Field theoretic renormalization study of interaction corrections to the universal ac conductivity of graphene
S. Teber A. V. Kotikov






















                     





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         e i m d Ok_ BTħμ_ B⟺A_μA_νF_μνF^μνψ̅ D p q k∂_μ∂_νD_μD_ν∂/∂ r∂/∂ x∂/∂ y∂/∂ zSchrödinger Heisenberg μ_ Bn^ o =  = ℒ
        
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                a]S. Teberb]and A. V. Kotikov[a]Sorbonne Université, CNRS, Laboratoire de Physique Théorique et Hautes Energies, LPTHE, F-75005 Paris, France[b]Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russiateber@lpthe.jussieu.frkotikov@theor.jinr.ruThe two-loop interaction correction coefficient to the universal ac conductivity of disorder-free intrinsic graphene is computed with the help of a field
        theoretic renormalization study using the Bogoliubov-Parasiuk-Hepp-Zimmermann prescription. Non-standard Ward identities imply that divergent subgraphs (related to Fermi velocity renormalization)
        contribute to the renormalized optical conductivity. Proceeding either via density-density or via current-current correlation functions, a single well-defined
        value is obtained: 𝒞= (19-6π)/12) = 0.01 in agreement with the result first obtained by Mishchenko and which is compatible with experimental uncertainties. Field theoretic renormalization study of interaction corrections to the universal ac conductivity of graphene
            [
            December 30, 2023
        ==============================================================================================================
        
        fmfsigma
        
        § INTRODUCTION
        
        
        Transport properties of graphene and similar planar Dirac liquids have been the subject of extensive studies for more than a decade
        now, see Ref. <cit.> for a review. Of central interest for charge transport is the conductivity, σ,
        which is in general a complicated function of, , frequency (), momentum (q⃗ ), temperature (T), chemical potential (μ) and scattering rates (Γ). 
        A remarkable feature of the ideal case of an intrinsic (μ=0) disorder-free graphene monolayer is that,
        despite the vanishing density of states at the Fermi points, the relativistic-like nature of the charge carriers <cit.> yields a non-zero universal 
        	ac conductivity in the collisionless limit (≫ T, Γ): [Throughout this paper, we work in units where ħ=k_B=1.]
        σ_0 = e^2/4  .
        
        This result, which was predicted long ago to hold for free Dirac fermions <cit.>, 
        agrees to within 1-2% with experiments in the optical regime, , at ∼ 1eV (visible range of the spectrum) <cit.>.
        Still in the case of free fermions, adding other factors, , T, μ, Γ, ..., is rather non-trivial but leads to results <cit.> 
        which fit quite well the experimental data <cit.>. This is rather surprising because the long-range Coulomb interaction among charge carriers 
        is not only unscreened but also supposed to be strong as witnessed by the fine structure constant of suspended graphene:
        
        _g = e^2/4 π v≈ 2.2   ,
        
        which is of the order of unity due to the fact that v ≈ c / 300 where v and c are the Fermi and light velocities, respectively. 
        Moreover, it is well known that Kohn's theorem <cit.> does not apply to pseudo-relativistic systems thereby allowing electron-electron interactions to affect Eq. (<ref>), 
        see also the recent Ref. <cit.> and references therein for more.
         There has therefore been extensive theoretical works during the past decade devoted to understanding the intriguing effect of interactions on the 
        optical conductivity of graphene in the collisionless limit, see,  e.g., Refs. <cit.>, see also <cit.> for a short review. 
        The latter can be defined via a density-density correlation function:
        
        σ(q_0) = - lim_q⃗ 0  q_0/|q⃗ |^2 Π^00(q_0,q⃗ )  ,
        
        where, in real time, Π^0 0 (t,q⃗ ) =  T ρ(t,q⃗ ) ρ(0,-q⃗ ), ρ is the charge density and T the time-ordering operator. 
        Equivalently, from current conservation, it can also be defined via a current-current correlation function (Kubo formula):
        
        σ̃(q_0) = 1/ q_0 K^11(q_0,q⃗=0 ) + K^22(q_0,q⃗=0 )/2  ,
        
        where, in real time, K^i j (t,q⃗ ) =  T j^i(t,q⃗ ) j^j(0,-q⃗ ) and j⃗ is the charge current. 
        In the case of weak short-range interactions, it was rigorously established that no interaction correction 
        arises <cit.>. [According to Ref. <cit.>, in the
        case of weak short-range interactions, interaction corrections cancel out to
        all orders in the renormalized expansion as shown in Ref. <cit.> as a
        consequence of Ward identities and the irrelevance, in a renormalization
        group sense, of the interaction (long-range interactions are, on the
        other hand, only marginally irrelevant).]
         However, no exact result is available in the more realistic case of long-range interactions.	
        Analytically, though interactions are strong, the problem is generally considered with the help of perturbation theory with a 
        focus on the lowest order corrections to Π^00(q) and K^ij(q):
        
              σ(q_0) = σ_0 ( 1 + 𝒞_g + (_g^2) )  , 
        
              σ̃(q_0) = σ_0 ( 1 + 𝒞̃_g + (_g^2) )  , 
        
        where the numbers 𝒞 and 𝒞̃ are the so-called first order interaction-correction coefficients.
        On physical grounds, one expects that 𝒞 = 𝒞̃, independent on the method used. 
        
        It turns out that, despite the apparent simplicity of the task, different theoretical results have been found in the literature during the past ten years, see Tab. <ref> 
        for a summary of some results. Following Refs. <cit.>, the main three values read:
        
        𝒞^(1)  = 25-6π/12≈ 0.512  ,
        
        𝒞^(2)  = 19-6π/12≈ 0.013  ,
        
        𝒞^(3)  = 11-3π/6≈ 0.263  .
        
        As can be seen from Tab. <ref>, the most commonly accepted result is, up to date, the value 𝒞^(2) of Mishchenko <cit.> since it has been recovered 
        by a majority of groups. Incidentally, this is also the only result, among those of Eqs. (<ref>), which is consistent with the experimental 
        uncertainties <cit.> as 𝒞^(2)_g ≈ 2%.
        
