Theoretical aspects of gauge-boson production at LEP2 and the NLC

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Triple Neutral Gauge Boson Couplings in Noncommutative Standard Model

2 Department
of Physics, National Taiwan University, Taipei, Taiwan 10764, R.O.C.
Abstract
It has been shown recently that the triple neutral gauge boson couplings are not uniquely determined in noncommutative extension of the Standard Model (NCSM). Depending on speciﬁc schemes used, the couplings are diﬀerent and may even be zero. To distinguish diﬀerent realizations of the NCSM, additional information either from theoretical or experimental considerations is needed. In this paper we show that these couplings can be uniquely determined from considerations of uniﬁcation of electroweak and strong interactions. Using SU (5) as the underlying theory and integrating out the heavy degrees of freedom, we obtain unique non-zero new triple γγγ , γγZ , γZZ , ZZZ , γGG, ZGG and GGG couplings at the leading order in the NCSM. We also brieﬂy discuss experimental implications. PACS numbers:

PACSnumbers1215Ff,1130Hv,1210Dm,1125Mj…

a r X i v :0803.2889v 2 [h e p -p h ] 14 J u l 2008Mapping Out SU (5)GUTs with Non-Abelian Discrete Flavor SymmetriesFlorian Plentinger ∗and Gerhart Seidl †Institut f¨u r Physik und Astrophysik,Universit¨a t W¨u rzburg,Am Hubland,D 97074W¨u rzburg,Germany(Dated:December 25,2013)We construct a class of supersymmetric SU (5)GUT models that produce nearly tribimaximal lepton mixing,the observed quark mixing matrix,and the quark and lepton masses,from discrete non-Abelian ﬂavor symmetries.The SU (5)GUTs are formulated on ﬁve-dimensional throats in the ﬂat limit and the neutrino masses become small due to the type-I seesaw mechanism.The discrete non-Abelian ﬂavor symmetries are given by semi-direct products of cyclic groups that are broken at the infrared branes at the tip of the throats.As a result,we obtain SU (5)GUTs that provide a combined description of non-Abelian ﬂavor symmetries and quark-lepton complementarity.PACS numbers:12.15.Ff,11.30.Hv,12.10.Dm,One possibility to explore the physics of grand uniﬁed theories (GUTs)[1,2]at low energies is to analyze the neutrino sector.This is due to the explanation of small neutrino masses via the seesaw mechanism [3,4],which is naturally incorporated in GUTs.In fact,from the perspective of quark-lepton uniﬁcation,it is interesting to study in GUTs the drastic diﬀerences between the masses and mixings of quarks and leptons as revealed by current neutrino oscillation data.In recent years,there have been many attempts to re-produce a tribimaximal mixing form [5]for the leptonic Pontecorvo-Maki-Nakagawa-Sakata (PMNS)[6]mixing matrix U PMNS using non-Abelian discrete ﬂavor symme-tries such as the tetrahedral [7]and double (or binary)tetrahedral [8]groupA 4≃Z 3⋉(Z 2×Z 2)and T ′≃Z 2⋉Q,(1)where Q is the quaternion group of order eight,or [9]∆(27)≃Z 3⋉(Z 3×Z 3),(2)which is a subgroup of SU (3)(for reviews see, e.g.,Ref.[10]).Existing models,however,have generally dif-ﬁculties to predict also the observed fermion mass hierar-chies as well as the Cabibbo-Kobayashi-Maskawa (CKM)quark mixing matrix V CKM [11],which applies especially to GUTs (for very recent examples,see Ref.[12]).An-other approach,on the other hand,is oﬀered by the idea of quark-lepton complementarity (QLC),where the so-lar neutrino angle is a combination of maximal mixing and the Cabibbo angle θC [13].Subsequently,this has,in an interpretation of QLC [14,15],led to a machine-aided survey of several thousand lepton ﬂavor models for nearly tribimaximal lepton mixing [16].Here,we investigate the embedding of the models found in Ref.[16]into ﬁve-dimensional (5D)supersym-metric (SUSY)SU (5)GUTs.The hierarchical pattern of quark and lepton masses,V CKM ,and nearly tribi-maximal lepton mixing,arise from the local breaking of non-Abelian discrete ﬂavor symmetries in the extra-dimensional geometry.This has the advantage that theFIG.1:SUSY SU (5)GUT on two 5D intervals or throats.The zero modes of the matter ﬁelds 10i ,5H,24H ,and the gauge supermul-tiplet,propagate freely in the two throats.scalar sector of these models is extremely simple without the need for a vacuum alignment mechanism,while of-fering an intuitive geometrical interpretation of the non-Abelian ﬂavor symmetries.As a consequence,we obtain,for the ﬁrst time,a realization of non-Abelian ﬂavor sym-metries and QLC in SU (5)GUTs.We will describe our models by considering a speciﬁc minimal realization as an example.The main features of this example model,however,should be viewed as generic and representative for a large class of possible realiza-tions.Our model is given by a SUSY SU (5)GUT in 5D ﬂat space,which is deﬁned on two 5D intervals that have been glued together at a common endpoint.The geom-etry and the location of the 5D hypermultiplets in the model is depicted in FIG.1.The two intervals consti-tute a simple example for a two-throat setup in the ﬂat limit (see,e.g.,Refs.[17,18]),where the two 5D inter-vals,or throats,have the lengths πR 1and πR 2,and the coordinates y 1∈[0,πR 1]and y 2∈[0,πR 2].The point at y 1=y 2=0is called ultraviolet (UV)brane,whereas the two endpoints at y 1=πR 1and y 2=πR 2will be referred to as infrared (IR)branes.The throats are supposed to be GUT-scale sized,i.e.1/R 1,2 M GUT ≃1016GeV,and the SU (5)gauge supermultiplet and the Higgs hy-permultiplets 5H and2neously broken to G SM by a 24H bulk Higgs hypermulti-plet propagating in the two throats that acquires a vac-uum expectation value pointing in the hypercharge direc-tion 24H ∝diag(−12,13,15i ,where i =1,2,3is the generation index.Toobtainsmall neutrino masses via the type-I seesaw mechanism [3],we introduce three right-handed SU (5)singlet neutrino superﬁelds 1i .The 5D Lagrangian for the Yukawa couplings of the zero mode fermions then readsL 5D =d 2θ δ(y 1−πR 1) ˜Y uij,R 110i 10j 5H +˜Y d ij,R 110i 5H +˜Y νij,R 15j5i 1j 5H +M R ˜Y R ij,R 21i 1j+h.c. ,(3)where ˜Y x ij,R 1and ˜Y x ij,R 2(x =u,d,ν,R )are Yukawa cou-pling matrices (with mass dimension −1/2)and M R ≃1014GeV is the B −L breaking scale.In the four-dimensional (4D)low energy eﬀective theory,L 5D gives rise to the 4D Yukawa couplingsL 4D =d 2θ Y u ij 10i 10j 5H +Y dij10i 5H +Y νij5i ∼(q i 1,q i 2,...,q i m ),(5)1i ∼(r i 1,r i 2,...,r im ),where the j th entry in each row vector denotes the Z n jcharge of the representation.In the 5D theory,we sup-pose that the group G A is spontaneously broken by singly charged ﬂavon ﬁelds located at the IR branes.The Yukawa coupling matrices of quarks and leptons are then generated by the Froggatt-Nielsen mechanism [21].Applying a straightforward generalization of the ﬂavor group space scan in Ref.[16]to the SU (5)×G A represen-tations in Eq.(5),we ﬁnd a large number of about 4×102ﬂavor models that produce the hierarchies of quark and lepton masses and yield the CKM and PMNS mixing angles in perfect agreement with current data.A distri-bution of these models as a function of the group G A for increasing group order is shown in FIG.2.The selection criteria for the ﬂavor models are as follows:First,all models have to be consistent with the quark and charged3 lepton mass ratiosm u:m c:m t=ǫ6:ǫ4:1,m d:m s:m b=ǫ4:ǫ2:1,(6)m e:mµ:mτ=ǫ4:ǫ2:1,and a normal hierarchical neutrino mass spectrumm1:m2:m3=ǫ2:ǫ:1,(7)whereǫ≃θC≃0.2is of the order of the Cabibbo angle.Second,each model has to reproduce the CKM anglesV us∼ǫ,V cb∼ǫ2,V ub∼ǫ3,(8)as well as nearly tribimaximal lepton mixing at3σCLwith an extremely small reactor angle 1◦.In perform-ing the group space scan,we have restricted ourselves togroups G A with orders roughly up to 102and FIG.2shows only groups admitting more than three valid mod-els.In FIG.2,we can observe the general trend thatwith increasing group order the number of valid modelsper group generally increases too.This rough observa-tion,however,is modiﬁed by a large“periodic”ﬂuctu-ation of the number of models,which possibly singlesout certain groups G A as particularly interesting.Thehighly populated groups would deserve further system-atic investigation,which is,however,beyond the scopeof this paper.From this large set of models,let us choose the groupG A=Z3×Z8×Z9and,in the notation of Eq.(5),thecharge assignment101∼(1,1,6),102∼(0,3,1),103∼(0,0,0),52∼(0,7,0),52↔4FIG.3:Eﬀect of the non-Abelian ﬂavor symmetry on θ23for a 10%variation of all Yukawa couplings.Shown is θ23as a function of ǫfor the ﬂavor group G A (left)and G A ⋉G B (right).The right plot illustrates the exact prediction of the zeroth order term π/4in the expansion θ23=π/4+ǫ/√2and the relation θ13≃ǫ2.The important point is that in the expression for θ23,the leading order term π/4is exactly predicted by thenon-Abelian ﬂavor symmetry G F =G A ⋉G B (see FIG.3),while θ13≃θ2C is extremely small due to a suppression by the square of the Cabibbo angle.We thus predict a devi-ation ∼ǫ/√2,which is the well-known QLC relation for the solar angle.There have been attempts in the literature to reproduce QLC in quark-lepton uniﬁed models [26],however,the model presented here is the ﬁrst realization of QLC in an SU (5)GUT.Although our analysis has been carried out for the CP conserving case,a simple numerical study shows that CP violating phases (cf.Ref.[27])relevant for neutri-noless double beta decay and leptogenesis can be easily included as well.Concerning proton decay,note that since SU (5)is bro-ken by a bulk Higgs ﬁeld,the broken gauge boson masses are ≃M GUT .Therefore,all fermion zero modes can be localized at the IR branes of the throats without intro-ducing rapid proton decay through d =6operators.To achieve doublet-triplet splitting and suppress d =5pro-ton decay,we may then,e.g.,resort to suitable extensions of the Higgs sector [28].Moreover,although the ﬂavor symmetry G F is global,quantum gravity eﬀects might require G F to be gauged [29].Anomalies can then be canceled by Chern-Simons terms in the 5D bulk.We emphasize that the above discussion is focussed on a speciﬁc minimal example realization of the model.Many SU (5)GUTs with non-Abelian ﬂavor symmetries,however,can be constructed along the same lines by varying the ﬂavor charge assignment,choosing diﬀerent groups G F ,or by modifying the throat geometry.A de-tailed analysis of these models and variations thereof will be presented in a future publication [30].To summarize,we have discussed the construction of 5D SUSY SU (5)GUTs that yield nearly tribimaximal lepton mixing,as well as the observed CKM mixing matrix,together with the hierarchy of quark and lepton masses.Small neutrino masses are generated only by the type-I seesaw mechanism.The fermion masses and mixings arise from the local breaking of non-Abelian ﬂavor symmetries at the IR branes of a ﬂat multi-throat geometry.For an example realization,we have shown that the non-Abelian ﬂavor symmetries can exactly predict the leading order term π/4in the sum rule for the atmospheric mixing angle,while strongly suppress-ing the reactor angle.This makes this class of models testable in future neutrino oscillation experiments.In addition,we arrive,for the ﬁrst time,at a combined description of QLC and non-Abelian ﬂavor symmetries in SU (5)GUTs.One main advantage of our setup with throats is that the necessary symmetry breaking can be realized with a very simple Higgs sector and that it can be applied to and generalized for a large class of uniﬁed models.We would like to thank T.Ohl for useful comments.The research of F.P.is supported by Research Train-ing Group 1147“Theoretical Astrophysics and Particle Physics ”of Deutsche Forschungsgemeinschaft.G.S.is supported by the Federal Ministry of Education and Re-search (BMBF)under contract number 05HT6WWA.∗********************************.de †**************************.de[1]H.Georgi and S.L.Glashow,Phys.Rev.Lett.32,438(1974);H.Georgi,in Proceedings of Coral Gables 1975,Theories and Experiments in High Energy Physics ,New York,1975.[2]J.C.Pati and A.Salam,Phys.Rev.D 10,275(1974)[Erratum-ibid.D 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Quartic Gauge Boson Couplings Results at LEP

a r X i v :h e p -e x /0105063v 1 22 M a y 2001QUARTIC GAUGE BOSON COUPLINGS RESULTS AT LEPM.MUSY CERNThe study of charged and neutral boson vertices has been performed in diﬀerent production channels at the LEP experiments.Decay rates and kinematic properties of these events are exploited to set constraints on the corresponding gauge couplings.1IntroductionThe study of quartic gauge boson couplings (QGCs)has become possible due to the recent theoretical developments on this topic.The presence of QGCs aﬀects the data collected for diﬀerent ﬁnal states at LEP.The continous increase of the centre-of-mass energy in e +e −collisions allowed for the precise measurement of the W-pair cross section and couplings.Now,also the study of radiative W-pair events,e +e −→W +W −γ,has become possible.The Standard Model (SM)predicts the existence of quartic gauge boson couplings leading to W +W −γproduction via s -channel exchange of a γor a Z boson as shown in Fig.1a.The contribution of these two quartic Feynman diagrams with respect to the other competing diagrams,mainly initial state radiation,is negligible at the LEP centre of mass energies.Nonetheless,the process leading to the W +W −γﬁnal state can be sensitive to anomalous contributions to the SM quartic vertices W +W −γγand W +W −Z γ.The theoretical framework of Ref.[1]is used for the parametrisation of such anomalous couplings.The existence of Anomalous QGCs would also aﬀect the e +e −→νe ¯νe γγprocess via the W +W −fusion Feynman diagram containing the W +W −γγvertex 2(see Figure 1b).In the SM the reaction e +e −→ν¯νγγproceeds predominantly through s -channel Z exchange and t -channel W exchange,with the two photons coming from initial state radiation,whereas the SM contribution from the W +W −fusion is again negligible at LEP.AQGCs would enhance the ν¯νγγproduction rate,especially for the hard tail of the photon energy distribution and for photons produced at large angles with respect to the beam direction.eeeeγγFigure 1:Feynman diagrams containing a four boson vertex leading to the (a)W +W −γ,(b)ν¯νγγ,and (c)Z γγﬁnal states.Finally,the Z γγproduction (see Fig.1c)can also be exploited to derive limits on AQGCs,as it will be discussed in the next paragraph.2QGCs Analyses at LEPTable 1shows the diﬀerent data sets used for the AQGCs analyses at LEP.L3W +W −γ189-202GeV189-202GeV189GeVZ γγ130-202GeV Opal E γ>10GeV|cos θγ|<0.94cos θfγ<0.9–√s (GeV)σW W γ (p b )00.10.20.30.40.5Figure 2:Measured cross section for the process e +e −→W +W −γ.The resulting cross sections for the L3experiment are:σW W γ(188.6GeV)=0.29±0.08±0.02pb σW W γ(194.4GeV)=0.23±0.10±0.02pb σW W γ(200.2GeV)=0.39±0.12±0.02pbwhile for Opal,the measurement at 189GeV gives σW W γ=0.136±0.037±0.008pb,where the ﬁrst error is statistic and the second systematic.All the obtained results are compatible with the SM expectations.Figure 2shows the L3measured cross section as a fuction of√The information from theν¯νγγtotal cross section,together with the information derived from the shape and normalisation of the photon spectra in W+W−γevents,produces the fol-lowing ADL combined limits on charged AQGCs using the data set of Table1(ﬁrst two lines):−0.022GeV−2<a0/Λ2<0.021GeV−2−0.043GeV−2<a c/Λ2<0.058GeV−2−0.022GeV−2<a n/Λ2<0.020GeV−2,in good agreement with the SM expectation of zero for each coupling.The Feynman diagram of Fig.1c involves only neutral bosons,and it is not present in the SM.Besides,as it has been recently indicated3,under more general assumptions the quartic couplings in the neutral sector,still named a0/Λ2,and a c/Λ2in Ref.[1],can be regarded as independent from those in the charged sector.For this reason,on the experiimental side,the convention was used to keep apart the measurements in these channels not performing any combinations,in the wait for a uniﬁed theoretical approach.To obtain limits on AQGCs in the neutral sector,the hadronic Zγγ→qqγγevents are used. The signal consists in an enhancement in the rate of high energy photons,while the background mainly comes from doubly radiative returns to the Z.Figure3shows the Zγγcross section as predicted by the EEZGG2program a.The same program is used to model AQGCs and toﬁt these couplings to the observed data distributions of P Tγ1(for L3),Eγand max|cosθγi|(for Opal). The following LEP combined bounds are derived4:−0.0048GeV−2<a0/Λ2<0.0056GeV−2−0.0052GeV−2<a c/Λ2<0.0099GeV−2All experimental observations are compatible with the SM predictions.3ConclusionsThe measurements of rare processes as Zγγ,W+W−γandν¯νγγconstitute an important test of the SM.Preliminary results on AQGCs were presented here,including new results in the W+W−γchannel up to the centre-of-mass energy of202GeV with a signiﬁcant increase in the ﬁnal precision.References1.J.W.Stirling and A.Werthenbach,Phys.Lett.B446,369.2.J.W.Stirling and A.Werthenbach,Eur.Phys.J.C14(2000)103.3.G.Belanger and F.Boudjema,Nucl.Phys.B288(1992)201.4.The Opal Collaboration,ICHEP abs572(2000),Opal PN452;The L3Collaboration,ICHEP abs.505(2000),CERN-EP/2000-006.Recoil Mass (GeV)E v e n t s /5 G e V246810121400.511.5√s/GeVOPAL Preliminaryσ(e +e -→q q γγ) /p bFigure 3:Cross section for the process e +e −→Z γγas a function of√。

