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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 Quantum Theory of Particles and 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) Higgs Boson(上帝粒子)at LHC Higgs Mass ~ 125 GeV HC, DM, DE At Planck 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 Quantum Field Theory and Symmetry Special Relativity 相对论 + Quantum Mechanics 量子力学 = Quantum Field Theory 量子场论 + Elementary Particle Physics 基本粒子物理 = Symmetry Principle 对称原理 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) 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. QED Freeman Dyson initiated the perturbative expansion of QED and proposed the renormalization of mass and coupling constant to treat the divergences 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 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 fields in the path integral It is due to very short proper-time between particle emission and absorption when the loop is thought of as a sum over particle paths 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 !? 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 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 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 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 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_c and 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. 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 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 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-Lasinio model), supersymmetric theories and chiral theories. (iv) The regularization should also be simple: It can provide practical calculations. 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 and renormalization 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 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】, radiatively induced Lorentz/CPT-violating Chern-Simons term in QED【6】, the QED trace anomaly【7】 【3】 J.W.Cui and 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:hepph/0509083]. 【6】Y.L.Ma and Y.L.Wu, “On the radiatively induced Lorentz and CPT violating ChernSimons 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]. 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 pseudoscalar mesons, 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]. The LORE method has been applied to clarify the issue【13】 raised by Gastmans, S.L. Wu and T.T. Wu. in the process H →γγ through 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’s analogy 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,”Consistency and 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] Loop Regularization(LORE) Method Concept of Irreducible Loop Integrals(ILIs) Scalar-type ILIs Tensor-type ILIs LORE Method Prescription of LORE method In ILIs, make the following replacement regulator mass coefficients With the conditions for regulator masses and coefficients Which is resulted from the requirement: Divergence power ≥ the space-time dimension vanishes Gauge Invariant Consistency Conditions Checking Consistency Conditions Checking Consistency Conditions Vacuum Polarization Fermion-Loop Contributions Gluonic Loop Contributions Proper Treatment on Divergent Integrals Lorentz decomposition & Naïve tensor manipulation Violating gauge symmetry Tensor manipulation and integration don’t commute for divergent integrals Direct Proof of Consistency Conditions Consider the zero components and convergent integration over zero momentum component 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 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 Explicit One Loop Feynman Integrals in LORE Compare to DR With Euler constant =0.577216… LORE is an Infinity-Free Regularization! Two intrinsic energy scales and play the roles of UVand IR-cut off, but physically meaningful as the CES and SES Interesting Mathematical Identities which lead the functions to the following explicit forms 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 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 ILIs and Bjorken-Drell’s Circuit Analogy Current conservation at vertex: Kirchhoff’s laws in the q-- internal currents flowing in electric circuit analogy: the circuit; p-- the external sum of voltage drop around currents entering it any closed loop is zero ILIs and Bjorken-Drell’s Circuit Analogy Ohm’s Law --- the resistance of the jth line 