Theoretical description of the neutron beta decay in the standard model at the level of 10^{-5}

In the framework of the Standard Model (SM) a theoretical description of the neutron beta decay is given at the level of 10^{-5}. The neutron lifetime and correlation coefficients of the neutron beta decay for a polarized neutron, a polarized electron and an unpolarized proton are calculated at the account for i) the radiative corrections of order O(\alpha E_e/m_N) ~ 10^{-5} to Sirlin's outer and inner radiative corrections of order O(\alpha/\pi), ii) the corrections of order O(E^2_e/m^2_N) ~ 10^{-5}, caused by weak magnetism and proton recoil, and iii) Wilkinson's corrections of order 10^{-5} (Wilkinson, Nucl. Phys. A377, 474 (1982)). These corrections define the SM background of the theoretical description of the neutron beta decay at the level of 10^{-5}, which is required by experimental searches of interactions beyond the SM with experimental uncertainties of a few parts of 10^{-5}.


I. INTRODUCTION
A contemporary level of sensitivity of about 10 −4 or even better for experimental investigations of the neutron beta decay [1][2][3] with a polarized neutron and unpolarized electron and proton [4][5][6][7] and with a polarized neutron, a polarized electron and an unpolarized proton [8] demand the theoretical description of the neutron beta decay within the Standard Model (SM) [9,10] at the level of 10 −5 . As has been shown in [11][12][13][14] Wilkinson's corrections [15] provide the SM contributions to the neutron lifetime and correlation coefficients of the neutron beta decay of order 10 −5 . Of course, they do not exhaust a complete set of the SM corrections of order 10 −5 .
In Refs. [16] and [17] we have calculated radiative corrections (RC) of order O(αE e /m N ) ∼ 10 −5 , i.e. O(αE e /m N ) RC, where α, E e and m N are the fine-structure constant [10], the electron energy and the nucleon mass, respectively, to Sirlin's outer and inner (see [18]) O(α/π) RC [19,20] (see also [21][22][23]), which are independent of the hadronic structure of the neutron (see [16] ) and induced by the hadronic structure of the neutron (see [17]), respectively. In turn, in [24] we have calculated a complete set of corrections of order O(E 2 e /m 2 N ) ∼ 10 −5 , caused by weak magnetism and proton recoil. Together with Wilkinson's corrections [15] (see also [11][12][13][14]) the corrections, calculated in [16,17,24], define the SM background of the theoretical description of the neutron beta decay at the level of 10 −5 .In this work we supplement this SM theoretical background of the neutron beta decay by the outer O(αE e /m N ) ∼ 10 −5 RC, caused by Sirlin's outer O(α/π) RC and the phase-volume of the neutron beta decay, calculated to next-to-leading order (NLO) in the large nucleon mass m N expansion (see Appendix D).
We would like to notice that recently [29,30] the correlation coefficients T (E e ), S(E e ) and U (E e ) have been investigated theoretically within the SM at the level of 10 −3 by taking into account i) the outer model-independent O(α/π) RC, calculated to leading order (LO) in the large nucleon mass m N expansion, ii) the O(E e /m N ) corrections, caused by weak magnetism and proton recoil, and iii) the corrections, caused by interactions beyond the SM [25], including the contributions of the second class currents or the G-odd correlations (as for G-parity invariance of strong interactions, we refer to the paper by Lee and Yang [42]) by Weinberg [43] (see also [44,45] and [12][13][14]).
The paper is organized as follows. In section II we give the analytical expressions for the correlation function ζ(E e ), which is responsible for the correct electron-energy spectrum of the neutron beta decay and correct value of the neutron lifetime, and the correlation coefficients X(E e ) for X = a, A, B, . . . , T and U , including i) the O(α/π) and O(αE e /m N ) ∼ 10 −5 RC, ii) the O(E e /m N ) and O(E 2 e /m 2 N ) ∼ 10 −5 corrections, caused by weak magnetism and proton recoil, and iii) Wilkinson's corrections of order of a few parts of 10 −5 , which we have calculated in Appendices A, B and C. The results, represented in section II, illustrate the SM theoretical description of the neutron beta decay at level of 10 −5 with a theoretical accuracy of a few parts of 10 −6 . In section III we discuss the obtained results and some problems of the analysis of the contributions of the neutron radiative beta decay. In Appendices A, B, C and D we give detailed calculations of the correlation function ζ(E e ) and correlation coefficients X(E e ) for X = a, A, B, . . . , T and U adduced in section II. In Appendix E we give the analytical expressions of the correlation function ζ(E e ) and correlation coefficients X(E e ) for X = a, A, B, . . . , U as functions of the electron energy E e and the axial coupling constant g A . For the practical applications and numerical analysis the correlation function ζ(E e ) and correlation coefficients X(E e ) for X = a, A, B, . . . , U are programmed in [46]. In Appendix F we give the contributions to the electron-energy and angular distribution of the neutron beta decay with correlations structures, which go beyond the standard correlation structures in Eq.(1) by Jackson et al. [25][26][27] and Ebel and Feldman [28].

