PHYSICAL REVIEW D, VOLUME 65, 045004 When the Casimir energy is not a sum of zero-point energies Luiz C. de Albuquerque* Faculdade de Tecnologia de Sa˜o Paulo–CEETEPS–UNESP, Prac¸a Fernando Prestes, 30, 01124-060 Sa˜o 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 ~Received 3 September 2001; published 15 January 2002! We compute the leading radiative correction to the Casimir force between two parallel plates in thelF4 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 tol3/2. DOI: 10.1103/PhysRevD.65.045004 PACS number~s!: 11.10.Kk, 11.10.Gh, 12.20.Ds e th tu i i a e in re in - th ec o- e e or ti- ri um b u di e ar ns tu e tifi- ical is tifi- om- ure e ot l d the re. s r of ry e of at- in e isar- nt del cal er- e- I. INTRODUCTION An important question in quantum field theory is the r sponse of the vacuum fluctuations to perturbations of space-time manifold: in the absence of a consistent quan theory of gravitation in four space-time dimensions one led to study vacuum fluctuations of matter or gauge fields the presence of an external~i.e., classical! gravitational field @1#. One may also ask how the properties of a field theory affected by the topology of space-time or by the presenc boundaries, which impose constraints on the fields. For stance, periodic boundary conditions on a spatial sector a key ingredient in compactification schemes of Kaluza-Kle theories@2#. Boundary conditions~BC! are also used to de scribe complicated physical systems in a simplified ma ematical framework. In the electromagnetic Casimir eff @3# one considers classical conductor plates~perfect mirrors!, with the field satisfying Dirichlet BC on them. The anal gous condition in the MIT bag model is the perfect confin ment of quarks and gluons to the interior of hadrons@4#. In thermal field theory, periodic or antiperiodic BC in th imaginary-time are the starting point of the Matsubara f malism @5#. Finally, the study of surface effects on the cri cal properties of a~magnetic, binary liquid, etc.! system leads in many cases to the analysis of scalar field theo subject to certain boundary conditions@6#. Although BC have been extensively studied in quant field theory models, there remains a lot of questions to answered. In this paper we will investigate some unus features of periodic and Dirichlet BC on one spatial coor nate. In the remainder of the Introduction we will give som motivations to the study of these particular types of bound conditions. Quantum field theories in compactified spaces~i.e., with periodic boundary conditions in some spatial directio! have been the subject of considerable interest in the litera @7–9#. The calculation of the effective potential in spontan *Email address: lclaudio@fatecsp.br †Email address: rmoritz@if.ufrj.br 0556-2821/2002/65~4!/045004~10!/$20.00 65 0450 - e m s n re of - a - t - - es e al - y re - ously broken symmetry theories shows that the compac cation process may introduce a mechanism for dynam symmetry restoration. Generation of a dynamical mass connected with the inclusion of a new scale, the compac cation radiusR. There is a complete mathematical analogy between c pactified field theory and thermal field theory~TFT! in the Matsubara formalism. In the latter, the inverse temperat b51/T is the compactification radius in the imaginary-tim direction. The well-known fact that thermal effects do n lead to new ultraviolet divergences in TFT~besides the usua ones found atT50) @5# applies as well to compactified fiel theories. On the other hand, the infrared properties of TFT are very different from the ones at zero temperatu The free energy of masslesslF4 theory in three spatial di- mensions develops new infrared divergences at orderl2 in perturbation theory@5#. The dominant infrared divergence come from then50 mode of the loop momenta. A prope treatment of the collective effects leads to a correction orderl3/2 to the free energy. The infrared behavior of the compactified field theo mimics the one at finite temperature, at least in the cas one spatial compactified coordinate. In a perturbative tre ment, then50 mode generates new infrared divergences the compactified version of thelF4 theory. This seems to be not so well-appreciated in the literature. To fill