{'fraction_non_alphanumeric': 0.04325974635979333, 'fraction_numerical': 0.02137153593236261, 'mean_word_length': 4.401560319675251, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Radiative corrections to parity-non-conservation in atoms

Recent progress in calculations of QED radiative corrections to parity nonconservation in atoms is reviewed. The QED vacuum polarization, the self-energy corrections and the vertex corrections are shown to be described very reliably by different methods used by different groups. All new calculations have recently converged to very close final values. Each separate radiative correction is very large, above 1 % for heavy atoms, but having different signs they partly compensate each other. Our results for the radiative corrections for all atoms are presented. The corrections are -0.54 % for 133 Cs, and -0.70% for 205 Tl, 208 Pb, and 209 Bi. The result for 133 Cs reconciles the most accurate atomic experimental data for the 6s-7s PNC amplitude in 133 Cs of Wood et al with the standard model.


I. INTRODUCTION
It has become clear quite recently that the QED corrections give a large contribution, of the order of ∼ 1%, to the parity nonconservation (PNC) amplitude in heavy atoms, which makes them important for an analysis of modern experimental data. The present paper summarizes recent progress in calculations of the QED radiative corrections to PNC. Over the last few years the experimental data of the Boulder group for the 6s − 7s PNC amplitude in 133 Cs, which remains the most accurate in the area of atomic PNC, were presumed to be able to challenge the standard model. The radiative corrections eliminate this possibility, their accurate determination brings the atomic experimental data for the mentioned transition in agreement with the standard model. The paper presents also a brief description of the methods used for calculations of the QED corrections. Some of them, that have been recently designed for applications to the PNC problem, may be of interest in other areas where QED corrections are studied.
Several atoms and several transitions have been studied experimentally in relation to PNC. As was mentioned, the most accurate experimental data was obtained for the PNC 6s − 7s PNC transition in 133 Cs. Studies of this transition initiated by Bouchiat and Bouchiat [1], were continued and developed in Paris [2,3,4,5,6,7] and in Boulder [8,9,10,11]. The latest work of Wood et al [11] reduced the experimental error to 0.35% that remains the best accuracy in atomic PNC experiments. These results inspired interest in very accurate atomic PNC calculations that, eventually, led to recognition of the fact that the QED corrections contribute at the necessary level of accuracy. Other experiments on PNC have been carried out for several transitions in several atoms. This includes the 6p 1/2 − 7p 1/2 transition in 205 Tl [12,13,14], and the 6p 1/2 − 6p 3/2 transition in 205 Tl [15,16,17], the 3 P 0 − 3 P 1 transition in 208 Pb [18,19,20] and two transitions in 209 Bi, 4 S 3/2 − 2 D 5/2 measured in pioneering works of Barkov and Zolotorev [21,22,23,24] and in [25,26,27] as well as the 4 S 3/2 − 2 D 3/2 transition [28,29,30]. The comprehensive theoretical background on PNC as well as a detailed account of events which led to a discovery of PNC in atoms (that was also the first observation of PNC in neutral currents) can be found in the book of Khriplovich [31].
Errors of final quantities that can be extracted from experimental results on PNC in atoms crucially depend on the accuracy of atomic calculations. The most difficult and cumbersome part of the theoretical research is the atomic structure calculations that take into account the many-electron nature of the atomic wave function in heavy atoms. A foundation for the modern theoretical description of many-electron correlations in the PNC problem was laid out in the works [32,33] which also give references to previous papers. The calculations of the many-electron correlations in the 6s − 7s transition in 133 Cs in these works gave very close results, with the error in both sets of calculations estimated as ∼ 1%. The problem was revisited in several recent publications that supported the results of [32,33] improving the error. Improvement of the theoretical error was initiated in Refs. [34] that compared a number of experimentally measured quantities, such as polarizabilities, dipole amplitudes and hyperfine constants, with theoretical predictions of Refs. [32,33], concluding that the theoretical error of these works was probably better than it had been anticipated, at the level of 0.4%. This conclusion allowed the authors of [34] to reveal a deviation that existed between the experimental data of [11] and the standard model (∼ 2σ). The following theoretical works [35,36,37] demonstrated that indeed, the theoretical error that originates from the manyelectron correlations can be reduced down to ∼ 0.5% for the 6s − 7s transition in 133 Cs, thus matching the experimental accuracy of Ref. [11]. The net result of the experimental and theoretical progress at this period of time was a pronounced deviation, of the order of 2σ, between predictions of the standard model and experimental data that was proved to be not related to uncertainties imposed by the complicated manyelectron nature of the problem.
The progress outlined above indicated that either the standard model needed improvement, or some unrecognized, underestimated factors should come into play. A careful analysis revealed that there was an overlooked factor well within the framework of the standard model, namely the conventional QED corrections to the PNC amplitude. Obviously, they have always been kept in mind, but for a long period of time they were estimated as negligible in the PNC problem. From the perspective of the present day situation the fact that they play an important role can be considered as a general statement. Historically, however, its realization stemmed from several works devoted to different, separate phenomena that contribute to the QED corrections. Derevianko [38] found that the Breit corrections to electron-electron interaction in heavy atoms are much larger than it had been anticipated, giving −0.6% for the 6s − 7s PNC amplitude in 133 Cs. This result was obtained numerically, and later confirmed in a number of papers [35,39,40] both numerically and analytically. The account of the Breit corrections put the experimental data of [11] much closer to the standard model, but this was only the beginning of an intriguing zigzag road of research. An analysis of the Breit corrections by Sushkov [40] revealed that one can expect the QED vacuum polarization to be as important as the Breit corrections. Johnson et al [41] for the first time proved that indeed, the vacuum polarization is very important. For the 6s − 7s PNC amplitude in 133 Cs atom they found that it gives 0.4%, as confirmed in [36,42,43]. From a purely pragmatic, numerical point of view the vacuum polarization mostly compensated for the contribution of the Breit corrections, bringing the experimental data for the 133 Cs atom back in contradiction with the standard model.
The most difficult and, correspondingly, challenging part of the QED corrections to the PNC amplitude is presented by the QED self-energy corrections. We are using this term here in a sense that embraces the vertex corrections as well. One could have anticipated these corrections to be large from the very beginning. The selfenergy corrections are known to give a large contribution to a number of phenomena in atomic physics, including the Lamb shift, the hyperfine interaction, and the energy shift due to the finite nuclear size (FNS). The latter phenomenon will play a particular role in the following discussion. There is no physical reason indicating that the PNC amplitude should be any different. The sign of the self-energy corrections is also predetermined. The QED self-energy is known to be mostly repulsive (this is definitely true for the s 1/2 state that should play a very important role) in contrast to the attractive vacuum polarization. Therefore the self-energy corrections should make the electron wave function smaller at the nucleus and, consequently, give a negative contribution to the PNC amplitude.
However, the aforementioned reasons were confronted by counter-arguments indicating that the self-energy corrections should be small. One reason, briefly mentioned in [41], relies on the fact that in the plane-wave approximation the self-energy corrections are small, of the order of 0.1% for 133 Cs atom, as was found by Marciano and Sirlin [44] and Lynn and Sandars [45]. One can argue that account of the nuclear Coulomb field should not produce any dramatic enhancement, and the self-energy corrections should, probably, remain small. The argument has its merit, it is known to be correct for the case of the vacuum polarization, where higher-order corrections are suppressed, see section II. However, the self-energy corrections behave differently. The well-studied case of the hyperfine interaction is an example. It is known from both analytical and numerical calculations, see Refs. [46,47] and references therein, that in this problem the manifestations of the Coulomb field in the self-energy corrections in heavy atoms are more pronounced than a contribution of the plane-wave approximation, the latter one is given by the known Z-independent Schwinger term α/2π, while the former rise quickly with the nuclear charge [76].
Another case, that looked even stronger, against a possible role of the self-energy corrections in the PNC problem was made by Milstein and Sushkov in Ref. [42]. They presented results of direct analytical calculations claiming that the self-energy corrections are positive and small, concluding that the experimental data on the 6s−7s PNC amplitude in the 133 Cs atom are in contradiction with the standard model. The impact of this work was strong, for a long period of time it was considered as a reliable argument indicating that the experimental data can challenge the standard model. The opposite opinion expressed in Ref. [36], which presented a model for rough estimation of the self-energy corrections, was not considered as convincing, probably because the proposed model neglected the vertex corrections and was not gauge invariant.
The analytical calculations of Ref. [42] seemed to prove that the self-energy corrections are small and positive, while some physical arguments mentioned above indicated otherwise. To find the way out of this controversy we proposed in Ref. [48] an approach that avoided difficult analytical calculations. This work derived a new equality, we will call it the chiral invariance identity due to reasons explained in section III A. It expressed the self-energy corrections for the PNC amplitude in terms of the self-energy corrections to the energy shifts due to FNS. The latter ones were calculated numerically in Refs. [49,50,51,52,53,54]. Thus the found equality allowed one for the first time to find the accurate self-energy corrections for the PNC amplitude for heavy atoms. As one could have anticipated, they were large and negative. In particular, for the 6s − 7s transition in133Cs we found −0.73 (20)% that brought the experimental data of Wood et al in agreement with the standard model. The uncertainty of this result was mainly due to errors of the numerical results of Ref. [52] that we used as input data.
