The two-loop massless (lambda/4!)phi(4) model in nontranslational invariant domain

M. Aparicio Alcalde, G. Flores Hidalgo, and N. F. Svaiter

Citation: Journal of Mathematical Physics 47, 052303 (2006); doi: 10.1063/1.2194632

View online: http://dx.doi.org/10.1063/1.2194632

View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/47/5?ver=pdfcov Published by the AIP Publishing

The two-loop massless

_„␭/4!…

␸

4

model in nontranslational

invariant domain

M. Aparicio Alcaldea兲

Centro Brasileiro de Pesquisas Físicas-CBPF, Rua Dr. Xavier Sigaud 150, Rio de Janeiro, RJ, 22290-180, Brazil

G. Flores Hidalgob兲

Instituto de Física Teórica-IFT/UNESP, Rua Pamplona 145, São Paulo, SP, 01405-900, Brazil

N. F. Svaiterc兲

Centro Brasileiro de Pesquisas Físicas-CBPF, Rua Dr. Xavier Sigaud 150, Rio de Janeiro, RJ, 22290-180, Brazil

共Received 9 December 2005; accepted 17 March 2006; published online 12 May 2006兲 We study the共␭/4!兲␸4_{massless scalar field theory in a four-dimensional Euclidean}

space, where all but one of the coordinates are unbounded. We are considering Dirichlet boundary conditions in two hyperplanes, breaking the translation invari-ance of the system. We show how to implement the perturbative renormalization up to two-loop level of the theory. First, analyzing the full two and four-point func-tions at the one-loop level, we show that the bulk counterterms are sufficient to render the theory finite. Meanwhile, at the two-loop level, we must also introduce surface counterterms in the bare Lagrangian in order to make finite the full two and also four-point Schwinger functions. © 2006 American Institute of Physics. 关DOI:10.1063/1.2194632兴

I. INTRODUCTION

In this paper we are interested to show how to implement the renormalization procedure up to two-loop level in the massless共␭/4!兲␸4_{scalar field theory, defined in a four-dimensional}

Euclid-ean space with one compactified dimension. Our aim is to shed light on the renormalization procedure in a system defined in a domain where translational symmetry is broken, which must be done, for example, in the high temperature dimensional reduced quantum chromodynamics 共QCD兲, defined in a finite region.

Quantum chromodynamics is a non-Abelian Yang-Mills theory with gauge group SU共3兲. Since it is assumed that the fermions of the theory transform according to the fundamental rep-resentation of the gauge group, each flavor of quark is a triplet of the color group SU共3兲. Gauge bosons transform according to the adjoint representation. The interaction between the quarks is mediated by the gluons. Due to the non-Abelian structure of the theory, the gluons couple not only with the quarks but have also cubic and quartic self-interaction. The self-interaction of the gluons provides the antiscreening of the color charge in QCD. This is responsible for asymptotic freedom and presumably confinement.

The confinement-deconfinement phase transition in QCD may occur in usual matter at suffi-ciently high temperature or if it is strongly compressed.1–3In ultrarelativistic heavy ion collisions, we expect that the plasma of quarks and gluons can be produced. We would like to stress that, although nonequilibrium processes occur in the quark-gluon plasma in the heavy ion collisions, for

a兲_{Electronic mail: aparicio@cbpf.br} b兲_{Electronic mail: gflores@ift.unesp.br} c兲_{Electronic mail: nfuxsvai@cbpf.br}

47, 052303-1

simplicity in a first approximation we can assume a static situation. Just after the collision hot and compressed nuclear matter is confined in a small region of the space and in such circumstances the volume and surface effects become very important.

In the above described physical situation there are two important points: the first one is that the thermodynamic limit of the infinite volume system cannot be used and therefore finite volume effects should be investigated and taken into account. The second point is that the quark-gluon plasma exists in a situation of high temperature, where using the Matsubara formalism to describe high temperature QCD, dimensional reduction must occur.4–7Dimensional reduction is based on the Appelquist-Carrazone decoupling theorem.8 From a more fundamental theory, the effective Lagrangian density of this theory can be obtained as some low-energy limit of the fundamental theory where the heavy modes have been removed. There are some interesting physical situations where the decoupling theorem can be used. First, for scalar fields without spontaneous symmetry breaking. Second, in quantum electrodynamics, where a derivative expansion of the photon effec-tive action can be obtained by integrating out the fermionic fields. Also in QCD at least in lowest order in perturbation theory the decoupling theorem works. The decoupling theorem is not valid, for example, in spontaneous broken gauge theories. It is important to stress that the nonvalidity of the decoupling theorem means that low-energy experiments can provide information about the high energy physics.

Going back to the heavy-ions collision situation, we can assume that the following scenario appears: in the situation where dimensional reduction occurs, we have an effective theory for the gluons field and also finite size effects for these bosonic fields. To shed light on the renormaliza-tion procedure in systems defined in domain where translarenormaliza-tional symmetry is broken, as for example, the high temperature dimensional reduced QCD, in this paper we are interested to investigate scalar models, impose classical boundary condition over the fields. We hope that this study will give us some insight over the most interesting and also more complicated situation as the one mentioned above. Therefore, in this paper we analyze how to implement the perturbative renormalization up to two-loop level of共␭/4!兲␸4 _{massless scalar field model defined in a}

four-dimensional Euclidean space with one compactified dimension.

Finite size effects and the presence of macroscopic structures in different field theory models have been extensively studied in the literature. The critical behavior of the O共N兲 model in the presence of a surface was a target of intense investigations.9The same O共N兲 model was studied in two different geometries: the periodic cube and the cylinder along one dimension共the time兲 and finite and periodic in the 共d−1兲 remaining dimensions by Brezin and Zinn-Justin.10 Finite size effects in QCD11and also in different field theory models have also been extensively studied in the literature. Assuming periodic or antiperiodic boundary conditions for bosonic and fermionic mod-els, respectively, the translation symmetry is maintained, and surface effects are avoided. There-fore, to avoid surface effects, quantum fields defined in manifolds with periodic or anti-periodic boundary conditions in the spatial section were preferred by many authors.12 Nevertheless, the case of boundaries conditions that break the translational symmetry deserves our attention.

In the case of hard boundary conditions as, for example, Dirichlet-Dirichlet 共DD兲 or Neumann-Neumann 共NN兲, the translational invariance is lacking. This fact makes the Feynman diagrams harder to compute than in an unbounded space. Moreover the renormalization program is implemented in a different way from unbounded or translational invariance systems since some surface divergence appears.13 For translational invariant systems, one can use the momentum space representation, which is a more convenient framework to analyze the ultraviolet divergences of a theory. Translational invariance is preserved for momentum conservation conditions. For nontranslational invariant systems a more convenient representation for the n-point Schwinger functions is a mixed momentum coordinate space.

