Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 47 (2014)

Lorentz invariant CPT violation†
M. Chaichian,∗1 K. Fujikawa,∗2 and A. Tureanu∗1A Lorentz invariant CPT violation, which may betermed as the long distance CPT violation in contrast
to the familiar short distance CPT violation, has been
recently proposed1) . This scheme is based on the nonlocal interaction vertex and characterized by the infrared divergent form factor. We show that Lorentz
covariant T ⋆ -product is consistently defined and theenergy-momentum conservation is preserved in perturbation theory if the path integral is suitably defined forthis non-local theory, although unitarity is generallylost. It is illustrated that T-violation is realized in thedecay and formation processes. It is also argued thatthe equality of masses and decay widths of the particle and anti-particle is preserved if the non-local CPT
violation is incorporated either directly or as perturbation by starting with the conventional CPT-even local
Lagrangian. However, we also explicitly show that thepresent non-local scheme can induce the splitting ofparticle and anti-particle mass eigenvalues if one considers a more general class of Lagrangians.
We study the specific realization of CPT violation
1L = ψ̄(x)[iγ µ ∂µ − M ]ψ(x) + ∂µ ϕ(x)∂ µ ϕ(x)21 22− m ϕ(x) + g ψ̄(x)ψ(x)ϕ(x) − V (ϕ)
2∫+ g1 ψ̄(x)ψ(x)d4 yθ(x0 − y 0 )δ((x − y)2 − l2 )ϕ(y)as a main theoretical model. This Lagrangian is formally hermitian and the term with a small real g1 andthe step function θ(x0 − y 0 ) stands for the CTP andT violating interaction; l is a real constant parameter.
It is interesting that the CPT and T violating term isreal in the present case. We define the interaction part
LI = g ψ̄(x)ψ(x)ϕ(x)∫+ g1 ψ̄(x)ψ(x) d4 yθ(x0 − y 0 )δ((x − y)2 − l2 )ϕ(y).
We treat this highly non-local Lagrangian in path integral as described in2) .Namely⟨0, +∞|0, −∞⟩J∫∫= Dψ̄DψDϕ exp{i d4 x[L0 + LI + LJ ]}(i)n ⟨T ⋆ ϕ(x1 )..ϕ(xN )∗1∗2d4 y1 LI (y1 )..

∫d4 yn LI (yn )⟩.

We use the T ⋆ -product which is essential to make thepath integral on the basis of Schwinger’s action principle consistent2) .
The present way to introduce CPT violation is basedon an extra form factor in momentum space as∫=

d4 xψ̄(x)ψ(x)∫∫d4 yθ(x0 − y 0 )δ((x − y)2 − l2 )ϕ(y)dp1 dp2 dq(2π)4 δ 4 (p1 + p2 + q)ψ̄(p1 )ψ(p2 )f (q)ϕ(q)withf (q) ≡∫d4 zθ(z 0 )δ(z 2 − l2 )eiqz

namely, CPT violation is realized by a form factor f (q)which becomes complex for time-like momentum. The
ordinary local field theory is characterized by f (q) = 1.
The above form factor is infrared divergent, and it isquadratically divergent in the present example. This
infrared divergence arises from the fact that we cannot divide Minkowski space into (time-like) domains
with finite 4-dimensional volumes in a Lorentz invariant manner. The Minkowski space is hyperbolic rather
than elliptic. CPT symmetry is related to the fundamental structure of Minkowski space, and thus it isgratifying that its possible breaking is also related tothe basic property of Minkowski space.
Based on this setting, we confirmed the followings:
1. The present model produces T-violation in the decay ϕ → ψ + ψ̄ and its reversed formation processψ + ψ̄ → ϕ.
2. The equality of masses and decay widths of theparticle and anti-particle is preserved if the non-local
CPT violation is incorporated either directly or as perturbation by starting with the conventional CPT-even
local Lagrangian.
Some of the more realistic applications of the present
CPT violation scheme to the particle-antiparticle mass
splitting, inparticular, the neutrino-antineutrino mass
splitting in the standard model have been already dicussed elsewhere3,4) .
Referenceswith the source term LJ = ψ̄(x)η(x) + η̄(x)ψ(x) +ϕ(x)J(x), and one may generate Green’s functions ina power series expansion of perturbation as†∫

Condensed from the article in Eur. Phys. J. C 73, 2349
(2013).
Department of Physics, University of Helsinki,RIKEN Nishina Center- 120 -1) M. Chaichian, A.D. Dolgov, V.A. Novikov and A.
Tureanu, Phys. Lett. B699 (2011) 177.
2) K. Fujikawa, Phys. Rev. D70 (2004) 085006.
3) M. Chaichian, K. Fujikawa and A. Tureanu, Phys.
Lett. B 712, 115 (2012).
4) M. Chaichian, K. Fujikawa and A. Tureanu, Phys.
Lett. B 718,178 (2012).