Supersymmetric Standard Model from String Theory
research statement的范例_jiangyun
Research StatementYun JiangPh.D.candidate in Physics at U.C.Davis2013LHC-TI Graduate FellowMy current research concentrates on Large Hadron Collider(LHC)phenomenology,especially that related to Higgs physics and dark matter.The phenomenology of the125.5GeV Higgs boson,which was discovered at the LHC,in the next-to-minimal supersymmetric standard model(NMSSM)and two-Higgs-doublet model(2HDM)has constituted the main part of my work towards my Ph.D. degree.I am in the process of expanding my research work to include the topics of extra dimensions and inflation of the early universe.Accomplished workWefirst assessed the extent to which various semi-constrained NMSSM(scNMSSM)scenarios with a∼125GeV lightest CP-even Higgs h1are able to describe the LHC signal.We found that enhancedγγrates are most natural when the h1has mass similar to the second lightest CP-even Higgs,h2,with enhancement particularly likely if the h1and h2are degenerate.To experimentally probe this possibility,we developed diagnostic tools that could discriminate whether or not there are two(or more)Higgs bosons versus just one contributing to the LHC signals at125.5GeV.In addition,we considered the case where the lightest Higgs h1provides a consistent description of the small LEP excess at98GeV whereas the heavier Higgs h2possesses the primary features of the LHC Higgs-like signals at125GeV.Besides the NMSSM studies,the2HDM,one of the simplest extensions of the Higgs sector,is another focus of my research work.We recently performed an exhaustive analysis for Type I and Type II models to address an important question:to what extent are the latest measurements of the125.5GeV Higgs-like signal at the LHC compatible with the2HDM,assuming that the observed 125.5GeV state is one of the two CP-even Higgs bosons?We also discussed the implications for future colliders,including expectations regarding other lighter or heavier Higgs bosons.In an earlier study,we examined the maximum Higgs signal enhancements that can be achieved in the2HDM in which either a single Higgs boson or multiple Higgs bosons have mass(es)near125.5GeV.We found that the constraints requiring vacuum stability,unitarity and perturbativity substantially restrict possibilities for signal enhancement.Furthermore,we extend the2HDM by adding a real gauge-singlet scalar(2HDMS),which couldbe stable under the extra Z2symmetry and thereby a possible dark matter(DM)candidate. Comparing with the simplest singlet extension this model has richer phenomenology.For heavy DM (mass above55GeV)which generates the desired relic abundance,the predicted cross section for DM-nucleon scattering is below the current LUX limit and even the future XENON1T projection. In contrast,this model can accommodate light DM,even if the constraint on Higgs invisible decay is taken into account,and describe the CDMS II and CoGeNT positive signal regions. More impressively,the tension with the LUX/SuperCDMS exclusion can be alleviated in the Type II2HDMS in which the DM-nucleon interaction could be isospin-violating.In the process of completing this project,we independently worked out the modelfiles for the FeynRules program and will make the model database publicly available soon.Ongoing projectsBased on the comprehensive studies we have accomplished,we focus on the light(pseudo)scalar Higgs boson region in the2HDM.We are also pursuing whether the current LHC8TeV-run data pushes the2HDM to the alignment limit and/or the decoupling limit.In the meanwhile we are developing a routine to simplify the calculation for gluon-fusion and bottom-quark associated production cross sections.Besides,we consider the decoupling2HDM to determine if the vacuum could be stable above the inflation or GUT scale,assuming the2HDM is a low-energy effective theory.If it is stable,then the inflation driven by the2HDM Higgs would be possible and a topic for future study.One of the most important extensions of the standard model(SM)is the inclusion of additional particle(s)that comprise the DM of the universe.So far a number of collaborations have been devoted to working on the direct detection of DM.They typically translate the limit on the event rate against recoil energy they directly detect into a limit on the DM-proton cross section as a function of DM mass.However,there are several standard assumptions hidden in this translation that might not be correct.In particular,it is normally assumed that DM has equal coupling to neutrons and protons.In fact,the tension between the null LUX/SuperCDMS exclusions and the positive signal regions favored by CDMS II and CoGeNT could be alleviated if the DM interactions with nucleons are allowed to violate isospin symmetry.Thus,we are now interested in exploring the possibilities of a light isospin-violating DM(IVDM)in the2HDMS and NMSSM even though such an isospin-violation effect in supersymmetry(SUSY)models has been claimed to be negligible.If present,such light annihilating IVDM may explain the origin of the excess of gamma rayflux from the galactic center,as indicated in the previous studies.Another project I am now involved in is warped DM.In view of the success of extra dimensions in resolving the hierarchy andflavor problems of the SM,we are studying DM in warped extra dimensions in particular with Randall-Sundrum like geometries.We consider the case that all SM fields live in the bulk.In our model thefirst Higgs excited state is a possible stable DM candidate due to the presence of a geometric KK-parity.Our focus is on phenomenological implications of the DM after imposing constraints from current experimental data.Future planIn the near future I will continue investigating LHC implications of various Higgs models beyond the SM both within and outside the framework of SUSY.Potential extensions to my previous studies include the future prospects of2HDM at the100TeV collider and the related analyses in the framework of phenomenological NMSSM,a version of NMSSM without GUT-scale unification assumptions.Additionally,dark matter physics and inflation of the early universe driven by the Higgs boson,Higgs portal DM,axion,etc.will be important topics of exploration in my post-doctoral research.It is well-known that Higgs inflation is unlikely to occur within the pure SM given the latest LHC measurement on the top quark mass.To remedy this issue,I am considering the additional loop contribution from Higgs portal interactions to raise the tensor-to-scalar ratio at the inflation scale.Another probable direction of my future work is in Higgs triplet and neutrino physics.I wish to construct a model that contains a LHC observed Higgs and a DM candidate and is also able to explain the neutrino mass by means of Type-II seesaw mechanism.Rather than being the end of the story,the discovery of the125.5GeV Higgs boson has marked a new era in particle physics.I anticipate that this discovery will provide a key window into theories beyond the SM,and that additional Higgs bosons and SUSY particles may well be found.A variety of ongoing experiments aimed at detecting dark matter will either provide further limits or succeed in detecting dark matter.Either way,DM models will be constrained and/or eliminated,thereby providing guidance to ongoing theoretical work.As a young researcher,I am fortunate to be in the midst of an exciting time and will certainly work extremely hard to contribute to our high energy physics community.。
Signals of Doubly-Charged Higgsinos at the CERN Large Hadron Collider
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.
Electron-Positron colliders
a r X i v :h e p -e x /0111070v 1 22 N o v 2001ELECTRON-POSITRON-COLLIDERSR.-D.HEUERInstitut f¨u r Experimentalphysik,Universit¨a t Hamburg,Luruper Chaussee 149,22761Hamburg GermanyE-mail:rolf-dieter.heuer@desy.deAn electron-positron linear collider in the energy range between 500and 1000GeV is of crucial importance to precisely test the Standard Model and to explore the physics beyond it.The physics program is complementary to that of the Large Hadron Collider.Some of the main physics goals and the expected accuracies of the anticipated measurements at such a linear collider are discussed.A short review of the different collider designs presently under study is given including possible upgrade paths to the multi-TeV region.Finally a framework is presented within which the realisation of such a project could be achieved as a global international project.1IntroductionA coherent picture of matter and forces has emerged in the past decades through in-tensive theoretical and experimental studies.It is adequately described by the Standard Model of particle physics.In the last few years many aspects of the model have been stringently tested,some to the per-mille level,with e +e −,ep and p ¯p machines making com-plementary contributions,especially to the determination of the electroweak bining the results with neutrino scatter-ing data and low energy measurements,the experimental analysis is in excellent concor-dance with the electroweak part of the Stan-dard Model.Also the predictions of QCD have been thoroughly tested,examples being precise measurements of the strong coupling αs and probing the proton structure to the shortest possible distances.Despite these great successes there are many gaps in our understanding.The clearest one is the present lack of any direct evidence for the dynamics of electroweak symmetry breaking and the generation of the masses of gauge bosons and fermions.The Higgs mechanism which generates the masses of the fundamental particles in the Standard Model,has not been experimentally estab-lished though the indirect evidence from pre-cision measurements is very strong.Even ifsuccessfully completed,the Standard Model does not provide a comprehensive theory of matter.There is no explanation for the wide range of masses of the fermions,the grand unification between the two gauge theories,electroweak and QCD,is not realised and gravity is not incorporated at the quantum level.Several alternative scenarios have been de-veloped for the physics which may emerge beyond the Standard Model as energies are increased.The Supersymmetric extension of the Standard Model provides a stable bridge from the presently explored energy scales up to the grand unification scale.Alternatively,new strong interactions give rise to strong forces between W bosons at high energies.Quite general arguments suggest that such new phenomena must appear below a scale of approximately 3TeV.Extra space dimen-sions which alter the high energy behaviour in such a way that the energy scale of gravity is in the same order as the electroweak scale are another proposed alternative.There are two ways of exploring the new scales,through attaining the highest possible energy in a hadron collider and through high precision measurements at the energy fron-tier of lepton colliders.This article is based on the results ofmany workshops on physics and detector studies for linear colliders.Much more can be found in the respective publications1,2,3,4 and on the different Web sites5,6,7,8.Many people have contributed to these studies and the references to their work can be found in the documents quoted above.2Complementarity of Lepton and Hadron MachinesIt is easier to accelerate protons to very high energies than leptons,but the detailed colli-sion process cannot be well controlled or se-lected.Electron-positron colliders offer a well√defined initial state.The collision energying generation of colliders.The physics case for such a machine will depend on the results from the LHC and the linear collider in the sub-TeV range.3Selected Physics TopicsIn this chapter,some of the main physics top-ics to be studied at a linear collider will bediscussed.Emphasis is given to the study of the Higgs mechanism in the Standard Model,the measurements of properties of su-persymmetric particles,and precision tests of the electroweak theory.More details about these topics as well as information about the numerous topics not presented here can be found in the physics books published in the studies of the physicspotential offuture lin-ear colliders 1,2,3,4.3.1Standard Model Higgs BosonThe main task of a linear electron-positron collider will be to establish experimentally the Higgs mechanism as the mechanism for generating the masses of fundamental parti-cles:•The Higgs boson must be discovered.•The couplings of the Higgs boson to gauge bosons and to fermions must be proven to increase with their masses.•The Higgs potential which generates the non-zero field in the vacuum must be reconstructed by determining the Higgs self-coupling.•The quantum numbers (J P C =0++)must be confirmed.The main production mechanisms for Higgs bosons in e +e −collisions are Higgs-strahlung e +e −→HZ and WW-fusion e +e −→νe ¯νe H ,and the corresponding cross-sections as a function of M H are depicted in figure 2for three different centre of mass en-ergies.With an integrated luminosity of 500Figure 2.The Higgs-strahlung and WW fusion pro-duction cross-sections as a function of M H for differ-ent√sof about 800GeV;for 1000fb −1an accuracyof 6%can be expected.The Higgs boson quantum numbers can be determined through the rise of the cross sec-tion close to the production threshold and through the angular distributions of the H and Z bosons in the continuum.Recoil Mass [GeV ]N u m b e r o f E v e n t s / 1.5 G e VFigure 3.The µ+µ−recoil mass distribution in theprocess e +e −→HZ →µ+µ−for M H =120GeV,500fb −1at√2of the self potential of the Higgs field V =λ(φ2−14λH4.The trilinearHiggs coupling λHHH =6λv can be mea-sured directly in the double Higgs-strahlung process e +e −→HHZ →q ¯q b ¯bb ¯b .The fi-nal state contains six partons resulting in a rather complicated experimental signature with six jets,a challenging task calling for ex-cellent granularity of the tracking device and the calorimeter 9.Despite the low cross sec-tion of the order of 0.2fb for M H =120GeV at√s =500GeV with an integrated luminosity of 1ab −1as shown in figure 6.Measurements of Higgs boson properties and their anticipated accuracies are sum-marised in table 1.In summary,the Higgs mechanism can be established in an unambiguous way at a high luminosity electron-positron collider with a centre-of-mass energy up to around one TeV as the mechanism responsible for the sponta-neous symmetry breaking of the electroweak interactions.3.2Supersymmetric ParticlesSupersymmetry (SUSY)is considered the most attractive extension of the Standardg c /g c (SM)g b /g b (S M )0.80.850.90.9511.051.11.151.2Figure 5.Higgs coupling determination:The con-tours for g b vs.g c for a 120GeV Higgs boson normalised to their Standard Model expectations as measured with 500fb −1.Model,which cannot be the ultimate the-ory for many reasons.The most impor-tant feature of SUSY is that it can explain the hierarchy between the electroweak scale of ≈100GeV,responsible for the W and Z masses,and the Planck scale M P l ≃1019GeV.When embedded in a grand-unified the-ory,it makes a very precise prediction of the electroweak mixing angle sin 2θW in excellent concordance with the precision electroweak measurement.In the following,only the min-imal supersymmetric extension to the Stan-dard Model (MSSM)will be considered and measurements of the properties of the super-symmetric particles will be discussed.Stud-ies of the supersymmetric Higgs sector can be found elsewhere 1,2,3,4.In addition to the particles of the Stan-dard Model,the MSSM contains their su-persymmetric partners:sleptons ˜l ±,˜νl (l =e,µ,τ),squarks ˜q ,and gauginos ˜g ,˜χ±,˜χ0.In the MSSM the multiplicative quantum num-ber R-parity is conserved,R p =+1for par-10012014016018000.20.10.3M H [GeV ]SM Double Higgs-strahlung: e + e - → ZHH σ [fb ]√s = 800 GeV√s = 500 GeVFigure 6.The cross-section for doubleHiggs-strahlung in the Standard Model at√120GeVmass 0.05%spin yes CPyes6%g HZZ 1%g HW W 2%g Hbb 2%g Hcc 10%g Hττ5%g Htt 6%λHHH∼30%ticles and R p =−1for sparticles.Spar-ticles are therefore produced in pairs and they eventually decay into the lightest spar-ticle which has to be stable.As an example,smuons are produced and decay through theprocess e +e −→˜µ+˜µ−→µ+µ−χ01χ01with χ01as the lightest sparticle being stable and,therefore,escaping detection.The mass scale of sparticles is only vaguely known.In most scenarios some spar-ticles,in particular charginos and neutrali-nos,are expected to lie in the energy region accessible by the next generation of e +e −200400600800Figure 7.Examples of mass spectra in mSUGRA,GMSB and AMSB models.colliders alsosupported bythe recentmea-surement of (g −2)µ10.Examples of massspectra for three SUSY breaking mechanisms (mSUGRA,GMSB,AMSB)are given in fig-ure 7.The most fundamental problem of super-symmetric theories is how SUSY is broken and in which way this breaking is communi-cated to the particles.Several scenarios have been proposed in which the mass spectra are generally quite different as illustrated in fig-ure 7.High precision measurements of the particle properties are therefore expected to distinguish between some of these scenarios.The study and exploration of Supersymmetry will proceed in the following steps:•Reconstruction of the kinematically ac-cessible spectrum of sparticles and the measurement of their properties,masses and quantum numbers•Extraction of the basic low-energy pa-rameters such as mass parameters,cou-plings,and mixings•Analysis of the breaking mechanism and reconstruction of the underlying theory.While it is unlikely that the complete spectrum of sparticles will be accessible at acollider with√Figure 9.Cross section near threshold for the processe +e −→˜χ+1˜χ−1,10fb−1per point.approach,the measured electroweak scaleSUSY parameters are extrapolated to high energies using these RGE’s.Due to the high precision of the measured input variables,only possible at the linear collider,an accurate test can be performed at which energy scale certain parameters be-come equal.Most interesting,the assump-tion of grand unification of forces requires the gaugino mass parameters M 1,M 2,M 3to meet at the GUT scale (figure 10(left)).Different SUSY breaking mechanisms predict different unification patterns of the sfermion mass parameters at high energy.With the high accuracy of the linear collider measure-ments these models can be distinguished as shown in figure 10for the case of mSUGRA (middle)and GMSB (right).In summary,the high precision studies of supersymmetric particles and their properties can open a window to energy scales far above the scales reachable with future accelerators,possibly towards the Planck scale where grav-ity becomes important.3.3Precision MeasurementsThe primary goal of precision measurements of gauge boson properties is to establish the