Projected Proca Field Theory: a One-Loop Study
R. F. Ozela Van Sérgio Alves E. C. Marino Leandro O. Nascimento J. F. Medeiros Neto Rudnei O. Ramos C. Morais Smith
hep-th cond-mat.supr-con hep-ph




 
Institute for Theoretical Physics, Utrecht University,
  3584 CC Utrecht, The Netherlands
Faculdade de Física, Universidade Federal do Pará,
  66075-110 Belém, PA,  Brazil

 
Faculdade de Física, Universidade Federal do Pará,
  66075-110 Belém, PA,  Brazil

 
Instituto de Física, Universidade Federal do Rio de Janeiro,
21941-972 Rio de Janeiro, RJ , Brazil

 
Faculdade de Física, Universidade Federal do Pará,
  68800-000 Breves, PA,  Brazil 
  
 
Faculdade de Física, Universidade Federal do Pará,
66075-110 Belém, PA, Brazil


 
Departamento de Física Teórica, Universidade do
  Estado do Rio de Janeiro, 20550-013 Rio de Janeiro, RJ, Brazil

 
Institute for Theoretical Physics, Utrecht University,
  3584 CC Utrecht, The Netherlands




 
The recent discovery of two-dimensional Dirac materials, such as
graphene and transition-metal-dichalcogenides, has raised questions
about the treatment of hybrid systems, in which electrons moving in a
two-dimensional plane interact via virtual photons from the
three-dimensional space. In this case, a projected non-local theory,
known as Pseudo-QED, or reduced QED, has shown to provide a correct
framework for describing the interactions displayed by these systems.
In a related situation, in planar materials exhibiting
a superconducting phase, the electromagnetic field has a typical exponential
decay that is interpreted as the photons having an effective
mass, as a consequence of the Anderson-Higgs mechanism. Here, we use an analogous projection to
that used to obtain the pseudo-QED to derive a Pseudo-Proca
equivalent model. In terms of this model, we unveil the main effects
of attributing a mass to the photons and to the quasi-relativistic
electrons. The one-loop radiative corrections  to the electron mass,
to the photon and to the electron-photon vertex are computed. 
We calculate the quantum corrections to the electron g-factor and show that it smoothly goes to zero in the limit when the photon mass is much larger than the electron mass. In addition, we correct the results obtained for graphene within Pseudo-QED in the limit when the photon mass vanishes.





Projected Proca Field Theory: a One-Loop Study
    C. Morais  Smith
    
==============================================






§ INTRODUCTION



The use of  Quantum Field Theories (QFT) to describe two-dimensional
systems has gained increased attention during the last years. This is
due to the great agreement obtained between the theoretical
predictions and the experimental data in many condensed-matter
systems. Examples range from the integer and the fractional quantum
Hall effects <cit.> to the  study of
transport in
graphene <cit.>,
excitonic properties of Trasition Metal Dichalcogenides
(TMDs) <cit.> and superconductivity in layered
materials <cit.>. 

Among those, a particular interest is devoted to the so-called Dirac
systems, which exhibit quasi-relativistic  dynamics, with massless
(e.g., graphene) or massive (e.g., silicene and TMDs) electrons moving with
the Fermi velocity. These Dirac materials then operate as QFT
laboratories <cit.>.

The description of the electron-electron interactions in planar
materials is rather involved because the photons mediating the
interactions live in (3+1)D, whereas the electron-dynamics is
constraint to a two-dimensional spatial plane. The appropriate theory
to capture this electron-photon dimensional mismatch is the so called
pseudo-quantum electrodynamics (PQED) <cit.> (sometimes also
named reduced-QED <cit.>). In this approach, a
projected electromagnetic field emulates the properties of the (3+1)D photons.

The PQED has been demonstrated to be unitary <cit.>, causal <cit.> and
has been successfully applied to describe several properties of
very thin systems (graphene-like structures) <cit.>, where the 2d approximation is a reasonable assumption. Among others, we highlight the Fermi velocity renormalization in the absence <cit.> or in the
presence <cit.> of a magnetic field in the vicinity of a
conducting plate <cit.> or in a cavity <cit.>. In
addition, it provided a theoretical description of the  Quantum Valley
Hall Effect, quantum corrections for the longitudinal conductivity in
graphene <cit.>, and of the corrections to the  electron's
g-factor due to interactions <cit.>. When  accounting
for massive electrons, the theory was shown to describe the excitonic
spectrum in  TMDs <cit.> and dynamical chiral symmetry
breaking <cit.>.  A dual  PQED type of model describing the
interaction between point vortex excitations and with some interesting
properties has also been recently constructed <cit.>.

