On the correlation coefficient $T(E_e)$ of the neutron beta decay, caused by the correlation structure invariant under discrete P, C and T symmetries

We analyze the correlation coefficient $T(E_e)$, which was introduced by Ebel and Feldman (Nucl. Phys. 4, 213 (1957)). The correlation coefficient $T(E_e)$ is induced by the correlation structure $(\vec{\xi}_n\cdot \vec{k}_{\bar{\nu}})(\vec{k}_e\cdot \vec{\xi}_e)/E_e E_{\bar{\nu}}$, where $\vec{\xi}_{n,e}$ are unit spin-polarization vectors of the neutron and electron, and $(E_{e,\bar{\nu}}, \vec{k}_{e,\bar{\nu}})$ are energies and 3-momenta of the electron and antineutrino. Such a correlation structure is invariant under discrete P, C and T symmetries. The correlation coefficient $T(E_e)$, calculated to leading order in the large nucleon mass m_N expansion, is equal to $T(E_e) = - 2 g_A(1 + g_A)/(1 + 3 g^2_A) = - B_0$, i.e. of order $|T(E_e)|\sim 1$, where $g_A$ is the axial coupling constant. Within the Standard Model (SM) we describe the correlation coefficient $T(E_e)$ at the level of $10^{-3}$ by taking into the radiative corrections of order $O(\alpha/\pi)$ or the outer model-independent radiative corrections, where $\alpha$ is the fine-structure constant, and the corrections of order $O(E_e/m_N)$, caused by weak magnetism and proton recoil. We calculate also the contributions of interactions beyond the SM, including the contributions of the second class currents.


I. INTRODUCTION
After the discovery by Chadwick in 1932 [1], neutron has started to play an unprecedentedly important role in particle and nuclear physics, astrophysics and cosmology [2]- [8]. The experimental analysis of correlation coefficients of the neutron beta decay allows to obtain precise information about the structure of the Standard Model (SM) interactions [9][10][11][12][13][14] and interactions beyond the SM [2,3,[15][16][17] at low and high energies [18][19][20][21][22][23][24]. For the first time the most general form of the electron-energy and angular distribution of the neutron beta decay for a polarized neutron, a polarized electron and an unpolarized proton was proposed by Jackson et al. [25][26][27]. Following Jackson et al. [25] the electron-energy and angular distribution of the neutron beta decay for a polarized neutron, a polarized electron and an unpolarized proton can be represented as follows d 5 λ n (E e , k e , kν, ξ n , ξ e ) dE e dΩ e dΩν = (1 + 3g 2 A ) 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) where we have used the notations in Refs. [28] - [38]. Then, g A and G V are the axial and vector coupling constants, respectively, [2,3,14,36,37], ξ n and ξ e are unit spin-polarization vectors of the neutron and electron [28,29,31] (see also [39]), respectively, dΩ e and dΩν are infinitesimal solid angles 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-energy spectrum [2,3], F (E e , Z = 1) is the relativistic Fermi function, describing the electron-proton final-state Coulomb interaction, is equal to (see, for example, [40,41] and a discussion in [29]) F (E e , Z = 1) = 1 + 1 2 γ 4(2r p m e β) 2γ Γ 2 (3 + 2γ) 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 [42]. The correlation coefficient b(E e ) is the Fierz interference term [43]. The structure and the value of the Fierz interference term may depend on interactions beyond the SM [43]. An information of a contemporary theoretical and experimental status of the Fierz interference term can be found in [44][45][46][47][48] (see also [32]). The correlation coefficients of the electron-antineutrino correlations a(E e ), the electron asymmetry A(E e ), the antineutrino asymmetry B(E e ) and others G(E e ), H(E e ), Q e (E e ) and K e (E e ) survive to leading order in the large nucleon mass m N expansion [2,3,29,31] and depend on the axial coupling constant g A only. The correlation coefficients a(E e ) and Q e (E e ) do not violate invariance under discrete symmetries, i.e. parity conservation (P-invariance), time-reversal invariance (T-invariance) and charge conjugation invariance (C-invariance) [11]. This is unlike the correlation coefficients A(E e ), B(E e ), G(E e ), H(E e ) and K e (E e ), which provide a quantitative information about effects of parity violation (P-odd effects) [26]. In turn, the correlation coefficients K n (E e ) and Q n (E e ) appear only to next-to-leading order in the large nucleon mass m N expansion. They are caused by the contributions of weak magnetism and proton recoil [49] (see also [41,50,51] and [28]) and measure the strength of P-odd effects. The correlation coefficients D(E e ), R(E e ) and L(E e ) characterize quantitatively the strength of effects of violation of time-reversal invariance (T-odd effects). In addition, the correlation coefficients D(E e ) and L(E e ) are responsible also for violation of charge invariance (C-odd effects), whereas the correlation coefficient R(E e ) defines also quantitatively effects of violation of parity invariance (P-odd effects) [25,26]. In the SM the correlation coefficients D(E e ), R(E e ) and L(E e ) are induced by the distortion of the Dirac wave function of the electron in the Coulomb field of the proton [26,27,52,53] (see also [29,31]). The radiative corrections of order O(α/π), where α is the fine-structure constant [14], to the neutron lifetime and correlation coefficients in Eq.