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 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 Evaluation of ILIs and UVDP Parameterization Loop momentum integral by diagonalizing the 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 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 subdiagrams are 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 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 Consistency and Advantage of LORE Method The LORE method naturally 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 proof on the consistency of LORE method with the observation of one-to-one correspondence of divergences between UVDP parameters and subdiagrams of Feynman diagrams Applicability of LORE Method Why the calculation of finite amplitude for the Feynman diagrams in the standard model still needs a consistent regularization method ??? Issue on Higgs Decay into Two Photons Hγγ 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 Naïve replacement in divergent integrals Question? W-boson contribution to 2 photon in unitary gauge Amplitudes of three diagrams in unitary gauge Divergent tensor-type & scalar-type integrals has an inconsistency relation in GWW paper Regularized Divergent ILIs in LORE have the consistency condition The difference is a finite part which is crucial to ensure gauge invariance by requiring the consistency condition The Consistency of LORE Method with Explicit Calculations at One-Loop Level Renormalization Constants of Non- Abelian gauge Theory and β Function of QCD in LORE Method Lagrangian of gauge theory Possible counter-terms Ward-Takahaski-Slavnov-Taylor Identities Gauge Invariance Two-point Diagrams Unlike DR which leads tadpole diagram to vanish, the LORE preserves original quadratic divergence of diagrams Three-point Diagrams Four-point Diagrams Ward-Takahaski-Slavnov-Taylor Identities Renormalization Constants in ξ gauge Quadratic divergences cancel, all renormalization constants satisfy Ward-Takahaski-Slavnov-Taylor identities Renormalization β Function Gauge Coupling Renormalization It reproduces the well-known QCD β function (GWP) Supersymmetry-Preserving LORE Method J.W. Cui, Y.Tang,Y.L. Wu Phys.Rev.D79:125008,2009 Supersymmetry Supersymmetry is a full symmetry of quantum theory Supersymmetry is priory to gauge symmetry for treating divergence The LORE is a supersymmetry- and gauge symmetry-preserving regularization Massless Wess-Zumino Model Lagrangian Ward identity In momentum space Check of Ward Identity With gamma matrix algebra in exact 4-dimension and translational invariance of integral momentum quadratic divergences cancel LORE method satisfies these conditions Massive Wess-Zumino Model Lagrangian Ward identity Check of Ward Identity With gamma matrix algebra in exact 4-dimension and translational invariance of integral momentum, thus quadratic divergences cancel LORE method satisfies these conditions WARD IDENTITY IN SUPERSYMMETRIC GAUGE THEORY Lagrangian (with source terms) Infinitesimal supersymmetric transformation Supersymmetric Ward identity Contribution from Figs. (1)-(4) With gamma matrix algebra in exact 4-dimension and translational invariance of integral momentum, the quadratic divergences cancel Transverse condition is satisfied in supersymmetric model with the Feynman gauge ξ = 1 Fermion self-energy diagram Fig. (5) Contribution from Fig. (6) Contribution from Fig. (7)-(9) Quadratic divergences cancel automatically due to SUSY without the need of consistency condition for the quadratic ILIs Ward identity in SUSY gauge model is satisfied with only the need of consistency condition for the logarithmic ILIs in LORE method In the general ξ gauge, there is a term proportion to With a_0 being defined via logarithmic divergent a_0 ≠ 1 will break the transverse condition, only when regularization scheme satisfies consistency condition with Transverse condition or Gauge symmetry can be maintained LORE method preserves not only Yang-Mills gauge symmetry, but also supersymmetry Renormalization of Massive Wess-Zumino Model The action of massive Wess-Zumino mode Non-renormalization theorem implies a single renormalization constant (only the first dynamical term in superpotential needs a counterterm) Renormalizations of fields, mass and coupling constant must satisfy non-vanishing one-loop divergent graphs Resulting renormalization of fields and masses With the counterterms Lagrangian