Mode Recombination Formula and Nonanalytic Term in Eﬀective Potential at Finite Temperature on Compactiﬁed Space

We develop a new formula called a mode recombination formula, and we can recast the eﬀective potential at ﬁnite temperature in one-loop approximation for fermion and scalar ﬁelds on the D -dimensional spacetime, S 1 τ × R D − ( p +1) × Q pi =1 S 1 i into a convenient form for discussing nonanalytic terms, which cannot be written in the form of any positive integer power of the ﬁeld-dependent mass squared, in the eﬀective potential. The formula holds irrespective of whether the ﬁeld is a fermion or a scalar and of boundary conditions for spacial S 1 i directions and clariﬁes the importance of zero modes in the Kaluza-Klein modes for the existence of the nonanalytic terms. The eﬀective potential is drastically simpliﬁed further to obtain the nonanalytic terms in easier and more transparent way. In addition to reproducing previous results, we ﬁnd that there exists no nonanalytic term for the fermion ﬁeld with arbitrary boundary condition for the spacial S 1 i direction, which is also the case for the scalar ﬁeld with the antiperiodic boundary condition for the spacial direction.


Introduction
Quantum field theory at finite temperature has provided a fundamental theoretical framework in high energy physics.In particular, the effective potential at finite temperature is a crucial tool to investigate physical phenomena involved with the order of the phase transition and its strength.
In the pioneering work by the Dolan-Jackiw [1], they found that there exists a nonanalytic term, which cannot be written in the form of any positive integer power of field-dependent mass squared, in the effective potential at finite temperature for a real scalar field.The nonanalytic term obtained by them has three-halves power of the mass squared, and it turns out to play an essential role to trigger the first order phase transition [2,3].Moreover, the magnitude of the term is related with the strength of the first order phase transition, which, in turn, put certain constraint on physical quantity such as the Higgs mass, for example, in the study of electroweak baryogenesis [4].Hence, the nonanalytic term in the effective potential is an important quantity that must be studied in detail.
Quantum field theory with compactified dimensions has been one of the attractive approaches for physics beyond the standard model.Orbifold compactification provides a framework for gauge-Higgs unification, where the Higgs field is unified into higher dimensional gauge fields [5,6].The order of the finite temperature phase transition in the gauge-Higgs unification has been studied in [7,8], and the first order phase transition actually occurs due to the nonanalytic term in the effective potential.It has been also shown that the quantum field theory at finite temperature with compactified dimensions can possess rich phase structures [9,10].Compactified dimensions also provide the theoretical framework for studying quantum field theory itself.For instance, from a point of view of dimensional reduction [11,12], models with several numbers of S 1 have been investigated.
In the previous paper [13], we studied all the possible nonanalytic terms in the effective potential at finite temperature in one-loop approximation for a real scalar field on the Ddimensional spacetime, S 1 τ × R D−(p+1) × p i=1 S 1 i , where the S 1 τ , R D−(p+1) , S 1 i stand for the Euclidean time direction, the D − (p + 1) dimensional flat Euclidean space, the spatial compactified direction, respectively.The effective potential contains the modified Bessel function of the second kind accompanied with multiple mode summations with respect to the winding mode associated with each S 1 .By using the integral representation for the modified Bessel function given by the inverse Mellin transformation [14,15] and the analytical extension for the mode summation [16], we recast the effective potential into the integral form on the complex plane 3 and performed the residue integration in order to obtain the nonanalytic terms.We found that the only nonanalytic power comes from a term M D−(p+1) which is not analytic in M 2 when the dimension of the flat Euclidian space, D − (p + 1) is odd.We obtained the coefficient of this term, but other nonanalytic terms proportional to log M were not studied.
In this paper, we develop a new formula called a mode recombination formula, which plays a central role for discussions in the present paper.The effective potential can be recast into the convenient form for studying the nonanalytic terms.The formula also clarifies that only the zero mode in the Kaluza-Klein mode associated with each S 1 is crucial for the existence of the nonanalytic terms in the case of the scalar field satisfying the periodic boundary condition for the spatial S 1 i (i = 1, • • • , p) direction.Then, the effective potential relevant for the nonanalytic terms can be simplified further drastically and is given in terms of the contribution of the single mode summation with respect to the winding mode associated with each S 1 .This is quite different from the previous paper, where it includes the contribution of the multiple mode summations.The integral form for the simplified effective potential on the complex plane is easy to perform the residue integration in order to obtain the nonanalytic terms.We reproduce the previous results in easier and more transparent way.