non-abelian nature of electroweak interac-tions.The gauge symmetries of the Stan-dard Model determine the form and the strength of the self-interactions of the elec-troweak bosons,the triple couplings W W γand W W Z and the quartic couplings.Devi-ations from the Standard Model expectations for these couplings could be expected in sev-eral scenarios,for example in models where there exists no light Higgs boson and where the W and Z bosons are generated dynam-ically and interact strongly at high scales.Also for the extrapolation of couplings to high scales to test theories of grand unifi-cation such high precision measurements are mandatory.For the study of the couplings between gauge bosons the best precision is reached at the highest possible centre of mass energies.These couplings are especially sen-sitive to models of strong electroweak sym-metry breaking.W bosons are produced either in pairs,e +e −→W +W −or singly,e +e −→W eνwith both processes being sensitive to the triple gauge couplings.In general the total errors estimated on the anomalous couplings are in the range of few ×10−4.Figure 11com-pares the precision obtainable for ∆κγand ∆λγat different machines.The measurements at a linear collider are sensitive to strong symmetry breaking be-yond Λof the order of 5TeV,to be com-pared with the electroweak symmetry break-ing scale ΛEW SB =4πv ≈3TeV.One of the most sensitive quantities to loop corrections from the Higgs boson is the effective weak mixing angle in Z boson de-cays.By operating the collider at ener-gies close to the Z -pole with high luminos-ity (GigaZ)to collect at least 109Z bosons in particular the accuracy of the measure-Figure 10.Extrapolation of SUSY parameters measured at the electroweak scale to high energies.10-410-310-2∆κγLEP TEV LHCTESLA TESLA 50080010-410-310-2∆λγLEP TEV LHCTESLA TESLA500800Figure parison of constraints on the anomalous couplings ∆κγand ∆λγat different machinesment of sin 2θleff can be improved by one or-der of magnitude wrt.the precision obtained today 11.With both electron and positronbeams longitudinally polarised,sin 2θleff can be determined most accurately by measur-ing the left-right asymmetry A LR =A e =2v e a e /(v 2e +a 2e )with v e (a e )being the vec-tor (axialvector)couplings of the Z boson tothe electron and v e /a e =1-4sin 2θleff for pure Z exchange.Particularly demanding is the precision of 2×10−4with which the po-larisation needs to be known to match the statistical accuracy.An error in the weakmixing angle of ∆sin 2θleff =0.000013can be expected.Together with an improved de-termination of the mass of the W boson toa precision of some 6MeV through a scan of the W W production threshold and with the measurements obtained at high energy run-ning of the collider this will allow many high precision tests of the Standard Model at the loop level.As an example,figure 12shows the variation of the fit χ2to the electroweak measurements as a function of M H for the present data and for the data expected at a linear collider.The mass of the Higgs bo-son can indirectly be constraint at a level of 5%.Comparing this prediction with the di-rect measurement of M H consistency tests of the Standard Model can be performed at the quantum level or to measure free parameters in extensions of the Standard Model.This is5101520101032000LCm hχ2Figure 12.∆χ2as a function of the Higgs boson mass for the electroweak precision data today (2000)and after GigaZ running (LC).of particular importance if M H >200GeV in contradiction to the current electroweak mea-surements.In summary,there is strong evidence for new phenomena at the TeV energy scale.Only the precision exploration at the linear collider will allow,together with the results obtained at the Large Hadron Collider,the understanding of the underlying physics and will open a new window beyond the centre-of-mass energies reachable.Whatever sce-nario is realized in nature,the linear collider will add crucial information beyond the LHC.There is global consensus in the high energy physics community that the next accelera-tor based project needs to be an electron-positron linear collider with a centre-of-mass energy of at least 500GeV.4Electron-Positron Linear CollidersThe feasibility of a linear collider has been successfully demonstrated by the operationof the SLAC Linear Collider,SLC.How-ever,aiming at centre-of-mass energies at the TeV scale with luminosities of the order of 1034cm −2s −1requires at least two orders of magnitude higher beam power and two orders of magnitude smaller beam sizes at the inter-action point.Over the past decade,several groups worldwide have been pursuing differ-ent linear collider designs for the centre-of-mass energy range up to around one TeV as well as for the multi-TeV range.Excel-lent progress has been achieved at various test facilities worldwide in international col-laborations on crucial aspects of the collider designs.At the Accelerator Test Facility at KEK 12,emittances within a factor two of the damping ring design have been achieved.At the Final Focus Test Beam at SLAC 13de-magnification of the beams has been proven;the measured spot sizes are well in agreement with the theoretically expected values.The commissioning and operation of the TESLA Test Facility at DESY 14has demonstrated the feasibility of the TESLA technology.In the following,a short review of the different approaches is given.4.1TeV rangeThree design studies are presently pursued:JLC 15,NLC 16and TESLA 17,centred around KEK,SLAC and DESY,respectively.Details about the design,the status of de-velopment and the individual test facilities can be found in the above quoted references as well as in the status reports presented at LCWS200018,19,20.A comprehensive sum-mary of the present status can be found in the Snowmass Accelerator R&D Report 21,here only a short discussion of the main features and differences of the three approaches will be given with emphasis on luminosity and en-ergy reach.One key parameter for performing the physics program at a collider is the centre-of-mass energy achievable.The energy reachof a collider with a given linac length and a certain cavityfilling factor is determined by the gradient achievable with the cavity tech-nology chosen.For normalconducting cavi-ties the maximum achievable gradient scales roughly proportional to the RF frequency used,for superconducting Niobium cavities, the fundamental limit today is around55 MV/m.The second key parameter for the physicsprogram is the luminosity L,given byL=n b N2e f rep(σ∗x+σ∗y)2.Choos-ing aflat beam size(σ∗x≫σ∗y)at the inter-action point,δE becomes independent of the vertical beam size and the luminosity can be increased by reducingσ∗y as much as possi-ble.Sinceσ∗y∝sn b N e f rep=ηP AC is obtained from themains power P AC with an efficiency η.Equation(1)can then be rewritten asL∝ηP AC s ǫy(2)High luminosity therefore requires high ef-ficiencyηand high beam quality with low emittanceǫy and low emittance dilution ∆ǫ/ǫ∝f6RF,which is largely determined by the RF frequency f RF of the chosen technol-ogy.The fundamental difference between the three designs is the choice of technology for the accelerating structures.The design of NLC is based on normalconducting cavities using f RF of11.4GHz(X-band),for JLC two options,X-band or C-band(5.7GHz)are pursued.The TESLA concept,developed by the TESLA collaboration,is using supercon-ducting cavities(1.3GHz).As an example for a linear collider facility,figure13shows the schematic layout of TESLA.Figure13.Schematic layout of TESLATable2compares some key parameters for the different technologies at√Figure 14.Evolution of superconducting cavity per-formance.The average gradient achieved with TESLA 9-cell cavities produced in industry (first test,no additional processing)is shown as dots.with N b bunches,the time ∆T b between bunches within a train which allows head on crossing of the bunches for TESLA but requires a crossing angle for the other de-signs.The design luminosity L ,beam power P beam and the required mains power P AC il-lustrate that for a given mains power the su-perconducting technology delivers higher lu-minosity.On the other hand the lower gradi-ent G acc requires a longer linac for the samecentre-of-massenergy reach.As can be seen from table 2the X-band machines call for a beam loaded (unloaded)gradient of some 50(70)MV/m for√s =500GeV,a gradient which is mean-while routinely achieved for cavities fabri-cated in industry as illustrated in figure 14.Table 2also contains the presently planned length of the facilities 17,16,22,23.AnFigure 15.Excitation curves of three electropolished single-cell cavities.Gradients well above 35MV/m are reached.upgrade in energy up to around one TeV seems possible for all designs.In the NLC case,more cavities would be installed within the existing tunnel,in the JLC case,the tunnel length would have to be increased to house more cavities.In the TESLA case,a gradient of around 35MV/m is neededto reach√Table parison of some crucial parameters at 500GeV for the different technologies under study,see text for details.NLCJLC-C51502820190337 1.4head on angle 20.7σ∗x/y [nm ]245/2.7318/4.3δE [%]4.73.93.42.64P beam [MW ]13.212.6P AC (linacs )[MW ]13222023.550.23316s of 3TeV,usinghigh frequency (30GHz)normalcon-ducting structures operating at very high ac-celerating fields (150MV/m).The present design calls for bunch separations of .67ns,a vertical spotsize of 1nm and beamstrahlung δE of 30%.For this promising concept a new test facility is under construction at CERN which should allow tests with full gradient starting in 2005.5RealisationThe new generation of high energy colliders most likely exceeds the resources of a coun-try or even a region.There is general consen-sus that the realisation has to be done in an international,interregional framework.One such framework,the so called Global Accel-erator Network (GAN),has been proposed to ICFA in March 2000.A short discussion of the principle considerations will be presented here,more details can be found in ref.25.The GAN is a global collaboration of lab-oratories and institutes in order to design,construct,commission,operate and main-tain a large accelerator facility.The model is based on the experience of large experi-mental collaborations,particularly in particle physics.Some key elements are listed below:•it is not an international permanent in-stitution,but an international project of limited duration;•the facility would be the common prop-erty of the participating countries;•there are well defined roles and obliga-tions of all partners;•partners contribute through components or subsystems;•design,construction and testing of com-ponents is done in participating institu-tions;•maintenance and running of the accel-erator would be done to a large extent from the participating institutions.The GAN would make best use of world-wide competence,ideas and resources,create a visible presence of activities in all partici-pating countries and would,hopefully,make the site selection less controversial.study general considerations of implementing a GAN and to study the technical considera-tions and influence on the design and cost of the accelerator.The reports of these working groups can be found on the web26.Their overall conclusion is that a GAN can be a fea-sible way to build and operate a new global accelerator,although many details still need to be clarified.6SummaryThere is global consensus about the next ac-celerator based project in particle physics.It has to be an electron-positron linear collider with an initial energy reach of some500GeV with the potential of an upgrade in centre-of-mass energy.The physics case is excellent, only a few highlights could be presented here. There is also global consensus that concur-rent operation with LHC is needed and fruit-ful.Therefore,a timely realisation is manda-tory.The technical realisation of a linear col-lider is now feasible,several technologies are either ripe or will be ripe soon.A fast consen-sus in the community about the technology is as a global project with the highest possible luminosity and a clear upgrade potential be-yond500GeV.AcknowledgmentsThe author would like to express his grati-tude to all people who have contributed to the studies of future electron-positron linear colliders from the machine design to physics and detector studies.Special thanks go to the organisers and their team for a very well or-ganised,inspiring conference as well as for the competent technical help in preparing this presentation.References1.J.A.Aguilar-Saavedra et al,TESLATechnical Design Report,Part III,Physics at an e+e−Linear Collider,DESY2001-011,ECFA2001-209,hep-ph/0106315.2.T.Abe et al,Linear Collider Physics Re-source Book for Snowmass2001,BNL-52627,CLNS01/1729,FERMILAB-Pub-01/058-E,LBNL-47813,SLAC-R-570,UCRL-ID-143810-DR,LC-REV-2001-074-US,hep-ex/0106055-583.K.Abe et al,Particle Physics Exper-iments at JLC,KEK-Report2001-11, hep-ph/0109166.4.Proceedings of LCWS,Physics and Ex-periments with Future Linear Colliders, eds A.Para,H.E.Fisk,(AIP Conf.Proc.,Vol578,2001).5.Worldwide Study of the Physics and De-tectors for Future e+e−Colliders/lc/6.ACFA Joint Linear Collider Physics andDetector Working Grouphttp://acfahep.kek.jp/7.2nd Joint ECFA/DESY Studyon Physics and Detectors for a Linear Electron-Positron Colliderhttp://www.desy.de/conferences/ecfa-desy-lc98.html8.A Study of the Physics and Detectors forFuture Linear e+e−Colliders:American Activities/lc/ameri-ca.html9.G.Alexander et al,TESLA TechnicalDesign Report,Part IV,A Detector for TESLA,DESY2001-011,ECFA2001-209.10.H.N.Brown et al.[Muon g-2Collabo-ration],Phys.Rev.Lett.86(2001)222711.J.Drees,these proceedings12.E.Hinode et al,eds.,KEK Internal95-4,1995,eds J.Urakawa and M.Yoshioka, Proceedings of the SLAC/KEK Linear Collider Workshop on Damping Ring, KEK92-6,199213.The FFTB Collaboration:BINP(Novosibirsk/Protvino),DESY, FNAL,KEK,LAL(Orsay),MPI Mu-nich,Rochester,and SLAC14.Proposal for a TESLA Test Facility,DESY TESLA-93-01,199215.KEK-Report97-1,1997.16.Zeroth Order Design Report for theNext Linear Collider,SLAC Report474,1996.2001Report on the Next Linear Collider,Fermilab-Conf-01-075-E,LBNL-47935,SLAC-R-571,UCRL-ID-14407717.J.Andruszkow et al,TESLA TechnicalDesign Report,Part II,The Accelerator, DESY2001-011,ECFA2001-20918.O.Napoly,TESLA Linear Collider:Sta-tus Report,in ref419.T.O.Raubenheimer,Progress in theNext Linear Collider Design,in ref4 20.Y.H.Chin et al Status of JLC Accelera-tor Development,in ref421.A.Chao et al,2001Snowmass Accelera-tor R&D Report,http://www.hep.anl.gov/pvs/dpb/Snowmass.pdf22.Y.H.Chin,private communication23.H.Matsumoto,T.Shintake,private com-munication24.I.Wilson,A Multi-TeV Compact e+e−Linear Collider,in ref425.F.Richard et al,TESLA Technical De-sign Report,Part I,Executive Summary, DESY2001-011,ECFA2001-209,hep-ph/0106314.26./directorate/icfa/icfa reports.html。
学理论物理推荐书目
理了一些曾经读过而且觉得很不错的理论物理参考书，希望能对想做或者正在做理论物理的人有点用。
１:经典力学/电动力学/统计力学/量子力学1.1: Greiner系列,其实不止四大力学，覆盖面从Mechanics到QCD,基本都不错，物理图像非常清晰明了。
还有Schwabl写的两本书 Quantum Mechanics&Advanced Quantum Mechanics，应该比传统的经典教材容易念一些。
1.2: 传统的经典教材，Landau系列,Goldstein的经典力学,Jackson的电动力学， Schiff，Sakurai的量子力学，不用多说了。
2:量子场论/标准模型2.1: 前面提到的Greiner系列，Mandl&Shaw的Quantum Field Theory,Ryder的Quantum Field Theory，Brown的Quantum Field Theory以及Bailin&Love的Introduction to Gauge Field Theory,比较容易念。
2.2: Peskin&schroeder，绝对经典。
2.3: Cheng&Li的Gauge Theory of Elementary Particle Physics，也是经典。
2.4: Itzykson&Zuber的Quantum Field Theory,Pokorski的Gauge Field Theories, 可能难一些，但是是非常好的参考书。
2.5: Muta的Foundations of Quantum Chromodynamics，通俗易懂。
2.6: Martin的讲义Phenomenology of Particle Physics，值得一看。
2.7: Boehm,Denner&Joos的Gauge Theories of the Strong and Electroweak Interaction, 做Particle Phenomenology的话绝对案头必备书目。
On the Interactions of Light Gravitinos
On the Interactions of Light GravitinosT.E.Clark1,Taekoon Lee2,S.T.Love3,Guo-Hong Wu4Department of PhysicsPurdue UniversityWest Lafayette,IN47907-1396AbstractIn models of spontaneously broken supersymmetry,certain light gravitino processes are governed by the coupling of its Goldstino components.The rules for constructing SUSY and gauge invariant actions involving the Gold-stino couplings to matter and gaugefields are presented.The explicit oper-ator construction is found to be at variance with some previously reported claims.A phenomenological consequence arising from light gravitino inter-actions in supernova is reexamined and scrutinized.1e-mail address:clark@2e-mail address:tlee@3e-mail address:love@4e-mail address:wu@1In the supergravity theories obtained from gauging a spontaneously bro-ken global N=1supersymmetry(SUSY),the Nambu-Goldstone fermion, the Goldstino[1,2],provides the helicity±1degrees of freedom needed to render the spin3gravitino massive through the super-Higgs mechanism.For a light gravitino,the high energy(well above the gravitino mass)interactions of these helicity±1modes with matter will be enhanced according to the su-persymmetric version of the equivalence theorem[3].The effective action de-scribing such interactions can then be constructed using the properties of the Goldstinofields.Currently studied gauge mediated supersymmetry breaking models[4]provide a realization of this scenario as do certain no-scale super-gravity models[5].In the gauge mediated case,the SUSY is dynamically broken in a hidden sector of the theory by means of gauge interactions re-sulting in a hidden sector Goldstinofield.The spontaneous breaking is then mediated to the minimal supersymmetric standard model(MSSM)via radia-tive corrections in the standard model gauge interactions involving messenger fields which carry standard model vector representations.In such models,the supergravity contributions to the SUSY breaking mass splittings are small compared to these gauge mediated contributions.Being a gauge singlet,the gravitino mass arises only from the gravitational interaction and is thus farsmaller than the scale √,where F is the Goldstino decay constant.More-2over,since the gravitino is the lightest of all hidden and messenger sector degrees of freedom,the spontaneously broken SUSY can be accurately de-scribed via a non-linear realization.Such a non-linear realization of SUSY on the Goldstinofields was originally constructed by Volkov and Akulov[1].The leading term in a momentum expansion of the effective action de-scribing the Goldstino self-dynamics at energy scales below √4πF is uniquelyfixed by the Volkov-Akulov effective Lagrangian[1]which takes the formL AV=−F 22det A.