Abstract
We consider reduced quantum electrodynamics ( {mathrm{RQED}}_{d_{gamma },{d}_e} ) a model describing fermions in a de-dimensional space-time and interacting via the exchange of massless bosons in dγ-dimensions (de ≤ dγ). We compute the two-loop mass anomalous dimension, γm, in general {mathrm{RQED}}_{4,{d}_e} with applications to RQED4,3 and QED4. We then proceed on studying dynamical (parity-even) fermion mass generation in {mathrm{RQED}}_{4,{d}_e} by constructing a fully gauge-invariant gap equation for {mathrm{RQED}}_{4,{d}_e} with γm as the only input. This equation allows for a straightforward analytic computation of the gauge-invariant critical coupling constant, αc, which is such that a dynamical mass is generated for αr> αc, where αr is the renormalized coupling constant, as well as the gauge-invariant critical number of fermion flavours, Nc, which is such that αc → ∞ and a dynamical mass is generated for N < Nc. For RQED4,3, our results are in perfect agreement with the more elaborate analysis based on the resolution of truncated Schwinger-Dyson equations at two-loop order. In the case of QED4, our analytical results (that use state of the art five-loop expression for γm) are in good quantitative agreement with those obtained from numerical approaches.
Highlights
The model of RQED4,3 captures some universal features of a broader range of socalled Dirac liquids that have been discovered experimentally during the last decade and are under active study such as, e.g., artificial graphene-like materials [19], surface states of topological insulators [20] and half-filled fractional quantum Hall systems [21]
This equation allows for a straightforward analytic computation of the gauge-invariant critical coupling constant, αc, which is such that a dynamical mass is generated for αr > αc, where αr is the renormalized coupling constant, as well as the gauge-invariant critical number of fermion flavours, Nc, which is such that αc → ∞ and a dynamical mass is generated for N < Nc
Our main formulas, (3.20), for γm and the field anomalous dimension γψ, take into account the non-commutativity of the εγ → 0 and εe → 0 limits and are valid in both cases of RQED4,3 and QED4
Summary
In Minkowski space, the RQEDdγ,de action [1, 22, 25] including (in order to compute γm) a bare (parity-even) fermion mass reads: S=. (2.3c) where the photon propagator is the reduced (de-dimensional) one obtained by integrating out the bulk dγ − de degrees of freedom and the corresponding reduced gauge-fixing parameter reads:. Where the fermion self-energy has the following parameterization appropriate to the massive case: Σ(p) = p/ ΣV (p2) + m ΣS(p2) ,. Anticipating the RPA analysis of section 4.4.2, let’s note that, in the case of RQED4,3, the transverse part of the photon propagator, (2.8b), reads: d⊥(p2) = 2 i −p2 1 −. With this perturbative setup, the renormalization constants are defined as:. In the MS-scheme, the renormalization constants are Laurent series in εγ.
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