Another topic that attracted much attention recently is the
superconductivity that shows up at low temperatures in bilayer
graphene twisted at the magic angle
(θ≈1.1^∘) <cit.>.
An interesting related question is how to treat a decaying magnetic field
due to the presence of superconductivity in that case. The
characteristic Meissner-Ochsenfeld magnetic field screening in
superconductors is usually described by  massive photons (e.g., as
described by using the Ginzburg-Landau equations in
three-spatial dimensions). Such a model, however, lacks the projection
component to describe planar relativistic condensed-matter systems,
which are accounted for within the PQED.

Here, we will follow steps similar to the ones that led to the
construction of the PQED model <cit.> and develop a theory to
describe electron-electron interactions through a massive vector (Proca) field.
We consider the Proca model without concern as to the origin of the mass term,
which in a more
fundamental theory comes from a Higgs-like mechanism.
 For a derivation of the model we use here, see Ref. <cit.>, which also shows that a Yukawa potential is generated in the static limit. 
Such model constrains only the matter (electrons)
current to the spatial xy plane. The corresponding quantum partition functional
is defined initially in 3+1-dimensions and then the third spatial
dimension is integrated out. This procedure is very much
analogous to the one that links the (3+1)D Maxwell model to
the 2+1-dimensions PQED model. 

This work is organized as follows. In Sec. <ref> we
present the model used in this work and we derive its planar
dimensional reduction  in a procedure analogous to that used to derive
the PQED model.  In Sec. <ref>, we compute the electron and
photon self-energies for the model, as well as the interaction vertex,
within the leading order (one-loop) level.  In this same section, we
also explicitly derive the g-factor for our model. A
comparison of our results with previous ones, when considering the
massless regime, is also performed. In Sec. <ref> we present
our conclusions. Some technical details of the calculation of the
g-factor are give in the App. <ref>.




§ PSEUDO PROCA MODEL



Let us first consider the (3+1)D Proca (P) model,
including the coupling  to a general conserved current J^μ. The quantum partition functional is


    Z_P[J^μ]  =  ∫ DA_μexp( i
    S_P),


where A_μ is a (3+1)D massive vector field and S_P is
the action, given by


    S_P=∫ d^4 χ(
    -1/4F_μν F^μν + m^2/2A^μ A_μ - eA_μJ^μ),


where F_μν = ∂_μ A_ν - ∂_ν A_μ and m is the vector field mass. Natural units (ħ=c=1) are considered for the
remainder of this section. The vector field propagator is directly
derived from Eq. (<ref>), 


    G^μν_P(χ - χ')=  ∫d^4k/(2π)^4e^-ik·(χ - χ')/k^2 - m^2 [η^μν-k^μ k^ν/m^2] ,


with χ and χ' representing points in the (3+1)D
spacetime and η^μν=diag(+,
-, -, -) is the metric tensor.  Integrating out the vector field in Eq. (<ref>), we obtain


    Z_P[J^μ]     =    Z_0
    e^-ie^22∫ d^4χ  d^4χ'  J_μ(χ)
      G^μν_P(χ - χ') J_ν(χ') ,


where Z_0 is a normalization constant, independent of
J^μ. Note that by the conservation of the current, ∂_μ
J^μ=0, the last term in Eq. (<ref>) (k^μ
k^ν/m^2) vanishes in Eq. (<ref>). 

By using the constrain of having only currents in the xy plane, we
can explicitly write the current J^μ as


    J^μ(χ) = {[ J^μ̂(t,  x,  y)δ(z),     μ = 0, 
    1,  2  ,;                   0,               μ = 3, ].


where the hat over an index notation is used to identify objects that
assume three values, i.e., μ̂=0,1,2. 

