Symmetry Restorations in the Singlet Scalar Yukawa Model Within the Auxiliary Field Method

Abstract

The aim of this work is to investigate the connection between thermal gap-coupled equations and the concept of symmetry restoration. For that, we consider the Yukawa model, a standard model for interactions between massless fermions mediated by a real self-interacting scalar field. To explore possible symmetry restoration, we study the thermal gap-coupled equations using the auxiliary field method (Hubbard–Stratonovich), and then we derive the effective action with thermal contributions through the background external fields method. With the thermal contributions for the effective action, we investigate the phase transitions and critical phenomena in an environment featuring mixing angles arising from the quantum description of composite states. Finally, we present the Dolan–Jackiw equations to determine the critical temperatures.

1. Introduction

The behavior of symmetries and symmetry breaking at high energy densities has been extensively studied across various systems from cosmology, Higgs dynamics and interactions, and high-energy quantum chromodynamics (QCD). The scalar singlet Yukawa model (YuM) serves as a prototype for the Standard Model, offering insights into the mechanism of mass generation for elementary particles. Alongside QCD, it enables the investigation of emergent hadron masses at significantly lower temperatures than those typical of the Standard Model or Higgs dynamics. These phenomena can also be explored through their energy density behavior, particularly at high temperatures.

As is widely acknowledged, the concept of gap equations first emerged in condensed matter physics, where at low temperatures, electrons can bind together through interactions (energy gap) induced by lattice vibrations (phonons), forming pairs that behave like composite bosons (Cooper pairs, superconductivity) [1,2,3,4,5,6]. Later, this composite boson phenomenology reappeared in nuclear physics with the Nambu–Jona–Lasinio model [7,8,9], which involves the breaking of chiral symmetry, a mechanism for dynamical mass generation in fermions, and the appearance of a Goldstone boson. Thus, the gap manifests in both the mass and, reciprocally, in the Hamiltonian. In general, effective chiral models are commonly explored through frameworks such as the linear and non-linear sigma models, particularly in the context of chiral perturbation theory [10,11,12,13]. Finally, the condensate phenomenology, together with chiral methodology, culminates in an effective description of nuclear physics derived from quantum chromodynamics (QCD) at low energy. In this scenario, gap equations are associated with the bound states of two fermions (pions) [14,15]. One way to approach this QCD-related problem is by employing auxiliary background fields [16]. In the functional integration formalism, this is achieved through the Hubbard–Stratonovich identities, which stem from the concept of bosonization [17,18,19].

The Higgs sector of the Standard Model offers the most elementary known mechanism of mass generation by means of spontaneous symmetry breaking (SSB). At larger distance scales, dynamical chiral symmetry breaking (DChSB) is an important effect that helps to endow hadrons with much larger masses than the contribution from the Higgs sector can describe. This is usually parameterized by a quark–antiquark (chiral) scalar condensate that is the order parameter of the DChSB and has a corresponding gap equation [15]. Thus, there are mechanisms of mass generation and symmetry breaking that can be straightforwardly analyzed by means of gap equations, usually describing order parameters, and reciprocally in the Hamiltonian. The auxiliary field method is interesting and usually applied for systems in which composite fields are useful and might correspond to dynamical degrees of freedom of the system [20,21,22,23]. It has been applied to the Yukawa model in some restricted way [24,25]. This method is suitable to investigate symmetry breakings and the behavior of regions of the phase diagram of a model. It is important to emphasize that the resulting theory, from the perspective of auxiliary fields, can only be considered an effective model or effective field theory, requiring further UV completion [26,27,28].

Different mechanisms for symmetry breaking have been proposed in the literature, such as radiative corrections [29,30,31], while thermal corrections might be responsible for its restoration [32,33,34,35,36]. Within the thermal approach, the thermal gap equations generally exhibit both effects, especially when the auxiliary field methodology is extended to finite temperatures. This approach is inspired by prior studies conducted in the context of 1/N expansions [37,38,39], and simplified Yukawa models involving fermions and scalar bosons [24,40,41,42,43,44,45,46,47]. The concepts of symmetry breaking and restoration also have significant applications in QCD and its low-energy models [48,49,50,51].

As is well known, the thermal behavior of the field interactions, and in general quantum dynamics, is of great interest in the discussion of phase diagrams with the corresponding behavior of involved symmetries, in which the concepts of stability and thermodynamic equilibrium might play a crucial role [52,53,54,55]. Within this thermal field context, one of the formalisms widely used to describe thermal dynamics was developed by Matsubara [56], building on Bloch’s work. This formalism defines the partition function and mean values using the density matrix, incorporating a Euclidean structure via Wick rotation to imaginary times. For relativistic fields, the Matsubara path integral approach was extensively studied by Fradkin [57,58,59]. General aspects of thermal quantum field theory can be found in recent literature [60,61,62].

Therefore, we investigate thermal effects for the scalar singlet scalar model using the auxiliary field method, as developed in [63], actually inspired by [64,65], for the interplay between different symmetry-breaking mechanisms in the Yukawa model. An analysis of symmetry restoration is performed by studying the renormalized thermal gap equations and the thermal contributions to the masses and coupling constants in the effective action. The paper is organized as follows: In Section 2.1, we study the thermal gap-coupled equations. In Section 3, we derive the effective action from thermal contributions using the external fields method. In Section 3.1, we investigate the concept of phase transition and critical phenomena. In Section 3.2, we explore the concepts of symmetry restoration and critical temperature. In Section 4, we present the conclusion and final remarks.

2. The Yukawa Model at Finite Temperatures

Using the Matsubara–Fradkin formulation, we can write from the density matrix $\widehat{\rho }$ the partition function Z

$$\begin{array}{ccc}Z=tr\widehat{\rho },\hfill & where\hfill & \widehat{\rho }=exp[-\beta (\widehat{H}-{\mu }_e\widehat{N})],\beta ={\displaystyle \frac{1}{kT}}.\hfill \end{array}$$

(1)

The Hamiltonian $\widehat{H}$ and global charge conservation $\widehat{N}$ are defined from the Lagrangian in [24,41,42,43]. So, by the Schwinger variational principle

$$\begin{array}{c}\delta (tr\widehat{\rho })=\delta S_E(tr\widehat{\rho })\hfill \end{array}$$

(2)

we write the generating functional of the scalar singlet Yukawa model:

$$\begin{array}{c}Z=N\int \mathcal{D}[\varphi ,\overline{q},q]exp[-S_E],\hfill \end{array}$$

(3)

in which the Euclidean action of the scalar singlet Yukawa model is given explicitly as

$$\begin{array}{c}{S}_E={\int }_0^{\beta }d{\tau }_x\int d^3\overrightarrow{x}[-{\mathcal{Z}}_q\overline{q}({\gamma }_{\mu }^E{\partial }_{\mu }^{{\mu }_e})q+{\mathcal{Z}}_{g}g\varphi \overline{q}q+{\mathcal{Z}}_{\varphi }{\displaystyle \frac{1}{2}}\varphi \Delta \varphi +{\mathcal{Z}}_m{m}^2{\varphi }^2-{\mathcal{Z}}_{\lambda }{\displaystyle \frac{\lambda }{4!}}{\varphi }^4],\hfill \end{array}$$

(4)

where a chemical potential ${\mu }_e$ and the renormalization constants were included, but renormalization labels (R) were omitted for simplicity. The following Euclidean convention was adopted for a given chemical potential:

$$\begin{array}{ccc}\hfill & \hfill & {\gamma }_0^E={\gamma }_0,{\partial }_{\mu }^{{\mu }_e}={\partial }_{\mu }+{\mu }_e{\delta }_{\mu 0}\hfill \\ \\ \hfill & \hfill & {\gamma }_j^E=-i{\gamma }_j,\Delta =-{\displaystyle \frac{{\partial }^2}{\partial {\beta }^2}}-{\nabla }^2.\hfill \end{array}$$

(5)

We can establish a set of rules that relates the quantum description (scattering matrix and transition amplitude) and the thermal description (density matrix and partition function) of a field theory due to a Wick rotation to imaginary time [58]

$$\begin{array}{ccc}\hfill & \hfill & (\mathrm{Minkowski}\mathrm{space})(t,\overrightarrow{x})\to (i\tau ,\overrightarrow{x})(\mathrm{Euclidean}\mathrm{space})\hfill \\ \\ \hfill & \hfill & \square \to -▵\hfill \\ \\ \hfill & \hfill & \int d^{4}x\to i{\int }_0^{\beta }d\tau \int d^3\overrightarrow{x}\hfill \\ \\ \hfill & \hfill & D_{\mu }={\partial }_{\mu }-ie{A}_{\mu }\to D_{\mu }^{(e,{\mu }_e)}={\partial }_{\mu }-ie{\mathcal{A}}_{\mu }^E-{\mu }_e{\delta }_{\mu 0}\hfill \\ \\ \hfill & \hfill & \delta <|\widehat{\mathcal{S}}|>=i\delta S<|\widehat{\mathcal{S}}|>\to \delta <|\widehat{\rho }|>=-\delta S_E<|\widehat{\rho }|>,\hfill \end{array}$$

(6)

wherein we point out the relations

$$\begin{array}{ccc}\hfill & \hfill & \widehat{\mathcal{S}}\to \widehat{\rho }\hfill \\ \\ \hfill & \hfill & S\to i{S}_E.\hfill \end{array}$$

(7)

As we can see, the equations derived from Schwinger’s variational principle preserve their form under Wick rotations. For this reason, the thermal equations will closely resemble the quantum equations structures throughout the article.

At the tree level, the usual conditions for the SSB and the emergence of the so-called scalar field condensate are the following:

$$\begin{array}{c}{\overline{\varphi }}_0^2={\displaystyle \frac{12m^2}{\lambda }}{\displaystyle \frac{{\mathcal{Z}}_m}{{\mathcal{Z}}_{\lambda }}},\hfill \end{array}$$

(8)

in which one needs $m^2>0$. These conditions will receive thermal corrections due to the quantization of the scalar and the fermion fields and it will be discussed again below.

