Pauli-Villars regularization of Kaluza-Klein Casimir energy with Lorentz symmetry

The Pauli-Villars regularization is appropriate to discuss the UV sensitivity of low-energy observables because it mimics how the contributions of new particles at high energies cancel large quantum corrections from the light particles in the effective field theory. We discuss the UV sensitivity of the Casimir energy density and pressure in an extra-dimensional model in this regularization scheme, and clarify the condition on the regulator fields to preserve the Lorentz symmetry of the vacuum state. Some of the conditions are automatically satisfied in spontaneously-broken supersymmetric models, but supersymmetry is not enough to ensure the Lorentz symmetry. We show that the necessary regulators can be introduced as bulk fields. We also evaluate the Casimir energy density with such regulators, and its deviation from the result obtained in the analytic regularization.


Introduction
The Casimir effect is a macroscopic quantum effect that has been observed in various experiments and the observed values are in good agreement with theoretical predictions [1][2][3][4][5].The Casimir energy is defined as the energy difference between the vacuum energy in a compact space, such as a space enclosed by conducting plates, and that in a non-compact space.The vacuum energy in quantum field theory (QFT) is generally divergent and must be regularized, such as the cutoff regularization, in which the cutoff scale Λ cut is set for the momenta of virtual particles in the loops.It is well-known that the Casimir energy remains finite even in the limit of Λ cut → ∞.The scale Λ cut is regarded as a scale at which the theory under consideration breaks down and is replaced with a more fundamental theory.The Casimir effect also plays an important role in extra-dimensional models.The quantum correction for the extra-dimensional models is Kaluza-Klein (KK) Casimir energy, which depends on the compactification scale m KK and determines the physical properties of the extra-dimensional models [6][7][8][9].Since the extra-dimensional models are nonrenormalizable, they should be regarded as effective theories of more fundamental ones, such as a string theory or quantum gravity (QG).Hence Λ cut can not be infinite, and may be close to m KK .In the latter case, the unknown ultraviolet (UV) physics can affect the KK Casimir energy.This indicates that the Casimir energy in the extra-dimensional models have regularization dependence [10][11][12][13][14][15], in contrast to the case of renormalizable theories, in which we can safely take the limit Λ cut → ∞.In particular, one of the authors suggests the Casimir energy receives a large correction from the UV physics when Λ cut is not far from m KK in the cutoff regularization scheme [15].
In 3 + 1-dimensional QFT, there is a significant discussion regarding the Lorentz symmetry violation in the regularization of vacuum energy.Indeed, when utilizing the cutoff regularization, the UV divergences break the Lorentz symmetry [16][17][18].If this Lorentz symmetry violation is considered as an actual physical phenomena, they could lead to significant cosmological issues [19].When we consider the Friedmann-Lematre-Robertson-Walker (FLRW) universe with the metric ds 2 = dt 2 − a 2 (t) δ ij dx i dx j , the semiclassical Friedmann equations for a flat universe with a vacuum state are given by where Λ cc is the cosmological constant, the hat denotes an operator, ρ is the energy density, p is the pressure, the dot denotes the time derivative, and H ≡ ȧ/a is the Hubble parameter.A combination of these equations leads to In the cutoff regularization, we have where δ if is the Kronecker delta with i = b, f for bosons and fermions respectively, and m is the mass.This shows that the UV divergences directly contribute to the dynamics of the universe. 1On the other hand, we should note that (1.4) for the fermionic contribution clearly violates the null energy condition (NEC).The NEC is known as a necessary condition to eliminate any pathological spacetime or unphysical geometry [21,22] and it states T µν n µ n ν ≥ 0, for any null light-like vector n µ .This is summarized as ρ + p ≥ 0 for the FLRW metric.In the context of the vacuum energy of the quantum fields and its regularization, there exist issues related to the breaking of Lorentz symmetry and the violation of the NEC.
In this paper, we explore the KK Casimir energy density and pressure from compact dimension.We particularly study the UV sensitivity of the KK Casimir energy.As we will show in the next section, the analytic regularization inherently omits the UV contributions, and the cutoff regularization violates the Lorentz symmetry in the vacuum state.Therefore, we adopt the Pauli-Villars regularization, which effectively demonstrates the cancellation of large quantum corrections by the contributions of high-energy virtual particles in the effective field theory.We further specify the necessary conditions on the regulator fields to preserve Lorentz symmetry. 2Although spontaneously-broken supersymmetric (SUSY) models satisfy some of these conditions, SUSY is not enough to ensure the Lorentz invariance of the vacuum state.
The rest of this paper is organized as follows.In Section 2, we review the analytic and cutoff regularizations of the KK Casimir energy density and pressure.We point out that these regularizations are not adequate to evaluate the UV sensitivity of the Casimir energy preserving the Lorentz symmetry.In Section 3, we consider the Pauli-Villars regularization to regularize the Casimir energy density, and provide the necessary conditions for regulator fields to preserve Lorentz symmetry.In Section 4, we numerically calculate the Casimir energy density and pressure in the Pauli-Villars regularization, and evaluate their dependence on the UV regulator mass scale.In Section 5, we conclude our work.

