International Workshop on Particle Physics and Cosmology

1. International Workshop on Particle Physics and Cosmology after Higgs and Planck 后希格斯与普郎克粒子物理与宇宙学国际研讨会 September 5-9 2013 Chongqing University of Posts and Telecommunications Chongqing China

2. Quantum Theoryof Particlesand Fields Yue-Liang Wu Kavli Institute for Theoretical Physics China (KITPC) State Key Laboratory of Theoretical Physics (SKLTP) ITP-CAS University of Chinese Academy of Sciences(UCAS)

3. HiggsBoson（上帝粒子）at LHC Higgs Mass ~ 125 GeV

4. HC, DM, DE At Planck

5. Higgs Boson, Dark Matter, Dark Energy & Inflation Mass Generation, WIMP, Vacuum Energy Appearance of Mass/Energy Scale Quantum Theory of Scalar & Gravitational Fields Quantum Structure of Quadratic Divergence

6. Quantum Field Theory and Symmetry Quantum Field Theory 量子场论 Quantum Mechanics 量子力学 Special Relativity 相对论 = + + Symmetry Principle 对称原理 Elementary Particle Physics 基本粒子物理 =

7. Basic Symmetry in Standard Model • Symmetry has played an important role in elementary particle physics • All known basic forces of nature: Electromagnetic, weak, strong & gravitational forces, are governed by the symmetries U(1)_Y x SU(2)_L x SU(3)_c x SO(1,3) • It has been found to be successfully described by quantum field theory (QFT)

8. Divergence Problem in QFTs QFTs cannot be defined by a straightforward perturbative expansion due to the presence of ultraviolet divergences. • The divergences appear when developing quantum electrodynamics (QED) in the 1930s by Max Born, Werner Heisenberg, Pascual Jordan, Paul Dirac. • The treatment of divergences was further described in the 1940s by Julian Schwinger, Richard Feynman, Shinichiro Tomonaga, and investigated systematically by Freeman Dyson.

9. QED Freeman Dyson initiated the perturbative expansion of QED and proposedthe renormalization of mass and coupling constantto treat the divergences

10. Freeman Dyson showed that these divergences or infinities are of a basic nature and cannot be eliminated by any formal mathematical procedures, such as the renormalization method

11. Origin of Divergence in QFTs The divergence arises from the calculations of Feynman diagrams with closed loops of virtual particles • It is because the integral region where all particles in the loop have large energies and momenta • It is caused from the very short wavelength or high frequency fluctuations of the fieldsin the path integral • It is due to very short proper-time between particle emission and absorptionwhen the loop is thought of as a sum over particle paths

12. Treatment on Divergence in QFT Treatment of divergences is the key to understand the quantum structure of field theory. • 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, string theory !?

13. Regularization Schemes in QFT • Cut-off regularization Keeping divergent behavior, direct presence of energy scales spoiling gauge symmetry, translational/rotational symmetries • Pauli-Villars regularization Introducing superheavy particles, applicable to U(1) gauge theory  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 &shortcomings

14. Dirac’s Criticism on QED Most physicists are very satisfied with the situation. They say: 'Quantum electrodynamics(QED) is a good theory and we do not have to worry about it any more.’ I must say that I am very dissatisfied with the situation, because this so-called 'good theory' does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small - not neglecting it just because it is infinitely great and you do not want it! • P.A.M. Dirac, “The Evolution of the Physicist‘s Picture of Nature,” in Scientific American, May 1963, p. 53. • Kragh, Helge ; Dirac: A scientific biography, CUP 1990, p. 184

15. Feynman’s Criticism on QED The shell game that we play ... is technically called 'renormalization'. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics(QED) is mathematically self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate. Feynman, Richard P. ; QED, The Strange Theory of Light and Matter, Penguin 1990, p. 128

16. Why Quantum Field Theory So Successful Folk’s theorem by Weinberg: Any quantum theory that at sufficiently low energy and 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 meaningful E << Mc QFTs

17. Why Quantum Field Theory So Successful Renormalization group Analysis 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 both charactering energy scale (CES) M_cand sliding energy scale(SES) μs which is not related to masses of particles. • Physical effects above SES μs can be integrated in the renormalized couplings and fields.

