††institutetext: Dzhelepov Laboratory of Nuclear Problems, JINR, 141980,
Dubna, Russia

# On underestimation of the inelastic interactions
in the direct dark matter search

V.A. Bednyakov<EMAIL_ADDRESS>
###### Abstract

In the paper explicit expressions are obtained for the event rates expected in
experiments aimed at direct detection of dark matter (DM) particles. These
expressions allow one to estimate the rates taking into account simultaneously
elastic (coherent) and inelastic (incoherent) channels of DM particle
interaction with target nuclei. The nonzero nuclear excitation energies are
used in the calculation of the inelastic scattering contributions. A strong
correlation between the excitation energy and the recoil energy of the excited
nucleus limits the possibility of the inelastic channel detection with a
number of nuclei. Together with the standard model of the DM distribution in
the Galaxy some other models are considered, which allow higher velocities of
the DM particle. As the recoil energy of the nucleus, $T_{A}$, increases, the
dominance of the elastic interaction channel is smoothly replaced by the
dominance of the inelastic one. Therefore, if a DM detector is set up to
detect only elastic scattering events, it starts to lose capability of
“seeing” anything. The only way to ”notice” the interaction remains the
$\gamma$ radiation from the deexcitation of the target nucleus. In the case of
spin-independent DM interaction, as $T_{A}$ increases, the inelastic
contribution quickly dominates, while the observable differential event rate
decreases quite insignificantly. If the DM particle interacts with (zero-spin)
nuclei only spin-dependently, the detectors focused on registration of the
elastic spin-dependent DM signal will see nothing, since the entire signal
”goes through” the inelastic channel. It looks like the desired DM interaction
could have a noticeable intensity, but the DM detector is unable to detect it.
Therefore, a setup aimed at the direct DM detection should register two
signals. The first is the nuclear recoil energy and the second is the
$\gamma$-quanta with a certain energy from the target nucleus deexitation. The
experiment will provide the complete information about the DM interaction.

##  1 Direct detection of galactic dark matter particles

Over almost a century, the presence of dark matter (DM) in the outer space
around us not only has been well confirmed by a variety of astrophysical
observations Bringmann:2012ez ; Sofue:2015xpa ; Feldmann:2013hqa ;
Madhavacheril:2014slf ; Feng:2010gw ; Famaey:2015bba ; Safdi:2023wsk , but
also has become one of the most intriguing problem of fundamental physics
Cebrian:2022brv ; Bernabei:2020mon ; Bernabei:2014maa ; ArkaniHamed:2008qn ;
Hoeneisen:2015rva ; Livio:2014gda . According to modern science, this extra,
non-luminous and non-baryonic matter, manifesting itself only gravitationally
so far, consists of particles yet unknown in the Standard Model (SM)
Bernabei:2022loo ; Slatyer:2021qgc ; Cooley:2021rws ; Bertone:2004pz ;
Drukier:1986tm ; Gelmini:2015zpa . For a long time one of the best candidates
for the DM particle was the so-called Weakly Interacting Massive Particle
(WIMP), since, rather weakly interacting and having a mass in the range from 1
GeV$/c^{2}$ to 1 TeV$/c^{2}$, it naturally satisfied the cosmological
requirements of the early universe. The search for WIMPs was carried out in
three main directions. The first is the direct detection, which is aimed at
registration of the DM particle interaction with ordinary matter (on the
Earth). The second is the indirect detection through detecting products of the
DM particle annihilation inside the outer space objects. The third is the
search for appearance of the DM candidates at modern accelerators with
sufficiently high energy and luminosity Bednyakov:2015uoa .

In the light of overwhelming recognition of the DM existence, the results of
all above-mentioned DM searches look very inconsistent. One the one hand, one
believes that WIMPs were not yet detected in these experiments Cebrian:2022brv
; Livio:2014gda . Indeed, during several decades intensive searches for the
WIMPs were carried out. Many restrictions were obtained on the cross section
of their interaction with nucleons, which were brought rather close to the
absolute minimum limit for GeV$/c^{2}$ DM masses, the so-called neutrino flour
Boehm:2018sux ; Papoulias:2018uzy ; Dai:2022lsj .

On the other hand, no one has yet been able to refute the results of the
DAMA/LIBRA collaboration. This is the only collaboration that has measured the
annual modulation signal corresponding to the presence of DM particles in our
Galaxy at the 13$\sigma$ confidence level Bernabei:2022loo ; Bernabei:2020mon
; Bernabei:2014maa ; Zurowski:2020dxe . The rotation of the Earth (with a DM
detector) around the Sun, when both of them move through the galactic DM
environment, causes very specific annual modulation of the intensity of the DM
interaction with nuclei Aboubrahim:2022lwb ; Kahlhoefer:2018knc ;
Bednyakov:2012cu ; Freese:1987wu . When the movement of the Earth is opposite
to the direction of the Sun, the relative speed of the DM particle and the
detector (and the intensity of their interaction) reaches a minimal value.
When the Earth moves in the same direction as the Sun, the relative speed and
the intensity of the DM-nucleus interaction are maximal. This phenomenon is
almost independent of the nature of the interaction. If the interaction is
possible, it is mainly determined by the distribution of the DM-particle
velocities in the Galaxy (more precisely, near the Earth). Although the
modulation component of the signal is noticeably smaller than the time-
averaged total signal of the interaction, its value can hardly be
overestimated. Nowadays it is the only available positive signature that is
caused by the galactic nature of the WIMPs. Only this signature and only
experiments aimed at the direct DM detection allow one to prove the presence
of DM in the space around us Bednyakov:2012cu .

In connection with the DAMA/LIBRA results, the following should be emphasized.
The modulation observed has the phase, amplitude and period very well
corresponding to our Galaxy DM. There is no other collaboration which is able
to observe this modulation at the same confidence level. The other above-
mentioned searches for DM particles are only able of discover a potential DM
candidate. To prove its belonging to the galactic DM population, one has to
observe the modulation signal induced right by this DM candidate.

Therefore, despite significant technical difficulties and serious systematic
uncertainties111One needs very accurate low-background and low-threshold
detectors, and strong protection against a variety of background processes.
One has very small interaction probability, and the low event rate. The
statistics must be collected over years. There are a lot of uncertainties in
the DM distribution both in the Galaxy and near the Earth. The interaction of
DM particles with one type of target nuclei does not guarantee any interaction
with other nuclei, for example, due to different nucleon and spin composition,
etc., direct DM experiments are inevitably important, because one is unable to
understand the nature of DM without them Spooner:2007zh ; Bednyakov:2015uoa ;
Bednyakov:2020njj ; Slatyer:2021qgc ; Cebrian:2022brv ; Aboubrahim:2022lwb ;
Lawrence:2022niq ; Cushman:2013zza ; Saab:2012th .

As a response to the long-term lack of positive results from the existing
experiments searching for the WIMP dark matter (excluding the DAMA/LIBRA),
many new, sometimes very exotic, alternative models and suggestions appeared
in the literature. They are concerning with compositions of DM
Nakayama:2019rhg ; Hurtado:2020vlj ; Du:2020ldo ; Baryakhtar:2020rwy ;
Majumdar:2021vdw ; Afek:2021vjy ; Lebedev:2023uzp ; Mohapatra:2023aei ;
Olle:2019kbo ; Belanger:2022gqc ; Dvali:2023xfz ; Kitajima:2023fun ;
Friedlander:2023jzw ; Mistele:2023wao ; Chiueh:2022krl ; Abbas:2023ion ;
Ralegankar:2023pyx ; Bai:2023mfi ; Marfatia:2022jiz ; Biswas:2023azl ;
Chu:2023zbo ; Arguuelles:2023ktd ; DeLuca:2023laa ; Hong:2022gzo ; Roy:2023zar
; Biggio:2023gtm ; Carney:2022gse , DM properties Giudice:2017zke ;
Zurowski:2020dxe ; Wang:2021jic ; Feng:2021hyz ; Emken:2021vmf ;
Granelli:2022ysi ; Filimonova:2022pkj ; Bell:2022yxn ; Freese:2023fcr ;
Baryakhtar:2022hbu ; DeRocco:2022rze ; Emken:2021lgc ; Proukakis:2023nmm ;
Biswas:2023eju ; Garcia:2023awt ; Das:2023vsq ; Maldonado:2023alg ;
Lee:2023hrg ; Arellano-Celiz:2019pax ; Wang:2023xgm , as well as very new and
sophisticated methods of the DM detection Tsuchida:2019hhc ; Coskuner:2021qxo
; Hamaide:2021hlp ; Boos:2022gtt ; Flambaum:2022zuq ; Fan:2022uwu ;
Blanco:2022cel ; Billard:2022cqd ; Araujo:2022wjh ; Li:2022acp ;
Badurina:2022ngn ; Endorsers:2022aie ; Chigusa:2023hms ; Ruzi:2023cvp ;
DarkSide-50:2022hin ; DarkSide:2022dhx ; Albakry:2023aby ; Saez:2023sjt ;
Kryemadhi:2022vuk ; Su:2022wpj ; Das:2022srn ; PALEOCCENE:2023rjj ;
ALPHA:2022rxj ; Delos:2023exh ; Catena:2023qkj ; Catena:2023awl ; Ruzi:2023mxp
; Ray:2023auh ; DuttaBanik:2023yxj ; Dhakal:2022rwn ; Kim:2020bwm ;
Leane:2020wob ; DarkSide-50:2023fcw .

In fact, the main interest of the DM-search community was shifted towards
”light” DM particles, whose masses are comparable with the masses of electrons
and/or nucleons Bardhan:2022ywd ; Krnjaic:2022ozp . The sensitivity to light
DM particles of detectors aimed at direct searches for WIMPs strongly
decreases with decreasing DM mass, because they can measure only the nuclear
recoil energy. For example, a DM particle with a mass of about 1 GeV/$c^{2}$
and a kinetic energy in the local Galaxy halo is not able to cause a nuclear
recoil above the threshold of 1 keV. Therefore one has the rapid decrease in
sensitivity Granelli:2022ysi . This problem with the light DM particles,
first, required new ideas and technologies for more sensitive detectors, at
least with significantly lower energy thresholds Billard:2022cqd ;
Proceedings:2022hmu ; Du:2020ldo ; EDELWEISS:2022ktt ; Edelweiss:2022bzh ;
EDELWEISS:2019vjv . Second, since one believes that the light DM particles can
effectively produce the recoil electrons with energy above their detection
threshold (about 0.186 keV for xenon XENON:2019gfn ), it was proposed to
register the light DM particles via their interactions with the atomic
electrons XENON:2019gfn ; PandaX-II:2021nsg ; Granelli:2022ysi . Third, this
problem has stimulated searches for possible ways to increase the kinetic
energy of the light DM particles.

One variant of the light DM energy increase is based on the accelerators with
high intensities, at which a sufficiently abundant production of the light DM
particles could be expected, for example, directly in beams of leptons or
protons, in the decays of mesons and baryons, by bremsstrahlung, etc.
Krnjaic:2022ozp ; Boos:2022gtt ; Kim:2017qdi . The other way is based on
”natural” mechanisms of almost relativistic DM particle production in the
modern universe. As a rule, one considers the streams of DM particles
accelerated in such a way as less intense, but more energetic. It became clear
that the light DM particles can be accelerated to (almost) relativistic
velocities due to their elastic interaction with the cosmic rays in the halo
of the Milky Way PhysRevLett.122.171801 ; Lei:2020mii ; Xia:2020apm ;
Cappiello:2019qsw ; Ema:2018bih ; Dent:2019krz ; PandaX-II:2021kai ;
Alvey:2022pad . A new DM particle acceleration mechanism, called the blazar-
boosted DM was proposed, where the DM particles can be significantly
accelerated and can have sufficiently large local densities due to their
interaction with high-energy proton jets of a blazar Wang:2021jic ;
Granelli:2022ysi ; Gondolo:1999ef . There are other mechanisms of the DM
particle acceleration, see for example, Agashe:2014yua ; Kouvaris:2015nsa ;
An:2017ojc ; Emken:2017hnp ; Granelli:2022ysi ; Xia:2022tid ; CDEX:2022fig ;
Bhowmick:2022zkj ; Bardhan:2022bdg ; Maity:2022exk ; Wang:2019jtk ;
Elor:2021swj ; Hu:2016xas ; TEXONO:2018nir . This allows one to overcome the
sensitivity limitations of the low-energy threshold detectors Berger:2022cab .