Regularizations
We take the following semiclassical treatment [24], which approximately combines QFT and general relativity (GR), and is expected to be reliable under conditions where QG is not important.We treat spacetime classically and use the expected value of the quantized stress-energy tensor in Einstein's equations.Hence, the quantum effect of matter fields on spacetime geometry can be approximately described by the Lorentz violation by dark energy can be formalized by the following expression: (1.5) Although some results suggest a slight phantom-like equation of state, w dark ≃ −1.03, several independent observations are broadly consistent with the cosmological constant value of w dark = −1.013+0.038 −0.043 [20].Thus, the vacuum must preserve the Lorentz symmetry with the accuracy, ρ dark + p dark ≲ O(10 −2 )(10 −3 eV) 4 .
semiclassical equations,3 where G µν is the Einstein tensor, Λ cc is the cosmological constant and ⟨T µν ⟩ is the expected value of the quantum stress-energy tensor.Phenomenologically, such treatment will suffice. 4t is known that the (quantum) vacuum is Lorentz invariant to a high accuracy from the observation [38,39].Therefore, the vacuum energy density ρ must give rise to an energy-momentum tensor in the 4D Minkowski spacetime of the form, where η µν = diag (−1, 1, 1, 1) is the Minkowski metric, and thus the quantum correction to the vacuum energy density is renormalized by the cosmological constant Λ cc .Note that (2.2) indicates that where p ≡ T vac 11 = T vac 22 = T vac 33 is the vacuum pressure.Therefore, the LHS of (2.3) measures the violation of the Lorentz symmetry.

Formal expressions for energy density and pressure
To simplify the discussion, we consider a real scalar theory in a flat 5-dimensional spacetime, and one of the spatial dimensions is compactified on S 1 /Z 2 .
where µ = 0, 1, • • • , 4, and M bulk is a bulk mass parameter.The coordinate of the compact dimension is denoted as y ≡ x 4 .The fundamental region of S 1 /Z 2 is chosen as 0 ≤ y ≤ πR, where R is the radius of S 1 .The real scalar field Φ is assumed to be Z 2 odd.Then the KK masses are given by The vacuum energy density and the vacuum pressure in the 4D effective theory are formally expressed as (2.6) These obviously diverge, and we need to regularize them.In the following, we review the analytic and the momentum-cutoff regularizations, and mention unsatisfactory points for our purpose.To make the relation between them clear, we introduce the cutoff for the KK mode number N cut , the momentum cutoff Λ cut and the complexified dimension d.Then, (2.6) is regularized as where µ is some scale to adjust the mass dimension.Naively, the cutoff scales for the 3D momentum and the fifth one are expected to be common.Thus we assume that m Ncut ≃ Λ cut , or more specifically Performing the ⃗ k-integral, we obtain where Γ(α) is the Euler gamma function, B z (α, β) is the incomplete beta function, and (2.10)

Review of conventional derivation
The most popular regularization scheme for the calculation of the Casimir energy is the combination of the dimensional regularization and the zeta-function regularization, which we call analytic regularization in this paper.
Thus, (2.9) become The infinite sum over the KK modes is evaluated by the zeta-function regularization technique [8,40,41].Using the formula (B.10) with (B.11) in Appendix, the energy density is expressed as where Mbulk ≡ RM bulk , and K α (z) is the modified Bessel function of the second kind.The first term diverges as d → 3, but it does not depend on R and is irrelevant to the stabilization of the extra dimension.Thus we simply neglect it.We require that the vacuum energy density in the decompactified limit R → ∞ vanishes [42].Thus the Casimir energy density, which is a function of R, is defined as Note that the subtraction should be performed for the 5D energy density since the second term is the quantity in the decompactified limit.Then, the second term in (2.13) is cancelled, and we obtain (2.15) We have taken the limit d → 3 at the last step.Similarly, the vacuum pressure is calculated as In the massless case M bulk = 0, (2.15) reduces to the well-known form, (2.18) Thus, the Lorentz symmetry and NEC are both preserved in this regularization.5