(1) were calculated in [54] - [62] (see also [50,51] and [28,29,31]). The corrections of order O(E e /m N ), caused by weak magnetism and proton recoil, were calculated in [49], [41,50,51] (see also [28,29,31]). So one may argue that within the SM the correlation coefficients in Eq.(1) were fully investigated theoretically to order 10 −3 , caused by radiative corrections of order O(α/π) and the corrections of order O(E e /m N ), induced by weak magnetism and proton recoil.
The electron-energy and angular distribution in Eq.(1) was supplemented by the term with the correlation structure ( ξ n · kν)( k e · ξ e )/E e Eν and the correlations coefficient T (E e ) by Ebel and Feldman [63]. Such a correlation structure is invariant under discrete P, C and T symmetries [11,25,26]. The main peculiarity of the correlation coefficient T (E e ) in comparison to other correlation coefficients, introduced by Ebel and Feldman [63] in addition to the correlation coefficients proposed by Jackson et al. [25] in Eq.(1), is to be finite with the absolute value of order of 1, i.e. |T (E e )| ∼ 1, to leading order in the large nucleon mass m N expansion. This paper is addressed to the detailed analysis of the structure of the correlation coefficient T (E e ).
The paper is organized as follows. In section II within the standard effective V − A theory of weak interactions [64] we calculate the correlation coefficient T (E e ) to leading order in the large nucleon mass m N expansion. Such a correlation coefficient was introduced by Ebel and Feldman [63], who calculated the correlation coefficient by using the effective phenomenological interactions by Jackson et al. [25,26] only. We show that the absolute value of T (E e ) is of order of 1, i.e. |T (E e )| ∼ 1. According to [25,26], the correlation structure responsible for T (E e ) does not violate invariance under discrete P, C and T symmetries [11]. In section III within the SM we give a detailed description of the correlation coefficient T (E e ) at the level of 10 −3 including the radiative corrections of order O(α/π), calculated to leading order in the large nucleon mass m N expansion, and the corrections of order O(E e /m N ), caused by weak magnetism and proton recoil. In section IV we calculate the contribution of interactions beyond the SM expressed in terms of the phenomenological coupling constants of the effective low-energy weak interactions proposed by Jackson et al. [25]. In section V we calculate the contribution of the second class currents or the G-odd correlations (regarding G-parity invariance of strong interactions, we refer to the paper by Lee and Yang [65]) by Weinberg [66]. In section VI we discuss the obtained results and perspectives of i) an experimental analysis of the correlation coefficient T (E e ) and ii) an improvement of the SM description of T (E e ) at the level of 10 −5 . In section VIII we give a detailed calculation of the contribution of the neutron radiative beta decay n → p + e − +ν e + γ to the correlation coefficient T (E e ) providing a removal of the dependence of the radiative corrections on the infrared cut-off. The analysis of the electron-energy and angular distribution of the neutron beta decay for a polarized neutron, a polarized electron and an unpolarized proton within the standard effective V − A theory of weak low-energy interactions, carried out even to leading order in the large nucleon mass expansion, shows that the set of correlation coefficients in Eq.(1), which survive in such a limit, is not complete. In order to show this we use the standard effective Lagrangian of V − A low-energy weak interactions [64] where G V and g A are the vector and axial coupling constants [2,3,14], 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 ) and γ 5 are the Dirac matrices [39]. In the non-relativistic approximation for the neutron and proton the amplitude of the neutron beta decay, calculated with the effective Lagrangian Eq.(3), is [28] M where ϕ n and ϕ p are Pauli wave functions of the neutron and proton, whereas u e and vν are Dirac wave functions of the electron and antineutrino, σ are Pauli 2×2 matrices of the neutron spin [39]. According to [28], the electron-energy and angular distribution of the neutron beta decay for a polarized neutron, a polarized electron and an unpolarized proton, calculated with the effective Lagrangian Eq.(3), is given by where we sum over polarizations of the massive fermions. The sum over polarizations of the massive fermions is defined by [28,29,31] pol.
