NLO radiative correction to the Casimir energy in Lorentz-violating scalar field theory

Violation of the Lorentz symmetry has important effects on physical quantities including field propagators. Therefore, in addition to the leading order, the sub-leading order of quantities may be modified. In this paper, we calculate the next to leading (NLO) radiative corrections to the Casimir energy in the presence of two perfectly conducting parallel plates for $\phi^4$ theory with a Lorentz-breaking extension. We do the renormalization and investigate these NLO corrections for three distinct directions of the Lorentz violation; temporal direction, parallel and perpendicular to the plates.


I. INTRODUCTION
The Casimir effect which is a physical manifestation of changes in the quantum vacuum fluctuations for different configurations, was discovered by H. B. G. Casimir in 1948 [1]. He showed the existence of this effect as an attractive force between two infinite parallel uncharged perfectly conducting plates in vacuum (for a general review on the Casimir effect, see Refs. [2,3]). Sparnaay [4] and Arnold et al [5] experimentally observed the Casimir force for such a configuration. Also, the other measurements, with greatly improved precisions, have been done for various geometries [6][7][8].
In addition to the leading Casimir energy, the next to leading order (NLO) radiative corrections to this effect is an exciting subject of discussion. The first endeavors to calculate the leading radiative corrections to the Casimir energy were reported in [9]. Also, many works on the radiative corrections to the Casimir energy for various cases exist in the literature (see for instance [10,11]). In the case of a real massive scalar field, NLO correction to the Casimir energy has been computed in [2,12]. We have also calculated one loop radiative corrections to the Casimir energy in [13].
In original quantum field theory (QFT), the Lorentz symmetry is preserved. However, there are some theories which present models with Lorentz symmetry violation (for example [14,15]). Naturally, Lorentz symmetry violation arises from, for example, existence of space-time anisotropy [16,17] or non-commutativity [18,19] or a spacetime varying coupling constant [20,21]. Investigations of Casimir effect with Lorentz-breaking symmetry for QED theory have been done (see please [22][23][24]). It has also been studied recently for a real massive scalar field in [25].
In this paper we calculate the NLO correction to the Casimir energy in an interacting scalar field theory, λφ 4 , with a Lorentz violating term. Our configuration is two perfectly conducting parallel plates. We work within the renormalized perturbation theory, therefore we need to reconsider the renormalization for this theory. Naturally, the counterterms needed for renormalization, are modified due to the existence of new Lorentz violating terms in the Lagrangian.
To take the physical result and resolve infinities problem, we use a well-known approach called Boyer method [26]; also is known as Box Renormalization Scheme (BRS). This method uses a completely physical approach by enclosing the whole system in a box of volume V = L 3 which finally may tend to infinity in such a way that difference between the zero point energies of two different configurations is calculated. It removes all ambiguities associated with appearance of the infinities without resorting to any other schemes such as analytic continuation approach. It is notable that, in BRS the substraction precedure in calculation of Casimir energy takes place in two physical configuration with similar nature, which is another advantage of BRS.
We organized our paper as follows: We introduce our model for Lorentz-breaking symmetry of the theory in section II . We shall see that energymomentum tensor and Klein-Gordon (KG) equation is modified. In section III, we survey renormalization of the related theory within a Lorentz-braeking case. In section IV we calculate the NLO radiative correction to the Casimir energy for φ 4 theory with Lorentz-breaking symmetry. We note that at this stage we consider the existence of Lorentz-symmetry parameter in two cases: 1. time-like (TL), and 2. space-like (SL). Finally, in last section we state our conclusions.

A. The Model
In this section, we present the Lorentz symmetry breaking for a scalar field theory due to an anisotropy of space-time. We do this by insertion an additional term in the KG Lagrangian density where m 0 is the bare mass and the dimensionless parameter c, which is much smaller than one, manifests the Lorentz symmetry breaking of the system by a coupling between the derivative of the scalar field φ and a constant four-vector u µ . Adding a self-interaction term to Eq. (1) we get where λ 0 is our bare coupling. The equation of motion for Lagrangian (1) reads as It is obvious that this modified KG equation, for c = 0 reverts to the original KG equation of motion with the following dispersion relation: The violation of Lorentz symmetry has vital consequences such as modification of dispersion relation which directly affects the propagator of the field. We consider this effect in three different cases. In the first case we assume that the Lorentz violation is in the time direction. The second and third are the SL Lorentz violations in the directions parallel (pl-SL) and perpendicular (pr-SL) to the plates.

