Lorentz Structure of Vector Part of Matrix Elements of Transitions n <-> p, Caused by Strong Low-Energy Interactions and Hypothesis of Conservation of Charged Vector Current

TL;DR

This paper investigates the Lorentz structure of neutron-proton transition matrix elements induced by the charged hadronic vector current, revealing the dynamical origin of current conservation and the absence of G-odd contributions due to G-parity invariance.

Contribution

It demonstrates that the conservation of the charged hadronic vector current arises from G-even contributions and shows G-odd terms with $q_{\mu}$ do not appear due to G-parity invariance.

Findings

Current conservation linked to G-even contributions.

G-odd Lorentz structures are absent in the matrix elements.

Dynamical origin of current conservation explained.

Abstract

We analyse the Lorentz structure of the matrix elements of the transitions "neutron <-->proton", induced by the charged hadronic vector current. We show that the term maintaining conservation of the charged hadronic vector current even for different masses of the neutron and proton (see T. Leitner et al., Phys. Rev. C 73, 065502 (2006) and A. M. Ankowski, arXiv:1601.06169 [hep-ph]) has a dynamical origin, related to the G-even first class current contribution. We show that because of invariance of strong low-energy interactions under the G-parity transformations, the G-odd contribution with the Lorentz structure $q_{\mu}$, where $q_{\mu}$ is a momentum transferred, does not appear in the matrix elements of the ``neutron <-> proton' transitions.

Equations

$$⟨p(k_p,{\sigma }_p)|V_{\mu }^{(+)}(0)|n(k_n,{\sigma }_n)⟩$$

$$q^{\mu }⟨p(k_p,{\sigma }_p)|V_{\mu }^{(+)}(0)|n(k_n,{\sigma }_n)⟩=0,$$

$$\displaystyle\langle p(k_{p},\sigma_{p})|V^{(+)}_{\mu}(0)|n(k_{n},\sigma_{n})\rangle=\bar{u}_{p}(k_{p},\sigma_{p})\Big{(}\gamma_{\mu}\,F_{1}(q^{2})+\frac{i\sigma_{\mu\nu}q^{\nu}}{2m_{N}}\,F_{2}(q^{2})+\frac{q_{\mu}\not{q}}{m^{2}_{N}}\,F_{4}(q^{2})\Big{)}u_{n}(k_{n},\sigma_{n}).$$

$${\mathcal{L}}_{\pi N}(x)=\bar{N}(x)(i{\gamma }^{\mu }{\partial }_{\mu }-m_N)N(x)+\frac{1}{2}{\partial }_{\mu }\vec{\pi }(x)\cdot {\partial }_{\mu }\vec{\pi }(x)-\frac{1}{2}{m}_{\pi }^2{\vec{\pi }}^2(x)+g_{\pi }\bar{N}(x)i{\gamma }^5\vec{\tau }\cdot \vec{\pi }(x)N(x).$$

$${\vec{V}}_{\mu }(x)=\frac{1}{2}\bar{N}(x)\vec{\tau }{\gamma }_{\mu }N(x)+\vec{\pi }(x)\times {\partial }_{\mu }\vec{\pi }(x),$$

$$(i{\gamma }^{\mu }{\partial }_{\mu }-m_N+g_{\pi }i{\gamma }^5\vec{\tau }\cdot \vec{\pi }(x))N(x)$$

$$(\square +m_{\pi }^2)\vec{\pi }(x)-g_{\pi }\bar{N}(x)i{\gamma }^5\vec{\tau }N(x)$$

$$\displaystyle\partial^{\mu}\Big{(}\frac{1}{2}\,\bar{N}(x)\vec{\tau}\gamma_{\mu}N(x)\Big{)}=\frac{1}{2}\,\partial^{\mu}\bar{N}(x)\gamma_{\mu}\vec{\tau}\,N(x)+\frac{1}{2}\,\bar{N}(x)\vec{\tau}\gamma_{\mu}\partial^{\mu}N(x)=-\vec{\pi}(x)\times g_{\pi}\,\bar{N}(x)i\gamma^{5}\vec{\tau}\,N(x).$$

$$\displaystyle\partial^{\mu}\Big{(}\vec{\pi}(x)\times\partial_{\mu}\vec{\pi}(x)\Big{)}=\vec{\pi}(x)\times g_{\pi}\,\bar{N}(x)i\gamma^{5}\vec{\tau}\,N(x).$$

$$\displaystyle V^{(+)}_{\mu}(x)=\bar{N}(x)\tau^{(+)}\gamma_{\mu}N(x)+2\,\varepsilon^{+bc}\,\pi^{b}(x)\partial_{\mu}\pi^{c}(x)=\bar{p}(x)\gamma_{\mu}n(x)+\sqrt{2}\,i\,\Big{(}\pi^{0}(x)\partial_{\mu}\pi^{-}(x)-\pi^{-}(x)\partial_{\mu}\pi^{0}(x)\Big{)},$$

$$⟨\mathrm{out},p({\vec{k}}_p,{\sigma }_p)|V_{\mu }^{(+)}(0)|\mathrm{in},n({\vec{k}}_n,{\sigma }_n)⟩,$$

$$⟨\mathrm{out},p({\vec{k}}_p,{\sigma }_p)|V_{\mu }^{(+)}(0)|\mathrm{in},n({\vec{k}}_n,{\sigma }_n)⟩=⟨\mathrm{in},p({\vec{k}}_p,{\sigma }_p)|\mathbb{S}{V}_{\mu }^{(+)}(0)|\mathrm{in},n({\vec{k}}_n,{\sigma }_n)⟩.$$

$$\mathbb{S}=\mathrm{T}{e}^{i\int d^{4}x{\mathcal{L}}_{\pi NN}(x)},$$

$${\mathcal{L}}_{\pi NN}(x)=g_{\pi }\bar{N}(x)i{\gamma }^5\vec{\tau }\cdot \vec{\pi }(x)N(x).$$

$$\displaystyle\langle{\rm in},p(\vec{k}_{p},\sigma_{p})|{\mathbb{S}}V^{(+)}_{\mu}(0)|{\rm in},n(\vec{k}_{n},\sigma_{n})\rangle=\langle{\rm in},p(\vec{k}_{p},\sigma_{p})|{\rm T}\Big{(}e^{\textstyle i\int d^{4}x\,{\cal L}_{\pi NN}(x)}V^{(+)}_{\mu}(0)\Big{)}|{\rm in},n(\vec{k}_{n},\sigma_{n})\rangle.$$

