Two-dimensional Yukawa interaction driven by a nonlocal-Proca quantum electrodynamics

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+1) dimensions. Thereafter, we consider that the matter field propagates only in (2+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+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 $e^{-m r}/r$ 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 Pseudoquantum 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.


I. INTRODUCTION
In the last decades, the interest of studying planar theories has increased in theoretical physics, mainly because of the discovery of new quantum effects, such as high-T c superconductivity, quantum Hall effect, and topological phase transitions [1].Furthermore, the emergence of both massless and massive Dirac excitations in twodimensional materials, such as graphene [2] and silicene [3], has built a bridge between high-energy and condensed matter physics.For instance, well-known effects have been experimentally verified, or proposed in reachable energy scales, see [4] for an experimental realization of Klein paradox in graphene.On the other hand, for quantum chromodynamics the interest relies on the possibility of studying confinement in simplest models [5].More recently, ultracold atomic gases offered a clean and highly controllable platform for the quantum simulation of bosonic and fermionic systems [6].Within such systems, static as well as dynamical properties of models in trapped geometries with short or long-range interactions [7] can be probed, for example via their collective dynamics, or via density or momentum correlations [8].Importantly, prototypical high energy physics models can be mapped into the low-energy, non-relativistic, many-body dynamics of ultracold atoms [9].Recently the experimental demonstration of a digital quantum simulation of the paradigmatic Schwinger model, a U (1)-Wilson lattice gauge theory [10] was shown.
Among two-dimensional models, Pseudo quantum electrodynamics (PQED) [11,12] (or Reduced quantum electrodynamics [13]) has attracted some attention.This model describes the electromagnetic interaction in a system where electrons are confined to the plane, but photons (or the intermediating particle) may propagate out of the plane.Despite its nonlocal nature, PQED is still unitary [14].Indeed, its main striking feature is that the effective action, for the matter field, remains the very same as the one provided by quantum electrodynamics in 3+1 dimensions, hence, unitarity is respected.In the static limit, it yields a Coulomb potential, which renormalizes the Fermi velocity in graphene [15], as it has been verified by experimental findings [16].In the dynamical description, it is expected to generate a set of quantizedenergy levels in graphene as well as an interaction driven quantum valley Hall effect [17] at low enough temperatures.Furthermore, chiral-symmetry breaking has been shown to take place for both zero and finite temperatures [18,19].It also occurs in the presence of a Gross-Neveu interaction, whose main effect is to decrease the critical coupling constant, yielding a better scenario for dynamical mass generation [20].Lower dimensional versions of PQED have been investigated in Ref. [21], aiming for applications in cold atoms and in Ref. [22] for applications in the realm of topological insulators.All of these works rely on the fact that PQED generates a long-range interaction in the static limit, namely, the Coulomb potential In the meson theory of Yukawa, the so-called Yukawa potential V (r) ∝ e −mr /r is a static solution of the motion equation (−∇ 2 +m 2 )V (r) = δ(r), where δ(r) is the Dirac delta function [23].Within the quantum-field-theory interpretation, we may claim that the mediating field has a mass term.Motivated by this well-known result, we shall use the paradigm of including a mass term, for the intermediate particle, in order to generate a shortrange interaction, i.e, the Yukawa potential in the plane.This potential has been applied to describe bound states [24,25], electron-ion interactions in a crystal [26], and interactions between dark energy and dark matter [27] among others.Nevertheless, a planar quantum-field theory accounting for this interaction, for both static and dynamical regime, has not been derived yet.
In this paper, we show how one may include an interaction length in PQED, yielding a short-range and nonlocal theory.The simplest method is to generate a mass term for the mediating particle.Hence, we consider both the massive Klein-Gordon field as well as the massive Stueckelberg model.For both cases, we have a Yukawa potential between static charges.Thereafter, we calculate the mass renormalization at one loop in the small-coupling limit.
The outline of this paper is the following: in Sec.II, we consider a Dirac field coupled to a scalar field, whose dynamics is given by the massive Klein Gordon model.In Sec.III, we introduce the gauge field model.Since a naive addition of a mass for the gauge field would break gauge invariance, we consider the well known Stueckelberg action.This model is a generalized version of Proca quantum electrodynamics, on which we perform the dimensional reduction.In Sec.IV, we compute the asymptotic behavior of the boson propagator at both small and large distances.In Sec.V, we show that, by tuning the mass of the intermediate field, one may control the sign of the quantum correction, generated by the electron-self energy.We also include one appendix, where we present the details about the electron-self energy.

