When the Casimir energy is not a sum of zero-point energies

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When the Casimir energy is not a sum of zero-point energies

Luiz C. de Albuquerque*

Faculdade de Tecnologia de Saõ Paulo–CEETEPS–UNESP, Praça Fernando Prestes, 30, 01124-060 Saõ Paulo, SP, Brazil R. M. Cavalcanti†

Instituto de Fı´sica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, 21945-970 Rio de Janeiro, RJ, Brazil

共Received 3 September 2001; published 15 January 2002兲

We compute the leading radiative correction to the Casimir force between two parallel plates in the␭⌽4 theory. Dirichlet and periodic boundary conditions are considered. A heuristic approach, in which the Casimir energy is computed as the sum of one-loop corrected zero-point energies, is shown to yield incorrect results, but we show how to amend it. The technique is then used in the case of periodic boundary conditions to construct a perturbative expansion which is free of infrared singularities in the massless limit. In this case we also compute the next-to-leading order radiative correction, which turns out to be proportional to␭3/2.

DOI: 10.1103/PhysRevD.65.045004 PACS number共s兲: 11.10.Kk, 11.10.Gh, 12.20.Ds

I. INTRODUCTION

An important question in quantum field theory is the re-sponse of the vacuum fluctuations to perturbations of the space-time manifold: in the absence of a consistent quantum theory of gravitation in four space-time dimensions one is led to study vacuum fluctuations of matter or gauge fields in the presence of an external共i.e., classical兲 gravitational field 关1兴. One may also ask how the properties of a field theory are affected by the topology of space-time or by the presence of boundaries, which impose constraints on the fields. For in-stance, periodic boundary conditions on a spatial sector are a key ingredient in compactification schemes of Kaluza-Klein theories 关2兴. Boundary conditions 共BC兲 are also used to de-scribe complicated physical systems in a simplified math-ematical framework. In the electromagnetic Casimir effect 关3兴 one considers classical conductor plates 共perfect mirrors兲, with the field satisfying Dirichlet BC on them. The analo-gous condition in the MIT bag model is the perfect confine-ment of quarks and gluons to the interior of hadrons 关4兴. In thermal field theory, periodic or antiperiodic BC in the imaginary-time are the starting point of the Matsubara for-malism关5兴. Finally, the study of surface effects on the criti-cal properties of a 共magnetic, binary liquid, etc.兲 system leads in many cases to the analysis of scalar field theories subject to certain boundary conditions关6兴.

Although BC have been extensively studied in quantum field theory models, there remains a lot of questions to be answered. In this paper we will investigate some unusual features of periodic and Dirichlet BC on one spatial coordi-nate. In the remainder of the Introduction we will give some motivations to the study of these particular types of boundary conditions.

Quantum field theories in compactified spaces共i.e., with periodic boundary conditions in some spatial directions兲 have been the subject of considerable interest in the literature 关7–9兴. The calculation of the effective potential in

spontane-ously broken symmetry theories shows that the compactifi-cation process may introduce a mechanism for dynamical symmetry restoration. Generation of a dynamical mass is connected with the inclusion of a new scale, the compactifi-cation radius R.

There is a complete mathematical analogy between com-pactified field theory and thermal field theory 共TFT兲 in the Matsubara formalism. In the latter, the inverse temperature

␤⫽1/T is the compactification radius in the imaginary-time direction. The well-known fact that thermal effects do not lead to new ultraviolet divergences in TFT共besides the usual ones found at T⫽0) 关5兴 applies as well to compactified field theories. On the other hand, the infrared properties of the TFT are very different from the ones at zero temperature. The free energy of massless ␭⌽4 theory in three spatial di-mensions develops new infrared divergences at order ␭2 in perturbation theory 关5兴. The dominant infrared divergences come from the n⫽0 mode of the loop momenta. A proper treatment of the collective effects leads to a correction of order␭3/2_{to the free energy.}

The infrared behavior of the compactified field theory mimics the one at finite temperature, at least in the case of one spatial compactified coordinate. In a perturbative treat-ment, the n⫽0 mode generates new infrared divergences in the compactified version of the␭⌽4theory. This seems to be not so well-appreciated in the literature. To fill this gap, we apply the resummation method developed by Braaten, Pisar-ski and others共in the context of TFT兲关10–12兴 to the com-pactified␭⌽4 theory.

