Induced higher-derivative Lorentz-violating Chern-Simons term at finite temperature
J. Leite T. Mariz W. Serafim
hep-th





Instituto de Física, Universidade Federal de Alagoas, 57072-270, Maceió, Alagoas, Brazil
julioleite,tmariz,wserafim@fis.ufal.br


Instituto de Física, Universidade Federal de Alagoas, 57072-270, Maceió, Alagoas, Brazil
julioleite,tmariz,wserafim@fis.ufal.br


Instituto de Física, Universidade Federal de Alagoas, 57072-270, Maceió, Alagoas, Brazil
julioleite,tmariz,wserafim@fis.ufal.br




In this work, we analyze the generation of the higher-derivative Lorentz-violating Chern-Simons term at zero temperature and at finite temperature. We use the method of derivative expansion and the Matsubara formalism in order to consider the finite temperature effects. The results show that at zero temperature the induced higher-derivative Chern-Simons term is nonzero; in contrast, when the temperature reaches infinity, the coefficients of the induced term vanish. In addition, we also briefly study the question of large gauge invariance of this higher-derivative term as we as the conventional Chern-Simons term. We compute the exact induced action for both terms at finite temperature, however, in a particular gauge field background, and observe that they are, in fact, invariant under large gauge transformation.


Induced higher-derivative Lorentz-violating Chern-Simons term at finite temperature
    W. Serafim
    December 30, 2023
===================================================================================





§ INTRODUCTION



Lorentz symmetry is one of the most tested symmetries of nature. Nevertheless, it is nowadays believed not to be an exact symmetry. In fact, it has been argued in <cit.> that tiny spontaneous breaking of Lorentz symmetry might arise in the context of string theory. Thus, if we add Lorentz-violating terms to the standard model, this new theory, namely the standard-model extension (SME) <cit.>, can be viewed as the low-energy limit of a physically relevant fundamental theory in which spontaneous Lorentz violation occurs. Many experiments and observations have been conducted to detect physical effects due to the Lorentz violation and some constraints on the Lorentz-violating terms were obtained, as we can see in <cit.>.

The most studied part of the SME presents renormalizable operators of mass dimension d=3 and d=4, consequently, the coefficients contracted with these operators have mass dimension d=1 or they are dimensionless, respectively. This part of the SME is known as the minimal SME. However, in the SME there are operators with mass dimension d ≥ 5, which are nonrenormalizable, contracted with coefficients of mass dimension d≤ -1. Although these operators are likely to be specially relevant in searches involving very high energies, providing new corrections, there are few studies on them. 

Recently, some studies on higher-derivative Lorentz-violating terms, described by higher mass dimension operator (d≥ 5), have been made, such as <cit.>. In particular, the work <cit.> on operators of arbitrary mass dimension in the QED extension have shown that the general form of the higher-derivative terms in the photon sector, is given by

    S_(d)=∫d^4x 𝒦^α_1 α_2 ⋯α_d_(d) A_α_1∂_α_3⋯∂_α_d A_α_2,

where d is the dimension of the tensor operator and 𝒦^α_1 α_2 ⋯α_d_(d) has mass dimension 4-d. 

On the one hand, for CPT-even terms, the first four indices of the coefficient 𝒦^α_1 α_2 ⋯α_d_(d) have the symmetries of the Riemann tensor, and there is total symmetry in the remaining d-4 indices. On the other hand, for CPT-odd terms, the coefficient 𝒦^α_1 α_2 ⋯α_d_(d) is antisymmetric in the three first indices and symmetric in the last d-3. 

In this work, we study the fermion sector of the quantum electrodynamics (QED) extended by a CPT-odd term, governed by the coefficient b_μ, namely b_μψ̅γ_5γ^μψ. From the fermion sector of this minimal QED extension, at zero temperature, we know that we can generate through radiative corrections the higher-derivative Chern-Simons term,

    S_(5)=∫d^4x 𝒦^μνραβ_(5) A_μ∂_ρ∂_α∂_β A_ν,

with 𝒦^μνραβ_(5)∝ϵ^μνρσ b_σ g^αβ, as can be seen by expanding the results in <cit.> or directly in <cit.>. Our goal here is to calculate the generation of the above term (<ref>) using the method of derivative expansion <cit.>, as a preparation for the finite temperature analysis, which has never been studied. As a complement, we also address the question of large gauge invariance of this higher-derivative term as well as the conventional Chern-Simons term, both at finite temperature. For issues related to the Chern-Simons term, L_(3) =𝒦^μνρ_(3) A_μ∂_ρ A_ν, see <cit.>, and references therein.

It is interesting to mention that the term (<ref>) arises when the Myers-Pospelov bosonic term <cit.> is radiatively induced from the fermion sector <cit.>. Therefore, effects of finite temperature on the higher-derivative Chern-Simons term are somehow related to the effects on the Myers-Pospelov term. Thus, studies of this nature deserve to be considered in order to better characterize the higher-derivative Lorentz-violating models. 

This paper has the following structure. In Sec. <ref>, we analyze the fermion sector of the Lorentz-violating QED with the coefficient b_μ, performing radiative corrections and using derivative expansion to induce the higher-derivative Chern-Simons term at zero temperature. In the end of the section, we look for constraints on the coefficient of the induced term imposed by experimental and observational data. In Sec. <ref>, we keep looking at the same theory of the previous section, but now we are interested on the behavior of the term (<ref>) at finite temperature. The issue of large gauge invariance is discussed in Sec. <ref>. Finally, we present a summary in Sec. <ref>.



