Test of the Standard Model in Neutron Beta Decay with Polarized Electron and Unpolarized Neutron and Proton

We calculate the correlation coefficients of the electron-energy and electron-antineutrino angular distribution of the neutron beta decay with polarized electron and unpolarised neutron and proton. The calculation is carried out within the Standard Model (SM) with the contributions, caused by the weak magnetism, proton recoil and radiative corrections of order of 10^{-3}, Wilkinson's corrections of order 10^{-5}$(Wilkinson, Nucl. Phys. A377, 474 (1982) and Ivanov et al., Phys. Rev. C95, 055502 (2017)) and the contributions of interactions beyond the SM. The obtained results can be used for the analysis of experimental data on searches of interactions beyond the SM at the level of 10^{-4} (Abele, Hyperfine Interact. 237, 155 (2016)). The contributions of G-odd correlations are calculated and found at the level of 10^{-5} in agreement with the results obtained by Gardner and Plaster (Phys. Rev. C87, 065504 (2013)) and Ivanov et al. (Phys. Rev. C98, 035503 (2018)).

The paper is organized as follows. In section II we write down the general expression for the electron-energy and electron-antineutrino angular distribution of the neutron β − -decay with polarized electron and unpolarized neutron and proton. In section III we discuss the renormalization procedure of the amplitude of the neutron β −decay, caused by the effective V − A weak interaction and radiative corrections, calculated to order O(α/π) in the one-photon exchange approximation. In section IV we calculate the renormalized electron-energy and electronantineutrino angular distribution to order O(E e /M ) and O(α/π), caused by the weak magnetism, proton recoil and radiative corrections, dependent on the infrared cut-off µ and obtained within the finite-photon mass regularization [1,5]. In section V using the Dirac wave function of the decay electron, distorted in the Coulomb field of the decay proton, we calculate the correlation coefficient L(E e ), responsible for time reversal violation. In section VI we calculate the electron-energy and electron-antineutrino angular distribution of the neutron radiative β − -decay with polarized electron and unpolarized neutron and proton. We use these results for a cancellation of the infrared divergences in the observable electron-energy and electron-antineutrino angular distribution of the neutron β − -decay with polarized electron and unpolarized neutron and proton. The results, obtained in this section can be also used for the experimental analysis of the neutron radiative β − -decay with polarized electron and unpolarized neutron and proton. In section VII we write down the observable electron-energy and electron-antineutrino angular distribution, calculated in the SM to 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). We show that the radiative corrections to the correlation coefficients H(E e ) and K(E e ) are defined by the functions (α/π) h (3) n (E e ) and (α/π) h (4) n (E e ), calculated for the first time in the present paper. The radiative corrections (α/π) h (3) n (E e ) and (α/π) h (4) n (E e ) are plotted in Fig. 3. In section VIII we adduce the analytical expressions for the correlation coefficients a(E e ), G(E e ), H(E e ), K e (E e ) and L(E e ), calculated in the SM to order 10 −3 , caused by the weak magnetism, proton recoil and radiative corrections. The obtained results can be used for the analysis of the experimental data on the neutron β − -decay with polarized electron and unpolarized neutron and proton. In section IX we discuss Wilkinson's corrections of order 10 −5 , which have not been taken into account for the calculation of the correlation coefficients in section VIII. They are caused by i) the proton recoil in the Coulomb electron-proton final-state interaction, ii) the finite proton radius, iii) the proton-lepton convolution and iv) the higher-order outer radiative corrections [8]. We calculate the contributions to the correlation coefficients, induced by the change of the Fermi function caused by the proton recoil in the electron-proton final-state Coulomb interaction. We plot these corrections in the electron-energy region 0.761 MeV ≤ E e ≤ 0.966 MeV in Fig. 4. We point out that Wilkinson's corrections of order 10 −5 , caused by ii) the finite proton radius, iii) the proton-lepton convolution and iv) the higher-order outer radiative corrections and calculated in [2], retain fully their shapes and values for the correlation coefficients analysed in the present paper. In sections X and XI we calculate the contributions to the correlation coefficients, caused by interactions beyond the SM [9]- [20] (see also [1,3]), and give the correlation coefficients in the form suitable for the analysis of experimental data on searches of contributions of interactions beyond the SM [21] (see also [1,3]). In section XII we discuss the obtained results and perspectives of the theoretical background to order 10 −5 , which goes beyond the scope of Wilkinson's corrections of order 10 −5 [2,22].