Symmetries in quantum field theory put very stringent conditions on the perturbative renormalization of a model and in its physical predictions. Lorentz invariance共rotational invariance in the Euclidean case兲 is of paramount importance in this respect. However, external conditions or dynamical effects may lead to its breakdown. There is a growing inter-est on effective field theories in which this occurs共e.g., non-commutative field theories 关13兴, anisotropic systems 关14兴, and Chern-Simons theories 关15兴兲. Theories defined in finite volumes or in the presence of macroscopic bodies 共as in the Casimir effect兲 may provide useful insights on the conse-*Email address: lclaudio@fatecsp.br

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quences of lack of Lorentz symmetry to the renormalization program.

Recently, there has been much effort in the computation of radiative corrections to the Casimir energy, specially in QED关16兴. One of the purposes of this paper is to discuss an alternative method to compute such corrections. For simplic-ity we work with the␭⌽4theory subject to Dirichlet bound-ary conditions on a pair of parallel plates. The method is based on a resummation of the perturbative series for the two-point Green function, and leads to a Klein-Gordon equa-tion in which the one-loop self-energy acts as an effective scalar potential. In four space-time dimensions this equation can be solved exactly in the massless case. The new set of 共resummed兲 eigenvalues contain radiative corrections of all orders in␭, and reduce to the free ones for ␭⫽0. The com-putation of the sum of effective zero-point energies, includ-ing non-perturbative corrections and renormalization issues, is discussed in detail.

The plan of the paper is as follows. In Sec. II we fix the conventions and discuss the resummation technique in the ␭⌽4_{theory with Dirichlet boundary conditions. We solve the} effective Klein-Gordon equation, and obtain the ‘‘improved’’ eigenvalues. The solution is used to obtain the resummed Casimir energy, including radiative corrections. In Sec. III we discuss the resummation of the vacuum energy in the case of periodic boundary conditions; this sheds some light on the results of Sec. II. In the Conclusion we discuss the drawbacks of this method as well as other minor points. Fi-nally, three Appendixes collect some mathematical results used in the paper.

II. DIRICHLET BOUNDARY CONDITIONS Boundary conditions breaking the full Lorentz invariance in general pose new problems to the renormalization pro-gram. For some geometries and boundary conditions 共de-pending also on the spin of the field兲 it may be necessary to introduce surface counterterms besides the bulk ones. For instance, in the MIT bag model the free energy is ill defined at one-loop due to an extra singularity which shows up as the surface is approached 关17兴. The standard recipe associates a free parameter to each distinct singular term, included as a new counterterm in the starting Lagrangian. If this procedure continues to all orders, with the consequent loss of predictive power, we say that the theory is non-renormalizable due to the boundary conditions.

In a remarkable paper, Symanzik gave strong arguments showing the renormalizability of the ⌽4 theory in the pres-ence of flat boundaries关18兴. In particular, he showed that the renormalized Casimir pressure for disjoint boundaries and Dirichlet BC is finite to all orders in perturbation theory. He also verified explicitly that no surface counterterms are needed in the computation of the two-loop vacuum energy. ⌽4_{-type theories are still renormalizable for more general} boundary conditions and surfaces, but at the price of intro-ducing surface counterterms 关6,18兴. We wish to point out here that many proposals have been made in order to avoid the surface-like singularities. These include, among others, the ‘‘softening’’ of the Dirichlet BC 关19,20兴 or treating the

boundaries as quantum mechanical objects with a nonzero position uncertainty 关21兴. However, this question is outside the scope of our present discussion.

A key ingredient in our computation of the Casimir en-ergy is the self-enen-ergy of the field, as it determines the shift in the single-particle energy levels. Since there is some dis-agreement among existent results in the literature 关7,8,18,25兴, we present its computation in some detail.

A. Self-energy

We work in D⫽(d⫹1)⫹1 dimensional Minkowski space-time, and define x␮⬅(t,x,z), with x⫽(x1, . . . ,xd). The renormalized Lagrangian reads 关ប⫽c⫽1, ␩_␮␯⫽diag (⫹,⫺, . . . ,⫺)兴 L⫽L0⫹LI⫽

再

1 2共⳵⌽兲 2_⫺1 2m 2_⌽2

冎

_⫹

再

_⫺ ␭ 4!⌽ 4_⫹L ct

冎

⌬F共x,x兲⫽

兺

n⫽⫺⬁ ⬁ 关⌬F (0)_{共2nl兲⫺⌬} F (0)_{共2z⫹2nl兲兴.} _共4兲

The term⌬_F(0)(0) contains the usual UV singularity. It can be removed, as usual, by a mass renormalization.