After the integration in z and z' space coordinates,
Eq. (<ref>) can be written as


    Z_PP[ J]     =    Z_0   e^-i
      e^22∫ d^3ζ  d^3ζ' J_μ̂(ζ) 
        G^μ̂ν̂_P(ζ - ζ')
      J_ν̂(ζ')  ,


with  ζ and ζ' denoting points in the 2+1-dimensions
spacetime and G^μ̂ν̂_P(ζ -
ζ') is given by


    G^μ̂ν̂_P(ζ - ζ') = i∫d^4k/(2π)^4η^μ̂ν̂/k^2 - m^2.e^-ik·(χ - χ')|_z = z' = 0,


where the momentum integration is over the energy-momentum four-vector
k^μ.

If we now perform the integration over the third component of k^μ, i.e.,
k_z, in Eq. (<ref>) and hence restrict the dynamics to the xy-space plane, we obtain


    G^μ̂ν̂_P(ζ - ζ') 
    = i/2∫d^3 k/(2π)^3η^μ̂ν̂ 
    e^-i k· (ζ-ζ')/√(k^2 - m^2),


with k now indicating a three-vector, stressing
the fact that we are now working in the reduced space (effectively, in
2+1-dimensions).
However, the above result can also be obtained if we start
from a completely 2+1-dimensions, in principle nonlocal, model from the very beginning. We will name this model the “Pseudo Proca” (PP) model, in analogy with the  case of the PQED model. Notice that the denominator of the propagator in Eq. (<ref>) is  significantly different from the original Proca model and this impacts directly the quantum corrections in a mixed dimensionality  physical system.

The Lagrangian density for this model is then expressed as


    L_PP    =     -1/2F_μ̂ν̂F^μ̂ν̂/√(- -
        m^2)  - eA_μ̂ J^μ̂ -m^2/2A^μ̂A_μ̂/√(- - m^2),


with J^μ̂ being the general 2+1-dimensions current
defined in Eq. (<ref>) and   the d'Alembertian operator
(which must be understood as a convolution).  The free propagator
associated with L_PP is simply given by 
Eq. (<ref>), as can be easily verified.
Thus, we immediately realize that 


    G^μ̂ν̂_PP(ζ-ζ') =
    G^μ̂ν̂_P(χ-χ')|_z = z' = 0


and, therefore, the quantum partition function of the Pseudo-Proca and that of the (3+1)D Proca models are
completely equivalent, as long as the currents of the latter are
constrained to a plane, such as in Eq. (<ref>).




§ RADIATIVE CORRECTIONS



In this section, we consider a soft symmetry-breaking term in the
Dirac action through the Fermi velocity v_F, in order to reproduce
the Dirac-like low-energy electronic dispersion. The Lagrangian
density in the now 2+1-dimensional Minkowski space is then
given by


    L   =    -1/2F_μνF^μν/√(-
            - m^2) -m^2/2A^μA_μ/√(- -
            m^2)
       +   ψ̅(i γ^0∂_0 +i
        v_Fγ^i∂_i - Mc^2)ψ
        -e ψ̅γ^0ψ A_0
       -   
        e βψ̅γ^iψ A_i,


with ψ a four-component Dirac spinor, β =v_F/c and
M the Dirac fermion mass. In the above equation and  from this
point on, we have explicitly retrieved the dimension of the speed of
light c (but still keeping ħ =1 units). For convenience, and
to avoid overloading the notation, we suppress the hat of the Lorentz
index. 

The Feynman rules for the model are obtained as
usual <cit.>. The interaction vertex is given by
Γ^α=-ie(γ^0,β γ^i ), the
fermion propagator is

    S_F(p) =i ( γ^0 p_0 + v_F γ^i p_i + M
       c^2p_0^2-v_F^2  p^2-M^2 c^4),

and the massive vector field propagator reads

    Δ_μν (p) =-i c η_μν/2√(p_0^2 - c^2  𝐩^2 - m^2 c^4).






 §.§ Electron self-energy







The electron self-energy at one-loop order is shown in
Fig. <ref> and the regularized amplitude is given by 


    -iΣ(p) = ∫d^D k/(2π)^DΓ^μ  S_F(p-k)
    Γ^νΔ_μν(k),


where D=d+1.  In what follows, all momentum integrations in the loop
radiative quantities are computed using standard dimensional
regularization procedure <cit.>, with e →μ^ϵ/2 e, d=2-ϵ and  μ is the dimensional
regularization energy scale. 

