Transcript On the Radiatively Induced Lorentz and CPT Violating Chern

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
Symmetry & Quantum Field Theory
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Symmetry has played an important role in physics
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CPT and Lorentz invariance are regarded as the
fundamental symmetries of nature.
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CPT invariance is the basic property of relativistic
quantum field theory for point particle.
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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)
Lorentz and CPT Violation in QFT
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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
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Real world has been found to be successfully
described by quantum field theories
EQED with constant vector
 mass
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What is the relation
?
Diverse Results
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Gauge invariance of axial-current
S.Coleman and S.L.Glashow, Phys.Rev.D59: 116008 (1999)
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Pauli-Villas regularization with
D.Colladay and V.A.Kostelecky, Phys. Rev. D58:116002 (1998).
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Gauge invariance and conservation of vector Ward identity
M.Perez-Victoria, JHEP 0104 032 (2001).
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Consistent analysis via dimensional regularization
G.Bonneau, Nucl. Phys. B593 398 (2001).
Diverse Results
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Based on nonperturbative formulation with
R.Jackiw and V.A.Kostelecky, Phys.Rev.Lett. 82: 3572 (1999).
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Derivative Expansion with dimensional regularization
J.M.Chung and P.Oh, Phys.Rev.D60: 067702 (1999).
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Keep full
dependence with
M.Perez-Victoria, Phys.Rev.Lett.83: 2518 (1999).
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Keep full
dependence with
M.Perez-Victoria, Phys.Rev.Lett.83: 2518 (1999).
Consistent Result
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Statement in Literature: constant vector K can
only be determined by experiment
Our Conclusion: constant vector K can consistently
be fixed from theoretical calculations
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for
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for
Regularization Scheme
Regularization scheme dependence
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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 ?
Regularization Methods
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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
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-JonaLasinio model and chiral perturbation theory).
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.
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.A18:2003, 5363-5420.

Yue-Liang Wu, SYMMETRY PRESERVING LOOP
REGULARIZATION AND RENORMALIZATION OF QFTS.
Mod.Phys.Lett.A19:2004, 2191-2204.
Why Quantum Field Theory
So Successful
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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.
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Indication: existence in any case a characterizing energy
scale (CES) M_c
At sufficiently low energy then means:
E << M_c  QFTs
Why Quantum Field Theory
So Successful
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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 .
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Implication: Existence of sliding energy scale (SES) μ_s
which is not related to masses of particles.
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The physical effects above the SES μ_s are integrated in
the renormalized couplings and fields.
Irreducible Loop Integrals (ILIs)
Loop Regularization (LORE)
Simple Prescription:
in ILIs, make the following replacement
With the conditions
So that
Gauge Invariant Consistency Conditions
Checking Consistency Condition
Checking Consistency Condition
Vacuum Polarization
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Fermion-Loop Contributions
Gluonic Loop Contributions
Cut-Off & Dimensional Regularizations
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Cut-off violates consistency conditions
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DR satisfies consistency conditions
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But quadratic behavior is suppressed in DR
Symmetry–preserving & Infinity-free
Loop Regularization (LORE)
With String-mode Regulators
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Choosing the regulator masses to have the
string-mode Reggie trajectory behavior
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Coefficients are completely determined
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from the conditions
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
Interesting Mathematical Identities
which lead the functions to the following simple forms
Renormalization Constants of Non- Abelian gauge
Theory and β Function of QCD in Loop Regularization
Jian-Wei Cui, Yue-Liang Wu, Int.J.Mod.Phys.A23:2861-2913,2008
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Lagrangian of gauge theory
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Possible counter-terms
Ward-Takahaski-Slavnov-Taylor Identities
Gauge Invariance
Two-point Diagrams
Three-point Diagrams
Four-point Diagrams
Ward-Takahaski-Slavnov-Taylor Identities
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Renormalization Constants
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All satisfy Ward-Takahaski-Slavnov-Taylor identities
Renormalization β Function
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Gauge Coupling Renormalization
which reproduces the well-known QCD β function (GWP)
Supersymmetry in Loop Regularization
J.W. Cui, Y.Tang,Y.L. Wu Phys.Rev.D79:125008,2009
Supersymmetry
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Supersymmetry is a full symmetry of
quantum theory
Supersymmetry should be Regularizationindependent
Supersymmetry-preserving regularization
Massless Wess-Zumino Model
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Lagrangian
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Ward identity
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In momentum space
Check of Ward Identity
Gamma matrix algebra in 4-dimension and
translational invariance of integral momentum
Loop regularization satisfies these conditions
Massive Wess-Zumino Model
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Lagrangian
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Ward identity
Check of Ward Identity
Gamma matrix algebra in 4-dimension and
translational invariance of integral momentum
Loop regularization satisfies these conditions
Triangle Anomaly
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Amplitudes
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Using the definition of gamma_5
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The trace of gamma matrices gets the most general and
unique structure with symmetric Lorentz indices
Anomaly of Axial Current
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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
Chiral Anomaly Based on Loop Regularization
Including the cross diagram, the final result is
Which leads to the well-known anomaly form
Anomaly Based on Various Regularizations
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Using the most general and symmetric trace formula for
gamma matrices with gamma_5.
In unit
Quantum Loop Induced CPT/Lorentz Violating
Chern-Simons Term in Loop Regularization
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Amplitudes of triangle diagrams
Contributions to Amplitudes
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Convergent contributions
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Divergent contributions
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 Logarithmic DV
 Linear DV
Contributions to Amplitudes
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Logarithmic Divergent Contributions
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Regularized result with LORE
Contributions to Amplitudes
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Linear divergent contributions
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Regularized result
Contributions to Amplitudes
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Total contributions arise from convergent part
Final Result
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Setting
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Final result is
Comments on Ambiguity
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Momentum translation relation of linear
divergent
Regularization after using the relation
Check on Consistency
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Ambiguity of results
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Inconsistency with U(1) chiral anomaly of
Conclusions
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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.
THANKS