Vacuum polarization of planar Dirac fermions by a superstrong Coulomb potential V.R. Khalilov a† and I.V. Mamsurov Faculty of Physics, Lomonosov Moscow State University, 119991, Moscow, Russia Westudythevacuumpolarization ofplanarcharged DiracfermionsbyastrongCoulomb poten- tial. Inducedvacuum charge density is calculated and analyzed at thesubcritical and supercritical Coulomb potentials for massless and massive fermions. For the massless case the induced vacuum charge density is localized at the origin when the Coulomb center charge is subcritical while it has 6 a power-law tail when the Coulomb center charge is supercritical. The finite mass contribution 1 into the induced charge due to the vacuum polarization is small and insignificantly distorts the 0 Coulomb potential only at distances of order of the Compton length. The induced vacuum charge 2 has a screening sign. As is known the quantum electrodynamics vacuum becomes unstable when n theCoulomb centercharge is increased from subcritical tosupercritical values. In thesupercritical a Coulomb potential the quantum electrodynamics vacuum acquires the charge due to the so-called J real vacuum polarization. We calculate the real vacuum polarization charge density. Screening of 8 theCoulombcenterchargearebrieflydiscussed. Weexpectthatourresultswillbehelpfulformore 2 deep understanding of the fundamental problem of quantum electrodynamics and can as a matter of principle betested in graphenewith a supercritical Coulomb impurity. ] h PACSnumbers: 12.20.-m,73.43.Cd,71.55.-i p Keywords: Vacuum polarization; Planar Dirac fermion; Induced vacuum charge; Supercritical Coulomb - t potential; Realvacuumpolarization n a u q [ 1 v 8 6 6 7 0 . 1 0 6 1 : v i X r a a Correspondingauthor † [email protected] 2 I. INTRODUCTION ThevacuumofthequantumelectrodynamicsandtheinducedvacuumpolarizationinastrongCoulomb field produced by a heavy atomic nucleus have been studied a long time [1–7]. When the nuclear charge Z e (eistheelectroncharge)isincreasedfromsubcriticaltosupercriticalvaluesthenthe lowestelectron | | energylevel(intheregularizedCoulombpotential)divesintothenegativeenergycontinuumandbecomes a resonance with complex “energy”E = E eiτ signaling the instability of the quantum electrodynamics | | vacuum in the supercritical range. The nuclear charge Z e for which the lowest energy level descends cr | | to the negative-energy continuum boundary m is called the critical charge for the ground state. The − critical charge is obviously related to the fine structure constant 1/137 and the number Z 170 [8]. cr ∼ New interest to these problems was revived in connection with the charged impurity problem in graphene because charged impurities can produce the supercritical Coulomb potential due to the cor- responding “effective fine structure constant” is large. In graphene, the electrons near the Fermi surface canbedescribedintermsofaneffectiveLorentz-invarianttheorywiththeirenergydeterminedbyDirac’s dispersion law for massless fermions [9–11], which allows to consider graphene as the condensed matter analog of the quantum electrodynamics in 2+1 dimensions [12, 13]. The existence of charged Fermi quasiparticles in graphene makes experimentally feasible to observe the vacuum polarization in strong Coulomb field but the massless case turn out to be rather more complicated since an infinite number of quasi-localized resonances emerges in the hole sector at the supercritical Coulomb potential [14–16]. Charged impurity screening produced by massless charged fermions in graphene in terms of vacuum polarization were investigated in [10, 11, 14, 15, 17–23]. For massless fermions the induced vacuum charge density is localized at the origin in the subcritical Coulomb potential [14, 19, 21] while it has the form c/r2, therefore, causing a modification of the supercriticalCoulomb potential [14, 15]. The vacuum polarizationof the massive chargedfermions canalso be of interest for graphene with Coulomb impurity [24]. For massive fermions the vacuum polarization charge density behaves differently from the massless ones. The dynamics of charged fermions in a Coulomb potential is governed by a singular Dirac