§ RADIATIVE CORRECTION AT ZERO TEMPERATURE



As our aim is to study the generation of higher-derivative CPT-odd terms (<ref>), we consider the fermion sector of the Lorentz-violating QED, given by

    ℒ=ψ̅(i∂-m-A-γ_5b)ψ,
 
where the three first terms are the usual ones, and the last and unusual term, which is governed by the coefficient b^μ, is that responsible for the Lorentz and CPT violation. 

In order to perform this task, we prefer to use the method of derivative expansion (for details see <cit.>), which is more suitable to calculate the sum-integrals of the finite-temperature approach.

Then, from (<ref>), the effective action S_ is defined as

    Z = ∫ D ψ̅ D ψ e^i∫ d^4 x ψ̅(i∂-m-A-γ_5b)ψ = e^i S_,

in which 

    S_=-iln(p-m- A-γ_5 b).
 
Now, we can easily rewrite the effective action as S_=S^(0)_+S^(1)_, where S_^(0)=-i ln(p-m- γ_5 b) and 

    S^(1)_=i ∑_n=1^∞1/n[1/p-m-γ_5bA]^n .

As the term S^(0)_ does not depend on the gauge field, we must focus only on the term S^(1)_. To generate the higher-derivative Chern-Simons term it is necessary to have two contributions of the gauge field. Thus, we take n=2 in the expression (<ref>) and use the following expansion for the fermionic propagator,

    1/p-m-γ_5b   =   1/p-m+1/p-mγ_5b1/p-m
          +1/p-mγ_5b1/p-mγ_5b1/p-m+⋯,

to single out contributions of first order in b_μ. It is worth emphasizing that the higher-derivative Chern-Simons term (and the Myers-Pospelov term) is also induced when we consider contributions of third order in b_μ, as it has been shown in <cit.>. Following with the derivative expansion approach <cit.>, we find

    S_b^(1,2)   =   i/2 [S(p)γ^μ S(p-k)γ_5 bS(p-k)γ^ν+S(p)γ_5 bS(p)γ^μ S(p-k)γ^ν ]A_μ A_ν,

where S(p)=(p-m)^-1. Finally, expanding the propagator up to third order in k_μ,

    S(p-k)   =   S(p)+S(p)kS(p)+S(p)kS(p)kS(p)+S(p)kS(p)kS(p)kS(p)+⋯,

we obtain

    S_(5)   =   i/2∫d^4k/(2 π)^4Π^μν_(5)  A_μ(k) A_ν(-k),

with

    Π^μν_(5)   =   ∫d^4 p/(2π)^4[S(p)γ^μ S(p)γ_5 bS(p)k S(p)kS(p)k S(p)γ^ν
          + S(p)γ^μ S(p)k S(p)γ_5 bS(p)k S(p)k S(p)γ^ν
          + S(p)γ^μ S(p)k S(p)k S(p)γ_5 bS(p)k S(p)γ^ν
          + S(p)γ^μ S(p)k S(p)k S(p)k S(p)γ_5 bS(p)γ^ν
          + S(p)γ_5 bS(p)γ^μ S(p)k S(p)k S(p)k S(p)γ^ν],

where we have calculated the trace over the coordinate and momentum spaces and considered cubic terms of k_μ. 

Before evaluating the integrals in (<ref>), we first calculate the trace over the Dirac matrices, so that we have

    Π^μν_(5)   =    4i ϵ^μνσρ b_σ k_ρ k^2∫d^4 p/(2π)^41/(p^2-m^2)^3 - 16i  ϵ^μνσρ b_σ k_ρ k^α k^β∫d^4 p/(2π)^4p_α p_β/(p^2-m^2)^4
          + 24i m^2 ϵ^μνσρ  b_σ k_ρ k^2 ∫d^4 p/(2π)^41/(p^2-m^2)^4 + 24i ϵ^μνρα k_ρ b^β k^2 ∫d^4 p/(2π)^4p_α p_β/(p^2-m^2)^4
          + 32i ϵ^μνραk_ρ (b· k)k^β∫d^4 p/(2π)^4p_α p_β/(p^2-m^2)^4 - 128i ϵ^μνσρ b_σ k_ρ k^α k^β∫d^4 p/(2π)^4p_α p_β/(p^2-m^2)^5
          - 128i ϵ^μνρα k_ρ b^β k^δ k^γ∫d^4 p/(2π)^4p_α p_β p_δ p_γ/(p^2-m^2)^5.

Now, by analyzing the above integrals we observe that, after a simple power counting, we do not need to use regularization schemes because all integrals are convergent. Therefore, we can solve (<ref>) directly by using well-known solutions in which the higher-derivative Chern-Simons term is given by

    Π^μν_(5)=-ϵ^μνσρ b_σ k_ρ k^2/12 m^2 π^2,

or through the following Lagrangian as

    ℒ_(5)=e^2/24 m^2 π^2ϵ^μνρσ b_σ A_μ∂_ρ A_ν,

with 𝒦^μνραβ_(5)= e^2/24 m^2 π^2ϵ^μνσρ b_σ g^αβ, where we have re-instaled the electron charge.

The higher-derivative Chern-Simons term (<ref>) is also induced through radiative corrections from another CPT-odd term of the SME, namely governed by the coefficient g^μνρ, when g^μνρ is totally antisymmetric <cit.>.