2.1. Scalar Field Sector and the Thermal Gap Equation

Let us consider the background field method that makes possible the computation of an effective action with quantum and thermal corrections for the spontaneously broken $Z_2$ symmetry. Writing the scalar field as $\varphi \to {\varphi }_0+\tilde{\varphi }$ where ${\varphi }_0$ is the condensate and $\tilde{\varphi }$ the fluctuation. To integrate out the scalar field, the Hubbard–Stratonovich auxiliary field identity is considered by means of the following unity integral in the generating functional Z that includes the renormalization factors

$$1=N^{\prime }\int D\Psi exp\{-{\int }_0^{\beta }d{\tau }_x\int d^3\overrightarrow{x}{\displaystyle \frac{4!}{\lambda }}[{\mathcal{Z}}_{\Psi }^{{\displaystyle \frac{1}{2}}}\Psi +{\mathcal{Z}}_{\lambda }^{{\displaystyle \frac{1}{2}}}{\displaystyle \frac{\lambda }{4!}}({\tilde{\varphi }}^2+2{\varphi }_0\tilde{\varphi }){]}^2\}$$

(9)

It yields the following equation

$$\begin{array}{ccc}Z\hfill & =\hfill & N\int D\overline{q}DqD\tilde{\varphi }D\Psi exp[-{\int }_0^{\beta }d{\tau }_x\int d^3\overrightarrow{x}\{-{\mathcal{Z}}_q\overline{q}({\gamma }_{\mu }^E{\partial }_{\mu }^{{\mu }_e})q+{\mathcal{Z}}_{g}g({\varphi }_0+\tilde{\varphi })\overline{q}q\hfill \\ \\ \hfill & +\hfill & {\mathcal{Z}}_{\varphi }\Delta {\varphi }_0\tilde{\varphi }+2Z_m{m}^2{\varphi }_0\tilde{\varphi }-{\mathcal{Z}}_{\lambda }{\displaystyle \frac{\lambda }{3!}}{\varphi }_0^3\tilde{\varphi }+4{\mathcal{Z}}_{\Psi }^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{\lambda }^{{\displaystyle \frac{1}{2}}}{\varphi }_0\Psi \tilde{\varphi }\hfill \\ \\ \hfill & +\hfill & {\displaystyle \frac{1}{2}}\tilde{\varphi }({\mathcal{Z}}_{\varphi }\Delta +2{\mathcal{Z}}_m{m}^2+4{\mathcal{Z}}_{\Psi }^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{\lambda }^{{\displaystyle \frac{1}{2}}}\Psi -4{\mathcal{Z}}_{\lambda }{\displaystyle \frac{\lambda }{4!}}{\varphi }_0^2)\tilde{\varphi }+{\displaystyle \frac{4!}{\lambda }}{\mathcal{Z}}_{\Psi }{\Psi }^2\}]exp(-{\Gamma }_0),\hfill \end{array}$$

(10)

in which ${\Gamma }_0$ is the effective action associated with the background

$${\Gamma }_0={\int }_0^{\beta }d{\tau }_x\int d^3\overrightarrow{x}[{\mathcal{Z}}_{\varphi }{\displaystyle \frac{1}{2}}{\varphi }_0\Delta {\varphi }_0+{\mathcal{Z}}_m{m}_R^2{\varphi }_0^2-{\mathcal{Z}}_{\lambda }{\displaystyle \frac{{\lambda }_R}{4!}}{\varphi }_0^4]$$

(11)

By following the same line of reasoning of Ref. [43], we apply the idea of current expansion in Equation (10) by means of the field redefinition

$$\begin{array}{cc}\hfill & \tilde{\varphi }({\tau }_x,\overrightarrow{x})\to {\tilde{\varphi }}_0({\tau }_x,\overrightarrow{x})-{\int }_0^{\beta }d{\tau }_y\int d^3\overrightarrow{y}G({\tau }_x,\overrightarrow{x};{\tau }_y,\overrightarrow{y})j({\tau }_y,\overrightarrow{y})\\ \mathrm{where}\hfill & j={\mathcal{Z}}_{g}g\overline{q}q+{\mathcal{Z}}_{\varphi }\Delta {\varphi }_0+2Z_m{m}^2{\varphi }_0-{\mathcal{Z}}_{\lambda }{\displaystyle \frac{\lambda }{3!}}{\varphi }_0^3+4{\mathcal{Z}}_{\Psi }^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{\lambda }^{{\displaystyle \frac{1}{2}}}{\varphi }_0\Psi ,\\ \hfill & G^{-1}({\tau }_x,\overrightarrow{x};{\tau }_y,\overrightarrow{y})=({\mathcal{Z}}_{\varphi }\Delta +2{\mathcal{Z}}_m{m}^2+4{\mathcal{Z}}_{\Psi }^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{\lambda }^{{\displaystyle \frac{1}{2}}}\Psi -4{\mathcal{Z}}_{\lambda }{\displaystyle \frac{\lambda }{4!}}{\varphi }_0^2)\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y}).\end{array}$$

(12)

We can write the partition function as

$$\begin{array}{ccc}Z\hfill & =\hfill & N\int D\overline{q}DqD{\tilde{\varphi }}_{0}D\Psi exp[-{\int }_0^{\beta }d{\tau }_x\int d^3\overrightarrow{x}\{-{\mathcal{Z}}_q\overline{q}({\gamma }_{\mu }^E{\partial }_{\mu }^{{\mu }_e})q+{\mathcal{Z}}_{g}g{\varphi }_0\overline{q}q+\hfill \\ \\ \hfill & +\hfill & {\displaystyle \frac{1}{2}}{\tilde{\varphi }}_0{G}^{-1}{\tilde{\varphi }}_0-{\displaystyle \frac{1}{2}}jGj+{\displaystyle \frac{4!}{\lambda }}{\mathcal{Z}}_{\Psi }{\Psi }^2]exp(-{\Gamma }_0).\hfill \end{array}$$

(13)

By integrating out the scalar field with the identity $detA=exp[trlnA]$, we write the effective potential and the gap equation, respectively

$$\begin{array}{ccc}\hfill & \hfill & -V_{eff}={\displaystyle \frac{1}{2}}ln[{\mathcal{Z}}_{\varphi }\Delta +{\mathcal{M}}_{\varphi }^2]\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y})+{\displaystyle \frac{4!}{{\lambda }_R}}{\mathcal{Z}}_{\Psi }{\Psi }_R^2,\hfill \\ \\ \hfill & \hfill & {\displaystyle \frac{\partial V_{eff}}{\partial \Psi }}{|}_{\Psi ={\Psi }_0}=0.\hfill \end{array}$$

(14)

In this equation, a total dressed mass for the scalar field was defined for the case that the fields $\varphi$ and $\Psi$ develop classical solutions from the corresponding gap equations. It can be written as

$$\begin{array}{c}{\mathcal{M}}_{\varphi }^2=2{\mathcal{Z}}_m{m}^2+4({\mathcal{Z}}_{\Psi }^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{\lambda }^{{\displaystyle \frac{1}{2}}}{\Psi }_0-{\mathcal{Z}}_{\lambda }{\displaystyle \frac{\lambda }{4!}}{\varphi }_0^2).\hfill \end{array}$$

(15)

The corresponding saddle point equation for the auxiliary field is a gap equation, and it is given by1:

$${\mathcal{Z}}_{\Psi }{\Psi }_0={\displaystyle \frac{1}{4!\beta }}\sum _{n=-\infty }^{n=\infty }\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{{\mathcal{Z}}_{\Psi }^{\frac{1}{2}}{\mathcal{Z}}_{\lambda }^{\frac{1}{2}}\lambda }{{\mathcal{Z}}_{\varphi }({\omega }_n^2+{\overrightarrow{k}}^2)+{\mathcal{M}}_{\varphi }^2}}$$

(16)

in which ${\omega }_n={\displaystyle \frac{2\pi n}{\beta }}$ is the Matsubara frequency.

By performing the sum in Matsubara frequencies, the following equation is obtained:

$${\mathcal{Z}}_{\Psi }{\Psi }_0={\displaystyle \frac{{\mathcal{Z}}_{\Psi }^{\frac{1}{2}}{\mathcal{Z}}_{\lambda }^{\frac{1}{2}}\lambda }{48{\pi }^2{\mathcal{Z}}_{\varphi }}}{\int }_0^{\infty }{\displaystyle \frac{{\overrightarrow{k}}^{2}dk}{\sqrt{[{\overrightarrow{k}}^2+\frac{{\mathcal{M}}_{\varphi }^2}{{\mathcal{Z}}_{\varphi }}]}}}[1+2\eta ]$$

(17)

where we see the appearance of Bose–Einstein distribution

$$\eta ={\displaystyle \frac{1}{exp(\beta \sqrt{[{\overrightarrow{k}}^2+\frac{{\mathcal{M}}_{\varphi }^2}{{\mathcal{Z}}_{\varphi }}]})-1}}.$$

(18)

By considering the emerging four-fermion vertex in Equation (13), we now assume that the kinetic part of the scalar field is suppressed by a large total thermal mass term such that the following local limit can be taken:

$$\begin{array}{ccc}{\displaystyle \frac{1}{2}}\int d{\tau }_x{d}^3\overrightarrow{x}d{\tau }_y{d}^3\overrightarrow{y}jGj\hfill & =\hfill & {\displaystyle \frac{1}{2}}\int d{\tau }_x{d}^3\overrightarrow{x}{\displaystyle \frac{{[{\mathcal{Z}}_{g}g\overline{q}q+{\mathcal{Z}}_{\varphi }\Delta {\varphi }_0+2Z_m{m}^2{\varphi }_0-{\mathcal{Z}}_{\lambda }\frac{\lambda }{3!}{\varphi }_0^3+4{\mathcal{Z}}_{\Psi }^{\frac{1}{2}}{\mathcal{Z}}_{\lambda }^{\frac{1}{2}}{\varphi }_0\Psi ]}^2}{({\mathcal{Z}}_{\varphi }\Delta +2{\mathcal{Z}}_m{m}^2+4{\mathcal{Z}}_{\Psi }^{\frac{1}{2}}{\mathcal{Z}}_{\lambda }^{\frac{1}{2}}\Psi -4{\mathcal{Z}}_{\lambda }\frac{\lambda }{4!}{\varphi }_0^2)}}\hfill \\ \\ \hfill & \cong \hfill & {\displaystyle \frac{1}{2}}{\int }_0^{\beta }d{\tau }_x\int d^3\overrightarrow{x}[r{(\overline{q}q)}^2+s(\overline{q}q)+u\Psi +v{\psi }^2+w]\hfill \end{array}$$

(19)

wherein

$$\begin{array}{ccc}r\hfill & =\hfill & {\displaystyle \frac{{\mathcal{Z}}_g^2{g}^2}{{\mathcal{M}}_{\varphi }^2}},\hfill \end{array}$$

(20)

$$\begin{array}{ccc}s\hfill & =\hfill & {\displaystyle \frac{{\mathcal{Z}}_{g}g}{{\mathcal{M}}_{\varphi }^2}}[{\mathcal{Z}}_{\varphi }\Delta {\varphi }_0+2Z_m{m}^2{\varphi }_0-{\mathcal{Z}}_{\lambda }{\displaystyle \frac{\lambda }{3!}}{\varphi }_0^3+4{\mathcal{Z}}_{\Psi }^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{\lambda }^{{\displaystyle \frac{1}{2}}}{\varphi }_0{\Psi }_0],\hfill \\ \\ u\hfill & =\hfill & {\displaystyle \frac{8{\mathcal{Z}}_{\Psi }^{\frac{1}{2}}{\mathcal{Z}}_{\lambda }^{\frac{1}{2}}{\varphi }_0^2}{{\mathcal{M}}_{\varphi }^2}}[{\mathcal{Z}}_{\varphi }\Delta {\varphi }_0+2Z_m{m}^2{\varphi }_0-{\mathcal{Z}}_{\lambda }{\displaystyle \frac{\lambda }{3!}}{\varphi }_0^3],\hfill \\ \\ v\hfill & =\hfill & {\displaystyle \frac{16{\mathcal{Z}}_{\Psi }{\mathcal{Z}}_{\lambda }{\varphi }_0^2}{{\mathcal{M}}_{\varphi }^2}},\hfill \\ \hfill w& =& {\displaystyle \frac{{[{\mathcal{Z}}_{\varphi }\Delta {\varphi }_0+2Z_m{m}^2{\varphi }_0-{\mathcal{Z}}_{\lambda }\frac{\lambda }{3!}{\varphi }_0^3]}^2}{{\mathcal{M}}_{\varphi }^2}}.\hfill \end{array}$$