II. THE SCALAR CASE
In this section, we perform a dimensional reduction of the Yukawa action in 3+1 dimensions.To generate an interaction length, we assume that the mediating particle is a massive real scalar field.Let us start with the Euclidean action in (3+1) dimensions, given by where g is a dimensionless coupling constant, ϕ is a real and massive Klein-Gordon field, and ψ is the Dirac field.First, we calculate the effective action for the matter field L eff [ψ].In order to do so, we define the generating function Integrating out ϕ in Eq. ( 2) yields where is the free scalar-field propagator, which yields the interaction between the matter field.The static interaction V (r) is provided by the Fourier transform of Eq. ( 4) at k 0 = 0 (no time dependence), namely, Eq. ( 5) is the well-known Yukawa potential, where the inverse of m is the interaction length of the model.This is just the consequence of the coupling gϕ ψψ in (3+1)D.
Next, we show how to generalize Eq. ( 3) and Eq. ( 5) for 2+1 dimensions.Here, the main purpose is to keep the Yukawa potential between the matter field.In other words, we say that the fermionic field is confined to the plane, but the bosonic field is not.This is a roughly approximation of the derivation of PQED [11].In order to do so, we assume that matter field is confined to the plane, i.e, Using Eq. ( 6) in Eq. ( 3), we obtain where Integrating over k z above, we find In the static limit, Eq. ( 9) yields the Yukawa potential.Indeed, as expected from our dimensional reduction.We may go beyond the static approximation by considering a nonlocal model with a propagator equal to Eq. ( 9).This is given by From Eq. ( 11), it is straightforward that the scalarfield propagator is G ϕ and the effective action for the matter field are the very same as in Eq. ( 9) and Eq. ( 7), respectively.

III-THE GAUGE-FIELD CASE
In this section, we consider that the matter field is coupled to a gauge field A µ , through a minimal coupling A µ j µ .The main steps are the same as in the previous calculation.Nevertheless, a massive term as m 2 A µ A µ breaks gauge invariance.Hence, we must be careful about how to introduce the mass, i.e, the length scale for interactions.For the sake of simplicty, we consider an abelian field A µ , for which we may use the Stueckelberg mechanism.This is a mechanism for generating mass for A µ without breaking gauge invariance [29].Before we perform the dimensional reduction, let us summarize this method.
First, we introduce a mass term into (3+1) QED, yielding the so-called Proca quantum electrodynamics, whose action is given by where e is the electric charge, j µ = ψγ µ ψ is the matter current, m 2 is a massive parameter for the gauge field, and λ is a gauge-fixing parameter.As expected, Indeed, gauge invariance is explicitly broken.Next, we introduce a scalar-field B(x) (the Stueckelberg field) in Eq. ( 13), hence, Eq. ( 15) is known as Stueckelberg action.Despite the mass for the gauge field, it is invariant under gauge transformation, namely, A µ → A µ + ∂ µ Λ, B → B − mΛ, and ψ → exp(−ieΛ)ψ [29].Furthermore, it still produces the Yukawa interaction between static charges.From Eq. ( 15), we may find We shall use Eq. ( 16) to compute the effective action among the matter current j µ .The generating functional is Integration over B yields where Now, for the sake of simplicity, we isolate the quadratic term in A µ , hence, Integrating out A µ , we get our desired effective action where Using Eq. ( 22) in Eq. ( 21) with charge conservation ∂ µ j µ = 0, we may conclude that, for a correct description, the gauge-field propagator is In this way, all the gauge-dependence vanishes in the effective action.
In order to obtain the projected theory in (2+1) dimensions, we consider that the current matter only propagates in the plane, therefore, Similarly to the previous case, this shall lead to which is the effective propagator in (2+1) dimensions.
Our main goal is to find the corresponding theory in (2+1) dimensions with the same effective action in Eq. ( 21).This model reads where λ is a gauge-fixing parameter.It is straightforward to show that, after integrating out A µ in Eq. ( 26), we obtain the same effective action as in Eq. ( 21) with the constraint in Eq. (24).For an arbitrary λ, the free gaugefield propagator reads There are two main features of Eq. ( 26): (a) The massive parameter m is no longer a pole of the gauge-field propagator, hence, it can not be thought as a mass and (b) Gauge-invariance is explicitly respected, i.e, there is no need to deal with Stueckelberg fields.Indeed, we could set B = 0 from the very beginning, which means starting with Proca quantum electrodynamics and, therefore, breaking of gauge invariance.Then, after dimensional reduction, the corresponding 3D theory is still the same as in Eq. ( 26) and that it is gauge invariant.