Fosco and Svaiter considered the anisotropic scalar model in a d-dimensional Euclidean space, where all but one of the coordinates are unbounded. Translational invariance along the bounded coordinate which lies in the interval 关0,L兴 is broken because the choice of boundary condition chosen for the hyperplanes at z = 0 and z = L. Two different possibilities of boundary conditions were considered:共DD兲 and also 共NN兲, and the renormalization of the two-point

func-tion was achieved in the one-loop approximafunc-tion.14Further, the renormalization of the four-point function was achieved in the one-loop approximation by Caicedo and Svaiter.15Finally Svaiter16 studied the renormalization of the共␭/4!兲␸4_{massless scalar field model in the one-loop}

approxi-mation in finite size systems assuming that the system is in thermal equilibrium with a reservoir. Also, still studying surface, edge, and corners effects, Rodrigues and Svaiter17analyzed first the renormalized vacuum fluctuations associated with a massless real scalar field, confined in the interior or a rectangular infinitely long waveguide. A closed form of the analytic continuation of the local zeta function in the interior of the waveguide was obtained and a detailed study of the surface and edge divergences was presented. Next, these authors18studied the renormalized stress tensor associated with an electromagnetic field in the interior of a rectangular infinitely long waveguide.

In this paper we will consider an interacting massless scalar model, in a four-dimensional Euclidean space, where the first three coordinates are unbounded and the last one lies in the interval关0,L兴. We analyze only DD boundary conditions. First, we present an algebraic expression in coordinate space for the free propagator which let us identify the divergences of the n-point Schwinger functions for the interacting theory. This algebraic expression agrees with the result obtained by Lukosz.19We would like to stress that instead of assuming hard boundary conditions, some authors assumed soft boundary conditions and also treated the boundary as a quantum mechanical object.20 Here, we prefer to keep hard classical boundary conditions.

The organization of the paper is as follows: In Sec. II we discuss the slab configurations, obtaining some important expressions for the free propagator in order to understand some proce-dures in the divergence identification. In Sec. III the regularization program is implemented in the one-loop approximation. In Sec. IV the regularization program is implemented in the two-loop approximation. Section V contains our conclusions. In the Appendix, an expression for the free propagator is introduced. Throughout this paper we useប=c=1.

II. CLASSICAL BOUNDARY CONDITIONS AND SOME PROPERTIES OF THE FREE PROPAGATOR

Let us consider a neutral scalar field with a共␭␸4兲 self-interaction, defined in a d-dimensional Minkowski space-time. The vacuum persistence functional is the generating functional of all vacuum expectation value of time-ordered products of the theory. The Euclidean field theory can be obtained by analytic continuation to imaginary time allowed by the positive energy condition for the relativistic field theory. In the Euclidean field theory, we have the Euclidean counterpart for the vacuum persistence functional, that is, the generating functional of complete Schwinger func-tions. The共␭␸4兲dEuclidean theory is defined by these Euclidean Green’s functions. The Euclidean

generating functional Z共h兲 is formally defined by the following functional integral:

Z共h兲 =

冕

关d␸兴exp

冉

− S0− SI+

冕

ddxh共x兲␸共x兲

冊

, 共1兲

where the action that describes a free scalar field is

S₀共␸兲 =

冕

ddx

冉

1

2共⳵␸兲

2₊1

2m0

2_␸2_共x兲

冊

_{, 共2兲}

and the interacting part, defined by the non-Gaussian contribution, is

SI共␸兲 =

冕

ddx

␭ 4!␸

4_{共x兲. 共3兲}

In Eq. 共1兲, 兩d␸兩 is a translational invariant measure, formally given by 兩d␸兩 =兿xd␸共x兲. The

Finally, h共x兲 is a smooth function that we introduce to generate the Schwinger functions of the theory by means of functional derivatives. Note that we are using the same notation for functionals and functions, for example, Z共h兲 instead of the usual notation Z关h兴.

In the weak-coupling perturbative expansion, we perform a formal perturbative expansion with respect to the non-Gaussian terms of the action. As a consequence of this formal expansion, all the n-point unrenormalized Schwinger functions are expressed in a power series of the bare coupling constant g0. Let us summarize how to perform the weak-coupling perturbative expansion

in the共␭␸4_兲

d theory. The Gaussian functional integral Z0共h兲 associated with the Euclidean

gener-ating functional Z共h兲 is

Z0共h兲 = N

冕

关d␸兴 exp

冉

−

1

2␸K␸+ h␸

冊

. 共4兲

We are using a compact notation and the first term on the right-hand side of Eq.共4兲 is given by ␸K␸=

冕

dd_x

冕

_dd_{y␸共x兲K共m}

0;x,y兲␸共y兲. 共5兲

The term that couples linearly the field with the external source is

h␸=

冕

dd_{x␸共x兲h共x兲. 共6兲}

As usual N is a normalization factor and the symmetric kernel K共m0; x , y兲 is defined by

K共m0;x,y兲 = 共− ⌬ + m02兲␦

d_{共x − y兲, 共7兲}

where⌬ denotes the Laplacian in the Euclidean space Rd. As usual, the normalization factor is defined using the condition Z0共h兲兩h=0= 1. ThereforeN=共det共−⌬+m0

2_兲兲1/2_{but, in the following, we}

are absorbing this normalization factor in the functional measure. It is convenient to introduce the inverse kernel, i.e., the free two-point Schwinger function G0共m0; x − y兲 which satisfies the identity

冕

dd_zG

0共m0;x − z兲K共m0;z − y兲 =␦d共x − y兲. 共8兲

Since Eq.共4兲 is a Gaussian functional integral, simple manipulations, performing only Gaussian integrals, gives

冕

关d␸兴e−S0+兰ddxh共x兲␸共x兲_{= exp}

冋

1 2

冕

d d_x

冕

_dd_yh共x兲G 0 共2兲_共m 0;x − y兲h共y兲

册

. 共9兲

Therefore, we have an expression for Z0共h兲 in terms of the inverse kernel G0 共2兲_共m

0; x − y兲, i.e., in

terms of the free two-point Schwinger function. This construction is fundamental to perform the weak-coupling perturbative expansion with the Feynman diagramatic representation of the pertur-bative series. The non-Gaussian contribution in a perturbation with regard to the remaining terms of the action. It is important to point out that the weak-coupling perturbative expansion can be defined in arbitrary geometries, and classical boundary conditions must be implemented in the two-point Schwinger function. Another way is to restrict the space of functions that appear in the functional integral.