Hamilto- nian that requires the supplementary definition in order for it to be treated as a self-adjoint quantum- mechanical operator. So, at first we need to determine the self-adjoint Dirac Hamiltonians and then to construct the correct Green function of the Dirac equation in a singular Coulomb potential. The self- adjoint Dirac Hamiltonians are not unique and can be specified by a self-adjoint extension parameter which implies additional nontrivial boundary conditions on the wave functions at the origin [25]. Here we study the vacuum polarization of planar charged Dirac fermions in a Coulomb potential. We express the induced charge density in the vacuum via the exact Green’s function, constructed from solutions of the self-adjoint two-dimensional Dirac Hamiltonians with a (subcritical and supercritical) Coulomb potential. To avoid misunderstanding, it should be noted that by Coulomb potential in 2+1 dimensions, we mean potential that decreases as 1/r with the distance from the source, having in mind that in a physical situation (e.g., in graphene), although the electrons move in a plane, their interaction with the external field of the Coulomb impurity occurs in a physical (three-dimensional) space and the electric fieldstrengthofthe impurityis a three-dimensional(nottwo-dimensional)vector. Therefore,the potential A (r) 1/r (and not A (r) logr, as would be the case in 2+1 dimensions) does not satisfy 0 0 ∼ ∼ the two-dimensional Poissonequation with a pointlike source at the origin. In order to see how the electron spin affects on the physical process under investigation, we also consider a superposition of Coulomb and Aharonov–Bohm (AB) potentials. Then, the two-dimensional Dirac Hamiltonian will contain the term characterizing the interaction potential of the electron spin magnetic moment with AB magnetic field H = (0, 0, H) = A = Bπδ(r) in the form seBδ(r)/r, ∇× − which is singular and must influence the behavior of solutions at the origin. Here s corresponds to the spin projection of a planar electron on the z (quantization) axis in three spatial dimensions. We note that such kind of point interaction also appears in several Aharonov–Bohm-likeproblems [26–28]. We shall adopt the units where c=~=1. II. GREEN’S FUNCTION FOR THE SELF-ADJOINT TWO-DIMENSIONAL DIRAC HAMILTONIANS It is well known that the Dirac γµ-matrix algebra is known to be represented in terms of the two- dimensional Pauli matrices γ0 = σ , γ1 = isσ , γ2 = iσ where the parameter s = 1 can be 3 1 2 ± introduced to label two types of fermions in accordance with the signature of the two-dimensionalDirac 3 matrices [29]; for the case of massive fermions it can be applied to characterizetwo states of the fermion spin (spin ”up” and ”down”) [30]. The Dirac Hamiltonian for a fermion of the mass m and charge e = e < 0, which contains a 0 − parameter s to describe the particle spin, in Coulomb (A (r) = Ze /r a/e r, A =0, A = 0, a > 0) 0 0 0 r ϕ ≡ and Aharonov–Bohm (A = 0, A = 0, A = B/r) potentials (r = x2+y2 and ϕ = arctan(y/x) are 0 r ϕ circular cylindrical coordinates) is p H =σ P sσ P +σ m e A (r), (1) D 1 2 2 1 3 0 0 − − whereP = i∂ eA isthegeneralizedfermionmomentumoperator(athree-vector). TheHamiltonian µ µ µ − − (1)shouldbedefinedasaself-adjointoperatorintheHilbertspaceofsquare-integrabletwo-spinorsΨ(r). The total Dirac momentum operator J = i∂/∂ϕ+sσ /2 commutes with H . Eigenfunctions of the 3 D − Hamiltonian (1) are (see, [31, 32]) 1 f(r) Ψ(t,r)= exp( iEt+ilϕ) , (2) √2πr g(r)eisϕ − (cid:18) (cid:19) whereE isthefermionenergy,l istheintegerquantumnumber. ThewavefunctionΨisaneigenfunction of the operator J with eigenvalue j = (l+s/2) in terms of the angular momentum l and ± f(r) hˇF(r)=EF(r), F(r)= , (3) g(r) (cid:18) (cid:19) where d l+µ+s/2 a hˇ =isσ +σ +σ m , µ e B. (4) 2 1 3 0 dr r − r ≡ The induced current density due to to vacuum polarization is determined by the three-vector j (r), µ which is expressed via the single-particle Green function of the Dirac equation as e dE jµ(r)= trG(r,r′;E)r=r′γµ, (5) −2 2πi | Z C whereCisthepathinthecomplexplaneofEenclosingallthesingularitiesalongtherealaxisEdepending upon the choice ofthe Fermi level E . The Green’s