Resonances in positron scattering on a supercritical nucleus and spontaneous production of $e^{+}e^{-}$ pairs

We re-examine the physics of supercritical nuclei, specially focusing on the scattering phase $\delta_{\varkappa}$ and its dependence on the energy $\varepsilon$ of the diving electronic level, for which we give both exact and approximate formulas. The Coulomb potential $Z\alpha/r$ is rounded to the constant $Z\alpha/R$ for $r<R$. We confirm the resonant behavior of $\delta_{\varkappa}$ that we investigate in details. In addition to solving the Dirac equation for an electron, we solve it for a positron, in the field of the same nucleus. This clarifies the interpretation of the resonances. Our results are compared with claims made in previous works.


I. INTRODUCTION
The Coulomb problem for a nucleus with charge Z > Z cr was recently analysed [1] by solving the Dirac equation for an electron in the external field of this nucleus. Because of the specificity of the Dirac equation that accounts simultaneously for electrons and positrons this problem gets connected to the scattering of positrons (holes in the Dirac sea) on the nucleus (see below). The behavior of the scattering amplitude was found to be very peculiar: it contains resonances and their energies, obtained from an analytical formula found in [1], correspond to poles of the S matrix located above the left cut, on the second (unphysical) sheet of the energy plane. The resonances in positron scattering were discussed in Refs. [2,3]. At Z < Z cr , the width γ vanishes, and this equation describes the usual bound states of electrons in the Coulomb field of the nucleus.
When Z > Z cr , γ = 0 makes these states quasistationary [4,5]. For electrons, as Z increases, the transition from bound states to resonant states corresponds to the diving of the bound states, which start at ε = +m, downwards into the lower continuum.
In the present paper, in order to clarify the situation, we will also study "the Dirac equation for positron". By this we mean here the standard Dirac equation with the substitution of electron charge e by −e. Now, as Z increases, bound states raise up from ε = −m and become resonant in the upper continuum.
For Z < Z cr , the interpretation of these bound states (also noted in [6] chapter 4.3) is the following. For obvious reasons they cannot be (e + N + ) bound states, but are just our previous (e − N + ) bound states. There is no more information in there 1 .
For Z > Z cr , we find that (e + N + ) resonances occur at the energies which now correspond to poles of the S matrix below the right cut of the energy plane, also, as it should be, on the second, unphysical, sheet. This result confirms the proposal made in [1] that the sign of the energy in (1) should be reversed. This change of sign we are accustomed to when dealing with holes in the lower continuum: the absence of an electron with energy −ε is then interpreted as the presence of a positron with energy ε.
It is now to be operated on the empty states of the energy levels that dive into the lower continuum. Our consideration of the Dirac equation for positrons therefore helps to clarify the nature and position of the resonances.
We, however, do not see any sensible objection to the occurrence of this process: an empty state diving into the lower continuum gets filled by one electron of the Dirac sea; the resulting hole in the sea is the positron that gets ejected by the nucleus the charge of which has become Z − 1. The characteristic time of this emission process is 1/γ, in agreement with the results obtained in [2,3,[6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].
Furthermore, spontaneous production of e + e − pairs was recently observed in the numerical solution of the Dirac equation in the case of heavy ion collisions [21,22].
The plan of the paper is as follows. In Section II, following [1] and using the Dirac equation, we study the scattering of states of the lower continuum on a supercritical nucleus. In addition to reproducing the approximate results obtained in [1] we get explicit results without using an expansion over the parameter m × R, where R is the nucleus radius. Such an expansion being good for electrons does not work for heavy particles, for example, muons [4,5]. In Section III, we use instead the Dirac equation for positrons (see above) and study the scattering of states of its upper continuum on a supercritical nucleus. We conclude in Section IV.

II. LOWER CONTINUUM WAVE FUNCTIONS AND SCATTERING PHASES IN THE COULOMB FIELD OF A SUPERCRITICAL NUCLEUS
The radial functions of the Dirac equation F (r) ≡ rf (r) and G(r) ≡ rg(r) are determined by the following differential equations [23][24][25]: 1 The reader can be convinced as regards this interpretation as follows. In the present simple formalism, which only uses the Dirac equation, the energy of a bare positron, which is obtained by simply taking the limit Z → 0, is found to be −m. Since the "production" of such a particle costs at least the energy +m, the result that is obtained can only be interpreted in term of an electron with energy −(−m) = +m. This is what we mean by the statement that "there is no more information". A more satisfying description of positrons can only be achieved in the framework of Quantum Field Theory, where creating (annihilating) an electron and annihilating (creating) a positron both occur in the expansion of the field operator ψ in terms of creation and annihilation operators.
