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1. Introduction
Two dimensional relativistic Dirac fermions have been the subject of active research for many decades. These are by no means merely flat cousins of quarks and leptons on the high energy physics realm, but rather these particles offer an opportunity to explore vast and intriguing phenomenology in other branches of physics as well (see, for instance, Ref.1). The pioneer work of Wallace2 on the band structure of graphene paved the way to consider relativistic Dirac fermions in solid state physics realms. The realization of the quantum anomaly emerging from the electromagnetic dynamics of Dirac fermions and photons in two spatial dimensions was first simulated in the seminal work of Semenoff3 under condensed-matter considerations, whereas one the first glimpses into topological matter was achieved from the dynamics of relativistic Dirac excitations in a Quantum Hall set up without Landau levels, where the role of parity anomaly is seen in the quantization of the conductivity as discussed by Haldane4. High temperature superconductivity in layered cuprates5-9 has also been naturally explained in therms of the dynamics of planar Dirac fermions in these systems. All of these ideas have highlighted the intricacies and interesting features of the interactions among Dirac particles constrained to move on a plane. But the first isolation of graphene membranes by the Cambridge group10 and others11 with the overwhelming evidence of the relativistic nature of charge carriers in this material12 have indeed boosted the interest toward these systems.
The relativistic behavior of this material is rooted in the organization of the carbon atoms in honeycomb lattices (see, for instance,13). This structure is conveniently represented by two overlapped triangular sublattices (with a bipartite unit cell), as represented in Fig. 1.
Figure 1 Crystallographic structure of the honeycomb array. In the left panel, the two overlapping triangular lattices are rep- resented by the red and green points. The primitive vectors
The index labelling these sublattices constitute a degree of freedom for the charge carriers. In the tight-binding approach, electrons are firmly bounded to the atom except for one electron per atom belonging to the p orbital. Considering that the charge carrier can only hop to the nearest neighbor (which means that only inter-lattice hopping is allowed) one gets an expression for the energy that vanishes in six points, two of them non-equivalent. Those are called the Dirac points K and K’, located precisely where the valence band touches the conduction band and where the Fermi level is found. One can easily verify that in the vicinity of each of the Dirac points, the dispersion relation becomes linear, a feature typically associated to relativistic fermions. Besides that, the operators associated to the sublattice degree of freedom acquire a Pauli matrices structure in such a way that the system can be represented by a Dirac equation, where the speed of the light is replaced by the Fermi velocity v F , which turns out to be v F ≃ 1×106 m/s13. For this reason, an extensive work to characterize and explore graphene has been developed in the framework of quantum field theory (QFT). Moreover, the analytic structure of the Dirac equation unveils the fact that electrons in graphene have a chirality degree of freedom, which is another feature associated to relativistic particles and which has important implications for the electronic transport phenomena in this material.
Although in graphene the charge carriers are constrained to the plane, the external electromagnetic fields fermions may interact with are not. Because of that, usual Quantum Electrodynamics fully defined in (2+1)D (QED3), does not provide a suitable description. Among other inaccuracies, QED3 yields to a logarithmic static interaction (see, for instance, Ref.14) rather than the expected Coulomb interaction15-17 between electrons in graphene. Instead of that, in a QFT approach, electromagnetic fields must be represented by Abelian gauge boson fields in (3+1)D and the interacting theory therefore must deal with the subtlety of combining particles moving in different dimensionality.
Mixed dimensional gauge theories have been proposed in the nineties aiming to explain the then recent discovery of the Quantum Hall effect18. After that, several aspects of these theories were developed, in both formal and applied contexts. The first appearance of theories that provide a suitable background to explore fermions in two-space dimension interacting with gauge fields in three-space dimension occurred almost simultaneously in Refs.16-17. The procedure adopted in both approaches was to perform a dimensional reduction of the gauge field, integrating out its third spatial component, obtaining an effective interaction in (2+1)D. Among other features, this dimensional reduction softens the infrared behavior of the gauge boson propagator, which behaves as ~ 1/q 2 rather than the usual pole-dependence 1/q 2. In Ref.16 the authors obtain an effective Lagrangian in (2+1)D which for all intents and purposes corresponds to the mixed dimensional theory described above, naming the theory pseudo-quantum Electrodynamics (PQED). In particular, the authors focus on tracing a correspondence between PQED and the Chern-Simons theory. Further formal aspects of PQED were developed in Refs.19-20. On the other side, in Ref.17, the discussion is focused on the renormalization group aspects of graphene, and it is shown that the mixed dimensional theory possess a fixed point when the Fermi velocity evolves to the speed of light. The renormalization group analysis of this theory applied to graphene was posteriorly extended in Refs. 21-24. Recently, renormalization group techniques applied to PQED could successfully describe experimental data on the renormalization of the band gap in other two-dimensional materials like diselenide (WSe2) and molybdenumm disulfide (MoS2)25.
A more general procedure that could also be applied to systems where the charged particles are constrained to a lower dimensional string was proposed a few years later in Ref.26. The theory resulting from this method was named reduced-quantum electrodynamics (RQED). Although the framework developed in Ref.26 could be used to construct theories in arbitrary dimensions for the fermions d e and gauge bosons d γ (which then allow to label the extended theory as RQED dγ,de ), the major goal in Ref.26 was to analyse the dynamical chiral symmetry breaking in systems where two-spatially dimensional fermions interact with three-spatially dimensional gauge fields, focusing on condensed matter systems i . For this purpose, the authors solve the Schwinger-Dyson equations within an improved rainbow-ladder approximation and found that the dynamics of chiral symmetry breaking is rich and nontrivial. Posteriorly, the renormalization of RQED to one- and two-loops was deeply investigated27-30. The scale invariance obtained in previous approaches was confirmed in the context of RQED.
Following Ref.26, some activity involving the chiral symmetry breaking of dimensionally reduced theories took place. In Ref.31 the Schwinger-Dyson formalism within the rainbow approximation was applied to conclude that there is a critical coupling α = π/8 (in the conventions adopted in this work) above which there is room for dynamical mass generation in the theory. The existence of a critical coupling was confirmed in Ref.32. Additionally, a critical number of fermion families was determined from the similar structure of the gap equation in QED3 and RQED.
Chiral symmetry breaking was also explored in different external conditions: in Refs.33,34 the dynamical symmetry breaking was studied at finite temperature. In Refs.35,37 the dynamical mass generation was studied in RQED and PQED coupled to a Chern-Simons term. In Ref.38, anisotropy associated to strained graphene was shown to slightly affect the critical coupling. Moreover, in Ref.39 perturbative aspects of RQED in curved space was analysed.
Additionally, analysis of the Landau-Khalatnikov-Fradkin transformations (LKFT)40,41 was performed in RQED. These are non-perturbative transformations that connect the Green’s functions in different gauges and give valuable information about the renormalization coefficients in multi-loop calculations. Using the knowledge previously obtained in QED4 and QED342-44 the LKFT for the fermion propagator in RQED were derived in Ref.45.
In this manuscript we review the foundations of Pseudo- and Reduced-QED and focus on several aspects of dynamical chiral symmetry breaking explored via Schwinger-Dyson equations. In Sec. 2 we deduce the Lagrangian of the theory reviewing formal development as locality, unitarity and scale invariance. In Sec. 3 we review the formalism of Schwinger-Dyson equations, analyse general aspects of the gap equations, deduce the gap equations for RQED in the vacuum, analysing the wave function renormalization and renormalization group results. Section 4 is dedicated to study chiral symmetry breaking in a medium, including finite temperature effects, interaction with a Chern-Simons term and possible effects of curved space. In Sec. 5 we discuss how LKF transformations can provide information about renormalizability of the theory. Finally in Sec. 6 we present an outlook of the subjects discussed in this work.