Furthermore, a new mechanism was proposed to explain how the light
nonrelativistic DM particles can produce detectable (nuclear) recoil energy.
It concerns the models of the dark sector with large enough mass difference
between the DM particles in the initial and final state Giudice:2017zke ;
Cui:2009xq ; Kang:2019uuj ; Bell:2021xff ; Bell:2022yxn ; Filimonova:2022pkj ;
Feng:2021hyz ; Baryakhtar:2020rwy ; Zurowski:2020dxe ; Smith-Orlik:2023kyl ;
Adams:2023pws . One believes that these energetic ”secondary” DM particles
will transfer sufficient energy to the nuclei and produce signals that exceed
the detection threshold. On this way one expects to lower the DM-mass limit to
a level of about 100 keV$/c^{2}$, which can be registered in the direct DM
experiment CDEX:2022fig .

In addition, it was found Lawrence:2022niq that the velocity distribution in
the galacto-centric reference frame has strong deviation from the Maxwell-
Boltzmann form. This indicates the presence of some high-speed DM substructure
with particle velocities as large as 800–1000 km/s with a noticeable increase
in the local density.

Summarizing, we emphasize the critical significance of the direct DM search
experiments. Let us stress that the range of kinetic energy of the DM
particles (with both small and large masses) is significantly expanded today
beyond limits of the standard DM halo model, which is based on the Maxwell-
Boltzmann distribution with the galaxy escape velocity of 540 km/s. Therefore,
taking into account the high importance of the topic Bednyakov:2015uoa ;
Bednyakov:2020njj ; Bednyakov:2012cu ; Feng:2022rxt , it seems premature to
consign to oblivion the traditional way of the direct DM detection without a
critical analysis of the generally accepted assumptions.

To this end, the first analysis was started in Bednyakov:2022dmc based on the
approach Bednyakov:2018mjd ; Bednyakov:2019dbl ; Bednyakov:2021ppn ;
Bednyakov:2021bty . It was shown that as the nuclear recoil energy $T_{A}$
increases, in the massive nonrelativistic $\chi$ particle scattering on a
nucleus a transition occures from dominance of the elastic $\chi A$
interaction to the dominance of the inelastic one. In this situation, as
$T_{A}$ increases, a device tuned only to the elastic $\chi A$ scattering
begins ”to go blind”, since the number of the elastic events strongly
decreases. They become gradually replaced by the inelastic events that the
device is not able to see. It was shown Bednyakov:2022dmc that this
”phenomenon” can strongly manifest itself in the direct DM search experiments,
whose results are interpreted in terms of the spin-independent and spin-
dependent interaction of the DM particle with nucleons. It sould be stressed
that both of these interactions are beyond the scope of the SM and may have a
completely unexpected character Bednyakov:2022dmc .

In addition, if the recoil energy of the nucleus, after the $\chi A$
scattering, turns out to be below the detection threshold, i.e.,
$T_{A}<T_{A}^{\min}$, then the elastic signal cannot be detected at all. For
this ”invisible” $T_{A}$, the only evidence of the $\chi A$ interaction is the
radiation from the nuclear deexcitation, i.e., the inelastic signal, although
its intensity at $T_{A}<T_{A}^{\min}$ can be an order of magnitude less than
the intensity of the elastic one Bednyakov:2022dmc . In general, it is not
possible to understand which process, elastic or inelastic, took place when
only the nuclear recoil energy $T_{A}$ is measured Bednyakov:2015uoa ;
Papoulias:2018uzy ; Boehm:2018sux ; Cooley:2021rws ; Cebrian:2022brv . In
other words, it is unclear whether the elastic or inelastic formula should be
used in data analysis.

Thus, the goal of this work is to study within formalism of Bednyakov:2018mjd
; Bednyakov:2019dbl ; Bednyakov:2021ppn ; Bednyakov:2022dmc some parameter
space (kinematic and physical) where the expected DM event rate caused by the
inelastic (incoherent) process $\chi A\to\chi A^{*}$ can compete with the
event rate caused by elastic (coherent) $\chi A\to\chi A$ scattering. Because
of the beyond-the-Standard-Model nature of the $\chi A$ interaction, this
parameter space could explain the ”blindness” of the DM detectors, intended
only for detection of the elastic DM scattering events.

##  2 Kinematics and cross section of ${\chi A}$ scattering

In the case of the interaction of two particles with the formation of two
particles,
$\chi(k)+A(P_{n})\rightarrow\chi(k^{\prime})+A^{(*)}(P^{\prime}_{m}),$ the
4-momenta of the incoming and the outgoing neutral massive lepton ($\chi$
particle) are denoted as $k=(k_{0}=E_{\chi},\bm{k})$ and
$k^{\prime}=(k^{\prime}_{0}=E_{\chi}^{\prime},\bm{k}^{\prime})$, and the
4-momenta of the initial and the final state of the nucleus are denoted,
respectively, as $P_{n}=(P^{0}_{n},\bm{P}_{n})$ and
$P^{\prime}_{m}=(P^{0}_{m},\bm{P}_{m})$ (Fig. 1, left). The total energy of
the state $|P_{n}\rangle$ is $P_{n}^{0}=E_{\bm{P}}+\varepsilon_{n}$, where
$\varepsilon_{n}$ is the internal energy of the $n$th quantum level of the
nucleus.

Figure 1: An example of $\chi A$-interactions due to the exchange of the
neutral $Z$-boson (left). Kinematics of this process in the laboratory frame
where the nucleus $A$ is at rest (right).

If the $\chi$ particle with mass $m_{\chi}$ and momentum $\bm{k}$ hits the
nucleus $A$ at rest along the $z$-axis and flies away at angle $\theta$ to the
$x$-axis with momentum $\bm{k}^{\prime}$, the 4-momenta can be written as

$\displaystyle k$ $\displaystyle=$
$\displaystyle\big{(}k_{0}=\sqrt{m^{2}_{\chi}+|\bm{k}|^{2}},0,0,k_{z}=|\bm{k}|\big{)},\qquad
P_{n}=\big{(}P_{n}^{0}=m_{A}+\varepsilon_{n},0,0,0\big{)},$ $\displaystyle
k^{\prime}$ $\displaystyle=$
$\displaystyle\big{(}k^{\prime}_{0}=\sqrt{m^{2}_{\chi}+|\bm{k}^{\prime}|^{2}},k^{\prime}_{x}=|\bm{k}^{\prime}|\sin\theta,0,k^{\prime}_{z}=|\bm{k}^{\prime}|\cos\theta\big{)},$
$\displaystyle P^{\prime}_{m}$ $\displaystyle=$
$\displaystyle\big{(}P_{m}^{0}=\varepsilon_{m}+\sqrt{m_{A}^{2}+(P_{m}^{x})^{2}+(P_{m}^{z})^{2}},P_{m}^{x},0,P_{m}^{z}\big{)}=$
$\displaystyle=$
$\displaystyle\big{(}P_{m}^{0}=\varepsilon_{m}+\sqrt{m_{A}^{2}+\bm{q}^{2}},-|\bm{k}^{\prime}|\sin\theta,0,|\bm{k}|-|\bm{k}^{\prime}|\cos\theta\big{)},$

where $m_{A}$ is the mass of the target nucleus, and $\varepsilon_{m}$ is the
excitation energy of the $m$th level of the nucleus. The 4-momentum
$q=(q_{0},\bm{q})$ transferred to the nucleus is related to these quantities
as follows:

$\displaystyle q^{2}$ $\displaystyle\equiv$
$\displaystyle(k-k^{\prime})^{2}=2\big{(}m^{2}_{\chi}-(kk^{\prime})\big{)}=2\big{(}m^{2}_{\chi}-\sqrt{(m^{2}_{\chi}+|\bm{k}^{\prime}|^{2})(m^{2}_{\chi}+|\bm{k}|^{2})}+|\bm{k}||\bm{k}^{\prime}|\cos\theta\big{)},$
$\displaystyle q_{0}$ $\displaystyle=$ $\displaystyle
k_{0}-k^{\prime}_{0}=P^{0}_{m}-P^{0}_{n}=\Delta\varepsilon_{mn}+T_{A},$ (1)
$\displaystyle\bm{q}^{2}$ $\displaystyle=$
$\displaystyle(\bm{k}-\bm{k}^{\prime})^{2}=(-|\bm{k}^{\prime}|\sin\theta)^{2}+(|\bm{k}|-|\bm{k}^{\prime}|\cos\theta)^{2}=|\bm{k}|^{2}+|\bm{k}^{\prime}|^{2}-2|\bm{k}||\bm{k}^{\prime}|\cos\theta.$

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

$T_{A}=\sqrt{m^{2}_{A}+\bm{q}^{2}}-m_{A}.$ (2)

The energy conservation law, i.e., $k_{0}+P_{n}^{0}=k^{\prime}_{0}+P_{m}^{0}$,
can be rewritten in the form

$k_{0}+m_{A}-\Delta\varepsilon_{mn}=\sqrt{m^{2}_{\chi}+|\bm{k}^{\prime}|^{2}}+\sqrt{m^{2}_{A}+|\bm{k}^{\prime}|^{2}+|\bm{k}|^{2}-2|\bm{k}||\bm{k}^{\prime}|\cos\theta},$
(3)

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

$\Delta\varepsilon_{mn}\equiv\varepsilon_{m}-\varepsilon_{n}.$ (4)

From expression (3) it follows that

$T_{A}(|\bm{k}^{\prime}|,\cos\theta)=k_{0}-k^{\prime}_{0}-\Delta\varepsilon_{mn}=\sqrt{m^{2}_{\chi}+|\bm{k}|}-\sqrt{m^{2}_{\chi}+|\bm{k}^{\prime}|^{2}}-\Delta\varepsilon_{mn}.$
(5)

In the Standard Halo Model Bertone:2004pz ; Drukier:1986tm ; Freese:1987wu ;
Lewin:1996rx , the DM-particle velocity near the Earth is estimated at
approximately $10^{-3}$ of the speed of light,
$|\bm{v}|={|\bm{k}|}/{m_{\chi}}\simeq 10^{-3}c\simeq 350~{}$km/s, with the
maximal (escape) velocity to be about 540 km/sec. As mentioned in the
introduction, the maximal DM velocity can be as large as 1000 km/sec.
Nevertheless, the kinetic energy of DM particles incident on the target
nucleus at rest

$T_{0}\equiv\dfrac{|\bm{k}|^{2}}{2m_{\chi}}=\dfrac{|\bm{v}|^{2}m_{\chi}}{2}\simeq
10^{-(6\div 4)}\dfrac{m_{\chi}c^{2}}{2}$ (6)

still remains in the region of tens of MeV, which is noticeably smaller than
the mass of the DM particle $m_{\chi}$, although, according to other
estimations, this energy can reach the GeV level. In the direct DM experiments
one usually has $m_{A}\geq 10$ GeV$/c^{2}$, the recoil energies of nuclei do
not exceed 200 keV, the typical difference between the excitation energies of
nuclei is in the region of several hundred keV. The characteristic scales of
these quantities are

$\displaystyle 10\leq m_{A}\leq 100\text{~{}GeV}/c^{2},\quad
1<m_{\chi}<10^{4}\text{~{}GeV}/c^{2},\quad 1\text{~{}keV}\leq T_{0}\leq
1\text{~{}GeV},$ (7) $\displaystyle T_{A}\leq
200\text{~{}keV},\quad|\bm{q}|\leq 0.2\div
0.3\text{~{}GeV}/c,\quad\Delta\varepsilon_{mn}\leq 1\text{~{}MeV}.$

Therefore, in what follows, it is enough to use the nonrelativistic
approximation

$k_{0}=(m^{2}_{\chi}+|\bm{k}|^{2})^{1/2}\simeq
m_{\chi}+\dfrac{|\bm{k}|^{2}}{2m_{\chi}}=m_{\chi}+T_{0},\quad\sqrt{m^{2}_{\chi}+|\bm{k}^{\prime}|^{2}}\simeq
m_{\chi}+\dfrac{|\bm{k}^{\prime}|^{2}}{2m_{\chi}},\quad\text{and}$
$T_{A}(|\bm{k}^{\prime}|,\cos\theta)\simeq\dfrac{\bm{q}^{2}}{2m_{A}}=\dfrac{|\bm{k}^{\prime}|^{2}+|\bm{k}|^{2}-2|\bm{k}||\bm{k}^{\prime}|\cos\theta}{2m_{A}}.$
(8)