Numerical estimates for the coefficient b_μ of the higher-derivative Chern-Simons term can be obtained from bounds on the coefficient 𝒦^μνραβ_(5) of the photon sector (see <cit.>, table XIX). From observational data related to the cosmic microwave background (CMB) polarization, we can estimate b∼ 10^-24. Moreover, from systems related to the astrophysical birefringence, we estimate that the coefficient is ∼ 10^-36. In these estimates we have considered m as being the electron mass, m≃0.5×10^-3GeV, and e≃10^-1 for the electron charge. In this way, we observe that we have found tiny values for the coefficient b_μ compatible with maximal sensitivities for the electron sector, table II of Ref. <cit.>. 



§ RADIATIVE CORRECTION AT FINITE TEMPERATURE



In this section, we are interested in the finite temperature behavior of the higher-derivative term (<ref>). For this, we take the Eq. (<ref>) and change it from Minkowski space to Euclidean space. We would like to emphasize here that the method of derivarive expansion used above, in general, is rather delicate at finite temperature, due to the fact that the limits k_0→0 and k⃗→0 do not commute, arising from the non-analyticity of thermal amplitudes <cit.>. In the expression (<ref>) we have considered the limits k_0→0 and k⃗→0. Then, by performing the following procedure: p_0 → i p_0 (g^μν→ -δ^μν), d^4p→ i d^4p_E, and p^2→ -p_0^2-p⃗^2 = -p^2_E, we obtain

    Π^μν_(5)   =    - 4 ϵ^μνσρ b^σ_E k^ρ_E k^2_E ∫d^4 p_E/(2 π)^41/(p^2_E + m^2)^3 + 16 ϵ^μνσρ b^σ_E k^ρ_E k^α_E k^β_E ∫d^4 p_E/(2 π)^4p^α_E p^β_E/(p^2_E + m^2)^4
          + 24 m^2 ϵ^μνσρ  b^σ_E k^ρ_E k^2_E ∫d^4 p_E/(2 π)^41/(p^2_E + m^2)^4 - 24 ϵ^μνρα k^ρ_E b^β_E k^2_E ∫d^4 p_E/(2 π)^4p^α_E p^β_E/(p^2_E + m^2)^4
          - 32 ϵ^μνραk^ρ_E (b_E· k_E)k^β_E ∫d^4 p_E/(2 π)^4p^α_E p^β_E/(p^2_E + m^2)^4 - 128 ϵ^μνσρ b^σ_E k^ρ_E k^α_E k^β_E ∫d^4 p_E/(2 π)^4p^α_E p^β_E/(p^2_E + m^2)^5
          + 128 ϵ^μνρα k^ρ_E b^β_E k^δ_E k^γ_E ∫d^4 p_E/(2 π)^4p^α_E p^β_E p^δ_E p^γ_E/(p^2_E + m^2)^5.


In order to effectively tackle the issue of the finite temperature behavior of the above expression, we separate the space and time components of the four-momentum p^σ_E=(p_0,p⃗) as p^σ_E→p̂^σ +p_0δ^σ 0, so that p̂^σ=(0,p⃗). Also, due to the symmetry of the integral under spacial rotations, it is possible to use the substitutions

    p̂^αp̂^β→p̂^2/3(δ^αβ-δ^α 0δ^β 0)

and

    p̂^αp̂^βp̂^δp̂^γ   →   p̂^4/15[(δ^αβ-δ^α 0δ^β 0)(δ^δγ-δ^δ 0δ^γ 0)+(δ^αδ-δ^α 0δ^δ 0)(δ^βγ-δ^β 0δ^γ 0)
          +(δ^αγ-δ^α 0δ^γ 0)(δ^βδ-δ^β 0δ^δ 0)].

After applying these changes into Eq. (<ref>), we find seven different tensorial structures. The first two structures, ϵ^μνσρ k^σ_E k^ρ_E b^0_E k^0_E and ϵ^μνσρ k^σ_E k^ρ_E (b_E · k_E), obviously vanish by antisymmetry of ϵ^μνσρ. In addition, the third contribution, ϵ^μν 0 σ k^0_E k^σ_E (b_E · k_E), goes to zero after the integration in the space components of p_E^α. Therefore, we are left with four different structures to analyze, namely ϵ^μνσρ b^σ_E k^ρ_E k^2_E, ϵ^μν 0 ρ b^0_E k^ρ_E k^2_E, ϵ^μνσρ b^σ_E k^ρ_E (k^0_E)^2 and ϵ^μν 0 ρ b^0_E k^ρ_E (k^0_E)^2. Thus, we have Π^μν_(5)=Π^μν_(a)+Π^μν_(b)+Π^μν_(c)+Π^μν_(c), with the following expressions for these four tensorial structures:

    Π^μν_(a)   =    -4 ϵ^μνσρ  b^σ_E k^ρ_E k^2_E ∫dp_0/2 π[ 32/15I_1(p_0,m)+32 m^2/3 I_2(p_0,m)-10/3 I_4(p_0,m). 
          - 6 m^2 I_5(p_0,m)+ I_6(p_0,m)], 
    Π^μν_(b)   =    8 ϵ^μν 0 ρ b^0_E k^ρ_E k^2_E ∫dp_0/2 π[ 16/15I_1(p_0,m)-16/3p_0^2 I_2(p_0,m) - I_4(p_0,m). 
          .+ 3p_0^2 I_5(p_0,m)], 
    Π^μν_(c)   =    -16  ϵ^μνσρ b^σ_E k^ρ_E  (k^0_E)^2∫dp_0/2 π[ -8/15I_1(p_0,m)+8/3(p_0^2-m^2)I_2(p_0,m) . 
          .+8 p_0^2 m^2 I_3(p_0,m) + 1/3 I_4(p_0,m)-p_0^2I_5(p_0,m)], 
    Π^μν_(d)   =    -128ϵ^μν 0 ρ b^0_E k^ρ_E (k^0_E)^2 ∫dp_0/2 π[ 1/5I_1(p_0,m)-2 p_0^2 I_2(p_0,m)+p_0^4 I_3(p_0,m)],

