On the structure of the correlation coefficients S(E_e) and U(E_e) of the neutron beta decay

In the standard effective V - A theory of low-energy weak interactions (i.e. in the Standard Model (SM)) we analyze the structure of the correlation coefficients S(E_e) and U(E_e), where E_e is the electron energy. These correlation coefficients were introduced to the electron-energy and angular distribution of the neutron beta decay by Ebel and Feldman ( Nucl. Phys. 4, 213 (1957)) in addition to the set of correlation coefficients proposed by Jackson et al. (Phys. Rev. 106, 517 (1957)). The correlation coefficients $S(E_e)$ and $U(E_e)$ are induced by simultaneous correlations of the neutron and electron spins and electron and antineutrino 3-momenta. These correlation structures do no violate discrete P, C and T symmetries. We analyze the contributions of the radiative corrections of order O(alpha/pi), taken to leading order in the large nucleon mass m_N expansion, and corrections of order O(E_e/m_N), caused by weak magnetism and proton recoil. In addition to the obtained SM corrections we calculate the contributions of interactions beyond the SM (BSM contributions) in terms of the phenomenological coupling constants of BSM interactions by Jackson et al. (Phys. Rev. 106, 517 (1957)) and the second class currents by Weinberg (Phys. Rev. 112, 1375 (1958)).


I. INTRODUCTION
The general form of the electron-energy and angular distribution of the neutron beta decay for polarized neutrons, polarized electrons and unpolarized protons were proposed by Jackson et al. [1] and Ebel and Feldman [2]. In the notations of Ref. [3] it looks like d 5 λ n (E e , k e , kν , ξ n , ξ e ) dE e dΩ e dΩν ∝ ζ(E e ) 1 + b(E e ) m e E e + a(E e ) k e · kν E e Eν + A(E e ) ξ n · k e E e + B(E e ) ξ n · kν Eν +K n (E e ) ( ξ n · k e )( k e · kν) E 2 e Eν + Q n (E e ) ( ξ n · kν)( k e · kν) E e E 2 ν + D(E e ) ξ n · ( k e × kν ) E e Eν + G(E e ) ξ e · k e E e +H(E e ) ξ e · kν Eν + N (E e ) ξ n · ξ e + Q e (E e ) ( ξ n · k e )( k e · ξ e ) (E e + m e )E e + K e (E e ) ( ξ e · k e )( k e · kν) (E e + m e )E e Eν +R(E e ) ξ n · ( k e × ξ e ) E e + L(E e ) ξ e · ( k e × kν) E e Eν + S(E e ) ( ξ n · ξ e )( k e · kν ) E e Eν + T (E e ) ( ξ n · kν)( ξ e · k e ) E e Eν +U (E e ) ( ξ n · k e )( ξ e · kν) E e Eν + V (E e ) ξ n · ( ξ e × kν ) Eν where ξ n and ξ e are unit 3-vectors of spin-polarizations of the neutron and electron, (E e , k e ) and (Eν , kν) are energies and 3-momenta of the electron and antineutrino, dΩ e and dΩν are infinitesimal solid angles in directions of 3-momenta of the electron and antineutrino, respectively.
