1、Yue-Liang WuKavli Institute for Theoretical Physics China Key Laboratory of Frontiers in Theoretical Physics Institute of Theoretical Physics, CAS Chinese Academy of Sciences,CPT and Lorentz Invariance and Violation in QFT,Symmetry & Quantum Field Theory,Symmetry has played an important role in phys
2、icsCPT and Lorentz invariance are regarded as the fundamental symmetries of nature. CPT invariance is the basic property of relativistic quantum field theory for point particle.All known basic forces of nature: electromagnetic, weak, strong & gravitational forces, are governed by the local gauge sym
3、metries:U(1)_Y x SU(2)_L x SU(3)_c x SO(1,3),Lorentz and CPT Violation in QFT,QFT may not be an underlying theory but EFTIn String Theory, Lorentz invariance can be broken down spontaneously.Lorentz non-invariant quantum field theoryExplicit、Spontaneous 、InducedCPT/Lorentz violating Chern-Simons ter
4、m constant vector,Induced CTP/Lorentz Violation,Real world has been found to be successfully described by quantum field theories EQED with constant vector mass What is the relation ?,Diverse Results,Gauge invariance of axial-currentS.Coleman and S.L.Glashow, Phys.Rev.D59: 116008 (1999) Pauli-Villas
5、regularization with D.Colladay and V.A.Kostelecky, Phys. Rev. D58:116002 (1998). Gauge invariance and conservation of vector Ward identityM.Perez-Victoria, JHEP 0104 032 (2001). Consistent analysis via dimensional regularization G.Bonneau, Nucl. Phys. B593 398 (2001).,Diverse Results,Based on nonper
6、turbative formulation withR.Jackiw and V.A.Kostelecky, Phys.Rev.Lett. 82: 3572 (1999). Derivative Expansion with dimensional regularizationJ.M.Chung and P.Oh, Phys.Rev.D60: 067702 (1999). Keep full dependence with M.Perez-Victoria, Phys.Rev.Lett.83: 2518 (1999).Keep full dependence with M.Perez-Vict
7、oria, Phys.Rev.Lett.83: 2518 (1999).,Consistent Result,Statement in Literature: constant vector K can only be determined by experiment Our Conclusion: constant vector K can consistently be fixed from theoretical calculationsfor for,Regularization Scheme,Regularization scheme dependence Ambiguity of
8、Dimensional regularization withproblemAmbiguity with momentum translation for linear divergent termAmbiguity of reducing triangle diagrams How to reach a consistent regularization scheme ?,Regularization Methods,Cut-off regularizationKeeping divergent behavior, spoiling gauge symmetry & translationa
9、l/rotational symmetriesPauli-Villars regularizationModifying propagators, destroying non-abelian gauge symmetry, introduction of superheavy particlesDimensional regularization: analytic continuation in dimensionGauge invariance, widely used for practical calculationsGamma_5 problem, losing scaling b
10、ehavior (incorrect gap eq.), problem to chiral theory and super-symmetric theory All the regularizations have their advantages and shortcomings,Criteria of Consistent Regularization,(i) The regularization is rigorous that it can maintain the basic symmetry principles in the original theory, such as:
11、 gauge invariance, Lorentz invariance and translational invariance(ii) The regularization is general that it can be applied to both underlying renormalizable QFTs (such as QCD) and effective QFTs (like the gauged Nambu-Jona-Lasinio model and chiral perturbation theory).,Criteria of Consistent Regula
12、rization,(iii) The regularization is also essential in the sense that it can lead to the well-defined Feynman diagrams with maintaining the initial divergent behavior of integrals. so that the regularized theory only needs to make an infinity-free renormalization.(iv) The regularization must be simp
13、le that it can provide the practical calculations.,Symmetry-Preserving Loop Regularization (LORE) with String Mode Regulators,Yue-Liang Wu, SYMMETRY PRINCIPLE PRESERVING AND INFINITY FREE REGULARIZATION AND RENORMALIZATION OF QUANTUM FIELD THEORIES AND THE MASS GAP. Int.J.Mod.Phys.A18:2003, 5363-542
14、0.Yue-Liang Wu, SYMMETRY PRESERVING LOOP REGULARIZATION AND RENORMALIZATION OF QFTS. Mod.Phys.Lett.A19:2004, 2191-2204.,Why Quantum Field Theory So Successful,Folks theorem by Weinberg:Any quantum theory that at sufficiently low energy and large distances looks Lorentz invariant and satisfies the cl
15、uster decomposition principle will also at sufficiently low energy look like a quantum field theory.Indication: existence in any case a characterizing energy scale (CES) M_cAt sufficiently low energy then means: E M_c QFTs,Why Quantum Field Theory So Successful,Renormalization group by Wilson or Gel
