§ INTRODUCTION: FROM NEUTRINO TO MASSIVE NEUTRAL LEPTON

In <cit.>, an approach to description of the
neutrino–nucleus $\nu A\to \nu A^{(*)}$ and antineutrino–nucleus $\bar\nu A\to \bar\nu A^{(*)}$
interactions was formulated and detailed for the case where the target nucleus $A^{}$
may remain in its initial state or undergo a transition to the excited state $A^{*}$ (with its integrity retained).
The approach relies on the description of the nucleus as a bound state of its constituent nucleons based on the
multiparticle wave function of the nucleus in the general form.
It was shown that an elastic interaction which preserves the initial nuclear quantum state leads to quadratic enhancement
of the corresponding observed cross section in terms of the number of nucleons.
At the same time, the total cross section of all other (possible) inelastic processes accompanied by a change in
the nuclear quantum state generally demonstrates only linear dependence on the number of nucleons.
In addition, it was demonstrated that the behavior of elastic and inelastic cross sections is determined by the
factors   $|F_{p/n}(\bm{q})|^2$ and  $(1-|F_{p/n}(\bm{q})|^2)$, respectively, where  $F_{p/n}(\bm{q})$  is the nuclear form factor of the proton/neutron normalized to unity.
These form factors govern a smooth transition from the regime of elastic (coherent) (anti)neutrino–nucleus scattering to
the inelastic (incoherent) regime.
In the general case, where the three-momentum $\bm{q}$   transferred to the nucleus is neither small nor large, the coherent and incoherent contributions to the total cross section should be considered simultaneously.

It was also noted that owing to their common nature resulting from weak neutral currents,
the elastic and inelastic neutrino processes $\nu(\bar\nu) A$ are experimentally indistinguishable
when the only observable is the target nucleus recoil energy.
Therefore, in experiments aimed (at rather high energies) to study coherent (anti)neutrino scattering, there can be an incoherent background indistinguishable from the signal when the $\gamma$ deexciting the nucleus are undetectable.
For example, if in the COHERENT experiment
<cit.> with the ${}^{133}\text{Cs}$  nucleus at the (anti)neutrino energy of
30–50 MeV nuclear deexcitation $\gamma$ quanta fail to be detected, the data from this
experiment contain the desired CE$\nu$NS events with an unavoidable inelastic (incoherent) admixture at a level of 15–20%.
On the other hand, the incoherent contribution can be directly measured (estimated) by purposeful detection of photons emitted by target nuclei excited in inelastic processes <cit.>
with the number of these photons being proportional to the inelastic-to-elastic $\nu A$ scattering ratio.
These photons should be time-correlated with the incident neutrino beam and have a range of energies typical of the nucleus $A$, which are usually much higher than the nuclear recoil energy, thus noticeably simplifying their detection.
Simultaneous detection of the recoil energy and the nuclear deexcitation
$\gamma$ energy will make it possible to separate the “pure” CE$\nu$NS contribution,
and, consequently, to study the nuclear structure and carry out the precision search for new physics in neutrino processes.
Obviously, the approach to the scattering of neutrinos and antineutrinos off nuclei used in
can be generalized to the case where a massive neutral particle is considered instead of a massless (anti)neutrino
and where the rather weak interaction of this particle with nucleons of the nucleus is described by the
phenomenological Lagrangian taking into account possible Lorentz structures (scalar, pseudoscalar, vector and axial vector).
The relevant importance of considering this case follows, for example, from the necessity of correctly understanding the balance of elastic and inelastic interaction in the problem of direct detection of massive weakly interacting neutral particles of dark matter via their (nonrelativistic) scattering from target nuclei of the corresponding detector.
Note that CE$\nu$NS-events considered in
are an irremovable background for experiments aimed at direct detection of dark matter particles under terrestrial conditions (see, for example, <cit.>,
since the main and so far, the only signature of both processes is the target nucleus recoil energy.
For this reason, it seems expedient to consider these two processes, which proved to be interrelated, within a single approach.
In addition, the above-mentioned analysis of coherence–incoherence balance in these processes at neutrino energies of 30 to 50 MeV suggests that the role of inelastic (incoherent) channels in the interaction of massive neutral particles with nuclei could well be more significant, especially in view of new physics effects.
The long absence of positive results[
Except the DAMA/LIBRA results<cit.>, which are almost ignored without adequate refutation.]
from a wide range of dark matter search experiments gave rise to a lot of new,
sometimes exotic suggestions about both the possible composition of dark matter itself
<cit.>, its unusual properties <cit.>,  and about more sophisticated methods of its detection
Nevertheless, due to its exceptional significance
it seems somewhat untimely to bury in oblivion the traditional method of directly detecting massive weakly
interacting dark-matter particles without critical analysis of fundamental assumptions and details accepted in this method.
Thus, the first objective of this work is to generalize the above massless neutrino approach
to the case of interaction between massive neutral weakly
interacting $\chi$-particles and nuclei $\chi A\to \chi A^{(*)}$ in the nonrelativistic approximation.
The second objective is to investigate those regions of parameters (including kinematic ones)
where the inelastic (incoherent) process $\chi A\to \chi A^{*}$ can noticeably compete with the
elastic (coherent) process $\chi A\to \chi A^{}$, which is traditionally believed to be generally dominant.
The presence of regions where the inelastic channel plays a leading part, especially in view of the fact that the character of
the $\chi A$ interaction is usually beyond the Standard Model, may explain “blindness” of dark matter detectors
tuned to search $\chi A$ scattering events only in the elastic channel.

§ KINEMATICS AND CROSS SECTION OF ELASTIC AND INELASTIC ${\CHI A}$ SCATTERING

In the interaction of two particles with the formation of two particles ($2\to 2$ process):

\begin{equation} \label{eq:2DM-Kinematics-2-to-2-Definition}
\chi(k)+A(P_n)\rightarrow \chi(k' )+A^{(*)}(P'_m),
\end{equation}

the four-momenta of the incident and outgoing neutral massive leptons ($\chi$ particles) are denoted by
$k=(k_0=E_\chi,\bm{k})$ and $k'=(k'_0=E_\chi',\bm{k}')$,
and the four-momenta of the initial and final nuclear states are
denoted by $P_n=(P^0_n,\bm{P}_n)$ and $P'_m=(P^0_m, \bm{P}_m)$  (see Fig. <ref>, left).
The total energy of the nuclear state $|P_n\rangle$  is $P_n^0 = E_{\bm{P}}+\varepsilon_n$,
where $\varepsilon_n$  is the internal energy of the $n$-th nuclear quantum state.

“Diagram" of the $\chi A$ interaction due to, for example, the neutral $Z$ boson exchange (left).
Kinematics of this process in the lab frame where the nucleus $A$ is at rest (right).

The expression for the cross section of process (<ref>) has the form (see, for example,
[The squared matrix element $|\mathcal{M}_{mn}|^2$ usually does not depend on the $\phi$-angle.
Integration is performed over it.]:

\begin{equation}\label{eq:2DM-Kinematics-CrossSection-2-to-2-Definintion}
d\sigma =-\frac{\delta(k_0+P^0_n-k'_0-P^0_m)|{\cal M}|^2|\bm{k}'|^2 d|\bm{k}'| d\cos\theta}{32\pi k'_0 P^0_m \sqrt{(k P_n)^2-m_\chi^2 m_A^2}}.
\end{equation}

If the $\chi$ particle with mass $m_\chi$  and momentum $\bm{k}$ impinges on the nucleus $A$ at rest along the $z$ axis
and moves away with the momentum $\bm{k}'$ at an angle $\theta$ with the $x$ axis (the $y$ axis can always be chosen to be
perpendicular to the scattering plane), the four-momenta can be written as

\begin{eqnarray*}
k&=&\Big(k_0=\sqrt{m^2_\chi+|\bm{k}|^2}, 0, 0, k_z=|\bm{k}|\Big),   \qquad
P_n =  \Big(P_n^0=m_A +\varepsilon_n, 0, 0, 0\Big),   \\
k'&=&\Big(k'_0=\sqrt{m^2_\chi +  |\bm{k}'|^2}, k'_x= |\bm{k}'|\sin\theta, 0, k'_z =  |\bm{k}'|\cos\theta\Big),  \\
P'_m&=&\Big(P_m^0=\varepsilon_m + \sqrt{m_A^2 + (P_m^x)^2 + (P_m^z)^2}, P_m^x, 0, P_m^z\Big)= \\
&=& \Big(P_m^0=\varepsilon_m + \sqrt{m_A^2 + \bm{q}^2}, - |\bm{k}'|\sin\theta, 0,  |\bm{k}|-  |\bm{k}'|\cos\theta\Big),
\end{eqnarray*}

where $m_A$  is the mass of the nucleus $A$, and $\varepsilon_m$ is the
excitation energy of the $m$-th level (state) of this nucleus.
Here allowances are made for conservation of the momentum components along the $z$ axis:
$ k'_z + P_m^z = k_z$ (or $P_m^z = |\bm{k}|-  |\bm{k}'|\cos\theta$) and the $x$ axis:
$k'_x + P_m^x = 0$ (or $P_m^x= -  |\bm{k}'|\sin\theta$),
and for the fact that the four-momentum transferred to the nucleus $q =(q_0, \bm{q})$ is related to these quantities as follows:

\begin{eqnarray} \nonumber \label{eq:2DM-Kinematics-momentum-transfer}
q^2&\equiv &(k-k')^2 = 2\big(m^2_\chi- (kk')\big) =
2 \big(m^2_\chi -\sqrt{(m^2_\chi +  |\bm{k}'|^2)(m^2_\chi +  |\bm{k}|^2)}  + |\bm{k}| |\bm{k}'|\cos\theta\big),
\\ q_0 &=& k_0 -k'_0 = P^0_m - P^0_n = \Delta\varepsilon_{mn}+ T_A,
\\ \bm{q}^2 &=& (\bm{k}-\bm{k}')^2 = (-|\bm{k}'|\sin\theta)^2+(|\bm{k}|-  |\bm{k}'|\cos\theta)^2=
|\bm{k}|^2 + |\bm{k}'|^2 - 2 |\bm{k}| |\bm{k}'|\cos\theta.
\nonumber\end{eqnarray}

The kinetic energy of the recoil nucleus motion $T_A$ is defined as

\begin{equation} \label{eq:2DM-Kinematics-Recoil-kinetic-energy}

The energy conservation law from (<ref>), i.e., the equality $k_0 + P_n^0=k'_0 + P_m^0$, can be recast in the form

\begin{equation}\label{eq:2DM-Kinematics-Kinematics-EnergyConservation}
k_0+m_A-\Delta\varepsilon_{mn}=\sqrt{m^2_\chi +  |\bm{k}'|^2}+ \sqrt{m^2_A+|\bm{k}'|^2+|\bm{k}|^2  - 2 |\bm{k}| |\bm{k}'|\cos\theta}
, \end{equation}

where the difference of the energies of the nuclear states  $|m\rangle$ and $|n\rangle$  is denoted as follows:

\begin{equation}\label{eq:2DM-Kinematics-delta_Epsilon}
\Delta\varepsilon_{mn} \equiv \varepsilon_m-\varepsilon_n.
\end{equation}

Equation (<ref>) in the laboratory frame gives dependence of the momentum $|\bm{k}'|$ on the angle  $\theta$  between the vectors $\bm{k}$ and $\bm{k}'$.
Since in the laboratory frame the nucleus before the interaction is supposed to be at rest and in a certain quantum state $|n\rangle$ and considering the above kinematic relations, cross section (<ref>) can be written as

\begin{equation} \label{eq:2DM-Kinematics-CrossSection-DM-n-2-DM-m}
\frac{d^2\sigma_{mn}}{d|\bm{k}'| d\cos\theta}
= \frac{-|{i\cal M}_{mn}|^2 |\bm{k}'|^2 }{2^5\pi\sqrt{m^2_\chi +  |\bm{k}'|^2}}
\frac{\delta\Big(k_0-\sqrt{m^2_\chi +  |\bm{k}'|^2}-\Delta\varepsilon_{mn}-T_A(|\bm{k}'|,\cos\theta)\Big)}
{(m_A+\varepsilon_m+T_A(|\bm{k}'|)) \sqrt{k^2_0(m_A+\varepsilon_n)^2 -m_\chi^2 m_A^2}}.
\end{equation}

Here the following expressions for the square of the scalar product are used:

$$(k P_n)^2=(k_0\, P^0_n)^2=k^2_0(m_A+\varepsilon_n)^2, \quad k_0=(m^2_\chi +  |\bm{k}|)^{1/2},
\ \text{~and~} \ q_0= \Delta\varepsilon_{mn}+ T_A(|\bm{k}'|, \cos\theta).$$

From the energy conservation delta function of (<ref>),
it also follows that

\begin{equation}\label{eq:2chiA-Kinematics-T_A-vs-T_0-etc}
=\sqrt{m^2_\chi +  |\bm{k}|}-\sqrt{m^2_\chi +  |\bm{k}'|^2}-\Delta\varepsilon_{mn}.
\end{equation}

As is known <cit.>,
the velocity of massive dark-matter particles near the Earth is estimated at
about $10^{-3}$  of the speed of light, i.e.,
$ |\bm{v}|={|\bm{k}|}/{m_\chi} \simeq 10^{-3}c\simeq 300\div 400~$km/s.
Therefore, the kinetic energy of these particles when incident on the target,

\begin{equation}\label{eq:2chiA-Kinematics-T_0}
T_0\equiv \dfrac{|\bm{k}|^2}{2 m_\chi} =\dfrac{|\bm{k}|^2}{m^2_\chi}\dfrac{m_\chi}{2 }
\simeq 10^{-6}  \dfrac{m_\chi c^2}{2 },
\end{equation}

ranges from 1 keV to 10 MeV, because it is now believed that the mass of a nonrelativistic particle of
(cold) dark matter is between few GeV/$c^2$ and a few tens of TeV/$c^2$.
In addition, for the square of the momentum transferred to the nucleus $q^2$ there is an approximation

\begin{eqnarray*}
q^2&=& - 2m^2_\chi \Bigg(\sqrt{\Big(1 +  \frac{|\bm{k}'|^2}{m^2_\chi}\Big)\Big(1 + \frac{|\bm{k}|^2}{m^2_\chi}}\Big)   -1 - \frac{ |\bm{k}| |\bm{k}'|}{m^2_\chi}\cos\theta\Bigg)  \simeq - \bm{q}^2  ,\\ \bm{q}^2 &=& (\bm{k}-\bm{k}')^2 = |\bm{k}|^2 + |\bm{k}'|^2 - 2 |\bm{k}| |\bm{k}'|\cos\theta \simeq  2 m_A T_A.
\end{eqnarray*}

The latter approximation follows from (<ref>) if $\bm{q}^2\ll m^2_A$,
which holds (very well) in problems of direct detection of dark matter, where it is usually assumed that
$m_A\ge 10$ GeV$/c^2$, and nuclear recoil energies are
no higher than 100–150 keV, i.e., $T_A \le 1.5\times 10^{-4}$ GeV.
A typical excitation energy difference is in the region of a few hundreds of keV, i.e., being undoubtedly
$\Delta\varepsilon_{mn}\le 1\,$MeV.
Thus, the problem of searching for and detecting dark-matter particles using
$\chi A$ scattering involves the following typical scales of quantities:

\begin{eqnarray}\nonumber \label{eq:2DM-Kinematics-DDMD-scales}
\frac{|\bm{k}|}{m_\chi} \simeq 10^{-3}, &\quad& \Delta\varepsilon_{mn}\le 1 \text{~MeV}, \quad
10 \le m_A \le 100 \text{~GeV}/c^2, \quad 1<m_\chi< 10^4 \text{~GeV}/c^2
,\\ && T_A \le 150 \text{~keV}, \quad  |\bm{q}| \le 0.2   \text{~GeV}/c,  \ \quad  1 \text{~keV} \le T_0 \le 10 \text{~MeV}

Therefore, it is sufficient to use the nonrelativistic approximation here.
It also seems applicable to problems of detecting candidates for dark-matter particles produced at accelerators
<cit.> or accelerated in space
when at their rather high energies $T_0$ the momentum $|\bm{q}|$ transferred to the nucleus
is still insufficient for complete breakup of the nucleus.
The $\chi A\to \chi A^{(*)}$ scattering cross section (<ref>) in the nonrelativistic approximation, i.e., at

$$k_0=(m^2_\chi +  |\bm{k}|^2)^{1/2}\simeq m_\chi + \dfrac{|\bm{k}|^2}{2 m_\chi} = m_\chi + T_0,
\quad  \sqrt{m^2_\chi +  |\bm{k}'|^2}\simeq m_\chi + \dfrac{|\bm{k}'|^2}{2 m_\chi},\quad \text{and}$$

\begin{equation}\label{eq:2DM-Kinematics-Kinetic-energy-of-nucleus-via-k-k'-nonrel}
T_A(|\bm{k}'|, \cos\theta) \simeq  \dfrac{\bm{q}^2}{2 m_A} = \dfrac{|\bm{k}'|^2+|\bm{k}|^2  - 2 |\bm{k}| |\bm{k}'|\cos\theta}{2m_A}
, \end{equation}

takes the form

\begin{equation}\label{eq:2DM-Kinematics-CrossSection-DM-n-2-DM-m-nonrel}
\frac{d^2\sigma_{mn}}{d|\bm{k}'| d\cos\theta}\!
= \frac{-|{i\cal M}_{mn}|^2 |\bm{k}'|^2 }{2^5\pi m_A |\bm{k}|\sqrt{1 + \dfrac{m_\chi}{m_A }\dfrac{\varepsilon_n}{T_0} }} \frac{\delta\Big(T_0\!-\!\dfrac{|\bm{k}'|^2}{2 m_\chi}\!-\!\Delta\varepsilon_{mn}\!-\! \dfrac{|\bm{k}'|^2\!+\!|\bm{k}|^2\! -\!2 |\bm{k}| |\bm{k}'|\cos\theta}{2m_A}
\Big)}{\big(m_A+\varepsilon_m+T_A(|\bm{k}'|, \cos\theta)\big)
\Big(m_\chi + \dfrac{|\bm{k}'|^2}{2 m_\chi} \Big)}. \qquad
\end{equation}

Here a transition is made from the relativistic expression for the initial particle flux

\begin{eqnarray}\label{eq:2DM-Kinematics-Flux-rel}
\sqrt{w}\equiv \sqrt{k^2_0(m_A+\varepsilon_n)^2 -m_\chi^2 m_A^2}
= m_A |\bm{k}|  \Big[1+\dfrac{\varepsilon_n }{m_A}\big(2+\dfrac{\varepsilon_n}{ m_A}\big)
+\dfrac{m^2_\chi \varepsilon_n }{|\bm{k}|^2 m_A} \big(2 +\dfrac{\varepsilon_n}{m_A}\big) \Big]^{1/2}
\end{eqnarray}

to its nonrelativistic version in the form

\begin{equation}\label{eq:2DM-Kinematics-Flux-unrel}
\sqrt{w} =  \sqrt{(m_\chi +T_0)^2(m_A+\varepsilon_n)^2-m_\chi^2 m_A^2}\simeq
m_A |\bm{k}|\sqrt{1 + \frac{m_\chi}{m_A }\frac{\varepsilon_n}{T_0}} = m_A |\bm{k}|.
\end{equation}

The last equality in (<ref>)  is valid if the nuclear state $|n \rangle$
is the ground state with the minimal energy, i.e.,  $\varepsilon_n=0$.