where the integrals I_1,2,3,4,5,6(p_0,m) are given by

    I_1,2,3(p_0,m)   =   ∫d^3p_E/(2 π)^3α_1,2,3/(p⃗^2+p_0^2+m^2)^5, 
    
    I_4,5(p_0,m)   =   ∫d^3p_E/(2 π)^3α_4,5/(p⃗^2+p_0^2+m^2)^4, 
    
    I_6(p_0,m)   =   ∫d^3p_E/(2 π)^31/(p⃗^2+p_0^2+m^2)^3,

with α_1=p⃗^4,α_2=α_4=p⃗^2 and α_3=α_5=1.

Now, we calculate the integrals over the space components p⃗. This can be done without using regularization schemes, because all the integrals are convergent. Then, we find

    Π^μν_(a)   =   m^2/ 8πϵ^μνσρ b^σ_E k^ρ_E  k^2_E [∫dp_0/2 π(p_0^2+m^2)^-5/2-2/3m^2∫dp_0/2 π(p_0^2+m^2)^-3/2], 
          +1/12πϵ^μνσρ b^σ_E k^ρ_E  k^2_E ∫dp_0/2 π(p_0^2+m^2)^-3/2
    Π^μν_(b)   =    -m^2/8 πϵ^μν 0 ρ b^0_E k^ρ_E k^2_E [∫dp_0/2 π(p_0^2+m^2)^-5/2-2/3m^2∫dp_0/2 π(p_0^2+m^2)^-3/2],
    Π^μν_(c)   =    m^2/πϵ^μνσρ b^σ_E k^ρ_E (k^0_E)^2[5 m^2/4∫dp_0/2 π(p_0^2+m^2)^-7/2-∫dp_0/2 π(p_0^2+m^2)^-5/2], 
    Π^μν_(d)   =    -m^2/πϵ^μν 0 ρ b^0_E k^ρ_E (k^0_E)^2 [5 m^2/4∫dp_0/2 π(p_0^2+m^2)^-7/2-∫dp_0/2 π(p_0^2+m^2)^-5/2].

Note that Π^μν_(b) and Π^μν_(d) cancel the corresponding time components b_E^0 of Π^μν_(a) and Π^μν_(c), respectively. Therefore, by summing all expressions, we obtain

    Π^μν_(5)   =    -1/m^2ϵ^μνσρ b^σ_E k^ρ_E  k^2_E B(m) - 1/m^2ϵ^μν i ρ b^i_E k^ρ_E k^2_E C_1(m)
          - 1/m^2ϵ^μν i ρ b^i_E  k^ρ_E (k^0_E)^2 C_2(m),

where


    B(m)    =    -m^2/12 π∫dp_0/2 π(p_0^2+m^2)^-3/2,
    
    C_1(m)    =   m^2/12π∫dp_0/2 π(p_0^2+m^2)^-3/2 - m^4/8 π∫dp_0/2 π(p_0^2+m^2)^-5/2, 
    
    C_2(m)    =   m^4/π∫dp_0/2 π(p_0^2+m^2)^-5/2 - 5 m^6/4 π∫dp_0/2 π(p_0^2+m^2)^-7/2,


with i=1,2,3.

The finite temperature behavior of the above terms can be obtained by using the Matsubara formalism. For this, we write p_0=(n +1/2) 2π/β, which are the Matsubara frequencies, and change the integral over p_0 into a sum, 1/2π∫dp_0→1/β∑_n, where T=β^-1 is the temperature of the system. Assuming that ξ=β m/2 π, we can write B(m)→ B(ξ), C_1(m)→ C_1(ξ), and C_2(m)→ C_2(ξ), so that


    B(ξ)    =    -ξ^2/24 π^2∑_n[(n+1/2)^2+ξ^2]^-3/2,
    C_1(ξ)    =   ξ^2/24π^2∑_n[(n+1/2)^2+ξ^2]^-3/2 - ξ^4/16π^2∑_n[(n+1/2)^2+ξ^2]^-5/2
    C_2(ξ)    =   ξ^4/2 π^2∑_n[(n+1/2)^2+ξ^2]^-5/2 - 5 ξ^6/8π^2∑_n[(n+1/2)^2+ξ^2]^-7/2.



As the above summations are all convergent, they can be easily numerically evaluated. Nevertheless, the asymptotic limits, zero and high temperatures, are not easy to obtain. One way to do this is to convert the sums into integrals, by using the expression <cit.>

    ∑_n[(n+b)^2+a^2]^-λ   =   √(π)Γ(λ - 1/2)/Γ(λ)(a^2)^λ-1/2+ 4sin(πλ) f_λ (a,b),

where 

    f_λ(a,b)= ∫_|a|^∞dz/(z^2-a^2)^λ Re ( 1/e^2π(z+ib)-1),

which is valid for Re λ < 1, asides from the poles at λ = 1/2, -1/2, -3/2, ⋯. However, by analyzing the equations (<ref>), (<ref>), and (<ref>), we have the values λ=3/2, λ=5/2 and λ=7/2, i.e. they are all out of range of validity. Luckily, we can use a recurrence relation, given by

    f_λ -1(a,b)=-1/2a^22λ-5/λ-2f_λ-2(a,b)-1/4a^21/(λ-3)(λ-2)∂^2/∂ b^2f_λ-3(a,b),

in order to decrease these values of λ. In this way, for λ=3/2 we must use the recurrence relation once, for λ=5/2  twice, and for λ=7/2 thrice. We must also write λ=3/2→ D/2, λ=5/2→ D/2+1, and λ=7/2→ D/2+2, to avoid the poles λ=1/2,-1/2 in the first term of the solution (<ref>), so that the expressions (<ref>) become