IV-ASYMPTOTIC BEHAVIOR OF BOTH SCALAR AND GAUGE-FIELD PROPAGATORS
In this section, we calculate the asymptotic expressions of G ϕ (x−y; m) and δ µν G 0µν (x−y; m), i.e, propagators in the space-time coordinates.We use the integral version in Eq. ( 8), which has an extra k z -integral.Hence, (28) Note that we have replaced k z by µ, since µ is just a parametric variable.Thereafter, we use Eq. ( 5) for solving the k-sphere integral, therefore, Next, after solving the µ-integral (see Ref. [28]), we have where K 1 is a modified Bessel function of the second kind.In the short-range limit m|x − y| 1, it yields whereas in the long-range limit m|x − y| 1, we find The gauge-field propagator may be calculated by following the very same steps.
We have described some general features of NPQED, in particular, its derivation, two-point functions, and interactions.Next, we shall explore quantum corrections by using perturbation theory.

V-PERTURBATION THEORY RESULTS
In this section, we calculate the renormalized electron mass m R of the model in Eq. ( 26) at one-loop approximation.The details about the calculation are shown in Appendix A. In particular, we would like to obtain its dependence on m, the mass term of the gauge field.Note that in our 3D model, this parameter must to be understood as the inverse of the interaction length.This, nevertheless, is the mass of the intermediate field that propagates in 4D.Thereafter a standard calculation, we obtain where z R ≡ m R /M 0 , z ≡ m/M 0 , and From Eq. (34), it is clear that f (z) = f (−z), therefore, the corrections are only dependent on the modulus of m.From now on, we assume α = 1/137.Next, we would like to address the effects of the m parameter on z R .Surprisingly, for |z| ≤ z c ≈ 1.2, the quantum correction αf (z)/2π is negative, while for |z| ≥ z c , they become positive and cross the free-energy level M 0 , see Fig. 1.
Let us calculate the energy gap of the renormalized state , where E + R > 0 and E − R < 0 are the positive and negative energies, see Appendix A. Since f (z) may be negative for |z| ≤ z c ≈ 1.2, we have that δ R E lies inside the energy gap 2M 0 .The FIG. 1: Quantum correction for the electron mass.We plot the quantity αf (z)/2π in Eq. ( 33), where z is the ratio between the inverse of interaction length m and the bare electron mass M0.For |z| ≤ zc ≈ 1.2, we have f (z) < 0, therefore, the renormalized energy gap δRE is less than the bare energy gap 2M0.On the other hand, |z| ≥ zc yields f (z) > 0 (shaded area), i.e, the renormalized energy gap is larger than the bare energy gap.Finally, for |z| = zc, we find f (zc) = 0 and the renormalized energy gap is equal to 2M0.
case z ≤ z c resembles the quantized energy levels calculated in Ref. [17] or Ref. [20], because that renormalized states are closer to the zero-energy level.Nevertheless, because we are in the small coupling limit, hence, we do not generate dynamical symmetry breaking, since z R → 0 when M 0 → 0.