The analysis of the distribution in Eq.(1) within the standard effective V − A theory of low-energy weak interactions [4][5][6][7] (i.e within the Standard Model (SM)), carried out to leading order in the large nucleon mass m N expansion [3], has shown that correlation coefficients a(E e ), A(E e ), B(E e ), G(E e ), H(E e ), N (E e ), Q e (E e ) and K e (E e ) of the electron-energy and angular distribution by Jackson et al. [1] and the correlation coefficient T (E e ), introduced by Ebel and Feldman [2], survive and depend on the axial coupling constant g A only [8][9][10], which appears in the effective V − A theory of low-energy weak interactions by renormalization of the hadronic axial-vector current by strong lowenergy interactions [5,11]. The function ζ(E e ) defines the contributions of different corrections to the neutron lifetime [16]. In the SM it is equal to unity in the leading order of the large nucleon mass m N expansion and at the neglect of radiative corrections [6,7] (see also [16]). In Refs. [12][13][14][15] (see also [16]) the radiative corrections of order O(α/π) (or so-called outer model-independent radiative corrections [17]) were calculated to leading order in the large nucleon mass m N expansion to the neutron lifetime and correlation coefficients a(E e ), caused by electron-antineutron 3-momentum correlations, A(E e ) and B(E e ), defining the electron-and antineutrino asymmetries, respectively. In turn, the outer radiative corrections of order O(α/π) were calculated to leading order in the large nucleon mass m N expansion to the correlation coefficients G(E e ), H(E e ), N (E e ), Q e (E e ), and K e (E e ) in [18,19] and to the correlation coefficient T (E e ) in [3]. These correlation coefficients are induced by correlations of the electron spin with a neutron spin and 3-momenta of the electron and antineutrino. The corrections of order O(E e /m N ), caused by weak magnetism and proton recoil, were calculated i) to the neutron lifetime and correlation coefficients a(E e ), A(E e ) and B(E e ) in [20,21] (see also [14][15][16]), ii) to the correlation coefficients G(E e ), H(E e ), N (E e ), Q e (E e ) and K e (E e ) in [18,19] and iii) to the correlation coefficient T (E e ) in [3]. The correlation coefficients D(E e ), R(E e ) and L(E e ), characterizing the strength of violation of time reversal invariance (T-odd effect) [22], are induced by the distortion of the Dirac wave function of the decay electron in the Coulomb field of the decay proton [23][24][25][26] (see also [19]). The correlation coefficient b(E e ) is the Fierz interference term [27]. It is assumed that the Fierz interference term is caused by interactions beyond the SM [27]. As regards the contemporary experimental and theoretical status of the Fierz interference term we refer to Refs. [28] - [34]. So one may conclude that the neutron lifetime and the correlation coefficients of the electron-energy and angular distribution of the neutron beta decay proposed by Jackson et al. [1] are investigated theoretically well in the SM at the level of 10 −4 − 10 −3 , caused by the outer radiative corrections of order O(α/π) and the corrections of order O(E e /m N ), induced by weak magnetism and proton recoil. This paper is addressed to the analysis of the structure of the correlation coefficients S(E e ) and U (E e ), introduced by Ebel and Feldman [2]. As has been shown in [3] these correlation coefficients do not survive to leading order in the large nucleon mass m N expansion in contrast to the correlation coefficient T (E e ).
The paper is organized as follows. In section II we adduce the analytical expressions for the correlation coefficients S(E e ) and U (E e ) in dependence of i) the radiative corrections of order O(α/π), calculated to leading order in the large nucleon mass m N expansion, and ii) the corrections of order O(E e /m N ), caused by weak magnetism and proton recoil. In section III we give the contributions of interactions beyond the SM, expressed in terms of the phenomenological coupling constants of the effective phenomenological BSM interactions by Jackson et al. [1] and the contributions of the second class currents by Weinberg [35]. In section IV we give the total expressions for the S(E e ) and U (E e ). We discuss the obtained results and the usage of these correlation coefficients for experimental searches of interactions beyond the SM. We point out that the obtained SM theoretical background of the correlation coefficients S(E e ) and U (E e ) at the level a few parts of 10 −4 should be very useful for experimental searches of contributions of interactions beyond the SM in the experiments with transversally polarized decay electrons [36]. In Appendices A and B of the Supplemental Material within the SM we give in details the calculations of the correlation coefficients S(E e ) and U (E e ) and the analysis of the correlation structure of the neutron radiative beta decay for polarized neutrons, polarized electrons, unpolarized protons and unpolarized photons.