this gap, w apply the resummation method developed by Braaten, P ski and others~in the context of TFT! @10–12# to the com- pactifiedlF4 theory. Symmetries in quantum field theory put very stringe conditions on the perturbative renormalization of a mo and in its physical predictions. Lorentz invariance~rotational invariance in the Euclidean case! is of paramount importance in this respect. However, external conditions or dynami effects may lead to its breakdown. There is a growing int est on effective field theories in which this occurs~e.g., non- commutative field theories@13#, anisotropic systems@14#, and Chern-Simons theories@15#!. Theories defined in finite volumes or in the presence of macroscopic bodies~as in the Casimir effect! may provide useful insights on the cons ©2002 The American Physical Society04-1 io tio in a li - i th ua iv tio et al d e he t e d’’ II th ig th F ul ce ro F e th a s ur tiv en e an e r rg ra tro vo er zero e n- hift dis- ure - in e pli- LUIZ C. DE ALBUQUERQUE AND R. M. CAVALCANTI PHYSICAL REVIEW D 65 045004 quences of lack of Lorentz symmetry to the renormalizat program. Recently, there has been much effort in the computa of radiative corrections to the Casimir energy, specially QED @16#. One of the purposes of this paper is to discuss alternative method to compute such corrections. For simp ity we work with thelF4 theory subject to Dirichlet bound ary conditions on a pair of parallel plates. The method based on a resummation of the perturbative series for two-point Green function, and leads to a Klein-Gordon eq tion in which the one-loop self-energy acts as an effect scalar potential. In four space-time dimensions this equa can be solved exactly in the massless case. The new s ~resummed! eigenvalues contain radiative corrections of orders inl, and reduce to the free ones forl50. The com- putation of the sum of effective zero-point energies, inclu ing non-perturbative corrections and renormalization issu is discussed in detail. The plan of the paper is as follows. In Sec. II we fix t conventions and discuss the resummation technique in lF4 theory with Dirichlet boundary conditions. We solve th effective Klein-Gordon equation, and obtain the ‘‘improve eigenvalues. The solution is used to obtain the resumm Casimir energy, including radiative corrections. In Sec. we discuss the resummation of the vacuum energy in case of periodic boundary conditions; this sheds some l on the results of Sec. II. In the Conclusion we discuss drawbacks of this method as well as other minor points. nally, three Appendixes collect some mathematical res used in the paper. II. DIRICHLET BOUNDARY CONDITIONS Boundary conditions breaking the full Lorentz invarian in general pose new problems to the renormalization p gram. For some geometries and boundary conditions~de- pending also on the spin of the field! it may be necessary to introduce surface counterterms besides the bulk ones. instance, in the MIT bag model the free energy is ill defin at one-loop due to an extra singularity which shows up as surface is approached@17#. The standard recipe associates free parameter to each distinct singular term, included a new counterterm in the starting Lagrangian. If this proced continues to all orders, with the consequent loss of predic power, we say that the theory is non-renormalizable due the boundary conditions. In a remarkable paper, Symanzik gave strong argum showing the renormalizability of theF4 theory in the pres- ence of flat boundaries@18#. In particular, he showed that th renormalized Casimir pressure for disjoint boundaries Dirichlet BC is finite to all orders in perturbation theory. H also verified explicitly that no surface counterterms a needed in the computation of the two-loop vacuum ene F4-type theories are still renormalizable for more gene boundary conditions and surfaces, but at the price of in ducing surface counterterms@6,18#. We wish to point out here that many proposals have been made in order to a the surface-like singularities. These include, among oth the ‘‘softening’’ of the Dirichlet BC@19,20# or treating the 04500 n n n c- s e - e n of l - s, he ed I e ht e i- ts - or d e a e e to ts d e y. l - id s, boundaries as quantum mechanical objects with a non