In the massless case the bulk free propagator gets simpli-fied, and it is possible to find a closed expression for ⌬(x) ⬅⌬F(x,x)⫺⌬F (0) (0). Using ⌬F (0) 共x;m⫽0兲⫽ ⌫

冉

D 2 ⫺1

冊

4␲D/2 兩x兩 2⫺D_, _共5兲 one finds, for D⫽4,

, 共6兲 where␺(x) is the digamma function关24兴. Equation 共6兲 can be simplified to ⌬共x;m⫽0兲⫽⫺ 1 16l2

冋

csc 2

冉

␲z l

冊

⫺ 1 3

册

共D⫽4兲. 共7兲 Let us take a closer look at the D⫽3 case, keeping m ⫽0. We obtain from Eqs. 共3兲 and 共4兲, after changing the summation variables and using the explicit form of K1/2(x),

⌬共x兲⫽₈1_␲_l

冋

2e⫺2mlS共2ml,1兲⫺e⫺2mzS

冉

2ml,z l

冊

⫺e⫺2m(l⫺z)_S

冉

_2ml,1_⫺z l

冊册

, 共8兲 where S共a,b兲⬅

兺

冊册

共D⫽3兲. 共11兲 The renormalized one-loop self-energy is given by ⌺(1)_(x)_{⫽(␭/2)⌬}

F(x,x)⫹␦m2. The mass counterterm is fixed by the condition liml→⬁⌺(1)⫽0. This amounts to re-move the contribution of the bulk free propagator from Eq. 共4兲. With this choice of mass renormalization we have ⌺(1)_(x)_{⫽(␭/2)⌬(x). Therefore, the self-energy is infrared} finite in the massless case also at D⫽3, in disagreement with Ref. 关25兴. 关However, it is infrared divergent in the case of Neumann boundary conditions. The propagator is then given by Eq. 共2兲 with the minus sign on its right-hand side 共RHS兲 replaced by a plus sign. As a consequence, the logarithmic terms in the massless limit of ⌬(x) do not cancel.兴

From now on, we shall focus our attention on the mass-less case at D⫽4.

B. Radiative corrections to the Casimir energy: A heuristic approach

Computations of the Casimir energy in the literature are restricted to perturbation theory. A non-perturbative calcula-tion would be a very interesting result. Our goal here is more modest. We will discuss the computation of the Casimir en-ergy in the approximation where the two-point Green func-tion is dressed with an arbitrary number of inserfunc-tions of the one-loop self-energy 共‘‘daisy’’ resummation兲. This approxi-mation contains the leading correction in a 1/N expansion. 共In our case, however, N⫽1. With this caveat in mind, let us proceed.兲 The Casimir energy will be given formally by

E⫽1

2

兺

␣ ␻␣, 共12兲

where ␻_␣ are the positive poles of the dressed two-point function G˜(2) _{in the frequency domain.}

To compute G˜(2)we note that it satisfies 关⳵x

2_⫹⌺(1)_共x兲兴G_˜(2)_共x,x

_⬘

_兲⫽⫺i_␦(4)_共x,x

_⬘

_兲. _共13兲 As usual, the solution to Eq. 共13兲 can be written as

G˜(2)共x,x

⬘

兲⫽⫺i

兺

␣

␾␣共x兲␾␣*共x

⬘

兲

⌳␣ , 共14兲

where ⌳_␣ and␾_␣(x) are the eigenvalues and共normalized兲 eigenfunctions, respectively, of the Klein-Gordon operator

⳵2_⫹⌺(1)_:

关⳵2_⫹⌺(1)_共x兲兴_␾

共n⫹s兲 2_⫹g 3

册

共n⫽1,2, . . . 兲. 共19兲 Before we proceed with the calculation of the Casimir energy, a remark is in order here. As we have seen, the 共renormalized兲 one-loop self-energy is a function of x. It may be tempting to interpret ⌺(1)(x) 共more generally, m2 ⫹⌺(1)_{(x)) as a position-dependent} _{共squared兲 mass of the} field. A problem would then occur in regions where ⌺(1)_(x)_{⬍0, for this could imply the presence of tachyons in} the theory. For that reason, Ford and Yoshimura 关7兴 argued that models which exhibits this behavior共such as the one we are considering兲 are unphysical. However, the analysis of Eq. 共16兲, summarized in Appendix A, shows that its solutions do not have imaginary frequencies as long as ␭⬍␭crit⫽8␲2 共which is anyway well outside the range of validity of per-turbation theory兲. The one-loop effective theory is consistent in this case. On the other hand, for ␭⬎8␲2the Schro¨dinger equation 共16兲 leads to an energy spectrum which is un-bounded from below, rendering the associated effective field theory ill-defined. This solves a long-standing problem of interpretation.