By substituting the propagators and vertices in Eq. (<ref>),
we obtain 

    -iΣ (p)  = (ie)^2 μ^ϵ c/2∫d^D
      k/(2π)^DN_1D_1 ,


where N_1 and D_1 are given, respectively, by

    N_1     =    v_F(p-k)_i  (γ^0 γ^i γ_0
     +β^2γ^j  γ^i γ_j ) 
       +    (p-k)_0
     (γ^0 γ^0 γ_0 +β^2γ^j γ^0 γ_j
     )  
       +    M c^2 (γ^0  γ_0
     +β^2γ^j γ_j )  , 
    
     D_1    =   [(p-k)_0^2-v_F^2 ( p- k)^2-M^2 c^4] 
       ×   √(k_0^2 - c^2   k^2 - m^2 c^4) .


Using, hereinafter, the gamma matrix properties
γ^0γ_0=1 and γ^jγ_j=2, we have that



    -iΣ (p)=  (ie)^2μ^ϵ c/2∫dk_0/2 πd^d k/(2π)^dγ^0 (p-k)_0(
      1+2β^2)  - v_F γ^i (p-k)_i  + ( 1 + 2β^2
      ) M c^2[(p-k)_0^2-v_F^2 (𝐩-𝐤)^2-M^2
        c^4] (√(k_0^2 - c^2  𝐤^2 - m^2 c^4)).



Using the Feynman parameterization,


    1/a√(b)=34∫_0^1 dx{(1-x)^-1/2/[a x + (1-x)  b]^3/2},


and performing the integration in k_0, we obtain that 
Eq. (<ref>) can be rewritten as


    -iΣ (p)  =(ie)^2μ^ϵ c/4π∫_0^1
    dx√(1-x)∫d^d  k/(2π)^dN_2D_2,


where N_2 and D_2 are defined, respectively, as 


    N_2    =   ( 1 + 2β^2 ) M c^2 +(
    1-2β^2)γ^0 p_0 (1-x) 
       -   (1-v^2_F x/ B) v_F γ^i p_i,
    
    D_2   =    B[ (  k- p v^2_F
        x/ B) ^2-  p^2
              v_F^4x^2/ B^2+  A/ B],

with A and B defined as


    A   =    p^2_0 x^2+ [ (M^2-m^2)c^4+ p^2
          v_F^2-p^2_0 ] x  + m^2c^4,
    
     B   =    c^2 [1- x  (1-β^2)].


The electron self-energy then becomes,


    -iΣ (p) = (ie)^2  cμ^ϵ/4π∫_0^1
    dx B√(1-x)∫d^d k/(2π)^dN_2  k^2-Δ ,


where 


    Δ =   p^2 v_F^4x^2/ B^2+ A/ B.


Performing the integration in the arbitrary dimension d=2-ϵ
in Eq. (<ref>) in the dimensional regularization procedure,
we obtain


    Σ (p)   =   (ie)^2  cμ^ϵ/16 π^2∫_0^1
    dx√(1-x)Γ(ϵ2) N_2  / B[Δ4 π]^-ϵ2
       =   (ie)^2 
      c/16 π^2∫_0^1  dx√(1-x)N_2  /
        B
       ×   [ 2/ϵ-γ_E -ln(Δ/4 πμ^2)  + O(ϵ) ]
    
       =   Σ_ finite (p) + Σ_ div(p),


where


    Σ_ finite (p)   =    -(ie)^2  c/16 π^2∫_0^1
    dx√(1-x)N_2  / B
       ×   [γ_E +ln(Δ/4 πμ^2) ]


and


    Σ_ div (p)= (ie)^2  c/8 π^2 ϵ∫_0^1
    dx√(1-x)N_2  / B


are the finite and divergent contributions to the electron
self-energy, respectively, in the pseudo-Proca QED.  One notices that
the divergent part of the electron self-energy, Eq. (<ref>),
is independent of the vector field mass and it is in fact exactly the same
result as that obtained in the PQED
case <cit.>. Consequently, the Fermi velocity
renormalization will also be the same.

