Precision analysis of electron energy spectrum and angular distribution of neutron beta decay with polarized neutron and electron

TL;DR

This paper provides a high-precision analysis of neutron beta decay, calculating correlation coefficients with order 10^(-3) accuracy within the Standard Model, aiding future experiments searching for new physics beyond it.

Contribution

It offers a detailed Standard Model calculation of neutron decay correlation coefficients including weak magnetism, recoil, and radiative corrections at next-to-leading order.

Findings

Correlation coefficients calculated with 10^(-3) precision.

Results enable planning of experiments probing beyond Standard Model interactions.

Provides theoretical framework for interpreting neutron decay measurements.

Abstract

We give a precision analysis of the correlation coefficients of the electron-energy spectrum and angular distribution of the beta decay and radiative beta decay of the neutron with polarized neutron and electron to order 10^(-3). The calculation of correlation coefficients is carried out within the Standard model with contributions of order 10^(-3), caused by the weak magnetism and proton recoil, taken to next-to-leading order in the large proton mass expansion, and with radiative corrections of order "alpha/pi ~ 10^(-3", calculated to leading order in the large proton mass expansion. The obtained results can be used for the planning of experiments on the search for contributions of order 10^(-4) of interactions beyond the Standard model.

Equations

$$G_0(E_e)=-1,N_0(E_e)=-\frac{m_e}{E_e}{A}_0,Q_{e0}(E_e)=-A_0,R_0(E_e)=-\alpha \frac{m_e}{k_e}{A}_0,A_0=-2\frac{\lambda (1+\lambda )}{1+3{\lambda }^2},$$

$$G_0$$

$$Q_{e0}$$

$$G_0$$

$$Q_{e0}$$

$$\frac{d^3{\lambda }_n(E_e)}{d{E}_{e}d{\Omega }_e}$$

$$\displaystyle+G(E_{e})\,\frac{\vec{\xi}_{e}\cdot\vec{k}_{e}}{E_{e}}+N(E_{e})\,\vec{\xi}_{n}\cdot\vec{\xi}_{e}+Q_{e}(E_{e})\,\frac{(\vec{\xi}_{n}\cdot\vec{k}_{e})(\vec{k}_{e}\cdot\vec{\xi}_{e})}{E_{e}(E_{e}+m_{e})}+R(E_{e})\,\frac{\vec{\xi}_{n}\cdot(\vec{k}_{e}\times\vec{\xi}_{e})}{E_{e}}\Big{\}},$$

$$\displaystyle F(E_{e},Z=1)=\Big{(}1+\frac{1}{2}\gamma\Big{)}\,\frac{4(2r_{p}m_{e}\beta)^{2\gamma}}{\Gamma^{2}(3+2\gamma)}\,\frac{\displaystyle e^{\,\pi\alpha/\beta}}{(1-\beta^{2})^{\gamma}}\,\Big{|}\Gamma\Big{(}1+\gamma+i\,\frac{\alpha}{\beta}\Big{)}\Big{|}^{2},$$

$$\displaystyle{\cal H}_{W}(x)=\frac{G_{F}}{\sqrt{2}}\,V_{ud}\,\Big{\{}[\bar{\psi}_{p}(x)\gamma_{\mu}(1+\lambda\gamma^{5})\psi_{n}(x)]+\frac{\kappa}{2M}\partial^{\nu}[\bar{\psi}_{p}(x)\sigma_{\mu\nu}\psi_{n}(x)]\Big{\}}[\bar{\psi}_{e}(x)\gamma^{\mu}(1-\gamma^{5})\psi_{\nu_{e}}(x)],$$

$$\zeta (E_e)$$

$$\zeta (E_e)A_W(E_e)$$

$$\zeta (E_e)G(E_e)$$

$$\zeta (E_e)N(E_e)$$

$$\zeta (E_e)Q_e(E_e)$$

$$\zeta (E_e)R(E_e)$$

$$G(E_e)$$

$$N(E_e)$$

$$Q_e(E_e)$$

$$R(E_e)$$

$$G(E_e)$$

$$N(E_e)$$

$$Q_e(E_e)$$

$$\displaystyle F(E_{e},Z=1)\to F(E_{e},Z=1)\,\Big{(}1-\frac{\pi\alpha}{\beta}\,\frac{E_{e}}{M}-\frac{\pi\alpha}{\beta^{3}}\,\frac{E_{0}-E_{e}}{M}\,\frac{\vec{k}_{e}\cdot\vec{k}_{\nu}}{E_{e}E_{\nu}}\Big{)},$$

