Dynamical Mass Generation in Pseudoquantum Electrodynamics with Gross-Neveu Interaction at Finite Temperature

PHYSICAL REVIEW D 103, 025018 (2021)

Dynamical mass generation in pseudoquantum electrodynamics with Gross-Neveu interaction at finite temperature

† ‡ Luis Fernández ,1,* Reginaldo O. Corrêa, Jr. ,2, Van Sergio ´ Alves ,1, ∥ Leandro O. Nascimento ,3,§ and Francisco Peña 4, 1Faculdade de Física, Universidade Federal do Pará, 66075-110 Belém, PA, Brazil 2Centro de Ciências e Planetário Sebastião Sodré da Gama, Universidade do Estado do Pará, Rodovia Augusto Montenegro km 03, 66640-000 Belém, PA, Brazil 3Faculdade de Ciências Naturais, Universidade Federal do Pará, C.P. 68800-000 Breves, PA, Brazil 4Departamento de Ciencias Físicas, Facultad de Ingeniería y Ciencias, Universidad de La Frontera, Avenida Francisco Salazar 01145, Casilla 54-D, Temuco, Chile

(Received 20 September 2020; accepted 2 January 2021; published 20 January 2021)

We describe the dynamical mass generation in pseudoquantum electrodynamics (PQED) coupled to the Gross-Neveu (GN) interaction, in (2 þ 1) dimensions, at both zero and finite temperatures. We start with a gapless model and show that, under particular conditions, a dynamically generated mass emerges. In order to do so, we use a truncated Schwinger-Dyson equation, at the large-N approximation, in the imaginary- time formalism. In the instantaneous-exchange approximation (the static regime), we obtain two critical parameters, namely, the critical number of fermions NcðTÞ and the critical coupling constant αcðTÞ as a function of temperature and of the finite cutoff Λ, which must be provided by experiments. In the dynamical regime, we find an analytical solution for the mass function Σðp; TÞ as well as a zero-external momentum solution for p ¼ 0. In the continuum theory Λ ¼ ∞, where scale-invariance is respected, it is shown that the model has a dynamically generated mass for any value of the coupling constant α. Furthermore, after calculating the effective potential for PQED, we prove that the dynamically generated mass is an energetically favorable solution in comparison to the massless phase. We compare our analytical results with numerical tests and a good agreement is found.

DOI: 10.1103/PhysRevD.103.025018

I. INTRODUCTION discussed the possibility of dynamical mass generation for massless Dirac particles, yielding a phase transition to a new In the last decades, the interest in studying two-dimen- quantum state of matter in which the chiral symmetry is sional theories has been increased since the experimental broken [18–21]. This is generated due to the electronic realization of graphene [1,2] and other related materials, interactions in the plane and it may occur even at finite such as the transition metal dichalcogenide monolayers [3]. temperatures. Because electrons in these materials obey a The main goal is to derive either new quantum phases of Dirac-like equation, therefore, some authors have also matter [4–7] or calculate renormalized parameters that considered the realization of dynamical mass generation change the electronic properties of these materials [8–13], in these two-dimensional systems [22–25]. opening possibilities for future technological applications The possibility of chiral symmetry breaking, i.e., (in particular spintronics [14], valleytronics [15], and dynamical mass generation in quantum electrodynamics electric-field tunning of energy bands [16,17]). Before this (QED) has been discussed in several Refs. [18–21,26–28] context, in the realm of high-energy physics, several works for both ð2 þ 1ÞD and ð3 þ 1ÞD cases, just to cite a few. Furthermore, this symmetry-broken phase also has been shown to occur due to four-fermion interactions (such as *[email protected] † Gross-Neveu and Thirring interactions) in Refs. [29–32].In [email protected] ‡ [email protected] this case, the fermionic field exhibits a massive phase due §[email protected] to the spontaneous symmetry breaking, described by the ∥ [email protected] so-called gap equation. This is usually calculated at the large-N expansion, where N is the number of fermion fields Published by the American Physical Society under the terms of and the gap equation is calculated at order of 1=N [29,32]. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to The dynamical mass generation is a typical effect of the author(s) and the published article’s title, journal citation, the nonperturbative regime, hence, it is common the and DOI. Funded by SCOAP3. application of the Schwinger-Dyson equations (SDEs)