Substituting the eigenfrequencies 共19兲 into Eq. 共12兲 we obtain the following expression for the Casimir energy per unit area: E⫽1₂␮1⫺2␯

兺

n⫽1 ⬁

冕

d2p 共2␲兲2

再

p 2_⫹␲ 2 l2

冋

共n⫹s兲 2_⫹g 3

册

冎

␯

冏

_␯⫽1/2. 共20兲 The formal sum over zero-point energies has been analyti-cally regularized; we shall set␯⫽1/2 at the end of the cal-culation. The factor ␮1⫺2␯, where ␮ is a mass parameter, keeps the RHS of Eq.共20兲 with the dimension of energy per unit area.

Integrating Eq.共20兲 over p, we get

E⫽␮ 1⫺2␯ 8␲ ⌫共⫺␯⫺1兲 ⌫共⫺␯兲

冉

␲ l

冊

2(␯⫹1) H

冉

⫺␯⫺1;s,g 3

冊

, 共21兲 where the functionH(z;s,a2) is defined as

H共z;s,a2_兲⬅

兺

n⫽1

⬁

关共n⫹s兲2_⫹a2_兴⫺z_. _共22兲 The series converges for Rz⬎1/2. The analytical continua-tion ofH(z;s,a2_{) to the whole complex z plane is performed} in Appendix B. Substituting the result into Eq. 共21兲 we ob-tain 关26兴 E⫽␮ 1⫺2␯ 8␲ ⌫共⫺␯⫺1兲 ⌫共⫺␯兲

冉

␲ l

冊

2(␯⫹1)

再

1 2

冋

共1⫹s兲 2_⫹g 3

册

␯⫹1 ⫹i

冕

0 ⬁ dtf␯共1⫹it兲⫺ f␯共1⫺it兲 e2␲t⫺1 ⫺ 共1⫹s兲2␯⫹3 2␯⫹3 ⫻F

冉

⫺␯⫺1,⫺␯⫺3 2;⫺␯⫺ 1 2;⫺ g 3共1⫹s兲2

冊冎

, 共23兲 where f_␯(x)⬅关(x⫹s)2⫹g/3兴␯⫹1and F(␣,␤;␥;z) is the hy-pergeometric function. From the definition of the latter it follows that E has a simple pole at ␯⫽1/2 共in fact, it has poles at␯⫽⫺3/2,⫺1/2,1/2,3/2, . . . ). This requires that E be renormalized before we set ␯⫽1/2. In general, this is done by subtracting fromE its value at l→⬁. Unfortunately, such a prescription does not work in the present case, since, ac-cording to Eq.共23兲, the Casimir energy per unit area has the form E⫽C(␯)/l2(␯⫹1).

One can obtain a hint on what is going wrong by noting that the residue ofE at␯⫽1/2 is of second order in g 共or ␭). This is consistent with the fact that we have worked with the one-loop two-point Green function, which is共formally兲 cor-rect only to first order in the coupling constant. Since the ␭⌽4 _{theory is perturbatively renormalizable in D}_{⫽4, one} may suspect that in order to obtain a finite共or at least renor-malizable兲 E to order ␭n one must work within an approxi-mation in which the two-point Green function is dressed with the n-loop self-energy. As we show below, this is not suffi-cient or necessary. In spite of that, the argument suggests that Eq. 共23兲 cannot be trusted beyond order ␭.

Expanding the RHS of Eq.共23兲 in a power series in ␭ and making␯→1/2, we obtain E⫽1 l3

冋

⫺ ␲2 1440⫹ ␭ 9216⫹•••

册

. 共24兲 The first term is the usual free Casimir energy共per unit area兲. The second term is the leading radiative correction to it. It overestimates the correct result 关18兴 by a factor of 2. This discrepancy occurs because the method we have used to compute the Casimir energy only works in the absence of interactions. To show this, we start by noting that one can define the Casimir energy as

E⫽

冕

dD⫺1x

具

0兩T00共x兲兩0

典

_⬘

_兲 ⫺1₂⌺(1)_共x兲G_˜(2)_共x,x兲⫹ ␭ 4!G (4)_{共x, . . . ,x兲}

册

_, _共29兲 where⌬G(2)⬅G(2)⫺G˜(2). While the first line of Eq.共29兲 is formally O(␭2), the second one is O(␭). This explains why the second term in Eq. 共24兲 is incorrect.