function G can be expanded in eigenfunctions ofthe F operator J. The radial parts (the doublets) of above eigenvalues must satisfy the two-dimensionalDirac equation (3). Then the radial partialGreen’s function G (r,r′;E) is given by (just as in 3+1 dimensions l [3]) 1 G (r,r′;E)γ0 = [Θ(r′ r)U (r)U†(r′)+Θ(r r′)U (r)U†(r′)], (6) l W(E) − R I − I R whereW(E)isthe(r-independent)Wronskian,definedbytwodoubletsV andF asWr(V,F)=Viσ F = 2 (v f f v ), where indexes denote upper and lower doublet components, and U (r) and U (r) are the 1 2 1 2 R I regula−r and irregular solutions of the radial Dirac equation (hˇ E)U(r) = 0; the regular (irregular) − solutions are integrable at r 0 (r ). We see that the problem is reduced to constructing the self-adjointradialHamiltonian→hˇ in the→H∞ilbert space ofdoublets F(r) square-integrableonthe half-line. SincetheinitialradialDiracoperatorisnotdeterminedasanuniqueself-adjointoperatortheadditional specification of its domain, given with the real parameter ξ (the self-adjoint extension parameter) is required in terms of the self-adjoint boundary conditions. Any correct doublet F(r) of the Hilbert space must satisfy the self-adjoint boundary condition [25, 33, 34] (F†(r)iσ F(r)) =(f¯f f¯f ) =0. (7) 2 r=0 1 2 2 1 r=0 | − | Physically,the self-adjointboundaryconditionsshowthatthe probabilitycurrentdensityisequaltozero at the origin. We shall apply as the solutions of the radial Dirac equation (4) the doublets represented in the form f (r,γ,E) f (r,γ,E) F = R ,F = I , (8) R g (r,γ,E) I g (r,γ,E) R I (cid:18) (cid:19) (cid:18) (cid:19) 4 where √m+E f (r,γ,E)= A M (x)+C M (x) , R x R aE/λ+s/2,γ R aE/λ−s/2,γ √m E(cid:0) (cid:1) C sγ aE/λ R g (r,γ,E)= − A M (x) C M (x) , = − , (9) R x R aE/λ+s/2,γ − R aE/λ−s/2,γ A ν+ma/λ R (cid:0) (cid:1) √m+E f (r,γ,E)= A W (x)+C W (x) , I x I aE/λ+s/2,γ I aE/λ−s/2,γ √m E (cid:0) (cid:1) C g (r,γ,E)= − A W (x) C W (x) , I =(ma/λ sν)s. (10) I x I aE/λ+s/2,γ − I aE/λ−s/2,γ A − I (cid:0) (cid:1) Here x=2λr, λ= m2 E2, γ = ν2 a2, ν = l+µ+s/2, (11) − − | | A ,A ,C ,C arenumericalcoefficpientsandthe Whitptaker functions M (x) andW (x) representthe R I R I a,b c,d regular and irregular solutions. For a2 ν2 γ is real, for a2 >ν2 γ =i√a2 ν2 iσ is imaginary. The quantities q = ν2 γ2 and ≤ − ≡ − q = ν γ = 0 are called the effective and critical charge, respectively; it is helpful also to determine c ⇔ p q = ν2 1/4 γ = 1/2. We note that all the fermion states are doubly degenerate with respect to u − ⇔ the spin parameter s at µ=0. p Inthe subcriticalrange,forq q (γ 1/2),onlysolutionsF (r) vanishing atr =0canbe chosenas u R ≤ ≥ the regular ones; they satisfy (7). For q < q < q (0 < γ < 1/2), the regular solutions U (r) satisfying u c R the self-adjoint boundary condition (7) should be chosen in the form of linear combination of F (r) and R F (r) [25, 32] I U (r)=F (r)+ξF (r) (12) R R I and the Wronskian is Γ(2γ) sγ Wr(F ,F ) W(E,γ)=(g f f g )= 2A A (13) R I R I R I R I ≡ − − Γ(γ+1/2 s/2 aE/λ)ν+ma/λ − − where Γ(z) is the Gamma function [35] and, therefore, in the subcritical range the single-particle Green function is completely determined. One can show that the contribution into the induced charge density coming for 0<γ <1/2 is small at any ξ, therefore it is enough to consider the case ξ =0. Thus, we can chose as the regular solutions the functions F (r) for all γ >0 R ∞ f f +g g trGν(r,r′;E)r=r′γ0 = I R I R. (14) | 2πsW(E,γ) s=±1l=−∞ X X Performing some simple calculations, we obtain ∞ 1 Γ(γ aE/λ) trGν(r,r′;E)|r=r′γ0 =−2πλ2r2 Γ(2−γ+1) (m2a/λ+E(x−2aE/λ−1))MaE/λ+1/2,γ(x)WaE/λ+1/2,γ(x)+ l=−∞ X (cid:2) d +m2a[(γ aE/λ)/λ]M (x)W (x)+Ex (M (x)W (x)) . (15) − aE/λ−1/2,γ aE/λ−1/2,γ dx aE/λ+1/2,γ aE/λ+1/2,γ (cid:21) We note that the singularities of G (r,r′;E) are simple poles associated with the discrete spectrum ν for m < E < m, and two cuts (( , m] and [m, )) associated with the continuous spectrum for − −∞ − ∞ E m [36]. | |≥ The path C in Eq. (5) may be deformed to run along the singularities on the real E axis as follows: C = C +C +C , where C is the path along the negative real E axis (ReE < 0) from to 0 − p + − −∞ turning around at E = 0 with positive orientation, C is a circle around