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。

Abelian projection and studies of gauge-variant quantities in lattice QCD without gauge fix

a rXiv:h e p-la t/961129v127Nov1996Abelian projection and studies of gauge-variant quantities in the lattice QCD without gauge ﬁxing Sergei V.SHABANOV 1Institute for Theoretical Physics,Free University of Berlin,Arnimallee 14,WE 2,D-14195,Berlin,Germany Abstract We suggest a new (dynamical)Abelian projection of the lattice QCD.It contains no gauge condition imposed on gauge ﬁelds so that Gribov copying is avoided.Conﬁgurations of gauge ﬁelds that turn into monopoles in the Abelian projection can be classiﬁed in a gauge invariant way.In the continuum limit,the theory respects the Lorentz invariance.A similar dynamical reduction of the gauge symmetry is proposed for studies of gauge-variant correlators (like a gluon propagator)in the lattice QCD.Though the procedure is harder for numerical simulations,it is free of gauge-ﬁxing artifacts,like the Gribov horizon and copies.1.One of the important features of the QCD conﬁnement is the existence of a stable chromoelectrical ﬁeld tube connecting two color sources (quark and antiquark).Numerical studies of the gluon ﬁeld energy density between two color sources leave no doubt that such a tube exists.However,a mechanism which could explain its stability is still unknown.It is believed that some speciﬁc conﬁgurations (or excitations)of gauge ﬁelds are re-sponsible for the QCD conﬁnement,meaning that they give a main contributions to the QCD string tension.Numerical simulations of the lattice QCD shows that Abelian (com-mutative)conﬁgurations of gauge potentials completely determine the string tension in the full non-Abelian gauge theory [1].This phenomenon is known as the Abelian dominance.Therefore one way of constructing eﬀective dynamics of the conﬁgurations relevant to the QCD conﬁnement is the Abelian projection [2]when the full non-Abelian gauge group SU(3)is restricted to its maximal Abelian subgroup (the Cartan subgroup)U(1)×U(1)by a gauge ﬁxing.Though dynamics of the above gauge ﬁeld conﬁguration cannot be gauge dependent,a right choice of a guage condition may simplify its description.There is a good reason,supported by numerical simulations [3],[4],to believe that the sought conﬁgurations turn into magnetic monopoles in the eﬀective Abelian theory,and the conﬁnement can be due to the dual mechanism [5]:The Coulomb ﬁeld of electric charges is squized into a tube,provided monopole-antimonopole pair form a condensate like the Cooper pairs in superconductor.It is important to realize that the existence of monopoles in the eﬀective Abelian theory is essentially due to the gauge ﬁxing,in fact,monopoles are singularities of thegaugeﬁxing.Note that monopoles cannot exist as stable excitations in pure gauge the-ory with simply connected group like SU(3).Since the homotopy groups of SU(3)and of U(1)×U(1)are diﬀerent(the one of SU(3)is trivial),a gauge condition restricting SU(3)to U(1)×U(1)should have singularities which can be identiﬁed as monopoles[2].A dynamical question is to verify whether all conﬁgurations of non-Abelian gaugeﬁelds relevant to the conﬁnement(in the aforementioned sense)are”mapped”on monopoles of the Abelian theory(the monopole domimance[4]).It appears that monopole dynam-ics may depend on the projection recipe[6].There are indications that some Abelian projections exhibit topological singularities other than magnetic monopoles[7].Though the lattice QCD is,up to now,the only relible tool for studying monopole dynamics,the true theory must be continuous and respect the Lorentz invariance.In this regard,Abelian projections based on Lotentz invariant gauge conditions play a dis-tinguished role.For example,the gauge can be chosen as follows D HµA offµ=0where D Hµ=∂µ+igA Hµ,A Hµare Cartan(diagonal)components of guage potentials Aµ,while A offµare its non-Cartan(oﬀ-diagonal)components.This gauge restricts the gauge sym-metry to the maximal Abelian(Cartan)subrgoup and is manifestly Lorentz invariant. The lattice version of the corresponding Abelian theory is known as the maximal Abelian projection.The above homotopy arguments can be implemented to this gauge to show that it has topological singularities and Gribov’s copying[9](in the continuum theory, zero boundary conditions at inﬁnity have to be imposed[10]).The Gribov copying makes additional diﬃculties for describing monopole dynamics(even in the lattice gluodynamics [11]).In this letter,a new(dynamical)Abelian projection is proposed.It involves no gauge condition to be imposed on gaugeﬁelds.The eﬀective Abelian theory appears to be non-local,though it can be made local at the price of having some additional(ghost)ﬁelds.All conﬁgurations of gaugeﬁelds that turn into magnetic monopoles in the eﬀective Abelian theory are classiﬁed in a gauge invariant way.The eﬀective Abelian theory fully respects the Lorentz symmetry and the Gribov problem is avoided.Another important aspect of the QCD conﬁnement is the absence of propagating color charges,meaning that a nonperturbative propagator of colored particles,gluons or quarks, has no usual poles in the momentum space.It has been argued that such a behavior of a gluon propagator in the Coulomb gauge could be due to an inﬂuence of the so called Gribov horizon on long-waveﬂuctuations of gaugeﬁelds[9],[12].The result obviously depends on the gauge chosen,which makes it not very reliable.The situation looks more controversial if one recalls that a similar qualitative behavior of the gluon propagator has been found in the study of Schwinger-Dyson equations[13]. In this approach,the Gribov ambiguities have not been accounted for.So,the speciﬁc pole structure of the gluon propagator occurred through a strong self-interaction of gauge ﬁelds.In this letter,we would also like to propose a method for how to study gauge-variant quantities,like a gluon propagator,in the lattice QCD,avoiding any explicit gaugeﬁxing. The method is,hence,free of all the aforementioned gaugeﬁxing artifacts.It gives a hope that dynamical contributions(self-interaction of gaugeﬁelds)to the pole structure of the gluon propagator can be separated from the kinematical(gauge-ﬁxing)ones.2.To single out monopoles in non-Abelian gauge theory,oneﬁxes partially a gauge so that the gauge-ﬁxed theory possesses an Abelian gauge group being a maximal Abelian subgroup of the initial gauge group.The lattice formulation of the Abelian projection has been given in[8].The idea is to choose a function R(n)of link variables Uµ(n),n runs over lattice sites, such thatR(n)→g(n)R(n)g−1(n)(1) under gauge transformations of the link variablesUµ(n)→g(n)Uµ(n)g−1(n+ˆµ),(2) where g(n)∈G,G is a compact gauge group,andˆµis a unit vector in theµ-direction.A gauge is chosen so that R becomes an element of the Cartan subalgebra H,a maximal Abelian subalgebra of a Lie algebra X of the group G.In a matrix representation,the gauge condition means that oﬀ-diagonal elements of R are set to be zero.Clearly,the gaugeﬁxing is not complete.A maximal Abelian subgroup G H of G remains as a gauge group because the adjoint action(1)of G H leaves elements R∈H untouched.A conﬁguration Uµ(n)contains monopoles if the corresponding matrix R(n)has two coinciding eigenvalues.So,by construction,dynamics of monopoles appears to be gauge-dependent,or projection-dependent.It varies from gauge to gauge,from one choice of R to another[6].Yet,the monopole singularities are not the only ones in some Abelian projections[7].In addition,Abelian projections may suﬀer oﬀthe Gribov ambiguities [11].To restrict the full gauge symmetry to its maximal Abelian part and,at the same time, to avoid imposing a gauge condition on link variables,we shall use a procedure similar to the one discussed in[14]in the framework of continuumﬁeld theory.A naive continuum limit of our procedure poses some diﬃculties.To resolve them,a corresponding operator formalism has to be developed.It has been done in[15]for a suﬃciently large class of gauge theories.Consider a complex Grassmannﬁeldψ(n)(a fermion ghost)that realizes the adjoint representation of the gauge group:ψ(n)→g(n)ψ(n)g−1(n),(3)ψ∗(n)→g(n)ψ∗(n)g−1(n).(4) Let the fermion ghost be coupled to gaugeﬁelds according to the actionS f= n,µtrDµψ∗(n)Dµψ(n),(5)where Dµψ(n)=ψ(n+ˆµ)−U−1µ(n)ψ(n)Uµ(n)is the lattice covariant derivative in the adjoint representation.We assume thatψ(n)=ψi(n)λi,whereλi is a matrix represen-tation of a basis in X normalized as trλiλj=δij,andψi(n)are complex Grassmann variables.The partition function of the fermion ghostﬁeld readsZ f(β)= n(dψ∗(n)dψ(n))e−βS f=detβD TµDµ,(6)where the integration over Grassmann variables is understood,and D Tµdenotesa trans-position of Dµwith respect to a scalar product induced by n,µtr in(5).Note that the action(5)can be written in the form S f= ψ∗D TµDµψ.Consider a pair of real Lie-algebra-valued scalarﬁeldsϕ(n)andφ(n)(boson ghosts) with an actionS b=1(2π)dim G e−βS b=(detβD TµDµ)−1.(10) We have the identityZ b(β)Z f(β)=1.(11) By making use of this identity,the partition function of gaugeﬁelds can be transformed to the formZ Y M(β)=v−L G µ,n dUµ(n)e−βS W Z b(β)Z f(β)=(12)=v−L G D UµDψ∗DψDϕDφe−β(S W+S b+S f),(13)where S W is the Wilson action of gaugeﬁelds,v G a volume of the group manifold G,L a number of lattice sites,and D denotes a product of correspondingﬁeld diﬀerentials over lattice sites.The eﬀective actionS eff=S W+S b+S f(14) is invariant under gauge transformations(2)–(4)and(8),(9).The factor v−L G is included to cancel the gauge group volume factorizing upon the integration overﬁeld conﬁgurations in(13).Now we may take the advantage of having scalarﬁelds in the adjoint representation and restrict the gauge symmetry to the Cartan subgroup without imposing gauge conditions on the link variables.We make a change of the integration variables in(13)φ(n)=˜g(n)h(n)˜g(n)−1,(15) where˜g(n)belongs to the coset space G/G H,dim G/G H=dim G−dim G H,and h(n)∈H.Other newﬁelds denoted˜Uµ(n),˜ϕand˜ψ∗,˜ψare deﬁned as the corresponding gauge transformations of the initialﬁelds with g(n)=˜g−1(n).No restriction on their values is imposed.Relation(15)determines a one-to-one correspondence between old and new variables if and only if˜g(n)∈G/G H and h(n)∈K+,where K+is the Weyl chamber in H.An element h of the Cartan subalgebra H belongs to the Weyl chamber K+⊂H if for any simple rootω,(h,ω)>0;(,)stands for an invariant scalar product in X.In a matrix representation of X,it is proportional to tr(see[16],pp.187-190).With the help of the adjoint transformation,any element of a Lie algebra can be brought to the Cartan subalgebra.Since the Cartan subalgebra is invariant under the adjoint action of the Cartan subgroup,˜g(n)must be restricted to the coset G/G H.There are discrete transformations in G/G H which form the Weyl group W[16].Any element of W is a composition of reﬂections in hyperplanes orthogonal to simple roots in H.Its action maps H onto H itself.The Weyl group is a maximal isomorphism group of H[16].Therefore, a one-to-one correspondence in(15)is achieved if h(n)∈H/W≡K+.Due to the gauge invariance of both the measure and exponential in(13),the integral over group variables˜g(n)is factorized and yields a numerical vector that,being divided by v L G,results in(2π)−Lr,r=dim H=rank G.This factor is nothing but a volume of the Cartan gauge group G H.The integration over h(n)inquires a nontrivial measure,and the integration domain must be restricted to the Weyl chamber K+.So,in(13)we havev−1G dφ(n)=(2π)−r K+dh(n)µ(n).(16) The measure has the form[17]µ(n)= α>0(h(n),α)2,(17)whereαranges all positive roots of the Lie algebra X.The Cartan subalgebra is isomor-phic to an r-dimensional Euclidean space.The invariant scalar product can be thought as an ordinary vector scalar product in it.Relative orientations and norms of the Lie algebra roots are determined by the Cartan matrix[16].The integration measure for the otherﬁelds remains unchanged.For example,G=SU(2),then r=1,µ=h2(n)where h(n)is a real number because H SU(2)is isomorphic to a real axis.The Weyl chamber is formed by positive h(n).The su(3)algebra has two simple rootsω1,2(r=2).Their relative orientation is determined by the Cartan matrix,(ω1,ω2)=−1/2,|ω1,2|=1.The Weyl chamber is a sector on a plane(being isomorphic to H SU(3))with the angleπ/3.The algebra has three positive rootsω1,2andω1+ω2.So,the measure(17)is a polynom of the sixth order.Its explicit form is given by(28).Theﬁeld h(n)is invariant under Abelian gauge transformationsg H(n)h(n)g−1H(n)=h(n),g H(n)∈G H.(18) Therefore,after integrating out the coset variables˜g(n)in accordance with(16),we represent the partition function of Yang-Mills theory as a partition function of the eﬀective Abelian gauge theoryZ Y M(β)=(2π)−Lr D˜Uµe−βS W F(˜U),(19)whereF(˜U)=(detβD TµDµ)1/2 K+ n(dh(n)µ(n))e−βS H,(20)S H=1/2 n,µtr h(n+ˆµ)−˜U−1µ(n)h(n)˜Uµ(n) 2.(21)To obtain(19),we have done the integral over both the Grassmann variables and the boson ghostﬁeld˜ϕ(n),which yields(detβD TµDµ)1/2.The function F(˜U)is invariant only with respect to Abelian gauge transformations,˜Uµ(n)→g H(n)˜Uµ(n)g−1H(n+ˆµ).It provides a dynamical reduction of the full gauge group to its maximal Abelian subgroup.Since no explicit gauge condition is imposed on the link variables˜Uµ(n),the theory do not have usual gaugeﬁxing deceases,like the Gribov copies or horizon.We shall call the Abelian projection thus constructed a dynamical Abelian projection.3.Making a coset decomposition of the link variables[8]˜Uµ(n)=U Hµ(n)U chµ(n),(22) where U Hµ(n)=exp u Hµ(n),u Hµ(n)∈H and U chµ(n)=exp u chµ(n),u chµ(n)∈X⊖H,we conclude that lattice Yang-Mills theory is equivalent to an Abelian gauge theory with the actionS A=S W−β−1ln F.(23) The link variables U chµ(n)play the role of chargedﬁelds,while U Hµ(n)represents”electro-magnetic”ﬁelds.In the naive continuum limit,U Hµbecome Abelian potentialsU Hµ(n)→exp n+ˆµndxµA Hµ,A Hµ∈H.(24)Note that theﬁeld h(n)carries no Abelian charge and does not interact with U Hµas easily seen from(22)and(21)because(U Hµ)−1(n)h(n)U Hµ(n)=h(n).Bearing in mind results on simulations of the Polyakov loop dynamics on the lattice, one should expect that the Coulombﬁeld of charges in the eﬀective Abelian theory is squeezed into stable tubes connecting opposite charges.A mechanism of the squeezing has to be found from a study of dynamics generated by(23).First,one should verify if the dual mechanism can occur in the eﬀective Abelian theory.In our approach,conﬁgurations U Hµ(n)containing monopoles can exist.Kinematical arguments for this conjecture are rather simple.Let G be SU(N).In a matrix represen-tation,the change of variables(15)becomes singular at lattice sites where theﬁeldφ(n) has two coinciding eigenvalues.This condition implies three independent conditions on components ofφ(n)which can be thought as equations for the singular sites.At each moment of lattice time,these three equations determine a set of spatial lattice vertices (locations of monopoles).Therefore on a four-dimensional lattice,the singular sites form world-lines which are identiﬁed with world-lines of monopoles[2].The new link variables˜Uµ(n)=˜g(n)Uµ(n)˜g−1(n+ˆµ)(25)inquires monopole singularities via˜g(n).Their density can be determined along the lines given in[8].So,monopole dynamics is the dynamics of conﬁgurationsφ(n)with two equal eigenval-ues in the full theory(13).If such conﬁgurations are dynamically preferable,then one can expect that in the dynamical Abelian projection,eﬀective monopoles and antimonopoles form a condensate.All monopole-creating conﬁgurations of the scalarﬁeldφ(n)can easily be classiﬁed in a gauge invariant way.First of all we observe that the change of variables(15)is singular if its Jacobian vanishes nµ(n)=0.(26) We have to classify all conﬁgurationsφ(n)which lead toµ(n)=0.The polynom(17)is invariant with respect to the Weyl group.According to a theorem of Chevalley[16],any polynom in H invariant with respect to W is a polynom of basis(elementary)invariant polynoms tr h l(n)with l=l1,l2,...,l r being the orders of independent Casimir operators of G[16].Therefore,µ(n)=P(tr h l1(n),tr h l2(n),...,tr h l r(n))==P(trφl1(n),trφl2(n),...,trφl r(n))=0.(27) Solutions of this algebraic equation determine all conﬁgurationsφ(n)which will create monopoles in the dynamical Abelian projection(19).For G=SU(3),we have r=2,l1= 2,l2=3and[18]1µsu(3)(n)=As follows from(21)and(22),the Abelianﬁeld U Hµ(n)and the Cartanﬁeld h(n) are decoupled because[U Hµ(n),h(n)]=0.So,in the full theory,we deﬁne Abelian link variables by the relation[Uφµ(n),φ(n)]=0.(29) The coset decomposition assumes the formUµ(n)=Uφµ(n)U chµ(n).(30) One can regard it as a deﬁnition of chargedﬁelds U chµ(n)for given Uµ(n)andφ(n).Consider a vector potential corresponding to Uφµ(n)as determined by(24).It has theformAφµ(n)=rα=1Bαµ(n)eφα(n),(31)where Bαµ(n)are real numbers,and Lie algebra elements eφα(n)form a basis in the Cartan subalgebra constructed in the following wayeφα=λi trλiφlα−1.(32) It is not hard to be convinced that[18][eφα,eφβ]=0.(33) Since for any group G one of the numbers lαis equal to2,one of the elements(32) coincides withφitself.The elements(32)are linearly independent in X becausedet Pαβ≡det tr eφαeφβ=const·P.(34) So,a generic elementφof X has a stationary group Gφ⊂G with respect to the adjoint action of G in X,gφφg−1φ=φ,gφ∈Gφ.This stationary group is isomorphic to the Cartan subgroup G H.All linear combinations of the elements(32)form a Lie algebra of Gφ∼G H.In fact,the basis(32)can be constructed without an explicit matrix representation ofλi.We recall that for each compact simple group G and its Lie algebra X,there existr=rank G=dim H symmetrical irreducible tensors of ranks lα,d i1,i2,...,i lα,invariant withrespect to the adjoint action of G in X.Clearly,(eφα)i=d ij1...j lα−1φj1···φjlα−1.Now it is easy to see that the Abelian potentials Bφµ(n)are singular at lattice sites whereφ(n)satisﬁes(27).Indeed,from(31)we getBαµ(n)=Pαβ(n)tr eφβ(n)Aφµ(n),(35) where PαβPβγ=δαγ.The determinant of the matrix Pαβ(n)vanishes at the sites where µ(n)=P(n)=0.At these sites,the inverse matrix Pαβ(n)does not exist,and theﬁelds Bαµ(n)are singular.For unitary groups SU(N),lα=2,3,...,N,the singular sites form lines in the four-dimensional lattice[2],[8].These lines are world-lines of monopoles.5.The above procedure of avoiding explicit gaugeﬁxing can be implemented to re-move the gauge arbitrariness completely and,therefore to study gauge-variant correlators,like the gluon propagator,or some other quantities requiring gaugeﬁxing on the lattice [20].The advantage of dynamical gaugeﬁxing is that it is free of all usual gaugeﬁxing dynamical artifacts,Gribov’s ambiguities and horizon[14].It is also Lorentz covariant.Recent numerical studies of the gluon propagator in the Coulomb gauge[19]show that it can beﬁt to a continuum formula proposed by Gribov[9].The same predictions were also obtained in the study of Schwinger-Dyson equations where no eﬀects of the Gribov horizon have been accounted for[13].The numerical result does not exclude also a simple massive boson propagator for gluons[19].So,the problem requires a further investigation.Gaugeﬁxing singularities(the Gribov horizon)occur when one parametrizes the topo-logically nontrivial gauge orbit space by Cartesian coordinates.So,these singularities are pure kinematical and depend on the parametrization(or gauge)choice.They may,how-ever,have a dynamical evidence in a gauge-ﬁxed theory[21].For example,a mass scale determining a nonperturbative pole structure of the gluon propagator in the infrared region(gluon conﬁnement)arises from the Gribov horizon[9],[12]if the Lorentz(or Coulomb)gauge is used.From the other hand,no physical quantity can depend on a gauge chosen.There is no gauge-invariant interpretation(or it has not been found yet) of the above mass scale.That is what makes the gluon conﬁnement model based on the Gribov horizon looking unsatisfactory.Here we suggest a complete dynamical reduction of the gauge symmetry in lattice QCD,which involves no gauge condition imposed on gaugeﬁelds and,hence,is free of the corresponding kinematical artifacts.For the sake of simplicity,we discussﬁrst the gauge group SU(2).Consider two auxiliary(ghost)complexﬁeldsψandφ,Grassmann and boson ones,respectively.Let they realize the fundamental representation of SU(2),i.e.they are isotopic spinors.The identity(11)assumes the formZ b(β)Z f(β)= Dφ+DφDψ+Dψe−β(S b+S f)=1,(36) where S f= n(∇µψ)+∇µψand S b=1/2 n(∇µφ)+∇µφ,and the lattice covariant deriva-tive in the fundamental representation is deﬁned by∇µφ(n)=φ(n+ˆµ)−U−1µ(n)φ(n). Inserting the identity(36)into the integral representation of the Yang-Mills partition func-tion(12),we obtain an eﬀective gauge invariant action.The ghostﬁelds are transformed asφ(n)→g(n)φ(n)andψ(n)→g(n)ψ(n).In the integral(13),we go over to new variables to integrate out the gauge group volumedφ+(n)dφ(n)=v su(2)∞dρ(n)ρ3(n),(37)whereφ(n)=˜g(n)χρ(n),χ+=(10),ρ(n)is a real scalarﬁeld,and˜g(n)is a generic element of SU(2).A new fermion ghostﬁeld and link variables˜Uµare related to the old ones via a gauge transformation with g(n)=˜g−1(n).Since the eﬀective action is gaugeinvariant,the integral over˜g(n)yields the gauge group volume v Lsu(2).We end up withthe eﬀective theoryZ Y M(β)= D˜Uµe−βS W F(˜U),(38)F(˜U)=(detβ∇+µ∇µ)1/2∞0 ndρ(n)ρ3(n) e−βS(ρ),(39)S(ρ)=1/2 n,µ ρ(n+ˆµ)−χ+˜U−1µ(n)χρ(n) 2.(40)The function(39)is not gauge invariant and provides the dynamical reduction of the SU(2) gauge symmetry.A formal continuum theory corresponding to(38)has been proposed and discussed in[14].Expectation values of a gauge-variant quantity G(U)are determined byG(U) ≡ F(U)G(U) W= D Uµe−βS W F(U)G(U).(41)For example,for the gluon two-point correlator one sets G(U)=Aµ(n)Aµ′(n′)where the gluon vector potential on the lattice reads2iaAµ(n)=Uµ(n)−U+µ(n)−1Though the integration domain is restricted in the sliced path integral(20),this re-striction will disappear in the continuum limit because of contributions of trajectories reﬂected from the boundary∂K+[17],[18].It is rather typical for gauge theories that a scalar product for physical states involves an integration over a domain with boundaries which is embedded into an appropriate Euclidean space.The domain can even be com-pact as,for example,in two-dimensional QCD[22].In the path integral formulation,this feature of the operator formalism is accounted for by appropriate boundary conditions for the transition amplitude(or the transfer matrix)rather than by restricting the integration domain in the corresponding path integral[22],[23].In turn,the boundary conditions are to be found from the operator formulation of quantum gauge theory[18],[22],[23].So,a study of the continuum limit requires an operator formulation of the dynamical reduction of a gauge symmetry,which has been done in[15].The dynamical Abelian projection can be fulﬁlled in the continuum operator formal-ism.The whole discussion of monopole-like singular excitations given in sections3and4 can be extended to the continuum theory.So,it determines Lorentz covariant dynamics of monopoles free of gaugeﬁxing artifacts.To study monopole dynamics in the continuum Abelian gauge theory,one has to introduce monopole-carrying gaugeﬁelds[24].AcknowledgementI express my gratitude to F.Scholtz for valuable discussions on dynamical gauge ﬁxing,to A.Billoire,A.Morel and V.K.Mitrjushkin for providing useful insights about lattice simulations,and D.Zwanziger and M.Schaden for a fruitful discussion on the Gribov problem.I would like to thank J.Zinn-Justin for useful comments on a dynamical evidence of conﬁguration space topology in quantumﬁeld theory.I am very grateful to H.Kleinert for a stimulating discussion on monopole dynamics.References[1]T.Suzuki and I.Yotsuyanagi,Phys.Rev.D42(1990)4257.[2]G.’t Hooft,Nucl.Phys.B190[FS3](1981)455.[3]H.Shiba,T.Suzuki,Phys.Lett.B333(1994)461.[4]J.D.Stack,S.D.Neiman and R.J.Wensley,Phys.Rev.D50(1994)3399.[5]S.Mandelstam,Phys.Rep.23(1976)245;’t Hooft,in:High Energy Physics,ed.M.Zichichi(Editrice Compositori,Bologna, 1976).[6]see,for example,L.Del Debbio,A.Di Giacomo,G.Pafutti and P.Pier,Phys.Lett.B 355(1995)255.[7]M.N.Chernodub,M.I.Polikarpov and V.I.Veselov,Phys.Lett.B342(1995)303.[8]A.S.Kronfeld,G.Schierholz and U.-J.Wiese,Nucl.Phys.B293(1987)461.[9]V.N.Gribov,Nucl.Phys.B139(1978)1.[10]I.M.Singer,Commun.Math.Phys.60(1978)7.[11]S.Hioki,S.Kitahara,Y.Matsubara,O.Miyamura,S.Ohno and T.Suzuki, Phys.Lett.B,271(1991)201.[12]D.Zwanziger,Nucl.Phys.B378(1992)525.[13]M.Stingl,Phys.Rev.D34(1986)3863.[14]F.G.Scholtz and G.B.Tupper,Phys.Rev.D48(1993)1792.[15]F.G.Scholtz and S.V.Shabanov,Supersymmetric quantization of gauge theories, FU-Berlin preprint,FUB-HEP/95-12,1995.[16]S.Helgason,Diﬀerential Geometry,Lie Groups,and Symmetric Spaces(Academic Press,NY,1978).[17]L.V.Prokhorov and S.V.Shabanov,Phys.Lett.B216(1989)341;pekhi34(1991)108.[18]S.V.Shabanov,Theor.Math.Phys.78(1989)411.[19]C.Bernard,C.Parrinello and A.Soni,Phys.Rev.D49(1994)1585.[20]Ph.de Forcrand and K.-F.Liu,Nucl.Phys.B(Proc.Suppl.)30(1993)521.[21]V.G.Bornyakov,V.K.Mitrjushkin,M.M¨u ller-Preussker and F.Pahl,Phys.Lett.B 317(1993)596.[22]S.V.Shabanov,Phys.Lett.B318(1993)323.[23]S.V.Shabanov,Phys.Lett.B255(1991)398;Mod.Phys.Lett.A6(1991)909.[24]H.Kleinert,Phys.Lett.B293(1992)168.。

