Neutron Dark Matter Decays and Correlation Coefficients of Neutron Beta Decays

As we have pointed out in (arXiv:1806.10107 [hep-ph]), the existence of neutron dark matter decay modes"n ->chi + anything", where"chi"is a dark matter fermion, for the solution of the neutron lifetime problem changes priorities and demands to describe the neutron lifetime"tau_n = 888.0(2.0)s", measured in beam experiments and defined by the decay modes"n ->p + anything", in the Standard Model (SM). The latter requires the axial coupling constant"lambda"to be equal to"lambda = - 1.2690"(arXiv:1806.10107 [hep-ph]). Since such an axial coupling constant is excluded by experimental data reported by the PERKEO II and UCNA Collaborations, the neutron lifetime"tau_n = 888.0(2.0)s"can be explained only by virtue of interactions beyond the SM, namely, by the Fierz interference term of order"b ~ - 10^{-2}"dependent on scalar and tensor coupling constants. We give a complete analysis of all correlation coefficients of the neutron beta decays with polarized neutron, taking into account the contributions of scalar and tensor interactions beyond the SM with the Fierz interference term"b ~ - 10^{-2}". We show that the obtained results agree well with the contemporary experimental data that does not prevent the neutron with the rate of the decay modes"n ->p + anything", measured in beam experiments, to have dark matter decay modes"n ->chi + anything".


I. INTRODUCTION
Recently Fornal and Grinstein [1] have proposed to explain the neutron lifetime anomaly, related to a discrepancy between experimental values of the neutron lifetime measured in bottle and beam experiments, through contributions of the neutron dark matter decay modes n → χ + γ an n → χ + γ * → χ + e − + e + , where χ is a dark matter fermion, γ and γ * are real and virtual photons, and (e − e + ) is the electron-positron pair. However, according to recent experimental data [2,3], the decay modes n → χ + γ and n → χ + e − + e + are suppressed. In [4] the experimental data on the decay mode n → χ + e − + e + in [3] have been interpreted as follows. An unobservability of the decay mode n → χ+ e − + e + , which is not mediated by a virtual photon, may also mean that the production of the electronpositron pair in such a decay is below the reaction threshold, i.e. a mass m χ of dark matter fermions χ obeys the constraint m χ > m n − 2m e , where m n and m e are masses of the neutron and electron (positron), respectively. Then, we have proposed that the neutron lifetime anomaly can be explained by the decay mode n → χ+ν e +ν e , where (ν eνe ) is a neutrino-antineutrino pair [4]. Since neutrino ν e and electron e − belong to the same doublet in the Standard Electroweak Model (SEM) [5], neutrino-antineutrino (ν eνe ) pairs couple to the neutron-dark matter current with the same strength as electron-positron (e − e + ) pairs [4]. For the UV completion of the effective interaction (nχℓl), where ℓ(l) is electron (positron) or neutrino(antineutrino), we have proposed a gauge invariant quantum field theory model with SU L (2) × U R (1) × U ′ R (1) × U ′′ L (1) gauge symmetry. Such a quantum field theory model contains the sector of the SEM (or the Standard Model (SM) sector) [5] with SU L (2) × U R (1) gauge symmetry and the dark matter sector with U ′ R (1) × U ′′ L (1) gauge symmetry. In the physical phase the dark matter sectors with U ′ R (1) and U ′′ L (1) symmetries are responsible for the UV completion of the effective interaction (nχℓl) [4] and interference of the dark matter into dynamics of neutron stars [6][7][8][9], respectively. The dark matter sector with U ′′ L (1) gauge symmetry we have constructed in analogue with scenario proposed by Cline and Cornell [9]. This means that dark matter fermions with mass m χ < m n couple to a very light dark matter spin-1 boson Z ′′ providing a necessary repulsion between dark matter fermions in order to give a possibility for neutron stars to reach masses of about 2M ⊙ [10], where M ⊙ is the mass of the Sun [5]. We have shown that in the physical phase the predictions of the dark matter sector with U ′ R (1) gauge symmetry do not contradict constraints on i) dark matter production in ATLAS experiments at the LHC, ii) the cross section for the low-energy dark matter fermion-electron scattering (χ + e − → χ + e − ) [11], and iii) the cross section for the low-energy dark matter fermion-antifermion annihilation into the electron-positron pairs (χ +χ → e − + e + ) [12]. We have also proposed that reactions n → χ + ν e +ν e , n + n → χ + χ, n + n → χ + χ + ν e +ν e and χ + χ → n + n, allowed in our model, can serve as URCA processes for the neutron star cooling [13][14][15].
