Quantum Field Theory: Symmetry, Renormalization, and Consistent Regularization

1. Quantum Structure of Field Theory Yue-Liang Wu State Key Laboratory of Theoretical Physics (SKLTP) Kavli Institute for Theoretical Physics China（KITPC） Institute of Theoretical Physics, Chinese Academy of Sciences 2012.04.12

2. Symmetry & Quantum Field Theory • Symmetry has played an important role in elementary particle physics • All known basic forces of nature: electromagnetic, weak, strong & gravitational forces, are governed by U(1)_Y x SU(2)_L x SU(3)_c x SO(1,3) • Which has been found to be successfully described by quantum field theories (QFTs)

3. Why Quantum Field Theory So Successful Folk’s theorem by Weinberg: Any quantum theory that at sufficiently low energyand large distances looks Lorentz invariant and satisfies the cluster decomposition principle will also at sufficiently low energy look like a quantum field theory. • Indication:existence in any case a characterizing energy scale (CES) Mc • So that at sufficiently low energy gets meaning: E << Mc  QFTs

4. Why Quantum Field Theory So Successful Renormalization group by Wilson/Gell-Mann & Low Allow to deal with physical phenomena at any interesting energy scale by integrating out the physics at higher energy scales. Allow to define the renormalized theory at any interesting renormalization scale . • Implication:Existence of sliding energy scale(SES) μs which is not related to masses of particles. • Physical effects above the SES μs are integrated in the renormalized couplings and fields.

5. How to Avoid Divergence • QFTs cannot be defined by a straightforward perturbative expansion due to the presence of ultraviolet divergences. • Regularization: Modifying the behavior of field theory at very large momentum so Feynman diagrams become well-defined quantities • String/superstring: Underlying theory might not be a quantum theory of fields, it could be something else.

6. Regularization Schemes • Cut-off regularization Keeping divergent behavior, spoiling gauge symmetry & translational/rotational symmetries • Pauli-Villars regularization Modifying propagators, destroying non-abelian gauge symmetry • Dimensional regularization: analytic continuation in dimension • Gauge invariance, widely used for practical calculations • Gamma_5 problem: questionable to chiral theory • Dimension problem: unsuitable for super-symmetric theory • Divergent behavior: losing quadratic behavior (incorrect gap eq.) All the regularizations have their advantages and shortcomings

7. Criteria of Consistent Regularization (i)The regularization is rigorous: It can maintain the basic symmetry principles in the original theory, such as: gauge invariance, Lorentz invariance and translational invariance (ii)The regularization is general: 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).

8. Criteria of Consistent Regularization (iii)The regularization is also essential: 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 simple: It can provide practical calculations.

9. Loop Regularization (LORE) Method 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, 5363-5420. • Yue-Liang Wu, SYMMETRY PRESERVING LOOP REGULARIZATION AND RENORMALIZATION OF QFTS.Mod.Phys.Lett.A19:2004, 2191-2204. • J.~W.~Cui and Y.~L.~Wu, Int. J. Mod. Phys. A 23, 2861 (2008) • J.~W.~Cui, Y.~Tang and Y.~L.~Wu, Phys. Rev. D 79, 125008 (2009) • Y.~L.~Ma and Y.~L.~Wu, Int. J. Mod. Phys. A21, 6383 (2006) • Y.~L.~Ma and Y.~L.~Wu, Phys. Lett. B 647, 427 (2007) • J.W. Cui, Y.L. Ma and Y.L. Wu, Phys.Rev. D 84, 025020 (2011) • Y.~B.~Dai and Y.~L.~Wu, Eur. Phys. J. C 39 (2004) S1 • Y.~Tang and Y.~L.~Wu, Commun. Theor. Phys. 54, 1040 (2010) • Y.~Tang and Y.~L.~Wu, JHEP 1111, 073 (2011), arXiv:1109.4001 [hep-ph]. • Y.~Tang and Y.~L.~Wu, arXiv:1012.0626 [hep-ph]. • D. Huang and Y.L. Wu, arXiv:1108.3603

10. Irreducible Loop Integrals (ILIs)

11. Loop Regularization (LORE) Method Simple Prescription: in ILIs, make the following replacement With the conditions So that

12. Gauge Invariant Consistency Conditions

13. Checking Consistency Condition

14. Checking Consistency Condition

15. Vacuum Polarization • Fermion-Loop Contributions

16. Gluonic Loop Contributions

17. Proper Treatment of Divergent Integrals Lorentz decomposition & Naïve tensor manipulation • Violating gauge symmetry • Tensor manipulation and integration don’t commute for divergent integrals

18. Direct Proof of Consistency Condition • Consider the zero components and convergent integration over zero momentum component

19. Cut-Off & Dimensional Regularizations • Cut-off violates consistency conditions • DR satisfies consistency conditions • But quadratic behavior is suppressed with opposite sign  0 when m 0

20. Symmetry–preserving Loop Regularization (LORE) With String-mode Regulators • Choosing the regulator masses to have the string-mode Reggie trajectory behavior • Coefficients are completely determined • from the conditions