Substituting Eq. (<ref>) in Eq. (<ref>), we immediately
read the one-loop contributions to the electron mass and the
wave-function renormalization terms. In particular, the one-loop
correction for the fermion mass at zero-external momenta is


    Σ_ finite^M (0)=    e^2  c/16 π^2( 1 +
    2β^2 ) M  ∫_0^1  dx√(1-x) 
       1  / 1- x  (1-β^2)    {γ_E +ln[M^2c^2 x+ m^2 c^2(1-x)/4 πμ^2 [1-x 
          (1-β^2)]] }.


The PQED result is obtained by simply setting m=0 in
Eq. (<ref>).

In Fig. <ref> we show the behavior of the ratio ΔΣ^M≡ [ Σ_ finite^M (0) - lim_m→ 0Σ_ finite^M (0)]/(M c^2) as a function of (m/M). We note that the electron self-energy correction increases with the vector field mass.













 §.§ The vector field self-energy






Let us now compute the vector field self-energy, shown in  Fig. <ref>, 
for the present model.  Explicitly, we have that


    iΠ^μν(p_0,p) = -∫d^D k/(2π)^D[ Γ^μ S_F(k+p)Γ^ν S_F(k)].


The manipulation of the Dirac algebra proceeds in a standard
fashion <cit.>. Separating the components of the polarization
tensor and performing the integration over k_0 and k in
Eq. (<ref>), we can write


    Π^00(p̅)   =   e^2/8π v_F[I_1+
          (p̅^2-2p_0^2)I_2/v_F.. + 
          Mc^2 I_3/v_F^2], 
    
    Π^ij(p̅)    =   e^2v_F/8π c^2[
          η^ijp̅^2/v_F^2-2p^ip^jI_2. 
       +   .η^ijI_1+Mc^2 I_3/v_F^2] ,
    
    Π^0j(p̅)   =    -e^2/4π v_F c p^0p^j
        I_2,

where for convenience we have introduced the notation p̅=√(p^2_0-v^2_F p^2) and I_1, I_2 and I_3 are given,
respectively, by


    I_1   =   1/v_F∫_0^1 dx √(x(1-x)p̅^2- M^2 c^4)
       =   -i/8v_Fp̅{ 4 Mc^2 p̅ + 4
      M^2c^4-p̅^2ln(2 Mc^2+p̅/2Mc^2-p̅)} ,
    
    
    
    I_2   =   v_F∫_0^1dxx(1-x)/√(x(1-x)p̅^2 -
        M^2c^4)
       =   i v_F/8p̅^3{
    4 Mc^2p̅ + 4 M^2c^4+p̅^2ln2Mc^2+p̅/2Mc^2-p̅}  , 
    
    
     
    I_3   =   ∫_0^1 v_F dx/√( x(1-x)p̅^2 - M^2c^4)  =
    i v_F/p̅ln(2Mc^2+p̅/2Mc^2-p̅) .



Notice that the use of the dimensional regularization scheme makes the
vector field self-energy explicitly finite.  Furthermore, as
at the one-loop level the vector field self-energy involves only the
electron propagator, the result turns out to be identical to that
computed in the context of the 2+1-D QED at one-loop order.




 §.§ Vertex correction and the g-factor



To have a complete one-loop analysis of relevant quantities, we also compute next
the one-loop interaction vertex radiative contribution
Γ^ν_ 1L as shown in Fig. <ref>. 






Explicitly, Γ^ν_1L is given by


    Γ^ν_ 1L =  ∫d^D k/(2π)^DΓ^αS_F(k + p') Γ^ν S_F(k + p)
    Γ^βΔ_αβ(k).



We will analyze the quantity M^i = u(p')Γ^i
u(p), which relates to the two external fermion lines, u(p')
and u(p), in the spatial component of the vertex diagram (because
the temporal part of it does not affect the g-factor), such
that we can use the Dirac equations to simplify the calculations
further.