(1)Here the Volkov-Akulov vierbein is defined as Aµν=δνµ+iF2λ↔∂µσν¯λ,withλ(¯λ)the Goldstino Weyl spinorfield.This effective Lagrangian pro-vides a valid description of the Goldstino self interactions independent of the particular(non-perturbative)mechanism by which the SUSY is dynam-ically broken.The supersymmetry transformations are nonlinearly realized on the Goldstinofields asδQ(ξ,¯ξ)λα=Fξα+Λρ∂ρλα;δQ(ξ,¯ξ)¯λ˙α= F¯ξ˙α+Λρ∂ρ¯λ˙α,whereξα,¯ξ˙αare Weyl spinor SUSY transformation param-eters andΛρ≡−i Fλσρ¯ξ−ξσρ¯λis a Goldstinofield dependent translationvector.Since the Volkov-Akulov Lagrangian transforms as the total diver-genceδQ(ξ,¯ξ)L AV=∂ρ(ΛρL AV),the associated action I AV= d4x L AV is SUSY invariant.The supersymmetry algebra can also be nonlinearly realized on the matter3(non-Goldstino)fields,generically denoted byφi,where i can represent any Lorentz or internal symmetry labels,asδQ(ξ,¯ξ)φi=Λρ∂ρφi.(2) This is referred to as the standard realization[6]-[9].It can be used,along with space-time translations,to readily establish the SUSY algebra.Under the non-linear SUSY standard realization,the derivative of a matterfield transforms asδQ(ξ,¯ξ)(∂νφi)=Λρ∂ρ(∂νφi)+(∂νΛρ)(∂ρφi).In order to elim-inate the second term on the right hand side and thus restore the standard SUSY realization,a SUSY covariant derivative is introduced and defined so as to transform analogously toφi.To achieve this,we use the transformation property of the Volkov-Akulov vierbein and define the non-linearly realized SUSY covariant derivative[9]Dµφi=(A−1)µν∂νφi,(3) which varies according to the standard realization of SUSY:δQ(ξ,¯ξ)(Dµφi)=Λρ∂ρ(Dµφi).Any realization of the SUSY transformations can be converted to the standard realization.In particular,consider the gauge covariant derivative,(Dµφ)i≡∂µφi+T a ij A aµφj,(4)4with a=1,2,...,Dim G.We seek a SUSY and gauge covariant deriva-tive(Dµφ)i,which transforms as the SUSY standard ing the Volkov-Akulov vierbein,we define(Dµφ)i≡(A−1)µν(Dνφ)i,(5) which has the desired transformation property,δQ(ξ,¯ξ)(Dµφ)i=Λρ∂ρ(Dµφ)i, provided the vector potential has the SUSY transformationδQ(ξ,¯ξ)Aµ≡Λρ∂ρAµ+∂µΛρAρ.Alternatively,we can introduce a redefined gaugefieldV aµ≡(A−1)µνA aν,(6) which itself transforms as the standard realization,δQ(ξ,¯ξ)V aµ=Λρ∂ρV aµ, and in terms of which the standard realization SUSY and gauge covariant derivative then takes the form(Dµφ)i≡(A−1)µν∂νφi+T a ij V aµφj.(7) Under gauge transformations parameterized byωa,the original gaugefield varies asδG(ω)A aµ=(Dµω)a=∂µωa+gf abc A bµωc,while the redefinedgaugefield V aµhas the Goldstino dependent transformation:δG(ω)V aµ= (A−1)µν(Dνω)a.For all realizations,the gauge transformation and SUSY transformation commutator yields a gauge variation with a SUSY trans-formed value of the gauge transformation parameter,δG(ω),δQ(ξ,¯ξ)=δG(Λρ∂ρω−δQ(ξ,¯ξ)ω).(8) 5If we further require the local gauge transformation parameter to also trans-form under the standard realization so thatδQ(ξ,¯ξ)ωa=Λρ∂ρωa,then the gauge and SUSY transformations commute.In order to construct an invariant kinetic energy term for the gaugefields, it is convenient for the gauge covariant anti-symmetric tensorfield strength to also be brought into the standard realization.The usualfield strengthF a αβ=∂αA aβ−∂βA aα+if abc A bαA cβvaries under SUSY transformations asδQ(ξ,¯ξ)F aµν=Λρ∂ρF aµν+∂µΛρF aρν+∂νΛρF aµρ.A standard realization of thegauge covariantfield strength tensor,F aµν,can be then defined asF aµν=(A−1)µα(A−1)νβF aαβ,(9) so thatδQ(ξ,¯ξ)F aµν=Λρ∂ρF aµν.These standard realization building blocks consisting of the gauge singlet Goldstino SUSY covariant derivatives,Dµλ,Dµ¯λ,the matterfields,φi,their SUSY-gauge covariant derivatives,Dµφi,and thefield strength tensor,F aµν, along with their higher covariant derivatives can be combined to make SUSY and gauge invariant actions.These invariant action terms then dictate the couplings of the Goldstino which,in general,carries the residual consequences of the spontaneously broken supersymmetry.A generic SUSY and gauge invariant action can be constructed[9]asI eff=d4x detA L eff(Dµλ,Dµ¯λ,φi,Dµφi,Fµν)(10)6where L effis any gauge invariant function of the standard realization basic building ing the nonlinear SUSY transformationsδQ(ξ,¯ξ)detA=∂ρ(ΛρdetA)andδQ(ξ,¯ξ)L eff=Λρ∂ρL eff,it follows thatδQ(ξ,¯ξ)I eff=0.It proves convenient to catalog the terms in the effective Lagranian,L eff, by an expansion in the number of Goldstinofields which appear when covari-ant derivatives are replaced by ordinary derivatives and the Volkov-Akulov vierbein appearing in the standard realizationfield strengths are set to unity. So doing,we expandL eff=L(0)+L(1)+L(2)+···,(11)where the subscript n on L(n)denotes that each independent SUSY invariant operator in that set begins with n Goldstinofields.L(0)consists of all gauge and SUSY invariant operators made only from light matterfields and their SUSY covariant derivatives.Any Goldstinofield appearing in L(0)arises only from higher dimension terms in the matter covariant derivatives and/or thefield strength tensor.Taking the light non-Goldstinofields to be those of the MSSM and retaining terms through mass dimension4,then L(0)is well approximated by the Lagrangian of the mini-mal supersymmetric standard model which includes the soft SUSY breaking terms,but in which all derivatives have been replaced by SUSY covariant ones and thefield strength tensor replaced by the standard realizationfield7strength:L(0)=L MSSM(φ,Dµφ,Fµν).(12) Note that the coefficients of these terms arefixed by the normalization of the gauge and matterfields,their masses and self-couplings;that is,the normalization of the Goldstino independent Lagrangian.The L(1)terms in the effective Lagrangian begin with direct coupling of one Goldstino covariant derivative to the non-Goldstinofields.The general form of these terms,retaining operators through mass dimension6,is given byL(1)=1[DµλαQµMSSMα+¯QµMSSM˙αDµ¯λ˙α],(13)Fwhere QµMSSMαand¯QµMSSM˙αcontain the pure MSSMfield contributions to the conserved gauge invariant supersymmetry currents with once again all field derivatives being replaced by SUSY covariant derivatives and the vector field strengths in the standard realization.That is,it is this term in the effective Lagrangian which,using the Noether construction,produces the Goldstino independent piece of the conserved supersymmetry current.The Lagrangian L(1)describes processes involving the emission or absorption of a single helicity±1gravitino.Finally the remaining terms in the effective Lagrangian all contain two or more Goldstinofields.In particular,L(2)begins with the coupling of two8Goldstinofields to matter or gaugefields.Retaining terms through mass dimension8and focusing only on theλ−¯λterms,we can writeL(2)=1F2DµλαDν¯λ˙αMµν1α˙α+1F2Dµλα↔DρDν¯λ˙αMµνρ2α˙α+1F2DρDµλαDν¯λ˙αMµνρ3α˙α,(14)where the standard realization composite operators that contain matter and gaugefields are denoted by the M i.They can be enumerated by their oper-ator dimension,Lorentz structure andfield content.In the gauge mediated models,these terms are all generated by radiative corrections involving the standard model gauge coupling constants.Let us now focus on the pieces of L(2)which contribute to a local operator containing two gravitinofields and is bilinear in a Standard Model fermion (f,¯f).Those lowest dimension operators(which involve no derivatives on f or¯f)are all contained in the M1piece.After application of the Goldstino field equation(neglecting the gravitino mass)and making prodigious use of Fierz rearrangement identities,this set reduces to just1independent on-shell interaction term.In addition to this operator,there is also an operator bilinear in f and¯f and containing2gravitinos which arises from the product of det A with L(0).Combining the two independent on-shell interaction terms involving2gravitinos and2fermions,results in the effective actionIf¯f˜G˜G =d4x−12F2λ↔∂µσν¯λf↔∂νσµ¯f9+C ffF2(f∂µλ)¯f∂µ¯λ,(15)where C ff is a model dependent real coefficient.Note that the coefficient of thefirst operator isfixed by the normaliztion of the MSSM Lagrangian. This result is in accord with a recent analysis[10]where it was found that the fermion-Goldstino scattering amplitudes depend on only one parameter which corresponds to the coefficient C ff in our notation.In a similar manner,the lowest mass dimension operator contributing to the effective action describing the coupling of two on-shell gravitinos to a single photon arises from the M1and M3pieces of L(2)and has the formIγ˜G˜G =d4xCγF2∂µλσρ∂ν¯λ∂µFρν+h.c.,(16)with Cγa model dependent real coefficient and Fµνis the electromagnetic field strength.Note that the operator in the square bracket is odd under both parity(P)and charge conjugation(C).In fact any operator arising from a gauge and SUSY invariant structure which is bilinear in two on-shell gravitinos and contains only a single photon is necessarily odd in both P and C.Thus the generation of any such operator requires a violation of both P and ing the Goldstino equation of motion,the analogous term containing˜Fµνreduces to Eq.(16)with Cγ→−iCγ.Recently,there has appeared in the literature[11]the claim that there is a lower dimensional operator of the form˜M2F2∂νλσµ¯λFµνwhich contributes to the single photon-102gravitino interaction.Here˜M is a model dependent SUSY breaking massparameter which is roughly an order(s)of magnitude less than √.¿Fromour analysis,we do notfind such a term to be part of a SUSY invariant action piece and thus it should not be included in the effective action.Such a term is also absent if one employs the equivalent formalism of Wess and Samuel [6].We have also checked that such a term does not appear via radiative corrections by an explicit graphical calculation using the correct non-linearly realized SUSY invariant action.This is also contrary to the previous claim.There have been several recent attempts to extract a lower bound on the SUSY breaking scale using the supernova cooling rate[11,12,13].Unfortu-nately,some of these estimates[11,13]rely on the existence of the non-SUSY invariant dimension6operator referred to ing the correct low en-ergy effective lagrangian of gravitino interactions,the leading term coupling 2gravitinos to a single photon contains an additional supression factor ofroughly Cγs˜M .Taking√s 0.1GeV for the processes of interest and using˜M∼100GeV,this introduces an additional supression of at least10−12in the rate and obviates the previous estimates of a bound on F.Assuming that the mass scales of gauginos and the superpartners of light fermions are above the core temperature of supernova,the gravitino cooling of supernova occurs mainly via gravitino pair production.It is interesting to11compare the gravitino pair production cross section to that of the neutrino pair production,which is the main supernova cooling channel.We have seen that for low energy gravitino interactions with matter,the amplitudes for gravitino pair production is proportional to1/F2.A simple dimensional analysis then suggests the ratio of the cross sections is:σχχσνν∼s2F4G2F(17)where GF is the Fermi coupling and√s is the typical energy scale of theparticles in a supernova.Even with the most optimistic values for F,thegravitino production is too small to be relevant.For example,taking √F=100GeV,√s=.1GeV,the ratio is of O(10−11).It seems,therefore,thatsuch an astrophysical bound on the SUSY breaking scale is untenable in mod-els where the gravitino is the only superparticle below the scale of supernova core temperature.We thank T.K.Kuo for useful conversations.This work was supported in part by the U.S.Department of Energy under grant DE-FG02-91ER40681 (Task B).12References[1]D.V.Volkov and V.P.Akulov,Pis’ma Zh.Eksp.Teor.Fiz.16(1972)621[JETP Lett.16(1972)438].[2]P.Fayet and J.Iliopoulos,Phys.Lett.B51(1974)461.[3]R.Casalbuoni,S.De Curtis,D.Dominici,F.Feruglio and R.Gatto,Phys.Lett.B215(1988)313.[4]M.Dine and A.E.Nelson,Phys.Rev.D48(1993)1277;M.Dine,A.E.Nelson and Y.Shirman,Phys.Rev.D51(1995)1362;M.Dine,A.E.Nelson,Y.Nir and Y.Shirman,Phys.Rev.D53,2658(1996).[5]J.Ellis,K.Enqvist and D.V.Nanopoulos,Phys.Lett.B147(1984)99.[6]S.Samuel and J.Wess,Nucl.Phys.B221(1983)153.[7]J.Wess and J.Bagger,Supersymmetry and Supergravity,second edition,(Princeton University Press,Princeton,1992).[8]T.E.Clark and S.T.Love,Phys.Rev.D39(1989)2391.[9]T.E.Clark and S.T.Love,Phys.Rev.D54(1996)5723.[10]A.Brignole,F.Feruglio and F.Zwirner,hep-th/9709111.[11]M.A.Luty and E.Ponton,hep-ph/9706268.13[12]J.A.Grifols,R.N.Mohapatra and A.Riotto,Phys.Lett.B400,124(1997);J.A.Grifols,R.N.Mohapatra and A.Riotto,Phys.Lett.B401, 283(1997).[13]J.A.Grifols,E.Masso and R.Toldra,hep-ph/970753.D.S.Dicus,R.N.Mohapatra and V.L.Teplitz,hep-ph/9708369.14。
MaxDEA
Detailed Contents
Chapter 1: Main Features of MaxDEA ..................................................8
1.1 Main Features ............................................................................................... 8 1.2 Models in MaxDEA...................................................................................... 9 1.3 What’s NEW ............................................................................................... 12 1.4 Compare MaxDEA Editions ..................................................................... 17
3.1 Import Data ................................................................................................ 19 3.2 Define Data ................................................................................................. 24 3.3 Set and Run Model..................................................................................... 25 3.4 Export Results ............................................................................................ 77
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)。
SCI论文摘要中常用的表达方法
（3）介绍应用、用途，常用词汇有use, apply, application等
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（1）展示研究结果，常用词汇有show, result, present等
（2）介绍结论，常用词汇有summary, introduce,conclude等
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（1）陈述论文的论点和作者的观点，常用词汇有suggest, repot, present, expect, describe等
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SCI摘要引言部分案例 attention
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SCI高被引摘要引言部分案例 词汇outline
? Author(s): TIERNEY, L SCI引用728次
引言部分 回顾研究背景常用词汇summarize
Abstract: This biennial review summarizes much of Particle Physics. Using data from previous editions, plus 1900 new measurements from 700 papers, we list, evaluate, and average measured properties of gauge bosons, leptons, quarks, mesons, and baryons. We also summarize searches for hypothetical particles such as Higgs bosons, heavy neutrinos, and supersymmetric particles. All the particle properties and search limits are listed in Summary Tables. We also give numerous tables, figures, formulae, and reviews of topics such as the Standard Model, particle detectors, probability, and statistics. A booklet is available containing the Summary Tables and abbreviated versions of some of the other sections of this full Review.
The Physics of Flavor is the Flavor of Physics
arXiv:hep-ph/0111051v1 6 Nov 2001
H. FRITZSCH Ludwig–Maximilians–Universit¨ at M¨ unchen, Sektion Physik, Theresienstraße 37, D–80333 M¨ unchen, Germany and CERN, CH-1211 Geneva 23
a Supported
in part by VW–Stiftung Hanotation, I still use today in the German language. Once Gell–Mann and I were driving to a lunch restaurant in Pasadena and passed by a Baskin and Robins icecream place, advertising 32 different flavors. Murray suddenly came up with the proposal to use the name “flavor”. I did not like this proposal at the beginning, translating it into German, where it means “Geschmack”, an expression one could hardly use for the description of a subatomic particle. Soon afterwards, however, I went along with it, expecially after realizing that in other languages the translation of “flavor” gives quite meaningful results. For example, in Italian the word “il sapore”, used e. g. in “il sapore del vino” could very well be used to distinguish the various degrees of freedom of the quarks. Compared to the present time, the flavor physics in those days was rather poor. Only three flavors, i. e. u, d, s, were known, and the basic parameters of flavor physics were the three quark masses and the Cabibbo angle. CP – violation was considered to be a peculiar phenomenon not intrinsically related to the flavor mixing. Today we see the sharp contures of the Standard Model1) in front of us, like the contures of the Wulingyan mountains seen from the Golden Whip Stream. The physics of flavor is at the same time the physics of the multitude of the free parameters of the theory. Even if we disregard possible neutrino masses, the minimal number of parameters is 18, among them the six masses of the quarks, the three lepton masses, and four flavor mixing parameters. Especially those 13 parameters are in the focus of flavor physics. In the Standard Model they arise in a way, which can hardly be considered satisfactory, even on low standards. They appear as the result of a direct coupling of the fermions to the “Higgs” field, a formal device without any predictive power, as far as those parameters are concerned. In my view, this mechanism of fermion mass generation is the least attractive corner of the Standard Model, and it is quite likely that this is the corner where the model might deviate from reality. Furthermore it might well be that the “Higgs” particle responsible for the generation of mass for the W – and the Z –bosons does not couple to the b–quark with a strength proportional to mb as expected in the Standard Model, in which case the “Higgs” particle would not decay predominantly into a ¯ bb–system, but into other particles, e. g. into two gluons or into γγ (see e. g. ref. (2)). More than a year ago we have entered the new millenium with a rather bizarre spectrum of the lepton and quark masses, which extends (in the absence of neutrino masses) from about 0.5 MeV (electron mass) to about 175000 MeV (t–mass), stretching over almost six orders of magnitude. On a logarithmic scale, the quark masses are nearly on straight lines, if plotted as functions of 2
高能物理中的超对称理论：探索超对称理论的实验验证与新物理预言
高能物理中的超对称理论：探索超对称理论的实验验证与新物理预言摘要超对称理论作为粒子物理学标准模型的拓展，为解决一系列未解之谜提供了潜在方案。
本文深入探讨超对称理论的核心概念、实验验证方法以及对新物理的预言。
通过分析大型强子对撞机（LHC）等实验的最新进展，本文旨在评估超对称理论的现状，并展望其在未来高能物理研究中的发展方向。
引言粒子物理学标准模型（Standard Model, SM）在描述基本粒子和相互作用方面取得了巨大成功，但仍存在一些未解之谜，如等级问题（Hierarchy Problem）、暗物质（Dark Matter）等。
超对称理论（Supersymmetry, SUSY）作为一种超越标准模型的新物理理论，为解决这些问题提供了可能的答案。
超对称理论预言每个标准模型粒子都有一个超对称伙伴，这些超对称粒子的存在可以解决等级问题，并为暗物质提供候选者。
然而，超对称理论尚未得到实验的直接验证，其正确性仍存在争议。
超对称理论的核心概念超对称理论的核心思想是在时空对称性的基础上引入一种新的对称性——超对称性。
超对称性将费米子（如电子、夸克）和玻色子（如光子、胶子）联系起来，认为它们是同一基本粒子的不同表现形式。
超对称理论预言每个标准模型粒子都有一个超对称伙伴，称为超粒子（Superpartner）。
例如，电子的超对称伙伴是标量电子（selectron），光子的超对称伙伴是光微子（photino）。
超对称理论的实验验证方法目前，寻找超对称粒子的主要实验方法是在高能粒子对撞机上进行实验。
大型强子对撞机（LHC）是目前能量最高的粒子对撞机，其对撞能量可以达到13 TeV，为寻找超对称粒子提供了理想的平台。
在LHC上寻找超对称粒子的主要策略包括：1. 直接寻找：通过分析对撞产生的粒子信号，寻找与超对称粒子质量和衰变模式相符的信号。
2. 间接寻找：通过测量标准模型粒子的性质，寻找超对称理论预言的偏差。
超对称理论的实验进展尽管LHC已经进行了多年的实验，但目前尚未发现超对称粒子的直接证据。
Journal of Electronic Imaging 13(3), 411–417 (July 2004).