21. Explicit One Loop Feynman Integrals With Two intrinsic mass scales and play the roles of UV- and IR-cut offas well asCES and SES

22. Interesting Mathematical Identities which lead the functions to the following simple forms

23. Renormalization Constants of Non- Abelian gauge Theory and β Function of QCD in Loop Regularization Jian-Wei Cui & Yue-Liang Wu Int. J. Mod. Phys. A 23, 2861 (2008) • Lagrangian of gauge theory • Possible counter-terms

24. Ward-Takahaski-Slavnov-Taylor Identities Gauge Invariance Two-point Diagrams

25. Three-point Diagrams

26. Four-point Diagrams

27. Ward-Takahaski-Slavnov-Taylor Identities • Renormalization Constants • All satisfy Ward-Takahaski-Slavnov-Taylor identities

28. Renormalization β Function • Gauge Coupling Renormalization which reproduces the well-known QCD β function (GWP)

29. Supersymmetry in Loop Regularization J.W. Cui,Y.Tang,Y.L. Wu Phys.Rev.D79:125008,2009 Supersymmetry • Supersymmetry is a full symmetry of quantum theory • Supersymmetry should be Regularization-independent • Supersymmetry-preserving Regularization

30. Massless Wess-Zumino Model • Lagrangian • Ward identity • In momentum space

31. Check of Ward Identity Gamma matrix algebra in 4-dimension and translational invariance of integral momentum Loop regularization satisfies these conditions

32. Massive Wess-Zumino Model • Lagrangian • Ward identity

33. Check of Ward Identity Gamma matrix algebra in 4-dimension and translational invariance of integral momentum Loop regularization satisfies these conditions

34. Triangle Anomaly • Amplitudes • Using the definition of gamma_5 • The trace of gamma matrices gets the most general and unique structure with symmetric Lorentz indices Y.L.Ma & YLW

35. Anomaly of Axial Current • Explicit calculation based on Loop Regularization with the most general and symmetric Lorentz structure • Restore the original theory in the limit which shows that vector currents are automatically conserved, only the axial-vector Ward identity is violated by quantum corrections

36. Chiral Anomaly Based on Loop Regularization Including the cross diagram, the final result is Which leads to the well-known anomaly form

37. Anomaly Based on Various Regularizations • Using the most general and symmetric trace formula for gamma matrices with gamma_5. • In unit Loop Regularization (LORE) Method

38. 量子引力对规范场的影响 规范耦合常数的幂次跑动渐近自由性质 基于单圈图和传统背景场的计算表明，微扰量子引力对规范场的修正是规范不变的结论与正规化无关，但计算结果与引力规范条件有关。在Harmonic 引力规范条件下，任何保证对称性 和理论原有发散行为的正规化方法，如圈正规化方法，得到规范耦合常数的幂次跑动渐近自由行为。而维数正规化和动量切断正规化都得到0的结果。 Y. Tang, YLW，JHEP 1111, 073 (2011), e-Print: arXiv:1109.4001 [hep-ph], Y. Tang, YLW， to be published in PLB, e-Print: arXiv:1012.0626 [hep-ph] Y. Tang, YLW， Commun.Theor.Phys. 54 (2010) 1040-1044, e-Print: arXiv:0807.0331 [hep-ph]

39. 基于引力规范条件无关的Vilkovisky-DeWitt背景场方法，任何保证对称性和理论原有发散行为的正规化方法，如圈正规化方法，得到规范条件无关的结论：量子引力的贡献使得规范耦合常数呈现幂次跑动渐近自由行为。而维数正规化得到0的结果，动量切断正规化得到相反结论。基于引力规范条件无关的Vilkovisky-DeWitt背景场方法，任何保证对称性和理论原有发散行为的正规化方法，如圈正规化方法，得到规范条件无关的结论：量子引力的贡献使得规范耦合常数呈现幂次跑动渐近自由行为。而维数正规化得到0的结果，动量切断正规化得到相反结论。

40. Loop Regularization Merging With Bjorken-Drell’s Circuit Analogy The divergence arises from zero resistance  Short Circuit which enables us to prove the validity of LORE to all orders

41. Loop Regularization Merging With Bjorken-Drell’s Circuit Analogy

42. Application to Two Loop Calculations by LORE in ϕ^4 Theory

44. Log-running to coupling constant at two loop level Power-law running of mass at two loop level

45. Application to Two Loop Calculations by LORE in ϕ^4 Theory One loop contribution with quadratic term to the scalar mass by the LORE method Two loop contribution with quadratic term to the scalar mass by the LORE method

46. Dynamically Generated Spontaneous Chiral Symmetry Breaking In Chiral Effective Field Theory

47. QCD Lagrangian and Symmetry Chiral limit: Taking vanishing quark masses mq→ 0. QCD Lagrangian has maximum global Chiral symmetry :

48. QCD Lagrangian and Symmetry • QCD Lagrangian with massive light quarks

49. Effective Lagrangian Based on Loop Regularization (LORE) Y.B.Dai and Y-L. Wu, Euro. Phys. J. C 39 s1 (2004) After integrating out quark fields by the LORE method

50. Dynamically Generated Spontaneous Symmetry Breaking