When represented according to the flux choices in Fig. <ref>,
and after applying a parameterization detailed in the
Appendix <ref>, the diagram is written  in its parameterized form
as


    M^i = u(p') ( ∫_0^1dx∫_0^1-xdy6
      π^2M  p^i
      v_F^2√(1-x-y) N_3 K)  u(p),


with


    N_3 = 2(1-x-y) γ^j ω_j p_i v_F^2 / D
    -[ω_i-2(x+y)^2 p_i] β^4/ D^2,


where D in the above equation is defined as


    D =[(1-β^2) (x+y) -1],

and


    K   =    D^2 (M^2 c^4-m^2c^4-ω_0^2) c^2 +
    [v_F^2 (p'-p)_i^2+p_0^2] x y		
       +   [(x+y)(1-x-y)c^2] (p'+p)_μ^2 +p_0^2 y^2,


where ω_0 = x p_0' + y p_0  and ω_i = x p' + y p.
Then, we detach the terms proportional to σ^i α from
Eq. (<ref>) using Gordon decomposition to select the terms
M_g^i that are relevant to the g-factor  (for more details,
see App. <ref>) 


    M^i_g    =   u(p'){∫_0^1 dx ∫_0^1-xdy/6
      π^2 M  v_F  σ^iα (p' -
      p)_αβ^2/ K√(1-x-y).
    
       ×   .  [ 2 (x+y)
          D-β^2(x+y)^2 D^2]} u(p),


where the subscribed g index was used to identify that we are only
accounting for the σ^iα=(i/2)[γ_α,γ_i] terms which contribute to the
gyromagnetic factor. 


To extract the g-factor, however, a few conditions are also
necessary, such as the low-energy approximation (q^2→0)
and the mass-shell condition (which for small energy values reads as
v_F γ^j p_j ≅ M c^2) <cit.>. After taking the above conditions, we
conveniently define a variable 


    K'   =   {[(x+y)^2-x-y] v_F^2-[(1-x-y)^2]c^2 } m^2
    c^4
       +    2{[(x+y)^2- x- y] c^2- 2(x+y)^2 v_F^2 } M^2 c^4,


so that the relevant parts of the vertex correction M_g^i are
reduced to

 
    M^i_g    =   u(p'){16 π^2∫_0^1 dx
    ∫_0^1-x dy   M^2 c^2 σ^iα q_α v_F^3  K'√(1-x-y).  
       ×   .
        [ 2(x+y)  D-(x+y)^2 v_F^2
          ]}u(p).


We can also identify the  g-factor correction form factor
F_2 by comparing Eq. (<ref>) to 


    M^i_g= i  e β u(p') (i2Mc^2 F_2 
        v_F  σ^iν q_ν) u(p) .



Therefore, F_2 is turned into an integral  related to the one obtained in Ref. <cit.>,
but in the present work a contribution depending on the effective vector field mass m also appears.  We then find

    F_2 = -i
    α^* β^32π R,

with α^* =e^2/4π v_F the effective fine-structure constant and

    R =R(β, m/M)=
    ∫_0^1 d x ∫_0^1-xdy/√(1-x-y)
    
    2 (1+2β^2)(x+y)+β^2(x+y)^2
    /m^2/M^2(1-x-y) D^2
       -     (x+ y)^2  D

 a mass-dependent parametric integral.


A few predictions emerge from the result in Eq. (<ref>) when looking at some of the limits achievable through the tweak of 
mass parameters: 
The mostly intuitive case occurs when the
vector field is heavy with respect to the electron mass,
m ≫ M, which implies that the parametric integral assumes the form

    R|_m≫ M≃M^2/m^2∫_0^1 d x ∫_0^1-xdy(x+y)/(1-x-y)^3/2
    
     2+4β^2+β^2(x+y)
    /[(1-β^2) (x+y) -1]^2

and, consequently, F_2=0, i.e., the g-factor vanishes.


The second relevant limit happens when m goes
to zero 
in the original unprojected model, which after the projection reproduces the usual PQED behavior, i.e.,

    m0 F_2= F_2^PQED .
 

 


 


As a consistency check, we take the limit of Eq. (<ref>) when (m/M) → 0 and obtain

    R=
    ∫_0^1 d x ∫_0^1-xdy/√(1-x-y)
    2+4β^2+β^2(x+y)
    / (x+ y) [1-(1-β^2) (x+y) ],

which has a similar structure to the parametric integral in Ref. <cit.>, except that the term 2 in the numerator of Eq. (<ref>) was missing  in that reference.
 Hence, our result not only extends the projection to the massive case, but also corrects the g-factor result for graphene found previously in Ref. <cit.>. 