The mode recombination formula holds irrespective of whether the field is a fermion or a scalar and of the boundary condition for the spatial The formula also provides a convenient form for studying the nonanalytic terms in the effective potential for the case of the fermion field with arbitrary boundary condition for the S 1 i direction.The zero mode for the Euclidean time direction is removed due to the antiperiodic boundary condition followed from the quantum statistics for the fermion.This changes the pole structure of the analytical extension for the mode summation with respect to the Matsubara mode compared with that of the scalar case.We find that there is no nonanalytic term for the case of the fermion.This immediately implies that the effective potential does not possess the nonanalytic term for the case of the real scalar field satisfying the antiperiodic boundary condition for at least one spatial S 1 i direction.This paper is organized as follows.We present the setup in the next section.We prove the mode recombination formula and present the convenient form of the effective potential for studying the nonanalytic terms in each case of the fermion and the scalar in the section 3. We reproduce the same result as the one in the previous paper in easier and more transparent way in the section 4. We also study the nonanalytic terms for the case of fermion with arbitrary boundary condition in the section 5 and of the real scalar with the antiperiodic boundary condition in the section 6.The final section is devoted to conclusions and discussions which also include the case for a higher dimensional gauge field.

Setup
Let us first present the setup for our discussions.We study nonanalytic terms in the effective potential at finite temperature in one-loop approximation for a real scalar (fermion) field on the D-dimensional spacetime, S 1 τ ×R D−(p+1) × p i=1 S 1 i .We employ the Euclidean time formulation for finite temperature quantum field theory and then the Euclidean time direction, whose coordinate is denoted by τ , is compactified on S 1 τ .The spatial p directions are compactified on the p numbers of S 1 and their coordinates are y i (i = 1, • • • , p).We denote the circumference of each S 1 i as L i (i = 0, 1, • • • , p) and L 0 stands for the inverse temperature T −1 .The R D−(p+1) is the D − (p + 1) dimensional flat Euclidean space whose coordinates are The Lagrangian is given by where the N stands for N = (τ, k, i), and φ (ψ) is the scalar (fermion) field whose bulk mass is m s (m f ).The g is the Yukawa coupling.
One needs to specify the boundary conditions for the S 1 τ and S 1 i (i = 1, • • • , p) directions.For a given field Φ(τ, x k , y i ), the boundary condition for the S 1 τ direction is specified by The parameter η 0 is definitely determined by quantum statistics to be 0 (periodic) for the scalar field or to be 1 2 (antiperiodic) for the fermion field.On the other hand, the boundary condition for the The parameter η i can take 0 or 1 2 for the real scalar field and can be arbitrary for the fermion field.
We employ the standard prescription to calculate the effective potential at finite temperature in one-loop approximation.Let us quickly review the calculations given in the previous paper [13].For those who are familiar with it, readers can directly go to the next section.By taking up the quadratic terms in the shifted Lagrangian around the constant field ϕ for the scalar field φ in Eq.(2.1), one needs to evaluate in order to obtain the effective potential on S 1 τ ×R D−(p+1) × p i=1 S 1 i in one-loop approximation.Here, the M(ϕ) is the field-dependent mass of the scalar (fermion) field, Hereafter, we denote M(ϕ) by M for simplicity.The p E denotes the D − (p + 1)-dimensional Euclidean momentum.The f is the fermion number that is 0 (1) for the boson (fermion) and the N is the on-shell degrees of freedom.The n 0 denotes the Matsubara mode arising from the S 1 τ , and the Kaluza-Klein mode

2) and (2.3).
Let us make use of the zeta-function regularization in order to evaluate Eq. (2.4).By defining then, the V eff is written as (2.7) Performing the p E integration with the formula and employing the Poisson summation we obtain Hereafter, we call m j (j = 0, 1, • • • , p) in Eq.(2.9) or Eq.(2.10) the winding modes, while n 0 and n i (i = 1, 2, • • • , p) in Eq.(2.9) or Eq.(2.4) the Matsubara and the Kaluza-Klein modes, respectively.