2. Reduced or Pseudo QED: A historical recount
2.1. The Pseudo- or Reduced-QED Lagrangian
The derivation of PQED in Ref.16 and RQED in Ref.26 follow a similar procedure, consisting of performing a dimensional reduction of standard QED, imposing the matter fields to be constrained to a lower dimensional space. Although the general case, where an Abelian gauge field in arbitrary dimension interacts with fermions constrained to a generic manifold, has been explored in Refs.26,27, here we focus on the particular situation where four-dimensional QED is reduced to a three-dimensional theory that represents fermions in a plane.
The starting point is the usual QED4 Lagrangian,
where the first term is the Maxwell term, j u is a matter current that couples to the gauge field A, e is the electric charge, ℒ M is a generic matter kinetic lagrangian and ℒ GF is a gauge fixing term.
In order to describe matter confined to a plane, the current must be defined in such a way that there is no dynamics in the third spatial coordinate,
An effective interaction can be obtained integrating Eq. (1) over the gauge field Aµ This results in the generating functional (in Euclidean space)
where □ denotes the d’Alambertian operator.
Considering the definition in Eq. (2) and the usual photon propagator,
the effective action can be written as
where the (Euclidean) kernel is given by
and now x and x’ are defined in three dimensions.
It is possible to construct a theory fully defined in (2+1)D that mimics this effective action. This can be done noticing that the kernel in Eq. (6) can be written as a three-dimensional integral,
where here the d’Alambertian operator is defined in (2+1)D and the label E denotes Euclidean space. This yields the following effective action,
It is straightforward to verify that the action in Eq. (8) can be obtained from the Lagrangian
This is the Pseudo or Reduced-QED Lagrangian.
A remarkable consequence of the dimensional reduction, is that the photon propagator of RQED becomes proportional to
Defining the most general form for the propagator,
it is possible to find the coefficients a(k2) and b(k2) (see e.g.26),
Note that this propagator has a softer infrared behavior than the photon propagator in QED4 and QED3. We notice that several groups differ in conventions by a global factor of 1/2 in this propagator.
The Feynman rules for RQED are represented in Fig. 2.
2.2. Coulomb interaction
A crucial feature of this theory is that it correctly reproduces the Coulomb interaction expected for charged fermions, no matter if they are constrained to the plane. Naively considering full QED3, where the gauge fields are also in lower dimension, yields to a logarithmic interaction that does not correspond to what is observed in graphene15-17.
This can be seen considering a pair of static point charges with associated current
The interaction energy associated to the current defined above interacting with a gauge field is given by
Rewriting it in terms of the gauge field propagator,
Integrating over t’ and ω and taking μ = v = 0 to obtain the potential between static points,
Considering the current (13) and inserting the propagator in (12), one obtains, apart from unphysical self-interaction terms, that
namely, the expected Coulomb interaction.
2.3. Scale invariance
In opposition to QED3, in RQED the electromagnetic coupling e2 is dimensionless, since it originates from the four-dimensional theory. This can also be seen doing a classical power counting of dimensions in (9). The theory is therefore classically scale invariant.
In Ref.17, a 1-loop calculation in a graphene motivated model, similar to RQED, showed that the system flows towards a Lorentz covariant point in the infrared. Perturbative calculations performed in the
The renormalized fields and parameters can be defined in terms of dimensionless renormalization constants,
where ψ is the fermion field, Ã is the reduced gauge field, a is a gauge fix parameter, Γµ is the fermion-photon vertex and the index r refers to renormalized. Because of finiteness of the vertex and electron charge, there is a relation between the renormalization constants,
In what follows we briefly review the 1-loop calculation of the β-function performed in Ref.27. The 1-loop corrections to RQED propagators and vertex is represented in Fig. 3. For fermion fields constrained to a space in de dimensions and gauge fields in dγ dimensions, the diagrams in Fig. 3 are defined as,
where
where G(v1, v2) is the massless one-loop propagator defined by
For the configuration relevant to graphene and further two-dimensional Dirac/Weyl materials, d γ = 4, d e = 3, d γ ℰe = 1/2 and ℰ→0. The β-function is defined as
and the renormalized coupling constant α is defined as
where μ -2ℰγ is a factor to compensate the dimension of the coupling.
Figure 3 RQED 1-loop corrections for a) the photon propagator, b) the fermion propagator and c) the vertex.
At this order, it can be obtained from (21) that ZΓZψ = 1, which indicates that the Ward identity is satisfied. Therefore, according to (19), the charge renormalization constant Zα depends exclusively on ZA. The particular
Together with Eqs. (23) and (24), it yields to
Since this theory has no anomalous dimension associated to the gauge field, and ℰ γ →0, the beta function vanishes.
A similar but more troublesome calculation can be performed at 2-loops27-30, showing that at this order the β-function remains vanishing.
In order to generalize this result to all orders, it is argued in Ref.30 that there is no renormalization of the gauge field since it would come from a non-local term in the free part of the action while counterterms must be local polynomials in the fields and their derivatives. Reminding that the charge renormalization depends only on Z A , this would imply in scale invariance at all-order. It is pointed out in Ref.46 however that this argument does not hold for the present theory, since it is valid only for renormalizable theories, which does not happen to be the case. For a discussion on this topic we refer to46.
An alternative procedure was proposed in order to generalize the result at 1-loop and show that it is valid at all orders46. First of all an effective theory for the gauge fields is obtained integrating out the fermions in the action, namely,
Invoking gauge symmetry, it is possible to show that
Expressing
it is possible to define the all-order expansion of the effective action as,
with
By power counting and taking into account that the Furry theorem holds for RQED, the only contribution where divergences may arise is for n=2. This contribution corresponds to the usual transverse photon self-energy that leads to the following finite correction26,
The effective action for the gauge field then reads
which shows that the contributions from higher orders in the field strength are sufficiently ultraviolet-suppressed and give only finite corrections. Given that, it is safe to say that the renormalization constant ZA = 1 in RQED to all orders.
2.4. Renormalization group - RQED as a fixed point of a nonrelativistic model
Although one of the main achievements of RQED is to correctly describe several aspects of graphene, one point that deserves special attention is that usually the calculations are performed in a relativistic framework, where the Fermi velocity is taken to be the speed of light, or in natural units vF →1. Since the actual value of the Fermi velocity vF/c ≈ 1/300, the standard procedure of recovering vF in the end of the calculation must be done with extreme caution.
Previous to27,30, a renormalization group analysis was performed considering a mixed dimensional non-relativistic model17, where the Fermi velocity is taken into account. This breaking of Lorentz invariance can be explicitly appreciated in the action,
Here, a procedure similar to the one adopted in RQED was performed, where the gauge field was previously integrated out in the third space coordinate. As expected, this yields to the same dependence of the photon propagator on 1/q found before.
In order to fully renormalize this model, the Fermi velocity must be also renormalized, and one more renormalization constant must be introduced, v0 = ZVZR. This explicitly modifies the fermion self-energy as:
The one-loop calculation within this model was the first to predict the absence of charge renormalization for this type of mixed dimensional theory. Besides that, the renormalization constant for the Fermi velocity was obtained17,24,
Defining an effective coupling
where g plays the role of a fine structure constant with v replacing the speed of light. The Fermi velocities at different energies are related by
It is possible to determine the existence of fixed points, imposing
The Callan-Symanzik equation allows for obtaining an explicit form for the β-function17, and (39) is satisfied for vF = 1.