Superquadric representation of automotive parts applying part decompositionYan ZhangAndreas KoschanMongi A.AbidiUniversity of TennesseeDepartment of Electrical and Computer Engineering334Ferris HallKnoxville,Tennessee37996-2100E-mail:yzhang@Abstract.Superquadrics are able to represent a large variety of objects with only a few parameters and a single equation.We present a superquadric representation strategy for automotive parts composed of3-D triangle meshes.Our strategy consists of two ma-jor steps of part decomposition and superquadricfitting.The origi-nalities of this approach include the following two features.First,our approach can represent multipart objects with superquadrics suc-cessfully by applying part decomposition.Second,superquadrics re-covered from our approach have the highest confidence and accu-racy due to the3-D watertight surfaces utilized.A novel,generic3-D part decomposition algorithm based on curvature analysis is also proposed.Experimental results demonstrate that the proposed part decomposition algorithm is able to segment multipart objects into meaningful single parts efficiently.The proposed superquadric rep-resentation strategy can then represent each individual part of the original objects with a superquadric model successfully.©2004 SPIE and IS&T.[DOI:10.1117/1.1762516]1IntroductionObject representation denotes representing real-world ob-jects with known graphic or mathematical primitives that can be recognized by computers.This research has numer-ous applications in areas including computer vision,com-puter graphics,and reverse engineering.An object can be represented by three levels of primitives in terms of the dimensional complexity:volumetric primitives,surface el-ements,and contours.The primitive selected to describe the object depends on the complexity of the object and the tasks involved.As the highest level primitives,volumetric primitives can better represent global features of an object with a significantly reduced amount of information com-pared with surface elements and contours.In addition, volumetric primitives have the ability to achieve the highest data compression ratio without losing the accuracy of the original data.The primarily used volumetric primitives in-clude generalized cylinders,geons,and superquadrics.1Su-perquadrics are a generalization of basic quadric surfaces and they can represent a large variety of shapes with only a few parameters and a single equation.An object initially represented by thousands of triangle meshes can be repre-sented by only a small set of superquadrics.This compact representation can be applied to object recognition to aid, for example,automated depalletizing of industrial parts or robot-guided bin picking of mixed nuclear waste in a haz-ardous environment.The quality control of both tasks can be enhanced by employing superquadrics.Furthermore,the registration of multiview data is indispensable to measure the size of partially occluded objects or their distances from each other in several image-based quality control tasks.Su-perquadrics can be used to efficiently register multiview range data of scenes with small overlap.2Most early research on superquadric representation con-centrated on representing single-part objects from single-view intensity or range images by assuming that the objects have been appropriately segmented.3–13This category of research focused on the data-fitting process,including ob-jective function selection,fitting criteria measurements,andPaper ORNL-007received Jul.30,2003;accepted for publication Feb.23,2004. 1017-9909/2004/$15.00©2004SPIE andIS&T.Fig.1Real range image of a multipart object obtained from Ref.18. Journal of Electronic Imaging13(3),411–417(July2004).Journal of Electronic Imaging/July2004/Vol.13(3)/411convergence analysis.For complex,multipart objects or scenes,there are two major types of approaches in the re-search literature.The first type of method incorporates an image segmentation step prior to the superquadric fitting.11–15The other type of method directly recovers su-perquadrics from a range image withoutpresegmentation.16–19Compared with superquadric repre-sentation of single-part object,these two types of methods can represent more complex objects and have wider appli-cations in related tasks including robotic navigation,object recognition,and virtual reality.However,existing super-quadric representation methods have several weaknesses.First,existing methods cannot handle arbitrary shapes or significant occlusions in the scene.Figure 1shows an ex-ample of the most complicated object that can be repre-sented by superquadrics appeared in the research literature.18We observe that the range image shown in Fig.1con-tains very few occlusions due to the simplicity of the ob-ject.In this case,an optimal viewpoint can easily be found from which each part of the object is visible.When an automotive part,i.e.,a complex,multipart object such as shown in Fig.2,is of interest,no existing methods can represent this object correctly because heavy occlusions are inevitable from any single viewpoint due to the complexity of the object.The second weakness of existing methods is that they utilize only single-view images.Again,for the automotive part shown in Fig.2͑a ͒,it is too difficult to find an optimal viewpoint from which all the parts are visible due to self-occlusions and occlusions,as shown in Fig.2͑b ͒.In addi-tion,the confidence of recovered superquadrics is low due to incomplete single-view data utilized and the accuracy of the recovered models highly depends on the viewpoint used to acquire the data.How complicated,multipart objects canbe represented by superquadrics with high confidence and accuracy remains unknown from the literature.In this paper,we propose an efficient strategy to repre-sent multipart objects with superquadrics.We also present a novel 3-D part decomposition algorithm based on curvature analysis to facilitate our superquadric representation strat-egy.Experiments are shown for automotive parts composed of 3-D triangulated surfaces.The remainder of this paper proceeds as follows.Section 2presents a superquadric representation approach for mul-tipart objects.Section 3proposes the 3-D part decomposi-tion algorithm for triangle meshes.The experimental results are presented in Sec.4and Sec.5concludes the paper.2Superquadric Representation of Multipart Objects Utilizing Part DecompositionA diagram for the proposed superquadric representation al-gorithm is illustrated in Fig.3.Beginning with a multipart object composed of triangle meshes,we propose a part de-composition algorithm to segment the meshes into single parts.Next,each single part is fitted with a superquadric model.Utilizing part decomposition,the difficult represen-tation problem of complicated objects is solved.We use 3-D triangulated surfaces reconstructed from multiview range images as input so that the recovered superquadrics have significantly higher confidence than those recovered from single-view images.In addition,our proposed algo-rithms are generic and flexible in the sense of triangle mesh handling ability since triangle meshes have been the stan-dard surface representation elements in many computer-related areas.A triangulation step is required only if un-structured 3-D point clouds areprovided.Fig.2Distributor cap:(a)photograph of the object,(b)rendering of 3-D triangulated surfaces scanned from view 1,and (c)rendering of 3-D triangulated surfaces scanned from view2.Fig.3Diagram of the proposed superquadric representation strategy utilizing part decomposition.Zhang,Koschan,and Abidi412/Journal of Electronic Imaging /July 2004/Vol.13(3)2.1Introduction to SuperquadricsA set of superquadrics with various shape factors is shown in Fig.4.The implicit definition of superquadrics is ex-pressed as 18F ͑x ,y ,z ͒ϭͫͩx a 1ͪ2/2ϩͩy a 2ͪ2/2ͬ2/1ϩͩz a 3ͪ2/1ϭ1,1,2͑0,2͒,͑1͒where (x ,y ,z )represents a surface point of the superquad-ric,(a 1,a 2,a 3)represent sizes in the (x ,y ,z )directions,and (1,2)represent shape factors.To represent a super-quadric model with global deformations in the world coor-dinate system,15parameters are needed.They are summa-rized as 18∧ϭ͑a 1,a 2,a 3,1,2,,,,p x ,p y ,p z ,k x ,k y ,␣,k ͒,͑2͒where the first 11parameters define a regular superquadric.Parameters k x and k y define the tapering deformations and ␣and k define the bending deformations.Most approaches define an objective function and find the superquadric pa-rameters through minimizing this objective function.The objective function used in this paper is 1G ͑∧͒ϭa 1a 2a 3͚i ϭ1N͓F 1͑x c ,y c ,z c ͒Ϫ1͔2.͑3͒The Levenberg-Marquardt method 20was implemented tominimize the objective function due to its stability and ef-ficiency.In addition,our superquadric fitting algorithm is able to recover superquadrics with global deformations from unstructured 3-D data points.3Curvature-Based 3-D Part DecompositionMany tasks in computer vision,computer graphics,and re-verse engineering involve objects or models.These tasks become extremely difficult when the object of interest is complicated,e.g.,it contains multiple parts.Part decompo-sition can simplify the original task performed on multipart objects into several subtasks each dealing with their con-stituent single,much simpler parts.21,22While a significant amount of research for part decomposition of 2-D intensity or 2.5-D range images has been conducted over the last 2decades,23–25little effort has been made on part segmenta-tion of 3-D data.26,27Therefore,a novel 3-D part decompo-sition algorithm is proposed in this paper.Figure 5illus-trates the difference between region segmentation and part decomposition.A scene consisting of a barrel on the floor is segmented into three surfaces by a region segmentation al-gorithm and two single-part objects by a part decomposi-tion algorithm.We can observe that the scene can be rep-resented by two superquadrics,which is consistent with the part decomposition result.Therefore,we conclude that part decomposition is more appropriate for high-level tasks such as volumetric primitives-based object representation and recognition.A diagram of the proposed part decomposition algorithm is shown in Fig.6.The proposed part decomposition consists of four major steps:Gaussian curvature estimation,boundary detection,region growing,and postprocessing.Boundaries between two articulated parts are composed of points with highly negative curvature according to the transversality regularity.21,22These boundaries are therefore detected by thresholding estimated curvatures for each vertex.A component-labeling operation is then performed to grow nonboundary vertices into parts.Finally,a postprocessing step is performed to assign nonlabeled vertices,including boundary vertices,to one of the parts and merge parts con-taining fewer vertices than a prespecified threshold into their neighbor parts.This part decomposition algorithm is summarized as follows.ˆAlgorithm 1…3-D part decomposition of triangle meshes …‰ˆInput:‰Triangulated surfaces.ˆStep 1.‰Compute Gaussian curvature for each vertex on the surface.ˆStep 2.‰Label vertices of highly negative curvatureasFig.4Superquadrics with various shapeparameters.Fig.6Diagram of the proposed 3-D part decomposition algorithm.Superquadric representation of automotive parts ...Journal of Electronic Imaging /July 2004/Vol.13(3)/413boundaries and the remaining vertices as seeds.ˆStep 3.‰Perform iterative region growing on each seed vertex.ˆStep 4.‰Assign nonlabeled vertices to parts and merge parts having less than a prespecified number of vertices into their neighboring parts.ˆOutput:‰Decomposed single parts.The major steps of this part decomposition algorithm are described respectively in the following sections.3.1Gaussian Curvature Estimation and BoundaryDetectionThe Gaussian curvature for each vertex on a triangulated surface is estimated by K ͑p ͒ϭ3͑2Ϫ͚i ϭ1N i ͒i ϭ1NA i␦2͑p Ϫp i ͒,͑4͒using the method proposed in Ref.28.Variable p representsthe point of interest,p i represents a neighboring vertex of the point p ,and A i represents the area of the triangle con-taining the point p .Variable i represents the interior angle of the triangle at p ,and ␦is the Dirac delta function.The triangles sharing the vertex p are illustrated in Fig.7.After Gaussian curvature is obtained for each vertex on the surface,a prespecified threshold is applied to label ver-tices as boundary or seed.Vertices of highly negative cur-vature are labeled as boundaries between two parts accord-ing to the transversality regularity,21while the rest are labeled as seeds.The threshold is critical and affects the performance of region growing.This threshold is deter-mined in a heuristic manner and depends on mesh resolu-tion.Two types of isolated vertices defined in this work according to their labels include:͑1͒a point that is labeled as boundary while all of its neighbors are labeled as seeds and ͑2͒a point that is labeled as a seed while all of its neighbors are labeled as boundary.The isolated vertices are removed by changing their labels to be the same as those of their neighbors.3.2Region Growing and PostprocessingAfter the vertices are labeled,a region-growing step is per-formed on each vertex labeled as seed.Figure 8showstriangle meshes around the point p .To illustrate the region growing process,a set of two-ring neighbor meshes around point p is shown in this figure.Region growing is performed as follows.Starting from a seed vertex p ,the region number 1is first assigned to the vertex.Second,all the neighbors p i initially labeled as seeds are then labeled with the same region number as the point p .The same labeling process is performed for each neighbor p i to label vertices p i j .This process terminates when the grown region is surrounded by boundary vertices,i.e.,the neighbors of the edge vertices of the region are all labeled as boundaries (Ϫ1).This process is repeated for every other vertex labeled as seed ͑0͒,but not for a vertex that has been grown and labeled with one of the region numbers (1,2,...,N ).After all the seed vertices are assigned new labels,a postprocessing is performed for each bound-ary vertex.Given a seed point x ,all its neighbors x i are first sorted in an ascending order based on their Euclidean distance to the point x .Next,a neighboring vertex x i ,which is the first point labeled with a region number ͑Ͼ0͒,is picked up.The boundary vertex x is then labeled the same as the vertex x i ,i.e.,the label of x is changed from Ϫ1to a region number (Ͼ0).Finally,with the exception of a few missing vertices,each vertex is labeled as 1,2,3,...,N ,the number of the parts.Missing vertices are usually located around boundaries between two articulated parts,and they are further assigned to parts during the post-processing step.Finally,a postprocessing step is performed to assign the nonlabeled vertices to parts.For example,the vertex p is an unlabeled vertex and needs further postprocessing.Assum-ing that p i (i ϭ1,2,...,N )represents a neighboring vertex of the point p ,the neighboring vertices are first selected if they have the same sign of curvature as that of the vertex p and belong to one of the segmented parts.Next,among those neighbor vertices,the vertex that has the smallest Euclidean distance to the vertex p is selected as a target vertex.For example,the vertex p 1is assumed to be the target vertex of the vertex p .Vertex p is assigned thesameFig.7Curvature estimation for the vertex p utilizing triangle meshinformation.Fig.8Region growing process for the vertex p .Zhang,Koschan,and Abidi414/Journal of Electronic Imaging /July 2004/Vol.13(3)Superquadric representation of automotive parts...Fig.5Region and part segmentation of a synthetic scene:(a)rendering of a synthetic scene consist-ing of a barrel on thefloor,(b)three segmented regions rendered in different colors,and(c)twocolors.decomposed parts rendered in differentview range images from the IVP Ranger system29and consists of37,171vertices and73,394tri-angles.The part decomposition results consist of two parts:(a)photograph of the original object,(b)rendering of the reconstructed mesh,(c)decomposed parts rendered in different colors,and(d)twocolors.recovered superquadrics rendered in differentmultiview range images from the IVP Ranger system29and consists of58,975vertices and117,036triangles.The part decomposition results consist of13parts:(a)photograph of the original object,(b)rendering of the reconstructed mesh,(c)decomposed parts rendered in different colors,and(d)colors.recovered superquadrics rendered in differentmultiview range images from the IVP Ranger system29and consists of58,784vertices and117,564triangles.The part decomposition results consist of nine parts:(a)photograph of the original object,(b)rendering of the reconstructed mesh,(c)decomposed parts labeled in different colors,and(d)recov-ered superquadrics rendered in different colors.Journal of Electronic Imaging/July2004/Vol.13(3)/415label as vertex p1,i.e.,the same segmented part.Further-more,parts composed of fewer vertices than a specified threshold are merged with adjacent regions.4Experimental ResultsExperimental results on superquadric representation for multipart,automotive objects including a disk brake,a dis-tributor cap,and a water neck are shown in this section. The meshes were reconstructed from multiview range im-ages scanned from the IVP Ranger System.29The recovered superquadrics were rendered in three dimensions using quad meshes.30Figure9shows the disk brake and its part decomposition and superquadric representation results.The reconstructed3-D triangulated surface shown in Fig.9͑b͒consists of37,171vertices and73,394triangles.Starting from this reconstructed mesh,our part decomposition algo-rithmfirst decomposed the disk brake into two single parts, as shown in Fig.9͑c͒.Each decomposed part was nextfit-ted to a superquadric model,as shown in Fig.9͑d͒.The decomposed parts and recovered superquadrics are ren-dered in different colors.We observe that our part decom-position algorithm successfully decomposed the disk brake into its constituent parts and the superquadric representa-tion strategy recovered correct superquadrics in terms of their size,shape,and pared to the original triangle mesh representation consisting of37,171vertices and73,394triangles,the recovered superquadrics describe the disk brake with only22parameters͑11parameters for each superquadric without global deformations͒.This low representation cost of superquadric representation can sig-nificantly benefit tasks including virtual reality,object rec-ognition,and robotic navigation.However,the hole at the center of the disk brake failed to be represented since su-perquadrics can only represent objects with genus of zero.19 Figure10shows the distributor cap and its part decom-position and superquadric representation results.The recon-structed mesh shown in Fig.10͑b͒consists of58,975ver-tices and117,036triangles and was decomposed into13 single parts,as shown in Fig.10͑c͒.We observe that this decomposition result is consistent with human perception. The recovered superquadrics shown in Fig.10͑d͒correctly represent the distributor cap.The recovered superquadric parameters and the ground truths for one of the small cyl-inders on top of the distributor cap are shown in Table1. We can observe that the recovered superquadric parameters for this cylinder have the correct size and shape informa-tion when compared with the ground truth parameters of the object.In addition,superquadrics represent the distribu-tor cap with only143floating numbers,while the original triangle mesh consists of58,975vertices and117,036tri-angles.Figure11shows the water neck and its part decomposi-tion