$$\frac{\delta A_W(E_e)}{A_W(E_e)}$$

$$\frac{\delta G(E_e)}{G(E_e)}$$

$$\frac{\delta N(E_e)}{N(E_e)}$$

$$\displaystyle L(E_{e},Z=1)=1+\frac{13}{60}\,\alpha^{2}-\alpha\,r_{p}\,E_{e}\,\Big{(}1-\frac{1}{2}\,\frac{m^{2}_{e}}{E^{2}_{e}}\Big{)}=1+1.154\times 10^{-5}-4.183\times 10^{-5}\,\frac{E_{e}}{E_{0}}+0.827\times 10^{-5}\,\frac{m_{e}}{E_{e}}.$$

$$\frac{\delta G(E_e)}{G(E_e)}=\frac{\delta N(E_e)}{N(E_e)}=\frac{\delta Q_e(E_e)}{Q_e(E_e)}=1-L(E_e,Z=1).$$

$$\displaystyle C(E_{e},Z=1)=1+\Big{[}\Big{(}-\frac{9}{20}\,\alpha^{2}+\frac{1}{5}\,m^{2}_{e}r^{2}_{p}-\frac{1}{5}\,E^{2}_{0}r^{2}_{p}\Big{)}+\Big{(}-\frac{1}{5}\,\alpha\,r_{p}\,E_{0}-\frac{2}{15}\,E^{2}_{0}r^{2}_{p}\Big{)}\,\frac{1-\lambda^{2}}{1+3\lambda^{2}}\Big{]}$$

$$\displaystyle+\Big{[}\Big{(}-\frac{3}{5}\,\alpha\,r_{p}\,E_{0}+\frac{2}{5}\,E^{2}_{0}\,r^{2}_{p}\Big{)}+\Big{(}\frac{1}{5}\,\alpha\,r_{p}\,E_{0}-\frac{2}{15}\,E^{2}_{0}\,r^{2}_{p}\Big{)}\,\frac{1-\lambda^{2}}{1+3\lambda^{2}}\Big{]}\,\frac{E_{e}}{E_{0}}+\frac{2}{15}\,m_{e}\,E_{0}\,r^{2}_{p}\,\frac{1-\lambda^{2}}{1+3\lambda^{2}}\,\frac{m_{e}}{E_{e}}$$

$$\displaystyle+\frac{2}{5}\,\Big{(}-1+\frac{1}{3}\,\frac{1-\lambda^{2}}{1+3\lambda^{2}}\Big{)}\,E^{2}_{0}r^{2}_{p}\,\frac{E^{2}_{e}}{E^{2}_{0}}=1-2.854\times 10^{-5}-1.238\times 10^{-5}\,\frac{E_{e}}{E_{0}}-0.018\times 10^{-5}\,\frac{m_{e}}{E_{e}}$$

$$-1.361\times 10^{-5}\frac{E_e^2}{E_0^2}.$$

$$\frac{\delta G(E_e)}{G(E_e)}=\frac{\delta N(E_e)}{N(E_e)}=\frac{\delta Q_e(E_e)}{Q_e(E_e)}=1-C(E_e,Z=1).$$

$$⟨N(E_e)⟩=0.07767(22),⟨R(E_e)⟩=0.000891(3)$$

$$\displaystyle\lambda_{n}(\vec{\xi}_{n},\vec{\xi}_{e})=\lambda_{n}\,\Big{(}1+\langle\bar{N}(E_{e})\rangle\,\vec{\xi}_{n}\cdot\vec{\xi}_{e}\Big{)},$$

$${\lambda }_n=(1+3{\lambda }^2)\frac{G_F^2|V_{ud}{|}^2}{2{\pi }^3}{f}_n(E_0,Z=1).$$

$$\displaystyle f_{n}(E_{0},Z=1)=\int^{E_{0}}_{m_{e}}(E_{0}-E_{e})^{2}\,\sqrt{E^{2}_{e}-m^{2}_{e}}\,E_{e}\,F(E_{e},Z=1)\,\Big{\{}\Big{(}1+\frac{\alpha}{\pi}\,g_{n}(E_{e})\Big{)}+\frac{1}{M}$$

$$\displaystyle\times\,\frac{1}{1+3\lambda^{2}}\Big{[}\Big{(}10\lambda^{2}-4(\kappa+1)\,\lambda+2\Big{)}\,E_{e}-\Big{(}2\lambda^{2}-2(\kappa+1)\,\lambda\Big{)}\,\Big{(}E_{0}+\frac{m^{2}_{e}}{E_{e}}\Big{)}\Big{]}\Big{\}}\,dE_{e}.$$

$$\displaystyle\bar{N}(E_{e})=N(E_{e})+\frac{1}{3}\,\Big{(}1-\frac{m_{e}}{E_{e}}\Big{)}\,Q_{e}(E_{e}).$$