16、l-Mann & LowAllow to deal with physical phenomena at any interesting energy scale by integrating out the physics at higher energy scales. To be able to define the renormalized theory at any interesting renormalization scale . Implication: Existence of sliding energy scale (SES) _s which is not relat
17、ed to masses of particles.The physical effects above the SES _s are integrated in the renormalized couplings and fields.,Irreducible Loop Integrals (ILIs),Loop Regularization (LORE),Simple Prescription: in ILIs, make the following replacementWith the conditionsSo that,Gauge Invariant Consistency Con
18、ditions,Checking Consistency Condition,Checking Consistency Condition,Vacuum Polarization,Fermion-Loop Contributions,Gluonic Loop Contributions,Cut-Off & Dimensional Regularizations,Cut-off violates consistency conditionsDR satisfies consistency conditionsBut quadratic behavior is suppressed in DR,S
19、ymmetrypreserving & Infinity-free Loop Regularization (LORE) With String-mode Regulators,Choosing the regulator masses to have the string-mode Reggie trajectory behaviorCoefficients are completely determinedfrom the conditions,Explicit One Loop Feynman Integrals,With,Two intrinsic mass scales and pl
20、ay the roles of UV- and IR-cut off as well as CES and SES,Interesting Mathematical Identities,which lead the functions to the following simple forms,Renormalization Constants of Non- Abelian gauge Theory and Function of QCD in Loop Regularization,Lagrangian of gauge theoryPossible counter-terms,Jian
21、-Wei Cui, Yue-Liang Wu, Int.J.Mod.Phys.A23:2861-2913,2008,Ward-Takahaski-Slavnov-Taylor Identities,Gauge Invariance,Two-point Diagrams,Three-point Diagrams,Four-point Diagrams,Ward-Takahaski-Slavnov-Taylor Identities,Renormalization ConstantsAll satisfy Ward-Takahaski-Slavnov-Taylor identities,Renor
22、malization Function,Gauge Coupling Renormalizationwhich reproduces the well-known QCD function (GWP),Supersymmetry in Loop Regularization,SupersymmetrySupersymmetry is a full symmetry of quantum theory Supersymmetry should be Regularization-independent Supersymmetry-preserving regularization,J.W. Cu
23、i, Y.Tang,Y.L. Wu Phys.Rev.D79:125008,2009,Massless Wess-Zumino Model,LagrangianWard identityIn momentum space,Check of Ward Identity,Gamma matrix algebra in 4-dimension and translational invariance of integral momentum Loop regularization satisfies these conditions,Massive Wess-Zumino Model,Lagrang
24、ianWard identity,Check of Ward Identity,Gamma matrix algebra in 4-dimension and translational invariance of integral momentum Loop regularization satisfies these conditions,Triangle Anomaly,AmplitudesUsing the definition of gamma_5The trace of gamma matrices gets the most general and unique structur
25、e with symmetric Lorentz indices,Anomaly of Axial Current,Explicit calculation based on Loop Regularization with the most general and symmetric Lorentz structureRestore the original theory in the limit which shows that vector currents are automatically conserved, only the axial-vector Ward identity
26、is violated by quantum corrections,Chiral Anomaly Based on Loop Regularization,Including the cross diagram, the final result is,Which leads to the well-known anomaly form,Anomaly Based on Various Regularizations,Using the most general and symmetric trace formula for gamma matrices with gamma_5. In u
27、nit,Quantum Loop Induced CPT/Lorentz Violating Chern-Simons Term in Loop Regularization,Amplitudes of triangle diagrams,Contributions to Amplitudes,Convergent contributionsDivergent contributions Logarithmic DV Linear DV,Contributions to Amplitudes,Logarithmic Divergent ContributionsRegularized resu
28、lt with LORE,Contributions to Amplitudes,Linear divergent contributionsRegularized result,Contributions to Amplitudes,Total contributions arise from convergent part,Final Result,SettingFinal result is,Comments on Ambiguity,Momentum translation relation of linear divergentRegularization after using t
29、he relation,Check on Consistency,Ambiguity of resultsInconsistency with U(1) chiral anomaly of,Conclusions,First applying for the regularization before using momentum translation relation of linear divergent integralLoop regularization is translational invariant Induced Chern-Simons term is uniqely determined when combining the chiral anomalyThere is no harmful induced Chern-Simons term for massive fermions.,谢谢！,THANKS,
