On the induction of the four-dimensional Lorentz-breaking non-Abelian
  Chern-Simons action
On the induction of the four-dimensional Lorentz-breaking non-Abelian
Chern-Simons action
M. Gomes,1 J. R. Nascimento,1, 2 E. Passos,2 A. Yu. Petrov,2 and A. J. da Silva1
1Instituto de F́ısica, Universidade de São Paulo
Caixa Postal 66318, 05315-970, São Paulo, SP, Brazil∗
2Departamento de F́ısica, Universidade Federal da Paráıba
Caixa Postal 5008, 58051-970, João Pessoa, Paráıba, Brazil†
Abstract
A four-dimensional Lorentz-breaking non-Abelian Chern-Simons like action is generated as a one-loop per-
turbative correction via an appropriate Lorentz-breaking coupling of the non-Abelian gauge field to a spinor
field. This term is shown to be regularization dependent but nevertheless it can be found unambiguously in
different regularization schemes at zero and at finite temperature.
Electronic address: mgomes,jroberto,ajsilva@fma.if.usp.br
Electronic address: jroberto,passos,petrov@fisica.ufpb.br
http://arxiv.org/abs/0704.1104v2
mailto:mgomes,jroberto,ajsilva@fma.if.usp.br
mailto:jroberto,passos,petrov@fisica.ufpb.br
During the last years, different aspects of the Lorentz symmetry breaking have been intensively
studied [1]. One of the theoretical consequences of this effect is the birefringence of light in
vacuum. After the formulation of the concept of noncommutativity of the space-time [2], which
implies in Lorentz symmetry breaking (see the discussion in [3]), The interest in this subject has
greatly increased . One of the implications of the Lorentz symmetry breaking is the possibility
of introducing a lot of new couplings in the Standard Model [4]. These terms may arise from
radiative corrections to some Lorentz-breaking field theories at zero [5, 6, 7, 8, 9, 10] and at finite
temperature [11, 12, 13, 14]. Alternatively, they may be induced from the deformation of the
canonical commutation relation through the use of the noncommutative fields method [15, 16].
Recently, the renormalizability of the Yang Mills (YM) theory with a four-dimensional non-
Abelian Lorentz-breaking Chern-Simons (CS) term was studied in [17]. The induction of such
Lorentz-breaking CS term starting from a pure YM was investigated within the noncommuta-
tive fields method in [16]. In the present work we show how the same CS term can be induced
through radiative corrections starting from a YM theory coupled with fermions in the presence of
an interaction of the fermions with a constant external field at zero and at finite temperture.
We start with the following model which represents a non-Abelian generalization of the spinor
electrodynamics with the Lorentz-breaking coupling
Lf = ψ
(i∂/−m− γ5b/)δij − gγµAaµ(Ω
ψj . (1)
Here bρ is a constant four-vector. The Aµ = A
a is a Yang-Mills field coupled to spinors ψ which
carry the isotopic indices, ψ = (ψi), with i taking values from 1 to N with N being the dimension
of the chosen representation of the Lie algebra. The Ωa = (Ωa)ij are the Lie group generators in
this representation satisfying the relations: [Ωa, Ωb] = ifabcΩc and Tr[ΩaΩb] = δab.
The one-loop effective action of the gauge field Aaµ is SYM+Sf [b,A] where SYM is the Yang-Mills
action and Sf [b,A] can be expressed in the form of the following functional trace:
Sf [b,A] = −iTr ln(i∂/−m− γ5b/− gγµAaµΩ
a). (2)
This functional trace can be rewritten as Sf [b,A] = Sf [b] + S
f [b,A], with the first term being
Sf [b] = −iTr ln(i∂/ − m − γ5b/). The nontrivial dynamics is concentrated in the second term
S ′f [b,A], which is given by the power series:
S ′f [b,A] = iTr
i∂/−m− γ5b/
gγµAaµΩ
. (3)
To make explicit the non-Abelian Chern-Simons term we should expand this expression up to the
third order in the gauge field:
S ′f [b,A] = S
f [b,A] + S
f [b,A] + . . . (4)
where
f [b,A] =
i∂/−m− γ5b/
γµAaµΩ
i∂/−m− γ5b/
γνAbνΩ
b], (5)
f [b,A] =
i∂/−m− γ5b/
γµAaµΩ
i∂/−m− γ5b/
γλAbλΩ
i∂/−m− γ5b/
γνAcνΩ
c]. (6)
and the elipsis stands for higher order terms in the gauge field.
