Report on the Detailed Calculation of the Effective Potential in Spacetimes with S^{1}\times R^{d} Topology and at Finite Temperature

# Report of the Detailed Calculation of the Effective Potential in Spacetimes with $S^1\times R^d$ Topology and at Finite Temperature

## Abstract

In this paper we review the calculations that are needed to obtain the bosonic and fermionic effective potential at finite temperature and volume (at one loop). The calculations at finite volume correspond to topology. These calculations appear in the calculation of the Casimir energy and of the effective potential of extra dimensional theories. In the case of finite volume corrections we impose twisted boundary conditions and obtain semi-analytic results. We mainly focus in the details and validity of the results. The zeta function regularization method is used to regularize the infinite summations. Also the dimensional regularization method is used in order to renormalize the UV singularities of the integrations over momentum space. The approximations and expansions are carried out within the perturbative limits. After the end of each section we briefly present applications associated to the calculations. Particularly the calculation of the effective potential at finite temperature for the Standard Model fields, the effective potential for warped and large extra dimensions and the topological mass creation. In the end we discuss on the convergence and validity of one of the obtained semi-analytic results.

Keywords: Effective potential, zeta regularization, Casimir energy, finite temperature, extra dimensions

## 1 Introduction

During the development of Quantum Field Theory, many quantitative methods have been developed. Some of the most frequently used techniques are one-dimensional infinite lattice sums [3, 34]. In this article we shall review the calculations associated with these summations, that appear in many important branches of Quantum Field Theory, three of which are, the physics of extra dimensions [81, 65, 67, 68, 66, 55, 88], the Casimir effect, [4]-[57], [3, 76, 82, 75, 61, 91, 89] and finally in field theories at finite temperature [60, 58, 73, 69, 3, 4, 82, 16, 34, 54]. In both three cases we shall compute the effective potential. The method we shall use involves the expansion of the potential in Bessel series and zeta regularization [3, 4, 34, 11]. We focus on the details of the calculation and we thing the paper will be a useful tool for the ones that want to study these theories.

### 1.1 Effective Potential in Theories with Large Extra Dimensions

In theories with large extra dimensions [81, 65, 67, 68, 66, 55, 88], the fields entering the Lagrangian are expanded in the eigenfunctions of the extra dimensions. Let us focus on theories with one extra dimension with the topology of a circle, namely of the type ( stands for the 4-dimensional Minkowski space). In the following we shall also discuss the orbifold compactification apart from the circle compactification we describe here. For circle compactifications, the harmonic expansion of the fields reads,

 ϕ(x,y)=∞∑n=−∞ϕn(x)ei2πnyL, (1)

where stands for the 4-dimensional Minkowski space coordinates, for the extra dimension and the radius of the extra dimension. We note that fields are periodic in the extra dimension namely, . One of the ways to break supersymmetry is the Scherk-Schwarz compactification mechanism. This is based on the introduction of a phase . For fermions we denote it and for bosons . Now the harmonic expansions for fermion and bosons fields read,

 ϕ(x,y)=∞∑n=−∞ϕn(x)ei2π(n+qF)yL, (2)

for fermions and,

 ϕ(x,y)=∞∑n=−∞ϕn(x)ei2π(n+qB)yL, (3)

for bosons. We can observe that the initial periodicity condition is changed. Using equations (2) and (3) we can find that the effective potential at one loop is equal to,

 V(ϕ)=12Tr∞∑n=−∞∫d4p(2π)4ln[p2+(n+qB)2L2+M2(ϕ)p2+(n+qF)2L2+M2(ϕ)] (4)

Note that fermions and bosons contribute to the effective potential with opposite signs. This is due to the fact that fermions are described by anti-commuting Grassmann fields. Also is a independent term and depends on the way that spontaneous symmetry breaking occurs. We shall not care for the particular form of this and we focus on the general calculation of terms like the one in equation (4).

### 1.2 The Casimir Energy

One of the most interesting phenomena in Quantum Field Theory is the Casimir effect (for a review see [3, 4, 10, 22, 34, 30]). It expresses the quantum fluctuations of the vacuum of a quantum field. It originates from the ”confinement” of a field in finite volume. Many studies have been done since H. Casimir’s original work [2]. The Casimir energy, usually calculated in these studies, is closely related to the boundary conditions of the fields under consideration [26, 29, 13, 14, 3, 4, 40, 41]. Boundary conditions influence the nature of the so-called Casimir force, which is generated from the vacuum energy.