and superquadric representation results.The recon-structed mesh shown in Fig.11͑b͒consists of58,784ver-tices and117,564triangles and was decomposed into nine single parts,as shown in Fig.11͑c͒.We observe that the decomposed parts are consistent with human perception. The recovered superquadrics shown in Fig.11͑d͒correctly represent the water neck.The recovered superquadric pa-rameters and the ground truths for the handle,the ball,and the small cylinder next to the handle of the water neck are shown in Table2.From this table,we observe that the recovered superquadric parameters have the correct size and shape information when compared with the ground truth parameters of the objects.Again,superquadrics repre-sent the water neck in a desirable accuracy with only99 parameters while the original triangle mesh consists of 58,784vertices and117,564triangles.5ConclusionsThis paper proposed a superquadric representation ap-proach for multipart objects.Superquadrics can represent objects in an acceptable accuracy with only a few param-eters,while other surface primitives and contours usually require thousands of representation elements.Such a com-pactness and low representation cost can significantly ben-efit tasks including virtual reality,object recognition,and robot navigation,e.g.,it enables these tasks to run in a real-time manner.The advantages of the proposed super-quadric representation approach include:͑1͒it can success-fully represent complicated,multipart objects byfirst de-composing them into single-part objects,and͑2͒the recovered superquadrics have the highest confidence and accuracy since the input we use are3-D triangulated sur-faces reconstructed from multiview range images.The in-completeness and ambiguities contained in single-view im-ages were eliminated during the multiview surface reconstruction process.We also proposed a3-D part de-composition algorithm to decompose compound objects represented by triangle meshes into their constituent single parts based on curvature analysis.Considering the fact that the triangle mesh has been a standard surface representation element in computer vision and computer graphics,the pro-posed part decomposition algorithm is generic,flexible,and can facilitate computer vision tasks such as shape descrip-tion and object recognition.Furthermore,the part decom-position algorithm can segment a large number of triangle meshes͑over100,000͒in only seconds on an SGI Octane workstation.Table1Recovered superquadric parameters and ground truths for one of the small cylinders shown in Fig.10(d)where the unit is millimeters.Parameters a1a2a312 Ground truths15.215.620.10.1 1.0 Superquadric parameters16.4515.6720.420.120.96Table2Recovered superquadric parameters and ground truths for the water neck shown in Fig.11(d)where the unit is millimeters. Object Parameters a1a2a312 Handle Ground truths39.739.417.60.1 1.0 Superquadric parameters40.2340.5866.830.130.98 Ball Ground Truths50.047.656.0 1.0 1.0 Superquadric parameters51.6247.5654.28 1.020.95 Cylinder Ground truths16.517.844.20.1 1.0 Superquadric parameters17.5617.9443.380.110.95Zhang,Koschan,and Abidi 416/Journal of Electronic Imaging/July2004/Vol.13(3)AcknowledgmentsThis work was supported by the University Research Pro-gram in Robotics under Grant No.DOE-DE-FG02-86NE37968,by the Department of Defense/U.S.Army Tank-automotive and Armaments Command/National Au-tomotive Center/Automotive Research Center Program R01-1344-18,and by the Federal Aviation Administration National Safe Skies Alliance Program R01-1344-48/49. 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multi-view range data,’’in puter Vision and Pattern Recognition,pp.159–166͑1994͒.15.H.Zha,T.Hoshide,and T.Hasegawa,‘‘A recursivefitting-and-splitting algorithm for3D object modeling using superquadrics,’’in Proc.14th Int.Conf.Pattern Recognition,pp.658–662͑1998͒. 16.Y.Hu and W.Wee,‘‘Robust3-D part extraction from range imageswith deformable superquadric models,’’in Signal Processing,Sensor Fusion and Target Recognition IV,Proc.SPIE2484,524–535͑1995͒.17.A.Leonardis,A.Jaklic,and F.Solina,‘‘Superquadrics for segmenta-tion and modeling range data,’’IEEE Trans.Pattern Anal.Mach.Intell.19,1289–1295͑1997͒.18.A.Jaklic,A.Leonardis,and F.Solina,Segmentation and Recovery ofSuperquadrics,Kluwer Academic Publishers,Boston͑2000͒.19.A.H.Barr,‘‘Superquadrics and angle-preserving transformation,’’IEEE put.Graph.1,11–23͑1981͒.20.W.Press,W.Vetterling,S.Teukolsky,and B.Flannery,NumericalRecipes in C:The Art of Scientific Computing,Cambridge Press,New York͑1992͒.21.D.Hoffman and W.Richards,‘‘Parts of recognition,’’Cognition18,65–96͑1984͒.22.A.Pentland,‘‘Part segmentation for object recognition,’’Neural Com-put.1,82–91͑1981͒.23.H.Rom and G.Medioni,‘‘Part decomposition and description of3Dshapes,’’in Proc.Int.Conf.Pattern Recognition,pp.629–632͑1994͒.24.M.Bennamoun,‘‘A contour-based part segmentation algorithm,’’inProc.Int.Conf.Acoustics,Speech,and Signal Processing,V ol.5,pp.41–44͑1994͒.25.K.Koara,A.Nishikawa,and O.Miyazaki,‘‘Hierarchical part decom-position method of articulated body contour,’’in Proc.Int.Conf.In-telligent Robots and Systems,V ol.3,pp.2055–2060͑2000͒.26.K.Wu and M.D.Levine,‘‘3D part segmentation using simulatedelectrical charge distributions,’’IEEE Trans.Pattern Anal.Mach.In-tell.19,1223–1235͑1997͒.27.A.Mangan and R.Whitaker,‘‘Partitioning3D surface meshes usingwatershed segmentation,’’IEEE put.Graph.5͑4͒, 308–321͑1999͒.28.C.Lin and M.Perry,‘‘Shape description using surface triangulation,’’in puter Vision:Representation and Control,pp.38–43͑1982͒.er Documentation:MAPP2500Ranger PCI System,Version1.6,Integrated Vision Products,Sweden͑2000͒.30.J.Wernecke,The Inventor Mentor:Programming Object-oriented3DGraphics with Open Inventor,Addison-Wesley,Reading,MA͑1994͒.Yan Zhang received her BS and MS de-grees in electrical engineering from Hua-zhong University of Science and Technol-ogy,China,in1994and1997,respectively,and her PhD degree in electrical engineer-ing from the University of Tennessee,Knoxville,in2003.Her research interestsinclude3-D image processing,computervision,and patternrecognition.Andreas Koschan received his MS de-gree in computer science and his PhD incomputer engineering from the TechnicalUniversity Berlin,Germany,in1985and1991,respectively.He is currently a re-search associate professor with the De-partment of Electrical and Computer Engi-neering,the University of Tennessee,Knoxville.His work has primarily focusedon color image processing and3-D com-puter vision including stereo vision and la-ser rangefinding techniques.He is a coauthor of two textbooks on 3-D image processing and a member of IS&T andIEEE.Mongi A.Abidi is a W.Fulton Professorwith the Department of Electrical and Com-puter Engineering,the University of Ten-nessee,Knoxville,which he joined in1986.Dr.Abidi received his MS and PhD degreesin electrical engineering in1985and1987,both from the University of Tennessee,Knoxville.His interests include image pro-cessing,multisensor processing,3-D imag-ing,and robotics.He has published over120papers in these areas and coedited the book Data Fusion in Robotics and Machine Intelligence(Academic Press,1992).He is the recipient of the1994to1995Chancellor’s Award for Excellence in Research and Creative Achievement and the2001Brooks Distinguished Professor Award.He is a member of the IEEE,the Computer Society,the Pattern Recognition Society, SPIE and the Tau Beta Pi,Phi Kappa Phi,and Eta Kappa Nu honor societies.He also received the First Presidential Principal Engineer Award prior to joining the University of Tennessee.Superquadric representation of automotive parts...Journal of Electronic Imaging/July2004/Vol.13(3)/417。
H--ggg(gqqbar) at Two Loops in the Large-M_t Limit
a r X i v :h e p -p h /9707448v 1 24 J u l 1997MSUHEP-70723July 1997hep-ph/9707448H →ggg (gq ¯q )at Two Loops in the Large-M t LimitCarl R.SchmidtDepartment of Physics and AstronomyMichigan State University East Lansing,MI 48824,USAAbstractWe present a calculation of the two-loop helicity amplitudes for the processes H →ggg and H →gq ¯q in the large-M t limit.In this limit the calculation can be performed in terms of one-loop diagrams containing an effective Hgg operator.These amplitudes are required for the next-to-leading order (NLO)corrections to the Higgs transverse momentum distribution and the next-to-next-to-leading order (NNLO)corrections to the Higgs production cross section via the gluon fusion mechanism.1IntroductionThe Higgs Boson H is the only particle of the Standard Model remaining to be discovered.Its role is to provide a simple mechanism to break the electroweak gauge symmetry and to give mass to the weak gauge bosons and the fermions. Of course,it is possible that nature uses more than a single scalar boson for this purpose,but still the Standard Model and its supersymmetric extension are the primary examples of the class of symmetry-breaking models which interact weakly. Therefore,the search for the Higgs boson is of the highest priority for the Large Hadron Collider(LHC)at CERN.The detection of the Higgs Boson above background at the LHC will be a chal-lenging task.In particular,if the mass of the Higgs is below∼140GeV,near the threshold for decay into W boson pairs,the detection of the Higgs is quite sub-tle.Although the largest production mechanism by far is gluon-gluon fusion,the equally large backgrounds require the use of the H→γγdecay channel,which has a branching ratio of O(10−3).To prepare for the search,we need the best theoret-ical predictions possible,and this means the inclusion of quantum chromodynamic (QCD)corrections to Higgs production and decay.Recent relevant reviews are given in reference[1].The Higgs production via gluon-gluon fusion proceeds at lowest order(LO) through a quark loop.This loop is dominated by the top quark,because the Higgs coupling is proportional to the quark mass.The two-loop next-to-leading order (NLO)QCD corrections have also been calculated[2],and they are quite large:1∼50−100%.An interesting feature of this NLO calculation is that it becomes much simpler in the limit of large top quark mass(M t→∞).In this limit,one can integrate out the heavy top quark loop,leaving behind an effective gauge-invariant Hgg coupling.Thus,the number of loops at each order is reduced byone.It has been shown that the NLO corrections in this large-M t limit give agood approximation to the complete two-loop result over a large range of Higgsmasses[3].The large NLO correction suggests that even higher orders still maybe important.A soft-gluon resummation in the large-M t limit has recently beenperformed,which gives an estimate of the next-to-next-to-leading order(NNLO)corrections[4].Meanwhile,other groups have considered less inclusive quantities,such as thetransverse momentum spectrum of the Higgs boson.This observable has beenconsidered at the one-loop Born level,both in the large-top-mass limit and withfull M t dependence[5].In addition,the effects of soft gluons have also been studied,which modify the spectrum at small Higgs p⊥[6].However,a NLO calculation hasnot been done.The realfive-point H→gggg amplitudes which are needed havebeen calculated by Dawson and Kauffman[7]in the large-M t limit,and recentlyKauffman et al.[8]have calculated thefive-point amplitudes with light externalquarks.In this paper we present the virtual corrections to the four-point H→ggg(gq¯q)amplitudes in the large-M t limit,which completes the set of amplitudes neededto study the Higgs p⊥spectrum at NLO.In this limit,the two-loop results canbe computed from effective one-loop diagrams.The large-M t approximation to2the Higgs p⊥spectrum will be good for some range of M H and Higgs p⊥,and furthermore this calculation offers a check of the complete M t-dependent result, when it should become available.Moreover,these amplitudes are necessary for a full NNLO calculation of the cross section in the large-M t limit[9].In addition to using the effective Higgs-gluon operator in the large-M t limit,we have also used several other techniques that have been found convenient in QCD loop calculations[10].These include the use of helicity spinors,color ordering,and background-field gauge.In section2we discuss the details of the calculation,while in section3we present the amplitudes and discuss various cross-checks.In section 4we summarize our conclusions.2Calculational DetailsIn the large-M t limit the top quark can be removed from the full theory,leaving a residual Higgs-gluon coupling term in the lagrangian of the effective theory:L eff=−13πH.(2)4πFollowing Mangano and Parke[11],we use the unconventional normalization for√the SU(3)generator matrices Tr(T a T b)=δab and[T a,T b]=i2from the helicity amplitudes below.3As suggested by string theory methods [10],we use the background-field gauge to calculate the one-particle irreducible parts of the Feynman diagrams.The gluon field G µis split into a background component B µand a quantum component Q µ,i.e,G µ=B µ+Q µ.Two reasonable choices for the gauge-fixing term areL (1)gf=−121−αsv(1+∆)(D B µQ µ)2(3)whereD B µQµ=∂µQ µ−(ig/√3πvg sbe written[12].(5)ǫ±(p)µ=± p±|γµ|q±2 q∓|p±The arbitrary reference vector q satisfies q2=0and q·p=0.A change in the reference vector just shiftsǫ(p)µby a term proportional to pµ,which drops out of the gauge-invariant helicity amplitude.There are two independent Hggg helicity subamplitudes,which at tree level arem0(1+,2+,3+)=−M4H.(6)[12][23][31]Here we have used the notation ij = p i−|p j+ and[ij]= p i+|p j− .These spinor products are antisymmetric and satisfy ij [ji]=2p i·p j≡S ij.All other subamplitudes can be obtained by invariance under cyclic permutations,charge conjugation,and parity.We also consider the process0→Hgq¯q with p H+p+q+¯q=0.The amplitude for a gluon and quarks with helicitiesλ,h,¯h and colors a,i,¯ıcan be writtenM=−αs2T a i¯ım(p,λ;q,h;¯q,¯h).(7)At tree level we have[p¯q]2m0(g+,q−,¯q+)=3The Helicity AmplitudesFigures 1and 2show representative box diagrams for the Hggg and Hgq ¯q ampli-tudes,respectively.The Feynman diagrams have been evaluated with the aid of the symbolic manipulation program MAPLE using the straightforward Passarino-Veltman reduction.For the sake of generality,we have regularized the loop in-tegrals by continuing the loop momenta to (4−2ǫ)dimensions,while taking the number of helicity states of the internal gluons to be (4−2δR ǫ).Thus,δR =1corresponds to the t’Hooft-Veltman scheme [13]and δR =0corresponds to the four-dimensional-helicity scheme [14].For the Hggg amplitudes we obtain m 1(1+,2+,3+)=m 0(1+,2+,3+)αs −M 2HǫN c U+1M 4Hm 1(1−,2+,3+)=m 0(1−,2+,3+)αs −M 2HǫN c U(9)+1S 223,where the prefactor isr Γ=Γ(1+ǫ)Γ2(1−ǫ)ǫ2−−M 2H−S 23ǫ−−M 2H2−ln−S 12−M 2H−ln−S 12−M 2H−ln−S 23−M 2H6−2Li2 1−S12M2H −2Li2 1−S314πrΓ 4πµ2N c V2+n f V3 ,(12) withV1=1−S gqǫ− −M2H6ǫ −M2H−M2Hln −S q¯q−M2Hln −S q¯qM2H −Li21−S gq M2H (13) +836+π22S q¯qǫ2+3−S q¯qǫ+ln −S gq−M2H+Li2 1−S gq M2H (14) +72−π22S q¯q3ǫ −M2H9.(15)In these expressions,N c=3is the number of colors,n f is the number of light fermions,and Li2is the dilogarithm function.The amplitudes are written for S ij<0and M2H<0,but they can be analytically continued to the physical region by letting−S ij=−S ij−iηand−M2H=−M2H−iηforη→0+.Note that these amplitudes are ultraviolet-unrenormalized amplitudes.Including the7renormalization gives the modificationm1→m1+(∆+3δg)m0,(16) where∆is thefinite renormalization of the effective Hgg operator,given in eq.(2) andδg is the gauge-coupling ing theǫαs6−n fcalculation of the Higgs p⊥spectrum in the large-M t limit,which is currently in progress[18].In addition,they are part of the set needed for a complete NNLO calculation of the total cross-section in this limit.AcknowledgementsWe would like to thank Vittorio Del Duca,Lance Dixon,Chris Glosser,and Jim Amundson for useful discussions.References[1]M.Spira,preprint CERN-TH/97-68,hep-ph/9705337(1997);Z.Kunszt,S.Moretti,and W.J.Stirling,Z.Phys.C74,479(1997);S.Dawson,preprint BNL-HET-SD-97-004,hep-ph/9703387(1997).[2]D.Graudenz,M.Spira,and P.Zerwas,Phys.Rev.Lett.70,1372(1993);M.Spira,A.Djouadi,D.Graudenz,and P.Zerwas,Nucl.Phys.B453,17 (1995).[3]S.Dawson,Nucl.Phys.B359,283(1991);A.Djouadi,M.Spira,and P.Zerwas,Phys.Lett.B264,440(1991).[4]M.Kr¨a mer, enen,and M.Spira,preprint CERN-TH/96-231,hep-ph/9611272(1997).9[5]R.K.Ellis,I.Hinchliffe,M.Soldate,and J.J.Van Der Bij,Nucl.Phys.B297,221(1988);U.Baur and E.Glover,Nucl.Phys.B339,38(1990).[6]I.Hinchliffe and S.Novaes,Phys.Rev.D38,3475(1988);R.Kauffman,Phys.Rev.D44,1415(1991);Phys.Rev.D45,1512(1992);C.-P.Yuan,Phys.Lett.B283,395(1992).[7]S.Dawson and R.Kauffman,Phys.Rev.Lett.68,2273(1992).[8]R.Kauffman,S.Desai,and D.Risal,Phys.Rev.D55,4005(1997).[9]A recent NNLO calculation in the large-M t limit of the crossed process,H→gg,has been done by K.Chetyrkin,B.Kniehl,and M.Steinhauser, Phys.Rev.Lett.79,353(1997).[10]For a review,see Z.Bern,L.Dixon,and D.Kosower,Ann.Rev.Nucl.Part.Sci.46,109(1996).[11]M.Mangano and S.Parke,Phys.Rep.200,301(1991).[12]Z.Xu,D.Zhang,and L.Chang,Nucl.Phys.B291,392(1987);R.Kleiss and W.Stirling,Nucl.Phys.B262,235(1985).[13]G.t’Hooft and M.Veltman,Nucl.Phys.B63,277(1973).[14]Z.Bern and D.Kosower,Phys.Rev.Lett.66,1669(1991);Z.Bern and D.Kosower,Nucl.Phys.B379,451(1992).10[15]S.Catani and M.Seymour,Nucl.Phys.B485,291(1997).[16]Z.Kunszt,A.Signer,and Z.Tr´o cs´a nyi,Nucl.Phys.B411,397(1994);S.Catani,M.Seymour,and Z.Tr´o cs´a nyi,Phys.Rev.D55,6819(1997).[17]Z.Bern,L.Dixon,D.Dunbar,and D.Kosower,Nucl.Phys.B425,217(1994).[18]C.Glosser and C.Schmidt,(to be published).11Fig.1:Box diagram for the Hggg amplitude.The dot represents the effective Hgg vertex in the large-M t limit.Fig.2:Box diagrams for the Hgq¯q amplitude.The dot represents the effective Hgg vertex in the large-M t limit.12。
Particle Codes(粒子的编码)
Particle Codes
The Particle Data Group particle code [PDG88,PDG92,PDG00] is used consistently throughout the program. Almost all known discrepancies between earlier versions of the PDG standard and the PYTHIA usage have now been resolved. The one known exception is the (very uncertain) classification of , with also affected as a consequence. There is also a possible point of confusion in the technicolor sector between and . The latter is retained for historical reasons, whereas the
37. Gauge bosons and other fundamental bosons, Table [*] . This group includes all the gauge and Higgs bosons of the Standard Model, as well as some of the bosons appearing in various extensions of it. They correspond to one extra U(1) and one extra SU(2) group, a further Higgs doublet, a graviton, a horizontal gauge boson (coupling between
GDS-1052U示波器使用说明书
产品介绍 .................................................. 12
主要特点 ......................................................................... 12 面板介绍 ......................................................................... 14
测量 ........................................................ 43
基本测量 ........................................................................ 43
激活通道........................................................................... 43 使用自动设置 ...................................................................44 运行和停止触发 ...............................................................45 改变水平位置和档位 ....................................................... 46 改变垂直位置和档位 ........................................................ 47 使用探棒补偿信号............................................................48
N=4 Super-Yang-Mills Theory, QCD and Collider Physics