The observable differential cross section of the process $\chi A\to\chi
A^{(*)}$ is the sum of the incoherent and coherent terms Bednyakov:2022dmc ,
which are expressed via scalar products
$(l_{s^{\prime}s},h^{p/n}_{r^{\prime}r})$ of the lepton and nucleon currents

$\displaystyle\frac{d\sigma_{s^{\prime}s}}{dT_{A}}(\chi A\to\chi A^{(*)})$
$\displaystyle=$
$\displaystyle\frac{d\sigma^{s^{\prime}s}_{\text{inc}}}{dT_{A}}(\chi_{s}A\to\chi_{s^{\prime}}A^{*})+\frac{d\sigma^{s^{\prime}s}_{\text{coh}}}{dT_{A}}(\chi_{s}A\to\chi_{s^{\prime}}A),\quad\text{where}$
(9) $\displaystyle\frac{d\sigma^{s^{\prime}s}_{\text{inc}}}{dT_{A}}$
$\displaystyle=$ $\displaystyle
c_{A}(T_{0},m_{A},m_{\chi})g_{\text{i}}\sum_{f=p,n}[1-|F_{f}(\bm{q})|^{2}]\Big{[}A^{f}_{+}\sum_{r^{\prime}=\pm}|(l_{s^{\prime}s},h^{\eta,f}_{r^{\prime}+})|^{2}+A^{f}_{-}\sum_{r^{\prime}=\pm}|(l_{s^{\prime}s},h^{\eta,f}_{r^{\prime}-})|^{2}\Big{]},\qquad$
$\displaystyle\frac{d\sigma^{s^{\prime}s}_{\text{coh}}}{dT_{A}}$
$\displaystyle=$ $\displaystyle
c_{A}(T_{0},m_{A},m_{\chi})g_{\text{c}}\Big{|}\sum_{f=p,n}F_{f}(\bm{q})[A^{f}_{+}(l_{s^{\prime}s},h^{\eta,f}_{++})+A^{f}_{-}(l_{s^{\prime}s},h^{\eta,f}_{--})]\Big{|}^{2},$
(10)

and $A^{f}_{\pm}$ is the number of $f$-type nucleons ($f=p,n$) with the spin
projection $\pm 1$ on some direction (for example, the arrival of the $\chi$
particle). In the experimental situation (or in calculations), the initial
external quantity is the recoil energy $T_{A}$. The $\chi$-particle emission
angle in the laboratory system as a function of $T_{A}$,
$\Delta\varepsilon_{mn}$ and $T_{0}$ has the form Bednyakov:2022dmc

$\displaystyle\cos\theta(T_{A})$ $\displaystyle=$
$\displaystyle\dfrac{m_{\chi}(2T_{0}-\Delta\varepsilon_{mn})-T_{A}(m_{\chi}+m_{A})}{2m_{\chi}\sqrt{T_{0}(T_{0}-\Delta\varepsilon_{mn}-T_{A})}}.$
(11)

In formulas (2) the universal common factor is introduced

$\displaystyle c_{A}\equiv c_{A}(T_{0},m_{A},m_{\chi})$ $\displaystyle\equiv$
$\displaystyle\dfrac{G^{2}_{F}m_{A}}{2^{6}\pi
m^{2}(2T_{0}m_{\chi})}=\dfrac{G^{2}_{F}m_{A}}{4\pi}\dfrac{1}{2^{4}m^{2}|\bm{k}^{l}_{\chi}|^{2}},\quad$
(12)

where $m$ is the nucleon mass and the dependence on the initial energy $T_{0}$
of the $\chi$ particle incident on the nucleus at rest is explicitly
distinguished, and the intensity of the hypothetical interaction of the $\chi$
particle with nucleons (proportional to $G^{2}_{F}$) is ”hidden” in the
corresponding scalar products. For all spin projections of the nucleon
($r^{\prime},r=\pm 1$) and the $\chi$ particle ($s^{\prime},s=\pm 1$), the
scalar products of the $\chi$ particle and nucleon weak currents are defined
as follows Bednyakov:2021pgs :

$\displaystyle(l^{w}_{s^{\prime}s},h^{w,f}_{r^{\prime}r})$ $\displaystyle=$
$\displaystyle\alpha_{f}(l^{v}_{s^{\prime}s}\,h^{v}_{r^{\prime}r})+\beta_{f}(l^{v}_{s^{\prime}s}\,h^{a}_{r^{\prime}r})+\gamma_{f}(l^{a}_{s^{\prime}s}\,h^{v}_{r^{\prime}r})+\delta_{f}(l^{a}_{s^{\prime}s}\,h^{a}_{r^{\prime}r}).$
(13)

In the nonrelativistic approximation, these scalar products look like

$\displaystyle(l^{w}_{\pm\pm},h^{w,f}_{\pm\pm})$ $\displaystyle=$
$\displaystyle
m^{2}_{c}(\alpha_{f}-\delta_{f}),\quad(l^{w}_{\pm\pm},h^{w,f}_{\mp\mp})=m^{2}_{c}(\alpha_{f}+\delta_{f}),\quad(l^{w}_{\pm\mp},h^{w,f}_{\mp\pm})=-2m^{2}_{c}\delta_{f},$
$\displaystyle(l^{w}_{\pm\mp},h^{w,f}_{\mp\mp})$ $\displaystyle=$
$\displaystyle\pm m^{2}_{s}e^{\mp
i\phi}(\alpha_{f}-\delta_{f}),\quad(l^{w}_{\pm\mp},h^{w,f}_{\pm\pm})=\pm
m^{2}_{s}e^{\mp i\phi}(\alpha_{f}+\delta_{f}),$ (14)
$\displaystyle(l^{w}_{\pm\pm},h^{w,f}_{\pm\mp})$ $\displaystyle=$
$\displaystyle\mp 2m^{2}_{s}e^{\mp
i\phi}\delta_{f},\quad(l^{w}_{\pm\pm},h^{w,f}_{\mp\pm})=(l^{w}_{\pm\mp},h^{w,f}_{\pm\mp})\simeq
0.$

Here the notations are introduced

$\displaystyle m^{2}_{c}\equiv 4m_{\chi}m\cos\frac{\theta}{2},\quad
m^{2}_{s}\equiv 4m_{\chi}m\sin\frac{\theta}{2}.$ (15)

It is convenient to rewrite formulas (2) in terms of the total number of
$f$-type nucleons, $A_{f}$, and the difference in the number of nucleons,
$\Delta A_{f}$, with the positive and negative spin projections

$A^{f}_{\pm}=\dfrac{1}{2}(A_{f}\pm\Delta A_{f}),\quad\text{where}\quad
A_{f}\equiv A^{f}_{+}+A^{f}_{-}\quad\text{and}\quad\Delta A_{f}\equiv
A^{f}_{+}-A^{f}_{-}.$ (16)

The cross sections for the nonrelativistic coherent $\chi A$ interaction have
the form

$\displaystyle\frac{d\sigma^{\mp\mp}_{\text{coh}}(\bm{q})}{g_{\text{c}}dT_{A}}$
$\displaystyle=$
$\displaystyle\cos^{2}\frac{\theta}{2}\dfrac{G^{2}_{F}m_{A}}{4\pi}\dfrac{m^{2}_{\chi}}{|\bm{k}^{l}_{\chi}|^{2}}\Big{[}\sum_{f=p,n}F_{f}(\bm{q}){A_{f}}\Big{(}\alpha_{f}\pm\delta_{f}\dfrac{\Delta
A_{f}}{A_{f}}\Big{)}\Big{]}^{2},$
$\displaystyle\frac{d\sigma^{\mp\pm}_{\text{coh}}(\bm{q})}{g_{\text{c}}dT_{A}}$
$\displaystyle=$
$\displaystyle\sin^{2}\frac{\theta}{2}\dfrac{G^{2}_{F}m_{A}}{4\pi}\dfrac{m^{2}_{\chi}}{|\bm{k}^{l}_{\chi}|^{2}}\Big{[}\sum_{f=p,n}F_{f}(\bm{q})A_{f}\Big{(}\alpha_{f}\mp\delta_{f}\dfrac{\Delta
A_{f}}{A_{f}}\Big{)}\Big{]}^{2},$ (17)
$\displaystyle\dfrac{d\sigma^{\text{total}}_{\text{coh}}(\bm{q})}{g_{\text{c}}dT_{A}}$
$\displaystyle=$
$\displaystyle\dfrac{1}{2}\sum_{s^{\prime}s}\dfrac{d\sigma^{s^{\prime}s}_{\text{coh}}}{g_{\text{c}}dT_{A}}=\dfrac{G^{2}_{F}m_{A}}{4\pi}\dfrac{m^{2}_{\chi}}{|\bm{k}^{l}_{\chi}|^{2}}\Big{(}\big{[}\sum_{f=p,n}\alpha_{f}A_{f}F_{f}(\bm{q})\big{]}^{2}+\big{[}\sum_{f=p,n}\delta_{f}{\Delta
A_{f}}F_{f}(\bm{q})\big{]}^{2}\Big{)}.$

The cross sections for the nonrelativistic incoherent $\chi A$ interaction are
the following:

$\displaystyle\frac{d\sigma^{\mp\mp}_{\text{inc}}(\bm{q})}{g_{\text{i}}dT_{A}}$
$\displaystyle=$
$\displaystyle\dfrac{G^{2}_{F}m_{A}}{4\pi}\dfrac{m^{2}_{\chi}}{|\bm{k}^{l}_{\chi}|^{2}}\sum_{f=p,n}\big{[}1-F^{2}_{f}(\bm{q})\big{]}A_{f}\times$
$\displaystyle\times\Big{\\{}\cos^{2}\frac{\theta}{2}\alpha^{2}_{f}+(1+\sin^{2}\frac{\theta}{2})\delta^{2}_{f}\pm\dfrac{2\Delta
A_{f}}{A_{f}}\delta_{f}\Big{[}\cos^{2}\frac{\theta}{2}\alpha_{f}+\sin^{2}\frac{\theta}{2}\delta_{f}\Big{]}\Big{\\}},$
$\displaystyle\frac{d\sigma^{\mp\pm}_{\text{inc}}(\bm{q})}{g_{\text{i}}dT_{A}}$
$\displaystyle=$
$\displaystyle\dfrac{G^{2}_{F}m_{A}}{4\pi}\dfrac{m^{2}_{\chi}}{|\bm{k}^{l}_{\chi}|^{2}}\sum_{f=p,n}\big{[}1-F^{2}_{f}(\bm{q})\big{]}A_{f}\times$
$\displaystyle\times\Big{\\{}\sin^{2}\frac{\theta}{2}\alpha^{2}_{f}+(1+\cos^{2}\frac{\theta}{2})\delta^{2}_{f}\mp\dfrac{2\Delta
A_{f}}{A_{f}}\delta_{f}\Big{[}\sin^{2}\frac{\theta}{2}\alpha_{f}+\cos^{2}\frac{\theta}{2}\delta_{f}\Big{]}\Big{\\}},$
$\displaystyle\frac{d\sigma^{\text{total}}_{\text{inc}}(\bm{q})}{g_{\text{i}}dT_{A}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{s^{\prime}s}\frac{d\sigma^{s^{\prime}s}_{\text{inc}}}{g_{\text{i}}dT_{A}}=\dfrac{G^{2}_{F}m_{A}}{4\pi}\dfrac{m^{2}_{\chi}}{|\bm{k}^{l}_{\chi}|^{2}}\sum_{f=p,n}A^{f}\big{[}1-F^{2}_{f}(\bm{q})\big{]}\big{[}\alpha^{2}_{f}+3\delta^{2}_{f}\big{]}.$

The measurable total cross section of the $\chi A$ interaction in the
nonrelativistic approximation has the form

$\displaystyle\dfrac{d\sigma^{\text{weak}}_{\text{nonrel}}}{dT_{A}}(\chi
A\to\chi A^{*})$ $\displaystyle=$
$\displaystyle\dfrac{G^{2}_{F}m_{A}}{4\pi}\dfrac{m^{2}_{\chi}}{|\bm{k}^{l}_{\chi}|^{2}}\Big{\\{}g_{\text{i}}\sum_{f=p,n}A^{f}\big{[}1-F^{2}_{f}(\bm{q})\big{]}\big{[}\alpha^{2}_{f}+3\delta^{2}_{f}\big{]}$
$\displaystyle+g_{\text{c}}\Big{(}\big{[}\sum_{f=p,n}\alpha_{f}A_{f}F_{f}(\bm{q})\big{]}^{2}+\big{[}\sum_{f=p,n}\delta_{f}{\Delta
A_{f}}F_{f}(\bm{q})\big{]}^{2}\Big{)}\Big{\\}}.$

Here following the notation is used:

$c_{A}(4m_{\chi}m)^{2}=(4m_{\chi}m)^{2}\dfrac{G^{2}_{F}m_{A}}{2^{6}\pi
m^{2}|\bm{k}^{l}_{\chi}|^{2}}=\dfrac{G^{2}_{F}m_{A}}{4\pi}\dfrac{m^{2}_{\chi}}{|\bm{k}^{l}_{\chi}|^{2}}.$
(20)

##  3 Expected event counting rate

In an experiment aimed at the direct DM detection (see, for example,
Lewin:1996rx ; Freese:1987wu ; Bednyakov:2015uoa ; Vergados:1996hs ;
Baudis:2012ig ; Tanabashi:2018oca ), one tries to measure the so-called event
counting rate, i.e., some number of useful events per unit of time. This value
is proportional to the cross section of the particle interaction multiplied by
their relative velocity, $v\sigma(v,...)$. It is further integrated over the
relative velocity $v$ with a distribution function $f(v)$, which determines
the probability to have velocity $v$ at the moment of the interaction.