    B(ξ)    =    -m^3-D/24π^2ξ^D-1∑_n[(n+1/2)^2+ξ^2]^-D/2,
    
    C_1(ξ)    =   m^3-D/24π^2ξ^D-1∑_n[(n+1/2)^2+ξ^2]^-D/2
          - m^3-D/16π^2ξ^D+1∑_n[(n+1/2)^2+ξ^2]^-D/2-1, 
    
    C_2(ξ)    =   m^3-D/2 π^2ξ^D+1∑_n[(n+1/2)^2+ξ^2]^-D/2-1
          - 5m^3-D/8π^2ξ^D+3∑_n[(n+1/2)^2+ξ^2]^-D/2-2.
 

Then, by using the recurrence relation (<ref>) and considering the sum (<ref>), after we take the limit D→3, we obtain

    Π^μν_(5)   =   1/m^2ϵ^μνσρ b_σ k_ρ k^2 B(ξ) + 1/m^2ϵ^μν i ρ b_i k_ρ  k^2 C_1(ξ) 
          +1/m^2ϵ^μν i ρ b_i k_ρ k_0^2 C_2(ξ),

with


    B(ξ)    =   1/6∫_|ξ|^∞dz √(z^2-ξ^2)tanh(π z)^2(π z)-1/12 π^2,
    C_1(ξ)    =   ξ^2/12∫_|ξ|^∞dztanh(π z)^2(π z)/√(z^2-ξ^2), 
    C_2(ξ)    =   π^2 ξ^2/6∫_|ξ|^∞dz√(z^2-ξ^2)^5(π z)[sinh(3π z)-11sinh(π z)],


where we have returned to the Minkowski space.

By analyzing the above equations we observe that when T→0 (ξ→∞) all the integrals vanish, so that Π^μν_(5) goes to -1/12 π^2 m^2ϵ^μνσρ b_σ k_ρ k^2. This is in fact the result of the temperature zero, obtained previously in (<ref>). On the other hand, when T→∞ (ξ→0) the expression (<ref>) goes to zero, as well as the expressions (<ref>) and (<ref>). This happens mainly because all the summations in (<ref>) are strongly suppressed by the temperature, thus suggesting that in general the operators of mass dimension d≥5 vanish in the limit of high temperature.

The plot of the functions B(ξ), C_1(ξ), and C_2(ξ), which can be numerically calculated from either expressions (<ref>) or (<ref>), are presented in Figs. <ref>, <ref>, and <ref>. It is interesting to note here that these functions are the same as those obtained in the context of the coefficient g^μνρ <cit.>. This is because under a certain fermion field redefinition <cit.> the totally antisymmetric component of g^μνρ is completely absorbed into coefficient b^μ.

 





§ LARGE GAUGE INVARIANCE



Let us now briefly discuss the question of large gauge invariance, at finite temperature, in the context of Lorentz- and CPT-violating QED. This issue has been extensively studied in three-dimensional Chern-Simons theories <cit.>. In our case, we want to examine the invariance of the higher-derivative Chern-Simons term (<ref>) under large gauge transformation, as well as the four-dimensional Chern-Simons term, namely,

    S_(3)=∫d^4x 𝒦^μνρ_(3) A_μ∂_ρ A_ν,

with 𝒦^μνρ_(3)∝ϵ^μνρσ b_σ, where both are dynamically generated by the effective action (<ref>). It would be interesting to consider the other terms with operators of mass dimension d=7,9,⋯, however, this is a challenging task. In fact, in the literature we find only models with Lorentz-violating operators of mass dimension d=5, for CPT-odd terms. In particular, we can argue that, as K_(d)∼ M_Pl^4-d, these operators with d≥ 7 are highly suppressed by the mass Planck M_Pl and, therefore, are even more negligible.

To perform this investigation, we continue to use the method of derivative expansion, however, in a particular gauge field background, where we have a vanishing electric field and a time-independent magnetic field, as follows

    A_0=A_0(t),  A_i=A_i(x,y,z).

It is worth mentioning that this specific choice of gauge background is consistent with the static limit, as shown in <cit.>, although it is not a static background. The point here is that in this static limit we can avoid the non-analyticity of thermal amplitudes, which can appear when the effective action is expanded in powers of derivatives. Furthermore, in this background (<ref>) we believe that the effective action can be calculated exactly. 

Finally, by considering the appropriate gauge transformation <cit.>, 

    A_μ→ A_μ + ∂_μΩ, Ω(t)=(-∫_0^t+t/β∫_0^β)dt' A_0(t'),

we obtain

    A_0 →a_0/β= 1/β∫_0^β dt  A_0(t),  A_i → A_i(x,y,z),

so that under large gauge transformation,

    a_0 → a_0 + 2π n,

where n is an integer. Then, by taking into account these constraints, we can rewrite the effective action (<ref>) in the Euclidean space as

    S_=-∑_nln(p⃗+ω̃_nγ^0+m-A⃗-γ_5 b_E),
 
in which we have introduced the definitions: p⃗=p^iγ^i, b_E=b^0γ^0+b^iγ^i,  and

    ω̃_n=ω_n-a_0/β=(n +1/2) 2π/β-a_0/β.