The most pronounced among them is the error for the self-energy corrections to the energy shift induced by the FNS in the p 1/2 wave. The authors of [52], as well as others in the relevant papers above, estimated their results as very reliable for heavy atoms Z ∼ 90, and less accurate for lighter atoms, making numerical calculations below Z = 55 difficult.
In order to verify and refine our results, there were performed analytical calculations of the leading, linear term in the Zα-expansion for the self-energy (and vertex) corrections for the PNC amplitude [55]. The result of this work confirmed that of [48], the self-energy corrections were found to be large and negative, in clear contradiction with Ref. [42]. Combining the data of [48], which is accurate for heavy atoms, with the analytical data of [55] that is reliable for light atoms, Ref. [55] found the selfenergy corrections for the PNC amplitude for all atoms. This analysis also allows one to reduce the impact of the above mentioned errors of the data extracted from [52]. The result for the 6s − 7s transition in 133 Cs was −0.9%, in good agreement with −0.73 (20)% of [48]. These two works, using different techniques, unambiguously demonstrated that the experimental data of Wood et al [11] support the standard model.
The contradiction that existed between this statement and conclusions of Ref. [42] was, fortunately, eradicated by the the work of Milstein et al [56] that appeared shortly after [55]. Milstein et al arrived at results that are close to ours [48,55]. For the 6s − 7s transition in 133 Cs they found the self-energy corrections to be −0.85%. A careful comparison of the results of the two groups presented in Ref. [57] revealed a similar agreement for all values of Z. In this latter work it was also proposed that the remaining small discrepancy is due to limitations of accuracy in calculations of Milstein et al that are based on Zα-expansion (Zα = 0.4 for Cs). This conclusion has recently been proved true by Milstein et al who refined their calculations in [58]. Now, for a vast area of the nuclear charges Z the results of Milstein et al for the self-energy corrections agree with ours very closely, the remaining deviation is below 9%. The deviation manifests itself more prominently for large Z, and we believe that it is still due to those terms of the Zα-expansion that remain unaccounted for by the analytical calculations of [58]. Thus, using several different techniques, the two teams arrived to one and the same result for the self-energy corrections. The fact that the results of Milstein et al converged to ours after a period of initial deviation makes the final results even more convincing.
The plan of this paper is to discuss in some detail the radiative corrections, summarizing all available methods and data. As an introduction to this study let us first present very briefly several known facts about relativistic corrections in general. The main relativistic correction is, obviously, the correction that is due to the relativistic nature of the Dirac equation that governs the electron propagation. It is usually taken care of in the initial approximation that relies on the Dirac-Hartree-Fock approach, using then the machinery of the many-body theory to account for the electron correlations. The next necessary step is to include the relativistic correction to the photon propagator that describes the electron-electron interaction. In the nonrelativistic approach this propagator is approximated by its Coulomb part. Relativistic effects can be taken care of perturbatively, using the Breit corrections for this propagator. This approximate approach is convenient because the Breit correction can also be incorporated in the initial Hartree-Fock mean field and then dealt with by conventional many-body methods. At the same time this approximation provides sufficient accuracy.
The next layer of corrections originates from the QED radiative corrections. Numerically, as well as parametrically, they may be comparable with a contribution of the Breit corrections, but it is convenient to consider them separately. The radiative corrections in the lowest order of the perturbation theory are divided into two classes. One of them constitutes the QED vacuum polarization, another one consists of the self-energy and the vertex corrections. These latter two corrections are often grouped together and called the e-line corrections.
The discussion below focuses entirely on the QED radiative corrections. This presumes that every other above mentioned relativistic correction has been reliably taken care of in the previous research on the atomic PNC problem, see the references discussed above. The present paper demonstrates that all radiative corrections have recently been taken into account and calculated reliably. Thus, combining the radiative corrections with the previous theoretical data one can be absolutely certain that every possible relativistic correction in manyelectron atoms is accounted for. This done, the theory comes up with an important conclusion [48,55]: the most accurate available experimental data for parity nonconservation of Wood et al [11] fully support the standard model.

II. VACUUM POLARIZATION AND RELATED PHENOMENA
A. Variation of the electron wave function due to short-range potentials The effective Hamiltonian H PNC induced by the Zboson exchange that describes the PNC part of the weak interaction between the atomic electrons and the nucleus can be presented in the following form (the relativistic units = c = 1, e 2 = α are used, if not stated otherwise) Here G F and Q W are the Fermi constant and the nuclear weak charge, and ρ(r) is the nuclear density. The PNC amplitude describes the matrix element that mixes two states of opposite parity for a valence atomic electron. According to Eq.(2.1) the electron wave function in this matrix element must be taken on the nucleus. Therefore the largest PNC amplitude arises due to the mixing of the s 1/2 and p 1/2 partial waves ψ n,s,1/2 |H PNC |ψ n ′ ,p,1/2 , here n, n ′ refer to the electron energy states, the electron wave functions are the Dirac spinors f nlj (r) and g nlj (r) are the large and small radial components of the spinor, Ω jlµ (n) and Ω jlµ (n) = −(σ · n) Ω jlm (n) are the spherical spinors, and l +l = 2j.
In applications the electron wave function is known in some reasonably good approximation (say, the Hartree-Fock-Dirac initial approximation plus contributions of the many-electron correlations). Inevitably there exist a number of perturbations that are not included in this approximation and can affect the electron wave function. The QED vacuum polarization is one of such perturbations, some others are considered below. In many cases the perturbation is localized in the vicinity of the nucleus, at distances r that are much smaller than the Bohr radius of the K-shell r ≪ 1/(αZm). This is definitely the case for the QED vacuum polarization localized at r ≤ 1/m. For this range of distances the behavior of the electron wave function is quite simple because it is governed mostly by the pure Coulomb field of the nucleus. One can anticipate therefore that the influence of the perturbation on the electron wave function can be presented in simple terms, directly via the perturbative potential. This anticipation proves true, there exists a simple way to account for the perturbation that was suggested in Ref. [43].
Note that it suffices to find the relative variation of the electron wave function induced by a perturbation. Moreover, for small distances r ≪ 1/(αZm) the relative variation does not depend significantly on r. Therefore it is sufficient to find the relative variations of the wave function at the origin, i. e. to find δf nlj (0)/f nlj (0) and δg nlj (0)/g nlj (0), where δf nlj (r), δg nlj (r) are the variations of the large and small components of the wave function due to the considered perturbation, and the relative variations at the origin are taken in the limit r → 0. The limit is necessary because either the large or small components of the wave function vanish at the origin, see Eq.(3.16) below. Ref. [43] found for the relative variations of the wave functions the following result Here and below the indices nlj for the wave functions are suppressed to simplify notation, a is a parameter with length dimension and k is a dimensionless coefficient is the perturbative potential. Parameters γ, κ are defined conventionally γ = (1 − (Zα) 2 ) 1/2 and κ = l(l+1)−j(j +1)−1/4 = ±(j +1/2). Relations (2.3), (2.4) are presented in absolute units, to make them more convenient for different applications below. The simple, clear formula (2.3) solves the problem formulated above, presenting a variation of the wave function at the origin in transparent terms, over the two first moments of the potential. It is instructive to consider Eq.(2.3) in the nonrelativistic limit that reads Here ψ(r) is the Schroedinger nonrelativistic wave function. Deriving Eq.(2.6) from (2.3) one uses the fact that according to (2.4) the parameter a goes to zero in the limit Zα → 0, while from (2.5) one finds k → 2/(2l + 1).
There is a simple short-cut derivation that leads to (2.6).
One expresses the variation of the Schroedinger electron wave function via the Green function G approximating the Green function by the Green function for free motion G (0) where G (0) l (r, r ′ ) is the Green function for the free motion in the l-th partial wave, n = r/r, n ′ = r ′ /r ′ , and the last identity in Eq.(2.8) takes into account the fact that the binding energy of the valence electron is negligible for short distances. Remembering that for nonrelativistic motion the wave function behaves like ψ l (r) ∝ Y lm (n) r l , r → 0 one immediately derives Eq.(2.6) from Eq.(2.7).
The point to be noted in this simple derivation is the dominance of the kinetic term in the nonrelativistic limit. For small distances the nonrelativistic kinetic energy behaves as ∝ 1/r 2 , being larger than the potential energy. That is why Eq.(2.8) holds in the nonrelativistic limit. In the relativistic approach the situation is different. The kinetic term behaves as ∝ 1/r, while the potential behaves as Zα/r. One has to expect henceforth that the contribution of the Coulomb potential energy to the final answer is to be suppressed compared with the contribution of the kinetic term only by a factor Zα. This is exactly what happens in Eq.(2.3). The second term ∝ kr on the right-hand side can be considered as a mere modification of the nonrelativistic result, while the first one ∝ a is the expected new term proportional to Zα. Thus the structure of Eq.(2.3) is predetermined. The calculations of Ref. [43] establish the coefficients in this relation as well as the dependence on higher powers of the Zα-expansion that appear through the parameter γ.
