Precision analysis of pseudoscalar interactions in neutron beta decays

We analyze the contributions of the one-pion-pole (OPP) exchange, caused by strong low-energy interactions, and the pseudoscalar interaction beyond the Standard Model (BSM) to the correlation coefficients of the neutron beta-decays for polarized neutrons, polarized electrons and unpolarized protons. The strength of contributions of pseudoscalar interactions is defined by the effective coupling constant C_ps = C^(OPP)_ps + C^(BSM)_ps. We show that the contribution of the OPP exchange is of order C^(OPP)_ps ~ - 10^(-5). The effective coupling constant C^(BSM)_ps of the pseudoscalar interaction BSM can be in principle complex. Using the results, obtained by Gonzalez-Alonso et al.( Prog. Part. Nucl. Phys. 104, 165 (2019)) we find that the values of the real and imaginary parts of the effective coupling constant C^(BSM)_ps are constrained by - 3.5x10^{-5}<ReC^(BSM)_ps<0 and ImC^(BSM)_ps<- 2.3x10^(-5), respectively. The obtained results can be used as a theoretical background for experimental searches of contributions of interactions BSM in asymmetries of the neutron beta-decays with a polarized neutron, a polarized electron and an unpolarized proton at the level of accuracy of a few parts of $10^{-5}$ or even better (Abele, Hyperfine Interact.237, 155 (2016)).

We analyze the contributions of the one-pion-pole (OPP) exchange, caused by strong low-energy interactions, and the pseudoscalar interaction beyond the Standard Model (BSM) to the correlation coefficients of the neutron β − -decays for polarized neutrons, polarized electrons and unpolarized protons. The strength of contributions of pseudoscalar interactions is defined by the effective coupling constant Cps = C  [13], where α, E e and M are the fine-structure constant [14], an electron energy and the nucleon mass, respectively. Such a SM theoretical background has allowed to make steps forwards investigations of contributions of interactions beyond the SM (BSM) of order 10 −4 or even smaller [40]. The analysis of interactions beyond the V − A effective theory of weak interactions [15][16][17][18][19] (see also [20,21]) in the neutron β − -decays with different polarizations of massive fermions has a long history and started in 50th of the 20th century and is continuing at present time [22]- [34] (see also [8,9,11]). The most general form of the Lagrangian of interactions BSM has been written in [22]- [27], including non-derivative vectorψ p γ µ ψ n , axial-vectorψ p γ µ γ 5 ψ n , scalarψ p ψ n , pseudoscalarψ p γ 5 ψ n and tensor ψ p σ µν ψ n nucleon currents coupled to corresponding lepton currents in the form of local nucleon-lepton current-current interactions, where {1, γ µ , γ µ γ 5 , γ 5 , σ µν } are the Dirac matrices [35], With respect to G-parity transformations [36], i.e. G = C e iπI2 , where C and I 2 are the charge conjugation and isospin operators [35], the vector, axial-vector, pseudoscalar and tensor nucleon currents are G-even and the scalar nucleon current is G-odd. According to the G-transformation properties of hadronic currents, Weinberg divided hadronic currents into two classes, which are G-even first class and G-odd second class currents [37], respectively. Thus, following Weinberg's classification the non-derivative vector, axial-vector, pseudoscalar and tensor nucleon currents in the interactions BSM, introduced in [22]- [27], are the first class currents, whereas the non-derivative scalar nucleon current is the second class one (see also [38]). The analysis of superallowed 0 + → 0 + nuclear beta transitions by Hardy and Towner [39] and González-Alonso proportional to the effective coupling constants C ′ ps and C ′′ ps were never calculated in literature. In the Appendix we give a detailed calculation of the contributions of pseudoscalar interactions caused by the OPP exchange and BSM to the correlation coefficients of the neutron β − -decays for a polarized neutron, a polarized electron and an unpolarized proton, completing the analysis of contributions of interactions BSM to the correlation coefficients of the neutron β − -decays carried out in [10][11][12].