II. CORRELATION COEFFICIENTS S(Ee) AND U (Ee) IN THE STANDARD MODEL
In the SM with the account for the contributions of the radiative corrections of order O(α/π) and the corrections of order O(E e /m N ), caused by weak magnetism and proton recoil, the neutron beta decay can be described by the standard effective V − A low-energy weak interaction [4,5] and electromagnetic interaction with the Lagrangian where L W (x) and L em (x) are the Lagrangian of the standard effective V − A low-energy weak interactions [4,5] (see also [16]) and the Lagrangian of electromagnetic interactions [22] respectively, where G V is the vector weak coupling constant, including the Cabibbo-Kobayashi-Maskawa (CKM) matrix element V ud [10], g A is the real axial coupling constant [8,9], ψ p (x), ψ n (x), ψ e (x) and ψ ν (x) are the field operators of the proton, neutron, electron and antineutrino, respectively, γ µ = (γ 0 , γ ), γ 5 and σ µν = i 2 (γ µ γ ν − γ ν γ µ ) are the Dirac matrices [22]; κ = κ p − κ n = 3.7059 is the isovector anomalous magnetic moment of the nucleon, defined by the anomalous magnetic moments of the proton κ p = 1.7929 and the neutron κ n = −1.9130 and measured in nuclear magneton [10], and m N = (m n + m p )/2 is the average nucleon mass; e is the electric charge of the proton, and A µ (x) is a 4-vector electromagnetic potential.
For the calculation of the correlation coefficients under consideration we use the amplitude of the neutron beta decay, calculated in [16] (see also [19] and the Supplemental Material). The detailed calculation we have carried out in the Supplemental Material. Below we adduce only the obtained results.
Analytical expressions for the correlation coefficients S(Ee) and U (Ee) in the Standard Model In Eq.(A-9) of Appendix A in the Supplemental Material we have defined the general expression for the structure part of the electron-energy and angular distribution of the neutron beta decay for a polarized neutron, a polarized electron and an unpolarized proton. According to this expression, we have shown that the contributions of the radiative corrections of order O(α/π), caused by one-virtual photon exchanges [12][13][14][15] (for the detailed calculations we refer to [16]) do not appear in the correlations coefficients S(E e ) and U (E e ), respectively. In Appendix B of the Supplemental Material we have shown that the neutron radiative beta decay n → p+e − +ν e +γ does not contribute to the correlation coefficients S(E e ) and U (E e ). It is well-known [37][38][39][40][41] (see also [12,13] and [16]) 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.
Thus (see Eq.(A-9)) the contributions, caused by the SM interactions, appear in the correlation coefficients S(E e ) and U (E e ) only due to weak magnetism and proton recoil. For the correlation coefficients ζ(E e ) (SM) S(E e ) (SM) and ζ(E e ) (SM) U (E e ) (SM) we have obtained the following analytical expressions where for the calculation of the corrections of order O(E e /m N ), caused weak magnetism and proton recoil, we have taken into account the contribution of the phase-volume of the neutron beta decay (see Eq.(A-3)). The correlation function ζ(E e ) (SM) was calculated in [15,16]. It is equal to unity at the neglect of the contributions of radiative corrections and corrections, caused by weak magnetism and proton recoil. Hence, the correlation coefficients S(E e ) (SM) and U (E e ) (SM) , including the SM contributions of order O(E e /m N ), are equal to Now we may move on to calculating the contributions of interactions beyond the SM.

III. CONTRIBUTIONS OF INTERACTIONS BEYOND THE STANDARD MODEL AND SECOND CLASS CURRENTS OF THE G-ODD CORRELATIONS
For the calculation of the contributions of interactions beyond the SM we use the effective phenomenological Lagrangian of BSM interactions proposed by Jackson et al. [1] (see also [42,43]). In turn, the account for the contributions of the second class currents or the G-odd correlations (see [44]) we follow Weinberg [35], Gardner and Zhang [45], and Gardner and Plaster [46] (see also [3,19,47] where C T andC T are the phenomenological tensor coupling constants of the effective phenomenological BSM interactions by Jackson et al. [1], and Ref 3 (0) and Reg 2 (0) are the phenomenological coupling constants of the induced scalar and tensor second class currents [35,45,46] (see also [28]), respectively. The contributions of the tensor BSM interactions by Jackson et al. [1] are linear in the phenomenological tensor coupling constants C T andC T . This agrees well with the result obtained by Ebel and Feldman [2]. However, in addition to the result obtained by Ebel and Feldman [2] we, following [48][49][50][51][52] (see also [3,16,18,19,47]), have taken the contributions of the phenomenological vector coupling constants C V andC V in the linear approximation, i.e.