The nonrelativistic Eq.(2.6) shows that the parameter that governs the variation of the wave function is mV (r)rdr that is almost an obvious result valid for a variety of quantum mechanical problems [61]. It is interesting that relativistic Eq.(2.3) shows that there exists another relativistic parameter maV (r)dr. It is suppressed compared with the nonrelativistic one only by a factor Zα that is not small for heavy atoms. This suppression can be well compensated for if the potential increases at small distances which makes maV (r)dr larger than m V (r)rdr. In this case the new relativistic parameter becomes dominant, as the examples below demonstrate. The singular rise of the potential at small distances is typically smeared out by the finite nuclear size. Dealing with such singular potentials it is important to keep in mind that the first term on the right-hand side of Eq.(2.3) is proportional to Z. Therefore it originates from those distances where the full strength of the electric field created by the nuclear charge is present, i. e. from the region outside the nuclear core r ≥ r N . Inside the nucleus the Coulomb field created by the nuclear charge diminishes, making this region less important. In order to estimate its contribution, one can introduce for distances r ≤ r N an effective nuclear charge Z → Z eff = (r/r N ) 3 Z that diminishes with the nuclear radius taking into account the decline of the electric field inside the nucleus. Correspondingly modifying the parameter a in Eq.
one can apply this equation for the region r ≤ r N . We will see that this approximate procedure gives sensible results for the vacuum polarization, see Eqs.(2.21)-(2.23). Let us apply Eq.(2.3) to the PNC problem, in which one needs to calculate the PNC matrix element ψ n,s,1/2 |H PNC |ψ n ′ ,p,1/2 . Both s 1/2 and p 1/2 wave functions are influenced by the perturbation V (r). Therefore the relative variation of the PNC amplitude can be found simply by adding variations of s 1/2 and p 1/2 states given in Eq.(2.3) Here δ ψ s, 1/2 |H PNC |ψ p, 1/2 is the variation of the PNC matrix element due to variations of the wave functions created by the potential V (r). Deriving this result we take into account that essential parameters for the s 1/2 and p 1/2 states are κ s = −1, κ p = 1, γ s = γ p = (1 − (Zα) 2 ) 1/2 ≡ γ and define k in (2.10) to be k ≡ k s = 6/ γ(4γ 2 − 1) . It was assumed in Eq.(2.3) that the perturbative potential is a scalar. In applications it is necessary also to deal with more sophisticated, tensor-type perturbations. The electric and magnetic fields created by the nuclear dipole, quadrupole and higher moments give examples of such perturbations. It is interesting that Eq.(2.3) can be generalized to describe the tensor-type perturbations in simple clear terms. Suppose that we have the valence electron described by the Dirac spinor ψ lj (r) (the energy of the valence electron is so low that we can neglect it, suppressing the irrelevant index n). Suppose that there is some tensor perturbation V (r). When we consider the PNC amplitude we need to evaluate the wave function at the nucleus. There are only two partial waves that remain large at the origin, s 1/2 and p 1/2 . This means that in the multipole expansion of the V (r)ψ lj (r) term in the Dirac equation only these two partial waves contribute. The angular structure of these partial waves is predetermined, compare Eq.(2.2), therefore we may write where [V (r)ψ lj (r)] L,1/2 means the admixture of the L 1/2 state in the term V (r)ψ lj (r). Eq.(2.11) shows that the contribution of the term V (r)ψ lj (r) to the Dirac equation is expressed in terms of the scalar functions F L,1/2 (r) and G L,1/2 (r) with L = 0, 1. We can introduce now some effective potentials V eff L (r), W eff L (r) that satisfy the following conditions Here f L,1/2 (r), g L,1/2 (r) are the known large and small components of the nonperturbed wave function for the valence electron in the L 1/2 partial wave. The procedure described shows that the term V (r)ψ lj (r) in the Dirac equation at small distances can be described by the effective potentials V eff L (r) and W eff L (r) (which can be different for the large and the small components for an arbitrary perturbation). Correspondingly, the variation of the wave function δψ lj (r) at small distances can be expressed via its multipole expansion in which only the s 1/2 and p 1/2 states are retained δψ lj (r) ≃ L=0,1 δf lj; L,1/2 (r)Ω 1/2,L,µ (n) iδg lj; L,1/2 (r)Ω 1/2,L,µ (n) . (2.14) Here the functions δf lj; L,1/2 , δg lj; L,1/2 describe the admixture of the L 1/2 states to the function ψ lj (r) that arise due to the tensor-type perturbation. They are to be found from the Dirac equation. We see that the complicated multipole problem is reduced to the problem with the scalar potential, the only distinction is that the role of the scalar potential plays the effective potentials V eff L (r), W eff L (r) defined above. However, this distinction makes no difference, we introduce the effective scalar potentials in the Dirac equation and solve it regardless of the nature of the potentials. We can rely on the discussion in [43] to derive formulas similar to Eq.(2.3). The result reads Remember that here δf lj; L,1/2 (0), δg lj; L,1/2 (0) describe the admixture of L 1/2 , L = 0, 1 states that arise in the wave function ψ lj (r) of the valence electron in the state with the quantum numbers lj under the influence of the tensor-type perturbation V (r), while f L,1/2 (r), g L,1/2 (r) describe the wave function of the valence electron in the state with the quantum numbers L 1/2 . The latter ones are the same as the ones used in Eqs.(2.12),(2.13) to define the effective potentials. This makes our results presented in Eqs.(2.15),(2.16) independent of the normalization of the wave functions f L,1/2 (r), g L,1/2 (r). In particular, they are independent of the energy of the valence electron (as they should be since this energy is too low to produce any significant effects for small distances). The quantum numbers in Eqs.(2.4),(2.5) that define the parameters a, k in the right-hand sides of Eqs.(2.15), (2.16) correspond to the state L 1/2 , L = 0, 1. We see that for an arbitrary tensor-type potential the result remains very simple. The only modification, compared with the simplest scalar case, is that one needs first to define the effective potentials using Eqs. (2.12),(2.13).
Summarizing, we described in this section simple useful procedures that express the variation of the valence electron wave function at the origin directly in terms of the perturbative potential.

B. QED vacuum polarization
The vacuum polarization in the lowest order of perturbation theory is described by the Uehling potential V VP [62] V VP (r) = − 2α 3π Importantly, the Uehling potential (2.17) is singular at the origin, its asymptotic for mr ≪ 1 reads the singularity is smeared out by the nuclear finite size r N . In Eq.(2.19) C = 0.577 . . . is the Euler constant. The function ln mr arises due to conventional scaling of the QED coupling constant e 2 that manifests itself at short distances. Alternatively, this fact is referred to as the unscreening of the nuclear charge at small distances. For large distances mr ≫ 1 the Uehling potential exponentially decreases In order to find the influence of the vacuum polarization on the PNC amplitude we substitute Eq.(2.17) into Eq.(2.10). Integration over the variable r is performed analytically resulting in the following expression for the relative correction to the PNC amplitude Here the function E 1 (x) is defined conventionally Eq.(2.22), which was first found in [43], gives the main contribution [77]. Deriving it one restricts the integration of the singular term maV VP in Eq.(2.10) to the region outside the nucleus r ≥ r N . The large Coulomb field of the nucleus makes this region important. Inside the nucleus r ≤ r N the electric Coulomb field diminishes, suppressing the contribution of this region. In order to estimate the contribution of small distances r ≤ r N we use a modification of Eq.(2.10) introducing into Eq.(2.3) the effective parameter a defined in Eq.(2.9). The resulting integral in Eq.(2.3) is saturated in the nuclear interior region in the vicinity of the nuclear surface, where the electric field remains large. This fact allows one to take the values of γ and V VP (r) directly at the nuclear surface. The described procedure results in estimate Eq.(2.23).
Eqs.(2.21)-(2.23) give the weak interaction matrix element for an arbitrary atom in a transparent analytical form without fitting parameters. Numerical results are easily obtained by a straightforward one-dimensional integration in Eq.(2.22). One only needs to specify the nuclear size that can be taken conventionally as r N = 1.1A 1/3 fm, where A is the atomic number [78]. Alternatively, the right-hand side of Eq.(2.21) can be calculated using an expansion in powers of mr N ≪ 1 that reads [43] .25) is the second power of the logarithmic factor in the term ∼ Z 2 α 3 ln 2 (mr N ). This strong, double-logarithm enhancement was first found by Milstein and Sushkov [42]. The derivation presented above makes the origin of this effect clear. The first power of the logarithmic function ln mr originates from the unscreening of the nuclear charge at small separations (2.19), while the second one appears from the relativistic parameter V (r)dr ∝ dr ln(mr)/r ∝ ln 2 1/mr N in Eq.(2.10).
In Fig. 1 we present the relative corrections due to the QED vacuum polarization δ (VP) to the PNC matrix element calculated with the help of Eqs. For all four atoms the nuclear interior region gives 10 % of the total correction, δ (VP, 2) = 0.1δ (VP) , which means that to make results accurate numerically this region needs to be taken care of.