II. AMPLITUDE OF THE NEUTRON β − -DECAY WITH CONTRIBUTIONS OF OPP EXCHANGE AND PSEUDOSCALAR INTERACTION BSM
Since the expected order of contributions of pseudoscalar interactions of about 10 −5 , we take them into account in the linear approximation additively to the corrections of order 10 −4 − 10 −3 calculated in [1]- [34]. In such an approximation and following [9,11,12] the amplitude of the neutron β − -decay we take in the form where G F and V ud are the Fermi couping constant and the Cabibbo-Kobayashi-Maskawa (CKM) matrix element [14]. Then, p( k p , σ p )|J (+) µ (0)|n( k n , σ n ) is the matrix element of the charged hadronic current J (+) µ (0) are the charged vector and axial-vector hadronic currents [15,18,19]. The fermions in the initial and final states are described by Dirac bispinor wave functions u n , u p , u e and vν of free fermions [9,62]. In the second term of Eq.(1) we take into account the contribution of the pseudoscalar interaction BSM [22]- [27] with two complex phenomenological coupling constants C P andC P in the notation of [9,11,12].
For the analysis of contributions of pseudoscalar interactions to the neutron β − -decays for a polarized neutron, a polarized electron and an unpolarized proton we define the matrix element p( k p , σ p )|J where λ is the axial coupling constant with recent experimental value λ = −1.27641 (45) stat. (33) syst. [41]. The first term in Eq.(1) is written in agreement with the standard V − A effective theory of weak interactions [15,18,19] (see also [20,21]). The term proportional to q µ γ 5 defines the contribution of the OPP exchange, caused by strong lowenergy interactions (see also [18]) with the π − -meson mass m π = 139.57061(24) MeV [14] and q = k p − k n = −k e − kν is a 4-momentum transfer. The OPP contribution is required by conservation of the charged hadronic axial-vector current in the chiral limit m π → 0 [18].
In the more general form the matrix element of the hadronic axial-vector current can be taken in the form accepted in the HBχPT [58][59][60]. This gives where G A (q 2 ) and G P (q 2 ) are the axial-vector form factor and the induced pseudoscalar form factor, respectively, at 0 ≤ q 2 ≤ ∆ 2 for the neutron β − -decay with ∆ = m n − m p . The invariant 4-momentum transfer squared q 2 vanishes, i.e. q 2 = 0, at the kinetic energy of the proton T p = E p − m p = ∆ 2 /2m n . In the chiral limit m π → 0 because of conservation of the charged hadronic axial-vector current [18] the form factors G A (q 2 ) and G P (q 2 ) are related by G P (q 2 ) = −(4M 2 /q 2 )G A (q 2 ). In turn, for a finite pion mass the pseudoscalar form factor G P (q 2 ) has been calculated in the two-loop approximation within HBχPT by Kaiser [60]. A precision analysis of the induced pseudoscalar form factor in the proton weak interactions has been also carried out by Gorringe and Fearing [61].
A. Pseudoscalar interaction BSM as induced by corrections to the pseudoscalar form factor, caused by strong low-energy interactions According to [58], the axial-vector form factor G A (q 2 ) can be rather good parameterized by a dipole form (see also [63]) where g A = −λ is the axial-coupling constant, and M A is the cut-off mass related to the mean square axial radius of the nucleon r 2 A as r 2 A = 12/M 2 A = 0.403(29) fm 2 with M A = 1.077(39) GeV extracted from charged pion electroproduction experiments [63]. In turn, the cut-off mass M A = 1.026 (17) GeV extracted from (quasi)elastic neutrino and antineutrino scattering experiments [63] gives r 2 A = 12/M 2 A = 0.440(16) fm 2 . In the approximation Eq.(4) the pseudoscalar form factor G P (q 2 ) acquires the following form [58] (see also [61]) where the correction to the OPP exchange is the Adler-Dothan-Wolfenstein (ADW) term [56,57]. The ADW-term induces the BSM-like pseudoscalar interaction with the coupling constants According to Eq. (11), this gives the contribution to the correlation coefficients of the neutron β − -decays equal to Re C Using the results, obtained by Kaiser [60] (see Eq.(7) of Ref. [60]) in the two-loop approximation in the HBχPT, the induced BSM-like pseudoscalar coupling constants are equal to where f π = 92.4 MeV is the charged pion leptonic (or PCAC) constant [58,60]. Since |ζ 0 | ∼ 1 [60], we get |C P | = |C P | ∼ 4.1 × 10 −5 . The contribution of C to the coupling constant Re C (BSM) ps (see Eq. (11)) is of order |Re C (BSM) ps | ∼ 9.6 × 10 −9 . This means that the SM strong low-energy interactions are able to induce the BSM-like pseudoscalar interaction with real coupling constants, the contributions of which are much smaller than the current experimental sensitivity of the neutron β − -decays [40]. Below we consider a more general pseudoscalar interaction BSM with complex phenomenological coupling constants C P andC P such as C P = −C P .