We are interested in studying finite size systems, where the translational invariance is broken. In this situation, we are analyzing the perturbative renormalization for the 共␭/4!兲␸4 _massless

scalar field model, in the two-loop approximation. Therefore, let us assume boundary conditions over the plates for the massless field ␸共x兲. For simplicity we are assuming Dirichlet-Dirichlet boundary conditions, i.e.,

␸共rជ,z兲兩z=0=␸共rជ,z兲兩z=L= 0, 共10兲

for the free field. Since the translational invariance is not preserved, let us use a Fourier expansion of the fields in the following form:

␸共rជ,z兲 = 1

共2␲兲共d−1兲/2

冕

dd−1p

兺

n

␾n共pជ兲eipជ,rជun共z兲, 共11兲

where the set un共z兲 are the orthonormalized eigenfunctions associated to the operator −d2/ dz2,

关−d/dz2_u

n共z兲=kn2un共z兲兴, and kn= n␲/ L, n = 1 , 2 , . . . . The orthonormal set corresponding to the

eigenfunctions of the Hermitian operator −d2_{/ dz}2 _{defined on a finite interval is given by}

un共z兲 =

冑

2

Lsin

冉

n␲z

L

冊

, n = 1,2, . . . . 共12兲

These eigenfunctions satisfy the completeness and orthonormality relations, i.e.,

兺

_n un共z兲un*共z

⬘

兲 =␦共z − z

⬘

兲 共13兲 and

冕

0 L dz un共z兲un⬘ * _{共z兲 =␦} n,n⬘. 共14兲

Since we are interested in performing the weak coupling expansion, let us first write the free two-point Schwinger function. This free two-point Schwinger function can be expressed in the following form: G₀共2兲共rជ,z,z

⬘

兲 = 1 共2␲兲d−1

冕

dd−1p

兺

n eipជ.rជ_u n共z兲un*共z

⬘

兲G0,n共pជ兲, 共15兲 where G0,n共pជ兲 is given by G0,n共pជ兲 = 共pជ2+ kn 2 + m2兲−1. 共16兲

Next, we will present some properties of the two-point free Schwinger function in order to understand the behavior of the interacting field theory in the presence of macroscopic structures. Therefore, in order to understand some procedures used in the identification of the divergences in the Schwinger functions that will appear in the next section, let us analyze some properties of the free two-point Schwinger function. Substituting Eq.共12兲 and Eq. 共16兲 in Eq. 共15兲 we get that the free propagator G₀共2兲共rជ₁− rជ₂, z₁, z₂兲 can be written as

G₀共2兲共rជ1− rជ2,z1,z2兲 = 2 L

兺

_n=1 ⬁ sin

冉

n␲z1 L

冊

sin

冉

n␲z2 L

冊

冕

dd−1p 共2␲兲d−1 eipជ.共rជ1−rជ2兲

冉

pជ2₊

冉

n␲ L

冊

2 + m2

冊

. 共17兲

The next step is to show that the two-point free Schwinger function can be written in terms of the variables: r12, z12

−

and finally z₁₂+, where r12=共兩rជ1− rជ2兩兲/L, z12 −

=共z1− z2兲/L, and z12 +

=共z1+ z2兲/L,

re-spectively. Working in the four-dimensional case and also in the massless situation, a straightfor-ward calculation共see the Appendix兲 gives us that G₀共2兲共rជ₁− rជ₂, z₁, z₂兲 can be written as

G₀共2兲共rជ1− rជ2,z1,z2兲 = 1 16␲2L2_k=−

兺

_⬁ ⬁

冦

1

冉

k −兩z12 −_兩 2

冊

2 +

冉

r12 2

冊

2− 1

冉

k −兩z12 +_兩 2

冊

2 +

冉

r12 2

冊

2

冧

. 共18兲

The former expression for the two-point Schwinger function was obtained also by Lukosz19using the image method. Performing the summations in Eq.共18兲 共see the Appendix兲, it is possible to find a closed expression for G₀共2兲共rជ1− rជ2, z1, z2兲. We get

G₀共2兲共rជ1− rជ2,z1,z2兲 = sinh共␲r12兲 16␲L2_r 12

冦

sin

共

␲z1 L

兲

sin

共

␲z2 L

兲

冋

sinh2

冉

␲r12 2

冊

+ sin 2

冉

␲z12− 2

冊

册冋

sinh 2

冉

␲r12 2

冊

+ sin 2

冉

␲z12+ 2

冊

册

冧

. 共19兲 It is not difficult to show that the two-point Schwinger function G₀共2兲共rជ₁− rជ₂, z₁, z₂兲 satisfies the following properties:

共i兲 The free two-point Schwinger function is not negative, i.e., G₀共2兲共rជ1− rជ2, z1, z2兲艌0, for

z1, z2苸关0,L兴 and rជ1, rជ2苸R3, since we are working in a Euclidean space.

共ii兲 The free two-point Schwinger function is zero when one of its points are evaluated on the boundaries

G₀共2兲共rជ₁− rជ₂,0,z₂兲 = G₀共2兲共rជ₁− rជ₂,L,z₂兲 = G₀共2兲共rជ₁− rជ₂,z₁,0兲 = G₀共2兲共rជ₁− rជ₂,z₁,L兲 = 0, since we are assuming Dirichlet boundary conditions.

共iii兲 The free two-point Schwinger function contain the usual bulk divergences, i.e., when 共rជ1, z1兲=共rជ2, z2兲, it is singular. From Eq. 共18兲 we can identify three singular terms.

Split-ting the free two-point Schwinger function in the singular and regular terms we have

G₀共2兲共rជ1− rជ2,z1,z2兲 = 1 4␲2L2

再

1 共z12−兲2+ r122 − 1 共z12+兲2+ r122 − 1 共2 − z12+兲2+ r122

冎

+ 1 4␲2L2

冦

_k=−

兺

_⬁ k⫽0 ⬁ 1 共2k − 兩z12 −_兩兲2_{+ r} 12 2 −

兺

k=−⬁ k⫽0 ⬁ 1 共2k − 兩z12 +_兩兲2_{+ r} 12 2

冧

. 共20兲

The first term on the right-side of the last equation, is singular only when rជ1= rជ and z2 1

= z2. This is the term that carries the usual bulk divergences. The second term is singular

only when z1= z2= 0 and rជ1= rជ2. The third term is singular only when z1= z2= L and rជ1

= rជ₂. These two terms mentioned previously carries surface divergences. Finally the two last terms do not have singularities.

共iv兲 When 兩rជ1− rជ2兩 /LⰇ1 the free propagator behaves like

G₀共2兲共rជ₁− rជ₂,z₁,z₂兲 ⬃ 1 2␲L2 e−␲r12 r12 sin

冉

␲z1 L

冊

sin

冉

␲z₂ L

冊

, 共21兲

which shows an exponential convergence behavior.