Full text

I Introduction

It is well–known that the neutron $\beta^{-}$–decay is a good laboratory for tests of the Standard model (SM) Abele2008 –Ivanov2013a . As has been pointed out in Ivanov2013 ; Ivanov2013a by example of the neutron $\beta^{-}$–decay with a polarized neutron and unpolarized proton and electron the weak magnetism and proton recoil corrections of order $E_{e}/m_{p}$, where $E_{e}$ and $m_{p}$ are the electron energy and proton mass, and radiative corrections of order $\alpha/\pi$, where $\alpha$ is the fine–structure constant PDG2014 , define a complete set of corrections to the correlation coefficients of order $10^{-3}$. These corrections provide a robust background for a search of contributions of order $10^{-4}$, induced by interactions beyond the Standard model (SM) Ivanov2013 ; Ivanov2013a . This paper is addressed a precision analysis of the neutron $\beta^{-}$–decay with polarized neutron and electron. The aim of this paper is to give a robust background to order $10^{-3}$ for the experimental search for contributions of order $10^{-4}$, caused by interactions beyond the SM. According to Ivanov2013 ; Ivanov2013a , for the realization of this aim we have to calculate in the SM the correlation coefficients of the electron–energy spectrum and angular distribution of the neutron $\beta^{-}$–decay with polarized neutron and electron by taking into account the contributions of the weak magnetism and proton recoil to next–to–leading order in the large proton mass expansion and radiative corrections of order $\alpha/\pi$ to leading order in the large proton mass expansion. These corrections make possible a meaningful search for contributions of order $10^{-4}$, caused by interactions beyond the SM for most of the correlation coefficients presented here. Of course, these corrections should be meaningful if the theoretical uncertainties of the correlation coefficients, calculated to leading order in the large proton mass expansion and without radiative corrections, should be small compared to $10^{-3}$. The correlation coefficients $G(E_{e})$, $N(E_{e})$, $Q_{e}(E_{e})$ and $R(E_{e})$ (see Eq.(3)), calculated to leading order in the large proton mass expansion and without radiative corrections, are equal to (see Eq.(II))

[TABLE]

where $m_{e}$ and $k_{e}=\sqrt{E^{2}_{e}-m^{2}_{e}}$ are the electron mass and its momentum, $\alpha=1/137.036$ is the fine–structure constant PDG2014 and $\lambda$ is the axial coupling constant Abele2008 ; Nico2009 . The dependence of the correlation coefficient $R_{0}(E_{e})$ on the fine–structure constant is caused by the Coulomb distortion of the electron wave function in the Coulomb field of the proton Jackson58 ; Konopinski . The most precise, published values for the electron asymmetry from a single experiment provide values for $A^{(\exp)}_{0}=-0.11933(34)$ Abele2008 and $A^{(\exp)}_{0}=-0.11996(58)$ Abele2013 (see also Abele2014 ), giving $\lambda=-1.2750(9)$ and $\lambda=-1.2767(16)$, respectively, and averaging over the electron–energy spectrum (see Eq.(D-59) of Ref.Ivanov2013 ) we obtain the following numerical values for the correlation coefficients in Eq.(1)

[TABLE]

One may see that the relative theoretical uncertainties of the correlation coefficients in Eq.(I) are of order $(3-5)\times 10^{-3}$. They are practically defined by the experimental uncertainties of the axial coupling constant. An improvement of the theoretical uncertainties of the correlation coefficients in Eq.(I) may only result through an improvement of the experimental uncertainty of the axial coupling constant $\lambda$. Hence, currently because of the theoretical uncertainties of order $(3-5)\times 10^{-3}$ a precision analysis of the correlation coefficients $G(E_{e})$, $N(E_{e})$, $Q_{e}(E_{e})$ and $R(E_{e})$ to order $10^{-3}$ seems to be meaningful only for the correlation coefficient $G(E_{e})$. Nevertheless, we calculate below the corrections, caused by the weak magnetism and proton recoil of order $1/m_{p}$ and radiative corrections of order $\alpha/\pi$, to all correlation coefficients. In order to push meaningful tests of the SM to the $10^{-4}$ level for the correlation coefficients $N(E_{e})$, $Q_{e}(E_{e})$ and $R(E_{e})$, an improvement of experimental uncertainties for the axial coupling constant is required in addition to the theoretical predictions we present in this work. We see this as an important challenge for the experimental characterization of the charged weak interaction.