a rXiv:h ep-th/04121v12O ct24SLAC–PUB–10739,IPPP/04/59,DCPT/04/118,UCLA/04/TEP/40Saclay/SPhT–T04/116,hep-th/0410021October,2004N =4Super-Yang-Mills Theory,QCD and Collider Physics Z.Bern a L.J.Dixon b 1 D.A.Kosower c a Department of Physics &Astronomy,UCLA,Los Angeles,CA 90095-1547,USA b SLAC,Stanford University,Stanford,CA 94309,USA,and IPPP,University of Durham,Durham DH13LE,England c Service de Physique Th´e orique,CEA–Saclay,F-91191Gif-sur-Yvette cedex,France1Introduction and Collider Physics MotivationMaximally supersymmetric (N =4)Yang-Mills theory (MSYM)is unique in many ways.Its properties are uniquely specified by the gauge group,say SU(N c ),and the value of the gauge coupling g .It is conformally invariant for any value of g .Although gravity is not present in its usual formulation,MSYMis connected to gravity and string theory through the AdS/CFT correspon-dence[1].Because this correspondence is a weak-strong coupling duality,it is difficult to verify quantitatively for general observables.On the other hand, such checks are possible and have been remarkably successful for quantities protected by supersymmetry such as BPS operators[2],or when an additional expansion parameter is available,such as the number offields in sequences of composite,large R-charge operators[3,4,5,6,7,8].It is interesting to study even more observables in perturbative MSYM,in order to see how the simplicity of the strong coupling limit is reflected in the structure of the weak coupling expansion.The strong coupling limit should be even simpler when the large-N c limit is taken simultaneously,as it corresponds to a weakly-coupled supergravity theory in a background with a large radius of curvature.There are different ways to study perturbative MSYM.One approach is via computation of the anomalous dimensions of composite,gauge invariant operators[1,3,4,5,6,7,8].Another possibility[9],discussed here,is to study the scattering amplitudes for(regulated)plane-wave elementaryfield excitations such as gluons and gluinos.One of the virtues of the latter approach is that perturbative MSYM scat-tering amplitudes share many qualitative properties with QCD amplitudes in the regime probed at high-energy colliders.Yet the results and the computa-tions(when organized in the right way)are typically significantly simpler.In this way,MSYM serves as a testing ground for many aspects of perturbative QCD.MSYM loop amplitudes can be considered as components of QCD loop amplitudes.Depending on one’s point of view,they can be considered either “the simplest pieces”(in terms of the rank of the loop momentum tensors in the numerator of the amplitude)[10,11],or“the most complicated pieces”in terms of the degree of transcendentality(see section6)of the special functions entering thefinal results[12].As discussed in section6,the latter interpreta-tion links recent three-loop anomalous dimension results in QCD[13]to those in the spin-chain approach to MSYM[5].The most direct experimental probes of short-distance physics are collider experiments at the energy frontier.For the next decade,that frontier is at hadron colliders—Run II of the Fermilab Tevatron now,followed by startup of the CERN Large Hadron Collider in2007.New physics at colliders always contends with Standard Model backgrounds.At hadron colliders,all physics processes—signals and backgrounds—are inherently QCD processes.Hence it is important to be able to predict them theoretically as precisely as possi-ble.The cross section for a“hard,”or short-distance-dominated processes,can be factorized[14]into a partonic cross section,which can be computed order by order in perturbative QCD,convoluted with nonperturbative but measur-able parton distribution functions(pdfs).For example,the cross section for producing a pair of jets(plus anything else)in a p¯p collision is given byσp¯p→jjX(s)= a,b1 0dx1dx2f a(x1;µF)¯f b(x2;µF)׈σab→jjX(sx1x2;µF,µR;αs(µR)),(1)where s is the squared center-of-mass energy,x1,2are the longitudinal(light-cone)fractions of the p,¯p momentum carried by partons a,b,which may be quarks,anti-quarks or gluons.The experimental definition of a jet is an in-volved one which need not concern us here.The pdf f a(x,µF)gives the prob-ability forfinding parton a with momentum fraction x inside the proton; similarly¯f b is the probability forfinding parton b in the antiproton.The pdfs depend logarithmically on the factorization scaleµF,or transverse resolution with which the proton is examined.The Mellin moments of f a(x,µF)are for-ward matrix elements of leading-twist operators in the proton,renormalized at the scaleµF.The quark distribution function q(x,µ),for example,obeys 10dx x j q(x,µ)= p|[¯qγ+∂j+q](µ)|p .2Ingredients for a NNLO CalculationMany hadron collider measurements can benefit from predictions that are accurate to next-to-next-to-leading order(NNLO)in QCD.Three separate ingredients enter such an NNLO computation;only the third depends on the process:(1)The experimental value of the QCD couplingαs(µR)must be determinedat one value of the renormalization scaleµR(for example m Z),and its evolution inµR computed using the3-loopβ-function,which has been known since1980[15].(2)The experimental values for the pdfs f a(x,µF)must be determined,ide-ally using predictions at the NNLO level,as are available for deep-inelastic scattering[16]and more recently Drell-Yan production[17].The evolu-tion of pdfs inµF to NNLO accuracy has very recently been completed, after a multi-year effort by Moch,Vermaseren and Vogt[13](previously, approximations to the NNLO kernel were available[18]).(3)The NNLO terms in the expansion of the partonic cross sections must becomputed for the hadronic process in question.For example,the parton cross sections for jet production has the expansion,ˆσab→jjX=α2s(A+αs B+α2s C+...).(2)The quantities A and B have been known for over a decade[19],but C has not yet been computed.Figure 1.LHC Z production [22].•real ×real:וvirtual ×real:וvirtual ×virtual:וdoubly-virtual ×real:×Figure 2.Purely gluonic contributionsto ˆσgg →jjX at NNLO.Indeed,the NNLO terms are unknown for all but a handful of collider puting a wide range of processes at NNLO is the goal of a large amount of recent effort in perturbative QCD [20].As an example of the im-proved precision that could result from this program,consider the production of a virtual photon,W or Z boson via the Drell-Yan process at the Tevatron or LHC.The total cross section for this process was first computed at NNLO in 1991[21].Last year,the rapidity distribution of the vector boson also be-came available at this order [17,22],as shown in fig.1.The rapidity is defined in terms of the energy E and longitudinal momentum p z of the vector boson in the center-of-mass frame,Y ≡1E −p z .It determines where the vector boson decays within the detector,or outside its acceptance.The rapidity is sensitive to the x values of the incoming partons.At leading order in QCD,x 1=e Y m V /√s ,where m V is the vector boson mass.The LHC will produce roughly 100million W s and 10million Z s per year in detectable (leptonic)decay modes.LHC experiments will be able to map out the curve in fig.1with exquisite precision,and use it to constrain the parton distributions —in the same detectors that are being used to search for new physics in other channels,often with similar q ¯q initial states.By taking ratios of the other processes to the “calibration”processes of single W and Z production,many experimental uncertainties,including those associated with the initial state parton distributions,drop out.Thus fig.1plays a role as a “partonic luminosity monitor”[23].To get the full benefit of the remarkable experimental precision,though,the theory uncertainty must approach the 1%level.As seen from the uncertainty bands in the figure,this precision is only achievable at NNLO.The bands are estimated by varying the arbitrary renormalization and factorization scales µR and µF (set to a common value µ)from m V /2to 2m V .A computation to all orders in αs would have no dependence on µ.Hence the µ-dependence of a fixed order computation is related to the size of the missing higher-order terms in the series.Althoughsub-1%uncertainties may be special to W and Z production at the LHC, similar qualitative improvements in precision will be achieved for many other processes,such as di-jet production,as the NNLO terms are completed.Even within the NNLO terms in the partonic cross section,there are several types of ingredients.This feature is illustrated infig.2for the purely gluonic contributions to di-jet production,ˆσgg→jjX.In thefigure,individual Feynman graphs stand for full amplitudes interfered(×)with other amplitudes,in order to produce contributions to a cross section.There may be2,3,or4partons in thefinal state.Just as in QED it is impossible to define an outgoing electron with no accompanying cloud of soft photons,also in QCD sensible observables require sums overfinal states with different numbers of partons.Jets,for example,are defined by a certain amount of energy into a certain conical region.At leading order,that energy typically comes from a single parton, but at NLO there may be two partons,and at NNLO three partons,within the jet cone.Each line infig.2results in a cross-section contribution containing severe infrared divergences,which are traditionally regulated by dimensional regula-tion with D=4−2ǫ.Note that this regulation breaks the classical conformal invariance of QCD,and the classical and quantum conformal invariance of N=4super-Yang-Mills theory.Each contribution contains poles inǫranging from1/ǫ4to1/ǫ.The poles in the real contributions come from regions ofphase-space where the emitted gluons are soft and/or collinear.The poles in the virtual contributions come from similar regions of virtual loop integra-tion.The virtual×real contribution obviously has a mixture of the two.The Kinoshita-Lee-Nauenberg theorem[24]guarantees that the poles all cancel in the sum,for properly-defined,short-distance observables,after renormal-izing the coupling constant and removing initial-state collinear singularities associated with renormalization of the pdfs.A critical ingredient in any NNLO prediction is the set of two-loop ampli-tudes,which enter the doubly-virtual×real interference infig.2.Such ampli-tudes require dimensionally-regulated all-massless two-loop integrals depend-ing on at least one dimensionless ratio,which were only computed beginning in 1999[25,26,27].They also receive contributions from many Feynman diagrams, with lots of gauge-dependent cancellations between them.It is of interest to develop more efficient,manifestly gauge-invariant methods for combining di-agrams,such as the unitarity or cut-based method successfully applied at one loop[10]and in the initial two-loop computations[28].i,ij+ i iFigure3.Illustration of soft-collinear(left)and pure-collinear(right)one-loop di-vergences.3N=4Super-Yang-Mills Theory as a Testing Ground for QCDN=4super-Yang-Mills theory serves an excellent testing ground for pertur-bative QCD methods.For n-gluon scattering at tree level,the two theories in fact give identical predictions.(The extra fermions and scalars of MSYM can only be produced in pairs;hence they only appear in an n-gluon ampli-tude at loop level.)Therefore any consequence of N=4supersymmetry,such as Ward identities among scattering amplitudes[29],automatically applies to tree-level gluonic scattering in QCD[30].Similarly,at tree level Witten’s topological string[31]produces MSYM,but implies twistor-space localization properties for QCD tree amplitudes.(Amplitudes with quarks can be related to supersymmetric amplitudes with gluinos using simple color manipulations.)3.1Pole Structure at One and Two LoopsAt the loop-level,MSYM becomes progressively more removed from QCD. However,it can still illuminate general properties of scattering amplitudes,in a calculationally simpler arena.Consider the infrared singularities of one-loop massless gauge theory amplitudes.In dimensional regularization,the leading singularity is1/ǫ2,arising from virtual gluons which are both soft and collinear with respect to a second gluon or another massless particle.It can be char-acterized by attaching a gluon to any pair of external legs of the tree-level amplitude,as in the left graph infig.3.Up to color factors,this leading diver-gence is the same for MSYM and QCD.There are also purely collinear terms associated with individual external lines,as shown in the right graph infig.3. The pure-collinear terms have a simpler form than the soft terms,because there is less tangling of color indices,but they do differ from theory to theory.The full result for one-loop divergences can be expressed as an operator I(1)(ǫ) which acts on the color indices of the tree amplitude[32].Treating the L-loop amplitude as a vector in color space,|A(L)n ,the one-loop result is|A(1)n =I(1)(ǫ)|A(0)n +|A(1),finn ,(3)where |A (1),fin nis finite as ǫ→0,and I (1)(ǫ)=1Γ(1−ǫ)n i =1n j =i T i ·T j 1T 2i 1−s ij ǫ,(4)where γis Euler’s constant and s ij =(k i +k j )2is a Mandelstam invariant.The color operator T i ·T j =T a i T a j and factor of (µ2R /(−s ij ))ǫarise from softgluons exchanged between legs i and j ,as in the left graph in fig.3.The pure 1/ǫpoles terms proportional to γi have been written in a symmetric fashion,which slightly obscures the fact that the color structure is actually simpler.We can use the equation which represents color conservation in the color-space notation, n j =1T j =0,to simplify the result.At order 1/ǫwe may neglect the (µ2R /(−s ij ))ǫfactor in the γi terms,and we have n j =i T i ·T j γi /T 2i =−γi .So the color structure of the pure 1/ǫterm is actually trivial.For an n -gluon amplitude,the factor γi is set equal to its value for gluons,which turns out to be γg =b 0,the one-loop coefficient in the β-function.Hence the pure-collinear contribution vanishes for MSYM,but not for QCD.The divergences of two-loop amplitudes can be described in the same for-malism [32].The relation to soft-collinear factorization has been made more transparent by Sterman and Tejeda-Yeomans,who also predicted the three-loop behavior [33].Decompose the two-loop amplitude |A (2)n as|A (2)n =I (2)(ǫ)|A (0)n +I (1)(ǫ)|A (1)n +|A (2),fin n,(5)where |A (2),fin n is finite as ǫ→0and I (2)(ǫ)=−1ǫ+e −ǫγΓ(1−2ǫ)ǫ+K I (1)(2ǫ)+e ǫγT 2i µ22C 2A ,(8)where C A =N c is the adjoint Casimir value.The quantity ˆH(2)has non-trivial,but purely subleading-in-N c ,color structure.It is associated with soft,rather than collinear,momenta [37,33],so it is theory-independent,up to color factors.An ansatz for it for general n has been presented recently [38].3.2Recycling Cuts in MSYMAn efficient way to compute loop amplitudes,particularly in theories with a great deal of supersymmetry,is to use unitarity and reconstruct the am-plitude from its cuts [10,38].For the four-gluon amplitude in MSYM,the two-loop structure,and much of the higher-loop structure,follows from a sim-ple property of the one-loop two-particle cut in this theory.For simplicity,we strip the color indices offof the four-point amplitude A (0)4,by decomposing it into color-ordered amplitudes A (0)4,whose coefficients are traces of SU(N c )generator matrices (Chan-Paton factors),A (0)4(k 1,a 1;k 2,a 2;k 3,a 3;k 4,a 4)=g 2 ρ∈S 4/Z 4Tr(T a ρ(1)T a ρ(2)T a ρ(3)T a ρ(4))×A (0)4(k ρ(1),k ρ(2),k ρ(3),k ρ(4)).(9)The two-particle cut can be written as a product of two four-point color-ordered amplitudes,summed over the pair of intermediate N =4states S,S ′crossing the cut,which evaluates toS,S ′∈N =4A (0)4(k 1,k 2,ℓS ,−ℓ′S ′)×A (0)4(ℓ′S ′,−ℓS ,k 3,k 4)=is 12s 23A (0)4(k 1,k 2,k 3,k 4)×1(ℓ−k 3)2,(10)where ℓ′=ℓ−k 1−k 2.This equation is also shown in fig.4.The scalar propagator factors in eq.(10)are depicted as solid vertical lines in the figure.The dashed line indicates the cut.Thus the cut reduces to the cut of a scalar box integral,defined byI D =4−2ǫ4≡ d 4−2ǫℓℓ2(ℓ−k 1)2(ℓ−k 1−k 2)2(ℓ+k 4)2.(11)One of the virtues of eq.(10)is that it is valid for arbitrary external states in the N =4multiplet,although only external gluons are shown in fig.4.Therefore it can be re-used at higher loop order,for example by attaching yet another tree to the left.N =41234=i s 12s 231234Figure 4.The one-loop two-particle cuts for the four-point amplitude in MSYM reduce to the tree amplitude multiplied by a cut scalar box integral (for any set of four external states).i 2s 12s121234+s 121234+perms Figure 5.The two-loop gg →gg amplitude in MSYM [11,39].The blob on theright represents the color-ordered tree amplitude A (0)4.(The quantity s 12s 23A (0)4transforms symmetrically under gluon interchange.)In the the brackets,black linesare kinematic 1/p 2propagators,with scalar (φ3)vertices.Green lines are color δab propagators,with structure constant (f abc )vertices.The permutation sum is over the three cyclic permutations of legs 2,3,4,and makes the amplitude Bose symmetric.At two loops,the simplicity of eq.(10)made it possible to compute the two-loop gg →gg scattering amplitude in that theory (in terms of specific loop integrals)in 1997[11],four years before the analogous computations in QCD [36,37].All of the loop momenta in the numerators of the Feynman di-agrams can be factored out,and only two independent loop integrals appear,the planar and nonplanar scalar double box integrals.The result can be writ-ten in an appealing diagrammatic form,fig.5,where the color algebra has the same form as the kinematics of the loop integrals [39].At higher loops,eq.(10)leads to a “rung rule”[11]for generating a class of (L +1)-loop contributions from L -loop contributions.The rule states that one can insert into a L -loop contribution a rung,i.e.a scalar propagator,transverse to two parallel lines carrying momentum ℓ1+ℓ2,along with a factor of i (ℓ1+ℓ2)2in the numerator,as shown in fiing this rule,one can construct recursively the external and loop-momentum-containing numerators factors associated with every φ3-type diagram that can be reduced to trees by a sequence of two-particle cuts,such as the diagram in fig.7a .Such diagrams can be termed “iterated 2-particle cut-constructible,”although a more compact notation might be ‘Mondrian’diagrams,given their resemblance to Mondrian’s paintings.Not all diagrams can be computed in this way.The diagram in fig.7b is not in the ‘Mondrian’class,so it cannot be determined from two-particle cuts.Instead,evaluation of the three-particle cuts shows that it appears with a non-vanishing coefficient in the subleading-color contributions to the three-loop MSYM amplitude.ℓ1ℓ2−→i (ℓ1+ℓ2)2ℓ1ℓ2Figure 6.The rung rule for MSYM.(a)(b)Figure 7.(a)Example of a ‘Mondrian’diagram which can be determined re-cursively from the rung rule.(b)Thefirst non-vanishing,non-Mondrian dia-grams appear at three loops in nonplanar,subleading-color contributions.4Iterative Relation in N =4Super-Yang-Mills TheoryAlthough the two-loop gg →gg amplitude in MSYM was expressed in terms of scalar integrals in 1997[11],and the integrals themselves were computed as a Laurent expansion about D =4in 1999[25,26],the expansion of the N =4amplitude was not inspected until last fall [9],considerably after similar investigations for QCD and N =1super-Yang-Mills theory [36,37].It was found to have a quite interesting “iterative”relation,when expressed in terms of the one-loop amplitude and its square.At leading color,the L -loop gg →gg amplitude has the same single-trace color decomposition as the tree amplitude,eq.(9).Let M (L )4be the ratio of this leading-color,color-ordered amplitude to the corresponding tree amplitude,omitting also several conventional factors,A (L ),N =4planar 4= 2e −ǫγg 2N c2 M (1)4(ǫ) 2+f (ǫ)M (1)4(2ǫ)−12(ζ2)2is replaced by approximately sixpages of formulas (!),including a plethora of polylogarithms,logarithms and=+f(ǫ)−12(ζ2)2+O(ǫ)f(ǫ)=−(ζ2+ǫζ3+ǫ2ζ4+...)Figure8.Schematic depiction of the iterative relation(13)between two-loop and one-loop MSYM amplitudes.polynomials in ratios of invariants s/t,s/u and t/u[37].The polylogarithm is defined byLi m(x)=∞i=1x i t Li m−1(t),Li1(x)=−ln(1−x).