Induced Chern-Simons modified gravity at finite temperature J. F. Assunção,1 T. Mariz,1 and A. Yu. Petrov2 1 Instituto de Fı́sica, Universidade Federal de Alagoas, 57072-900, Maceió, Alagoas, Brazil∗ 2 Departamento de Fı́sica, Universidade Federal da Paraı́ba, Caixa Postal 5008, 58051-970, João Pessoa, Paraı́ba, Brazil† Abstract We calculate the linearized four-dimensional gravitational Chern-Simons term at the finite temperature case and explicitly demonstrate that its transversal part matches the known result for the axial magnetic conductivity. PACS numbers: 11.30.Cp, 04.60.-m, 11.10.Wx ∗ † Electronic address: jfassuncao,[email protected] Electronic address: [email protected] 1 As it is known, the Lorentz and CPT symmetries are violated [1–3] when we add the fourdimensional gravitational Chern-Simons term, in the weak (linearized) gravity case looking like Lb = 14 bλ hµν αµλρ ∂ ρ (∂γ ∂ γ hαν − ∂ν ∂γ hγα ), to the Einstein-Hilbert Lagrangian [4–7]. Recently, it has been shown in [7] that this term displays the ambiguity, similar to that one of the Carroll-FieldJackiw term Lb = 12 bµ µνλρ Fνλ Aρ [8–10]. In this work we have interested in to study the generation of the gravitational Chern-Simons term at finite temperature. The main motivation for this is the emergence of chiral magnetic ~ which involves the contribution to conductivity dependent of the temperature current ~j = σ B, T [11]. This has been argued in [11] as a possibility to investigate evidence of the gravitational anomaly. Let us then consider the fermionic action given by Z ↔ i µ Sψ = d4 x e e a ψ̄γ aDµψ − mψ̄ψ − bµ eµ a ψ̄γ a γ5 ψ , 2 (1) where eµa is the tetrad, e ≡ det eµa , and Dµ ψ = ∂µ ψ − iωµ ψ (Dµ ψ̄ = ∂µ ψ̄ + iωµ ψ̄), with ωµ = 1 bc 4 ωµbc σ being the spin connection, and σ bc = 2i [γ b , γ c ]. Note that in the term bµ eµ a ψ̄γ a γ5 ψ, we have the Lorentz-violating coefficient bµ and the CPT-violating operator eµ a ψ̄γ a γ5 ψ. In order to obtain the effective action, we must consider the fermionic generating functional Z Zψ = DψDψ̄eiSψ = eiSeff , (2) so that after we perform the fermionic integration, we get i µ a↔ µ a µ a e e a γ ∂ µ −e m − e bµ e a γ γ5 + e e a γ ωµ , Seff = −iTr ln 2 (3) where Tr stands for the trace over Dirac matrices as well as trace over the integration in momentum and coordinate spaces. Now, we use the weak field approximation, in which the tetrad and the connection are expressed in terms of the metric fluctuation hµν as eµa = ηµa + 12 hµa and ωµab = 21 (∂b hµa − ∂a hµb ). Thus, we have ↔ i i µ ν ν ν λ ρ µ / = −iTr ln i6 ∂ − m − b γ5 − hµν γ ∂ + (hµν ∂λ hρ − hρ ∂λ hµν )γ γ γ , 4 32 Seff (4) Finally, in order to single out the quadratic terms in hµν within the effective action, we rewrite ∞ P (0) (n) the expression (4) as Seff = Seff + Seff , where n=1 (n) Seff 1 = iTr n n ↔ 1 i i µ ν ν ν λ ρ µ hµν γ ∂ − (hµν ∂λ hρ − hρ ∂λ hµν )γ γ γ . 