2470-0010=2021=103(2)=025018(14) 025018-1 Published by the American Physical Society LUIS FERNÁNDEZ et al. PHYS. REV. D 103, 025018 (2021) for calculating the critical parameters that describe the interaction, given by the Gross-Neveu (GN) action [37]. phase transition. Therefore, in the Euclidean space-time, the action of the The SDEs are an infinite set of coupled-integral equa- model reads tions, relating all of the interacting Green functions of the μν model [19,20,33]. Fortunately, within some approximative 1 FμνF μ L ¼ pffiffiffiffiffiffiffiffi þ iψ¯ ∂ψ þ eψ¯ γ ψ Aμ scheme, it is possible to find a truncated set of equations for 2 −□ a a a a calculating the desired Green functions, in particular, the μ ν G0 2 ξ ∂ ∂ electron-self energy that provides the chiral symmetry þ − ðψ¯ ψ Þ − Aμ pffiffiffiffiffiffiffiffi Aν; ð1Þ 2 a a 2 −□ breaking. For pseudo-quantum electrodynamic (PQED) [23,34], also called reduced quantum electrodynamics where Fμν is the field intensity tensor of Aμ, which is our [35], the dynamical mass generation has been studied both ξ at finite temperatures [36] and at zero temperature with the gauge field, and is the gauge-fixing parameter. On the ψ presence of the Gross-Neveu interaction [31]. Nevertheless, other hand, a is a four-component Dirac field, with the 1 … the effect of the Gross-Neveu interaction in PQED at finite flavor index a ¼ ; ;N. The dimensionless coupling γ temperatures has not been considered until now. constant e is the electric charge, μ are the Dirac matrices In this work, we describe the dynamical mass generation in the four-rank representation, whose algebra is defined by in PQED coupled to a Gross-Neveu interaction at finite fγμ; γνg¼−2δμν. The coupling constant G0 describes the temperature. We use the Matsubara formalism in order to strength of the GN interaction and has unit of inverse of include the effect of the thermal bath into the Schwinger- mass in the natural system of units ðℏ ¼ c ¼ 1Þ. Dyson equation for the electron. This is dependent on both In the perturbative regime at zero temperature, the model the gauge-field propagator and the auxiliary-field propa- in Eq. (1) for massive fermions has been used to describe the gator, obtained after we use a Hubbard-Stratonovich trans- renormalization of the band gap for WSe2 and MoS2 [13]. formation into the four-fermion interaction. These two This result, nevertheless, requires a bare-mass term such as ¯ bosonic propagators are calculated in order of 1=N, which m0ψ aψ a which is renormalized at one-loop and provides a is consistent with our assumption of strong-coupling limit. beta function for the mass in terms of the RG scale. From this Thereafter, we use this result into the Schwinger-Dyson result, one concludes that the renormalized mass is depen- equation for the electron self-energy and calculate the full dent on the electronic density and a good agreement with electron propagator in the dominant order of 1=N in the experimental data has been found in Ref. [13]. nonperturbative limit. From this result, we conclude that a In the nonpertubative limit, the mass term is generated mass function is dynamically generated whether the num- due to interactions even if we start with m0 → 0 [31]. Here, ber of fermions is less than a temperature-dependent critical we generalize this result by including a thermal bath of parameter NcðTÞ for a finite cutoff Λ. In the continuum temperature T. theory Λ ¼ ∞, the mass generation occurs at any value of From Eq. (1), we find the gauge-field propagator, α. At large temperatures, we find that NcðTÞ ≪ 1, hence, namely, the dynamically generated mass vanishes and the system is 1 1 pμpν in the gapless phase. Δ pffiffiffiffiffi δ − 1 − 0;μνðpÞ¼ μν 2 ð2Þ This paper is organized as follow. In Sec. II. we show our 2 p2 ξ p model and perform the large-N approximation. In Sec. III. we write the truncated set of Schwinger-Dyson equations and the fermion propagator within the unquenched-rainbow approximation. In Sec. IV. we calculate the mass function in the static regime and 1 − obtain the critical parameters for the phase transition. In S0;FðpÞ¼ μ : ð3Þ γ pμ Sec. V. we use the zero-mode approximation in the dynamical regime for calculating the mass function. In Before we discuss the vertex interactions, let us apply the Sec. VI. we summarize and discuss our main results. We Hubbard-Stratonovich transformation, which converts the also include four appendixes, where we give details about four-fermion interaction into a Yukawa-type interaction by the angular integral, kernel expansion, the numerical results including an auxiliary field φ and the coupling constant for the mass function, and show that the dynamically g ¼ G0N. In this case, we replace the four-fermion inter- generated mass is the minimum of the effective potential. action by the following scheme II. PSEUDOQUANTUM ELECTRODYNAMICS 1 2 gffiffiffiffi WITH GROSS-NEVEU INTERACTION L → L þ φ − p ψ¯ aψ a ; ð4Þ GN GN 2g N We consider N fermion fields constrained to the plane, L − ψ¯ ψ 2 2 whose interaction is described by pseudoquantum electro- where GN ¼ G0ð a aÞ = . Therefore, using Eq. (4) in dynamics (PQED) [34]. Furthermore, we assume a contact Eq. (1), we find

025018-2 DYNAMICAL MASS GENERATION IN PSEUDOQUANTUM … PHYS. REV. D 103, 025018 (2021)

φ φ2 A. Auxiliary field L − ffiffiffiffi ψ¯ ψ GN ¼ p a a þ : ð5Þ N 2g In Fig. 1, we show the diagrammatic representation of the Schwinger-Dyson equation for the auxiliary field. Equation (5) must be supplemented by the motion equation Its analytical expression is given by [31] of φ, given by Δ−1 Δ−1 − Π φ ðpÞ¼ 0;φ ðpÞ; ð9Þ g ffiffiffiffi −1 φ ¼ p ψ¯ aψ a; ð6Þ Δ φ Π N where φ is the full propagator of and ðpÞ is given by the fermionic loop, hence Z which proves that the transformation does not change the 1 d3k Π p − 1S p − k Γ p; k S k ; dynamics of the model at classical level. Furthermore, the ð Þ¼ Tr 2π 3 Fð Þ ð Þ Fð Þ ð10Þ bare auxiliary-field propagator is N ð Þ

where SFðpÞ is the full fermion propagator.pffiffiffiffiffi In the lowest 1 2 Δ order of 1=N, we find that ΠðpÞ¼− p g0, with g0 ¼ 0;φ ¼ 1 ; ð7Þ =g 1=4 [21]. Therefore, the full propagator of the auxiliary field is given by and Eq. (1) reads 1 1 Δ pffiffiffiffiffi μν φðpÞ¼ −1 2 : ð11Þ 1 FμνF μ g0 ðgg0Þ þ p L ¼ pffiffiffiffiffiffiffiffi þ iψ¯ ∂ψ þ eψ¯ γ ψ Aμ 2 −□ a a a a φ φ2 ξ ∂μ∂ν − ffiffiffiffi ψ¯ ψ − ffiffiffiffiffiffiffiffi p a a þ Aμ p Aν: ð8Þ B. Gauge field N 2g 2 −□ In Fig. 2, we show the diagrammatic representation of the Schwinger-Dyson equation for the gauge-field propa- The Yukawa-type vertex functionffiffiffiffi is easily obtained from p gator [31]. Eq. (8) and is given by −1= N. On the other hand, for Its analytical expression is given by summing the self-energies in the large-N expansion for 2 PQED, we shall replace e → λ=N, where λ is taken fixed −1 −1 Δμν ðpÞ¼Δ ðpÞ − ΠμνðpÞ; ð12Þ at large N. This allow us to sum over all of the diagram- 0;μν matic contributions in order of 1=N, which is an infinite sum, unlike the standardpffiffiffiffiffiffiffiffiffi perturbation in e. Therefore, the PQED vertex reads λ=Nγμ, describing the electromagnetic interaction.