Compare these results with other results reported recently. Johnson, Bednyakov and Soff [41] calculated for the first time the correction due to the vacuum polarization for the 6s − 7s PNC amplitude in 133 Cs and found it to be 0.4%. The result of [41] includes, along with the variation of the weak matrix element, variations of the dipole matrix element and the corresponding energy denominators that, combined together, describe the s 1/2 − s 1/2 amplitude measured experimentally. Dzuba et al [36] confirmed this result, calculating the correction 0.41%, and supplied more details providing separate variations for all three quantities mentioned above. They found that variations of the dipole matrix elements and the energy denominators induced by the Lamb shift corrections, being not small, compensate each other almost completely. Thus the variation of the PNC amplitude proves to be equal to the variation of the PNC matrix element. The numerical result for thallium found in [66] gives a correction 0.94%, to be compared with 0.93% mentioned above. Calculations of Milstein et al in Ref. [42] were restricted by the logarithmic accuracy that was further improved by using some constant as a fitting parameter to reproduce 0.4% for 133 Cs, in line with [41]. For Tl Ref. [42] finds 0.9%. This is slightly less than the above mentioned prediction of Eq.(2.21) (0.93%). This (insignificant) discrepancy probably arises because the procedure adopted in [42] does not account for the increase of the contribution of the nuclear interior region with the nuclear charge. We conclude that all reported calculations for the vacuum polarization correction are in very good agreement, distinctions between results derived by different methods and by different groups are all below 5%.
Let us discuss now possible corrections to the results discussed above. One of them originates from a modification of the vacuum polarization due to the finite nuclear size. The corresponding generalization of the Uehling potential was given by Fullerton and Rinker [63]. In order to estimate its contribution we need to remember that the main ∝ ln 2 mr N term in Eq.(2.25) originates from the integration well outside of the nuclear radius where the Uehling potential is very close to the one of Fullerton and Rinker. Therefore this main part of the correction is not changed. Modifications could arise only for the integration in the vicinity of the nuclear surface, but this region by itself gives only a fraction (∼ 10%) of the total correction, while the relative difference of the Uehling potential and the Fullerton and Rinker potential on the nuclear surface is small. For all atoms it remains ≤ 5%, as we found by direct numerical comparison of the Uehling potential with the Fullerton and Rinker potential. We conclude that the modification of the vacuum polarization due to the finite nuclear size is irrelevant.
Another possible modification can originate from the QED higher-order corrections that contribute to the vacuum polarization. One of them stems from the fact that the Landau pole requires that the linear in ln mr term in Eq.(2.19) be followed by higher-order logarithmic terms. However, the corresponding series runs in powers of Zα(α ln mr) n , n = 1, 2 . . . , giving small contributions for higher order terms n > 1.
Consider now a modification of the Uehling potential that is due to the fact that the Coulomb field of the nucleus disturbs the virtual electron-positron pair. This effect is conveniently described with the help of the Wichmann-Kroll potential [64] that, compared to Eq.(2.17), describes the next term in the Zα-expansion. For estimates it is sufficient to take the Wichmann-Kroll potential at small distances, where its asymptotic mr ≪ 1 reads [65]: Using Eq.(2.10) one immediately finds from Eq.(2.26) the ratio of the correction induced by the Wichmann-Kroll potential to the Uehling correction , (2.27) as was found in [43]. For 133 Cs this ratio is about −0.007, demonstrating that the correction due to the Wichmann-Kroll potential is very small. Thus all higher order QED terms in the vacuum polarization are negligible. Alongside the vacuum polarization there exists also another polarization effect, namely the polarization of the nucleus. One can estimate its influence on the electron wave functions and the PNC matrix element with the help of the conventional polarization potential V NP that is created by the nucleus Here α d is the static nuclear dipole polarizability. It can be estimated roughly assuming that the total oscillator strength for the nuclear dipole excitations is located at the energy of nuclear giant dipole resonance Ω d . Then, using the conventional sum rule for the oscillator strengths one finds where m p is the proton mass. Since the potential (2.28) increases at small separations the main contribution in Eq.(2.10) is given by the first term in Eq.(2.10) which should be integrated outside the nuclear radius r ≥ r N , as was explained above. Having this in mind, one substitutes (2.28) in (2.10) finding an estimate for the influence of the nuclear polarization on the PNC matrix element For the 133 Cs atom we take Ω d ≃ 15 MeV obtaining δ (NP) ≃ 0.01% that shows that the nuclear polarization is negligible. This conclusion agrees with estimates of Milstein and Sushkov [42], who proposed that the effect is of the order of ∼ 0.1Z 2/3 α 2 [80], which numerically is close to Eq.(2.30). Summarizing, we demonstrated above that the QED vacuum polarization gives a large positive correction (∼ 1%) to the PNC amplitude. Contributions of higher order QED corrections are much smaller. This means that the influence of the nuclear Coulomb potential on the virtual electron-positron pair is not pronounced, possibly due to opposite signs of the virtual particles. We will see below that for a single electron the situation is different, the Coulomb field does play an important role for the self-energy correction, see Eq.(3.29). The influence of the finite nuclear size on the vacuum polarization was found to be nonessential, the polarization can be well described by the simplest Uehling potential. Also negligible is the polarization of the nucleus. We presented above in some detail analytical methods of calculation, paying less attention to purely numerical methods that have played a very important role in the problem, see [36,41,66]. Omissions of details in our presentation of numerical methods is justified by the fact that the vacuum polarization is described by a sufficiently simple potential, which can be incorporated in the initial Hartree-Fock-Dirac approximation for the atomic electrons, and then dealt with by conventional methods of many-body theory. Thus the numerical approach follows the mainstream of atomic structure calculations discussed in some detail elsewhere, see for example [32,33]. Overall, the polarization is a well understood, robust effect that is well described by Eqs.(2.21)-(2.23).

A. Chiral Invariance and QED corrections
Let us show, following Ref. [48], that there exists an interesting and useful relation that gives the QED radiative corrections to the PNC matrix element via similar radiative corrections to another quantity, the energy shifts of the atomic electron induced by the finite nuclear size (FNS). This relation can be written as where δ PNC, sp is the relative radiative correction to the PNC matrix element between the s 1/2 and p 1/2 orbitals The energy difference between the considered opposite parity orbitals ∼ 1 eV is much lower than a typical excitation energy ∼ m = 0.5 MeV that governs the radiative corrections. We can therefore neglect this difference assuming that E s,1/2 ≃ E p,1/2 . This assumption makes the correction ψ s, 1/2 |H PN C |ψ p, 1/2 rad gauge invariant. To make this argument more transparent, one can use the Coulomb approximation for the atomic field and consider the degenerate Coulomb ns 1/2 and np 1/2 levels.
The problem of gauge invariance can be viewed from another perspective. One can consider the full PNC amplitude. Alongside the PNC matrix element it includes also the conventional E1 amplitude and the corresponding energy denominator. The PNC amplitude describes the events on the mass shell, the photon frequency in the E1 transition equals the excitation energy of the atom. Therefore the amplitude is definitely gauge invariant. Considering the radiative corrections in the lowest order, one can separate them into corrections to the PNC matrix element, the Lamb-shift corrections to the energies, and the corrections to the E1 transition. Their sum is definitely gauge invariant. The point is that the latter two corrections are known to compensate each other almost completely [36]. The only remaining corrections are the ones which are associated with the PNC matrix element. Thus these latter corrections are gauge invariant. To make this discussion complete, let us mention also that in the perturbation theory approach outlined in the next section III B the gauge invariance of the PNC matrix element is self-evident. There is no doubt, therefore, that one can assume that the radiative corrections to the PNC matrix element are gauge invariant.
The quantities δ FNS, s and δ FNS, p in Eq. The FNS energy shifts can be presented as matrix elements of the potential V FNS (r) that describes the deviation of the nuclear potential from the pure Coulomb one inside the nucleus due to the spread of the nuclear charge E FNS, l = ψ l, 1/2 |V FNS |ψ l, 1/2 , l = 0, 1.
(3.4) Equality (3.1) may be established for the sum of all QED radiative corrections (∼ Zα 2 f (Zα)), or specified for any gauge invariant class of them. We will discuss first the self-energy and vertex corrections, restricting our consideration to the lowest order of perturbation theory presented by the Feynman diagrams in Fig.2, calling them the e-line corrections. Later we will switch our attention to the vacuum polarization and demonstrate that it satisfies (3.1) as well.