position uncertainty@21#. However, this question is outsid the scope of our present discussion. A key ingredient in our computation of the Casimir e ergy is the self-energy of the field, as it determines the s in the single-particle energy levels. Since there is some agreement among existent results in the literat @7,8,18,25#, we present its computation in some detail. A. Self-energy We work in D5(d11)11 dimensional Minkowski space-time, and definexm[(t,x,z), with x5(x1, . . . ,xd). The renormalized Lagrangian reads@\5c51, hmn5diag (1,2, . . . ,2)# L5L01LI5H 1 2 ~]F!22 1 2 m2F2J 1H 2 l 4! F41LctJ , ~1! with Lct the counterterm Lagrangian. We impose Dirichlet BC on a pair of plates atz50 and z5 l : F(z50)5F(z5 l )50. The bare Feynman propaga tor with Dirichlet BC may be written as an expansion multiple reflections@22#: DF~x,x8!5 ( n52` ` @DF (0)~xn2x18 !2DF (0)~xn2x28 !#, ~2! wherexn5(t,x,z12nl), x68 5(t8,x8,6z8), andDF (0) is the bulk free propagator, which forD.2 is given by@23# DF (0)~x! 5 1 ~2p!D/2 S m A2x21 i e D (D22)/2 K (D22)/2~mA2x21 i e!, ~3! with A2x21 i e5 iAx2 if x2.0 (e→01). Actually, what we are interested in isDF(x,x). It follows from Eq. ~2! that DF~x,x!5 ( n52` ` @DF (0)~2nl !2DF (0)~2z12nl !#. ~4! The termDF (0)(0) contains the usual UV singularity. It can b removed, as usual, by a mass renormalization. In the massless case the bulk free propagator gets sim fied, and it is possible to find a closed expression forD(x) [DF(x,x)2DF (0)(0). Using DF (0)~x;m50!5 GS D 2 21D 4pD/2 uxu22D, ~5! one finds, forD54, 4-2 t ll c it by s q ve d of ve m ss- re la- ore n- c- the int tor e rit- nt . tor WHEN THE CASIMIR ENERGY IS NOT A SUM OF . . . PHYSICAL REVIEW D 65 045004 D~x;m50!5 1 16p2l 2 F2 c8~1!2c8S z l D2c8S 12 z l D G , ~6! wherec(x) is the digamma function@24#. Equation~6! can be simplified to D~x;m50!52 1 16l 2 Fcsc2S pz l D2 1 3G ~D54!. ~7! Let us take a closer look at theD53 case, keepingm Þ0. We obtain from Eqs.~3! and ~4!, after changing the summation variables and using the explicit form ofK1/2(x), D~x!5 1 8p l F2e22mlS~2ml,1!2e22mzSS 2ml, z l D 2e22m( l 2z)SS 2ml,12 z l D G , ~8! where S~a,b![ ( n50 ` e2an n1b . ~9! The massless limit must be taken with care, as each of series in Eq.~8! is logarithmically divergent. As we sha show, the divergent terms cancel in the complete formula~8!. Indeed, the asymptotic limit ofS(2ml,b) asm→0 is given by S~2ml,b!5 ( n50 ` e22mlnS 1 n1b 2 1 n11D1 ( n50 ` e22mln n11 ; m→0 2g2c~b!2 ln~2ml!1O~ml!, ~10! whereg50.577 . . . is theEuler constant. The logarithmi terms cancel in Eq.~8!, so that we can now take the lim m→0 safely, obtaining D~x;m50!5 1 8p l F2g1cS z l D1cS 12 z l D G ~D53!. ~11! The renormalized one-loop self-energy is given S (1)(x)5(l/2)DF(x,x)1dm2. The mass counterterm i fixed by the condition liml→`S (1)50. This amounts to re- move the contribution of the bulk free propagator from E ~4!. With this choice of mass renormalization we ha S (1)(x)5(l/2)D(x). Therefore, the self-energy is infrare finite in the massless case also atD53, in disagreement with Ref. @25#. @However, it is infrared divergent in the case Neumann boundary conditions. The propagator is then gi by Eq. ~2! with the minus sign on its right-hand side~RHS! replaced by a plus sign. As a consequence, the logarith terms in the massless limit ofD(x) do not cancel.# 04500 he . n ic From now on, we shall focus our attention on the ma less case atD54. B. Radiative corrections to the Casimir energy: A heuristic approach Computations of the Casimir energy in the literature a restricted to perturbation theory. A non-perturbative calcu tion would be a very interesting result. Our goal here is m modest. We will discuss the computation of the Casimir e ergy in the approximation where the two-point Green fun tion is dressed with an arbitrary number of insertions of one-loop self-energy~‘‘daisy’’ resummation!. This approxi- mation contains the leading correction in a 1/N expansion. ~In our case, however,N51. With this caveat in mind, let us proceed.! The Casimir energy will be given formally by E5 1 2 ( a va , ~12! where va are the positive poles of the dressed two-po function G̃(2) in the frequency domain. To computeG̃(2) we note that it satisfies @]x 21S (1)~x!