The function ζ(E e ) and the correlation coefficients a(E e ) and G(E e ) have been calculated in [1][2][3]. They are defined by the contributions of order 10 −3 of the SM interactions, Wilkinson's corrections of order 10 −5 and interactions beyond the SM (see [1][2][3] and [30]). In this paper we calculate the correlation coefficients H(E e ), K e (E e ) and L(E e ), where the correlation coefficient L(E e ) is responsible for violation of invariance under transformation of time reversal. We calculate i) a 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), ii) Wilkinson's corrections of order 10 −5 [8] (see also [1,2]), iii) contributions of interactions beyond the SM [11]- [14] (see also [1,3]) and iv) second class contributions or G-odd correlations [19,20]) (see also [3]).
In the SM of electroweak interactions the neutron β − -decays, defined in the one-loop approximation with onevirtual-photon exchanges, are described by the following interactions Here L W (x) is the effective Lagrangian of low-energy V − A interactions with a real axial coupling constant λ = −1.2750(9) [26] (see also [1,2]) where ψ 0p (x), ψ 0n (x), ψ 0e (x) and ψ 0ν (x) are bare field operators of the proton, neutron, electron and antineutrino, respectively, G 0F is a bare Fermi weak coupling constant, and γ µ = (γ 0 , γ ) and γ 5 are the Dirac matrices [31]; κ = κ p − κ n = 3.7058 is the isovector anomalous magnetic moment of the nucleon, defined by the anomalous magnetic moments of the proton κ p = 1.7928 and the neutron κ n = −1.9130 and measured in nuclear magneton [4], and M = (m n + m p )/2 is the average nucleon mass. For the calculation of the radiative corrections to order O(α/π) the Lagrangian of the electromagnetic interaction L em (x) we take in the following form [22] where µ (x) is the electromagnetic field strength tensor of the bare (unrenormalized) electromagnetic field operator A (0) µ (x); ψ 0e (x) and ψ 0p (x) are bare operators of the electron and proton fields with bare masses m 0e and m 0p , respectively; −e 0 and +e 0 are bare electric charges of the electron and proton, respectively. Then, ξ 0 is a bare gauge parameter. After the calculation of the one-loop corrections of order O(α/π) a transition to the renormalized field operators, masses and electric charges is defined by the Lagrangian where A µ (x), ψ e (x) and ψ p (x) are the renormalized operators of the electromagnetic, electron and proton fields, respectively; m e and m p are the renormalized masses of the electron and proton; e is the renormalized electric charge; and ξ is the renormalized gauge parameter. The Lagrangian δL em (x) contains a complete set of the counterterms [32], FIG. 1: The Feynman diagrams, defining the main contribution of the radiative corrections of order O(α/π), caused by one-virtual photon exchanges, to the neutron β − -decay (see Sirlin [5]). where Z 3 , Z are the renormalization constants of the proton field operator ψ p and the proton-proton-photon (ppγ) vertex, respectively. Then, (−e) and (+e), m e and m p and δm e and δm p are the renormalized electric charges and masses and the mass-counterterms of the electron and proton, respectively. Rescaling the field operators [32,33] and denoting m e + δm e = m 0e , m p + δm p = m 0p and Z ξ ξ = ξ 0 we arrive at the Lagrangian Because of the Ward identities Z = +e 0 . This brings Eq.(9) to the form of Eq.(5). We would like to emphasize that to order O(α/π) the renormalization constant Z 3 is equal to unity because of the absent of closed fermion loops [31][32][33], i.e., Z 3 = 1. This means that in such an approximation the bare electric charge e 0 coincides with the renormalized electric charge e, i.e. e 0 = e. After the rescaling of the proton and electron field operators Eq.(8) the Lagrangian of V − A weak interactions Eq.(4) takes the form where 2 G 0F is the Fermi weak coupling constant renormalized by electromagnetic interactions to order O(α/π). The bare neutron ψ 0n (x) and antineutrino ψ 0ν (x) field operators are not renormalized by electromagnetic interactions and coincide with the field operators ψ n (x) and ψ ν (x), respectively, i.e. ψ 0n (x) = ψ n (x) and ψ 0ν (x) = ψ ν (x).