$$|\mathrm{in},n({\vec{k}}_n,{\sigma }_n)⟩$$

$$⟨\mathrm{in},p({\vec{k}}_p,{\sigma }_p)|$$

$$[a_{n,\mathrm{in}}({\vec{k}}_n^{{}^{\prime }},{\sigma }_n^{\prime }),a_{n,\mathrm{in}}^{†}({\vec{k}}_n,{\sigma }_n)]$$

$$[a_{n,\mathrm{in}}({\vec{k}}_n^{{}^{\prime }},{\sigma }_n^{\prime }),a_{n,\mathrm{in}}({\vec{k}}_n,{\sigma }_n)]$$

$$[a_{p,\mathrm{in}}({\vec{k}}_p^{{}^{\prime }},{\sigma }_p^{\prime }),a_{p,\mathrm{in}}^{†}({\vec{k}}_p,{\sigma }_p)]$$

$$[a_{p,\mathrm{in}}({\vec{k}}_p^{{}^{\prime }},{\sigma }_p^{\prime }),a_{p,\mathrm{in}}({\vec{k}}_p,{\sigma }_p)]$$

$$\displaystyle\langle{\rm in},p(\vec{k}_{p},\sigma_{p})|{\mathbb{S}}V^{(+)}_{\mu}(0)|{\rm in},n(\vec{k}_{n},\sigma_{n})\rangle={{}_{N}}\langle{\rm in},p(\vec{k}_{p},\sigma_{p})|\langle 0_{\pi}|{\rm T}\Big{(}e^{i\int d^{4}x\,{\cal L}_{\pi NN}(x)}V^{(+)}_{\mu}(0)\Big{)}|0_{\pi}\rangle|{\rm in},n(\vec{k}_{n},\sigma_{n})\rangle_{N}.$$

$$\displaystyle\langle 0_{\pi}|{\rm T}\Big{(}e^{i\int d^{4}x\,{\cal L}_{\pi NN}(x)}V^{(+)}_{\mu}(0)\Big{)}|0_{\pi}\rangle=$$

$$\displaystyle={\rm T}\Big{(}\frac{1}{2}\,\bar{N}(0)\tau^{(+)}\gamma_{\mu}N(0)\int D\vec{\pi}\,e^{\textstyle i\int d^{4}x\,(\frac{1}{2}\,\partial_{\alpha}\vec{\pi}(x)\cdot\partial^{\alpha}\vec{\pi}(x)-\frac{1}{2}\,m^{2}_{\pi}\,\vec{\pi}^{\,2}+g_{\pi}\bar{N}(x)i\gamma^{5}\vec{\tau}\cdot\vec{\pi}(x)\,N(x))}$$

$$\displaystyle+2\,\varepsilon^{+bc}\int D\vec{\pi}\,\pi^{b}(z)\partial_{\mu}\pi^{c}(z)\,e^{\textstyle i\int d^{4}x\,(\frac{1}{2}\,\partial_{\alpha}\vec{\pi}(x)\cdot\partial^{\alpha}\vec{\pi}(x)-\frac{1}{2}\,m^{2}_{\pi}\,\vec{\pi}^{\,2}+g_{\pi}\bar{N}(x)i\gamma^{5}\vec{\tau}\cdot\vec{\pi}(x)\,N(x))}\Big{|}_{z=0}\Big{)}.$$

$$\int D\vec{\pi }{e}^{i\int d^{4}x(\frac{1}{2}{\partial }_{\alpha }\vec{\pi }(x)\cdot {\partial }^{\alpha }\vec{\pi }(x)-\frac{1}{2}{m}_{\pi }^2{\vec{\pi }}^2+g_{\pi }\bar{N}(x)i{\gamma }^5\vec{\tau }\cdot \vec{\pi }(x)N(x))}=$$

$$=\int D\vec{\pi }{e}^{i\int d^{4}x(-\frac{1}{2}\vec{\pi }\cdot (\square +m_{\pi }^2-i0)\vec{\pi }+g_{\pi }\bar{N}(x)i{\gamma }^5\vec{\tau }\cdot \vec{\pi }(x)N(x))}.$$

$$\vec{\pi }(x)\to \vec{\pi }(x)+\frac{1}{{\square }_x+m_{\pi }^2-i0}{g}_{\pi }\bar{N}(x)i{\gamma }^5\vec{\tau }\cdot N(x)=\vec{\pi }(x)+\int d^{4}y\Delta (x-y)g_{\pi }\bar{N}(y)i{\gamma }^5\vec{\tau }N(y),$$

$$\int D\vec{\pi }{e}^{i\int d^{4}x(\frac{1}{2}{\partial }_{\alpha }\vec{\pi }(x)\cdot {\partial }^{\alpha }\vec{\pi }(x)-\frac{1}{2}{m}_{\pi }^2{\vec{\pi }}^2+g_{\pi }\bar{N}(x)i{\gamma }^5\vec{\tau }\cdot \vec{\pi }(x)N(x))}=$$

$$=e^{i\frac{1}{2}{g}_{\pi }^2\int d^{4}x{d}^{4}y\bar{N}(x)i{\gamma }^5\vec{\tau }N(x)\cdot \Delta (x-y)\bar{N}(y)i{\gamma }^5\vec{\tau }N(y)}.$$

$$\displaystyle\varepsilon^{abc}\int D\vec{\pi}\,\pi^{b}(z)\partial_{\mu}\pi^{c}(z)\,e^{\textstyle i\int d^{4}x\,(\frac{1}{2}\,\partial_{\alpha}\vec{\pi}(x)\cdot\partial^{\alpha}\vec{\pi}(x)-\frac{1}{2}\,m^{2}_{\pi}\,\vec{\pi}^{\,2}+g_{\pi}\bar{N}(x)i\gamma^{5}\vec{\tau}\cdot\vec{\pi}(x)\,N(x))}\Big{|}_{z=0}\to$$

$$\displaystyle\varepsilon^{abc}\int D\vec{\pi}\,\pi^{b}(z)\partial_{\mu}\pi^{c}(z)\,e^{\textstyle i\int d^{4}x\,(\frac{1}{2}\,\partial_{\alpha}\vec{\pi}(x)\cdot\partial^{\alpha}\vec{\pi}(x)-\frac{1}{2}\,m^{2}_{\pi}\,\vec{\pi}^{\,2}+g_{\pi}\bar{N}(x)i\gamma^{5}\vec{\tau}\cdot\vec{\pi}(x)\,N(x)+\vec{J}(x)\cdot\vec{\pi}(x))}\Big{|}_{z=0}\to$$