巴兹尔·伯恩斯坦教育、符号控制与认同英文版

巴兹尔·伯恩斯坦.教育、符号控制与认同英文版全文共3篇示例，供读者参考篇1Basil Bernstein: Education, Symbolic Control, and IdentityIntroductionBasil Bernstein was a British sociologist known for his work in the field of education. He developed the theory of symbolic control, which focuses on how educational systems create and maintain social inequality through the use of language and symbols. This theory also examines how individuals' identities are shaped by their access to different forms of knowledge and discourse. In this paper, we will explore Bernstein's theory of symbolic control and its implications for education and identity formation.Symbolic ControlAccording to Bernstein, symbolic control refers to the ways in which language and symbols are used to regulate behavior and knowledge within educational systems. He argued that different forms of knowledge are valued differently in society, and that those who have access to the dominant forms ofknowledge are able to exert control over others. This control is exercised through the use of language and symbols that reinforce existing power structures and social hierarchies.In Bernstein's theory, there are two types of educational codes: the elaborated code and the restricted code. The elaborated code is associated with formal education and is characterized by complex language and abstract concepts. Those who have access to this code are typically from more privileged backgrounds and are able to navigate the demands of the educational system more easily. In contrast, the restricted code is associated with informal education and is characterized by simpler language and concrete concepts. Those who use this code are often marginalized in educational settings and may struggle to succeed within the system.Implications for EducationBernstein's theory of symbolic control has important implications for education policy and practice. It suggests that educational systems can perpetuate social inequality by privileging certain forms of knowledge over others. In order to create a more equitable system, educators must be aware of the ways in which language and symbols are used to control behavior and knowledge within educational settings.One way to address this issue is to promote linguistic diversity within the classroom. By valuing the different forms of language and discourse that students bring to the educational environment, educators can create a more inclusive learning environment that allows all students to succeed. This can also help to break down the barriers that exist between different social groups and promote greater social cohesion within the school community.Implications for Identity FormationBernstein's theory also has implications for how individuals construct their identities within society. He argued that individuals' access to different forms of knowledge and discourse shapes their sense of self and their place in the world. Those who have access to the dominant forms of knowledge may develop a strong sense of self-efficacy and confidence, while those who are marginalized may struggle to develop a positive self-concept.In order to promote positive identity formation, educators must be mindful of the ways in which language and symbols shape students' perceptions of themselves and others. By creating a supportive and inclusive learning environment thatvalues linguistic diversity, educators can help students to develop a strong sense of self and a positive identity.ConclusionIn conclusion, Basil Bernstein's theory of symbolic control provides a valuable framework for understanding how educational systems create and maintain social inequality. By examining the ways in which language and symbols are used to regulate behavior and knowledge within educational settings, educators can work to create a more equitable system that promotes social justice and positive identity formation. By valuing linguistic diversity and promoting inclusive practices, educators can help all students to succeed and thrive within the educational environment.篇2Basil Bernstein: Education, Symbolic Control, and IdentityIntroductionBasil Bernstein was a prominent British sociologist known for his work on education, symbolic control, and identity. His theoretical framework has been influential in exploring the role of education in shaping social hierarchy, linguistic codes, and cultural practices. In this document, we will examine Bernstein'skey ideas and their implications for understanding the relationship between education, symbolic control, and identity.Basil Bernstein's Theory of EducationBernstein's theory of education is grounded in the idea that educational systems reflect and perpetuate existing social structures. He argued that there are two types of educational codes that shape how knowledge is transmitted and acquired: the restricted code and the elaborated code. The restricted code is associated with working-class communities and is characterized by its reliance on implicit meanings, shared cultural knowledge, and context-specific communication. In contrast, the elaborated code is associated with middle-class communities and is characterized by its explicit rules, formal language, and abstract reasoning.According to Bernstein, the educational system favors the elaborated code, which can present a barrier for working-class students who are more familiar with the restricted code. This can lead to educational inequality and perpetuate social hierarchies. Bernstein's theory highlights how language and communication styles are linked to social class and how they shape educational outcomes.Symbolic Control and IdentityBernstein also examined the concept of symbolic control, which refers to the ways in which power is exercised through language, knowledge, and cultural symbols. He argued that symbolic control plays a key role in shaping social relations and maintaining social order. Through language and cultural practices, dominant groups can assert their authority and reinforce existing power dynamics.Symbolic control is closely linked to identity formation, as individuals internalize the values, norms, and beliefs of the dominant culture. Education plays a central role in this process, as it transmits cultural knowledge and shapes individual identities. Bernstein's work highlights how education is not only about acquiring knowledge but also about socializing individuals into specific cultural identities and roles.Implications for EducationThe implications of Bernstein's theory for education arefar-reaching. It suggests that educational systems can either reproduce or challenge existing social inequalities, depending on how they are structured. To promote greater equity and inclusivity, educators need to be aware of the different language codes and cultural practices that students bring to the classroom. They should also strive to create a learning environment that isresponsive to the diverse backgrounds and experiences of students.Furthermore, Bernstein's theory calls attention to the ways in which power operates through language and symbols in education. Educators should be critical of how knowledge is constructed, represented, and transmitted in the curriculum. They should also encourage students to question dominant narratives and challenge systems of oppression and discrimination.ConclusionIn conclusion, Basil Bernstein's work on education, symbolic control, and identity offers valuable insights into the complex relationship between education and society. His theory highlights the role of language, communication styles, and cultural practices in shaping educational outcomes and social hierarchies. By understanding and addressing the issues raised by Bernstein, educators can work towards creating a more inclusive and equitable educational system that empowers all students to thrive.篇3Basil Bernstein: Education, Symbolic Control, and IdentityIntroductionBasil Bernstein was a British sociologist known for his work on the sociology of education. His theories regarding the role of education in shaping social identity and maintaining social order have had a significant impact on educational research and practice. This paper will explore Bernstein's concept of symbolic control in education and its influence on the formation of social identity.Symbolic Control in EducationBernstein's theory of symbolic control posits that education functions as a mechanism through which dominant social groups maintain control over society by regulating access to knowledge and cultural capital. According to Bernstein, the educational system is a key site of symbolic control, where the dominant ideology is perpetuated through the transmission of symbolic codes and rules that dictate acceptable forms of knowledge and behavior.Bernstein identified two types of symbolic control in education: restricted and elaborated codes. Restricted codes are characterized by their reliance on implicit, context-specific meanings and are typically associated with working-class cultural practices. Elaborated codes, on the other hand, are characterizedby their explicit, generalizable meanings and are associated with middle-class cultural practices. In Bernstein's view, the dominant social group dictates which code is considered legitimate within the educational system, thereby perpetuating social inequalities.Implications for Social IdentityThe concept of symbolic control has important implications for the formation of social identity within the educational system. Bernstein argued that the symbolic codes and rules enforced in schools act as a form of cultural capital that confers status and privilege on those who are able to conform to them. As a result, students from working-class backgrounds may struggle to succeed in the educational system because their cultural practices do not align with the dominant code.Moreover, Bernstein's theory suggests that the educational system plays a role in shaping individuals' sense of self and social identity. By prescribing certain forms of knowledge and behavior as legitimate, the system reinforces existing social hierarchies and identities. Students who do not conform to the dominant code may experience feelings of marginalization and alienation, leading to a diminished sense of self-worth and belonging.Implications for Educational PracticeBernstein's theory of symbolic control challenges educators to critically examine the ways in which the educational system perpetuates social inequalities and reinforces dominant ideologies. To create a more inclusive and equitable learning environment, educators must be mindful of the ways in which cultural capital is distributed within the classroom and work to provide all students with access to the symbolic codes and rules valued by the dominant culture.One way educators can address the issues raised by Bernstein's theory is by incorporating a variety of cultural practices and perspectives into the curriculum. By exposing students to a diverse array of symbolic codes and rules, educators can help to broaden students' understanding of what constitutes legitimate knowledge and behavior, thereby fostering a more inclusive and tolerant learning environment.ConclusionBasil Bernstein's theory of symbolic control in education provides valuable insights into the ways in which the educational system functions as a mechanism of social control and identity formation. By examining the distribution of cultural capital within the educational system, educators can work to create a more equitable and inclusive learning environment that recognizesand values the diversity of students' cultural backgrounds and experiences. Through a critical engagement with Bernstein's theory, educators can play a role in challenging and transforming the structures of power and privilege that shape students' educational experiences.。