From this, one can conclude that the Fermi velocity grows as the energy decreases. Furthermore, the β-function has a non-trivial zero when the Fermi velocity reaches the speed of light. This limit corresponds to a Lorentz invariant weak coupling model whose coupling corresponds to the fine structure constant of QED. RQED therefore can be seen as a fixed point of the renormalization group to which the system represented by (34) flows. A detailed extension of renormalization group aspects of this mixed dimensional theory was performed in Refs. 21-24.
2.5. Causality and unitarity
As it usually happens to theories where some degrees of freedom are integrated out and the resulting theory is represented by an effective Lagrangian, RQED is nonlocal. Because of this, it becomes important to check if the theory respects causality and unitarity.
In fact, it has been shown that this mixed dimension theory respects both conditions. In Ref.19, the behavior of the classical Green’s function, advanced and retarded, is investigated inside and outside the light-cone. Generalizing the theory, considering a generic power α for the d’Alambertian operator in the denominator of the pure gauge term in the Lagrangian (9), it is shown that the Green’s function vanishes outside the light-cone. This implies that this family of theories respects causality for any value of the parameter α, including RQED, for which α = 1/2. Furthermore, for this theory the Green’s function vanishes inside the light-cone as well, being finite only on its surface. This means that RQED is constrained by a more strict condition than causality, it obeys Huygens principle19,47. While causality is found for any α, the Huygens principle applies exclusively to RQED, not holding for its cousin QED3.
Working with a more restrict generalization, where the power of the d’Alambertian operator in the RQED Lagrangian α = ENT#091;0,1), the validity of unitarity was explored in Ref.20. To this end, the authors invoke the optical theorem as follows. The S-matrix, that relates the initial and the final state in a scattering process, must be unitary so that the theory is unitary. Representing the S-matrix as S = 1 + iT, this implies that
which is the optical theorem. Evaluating the operator in the above equation between initial and final states, it can be represented in terms of the propagator,
where a phase space factor was introduced to ensure a correct dimensionality. For i→ƒ the propagator in (41) becomes the Feynman propagator and performing a Fourier transform, (40) yields to
Here Τ comes from integrating the phase space factor and is the characteristic time of the system, and γ = -2(1- α). The generalized Feynman propagator can be easily obtained from the generalized Lagrangian,
A straightforward algebra shows that the optical theorem (40) is obeyed exclusively for α = 0, which corresponds to QED3, and for RQED, where α = 1/2, as long as a particular relation between Τ and ℰ is respected. Therefore, unitarity holds for these two theories, while for other values of the parameter α it does not.
All this reasoning was obtained for the free theory. In the case of interacting theory a similar procedure can be performed considering a dressed propagator in the random phase approximation (RPA),
Inserting in the above equation the 1-loop expression for Πμν calculated in Ref.48, yields to a result similar to the one for the free theory. Results beyond RPA were also obtained using the 2-loop expression calculated in Ref.27. In this case, the functional form for the propagator remains the same, the only modification being the coefficients. Therefore, up to 2-loops the optical theorem still holds, fulfilling the unitarity condition.
3. Chiral symmetry breaking
3.1. Schwinger-Dyson equations
Schwinger-Dyson equations (SDEs) are the field equations of a given quantum field theory (see, for instance49). These conform an infinite tower of relations among the Green functions involved: n-point functions are related to other n-point and higher-point functions, each verifying its own SDE. In their formal derivation, no assumption is made regarding the strength of the coupling constant(s) of the theory. Therefore, these equations are non-perturbative in nature and provide a useful tool to understand phenomena like bound-states, chiral symmetry breaking and confinement. The only systematic scheme to truncate the infinite tower of SDEs is perturbation theory, but in this scheme none of the above mentioned phenomena can be addressed reliably. In gauge theories such as QCD and QED, symmetry-preserving truncations of SDEs have made a tremendous development in addressing several non-perturbative aspects of these theories, which would otherwise require other frameworks to be addressed, including lattice field theory and effective model considerations.
SDEs are formally derived from the observation of the vanishing of the functional derivatives of the connected Green functions generating functional with respect to the fields. An alternative diagrammatic derivation of these equations can be derived directly from the Feynman rules of the theory under study. In QED, for instance (see Ref.50), the perturbative expansion of the fermion propagator, depicted in Fig. 4 shows three types of corrections, those to the fermion propagator itself (first row of corrections), those to the gauge boson propagator (second row of corrections) and the third kind corresponds to vertex corrections (third row of corrections).
Figure 4 Radiative corrections to the fermion propagator in QED. These include corrections to the fermion propagator itself, to the gauge boson propagator and vertex corrections.
The infinite resummation of diagrams is better carried out by defining the fermion self-energy Σ(p), which is represented in the diagram in Fig. 5 and upon which the perturbative expansion of the fermion propagator contains all the radiative corrections shown in Fig. 4. In terms of Σ(p), the perturbative expansion of the fermion propagator is shown in Fig. 6.
Figure 5 Fermion self-energy. Blobs over the different parts of the diagram indicate that all perturbative corrections to these pares are already taken into account.
It corresponds to the expansion
The last line corresponds to the SDE for the fermion propagator depicted in Fig. 7.
It is convenient to re-write the above equation in terms of the inverse fermion propagator, which then becomes
and diagrammatically can be depicted as in Fig. 8.
This equation involves the full photon propagator and fermion-photon vertex. The former, by a similar reasoning, can be seen to obey its own SDE depicted in the diagram in Fig. 9.
This two-point function is coupled to the fermion propagator and the fermion-photon vertex which verifies the SDE shown in Fig. 10.
This diagram shows that the three-point function is coupled to two-point and four-point functions, hence illustrating the structure of the infinite tower of SDE.
At first glance, it seems impossible to envisage a non-perturbative truncation of the tower of SDEs without compromising the reliability of the predictions hence derived. Nevertheless, because of the gauge symmetry, a number of relations exist in which a (n + 1)-point Green function can be written on terms of n-point function. That is the case of Ward identities in QED51-54. Thus, one can attempt to propose a symmetry-preserving truncation and explore general features of the non-perturbative solution to the infinite tower of equations.
Below we shall review a favorite truncation to the SDE for the fermion propagator or gap equation, the so-called rainbow-ladder truncation. We also review how to improve of this truncation by including vacuum polarization effects and vertex corrections.
3.2. Structure of the gap equation
Let us consider the parity preserving ordinary version of QED. The SDE for the fermion propagator in an arbitrary number space-time dimensions d is given by
where α = e2 / (4π) is the (dimensionless) fine-structure constant and e is the electric charge. The most general form of this propagator is commonly expressed as
where F(p2) is the wavefunction renormalization function and M(p2) is the mass function. The tree level values of these functions are F(p2) = 1 and M(p2) = m0 where m0 is the fermion bare mass. Furthermore, Δμν (p - k) represents the full gauge boson propagator and Γμ (k,p) the full fermion-boson vertex. By neglecting vacuum polarization effects, i.e., working within the quenched approximation, we replace the full photon propagator Δμν (q) by its bare counterpart
where ξ is the covariant gauge parameter. In terms of diagrams, it amounts to consider Fig. 11.
