an excuse forexploitation or underpayment. It is important to value the work of creative professionals and provide them with fair compensation for their talents.5. Constraints can also be self-imposed, such assetting unrealistic expectations or being too hard on ourselves. These constraints can lead to burnout and a lack of self-care. It is important to recognize when we are putting too much pressure on ourselves and take a step back to prioritize our mental and physical well-being.6. Ultimately, constraints are a part of life and cannot be avoided. It is up to us to decide how we respond to them and use them to our advantage. By embracing constraints as a challenge and recognizing when they are limiting us, we can learn and grow as individuals.。
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February 2008
∗
E-mail addresses: alejandro.ibarra@desy.de, cristoforo.simonetto@ph.tum.de
1
Introduction
The renormalizable part of the Standard Model Lagrangian is invariant under four global U (1) symmetries, namely baryon number, B , and the three family lepton numbers, Le , Lµ and Lτ . This invariance has been for many years considered accidental and expected to be broken by additional terms in the Lagrangian, possibly nonrenormalizable. Whereas experiments searching for proton decay have not provided yet any evidence for baryon number violation, it is nowadays a firmly established experimental fact that the three family lepton numbers are violated in the neutrino sector. Namely, the disappearance of electron neutrinos on their way from the Sun indicated by the Homestake chlorine detector [1], SAGE [2], GALLEX/GNO [3,4], Kamiokande [5], Super-Kamiokande [6] and Borexino [7], and unequivocally confirmed by SNO [8], proves the violation of Le . This is further supported by the disappearance of electron antineutrinos observed by the reactor experiment KamLAND [9]. On the other hand, the atmospheric neutrino anomaly discovered by Kamiokande and IMB [10], and explained by Super-Kamiokande [11], Soudan2 [12] and MACRO [13] as an oscillation of muon neutrinos into a different neutrino species, proves the violation of Lµ . The disappearance of muon neutrinos reported by the long baseline accelerator experiments K2K [14] and MINOS [15] supports this conclusion. Finally, the observation of tau neutrino appearance in the atmospheric neutrino flux by Super-Kamiokande [16] indicates the violation of Lτ . The most economical way to accommodate the family lepton number violation in the Standard Model is by adding to the leptonic Lagrangian a dimension five operator [17]
2
(2)
which gives BR(τ → µγ ) ∼ 10−54 , BR(µ → eγ ) ∼ 10−57 , BR(τ → eγ ) ∼ 10−57 , far below the sensitivity of present and projected experiments (see Table 1)1 . Nevertheless, the Lagrangian Eq. (1) describes just an effective theory and new degrees of freedom are expected to arise above the scale Λ. The interactions of the new degrees of freedom with the lepton fields are likely to contain additional sources of flavour violation that can enhance the branching ratios of the rare decays by many orders of magnitude, bringing them to the reach of future experiments. For this reason, rare lepton decays are considered very powerful probes for physics beyond the Standard Model. The supersymmetric (SUSY) see-saw mechanism is probably the best motivated high energy theory to generate small neutrino masses [25] . In this framework the particle content of the Minimal Supersymmetric Standard Model (MSSM) is extended with three right-handed neutrino superfields, νRi , i = 1, 2, 3, singlets under the Standard Model gauge group. Imposing R-parity conservation, the leptonic superpotential reads: 1 c c c Wlep = ec Ri Yeij Lj Hd + νRi Yν ij Lj Hu − νRi Mij νRj , 2 (3)
Abstract It is now a firmly established fact that all family lepton numbers are violated in Nature. In this paper we discuss the implications of this observation for future searches for rare tau decays in the supersymmetric see-saw model. Using the two loop renormalization group evolution of the soft terms and the Yukawa couplings we show that there exists a lower bound on the rate of the rare process µ → eγ of the form BR(µ → eγ ) > ∼ C × BR(τ → µγ )BR(τ → eγ ), where C is a constant that depends on supersymmetric parameters. Our only assumption is the absence of cancellations among the high-energy see-saw parameters. We also discuss the implications of this bound for future searches for rare tau decays. In particular, for large regions of the mSUGRA parameter space, we show that present B -factories could discover either τ → µγ or τ → eγ , but not both.
αij Λ
H 0 2 , being the smallness of neutrino masses attributed to small values of
the couplings αij and/or to a large value of Λ. In this minimal framework, lepton flavour violation is generated by the same operator that generates neutrino masses. Therefore, the rate of any lepton flavour violating pro1
DESY 08-018 TUM-HEP-682/08
arXiv:0802.3858v2 [hep-ph] 3 Mar 2008
Constraints on the rare tau decays from µ → eγ in the supersymmetric see-saw model
Alejandro Ibarra1 and Cristoforo Simonetto2