Having assumed that the results of the experimental data [2,3] can be also interpreted as a production of electronpositron pairs below reaction threshold of the decay mode n → χ + e − + e + , we have proposed to search for traces of dark matter fermions induced by the nχ e − e + interaction in the low-energy electron-neutron inelastic scattering e − +n → χ+e − . Such a reaction can be compared experimentally with low-energy electron-neutron elastic scattering e − + n → n + e − [16]- [20]. The differential cross section for the reaction e − + n → χ + e − possesses the following properties: i) it is inversely proportional to a velocity of incoming electrons, ii) it is isotropic relative to outgoing electrons and iii) momenta of outgoing electrons are much larger than momenta of incoming electrons. Because of these properties of the differential cross section for the reaction e − + n → χ + e − can be in principle distinguished above the background of the elastic electron-neutron scattering e − + n → n + e − . In order to have more processes with particles of the SM in the initial and final states allowing to search dark matter in terrestrial laboratories we have proposed to search dark matter fermions by means of the electrodisintegration of the deuteron into dark matter fermions and protons e − +d → χ+p+e − close to threshold [21], induced by the electron-neutron inelastic scattering e − +n → χ+e − with energies of incoming electrons larger than the deuteron binding energy, which of about |ε d | ∼ 2 MeV. We have calculated the triple-differential cross section for the reaction e − + d → χ + p + e − close to threshold, and proposed to detect dark matter fermions from the electrodisintegration of the deuteron e − + d → χ + p + e − above the background e − + d → n + p + e − by detecting outgoing electrons, protons and neutrons in coincidence. A missing of neutron signals at simultaneously detected signals of protons and outgoing electrons should testify an observation of dark matter fermions in the final state of the electrodisintegration of the deuteron close to threshold.
As has been pointed out in [4], the acceptance of existence of the neutron dark matter decay modes n → χ+anything is not innocent and demands to pay the following price. Indeed, the neutron lifetime time τ n = 879.6(1.1) s, calculated in the SM [22] for the axial coupling constant λ = −1.2750 (9) [31] by taking into account the complete set of corrections of order 10 −3 , caused by the weak magnetism and proton recoil, taken to next-to-leafing order in the large nucleon mass M expansion, and radiative corrections of order O(α/π), where α is the fine-structure constant [5], agrees well with the world averaged lifetime of the neutron τ n = 880.1(1.0) s [5], and the neutron lifetime τ n = 879.6(6), averaged over the experimental values measured in bottle experiments [24]- [29] included in the Particle Date Group (PDG) [5]. It agrees also with the value τ n = 879.4(6) s and the axial coupling constant λ = −1.2755 (11), obtained by Czarnecki et al. [30] by means of a global analysis of the experimental data on the neutron lifetime and axial coupling constant. At first glimpse such an agreement rules out fully any dark matter decay mode n → χ + anything of the neutron. For a possibility of the neutron to have any dark matter decay mode n → χ+ anything the SM should explain the neutron lifetime τ n = 888.0(2.0) s, measured in beam experiments, instead of to explain the neutron lifetime τ n = 879.6(6) s, measured in bottle ones. As has been shown in [4], using the analytical expression for the neutron lifetime (see Eq. (41) and (42) of Ref. [22]) the value τ n = 888.0(2.0) s can be fitted by the axial coupling constant equal to λ = −1.2690. Since such a value of the axial coupling constant is ruled out by recent experiments [31]- [34] and a global analysis by Czarnecki et al. [30], so the hypothesis of the existence of the neutron dark matter decay modes should state that the SM, including a complete set of corrections of order 10 −3 caused by the weak magnetism, proton recoil and radiative corrections [35] (see also [22]), is not able to describe correctly the rate and correlation coefficients of the neutron decay modes n → p + anything. Hence, the theoretical description of the neutron lifetime, measured in beam experiments, should go beyond the SM. Indeed, keeping the value of the axial coupling constant equal to λ = −1.2750 or so [31]- [34] and having accepted the existence of the dark matter decay modes n → χ + anything we have also to accept a sufficiently large contribution of the Fierz interference term b [36], dependent on the scalar and tensor coupling constants of interactions beyond the SM [37]- [50] (see also [35] and [22]). Using the results obtained in [22], the neutron lifetime τ n = 888.0 s can be fitted by the axial coupling constant λ = −1.2750, the Cabibbo-Kobayashi-Maskawa (CKM) matrix element V ud = 0.97420 [5] and the Fierz interference term b ∼ −0.014 calculated at the neglect of the quadratic contributions of scalar and tensor coupling constants of interactions beyond the SM [4]. Thus, in order to confirm a possibility for the neutron to have any dark matter decay modes n → χ + anything we have to show that a tangible influence of the Fierz interference term b ∼ −0.014 is restricted only by the rate 1/τ n = 1/888.0 s −1 = 1.126 × 10 −3 s −1 of the neutron decay modes n → p + anything, measured in beam experiments, and such a term does not affect the correlation coefficients of the electron-energy and angular distributions of the neutron β − -decay. This paper is addressed to the analysis of the contributions of scalar and tensor interactions beyond the SM to the rate of the neutron decay modes n → p + anything, measured in beam experiments, and correlation coefficients of the neutron β − -decay with polarized neutron, polarized electron and unpolarized proton. We take into account the contributions of the SM, including a complete set of corrections of order 10 −3 , caused by the weak magnetism and proton recoil, calculated to next-to-leading order in the large nucleon mass M expansion [51,52] (see also [35] and [22,53]), and radiative corrections of order O(α/π), calculated to leading order in the large nucleon mass expansion [54]- [57] (see also [35] and [22,53]). We search anyone solution for the values of scalar and tensor coupling constants of interactions beyond the SM allowing to fit the rate 1/τ n = 1.126 × 10 −3 s −1 of the neutron decay modes n → p + anything, measured in beam experiments, and the experimental data on the correlation coefficients of the neutron β − -decay under consideration. The existence of such a solution for the values of scalar and tensor coupling constants of interactions beyond the SM should imply an allowance for the neutron to have the dark matter decay modes n → χ + anything. The paper is organized as follows. In section II we give the electron-energy and angular distribution of the neutron β − -decay with polarized neutron, polarized electron and unpolarized proton. We write down the expressions for the correlation coefficients including the contributions of the SM corrections of order 10 −3 , caused by the weak magnetism and proton recoil to next-to-leading order in the large nucleon mass expansion, and radiative corrections of order O(α/π), and the contributions of scalar and tensor interactions beyond the SM, calculated to leading order in the large nucleon mass M expansion. In section III we analyse the rate 1/τ n = 1.126 × 10 −3 s −1 of the neutron decay modes n → p + anything, measured in beam experiments, and fit it by the contribution of the Fierz interference term by taking into account the quadratic contributions of scalar and tensor interactions beyond the SM. In section IV we define the correlation coefficients for the real scalar and tensor coupling constants settingC S = −C S andC T = −C T . In section V we propose the solution C S = −C S = −C T =C T = −8.79 × 10 −3 for real scalar and tensor coupling constants, and define the correlations coefficients in terms of the coupling constant C S and the Fierz interference term b = −1.44 × 10 −2 . We show that the contributions of quadratic terms C 2 S are of the standard order 10 −4 . In turn, the contributions of linear terms are of order 10 −2 − 10 −3 . In section VI we analyse the contributions of the Fierz interference term to the electron and antineutrino asymmetries, defined by the neutron spin and electron and antineutrino 3-momentum correlations, respectively, and to the asymmetry, caused by the correlations of the electron and antineutrino 3-momenta. We show that the contribution of the Fierz interference term b = −1.44 × 10 −2 does not contradict the experimental data on the measurements of the correlation coefficients A 0 , B 0 and also a 0 , defined to leading order in the large nucleon mass M