#G̃(2)~x,x8!52 i d (4)~x,x8!. ~13! As usual, the solution to Eq.~13! can be written as G̃(2)~x,x8!52 i( a fa~x!fa* ~x8! La , ~14! whereLa and fa(x) are the eigenvalues and~normalized! eigenfunctions, respectively, of the Klein-Gordon opera ]21S (1): @]21S (1)~x!#fa~x!5Lafa~x!. ~15! Since S (1)(x) is a function ofz alone, we can reduce th above equation to an ordinary differential equation by w ing f(x)5e2 ivt1 ip•xw(z): H 2 d2 dz2 2s22 p2g l 2 Fcsc2S pz l D2 1 3G J w~z!50, ~16! wheres2[L1v22p2 and g[l/32p2. Now we make the change of variablez5 ly /p and get S d2 dy2 1k21 g sin2 y D w~y!50 S k2[ l 2s2 p2 2 g 3D . ~17! Equation ~17! may be viewed as the time-independe Schrödinger equation for a particle of massm̃51/2 moving in the potentialV(y)52g csc2 y ~inverted Poschl-Teller!, with energyE5k2. Its solution is discussed in Appendix A In particular, it is shown thatk25(n1s)2 (n51,2, . . . ), with s[ 1 2 (211A124g). From the definition ofk2 ands2 it follows that the eigenvalues of the Klein-Gordon opera have the form 4-3 i th re in q do r- nt n- e o e ly - r, r - it ac- he ng he he - ith fi- that . It d to of an we we LUIZ C. DE ALBUQUERQUE AND R. M. CAVALCANTI PHYSICAL REVIEW D 65 045004 L52v21p21 p2 l 2 F ~n1s!21 g 3G ~n51,2, . . .!. ~18! From Eqs.~14! and~18! it follows that the~positive! poles of G̃(2) are given by vn~p!5Ap21 p2 l 2 F ~n1s!21 g 3G ~n51,2, . . .!. ~19! Before we proceed with the calculation of the Casim energy, a remark is in order here. As we have seen, ~renormalized! one-loop self-energy is a function ofx. It may be tempting to interpretS (1)(x) ~more generally, m2 1S (1)(x)) as a position-dependent~squared! mass of the field. A problem would then occur in regions whe S (1)(x),0, for this could imply the presence of tachyons the theory. For that reason, Ford and Yoshimura@7# argued that models which exhibits this behavior~such as the one we are considering! are unphysical. However, the analysis of E ~16!, summarized in Appendix A, shows that its solutions not have imaginary frequencies as long asl,lcrit58p2 ~which is anyway well outside the range of validity of pe turbation theory!. The one-loop effective theory is consiste in this case. On the other hand, forl.8p2 the Schro¨dinger equation ~16! leads to an energy spectrum which is u bounded from below, rendering the associated effective fi theory ill-defined. This solves a long-standing problem interpretation. Substituting the eigenfrequencies~19! into Eq. ~12! we obtain the following expression for the Casimir energy p unit area: E5 1 2 m122n ( n51 ` E d2p ~2p!2 H p21 p2 l 2 F ~n1s!21 g 3G J nU n51/2 . ~20! The formal sum over zero-point energies has been ana cally regularized; we shall setn51/2 at the end of the cal culation. The factorm122n, wherem is a mass paramete keeps the RHS of Eq.~20! with the dimension of energy pe unit area. Integrating Eq.~20! over p, we get E5 m122n 8p G~2n21! G~2n! S p l D 2(n11) HS 2n21;s, g 3D , ~21! where the functionH(z;s,a2) is defined as H~z;s,a2![ ( n51 ` @~n1s!21a2#2z. ~22! The series converges forRz.1/2. The analytical continua tion of H(z;s,a2) to the whole complexz plane is performed in Appendix B. Substituting the result into Eq.~21! we ob- tain @26# 04500 r e . ld f r ti- E5 m122n 8p G~2n21! G~2n! S p l D 2(n11)H 1 2 F ~11s!21 g 3Gn11 1 i E 0 ` dt f n~11 i t !2 f n~12 i t ! e2pt21 2 ~11s!2n13 2n13 3FS 2n21,2n2 3 2 ;2n2 1 2 ;2 g 3~11s!2D J , ~23! wheref n(x)[@(x1s)21g/3#n11 andF(a,b;g;z) is the hy- pergeometric function. From the definition of the latter follows that E has a simple pole atn51/2 ~in fact, it has poles atn523/2,21/2,1/2,3/2, . . . ).This requires thatE be renormalized before we setn51/2. In general, this is done by subtracting fromE its value atl→`. Unfortunately, such a prescription does not work in the present case, since, cording to Eq.~23!, the Casimir energy per unit area has t form E5C(n)/ l 2(n11). One can obtain a hint on what is going wrong by noti that the residue ofE at n51/2 is of second order ing ~or l). This is consistent with the fact that we have worked with t one-loop two-point Green function, which is~formally! cor- rect only to first order in the coupling constant. Since t lF4 theory is perturbatively renormalizable inD54, one may suspect that in order to obtain a finite~or at least renor- malizable! E to orderln one must work within an approxi mation in which the two-point Green function is dressed w the n-loop self-energy. As we show below, this is not suf cient or necessary. In spite of that, the argument suggests Eq. ~23! cannot be trusted beyond orderl. Expanding the RHS of Eq.