It is useful to separate each summation with respect to m j in Eq.(2.10) into the zero mode (m j = 0) and the nonzero ones (m j = 0) and to express Eq.(2.10) into the form where 12) The prime of the summation in ′ ∞ m j =−∞ means that the zero mode (m j = 0) is removed.The F (0) in Eq.(2.11) corresponds to the contribution from all the zero modes m 0 = m 1 = • • • = m p = 0 in Eq.(2.10) and is found to be It must be understood that F (0)D (M) is regularized by the dimensional regularization for D = even.On the other hand, for D = odd, it yields[13] By using the formula where the K ν (z) is the modified Bessel function of the second kind, Eq.(2.12) for n ≥ 1 becomes (2.16) The effective potential is given by the modified Bessel function of the second kind accompanied with the multiple mode summation with respect to the winding mode m i j (j = 1, • • • , n) associated with the S 1 i j .The winding mode is derived from the Matsubara/Kaluza-Klein mode n i j through the Poisson summation (2.9).We will use the inverse process in the next section, which is employed to prove a mode recombination formula.The results in this section have already been obtained in the previous paper (see Eq.(2.10) of Ref. [13]) and they are the starting point for the discussion in the present paper.

Mode Recombination Formula
In this section, we present and prove the formula called a mode recombination formula.The formula plays a crucial role to obtain a new form of the effective potential, which is different from Eq.(2.11) and is convenient for discussing the nonanalytic terms.These terms can be obtained from the effective potential by use of the mode recombination formula in easier and more transparent way, as we will see in the next section.
Let us recall Eq.(2.11) and first write it as where we have separated the n = 0 term from the summation in Eq.(2.11) for later convenience.We focus on the scale L 0 in L i j (i j = 0, • • • , p) and separate the term associated with L 0 from the terms without L 0 on the right-hand side of Eq.(3.1) as Let us show that the second and third terms on the right-hand side of Eq.(3.2) can be combined into a single expression where Here, the n 0 is the original Matsubara mode appeared in Eq.(2.4).
Let us prove Eq.(3.3).By using Eq.(2.16), the two terms on the left-hand side of Eq.(3.3) are combined into a single expression, including the m 0 = 0 mode 5 , as (3.5) 4 Let us comment that the effective potential obtained by Dolan-Jackiw, Eq.(3.13c) in their paper [1], is reduced to to the series expansion of the logarithm in Eq.(3.13c).The mode in the series expansion plays the same role with the winding mode m 0 in F (1)D L0 (M ). 5 Let us note that this is allowed because Eq.(2.16) is regularized with respect to With the help of the formula (2.15), the modified Bessel function of the second kind in Eq.(3.5) is written in the integral form, and we inversely use the Poisson summation (2.9) for the m 0 mode, which is then converted into the Matsubara mode n 0 .Thus, we obtain where we have used Eq.(2.15) again to rewrite the integral from into the Bessel function in the last equality.Eq.(3.6) is nothing but the right-hand side of Eq.( 3.3) and we have proved Eq.(3.3).
We shall call Eq.( 3.3) the mode recombination formula6 in the present paper.Let us note that the Matsubara mass squared, the second term in Eq. (3.4), is recovered by the inverse use of the Poisson summation (2.9).From Eqs.(3.2) and (3.3), we obtain an important relation (3.7) Some comments are in order.It is important to note that the first term on the righthand side of Eq.(3.7), which is the contribution of the single mode summation with respect to the winding mode m 0 associated with the S 1 τ having the focused scale L 0 , is separated from the contributions of the multiple mode summations in the process of deriving the mode recombination formula.The Matsubara mass squared, the second term in Eq.(3.4), arises after using inversely the Poisson summation.As is clear from the discussions given above, the formula (3.7) holds irrespective of whether the field is the scalar or fermion and also of the boundary condition for the spatial ) is effectively reduced to D − 1.This is interpreted as that the particle with mass squared M 2 having the Matsubara mass squared ( 2π L 0 ) 2 (n 0 + η 0 ) 2 to be considered on the D − 1 dimensional spacetime, R D−(p+1) × p i=1 S 1 i .One can choose another scale, say From Eq.(3.7), the effective potential (3.1) is written by We repeat the same discussion for the third term in Eq.(3.8) by focusing next on the scale L 1 .