It is important to note that Eqs.共12兲 and 共26兲 would lead to distinct results even if we had worked with the complete two-point Green function. The difference between them would then be given by

⌬E⫽

冕

dD⫺1x

再

⫺1 2

冕

d

D_y_{⌺共x,y兲G}(2)_共y,x兲 ⫹_4!␭ G(4)共x, . . . ,x兲

冎

. 共30兲 A perturbative evaluation of the above expression shows that ⌬E would still be of order ␭. Physically, this discrepancy is due to the fact that, in contrast with the free theory, the interacting theory is not equivalent to a collection of inde-pendent harmonic oscillators. The sum of zero-point ener-gies, Eq.共12兲, takes into account only the Lamb shift on the single-particle energy levels caused by the interaction; the difference ⌬E accounts for the residual interaction among the 共anharmonic兲 oscillators.

The above discussion also shows that the Casimir energy is not determined solely by the two-point Green function, but also 共in the ␭⌽4 theory兲 by the four-point function. In par-ticular, in order to consistently remove the O(␭2) UV singu-larity in Eq.共23兲 one must not only obtain G(2)to that order, but also G(4) to O(␭). These ideas will be illustrated in the next section in the simpler case of periodic boundary condi-tions.

III. PERIODIC BOUNDARY CONDITIONS A. Conventional perturbation theory

ddp 共2␲兲d

e⫺i␻n(p)兩t⫺t⬘兩⫹ip•(x⫺x⬘)⫹iqn(z⫺z⬘)

␻n共p兲

, 共31兲

where p␮⫽(␻,p,qn), qn⫽2␲n/R, and ␻n(p)

⬅

冑

p2⫹q_n2⫹m2. Since ⌬F(x,x

⬘

)⫽⌬F(x⫺x

⬘

), such bound-ary conditions do not break translational invariance.

The renormalized vacuum energy density may be com-puted from Eqs.共25兲–共27兲, but its perturbative expansion is more easily derived from the vacuum persistence amplitude:

␧⫽ lim

T→⬁ i

VTln

冋

冕

where A(␣) and B(␣) are given by Eqs. 共C8兲 and 共C9兲, respectively. Only A(␣) depends on R, and vanishes when R→⬁.

⫹␭

冋

₁₁₅₂1 ⫺₁₉₂mR_␲⫹•••

册冎

⫹••• . 共40兲 Analogously, we obtain for the one-loop self-energy

⌺(1)_⫽ ␭ 共2␲兲D/2

冉

m R

冊

(D⫺2)/2 F

冉

D 2 ⫺1;mR

冊

⫽ ␭ R2

冋

1 24⫺ mR 8␲⫹ 共mR兲2 16␲2 ln共mR兲⫹•••

册

共D⫽4兲. 共41兲 The first term does not depend on m, and is sometimes called ‘‘topological mass’’共squared兲. For reasons discussed in 关1兴, we prefer the name ‘‘compactification mass’’ for M ⬅(␭/24R2₎1/2_.

B. Resummed perturbation theory

From now on, let us focus the discussion on the massless theory (m⫽0) in D⫽4. In this case, the second and higher order terms in the perturbative expansion of ␧ are plagued with IR divergences. A qualitative analysis shows that the most IR divergent diagrams are the ‘‘ring’’ 共or ‘‘daisy’’兲 ones. 共These are just the diagrams with the greatest number of insertions of the one-loop self-energy in each order of perturbation theory.兲 As in the case of TFT 关5兴, it is possible to sum these diagrams to all orders. The result is finite in the IR and is nonanalytic in␭, as we show below.

To avoid overcounting of diagrams in higher order calcu-lations, it is convenient to redefine the free and the interact-ing parts of the Lagrangian by addinteract-ing and subtractinteract-ing to it the compactification mass term 1

2M 2_⌽2 _{关10–12兴:} L⫽L˜₀_⫹L˜_I_⫽

再

1 2共⳵⌽兲 2_⫺1 2M 2_⌽2

冎

⫹

再

⫺ ␭ 4!⌽ 4_⫹1 2M 2_⌽2_⫹L ct

冎

. 共42兲 The free propagator 共in momentum space兲 is now given by

⌬˜F共p兲⫽ i

p2⫺M2⫹i⑀. 共43兲 It coincides with the propagator of the original theory in the daisy approximation.

