On the Hypothesis of Exact Conservation of Charged Weak Hadronic Vector Current in the Standard Model

1. Introduction

The neutron lifetime with the account for the complete set of corrections of order ${10}^{-3}$, caused by the weak magnetism, proton recoil and electromagnetic interaction, has been calculated in [1]. The theoretical value ${\tau }_n=879.6(1.1)\mathrm{s}$ agrees well with the world averaged one ${\tau }_n=880.2(1.0)\mathrm{s}$[2] and recent experimental value ${\tau }_n=880.2(1.2)\mathrm{s}$[3]. The theoretical uncertainty $\pm 1.1$ is fully defined by the experimental uncertainties of the axial coupling constant $\lambda =-1.2750\left(9\right)$[4] and the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $\left|V_{ud}\right|=0.97425\left(22\right)$[5], which agrees well with a new value $\left|V_{ud}\right|=0.97417\left(21\right)$[2] reported by Hardy and Tower [6]. Both values of the CKM matrix elements have been extracted from the $0^{+}\to 0^{+}$transitions with the errors dominated by the theoretical uncertainties caused by nuclear Coulomb distortion and radiative corrections [2,6].

According to recent analysis by Hardy and Tower [6] (see also [7]), the effect of conservation of the vector current (CVC) in the $0^{+}\to 0^{+}$transitions (or in the pure Fermi transitions) is being observed at the level of $1.2\times {10}^{-4}$. In turn, as has been found by Naviliat-Cuncic and Severijns [8], in the mirror decays of ¹⁹Ne, ²¹Na, ²⁹P, ³⁵Ar, and ¹⁷ K, caused by the Gamow-Teller mirror transitions, a new independent test of the CVC effect may be performed at the level of $4\times {10}^{-3}$[7]. Recently, the CVC effect has been investigated by Ankowski [9] and Giunti [10] in the inverse $\beta$-decay ${\overline{\nu }}_e+p\to n+e^{+}$.

2. Precision Analysis of Neutron Lifetime to order ${10}^{-4}$

This work is addressed to the analysis of the reliability of the CVC hypothesis or the CVC effect in the neutron lifetime. The effective low-energy Lagrangian of $V-A$ weak interactions takes the standard form [1,11]

$${\mathcal{L}}_{\mathrm{W}}\left(x\right)=-{\displaystyle \frac{G_F}{\sqrt{2}}}{V}_{ud}{J}_{\mu }\left(x\right)\left[{\overline{\psi }}_e\left(x\right){\gamma }^{\mu }\left(1-{\gamma }^5\right){\psi }_{{\nu }_e}\left(x\right)\right]$$

(1)

where $G_F=1.1664\times {10}^{-11}{\mathrm{MeV}}^{-2}$ and $V_{ud}=0.97417\left(21\right)$ are the Fermi weak constant and the Cabibbo-KobayashiMaskawa (CKM) matrix element [2], respectively, $J_{\mu }\left(x\right)$ is the hadronic $V-A$ current, ${\psi }_e\left(x\right)$ and ${\psi }_{{\nu }_e}\left(x\right)$ are the operators of the electron and electron neutrino (antineutrino) fields. The amplitude of the neutron ${\beta }^{-}$-decay $n\to p+e^{-}+{\overline{\nu }}_e$ is equal to (for more details see Appendix A)

$$M\left(n\to p{e}^{-}{\overline{\nu }}_e\right)=-{\displaystyle \frac{G_F}{\sqrt{2}}}{V}_{ud}\left.〈p\left({\overrightarrow{k}}_p,{\sigma }_p\right)\right|J_{\mu }\left(0\right)\left|n\left({\overrightarrow{k}}_n,{\sigma }_n\right)\right.〉\left[{\overline{u}}_e\left({\overrightarrow{k}}_e,{\sigma }_e\right){\gamma }^{\mu }\left(1-{\gamma }^5\right)v_{\nu }\left(\overrightarrow{k},+{\displaystyle \frac{1}{2}}\right)\right]$$