II. CORRELATION FUNCTION AND COEFFICIENTS OF THE ELECTRON-ENERGY AND ANGULAR DISTRIBUTION EQ.(1)
In Appendices A, B, C and D at the level of 10 −5 with a theoretical accuracy of a few parts of 10 −6 we give a detailed SM calculation of the correlation function ζ(E e ) and correlation coefficients in Eq.(1), the correlation structures of which are invariant under time-reversal transformation, i.e. T-even. According to our analysis carried out in Appendices A, B, C and D, the correlation function ζ(E e ) and correlation coefficients can be represented in the following form ζ(E e ) = ζ(E e ) RC + ζ(E e ) RC−PhV + ζ(E e ) WP + ζ(E e ) WC , X(E e ) = X(E e ) RC + X(E e ) RC−PhV + X(E e ) WP + X(E e ) WC , where X = a, A, B, K n , Q n , G, H, N, K e , Q e , S, T and U . Then, Y (E e ) RC , Y (E e ) RC−PhV , Y (E e ) WP and Y (E e ) W for Y = ζ, X are i) the sum of the outer O(α/π) RC, calculated to LO in the large nucleon mass m N expansion, and O(αE e /m N ) ∼ 10 −5 RC, which are treated as NLO corrections in the large nucleon mass m N expansion to the outer and inner O(α/π) RC and denoted as Y RC−NLO (see Appendix A), ii) the outer O(αE e /m N ) RC, caused by the outer O(α/π) RC and the phase-volume of the neutron beta decay taken to NLO in the large nucleon mass expansion, iii) the sum of the O(E e /m N ) and O(E 2 e /m 2 N ) ∼ 10 −5 corrections, caused by weak magnetism and proton recoil, and iv) Wilkinson's corrections of order 10 −5 . For the practical applications and numerical analysis the analytical expressions of the correlation function ζ(E e ) and correlation coefficients X(E e ) for X = a, A, B, K n , Q n , G, H, N, K e , Q e , S, T and U are programmed in [46].
In order to illustrate the SM description of the neutron beta decay at the level 10 −5 we represent the correlation function ζ(E e ) and correlation coefficients a(E e ), A(E e ), B(E e ), . . . , U (E e ) with the contributions of the corrections, caused by weak magnetism and proton recoil of order O(E e /m N ) and O(E 2 e /m 2 N ) in Appendix B, and Wilkinson's corrections in Appendix C, as functions of the variable E e /E 0 . According to our calculation in [46], we get where the numerical coefficients are calculated at the axial coupling constant g A = 1.2764 [5]. The contributions of the corrections of the O(αE e /m N ) RC are plotted in [46]. Then, the analytical expressions of ζ(E e ) RC and X(E e ) RC for X = a, A, . . . , T and U are given in Appendix A (see Eq.(A-20)) and in [46]. At α = 0 the correlation function ζ(E e ) RC and the correlation coefficients X(E e ) RC for X = a, A, . . . , U reduce to their values, calculated to LO in the large nucleon mass m N expansion (see Appendix A). The outer RC ζ(E e ) RC−PhV and X(E e ) RC−PhV are calculated in Appendix D (see Eq.(D-10)).
In addition to the correlation function ζ(E e ) and the correlation coefficients in Eq.(4) we give the correlation coefficient [15] that measures the electron (beta) asymmetry of the neutron beta decay [5]: -20)). The correlation function ζ(E e ) and the correlation coefficients X(E e ) for X = a, A, B, . . . , U in Eq.(4) and A (β) (E e ) in Eq.(5) describe the neutron beta decay for a polarized neutron, a polarized electron and an unpolarized proton at the level of 10 −5 in the framework of the SM with a theoretical accuracy of a few parts of 10 −6 .