共v兲 The integral of the variable 兵rជ, z其 on a neighborhood around 兵rជ

⬘

, z

⬘

其 of the free propagator is finite, i.e.,兰Rd3r dzG共2兲₀ 共rជ− rជ

⬘

, z , z

⬘

兲⬍⬁. See Fig. 1.

Property共v兲 allows us to show that the external legs of the Feynman diagrams do not create divergences. Let us suppose we have the integral corresponding to some Feynman diagram,

冕

R

d3_{r dz G} 0

共2兲_{共rជ− rជ}

_⬘

_,z,z

_⬘

_兲F共rជ

_⬘

_,z

_⬘

_{兲, 共22兲}

where G₀共2兲共rជ− rជ

⬘

, z , z

⬘

兲 is some external leg and F共rជ

⬘

, z

⬘

兲 describes the remainder part of the diagram. Now in order to proceed we must use the following statement: for two continuous and positives functions f共xជ兲 and g共xជ兲 defined in a finite region R with the exception of the point xជ1

where f共xជ兲 diverges, then the integral I=兰Rddxf共xជ兲g共xជ兲 is finite, if and only if I

⬘

=兰Vddxf共xជ兲 is

finite on some neighborhoods V of the point xជ1. With the property共v兲 and the statement before we

can see that external legs from the Feynman diagrams do not generate divergences.

III. REGULARIZED TWO- AND FOUR-POINT SCHWINGER FUNCTIONS AT ONE-LOOP ORDER

In this section we identify the divergent contribution in the two- and four-point Schwinger function at one-loop level. Essentially we use Eq.共20兲 in the 1PI diagrams of the Green functions considering their external legs, and the integrations in the coordinate space. We write Eq.共20兲 as

G₀共2兲共rជ1− rជ2,z1,z2兲 = 1 4␲2L2

冋

1 共z12−兲2+ r122 − 1 共z12+兲2+ r122 − 1 共2 − z12+兲2+ r122 + h共r12,z1,z2兲

册

, 共23兲 where h共r12, z1, z2兲 is given by h共r12,z1,z2兲 = 1 4_k=−

兺

_⬁ k_⫽0 ⬁ 1

冉

k −兩z12 −_兩 2

冊

2 +

冉

r12 2 2

冊

−1 4_k=−

兺

_⬁ k_⫽0,1 ⬁ 1

冉

k −z12 + 2

冊

2 +

冉

r12 2 2

冊

. 共24兲

From the property共iii兲 we see that the three first contributions on the right-hand side of Eq. 共23兲 have singularities. Otherwise, the last term is finite in the whole domain where we defined the model. After this brief introduction, we are able to study the interacting theory. Let us start analyzing the tadpole diagram, displayed in Fig. 2, from which we can write the expression for the

FIG. 1. The integral of the variable兵rជ, z其 on a neighborhood around 兵rជ⬘; z⬘其.

one-loop two-point Schwinger function G₁共2兲共rជ1− rជ2, z1, z2兲. We have that G₁共2兲共rជ1− rជ2,z1,z2兲 = ␭ 2

冕

d 3_{r dz G} 0 共2兲_共rជ 1− rជ,z1,z兲G0共2兲共0,z,z兲G共2兲0 共rជ2− rជ,z2,z兲. 共25兲

In the following we are generalizing the results obtained by Fosco and Svaiter.14 Let us begin studying the quantity G₀共2兲共0,z,z兲 that appears in the tadpole defined in Eq. 共25兲. From Eq. 共20兲 we get that G₀共2兲共0,z,z兲 can be written as

G₀共2兲=共0,z,z兲 = 1 4␲2_L2

冤

A − 1 共2z/L兲2− 1 共2 − 2z/L兲2+

兺

k=−⬁ k⫽0 ⬁ 1 共2k兲2−

兺

k=−⬁ k⫽0,1 ⬁ 1 共2k − 2z/L兲2

冥

, 共26兲 where A is given by A = lim 共z1,rជ1兲→共z2,rជ2兲 L2 共z1− z2兲2+兩rជ1− rជ2兩2 = lim ⌳→⬁ L2S4 8␲2⌳ 2_{, 共27兲}

Sd=关2␲d/2/⌫共d/2兲兴 and ⌳ is an ultraviolet cutoff. In the same way, from Eq. 共26兲 by performing

the summations, we-get for G₀共2兲共0,z,z兲,

G₀共2兲共0,z,z兲 = 1 4␲2L2

冋

A + ␲2 12− ␲2 4 1 sin2共␲z/L兲

册

. 共28兲

Substituting Eq.共27兲 in Eq. 共28兲 we obtain

G₀共2兲共0,z,z兲 = lim ⌳→⬁ S4 32␲4⌳ 2₊ 1 48L2− 1 16L2 1 sin2共␲z/L兲. 共29兲

The first term in Eq.共29兲 is a bulk divergence. Substituting Eq. 共29兲 in Eq. 共25兲 we get

G₁共2兲共rជ₁− rជ₂,z₁,z₂兲 = lim ⌳→⬁ ␭S4 64␲4⌳ 2

冕

R d3_{r dz G} 0 共2兲_共rជ 1− rជ,z1,z兲G0共2兲共rជ2− rជ,z2,z兲 + ␭ 96L2

冕

R d3r dz G₀共2兲共rជ1− rជ,z1,z兲G0共2兲共rជ2− rជ,z2,z兲 − ␭ 32L2

冕

R d3r dzG0 共2兲_共rជ 1− rជ,z1,z兲G0共2兲共rជ2− rជ,z2,z兲 sin2共␲z/L兲 共30兲

The first term on the right-hand side carries a bulk divergence. The second term is finite. To see this we analyze the integral by sectors. Therefore we have

冕

R d3r dz G₀共2兲共rជ1− rជ,z1,z兲G0共2兲共rជ2− rជ,z2,z兲 =

冕

R₁ +

冕

R₂ +

冕

R₃ +

冕

R₄ +

冕

R₅ , 共31兲

where each integral is defined in different regions displayed in Fig. 3, where the points共rជ1, z1兲 and

共rជ2, z2兲 are the centers of the regions R1and R2, respectively. Using the property共v兲 we have that

the integrals on R₁ and R₂ are finite. Since the free propagators G₀共2兲共rជ₁− rជ, z₁, z兲 and G₀共2兲共rជ₂ − rជ, z₂, z兲 presented in Eq. 共31兲 do not have divergences on R₃and this region is compact, then the integral on R₃is finite. The integrals defined in regions R₄ and R₅ also are finite since from the property共iv兲 the propagator decreases exponentially when one of its points becomes far from the

other. Thus the integral defined by Eq.共31兲 is finite. Finally we must study the third integral on the right-hand side of Eq.共30兲. Note that the term 1/sin2共␲z / L兲 diverges when z is evaluated on the

boundaries.

