Two-Dimensional Yukawa Interactions from Nonlocal Proca Quantum Electrodynamics

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PHYSICAL REVIEW D 97, 096003 (2018)

Two-dimensional Yukawa interactions from nonlocal Proca quantum electrodynamics

Van Sergio ´ Alves,1 Tommaso Macrì,2,3,4 Gabriel C. Magalhães,1 E. C. Marino,5 and Leandro O. Nascimento3,6 1Faculdade de Física, Universidade Federal do Pará, Av. Augusto Correa 01, Bel´em, PA, 66075-110, Brazil 2Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, Natal-RN, Brazil 3International Institute of Physics, Campus Universitário Lagoa Nova, C.P. 1613, Natal RN, 59078-970, Brazil 4Dipartimento di Scienze Fisiche e Chimiche, Universit`a dell’Aquila, via Vetoio, I-67010 Coppito-L’Aquila, Italy 5Instituto de Física, Universidade Federal do Rio de Janeiro, C.P. 68528, Rio de Janeiro, RJ, 21941-972, Brazil 6Faculdade de Ciências Naturais, Universidade Federal do Pará, C.P. 68800-000, Breves, PA, Brazil

(Received 18 December 2017; published 3 May 2018)

We derive two versions of an effective model to describe dynamical effects of the Yukawa interaction among Dirac electrons in the plane. Such short-range interaction is obtained by introducing a mass term for the intermediate particle, which may be either scalar or an abelian gauge field, both of them in (3 þ 1) dimensions. Thereafter, we consider that the fermionic matter field propagates only in (2 þ 1) dimensions, whereas the bosonic field is free to propagate out of the plane. Within these assumptions, we apply a mechanism for dimensional reduction, which yields an effective model in (2 þ 1) dimensions. In particular, for the gauge-field case, we use the Stueckelberg mechanism in order to preserve gauge invariance. We refer to this version as nonlocal-Proca quantum electrodynamics (NPQED). For both scalar and gauge cases, the effective models reproduce the usual Yukawa interaction in the static limit. By means of perturbation theory at one loop, we calculate the mass renormalization of the Dirac field. Our model is a generalization of Pseudo quantum electrodynamics (PQED), which is a gauge-field model that provides a Coulomb interaction for two-dimensional electrons. Possibilities of application to Fermi-Bose mixtures in mixed dimensions, using cold atoms, are briefly discussed.

DOI: 10.1103/PhysRevD.97.096003

I. INTRODUCTION Klein paradox in graphene. On the other hand, for quantum chromodynamics the interest relies on the possibility of In the last decades, the interest of studying planar theories has increased in theoretical physics, mainly studying confinement in simpler models [5]. More recently, because of the discovery of new quantum effects, such ultracold atomic gases offered a clean and highly control- lable platform for the quantum simulation of bosonic and as high-Tc superconductivity, quantum Hall effect, and topological phase transitions [1]. Furthermore, the emer- fermionic systems [6]. Within such systems, static as well gence of both massless and massive Dirac excitations in as dynamical properties of models in trapped geometries two-dimensional materials, such as graphene [2] and with short or long-range interactions [7] can be probed, for silicene [3], has built a bridge between high-energy and example via their collective dynamics, or via density or condensed matter physics. For instance, well-known effects momentum correlations [8]. Importantly, prototypical high have been experimentally verified, or proposed in reachable energy physics models can be mapped into the low-energy, energy scales, see [4] for an experimental realization of nonrelativistic, many-body dynamics of ultracold atoms [9]. Recently the experimental demonstration of a digital quantum simulation of the paradigmatic Schwinger model, a Uð1Þ-Wilson lattice gauge theory [10] was shown. Published by the American Physical Society under the terms of Among two-dimensional models, pseudo quantum the Creative Commons Attribution 4.0 International license. electrodynamics (PQED) [11,12] (or reduced quantum Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, electrodynamics [13]) has attracted some attention. This and DOI. Funded by SCOAP3. model describes the electromagnetic interaction in a system