We remark that in a loop expansion of the vacuum energy 共or of any other quantity兲 each insertion of the mass term in

L˜

I is to be formally counted as one loop, like the mass counterterm—otherwise takingL˜0 as the new free Lagrang-ian would not cure the IR divergence problem关27兴.

The one-loop approximation to the vacuum energy is given by ␧˜(1)_⫽ 1 2R n⫽⫺⬁

兺

⬁

冕

d2p 共2␲兲2共p 2_⫹q n 2_⫹M2_兲1/2_. _共44兲 Using the results of Sec. III A and of Appendix C we obtain

␧˜(1)_{共R兲⫽⫺} M 2 2␲2R2F共2;MR兲 ⫹ lim ⑀→0 ␮⑀_M4⫺⑀ 24⫺⑀␲(3⫺⑀)/2 ⌫

冉

⫺2⫹⑀ 2

冊

⌫

冉

⫺1₂

冊

⫽ 1 R4

冋

⫺ ␲2 90⫹ ␭ 576⫺ ␭3/2 288

冑

6␲⫹O

冉

␭2 ⑀

冊

册

. 共45兲 As in the Dirichlet BC case, the O(␭) term in the one-loop approximation is twice the value obtained in

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␭2 ⑀

冊册

. 共48兲

Thus, to order␭2 we finally obtain ␧共R兲⫽ 1 R4

冋

⫺ ␲2 90⫹ ␭ 1152⫺ ␭3/2 288

冑

6␲⫹O

冉

␭2 ⑀

冊

册

. 共49兲

This agrees to order␭ with the result found in Sec. III A 共in the limit m→0). Besides, we have obtained a correction of order␭3/2. This nonanalyticity in ␭ is a consequence of the fact that the loop expansion in the rearranged Lagrangian is equivalent to a resummation of an infinite number of graphs in the conventional perturbation expansion.

Finally, we note that the UV singularities in the resummed theory depend on R, via their dependence on M. For instance, in the computation of ␧˜(1) a singular term of the form M4/⑀⬃␭2/⑀R4 appears in the limit ⑀→0. In the analogous case of TFT it can be shown that the UV singularity present in the one-loop free energy cancels against two- and three-loop contributions in the resummed theory, including a cou-pling constant renormalization counterterm 关12,28兴. These contributions on their turn introduce new singularities at O(␭3), which are cancelled by including higher order graphs in the resummed theory, and so on. The situation is exactly the same in our case, so we can safely neglect the O(␭2) term in Eq. 共49兲.

IV. CONCLUSION

In this paper we have discussed the computation of radia-tive corrections to the Casimir energy of the massless ␭⌽4 theory confined between two parallel plates. The case of Di-richlet boundary conditions at the plates was discussed in Sec. II. We obtained an analytical expression for the one-loop self-energy ⌺(1)_{(x) both in D}_{⫽3 and D⫽4. The} former was shown to be free of IR singularities, in contrast with the claim made in关25兴.

In the ‘‘daisy’’ resummation of the two-point Green func-tion one is led to solve a Klein-Gordon equafunc-tion with ⌺(1)_{(x) acting as an effective scalar potential. We were able} to solve this equation in four dimensions. In spite of ⌺(1) being negative everywhere, there are no tachyonic modes if the coupling constant␭ is smaller than ␭crit⫽8␲2. We then computed the sum of the eigenenergies of the Klein-Gordon operator. Expanding the result in a power series in ␭ one discovers that the O(␭) correction does not agree with the result of conventional perturbation theory, and the correction of order␭2 _{contains a UV singularity which apparently} can-not be renormalized away. The first problem was shown to occur because the sum of zero-point energies does not take into account all the contributions to the vacuum energy in a theory with interaction. As for the second problem, it was argued that the consistent renormalization of the Casimir en-ergy at a given order requires that one takes into account all diagrams to that order. This conjecture is supported by the fact that the Dirichlet BC共in the case of flat boundaries兲 do not spoil the perturbative renormalizability of the ␭⌽₄4 theory 关18兴.

