Dimensional reduction of a finite-size scalar field model at finite temperature

We investigate the process of dimensional reduction of one spatial dimension in a thermal scalar field model defined in $D$ dimensions (inverse temperature and $D-1$ spatial dimensions). We obtain that a thermal model in $D$ dimensions with one of the spatial dimensions having a finite size $L$ is related to the finite temperature model with just $D-1$ spatial dimensions and no finite size. Our results are obtained for one-loop calculations and for any dimension $D$. For example, in $D=4$ we have a relationship between a thin film with thickness $L$ at finite temperature and a surface at finite temperature. We show that, although a strict dimensional reduction is not allowed, it is possible to define a valid prescription for this procedure.


I. INTRODUCTION
Dimensional reduction states that considering a model in D dimensions we can obtain -following some prescription -a theory with fewer dimensions. One approach is to take a model with a compactified dimension and investigate its behavior when the size of the compactified dimension is reduced to zero.
Let us consider the case of a finite-temperature field theory using the imaginary-time formalism. Temperature is introduced by compactifying the imaginary-time variable using periodic or antiperiodic boundary conditions, respectively, for bosons or fermions. In this context, dimensional reduction would be equivalent to a high-temperature limit. This idea was proposed long ago by Appelquist and Pisarski [1] and later formulated in more detail by Landsman [2]. This idea is currently accepted and understood [3,4]. However, up to now, we think that there is a lack of investigation on this topic when one is interested in a greater number of compactifications.
We might ask how a system in arbitrary D dimensions with two compactified dimensions, one corresponding to the inverse temperature β = 1/T and the other with a finite-size L, behave when L → 0. However, strictly speaking, in the context of both first-order and second-order phase transitions, it can be shown that the limit of L → 0 cannot be fully attained [5]. There are many works in the context of phase transitions in thin films using different models (Ginzburg-Landau, Nambu-Jona-Lasinio, Gross-Neveu and others), see Refs. [6][7][8][9][10][11][12][13][14][15][16], that indicate the existence of a minimal thickness below which no phase transition occur. There is also, see for instance Ref. [6], a comparison between a phenomenological model for superconducting thin films using a Ginzburg-Landau model and experimental results that strengthen the indication. Therefore, this is a direct indication that the physics of surfaces and thin films are different and one could not achieve a surface from a thin film.
In this article, we investigate the problem of dimensional reduction from a mathematical physics perspective, for example, to investigate the relationship between systems in the form of "films" and "surfaces". Here this is generalized so that we investigate the relation between a D-dimensional and a D − 1-dimensional scenario.
To study a quantum field theory at finite temperature and finite size we use the formalism of quantum field theory in spaces with toroidal topologies [5,17]. As a first investigation, we consider a scalar field model at the one-loop level. We obtain a remarkably simple relationship between the following situations: • Starting with a space in D dimensions, we consider two compactifications: one of the dimensions corresponding to the inverse temperature 1/T , the other one to a finite size L, and the other D − 2 dimensions remaining of infinite size.
• Another possibility is to eliminate from the beginning one spatial dimension, and starting in a space with D − 1 dimensions, we consider one compactification corresponding to the inverse temperature 1/T . This should, for example, entail a relation between surfaces and thin films for D = 4. Furthermore, from a mathematical physics perspective, we also investigate the possibility of fractal dimensions.