Under this approximation, the gap equation can be solved with a suitable choice of the fermion-boson vertex. The rainbow-ladder approximation (see50) corresponds to the perturbation-theory inspired choice Γμ (k,p) = γμ, which allows to decouple the SDE for the fermion propagator from the infinite tower, as it corresponds to the diagram in Fig. 12.
Chiral symmetry breaking is usually approached starting with m0 = 0. Thus, the natural consequence of this phenomenon corresponds to the dynamical appearance of fermion mass where there was none. Upon inserting the fermion and gauge boson propagators and fermion-photon vertex in the gap equation, after multiplying it by
Let us explore the solutions to this coupled set of equations in different space-time dimensions.
3.2.1. Gap equation in d = 4
It is interesting to consider the scenario of chiral symmetry breaking in ordinary QED. After taking the traces and Wick rotate to Euclidean space Eqs. (50), by switching to hyper-spherical coordinates, angular integrals can be performed analytically. Radial integrations are divergent. However, regulating these integrals with a hard cut-off in the momentum, we reach to the following pair equations to be solved self-consistently,
where Θ(x) is the Heaviside step function. In Landau gauge, ξ = 0, we have that F(p2) = 1 and thus we have to solve the following non-linear integral equation for the mass function,
This equation has no analytical solution. Nevertheless, we can gain some insight by transforming it into the differential equation
For p2 >> M2(p2), the above linearizes to
which is restricted to the boundary conditions
The infrared cut-off κ2 is introduced to preserve the non-invariance of the original under M(p2) → cM (p2) with c constant as p2 → 0. Physically, it serves to quantify the amount of mass that is dynamically generated. We look for solutions to Eq. (54) of the form
which after substituting in Eq. (54) yields
Solving for s, we have that
Thus, depending on the value of α compared to the value of αc = π / 3, we could observe an oscillatory behavior or a power-law solution. Consistency with boundary conditions demands α > αc. Moreover, the infrared and ultraviolet cut-offs are constrained as
This behavior is called the Miransky scaling law which states that the chiral symmetry breaking in QED corresponds to a conformal phase transition55. Thus, we conclude that in order for chiral symmetry breaking to be broken in ordinary QED, the coupling α must exceed a critical value.
3.2.2. Gap equation in QED3
A similar reasoning can be followed to explore the scenario of chiral symmetry breaking in QED restricted to a plane, namely, QED3. The main difference in this case is that the electric charge e2 has mass dimension one and hence serves as a natural scale for all mass scales in the theory. Furthermore, QED3 is super-renormalizable, and thus there is no need to regulate the integrals. In the rainbow approximation and in Landau gauge the gap equation, reduces to (see Ref.56)
At this point, although the integrals are finite, one can introduce an ultraviolet regulator Λ in the integral with Λ >> e2 and consider at the end the limit Λ → ∞ maintaining e2/Λ fixed. Proceeding in this way, after expanding the logarithm for k >> p and p >> k, the integral reduces to
This equation can be transformed into the differential equation
subjected to the boundary conditions
Again, in the regime where p2 >> M2(p), the differential equation (62) linearizes in the following form
which admits the general solution
where Jv (x) and Yv (x) are Bessel functions of the first and second kind of order v, respectively. UV boundary condition demands that C2 = 0. The constant C1 cannot be fixed from the IR conditions. Nevertheless, noticing again that the nonlinear is non-invariant under M(p) → cM(p) with c a constant, we require to change that boundary condition by demanding that
in the IR, where κ is a regulator quantifying the amount of mass being generated. Such boundary condition allows to write the solution to the gap equation as57
With
It is interesting that this solution is positive definite for
Figure 13 Multiple solutions to the gap equation in QED3. The solid black curve corresponds to
This pattern continues ad infinitum and emerges from the analytical properties of the gap equation, which corresponds to a Hammerstein equation of the first kind. This structure is particular of the truncation and is modified when some of the assumptions are removed from consideration. See Ref.57 fir further discussion.
3.2.3. Gap equation in QED3 including vacuum polarization effects
Including vacuum polarization effects in this model turns out interesting. Let us consider the scenario where we include a large number N of massless fermion families circulating in loops. This amounts to ressuming an infinite number of planar diagrams as shown in Fig. 14 such that one considers the unquenched approximation depicted in Fig. 15.
In this case, the non-pertubative gauge boson propagator can be calculated exactly. In an arbitrary covariant gauge, it reads58-60
where
Once more, by expanding the logarithm for κ >> p and p >> κ, we have
This expression is equivalent to the differential equation
subjected to the boundary conditions
In the large-coupling regime
Moreover, in the momentum domain
Equation (75) admits a power law solution of the form
which upon substitution into (75) yields
Thus, the solution to the linearized equation is
Again, we notice the existence of a critical number of fermion families, Nc = 32/π2 that distinguishes an oscillatory from a decaying behavior of the mass function. Consistency with boundary conditions suggests that Nc > 32/π2 and the amount of dynamically generated mass, quantified as M(p = 0) also follows a Miransky scaling law of the form
Thus, this scenario has a similar scaling behavior as ordinary QED. Furthermore, vacuum polarization effects translate to the existence of a critical (large) number of fermion families above which chiral symmetry breaking is no longer possible and the chiral symmetry restoration corresponds to a conformal phase transition55.
It is interesting that the gauge dependence of the mass function and wavefunction renormalization have been considered in covariant gauges61 and up to the next-to-leading order of the approximation62-65. These ressumations carried out in an arbitrary non-local gauge do not spoil the existence of the critical value Nc for the restoration of chiral symmetry and only refine the value of this critical number Nc. The conclusion up to date is that such number is an integer Nc ≤ 3.
3.2.4 Gap equation in RQED
So far we have reviewed different scenarios for chiral symmetry breaking in ordinary QED. The gap equation in this theory is such that the wavefunction renormalization is trivial in Landau gauge. The scenario of criticality emerges in d = 4 as the need for the coupling to exceed a critical value for the theory to be able to break dynamically the chiral symmetry. In three dimensions, the quenched theory does not exhibit this feature. Chiral symmetry breaking can be broken for arbitrary values of the coupling. Nevertheless, including vacuum polarization effects, it is observed that if the number of fermion families circulating in loops exceeds a critical value, chiral symmetry cannot be broken. This statement is valid up to 1/N2 and in arbitrary non-local gauge. An interesting observation is that when fermions in loops remain massless, there is a softening of the infrared pole in the propagator 1/q2 as the gauge boson momentum q→0 to a 1/q softer behavior. This behavior is similar to the tree-level gauge boson propagator in RQED
Notice that in this case, the gauge parameter ζ is not the same as ξ in the ordinary theory, but here it has to be renormalized because fermions and bosons live in different dimensions. In fact, ζ = ξ / 2, and hence ζ = 0 corresponds to Landau gauge.
Following the reasoning of the previous subsections, we truncate the gap equation in the rainbow approximation as in Ref.31. In an arbitrary covariant gauge, it is equivalent to the following coupled system of equations (see66)
with the dimensionless coupling α = e2 / 4π as usual. It is evident from the above expressions that for ζ = 0, F(p) ≠ 1. Nevertheless, as a first approximation, let us take F(p) = 1 and explore the solution of the gap equation. Then, the gap equation becomes the non-linear integral equation for the mass function31,66
This expression can be straightforwardly converted into the following differential equation
restricted to the boundary conditions
Upon linearizing the differential equation 83 when p >> M(p), we can again write the resulting equation in the Euler-Cauchy form
which, admits a general solution of the form
where
As in previous cases, the non-invariance of the gap equation under scalings M(p) → cM(p) where c is a constant demands the introduction of an infrared cut-off κ and the modification of the infrared boundary condition to
Thus, defining αc = π / 8, boundary conditions demand that α > αc. Furthermore, the dynamical mass obeys the Miransky scaling
with A = -2π and ∂ = -4, hence indicating that the chiral symmetry is broken in a conformal phase transition provided the coupling exceeds a critical value.