The intermediate electron states in diagrams of Fig. 2 are described by the propagator G = (γ µ p µ +γ 0 U −m) −1 , with p µ = (ǫ, −i∇), where ǫ = m + E ≃ m is the relativistic electron energy, U = U (r) is the atomic potential that includes the potential created by the nucleus with FNS and, generally speaking, the screening potential of atomic electrons. The latter one is not important at small distances, but it still can be accounted for by the formalism. For the relativistic propagator we use the same symbol G that was applied previously for the nonrelativistic Green function in Eq.(2.7), which should not produce confusion. At small distances the exchange potential is negligible, therefore the potential is local. The external legs of the diagrams describe the wave functions ψ s,1/2 (r) and ψ p,1/2 (r) for the considered s 1/2 and p 1/2 levels. Let us note first that in the region of short distances the following relations hold 5) ψ s,1/2 (r) = cγ 5 ψ p,1/2 (r) . (3.6) They follow from the chiral invariance that manifests itself at small distances. For r ≪ 1/m the mass term in the Dirac equation becomes smaller than the kinetic term (∝ 1/r). For even smaller distances r ≤ Zα/m the potential (Zα/r) also exceeds the mass term. Therefore for small distances we can neglect the mass m ⇒ 0, using the resulting chiral invariance for our advantages. Since Eqs.(3.5),(3.6) will be important for us, we examine them in some detail. In order to prove Eq.(3.5) one neglects the mass term in the electron propagator assuming that Then, the identity γ 5 γ µ = −γ µ γ 5 leads to G −1 γ µ = −γ µ G −1 that results in (3.5). In order to verify Eq. (3.6) let us write down the conventional Dirac equations for the large f and small g components of the electron wave function (2.2) Here we omitted the indexes njl, distinguishing the states by the parameter κ that takes the values −1 and 1 for the s 1/2 and p 1/2 states respectively. For small separations one can neglect in Eqs.(3.8),(3.9) the mass term m. After that we observe that the equations become invariant under a substitution Remember now that according to Eq.(2.2) the s 1/2 and p 1/2 wave functions have the form ψ s,1/2 = f s,1/2 u i g s,1/2 (−σ · n) u , (3.11) ψ p,1/2 = f p,1/2 (−σ · n) u i g p,1/2 u .
Here the two component spinor u = u α , α = 1, 2 depends on the projection of the total momentum µ, u α = δ α,1 for µ = 1/2, and u α = δ α,2 for µ = −1/2. It follows from Eq.(3.11) that where we took into account that in the standard representation considered for the Dirac matrixes  6). The constant c in (3.6), which depends on the normalization conditions, will be irrelevant for us because we are interested in relative quantities that appear in Eq.(3.1) where this constant will be canceled out.
It is instructive to look at numerical verifications of Eq.(3.6) shown in Fig.3. Supporting the proof given above, they provide also details that are cumbersome for analytic treatment. Fig.3 (a) shows that Eq.(3.14), which according to the discussion above holds at small distances r ≪ 1/m, remains approximately valid at sufficiently large distances, up to (and even beyond) the Compton radius r ≤ 1/m. In contrast, Eq.(3.15) is applicable only at small r ≪ 1/m as shown in Fig. 3 (b), being violated at r ∼ 1/m, as one can see from Fig. 3(a). We can remember, however, that the dominant contribution to the PNC matrix element arises from the f s,1/2 and g p,1/2 functions, which are both large for r ≤ 1/m, much larger than the other two functions g s,1/2 , f p,1/2 . Since these large functions satisfy Eq.(3.14) even at sufficiently large distances r ≤ 1/m, one can also rely in this region on Eq.(3.6). We will exploit this opportunity at some later stage below, but our prime interest is related to the distances as small as the nuclear radius r ≤ r N ≪ Zα/m, where we verified beyond doubt that Eqs.(3.5),(3.6) hold, see Fig.3 (b).
Consider the diagrams (b),(c) in Fig. 2 that describe the self-energy correction for the PNC amplitude. The thick lines in this figure represent the electron propagation in the atomic field, the dot vertex is the PNC interaction described by Eq. (2.1), and the dashed line shows the photon. Let us specify that the left and right legs of these diagrams describe the s 1/2 and p 1/2 states respectively. The PNC interaction (2.1) is located at small distances r N ≪ 1/m. Let us use this fact and, relying on Eqs.(3.5)(3.6), transform the PNC matrix element. To simplify the notation it is convenient to present the Hamiltonian of the weak interaction Eq.(2.1) in the form is a scalar factor in the PNC interaction. Consider first the diagram (b). The ψ p,1/2 (r) function in this diagram enters the PNC interaction that makes its argument r localized inside the nucleus, r ≤ r N . This allows us to use Eq.(3.6) deriving Here and below we use the symbol of identity (≡) that simply means dropping the irrelevant constant c introduced in Eq.(3.5) (that is canceled out when the relative contribution of the correction is considered anyway). Eq.(3.18) uses conventional notation, Σ and G are the self-energy operator and the electron propagator mentioned above. The right-hand side of this relation shows that we can look at the diagram (b) as the one that describes the diagonal s 1/2 − s 1/2 transition induced by an effective scalar potential V PNC . Consider now the same diagram (b), but for another physical quantity, for the FNS energy shift for the s 1/2 level. Its contribution to the energy reads Eqs.(3.18), (3.19) give the contribution of one and the same diagram (b) to different quantities, namely the PNC amplitude and FNS energy shifts. Comparing them one observes their similarity. In both cases the diagram describes the s 1/2 − s 1/2 transitions that are induced by the short-range scalar potentials. The only distinction comes from the fact that the potentials V PNC (r) and V FNS (r) possess different profiles. This latter difference, however, proves to be irrelevant. In order to see this we need to discuss the integral with a short-range potential that arises in the diagram. Its integrand includes the potential (i. e. V PNC (r) for the PNC amplitude and V FNS (r) for the FNS energy shift), the electron wave function ψ s,1/2 (r) and the electron propagator G(r, r ′ ). The wave function and the propagator inside the nucleus are almost constants, see discussion of this fact below. Taking them out of the integration we observe that the considered sophisticated integral is simplified, being reduced to the integral over the potential itself V PNC (r)dr. It is easy to see that the same integral arises in the non-perturbed matrix element. Therefore in the relative contribution of the correction this integral is canceled out, and the result does not depend on the potential at all. This conclusion is valid for any short-range potential. Thus the relative contribution of the diagram (b) proves to be one and the same for the PNC and the FNS problems. This means that the relative contribution of the diagram (b) to the PNC matrix element equals half of its relative contribution to the energy shift of the s 1/2 state due to FNS. The mentioned factor 1/2 takes into account the fact that the energy shift is induced by the two diagrams (b) and (c) in Fig. 2 that are identical for the FNS problem.
Before proceeding further, let us discuss the behavior of the electron wave functions inside the nucleus which was exploited in the above derivation. Note that inside the nucleus the large functions f s,1/2 (r) and g p,1/2 (r) exhibit smooth, almost constant behavior, as demonstrates Fig. 3. The reason for this effect stems from the fact that inside the nucleus the electric field rapidly diminishes with the radius. The functions g s,1/2 (r) and f p,1/2 (r), show some variation, but it remains very mild. Assessing consequences of this smooth variation one can recall that, firstly, the integration volume r 2 dr enhances the contribution of the nuclear surface region where these functions are very smooth indeed, as shows Fig. 3 (b). Secondly, their role is not prominent anyway, simply because they are smaller than the functions f s,1/2 (r) g p,1/2 (r). Therefore they can be well approximated by constants. Thus both s 1/2 and p 1/2 states are well described by constanttype functions inside the nucleus. This fact was used above for the s 1/2 state, the argument is repeated below for the p 1/2 state. We also presumed above that the propagator G(r, r ′ ) does not depend on the variable r, when r ≤ r N . This statement follows from the simple fact that the coordinate r ′ gives the location of the point where the radiation of the photon takes place, therefore r ′ ∼ 1/m ≫ r. The smallness of r compared with r ′ justifies its neglect in the propagator.
Examining the diagram Fig.2 (c) we use the same arguments as we exploited above for the (b) diagram proving that its relative contribution to the PNC amplitude equals half of its relative contribution to the energy shift due to FNS of the p 1/2 state. Combining our results for the diagrams (b) and (c) we conclude that the self-energy corrections (superscript SE) described by these diagrams comply with Eq.(3.1) (3.20) The given derivation does not use any specific gauge condition, therefore it is gauge invariant. Consider now the vertex correction (a) in Fig. 2. An analytical expression for it can be written in the conventional form Here D µν (q) is the photon propagator, q µ = (ω, q) is the four-vector of the photon momentum. The range of distances r rad where the radiation processes take place is of the order of the Compton radius, r rad ∼ m −1 . This ensures that r, r ′ ≫ r 0 because the PNC interaction is localized at r 0 ≤ r N . We can simplify therefore Eq.(3.21) factorizing the interaction over r 0 where we put r 0 = 0 in the arguments of the propagators. We observe that the potential arises only in a factor V PNC (r 0 ) dr 0 that is separated from all other elements of the diagram. The same factor appears in the main term of the PNC amplitude. Therefore the relative contribution of the vertex correction does not depend on the potential at all. Remember that the same property is exhibited by the self-energy correction discussed above.