Adler-Bardeen theorem and vanishing of the gauge beta function

a rXiv:h ep-th/932123v125Fe b1993UNIVERSIT ´E DE GEN `EVE SCHOLA GEVENENSIS MDLIX Adler-Bardeen theorem and vanishing of the gauge beta function O.Piguet 1and S.P.Sorella 1D´e partement de Physique Th´e orique 24,quai Ernest Ansermet CH –1211Gen`e ve 4(Switzerland)UGVA—DPT 1992/07–774AbstractThe proof of the non-renormalization theorem for the gauge anomaly of four-dimensional theories is extended to the case of models with a vanishing one-loop gauge beta function.1IntroductionThe non-renormalization theorem[1,2,3,4,5,6,7]of the four-dimensional gauge an-omaly[8]is of fundamental importance for the construction of consistent high energy physics models.This theorem states that the anomaly coeﬃcient vanishes at all orders of perturbation theory if it vanishes in the one-loop approximation.The original proof of Bardeen[2]of the theorem for the non-Abelian gauge anomaly is based on an analysis of Feynman graphs:one shows that if the one-loop triangle anomaly cancels,then there exists a gauge invariant regularization valid to all ter on it was recognized by Zee[3]in the Abelian case,and by Costa et al.[5]in the non-Abelian case,that it is possible to give a proof based on the combined use of the gauge(or BRS)Ward identities and of the Callan-Symanzik equation.In the same time Lowenstein and Schroer[4],and later on Bandelloni et al.[6],achieved,with the quantum action principle[9]as the main tool,an algebraic,regularization independent,version of the previous proof.The main advantage of a regularization independent proof is that it can be naturally extended to more sophisticated theories,e.g.supersymmetric gauge theories and topolog-ical theories,for which no regularization preserving all the symmetries is available.The regularization independent proofs given up to the present time[4,6,7],as well as the proof given in[5],based on dimensional regularization,although very general,have their domain of validity restricted by the”technical”assumption that the one-loop beta function for the gauge coupling[10]should not identically vanish.Even if this assumption covers a very large class of models including the standard model,there is a wide set of interesting theories for which the one-loop gauge beta function do indeed vanish.This set includes in particular gauge models with N=1supersymmetry which may have some relevance in the construction of grand uniﬁed theories[11].Moreover the supersymmetric gauge models with a vanishing one-loop gauge beta function[12,13,14]are the starting point towards the construction of ultravioletﬁnite theories[15].It is therefore needed to have a proof which also applies to the case of a one-loop vanishing gauge beta function. This is the aim of the present paper.The demonstration follows the diﬀerential geometry setup of the descent equations which are known to characterize the anomaly[16,17,18].It is the continuation of a previ-ous work of the authors[7],where a completely algebraic proof of the non-renormalization theorem was given in the case of a non-vanishing one-loop gauge beta function.The main ingredient,as shown in[7],is the vanishing of the anomalous dimensions of the diﬀerential form operators which are solutions of the descent equations.In the proof one has to use the ghost equation shown in[19],which controls the coupling of the Faddeev-Popov ghost c.However this equation holds only in the Landau gauge,and we will therefore have to present our arguments in this particular gauge.The extension of the non-renormalization theorem to a general linear covariant gauge can be easily performed by following the techniques of extended BRS invariance[20],as it was done in[21].Let usﬁnish this introduction by some remarks.The proof we are going to present here concerns the non-supersymmetric theories for simplicity,the generalization to the supersymmetric case being apparently straightfor-ward.There is indeed a supersymmetric version of the descent equations which allows for an algebraic set up analog to the non-supersymmetric one and which leads to a unique characterization of the anomaly[22,23].Our proof covers the cases of theories for which the gauge beta function does not vanish to higher than one-loop order.It does not hold,as it stands,in cases of higher order vanishing gauge beta function.This proof in particular would not apply to the topological theories which have vanish-ing beta functions to all orders[24],but to the present time there is no known exemple of such a theory having a gauge anomaly,given as a non-trivial solution of the Wess-Zumino consistency conditions[25,24].It is however relevant for the construction ofﬁnite supersymmetric gauge theories[15]. Indeed such a construction starts with a model whose gauge beta function vanishes only in the one-loop approximation and depends on a certain number of independent couplings (a gauge and a few Yukawa couplings).It is at this stage that the non-renormalization theorem is needed.The all order vanishing of the whole set of beta functions is then ensured by requiring the Yukawa couplings to be a function of the gauge coupling constant, according to the”reduction of coupling constants”theory of Zimmermann[26].2Properties of Yang-Mills theories in the Landau gaugeThe purpose of this section is to give a brief summary of the algebraic properties which characterize a four-dimensional gauge theory quantized in the Landau gauge[19,7].Let us consider a massless gauge theory whose complete classical actionΣ,using the same notations of ref.[7],reads:Σ=Σinv+Σgf+Σext,(2.1) whereΣinv,Σgf andΣext are respectively the gauge invariant action,the Landau gauge ﬁxing term and the externalﬁeld dependent part.They are given by:Σinv= d4x −1Σext = d 4x −Ωaµ(D µc )a +12f abc c b c c ,s ¯c a=b a ,sb a=0,sφ=−ic a T a φ,(2.5)is expressed by the classical Slavnov identity:S (Σ)= d 4x δΣδA a µ+δΣδc a +δΣδφ+b a δΣ12A b νA c ρA d σ ,(2.8)D abcd =d abn f ncd +d acn f ndb +d adn f nbc ,(2.9)where d abc is the totally symmetric invariant tensor and Γis the vertex functionalΓ=Σ+O (¯h ).(2.10)One has to note thateq.(2.7)implies that the gauge anomaly is absent at the one loop level,i.e.we consider the case in which the coeﬃcient r of the one loop triangle diagram is equal to zero,due to an appropriate choice of the matter ﬁeld representation [8].In such a situation the Adler-Bardeen theorem [8,5,6,7]states that the coeﬃcient r in (2.7)identically vanishes.The vertex functional Γ,besides the anomalous Slavnov identity (2.7),is known to obey:i)the Landau gauge-ﬁxing condition and the antighost equation[29]δΓδ¯c a +∂δΓδϕ=0,(2.12)iii)the ghost equation[19]d4x δΓδb c =∆a,(2.13)∆a= d4x f abcΩbµA cµ−f abcσb c c+iY T aφ .(2.14)These conditions,together with(2.7),allow to write a Callan-Symanzik equation which is Slavnov invariant up to the order¯h n[7],i.e.:CΓ= µ∂∂g+¯h iβi∂δA aµ−b aδδ¯c a−Ωaµδδφ−Yδδc a−σaδΓ[A ·Γ]=A +O (¯h A ),(2.19)which possesses the following properties:C [A ·Γ]=¯h S Γ[ˆ∆·Γ]+O (¯h n +1),(2.20)S Γ[A ·Γ]=O (¯h n ),(2.21)where S Γis the linearized Slavnov operatorS Γ= d 4x δΓδA a µ+δΓδΩaµ+δΓδc a +δΓδσa+δΓδφ+δΓδY +b a δwe get,to the lowest order (i.e.order n +1)in ¯h ,the equationβ(1)g ∂r∂λi A +µ∂B∂µ=0,(3.4)and that the gauge anomaly A cannot be written as a local S Σ-variation,it follows that(3.3)is equivalent to the two conditions:β(1)g ∂r ∂λi =0,(3.5)andS Σ(∆n +1c −r ˆ∆)=0.(3.6)In the case in which the one loop gauge beta function β(1)g does not identically vanish,(3.5)implies the Adler-Bardeen theorem [5,6,7].However,for the time being,we keep (3.5)just as an algebraic equation in view of the fact that we will allow the coeﬃcientβ(1)g to vanish.In this case eq.(3.5)implies only that r does not depend on the self matter couplings λi .Let us turn now to the analysis of the second condition (3.6).This equation showsthat the diﬀerence (∆n +1c −r ˆ∆),being a Slavnov invariant quantity,can be expanded in terms of the elements of the invariant basis [29]:∂Σ∂λi ,N A Σ,N φΣ,N c Σ .(3.7)This amounts to rewrite the Callan-Symanzik equation (2.15)asC Γ+¯h n +1γc N c Γ=r ¯h n +1ˆ∆+O (¯h n +2),(3.8)where the ghost-anomalous dimension has reappeared,in agreement with the fact that its absence is ensured only up to the order ¯h n [7].Finally,repeating the same argument as in [6],the Callan-Symanzik equation (3.8)extends to the order ¯h n +2asCΓ+¯h n+1γc N cΓ=r¯h n+1[ˆ∆·Γ]+¯h n+2∆n+2c+O(¯h n+3),(3.9)where∆n+2c is a local integrated functional.The interesting feature of this equation isthat the general local polynomial∆n+1c of eq.(2.15)has been replaced by the term[ˆ∆·Γ],which is the same quantity as the one appearing in the Callan-Symanzik equation for the anomaly insertion(2.20).This step will turn out to be very useful in the discussion of the model at the order¯h n+2.4Order¯h n+2This section is devoted to the analysis of the anomalous Slavnov identity at the order ¯h n+2,i.e.to the algebraic characterization of the local polynomial B in eq.(3.1).To do this we will use property(2.21)which shows that the anomaly insertion[A·Γ] is Slavnov invariant up to the order¯h n.Applying the linearized operator SΓ(2.22)to both sides of equation(3.1)and making use of(2.21)and of the exact relationSΓS(Γ)=0,(4.1) weﬁnd,to the lowest order in¯h(remember that n≥2),the equationSΣB=0.(4.2) This condition implies that the local polynomial B is Slavnov invariant with ghost number one,and then can be written asB=ˆr A+SΣQ,(4.3) where A is the gauge anomaly(2.8),ˆr is an arbitrary coeﬃcient and Q a local integrated polynomial of dimension four and ghost number zero.Moreover,since B appears in the Slavnov equation(3.1)at the order¯h n+1,it follows that the cohomological trivial term SΣQ can be reabsorbed in the eﬀective actionΓas a local counterterm without aﬀecting properties(2.20),(2.21)and the Callan-Symanzik equation(3.9).The Slavnov identity(3.1)becomes then:S(Γ)=r¯h n[A·Γ]+ˆr¯h n+1A+O(¯h n+2),(4.4) and extends to the order¯h n+2asS(Γ)=r¯h n[A·Γ]+ˆr¯h n+1[A·Γ]+¯h n+2ˆB+O(¯h n+3),(4.5)whereˆB is an integrated local polynomial of ultraviolet dimension four and ghost number one.It is important to note that we cannot iterate the previous arguments to characterize ˆB,i.e.property(2.21)allows to characterize only the order¯h n+1.Commuting now the Callan-Symanzik equation(3.9)with the Slavnov identity(4.5) and using eqs.(2.20),(3.5)and the algebraic relations:N c S(Γ)=SΓN cΓ,(4.6)N c[A·Γ]=A+O(¯h),(4.7) we get,to the lowest order(i.e.n+2)in¯h,the equationβ(2)g∂r∂λi+β(1)g∂ˆr∂λi A+µ∂ˆB+ iβ(2)i∂r∂g+ iβ(1)i∂ˆr∂g+ iβ(1)i∂r∂gβ(2)g∂r∂λi+β(1)g∂ˆr∂λi =0.(5.2) To discuss the consequencies of these equations on the coeﬃcients(r,ˆr)let us consider ﬁrst the case in which the one loop gauge beta functionβ(1)g is nonvanishing.In this case,as shown in[5,6,7],eq.(5.1)implies the Adler-Bardeen theorem,i.e.that r=0.Equation(5.2)reduces to:β(1)g ∂ˆr∂λi=0,(5.3)from which it follows that alsoˆr vanishes;improving then the validity of the Slavnov identity(4.5)to all orders of perturbation theory by induction.Let us consider now the case in whichβ(1)g=0,β(2)g=0.(5.4) Equations(5.1),(5.2)become:iβ(1)i∂r∂g+ iβ(2)i∂r∂λi=0.(5.6)Eq.(5.5)implies that r is independent from the self matter couplingsλi.It follows then that eq.(5.6)reads:β(2)g ∂r∂λi=0,(5.7)which is easily seen to imply that r=0,howing to the fact that r depends only on the gauge coupling g and that the two loop gauge beta functionβ(2)g[30]is not identically zero for vanishing self matter couplings.This concludes the proof of the Adler-Bardeen theorem in the case of vanishing one loop gauge beta function.References[1]S.L.Adler and W.A.Bardeen,Phys.Rev.182(1969)1517;[2]W.A.Bardeen,Renormalization of Yang-Millsﬁelds and application to particlephysics,C.N.R.S.(Marseille)report72/p470,29(1972);W.Bardeen,proceedings of the16th International Conference on High Energy Physics,Fermilab1972,vol.II,p.295;[3]A.Zee,Phys.Rev.Lett.29(1972)1198;[4]J.H Lowenstein and B.Schroer,Phys.Rev.D7(191975)1929;[5]G.Costa,J.Julve,T.Marinucci and M.Tonin,Nuovo Cim.38A(1977)373;[6]G.Bandelloni,C.Becchi,A.Blasi and R.Collina,Comm.Math.Phys.72(1980)239;[7]O.Piguet and S.P.Sorella,Nucl.Phys.B381(1992)373;[8]S.L.Adler,Phys.Rev.117(1969)2426;J.S.Bell and R.Jackiw,Nuovo Cim.60(1969)47;J.Schwinger,Phys.Rev.82(1951)664;W.A.Bardeen,Phys.Rev.184(1969)1848;[9]Y.M.P Lam,Phys.Rev.D6(1972)2145,2161;T.E.Clark and J.H.Lowenstein,Nucl.Phys.B113(1976)109;[10]D.J.Gross and F.Wilczek,Phys.Rev.D8(1973)3633;[11]L.E.Ibanez,Grand uniﬁcation,supersymmetry,superstrings:an introduction tophysics beyond the standard model,lectures given at the1988ASI School,ST.Croix, Virgin Islands,preprint CERN-TH-5237/88;[12]A.J.Parkes and P.C.West,Phys.Lett.B138(1984)99;Nucl.Phys.B256(1985)340;D.R.T.Jones and L.Mezincescu,Phys.Lett.B138(1984)293;[13]S.Hamidi,J.Patera and J.H.Schwarz,Phys.Lett.B141(1984)349;[14]S.Rajpoot and J,G.Taylor,Int.J.Theor.Phys.25(1986)117;[15]O.Piguet and K.Sibold,Phys.Lett.B177(1986)373;C.Lucchesi,O.Piguet and K.Sibold,Helv.Phys.Acta61(1988)321;[16]J.Manes,R.Stora and B.Zumino,Comm.Math.Phys.102(1985)157;B.Zumino,Nucl.Phys.B253(1985)477;[17]M.Dubois-Violette,M.Talon and C.M.Viallet,Comm.Math.Phys.102(1985)105;Ann.Inst.Henri Poincar´e44(1986)103;[18]F.Brandt,N.Dragon and M.Kreuzer,Phys.Lett.B231(1989)263;Nucl.Phys.B332(1990)224,250;F.Brandt,PhD Thesis,University of Hanover(1991),unpublished;[19]A.Blasi,O.Piguet and S.P.Sorella,Nucl.Phys.B356(1991)154;[20]O.Piguet and K.Sibold,Nucl.Phys.B253(1985)517;[21]C.Lucchesi,O.Piguet and K.Sibold,Int.J.Mod.Phys.A2(1987)385;[22]O.Piguet and 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Supersymmetric Gauge Theories and Gravitational Instantons