In order to deal with the case Zα > 1 the Coulomb potential should be regularised at r = 0 [26]. To do this we shall approximate the nucleus as a homogeneous charged sphere with radius R (the so-called rectangular cutoff). Thus, the potential in which the Dirac equation should be solved looks like: At small distances r < R, substituting expression (4a) into (3), we obtain the Dirac equation with a constant potential, the solution of which is expressed through Bessel functions. In order to obtain finite f and g at r = 0 among the two sets of solutions the one with a positive index of the Bessel function should be selected 2 : where β = (ε + Zα/R) 2 − m 2 . Upper (lower) signs should be taken for κ < 0 (κ > 0). For r > R, we need the solution of the Dirac equation for the Coulomb potential. We introduce the standard quantity λ which, for −m < ε < m, equals λ = (m − ε)(m + ε) ≡ −ik, where k is the electron momentum. Here we have to make an important remark. Since later we are going to look for resonances in the complex ε plane, we must carefully define the square roots used here. Each of them, √ m − ε and √ m + ε, are defined on two Riemann sheets of the complex ε plane. To avoid ambiguous expressions let us introduce a uniquely defined function sqrt(z) as follows: sqrt |z|e iArg(z) = |z|e iArg(z)/2 , for Arg(z) ∈ (−π; π].
For example It is therefore the first branch of the function √ z with the cut (−∞; 0). The second branch is given by −sqrt(z). This definition is also very convenient because the square root is defined in this way in many numerical tools for computers.
Switching branches of both square roots, √ m − ε and √ m + ε, leads to the same value of λ. Therefore, λ is defined on the two Riemann sheets according to: with two cuts originating, respectively, from each of the square roots (see Fig. 1) 3 . From general arguments of scattering theory, we know that electron bound states are located at real ε in the interval 2 Any solution with a negative index of the Bessel function is not normalizable, so it should be discarded. 3 The procedure used in [1] amounts to stating that, below the left cut, λ = −i (m − ε)(−m − ε). So doing, √ −m − ε is defined with the same cut (−∞; −m) as √ m + ε, with positive values below the cut. With such a definition, −i √ −m − ε = sqrt (m + ε) everywhere on the physical sheet, not only below the left cut. There is no need to rewrite formulas in this way since, when numerical outputs are needed, we should return to the original definition (8). Let us note that on the first sheet formulas (17) and (26) from [1] are exactly the same. −m < ε < m. Unbound electron states are located above the right cut and unbound positron states below the left cut.
In what follows we shall use the following conventions for the " √ " symbol: It does not matter which root changes sign when we go to the second sheet since we can always also change the signs of both. We are looking for solution written in the standard form [25] 4 : where τ = (Zα) 2 − κ 2 , ρ = 2λr = −2ikr, Q 1 and Q 2 are determined by differential equations, the solutions of which are Kummer confluent hypergeometric functions 1 F 1 (α, β, z) (also sometimes noted F (α, γ, z) like in [25]). In textbooks dealing with the case Zα < 1, R = 0, only solutions regular at r = 0 are considered. We must instead here take into account both type of solutions of the equations for Q 1 and Q 2 . The formulas for the Q i are derived in Appendix A. From (A6) to (A8) we get: where C and D are arbitrary coefficients 5 . The scattering phase δ κ (ε, Z) is determined by investigating the behavior of the wave function at large r. To this purpose, the asymptotic expansion of 1 F 1 at large |z| is very useful.
Using the asymptotic expansion (13) for the Kummer functions occurring in (12) gives: where the upper sign corresponds to F and the lower sign corresponds to G.
The ratio yields the Coulomb logarithm (for real ε below the left cut it gives exp 2iZαε k ln (2kr) ). Since the latter does not contribute to the differential scattering cross section at nonzero angle θ, we will omit this term in our further calculations.