Nevertheless this integral is convergent, because the products of G₀共2兲共rជ₁− rជ, z₁, z兲 and G₀共2兲 ⫻共rជ2− rជ, z2, z兲 take away the divergence. Using Eq. 共19兲, we have that the third integral on the

right-hand side of Eq.共30兲 is finite. Therefore the one-loop two-point Schwinger function only has bulk divergence.

Our next step is to analyze the four-point Schwinger function in the one-loop level共see Fig. 4兲. Since the free propagator only has singularities when its two points are equal or also when the two points joined are evaluated at the boundaries, we continue our analysis of the integrals only in the domains where the two external points of the free propagators take the same values. The complete four-point function at one-loop level is given by

G₁共4兲共rជ1,z1,rជ2,z2,rជ3,z3,rជ4,z4兲 = ␭2 2

冕

d d−1_r

冕

_dd−1_r

_⬘

冕

0 L dz

冕

0 L dz

⬘

G₀共2兲共rជ1− rជ,z1,z兲G0共2兲共rជ2− rជ,z2,z兲 ⫻

关

G₀共2兲共rជ− rជ

⬘

,z,z

⬘

兲

兴

2G₀共2兲共rជ3− rជ

⬘

,z3,z

⬘

兲G0共2兲共rជ4− rជ

⬘

,z4,z

⬘

兲. 共32兲

For simplicity, in Fig. 5 we define three different regions between the boundaries. The first one, R1

is concerned when 兵rជ

⬘

, z

⬘

其 is close to 兵rជ, z其. In this region the contribution coming from 关G₀共2兲共rជ − rជ

⬘

, z , z

⬘

兲兴2_{is singular. Nevertheless, we still must analyze if this divergent behavior will appear}

in the integral defined by Eq.共32兲. We will show that the singularities will appear only as bulk divergences. In the region R2 共z,z

⬘

→0 and rជ

⬘

→rជ兲 the term 关G0

共2兲_{共rជ− rជ}

_⬘

_{, z , z}

_⬘

_兲兴2 _{is also divergent.}

As we will see, this divergent behavior disappears when we compute the complete four-point function at one-loop order, defined by Eq.共32兲. In the region R3共z,z

⬘

→L and rជ

⬘

→rជ兲 the situation

is identical as in the region R2. Using the same argument that we used before to analyze the

convergence of the integral denned by Eq. 共22兲, we can study the convergence of the integral defined by Eq.共32兲 with the amputated external legs. Therefore we must study Eq. 共32兲 with the external legs amputated. Therefore we must study the quantity 兰d3r dz d3r

⬘

dz

⬘

关G₀共2兲共rជ

⬘

− rជ, z

⬘

, z兲兴2. Substituting Eq. 共23兲 in the former equation we get

FIG. 3. Regions of integration Ri.

冕

d3r dz d3r

⬘

dz

⬘

关

G共2兲₀ 共rជ

⬘

− rជ,z

⬘

,z兲

兴

2= 1

共4␲2_L2_兲2

共

I1+ I2+ I3+ I4+ I5

兲

+ finite part, 共33兲

where the integrals Ii, i = 1 , 2 , . . . , are given by

I1=

冕

d3r

⬘

dz d3r

⬘

dz

⬘

1

关

共z12 −_兲2_{+ r} 12 2

兴

2, 共34兲 I2=

冕

d3r dz d3r

⬘

dz

⬘

冋

− 1

关

共z12+兲2+ r122

兴

2 + 1

关

共2 − z12+兲2+ r122

兴

2

册

, 共35兲 I3=

冕

d3r dz d3r

⬘

dz

⬘

冋

− 2

关

共z12−兲2+ r122

兴关

共z12+兲2+ r122

兴

− 2

关

共z12−兲2+ r122

兴关

共2 − z12+兲2+ r122

兴

册

共36兲 I4=

冕

d3r dz d3r

⬘

dz

⬘

2

关

共z12+兲2+ r122

兴关

共2 − z12+兲2+ r122

兴

, 共37兲 I5=

冕

d3r dz d3r

⬘

dz

⬘

冋

1 共z12−兲2+ r122 − 1 共z12+兲2+ r122 − 1 共2 − z12+兲 + r122

册

h共r12,z1,z2兲. 共38兲

Let us investigate each term of Eq.共33兲. The integral I1must be analyzed only in the region R1.

For this purpose we need an auxiliary result. We can prove that a continuous and positive function

f共x兲 which does not have singularities except for x=0, and M =兰_−⑀ជ⑀ជ ddxf共w2兲 where w2=兩wជ兩2, then there exist⑀

⬘

such that M = Sd兰0⑀⬘dw wd−1f共w2兲 where⑀⬍⑀

⬘

⬍

冑

d⑀. Then we get

I₁=

冕

R₁ d3_r

_⬘

_dz

_⬘

1 关共z12 −_兲2_{+ r} 12 2 _兴2=

冕

r ជ−⑀ជ r ជ+⑀ជ d3_r

_⬘

冕

z−⑀ z+⑀ dz

⬘

1 关共z − z

⬘

兲2_{+兩rជ− rជ}

_⬘

_兩2_兴2 =

冕

−⑀ជ ⑀ជ _d4_w w4 = S4

冕

₀ ⑀_⬘ dww 3 w4= S4ln w

兩

0 ⑀_{⬘. 共39兲}

Therefore I₁ contributes with a bulk divergence of the type as the one that appears in the theory without boundaries. In the usual renormalization procedure, the contribution coming from I₁ can be eliminated by the usual counterterms. Concerning the contribution coming from I₂we have that the first term 1 /关共z₁₂+兲2_{+ r}

12

2_兴2 _{is not singular in the region R}

1. In the region R2, using the same

auxiliary result that we used before, we can obtain an upper bound to the contribution coming from this term. We get

冕

R₂ d3r dz d3r dz

⬘

1 关共z + z

⬘

兲2_{+兩rជ− rជ}

_⬘

_兩2_兴2⬍

冕

d 3_r

冕

r ជ−⑀ជ r ជ+⑀ជ d3r

⬘

冕

0 ⑀

冕

0 ⑀ dz dz

⬘

1 关z2_{+ z}

_⬘

2_{+兩rជ− rជ}

_⬘

_兩2_兴2 ⬍1 4

冕

d 3_r

冕

−⑀ជ ⑀ជ d5w1 w2= 1 12S5⑀

⬘

3

冕

R⬘ d3r. 共40兲

Since the region R

⬘

傺R₂ is finite this integral is convergent. Next, let us analyze the term 1 /关共2 − z₁₂+兲2_{+ r}

