Casimir Energy in (2 + 1)-Dimensional Field Theories

We explore the dependence of vacuum energy on the boundary conditions for massive scalar fields in (2 + 1)-dimensional spacetimes. We consider the simplest geometrical setup given by a two-dimensional space bounded by two homogeneous parallel wires in order to compare it with the non-perturbative behaviour of the Casimir energy for non-Abelian gauge theories in (2 + 1) dimensions. Our results show the existence of two types of boundary conditions which give rise to two different asymptotic exponential decay regimes of the Casimir energy at large distances. The two families are distinguished by the feature that the boundary conditions involve or not interrelations between the behaviour of the fields at the two boundaries. Non-perturbative numerical simulations and analytical arguments show such an exponential decay for Dirichlet boundary conditions of SU(2) gauge theories. The verification that this behaviour is modified for other types of boundary conditions requires further numerical work. Subdominant corrections in the low-temperature regime are very relevant for numerical simulations, and they are also analysed in this paper.


Introduction
The role of boundary effects in quantum field theory is fundamental for many quantum phenomena.One of the earliest applications was the Casimir effect [1].A quantum field, when confined between two solid bodies, generates a dependence of the renormalized vacuum energy on the boundary conditions at the interfaces of the bodies.This dependence of the vacuum energy generates a force between them which depends on the nature of the boundary conditions of the quantum fields.Although this effect is very tiny, it has been experimentally measured in different setups.[2,3,4,5,6,7].
A remarkable effort has been made in understanding and computing the Casimir effect for different models and setups.Some relevant results were obtained in Ref. [8], where the Casimir vacuum energy at zero temperature was computed for general boundary conditions and arbitrary dimensions for massless scalar fields using heat kernel methods.These results were later extended to finite temperatures in 3+1 dimensions [9].
Less known are the characteristics of the effect for interacting theories [10].Quite recently, the behaviour of the Casimir energy has been investigated in 2+1 dimensional Yang-Mills theories, where some reparametrization of gauge fields in terms of scalar fields allows an analytic approach to the problem [11,12,13].Numerical simulations with Dirichlet boundary conditions on gauge fields confirm the results of this analytic approach [14].
For SU (2) gauge theories the analytic approach is based on the description of gauge fields in terms of a massive scalar field, whose mass depends on the gauge coupling that in 2+1 dimensions is not dimensionless as in 3+1 dimensions.In that case, the Casimir energy of the strongly interacting gauge theories with Dirichlet boundary conditions does coincide with the Casimir energy of a scalar field with a magnetic mass m = g 2 π , where g is the gauge coupling constant.
Some numerical simulations are in progress with different boundary conditions on gauge fields [15] to test if the relation between Casimir energy of massive fields and Yang Mills theory is robust under the change of boundary conditions.Making comparisons with what happens for scalar fields requires to know the behaviour of Casimir energy of the massive scalars for different families of boundary conditions.
In this paper, we study the vacuum energy for massive scalar fields with general boundary conditions in a two dimensional setup bounded by two homogeneous parallel wires by using a regularization scheme similar to the one used in [16,17] for massless theories.To compare with the lattice gauge theories results, it is necessary to work at finite temperature; thus, it is important to understand how the thermal fluctuations affect the Casimir energy at low temperatures both in 2+1 dimensional SU (2) gauge theories and massive scalar field theories in order to have some analytical background reference to results to.
Independently of these motivations, some interest has been raised recently on the applications of the thermal Casimir effect in nano-electronic devices [18,19] or the appearance of negative self entropy related to this effect [20,21,22] which boosted the interest on new aspects of these thermal effects.