A ”traditional estimate” of the event rate is based on a rather substantial
expected flow of the DM particles through the Earth Baudis:2012ig , which for
the DM mass $m_{\rm{DM}}=100$ GeV/$c^{2}$ is

${\Phi}_{\rm{DM}}=\frac{{\rho}_{\rm{DM}}}{m_{\rm{DM}}}\langle
v\rangle\cong\dfrac{6.6\times 10^{4}}{{\rm{sm^{2}~{}{\rm
sec}}}}\Big{[}\frac{{\rho}_{\rm{DM}}}{0.3~{}{\rm{GeV/sm^{3}}}}\Big{]}\Big{[}\frac{100~{}{\rm{GeV}}}{m_{\rm{DM}}}\Big{]}\Big{[}\frac{\langle
v\rangle}{220~{}{\rm{km/sec}}}\Big{]}.$

Therefore, for a target with the atomic number $A=$ 100 and typical cross
section at a level of $10^{-38}~{}\rm{sm^{2}}$ the rate has the following
rather small value:

$R=\sigma\frac{N_{\rm{Av}}}{A}{\Phi}_{\rm{DM}}\cong
0.13~{}\frac{\rm{events}}{\rm{kg\,year}}\Big{[}\frac{100~{}{\rm{g/mol}}}{A}\Big{]}\Big{[}\frac{{\rho}_{\rm{DM}}}{0.3~{}{\rm{GeV/sm^{3}}}}\Big{]}\Big{[}\frac{100~{}{\rm{GeV}}}{m_{\rm{DM}}}\Big{]}\Big{[}\frac{\langle
v\rangle}{220~{}{\rm{km/sec}}}\Big{]}\Big{[}\frac{\sigma}{10^{-38}~{}{\rm{cm^{2}}}}\Big{]}.$

Here $\langle v\rangle$ and ${\rho}_{\rm{DM}}$ are the averaged velocity and
density of the DM particles near the Earth, ${\sigma}$ is the cross section
for the DM particle–nucleon interaction. $N_{\rm{Av}}=6.02\times
10^{23}~{}{\rm{mol}}^{-1}$ is the Avogadro number (the number of protons in
one gram of protons), $A$ is the mass of one mole, i.e., it is a weight of
$6.02\times 10^{23}$ nuclei of the target material.

In fact, the integral counting event rate is determined in the form

$R(T_{A}^{\min})=n_{\chi}\,N_{A}\int_{v_{\min}}^{v_{\max}}\\!\\!f(v)dv\int_{T_{A}^{\min}}^{T_{A}^{\max}}\\!\frac{v\,d\sigma}{dT_{A}}(v,T_{A})\,dT_{A}.$
(21)

As a function of the nuclear recoil energy,
$T_{A}\in(T_{A}^{\min},T_{A}^{\max})$, the differential event rate is

$\frac{dR(T_{A}^{\min})}{dT_{A}}=n_{\chi}\,N_{A}\int_{v_{\min}}^{v_{\max}}\\!\\!f(v)\,\frac{v\,d\sigma}{dT_{A}}(v,T_{A})\,dv.$
(22)

Here $N_{A}$ is the number of scattering centers (the number of $A$-type
nuclei) in the target. If the target mass is one gram,
$N_{A}=\dfrac{N_{\rm{Av}}}{A}$. Furthermore,
$n_{\chi}\equiv\dfrac{\rho_{\chi}}{m_{\chi}}$ is the number density of the
$\chi$ particles, where $\rho_{\chi}\simeq 0.35\,\text{GeV}/$cm3 is the
density of the halo (assumed to consist entirely of the $\chi$ particles)
around the Earth, $T_{A}^{\min}$ is the minimal recoil energy of the nucleus
(the lower energy threshold), $v_{\min}$ is the minimal relative velocity of
the $\chi$ particle and nucleus at which the recoil energy of the
$T_{A}^{\min}$ is still possible

$v_{\min}=\sqrt{T_{A}^{\min}\dfrac{(m_{A}+m_{\chi})^{2}}{2m_{A}m^{2}_{\chi}}}=\sqrt{\dfrac{2T_{A}^{\min}}{m_{\chi}\mu_{A}}},\qquad\text{where}\qquad\mu_{A}\equiv\frac{4m_{\chi}m_{A}}{(m_{\chi}+m_{A})^{2}}.$
(23)

When the relative velocity $v<v_{\min}$, any event registration is not
possible. Further, $v_{\max}=v_{\text{esc}}$ is the maximal velocity of the
$\chi$ particle for a given DM distribution. Then the maximal kinetic energy
of the $\chi$ particle with mass $m_{\chi}$ and the maximal kinetic energy
(recoil) of the nucleus with mass $m_{A}$, caused by the elastic $\chi$
particle–nucleus collision are

$T_{\chi}^{\max}=\dfrac{m_{\chi}v^{2}_{\max}}{2}\quad\text{and}\quad
T_{A}^{\max}=\dfrac{q^{2}_{\max}}{2m_{A}}=\dfrac{2m_{A}v^{2}_{\max}m^{2}_{\chi}}{(m_{A}+m_{\chi})^{2}}=T_{\chi}^{\max}\mu_{A}.$
(24)

In the Standard Halo Model (SHM) Kang:2019uuj ; Evans:2018bqy ;
Zurowski:2020dxe the $\chi$-particle velocities near the detector are
distributed as follows:

$\displaystyle f_{\rm SHM}(\bm{w})$ $\displaystyle=$
$\displaystyle\frac{1}{(\pi
v_{0}^{2})^{3/2}}\exp\Big{[}\frac{-(\bm{w}-\bm{v}_{\oplus})^{2}}{v_{0}^{2}}\Big{]}\text{~{}~{}and~{}~{}}$
(25) $\displaystyle v_{\rm esc}$ $\displaystyle=$
$\displaystyle|\bm{v}|_{\max}=|\bm{w}+\bm{v}_{\oplus}|_{\max}\simeq
540~{}\frac{\rm km}{\rm sec},$

where $v_{0}\simeq 220$ km/sec is the circular velocity in the halo. It is
usually assumed Freese:1987wu that the speed of the Sun $v_{\odot}\equiv
v_{\rm Sun}\simeq v_{\text{circ}}\equiv v_{0}=\bar{v}\sqrt{\frac{2}{3}}\simeq
232\pm 20$ km/sec, where $\bar{v}$ is the velocity dispersion in the halo
($\simeq 270\pm 25$ km/s). The Earth’s velocity relative to the Galactic halo
is $\bm{v}_{\oplus}\neq 0$.

For the one-dimensional DM velocity relative to the Earth (and relative to the
detector) $w=|\bm{w}|$ one has the following distribution function
Freese:1987wu :

$\displaystyle f_{\rm SHM}(w)dw\equiv f_{\rm
SHM}(x)dx,\text{~{}where~{}}f_{\rm
SHM}(x)=\frac{4x^{2}}{\sqrt{\pi}}e^{-(x^{2}+\eta^{2})}\frac{\mbox{sinh}(2x\eta)}{2x\eta},\
\int\\!\\!f_{\rm SHM}(x)dx=1.\qquad$ (26)

Here $x=\dfrac{w}{v_{0}}$ is the dimensionless velocity of the DM particle
relative to the Earth, $\eta=\dfrac{v_{\oplus}}{v_{0}}$ is the dimensionless
velocity of the Earth relative to the Galactic halo. It depends on the time
$\eta=\eta(t)$. Since the Earth has its annual revolution around the Sun, one
has $\langle\eta(t)\rangle_{\text{year}}=\eta_{0}\simeq 1.05.$

It is currently believed that the SHM distribution (25) does not accurately
describe the DM distribution in the Galaxy. There are anisotropic structures,
such as DM particle flows, due to impact of nearby galaxies OHare:2018trr . In
what follows, we will use the following ”simplified-generalized” expression
from OHare:2018trr :

$f_{\rm str}(\bm{w})=\frac{1}{\pi^{3/2}(\epsilon
v_{0})^{3}}\exp\Big{[}-\frac{\big{(}\bm{w}-(\bm{v}_{\oplus}+\bm{v}_{\rm
str})\big{)}^{2}}{(\epsilon
v_{0})^{2}}\Big{]},\quad\text{where}\quad|\bm{v}_{\rm str}|\simeq
300\text{~{}km/sec}.$ (27)

In this ”external-DM-flow” distribution the vector sum of the Earth velocity
and the flow velocity relative to the Galactic halo222We neglect here their
time dependences., $\bm{v}_{\oplus}+\bm{v}_{\rm str}$, plays the role of the
Earth’s velocity $\bm{v}_{\oplus}$ from (25). Therefore, by analogy with
formula (26) one can write

$\displaystyle f_{\rm str}(w)dw\equiv f_{\rm
str}(y)dy,\text{~{}where~{}}f_{\rm
str}(y)=\frac{4}{\sqrt{\pi}}y^{2}e^{-(y^{2}+\xi^{2})}\frac{\mbox{sinh}(2y\xi)}{2y\xi}\text{~{}and~{}~{}}\int
f_{\rm str}(y)dy=1,\qquad$ (28)

where $y=\dfrac{w}{\epsilon v_{0}}$ is the dimensionless velocity of the DM
particle (belonging to the external flow) relative to the Earth, and
${\xi}\equiv\dfrac{|\bm{v}_{\oplus}+\bm{v}_{\rm str}|}{\epsilon v_{0}}$ is the
dimensionless sum of the Earth and flow velocities. For simplicity, we assume
that ${\xi}=\dfrac{{v}_{\oplus}+{v}_{\rm str}}{\epsilon v_{0}}$. The complete
(one-dimensional) normalized to unity DM distribution (of $\chi$ particles) in
terms of the dimensionless velocity relative to the Earth has the form

$\displaystyle f_{\rm SHM+str}(w)dw$ $\displaystyle=$
$\displaystyle\Big{[}\Big{(}1-\frac{\rho_{\rm str}}{\rho}\Big{)}f_{\rm
SHM}(w)+\frac{\rho_{\rm str}}{\rho}f_{\rm
str}(w)\Big{]}dw\equiv\Big{(}1-\frac{\rho_{\rm str}}{\rho}\Big{)}f_{\rm
SHM}(x)dx+\frac{\rho_{\rm str}}{\rho}f_{\rm str}(y)dy.$

With allowance for $y=\dfrac{x}{\epsilon}$ and $\xi\equiv\dfrac{\eta_{\rm
str}}{\epsilon}$, this expression can be rewritten in the final form

$\displaystyle f(x)dx$ $\displaystyle\equiv$ $\displaystyle f_{\rm
SHM+str}(w)dw=\gamma_{1}f_{\rm SHM}(x)dx+\gamma_{\eta}f_{\rm
str}(x)dx,\text{~{}~{}where~{}now}$ (29) $\displaystyle f_{\rm str}(x)$
$\displaystyle=$
$\displaystyle\frac{4x^{2}}{\sqrt{\pi}}e^{-\dfrac{x^{2}+\eta^{2}_{\rm
str}}{\epsilon^{2}}}\frac{\mbox{sinh}(2x\eta_{\rm
str}/\epsilon^{2})}{2x\eta_{\rm
str}}\qquad\text{and}\qquad\gamma_{1}+\gamma_{\eta}=1.$

In formula (29), one has a small generalization by means of weights of the
distributions, the sum of which is equal to one. In what follows, we will use
expression (29) for the function $f(v)$ from the counting event rate (22).