2470-0010=2018=97(9)=096003(9) 096003-1 Published by the American Physical Society VAN SERGIO´ ALVES et al. PHYS. REV. D 97, 096003 (2018) where electrons are confined to the plane, but photons (or Yukawa potential between static charges. This is exactly the the intermediating particle) may propagate out of the plane. kind of interaction that has been studied in Ref. [28–30] Despite its apparent nonlocal nature, PQED is still unitary in the context of ultracold atomic gases. Thereafter, we and local [14]. Indeed, its main striking feature is that the calculate the mass renormalization at one loop in the small- effective action, for the matter field, remains the very same coupling limit. as the one provided by quantum electrodynamics in 3 þ 1 The outline of this paper is the following: in Sec. II,we dimensions, hence, unitarity is respected. In the static limit, consider a Dirac field coupled to a scalar field, whose it yields a Coulomb potential, which renormalizes the dynamics is given by the massive Klein Gordon model. In Fermi velocity in graphene [15], as it has been verified Sec. III, we introduce the gauge field model. Since a naive by experimental findings [16]. In the dynamical descrip- addition of a mass for the gauge field would break gauge tion, it is expected to generate a set of quantized-energy invariance, we consider the well known Stueckelberg levels in graphene as well as an interaction driven quantum action. This model is a generalized version of Proca valley Hall effect [17] at low enough temperatures. quantum electrodynamics, on which we perform the dimen- Furthermore, chiral-symmetry breaking has been shown sional reduction. In Sec. IV, we compute the asymptotic to take place for both zero and finite temperatures [18,19]. behavior of the boson propagator at both small and large It also occurs in the presence of a Gross-Neveu interaction, distances. In Sec. V, we show that, by tuning the mass of the whose main effect is to decrease the critical coupling intermediate field, one may control the sign of the quantum constant, yielding a better scenario for dynamical mass correction, generated by the electron-self energy. We also generation [20]. Lower dimensional versions of PQED include one Appendix, where we present the details about have been investigated in Ref. [21], aiming for applications the electron-self energy as well as the corrected gauge-field in cold atoms and in Ref. [22] for applications in the realm propagator at one loop in perturbation theory. of topological insulators. All of these works rely on the fact that PQED generates a long-range interaction in the static II. THE SCALAR CASE limit, namely, the Coulomb potential VCðrÞ ∝ 1=r. In the meson theory of Yukawa, the so-called Yukawa In this section, we perform a dimensional reduction of 3 1 potential VðrÞ ∝ e−mr=r is a static solution of the Green the Yukawa action in þ dimensions. To generate an 2 2 interaction length, we assume that the mediating particle is function equation ð−∇ þ m ÞVðrÞ¼δðrÞ, where δðrÞ is a massive real scalar field. Let us start with the Euclidean the Dirac delta function [23]. Within the quantum-field- action in (3 1) dimensions, given by theory interpretation, we may claim that the mediating field þ has a mass term. Motivated by this well-known result, we 1 1 L ∂ φ∂μφ 2φ2 φψψ¯ ψ¯ ∂ − ψ shall use the paradigm of including a mass term, for the 4D ¼ 2 μ þ2m þg þ ði= M0Þ ; ð1Þ intermediate particle, in order to generate a short-range interaction, i.e., the Yukawa potential in the plane. This where g is a dimensionless coupling constant, φ is a real potential has been applied to describe bound states [24,25], and massive Klein-Gordon field, and ψ is the Dirac field. electron-ion interactions in a crystal [26], and interactions First, we calculate the effective action for the matter L ψ between dark energy and dark matter [27] among others. field eff ½ . In order to do so, we define the vacuum Nevertheless, a planar quantum-field theory accounting for functional Z this interaction, for both static and dynamical regime, has Z not been derived yet. φ ψ¯ ψ − ψ φ In this paper, we show how one may include an Z ¼ D D D expf S½ ; g interaction length in PQED, yielding a short-range and Z R 4 −φ −□ 2 φ 2− φψψ¯ ψ nonlocal theory. The simplest mechanism is to consider that ¼ Dψ¯ DψDφe d x½ ð þm Þ = g þO½ the mediating particle, i.e., the boson field, is massive. This Z parameter has to be a consequence either of the coupling ψ¯ ψ − ¼ D D expf Seffg: ð2Þ with a Higgs field in the broken phase or some sort of intrinsic parameter in the system. The latter case can indeed φ be realized experimentally in ultracold Bose-Fermi mix- Integrating out in Eq. (2) yields tures in mixed dimensions. The typical setup consists of a 2 Z Bose-Einstein condensate trapped in a three-dimensional L ψ − g 4 4 ψψ¯ Δ − ψψ¯ eff½ ¼ d xd yð ÞðxÞ φðx yÞð ÞðyÞ; ð3Þ harmonic potential weakly interacting with a fermionic gas 2 confined in a two-dimensional periodic potential (optical lattice). The bosonic field mass corresponds to the atomic where Z mass of the atoms in a Bose-Fermi mixture. Here, we 4 −ikðx−yÞ −□ 2 −1 ≡ Δ − d k e consider both the massive Klein-Gordon field as well as the ð þ m Þ φðx yÞ¼ 4 2 2 ð4Þ massive Stueckelberg model. For both cases, we have a ð2πÞ k þ m