IV. ELECTRON-ENERGY AND ELECTRON-ANTINEUTRINO ANGULAR DISTRIBUTION WITH RADIATIVE CORRECTIONS CAUSED BY ONE-VIRTUAL PHOTON EXCHANGES
Using the results, obtained in [1], the renormalized amplitude of the neutron β − -decay with contributions, caused by the weak magnetism and proton recoil, calculated to next-to-leading order O(E e /M ) in the large nucleon mass expansion, and radiative corrections to order O(α/π), defined by the Feynman diagrams in Fig. 1 and calculated to leading order in the large nucleon mass expansion, takes the form (see Eq.(D-52) of Ref. [1]) 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, andλ = λ(1 − E 0 /2M ) and k p = − k e − k ν is the proton 3-momentum in the rest frame of the neutron. The functions where µ is a photon mass, which should be taken in the limit µ → 0, and Li 2 (x) is the Polylogarithmic function. A photon mass µ is used for Lorentz invariant regularization of infrared divergences of radiative corrections [5]. The constant C W Z , defined by the contributions of the W -boson and Z-boson exchanges and the QCD corrections [34] (see also [35,36]), is equal to C W Z = 10.249 (see also discussion below Eq.(D-58) of Ref. [1]). The squared absolute value of the matrix element Eq.(12), summed over polarizations of massive fermions, we calculate for polarized electron and unpolarized neutron and proton [2]. We get (see also Eq.(A- 16) where ζ µ e = (ζ 0 e , ζ e ) is the 4-vector of an electron polarization defined by [2] ζ It obeys the constraints ζ 2 e = −1 and k e · ζ e = 0, where ξ e is a unit vector of the electron polarization [31]. We would like to emphasize that in Eq.(13) following Sirlin [5] we have neglected the contributions of order O(αE e /πM ). Having calculated the traces over Dirac matrices we obtain where we have used a relation E e + E ν = E 0 . Now we have to take into account the contribution of the phase-volume [1] and multiply Eq.(15) by the function This gives where we have denoted a 0 = (1 − λ 2 )/(1 + 3λ 2 ) and The use of the Dirac wave function of a free decay electron leads to a vanishing correlation coefficientL(E e ) = 0. In order to get a non-vanishing correlation coefficientL(E e ) we have to use the Dirac wave function of a decay electron, distorted in the Coulomb field of the decay proton [24,25,37].
For the calculation of the correlation coefficient we use the Dirac wave function of the electron, distorted by the Coulomb proton-electron final state interaction. It is equal to [24,25,37] where We normalize the wave function Eq.
, keeping the contributions of order O(α) we have to set γ = 0. The contribution of the Coulomb distortion to the right-hand-side (r.h.s) of Eq. (15), multiplied by the contribution of the phase-volume Eq. (15) is defined by the trace We would like to emphasize that the contribution of the Coulomb distortion of the Dirac wave function of a decay electron to the correlation coefficient comes from the traces of V ×V and A×A products only, i.e. tr{V ×V +A×A} ∼ (1 − λ 2 ). Thus, we get The correlation coefficientζ(E e )L(E e ) is equal tõ where we have set Z = 1. Thus, the electron-energy and electron-antineutrino angular distribution of the neutron β − -decay with polarized electron and unpolarized neutron and proton is The radiative corrections to the correlation coefficients, defined by the function f β − c (E e , µ), depend on the infrared cut-off µ. In order to remove such a dependence we have to add the contribution of the neutron radiative β − -decay [5](see also [1,2]).

VI. NEUTRON RADIATIVE β − -DECAY WITH POLARIZED ELECTRON AND UNPOLARIZED NEUTRON AND PROTON
Following [1,2] (see also [22,30]) the energy and angular distribution of the neutron radiative β − -decay with polarized electron and unpolarized neutron and proton is where dΩ e , dΩ ν and dΩ γ are elements of the solid angels of the electron, antineutrino and photon, respectively. Then, where ε * λ (q) (or ε λ ′ (q)) and q = (ω, q ) = (ω, ω n q ) are the polarization vector and 4-momentum of the photon obeying the constraints ε * λ (q) · q = 0 (or ε λ ′ (q) · q = 0) and q 2 = 0, n q = q/ω is a unit vector and λ(λ ′ ) = 1, 2 defines physical polarization states of the photon. In Eq.(25) the traces over Dirac matrices in the covariant form are defined by where a = k e and a = m e ζ e , and ε ανµβ is the Levi-Civita tensor defined by ε 0123 = 1 and ε ανµβ = −ε ανµβ [31]. Plugging Eq.(26) into Eq. (25), using the Coulomb gauge [1,2] (see also [22,30]) and summing over photon polarizations we obtain the following expression for the energy and angular distribution of the neutron radiative β − -decay The integration over directions of the photon momentum we carry out using the results obtain in the Appendix of Ref. [2]. As result the energy and angular distribution Eq.(27) takes the form The first three correlation coefficients agree well with the results, obtained in [1] (see Eq.(B-11) of Ref. [1]) and [2] (see Eq.(A-5) of Ref. [2]). Having integrated over the photon energy in the region ω min ≤ ω ≤ E 0 − E e , where ω in is an infrared cut-off [1], we arrive at the expression The functions g (1) β − c γ (E e , ω min ) and g (2) β − c γ (E e , ω min ) have been calculated in [1,2], whereas the functions g (5) β − c γ (E e , ω min ) and g (6) β − c γ (E e , ω min ) are defined by the integrals The results of the integration are equal to Now we are able to define the electron-energy and electron-antineutrino angular distribution for the neutron β −decay with polarized electron and unpolarized neutron and proton, where the correlation coefficients are calculated to 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).