$$\displaystyle\varepsilon^{abc}\int D\vec{\pi}\,\pi^{b}(z)\partial_{\mu}\pi^{c}(z)\,e^{\textstyle i\int d^{4}x\,(-\frac{1}{2}\,\vec{\pi}(x)\cdot(\Box+m^{2}_{\pi}-i0)\vec{\pi}(x)+g_{\pi}\bar{N}(x)i\gamma^{5}\vec{\tau}\cdot\vec{\pi}(x)\,N(x)+\vec{J}(x)\cdot\vec{\pi}(x))}\Big{|}_{z=0}.$$

$$\displaystyle\varepsilon^{abc}\int D\vec{\pi}\,\pi^{b}(z)\partial_{\mu}\pi^{c}(z)\,e^{\textstyle i\int d^{4}x\,(\frac{1}{2}\,\partial_{\alpha}\vec{\pi}(x)\cdot\partial^{\alpha}\vec{\pi}(x)-\frac{1}{2}\,m^{2}_{\pi}\,\vec{\pi}^{\,2}+g_{\pi}\bar{N}(x)i\gamma^{5}\vec{\tau}\cdot\vec{\pi}(x)\,N(x))}\Big{|}_{z=0}\to$$

$$\to {\epsilon }^{abc}(-i)\frac{\delta }{\delta J^b(z)}\frac{\partial }{\partial z^{\mu }}(-i)\frac{\delta }{\delta J^c(z)}$$

$$\displaystyle\,\int D\vec{\pi}\,e^{\textstyle i\int d^{4}x\,(-\frac{1}{2}\,\vec{\pi}(x)\cdot(\Box+m^{2}_{\pi}-i0)\vec{\pi}(x)+g_{\pi}\bar{N}(x)i\gamma^{5}\vec{\tau}\cdot\vec{\pi}(x)\,N(x)+\vec{J}(x)\cdot\vec{\pi}(x))}\Big{|}_{z=0,\vec{J}=0}.$$

$$\vec{\pi }(x)$$

Full text

I Introduction

In the paper by Leitner et al. Leitner2006 (see also Leitner2006a ; Leitner2006b ) the matrix element of the transition “neutron $\longrightarrow$ proton” or $n\to p$, induced by the charged hadronic vector current $V^{(+)}_{\mu}(0)$, has been written in the following form

[TABLE]

where $\bar{u}_{p}(k_{p},\sigma_{p})$ and $u_{n}(k_{n},\sigma_{n})$ are the Dirac bispinor wave functions of the free proton and neutron in the final and initial states of the transition $n\to p$, $m_{N}=(m_{p}+m_{n})/2$ is a nucleon mass or an averaged nucleon mass, expressed in terms of the proton $m_{p}$ and neutron $m_{n}$ masses, $\eta_{\mu\nu}$ is the metric tensor of the Minkowski spacetime, and $\gamma_{\mu}$ and $\sigma_{\mu\nu}=\frac{i}{2}(\gamma_{\mu}\gamma_{\nu}-\gamma_{\nu}\gamma_{\mu})$ are the Dirac matrices Itzykson1980 . Then, $q=k_{p}-k_{n}$ is the momentum transferred, and $F_{1}(q^{2})$ and $F_{2}(q^{2})$ are the form factors. The second term in Eq.(1) describes the contribution of the weak magnetism. The right-hand-side (r.h.s.) of Eq.(1) vanishes after multiplication by a momentum transferred $q^{\mu}$, i.e.

[TABLE]

even for $m_{p}\neq m_{n}$. Such a property of the matrix element of the transition $n\to p$ testifies conservation of the charged hadronic vector current $V^{(+)}_{\mu}(x)$, but only in the sense of the vanishing matrix element $\langle p(k_{p},\sigma_{p})|\partial^{\mu}V^{(+)}_{\mu}(x)|n(k_{n},\sigma_{n})\rangle=0$. This, of course, should not contradict the hypothesis of conservation of the vector current or the CVC hypothesis by Feynman and Gell-Mann Feynman1958 . Recently Ivanov2017c we have shown that the term $(-q_{\mu}\not{q}/q^{2})\,F_{1}(q^{2})$ is the contribution of the first class current Weinberg1958 ; Lee1956 .

This letter is addressed to the analysis of the dynamical nature of the term with the Lorentz structure $q_{\mu}\not{q}$. As has been proposed in Ivanov2017c , the vector part of the matrix element of the transition $n\to p$, caused by the contributions of the first class current only, should be taken in the following general form

[TABLE]

Below we show that the appearance of the term with the Lorentz structure $q_{\mu}\not{q}$ is fully caused by strong low–energy interactions.

The paper is organized as follows. In section II we propose for the analysis of the dynamical nature of the term with the Lorentz structure $q_{\mu}\not{q}$ to use a strongly coupled $\pi N$–system with the linear pion–nucleon pseudoscalar interaction. We show that only the total hadronic isovector vector current, being the sum of the nucleon and mesonic currents, can be locally conserved. In section III we derive the Lorentz structure of the matrix element of the transition $n\to p$ using the path–integral technique. In section IV we discuss the obtained results. In Appendix A we calculate the cross sections for the inelastic electron neutrino–neutron scattering and for the inverse $\beta$–decay. In order to illustrate the influence of the contributions of the term with the Lorentz structure $(-q_{\mu}\not{q}/q^{2})\,F_{1}(q^{2})$ we neglect the contributions of the weak magnetism and recoil of the outgoing nucleon, and the radiative corrections. In Fig. 1 we plot the relative contributions of the term $(-q_{\mu}\not{q}/q^{2})\,F_{1}(q^{2})$. We show that the processes of the inelastic electron neutrino–neutron scattering and of the inverse $\beta$–decay are insensitive to the contributions of the term, responsible for the vanishing of the matrix elements $\langle p|\partial^{\mu}V^{(+)}_{\mu}(0)|n\rangle=\langle n|\partial^{\mu}V^{(-)}_{\mu}(0)|p\rangle=0$ for different masses of the neutron and proton. In Appendix B we analyse the dynamical nature of the Lorentz structure of the matrix element $\langle p|A^{(+)}_{\mu}(0)|n\rangle$ of the transition $n\to p$, caused by the charged hadronic axial–vector current. We show that the linear pion–nucleon pseudoscalar interaction, used for the analysis of the dynamical nature of the Lorentz structure of the charged hadronic vector part of the transition $n\to p$, allows to reproduce fully the standard Lorentz structure of the axial–vector part of the hadronic $n\to p$ transition Leitner2006 .