expansion [23,60]. In section VII we analyse the averaged values of the correlation coefficients N (E e ) and R(E e ). The correlation coefficient N (E e ) defines the neutron-electron spinspin ξ n · ξ e correlations. In turn, the correlation coefficient R(E e ) is caused by the P-odd and T-odd correlations defined by ξ n · ( k e × ξ e ), where ξ n and ξ e are the unit vectors of the neutron and electron polarizations, and k e is the electron 3-momentum. We show that the averaged value of the correlation coefficient N (E e ) , obtained in this paper, agrees with the experimental value within two standard deviations. In turn, the averaged value of the correlation coefficient R(E e ) acquires the relative contributions of order 10 −4 , caused by interactions beyond the SM. However, the solution C S = −C S = −C T =C T = −8.79 × 10 −3 , being valid for the neutron β − -decay, fails when it is applied to the analysis of the superallowed 0 + → 0 + transitions. Indeed, as has been found by Hardy and Towner [58] and González-Alonso et al. [59], the scalar coupling constant should obey the constraints |C S | ≃ 0.0014 (13) and |C S | ≃ 0.0014 (12), respectively. Since in the superallowed 0 + → 0 + transitions the scalar coupling constant is commensurable with zero, in section VIII we propose the solutionC S = −C S = 0 and C T = −C T = 1.11 × 10 −2 with the Fierz interference term b = −1.44×10 −2 . The Fierz interference term has the same value for two different solutions, since it is well-defined in the linear approximation. We show that the correlation coefficients and asymmetries of the neutron β − -decay, calculated for the solutionC S = −C S = 0 and C T = −C T = 1.11 × 10 −2 and the Fierz interference term b = −1.44 × 10 −2 , do not contradict contemporary experimental data. In section IX we discuss the obtained results. We argue that the obtained agreement between theoretical values of the correlation coefficients, defined by the contributions of the SM to order 10 −3 , the Fierz interference term b = −1.44 × 10 −2 and other linear and quadratic coupling constants of scalar and tensor interactions beyond the SM, implies an allowance for the neutron to have dark matter decay modes n → χ + anything.

II. ELECTRON-ENERGY AND ANGULAR DISTRIBUTION OF NEUTRON β − -DECAY WITH POLARIZED NEUTRON, POLARIZED ELECTRON AND UNPOLARIZED PROTON
The electron-energy and angular distribution of the neutron β − -decay with polarized neutron and electron and unpolarized proton takes the form where G F = 1.1664 × 10 −11 MeV −2 and V ud = 0.97420 (21) are the Fermi weak coupling constant and the Cabibbo-Kobayashi-Maskawa (CKM) matrix element [5], respectively, λ = −1.2750 (9) is a real axial coupling constant [23], MeV is the end-point energy of the electron spectrum, calculated for m n = 939.5654 MeV, m p = 938.2720 MeV and m e = 0.5110 MeV [5], ξ n and ξ e are unit polarization vectors of the neutron and electron, respectively, F (E e , Z = 1) is the relativistic Fermi function [40,[61][62][63] F (E e , Z = 1) = 1 + r p is the electric radius of the proton. In the numerical calculations we use r p = 0.841 fm [64]. The Fermi function F (E e , Z = 1) describes final-state Coulomb proton-electron interaction. Then, b is the Fierz interference term [36,38]- [50] (see also [35] and [22]). The infinitesimal solid angles dΩ e = sin ϑ e dϑ e dϕ e and dΩ ν = sin ϑ ν dϑ ν dϕ ν are defined relative to the 3-momenta k e and k ν of the decay electron and antineutrino, respectively.
The correlation coefficients ζ(E e ), a(E e ) and so on, taking into account the contributions of the SM and scalar and tensor interactions beyond the SM [22,50], and the Fierz interference term b are given by where the correlation coefficients X (SM) (E e ) for X = ζ, a, A and so on are calculated within the SM including the complete set of corrections caused by the weak magnetism and proton recoil of order O(E e /M ) and radiative corrections of order O(α/π) [22,53]. In turn, the correlation coefficients b F and X (BSM) are defined by [22,50] b For the subsequent analysis it is convenient to rewrite the correlation coefficients Eq.(4) in terms of real and imaginary parts of scalar and tensor coupling constants. We get Now we may proceed to the analysis of contributions of scalar and tensor interactions beyond the SM to the rate of the neutron decay modes n → p + anything, measured in beam experiments, and correlation coefficients under consideration.