Cutoff sensitivity in analytic regularization
Although the formula (2.15) or (2.16) is useful because of its rapid convergent property, the analytic continuation processes make it difficult to see how the divergent terms are removed.It is well-known that this regularization only captures the logarithmic divergences, and is insensitive to the power-law divergences of Λ cut .To see the situation, let us review the procedure we have performed in (2.11) in more detail.
As long as Λ cut is kept finite, the incomplete beta functions in (2.9) are well-defined for any values of the dimension d.Before taking the limit Λ cut → ∞, let us consider a case that d < −1.Then, using (A.6) in the Appendix, the incomplete beta functions are expanded as Since all the powers in RHS are positive for d < −1, we can safely take the limit Λ cut → ∞ (i.e., ϵ n → 0), and drop all ϵ n -dependent terms.After dropping them, we can move d to a value close to 3.This is what we have done in (2.11).However, if we keep the ϵ n -dependent terms when we move d to a value close to 3, the second and the third terms in RHS of (2.19) have negative powers, and correspond to the quartic and quadratic divergences, respectively. 6Therefore, what we have done in (2.11) is just dropping the quartic and quadratic divergent terms by hand.
A similar prescription has been performed when we apply the zeta-function regularization for the infinite sum over the KK modes.If we keep the cutoff Λ cut finite, the incomplete functions in (2.9) depend on the KK level n, and cannot be factored out from the summation over n.Therefore, it is not easy to perform the exact calculation of (2.9).Hence we investigate the following expression instead.
where a ≡ 1/N cut is a tiny positive constant.Instead of the sharp cutoff at n = N cut , we introduce the damping factor e −a 2 n 2 , which suppresses the contribution of heavy KK modes with m n > Λ cut . 7Then, (2.20) is rewritten as where U (α, β; M 2 ) is defined in (B.1) in Appendix.According to the expression (B.3) with (B.4) and (B.12), this has the following terms.
where C i (β; M 2 ) (i = 1, 2, 3) are defined in (B.13), and and the ellipsis denotes terms that appeared in (2.13) and irrelevant terms that will vanish in the limit of Λ cut → ∞ when d = 3.In the limit of d → 3 keeping Λ cut finite, the terms shown in (2.22) represent power-law divergent terms up to quintic in Λ cut .This is expected because we are considering the 5D theory.In the derivation of (2.13), we have taken the limit of Λ cut → ∞ for d < −2, where all terms shown in (2.22) vanish.However, this treatment is equivalent to just dropping those terms by hand.Therefore, the analytic regularization is inappropriate for studying the UV sensitivity of the Casimir energy density or pressure.

Cutoff regularization
Next, we consider the cutoff regularization.Take the limit d → 3, keeping Λ cut finite, in (2.9).Then we obtain (2.24) The sum of the vacuum energy density and pressure is where the logarithmic terms exactly cancel but the cut-off divergences remain.Namely, the Lorentz symmetry is violated in this regularization.If Φ is replaced with a 5D fermion, an overall minus sign appears in the above expressions.Hence NEC is also violated in that case.
For a light mode with m n ≪ Λ cut , its contribution to the vacuum energy and the vacuum pressure can be expanded as

.26)
After summing over the KK modes, the leading terms of where N cut is defined in (2.8).Since these are proportional to R, they are canceled in the Casimir energy density and pressure defined in (2.14) and (2.16), respectively.However, the other terms remain and violate the Lorentz symmetry.Thus the cutoff regularization is considered to be problematic for the calculation of the Casimir energy [16][17][18].
From the physical point of view, contributions of massive KK modes near the cutoff scale Λ cut should be suppressed by UV physics.In the previous work [15], we introduced a damping function, such as or where A ≳ 10 is a positive constant that controls the steepness around the cutoff scale, and inserted it into the expression (2.9) as Then we obtain a finite value for the Casimir energy density, which agrees with the value obtained by (2.16). 8The cutoff regularization considered in this subsection corresponds to the limit of A → ∞ in (2.29).It is known that the regularization with such a sharp cutoff provides a divergent Casimir energy density, and should not be applied to the calculations for the Casimir energy density and pressure [15].