One immediately sees that the term, aside from the scale L 0 , can be written as where we have defined and have used Eq.(3.3) with L 0 , D − 1, M and M (0) being replaced by L 1 , D − 2, M (0) and M (0,1) , respectively in the last equality.We observe that the first term on the right-hand side of Eq.(3.9), the contribution of the single mode summation with respect to the winding mode m 1 associated with the S 1 1 having the focused scale L 1 , is separated from the contributions of the multiple mode summations.The Kaluza-Klein mass squared ( 2π L 1 ) 2 (n 1 + η 1 ) 2 turns out to be added to M 2 (0) through the formula (3.3) in Eq.(3.10).We repeat the same discussion given above by focusing on the scales L 2 , L 3 , • • • , L p−1 sequentially for the multiple mode summation obtained at each step such as the second term on the right-hand side of Eq.(3.7) (or Eq.(3.9)).We finally find that the effective potential is recast into where we have defined (3.12) The second term F (1)D L 0 (M) in the first line of Eq.(3.11) is incorporated into the summation with respect to k as the k = 0 term in Eq. (3.11), where it is understood that we set (3.13) Eq. (3.11) is the effective potential rewritten by using successively the mode recombination formula (3.3) and gives the starting point to discuss the nonanalytic terms in the present paper.Let us note that Eq.(3.11) holds for any boundary condition for the spatial S 1 i direction, reflecting the fact that the mode recombination formula holds irrespective of the boundary condition for any spatial direction.
It may be instructive to mention that the effective potential (3.1) is invariant under the exchange of the scales L i and L j (i = j).The invariance, however, does not become manifest in the new form of the effective potential (3.11).Instead of losing the manifest invariance, the effective potential has a remarkable feature that except for the first term in Eq. (3.11), all the contributions to the effective potential are given by the single mode summation with respect to the winding mode m k (k = 0, • • • , p) associated with the S 1 k , even though the M 2 has the Kaluza-Klein (Matsubara) mass squared, the second term in Eq.(3.12).One can further simplify Eq.(3.11) by taking account of the discussion on the zero modes in Eq.(3.12), as we will see in the next section.
We shall study the nonanalytic terms for the case of the fermion field as well as the scalar one in the present paper.It may be appropriate here to present the convenient form of the effective potential for the discussion in the case of the fermion.One can show that the first and the second terms in Eq.(3.8) are combined into a single expression Then, the effective potential (3.8) is written as or where the first term in Eq.(3.15) is incorporated into the summation with respect to n as the n = 0 term in Eq. (3.16).
Let us prove Eq. (3.14).By setting m i 1 = • • • = m in = 0, n = 0 and dropping the summation where we have used Eq.(2.8) in the last equality.From Eq.(2.13), this is nothing but the right-hand side of Eq.(3.14) and we have proved Eq.(3.14).
We will use Eqs.(3.15) and (3.16) in section 5, where we discuss the nonanalytic terms in the effective potential for the case of the fermion.It can be said that Eq.(3.14) is also the mode recombination formula and is regarded as the n = 0 case in Eq. (3.3).Let us note that we can always write the effective potential as Eq.(3.16) for any boundary condition of the spatial S 1 i direction.For later convenience, we introduce the integral representation for the modified Bessel function of the second kind on the complex plane [14,15] The constant c should be understood to be a point located on the real axis which is greater than all the poles of the gamma functions in the integrand.By using the formula (3.18), one obtains the integral form of the right-hand side of Eq.(3.3) with n ≥ 1 as where we have defined and M (0) is given by Eq.(3.4).Let us note that the n = 0 case in Eq.(3.3) is given by Eq.(3.17).
We have succeeded in representing the effective potential in the various forms like Eqs.(3.8), (3.11), (3.15) and (3.16).It may be helpful to comment on which forms are useful for the analysis of the scalar or fermion field.The expressions (3.8) and (3.11) of the effective potential will be used for the scalar with the periodic boundary condition.On the other hand, the expressions (3.15) and (3.16) turn out to be useful for the fermion and the scalar with the antiperiodic boundary condition.