18. Why Quantum Field Theory So Successful More Indications Based on RG Analysis: • Any QFT can be defined fundamentally with the meaningful energy scale that has some physical significance. • Whatever the Lagrangian of QFTs was at the fundamental scale, as long as its couplings are sufficiently weak, it can be described at the interesting energy scales by a renormalizable effective Lagrangian of QFTs. Explanation to the renormalizability of QFTs and SM • Electroweak interaction with spontaneous symmetry breaking has been shown to be a renormalizable theory • by t Hooft & Veltman • QCD as the Yang-Mills gauge theory has been shown to have an interesting property of asymptotic freedom • by Gross, Wilzck, Politz

19. Treatment on Divergence with Meaningful Regularization Scheme (i)The regularization should be essential: It can lead to the well-defined Feynman diagrams with physically meaningful energy scales to maintain the initial divergent behavior of integrals, so that the regularized theory only needs to make an infinity-free renormalization. (ii) The regularization should be rigorous: It can maintain the basic symmetry principles in the original theory, such as: gauge invariance, Lorentz invariance and translational invariance

20. Treatment on Divergences with Meaningful Regularization Scheme (iii)The regularization should be general: It can be applied to the underlying renormalizable QFTs (such as QCD), effective QFTs (like the gauged Nambu-Jona-Lasiniomodel), supersymmetrictheories and chiral theories. (iv)The regularization should also be simple: It can provide practical calculations.

21. Loop Regularization (LORE) Method The Loop Regularization method(LORE) 【1】【2】 realized in 4D space-time has been shown to satisfy all mentioned properties 【1】Yue-Liang Wu, “Symmetry principle preserving and infinity free regularization andrenormalization of quantum field theories and the mass gap” Int.J.Mod.Phys.A18:2003, 5363-5420. 【2】Yue-Liang Wu, “Symmetry-preserving loop regularization and renormalization of QFTs”Mod.Phys.Lett.A19:2004, 2191-2204. • The key concept of LORE is the introduction of the irreducible loop integrals(ILIs)which are evaluated from the Feynman diagrams • The crucial point in LORE method is the presence of two intrinsic energy scales introduced via the string-mode regulators in the regularization prescription acting on the ILIs. • These two intrinsic energy scales have been shown to play the roles of ultraviolet (UV) cut-off and infrared (IR) cut-off to avoid infinities without spoiling symmetries in original theory, and become meaningful as charactering energy scale and sliding energy scale

22. The LORE method has been proved with explicit calculations at one loop level that it can preserve non-Abelian gauge symmetry 【3】and supersymmetry【4】 • The LORE method can provide a consistent calculation for the chiral anomaly【5】,radiativelyinduced Lorentz/CPT-violating Chern-Simons term in QED【6】, the QED trace anomaly【7】 【3】J.W.Cuiand Y.L.Wu,One-Loop Renormalization of Non-Abelian Gauge Theory and \beta Function Based on Loop Regularization Method,’’Int. J. Mod. Phys. A 23, 2861 (2008) [arXiv:0801.2199] 【4】J.W.Cui, Y.Tang and Y.L.Wu, “Renormalization of Supersymmetric Field Theories in Loop Regularization with String-mode Regulators” Phys. Rev. D 79, 125008 (2009) [arXiv:0812.0892 [hep-ph]]. 【5】Y.L.Ma and Y.L.Wu, “Anomaly and anomaly-free treatment of QFTs based on symmetry-preserving loop regularization”Int. J. Mod. Phys. A 21, 6383 (2006) [arXiv:hep-ph/0509083]. 【6】Y.L.Maand Y.L.Wu, “On the radiatively induced Lorentz and CPT violating Chern-Simons term” Phys. Lett. B 647, 427 (2007) [arXiv:hep-ph/0611199]. 【7】J.W. Cui, Y.L. Ma and Y.L. Wu, “Explicit derivation of the QED trace anomaly in symmetry-preserving loop regularization at one-loop level” Phys.Rev. D 84,025020 (2011), arXiv:1103.2026 [hep-ph].