B. Non-relativistic approximation for the amplitude of the neutron β − -decay Eq. (1) In the non-relativistic approximation for the neutron and proton the amplitude of the neutron β − -decay in Eq.(1) takes the form where ϕ j for j = p, n are the Pauli spinorial wave functions of non-relativistic neutron and proton, and k p = − k e − kν is a 3-momentum of the proton.

III. ELECTRON-ENERGY AND ANGULAR DISTRIBUTION OF THE NEUTRON β − -DECAY FOR POLARIZED NEUTRON, POLARIZED ELECTRON, AND UNPOLARIZED PROTON
The electron-energy and angular distribution of the neutron β − -decays for a polarized neutron, a polarized electron and an unpolarized proton has been written by Jackson et al. [24]. It reads where we have followed the notation [9][10][11][12]. The last three terms in Eq.(9) are caused by the contributions of the proton recoil calculated to order O(E e /M ) [8][9][10][11][12]. Then, ξ n and ξ e are unit polarization vectors of the neutron and electron, respectively, dΩ e and dΩν are infinitesimal solid angels in the directions of electron k e and antineutrino kν 3-momenta, respectively, E 0 = (m 2 n − m 2 p + m 2 e )/2m n = 1.2926 MeV is the end-point energy of the electron spectrum, F (E e , Z = 1) is the relativistic Fermi function equal to [64]- [66] (see also [4,[9][10][11][12]) where β = k e /E e = E 2 e − m 2 e /E e is the electron velocity, γ = √ 1 − α 2 − 1, r p is the electric radius of the proton. In the numerical calculations we use r p = 0.841 fm [67]. The function ζ(E e ) contains the contributions of radiative corrections of order O(α/π) and corrections from the weak magnetism and proton recoil of order O(E e /M ), taken in the form used in [8][9][10][11][12]. Then, b is the Fierz interference term defined by the contributions of interactions beyond the SM [55]. The analytical expressions for the correlation coefficients a(E e ), A(E e ) and so on, calculated within the SM with the account for radiative corrections of order O(α/π) and corrections caused by the weak magnetism and proton recoil of order O(E e /M ) together with the contributions of Wilkinson's corrections [4], are given in [9][10][11][12].
A. Corrections to the correlation coefficients of the electron-energy and angular distribution of the neutron β − -decays caused by pseudoscalar interactions In the Appendix we calculate the contributions of the OPP exchange and the pseudoscalar interaction BSM to the correlation coefficients of the electron-energy and angular distribution of the neutron β − -decays for a polarized neutron, a polarized electron and an unpolarized proton. The corrections to the correlation coefficients and the correction to the electron-energy and angular distribution are given in the Appendix in Eqs.(A-5) and (A-6), respectively. The strength of these corrections (see Eq.(A-5)) is defined by the effective coupling constants C ′ ps and C ′′ ps , which are the real and imaginary parts of the effective coupling constant C ps given by where C (OPP) ps and C (BSM) ps are the effective coupling constants caused by the OPP exchange and the pseudoscalar interaction BSM, respectively. The numerical values are calculated for λ = −1.27641 [41], m e = 0.5110 MeV, m π = 139.5706 MeV [14], E 0 = (m 2 n −m 2 p +m 2 e )/2m n = 1.2926 MeV and M = (m n +m p )/2 = 938.9188 MeV [14], respectively. According to our analysis (see Eqs. (6) and (7)), a real part of the phenomenological coupling constant C (BSM) ps can be partly induced by the SM strong low-energy interactions through the ADM-term (see Eq. (6)) and Kaiser's two-loop corrections, calculated within the HBχPT (see Eq. (7)).
The corrections, caused by pseudoscalar interactions (see Eq.(A-5) and Eq.(A-6)), to the electron-energy and angular distribution of the neutron β − -decays for a polarized neutron, a polarized electron and an unpolarized proton, taken together with the electron-energy and angular distributions calculated in [8][9][10][11][12] can be used as a theoretical background for experimental searches of contributions of interactions BSM of order 10 −4 or even smaller [40].