Although we could not find an analytical solution for the integral in Eq. (<ref>),  some numerical solutions for R are shown in Fig. <ref> for different mass ratios (m/M).
These  numerical values of R, when substituted into Eq. (<ref>), allow us to verify how small changes in β affect the g-factor in each case.

To show that the g-factor associated with the system is going to decrease faster with the mass when v_F gets closer to c, we plot
in Fig. <ref> the numerical values of R for a few different values of β, but now as a function of the mass ratio (m/M).




§ CONCLUSIONS



In this work, we have provided a generalization of the dimensional
reduction, developed in Ref. <cit.>, for the case of a
(3+1)D massive vector field. By doing so, we have
obtained an effective planar model that retains the fundamental
physical properties of the Proca electrodynamics, which is taken as an 
effective model for describing massive (via the Anderson-Higgs-Meissner mechanism)
 photons in a material.
 We have then
evaluated the one-loop radiative corrections for the electron and
vector field self-energies for this model. 

We observed that the divergent
part of the electron self-energy is exactly identical to the one
obtained in the context of the PQED (massless) model. Therefore, from
the renormalization group perspective, the renormalization of the
Fermi velocity does not depend on the mass of the vector field and,
consequently, it is identical  to the one obtained for the PQED
model <cit.>, even though the effective mass for the electron is seen
to increase with the vector field mass.


For the vertex diagram, we have found that the parametric integral in its form factor —
and thus this theory's predicted g-factor — decreases with both the value of β and the ratio (m/M), effectively switching  the magnetic coupling off of the system for very large values of m.

This work contributes to the proper description of planar effective models, which are known to be intrinsically related to condensed-matter physical systems, by investigating the effects of attributing a mass to the interaction field.




R.F.O. is partially supported by Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior – Brasil
(CAPES), finance code 001, and by CAPES/NUFFIC, finance code 0112;
V.S.A. and L.O.N. are partially supported by research grants from Conselho
Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and
by CAPES/NUFFIC, finance code 0112;  V.S.A. acknowledges the Institute
for Theoretical Physics of Utrecht University for the kind hospitality
and M. Gomes for very fruitful discussions; E.C.M. is partially
supported by both CNPq and Fundação Carlos Chagas Filho de
Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ).  R.O.R is
partially supported by research grants from CNPq, grant
No. 302545/2017-4 and FAPERJ, grant No.  E –
26/202.892/2017. J.F.M.N. and R.F.O. are also grateful to M.C. Lima and G.C. Magalhães for interesting discussions about the model.




§ VERTEX CORRECTION



To shed more light on the g-factor corrections that come from
the vertex developed in
Subsec. <ref>, we make the steps
that differ from the usual QED more explicit. The subtlety on the
calculation that appears for the model discussed here comes from the
fact that it is anisotropic and has a different photon propagator (the
planar projection of a massive vector field). 

Starting from the vertex structure in
Subsec. <ref>, 


    Γ^ν_ 1L =   ∫d^D k/(2π)^DΓ^αS_F(k +
        p')Γ^ν S_F(k + p)_ρ^νΓ^βΔ_αβ(k) ,


we define an auxiliary variable ρ^ν and expand
 the indices α and β to get


    Γ^ν_ 1L =   ∫d^D k/(2π)^D[
      Γ^0ρ^νΓ^0Δ_00(k)+    Γ^lρ^νΓ^nΔ_ln(k)].