III. DISCUSSION
We have given a SM theoretical description of the neutron beta decay for a polarized neutron, a polarized electron and an unpolarized proton at the level of 10 −5 with a theoretical accuracy of a few parts of 10 −6 . To the wellknown O(α/π) RC [19][20][21][22][23] (see also [11,12,14,29,30]) and O(E e /m N ) corrections [47] and [15,22,23] (see also [11,12,14,29,30]) we have added i) the inner O(αE e /m N ) ∼ 10 −5 RC [16,17], which are treated as NLO corrections in the large nucleon mass m N expansion to Sirlin's outer and inner O(α/π) RC, calculated to LO in the large nucleon mass m N , expansion, ii) the outer O(αE e /m N ) ∼ 10 −5 RC, induced by Sirlin's outer O(α/π) RC and the phase-volume of the neutron beta decay, calculated to NLO in the large nucleon mass m N expansion, iii) the O(E 2 e /m 2 N ) ∼ 10 −5 corrections [24], caused by weak magnetism and proton recoil, and iv) Wilkinson's corrections [15] (see also [11,12,14]) of order 10 −5 . As has been shown in Eq.(4), all of these corrections define the SM background of the theoretical description of the neutron beta decay at the level of 10 −5 with a theoretical accuracy of about a few parts of 10 −6 [46].
Having accepted the value of the axial coupling constant g A = 1.2764 [5,6], the correlation function ζ(E e ) and correlation coefficients, given in Eq.(4) and Eq.(5), can be used as the SM theoretical background of the neutron beta decay for experimental searches of contributions of interactions beyond the SM with experimental uncertainties of a few parts of 10 −5 [4,7,8] (see also [12,24]). Because of Wilkinson's corrections, induced by the proton recoil in the electron-proton final-state Coulomb interaction (see Eq.(C-8) in Appendix C), and the outer O(αE e /m N ) RC (see Eq.(D-10) in Appendix D) the correlation function ζ(E e ) and correlation coefficients in Eq.(4) and Eq.(5) are well defined in the experimental electron-energy region 0.811 MeV ≤ E e ≤ 1.211 MeV [5].
In Appendix E we give the analytical expressions for the correlation function ζ(E e ) and correlation coefficients X(E e ) for X = a, A, B, . . . , U as functions of the electron energy E e and the axial coupling constant g A . These expressions can be used as a SM theoretical background for processing experimental data on the neutron lifetime, the electron-antineutrino angular correlations, and electron and antineutrino asymmetries with experimental uncertainties of about a few parts of 10 −5 . Such a SM theoretical background and experimental data, obtained with experimental uncertainties of about a few parts of 10 −5 , should allow to improve the currently available experimental value of the axial coupling constant g A [5,6]. They can be also used for searches of contributions of interactions beyond the SM in experiments with polarized neutrons and electrons [8].
We have also to emphasize that for the correct description of the neutron lifetime one has to add the inner radiative corrections ∆ V R and ∆ A R of order O(α/π), defined by the Feynman γW -box diagrams, to the rates of the neutron beta decay and superallowed nuclear beta decays, which have been calculated to LO in the large nucleon mass m N expansion in [48][49][50][51][52][53][54][55][56]. These corrections are very important for the correct extraction of the value of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element V ud .
Finally we would like to discuss the problem of the removal of infrared divergences for the calculation of the outer RC in the neutron beta decay. Since the virtual photon exchange leads to the dependence of the amplitude of the neutron beta decay on the infrared cut-off µ, which is an infinitesimal photon mass µ in the covariant regularization [19,[57][58][59][60][61], one has to take into account the contribution of the neutron radiative beta decay [19,[57][58][59][60][61]. For this aim the energy and angular distribution of the rate of the neutron radiative decay should be summed with the energy and angular distribution of the rate of the neutron beta decay [19,[57][58][59][60][61]. In case of the investigation of the electronenergy and angular distribution of the neutron beta decay (see, for example, Eq.(1)), the standard procedure for the calculation of distributions for both the neutron beta decay and neutron radiative beta decay is to integrate, first, over the proton 3-momenta and then over the energy of the antineutrino [19,[21][22][23][57][58][59][60][61][62][63][64][65][66][67][68][69] (see also [11-14, 29, 30, 40]). In the rest frame of the neutron and after the integration of the proton 3-momentum, the latter appears in the distributions in the form k p = − k e − kν e imposed by momentum conservation, where k e and kν e are 3-momenta of the electron and antineutrino, respectively. In this case for the calculation of the electron-energy and angular distribution of the neutron radiative beta decay the integration over directions of the photon 3-momentum takes into account correlations of the photon 3-momentum with 3-momenta of the electron and antineutrino and implicitly with the proton 3-momentum (for the details of these calculations we refer to [11,12,14,29,30]). The electron-energy and angular distribution Eq.(1) is usually used for the measurements of the electron (beta) asymmetry, which is characterized by the correlation coefficient A (β) (E e ) [5]. In these measurements the electron asymmetry defines the asymmetry of the emission of decay electrons relative to the neutron spin polarization into solid angles related by the polar angle θ → π − θ [5,[70][71][72][73] (for the details of the calculation we refer to [11]). The electron-energy and angular distribution Eq.(1) can be also applied to the measurement of the antineutrino asymmetry, which is practically defined by the correlation coefficient B(E e ). Formally, the antineutrino asymmetry B exp (E e ) defines the asymmetry of the emission of the antineutrino relative to the neutron spin polarization into solid angles related by the polar angle θ → π − θ. However, in experiments [7,[74][75][76] because of the electroneutrality of the antineutrino such an asymmetry is equivalent to the asymmetry of the emission of the electron-proton pairs into the solid angles related by the polar angle θ → π − θ. For the first time, the asymmetry B exp (E e ) has been calculated by Glück et al. [66,69] in terms of the correlation coefficients a(E e ), A(E e ) and B(E e ) (for the details of the calculation we refer to [11]).