On the one hand, the nuclear recoil kinetic energy $T_A$ from (<ref>)
enters into the relation

\begin{equation}\label{eq:2chiA-Kinematics-T_A-vs-T_0-etc-nonrel}
T_A(|\bm{k}'|,\cos\theta)=T_0 - \dfrac{|\bm{k}'|^2}{2 m_\chi} - \Delta\varepsilon_{mn}

On the other hand, according to (<ref>), it is a function of
two independent variables $|\bm{k}'|$ and $\cos\theta$, and, therefore, the energy-conserving delta function in
also simultaneously depends on these two variables.
The expression for the nonrelativistic cross section (<ref>),
like the initial formula (<ref>), can be integrated either over
$|\bm{k}'|$ or over $\cos\theta$ using this delta function
(connecting the independent $|\bm{k}'|$ and $\cos\theta$).
As a result, the delta function disappears, and there remains only one independent variable that determines the
differential cross section for the process of our interest.
It becomes more convenient to use the energy conservation delta function for integrating the differential cross section
(<ref>) over $\cos\theta$.
When $|\bm{k}'|$ is not yet connected to $\cos\theta$, the delta function from
<ref>) has the form

\begin{equation}\label{eq:2DM-Kinematics-Delta-function-4-cosT}
\delta\Big(T_0\!-\dfrac{|\bm{k}'|^2}{2 m_\chi}\!-\Delta\varepsilon_{mn}\!-\dfrac{|\bm{k}'|^2\!+|\bm{k}|^2\!  - 2 |\bm{k}| |\bm{k}'|\cos\theta}{2m_A} \Big)\equiv \delta(f(\cos\theta))=\frac{m_A\delta(\cos\theta-\cos\theta_i)}{|\bm{k}'| |\bm{k}|}

\begin{equation}\label{eq:2DM-Kinematics-CosT-vs-k-prime-from-Delta-function}
\text{where}\qquad
\cos\theta_i =-\dfrac{2m_A\Big(T_0- \dfrac{|\bm{k}'|^2}{2 m_\chi}-\Delta\varepsilon_{mn}\Big)
-|\bm{k}'|^2-|\bm{k}|^2}{2 |\bm{k}| |\bm{k}'|}\qquad
\end{equation}

is the solution of the nonrelativistic equation for energy conservation $ f(\cos\theta_i)=0$.
“Inverse” of formula (<ref>) has the form

\begin{eqnarray*}
|\bm{k}'| =|\bm{k}| \frac{r \cos\theta_i + \sqrt{1-r^2 \sin^2\theta_i-\alpha (1+r) }}{1+r}
, \quad \text{where}\quad
r=\dfrac{m_\chi}{m_A}, \quad \alpha=\dfrac{\Delta\varepsilon_{mn}}{T_0}

Integration of the cross section (<ref>)
over $\cos\theta$  using (<ref>) leads to the expression

\begin{eqnarray} \label{eq:2DM-Kinematics-dSigma-po-dk-prime-nonrel}
\frac{d\sigma_{mn}}{d|\bm{k}'|}  &=&
\frac{1}{2^5\pi |\bm{k}|^2\sqrt{1 + \dfrac{m_\chi}{m_A }\dfrac{\varepsilon_n}{T_0} }}
\frac{-|{i\cal M}_{mn}|^2 |\bm{k}'| }{(m_A+\varepsilon_m+T_A(|\bm{k}'|, \cos\theta_i))}
\frac{1}{\Big(m_\chi + \dfrac{|\bm{k}'|^2}{2 m_\chi} \Big) }
, \end{eqnarray}

where dependence of $\cos\theta_i$ on  $|\bm{k}'|$ is given by formula
The next step in the transformation of the formula for the cross section is a transition to differentiation
with respect to the observable $\dfrac{d \sigma_{mn}}{d T_A} = \dfrac{d \sigma_{mn}}{d|\bm{k}'|} \dfrac{d |\bm{k}'|}{d T_A} $; i.e.,
the transition Jacobian $\dfrac{d |\bm{k}'|}{d T_A} $ should be found.
This can be done in two ways.
One is to explicitly (and tediously) calculate the kinetic energy derivative from formula (<ref>)

\begin{equation}\label{eq:2DM-Kinematics-Diff-Kinetic-energy-of-nucleus-via-k-k'-nonrel}
\dfrac{d T_A(|\bm{k}'|, \cos\theta(|\bm{k}'|))}{d |\bm{k}'|}
= \dfrac{1}{m_A} \Big\{|\bm{k}'| - |\bm{k}|\cos\theta - |\bm{k}| |\bm{k}'| \dfrac{d \cos\theta}{d |\bm{k}'| } \Big\}
\end{equation}

by substituting into (<ref>)
expression (<ref>)  for $\cos\theta_i$ as a function of $|\bm{k}'|$.
The other is to simply differentiate the energy conservation law (<ref>)
written as

\begin{equation}\label{eq:2DM-Kinematics-Chi-prime-energy-nonrel}
\dfrac{|\bm{k}'|^2}{2 m_\chi}=T_0-\Delta\varepsilon_{mn}-T_A.
\end{equation}

In both cases, there arises a simple formula

\begin{equation} \label{eq:2DM-Kinematics-Diff-Kinetic-energy-of-nucleus-via-k-k'-nonrel-final}
\dfrac{d T_A(|\bm{k}'|)}{d |\bm{k}'|} = -\dfrac{|\bm{k}'|}{m_\chi}.
\end{equation}

As a result, in view of
(<ref>), (<ref>) and
the $\chi A_n\to \chi A_m $ cross section (<ref>)
(in the nonrelativistic approximation) takes the form

\begin{eqnarray}                                 \label{eq:2DM-Kinematics-dSigma-po-d-T-A-nonrel}
\dfrac{d \sigma_{mn}}{d T_A} \big(\chi A_n\to \chi A_m\big)
&=&\dfrac{|{i\cal M}_{mn}|^2  }{2^6\pi T_0 m_\chi m_A } C^{\rm nonrel}_{mn}(T_A) ,
\end{eqnarray}

which involves the introduced (kinematic) coefficient of about a unity

\begin{eqnarray}\label{eq:2DM-Kinematics-dSigma-po-d-T-A-nonrel-Coeff}
C^{\rm nonrel}_{mn}(T_A)&=& \dfrac{1}{\sqrt{1 + \dfrac{m_\chi}{T_0}\dfrac{\varepsilon_n}{m_A } }}
\frac{1}{1+\dfrac{\varepsilon_m+T_A}{m_A}}
\frac{1}{1  + \dfrac{T_0-\Delta\varepsilon_{mn}-T_A}{m_\chi} }
\equiv C^{}_{2,nm}
\simeq O(1). \qquad
\end{eqnarray}

Considering relations  (<ref>),
it can be shown that this coefficient is really little different from unity and almost independent of either
$T_A$, or subscripts $n$  $m$.
In the experimental situation (or in calculations of the expected cross section) the initial external quantity
is recoil energy $T_A$.
Therefore, specifying a certain value of $T_A$, initial $\chi$-particle energy $T_0$, and nuclear
characteristic  $\Delta\varepsilon_{mn}$, one finds from the energy
conservation law (<ref>)
the kinetic energy of the outgoing  $\chi$ particle  (<ref>) or
$|\bm{k}'(T_A)|^2 = 2 m_\chi (T_0-\Delta\varepsilon_{mn}-T_A)$.
With this known value of $|\bm{k}'|^2$ and “its generating” value of $T_A$,
one derives from the definition of the nonrelativistic nuclear recoil energy,
formula (<ref>),
an expression for the outgoing angle of the $\chi$-particle in the lab frame
as a function of $T_A$, $\Delta\varepsilon_{mn}$ , and $T_0$ as follows:

\begin{eqnarray} \label{eq:2chiA-Kinematics-CosT-from-k2prime-and-T_A}
\cos\theta(T_A) &=&  \dfrac{|\bm{k}'|^2+|\bm{k}|^2-2 m_A T_A}{2 |\bm{k}|  |\bm{k}'|}  = \dfrac{m_\chi( 2 T_0- \Delta\varepsilon_{mn})  - T_A (m_\chi +m_A)}{2 m_\chi\sqrt{ T_0 (T_0-\Delta\varepsilon_{mn}-T_A)}}

§ THE AMPLITUDE OF ${\CHI}$ PARTICLE-NUCLEUS SCATTERING

The formalism of obtaining the amplitude of the scattering of a massive weakly interacting neutral ${\chi}$ particle from a nucleus
(as a compound complex system) is a generalization of the approach
first proposed in <cit.> for the description of the neutrino-nucleus scattering.
To construct the scattering amplitude (Fig <ref>, left),
we will describe the initial and the final nucleus (as a bound compound system) by the wave function $|P_{l}\rangle$.
It corresponds to a nucleus with four-momentum $P_{l}$, in a certain $l$-th internal quantum state  ($l=n,m$)
and is a superposition of free nucleons $|\{p\}\rangle$, “weighted” with
the general wave function of the bound state $\tilde{\psi}'_n(\{p\})$.
The latter is a product of the wave function $\widetilde{\psi}_n(\{p{^\star}\})$,
describing the internal structure of the nucleus in its rest frame (the corresponding momenta are marked by
the superscript $\star$)  and the wave function $\Phi_n(p)$
responsible for the motion of a nucleus as a whole with
$\bm{p}=\sum^A_{i=1}\bm{p}_i$ and the nuclear spin projection  $s$

\begin{equation}		\label{eq:3chiA-ScatteringAmplitude-1}
\widetilde{\psi}'_n(\{p\}) = \widetilde{\psi}_n(\{p{^\star}\})\Phi_n(p),
\quad \text{where} \quad p=(\bm{p},s).
\end{equation}

The “internal” wave function depends on $A-1$ three-momenta since the sum of all three-momenta
is equal to the total momentum of a nucleus as an entire object.
Thus, for the state $|P_n\rangle$  we will use the (antinsymmetrized) expression

\begin{equation}\label{eq:3chiA-ScatteringAmplitude-nucleus_state}
|P_n\rangle=\int\left(\prod^{A}_{i}d\widetilde{\bm{p}}^\star_i\right)\frac{\widetilde{\psi}_n(\{p^\star\})}{\sqrt{{A!}}} \Phi_n(p)|\{p^\star\}\rangle, \quad\text{where}\quad  d\bm{\widetilde{p}}^\star_i \equiv \frac{d\bm{p}^\star_i}{(2\pi)^3 \sqrt{2E_{\bm{p}^\star_i}}}
, \end{equation}

and the function

\begin{equation}\label{eq:3chiA-ScatteringAmplitude-Outer-Nuclear-state}
\Phi_n(p)=(2\pi)^3\sqrt{2P_n^0}\delta^3(\bm{p}-\bm{P})
\end{equation}

corresponds to a nucleus with a certain three-momentum $\bm{P}$  and energy
$P^0_n=E_{\bm{p}}+\varepsilon_n$, involving the nuclear
excitation energy $\varepsilon_n$.
The symbol $\{p^\star\}$ denotes the notation  $\{p^\star\}\equiv (p^\star_1\dots p^\star_n)$,
where $p^\star_i$  is the four-momentum of the $i$-th nucleon in the center-of-mass
system of the nucleus (at rest).
For nuclear states $|n\rangle$ describing a nucleus at rest in the $n$-th internal quantum state (nth level),
the conventional normalization condition is

\begin{equation} \label{eq:3chiA-ScatteringAmplitude-nucleus_state_norm}
\langle m|n\rangle \equiv\int\left(\prod^{A}_{i}\frac{d\bm{p}^\star_i}{(2\pi)^3}\right)\widetilde{\psi}_n(\{p^\star\})\widetilde{\psi}^*_m(\{p^\star\})(2\pi)^3 \delta^3(\sum^A_{i=1} \bm{p}^\star_i)
\end{equation}

For the nuclear wave functions  $|P_n\rangle$ from (<ref>), it
gives a simple normalization condition

\begin{equation}\label{eq:3chiA-ScatteringAmplitude-nucleus_state_norm_1}
\langle P'_m|P_n\rangle = (2\pi)^3 2P^0_n \delta^3(\bm{P}-\bm{P}')\delta_{nm}.
\end{equation}

The state $|n\rangle$ satisfying (<ref>)
can be formally defined as

\begin{equation}\label{eq:3chiA-ScatteringAmplitude-n_state}
|n\rangle = \int \left(\prod_{i=1}^{A} d\widetilde{\bm{p}}_i^\star\right) \frac{\widetilde\psi_n(\{p{^\star}\}) }{\sqrt{A!}} \Bigg[(2\pi)^3\delta^3\left(\sum_{i=1}^A\bm{p}_i^\star\right)\Bigg]^{1/2}|\{p^\star\}\rangle.
\end{equation}

Based on the wave function (<ref>), one can calculate the amplitude (probability)
of the scattering of a massive neutral $\chi$ particle from a nucleus as a system of mutually interacting nucleons,
assuming that the external interaction is between the (pointlike) $\chi$ particle and the structureless nucleons.
It is an acceptable hypothesis, since the $\chi$ particle with an energy below 50–100 MeV is usually incapable of
“penetrating into a nucleon and seeing quarks.”
Therefore, the effective four-fermion interaction Lagrangian can be used with a sufficient accuracy
In the problem of direct detection of dark-matter particles,
the effective Lagrangian involving only the axial-vector (or spin-dependent) and scalar (spin-independent) interactions
between the $\chi$ particle and nucleons is normally used (see, for example,
It can be written as a product of two currents

\begin{equation}		\label{eq:3chiA-ScatteringAmplitude-CohDM-Effective-Axial-and-Scalar-chi-N-Lagrangian}
\mathcal{L}(x)=\dfrac{G_F}{\sqrt 2} L_\mu(x)H^\mu(x).
\end{equation}

Here the current operators of the $\chi$ particle and the nucleons of the nucleus are given
in terms of normal products of quantum-field operators $\psi_\chi(x)$ and $\psi_{n,p}(x)$

\begin{eqnarray}			\label{eq:3chiA-ScatteringAmplitude-Lepton-S+A-Operator}
L_\mu(x)&=&\chi_A \normord{ \overline{\psi}_\chi(x)\gamma_\mu\gamma_5\psi_\chi(x)}
+\chi_S \normord{ \overline{\psi}_\chi(x)\psi_\chi(x)},  \\
\label{eq:3chiA-ScatteringAmplitude-Nucleon-S+A-Operator}
H^\mu(x)&=&\sum_{f=n,p}\big[ h_A^f \normord{ \overline{\psi}_f(x)\gamma^\mu\gamma_5\psi_f(x)}
+ h^f_S \normord{ \overline{\psi}_f(x) \psi_f(x)} \big],
\end{eqnarray}

The effective coupling constants
$$c_{A,S}^f \equiv \chi^{}_{A,S} h^f_{A,S}$$

specify the intensity of the interaction between $\chi$ particles and nucleons (relative to the Fermi constant $G_F$).
In Section <ref>,
the effective Lagrangian (<ref>)
is generalized to the case of other admissible Lorentz-invariant structures.
With the Lagrangian of the form (<ref>), the $\mathbb{S}$-matrix $\langle P'_m,k'|\mathbb{S}|P_n,k\rangle$
describing the probability for the transition of the nucleus and the $\chi$ particle
from the initial state $|P_n,k\rangle$  to the final state $\langle P'_m,k'|$
due to their interaction is written in the first order with respect to the Fermi constant $G_F$ in a standard form

\begin{equation}		\label{eq:3chiA-ScatteringAmplitude-CohDM-A+S-MatrixElevent}
\langle P'_m,k'|\mathbb{S}|P_n,k\rangle= (2\pi)^4\delta^4(q+P_n-P'_m)i\mathcal{M}_{mn}
=\dfrac{i G_{\rm F}}{\sqrt{2}} \int d^4 x \, H^\mu_{nm}(x) \,  L_\mu^\chi(x)
, \end{equation}

where $H^\mu_{mn}(x)\equiv \langle P'_m| H^\mu(x)|P_n\rangle$  is the amplitude of the
probability (matrix element) of the transition of the nucleus from the state $|P_n\rangle$ to the state
$\langle P'_m|$ due to the hadronic current of the form $H^\mu(x)$.
Substituting operator expressions for the currents
$H^\mu_{nm}(x)$ and $L_\mu^\chi(x)$ into the right-hand side of
considering the explicit form of the nuclear functions  (<ref>), performing the appropriate calculations, taking the integral over $d x^4$ responsible for energy
conservation at the $\chi$-particle-nucleon level, and
introducing the designation for the leptonic current (or $\chi$ particle current)

\begin{equation}\label{eq:3chiA-ScatteringAmplitude-CohDM-LeptonicCurrent}
l_\mu(k',k,s',s) \equiv \overline{u}_\chi(\bm{k}',s') O_{\mu} u_\chi (\bm{k},s)

we arrive at the following expression for the matrix element from

\begin{eqnarray}  \label{eq:3chiA-ScatteringAmplitude-M-Element-via-H-and-L}
i \mathcal{M}_{mn}=\dfrac{i G_{\rm F}}{\sqrt{2}}  \sqrt{4P^{0'}_m P^0_n}\, \,  l_\mu(k',k,s',s)\, h^\mu_{mn} (\bm{q})

The hadronic current $h^\mu_{mn}(\bm{q})=\langle m|H^\mu(0)|n\rangle$
in (<ref>)
defined in terms of nuclear state functions of the nucleus at rest has the form
[Derivation of formula (<ref>)
does not depend on the explicit form of combinations of $\gamma$-matrices $O^\mu_k$
(see <cit.>).

\begin{eqnarray}\nonumber \label{eq:3chiA-ScatteringAmplitude-h-mu-mn-with-delta-p_k}
h^\mu_{mn}(\bm{q})&=& \sum^A_{k}
\frac{ \overline{u}(\bm{\bar{p}}^\star_k+\bm{q},r'_{k})\, O^\mu_k\,  u(\bm{\bar{p}^{\star}}_k,r_{k})}
\times\\&&\times
\int \prod^{A}_{i=1}\frac{d\bm{p}^\star_i   \delta\big(f(\bm{p^{\star}_k})\big)}{(2\pi)^3}
\widetilde{\psi}^{*}_m(\{p^{(k)}_\star\},\bm{p^{\star}}_k\!+\!\bm{q} )\widetilde{\psi}_n(\{p^\star\})
(2\pi)^{3} \delta^3(\sum^A_{i=1}\bm{p}^\star_i)
, \end{eqnarray}

where  $\bm{\bar{p}^{\star}_k}(\bm{q})$ is the solution of the equation $f(\bm{\bar{p}^{\star}_k}) = 0$,
which depends on $\bm{q}$ and is determined by the condition

\begin{equation}\label{eq:3chiA-ScatteringAmplitude-Energy-and-Identity-Conservation}
\delta\big(f(\bm{p^{\star}_k})\big) \equiv \delta( - T_A - \Delta\varepsilon_{mn}+\sqrt{m^2+ {\bm{p^{\star}_k}}^2}-\sqrt{m^2+ (\bm{p^{\star}}_k+\bm{q})^2})\ne 0.
\end{equation}

It arises from the necessity that the energy conservation
law be simultaneously obeyed at the levels of the
nucleus and a single nucleon. Technically, it is because
the $q_0$-dependent delta function responsible for energy
conservation at the level of the entire nucleus

$$\delta(q_0+P_{0,n}-P'_{0,m}) = \delta(q_0 - T_A - \Delta\varepsilon_{mn})
=\delta\big(q_0 - \frac{\bm{q}^2}{2m_A} - \Delta\varepsilon_{mn}\big),

which is on the left in (<ref>), should be “balanced” by
the delta function of energy conservation at the level of a single (active) nucleon
[Hereinafter, unless stated otherwise, the proton mass is taken to
be equal to the neutron mass $m=m_p=m_n$.]

$$\delta(q_0+p^{\star}_{0,k}-p^{'\star}_{0,k})= \delta(q_0+ \sqrt{m^2+  {\bm{p^{\star}_k}}^2} -  \sqrt{m^2+ (\bm{p^{\star}}_k+\bm{q})^2}),

which naturally arises on the right in (<ref>)
after integration over $d x^4$ and depends on the same variable $q_0$.
Therefore, relation (<ref>)
is the condition for energy conservation and preservation of integrity of the
nucleus  met simultaneously

\begin{equation}\label{eq:3chiA-ScatteringAmplitude-Energy-and-Identity-Conservation-1}
\displaystyle\sqrt{m^2+(\bar{\bm{p}}+\bm{q})^2}-\sqrt{m^2+\bar{\bm{p}}^2}
=\frac{\bm{q}^2}{2m_A}+\Delta\varepsilon_{mn} = T_A+\Delta\varepsilon_{mn},
\end{equation}

that “chooses” the active nucleon momentum $\bar{\bm{p}}= ( p_L,  p_T)$
dependent on $\bm{q}$ in the form <cit.>:

\begin{equation}\label{eq:3chiA-ScatteringAmplitude-Energy-and-Identity-Conservation-Solution}
p_L = -\frac{|\bm{q}|}{2}\Bigg[1-\sqrt{\beta}\sqrt{1+\frac{4m_{T}^2}{\bm{q}^2(1-\beta)}}\Bigg], \quad \text{where} \quad \beta=\frac{(T_A+\Delta\varepsilon_{mn})^2}{\bm{q}^2}, \quad  m_{T}^2 = m^2+p_T^2
.  \end{equation}

According to the condition for preservation of integrity of the nucleus
in the rest frame of the
nucleus the active nucleon “meets” the $\chi$ particle with the nonzero momentum $p_l = (m,0,0,p_L)$  from
incident on the nucleus (along the $z$ axis), and therefore
the $s$ invariant in the rest frame of the nucleus (lab frame) should be recalculated

\begin{eqnarray}\label{eq:42chiA-CrossSection-ScalarProducts-ChiEta-All-s-with-T_A-recoil}
s&=& (k_l+p_l)^2
m^2_\chi +m^2 + 2 m m_\chi\Bigg\{ \sqrt{1 + \dfrac{2 T_0}{m_\chi }}
\sqrt{1+ \dfrac{p_L^2}{m^2} } - \dfrac{|\bm{k}^l_\chi |}{m_\chi} \dfrac{p_L}{m}\Bigg\}.
\end{eqnarray}