In Eq. (<ref>), the non-diagonal terms of the vector field propagator
were dropped because the metrics inside them vanish. Explicitly
substituting the Feynman rules given in Sec. <ref>, we find



    Γ^ν_ 1L = - ie^2 c/2μ∫d^D k/(2π)^D
    {γ^0(
    γ^0 (k + p')_0 + v_F γ^i (k + p')_i + M c^2(k +
      p')_0^2-v_F^2  (k + p')^2-M^2 c^4).
    Γ^ν( γ^0 (k + p)_0 + v_F γ^i (k +
      p)_i + M c^2(k + p)_0^2-v_F^2  (k + p)^2-M^2 c^4)
    γ^0[1/√(k_0^2 - c^2  𝐤^2 - m^2
          c^4)]
    +
    β^2γ^l( γ^0 (k +
        p^')_0 + v_F γ^i (k + p')_i + M c^2(k +
        p^')_0^2-v_F^2  (k + p')^2-M^2 c^4)
    Γ^ν( γ^0 (k + p)_0 + v_F γ^i (k +
        p)_i + M c^2(k + p)_0^2-v_F^2  (k + p)^2-M^2 c^4)
    .  γ^n[η_ln/√(k_0^2 - c^2  𝐤^2 - m^2 c^4)] }.



To handle this unusual photon propagator, which contains a square root and a mass term, we apply a slightly modified Feynman
parameterization,

    1/a b d^1/2=3/4∫_0^1 d x ∫_0^1-x
    d y (1-x-y)^-1/2/[a x+b y+d(1-x-y)]^5/2

on the denominator of each term in Eq. (<ref>). Assuming 


    a   =   (k + p')_0^2-v_F^2  (k + p')^2-M^2 c^4,
     b   =   (k +
      p)_0^2-v_F^2  (k + p)^2-M^2 c^4,
     d   =   3k_0^2 - c^2  𝐤^2 - m^2 c^4,

we find that the denominator of Eq. (<ref>) is given by


    1/a b c^1/2=3/4∫_0^1 d x ∫_0^1-x d y
     (1-x-y)^-1/2
    { k_0^2(1+x+y)+[ 2k_0 p_0^' + p^'_0^2
    ] x 
     [2k_0 p_0+ p_0^2  - v_F^2 (k + p)^2  - M^2 c^4
    ]y+
     - c^2  𝐤^2 - m^2 c^4 (1-x-y)}^-5/2.


Now, we complete the k_0 square, make a shift k_0→ k_0- ω_0
(with ω_0 = x p_0' + y p_0) on it, and solve the integration
over k_0 as done in standard anisotropic procedures (at this point,
we also drop the odd terms in k_0). 

Then, we follow essentially the same steps for 𝐤, but using
the shift 𝐤→𝐤-ωβ^2/ D^2, with D
=[(1-β^2) (x+y) -1] and  ω =
x 𝐩' + y 𝐩. Finally, we perform the integration
over 𝐤 using dimensional regularization <cit.> to
obtain Eq. (<ref>), namely:


    Γ^i_ 1L = ∫_0^1dx∫_0^1-xdy6
      π^2[N_3 K]  M  p^i
      v_F^2√(1-x-y),


where, for convenience, we defined the numerator result for the
integration in 𝐤  as 


    N_3 =. 2(1-x-y) γ^j ω_j p_i v_F^2 /
        D.  -.[ω_i-2(x+y)^2 p_i] β^4/
        D^2.,	
     D =[(1-β^2) (x+y)
      -1],


and its denominator as 


    K = D^2 (M^2 c^4-m^2c^4-ω_0^2) c^2 +
    [v_F^2 (p'-p)_i^2+p_0^2] x y		
    
    +[(x+y)(1-x-y)c^2] (p'+p)_μ^2	 +p_0^2 y^2  .


Using the (Gordon) decomposition identity

    u(p')  γ ^μ u(p) =u (p')
     [(p'+p)^μ/2Mc^2+ i σ^ νμ q_ν/2Mc^2] u (p)  ,

where u(p') and u(p) are spinorial solutions to the Dirac
equations,  q_ν is the transferred momentum given by q_ν = (p'
- p)_ν, and    σ_νμ is defined as σ_νμ=(i/2)[γ_μ,γ_ν], we stablish which of
the terms of M^i will be relevant to the g-factor. In other
words, we select only the terms proportional to σ^i α
from Eq. (<ref>) and leave aside the terms that would have a
(p'+p)^μ to define 

    M^i_g = u(p'){∫_0^1 dx ∫_0^1-xdy/6
      π^2 M  v_F  σ^iα (p' -
      p)_αβ^2/ K√(1-x-y).  
    .
    [ 2 (x+y) D-β^2(x+y)^2
            D^2]} u(p),

 as it appears in  Eq. (<ref>)






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