The total (summed over all spin indices of the $\chi$ particle) differential
cross section included in (21) and (22), according to (9) and (2), is the sum
of the coherent and incoherent contributions to the $\chi A$ cross section

$\frac{d\sigma}{dT_{A}}(v,T_{A})\equiv\frac{d\sigma_{\text{i(nc)}}}{dT_{A}}(\chi_{s}A\to\chi_{s^{\prime}}A^{*})+\frac{d\sigma_{\text{c(oh)}}}{dT_{A}}(\chi_{s}A\to\chi_{s^{\prime}}A).$

Then the total event rate, integrated over the nuclear recoil energy $T_{A}$
in the interval $T_{A}^{\min}\leq T_{A}\leq T_{A}^{\max}$, can be given as the
sum of the coherent and incoherent event rates

$\displaystyle R(T_{A}^{\min},T_{A}^{\max})$ $\displaystyle\equiv$
$\displaystyle
R_{\text{c(oh)}}(T_{A}^{\min},T_{A}^{\max})+R_{\text{i(nc)}}(T_{A}^{\min},T_{A}^{\max}),\quad\text{where}$
(30) $\displaystyle R_{\text{c/i}}(T_{A}^{\min},T_{A}^{\max})$
$\displaystyle\equiv$
$\displaystyle\int_{T_{A}^{\min}}^{T_{A}^{\max}}dT_{A}\frac{dR(T_{A})_{\text{c/i}}}{dT_{A}},\text{~{}~{}and~{}~{}}$
$\displaystyle\frac{dR(T_{A})_{\text{c/i}}}{dT_{A}}$ $\displaystyle=$
$\displaystyle
n_{\chi}v_{0}N_{A}\int_{x_{\min}}^{x_{\max}}f(x)xdx\frac{d\sigma_{\text{c/i}}(xv_{0},T_{A})}{dT_{A}}.$

For the differential event counting rates one has

$\displaystyle\frac{dR(T_{A})_{\text{c/i}}}{n_{\chi}v_{0}N_{A}dT_{A}}$
$\displaystyle=$
$\displaystyle\int_{x_{\min}}^{x_{\max}}\Big{[}\gamma_{1}f_{\rm
SHM}(x)+\gamma_{\eta}f_{\rm
str}(x)\Big{]}xdx\frac{d\sigma_{\text{c/i}}(xv_{0},T_{A})}{dT_{A}}=$
$\displaystyle=$ $\displaystyle\gamma_{1}\int_{x_{\min}}^{x^{\max}_{1}}f_{\rm
SHM}(x)dx\frac{xd\sigma_{\text{c/i}}(x,T_{A})}{dT_{A}}+\gamma_{\eta}\int_{x_{\min}}^{x^{\max}_{\eta}}f_{\rm
str}(x)dx\frac{xd\sigma_{\text{c/i}}(x,T_{A})}{dT_{A}}.$

There are the following relations according to (23):

$\displaystyle x_{\min}$ $\displaystyle=$
$\displaystyle\dfrac{w_{\min}}{v_{0}}=\Big{[}\dfrac{T_{A}^{\min}}{T_{A}^{0}}\Big{]}^{1/2},\text{~{}~{}where~{}~{}}T_{A}^{0}\equiv\mu_{A}\dfrac{m_{\chi}v_{0}^{2}}{2},$
$\displaystyle x^{\max}_{1}$ $\displaystyle=$ $\displaystyle\dfrac{w^{\rm
SHM}_{\max}}{v_{0}}=\dfrac{v_{\rm esc}+v_{\oplus}}{v_{0}}=\dfrac{v_{\rm
esc}}{v_{0}}+\eta_{1},\text{~{}~{}where~{}~{}}\eta_{1}=\dfrac{v_{\oplus}}{v_{0}}\simeq
1\text{~{}~{}and~{}~{}}$ (32) $\displaystyle x^{\max}_{\eta}$ $\displaystyle=$
$\displaystyle\dfrac{w^{\rm str}_{\max}}{v_{0}}=\dfrac{v_{\rm
esc}+{v}_{\oplus}+{v}_{\rm str}}{v_{0}}=\dfrac{v_{\rm
esc}}{v_{0}}+\eta,\text{~{}~{}where~{}~{}}\eta=\dfrac{v_{\oplus}+{v}_{\rm
str}}{v_{0}}.$

Here $x^{\max}_{1,\eta}$ corresponds to the maximal collision velocity between
the $\chi$ particle and the target, i.e., when the escape velocity and the
velocity of the Sun (together with the external DM stream velocity) fold in
the opposite direction.

If one assumes that the interaction of the massive $\chi$ lepton with the
nucleons has the form of the weak interactions (13), the total cross section
of the $\chi A$ scattering in the nonrelativistic approximation is given by
formula (2). It explicitly contains the coherent and incoherent contributions
to the $\chi A$ cross section as follows:

$\displaystyle\dfrac{d\sigma^{\text{weak}}_{\text{i(nc)}}(x,T_{A})}{dT_{A}}$
$\displaystyle=$
$\displaystyle\dfrac{G^{2}_{F}m_{A}}{4\pi}\dfrac{1}{x^{2}v_{0}^{2}}\Big{\\{}g_{\text{i}}\sum_{f=p,n}A^{f}\big{[}1-F^{2}_{f}(\bm{q})\big{]}\big{[}\alpha^{2}_{f}+3\delta^{2}_{f}\big{]}\Big{\\}}_{\text{i}},$
(33)
$\displaystyle\dfrac{d\sigma^{\text{weak}}_{\text{c(oh)}}(x,T_{A})}{dT_{A}}$
$\displaystyle=$
$\displaystyle\dfrac{G^{2}_{F}m_{A}}{4\pi}\dfrac{1}{x^{2}v_{0}^{2}}\Big{\\{}g_{\text{c}}\big{[}\sum_{f=p,n}\alpha_{f}A_{f}F_{f}(\bm{q})\big{]}^{2}+\big{[}\sum_{f=p,n}\delta_{f}{\Delta
A_{f}}F_{f}(\bm{q})\big{]}^{2}\Big{\\}}_{\text{c}}.$

Here it is used that
$\dfrac{m^{2}_{\chi}}{|\bm{k}^{l}_{\chi}|^{2}}=\dfrac{1}{x^{2}v_{0}^{2}}$,
since in the nonrelativistic approximation the momentum of the $\chi$ lepton
incident on the nucleus at rest is
$|\bm{k}^{l}_{\chi}|^{2}=m^{2}_{\chi}w^{2}$. As a result, the differential
rates of the coherent and incoherent events (3) take the form

$\displaystyle\frac{dR(T_{A})_{\text{c/i}}}{n_{\chi}v_{0}N_{A}dT_{A}}$
$\displaystyle=$ $\displaystyle\gamma_{1}\int_{x_{\min}}^{x^{\max}_{1}}f_{\rm
SHM}(x)dx\frac{xd\sigma_{\text{c/i}}(x,T_{A})}{dT_{A}}+\gamma_{\eta}\int_{x_{\min}}^{x^{\max}_{\eta}}f_{\rm
str}(x)dx\frac{xd\sigma_{\text{c/i}}(x,T_{A})}{dT_{A}}=$ $\displaystyle=$
$\displaystyle\dfrac{G^{2}_{F}m_{A}}{4\pi
v_{0}^{2}}\Big{\\{}..\Big{\\}}_{\text{c/i}}\Big{[}\gamma_{1}\int_{x_{\min}}^{x^{\max}_{1}}f_{\rm
SHM}(x)\dfrac{dx}{x}+\gamma_{\eta}\int_{x_{\min}}^{x^{\max}_{\eta}}f_{\rm
str}(x)\dfrac{dx}{x}\Big{]}.$

Finally, taking into account $\big{\\{}...\big{\\}}$-parentheses from (33) and
notation (3), one arrives at

$\displaystyle\frac{dR(T_{A})_{\text{c/i}}}{dT_{A}}$ $\displaystyle=$
$\displaystyle
R_{0}(A,\chi)\Big{\\{}g_{\text{c/i}}...\Big{\\}}_{\text{c/i}}\Big{[}\gamma_{1}S_{1}(x_{\min},x^{\max}_{1})+\gamma_{\eta}S_{\eta}(x_{\min},x^{\max}_{\eta})\Big{]}.$
(34)

Here, the auxiliary notation is introduced

$\displaystyle R_{0}(A,\chi)$ $\displaystyle\equiv$
$\displaystyle\dfrac{G^{2}_{F}m_{A}}{4\pi}\dfrac{n_{\chi}N_{A}}{v_{0}},$ (35)

which sets the scale of the event rate, and the DM halo contributions are
given as

$\displaystyle S_{1}(x_{\min},x^{\max}_{1})$ $\displaystyle\equiv$
$\displaystyle\int_{x_{\min}}^{x^{\max}_{1}}f_{\rm
SHM}(x)\frac{dx}{x}=S(x_{\min},\eta_{1},1)-S(x^{\max}_{1},\eta_{1},1),$ (36)
$\displaystyle S_{\eta}(x_{\min},x^{\max}_{\eta})$ $\displaystyle\equiv$
$\displaystyle\int_{x_{\min}}^{x^{\max}_{\eta}}f_{\text{str}}(x)\frac{dx}{x}=S(x_{\min},\eta,\epsilon)-S(x^{\max}_{\eta},\eta,\epsilon).$
(37)

Here the common distribution function is introduced

$S(x,\eta,\epsilon)\equiv\dfrac{\epsilon}{2\eta}\Big{\\{}\text{erf}\big{(}\dfrac{x+\eta}{\epsilon}\big{)}-\text{erf}\big{(}\dfrac{x-\eta}{\epsilon}\big{)}\Big{\\}}=\dfrac{\epsilon}{\sqrt{\pi}\eta}\int_{\frac{x-\eta}{\epsilon}}^{\frac{x+\eta}{\epsilon}}e^{-t^{2}}dt.$
(38)

The common value of $x_{\min}$ from (3) is proportional to the square root of
the recoil energy threshold $T_{A}^{\min}$. In the elastic $\chi A$
interaction, according to (24) and (3), $x^{\max}_{\eta}$ determines the
maximal nuclear recoil energy as a function of the maximal DM velocity

$T_{A,\eta}^{\max}=\mu_{A}\dfrac{m_{\chi}(v_{0}x^{\max}_{\eta})^{2}}{2}=\mu_{A}T_{\chi,\eta}^{\max}.$
(39)

In the inelastic $\chi A$ collision (when $\Delta\varepsilon_{mn}>0$) the
maximal nuclear recoil energy can be obtained by substituting the maximal
energy of the DM particle (24) into formula (11) and taking into account that
the $\chi$ particle, incident on the nucleus, ”is reflected” strictly in the
opposite direction, i.e., from the following condition:

$\cos\theta(T_{A^{*}})=\dfrac{m_{\chi}(2T_{\chi}^{\max}-\Delta\varepsilon_{mn})-T_{A^{*}}(m_{\chi}+m_{A})}{2m_{\chi}\sqrt{T_{\chi}^{\max}(T_{\chi}^{\max}-\Delta\varepsilon_{mn}-T_{A^{*}})}}=-1.$

The solution to this equation is

$T_{A^{*},\eta}^{\max}(r,\Delta\varepsilon_{mn})=T_{\chi,\eta}^{\max}\dfrac{\mu_{A}}{2}\Big{[}1+\sqrt{1-\dfrac{(r+1)\Delta\varepsilon_{mn}}{T_{\chi,\eta}^{\max}}}\Big{]}-\dfrac{{r}\Delta\varepsilon_{mn}}{r+1},\quad\text{where~{}~{}}r\equiv\dfrac{m_{\chi}}{m_{A}}.$
(40)