In turn, for the measurements of the electron-antineutrino angular correlations [77][78][79] and the proton recoil asymmetry, defined by the correlation coefficient C [80], one has to use the electron-proton-energy and angular distribution (or the proton-energy and angular distribution) [65][66][67][68][69]. For the calculation of the electron-proton-energy and angular distribution one has to integrate over the antineutrino 3-momentum. Then, having integrated over the electron energy one obtains the proton-energy and angular distribution (for the details of the calculation we refer to [11]). The same procedure should be used for the neutron radiative beta decay [11]. In case of the neutron radiative beta decay for the calculation of the electron-proton-photon-energy and angular distribution one deals with direct photon-proton correlations [65][66][67][68][69]. However, as has been shown in [81], the contributions of these correlations do not destroy the radiative corrections, defined by the functions (α/π)ḡ n (E e ) and (α/π) f n (E e ). As has been found in [81], the contributions of the photon-proton correlations in the neutron radiative beta decay to the proton recoil asymmetry C are of order of 10 −4 . They make the contributions of the radiative corrections to the proton recoil asymmetry C symmetric with respect to a change A 0 ↔ B 0 , where A 0 and B 0 are the correlation coefficients A(E e ) and B(E e ) calculated to LO in the large nucleon mass m N expansion [1,2]. They depend on the axial coupling constant only (see also Eq.(A-16) in Appendix A). We are planning to carry out the analysis of the electron-proton-energy and angular distributions of the neutron beta decay at the SM theoretical level of about 10 −5 in our forthcoming publication.
We would like to note that for practical applications and numerical analysis of the correlation function ζ(E e ) and correlation coefficients a(E e ), A(E e ), B(E e ), . . . , U (E e ) we have programmed their analytical expressions in [46]. We have carried out numerical calculations for the axial coupling constant g A = 1.2764 [5] and plotted the O(αE e /m N ) corrections.
Appendix A: The electron-energy and angular distribution of the neutron beta decay with the account for the radiative corrections of order O(αEe/mN ) According to [11,29,30], the electron-energy and angular distribution of the neutron beta decay for a polarized neutron, a polarized electron and an unpolarized proton in Eq.(1) is determined by where we sum over polarizations of the massive fermions. The function Φ n ( k e , kν) defines the contribution of the phase-volume of the neutron beta decay [11,24]. It is equal to [11,24] Φ n ( k e , kν) taken to NLO in the large nucleon mass m N expansion. The amplitude of the neutron beta decay, taking into account the radiative corrections of order O(α/π) and O(αE e /m N ), is defined by [11,16,17] The functions U j for j = 1, 2, . . . , 8 are given by where β = k e /E e = 1 − m 2 e /E 2 e is the electron velocity, the function f β − c (E e , µ), where µ is a covariant infrared cut-off having a meaning of a photon mass [19,57,58], was calculated by Sirlin [19] to LO in the large nucleon m N expansion (for the details of the calculation of the function f β − c (E e , µ) we refer to [11]). It defines so-called outer model-independent radiative corrections [18]. Then, the functions and h A (E e ) determine the inner radiative corrections dependent on the axial coupling constant g A to Sirlin's outer radiative corrections O(α/π), calculated to NLO in the large nucleon mass m N expansion in [16]. In turn, the functionsḡ st (E e ) andf st (E e ) describe the inner radiative corrections, caused by the hadronic structure of the neutron and calculated to NLO in the large nucleon m N mass expansion in [17] as NLO corrections to Sirlin's inner radiative corrections O(α/π), caused by the hadronic structure of the neutron and calculated to LO in the large nucleon mass expansion [19,20]. The analytical expressions of these functions are equal to [11,16,17] where Li 2 (z) is the PolyLogarithmic function [83], and For the subsequent analysis of radiative corrections we follow [29] (see also [11]) and represent the function f β − c (E e , µ) as follows βγ (E e , µ), where 2ḡ n (E e ) is Sirlin's function, defining the outer radiative corrections of order O(α/π) to the neutron lifetime [19]. The function g (1) βγ (E e , µ) can be removed by the contribution of the neutron radiative beta decay n → p + e − +ν e + γ with a real photon γ, which should be added, according to Berman [57] and Kinoshita and Sirlin [58] (see also Sirlin [19]), for the removal of the dependence of the neutron lifetime on the infrared cut-off. For the detailed calculation of the function g (1) βγ (E e , µ) and as well as the function g (1) βγ (E e , ω min ), describing the contributions of the neutron radiative beta decay n → p + e − +ν e + γ to the neutron lifetime, where ω min is a non-covariant infrared cut-off having a meaning of the photon-energy threshold of the detector, we refer to [11]. We adduce here the analytical expressions of these functions for completeness (see [11]) The hermitian conjugate amplitude of the neutron beta decay Eq.(A-3) is equal to We use this amplitude for the calculation of the square of absolute value of the amplitude Eq.(A-3), summed over polarizations of massive particles. It is equal to pol.