a rXiv:h ep-th/987237v13J ul1998CALT-68-2190SUPERSYMMETRIC GAUGE THEORIES AND GRA VITATIONAL INSTANTONS SERGEY A.CHERKIS California Institute of Technology,Pasadena CA 91125,USA E-mail:cherkis@ Various string theory realizations of three-dimensional gauge theories relate them to gravitational instantons 1,Nahm equations 2and monopoles 3.We use this correspondence to model self-dual gravitational instantons of D k -type as moduli spaces of singular monopoles,ﬁnd their twistor spaces and metrics.This work provides yet another example of how string theory unites seem-ingly distant physical problems.(See references for detailed results.)The central object considered here is supersymmetric gauge theories in three di-mensions.In particular,we shall be interested in their vacuum structure.The other three problems that turn out to be closely related to these gauge theories are:•Nonabelian monopoles of Prasad and Sommerﬁeld,which are solutions of the Bogomolny equation ∗F =D Φ(where F is the ﬁeld-strength of a nonabelian connection A =A 1dx 1+A 2dx 2+A 3dx 3and Φis a nonabelian Higgs ﬁeld).•An integrable system of equations named after Nahm dT i2εijk [T j ,T k ],(1)for T i (s )∈u (n ).These generalize Euler equations for a rotating top.•Solutions of the Euclideanized vacuum Einstein equation called self-dual gravitational instantons ,which are four-dimensional manifolds with self-dual curvature tensorR αβγδ=1The latter provide compactiﬁcations of string theory and supergravity that preserve supersymmetry and are of importance in euclidean quantum gravity.The compact examples are delivered by a four-torus and K3.The noncom-pact ones are classiﬁed according to their asymptotic behavior and topology.Asymptotically Locally Euclidean (ALE)gravitational instantons asymptoti-cally approach R 4/Γ(Γis a ﬁnite subgroup of SU (2)).These were classiﬁed by Kronheimer into two inﬁnite (A k and D k )series and three exceptional (E 6,E 7and E 8)cases according to the intersection matrix of their two-cycles.Asymp-totically Locally Flat (ALF)spaces approach the R 3×S 1 /Γmetric.(To be more precise S 1is Hopf ﬁbered over the two-sphere at inﬁnity of R 3.)Sending the radius of the asymptotic S 1to inﬁnity we recover an ALE space of some type,which will determine the type of the initial ALF space.For example,the A k ALF is a (k+1)-centered multi-Taub-NUT space.Here we shall seek to describe the D k ALF space.M theory on an A k ALF space is known to describe (k+1)D6-branes of type IIA string theory.Probing this background with a D2-brane we obtain an N =4U (1)gauge theory with (k+1)electron in the D2-brane worldvolume.As the D2-brane corresponds to an M2-brane in M theory,a vacuum of the above gauge theory corresponds to a position of the M2-brane on the A k ALF space we started with.Thus the moduli space of this gauge theory is the A k ALF.Next,considering M theory on a D k ALF one recovers 4k D6-branes parallel to an orientifold O 6−.On a D2-brane probe this time we ﬁnd an N =4SU(2)gauge theory with k matter multiplets.Its moduli space is the D k ALF.So far we have related gauge theories and gravitational instantons .D3NS5NS5pp p k D3Figure 1:The brane conﬁguration corresponding to U (2)gauge theory with k matter mul-tiplets on the internal D3-branes.There is another way of realizing these gauge theories.Consider the Chalmers-Hanany-Witten conﬁguration in type IIB string theory (Figure 1).2In the extreme infrared limit the theory in the internal D3-branes will appear to be three-dimensional with N=4supersymmetry.This realizes the gauge theory we are interested in.A vacuum of this theory describes a particular position of the D3-branes.In the U(2)theory on the NS5-branes the internal D3-branes appear as nonabelian monopoles,while every external semiinﬁnite D3-brane appears as a Dirac monopole in the U(1)of the lower right corner of the U(2).Thus the moduli space of such monopole conﬁgurations of non-abelian charge two and with k singularities is also a moduli space of the gauge theory in question.Another way of describing the vacua of the three-dimensional theory on the D3-branes is by considering the reduction of the four-dimensional theory on the interval.For this reduction to respect enough supersymmetry theﬁelds (namely the Higgsﬁelds of the theory on the D3-branes)should depend on the reduced coordinate so that Nahm Equations(1)are satisﬁed.Thus the Coulomb branch of the three-dimensional gauge theory is described as a moduli space of solutions to Nahm Equations.At this point we have two convenient descriptions of D k ALF space as a moduli space of solutions to Nahm equations and as a moduli space of sin-gular monopoles.It is the latter description that we shall make use of here. Regular monopoles can be described5by considering a scattering problem u·( ∂+ A)−iΦ s=0on every lineγin the three-dimensional space di-rected along u.The space of all lines is a tangent bundle to a sphere T=TP1. Let(ζ,η)be standard coordinates on T,such thatζis a coordinate on the sphere andηon the tangent space.Then the set of lines on which the scat-tering problem has a bound state forms a curve S∈T.S is called a spectral curve and it encodes the monopole data we started with.In case of singular monopoles some of the linesγ∈S will pass through the singular points.These lines deﬁne two sets of points Q and P in S,such that Q and P are conjugate to each other with respect to the change of orientation of the lines.Analysis of this situation1shows,that in addition to the spectral curve,we have to consider two sectionsρandξof the line bundles over S with transition functions eµη/ζand e−µη/ζcorrespondingly.Alsoρvanishes at the points of Q andξat those of P.Since we are interested in the case of two monopoles the spectral curve is given byη2+η2(ζ)=0whereη2(ζ)=z+vζ+wζ2−¯vζ3+¯zζ4.z,v and w are the moduli.z and v are complex and w is real.The sectionsρand ξsatisfyρξ= k i=1(η−P i(ζ)),where P i are quadratic inζwith coeﬃcients given by the coordinates of the singularities.The above equations provide the description of the twistor space of the singular monopole moduli space.3Knowing the twistor space one can use the generalized Legendre transform techniques6toﬁnd the auxiliary function F of the moduliF(z,¯z,v,¯v,w)=1ζ3+2 ωr dζ√ζ2−a1ζ2(√−η2−za(ζ)).(2)Imposing the consistency constraint∂F/∂w=0expresses w as a function of z and v.Then the Legendre transform of FK(z,¯z,u,¯u)=F(z,¯z,v,¯v)−uv−¯u¯v,(3) with∂F/∂v=u and∂F/∂¯v=¯u,gives the K¨a hler potential for the D k ALF metric.This agrees with the conjecture of Chalmers7.AcknowledgmentsThe results presented here are obtained in collaboration with Anton Kapustin. This work is partially supported by DOE grant DE-FG03-92-ER40701. References1.S.A.Cherkis and A.Kapustin,“Singular Monopoles and GravitationalInstantons,”hep-th/9711145to appear in Nucl.Phys.B.2.S.A.Cherkis and A.Kapustin,“D k Gravitational Instantons and NahmEquations,”hep-th/9803112.3.S.A.Cherkis and A.Kapustin,“Singular Monopoles and SupersymmetricGauge Theories in Three Dimensions,”hep-th/9711145.4.A.Sen,“A Note on Enhanced Gauge Symmetries in M-and String The-ory,”JHEP09,1(1997)hep-th/9707123.5.M.Atiyah and N.Hitchin,The Geometry and Dynamics of MagneticMonopoles,Princeton Univ.Press,Princeton(1988).6.N.J.Hitchin,A.Karlhede,U.Lindstr¨o m,and M.Roˇc ek,“Hyperk¨a hlerMetrics and Supersymmetry,”Comm.Math.Phys.108535-589(1987), Lindstrom,U.and Roˇc ek,M.“New HyperK¨a hler metrics and New Su-permultiplets,”Comm.Math.Phys115,21(1988).7.G.Chalmers,“The Implicit Metric on a Deformation of the Atiyah-Hitchin Manifold,”hep-th/9709082,“Multi-monopole Moduli Spaces for SU(N)Gauge Group,”hep-th/9605182.4。

Standard Model

1 Historical background
The Standard Model of elementary particles (more schematic depiction), with the three generations of matter, gauge bosons in the fourth column, and the Higgs boson in the ﬁfth.
3.1 Fermions
quarks being very strongly bound to one another, forming color-neutral composite particles (hadrons) containing either a quark and an antiquark (mesons) or three quarks (baryons). The familiar proton and the neutron are the two baryons having the smallest mass. Quarks also carry electric charge and weak isospin. Hence they interact with other fermions both electromagnetically and via the weak interaction.
Standard Model
This article is about the Standard Model of particle physics. For other uses, see Standard model (disambiguation). This article is a non-mathematical general overview of the Standard Model. For a mathematical description, see the article Standard Model (mathematical formulation). For the Standard Model of Big Bang cosmology, Lambda-CDM model. The Standard Model of particle physics is a theory con-

Simple theoretical tools for low dimension Bose gases

µ is the chemical potential, and the diﬀerential operator h0 includes the kinetic energy operator and the trapping potential U (r): h0 = − ¯2 h ∆ + U (r) 2m (1.2)
J. Phys. IV France 1 (2008)
c EDP SciXiv:cond-mat/0407118v2 [cond-mat.other] 6 Sep 2005
Simple theoretical tools for low dimension Bose gases
Pr1-2
JOURNAL DE PHYSIQUE IV
It is convenient to expand the ﬁeld operator on the orthonormal basis of the eigenmodes φα (r) of h0 with eigenenergy ǫα : ˆ(r) = ψ φα (r)ˆ aα (1.5)
so that we recover the condition for a non-degenerate regime ρλd ≪ 1. (1.17)
• degenerate regime: |µ| ≪ kB T so that the ground mode of the gas has a large occupation number: kB T ≫1 (1.18) n0 ≃ |µ| where we have expanded the exponential in the Bose law to ﬁrst order in its argument. In this case several modes have a large occupation number, as soon as |µ| ≪ kB T , because condition (1.13) holds. • the ground mode is more populated than any other mode: this regime is reached when the ground mode is more populated that the ﬁrst excited mode: n0 ≃ which results in |µ| ≪ ǫ1 . (1.20) It is the right time to recall the phenomenon of saturation of the total occupation of the excited modes, a key consequence of the Bose law: for a given temperature the maximal value of the excited modes population is N′ ≡

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。

Gravitational Instantons from Gauge Theory

NC instantons, is related to the size of a minimal two-
sphere known as a “Bolt” in the gravitational instantons.
Substituting the ansatz (11) into Eq.(12), we get
A general strategy was suggested in [6] to solve the self-duality equation (9). For example, let us consider the anti-self-dual (ASD) case. Take a general ansatz for the ASD Fµν as follows
Consider a NC spacetime deﬁned by
[yµ, yν ]⋆ = iθµν
(1)
with a constant 4 × 4 matrix θµν . Gauge theories can
be constructed on this NC spacetime. For example, the action for NC U (1) gauge theory in ﬂat Euclidean R4 is
The equation (13) was solved in [6] for the single instanton case. It was found there that the eﬀective metric (6) for the single U (1) instanton is related to the Eguchi-Hanson (EH) metric [8], the simplest ALE space, and that the family of the EH space is parameterized by the instanton number. In this paper we will show that the connection between NC U (1) instantons and hyperKa¨hler geometries is more general. More precisely, we will see that the NC self-duality equation (4) is mapped