From the general formula it follows that the ratio of the remaining coefficients define the scattering phase δ κ (on the real axis below the left cut e −ρ/2 ≡ e ikr corresponds to the outgoing wave and e ρ/2 ≡ e −ikr corresponds to the incoming wave): where The resonance of the scattering amplitude corresponds to the pole of the S-matrix element S ≡ e 2iδ and from (19) we immediately get an equation for the position of this pole in the ε-plane: In what follows we will match the solutions at r < R and r > R to obtain the ratio C/D, such that we can calculate the phase δ κ and find the poles of the S matrix which correspond to the energy levels. This procedure can be performed both exactly and approximately.

A. Exact results
With the help of the exact formulas (5), (11), and (12) we get the ratio C/D from matching F/G at r = R + 0 and r = R − 0: where and The numerical evaluation of the square roots in (23) and of k in (26) for real ε is somewhat tricky since one should carefully choose the side of the cut to use. Due to the definition (6)-(7) of the sqrt() function the expression sqrt (m + ε) gives, for real ε, the values above the cut such that −sqrt (m + ε) should be used. It corresponds formally to calculating the scattering phase on the second (unphysical) sheet. The same holds for k. For any real ε it is also possible to use k = sqrt (ε 2 − m 2 ) which chooses the correct side of the cut; then . With the help of (22) we can calculate the scattering phase δ κ defined by (19). In the domain ε < −m, δ κ (ε, Z) gives the scattering phase of a positron with energy ε p = −ε > m on the nucleus (for real ε < −m we get Arg [ρ] < 0). Its dependence on ε p for κ = −1 and Z = 232 is shown in Fig. 2 (compare with Fig. 3 of [1]). The scattering phase δ κ exhibits a resonance behavior; it goes through π/2 at ε p /m ≈ 5.06.
We obtain the equation for the position of the poles by substituting (22) into (21) Γ (−2iτ ) The solutions of (27) can be found by scanning the complex ε plane. This is the method that we used to find the exact positions 6 of the S-matrix poles (see Fig. 3 and Table I). The energies ε of the quasistationary states are located above the left cut on the second sheet of the complex ε plane 7 :

B. Approximate results
In [1] the approximation 1/R ε, m was used. In this section we are going to reproduce their results and compare them to the exact ones.
Being interested in the case Zα 1 and taking into account the smallness of the nucleus radius in comparison with the electron Compton wavelength 1/m we obtain that β ≈ Zα/R in (5).
The solution of the system (3) at r > R should match (5) at r = R, in particular the ratio F/G of both solutions at r = R should coincide. Substituting (4b) in (3) at r → 0 we easily get where σ = √ κ 2 − Z 2 α 2 and η σ and η −σ are arbitrary constants. Matching the ratios F/G from (29) and (5) at r = R we obtain which coincides with Eq. (13) from [1]. In the case Zα > |κ| one should substitute σ by iτ (where, as before, τ = √ Z 2 α 2 − κ 2 ): The modulus of the r.h.s of (31) can be easily checked to be unity, this is why we can rewrite it as an exp (2iθ) with real θ. The expansion of (11) at small ρ contains terms ∼ ρ iτ and ρ −iτ . Comparing this expansion with (29) and substituting σ → iτ , η σ → η τ , η −σ → η −τ yields: Getting an equation for C/D needs matching (31) with η τ /η −τ obtained from (32): Two sets of approximations were made in deriving (33): i. to get η τ /η −τ at r = R − 0 we replaced βR with Zα and used (29) which was itself derived for Zα/r ε, m; ii. to get η τ /η −τ at r = R + 0 6 Due to the unitarity of the S matrix, there is a zero of e 2iδκ at ε = −ξ − i 2 γ that is symmetric to the pole ε = −ξ + i 2 γ with respect to the real axis. It corresponds to incoming waves instead of the outgoing waves that we selected. 7   we expanded (11) and (12) at ρ 1. For m · R = 0.031 one cannot expect an accuracy better than 3% and, with growing |ε| it can even get worse. The accuracy of the final result is not easy to guess from the start, and the best way is to compare it with the exact solution which was found in the previous subsection. Note that all results in [1] are based on the asymptotic behavior (29) and are therefore approximate by default.
Let us note that on the real axis of ε the expression for the scattering phase δ κ can be written in the same form as in [1] (see Appendix B).