VII. DISCUSSION
PQED has been applied to describe the interactions of two-dimensional electrons, in particular, graphene in the strong-coupling regime.The main results rely on the fact that electrons do interact trough the Coulomb potential, which has an infinite range.Here, nevertheless, we consider a scenario where interactions have a finite range.We have applied the very same procedure for deriving PQED [11], but considering a massive photon in (3+1) dimensions.Our main result is that, after performing the dimensional reduction, one obtains that the matter field interacts through a Yukawa potential in the static limit.Similar to PQED, it yields a nonlocal theory in both space and time.Since the matter field is not relevant for the dimensional reduction, this model may be generalized to describe the Yukawa interaction between other kind of particles.Although our derivation follows standard steps of QED literature, we believe that they shall be relevant, in particular, for applications in condensed matter physics and cold-atom systems.
In Ref. [30], the authors proposed a realization of a cold-atom system made of fermions in two-dimensions with bosons in three-dimensions.This has been called Fermi-Bose mixture in mixed dimensions.Accordingly to their theoretical model, the static interaction between the fermionic particles is given by the Yukawa potential.Although, in their approach, retardation effects are neglected, we may make use of NPQED for investigating these new regime.Indeed, besides anomalies [17], it has been shown that a full dynamical description provides new results for PQED in application to exciton spectrum in transition-metal dichalcogenides [32].
We finally discuss the importance of the order in which the following two operations are implemented, namely: a) the mass generation for the gauge field; b) the dimensional reduction.Interestingly, the result is sensitive to the order in which the inclusion of a mass and the dimensional reduction are performed.As a matter of fact, by doing "a" before "b", we have shown that the Yukawa potential V (r) = e −mr /4πr is obtained from the static limit of the gauge-field propagator, given by Eq. ( 27).This is proportional to 1/(2 p 2 + m 2 ).
Conversely, let us start with a massless gauge field, hence, by applying the dimensional reduction, we arrive at the PQED model [11], whose propagator is proportional to 1/(2 p 2 ).Suppose we now couple the PQED model, to a Higgs field in the broken phase, such that a massive term is generated to the gauge field.This changes the propagator from 1/(2 p 2 ) to 1/(2 p 2 +m), which clearly shows that we shall obtain a different model from the one associated to Eq. (26).The potential is now given by a combination of Coulomb and Keldysh [31] potentials, i.e, V (r) ∝ 1/r − m [H 0 (mr) − Y 0 (mr)], where H 0 (mr) and Y 0 (mr) are Struve and Bessel functions, respectively.This potential does not decay exponentially at large distances as the Yukawa potential, rather, it has a power-law decay.We conclude, therefore, that we must to be careful about how one wishes to use quantum electrodynamics in applications for lower-dimensional systems, because of this sensitive relation between the dimensional reduction and phenomenological parameters, such as a mass for the gauge field.In this Appendix, we show some details about the electron-self energy.First, let us write the Feynman rules of Eq. ( 26) in Euclidean space.The free electronpropagator reads the gauge-field propagator (in the Feynman gauge λ = 1) is and the vertex interaction is Γ µ = γ µ e.The Euclidean matrices satisfy {γ µ , γ ν } = −2δ µν .From Eq. ( 35), we have that the pole of the free Dirac electron is p 2 E = −M 2 0 , where p 2 E is the Euclidean momentum.For calculating the physical mass, one must to return to the Minkowski space, using p 2 E → −p 2 M , such that p 2 M = M 2 0 are the physical poles.
The corrected electron propagator S F is The electron-self energy reads Eq. (39) has a linear divergence, therefore, we need to use a regularization scheme.We choose to use the usual dimensional regularization = 3 − D, where D is an arbitrary dimension, which we shall consider D → 3 in the very end of the calculation.After application of standard methods, we find where µ is an arbitrary massive parameter, generated by the prescription e → eµ , with = D − 3, where D is the dimension of the space-time.This is a standard step in the dimensional regularization.The constant C is given by and the parametric integral reads Eq. ( 40) has both finite and a regulator-dependent terms, which must be eliminated by some renormalization scheme.For the sake of simplicity, we choose the minimal subtraction procedure, which essentially avoid the poles by introducing counter-terms in the original action.Hence, where CT stands for counter-terms, and Using Eq. (44) in Eq. ( 38), we find Multiplying Eq. ( 47) by (1 + A(p)), we have Next, we define the renormalized matter field ψ R , namely, Therefore, The physical propagator reads with Using Eqs.(41), ( 46), (45) in Eq. (51), we find Eq. (51) yields the so-called mass function M (p), which is, essentially, the momentum-dependent part of the electron-self energy that renormalizes the electron mass [33].In Fig. 2, we show that by using different values of m/M 0 , we may generate quantum corrections that either increase or decrease the renormalized mass in comparison with the bare value M 0 .To clarify this result, we shall calculate the renormalized mass m R .The pole of Eq. ( 50) is given by the solution of p 2 E = −M 2 (p 2 E ), which, in the Minkowski space, yields  (53) Considering z R ≡ m R /M 0 and z ≡ m/M 0 , we have Eq.(33) and Eq.(34).In the scalar-field case, given by Eq. ( 11), the critical point z c is the same.This is not surprising because of the similarities of the electron-self energy and the bosonic propagators.
APPENDIX A: ELECTRON-SELF ENERGY:ISOTROPIC CASE

m 2 R
= M 2 (−m 2 R ), where m 2 R is the renormalized mass.Note that we have applied p 2 E → −m 2 R for calculating the pole in the Minkowski space.Furthermore, because we are at one-loop approximation, we may useM (−m 2 R ) = M (−M 2 0 ).Therefore, the renormalized masses are m R = ±M (−m R ) = E ± R , where E + R = +|M (−m R )| and E − R = −|M (−m R )| are positive and negative solutions, respectively.Using Eq. (51) with m 2 R = M (−M 2 0 ) and e 2 = 4πα, we find