Effective action of a Massive Scalar Field in 2+1 Dimensions
We consider a free scalar massive field in 2+1 dimensions confined between two homogeneous infinite wires separated by a distance L. Depending on the structure of the wires the quantum fields have to satisfy some conditions on the boundary wires.Moreover, finite temperature T ̸ = 0 effects can be described in the Euclidean formalism by compactification of the Euclidean time direction into a circle of radius β 2π = 1 2πT .In this case, the partition function can be written as the following determinant where m is the mass of the fields, ∇ 2 the spatial Laplacian and ∂ 0 the Euclidean time derivative.As already mentioned, the boundary conditions are periodic in time ψ(t + β, x) = ψ(t, x), and because of the homogeneity of the boundary wires the spatial boundary conditions can be given in terms of of 2 × 2 unitary matrices U ∈ U(2) [23] φ where δ is an arbitrary scale parameter and are the boundary values φ(±L/2) = ψ(t, x 1 , ±L/2) of the fields ψ and their outward normal derivatives φ(±L/2) = ±∂ 2 ψ(t, x 1 , ±L/2) on the wires.From now on we will assume that δ = 1 for simplicity.
In the standard parametrization of U(2) matrices in terms of an unit vector n ∈ S 2 and Pauli matrices σ,the space of boundary conditions that preserve the non-negativity of the spectrum of the operator −∇ 2 is restricted by the inequalities 0 ≤ α ± γ ≤ π 1 .[8].
The determinant of the second order differential operator −∂ 2 0 − ∇ 2 + m 2 in equation ( 1) is ultraviolet (UV) divergent but can be regularized by means of zeta regularization method [24,25].The effective action is defined by the logarithm of the partition function which can be expressed as in terms of the zeta function where we have introduced the scale parameter µ, which encodes the standard ambiguity of zeta function renormalization techniques 2 , to make the zeta function dimensionless.This ambiguity will be fixed by the renormalization scheme prescription.Actually, the scale parameter µ can be seen as an explicit implementation of the renormalization group.
In our case of a massive scalar field confined between two infinite wires, the eigenvalues of operator −∂ 2 0 − ∇ 2 are given by the sum of the square of the temporal modes 2πl/β associated to the Matsubara frequencies, the continuous spatial modes k and the discrete spatial modes k i that depend on the boundary conditions imposed by the boundary wires 1 Moreover, since the scalar field is real, the second component of the unit vector n has to vanish, i.e., n2 = 0 2 See e.g.[26,27] for a detailed discussion and comparison with other renormalization methods.
Thus, the zeta function in this case reads as follows where A is the length of the wires.Now we can integrate the continuous spatial modes using the analytic extension of the zeta function It was shown in [8] that for homogeneous boundary conditions along the wires the discrete spatial modes are given by the zeros of the spectral function in the following way where the integral is defined along the contour of a thin strip enclosing the positive real axis where all the zeros of the spectral function h U (k) are located.
All ultraviolet divergences arise in the zero temperature limit of the vacuum energy and the removal of such a divergences require a consistent prescription method (renormalization scheme) with a clear physical meaning.They appear in the leading terms of the zero temperature expansion that has the following asymptotic behaviour in the large L limit [8,16] where E 0 is the vacuum energy, C 0 (m) the divergent bulk vacuum energy density, C 1 (m) the divergent energy density of the boundary wires and C c (m, L) is the the finite coefficient of the Casimir energy.One renormalization prescription which allows us to get rid of all these divergences consists on the redefinition of the renormalized effective action as [16,17] where in terms of an auxiliary length L 0 .Notice that the physical condition which fixes this renormalization scheme is the complete removal of the spurious contributions to the bulk and the boundary vacuum energies, leaving only Casimir energy terms and non linear in β temperature dependent contributions to the effective action.These are precisely the physical requirements that fix the renormalization scheme prescription.
The sum over Matsubara modes can be explicitly computed in the low temperature regime.
3 Low Temperature regime In the low temperature limit βm ≫ 1, we can not express the result as an infinite series of 1 β .This means that we have to deal first with the Matsubara modes and later with the boundary modes.We start by rewriting (9) as follows Now, we use the Mellin transform and apply the Poisson formula for the sum over l modes We can compute the integral where we have obtained a term (l = 0) that has a linear dependence on β, and the rest of terms can be expressed in terms of the modified Bessel function of second type K ν .Let us focus on the first term which is the zero temperature one, by replacing the sum of boundary modes by an integral modulated by the spectral function (10) we have Thus, the zero temperature term of the renormalized zeta ( 14) is As it was explained previously, this combination cancels the UV divergences on the integral, thus, the only divergent terms left are the ratio of two Gamma functions whose asymptotic behaviour in the small s expansion is which allows to calculate the derivative Thus, we have Since the integrand is holomorphic, we can extend the integration contour to an infinite semicircle limited by the imaginary axis on its left.Also, because the integration over the semicircle is zero, we can reduce the integral to the imaginary axis Taking into account that the integrand is parity odd, the integral would vanish if it were not by the contribution of the branching point k = m of the logarithm log(m 2 − k 2 ), which gives a factor 2πi for the interval (m, ∞) which is absent in the interval (−∞, −m).Thus, the expression reduces to Since the integral domain begins at m, we can take the limit L 0 → ∞ on the spectral functions by noticing that lim If we define the result in terms of the limit for the spectral function we get a simplified formula to