where ζ e is the 4-vector of the spin-polarization of the electron. It is defined by [39] ζ e = (ζ 0 e , ζ e ) = k e · ξ e m e , ξ e + ( k e · ξ e ) k e m e (E e + m e ) .
The 4-vector of the spin-polarization of the electron ζ e is normalized by ζ 2 e = −1. It obeys also the constraint k e · ζ e = 0 [39]. Calculating the traces over the nucleon degrees of freedom and using the properties of the Dirac matrices [39] where η µν is the metric tensor of the Minkowski space-time, ε ανµβ is the Levi-Civita tensor defined by ε 0123 = 1 and ε ανµβ = −ε ανµβ [39], we transcribe the right-hand-side (r.h.s.) of Eq.(6) into the form [28,29,31] pol.
where a 0 , A 0 and B 0 are defined in terms of the axial coupling constant g A only [2, 3] Having calculated the traces over lepton degrees of freedom, we arrive at the expression pol.
Plugging Eq.(11) into Eq. (5) we obtain the electron-energy and angular distribution of the neutron beta decay, calculated to leading order in the large nucleon mass m N expansion . We get From the comparison of Eq.(12) with Eq.(1) we determine the following correlation coefficients in the electron-energy and angular distribution, calculated to leading order in the large nucleon mass m N expansion: The correlation structure of the last term in Eq.(12) was introduced by Ebel and Feldman [63] with the correlation coefficient T (E e ). According to our analysis carried out above to leading order in the large nucleon mass expansion, it is equal to T (E e ) = −B 0 . The correlation structure responsible for the correlation coefficient T (E e ) is invariant under discrete P, C and T symmetries [11,25,26,39].
Having calculated the correlation coefficient T (E e ) to leading order in the large nucleon mass expansion and without radiative corrections, we may proceed to the analysis of the structure of this correlation coefficient by taking into account the radiative corrections of order O(α/π) or so-called outer model-independent radiative corrections [67] and corrections of order O(E e /m N ), caused by weak magnetism and proton recoil. This should give within the SM the theoretical description of the correlation coefficient T (E e ) at the level of 10 −3 .

III. THE CORRELATION COEFFICIENT T (Ee) TO ORDER 10 −3
The procedure of the calculation 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, has been expounded well in [28,29,31]. For the calculation of these corrections we use the following effective Lagrangian [28,29,31] In comparison to the effective Lagrangian Eq.(3), we have added i) in the hadronic current an additional term with the Lorentz structure, defined the Dirac matrix σ µν = i 2 (γ µ γ ν − γ ν γ µ ) [39], where κ = κ p − κ n = 3.7059 is the isovector anomalous magnetic moment of the nucleon, induced by the anomalous magnetic moments of the proton κ p = 1.7929 and the neutron κ n = −1.9130 and measured in nuclear magneton [14], and ii) the electromagnetic interactions for the proton and electron, where e is the electric charge of the proton and A µ (x) is the 4-vector potential of the electromagnetic field.