Nonanalytic terms for scalar field with periodic boundary condition
In this section, we derive the nonanalytic terms of the effective potential for a real scalar (f = 0, N = 1, η 0 = 0) with the periodic boundary condition in all the spatial directions (η 1 = • • • = η p = 0), although the results have already been obtained in the previous paper [13].The purpose of this section is to show that our formulation presented in this paper is easier and more transparent to obtain the nonanalytic terms than that in the previous paper.To this end, we examine the right-hand side of Eq.(3.3).From Eq.(3.19), we have where we have defined and have separated the zero mode (n 0 = 0) from the nonzero modes (n 0 = 0).
We show that the second term in Eq.(4.1) never has nonanalytic terms, in other words, all the terms are given by positive integer powers of the mass squared M 2 .The second term in Eq.(4.1) is written, after changing the variable t = t − D−1 2 , as where we have again denoted t by t.Let us recall the following formula used in the previous paper [13,16] Γ(t) where we have used Eq.(3.18) in the last equality.
We apply the formula (4.4) with L 0 = 1 and z = M L 0 2π to Eq.( 4.3), which becomes We deform the integration path with respect to t 1 in Eq.(4.5) in such a way that it encloses all the poles in the integrand and perform the t 1 integration by the residue theorem.Among the poles ) and is frequently used throughout our discussions.The residue integral from the pole t 1 = t − 1 2 with ζ(0) = −1/2 cancels the second term in Eq.(4.5) and the integral from the pole t 1 = t of ζ(2t 1 − 2t + 1) with Γ(1/2) = √ π does the first terms in Eq.(4.5).We are left with the contribution from the pole , so that we find that the second term in Eq.(4.1) becomes Even though there would appear the poles in the multiple mode summations S(n) (t + D−1 that contribute to the t integration in Eq.(4.6) by the residue theorem, it does not affect the power of the mass squared (M 2 ) n in Eq.(4.6).Thus, Eq.(4.6) has only positive integer powers of the mass squared M 2 , i.e. (M 2 ) n.This observation implies that only the first term in Eq.(4.1) potentially can have the nonanalytic terms, so that from Eq.(3.3), we obtain the important relation The abbreviation denoted by "n.a." in Eq.(4.7) means nonanalytic terms, which is used throughout the paper.It should be stressed that only the zero mode (n 0 = 0) in M (0) is relevant for the existence of the nonanalytic terms in the mode recombination formula (3.3).
The relation (4.7) is crucial for the continuing discussion given below.
The relevant part for the nonanalytic terms in the effective potential (3.8) is given by replacing M (0) by M without the summation because of Eq.(4.7).Considering the discussion led to Eq.(4.7), we understand that the relevant part for the nonanalytic terms in Eq.(3.9) is given by replacing the M (0) , M (0,1) by M, M (1) , respectively and dropping the n 0 summation.By repeating the same discussion as above and applying it to the effective potential (3.11), the relevant part of the nonanalytic terms in the effective potential is given by picking up only the zero modes in It turns out that the relevant part of the nonanalytic terms in the effective potential drastically reduces to the simple expression (4.8) and is given by the contribution of the single mode summation of the winding mode m k (k = 0, • • • , p) associated with the S 1 k .This is quite different from that given in the previous paper where we have analyzed the multiple mode summations in order to obtain the nonanalytic terms.It should be emphasized that Eq.(4.8) is obtained as the consequence of taking account of the zero mode (n 0 = • • • = n p = 0) alone in the Matsubara and Kaluza-Klein modes.
Let us confirm the results obtained in the previous paper by using Eq.(4.8).We will soon recognize that the calculations based on Eq.(4.8) are easier and more transparent compared with that in the previous paper.We first study the nonanalytic terms of F , (4.9) where we have used the definition of the zeta function ζ(z) = ∞ m=1 m −z .Since the scale dependence on M in Eq.(4.9) is (M 2 ) yields analytic terms (M 2 ) ℓ , which are outside of our interest.Only the pole t = 0 among the poles t = −m (m = 0, 1, • • • , ) of Γ(t) contributes to the residue integration because of the property ζ(−2m) = 0 (m = 1, 2, • • • ).Hence, there are two poles, t = 0 of Γ(t) and t = 1 2 of ζ(2t) that contribute to the residue integration in Eq.(4.9) for obtaining the nonanalytic terms 7 .