Temperature dependent terms
Let us now compute the terms with l ̸ = 0 of the zeta function −1+s Since the Bessel special function K 1 decreases exponentially as the argument grows, both sums are finite, thus, the only divergent contribution is the Gamma function, which after derivation gives and we can we rewrite the sum of the discrete eigenvalues by means of the spectral formula (10) Thus, the temperature dependent terms of the renormalized zeta function ( 14) have the following form (32) In a similar manner as we did for the l = 0 term, since the integrand is also holomorphic we can extend the contour to an infinite semi-circle limited by the imaginary axis.Because the integral is zero on the semicircle, we can reduce the integral just to the imaginary axis Because the integrand is odd, the contribution of (−m, m) is zero, whereas the branching point of √ m 2 − k 2 introduces a change of sign on the integrand on (−∞, −m) and also in the argument of the Bessel function.Using that K 1 (z) = K 1 (z), the real part of the integrals between (−∞, −m) and (m, ∞) is twice one of the integrals, whereas the imaginary part cancels out.In summary, the integral can be reduced to We can take the limit L 0 → ∞ using (26) as we did for the l = 0 term and the integral is simplified to

Casimir energy
The Casimir energy can be derived from the terms we have just computed in the previous sections.We can easily compute the free energy with the effective action simply as F = S eff /β.This free energy has two different contributions [22], the non temperature dependent part (l = 0) which corresponds to the Casimir energy of the system and the temperature dependent part .
(37) Both terms of the free energy decrease to zero as the distance between the wires L grows to infinite, which is the expected behaviour.The temperature dependent term also vanishes F l̸ =0 U → 0 when the temperature also does (β → ∞).

Asymptotic behaviour
Let us now analyse the behaviour of the Casimir energy when mL → ∞.First, we rewrite the hyperbolic functions of the spectral function as where A and B are We can use this expression to approximate the logarithm of the quotient of spectral functions as where A ′ = (n 1 sin(γ)A) 2 , and we expanded the logarithm in powers of e −kL .Now we introduce this expression on the integral of the Casimir energy formula We can expand this integral as a power series of 1/L for each exponential order by integrating by parts as follows and iterate this process since all derivatives of g(α, γ, n 1 , k) are regular in [m, ∞].Thus, the Casimir energy is given by where the coefficients corresponding to the leading order in the exponential expansion are of the form This means that when n 1 sin γ = 0 all the terms that behave as e −mL vanish and the leading contribution will be of the order of e −2mL .Thus, we have two different families of boundary conditions with different asymptotic behaviour depending on whether the matrix U that defines the boundary conditions depends or not on σ 1 .This is the most important of this paper because it classifies the boundary conditions in two families.The difference between the two families is the rate of the exponential decay of the Casimir energy (45).
The physical characterization of the two families of boundary conditions with different exponential decays is the vanishing or not of tr(U σ 1 ).The non-vanishing case corresponds to boundary conditions which connect the values of the fields or its normal derivatives at the two boundary wires, whereas the vanishing case corresponds to families of boundary conditions which only constraints the values of the fields or its normal derivatives at each boundary separately.
The result is obtained for a massive free field bosonic theory.If the observed rate of decay in gauge theories has the same behaviour it will provide a strong evidence of the scenario that describes the dynamics of gauge theories in 2+1 dimensions in terms of a bosonic massive scalar field.

Special cases of boundary conditions
Let us analyse some particular cases where the integral of the Casimir energy can be analytically computed and which are of special interest for their potential implementation for gauge fields.An alternative derivation based on the explicit calculation of the spectrum of spatial Laplacian for these cases is postponed to Appendix A. The derivative of the logarithm of the quotient of spectral functions is

Dirichlet and Neumann boundary conditions
We can integrate the Casimir energy formula (36) which, in the massless limit gives But in the asymptotic very large mL ≫ 1 limit the Casimir energy has a fast exponential decay e −2mL as predicted by the fact that −Tr I σ 1 = 0.
The temperature dependent terms of the free energy cannot be analytically computed but from the asymptotic expansion of the term of the integrand, it can be shown that they have the same exponential decay with mL as the Casimir energy (47).
Neumann boundary conditions correspond to the case where normal derivative of the fields vanish at the boundary wires φ(L/2) = φ(−L/2) = 0.They are parameterized by the unitary matrix U N = I.The derivative of the logarithm of the quotient of spectral functions is the same as for Dirichlet boundary conditions (46), which tell us that the free energy has the same value,