Using the amplitude of the neutron beta decay (see Eq.(D-52) in Appendix D of [28] with a replacementλ = −g A , whereg A = g A (1 − E 0 /2m N )), following [28,31] and skipping intermediate calculations we obtain where the function f β − c (E e , µ) was calculated by Sirlin [54] (for the calculation of the function f β − c (E e , µ) in details we refer to [28]), µ is a covariant infrared cut-off having a meaning of the photon mass [54]. For the definition of the function ζ(E e ) we refer to [25,26], where the function ζ(E e ) was calculated to leading order in the large nucleon mass m N expansion. However, below we shall use the function ζ(E e ), calculated in [28] to order 10 −3 by taking into account 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 (see also [51]). According to [67], the function f β − c (E e , µ) defines the contribution of model-independent or outer radiative corrections. Following [28] the function f β − c (E e , µ) can be rewritten as follows where 2g n (E e ) is Sirlin's function, defining the outer radiative corrections of order O(α/π) to the neutron lifetime [54]. The function g corresponds to the contribution of the neutron radiative beta decay n → p + e − +ν e + γ with a real photon γ, which should be added, according to Berman [68] and Kinoshita and Sirlin [69] (see also Sirlin [54]), for the removal of the dependence of the neutron lifetime on the infrared cut-off. Following [28] the function g where q 0 = q 2 + µ 2 , q = | q | and v = q/q 0 . The analytical expression of the function g , which was calculated in [28], we adduce in section VIII for completeness. Then, in Eq. (17) we integrate over directions of the 3-momentum q of the photon [28]. As has been shown in [28,29,31] for the calculation of the radiative corrections to the correlation coefficients the function g (1) β − c γ (E e , µ) can be also regularized by a non-covariant infrared cut-off ω min . Setting µ = 0 in Eq. (17) and integrating over directions of the 3-momentum of the photon we arrive at the integral [28] (see Eq.(B-15) in Appendix B in Ref. [28]) where ω min is a non-covariant infrared cut-off, which can be also treated as the photon-energy threshold of the detector [28].
We would like to emphasize that the function g (1) β − c γ (E e , µ) removes the dependence on the infrared cut-off µ only in the neutron lifetime or in the function ζ(E e ) [28]. The dependence of the radiative corrections on the infrared cut-off in the correlation coefficients of the neutron beta decay should be removed by the contributions of the corresponding correlation coefficients in the neutron radiative beta decay [28,29,31]. The correlation coefficient of the neutron radiative beta decay, caused by the correlation structure ( ξ n · kν)( ξ e · k e )/E e Eν, we have calculated in section VIII. It is given by the function g Since, according to [28,29,31], the difference lim µ→0 g we may rewrite Eq.(19) as follows Using Eqs. (18) and (A-11), taking the limit ω min → 0 and integrating over the photon energy ω in the region 0 ≤ ω ≤ E 0 − E e (see, for example, [28]) we define the correlation coefficient ζ(E e )T (E e ) by taking into account the outer radiative corrections of order O(α/π), calculated to leading order in the large nucleon mass m N expansion, and the corrections of order O(E e /m N ), caused by weak magnetism and proton recoil where the function f n (E e ) is equal to (see also [28,29]) For the first time the function f n (E e ), defining the radiative corrections (α/π) f n (E e ) to the correlation coefficients of the electron-antineutrino correlations a(E e ) and of the electron asymmetry A(E e ), was calculated by Shann [55] (see also Eq.(D-58) in Appendix D in Ref. [28]). Using the correlation function ζ(E e ), calculated in [28] (see Eq. (6) with the replacement λ = −g A ), we define the This is a complete description of the correlation coefficient T (E e ) at the level of 10 −3 including the outer radiative corrections of order O(α/π), calculated to leading order in the large nucleon mass m N expansion, and the corrections of order O(E e /m N ), caused by weak magnetism and proton recoil.

IV. CONTRIBUTIONS OF INTERACTIONS BEYOND THE STANDARD MODEL [25]
For the calculation of the contributions of interactions beyond the SM we use the effective phenomenological Lagrangian proposed in [25] (see also [63,71,72]). It reads where we have followed the notations in [28]. The effective phenomenological Lagrangian L BSM (x) reduces to the standard effective Lagrangian L W (x) of V −A weak low-energy interactions Eq.(3) by the replacement C V = −C V = 1, C A = −C A = g A and C S =C S = C P =C P = C T =C T = 0. Following [28] (see Appendix G in Ref. [28]) and skipping intermediate calculations, carried out in the approximation of the leading order in the large nucleon mass m N expansion, we get where the factor ξ is equal to [25,28] Our result in Eq.(26) agrees well with the result obtained by Ebel and Feldman [63] but without contributions of imaginary parts of the phenomenological scalar and tensor coupling constants, which are proportional to the factor αm e /k e , caused by the distortion of the Dirac wave function of the electron in the Coulomb field of the proton [26]. In this connections, we wold like to remind that, according to [44,45], the phenomenological scalar coupling constants should be zero, i.e. C S =C S = 0. Then, we would like to notice that the contribution of the phenomenological pseudoscalar coupling constants C P andC P vanishes to leading order in the large nucleon mass expansion (see, for example, [34]).