Value of color charges and structure of gauge bosons

a r X i v :h e p -p h /0410207v 1 14 O c t 2004Value of color charges and structure of gaugebosonsAmjad Hussain Shah GilaniNational Center for Physics,Quaid-i-Azam University,Islamabad 45320,PakistanAbstractThe values of color and anticolor charges are proposed.The struc-ture of gluons is predicted relative to their color and anticolor charges.It is shown that the gauge bosons of lower order theories can be used as it is for higher order theories.Various mass relations between the gauge bosons are also given.1IntroductionSince the birth of Quantum Chromodynamics (QCD),no attention was paid to ﬁnd out the numerical value of the color charges.The color combinations were given in the literature with the help of SU(3)on the anology of quark combinations and there was not clear deﬁnition for color and anticolor.It was observed that the gluons cannot carry a combination of a color and anticolor charge,because this combination violate the group property [1].Gilani [1]pointed out that gluons will carry either color charges or anticolor charges.He claimed,with the help of set theory,that there will be six colored gluons,one color singlet gluon and one massless neutral gluon.He also claimed that seven gluons will be massive and only one massless gluon.Relations between color charges and anticolor charges are also deﬁned in Ref.[1].The paper is organized as follows:In sec.2,the values of the color charges are predicted and obtained the values of respective anticolor charges from color charges as deﬁned in Ref.[1].Charged gluons structure is pre-dicted in Sec. 3.Charged gluons masses are predicted in Sec. 4.Pictorial1representation of the structure of electroweak gauge bosons and gluons is presented in Sec.5.Position of massive gluon is explained in Sec.6.Section 7is devoted to Higgs.Relation between electroweak gauge bosons and QCD gauge bosons(gluons)is discussed in Sec.8.Expected Higgs mass is pre-dicted in Sec.9by using the existing value of W-boson mass(m W).Quark color charges and some decays of mesons and baryons are presented in Sec.10.Finally,the results are summarized in Sec.11.2Can we give values to color charges?The electroweak charges are two i.e.+1,−1.The question is:Is there any mathematical relation which has such a solution that we obtain a solution set as{+1,−1}?The equation isx2=1(1)x={+1,−1}(2)This shows that the electroweak charges are the result of square roots of unity.In QCD,there are three charges and if we take the cube roots of unity,we obtainx3=1(3)x= +1,−132+i√2(6)b≡−132+i√2(8)2¯g ≡b +r =+132+i√2 g −(12)g b ≡−132+i √2 g +(14)g ¯g ≡+13We give the positive or negative charge to gluons only seeing the sign of the real part,see for example Eqs.(12–15).If you are not convinced at this stage why we give the same charge to the hybrid gluons [Eqs.(12–15)]as the sign their real part,you will be convinced after doing exercise of decay processes in Sec.10.4Masses of charged gluonsAs in electroweak theory,the masses of charged vector gauge bosons are equal.On same analogy we suppose that m g +=m g −=m g .Therefore,m g r =+m g + =m g(17)m g g =−132−i√2 m g −=m g (19)m g ¯b=+132−i√2m g +=m g (21)m g ¯r =−m g −=m g(22)Equations (17–22)give the masses of gluons and we have found that themasses of color charged gluons and anticolor charged gluons are equal,i.e.m g r =m g g =m g b =m g ¯b=m g ¯g =m g ¯r =m g .5Pictorial representation of electroweak gauge bosons and gluonsThe electroweak gauge bosons are γ,W +,W −and Z 0.Among these four gauge bosons,only photon (γ)is massless while the remaining three are massive.The gauge bosons are predicted only by set theory but not by special unitary groups i.e.SU(2)or SU(3)[1].The electroweak charges can be described by Eq.(1)and we obtain a set of its roots as given in Eq.4(2).A set has proper and improper subsets.In this case(2),we have two component set,so we have two proper subsets and two improper subsets,i.e., {+1,−1}⇒{{},{+1},{−1},{+1,−1}}The subsets{+1},{−1}are proper subsets of the set of charges while{}, {+1,−1}are improper subsets.If we plot all these,then weﬁx the empty subset at the origin and{+1}on the x-axis at+1position and{−1}at the −1position while the{+1,−1}gets the position above the empty subset along the z-axis.We have shown the electroweak gauge bosons in Fig. 2. How the Z0vector gauge boson gets the position above the photon(γ),we will explain it in one of the next sections.It was claimed that the gluons does not obey the SU(3)but can be pre-dicted by set theory[1].The value of the color charges is explained in section 2and ploted in Fig.1.We have sketched a pattern of the gluons with respect to their charges as shown in Fig.3.6Position of the massive neutral gluonThe massive color-singlet(neutral)gluon G0is placed over the massless gluon as shown in Fig.3just like the neutral vector boson Z0is placed over the photon(γ),see Fig.2.The question is,how?and why?To answer this question,let us join the color points by straight lines which√results in an equilateral triangle∆rgb having the length of each side3as shown in Fig. 4.Take the apex r of triangle ∆rgb and¯r of the other triangle∆¯b¯g¯r,and join them togather,which meet on the z-axis as shown in Fig. 5.The points r and¯r meet on the z-axis at the point z.On the same grounds,if we take apexes g and¯g of the respective triangles,which meet again at the same point z on the z-axis,similar the case will be for the apexes b and¯b.Now the question is,what√is the value of z=?Let us any triangle∆ozr,the side|or|=1,|rz|=2.This shows√that the point z is7Higgs:Is there any?Following the discussion given in Sec.6,in triangle∆ozr,the side|or|gives the size of the charged gluon i.e.W r(g r)[see Eq.(11)]and the side|oz| gives the size of the color singlet gluon(G0).The remaining third side of the triangle|zr|will give the size of Higgs.We get six Higgs of equal size.Among these Higgs,three carry color charge and three carry anticolor charge.The sides of the triangle∆ozr have certain ratio between each other,i.e.|or|:|oz|:|rz|=1:√3From the above ratio between the side of triangle,we can predict the masses of the color singlet gluon(G0)and the Higgs(H i)asm G0=√2mW,(23)m Hi =√3mW,(24)respectively.8Relation between electroweak and QCD gauge bosonsIs there any relation between electroweak and QCD theories?This ques-tion raised by many scientists who wrote articles entitled:‘Theory of Every Thing’.Keeping in mind the above discussion,we conclude that QCD is nothing but a higher order electroweak theory.Electroweak theory is second order theory and QCD is third order theory because the structure of their charges obtained from Eqs.(1)and(3)respectively.The position of r=+1 and¯r=−1charges is exactly same as the respective charges in electroweak theory.Due to this similarity,we suppose thatg+=W+,g−=W−(25) Therefore,we can modify the Eqs.(11–16)asW r=g r≡+W+,(26)W g=g g≡ −13W b =g b ≡−132+i√2 W +,(29)W ¯g =g ¯g ≡+133m W ,i =r,g,b,¯b,¯g ,¯r (33)and color singlet gluon G 0mass ism G 0=m Z 0=√|or |2+|oz |2m H =where m W=80.423GeV and m Z=91.1876GeV[2].Whereas,from Eq.(33)m H=√3e for up-type quarksand−1whereW b =−132+i√2W −,(44)andρ¯g =132+i√2ρ+.(46)This shows that by solving the decays like ¯B 0b ¯g ¯d g→W r ρ¯ru b ¯d gand B 0 ¯b g d ¯g→W ¯r ρr ¯u¯b d ¯g ,we can solve all the remaining decays easily.We cannot ignore the possibility that the up-type quarks serve as quarks and their respective down-type quarks serve as anti-quarks but at the moment we are not sure.If such a possibility exists then the quarks will be reducedto three (i.e.u ,c ,t )and their anti-quarks (i.e.¯u ≡d ,¯c ≡s ,¯t≡b ),then the life will become too much simple.We will now focus on the hybrid states like given in Eqs.(45,46),if such states are virtual and further decay.Thenρ¯bu r ¯dg→W ¯b ρd ¯g ¯dg.Following B 0decay given in Eq.(38)¯Bb ¯g ¯d g→W b ρ¯bu r ¯dg→W bW ¯b ρ0d ¯g ¯dg→Z 0ρd ¯g ¯dg(47)So,in the above decayb ¯g →W b u r →W b W ¯b d ¯g→Z 0d ¯g .(48)For three quark baryon states,we will write the combinations asu b d ¯b +u g d ¯g 2 d ¯r =(udd )−,uru b d ¯b +u g d ¯g 2=(uud )+,(49)9In the above examples,combination of type u r d¯r d¯r or u r u r d¯r does not exist because of repeated anticolor or color index,repectively.We are just giving here simple examples.The rest of the states we can make likeu g d¯b+u b d¯g2d¯r=(udd)−,(50) etc.IfΛ0b→W+Λ−,thenΛ0b b¯g d¯b d¯r →W rΛ¯r u b d¯b d¯r ,Λ0b b¯b d¯g d¯r →W rΛ¯r(u g d¯g d¯r)(51) where b¯g→W r u b and b¯b→W r u g or vice versa.Let us concentrate upon the above examples of meson decays(37–42)and baryon decays(51).If we consider down-type quarks as antiquarks of up-type quarks then in the baryon caseΛ0b is composed of three down-type quarks while theΛ¯r is composed of one up-type quark and two down-type quarks.Whereas the baryons are built up of either quarks or antiquarks.By keeping this view,we cannot consider down-type quarks as antiquarks of up-type quarks.11ConclusionsIn this article,the value of the color charges are given.The structure of the colored,anticolored and color singlet gluons are proposed.Their mass relations are also given.It is shown that the gauge boson in the electroweak, QCD and gravity theories are not diﬀerent but they are linked to one another. The gauge bosons of lower order theories serve the purpose for higher order theories.The proof of the claims of Ref.[1]are given in a systematic way.Quark fractional charges(i.e.+23e for up-typequarks)are totally rejected.Only color charges are given to up-type quarks and anticolor charges to down-type quarks.This is explained by applying to meson and baryon decays.Acknowledgements:Author thanks to the friends and collegues for their encourging and/or discourging comments on my previous article[1]. Author also thanks to those who keep silence.References[1]A.H.S.Gilani,Are gluons massive?[hep-ph/0404026v2]10[2]S.Eidelman et al.,(Particle Data Group)Phys.Lett.B592,1(2004) 12Figure Captions1.The plot of cube roots of unity,i.e.the color and anti color charges.2.Electroweak gauge bosonsγ,W+,W−and Z0.Photon(γ)gets positionat the origin as it is massless while the others relative to their charge positions.3.The gluons are plotted relative to their charge positions.The masslessgluon gets the position at origin.4.The color points are joined by lines which form triangle∆rgb and byjoining anticolor points another triangle is formed∆¯b¯g¯r.Both the√triangles are equilateral triangles having length of each sideFigure1Figure2Figure3:Figure4Figure5。

Off-Diagonal Long-Range Order Meissner Effect and Flux Quantization

a r X i v :c o n d -m a t /9411025v 1 5 N o v 1994Oﬀ-Diagonal Long-Range Order:Meissner Eﬀect and FluxQuantizationH.T.Nieh and Gang Su ∗Institute for Theoretical Physics State University of New York at Stony Brook Stony Brook,New York 11794,USA Bao-Heng Zhao Department of Physics,Graduate School Chinese Academy of Sciences,P.O.Box 3908Beijing 100039,China Abstract There has been a proof by Sewell that the hypothesis of oﬀ-diagonal long-rangeorder in the reduced density matrix ρ2implies the Meissner eﬀect.We present in this note an elementary and straightforward proof that not only the Meissner eﬀect but also the property of magnetic ﬂux quantization follows from the hypothesis.It is explicitly shown that the two phenomena are closely related,and phase coherence is the origin for both.PACS numbers:05.30.-d;74.20.-z;74.25.HaIntroductionPenrose and Onsager1ﬁrst proposed that the mathematical statement of Bose-Einstein condensation in the case of interacting boson systems is the existence of oﬀ-diagonal long-range order(ODLRO)in the reduced density matrixρ1.Yang2later made a thorough study of the subject and generalized the concept to interacting fermion systems.It is now generally accepted that the superconducting state of electrons is characterized by the existence of oﬀ-diagonal long-range order in the reduced density matrixρ2.In view of the fundamental importance of the concept of ODLRO,it is important that there be a rigorous proof of the assertion that ODLRO in the reduced density matrixρ2 implies both the Meissner eﬀect and the quantization of magneticﬂux,which are the basic characteristic properties of superconducting states.There has been a recent proof by Sewell3 of the Meissner eﬀect as a consequence of ODLRO,under certain simplifying assumptions. We shall extend in this note the proof to the case of magneticﬂux quantization.The proof we present is simple and elementary,and clearly shows that both the Meissner eﬀect and theﬂux quantization are closely related phenomena,and the underlying origin for both is the coherence of phases.System in Uniform Magnetic FieldWe consider a system of N electrons inside a crystal,which is placed in a uniform external magneticﬁeld.We make the simplifying assumption that the magneticﬁeld B inside the material is also uniform,with the corresponding vector potential,given by A(r),which is of the general form:A(r)=A0(r)+∇φ(r),(1) where A0(r)=12m [¯hcA(r j)]2+1depends on the distance between the two electrons.Letψn(r1,r2,···,r N),n=1,2,...,represent a complete orthonormal set of single-valued and properly antisymmetrized eigen-functions of the HamiltonianHψn(r1,r2,···,r N)=E nψn(r1,r2,···,r N).(3)We neglect the edge eﬀects near the boundaries,and assume the usual periodic boundary conditions for the wave functions.The phenomenon ofﬂux penetration near boundaries is beyond the scope of the present idealized considerations.Two Complete Sets of EigenfunctionsNow,consider a space displacementr j→r j−a,under whichV(r jk)→V(r jk),A(r j)→A(r j−a)=A(r j)+∇j[a·A0(r j)+φ(r j−a)−φ(r j)].(4) We observe that a space displacement induces a gauge transformation in the vector potential. It is this property that makes it possible for a simple proof of theﬂux quantization and the Meissner eﬀect as consequences of the ODLRO hypothesis.The lattice potential V c(r j)has the symmetry of the lattice,and is invariant under a displacement only if the displacement respects the symmetry of the crystal lattice.Weﬁrst consider the case where the lattice potential V c(r j)in the Hamiltonian is omitted, and will come back later to consider the case where it is not.Without the lattice potential, the eigen-equation(3),under the displacement r j→r j−a,becomes{ j1i∇j+e2 j,k V(r jk)}ψn(r1−a,r2−a,···,r N−a)=E nψn(r1−a,r2−a,···,r N−a).(5)Noting that a displacement induces a gauge transformation,as in(4),in the vector potential, we can absorb the gauge into a phase for the electron wave function to obtain{ j1i∇j+e2 j,k V(r jk)}exp[iec¯h jχa(r j)]ψn(r1−a,r2−a,···,r N−a),(6) whereχa(r j)=a·A0(r j)+φ(r j−a)−φ(r j).(7) It is seen thatψ′n(r1,r2,···,r N),given byieψ′n(r1,r2,···,r N)=exp[1(N−2)!1(N−2)!×ψ′n(r′1,r′2,r3,···,r N)ψ′n∗(r1,r2,r3,···,r N),(10) Expressing the primed eigenfunctions in the above equation in terms of the original eigen-functions,we obtainieρ2(r′1,r′2;r1,r2)=exp{1(N−2)![χa(r′1)+χa(r′2)−χa(r1)−χa(r2)]}ρ2(r′1−a,r′2−a;r1−a,r2−a).c¯h(11) This equation,as we shall see,is of basic importance.Oﬀ-Diagonal Long-Range OrderWe can represent the reduced density matrix element in the spectral form:ρ2(r′1,r′2;r1,r2)= sαsΦs(r′1,r′2)Φ∗s(r1,r2),(12) whereαs are eigenvalues,andΦs(r1,r2)the properly normalized eigenfunctions ofρ2:Φ∗s(r1,r2)Φs(r1,r2)d r1d r2=1.We are interested in the property ofρ2in the oﬀ-diagonal long-range(ODLR)limit:|r′j−r k|→∞,j,k=1,2while keeping|r1−r2|and|r′1−r′2|ﬁnite.The hypothesis of ODLRO inρ2is the statement that in the ODLR limit the reduced density matrix element factorizes in the formρ2(r′1,r′2;r1,r2)→α0Φ(r′1,r′2)Φ∗(r1,r2),(13)whereα0is the largest eigenvalue of O(N),andΦthe corresponding eigenfunction2.Applying the factorization property of the ODLRO hypothesis to both sides of the relation (11),we obtain,in the ODLR limit,Φ(r′1,r′2)Φ∗(r1,r2)=exp{iec¯h[χa(r1)+χa(r2)]}Φ(r1−a,r2−a),(15) where f(a)is a possible displacement dependent,but position independent,phase factor6, with f(0)=1.Thus,the functionΦ,which is to be identiﬁed with the Ginsburg-Landau wave function,at diﬀerent space points is related by a phase factor.Repeating successively for inﬁnitesimal displacements along a closed path C,and picking up products of phase factors and the path-dependent f-factors,we obtain from(15)the relationΦ(r1,r2)=F(C)exp{i2ec¯h[χa(r1)+χb(r1−a)+χa(r2)+χb(r2−a)]}=exp{iec¯h(a×b)A(r)·d r}=1,(18)where the loop path of integration is a parallelogram formed by a and b.Therefore,B·(a×b)=nchc¯hCA(r)·d r}Φ(r1,r2).(22)Consequently,there is the quantization conditionC A(r)·d r=ncheﬀect.When C is not simply-connected,like a path winding around an inaccessible tunnel, the path can not be deformed across this inaccessible region.Then the quantization condition (23)implies that the totalﬂux8passing through the loop C can only have quantized values. We note that in our idealized considerations the loop path C is away from the boundaries.We can prove F(C)=1more explicitly by observing some simple properties of f(a):1.As f(a)is independent of r,we can choose to explore the properties of f(a)by considering displacements in regions where B=0.2.In any region of B=0,consider the succession of two inﬁnitesimal displacements a and b,as well as the displacement a+b.Because B=0in the region enclosed by the three vectors a,b and a+b,it is clear from(15)thatf(a+b)=f(a)f(b).(24)(24)oﬀers another more explicit proof of(21).Recall that F(C)is the product of the f-factors along the loop path.Hence,for a closed loop,on account of(24),we again obtainF(C)=f(a)f(b)f(c)···=f(a+b+c+···)=f( C d r)=f(0)=1.Inclusion of Lattice PotentialThe lattice potential V c(r),which is not invariant under arbitrary displacements,has so far been omitted in our consideration.We now consider the case when it is included.Let the symmetry of the lattice be deﬁned by three basic lattice vectors:e1,e2,e3.The lattice potential V c(r)is then invariant under the space displacementr j→r j−△r,(25)if9△r=n1e1+n2e2+n3e3,n1,n2,n3=integers.(26) Under such displacements,the total Hamiltonian is invariant,except for an induced gauge term in the vector potential,like that in(4).This is exactly like the situation we had when the lattice potential was omitted,except,of course,that the displacements have to be restricted to those in(26).All previous arguments apply,leading to the conclusion,similar to(19),thatB·(e j×e k)=n jk chchV2e∼109G,(29) according to(28).This is many orders stronger than the typical criticalﬁeld strengths of less than103G,beyond which the superconducting state no longer exists,for supercon-ductors.Therefore,to be consistent with(28),the only possible value of B is zero for a superconducting state.The same reasoning that lead to the quantization condition(23)applies,and leads toC A(r)·△r=nchFootnotes1O.Penrose,Phil.Mag.42,1373(1951);O.Penrose and L.Onsager,Phys.Rev.104, 576(1956).2C.N.Yang,Rev.Mod.Phys.34,694(1962).3G.L.Sewell,J.Stat.Phys.61,415(1990).4In proving theﬂux quantization,we shall need to consider the case where there are regions inaccessible to the electrons,like a tunnel.In these inaccessible regions,the magneticﬁeld does not necessarily vanish.In order to guarantee the single-valuedness of phaseχa(r j)in(7),we need to restrict to inﬁnitesimal a,so thatφ(r j−a)−φ(r j)is single-valued,althoughφ(r j−a)andφ(r j)individually are not.5HereΦ(r1,r2)has been renormalized,i.e.,a factor√。

马哲教学案例

第一节案例分析专题1：物质概念[相关观点]观点1 德谟克利特的“原子”古希腊哲学家、原子论者德谟克利特认为，世界万物包括人的灵魂在内，都是由原子这种不可分割的物质粒子组成的。