Now, following the procedure performed in Ref. <cit.>, let us first derivate the effective action (<ref>) with respect to a_0, and later consider only the linear term in A^i, so that we can write

    ∂ S_^(1)/∂ a_0 = ∑_n  1/p⃗+ω̃_nγ^0+m-γ_5 b_EA⃗1/p⃗+ω̃_nγ^0+m-γ_5 b_Eγ^0.

As we are interested only in contributions of first order in b_μ, we single out the expression

    ∂ S_b^(1)/∂ a_0 = ∑_n   [S_E(p⃗)γ^i S_E(p⃗-i∂⃗)γ_5 b_E S_E(p⃗-i∂⃗)γ^0+S_E(p⃗)γ_5 b_E S_E(p⃗)γ^i S_E(p⃗-i∂⃗)γ^0 ]A^i,

where S_E(p⃗)=(p⃗+ω̃_nγ^0+m)^-1. Note that the above equation is similar to Eq. (<ref>), and therefore the calculations of Eq. (<ref>) are similar to those that have been performed previously.

In order to generate the Lorentz-violating Chern-Simons term (<ref>), we expand the propagator S_E(p⃗-i∂⃗) up to first order in ∂^k, so that we obtain

    ∂ S_(3)/∂ a_0 = i ∫ d^3x Π_(3)^i0 A^i

with

    Π_(3)^i0   =   ∑_n ∫d^3 p_E/(2π)^3[S_E(p⃗)γ_5 b_E S_E(p⃗)γ^i S_E(p⃗)∂⃗ S_E(p⃗)γ^0 
          + S_E(p⃗)γ^i S_E(p⃗)∂⃗ S_E(p⃗)γ_5 b_E S_E(p⃗)γ^0 
          + S_E(p⃗)γ^i S_E(p⃗)γ_5 b_E S_E(p⃗)∂⃗ S_E(p⃗)γ^0].

Hereafter, we calculate the trace over the Dirac matrices and thus find the expression

    Π_(3)^i0   =    -12i ϵ^i0jk b^j ∂^k ∑_n∫d^3 p_E/(2 π)^31/(p⃗^2 + ω̃_n^2 + m^2)^2
          -16i ϵ^0jkl b^j ∂^k ∑_n∫d^3 p_E/(2 π)^3p^ip^l/(p⃗^2 + ω̃_n^2 + m^2)^3
          +16i ϵ^ijk0 b^j ∂^k ∑_n∫d^3 p_E/(2 π)^3ω̃_n^2/(p⃗^2 + ω̃_n^2 + m^2)^3
          +16i ϵ^i0jl b^j ∂^k ∑_n∫d^3 p_E/(2 π)^3p^k p^l/(p⃗^2 + ω̃_n^2 + m^2)^3,

where we have moved γ_5 to the very end of every expression, before calculating the trace. The next step is to calculate the space integral, by promoting the three- to D-dimension and using the substitution p^ip^l→p⃗^2δ^il/D, which leads to the following result:

    Π_(3)^i0   =     -iϵ^i0jk b^j ∂^k 2^1-Dm π^-D/2(μ^2)^3/2-D/2Γ(2-D/2)
          ×∑_n[(D-3)/(ω̃_n^2+m^2)^2-D/2-(D-4)m^2/(ω̃_n^2+m^2)^3-D/2].

To perform the above summations, we cannot readily take the limit D→3 in the expression because the first sum exhibits singularity. Therefore, we must use the Eq. (<ref>) in order to isolate the corresponding divergent term, which becomes finite due to the factor (D-3) in the numerator. For the second sum, when D→3, we have λ→ 3/2, i.e. we also want to use here the recurrence relation (<ref>). Following these procedures, we arrive at

    Π_(3)^i0 = -iϵ^i0jk b^j ∂^k F'(ξ,a_0),

where

    F'(ξ,a_0) = ∫_|ξ|^∞ dz √(z^2-ξ^2) sinh(2π z)2cos(a_0)cosh(2π z)-cos(2a_0)+3/[cos(a_0)+cosh(2π z)]^3,

or, in other words,

    F'(ξ,a_0) = ∂ /∂ a_0F(ξ,a_0) = ∂ /∂ a_0∫_|ξ|^∞ dz √(z^2-ξ^2) 2sin(a_0)sinh(2π z)/[cos(a_0)+cosh(2π z)]^2.

Therefore, after returning to the Minkowski space, we obtain the Lorentz-violating Chern-Simons action,

    S_(3) = ∫ d^3x ϵ^i0jk b_j ∂_k A_i  F(ξ,a_0),

with

    F(ξ,a_0) = ∫_|ξ|^∞ dz √(z^2-ξ^2) 2sin(a_0)sinh(2π z)/[cos(a_0)+cosh(2π z)]^2,

which, as expected, is invariant under large gauge transformation (<ref>). Note that a perturbative expansion in terms of a_0 yields the usual perturbative result,

    F(ξ,a_0) = a_0 ∫_|ξ|^∞ dz √(z^2-ξ^2) ^2(π z)tanh(π z) +  O(a_0^2),

that it was obtained previously in <cit.>, and in <cit.> from the derivative term ψ̅i2g^κλμσ_κλ(∂_μ+iA_μ)ψ, governed by the coefficient g^κλμ.