The first experimental analysis of the neutron $\beta^{-}$–decay $n\to p+e^{-}+\bar{\nu}_{e}$ with polarized neutron and electron has been undertaken by Kozela et al. Kozela2009 ; Kozela2012 , where the correlation coefficients $R(E_{e})$ and $N(E_{e})$ of the electron energy spectrum and angular distribution of the neutron $\beta^{-}$–decay with polarized neutron and with the electron, polarized transverse to its 3–momentum, were measured. The correlation coefficient $R(E_{e})$ describes the correlation between the neutron polarization and the polarization and 3-momentum of the electron $\vec{\xi}_{n}\cdot(\vec{k}_{e}\times\vec{\xi}_{e})$, where $\vec{\xi}_{n}$ and $\vec{\xi}_{e}$ are unit polarization vectors of the neutron and electron, respectively, and $\vec{k}_{e}$ is the electron 3–momentum. The correlation coefficient $R$ characterizes quantitatively a T–odd and a P–odd effect, caused by violation of time reversal invariance and invariance under parity transformation. The correlation coefficient $N(E_{e})$ is a quantitative characteristic of the correlation between the neutron and electron polarization $\vec{\xi}_{n}\cdot\vec{\xi}_{e}$. One may see that the experimental values $R_{\exp}=0.004\pm 0.012\pm 0.005$ and $N_{\exp}=0.067\pm 0.011\pm 0.004$, measured by Kozela et al. Kozela2012 , do not contradict the predictions of the SM, given by Eq.(I), within the experimental uncertainties. The primary goal of the nTRV Collaboration through these measurements of $R(E)$ and $N(E)$ was to make a useful probe for contributions from interactions beyond the SM (see Eqs. (7) - (11) of Ref. Kozela2012 ). Such tests require a search for deviations from the expected SM values for these correlations. If the precision level is to be greatly improved, the theoretical predictions for the SM values must also be refined to include contributions from the weak magnetism, proton recoil and radiative corrections. In this way, one can produce corrections to order $10^{-3}$ in the correlation coefficients, and open a path to a search for traces of interactions beyond the SM to $10^{-4}$.

The paper is organized as follows. In section II we give the electron–energy spectrum and angular distribution of the $\beta^{-}$–decay of the neutron with polarized neutron and electron. The correlation coefficients $G(E_{e})$, $N(E_{e})$, $Q_{e}(E_{e})$ and $R(E_{e})$ are calculated in the SM with the contributions of the weak magnetism and proton recoil to next–to–leading order in the large proton mass expansion and with radiative corrections of order $\alpha/\pi$, calculated to leading order in the large proton mass expansion Ivanov2013 . In section III we discuss some corrections of order $10^{-5}$ to the correlation coefficients beyond those, which are calculated in section II and which have been analysed and discussed by Wilkinson Wilkinson1982 . In section IV we discuss the obtained results and the experimental observables. In the Appendix we calculate the photon–electron energy spectrum, the electron–energy spectrum and angular distributions of the radiative $\beta^{-}$–decay of the neutron with polarized neutron and electron.

II Electron–energy spectrum and angular distribution

The electron–energy spectrum and angular distribution of the neutron $\beta^{-}$–decay with polarized neutron and electron takes the form SPT1 (see also SPT4 and Ivanov2013 )

[TABLE]

[TABLE]

where $G_{F}=1.1664\times 10^{-11}\,{\rm MeV}^{-2}$ is the Fermi weak constant, $V_{ud}=0.97428(15)$ is the Cabibbo-Kobayashi–Maskawa (CKM) matrix element PDG2014 , $\lambda$ is a real axial coupling constant, $E_{0}=(m^{2}_{n}-m^{2}_{p}+m^{2}_{e})/2m_{n}=1.2927\,{\rm MeV}$ is the end–point energy of the electron spectrum, calculated for $m_{n}=939.5654\,{\rm MeV}$, and $m_{p}=938.2720\,{\rm MeV}$ and $m_{e}=0.5110\,{\rm MeV}$ PDG2014 , $\vec{\xi}_{n}$ and $\vec{\xi}_{e}$ are unit polarization vectors of the neutron and electron, respectively, $F(E_{e},Z=1)$ is the relativistic Fermi function Konopinski ; Ivanov2013

[TABLE]

where $\beta=k_{e}/E_{e}=\sqrt{E^{2}_{e}-m^{2}_{e}}/E_{e}$ is the electron velocity, $\gamma=\sqrt{1-\alpha^{2}}-1$, $r_{p}$ is the electric radius of the proton. In the numerical calculations we will use $r_{p}=0.875\,{\rm fm}$ LEP . The Fermi function Eq.(4) describes the contribution of the electron–proton final–state Coulomb interaction. Since it is defined by the exact solution of the Dirac equation for the electron, moving in the Coulomb field of the proton Konopinski , it cannot introduce additional uncertainties to the approximate contributions, caused by the weak magnetism, proton recoil and radiative corrections. We would like to emphasize that the Fermi function Eq.(4) gives a contribution to the phase space factor of the neutron of about $3.32\,\%$. The use of the approximate expression $F(E_{e},Z=1)=1+\alpha\,\pi/\beta$ Wilkinson1982 diminishes the contribution of the Coulomb electron–proton final–state interaction at the level of $8.75\times 10^{-4}$. This justifies the use of the exact Fermi function Eq.(4) for the precision analysis of the neutron $\beta^{-}$–decay.