原子是永恒的，由原子所组成的整个自然界、整个世界也是永恒的。

原子本身不变化，但能运动。

原子的数量是无限多的，但它们之间没有性质的不同，仅有大小、形状、次序和位置的区别。

世界万物由于构成它们的原子在大小、形状、次序和位置上的不同，而形成千差万别的性质。

虚空是原子运动的地方，是“非存在”。

原子在虚空中互相碰撞，形成旋涡运动，从而互相结合形成万物。

原子分离，事物就灭亡。

观点2 霍布斯的“物体”“物体”是17世纪英国著名哲学家霍布斯哲学体系的基本范畴。

霍布斯认为，世界上真实存在着的只有物质所构成的物体。

他说：“哲学的对象，或者哲学所处理的材料，乃是每一个这样的物体：这种物体我们可以设想它有产生，并且可以通过对它的思考，指导它同别的物体加以比较，或者是，这种物体是可以加以组合与分解的，也就是说，它的产生或特性我们是可以认识的”。

霍布斯给物体下了一个明确的定义。

他说：“物体是不依赖于我们思想的东西，与空间的某个部分相合或具有同样的广延”。

霍布斯认为物体具有如下特征：它不依赖于我们思想而客观存在；占有一定的空间；能为人们所认识。

霍布斯所说的物体概念基本上接近马克思主义哲学的物质概念。

霍布斯还认为，经院哲学所谓的“无形体的实体”，如同说“圆的方形”一样荒谬。

观点3 “存在就是被感知”18世纪英国经验主义哲学家贝克莱在其《人类知识原理》一书中提出了“存在就是被感知”的著名论断。

他认为，知识起源于感觉，知识的对象就是观念。

我们所能知道的只是观念，而不是观念之外的事物。

观念不是对客观事物的反映，相反，外在事物是“一些观念的集合”，离开了感觉或经验的“纯客观存在”是不可理喻的。

他还认为，物体的广延、形状、运动完全是相对的，是随着感觉器官的结构或位置的变化而相应变化的，因此，它们完全依赖观点4 莱布尼茨的“单子论”德国哲学家、唯理论的代表人物莱布尼茨在世界本原问题上提出了著名的单子论。

原子核物理专业词汇中英文对照表

原子核物理专业词汇中英文对照表absorption cross-section 吸收截面activity radioactivity 放射性活度activity 活度adiabatic approximation 浸渐近似allowed transition 容许跃迁angular correlation 角关联angular distribution 角分布angular—momentum conservation 角动量守恒anisotropy 各项异性度annihilation radiation 湮没辐射anomalous magnetic moment 反常极矩anti neutrino 反中微子antiparticle 反粒子artificial radioactivity 人工放射性atomic mass unit 原子质量单位atomic mass 原子质量atomic nucleus 原子核Auger electron 俄歇电子bag model 口袋模型baryon number 重子数baryon 重子binary fission 二分裂变binging energy 结合能black hole 黑洞bombarding particle 轰击粒子bottom quark 底夸克branching ration 分支比bremsstrahlung 轫致辐射cascade radiation 级联辐射cascade transition 级联跃迁centrifugal barrier 离心势垒chain reaction 链式反应characteristic X-ray 特征X射线Cherenkov counter 切连科夫计数器collective model 集体模型collective rotation 集体转动collective vibration 集体振动color charge 色荷complete fusion reaction 全熔合反应complex potential 复势compound—nucleus decay 复合核衰变compound—nucleus model 复合核模型compound nucleus 复合核Compton effect 康普顿效应Compton electron 康普顿电子Compton scattering 康普顿散射conservation law 守恒定律controlled thermonuclear fusion 受控热核聚变cosmic ray 宇宙射线Coulomb barrier 库仑势垒Coulomb energy 库伦能Coulomb excitation 库仑激发CPT theorem CPT定理critical angular momentum 临界角动量critical distance 临界距离critical mass 临界质量critical volume 临界体积damped oscillations 阻尼震荡damped vibration 阻尼震荡damped wave 阻尼波damper 减震器damping factor 衰减系数damping 衰减的damp proof 防潮的damp 湿气danger coefficient 危险系数danger dose 危险剂量danger range 危险距离danger signal 危险信号data acquisition and processing system 数据获得和处理系统data base 数据库data communication 数据通信data processing 数据处理data 数据dating 测定年代daughter atom 子体原子daughter element 子体元素daughter nuclear 子核daughter nucleus 子体核daughter nuclide 子体核素daughter 蜕变产物dd reaction dd反应deactivation 去活化dead band 不灵敏区dead time correction 死时间校正dead time 失灵时间deaerate 除气deaeration 除气deaerator 除气器空气分离器deaquation 脱水debris activity 碎片放射性debris 碎片de broglie equation 德布罗意方程de broglie frequency 德布罗意频率de broglie relation 德布罗意方程de broglie wavelength 德布罗意波长de broglie wave 德布罗意波debye radius 德拜半径debye temperature 德拜温度decade counter tube 十进计数管decade counting circuit 十进制计数电路decade counting tube 十进管decade scaler 十进位定标器decagram 十克decalescence 相变吸热decalescent point 金属突然吸热温度decarburization 脱碳decascaler 十进制定标器decatron 十进计数管decay chain 衰变链decay coefficient 衰变常数decay constant 衰变常数decay constant 衰变常量decay energy 衰变能decay factor 衰变常数decay fraction 衰变分支比decay heat removal system 衰变热去除系统decay heat 衰变热decay kinematics 衰变运动学decay out 完全衰变decay period 冷却周期decay power 衰减功率decay rate 衰变速度decay scheme 衰变纲图decay series 放射系decay storage 衰变贮存decay table 衰变表decay time 衰变时间decay 衰减decelerate 减速deceleration 减速decigram 分克decimeter wave 分米波decommissioning 退役decompose 分解decomposition temperature 分解温度decomposition 化学分解decontaminability 可去污性decontamination area 去污区decontamination factor 去污因子decontamination index 去污指数decontamination 净化decoupled band 分离带decoupling 去耦解开decrease 衰减decrement 减少率deep dose equivalent index 深部剂量当量指标deep inelastic reaction 深度非弹性反应deep irradiation 深部辐照deep therapy 深部疗de excitation 去激发de exemption 去免除defectoscope 探伤仪defect 缺陷definition 分辨deflecting coil 偏转线圈deflector 偏转装置deformation energy 变形能deformation of irradiated graphite 辐照过石墨变形deformation parameter 形变参量deformation 变形deformed nucleus 变形核deformed region 变形区域deform 变形degassing 脱气degas 除气degeneracy 简并degenerate configuration 退化位形degenerate gas 简并气体degenerate level 简并能级degenerate state 简并态degeneration 简并degradation of energy 能量散逸degradation 软化degraded spectrum 软化谱degree of acidity 酸度degree of burn up 燃耗度degree of purity 纯度dehumidify 减湿dehydrating agent 脱水剂dehydration 脱水deionization rate 消电离率deionization time 消电离时间deionization 消电离delay circuit 延迟电路delayed alpha particles 缓发粒子delayed neutron 缓发中子delayed proton 缓发质子deliquescence 潮解deliquescent 潮解的demagnetization 去磁denitration 脱硝density gradient instability 密度梯度不稳定性density of electrons 电子密度deoxidation 脱氧deoxidization 脱氧departure from nucleate boiling ratio 偏离泡核沸腾比departure from nucleate boiling 偏离泡核沸腾depleted fuel 贫化燃料deposit dose 地面沉降物剂量deposited activity 沉积的放射性deposition 沉积deposit 沉淀depression 减压depressurization accident 失压事故depressurizing system 降压系统desalinization 脱盐desalting 脱盐descendant 后代desorption 解吸detailed balance principle 细致平衡原理detection of radiation 辐射线的探测detonation 爆炸deuteride 氘化物deuterium alpha reaction 氘反应deuterium 重氢deuton 氘核deviation 偏差dew point 露点dextro rotatory 右旋的diagnostic radiology 诊断放射学diagnostics 诊断diagram 线图diamagnetism 反磁性diameter 直径diamond 稳定区；金刚石diaphragm 薄膜diatomic gas 双原子气体diatomic molecule 二原子分子dielectric 电介质differential control rod worth 控制棒微分价值differential cross section 微分截面diffraction spectrometer 衍射谱仪diffraction spectrum 衍射光谱diffraction 衍射diffuse 扩散diffusion stack 务马堆diffusion theory 扩散理论diffusion time 扩散时间diffusion 扩散dilution 稀释dipole 偶极子dirac equation 狄拉克方程direction 方向discharge 放电discrete 离散的disintegrate 蜕衰disintegration 蜕变dislocation 位错disorder 无序dispersion 分散displacement current 位移电流displace 位移；代替dissociation 离解dissolution 溶解distillation 蒸馏distortion 畸变divergence 发散domain 磁畴Dopper effect 多普勒效应dose albedo 剂量反照率dose build up factor 剂量积累因子dose equivalent 剂量当量dose rate 剂量率dose 剂量down quark 下夸克dry out 烧干duality 二重性duct 管dysprosium 镝endothermic reaction 吸能反应energy conservation 能量守恒even—even nucleus 偶偶核exchange force 交换力excited state 激发态exothermic reaction 放能反应exposure 照射量。

geroscience杂志格式

geroscience杂志格式Gerontology: Understanding the Impact of Aging [Geroscience Journal Format]Introduction to Gerontology:Gerontology is the field of study that focuses on understanding the biological, psychological, and sociological aspects of aging. It is an interdisciplinary science that aims to explore the impact of aging on individuals and societies as a whole. By examining various factors related to aging, gerontology seeks to improve the quality of life for older adults and address the challenges that come with aging populations.Understanding the Biological Aspects of Aging:One of the main areas of research in gerontology is the study of biological aging. This involves exploring the changes that occur in the body as individuals grow older. Scientists examine cellular and molecular processes that contribute to aging, such as telomere shortening and oxidative stress. By understanding these mechanisms, researchers hope to find ways to slow down the agingprocess and develop interventions that promote healthy aging.Psychological Effects of Aging:In addition to exploring the biological aspects of aging, gerontology also seeks to understand the psychological effects of growing older. Mental health and cognitive functioning are critical areas of concern, as older adults may be more prone to conditions such as depression, anxiety, and Alzheimer's disease. Gerontologists study factors that contribute to overall well-being in older adults, including social support, cognitive stimulation, and cultural influences. They also examine the impact of life events, such as retirement and loss of loved ones, on psychological health.Sociological Implications of Aging:The field of gerontology also looks at the social implications of aging, as the global population continues to age rapidly. Demographic shifts have significant societal consequences, creating challenges for healthcare systems, retirement policies, and intergenerational relationships. Gerontologists study the social dimensions of aging, including socio-economic disparities amongolder adults, social isolation, and caregiving dynamics. By understanding the social aspects of aging, researchers can develop policies and interventions that address the needs of older individuals and promote healthier aging communities.The Emergence of Geroscience:In recent years, a new field called geroscience has emerged within gerontology. Geroscience focuses on understanding the common mechanisms of aging that contribute to multiple age-related diseases. By identifying these underlying processes, researchers aim to develop interventions that target aging itself, rather than individual diseases. This interdisciplinary approach has the potential to revolutionize healthcare by shifting the focus from reactive disease treatment to proactive aging interventions.Conclusion:Gerontology is a multidisciplinary field that encompasses the study of biological, psychological, and sociological aspects of aging. By understanding the impact of aging on individuals and society as awhole, gerontologists aim to improve the quality of life for older adults and address the challenges presented by aging populations. The emergence of geroscience highlights the importance of understanding the common mechanisms of aging and developing interventions that target the aging process itself. In the coming years, continued research in gerontology and geroscience will contribute to advancements in healthcare and promote healthier aging for all.。

弦论在数学物理的一些应用 - USTC, ICTS

弦论是上世纪60，70年代研究强作用力的一个候补理论。后来Green, Schwarz等人发现弦论不适合描述强作用力，而是一个自洽的量子引力的候补理论。弦论原则上可以解决标准模型的缺陷。弦论只有在10维时空，引入超对称性(Supersymmetry) 才能自洽（简称超弦）。要低能量得到和标准模型比较接近的4维时空，N=1超对称理论，我们需要一个6维的紧致空间，Calabi-Yau 流形。
在五十年代，大量的新粒子被发现。那些粒子是基本粒子，那些粒子是有基本粒子所组成的呢？
在五十年代，大量的新粒子被发现。那些粒子是基本粒子，那些粒子是有基本粒子所组成的呢？一个例子：电子是一个基本粒子，而中子和质子则是由更小的粒子：夸克所组成的。
在五十年代，大量的新粒子被发现。那些粒子是基本粒子，那些粒子是有基本粒子所组成的呢？一个例子：电子是一个基本粒子，而中子和质子则是由更小的粒子：夸克所组成的。
在现代的科学研究中，尚未发现由更小的结构组成的基本粒子包括电子，夸克，光子等粒子。
在现代的科学研究中，尚未发现由更小的结构组成的基本粒子包括电子，夸克，光子等粒子。因为探测粒子的基本结构需要把粒子加速到很高的能量，然后进行对撞，所以粒子物理也被称为高能物理。
粒子物理学：理论
粒子物理学：理论
在规范理论中，规范玻色子(Gauge Boson)的质量为零。标准模型的规范玻色子中，光子和胶子没有质量，但是 Z 和 W±有质量。
在规范理论中，规范玻色子(Gauge Boson)的质量为零。标准模型的规范玻色子中，光子和胶子没有质量，但是 Z 和 W±有质量。这个问题的解决方法是引入一个新的粒子：Higgs 子. 它的引入使得规范对称性破缺了， Z 和 W±变得有质量了。