12 2_兴2 _{of I}

2 in the region R3. Since the behavior of the field in each plates 共for z=0 and

z = L兲 is the same, then the analysis follows the same lines as previous ones and therefore this

contribution is also finite. To study I3, we consider first the term 2 /关共z12 −_兲2_{+ r} 12 2_兴关共z 12 +_兲2_{+ r} 12 2_{兴. This}

expression must be studied in the regions R1 and R2, respectively. In R1 we can see that the

convergence of

共41兲 depends on the convergence of

冕

R₁

d3r dz d3r

⬘

dz

⬘

1

共z − z

⬘

兲2_{+兩rជ− rជ}

_⬘

_兩2. 共42兲

From the above arguments we have that Eq.共42兲 can be written as

冕

−⑀ជ ⑀ជ _d4_w w2 = S4

冕

0 ⑀⬘ dw w =S4⑀

⬘

2 2 , 共43兲

thus Eq.共42兲 gives a finite contribution. Now we consider the first term of I3in the region R2. For

this purpose we will use the following property. Let us take a continuous and positive function f共x兲 which does not have singularities except for x = 0, and N =兰₀⑀ជ兰₀⑀ជdl_ydm_zf共y2_{+ z}2_{兲 then there exist⑀}

_⬘

in such a way that N =共S_l+m+2/ S_l+1S_m+1兲兰₀⑀⬘dw wl+m+1_f共w2_{兲 where ⑀}

_⬘

_{⬎0. Using this property, we}

have for the first term of Eq.共35兲, in the region R₂, that

冕

d3r

冕

0 ⑀ dz

冕

z z+⑀ dz

⬘

冕

r ជ−⑀ជ r ជ+⑀ជ d3r

⬘

1 关共z − z

⬘

兲2_{+兩rជ− rជ}

_⬘

_兩2_{兴关共z + z}

_⬘

_兲2_{+兩rជ− rជ}

_⬘

_兩2_兴 ⬍

冕

d3r

冕

0 ⑀ dz

冕

0 ⑀ du

冕

−⑀ជ ⑀ជ d3v 1 共u2_+v2_兲共z2_{+ u}2_+v2_兲 ⬍1 4S4

冕

d 3_r

冕

0 ⑀ dz

冕

0 ⑀⬘ dw w 共z2_{+ w}2_兲= S3S4 2S1S2 ⑀

⬙

. 共44兲

Therefore the first term of I3 is also finite in R2. The second term 2 /关共共z12−兲2+ r122兲共共2−z12+兲2

+ r₁₂2 兲兴 in I3must be analyzed also in the regions R1and R3. This analysis follows the same lines

as the last case, therefore the contribution coming from this term is also finite. We have now to study the term I₄. Note that 2 /关共共z₁₂+兲2_{+ r}

12 2 _{兲共共2−z} 12 +_兲2_{+ r} 12 2_{兲兴 must be analyzed}

in the regions R₂and R₃, respectively. Let us start with the region R₂. Using previous arguments we have that the convergence of I₄ depends on the convergence of the following expression:

冕

0 ⑀

冕

0 ⑀ dz dz

⬘

冕

r ជ−⑀ជ r ជ+⑀ជ d3r

⬘

1 z2+ z

⬘

2+兩rជ− rជ

⬘

兩2= S₅ 4

冕

₀ ⑀⬘ dww 4 w2= S₅ 12⑀

⬘

3_{, 共45兲}

which is finite. In the region R3 our analysis follows the same lines as in the region R2, thus the

integral in the region. R3is also finite.

Using the same argument that we used before, it is not difficult to show that the contribution coming from I5 is also finite. We conclude that the integrals given by Eq. 共32兲 only have bulk

divergences. In this way we can conclude that at the one-loop level the bulk counterterms are sufficient to render the complete connected Schwinger functions finite. In the next section we will identify the divergent contribution in the connected two-point Schwinger functions at the two-loop order.

IV. THE DIVERGENCES IN THE TWO-POINT SCHWINGER FUNCTIONS AT TWO-LOOP LEVEL

In this section we will generalize some results obtained by Fosco and Svaiter14 and also by Caicedo and Svaiter.15 We will identify the divergent contribution in the connected two-point Schwinger functions at the two-loop order. The diagrams that we are interested to analyze are displayed in Fig. 6. The expression that corresponds to Fig. 6共a兲 is given by

␭2 4

冕

d 3_r

_⬘

_dz

_⬘

_d3_{r dz G} 0 共2兲_共rជ 1− rជ

⬘

,z1,z

⬘

兲

关

G0共2兲共rជ

⬘

− rជ,z

⬘

,z兲

兴

2 G₀共2兲共0,z,z兲G₀共2兲共rជ2− rជ

⬘

,z2,z

⬘

兲. 共46兲 Since the external legs in Eq.共46兲 do not contribute to generate divergences, let us consider only the following integral:

冕

d3r dz关G₀共2兲共rជ

⬘

− rជ,z

⬘

,z兲兴2G₀共2兲共0,z,z兲. 共47兲

Replacing Eq.共29兲 in Eq. 共47兲 we get

冕

d3r dz关G₀共2兲共rជ

⬘

− rជ,z

⬘

,z兲兴2G₀共2兲共0,z,z兲 = lim ⌳→⬁ S4 32␲4⌳ 2

冕

_d3_{r dz关G} 0 共2兲_共rជ

_⬘

_{− rជ,z}

_⬘

_,z兲兴2 + 1 48L2

冕

d 3_{r dz关G} 0 共2兲_共rជ

_⬘

_{− rជ,z}

_⬘

_,z兲兴2 − 1 16L2

冕

d 3_{r dz关G} 0 共2兲_共rជ

_⬘

_{− rជ,z}

_⬘

_,z兲兴2 1 sin2共␲z/L兲. 共48兲 The first term and the second one in Eq.共48兲 can be renormalized introducing only bulk counter-terms. The most interesting behavior appears in the last term of this equation. Note that this unrenormalized quantity contains only bulk divercences, since the contribution coming from

关G₀共2兲共rជ

⬘

− r¯ , z

⬘

, z兲兴2 _{cancels the surface divergent behavior generated by the 1 / sin}2_{共␲z / L兲 term.}

Nevertheless, after the introduction of a bulk counterterm to render the contribution 关G₀共2兲共rជ

⬘

− rជ, z

⬘

, z兲兴2_{finite between the plates, surface divergences appear. Thus this surface divergence must}

be renormalized. After the introduction of surface and bulk counterterms, the finite contribution coming from the last term of Eq.共48兲, up to a finite renormalization constant, is given by

1 16L2

冕

d 3_{r dz}

冋

_共G 0 共2兲_共rជ

_⬘

_{− rជ,z}

_⬘

_,z兲兲2₋ 1 共4␲2_L2_兲2 1 关共z12−兲2+ r122 兴2

册

冋

1 sin2共␲z/L兲− L2 共␲z兲2− L2 ␲2_{共L − z兲}2

册

. 共49兲 Therefore this term contains an overlapping between bulk and surface counterterms.