096003-2 TWO-DIMENSIONAL YUKAWA INTERACTIONS FROM … PHYS. REV. D 97, 096003 (2018) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is the free scalar-field propagator, which yields the inter- 2 −□ þ m2 d3k 2 k2 þ m2 K½□ ≡ ≡ eikx : ð12Þ action between the matter field. The static interaction VðrÞ −□ ð2πÞ3 k2 is provided by the Fourier transform of Eq. (4) at k0 ¼ 0 (no time dependence), namely, From Eq. (11), it is straightforward that the scalar-field Z d3k e−ik:r e−mr propagator is Gφ and the effective action for the matter field VðrÞ¼ ¼ : ð5Þ are the very same as in Eqs. (9) and (7), respectively. ð2πÞ3 k2 þ m2 4πr

Equation (5) is the well-known Yukawa potential, where III. THE GAUGE-FIELD CASE m the inverse of is the interaction length of the model. This In this section, we consider that the matter field is is just the consequence of the coupling gφψψ¯ in ð3 þ 1ÞD. coupled to a gauge field Aμ, through a minimal coupling Next, we show how to apply the procedure of dimen- μ A jμ. The main steps are the same as in the previous sional reduction of PQED to Eqs. (3) and (5). In other 2 μ words, we consider the case where the fermionic field calculation. Nevertheless, a massive term as m AμA breaks is confined to the plane, but the bosonic field is not. This is gauge invariance. Hence, we must be careful about how to an approximation of the derivation of PQED [11]. In order introduce the mass, i.e., the length scale for interactions. to do so, we assume that matter field is confined to the For the sake of simplicity, we consider an Abelian field Aμ, plane, i.e., for which we may use the Stueckelberg mechanism. This is a mechanism for generating mass for Aμ without breaking

ψψ¯ ðxÞ¼ψψ¯ ðx0;x1;x2Þδðx3Þ: ð6Þ gauge invariance [31]. Before we perform the dimensional reduction, let us summarize this method. 3 1 Using Eq. (6) in Eq. (3), we obtain First, we introduce a mass term into ( þ ) QED, yielding the so-called Proca quantum electrodynamics, Z 2 whose action is given by L − g 3 3 ψψ¯ − ψψ¯ eff ¼ 2 d xd yð ÞðxÞGφðx yÞð ÞðyÞ; ð7Þ 1 2 L μν m μ μ 4D ¼ FμνF þ AμA þ eA jμ where Gφðx − yÞ¼Δφðx − y; x3 ¼ 0;y3 ¼ 0Þ is the effec- 4 2 tive scalar-field propagator in (2 þ 1) dimensions, given by λ μ 2 − ð∂μA Þ þ ψ¯ ði=∂ − M0Þψ; ð13Þ Z Z 2 d3k dk 1 − −ikðx−yÞ z Gφðx yÞ¼ 3 e 2π 2 2 2 : ð8Þ ψγ¯ ψ ð2πÞ ð Þ k þ kz þ m where e is the electric charge, jμ ¼ μ is the matter current, m2 is a massive parameter for the gauge field, and λ Integrating over kz above, we find is a gauge-fixing parameter. As expected, Z 3 −ikðx−yÞ − d k e ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 m2 1 Gφðx yÞ¼ 3 p : ð9Þ μ → μ μ∂ ∂ 2 ð2πÞ 2 k2 þ m2 2 AμA 2 AμA þ mA μB þ 2 ð μBÞ ð14Þ