        
        
        
        
        
        
        At this point, we note that there is a limit where the result for the interaction correction coefficient does not raise any doubt: the deep infra-red (IR) limit corresponding 
        to the Lorentz-invariant fixed point <cit.>. The later arises from the long-range Coulomb interaction among the Dirac fermions which
        enforces the flow of the Fermi velocity, v ≈ c/300 at the present experimentally accessible scales, to the velocity of light, c, in the IR with a corresponding flow of
        the fine structure constant, _g ≈ 2.2, to the usual fine structure constant, ≈ 1/137, in the IR. The IR fixed point therefore corresponds to an
        ultra-relativistic limit with fully-retarded interactions (v/c  1). The corresponding conductivity reads <cit.>:
        
        σ̃(q_0) = σ_0 ( 1 + 𝒞^*  + (^2) )  ,     𝒞^* = 92-9π^2/18π  .
        
        Of course, at the fixed point = 1/137 and the product 𝒞^* ≈ 10^-4 is very small leading to almost unobservable effects. However, it is interesting to note that
        𝒞^* = 0.056, a value which is of the same order of magnitude as 𝒞^(2). [The interest in comparing 𝒞^(2) to 𝒞^* 
        (rather than 𝒞^(2)α_g to 𝒞^*α) comes from the fact that there is a more general model (model I in the terminology of Ref. <cit.>) 
        describing interactions in graphene which is valid for arbitrary v/c. So there is actually a non-trivial interaction correction  function𝒞(v/c) which encodes relativistic corrections (the dependence of the fine structure constant on v is trivial). Presently, only the limiting
        values 𝒞(v/c → 0) = 𝒞^(2) (see the proof in the following pages) and 𝒞(v/c → 1) = 𝒞^* are
        known. It is surprising that these values are of the same order as if 𝒞(v/c) was only weakly dependent on v/c. A study of 𝒞(v/c) for arbitrary
        v/c is beyond the scope of the present paper and we leave it for our future investigations.] The result of Eq. (<ref>) was derived in 
        Refs. <cit.> with the help of multi-loop techniques together with conventional renormalization. 
        In Ref. <cit.>, see also the review <cit.>, the 
        Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ), see Refs. <cit.> as well as Ref. <cit.> for a textbook, 
        prescription was used. [Let's also note the more recent Hopf algebraic formulation <cit.> of renormalization (see Ref. <cit.> for a
        recent review). Its application to our model is beyond the scope of our present study.] 
        This is a powerful renormalization prescription, which allows to systematically construct counter-terms on a diagram by diagram basis and can be generalized to higher orders.
        It was then found in Refs. <cit.> 
        that, at the fixed point, divergent subgraphs (due to wave function renormalization) 
        cancel each other due to standard, , similar to those appearing in relativistic quantum electrodynamics (QED), Ward identities yielding the fixed point value of Eq. (<ref>). 
        Let's also remark that at the fixed point ≪ 1 and the result (<ref>) is reliable. But clearly, at higher energy scales such as those relevant to experiments, results based on the loop 
        expansion (<ref>) may receive strong corrections from higher orders and the perturbative approach is rightly questionable. 
        A way to overcome this difficulty may be to reorder the perturbative series in the form of a 1/N-expansion or, in other words, to perform a random phase approximation-like resummation, see, ,
        Ref. <cit.> for an attempt to carry out next-to-leading order (NLO) computations. 
        This is a very interesting suggestion which certainly requires further study. In the following, we shall pursue a more modest goal
        and restrict ourselves to the study of Eq. (<ref>) without any additional resummation. The reason is that, in our opinion, the problem of an  accurate evaluation of NLO
        corrections cannot reasonably be addressed before a  full understanding of the first few orders of the loop expansion is achieved.
        
        In the present paper, we reconsider the computation of the first order interaction correction to the minimal conductivity of graphene in the standard but more subtle non-relativistic 
        limit with instantaneous interactions (v/c  0). Our analysis clarifies the fact that the origin of the discrepancy between the different results
        found in this limit, the 𝒞^(i) (i=1,2,3) of Eqs. (<ref>), does not lie in the regularization method but, more simply, in the renormalization procedure itself. This was first revealed 
        in Ref. <cit.> with a proof obtained with the help of conventional renormalization. 
        In the following, we will give a stronger proof based on the BPHZ renormalization prescription. In particular, we will explicitly show that, in the present non-relativistic case, 
        non-standard Ward identities imply that divergent subgraphs (related to Fermi velocity renormalization) do contribute to the renormalized optical conductivity.
        Proceeding either via density-density or current-current correlation functions and properly taking into account of these counter-terms will provide a clear explanation 
        for why radiative corrections to the optical conductivity of graphene are finite and perfectly well determined. [We are therefore
        not in the field theoretic situation reviewed by Jackiw in Ref. <cit.>.]
        Anticipating the conclusion, we will show that our approach is in favour of the result first derived by Mishchenko <cit.>. Our final result reads:
        
        𝒞_r = 𝒞̃_r = 𝒞 + 𝒞' = 𝒞^(2),     𝒞 = 𝒞^(3),   𝒞' = -1/4  ,
        
        where r stands for renormalized and 𝒞' originates from one-loop counterterms.
        
        The paper is organized as follows. In Sec. <ref>, we introduce the basic model, the corresponding Feynman rules and the notations that will be used throughout the text.
        In Sec. <ref>, we briefly study the model at one-loop (self-energies and Ward identities) introducing the basic ingredients required for the two-loop computation. 
        In Sec. <ref>, we focus on the optical conductivity at two-loop and compute the counter-terms with the help of the BPHZ prescription.
        In Sec. <ref> we summarize our results and conclude. Some useful integrals are provided in App. <ref> and
        App. <ref> contains some additional illustration of the singular subgraphs in two-loop diagrams.
        
        
        § MODEL, FEYNMAN RULES AND RENORMALIZATION
        
        
        The effective low-energy action (model II in the terminology of Ref. <cit.>) that we wish to consider reads:
        
        
        S = ∫ t ^2 x ψ̅_σ[ γ^0 ( ∂_t -eA_0 ) +  v γ⃗·∇⃗ ] ψ^σ 
        + 1/2 ∫ t ^3 x ( ∇⃗ A_0 )^2   ,
        
        where v and e are the bare Fermi velocity and charge, respectively,
        ψ_σ is a four-component spinor field describing a fermion of specie σ (σ=1,⋯,N_F and for graphene N_F=2) and
        A_0 is the gauge field mediating the instantaneous Coulomb interaction.
        The Dirac matrices, γ^μ = (γ^0,γ⃗ ) satisfy the usual algebra {γ^μ,γ^ν} = 2g^μν,
        with metric tensor g^μν = diag(+,-,-). 
        