Pauli-Villars regularization
As mentioned in Sec.2.3, the contributions of massive KK modes near Λ cut should be suppressed by the UV physics, such as contributions of new particles with masses of O(Λ cut ).Such contributions can be mimicked by the Pauli-Villars regulators.However, a single regulator that has opposite statistics and a large mass M reg is not enough to suppress contributions of the KK modes heavier than M reg . 9Hence, for each KK mode with mass m n , we introduce k species of regulators.Then, its contributions to the Casimir energy density and pressure are modified as where ρ and p are defined in (2.26), M i and an integer c i denote the mass and the degree of freedom for the i-th regulator, respectively.We assume that all M i (i = 1, 2, • • • , k) are of O(M reg ).Note that we introduce both bosonic (c i < 0) and fermionic (c i > 0) regulators.Then, using the expanded expressions in (2.26), we have If we require the integers c i (i = 1, 2, • • • , k) to satisfy [17,23] 10 the Lorentz-violating terms are canceled, and obtain in the limit of Λ cut → ∞.Hence we have The first condition in (3.3) is the requirement of the balance between the bosonic and fermionic degrees of freedom.The second one has the same form as the supertrace mass formula in a model that has spontaneously broken supersymmetry (SUSY) [43].Namely, the first two conditions in (3.3) are automatically satisfied in such a model.To preserve the Lorentz symmetry, however, the third condition is also necessary.It is intriguing to discuss the possibility of constructing a SUSY model in which all conditions in (3.3) are satisfied [23].
To suppress the contributions of the massive KK modes heavier than M reg , we should also require that This is rewritten as In the case of we can solve (3.3), and obtain For c 2 = 3, we have 4 bosonic and 4 fermionic degrees of freedom in total and can be embedded into a chiral multiplet in a (spontaneously broken) SUSY model.In the following, we consider the case of (3.8) with c 2 = 3 as a specific example.We assume that M 2 i (i = 1, 2, 3) are functions of m 2 n and M 2 reg .In solving (3.7), we are interested in the KK modes with m n ≫ M reg .Thus, we expand M 2 1 as where δ ≡ M 2 reg /m 2 n .Using this expression and (3.9), we can expand the LHS in (3.7) as where the coefficients and C 3 automatically vanish, and do not give any constraints on α, β 1 and β 2 .The coefficient C 4 is a function of only α, and the solution of C 4 = 0 is α = 1.Under the condition α = 1, we can easily see that both C 5 and C 6 vanish identically.Therefore, we can take β 2 = 0, and assume that as a solution to (3.7).If we rescale M 2 reg , we can always set β 1 = 1.As a result, we can choose a solution of (3.7) (and (3.3)) as This result indicates that the regulators can be regarded as the KK modes for 5D bulk fields.In fact, if we introduce one fermionic 5D field with the (squared) bulk mass M 2 bulk + M 2 reg , three fermionic 5D fields with M 2 bulk + 1 3 M 2 reg , and three bosonic 5D fields with M 2 bulk + 2 3 M 2 reg , the conditions (3.3) and (3.6) are satisfied for each KK mode.
Before ending this section, we comment on the relation to analytic regularization.In that regularization, ρ n and p n are read off from (2.12) as where γ E is the Euler-Mascheroni constant.We have used (A.3) at the last equality.
After the minimal subtraction, we have To match (3.2) with this result, a further additional condition has to be imposed [17].
Therefore, the number of the regulator species has to be chosen as k ≥ 4.Then, (3.2) agrees with (3.16) in the limit of Λ cut → ∞.However, we do not have a simple solution of (3.3) when k ≥ 4. For example, if we assume that k = 4 and c 3 = −c 4 , we obtain from (3.3) where Plugging this into (3.17) and solving it, we can express M 2 2 in terms of M 2 1 in principle.As a result, M 2 i (i = 2, 3, 4) can be expressed as functions of M 2 1 and m 2 n .However, we do not have analytic expressions for them in general.
As we will see in the next section, even if the condition (3.17) is not imposed, the result well agrees with the one obtained in the analytic regularization (2.15) as long as

