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

Abstract

We investigate the reliability of the conservation of the vector current (CVC) hypothesis in the neutron beta decay (n ${\beta }^{-}$ decay). We calculate the contribution of the phenomenological term, responsible for the CVC in the hadronic current of the n ${\beta }^{-}$ decay (or the CVC effect), to the neutron lifetime. We show that the CVC effect increases the neutron lifetime with a relative contribution of $8.684\times {10}^{-2}$. This leads to the increase of the neutron lifetime by 76.4 s with respect to the world averaged value ${\tau }_n=880.2\left(1.0\right)\mathrm{s}$ from the Particle Data Group. We show that since in the Standard Model there are no interactions that are able to cancel such a huge increase in the neutron lifetime, we have to turn to the interactions beyond the Standard Model, the contribution of which to the neutron lifetime reduces to the Fierz interference term $b_F$ only. Cancelling the CVC effect at the level of the experimental accuracy, we obtain $b_F=0.1219\left(12\right)$. If this value cannot be accepted for the Fierz interference term, the CVC effect induces irresistible problems for description and understanding of the n ${\beta }^{-}$ decay.

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 ${\tau }_n=879.6\left(1.1\right)\mathrm{s}$ agrees well with the world averaged one ${\tau }_n=880.2\left(1.0\right)\mathrm{s}$ [2] and recent experimental value ${\tau }_n=880.2\left(1.2\right)\mathrm{s}$ [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 ¹⁷K 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 ${\overline{\nu }}_e+p\to n+e^{+}$. The main controversy between M. Ankowski’s work and C. Giunti’s reply is given by formulations of current conservation in the context of the hadronic current in neutron beta decay. Ankowski employs a phenomenological approach to adapt conservation terms that are ordinarily used in virtual transitions, which Giunti argues is inconsistent. Giunti states that this implies a new re-assessment of all past results with the new current form with a set of parameters that were derived from the assumption of a more conventional current conservation approach. Here we try to follow this suggestion by analyzing the neutron lifetime, taking into account the broken isospin symmetry and resulting mass difference of the u and d quarks, respectively, proton and neutron.

2. Precision Analysis of Neutron Lifetime to Order ${\mathbf{10}}^{-\mathbf{4}}$

This study aims to evaluate the reliability of the CVC hypothesis, or CVC effect, regarding neutron lifetime. The effective low-energy Lagrangian for $V-A$ weak interactions is formulated as [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–Kobayashi–Maskawa (CKM) matrix element [2], respectively, $J_{\mu }\left(x\right)$ is the hadronic $V-A$ current, ${\psi }_e\left(x\right)$ and ${\psi }_{{\nu }_e}\left(x\right)$ represent the operators of the electron and electron neutrino (antineutrino) fields. The amplitude of the n ${\beta }^{-}$ 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)

Here $\left|p\left({\overrightarrow{k}}_p,{\sigma }_p\right)\right.〉,\left|n\left({\overrightarrow{k}}_n,{\sigma }_n\right)\right.〉$ represent the wave functions of the free p and n 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 $v_{\nu }\left({\overrightarrow{k}}_{\nu },+{\displaystyle \frac{1}{2}}\right)$ represent the Dirac wave functions of the free e 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]. In the Standard Model, the matrix element of the hadronic current we take the 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)