We still must analyze the sunset diagram. The expression corresponding to Fig. 6共b兲 is given by ␭2 6

冕

d 3_r

_⬘

_dz

_⬘

_d3_{r dz G} 0 共2兲_共r 1 ជ − rជ

⬘

,z1z

⬘

兲

关

G0共2兲共rជ

⬘

− rជ,z

⬘

,z兲

兴

3 G₀共2兲共rជ2− rជ

⬘

,z2,z

⬘

兲. 共50兲

Again, the external legs do not contribute to generate divergences, and therefore let us study the amputated diagram, i.e., without external legs. We have

冕

d3r dz d3r

⬘

dz

⬘

关G共2兲₀ 共rជ

⬘

− rជ,z

⬘

,z兲兴3= 1

共4␲2_L2_兲3共I1+ I2+ ¯ + I12兲 + finite part, 共51兲

where I1=

冕

d3r dz d3r

⬘

dz

⬘

1 关共z12−兲2+ r122 兴3 , 共52兲 I₂=

冕

d3_{r dz d}3_r

_⬘

_dz

_⬘

1 关共z12 +_兲2_{+ r} 12 2_兴3, 共53兲 I₃=

冕

d3_{r dz d}3_r

_⬘

_dz

_⬘

1 关共2 − z12 +_兲2_{+ r} 12 2_兴3, 共54兲 I₄=

冕

d3_{r dz d}3_r

_⬘

_dz

_⬘

2 关共z12 −_兲2_{+ r} 12 2 _兴2_关共z 12 +_兲2_{+ r} 12 2 _兴, 共55兲 I₅=

冕

d3r dz d3r

⬘

dz

⬘

2 关共z12 −_兲2_{+ r} 12 2 _兴2_{关共2 − z} 12 +_兲2_{+ r} 12 2_兴, 共56兲 I₆=

冕

d3r dz d3r

⬘

dz

⬘

2 关共z12 −_兲2_{+ r} 12 2 _兴关共z 12 +_兲2_{+ r} 12 2_兴2, 共57兲 I7=

冕

d3r dz d3r

⬘

dz

⬘

2 关共z12 −_兲2_{+ r} 12 2 _{兴关共2 − z} 12 +_兲2_{+ r} 12 2 _兴2, 共58兲 I8=

冕

d3r dz d3r

⬘

dz

⬘

2 关共z12 +_兲2_{+ r} 12 2 _兴2_{关共2 − z} 12 +_兲2_{+ r} 12 2_兴, 共59兲

I9=

冕

d3r dz d3r

⬘

dz

⬘

2 关共z12+兲2+ r122 兴关共2 − z12+兲2+ r122 兴2 , 共60兲 I10=

冕

d3r dz d3r

⬘

dz

⬘

6 关共z12 −_兲2_{+ r} 12 2 _兴关共z 12 +_兲2_{+ r} 12 2_{兴关共2 − z} 12 +_兲2_{+ r} 12 2_兴, 共61兲 I11=

冕

d3r dz d3r

⬘

dz

⬘

冋

1 共z12 −_兲2_{+ r} 12 2 − 1 共z12 +_兲2_{+ r} 12 2 − 1 共2 − z12 +_兲2_{+ r} 12 2

册

2 h共r₁₂,z₁,z₂兲, 共62兲 I12=

冕

d3r dz d3r

⬘

dz

⬘

冋

1 共z12−兲2+ r122 − 1 共z12+兲2+ r122 − 1 共2 − z12+兲2+ r122

册

h2共r12,z1,z2兲, 共63兲

Let us analyze each contribution coming from each term of Eq.共51兲. The first integral I1given by

Eq.共52兲 is divergent in R1. In general we can show that

冕

R₁

d3r

⬘

dz

⬘

1 关共z12−兲2+ r122兴n

=

再

finite, n⬍ 2,

⬁ , n艌 2.

冎

共64兲

Using the above result we can see that the integrals I₃, I₄and the first integral of I₁₁are divergent. These integrals contain bulk divergences which must be removed introducing bulk counterterms. Next let us analyze the contribution coming from the integral I₂in the region R₂. Using previous arguments and considering the external legs we get

冕

R₂ d3r dz d3r

⬘

dz

⬘

zz

⬘

关共z12 +_兲2_{+ r} 12 2_兴3⬍

冕

d 3_r

冕

−⑀ជ ⑀ជ

冕

0 ⑀

冕

0 ⑀ d3u dz dz

⬘

zz

⬘

共z2_{+ z}

_⬘

2_{+ w}2_兲3⬍ S₇ S₂2⑀

⬘

冕

d 3_r. 共65兲 Therefore this term gives a finite contribution to Eq.共50兲. The contribution from the integral I6to

the integral must be studied in region R2. In this case we must consider the external legs, and the

property: let us take a function f共x,y兲 positive which does not have singularities except for 共x,y兲=共0,0兲,I=兰0⑀兰0⑀dx dy f共x,y兲 then, I⬍兰0⑀dx兰x

x+⑀_{dy f共x,y兲+兰} 0 ⑀_dy兰 y y+⑀_{dy f共x,y兲, we get}

冕

R₂ d3r dz d3r

⬘

dz

⬘

zz

⬘

关共z12−兲2+ r122 兴关共z12+兲2+ r122兴2 ⬍ 2

冕

d3_r

冕

−⑀ជ ⑀ជ d3_w

冕

0 ⑀ dz

冕

0 ⑀ du z共z + u兲 共u2_{+ w}2_兲共u2_{+ z}2_{+ w}2_兲2. 共66兲

From the above arguments we have that the contribution from the integral I6 is smaller than

S4

冕

d3r

冕

0 ⑀ dz

冕

0 ⑀_⬘ ds z 2_s 共s2_{+ z}2_兲2+ 2S3

冕

d3r

冕

0 ⑀ dz

冕

0 ⑀ du

冕

0 ⑀_⬘ ds zus 2 共u2_{+ s}2_兲共u2_{+ s}2_{+ z}2_兲2

⬍S4S5 S2S3 ⑀

⬙

冕

d3r + 2 共S5兲 2 共S2兲2S3

冕

d3r

冕

0 ⑀_⵮ dww 4 w4⬍

冉

S4S5 S2S3 ⑀

⬙

+ 2 共S5兲 2 共S2兲2S3 ⑀

⵮

冊

冕

d3r. 共67兲

We conclude that the integral I6is finite. Also integrating the contribution coming from the term

I8 on R2we get

共68兲 Using the fact that the integral兰R₂d3r dz d3r

⬘

dz

⬘

1 /关共z12

+_兲2_{+ r} 12

2 _兴2_{is finite, we have that this}

inte-gral also is convergent in R₂. The contribution from the term I₁₀on R₂is given by

共69兲 Since the integral兰R₂d3r dz d3r

⬘

dz

⬘

1

关共z12−兲2+r122兴关共x12+兲+r122兴 is finite, then the integral defined by Eq.共69兲

is convergent in R2. The contribution coming from the terms I11contain only a bulk divergence.