(21)

As a result, the following four-fermion interaction effective model is obtained:

$$\begin{array}{ccc}Z\hfill & =\hfill & N\int D\overline{q}DqD{\tilde{\varphi }}_{0}D\Psi exp[-{\int }_0^{\beta }d{\tau }_x\int d^3\overrightarrow{x}\{-{\mathcal{Z}}_q\overline{q}({\gamma }_{\mu }^E{\partial }_{\mu }^{{\mu }_e})q+[{\mathcal{Z}}_{g}g{\varphi }_0-{\displaystyle \frac{s}{2}}]\overline{q}q+{\mathcal{Z}}_{C_F}{C}_F{(\overline{q}q)}^2\hfill \\ \\ \hfill & \hfill & +{\displaystyle \frac{1}{2}}{\tilde{\varphi }}_0{G}^{-1}{\tilde{\varphi }}_0-{\displaystyle \frac{u}{2}}\Psi +({\displaystyle \frac{4!}{\lambda }}{\mathcal{Z}}_{\Psi }-{\displaystyle \frac{v}{2}}){\Psi }^2-{\displaystyle \frac{w}{2}}]exp(-{\Gamma }_0)\hfill \end{array}$$

(22)

in which we define the renormalized Fermi constant $C_F=-{\displaystyle \frac{1}{2}}{\displaystyle \frac{g^2}{{\mathcal{M}}_{\varphi }^2}}$ with ${\mathcal{Z}}_{C_F}={\mathcal{Z}}_g^2$. If we want to study the non-local limit, the Fermi constant would be

$$\begin{array}{ccc}{C}_F\hfill & =\hfill & -{\displaystyle \frac{1}{2}}{g}^2\int d{\tau }_y{d}^3\overrightarrow{y}{\displaystyle \frac{1}{\beta }}{\displaystyle \sum _{n=-\infty }^{n=\infty }}\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{1}{{\mathcal{Z}}_{\varphi }({\omega }_n^2+{\overrightarrow{k}}^2)+{\mathcal{M}}_{\varphi }^2}}exp[i{\omega }_n({\tau }_x-{\tau }_y)+i\overrightarrow{k}·(\overrightarrow{x}-\overrightarrow{y})]\hfill \end{array}$$

(23)

and this will influence, for example, the future discussion of symmetry restoration and critical temperature. Also note that we only consider cases in which $\Delta {\varphi }_0=0$.

2.2. Background Fermions and Fermion–Antifermion Condensate

In the same way as the scalar field was treated, a background fermion current ${(\overline{q}q)}_0$ is introduced by means of the shift $\overline{q}q\to {(\overline{q}q)}_0+(\tilde{\overline{q}}\tilde{q})$ where $(\tilde{\overline{q}}\tilde{q})$ are the fluctuations. To integrate our fermion field, the following unit integral is introduced in Equation (22)

$$1=N″\int DSexp[{\int }_0^{\beta }d{\tau }_x\int d^3\overrightarrow{x}{\displaystyle \frac{1}{4C_F}}{({\mathcal{Z}}_S^{{\displaystyle \frac{1}{2}}}S+2{\mathcal{Z}}_{C_F}^{{\displaystyle \frac{1}{2}}}{C}_F{(\tilde{\overline{q}}\tilde{q})}^2)}^2].$$

(24)

We are left with the following functional generator

$$\begin{array}{ccc}\hfill & \hfill & Z=N\int D\tilde{\overline{q}}D\tilde{q}DSD{\tilde{\varphi }}_{0}D\Psi exp[-{\int }_0^{\beta }d{\tau }_x\int d^3\overrightarrow{x}\{\tilde{\overline{q}}(-{\mathcal{Z}}_q{\gamma }_{\mu }^E{\partial }_{\mu }^{{\mu }_e}+{\mathcal{Z}}_{g}g{\varphi }_0-{\displaystyle \frac{s}{2}}-{\mathcal{Z}}_S^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{C_F}^{{\displaystyle \frac{1}{2}}}S+\hfill \\ \\ \hfill & \hfill & +2{\mathcal{Z}}_{C_F}{C}_F{(\overline{q}q)}_0)\tilde{q}+{\displaystyle \frac{1}{2}}{\tilde{\varphi }}_0({\mathcal{Z}}_{\varphi }\Delta +2{\mathcal{Z}}_m{m}^2+4{\mathcal{Z}}_{\Psi }^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{\lambda }^{{\displaystyle \frac{1}{2}}}\Psi -4{\mathcal{Z}}_{\lambda }{\displaystyle \frac{\lambda }{4!}}{\varphi }_0^2){\tilde{\varphi }}_0-{\displaystyle \frac{1}{4C_F}}{\mathcal{Z}}_S{S}^2+\hfill \\ \\ \hfill & \hfill & -{\displaystyle \frac{u}{2}}\Psi +({\displaystyle \frac{4!}{\lambda }}{\mathcal{Z}}_{\Psi }-{\displaystyle \frac{v}{2}}){\Psi }^2-{\displaystyle \frac{w}{2}}]exp(-{\Gamma }_0)\},\hfill \end{array}$$

(25)

in which

$$\begin{array}{ccc}{\Gamma }_0\hfill & =\hfill & {\int }_0^{\beta }d{\tau }_x\int d^3\overrightarrow{x}\{{\mathcal{Z}}_{\varphi }{\displaystyle \frac{1}{2}}{\varphi }_0\Delta {\varphi }_0+{\mathcal{Z}}_m{m}^2{\varphi }_0^2-{\mathcal{Z}}_{\lambda }{\displaystyle \frac{\lambda }{4!}}{\varphi }_0^4+\hfill \\ \\ \hfill & \hfill & -{\mathcal{Z}}_q{\overline{q}}_0({\gamma }_{\mu }^E{\partial }_{\mu }^{{\mu }_e})q_0+[{\mathcal{Z}}_{g}g{\varphi }_0-{\displaystyle \frac{s}{2}}]{\overline{q}}_0{q}_0+{\mathcal{Z}}_{C_F}{C}_F{({\overline{q}}_0{q}_0)}^2\}.\hfill \end{array}$$

(26)

It is possible to define a total dressed mass for the fermionic field in the case that the fields $\varphi$ and S develop classical solutions for the corresponding gap equations, which, in this case, can be written as

$${\mathcal{M}}_q={\mathcal{Z}}_{g}g{\varphi }_0-{\displaystyle \frac{\tilde{S}}{2}}-{\mathcal{Z}}_S^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{C_F}^{{\displaystyle \frac{1}{2}}}{S}_0$$

(27)

where $s\to {\displaystyle \frac{{\mathcal{Z}}_{g}g}{{\mathcal{M}}_{\varphi }^2}}[2Z_m{m}^2{\varphi }_0-{\mathcal{Z}}_{\lambda }{\displaystyle \frac{\lambda }{3!}}{\varphi }_0^3+4{\mathcal{Z}}_{\Psi }^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{\lambda }^{{\displaystyle \frac{1}{2}}}{\varphi }_0{\Psi }_0]$

By performing the integration of fermion fields, we have for the generating functional of the Yukawa model the following equation:

$$\begin{array}{ccc}\hfill & \hfill & Z=N\int DSD\Psi det[F^{-1}]det{[B^{-1}]}^{-{\displaystyle \frac{1}{2}}}\times \hfill \\ \\ \hfill & \hfill & exp[{\int }_0^{\beta }d{\tau }_x\int d^3\overrightarrow{x}\{{\displaystyle \frac{1}{4C_F}}{\mathcal{Z}}_{C_F}{S}^2+{\displaystyle \frac{u}{2}}\Psi -[{\displaystyle \frac{4!}{{\lambda }_R}}{\mathcal{Z}}_{\Psi }-{\displaystyle \frac{v}{2}}]{\Psi }^2+{\displaystyle \frac{w}{2}}\}exp(-{\Gamma }_0)\hfill \end{array}$$

(28)

where

$$\begin{array}{ccc}\hfill & \hfill & F^{-1}=-{\mathcal{Z}}_q{\gamma }_{\mu }^E{\partial }_{\mu }^{{\mu }_e}+{\mathcal{M}}_q\hfill \\ \\ \hfill & \hfill & B^{-1}={\mathcal{Z}}_{\varphi }\Delta +{\mathcal{M}}_{\varphi }^2\hfill \end{array}$$

(29)

Now let us compute the effective potential $V_{eff}$ from Equation (28) and obtain the coupled gap equations for the dynamic mass generations. By means of the identity $detA=exp[trlnA]$ we can write

$$\begin{array}{ccc}-V_{eff}\hfill & =\hfill & -trln[-{\mathcal{Z}}_q{\gamma }_{\mu }^E{\partial }_{\mu }^{{\mu }_e}+{\mathcal{M}}_q]\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y})+{\displaystyle \frac{1}{4C_F}}{\mathcal{Z}}_{C_F}{S}^2\hfill \\ \\ \hfill & \hfill & +{\displaystyle \frac{1}{2}}ln[{\mathcal{Z}}_{\varphi }\Delta +{\mathcal{M}}_{\varphi }^2]\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y})+{\displaystyle \frac{u}{2}}\Psi -[{\displaystyle \frac{4!}{\lambda }}{\mathcal{Z}}_{\Psi }-{\displaystyle \frac{v}{2}}]{\Psi }^2+{\displaystyle \frac{w}{2}}\hfill \end{array}$$

(30)

and as a consequence, the extremal points for the auxiliary fields $(\Psi ={\Psi }_0;S=S_0)$ lead to the following gap equations

$$\begin{array}{ccc}{\displaystyle \frac{\partial V_{eff}}{\partial \Psi }}{|}_{\Psi ={\Psi }_0}\hfill & =\hfill & tr{\displaystyle \frac{2{\mathcal{Z}}_{\Psi }^{\frac{1}{2}}{\mathcal{Z}}_{\lambda }^{\frac{1}{2}}}{[-{\mathcal{Z}}_q{\gamma }_{\mu }^E{\partial }_{\mu }^{{\mu }_e}+{\mathcal{M}}_q]}}{\displaystyle \frac{[s+{\mathcal{Z}}_{g}g{\varphi }_0]}{{\mathcal{M}}_{\varphi }^2}}\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y})-{\displaystyle \frac{2{\mathcal{Z}}_{\Psi }^{\frac{1}{2}}{\mathcal{Z}}_{\lambda }^{\frac{1}{2}}{\mathcal{Z}}_{C_F}}{g^2}}{S}_0^2\hfill \\ \\ \hfill & \hfill & +{\displaystyle \frac{2{\mathcal{Z}}_{\Psi }^{\frac{1}{2}}{\mathcal{Z}}_{\lambda }^{\frac{1}{2}}}{[{\mathcal{Z}}_{\varphi }\Delta +{\mathcal{M}}_{\varphi }^2]}}\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y})-({\displaystyle \frac{2{\mathcal{Z}}_{\Psi }^{\frac{1}{2}}{\mathcal{Z}}_{\lambda }^{\frac{1}{2}}{\Psi }_0}{{\mathcal{M}}_{\varphi }^2}}-{\displaystyle \frac{1}{2}})u-2[{\displaystyle \frac{4!}{\lambda }}{\mathcal{Z}}_{\Psi }+({\displaystyle \frac{{\mathcal{Z}}_{\Psi }^{\frac{1}{2}}{\mathcal{Z}}_{\lambda }^{\frac{1}{2}}}{{\mathcal{M}}_{\varphi }^2}}-{\displaystyle \frac{1}{2}})v]{\Psi }_0\hfill \\ \\ \hfill & \hfill & -{\displaystyle \frac{2{\mathcal{Z}}_{\Psi }^{\frac{1}{2}}{\mathcal{Z}}_{\lambda }^{\frac{1}{2}}w}{{\mathcal{M}}_{\varphi }^2}}=0,\hfill \\ \\ {\displaystyle \frac{\partial V_{eff}}{\partial S}}{|}_{S=S_0}\hfill & =\hfill & tr{\displaystyle \frac{{\mathcal{Z}}_S^{\frac{1}{2}}{\mathcal{Z}}_{C_F}^{\frac{1}{2}}}{[-{\mathcal{Z}}_q{\gamma }_{\mu }^E{\partial }_{\mu }^{{\mu }_e}+{\mathcal{M}}_q]}}+{\displaystyle \frac{1}{2C_F}}{\mathcal{Z}}_{C_F}{S}_0=0.\hfill \end{array}$$