The correlation coefficients of the electron–energy spectrum and angular distribution Eq.(3) we calculate with the Hamiltonian of $V-A$ weak interactions and the weak magnetism Ivanov2013

[TABLE]

where $\psi_{p}(x)$, $\psi_{n}(x)$, $\psi_{e}(x)$ and $\psi_{\nu_{e}}(x)$ are the field operators of the proton, neutron, electron and anti-neutrino, respectively, $\gamma^{\mu}$, $\sigma^{\mu\nu}=\frac{i}{2}(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu})$ and $\gamma^{5}$ are the Dirac matrices; $\kappa=\kappa_{p}-\kappa_{n}=3.7058$ is the isovector anomalous magnetic moment of the nucleon, defined by the anomalous magnetic moments of the proton $\kappa_{p}=1.7928$ and the neutron $\kappa_{n}=-1.9130$ and measured in nuclear magneton PDG2014 , and $2M=m_{n}+m_{p}$ is the average nucleon mass.

The coefficients $\zeta(E_{e})$ and $A_{W}(E_{e})$ have been calculated in Gudkov2006 ; Ivanov2013 . They read

[TABLE]

where the correlation coefficients $A(E_{e})$ and $Q_{n}(E_{e})$ are given in Gudkov2006 ; Ivanov2013 . The correlation coefficient $A_{W}(E_{e})$ without the contribution of the radiative corrections, defined by the function $f_{n}(E_{e})$, has been calculated by Wilkinson Wilkinson1982 . The radiative corrections $g_{n}(E_{e})$ and $f_{n}(E_{e})$ (see Ivanov2013 ) are in analytical agreement with the radiative corrections, obtained by Sirlin et al. Sirlin1967 and Gudkov et al. Gudkov2006 , respectively (where the function $f_{n}(E_{e})$ was calculated for the first time by Shann Shann1971 ).

Using the results, obtained in Ivanov2013 (see Appendix A of Ref.Ivanov2013 ), for other correlation coefficients in Eq.(3) we get the expressions

[TABLE]

[TABLE]

The functions $f_{n}(E_{e})$, $h^{(\ell)}_{n}(E_{e})$ for $\ell=1,2$ describe the radiative corrections of order $\alpha/\pi$. They are calculated in the Appendix (see Eq.(VI)) and plotted in Fig 1. In the electron energy region $m_{e}\leq E_{e}\leq E_{0}$ they vary over the regions $2.81\times 10^{-3}\geq(\alpha/\pi)\,f_{n}(E_{e})\geq 6.24\times 10^{-4}$, $-6.04\times 10^{-4}\geq(\alpha/\pi)\,h^{(1)}_{n}(E_{e})\geq-3.37\times 10^{-3}$ and $5.07\times 10^{-3}\geq(\alpha/\pi)\,h^{(2)}_{n}(E_{e})\geq 2.20\times 10^{-3}$, respectively.

The term proportional to the fine–structure constant $\alpha$ in the correlation coefficient $\zeta(E_{e})R(E_{e})$ is induced by the Coulomb distortion of the Dirac bispinor wave function of the electron Jackson58 ; Konopinski .

Keeping the contributions of the terms of order of $1/M$ inclusively the correlation coefficients under consideration take the form

[TABLE]

where we have neglected the terms of order $(\alpha/\pi)(E_{e}/M)<3\times 10^{-6}$. The correlation coefficients Eq.(II) are defined by a complete set of contributions to order $10^{-3}$, caused by the weak magnetism and proton recoil corrections of order $1/M$ and radiative corrections of order $\alpha/\pi$. For example, at $\lambda=-1,2750(9)$ and $E_{0}=1.2927\,{\rm MeV}$ we get

[TABLE]

where the correlation coefficient $A_{0}$ is factorized out of the brackets of the correlation coefficients $N(E_{e})$ and $Q_{e}(E_{e})$. The obtained results provide a robust theoretical background to order $10^{-3}$ for planning experiments on the search for contributions of order $10^{-4}$ of interactions beyond the SM. The appearance of the term of order $10^{-5}$ is caused by an occasional cancellation of different contributions.

III Wilkinson’s analysis of higher order corrections

In this section we discuss the contibutions of higher order corrections, which are not calculated in section II. These corrections were calculated by Wilkinson Wilkinson1982 and we apply them to the analysis of the correlations coefficients $G(E_{e})$, $N(E)$, $Q_{e}(E_{e})$ and $R(E_{e})$, respectively. According to Wilkinson Wilkinson1982 , the higher order corrections with respect to those calculated in section II 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.