Otherwise, the contributions coming from the terms I12are finite. We conclude that we need only

bulk counterterms to render the integral defined by Eq.共50兲 finite. The same analysis can be done for the four-point Schwinger function in the two-loop approximation. We obtained that only bulk divergences appear in the full four-point function.

V. CONCLUSIONS

In this paper we are interested to show how to implement the renormalization procedure in systems where the translational invariance is broken by the presence of macroscopic structures. For the sake of simplicity we are studying the self-interacting massless scalar field theory in a four-dimensional Euclidean space. We impose that one coordinate is defined in a compact domain, introducing two parallel mirrors where we are assuming Dirichlet-Dirichlet boundary conditions. Note that although that there are some similarities with the finite temperature field theory using the Matsubara formalism, in thermal systems there appears only bulk divergences, as for example, in the case of the system where we assume periodic boundary conditions. In nontranslational invari-ant systems, in general, to render the theory finite it is necessary to introduce surface counterterms. In this work we generalize some results obtained by Fosco and Svaiter14and also by Caicedo and Svaiter.15 We identify the divergences of the Schwinger functions in the massless self-interacting scalar field theory up to the two-loop approximation. First, analyzing the full two- and four-point Schwinger functions at the one-loop level, we show that the bulk counterterms are sufficient to render the theory finite. Second, at the two-loop level, we must introduce surface counterterms in the bare Lagrangian in order to make finite the full two- and also four-point Schwinger functions. The most interesting behavior appears in the “double scoop” diagram given by Eq.共46兲. The amputated diagram is given by Eq. 共47兲 and we are interested in the last term of Eq. 共48兲. This unrenormalized quantity contains only bulk divergences. Nevertheless, after the introduction of a bulk counterterm to render the contribution finite between the plates, surface divergences appear. Thus this surface divergence must be renormalized. Therefore this term con-tains an overlapping between bulk and surface counterterms. This procedure can be generalized to the n-loop level. The inclusion of the counterterm in the Lagrangian up to two-loop level with the full renormalized action and the general algorithm to identify the surface and bulk conterterms in the n-loop level will be left to future work.

ACKNOWLEDGMENTS

One of the authors共M.A.A.兲 thanks Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro 共FAPERJ兲 and also thanks Conselho Nacional do Desenvolvimento Cientifico e Tecnológico do Brazil共CNPq兲, one of the authors 共G.F.H.兲 thanks FAPESP, Grant No. 02/09951-3, for full support, and one of the authors共N.F.S.兲 thanks CNPq for partial support.

APPENDIX

In this Appendix we will derive a useful representation for the free two-point Schwinger function. Starting from Eq.共17兲, we have that G₀共2兲共rជ1− rជ2, z1, z2兲 is given by

G₀共2兲共rជ1− rជ2,z1,z2兲 = 2 共2␲兲d−1_L

兺

n=1 ⬁

冕

dd−1p sin

冉

n␲z1 L

冊

sin

冉

n␲z2 L

冊

eipជ·共rជ1− rជ2兲

冋

pជ2+

冉

n␲ L

冊

2 + m2

册

. 共A1兲 Using the variables u =共z1− z2兲/L and v=共z1+ z2兲/L defined, respectively, in the region

u苸关−1,1兴 and v苸关0,2兴, and also making use of a trigonometric identity and performing the sum

that appears in Eq.共A1兲 we obtain21

G₀共2兲共rជ1− rជ2,z1,z2兲 = 1 2

冕

dd−1p 共2␲兲d−1 eipជ·共rជ1− rជ2兲 共pជ2_{+ m}2_兲1/2

冋

cosh共L共1 − 兩u兩兲共pជ2+ m2兲1/2兲 sinh共L共pជ2_{+ m}2_兲1/2_兲 −cosh共L共1 − v兲共pជ 2_{+ m}2_兲1/2_兲 sinh共L共pជ2_{+ m}2_兲1/2_兲

册

. 共A2兲

Taking m = 0 , d = 4, and integrating the angular part, it is possible to show that G₀共2兲共rជ1− rជ2, z1, z2兲

can be written as

G₀共2兲共rជ₁− rជ₂,z₁,z₂兲 = − i 2共2␲兲2_r

_⬘

_L2

冕

0 ⬁

dx共eixr⬘_{− e}−ixr⬘_兲

冋

cosh共共1 − 兩u兩兲x兲

sinh x −

cosh共共1 − v兲x兲 sinh x

册

,

共A3兲 where the variable r

⬘

is defined by r

⬘

⬅共兩rជ1− rជ2兩兲/L. Making use of the following integral

repre-sentation of the product between the gamma function and the Riemann zeta function21

冕

0 ⬁ dxx z−1_e−␤x epx_{− 1} = ⌫共z兲 pz ␨

冉

z, ␤ p+ 1

冊

, 共A4兲

where Re共z兲⬎1,Re共␤/ p兲⬎−1 and the Riemann zeta function␨共z,q兲 is defined by ␨共z,q兲 =

兺

k=0

⬁

1

共k + q兲z, q⫽ 0,− 1,− 2, ... , 共A5兲

then, it is possible to write G₀共2兲共rជ1− rជ2, z1, z2兲 as

G₀共2兲共rជ1− rជ2,z1,z2兲 = 1 16␲2L2

冤

k=−

兺

⬁ ⬁ 1

冉

k −兩u兩 2

冊

2 +

冉

r

⬘

2

冊

2−

兺

k=−⬁ ⬁ 1

冉

k −v 2

冊

2 +

冉

r

⬘

2

冊

2

冥

. 共A6兲

兺

k=−⬁ ⬁ 1 共k − z兲2_{+ r}2= ␲ 2r sinh共2␲r兲

sinh2共␲r兲 + sin2共␲z兲, 共A7兲

we obtain the expression for the two-point Schwinger function that we need to proceed in our analysis. Using the above equation in Eq.共A6兲 we get

G₀共2兲共rជ1− rជ2,z1,z2兲 = sinh共␲r

⬘

兲 16␲L2_r

_⬘

冋

sin

共

␲z1 L

兲

sin

共

␲z2 L

兲

关

sinh2

共

␲r⬘ 2

兲

+ sin2

共

␲u 2

兲

兴关

sinh2

共

␲r⬘ 2

兲

+ sin2

共

␲v 2

兲

兴

册

. 共A8兲

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