the bound states singularities p with m<E <0 (if we chose E = m), and C is the path along the positive real E axis (ReE >0) F + − − from to 0 but with negative orientation (i.e. clockwise path) turning around at E =0 [3, 37]. ∞ The contour of integration C with respect to E can be deformed to coincide with the imaginary axis and we obtain as a result: ∞ dE j (r)= e trG (r,r,iE)γ0. (16) 0 ν − 2π Z −∞ 5 Let us represent µ = [µ]+α n+α, where n = 0,1,2,... for µ > 0, n = 1, 2, 3,... for µ < 0 and ≡ − − − 1 > α 0; denote ν = l α+1/2, γ = ν2 a2, where here and in all formulas below l l+n. ≥ ± ± ± ±− ≡ Since signs of e and B are fixed it is enough tqo consider only the case µ>0. Then, by means of formula [35] ∞ xΓ(2γ+1) M (x)W (x)= e−xcoshs[coth(s/2)]2aE/λ±1I (xsinhs)ds, (17) aE/λ±1/2,γ aE/λ±1/2,γ Γ(1/2+γ aE/λ 1/2) 2γ − ∓ Z 0 after long calculations, we represent the induced charge density in the form ∞ ∞ ∞ 2e j (r)= dE dte−2λrcotht 2acos(2aE/λ)cotht(I (2λr/sinht)+I (2λr/sinht)) 0 −π2r 2γ+ 2γ− − Xl=0Z0 Z0 (cid:0) 2Er sin(2aE/λ)(I′ (2λr/sinht)+I′ (2λr/sinht))(1.8) − sinht 2γ+ 2γ− (cid:19) where λ = √m2+E2, I (z) is the modified Bessel function of the first kind and the prime (here and µ below) denotes the derivative of function with respect to argument. We note that j is odd with respect 0 to charge e. III. RENORMALIZED INDUCED VACUUM CHARGE DuetothemassmtherenormalizationofEq. (18)canbeperformedaswellasinthemassivequantum electrodynamics and it also is convenient to do the renormalization in momentum space: ∞ ∞ ∞ ∞ 2e sinht j (z) ρ(z)= dreiq·rj (r)= dx dt dy e−ycoshtJ (zysinht/2b)f(y,t), 0 0 0 ≡ π 2b Z Xl=0Z0 Z0 Z0 xy f(y,t)= sin(ct)(I′ (y)+I′ (y)) 2acothtcos(ct)(I (y)+I (y)). (19) b 2γ+ 2γ− − 2γ+ 2γ− Here z = q /m, x=E/m, b=√1+x2, y =2bmr/sinht, c=2ax/b. | | We can satisfy the obvious physical requirement of vanishing of the total induced charge, because the induced charge density diminishes rapidly at distances r 1/m. Since the presence of external fields do ≫ notgiverisetoadditionaldivergencesinexpressionsofperturbationtheoryitisenough(andconvenient) tocarryouttherenormalizationinthesubcriticalrange. Then,calculationswhichweneedtoperformare similar to those described in [4, 5, 21, 38]. We introduce the renormalized induced charge in momentum representation as ρ˜(z) = lim [ρ(z) lim ρ(z)] with an ultraviolet cutoff E < Λ. Because the Λ→∞ z→0 − | | mass m is the only dimensionful parameter in the Green function the resulting dimensionless function ρ˜(z) can depend only on the ratio q/m. As a<1/2,the terms of different order in a behave differently. We can see it in terms of perturbation theory. Indeed, the linear in a term corresponds to the diagram of the polarization operator in the one-loop approximation and its renormalization coincides with the usual procedure of renormalizing the polarization operator. The terms proportional to a3 correspond to diagrams of the type of photon scattering by photon and, in difference on the case of the 3D quantum electrodynamics (see [4–6, 38]) they are finite. However their regularization must still be carried out in the considered case due to the requirements of gauge invariance, which, in particular, determine the behaviors of the scattering amplitude at small q /m. | | Massless case. We shall first consider the more complicated case with m = 0. It will be noted that the massless fermions do not have spin degree of freedom in 2+1 dimensions [39], nevertheless, solutions of the Dirac equation for massless fermions in the superposition of Coulomb and AB potentials keep the introduced spin parameter. The leading term of the induced charge density at the limit m 0 (or → z ) is a constant Q = lim ρ˜(z). So Q is the induced charge density localized in the point ind m→0 ind → ∞ r=0 and, therefore, the total induced charge density in coordinate space has the form ρ˜(r)=Q δ(r)+ρ (r), (20) ind dist where ρ (r) is the so-called distributed charge density. The induced and distributed charge densities dist have opposite signs. The total distributed charge ρ (r)dr is equal to Q . dist ind − R 6 Wehavecarriedoutlongcalculationsandgottherenormalizedinducedchargeinthesubcriticalregime that is exact in the