        From Eq. (<ref>), the bare momentum space fermion propagator reads:
        
        
        
        S_0(p) = i/p^2  ,      = γ^μ p_μ = γ^0 p_0 - v γ⃗·p⃗  ,
        
        where we use pseudo-relativistic notations with p^μ=(p^0,v p⃗ ).
        Because of the absence of any retardation effect (non-relativistic limit corresponding to v/c  0), the effective photon propagator reduces to the instantaneous Coulomb interaction and reads:
        
        
        V_0(q⃗ ) = /2 (|q⃗ |^2)^1/2  .
        
        Moreover, vector photons decouple and the bare vertex reduces to the temporal part: 
        
        
        - e Γ_0^0 = - e γ^0   .
        
        For our future purposes, it will be nevertheless convenient to define Γ^μ = (Γ^0, Γ⃗ ) such that Γ_0^μ = γ^μ = (γ^0, γ⃗ )
        and employ the following graphical notations:
        
              - e γ^0 =   20mm(20,10)
                  
                  pei,eobosonp,vvanillaei,vfermionv,eodecor.shape=circle,decor.filled=empty,decor.size=2thickv  , 
        
              - e γ⃗ =   20mm(20,10)
                  
                  pei,eobosonp,vvanillaei,vfermionv,eo
        	decor.shape=circle,decor.filled=shaded,decor.size=2thickv
                  
                  , 
        
              - e γ^μ =   20mm(20,10)
                  
                  pei,eobosonp,vvanillaei,vfermionv,eov  .
        
        
        In the following, we will use conventional dimensional regularization <cit.>. The above Feynman rules stay the same but momenta, 
        Dirac matrices and metric tensor are extended to span a D_e-dimensional space (keeping [  1] = 4 N_F) with 
        
        
        D_e = 2 - 2_γ  .
        
        All bare parameters and fields are then related to renormalized ones via renormalization constants in a standard way: 
        
        
              ψ = Z_ψ^1/2ψ_r,      A_0 = Z_A_0^1/2 A_0r, 
        	
              e = Z_e e_r μ^_γ,      v = Z_v v_r  ,
        
        where:
        
        
        Z_e = Z_Γ^0/Z_ψ Z_A_0^1/2  ,
        
        and μ is the so-called renormalization scale in the minimal subtraction (MS) scheme. The latter
        is related to the corresponding parameter μ in the modified minimal subtraction (MS) scheme with the help of:
        
        μ^ 2 = 4π e^-γ_Eμ^2  ,
        
        where γ_E is Euler's constant. The attractive feature of the MS scheme is that the renormalization constants take the simple form:
        
        
        Z_x(_gr) = 1 + δ Z_x (_gr) = 1 + ∑_l=1^∞∑_j=1^l Z_x^(l,j) _gr^l/_γ^j  ,
        
        where x ∈{ψ,A,e,v,Γ^μ}, _gr = e_r^2/(4π v_r) is the renormalized coupling constant and
        l runs over the number of loops at which ultra-violet (UV) singularities are subtracted. The Z_x do not depend on momentum; furthermore,
        the dependence on μ is only through _gr. So the Z_x depend only on _gr(μ) and .
        The basic correlation functions of the model are renormalized as follows:
        
              S(k) = Z_ψ S_r(k)  , 
        	
              V(q⃗ ) = Z_A_0 V_r(q⃗ )  , 
        	
              Γ^μ(k,q) = Z_Γ^μ^-1Γ_r^μ(k,q)  ,
        
        where the notation implicitly assumes that, besides the Z_x, only the renormalized quantities depend on μ.
        
        For the model of Eq. (<ref>), once integrated over the third spacial direction the effective action of the free gauge field becomes non-local.
        This implies that the gauge field is not renormalized: Z_A_0=1 <cit.>.
        As a consequence, the charge is not renormalized: Z_e=1. The renormalization of the coupling constant is therefore entirely due to the renormalization of the velocity which is the only
        running parameter of the model:
        
        _g = Z_ _gr,      Z_ = Z_v^-1  .
        
        The velocity β-function is then defined as:
        
              β_v(_gr) = μ∂ v_r/∂μ|_B = ∑_l=0^∞β_v,l_gr^l+1  , 
        	
              β_v,l = 2 v_r (l+1) Z_v^(l+1,1)  ,
        
        where the subscript B indicates that bare parameters, which do not depend on μ, are fixed.
        
        
        
        
        
        § ONE-LOOP ANALYSIS
        
        
        We start by analyzing model of Eq. (<ref>) at one-loop. 
        
        
        
         §.§ One-loop fermion and photon self-energies
        
        
        Let's first recall some elementary results for clarity, see Ref. <cit.> for more details. The one-loop fermion self-energy, Fig. <ref>a, is defined as:
        
        
        -iΣ_1(k) = ∫ [^d_e q]  (- e γ^0)  S_0(k+q)  (- e γ^0) V_0(q)  ,
        
        where d_e = 1 + D_e is the space-time dimension. The following parametrization is useful:
        
        Σ(k) = γ^0 k_0  Σ_(k^2) - v γ⃗·k⃗  Σ_k(k^2)  ,
        
        which is such that:
        
        Σ_(k^2) = [γ^0 k_0 Σ(k)]/4N_F k_0^2  ,   Σ_k(k^2) = [γ⃗·k⃗ Σ(k)]/4N_F v |k⃗ |^2  .
        
        Using this parametrization together with the standard rules for integrating massless Feynman diagrams straightforwardly yields:
        
        
        
           Σ_1k(|k⃗ |^2) =  -_gr/8 (μ^ 2/|k⃗ |^2)^_γ  
        e^2γ_E_γ G(1/2,1/2)   ,
        
        
        
        
        where G(α,β) is defined in Eq.(<ref>) and Σ_1(k^2) =0.
         The factor G(1/2,1/2) contains the pole 1/_γ at _γ→ 0 and is such that:
        
        
        
        
        G(1/2,1/2)=  Γ(1+_γ)  ( 1/_γ + 4 log 2 + (_γ) )   .
        
        
        Performing 
        
        the _γ expansion <cit.> then yields: 
        
        
        
        Σ_1k(|k⃗ |^2) =  -_gr/8 ( 1/_γ - L_k + 4 log 2 + (_γ) )   ,
        
        
        
        
        
        where L_k = log(|k⃗ |^2/μ^ 2). From the above results, we see that there is a momentum but no frequency-dependence of 
        the one-loop fermion self-energy due the instantaneous nature of the interaction. 
        At one-loop, there is therefore a Fermi velocity renormalization <cit.> but no wave-function renormalization: 
        
              Z_1v = 1 + δ Z_1v,     δ Z_1v= -_gr/8 _γ  ,
        
              Z_1ψ = 1 + δ Z_1ψ,     δ Z_1ψ = 0  .
        