32 i6 ∂ − m − /bγ5 4 2 (5) As our goal is the generation of the gravitational Chern-Simons action, we will single out above terms in which are second order in hµν and first order in bµ . Firstly, let us analyze the term coming from n = 1, given by (1) SCS = iTr 1 i 1 /bγ5 (hµν ∂λ hρ ν − hρ ν ∂λ hµν )γ λ γ ρ γ µ . i6 ∂ − m i6 ∂ − m 32 Then, after we carry out the traces over the integration in spaces, we obtain Z (1) SCS [h] = i d4 x hµν Πµνρσ hρσ , a (6) (7) with Πµνρσ a 1 = tr 16 Z d4 p / ρ γ µ η νσ . S(p)/bγ5 S(p)∂γ (2π)4 (8) where the symbol tr means that the trace is only over Dirac matrices and S(p) = (p / − m)−1 . Now, let us consider the terms coming from n = 2, given by (2) SCS [h] = ↔ ↔ i 1 1 i 1 i /bγ5 Tr hµν γ µ∂ µ hαβ γ α∂ β 2 i6 ∂ − m i6 ∂ − m 4 i6 ∂ − m 4 ↔ ↔ 1 i 1 1 i i /bγ5 hµν γ µ∂ µ hαβ γ α∂ β. + Tr 2 i6 ∂ − m 4 i6 ∂ − m i6 ∂ − m 4 (9) By using the key identity of the derivative expansion approach hµν (x)S(p) = S(p − i∂)hµν (x), in order to disentangle the traces over xµ and pµ , we arrive at Z i (2) SCS [h] = + Πµνρσ hρσ , d4 x hµν Πµνρσ c b 2 (10) where Πµνρσ b 1 = tr 16 d4 p S(p)/bγ5 S(p)γ µ (2pν − i∂ ν )S(p − i∂)γ ρ (2pσ − i∂ σ ) (2π)4 (11) d4 p S(p)γ µ (2pν − i∂ ν )S(p − i∂)/bγ5 S(p − i∂)γ ρ (2pσ − i∂ σ ). (2π)4 (12) Z and Πµνρσ = c 1 tr 16 Z By considering the whole expression for the vacuum polarization tensor given by Πµνρσ = Πµνρσ + 21 Πµνρσ + 12 Πµνρσ , we have a c b " # r 2 2 2 µρκλ 1 12m 4m 2m 12m −1 2 νσ ν σ √ − Πµνρσ = 1 + − 1 csc k g − k k bκ kλ . (13) 192π 2 k2 k2 k2 k2 The tensor Πµνρσ will not contribute to the gravitational Chern-Simons term, actually it yields a the one-derivative term treated in [12]. We note that this term, first, breaks the gauge 3 symmetry possessing only some restricted gauge symmetry as was argued in [12], second, vanishes in the zero mass limit. Thus, let us focus on the tensors Πµνρσ and Πµνρσ . c b To calculate the trace over the Dirac matrices, we first use the ciclic property of the trace, to move γ5 matrix to the end of the expression, so that finally we obtain Z µ4−D dD p (2p − k)ν (2p − k)σ h 2 µνρσ Πb = − (p − m2 )µρκλ bκ (kλ + pλ ) 4 (2π)D (p2 − m2 )2 (p21 − m2 ) i +2 pµ ρκλτ bκ kλ pτ − pρ µκλτ bκ kλ pτ − (p · k)µρκλ bκ pλ , Z dD p (2p − k)ν (2p − k)σ h 2 µ4−D (p1 − m2 )µρκλ bκ (2kλ − pλ ) Πµνρσ = − c 4 (2π)D (p2 − m2 )(p21 − m2 )2 i +2 pµ1 ρκλτ bκ kλ pτ − pρ1 µκλτ bκ kλ pτ − (p1 · k)µρκλ bκ p1λ , (14a) (14b) where we have considered i∂µ → kµ , and pµ1 = pµ − k µ . This procedure is similar to the one used in the ’t Hooft and Veltman prescription. By power counting, the terms (p2 − m2 )pν pσ µρκλ bκ pλ and (p21 − m2 )pν pσ µρκλ bκ pλ are cubically divergent. These terms are also not transversal, however, they cancel each other already in the integrand, when we consider Πµνρσ +Πµνρσ . So, the remaining c b divergences are at most quadratic, and thus gauge invariance is restored. It is interesting to carry out the separation 3 (p2 − m2 )µρκλ bκ kλ 1 (p21 − m2 )µρκλ bκ kλ (p2 − m2 )µρκλ bκ kλ = − , 2 (p2 − m2 )2 (p21 − m2 ) 2 (p2 − m2 )(p21 − m2 )2 (p2 − m2 )2 (p21 − m2 ) + Π̃µνρσ , where + Πµνρσ = Π̃µνρσ , in which we can rewrite (14), such that Πµνρσ in Πµνρσ c c b b b Z µ4−D dD p (2p − k)ν (2p − k)σ 3 2 = − Π̃µνρσ (p − m2 )µρκλ bκ kλ b 4 (2π)D (p2 − m2 )2 (p21 − m2 ) 2 i +2 pµ ρκλτ bκ kλ pτ − pρ µκλτ bκ kλ pτ − (p · k)µρκλ bκ pλ , (15) and Π̃µνρσ (k) = Π̃ρνµσ (−k). Thus, we need only to calculate Π̃µνρσ , so that after using the Feynman c b b parametrization, we have Π̃µνρσ = b Z 1 Z µ4−D dD p [2pν + (2x − 1)k ν ][2pσ + (2x − 1)k σ ] dx(x − 1) 4 (2π)D [p2 − m2 − x(x − 1)k 2 ]3 0 n h × 3[(p + xk)2 − m2 ]µρκλ bκ kλ + 4 (pµ + xk µ )ρκλτ bκ kλ pτ io −(pρ + xk ρ )µκλτ bκ kλ pτ − (p · k + xk 2 )µρκλ bκ (pλ + xkλ ) . (16) As we are interested in considering the finite temperature effects, we carry out the Wick rotation and split the internal momentum pµ into its spatial and temporal components. For this, we take into account the replacements: g µν → −δ µν , i.e., p2 → −δ µν pµ pν , and so on, Z Z Z dd p~ dp0 dD p 4−D 3−d →µ i , µ D d (2π) 2π (2π) 4 (17) and pµ → p~µ + p0 uµ , to separate the integration variables, with p~µ = (0, p~) and uµ = (1, 0, 0, 0). Besides, let us assume from now on the system to be in thermal equilibrium with a temperature T = β −1 , so that the antiperiodic (periodic) boundary conditions for fermions (bosons) lead to discrete values of p0 = (2n + 1) βπ (k0 = 2πl β ), where n (respectively l) is an integer. Thus, we can rewrite (16), as follows: iµ3−d 4 Π̃µνρσ (k) = b Z 1 dx(x − 1) 0 Z 1X dd p~ n 3[(~ p + x~k)2 + p20 + m2 ]µρκλ bκ kλ β n (2π)d µ +4 [~ p + (p0 − xk0 )uµ + xk µ ] ρκλγ bκ kλ [~ p γ + (p0 − xk0 )uγ ] −4 [~ p ρ + (p0 − xk0 )uρ + xk ρ ] µκλγ bκ kλ [~ p γ + (p0 − xk0 )uγ ] o −4[~ p · ~k + (p0 − xk0 )k0 + xk 2 ]µρκλ bκ [~ pλ + (p0 − xk0 )uλ + xkλ )] × [2~ p ν + (2x − 1)k ν + 2(p0 − xk0 )uν ][2~ p σ + (2x − 1)k σ + 2(p0 − xk0 )uσ ] . (18) [~ p 2 + (p0 − xk0 )2 + m2 + x(1 − x)k 2 ]3 Now, in order to select the tensorial structures of the above result (18), let us consider the expressions p~ 2 µν (g − uµ uν ), d h p~ 4 → (g µν − uµ uν )(g αβ − uα uβ ) + (g µα − uµ uα )(g νβ − uν uβ ) d(d + 2) i +(g µβ − uµ uβ )(g αν − uα uν ) , p~ µ p~ ν → p~ µ p~ ν p~ α p~ β (19a) (19b) as well as the identity g µρ νσαλ − g νρ µσαλ − g λρ µσνα + g ρσ µναλ − g αρ µνσλ = 0, (20) so that we have Π̃µνρσ = A(k, m)g νσ µρκλ bκ kλ − B(k, m)g νσ bi µρiλ kλ + C(k, m)k ν k σ µρκλ bκ kλ b −D(k, m)k ν k σ bi µρiλ kλ + [E(k, m) − 2J(k, m)] (k ν uσ + k σ uν )µρκλ bκ kλ + [F (k, m) − 4B(k, m)] uν uσ µρκλ bκ kλ − G(k, m)(k ν uσ + uν k σ )bi µρiλ kλ h i −H(k, m)uν uσ bi µρiλ kλ + I(k, m) bν µρσλ kλ + bσ µρνλ kλ h i +B(k, m) −(uν bσ + uσ bν )µρλτ kλ uτ + b0 (uν µρσλ kλ + uσ µρνλ kλ ) h i +J(k, m) −(k ν bσ + k σ bν )µρλτ kλ uτ + b0 (k ν µρσλ kλ + k σ µρνλ kλ ) . (21) where Z d−4 Γ 1 − d2 X 1 A(k, m) = dx (1 − x) (p0 − xk0 )2 + M 2 2 d/2 4(4π) 0 p0 × 4 (p0 − xk0 )2 + M 2 + (2 − d)xk 2 , 5 (22) d 2 Γ 1− B(k, m) = XZ d 2 1 d−4 dx (1 − x) (p0 − xk0 )2 + M 2 2 0 2(4π) p0 × (d − 1)(p0 − xk0 )2 + M 2 , Z d−6 Γ 3 − d2 X 1 dx (1 − 2x)2 M 2 (p0 − xk0 )2 + M 2 2 , C(k, m) = d/2 4(4π) 0 p (23) (24) 0 D(k, m) = Γ 2− d 2 XZ 1 d d−6 dx (1 − x)(1 − 2x)2 (p0 − xk0 )2 + M 2 2 4(4π) 2 p0 0 × (d − 3)(p0 − xk0 )2 + M 2 , (25) Z d−6 Γ 3 − d2 2 X 1 2 2 2 dx (1 − x)x(2x − 1)(p − xk ) (p − xk ) + M k E(k, m) = , 0 0 0 0 2(4π)d/2 