parameter a: Q =Q (e a,α)+Q (e a,α). (21) ind 1 0 r 0 Here ∞ 2ea l+1/2 Q (e a,α)= (l+1/2+α)ψ′(l+1/2+α)+(l+1/2 α)ψ′(l+1/2 α) 2 (22), 1 0 π − − − − (l+1/2)2 α2 l=0(cid:18) − (cid:19) X ∞ 2e 1 Q (e a,α)= Im ln(Γ(γ ia)Γ(γ ia))+ ln((γ ia)(γ ia)) r 0 + − + − π − − 2 − − − l=0 (cid:20) X l+1/2 ((γ ia)ψ(γ ia)+(γ ia)ψ(γ ia))+ia − +− +− −− −− (l+1/2)2 α2 − − ia((l+1/2+α)ψ′(l+1/2+α)+(l+1/2 α)ψ′(l+1/2 α))], (23) − − − where ψ(z)is the logarithmicderivativeofGamma function [35]. The expression(21)is exactina inthe subcritical range and α, is odd (even) with respect to a (α). It is in agreement with result obtained in [21] for α = 0; the coefficient of the a3 term at α = 0 was also found in perturbation theory [19]. The inducedvacuumchargeQ isnegative;thusithasascreeningsign,leadingtoadecreaseoftheeffective ind charge of Coulomb center. For α 1, we find ≪ Q (e a,α)=eaπ/4+eaπ(2ln2+1 π2/4)α2 eaπ(0.25 0.04α2). (24) 1 0 − ≈ − Thisexpressionreflectsthelinearone-looppolarizationcontribution. ThefirstterminEq. (24)coincides with result obtained in [15, 19, 21]. We note that the contribution into Q (e a,α) from AB potential 1 0 arisesinthepresenceofCoulombfieldonly,issmallandhasoppositesigncomparedwithapureCoulomb one. Since Eq. (21) is even with respect to µ, it is clear that the fermion spin does not contribute in the induced vacuum charge in the subcritical regime. Moreover, such a (spin) problem has a physical meaning in the subcritical range only, when the expression for induced vacuum charge can be expanded in ascending power series of a,α. In the supercritical range, we shall consider the only Coulomb problem putting α = 0 in γ,ν. Now γ = iσ, therefore, we need directly to determine the Green’s function specified by boundary conditions (7). WestraightforwardconstructtheGreenfunctionintheform(6)inwhichtheregularsolutionsU (r), R satisfying (7), have to be chosenin the form of linear combinationof the functions F (r) and F (r). For R I this range,the abovetwosolutionsF (r) andF (r) becomeoscillatorywiththe imaginaryexponentand R I it is convenient to use in this range the self-adjoint extension parameter θ [32, 36], related to ξ by −2iσ A 2λ ν+a(m+E)/λ+isσ Γ(2iσ) Γ( 2iσ) R =e2iθ − (25,) ξA E ν+a(m+E)/λ isσΓ(1/2 s/2 aE/λ+iσ) − Γ(1/2 s/2 aE/λ iσ) I (cid:18) 0(cid:19) − − − − − − where π θ 0 and a positive constant E gives an energy scale. 0 ≥ ≥ The Green’s function has a discontinuity, which is solely associated with the appearance of its singu- larities situated on a second (unphysical) sheet ReE < 0,ImE < 0 of the complex plane E at q > q ; c these singularities are determined by complex roots of equation W(E,iσ) = 0 and describe the infinite (formasslessfermions)numberofquasistationarystateswithcomplex”energies”E = E eiτ withτ >π; | | for σ 1 their energy spectrum was found in [36]: ≪ E ReE =E cos(τ)exp( k/2σ+θ/σ+πcothπa/2a), (26) k,θ 0 ≡ − where τ 1/2a+Imψ(ia)+π/2. These quasi-localized resonances have negative energies, thus they ≈ are situated in the hole sector. For σ 1 the imaginary part ImE = tanτE ReE is very small k,θ ≪ ≪ and, therefore, the resonancesare practically stationarystates [36]. Physically,the self-adjointextension parametercanbeinterpretedintermsofthecutoffradiusRofaCoulombpotential. Forthis,forexample, we can compare Eq. (26) with the spectrum of supercritical resonances in the cutoff Coulomb potential [15, 16] and approximately derive θ σ[c(a)+lnE R], where c(a) does not depend on R. We note 0 ∼ that the cutoff radius R rather relates to a renormalized critical coupling that is also characterized by a logarithmic singularity at mR 1 in massive case [16, 31]. ≪ 7 The simplest wayto include these resonancesin the induced chargedensity is to carryout the integral inE from to0alongthepathS takingintoaccountthesingularitiesonthesecondsheet. Aftersome −∞ calculations, we representthe induced charge(electron) density (5) as the sum of contributions from the subcritical and supercritical ranges, which have to be treated separately ∞ dE dE f (r,γ,E)f (r,γ,E)+g (r,γ,E)g (r,γ,E) j (r)= e trG (r,r,E)γ0 = e I R I R 0 − 8π2i ν − 8π2i sW(E,γ) − Z CZ sX=±1l=X−∞ dE ξ(f2(r,iσ,E)+g2(r,iσ,E)) e I I =Q (r)+j (r). (27) − 8π2i sW(E,iσ) ind