Regulator-mass dependence of Casimir energy
In this section, we will numerically calculate the Casimir energy density and pressure in the Pauli-Villars regularization, and evaluate their dependence on the regulator mass scale M reg .As a specific example, we choose the regulator masses as (3.14).In this case, the energy density and pressure for the vacuum are expressed as where a ≡ (M reg R) −1 , and where 11 In order to evaluate ∆(a), the Euler-Maclaurin formula is useful [44][45][46].Then we obtain where B 2p are the Bernoulli numbers, q is an integer greater than 1, and q! a q F (q) (ax), (4.7) 11 We have used that See Sec.3.3 of Ref. [15] for details. with the Bernoulli polynomial B q (x).At the last step in (4.6), we have used that Here we set q = 2.Then, noting that F (1) (0) = 0 from (C.1), (4.6) becomes where the explicit form of F (2) (x) is shown in (C.1) in Appendix, and To see the deviation of the Casimir energy (4.3) from the one obtained in the analytic regularization (2.15), we define where ⟨0 | ρ | 0⟩ anal Casimir denotes (2.15).Fig. 2 shows the ratio r cas as a function of a = m KK /M reg , where m KK ≡ 1/R is the KK mass scale.We can see that the result obtained by the Pauli-Villars regularization well agrees with that of the analytic regularization as long as the compactification scale m KK is well below the regulator mass scale M reg .
Before ending the section, one comment is in order.The above results can also be expressed by using the analytic regularized formula (2.15).As mentioned below (3.14), the current choice of the Pauli-Villars regulators can be understood as 5D fields.Thus, the Casimir energy density in (4.1) is also expressed as where Thus, (4.11) can be rewritten as where Mbulk = RM bulk , Mreg = RM reg , and The function ∆( M1 , M2 ) is exponentially suppressed when M1 ≪ M2 , but becomes non-negligible when M2 = O( M1 ).Since the infinite summation in (4.13) or (4.15) converges much faster than the KK summation, this expression is convenient to the numerical computation.

Discussions and Conclusions
We studied the dependence of the Casimir energy density on the UV dynamics in the context of a 5D model with a compact dimension.In contrast to renormalizable theories, a non-renormalizable theory, such as our 5D model, should be regarded as an effective theory, and be replaced by a more fundamental theory at some high energy scale M UV .A typical situation is that some new particles appear at a scale around M UV , and cancel quantum corrections from the light fields in the 5D effective theory.If M UV is not far from m KK , the existence of the new particles can affect lowenergy observables, such as the Casimir energy density.We have evaluated such effects on the Casimir energy density (and pressure).The most popular way of calculating the Casimir energy is the method using the analytic regularization because the resultant expression is convenient for the numerical evaluation and the regularization preserves various symmetries, including the Lorentz symmetry.However, this regularization removes the power-law divergences by hand, and thus is inappropriate for our purpose, as we showed in Sec.2.2.2.Instead of this, we work in the Pauli-Villars regularization, which mimics the situation that new particles cancel the quantum corrections from the light particles.To preserve the Lorentz symmetry of the vacuum, we have to prepare more than one regulator for each mode, and their masses and the degrees of freedom have to satisfy some conditions (see (3.3) and (3.6)).It should be noticed that two of them are automatically satisfied in a (spontaneously broken) SUSY model.The result in (3.14) indicates that the Pauli-Villars regulators can be regarded as the KK modes for 5D bulk fields.In a case that the model is embedded into a (spontaneously broken) SUSY 5D theory, the scalar field Φ and the bulk regulators should be embedded into a 5D SUSY multiplet.Thus, the example of the regulators considered in Sec. 3 must be modified.Needless to say, the deviation from the result in the analytic regularization depends on the choice of the Pauli-Villars regulators.Still, our example shows a typical order of magnitude for the deviation.
If we do not impose the condition (3.17), it is not guaranteed that the resultant Casimir energy density (or pressure) agrees with the one obtained by the analytic regularization.We numerically evaluate them and confirm that they well agree with each other even if (3.17) is not satisfied, as loong as the KK mass m KK and the bulk mass M bulk are smaller than all the regulator masses.

B Formulae for zeta function regularization
In order to evaluate the regularized sums in (2.9), we define where a is a tiny positive constant, and Using the formula (A.8), this is expressed as Here note that S δ (β; M 2 ) can be rewritten as ) where ζ(s) is the Riemann zeta function.From (2.15) and (2.16), we can see that the sum of the KK Casimir energy density and pressure are exactly zero, ⟨0 | ρ + p | 0⟩ Casimir = 0 .

. 2 )Fig. 1 Figure 1 .
Fig.1shows the profile of the function F (x) for various values of Mbulk .We can see that the contribution of the KK modes damps around x = 1, which corresponds to the regulator mass scale M reg .According to(2.14), the Casimir energy and pressure are given by