The variables r, r ′ describe the radiation processes and therefore are expected to be of the order of the Compton radius r, r ′ ∼ 1/m. Let us consider the implications that arise if one presumes that the region of small separations gives the dominant contribution to the integral in Eq.(3.21). We can engage in this region Eqs.(3.5),(3.6). The first of them allows one to exchange the γ 5 matrix in Eq.(3.23) with the propagator G. One can also transpose it with the vertexes γ 5 γ µ = −γ µ γ 5 . Using several such transpositions we can bring the γ 5 matrix from its central position in expression (3.23) to the side, where it hits the external wave function. After that we use Eq.(3.6) for this function. Repeating the procedure twice, one time shifting γ 5 to the left, and another one to the right we deriveψ To simplify notation we dropped here the total momentum 1/2 from the indexes in the wave functions (i. e. ψ p ≡ ψ p,1/2 etc), and omitted the energy variable in the propagators (this variable equals ǫ + ω for all propagators). The point of these transformation is that the γ 5 matrix disappears from the right-hand sides of Eqs.(3.25), (3.26). Substituting these two expressions back in Eqs.(3.23), (3.21) we find that the relative contribution of the vertex diagram (a) to the PNC matrix element can be expressed either as a matrix element of the s 1/2 − s 1/2 transition, or the matrix element of the p 1/2 − p 1/2 transition. We recognize in these two matrix elements the relative contributions to the energy shifts due to FNS for the s 1/2 and p 1/2 states respectively. This latter conclusion takes advantage of the fact discussed above that the profile of the short-range potential does not influence the relative correction, one can conveniently choose the potential to be V PNC or V FNS . This discussion demonstrates that the vertex (superscript V) corrections to the PNC amplitude and to the energy shifts due to FNS prove to be approximately equal The derivation presented above exploits Eq.(3.24) presuming that the small distances are dominant in the vertex correction. There are reasons indicating that this assumption does not put restrictions on the derived result (3.1). As a first attempt one can try to exploit the behavior of the wave functions f s,1/2 , g p,1/2 . Fig. 3 shows that they approximately satisfy Eq.(3.6) even at sufficiently large separations, of the order of the Compton radius r ≤ 1/m. This shows that Eq. FNS, s also remains approximately valid. This is a positive result. However, this fact alone is not sufficient. The second equality in Eq.(3.27) proves to be much more sensitive to the essential integration region because the wave functions f p,1/2 , g s,1/2 fail to satisfy Eq.(3.6) at r ∼ 1/m. We need therefore to be more careful.
Let us recall at this stage that the range of distances r rad where the radiation processes take place depends on the chosen gauge for the electromagnetic field. It is well known that the usual length-gauge favors larger distances, while the velocity and, especially, acceleration forms, make the smaller distances contribute more to the radiation process. These known examples show that the region that contributes to the radiation process is not gauge invariant. This is important. If the radiation region is not gauge invariant, then we are able to choose a gauge in which this region is located close to the nucleus, inside the zone defined by Eq. (3.24). In this gauge the above derivation is valid and Eq.(3.27) for the vertex is correct. Note that this argument does not rely on an explicit form for the necessary gauge. It suffices to acknowledge only that the radiation region is not gauge invariant. Summarizing, there exists a gauge (more accurately, a family of gauges) in which Eq.(3.27) holds.
Recall now that Eq.(3.20) for the self-energy remains valid in any gauge, in particular in the one discussed above. In this gauge we can combine together the selfenergy and vertex corrections (3.20), (3.27) proving the validity of their sum Eq.(3.1). The three quantities in this latter relation are all well-defined, gauge invariant physical observables. The fact that Eq.(3.1) is derived in one particular gauge means that it is valid in any gauge, as was proposed in [48].
Let us summarize the above discussion. We argued that the calculations can be organized in such a way that all important events happen at small distances. In this region the chiral invariance holds, resulting in Eqs.(3.5),(3.6) and, consequently, in Eq.(3.1). Thus this latter equality expresses the fundamental and simple fact. At small, nuclear distances the chiral invariance governs the problem. Consequently, Eq.(3.1) that follows from it, can be called the chiral invariance identity.
Let us estimate the accuracy of Eq.(3.1). The above derivation used the gauge in which the radiation processes for the diagram (a) take place mostly at small distances r rad < m −1 . They should also take place outside the nucleus r N < r rad , as is necessary to justify our presumption that the shape of the potential inside the nucleus is irrelevant. From the last two inequalities we find that the derivation relies on a parameter ξ = mr N ∼ 0.01. This determines the magnitude of the error (few per cent) of Eq.  Note that we do not consider above the radiative corrections of the order ∼ α/π which appear in the plane wave approximation. These contributions have been included into the radiative corrections to the weak charge Q W (and the renormalization of the charge and electron mass in the case of FNS energy shifts). Correspondingly, we subtract the contribution of the plane waves from Eq.(3.1), considering only the part of the corrections that depends on the atomic potential ∼ Zα 2 f (Zα). For heavy atoms this subtlety is insignificant numerically because the considered Z-dependent part of the correction is bigger than the omitted Z-independent one, as we will see below.
The chiral invariance identity (3.1) dives the e-line corrections to the PNC matrix element, which are difficult to calculate, in terms of the corrections to the FNS energy shifts that have been well-studied both numerically, by Johnson and Soff [49], Blundell [50], Cheng et al [52] and Lindgren et al [53], and analytically, by Pachucki [67] and Eides and Grotch and Eides et al [68]. Ref. [52] presents the e-line radiative corrections to the FNS energy shifts for 1s 1/2 , 2s 1/2 and 2p 1/2 levels in hydrogenlike ions with atomic charges Z = 60, 70, 80, 90. Eq. (3.1) contains relative corrections, therefore we needed to calculate the FNS energy shifts E FNS . This was done in [48] by solving the Dirac equation with the conventional Fermi-type nuclear distribution ρ(r) = ρ 0 /{1+exp[(r−a)/c]}. Parameters a, c were taken the same as in [52], namely a = 0.523 fm and c chosen to satisfy R rms = 0.836A 1/3 + 0.570 fm.
Using the results of [52] and this calculation we obtained in [48] the relative radiative corrections shown in Fig. 4. In order to include the interesting case Z = 55 and to account for all values of 55 ≤ Z ≤ 90 we used interpolating formulae presented in [52], as well as data of [54]. The relative corrections for the 1s and 2s levels are approximately the same size. This indicates that the radiative processes responsible for the correction take place at separations much smaller than the K-shell radius, r ≪ (Zαm) −1 , which is consistent with the assumption r ≤ 1/m above. For these separations we can assume that, firstly, the screening of the nuclear Coulomb field in manyelectron atoms does not produce any significant effect, and, secondly, the relative corrections do not depend on the atomic energy level because for small separations all atomic ns 1/2 -wave functions exhibit similar behavior. These arguments remain valid for the p 1/2 states as well, permitting us to presume that the results shown in Fig. 4 for the 2s 1/2 -levels and 2p 1/2 -levels of hydrogenlike ions remain valid for s 1/2 and p 1/2 states of the valence electron in a manyelectron atom. We obtain the e-line radiative corrections for the PNC matrix element using the identity (3.1) that expresses them via the found corrections to the FNS energy shifts. The found PNC corrections, presented in Fig. 4 by the dotted line, are negative and large. For the 133 Cs atom the correction is −0.73(20)%, for Tl it is −1.6%. The result for 133 Cs is important because it indicated for the first time that the radiative corrections reconcile the experimental data of Wood et al [11] with the standard model [48]. A notable feature of the self-energy corrections presented in Fig. 4 is their close similarity. The corrections to the FNS energy shifts for both s 1/2 and p 1/2 states, as well as for the PNC amplitude, have large negative contributions from the QED self-energy, which monotonically and very smoothly increase with the nuclear charge. This observation brings us back to the point mentioned in section I. The large and negative contribution of the self-energy for heavy atoms is exactly what one should expect from it. Anything else would be a surprise. In order to verify this point one can try to apply the approach outlined to some other problem, for example to the hyperfine interaction HFI which has been carefully examined previously, see detailed discussions in [46,47] and references therein [81].
With this purpose one can try to derive the chiral invariance identity for the HFI. The problem is that the HFI has a long-range tail ∼ 1/r 3 that is prominent in the region where the chiral invariance is violated. However, if one believes that convergence of the HFI matrix elements is fast enough, then the following equality that was proposed in [48] should hold at least approximately. Here δ ′ HFI, s is the radiative correction to the HFI for s 1/2 -levels, the primed notation indicates that the Z-independent Schwinger term α/(2π) should be excluded. For heavy atoms this subtlety is not important, since the contribution from the Coulomb-induced corrections is much stronger than the Z-independent Schwinger term. Fig.4 shows the e-line contribution to δ ′ HFI, s that was extracted in [48] from data of [52] using interpolation for all values of Z considered there. It agrees semiquantitatively with Eq.(3.28), the deviation is less than 33 %. We observe again the same trend, the self-energy correction is large and negative, in accordance with the clear physical reasons mentioned in section I. Another positive conclusion is that chiral invariance identities similar to (3.28) can be applicable even for those potentials that spread out of the nuclear core.