Since the nonanalytic terms are given by positive odd integer powers of M, whether the pole of t = 0 or t = 1/2 can produce a nonanalytic term depends on whether D and k are even or odd.For (D, k) = (even, even) or (odd, odd), the nonanalytic terms in Eq.(4.9) is given by the pole t = 1 2 as while for (D, k) = (even, odd) or (odd, even), the pole t = 0 yields the nonanalytic term Here, we have used We are ready to calculate the nonanalytic terms in the effective potential based on Eq.(4.8) by using Eqs.(4.10) and (4.11).It is convenient to study the terms for each case of even/odd D and p + 1.Let us remind that p + 1 is the total number of S 1 .
In terms of Eqs.(4.13) and (4.14), the nonanalytic term in Eq.(4.8) is given by where we have used Eq.(4.15) and the fact that from Eq.(2.13), the F (0)D (M) does not possess any nonanalytic term for D = even.The B p is given by Eq.(4.14) with n = p 2 to yield, by noting that (−1)(−1) for p = even, 4.2 (D, p + 1) = (even, even) In this case, the nonanalytic term in Eq.(4.8) is written by using Eqs.(4.13) and (4.14) as where we have used Eq.(4.15) in the last equality.The effective potential does not have any nonanalytic term in this case.

(D, p + 1) = (odd, odd)
For the case of D = odd, from Eqs.(4.10) and (4.11), it is convenient to introduce respectively.We see that the relation holds.It should be understood that in Eq.(4.21) we have defined L 0 (M).One immediately confirms that the C −1 = −D 0 is satisfied by using the explicit expressions for F (0)D (M) in Eq.(2.14) and F (1)D L 0 (M) [13] in Eq.(4.11) for D = odd.
In terms of Eqs.(4.19) and (4.20), the nonanalytic part of the effective potential in Eq.(4.8) is calculated as thanks to Eq.(4.21).
The results (4.17), (4.18), (4.22) and (4.24) are found to exactly agree with those given in the previous paper, as they should be.Even though they have already been obtained in the previous paper, our formalism developed in this paper is easier and more transparent to derive the nonanalytic terms of the effective potential, and furthermore makes it possible to analyze the case of the fermion field, as we will see in the next section.In addition to it, a new insight on the nonanalytic term is obtained.The nonanalytic term Eq.(4.17) (or Eq.(4.24)) depends on each scale, L 0 , • • • , L p of all the S 1 's.This may reflect the fact that only the zero modes in the Matsubara and Kaluza-Klein modes are relevant for the existence of the nonanalytic terms, as shown in Eq.(4.8).

Nonanalytic terms for fermion field
Let us study the nonanalytic terms in the effective potential for the case of the fermion field (f = 1, η 0 = 1 2 ) satisfying arbitrary boundary condition for the spatial S 1 i (i = 1, 2, • • • , p) direction.The effective potential (3.15) rather than Eq.(3.11) is appropriate to discuss the terms in this case.
We first consider the n ≥ 1 term, the second one in Eq. (3.15), whose integral form on the complex plane is given from Eq.(3.19) by where we have changed the variable t = t − D−1 and have denoted t by t again.
One needs to develop the analytical extension for the mode summation with respect to n 0 in Eq.(5.1).To this end, we employ the formula (2.8) and the Poisson summation (2.9).Then, we find ( By separating m 0 = 0 and m 0 = 0 modes in Eq.( 5.2) and by using Eqs.(2.8) and (2.15), the right-hand side of Eq.(5.2) becomes where we have used the property K −ν (z) = K ν (z) and the formula for later convenience.We further recast Eq.( 5.3) into the integral form on the complex plane by Eq.(3.18).Then, we obtain This is an analytical extension for the mode summation with respect to n 0 in Eq.(5.1).Eq.(5.5) does not have the term corresponding to the first term in Eq.(4.4) because of the lack of the zero mode due to the antiperiodic boundary condition for the fermion field in the Euclidean time direction.