III.1 Proton recoil corrections, caused by the

Coulomb electron–proton final–state interaction

As has been found by Ivanov et al. Ivanov2013 proton recoil, caused by the Coulomb electron–proton final–state interaction, leads to the following change of the Fermi function $F(E_{e},Z=1)$ (see Appendix H of Ref.Ivanov2013 )

[TABLE]

where we have taken only the leading order $\alpha/M$ contributions. Then, $E_{\nu}=E_{0}-E_{e}$ and $\vec{k}_{\nu}$ are the energy and 3–momentum of the electron antineutrino. As has been shown in Ivanov2013 the contribution of the proton recoil, caused by the final–state Coulomb electron–proton interaction Eq.(10), to the function $\zeta(E_{e})$ agrees well with the result, obtained by Wilkinson Wilkinson1982 . For the calculation of the corrections to the correlation coefficients $A_{W}(E_{e})$, $G(E_{e})$, $N(E_{e})$ and $Q_{e}(E_{e})$, caused by the change of the Fermi function Eq.(10), we have to take into account the contributions of the correlation corfficients $\zeta(E_{e})$, $a(E_{e})(\vec{k}_{e}\cdot\vec{k}_{\nu}/E_{e}E_{\nu})$, $B(E_{e})(\vec{\xi}_{n}\cdot\vec{k}_{\nu}/E_{\nu})$, $H(E_{e})(\vec{\xi}_{e}\cdot\vec{k}_{\nu}/E_{\nu})$ and $K_{e}(E_{e})(\vec{\xi}_{e}\cdot\vec{k}_{e})(\vec{k}_{e}\cdot\vec{k}_{\nu})/(E_{e}+m_{e})E_{e}E_{\nu}$ and then to integrate over the directions of the anineutrino 3–momentum $\vec{k}_{\nu}$ Ivanov2018 . As a result we get

[TABLE]

In the experimental electron energy region $0.761\,{\rm MeV}\leq E_{e}\leq 0.966\,{\rm MeV}$ the corrections to the correlation coefficients are plotted in Fig. 2. They vary in the following limits $9.0\times 10^{-5}\geq\delta A_{W}(E_{e})/A_{W}(E_{e})\geq 3.8\times 10^{-5}$, $5.1\times 10^{-7}\geq\delta G(E_{e})/G(E_{e})\geq 1.3\times 10^{-7}$ and $-6.3\times 10^{-7}\leq\delta X(E_{e})/X(E_{e})\leq-3.5\times 10^{-7}$ for $X(E_{e})=N(E_{e})$ and $Q_{e}(E_{e})$, respectively.

III.2 Corrections, caused by finite proton–radius

According to Wilkinson Wilkinson1982 , the finite proton–radius correction to the phase–space factor of the neutron $\beta^{-}$–decay takes the form

[TABLE]

The contribution of the function $L(E_{e},Z=1)$ can be absorbed by the function $\zeta(E_{e})$ and through the expansions Eq.(II) may provide equal corrections to the correlation coefficients $G(E_{e})$, $N(E_{e})$ and $Q_{e}(E_{e})$

[TABLE]

The contribution of the finite proton–radius corrections to the neutron lifetime is at the level of $10^{-5}$.

III.3 Corrections, caused by lepton–nucleon convolution

As has been pointed out by Wilkinson Wilkinson1982 , the wave functions of the electron and electron antineutrino, calculated at the center of the nucleon, are not constant and may undergo a distortion in the nucleon volume that may lead to a convolution of the decay rate. Such an effect Wilkinson has described by the function $C(E_{e},Z=1)$. Following Wilkinson Wilkinson1982 we obtain the function $C(E_{e},Z=1)$ in the form

[TABLE]

Because of the expansion Eq.(II) the corrections, caused by the lepton–nucleon convolution, to the correlation coefficients $G(E_{e})$, $N(E_{e})$ and $Q_{e}(E_{e})$ are equal and given by

[TABLE]

The contribution of the function $C(E_{e},Z=1)$ to the neutron lifetime is at the level of $10^{-5}$.