supcr ZS l,sX:ν<a For the supercritical range γ = iσ, 0 θ π, the sum in second term j is taken over l of a2 > supcr (l+s/2)2. Then the paths C,S can be≥defo≥rmed to coincide with the imaginary axis E. The first term (Q (r)) in Eq. (27) was calculated and explicitly represented by Eqs. (20) and (21). ind Thesecondtermisconvergentanditscontributiontotheinducedchargedensitycanbedirectlyevaluated at m=0. Having performed simple calculations we leads j to supcr 0 e sνs+1 dE j (r)= Γ(iσ+(1 s)/2 iaE/E ) supcr 8π2r2 σΓ(2iσ)Γ( 2iσ) Eω(σ) − − | | × l,sX:ν<a − −Z∞ Γ( iσ+(1 s)/2 iaE/E )W (2E r)W (2E r), (28) iaE/|E|+s/2,iσ iaE/|E|−s/2,iσ × − − − | | | | | | where −2iσ 2E ν+iaE/E +isσ Γ(2iσ) Γ( iσ+(1 s)/2 iaE/E ) ω(σ)=1 e2iθ | | | | − − − | | . (29) − E ν+iaE/E isσΓ( 2iσ) Γ(iσ+(1 s)/2 iaE/E ) (cid:18) 0 (cid:19) | |− − − − | | Rewrite (2E /E )−2iσ as exp( 2iσln(E /E )). As far as the integrand (28) decreases exponentially 0 0 | | − | | at E 1/r and strongly oscillate at E 0, the main contribution to the integral over E is given by | |≫ | |→ the region E 1/r. So in order to evaluate j we replace E by 1/r in the log-periodic term of the supcr | |∼ | | integrand (29) and obtain e sνs+1Γ(iσ+(1 s)/2+ia) j (r)= − Γ( iσ+(1 s)/2+ia) supcr −8π2r2 σω (σ)Γ(2iσ)Γ( 2iσ) − − × − l,s:ν<a − X ∞ dE W (2Er)W (2Er), (30) × E −ia+s/2,iσ −ia−s/2,iσ Z 0 where ν ia+isσ Γ(2iσ) Γ( iσ+(1 s)/2+ia) ω (σ)=1 e2iθ+2iσln(E0r) − − − . (31) − − ν ia isσΓ( 2iσ) Γ(iσ+(1 s)/2+ia) − − − − Becauseofthecomplexsingularitiesonthe unphysicalsheetatq >q ,theGreen’sfunctionandj (r) c supcr are complex though for σ 1 their imaginary parts are small. In terms of the physics the complex ≪ Green’sfunctionprobablyreflectsthelackofstabilityofchosen(forconstructingGreenfunction)neutral vacuum for q >q (see, also [3]). c Now we can integrate in Eq. (30) using formula [35] ∞ dE π W (2Er)W (2Er)= E −ia+s/2,iσ −ia−s/2,iσ ssin(2πiσ) × Z 0 1 1 (32) × Γ((1 s)/2+ia+iσ)Γ((1+s)/2+ia iσ) − Γ((1 s)/2+ia iσ)Γ((1+s)/2+ia+iσ) (cid:20) − − − − (cid:21) and after simple transformations we finally find the induced charge density in the supercritical range as e σ jr (r)= Re . (33) supcr 2π2r2 ω (σ) − l,s:ν<a X 8 The main effect, arisingatthe supercriticalregime,is that the induced vacuum polarizationfor nonin- teractingmasslessfermionshasapowerlawform( c/r2)whosecoefficientislog-periodicfunctionswith ∼ respect to the distance from the origin. In the subcritical regime the induced vacuum charge is localized at origin and exhibits no long range tail. As an example, we consider Eq. (33) for 1/2<a<3/2, when just the lowest l= 1,0 channels are supercritical, and find − e 2 Aze2iθ+2iσ0ln(E0r)+iψ jr (r)= σ Re −| | , (34) supcr π2r2 0 1 Aze2iθ+2iσ0ln(E0r)+iψ + A2[(a σ0)/(a+σ0)]e4iθ+4iσ0ln(E0r)+2iψ −| | | | − where Γ(2iσ )Γ( iσ +ia) a σ A= 0 − 0 , z =2 − 0, σ = a2 1/4, 0 Γ( 2iσ )Γ(iσ +ia) a − 0 0 − p ∞ 2σ 2σ 2nσ 0 0 0 ψ ArgA= π 2 σ + 2arctan +arctan . ≡ − − C 0 n − n n2+1/4 n=1(cid:18) (cid:19) X Here =0.57721is Euler’s constant. For small σ 1, Eq. (34) takes the simplest form 0 C ≪ eσ jr (r)= 0 . (35) supcr π2r2 It is of importance that the induced chargedensity jr (r) (35) atσ 1 does notcontainatallthe supcr 0 ≪ self-adjoint extension parameter θ. From the physical point of view, when the Coulomb center charge is suddenly increased from subcritical to supercritical values, the transition will occur from the subcritical range to the supercritical one, and then a small change in q such that q > q leads to a sudden change c in the character of a physical phenomenon due to emerging of infinitely many resonances with negative energies. However, the character of a physical phenomenon itself must be due only to physical (but not mathematical) reasons. We also note that the expression (35) is in agreement with results obtained in [15]fortheproblemofvacuumpolarizationofsupercriticalimpuritiesingraphenebymeansofscattering phase analysis. For large a 1,σ a l2/2a, the induced charge density can approximately be represented as ≫ ≈ − e Rej (r)= a2 l2, Imj (r)=0. (36) supcr π2r2 − supcr Xl<ap IV. SCREENING OF COULOMB CENTER CHARGE Theexpressionsforinducedchargedensities,foundfornoninteractingfermions,canbeusedtodescribe