In other words, the invariant energy $s$, in terms of which scalar products are calculated in the lepton–
nucleon center-of-mass system, depend not only on the kinetic energy of the incident $\chi$ particle
$T_0$ but also on the kinetic energy of the nuclear recoil $T_A$,
since $\bm{q}^2\simeq 2 m_A T_A$.
Note that this correction is relatively small, because $\dfrac{p_L}{m} \le  0.1$
<cit.> and is ignored in what follows.
Arguments of the wave functions of the bound nuclear states $\widetilde{\psi}_m^*(\{p^{(k)}_\star\} )$
and $\widetilde{\psi}_n(\{p_\star\})$ in (<ref>)  are
expressions defined as $\{p_\star\}=(p_1^\star\dots p_A^\star)$, where the $i$-th element is a pair
$p_i^\star \equiv (\bm{p}_i^\star,r_i)$ of the three-momentum (in the rest frame of the nucleus) and the spin of
the $i$-th nucleon.
The expression $\{p^{(k)}_\star\}$  coincides with $\{p_\star\}$ except for the $k$-th element which is
where $\bm{q}$  is the $\chi$ particle momentum transferred (to the nucleus).
The hadronic current $h^\mu_{mn}(\bm{q})$ defined by formula
(<ref>) is a sum of terms like
$\bar{u}(\bm{p}_k+\bm{q},s_k) O^\mu_k u(\bm{p}_k,r_k)$ corresponding
to scattering of the $\chi$ particle by the single $k$-th nucleon, which had the three-momentum $\bm{p}_k$
and the spin projection on a certain preferred direction $r_k$ before the interaction.
The amplitude of the probability to find this nucleon in the initial nuclear state $|P_n\rangle$
with exactly these quantum numbers is governed by
the wave function $\tilde{\psi}_n(\{p_\star\})$ given in the momentum
representation and depending on the nucleon momenta in the rest frame of the nucleus.
A scattered
(“interaction-stricken,” active) nucleon has a three-momentum $ \bm{p}_k+\bm{q}$
and, generally, an arbitrary spin projection on the preferred direction $s_k$.
The corresponding amplitude of the probability to find the nucleon in the final nucleus $|P_m\rangle$ with exactly these quantum numbers is given by the expression $\tilde{\psi}^*_m(\{p^{(k)}_\star\})$.
All three-momenta of the other spectator nucleons (not participating in the interaction with the
$\chi$ particle) remain unchanged and drop from consideration due to the normalization rules of nuclear wave functions.
To transform the hadronic current  (<ref>),
we assume, as in <cit.>,
that the wave function $\widetilde{\psi}_n$  can be written as a product of the momentum
$\widetilde{\psi}_n$  and spin $\chi_n$ components

\begin{equation}\label{eq:3chiA-ScatteringAmplitude-CohDM-factorize_spin}
\widetilde{\psi}_n(\{p^\star\}) =\widetilde{\psi}_n(\{\bm{p}_\star\})\chi_n(\{r\}),
\end{equation}

the former being dependent on the momentum variables  $\{\bm{p}_\star\}=(\bm{p}_1^\star\dots{} \bm{p}_A^\star)$, and the latter on the spin variables $\{r\}=(r_1\dots r_A)$.
It is evident from (<ref>) that it involves products of spin functions

\begin{equation}				\label{eq:3chiA-ScatteringAmplitude-spin-products}
\lambda^{mn}(r',r) \equiv \lambda^{mn}_{r'r} \equiv \chi^*_m(\{r^{(k)}\})\chi_n(\{r\})
, \end{equation}

where $\{r^{(k)}\}$ coincides with $\{r\}$, except for the $k$-th element, which is equal to $r'_k$.
Obviously, after the interaction of the active ($k$-th) nucleon with the $\chi$ particle its spin state either
does not change (index $r'_k=r_k$) or changes ($r'_k \ne r_k$).
In the former case, according to the normalization condition
(<ref>), the product of spin functions
(<ref>) takes the form  $\chi_m^{*}(\{r\})\chi_n(\{r\}) = \delta_{nm}$,
which means that the nucleus did not change, i.e., after the interaction it remained in its initial state  ($m=n$).
In the latter case, the product of spin functions (<ref>)
will be different from zero only if the index $r'_k$, not equal to exactly $r_k$ corresponds to the definition
of the spin state of $|m\rangle $ the nucleus  $A$
[To be in correspondence to <cit.>,
the product $\chi^*_m(\{r^{(k)}\})\chi_n(\{r\})$, like $\chi_n^{*}(\{r\})\chi_n(\{r\})$, will be considered equal to
unity. Validity of this assumption is a subject for a separate study.].
In other words, in the cases of our interest the product of the spin wave functions
from (<ref>)
may be thought of as being almost independent of the indices $m,n$:

\begin{equation} \label{eq:3chiA-ScatteringAmplitude-CohDM-spin_functions_norm2}
\lambda^{mn}(r',r)= \delta_{mn}\delta_{r'r}+(1-\delta_{mn})(1-\delta_{r'r})
. \end{equation}

Considering these assumptions and $\bar{\bm{p}}= f(\bm{q})$, from
expression  (<ref>) is recast as

\begin{equation}	\label{eq:3chiA-ScatteringAmplitude-h-mu-mn-with-delta-p_k-spin-factorization}
\begin{aligned}
h^\mu_{mn}(\bm{q})& =  \sum_{k=1}^{A}  \frac{\bar{u}(\bar{\bm{p}}+\bm{q},r'_k) O^\mu_k u(\bar{\bm{p}},r_k)}{\sqrt{4E_{\bar{\bm{p}}}E_{\bar{\bm{p}}+\bm{q}}}}
\lambda^{mn}(r',r)  \times
\\ & ~~ \times\int \Bigg[\prod^{A}_{i=1}\frac{d\bm{p}^\star_i  }{(2\pi)^3}\Bigg] \delta\big(f(\bm{p^{\star}_k})\big)
\widetilde{\psi}_m^*(\{\bm{p}^{(k)}_\star\},\bm{p^{\star}}_k\!+\!\bm{q} ) \widetilde{\psi}_n(\{\bm{p}_\star\})
(2\pi)^3 \delta^3\big(\sum_{l=1}^A \bm{p}^\star_l\big)
. \end{aligned}
\end{equation}

The multidimensional integral in this formula can be written as a matrix element of the operator
$\hat{\bm{X}}_k$ that implements the three-dimensional shift of the
$k$-th nucleon

\begin{equation}\label{eq:3chiA-ScatteringAmplitude-f^k_mn-definition}
f^k_{mn}(\bm{q})\equiv\langle m|e^{i\bm{q}\hat{\bm{X}}_{k}}|n\rangle  =
\int \Bigg[\prod^{A}_{i=1}\frac{d\bm{p}^\star_i  }{(2\pi)^3}\Bigg] \delta\big(f(\bm{p^{\star}_k})\big)
\widetilde{\psi}_m^*(\{\bm{p}^{(k)}_\star\}) \widetilde{\psi}_n(\{\bm{p}_\star\})
(2\pi)^3 \delta^3\big(\sum_{l=1}^A \bm{p}^\star_l\big), \qquad
\end{equation}

where the delta function $\delta\big(f(\bm{p^{\star}_k})\big)$ ensures preservation of the integrity of the nucleus after the shift of the active $k$-th-nucleon momentum by the operator  $\hat{\bm{X}}_k$
due to an external action.
As a result, in view of (<ref>) and
one obtains the following expression for the matrix element
defining the probability of the $\chi_s A_n \to \chi_{s'} A_m$ process:

\begin{equation}			\label{eq:3chiA-ScatteringAmplitude-CohDM-matrix_element}
i\mathcal{M}^{s' s,r'r }_{mn}(\bm{q}) = i\frac{G_F}{\sqrt{2}} \frac{m_A}{m}C_{1,mn}^{1/2} \sum_{k=1}^{A}
f^k_{mn}(\bm{q})  \lambda^{mn}(r',r) (l_{s's},h^k_{r'r}),  \quad\text{where}
\end{equation}

\begin{equation}\label{eq:3chiA-ScatteringAmplitude-CohDM-ScalarProduct-lh}
(l_{s's}, h^k_{r'r})\equiv l_\mu(k',k,s',s)\  \bar{u}(\bar{\bm{p}}+\bm{q},r'_k) O^\mu_k u(\bar{\bm{p}},r_k)
\end{equation}

is the scalar product of the lepton and $k$-th-nucleon currents, which involves all the specificity of the interaction
between them.
Formula (<ref>) takes into account
that the general kinematic factor, which arises as we
progress, can be rewritten by “separating from it” the
leading factor  $m_A/m$, where $m$ and $m_A$  are the nucleon
and nuclear masses, respectively, and introducing a correction  $C_{1,mn}$
that is close to unity with a good accuracy,

\begin{equation}	\label{eq:3chiA-ScatteringAmplitude-CohDM-C1_def}
C_{1,mn}^{1/2} =  \sqrt{\frac{P^0_n}{m_A}\frac{P^{'0}_m}{m_A}\frac{m}{E_{\bar{\bm{p}}}}\frac{m}{E_{\bar{\bm{p}}+\bm{q}}}}   \sim O(1).
\end{equation}

Thus, the matrix element, or the probability amplitude,
for the process  $\chi A_n\to \chi A^{(*)}_m$ (<ref>)
is a sum of individual lepton-nucleon probability amplitudes
proportional to the corresponding scalar products of
currents $(l_{s's}, h^k_{r'r})$ weighted with two factors, each of which is little different from unity.
$\lambda^ {mn}(r',r)$  (see. (<ref>))
is almost independent of the nuclear states $|n\rangle$ and $|m\rangle$,
but it “controls” (as will be shown below) the initial $r$ and final $r'$ values of the (doubled) projection
of the spin of the active nucleon of a nucleus on a preselected axis.
The other factor $f^k_{mn}(\bm{q})$ defines the nucleon form factor of the nuclear transition from the
$|n\rangle$-state to the $|m\rangle$-state.
The expression $\langle m|e^{i\bm{q}\hat{\bm{X}}_{k}}|n\rangle$ from
(<ref>) is a generalization of the factor  $e^{i\bm{q} \bm{x}_k}$
used by Freedman <cit.>.
The fundamental difference between $\langle m|e^{i \bm{q}\hat{\bm{X}}_k} |n\rangle$ and $e^{i\bm{q}\bm{x}_k}$
is that when $e^{i\bm{q}\bm{x}_k}$ “works,” the $k$-th nucleon is explicitly considered to have a quite definite position in the nucleus specified by the coordinate $\bm{x}_k$ (“nailed” at the point $\bm{x}_k$).
However, scattering by one free nucleon cannot lead to the nucleus moving as a single entity,
since nucleons should be bound to one another, otherwise the nucleus will break up.
The wave functions of the bound nuclear state used here do not rely on the assumption that spatial positions of nucleons in a nucleus are fixed, and therefore the matrix element
$\langle m|e^{i \bm{q}\hat{\bm{X}}_k} |n\rangle$
does not depend on the number $k$
Moreover, $f^k_{mn}(\bm{q})=\langle m|e^{i\bm{q}\hat{\bm{X}}_{k}}|n\rangle$ is the key quantity for
establishing interrelation of the coherent and incoherent
regimes in the $\chi$-nucleus scattering (Section <ref>).

§ CROSS SECTIONS OF ${\CHI}$ PARTICLE-NUCLEUS SCATTERING

§.§ Coherent and Incoherent contributions
to the  ${\chi A\to \chi A^{(*)}}$ scattering cross section

The  observed differential cross section for the process $\chi A\to \chi A^{(*)}$ can be obtained by averaging the
differential cross section that determines the nuclear transition from the state $|n\rangle$
to the state $|m\rangle$ (formulas (<ref>) and
(<ref>)) over all potentially possible initial (internal)
states $|n\rangle$ of the target nucleus and summing over all
allowed final states $|m\rangle$ of the recoil nucleus

\begin{equation}\label{eq:41chiA-CrossSection-Coh-vs-InCoh-CrossSection-definition}
\frac{d\sigma}{dT_A}(\chi A \to \chi A^{(*)}) = \sum_{n,m}\omega_n \frac{d\sigma_{mn}}{dT_A}(\chi A \to \chi A^{(*)}), \quad \text{where}\quad  \sum_n \omega_n=1
\end{equation}

is the total sum of probabilities of all possible initial states of the nucleus $A$.
The matrix element of the $\chi A \to \chi A^{(*)}$ process in (<ref>)
was given by formula (<ref>).
With summation taken in it over “internal” spin indices of the (active) nucleon and with “external” spin
indices of the $\chi$ particle retained, formula (<ref>)  takes the form

\begin{eqnarray}\label{eq:41chiA-CrossSection-MatrixElement-with-s-sprime}
i\mathcal{M}_{mn}^{s's}(\bm{q})  = i\frac{G_F}{\sqrt{2}} \frac{m_A}{m}C_{1,mn}^{1/2} \sum_{k=1}^{A} \sum_{r',r} f^k_{mn}(\bm{q}) (l_{s's},h^k_{r'r}) \lambda^{mn}(r',r)
. \end{eqnarray}

Recall that a similar matrix element for the (anti)neutrino–nucleus scattering was considered
at fixed values of lepton helicity (spin projection on momentum direction).
Since the neutrino helicity is always negative, $s'=s=-1$, and the antineutrino helicity
is always positive, $s'=s=+1$, the subscripts $s', s$ have always been specified for the (anti)neutrino
matrix element like  (<ref>) “from the outside.”
By following this logic, the form   (<ref>)
was determined for the matrix element.
It corresponds to the $\chi A$ interaction when the $\chi$ particle has fixed values of its initial $s$
and final $s'$ spin projection on a certain preferred direction.
Summation over the active nucleon spin projections $\sum_{r'r}$  is made “inside” the matrix element (at the
level of nucleon amplitudes) rather than at the level of the square of the matrix element, as is usually done for
real (or potentially detectable) final-state particles.
This summation is justified by impossibility of determining the active nucleon spin before and after interaction.
For example, the matrix element $i\mathcal{M}_{mn}^{-+}$ describes a process in which the
$\chi$ particle has the spin projection $s=+1$ upon incidence and $s'=-1$ for the outgoing one.
Since we cannot know (even in general) the spin projection of the active nucleon when it
“meets” and “sends away” the particle, we have to take into consideration all possible spin projections of
this active nucleon, i.e., to sum over $r=\pm$ and $r'=\pm$.
Substituting (<ref>)
into cross-section expression  (<ref>),
we obtain

\begin{eqnarray}\label{eq:41chiA-CrossSection-Coh-vs-InCoh-CrossSection-with-s-sprime-and-Tmns}
\frac{d\sigma_{s's}}{d T_A}(\chi A \to \chi A^{(*)})
&=& \dfrac{G^2_F  m_A}{2^7\pi m^2 T_0 m_\chi} \Big[T^{s's}_{m=n} + T^{s's}_{m\ne n}\Big],  \quad\text{where}
\end{eqnarray}

\begin{eqnarray} \label{eq:41chiA-CrossSection-Coh-vs-InCoh-CrossSection-Term-nn-definition-with-s-sprime}
T^{s's}_{m=n}\!\!&=&\!\!\sum^A_{k,j}\sum_{n}\omega_n\Big[ C^{}_{1,nn}C^{\rm nonrel}_{nn}\, f^k_{nn}f^{j*}_{nn}
\sum_r  (l_{s's},h^k_{rr})\sum_{x} (l_{s's},h^j_{xx})^{*}\Big],
\\
\label{eq:41chiA-CrossSection-Coh-vs-InCoh-CrossSection-Term-mn-definition-with-s-sprime}
T^{s's}_{m\ne n}\!\!&=&\!\!\sum^A_{k,j}\sum_{n}\omega_n  \Big[ \sum_{m\ne n} C^{}_{1, mn}C^{\rm nonrel}_{mn}\, f^k_{mn} f^{j*}_{mn}  \sum_{r'r}
\lambda^{mn}_{r'r}
(l_{s's}, h^k_{r'r})
\Big(\sum_{x'x}\lambda^{mn}_{x'x}(l_{s's}, h^j_{x'x}) \Big)^{\dag} \Big].\qquad
\end{eqnarray}

Here $T_{m=n}$  is the contribution to the cross section that corresponds to preservation of the initial nuclear
state while the spin projection of the interacting (active) nucleon does not change.
According to (<ref>), $\lambda^{nn}({r',r})=\delta_{r'r}$, and spin amplitudes drop from further consideration.
The crucial part in (<ref>)
is played by the scalar products of four-vectors of the leptonic and
nucleon currents $(l_{s's},h^{p/n}_{r'r})$ defined by formua (<ref>).
Introducing the notations

\begin{eqnarray}\label{eq:41chiA-CrossSection-Coh-vs-InCoh-g-factors}
g^{p/n}_\text{coh}= C^{p/n}_{1,nn}  C^{\rm nonrel}_{nn} \qquad\text{and}\qquad
g^{p/n}_\text{inc}= C^{p/n}_{1,mn} C^{\rm nonrel}_{mn}

and considering the form of formulas (<ref>) and
(<ref>),  we can assume with a good degree of confidence that
these quantities have constant values of about unity and are almost independent of the nuclear recoil
energy $T_A$  and nuclear indices $n, m$.
In view of this fact, summation over the index  $n$ in
gives rise to form factors averaged over all initial nuclear states, i.e.,

\begin{equation}\label{eq:41chiA-CrossSection-Coh-vs-InCoh-form-factors-with-s-sprime}
\sum_n \omega_n f_{nn}^k f_{nn}^{j*}= \left\{\begin{matrix}
|F_{p/n}(\bm{q})|^2,                   &\text{ when }(k,j)=(p,p) \text{~~or when~~} (k,j)=(n,n);\\
F_{p}(\bm{q})F_{n}^*(\bm{q}),   &\text{ when }(k,j) = (p,n);\\
F_{n}(\bm{q})F_{p}^*(\bm{q}),   &\text{ when }(k,j)= (n,p).\\
\end{matrix}\right.
\end{equation}

In other words, the left-hand side of this relation is equal to the square of the modulus of the nuclear proton
or neutron form factor  $|F_{p/n}(\bm{q})|^2$ if the summation indices $k$ and $j$ correspond to either a pair of protons or a pair of neutrons.
If these indices do not simultaneously correspond to a pair of identical nucleons, we
have a product of the nuclear proton and neutron form
factors in the form $F_{p}(\bm{q})F_{n}^*(\bm{q})$ or $F_{n}(\bm{q})F_{p}^*(\bm{q})$.
Therefore, expression (<ref>)
can be written as a square of the modulus of the sum of the proton and neutron contributions

\begin{eqnarray}\nonumber
\label{eq:41chiA-CrossSection-Coh-vs-InCoh-cross_section_coherent_term-1-with-s-sprime}
T^{s's}_{m=n}(\bm{q}) &=&\Big|\sum^Z_{k}\sqrt{g^p_\text{coh}}\sum_{r}  (l_{s's},h^p_{rr}(\bm{q}))F_p(\bm{q})
+\sum^N_{j}\sqrt{g^n_\text{coh}} \sum_{r}  (l_{s's},h^n_{rr}(\bm{q}))F_n(\bm{q})\Big|^2
= \\&&= \Big|\sum_{f=p,n}\sqrt{g^f_\text{coh}}\sum^{A_f}_{k=1}\sum_{r} (l_{s's},h^f_{rr}(\bm{q}))F_f(\bm{q})\Big|^2.
\end{eqnarray}

Here  $A_f$ denotes the total number of nucleons of the type $f=p, n$  in the nucleus.
The second term in the cross section formula (<ref>), $T_{m\ne n}$, specified by
(<ref>), involves summation over the nuclear indices $m,n$.
To make this summation, we make use of the condition
(<ref>)  of $\lambda^{mn}(r',r)$
being independent of the indices $m,n$ when  ${m\ne n}$.
We will (temporarily) consider <cit.>,
that they are equal to their averages for protons and neutrons and that the following
“normalization condition” applies to them:

\begin{equation}\label{eq:41chiA-CrossSection-Coh-vs-InCoh-NuclearSpinAmplitudeProduct}
\lambda^{mn}_{r'r} \simeq \lambda^{p/n}_{r'r} \quad \text{and}\quad \lambda^{f}_{r'r} [\lambda^{f}_{x'x}]^* \equiv \delta_{r'x'} \delta_{rx}|\lambda^{f}_{r'r}|^2.
\end{equation}

Then summation over $m,n$ can be made in
since the products of the spin wave functions $\lambda^{f}_{r'r} [\lambda^{f}_{x'x}]^*$
can be taken outside the sum sign.
If the indices $k$ and $j$ in
“indicate” the same nucleon, e.g., the proton, summation yields

\begin{eqnarray}\label{eq:41chiA-CrossSection-Coh-vs-InCoh-cross_section_incoherent_term1}
\sum_n \omega_n\sum_{m\ne n} f_{mn}^k f_{mn}^{k*} &=&\sum_n \omega_n \Big[\sum_{m} f_{mn}^k f_{mn}^{k*} -f_{nn}^k f_{nn}^{k*}\Big] = \\ &=&\sum_n \omega_n \Big[ \langle n|e^{i\bm{q}\bm{X}_k}\sum_{m}|m\rangle\langle m|e^{-i\bm{q}\bm{X}_k}|n\rangle\Big] -|F_{p}(\bm{q})|^2 =1-|F_{p}(\bm{q})|^2
\nonumber .\end{eqnarray}

If $k\ne j$, but they still indicate protons (index $p$), it can be written that

\begin{equation}		 \label{eq:41chiA-CrossSection-Coh-vs-InCoh-cross_section_incoherent_term2}
\sum_n \omega_n\sum_{m\ne n} f_{mn}^k f_{mn}^{j*}  = \langle\text{cov}(e^{i\bm{q}\hat{\bm{X}}_k}, e^{-i\bm{q}\hat{\bm{X}}_j})\rangle_p,
\end{equation}

where the covariation operator of the shift operators  $e^{-i\bm{q}\hat{\bm{X}}_j}$
and $e^{i\bm{q}\hat{\bm{X}}_k}$  with respect to the state $|n\rangle$  is introduced in the form

\begin{equation} \label{eq:41chiA-CrossSection-Coh-vs-InCoh-nn-Covariance}
\text{cov}_{nn}(e^{i\bm{q}\hat{\bm{X}}_k},e^{-i\bm{q}\hat{\bm{X}}_j}) \equiv \langle n|e^{i\bm{q}\hat{\bm{X}}_k}\, e^{-i\bm{q}\hat{\bm{X}_j}} |n\rangle - \langle n|e^{i\bm{q}\hat{\bm{X}}_k}|n\rangle \langle n|e^{-i\bm{q}\hat{\bm{X}}_j}|n\rangle.
\end{equation}