B. Propagator in Bounded Space
To calculate radiative corrections to any physical quantity, including Casimir energy, we need to know the exact form of propagator. In this subsection we first derive the propagator, suitable for Casimir effect problem, in the context of standard quantum field theory (without any Lorentz-violating term). Our configuration is two parallel plates located at z = ±a/2 perpendicular to z-axis with a separation a. We suppose the fields satisfy Dirichlet boundary conditions (DBCs) on the plates, Being d the dimension of space-time, the field φ is defined with quantized modes as where k ⊥ and k n = nπ a denote the momenta parallel and perpendicular to the plates, respectively. Here, a † n and a n are creation and annihilation operators, respectively, with the following commutation relations: [a n , a † n ] = δ n,n , [a n , a n ] = [a † n , a † n ] = 0, and a|0 = 0 defines the vacuum state in the presence of boundary conditions. One may easily find Feynman Green's function of the KG equation as We then find Euclidean Green's function by the following definitions: which finally leads to (we need only G F (x, x) in our calculations)

TL vector case
Choosing the four-vector to be TL, u µ = (1, 0, 0, 0), the second term in Eq. (3) becomes c ∂ 2 0 . Hence, the dispersion relation (4) takes the form Therefore, we can find the propagator for this case by replacing where k = (ω , k ⊥ E ) . Performing the angular integration, finally we have where the solid angle , with Γ(x) being the Gamma function, corresponds to the area of a unit sphere in d dimensions.

SL vector case
In SL case we choose three distinct directions for four-vector u µ ; u µ = (0, 1, 0, 0), u µ = (0, 0, 1, 0) and u µ = (0, 0, 0, 1). In this case the Lorentz-breaking term in (3) is −c∂ 2 i with i = x, y or z. There is no difference between the physics of the first two vectors (pl-SL case), which are parallel to the plates, and the dispersion relations for both cases are also the same. Choosing u µ = (0, 0, 1, 0) for instance, Eq. (4) becomes Changing the variables k = (ω, k ⊥ E ) with k y = √ 1 − c k y , in a similar manner to the TL case, the Green's function is derived as: Now, for the last case (pl-SL), u µ = (0, 0, 0, 1) is normal to the plates and Eq. (4) becomes In this case, the Euclidean Feynman propagator is derived as where k n = nπ/a with a = a/ √ 1 − c. For the future use, in the case of free space without plates, we note that the propagator for a Lorentz symmetry breaking theory becomes where + (−) is used for TL (SL) vector case.

III. RENORMALIZATION UP TO ORDER λ
At the level of quantum corrections, all unphysical quantities such as m 0 and λ 0 need to be renormalized. Therefore, we need to do a renormalization procedure to extract the physical m and λ from the bare parameters m 0 and λ 0 (see [27]). Here, we work within the standard renormalized perturbation theory. In the Lagrangian (2), after rescaling the fields by a field strength renormalization Z, namely φ = Z 1 2 φ r we have where δ m = m 2 0 Z − m 2 , δ λ = λ 0 Z 2 − λ and δ Z = Z − 1 are the counterterms. Then, we have two new Feynman rules from the above Lagrangian where + (−) along with µ = 0 (µ = i) are used for TL (SL) vector case (for more details see [28]). The counterterms are totally fixed by two renormalization conditions: = i p 2 − m 2 + (terms regular at p 2 = m 2 ).
From the first renormalization condition it is obvious that δ λ = O(λ 2 ). The second renormalization condition which gives the physical mass m, up to order λ, can be written as where we have used Eq. (17). Therefore, δ Z up to order O(λ) is zero, and

IV. RADIATIVE CORRECTION TO THE CASIMIR ENERGY
In order to calculate the radiative correction to the Casimir energy we use box renormalization scheme (BRS) [26]. In this approach, we first compare the energies in two various configurations: when the plates are at ±a/2 as compared to ±b/2. We confine each configuration in a box with edges are located at ±L/2 in all directions (see figure 1). Now, the Casimir energy is defined as where, The radiative corrections to the zero point energy in the (for example) a1 part, i.e. z ∈ [ −a 2 , a 2 ], are where |Ω is the vacuum state in the presence of interaction. Up to order λ we have where G a1 (x, x) is the propagator of the real scalar field in region a1 (we drop the subscript 'F' for simplicity).