0 p 0 F (k, m) = −Γ 2 − d 2 k d 2 2 XZ 1 d−6 (1 − x)x (p0 − xk0 )2 + M 2 2 0 2(4π) p0 × (d − 3)(p0 − xk0 )2 + M 2 , Z d−6 Γ 2 − d2 X 1 2 2 2 G(k, m) = dx (1 − x)(1 − 2x)(p − xk ) (p − xk ) + M 0 0 0 0 2(4π)d/2 p 0 0 × (d − 1)(p0 − xk0 )2 + 3M 2 , 1 d −1 d (x − 1) 3Γ 1 − (p0 − xk0 )2 + M 2 2 H(k, m) = d 2 2(4π) 2 p0 0 d −2 d −12Γ 2 − (p0 − xk0 )2 (p0 − xk0 )2 + M 2 2 2 d d 4 2 2 2 −3 (p0 − xk0 ) (p0 − xk0 ) + M +4Γ 3 − , 2 XZ 1 (26) (27) (28) Z d−2 Γ 1 − d2 X 1 2 2 2 dx (1 − x) (p − xk ) + M I(k, m) = , 0 0 2(4π)d/2 p 0 (29) Z d−4 Γ 2 − d2 X 1 J(k, m) = dx (1 − x)(1 − 2x)(p0 − xk0 ) (p0 − xk0 )2 + M 2 2 , d/2 2(4π) 0 p (30) 0 0 with M 2 = m2 −x(x−1)k 2 . Note that, in the zero temperature limit that is, if the summation R k2 2 and over integer n is replaced by the integral dp0 /(2π), we obtain A = − 192π 2 = −Ck B = 0 = D = E = F , which matches the known zero-temperature result [5]. 6 Now, by manipulating these expressions, we found the following relations, k 2 E(k, m) + k0 F (k, m) = 2k0 B(k, m), (31a) k 2 G(k, m) + k0 H(k, m) = 0, (31b) k 2 J(k, m) + k0 B(k, m) = 0, (31c) k 2 D(k, m) + k0 G(k, m) = −B(k, m), (31d) A(k, m) + k 2 C(k, m) + k0 E(k, m) = 2k0 J(k, m) + f (m), (31e) d−1 2 d+1 2(4π) 2 where f (m) = − (m 2) Γ 1−d 2 . From these relations, the tensor (21) can be reewriten as we see below, k ν k σ µρκλ kν kσ νσ νσ = A(k, m) g − 2 bκ kλ − B(k, m) g − 2 bi µρiλ kλ k k k0 ν k0 σ kν kσ ν σ + [F (k, m) − 4B(k, m)] µρκλ bκ kλ + f (m) 2 µρκλ bκ kλ k − u k − u 2 2 k k k i h k0 σ k0 ν ν σ µρiλ ν µρσλ σ µρνλ k − u k − u b k + I(k, m) b k + b k −H(k, m) i λ λ λ k2 k2 h i +B(k, m) −(uν bσ + uσ bν )µρλτ kλ uτ + b0 (uν µρσλ kλ + uσ µρνλ kλ ) (32) h i k0 −B(k, m) 2 −(k ν bσ + k σ bν )µρλτ kλ uτ + b0 (k ν µρσλ kλ + k σ µρνλ kλ ) , k ν σ where the structures g νσ − k kk2 and kk02 k ν − uν are gauge invariant. In fact, from the topologiΠ̃µνρσ b cal structures that contributes to generate the gravitational CS term only the term mass dependent, ν σ f (m) k kk2 , is not gauge invariant, i.e., the gauge invariance is manifest only when the fermionic mass vanishes, as we can see in the ref. [13] in the Eq. (3.11) and in the ref. [5] in the Eq. (30). To carry out the above summations, we use the expression [14] √ X πΓ(λ − 1/2) 2 2 −λ (n + b) + a = + 4 sin(πλ)fλ (a, b) Γ(λ)(a2 )λ−1/2 n (33) with Z ∞ fλ (a, b) = |a| ! dz 1 Re 2π(z+ib) . (z 2 − a2 )λ e −1 (34) This solution is valid only for λ < 1, aside from the poles at λ = 1/2, −1/2, −3/2, · · · . However, this restriction can be circumvented when we use the recurrence relation fλ (a, b) = − 1 2λ − 3 1 1 ∂2 f (a, b) − fλ−2 (a, b) λ−1 2a2 λ − 1 4a2 (λ − 2)(λ − 1) ∂b2 once, twice, and so on, until λ is placed in the range of validity. 7 By using the above considerations and also taking into account m = 0 = k0 , we obtain Z 1 d−1 β 1−d Γ 1 − d2 Γ 12 − d2 πd 2 2 dx (5 − 4x − d) M sin + A = (d+1)/2 (4−d)/2 2 4(4π) 2π 0 Z 1 Z ∞ d −1 × dx (1 − x) dζ [tanh(πζ) − 1] ζ 2 2 0 0 k2 T2 = − − , 192π 2 48 β 1−d Γ 1 − B = 4π = − F = (d − 1) sin 2− d2 πd 2 1 Z ∞ Z dζ (tanh(πζ) − 1) ζ 2 dx (1 − x) d −1 2 0 0 T2 , 48 8(4π) = 0, (36) d 2 d −1 2 Γ 1− H = d 2 (35) sin β 2π 3−d k2 1 Z Z ∞ dζ ζ d−2 tanh(πζ)sech2 (πζ) dx x(1 − x) 0 0 (37) β 3−d Γ 1 − d 2 2− d2 + πd 2 sin πd 2 8π (1 − d2 )β 1−d Γ 1 − d2 4π 2− d2 k2 1 Z dx (1 − x)2 x sin πd 2 ∞ dζ tanh(πζ)sech2 (πζ)ζ d−2 0 0 Z Z 1 Z dx (1 − x) 0 ∞ dζ (tanh(πζ) − 1)ζ d−2 0 T2 = . 