The averaging symbol $\langle\dots\rangle$ in
is defined as

\begin{equation}\label{eq:41chiA-CrossSection-Coh-vs-InCoh-Averaged-Covariance}
\langle\text{cov}(e^{i\bm{q}\hat{\bm{X}}_k},e^{-i\bm{q}\hat{\bm{X}}_j})\rangle_p
\equiv \sum_n \omega_n \text{cov}_{nn}(e^{i\bm{q}\hat{\bm{X}}_k},e^{-i\bm{q}\hat{\bm{X}}_j}).
\end{equation}

Expression (<ref>)
becomes zero at both small and
large transferred momenta

\begin{equation} \label{eq:041chiA-CrossSection-Coh-vs-InCoh-cross_section_incoherent_term3}
\lim_{\bm{q}\to 0}\langle\text{cov}(e^{-i\bm{q}\hat{\bm{X}}_j},e^{i\bm{q}\hat{\bm{X}}_k})\rangle_p = 0,\quad
\lim_{\bm{q}\to \infty}\langle\text{cov}(e^{-i\bm{q}\hat{\bm{X}}_j},e^{i\bm{q}\hat{\bm{X}}_k})\rangle_p=0.
\end{equation}

A similar consideration applies to neutrons and is also generalized to the common case of protons and
neutrons. If, for example, $k$ corresponds to a proton
and $j$ to a neutron, ($k\ne j$), their covariation is

\begin{equation} \label{eq:41chiA-CrossSection-Coh-vs-InCoh-cross_section_incoherent_term4}
\sum_n \omega_n\sum_{m\ne n} f_{mn}^k f_{mn}^{j*} =
\langle\text{cov}(e^{i\bm{q}\hat{\bm{X}}_k},e^{-i\bm{q}\hat{\bm{X}}_j})\rangle_{pn}.
\end{equation}

It is believed that when correlations between nucleons in a nucleus are rather weak, covariation functions
like (<ref>)
can be ignored.
For example, in nuclear shell models where multiparticle wave functions of nuclei
are constructed in the form of a product of single-particle
wave functions <cit.>, covariation
identically vanishes.
Counting in favor of smallness of covariation
is the fact that the inelastic scattering cross section can be approximated with a good accuracy
by the linear dependence on the number of scatterers.
Thus, covariation contributions to the observed cross section $\displaystyle\frac{d\sigma}{d T_A}$ like
(<ref>) and
are ignored below, since all covariation contributions are taken to be zero.
This simplifies further calculations.
Considering the above assumptions, relations
and the assumption that at $k\ne j$ all correlators of the type of
(<ref>) and
(<ref>) vanish

=\langle\text{cov}(e^{i\bm{q}\hat{\bm{X}}_k},e^{-i\bm{q}\hat{\bm{X}}_j})\rangle_{pn} =0, $$

the term (<ref>)
can be finally written as a sum over protons and neutrons

\begin{eqnarray}\label{eq:41chiA-CrossSection-Coh-vs-InCoh-CrossSection-Term-mn-with-s-sprime-final}
T^{s's}_{m\ne n} &=& \sum_{f=p,n}g^f_\text{inc}\big[1-|F_f(\bm{q})|^2\big]
\sum^{A_f}_{k=1} \sum_{r'r}|\lambda^{f}_{r'r}|^2\big| (l_{s's}, h^f_{r'r}(\bm{q})) \big|^2

Thus, the measured differential cross section
for the process $\chi A\to \chi A^{(*)}$ can be written in the form
of two fundamentally different terms

\begin{eqnarray}\label{eq:41chiA-CrossSection-Coh-vs-InCoh-CrossSection-TwoTerms-with-s-sprime}
\frac{d\sigma_{s's}}{d T_A}(\chi A\to \chi A^{(*)})&=&
\Big[\dfrac{G^2_F  m_A}{2^7\pi m^2 T_0 m_\chi}\Big]\Big\{T^{s's}_{m=n} + T^{s's}_{m\ne n}\Big\}
\equiv  \frac{d\sigma^{s's}_\text{inc}}{d T_A} + \frac{d\sigma^{s's}_\text{coh}}{d T_A},
\quad \text{where}\\ \nonumber
\frac{d\sigma^{s's}_\text{inc}}{d T_A} &=& c_A(T_0, m_A, m_\chi) g_\text{i}\!\sum_{f=n,p}\sum_{k=1}^{A_f} \sum_{r'r}  |\lambda^f_{r'r}|^2 |(l_{s's},h^f_{r'r}(\bm{q}))|^2[1-|F_f(\bm{q})|^2],
\\ \nonumber \frac{d\sigma^{s's}_\text{coh}}{d T_A} &=& c_A(T_0, m_A, m_\chi)
g_\text{c}\Big|\sum_{f=n,p}\sum_{k=1}^{A_f} \sum_{r} (l_{s's}, h^f_{rr}(\bm{q}))F_f(\bm{q})\Big|^2

Kinematic correction coefficients $g_\text{c/i}$  in expressions
(<ref>) are determined by averaging parameters from (<ref>)
over nucleons and factoring them out from the double summation  $\sum_{nm}$.
Squares of nuclear form factors $|F_{p/n}|^2$  for protons and neutrons are defined by
In formulas (<ref>)
the following universal common factor is introduced:

\begin{eqnarray}  \label{eq:43chiA-CrossSection-via-ScalarProducts-c_A-definition}
c_A \equiv c_A(T_0, m_A, m_\chi) &\equiv&  \dfrac{G^2_F  m_A}{2^6\pi m^2 (2T_0 m_\chi)}
=\dfrac{G^2_F  m_A}{4\pi }\dfrac{1}{2^4 m^2 |\bm{k}^l_\chi|^2 },\quad
\end{eqnarray}

where the dependence on the initial energy $T_0$  of the $\chi$ particle incident on the nucleus
at rest is explicitly separated out, and the intensity of the hypothetical interaction
of the $\chi$ particle with nucleons (proportional to $G^2_F$) is “hidden deep” in the corresponding scalar products of currents.
When obtaining the main formulas of this section
a simplifying assumption was made that all correlation contributions like
(<ref>)  and
(<ref>) can be neglected.
Another assumption was the use of the normalization
condition (<ref>) for spin amplitudes.
Note that the first and second terms in
(<ref>) and
correspond to the elastic and inelastic $\chi$-nucleus interactions, respectively.
The first term (when $\sum_{n=m}$) contains both indices $k$ and $j$,
which leads to quadratic dependence of the cross section on the number of nucleons in the nucleus.
The second term (when $\sum_{n\ne m}$) is proportional to  $\delta_{kj}$ in good approximation, which automatically leads to linear dependence of the cross section on the number of nucleons in the nucleus.
Summation at the level of amplitudes of scattering by individual nucleons (which is traditionally
associated with the notion of coherence) is explicitly
seen in the first term of (<ref>). Here each nucleon (proton and/or neutron) enters the sum being weighted
with its corresponding form factor $F_{p/n}(\bm{q})$.
This term corresponds to the case where the nucleus does not
change its spin state (spin projection on a given direction),
which is evident from the character of summation
over spin projections of nucleons participating in
the interaction, i.e., $\sum_r (l, h^{p/n}_{rr})$.
All active nucleons retain the spin orientation after the interaction ($r'\equiv r$).
The second term in (<ref>)
is traditionally associated with the incoherent contribution to the cross section.
It involves the square of the product of the nuclear
spin amplitudes  $|\lambda^{p/n}_{r'r}|^2$, which, according to the
adopted condition  (<ref>),
should be equal to unity with a good accuracy

\begin{equation}\label{eq:41chiA-CrossSection-Coh-vs-InCoh-AllNuclearSpinsEquialTo-1}|\lambda^{p/n}_{r'r}|^2\to 1.\end{equation}

Thus, differential cross sections
for the scattering of a massive $\chi$ particle from a nucleus $\chi_s A\to \chi_{s'} A^{(*)}$
(with its spin index changed from $s$ to $s'$) in terms of scalar products of leptonic and nucleon
currents  $(l_{s's},h^f_{r'r})$ take the form

\begin{eqnarray}  \nonumber
\label{eq:41chiA-CrossSection-Coh-vs-InCoh-CrossSections-via-ScalarProducts-with-s-sprime}
\frac{d\sigma^{s's}_\text{inc}}{d T_A} &=&
c_A(T_0, m_A, m_\chi) g_\text{i} \sum_{f=p,n}[1-|F_f(\bm{q})|^2]\sum^{A_f}_{k=1}
\Big[ \sum^{}_{r'=\pm}|(l_{s's}, h^{\eta,f}_{r'+})|^2+\sum^{}_{r'=\pm}|(l_{s's},h^{\eta,f}_{r'-})|^2 \Big], \qquad
\\
\frac{d\sigma^{s's}_\text{coh}}{d T_A} &=&
c_A(T_0, m_A, m_\chi) g_\text{c}
\Big| \sum_{f=p,n}F_f(\bm{q}) \sum^{A_f}_{k=1}[(l_{s's},h^{\eta,f}_{++})+(l_{s's},h^{\eta,f}_{--})] \Big|^2

where sums over initial nucleon spin projections on the $\chi$ particle motion direction
are explicitly separated out, which is marked by the index $\eta$ in hadronic currents in scalar products.
Since scalar products $(l_{s's},h^f_{r'r})$ depend only on the type of the active nucleon (index $f$
indicates a proton or a neutron) and do not depend on the active nucleon number in the nucleus (summation
index $k$), simple summation over this index can be taken in formulas
i.e., separate summation over all protons and all neutrons in the nucleus (considering their spin projections).
Then formulas (<ref>)
take the form

\begin{eqnarray}  \nonumber
\label{eq:41chiA-CrossSection-Coh-vs-InCoh-CS-with-s-sprime-for-proton-neutron}
\frac{d\sigma^{s's}_\text{inc}}{d T_A} &=& c_A(T_0, m_A, m_\chi) g_\text{i} \sum_{f=p,n}[1-|F_f(\bm{q})|^2]
\Big[A^f_+  \sum^{}_{r'=\pm}|(l_{s's}, h^{\eta,f}_{r'+})|^2+A^f_- \sum^{}_{r'=\pm}|(l_{s's},h^{\eta,f}_{r'-})|^2 \Big], \qquad
\\
\frac{d\sigma^{s's}_\text{coh}}{d T_A} &=&c_A(T_0, m_A, m_\chi) g_\text{c}
\Big| \sum_{f=p,n}F_f(\bm{q})  [A^f_+(l_{s's},h^{\eta,f}_{++})+ A^f_-(l_{s's},h^{\eta,f}_{--})] \Big|^2
, \end{eqnarray}

where  $A^f_\pm$ is the number of $f$-type nucleons ($f=p,n$) with the spin projection $\pm 1$ on the preferred direction
(e.g., of the arrival of $\chi$ particle).
To finish derivation of the formula for the differential cross section of the process $\chi A\to \chi A^{(*)}$,
one should have, according to
explicit expressions for scalar products $(l_{s's},h^{p/n}_{r'r})$ of interacting leptonic and nucleon currents.
These quantities were obtained in <cit.> and are given in Section

§.§ Set of scalar products for ${\chi A\to \chi A^{(*)}}$ scattering

In this section, we give scalar products for all combinations of $\chi$-lepton and nucleon currents corresponding
to all possible lepton helicities and all possible nucleon spin projections on the specified direction.
All scalar products are calculated in the lepton and (active) nucleon center-of-mass system (c.m.s.) in
the so-called mixed $\chi\eta$-basis, where $\chi$ particle currents are taken in the helicity basis
(by analogy with neutrino), and nucleon currents are taken in the $\eta$-basis
(or $\sigma_3$-basis, i.e., quantized with respect to the direction of the momentum of the incident particle).
The procedure for obtaining these scalar products is detailed in <cit.>.
They are needed for self-consistent calculations
of coherent and incoherent $\chi A\to \chi A^{(*)}$ scattering
cross sections and are defined by the expressions

\begin{equation*}
(l^{i}_{s's} \, h^{k}_{r'r})  \equiv \sum^{4}_{\mu,\nu} J^{i,\mu}_{s's}(\bm{k}')\, g_{\mu\nu}\, J^{k,\nu}_{r'r}(\bm{p}'),
\end{equation*}

where subscripts $s$ and $r$  denote fixed values of the initial lepton and nucleon spin states,
and subscripts  $s'$ and $r'$ denote the respective final states.
Indices $i$ and $k$ designate vector ($J^{v,\mu}\equiv V^\mu$),
axial vector ($J^{a,\mu}\equiv A^\mu$), scalar ($J^{s}\equiv S$),
and pseudoscalar ($J^{p}\equiv P $)
leptonic (argument $\bm{k}'$) and nucleon (argument $\bm{p}'$) currents.
For example, the expression for the
scalar product of the axial-axial currents is

\begin{equation}\label{eq:42chiA-CrossSection-ScalarProducts-ChiEta-All-A-A}
(l^{a}_{s's} \, h^{a}_{r'r})  = \sum^{}_{\mu,\nu} A^\mu_{s's}(\bm{k}')\, g_{\mu\nu} \, A^\nu_{r'r}(\bm{p}').
\end{equation}

Scalar products are expressed in terms of nucleon and $\chi$ particle masses

$$m, \quad m_\chi, \quad \text{and \ parameters} \quad
\lambda_\pm=\sqrt{E_p\pm m}, \qquad \xi_\pm=\sqrt{E_\chi\pm m_\chi} ,$$

and the c.m.s. angle of elastic scattering of a $\chi$ lepton from a nucleon,
i.e., the angle $\theta$  between the direction of the initial lepton momentum  $\bm{k}$
and the direction of the final lepton momentum $\bm{k}'$,
where $\bm{k}+\bm{p}=\bm{k}'+\bm{p}'=0$;
in addition, the following are valid:

\begin{eqnarray}\nonumber \label{eq:42chiA-CrossSection-ScalarProducts-ChiEta-All-Energies-etc}
E_\chi &\equiv& \sqrt{m^2_\chi+|\bm{k}|^2} = \sqrt{m^2_\chi+|\bm{k}'|^2} = \frac{s+m^2_\chi-m^2}{2\sqrt{s}}, \\
E_p &\equiv& \sqrt{m^2+|\bm{p}|^2} =\sqrt{m^2+|\bm{p}'|^2}  = \frac{s+m^2-m^2_\chi}{2\sqrt{s}}
, \\ |\bm{p}| &=& \sqrt{E^2_p-m^2} = \lambda_+ \lambda_-
=|\bm{k}| = \sqrt{E^2_\chi- m^2_\chi} = \xi_+ \xi_-
= \dfrac{\lambda(s, m^2, m^2_\chi)}{2\sqrt{s}}
\nonumber

Here  $p$ and $k$  are the four-momenta of the nucleon and the $\chi$ particle
[The kinematic $\lambda$-function is defined by the expression
$\lambda^2(s, m^2, m^2_\chi)\equiv %s^2+m^4+m^4_\chi -2 s m^2 - 2s m^2_\chi - 2 m^2 m^2_\chi=
\big(s-(m+m_\chi)^2\big)\big(s-(m-m_\chi)^2\big)$.].
The invariant square of the c.m.s. total energy has the form

\begin{eqnarray}\label{eq:42chiA-CrossSection-ScalarProducts-ChiEta-s-in-CMS-lepton-nucleon}
s&=&(p+k)^2 = (p_0+k_0)^2 - (\bm{p}+\bm{k})^2 \big|_{\text{c.m.s.}}= (k_0+p_0)^2 = (E_\chi+ E_p)^2
\end{eqnarray}

In the lab frame, where the nucleon is at rest, $p_l =(m, \bm{p}_l=\bm{0})$, it is

\begin{eqnarray}\label{eq:42chiA-CrossSection-ScalarProducts-ChiEta-s-in-CLab-lepton-nucleon}
s&=& (k_l+p_l)^2\big|_{\text{Lab}} = m^2_\chi +m^2 + 2k_lp_l
=m^2_\chi +m^2 + 2 m\sqrt{m^2_\chi + [\bm{k}^l_\chi]^2}

From (<ref>)
and (<ref>)
there follows dependence of the momentum $\bm{k}$ and kinetic energy $T_* = \dfrac{|\bm{k}^2|}{2 m_\chi}$
of the incident $\chi$ particle in the c.m.s. on its momentum
$\bm{k}^l_\chi \equiv \bm{k}^{}_l$  in the lab frame

\begin{equation}\label{A1DM-CrossSectionsEvaluations-Helper-k-from-Lab-to-CMS}
|\bm{k}|=   |\bm{k}^l_\chi| \dfrac{m}{\sqrt{s} } \quad \text{and}\quad  T_* = T_0 \dfrac{m^2}{s},
\quad \text{where}\quad T_0 = \dfrac{|\bm{k}^l_\chi|^2}{2 m_\chi}

The scattering angle appears in the three-momentum transferred to the nucleon

\begin{equation}\label{eq:42chiA-CrossSection-ScalarProducts-ChiEta-q2}
\bm{q}^2=(\bm{k}-\bm{k}')^2 = |\bm{k}|^2 + |\bm{k}'|^2 - 2 |\bm{k}| |\bm{k}'|\cos\theta
= 2 |\bm{k}|^2(1-\cos\theta) \equiv \bm{q}^2_{\max} \sin^2\dfrac{\theta}{2}.
\end{equation}

Hence, in view of the nonrelativistic approximation $\bm{q}^2 \simeq  2 m_A T_A$,  there follows

\begin{equation}\label{eq:42chiA-CrossSection-ScalarProducts-ChiEta-sin2-via-TTmax}
\sin^2\dfrac{\theta}{2} = \dfrac{\bm{q}^2}{\bm{q}^2_{\max}}
\simeq \dfrac{T_A^{}}{T_A^{\max}},
\quad \text{where}\quad T_A^{\max}  \simeq \dfrac{4 m_\chi m^2 T_0}{s m_A}

Then, knowing $m_\chi$, nucleon mass $m$, and $T_0$ in the lab frame, we can find $s$

\begin{equation} \label{A1DM-CrossSectionsEvaluations-Helper-s-invariant-in-CLab}
s= (k_l+p_l)^2\big|_{\text{Lab}} =m^2_\chi +m^2 + 2 m m_\chi \sqrt{1 + \dfrac{2 T_0}{m_\chi }},
\end{equation}

and calculate (in the lepton and nucleon c.m.s.) all necessary energies and momenta
After that, considering the determined angle $\theta$
and the parameters $T_0$  (initial kinetic energy of the $\chi$ particle)
and $T_A$  (kinetic energy of the nuclear recoil) specified by the lepton-nucleus scattering conditions,
the kinematics of the (internal) elastic 2-in-2 scattering of a lepton by a nucleon in their c.m.s.
and then all scalar products are completely determined.
Below, expressions for all scalar products from <cit.> are given.
According to (<ref>), a set of scalar products of
axial vector currents $(l^a,h^a)$ has the form

\begin{eqnarray*}
(l^a_{\pm\pm}\, h^a_{\pm\pm}) &=& -  4\cos\frac{\theta}{2} \Big[\xi_+ \xi_-  \lambda_+ \lambda_- \cos^2\frac{\theta}{2} + (m_\chi + \xi^2_-)(m + \lambda^2_- \cos^2\frac{\theta}{2}) \Big]
, \\ (l^a_{\pm\pm}\, h^a_{\mp\mp})&=& + 4 \cos\frac{\theta}{2} \Big[\xi_+ \xi_-\lambda_+ \lambda_-  \cos^2\frac{\theta}{2}   +(m_\chi + \xi^2_-) \big(m + \lambda^2_- (1 + \sin^2\frac{\theta}{2}) \big) \Big]
,\\  (l^a_{\pm\pm}\, h^a_{\pm\mp})&=& \mp 4 \sin\frac{\theta}{2} \Big[ \xi_+ \xi_-  \lambda_+ \lambda_- \cos^2\frac{\theta}{2}    +(m_\chi + \xi^2_-)\big(2m + \lambda^2_- (1+\sin^2\frac{\theta}{2}\big) \Big]e^{\mp i\phi}
, \\ (l^a_{\pm\pm}\, h^a_{\mp\pm}) &=&  \mp 4 \sin\frac{\theta}{2}\Big[\xi_+ \xi_-  \lambda_+ \lambda_-  \cos^2\frac{\theta}{2} + (m_\chi + \xi^2_-) \lambda^2_- \cos^2\frac{\theta}{2} \Big]  e^{\pm i\phi}
,\\  (l^a_{\pm\mp}\, h^a_{\mp\mp})&=& \mp 4m_\chi\big(m + \lambda^2_- \cos^2\frac{\theta}{2}\big) \sin\frac{\theta}{2}e^{\mp i\phi}
,\\ (l^a_{\pm\mp}\, h^a_{\pm\pm})&=& \mp 4m_\chi \sin\frac{\theta}{2} \big(\lambda^2_-  \cos^2\frac{\theta}{2} - m\big) e^{\mp i\phi}
, \\ (l^a_{\pm\mp}\, h^a_{\mp\pm}) &=& - 4m_\chi \cos\frac{\theta}{2} \big(2m + \lambda^2_- \sin^2\frac{\theta}{2}\big)
, \quad  (l^a_{\pm\mp}\, h^a_{\pm\mp})=  + 4m_\chi \lambda^2_-  \cos\frac{\theta}{2}\sin^2\frac{\theta}{2}  e^{\mp 2 i \phi}