By setting z = M L 0 2π and inserting Eq.(5.5) into Eq.(5.1), we have We perform the residue integration with respect to t 1 by deforming the integration path in such a way that it encloses all the poles in the integrand.Among the poles 2 ), only the pole t 1 = t − 1 2 is relevant to the residue integration thanks to the property ζ(−2ℓ) = 0 (ℓ = 1, 2, • • • ).In addition to it, the poles t 1 = t of ζ(2t 1 − 2t + 1) and t 1 = −n (n = 0, 1, • • • ) of Γ(t 1 ) contribute to the t 1 integration by the residue theorem.It is easy to see that the contributions from the residue integration of the pole t 1 = t − 1 2 in the second and third terms of Eq.(5.6) cancel the first term in Eq. (5.6) and that the contributions from the residue integration of the pole t 1 = t of ζ(2t 1 − 2t + 1) in the second and the third terms of Eq.(5.6) are canceled each other.Thus, what is left is the contribution of the residue integration of the pole t 1 = −n (n = 0, 1, • • • ) of Γ(t 1 ) alone, which is given by (5.7) One observes that the dependence of the power on M in Eq.(5.7) is given by the positive integer power of the mass squared (M 2 ) n, so that there exists no nonanalytic term in the second term in Eq.(3.15) (5.8) We note that the multiple mode summations in S (n) (t+ D−1 2 ; L i 1 , • • • , L in ) could produce poles that contribute to the t integration in Eq.(5.7), but they do not change the power of (M 2 ) n because of the nonexistence of M in the summations.It is important to note that Eq.(5.7) holds irrespective of the boundary condition for the spatial S 1 i (i = 1, 2, • • • , p) direction and thus, so does Eq.(5.8).
Let us next study the n = 0 term, the first one in Eq.(3.15), which is given by Eq.(3.17) with f = 0 and η 0 = 1 2 .We make use of the analytical extension (5.4) in order to calculate the mode summation with respect n 0 in Eq. (3.17).By putting t = − D−1 2 , z = M L 0 2π in Eq.(5.5), the relevant part of Eq.(3.17) becomes (5.9) We again evaluate the t 1 integration by the residue theorem by deforming the integration path in such a way that it encloses all the poles in the integrand.The nonanalytic terms8 in the second and the third terms in Eq.(5.9) are given by the residue integration of the poles that have the minus half odd integer values of t 1 , as seen from the scale dependence on M in Eq.(5.9).
For D = even, the relevant pole is t , from which the second and the third terms in Eq.(5.9) yield The first term in Eq.(5.9) is analytic for D = even, so that there is no nonanalytic term in Eq.(5.9) for D = even.
On the other hand, for D = odd, the relevant pole for the nonanalytic terms is given only by the pole t 1 = − D 2 among the poles Then, the second and the third terms in Eq.(5.9) lead to which cancels the first term in Eq.(5.9).Thus, Eq.(5.9) has no nonanalytic term for D = odd.
We have shown that the first term in Eq.(3.15) does not possess the nonanalytic term = 0. (5.12) We arrive at the important conclusion from Eqs.(5.8) and (5.12) that the effective potential (3.16) has no nonanalytic term for the case of the fermion It should be emphasized that Eq.(5.13) holds irrespective of the boundary condition for the spatial 6 Nonanalytic terms for scalar field with antiperiodic boundary condition Taking account of the result obtained in the previous section, we can also study the case of the scalar field with some of the boundary condition for the spatial S 1 i (i = 1, • • • , p) direction being antiperiodic.In this case, by regarding the direction that has the antiperiodic boundary condition as the Euclidean time direction, the effective potential essentially has the same with that of the case for the fermion field.Hence, we conclude that there is no nonanalytic term in the effective potential if the scalar field satisfies at least one antiperiodic boundary condition for the spatial S 1 i direction.

Conclusions and discussions
We have studied the nonanalytic terms, which cannot be written in the form of any positive integer power of field-dependent mass squared, in the effective potential at finite temperature in one-loop approximation for the fermion and scalar fields on the D-dimensional spacetime, S 1 τ × R D−(p+1) × p i=1 S 1 i .In doing it, we have developed the new formula called the mode recombination formula (3.3), which holds irrespective of whether the field is a fermion or a scalar and of the boundary condition for the spatial S 1 i direction.The effective potential has been recast into the new forms (3.11) and (3.16) by using the formula, which is convenient to study the nonanalytic terms for both cases of the fermion and the scalar.