In the static limit, Eq. (9) yields the Yukawa potential. under Aμ → Aμ þ ∂μB=m. Indeed, gauge invariance is Indeed, explicitly broken. Z 2 −ik:r −mr Next, we introduce a scalar-field BðxÞ (the Stueckelberg d k effiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e VðrÞ¼ 2 p ¼ ; ð10Þ field) in Eq. (13), hence, ð2πÞ 2 k2 þ m2 4πr 2 2 1 ∂μB as expected from our dimensional reduction. L μν m − μ 4D ¼ FμνF þ Aμ þ eA jμ We may go beyond the static approximation by consid- 4 2 m λ ering a nonlocal model with a propagator equal to Eq. (9). − ∂ μ 2 ψ¯ ∂ − ψ This is given by 2 ð μA Þ þ ði= M0Þ : ð15Þ

1 1 μ 2 2 Equation (15) is known as Stueckelberg action. Despite L3D ¼ ∂ φK½□∂μφ þ m φ þ gφψψ¯ þ ψ¯ ði=∂ − M0Þψ; 2 2 the mass for the gauge field, it is invariant under gauge ð11Þ transformation, namely, Aμ → Aμ þ ∂μΛ, B → B − mΛ, and ψ → expð−ieΛÞψ [31]. Furthermore, it still produces with the Yukawa interaction between static charges.

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μ From Eq. (15), we may find j ðx0;x1;x2Þδðx3Þ; μ ¼ 0; 1; 2; jμðxÞ¼ ð24Þ 0; μ 3 2 2 ¼ . 1 ∂μB L μν m μ ð Þ ∂μ 4D ¼ FμνF þ AμA þ þ mAμ B 4 2 2 Similarly to the previous case, this shall lead to λ μ − ∂ μ 2 ψ¯ ∂ − ψ þ eA jμ ð μA Þ þ ði= M0Þ : ð16Þ Z 3 2 δμν d k −ikðx−yÞ G0 μν ¼ e pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð25Þ ; 2π 3 2 2 2 We shall use Eq. (16) to compute the effective action for the ð Þ k þ m matter associated to the matter current jμ. The vacuum 2 1 functional is which is the effective propagator in ( þ ) dimensions. Our main goal is to find the corresponding theory in Z (2 þ 1) dimensions with the same effective action in Z ¼ Dψ¯ DψDAμDB expf−S½ψ;Aμ;Bg: ð17Þ Eq. (21). This model reads

1 Integration over B yields L □ μν λ μ∂ □ ∂ ν μ 3D ¼ 2 FμνK½ F þ A μK½ νA þ eA jμ 2 1 λ þ ψ¯ ði=∂ − M0Þψ; ð26Þ eff μν m μ □ μ 2 L FμνF AμA ∂μA 4D ¼ 4 þ 2 þ 2 ð Þ μ λ þ eA jμ þ ψ¯ ði=∂ − M0Þψ; ð18Þ where is a gauge-fixing parameter. It is straightforward to show that, after integrating out Aμ in Eq. (26), we obtain the where same effective action as in Eq. (21) with the constraint in Eq. (24). For an arbitrary λ, the free gauge-field propagator 2 −1 reads λ□ ¼ −λ þ m □ : ð19Þ 1 1 − λ Now, for the sake of simplicity, we isolate the quadratic ð Þ kμkν G0 μν k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δμν : 27 ; ð Þ¼ 2 2 þ λ 2 ð Þ term in Aμ, hence, 2 k þ m k