        From Eq. (<ref>) we see that the velocity beta-function is negative: 
        
        β_v = - v_r _gr/4 + (_gr^2)  ,
        
        implying that Fermi velocity grows in the infrared <cit.>.
        Graphically, these results can be summarized as follows:
        
        δ Z_1v = 𝒦[ Σ_1k(k^2) ]   =   𝒦[  15mm(15,15)
              
              inoutplainin,viplain,tension=0.2vi,voboson,left,tension=0.2vi,voplainvo,outdecor.shape=circle,decor.filled=empty,decor.size=2thickvi,vo  ]   =   - _gr/8 _γ  ,
        
        where the pseudo Lorentz structure of the diagram in the brackets has been projected out, , the displayed graph corresponds to Σ_1k(k^2) and not Σ_1(k), and 
        the 𝒦 operator is defined as:
        
        𝒦 ( ∑_n=-∞^+∞c_n/_γ^n) = ∑_n=1^+∞c_n/_γ^n  .
        
        Similar notations and conventions will be used in the following.
        
        The one-loop photon self-energy, Fig. <ref>b, is defined in the usual way as:
        
        Π_1^μν(q) = - ∫ [^d_ek] [ (- e γ^μ) S_0(k+q) (- e γ^ν) S_0(k) ]   .
        
        Focusing on Π^00, performing the integrations and the _γ expansion <cit.> yields:
        
        Π_1^00(q_0,q⃗ 0) = -N_F e_r^2/8 |q⃗ |^2/ q_0 ( 1 - (1 + L_q_0) _γ + (_γ^2) )  ,
        
        which is finite as expected and where L_q_0=log(-q_0^2/(4v^2 μ^ 2)).
        Combining Eqs. (<ref>) and (<ref>), we arrive at the (renormalized) one-loop conductivity: 
        
        σ_1r(q_0) = N_F e_r^2/8  , 
        
        which, for N_F=2, corresponds to the well known universal ac conductivity σ_0, Eq. (<ref>).
        
        We may proceed in a similar way with the help of the Kubo formula Eq. (<ref>). 
        The transversality of Π^μν allows to parametrize it as follows:
        
        Π^μν(q) = (g^μνq^2 - q^μ q^ν) Π(q^2),   Π(q^2) = - Π^μ_μ(q)/(d_e-1)(-q^2)  .
        
        Then:
        
        σ̃(q_0) =  q_0   K(q_0),      K(q_0) = v^2 Π(q_0^2,|q⃗ |^2  0)   .
        
        Performing the integrations and the _γ expansion <cit.> yields:
        
        
        K_1(q_0) =  N_F   e_r^2/8   q_0 ( 1 - (1 + L_q_0) _γ + (_γ^2) )  . 
        
        Combining Eqs. (<ref>) and (<ref>), we arrive once again at: 
        
        σ̃_1r(q_0) = N_F e_r^2/8  , 
        
        in perfect agreement with Eq. (<ref>).
        
        
        
         §.§ One-loop fermion-photon vertex and Ward identities
        
        
        At this point it is also instructive to look at the vertex part: Γ^μ = γ^μ + Λ_1^μ + (_g^2) where Λ_1^μ is the 
        one-loop correction, see Fig. <ref>a. The latter is defined as:
        
        
        - e Λ_1^μ(k,q) = ∫ [^d_ep]  V_0(p-k) (- eγ^0) S_0(p+q)(-ieγ^μ) S_0(p)(- e γ^0)  .
        
        We first consider the temporal part, Fig. <ref>b. In order to single out it's UV divergent part we will evaluate it for k=q=0. 
        Performing the trace and going to euclidean space (q_0= q_E0) yields:
        
        Λ_1^0(0,0) = μ^2_γ e_r^2/2 γ^0 ∫ [^D_ep] [ p_E0] p_E0^2 - v^2 |p⃗ |^2/[p_E0^2 + v^2 |p⃗ |^2]^2 |p⃗ | = 0  ,
        
        where the frequency integral vanishes identically, see App. <ref> for some useful integrals. 
        The temporal part of the vertex is therefore not renormalized at one-loop.  Graphically, this result can be summarized as follows:
        
        δ Z_1Γ^0 = - 𝒦[ Λ_1^0/γ^0 ]   =   -
        𝒦[  15mm(15,15)
              
              ine1,e2bosonin,viplaine1,v1plain,tension=0.7v1,viplain,tension=0.7vi,v2plainv2,e2boson,right,tension=0.7v1,v2decor.shape=circle,decor.filled=empty,decor.size=2thickvi,v1,v2  ]    = 0   .
        
        Together with Eq. (<ref>), this implies that: Z_1ψ = Z_1Γ^0 = 1
        and the Ward identity, Z_ψ = Z_Γ^0, is (trivially) satisfied at one-loop.
        
        Let's now turn to the vector part of the vertex, see Fig. <ref>c, focusing again of the case where k=q=0.
        Performing the trace and going to euclidean space yields:
        
        Λ⃗_1(0,0) = μ^2_γ e_r^2/2 γ⃗ ∫ [^D_ep] [ p_E0] p_E0^2 + v^2 |p⃗ |^2 (D_e-2)/D_e/[p_E0^2 + v^2 |p⃗ |^2]^2 |p⃗ |  ,
        
        where now the frequency integral is non-zero. Performing the remaining integrations, we arrive at:
        
        Λ⃗_1(0,0) = γ⃗  _g/8 (μ^ 2/m^2)^_γ e^γ_E_γ Γ(1+_γ)/_γ  ,
        
        which shows that the vector part of the vertex is UV singular (m is just an arbitrary IR regulator). Extracting the pole part, the corresponding renormalization constant together with its graphical representation
        are given by:
        
        δ Z_1 Γ⃗  = - 𝒦[ Λ⃗_1/γ⃗ ]   =   -
        𝒦[  15mm(15,15)
              
              ine1,e2bosonin,viplaine1,v1plain,tension=0.7v1,viplain,tension=0.7vi,v2plainv2,e2boson,right,tension=0.7v1,v2decor.shape=circle,decor.filled=shaded,decor.size=2thickvidecor.shape=circle,decor.filled=empty,decor.size=2thickv1,v2  ]    = - _gr/8 _γ  .
        