(31)

Solving the previous equations may not be an easy task. Following the line of reasoning of [63], we could investigate the limit in which m is very large (and so ${\mathcal{M}}_{\varphi }^2$). In this case, the previous coupled gap equations are simplified and can be re-written as follows:

$$\begin{array}{ccc}\hfill & \hfill & -{\displaystyle \frac{2{\mathcal{Z}}_{\Psi }^{\frac{1}{2}}{\mathcal{Z}}_{\lambda }^{\frac{1}{2}}{\mathcal{Z}}_{C_F}}{g^2}}{S}_0^2+{\displaystyle \frac{2{\mathcal{Z}}_{\Psi }^{\frac{1}{2}}{\mathcal{Z}}_{\lambda }^{\frac{1}{2}}}{[{\mathcal{Z}}_{\varphi }\Delta +{\mathcal{M}}_{\varphi }^2]}}\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y})+{\displaystyle \frac{u}{2}}-[{\displaystyle \frac{4!}{\lambda }}{\mathcal{Z}}_{\Psi }-{\displaystyle \frac{v}{2}}]\Psi =0\hfill \\ \\ \hfill & \hfill & tr{\displaystyle \frac{i{\mathcal{Z}}_S^{\frac{1}{2}}{\mathcal{Z}}_{C_F}^{\frac{1}{2}}}{[-{\mathcal{Z}}_q{\gamma }_{\mu }^E{\partial }_{\mu }^{{\mu }_e}+{\mathcal{M}}_q]}}\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y})+{\displaystyle \frac{1}{2C_F}}{\mathcal{Z}}_{C_F}{S}_0=0.\hfill \end{array}$$

(32)

With the purpose of seeking a solution to these equations, we write them in the momentum representation:

$$\begin{array}{ccc}\hfill & \hfill & {\mathcal{Z}}_{\Psi }^{{\displaystyle \frac{1}{2}}}{\Psi }_0=\theta +{\displaystyle \frac{1}{4!\beta }}{\displaystyle \sum _{n=-\infty }^{n=\infty }}\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{{\mathcal{Z}}_{\Psi }^{\frac{1}{2}}{\mathcal{Z}}_{\lambda }^{\frac{1}{2}}\lambda }{{\mathcal{Z}}_{\varphi }({\omega }_n^2+{\overrightarrow{k}}^2)+{\mathcal{M}}_{\varphi }^2}}+{\displaystyle \frac{{\mathcal{Z}}_{\lambda }^{\frac{1}{2}}\lambda {\mathcal{Z}}_{C_F}}{24g^2}}{S}_0^2,{\omega }_n={\displaystyle \frac{2\pi n}{\beta }},\hfill \\ \\ \hfill & \hfill & {\mathcal{Z}}_{C_F}^{{\displaystyle \frac{1}{2}}}{S}_0={\displaystyle \frac{1}{\beta }}{\displaystyle \sum _{n=-\infty }^{n=\infty }}\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{8{\mathcal{Z}}_S^{\frac{1}{2}}{C}_F{\mathcal{M}}_q}{[{\mathcal{Z}}_q^2{({\vartheta }_n+i{\mu }_e)}^2+{\overrightarrow{k}}^2+{\mathcal{M}}_q^2]}},{\vartheta }_n={\displaystyle \frac{(2n+1)\pi }{\beta }},\hfill \end{array}$$

(33)

in which $\theta$ is the thermal spontaneous symmetry-breaking contribution with the ${\Psi }_0$ dependence of ${\mathcal{M}}_{\varphi }$

$$\theta ={\displaystyle \frac{{\mathcal{Z}}_{\lambda }^{\frac{1}{2}}\lambda }{12{\mathcal{M}}_{\varphi }^2}}[2Z_m{m}_R^2{\varphi }_0^2-{\mathcal{Z}}_{\lambda }{\displaystyle \frac{{\lambda }_R}{3!}}{\varphi }_0^4]+{\displaystyle \frac{{\mathcal{Z}}_{\Psi }^{\frac{1}{2}}{\mathcal{Z}}_{\lambda }\lambda {\varphi }_0^2}{6{\mathcal{M}}_{\varphi }^2}}\to 0.$$

(34)

Here, we can see a set of rules that relates the quantum description and the thermal description of a field theory due to the Wick rotation to imaginary times. Just compare, for example, the quantum gap-coupled equations and the thermal gap-coupled equations, seen in Equation (33), respectively. The rules in the momentum representation are the following [58,63]

$$\begin{array}{ccc}\hfill & \hfill & k_{\mu }=(\omega ,\overrightarrow{k})\to k_{\mu }^E=(i\omega ,\overrightarrow{k}),\hfill \\ \\ \hfill & \hfill & {\gamma }_{\mu }{k}^{\mu }\to {\gamma }_{\mu }^E(k_{\mu }^E+{\mu }_e{\delta }_{\mu 0}),\hfill \\ \\ \hfill & \hfill & \int {\displaystyle \frac{d^{4}k}{{(2\pi )}^4}}\to {\displaystyle \frac{1}{\beta }}{\displaystyle \sum _{n=-\infty }^{n=\infty }}\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}.\hfill \end{array}$$

(35)

So, at this moment, let us put the fermion sector of the previous coupled equations in a more explicit form

$${\mathcal{Z}}_{C_F}^{{\displaystyle \frac{1}{2}}}{S}_0=4{\mathcal{Z}}_S^{{\displaystyle \frac{1}{2}}}{C}_F{\displaystyle \frac{1}{\beta }}\sum _{n=-\infty }^{n=\infty }\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{\partial }{\partial {\mathcal{M}}_q}}ln\{{(\beta {\mathcal{Z}}_q)}^2[{({\vartheta }_n+i{\mu }_e)}^2+{\displaystyle \frac{{\vartheta }^2}{{\mathcal{Z}}_q^2}}]\},{\vartheta }^2={\overrightarrow{k}}^2+{\mathcal{M}}_q^2.$$

(36)

Therefore, we can make the sum and write

$${\mathcal{Z}}_{C_F}^{{\displaystyle \frac{1}{2}}}{S}_0=4{\mathcal{Z}}_S^{{\displaystyle \frac{1}{2}}}{C}_F{\displaystyle \frac{1}{\beta }}{\displaystyle \frac{\partial }{\partial {\mathcal{M}}_q}}\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}\left[\beta \vartheta +ln\{1+exp[-\beta (\vartheta -{\mathcal{Z}}_q{\mu }_e)]+ln\{1+exp[-\beta (\vartheta +{\mathcal{Z}}_q{\mu }_e)]\}\right],$$

(37)

and so, we have

$${\mathcal{Z}}_{C_F}^{{\displaystyle \frac{1}{2}}}{S}_0=2{\mathcal{Z}}_S^{{\displaystyle \frac{1}{2}}}{C}_F\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{{\mathcal{M}}_q}{\sqrt{{\overrightarrow{k}}^2+{\mathcal{M}}_q^2}}}[1-{\displaystyle \frac{exp[-\beta (\vartheta -{\mathcal{Z}}_q{\mu }_e)]}{1-exp[-\beta (\vartheta -{\mathcal{Z}}_q{\mu }_e)]}}-{\displaystyle \frac{exp[-\beta (\vartheta +{\mathcal{Z}}_q{\mu }_e)]}{1+exp[-\beta (\vartheta +{\mathcal{Z}}_q{\mu }_e)]}}].$$

(38)

We can write the previous equation in the following form

$${\mathcal{Z}}_{C_F}^{{\displaystyle \frac{1}{2}}}{S}_0=2{\mathcal{Z}}_S^{{\displaystyle \frac{1}{2}}}{C}_F\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{{\mathcal{M}}_q}{\sqrt{{\overrightarrow{k}}^2+{\mathcal{M}}_q^2}}}[1-{\eta }_{+}-{\eta }_{-}],$$

(39)

where we see the appearance of Fermi–Dirac distribution with chemical potential

$${\eta }_{\pm }={\displaystyle \frac{1}{exp[\beta (\vartheta \mp {\mathcal{Z}}_q{\mu }_e)]+1}}.$$

(40)

Finally, with the previous results, we have the following set of coupled thermal gap equations

$$\begin{array}{ccc}{\mathcal{Z}}_{\Psi }^{{\displaystyle \frac{1}{2}}}{\Psi }_0\hfill & =\hfill & {\displaystyle \frac{{\mathcal{Z}}_{\Psi }^{\frac{1}{2}}{\mathcal{Z}}_{\lambda }^{\frac{1}{2}}\lambda }{12{\mathcal{Z}}_{\varphi }}}{\int }_0^{\infty }{\displaystyle \frac{{\overrightarrow{k}}^{2}dk}{\sqrt{[{\overrightarrow{k}}^2+\frac{{\mathcal{M}}_{\varphi }^2}{{\mathcal{Z}}_{\varphi }}]}}}coth[{\displaystyle \frac{\beta }{8}}\sqrt{({\overrightarrow{k}}^2+{\displaystyle \frac{{\mathcal{M}}_{\varphi }^2}{{\mathcal{Z}}_{\varphi }}})}]+{\displaystyle \frac{{\mathcal{Z}}_{\lambda }^{\frac{1}{2}}\lambda {\mathcal{Z}}_{C_F}}{24g^2}}{S}_0^2,\hfill \\ \\ {\mathcal{Z}}_{C_F}^{{\displaystyle \frac{1}{2}}}{S}_0\hfill & =\hfill & 2{\mathcal{Z}}_S^{{\displaystyle \frac{1}{2}}}{C}_F\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{{\mathcal{M}}_q}{\sqrt{{\overrightarrow{k}}^2+{\mathcal{M}}_q^2}}}[1-2exp[-{\displaystyle \frac{\beta }{2}}(\vartheta -{\mathcal{Z}}_q{\mu }_e)]tanh{\displaystyle \frac{\beta }{2}}(\vartheta -{\mathcal{Z}}_q{\mu }_e)]+\hfill \\ \\ \hfill & \hfill & -2exp[-{\displaystyle \frac{\beta }{2}}(\vartheta +{\mathcal{Z}}_q{\mu }_e)]tanh{\displaystyle \frac{\beta }{2}}(\vartheta +{\mathcal{Z}}_q{\mu }_e)].\hfill \end{array}$$

(41)

Note that m was assumed to be very large, although the limit of (${\mathcal{M}}_{\varphi }^2\to \infty$) would lead to $C_F=-{\displaystyle \frac{g^2}{m^2}}\to 0$. In this case, there is no dynamical mass generation mechanism, and Equation (33) can be written with $S_0=0$. Only a thermal gap equation associated with the restoration of $Z_2$ symmetry would remain.