The approach outlined raises two points. Firstly, it is interesting to have an estimate for the self-energy correction in simple terms, that would not appeal to sophisticated numerical calculations. Secondly, it is important to derive the self-energy corrections for lighter elements. Direct numerical calculations are difficult for them because the error rapidly increases for lighter atoms. The main source for the error is the FNS energy shift for the p 1/2 state. The lightest element for which the self-energy corrections have been calculated is Cs, where the error for the FNS energy shift in the p 1/2 wave was ≃ 100% [54]. Exactly this error results in the uncertainty of the above mentioned result −0.73 (20)%. For heavier atoms the numerical errors of [52] rapidly decrease. Therefore the main error in our result for very heavy (Z ≥ 80) atoms is associated with the validity of Eq.(3.1), that was estimated above as a few percent.
The estimate for the magnitude of the effect and calculations for lighter elements can be conveniently performed with the help of the Zα-expansion considered in the next section.

B.
Perturbation theory in powers of Zα Let us consider the perturbation theory for the e-line corrections. In the initial approximation one describes the electron using plane waves and consequently including in the perturbation theory three types of processes. Firstly, the PNC interaction, secondly, the e-line corrections, and, thirdly, the Coulomb interaction. Taking the nuclear Coulomb field as a perturbation one can formulate the perturbation theory in powers of the nuclear charge Z. The parameter that governs the corresponding expansion is α(Zα) n , where n is the order of the perturbation theory. This statement follows from a conventional, well known result [61]. In the Coulomb problem the perturbation runs in powers of Ze 2 / v, where v is a typical velocity of the electron. For short distances v ≃ 1, thus Ze 2 / v = Zα. The type of expansion considered typically includes some additional large logarithmic factors, as specified below. In the lowest order of perturbation theory the simplest, linear in Z correction is ∝ Zα 2 . Thus formulated problem requires calculation of the relevant coefficients that can be achieved using the corresponding Feynman diagrams.
The Feynman diagrams for the e-line corrections in the first order of perturbation theory over the Coulomb field are presented in Fig. 5. The direct calculation of these diagrams for the PNC amplitude gives the following result for the relative contribution of the e-line corrections to the PNC amplitude δ (e−line, 1) PNC,sp = −1.97 Zα 2 . (3.29) It was first obtained in [55] that had an amusing history [69]. The index 1 in the superscript of (3.29) reminds one that this is the linear in Z correction. For the 133 Cs atom Eq.(3.29) gives −0.6%, which is in line with the result −0.73 (20)% that follows from the chiral invariance identity (3.1) as discussed in the previous section. An earlier attempt of Milstein and Sushkov [42] gave the coefficient ≈ +0.1, instead of −1.97 in Eq.(3.29). This distinction had important implications. Based on their result the authors of [42] claimed that there existed the contradiction between the experimental data of [11] and the standard model, while Eq.(3.29) indicated that the experimental data agree with the model. Fortunately, the controversy was soon resolved by Milstein et al in the following Ref. [56] that agreed with (3.29) and found also an analytical expression for the coefficient in this equation that reads −(23/4 − 4 log 2) = −1.970. This result has also been confirmed in recent Ref. [70]. At this point it is worth returning to the problem of gauge invariance. It is easy to demonstrate by conventional methods that the sum of all the diagrams in Fig.  5 is gauge invariant. This fact should be compared with the discussion of the gauge invariance in Section III A, see after Eq.(3.1). However, for practical applications one often fixes the gauge because this allows one to simplify lengthy analytical calculations. This is the way the calculations were performed in [55,56,70]. Since these works employed different gauges, the first one used the Feynman gauge while the second and third ones relied on the Fried-Yennie gauge (also called the Yennie gauge), their mutual agreement provides an additional helpful check of the validity of the final result Eq.(3.29).
Eq.(3.29) provides a simple transparent estimate for the contribution of the e-line corrections. One point to be noted is a strong dependence of the result on the nuclear charge that makes the correction large for heavy atoms; remember the discussion of this point in Section I. This behavior of the self-energy correction differs qualitatively from what we saw for the vacuum polarization in section II, where the higher-orders in the Zα-expansion were found small. A notable feature in Eq.(3.29) is the large coefficient ∼ −2.0 on the right-hand side. Naively one could expect this coefficient to be smaller, of the order of ∼ 1/π. It is interesting that a similar "numerical enhancement" occurs for the e-line radiative correction for the energy shift that is due to the finite nuclear size (FNS). This correction was examined analytically in Refs. [67,68], and more recently in Ref. [56]. The result can be written as where the analytical expression was derived in [56]. We see that indeed, the e-line corrections to the FNS energy shift are governed by the large coefficient ∼ −3.0 in (3.30), similar to (3.29) for the PNC amplitude. Fig.  6 examines this similarity in more detail. It shows data available for relative e-line corrections for the two problems mentioned above, namely for the PNC amplitude and FNS energy shifts. The linear in Z approximations (3.29),(3.30) (that are valid for sufficiently small values of Z) are compared in this figure with results that follow from numerical calculations of Ref. [52]. We observe a very close quantitative similarity between corrections to PNC and FNS. In both cases the linear approximations (3.29) and (3.30) predict large negative corrections, which qualitatively agrees with results based on numerical calculations for heavy atoms. The numerical validity of the linear approximations seems to be limited by the region below Z = 55, for higher Z they underestimate the effect. The Cs atom lies on the border, where results of the small-Z and large-Z approaches agree reasonably well. Numerical data used in Eq.(3.1) incorporates an error that increases for smaller values of Z, see Tables III and  IV of [52]. In order to reduce the impact of this error we can combine together Eqs.(3.1), (3.29). One can approximate all nonlinear terms omitted in (3.29) by the simplest quadratic function and choose the corresponding coefficient to reproduce the results based on numerical data for very large Z, Z ∼ 90, where the numerical errors are small. This approach gives the following interpolating formula [55] for the corrections to the PNC amplitude δ (e−line, int) PNC = −1.97 Zα 2 ( 1 + 1.55 Zα ) .
(3.31) Fig. 6 shows that the data available for large-Z and small-Z regions is very smooth. Therefore any reasonable interpolation would produce a pattern close to Eq.(3.31).
We can be certain therefore that ( 60% that agrees with the prediction of (3.1) discussed above (as it should, since the data of [52] incorporates only very small errors for heavy atoms).
It should be noted that Eq.(3.1) does not rely on the perturbation theory in powers of Zα, effectively including all nonlinear corrections. This makes it accurate even at large values of the nuclear charge. It should therefore be considered as a formula that provides a convenient short-cut presentation of the nonlinear result. A similar formula was suggested in [55] to reconcile the data for the e-line corrections to the energy shifts due to FNS for the s 1/2 state. The linear in Z term of Eq.(3.30) and numerical data of [52]  An alternative method to include the nonlinear corrections was developed in Refs. [56,58] that relies on direct analytical calculations. This approach faces a difficulty because even in the next-to-leading order ∼ Z 2 α 3 the calculations become complicated. Ref. [42] showed, however, that in this order there appears a large logarithmic factor that makes the correction proportional to ∼ Z 2 α 3 ln 1/mr N . Calculations with the logarithmic accuracy, when some constant is neglected, are much more feasible. The elegant result for the thus calculated second-order correction (index 2 in the superscript) was derived in [56] δ e−line,2 PNC,sp = − The final result of [56] is given by the sum of the linear term Eq.(3.29) and the second order correction Eq.(3.33). It is compared in Fig.  7 with predictions of Eqs. (3.1) and (3.31). For the sake of completeness the figure also shows the previous results of Ref. [42]. For all values of the nuclear charges the results of [56] are close to our results derived from Eqs.(3.1) and (3.31), as was emphasized in [57]. Recently Milstein et al [58] refined their arguments, demonstrating that the logarithmic factor similar to the one in Eq.(3.33) exists in all higher order terms of the Zα-expansion. Ref. [58] found the corresponding contribution of the higher-order terms numerically. This latest calculation, also shown in Fig.7, is even closer to our results given by Eqs.(3.1) and (3.31). The deviation from (3.31) is below 9% for a wide range of atomic charges. Ref. [58] claims their error to be of the order of ∼ 5%; our estimate for the accuracy of Eq.(3.1), which limits the error at large Z, is a few percent, see Section III A. Within these errors the results of the two groups completely agree. The fact that the remaining discrepancy increases for heavy atoms indicates probably that it is due to terms of the Zα-expansion still unaccounted for in [58]. Overall, Fig.7 shows good agreement of the recent results obtained by the two groups using different approaches. The sharp contradiction that existed during the initial stages of this research (compare the dashed-dotted line with the thick-dotted line) makes the latest convergence even more satisfying and trustworthy .
There is an interesting physical link between the chiral invariance identity of Eq.(3.1) and the perturbation theory approach of Eqs.(3.29), (3.33). Remember that in deriving Eq.(2.3) in the previous section we focused our attention at one stage of the consideration at small distances r ≤ Zα/m. The claim was that if this region is proved to comply with Eq.(2.3), then the outer region r ≥ Zα/m would inevitably comply with it as well, see discussion after Eq.(3.24). A similar philosophy lies behind the logarithmic approximation in Eq. (3.33). The integration that leads to the logarithmic function in this equation is saturated at small distances. The omitted constant is related to larger distances, but it is of lesser importance and can, in the simplest approximation, be neglected. Thus in both approaches the region of small distances proves to be the most important.