In this paper we shall concentrate on the computation of the effective potential (Casimir Energy) of bosonic and fermionic fields in a space time with the topology [3, 4, 10, 21, 25, 27, 28, 34]. Fermionic and bosonic fields in spaces with non trivial topology are allowed to be either periodic or anti-periodic in the compact dimension. The forms of the potential to be studied are,

 1L∫dkd(2π)d∞∑n=−∞ln[4π2n2L2+k2+m2], (5)

and the fermionic one,

 1L∫dkd(2π)d∞∑n=−∞ln[(2n+1)2π2L2+k2+m2]. (6)

We shall study them also in the cases and , which are of particular importance in physics since they correspond to three and four total dimensions. Both have many applications in solid state physics and cosmology [10, 3]. Also we shall generalize to the case with fermions and bosons obeying general boundary conditions also in dimensions. This is identical from a calculational aspect with the effective potential of theories with extra dimensions [55, 67]. So computing one of the two gives simultaneously the other. The expression that is going to be studied thoroughly is,

 1L∫dkd(2π)d∑ln[((n+ω)2πL)2+k2+m2]= (7) ∫dkd+1(2π)d+1ln[k2+a2] +1L∫dkd(2π)dln[1−e−2(aL2−iπω)]+1L∫dkd(2π)dln[1−e−2(aL2+iπω)].

The calculations shall be done in dimensions, quite general, and the application to every dimension we wish, can be done easily. The only constraint shall be if is even or odd. We shall make that clear in the corresponding sections and treat both cases in detail.

### 1.3 Field Theories at Finite Temperature

The calculations used in finite temperature field theories are based on the imaginary time formalism [58, 60, 3, 34, 4]:

 t→iβ, (8)

with =. The eigenfrequencies of the fields that appear to the propagators are discrete and are summed in the partition function. These are affected from the boundary conditions used for fermions and bosons [3, 4]. Bosons obey only periodic and fermions antiperiodic boundary conditions at finite temperature, as we shall see (this is restricted and dictated by the KMS relations [60]). Indeed for bosons the boundary conditions are:

 φ(x,0)=φ(x,β), (9)

where stands for space coordinates, and the fermionic boundary conditions are,

 ψ(x,0)=−ψ(x,β), (10)

In most calculations involving bosons, we are confronted with the following expression:

 T∫dk3(2π)3∞∑n=−∞ln[4π2n2T2+k2+m2], (11)

while the fermionic contribution is,

 T∫dk3(2π)3∞∑n=−∞ln[(2n+1)2π2T2+k2+m2], (12)

and stands for the Euclidean momentum:

 k2=k21+k22+k23, (13)

while is the field mass. In the next sections we deal with the two above contributions in dimensions and we specify the results for and .

## 2 Bosonic Contribution at Finite Temperature

We will compute the following expression,

 S1=T∫dk3(2π)3∞∑n=−∞ln[4π2n2T2+k2+m2] (14)

In the following we generalize in dimensions. This will give us the opportunity to deal other cases apart from the . Consider the sum:

 So=∞∑n=−∞14π2n2T2+a2=14π2T2∞∑n=−∞1n2+a24π2T2, (15)

where,

 a2=k2+m2 (16)

Integrating over ,

 ∞∑n=−∞14π2n2T2+a2, (17)

we get:

 ∫ ∞∑n=−∞da24π2n2T2+a2=∑ln[4π2n2T2+a2]. (18)

Now,

 ∞∑n=−∞14π2n2T2+a2=24aTcoth(a2T), (19)

thus equation (18) becomes,

 ∫ ∞∑n=−∞da24π2n2T2+a2 = ∫24aT(coth(a2T)da2 (20) = 2ln(sinh[a2T]).

Using the relation [1],

 ln(sinhx)=ln(12[ex−e−x])=x+ln[1−e−2x]−ln[2], (21)

and upon summation,

 ln(sinha2T)=a2T+ln[1−e−aT]−ln[2], (22)

and,

 ln(sinha2T)=a2T+ln[1−e−aT]−ln[2]. (23)

Summing equations (22) and (23) we obtain,

 ∫∞∑n=−∞da24π2n2T2+a2=2ln(sinha2T)=aT+2ln[1−e−aT]−2ln[2]. (24)

Finally the result is [58, 60, 3, 34]:

 ∞∑n=−∞ln[4π2n2T2+a2]=aT+2ln[1−e−aT]−2ln[2] . (25)

Upon using,

 ∑ln[(n+ω)24π2T2+a2)(n+ω)24π2T2+b2)]=2(a−b), (26)

equation (25) becomes,

 ∑ln[4π2n2T2+a2]=12πT∞∫−∞dxln[x2+a2]+2ln[1−e−aT]. (27)

Finally we have,

 T∫dk3(2π)3∑ln[(2πnT)2+k2+m2] =∫dk3(2π)3∞∫−∞dx2πln[x2+a2] (28) +2T∫dk3(2π)3ln[1−e−aT].