正负电子对撞机上三规范粒子的伴随产生

正负电子对撞机上三规范粒子的伴随产生李小龙;武悦;吕立庭;宋昴【摘要】在标准型的理论框架下研究了国际直线对撞机(ILC)上W+W-Z、ZZZ的产生过程,给出了这两个过程在不同对撞能量下的截面,结果显示这两个过程的反应截面都很大,有足够的事件进行实验分析.还给出了这两个过程在能量为500 GeV 时末态W和Z粒子的横动量分布.【期刊名称】《宜宾学院学报》【年(卷),期】2014(014)006【总页数】3页(P30-32)【关键词】国际直线对撞机;规范耦合;标准模型【作者】李小龙;武悦;吕立庭;宋昴【作者单位】安徽大学物理与材料科学学院,安徽合肥230039;安徽大学物理与材料科学学院,安徽合肥230039;安徽大学物理与材料科学学院,安徽合肥230039;安徽大学物理与材料科学学院,安徽合肥230039【正文语种】中文【中图分类】O572标准模型是建立在SU(2)×U(1)对称群下的规范理论.通过电弱对称性的破缺，从而使基本粒子获得了质量[1-6].规范对称性对规范粒子的三线和四线耦合给出了严格的限制，任何超出标准模型的反常耦合都会在实验上引起大的偏差.多个规范粒子的伴随产生非常适合研究规范粒子的自耦合，尤其对于四线耦合，只有末态三个以上规范粒子伴随产生才会出现四线耦合的顶点.如果存在超出标准模型反常的规范耦合，实验上探测到的事例将与标准模型的预言有很大不同.因此，给出标准模型框架下多规范粒子在高能对撞机上伴随产生过程的理论预言是非常有意义的工作.计划建造的国际直线对撞机(ILC)是对撞能量在200到500 GeV的正负电子对撞机，升级以后可以达到1 TeV[7].相比于强子对撞机，正负电子对撞机的背景非常干净，对撞能量可以调节，并且正负电子的束流可以极化.在标准模型以及许多超出标准模型的其他模型下，研究正负电子对撞机上三规范粒子伴随产生和反常耦合的工作已经有很多[8-11].本文将在标准型的理论框架下研究ILC上W+W-Z、ZZZ的产生过程.在标准模型中，四线耦合只有W+W-AA、W+W-ZZ、W+W-AZ、W+W-W+W-四种耦合形式.在e+e¯→W+W-Z、ZZZ过程中，涉及到W+W-AA、W+W-ZZ、W+W-AZ这三种耦合.对应顶点的费曼规则为：采用FeynArts 3.3[12]程序包产生对应的费曼图和对应的费曼幅度，然后调用FormCalc 5.3[13]程序包进行费曼幅度的化简和γ矩阵的收缩，最后转化为Fortran程序进行数值运算.计算中采用’t Hooft-Feynman规范.计算过程可以表示为：这两个过程的微分截面可以表示为：其中，M代表各个过程所有的费曼幅度相加，1/4是对初态粒子的自旋求平均.Σ表示对所有的初末态粒子的自旋求平均.e+e¯→ZZZ的过程，由于末态是三个全同粒子，整个反应截面还需要除以3的阶乘.三体末态的相空间矩阵元dΦ3定义为：在数值计算中选取下面的相关参数[14]:图1(a,b)分别给出了e+e-→W+W-Z、ZZZ反应截面随着质心系能量变化的曲线.从图中可看出，随着质心系能量的增加，这两个过程的反应截面变化趋势是不一样的.当能量从300 GeV增加到1 000 GeV时，e+e-→W+W-Z过程的截面从36.24 fb增加到65.19 fb；对于e+e-→ZZZ过程，总截面不是单调增加，而是先增大后减小，在能量约等于550 GeV的地方有最大值，并且这个过程的总截面要比e+e-→W+W-Z过程的截面小几十倍.当然，由于ILC的年积分亮度非常高，约每年100 fb-1，这两个过程都可以收集到足够的事例.对于e+e-→W+W-Z来说，每年可以收集几千个事例；对于e+e-→ZZZ过程也可以收集到一百多个事例.这对于检验标准模型，或者给出是否有超出标准模型的新物理都是非常重要的.对于e+e-→W+W-Z过程，随着能量的增加，产生的事例也越多，这对于能量不断提高的ILC来说是有利于实验的探测的.而对于e+e-→ZZZ过程，能量的增加并不是探测这个过程最好的方式，因为这个过程是一个S道占优的过程，反应截面正比于质心系能量的倒数，随着能量的增加反应截面反而会减小.为了清楚给出图1中的结果，表1列出了能量为300 GeV、500 GeV、800 GeV和1 000 GeV四个能量时对应的反应截面，并且给出了对应的数值运算的积分误差.三规范玻色子产生是检验标准模型非常重要的过程，然而由于之前的大型正负电子对撞机LEP的能量最高只有200 GeV，没有达到产生三规范粒子的阈值，所以不能用来研究这些过程，即将建造的国际直线对撞机ILC正是为了弥补LEP的不足而设计的能量为500 GeV的正负电子对撞机.这些过程在ILC上将有足够多的事例可以产生，并且用来检验标准模型和发现一些新物理.图2给出了ILC上质心系能量为500 GeV时，e+e-→W+W-Z、ZZZ过程末态粒子W和Z玻色子的横动量分布.图2(a)分别给出了e+e-→W+W-Z中W和Z玻色子的横动量分布.由于在标准模型中CP守恒，W+和W-粒子的分布是相同的，因此这里不再区分W+和W-.图2 (b)给出了e+e-→ZZZ中Z玻色子的横动量分布，其中三个Z粒子是全同粒子，分布也应该相同，只需要给出其中一个的分布就可以了.从图中可以看出，W和Z玻色子的横动量分布都在横动量约等于50GeV处出现极大值.这是由于在横动量很小或很大的区域，相空间比较小，粒子在这些地方产生的几率也要小，而横动量为50 GeV附近是相空间最大的地方，产生的粒子也应该最多.横动量是一个重要的可观测量，它的分布与选取的参考系无关，对于理论计算与实验观测都非常方便.通过给出末态粒子横动量的分布，实验上不仅可以对比总截面的大小，还可以与不同横动量处的微分截面相比较.这也为实验观测提供了理论依据.本文在标准型的理论框架下，研究了国际直线对撞机(ILC)上W+W-Z、ZZZ的产生过程,计算了这两个过程在对撞能量从300 GeV到1TeV的总截面，并且绘制了这两个过程末态粒子W和Z玻色子的横动量分布.理论计算表明，这两个过程的反应截面比较大，在ILC上将有足够多的事例产生，为检验标准模型的规范粒子四线耦合和寻找超出标准模型的新物理提供了理论依据.【相关文献】[1]Glashow S L.Partial-symmetries of weak interactions[J].Nucl Phys, 1961,22(4):579-588.[2]Weinberg S.A model of leptons[J].Phys Rev Lett,1967,19:1264.[3]Politzer H D.Asymptotic freedom:an approach to strong interactions[J]. PhysRep,1974,14:129-180.[4]Englert F,Brout R.Broken symmetry and the mass of gauge vector mesons[J].Phys Rev Lett,1964,13:321.[5]Higgs P W.Broken symmetries,massless particles and gauge fields[J]. PhysLett,1964,12(2):132-133.[6]Higgs P W.Broken symmetries and the masses of gauge bosons[J].Phys RevLett,1964,13:508.[7]Barish B,Brau J E.The international linear collider[J].Int J Mod PhysA,2013,28(27):1330039.[8]Sun W,Ma W G,Zhang R Y,et al.Full electroweak one-loop corrections to W+W-Z0 production at the ILC[J].Phys Lett B,2009,680: 321-327.[9]Su J J,Ma W G,Zhang R Y,et plete one-loop electroweak corrections to ZZZ production at the ILC[J].Phys Rev D,2008,78:016007.[10]Han T,He H J,Yuan C P.Quartic gauge boson couplings at linear colliders:Interplay of WWZ/ZZZ production and WW fusion[J].Phys Lett B,1998,422:294.[11]Jiang R C,Li X Z,Ma W G,et al.Triple Z0-boson production in large extra dimensions model at ILC[J].Chin Phys Lett,2012,29:111101.[12]Hahn T.Generating Feynman diagrams and amplitudes with FeynArts 3 [J].Comput Phys Commun,2001,140:418-431.[13]Hahn T,Perez-Victoria M.Automatized one-loop calculations in 4 and d dimensions[J].Comput Phys Commun,1999,118:153-165.[14]Amsler C,Doser M,Antonelli M,et al.Review of Particle Physics[J]. Phys LettB,2008,667:1-6.。

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a r X i v :h e p -p h /9612295v 1 9 D e c 1996THEORETICAL ASPECTS OF GAUGE-BOSON PRODUCTIONAT LEP2AND THE NLCW.BEENAKKERInstituut–Lorentz,P.O.Box 9506,NL–2300RA Leiden,The NetherlandsA report is given on the theoretical aspects of gauge-boson production processes at LEP2and the next linear collider (NLC).A short discussion is given on the gauge-boson-related physics issues that play a role at both coliders and on the outcome of the various LEP2/NLC studies that have taken place during the last few years.The main emphasis of this report is on the question of preserving gauge-invariance when dealing with unstable gauge bosons.In this context a strategy is proposed for treating radiative corrections.1Physics issues at LEP2Recently LEP2,the second stage of the LEP pro-gram,has started operation.After a ﬁrst short run around 130GeV,the energy has been increased to the nominal threshold for the production of W-boson pairs.This opens the possibility of accu-rately measuring the W-boson mass M W .Two methods are advocated for this measurement.The ﬁrst procedure involves a measurement of the to-tal W-pair cross-section σ(e +e −→W +W −→4f +nγ)at an energy close to 161GeV,i.e.,just above the nominal threshold.In this energy re-gion the cross-section σ(e +e −→4f +nγ)is very sensitive to M W through phase-space.Since for this measurement a precise knowledge of the total cross-section is required,this method is model de-pendent.During last year’s LEP2working-group studies the conclusion was reached that an ex-perimental error on the cross-section of the order of 5–6%is feasible (for an integrated luminosity of 50pb −1/exp).1This translates into an envis-aged experimental error of about 100MeV on the W-boson mass.From the theoretical side a sys-tematic error of the order of 2%is expected.1,2This uncertainty originates from the implementa-tion of initial-state (photon)radiation (ISR),from missing O (α)corrections,and from the depen-dence of the cross-section on the unknown Higgs-boson mass 3.The second procedure is a model-independent one.It consists in a reconstruction of the Breit–Wigner resonance shapes of the W bosons from the measured momenta of the de-cay products.For this method higher energies (>∼175GeV)are required in order to have a suf-ﬁciently large W-pair production rate.The ex-pected experimental error on the reconstructed W-boson mass is 40–50MeV for an integrated lumi-nosity of 500pb −1/exp.1From the theoretical sidetwo main sources of systematic uncertainties can be identiﬁed.First of all a precise knowledge of ISR is required,in view of the fact that the W-pair energy will be diﬀerent from the laboratory en-ergy of the incoming electrons and positrons in the presence of energy losses in the initial state.This particular uncertainty seems to be under control,amounting to roughly 10MeV from estimates of the average energy loss.2In the hadronic–leptonic decay channels this is the main uncertainty.In the purely hadronic (four-jet)decay channels,frag-mentation eﬀects can add another 50MeV to the uncertainty.1As the average space–time distance between the two decaying W bosons is smaller than 0.1fm,i.e.,less than the typical hadronic size of 1fm,the fragmentation of the two W bosons is not independent (“color rearrangement”).In view of the overlapping hadronization regions also coherence eﬀects are possible between identical low-momentum bosons stemming from diﬀerent W bosons (“Bose–Einstein correlations”).Both frag-mentation eﬀects are strongly model-dependent,so further theoretical and experimental studies are needed before conclusive statements can be made.The envisaged precision of 40–50MeV in the determination of M W at LEP2constitutes a signif-icant improvement on the present hadron-collider measurements (∆M W ≈125MeV)4.Since the mass of the W boson is one of the key parame-ters of the electroweak theory,such an improved accuracy makes the tests on the Standard Model (SM)of electroweak interactions more stringent.In order to facilitate the extraction of this piece of experimental information,a new input-parameter scheme was introduced.2Besides the usual LEP1input parameters:α,G µ,M Z and the light-fermion masses,also MWis treated as input(ﬁt parameter).From muon decay the mass of the top quark can be calculated as a function of the Higgs-boson mass MHand the strong couplingαs:Gµ=απ/√M2W(1−M2W/M2Z)1in the construction of lowest-order LEP2/NLCMonte Carlo generators.Theﬁrst one involves the systematic replacement1/(k2−M2)→1/(k2−M2+iMΓ),also for k2<0.HereΓdenotesthe physical width of the gauge boson with mass M and momentum k.This scheme is called the ‘ﬁxed-width scheme’.As in general the reso-nant diagrams are not gauge-invariant by them-selves,this substitution will destroy gauge invari-ance.Moreover,it has no physical motivation, since in perturbation theory the propagator for space-like momenta does not develop an imagi-nary part.Consequently,unitarity is violated in this scheme.To improve on the latter another approach can be adopted,involving the use of a running width iMΓ(k2)instead of the constant one iMΓ(‘running-width scheme’).This,how-ever,still cannot cure the problem with gauge in-variance.At this point one might ask oneself the legit-imate question whether the gauge-breaking terms are numerically relevant or not.After all,the gauge breaking is caused by theﬁnite decay width and is,as such,in principle suppressed by pow-ers ofΓ/M.From LEP1we know that gauge breaking can be negligible for all practical pur-poses.However,the presence of small scales can amplify the gauge-breaking terms.This is for in-stance the case for almost collinear space-like pho-tons or longitudinal gauge bosons at high ener-gies,involving scales of O(p2B /E2B)(with pBthemomentum of the involved gauge boson).In these situations the external current coupled to the pho-ton or to the longitudinal gauge boson becomes ap-proximately proportional to pB.In other words,in these regimes sensible theoretical predictions are only possible if the amplitudes with external cur-rents replaced by the corresponding gauge-boson momenta fulﬁll appropriate Ward identities.In order to substantiate these statements,a truly gauge-invariant scheme is needed.It should be stressed,however,that any such scheme is arbi-trary to a greater or lesser extent:since the Dyson summation must necessarily be taken to all orders of perturbation theory,and we are not able to com-pute the complete set of all Feynman diagrams to all orders,the various schemes diﬀer even if they lead to formally gauge-invariant results.Bearing this in mind,we need some physical motivation for choosing a particular scheme.In this context two options can be mentioned,which fulﬁll the criteria of gauge invariance and physical motivation.The ﬁrst option is the so-called‘pole scheme’.7,8,9Inthis scheme one decomposes the complete ampli-tude according to the pole structure by expand-ing around the poles(e.g.f(k2)/(k2−M2)= f(M2)/(k2−M2)+ﬁnite terms).As the phys-ically observable residues of the poles are gauge-invariant,gauge invariance is not broken if theﬁ-nite width is taken into account in the pole terms ∝1/(k2−M2).It should be noted,however, that there exists some controversy in the litera-ture about the‘correct’procedure for doing this and about the range of validity of the pole scheme, especially in the vicinity of thresholds.The second option is based on the philosophy of trying to de-termine and include the minimal set of Feynman diagrams that is necessary for compensating the gauge violation caused by the self-energy graphs. This is obviously the theoretically most satisfying solution,but it may cause an increase in the com-plexity of the matrix elements and a consequent slowing down of the numerical calculations.For the gauge bosons we are guided by the observa-tion that the lowest-order decay widths are exclu-sively given by the imaginary parts of the fermion loops in the one-loop self-energies.It is therefore natural to perform a Dyson summation of these fermionic one-loop self-energies and to include the other possible one-particle-irreducible fermionic one-loop corrections(“fermion-loop scheme”).10 For the LEP2process e+e−→4f this amounts to adding the fermionic triple gauge-boson vertex corrections.The complete set of fermionic con-tributions forms a gauge-independent subset and obeys all Ward identities exactly,even with re-summed propagators.11As mentioned above,the validity of the Ward identities guarantees a proper behavior of the cross-sections in the presence of collinear photons and at high energies in the pres-ence of longitudinal gauge-boson modes.On top of that,within the fermion-loop scheme the ap-propriately renormalized matrix elements for the generic LEP2process e+e−→4f can be formu-lated in terms of eﬀective Born matrix elements, using the familiar language of running couplings.11A numerical comparison of the variousschemes10,11conﬁrms the importance of not vi-olating the Ward identities.For the LEP2pro-cess e+e−→e−¯νe u¯d,a process that is particu-larly important for TGC studies,the impact of violating the U(1)electromagnetic gauge invari-ance was demonstrated.10Of the above-mentioned schemes only the running-width scheme violates U(1)gauge invariance.The associated gauge-breaking terms are enhanced in a disastrous way by a factor of O(s/m2e),in view of the fact that the electron may emit a virtual(space-like)pho-ton with p2γas small as m2e.A similar observa-tion can be made at high energies(NLC)when some of the intermediate gauge bosons become eﬀectively longitudinal.There too the running-width scheme renders completely unreliable re-sults.11In processes involving more intermediate gauge bosons,e.g.e+e−→6f,also theﬁxed-width scheme breaks down at high energies as a result of breaking SU(2)gauge invariance.3.2Radiative correctionsBy employing the fermion-loop scheme all one-particle-irreducible fermionic one-loop corrections can be embedded in the tree-level matrix ele-ments.This results in running couplings,prop-agator functions,vertex functions,etc.However, there is still the question about the bosonic cor-rections.A large part of these bosonic correc-tions,as e.g.the leading QED corrections,factor-ize and can be treated by means of a convolution, using the fermion-loop-improved cross-sections in the integration kernels.This allows the inclusion of higher-order QED corrections and soft-photon exponentiation.In this way various important ef-fects can be covered,as e.g.the large negative soft-photon corrections near the nominal W-pair threshold,the distortion of angular distributions as a result of hard-photon boost eﬀects,and the average energy loss due to radiated photons.2,12 Nevertheless,the remaining bosonic corrections can be large,especially at high energies.2,12 In order to include these corrections one might attempt to extend the fermion-loop scheme.In the context of the background-ﬁeld method a Dyson summation of bosonic self-energies can be per-formed without violating the Ward identities.13 However,the resulting matrix elements depend on the quantum gauge parameter at the loop level that is not completely taken into account.As mentioned before,the perturbation series has to be truncated;in that sense the dependence on the quantum gauge parameter could be viewed as a parametrization of the associated ambiguity.As a more appealing strategy one might adopt a hybrid scheme,adding the remaining bosonic loop corrections by means of the pole scheme.This is gauge-invariant and contains the well-known bosonic corrections for the production of on-shell gauge bosons(in particular W-boson pairs). Moreover,if the quality of the pole scheme were to degrade in certain regions of phase-space,the associated error is reduced by factors ofα/π.It should be noted that the application of the pole scheme to photonic corrections requires some spe-cial care,because in that case terms proportional to log(k2−M2)/(k2−M2)complicate the pole expansion.9,12AcknowledgmentThis research has been supported by a fellowship of the Royal Dutch Academy of Arts and Sciences.References1.Z.Kunszt et al.,in Physics at LEP2,eds.G.Altarelli,T.Sj¨o strand and F.Zwirner,(CERN96-01,Gen`e ve,1996)Vol.1,p.141,hep-ph/9602352.2.W.Beenakker et al.,in Physics atLEP2,eds.G.Altarelli,T.Sj¨o strand andF.Zwirner,(CERN96-01,Gen`e ve,1996)Vol.1,p.79,hep-ph/9602351.3.W.Beenakker and G.J.van Oldenborgh,Phys.Lett.B381(1996)248.4.M.Rijssenbeek,these proceedings.5.G.Gounaris et al.,in Physics at LEP2,eds.G.Altarelli,T.Sj¨o strand and F.Zwirner,(CERN96-01,Gen`e ve,1996)Vol.1,p.525,hep-ph/9601233.6.P.M.Zerwas(ed.),e+e−collisions at500GeV:The physics potential(DESY93-123C,Hamburg,1993).7.M.Veltman,Physica29(1963)186.8.R.G.Stuart,Phys.Lett.B262(1991)113.9.A.Aeppli,G.J.van Oldenborgh andD.Wyler,Nucl.Phys.B428(1994)126.10.E.N.Argyres et al.,Phys.Lett.B358(1995)339.11.W.Beenakker et al.,hep-ph/9612260.12.W.Beenakker and A.Denner,Int.J.Mod.Phys.A9(1994)4837.13.A.Denner and S.Dittmaier,Phys.Rev.D54(1996)4499.。