2.3. Thermal Corrections to Higgs-Type SSB

It is also relevant to consider again the equation that defines the expected value of the scalar field ${\varphi }_0$ with thermal corrections. Then, a corrected gap equation is given by:

$$\begin{array}{ccc}\hfill & \hfill & {\displaystyle \frac{\partial S_{eff}}{\partial \varphi }}{|}_{{\varphi }_0}\to \left(m^2-{\displaystyle \frac{\lambda }{12}}{\varphi }_0^2-{\displaystyle \frac{i\lambda }{6[\Delta +M_{\varphi }^2]}}\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y})\right)2{\varphi }_0=0.\hfill \end{array}$$

(42)

The solutions for this equation can be written as:

$$\begin{array}{c}{\varphi }_0=0,{\varphi }_0^2=\left({\displaystyle \frac{12m^2}{\lambda }}-{\displaystyle \frac{2i}{[\Delta +M_{\varphi }^2]}}\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y})\right),\hfill \end{array}$$

(43)

in which the non-trivial solutions can be written in the momentum representation

$$\begin{array}{c}{\varphi }_0^2={\displaystyle \frac{12m^2}{\lambda }}-{\displaystyle \frac{1}{\beta }}{\displaystyle \sum _{n=-\infty }^{n=\infty }}\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{2i}{({\omega }_n^2+{\overrightarrow{k}}^2)+2m^2+4({\Psi }_0+\frac{\lambda }{4!}{\varphi }_0^2)}}.\hfill \end{array}$$

(44)

This last equation can be written as:

$$\begin{array}{c}{\varphi }_0^2={\overline{\varphi }}_0^2-{\displaystyle \frac{48}{\lambda }}{\Psi }_0.\hfill \end{array}$$

(45)

This equation makes explicit the role of the auxiliary field $\Psi$ for the SSB in a thermal environment.

So, the previous renormalized thermal gap equations for the Higgs-type SSB are expressed by the general result

$$\begin{array}{ccc}\hfill & \hfill & {\varphi }_0^2={\displaystyle \frac{{\mathcal{Z}}_m}{{\mathcal{Z}}_{\lambda }}}{\displaystyle \frac{12m^2}{\lambda }}-{\displaystyle \frac{{\mathcal{Z}}_{\Psi }^{\frac{1}{2}}}{{\mathcal{Z}}_{\lambda }^{\frac{1}{2}}}}{\displaystyle \frac{48{\Psi }_0}{\lambda }}\hfill \\ \\ \hfill & \hfill & {\Psi }_0=-{\displaystyle \frac{i{\mathcal{Z}}_{\lambda }^{\frac{1}{2}}{\lambda }_R}{24{\mathcal{Z}}_{\Psi }^{\frac{1}{2}}[{\mathcal{Z}}_{\varphi }\Delta +M_{\varphi }^2]}}\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y})\hfill \\ \\ \hfill & \hfill & M_{\varphi }^2=2{\mathcal{Z}}_m{m}_R^2+4[{\mathcal{Z}}_{\Psi }^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{\lambda }^{{\displaystyle \frac{1}{2}}}{\Psi }_0+{\mathcal{Z}}_{\lambda }{\displaystyle \frac{{\lambda }_R}{4!}}{\varphi }_0^2],\hfill \end{array}$$

(46)

wherein the notation of the renormalization factors ${\mathcal{Z}}_i$ is to remind us that now the objects are functions of dressed quantities and counter-terms.

3. Effective Action from Thermal Contributions by External Fields Methods

Following the same line of reasoning as the quantum case in Ref. [63], let us study the thermal contributions to the masses and coupling constants of the effective action for the auxiliary fields as quasiparticles of the model by the external field methods. Therefore with the effective action $\Theta$, given by Equation (25), with the expansion in terms of the fluctuations ($\Psi \to {\Psi }_0+\tilde{\Psi }$, $S\to S_0+\tilde{S}$), we can write $\Theta ={\Theta }_F+{\Theta }_B+{\Theta }_0$ in which

$$\begin{array}{ccc}\hfill & \hfill & {\Theta }_F=-\int d^{4}xtrln\{1+\tilde{F}[-{\mathcal{Z}}_S^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{C_F}^{{\displaystyle \frac{1}{2}}}\tilde{S}+2{\mathcal{Z}}_{C_F}{C}_F{(\overline{q}q)}_0]\}\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y}),\hfill \\ \\ \hfill & \hfill & {\Theta }_B={\displaystyle \frac{1}{2}}\int d^{4}xln\{1-2\tilde{B}[{\mathcal{Z}}_{\Psi }^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{\lambda }^{{\displaystyle \frac{1}{2}}}\tilde{\Psi }+{\mathcal{Z}}_{\lambda }{\displaystyle \frac{\lambda }{4!}}{\varphi }_0^2]\}\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y}).\hfill \end{array}$$

(47)

The following quantities were defined:

$$\begin{array}{ccc}\hfill & \hfill & {\tilde{F}}^{-1}=-{\mathcal{Z}}_q{\gamma }_{\mu }^E{\partial }_{\mu }^{{\mu }_e}+{\tilde{\mathcal{M}}}_q,{\tilde{\mathcal{M}}}_q={\mathcal{Z}}_{g}g{\varphi }_0-{\displaystyle \frac{s}{2}}-{\mathcal{Z}}_S^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{C_F}^{{\displaystyle \frac{1}{2}}}{S}_0,\hfill \\ \\ \hfill & \hfill & {\tilde{B}}^{-1}={\mathcal{Z}}_{\varphi }\Delta -{\tilde{\mathcal{M}}}_{\varphi }^2,{\tilde{\mathcal{M}}}_{\varphi }^2=2{\mathcal{Z}}_m{m}^2+4{\mathcal{Z}}_{\Psi }^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{\lambda }^{{\displaystyle \frac{1}{2}}}{\Psi }_0,\hfill \\ \\ \hfill & \hfill & {\Theta }_0=-\int d^{4}xtr\left(ln({\tilde{F}}^{-1})-{\displaystyle \frac{ln({\tilde{B}}^{-1})}{2}}\right).\hfill \end{array}$$

(48)

By assuming the resulting effective thermal masses are large, we can apply large mass expansions

$$\begin{array}{ccc}{\Theta }_F\hfill & =\hfill & -\int d^{4}xtr\tilde{F}[-{\mathcal{Z}}_S^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{C_F}^{{\displaystyle \frac{1}{2}}}\tilde{S}+2{\mathcal{Z}}_{C_F}{C}_F{(\overline{q}q)}_0]\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y})+\hfill \\ \\ \hfill & \hfill & +{\displaystyle \frac{1}{2}}\int d^{4}xtr\{\tilde{F}[-{\mathcal{Z}}_S^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{C_F}^{{\displaystyle \frac{1}{2}}}\tilde{S}+2{\mathcal{Z}}_{C_F}{C}_F{(\overline{q}q)}_0]\tilde{F}[-{\mathcal{Z}}_S^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{C_F}^{{\displaystyle \frac{1}{2}}}\tilde{S}+2{\mathcal{Z}}_{C_F}{C}_F{(\overline{q}q)}_0]\}\times \hfill \\ \\ \hfill & \hfill & \delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y}),\hfill \end{array}$$

(49)

$$\begin{array}{ccc}{\Theta }_B\hfill & =\hfill & -\int d^{4}x\tilde{B}[{\mathcal{Z}}_{\Psi }^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{\lambda }^{{\displaystyle \frac{1}{2}}}\tilde{\Psi }+{\mathcal{Z}}_{\lambda }{\displaystyle \frac{{\lambda }_R}{4!}}{\varphi }_0^2]\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y})+\hfill \\ \\ \hfill & \hfill & -\int d^{4}x\tilde{B}[{\mathcal{Z}}_{\Psi }^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{\lambda }^{{\displaystyle \frac{1}{2}}}\tilde{\Psi }+{\mathcal{Z}}_{\lambda }{\displaystyle \frac{\lambda }{4!}}{\varphi }_0^2]\tilde{B}[{\mathcal{Z}}_{\Psi }^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{\lambda }^{{\displaystyle \frac{1}{2}}}\tilde{\Psi }+{\mathcal{Z}}_{\lambda }{\displaystyle \frac{\lambda }{4!}}{\varphi }_0^2]\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y}).\hfill \end{array}$$

(50)

The above leading terms can be put into effective action

$$\begin{array}{ccc}\hfill & \hfill & {\Gamma }_{eff}={\int }_0^{\beta }d{\tau }_x\int d^3\overrightarrow{x}\{{\mathcal{Z}}_{\varphi }{\displaystyle \frac{1}{2}}{\varphi }_0\Delta {\varphi }_0+[{\mathcal{Z}}_m{m}^2+\delta m]{\varphi }_0^2+[-{\mathcal{Z}}_{\lambda }{\displaystyle \frac{\lambda }{4!}}+\delta \lambda ]{\varphi }_0^4+\hfill \\ \\ \hfill & \hfill & -{\mathcal{Z}}_q{\overline{q}}_0({\gamma }_{\mu }^E{\partial }_{\mu }^{{\mu }_e})q_0+[{\mathcal{Z}}_{g}g{\varphi }_0-{\displaystyle \frac{s}{2}}+\delta M]{\overline{q}}_0{q}_0+[{\mathcal{Z}}_{C_F}{C}_F+\delta C_F]{({\overline{q}}_0{q}_0)}^2\}\hfill \end{array}$$

(51)

wherein we can see the thermal contributions to the masses and coupling constants in the zero-momentum transfer limit:

$$\delta M=-2{\mathcal{Z}}_{C_F}{C}_{F}tr\tilde{F}\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y})=8{\mathcal{Z}}_{C_F}{C}_F{\displaystyle \frac{1}{\beta }}\sum _{n=-\infty }^{n=\infty }\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{{\tilde{\mathcal{M}}}_q}{[{\mathcal{Z}}_q^2{({\vartheta }_n+i{\mu }_e)}^2+{\overrightarrow{k}}^2+{\tilde{\mathcal{M}}}_q^2]}},$$

(52)

$$\begin{array}{ccc}\hfill & \hfill & \delta C_F=2{[{\mathcal{Z}}_{C_F}{C}_F]}^{2}tr(\tilde{F}\tilde{F})\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y})=-8{[{\mathcal{Z}}_{C_F}{C}_F]}^2\times \hfill \\ \\ \hfill & \hfill & {\displaystyle \frac{1}{\beta }}{\displaystyle \sum _{n=-\infty }^{n=\infty }}\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{{\mathcal{Z}}_q^2{({\vartheta }_n+i{\mu }_e)}^2+{\overrightarrow{k}}^2-{\tilde{\mathcal{M}}}_q^2}{{[{\mathcal{Z}}_q^2{({\vartheta }_n+i{\mu }_e)}^2+{\overrightarrow{k}}^2+{\tilde{\mathcal{M}}}_q^2]}^2}},\hfill \end{array}$$