~23! in a power series inl and makingn→1/2, we obtain E5 1 l 3 F2 p2 1440 1 l 9216 1•••G . ~24! The first term is the usual free Casimir energy~per unit area!. The second term is the leading radiative correction to it overestimates the correct result@18# by a factor of 2. This discrepancy occurs because the method we have use compute the Casimir energy only works in the absence interactions. To show this, we start by noting that one c define the Casimir energy as E5E dD21x^0uT00~x!u0&, ~25! whereTmn is the energy-momentum tensor. In the case are considering~the masslesslF4 theory inD54), we have T005 1 2 ~]0F!21 1 2 ~¹W F!21 l 4! F4. ~26! Moving the differential operators outside the brackets, can rewrite the vacuum expectation value ofT00 in terms of n-point Green functionsG(n): 4-4 - te m that s he e- er- e the ng rgy ut e di- WHEN THE CASIMIR ENERGY IS NOT A SUM OF . . . PHYSICAL REVIEW D 65 045004 ^0uT00~x!u0&5 lim x8→x 1 2 ~]0]081¹W •¹W 8!G(2)~x,x8! 1 l 4! G(4)~x, . . . ,x!. ~27! On the other hand, using Eq.~13! and the spectral represen tation of G̃(2), Eq. ~14!, one can easily show that 1 2 ( a va5E dD21x lim x8→x 1 2 @]0]081¹W •¹W 8 1S (1)~x!#G̃(2)~x,x8!. ~28! It follows that the sum of~one-loop! zero-point energies differs from the true vacuum energy by DE5E dD21xF lim x8→x 1 2 ~]0]081¹W •¹W 8!DG(2)~x,x8! 2 1 2 S (1)~x!G̃(2)~x,x!1 l 4! G(4)~x, . . . ,x!G , ~29! whereDG(2)[G(2)2G̃(2). While the first line of Eq.~29! is formally O(l2), the second one isO(l). This explains why the second term in Eq.~24! is incorrect. It is important to note that Eqs.~12! and~26! would lead to distinct results even if we had worked with the comple two-point Green function. The difference between the would then be given by m s d - it 04500 DE5E dD21xH 2 1 2E dDyS~x,y!G(2)~y,x! 1 l 4! G(4)~x, . . . ,x!J . ~30! A perturbative evaluation of the above expression shows DE would still be of orderl. Physically, this discrepancy i due to the fact that, in contrast with the free theory, t interacting theory is not equivalent to a collection of ind pendent harmonic oscillators. The sum of zero-point en gies, Eq.~12!, takes into account only the Lamb shift on th single-particle energy levels caused by the interaction; differenceDE accounts for the residual interaction amo the ~anharmonic! oscillators. The above discussion also shows that the Casimir ene is not determined solely by the two-point Green function, b also ~in the lF4 theory! by the four-point function. In par- ticular, in order to consistently remove theO(l2) UV singu- larity in Eq. ~23! one must not only obtainG(2) to that order, but alsoG(4) to O(l). These ideas will be illustrated in th next section in the simpler case of periodic boundary con tions. III. PERIODIC BOUNDARY CONDITIONS A. Conventional perturbation theory The free Feynman propagator for the fieldF obeying periodic boundary conditions in thez-direction, F(t,x,z 1R)5F(t,x,z), is given by DF~x,x8!5 i RE dv 2p ( n52` ` E ddp ~2p!d e2 ipm(xm2xm8 ) v22p22qn 22m21 i e 5 1 2R ( n52` ` E ddp ~2p!d e2 ivn(p)ut2t8u1 ip•(x2x8)1 iqn(z2z8) vn~p! , ~31! op where pm5(v,p,qn), qn52pn/R, and vn(p) [Ap21qn 21m2. SinceDF(x,x8)5DF(x2x8), such bound- ary conditions do not break translational invariance. The renormalized vacuum energy density may be co puted from Eqs.~25!–~27!, but its perturbative expansion i more easily derived from the vacuum persistence amplitu «5 lim T→` i VT lnF E DF expS i E dDxLD G1L. ~32! The last term in Eq.~32! is fixed by the renormalization condition limR→`«(x)50. Due to the translational invari ance the vacuum energy density does not depend onx. ~A remark on notation:« denotes the Casimir energy per un volume, while E denotes the Casimir energy per unitarea. They are related, in the case of periodic BC, by«5E/R.! - e: To first order inl, we obtain from Eq.