III. TRUNCATED SCHWINGER-DYSON FIG. 1. The Schwinger-Dyson equation for the auxiliary field. EQUATION AT FINITE TEMPERATURES The left-hand side is the inverse of the full propagator of the In this section we present the Schwinger-Dyson equa- auxiliary field, the first term in the right-hand side corresponds to Π tions, obtained from Eq. (8), that describes the quantum the bare auxiliary-field propagator, and the second term ðpÞ is Γ → 1 corrections for the two-point functions. In principle, this is the quantum correction. The function ½p; k corresponds to the approximated vertex. a very complicated set of integral equations for all of the full Green functions of the model. From now on, we assume the ladder approximation, also called rainbow approach [38], which consists of neglecting quantum corrections to the vertex functions. It is worthwhile to note that this may be corrected by the Ball-Chiu vertex [39] when one wishes to preserve the Ward-Takahashi identity. Because we are interested in dynamical mass generation FIG. 2. The Schwinger-Dyson equation for the gauge-field for the electrons, we need to find a closed solution for propagator. The left-hand side is the inverse of the full propagator both gauge-field and auxiliary-field propagators. These of the gauge field, the first in the right-hand side corresponds to shall be given by the large-N approximation, thus the the bare gauge-field propagator, and the second term ΠμνðpÞ is bosonic-field propagators are calculated in the unquenched the exact vacuum polarization tensor. The function Γμ½p; k → γμ approximation [40]. corresponds to the approximated vertex.

025018-3 LUIS FERNÁNDEZ et al. PHYS. REV. D 103, 025018 (2021) Z Δ−1 Π 1 3 where μν is the full propagator and μν is the polarization g d k Ξ ðpÞ¼ 1S ðkÞΓðk; pÞΔφðp − kÞ tensor due to the electromagnetic interaction. This is N ð2πÞ3 F given by Z 1 d3k þ − 1Tr½S ðkÞΓðk; q ¼ 0ÞΔφðq ¼ 0Þ: Z 3 2π 3 F λ d k N ð Þ ΠμνðpÞ¼− Tr γμS ðpþkÞΓνðp;kÞS ðkÞ: ð13Þ N ð2πÞ3 F F ð19Þ

Similarly to the previouspffiffiffiffiffi case, in the lowest order of 1=N, In order to find the full fermion propagator, it is 2 2 we find Πμν ¼ λ p =8Pμν, where Pμν ¼ δμν − pμpν=p . convenient to decompose this Green function into its Therefore, the full propagator reads irreducible parts. Hence, we adopt the following ansatz

−1 μ α αβ −1 S ðpÞ¼−γ pμAðpÞþΣðpÞ; ð20Þ Δμν ¼ Δ0;μα½δν − Π Δ0;βν : ð14Þ F

Using Eqs. (2) and (13) in Eq. (14), we find where AðpÞ yields the wave function renormalization and ΣðpÞ is called mass function. Note that ΣðpÞ ≠ 0 implies a dynamical mass generation [21]. In order to obtain the mass pffiffiffiffiffiPμν ΔμνðpÞ¼ ð15Þ function, we take the trace operation in both sides of p2 2 λ ð þ 8Þ Eq. (16). Thereafter, we substitute Eqs. (3), (18), (19), and (20), within the rainbow approximation, Γνðp; kÞ → γν and ξ ∞ in the Landau gauge, i.e., with ¼ . Γðp; kÞ → 1, we obtain Z C. Matter field λ d3k δμνΣðkÞ ΣðpÞ¼ Δμνðp − kÞ In Fig. 3, we show the diagrammatic representation of N ð2πÞ3 k2A2ðkÞþΣ2ðkÞ Z the Schwinger-Dyson equation for the fermions field [31]. 1 d3k ΣðkÞ Its analytical expression is given by þ Δφðp − kÞ N ð2πÞ3 k2A2ðkÞþΣ2ðkÞ Z −1 −1 3 Σ S ðpÞ¼S ðpÞ − ΞðpÞ; ð16Þ d k ðkÞ F 0;F − Δφð0Þ : ð21Þ ð2πÞ3 k2A2ðkÞþΣ2ðkÞ where SF is the full fermion propagator and Ξ is the electron self-energy, given by On the other hand, for calculating the wave function renormalization, we multiply Eq. (16) by p and, after ΞðpÞ¼ΞλðpÞþΞgðpÞ; ð17Þ calculating the trace operation, we obtain Z λ λ 3 where Ξ is the PQED contribution and Ξg is the GN 1 − d k δβμδαν − δβαδμν AðpÞ¼ 2 3 ð contribution, namely, Np ð2πÞ

Z βν μα pβkαAðkÞ λ 3 þδ δ Þ Δμνðp − kÞ Ξλ d k γμ Γν ; Δ − k2A2ðkÞþΣ2ðkÞ ðpÞ¼ 3 SFðkÞ ðk pÞ μνðp kÞ; ð18Þ N ð2πÞ Z 3 βα 1 d k δ pβkαAðkÞ þ Δφðp − kÞ: ð22Þ Np2 ð2πÞ3 k2A2ðkÞþΣ2ðkÞ

It is clear that Eqs. (21) and (22) are a coupled set of equations for ΣðpÞ and AðpÞ. However, in the dominant order 1=N, we may replace AðpÞ → 1 in Eq. (21). This is also in agreement with the rainbow approximation. Next, we use the Matsubara formalism for introducing temperature effects into Eq. (21). In the imaginary-time formalism, the main step is to change the time-component integrals into a sum over Matsubara frequencies [41], i.e., FIG. 3. The Schwinger-Dyson equation for the electron. The we must perform left-hand side is the inverse of the full propagator of the fermion Z field, the first term in the right-hand side corresponds to the bare 1 X∞ dk0 k β → k β Dirac propagator, and the other terms in the right-hand side are 2π fðk0; ; Þ β fðn; ; Þ; ð23Þ the electron self-energy ΞðpÞ. n¼−∞

