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Yue-Liang Wu

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Title: Yue-Liang Wu


1
CPT and Lorentz Invariance and Violation in QFT
  • Yue-Liang Wu
  • Kavli Institute for Theoretical Physics China
  • Key Laboratory of Frontiers in Theoretical
    Physics
  • Institute of Theoretical Physics, CAS
  • Chinese Academy of Sciences

2
Symmetry Quantum Field Theory
  • Symmetry has played an important role in physics
  • CPT and Lorentz invariance are regarded as the
    fundamental symmetries of nature.
  • CPT invariance is the basic property of
    relativistic quantum field theory for point
    particle.
  • All known basic forces of nature
    electromagnetic, weak, strong gravitational
    forces, are governed by the local gauge
    symmetries
  • U(1)_Y x SU(2)_L x SU(3)_c x SO(1,3)

3
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

4
Induced CTP/Lorentz Violation
  • Real world has been found to be successfully
    described by quantum field theories
  • EQED with constant vector
  • ?
    mass
  • What is the relation
  • ?

5
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.
    D58116002 (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).

6
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).

7
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
  • for
  • for

8
Regularization Scheme
  • Regularization scheme dependence
  • Ambiguity of Dimensional regularization with
  • problem
  • Ambiguity with momentum translation for
  • linear divergent term
  • Ambiguity of reducing triangle diagrams
  • How to reach a consistent regularization scheme
    ?

9
Regularization Methods
  • Cut-off regularization
  • Keeping divergent behavior, spoiling gauge
    symmetry translational/rotational symmetries
  • Pauli-Villars regularization
  • Modifying propagators, destroying non-abelian
    gauge symmetry, introduction of superheavy
    particles
  • Dimensional regularization analytic continuation
    in dimension
  • Gauge invariance, widely used for practical
    calculations
  • Gamma_5 problem, losing scaling behavior
    (incorrect gap eq.),
  • problem to chiral theory and super-symmetric
    theory
  • All the regularizations have their advantages
    and shortcomings

10
Criteria of Consistent Regularization
  • (i) The regularization is rigorous that it can
    maintain the basic symmetry principles in the
    original theory, such as gauge invariance,
    Lorentz invariance and translational invariance
  • (ii) The regularization is general that it can be
    applied to both underlying renormalizable QFTs
    (such as QCD) and effective QFTs (like the gauged
    Nambu-Jona-Lasinio model and chiral perturbation
    theory).

11
Criteria of Consistent Regularization
  • (iii) The regularization is also essential in the
    sense that it can lead to the well-defined
    Feynman diagrams with maintaining the initial
    divergent behavior of integrals. so that the
    regularized theory only needs to make an
    infinity-free renormalization.
  • (iv) The regularization must be simple that it
    can provide the practical calculations.

12
Symmetry-Preserving Loop Regularization
(LORE) with String Mode Regulators
  • Yue-Liang Wu, SYMMETRY PRINCIPLE PRESERVING AND
    INFINITY FREE REGULARIZATION AND RENORMALIZATION
    OF QUANTUM FIELD THEORIES AND THE MASS GAP.
  • Int.J.Mod.Phys.A182003, 5363-5420.
  • Yue-Liang Wu, SYMMETRY PRESERVING LOOP
    REGULARIZATION AND RENORMALIZATION OF QFTS.
    Mod.Phys.Lett.A192004, 2191-2204.

13
Why Quantum Field Theory So
Successful
  • Folks 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) M_c
  • At sufficiently low energy then means
  • E ltlt M_c ? QFTs

14
Why Quantum Field Theory
So Successful
  • Renormalization group by Wilson or Gell-Mann
    Low
  • Allow to deal with physical phenomena at any
    interesting energy scale by integrating out the
    physics at higher energy scales.
  • To be able to define the renormalized theory
    at any interesting renormalization scale .
  • Implication Existence of sliding energy scale
    (SES) µ_s which is not related to masses of
    particles.
  • The physical effects above the SES µ_s are
    integrated in the renormalized couplings and
    fields.