IV. CORRELATION COEFFICIENTS
The aim of this paper is to find anyone plausible solution for the scalar and tensor coupling constants of interactions beyond the SM, which should be compatible with present time accuracy of the definition of the correlation coefficients of the neutron β − -decay. As a first step of the analysis of the contributions of scalar and tensor interaction beyond the SM we follow [44] and [50] and set i) real the scalar and tensor coupling constants and ii)C j = −C j for j = S.T . This gives and The total correlation coefficients are equal to Now we may proceed to analysing possible solutions for the scalar and tensor coupling constants.

V. CORRELATION COEFFICIENTS AND SCALAR AND TENSOR COUPING CONSTANTS. SOLUTION 1
The simplest solution, which sticks out a mile, is C S = −C T . Setting C S = −C T we transcribe the correlation coefficients Eq.(10) into the form At C S = −C T Eq.(8) reduces to the quadratic algebraical equation with the solution where we have chosen only the solution obeying the constraint |C S | ≪ 1. In the linear approximation we get Plugging Eq.(13) into Eq.(11) we obtain the following correlation coefficients, corrected by the contributions of scalar and tensor interactions beyond the SM: where we have used b N = −b F ≃ −b. Thus, we have shown that there exists the solution C S = −C S = −C T =C T = −8.79 × 10 −3 for real scalar and tensor coupling constants, which determines reasonable contributions of interactions beyond the SM to correlation coefficients a(E e ) and A(E e ) of order 10 −4 [65] and to the correlation coefficients G(E e ), N (E e ) and Q e (E e ) of order 10 −3 . In turn, the correlation coefficient B(E e ) ∼ 1 acquires the correction of order 10 −2 . Now we have to compare the obtained results with the experimental data. For this aim we have to analyse the asymmetries of the neutron β − -decay and the averaged values of correlation coefficients. The most sensitive asymmetry of the neutron β − -decay is the electron asymmetry, caused by correlations of the neutron spin ξ n and the electron 3-momentum k e and described by the scalar product ξ n · k e . The experimental electron asymmetry A exp (E e ) of electrons emitted forward and backward with respect to the neutron spin ξ e into the solid angle ∆Ω 12 = 2π(cos θ 1 − cos θ 2 ) with 0 ≤ ϕ ≤ 2π and θ 1 ≤ θ e ≤ θ 2 is equal to [22,50] A exp (E e ) = 1 2 β A(E e )P n (cos θ 1 + cos θ 2 ), (15) where P n = | ξ n | ≤ 1 is the neutron spin polarization, and the correlation coefficient A(E e ) is [22,50] A(E e ) = A The correlation coefficient A where the function f n (E e ) defines the radiative corrections, calculated by Shann [55], and the function A W (E e ) has been calculated by Bilen'kii et al. [51] and by Wilkinson [52] by taking into account the contributions of order O(E e /M ) caused by the weak magnetism and proton recoil (see also [22]). Following [31] we plot in Fig. 1 the function − 1 2 βA(E e ) in the electron energy region m e ≤ E e ≤ E 0 , where the electron asymmetry calculated in the SM at b = 0 is given by the blue curve, whereas the electron asymmetry calculated with the account for the contributions of interactions beyond the SM at b = −1.44 × 10 −2 is presented by the red curve. One may see that these two theoretical curves cannot be practically distinguished in experiments. The results presented by the theoretical curves in Fig. 1 can be also confirmed by the following estimates.
We propose to estimate the correlation coefficient A(E e ), calculated to leading order in the large nucleon mass M expression and at the neglect of the radiative corrections. The result is equal to  [34].