In order to explore the effect of the wavefunction renormalization in the result89, particularly in relation with the gauge dependence, below we explore some variants of the rainbow-ladder truncation following66.
3.2.5. Effect of the wavefunction renormalization
To explore the sensitivity of the truncation to the effect of the wavefunction renormalization function in connection with the gauge parameter dependence of the critical coupling and the dynamical mass, we conduct the following exercise:
First, we impose F(p) = 1 in all covariant gauges and solve the equation
for various values of the gauge parameter ζ.
As a second variant, we solve the coupled system of equations in (81) for various values of the gauge fixing parameter.
Next we impose the Ward-Takahashi identity (WTI),
which allow to split the vertex into its longitudinal and transverse pieces,
where
the gap equation can be written as
with q = k - p. In the final term we replace the identity (91) and upon taking traces after multiplying by 1 and
Then we solve the coupled system of equations for various values of ζ.
As a final variant of the gap equation, we use the Ball and Chiu vertex67, which is explicitly constructed to verify the Ward identity and is written as follows
This is a standard choice for the longitudinal piece of the vertex and it is explicitly constructed to satisfy these relations. Nevertheless, the additional terms to the central part exhibit spurious kinematic singularities as k2 → p2. These spurious singularities do not appear in perturbation theory and are unwanted in a non-perturbative construction of the vertex. These actually are cancelled by more educated ansatze of the vertex which include the transverse piece unconstrained by these identities. Thus, for our purposes, by keeping only the central part, which is
we guarantee that the ward identities are satisfied up to spurious singular terms which are not expected to appear in a complete form of the vertex.
Then, the gap equation corresponds to the following coupled system of equations
In Fig. 16 we compare the findings of the critical coupling in different gauges for the variants of the truncation described above.
At first glance, it might seem surprising that the simplest scenario of neglecting wavefunction renormalization effects and solving the mass function alone yields the least gauge parameter dependent results. This is better understood from a renormalization group analysis, that exploits the similarities of the 1/N approximation in QED3 and RQED itself, as discussed below.
3.3. Renormalization Group Analysis
It is remarkable that RQED becomes an infra-red Lorentz invariant fixed-point for a general theory of mixed dimensions for fermions and gauge bosons, RQED dγ,de 17, because it allows to analyze the critical behavior of the general theory, regarding chiral symmetry breaking, around such point. The important observation is that as we approach this fixed point, the Fermi velocity of fermion fields tends to the speed of light in vacuum and at the same time, the coupling constant approaches the fine-structure constant of QED. Therefore, one can exploit these facts to compare the critical structure of RQED and QED332,62-65,68 This is particularly relevant to understand the gauge dependence of the critical numbers associated to the chiral transition.
In order to make explicit the mapping between QED3 and RQED, we start by expressing the gauge boson propagator in RQED as32
where we recall that the gauge fixing parameter ζ of RQED is half of the corresponding parameter of QED. On the other hand, the leading behavior of the gauge boson propagator (69) of QED3 in the large N approximation has the form
Thus, identifying
we can simply map the solutions of the gap equation in RQED to those of QED3 in the 1/N approximation.
For instance, the gauge parameter dependence of the critical number of fermion families for chiral symmetry restoration in QED3 was explored in Refs.32,68. The authors find that the critical condition can be written as
Then, from (100) we can straightforwardly find that for RQED, this result is translated to the critical coupling
which shows a strong gauge dependence of the critical coupling, although it does not depend on the flavor number N. This result comes up as a consequence of a partial re-summation of diagrams in the approximation. It can be improved, for instance, by considering a RPA calculation of the vacuum polarization tensor. In the RPA, the gauge boson propagator has the leading behavior
Thus, by redefining the coupling as
the gauge dependence of the critical coupling is now
with
At next-to-leading order, one observes that the gauge dependence of the critical coupling has the form
where
and
with the coefficients
This is a rather intricate form of the coupling which nevertheless improves upon the leading order gauge parameter dependence. The RPA still helps to reduce the resudial gauge dependence of the critical coupling by redefining
Thus, we observe that in this framework, the appropriate re-interpretation of the coupling permits a less severe gauge dependence of the critical value of the coupling to trigger chiral symmetry breaking.
4. Chiral symmetry breaking in a medium
Mixed-dimensional theories allow natural extensions of the QFT formalism to incorporate the effects of (classical and quantum) external agents like a heat bath and/or external electromagnetic, strain or gravitational fields, among others. Generally speaking, these effects have the potential to enhance or inhibit phase transitions. In the case of the chiral transition, a heat bath is known to have the effect of restoring chiral symmetry. The imaginary-time formalism of thermal field theory (TFT) is a natural extension to consider in this case. On the contrary, electromagnetic fields promote the breaking this symmetry. In particular, in 3D materials, the configuration of parallel electric and magnetic fields gives rise to a special configuration that can be understood as the presence of a Chern-Simons (CS) term in the electromagnetic Lagrangian, with the added possibility of parity and time reversal symmetry breaking. In 2D systems, the presence of such a term for fermion fields is realized as the possibility of mass terms different from the ordinary Dirac mass.
A Haldane term of this type has interesting effects on the chiral phase transition in ordinary QED3. Curvature effects are also important in modifying the electric and optic properties of the materials and are generally expected to occur in materials with defects. In this section, we review the effect of a thermal bath, the role of a Chern-Simons term, the influence of strain and the formulation of RQED on curved space. We specialize the discussion in how these effects impact the scenario for chiral symmetry breaking.
4.1. Finite temperature effects
The scenario of chiral symmetry breaking in RQED has been considered at finite temperature. Let us revisit the calculations in Refs.26,31,33,34. We start our discussion from the imaginary time formalism of TFT. We introduce the fermionic Matsubara frequencies ω n = (2n +1) πT and replace of any integral over the temporal component of an arbitrary three-vector p = (p0, p) by the summation over frequencies,
Under this prescription, working in the Landau-like gauge ζ = 0 and neglecting wavefunction renormalization effects, the gap equation takes the form33
where Mn (p) = M(ωn, p) is the mass function associated to every Matsubara frequency and the notation ?Λ indicates that divergent integrals are to be regularized with an ultraviolet cut-off Λ.