(2)

where $\left|p\left({\overrightarrow{k}}_p,{\sigma }_p\right)\right.〉,\left|n\left({\overrightarrow{k}}_n,{\sigma }_n\right)\right.〉$ are the wave functions of the free proton and neutron with 3-momenta ${\overrightarrow{k}}_p$ and ${\overrightarrow{k}}_n=\overrightarrow{0}$ and polarizations ${\sigma }_p=\pm 1$ and ${\sigma }_n=\pm 1$, respectively. Then, ${\overline{u}}_e\left({\overrightarrow{k}}_e,{\sigma }_e\right)$ and $v_{\nu }\left({\overrightarrow{k}}_{\nu },+{\displaystyle \frac{1}{2}}\right)$ are the Dirac wave functions of the free electron and electron antineutrino with $3-$ momenta ${\overrightarrow{k}}_e$ and ${\overrightarrow{k}}_{\nu }$ and polarizations ${\sigma }_e=\pm 1$ and $+{\displaystyle \frac{1}{2}}$ [1,11], respectively. In the Standard Model (SM) the matrix element of the hadronic current we take in the following form

$$\begin{array}{cc}\hfill & \left.〈p\left({\overrightarrow{k}}_p,{\sigma }_p\right)\right|J_{\mu }\left(0\right)\left|n\left({\overrightarrow{k}}_n,{\sigma }_n\right)\right.〉=\hfill \\ \hfill & {\overline{u}}_p\left({\overrightarrow{k}}_p,{\sigma }_p\right)\left[\left({\gamma }_{\mu }-{\displaystyle \frac{q_{\mu }\widehat{q}}{q^2}}\right)+{\displaystyle \frac{\kappa }{2M}}i{\sigma }_{\mu \nu }{q}^{\nu }+\lambda \left(-{\displaystyle \frac{2M{q}_{\mu }}{q^2-m_{\pi }^2}}+{\gamma }_{\mu }\right){\gamma }^5\right]u_n\left({\overrightarrow{k}}_n,{\sigma }_n\right)\hfill \end{array}$$

(3)

which is similar to that used by Nowakowski et al. [12] and Leitner et al. [13], where ${\overline{u}}_p\left({\overrightarrow{k}}_p,{\sigma }_p\right)$ and $u_n\left({\overrightarrow{k}}_n,{\sigma }_n\right)$ are the Dirac wave functions of the free proton and neutron. Then, $-q_{\mu }\widehat{q}/q^2$, where $q=k_p-k_n$ is a $4-$ momentum transferred, is the phenomenological term responsible for the CVC in the neutron ${\beta }^{-}$-decay. The term $\kappa /2M$, where $2M=m_n+m_p$ and $m_n=939.5654\mathrm{MeV}$ and $m_p=938.2720\mathrm{MeV}$ are the neutron and proton masses, defines the contribution of the weak magnetism, where $\kappa ={\kappa }_p-{\kappa }_n=3.7058$ 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 [2]. The contribution of the axial current is given by the last term in Eq.(3), where $\lambda =-1.2750\left(9\right)$ is the axial coupling constant [4] (see also [1,11]) and $m_{\pi }$ is the charged pion mass [2]. In the limit $m_{\pi }\to 0$ (or in the chiral limit) the axial current is also conserved [14,15]. Skipping standard calculations [1] we arrive at the rate of the neutron ${\beta }^{-}$-decay given by

$${\displaystyle \frac{1}{{\tau }_n}}={\displaystyle \frac{1}{{\tau }_n^{\left(\mathrm{SM}\right)}}}\left(1+{\displaystyle \frac{f_n^{\left(\mathrm{CVC}\right)}}{f_n}}\right)$$

(4)

where ${\tau }_n^{\left(\mathrm{SM}\right)}=879.6(1.1)\mathrm{s}$ is the theoretical value of the neutron lifetime, calculated in [1] for $\lambda =-1.2750\left(9\right)$. It agrees perfectly well with the world averaged value ${\tau }_n=880.2(1.0)\mathrm{s}$[2] and recent experimental one ${\tau }_n=880.2(1.2)\mathrm{s}$[3]. Then, the phase space factor $f_n$ of the neutron, calculated order $O(1/M)$ and $O(\alpha /\pi )$ caused by the contributions of the weak magnetism and proton recoil and the radiative corrections, respectively, is equal to $f_n=6.116\times {10}^{-2}{\mathrm{MeV}}^5$. The phase space factor of the neutron $f_n^{\left(\mathrm{CVC}\right)}$, caused by the CVC effect, is given by the expression