The positions of the S matrix poles are defined by the same equality (21) with C/D given by (33): where the l.h.s. is defined by (31). The r.h.s. coincides with Eq. (26) from [1]. The exact expression (27) is, of course, more complicated, but, anyhow, special functions have to be evaluated numerically in both cases. The accuracy of Re[ε] obtained by the approximate procedure is quite reasonable (see Fig. 3 and Table I); however it is much worse for Im[ε], for example ≈ 15% at Z = 186. This is why it is worth getting the exact values of the energies ε.
The question that we want to address now is the origin of the resonance and how it transforms for Z < Z cr .  (27)) and the round markers are for the approximate ones calculated with the help of (34). The correspondence between color and Z is shown in the legend (the real part of the energy is monotonically decreasing). At Z = Z cr the bound states become resonances with positive Im[ε].
At Z < Z cr the resonances become bound states, the energies of which are determined by the same type of matching at r = R as before (it is convenient to replace now, in (34), k by iλ, since on the real axis, for −m < ε < +m, λ is real positive): Last, for Zα < |κ| we must change iτ into σ = √ κ 2 − Z 2 α 2 : Let us consider for example Zα < 1, for which taking a point-like nucleus is reliable. At the limit R → 0, the r.h.s. of (30) becomes infinite. Therefore, the spectrum of the Dirac equation is given by the poles of (36). They are given by the poles of Γ 1 + σ − Zαε to which must be added, for κ < 0, the zero of the last term in the denominator of (36) 8 : The electron bound states at Z < Z cr become therefore resonances at Z > Z cr ; the poles of the S matrix corresponding to the latter describe positron-nucleus scattering. The trajectory of the ground state energy with growing Z is shown in Fig. 3 (see also Table I).
Let us notice the unusual signs of both real and imaginary parts of the resonance energy. It was suggested in [1] that the sign of the energy should be reversed, under the claim that the corresponding state is a resonance in the positron-nucleus system. Such a sign reversal is usual for holes in the lower continuum of the Dirac equation: the absence of an electron of energy −ε is equivalent to the presence of a positron with energy ε. Advocating for the same procedure in the case at hands looks a priori suspicious since the resonances that we found originate from electron bound energy levels (however, also empty) that dive from ε = +m downwards into the lower continuum (and will return upwards to +m if Z decreases). An interpretation of the phenomenon in terms of electrons looks 8 This term is proportional to the sum It is easy to check that, when the first term vanishes, so does the second. Their sum increasing monotonically when ε increases can vanish only once. therefore more intuitive. In order to resolve this (apparent) puzzle, we shall solve in the next section the Dirac equation for positrons, which describes the scattering of a positron in the upper continuum on a nucleus.

III. THE DIRAC EQUATION FOR POSITRONS: UPPER CONTINUUM WAVE FUNCTIONS AND SCATTERING PHASES IN THE COULOMB FIELD OF A SUPERCRITICAL NUCLEUS
Changing the sign of Zα in (4), we get instead of (3) Notice that (3) gives (39) by the set of transformations κ → −κ, ε → −ε, F →G and G →F . The states in the upper continuum (ε > m) describe positron scattering on a nucleus. Since a positron cannot form a bound state with a positively charged nucleus, one could think that no resonance at Z > Z cr will occur, nor the resonant behavior of the scattering phase found in [1] and reproduced in Section II.
The central issue is therefore to investigate whether bound states and resonances arise or not in the Dirac equation for positrons (39).
Solving (39) at r < R we obtain F G = const · β r · ±J ∓(1/2+κ) (βr) whereβ = (ε − Zα/R) 2 − m 2 and, at small distances, where the solution (41) will be used,β ≈ β ≈ Zα/R. The upper (lower) signs in (41) should be taken for κ < 0 (κ > 0). Note that the sign ofF is opposite to that of F in (5), while the signs ofG and G coincide. Substituting in (39) the Coulomb potential (40a) and going to the limit r → 0 we get: Note that the sign ofG is opposite to that of G in (29), while the signs ofF and F coincide. Thus, when matching the ratios ofF /G from (41) and (42) at r = R we obtain equations identical to (30), (31) with the change η →η. Like in (11), we look for solutions of the form where as before ρ = 2λr = −2ikr. The expressions forQ 1 andQ 2 are given by (12), where Zα should be substituted by −Zα: Substituting (44) into (43) at the limit r → 0, we reproduce (42) for Matching Eqs. (46) and (31) yields an equation for the energies of the resonant states. After the substitution of (κ, ε) by (−κ, −ε), it coincides with the similar equation that we obtained in Section II. Thus, resonances also arise as solutions of the Dirac equation for positrons, at energies ε = ξ − i 2 γ, ξ > m, γ > 0 9 . After making the same substitutions as in Section II, we get equations that coincide with (35), (36)). This clears the mystery concerning the resonances that we have found there. Positrons states of negative energies should be interpreted in terms of electrons. At Z < Z cr we just found electronnucleus bound states -with growing Z, the energy of the bound particle moves from −m (at Z = 0) to +m (see also footnote 1) and, at Z > Z cr it becomes complex and located on the second sheet below the right cut.