II. THE MODEL
In this article, we take a scalar field theory with a quartic interaction in D dimensions with euclidean action We shall only discuss the 1-loop Feynman amplitude I with ρ propagators and zero external momenta, In particular, ρ = 1 corresponds to the tadpole contribution to the effective mass and ρ = 2 corresponds to the firstorder correction to the coupling constant. We introduce periodic boundary conditions on d < D coordinates. The compactification of the imaginary time introduces the inverse temperature β = 1/T ≡ L 0 and the compactification of the spatial coordinates introduces the characteristic lengths L i . So that the amplitude becomes We compute the remaining integrals on the (D − d)-dimensional subspace using dimensional regularization. The remaining infinite sum can be identified as an Epstein-Hurwitz zeta function [18] and leads -after an analytic continuation -to the sum over modified Bessel functions of the second kind K ν (x); see Refs. [5,19] for further details. The function I D,d ρ (M 2 ; L α ) reads in the case of d = 2 The double sum in Eq. (7) is known and has the analytical extension [18] ∞ n0,n1=1 3 After substituting Eq. (8) in Eq. (7) and also taking into account Eq. (6) we finally obtain the integral representation in the complex plane for the function W ν as the sum of these terms: The next step is to determine position of the poles of the functions in the complex-plane integrals and compute their residues. We see that the positions of the poles depend on the value of ν. We compute the integrals for the following specific cases: • integer ν, related to an even number of dimensions D; • half-integer ν, related to an odd number of dimensions D; • and, for completeness, other real values of ν, which can be thought of as related to fractal dimensions D.
We consider during this article two different scenarios. The first case considers a model in D dimensions with two compactified dimensions, one related to the inverse temperature β = 1/T and the other with a finite size L. For this one we need the function W ν , as defined in Eq. (9), and obtain Eq. (4). On the other situation, we consider a model in D − 1 dimensions with just one compactified dimension which is related to the inverse temperature. We see that, the amplitude requires just the knowledge of W (1) ν , To avoid a lengthy exposition we exhibit for each case only the final expressions. For integer ν the function W and W ν reads Here the notation a 2 indicates the unsigned Stirling number of the first kind: In the above expressions, the functions S (1) , S (2) , S (3) and S (4) are defined by For half-integer values of ν (ν = µ + 1/2, for integer µ) we obtain and Finally, for other real values of the index ν that are neither integer nor half-integer we have and We have managed so far to take into account both situations in which we have one or two compactified dimensions. Let us remember that we are interested in the one-loop Feynman diagram with ρ internal lines and at zero external momentum. The amplitude in the scenario with two compactifications, one related to the inverse temperature β = 1/T and another to a finite size L, is given by Eq. (4). And the amplitude in the scenario in which there are D−1 dimensions and just one compactification related to the inverse temperature β is given by Eq. (10) In the following, we investigate, for any value of ν, the relationship between both scenarios. The contribution from Γ(−ν) in amplitude is divergent for integer values of ν ≥ 0. To avoid the presence of poles in physical quantities we employ the modified minimal subtraction (M S) scheme [21] and the function Γ(−ν) is replaced by Γ (−ν) such that

III. DIMENSIONAL REDUCTION
We have considered above a class of one-loop Feynman diagrams with ρ internal lines at a dimension D. Their contributions involve the above defined functions W ν , where the index ν is given by ν = D/2 − ρ. In the previous section, we have managed to obtain the final version of W ν in terms of some analytical functions and sums over the Riemann zeta function in the argument. This was done for the specific cases of integer ν (useful for even dimensions), half-integer values of ν (for odd dimensions) and for other real values of ν (for completeness, and which can be considered for models in a fractal dimension).
Taking the situation with D dimensions and letting two of them be compactified, which introduces the temperature 1/β and a finite length L, one might ask how the function behaves as one takes the limit L → 0. This can be interpreted as a "dimensional reduction". In general, what happens is that the function W ν (β, L) diverges as L → 0. However, if we interpret the procedure of "dimensional reduction" as taking the dominant contribution 1 in β in the limit L → 0 and ignore the remaining dependence on the finite length, we obtain the relation This relation holds for any real value of D, which will be shown in the following subsections. The divergent behavior of I in Eq. (22) as L goes to zero depends on the quantity D/2 − ρ.

A. Integer ν, even D
To investigate the so-called dimensional reduction we first consider the case with integer values of ν, which corresponds to even dimensions D. The amplitude of a one-loop Feynman diagram in a scenario with both finite temperature and finite size is given by Eq. (4). To study its behavior we substitute Eq. (12) in Eq. (4) and split its contributions coming from three functions F 1 , G 1 , H 1 , such that, The function H 1 is the contribution that vanishes in the L → 0 limit: Note that the zero-temperature contribution was consistently subtracted. The divergent behavior as L goes to zero is given by G 1 , Lastly, the function F 1 is the contribution that survives during the dimensional reduction and does not diverge, Therefore, as the L → 0 limit is taken the contribution H 1 vanishes and the divergent behavior in L is given by the function G 1 : . (27) On the other hand, let us consider the second case in which we start from a scenario with one less dimension, D − 1. The amplitude of the one-loop Feynman diagram is simply Eq. (10), here written as And, for even dimensions D, we get that D 2 − 1 2 − ρ is a half-integer. So, we use Eq. (17) with µ = D 2 − ρ − 1 and obtain Therefore, Eq. (28) becomes , by direct comparison with F 1 we get Then we find the relation The asymptotic behavior of the function G 1 as L → 0 is B. Half-integer ν, odd D Now we consider odd dimensions D, meaning half-integer values of ν = µ + 1/2. For reference, µ = D−1 2 − ρ. Again, we split the contributions in three functions, F 2 , G 2 , H 2 , and after substitution of Eq. (18) in Eq. (4) the amplitude is given by with W (1) given by Eq. (11), that is, Once again, we must be careful with the zero-temperature contribution, as a modified minimal subtraction scheme is assumed, see Eq. (21). We obtain, after the cancellation with the zero-temperature contribution, Therefore, we get the relation The only difference compared to the scenario of integer values of ν is the divergent behavior of G 2 , in this case we have C. Other real ν, non-integer D To complete the analysis, we consider other real values of ν, that allow to take into account non-integer values of the dimension D. We follow the same procedure, defining three functions, (−1) k Γ(k + 1)