III.4 Higher–order outer radiative corrections

The energy–independent radiative corrections of order $O(\alpha^{2})$ and $O(\alpha^{3})$ have been calculated by Wilkinson Wilkinson1982 . The contribution of these corrections to the phase–space factor Wilkinson was defined by $J(Z=1)$. Using the results, obtained by Wilkinson Wilkinson1982 , we get $J(Z=1)=1+3.917\times 10^{-4}$. Of course, such corrections give equal contributions to the correlation coefficients $\delta G(E_{e})/G(E_{e})=\delta N(E_{e})/N(E_{e})=\delta Q_{e}(E_{e})/Q_{e}(E_{e})=-3.917\times 10^{-4}$. In principle, they should be taken into account for an experimental search of contributions of order $10^{-4}$ of interactions beyond the SM. The factor $J(Z=1)$ changes the neutron lifetime by $0.3\,{\rm s}$, which is, of course small, compared to the current experimental accuracy of the neutron lifetime $\tau_{n}=880.2(1.2)\,{\rm s}$ Arzumanov2015 (see also the world averaged value $\tau_{n}=880.2(1.0)\,{\rm s}$ PDG2016 ).

IV Conclusion

We have calculated the correlation coefficients of the electron–energy spectrum and angular distribution of the $\beta^{-}$–decay of the neutron with polarized neutron and electron. We have performed the calculation within the SM with $V-A$ weak interactions by taking into account the contributions of the weak magnetism and proton recoil to next–to–leading order in the large proton mass expansion and the radiative corrections of order $\alpha/\pi$, calculated to leading order in the large proton mass expansion. Such an approximation provides a theoretical background for the analysis of contributions of order $10^{-4}$ of interactions beyond the SM Ivanov2013 ; Ivanov2013a .

The correlation coefficients $N(E_{e})$ and $R(E_{e})$, given by Eq.(II), averaged over the electron energy spectrum (see Eq.(D-59)) and calculated at $\lambda=-1.2750(9)$, are equal to

[TABLE]

The recent experimental data $N_{\exp}=0.067\pm 0.011\pm 0.004$ and $R_{\exp}=0.004\pm 0.012\pm 0.005$ Kozela2012 do not contradict the predictions of the SM within the experimental uncertainties.

Using the electron energy spectrum and angular distribution Eq.(3) we can define the rate of the $\beta^{-}$–decay of the neutron in dependence of the neutron and electron polarisations

[TABLE]

where $\lambda_{n}$ is the $\beta^{-}$–decay rate of the neutron, defining the lifetime of the neutron $\tau_{n}=1/\lambda_{n}$, and equal to Ivanov2013

[TABLE]

The Fermi integral $f_{n}(E_{0},Z=1)$ is given by Ivanov2013

[TABLE]

The correlation coefficient $\bar{N}(E_{e})$ is defined by the expression

[TABLE]

For the experimental observation of the correlation coefficient $\bar{N}(E_{e})$ we propose to analyse the asymmetry

[TABLE]

where $P_{n}$ and $P_{e}$ are the neutron and electron polarisations. The asymmetry $P(\vec{\xi}_{n},\vec{\xi}_{e})$ can be measured for the polarized neutron and electron with parallel and antiparallel spins. Averaging $\bar{N}(E_{e})$ over the electron energy spectrum (see Eq.(D-59) in Appendix D of Ref. Ivanov2013 ) we get $\langle\bar{N}(E_{e})\rangle=0.0911$. Thus, the theoretical prediction for asymmetry $P(\vec{\xi}_{n},\vec{\xi}_{e})$, obtained in the SM with the weak magnetism, proton recoil and radiative corrections, is

[TABLE]

Our results should provide a necessary background for the measurement of the contributions of order $10^{-4}$ to the $\beta^{-}$–decay of a polarized neutron with a polarized electron, caused by interactions beyond the SM Ivanov2013 ; SUSY .

The radiative corrections to the correlation coefficients $G(E_{e})$, $N(E_{e})$ and $Q_{e}(E_{e})$ are given by the functions $f_{n}(E_{e})$, $h^{(1)}_{n}(E_{e})$ and $h^{(2)}_{n}(E_{e})$, calculated in the Appendix. The photon–electron and electron–energy spectra and angular distributions of the radiative $\beta^{-}$–decay of the neutron with polarized neutron and electron, obtained in the Appendix, may be used for future experiments on the radiative $\beta^{-}$–decay of the neutron Nico2006 .

We would like to emphasize that the radiative corrections, described by the functions $h^{(1)}_{n}(E_{e})$ and $h^{(2)}_{n}(E_{e})$, and the photon–electron and electron–energy spectra and angular distributions of the radiative $\beta^{-}$–decay of the neutron with polarized neutron and electron have been never calculated in literature before.

Completing our discussion of these corrections, we would like to make three comments: i) for predictions of $10^{-4}$ precision, it is apparent that the higher order outer radiative corrections, discussed in secion III, should be included, ii) for an experimental search for interactions beyond the SM, a ”discovery” experiment with the required 5$\sigma$ sensitivity will require experimental uncertainties of a few parts in $10^{-5}$, and iii) the correction $\delta G(E_{e})$, caused by the contribution of the proton recoil to the Fermi function, can be also changed by the contributions of the correlation coefficients $H(E_{e})$ and $K_{e}(E_{e})$ of the electron–energy and electron–antineutrino angular distribution of the neutron $\beta^{-}$–decay with polrized electron and unpolarized neutron and proton.