1 eff μ 2 ν There are two main features of Eq. (26): (a) The massive L A −□ m δμν λ□∂μ∂ν A 4D ¼ 2 ½ð þ Þ þ parameter m is no longer a pole of the gauge-field μ propagator, hence, it can not be thought as a mass and þ eA jμ þ ψ¯ ði=∂ − M0Þψ: ð20Þ (b) Gauge-invariance is explicitly respected, i.e., there is no need to deal with Stueckelberg fields. Indeed, we could set Integrating out Aμ, we get our desired effective action B ¼ 0 from the very beginning, which means starting with 2 Proca quantum electrodynamics and, therefore, breaking of eff e μ ν L j − j x Δμνj y ψ¯ i=∂ − M0 ψ; gauge invariance. Then, after dimensional reduction, the 4D½ ¼ 2 ð Þ ð Þþ ð Þ ð21Þ corresponding 3D theory is still the same as in Eq. (26) and that it is gauge invariant. where 1 λ IV. ASYMPTOTIC BEHAVIOR OF BOTH SCALAR Δ δ − □ ∂ ∂ μν ¼ 2 μν 2 μ ν : ð22Þ −□ þ m −□ þ m − λ□ AND GAUGE-FIELD PROPAGATORS In this section, we calculate the asymptotic expressions μν Using Eq. (22) in Eq. (21) with charge conservation of Gφðx − y; mÞ and δ G0μνðx − y; mÞ, i.e., propagators in ∂ μ 0 μj ¼ , we may conclude that, for a correct description, the space-time coordinates. We use the integral version in the gauge-field propagator is Eq. (8), which has an extra kz-integral. Hence, Z 4 Z Z d k δμν 3 Δ − −ikðx−yÞ d k þ∞ dμ 1 μνðx yÞ¼ 4 e 2 2 : ð23Þ − ; −ikðx−yÞ ð2πÞ k þ m Gφðx y mÞ¼ 3 e 2 2 2 : ð2πÞ −∞ ð2πÞ k þ μ þ m In this way, all the gauge-dependence vanishes in the ð28Þ effective action. In order to obtain the projected theory in (2 þ 1) Note that we have replaced kz by μ, since μ is just a dimensions, we consider that the current matter only parametric variable. Thereafter, we use Eq. (5) for solving propagates in the plane, therefore, the k-sphere integral, therefore,

096003-4 TWO-DIMENSIONAL YUKAWA INTERACTIONS FROM … PHYS. REV. D 97, 096003 (2018) Z ffiffiffiffiffiffiffiffiffiffiffiffiffi þ∞ μ 1 p d − ðμ2þm2Þjx−yj Gφðx−y;mÞ¼ e : ð29Þ −∞ ð2πÞ4πjx−yj

Next, after solving the μ-integral (see Ref. [32]), we have m Gφðx − y; mÞ¼ K1ðmjx − yjÞ; ð30Þ 4π2jx − yj where K1 is a modified Bessel function of the second kind. In the short-range limit mjx − yj ≪ 1, it yields 1 Gφðx − y; mÞ ≈ ; ð31Þ 4π2jx − yj2 whereas in the long-range limit mjx − yj ≫ 1, we find rffiffiffiffiffiffiffi FIG. 1. Quantum correction for the electron mass. We plot the mπ e−mjx−yj quantity αfðzÞ=2π in Eq. (33), where z is the ratio between the Gφðx − y; mÞ ≈ : ð32Þ 2 4π2jx − yj3=2 inverse of interaction length m and the bare electron mass M0. For jzj ≤ zc ≈ 1.2,wehavefðzÞ < 0, therefore, the renormalized δ 2 The gauge-field propagator may be calculated by following energy gap RE is less than the bare energy gap M0. On the other hand, jzj ≥ zc yields fðzÞ > 0 (shaded area), i.e., the renormal- the very same steps. ized energy gap is larger than the bare energy gap. Finally, for We have described some general features of NPQED, in jzj¼zc, we find fðzcÞ¼0 and the renormalized energy gap is particular, its derivation, two-point functions, and inter- equal to 2M0. actions. Next, we shall explore quantum corrections by using perturbation theory. − 0 ER < are the positive and negative energies, see Appendix A. Since fðzÞ may be negative for V. PERTURBATION THEORY RESULTS jzj ≤ zc ≈ 1.2, we have that δRE lies inside the energy 2 ≤ In this section, we calculate the renormalized electron gap M0. The case z zc resembles the quantized energy levels calculated in Ref. [17] or Ref. [20], because that mass MR of the model in Eq. (26) at one-loop approximation. The details about the calculation are shown in renormalized states are closer to the zero-energy level. Appendix A. In particular, we would like to obtain its Nevertheless, because we are in the small coupling limit, dependence on m, the mass term of the gauge field. Note hence, we do not generate dynamical symmetry breaking, → 0 → 0 that in our 3D model, this parameter must to be understood since ZR when M0 . as the inverse of the interaction length. This, nevertheless, is the mass of the intermediate field that propagates in 4D. VI. DISCUSSION Thereafter a standard calculation, we obtain PQED has been applied to describe the interactions of α two-dimensional electrons, in particular, graphene in the 1 ZR ¼ þ 2π fðzÞ; ð33Þ strong-coupling regime. The main results rely on the fact that electrons do interact trough the Coulomb potential, where ZR ≡ MR=M0, z ≡ m=M0, and which has an infinite range. Here, nevertheless, we consider Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a scenario where interactions have a finite range. We have 1 2 þ x 2 applied the very same procedure for deriving PQED [11], fðzÞ¼ dx pffiffiffiffiffiffiffiffiffiffiffi ln ð1 − xÞðz − xÞþx : ð34Þ 0 1 − x but considering a massive photon in (3 þ 1) dimensions. Our main result is that, after performing the dimensional From Eq. (34), it is clear that fðzÞ¼fð−zÞ, therefore, reduction, one obtains that the matter field interacts through the corrections are only dependent on the modulus of m. a Yukawa potential in the static limit. Similar to PQED, it From now on, we assume α ¼ 1=137. Next, we would yields a nonlocal theory in both space and time. Since the like to address the effects of the m parameter on ZR. matter field is not relevant for the dimensional reduction, Surprisingly, for jzj ≤ zc ≈ 1.2, the quantum correction this model may be generalized to describe the Yukawa αfðzÞ=2π is negative, while for jzj ≥ zc, they become interaction between other kind of particles. Although our positive and cross the free-energy level M0, see Fig. 1. derivation follows standard steps of QED literature, we Let us calculate the energy gap of the renormalized believe that they shall be relevant, in particular, for δ þ − − α 2π − − − state RE ¼ ER ER ¼ M0 þ M0 fxðzÞ= ð M0 applications in condensed matter physics and cold-atom α 2π 2 α π þ 0 M0 fðzÞ= Þ¼ M0 þ M0 fðzÞ= , where ER > and systems.