(14)It appears with degree m up to4at thefinite,orderǫ0,level;and up to degree4−i in the O(ǫ−i)terms.In the case of MSYM,identities relating these polylogarithms are needed to establish eq.(13).Although the O(ǫ0)term in eq.(13)is miraculously simple,as noted above the behavior of the pole terms is not a miracle.It is dictated in general terms by the cancellation of infrared divergences between virtual corrections and real emission[24].Roughly speaking,for this cancellation to take place,the virtual terms must resemble lower-loop amplitudes,and the real terms must resemble lower-point amplitudes,in the soft and collinear regions of loop or phase-space integration.At the level of thefinite terms,the iterative relation(13)can be understood in the Regge/BFKL limit where s≫t,because it then corresponds to expo-nentiation of large logarithms of s/t[40].For general values of s/t,however, there is no such argument.The relation is special to D=4,where the theory is conformally invariant. That is,the O(ǫ1)remainder terms cannot be simplified significantly.For ex-ample,the two-loop amplitude M(2)4(ǫ)contains at O(ǫ1)all three independent Li5functions,Li5(−s/u),Li5(−t/u)and Li5(−s/t),yet[M(1)4(ǫ)]2has only the first two of these[9].The relation is also special to the planar,leading-color limit.The subleading color-components of thefinite remainder|A(2),finn defined by eq.(5)show no significant simplification at all.For planar amplitudes in the D→4limit,however,there is evidence that an identical relation also holds for an arbitrary number n of external legs, at least for certain“maximally helicity-violating”(MHV)helicity amplitudes. This evidence comes from studying the limits of two-loop amplitudes as two of the n gluon momenta become collinear[9,38,41].(Indeed,it was by analyzing these limits that the relation for n=4wasfirst uncovered.)The collinear limits turn out to be consistent with the same eq.(13)with M4replaced by M n everywhere[9],i.e.M(2)n(ǫ)=12(ζ2)2+O(ǫ).(15)The collinear consistency does not constitute a proof of eq.(15),but in light of the remarkable properties of MSYM,it would be surprising if it were not true in the MHV case.Because the direct computation of two-loop amplitudes for n>4seems rather difficult,it would be quite interesting to try to examine the twistor-space properties of eq.(15),along the lines of refs.[31,42].(The right-hand-side of eq.(15)is not completely specified at order1/ǫandǫ0for n>4.The reason is that the orderǫandǫ2terms in M(1)n(ǫ),which contribute to thefirst term in eq.(15)at order1/ǫandǫ0,contain the D=6−2ǫpentagon integral[43],which is not known in closed form.On the other hand, the differential equations this integral satisfies may suffice to test the twistor-space behavior.Or one may examine just thefinite remainder M(L),finn definedvia eq.(5).)It may soon be possible to test whether an iterative relation for planar MSYM amplitudes extends to three loops.An ansatz for the three-loop planar gg→gg amplitude,shown infig.9,was provided at the same time as the two-loop re-sult,in1997[11].The ansatz is based on the“rung-rule”evaluation of the iterated2-particle cuts,plus the3-particle cuts with intermediate states in D=4;the4-particle cuts have not yet been verified.Two integrals,each be-ginning at O(ǫ−6),are required to evaluate the ansatz in a Laurent expansion about D=4.(The other two integrals are related by s↔t.)The triple ladder integral on the top line offig.9was evaluated last year by Smirnov,all the way through O(ǫ0)[44].Evaluation of the remaining integral,which contains a factor of(ℓ+k4)2in the numerator,is in progress[45];all the terms through O(ǫ−2)agree with predictions[33],up to a couple of minor corrections.5Significance of Iterative Behavior?It is not yet entirely clear why the two-loop four-point amplitude,and prob-ably also the n-point amplitudes,have the iterative structure(15).However, one can speculate that it is from the need for the perturbative series to=i3s12s212+s223+2s12(ℓ+k4)+2s23(ℓ+k1)21Figure9.Graphical representation of the three-loop amplitude for MSYM in the planar limit.be summable into something which becomes“simple”in the planar strong-coupling limit,since that corresponds,via AdS/CFT,to a weakly-coupled supergravity theory.The fact that the relation is special to the conformal limit D→4,and to the planar limit,backs up this speculation.Obviously it would be nice to have some more information at three loops.There have been other hints of an iterative structure in the four-point correlation func-tions of chiral primary(BPS)composite operators[46],but here also the exact structure is not yet clear.Integrability has played a key role in recent higher-loop computations of non-BPS spin-chain anomalous dimensions[4,5,6,8].By imposing regularity of the BMN‘continuum’limit[3],a piece of the anoma-lous dimension matrix has even been summed to all orders in g2N c in terms of hypergeometric functions[7].The quantities we considered here—gauge-invariant,but dimensionally regularized,scattering amplitudes of color non-singlet states—are quite different from the composite color-singlet operators usually treated.Yet there should be some underlying connection between the different perturbative series.6Aside:Anomalous Dimensions in QCD and MSYMAs mentioned previously,the set of anomalous dimensions for leading-twist operators was recently computed at NNLO in QCD,as the culmination of a multi-year effort[13]which is central to performing precise computations of hadron collider cross sections.Shortly after the Moch,Vermaseren and Vogt computation,the anomalous dimensions in MSYM were extracted from this result by Kotikov,Lipatov,Onishchenko and Velizhanin[12].(The MSYM anomalous dimensions are universal;supersymmetry implies that there is only one independent one for each Mellin moment j.)This extraction was non-trivial,because MSYM contains scalars,interacting through both gauge and Yukawa interactions,whereas QCD does not.However,Kotikov et al.noticed, from comparing NLO computations in both leading-twist anomalous dimen-sions and BFKL evolution,that the“most complicated terms”in the QCDcomputation always coincide with the MSYM result,once the gauge group representation of the fermions is shifted from the fundamental to the adjoint representation.One can define the“most complicated terms”in the x-space representation of the anomalous dimensions—i.e.the splitting kernels—as follows:Assign a logarithm or factor ofπa transcendentality of1,and a polylogarithm Li m or factor ofζm=Li m(1)a transcendentality of m.Then the most complicated terms are those with leading transcendentality.For the NNLO anomalous dimensions,this turns out to be transcendentality4.(This rule for extracting the MSYM terms from QCD has also been found to hold directly at NNLO,for the doubly-virtual contributions[38].)Strikingly,the NNLO MSYM anomalous dimension obtained for j=4by this procedure agrees with a previous result derived by assuming an integrable structure for the planar three-loop contribution to the dilatation operator[5].7Conclusions and OutlookN=4super-Yang-Mills theory is an excellent testing ground for techniques for computing,and understanding the structure of,QCD scattering amplitudes which are needed for precise theoretical predictions at high-energy colliders. One can even learn something about the structure of N=4super-Yang-Mills theory in the process,although clearly there is much more to be understood. Some open questions include:Is there any AdS/CFT“dictionary”for color non-singlet states,like plane-wave gluons?Can one recover composite operator correlation functions from any limits of multi-point scattering amplitudes?Is there a better way to infrared regulate N=4supersymmetric scattering amplitudes,that might be more convenient for approaching the AdS/CFT correspondence,such as compactification on a three-sphere,use of twistor-space,or use of coherent external states?Further investigations of this arena will surely be fruitful.AcknowledgementsWe are grateful to the organizers of Strings04for putting together such a stim-ulating meeting.This research was supported by the US Department of En-ergy under contracts DE-FG03-91ER40662(Z.B.)and DE-AC02-76SF00515 (L.J.D.),and by the Direction des Sciences de la Mati`e re of the Commissariat `a l’Energie Atomique of France(D.A.K.).。
Search for the Rare Decays JPsi -- Ds- e+ nu_e, JPsi -- D- e+ nu_e, and JPsi -- D0bar e+ e-
a r X i v :h e p -e x /0604005v 1 3 A p r 2006Search for the Rare Decays J/ψ→D −s e +νe ,J/ψ→D −e +νe ,and J/ψ→11Nankai University,Tianjin300071,People’s Republic of China12Peking University,Beijing100871,People’s Republic of China13Shandong University,Jinan250100,People’s Republic of China14Sichuan University,Chengdu610064,People’s Republic of China15Tsinghua University,Beijing100084,People’s Republic of China16University of Hawaii,Honolulu,HI96822,USA17University of Science and Technology of China,Hefei230026,People’s Republicof China18Wuhan University,Wuhan430072,People’s Republic of China19Zhejiang University,Hangzhou310028,People’s Republic of Chinaa Current address:Purdue University,West Lafayette,IN47907,USAb Current address:DESY,D-22607,Hamburg,Germanyc Current address:Laboratoire de l’Acc´e l´e rateur Lin´e aire,Orsay,F-91898,Franced Current address:University of Michigan,Ann Arbor,MI48109,USAD0e+e−+c.c.in a sample of5.8×107J/ψevents collected with the BESII detector at the BEPC.No excess of signal above background is observed, and90%confidence level upper limits on the branching fractions are set:B(J/ψ→D−s e+νe+c.c.)<4.8×10−5,B(J/ψ→D−e+νe+c.c.)<1.2×10−5,and B(J/ψ→1INTRODUCTIONHadronic,electromagnetic,and radiative decays of the J/ψhave been widely studied.However there have been few searches for rare weak J/ψdecay pro-cesses.Kinematically,the J/ψcannot decay into a pair of charmed D mesons, but can decay into a single D meson.Searches for weak decays of J/ψto single D or D s mesons provide tests of standard model(SM)theory and serve as a probe of new physics[1],such as TopColor models,the minimal supersym-metric standard model with or without R-parity,and the two Higgs doublet model[2].The branching fractions of J/ψdecays to single D or D s mesons are pre-dicted to be about10−8or smaller[3]in the SM.Theflavor changing neu-tral current(FCNC)process c→u occurs in the standard model only at the loop level where it is suppressed by the GIM mechanism.Fig.1shows the dominant Feynman diagrams within the standard model for the decaysJ/ψ→D −s e +νe ,J/ψ→D −e +νe ,and J/ψ→D 0e +e −in a sample of 5.8×107J/ψevents collected with the Beijing Spectrometer (BESII)[4]de-tector at the Beijing Electron-Positron Collider (BEPC)[5].Throughout this paper the charge conjugate states are implicitly included.c–c–c sc –c–c dc–c–c u WFig.1.Typical Feynman diagrams for (a)J/ψ→D −s e +νe ,(b)J/ψ→D −e +νe,and (c)J/ψ→1+p 2(p in GeV/c )and a dE/dx resolution for Bhabhaelectrons of ∼8%are obtained.An array of 48scintillation counters surround-ing the MDC measures the time of flight (TOF)of charged particles with a resolution of about 200ps for hadrons.Outside the TOF counters,a 12radia-tion length,lead-gas barrel shower counter (BSC),operating in self quenching streamer mode,measures the energies and positions of electrons and photons over 80%of the total solid angle with resolutions of σE /E =0.21/√In this analysis,a GEANT3-based Monte Carlo program(SIMBES)with de-tailed consideration of the detector performance is used.The consistency be-tween data and Monte Carlo has been checked in many physics channels from J/ψandψ(2S)decays,and the agreement is reasonable,as described in detail in Ref.[7].3Event SelectionEach charged track is required to be wellfitted to a helix that is within the polar angle region|cosθ|<0.8and to originate from the beam interaction region,which is defined to be within2cm of the beam line in the transverse plane and within20cm of the interaction point along the beam direction.For particle identification,confidence levels(CL)are calculated for each particle hypothesis using combined time-of-flight[8]and MDC energy loss information. Pions and kaons are identified by requiring the confidence level for the desired hypothesis to be greater than0.1%and,further,by requiring the normalized weights,defined as CLα/(CLπ+CL K),whereαdenotes the desired particle,to exceed0.7.For electron identification,the dE/dx,TOF,and BSC information are combined to form a particle identification confidence level for the electron hypothesis.An electron candidate is required to have CL e>1.0%and satisfy CL e/(CL e+CLπ+CL K)>0.85.An isolated neutral cluster is considered to be a photon candidate when the angle between the nearest charged track and the cluster is greater than18◦, thefirst hit is in the beginning six radiation lengths,the difference between the angle of the cluster development direction in the BSC and the photon emission direction is less than37◦,and the energy deposit in the shower counter is greater than60MeV.e+νe and J/ψ→D−e+νe3.1J/ψ→D−sBoth J/ψ→D−s e+νe and J/ψ→D−e+νe candidate decays must contain an e+.D−s mesons are reconstructed in two modes D−s→φπ−and K−K∗0, withφ→K+K−and K∗0→K+π−.Theφcandidates are reconstructed from two oppositely-charged kaons and must have an invariant mass|M KK−1.02|<0.015GeV/c2.The K∗0candidates are constructed from K+andπ−candidates and are required to have an invariant mass in the range|M Kπ−0.896|<0.060GeV/c2.D−candidates are reconstructed in the mode D−→K+π−π−.Candidate events are required to have four charged tracks which satisfy charged trackselection criteria,and the total charge of the tracks is required to be zero.The polarization of the K ∗0meson in D −s decay is also utilized to reject back-grounds by imposing a selection requirement on the helicity angle θH .The helicity angle is defined as the angle between one of the decay products of the K ∗0and the direction of the flight of K ∗0in the K ∗0rest frame.Back-ground events are distributed uniformly in cos θH since they originate from random combinations,while signal events are distributed as cos 2θH .The K ∗0candidates are required to have |cos θH |>0.4.Fig.2shows the invariant mass distributions of K +K −and K +π−systems with arrows indicating the selection of φand K ∗(892)candidates.00.511.522.533.50.981 1.02 1.041.06 1.08m KK (GeV/c 2)E n t r i e s / 0.002 G e V246810120.70.80.91 1.1m K π (GeV/c 2)(b)(a)E n t r i e s / 0.010 G e VFig. 2.(a)K +K −and (b)K +πinvariant mass distributions of D −s →φπ−,φ→K +K −and D −s→K −K ∗0,K ∗0→K +π−candidates.The signal regions for individual channels are indicated by arrows in the plot.The angle between the identified electron and the nearest charged track is required to be larger than 12◦to reject backgrounds from gamma conver-sions and π0Dalitz decays.In order to reject backgrounds from J/ψdecaying to states with extra neutral particles,the number of isolated photons is con-strained to be zero.The requirements for an isolated photon are more stringent than those for a good photon in order to increase the detection efficiency.An isolated photon is a photon with the angle between the nearest charged track and the cluster greater than 22◦and the difference between the angle of the cluster development direction in the BSC and the photon emission direction less than 60◦,and the energy deposit in the shower counter is greater than 0.1GeV.The four-momentum of the neutrino,p ν=(E miss ,−−→p miss),is inferred from the difference between the net four momentum of the J/ψparticle,p J/ψ=(M J/ψ,0,0,0,),and the sum of the four-momentum of all detected parti-cles in the event.P miss is required to be larger than 0.2GeV/c and lessthan 0.9GeV/c for J/ψ→D −s e +νe and larger than 0.2GeV/c and less than 1.0GeV/c for J/ψ→D −e +νe to reduce background from J/ψdecay to 4-prong final states with misidentified particles but with no missing par-ticles.We further require the absolute value of U miss ,which is defined as U miss =E miss −P miss ,to be less than 0.1GeV,to reject backgrounds fromJ/ψdecaying to K 0L ,η,and partially missing π0final states,which were not rejected by prior criteria.After all requirements,the invariant mass distribu-tions of D −s →K +K −π−and D −→K +π−π−are shown in Figs.3(a)and (b),respectively.012340240123m (GeV/c 2)m (GeV/c 2)m K π , m K πγγ , m K πππ (GeV/c 2)E n t r i e s / 0.020 G e VFig.3.Invariant mass distribution of (a)D −s→K +K −π−,(b)D −→K +π−π−,and (c)D 0e +e −D 0→K +π−,D 0→K +π−π−π+.Inconstraint(4C)energy-momentum conservation kinematicfits are performed under the J/ψ→e+e−K+π−or the J/ψ→e+e−K+π−π−π+hypotheses, and theχ24C is required to be less than15.The J/ψ→e+e−K+π−γγhy-pothesis is subjected to afive-constraint(5C)fit where the invariant mass of theγγpair associated with theπ0is constrained to mπ0,andχ25C< 15is required.ForD0→K+π−,K+π−π0,K+π−π−π+invariant mass distribution is shown in Fig3(c).4Monte Carlo SimulationMonte Carlo(MC)simulation is used for the determination of mass resolutions and detection efficiencies.We simulate J/ψ→D−s e+νe,J/ψ→D−e+νe,and J/ψ→D0are taken from the world average values[9].The detection efficiencies and branching fractions obtained are listed in Table1. Table1Detection efficiencies and branching fractions[9].εiD−s→φπ−(φ→K+K−)(3.6±1.1)%×(49.1±0.6)%4.29%D−→K+π−π−(9.2±0.6)%D0→K+π−(3.80±0.09)%D0→K+π−π0(13.0±0.8)%D0→K+π−π−π+(7.46±0.31)%The photon detection efficiency has been studied with several different meth-ods using J/ψ→ρ0π0decays[10];the difference between data and Monte Carlo simulation is about2%for each photon.We estimate a systematic error of4%forD0,φ,and K∗0are taken from the PDG[9].The statistical error of the Monte Carlo sample is taken into account.The total number of J/ψevents is(57.7±2.7)×106[11],determined from inclusive4-prong hadronic final states,and the uncertainty,4.7%,is taken as a systematic error.The systematic errors from all sources,as well as the total,are listed in Table2. Table2Summary of the systematic errors.J/ψ→D−s e+νe J/ψ→D−→K+π−π−D0→K+π−K+π−π−π+18.6%20.6%2.0%0.0%0.0%5.0%10.0%4.5% 3.0% 6.0%25% 2.4% 4.2%5.1% 2.7%4.7%32.7%23.8%24.6%D0e+e−events above background is observed.The upper limit on the branching fractions of these decay modes are calculated usingn obs ULB<decay modes,and N is equal to2.The systematic error in the measurement is taken into consideration by introducing1−σsys in the denominator of the branching fraction calculation.We obtain upper limits for the observed number of events at90%confidence level of3.55for J/ψ→D−s e+νe,4.64for J/ψ→D−e+νe and3.07for J/ψ→D0e+e−+c.c..J/ψ→D−s e+νe+c.c.J/ψ→n obs UL 4.64εB7.98×10−3Sys.Err.15.9%B(90%C.L.)<1.2×10−5In summary,we have searched for the decays J/ψ→D−s e+νe,J/ψ→D−e+νe, and J/ψ→D0e+e−are not inconsistent with the standard model.The BES collaboration thanks the staffof BEPC and computing center for their hard efforts.This work is supported in part by the National Natural Sci-ence Foundation of China under contracts Nos.10491300,10225524,10225525, 10425523,the Chinese Academy of Sciences under contract No.KJ95T-03, the100Talents Program of CAS under Contract Nos.U-11,U-24,U-25,and the Knowledge Innovation Project of CAS under Contract Nos.U-602,U-34 (IHEP),the National Natural Science Foundation of China under Contract No.10225522(Tsinghua University),and the Department of Energy under Contract No.DE-FG02-04ER41291(U Hawaii).00.050.102468101200.050.102468101214161800.0250.050.0750.10246810Number of events(a)(b)(c)L i k e l i h o o dFig. 4.Likelihood distributions for the observed number of events of (a)J/ψ→D −s e +νe ,(b)J/ψ→D −e +νe,and (c)J/ψ→。
超对称理论导论
• Discrete symmetries: C, P, and T.