All renormalized vertices do lead to a single renormalization Gravitational Contributions to Gauge Green’s Functions and Asymptotic Free Power-Law Running of Gauge Coupling Gravitational Contributions to Gauge Coupling Robinson and Wilczek was the first to calculate the gravitational contributions to gauge coupling and show the power-law running in hamornic gauge condition by using cut-off regularization approach Pietrykowski noticed that RW result is gauge condition dependent; Toms calculated in DR with using gauge-condition independent formalism based on Vilkovisky-DeWitt’s background field method; Ebert et al. performed a diagrammatic calculation of two- and three-point Green’s functions in the harmonic gauge by using both cut-off and DR schemes Conclusion: No power-law running from gravitational contributions 【1】S. P. Robinson and F. Wilczek Phys. Rev. Lett. 96, 231601 (2006) 【2】A. R. Pietrykowski, Phys. Rev. Lett. 98, 061801 (2007). 【3】D. J. Toms, Phys. Rev. D 76, 045015 (2007). 【4】D. Ebert, J. Plefka and A. Rodigast, Phys. Lett. B660, 579(2008). Gravitational Contributions to Gauge Theories and Asymptotic Free Power-Law Running of Gauge Coupling Conclusion I: by checking all the calculations: the results are not only gauge condition dependent but also regularization scheme dependent (DRsuppresses quadratic divergent behavior; Cut-offdoesn’t satisfy consistency condition for gauge invariance) Conclusion II: gravitational contributions lead to asymptotically free power-law running gauge coupling in the harmonic gauge condition when using the LORE method to carry out the same calculations with both the diagrammatic and traditional background-field methods Conclusion III: Gauge coupling is power-law running and asymptotically free due to gravitational contributions when using the gauge condition independent Vilkovisky-DeWitt formalism of background field method and consistency condition of quadratic ILIs 【1】 Y. Tang and Y. L. Wu, Comm. Theo. Phys. 54, 1040(2010), arXiv:0807.0331 【2】 Y. Tang and Y. L. Wu, Comm. Theor. Phys.57, 629 (2012), arXiv:1012.0626 [hep-ph] More other discussions J. E. Daum, U. Harst and M. Reuter, JHEP 1001,084(2010). Feng Wu and Ming Zhong, Phys. Lett. B659, 694(2008), Phys. Rev. D 78, 085010 (2008). A. Rodigast and T. Schuster, Phys. Rev. D 79, 125017(2009), Phys. Rev. Lett. 104, 081301 (2010). O. Zanusso, L. Zambelli, G.P. Vacca and R. Percacci, Phys.Lett. B689, 90(1010). Paul T. Mackay and David J. Toms, Phys. Lett. B684, 251(2010). M. M. Anber, J. F. Donoghue and M. El-Houssieny, Phys. Rev. D83, 124003 (2011). [arXiv:1011.3229 [hep-th]]. E. Gerwick, Eur. Phys. J. C71, 1676 (2011). [arXiv:1012.1118 [hep-ph]]. John Ellis and Nick E. Mavromatos, arXiv:1012:4353 [hep-th]. S. Folkerts, D. F. Litim, J. M. Pawlowski, [arXiv:1101.5552 [hep-th]], D. F. Litim, [arXiv:1102.4624 [hep-th]]. D. J. Toms, Phys. Rev. Lett. 101, 131301 (2008), Phys. Rev. D 80, 064040(2009). D. J. Toms, Nature 468, 56 (2010). Vilkovisky-DeWitt Effective Action (Gauge Condition Independent) Background field Quantum field Faddeev-Popov factor Gauge condition Gauge invariance Landau-DeWitt gauge condition Effective actions Application to Gravity-Gauge System Two fields Background Metric g_ij [φ] on the field space Landau-DeWitt gauge conditions Effective action Gauge fixed term Total graviton’s contribution to the effective action Renormalized gauge action Gravitational correction to the β function β Function Correction From Gravitational Contributions One-loop gravitational contributions concern the tensor-type and scalar-type quadratic divergences, their consistency condition and quadratic divergence behavior are crucial for β function correction Quadratic divergence behavior Power-Law running Landau-DeWitt gauge condition Consistency condition of gauge invariance Asymptotic free power-law running Cosmological constantΛeffect No quadratic effects due to DR In Dimensional Regularization In the cut-off regularization no asymptotic freedom For an independent check, revisit the traditional background field method in the harmonic gauge by taking the following parameters In the cut-off regularization With inconsistency condition With consistency condition of ILIs or in LORE method accidental cancellation Asymptotic free power-law running Diagrammatical Calculations Check of gauge invariance: Two-point, three-point, fourpoint Green functions by using the traditional