15
Irreducible Loop Integrals (ILIs)
16
Loop Regularization (LORE)
  • Simple Prescription
  • in ILIs, make the following replacement
  • With the conditions
  • So that

17
Gauge Invariant Consistency Conditions
18
Checking Consistency Condition
19
Checking Consistency Condition
20
Vacuum Polarization
  • Fermion-Loop Contributions

21
Gluonic Loop Contributions
22
Cut-Off Dimensional Regularizations
  • Cut-off violates consistency conditions
  • DR satisfies consistency conditions
  • But quadratic behavior is suppressed in DR

23
Symmetrypreserving Infinity-free
Loop Regularization (LORE)
With String-mode Regulators
  • Choosing the regulator masses to have the
    string-mode Reggie trajectory behavior
  • Coefficients are completely determined
  • from the conditions

24
Explicit One Loop Feynman Integrals
With
Two intrinsic mass scales and
play the roles of UV- and IR-cut off as well as
CES and SES
25
Interesting Mathematical Identities
which lead the functions to the following
simple forms
26
Renormalization Constants of Non- Abelian gauge
Theory and ß Function of QCD in Loop
Regularization
Jian-Wei Cui, Yue-Liang Wu, Int.J.Mod.Phys.A2328
61-2913,2008
  • Lagrangian of gauge theory
  • Possible counter-terms

27
Ward-Takahaski-Slavnov-Taylor Identities
Gauge Invariance
Two-point Diagrams
28
Three-point Diagrams
29
Four-point Diagrams
30
Ward-Takahaski-Slavnov-Taylor Identities
  • Renormalization Constants
  • All satisfy Ward-Takahaski-Slavnov-Taylor
    identities

31
Renormalization ß Function
  • Gauge Coupling Renormalization
  • which reproduces the well-known QCD ß function
    (GWP)

32
Supersymmetry in Loop Regularization
J.W. Cui, Y.Tang,Y.L. Wu Phys.Rev.D79125008,2009
  • Supersymmetry
  • Supersymmetry is a full symmetry of quantum
    theory
  • Supersymmetry should be Regularization-independent
  • Supersymmetry-preserving regularization

33
Massless Wess-Zumino Model
  • Lagrangian
  • Ward identity
  • In momentum space

34
Check of Ward Identity
Gamma matrix algebra in 4-dimension and
translational invariance of integral
momentum Loop regularization satisfies these
conditions
35
Massive Wess-Zumino Model
  • Lagrangian
  • Ward identity

36
Check of Ward Identity
Gamma matrix algebra in 4-dimension and
translational invariance of integral
momentum Loop regularization satisfies these
conditions
37
Triangle Anomaly
  • Amplitudes
  • Using the definition of gamma_5
  • The trace of gamma matrices gets the most general
    and unique structure with symmetric Lorentz
    indices

38
Anomaly of Axial Current
  • Explicit calculation based on Loop Regularization
    with the most general and symmetric Lorentz
    structure
  • Restore the original theory in the limit
  • which shows that vector currents are
    automatically conserved, only the axial-vector
    Ward identity is violated by quantum corrections

39
Chiral Anomaly Based on Loop Regularization
Including the cross diagram, the final result is
Which leads to the well-known anomaly form
40
Anomaly Based on Various Regularizations
  • Using the most general and symmetric trace
    formula for gamma matrices with gamma_5.
  • In unit

41
Quantum Loop Induced CPT/Lorentz Violating
Chern-Simons Term in Loop Regularization
  • Amplitudes of triangle diagrams

42
Contributions to Amplitudes
  • Convergent contributions
  • Divergent contributions
  • ? Logarithmic DV ? Linear
    DV

43
Contributions to Amplitudes
  • Logarithmic Divergent Contributions
  • Regularized result with LORE

44
Contributions to Amplitudes
  • Linear divergent contributions
  • Regularized result

45
Contributions to Amplitudes
  • Total contributions arise from convergent part

46
Final Result
  • Setting
  • Final result is

47
Comments on Ambiguity
  • Momentum translation relation of linear divergent
  • Regularization after using the relation

48
Check on Consistency
  • Ambiguity of results
  • Inconsistency with U(1) chiral anomaly of

49
Conclusions
  • First applying for the regularization before
    using momentum translation relation of linear
    divergent integral
  • Loop regularization is translational invariant
  • Induced Chern-Simons term is uniqely determined
    when combining the chiral anomaly
  • There is no harmful induced Chern-Simons term for
    massive fermions.

50
THANKS
??!
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