In the case of periodic boundary conditions in one spatial direction we have argued that the infrared properties of the model are analogous to the one in thermal field theory. In order to define a consistent 共i.e., IR finite兲 perturbative ex-pansion one has to include the screening effects due to col-lective excitations. A solution to this problem was proposed by Braaten, Pisarski and others in thermal field theory关10– 12兴. It consists in the resummation of an infinite class of diagrams, which gives the field an effective mass. This can be done in a systematic way using the Braaten-Pisarski re-summation method. This was illustrated with the calculation of the leading and next-to-leading order radiative corrections to the Casimir energy. Besides, our calculation shows that the resummed weak coupling expansion of the Casimir en-ergy in the case of periodic BC contains fractional powers of ␭, in contrast to the case of Dirichlet boundary conditions.

We note that calculations of radiative corrections to the Casimir energy via the resummation of zero-point energies have appeared recently in the literature关29兴, without paying due attention to the subtleties of the resummed perturbation theory. As we have shown, this may lead to inconsistencies in the results关30兴.

Finally, we hope that the techniques discussed here may be useful in investigations of Kaluza-Klein compactification scenarios, as well as in the study of surface critical phenom-ena.

ACKNOWLEDGMENTS

The authors acknowledge the financial support from FAPESP under grants 00/03277-3 共L.C.A.兲 and 98/11646-7 共R.M.C.兲, and from FAPERJ 共R.M.C.兲. They also acknowl-edge the kind hospitality of the Departamento de Fı´sica Matema´tica, Universidade de Sa˜o Paulo, where this work was initiated.

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APPENDIX A We discuss the solution to the equation

冉

d2 d y2⫹k

2_⫹ g

sin2y

冊

␸共y兲⫽0, 共A1兲 with Dirichlet boundary conditions at y⫽0 and y⫽␲.

Let us first consider the asymptotic behavior of its solu-tions near one of the boundaries 共say, at y⫽0). To this end, we can replace Eq.共A1兲 by

冉

d2 d y2⫹

g

y2

冊

␸共y兲⫽0. 共A2兲 The most general solution to Eq.共A2兲 is

␸共y兲⫽A ys_⫹_{⫹B y}s_⫺_, _共A3兲 where s_⫾⬅12(1⫾

冑

1⫺4g). If g⬍1/4, the boundary

condi-tion␸(0)⫽0 is not enough to fix the relation between A and B, as both ys⫹ _{and y}s_⫺ _{vanish at y}_{⫽0. To resolve this} inde-terminacy, we follow 关31兴 and regularize the potential near the origin: V_R(y )⫽⫺g/y2 _{for y}_{⬎a, and V}

R( y )⫽⫺g/a2for y⬍a. At the end, we shall take the limit a→0.

For y⬎a, the solution is given by Eq. 共A3兲. For y⬍a, the solution which satisfies the boundary condition at the origin is ␸( y )⫽C sin(

冑

g y /a). Continuity of ␸( y ) and its deriva-tive at y⫽a implies the relation B/A⬃as⫹⫺s⫺ as a→0, i.e., only the solution with the faster decay at the origin survives when the regularization is removed. If g⬎1/4, s_⫹⫺s_⫺ is purely imaginary and lima→0B/A does not exist. This sets a critical value to g, namely gcrit⫽1/4, above which the ‘‘Hamiltonian’’ H⫽⫺d2_{/d y}2_⫺g/y2 _{is unbounded from} be-low 关31兴.

Let us return to the complete equation 共A1兲. It is conve-nient to make some changes of variables. First, we set y ⫽ix⫹␲/2 and define s⬅12(⫺1⫹

冑

1⫺4g) 关so that s(s

⫹1)⫽⫺g兴. This transforms Eq. 共A1兲 into

冋

⫺ d 2 dx2⫹k

2_⫺s共s⫹1兲

cosh2_x

册

␸共x兲⫽0. 共A4兲 Then, we make ␰⫽tanh x and obtain

d d␰

冋

共1⫺␰ 2_兲d␺ d␰

册

⫹

冋

s共s⫹1兲⫺ k2 1⫺␰2

册

␸共␰兲⫽0. 共A5兲 Finally, we put ␸(␰)⫽(1⫺␰2)k/2w(␰), followed by the change of variable␰⫽2u⫺1, to get

u共1⫺u兲d 2_w du2⫹关1⫹k⫺2共1⫹k兲u兴 dw du⫺共k⫺s兲共k⫹s⫹1兲w ⫽0. 共A6兲

Equation共A6兲 is the hypergeometric differential equation 关24兴 with parameters ␣⫽k⫺s, ␤⫽k⫹s⫹1, and␥⫽1⫹k. Its general solution may be written as

w共u兲⫽A共1⫺u兲⫺kF共⫺s,s⫹1;1⫹k;u兲

⫺ i 2y

冊

⫺s⫺1 . 共A10兲

Recalling the analysis of Eq. 共A2兲, we impose that ␸( y ) ⬃ys_⫹1 _{as y}_{→0. This can be accomplished by taking B}