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In Ref. [28–30], the authors proposed a scheme appli- ACKNOWLEDGMENTS cable to a cold-atom mixture of bosons interacting with We are grateful to Rodrigo Pereira for very interesting polarized fermions to study possible p−wave pairing in the and stimulating discussions. G. C. M. acknowledges fermionic component induced by the bosons. An essential CAPES for financial support. T. M. acknowledges CNPq ingredient of these proposals is the mixed dimensionality, for support through Bolsa de produtividade em Pesquisa namely the two-dimensional confinement of the fermionic n.311079/2015-6. L. O. N. acknowledges the financial species and the three-dimensional one for the bosons, support from MEC/UFRN. E. C. M. is partially supported reducing considerably trap losses from three-body effects. by CNPq and FAPERJ. Interestingly, the static interaction between the fermionic particles results into a Yukawa potential allowing a notable increase in the critical temperature for the appearance of APPENDIX: PERTURBATIVE RESULTS fermionic superfluidity. In this context, our study based on FOR GAUGE THEORY the NPQED may certainly be useful for the investigation of novel quantum phases starting from a fully relativistic field- 1. Electron-self energy theory approach that can be simulated in condensed matter In this Appendix, we show some details about the platforms. Similarly, more recently it has been shown that a electron-self energy. First, let us write the Feynman rules full dynamical description provides new results for PQED of Eq. (26) in Euclidean space. The free electron- with applications to the exciton spectrum in transition- propagator reads metal dichalcogenides [33].