23. The LORE method allows us to derive the dynamically generated spontaneous chiral symmetry breaking of the low energy QCD【8】 for understanding the dynamical quark masses and the mass spectra of light scalar and pseudoscalarmesons, as well the chiral symmetry restoration at finite temperature【9】 • The LORE method enables us to consistently carry out calculations on quantum gravitational contributions to gauge theories with asymptotic free power-law running【10–12】. 【8】Y.B.Dai and Y.L.Wu,"Dynamically spontaneous symmetry breaking and masses of lightest nonet scalar mesons as composite Higgs bosons,’’ Eur. Phys. J. C 39 (2004) S1 [arXiv:hep-ph/0304075]. 【9】D. Huang and Y.L. Wu, “Chiral Thermodynamic Model of QCD and its Critical Behavior in the Closed-Time-Path Green Function Approach”, arXiv:1110.4491 [hep-ph] 【10】Y.Tang and Y.L.Wu, “Gravitational Contributions to the Running of Gauge Couplings”, Commun. Theor. Phys. 54, 1040 (2010) [arXiv:0807.0331 [hep-ph]]. 【11】Y.Tang and Y.L.Wu, “Quantum Gravitational Contributions to Gauge Field Theories” Commun. Theor. Phys.57, 629 (2012), arXiv:1012.0626 [hep-ph] 【12】Y.Tang and Y.L.Wu, “Gravitational Contributions to Gauge Green's Functions and Asymptotic Free Power-Law Running of Gauge Coupling” JHEP 1111, 073 (2011), arXiv:1109.4001 [hep-ph].

24. The LORE methodhas been applied to clarify the issue【13】 raised by Gastmans, S.L. Wu and T.T. Wu. in the process H →γγthrough a W-boson loop in the unitary gauge, and show that a finite amplitude still needs a consistent regularization for cancellation between tensor and scalar type divergent integrals • The LORE method has been applied to demonstrate consistently and explicitly the general structure of QFTs through higher-loop order calculations【14-15】. • In the LORE method, the evaluation of ILIs naturally merges to the Bjorken-Drell’sanalogy between the Feynman diagrams and electric circuits【14-15】. 【13】D.Huang,Y.Tang and Y.L.Wu “Note on Higgs Decay into Two Photons H→γγ”, Commun.Theor.Phys. 57 (2012) 427-434， arXiv:1109.4846[hep-ph] 【14】D.Huang and Y.L. Wu,”Consistencyand Advantage of Loop Regularization Method Merging with Bjorken-Drell's Analogy Between Feynman Diagrams and Electrical Circuits”, Eur.Phys.J. C72 (2012) 2066 , arXiv:1108.3603[hep-ph] 【15】D. Huang, L.F. Li and Y.L. Wu, Consistency of Loop Regularization Method and Divergence Structure of QFTs Beyond One-Loop Order, Eur.Phys.J. C73 (2013) 2353, arXiv:1210.2794[hep-ph]

25. Loop Regularization(LORE) Method Concept of Irreducible Loop Integrals(ILIs) Scalar-type ILIs Tensor-type ILIs

26. LORE Method Prescription of LORE method In ILIs, make the following replacement With the conditions for regulator masses and coefficients Which is resulted from the requirement: regulator mass coefficients Divergence power ≥ the space-time dimension vanishes

27. Gauge Invariant Consistency Conditions

28. Checking Consistency Conditions

29. Checking Consistency Conditions

30. Vacuum Polarization • Fermion-Loop Contributions

31. Gluonic Loop Contributions

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

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

34. Cut-Off & Dimensional Regularizations • Cut-off violates consistency conditions • DR satisfies consistency conditions • Quadratic behavior is suppressed and the sign is opposite  0 when m 0, namely