        At one-loop, this result is consistent with the Ward identity: Z_Γ⃗  = Z_v Z_ψ which may be graphically represented as:
        
        𝒦[  15mm(15,15)
              
              inoutplainin,viplain,tension=0.2vi,voboson,left,tension=0.2vi,voplainvo,outdecor.shape=circle,decor.filled=empty,decor.size=2thickvi,vo  ]   =   -
        𝒦[  15mm(15,15)
              
              ine1,e2bosonin,viplaine1,v1plain,tension=0.7v1,viplain,tension=0.7vi,v2plainv2,e2boson,right,tension=0.7v1,v2decor.shape=circle,decor.filled=shaded,decor.size=2thickvidecor.shape=circle,decor.filled=empty,decor.size=2thickv1,v2  ]    .
        
        Notice that the peculiar Ward identities Eqs. (<ref>) and (<ref>) are rather unusual with respect to those which can be found in the case of usual (Lorentz-invariant) QEDs.
        This will play a crucial role in deriving the correct interaction correction to the optical conductivity in the non-relativistic limit.
        
        
        
        
        § OPTICAL CONDUCTIVITY AT TWO LOOPS
        
        
        
        
        
        We now proceed on computing the 2-loop corrections displayed on Fig. <ref>:
        Π_2^μν(q) = 2 Π_2a^μν(q) + Π_2b^μν(q)
        where Π_2a is the so-called self-energy correction and Π_2b is the so-called vertex correction. The latter are defined in the usual way as:
        
           Π_2a^μν(q) =
        - ∫ [^d_ek ] [ (- eγ^ν)  S_0(k+q) (- eγ^μ)  S_0(k)( -Σ_1(k) ) S_0(k) ]  ,
        
           Π_2b^μν(q) = - ∫ [^d_ek] [ (- eγ^ν)  S_0(k+q) (- e Λ_1^μ(k,q))  S_0(k) ]  ,
        
        where the fermion self-energy was defined in Eq. (<ref>) and the fermion-photon vertex in Eq. (<ref>).
        For completeness, the diagrams associated to Π^00(a) are displayed on Fig. <ref>. Our convention for the conductivity will be that:
        
           σ_2 = σ_2a + σ_2b  , 
        	
           σ_2a = - lim_q⃗ 0  q_0/|q⃗ |^2 2  Π_2a^00(q_0,q⃗ )  , 
        	
           σ_2b = - lim_q⃗ 0  q_0/|q⃗ |^2 Π_2b^00(q_0,q⃗ )  ,
        
        and similarly for σ̃_2.
        
        
        
         §.§ Density-density correlation function approach
        
        
        The computation of the self-energy diagrams displayed on Fig. <ref> has been performed
        in Ref. <cit.> (see also App. B). We briefly recall the final results for clarity:
        
           2 Π_2a^00(q_0,q⃗ 0) = - N_F  e_r^2/8 |q⃗ |^2/ q_0 _gr/2  ,
        
           Π_2b^00(q_0,q⃗ 0) = -N_F  e_r^2/8 |q⃗ |^2/ q_0 _gr 8-3 π/6  ,
        
        where, with two-loop accuracy, the coupling constant is the renormalized one.
        Combining these results with Eqs. (<ref>) yields:
        
           σ_2a(q_0) = N_F  e_r^2/8 _gr/2  ,
        
           σ_2b(q_0) = N_F  e_r^2/8 _gr 8-3 π/6  ,
        
        from which the total (bare) optical conductivity reads:
        
        σ_2(q_0) = N_F  e_r^2/8 _gr 11-3 π/6  ,
        
        with a (bare) interaction correction coefficient corresponding to 𝒞^(3).
        
        We now proceed on computing the renormalized conductivity using the BPHZ prescription. According to the latter, the renormalized diagrams contributing to the density-density 
        correlation function, Π_2  r^00(q) (= a,b), are related to the bare ones as follows:
        
           Π_2  r^00 = ℛ Π_2 ^00 = (1 - 𝒦) ℛ' Π_2 ^00   (=a, b)  ,
        	
           ℛ' Π_2 ^00 = Π_2^00 + Π_2'^00  ,
        
        where ℛ' is the so-called incomplete R-operation because it subtracts only the subdivergences and Π_2'^00 may be graphically represented as:
        
           Π_2a'^00 = -
        𝒦 [ 15mm(15,15)
              inoutplainin,viplain,tension=0.2vi,voboson,left,tension=0.2vi,voplainvo,outdecor.shape=circle,decor.filled=empty,decor.size=2thickvi,vo ] ⋆ 15mm(15,15)
              inoutbosonin,veplain,right,tension=0.2ve,vwplain,right,tension=0.2vw,vebosonvw,outdecor.shape=circle,decor.filled=empty,decor.size=2thickve,vw   , 
        
           Π_2b'^00 = -2
        𝒦 [ 15mm(15,15)
              ine1,e2bosonin,viplaine1,v1plain,tension=0.7v1,viplain,tension=0.7vi,v2plainv2,e2boson,right,tension=0.7v1,v2decor.shape=circle,decor.filled=empty,decor.size=2thickvi,v1,v2 ] ⋆ 15mm(15,15)
              inoutbosonin,veplain,right,tension=0.2ve,vwplain,right,tension=0.2vw,vebosonvw,outdecor.shape=circle,decor.filled=empty,decor.size=2thickve,vw   .
        
        The peculiarity of the present non-relativistic theory is that the one-loop fermion self-energy and fermion-photon vertex subgraphs appearing in Eq. (<ref>) are not related by a Ward identity
        and therefore do not cancel each other (contrarily to what happens in usual QED). The case of Π_2b^00 is trivial: this diagram is finite overall so 
        𝒦Π_2b^00=0 and, from Eq. (<ref>), its subgraphs is also finite so: Π_2b'^00=0. This diagram is therefore absolutely convergent in 
        Weinberg's sense <cit.> so that:
        
        σ_2br(q_0) =  σ_2b = N_F  e_r^2/8 _gr 8-3 π/6  .
        