12 (38) Thus, the contribution from expression (32) to CS gravitational term, which we denote as Π̃µνρσ → Π̃µνρσ b bCS , can be rewritten as k2 k ν k σ µρκλ T2 kν kσ µνρσ νσ νσ Π̃bCS = − g − 2 bκ kλ − g − 2 b0 µρκλ uκ kλ 192π 2 k 48 k T 2 k0 ν k0 σ ν σ + k −u k − u b0 µρκλ uκ kλ . 12 k 2 k2 Note that the above first term, g νσ b0 µρκλ uκ kλ , with the coefficient T2 12 , (39) is in fact the one that contributes for the conductivity of the chiral magnetic current [11]. In addition, the tensor Π̃µνρσ bCS is gauge invariant. We note that, besides of this term, we have also another term proportional to T 2 . This, however, is not unusual – one should remind that the gravitational Chern-Smons term is strongly ambiguous, and, as it is known, an ambiguity arising from the Lorentz symmetry breaking takes place both at zero temperature and finite temperature. Moreover, in principle two calculation schemes for a certain integral giving the same result at zero temperature case can yield different results at 8 the finite temperature, see f.e. [15]. Therefore, we conclude that our gravitational Chern-Simons term becomes ambiguous as the finite temperature. By returning to the coordinate space through the replacement, k µ → i∂ µ , we obtain Z 1 T 2 µρκλ ∂ν ∂σ (2) µρκλ ν ν σ 4 ν bκ ∂λ (hρ − ∂ ∂ hρσ ) + SCS [h] = d x hµν − b0 uκ ∂λ hρ − hρσ 192π 2 48 T 2 µρκλ ∂0 ∂ σ ∂0 ∂ ν ν σ − b0 . (40) uκ ∂λ −u hρσ − u hρσ 12 We note that the nonlocality of our result is a price we pay for the gauge invariance. The similar situation was found to occur for the one-derivative term in [12]. Concerning the impact of the modified value for the gravitational Chern-Simons term for the conductivity, we note the following. The complete result is presented by expression (40), and only the u0 component of the vector uµ differs from 0. To extract the physical content of the extra terms, we take into account only the transversal components of hij keeping all other components to be zero, cf. [4], which we can do since the additive terms are gauge invariant. Afterwards, we see that all terms proportional to uµ hµν vanish since h0µ is kept to be zero, and due to the transversality we can put ∂µ hµν ' ∂i hij δ jµ = 0. As a consequence, the transversal part of the last term of (40) which introduces difference from the result found in [11] completely vanishes. Let us discuss our results. We find that the four-dimensional gravitational CS term turns out to be finite despite the initial expression for it strongly (cubically) diverges. As it is well known, in a Lorentz-breaking theory such a behavior can signalize about ambiguity of the result. Just this situation occurs in our case: while in the zero temperature limit, our result matches that one obtained in [5], at the finite