dimensional supersymmetry algebra is given in the Appendix; here we will be content with checking some of the features of this algebra. The anticommutator of the QA α with their adjoints is: { QA ˙ } α , QβB =
,
A where the CB are complex Lorentz scalar coefficients. Taking the adjoint of the left-hand side of Eq. 2, using m σα ˙ β † m = σβ α ˙
, , (3)
† QA α
= Qα ˙
A
A tells us that CB is a hermitian matrix. Furthermore, since {Q, Q} is a positive A definite operator, CB is a positive definite hermitian matrix. This means that A A we can always choose a basis for the QA α such that CB is proportional to δB . The factor of two in Eq. 2 is simply a convention. The SUSY generators QA α commute with the translation generators:
generalized maxwell model
Generalized Maxwell Model1. IntroductionThe Maxwell model is a linear viscoelastic model used to describe the rheological behavior of viscoelastic materials. It consists of a spring and a dashpot in parallel, and ismonly used to model the behavior of polymers, gels, and otherplex fluids. In this article, we will explore the generalized Maxwell model, which is an extension of the original Maxwell model and provides a more accurate representation of the viscoelastic properties of materials.2. The Maxwell modelThe Maxwell model, first proposed by James Clerk Maxwell in the 19th century, consists of a spring and a dashpot in parallel. The spring represents the elastic behavior of the material, while the dashpot represents the viscous behavior. The constitutive equation of the Maxwell model is given by:σ(t) = Eε(t) + ηdε(t)/dtWhere σ(t) is the stress, ε(t) is the strain, E is the elastic modulus, η is the viscosity, and dε(t)/dt is the rate of strain. The Maxwellmodel is simple and easy to understand, but it fails to capture the nonlinear viscoelastic behavior of many materials.3. The generalized Maxwell modelTo ovee the limitations of the original Maxwell model, the generalized Maxwell model introduces multiple springs and dashpots in parallel, each with its own elastic modulus and viscosity. This allows for a more accurate representation of theplex viscoelastic behavior of materials. The constitutive equation of the generalized Maxwell model is given by:σ(t) = ∑(Eiε(t) + ηidε(t)/dt)Where the summation is taken over all the springs and dashpots in the model, and Ei and ηi are the elastic moduli and viscosities of the individual elements. By including multiple elements with different relaxation times, the generalized Maxwell model can accurately describe the behavior of materials with nonlinear viscoelastic properties.4. Applications of the generalized Maxwell modelThe generalized Maxwell model has found wide applications in various fields, including polymer science, biomedicalengineering, and materials science. It has been used to study the viscoelastic behavior of polymers, gels, and foams, and to design materials with specific viscoelastic properties. In biomedical engineering, the model has been used to study the mechanical behavior of soft tissues and to develop new biomaterials for tissue engineering. In materials science, the model has been used to characterize the viscoelastic properties ofposites and to optimize their performance.5. Comparison with other viscoelastic modelsThe generalized Maxwell model is just one of many viscoelastic models used to describe the rheological behavior of materials. Other popular models include the Kelvin-Voigt model, the Burgers model, and the Zener model. Each of these models has its own advantages and limitations, and the choice of model depends on the specific material and the behavior of interest. The generalized Maxwell model is particularly useful for materials withplex viscoelastic behavior, as it allows for a more detailed description of the relaxation processes.6. ConclusionIn conclusion, the generalized Maxwell model is a powerful tool for describing the viscoelastic behavior of materials. Byextending the original Maxwell model to include multiple springs and dashpots, the generalized Maxwell model provides a more accurate representation of the nonlinear viscoelastic properties of materials. It has found wide applications in various fields and has contributed to our understanding of the mechanical behavior ofplex fluids and solids. As our knowledge of viscoelastic materials continues to grow, the generalized Maxwell model will undoubtedly remain an important tool for researchers and engineers alike.。
纳芯微-NSM2012 高精度霍尔电流传感器 数据手册说明书
NSM2012 基于霍尔原理，高精度，具有共模磁场抑制，可达3000V隔离的电流传感器Datasheet (CN) 1.0Product OverviewNSM2012是一款集成路径电流传感器，具有1.2mΩ极低的导通电阻，减少了芯片上的热损耗。
纳芯微创新的隔离技术以及信号调理设计能够满足高隔离等级的同时感测流过内部B usbar的电流。
内部采用差分霍尔对，因此对外部杂散磁场有很强的抵御能力。
NSM2012支持比例输出和固定输出模式，固定模式方便客户ADC差分采样Vref以及Vout的电压以减少外部共模干扰（比如温度等）。
对比同样Shunt+隔离运放的电流采样方式，NSM2012省去了原边供电并且Layout简单方便，同时具有极高隔离耐压以及Lifetime稳定性。
在高边电流检测应用中只需用一颗NSM2012即可达到600V pk工作电压，无需加任何保护器件即可耐受6kV浪涌电压。
由于NSM2012内部精确的温度补偿算法以及出厂精度校准，此电流传感器在全温度工作范围都可以保持很好的精度，客户无需做二次编程。
支持3.3V/5V供电电压（不同供电版本）。
Key Features•高带宽以及快速响应时间•400kHz带宽• 1.5us响应时间•高精度电流测量•差分霍尔检测可抵御外界杂散磁场•满足UL标准的高隔离等级•耐受隔离耐压（V ISO）:3000Vrms•最大浪涌隔离耐压（V surge）:6kV•CMTI > 100V/ns•CTI（I）•爬电距离/电气间距：4mm•纳芯微创新的斩波以及旋转电流激励技术使得零点温漂很小•比例输出或者固定输出•工作温度：-40℃ ~ 125℃•原边导通电阻：1.2mΩ•SOIC8封装•满足UL62368/EN62368安规认证•ROHSApplications•光伏•工业电源•电机控制•OBC/DCDC/PTC Heater•充电桩Device InformationFunctional Block DiagramsFigure 1. NSM2012 Block DiagramINDEX1. PIN CONFIGURATION AND FUNCTIONS (3)2. ABSOLUTE MAXIMUM RATINGS (4)3. ISOLATION CHARACTERISTICS (4)4. SPECIFICATIONS (5)4.1C OMMON C HARACTERISTICS (TA=-40°C TO 125°C,VCC=5V OR 3.3V, UNLESS OTHERWISE SPECIFIED) (5)4.2NSM2012-30B3R-DSPR C HARACTERISTICS (TA=-40°C TO 125°C,VCC=3.3V, UNLESS OTHERWISE SPECIFIED) (6)4.3NSM2012-30B5R-DSPR C HARACTERISTICS (TA=-40°C TO 125°C,VCC=5V, UNLESS OTHERWISE SPECIFIED) (6)4.4NSM2012-10U5R-DSPR C HARACTERISTICS (TA=-40°C TO 125°C,VCC=5V, UNLESS OTHERWISE SPECIFIED) (7)4.5NSM2012-20B5R-DSPR C HARACTERISTICS (TA=-40°C TO 125°C,VCC=5V, UNLESS OTHERWISE SPECIFIED) (8)4.6T YPICAL P ERFORMANCE C HARACTERISTICS (9)NSM2012-30B3R-DSPR[1] (9)NSM2012-30B5R-DSPR[1] (9)NSM2012-10U5R-DSPR[1] (10)NSM2012-20B5R-DSPR[1] (11)5. FUNCTION DESCRIPTION (13)5.1.O VERVIEW (13)5.2.NSM2012R版本(单端比例输出) (13)5.3.NSM2012F版本(固定输出版本) (13)5.4.NSM2012专业术语定义 (13)6. APPLICATION NOTE (16)6.1.典型应用电路 (16)6.2.PCB L AYOUT (16)6.3.热评估实验 (17)7. PACKAGE INFORMATION (18)8. ORDER INFORMATION (19)9. TAPE AND REEL INFORMATION (20)10. REVISION HISTORY (21)1. Pin Configuration and FunctionsR Version F VersionFigure 1.1 NSM2012 PackageTable 1.1 NSM2012 Pin Configuration and Description2.Absolute Maximum Ratings3.Isolation Characteristics4.Specifications[1]: 被设计保证。
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B. B. Deo
Physics Department, Utkal University, Bhubaneswar-751004, India. N=1, D=4 Superstring possessing SO(6) ⊗ SO(5) gauge symmetry is constructed from the open bosonic string in twenty six dimension. Without breaking supersymmetry, the gauge symmetry of the model descends to the supersymmetric standard model of the electroweak scale in four flat dimension.
(5) (6) (7)
δψ = −ie ρ ∂α X ǫ δφµ,k = iek ρα ∂α X µ ǫ.
j ǫ is a constant anticommuting spinor. ej , ek are unit component of c-number row vectors with ej el = δl , ej ej =6, ek ek =5. The commutors of two supersymmetric transformations, lead to a translation with the coefficient aα = 2iǫ ¯1 ρα ǫ2 provided µ ψj = ej Ψ µ , µ φµ k = ek Ψ
The upper indices i,j refer to a row and lowers to a column and ρ0 = and ρ1 = Dropping indices, ¯ = ψ † ρo , ψ ¯ = φ† ρo φ (4) 0 i i 0 (3) 0 −i i 0 (2)
3 The principal success has been to achieve the actions contained in equation(1) or equation(21) which possess SO(6) ⊗ SO(5) gauge supersymmetry besides the SO(3,1). To descend to the standard model group SUC (3) ⊗ SUL (2) ⊗ UY (1), it is normally done by introducing Higgs which breaks gauge symmetry and supersymmetry. However, one uses the method of symmetry breaking by using Wilson’s lines, supersymmetry remains in tact but the gauge symmetry is broken. This Wilson loop is Uγ = P exp(
i ¯µ ρ(α ∂β ) Ψµ = 0 Tαβ = ∂α X µ ∂β Xµ − Ψ 2 In a light cone basis, the vanishing of the light cone components are J± = ∂± Xµ Ψµ ± = 0 and i µj i T±± = ∂± X µ ∂± Xµ + ψ± ∂± ψ±µ,j − φµk ∂± φ±µ,k 2 2 ± where ∂± = 1 2 (∂τ ± ∂σ ). Using the equation(8), the component constraints are
(8)
and
µ Ψµ = ej ψj − ek φµ k
(9)
It is easy to verify that δX µ = ǫ ¯Ψµ , and [δ1 , δ2 ]X µ = aα ∂α X µ , [δ1 , δ2 ]Ψµ = aα ∂α Ψµ (11) δ Ψµ = −i ǫ ρα ∂α X µ (10)
µ ¯ µ Πµ α = ∂α X − iθΓ ∂α θ.
(20)
1 2π
d2 σ
√ αβ ¯Γµ ∂β θ gg Πα Πβ + 2iǫαβ ∂α X µ θ
(21)
(22)
This is the N=1 and D=4 superstring originating from the D=26 bosonic string. It is difficult to quantise this action covariantly. It is better to use NS-R [6] formulation with G.S.O projection [7]. This has been done in reference [8] where the explicit elimination of ghosts, modular invariance and a derivation of Einstein’s field equation have been presented.
Besides SO(3,1), the action (1) is invariant under SO(6) ⊗ SO(5). It is also invariant under the transformation
µ − ek φµ δX µ = ǫ ¯(ej ψj k ), µ,j j α µ
6 5
θα =
j =1
ej θjα −
ek θkα
k=1
(19)
With the usual Dirac matrices Γµ , since the identity ¯2 Γµ ψ3] = 0 Γµ ψ[1 ψ is satisfied due to the Fierz transformation in four dimension, the Green Schwarz action [5] for N=1 is S= where
PACS numbers: 11.25-w,11.30Pb,12.60.Jv
arXiv:hep-th/0301041v1 8 Jan 2003
The model begins from the Nambu-Goto [1] open bosonic string in the world sheet (σ, τ ) which makes sense in 26-dimension. Following Mandelstams’s [2], proof of equivalence between one boson to two fermionic modes, one can rewrite the action as the sum of the four bosonic coordinates X µ of SO(3,1) and forty four fermions which are scalars in the world sheet having symmetry SO(44). To have an action with SO(3,1) invariance and noting that Majorana Lorentz fermions are in bosonic representation of SO(3,1), the forty four fermions are grouped into eleven Lorentz vectors. The action, so obtained, is not supersymmetric. The eleven vectors have to be further divided into two species; ψ µ,j , j=1,2,..6 and φµ,k ,k =7,8..11. For the group of six, the the positive and negative parts are ψ µ,j = ψ (+)µ,j + ψ (−)µ,j whereas for the group of five, allowing the freedom of phase of creation operators for Majorana fermions ,φµ,k = φ(+)µ,k − φ(−)µ,k . The action is now S=− 1 2π ¯µ,j ρα ∂α ψµ,j + i φ ¯µ,k ρα ∂α φµ,k d2 σ ∂α X µ ∂ α Xµ − i ψ (1)
µ,j ∂± Xµ ψ± = ∂± Xµ ej Ψµ ± = 0,
(14)
2...6.
(17)
k µ ∂± Xµ φµk ± = ∂± Xµ e Ψ± = 0,
k = 7, 8, ..11.
(18)
Equations (16), (17) and (18) constitute 13 constraints and eliminates all the metric ghosts from the Fock space. The action in equation (12) is not space time supersymmetric. However,, in the fermionic representation SO(3,1) fermions are Dirac spinor with four components α. We construct Dirac spinor like equation (9) as the sum of component spinor