In the nonrelativistic approximation
[Formally due to the condition $\xi^2_- =E_\chi -m_\chi \to 0\quad \text{and} \quad \lambda^2_- =E-m\to 0$.]

these scalar products are greatly simplified

\begin{eqnarray}\nonumber\label{eq:42chiA-CrossSection-ScalarProducts-ChiEta-Axial-Nonrel}
(l^a_{\pm\pm}\, h^a_{\pm\pm})&=&  - m^2_c ,\qquad (l^a_{\pm\pm}\, h^a_{\mp\mp})=  + m^2_c
, \qquad (l^a_{\pm\mp}\, h^a_{\mp\pm}) =  - 2 m^2_c
, \\(l^a_{+-}\, h^a_{\mp\mp})&=&  \mp m^2_s e^{-i \phi} ,\quad (l^a_{-+}\, h^a_{\mp\mp})=  \mp  m^2_s e^{+i \phi} ,\quad (l^a_{\pm\pm}\, h^a_{\pm\mp})=  \mp 2 m^2_s e^{\mp i \phi}
\nonumber ,\\ (l^a_{\pm\pm}\, h^a_{\mp\pm})&=&(l^a_{\pm\mp}\, h^a_{\pm\mp})= 0

Here the following convenient designations for fur- ther use are introduced:

\begin{equation}\label{eq:42chiA-CrossSection-ScalarProducts-ChiEta-All-nonrel-parameters}
\displaystyle  m^2_c \equiv  4 m_\chi m  \cos\frac{\theta}{2},  \quad m^2_s \equiv  4 m_\chi m  \sin\frac{\theta}{2}.
\end{equation}

A set of scalar products of vector currents
$ (l^{v}_{s's} \, h^{v}_{r'r}) = V^{\mu} _{s's}(\bm{k}') g_{\mu\nu} V^{\nu }_{r'r}(\bm{p}')$
in the nonrelativistic approximation is

\begin{eqnarray*}
(l^v_{\pm\pm} h^v_{\pm\pm})&=& (l^v_{\pm\pm} h^v_{\mp\mp})= m^2_c
, \quad  (l^v_{+-} h^v_{\mp\mp})= + m^2_s e^{-i\phi} ,\quad  (l^v_{-+} h^v_{\mp\mp})= - m^2_s e^{+i\phi}
,\\ (l^v_{\pm\pm} h^v_{+-})&=&  (l^v_{\mp\mp} h^v_{-+})=
(l^v_{+-} h^v_{\mp\pm})= (l^v_{-+} h^v_{\pm\mp}) = 0

Scalar products of vector-axial and axial vector currents in the nonrelativistic approximation com- pletely vanish

$$(l^v_{s's}\, h^a_{r'r})_{\text{nonrel}}  = V^{\mu}_{s's}(\bm{k}') \,g_{\mu \nu}\, A^{\nu}_{r'r}(\bm{p}') \simeq 0.$$

$$ (l^a_{s's}\, h^v_{r'r})_{\text{nonrel}}  = A^{\mu}_{s's}(\bm{k}') \,g_{\mu \nu}\, V^{\nu}_{r'r}(\bm{p}') \simeq 0.$$

In the nonrelativistic approximation, only scalar products of scalar currents
$(l^s_{s's}\, h^s_{r'r})^{} = S_{s's}(\bm{k}') S_{r'r}(\bm{p}')$ corresponding to preservation of the nucleon
spin projection on the chosen direction “survive,” while all the other products of scalars vanish

\begin{eqnarray}\nonumber \label{eq:42chiA-CrossSection-ScalarProducts-ChiEta-Scalar-Nonrel}
(l^s_{\pm\pm}\, h^s_{\pm\pm})^{}&=& (l^s_{\pm\pm}\, h^s_{\mp\mp})^{}= m^2_c
, \quad (l^s_{+-}\, h^s_{\mp\mp})^{}=   +m^2_s e^{-i\phi},  \quad (l^s_{-+}\, h^s_{\mp\mp})^{}=  -m^2_se^{+i\phi}
, \\ (l^s_{-+}\, h^s_{\pm\mp})^{}&=&(l^s_{+-}\, h^s_{\mp\pm})^{}   =
(l^s_{\mp\mp}\, h^s_{+-})^{}=(l^s_{\mp\mp}\, h^s_{-+})^{}= 0

These expressions entirely coincide with the similar ones for vector currents. All scalar products of scalar- pseudoscalar currents, pseudoscalar-scalar currents, and pseudoscalar currents vanish in the nonrelativistic approximation

\begin{eqnarray*}
(l^s_{s's}\, h^p_{r'r})^{} &=& S_{s's}(\bm{k}') P_{r'r}(\bm{p}')=(l^p_{s's}\, h^s_{r'r})_{\text{nonrel}} =0
, \\ (l^p_{s's}\, h^s_{r'r})^{} &=& P_{s's}(\bm{k}') S_{r'r}(\bm{p}')=(l^s_{s's}\, h^p_{r'r})_{\text{nonrel}} = 0
,\\ (l^p_{s's}\, h^p_{r'r})^{} &=& P_{s's}(\bm{k}') P_{r'r}(\bm{p}')=(l^p_{s's}\, h^p_{r'r})_{\text{nonrel}} = 0

Scalar products of the leptonic ($\chi$ particle) and nucleon currents for any spin projections of the nucleon
($r',r=\pm1$)  and the $\chi$ particle ($s',s=\pm1$,
which correspond to the neutral weak current interaction of a massive neutral $\chi$ particle
with a nucleon, are defined as <cit.>

\begin{eqnarray}\label{eq:42chiA-CrossSection-ScalarProducts-ChiEta-All-WeakGeneralCurrent-definition}
(l^w_{s's}, h^{w,f}_{r'r})   &=& \alpha_f (l^v_{s's}\, h^v_{r'r}) + \beta_f (l^v_{s's}\, h^a_{r'r})
+ \gamma_f (l^a_{s's}\, h^v_{r'r})  +  \delta_f  (l^a_{s's}\, h^a_{r'r}) ,
\end{eqnarray}

\begin{eqnarray*}
\text{where} \quad  \alpha_f = \chi_V h^f_V  = + g_V^f ,\quad  \beta_f = \chi_V h^f_A = - g_A^f
, \quad \gamma_f = \chi_A  h^f_V = - g_V^f ,\quad \delta_f = \chi_A h^f_A = + g_A^f
. \end{eqnarray*}

The index $f$ denotes a neutron or a proton.
On the right are the values of these constants for the neutrino Standard Model (since in the neutrino
$\chi_V  = - \chi_A = 1$)).
In the nonrelativistic approximation, scalar products for the weak neutral current of a massive
particle (<ref>)  have the form

\begin{eqnarray} \nonumber \label{eq:42chiA-CrossSection-ScalarProducts-ChiEta-All-Weak-Nonrel}
(l^w_{\pm\pm}, h^{w,f}_{\pm\pm}) &=&   m^2_c (\alpha_f - \delta_f)
, \quad (l^w_{\pm\pm}, h^{w,f}_{\mp\mp}) =  m^2_c (\alpha_f + \delta_f)
,\quad  (l^w_{\pm\mp}, h^{w,f}_{\mp\pm}) = - 2 m^2_c \delta_f
,\\ (l^w_{\pm\mp}, h^{w,f}_{\mp\mp}) &= &  \pm m^2_s e^{\mp i\phi} (\alpha_f-\delta_f)
, \quad  (l^w_{\pm\mp}, h^{w,f}_{\pm\pm}) = \pm m^2_s e^{\mp i\phi} (\alpha_f+\delta_f)
, \\  (l^w_{\pm\pm}, h^{w,f}_{\pm\mp})  &= &  \mp 2m^2_s e^{\mp i \phi} \delta_f
,\quad (l^w_{\pm\pm}, h^{w,f}_{\mp\pm}) =  (l^w_{\pm\mp}, h^{w,f}_{\pm\mp})^{}\simeq  0
.  \nonumber  \end{eqnarray}

The set of scalar products corresponding to the $\chi$ particle-nucleon interaction, which is given in this section, is a basis for calculating coherent (elastic) and incoherent (inelastic) cross sections of the $\chi$ particle–nucleus interaction in the nonrelativistic approximation.

§.§ Expansion of scalar products in cross sections of ${\chi A\to \chi A^{(*)}}$ scattering

According to the results of Section <ref>,
the expression for the observed differential cross section
$\displaystyle \frac{d\sigma}{dT_A}(\chi A\to \chi A^{(*)})$  involves the incoherent (inelastic)
and coherent (elastic) terms.
They are given by formulas (<ref>)
written below in the following form:

\begin{eqnarray} \nonumber\label{eq:43chiA-CrossSection-via-ScalarProducts-in-Apm}
\frac{d\sigma^{s's}_\text{coh}(\bm{q})}{g_\text{c}c_A d T_A} &=&
\Big| \sum_{f=p,n}F_f(\bm{q})  [A^f_+(l_{s's},h^{\eta,f}_{++})+ A^f_-(l_{s's},h^{\eta,f}_{--})] \Big|^2
\frac{d\sigma^{s's}_\text{inc}(\bm{q})}{g_\text{i} c_A d T_A} &=&
\sum_{f=p,n}[1-|F_f(\bm{q})|^2] \Big[A^f_+  \sum^{}_{r'=\pm}|(l_{s's}, h^{\eta,f}_{r'+})|^2+A^f_- \sum^{}_{r'=\pm}|(l_{s's},h^{\eta,f}_{r'-})|^2 \Big]

Here $A^f_\pm$  is the number of $f$-type nucleons with the  spin projection  $\pm 1$ on the preferred direction
(e.g., that of the $\chi$ particle arrival), and the universal common factor
$c_A \equiv c_A(T_0, m_A, m_\chi) $ from (<ref>)
is introduced for convenience.
Helicities of the $\chi$ particle (in a more general form, spin projections on a certain direction) in the initial $s$ and final $s'$
states are considered to be fixed in (<ref>).
Later, averaging (summation) can be performed over them.
Formulas  (<ref>)  can be conveniently written in terms
of the total number of  $f$-type nucleons, $A_f$, and the difference in the number of nucleons,  $\Delta A_f $,
having a positive and a negative spin projection on the preferred direction. With a simple transformation

\begin{equation}\label{eq:43chiA-CrossSection-via-ScalarProducts-to-A_f-and-DeltaA_f}
A^f_\pm =\dfrac12 (A_f\pm \Delta A_f),
\quad\text{where}\quad  A_f\equiv A^f_+ + A^f_- \quad\text{and}\quad   \Delta A_f \equiv A^f_+ - A^f_-

cross sections  (<ref>)
take a key form for further consideration

\begin{eqnarray} \nonumber  \label{eq:43chiA-CrossSection-via-ScalarProducts-both-CS-main}
\frac{d\sigma^{s's}_\text{coh}(\bm{q})}{g_\text{c} c_A  d T_A}&=&
\Big| \sum^{}_{f=p,n} F_{f}(\bm{q}) \dfrac{A_f}{2}  Q^f_{s's}  \Big|^2
,\\  \frac{d\sigma^{s's}_\text{inc}(\bm{q})}{g_\text{i} c_A  d T_A} &=&
\sum_{f=p,n}  \big[1-F^2_f(\bm{q})\big] \dfrac{ A_f  }{2}   \Big[ \, Q^{s's}_+  + \dfrac{\Delta A_f }{A_f } \, Q^{s's}_-  \Big]

Thus, the coherent (elastic) and incoherent (inelastic) $\chi A$ cross sections are defined by the following
respective combinations of scalar products:

\begin{equation}\label{eq:43chiA-CrossSection-via-ScalarProducts-CohCS-factors}
Q_f^{s's}\equiv Q^f_{s's}\equiv \hat Q^{s's}_{+} + \dfrac{\Delta A_f}{A_f} \hat Q^{s's}_{-} ,
\quad \text{where} \quad  \hat Q^{s's}_{\pm}\equiv (l_{s's},h^{f}_{++}) \pm (l_{s's},h^{f}_{--})

\begin{equation}\label{eq:43chiA-CrossSection-via-ScalarProducts-InCohCS-factors}
\text{and}\quad  Q^{s's}_\pm\equiv
\sum^{}_{r'=\pm}|(l_{s's}, h^{f}_{r'+})|^2 \pm  \sum^{}_{r'=\pm}|(l_{s's},h^{f}_{r'-})|^2

In  (<ref>),
the superscripts and the subscript respectively correspond to the $\chi$-particle spin projections
and the nucleon spin projection on the preferred direction.
It is seen that $Q^{s's}_+$  is the total sum of squares of scalar products over all nucleon spin projections
for a fixed pair of lepton helicities $s'$ and $s$.
As a rule, if the nucleus has the total zero spin, then $ \Delta A_f= 0$.
Note also nearly always  ${\Delta A_f }\ll {A_f }$ holds (except probably
for the lightest nuclei).
Since the emphasis in this work is on the nonrelativistic approximation,
recall that the nonrelativistic approximation corresponds to the case when the condition
of the smallness of the kinetic energy of the $\chi$ particle compared to its rest energy  (at $c=1$) is satisfied, i.e.,

\begin{equation}\label{eq:43chiA-CrossSection-via-ScalarProducts-NonREL-definition}
T_0\equiv \dfrac{|\bm{k}^l_\chi |^2}{2 m_\chi} \simeq 10^{-6} c^2  m_\chi  \ll m_\chi \quad \text{or}\quad
\dfrac{T_0}{ m_\chi}  \simeq \dfrac{ |\bm{k}^l_\chi|^2}{ m^2_\chi} \simeq 10^{-6} \ll 1.
\end{equation}

In this approximation, the square of the total energy of the lepton and the (active) nucleon in the lab frame
(where the nucleus/nucleon is at rest) has the form

$$ s =  m^2_\chi +m^2 + 2 m m_\chi\sqrt{1 + [\bm{k}^l_\chi]^2/m^2_\chi} \equiv (m_\chi +m) ^2 + 2 m T_0. $$

Here the second term proportional to $T_0$, is much smaller than the first one
(and is only retained as a leading-order contribution with respect to the smallness parameter $T_0/m$).

§.§.§ Scalar ${\chi A}$  interaction

By scalar interaction is meant interaction of a $\chi$ particle with nucleons that leads to scalar products of the scalar
leptonic and scalar nucleon currents in the form

\begin{equation}\label{eq:43chiA-CrossSection-via-ScalarProducts-Scalar-ScalarProduct-deff}
(l^s_{s's}, h^{s,f}_{r'r})=c_{S}^f\, \overline{u}_\chi (s',k') u_\chi (s,k) \times \overline{u}_f (r',p') u_f (r,p) = c_{S}^f\, S_{s's}(\bm{k}') S_{r'r}(\bm{p}'),
\end{equation}

where effective coupling constants $\ c_{S}^f \equiv \chi^{}_{S} h^f_{S}$ from
(<ref>) and
specify intensity of this interaction (in terms of $G_{\rm F}$).
In the nonrelativistic approximation, these scalar products are given by formulas
Calculation of coherent cross sections requires the following combinations of scalar products:

\begin{eqnarray*}
\hat Q^{\mp\mp}_{+}&=& (l_{\mp\mp},h^{n}_{++}) + (l_{\mp\mp},h^{n}_{--})= m^2_c + m^2_c = 2 m^2_c
, \\  \hat Q^{\mp\mp}_{-}&=& (l_{\mp\mp},h^{n}_{++})- (l_{\mp\mp},h^{n}_{--})= m^2_c - m^2_c =0
, \\   \hat Q^{\mp\pm}_{+}&=& (l_{\mp\pm},h^{n}_{++}) + (l_{\mp\pm},h^{n}_{--}) = \mp m^2_se^{\pm i\phi} + (\mp m^2_se^{\pm i\phi}) = \mp 2m^2_se^{\pm i\phi}
, \\  \hat Q^{\mp\pm}_{-}&=& (l_{\mp\pm},h^{n}_{++})- (l_{\mp\pm},h^{n}_{--}) = \mp m^2_se^{\pm i\phi}
- (\mp m^2_se^{\pm i\phi}) =0

With these combinations, coherent scattering cross sections are calculated from
(<ref>) and
(<ref>) in terms of the quantities

\begin{eqnarray*}
Q_f^{\mp\mp} = \hat Q^{\mp\mp}_{+} + \dfrac{\Delta A_f}{A_f}  \hat Q^{\mp\mp}_{-}  = 2 m^2_c
\quad\text{and}\quad Q_f^{\mp\pm} = \hat Q^{\mp\pm}_{+} + \dfrac{\Delta A_f}{A_f}  \hat Q^{\mp\pm}_{-}
= \mp 2m^2_se^{\pm i\phi}

A set of nonrelativistic cross sections for the coherent $\chi A$ scattering in the scalar-current interaction
channel with the $\chi$-particle helicity unchanged (first formula)  and helicity flip (second formula) can be written as

\begin{eqnarray}\nonumber\label{eq:42chiA-CrossSectionsEvaluations-Scalar-ChiEta-nonrel-CohCS}
\frac{d\sigma^{\mp\mp}_\text{coh}(\bm{q})}{d T_A} &=&c_A    \big| \sum^{}_{f=p,n} F_{f}(\bm{q}) {A_f} c_S^{f} m^2_c   \big|^2 = \dfrac{G^2_F  m_A }{4\pi}\cos^2\frac{\theta}{2}  \dfrac{m^2_\chi  }{|\bm{k}^l_\chi|^2}
\big[ \sum^{}_{f=p,n} F_{f}(\bm{q}) {A_f} c_S^{f} \big]^2
,\\ \frac{d\sigma^{\mp\pm}_\text{coh}(\bm{q})}{ d T_A} &=& c_A \big| \sum^{}_{f=p,n} F_{f}(\bm{q}) {A_f} c_S^{f} m^2_s \big|^2 =\dfrac{G^2_F  m_A }{4\pi}\sin^2\frac{\theta}{2}  \dfrac{m^2_\chi  }{|\bm{k}^l_\chi|^2 }
\big[ \sum^{}_{f=p,n} F_{f}(\bm{q}) {A_f} c_S^{f} \big]^2,\qquad
\\ \dfrac{d\sigma^{\text{total}}_\text{coh}(\bm{q})}{d T_A}&=&
\dfrac12 \sum^{}_{s's}\dfrac{d\sigma^{s's}_\text{coh}}{d T_A}
=\dfrac{G^2_F  m_A }{4\pi} \dfrac{m^2_\chi  }{|\bm{k}^l_\chi|^2 }
\big[ \sum^{}_{f=p,n} F_{f}(\bm{q}) {A_f} c_S^{f} \big]^2
\nonumber.\end{eqnarray}

The last formula gives the total (averaged over initial lepton helicities and summed over final lepton helicities)
coherent cross section for the $\chi A$ scattering due to interaction of scalar currents in the nonrelativistic limit.
Here we use

\begin{equation}\label{43chiA-CrossSection-via-ScalarProducts-c_A+}
c_A (4m_\chi m)^2 = (4m_\chi m)^2 \dfrac{ G^2_F  m_A}{2^6\pi m^2 |\bm{k}^l_\chi|^2 }
= \dfrac{ G^2_F  m_A}{4\pi}\dfrac{m^2_\chi }{|\bm{k}^l_\chi|^2 },
\end{equation}
and it is considered that  $g_\text{c}\simeq 1$.
Incoherent $\chi A$-cross sections are defined by general formulas
(<ref>) and
They depend only on squares of scalar products of scalar currents through the parameters

\begin{eqnarray*}
Q^{s's}_\pm &\equiv&  S_+^{s's} \pm S_-^{s's},  \quad \text{where} \quad
S_\pm^{s's} \equiv \sum^{}_{r'=\pm}|(l_{s's}, h^{f}_{r'\pm})|^2 = |(l_{s's}, h^{f}_{+\pm})|^2+|(l_{s's}, h^{f}_{-\pm})|^2

In the nonrelativistic limit, for the squares of these scalar products we have

\begin{eqnarray}\nonumber\label{eq:42chiA-CrossSectionsEvaluations-Scalar-ChiEta-nonrel-ScalarProducts^2}
|(l^s_{--}\, h^s_{--})|^2&=& |(l^s_{++}\, h^s_{--})|^2=|(l^s_{--}\, h^s_{++})|^2=|(l^s_{++}\, h^s_{++})|^2 =  m^4_c
, \\ |(l^s_{+-}\, h^s_{--})|^2&=&  |(l^s_{+-}\, h^s_{++})|^2 =|(l^s_{-+}\, h^s_{--})|^2=|(l^s_{-+}\, h^s_{++})|^2 =  m^4_s
, \\ \nonumber  |(l^s_{-+}\, h^s_{+-})|^2&=&|(l^s_{+-}\, h^s_{-+})|^2  =
|(l^s_{-+}\, h^s_{-+})|^2 =|(l^s_{+-}\, h^s_{+-})|^2 =  0 , \\\nonumber
|(l^s_{--}\, h^s_{+-})|^2&=&|(l^s_{++}\, h^s_{+-})|^2=|(l^s_{--}\, h^s_{-+})|^2=|(l^s_{++}\, h^s_{-+})|^2 = 0 \nonumber