VII. ELECTRON-ENERGY AND ELECTRON-ANTINEUTRINO ANGULAR DISTRIBUTION OF NEUTRON β − -DECAY WITH POLARIZED ELECTRON AND UNPOLARIZED NEUTRON AND PROTON TO ORDER 10 −3
Summing the electron-energy and electron-antineutrino angular distributions Eq.(24) and Eq.(29) we obtain the electron-energy and electron-antineutrino angular distribution of λ n = λ β − c + λ β − γ equal to The correlation coefficients are equal to ζ(E e ) = 1 + α π g n (E e ) + 1 M The radiative corrections of order O(α/π) to the correlation coefficients are defined by the function g n (E e ) and the functions The functions g n (E e ) and f n (E e ) have been calculated by Sirlin [5] and Shann [6] (see also [7] and Appendices B, C, D, E and F in Ref. [1]), respectively. The contributions of the electroweak-boson exchanges and QCD corrections to the function g n (E e ) have been calculated in [34][35][36]. The radiative corrections (α/π) f n (E e ), (α/π) h n (E e ) and (α/π) h (4) n (E e ) are plotted in Fig. 3 in the electron-energy region m e ≤ E e ≤ E 0 . As has been pointed out in [1], the result of the calculation of the integral    logarithmically divergent in the infrared region of photon energy depends on the regularization procedure, where κ IR is an infrared parameter. Using the infrared cut-off regularization κ IR = ω min ≤ ω ≤ (E 0 − E e ), where ω min may be also treated as a photon-energy threshold of the detector, we get In turn, the use of the finite photon-mass (FPM) regularization where q 0 = ω 2 + µ 2 and v = q/q 0 are energy and velocity of a photon with mass µ, gives one (see Eq.(B-26) of Ref. [1]) where Li 2 (x) is a Polylogarithmic function [38]. The use of the FPM regularization, which is a Lorentz invariant regularization, is important for the calculation of the function g n (E e ), defining the radiative corrections to the neutron lifetime [5]. It is required by gauge invariance of radiative corrections and by the Kinoshita-Lee-Nauenberg theorem [5] (see also [1]). In turn, for the calculation of the functions f n (E e ) and h (ℓ) n (E e ), where ℓ = 1, 2 [2] and ℓ = 3, 4 (see Eq.(34)), the contributions of the integral J(β, κ IR ) cancel themselves in the differences lim κIR→0 [g  The correlation coefficients a(E e ) and G(E e ) have been calculated in [1] and [2], respectively. They are equal to a(E e ) = 1 + α π f n (E e ) a 0 + 1 M For the correlation coefficients H(E e ) and K e (E e ) we obtain the following expressions and The obtained correlation coefficients are calculated to order 10 −3 , taking into account the complete set of corrections of order O(E e /M ) and O(α/π), caused by the weak magnetism, proton recoil and one-photon exchanges, respectively.

IX. WILKINSON'S CORRECTIONS
According to Wilkinson [8], the higher order corrections with respect to those calculated in section VIII should be caused by i) the proton recoil in the Coulomb electron-proton final-state interaction, ii) the finite proton radius, iii) the proton-lepton convolution and iv) the higher-order outer radiative corrections.