V Acknowledgements

The results, obtained in this paper, were reported at the “Satellite workshop on symmetries in light and heavy flavour”, which was held on 7 - 8 November 2016 at Max Planck Institute for Astrophysics (MPA), München, Germany. The work of A. N. Ivanov was supported by the Austrian “Fonds zur Förderung der Wissenschaftlichen Forschung” (FWF) under contracts I689-N16, I862-N20, P26781-N20 and P26636-N20, “Deutsche Förderungsgemeinschaft” (DFG) AB 128/5-2 and by the ÖAW within the New Frontiers Groups Programme, NFP 2013/09. The work of R. Höllwieser was supported by the Erwin Schrödinger Fellowship program of the Austrian Science Fund FWF (“Fonds zur Förderung der wissenschaftlichen Forschung”) under Contract No. J3425-N27. The work of M. Wellenzohn was supported by the MA 23 (FH-Call 16) under the project “Photonik - Stiftungsprofessur für Lehre”.

VI Appendix A: Photon–electron and electron energy spectra

and angular distributions of radiative $\beta^{-}$–decay of neutron with polarized neutron and electron

Using the results, obtained in Ivanov2013 , the photon–electron spectrum and angular distribution of the radiative $\beta^{-}$–decay of the neutron with polarized neutron and electron takes the form

[TABLE]

where $\beta=\sqrt{E^{2}_{e}-m^{2}_{e}}/E_{e}$ is the electron velocity and $\omega$ is the photon energy, the vector $\vec{n}$ is directed along the photon 3–momentum, $d\Omega_{e}$ and $d\Omega_{\gamma}$ are the elements of the solid angles of the electron and the photon, respectively. The 4–vector of the electron polarization $\zeta^{\mu}_{e}=(\zeta^{0}_{e},\vec{\zeta}_{e})$ is defined by

[TABLE]

It obeys the constraints $\zeta^{2}_{e}=-1$ and $k_{e}\cdot\zeta_{e}=0$. For the derivation of the electron energy spectrum and angular distribution it is convenient to rewrite Eq.(VI) as follows

[TABLE]

The integration over the directions of $\vec{n}$ we carry out with the following auxiliary integrals

[TABLE]

As a result the photon–electron energy spectrum and angular distribution takes the form

[TABLE]

In terms of the irreducible scalar products the photon–electron energy spectrum and angular distribution of the radiative $\beta^{-}$–decay of the neutron reads

[TABLE]

[TABLE]

Integrating over the photon energy over the region $\omega_{\rm min}\leq\omega\leq E_{0}-E_{e}$ we obtain the electron energy spectrum and angular distribution

[TABLE]

where the functions $g^{(i)}_{\beta^{-}_{c}\gamma}(E_{e},\omega_{\rm min})$ for $i=1,2,3,4$ are defined by the integrals

[TABLE]

In terms of the functions $g^{(i)}_{\beta^{-}_{c}\gamma}(E_{e},\omega_{\rm min})$, depending on the infrared cut–off $\omega_{\rm min}$, for $i=1,2,3,4$ we determine the functions $f_{n}(E_{e})$ and $h^{(\ell)}_{n}(E_{e})$ for $\ell=1,2$, which do not depend on the infrared cut–off $\omega_{\rm min}$. They are

[TABLE]

For the calculation of the functions $f_{n}(E_{e})$ and $h^{(\ell)}_{n}(E_{e})$ for $\ell=1,2$, defining the radiative corrections to the correlation coefficients $A_{W}(E_{e})$, $G(E_{e})$, $N(E_{e})$ and $Q_{e}(E_{e})$, respectively, we have to take into account the contribution of the virtual photon exchanges, inducing the scalar and tensor weak nucleon–lepton coupling constant Ivanov2013 (see Appendix B).

Finally we would like to note that for the calculation of the radiative corrections, defined by the functions $f_{n}(E_{e})$ and $h^{(\ell)}_{n}(E_{e})$ for $\ell=1,2$, the final result does not depend on the regularization procedure. Indeed, one may use the infrared cut–off $\omega_{\rm min}$, which may be identified with the experimental threshold energy of photons, and the finite–photon mass (FPM) regularization Sirlin1967 (see also Gudkov2006 ; Ivanov2013 ). In turn the function $g_{n}(E_{e})$ has to be calculated with the FPM regularization in order to satisfy gauge invariance and the Kinoshita–Lee–Nauenberg theorem Sirlin1967 (see also Ivanov2013 ).