025018-4 DYNAMICAL MASS GENERATION IN PSEUDOQUANTUM … PHYS. REV. D 103, 025018 (2021) with X∞ 1 1 U ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð29Þ 2 1 2 2 2 2 n¼−∞ ð n þ Þ þ A B þ n þ C kμ ¼ðk0; kÞ → ðωn; kÞ; ð24Þ 2 2 0 and where we conclude that ðA ;C Þ > and, because we consider g>0 [37,42], we also find that B>0. 8 Next, we apply the Cauchy integral test [43], where the < ð2nþ1Þπ β for fermions; sum is written as ωn ¼ : 2nπ for bosons; β X∞ ðþÞ U ¼ 2 un þ u0; ð30Þ −1 where ωn are the Matsubara frequencies, T ¼ β is the n¼1 temperature of the thermal bath, and n is the vibration mode ðþÞ related to each momentum component. Note that we are with un ¼ðun þ u−nÞ=2. The test imposes that whether considering the Boltzmann constant as kB ¼ 1, in the uðxÞ¼ux is positive, continuous, and decreasing in the natural system of units. After including the thermal bath interval ½1; ∞, hence, the integral over uðxÞ is finite and, in Eq. (21),wehave therefore, convergence is derived. This is exactly our case Z and one may easily conclude that X Σ k Σ p nð ;TÞ mð ;TÞ¼C1 2 2 2 2 2 ð2n þ 1Þ π T þ k þ Σnðk;TÞ uðxÞ > 0; ∀ xjx ∈ ½1; ∞; ð31Þ 1 × pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ðn − mÞ2π2T2 þðp − kÞ2 u0ðxÞ < 0; ð32Þ XZ Σnðk;TÞ C2 þ 2 1 2π2 2 k2 Σ2 k limuðxÞ¼uðaÞ; ∀ aja ∈ ½1; ∞: ð33Þ ð n þ Þ T þ þ nð ;TÞ x→a 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi × −1 2 2 2 2 ; Therefore, Eq. (30) is a convergent series for all n. ðgg0Þ þ 4ðn − mÞ π T þðp − kÞ ð25Þ IV. STATIC REGIME p0 =0 where The static regime, also called instantaneous-exchange Z ∞ Z approximation [44], consists of neglecting all of the time- X X d2k → T ; components of the bosonic-field propagator in Eq. (21), 2π 2 ð26Þ n¼−∞ ð Þ with the consideration that the interaction vertex being only 0 2λ γ in Eq. (18). This is usually realistic in cases where the C1 ; 27 electron velocity is much less than the light velocity. In ¼ 2 λ ð Þ ð þ 8ÞN these systems, it is possible to show that charged particles and interacts through the Coulomb potential. This is given by the Fourier transform of the gauge-field propagator in 1 Eq. (2) after we perform pμ ¼ðp0; pÞ → ð0; pÞ, where C2 ¼ : ð28Þ 0 p ≡ 0 g0N we will use the following notation ð ; Þ ð ;PÞ. Next, let us consider both zero and finite temperature cases for Σ p Σ p From now on, we assume that mð ;TÞ¼ ð ;TÞ and, calculating ΣðP; TÞ. therefore, the mass function only depends on the dominant vibrational mode m ¼ 0. The standard procedure for calculating the mass function is to convert the integral A. Zero temperature case equation into a differential equation for Σðp;TÞ. It turns out After considering the static regime with T → 0 in that there exist two main operations, namely, the sum over Eq. (25), we find n and the angular integral that must be performed before Z finding this result. We shall explore some approximations 3 Σ Σ C1 d k ðKÞ Δ − for these operations. ðPÞ¼ 3 2 2 2 ðP KÞ 2 ð2πÞ k0 þ K þ Σ ðKÞ Before performing further approximations, let us show Z 3 Σ that the sum over the vibrational modes n is convergent. d k ðKÞ Δ − þ C2 3 2 2 2 φðP KÞ; ð34Þ The convergence of this sum for PQED at finite temper- ð2πÞ k0 þ K þ Σ ðKÞ ature has been shown in [36]. In Eq. (25), the sum over n 3 2 may be written as where d k ¼ dk0d K,

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1 Δ − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Equation (42) is also supplemented by two equations, φðP KÞ¼ −1 2 ; ð35Þ ðgg0Þ þ ðP − KÞ namely, limP2Σ0ðPÞ¼0; ð44Þ and P→0 1 ΔðP − KÞ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð36Þ which provides the infrared behavior (IR) and ðP − KÞ2 lim ½PΣ0ðPÞþΣðPÞ ¼ 0; ð45Þ P→Λ The integral over k0 in Eq. (34) is easily solved and the angular integral may be solved by using the identity which shows that the mass function is expected to vanish at Z large P, which is our ultraviolet (UV) condition. 2π pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4K x0 2 2 θΔ − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið Þ −2 Using the approximation P= P þΣ ðPÞ≈1 in Eq. (42), d φðP KÞ¼ 2 þ O½ðgg0Þ ; ð37Þ 0 ðP − KÞ which holds for large-external momentum, we find the analytical solution where Kðx0Þ is the complete elliptic integral of the first 2 γ γ kind with x0 ≡ −4PK=ðP − KÞ . Note that the angular ΣðPÞ¼A1P þ A2P ; ð46Þ integral for the gauge-field term is very similar and that we pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −2 will neglect the terms in order of O½ðgg0Þ . Replacing where γ ¼ −1=2 þ i=2 Nc=N − 1, and ðA1;A2Þ are arbi- Eq. (37) in Eq. (34), we find trary constants. Z The analytical solution in Eq. (46) must be in agreement Λ Σ K Σ 4 C1 dK KpðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKÞ ðK;PÞ with both IR and UV conditions in Eq. (44) and Eq. (45), ðPÞ¼ þ C2 2 2 2 ; ð38Þ 2 0 ð2πÞ 2 K þ Σ ðKÞ respectively. This is a standard step in order to calculate the critical behavior of ΣðPÞ. First, we may consider the limit where the kernel KðK;PÞ reads where Λ ¼ ∞, which represents the continuum theory. Thereafter, it is easy to show that ΣðPÞ obeys both IR and Kðx0Þ UV conditions for any value of N. Therefore, because Nc in KðK;PÞ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð39Þ λ 4πα ðP − KÞ2 Eq. (43) is dependent on ¼ N, we also conclude that the dynamical mass generation occurs for any value of α λ Next, we approximate the kernel for its lowest order (or ). This continuum limit has been discussed for PQED terms, namely, (in the dynamical limit) in Ref. [23] and for QED4 it has been reviewed in Ref. [19], where the same conclusion is π π derived. KðK;PÞ ≈ ΘðP − KÞþ ΘðK − PÞ: ð40Þ 2P 2K Next, we assume that the system has a finite cutoff Λ ∝ 1=a, where a ≈ 1 Å is the lattice parameter. This is Note that ΘðP − KÞ is the standard step function. Using motivated by an effective description of electrons in Eq. (40) in Eq. (38),wehave condensed matter physics [2–4,7,12]. In this case, the IR Z condition in Eq. (44) is also respected for any value of N. ðC1=2 þ C2Þ P ΣðKÞ Σ P KdK pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi On the other hand, the analysis of the UV condition in ð Þ¼ 4π 2 2 0 P K þ Σ ðKÞ Eq. (45) is more subtle. Indeed, for the sakep offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi simplicity, Z Λ Σ let us define γ ¼ −1=2 þ iR=2 with R ≡ Nc=N − 1 in pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðKÞ þ dK 2 2 : ð41Þ Eq. (46). Clearly, we have two cases, i.e., (i) where R is P K þ Σ ðKÞ imaginary and N>Nc and (ii) where R is real and N