B. Estimates of the real and imaginary parts of the phenomenological coupling constant CP −CP According to [30], the phenomenological coupling constant C P −C P can be defined as follows where ǫ P is a complex effective coupling constant of the four-fermion local weak interaction of the pseudoscalar quark currentūγ 5 d, where u and d are the up and down quarks, with the left-handed leptonic currentl(1 − γ 5 )ν ℓ [28] - [31] (see also [34,53]). Then, g P is the matrix element p|ūγ 5 d|n = g Pūp γ 5 u n caused by strong low-energy interactions, whereū p and u n are the Dirac wave functions of a free proton and neutron, respectively. According to González-Alonso and Camalich [53], one gets g P = 349(9) (see Eq.(13) of Ref. [53]).  Following [53] and using the constraint |ǫ P | < 5.8 × 10 −3 , obtained at 90 % C.L. from the experimental data on the search for an excess of events with a charged lepton (an electron or muon) and a neutrino in the final state of the pp collision with the centre-of-mass energy of √ s = 8 TeV with an integrated luminosity of 20 fb −1 at LHC [68], we get |Re(C P −C P )| < 4.1. In this case the pseudoscalar interaction BSM can dominate in the effective coupling constant C ′ ps in comparison to the OPP exchange, which is of order |C (OPP) ps | ∼ 10 −5 . In turn, the analysis of the leptonic decays of charged pions, carried out in [34] (see Eq.(113) and a discussion on p.51 of Ref. [34]), taken together with the results, obtained in [69], gives one Re ǫ P = (0.4 ± 1.3) × 10 −4 and, correspondingly, Re(C P −C P ) = 0.03 ± 0.09. Such an analysis implies that the phenomenological coupling constants Re(C P −C P ) and C (BSM) ps are commensurable with zero. This leads to a dominate role of the OPP exchange in the effective coupling constant C ′ ps equal to C ′ ps = −1.47 × 10 −5 . Then, following the assumption ǫ P = 2m e (m u + m d )/m 2 π ∼ 4 × 10 −4 [34],which is also related to the analysis of the leptonic decays of charged pions (see a discussion below Eq.(112) of Ref. [34]), we get Re(C P −C P ) ∼ 0.3 and Re C (BSM) ps ∼ −3.5 × 10 −5 . As a result, according to the assumption ǫ P = 2m e (m u + m d )/m 2 π ∼ 4 × 10 −4 , the contribution of the pseudoscalar interaction BSM to the effective coupling constant C ′ ps should be of order 10 −5 , that makes it commensurable with the contribution of the OPP exchange.
Since the constraint |ǫ P | < 5.8 × 10 −3 [53] disagrees with the constraints following from the analysis of the leptonic decays of charged pions [34,69], one may conclude that the phenomenological coupling constant Re(C P −C P ) should be constrained by 0 Re(C P −C P ) 0.3. This leads to the effective coupling constant Re C | ∼ 10 −5 or even smaller. The imaginary part Im(C P −C P ) = 2g P Im ǫ P we estimate using the upper bound Im ǫ P < 2.8 × 10 −4 , obtained at 90 % C.L. in [29] (see also Eq.(114) of Ref. [34]). We get Im(C P −C P ) < 0.3. The effective coupling constant C ′′ ps = Im C (BSM) ps is restricted by C ′′ ps = Im C (BSM) ps < −2.3 × 10 −5 . Since the contribution of the OPP exchange is real, the effective coupling constant C ′′ ps , constrained by C ′′ ps < −2.3 × 10 −5 , is fully defined by the pseudoscalar interaction BSM.
In Table I we adduce the constraints on the real and imaginary parts of the phenomenological coupling constant C P −C P and on the effective coupling constant C (BSM) ps , which may follow from the results obtained in [34,53,69].

IV. DISCUSSION
The corrections of order 10 −5 , calculated within the SM, are needed as a SM theoretical background for experimental searches of interactions beyond the SM in terms of asymmetries and correlation coefficients of the neutron β − -decays [10][11][12]. An experimental accuracy of about a few parts of 10 −5 or even better, which is required for experimental analyses of interactions BSM of order 10 −4 , can be reachable at present time [40]. In this paper we have continued the analysis of corrections of order 10 −5 to the correlation coefficients of the neutron β − -decays, which we have begun in [10][11][12][13]. In this paper we have taken into account the contributions of strong low-energy interactions in terms of the OPP exchange and the contributions of the pseudoscalar interaction BSM [22]- [27], and calculated corrections to the correlation coefficients of the electron-energy and angular distribution of the neutron β − -decay for a polarized neutron, a polarized electron and an unpolarized proton.