V. CONTRIBUTIONS OF THE SECOND CLASS CURRENTS OR THE G-ODD CORRELATIONS
The G-parity transformation, i.e. G = C e iπI2 , where C and I 2 are the charge conjugation and isospin operators, was introduced by Lee and Yang [65] as a symmetry of strong interactions. According to the properties of hadronic currents under G-transformation, Weinberg divided hadronic currents into two classes, which are G-even first class and G-odd second class currents [66], respectively. Following Weinberg [66], Gardner and Zhang [73], and Gardner and Plaster [74] (see also [29,31]) the G-odd contributions to the matrix element of the hadronic n → p transition, caused by the hadronic current, in the V − A theory of weak interactions can be taken in the form where J (+) µ (0),ū p ( k p , σ p ) and u n ( k n , σ n ) are the Dirac wave functions of the proton and neutron [30]. Then, f 3 (0) and g 2 (0) are the phenomenological coupling constants defining the strength of the second class currents in the weak decays. Following [30,31] and skipping intermediate calculations we get the contribution of the second class currents or the G-odd correlations to the correlation coefficient T (E e ) Apart from the dependence of T (G−odd) (E e ) on the axial coupling constant g A , the contribution of the second class currents or the G-odd correlations to the correlation coefficient T (E e ) is represented by the phenomenological coupling constant Reg 2 (0) only.

VI. DISCUSSION
We have analyzed the correlation coefficient T (E e ), caused by the correlation structure ( ξ n · kν )( ξ e · k e )/E e Eν invariant under discrete P, C and T symmetries. Such a correlation structure was introduced by Ebel and Feldman [63] in addition to the set correlation structures proposed by Jackson et al. [25,26]. The correlation coefficient T (E e ), calculated to leading order in the large nucleon mass m N expansion within the standard effective V − A theory of weak interactions [64], is equal to T (E e ) = −B 0 , where B 0 ∼ 1 [2,3]. Having calculated the correlation coefficient T (E e ) to leading order in the large nucleon mass m N expansion, we have given within the SM a complete description of the correlation coefficient T (E e ) at the level of 10 −3 by taking into account the outer model-independent radiative corrections of order O(α/π), calculated to leading order in the large nucleon mass expansion, and the corrections of order O(E e /m N ), caused by weak magnetism and proton recoil. In addition we have calculated the contributions of interactions beyond the SM, expressed in terms of i) the phenomenological coupling constants of the effective phenomenological interactions proposed by Jackson et al. [25], and ii) the phenomenological coupling constants of the second class currents, measuring the strength of G-odd correlations [66,73,74] (see also [30,31]). We have found that in the linear approximation for the phenomenological vector, axial-vector, scalar and tensor coupling constants [18][19][20][21][22] (see also [28,29,31]) the contribution of interactions beyond the SM, defined by the effective Lagrangian Eq. (25), is equal to zero, i.e. T (BSM) (E e ) = 0. As a result, in such an approximation the interactions beyond the SM are represented in the correlation coefficient T (E e ) by the second class currents or G-odd correlations only. Our result in Eq. (26) for the contribution of interactions beyond the SM, defined by the phenomenological coupling constants of the phenomenological interactions proposed by Jackson et al. [25], agrees well with the result obtained by Ebel and Feldman [63] but without contributions of the imaginary parts of the phenomenological scalar and tensor coupling constants proportional to the factor αm e /k e , caused by the distortion of the Dirac wave function of the electron in the Coulomb field of the proton. Then, we would like to remind that, according to [44,45], the contributions of the phenomenological scalar coupling constants should vanish.