025018-6 DYNAMICAL MASS GENERATION IN PSEUDOQUANTUM … PHYS. REV. D 103, 025018 (2021) Z this massive parameter obeys a Miransky-scaling law, Λ Σ K Kβ K;P Σ 4 C1 K pð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÞ ð Þ ðP; TÞ¼ þ C2 dK 2 ; namely, 2 0 ð2πÞ 2 K2 þ Σ2ðKÞ

2þR −2nπ=R ð52Þ MΛ ¼ Λe e ; ð48Þ

where the temperature-dependent kernel KβðK; PÞ reads where n is an integer. The typical values of MΛ ensures that ΣðPÞ obeys the UV condition and, therefore, we find a Kðx0Þ massive phase only for N α N

Σ γðTÞ γðTÞ B. Finite temperature ðP; TÞ¼A1ðTÞP þ A2ðTÞP ; ð57Þ

In order to describe temperature effects, we include the where the arbitrary constants ðA1;A2Þ may be dependent on Matsubara frequencies in Eq. (34). Thereafter, we solve the the temperature. Because both the IR/UV conditions and angular integral using Eq. (37) while for the Matsubara sum the momentum dependence of the solution ΣðP; TÞ are the we use the identity same as in the zero-temperature case, it follows the same critical behavior discussed before. In particular, for finite Λ, X 1 1 1 − 2 ϵ the dynamical mass generation only occurs for N

Λ=T Λ 1 4½4ð1 − e ÞþNπð1 þ eT Þ ϵ α ðTÞ¼− Λ : ð58Þ nFð KÞ¼ βϵ ð51Þ c Λ=T e K þ 1 Nπ½12ð1 − e ÞþNπð1 þ eT Þ is the Fermi-Dirac distribution. After using these proper- Note that for T → 0, Eq. (58) yields Eq. (49), as expected. ties, we find In terms of α, the dynamical mass generation only occurs

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After a simple changing of variables, the angular integral of the PQED sector (see details in Appendix B) reads

4 IPQED ; n ðp; k TÞ¼ n K½xn; ð59Þ wp;k

where, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 4 2π2 2 − 2 wp;k ¼ n T þðp kÞ ; ð60Þ

and FIG. 4. The critical parameters for the dynamical mass generation. We plot both Eq. (56) (the common line) and Eq. (58) Λ 10 4pk (the inset) with ¼ (units of energy). For the common line we x ¼ − : ð61Þ also use λ ¼ 8.0, while for the inset we use N ¼ 2. Note that for n 4n2π2T2 þðp − kÞ2 small enough temperatures, the critical coupling constants remain almost unchanged. Similarly, for the GN sector, we find when α > α ðTÞ and N

V. DYNAMICAL REGIME p0 ≠ 0 where KðxnÞ and Πðyn;xnÞ are the elliptic integral of the In this section we recover the retardation effects by first and third kind, respectively. In particular, note that for ≠ 0 0 assuming p0 in Eq. (21).ForT ¼ , this regime has g ¼ 0, we have that Πð0;xnÞ¼KðxnÞ, hence, the kernel of been discussed in several contexts. the GN sector vanishes, as expected. The integral representation for these functions are Z π=2 1 θ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi KðxnÞ¼ d 2 ð64Þ 0 1 − xn sin ðθÞ

and Z π=2 1 Π θ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðyn;xnÞ¼ d 2 2 : ð65Þ 0 ½1 − yn sin ðθÞ 1 −xn sin ðθÞ

The resulting kernel, after using the angular integral, is a very complicated function of n. This preclude us of solving the whole Matsubara sum, as we have did in the Σ FIG. 5. The analytical and numerical solutions for ðP; TÞ. The static regime. Therefore, we shall investigate the lowest dashed line is the numerical solution of the integral equation order contribution, provided by the zero-mode n ¼ 0 given by Eq. (52) with N ¼ 2. The common line is the analytical approximation. solution given by Eq. (57) with N ¼ 2 and A1ðTÞ¼A2ðTÞ ¼ 0.06 þ 0.13i. We have used Λ ¼ 10 (units of energy), λ ¼ 8.0, and T ¼ 0.1 (units of Λ) for all of the curves. We include an inset VI. ZERO-MODE APPROXIMATION in order to facilitate the visualization of the numerical solution. Note that the analytical solution is in good agreement with the In this section we consider the fundamental vibrational numerical results. mode n ¼ 0 in Eq. (25). Using Eqs. (64) and (65), we find