screeningofthe Coulombcenterchargeinaninteractingfermionsystem. Notice thatthe induced charge has a screening sign, leading to a decrease of the effective Coulombcenter charge. The stronglylocalized distribution of the induced charge in the subcritical regime implies that the Coulomb charge merely renormalizes the strength of the Coulomb center leading to the replacement a a in the Coulomb eff → potentialwherea arerealsolutionsofcorrespondingequationtakingintoaccountalsoelectron-electron eff interactions in the Hartree approximation [21] a =a e[Q (e a )+Q (e a )]. (37) eff 1 0 eff r 0 eff − It is essential that Eq. (37) do not have solutions with a 1/2 for a = e2, which means that the eff ≥ 0 effectiveCoulombcenterchargeremainssubcritical[21]. Nevertheless,forCoulombcentercharge2e ,3e 0 0 and higher, the effective charge can become supercritical at certain values of e2 [21]. 0 In the the supercritical regime, the induced charge density (34) causes a modification of Coulomb potential at large distances and since the infinite number of quasistationary states emerge they should contributesignificantlytoshieldtheCoulombcenter. Itisclearthatatleastforsmallσ aplanarelectron at some distance r from the Coulomb center feels the effect of an effective point charge consisting of the “bare” charge of the Coulomb center subtracted from the induced screening charge within the annulus r ,r, r <r. At first we can found the induced charge within the annulus Q integrating Eq. (35) 0 0 e σ r 0 0 Q(r)= 2 ln (38) − π r 0 9 and then treat Q(r) as an effective point charge. Since the logarithmic term represents the renormalization of the (supercritical) charge of Coulomb center, we can write instead of Eq. (38) a differential equation, which defines a self-consistent renormal- ization of the effective coupling g a like the differential equation of the renormalization group (see, eff ≡ for instance, [11, 15]): dg e2σ = 2 0 0. (39) dln(r/r ) − π 0 In this way, it can be seen that the effective coupling g will tend to the constant 1/2 within a finite distance r=r0e−(2π/e20)ln[2g+√4g2−1] from the Coulomb center. The renormalization group treatment is applicable when the right-hand side of Eq. (39) is small, i.e. for σ 1. Therefore for σ 1, the vacuum of planar charged electrons with Coulomb potential 0 0 ≪ ≪ self-consistently rearranges itself so that electrons at distances r > r never feel a supercritical effective 0 coupling irrespective of the “bare” supercritical Coulomb center charge (see, [11, 15]). It is well to note that in the convenient quantum electrodynamics the vacuum polarization charge in super-heavynucleibehavesinsuchawayastoreducethesupercriticalchargeofnucleustothethreshold value [40] (see also [41], where the problem was investigated for super-heavy nuclei in the presence of a superstrong constant uniform magnetic field). V. VACUUM POLARIZATION OF PLANAR CHARGED MASSIVE FERMIONS We now briefly address to the vacuum polarizationinduced by the Coulomb potential in massive case. Ifthe Coulombcenterchargeissubcriticalthe massivecasehasa welldefinedinfinite spectrumofbound solutions situated on the physical sheet, which for γ 1/2,a<1/2,ξ =0 is [31] ≥ k+√ν2 a2 E =m − , ν =l+1/2; k,l=0,1,2..., (40) k,l [k+√ν2 a2]2+a2 − q We see that all the energy levels are doubly degenerate with respect to s. It can be easily shown that the spectrum accumulates at the point E = m, and its asymptotic form as n = k + l is → ∞ given by the nonrelativistic formula ǫ = m E = ma2/n2. The problem of finding the spectra of n n − self-adjoint extensions of the radial Hamiltonian in the Coulomb and Aaronov-Bohm potentials in 2+1 dimensions was solved in [33] where, in particular, it was shown that the spectrum accumulates at the point E = m and is described by the same asymptotic formula (without AB potential), independent of ξ, i.e. ǫ =m E =ma2/n2. n n,ξ − Inthemassivecasethevacuumpolarizationofplanarchargedfermionsmanifestsitselfbymodifyingthe Coulomb potential. Therefore, it is rewarding to calculate the polarization corrections to the Coulomb potential. As applied to the vacuum polarization we shall assume that none of the bound levels are occupied. Ifa 1we