$$\begin{array}{cc}\hfill f_n^{\left(\mathrm{CVC}\right)}=& {\displaystyle \frac{1}{1+3{\lambda }^2}}{\int }_{m_e}^{E_0}d{E}_e{k}_e{\left(E_0-E_e\right)}^{2}F\left(E_e,Z=1\right)\int {\displaystyle \frac{d{\Omega }_{e\nu }}{4\pi }}\left\{-{\displaystyle \frac{2m_e^2\Delta }{m_e^2+2E_e\left(E_0-E_e\right)-2{\overrightarrow{k}}_e·{\overrightarrow{k}}_{\nu }}}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{m_e^2{\Delta }^2}{{\left(m_e^2+2E_e\left(E_0-E_e\right)-2{\overrightarrow{k}}_e·{\overrightarrow{k}}_{\nu }\right)}^2}}\left(E_e-{\displaystyle \frac{{\overrightarrow{k}}_e·{\overrightarrow{k}}_{\nu }}{E_{\nu }}}\right)\right\}\hfill \end{array}$$

(5)

where $\Delta =m_n-m_p,E_0=\left(m_n^2-m_p^2+m_e^2\right)/2m_n=1.2927\mathrm{MeV}$ is the end-point energy of the electron energy spectrum of the neutron ${\beta }^{-}$-decay [1], $k_e=\sqrt{E_e^2-m_e^2}$ is an absolute value of the electron 3 -momentum, $F\left(E_e,Z=1\right)$ is the relativistic Fermi function describing the proton-electron Coulomb final-state interaction [1]. Then, $d{\Omega }_{e\nu }$ is an element of the solid angle of the electron-antineutrino momentum correlations. Since we analyse the main contribution of the CVC effect, the integrand of the phase space factor $f^{\left(CVC\right)}$ is calculated to leading order in the large nucleon mass expansion. Because of the numerical value $f_n^{\left(\mathrm{CVC}\right)}=-4.887\times {10}^{-3}{\mathrm{MeV}}^5$ the contribution of the CVC effect to the rate of the neutron ${\beta }^{-}$-decay is equal to $f_n^{\left(\mathrm{CVC}\right)}/f_n=-7.991\times {10}^{-2}$. This corresponds to the relative correction to the neutron lifetime $\Delta {\tau }_n^{\left(\mathrm{CVC}\right)}/{\tau }_n=8.684\times {10}^{-2}$ that gives $\Delta {\tau }_n^{\left(\mathrm{CVC}\right)}=76.4\mathrm{s}$. Unfortunately, such a huge increase of the lifetime by the CVC effect cannot be accepted for the neutron and should be substantially suppressed for the correct agreement with recent experimental data ${\tau }_n=880.2(1.2)\mathrm{s}$[3] and world averaged value ${\tau }_n=880.2(1.0)\mathrm{s}$[2]. In this connection it is important to emphasize that in the SM there are no contributions, which are able to diminish such a huge increase of the neutron lifetime, induced by the phenomenological term $-q_{\mu }\widehat{q}/q^2$ responsible for the CVC in the neutron ${\beta }^{-}$-decay [16,17,18]. Indeed, the contribution of the pseudoscalar term for a physical mass of the charged pion $m_{\pi }=139.570\mathrm{MeV}$ decreases the neutron lifetime at the level of $\Delta {\tau }_n^{\left(\pi \right)}/{\tau }_n=4.691\times {10}^{-6}$. However, in the chiral limit $m_{\pi }\to 0$ the contribution of the charged pion may only aggravate the problem. Hence, in order to reduce a huge contribution of the CVC effect at the level of $f_n^{\left(\mathrm{CVC}\right)}/f_n=-7.991\times {10}^{-2}$ to the level of $1.2\times {10}^{-4}$ one has to turn to interactions beyond the SM.

