Author(s): <p>Emerson B. S. Corrêa*, Carlos A Bahia and Michelli S. R. Sarges</p>
We study four expressions involving the integrals of Jacobi’s theta functions. From Poisson’s summation formula, we write the integrals of the functions θi, (i = 1,2,3,4) in terms of modified Bessel functions of the second kind. For the integrals of θ1, θ2 and θ3, we get expressions with real arguments, but for the integral of θ4, we find an expression with imaginary argument. In addition, we apply our results to the description of two kinds of interacting quantum systems: boson gas and fermion gas both under a thermal bath and an external magnetic field.
Almost 200 years ago, Jacobi investigated some functions known today as theta functions: θi(z; q) with (z,q) E C, |q| < 1 (i = 1,2,3,4) and demonstrated their main properties [1, 2]. In the last years, Jacobi theta functions have been used in the context of heat conduction theory, for problems related to orthogonalization and interpolation, in probability laws, in a class of hypergeometric integrals, in the complete elliptic integral of the first kind K(k) and in quantum field theory (QFT) at finite temperature with boundary conditions [3-10].
Through Schwinger’s proper-time representation for the Feynman propagator, it is possible to write the integrals on the proper-time s in terms of integrals on the Jacobi theta functions. However, these integrals are only tractable from a numerical point of view (up to our knowledge). Indeed, the integrals involving the Jacobi theta functions that appear in QFT acquire infinite values for the argument s -> 0, thus requiring a regularization process of these integrals, for a later solution numeric [11-13].
The integrals of θi [f(s); g(s)] with s in the range [0,+∞] which we will compute in this note, play an important role in effective models of quantum chromodynamics (QCD) (for a review these effective models of QCD, see for example the Refs. [14-16]). This protagonism arises when we apply the generalized Matsubara formalism to the model, which allows us to introduce the spatial boundaries, temperature, chemical potential, and magnetic effects in the quantum system (the reader can find a review of generalized Matsubara formalism in) [17, 18].
In this contribution, we produce a closed form for these integrals, in terms of well-behaved Bessel functions of the second type. This will be done considering the physical interest of applying our results in Quantum Mechanics (QM) with and without magnetic effects. For this reason, we will define the functions f(s) and g(s) as (= abs) and [= exp(-a2 s)], respectively. This choice allows us to include in the systems, thermodynamic variables such as temperature and chemical potential by the formalism of imaginarytime, for boson fields (even frequencies) and for fermion fields (odd frequencies).
This approach establishes the QFT in the topology 1 D = S1 xRD-1, where the S1 represents a compact in the imaginary-time direction and there is no restriction on the remaining spatial coordinates (D - 1). From the boundary conditions for the bosonic or fermionic fields at coordinate , we can demonstrate that S1 defines a circle with radius β/2π where β is the inverse of system temperature, that is, β = 1/kBT. Therefore, in the imaginary-time formalism, E [0,β].
The paper is structured as follows: In Section II, we obtain explicit expressions for the integrals of all Jacobi’s theta functions in terms of exponentially decaying functions Kv (x). In Section III, we apply the results in two different and self-interacting quantum systems: the first one formed by a gas of bosons and the other one composed by a gas of fermions. We shall define the Hamiltonian density for the cited systems and use some QFT results, in particular, the expressions for the Feynman propagator when the quantum field is under an external magnetic field. In this scenario, the so-called Landau levels arise (quantized energy levels in the plane orthogonal to the applied external magnetic field). Also, we explore the findings in the context of phase transition and broken/ restoration of symmetry. We conclude this work and make some considerations about it, in Section IV. At physical applications, we considered a four-dimensional Euclidean space and the natural unit system such that c = h = kB = 1.
Let us recall the definitions of the four theta functions that can be found in Ref [1], chapter 21, namely
Remembering that the gamma function of argument v and the modified Bessel function of the second kind of order v, have the following representations [20]
In the next section, we apply I2 and I3 to an interacting quantum gas of fermions and bosons, respectively
In this Section, we will apply the results found previously to two different systems of interacting quantum particles: boson gas and fermion gas both defined on four-dimensional Euclidean space and under a constant magnetic field.
Let us apply our findings to a boson system described by the scalar field θ under the interaction (λ0 /4!)θ4 in a four-dimensional Euclidean space. The Hamiltonian density is given by
Again, let us use Eq. (17) to perform the magnetic and thermal corrections on the mass parameter . In this case, the Hamiltonian density at zero temperature is given by
We observe that the critical temperature is obtained when the boson system has zero mass parameter, i.e., m(Tc,w) = 0 (see Ref [23]).
For performing the phase structure analysis of the system we fix mσ = 2m0 , being mσ the scalar particle whose value used in this paper is mσ = 0.500 GeV [24]. For the graphics, we used λ = 0.17. In Figure 1, we show the behavior of the mass parameter m as a function of temperature for the boson gas in interaction under several values of an external magnetic field.