The relative corrections to the correlation coefficients ζ(E e ), a(E e ), G(E e ), H(E e ) and K e (E e ), caused by the proton recoil in the final state electron-proton Coulomb interactions, are equal to  Table I. The proton recoil corrections to the correlation coefficient a(E e ), caused by the electron-proton final-state Coulomb interactions, are of order 10 −4 and should be taken into account for the analysis of the experimental data on searches of contributions of interactions beyond the SM at the level of 10 −4 [21]. In turn, Wilkinson's corrections, caused by ii) the finite proton radius, iii) the proton-lepton convolution and iv) the higher-order outer radiative corrections, retain their expression for calculated in [2]

X. ELECTRON-ENERGY AND ELECTRON-ANTINEUTRINO ANGULAR DISTRIBUTION BEYOND THE SM
For the calculation of contributions of interactions beyond the SM we use the effective low-energy Hamiltonian of weak nucleon-lepton four-fermion local interactions, taking into account all phenomenological couplings beyond the SM [9]- [20] in the notations of [1,3]: This is the most general form of the effective low-energy weak interactions, where the phenomenological coupling constants C i andC i for i = V, A, S, P and T can be induced by the left-handed and right-handed hadronic and leptonic currents [9]- [14]. They are related to the phenomenological coupling constants, analogous to those which were introduced by Herczeg [13], as follows where the index h means that the phenomenological coupling constants are introduced at the hadronic level but not at the quark level as it has been done by Herczeg [13]. In the SM the phenomenological coupling constants C i andC i for i = V, A, S, P and T are equal to . The phenomenological coupling constants a h ij , A h ij and α h jj for i(j) = L or R are induced by interactions beyond the SM.
The contribution of interactions beyond the SM, given by the Hamiltonian of weak interactions Eq. (6), to the amplitude of the neutron β − -decay, calculated to leading order in the large nucleon mass expansion, takes the form The hermitian conjugate amplitude is The contributions of interactions with the strength, defined by the phenomenological coupling constants C P andC P , may appear only of order O(C P E e /M ) and O(C P E e /M ) and can be neglected to leading order in the large nucleon mass expansion. We have also neglected the contributions of the neutron-proton mass difference. The squared absolute value of the amplitude Eq.(8), summed over polarizations of massive fermions, is equal to pol.
The structure of the correlation coefficients in Eq.(47) agrees well with the structure of the corresponding expressions obtained in [11]. In the linear approximation for coupling constants of vector and axial-vector interactions beyond the SM [1] we get pol.
where we have replaced C j andC j with j = V, A by and neglected also the contributions of the products δC j C k , δC j C k and so on for j = V, A and k = S, T . Following [16,17](see also [1]) we have absorbed the contributions the vector and axial vector interactions beyond the SM by the axial coupling constant λ and the CKM matrix element V ud . Thus, the electron-energy and electron-antineutrino angular distribution Eq.(1), taking into account the contributions of interactions beyond the SM, can be transcribed into the form where the indices "SM" and "BSM" mean "Standard Model" and "Beyond Standard Model", respectively. The correlation coefficient ζ (SM) (E e ) is given in Eq. (32). The Fierz interference term b and the correlation coefficients X eff (E e ) with X = a, G, H and K e are defined by where the correlation coefficients with index "SM" are adduced in Eqs.(39) - (41). They should be also supplemented by Wilkinson's corrections Eq. (42) and those obtained in [2] (see Chapter III of Ref. [2]). The correlation coefficients b F and the correlation coefficients with index "BSM" are given by b F = 1 1 + 3λ 2 Re (C S −C S ) + 3λ (C T −C T ) , The correlation coefficient X eff (E e ) with X = a, G, H and K e are given in the form suitable for the analysis of experimental data of experiments on the searches of interactions beyond the SM [21]. The structure of the correlation coefficients in Eq.(51) agrees well with the structure of corresponding expressions calculated in [11]. The averaged values of the correlation coefficients X eff (E e ) with X = a, G, H and K e can be obtained with the electron-energy density [3] where the electron-energy density ρ The contributions of the G-odd correlations to the squared absolute value of the amplitude of the neutron β − -decay of polarized electron and unpolarized neutron and proton, summed over polarizations of massive fermions, are equal to Following Gardner and Plaster [20] and setting f 3 (0) = 0 and |Reg 2 (0)| < 0.01 we obtain the contributions of the G-odd correlations at the level of 10 −5 . Of course, the same order of magnitude of the G-odd correlations one may get also for |Ref 3 (0)| < 0.01 [3].

XII. DISCUSSION
We have analysed the electron-energy and electron-antineutrino angular distribution of the neutron β − -decay with polarized electron and unpolarized neutron and proton. The correlation coefficients are calculated in the SM to 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) Eqs.(39) - (41). The radiative corrections to the correlation coefficients H(E e ) and K e (E e ) are defined by the functions (α/π) h