3. Fierz Interference Term

It is well-know [20-23][20,21,22,23](see also the review papers [4,19]) that the simplest contribution of interactions beyond the SM to the neutron ${\beta }^{-}$-decay is the Fierz interference term $b_F{m}_e/E_e$, caused by scalar and tensor interactions beyond the SM. Below for the analysis of the contribution of the Fierz interference term we use the results, obtained in [1]. As has been shown in [1] all possible interactions beyond the SM [22] give the contribution to the neutron lifetime only in the form of the Fierz interference term. As a result, the Fierz interference term changes the rate of the neutron ${\beta }^{-}$-decay as follows

$${\displaystyle \frac{1}{{\tau }_n}}={\displaystyle \frac{1}{{\tau }_n^{\left(\mathrm{SM}\right)}}}\left(1+b_F{〈{\displaystyle \frac{m_e}{E_e}}〉}_{\mathrm{SM}}\right)$$

(6)

where ${〈m_e/E_e〉}_{\mathrm{SM}}=0.6556$ is calculated with the electron energy spectrum density Eq.(D-59) of Ref. [1]. Taking into account the contribution of the CVC effect we get

$${\displaystyle \frac{1}{{\tau }_n}}={\displaystyle \frac{1}{{\tau }_n^{\left(\mathrm{SM}\right)}}}\left(1+{\displaystyle \frac{f_n^{\left(\mathrm{CVC}\right)}}{f_n}}+b_F{〈{\displaystyle \frac{m_e}{E_e}}〉}_{\mathrm{SM}}\right)$$

(7)

where the right-hand-side (r.h.s.) of Eq.(7) contains a complete set of phenomenological contributions within the SM and contributions beyond the SM. In the obtained expression for the neutron lifetime, given by Eq.(7), effectively the contribution of the CVC effect and the Fierz interference term can be kept at the required level of $1.2\times {10}^{-4}$ if the Fierz interference term is equal to $b_F=0.12189\left(12\right)$.

4. Conclusions

We have analyzed the CVC hypothesis in the neutron ${\beta }^{-}$-decay and calculated the contribution of the CVC effect, i.e. the contribution of the phenomenological term $-q_{\mu }\widehat{q}/q^2$ responsible for the CVC of the hadronic weak current of the neutron ${\beta }^{-}$-decay. We have shown that the CVC effect gives a relative contribution to the rate of the neutron ${\beta }^{-}$-decay at the level of $f_n^{\left(\mathrm{CVC}\right)}/f_n=-7.991\times {10}^{-2}$, which is one order of magnitude large compared to the level of $4\times {10}^{-3}$ of the CVC test in the Gamow-Teller mirror transitions reported by Naviliat-Cuncic and Severijns [8]. As a result, the phenomenological term $-q_{\mu }\widehat{q}/q^2$, providing the CVC of the hadronic current in the neutron ${\beta }^{-}$-decay, changes the lifetime of the neutron by $\Delta {\tau }_n=76.4\mathrm{s}$. Since in the SM there are no interactions, which are able to cancel such a huge increase of the neutron lifetime, we have turned to interactions beyond the SM. As has been shown in [1], the contributions of all possible interactions beyond the SM [22], which may affect the energy spectra and angular distributions of the neutron ${\beta }^{-}$-decay, reduce themselves to the contribution to the neutron lifetimes in the form of the Fierz interference term only. Keeping the effective contribution, caused by the CVC effect and the Fierz interference term, at the level of the experimental uncertainty of the neutron lifetime $1.2\times {10}^{-3}$ we have obtained the Fierz interference term $b_F=0.1219\left(12\right)$, which seems to be also huge in comparison with the results $b_F<0.01$ Herczeg [24], $b_F=0.0032$ (23) Faber et al. [25], $b_F=-0.0028\left(26\right)$ Hardy and Tower [6] (see also discussion below Eq.(7) of Ref.[6]), and and $\left|b_F\right|<0.03$, reported by H. Saul on behalf of the PERKEO III Collaboration [26]. Thus, if the value of the Fierz interference term $b_F=0.1219\left(12\right)$ is not acceptable one may conclude that the phenomenological realization of the CVC hypothesis in the neutron ${\beta }^{-}$-decay in terms of the contribution $-q_{\mu }\widehat{q}/q^2$ induces irresistible problems for description and understanding of the neutron ${\beta }^{-}$-decay.