025018-8 DYNAMICAL MASS GENERATION IN PSEUDOQUANTUM … PHYS. REV. D 103, 025018 (2021) Z ∞ Σ Σ C1 kKðx0Þ T ðk; TÞ A. Large-external-momentum approximation ðp; TÞ¼ 2 dk 2 2 2 2 π 0 kp − kk π T þ k þ Σ ðk; TÞ Z In this section we investigate the linearized version of ∞ ≫ Σ C2 k TΣðk; TÞ Eq. (69) by assuming p T; ðpÞ. In this case, the þ 2 dk 2 2 2 2 behavior of the generated mass is given by π 0 kp − kk π T þ k þ Σ ðk; TÞ Π − ðy0;x0Þ 3Σ00 2 2 Σ0 Σ 0 × Kðx0Þ 2 2 2 ð66Þ p ðpÞþ½ p þ pb ðpÞþa ðpÞ¼ ; ð72Þ 1 − g g0ðp − kÞ which yields For calculating an analytical solution of Eq. (66),we F1 a b write the elliptic integral as the hypergeometrics Appell . b1F1½1 − ; 2; aþb b p 20 b b Because F1 has an expansion in its parameters, we may find ΣðpÞ¼A1 þ A2G − ; ð73Þ 12 0 1 a simplified version of the kernel (see Appendix C for more p p ; details). Hence, Eq. (66) yields where 1F1 is the confluent hypergeometric function and G Z Λ is the Meijer G-function [46]. The sign of this Meijer C1T Θðp − kÞ Θðk − pÞ ΣðpÞ¼ dkfðkÞ þ G-function changes for different values of the external 2π 0 p k Z momentum. Because we have not observed such behavior Λ C2T Θðp − kÞ Θðk −pÞk in the numerical results of Eq. (66), from now on, we shall þ dkfðkÞ þ 2 1 ; ð67Þ 0 2π 0 p k − 2 take A2 ¼ for the sake of agreement with the integral ðgg0Þ equation. In Fig. 6, we plot our analytical solution in Eq. (73) and compare this with the numerical results of the where fðkÞ ≡ kΣðkÞ=½ðπTÞ2 þ k2 þ Σ2ðkÞ. By taking integral equation in Eq. (66). derivatives in respect to p in both sides of Eq. (67), we find Z bΣ p a p B. Zero-external-momentum approximation Σ0 ð Þ − ðpÞþ 2 2 2 ¼ 2 dkfðkÞ; ð68Þ ðπTÞ þ p þ Σ ðpÞ p 0 In this section, we consider the limit Σðp ¼ 0;TÞ¼ mðTÞ in Eq. (25), using the zero-frequency mode. In this 2π with a ¼ðC1 þ C2ÞT=2π and b ¼ C2T=2π, both constants case, the angular integral provides a constant factor of . with dimension of mass. Next, we neglect the nonlinear Hence, we find terms by using ðπTÞ2 þ p2 þ Σ2ðpÞ ≈ ðπTÞ2 þ p2, which Z 1 Λ 1 is expected to be reasonable at large p [36,45]. After 1 ¼ dk 2 2 2 2 4π 0 π T þ k þ m ðTÞ deriving Eq. (68) in respect to p, we obtain the differential equation C2Tk × C1T þ −1 : ð74Þ ðgg0Þ þ k p3b p3Σ00ðpÞþ 2p2 þ Σ0ðpÞ ðπTÞ2 þ p2 2p2b 2p4b þ − þ a ΣðpÞ¼0; ð69Þ ðπTÞ2 þ p2 ððπTÞ2 þ p2Þ2 which is supplemented by two conditions for the infrared and ultraviolet regimes. These are given by

limΣðpÞ¼0 ð70Þ p→Λ and Σ Σ FIG. 6. The analytical and numerical solutions for Σ p; T in 2 d ðpÞ b ðpÞ 0 ð Þ limp þ 2 2 ¼ : ð71Þ the zero-mode approximation. The dashed line is the numerical p→0 dp ðπTÞ þ p solution of the integral equation given by Eq. (66) with N ¼ 2. The common line is the analytical solution given by Eq. (73) with Unfortunately, Eq. (69) has not an analytical solution for N ¼ 2 and A1ðTÞ¼0.6. We have used Λ ¼ 10 (units of energy), arbitrary values of the set ðp; TÞ of variables. Nevertheless, λ ¼ 3.0, and T ¼ 0.1 (units of Λ) for all of the curves. Note that we may consider a possible linearized version for large the analytical solution is in good agreement with the numerical external momentum. results only at large-p limit.

025018-9 LUIS FERNÁNDEZ et al. PHYS. REV. D 103, 025018 (2021)

After calculating the integral over k, we have We have shown that a dynamical mass is generated and, therefore, it is expected to observe a massive phase. arctan pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΛ Nevertheless, one also needs to prove that this is indeed a C1T π2T2þm2ðTÞ 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi favorable phase in comparison to the symmetric phase. It 2π m2ðTÞþπ2T2 turns out that PQED has the same vertex interaction as in pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ 8 QED3, namely, eγ , which implies the same Schwinger- < π2T2 þ m2ðTÞ arctan pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΛ C2T π2T2þm2ðTÞ Dyson equations, except by the different gauge-field þ 2π : π2T2 m2 T 1 2 propagator. Therefore, we can safely follow the same steps ½ þ ð Þþðgg0Þ in Ref. [47] for QED3 and conclude that the nontrivial ) π2T2þm2ðTÞþΛ2 solution is the minimum of the effective potential (See gg0 ln½ 2 2 2 2 ðπ T þm ðTÞÞð1þgg0ΛÞ Appendix D for more details). þ 2 2 2 2 : ð75Þ 2½ðgg0Þ ðπ T þ m ðTÞÞ þ 1 VII. SUMMARY AND OUTLOOK Equation (75) is our gap equation for the dynamical mass as a function of temperature and the coupling constants. In this work, we investigate the influence of temperature Similarly to the BCS model, we define the critical temper- on the dynamical mass generation for fermions in PQED ature as the point T ¼ T in which m vanishes. Here, with a Gross-Neveu interaction. In order to do so, we used c the Schwinger-Dyson equation for the electron in the nevertheless, we define that C1T and C2T are fixed in order 1 to be in agreement with the fact that m ≠ 0 for T ¼ 0. dominant order =N, neglecting quantum corrections to In Fig. 7, we plot the numerical solutions of mðTÞ for the vertices at finite temperature. In the static regime, we both g 0 and g 1.0, which proves that the GN were able to solve the whole Matsubara sum. Thereafter, ¼ ¼ α interaction increases the gapped phase. Furthermore, in we calculated the critical coupling constant c and the Fig. 8, we show that the GN interaction increases the critical numbers of fermions Nc as a function of T and the Λ critical temperature, as expected. cutoff . From these results, we concluded that the system goes to a gapless phase whether we increase the ratio T=Λ. In the continuum theory Λ ¼ ∞, where scale-invariance is respected, the dynamical mass generation occurs for any value of α [23]. Furthermore, we obtained an analytical solution for the mass function ΣðpÞ, which is in good agreement with our numerical results. In the dynamical regime, we used the zero-mode approximation in the sum over the Matsubara frequencies, which allowed us to calculate the analytical solution ΣðpÞ. This function agrees with our numerical results for large- external momentum, while for small-external momentum FIG. 7. The numerical result of the mass function given by some deviations have been found. Moreover, we found a Eq. (75). We use Λ ¼ 10, N ¼ 2 and λ ¼ 3 in the blue points with gap equation that provides the value of the mass function g ¼ 0, and in the square point g ¼ 1. Where, we see the augment in the zero-external momentum approximation, i.e., of the critical temperature due at the GN-coupling. Σðp ¼ 0;TÞ → mðTÞ. In this case, the numerical tests show that the critical temperature increases as we increase the strength of the fermionic self-interaction. Several generalizations of this work may be performed. For instance, one would investigate the main effects of vertices quantum corrections into the critical parameters for the mass generation. In particular, it would be relevant to find a way to increase the gapped phase for finite temperature. In this phase, the competition between α and g yields a tunable mass which may be relevant for applications in two-dimensional materials [13].