II Hadronic vector current of strongly coupled pion–nucleon

system

As an example of strongly coupled system we consider the $\pi N$–system with the simplest linear pseudoscalar interaction BD1967 . The Lagrangian of such a system is given by BD1967

[TABLE]

Here $N(x)$ is the nucleon isospin doublet with components $(p(x),n(x))$, where $p(x)$ and $n(x)$ are the proton and neutron field operators, $\vec{\pi}(x)=(\pi^{+}(x),\pi^{0}(x),\pi^{-}(x))$ is the pion field operator, $m_{N}$ and $m_{\pi}$ are the nucleon and pion masses, $g_{\pi}$ is the pion–nucleon coupling constant, $\gamma^{5}$ is the Dirac matrix Itzykson1980 , and $\vec{\tau}=(\tau_{1},\tau_{2},\tau_{3})$ is the Pauli isospin matrix BD1967 .

The Lagrangian Eq.(4) is invariant under global isospin transformations BD1967 . This, according to Feynman and Gell–Mann Feynman1958 , leads to the isovector hadronic vector current of the $\pi N$ system given by

[TABLE]

local conservation of which one may check using the equations of motion. The Dirac equation for the nucleon and the Klein–Gordon equation for the pions are given by

[TABLE]

Using the Dirac equation for the nucleon Eq.(II) one may show that the nucleon part of the isovector hadronic vector current Eq.(5) is not conserved

[TABLE]

Hence, in the strongly coupled $\pi N$–system a strong non–conservation of the nucleon part of the isovector hadronic vector current is caused by strong low–energy interactions but not by isospin violation. The divergence of the mesonic part of the isovector hadronic vector current is equal to

[TABLE]

Summing up the contributions Eqs.(7) and (8) we get $\partial^{\mu}\vec{V}_{\mu}(x)=0$. This means that in the strongly coupled $\pi N$–system only the total hadronic isovector vector current, being the sum of the nucleon and mesonic currents, can be locally conserved.

III Dynamical Lorentz structure of the matrix element of the

$n\to p$ transition, caused by the hadronic vector current Eq.(5)

The charged hadronic vector current responsible for the hadronic $n\to p$ transition is equal to BD1967

[TABLE]

where $\tau^{(+)}=(\tau^{1}+i\tau^{2})/2$, $\varepsilon^{+bc}=(\varepsilon^{1bc}+i\,\varepsilon^{2bc})/2$ and $\varepsilon^{abc}$ is the Levi-Civita isotensor BD1967 . Now we may calculate the matrix element

[TABLE]

where $\langle{\rm out},p(\vec{k}_{p},\sigma_{p})|$ and $|{\rm in},n(\vec{k}_{n},\sigma_{n})\rangle$ are the wave functions of the free proton and neutron in the final (i.e. out–state at $t\to+\infty$) and initial (i.e. in–state at $t\to-\infty$) states, respectively Itzykson1980 . Using the relation $\langle{\rm out},p(\vec{k}_{p},\sigma_{p})|=\langle{\rm in},p(\vec{k}_{p},\sigma_{p})|{\mathbb{S}}$, where ${\mathbb{S}}$ is the S–matrix, we rewrite the matrix element Eq.(10) as follows

[TABLE]

Since the transition $n\to p$ is fully induced by strong low–energy interactions, we define the S–matrix only in terms of strong low–energy interactions. For simplicity we propose to use only $\pi N$–system, a dynamics of which is determined by the Lagrangian Eq.(4). The corresponding S–matrix is given by Itzykson1980

[TABLE]

where ${\rm T}$ is a time–ordering operator and ${\cal L}_{\pi NN}(x)$ is equal to

[TABLE]

Plugging Eq.(12) into Eq.(11) we get

[TABLE]

The wave functions of the neutron and proton we determine in terms of the operators of creation (annihilation)

[TABLE]

The operators $(a^{\dagger}_{n,\rm in}(\vec{k}_{n},\sigma_{n}),a_{p,\rm in}(\vec{k}_{p},\sigma_{p})$ and $(a_{n,\rm in}(\vec{k}_{n},\sigma_{n}),a^{\dagger}_{p,\rm in}(\vec{k}_{p},\sigma_{p})$ obey standard anticommutation relations Itzykson1980

[TABLE]

The vacuum wave function we define as follows $|0\rangle=|0_{N}\rangle|0_{\pi}\rangle$, where $|0_{N}\rangle$ and $|0_{\pi}\rangle$ are the vacuum wave functions of a nucleon and mesons, respectively. Since there are no mesons in the initial and final states of the transition $n\to p$, the matrix element Eq.(14) we may rewrite as follows

[TABLE]

The wave functions ${{}_{N}}\langle{\rm in},p(\vec{k}_{p},\sigma_{p})|$ and $|{\rm in},n(\vec{k}_{n},\sigma_{n})\rangle_{N}$ mean that the operators $(a^{\dagger}_{n,\rm in}(\vec{k}_{n},\sigma_{n}),a_{p,\rm in}(\vec{k}_{p},\sigma_{p})$ act only on the nucleon vacuum wave function $|0_{N}\rangle$. The vacuum expectation value \langle 0_{\pi}|{\rm T}\Big{(}e^{\textstyle i\int d^{4}x\,{\cal L}_{\pi NN}(x)}V^{(+)}_{\mu}(0)\Big{)}|0_{\pi}\rangle we calculate using the path–integral technique Bogoliubov1959 . We rewrite the vacuum expectation value as follows

[TABLE]

The integrals are Gaussian. The calculation of the first integral runs as follows. We transcribe it into the form

[TABLE]

Then, we make a shift

[TABLE]

where $\Delta(x-y)$ is the $\pi$–meson propagator BD1967 . The result of the integration is

[TABLE]

For the integration of the pionic part of the charge hadronic vector current we use the following procedure. We rewrite the path–integral, given by the second term in the r.h.s. of Eq.(III), with an external source $\vec{J}(x)$ of the $\pi$–meson field:

[TABLE]

Then, the pionic fields in the integrand we replace by functional derivatives with respect to the external source:

[TABLE]

For the calculation of the integral over $\vec{\pi}$ we make a change of variables

[TABLE]

As a result, for the integral over $\vec{\pi}$ we obtain the following expression

[TABLE]

Plugging Eq.(III) into Eq.(III) and calculating the functional derivatives with respect to external sources we arrive at the expression

[TABLE]

Thus, after the calculation of the vacuum expectation value Eq.(III) the matrix element Eq.(11) of the transition $n\to p$ becomes equal to

[TABLE]

As the first step towards the analysis of the Lorentz structure of the matrix element of the transition $n\to p$, given by Eq.(III), we propose to consider the contributions of order $g^{2}_{\pi}$. We understand that the value of the coupling constant $g_{\pi}$ is sufficiently large. Nevertheless, the Lorentz structure of the matrix element Eq.(III) can be fully understood to order $g^{2}_{\pi}$ Landau1959 .