The analogous estimate we may make for the correlation coefficient a(E e ), describing electron-antineutrino 3momentum correlations, defined by the scalar product k e · k ν . To leading order in the large nucleon mass M expansion we get where The antineutrino asymmetry of the neutron β − -decay is caused by correlations of the neutron spin ξ n and antineutrino 3-momentum k ν , defined by the scalar product ξ n · k ν . Following [66] we define the antineutrino asymmetry B exp (E e ) as follows [22] This expression determines the asymmetry of the emission of the antineutrinos into the forward and backward hemisphere with respect to the neutron spin, where N ∓∓ (E e ) is the number of events of the emission of the electron-proton pairs as functions of the electron energy E e . The signs (++) and (−−) show that the electron-proton pairs were emitted parallel (++) and antiparallel (−−) to a direction of the neutron spin. This means that antineutrinos were emitted antiparallel (++) and parallel (−−) to a direction of the neutron spin. The number of events N −− (E e ) and N ++ (E e ) are defined by the electron-energy and angular distribution of the neutron β − -decay, integrated over the forward and backward hemisphere relative to the neutron spin, respectively [22]. The analytical expression for B exp (E e ) is given by [22] B exp (E e ) = 2P 3 , r ≤ 1 (21) and B exp (E e ) = 2P 3 where r = k e /E ν = k e /(E 0 − E e ) [66] (see also [22]). In the neutron β − -decay with polarized neutron and electron and unpolarized proton the averaged values have been measured only for the correlation coefficients N (E e ) and R(E e ): N exp = N (E e ) = 0.067 ± 0.011 ± 0.004 and R exp = R(E e ) = 0.004 ± 0.012 ± 0.005 [67]. Using Eq. (14) and the results, obtained in [50,53], we get N (E e ) = 0.07685 , R(E e ) = 0.00089.
The averaged value of the correlation coefficient N (E e ) = 0.07685 agrees with the experimental one within one standard deviation. In turn, as it follows from Eq. (14) the averaged value of the correlation coefficient R(E e ) can acquire only the relative correction of order 10 −4 , caused by scalar and tensor interactions beyond the SM. This does not contradict the experimental data by [67].

VIII. CORRELATION COEFFICIENTS AND SCALAR AND TENSOR COUPLING CONSTANTS. SOLUTION 2
The solution, implying C S = −C S = −C T =C T = −8.79 × 10 −3 and giving the Fierz interference term b = −1.44×10 −2 , contradicts the constraints on the scalar coupling constant C S extracted from the superallowed 0 + → 0 + transitions [58,59]. According to [58,59], the scalar coupling constant C S should obey the constraints |C S | ≃ 0.0014(13) [58] and |C S | ≃ 0.0014 (12) [59], respectively. Since the analysis of the superallowed 0 + → 0 + transitions implies that the real scalar coupling constant C S should be commensurable with zero, in this section we propose another solution setting C S = −C S = 0 and C T = −C T . In this case the correlation coefficients in Eq.(10) and b F , b E and b N take the form At C S = 0 the algebraic equation Eq.(8) has the following solution where we have chosen only the solution obeying the constraint |C T | ≪ 1. In the linear approximation we get It is obvious that the correlation coefficients Eq. (27) and corresponding asymmetries of the neutron β − -decay do not contradict contemporary experimental data. Our results, obtained above for the asymmetries of the neutron β − -decay with polarized neutron and unpolarized proton and electron are not practically changed. Better agreement we obtain for the averaged value of the correlation coefficient N (E e ) of the neutron-electron spin-spin correlations in the neutron β − -decay with polarized neutron and electron and unpolarized proton. We get N (E e ) = 0.07185, which agrees well with the experimental value N exp = N (E e ) = 0.067 ± 0.011 ± 0.004 [67].

IX. DISCUSSION
The main aim of this paper is to show that the fit of the rate of the neutron decay modes n → p + anything, measured in beam experiments, by the Fierz interference term b = −1.44 × 10 −2 does not contradict contemporary experimental data on the values of the correlation coefficients of the neutron β − -decay with polarized neutron, polarized electron and unpolarized proton. We have found that there exist as minimum two solutions of our interest for real scalar and tensor coupling constants: 1) C S = −C S = −C T =C T = −8.79 × 10 −3 and 2) C S = −C S = 0 and C T = −C T = 1.11 × 10 −2 . These solutions define the Fierz interference term b = −1.44 × 10 −2 , and their contributions of order 10 −4 − 10 −2 to the correlation coefficients do not contradict contemporary experimental data on the correlation coefficients and asymmetries of the neutron β − -decay with polarized neutron, polarized electron and unpolarized proton.