Remembering that,

 a2=k2+m2, (29)

the first integral of equation (28) is the one loop contribution to the effective potential at zero temperature. The 4-momentum is:

 K2=k2+x2. (30)

Writing the above in dimensions (in the end we take to come back to four dimensions) we get,

 T∫dkd(2π)d∑ln[4π2n2T2+k2+m2] =∫dkd+1(2π)d+1ln[k2+a2] (31) +2T∫dkd(2π)dln[1−e−aT].

The temperature dependent part has singularities stemming from the infinite summations. These singularities are poles of the form [3, 34, 4]:

 1ϵ, (32)

where the dimensional regularization variable (). As we shall see, by using the zeta regularization [3, 4, 82, 34, 11] these will be erased. In the following of this section we focus on the calculation of the temperature dependent part. Let,

 Vboson=2T∫dkd(2π)dln[1−e−aT]=2T∫dkd(2π)dln[1−e−aT]. (33)

By using [1],

 ln[1−e−aT]=−∞∑q=1e−aTqq. (34)

we obtain,

 Vboson = 2T∫dkd(2π)dln[1−e−aT]=−2T∫dkd(2π)d∞∑q=1e−aTqq (35) = −2∞∑q=1T∫dkd(2π)de−aTqq

and remembering,

 a=√k2+m2, (36)

by integrating over the angles we get,

 Vboson =−2∞∑q=1T∫dkd(2π)de−√k2+m2Tqq (37) =−2∞∑q=1T∫∞−∞dk(2π)dkd−1(2π)d2Γ(d2)e−√k2+m2Tqq =−2∞∑q=1T(2π)d2Γ(d2)q(2π)d∫∞−∞dkkd−1e−√k2+m2Tq.

The integral,

 ∫∞−∞dkkd−1e−√k2+m2Tq , (38)

equals to [1],

 ∫∞−∞dkkd−1e−√k2+m2Tq=2d2−1(√π)−1(qT)12−d2md+12Γ(d2)Kd+12(mqT). (39)

So can be written:

 Vboson = −2∞∑q=12d2−1(2π)d(2π)d+12md+1Kd+12(mqT)(Tmq)d+12 (40) = −∞∑q=11(2π)d(2π)d−12md+1Kd+12(mqT)(mq2T)d+12.

The function [1],

 Kν(z)(z2)ν=12∞∫0e−t−z24ttν+1dt , (41)

is even under the transformation . Thus equation (40) becomes:

 Vboson =−∞∑q=11(2π)d(2π)d−12md+1Kd+12(mqT)(mq2T)d+12 (42) =−12∞′∑q=−∞1(2π)d(2π)d−12md+1Kd+12(mqT)(mq2T)d+12.

(The symbol in the summation denotes omission of the zero mode term ). By using,

 Kν(z)(z2)ν=12∞∫0e−t−z24ttν+1dt, (43)

we get,

 Vboson=−141(2π)d(2π)d−12md+1∫∞0dte−t∞′∑q=−∞e−(mqT)24ttd+12+1. (44)

Let . Using the Poisson summation formula [11, 34, 3, 4] we have,

 ∞∑q=−∞e−λq2=√πλ∞∑k=−∞e−4π2k24λ, (45)

and omitting the zero modes we obtain:

 1+∞′∑q=−∞e−λq2=√πλ(  1 + ∞′∑k=−∞e−4π2k24λ). (46)

Finally,

 ∞′∑q=−∞e−λq2=√πλ( 1 + ∞′∑k=−∞e−4π2k24λ) − 1, (47)

and replacing in we take,

 Vboson= (48) −141(2π)d(2π)d−12md+1∫∞0dte−t(√πλ( 1 + ∑∞′k=−∞e−4π2k24λ) − 1td+12+1).

Set,

 ν=d+12. (49)

 Vboson = −141(2π)d(2π)d−12md+1(∫∞0dte−t√πλ tν+1)− (50) 141(2π)d(2π)d−12md+1∫∞0dte−t(√πλ( ∑∞′k=−∞e−4π2k24λ) tν+1) +141(2π)d(2π)d−12md+1∫∞0dte−t(1tν+1).

Also by setting,

 a=mT, (51)

equation (50) becomes (with ),

 Vboson =−141(2π)d(2π)d−12md+1(∫∞0dte−t√πt2atν+1) (52) −141(2π)d(2π)d−12md+1∫∞0dte−t(√πt2( ∑∞′k=−∞e−4π2k2a2t) atν+1) +141(2π)d(2π)d−12md+1∫∞0dte−t(1tν+1).

From this, after some calculations we obtain:

 Vboson =−12√π(2π)da(2π)d−12md+1(∫∞0dte−t t−ν−12) (53) −12√π(2π)da(2π)d−12md+1∫∞0dte−t(√πt2(∑∞′k=−∞e−4π2k2a2t) atν+12) +141(2π)d(2π)d−12md+1∫∞0dte−t(1tν+1).