(53)

$$\delta m=-{\mathcal{Z}}_{\lambda }{\displaystyle \frac{\lambda }{4!}}\tilde{B}\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y})=-{\displaystyle \frac{1}{24}}{\mathcal{Z}}_{\lambda }\lambda {\displaystyle \frac{1}{\beta }}\sum _{n=-\infty }^{n=\infty }\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{1}{{\mathcal{Z}}_{\varphi }({\omega }_n^2+{\overrightarrow{k}}^2)+{\tilde{\mathcal{M}}}_{\varphi }^2}},$$

(54)

$$\delta \lambda ={[{\mathcal{Z}}_{\lambda }{\displaystyle \frac{\lambda }{4!}}]}^2\tilde{B}\tilde{B}\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y})={[{\displaystyle \frac{{\mathcal{Z}}_{\lambda }\lambda }{4!}}]}^2{\displaystyle \frac{1}{\beta }}\sum _{n=-\infty }^{n=\infty }\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{1}{{[{\mathcal{Z}}_{\varphi }({\omega }_n^2+{\overrightarrow{k}}^2)+{\tilde{\mathcal{M}}}_{\varphi }^2]}^2}}.$$

(55)

We can see, too, from Equation (49) that the emergence of terms in the effective action suggests a composite fermion–antifermion particle $\tilde{S}$. These terms can be resolved as follows:

$${\displaystyle \frac{1}{2}}{\mathcal{Z}}_S{\mathcal{Z}}_{C_F}tr[\tilde{F}\tilde{S}\tilde{F}\tilde{S}]\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y})=-\alpha \tilde{S}\Delta \tilde{S}+\beta {\tilde{S}}^2$$

(56)

wherein

$$\begin{array}{ccc}\hfill & \hfill & \alpha ={\mathcal{Z}}_S{\mathcal{Z}}_{C_F}{\mathcal{Z}}_q^2{\displaystyle \frac{1}{2\beta }}{\displaystyle \sum _{n=-\infty }^{n=\infty }}\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{1}{{[{\mathcal{Z}}_q^2{({\vartheta }_n+i{\mu }_e)}^2+{\overrightarrow{k}}^2+{\tilde{\mathcal{M}}}_q^2]}^2}}\hfill \\ \\ \hfill & \hfill & \beta =-{\mathcal{Z}}_S{\mathcal{Z}}_{C_F}{\mathcal{Z}}_q^2{\displaystyle \frac{1}{2\beta }}{\displaystyle \sum _{n=-\infty }^{n=\infty }}\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{{\tilde{\mathcal{M}}}_q^2}{{[{\mathcal{Z}}_q^2{({\vartheta }_n+i{\mu }_e)}^2+{\overrightarrow{k}}^2+{\tilde{\mathcal{M}}}_q^2]}^2}},\hfill \end{array}$$

(57)

It also results in an interaction term of this field $\tilde{S}$ with an external background ${(\overline{q}q)}_0$

$$\begin{array}{ccc}\hfill & \hfill & 2{\mathcal{Z}}_S^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{C_F}^{{\displaystyle \frac{3}{2}}}{C}_F^{R}tr[\tilde{F}\tilde{F}]{\delta }^4(x-y)\tilde{S}{(\overline{q}q)}_0=2{\mathcal{Z}}_S^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{C_F}^{{\displaystyle \frac{3}{2}}}{C}_F\times \hfill \\ \\ \hfill & \hfill & {\displaystyle \frac{1}{\beta }}{\displaystyle \sum _{n=-\infty }^{n=\infty }}\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{{\mathcal{Z}}_q^2{({\vartheta }_n+i{\mu }_e)}^2+{\overrightarrow{k}}^2-{\tilde{\mathcal{M}}}_q^2}{{[{\mathcal{Z}}_q^2{({\vartheta }_n+i{\mu }_e)}^2+{\overrightarrow{k}}^2+{\tilde{\mathcal{M}}}_q^2]}^2}}\tilde{S}{(\overline{q}q)}_0.\hfill \end{array}$$

(58)

In the same way, from Equation (50), terms associated with a composite particle $\tilde{\Psi }$ that would be a two-boson bound state are given by

$$\begin{array}{ccc}\hfill & \hfill & -{\mathcal{Z}}_{\Psi }{\mathcal{Z}}_{\lambda }\tilde{B}\tilde{\Psi }\tilde{B}\tilde{\Psi }\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y})=-{\mathcal{Z}}_{\Psi }{\mathcal{Z}}_{\lambda }\tilde{\Psi }[{\mathcal{Z}}_{\varphi }\Delta -{\tilde{\mathcal{M}}}_{\varphi }^2]\tilde{\Psi }\times \hfill \\ \\ \hfill & \hfill & -{\displaystyle \frac{1}{{[{\mathcal{Z}}_{\varphi }\Delta -{\tilde{\mathcal{M}}}_{\varphi }^2]}^2}}\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y})=ϵ\tilde{\Psi }\Delta \tilde{\Psi }+\sigma {\tilde{\Psi }}^2\hfill \end{array}$$

(59)

wherein

$$\begin{array}{ccc}\hfill & \hfill & ϵ=-{\mathcal{Z}}_{\Psi }{\mathcal{Z}}_{\lambda }{\mathcal{Z}}_{\varphi }{\displaystyle \frac{1}{\beta }}{\displaystyle \sum _{n=-\infty }^{n=\infty }}\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{1}{{[{\mathcal{Z}}_{\varphi }({\omega }_n^2+{\overrightarrow{k}}^2)+{\tilde{\mathcal{M}}}_{\varphi }^2]}^2}}\hfill \\ \\ \hfill & \hfill & \sigma ={\mathcal{Z}}_{\Psi }{\mathcal{Z}}_{\lambda }({\mathcal{Z}}_m{m}^2+2{\mathcal{Z}}_{\Psi }^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{\lambda }^{{\displaystyle \frac{1}{2}}}{\Psi }_0){\displaystyle \frac{1}{\beta }}{\displaystyle \sum _{n=-\infty }^{n=\infty }}\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{1}{{[{\mathcal{Z}}_{\varphi }({\omega }_n^2+{\overrightarrow{k}}^2)+{\tilde{\mathcal{M}}}_{\varphi }^2]}^2}}.\hfill \end{array}$$

(60)

The interaction of this field $\tilde{\Psi }$ with an external background ${\varphi }_0^2$ is obtained as

$$\begin{array}{ccc}\hfill & \hfill & -2{\mathcal{Z}}_{\Psi }^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{\lambda }^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{\lambda }{\displaystyle \frac{\lambda }{4!}}\tilde{\Psi }{\varphi }_0^2\tilde{B}\tilde{B}\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y})=\hfill \\ \\ \hfill & \hfill & =-{\displaystyle \frac{1}{12}}{\mathcal{Z}}_{\Psi }^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{\lambda }^{{\displaystyle \frac{3}{2}}}\lambda \tilde{\Psi }{\varphi }_0^2{\displaystyle \frac{1}{\beta }}{\displaystyle \sum _{n=-\infty }^{n=\infty }}\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{1}{{[{\mathcal{Z}}_{\varphi }({\omega }_n^2+{\overrightarrow{k}}^2)+{\tilde{\mathcal{M}}}_{\varphi }^2]}^2}}.\hfill \end{array}$$

(61)

The contribution to the background effective action of the previous field interactions with the external background can be formalized in the following equation2

$$\tilde{Z}=\int D\tilde{S}D\tilde{\Psi }exp\{-{\int }_0^{\beta }d{\tau }_x\int d^3\overrightarrow{x}[\tilde{S}(-\alpha \Delta +\beta )\tilde{S}+{{\rm Y}}_F{C}_F{(\overline{q}q)}_0\tilde{S}+\tilde{\Psi }(ϵ\Delta +\lambda )\tilde{\Psi }+{{\rm Y}}_B\lambda {\varphi }_0^2\tilde{\Psi }]-\delta G\tilde{S}\tilde{\Psi }\},$$

(62)

in which a mixing-type term was found as

$$\begin{array}{ccc}\delta G\hfill & =\hfill & {({\mathcal{Z}}_S{\mathcal{Z}}_{C_F}{\mathcal{Z}}_{\lambda }{\mathcal{Z}}_{\Psi })}^{{\displaystyle \frac{1}{2}}}tr({\tilde{S}}_B{\tilde{S}}_F){\delta }^4(x-y)\hfill \\ \hfill & =\hfill & -4{({\mathcal{Z}}_S{\mathcal{Z}}_{C_F}{\mathcal{Z}}_{\lambda }{\mathcal{Z}}_{\Psi })}^{{\displaystyle \frac{1}{2}}}{\displaystyle \frac{1}{\beta }}{\displaystyle \sum _{n=-\infty }^{n=\infty }}\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{1}{{[{\mathcal{Z}}_{\varphi }({\omega }_n^2+{\overrightarrow{k}}^2)+{\tilde{M}}_{\varphi }^2]}^2}}{\displaystyle \frac{{\tilde{M}}_q}{[{\mathcal{Z}}_q^2({\omega }_n^2+{\overrightarrow{k}}^2)+{\tilde{M}}_q^2]}}.\hfill \end{array}$$

(63)

By considering the only quadratic terms in the discussion above, we can see the external background as the source of the fields and make the current expansion

$$\begin{array}{ccc}\hfill & \hfill & \tilde{S}\to \overline{S}-{\displaystyle \frac{1}{2}}\int d^{4}yV({\tau }_x,\overrightarrow{x};{\tau }_y,\overrightarrow{y})j_F\hfill \\ \\ \hfill & \hfill & j_F={{\rm Y}}_F{C}_F{(\overline{q}q)}_0\hfill \\ \\ \hfill & \hfill & V^{-1}({\tau }_x,\overrightarrow{x};{\tau }_y,\overrightarrow{y})=(-\alpha \Delta +\beta )\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y}),\hfill \end{array}$$

(64)

$$\begin{array}{ccc}\hfill & \hfill & \tilde{\Psi }\to \overline{\Psi }-{\displaystyle \frac{1}{2}}\int d^{4}yW({\tau }_x,\overrightarrow{x};{\tau }_y,\overrightarrow{y})j_B\hfill \\ \\ \hfill & \hfill & j_B={{\rm Y}}_B\lambda {\varphi }_0^2\hfill \\ \\ \hfill & \hfill & W^{-1}=(ϵ\Delta +\lambda )\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y}).\hfill \end{array}$$

(65)

So, we have the result

$$\tilde{Z}=det{(-\alpha \Delta +\beta )}^{-{\displaystyle \frac{1}{2}}}det{(ϵ\Delta +\lambda )}^{-{\displaystyle \frac{1}{2}}}exp\{-\int \int d{\tau }_x{d}^3\overrightarrow{x}d{\tau }_y{d}^3\overrightarrow{y}[-{\displaystyle \frac{1}{4}}{j}_{F}V{j}_F-{\displaystyle \frac{1}{4}}{j}_{B}W{j}_B]\}$$