Having calcuated the traces over leptonic degrees of freedom we arrive at the expression pol.
In terms of the irreducible correlation structures the r.h.s. of Eq.(A-13) is given by pol.
where the index RC means that these corrections are defined by the outer radiative corrections of order O(α/π), calculated to LO in the large nucleon mass expansion [12,14,19,21], and the inner radiative corrections of order O(αE e /m N ) [16,17]. The correlation function ζ(E e ) RC and the correlation coefficients ζ(E e ) RC X(E e ) RC for X = b, a, A, B, . . . , T and U are equal to where the correlation coefficients a 0 , A 0 and B 0 depend only on the axial coupling constant g A [1, 2] (see also [29]) Using the definitions of the functions U j for j = 1, 2, . . .
βγ (E e , µ) + βγ (E e , µ) + βγ (E e , µ) + βγ (E e , µ) + βγ (E e , µ) βγ (E e , µ) βγ (E e , µ) − Taking into account the contribution of the neutron radiative beta decay [11,12,14,29,30] we obtain the correlation function ζ(E e ) RC and correlation coefficients ζ(E e ) RC X(E e ) RC for X = b, a, A, B, . . . , T and U in Eq.(A-17) in the form The functionsḡ n (E e ) and f n (E e ) have been calculated by Sirlin [19] and Shann [21] (see also [11,12,29]), whereas the function h n (E e ) and h (2) n (E e ) have been calculated in [12,14]. They are equal tō The radiative corrections of order O(α/π) and O(αE e /m N ) to the neutron lifetime and the correlation coefficients of the electron-energy and angular distribution Eq.(1) are given by The correlation function ζ(E e ) RC and correlation coefficients X(E e ) RC contain a complete set of outer radiative corrections of order O(α/π) [12,14,19,21], calculated to LO in the large nucleon mass m N expansion, and radiative corrections of order O(αE e /m N ) [16,17], obtained as NLO corrections in the large nucleon mass m N expansion to Sirlin's outer and inner radiative corrections, calculated to LO in the large nucleon mass m N expansion. For α = 0 the correlation function ζ(E e ) RC and the correlation coefficients X(E e ) RC acquire their expressions, calculated to LO in the large nucleon mass m N expansion [1,2](see also [29]). We have plotted the Y (E e ) RC−NLO corrections for Y = ζ, a, A, B, . . . U and A (β) (E e ) (see Appendix E) in [46].