We finally discuss the importance of the order in which 1 −ðp= þ M0Þ the following two operations are implemented, namely: S0F ¼ ¼ 2 2 ; ðA1Þ p= − M0 p þ M (a) the inclusion of a mass for the gauge field; (b) the 0 dimensional reduction. Interestingly, the result is sensitive the gauge-field propagator (in the Feynman gauge λ 1)is to the order in which the inclusion of a mass and the ¼ dimensional reduction are performed. As a matter of fact, δ by doing “a” before “b,” we have shown that the Yukawa pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiμν G0;μνðpÞ¼ 2 2 ; ðA2Þ potential VðrÞ¼e−mr=4πr is obtained from the static limit 2 p þ m of the gauge-field propagator,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi given by Eq. (27). This is 2 2 Γμ γμ proportional to 1=ð2 p þ m Þ. and the vertex interaction is ¼ e. The Euclidean Conversely, starting with a massless gauge field, hence, matrices satisfy fγμ; γνg¼−2δμν. From Eq. (A1),wehave 2 − 2 by applying the dimensional reduction, we arrive at the that the pole of the free Dirac electron is pE ¼ M0, where 2 PQEDpffiffiffiffiffi model [11], whose propagator is proportional to pE is the Euclidean momentum. For calculating the 1=ð2 p2Þ. If we now couple the PQED model, to a Higgs physical mass, one must return to the Minkowski space, 2 → − 2 2 2 field in the broken phase, such that a massive term is using pE pM, such that pM ¼ M0 are the physical generatedp toffiffiffiffiffi the gauge field,pffiffiffiffiffi this changes the propagator poles. from 1=ð2 p2Þ to 1=ð2 p2 þ mÞ, which clearly shows The corrected electron propagator SF is that we shall obtain a different model from the one Σ associated to Eq. (26). The potential is now given by a SF ¼ S0F þ S0Fð ÞS0F þ ðA3Þ combination of Coulomb and Keldysh [34] potentials, namely, therefore,

S−1 ¼ S−1 − ΣðpÞ: ðA4Þ 2 π F 0F e 1 − r H r − r VðrÞ¼4π 2 0 Y0 ; ð35Þ r r0 r0 r0 The electron-self energy reads

Z 3 where r0 ≡ 2=m, H0ðmrÞ, and Y0ðmrÞ are Struve and 2 d k μ ν ΣðpÞ¼e Γ S0 ðkÞΓ G0 μνðp − kÞ: ðA5Þ Bessel functions, respectively. The potential in Eq. (35) ð2πÞ3 F ; does not decay exponentially at large distances as the Yukawa potential, rather, it has a power-law decay. We Equation (A5) has a linear divergence, therefore, we conclude, therefore, that we must be careful about how one need to use a regularization scheme. We choose to use the wishes to use quantum electrodynamics in applications for usual dimensional regularization ϵ ¼ 3 − D, where D is an lower-dimensional systems, because of this sensitive rela- arbitrary dimension, which we shall consider D → 3 in the tion between the dimensional reduction and phenomeno- very end of the calculation. After application of standard logical parameters, such as a mass for the gauge field. methods, we find