Let us finally consider the generation of the higher-derivative Chern-Simons term (<ref>), by expanding the propagator S_E(p⃗-i∂⃗), in Eq. (<ref>), up to three order in ∂^k. As we have mentioned, the corresponding action is similar to Eq. (<ref>), however, given by the expression

    ∂ S_(5)/∂ a_0 = i ∫ d^3x Π_(5)^i0 A^i,

where

    Π^i0_(5)   =    4i ϵ^i0jk b^j ∂^k ∇⃗^2 ∑_n ∫d^3 p_E/(2 π)^31/(p⃗^2 + ω̃_n^2 + m^2)^3
          - 16i ϵ^i0jk b^j ∂^k ∂^l ∂^m ∑_n ∫d^3 p_E/(2 π)^3p^l p^m/(p⃗^2 + ω̃_n^2 + m^2)^4
          - 24i m^2 ϵ^i0jk  b^j ∂^k ∇⃗^2 ∑_n ∫d^3 p_E/(2 π)^31/(p⃗^2 + ω̃_n^2 + m^2)^4
          - 24i ϵ^i0jk b^l ∂^k ∇⃗^2 ∑_n ∫d^3 p_E/(2 π)^3p^k p^l/(p⃗^2 + ω̃_n^2 + m^2)^4
          + 128i ϵ^i0jk b^j ∂^k ∂^l ∂^m ∑_n ∫d^3 p_E/(2 π)^3p^l p^m/(p⃗^2 + ω̃_n^2 + m^2)^5
          - 128i ϵ^i0kl b^j ∂^k  ∂^m ∂^n ∑_n ∫d^4 p_E/(2 π)^4p^j p^l p^m p^n/(p⃗^2 + ω̃_n^2 + m^2)^5.

In the above expression we must also use the substitution p^j p^l p^m p^n →p⃗^4(δ^jlδ^mn+δ^jmδ^ln+δ^jnδ^ml)/[D(D+2)], so that after we evaluate the space integrals and the sums, we obtain 

    Π^i0_(5) = -i/m^2ϵ^i0jk b^j ∂^k ∇⃗^2 G'(ξ,a_0),

with

    G'(ξ,a_0) = 1/12∫_|ξ|^∞ dz 2z^2-ξ^2/√(z^2-ξ^2) sinh(2π z)2cos(a_0)cosh(2π z)-cos(2a_0)+3/[cos(a_0)+cosh(2π z)]^3 -1/12 π^2,

which can be written as

    G'(ξ,a_0) = ∂ /∂ a_0G(ξ,a_0) = ∂ /∂ a_0(1/12∫_|ξ|^∞ dz 2z^2-ξ^2/√(z^2-ξ^2) 2sin(a_0)sinh(2π z)/[cos(a_0)+cosh(2π z)]^2-a_0/12 π^2).

Then, the higher-derivative Chern-Simons action takes the form

    S_(5) = ∫ d^3x ϵ^i0jk b_j ∂_k ∇⃗^2A_i  G(ξ,a_0),

where

    G(ξ,a_0) = 1/12∫_|ξ|^∞ dz 2z^2-ξ^2/√(z^2-ξ^2) 2sin(a_0)sinh(2π z)/[cos(a_0)+cosh(2π z)]^2-a_0/12 π^2,

which is also invariant under large gauge transformation (<ref>). Observe that by expanding the above expression, up to first order in a_0, Eqs. (<ref>) and (<ref>) are readily recovered. This complete our analysis.



§ SUMMARY



In this work, we study the radiative generation of the higher-derivative Lorentz-violating Chern-Simons term, at zero temperature and at finite temperature. At zero temperature, we use the method of derivative expansion, as a preparation for the finite temperature analysis. We also present numerical estimations for the coefficient b^μ, which are compatible with maximal sensitivities for the electron sector. At finite temperature, by using the Matsubara formalism, we observe that the higher-derivative Chern-Simons term vanishes when the temperature goes to infinity. This behavior has already been observed in the context of the coefficient g^μνρ <cit.>, suggesting, therefore, that in general the operators of mass dimension d≥5 vanish in the limit of T→∞. In the analysis of large gauge invariance, we compute the exact effective actions, Eqs. (<ref>) and (<ref>), at finite temperature and in a specific gauge field background. We observe that, in fact, the conventional Chern-Simons (<ref>) and the higher-derivative Chern-Simons (<ref>) actions are invariant under large gauge transformation. 



Acknowledgements. This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).




Kos1V. A. Kostelecky and S. Samuel, Phys. Rev. D 39, 683 (1989).
Kos2D. Colladay and V. A. Kostelecky, Phys. Rev. D 55, 6760 (1997) [hep-ph/9703464].
Kos3D. Colladay and V. A. Kostelecky, Phys. Rev. D 58, 116002 (1998) [arXiv:hep-ph/9809521].
Kos4V. A. Kostelecky and N. Russell, Rev. Mod. Phys.  83 (2011) 11 [arXiv:0801.0287 [hep-ph]].
GamR. Gambini and J. Pullin, Phys. Rev. D 59, 124021 (1999) [gr-qc/9809038].
Alf J. Alfaro, H. A. Morales-Tecotl and L. F. Urrutia, Phys. Rev. D 65, 103509 (2002) [hep-th/0108061].
SudD. Sudarsky, L. Urrutia and H. Vucetich, Phys. Rev. Lett.  89, 231301 (2002) [gr-qc/0204027].
MyeR. C. Myers and M. Pospelov, Phys. Rev. Lett.  90, 211601 (2003) [arXiv:hep-ph/0301124].
BolP. A. Bolokhov, S. Nibbelink Groot and M. Pospelov, Phys. Rev. D 72 (2005) 015013 [hep-ph/0505029].
JacT. Jacobson, S. Liberati and D. Mattingly, Annals Phys.  321 (2006) 150 [astro-ph/0505267].
Bol1P. A. Bolokhov and M. Pospelov, Phys. Rev. D 77 (2008) 025022 [hep-ph/0703291 [HEP-PH]].
ReyC. M. Reyes, L. F. Urrutia and J. D. Vergara, Phys. Rev. D 78, 125011 (2008) [arXiv:0810.5379 [hep-ph]].
Kos5V. A. Kostelecky and M. Mewes, Phys. Rev.  D 80, 015020 (2009) [arXiv:0905.0031 [hep-ph]].
Rey2C. M. Reyes, Phys. Rev.  D 82, 125036 (2010) [arXiv:1011.2971 [hep-ph]]. 
MarT. Mariz, Phys. Rev.  D83, 045018 (2011) [arXiv:1010.5013 [hep-th]].
Lei J. Leite and T. Mariz, Europhys. Lett.  99, 21003 (2012) [arXiv:1110.2127 [hep-th]].
Kos6A. Kostelecky and M. Mewes, Phys. Rev. D 85, 096005 (2012) [arXiv:1112.6395 [hep-ph]].
BriF. A. Brito, M. S. Guimaraes, E. Passos, P. Sampaio and C. Wotzasek, “The 4D-2D projection of Lorentz-violating Myers-Pospelov QED,” arXiv:1210.2369 [hep-th].



