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

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)

　　　　　

Quantum Theory of Particles and Fields

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

Elementary Particle Physics基本粒子物理

QuantumMechanics量子力学

SpecialRelativity相对论 +

=

Quantum Field Theory and Symmetry

Symmetry Principle对称原理

=Quantum

Field Theory量子场论+

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

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

Origin of Divergence in QFTs

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

(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.

Treatment on Divergences with Meaningful Regularization Scheme

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:hep-ph/0509083].【 6 】 Y.L.Ma and 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].

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 methodIn 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

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 conditionsDivergence power ≥ the space-time dimension vanishes

Explicit One Loop Feynman Integrals in LORE

With

Two intrinsic energy scales and play the roles of UV- and IR-cut off, but physically meaningful as the CES and SES

Euler constant =0.577216…

Compare to DR

LORE is an Infinity-Free Regularization!

Interesting Mathematical Identities

which lead the functions to the following explicit forms

General Evaluation of ILIs & UVDP Parameterization

Overall and vertex momentum conservations of Feynman diagrams

General structure of Feynman integral

Internal momentum (k_i) decomposition with loop momentum (l_r)and the undetermined internal currents flowing q_j

ILIs are resulted from the following conditions

Evaluation of ILIs and UVDP Parameterization

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

Kirchhoff’s laws in the electric circuit analogy: sum of voltage drop around any closed loop is zero

Current conservation at vertex:q-- internal currents flowing in the circuit; p-- the external currents entering it

--- the resistance of the jth line or

Ohm’s Law

--- the conductance of the jth line--- the displacement between two pointsEquation of motion for a free

particle

--the causal propagation of a particle--the causality of Feynman propagator

ILIs and Bjorken-Drell’s Circuit Analogy

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

Loop momentum integral by diagonalizing the quadraticmomentum terms with an orthogonal transformation O

-- the eigenvalues of the matrix M--functions of UVDP parameters v_i

Feynman integrals are evaluated into ILIs

Evaluation of ILIs and UVDP Parameterization

The momentum integral on in ILIs reflects the overall divergence of the Feynman diagram

(k-1) internal loop momentum integrals are convergent

For the condition:

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

ILIs

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

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

Consistency and Advantage of LORE Method

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

Question?

Not a divergent scalar-type ILI

The divergent tensor-type ILI

Naïve replacement in divergent integrals

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 paperRegularized Divergent ILIs in LORE have the consistency conditionThe 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

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

J.W. Cui, Y.Tang,Y.L. Wu Phys.Rev.D79:125008,2009

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 cancelTransverse 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

Transverse condition or Gauge symmetry can be maintained

In the general ξ gauge, there is a term proportion to

a_0 ≠ 1 will break the transverse condition, only when regularization scheme satisfies consistency condition with

With a_0 being defined via logarithmic divergent

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

All renormalized vertices do lead to a single renormalization constant

With the counterterms Lagrangian

Resulting renormalization of fields and masses

Gravitational Contributions to Gauge Green’s Functions and Asymptotic Free Power-Law Running of Gauge Coupling

【 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).

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

Gravitational Contributions to Gauge Coupling

【 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]

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

Gravitational Contributions to Gauge Theories and Asymptotic Free Power-Law Running of Gauge Coupling

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

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).

More other discussions

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

Metric g_ij [φ] on the field space

Landau-DeWitt gauge conditions

Gauge fixed termEffective action

Two fields Background

Total graviton’s contribution to the effective action

Renormalized gauge action

Gravitational correction to the β function

β Function Correction From Gravitational Contributions

Cosmological constantΛeffect

Landau-DeWitt gauge condition

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

Consistency condition of gauge invariance

Asymptotic free power-law running

In Dimensional RegularizationIn the cut-off regularization

no asymptotic freedomFor an independent check, revisit the traditional

background field method in the harmonic gauge by taking the following parameters

In the cut-off regularizationWith inconsistency conditionWith consistency condition of ILIs or in LORE method

Asymptotic free power-law running

accidental cancellation

No quadratic effects due to DR

Diagrammatical CalculationsCheck of gauge

invariance:Two-point, three-point, four-point Green functions by using the traditional background field method in harmonic gauge condition With Counter termsIn DR & cut-off reg.In LORE methodSatisfy Slavnov-Taylor-Ward identities gauge invariant

The β function

U(1) gauge

Asymptotic free power-law running

Recover the result via traditional background field method calculation

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]

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

overall divergenceDivergence of subdiagramαγ

Counter part

Treatment of UVDP parameter 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

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

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

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.

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.

Counter part

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 axial-vector 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

Naïve dilation Ward identity

Dilation transformation or scaling transformation

Current of dilation transformation & Energy-momentum Tensor

QED Lagrangian

Quantum corrections to the dilation Ward identityVacuum polarization

Consistency condition

Three-point function

With consistency condition

Dilation Ward identity violated by quantum corrections

Dilation Ward identity

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

eQED with constant vector

CPT/Lorentz violating Chern-Simons term

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 Chern-Simons term is uniqely determined when combining the chiral anomalyThere 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 SymmetryQCD 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

Quadratic Term by the LORE method

Composite Higgs Potential

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

Carrying out momentum integration by the LORE method

Applying the Schwinger Closed-Time-Path Green Function (CTPGF) Formalism to the Quark Propagators

The propagator of quark fields

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

Critical temperature is given by the Quadratic Term in the LORE method

T Tc

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

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

CONCLUSIONS

THANKS