(66)

and see the contributions to the vertices $C_F$ and $\lambda$, respectively

$$\begin{array}{ccc}\hfill & \hfill & -{\displaystyle \frac{1}{4}}{j}_{F}V{j}_F=\delta C_F^V{(\overline{q}q)}_0^2,\delta C_F^V=-{({{\rm Y}}_F{C}_F^R)}^2{\displaystyle \frac{\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y})}{(-\alpha \Delta +\beta )}}\hfill \\ \\ \hfill & \hfill & -{\displaystyle \frac{1}{4}}{j}_{B}W{j}_B=\delta {\lambda }^W{\varphi }_0^4,\delta {\lambda }^W=-{({{\rm Y}}_B{\lambda }_R)}^2{\displaystyle \frac{\delta ({\tau }_x-{\tau }_y){\delta }^3(\overrightarrow{x}-\overrightarrow{y})}{(ϵ\Delta +\lambda )}}.\hfill \end{array}$$

(67)

Therefore the effective action is written as

$$\begin{array}{ccc}\hfill & \hfill & {\Gamma }_{eff}={\int }_0^{\beta }d{\tau }_x\int d^3\overrightarrow{x}\{{\mathcal{Z}}_{\varphi }{\displaystyle \frac{1}{2}}{\varphi }_0\Delta {\varphi }_0+[{\mathcal{Z}}_m{m}^2+\delta m]{\varphi }_0^2+[-{\mathcal{Z}}_{\lambda }{\displaystyle \frac{\lambda }{4!}}+\delta \lambda +\delta {\lambda }^W]{\varphi }_0^4+\hfill \\ \\ \hfill & \hfill & -{\mathcal{Z}}_q{\overline{q}}_0({\gamma }_{\mu }^E{\partial }_{\mu }^{{\mu }_e})q_0+[{\mathcal{Z}}_{g}g{\varphi }_0-{\displaystyle \frac{s}{2}}+\delta M]{\overline{q}}_0{q}_0+[{\mathcal{Z}}_{C_F}{C}_F+\delta C_F+\delta C_F^V]{({\overline{q}}_0{q}_0)}^2\}+\hfill \\ \\ \hfill & \hfill & -{\displaystyle \frac{1}{2}}trln(-\alpha \Delta +\beta )-{\displaystyle \frac{1}{2}}trln(ϵ\Delta +\lambda ).\hfill \end{array}$$

(68)

The interesting thing to see is the link between the fields [$\tilde{\Psi }$; $\tilde{S}$] and the interactions [${\varphi }_0^4$; ${({\overline{q}}_0{q}_0)}^2$], respectively, fruit of the condensates [${\varphi }_0{\varphi }_0$; $({\overline{q}}_0{q}_0)$], in which we see the condensates as sources for this fields and they contribute to the vertices [$\delta {\lambda }^W$, $\delta C_F^V$].

3.1. Phase Transitions and Critical Phenomena

The mixing interaction S and $\Psi$ suggests one can rotate the system of states to diagonalize the corresponding masses. Let us define a mixing angle ${\theta }_{22}$ that allows the rotation of the auxiliary fields of 2-fermion states to 2-boson states. Following the steps of the quantum description [63], we have in a thermal environment the two conditions of final mass eigenstates and normalization

$$\begin{array}{ccc}{m}_{\tilde{S}}^2\hfill & =\hfill & -{\displaystyle \frac{\beta }{\alpha }}+2\delta Gtan({\theta }_{22}),m_{\tilde{\Psi }}^2={\displaystyle \frac{\sigma }{ϵ}}+2\delta Gtan({\theta }_{22}),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill & \hfill & tan(2{\theta }_{22})={\displaystyle \frac{\delta G}{\sigma +\beta }},\hfill \\ \\ \hfill & \hfill & ϵ-\alpha =0.\hfill \end{array}$$

(69)

The bound-state conditions for both the composite scalar field $\tilde{\Psi }$ and for the composite fermion–antifermion state $\tilde{S}$ can be written simply considering the condition of a pole for real positive masses

$$\begin{array}{ccc}\hfill & \hfill & {\mathcal{Z}}_{\varphi }{m}_{\tilde{\Psi }}^2={\tilde{M}}_{\varphi }^2-[{\displaystyle \frac{4!}{{\mathcal{Z}}_{\lambda }{\lambda }_R}}-{\displaystyle \frac{4{\varphi }_0^2}{M_{\varphi }^2}}]{\displaystyle \frac{1}{\frac{1}{\beta }{\sum }_{n=-\infty }^{n=\infty }\int \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}\frac{1}{{[{\mathcal{Z}}_{\varphi }{({\omega }_n^2+{\overrightarrow{k}}^2)}^2+{\tilde{M}}_{\varphi }^2]}^2[{\mathcal{Z}}_{\varphi }{({\omega }_n^2+{\overrightarrow{k}}^2)}^2-{\tilde{M}}_{\varphi }^2]}}}+2\delta Gtan({\theta }_{22}),\hfill \\ \\ \hfill & \hfill & {\mathcal{Z}}_q^2{m}_{\tilde{S}}^2={\displaystyle \frac{{\mathcal{Z}}_{\Psi }}{{\mathcal{Z}}_{C_F}}}{\tilde{M}}_q^2+{\displaystyle \frac{1}{4{\mathcal{Z}}_{C_F}{C}_F^R}}{\displaystyle \frac{1}{\frac{1}{\beta }{\sum }_{n=-\infty }^{n=\infty }\int \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}\frac{1}{{[{\mathcal{Z}}_q^2{({\omega }_n^2+{\overrightarrow{k}}^2)}^2+{\tilde{M}}_q^2]}^2}}}+2\delta Gtan({\theta }_{22}).\hfill \end{array}$$

(70)

The nonexistence of composite particles and the definition of a critical temperature in a mixing states environment can be described by the following equations:

$$\begin{array}{ccc}\hfill & \hfill & {\tilde{M}}_{\varphi }^2-{\displaystyle \frac{4!}{{\mathcal{Z}}_{\lambda }{\lambda }_R}}[1-{\displaystyle \frac{(\frac{{\mathcal{Z}}_m}{{\mathcal{Z}}_{\lambda }}12m_R^2-{\mathcal{Z}}_{\Psi }^{\frac{1}{2}}{\mathcal{Z}}_{\lambda }^{\frac{1}{2}}48{\Psi }_0)}{12M_{\varphi }^2}}]{\displaystyle \frac{1}{\frac{1}{\beta }{\sum }_{n=-\infty }^{n=\infty }\int \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}\frac{1}{{[{\mathcal{Z}}_{\varphi }{({\omega }_n^2+{\overrightarrow{k}}^2)}^2+{\tilde{M}}_{\varphi }^2]}^2[{\mathcal{Z}}_{\varphi }{({\omega }_n^2+{\overrightarrow{k}}^2)}^2-{\tilde{M}}_{\varphi }^2]}}}+\hfill \\ \\ \hfill & \hfill & +2\delta Gtan({\theta }_{22})=0,\hfill \end{array}$$

(71)

$${\displaystyle \frac{{\mathcal{Z}}_{\Psi }}{{\mathcal{Z}}_{C_F}}}{\tilde{M}}_q^2-{\displaystyle \frac{1}{2}}{\displaystyle \frac{M_{\varphi }^2}{{\mathcal{Z}}_m{m}_R^2}}{\varphi }_0^2{\displaystyle \frac{1}{\frac{1}{\beta }{\sum }_{n=-\infty }^{n=\infty }\int \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}\frac{1}{{[{\mathcal{Z}}_q^2{({\omega }_n^2+{\overrightarrow{k}}^2)}^2+{\tilde{M}}_q^2]}^2}}}+2\delta Gtan({\theta }_{22})=0.$$

(72)

3.2. Symmetry Restoration and Critical Temperature

At this moment, with the previous structure, we can explore the concept of restoration of symmetry due to the thermal contributions. First, let us analyze the restoration of $Z_2$ symmetry. So, if we look in Equation (68), we write the mass with its thermal contribution in the effective action. As the thermal contribution has an opposite sign, it could cancel out the mass, restoring the symmetry

$${\mathcal{Z}}_m{m}^2+\delta m=0.$$

(73)

With the help of Equation (54), we re-write the previous equation in an explicit form

$${\mathcal{Z}}_m{m}^2-{\displaystyle \frac{1}{24}}{\mathcal{Z}}_{\lambda }\lambda {\displaystyle \frac{1}{\beta }}\sum _{n=-\infty }^{n=\infty }\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{1}{{\mathcal{Z}}_{\varphi }({\omega }_n^2+{\overrightarrow{k}}^2)+{\tilde{\mathcal{M}}}_{\varphi }^2}}=0.$$

(74)

For simplicity, forgetting the renormalization procedure ${\mathcal{Z}}_i=1$ we have

$$\begin{array}{ccc}\hfill & \hfill & m^2-{\displaystyle \frac{\lambda }{24\beta }}{\displaystyle \sum _{n=-\infty }^{n=\infty }}\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{1}{({\omega }_n^2+{\overrightarrow{k}}^2)+{\tilde{\mathcal{M}}}_{\varphi }^2}}=0,{\omega }_n={\displaystyle \frac{2\pi n}{\beta }},\hfill \\ \\ \hfill & \hfill & {\tilde{\mathcal{M}}}_{\varphi }^2=2m^2+4{\Psi }_0\hfill \end{array}$$

(75)

Therefore making the sum [analogous to the gap equation], we are led to the result

$$m^2-{\displaystyle \frac{\lambda }{24}}{\displaystyle \frac{1}{4{\pi }^2}}{\int }_0^{\infty }{\displaystyle \frac{{\overrightarrow{k}}^{2}dk}{\sqrt{[{\overrightarrow{k}}^2+{\tilde{\mathcal{M}}}_{\varphi }^2]}}}[1+{\displaystyle \frac{2}{exp(\beta \sqrt{[{\overrightarrow{k}}^2+{\tilde{\mathcal{M}}}_{\varphi }^2]})-1}}]=0$$

(76)

and with the change in variable $x=\beta k$ we obtain the expression

$$m^2-{\displaystyle \frac{\lambda }{24}}{\displaystyle \frac{1}{4{\pi }^2{\beta }^2}}{\int }_0^{\infty }{\displaystyle \frac{x^{2}dx}{\sqrt{[x^2+{\beta }^2{\tilde{\mathcal{M}}}_{\varphi }^2]}}}[1+{\displaystyle \frac{2}{exp(\beta \sqrt{[x^2+{\beta }^2{\tilde{\mathcal{M}}}_{\varphi }^2]})-1}}]=0,$$

(77)

wherein throwing away the term that does not depend on the temperature because it would be absorbed in the counter-term $\delta {\mathcal{Z}}_{\lambda }$ we finally arrived at the Dolan–Jackiw equation [32]

$$\begin{array}{ccc}\hfill & \hfill & m^2-{\displaystyle \frac{\lambda }{24}}{\displaystyle \frac{1}{2{\pi }^2{\beta }^2}}{\int }_0^{\infty }{\displaystyle \frac{x^{2}dx}{\sqrt{[x^2+{\beta }^2{\tilde{\mathcal{M}}}_{\varphi }^2]}}}{\displaystyle \frac{1}{exp(\beta \sqrt{[x^2+{\beta }^2{\tilde{\mathcal{M}}}_{\varphi }^2]})-1}}=\hfill \\ \\ \hfill & \hfill & =m^2-{\displaystyle \frac{\lambda }{24}}[{\displaystyle \frac{1}{12{\beta }^2}}-{\displaystyle \frac{{\tilde{\mathcal{M}}}_{\varphi }}{4\pi \beta }}+\mathcal{O}({\tilde{\mathcal{M}}}_{\varphi }^{2}ln{\tilde{\mathcal{M}}}_{\varphi }^2)]=0.\hfill \end{array}$$

(78)

So from the above equation, it is possible to conclude that in the vicinity of the critical point does not exist a gap ${\Psi }_0\to 0$, and we have a critical temperature

$$m^2-{\displaystyle \frac{\lambda }{24}}{\displaystyle \frac{1}{12{\beta }_c^2}}=0.$$

(79)

This result is important and says to us that the logarithmic expansion for large temperature, discussed in the previous section, where we compute the thermal effective action, is working.

