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)

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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

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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).

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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).

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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

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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
  ?

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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

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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).

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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.

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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.

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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

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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.

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Irreducible Loop Integrals (ILIs)
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Loop Regularization (LORE)

• Simple Prescription
• in ILIs, make the following replacement
• With the conditions
• So that

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Gauge Invariant Consistency Conditions
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Checking Consistency Condition
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Checking Consistency Condition
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Vacuum Polarization

• Fermion-Loop Contributions

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Gluonic Loop Contributions
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Cut-Off Dimensional Regularizations

• Cut-off violates consistency conditions
• DR satisfies consistency conditions
• But quadratic behavior is suppressed in DR

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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

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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
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Interesting Mathematical Identities
which lead the functions to the following
simple forms
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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

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Ward-Takahaski-Slavnov-Taylor Identities
Gauge Invariance
Two-point Diagrams
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Three-point Diagrams
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Four-point Diagrams
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Ward-Takahaski-Slavnov-Taylor Identities

• Renormalization Constants
• All satisfy Ward-Takahaski-Slavnov-Taylor
  identities

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Renormalization ß Function

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

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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

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Massless Wess-Zumino Model

• Lagrangian
• Ward identity
• In momentum space

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

• Lagrangian
• Ward identity

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Check of Ward Identity
Gamma matrix algebra in 4-dimension and
translational invariance of integral
momentum Loop regularization satisfies these
conditions
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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

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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

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Chiral Anomaly Based on Loop Regularization
Including the cross diagram, the final result is
Which leads to the well-known anomaly form
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Anomaly Based on Various Regularizations

• Using the most general and symmetric trace
  formula for gamma matrices with gamma_5.
• In unit

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Quantum Loop Induced CPT/Lorentz Violating
Chern-Simons Term in Loop Regularization

• Amplitudes of triangle diagrams

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Contributions to Amplitudes

• Convergent contributions
• Divergent contributions
• ? Logarithmic DV ? Linear
  DV

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Contributions to Amplitudes

• Logarithmic Divergent Contributions
• Regularized result with LORE

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Contributions to Amplitudes

• Linear divergent contributions
• Regularized result

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Contributions to Amplitudes

• Total contributions arise from convergent part

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Final Result

• Setting
• Final result is

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Comments on Ambiguity

• Momentum translation relation of linear divergent
• Regularization after using the relation

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Check on Consistency

• Ambiguity of results
• Inconsistency with U(1) chiral anomaly of

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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.

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THANKS
??!