If $T_{\chi,\eta}^{\max}\gg(r+1)\Delta\varepsilon_{mn}$ (or
$\Delta\varepsilon_{mn}=0$), then
$T_{A^{*},\eta}^{\max}(r,\Delta\varepsilon_{mn})$ transforms back in
(39)333With $\mu_{A}=\dfrac{4r}{(1+r)^{2}}$, one has
$T_{A^{*}}^{\max}=T_{\chi}^{\max}\mu_{A}-\dfrac{2r\Delta\varepsilon_{mn}}{1+r}=T_{\chi}^{\max}\mu_{A}\Big{(}1-\dfrac{\Delta\varepsilon_{mn}}{T_{\chi}^{\max}}\dfrac{(r+1)}{2}\Big{)}\simeq
T_{\chi}^{\max}\mu_{A}.$. Solution (40) exists only when

$T_{\chi,\eta}^{\max}\geq\Big{(}\dfrac{m_{\chi}}{m_{A}}+1\Big{)}\Delta\varepsilon_{mn}\quad\text{or}\quad(x^{\max}_{\eta})^{2}\dfrac{v_{0}^{2}}{2}\geq\Big{(}1+\dfrac{m_{A}}{m_{\chi}}\Big{)}\dfrac{\Delta\varepsilon_{mn}}{m_{A}}.$
(41)

If $T_{\chi,\eta}^{\max}=(r+1)\Delta\varepsilon_{mn}$, the ”minimal” value of
the recoil energy is reached

$T_{A^{*},\eta}^{\max}(r,\Delta\varepsilon_{mn})=T_{\chi,\eta}^{\max}\dfrac{\mu_{A}}{4},$

which is a unique value of the nuclear recoil energy, at which excitation of
the nucleus with energy $\Delta\varepsilon_{mn}$ is still possible. In other
words, if for a nucleus with mass $m_{A}$ and excitation energy
$\Delta\varepsilon_{mn}$ and a $\chi$ particle with mass $m_{\chi}$ and
maximal kinetic energy $T_{\chi,\eta}^{\max}$ condition (41) is not satisfied,
the inelastic process is impossible. In this situation, the DM particle does
not have enough energy to excite the nucleus and cause its motion. Because of
the momentum conservation law, excitation of the nucleus by impact must be
accompanied by some motion of the excited nucleus, i.e., by the nuclear recoil
kinetic energy.

Considering (34), the coherent and incoherent event rates have two terms
corresponding to different DM distributions with different maximal nuclear
recoils

$\displaystyle\frac{dR(T_{A})_{\text{c}}}{g_{\text{c}}R_{0}()dT_{A}}$
$\displaystyle=$
$\displaystyle\gamma_{1}S_{1}(x_{\min},x^{\max}_{1})\Big{\\{}\big{[}\sum_{f=p,n}\alpha_{f}A_{f}F_{f}(\bm{q})\big{]}^{2}+\big{[}\sum_{f=p,n}\delta_{f}{\Delta
A_{f}}F_{f}(\bm{q})\big{]}^{2}\Big{\\}}\Theta(T_{A,\eta_{1}}^{\max}-T_{A})$
$\displaystyle+\gamma_{\eta}S_{\eta}(x_{\min},x^{\max}_{\eta})\Big{\\{}\big{[}\sum_{f=p,n}\alpha_{f}A_{f}F_{f}(\bm{q})\big{]}^{2}+\big{[}\sum_{f=p,n}\delta_{f}{\Delta
A_{f}}F_{f}(\bm{q})\big{]}^{2}\Big{\\}}\Theta(T_{A,\eta}^{\max}-T_{A}),$
$\displaystyle\frac{dR(T_{A})_{\text{i}}}{g_{\text{i}}R_{0}()dT_{A}}$
$\displaystyle=$
$\displaystyle\gamma_{1}S_{1}(x_{\min},x^{\max}_{1})\Big{\\{}\sum_{f=p,n}A^{f}\big{[}1-|F_{f}(\bm{q})|^{2}\big{]}\big{[}\alpha^{2}_{f}+3\delta^{2}_{f}\big{]}\Big{\\}}\Theta(T_{A^{*},\eta_{1}}^{\max}-T_{A})$
$\displaystyle+\gamma_{\eta}S_{\eta}(x_{\min},x^{\max}_{\eta})\Big{\\{}\sum_{f=p,n}A^{f}\big{[}1-|F_{f}(\bm{q})|^{2}\big{]}\big{[}\alpha^{2}_{f}+3\delta^{2}_{f}\big{]}\Big{\\}}\Theta(T_{A^{*},\eta}^{\max}-T_{A}).$

Here $T_{A^{(*)},\eta_{1},\eta}^{\max}$ are defined by formula (40), and
$T_{A}$ is the measurable recoil energy for both excited and unexcited nuclei.
It is related to the momentum transferred to the nucleus by formulas (5) and
(8), or $\bm{q}^{2}\simeq 2m_{A}T_{A}$. Recall that any detector aimed at
registration of the nuclear recoil energy is unable to distinguish the
”elastic” recoil $T_{A}$ from the ”inelastic” recoil $T_{A^{*}}$.

Formulas (3) are the most general expressions for differential event rates in
the case of the nonrelativistic weak $\chi A$ interaction and two different DM
distributions.

Let us assume (for illustration) that the correction factors are
$g_{\text{c}}=g_{\text{i}}\simeq 1$ Bednyakov:2023bbg and the nuclear form
factors are the same for protons and neutrons Bednyakov:2022dmc

$F(\bm{q})\equiv F_{p}(\bm{q})=F_{n}(\bm{q}).$ (43)

Then after introducing generalized (in)coherent weak charges of nuclei

$Q_{c}(A)\equiv\big{[}\sum_{f=p,n}\alpha_{f}A_{f}\big{]}^{2}+\big{[}\sum_{f=p,n}\delta_{f}{\Delta
A_{f}}\big{]}^{2}\quad\text{and}\quad
Q_{i}(A)\equiv\sum_{f=p,n}A^{f}\big{[}\alpha^{2}_{f}+3\delta^{2}_{f}\big{]}$
(44)

and $T_{A}$-dependent (in)coherent form factor functions

$\Phi(T_{A})_{c}\equiv|F(\bm{q})|^{2}\text{~{}~{}and~{}~{}}\Phi(T_{A})_{i}\equiv
1-|F(\bm{q})|^{2},$ (45)

general expression (3) for the event counting rates takes the form

$\displaystyle\frac{dR(T_{A})_{\text{c/i}}}{dT_{A}}$ $\displaystyle=$
$\displaystyle R_{0}(A,\chi)Q_{\rm c/i}(A)\Phi(T_{A})_{\rm c/i}C_{\rm
c/i}(x_{\min},\eta,\epsilon,T_{A}),\quad\text{where}$ (46) $\displaystyle
C_{\rm c/i}(x_{\min},\eta,\epsilon,T_{A})$ $\displaystyle\equiv$
$\displaystyle\gamma_{1}S_{1}(x_{\min},x^{\max}_{1})\Theta(T_{A/A^{*},1}^{\max}-T_{A})+\gamma_{\eta}S_{\eta}(x_{\min},x^{\max}_{\eta})\Theta(T_{A/A^{*},\eta}^{\max}-T_{A}).$

Since $x^{\max}_{\eta}>x^{\max}_{1}$ and $T_{A,\eta}^{\max}>T_{A,1}^{\max}$,
then, as the recoil energy $T_{A}$ increases, the role of the second term
increases, and for $T_{A,1}^{\max}<T_{A}<T_{A,\eta}^{\max}$, one will have the
only DM stream distribution (the second term). Futhermore, the incoherent-to-
coherent event rate ratio is as follows:

$\displaystyle{\dfrac{dR_{\text{i}}}{dT_{A}}}/{\dfrac{dR_{\text{c}}}{dT_{A}}}=\dfrac{Q_{i}(A)\Phi(T_{A})_{\rm
i}}{Q_{c}(A)\Phi(T_{A})_{\rm c}}\dfrac{C_{\rm
i}(x_{\min},\eta,\epsilon,T_{A})}{C_{\rm c}(x_{\min},\eta,\epsilon,T_{A})}.$
(47)

The total event rates (30), after integration of the differential rates (3),
take the form

$\displaystyle\frac{R(T_{A}^{\min})_{\text{c}}}{g_{\text{c}}R_{0}(A,\chi)}$
$\displaystyle=$
$\displaystyle\gamma_{1}S_{1}(x_{\min},x^{\max}_{1})\int_{T_{A}^{\min}}^{T_{A,1}^{\max}}\\!\\!dT_{A}\Big{\\{}\big{[}\sum_{f=p,n}\alpha_{f}A_{f}F_{f}(\bm{q})\big{]}^{2}+\big{[}\sum_{f=p,n}\delta_{f}{\Delta
A_{f}}F_{f}(\bm{q})\big{]}^{2}\Big{\\}}$
$\displaystyle+\gamma_{\eta}S_{\eta}(x_{\min},x^{\max}_{\eta})\int_{T_{A}^{\min}}^{T_{A,\eta}^{\max}}\\!\\!dT_{A}\Big{\\{}\big{[}\sum_{f=p,n}\alpha_{f}A_{f}F_{f}(\bm{q})\big{]}^{2}+\big{[}\sum_{f=p,n}\delta_{f}{\Delta
A_{f}}F_{f}(\bm{q})\big{]}^{2}\Big{\\}},$
$\displaystyle\frac{R(T_{A}^{\min})_{\text{i}}}{g_{\text{i}}R_{0}(A,\chi)}$
$\displaystyle=$
$\displaystyle\gamma_{1}S_{1}(x_{\min},x^{\max}_{1})\int_{T_{A}^{\min}}^{T_{A^{*},1}^{\max}}\\!\\!dT_{A}\Big{\\{}\sum_{f=p,n}A^{f}\big{[}1-F^{2}_{f}(\bm{q})\big{]}\big{[}\alpha^{2}_{f}+3\delta^{2}_{f}\big{]}\Big{\\}}$
$\displaystyle+\gamma_{\eta}S_{\eta}(x_{\min},x^{\max}_{\eta})\int_{T_{A}^{\min}}^{T_{A^{*},\eta}^{\max}}\\!\\!dT_{A}\Big{\\{}\sum_{f=p,n}A^{f}\big{[}1-F^{2}_{f}(\bm{q})\big{]}\big{[}\alpha^{2}_{f}+3\delta^{2}_{f}\big{]}\Big{\\}}.$

Since $T_{A^{*},\eta}^{\max}<T_{A,\eta}^{\max}$, for sufficiently large
nuclear excitation energies $\Delta\varepsilon_{mn}$, i.e., when the condition
$T_{\chi,\eta}^{\max}\gg(r+1)\Delta\varepsilon_{mn}$ is no more satisfied, the
integration of the incoherent contributions is carried out over a narrower
interval of the nuclear recoil energy.

With allowance for approximations (43)–(45), formulas (3) are simplified

$\displaystyle R(T_{A}^{\min})_{\text{c/i}}$ $\displaystyle=$ $\displaystyle
R_{0}(A,\chi)\gamma_{1}S_{1}(x_{\min},x^{\max}_{1})Q_{\rm
c/i}(A)\int_{T_{A}^{\min}}^{T_{A^{(*)},1}^{\max}}\\!\\!dT_{A}\Phi(T_{A})_{\rm
c/i}$ (49)
$\displaystyle+R_{0}(A,\chi)\gamma_{\eta}S_{\eta}(x_{\min},x^{\max}_{\eta})Q_{\rm
c/i}(A)\int_{T_{A}^{\min}}^{T_{A^{(*)},\eta}^{\max}}\\!\\!dT_{A}\Phi(T_{A})_{\rm
c/i}.$

The total incoherent-to-cohernet rate ratio as a function of the threshold
$T_{A}^{\min}$ is

$\displaystyle\dfrac{R_{\text{i}}}{R_{\text{c}}}(T_{A}^{\min})=\dfrac{Q_{i}(A)}{Q_{c}(A)}\\!\dfrac{\gamma_{1}S_{1}(x_{\min},x^{\max}_{1})\int_{T_{A}^{\min}}^{T_{A^{*},1}^{\max}}\\!\\!dT_{A}\Phi(T_{A})_{\rm
i}+\gamma_{\eta}S_{\eta}(x_{\min},x^{\max}_{\eta})\int_{T_{A}^{\min}}^{T_{A^{*},\eta}^{\max}}\\!\\!dT_{A}\Phi(T_{A})_{\rm
i}}{\gamma_{1}S_{1}(x_{\min},x^{\max}_{1})\int_{T_{A}^{\min}}^{T_{A,1}^{\max}}\\!\\!dT_{A}\Phi(T_{A})_{\rm
c}+\gamma_{\eta}S_{\eta}(x_{\min},x^{\max}_{\eta})\int_{T_{A}^{\min}}^{T_{A,\eta}^{\max}}\\!\\!dT_{A}\Phi(T_{A})_{\rm
c}}.\qquad$ (50)

One can notice (from these formulas) that both coherent and incoherent event
rates depend on the same energy threshold $T_{A}^{\min}$. It is the minimal
recoil nuclear kinetic energy, at which a measurable effect (ionization or
heat release) is still caused in the detector. When $T_{A}<T_{A}^{\min}$, this
effect is not visible at the background level, and no event can be registered.