Appendix B: The corrections of order O(Ee/mN ) and O(E 2 e /m 2 N ), caused by weak magnetism and proton recoil, to next-to-leading and next-to-next-to-leading order in the large nucleon mass mN expansion The corrections to the structure function ζ(E e ) WP and the correlation coefficients X(E e ) WP for X = a, A, B, . . . , T and U , caused by weak magnetism and proton recoil, we define as follows The corrections ζ(E e ) NLO and X(E e ) NLO , which are in principle of order 10 −3 [11,12,14] have been calculated in [47] and [11,12,14,15,22,23,29,30] (see also [24]). The analytical expressions of these corrections are given by For the definition of the O(E 2 e /m 2 N ) corrections it is convenient to adduce the following expressions, calculated in [11,12,14]: whereK n (E e ) NLO = K n (E e ) NLO andQ n (E e ) NLO = Q n (E e ) NLO . Using the results obtained in [24], we get the following analytical expressions for the N 2 LO corrections ζ(E e ) N 2 LO and X(E e ) N 2 LO for X(E e ) = a(E e ), A(E e ), . . . , U (E e ). They are given by The terms, proportional to E 2 0 /M 2 V and E 2 0 /M 2 A are induced by the vector and axial-vector form factors of the neutron beta decay [84] (see also [24]). The slope-parameters M V and M A are related to the charge radius of the proton r p = 0.841 fm [33] and the axial radius r A = 0.635 fm of the nucleon [85] (see also [86] [46]. The numerical analysis of relative contributions has been carried out for the axial coupling constant g A = 1.2764 [5], the value of which agrees well with the recommended value of the axial coupling constant obtained by means of the global analysis of the experimental data on the axial coupling constant by Czarnecki et al. [6]. Appendix C: Wilkinson's corrections of order 10 −5 to the neutron beta decay According to Wilkinson [15], the corrections, additional to those calculated in Appendices A and B, should be caused by i) the proton recoil in the electron-proton final-state Coulomb interaction, ii) the finite proton radius, iii) the proton-lepton convolution and iv) the higher-order outer radiative corrections. These corrections to the neutron lifetime and the correlation coefficients a(E e ), A(E e ), . . . , K e (E e ) have been calculated in [11,12,14]. The contributions of the proton recoil, caused by the phase-volume of the neutron beta decay proportional to 1/m N and 1/m 2 N , which are defined in [15] (see also [87]) by the function S(E e , E 0 , m N ), we have taken into account for the calculation of the NLO and N 2 LO corrections in the large nucleon mass m N expansion induced by weak magnetism and proton recoil (see Appendix B).

Wilkinson's corrections, induced by proton recoil in the Coulomb electron-proton final-state interaction
For the calculation of the contribution of the proton recoil in the Fermi function we replace the electron velocity β by a velocity of a relative motion of the electron-proton pair [11]. To NLO in the large nucleon mass m N expansion a velocity of a relative motion of the electron-proton pair is defined by To LO in the large nucleon mass m N expansion the second term in r.h.s. of Eq.(C-1) vanishes, and a velocity of a relative motion of the electron-proton pair reduces to an electron velocity. As has been shown in [11] the Fermi function F (E e , Z = 1) with a replacement β → v rel. , caused by a relative motion of the electron-proton pair in the final state of the neutron beta decay, undergoes the following change (see Appendix H of Ref. [11]) where we have taken into account the NLO terms in the large nucleon mass m N expansion. The corrections, caused by the proton recoil in the electron-proton final-state interactions to the neutron lifetime ζ(E e ) WF and the correlation coefficients X(E e ) WF for X = a, A, B, . . . , T and U are equal to In comparison with the result, obtained in [12], an additional term in Q e (E e ) WF appears because of the contribution off the correlation coefficient T (E e ). The contributions of Wilkinson's corrections, caused by proton recoil in the electron-proton final-state Coulomb interaction, to the electron-energy and angular distribution of the neutron beta decay with correlation structures beyond the standard correlation structures by Jackson et al. [25] and Ebel and Feldman [28] (see Eq.(1)) are given in Appendix F. We would like to emphasize that Wilkinson's corrections, induced by proton recoil in the final-state Coulomb electron-proton interaction are well defined in the experimental electronenergy region 0.811 MeV ≤ E e ≤ 1.211 MeV [5].
Wilkinson's corrections, induced by i) the finite proton-radius rp, ii) the lepton-nucleon convolution and iii) the higher-order outer radiative corrections The corrections under consideration, caused by i) the finite proton-radius r p , ii) the lepton-nucleon convolution and iii) the higher-order outer radiative corrections, are defined by the functions L(E e , Z = 1), C(E e , Z = 1) and J(Z = 1), respectively [15] (see also [12]). According to Wilkinson [15], the contribution of J(Z = 1) is equal to J(Z = 1) = 1 + 3.92 × 10 −4 (see also [12]). The corrections L(E e , Z = 1) and C(E e , Z = 1), adapted for the neutron beta decay, are determined by [12]: For the calculation of the numerical values of the constant terms and coefficients in front of the powers of E e /E 0 we use r p = 0.841 fm [33]. Following Wilkinson [15] we define Wilkinson's correction, caused by i) the finite proton-radius r p , ii) the lepton-nucleon convolution and iii) the higher-order outer radiative corrections, as follows The contributions of Wilkinson's corrections, caused by i) the finite proton-radius r p , ii) the lepton-nucleon convolution and iii) the higher-order outer radiative corrections, to the correlation coefficients are equal to Now we may obtain the total contributions of Wilkinson's corrections to the correlation function ζ(E e ) and the correlation coefficients. We would like to emphasize that the correction ζ(E e ) WR does not depend practically on the value of the axial coupling constant g A .