III.1 Lorentz structure of the matrix element Eq.(III)

to order $g^{2}_{\pi}$, determined by the mesonic part of the charged hadronic vector current Eq.(9)

To order $g^{2}_{\pi}$ the contribution of the mesonic part of the charged hadronic vector current is given by the expression

[TABLE]

where $S_{F}(x-y)$ is the nucleon propagator BD1967 . For the derivation of Eq.(III.1) we have used the relation $\varepsilon^{+bc}\tau^{b}\tau^{c}=2i\tau^{(+)}$. In the momentum representation the r.h.s. of Eq.(III.1) reads

[TABLE]

The integral is symmetric with respect to the transformation $k_{p}\longleftrightarrow k_{n}$. This means that the momentum integral possesses the following Lorentz structure

[TABLE]

which can be confirmed by a direct calculation of the integral, where $P=(k_{p}+k_{n})/2$ and $a$, $b$ and $c$ are coefficients, which can be determined by a direct calculation of the integral. The symmetry of the integral with respect to the transformation $k_{p}\longleftrightarrow k_{n}$ testifies that the term with the Lorentz structure $q_{\mu}=(k_{p}-k_{n})_{\mu}$, which is antisymmetric with respect to the transformation $k_{p}\longleftrightarrow k_{n}$, does not appear in the matrix element of the transition $n\to p$ in agreement with a suppression of the contributions of the second class currents Weinberg1958 ; Lee1956 .

As a consequence of the relations $m^{2}_{N}\gg m^{2}_{\pi}\gg q^{2}$ one may perform the calculation of the momentum integral Eq.(30) in the heavy baryon approximation Ericson2006 . Since we are interested in the term with the Lorentz structure $q_{\mu}\not{q}$ only, skipping standard intermediate steps of the calculations for the coefficient $c$ we obtain the following result

[TABLE]

where we have neglected the contributions of order $O(1/m^{2}_{N})$. Then, using the Dirac equations $\bar{u}_{p}\not{P}u_{n}=(m_{p}+m_{n})/2=m_{N}$ that does not violate the property of the term with the Lorentz structure $P_{\mu}\not{P}$ to be a contribution of the first class current Ivanov2017c and the Gordon identity Itzykson1980

[TABLE]

we transcribe Eq.(III.1) into the form

[TABLE]

where $A_{\pi}=a+m_{N}b$, $B_{\pi}=-m_{N}b$ and $C_{\pi}=m^{2}_{N}c$.

III.2 Lorentz structure of the matrix element Eq.(III)

to order $g^{2}_{\pi}$, determined by the nucleon part of the charged hadronic vector current Eq.(9)

To order $g^{2}_{\pi}$ the dynamical contribution of the nucleon part of the charge hadronic vector current to the matrix element of the transition $n\to p$, given by Eq.(III), is determined by the matrix element

[TABLE]

where we have used the relations $\vec{\tau}^{\,2}=3$ and $\vec{\tau}\cdot\tau^{(+)}\vec{\tau}=-\tau^{(+)}$. The contributions of the first two terms in Eq.(III.2) can be removed by renormalization of the masses and wave functions of the neutron and proton, respectively Matthews1951 ; Schweber1962 . Thus, a non–trivial contribution comes from the third term only. In the momentum representation it reads

[TABLE]

This integral is also symmetric with respect to the transformation $k_{p}\longleftrightarrow k_{n}$, so it should also have a structure

[TABLE]

where the coefficients $a^{\prime}$, $b^{\prime}$ and $c^{\prime}$ are determined by a direct calculation of the integral. Thus, the contribution of the term with the Lorentz structure $q_{\mu}=(k_{p}-k_{n})_{\mu}$, which is antisymmetric with respect to the transformation $k_{p}\longleftrightarrow k_{n}$, does not appear in the nucleon part of the charge hadronic vector current. A direct calculation of the integral in Eq.(36) gives the following value of the coefficient $c^{\prime}$: $c^{\prime}=-1/24m^{2}_{N}$. Using the Dirac equations $\bar{u}_{p}\not{P}u_{n}=m_{N}$ that does not violate the property of the term with the Lorentz structure $P_{\mu}\not{P}$ to be a contribution of the first class current Ivanov2017c and the Gordon identity Eq.(32) we transcribe the r.h.s. of Eq.(III.2) into the form

[TABLE]

where $A_{N}=(a^{\prime}+m_{N}b^{\prime})/4$, $B_{N}=-m_{N}b^{\prime}/4$ and $C_{N}=m^{2}_{N}c^{\prime}$.

Summing up the contributions of the nucleon and mesonic parts of the charged hadronic vector current for the vector part of the matrix element of the transition $n\to p$ Eq.(III), calculated to order $g^{2}_{\pi}$, we obtain the expression

[TABLE]

Thus, we have shown that the matrix element of the transition $n\to p$, calculated to order $g^{2}_{\pi}$, can be expressed in terms of three Lorentz structures $\gamma_{\mu}$, $i\sigma_{\mu\nu}q^{\nu}$ and $q_{\mu}\not{q}$, which are induced by the first class current Ivanov2017c . Indeed, the isovector hadronic vector current Eq.(5) has a positive $G$–parity and belongs to the first class current Ivanov2017c

[TABLE]

where $T$ is a transposition and $C$ is the charge conjugation matrix Itzykson1980 . For the derivation of the relation Eq.(III.2) we have used the relations $C\gamma_{\mu}C=\gamma^{T}_{\mu}$, $i\tau_{2}\vec{\tau}i\tau_{2}=\vec{\tau}^{\,T}$ and $N^{T}(x)\bar{N}^{T}(x)=-\bar{N}(x)N(x)$ Itzykson1980 and $G\vec{\pi}(x)G^{-1}=-\vec{\pi}(x)$ Weinberg1958 ; Lee1956 .