~32! the well- known results@«5« (0)1« (1)1•••, « (n)5O(ln)# « (0)5 1 2R ( n52` ` E ddp ~2p!d vn~p!1L (0), ~33! « (1)5 l 8 @DF~0!#21 1 2 dm2DF~0!1L (1), ~34! wheredm2 is the one-loop mass counterterm. The one-lo self-energy is given by S (1)5 l 2 DF~0!1dm2. ~35! 4-5 en e d less er he ber of the u- ct- it he rgy in ss - is in n- LUIZ C. DE ALBUQUERQUE AND R. M. CAVALCANTI PHYSICAL REVIEW D 65 045004 In order to compute the quantities above, it is conveni to define the function Ce~a![ me 2R ( n52` ` E dd2ep ~2p!d2e ~p21qn 21m2!a, ~36! where m is an arbitrary mass scale. We then haveDF(0) 5 lime→0 Ce(21/2) and« (0)5 lime→0@Ce(1/2)1L (0)#. The computation ofCe is discussed in Appendix C. Ther we show thatCe may be written~in the limit e→0) as the sum of two terms, namely lim e→0 Ce~a!5A~a!1B~a!, ~37! where A(a) and B(a) are given by Eqs.~C8! and ~C9!, respectively. OnlyA(a) depends onR, and vanishes when R→`. Before computing« (0) and« (1) we give our renormaliza- tion conditions. To first order inl two conditions are re- quired. We fixL and dm2 by the conditions limR→` «(R) 50 and limR→` S (1)(R)50, respectively. This givesL (0) 52B(1/2), L (1)5(l/8)@B(21/2)#2, and dm25 2(l/2)B(21/2). It follows that (e→0) « (0)~R!5A~1/2!52 2 ~2p!D/2 S m RD D/2 FS D 2 ;mRD , ~38! « (1)~R!5 l 8 @A~21/2!#2 5 l 2~2p!D S m RD D22FFS D 2 21;mRD G2 , ~39! whereF(s;a) is defined in Eq.~C5!. Taking D54 and ex- panding in powers ofmR @Eqs.~C6! and ~C7!# we thus ob- tain «5 1 R4 H F2 p2 90 1 ~mR!2 24 2 ~mR!3 12p 1•••G 1lF 1 1152 2 mR 192p 1•••G J 1••• . ~40! Analogously, we obtain for the one-loop self-energy S (1)5 l ~2p!D/2 S m RD (D22)/2 FS D 2 21;mRD 5 l R2 F 1 24 2 mR 8p 1 ~mR!2 16p2 ln~mR!1•••G ~D54!. ~41! The first term does not depend onm, and is sometimes calle ‘‘topological mass’’~squared!. For reasons discussed in@1#, we prefer the name ‘‘compactification mass’’ forM [(l/24R2)1/2. 04500 t B. Resummed perturbation theory From now on, let us focus the discussion on the mass theory (m50) in D54. In this case, the second and high order terms in the perturbative expansion of« are plagued with IR divergences. A qualitative analysis shows that t most IR divergent diagrams are the ‘‘ring’’~or ‘‘daisy’’ ! ones.~These are just the diagrams with the greatest num of insertions of the one-loop self-energy in each order perturbation theory.! As in the case of TFT@5#, it is possible to sum these diagrams to all orders. The result is finite in IR and is nonanalytic inl, as we show below. To avoid overcounting of diagrams in higher order calc lations, it is convenient to redefine the free and the intera ing parts of the Lagrangian by adding and subtracting to the compactification mass term12 M2F2 @10–12#: L5L̃01L̃I5H 1 2 ~]F!22 1 2 M2F2J 1H 2 l 4! F41 1 2 M2F21LctJ . ~42! The free propagator~in momentum space! is now given by D̃F~p!5 i p22M21 i e . ~43! It coincides with the propagator of the original theory in t daisy approximation. We remark that in a loop expansion of the vacuum ene ~or of any other quantity! each insertion of the mass term L̃I is to be formally counted as one loop, like the ma counterterm—otherwise takingL̃0 as the new free Lagrang ian would not cure the IR divergence problem@27#. The one-loop approximation to the vacuum energy given by «̃ (1)5 1 2R ( n52` ` E d2p ~2p!2 ~p21qn 21M2!1/2. ~44! Using the results of Sec. III A and of Appendix C we obta «̃ (1)~R!52 M2 2p2R2 F~2;MR! 1 lim e→0 meM42e 242ep (32e)/2 GS 221 e 2D GS 2 1 2D 5 1 R4 F2 p2 90 1 l 576 2 l3/2 288A6p 1OS l2 e D G . ~45! As in the Dirichlet BC case, theO(l) term in the one- loop approximation is twice the value obtained in conve 4-6 loo . o e n h ed en re u a h ct di D d ne as - th ble s if don e ion n- to take n a as en- t all the tial the In col- ed of can i re- ion ns that en- of s. he ies tion ies ay n m- m WHEN THE CASIMIR ENERGY IS NOT A SUM OF . . . PHYSICAL REVIEW D 65 045004 tional perturbation theory@Eq. ~40! with m50#. In order to reproduce the latter one has to take into account the two- contribution to«, given by «̃ (2)5 l 8 @D̃F~0!#22 1 2 M2D̃F~0!, ~46! where D̃F(x)5DF(x;m5M ). Using again results of Sec III A and of Appendix C we obtain D̃F~0!5 M 2p2R F~1;MR!1 lim e→0 meM22e 242ep (32e)/2 GS 211 e 2D GS 1 2D 5 1 R2 F 1 12 2 l1/2 8pA6 1OS l e D G . ~47! Substituting this into Eq.