Several considerations are at hand. First of all, as customary in TFT, one would be tempted to perform the sum over Matsubara frequencies and introduce the cut-off in momentum integrals. Nevertheless, because these frequencies appear inside the square-root, the standard techniques to perform such sums are hard to implement. First of all, the convergence of the sum
is ensured (see Ref.33) provided the sum in the last row of the above expression converges. In Eq. (33), it is assumed that A2, B2 > 0 and
which is convergent and satisfies that
Then, because V is convergent, so it is
Alternatively, one can perform the momentum integrals first and then the sum over Matsubara frequencies. In this case, one can assume that the mass function is the same for all the frequencies such that the gap equation can be written as33
where
K(x) denoting an elliptic function and
Considering only the zeroth Matsubara frequency n = 0, an approximation that is commonly used in high-temperature regime studies, the gap equation reduces to33
Expanding kK(x0) / |k - p| for k >> p and k << p, the gap equation becomes
Upon linearization, the resulting integral equation can be cast in the form of the differential equation
subject to the boundary conditions
where Tc would correspond to the critical temperature for chiral symmetry restoration, which will be defined from the analytical behavior of the mass function. The general solution to Eq. (121) is expressed as
where J1(x) and Y1(x) are Bessel functions of the first and second kind. UV boundary condition imply C2 = 0. At the same time, the IR boundary condition imposes the following relation
with
exceeds the critical value of the coupling in vacuum33. It has it s origin in the fact that there is no evidence that the mass function M(p) vanishes at the critical temperature. Therefore, a more refined treatment is called for.
One can improve on the above result recalling that the phenomenon of dynamical chiral symmetry breaking is infrared. Therefore, one can simply neglect the external momentum in the gap equation (119), which then becomes33
where we have introduced the shorthand notation m(T) = M(0, T). Retaining
Keeping the positive solution alone, one can derive the critical coupling for chiral symmetry breaking of the form
Nevertheless, this expression is oblivious to the critical value of α at T = 0.
A different view of the gap equation can be achieved as follows34. Let us introduce dimensionless quantities,
such that the gap Eq. (112) can be written as
where θ is the angle between σ and ρ
ii Notice that in this case, the cut-off is not introduced in momentum integrals, but in the number of frequencies Nƒ that are summed up. Whithin the so-called constant mass approximation69, all mass functions involved in the gap equation can be replaced with their values at zero momentum. Denoting
Fixing the the cut-off Nƒ such that
we are allowed to write all temperature scales as
Specializing on the zeroth Matsubara frequency m = 0, observing that Sn/S0 ≤ 1, and recalling our assumption that near the critical point the temperature-dependent Sm (Ť)→0 we have that34
The sum over n is finite for every value of temperature. This allows to obtain the behavior of the critical coupling, αc, as shown in Fig. 17.
The behavior of the critical coupling as a function of temperature is consistent with
with a,b real numbers. This behavior exhibits the correct behavior of
It is remarkable that within the finite temperature framework, the gauge dependence of the critical temperature and coupling is a pending matter up to date.
4.2. Chiral symmetry breaking with a Chern-Simons term
Physics on two spatial dimensions is very interesting in its own right. For electromagnetic interactions, there exists the possibility to include a Chern-Simons (CS) term in the gauge sector through the Lagrangian
where ϑ is the CS coefficient (see, for instance, Ref.70). Such a term induces a topological mass to the gauge bosons. It also induces fractional (anyon) statistics for matter particles as well as parity and time-reversal symmetry breaking of the theory, among other interesting effects. It is through this term that, for instance, the fractional quantum Hall effect can be explained and the emergence of the transverse conductivity without Landau levels can be explained.
The scenario of chiral symmetry breaking in QED3 with a CS has been explored by solving the gap equation in different truncation schemes71-74. The CS coefficient acts as an effective vacuum polarization such that when a large number of fermion families are considered, besides the existence of a critical number of families N c that restores chiral symmetry, there exists a critical value for the CS coefficient ϑ c that also induces chiral symmetry restoration. The order of the phase transition in this case changes to order one.
In RQED, the impact of the dimensionless parameter ϑ has been explored by our group (for a study exploring the effects of a modified Chern-Simons term containing a dimensionful parameter see75 and for other approaches involving time reversal symmetry breaking see76). The starting point is the addition of the CS term (136) in the RQED Lagrangian. The effective theory under consideration has the structure36
where all the quantities carry their usual meaning. The tree-level gauge boson propagator derived from the above Lagrangian is
From here it is immediate to notice the effect of the CS coefficient as an effective dielectric constant37 that has the potential to modify the conditions for chiral symmetry breaking. Moreover, for fermions, it is convenient to introduce their right- and left-handed projections, ψ± = Χ±ψ, defined from the chiral matrix τ = [Y3, Y5]/2. In this form, the chiral projectors are defined such that
which fulfill the following properties
The advantage of working with these projectors is the following: In order to explore the general scenario of chiral symmetry breaking and its consequence, the dynamical generation of fermion masses, we consider the possibility of emergence of an ordinary Dirac mass term
These masses however do not correspond to poles in the fermion propagator. The operators χ± project the upper and lower two-component spinors, making explicit that the would-be poles of each fermion species correspond to combination of Dirac and Haldane mass terms. Explicitly, upon acting with the chiral projectors, the matter Lagrangian can be written as
where M+ = me + m0 and M- = me - m0.
As mentioned earlier, the presence of a CS term generates a mass for the photon in QED3, where ϑ has mass dimension one. Such a mass is known as topological mass (in this case no real topology is involved and the terminology comes from historical reasons). In QED3, radiative corrections may give a contribution to this mass. For the case of QED3 incremented with a CS term, the Coleman-Hill theorem states that only one-loop radiative corrections may be non-vanishing, with the other corrections being identically zero to all orders. We note that the argument to demonstrate this theorem is valid for a theory with massive fermions. Remarkably, it was shown that the Coleman and Hill theorem is also valid for RQED77 in spite of the fact that the CS coefficient is dimensionless. Therefore, the only possible correction to the photon propagator comes from the one-loop diagram Fig. 3a).
In order to address the issue of chiral symmetry and parity breaking, in Ref.35 a robust truncation of the SDE was incorporated. To this end, the possibility of Fermi velocity renormalization by splitting the fermion momentum P = (p0, p) into its temporal p0 and spatial p components such that the propagator reads
Tha gap equation is truncated including a gauge-boson-fermion vertex dressing model and vacuum polarization effects. The former is based in the central Ball-Chiu vertex, which in this formalism takes the explicit form
where
whereas the latter are included modelling a Coulomb-like approximation. Noticing that the most general form of the gauge boson propagator can be written as
where the involved tensors are
with
It turns out that these vacuum polarization effects enter into the fermion gap equation only through the combinations GL and GDE = GD + GE. Then, the gap equation becomes a system of equation for the six unknown functions
The gap equation supports solutions which break parity, associated to the generation of a Haldane mass.
Non-trivial parity-preserving solutions are found that, nevertheless break chiral symmetry.
The Dirac dynamical mass reduces as θ increases. This confirms that the CS coefficient behaves as an effective dielectric constant, as was proposed in39.
A point-wise evolution of the dynamical mass in terms of the coupling and the CS term was considered by some of us in Ref.36. Assuming that Fermi velocity renormalization has already been carried out and thus no need to split the spatial and temporal components of the momentum, a rainbow-ladder truncation of the gap equation was adopted with the additional simplification of neglecting the wavefunction renormalization effects setting F ±(p) = 1 in Landau gauge. The gap equation is found equivalent to the following uncoupled system of non-linear integral equations for the left and right handed mass functions
Inspired by the reasoning of the previous section, the gap equation can be linearized and reduced to the following differential equation
with the boundary conditions
where κ is the IR regulator introduced as a scale that quantifies the amount of mass being dynamically generated. The general solution to Eq. (150) is
where
and
We notice that when ϑ = 0, ƒ(ϑ, p) = 0 and we obtain precisely the solution discussed in (89) which demands α > αc = π/8. For ϑ ≠ 0, we observe the role of the CS coefficient ϑ as an effective dielectric constant. The function ƒ(ϑ, p) describes deviations from the Miransky scaling by virtue of the CS coefficient.