Since, the coefficient $C_{N}$ is much smaller than the coefficient $C_{\pi}$, the contribution of the Lorentz structure $q_{\mu}\not{q}$ to the matrix element of the transition $n\to p$ is practically determined by the mesonic part of the charged hadronic vector current Eq.(9). Of course, the coefficients $A_{\pi}$ and $A_{N}$ can depend on the ultra–violet cut–off $\Lambda$. However, such a dependence can be removed by renormalization of the coupling constant $g^{2}_{\pi}$ Matthews1951 ; Schweber1962 .

As strong low–energy interactions are invariant under the $G$–parity transformation Lee1956 (see also Ivanov2017c )

[TABLE]

where we have used the relation $C\gamma^{5}C=-\gamma^{5T}$ Itzykson1980 , and the terms with the Lorentz structures $\gamma_{\mu}$, $i\sigma_{\mu\nu}q^{\nu}$ and $q_{\mu}\not{q}$ possess the positive $G$–parity, i.e. they are the contributions of the first class current Ivanov2017c , the term with the Lorentz structure $q_{\mu}$, having a negative $G$–parity Ivanov2017c , should not appear in the matrix element of the transition $n\to p$ Eq.(III) to any order of $g^{2}_{\pi}$–expansion. This allows to write Ivanov2017c

[TABLE]

where $F_{1}(q^{2})$, $F_{2}(q^{2})$ and $F_{4}(q^{2})$ are form factors, calculated to all orders of $g^{2}_{\pi}$–expansion.

IV Conclusive discussions

We have analysed the Lorentz structure of the matrix element of the transition $n\to p$, caused by the charged hadronic vector current. We have shown that in addition to the standard terms with the Lorentz structure $F_{1}(q^{2})\,\gamma_{\mu}$ and $F_{2}(q^{2})\,i\sigma_{\mu\nu}q^{\nu}/2m_{N}$, caused by the contributions of the electric charge distribution and the weak magnetism inside the hadron, one obtains the term with the structure $F_{4}(q^{2})\,q_{\mu}\not{q}/m^{2}_{N}$. Using the simplest model of strongly coupled $\pi N$–system with the linear pion–nucleon pseudoscalar interaction we have shown that the contribution of the term with the Lorentz structure $F_{4}(q^{2})\,q_{\mu}\not{q}/M^{2}$ is practically induced by the mesonic part of the hadronic isovector vector current. We have also shown that the term with the Lorentz structure $F_{3}(q^{2})\,q_{\mu}/m_{N}$, caused by the second class current Weinberg1958 ; Lee1956 , cannot be, in principle, induced by strong low–energy interactions invariant under $G$–parity transformations.

A requirement of conservation of the charged hadronic vector current even for different masses of the hadrons in the initial and final states (see Leitner2006 ; Leitner2006a ; Leitner2006b ; Paschos2006a ; Paschos2006b and so on) in the sense of the vanishing of the matrix element $\langle p|\partial^{\mu}V^{(+)}_{\mu}(0)|n\rangle=0$ of the hadronic $n\to p$ transition leads to the relation $F_{4}(q^{2})=-(m^{2}_{N}/q^{2})\,F_{1}(q^{2})$. Such a relation leads to the appearance of the term $(-q_{\mu}\not{q}/q^{2})\,F_{1}(q^{2})$ in the matrix element of the hadronic $n\to p$ transition.

For simplicity we have restricted our analysis by the simplest theory of $\pi N$ strong interactions described by the Lagrangian Eq.(4) with the linear pseudoscalar $\pi NN$–interaction BD1967 ; Matthews1951 ; Schweber1962 . However, one may assert that the obtained result, i.e. the existence of the term with the Lorentz structure $q_{\mu}\not{q}$ and the suppression of the term with the Lorentz structure $q_{\mu}$, which are the contributions of the first and second class currents, respectively, should be valid in any theory of strong low–energy interactions Weinberg1968 ; Weinberg1968a ; Scherer2012 , which are invariant under $G$–transformations Lee1956 (see also Weinberg1958 ). Our assertion is based only on $G$–invariance of such theories. Indeed, it is hardly possible to perform analytical calculations, which are similar to those we have carried out in this paper, within such complicated non–linear theories of meson–nucleon low–energy interactions as Weinberg1968 ; Weinberg1968a and Chiral perturbation theory Scherer2012 .

In Appendix A we have shown that the cross sections for the electron neutrino–neutron scattering and for the inverse $\beta$–decay, calculated in the non–relativistic approximation with respect to the outgoing hadron, are insensitive to the contributions of the term $(-q_{\mu}\not{q}/q^{2})\,F_{1}(q^{2})$. That is why one may assert that it is important to search for processes, which are sensitive to the contributions of the term $(-q_{\mu}\not{q}/q^{2})\,F_{1}(q^{2})$.

In Appendix B we have analysed the dynamical nature of the Lorentz structure of the matrix element $\langle p|A^{(+)}_{\mu}(0)|n\rangle$ of the transition $n\to p$, caused by the charged hadronic axial–vector current $A^{(+)}_{\mu}(0)$. We have shown that the low–energy pion–nucleon interaction Eq.(13) allows to reproduce fully the standard Lorentz structure of the axial–part of the hadronic $n\to p$ transition Leitner2006 .

Of course, our results, obtained for the hadronic $n\to p$ transition Leitner2006 , are fully valid for the hadronic $p\to n$ transition Ankowski2016 .

V Acknowledgements

This work was supported by the Austrian “Fonds zur Förderung der Wissenschaftlichen Forschung” (FWF) under Contracts I689-N16, P26781-N20 and P26636-N20 and “Deutsche Förderungsgemeinschaft” (DFG) AB 128/5-2.

VI Appendix A: Cross sections for the inelastic electron

neutrino–neutron scattering and for the inverse $\beta$–decay

In this Appendix we calculate the cross sections for the inelastic scattering $\nu_{e}+n\to p+e^{-}$ and the inverse $\beta$–decay $\bar{\nu}_{e}+p\to n+e^{+}$ by taking into account the contributions of the term $-q_{\mu}\not{q}/q^{2}$ responsible for the constraint $\langle h^{\prime}|\partial^{\mu}V^{(\pm)}_{\mu}(0)|h\rangle=0$ even for different masses of incoming $h$ and outgoing $h^{\prime}$ hadrons. Below the contributions of such a term we call the contributions of Exact Conservation of the charged weak hadronic Vector Current or the ECVC effect.