Now, let us explore the partial restoration of the chiral symmetry associated with the spontaneous symmetry breaking and set aside for now the chiral mass that is generated dynamically. So, if we look in Equation (68), we write

$$\begin{array}{ccc}\hfill & \hfill & {\mathcal{Z}}_{g}g{\varphi }_0-{\displaystyle \frac{s}{2}}+\delta M=0,\hfill \\ \\ \hfill & \hfill & s={\displaystyle \frac{{\mathcal{Z}}_{g}g}{{\mathcal{M}}_{\varphi }^2}}[{\mathcal{Z}}_{\varphi }\Delta {\varphi }_0+2Z_m{m}^2{\varphi }_0-{\mathcal{Z}}_{\lambda }{\displaystyle \frac{\lambda }{3!}}{\varphi }_0^3+4{\mathcal{Z}}_{\Psi }^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{\lambda }^{{\displaystyle \frac{1}{2}}}{\varphi }_0{\Psi }_0].\hfill \end{array}$$

(80)

With the help of Equation (52), we re-write the previous equation in an explicit form

$$\begin{array}{ccc}\hfill & \hfill & {\mathcal{Z}}_{g}g{\varphi }_0-{\displaystyle \frac{s}{2}}+8{\mathcal{Z}}_{C_F}{C}_F{\displaystyle \frac{1}{\beta }}{\displaystyle \sum _{n=-\infty }^{n=\infty }}\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{{\tilde{\mathcal{M}}}_q}{[{\mathcal{Z}}_q^2{({\vartheta }_n+i{\mu }_e)}^2+{\overrightarrow{k}}^2+{\tilde{\mathcal{M}}}_q^2]}}=0,{\vartheta }_n={\displaystyle \frac{(2n+1)\pi }{\beta }},\hfill \\ \\ \hfill & \hfill & {\tilde{\mathcal{M}}}_q={\mathcal{Z}}_{g}g{\varphi }_0-{\displaystyle \frac{s}{2}}-{\mathcal{Z}}_S^{{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_{C_F}^{{\displaystyle \frac{1}{2}}}{S}_0.\hfill \end{array}$$

(81)

So, without the renormalization procedure ${\mathcal{Z}}_i=1$, and taking the limit of large mass m, we have

$${\displaystyle \frac{1}{2}}g{\varphi }_0+8C_F{\displaystyle \frac{1}{\beta }}\sum _{n=-\infty }^{n=\infty }\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{{\tilde{\mathcal{M}}}_q}{[{({\vartheta }_n+i{\mu }_e)}^2+{\overrightarrow{k}}^2+{\tilde{\mathcal{M}}}_q^2]}}=0$$

(82)

Therefore, making the sum [analogous as the gap equation, Equation (36)], we are led to the result

$${\displaystyle \frac{1}{2}}g{\varphi }_0+4C_F{\displaystyle \frac{1}{\beta }}\sum _{n=-\infty }^{n=\infty }\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{\partial }{\partial {\tilde{\mathcal{M}}}_q}}ln\{{(\beta {\mathcal{Z}}_q)}^2[{({\vartheta }_n+i{\mu }_e)}^2+{\displaystyle \frac{{\vartheta }^2}{{\mathcal{Z}}_q^2}}]\}=0,{\vartheta }^2={\overrightarrow{k}}^2+{\tilde{\mathcal{M}}}_q^2,$$

(83)

therefore, we can make the sum and write

$${\displaystyle \frac{1}{2}}g{\varphi }_0+2C_F\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi )}^3}}{\displaystyle \frac{1}{\sqrt{{\overrightarrow{k}}^2+{\tilde{\mathcal{M}}}_q^2}}}[1-{\displaystyle \frac{1}{exp[\beta (\vartheta -{\mu }_e)]+1}}-{\displaystyle \frac{1}{exp[\beta (\vartheta +{\mu }_e)]+1}}]=0.$$

(84)

If we make a change in variable $x=\beta k$, take the chemical potential ${\mu }_e=0$ and throw away the term that does not depend on the temperature because it would be absorbed in the counter-term $\delta {\mathcal{Z}}_{C_F}$, we obtain the expression

$${\displaystyle \frac{1}{2}}g{\varphi }_0-2C_F{\displaystyle \frac{{\tilde{\mathcal{M}}}_q}{2{\pi }^2{\beta }^2}}{\int }_0^{\infty }{\displaystyle \frac{x^{2}dx}{\sqrt{[x^2+{\beta }^2{\tilde{\mathcal{M}}}_q^2]}}}{\displaystyle \frac{1}{exp(\sqrt{[x^2+{\beta }^2{\tilde{\mathcal{M}}}_q^2]})+1}}=0.$$

(85)

The previous integral can be solved for small $\beta$ [32]

$$\begin{array}{ccc}\hfill & \hfill & {\int }_0^{\infty }{\displaystyle \frac{x^{2}dx}{\sqrt{[x^2+{\beta }^2{\tilde{\mathcal{M}}}_q^2]}}}{\displaystyle \frac{1}{exp(\sqrt{[x^2+{\beta }^2{\tilde{\mathcal{M}}}_q^2]})+1}}=-{\displaystyle \frac{2}{{\beta }^2}}{\displaystyle \frac{\partial }{\partial {\tilde{\mathcal{M}}}_q^2}}\times \hfill \\ \\ \hfill & \hfill & {\int }_0^{\infty }{x}^{2}dxln(\sqrt{exp(-[x^2+{\beta }^2{\tilde{\mathcal{M}}}_q^2]})+1),\hfill \end{array}$$

(86)

in which we can use the identity

$$\begin{array}{ccc}\hfill & \hfill & {\int }_0^{\infty }{x}^{2}dxln(\sqrt{exp(-[x^2+{\beta }^2{\tilde{\mathcal{M}}}_q^2]})+1)={\displaystyle \frac{14{\pi }^4}{760}}-{\displaystyle \frac{2{\pi }^2{\tilde{\mathcal{M}}}_q^2{\beta }^2}{48}}-{\displaystyle \frac{{\tilde{\mathcal{M}}}_q^4{\beta }^4}{32}}ln({\tilde{\mathcal{M}}}_q^2{\beta }^2)+\hfill \\ \\ \hfill & \hfill & -{\displaystyle \frac{{\tilde{\mathcal{M}}}_q^4{\beta }^{4}c}{16}}+\mathcal{O}({\tilde{\mathcal{M}}}_q^6{\beta }^6)\hfill \end{array}$$

(87)

and so, from the previous results, we have the equation for critical temperature

$$\begin{array}{ccc}\hfill & \hfill & {\displaystyle \frac{1}{2}}g{\varphi }_0+{\displaystyle \frac{C_F{\tilde{\mathcal{M}}}_q}{12{\beta }^2}}=0\hfill \\ \\ \hfill & \hfill & {\tilde{\mathcal{M}}}_q={\displaystyle \frac{1}{2}}g{\varphi }_0-S_0.\hfill \end{array}$$

(88)

Here, the gap $S_0$ does not go to zero because we have yet to obtain the chiral mass generated dynamically.

Roughly speaking (dimensional analysis basically), with the critical behavior from Equations (79) and (88), we can extract some outcomes associated with critical temperature and symmetry restoration. In this case, less than constants and numerical factors, we have two critical temperatures

$$\begin{array}{ccc}\hfill & \hfill & T_c^1\propto {\displaystyle \frac{m}{\sqrt{\lambda }}}\hfill \\ \\ \hfill & \hfill & T_c^2\propto {\displaystyle \frac{m}{g}}\hfill \end{array}$$

(89)

where in the limit of large m we use that $C_F=-{\displaystyle \frac{g^2}{4m^2}}$. If $\sqrt{\lambda }\gg g$, we have first the restoration of $Z_2$ symmetry and then ${\varphi }_0=0\Rightarrow T_c^2=0$. On the other hand, if $g\gg \sqrt{\lambda }$, we have first the partial restoration of the chiral symmetry and later the restoration of the $Z_2$ symmetry.

4. Outcomes and Final Comments

Throughout the text, we have studied the thermal aspects of symmetry breakings (dynamic or spontaneous) with functional techniques approach in the language of background external fields methods (condensates) for the Yukawa interaction between fermions and scalars with self-interaction.

When we investigate the thermally coupled gap equations with the Hubbard–Stratonovich auxiliary field identity, what is natural to see is the appearance of the Bose–Einstein and Fermi–Dirac distributions in the gap equations, seen in Equations (17) and (39). Gap equations for auxiliary fields were displayed in usual forms in which the thermal fluctuations can be separated from the UV divergent part. Restoration of corresponding symmetries may manifest in the high-temperature regime. In the study of the thermal contributions to the effective action, what must be reasonably justified is the logarithmic expansion, which the leading terms lead us to the effective action in Equation (51). With this effective action, we discuss the collective behavior connected with the concept of quasiparticles. This may be associated with the emergence of composite particles as a combination of two bosons and two fermions, respectively, formally given by the interactions of these particles with the external background in Equation (62). Finally, with the effective action in Equation (68), we can also explore the concept of symmetry restoration, and we found the critical temperatures related to the spontaneous symmetry breaking of $Z_2$ and chiral symmetry in Equation (88), justifying the previous logarithmic high T expansion because we arrive at Dolan–Jackiw results. With the critical behavior, we extracted some outcomes. In this case, less than constants and numerical factors, we have two critical temperatures in Equation (89).

Considering the auxiliary field methodology recently developed [63,64,65], we extend this technique to finite temperatures by employing the Schwinger variational principle to derive the Matsubara–Fradkin formalism and explore the concept of symmetry restoration, revisiting the study of critical temperature from a different perspective. In addition to recovering known results from the literature, we present new findings, such as the interpretation of condensates as sources for the fields in Equation (68) for the effective action, the contribution of the gap to the particle masses and their mixing states in a thermal environment, and the existence of composite particles, as shown in Equation (70). All these results contribute to and enhance the analysis of critical temperature and symmetry restoration. Although our focus has been on recovering the Dolan–Jackiw results, Equations (75) and (81) are general expressions.

As a final comment, we could explore in Section 3.1 how to implement the quantum corrections to the masses and study the general critical temperature equations. A broader analysis to find the Dolan–Jackiw equations for the case in which the chemical potential is different from zero ${\mu }_e\ne 0$ would open the possibility of obtaining all the effective parameters in Section 3. There are alternative applications in the literature where the methodology presented in this article can be employed, such as in quark interactions, by extending the work carried out in [64]. These matters need to be analyzed and require elaboration.