We have clarified the importance of the zero mode in the Kaluza-Klein mode for the existence of the nonanalytic terms through the mode recombination formula.This has drastically simplified the relevant part of the effective potential for calculating the nonanalytic terms for the case of the scalar field satisfying the periodic boundary condition for the spatial S 1 i direction.The effective potential (4.8) is given in terms of the single mode summation of the winding mode for each S 1 and the integral form for the potential on the complex plane is easy to perform the residue integration.We have correctly reproduced the nonanalytic term (4.17) (and (4.24)) in easier and more transparent way.
The mode recombination formula has also provided the convenient form for studying the nonanalytic terms in the effective potential for the case of the fermion field satisfying arbitrary boundary conditions for the spatial S 1 i (i = 1, • • • , p) directions.The antiperiodicity for the Euclidean time direction for the fermion has resulted the quite different pole structure of the analytical extension for the mode summation with respect to the Matsubara mode compared with that of the scalar.We have found that there is no nonanalytic term in the effective potential for the case of the fermion.The result for the case of the fermion has immediately led to the conclusion that there also exists no nonanalytic term for the case of the scalar field satisfying the antiperiodic boundary condition for at least one spatial S 1 i direction.We have obtained some insight on the nonanalytic terms in the effective potential.We have found that the nonanalytic term can appear only when there exists the zero mode associated with each of all the S 1 's in the Matsubara and Kaluza-Klein modes.This observation may explain that the nonanalytic term depends on all the scales L 0 , L 1 , • • • , L p like in Eq.(4.17) (or Eq.(4.24)) and further that the effective potential has no nonanalytic term for the fermion due to the lack of the zero mode for the Euclidean time direction irrespective of the absence or presence of the zero mode for the spatial S 1 i (i = 1, 2, • • • , p) direction.Equipped with the result obtained for the cases of the scalar, we can also mention about the nonanalytic terms for the case of a higher dimensional gauge field on S 1 τ ×R D−(p+1) × p i S 1 i .Let us consider the D-dimensional gauge field A M (τ, x k , y i ), whose component gauge fields are written as A M (τ, x k , y i ) = A τ (τ, x k , y i ), A k (τ, x k , y i ), A i (τ, x k , y i ) . (7.1) We need to specify the boundary condition for the Euclidean time direction and the spatial S 1 i direction.The boundary condition for the Euclidean time direction must be periodic, i.e.
A M (τ + L 0 , x k , y i ) = +A M (τ, x k , y i ) (7.2) because of the quantum statistics.One can choose the boundary conditions of the A k (τ, x k , y i ) and the A i (τ, x k , y i ) for the spatial S 1 i (i = 1, • • • , p) direction to be 0 or twisted under the assumption that the Lagrangian density must be single valued.The gauge field A M (τ, x k , y i ) can be massive through the Higgs mechanism and it has the field-dependent mass such as M(ϕ).If we restrict the twisted boundary condition for the spatial S 1 i direction to the antiperiodic boundary condition, the nonanalytic term in the effective potential for each gauge field in Eq.(7.1) is reduced to the cases of the scalar field studied in the sections 4 and 6.Thus, the result has already been obtained 9 .
In addition to the nonanalytic terms we have studied, there is another type of the nonanalytic term, log M in the effective potential, as mentioned in the sectionse 4. Such the term arises from the double pole of the integrand, for example, in the residue integration of Eq.(4.9).In order to understand the whole nonanalytic structure of the effective potential with respect to M, one has to study such a term extensively as well.Our analyses have been carried out at the level of one-loop approximation for the effective potential at finite temperature.One may wonder what type of the nonanalytic terms besides the one obtained in the paper can emerge beyond the one-loop calculation.In connection with higher loop calculations, we are also interested in the behavior of the log M term, which actually stands for genuine quantum effects.These are under investigation and will be reported elsewhere.
the first equality of Eq.(3.6), the left-hand side of Eq.(3.14) is given by