ACKNOWLEDGMENTS L. F. is partially supported by Coordenação de FIG. 8. Behavior of critical temperature for different values of Aperfeiçoamento de Pessoal de Nível Superior Brasil the GN coupling constant given by Eq. (75) numerically treated. (CAPES), finance code 001. V.S. A. andL. O. N. are partially Where we use Λ ¼ 10, N ¼ 2 and λ ¼ 3. supported by Conselho Nacional de Desenvolvimento

025018-10 DYNAMICAL MASS GENERATION IN PSEUDOQUANTUM … PHYS. REV. D 103, 025018 (2021) Z Científico e Tecnológico (CNPq) and by CAPES/NUFFIC, X∞ ∞ Σ Σ dk k ðkÞ finance code 0112. F. P. acknowledge the financial support ðpÞ¼C1T 2 2 2 2 2 2 −∞ 0 ð2πÞ ð2n þ 1Þ π T þ k þ Σ ðkÞ from Dirección De Investigación De La Universidad De La Z n¼ 2π 1 Frontera Grant No. DI20-0005 of the Dirección de × dθ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Investigación y Desarrollo, Universidad de La Frontera. 0 4n2π2T2 þ p2 þ k2 − 2kpcosðθÞ X∞ Z ∞ dk kΣðkÞ þ C2T 2 2 2 2 2 2 0 ð2πÞ ð2n þ 1Þ π T þ k þ Σ ðkÞ APPENDIX A: THE NUMERICAL RESULTS Z n¼−∞ 2π 1 In this Appendix we briefly review the main steps in θ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi × d 1 2 2 2 2 2 : 0 4n π T p k − 2kpcos θ order to find our numerical results. For the sake of gg0 þ þ þ ð Þ simplicity, note that Eq. (25) is a kind of Fredholm integral ðB1Þ equation of first kind, defined by Z b After we use the identity cos θ 1–2 sin2 θ=2 and a ZðxÞ¼ dyKðx; y; ZðyÞÞZðyÞ; ða ≤ x ≤ bÞðA1Þ ð Þ¼ ð Þ a change of variable (θ → 2θ) in the angle, it is possible to write the PQED angular integral as where Kðx; y; ZðyÞÞ is the kernel of the integral equation and ZðxÞ is an unknown function. Throughout this work we Z 4 π=2 1 4 convert this into a differential equation, allowing us to find IPQED dθpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K x ; B2 n ¼ ωn 2 ¼ ωn ð nÞ ð Þ analytical solutions for each approximation, namely, p;k 0 1−xn sin ðθÞ p;k Eq. (57) in the static approximation and Eq. (73) in the zero-mode approximation. These solutions, nevertheless, where ωn and x are given in Eqs. (60) and (61), are not expected to well describe the behavior of the mass p;k n function for small external momentum. respectively. The angular integral corresponds to an ellip- For properly describing the behavior of ΣðpÞ for any p, tical integral of the first kind, whereas the angular integral we may find numerical results of both Eq. (52) and of the GN sector reads Eq. (67). It is easy to conclude that these equations are Z essentially a kind of Fredholm integral equation whether 2π 1 IGN θ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi we adjust the parameters, variables, and kernel. Next, let us n ¼ d 1 : ðB3Þ 0 þ ωn 1 − x sin2ðθÞ summarize our main steps. First, we use the trapezoidal gg0 p;k n quadrature rule for calculating the integral over the kernel, hence, After some algebra, it is possible to write Eq. (B3) as Z the sum of three integrals, using the cosine double b θ 1–2 2 θ 2 IGN dyKðx; y; ZÞZðyÞ angle identity (cosð Þ¼ sin ð = Þ), therefore, n is a given by −1 h MX → ½Kðyi;x;ZiÞZi þ Kðyi 1;x;Zi 1ÞZi 1; ðA2Þ Z 2 þ þ þ 2 π 1 i¼1 I1 GN θ ð Þn ¼ d 2 ; gg0y 0 1 − y sin θ nZ n ð Þ where h is the size of each interval and yi is the iterative 2 π 1 I2 GN θ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi variable (plays the role of loop momentum k). Setting M ¼ ð Þn ¼ n d 2 ; ω 0 1 − x sin ðθÞ 300 as the numbers intervals, hence, h ¼ða − bÞ=ðM − 1Þ, p;k Z n −3 π with a ¼ 10 (plays the role of the Λ cutoff), and b ¼ 10 3 2 1 I GN − dθ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð Þn ¼ ωn 2 2 in order to avoid divergences. Furthermore, note that p;k 0 ð1 − y sin ðθÞÞ 1 − x sin ðθÞ −3 n n 10 ≤ ðx; yiÞ ≤ 10. Using Eq. (A2) in Eq. (A1), we obtain a set of algebraic equations for all of Zi from x ¼ b to I 1 GN x ¼ a. Finally, after solving this set of equations, we find where the integral ð Þn vanishes and, after properly I2 GN I 3 GN Figs. 5 and 6. writing the integrals ð Þn and ð Þn , we obtain 4 Πðy ;x Þ APPENDIX B: ANGULAR INTEGRAL IGN ¼ Kðx Þ − n n ; ðB4Þ n ωn n 1 − 2 2 ωn 2 p;k g g0ð Þ The mass function given by Eq. (25) has two main p;k contributions, generated by PQED and GN interactions. In the polar system of coordinates, it yields with yn being defined in Eq. (63).