Wilkinson's corrections to the neutron beta decay
Summing the contributions in Eqs. (C-3), (C-6) and (C-7) we define total Wilkinson's corrections to the neutron lifetime and the correlation coefficients of the neutron beta decay where the ellipsis denotes the contributions, which are not important for the aim of this Appendix. In terms of irreducible correlation structures and using Eq.(A-7) we transcribe the r.h.s. of Eq.(D-1) into the form pol.
βγ (E e , µ) βγ (E e , µ) βγ (E e , µ) ξ n · ξ e − A 0 1 + βγ (E e , µ) βγ (E e , µ) ( ξ e · k e )( k e · kν) (E e + m e )E e Eν βγ (E e , µ) The electron-energy and angular distribution of the neutron beta decay, taking into account the radiative corrections of order O(α/π) and the NLO corrections in the large nucleon mass m N expansion, induced by the phase-volume, is given by βγ (E e , µ) + a 0 1 + βγ (E e , µ) βγ (E e , µ) ( ξ n · k e )( k e · ξ e ) (E e + m e )E e − a 0 1 + α π ḡ n (E e ) + (1 βγ (E e , µ) ( ξ e · k e )( k e · kν) βγ (E e , µ) In order to remove the dependence of the electron-energy and angular distribution of the neutron beta decay on the infrared cut-off µ we have to take into account the contribution of the neutron radiative beta decay n → p+e − +ν e +γ, where γ is a real photon. It is well-known [57][58][59][60][61] (see also [19,21] and [11,12,14]) that the contribution of the neutron radiative beta decay is extremely needed for cancellation of the infrared divergences in the radiative corrections of order O(α/π), caused by one-virtual photon exchanges. For the removal of the infrared dependence we use the following electron-photon-energy and angular distribution, calculated in [30] (see Eq.B-14) in Ref. [30]): where Φ nγ ( k e , kν e , ω) is the contribution of the phase-volume of the neutron radiative beta decay [82] Φ nγ ( k e , kν e , ω) The contribution of the phase-volume Eq.(D-6) is the rest of the expression, calculated to NLO in the large nucleon mass m N expansion and the integration over the directions of the photon 3-momentum [82]. As has been shown in [82], the contributions of the terms O(ω/m N ) are of order of 10 −6 and even smaller. So, the O(αE e /m N ) corrections can be induced only by the second term in Eq.(D-6).
One can see that in the electron-energy region m e ≤ E e < E 0 the functions 3(α/π)(E e /m N )ḡ n (E e ), 3(α/π)(E e /m N )h (1) n (E e ) and 3(α/π)(E e /m N )h (2) n (E e ) are of order of a few parts of 10 −5 , whereas the function 3(α/π)(E e /m N )f n (E e ) is of order of a few parts of 10 −6 . However, the functions 3(α/π)(E e /m N )h n (E e ) and 3(α/π)(E e /m N )X 0 h (j) n (E e ) become of order of a few parts of 10 −6 . Hence, the outer radiative corrections of order O(αE e /m N ) are defined by the function 3(α/π)(E e /m N )ḡ n (E e ) only.
As result, the correlation function ζ(E e ) and the correlation coefficients a(E e ), Q n (E e ), Q e (E e ) and K e (E e ) acquire the following outer or model-independent O(αE e /m N ) radiative corrections : (D-10) These radiative corrections are of order of a few parts of 10 −5 in the experimental electron-energy region 0.811 MeV ≤ E e ≤ 1.211 MeV [5]. They are plotted in [46]. The analytical expression of the functionḡ n (E e ) is given in Eq.(A- 19). We have to notice that there is the contribution, proportional to 3(α/π)(E e /m N )ḡ n (E e ), to the electron-energy and angular distribution of the neutron beta decay with the correlation structure beyond the standard correlation structures in Eq.(1) (see the last term in Eq. (F-2)). In this Appendix we give the analytical expressions for the correlation function ζ(E e ) and the correlation coefficients X(E e ) for X = a, A, B, . . . , U and also for the correlation coefficient A (β) (E e ) = A(E e ) + 1 3 Q n (E e ) as functions of the electron energy E e and the axial coupling constant g A . The correlation function ζ(E e ) and correlation coefficients are calculated with a theoretical accuracy of about 10 −6 .
For the correlation function ζ(E e ) we obtain the following expression Hereḡ n (E e ) is Sirlin's function [19] (see also Eq.(A-19. Then, the term proportional to E 0 /m N defines the well-known corrections O(E e /m N ), caused by weak magnetism and proton recoil (see, for example, [11,23] The terms, proportional to E 2 0 /M 2 V and E 2 0 /M 2 A , appear from the contributions of the vector and axial-vector formfactors of the neutron beta decay. For numerical analysis we use M V = 813 MeV and M A = 1077 MeV [24]. The first three terms in ζ(E e ) WC do not depend practically on the axial coupling constant g A .
