Download Quantum Theory of Particles and Fields

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 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
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
(DRsuppresses quadratic divergent behavior;
Cut-offdoesn’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
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
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
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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