Periodic boundary conditions
Periodicity of the fields and anti periodicity of their normal derivatives at the boundaries φ(L/2) = φ(−L/2), φ(L/2) = − φ(−L/2) correspond to periodic boundary conditions associated to the unitary matrix U P = σ 1 .Notice that by definition periodic boundary conditions do relate the boundary values of the fields of one boundary with the values of the fields at the other one.In this case, the derivative of the logarithm of the quotient of spectral functions is Thus, the Casimir energy is and in the massless limit becomes Notice that, in this case the exponential decay of the Casimir energy e −mL in the asymptotic limit mL → ∞ is slower than that observed in Dirichlet or Neumann boundary conditions, as corresponds to the fact that Tr σ 1 σ 1 = 2 ̸ = 0.The rest of terms of the free energy do share the same behaviour.

Anti-Periodic boundary conditions
Anti-periodic boundary conditions correspond to the values and normal derivatives of the field at the boundary wires satisfying that φ(L/2) = −φ(−L/2), φ(L/2) = φ(−L/2), and the associated unitary matrix is U A = −σ 1 .Again in this case, the boundary conditions relate the boundary values of the fields of one boundary with the boundary values at the other one.In this case the derivative of the logarithm of the quotient of spectral functions is Thus, the Casimir energy is which in the massless limit agrees with the well known results Notice that in this case the exponential decay of the Casimir energy e −mL is similar to the cae of periodic boundary conditions, as corresponds to the fact that −Tr σ 1 σ 1 = −2 ̸ = 0.The rest of the terms of the free energy do have the same exponential decay because 1 − tanh(kL/2) ≈ e −kL .

Zaremba boundary conditions
Zaremba boundary conditions correspond to the case where on wire has Dirichlet boundary conditions whereas the other Neumann boundary conditions.In our parametrization U Z = ±σ 3 , and the derivative of the spectral function is The Casimir energy is that in the massless limit reduces to The temperature dependent part of the free energy is In both cases the exponential suppression e −2mL does coincide with that of Dirichlet or Neumann boundary conditions and again in this case the boundary conditions do not relate the boundary values of the fields of one boundary with the values at the other one.By plotting the temperature dependent part of the free energy F l̸ =0 U (see Figure 2), it can also be seen how these terms exponentially decay to zero as mL grows.The physical difference between the two families giving rise to different asymptotic behaviours: the families with faster decay (Dirichlet, Neumann, Zaremba) are conditions imposed on each boundary wire separately, whereas in the second family (periodic, antiperiodic, pseudo periodic) with slower decay rate the boundary conditions involve a relationship between the values of the fields in both wires, establishing a interconnection between them.

Conclusions
We have shown the existence of two types of boundary conditions which give rise to different regimes of exponential decay of the Casimir energy at large distances for scalar field theories.The two types are distinguished by the feature that the boundary conditions involve or not inter-connections between the behaviour of the fields at the two boundaries.
The fast exponential decays of the Casimir energy associated to all massive fields makes it negligiable when compared with the contribution of massless fields coming from electrodynamics.This means that there is no hope of measuring its effects experimentally.However, from a conceptual viewpoin it can become of crucial importance to understand the confining infrarred behaviour of non-abelian gauge theories if this regime can be effectively driven by a massive scalar field.
Indeed, analytic arguments [11,12] and non-perturbative numerical simulations [14] show that there is a similar exponential decay in gauge theories with Dirichlet boundary conditions.
The verification that such a behaviour is modified for other types of boundary conditions would provide further evidences to the claim that the low energy behaviour of non-abelian SU(2) gauge theories is governed by an effective scalar field with a fixed non-vanishing mass.The remarkable feature is that the mass of this scalar field is considerably smaller than the lowest mass of the glueball spectrum [11,12].
In particular the confirmation of the existence of the two regimes for different boundary conditions will be crucial for the verification of this conjecture.Numerical simulations are in progress to clarify this issue.
Dirichlet boundary conditions corresponds to the physical case of fields vanishing at both boundary wires φ(L/2) = φ(−L/2) = 0; in our parametrization (4) they are given by U D = −I.Notice that these boundary conditions do not relate the boundary values of the fields of one boundary with the boundary values at the other one.

2 − 10 − 1 − 10 − 4 − 10 − 7 − 10 2 )Figure 1 :
Figure 1: Dependence of the Casimir energy in logarithmic scale as a function of the effective distance mL between the two boundary wires for different boundary conditions.

Figure 2 :
Figure 2: Free energy behaviour of the temperature dependent part in logarithmic scale as a function of the effective distance mL between the two boundary wires for different boundary conditions, with mβ = 1