By using [1],

 1(x2+a2)μ+1=1Γ(μ+1)∫∞0dte−(x2+a2)ttμ, (54)

we finally have:

 Vboson = −12√π(2π)da(2π)d−12md+1Γ(−ν−12+1) (55) −12√π(2π)da(2π)d−12md+1Γ(−ν−12+1) ×[∞′∑k=−∞(1+(2πka)2)ν+12−1] +141(2π)d(2π)d−12md+1Γ(−ν).

The sum,

 ∞′∑k=−∞(1+(2πka)2)ν+12−1, (56)

is invariant under the transformation . Thus we change the summation to,

 2∞∑k=1(1+(2πka)2)ν+12−1. (57)

Replacing the above to after some calculations we get:

 Vboson =−12√π(2π)da(2π)d−12md+1Γ(−ν−12+1) (58) −√π(2π)da(2π)d−12md+1Γ(−ν−12+1)(a2)12−ν ×[∞∑k=1(a2+4π2k2)ν+12−1] +141(2π)d(2π)d−12md+1Γ(−ν).

We use the binomial expansion (in the case that is even) or the Taylor expansion (in the case odd) [1]:

 (a2+b2)ν−12=σ∑l=0(ν−12)!(ν−12−l)!l!(a2)l(b2)ν−12−l. (59)

If is even, then equals to,

 σ=ν−12. (60)

If is odd then . We shall deal both cases. Replacing the sum into ,

 Vboson =−12√π(2π)da(2π)d−12md+1Γ(−ν−12+1) (61) +141(2π)d(2π)d−12md+1Γ(−ν) −√π(2π)da(2π)d−12md+1Γ(−ν−12+1)(a2)12−ν ×[∞∑k=1σ∑l=0((2π)2)ν−12−l(ν−12)!(ν−12−l)!l!(a2)l(k2)ν−12−l].

The last expression shall be the initial point for the following two subsections.

A much more elegant computation involves the analytic continuation of the Epstein-zeta function [3, 11, 4, 34, 57, 56, 76, 72, 78, 77]. In a following section we shall present the Epstein zeta functions in much more detail. In our case, relation (58) can be written in a much more elegant way, using the one dimensional Epstein zeta function,

 Zm21(ν,w,α)=∞∑n=1[w(n+α)2+m2]−ν, (62)

In our case, . Particularly one can make the relevant substitutions in the sum,

 ∞∑k=1(a2+4π2k2)ν+12−1], (63)

in terms of the one dimensional Epstein zeta function, (62).

#### The Chowla-Selberg Formula

It worths mentioning at this point a very important formula related with the Bessel sums [3, 4, 34] of relation,

 Vboson =−∞∑q=11(2π)d(2π)d−12md+1Kd+12(mqT)(mq2T)d+12 (64) =−12∞′∑q=−∞1(2π)d(2π)d−12md+1Kd+12(mqT)(mq2T)d+12.

Apart from the inhomogeneous Epstein zeta [3, 11, 4, 34, 57, 56, 76, 72, 78, 77], there exist in the literature a generalization of the inhomogeneous Epstein zeta function, namely the extended Chowla-Selberg formula [3], which we briefly describe at this point. We start with a two dimensional generalization of the Epstein zeta function,

 E(s;a,b,c;q)=∑n,mϵZ=(am2+bmn+cn2+q)−q. (65)

In the following is equal to,

 Q(m,n)=am2+bmn+cn2, (66)

and also is,

 Δ=4ac−b2. (67)

Following [3], relation (65), can be written as,

 E(s;a,b,c;q)=2ζEH(s,q/a)a−s+22s√πas−1Γ(s)√a∞∑n=1Γ(s−1/2)ζEH(s−1/2,4aq/Δ) (68) +22s√πas−1Γ(s)√a∞∑n=1ns−1/2cos(nπb/a)∑d/nd1−2s(Δ+4aqd2)1/4−s/2Ks−1/2(πna√Δ+4aqd2).

In the above relation, the summation is over the powers of the divisors of . Also stands for,

 ζEH(s;p)=−p−s2+√πΓ(s−1/2)2Γ(s)p−s+1/2+2πsp−s/2+1/4Γ(s)∞∑n=1ns−1/2Ks−1/2(2πn√p). (69)

Relation (68) has very attractive features. Most importantly the exponential convergence. We just mention this here for completeness and because (68) is very important. For more details see the detailed description of [3]. Our case is a special case of the extended Chowla-Selberg formula.

#### The Case d odd

As stated before in the odd case, . Then is:

 Vboson =−12√π(2π)da(2π)d−12md+1Γ(−ν−12+1) (70) +141(2π)d(2π)d−12md+1Γ(−ν) −√π(2π)da(2π)d−12md+1Γ(−ν−12+1