temperature we have, besides of the term discussed in [11], one more term which is also proportional to T 2 and hence can naturally be treated as a consequence of ambiguity of the gravitational Chern-Simons term. We note that in the case of the presence of an ambiguity, the results found to coincide at zero temperature within two calculation schemes, can be different within these schemes at the finite temperature. At the same time, our results can be interpreted in one more way – introducing the finite temperature can be naturally interpreted as one more regularization, and while the cancellation of divergences is common in different calculation schemes, the finite parts can differ. However, the physical (transversal) parts of our result and the result obtained in [11] are the same since the transversal part of the extra term of (40) completely vanishes, therefore the physical values, in particular the axial magnetic conductivity, exactly coincide. Also, we note that our result for the gravitational CS term was 9 obtained in the zero mass limit, while for the non-zero mass the gauge symmetry is broken. This matches the zero-temperature results [5, 12]. We conclude that our result is consistent with the previous studies. [1] D. Colladay and V. A. Kostelecky, Phys. Rev. D 55, 6760 (1997) [hep-ph/9703464]. [2] D. Colladay and V. A. Kostelecky, Phys. Rev. D 58, 116002 (1998) [hep-ph/9809521]. [3] V. A. Kostelecky, Phys. Rev. D 69, 105009 (2004) [hep-th/0312310]. [4] R. Jackiw and S. Y. Pi, Phys. Rev. D 68, 104012 (2003) [gr-qc/0308071]. [5] T. Mariz, J. R. Nascimento, E. Passos and R. F. Ribeiro, Phys. Rev. D 70, 024014 (2004) doi:10.1103/PhysRevD.70.024014 [hep-th/0403205]. [6] T. Mariz, J. R. Nascimento, A. Y. Petrov, L. Y. Santos and A. J. da Silva, Phys. Lett. B 661, 312 (2008) doi:10.1016/j.physletb.2007.10.089 [arXiv:0708.3348 [hep-th]]. [7] M. Gomes, T. Mariz, J. R. Nascimento, E. Passos, A. Y. Petrov and A. J. da Silva, Phys. Rev. D 78, 025029 (2008) doi:10.1103/PhysRevD.78.025029 [arXiv:0805.4409 [hep-th]]. [8] R. Jackiw and V. A. Kostelecky, Phys. Rev. Lett. 82, 3572 (1999) doi:10.1103/PhysRevLett.82.3572 [hep-ph/9901358]. [9] J. M. Chung, Phys. Rev. D 60, 127901 (1999) doi:10.1103/PhysRevD.60.127901 [hep-th/9904037]. [10] R. Jackiw, Int. J. Mod. Phys. B 14, 2011 (2000) doi:10.1142/S021797920000114X [hep-th/9903044]. [11] M. N. Chernodub, A. Cortijo, A. G. Grushin, K. Landsteiner and M. A. H. Vozmediano, Phys. Rev. B 89, no. 8, 081407 (2014) doi:10.1103/PhysRevB.89.081407 [arXiv:1311.0878 [hep-th]]. [12] A. F. Ferrari, M. Gomes, J. R. Nascimento, E. Passos, A. Y. Petrov and A. J. da Silva, Phys. Lett. B 652, 174 (2007) doi:10.1016/j.physletb.2007.07.013 [hep-th/0609222]. [13] L. Bonora, M. Cvitan, P. Dominis Prester, B. Lima de Souza and I. Smolic, JHEP 1605 (2016) 072 doi:10.1007/JHEP05(2016)072 [arXiv:1602.07178 [hep-th]]. [14] L. H. Ford, Phys. Rev. D 21, 933 (1980). [15] T. Mariz, J. R. Nascimento, A. Y. Petrov and W. Serafim, Phys. Rev. D 90, no. 4, 045015 (2014) doi:10.1103/PhysRevD.90.045015 [arXiv:1406.2873 [hep-th]]. 10