Of course, the correlation function ζ(E e ) in Eq.(E-1) should be supplemented by the inner radiative corrections ∆ V R and ∆ A R of order O(α/π), caused by the Feynman γW − -box diagrams and calculated in [48][49][50][51][52][53][54][55][56]). For the correlation coefficient a(E e ) we obtain the following expression a(E e ) = a 0 1 + α π f n (E e ) + 1 where the function f n (E e ) has been calculated by Shann [21] (see also Eq.(A-19) and [11,12]) and the coefficients a 1 , a 2 and a 3 are equal to [11] a 1 = 4g A (g 2 A + 1) g A + (κ + 1) , The corrections a(E e ) RC−NLO , a(E e ) RC−PhV , a(E e ) N 2 L0 and a(E e ) WC , defined by the O(αE e /m N ) inner and outer radiative corrections, the O(E 2 e /m 2 N ) corrections, caused by weak magnetism and proton recoil, and Wilkinson's corrections, respectively, are given by For the correlation coefficient A(E e ) we obtain the following expression where the function f n (E e ) has been calculated by Shann [21] (see also Eq.(A-19)) and the coefficients A 1 , A 2 and A 3 are given by [11] The corrections A(E e ) RC−NLO , A(E e ) N 2 L0 and A(E e ) WC , defined by the O(αE e /m N ) inner radiative corrections, the O(E 2 e /m 2 N ) corrections, caused by weak magnetism and proton recoil, and Wilkinson's corrections, respectively, are equal to For the correlation coefficient B(E e ) we obtain the following expression where the coefficients B 1 , B 2 and B 3 are given by [11]   The coefficients of Wilkinson's term in the parentheses do not practically depend on the axial coupling constant g A . For the correlation coefficient K n (E e ) we obtain the following expressions K n (E e ) = 1 1 + 3g 2 A E e m N 5g 2 A + (κ − 4) g A − (κ + 1) + K n (E e ) RC + K n (E e ) N 2 LO + K n (E e ) WC . (E-12) The corrections K n (E e ) RC−NLO , K n (E e ) N 2 L0 and K n (E e ) WC , defined by the O(αE e /m N ) inner radiative corrections, the O(E 2 e /m 2 N ) corrections, caused by weak magnetism and proton recoil, and Wilkinson's corrections, respectively, are equal to (E-13) For the correlation coefficient Q n (E e ) we obtain the following expression Q n (E e ) = 1 1 + 3g 2 A E 0 m N g 2 A + (κ + 2)g A + (κ + 1) − 7g 2 A + (κ + 8) g A + (κ + 1) E e E 0 + Q n (E e ) RC−NLO + Q n (E e ) RC−PhV + Q n (E e ) N 2 LO + Q n (E e ) WC , (E-14) where the corrections Q n (E e ) RC−NLO , Q n (E e ) RC−PhV , Q n (E e ) N 2 L0 and Q n (E e ) WC , defined by the O(αE e /m N ) inner and outer radiative corrections, the O(E 2 e /m 2 N ) corrections, caused by weak magnetism and proton recoil, and Wilkinson's corrections, respectively, are equal to Q n (E e ) RC−NLO = 0, Q n (E e ) RC−PhV = −3 B 0 α π E e m Nḡ n (E e ), Now we are able to give the analytical expression for the correlation coefficient A (β) (E e ) defined by [15] A (β) (E e ) = A(E e ) + 1 3 Q n (E e ). (E-16) This correlation coefficient is responsible for the electron (beta) asymmetry in the neutron beta decay [5] (see also [11]). The correlation coefficient A (β) (E e ) we give in the following form [15] (see also [11]) The function f n (E e ) is given in Eq. (A-19). The coefficients A are equal to [15] (see also [11]) The functions ζ(E e ) NLO , a(E e ) NLO , A(E e ) NLO , B(E e ) NLO and Q n (E e ) NLO are given in Eq.(B-2). For the experimental analysis of the antineutrino asymmetry in the neutron beta decay one has to use Eqs. (27) and (28) in Ref. [11] and the correlation coefficients a(E e ), A(E e ), B(E e ), K n (E e ) and Q n (E e ) given in this Appendix. For the account for the contribution of the Fierz interference term b in the antineutrino asymmetry one may use Eqs. (19) and (20) in Ref. [40], where the correlation coefficients X(E e ) are replaced by X(E e )/(1 + bm e /E e ) for X(E e ) = a(E e ), A(E e ), B(E e ), K n (E e ) and Q n (E e ), respectively.