Title: Yue-Liang Wu
1CPT 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
4Induced 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
10Criteria 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).
11Criteria 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.
13Why 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.
15Irreducible 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
18Checking Consistency Condition
19Checking Consistency Condition
20 Vacuum Polarization
- Fermion-Loop Contributions
21 Gluonic Loop Contributions
22Cut-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
24Explicit 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
25Interesting 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
27Ward-Takahaski-Slavnov-Taylor Identities
Gauge Invariance
Two-point Diagrams
28 Three-point Diagrams
29Four-point Diagrams
30Ward-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)
32Supersymmetry 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
35Massive Wess-Zumino Model
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
39Chiral 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
41Quantum 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
43Contributions to Amplitudes
- Logarithmic Divergent Contributions
- Regularized result with LORE
44Contributions to Amplitudes
- Linear divergent contributions
- Regularized result
45Contributions to Amplitudes
- Total contributions arise from convergent part
46 Final Result
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.
50THANKS
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