        The case of Π_2a^00 is more interesting: this diagram is also finite overall, so: 𝒦Π_2a^00=0. However, its subgraph is divergent, see Eq. (<ref>),
        and needs to be subtracted. In order to compute Π_2a'^00 we go to the integral representation of Eq. (<ref>) as the ⋆ operation
        does not reduce to a simple multiplication (the diagram is not logarithmic and care must be taken in projecting out its pseudo-Lorentz structure). 
        Performing the trace, going to euclidean space, integrating over frequencies and taking the q⃗ 0 limit
        leads to:
        
        Π_2a'^00(m,q⃗ 0) =  -N_F   e_r^2/4v μ^2_γ |q⃗ |^2 D_e-1/D_e ∫ [^D_e k] 𝒦[ Σ_1k(|k⃗^ 2|) ] |k⃗ |^2 - m^2/|k⃗ |  [|k⃗ |^2 + m^2]^2  ,
        
        where m = q_E0/2 v. Substituting Eq. (<ref>) and computing the integral using the formulas of App. <ref> yields:
        
        Π_2a'^00(m,q⃗ 0) = N_F e_r^2/16π v |q⃗ |^2/m _gr/8_γ (μ^ 2/m^2)^_γ (D_e-1) (D_e-2)/D_e e^2γ_E_γ
         B(1,1/2)  ,
        
        which is finite in the limit _γ 0 and reads:
        
        Π_2a'^00(q_0,q⃗ 0) = N_F e_r^2/8 |q⃗ |^2/ q_0 _gr/8  .
        
        The fact that both Π_2a'^00 and Π_2a^00 are finite implies that 𝒦 ℛ' Π_2 a^00=0 which was expected since there is no global divergence to subtract.
        However, the subtraction of the subdivergence brings a  finite contribution to the renormalized function:
        
        
        2Π_2ar^00(q_0,q⃗ 0) = 2Π_2a^00 + 2Π_2a'^00 
        = -N_F e_r^2/8 |q⃗ |^2/ q_0 _gr ( 1/2 - 1/4)  .
        
        The corresponding renormalized conductivity reads:
        
        σ_2ar(q_0) = N_F  e_r^2/8 _gr ( 1/2 - 1/4)  ,
        
        and is decreased with respect to its bare value Eq. (<ref>) by a quantity:  𝒞'=-1/4. 
        Hence,  the total two-loop renormalized conductivity reads:
        
        σ_2r(q_0) = σ_2ar + σ_2br = N_F  e_r^2/8 _gr ( 11-3 π/6 - 1/4) 
        = N_F  e_r^2/8 _gr 19-6π/12  , 
        
        in accordance with the advertised result for the interaction correction coefficient, Eq. (<ref>).
        
        
        
        
         §.§ Current-current correlation function approach
        
        
        We now proceed in a similar way with the self-energy diagrams displayed on Fig. <ref>. The latter
        were computed in Ref. <cit.> (see also App. B). We briefly recall the final results for clarity:
        
           2K_2a(q_0) = N_F e_r^2/8 q_0 _gr/4 ( -1/_γ + 3 + 2 L_q_0 - 4 log 2 + (_γ) )  ,
        
           K_2b(q_0) = -2K_2a(q_0)
        	+N_F e_r^2/8  q_0 _gr 11 - 3π/6  ,
        
        where again the coupling constant can be replaced by the renormalized one with two-loop accuracy and L_q_0=log(-q_0^2/(4v^2 μ^ 2)). Notice
        that the diagrams are now individually divergent contrarily to those of Eqs. (<ref>).
        Singular terms cancel from the sum of Eqs. (<ref>) and (<ref>). Combining these equations with Eq. (<ref>)
        yields the total (bare) optical conductivity at two-loops: 
        
        σ̃_2(q_0) = N_Fe_r^2/8 _gr 11-3π/6  ,
        
        with a (bare) interaction correction coefficient corresponding once again to 𝒞^(3).
        
        We now proceed on computing the renormalized conductivity using the BPHZ prescription.
        Similarly to the density-density case, the renormalized diagrams contributing to the current-current correlation function, K_2  r(q_0) (= a,b), are related to the bare ones as follows:
        
           K_2  r = ℛ K_2  = (1 - 𝒦) ℛ' K_2    (=a, b)  , 
        	
           ℛ' K_2  = K_2 + K_2'  ,
        
        where K_2' may be graphically represented as:
        
           K_2a' = -
        𝒦 [ 15mm(15,15)
              inoutplainin,viplain,tension=0.2vi,voboson,left,tension=0.2vi,voplainvo,outdecor.shape=circle,decor.filled=empty,decor.size=2thickvi,vo ] ⋆ 15mm(15,15)
              inoutbosonin,veplain,right,tension=0.2ve,vwplain,right,tension=0.2vw,vebosonvw,outdecor.shape=circle,decor.filled=full,decor.size=2thickve,vw  , 
        
           K_2b' = -2
        𝒦 [ 15mm(15,15)
              ine1,e2bosonin,viplaine1,v1plain,tension=0.7v1,viplain,tension=0.7vi,v2plainv2,e2boson,right,tension=0.7v1,v2decor.shape=circle,decor.filled=full,decor.size=2thickvidecor.shape=circle,decor.filled=empty,decor.size=2thickv1,v2 ] ⋆ 15mm(15,15)
              inoutbosonin,veplain,right,tension=0.2ve,vwplain,right,tension=0.2vw,vebosonvw,outdecor.shape=circle,decor.filled=full,decor.size=2thickve,vw   .
        
        Contrarily to the density-density case, both K_2a and K_2b need to be properly renormalized. Indeed, in the present case, both one-loop fermion self-energy and fermion-photon vertex 
        subgraphs appearing in Eqs. (<ref>) are singular. One may wonder at this point if these contributions will cancel each other due to the Ward identity Eq. (<ref>).
        As will be shown in the following, this is not the case. The proof requires some care as the ⋆ operation does not (necessarily) reduce to a simple multiplication. 
        Notice for example that only the vector part of the fermion-photon vertex subgraph is singular; but the full vertex appears in Eq. (<ref>).
        So, one needs to be careful in projecting out only the vector component. In order to do this we will use the integral representation of Eqs. (<ref>)
        
        We first consider the integral representation of K_2a'. Performing the trace, going to euclidean space, integrating over frequencies and taking the q⃗ 0 limit
        leads to:
        
        
        	K_2a'(m) = N_F e_r^2/4v m^2 D_e-1/D_e μ^2_γ ∫ [^D_ek] 𝒦[ Σ_1k(|k⃗^ 2|) ] |k⃗ | (|k⃗ |^2 - m^2)/[|k⃗ |^2 + m^2]^2  .
        
        Substituting Eq. (<ref>) and computing the integral yields:
        
        
        	K_2a'(m) = N_F e_r^2/16π v m _gr/8 _γ (μ^ 2/m^2)^2_γ (D_e-1) e^γ_E_γ B(1,-1/2)  .
        