~46! yields «̃ (2)~R!5 1 R4 F2 l 1152 1OS l2 e D G . ~48! Thus, to orderl2 we finally obtain «~R!5 1 R4 F2 p2 90 1 l 1152 2 l3/2 288A6p 1OS l2 e D G . ~49! This agrees to orderl with the result found in Sec. III A~in the limit m→0). Besides, we have obtained a correction orderl3/2. This nonanalyticity inl is a consequence of th fact that the loop expansion in the rearranged Lagrangia equivalent to a resummation of an infinite number of grap in the conventional perturbation expansion. Finally, we note that the UV singularities in the resumm theory depend onR, via their dependence onM. For instance, in the computation of«̃ (1) a singular term of the form M4/e;l2/eR4 appears in the limite→0. In the analogous case of TFT it can be shown that the UV singularity pres in the one-loop free energy cancels against two- and th loop contributions in the resummed theory, including a co pling constant renormalization counterterm@12,28#. These contributions on their turn introduce new singularities O(l3), which are cancelled by including higher order grap in the resummed theory, and so on. The situation is exa the same in our case, so we can safely neglect theO(l2) term in Eq.~49!. IV. CONCLUSION In this paper we have discussed the computation of ra tive corrections to the Casimir energy of the masslesslF4 theory confined between two parallel plates. The case of richlet boundary conditions at the plates was discusse Sec. II. We obtained an analytical expression for the o loop self-energyS (1)(x) both in D53 and D54. The former was shown to be free of IR singularities, in contr with the claim made in@25#. 04500 p f is s t e- - t s ly a- i- in - t In the ‘‘daisy’’ resummation of the two-point Green func tion one is led to solve a Klein-Gordon equation wi S (1)(x) acting as an effective scalar potential. We were a to solve this equation in four dimensions. In spite ofS (1) being negative everywhere, there are no tachyonic mode the coupling constantl is smaller thanlcrit58p2. We then computed the sum of the eigenenergies of the Klein-Gor operator. Expanding the result in a power series inl one discovers that theO(l) correction does not agree with th result of conventional perturbation theory, and the correct of orderl2 contains a UV singularity which apparently ca not be renormalized away. The first problem was shown occur because the sum of zero-point energies does not into account all the contributions to the vacuum energy i theory with interaction. As for the second problem, it w argued that the consistent renormalization of the Casimir ergy at a given order requires that one takes into accoun diagrams to that order. This conjecture is supported by fact that the Dirichlet BC~in the case of flat boundaries! do not spoil the perturbative renormalizability of thelF4 4 theory @18#. In the case of periodic boundary conditions in one spa direction we have argued that the infrared properties of model are analogous to the one in thermal field theory. order to define a consistent~i.e., IR finite! perturbative ex- pansion one has to include the screening effects due to lective excitations. A solution to this problem was propos by Braaten, Pisarski and others in thermal field theory@10– 12#. It consists in the resummation of an infinite class diagrams, which gives the field an effective mass. This be done in a systematic way using the Braaten-Pisarsk summation method. This was illustrated with the calculat of the leading and next-to-leading order radiative correctio to the Casimir energy. Besides, our calculation shows the resummed weak coupling expansion of the Casimir ergy in the case of periodic BC contains fractional powers l, in contrast to the case of Dirichlet boundary condition We note that calculations of radiative corrections to t Casimir energy via the resummation of zero-point energ have appeared recently in the literature@29#, without paying due attention to the subtleties of the resummed perturba theory. As we have shown, this may lead to inconsistenc in the results@30#. Finally, we hope that the techniques discussed here m be useful in investigations of Kaluza-Klein compactificatio scenarios, as well as in the study of surface critical pheno ena. ACKNOWLEDGMENTS The authors acknowledge the financial support fro FAPESP under grants 00/03277-3~L.C.A.! and 98/11646-7 ~R.M.C.!, and from FAPERJ~R.M.C.!. They also acknowl- edge the kind hospitality of the Departamento de Fı´sica Matemática, Universidade de Sa˜o Paulo, where this work was initiated. 