This link between the chiral invariance identity and the perturbation theory shows that Eq.(3.1) should be valid when one restricts consideration to the logarithmic approximation. This means that the term ∝ Z 2 α 3 ln mr N should exist not only in the PNC problem, see Eq.(3.33), but in the FNS problem for the s 1/2 and p 1/2 partial waves as well. Eq.(3.1) predicts a linear relation between the three coefficients of these three terms. Ref. [56] demonstrated that the three coefficients mentioned are all equal, in compliance with Eq.(3.1). This fact can be considered either as a consequence of Eq.(3.1), or, alternatively, as its independent verification, being fruitful either way.
The picture of reliability presented above based on the complete accord of all available data for the self-energy corrections in atoms has recently been put to test again by Milstein et al. They have claimed in Ref. [59] that for small values of the nuclear charge Z ≃ 1 the self-energy corrections for the energy shift induced by the FNS in the p 1/2 partial wave are positive and very large, being two orders of magnitude greater than anticipated previously for the hydrogen atom. For the p 3/2 state the proposed enhancement is even stronger, four orders of magnitude. This conclusion, if correct, could be related to the PNC problem for light elements through the equality of [48] mentioned above that binds together the radiative corrections for the PNC and FNS problems. It needs, however, to be pointed out that a questionable approach is applied in [59]. It uses the small-momenta expansion for the vertex operator. When this technique is used for the shortrange potential there arises a contradiction because the short-range potential needs large momenta. Attempts to use a similar line of arguments for calculations of radiative corrections have been made previously, but it was recognized that they strongly overestimate the effect, by orders of magnitude. A necessity to be very careful, to avoid using the small-momenta asymptotic dealing with the short-range potentials, was clearly stated long time ago by Lepage et al [60]. One is forced to presume therefore that conclusions of [59] are not convincing (probably they should be even called preliminary, if one uses the latter term as in Ref. [42]). Fortunately, these results do not influence either the above mentioned good numerical agreement that was achieved for heavy atoms, or the analytical agreement between the two groups within the Zα-expansion that defines the PNC in light elements.
Discussions in this section are almost entirely devoted to analytical methods, the only exception was made for the numerical data available for corrections to the energy shifts due to the FNS. The main reason for this is that direct numerical calculations for the self-energy corrections for the PNC amplitude have not yet been reported. Presumably the difficulties in this approach are significant, though not being experts in this area we would not speculate on the details. We wish to mention, however, that developments in this direction are quite desirable due to several reasons. One of them, purely theoretical, has come into existence only recently. Numerical studies can determine independently the accuracy of the chiral identity (3.1). Knowing the limitations of this relation in the PNC problem, one would be able to estimate the accuracy of other relations of this type that can be derived and used in similar problems in the future.
Summarizing, we discussed above two different methods for calculation of the self-energy corrections, one related to the chiral invariance identity (3.1), the other one based on the Zα-expansion. They give close results proving that the self-energy corrections to the PNC amplitude are large and negative. Most importantly, this fact brings the experimental data for the 6s − 7s PNC amplitude in 133 Cs into agreement with the standard model. (4.1) We see that both the vacuum polarization and the self-energy corrections rise quickly with the nuclear charge, but their opposite signs make the total correction smoother. We find δ with the theoretical error 0.5%. This value includes the correlation and the Breit corrections, as well as the neutron skin corrections (superscript C+B+N), but does not take into account the radiative corrections [83]. The neutron nuclear radius is slightly larger than its proton radius. This provides a possibility for the electron to interact with neutrons in the region outside of the proton core, where the wave function is slightly smaller than inside the nucleus. The corresponding reduction of the PNC amplitude is called the neutron skin correction in PNC. According to estimates of Derevianko [73] the neutron skin correction is approximately −0.2% for Cs. The calculations of Ref. [37] took this effect into account.
Eq.(4.3) is consistent with Q W ( 133 Cs) = −72.21 ± (0.28) expt ± (0.34) theor that was adopted in [41] by taking the average of the results of Refs. [32,33,35], and accepting the theoretical error 0.4% proposed in [34]. This value for the weak charge includes the vacuum polarization correction 0.4%. Extracting it and taking into account −0.2% for the nuclear skin correction one obtains −72.35 that is close to Eq.(4.3).
The radiative corrections derived from results presented in Fig. 8 are −0.54 ± (0.10)%; the error reflects the uncertainty of the self-energy radiative correction. Eq. where Total = Correlations + Breit corrections + Neutron skin correction + QED radiative corrections. The agreement with the standard model given in Eq.(4.2) is good. It is so good, in fact, that we hasten to remind the reader a historical aspect of the problem. The theory has come up with Eq.(4.4) after a turbulent period of research during which the experimental data were widely anticipated to be in contradiction with the standard model. This makes the found "unexpected"agreement more objective.
Let us consider now the case of the thallium atom that was studied experimentally in [12,13,14,15,16,17], the calculations were performed in [74,75], that give close results for the many-electron correlation. The corrections to the result of [74] should include contributions of the Breit interaction -0.98% [72]. As was mentioned, the QED radiative corrections for Tl give −0.70%. If we adopt the neutron skin correction −0.2%, then the total theoretical result reads  The QED corrections for parity nonconservation have recently emerged as an important ingredient in the analyses of modern experimental data. The improvement in the experimental accuracy, particularly for the 6s − 7s PNC amplitude in 133 Cs, and the progress of atomic structure calculations revealed a possible gap between the experiment data and atomic structure calculations on one side and the standard model on the other.
The QED corrections, which embrace the Breit corrections and the radiative corrections, reconcile the experimental data with the standard model. The area has evolved and developed very rapidly. Over the past couple of years, even over a year, the situation has changed dramatically. A short time ago one could be seriously contemplating the possibility to modify the standard model to accommodate a possible deviation of atomic experimental data with the standard model. Presently it has become clear that modifications of the model are to be postponed, one simply needs to calculate everything accurately. Everything here is essential. Different parts of the QED corrections are all very large and have different signs. One must be certain that all parts are included and accounted for properly. That is why we spent some time and space above presenting and comparing all available results for the radiative corrections. The conclusion is that all data for the radiative corrections fit together well. For each possible correction several different methods of calculation were employed by different groups with the same final conclusions. The self-energy corrections, that proved to be by far the most difficult ones, are known presently with an uncertainty of better than 10 %, which is sufficient for the present day experimental accuracy. One can contemplate a significant reduction of this error, if required.
The calculations of the radiative corrections for PNC were based on vast expertise accumulated in other related research areas that include the Lamb shift, the hyperfine interaction, the nuclear finite-size correction. In turn, some methods of calculations that have recently evolved in relation to PNC may find applications in other areas. One of them is the chiral invariance identity Eq.(3.1), that was originally applied to express radiative corrections to PNC in terms of the corrections to energy shifts due to the finite nuclear size. Similar identities can be derived for other related problems. One can also mention a simple convenient expression that gives the variation of the electron wave function at the atomic nucleus in terms of the two first moments of the perturbative potential Eq.(2.3).
In conclusion, large QED radiative corrections to the parity nonconservation amplitude in heavy atoms reconcile the accurate atomic experimental data on parity nonconservation for the 6s − 7s PNC amplitude in 133 Cs of Wood et al [11] with the standard model.  Solid and dotted lines -f s,1/2 and g p,1/2 wave functions that are large for this normalization, thick dashed and thin dashed lines −g s,1/2 and f p,1/2 wave functions that are much smaller, (a) the region of large distances r ∼ rC shows good coincidence of f s,1/2 with g p,1/2 , (b) the region in the vicinity of the nucleus, the nuclear radius is rN ≃ 0.016 rC ≃ 6.1 fm, where −g s,1/2 coincides with f p,1/2 . The shown results illustrate Eq.(3.6) based on chiral invariance.
FIG. 5: The lowest order Feynman diagrams for the QED self-energy and vertex corrections, called e-line corrections in the text, to the PNC matrix element. The thin line describes propagation of a free electron. For each diagram one of the wavy legs shows the Coulomb interaction with the nucleus, another one -the weak PNC interaction with the nucleus (which was shown by a large dot in Fig. 2 ). Each diagram represents all possible Feynman diagrams with the given topological structure.  [48], by the linear approximation Eq.(3.29) of [55], and by the interpolating Eq.(3.31) of [55] (all same as in Fig. 7). The thin dashed line, thick dashed line, and the dashed-dotted line -the e-line corrections to the FNS energy shifts in the linear approximation (3.30), from results of [51], and the interpolating formula (3.32). Note the similarity between the PNC and FNS problems.  [48], of the linear approximation (3.29) of [55], and the interpolating Eq.(3.31) of [55] respectively, as in Fig. 6. Dashed-dotted line, thin dashed line and thick dashed line -results of Milstein et al from [42,56,58] respectively. Note the convergence of the latest results of the two groups, which strongly deviated at the initial stage.