IV. DISCUSSION
We have analyzed the structure of the correlation coefficients S(E e ) and U (E e ), introduced by Ebel and Feldman [2] in addition to the set of correlation coefficients proposed by Jackson et al. [1]. Summing up the SM contributions, caused by weak magnetism and proton recoil only, and contributions beyond the SM we obtain the following expressions For the axial couping constant g A = 1.2764 [8] the correlation coefficients S(E e ) and U (E e ) are given by where we have also used m e = 0.5110 MeV and m N = (m n + m p )/2 = 938.9188 MeV [10].
We would like to notice that the correlation structures of the correlation coefficients S(E e ) and U (E e ) and as well as the correlation coefficients T (E e ) are even with respect to parity transformation (P-even), charge conjugation (C-even) and time reversal transformation (T-even). However, in contrast to the correlation coefficient T (E e ), the absolute value of which is of about |T (E e )| ∼ 1, the absolute values of the correlation coefficients S(E e ) and U (E e ) are of a few orders of magnitude smaller. It is also important to mention that unlike the correlation coefficient T (E e ) the correlation coefficients S(E e ) and U (E e ) do not depend on the electron energy E e .
The correlation coefficients S(E e ) and U (E e ) can, in principle, be investigated in experiments with both longitudinally and transversally polarized decay electrons [36] (see also [3]). However, a successful results for searches of interactions beyond the SM one may expect only from experiments with experimental uncertainties of about a few parts of 10 −5 . In this case any deviation of the correlation coefficient S(E e ) from −2.83 × 10 −4 , caused by weak magnetism and proton recoil, should testify a presence of interactions beyond the SM. Since most likely f 3 (0) = 0 [28,29] (see also [53]) the contributions of the phenomenological tensor BSM interactions by Jackson et al. [1], proportional to Re(C T −C T ), can be distinguished from the contributions of the tensor second class current, defined by the phenomenological tensor coupling constant Reg 2 (0), only after the measurement of the contribution of the phenomenological tensor second class currents to the correlation coefficient T (E e ) [3]. In case of f 3 (0) = 0 and if in the neutron beta decay the absolute value of the Fierz interference term b could be of order 10 −2 (see, for example, [30,32] Following [16,18,19] (see also [3]) we define the electron-energy and angular distribution of the neutron beta decay for a polarized neutron, a polarized electron, and an unpolarized proton as follows where the sum is over polarizations of massive fermions. Then, F (E e , Z = 1) is the relativistic Fermi function, describing the electron-proton final-state Coulomb interaction, is equal to (see, for example, [54] (see also [21] and a discussion in [18]) 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 [55]. The function Φ n ( k e , kν ) defines the contribution of the phase-volume of the neutron beta decay [16,56]. It is equal to [16,56] Φ n ( k e , kν) taken to next-to-leading order in the large nucleon mass m N expansion. The amplitude of the neutron beta decay M (n → pe −ν e ), taking into account the contribution of the corrections, caused by one-virtual photon exchanges, weak magnetism and proton recoil, was calculated in [16] (see also [19]). It is given by where ϕ p and ϕ n are Pauli spinorial wave functions of the proton and neutron, u e and v ν are Dirac wave functions of the electron and electron antineutrino, σ are the Pauli 2 × 2 matrices, andg A = g A (1 − E 0 /2m N ), E 0 = (m 2 n − m 2 p + m 2 e )/2m n = 1.2926 MeV is the end-point energy of the electron-energy spectrum of the neutron beta decay [6,7,10], and k p = − k e − k ν is the proton 3-momentum in the rest frame of the neutron. The functions f β − c (E e , µ) and g F (E e ) were calculated by Sirlin [12] (see also Eq.(D-51) of Ref. [16] and ), µ is a covariant infrared cut-off introduced as a finite virtual photon mass [12] (see also [37][38][39][40][41]). The function g F (E e ) (see Eq.(D-44) of Ref. [16]) is equal to It is defined by the contributions of one-virtual-photon exchanges [12] (see also [16]). Using Eq.(A-4) for the square of the absolute value of the amplitude M (n → pe −ν e ), summed over polarizations of massive fermions, we obtain the following expression pol.