This approach aligns with the methods used in [12,13], where $\overline{u}p\left(\overrightarrow{k}p,{\sigma }_p\right)$ and $u_n\left(\overrightarrow{k}n,\sigma n\right)$ represent the Dirac wave functions for a free p and n, respectively. The term $-q_{\mu }\widehat{q}/q^2$, where $q=k_p-k_n$ represents the transferred 4-momentum, serves as the phenomenological component responsible for conserving the vector current (CVC) in n ${\beta }^{-}$ decay. The weak magnetism contribution is represented by the term ${\displaystyle \frac{\kappa }{2M}}$, where $2M=m_n+m_p$ with n and p masses $m_n=939.5654\mathrm{MeV}$ and $m_p=938.2720\mathrm{MeV}$. Here, $\kappa ={\kappa }_p-{\kappa }_n=3.7058$ denotes the isovector anomalous magnetic moment of the nucleon, defined by the anomalous magnetic moments of the p (${\kappa }_p=1.7928$) and p (${\kappa }_n=-1.9130$), measured in nuclear magnetons [2]. The axial current contribution is given by the last term in Equation (3), where $\lambda =-1.2750\left(9\right)$ represents the axial coupling constant [4] (also referenced in [1,11]), and $m_{\pi }$ denotes 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 n ${\beta }^{-}$ 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 ${\tau }_n^{\left(\mathrm{SM}\right)}=879.6\left(1.1\right)\mathrm{s}$ 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 ${\tau }_n=880.2\left(1.0\right)\mathrm{s}$ [2] and the recent experimental one ${\tau }_n=880.2\left(1.2\right)\mathrm{s}$ [3]. Then, the phase space factor $f_n$ for the n, computed to order $O(1/M)$ and $O(\alpha /\pi )$ due to the contributions of weak magnetism, p recoil, and radiative corrections, respectively, is given by $f_n=6.116\times {10}^{-2}{\mathrm{MeV}}^5$. The phase space factor associated with the n, denoted as $f_n^{\left(\mathrm{CVC}\right)}$, due to the influence of the CVC effect, is represented by the following 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 n ${\beta }^{-}$ 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. To examine the primary impact of the Conserved Vector Current (CVC) effect, we compute the integrand of the phase space factor $f^{\left(CVC\right)}$ using a leading-order approach within the framework of a large nucleon mass expansion. Given the calculated value of $f_n^{\left(\mathrm{CVC}\right)}=-4.887\times {10}^{-3}{\mathrm{MeV}}^5$, the CVC effect’s impact on the n ${\beta }^{-}$ decay rate is found to be $f_n^{\left(\mathrm{CVC}\right)}/f_n=-7.991\times {10}^{-2}$. 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 in 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 ${\tau }_n=880.2\left(1.2\right)\mathrm{s}$ [3] and world averaged value ${\tau }_n=880.2\left(1.0\right)\mathrm{s}$ [2]. In this connection it is important to emphasize that in the Standard Model there are no contributions that 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 n ${\beta }^{-}$ decay [16,17,18]. Indeed, the impact 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 Standard Model.

3. Fierz Interference Term

It is widely recognized [19,20,21,22] (and discussed in review articles such as [4,23]) that the Fierz interference term, $b_F{m}_e/E_e$, which arises from tensor and scalar interactions beyond the Standard Model, represents the simplest contribution of such interactions to the n ${\beta }^{-}$ decay. Below, for the analysis of the impact of the Fierz interference term, we use the results obtained in [1]. As has been shown in [1] all possible interactions beyond the Standard Model [21] give the contribution to the n lifetime only in the form of the Fierz interference term. As a result, the Fierz interference term changes the rate of the n ${\beta }^{-}$ 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)

The value ${〈m_e/E_e〉}_{\mathrm{SM}}=0.6556$ is computed using the electron energy spectrum density as presented in Equation (D-59) of Ref. [1]. Considering the contribution from the CVC effect, we obtain the following result:

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

(7)

where the right-hand side of Equation (7) contains a complete set of phenomenological contributions within the Standard Model and contributions beyond the Standard Model. In the obtained expression for the neutron lifetime, given by Equation (7), the contribution from the CVC effect and the Fierz interference term can be maintained at the required level of $1.2\times {10}^{-4}$, provided that the Fierz interference term is set to $b_F=0.12189\left(12\right)$.

4. Conclusions

We have analyzed the CVC hypothesis in the n ${\beta }^{-}$ decay and calculated the impact of the CVC effect, i.e., the impact of the phenomenological term $-q_{\mu }\widehat{q}/q^2$ responsible for the CVC of the hadronic weak current of the n ${\beta }^{-}$ decay. We have shown that the CVC effect gives a relative contribution to the rate of the n ${\beta }^{-}$ 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 with 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 n ${\beta }^{-}$ decay, changes the lifetime of the n by $\Delta {\tau }_n=76.4\mathrm{s}$. Because there are no interactions in the Standard Model, which are able to cancel such a large increase of the n lifetime, we have turned to interactions beyond the Standard Model. As has been shown in [1], the contributions of all possible interactions beyond the Standard Model [21], which may affect the energy spectra and angular distributions of the n ${\beta }^{-}$ decay, reduce themselves to the contribution to the n 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 Equation (7) of Ref. [6]), and and $\left|b_F\right|<0.03$, reported by H. Saul on behalf of the PERKEO III Collaboration [26]. If the value of the Fierz interference term $b_F=0.1219\left(12\right)$ is not acceptable, it can be concluded that using the phenomenological form of the CVC hypothesis in n ${\beta }^{-}$ decay, represented by the term $-q_{\mu }\widehat{q}/q^2$, introduces significant challenges for accurately describing and understanding the decay process.