canestimate thesepolarizationcorrectionsinthe firstorderina. Forthreespatial ≪ dimensions, the potential taking into account the polarization corrections of the first order in a to the Coulomb potential is the Uehling-Serber potential. In terms of perturbation theory, these corrections correspond to the polarization operator in the lowest order in interaction. Performing the integrations and summation in Eq. (19) with taking only the linear in a terms into account, for the renormalized induced Coulomb center charge, we obtain a Π( q2) Q (q )= − , (41) m | | −e q 0 | | where, as it should be, e2 4m2 q2 q2 Π( q2)= 0 − arctan 2m − 8π q2 r4m2 − ! p 10 is the polarization operator in the first order of perturbation theory. After some transformations the induced charge distribution Q (r) am /e (here am is the effective coupling) takes the form in the m ≡ eff 0 eff coordinate space: ∞ dx Q (r)= e a e−2mrx. (42) m 0 − x3√x2 1 Z1 − The integral is calculated in limits mr 1 and mr 1 and as a result we find ≪ ≫ π Q (r) e a Cmr , mr 1,1 C mr, (43) m 0 ≈− 4 − ≪ ≫ ≫ h i where the first term on the right of Eq. (43) was already calculated (see Eq. (22)), and 4π Q (r) e a e−2mr, mr 1. (44) m 0 ≈− mr ≫ r We seethatevenatsmalldistancesfromthe Coulombcenter,thefinite masscontributiontotheinduced vacuumchargeissmallandinsignificantlydistortstheCoulombpotentialonlyatdistancesoftheCompton length r 1/m. The induced charge has a screening sign. ∼ In the supercritical regime the finite mass contribution to the vacuum polarization easier to estimate, at least when σ a2 1/4 1. Indeed, if the Coulomb potential charge is suddenly increased 0 ≡ − ≪ from subcritical to supercritical values then the only lowest energy level dives into the negative energy continuum and becompes a resonance with “complex energy”E = E eiτ. There appears the pole on the | | unphysical sheet τ > π, counted now as a “hole” state. Using results of Ref. [33], one can show the energy of dived state ReE = (m+ǫ),ǫ +0, is determined by the following transcendental equation − → argΓ(2iσ ) σ Reψ( iz) (σ /2)ln(8ǫ/m)+arctan[σ (1 2a2ǫ/m)]= θ, (45) 0 0 0 0 − − − − − where z = ma2/2ǫ. This resonance is spread out over an energy range of order Γ me−√2mπa2/ǫ g ∼ and strongly distort around the Coulomb center. The resonance is sharply defined state with diverging p lifetime (Γ )−1 e√2mπa2/ǫ/m. Thus, the resonance is practically a bound state. g ∼ The diving point for the energy level defines and depends upon the parameter θ. This diving of boundlevelsentailsacompleterestructuringofthequantumelectrodynamicsvacuuminthesupercritical Coulomb field [1, 3]. As a result, the QED vacuum acquires the charge, thus leading to the concept of a charged vacuum in supercritical fields due to the real vacuum polarization [1, 3]. As was shown in [3] the contribution to the Green’s function from the only pole on the second sheet contains the only term associated with the former lowest bound state: Γ Θ( m E) G (r,r′;E)=i g − − ψcr(r)[ψcr(r′)]†, (46) r (E E )2+Γ2/4 0 0 − 0 g where Θ(z) is the step function and ψcr(r) is the ground state of the Dirac Hamiltonian at a=a (the 0 cr critical state) with energy E within the gap m E <m but close to m. The critical charge a is 0 0 cr − ≤ − defined as the condition for the appearance of the imaginary part of “the energy”. It is important that the Green function of the type (46) eliminates the lack of stability of neutral vacuum for a > a (see, cr [3]). Then, the real vacuum polarization charge density can be determined by e dE j0real(r)≡− 20 2πitrG(r,r′;E)|r=r′γ0, (47) Z R wherethepathRsurroundsthesingularityontheunphysicalsheet. Integrating(47)weobtainjreal(r)= 0 e ψcr(r)2. − 0| 0 | We see that the space density of the real vacuum polarization is real quantity and approximately described with the modulus squared of the fermion wave function in the critical state: jreal(r) e m2[2(lnmr)2 2(lnmr)/a +1/a2 ], mr 1 0 ∼− 0 − cr cr ≪ and jreal(r) e me−2√r/l/r, l =1/√2mǫ ,mr 1, 0 ∼− 0 0 ≫ where ǫ depends upon a and the extension parameter θ. 0 cr Thetotalinducedchargedensityinmassivecasewithtakingintoaccounttherealvacuumpolarization (47) can be estimated as the sum: Q (r)m2+jreal. m 0