It is therefore not necessary to solve the Dirac equation for positrons as we did in this section. It is enough to note that, after substitution of ε by −ε, κ by −κ, F byG and G byF , Eq. (3) becomes (39) with V (r) converted toṼ (r). In this way, the formulas of Section III can be directly deduced from the ones of Section II.

IV. CONCLUSIONS
In Sections II and III, the scattering of positrons on a supercritical nucleus was studied. It has the spectacular resonance behavior discovered in [1][2][3]. In the present paper, results with an exact dependence on the parameter m × R have been obtained on both sheets of the complex energy plane in the form convenient for numerical evaluation. However, one can hardly hope to study this phenomenon experimentally: even if a supercritical nucleus can be produced in heavy ions collisions, its life time will be so short that one cannot scatter a positron on it, not to mention the still bigger challenge of making a target with supercritical nuclei. Let us note that since the elastic scattering matrix was found to be unitary (the scattering phase is real) there are no inelastic processes in the positron scattering on supercritical nucleus.
More realistic is the hope to detect the emission of positrons from a short-lived supercritical nucleus eventually produced in heavy ions collisions. Indeed we do not agree with the claim made in the abstract of [1] (and in contradiction with [16] in particular) that the spontaneous production of e + e − pairs from a supercritical nucleus does not occur. On the contrary, we believe that the resonance found in [1] in the system positron-supercritical nucleus is precisely the signal for pair production. It occurs when, as Z grows, an empty electron level dives into the lower continuum of the Dirac equation. In the absence of the nucleus, this empty state in the lower continuum would just mean the presence of a positron. The presence of the nucleus makes the energy of this state complex, and its lifetime is precisely 1/γ. In this lapse of time, an electron from the sea with the same energy −ξ located far from the nucleus can penetrate in its vicinity. It partially screens the charge of the nucleus and, at the same time, an empty electron state arises in the Dirac sea. This is the positron which gets repulsed to infinity by the nucleus.
Let us suppose that solutions of the Dirac equation we get are approximately valid also when an electron screens nuclear potential, being embedded in the lower continuum. It means our solutions for the resonance energy and width are almost valid. It well can be so, since electric charge of one electron is small and it is situated far from nucleus, r ≈ 1/m. So, the obtained width (imaginary part of energy) is the lifetime of positron in the vicinity of nucleus, which is already surrounded by diving electron. Therefore this is the lifetime of the system of nucleus, electron and positron with respect to positron emission to infinity, so it is an average time of e + e − pair production (in reality two independent pairs are produced because of electron spin degeneracy).
The potential barrier which holds the positron in the vicinity of the nucleus is shown in Fig. 2 of [16]; its penetration time is given by the analytical formulas (4.14, 4.15), and the results of numerical calculations are shown in Fig. 13 of the same review paper. We reproduced the curve shown in Fig. 13 from the dependence γ(Z) that we obtained in Section II for the energy of the Gamov (quasistationary) state.
Let us finally mention that we agree with the description of the stable states of a supercritical nucleus made in Section 6 of [1]: empty states in the upper continuum, empty discrete levels, and occupied states in the lower continuum. The levels of the lower continuum that get occupied by electrons after the diving process form the so-called "charged vacuum"; it has charge −n, where n is the number of these levels. The n positrons that get emitted compensate for this negative charge. A supercritical nucleus is no longer naked and its electric charge is partially screened by these electrons.
Indirect evidence of such a phenomenon is found in graphene physics [27,28].