The amplitudes of the inelastic scattering $\nu_{e}+n\to p+e^{-}$ and the inverse $\beta$–decay $\bar{\nu}_{e}+p\to n+e^{+}$ we define in the non–relativistic approximation for the outgoing nucleon. They are equal to

[TABLE]

and

[TABLE]

where $G_{F}$ and $V_{ud}$ are the Fermi weak coupling constant and the Cabibbo–Kobayashi–Maskawa (CKM) matrix element PDG2016 , $\bar{u}_{-}(\vec{k}_{-},\sigma_{-})$ and $u_{\nu}(\vec{k}_{\nu},-\frac{1}{2})$ are the Dirac wave functions of the free electron and electron neutrino with 3–momenta $\vec{k}_{-}$ and $\vec{k}_{\nu}$ and polarizations $\sigma_{e}=\pm 1$ and $-\frac{1}{2}$ Ivanov2013 ; Ivanov2014 ; Ivanov2013a , respectively, and $(\gamma_{\mu},\gamma^{5})$ are the Dirac matrices. Then, $\bar{v}_{\bar{\nu}}(\vec{k}_{\bar{\nu}},+\frac{1}{2})$ and $v_{+}(\vec{k}_{+},\sigma_{+})$ are the Dirac wave functions of the electron antineutrino and positron with 3–momenta $\vec{k}_{\bar{\nu}}$ and $\vec{k}_{+}$ and polarizations $\sigma_{+}=\pm 1$ and $+\frac{1}{2}$ Ivanov2013 ; Ivanov2014 ; Ivanov2013a , respectively. The matrix elements $\langle p(\vec{k}_{p},\sigma_{p})|J^{(+)}_{\mu}(0)|n(\vec{k}_{n},\sigma_{n})\rangle$ and $\langle n(\vec{k}_{n},\sigma_{n})|J^{(-)}_{\mu}(0)|p(\vec{k}_{p},\sigma_{p})\rangle$ of the hadronic $n\to p$ and $p\to n$ transitions we define as follows Leitner2006

[TABLE]

and

[TABLE]

where $J^{(\pm)}_{\mu}(0)=V^{(\pm)}_{\mu}(0)-A^{(\pm)}_{\mu}(0)$, $u_{n}(\vec{k}_{n},\sigma_{n})$ and $u_{p}(\vec{k}_{p},\sigma_{p})$ are the Dirac wave functions of the free neutron and proton with 3–momenta and polarizations $(\vec{k}_{n},\sigma_{n}=\pm 1)$ and $(\vec{k}_{p},\sigma_{p}=\pm 1)$. Then, $F_{1}(q^{2})$ and $F_{A}(q^{2})$ are the vector and axial–vector form factors Leitner2006 . The vector parts of the matrix elements Eqs.(A-3) and (A-4) obey the constraints

[TABLE]

even for different masses of the neutron and proton. In the matrix elements Eqs.(A-3) and (A-4) we have neglected the contributions of the weak magnetism and one–pion exchange Leitner2006 . In the approximation, when the squared momentum transferred $q^{2}=(\pm k_{p}\mp k_{n})^{2}$ is much smaller than the scales $M^{2}_{V}$ and $M^{2}_{A}$ defining the effective radii of the vector and axial–vector form factors, the matrix elements Eqs.(A-3) and (A-4) can be reduced to the form

[TABLE]

and

[TABLE]

where $\lambda=-1.2750(9)$ is the axial coupling constant Abele2008 (see also Ivanov2013 ; Ivanov2014 ; Ivanov2013a ; Ivanov2017e ).

In order to illustrate the contribution of the term $-q_{\mu}\not{q}/q^{2}$ responsible for the fulfilment of the constraints Eq.(A-5), we neglect the contributions of the weak magnetism, recoil and radiative corrections Ivanov2013a . Skipping intermediate standard calculations Ivanov2013a we obtain the following cross sections for the inelastic electron neutrino–neutron scattering $\sigma(E_{\nu})$ and the inverse $\beta$–decay $\sigma(E_{\bar{\nu}})$:

[TABLE]

where $E_{-}=E_{\nu}+\Delta$, $k_{-}=\sqrt{E^{2}_{-}-m^{2}_{e}}$ and $\beta_{-}=k_{-}/E_{-}$ are the energy, momentum and velocity of the electron, and

[TABLE]

where $E_{+}=E_{\bar{\nu}}-\Delta$ and $k_{+}=\sqrt{E^{2}_{+}-m^{2}_{e}}$ are the energy and momentum of the positron. The cross sections $\sigma_{0}(E_{\nu})$ and $\sigma_{0}(E_{\bar{\nu}})$ are given by Ivanov2013a

[TABLE]

In the inelastic electron neutrino–neutron scattering and the inverse $\beta$–decay the energies of neutrino and antineutrino vary in the regions $E_{\nu}\geq 0$ and $E_{\bar{\nu}}\geq(E_{\bar{\nu}})_{\rm thr}=((m_{n}+m_{e})^{2}-m^{2}_{p})/2m_{p}=1.8061\,{\rm MeV}$ Ivanov2013a . The terms dependent on $\Delta$ are caused by the ECVC effect. The relative contributions of the ECVC effect to the cross sections under consideration we define as follows

[TABLE]

and

[TABLE]

where $R_{\nu}(E_{\nu})=\Delta\sigma(E_{\nu})/\sigma_{0}(E_{\nu})$, $R_{\bar{\nu}}(E_{\bar{\nu}})=\Delta\sigma(E_{\bar{\nu}})/\sigma_{0}(E_{\bar{\nu}})$ with $\Delta\sigma(E_{\nu})=\sigma(E_{\nu})-\sigma_{0}(E_{\nu})$ and $\Delta\sigma(E_{\bar{\nu}})=\sigma(E_{\bar{\nu}})-\sigma_{0}(E_{\bar{\nu}})$, respectively. The cross sections Eq.(A-8) and Eq.(A-9) are calculated in the laboratory frame in the non–relativistic approximation for outgoing hadrons. Since the most important region of the antineutrino energies for the inverse $\beta$–decay is $2\,{\rm MeV}\leq E_{\bar{\nu}}\leq 8\,{\rm MeV}$ Ivanov2013a , in Fig. 1 we plot $R_{\nu}(E_{\nu})$ and $R_{\bar{\nu}}(E_{\bar{\nu}})$ for $E_{\nu}$ and $E_{\bar{\nu}}$ varying over the regions $2\,{\rm MeV}\leq E_{\nu}\leq 8\,{\rm MeV}$ and $2\,{\rm MeV}\leq E_{\bar{\nu}}\leq 8\,{\rm MeV}$, respectively.