background field method in harmonic gauge condition With Counter terms In DR & cut-off reg. In LORE method Satisfy Slavnov-TaylorWard identities gauge invariant The β function Recover the result via traditional background field method calculation U(1) gauge General Conclusion: Asymptotic Free Power Law Running of Gauge Coupling due to gravitational effects Gauge condition independent Regularization scheme independent Y. Tang and Y.L. Wu, JHEP 1111, 073 (2011), arXiv:1109.4001 [hep-ph] Asymptotic free power-law running The Consistency of LORE Method with Explicit Calculations at Two-Loop Level ’t Hooft & Veltman: A general two-loop order Feynman diagram can be reduced to the general αβγ integrals Evaluating General αβγ integral into ILIs: UVDP parametrization get rid of the cross terms of momenta Treatment of Overlapping Divergence in UVDP Parameter Space Divergence of subdiagramαγ Treatment of UVDP parameter divergence Counter part overall divergence overall quadratic divergence Similar for Circuit 2 and Circuit 3 Application to Two Loop Calculations by LORE in ϕ^4 Theory Log-running to coupling constant at two loop level β-function for the renormalized coupling constant λ Power-law running of mass at two loop level 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 Consistency of LORE Beyond One Loop Quantum Structure of Quadratic Divergent LORE Beyond One Loop Quadratic term is a harmful divergence, and also breaks underlying gauge invariance and its associated Ward identity. Counter part Quadratic term even combined with their corresponding counterterm insertion diagrams is still a harmful divergence, and also breaks underlying gauge invariance and its associated Ward identity. Two-loop vertex corrections Quadratic term is a harmful divergence for each diagram Counter part Again quadratic term even combined with their corresponding counterterm is still a harmful divergence for each diagram Sum up over all the relevant diagrams Quadratic harmful divergences cancel for the final result, recover the gauge invariance and locality Quadratic divergences are canceled, which is crucial to guarantee that photon does not obtain a mass from quantum fluctuation and that the whole theory remains gauge invariance. Two loop Harmful divergences like vanish, only when summing up over all relevant loop diagrams, which is expected as these terms are nonlocal and cannot be eliminated by any counter terms in the original Lagrangian which are local. Quantum Anomaly Based on LORE Method Triangle Anomaly Amplitudes Using the definition of gamma_5 Trace of gamma matrices gets the most general and unique structure with symmetric Lorentz indices Y.L.Ma & YLW Anomaly of Axial Current Explicit calculation based on LORE method with the most general and symmetric Lorentz structure Restore the original theory in the limit Vector currents are automatically conserved, only the axialvector Ward identity is violated by quantum corrections Chiral Anomaly Based on LORE Method 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 unit Loop Regularization (LORE) Method Trace Anomaly Based on LORE Dilation transformation or scaling transformation Current of dilation transformation & Energy-momentum Tensor Naïve dilation Ward identity QED Lagrangian Quantum corrections to the dilation Ward identity Vacuum polarization Consistency condition Three-point function With consistency condition Dilation Ward identity Dilation Ward identity violated by quantum corrections Trace anomaly/anomaly Ward identity in operator formalism Lorentz and CPT Violation in QFT QFT may not be an underlying theory but EFT In String Theory, Lorentz invariance can be broken down spontaneously. Lorentz non-invariant quantum field theory Explicit、Spontaneous 、Induced CPT/Lorentz violating Chern-Simons term constant vector Induced CTP/Lorentz Violation CPT/Lorentz violating Chern-Simons term constant vector eQED with constant vector mass What is the relation ? Diverse Results Gauge invariance of axial-current S.Coleman and S.L.Glashow, Phys.Rev.D59: 116008 (1999) Pauli-Villas regularization with D.Colladay and V.A.Kostelecky, Phys. Rev. D58:116002 (1998). Gauge invariance and conservation of vector Ward identity M.Perez-Victoria, JHEP 0104 032 (2001). Consistent analysis via dimensional regularization G.Bonneau, Nucl. Phys. B593 398 (2001). Diverse Results Based on nonperturbative formulation with R.Jackiw and V.A.Kostelecky, Phys.Rev.Lett. 82: 3572 (1999). Derivative Expansion with dimensional regularization J.