In addition to the results, concerning the corrections caused by pseudoscalar interactions to the electron-energy and angular distributions of the neutron β − -decay for a polarized neutron and unpolarized electron and proton, obtained in [47]- [54] and especially by Harrington [47] and Holstein [51], we have calculated corrections to the correlation coefficients, caused by correlations with the electron spin, i.e. for a polarized neutron and a polarized electron with an unpolarized proton.
We have shown that the energy independent contributions to the pseudoscalar form factor [56][57][58][59][60], related to the Adler-Dothan-Wolfenstein (ADM) term Eq.(6) and to the chiral corrections Eq. (7), calculated by Kaiser [60] in a two-loop approximation within the HBχPT, are able in principle to be responsible for sufficiently small real parts of the phenomenological coupling constants C P andC P and at the level of 10 −6 − 10 −8 of the effective coupling constant C (BSM) ps . In turn, the isospin breaking corrections of order 10 −5 , calculated by Kaiser within the HBχPT [46] to the vector coupling constant of the neutron β − -decay, should be taken into account for a correct description of the neutron lifetime at the level of 10 −5 .
As has been shown in [30] the phenomenological coupling constant C P −C P , introduced at the hadronic level [22]- [27], can be related to the effective coupling constant ǫ P of the pseudoscalar interaction of the up and down quarks with left-handed leptonic current by C P −C P = 2g P ǫ P , where g P = 349(9) [53] is the matrix element of the pseudoscalar quark current caused by strong low-energy interactions. Using the relation C P −C P = 2g P ǫ P [30] we have estimated the real and imaginary parts of the phenomenological coupling constant C P −C P . Having summarized the results, concerning the constraints on the parameter ǫ P , obtained in [29,34,53,69], and taking into account that g P = 349(9) [53], we have got 0 Re(C P −C P ) 0.3 and Im(C P −C P ) < 0.2. Such an estimate agrees well with the analysis of the contributions of the pseudoscalar interaction BSM to the lifetimes of charged pions [34].
For the effective coupling constants Re C The analysis of contributions of pseudoscalar interactions to the electron-energy and angular distributions of weak semileptonic decays of baryons has a long history [47]- [54] (see also [4,34]). That is why it is important to make a comparative analysis of the results obtained in our work with those in [4,34,47]- [54]. For the first time the contributions of pseudoscalar interactions to the correlation coefficients of electron-energy and angular distributions for weak semileptonic decays of baryons for polarized parent baryons and unpolarized decay electrons and baryons were calculated by Harrington [47]. In the notation of Jackson et al. [24] Harrington calculated the contributions of the induced pseudoscalar form factor to the Fierz interference term b(E e ) [55] and to the correlation coefficients a(E e ), A(E e ), B(E e ) and D(E e ), caused by electron-antineutrino angular correlations and correlations of the neutron spin with electron and antineutrino 3-momenta, respectively. The corresponding contributions of pseudoscalar interactions can be obtained from Eqs.(9) -(13) of Ref. [47] keeping the leading terms in the large baryon mass expansion. They read where the first term describes the contribution of pseudoscalar interactions to the Fierz-like interference term [55]. The analogous corrections can be extracted from the expressions, calculated by Holstein [51] (see Appendix B of Ref. [51]). The corrections of pseudoscalar interactions to the Fierz-like interference term δb ps (E e ) and correlation coefficients δa ps (E e ), δA ps (E e ), δB ps (E e ) and δD ps (E e ), calculated in Eqs.(A-5) and (A-6), agree well with those calculated by Harrington [47] (see Eq. (13)). Since in [4,34,[48][49][50][51][52][53][54] the electron-energy and angular distributions were analyzed for weak semileptonic decays either for polarized parent baryons and unpolarized decay electrons and baryons or for unpolarized parent baryons and unpolarized decay electrons and baryons the overlap of our results with those obtained in [4,[48][49][50][51][52][53][54] is at the level of the corrections shown in Eq. (13). Indeed, the contribution of the Fierz-like interference term δb ps (E e ) in Eq.(A-4) agrees well with the result, obtained by Wilkinson [4] and by González-Alonso and Camalich [53] where the term proportional to g A g IP , describing the contribution of the OPP exchange with g IP = 2g A M/m 2 π , was calculated by Wilkinson (see Table 1 and a definition of g IP on p.479 of Ref. [4]), whereas the second term, caused by the contribution of the pseudoscalar interaction BSM and where we have taken into account the relation C P −C P = 2g P ǫ P [30], was calculated by González-Alonso and Camalich [53] (see Eqs. (16) and (17) of Ref. [53])).