35. LORE Method With String-mode Regulators • Choosing the regulator masses to have the string-mode Reggie trajectory behavior with the conditions to recover original integrals and make regulator independent result • Coefficients are completely determined from the required conditions Divergence power ≥ the space-time dimension vanishes

36. Explicit One Loop Feynman Integrals in LORE Compare to DR Eulerconstant =0.577216… With LORE is an Infinity-Free Regularization! Two intrinsic energy scales and play the roles of UV- and IR-cut off, but physically meaningful as the CES and SES

37. Interesting Mathematical Identities which lead the functions to the following explicit forms

38. General Evaluation of ILIs & UVDP Parameterization General structure of Feynman integral Overall and vertex momentum conservations of Feynman diagrams Internal momentum (k_i) decomposition with loop momentum (l_r) and the undetermined internal currents flowing q_j

39. Evaluation of ILIs and UVDP Parameterization ILIs are resulted from the following conditions Writing the above conditions and momentum conservation into a more heuristic form which determine currents flowing q_j

40. ILIs and Bjorken-Drell’s Circuit Analogy Current conservationat vertex: q-- internal currents flowing in the circuit; p-- the external currents entering it Kirchhoff’s laws in the electric circuit analogy: sum of voltage droparound any closed loop is zero

41. ILIs and Bjorken-Drell’s Circuit Analogy Ohm’s Law --- the resistance of the jthline or --- the conductance of the jth line --- the displacement between two points Equation of motion for a free particle --the causal propagation of a particle --the causality of Feynman propagator

42. LORE Method Merging With Bjorken-Drell’s Circuit Analogy Divergence of loop integral arises from infinite conductance Zero Resistance  Short Circuit Circuit analogy helps to treat properly all divergences in LORE

43. Evaluation of ILIs and UVDP Parameterization Loop momentum integral by diagonalizingthe quadratic momentum terms with an orthogonal transformation O -- the eigenvalues of the matrix M --functions of UVDP parameters v_i Feynman integrals are evaluated into ILIs

44. For the condition: (k-1) internal loop momentum integrals are convergent ILIs UV divergences for the loop integrals over l_(r) (r = 1…k −1) in the original subdiagramsare characterized by zero eigenvaluesλ_(r) → 0 (r =1…k − 1) of the matrix M The momentum integral on in ILIs reflects the overall divergence of the Feynman diagram

45. Each zero eigenvalue λ_(r) → 0 • infinity values of parameters •  singularity for parameter integrals Divergence in UVDP-parameter space corresponds to Divergence of subdiagram in momentum space Regularized 1-fold ILIs for overall divergence of Feynman diagram

46. Consistency and Advantage of LORE Method • The LORE methodnaturally merges with Bjorken-Drell’s analogy between Feynman diagrams and electric circuits, and enables us to make a systematic procedure to all orders of Feynman diagrams • The LORE method has been realized in 4D space-time without modifying original Lagrangian, so it cannot be proved in the Lagrangian formalism to all orders • The Concept of ILIs and the Circuit Analogy of Feynman diagrams in LORE provides a diagrammatic approach for a general proofon the consistency of LORE method with the observation of one-to-one correspondence of divergences between UVDP parameters and subdiagrams of Feynman diagrams

47. Applicability of LORE Method Why the calculation of finite amplitude for the Feynman diagrams in the standard model still needs a consistent regularization method ???

48. Issue on Higgs Decay into Two Photons Hγγ Issues on Dimensional Regularization calculation for Higgs decay into two photon in unitary gauge by R. Gastmans, S.L. Wu and T.T. Wu The divergent tensor-type ILI Not a divergent scalar-type ILI Question? Naïve replacement in divergent integrals

49. W-boson contribution to 2 photon in unitary gauge

50. Amplitudes of three diagrams in unitary gauge