_⬘

⫽0 and k⫽⫺(n⫹s)(n⫽1,2, . . . ), that is

␸共y兲⫽A

⬘

ei(n⫹s)yF

冉

⫺s,s⫹1;⫺s⫺n⫹1;i e

⫺iy

2 sin y

冊

. 共A11兲 The boundary condition at y⫽␲ is also satisfied by this so-lution, since ␸(y )⬃(␲⫺y)s⫹1 as y→␲. Hence, Eq.共A11兲 is an acceptable solution to Eq. 共A1兲. 关Remark: we could also have taken A

⬘

⫽0 and k⫽n⫹s (n⫽1,2, . . . ) in Eq. 共A10兲, but this leads to the same values of k2_{and to the same} solutions given by Eq.共A11兲.兴

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APPENDIX B

The goal here is to obtain the analytical continuation of the functionH(z;s,a2), defined in Eq.共22兲, to the whole complex z plane. To this end, we shall use the Plana summation formula 关32兴

兺

k_⫽M N f共k兲⫽1 2关 f 共M 兲⫹ f 共N兲兴⫹

冕

M N f共x兲dx⫺i

冕

0 ⬁

d yf共N⫹iy兲⫺ f 共M⫹iy兲⫺ f 共N⫺iy兲⫹ f 共M⫺iy兲

e2␲y⫺1 . 共B1兲

In the present case, we choose M⫽1, N⫽⬁, and f (x) ⫽关(x⫹s)2_⫹a2_兴_⫺z_{. In order to apply the Plana formula} some conditions have to be satisfied 关32兴. First, we assume that Rz⬎12, so that the series in Eq. 共22兲 converges

abso-lutely. Then, it can be shown that the function f (␶⫹it) is holomorphic for ␶⭓1 for any t, and that lim_t_→⫾⬁e⫺2␲兩t兩f (␶⫹it)⫽0 uniformly in the interval 1⭐␶ ⬍⬁. In addition, lim␶→⬁兰⫺⬁⬁ dt e⫺2␲兩t兩兩 f (␶⫹it)兩⫽0. Under

these conditions, we have

H共z;s,a2_兲⫽1 2关共s⫹1兲 2_⫹a2_兴⫺z_⫹

冕

1 ⬁ dx 关共x⫹s兲2_⫹a2_兴z ⫹i

冕

0 ⬁ dtf共1⫹it兲⫺ f 共1⫺it兲 e2␲t⫺1 . 共B2兲 The first integral can be computed in closed form:

冕

1 ⬁ dx 关共x⫹s兲2_⫹a2_兴z ⫽共1⫹s兲 1⫺2z 2z⫺1 F

冉

z,z⫺ 1 2;z⫹ 1 2;⫺

冉

a 1⫹s

冊

2

冊

. 共B3兲 Equations 共B2兲 and 共B3兲 give the analytic continuation of

H(z;s,a2_{) to the whole complex z-plane. From the definition} of the hypergeometric function it follows thatH(z;s,a2) has simple poles at z⫽1/2,⫺1/2,⫺3/2, . . . .

APPENDIX C Consider the function

⌿⑀共␣兲⬅␮ ⑀ 2Rn⫽⫺⬁

兺

⬁

冕

dd⫺⑀p 共2␲兲d⫺⑀

冋

p 2_⫹

冉

2␲n R

冊

2 ⫹m2

册

冊册

, 共C4兲 where F共s;a兲⬅

兺

n⫽1 ⬁ n⫺sKs共na兲, 共C5兲 with Ks(x) the modified Bessel function of the second kind. The following expansions, valid for aⰆ1, will be useful 关34兴: F共1;a兲⫽␲ 2 6a⫺ ␲ 2⫹O共a ln a兲, 共C6兲 F共2;a兲⫽ ␲ 4 45a2⫺ ␲2 12⫹ ␲a 6 ⫹O共a 2_兲. 共C7兲 Using the analytical continuation given in Eq. 共C4兲, one may write ⌿_⑀(␣) 共in the limit ⑀→0) as the sum of two terms, namely, lim_⑀→0⌿_⑀(␣)⫽A(␣)⫹B(␣), where

⌫共⫺␣兲 . 共C9兲 Note that only A(␣) depends on R.

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