Then one obtains that $\ Q^{\mp\mp}_+ = 2 m^4_c,\ Q^{\mp\pm}_+ = 2 m^4_s, \ Q^{\mp\mp}_- = Q^{\mp\pm}_- = 0.$
As a result, the set of cross sections for incoherent $\chi A$ scattering due to interaction of scalar currents
with $\chi$-particle helicity unchanged (first formula) and helicity flip (second formula)
according to the formula in (<ref>),
and the total $\chi A$ cross section (last line) are as follows:

\begin{eqnarray}\nonumber \label{eq:42chiA-CrossSectionsEvaluations-Scalar-ChiEta-nonrel-InCohCS}
\frac{d\sigma^{\pm\pm}_\text{inc}(\bm{q})}{d T_A}  &=& c_A   \sum_{f=p,n} A^f  \big[1-F^2_f(\bm{q})\big] [c_S^f]^2 m^4_c   = \dfrac{G^2_F  m_A }{4\pi}\cos^2\frac{\theta}{2}  \dfrac{m^2_\chi  }{|\bm{k}^l_\chi|^2}
\sum_{f=p,n} A^f  [c_S^f]^2\big[1-F^2_f(\bm{q})\big]
, \\ \nonumber
\frac{d\sigma^{\pm\mp}_\text{inc}(\bm{q})}{d T_A} &=& c_A   \sum_{f=p,n}A^f \big[1-F^2_f(\bm{q})\big]  [c_S^f]^2m^4_s = \dfrac{G^2_F  m_A }{4\pi}\sin^2\frac{\theta}{2}  \dfrac{m^2_\chi  }{|\bm{k}^l_\chi|^2}
\sum_{f=p,n} A^f [c_S^f]^2 \big[1-F^2_f(\bm{q})\big]
\dfrac{d\sigma^{\text{total}}_\text{inc}}{d T_A}&=&\dfrac12 \sum_{s',s} \dfrac{d\sigma^{s's}_\text{inc}}{d T_A}= \dfrac{G^2_F  m_A }{4\pi}  \dfrac{m^2_\chi  }{|\bm{k}^l_\chi|^2} \sum_{f=p,n} A^f [c_S^f]^2 \big[1-F^2_f(\bm{q})\big]

Here relation (<ref>) is considered, and we take $g_\text{i} \simeq 1$.
Formulas (<ref>) and
give a general form of the total set of expressions for cross sections of the $\chi A$ scattering due to the interaction
of the scalar leptonic and scalar nucleon currents in the nonrelativistic approximation.
The total (experimentally measured) cross section of this interaction is expressed in terms
of the sum of the coherent and incoherent cross sections

\begin{eqnarray}\label{eq:42chiA-CrossSectionsEvaluations-Scalar-ChiEta-nonrel-Full}
\dfrac{d\sigma^{\text{scalar}}_{\text{nonrel}}}{d T_A}(\chi A \to \chi A^*)&=&\dfrac12\sum^{}_{s's}\dfrac{d\sigma^{s's}_\text{inc}}{d T_A} + \dfrac12\sum^{}_{s's}\dfrac{d\sigma^{s's}_\text{coh}}{d T_A}
=\\&=& \dfrac{G^2_F  m_A }{4\pi}  \dfrac{m^2_\chi  }{|\bm{k}^l_\chi|^2}\Big\{\sum_{f=p,n} A^f [c_S^f]^2 \big[1-F^2_f(\bm{q})\big]+ \big[ \sum^{}_{f=p,n} F_{f}(\bm{q}) {A_f} c_S^{f} \big]^2 \Big\}
\nonumber. \end{eqnarray}

On the assumption of identity of nuclear proton and neutron form factors, $F_{p}(\bm{q})=F_{n}(\bm{q})\equiv F(\bm{q})$,
and isoscalar character of the scalar interaction, $c_S^n=c_S^p\equiv c_S^{}$,
simple formulas for measured cross sections are obtained, which involve terms proportional
to $A^2=[A_p+A_n]^2$ and $A=A_p+A_n$, respectively,

\begin{eqnarray*}\nonumber
\dfrac{d\sigma^{\text{total}}}{d T_A}(\chi A \to \chi A^*)&=&
\dfrac{G^2_F  m_A }{4\pi}c_S^2  \dfrac{m^2_\chi  }{|\bm{k}^l_\chi|^2}
\Big\{ F^2(\bm{q}) A^2 + [1-F^2(\bm{q})] A  \Big\}
,\\ \dfrac{d\sigma^{\mp\mp}}{d T_A}(\chi A \to \chi A^*)&=&\cos^2\frac{\theta}{2}
\dfrac{d\sigma^{\text{total}}}{d T_A}(\chi A \to \chi A^*)
, \\ \dfrac{d\sigma^{\mp\pm}}{d T_A}(\chi A \to \chi A^*)&=&\sin^2\frac{\theta}{2}
\dfrac{d\sigma^{\text{total}}}{d T_A}(\chi A \to \chi A^*)

The ratio between the total (differential) cross sections for the incoherent
(<ref>) and coherent
$\chi A$ scattering due to scalar interaction in the nonrelativistic approximation has a simple form

\begin{eqnarray}\label{eq:43chiA-CrossSectionsEvaluations-Scalar-ChiEta-nonrel-InCoh-2-Coh-total-ratio}
R_{\rm scalar}(m_A, T_A)\equiv  \dfrac{\dfrac{d\sigma^{\text{total}}_\text{inc}}{d T_A}(\chi A\to \chi A^{*})}{\dfrac{d\sigma^{\text{total}}_\text{coh}}{d T_A}(\chi A\to \chi A^{})} &=&\dfrac{1-F^2(\bm{q})}{F^2(\bm{q})A}

This ratio is obviously equal to unity, i.e., the incoherent and coherent contributions to the cross section
are equal when $F^2(\bm{q})=(A+1)^{-1}.$
For example, in the case of a hydrogen target ($A=1$),  a “drop” to $1/2$ for
the square of the nuclear form factor is sufficient for the scalar-isoscalar
coherent contribution to the cross section to become equal to the incoherent contribution.

§.§.§ Axial vector  ${\chi A}$ interaction

By axial vector (or axial) interaction is meant interaction of a $\chi$ particle with nucleons that leads to the
following form of scalar products (<ref>):

\begin{eqnarray}\label{eq:43chiA-CrossSection-via-ScalarProducts-Axial-ScalarProduct-deff}
(l^a_{s's}, h^{a,f}_{r'r})\!=\!\chi^{}_{A} \overline{u}_\chi (s',k')\gamma_\mu\gamma_5 u_\chi (s,k)
h^f_{A} \overline{u}_f (r',p') \gamma^\mu\gamma_5 u _f (r,p)
\!=\!c_{A}^f \sum^{}_{\mu,\nu}\!A^\mu_{s's}(\bm{k}')g_{\mu\nu}A^\nu_{r'r}(\bm{p}') ,\quad
\end{eqnarray}

where the effective coupling constants $\ c_{A}^f \equiv \chi^{}_{A} h^f_{A}$ specify intensity of this interaction.
All necessary scalar products are given in Section <ref>.
In the nonrelativistic approximation, combinations
(<ref>)  of scalar products of the axial
$\chi A$ interaction from (<ref>)
involved in the coherent cross sections are

\begin{eqnarray*}
\hat Q^{\mp\mp}_{+}&=&  (l^a_{\mp\mp},h^{a}_{++}) + (l^a_{\mp\mp},h^{a}_{--}) = \pm m^2_c +(\mp m^2_c) = 0
,\\ \hat Q^{\mp\mp}_{-}&=& (l^a_{\mp\mp},h^{a}_{++})- (l^a_{\mp\mp},h^{a}_{--}) = \pm m^2_c - (\mp m^2_c) = \pm 2 m^2_c
,\\  \hat Q^{\mp\pm}_{+}&=& (l^a_{\mp\pm},h^{a}_{++}) + (l^a_{\mp\pm},h^{a}_{--})=  m^2_s e^{\pm i \phi}
+ (-  m^2_s e^{\pm i \phi}) = 0
,\\  \hat Q^{\mp\pm}_{-}&=& (l^a_{\mp\pm},h^{a}_{++})- (l^a_{\mp\pm},h^{a}_{--})=  m^2_s e^{\pm i \phi} - (-  m^2_s e^{\pm i \phi}) = 2 m^2_s e^{\pm i \phi}

Hence, for all possible lepton helicity combinations we have four quantities from

\begin{eqnarray*}
Q_f^{\mp\mp} &=& \pm  2 m^2_c\frac{\Delta A_f}{A_f}  \quad
\text{and}\quad Q_f^{\mp\pm}=  2 m^2_s e^{\pm i \phi} \frac{\Delta A_f}{A_f}
. \end{eqnarray*}

As a result, coherent $\chi A$ cross sections (<ref>)
due to interaction of axial currents with the
$\chi$-particle helicity unchanged (upper formula) and helicity flip (middle formula),
and the total coherent cross section (lower formula) in the nonrelativistic approximation take the form

\begin{eqnarray}\nonumber\label{eq:43chiA-CrossSectionsEvaluations-Axial-ChiEta-nonrel-CohCS}
\frac{d\sigma^{\mp\mp}_\text{coh}(\bm{q})}{g_\text{c} d T_A}&=&
c_A m^4_c \Big[\sum^{}_{f=p,n} c_{A}^f F_{f} \Delta A_f  \Big]^2 = \cos^2\frac{\theta}{2}
\dfrac{ G^2_F  m_A}{4\pi}\dfrac{m^2_\chi }{|\bm{k}^l_\chi|^2 }
\big[c_{A}^p F_p(\bm{q}) \Delta A_p +c_{A}^n F_n(\bm{q}) \Delta A_n \big]^2
, \\  \frac{d\sigma^{\mp\pm}_\text{coh}(\bm{q})}{g_\text{c} d T_A}&=& \sin^2\frac{\theta}{2}
\dfrac{ G^2_F  m_A}{4\pi}\dfrac{m^2_\chi }{|\bm{k}^l_\chi|^2 }
\big[ c_{A}^p F_p(\bm{q}) \Delta A_p +c_{A}^n F_n(\bm{q}) \Delta A_n \big]^2
,\\. \dfrac{d\sigma^{\text{axial}}_\text{coh}(\bm{q})}{g_\text{c} d T_A}&=&\dfrac12 \sum\dfrac{d\sigma^{s's}_\text{coh}}{g_\text{c} d T_A}
= \dfrac{ G^2_F  m_A}{4\pi}\dfrac{m^2_\chi }{|\bm{k}^l_\chi|^2 }  \big[\sum^{}_{f=p,n} c_{A}^f \Delta A_f  F_{f}(\bm{q}) \big]^2
\nonumber .\end{eqnarray}

In the formula for incoherent cross sections
(<ref>) and
in the nonrelativistic approximation there only remains dependence on the following squares of scalar products:

\begin{eqnarray*}
|(l^a_{++}\, h^a_{--})|^2&=&|(l^a_{--}\, h^a_{++})|^2 =
|(l^a_{++}\, h^a_{++})|^2 = |(l^a_{--} h^a_{--}) |^2 = m^4_c
,\\|(l^a_{++}\, h^a_{-+})|^2 &=&|(l^a_{--}\, h^a_{+-})|^2 =
|(l^a_{+-}\, h^a_{+-})|^2 = |(l^a_{-+}\, h^a_{-+})|^2 = 0
, \\|(l^a_{+-}\, h^a_{--})|^2 &=&|(l^a_{-+}\, h^a_{++})|^2=
|(l^a_{+-}\, h^a_{++})|^2 =|(l^a_{-+}\, h^a_{--})|^2  = m^4_s
, \\ |(l^a_{++}\, h^a_{+-})|^2&=&|(l^a_{--}\, h^a_{-+})|^2  = 4 m^4_s
,\qquad  |(l^a_{+-}\, h^a_{-+})|^2 = |(l^a_{-+}\, h^a_{+-})|^2 = 4 m^4_c
, \end{eqnarray*}

based on which expressions (<ref>)
take the form

\begin{eqnarray*}
Q^{\mp\mp}_+ &=& 2 (m^4_c + 2 m^4_s), \quad Q^{\mp\pm}_+ = 2(2 m^4_c+  m^4_s),
\quad Q^{\mp\mp}_- = \pm 4 m^4_s,  \quad Q^{\mp\pm}_-  = \mp 4 m^4_c

As a result, cross sections of the incoherent $\chi A$ interaction
due to axial currents with no change in the $\chi$-particle helicity (first formula)
and helicity flip (second formula) and the total $\chi A$ cross section (last formula)
in the nonrelativistic approximation are expressed as follows:

\begin{eqnarray} \nonumber\label{eq:43chiA-CrossSectionsEvaluations-Axial-ChiEta-nonrel-InCohCS}
\frac{d\sigma^{\mp\mp}_\text{inc}(\bm{q})}{d T_A}&=&
\dfrac{ G^2_F  m_A}{4\pi}\dfrac{m^2_\chi }{|\bm{k}^l_\chi|^2 } g_\text{i}
\sum_{f=p,n} [c^f_A]^2 A_f \big[1-F^2_f(\bm{q})\big]
\Big[1+\sin^2\frac{\theta}{2} \pm \sin^2\frac{\theta}{2} \dfrac{2\Delta A_f }{A_f } \Big]
, \\ \frac{d\sigma^{\mp\pm}_\text{inc}(\bm{q})}{d T_A}&=&
\dfrac{ G^2_F  m_A}{4\pi}\dfrac{m^2_\chi }{|\bm{k}^l_\chi|^2 } g_\text{i}
\sum_{f=p,n} [c^f_A]^2 A_f\big[1-F^2_f(\bm{q})\big]
\Big[1+\cos^2\frac{\theta}{2}\mp \cos^2\frac{\theta}{2} \dfrac{2\Delta A_f }{A_f } \Big]
,\qquad \\ \dfrac{d\sigma^{\text{axial}}_\text{inc}}{d T_A} &=&
\dfrac12 \sum_{s',s} \dfrac{d\sigma^{s's}_\text{inc}}{d T_A}=
\dfrac{ G^2_F  m_A}{4\pi}\dfrac{m^2_\chi }{|\bm{k}^l_\chi|^2 }
g_\text{i} \sum_{f=p,n} [c^f_A]^2 A_f\big[1-F^2_f(\bm{q})\big] \, 3
\nonumber .\end{eqnarray}

The experimentally measured total cross section of the $\chi A$ scattering due to interaction of pure
axial leptonic and nucleon currents has the form of a sum of two terms

\begin{eqnarray}\label{eq:43chiA-CrossSectionsEvaluations-Axial-ChiEta-nonrel-Full}
\dfrac{d\sigma^{\text{axial}}_{\text{nonrel}}}{d T_A}(\chi A \to \chi A^*)&=&\dfrac12\sum^{}_{s's}\dfrac{d\sigma^{s's}_\text{inc}}{d T_A} + \dfrac12 \sum^{}_{s's}\dfrac{d\sigma^{s's}_\text{coh}}{d T_A}
=\\&=& \dfrac{G^2_F  m_A }{4\pi}  \dfrac{m^2_\chi  }{|\bm{k}^l_\chi|^2}
\Big\{3 g_\text{i}  \sum_{f=p,n} [c^f_A]^2  A_f\big[1-F^2_f(\bm{q})\big]
+  g_\text{c} \big[\sum^{}_{f=p,n} c_{A}^f \Delta A_f  F_{f}(\bm{q}) \big]^2 \Big\}
\nonumber. \end{eqnarray}

The ratio of the axial total cross sections for the incoherent and coherent $\chi A$ scattering
in the nonrelativistic approximation is

\begin{eqnarray}\label{eq:43chiA-CrossSectionsEvaluations-Axial-ChiEta-nonrel-InCoh-2-Coh-total-ratio}
\dfrac{\dfrac{d\sigma^{\text{total}}_\text{inc}}{d T_A}(\chi A\to \chi A^{*})}{\dfrac{d\sigma^{\text{total}}_\text{coh}}{d T_A}(\chi A\to \chi A)}
&=&3 \dfrac{g_\text{i}}{g_\text{c}}
\dfrac{\sum_{f=p,n} [c^f_A]^2  A_f\big[1-F^2_f(\bm{q})\big]}{\big[\sum^{}_{f=p,n} c_{A}^f \Delta A_f  F_{f}(\bm{q}) \big]^2 }
\simeq \dfrac{ 3A [1-F^2(\bm{q})]}{F^2(\bm{q})  \big[ \Delta A_p +\Delta A_n \big]^2}
\end{eqnarray}

The last approximation in this formula corresponds to the assumption that nuclear proton and neutron form factors are identical,
i.e., $F_{p}(\bm{q})=F_{n}(\bm{q})\equiv F(\bm{q})$,
and the axial vector interaction under consideration is isoscalar, i.e., $c_A^n=c_A^p$
(and also $g_\text{i}\simeq g_\text{c}$).
It is evident from relation (<ref>)
that in the case of pure axial interaction of leptonic and nucleon currents in the nonrelativistic approximation,
the inelastic (incoherent) cross section can be considerably larger than the elastic (coherent) one,
except probably for very light nuclei with a large spin, when $\Delta A_f\simeq A_f$.
The coherent cross section completely vanishes for spin-zero nuclei.
The quantity $\Delta \equiv \Delta A_p +\Delta A_n$
actually plays the role of the total nuclear spin, thus governing the effect of coherence.
Therefore, it is only in the region of (quite) small transfers where the form factor is little different from
unity that the inelastic cross section can be considered suppressed.
Even when $|F_{f}(\bm{q})|^2 \simeq 1/2 $, the
inelastic cross section dominates over the elastic one by almost two orders of magnitude.
Concluding this section, we note that unlike the case in the nonrelativistic approximation
in the relativistic case, coherent $\chi A$ cross sections for pure axial interaction do not disappear for spin-zero nuclei
(more specifically, when $\Delta A_f=0$).

§.§.§ Weak ${\chi A}$ interaction

Scalar products $(l^w_{s's}, h^{w}_{r'r})$ corresponding to the interaction of a massive
neutral $\chi$ particle with a nucleon in the neutral weak current channel were given by expression
In the nonrelativistic approximation, they are expressed by formulas
According to the definition of the coherent $\chi A$ scattering cross sections
it is necessary to calculate auxiliary quantities (<ref>),
which, in this approximation, are as follows:

\begin{eqnarray*}
\hat Q^{\mp\mp}_{+}&=& (l^w_{\mp\mp},h^{w}_{++}) + (l^w_{\mp\mp},h^{w}_{--})
=  m^2_c (\alpha \pm \delta) + m^2_c (\alpha \mp \delta) = + 2 m^2_c \alpha
\hat Q^{\mp\mp}_{-}&=& (l^w_{\mp\mp},h^{w}_{++}) - (l^w_{\mp\mp},h^{w}_{--})
= m^2_c (\alpha \pm \delta) - m^2_c (\alpha \mp \delta) = \pm  2 m^2_c \delta
\hat Q^{\mp\pm}_{+}&=& (l^w_{\mp\pm},h^{w}_{++})+ (l^w_{\mp\pm},h^{w}_{--})
= \mp m^2_s e^{\pm i\phi} (\alpha\mp \delta) + (\mp m^2_s  e^{\pm i\phi}  (\alpha\pm \delta ))
= \mp 2 m^2_s e^{\pm i\phi} \alpha
\hat Q^{\mp\pm}_{-}&=& (l^w_{\mp\pm},h^{w}_{++})- (l^w_{\mp\pm},h^{w}_{--})
= \mp m^2_s e^{\pm i\phi} (\alpha\mp \delta) -(\mp m^2_s  e^{\pm i\phi}  (\alpha\pm \delta )) = +2  m^2_s e^{\pm i\phi} \delta

Then for four auxiliary quantities from (<ref>)
one has

\begin{eqnarray*}
Q_f^{\mp\mp}&=& \hat Q^{\mp\mp}_{+} + \dfrac{\Delta A_f}{A_f} \hat Q^{\mp\mp}_{-}
= 2 m^2_c \alpha_f +\dfrac{\Delta A_f}{A_f} (\pm  2 m^2_c \delta_f)
= 2 m^2_c \Big[\alpha_f \pm   \delta_f \dfrac{\Delta A_f}{A_f} \Big]
,\\ Q_f^{\mp\pm}&=&\hat Q^{\mp\pm}_{+} + \dfrac{\Delta A_f}{A_f} \hat Q^{\mp\pm}_{-}
= \mp 2 m^2_s e^{\pm i\phi} \alpha_f  +\dfrac{\Delta A_f}{A_f} 2  m^2_s e^{\pm i\phi} \delta_f
= \mp 2 m^2_s e^{\pm i\phi} \Big[\alpha_f  \mp \delta_f \dfrac{\Delta A_f}{A_f} \Big]

As a result, based on (<ref>),
we obtain a set of coherent $\chi A$ cross sections for the nonrelativistic interaction of weak currents

\begin{eqnarray}\nonumber\label{eq:43chiA-CrossSections-via-ScalarProducts-Weak-nonrel-CohSC-general}
\frac{d\sigma^{\mp\mp}_\text{coh}(\bm{q})}{g_\text{c} d T_A}&=&  \cos^2\frac{\theta}{2}
\dfrac{G^2_F  m_A}{4\pi}\dfrac{ m^2_\chi }{|\bm{k}^l_\chi|^2 }
\Big[ \sum^{}_{f=p,n} F_{f}(\bm{q}) {A_f} \Big(\alpha_f \pm   \delta_f \dfrac{\Delta A_f}{A_f} \Big)\Big]^2
\frac{d\sigma^{\mp\pm}_\text{coh}(\bm{q})}{ g_\text{c} d T_A}&=&\sin^2\frac{\theta}{2}
\dfrac{G^2_F  m_A}{4\pi} \dfrac{m^2_\chi}{ |\bm{k}^l_\chi|^2 }
\Big[ \sum^{}_{f=p,n} F_{f}(\bm{q}) A_f \Big( \alpha_f  \mp \delta_f \dfrac{\Delta A_f}{A_f} \Big)  \Big]^2
\dfrac{d\sigma^{\text{total}}_\text{coh}(\bm{q})}{ g_\text{c} d T_A}&=&  \dfrac12 \sum^{}_{s's}\dfrac{d\sigma^{s's}_\text{coh}}{ g_\text{c} d T_A} =  %  g_\text{c}
\dfrac{G^2_F  m_A}{4\pi}\dfrac{m^2_\chi}{|\bm{k}^l_\chi|^2 }
\Big(\big[\sum^{}_{f=p,n}\alpha_f A_f F_{f}(\bm{q}) \big]^2+ \big[\sum^{}_{f=p,n}\delta_f {\Delta A_f} F_{f}(\bm{q}) \big]^2  \Big)
\nonumber

The last formula in (<ref>)
corresponds to the total coherent $\chi A$ cross section
(averaged over the initial lepton helicities and summed over the final helicities).