This “logic” works well for the coherent $\chi A$ interaction, when the recoil
energy of the nucleus, $T_{A}$, is the only detectable ”signal” that the
(elastic) interaction occurred. This energy coincides with the energy, $\Delta
E_{\chi}$, transferred to the nucleus by the incident $\chi$ particle (lost by
the particle). Therefore, if $T_{A}<T_{A}^{\min}$, one always has the coherent
rate $R_{\text{c}}(T_{A}^{\min})=0$. This is exactly the meaning of the value
$T_{A}^{\min}$.

However, in the case of the incoherent rate $R_{\text{i}}(T_{A}^{\min})$, in
addition to the recoil nuclear energy, $T_{A}$, which is registered or even
not registered above the threshold $T_{A}^{\min}$, the other mandatory
signature of the incoherent (inelastic) event should be the nuclear
deexcitation Bednyakov:2021ppn ; Bednyakov:2021bty ; Bednyakov:2018mjd . It
can manifest itself as emission of $\gamma$-quanta or other (massive)
particles with the total energy $\Delta\varepsilon_{mn}$, which is the
difference between the $m$th and $n$th energy levels of the target nucleus.

This deexitation signature enhances the role of the incoherent rate
$R_{\text{i}}(T_{A}^{\min})$. The requirement for the rate absence,
$R_{\text{i}}(T_{A}<T_{A}^{\min})=0$, when the nuclear recoil energy is below
the threshold $T_{A}<T_{A}^{\min}$, is removed, since detection of the
deexcitation energy $\Delta\varepsilon_{mn}$ is independent of the recoil
energy $T_{A}$. The set of the deexcitation energies is fixed for the
individual nucleus and, as a rule, these energies are large enough. Therefore,
any detection of deexcitation energy specific for a given nucleus,
$\Delta\varepsilon_{mn}$, outside the recoil energy range
$T_{A}^{\min}<T_{A}\leq 100\div 200\text{~{}keV}$444Where the expected
coherent rate, $R_{\text{coh}}$, vanishes in sufficiently massive nuclei., can
be considered as a new possibility for the direct DM detection at the Earth.
To this end, a traditional low-background and low-threshold DM detector should
be surrounded by detector(s) aimed at registration of $\gamma$-quanta, whose
energy spectrum coincides with the deexcitation spectrum of the target
nucleus. It is obvious that there should be no other external sources of these
nuclei excitation (such as an electron beam or cosmic rays). Furthermore, the
annual variation (modulation) of this deexcitation signal Sahu:2020kwh ;
Sahu:2020cfq , would be an (extra) important signature of the (incoherent)
interaction of DM particles with nuclear matter.

##  4 Numerical estimates and discussions

### Dark matter distributions and kinematics

One can make numerical estimates of values from (3). It is always
$x_{\min}>0$. Further

$\displaystyle x^{\max}_{1}=\dfrac{v_{\rm
esc}}{v_{0}}+\eta_{1}\simeq\dfrac{550+220}{220}=3.5,\text{~{}~{}and~{}~{}}x^{\max}_{\eta}=\dfrac{v_{\rm
esc}}{v_{0}}+\eta_{1}+\eta_{\rm str}=3.5+\dfrac{300}{220}\simeq 4.9,\quad$
(51)

where $\eta_{1}=\dfrac{v_{\oplus}}{v_{0}}\simeq 1$ and $\eta_{\rm
str}=\xi\epsilon=\dfrac{{v}_{\rm str}}{v_{0}}\simeq 1.4$. Then555Due to
$\text{erf}(z+1)-\text{erf}(z-1)=\dfrac{2}{\sqrt{\pi}}\big{[}\int_{0}^{z+1}e^{-t^{2}}dt-\int_{0}^{z-1}e^{-t^{2}}dt\big{]}=\dfrac{2}{\sqrt{\pi}}\int_{z-1}^{z+1}e^{-t^{2}}dt$.
one can write

$\displaystyle S_{1}(x_{\min},x^{\max}_{1})$ $\displaystyle=$
$\displaystyle\dfrac{\text{erf}[x_{\min}+1]-\text{erf}[x_{\min}-1]}{2}-\dfrac{\text{erf}[x^{\max}_{1}+1]-\text{erf}[x^{\max}_{1}-1]}{2},$
$\displaystyle S_{\eta}(x_{\min},x^{\max}_{\eta},\epsilon)$ $\displaystyle=$
$\displaystyle\dfrac{\text{erf}[\dfrac{x_{\min}+\eta}{\epsilon}]-\text{erf}[\dfrac{x_{\min}-\eta}{\epsilon}]}{2\eta/\epsilon}-\dfrac{\text{erf}[\dfrac{x^{\max}_{\eta}+\eta}{\epsilon}]-\text{erf}[\dfrac{x^{\max}_{\eta}-\eta}{\epsilon}]}{2\eta/\epsilon}.$

With allowance for the general distribution function $S(x,\eta,\epsilon)$ from
(38), one has

$\displaystyle S_{1}(x_{\min},x^{\max}_{1})$ $\displaystyle=$ $\displaystyle
S(x_{\min},1,1)-S(x^{\max}_{1},1,1),$ $\displaystyle
S_{\eta}(x_{\min},x^{\max}_{\eta},\epsilon)$ $\displaystyle=$ $\displaystyle
S(x_{\min},\eta,\epsilon)-S(x^{\max}_{\eta},\eta,\epsilon).$

Figure 2: Dependence of the error function erf$(x)$ and function (38) on $x$
for $\eta=\\{1;\ 2;\ 3;\ 4\\}$ and $\epsilon=\\{0.5;\ 1;\ 2;\ 3\\}$. Moreover,
$x^{\max}_{\eta}(\eta)=\\{3.5;\ 4.5;\ 5.5;\ 6.5\\}$.

Figure 2 shows function (38) as a function of $x$ for $\eta=1;2;3;4$ and
$\epsilon=0.5;1;2;3$. Since $x^{\max}_{1}\equiv x^{\max}_{\eta=1}\simeq 2.5+1$
and $x^{\max}_{\eta}=2.5+\eta$, the maximal values of $x^{\max}_{\eta}$ as a
function of $\eta$ are as follows: $\\{3.5;\ 4.5;\ 5.5;\ 6.5\\}$.

It can be seen that for $x\leq 1$ the function (blue curve) $S(x_{\min}\leq
1,1,1)\geq 0.5$, while for $x\simeq 3.0\leq x^{\max}_{1}$ the function
$S(x_{\max}>3,1,1)\leq 0.001$. Therefore, due to the monotonic decrease of
this function, one can neglect its contribution when $x>x^{\max}_{1}=3.5$,
i.e., one can accept that $S(x>x^{\max}_{1}=3.5,1,1)\simeq 0$, and the
following approximation is always valid

$S_{1}(x_{\min},x^{\max}_{1})=S(x_{\min},1,1).$

One can see from Fig. 2 that the function $S(x,\eta,\epsilon)$ for
$\eta,\epsilon\neq 1$ does not vanish as fast as $S(x>3.5,1,1)$. Therefore,
when $\eta>1$, we will further use the expression

$\displaystyle S_{\eta}(x_{\min},x^{\max}_{\eta},\epsilon)$ $\displaystyle=$
$\displaystyle
S(x_{\min},\eta,\epsilon)-S({2.5}+\eta,\eta,\epsilon)\equiv\Delta
S(x_{\min},\eta,\epsilon),$ (52)

which becomes $S(x_{\min},1,1)$ for $\eta,\epsilon=1$.

In this approximation formula (46) for the differential event rates has the
form

$\displaystyle\frac{dR(T_{A})_{\text{c/i}}}{dT_{A}}$ $\displaystyle=$
$\displaystyle R_{0}(A,\chi)Q_{\rm c/i}(A)\Phi(T_{A})_{\rm c/i}C_{\rm
c/i}(x_{\min},\eta,\epsilon,T_{A}),\text{~{}~{}~{}where~{}~{}~{}}$ (53)
$\displaystyle C_{\rm c/i}(x_{\min},\eta,\epsilon,T_{A})$ $\displaystyle=$
$\displaystyle\Big{\\{}\gamma_{1}S(x_{\min},1,1)\Theta(T_{A/A^{*},1}^{\max}-T_{A})+\gamma_{\eta}\Delta
S(x_{\min},\eta,\epsilon)\Theta(T_{A/A^{*},\eta}^{\max}-T_{A})\Big{\\}}.$

The expressions for the maximal nuclear recoil energy used in (53) are given
below in formula (64). Similarly, for the total event rates from (49) one has

$\displaystyle\dfrac{R(T_{A}^{\min})_{\text{c/i}}}{R_{0}(A,\chi)}$
$\displaystyle=$ $\displaystyle\gamma_{1}S(x_{\min},1,1)Q_{\rm
c/i}(A)\int_{T_{A}^{\min}}^{T_{A^{(*)},1}^{\max}}dT_{A}\Phi(T_{A})_{\rm
c/i}\Theta(T_{A^{(*)},1}^{\max}-T_{A})$ $\displaystyle+$
$\displaystyle\gamma_{\eta}\Delta S(x_{\min},\eta,\epsilon)Q_{\rm
c/i}(A)\int_{T_{A}^{\min}}^{T_{A^{(*)},\eta}^{\max}}\\!\\!dT_{A}\Phi(T_{A})_{\rm
c/i}\Theta(T_{A^{(*)},\eta}^{\max}-T_{A}).$

This event rate depends on the detection threshold $T_{A}^{\min}$ through the
relation $x_{\min}=\Big{[}\dfrac{T_{A}^{\min}}{T_{A}^{0}}\Big{]}^{1/2}$.
According to (3),666Note that
$\dfrac{m_{\chi}c^{2}}{2}\Big{(}\dfrac{v_{0}}{c}\Big{)}^{2}=\dfrac{1}{2}\Big{[}\dfrac{m_{\chi}}{1\text{~{}GeV}}\Big{]}\Big{[}\dfrac{220\text{~{}km/sec}}{300\cdot
10^{3}\text{~{}km/sec}}\Big{]}^{2}10^{6}\text{keV}\simeq
0.269\Big{[}\dfrac{rm_{A}}{1\text{~{}GeV}}\Big{]}\text{keV}$.  one has

$\displaystyle T_{A}^{0}$ $\displaystyle\equiv$
$\displaystyle\mu_{A}\dfrac{m_{\chi}v_{0}^{2}}{2}=\dfrac{4r}{(1+r)^{2}}\dfrac{m_{\chi}c^{2}}{2}\Big{(}\dfrac{v_{0}}{c}\Big{)}^{2}\simeq\dfrac{4r^{2}0.269}{(1+r)^{2}}\Big{[}\dfrac{m_{A}}{1\text{~{}GeV}}\Big{]}\text{keV}.$

Therefore, the dimensionless minimal velocity of the DM particle (relative to
the nucleus at rest) is

$\displaystyle x_{\min}$ $\displaystyle\simeq$
$\displaystyle\sqrt{\dfrac{\Big{[}\dfrac{T_{A}^{\min}}{1\,\text{keV}}\Big{]}}{\dfrac{1.076r^{2}}{(1+r)^{2}}\Big{[}\dfrac{m_{A}}{1\text{~{}GeV}}\Big{]}}}\simeq\dfrac{(1+r)}{1.037r}\sqrt{\dfrac{\Big{[}\dfrac{T_{A}^{\min}}{1\,\text{keV}}\Big{]}}{\Big{[}\dfrac{m_{A}}{1\text{~{}GeV}}\Big{]}}}.$
(55)

Figure 3: Dependence of $x_{\min}$ on $m_{A}$ and $m_{\chi}$ at the threshold
$T^{\min}_{A}=1\,$keV. For fixed $m_{A}$ and $m_{\chi}$, the value of
$x_{\min}$ is proportional to the square root of $T^{\min}_{A}$ according to
(55).