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-Victoria, 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 calculations Regularization Scheme Regularization scheme dependence Ambiguity of Dimensional regularization with problem Ambiguity with momentum translation for linear divergent term Ambiguity of reducing triangle diagrams Explicit Calculation Based on LORE Method Amplitudes of triangle diagrams Contributions to Amplitudes Convergent contributions Divergent contributions Logarithmic DV Linear DV Contributions to Amplitudes Logarithmic Divergent Contributions Regularized result with LORE Contributions to Amplitudes Linear divergent contributions Regularized result Contributions to Amplitudes Total contributions arise from convergent part Final Result Setting Final result is Induced ChernSimons term is uniqely determined when combining the chiral anomaly There is no harmful induced Chern-Simons term for massive fermions. Comments on Ambiguity Momentum translation relation of linear divergent Make Regularization after using the relation Check on Consistency Ambiguity of results Inconsistency with U(1) chiral anomaly of Must applying for the regularization before using momentum translation relation of linear divergent integral Dynamically Generated Spontaneous Symmetry Breaking of QCD Based on LORE Method QCD Lagrangian and Symmetry QCD Lagrangian of light quarks Effective Lagrangian Based on LORE Method Y.B. Dai and Y-L. Wu, Euro. Phys. J. C 39 s1 (2004) Integrating out quark fields by using the LORE method Dynamically Generated Spontaneous Symmetry Breaking Composite Higgs Potential Quadratic Term by the LORE method Dynamically Generated Spontaneous Symmetry Breaking M_c meaningful characterizing energy scale Scalars as Partner of Pseudoscalars & The Lightest Composite Higgs Bosons Scalar mesons: Pseudoscalar mesons : Mass Formula Pseudoscalar mesons : Mass Formula Predictions for Mass Spectra & Mixings M_c ~ 1 GeV Nonperturbative energy scale μ_s ~ 300 MeV QCD energy scale Predictions QCD Phase Transition with Chiral Symmetry Restoration Based on LORE Method Consider two flavor without instanton effects After integrating out quark fields The propagator of quark fields Applying the Schwinger Closed-Time-Path Green Function (CTPGF) Formalism to the Quark Propagators Carrying out momentum integration by the LORE method Effective Lagrangian of Chiral Thermodynamic Model of QCD at the lowest order with Finite Temperature Both logarithmic and quadratic integrals depend on Temperature by LORE method Dynamically generated effective composite Higgs potential of mesons at finite temperature based on LORE Method Thermodynamic Gap Equation Assumption: The scale of NJL four quark interaction due to NP QCD has the same T-dependence as quark condensate Critical temperature for Chiral Symmetry Restoration at T Tc Critical temperature is given by the Quadratic Term in the LORE method Input Parameters Output Predictions Critical Temperature of chiral symmetry restoration Thermodynamic Behavior of Physical Quantities Thermodynamic VEV Thermodynamic Behavior of Physical Quantities CONCLUSIONS The LORE method is a kind of infinity-free and symmetry-preserving regularization scheme The LORE method introduces two intrinsic energy scales M_c & μ_s which become physically meaningful to play the role as the characterizing energy scale M_c and sliding energy scales μ_s The LORE method realized in exact dimension of original theory is applicable to the underlying, effective, supersymmetric and chiral QFTs CONCLUSIONS The LORE method with the consistency conditions can give the sensible results which satisfy all the requirements: gauge invariance and locality. The LORE method only requires the use of consistency conditions at one-loop level and does not need to introduce additional higher order consistency conditions. The concept of ILIs and the electric circuit analogy of Feynman diagrams enable us to apply the LORE method to all order by a diagramatic way CONCLUSIONS Quantum structure of quadratic term is crucial to understand symmetry and symmetry breaking Proper treatment of quadratic divergence is important to understand the quantum structure of QFTs Quantization of gravity is the key to understand eventually the unification of forces and the darkness of universe THANKS