It is evident from formulas (<ref>)
that the common factor $\cos^2\dfrac{\theta}{2}\simeq 1-\dfrac{T_A}{T_A^{\max}}$ from the first formula
typical of coherent (anti)neutrino scattering corresponds to the case where helicity of the incident
$\chi$ particle remains unchanged after interaction (which is always the case for neutrinos and antineutrinos).
When the $\chi$-particle helicity changes (which is possible owing to the nonzero mass), in the second formula
there arises another common factor $\sin^2\dfrac{\theta}{2}\simeq \dfrac{T_A}{T_A^{\max}}$,
which cancels out $\cos^2\dfrac{\theta}{2} $
in the total coherent cross section (last formula) in
negating the kinematic dependence of the common factor of the total coherent cross section
on the external variable $T_A$, which occurs in this place.
The following “simplifications” of formulas (<ref>)
for coherent $\chi A$ cross sections are possible.
(i) Nuclear form factors of protons and neutrons are identical, i.e., $F_p(\bm{q})=F_n(\bm{q})=F(\bm{q})$.

\begin{eqnarray}\nonumber\label{eq:43chiA-CrossSections-via-ScalarProducts-Weak-nonrel-CohSC-Fp=Fn}
\frac{d\sigma^{\mp\mp}_\text{coh}(\bm{q})}{g_\text{c} d T_A}&=&
\dfrac{G^2_F  m_A}{4\pi}\dfrac{ m^2_\chi }{|\bm{k}^l_\chi|^2 }\cos^2\frac{\theta}{2}  F^2(\bm{q})
\Big[ \sum^{}_{f=p,n} {A_f} \Big(\alpha_f \pm   \delta_f \dfrac{\Delta A_f}{A_f} \Big)\Big]^2
,\\ \frac{d\sigma^{\mp\pm}_\text{coh}(\bm{q})}{g_\text{c} d T_A}&=&
\dfrac{G^2_F  m_A}{4\pi} \dfrac{m^2_\chi}{ |\bm{k}^l_\chi|^2 }  \sin^2\frac{\theta}{2} F^2(\bm{q})
\Big[ \sum^{}_{f=p,n} A_f \Big( \alpha_f  \mp \delta_f \dfrac{\Delta A_f}{A_f} \Big)  \Big]^2
,\\  \nonumber
\dfrac{d\sigma^{\text{total}}_\text{coh}(\bm{q})}{g_\text{c} d T_A}&=&  \dfrac12 \sum^{}_{s's}\dfrac{d\sigma^{s's}_\text{coh}}{g_\text{c} d T_A} =   \dfrac{G^2_F  m_A}{4\pi}\dfrac{m^2_\chi}{|\bm{k}^l_\chi|^2 }  F^2(\bm{q})
\Big(\big[\sum^{}_{f=p,n}\alpha_f A_f \big]^2+ \big[\sum^{}_{f=p,n}\delta_f {\Delta A_f}\big]^2  \Big)
=\\ &=&   \dfrac{G^2_F  m_A}{4\pi}\dfrac{m^2_\chi}{|\bm{k}^l_\chi|^2 }  F^2(\bm{q})
\Big[ \sum^{}_{f=p,n} A_f^2 \big\{\alpha^2_f  +\delta^2_f \dfrac{\Delta A^2_f}{A_f^2}\big\}
+2A_p A_n \big\{\alpha_p \alpha_n  +\delta_p \delta_n \dfrac{\Delta A_p}{A_p} \dfrac{\Delta A_n}{A_n} \big\}   \Big] \nonumber

The target nucleus has a zero total spin (more specifically, $\Delta A_f=0$)
Then nonrelativistic weak coherent $\chi A$ cross sections  (at $\Delta A_f=0$)
have the form

\begin{eqnarray} \nonumber \label{eq:43chiA-CrossSections-via-ScalarProducts-Weak-nonrel-CohSC-spinless}
\frac{d\sigma^{\mp\mp}_\text{coh}(\bm{q})}{g_\text{c} d T_A}&=&\cos^2\frac{\theta}{2}
\dfrac{d\sigma^{\text{total}}_\text{coh}(\bm{q})}{g_\text{c} d T_A}
, \quad  \frac{d\sigma^{\mp\pm}_\text{coh}(\bm{q})}{g_\text{c} d T_A}=  \sin^2\frac{\theta}{2}
\dfrac{d\sigma^{\text{total}}_\text{coh}(\bm{q})}{g_\text{c} d T_A}
, \quad \text{where }\\ \dfrac{d\sigma^{\text{total}}_\text{coh}(\bm{q})}{g_\text{c} d T_A}&=&
\dfrac{G^2_F  m_A}{4\pi} \dfrac{m^2_\chi}{ |\bm{k}^l_\chi|^2 } [\alpha_p A_p F_p (\bm{q}) +\alpha_n A_n F_n (\bm{q})  ]^2

It thus follows that in the nonrelativistic case only the “vector component” of the weak current
proportional to parameters $\alpha_f$ “works” in the coherent $\chi A$ cross section on the nucleus with the zero total spin
while dependence on $\delta_f$ entirely disappears.
Note also that in the nonrelativistic approximation there is no contribution from interference
of pure vector and pure axial vector components of the weak interaction.
To calculate the nonrelativistic incoherent cross section for the $\chi A$ scattering due to the weak interaction,
one should use general expressions (<ref>),  where
the key factors $Q^{s's}_\pm$ from (<ref>)
are defined in terms of squares of scalar products of the weak current in the nonrelativistic approximation
With these squares of scalar products, the factors $Q^{s's}_\pm$  are as follows:

\begin{eqnarray*}
Q^{\mp\mp}_+  &=&  \sum^{}_{r'r=\pm} |(l_{\mp\mp}, h^{f}_{r'r})|^2
= 2(4m_\chi m)^2\Big[\cos^2\frac{\theta}{2}\alpha^2 + (1+\sin^2\frac{\theta}{2})\delta^2\Big]
,\\Q^{\mp\pm}_+ &=& \sum^{}_{r'r=\pm} |(l_{\mp\pm}, h^{f}_{r'r})|^2
= 2(4m_\chi m)^2\Big[\sin^2\frac{\theta}{2}\alpha^2 + (1+\cos^2\frac{\theta}{2} )\delta^2\Big]
,\\ Q^{++}_- &=& - Q^{--}_-  = \sum^{}_{r'=\pm} |(l_{++}, h^{f}_{r'+})|^2-\sum^{}_{r'=\pm} |(l_{++}, h^{f}_{r'-})|^2
= - 4 \delta (4m_\chi m)^2 \Big[\cos^2\frac{\theta}{2} \alpha  +  \sin^2\frac{\theta}{2}\delta \Big]
\\ Q^{+-}_- &=& - Q^{-+}_- = \sum^{}_{r'=\pm} |(l_{+-}, h^{f}_{r'+})|^2-\sum^{}_{r'=\pm} |(l_{+-}, h^{f}_{r'-})|^2
= 4 \delta (4m_\chi m)^2 \Big[\sin^2\frac{\theta}{2} \alpha  +  \cos^2\frac{\theta}{2}\delta \Big]

As a result, after substitution of the above expressions into
the cross sections of the incoherent $\chi A$ scattering in the nonrelativistic approximation take the form

\begin{eqnarray} \nonumber \label{eq:43chiA-CrossSections-via-ScalarProducts-Weak-nonrel-InCohSC-general}
\frac{d\sigma^{\mp\mp}_\text{inc}(\bm{q})}{g_\text{i} d T_A}&=&
\dfrac{ G^2_F  m_A}{4\pi}\dfrac{m^2_\chi }{|\bm{k}^l_\chi|^2} \sum_{f=p,n}\big[1-F^2_f(\bm{q})\big] A_f
\times\\&&\times  \nonumber
\Big\{\cos^2\frac{\theta}{2}\alpha^2_f + (1+\sin^2\frac{\theta}{2})\delta^2_f
\pm\dfrac{2\Delta A_f }{A_f}\delta_f\Big[\cos^2\frac{\theta}{2} \alpha_f +\sin^2\frac{\theta}{2}\delta_f \Big]\Big\}
, \\   \frac{d\sigma^{\mp\pm}_\text{inc}(\bm{q})}{g_\text{i} d T_A}&=&
\dfrac{ G^2_F m_A}{4\pi}\dfrac{m^2_\chi }{|\bm{k}^l_\chi|^2}\sum_{f=p,n}\big[1-F^2_f(\bm{q})\big]  A_f
\times\\&&\times \nonumber
\Big\{\sin^2\frac{\theta}{2}\alpha^2_f + (1+\cos^2\frac{\theta}{2} )\delta^2_f\mp\dfrac{2\Delta A_f}{A_f }\delta_f\Big[\sin^2\frac{\theta}{2} \alpha_f+ \cos^2\frac{\theta}{2}\delta_f\Big]\Big\}
,\\  \frac{d\sigma^{\text{total}}_\text{inc}(\bm{q})}{g_\text{i} d T_A} &=&\frac12 \sum^{}_{s's} \frac{d\sigma^{s's}_\text{inc}}{g_\text{i} d T_A}   = \dfrac{ G^2_F m_A}{4\pi}\dfrac{m^2_\chi }{|\bm{k}^l_\chi|^2}
\sum_{f=p,n} A^f \big[1-F^2_f(\bm{q})\big]  \big[\alpha^2_f + 3 \delta^2_f \big]
\nonumber   .\end{eqnarray}

It is seen that there are not any “interference” terms proportional to the weak-current coupling constants
$\beta$ and $\gamma$.
Only dependence on the pure vector and pure axial vector coupling constants $\alpha$
and $\delta$  “survives”.
Further, in the total incoherent $\chi A$ cross section, dependence on the nuclear spin factor $\Delta A_f$
disappears even it is not zero.
Note that for $\alpha_f=0$ and $\delta_f=1$
(i.e., in the absence of the vector contribution) these expressions entirely coincide with the similar formulas for pure axial-axial interactions (<ref>).
The experimentally measured total cross section for the $\chi A$ scattering due to interaction of weak
leptonic and nucleon currents in the nonrelativistic approximation is

\begin{eqnarray}\label{eq:43chiA-CrossSections-via-ScalarProducts-Weak-nonrel-Full}
\dfrac{d\sigma^{\text{weak}}_{\text{nonrel}}}{d T_A}(\chi A \to \chi A^*)&=&
\dfrac{G^2_F  m_A }{4\pi}  \dfrac{m^2_\chi  }{|\bm{k}^l_\chi|^2}
\Big\{g_\text{i} \sum_{f=p,n} A^f \big[1-F^2_f(\bm{q})\big]  \big[\alpha^2_f + 3 \delta^2_f \big]
\\&& +g_\text{c} \Big(\big[\sum^{}_{f=p,n}\alpha_f A_f F_{f}(\bm{q}) \big]^2+ \big[\sum^{}_{f=p,n}\delta_f {\Delta A_f} F_{f}(\bm{q}) \big]^2  \Big)\Big\}
\nonumber. \end{eqnarray}

Let us consider two “simplifications” of the incoherent $\chi A$ cross sections
(i) Nuclear form factors of neutrons and protons are identical.
Unlike the coherent case, here it is not much of simplification, since dependence on the form factor is simply taken
outside the sign of summation over $f$, and the following formulas are obtained:

\begin{eqnarray} \nonumber  \label{eq:43chiA-CrossSections-via-ScalarProducts-Weak-nonrel-InCohSC-Fp=Fn}
\frac{d\sigma^{\mp\mp}_\text{inc}(\bm{q})}{g_\text{i} d T_A}&=&
\dfrac{ G^2_F  m_A}{4\pi}\dfrac{m^2_\chi }{|\bm{k}^l_\chi|^2}[1-F^2(\bm{q})]  \sum_{f=p,n}A_f
\times\\&&\times  \nonumber
\Big\{\cos^2\frac{\theta}{2}\alpha^2_f + (1+\sin^2\frac{\theta}{2})\delta^2_f
\pm\dfrac{2\Delta A_f }{A_f}\delta_f\Big[\cos^2\frac{\theta}{2} \alpha_f +\sin^2\frac{\theta}{2}\delta_f \Big]\Big\}
, \\  \frac{d\sigma^{\mp\pm}_\text{inc}(\bm{q})}{g_\text{i} d T_A}&=&
\dfrac{ G^2_F m_A}{4\pi}\dfrac{m^2_\chi }{|\bm{k}^l_\chi|^2}[1-F^2(\bm{q})]   \sum_{f=p,n}A_f
\times\\&&\times \nonumber
\Big\{\sin^2\frac{\theta}{2}\alpha^2_f + (1+\cos^2\frac{\theta}{2} )\delta^2_f\mp\dfrac{2\Delta A_f}{A_f }\delta_f\Big[\sin^2\frac{\theta}{2} \alpha_f+ \cos^2\frac{\theta}{2}\delta_f\Big]\Big\}
,\\   \frac{d\sigma^{\text{total}}_\text{inc}(\bm{q})}{g_\text{i} d T_A} &=&
\dfrac{ G^2_F m_A}{4\pi}\dfrac{m^2_\chi }{|\bm{k}^l_\chi|^2}[1-F^2(\bm{q})]
\sum_{f=p,n} A^f \big[\alpha^2_f + 3 \delta^2_f \big]
\nonumber   .\end{eqnarray}

(ii) If the target nucleus has a zero total spin,  $\Delta A_f =0$, the formulas are as follows:

\begin{eqnarray} \nonumber  \label{eq:43chiA-CrossSections-via-ScalarProducts-Weak-nonrel-InCohSC-spinless}
\frac{d\sigma^{\mp\mp}_\text{inc}(\bm{q})}{g_\text{i} d T_A}&=&
\dfrac{ G^2_F  m_A}{4\pi}\dfrac{m^2_\chi }{|\bm{k}^l_\chi|^2} \sum_{f=p,n}\big[1-F^2_f(\bm{q})\big] A_f
\big\{\cos^2\frac{\theta}{2}\alpha^2_f + (1+\sin^2\frac{\theta}{2})\delta^2_f\big\}
, \\   \frac{d\sigma^{\mp\pm}_\text{inc}(\bm{q})}{g_\text{i} d T_A}&=&
\dfrac{ G^2_F m_A}{4\pi}\dfrac{m^2_\chi }{|\bm{k}^l_\chi|^2}\sum_{f=p,n}\big[1-F^2_f(\bm{q})\big]  A_f
\big\{\sin^2\frac{\theta}{2}\alpha^2_f + (1+\cos^2\frac{\theta}{2} )\delta^2_f\big\}
,\\  \frac{d\sigma^{\text{total}}_\text{inc}(\bm{q})}{g_\text{i} d T_A} &=&
\dfrac{ G^2_F m_A}{4\pi}\dfrac{m^2_\chi }{|\bm{k}^l_\chi|^2}\sum_{f=p,n} A^f \big[1-F^2_f(\bm{q})\big]  \big[\alpha^2_f + 3 \delta^2_f \big]
\nonumber   .\end{eqnarray}

The ratio of the total nonrelativistic cross sections for the incoherent
(<ref>) and coherent
$\chi A$ scattering due to the weak interaction is

\begin{eqnarray}\label{eq:43chiA-CrossSectionsEvaluations-Weak-ChiEta-nonrel-InCoh-2-Coh-total-ratio}
\dfrac{\dfrac{d\sigma^{\text{total}}_\text{inc}}{d T_A}(\chi A\to \chi A^{*})}{\dfrac{d\sigma^{\text{total}}_\text{coh}}{d T_A}(\chi A\to \chi A^{})}=  \dfrac{g_\text{i}  [1-F^2(\bm{q})]  \sum_{f=p,n} A^f \big[\alpha^2_f + 3 \delta^2_f \big]}
{g_\text{c} F^2(\bm{q})\Big[ \sum^{}_{f=p,n} A_f^2 \big\{\alpha^2_f  +\delta^2_f \dfrac{\Delta A^2_f}{A_f^2}\big\}
+2A_p A_n \big\{\alpha_p \alpha_n  +\delta_p \delta_n \dfrac{\Delta A_p}{A_p} \dfrac{\Delta A_n}{A_n} \big\} \Big]}

This ratio “goes into” a pure axial case of the $\chi A$ interaction
(when $\alpha_f=0$, see (<ref>))
and a pure scalar case of the $\chi A$ interaction (when $\delta_f=0$, see
coinciding with a pure vector case in the nonrelativistic approximation.
This section concludes the main part of this work.
Here expressions are obtained for the cross sections of the elastic and inelastic scattering
of the massive neutral (weakly interacting) $\chi$ particle off the nucleus $\chi A\to \chi A^{(*)}$
when the latter retains its integrity
(i.e., at $\chi$ particle energies that are usually below 100 MeV).
The pure scalar, axial vector, and generalized weak interaction of the $\chi$ particle with nucleons
of the nucleus is considered in the nonrelativistic approximation.
Practically, these new formulas are of interest for correctly understanding the balance of coherence and
incoherence in problems of the direct search for galactic dark-matter particles
when the signature of the desired events is the acts of interaction of these particles with nuclear targets.
This very issue is investigated in a separate work <cit.>.
Theoretically, new expressions for the scattering of massive leptons are of interest as
a generalization of the above formulas
to the case of coherent and incoherent scattering of massless neutrinos and antineutrinos from nuclei.

§ NUMERICAL ESTIMATIONS AND DISCUSSIONS OF ${\CHI}$ PARTICLE–NUCLEUS SCATTERING

A total set of expressions for cross sections of the coherent and incoherent scattering due to weak
${\chi A}$ interaction in the nonrelativistic approximation is given by formulas
and (<ref>).
For numerical estimation, we will use the expression for the experimentally measured
$\chi A$ scattering cross section, which is a sum of two cross sections averaged over the initial and
summed over the final $\chi$-lepton spin projections
[For simplicity, we consider in all further calculations that   ${g_\text{i}} \simeq {g_\text{c}} \simeq 1$.]:

\begin{eqnarray}\nonumber\label{eq:50chiA-Results-CrossSection-Weak-nonrel-Full}
\dfrac{d\sigma^{\text{weak}}_{\text{nonrel}}}{d T_A}(\chi A \to \chi A^*)&=&
\dfrac12 \sum^{}_{s's} \frac{d\sigma^{s's}_\text{inc}}{d T_A}(\chi A\to \chi A^{*}) +
\dfrac12 \sum^{}_{s's}\dfrac{d\sigma^{s's}_\text{coh}}{d T_A}(\chi A\to \chi A^{})
=\\&=& \dfrac{G^2_F  m_A }{4\pi}  \dfrac{m^2_\chi  }{|\bm{k}^l_\chi|^2}\Big\{\big[1-F^2(\bm{q})\big]  \sum_{f=p,n} A_f  \big[\alpha^2_f + 3 \delta^2_f \big]
\\&&\qquad +F^2(\bm{q})\Big[ \big[\sum^{}_{f=p,n}\alpha_f A_f \big]^2+ \big[\sum^{}_{f=p,n}\delta_f {\Delta A_f} \big]^2  \Big]\Big\}
\nonumber. \end{eqnarray}

We will use the form of the Helm nuclear form factor <cit.> for protons and neutrons:

$$F_{p/n}(\bm{q})=F(\bm{q})  = 3 \dfrac{j_1(qR_0)}{qR_0}e^{-\dfrac{(s q)^2}{2}},
\quad  j_1(x) = \dfrac{\sin x}{x^2} - \dfrac{\cos x}{x},  \ s = 0.9 \text{~fm}, \  R_0 = 1.14 A^{1/3} \text{~fm},$$

and the following designation and values
[Constant quantities:  $1 \text{~fm} =  \dfrac{1}{0.1973 \text{~GeV} } =  \dfrac{1}{197.3 \text{~MeV}}$,
$\mbox{~GeV} = 10^3 \mbox{~MeV}, \  \mbox{MeV} = 1$.]:

$$ \bm{q}^2 \simeq 2 m_A T_A,  \quad
G_{\text{F}}=1.166 378 \times 10^{-5}/\mbox{GeV}^2 \text{~~and~~}m = 0.938  \mbox{~GeV} \simeq 1 \mbox{~GeV}.