Our numerical analysis of the relative contributions of the ECVC effect to the cross sections for the inelastic electron neutrino–neutron scattering and for the inverse $\beta$–decay shows that these processes are not sensitive to the ECVC effect. Indeed, the contribution of the ECVC effect to the cross section for the inelastic electron neutrino–neutron scattering is smaller than $0.7\,\%$ at $E_{\nu}\simeq 2\,{\rm MeV}$ and decreases by about two orders of magnitude at $E_{\nu}\simeq 8\,{\rm MeV}$. The cross section for the inverse $\beta$–decay, applied to the analysis of the deficit of positrons induced by reactor electron antineutrinos Ivanov2013a ; Mention2013 , should be averaged over the reactor electron antineutrino energy spectrum, which has a maximum at $E_{\bar{\nu}}\simeq 4\,{\rm MeV}$. According to Fig. 1, the contribution of the ECVC effect should decease the yield of positrons $Y_{e^{+}}$ by about $0.5\,\%$. Since such a contribution is smaller than the experimental error bars $Y_{e^{+}}=0.943(23)$ Mention2013 , one may argue that the inverse $\beta$–decay is insensitive to the contribution of the ECVC effect.

VII Appendix B: The Lorentz structure of the matrix element of

the hadronic $n\to p$ transition, caused by the charged hadronic axial–vector current

In this Appendix we analyse the Lorentz structure of the axial–vector part of the hadronic $n\to p$ transition, induced by the charged hadronic axial–vector current

[TABLE]

The matrix element of our interest is

[TABLE]

where $\langle{\rm out},p(\vec{k}_{p},\sigma_{p})|$ and $|{\rm in},n(\vec{k}_{n},\sigma_{n})\rangle$ are the wave functions of the free proton and neutron in the final (i.e. out–state at $t\to+\infty$) and initial (i.e. in–state at $t\to-\infty$) states, respectively Itzykson1980 . Using the relation $\langle{\rm out},p(\vec{k}_{p},\sigma_{p})|=\langle{\rm in},p(\vec{k}_{p},\sigma_{p})|{\mathbb{S}}$, where ${\mathbb{S}}$ is the S–matrix, we rewrite the matrix element Eq.(B-2) as follows

[TABLE]

Since the transition $n\to p$ is fully induced by strong low–energy interactions, we define the S–matrix only in terms of strong low–energy interaction described by the Lagrangian Eq.(4). It is given by (see Eq.(12). Plugging Eq.(12) into Eq.(B-3) we get

[TABLE]

After the integration over the pion–fields we arrive at the expression

[TABLE]

To order $g^{2}_{\pi}$ the dynamical contribution of the nucleon part of the charge hadronic axial–vector current to the matrix element of the transition $n\to p$, given by Eq.(B-5), is determined by the matrix element

[TABLE]

In the momentum representation we get

[TABLE]

where $q=k_{p}-k_{n}$. The contribution proportional to $\delta^{(4)}(0)$ can be in principle removed by using the normal–ordered form of the four–nucleon operator of interaction Itzykson1980 . Indeed, replacing $\bar{N}(x)i\gamma^{5}\vec{\tau}N(x)\cdot\Delta(x-y)\,\bar{N}(y)i\gamma^{5}\vec{\tau}N(y)$ by $:\bar{N}(x)i\gamma^{5}\vec{\tau}N(x)\cdot\Delta(x-y)\,\bar{N}(y)i\gamma^{5}\vec{\tau}N(y):$ the vacuum expectation value of the operator $:\bar{N}(x)i\gamma^{5}\vec{\tau}N(x)\cdot\Delta(x-y)\,\bar{N}(y)i\gamma^{5}\vec{\tau}N(y):$ is equal to zero. The contribution of the momentum integral of the second term is divergent and proportional to $q_{\mu}$. As a result, the matrix element Eq.(VII we may define as follows

[TABLE]

Thus, for the matrix element Eq.(B-5), calculated to order $g^{2}_{\pi}$, we obtain the following expression

[TABLE]

Following Landau1959 we may argue that the Lorentz structure of the matrix element Eq.(B-5), calculated to order $g^{2}_{\pi}$, should be valid to all order of the $g^{2}_{\pi}$–expansion. The latter is, of course, because of invariance of strong low–energy interactions under the $G$–transformations. Thus, the matrix element Eq.(B-5) should have the following Lorentz structure, induced by the first class axial–vector current Weinberg1958

[TABLE]

where $F_{A}(q^{2})$ and $F_{P}(q^{2})$ are the axial–vector and pseudoscalar form factors Leitner2006 . Taking into account the PCAC hypothesis (or the hypothesis of Partial Conservation of Axial–vector Current) Adler1968 ; DeAlfaro1973 we may rewrite the r.h.s. of Eq.(B-10) as follows

[TABLE]

where we have set $F_{A}(q^{2})=-F_{P}(q^{2})$. At $q^{2}=0$ we set $F_{A}(q^{2})=\lambda$, where $\lambda=-1.2750(9)$ is the axial coupling constant Abele2008 (see also Ivanov2013 ; Ivanov2014 ; Ivanov2013a ; Ivanov2017e ). In the chiral limit $m_{\pi}\to 0$ the matrix element Eq.(B-11) multiplied by the 4–momentum transferred $q^{\mu}$ vanishes

[TABLE]

even for different masses of the neutron and proton, according to the PCAC hypothesis Adler1968 ; DeAlfaro1973 pointing out an operator relation $\partial^{\mu}\vec{A}_{\mu}(x)=m^{2}_{\mu}F_{\pi}\,\vec{\pi}(x)$, where $F_{\pi}$ is the pion–decay constant PDG2016 . In the chiral limit $m_{\pi}\to$ we get $\partial^{\mu}\vec{A}_{\mu}(x)=0$ agreeing well with Eq.(B-12).