4-7 lu - r gi e a - t n - - LUIZ C. DE ALBUQUERQUE AND R. M. CAVALCANTI PHYSICAL REVIEW D 65 045004 APPENDIX A We discuss the solution to the equation S d2 dy2 1k21 g sin2 y D w~y!50, ~A1! with Dirichlet boundary conditions aty50 andy5p. Let us first consider the asymptotic behavior of its so tions near one of the boundaries~say, aty50). To this end, we can replace Eq.~A1! by S d2 dy2 1 g y2D w~y!50. ~A2! The most general solution to Eq.~A2! is w~y!5A ys11B ys2, ~A3! wheres6[ 1 2 (16A124g). If g,1/4, the boundary condi tion w(0)50 is not enough to fix the relation betweenA and B, as bothys1 andys2 vanish aty50. To resolve this inde- terminacy, we follow@31# and regularize the potential nea the origin:VR(y)52g/y2 for y.a, andVR(y)52g/a2 for y,a. At the end, we shall take the limita→0. For y.a, the solution is given by Eq.~A3!. For y,a, the solution which satisfies the boundary condition at the ori is w(y)5C sin(Agy/a). Continuity of w(y) and its deriva- tive at y5a implies the relationB/A;as12s2 asa→0, i.e., only the solution with the faster decay at the origin surviv when the regularization is removed. Ifg.1/4, s12s2 is purely imaginary and lima→0B/A does not exist. This sets critical value to g, namely gcrit51/4, above which the ‘‘Hamiltonian’’ H52d2/dy22g/y2 is unbounded from be low @31#. Let us return to the complete equation~A1!. It is conve- nient to make some changes of variables. First, we sey 5 ix1p/2 and defines[ 1 2 (211A124g) @so that s(s 11)52g#. This transforms Eq.~A1! into F2 d2 dx2 1k22 s~s11! cosh2 x Gw~x!50. ~A4! Then, we makej5tanhx and obtain d dj F ~12j2! dc dj G1Fs~s11!2 k2 12j2Gw~j!50. ~A5! Finally, we put w(j)5(12j2)k/2w(j), followed by the change of variablej52u21, to get u~12u! d2w du2 1@11k22~11k!u# dw du 2~k2s!~k1s11!w 50. ~A6! Equation~A6! is the hypergeometric differential equatio @24# with parametersa5k2s, b5k1s11, andg511k. Its general solution may be written as 04500 - n s w~u!5A~12u!2kF~2s,s11;11k;u! 1B u2kF~2s,s11;12k;u!, ~A7! whereF(a,b;g;z) is the hypergeometric function. Return ing to the variabley and the functionw(y), we have w~y!5A8e2 ikyFS 2s,s11;11k; i e2 iy 2 sinyD 1B8eikyFS 2s,s11;12k; i e2 iy 2 sinyD . ~A8! The asymptotic behavior ofw(y) asy→0 may be extracted from limz→0 F(a,b;g;z)51, after using the relation @valid for uarg(2z)u,p, uarg(12z)u,p, a2bÞ0,61, 62, . . .# F~a,b;g;z! 5~2z!2a G~g!G~b2a! G~g2a!G~b! FS a,11a2g;11a2b; 1 zD 1~2z!2b G~g!G~a2b! G~g2b!G~a! FS b,11b2g;11b2a; 1 zD . ~A9! In this way, w~y! ; y→0 A8FS 2s,s11;11k; i 2yD 1B8FS 2s,s11;12k; i 2yD ;FA8 G~11k! G~11k1s! 1B8 G~12k! G~12k1s!GG~2s11! G~s11! 3S 2 i 2yD s 1FA8 G~11k! G~k2s! 1B8 G~12k! G~2k2s!G 3 G~22s21! G~2s! S 2 i 2yD 2s21 . ~A10! Recalling the analysis of Eq.~A2!, we impose thatw(y) ;ys11 as y→0. This can be accomplished by takingB8 50 andk52(n1s)(n51,2, . . . ),that is w~y!5A8ei (n1s)yFS 2s,s11;2s2n11; i e2 iy 2 sinyD . ~A11! The boundary condition aty5p is also satisfied by this so lution, sincew(y);(p2y)s11 as y→p. Hence, Eq.~A11! is an acceptable solution to Eq.~A1!. @Remark: we could also have takenA850 and k5n1s (n51,2, . . . ) in Eq. ~A10!, but this leads to the same values ofk2 and to the same solutions given by Eq.~A11!.# 4-8 WHEN THE CASIMIR ENERGY IS NOT A SUM OF . . . PHYSICAL REVIEW D 65 045004 APPENDIX B The goal here is to obtain the analytical continuation of the functionH(z;s,a2), defined in Eq.~22!, to the whole complex z plane. To this end, we shall use the Plana summation formula@32# ( k5M N f ~k!5 1 2 @ f ~M !1 f ~N!#1E M N f ~x!dx2 i E 0 ` dy f ~N1 iy !2 f ~M1 iy !2 f ~N2 iy !1 f ~M2 iy ! e2py21 . ~B1! a - f d. In the present case, we chooseM51, N5`, and f (x) 5@(x1s)21a2#2z. In order to apply the Plana formul some conditions have to be satisfied@32#. First, we assume that Rz. 1 2 , so that the series in Eq.~22! converges abso lutely. Then, it can be shown that the functionf (t1 i t ) is holomorphic for t>1 for any t, and that limt→6`e22putu f (t1 i t )50 uniformly in the interval 1