096003-6 TWO-DIMENSIONAL YUKAWA INTERACTIONS FROM … PHYS. REV. D 97, 096003 (2018) μ 1 2 1 − Σ 2 2 p − 6 −1 ½p þ MðpÞ ðpÞ¼ CI þ C ln þ M0 ; ðA6Þ SRF ¼ ¼ 2 2 ; ðA16Þ M0 ϵ 3 p − MðpÞ p þ M ðpÞ where μ is an arbitrary massive parameter, generated by with the prescription e → eμϵ=2, with ϵ ¼ D − 3, where D is the dimension of the space-time, as usually done in the MðpÞ¼M0 þ BðpÞþM0AðpÞ: ðA17Þ dimensional regularization. The constant C is given by Using Eqs. (A7), (A12), and (A11) in Eq. (A17), we find 2 e Z C ¼ 16π2 ðA7Þ α 1 ð2 þ xÞ Δ MðpÞ¼M0 þ M0 dx pffiffiffiffiffiffiffiffiffiffiffi ln : ðA18Þ 2π 0 1 − x M0 and the parametric integral reads Z Equation (A17) yields the so-called mass function MðpÞ, 1 pðx − 1Þþ3M0 Δ which is, essentially, the momentum-dependent part of the I ¼ dx pffiffiffiffiffiffiffiffiffiffiffi ln ; ðA8Þ 0 1 − x M0 electron-self energy that renormalizes the electron mass [35]. In Fig. 2, we show that by using different values of with m=M0, we may generate quantum corrections that either qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi increase or decrease the renormalized mass in comparison 2 2 2 with the bare value M0. To clarify this result, we shall Δ ¼ p xð1 − xÞþM0x þ m ð1 − xÞ: ðA9Þ calculate the renormalized mass MR. The pole of Eq. (A16) is given by the solution of Equation (A6) has both finite and a regulator-dependent 2 − 2 2 pE ¼ M ðpEÞ, which, in the Minkowski space, yields terms, which must be eliminated by some renormalization 2 2 − 2 2 mR ¼ M ð MRÞ, where MR is the renormalized mass by scheme. For simplicity, we choose the minimal subtraction 2 → − 2 applying the substitution pE MR for calculating the procedure, which avoids the poles by introducing counter- pole. Furthermore, since we are at one-loop approximation, terms in the original action. Hence, − 2 − 2 we may use Mð MRÞ¼Mð M0Þ. Therefore, the renor- − þ R malized masses are MR ¼Mð MRÞ¼ER , where ER ¼ Σ ðpÞ¼ lim ðΣðp;μ;ϵÞ−CTÞ¼AðpÞp=þBðpÞ; ðA10Þ − − − − μ;ϵ→0 þjMð MRÞj and ER ¼ jMð MRÞj are positive and negative solutions, respectively. Using Eq. (A17) with 2 − 2 2 4πα where CT stands for counterterms, MR ¼ Mð M0Þ and e ¼ , we find Z "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 1 Z ðx − 1Þ Δ α 1 2 2 AðpÞ¼2C dx pffiffiffiffiffiffiffiffiffiffiffi ln ; ðA11Þ jMRj 1 ffiffiffiffiffiffiffiffiffiffiþ x 1 − m − 0 1 − x M0 ¼ þ dxp ln ð xÞ 2 x þ x : M0 2π 0 1 − x M0 and ðA19Þ Z 1 3M0 Δ BðpÞ¼2C dx pffiffiffiffiffiffiffiffiffiffiffi ln : ðA12Þ 0 1 − x M0

Using Eq. (A10) in Eq. (A4), we find

−1 1 − − SF ðpÞ¼½ð AðpÞÞp ðM0 þ BðpÞÞ: ðA13Þ

Multiplying Eq. (A13) by ð1 þ AðpÞÞ, we have

1 −1 − − 1 ð þ AðpÞÞSF ðpÞ¼ i½p ðM0 þ BðpÞÞð þ AðpÞÞ: ðA14Þ

Next, we define the renormalized matter field ψ R, namely, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 FIG. 2. Mass function MðpÞ in Eq. (A17). For this plot we set ψ ψ ψ 1 − ψ R ¼ Z ¼ AðpÞ : ðA15Þ M0 ¼ 1 and α ¼ 1=137. Dashed line: m=M0 ¼ 1.5, full line: m=M0 ¼ 0.5. Note that Mðp ¼ 0Þ >M0 for the dashed line, but Therefore, the physical propagator reads Mðp ¼ 0Þ

096003-7 VAN SERGIO´ ALVES et al. PHYS. REV. D 97, 096003 (2018)

Considering ZR ≡ MR=M0 and z ≡ m=M0, we obtain where mR is the renormalized mass given by Eq. (33) and Eq. (34). In the scalar-field case, given by Eq. (11), the critical point zc is the same. This is not surprising because of the similarities of the electron-self 2 2 2Π energy and the bosonic propagators. mR ¼ m þ e ðp ¼ mRÞ; ðA21Þ Equation (A19) shows that the renormalized electron mass is dependent on m. It is reasonable to calculate whether m itself is renormalized in one loop. For consistency, the renormalization of the boson mass m in the Stueckelberg where ΠðpÞ is given by [36] theory, in Eq. (16), would imply a renormalized mass parameter in the reduced model, given by Eq. (26). Within the Landau gauge λ ¼ ∞, the corrected gauge-field Z 2 1 2 2 1 propagator with the insertion of polarization tensor reads Π p 1 − − Q − γ − ðpÞ¼ 2 dxxð xÞ ln 2 ; ðA22Þ 2π 0 ϵ 4πμ 2 δμν GμνðpÞ¼ ; p2 þ m2 þ e2ΠðpÞ δ μν with Q2 M2 x 1 − x p2. Thus, the bare mass of the ¼ 2 2 ; ðA20Þ ¼ 0 þ ð Þ p þ mR gauge field is renormalized in one-loop approximation.

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