In turn, the contributions of pseudoscalar interactions to the correlation coefficients, induced by correlations with the electron spin, were not calculated in [4,34,[47][48][49][50][51][52][53][54]. Thus, the calculation of contributions of pseudoscalar interactions to the correlation coefficients, induced by correlations with the electron spin, distinguishes our results from those obtained in [4,34,[47][48][49][50][51][52][53][54]. However, we would like to notice that in the book by Behrens and Bühring [52] there is a capture entitled "Electron polarization", concerning an analysis of a polarization of decay electrons in beta decays. In this capture the authors propose a most general density matrix, which can be applied to a description of energy and angular distributions for beta decays by taking into account a polarization of decay electrons (see Eq.(7.6) and Eq.(7.7) of Ref. [52]). Of course, by using such a general density matrix and the technique, developed by Biedenharn and Rose [70], one can, in principle, calculate contributions of pseudoscalar interactions to the correlation coefficients induced by correlations with the electron spin. Nevertheless, the calculation of these corrections were not performed in [52]. The authors applied such a general density matrix to a calculation of a general formula for a value of a longitudinal polarization of decay electrons in beta decays only (see Eq.(7.151) of Ref. [52]). Thus, we may assert that all corrections of pseudoscalar interactions to the correlation coefficients, induced by correlations with the electron spin (see Eq.(A-5)), and also other terms proportional to the coupling constants C ′ ps and C ′′ ps in Eq.(A-6) are new in comparison to the results, obtained in [4,34,47]- [54] and were never calculated in literature. Moreover, a theoretical accuracy O(αE 0 /πM ) ∼ 10 −6 and O(E 2 0 /M 2 ) ∼ 10 −6 of the calculation of a complete set of corrections of order 10 −3 [9][10][11][12] including radiative corrections of order O(α/π) and corrections of order O(E 0 /M ), caused by the weak magnetism and proton recoil, makes the contributions of corrections of order 10 −5 , induced by pseudoscalar interactions, observable in principle and important as a part of theoretical background for experimental searches of contributions of interactions BSM in asymmetries of the neutron β − -decays with a polarized neutron, a polarized electron and an unpolarized proton [40].
Thus, in this work we have calculated the contributions of pseudoscalar interactions, induced by the OPP exchange and BSM, to the complete set of correlation coefficients of the electron-energy and angular distribution of the neutron β − -decays for a polarized neutron, a polarized electron and an unpolarized proton. The corrections to the Fierz interference term b(E e ), the correlation coefficients a(E e ), A(E e ), B(E e ) and D(E e ), caused by electron-antineutrino angular correlations and correlations of the neutron spin with electron and antineutrino 3-momenta, respectively, and as well as the correlation coefficients, induced by correlations with the electron spin such as G(E e ), N (E e ) and so on, and also corrections, given by the terms proportional to the effective coupling constants C ′ ps and C ′′ ps in Eq.(A-6), are calculated by using one of the same theoretical technique. The agreement of the corrections to the Fierz interference term b(E e ) and the correlation coefficients a(E e ), A(E e ), B(E e ) and D(E e ) with the results obtained in [4,34,47]- [54] may only confirm a correctness of our results.
The obtained corrections (see Eq.(A-5) and Eq.(A-6)), caused by the OPP exchange and the pseudoscalar interaction BSM, complete the analysis of contributions of interactions BSM to the correlation coefficients of the neutron β −decays for a polarized neutron, a polarized electron and an unpolarized proton carried out in [9][10][11][12]. For experimental accuracies of about a few parts of 10 −5 or even better [40] the exact analytical expressions of these corrections can be practically distinguished from the contributions of order 10 −5 , caused by the second class hadronic currents or G-odd correlations, calculated by Gardner and Plaster [33] and Ivanov et al. [11,12]. Appendix A: Calculation of corrections caused by pseudoscalar interactions to the electron-energy and angular distribution of the neutron β − -decays for a polarized neutron, a polarized electron and an unpolarized proton A direct calculation of the corrections, caused by the OPP exchange and the pseudoscalar interaction BSM [9], to the electron-energy and angular distribution of the neutron β − -decays for a polarized neutron, a polarized electron and an unpolarized proton yields .