Fixing the value of the magnetic field, we see that the finite chemical potential tends to decrease the critical temperature of the system. That is, higher chemical potential values contribute to the system undergoing a phase transition at lower temperatures than in the case of zero chemical potential.
On the other hand, by fixing µ and taking different values of w, the critical temperature of the system increases, for higher external magnetic field strengths. The effect of the external magnetic field on the system is to facilitate the phase transition. This is the phenomenon of magnetic catalysis for the charged scalar field.
To complete the analysis of the phase structure of the model, we investigate the free energy of the system in Figures 2 and 3. In Figure 2, we note that Tc (w = 0) < Tc (w = 0.5). In particular, the temperature T = 0.400 GeV at which the system is in the symmetric phase for w = 0 GeV2 becomes a temperature that guarantees symmetry breaking for w = 0.5 GeV2 . The same analysis is valid for Figure 3, however, the effect of µ = 0.200 GeV is to attenuate the Tc value shown in Figure 2.
Figure 1: Mass parameter m as a function of temperature for zero (left plot) and non-zero (right plot) chemical potential and several values of the external magnetic field. We used, in [GeV]2 , w = eB = 1.0; 0.5; 0 for full, dashed and dot-dashed curves, respectively. Colors at the online version.
Figure 2: Free energy at zero chemical potential. On the left, we have zero magnetic field and on the right, we have a magnetic background. Note the increase of critical temperature, in [GeV], for augmentation magnetic field in [GeV]2 : Tc (w = 0) < Tc (w = 0.5), i.e., the magnetic catalysis phenomenon. Colors at the online version.
Another application of our results arises when we consider a four-fermion interacting model type Nambu-Jona-Lasinio describing quarks with punctual interaction, whose density Hamiltonian is given by [14-16]
Figure 3: Free energy at a finite chemical potential. On the left, we have zero magnetic field and, on the right, we have a magnetic background. The present panels show attenuation of critical temperature values concerning Figure 2, due to µ = 0.200 [GeV]. Again, we have the magnetic catalysis phenomenon. Colours at the online version.
where m0 is the current mass quark and Gs correspond to selfcoupling in the scalar channel. Using the mean field approximation, the gap equation reads
Under a magnetic field in the z-direction, the Hamiltonian density becomes
Figure 4: Constituent quark mass as a function of temperature for zero (left plot) and finite (right plot) chemical potential and several values of an external magnetic field. We have used, in [GeV]2 , w = eB = 0; 0.5; 1.0 for full, dashed (large) and dashed (small) curves, respectively. Colors at the online version.
In the Figures. 4 and 5 are analyzed the gap equation and thermal mass gradient of the fermion model. We use the parameters that fit the mass and pion decay constant in vacuum, namely mπ = 0.138 GeV and fπ = 0.092 GeV:
In Figure 4 we note that the chemical potential has little influence on the fermionic system. Furthermore, for temperatures close to zero, the system does not depend on the chemical potential, since the constituent quark mass presents practically the same values for the fixed magnetic field.
In Figure 5 we plot the thermal mass gradient. The peak of the curve indicates the temperature at which the chiral phase transition occurs [25]. As in the bosonic case, the finite chemical potential tends to lower the transition temperature of the system. Still in Figure 5, but considering the dependence on the external field, we can observe that the chiral transition is stimulated by increasing magnetic fields. Thus, we have the magnetic catalysis phenomenon again, now for fermions.
Figure 5: The thermal mass gradient versus temperature at zero (left plot) and non-zero (right plot) chemical potential and the same values of the external magnetic field showed in Figure 4. We have Tc (w = 0) < Tc (w ?= 0), according with the magnetic catalysis. Colors at the online version.
In this paper, we study four integrals of Jacobi’s theta functions. By the Poisson sum formula and the integral representations of the modified Bessel function of the second kind and gamma function, we were able to express the solutions of Ii, with i = 1,2,3,4 in terms of Kv. We also apply the expressions found in a quantum gas of bosons and in a quantum gas of fermions, both self-interacting and under an external magnetic field. Due to KMS conditions, we used periodic boundary conditions in imaginary-time coordinates for bosons, which meant θ3 in the heated gas. On the other hand, the KMS conditions applied to the fermions meant antiperiodic boundary conditions in the imaginary-time coordinates of this system. In this last case, we express the thermal effects by functions θ2.
From the phase structure of the systems, we observe the same behaviour of the parameters of mass (bosons) and constituent mass of quarks (fermions) for chemical potential zero and different from zero, namely: finite chemical potential makes the transition temperatures lower. In contrast, the effect due to the magnetic background is to increase the transition temperatures as the external field increases. This is the magnetic catalysis phenomenon.
Another point to highlight is in relation to the first term of Eqs. (13) and (15). These terms have a singularity of the type:
which is divergent for d = 3/2 or d = 1/2 as used in this note. For this reason, we evaluated it numerically
Jacobi theta functions are an interesting research topic with great potential for application in QM and QFT. We look forward to continuing to work on these issues in other models and backgrounds.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This paper is dedicated to the birth of Maria Clara.
The authors declare there is no conflict of interest.