Using the above expressions, it is easy now to verify that the one loop effective action expanded
up to first order in bµ may be written as
S ′f [b,A] =
d4x kρǫ
ρµνλ(∂λA
igAaµA
abc) (7)
where kρ is
kρ = 2ig
(2π)4
2 + 3m2)− 4pρ(b · p)
(p2 −m2)3
. (8)
This result exactly reproduces the structure of the non-Abelian Lorentz-breaking Chern-Simons
term described in [17]. One can observe that the expressions (7,8), after reduction to the Abelian
case, coincide with the known Abelian results [14, 18]. Apparently, there is a relation between the
induced Lorentz-breaking Chern-Simons term and Adler-Bell-Jackiw anomaly as both situation
are observed for the well known triangle graph. This issue has been discursed in Ref.[6]. Also, the
interesting discussion of the problem of ambiguities in the Lorentz-breaking theories is presented in
[7]. By power counting, the momentum integral in expression (8) involves terms with logarithmic
divergence so that different regularization prescriptions will produce diverse outcomes. Lorentz
preserving regularizations, more precisely any regularization in which we can make: pµpν →
will produce finite results. By adopting the method of dimensional regularization [19], the above
integral is promoted to D dimensions and a straightforward calculation yields
kρ = 2ig
(2π)D
(p2 −m2)3
)p2 + 3m2]
4g2 (4−D) Γ((4−D)/2)
Γ(3)(4π)D/2
bρ, (9)
which coincides with the result found in [18] for the Abelian situation. If, instead of dimensional
regularization, the integral in Eq. (8) is kept in four dimensions the regularization enforced the
replacement
kρ = 6ig
2m2bρ
(2π)4
(p2 −m2)3
bρ, (10)
which now agrees with the Abelian result obtained in [20].
To develop calculations in the finite temperature case, let us now assume that the system is in
the state of thermal equilibrium at a temperature T = 1/β. In this case, we can use the Matsubara
formalism for fermions, which consists in taking p0 ≡ iωn = (n + 1/2)2πiβ and replacing the
integration over the zeroth component of the momentum by a discrete sum (1/2π)
dp0 → iβ
Thus, the Eq. (7) can be written as
S ′f [b,A] =
d4x kρ(β)ǫ
ρµνλ(∂λA
igAaµA
abc). (11)
Hereafter all expressions are in the Euclidean space (all greek indices run from 1 to 4). The vector
kρ(β) is given by
kρ(β) =
(2π)3
bρ(3m
2 − p2) + 4pρ(b · p)
(p2 +m2)3
. (12)
By extending the ~p integration to d dimensions it follows that the time-like component of kρ(β) is
k4(β) =
(2π)d
3M2n − ~p 2
(~p 2 +M2n)
, (13)
where M2n = ω
2. Using the prescription of dimensional regularization [19], we have
k4(β) = −
(4π)d/2
[dΓ(2 − d/2) − 6Γ(3− d/2)]
(M2n)
2−d/2
λ−1/2
(3− d) Γ(λ)
[(n+ 1/2)2 + a2]λ
, (14)
where a = mβ/2π and λ = 2− d/2. At this point the following identity [21]:
[(n + b)2 + a2]λ
πΓ(λ− 1/2)
Γ(λ)(a2)λ−1/2
+ 4 sin(πλ)
(z2 − a2)λ
exp 2π(z + ib)− 1
, (15)
valid for 1/2 < λ < 1 can be used to get
k4(β) =
+4 (3− d)(a2)
2 Γ(λ) sin(πλ)
(z2 − a2)λ
exp 2π(z + ib)− 1
. (16)
In d = 3 this gives
k4(β) =
b4, (17)
i.e., the same result (9) without any dependence on the temperature, which agrees with the one
obtained in [22] for the Abelian situation. If instead of (12) we use (10) as the starting point for
the computation of finite temperature effects we get
k4(β) = b4(
F (a)), (18)
where
F (a) =
dz(z2 − a2)1/2
tanh(πz)
cosh2(πz)
, (19)
has the following asymptotics: F (a → ∞) → 0 (T → 0) and F (a → 0) → 1/2π2 (T → ∞), see
Fig.1.