        The _γ-expansion reads:
        
        
        2K_2a'(q_0) = N_F e_r^2/8 q_0 _gr/4 ( 1/_γ - 2 - L_q_0 + (_γ) )  .
        
        Adding the contribution of 2K_2a the pole terms cancel each other and we find that:
        
        
        2 K_2 ar = 2 ℛ' K_2 a = N_F e_r^2/8 q_0 _gr/4 (1 + L_q_0 - 4 log 2 + (_γ) )  .
        
        The fact that 𝒦ℛ' K_2 a = 0 means that the overall counter-term is zero in accordance with the fact that the only singularity which needs to be subtracted is the
        one associated with the fermion self-energy subgraph. From Eq. (<ref>), we may now derive the expression for the renormalized conductivity associated with 
        Fig. <ref>a:
        
        σ̃_2ar(q_0) = N_Fe_r^2/8 _gr/4 (1 + L_q_0 - 4 log 2 + (_γ) )  .
        
        
        We now consider the case of  K_2b'. In order to compute the latter, we find it convenient to go back to the initial definition Eq. (<ref>) replacing
        Λ_1^μ by γ⃗ 𝒦[ Λ⃗_1/γ⃗ ] with the appropriate γ-matrix contractions. Performing the trace, going to euclidean space, 
        integrating over frequencies and taking the q⃗ 0 limit leads to:
        
        
        	K_2b'(m) = - N_F e_r^2/v m^2 D_e-1/D_e μ^2_γ 𝒦[ Λ⃗_1/γ⃗] ∫ [^D_ek] |k⃗ |/|k⃗ |^2 + m^2  .
        
        The integral is easily computed with the rules for evaluation semi-massive tadpoles, see App. <ref>, and the result reads:
        
        
        	K_2b'(m) = - N_F e_r^2/4π v m 𝒦[ Λ⃗_1/γ⃗ ] (μ^ 2/m^2)^_γ D_e-1/D_e e^γ_E_γ B(1,-1/2)  .
        
        Substituting Eq. (<ref>) and performing the _γ-expansion then yields:
        
        
        K_2b'(q_0) = N_F e_r^2/8  q_0 _gr/4 ( -1/_γ + 1 + L_q_0 + (_γ) )  .
        
        Adding the contribution of K_2b, the pole terms cancel each other and we find that:
        
        
        	K_2 br = 2 ℛ' K_2 b = N_F e_r^2/8  q_0 _gr 11 - 3π/6
        	+ N_F e_r^2/8 q_0 _gr/4 (-2 - L_q_0 + 4 log 2 + (_γ) )  .
        
        The fact that 𝒦ℛ' K_2 b = 0 means that the overall counter-term is zero in accordance with the fact that the only singularity which needs to be subtracted is the
        one associated with the vector photon-fermion vertex subgraph. From Eq. (<ref>), we may now derive the expression for the renormalized conductivity associated with
        Fig. <ref>b:
        
        σ̃_2br(q_0) = N_F e_r^2/8 _gr 11 - 3π/6 + N_Fe_r^2/8 _gr/4 (-2 - L_q_0 + 4 log 2 + (_γ) )  .
        
        
        Hence, adding Eqs. (<ref>) and (<ref>), the total two-loop renormalized conductivity reads:
        
        σ̃_2r(q_0) = σ̃_2ar + σ̃_2br = N_F  e_r^2/8 _gr ( 11-3 π/6 - 1/4) 
        	= N_F  e_r^2/8 _gr 19-6π/12  ,
        
        in accordance with the advertised result for the interaction correction coefficient, Eq. (<ref>), 
        and therefore with the result obtained from the density-density correlation function, Eq. (<ref>).
        
        
        
        
        § CONCLUSION
        
        
        In this paper, we have elaborated a field theoretic renormalization approach to the computation of electron-electron interactions in graphene and related planar Dirac 
        liquids in the non-relativistic limit (v/c  0) with instantaneous Coulomb interaction. As we have seen, the broken Lorentz invariance brings some peculiarities 
        with respect to the relativistic case (non-standard Ward identities) as well as some complications (semi-massive Feynman diagrams instead of the massless
        ones). Focusing on the optical conductivity of these materials, we have provided a clear proof that radiative corrections affecting this experimentally 
        relevant observable are  finite and  well determined. Our proof makes use of the powerful BPHZ prescription. It confirms the validity of our previous approach 
        based on conventional renormalization <cit.> but is considerably more robust and allowed us to perform a refined diagram by diagram analysis.  
        Both the density (where individual diagrams are overall finite) and Kubo formula (where individual diagrams are explicitly singular) 
        approaches were shown to yield a single well-defined result for the two-loop interaction correction to the conductivity: 𝒞^(2) = (19-6π)/12 ≈ 0.013, 
        a result first found by Mishchenko <cit.> and which is compatible with experimental uncertainties <cit.>.
        At this point, let's recall that Mishchenko's analysis <cit.> warns against the use of the Kubo formula with a hard cut-off regulator. Reassuringly, 
        in dimensional regularization no such problem is encountered and both density-density and current-current approaches can be safely used. Our formalism, can be  
        extended to higher orders, [As we have discussed in the Introduction, for coupling constant _g ≈ 1, perturbation theory is highly questionable. 
        For the polarization operator, the appearance of a small numerical constant 𝒞^(2) in factor of α_g brings a small parameter 𝒞^(2)α_g.  This may restore the validity of perturbation theory for the optical conductivity provided that higher 
        order terms may be neglected as well. However, beyond first order, the value of 𝒞_n 
        at order n is unknown and it is therefore an open question as to whether or not 𝒞_nα_g^n is small.] 
         other quantities and/or systems of other dimensionality, , the optical conductivity of 3D Dirac materials.
        It also constitutes a solid base on which the perturbation theory may be optimized in order to deal with the strong-coupling 
        problem. [The computation of NLO corrections in Ref. <cit.> was approximate as an NLO diagram with two-loop polarization insertion was neglected (see lines below
        Eq. (9) in that paper). It turns out that this important diagram is the one proportional to 𝒞α_g. In another context, this kind of diagram was computed
        exactly: see diagram 1 in Fig. 2 of Ref. <cit.> together with Eqs. (19) and (20) in this reference as well as the sentence below
        Eq. (19) in Ref. <cit.> where Π̂ is a number analogous to 𝒞. The fact that this number is small (as in Ref. <cit.> 
        where it is actually proportional to 𝒞^*) or large (as in Ref. <cit.>) strongly impacts the results. Though this would require a more careful analysis,
        our proof that 𝒞 = 𝒞^(2) is indeed small for graphene may further justify
        the neglect of the corresponding diagram in Ref. <cit.> which may in turn insure the good convergence of the RPA observed in Ref. <cit.>.]
        We leave these issues for our future investigations.
        