The numerical solution to the non-linear Eqs. (149) are shown in Figs. 18 and 19 for ϑ = ηϑc with varying 0 ≤ η ≤ 1 but keeping α = 1.07 αc fixed. These numbers are consistent with a critical value of ϑc = 6.6 × 10-14. For small values of p, we observe that both masses are finite and approximately flat as p → 0. For larger momentum, the height of the mass functions decreases. This happens for different values of ϑ provided α > αc. Furthermore, Figs. 20 and 21 show that the dynamical masses depend on the magnitude of the CS coefficient in opposite forms for right- and left-handed fermions. While M_(0) increases M+(0) decreases as ϑ grows bigger. At the critical value ϑc, M+(0) flips sign abruptly, signaling a first order phase transition.
Figure 18 Mass function M_(p) plotted as a function of momentum. The different curves correspond to different values of the CS parameter ϑ = ηϑc with 0 ≤ η ≤ 1 and fixed value of the coupling α = 1.07 αc.
Figure 19 Mass function M+(p) plotted as a function of momentum. The different curves correspond to different values of the CS parameter ϑ = ηϑc with 0 ≤ η ≤ 1 and fixed value of the coupling α = 1.07 αc.
Figure 20 The absolute value of the height of the right-handed fermion mass function M+(0) plotted as a function of the CS parameter ϑ. Above ϑc the mass M+(0) becomes negative and here we flipped its sign for a better visualization. The different curves correspond to different values of the coupling α = xαc.
Figure 21 Height of the left-handed fermion mass function M_(0) plotted as a function of the CS parameter ϑ. The different curves correspond to different values of the coupling α = xαc.
Notice that the heights M +(0) of the mass functions parametrize how much mass has been generated. These hights are depicted in Figs. 20 and 21 for a constant value of α = xαc. Above ϑc, M+(0) undergoes a discontinuity and jumps assuming negative values, as shown in Fig. 22, where we have flipped the sign after the discontinuity to lead the eye. The blue dots stand for the absolute value of M+ above ϑc, where we observe that the two masses become the mirror image of one another. We notice that since the curves are re-scaled, they coincide for any value of x > 1.
Figure 22 M+(0) re-scaled by their value at ϑ = 0. The solid line corresponds to M+(0) and dashed line M_(0). The red dots are the absolute value |M+(0)| above the critical ϑ.
The Dirac and Haldane masses that appear in the Lagrangian (141) can be obtained summing up or subtracting M+(0), respectively. From Figs. 23 and 24, we notice that both these masses are non-vanishing for a large range of values of ϑ, clearly showing that CSB can occur within this model. For small ϑ, the Dirac mass me, associated to chiral symmetry breaking, presents a plateau, leaping at a critical ϑ to a value several orders of magnitude smaller. After that, me increases again and finally drops to zero. We interpret the discontinuity as an attempt to restore chiral symmetry, associated to the first piece in Eq. (149). That would be the critical point if ƒ(ϑ, p) as absent. However, as one moves towards the right of the discontinuity, the CS contribution ƒ(ϑ, p) becomes dominant and me starts to be significant again. For even larger ϑ the Dirac mass drops and the real critical ϑ, for which chiral symmetry is restored, is around ϑ = 1. In turn, parity breaking is encoded in m0. As the CS parameter becomes dominant, the Haldane mass increases, never restoring the symmetry.
As in the finite temperature case, the gauge dependence of these findings represent an open avenue to explore73,74. Furthermore, feedback effects of the dynamical mass and the CS coefficient ϑ need to be incorporated by solving the SDE for the fermion and photon propagators simultaneously.
Figure 23 Dynamical mass term, even under parity transformation, as a function of the CS parameter ϑ.
4.3. Anisotropy effects on chiral symmetry breaking
Straintronics has emerged as the field in which the electrical properties of graphene and other materials are manipulated through mechanical deformations (see, for instance, Ref.78). Effects of anisotropy (strain) in the gap equation have been recently considered in Ref.38. By writing the fermion propagator of RQED with strain as
with
the gap equation is considered as a function of the anisotropy parameter η = ν1/ ν2. Including one-loop vacuum polarization effects and a suitable generalization of the Ball-Chiu vertex, the authors of Ref.38 solved the gap equation and searched for the effect of η on the critical coupling. They observe that the net effect is to increase the value of αc for chiral symmetry breaking.
4.4. Chiral symmetry breaking in curved space
It is well known that for 2D materials, electric and optical properties are modified by impurities and deformation. The latter can be introduced in a framework in which these mechanical deformations can be incorporated in terms of curved geometry of the underlying space-time in the equations of motion. In this regard, the dynamics of gauge bosons and electrons in mixed space-time dimensions has been considered in a curved-space background for the latter (and flat extra dimensions for the former)39. The starting point is the action
where
with i, j = 1,2 and thus
Considering the representation
where in the case of graphene xdγ-de corresponds to the third spatial dimension. In fact, assuming that all extra dimensions are flat, the action of this theory is very similar to the corresponding action of QED is curved space39, namely,
where
where
where R(x’) is the Ricci scalar. For graphene,
The one-loop renormalization of the fermion propagator has been discused in Ref39. All divergent terms are the same as in curved-space ordinary QED. Curvature effects are only seen in the finite parts of the self-energies. As for the polarization tensor, such a term is found finite, hence making the beta function null at the one-loop level. Furthermore, the curvature effects in the case of graphene show up as an effective chemical potential of the form
5. Landau-Khalatnokov-Fradkin transformations
Gauge symmetry lays at the very foundation of the description of fundamental interactions. It can manifest in different ways at many levels. For the Green functions of QED, different sets of relations among them can be derived from the fundamental symmetry of the theory, as for example Ward-,51 Ward-Green-Takahashi-51-53 and Transverse Ward-identities54,79-82 which relate (n + 1)-point to n-point functions in constructions resembling divergence and curl of currents. In a different setting, the so-called Landau-Khalatnikov-Fradkin transformations (LKFT)40,41 which have been analyzed in different versions of QED and extended to non-Abelian gauge theories like QCD, have been derived from different arguments in the past decades83-87. These transformations are non-perturbative in nature and hence have the nature to address the issue of gauge invariance in perturbative and non-perturbative studies of field theories.
A widely used example of LKFT has been in the case of the fermion propagator. It has been used to establish the multiplicative renormalizability of spinor and scalar QED in different space-time dimensions88-91. The main lesson is that this transformation fixes some of the coefficients of the perturbative expansion of the fermion propagator at all orders. In the case of RQED, given that WTI are satisfied, it turns interesting that the LKFT serves as a tool to establish general features of multi-loop calculations as well as non-perturbative calculations in connection with chiral symmetry breaking. In what follows of this section, we review the LKFT in RQED.
5.1. LKFT for the fermion propagator in equal dimensional QED
We start from the most general form of the Dirac fermion propagator in arbitrary space-time dimensions and in momentum space and relate it to its coordinate space representation
through a Fourier transformation. We have additionally added the gauge parameter label ξ as we are interested in the form of these propagators for different gauges.