In Eq.(A-9) the second term on the third line from above, proportional to m e /m N , and last four lines define the contributions of order O(E e /m N ) of weak magnetism and proton recoil to the correlation coefficients of the neutron beta decay. Having calculated the traces over leptonic degrees of freedom, taking into account the contribution of the phase-volume Eq. (A-3) and keeping only the contributions with the correlation structures, inducing the correlation coefficients S(E e ) and U (E e ), we obtain the SM corrections, caused by weak magnetism and proton recoil only, which we give in Eq.(5).
Appendix B: The electron-photon-energy and angular distribution of the neutron radiative beta decay for polarized neutrons, polarized electrons and unpolarized protons and photons Following [3,16,18,19] we define the electron-photon-energy and angular distribution of the neutron radiative beta decay for a polarized neutron, a polarized electron, a polarized photon and an unpolarized proton as follows where we sum over polarizations of massive fermions. Since we calculate the contribution of the neutron radiative beta decay to leading order in the large nucleon mass m N expansion, the contribution of the phase volume of the decay is equal to unity. The photon state is determined by the 4-momentum q µ = (ω, q ) and the 4-vector of polarization ε µ (q) λ with λ = 1, 2, obeying the constraints ε * (q) λ ′ · ε λ (q) = −δ λ ′ λ and q · ε λ (q) = 0. In the tree-approximation and to leading order in the large nucleon mass m N expansion the amplitude of the neutron radiative beta decay is equal to [16] M (n → p e −ν e γ) λ = e G V m n ω The hermitian conjugate amplitude is determined by where n = q/ω, Q = 2(ε * · k e ) +ε * q andQ = γ 0 Q † γ 0 = 2(ε · k e ) +qε. Then, ϕ n and ϕ p are the Pauli wave functions of the neutron and proton, u e and vν are the Dirac wave functions of the electron and antineutrino, respectively. The sum over polarizations of the massive fermions is equal to [16,18,19] pol.
Having calculated the traces over the nucleon degrees of freedom and using the properties of the Dirac matrices Eq.(A-8) we transcribe the r.h.s. of Eq.(B-4) into the form [16,18,19] pol.
Plugging Eq.(B-9) into Eq.(B-1) we obtain the electron-energy and angular distribution for a polarized neutron, a polarized electron, an unpolarized proton and a polarized photon: . (B-10) Summing up over polarizations of the photon we get where we have used that β · ζ e = ζ 0 e . The next step is to average over directions of the 3-momentum q = ω n of the real photon. This gives The integration over the directions of the vector n we carry out by using the results obtained in [18]. We get d 6 λ β − c γ (E e , ω, k e , kν, ξ n , ξ e ) dωdE e dΩ e dΩν = (1 + 3g 2 A ) It is seen that the terms with the correlation structures ( ξ n · ξ e )( k e · kν) and ( ξ n · k e )( ξ e · kν ), inducing the correlation coefficients S(E e ) and U (E e ), respectively, do not appear in the electron-energy and angular distribution of the neutron radiative beta decay for polarized neutrons, polarized electrons, unpolarized protons and unpolarized photons. This confirms the results, obtained in Appendix A, that there are no contributions of the radiative corrections of order O(α/π), caused by one-virtual photon exchanges, to the correlation coefficients S(E e ) and U (E e ), respectively.