The results of such an analysis of the rate of the neutron decay modes n → p + anything, measured in beam experiments, and correlation coefficients of the neutron β − -decay can be interpreted as an allowance for the neutron to have the dark matter decay modes n → χ + anything, which have been analysed in [4] in the physical phase of a quantum field theory model with The traces of the dark matter fermions χ with mass m χ < m n can be searched in terrestrial laboratories through the measurements of the differential cross section for the low-energy inelastic electron-neutron scattering e − + n → χ + e − [4] and the triple-differential cross section for the electrodisintegration of the deuteron into dark matter fermions and protons e − + d → χ + p + e − close to threshold [21].
Our analysis of the rate of the neutron decay modes n → p + anything, measured in beam experiments, and the correlation coefficients of the neutron β − -decay, taking into account the complete set of corrections of order 10 −3 , caused by the weak magnetism and proton recoil of order O(E e /M ) and radiative corrections of order O(α/π), and the Fierz interference term b = −1.44 × 10 −2 , does diminish an important role of the SM corrections of order 10 −5 , which has been pointed out in [50,53,69,70]. As has been shown in [53] the SM corrections of order 10 −5 concern i) Wilkinson's corrections [52], i.e. the higher order corrections caused by 1) the proton recoil in the Coulomb electronproton final-state interaction, 2) the finite proton radius, 3) the proton-lepton convolution and 4) the higher-order outer radiative corrections, and then ii) the higher order corrections defined by 1) the radiative corrections of order O(α 2 /π 2 ), calculated to leading order in the large nucleon mass expansion, 2) the radiative corrections of order O(αE e /M ), calculated to next-to-leading order in the large nucleon mass expansion, which depend strongly on contributions of hadronic structure of the nucleon [69,70], and 3) the corrections of the weak magnetism and proton recoil of order O(E 2 e /M 2 ), calculated to next-to-next-to-leading order in the large nucleon mass expansion [50,69]. These theoretical corrections should provide for the analysis of experimental data of "discovery" experiments the required 5σ level of experimental uncertainties of a few parts in 10 −5 [53].
An important role of strong low-energy interactions and contributions of hadronic structure of the nucleon for a correct gauge invariant calculation of radiative corrections of order O(αE e /M ) and O(α 2 /π 2 ) as functions of the electron energy E e has been pointed out in [50,69]. This agrees well with Weinberg's assertion about important role of strong low-energy interactions in decay processes [71]. A procedure for the calculation of these radiative corrections to the neutron β − -decays with a consistent account for contributions of strong low-energy interactions, leading to gauge invariant observable expressions dependent on the electron energy E e determined at the confidence level of Sirlin's radiative corrections [54], has been proposed in [69,70].
The calculation of the SM corrections of order 10 −5 should also give a theoretical background for the experimental analysis of the corrections caused by the second class currents [72], which has been analysed in the neutron β − -decay with polarized neutron and unpolarized proton and electron by Gardner and Zang [48] and Garner and Plaster [49], and in the neutron β − -decay with polarized neutron and electron and unpolarized proton by Ivanov et al. [50].
Finalizing our discussion we would like to emphasize that perspectives of development of investigations of the neutron β − -decays with the contribution of the Fierz interference term b = −1.44 × 10 −2 become real only in case of discovery of the neutron dark matter decay modes n → χ + anything in terrestrial laboratories by measuring the differential cross sections for the inelastic low-energy electron-neutron scattering e − + n → χ + e − and for the electrodisintegration of the deuteron into dark matter fermions and protons e − + d → χ + p + e − close to threshold. These processes are induced by the same interaction nχe − e + having the strength of the interaction nχν eνe responsible for the dark matter decay mode n → χ + ν e +ν e , allowing to explain the neutron lifetime anomaly. Of course, indirect confirmations of existence of the neutron dark matter decay mode n → χ + ν e +ν e through the evolution of neutron stars and neutron star cooling should also testify a revision of the experimental data on the neutron β − -decays with substantially improved accuracies of the measurements of correlation coefficients and asymmetries allowing to feel the contribution of the Fierz interference term b = −1.44 × 10 −2 . Such a value of the Fierz interference term is required by the necessity to fit the rate of neutron decay modes n → p + anything, measured in beam experiments. It is obvious that in this connection a relative accuracy of measurements of the rate of the neutron decay modes n → p + anything (or the neutron lifetime τ n = 888.0(2.0) s) in beam experiments should be improved up to a few parts of 10 −4 or even better.