According to the definition of the scalar product for weak currents of the general form

\begin{equation}\label{eq:50chiA-Results-WeakScalarProdict-Definotion}
(l^w_{s's}, h^{w,f}_{r'r}) = \alpha_f (l^v_{s's}\, h^v_{r'r}) + \beta_f (l^v_{s's}\, h^a_{r'r})
+ \gamma_f (l^a_{s's}\, h^v_{r'r})  +  \delta_f  (l^a_{s's}\, h^a_{r'r}),
\end{equation}

the effective coupling constants having the form

$$\alpha_f = \chi_V h^f_V,\quad  \beta_f = \chi_V h^f_A, \quad \gamma_f = \chi_A  h^f_V ,\quad \delta_f = \chi_A h^f_A,

are free parameters. In the SM, for neutrinos there is $\chi_V  = - \chi_A = 1$, and for nucleons there is

h_V^p=g_V^p =  \frac12 -2 \sin^2\theta_W, \quad h_V^n=g_V^n = - \frac12;
\quad h_A^p = g_A^p =   \frac{g_A}{2}, \quad h_A^n = g_A^n =  - \frac{g_ A}{2}.

Then the effective parameters in the SM are

$$\alpha_f = \chi_V h^f_V  = + g_V^f, \quad \delta_f = \chi_A h^f_A = + g_A^f,
\quad \gamma_f = \chi_A  h^f_V = - g_V^f, \quad  \beta_f = \chi_V h^f_A = - g_A^f .$$

Considering that $\sin^2\theta_W = 0.23865$ and $g_A = 1.27$,
the numerical values obtained in the SM are

\begin{eqnarray}\nonumber \label{eq:50chiA-Results-SM-alpha+delta}
\alpha_p  &=&  + \frac12 -2 \sin^2\theta_W  = - \gamma_p  \simeq 0.02
,\quad \delta_p  =  \frac{g_A}{2} = - \beta_p  \simeq 0.64
; \\  \alpha_n  &=& - \frac12  = - \gamma_n  = - 0.5
,\qquad \qquad\qquad \delta_n  =  - \frac{g_ A}{2} = - \beta_n  \simeq  - 0.64
. \end{eqnarray}

In the nuclear rest frame, calculations of cross sections and their ratios involve only variation of the
kinetic (detected) nuclear recoil energy $T_A$ in the interval  from $T_A^{\min}= \epsilon_A^{\text{thre}} > 0$
to a certain maximal value $T_A^{\max}$ for this nucleus,
which is determined by the condition of the square of the nuclear form factor $|F(T_A)|^2$
becoming zero.
The energy of the $\chi$ particle incident on the nucleus (at rest)  $T_0 = \dfrac{|\bm{k}^l_\chi|^2}{2 m_\chi}$,
or the square of the momentum $|\bm{k}^l_\chi|^2$,
which enters into  (<ref>)
and (<ref>),
is obtained from the relation between $T_0$ and $T_A$
through the $\chi$ particle escape angle $\theta$ in the lab frame

\begin{eqnarray}\label{eq:50chiA-Kinematics-CosT-from-k2prime-and-T_A}
\cos\theta(T_A) &=&   \dfrac{m_\chi( 2 T_0- \Delta\varepsilon_{mn})  - T_A (m_\chi +m_A)}{2 m_\chi\sqrt{ T_0 (T_0-\Delta\varepsilon_{mn}-T_A)}}

The maximum $T_A$  is achieved when the incident $\chi$ particle is “reflected” strictly in the opposite
direction, i.e., when  $\cos\theta(T_A)=-1$.
Then, from this condition there arises the minimum value for the elastic case (when $\Delta\varepsilon_{mn}=0$)

\begin{equation}\label{eq:50chiA-Kinematics-T_0-from-T_A}
T_0= T_A  \dfrac{(m_\chi+m_A)^2} { 4m_\chi m_A }  = T_A  \dfrac{(1+r)^2} { 4r }
, \quad \text{where}\quad  r\equiv  \dfrac{m_\chi}{m_A}
, \end{equation}

at which this value of $T_A$  (maximum efficiency of momentum–energy transfer) is still possible.
The ratio of the total nonrelativistic cross sections for the incoherent
(<ref>) and coherent
$\chi A$ scattering due to weak interaction is as follows:

\begin{eqnarray}\nonumber \label{eq:50chiA-Results-Inc2CohRatio-Weak}
R^{\text{total}}_w(A,\bm{q}) &\equiv&
\dfrac{\dfrac{d\sigma^{\text{total}}_\text{inc}}{d T_A}(\chi A\to \chi A^{*})}{\dfrac{d\sigma^{\text{total}}_\text{coh}}{d T_A}(\chi A\to \chi A^{})}
=\dfrac{ [1-F^2(\bm{q})]  \sum_{f=p,n} A^f \big[\alpha^2_f + 3 \delta^2_f \big]}{F^2(\bm{q})\Big[
\big[\sum^{}_{f=p,n}\alpha_f A_f \big]^2+ \big[\sum^{}_{f=p,n}\delta_f {\Delta A_f} \big]^2 \Big]}
=\\&=& A R_A(\bm{q}) \dfrac{ A^p \big[\alpha^2_p+ 3 \delta^2_p \big] + A^n \big[\alpha^2_n + 3 \delta^2_n \big]}
{\big[\alpha_p A_p+\alpha_n A_n \big]^2+ \big[\delta_p {\Delta A_p}+\delta_n {\Delta A_n} \big]^2}

Here the following designation for the ratio of the pure “nuclear parts” of
formula <ref>)  is introduced:

\begin{equation} \label{eq:50chiA-Results-Inc2CohRatio-A}
R_A(\bm{q})=R_A(T_A)\equiv \dfrac{1-F^2_A(\bm{q}) }{A F^2_A(\bm{q})}=\dfrac{1-F^2_A(T_A) }{A F^2_A(T_A)}.
\end{equation}

These quantities together with the square of the corresponding form factors and the modulus
of the transferred momentum are shown in
Fig. <ref>  as a function of the nuclear recoil energy $T_A$.

“Nuclear” ratios $R_{A}(T_A)$ (<ref>) (top row),
squares of nuclear form factors $|F_A(T_A)|^2$  (middle row), and absolute values of the momentum transferred
to the nucleus $|\bm{q}|$  (bottom row) for a few target nuclei as a function of nuclear recoil energy $T_A$.

It is seen in Fig. <ref>  that the square of the nuclear form factor  $|F(T_A)|^2$
monotonically decreases with increasing energy transfer to the nucleus (kinematic nuclear recoil energy) $T_A$.
Once $|F(T_A)|^2$  is sufficiently close to zero, the “coherence effect” (together with the coherent contribution
to the total cross section) weakens fast and totally disappears when
$|F(T_A)|^2\simeq 0$
[As was already mentioned, for each of the nuclei this moment determines $T_A^{\max}$
Tmax in the corresponding plots.].
Simultaneously, the ratio $R_{A}(T_A)$  rapidly increases, which actually means a
transition to total domination of the “incoherence effect”
(together with the incoherent contribution to the total cross section).
For different nuclei this occurs at different $T_A$  and $q$ that are,
roughly speaking, inversely proportional to the atomic number of the nucleus $A$.
This is how the nucleus “works” in the “pure form,” regulating the transition from coherence to incoherence.
The second factor in (<ref>) written in the form

\begin{eqnarray}\label{eq:50chiA-Results-Inc2CohRatio-WeakCharge}
R_I(A,\alpha,\delta) &\equiv& (A_p+A_n) \dfrac{ A^p (\alpha^2_p+ 3 \delta^2_p) + A^n (\alpha^2_n + 3 \delta^2_n )}
{(\alpha_p A_p+\alpha_n A_n)^2+ (\delta_p {\Delta A_p}+\delta_n {\Delta A_n} )^2}

accumulates the entire dependence on the character of interaction of the $\chi$ lepton with nucleons.
It is determined by the balance of the weak-current coupling constants $\alpha_{p/n}$
and $\delta_{p/n}$, “weighted” with the proton-neutron structure of a particular nucleus.
It can both enhance and weaken the “pure” nuclear structure effect shown in Fig.  <ref>.
Its SM values for a few target nuclei are listed in Table <ref>.

$^A$Nucleus$(Z,N)$    $Z_+$    $ Z_-$    $\Delta Z$     $N_+$     $N_-$    $\Delta N$      Spin
${}^{4}\text{He}(2,2)$    1     1     0     1     1     0           23.44

${}^{12}\text{C}(6,6)$    3     3     0     3     3     0           23.44

${}^{19}\text{F}(9,10)$    5    4    1     5     5     0        $\frac12=\frac{ \Delta Z}{2}$    20.71

${}^{40}\text{Ar}(18,22)$     9     9     0     11     11     0         19.22

${}^{74}\text{Ge}(32,42)$     16     16     0     21     21     0          18.00

${}^{127}\text{I}(53,74)$   27   26   1   39   35   4   $\frac{5}{2}=\frac{\Delta Z}{2}+\frac{\Delta N}{2}$   17.01

${}^{129}\text{Xe}(54,75)$   27   27   0   38   37   1    $\frac{1}{2} =\frac{\Delta Z}{2}+\frac{\Delta N}{2}$    17.13

${}^{131}\text{Xe}(54,77)$   27   27   0   40   37   3   $\frac{3}{2}=\frac{\Delta Z}{2}+\frac{\Delta N}{2}$     16.72

${}^{133}\text{Cs}(55,78)$   28   27   1   46   32   6   $\frac{7}{2}=\frac{\Delta Z}{2}+\frac{\Delta N}{2}$     16.07

Characteristics of some target nuclei and their corresponding quantities $R_I(A,\alpha,\delta)$
with $\alpha_{p/n},\delta_{p/n}$ within the SM from (<ref>).
Here $Z=A_p,\ N=A_n$,  $\Delta A_p \equiv \Delta Z = Z_+ - Z_-$, and $ \Delta A_n \equiv \Delta N= N_+ - N_-$.
The subscripts $\pm$ indicate direction of nucleon spins relative to a chosen quantization axis.

Illustrated dependence of the kinematic factors  $\rho$ and $\mu_A$ on the ratio

The influence of the “nuclear charge” factor (<ref>)
from Table  <ref>, which enhances the “incoherence effect” more than tenfold,
is demonstrated in Fig <ref>,
(upper panel of two plots), where ratios  (<ref>)  for the
effective coupling constants $\alpha_{p/n}$ and  $\delta_{p/n}$ from the SM are shown.

Ratio $ R^{\text{total}}_{\text{SM}}(A,T_A)$ (<ref>)
of the total nonrelativistic cross sections for the incoherent and coherent $\chi A$ scattering due to weak
SM interaction with the parameters from (<ref>) (top row),
squares of nuclear form factors $|F(T_A)|^2$ (middle row),
and measured total differential cross sections  (<ref>)
(bottom row) for a few target nuclei as a function of the recoil energy $T_A$.

Dependences of the nuclear form factors on $T_A$ in the middle panel of Fig. <ref>
are given as “reference plots.”
They allow judging the maximum value of  $T_A$ for a particular nucleus.
Two lower pots in Fig. <ref>
illustrate $T_A$ dependence of the measured total cross sections
(<ref>) calculated by the formula

\begin{eqnarray}\label{eq:50chiA-Results-CrossSection-Weak-nonrel-via-Coh+R}
\dfrac{d\sigma^{\text{weak}}_{\text{nonrel}}}{\rho d T_A}(\chi A \to \chi A^*)&=&
\dfrac{G^2_F  m_A^2 }{2\pi  T_A^{\max} }
\big\{ F^2(T_A)\big[(\alpha_p A_p + \alpha_n A_n)^2+ (\delta_p \Delta A_p +\delta_n \Delta A_n)^2 \big] \quad
\\&&\qquad \qquad
+(1-F^2(T_A))\big[ A^p (\alpha^2_p+ 3 \delta^2_p) + A^n (\alpha^2_n + 3 \delta^2_n )\big]\big\}
\nonumber
,  \end{eqnarray}

where $\rho \equiv  \dfrac{r^2 }{(1+r)^2}$ (see. Fig. <ref>)
considering the definition $r=\dfrac{m_\chi}{m_A}$ from (<ref>),
“hides” the entire kinematic dependence on the $\chi$-lepton mass.
Indeed, the right-hand side of (<ref>)
does not depend on $m_\chi$.
The explicit form of this expression is due to the fact that the calculation of cross sections
(110) requires knowing not only $T_A$ but also $T_0$,  whose minimum value for the particular  $T_A$
is given by formula (<ref>).
Therefore, in expression (<ref>),
the key dimensional factor of all cross sections from (<ref>)

\dfrac{G^2_F  m_A }{4\pi}  \dfrac{m^2_\chi  }{|\bm{k}^l_\chi|^2} = \dfrac{G^2_F  m_A }{4\pi}  \dfrac{m_\chi  }{2 T_0}=\dfrac{G^2_F  m_A^2 }{2\pi T_A } \rho

involves one universal one universal $T_0^{\min 4\max}$, that allows achieving the maximum value $T_A^{\max}$ presented in the
plots and defined from the condition $F_A(T_A^{\max})\simeq 0$:

\begin{equation}\label{eq:50chiA-Kinematics-T_0min-from-T_A}
T_0^{\min 4\max}=  \dfrac{T_A^{\max}}{\mu_A}, \quad \text{where}\quad  \mu_A \equiv  \dfrac{ 4r }{  (1+r)^2}
\quad \text{(see. Fig.~\ref{fig:50chiA-SimplePolot-r})}
. \end{equation}

This choice of the initial energy $T_0$ guaranteeing
achievability of $T_A^{\max}$ seems quite sufficient for illustrating the behavior of $\chi A$-cross sections as a function of  $T_A$.
From Fig. <ref>,
obtained for the SM effective coupling constants (<ref>)
at the (auxiliary simplifying) condition (<ref>),
one can see the following.
When squares of nuclear form factors of heavy nuclei decay to zero with increasing kinetic nuclear recoil energy $T_A$
(coherence is almost entirely lost, right middle panel),
the measured total cross section (sum of coherent and incoherent ones, right bottom panel)
decreases by about an order of magnitude and is saturated mainly by the incoherent contribution (top right panel).
Note also that in the case of these very nuclei, in already the half of the $T_A$
interval (at 40 keV) the incoherent contribution reaches the coherent one as
the measured total cross section decreases by about a factor of five, which seems insignificant.
The situation is quite different for light nuclei (Fig. <ref>).
Here “coherence” is preserved long enough (the lighter the nucleus, the longer).

Ratio $ R^{\text{total}}_{\text{SM}}(A,T_A)$ (<ref>)
of the total nonrelativistic cross sections for the incoherent and coherent $\chi A$ scattering due to weak
SM interaction with the parameters from (<ref>)
(top row), squares of nuclear form factors $|F(T_A)|^2$ (middle row), and measured
total differential cross sections  (<ref>)
(bottom row) for a few light target nuclei as a function of the recoil energy $T_A$.

Squares of nuclear form factors decay slowly, reaching the region $F_A(T_A)\simeq 0$
at rather high recoil energies $T_A$ (the lighter the nucleus, the higher $T_A^{\max}$).
On the one hand, these high $T_A$ should probably be readily observable, but on the
other hand, they could already be incompatible with preservation of nuclear integrity ($\bm{q}\ge 300$ MeV/$c$)
The total cross sections are noticeably smaller than for heavy nuclei and vary weakly.
An intermediate situation is for nuclei like Ga–Ge ($A\simeq 70$).
It is similar in terms of the character of dependence on $T_A$ to the case
of heavy nuclei with the main difference being the
magnitude of the recoil energy. Here it is noticeably higher ($T_A^{\max}\simeq 200~$keV).
Figures <ref> and <ref>
illustrate the main property of the approach
as the nuclear recoil energy $T_A$ increases, a smooth, well-controlled transition takes
place from domination of elastic $\chi A$ interaction to domination of inelastic
$\chi A$ interaction in nonrelativistic scattering of a massive $\chi$ particle off a nucleus.
Since the results of the experiments on the search for dark matter are usually interpreted in terms of
spin-independent and spin-dependent cross sections of interaction of a dark-matter particle with nucleons, we
will consider in more detail consequences of the approach
<cit.>  for these two types of interaction.
Scalar, or spin-independent
in terms of the direct search for dark matter, interaction in the nonrelativistic
approximation has the same form as a pure vector interaction, when only the effective coupling constants
$\alpha_p$ and $\alpha_n$ remain different from zero in the general formula for the scalar product
Then both axial coupling constants should be set equal to zero, $\delta_p=\delta_n=0$,
in expression (<ref>),
which, after a change of notation  $a_p \equiv \alpha_p$ and $a_n \equiv \alpha_n$,
takes the form

\begin{eqnarray} \label{eq:50chiA-Results-Inc2CohRatio-Scalar}
R^{\text{total}}_{\text{scalar}}(A,T_A) &=&R_A(T_A) (A_p+A_n)\dfrac{A_p a^2_p+ A_n a^2_n}{(A_p a_p+A_n a_n  )^2}

In the isoscalar case, where $a_p= a_n$, dependence on these constants is completely cancelled,
and this formula takes the form of expression  (<ref>):
$R^{\text{total}}_{\text{scalar}}(A,T_A) =  R_A(T_A)$.
It is shown in Fig. <ref>
for different nuclei with the corresponding total cross sections.

Ratio $ R^{\text{total}}_{\text{Scalar}}(A,T_A)$
(<ref>)  of the total nonrelativistic cross sections for the incoherent and $\chi A$ coherent  scattering due to weak scalar interaction when $a_p= a_n$,
(dependence on the square of these parameters is preserved only in the cross section) (top row)
and measured total differential cross sections (<ref>)
(bottom row) for a few target nuclei as a function of the recoil energy $T_A$.

If one of the parameters is much smaller than the other, as is, for example, in the SM,
where $\alpha_p \ll \alpha_n$ (or for simplicity, $\alpha_p \simeq 0$), then

$$ R^{\text{total}}_{\text{scalar}}(A,T_A)= R_A(T_A) \dfrac{A_p+A_n}{A_n}=\dfrac{1-F^2_A(T_A) }{A_n F^2_A(T_A)}, $$

which does not depend on $\alpha_n $ either and only differs from $R_A(T_A)$ by the factor of about 2.
In the anti-isoscalar case, where $a_n= - a_p$ and $p =\dfrac{A_n}{A_p}$, one obtains the following:

$$ R^{\text{total}}_{\text{scalar}}(A,T_A)=R_A(T_A)(A_p+A_n)\dfrac{A_p a^2_p+ A_n a^2_p}{(A_p a_p-A_n a_p  )^2}=R_A(T_A)\Big(\dfrac{p+1}{p-1}\Big)^2.

It is evident from Table <ref>  that in this case there is not any coherent contribution for the lightest nuclei, since the number of protons coincides with the number of neutrons, and $p=1$.
Further, the heavier the target nucleus, the larger $p$, and the smaller the difference
of $R^{\text{total}}_{\text{scalar}}(A,T_A)$ from $R_A$ presented in
Fig. <ref>.

Ratio $ R^{\text{total}}_{\text{Scalar}}(A,T_A)$
(<ref>) of the total nonrelativistic cross sections for the incoherent and coherent $\chi A$ scattering due to a weak scalar interaction when $a_p= - a_n$  (top row)
and measured total differential cross sections  (<ref>) (bottom row) for a few target nuclei as a function of the nuclear recoil energy $T_A$.

Figures <ref> and
<ref> reveal the “main trend”: as $T_A$
increases, the “content” of the total measured cross section smoothly changes —
the elastic (coherent) contribution is replaced by the inelastic (incoherent) one.
The cross section itself decreases, but by no more than an order of magnitude, which, to our mind, barely
reduces the probability of its measurement (detection).
That is, the total expected number of events to be detected falls within the capability limits of a typical experiment.
However, these events change their origin.
If a detector is set up to detect only elastic scattering, it grows incapable of sensing anything at all as
$T_A$ increases (elastic processes become fewer and fewer in number).
At the same time, the number of inelastic processes noticeably increases, while being undetectable
by the detector (say, because of its inability to detect photons from nuclear deexcitation).
A situation arises where desired interactions occur but the detector cannot see them.
Note, that contrary to the statements in
<cit.> and <cit.>
[Where it is stated that the scalar interaction is noticeably weaker
than the spin-dependent one, because the initial and final
nuclear states change, and only the spin-dependent interaction
should be considered.],
the scalar $\chi A$ interaction in the considered version may well make a noticeable
contribution to inelastic $\chi A$ scattering.
Let us turn to the pure axial vector $\chi A$ interaction
referred to as spin-dependent in terms of the direct search for dark matter.
Then, the proton and neutron vector effective coupling constants in general formulas
(<ref>) and (<ref>)  are $\alpha_p=\alpha_n=0$,  and formula (<ref>)  takes the form

\begin{eqnarray} \label{eq:50chiA-Results-Inc2CohRatio-Axial}
R^{\text{total}}_{\text{axial}}(A,T_A)  &=& 3 R_A(T_A)(A^p + A^n)
\dfrac{A^p \delta^2_p + A^n \delta^2_n} {(\delta_p \Delta A_p + \delta_n \Delta A_n)^2}
. \end{eqnarray}

From the above and from formula (<ref>)
it is evident that for spin-zero nuclei (more specifically, when $\Delta A_{p/n}=0$)
there is no coherent contribution to the total cross section from the pure axial vector interaction,
and this ratio loses its meaning.
Since the axial constants are connected by the relation $\delta_n=-\delta_p$,
in the SM, the ratio

$$  R^{\text{total}}_{\text{SM, axial}}(A,T_A)= \dfrac{3 R_A(T_A) A^2} {(\Delta A_p - \Delta A_n)^2}
=\dfrac{1-F^2_A(T_A) }{F^2_A(T_A)}\dfrac{3 A} {(\Delta A_p - \Delta A_n)^2}

does not depend on the axial coupling constant $\delta_p$ and,