        
        
        
        The work of A.V.K. was supported in part by the Russian Foundation for
        Basic Research (Grant No.16-02-00790-a).
        
        
        
        § MASTER INTEGRALS
        
        
        All integrals for counter-terms in the main text can be computed with the help of two basic integrals that we recall here.
        
        The first one in the one-loop massless propagator-type integral with n≤ 2 which reads <cit.>:
        
           ∫ [^D q] q^μ_1… q^μ_n/[q^2]^[(q-k)^2]^β  =
        (k^2)^D/2--β/(4π)^D [k^μ_1… k^μ_n  G^(n,0)_0(,β)  + δ^2_n g^μ_1 μ_2/D  G^(1,1)_1(,β)],    
           G^(n,m)_i(,β) = a_n()a_m(β)/a_n+m-i
        (+β-D/2-i),
        
           a_n() = Γ(D/2- + n)/Γ()  ,
        
        
        where [^D q] = ^D q / (2π)^D and  δ^2_n is the Kronecker
        symbol. The simplified notation: G(,β)=G^(0,0)_0(,β), will also be used.
        
        The second one is the one-loop semi-massive tadpole diagram which reads:
        
           ∫[^D k]/[k^2]^[k^2+m^2]^β =
        (m^2)^D/2--β/(4π)^D/2 B(β,)  ,
        
           B(β,) = Γ(D/2-) Γ( + β - D/2)/Γ(D/2) Γ(β)  .
        
        
        
        
        § PROPERTIES OF TWO-LOOP SELF-ENERGIES
        
        
        Here we present some peculiar properties of the two-loop self-energies Π_2a^00(q_0,q⃗ 0) and K_2a(q_0).
        
        
        
         §.§ Π_2a^00(q_0,q⃗ 0)
        
        
        Performing the trace, going to euclidean space, integrating over frequencies and taking the q⃗ 0 limit
        leads to  <cit.>:
        
        Π_2a^00(m,q⃗ 0) =  N_F   e_r^2/4v μ^2_γ |q⃗ |^2 D_e-1/D_e ∫ [^D_e k] Σ_1k(|k⃗^ 2|)  |k⃗ |^2 - m^2/|k⃗ |  [|k⃗ |^2 + m^2]^2  ,
        
        where, as in the main text, m = q_E0/2 v. Notice that this result can be represented in the following 
        
        form:
        
        Π_2a^00(m,q⃗ 0) =  N_F   e_r^2/4v μ^2_γ |q⃗ |^2 D_e-1/D_e Σ_1k(|k⃗^ 2|=1)   J_+(â=1)   ,
        
        where
        
             J_±(â) =
        ∫ [^D_e k] |k⃗ |^2 - m^2/[|k⃗ |^2]^± 1/2 + â_γ  [|k⃗ |^2 + m^2]^2  .
        
        The one-loop self-energy Σ_1k(|k⃗^ 2|) is singular at _γ→0 (see (<ref>)), but the integral
         J_+(â) ∼ O(_γ) in this limit. Indeed:
        
        
          J_+(â) ∼ B(2,-1/2 + â_γ) - B(2,1/2 + â_γ) 
        = -2(â+1) _γ   B(1,1/2 + â_γ)  ,
        
        with B(β,) defined in Eq. (<ref>).
        
        So, the result for Π_2a^00(q_0,q⃗ 0) has the following form
        
        Π_2a^00(m,q⃗ 0) = N_F e_r^2/16π v |q⃗ |^2/m _gr/8 (μ^ 2/m^2)^2_γ (D_e-1) 2_γ/D_e e^2γ_E_γ G(1/2,1/2) B(1,1/2+_γ)  ,
        
        which is finite in the limit _γ 0 and reads <cit.>:
        
        Π_2a^00(q_0,q⃗ 0) = N_F e_r^2/16 |q⃗ |^2/v m _gr/8  .
        
        
        The last integrations in Π_2a^00(q_0,q⃗ 0) (see Eqs. (<ref>) and (<ref>)) and in
        Π_2a'^00(q_0,q⃗ 0) (see Eq. (<ref>)) are related with the tadpoles J_+(â=1) and
        J_+(â=0), respectively. Since at _γ→ 0J_+(â) ∼ (â+1) _γ (see Eq. (<ref>)),
        then the counter-term Π_2a'^00(q_0,q⃗ 0) equals half the value of Π_2a^00(q_0,q⃗ 0) and thus the final
        result of Π_2ar^00(q_0,q⃗ 0) in (<ref>) is two times less than the one of Π_2a^00(q_0,q⃗ 0).
        
        
        
        
         §.§ K_2a(q_0)
        
        
        Performing the trace, going to euclidean space, integrating over frequencies and taking the q⃗ 0 limit
        leads to <cit.>:
        
        
        K_2a(m) = -N_F e_r^2/4v m^2 D_e-1/D_e μ^2_γ ∫ [^D_ek] Σ_1k(|k⃗^ 2|)  |k⃗ | (|k⃗ |^2 - m^2)/[|k⃗ |^2 + m^2]^2  .
        
        As it was for Π_2a^00(q_0,q⃗ 0),
        
        this result can be represented in the following form:
        
        
        K_2a(m) =  - N_F   e_r^2/4v μ^2_γ |q⃗ |^2 D_e-1/D_e Σ_1k(|k⃗^ 2|=1)   J_-(â=1)   .
        
        The one-loop self-energy Σ_1k(|k⃗^ 2|) is singular at _γ→ 0 (see (<ref>)) and the integral
         J_-(â) is finite in this limit:
        
        
          J_-(â) ∼ B(2,-3/2 + â_γ) - B(2,-1/2 + â_γ)
        = 2(1-(â+1) _γ)    B(1,-1/2 + â_γ)  .
        
        So, the result for K_2a(m) has the following form:
        
        
        
        K_2a(m) =  N_F e_r^2/16π v m _gr/8 (μ^ 2/m^2)^2_γ 2(D_e-1)^2/D_e  e^γ_E_γ  G(1/2,1/2) B(1,-1/2+_γ)  ,
        
        and its _γ-expansion reads <cit.>:
        
        
        K_2a(q_0) = -N_F e_r^2/16 vm _gr/8 ( 1/_γ - 3 +4ln 2 -2 L_q_0 + (_γ) )  .
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