A. TL & pl-SL vector cases
To calculate the first term in Eq. (26), E (1),F a1 , using Eqs. (12) and (22) and carrying out the spatial integration, one obtains the correction to the vacuum energy in region a1, up to O(λ), as: where k a1,n = nπ a . Integrating over momentum k yields where ω a1,n = (m 2 + k 2 a1,n ) 1/2 . This is one of the four terms (related to the a1 region) that contribute to the NLO radiative correction for Casimir energy Eq. (23). To derive the Casimir energy from Eq. (28), we apply Abel-Plana summation formula [29], with, We note that the g(0) term vanishes. Also the second term on the right hand side of Eq. (29), with respect to suitable changing of variables in the four integrals below, vanishes: Finally, we calculate branch-cut terms in Eq. (29).
where K n (x) is the modified Bessel function of order n. To calculate the integral, we have used the identity Therefore, we obtain wherem = ma is a dimensionless parameter. Then, according to Eq. (23) the contribution of the Eq. (28) to Casimir energy is Taking the limits, only the first term survives. Finally, we take the limit d → 4, where K q (x) = ∂ ∂q K q (x) and γ is the Euler-Mascheroni number. The contribution of the second term in Eq. (26) to the Casimir energy, E (1),S a1 , without Lorentz violating terms, have been calculated in Ref. [11] using BRS: But, when we have a TL (pl-SL) Lorentz breaking term, an extra factor 1 √ 1+c ( 1 √ 1−c ), as we see in Eq. (12) (Eq. (14)), is multiplied to the propagator. Therefore to derive the Casimir energy contribution we only need to multiply the factor 1 1+c ( 1 1−c ) to Eq. (38). Accordingly, using Eqs. (38) and (36), we can write NLO radiative correction to the Casimir enegy as From this result it is obvious that the influence of the Lorentz-symmetry breaking parameter appears only in a factor. Two special limits are interesting to calculate; the large mass ma 1, and small mass m → 0 limits: with + (−) for TL (pl-SL) case.

B. pr-SL vector case
For the pr-SL vector case, u µ = (0, 0, 0, 1), we do not need to do new calculation. In this case, applying Eq. (16) leads us to the following expression for Eq. (28): where ω a1,n = (m 2 + k 2 a1,n ) 1/2 . Therefore, the Eq. (32) becomes and hence, we get Now, we use the above equation to compute Eq. (35), and take the limit d → 4, to get Similary, for the second term in Eq. (26), now the Eq. (37) becomes Therefore the result for the radiative correction of Casimir energy for the pr-SL vector case can be written as We can also compute the large mass and massless limits: In figure 2, we have illustrated the variation of the ratio between the first order radiative corrections and leading terms, E (1) Cas. /E (0) Cas. , in terms of plates separation, for three distinct cases TL, pl-SL and pr-SL. We have also plotted this ratio in terms of Lorentz violating parameter c in figure 3.

V. CONCLUSION
In this paper we have calculated the next to leading order radiative correction to the Casimir energy for φ 4 theory with Lorentz-breaking symmetry in the context of renormalized perturbation theory. Our approach to calculate this energy is box renormalization method introduced firstly by Boyer [26] and used for example in [11,[30][31][32]. The  violation of symmetry breaking can be appeared in the Lagrangian by insertion of a term which couples the derivative of a field to a constant vector u µ . This additional term in the Lagrangian modifies the dispersion relation and accordingly propagators of the fields. Therefore, in addition to the leading terms of physical quantities, all their subleading corrections are also affected. In three separate cases of the Lorentz violation, violation in the time direction (TL), in the directions parallel (pl-SL) and perpendicular (pr-SL) to the plates, the leading terms of Casimir energy for φ 4 theory have been recently calculated in [25]. Here, we have investigated NLO corrections. We have plotted our