025018-11 LUIS FERNÁNDEZ et al. PHYS. REV. D 103, 025018 (2021) Z ∞ Σ Finally, using these results, we find the mass function, Σ C1T k ðkÞ ðpÞ¼ 2 dk 2 2 2 2 namely, ð2πÞ 0 π T þ k þ Σ ðkÞ h 2π 2πi Z × Θðp − kÞ þ Θðk − pÞ X∞ ∞ Σ p k Σ C1T 4KðxnÞ k ðkÞ Z ðpÞ¼ 2 dk n 2 2 2 2 2 ∞ Σ 2π 0 ω 2n 1 π T k Σ k C2T k ðkÞ ð Þ n¼−∞ p;k ð þ Þ þ þ ð Þ dk Z þ 2π 2 π2 2 2 Σ2 X∞ ∞ ð Þ 0 T þ k þ ðkÞ C2T 4 Πðy ;x Þ − n n 2 þ 2 dk n KðxnÞ 2 2 n 2 2π 2πðgg0Þ k 2π 0 ω 1− ω Θ − Θ − ð Þ n¼−∞ p;k g g0ð p;kÞ × ðp kÞ þ ðk pÞ 2 2 ; ðC6Þ p ½ðgg0Þ k − 1 kΣðkÞ × : ðB5Þ ð2nþ1Þ2π2T2 þk2 þΣ2ðkÞ which is Eq. (67).

APPENDIX D: THE EFFECTIVE POTENTIAL In this Appendix we show that the dynamically gen- APPENDIX C: THE KERNEL EXPANSION erated mass is energetically favorable in comparison to the The complete elliptic integral of the first and third kinds symmetric phase (massless fermions). may be written in terms of hypergeometric function of two In general grounds, one must use the solution of the variables, for jxnj < 1 and jynj < 1 [46], we have Schwinger-Dyson equation into the Conrwall-Jackiw- Tomboulis (CJT) potential [48] and find its minimum. This potential reads V ∝−ln Z, where Z is the partition π 1 1 CJT function, and its quantum dynamics is given by a sum of KðxnÞ¼ F1 ; ; 1; 1;xn; 0 ; ðC1Þ 2 2 2 2-particle-irreducible (2PI) graphs, which depends on all of the full propagators and vertex interactions. Therefore, we may not calculate VCJT for PQED, unless we assume a set π 1 1 of approximations. For QED3, within the large-N limit, this Π ; 1; 1 ðyn;xnÞ¼2 F1 2 2 ; ;xn;yn : ðC2Þ has been calculated in Ref. [47], where it is shown that a dynamically generated mass m ≠ 0 is the minimum of the CJT potential. Interesting, one could follow similar steps The series of the Hypergeometric Appell F1 is given by for PQED and find the same conclusion, because the only difference would be in the amplitude of the gauge-field propagator and this only changes the value of m (and the X∞ ðαÞ ðβÞ ðβ0Þ 0 lþq l q l q F1ðα; β; β ; γ; x ;y Þ¼ x y : ðC3Þ critical parameters). n n γ l!q! n n l;q¼0 ð Þlþq Next, we apply the bilocal-field representation proposed by Cahill-Roberts in Ref. [49] in order to provide an extra derivation of such result. For the sake of simplicity, we The standard procedure in the treatment of the SDEs is to consider the rainbow-quenched PQED, which consists of consider two regions (infrared and ultraviolet) in the neglecting quantum corrections to both the vertex inter- external momentum. After considering the zero-mode actions and the gauge-field propagator in Eq. (1). After frequency n ¼ 0, we find using the bilocal-field representation for PQED, we find R 3 − d ðx−yÞΓ ½Bμðx;yÞ −Ω 1 1 ≈ N CR ≈ N VCJT F1ð ; ;1;1;x0;0Þ 2π 2π Z e e ; ðD1Þ 2π 2 2 ¼ Θðp − kÞ þ Θðk − pÞ ðC4Þ jp −kj p k where Bμðx; yÞ is a bilocal-field satisfying the auxiliary 2 condition hBμðx; yÞi ¼ −e hΔðx − yÞψ¯ ðxÞγμψðyÞi, N is a and normalization constant, and Ω is the space-time volume. Furthermore, the Cahill-Roberts effective action reads 2π 1 1 1 ; 1 1; 1 μ F1ð2 2 ; ;x0;y0Þ B ðx; yÞBμðy; xÞ F1 ; ; 1; 1;x0; 0 − Γ − ∂ − γμ − 2 2 1 − 2 2 ωn 2 CR½Bμ¼ 2 Tr ln½i Bμ; ðD2Þ jp kj g g0ð p;kÞ e Δðx − yÞ 2π 2π 2 Θ − Θ − ðgg0Þ k ¼ ðp kÞ þ ðk pÞ 2 2 : ðC5Þ where Δðx − yÞ is the amplitude of the gauge-field propa- p ½ðgg0Þ k − 1 gator, obtained from Δ0;μνðx−yÞ¼Δðx−yÞPμν in Eq. (2). μ Note that γ Bμ ¼ Ξðx; yÞ is the electron self-energy, as Therefore, the mass function reads defined in Eq. (16). From Eq. (D1), we conclude that the

025018-12 DYNAMICAL MASS GENERATION IN PSEUDOQUANTUM … PHYS. REV. D 103, 025018 (2021) vacuum of the theory is given by the minimum of VCJT. dynamically generated mass. Using this Schwinger-Dyson Furthermore, note that Eq. (D2) is one term of the effective equation, for eliminating Δðx − yÞ in Eq. (D2), we find action derived in Ref. [50] for QED3 (see Appendix B). Z From the Fourier transform of the motion equation 1 ∞ 2 2 2 m − 1 m δΓ δ 0 VCJT½m¼ 2 k dk 2 2 ln þ 2 : ðD3Þ CR½Bμ= Bμ ¼ , we find the standard Schwinger- π 0 k þ m k Dyson equation for the electron self-energy ΞðpÞ. Next, we consider a constant-field configuration within the It turns out that x=ð1 þ xÞ − lnð1 þ xÞ is a negative value rainbow-quenched approximation, i.e., we decompose for any positive x ≡ m2=k2, hence, the dynamical mass μ the electron self-energy such that γ BμðpÞ¼ΞðpÞ¼ generation m ≠ 0 is energetically favorable to m ¼ 0,as ¯ μ ¯ ð1−AÞγ pμ þΣðpÞ≈m, where we have used that A ¼ 1 expected. It is worth to mention that Eq. (D3) also holds for (in order to neglect the renormalization of the electron QED3 (with a different m), as it has been shown in Eq. (4.9) wave function) and ΣðpÞ ≈ Σð0Þ¼m, which represents the of Ref. [47].

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