JacKosR. Jackiw and V. A. Kostelecky, Phys. Rev. Lett.  82, 3572 (1999) [arXiv:hep-ph/9901358].
Mar2T. Mariz, J. R. Nascimento and A. Y. Petrov, Phys. Rev. D 85, 125003 (2012) [arXiv:1111.0198 [hep-th]].
BonG. Bonneau, Nucl. Phys.  B 593, 398 (2001) [arXiv:hep-th/0008210].


Ait I. J. R. Aitchison and C. M. Fraser, Phys. Rev. D 31, 2605 (1985).
FraC. M. Fraser, Z. Phys. C 28, 101 (1985).
VaiA. I. Vainshtein, V. I. Zakharov, V. A. Novikov and M. A. Shifman, Sov. J. Nucl. Phys.  39, 77 (1984) [Yad. Fiz.  39, 124 (1984)].
BabK. S. Babu, A. K. Das and P. Panigrahi, Phys. Rev. D 36, 3725 (1987).

ColS. R. Coleman and S. L. Glashow, Phys. Rev.  D 59, 116008 (1999) [arXiv:hep-ph/9812418].
ChuJ. M. Chung and P. Oh, Phys. Rev.  D 60, 067702 (1999) [arXiv:hep-th/9812132].
PerM. Perez-Victoria, Phys. Rev. Lett.  83, 2518 (1999) [arXiv:hep-th/9905061].
ChaM. Chaichian, W. F. Chen and R. Gonzalez Felipe, Phys. Lett.  B 503, 215 (2001) [arXiv:hep-th/0010129].
AdaC. Adam and F. R. Klinkhamer, Nucl. Phys.  B 607, 247 (2001) [arXiv:hep-ph/0101087].
AndA. A. Andrianov, P. Giacconi and R. Soldati, JHEP 0202, 030 (2002) [arXiv:hep-th/0110279].
BelH. Belich, Jr., M. M. Ferreira, Jr., J. A. Helayel-Neto and M. T. D. Orlando, Phys. Rev. D 67, 125011 (2003) [Erratum-ibid. D 69, 109903 (2004)] [hep-th/0212330].
Mar3T. Mariz, J. R. Nascimento, E. Passos, R. F. Ribeiro and F. A. Brito, JHEP 0510, 019 (2005) [arXiv:hep-th/0509008].

Baz D. Bazeia, T. Mariz, J. R. Nascimento, E. Passos and R. F. Ribeiro, J. Phys. A 36, 4937 (2003) [hep-th/0303122].

DasA. K. Das and M. B. Hott, Phys. Rev. D 50, 6655 (1994) [hep-ph/9407283].

FordL. H. Ford, Phys. Rev. D 21, 933 (1980).
McDD. Colladay and P. McDonald, J. Math. Phys.  43, 3554 (2002) [hep-ph/0202066].

DunG. V. Dunne, K. -M. Lee and C. -h. Lu, Phys. Rev. Lett.  78, 3434 (1997) [hep-th/9612194].
Des1S. Deser, L. Griguolo and D. Seminara, Phys. Rev. Lett.  79, 1976 (1997) [hep-th/9705052].
Des2S. Deser, L. Griguolo and D. Seminara, Phys. Rev. D 57, 7444 (1998) [hep-th/9712066].
Fos1C. Fosco, G. L. Rossini and F. A. Schaposnik, Phys. Rev. Lett.  79, 1980 (1997) [Erratum-ibid.  79, 4296 (1997)] [hep-th/9705124].
Fos2C. D. Fosco, G. L. Rossini and F. A. Schaposnik, Phys. Rev. D 56, 6547 (1997) [hep-th/9707199].
Fos3I. J. R. Aitchison and C. D. Fosco, Phys. Rev. D 57, 1171 (1998) [hep-th/9709035].
Bra1F. T. Brandt, A. K. Das and J. Frenkel, Phys. Rev. D 62, 085012 (2000) [hep-ph/0005150].
Bra2F. T. Brandt, A. K. Das, J. Frenkel, S. Pereira and J. C. Taylor, Phys. Rev. D 64, 065018 (2001) [hep-th/0103221].
CerL. Cervi, L. Griguolo and D. Seminara, Phys. Rev. D 64, 105003 (2001) [hep-th/0104022]