Summing up all corrections we obtain the following expression for the correlation coefficient T (E e ): The correlation coefficient T (E e ), calculated for g A = 1.2764 [36], E 0 = 1.2926 MeV, m N = 938.9188 MeV, m e = 0.5110 MeV and κ = 3.7059 [14], is equal to According to [73,74] and [30,31], in the correlation coefficient T (E e ) the contribution of the second class currents may be estimated at the level of a few parts of 10 −5 and even smaller. Thus, the numerical analysis of the correlation coefficient T (E e ) shows that such a correlation coefficient can be a nice tool for experimental probes of contributions of the second class currents or G-odd correlations in terms of the phenomenological coupling constant Reg 2 (0) in Eq. (28). However, it is obvious that successful experimental searches of such contributions it is required the description of the correlation coefficient T (E e ) within the SM at the level of 10 −5 and as well as experimental uncertainties at the level of a few parts of 10 −5 . We are planning to carry out such a theoretical description of the correlation coefficient T (E e ) in our forthcoming publication by taking into account the results, obtained in [33,35,38] and also in [41] in terms of Wilkinson's corrections (see also [28,29,31]). A rather complicated correlation structure ( ξ n · kν )( ξ e · k e )/E e Eν responsible for the correlation coefficient T (E e ), which entangles the spin-polarization vectors of the neutron and electron and 3-momenta of the electron and antineutrino, makes difficult its experimental analysis. Indeed, the experimental investigation of the correlation coefficient T (E e ) should be performed for polarized neutrons and longitudinally polarized decay electrons. This is unlike the experiments i) on the neutron beta decay for polarized neutrons and transversally polarized electrons [17,24,75,76] and ii) on the nuclear beta decays for unpolarized nuclei and longitudinally polarized decay positrons [12,[77][78][79]. Moreover, the dependence of the correlation structure on the antineutrino 3-momentum demands a simultaneous detection of decay electrons and protons, i.e. electron-proton pairs, similar to the measurements of the antineutrino asymmetry [81][82][83]. The theoretical analysis of the antineutrino asymmetry in the neutron beta decay, related to the correlation coefficient B(E e ), was carried out by Glück et a. [80] (see also [28]). We are planning to perform an analogous theoretical analysis of the asymmetry, related to the correlation coefficient T (E e ), in our forthcoming publication.

VII. ACKNOWLEDGEMENTS
We thank Hartmut Abele for discussions stimulating this work. We thank Vladimir Gudkov for calling our attention to the result obtained by Ebel and Feldman [63], which we have overlooked. The work of A. N. Ivanov was supported by the Austrian "Fonds zur Förderung der Wissenschaftlichen Forschung" (FWF) under contracts P31702-N27 and P26636-N20, and "Deutsche Förderungsgemeinschaft" (DFG) AB 128/5-2. The work of R. Höllwieser was supported by the Deutsche Forschungsgemeinschaft in the SFB/TR 55. The work of M. Wellenzohn was supported by the MA 23 (FH-Call 16) under the project "Photonik -Stiftungsprofessur für Lehre".

VIII. THE SUPPLEMENTAL MATERIAL
Appendix A: The electron-photon-energy and angular distribution of the neutron radiative beta decay for a polarized neutron, a polarized electron and unpolarized proton and photon In order to remove the dependence of the radiative corrections to the neutron lifetime and correlation coefficients of the neutron beta decay on the infrared cut-off µ we have to add the contribution of the neutron radiative beta decay n → p + e − +ν e + γ [68,69] (see also [50,51,54,55] and [28,29,31]). Following [28,29,31] 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. 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. The sum over polarizations of the massive fermions is defined by [28,29,31] pol.
where the ellipsis denotes the contributions of the terms, which possess the correlation structures different to the correlation structure responsible for the correlation coefficient T (E e ). In Eq.(A-4) in the covariant form the traces over Dirac matrices were calculated in [28,31]. The result is The r.h.s. of Eq.(A-5) we calculate in the physical gauge ε λ = (0, ε λ ) [28,29,31,70], where the polarization vector ε λ obeys the constraints In the physical gauge ε λ = (0, ε λ ) we obtain for the r.h.s. of Eq.(A-5) the following expression pol.