0.0035
0.004
0.0045
0.005
0.0055
0.006
0 0.2 0.4 0.6 0.8 1 1.2
FIG. 1: The function F (a) is different from zero everywhere. At zero temperature (β → ∞), the function
tends to a nonzero value 1
Let us now consider the space part, ki(β), of the vector kρ(β). In this case, the expression (12)
can be rewritten as:
ki(β) =
(2π)3
bi(3m
2 − p2) + 4pi(b · p)
(p2 +m2)3
, (20)
Then, considering this expression formally in d space dimensions, we can replace pipj by
δij ,
hence we get
ki(β) =
(2π)d
4m2 − (d−4
)~p2 −M2n
(~p2 +M2n)
, (21)
which now furnishes
ki(β) =
4m2g2
Γ(3− d
(4π)d/2
(m2 + ω2n)
(a2)λ−1/2Γ(λ)
[(n+ 1
)2 + a2]λ
, (22)
where we have introduced λ = 3 − d
. We cannot apply the relation (15) for d = 3, because the
integral in that expression does not converge. Thus, let us perform the analytic continuation of
that relation; we obtain [13]
(z2 − a2)λ
exp 2π(z + ib)− 1
3− 2λ
(z2 − a2)λ−1
× (23)
exp 2π(z + ib)− 1
(2− λ)(1 − λ)
(z2 − a2)λ−2
exp 2π(z + ib)− 1
Thus for d = 3 the Eq. (22) takes the form
ki(β) = bi(
F (a)), (24)
where F (a) was defined in (19). Thus, we see that at high temperature the Chern-Simons coef-
ficient is twice its value at zero temperature, i.e., ki(β → 0) = 12π2 . On the other hand, at zero
temperature, one recovers the result ki(β → ∞) = 14π2 .
We have generated the non-Abelian Lorentz-breaking Chern-Simons term via the Lorentz-
breaking coupling of the Yang-Mills field with the spinor field at zero and at finite temperature.
The essential property of the result is that within the framework of dimensional regularization this
term turns out to be finite. We note that the derivative expansion approach naturally allows to
preserve the gauge invariance for the quantum corrections. It is natural to expect that at least
some of other Lorentz-breaking terms which existence was predicted in [4] also can be generated
via appropriate couplings of the gauge or gravity fields with some matter fields.
We have also obtained the coefficient kρ for the non-Abelian Lorentz-breaking Chern-Simons
term at the finite temperature. We found that the results for this term turn out to be dependent
on the regularization scheme both at zero and at finite temperature (in a particular regularization
scheme the time-like component was found to be temperature independent). Considering the
dependence on the regularization scheme, one should note that the momentum integral determining
the value of the vector kρ is formally superficially divergent, thus dependence of its finite part on
the renormalization procedure is very natural. However, in the regularization schemes suggested
in the paper the divergent part identically disappears as a consequence of the rotational invariance
of the relevant integrands.
Acknowledgements. This work was partially supported by Conselho Nacional de Desenvolvi-
mento Cient́ıfico e Tecnológico (CNPq) and Fundação de Amparo à Pesquisa do Estado de São
Paulo (FAPESP). The work by A. Yu. P. has been supported by CNPq-FAPESQ DCR program,
CNPq project No. 350400/2005-9.
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