For theories of fermions and bosons in the same space-time dimensions, the momentum space free gauge boson propagator
such that the longitudinal portion of the propagator, inversely proportional to q 4, describes how this Green function changes in different gauges, being proportional to the gauge fixing parameter. The LKFT emerges precisely from this part of the propagator40,41,83-87. These transformations are more clearly written in coordinate space. For the fermion propagator, the LKFT states that the fermion propagator in an arbitrary covariant gauge S(x; ξ) is related to its form in Landau gauge S(x; 0) through the transformation
where
e denoting the electric charge, and μ is an energy scale that renders e dimensionless in d = 4, but yields a dimensionful coupling e2 in QED3. Performing the momentum integration, Δd (x) is explicitly given by43
where α = e2/(4 π) is the coupling constant, and Γ(z) is the Euler Gamma function.
To taste a feeling of the structure of LKFT, let us take F(p; 0) = 1 and M(p; 0) = 0, namely, the massless fermion propagator in Landau gauge as a seed of LKFT.
For d = 3, the transformation reveals that the wavefunction renormalization in an arbitrary covariant gauge is (see, for instance, Ref.42,43)
with
5.2. The LKF transformation for RQED
Let us consider the free gauge boson propagator over bulk dimensions in RQED. It is of the same form as in Eq. eqn:phototond, but when considered reduced to the de -dimensional brane, it becomes27,29
with,
where εe = (dγ - de)/2 and
we define
With
This is the general form of LKFT for the fermion propagator in RQEDdγ - de first presented in REF.45. Note that when dγ - de = d, Eq. (173) matches the usual form of the LKFT Eq. (166) in any dimension d. For graphene and other 2D materials,
where the limit ∈→1/2 is understood. Expanding Eq. (176) around ∈ = 1/2, defining δ = ∈ = -1/2, we get
Since the transformation function, Eq. (177), diverges at x = 0, we introduce a cutoff xmin and consider
where we have defined v = ξα / (4 π), and the dimensionless coupling constant
With Eq. (178), we can obtain the non-perturbative structure of the fermion propagator in RQED starting with the tree-level massless case in Landau gauge. In coordinate space, we have
whereas in momentum space,
This is the non-perturbative form of the fermion propagator in RQED in any covariant gauge ξ. The power-law behavior of this two-point function is consistent with the multiplicative-renormalizable character of the theory. We notice that a similar behavior is found for the propagator in ordinary QED.
Next, expanding this result in powers of α, we get
with
where ψ(z) is the digamma function, ζ(s) is the Riemann Zeta function, and we have made use of the identity ψ(3/2) = γE - ln(4) + 2.
We make a comparison against the exact one-loop calculation of the wavefunction renormalization28,29,
where
and μ is the renormalization mass scale. This correction has a contribution that is proportional to the gauge parameter and one that remains finite in Landau gauge. Thus, we can only compare against the former from the LKFT result. F1 defined in is equivalent to
At
with the identification Eq. (186), as expected. Thus, we have shown that there is full consistency between our LKFT result, Eq. (181), and the perturbative result, Eq. (184), up to order α2. We hence predict the form of all the coefficients of the form (αξ)j in the all order perturbative expansion from our LKFT result Eq. (180).
5.3. Extensions of the LKF transformation
It is interesting to notice that the LKF transformation can be represented directly in momentum space. In Refs.92-95 it is discussed the form in which the wavefunction renormalization is related in two gauges labeled by the parameters ξ and η, respectively. Assuming that in gauge η the multiloop expansion of the wavefunction renormalization is
with ∈ = 2 - d/2 and ũ is a renormalization scale, the LKFT implies that
where the new coefficients
With the appropriate choice of the renormalization scale ũ, the multiloop expansion of the fermion propagator can be obtained in an easier form.
Moreover, taking advantage of the above mentioned multiplicative renormalizability of the theory, by factorizing
one can see from LKFT that
Thus, for the fermion anomalous dimension
where β(α) is the beta function and γ(α) the gauge boson anomalous dimension, given explicitly as
where β1 = 0 for graphene. From here its is evident that the fermion anomalous dimension in two gauges, namely,
i.e., the gauge dependence is encoded in the one-loop correction alone, whereas higher loops are gauge invariant.
The two-loop correction to the mass anomalous dimension γm has been explored very recently in68. In this work, the authors consider a theory in which the gauge boson lives in 4 dimensions, but the fermion fields are allowed to live in a general dimension de. From the multiplicative renormalizability of the fermion mass,
where the renormalization constant Zm is obtained from the relation
where Σs and Σv are the scalar and vector projections of the fermion self-energy. Furthermore, the mass anomalous dimension runs with the energy scale μ as
Using the
where N is the number of fermion families and ζ2 is the Euler function. This mass anomalous function is completely gauge invariant, and thus the critical coupling and/or critical number of fermion flavor derived from γm are also gauge invariant.
6. Outlook
In this review article we have revisited several aspects of a mixed-dimensional theory of electron and gauge boson interactions aiming to describe the new family of 2D Dirac material like graphene and its cousins. Quantum field theories in mixed dimensions have been discussed theoretically for over 4 decades in literature, but the isolation of graphene has indeed boosted the interest of these highly non-local yet well behaved theories. The fact that RQED, the theory we have specialized in, is a fixed point in the renormalization group allows to extend general conclusions from the behavior of the Green’s functions in RQED. From the historical point of view, we have revisited the formulation of the theory and the most fundamental aspects that render the theory suitable for phenomenological description of the properties of a variety of Dirac materials. The fact that the theory allows a standard 1/r fall-off for static Coulomb interactions to the observation of the scale invariance of the theory makes it already interesting to explore. Fundamental aspects of causality and unitarity have been revisited along with perturbation theory studies of the propagators and vertices.
An interesting aspect of this formulation is the possibility to carry out non-perturbative calculations in particular those connected with the chiral symmetry of the theory which might be broken when the coupling is strong enough. Schwinger-Dyson equations and renormaization group analysis exploit the structure of the gap equation as compared to other formulations of QED3 including vacuum polarization effects allow to extract critical properties of the chiral transition in RQED. Still a number of improvements and refining of the truncations to the infinite tower of SDE in particular in connection with the gauge invariance of the predictions still awaits for a formal answer.
The influence of external agents has also been considered in this theory. Finite temperature effects indicate a relation of the transition temperature competing against the strong coupling which by virtue of the heat bath diminishes, hence restoring chiral symmetry. Findings of different groups are still controversial, hence pointing toward the need of a careful revision of the working hypothesis. Moreover, the influence of a chemical potential is still required to have a more realistic description of the physical system and eventual test in the laboratory. In this sense, an effort was made in96 to address the role of chemical potential in a configuration with spin-orbit coupling. The phase diagram in the plane of temperature and chemical potential of RQED, similar to that of QCD, is a pending assignment for the interested community.
The possibility of parity and time reversal symmetry breaking has been addressed by different groups by including the CS term in the fundamental Lagrangian. The existence of parity and time reversal breaking solution is remarkable in this theory. The effect of the CS coefficient theta as an effective dielectric constant is seen to impact the chiral symmetry breaking pattern inasmuch as the effective screening might restore chiral symmetry but still break the discrete symmetries in the theory.
An interesting feature is the modeling of elastic deformation and anisotropy in relation to the dynamics of fermions in curved space. The effects of curvature are seen as density effects through a definition of a position dependent chemical potential.
LKF tranformations are a key ingredient from gauge invariance that have a fundamental role in perturbative and non-perturbative studies in RQED. In the former case, the closed connection between RQED and QED3 in the large N limit serves as a benchmark for perturbative calculations in improving the calculation of vacuum polarization effects.
Many features are still pending to address in this theory. Some of which include the influence of magnetic and electric field in different configurations and potential connections with other phenomenology are important features that perhaps the community will address in the future.