Comparing Fig. 3 with Fig. 2, one can conclude that for very small $m_{\chi}$
and sufficiently large $m_{A}$ the function $S(x,\eta,\epsilon)$ from (38)
vanishes (because of the large value of $x=x_{\min}\geq 8$) for almost all
reasonable values of parameters. The latter means that both coherent and
incoherent differential event rates vanish as well according to formulas (53).

Expression (4) for the total event rates also depends on the maximal
dimensionless velocities of the DM distributions, $x^{\max}_{1}\simeq 3.5$ and
$x^{\max}_{\eta}=2.5+\eta$ (i.e., on the flow velocity $\eta$), via relations
(39) and (40).

Therefore, to calculate the event counting rates with formulas (53) and (4),
one has to define the following values:

$m_{\chi},\ n_{\chi}=\rho/m_{\chi},\ v_{0}^{2},\quad\gamma_{1},\
\gamma_{\eta},\ \eta,\ \epsilon,\text{~{}~{}as well
as~{}~{}}m_{A},N_{A},T_{A}^{\min},\Delta\varepsilon_{mn},$

and use the common parameters $G_{\text{F}}=1.166\times
10^{-5}\mbox{GeV}^{-2}$ and $m=0.938\mbox{~{}GeV}\simeq 1\mbox{~{}GeV}.$

To estimate the $T_{A}$-dependent coherent and incoherent form-factor
functions (45), the Helm nuclear form factor parameterization PhysRev.104.1466
is used

$F(\bm{q})=3\dfrac{j_{1}(qR_{0})}{qR_{0}}e^{-\dfrac{(sq)^{2}}{2}},\quad
j_{1}(x)=\dfrac{\sin x}{x^{2}}-\dfrac{\cos x}{x},\ s=0.9\text{~{}fm},\
R_{0}=1.14A^{1/3}\text{~{}fm}.$

According to the definition (13) of the weak current scalar product
Bednyakov:2021pgs , the effective coupling constants of the form
$\alpha_{f}=\chi_{V}h^{f}_{V},\ \beta_{f}=\chi_{V}h^{f}_{A},\
\gamma_{f}=\chi_{A}h^{f}_{V},\ \delta_{f}=\chi_{A}h^{f}_{A}$ are free
parameters. In the SM, neutrinos have $\chi_{V}=-\chi_{A}=1$, and nucleons
have

$h_{V}^{p}=g_{V}^{p}=\frac{1}{2}-2\sin^{2}\theta_{W},\quad
h_{V}^{n}=g_{V}^{n}=-\frac{1}{2};\quad
h_{A}^{p}=g_{A}^{p}=\frac{g_{A}}{2},\quad
h_{A}^{n}=g_{A}^{n}=-\frac{g_{A}}{2}.$

Then the effective parameters in the SM are the following:

$\alpha_{f}=\chi_{V}h^{f}_{V}=+g_{V}^{f},\quad\delta_{f}=\chi_{A}h^{f}_{A}=+g_{A}^{f},\quad\gamma_{f}=\chi_{A}h^{f}_{V}=-g_{V}^{f},\quad\beta_{f}=\chi_{V}h^{f}_{A}=-g_{A}^{f}.$

With allowance for $\sin^{2}\theta_{W}=0.239$ and $g_{A}=1.27$, one has the
following numerical values:

$\displaystyle\alpha_{p}$ $\displaystyle=$
$\displaystyle+\frac{1}{2}-2\sin^{2}\theta_{W}=-\gamma_{p}\simeq
0.02,\quad\delta_{p}=\frac{g_{A}}{2}=-\beta_{p}\simeq 0.64;$
$\displaystyle\alpha_{n}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}=-\gamma_{n}=-0.5,\qquad\qquad\qquad\delta_{n}=-\frac{g_{A}}{2}=-\beta_{n}\simeq-0.64.$
(56)

These parameters determine the generalized weak nuclear charges from (44)

$\displaystyle Q_{c}(A)$ $\displaystyle=$
$\displaystyle\big{[}\sum_{f=p,n}\alpha_{f}A_{f}\big{]}^{2}+\big{[}\sum_{f=p,n}\delta_{f}{\Delta
A_{f}}\big{]}^{2}=\big{[}\alpha_{p}A_{p}+\alpha_{n}A_{n}\big{]}^{2}+\big{[}\delta_{p}{\Delta
A_{p}}+\delta_{n}{\Delta A_{n}}\big{]}^{2},$ $\displaystyle Q_{i}(A)$
$\displaystyle=$
$\displaystyle\sum_{f=p,n}A^{f}\big{[}\alpha^{2}_{f}+3\delta^{2}_{f}\big{]}=A^{p}\big{[}\alpha^{2}_{p}+3\delta^{2}_{p}\big{]}+A^{n}\big{[}\alpha^{2}_{n}+3\delta^{2}_{n}\big{]}.$
(57)

In calculation of the cross sections and their ratios in the rest frame of the
nucleus, one varies only the nuclear recoil energy $T_{A}$ in the interval
from $T_{A}^{\min}=\epsilon_{A}^{\text{threshold}}$ to the maximal
$T_{A}^{\max}$, which is determined by the DM distribution from condition
(40). With allowance for (43)–(45) and ”cosmic factors”

$\displaystyle C_{\rm
i/c}(x_{\min},\eta,\epsilon,T_{A})\\!=\\!\gamma_{1}S(x_{\min},1,1)\Theta(T_{A^{*}/A,1}^{\max}-T_{A})+\gamma_{\eta}\Delta
S(x_{\min},\eta,\epsilon)\Theta(T_{A^{*}/A,\eta}^{\max}-T_{A}),\quad$ (58)

ratio (47) between the differential incoherent and coherent rates becomes

$\displaystyle R_{\rm I}(A,T_{A})$ $\displaystyle=$
$\displaystyle{\dfrac{dR_{\text{i}}}{dT_{A}}}/{\dfrac{dR_{\text{c}}}{dT_{A}}}=R_{\rm
I}(A)R_{A}(T_{A})R_{C}(x_{\min},\eta,\epsilon,T_{A}),\quad\text{where}$ (60)
$\displaystyle R_{\rm I}(A)=\dfrac{AQ_{\rm i}(A)}{Q_{\rm c}(A)},\quad
R_{A}(T_{A})=\dfrac{1-|F(T_{A})|^{2}}{A|F(T_{A})|^{2}}\quad\text{and}$
$\displaystyle R_{C}(x_{\min},\eta,\epsilon,T_{A})=\dfrac{C_{\rm
i}(x_{\min},\eta,\epsilon,T_{A})}{C_{\rm c}(x_{\min},\eta,\epsilon,T_{A})}.$

With (43)–(45), the ratio of the incoherent total event rate to the coherent
one (50) takes the form

$\displaystyle\dfrac{R_{\text{i}}}{R_{\text{c}}}(T_{A}^{\min})$
$\displaystyle=$
$\displaystyle\dfrac{Q_{i}(A)}{Q_{c}(A)}\Theta(T_{A}-T_{A}^{\min})\times$
$\displaystyle\times$
$\displaystyle\dfrac{\gamma_{1}S(x_{\min},1,1)\int_{T_{A}^{\min}}^{T_{A^{*},1}^{\max}}dT_{A}\Phi_{\rm
i}(T_{A})+\gamma_{\eta}\Delta
S(x_{\min},\eta,\epsilon)\int_{T_{A}^{\min}}^{T_{A^{*},\eta}^{\max}}dT_{A}\Phi_{\rm
i}(T_{A})}{\gamma_{1}S(x_{\min},1,1)\int_{T_{A}^{\min}}^{T_{A,1}^{\max}}dT_{A}\Phi_{\rm
c}(T_{A})+\gamma_{\eta}\Delta
S(x_{\min},\eta,\epsilon)\int_{T_{A}^{\min}}^{T_{A,\eta}^{\max}}dT_{A}\Phi_{\rm
c}(T_{A})}.$

The expression $R_{A}(T_{A})$ from (60) does not depend on the interaction, it
completely defines the dependence of $R_{\rm I}(A,T_{A})$ from (60) on
$T_{A}$. However, in contrast to the nonrelativistic differential incoherent-
to-coherent cross section ratio Bednyakov:2022dmc , which has the form

$\displaystyle\dfrac{\dfrac{d\sigma^{\text{total}}_{\text{inc}}}{dT_{A}}(\chi
A\to\chi A^{*})}{\dfrac{d\sigma^{\text{total}}_{\text{coh}}}{dT_{A}}(\chi
A\to\chi A)}=R_{I}(A)R_{A}(T_{A}),\qquad$ (62)

the ratio of event rates from (60) has the meaning only for the recoil
energies in the interval

$T_{A}^{\min}\leq T_{A}\leq T_{A^{*},\eta}^{\max}(r,\Delta\varepsilon_{mn}).$
(63)

If $T_{A}>T_{A^{*},\eta}^{\max}(r,\Delta\varepsilon_{mn})$, both differential
and total inelastic event rates vanish (the inelastic process is not
possible). If $T_{A}<T_{A}^{\min}$, no recoil energy can be measured. Taking
into account that with $r$ from (40) the ”elastic” nuclear recoil energy has
the form

$T_{A,\eta}^{\max}(r,0)\equiv
T_{A^{*},\eta}^{\max}(r,\Delta\varepsilon_{mn}=0)=\dfrac{4T_{\eta}^{\max}r}{(r+1)^{2}},$

the right side of condition (63) can be expressed as follows:

$\displaystyle T_{A^{*},\eta}^{\max}(r,\Delta\varepsilon_{mn})$
$\displaystyle=$ $\displaystyle
T_{A,\eta}^{\max}(r,0)\dfrac{1-\omega/2+\sqrt{1-\omega}}{2}\leq
T_{A,\eta}^{\max}(r,0),$ (64) $\displaystyle\text{where~{}~{}}\omega$
$\displaystyle\equiv$
$\displaystyle\dfrac{(r+1)\Delta\varepsilon_{mn}}{T_{\eta}^{\max}},\text{~{}~{}and~{}according
to footnote \ref{footnote:T0chi}},$ $\displaystyle T_{\eta}^{\max}$
$\displaystyle=$ $\displaystyle(x^{\max}_{\eta})^{2}T^{0}_{\chi}\simeq
0.27\big{(}2.5+\eta\big{)}^{2}\Big{[}\dfrac{rm_{A}}{1\text{~{}GeV}}\Big{]}\text{keV},\text{~{}~{}and~{}~{}}\eta\geq
1.$

One can see that the maximal recoil energy in the elastic collision is always
larger than the same value in the inelastic collision. Formula (64) has the
meaning, i.e., the value $T_{A^{*},\eta}^{\max}(r,\Delta\varepsilon_{mn})$ is
a real (non-negative) number, and only under condition (41) is written as777If
$\omega>1$, the incoherent process with these $m_{A}$, $m_{\chi}$, $\eta$ and
$\Delta\varepsilon_{mn}$ is impossible.

$\displaystyle\omega$ $\displaystyle=$
$\displaystyle(r+1)\dfrac{\Delta\varepsilon_{mn}}{T^{\max}_{\eta}}\leq
1\text{~{}~{}~{}or~{}~{}~{}}0.27\big{(}2.5+\eta\big{)}^{2}\geq\Big{(}1+\dfrac{1}{r}\Big{)}\dfrac{\Big{[}\dfrac{\Delta\varepsilon_{mn}}{1\text{~{}keV}}\Big{]}}{\Big{[}\dfrac{m_{A}}{1\text{~{}GeV}}\Big{]}}.$
(65)

The case $\omega=1$ means that only the maximal (for given DM distribution)
energy of the DM particle, $T_{\eta}^{\max}$, is enough to excite the target
nucleus at the level with energy $\Delta\varepsilon_{mn}$, provided the mass
ratio of the DM particle and the nucleus is $r$. Moreover, this maximal energy
is directly connected with the nuclear excitation energy,
$T_{\eta}^{\max}=(r+1)\Delta\varepsilon_{mn}$. In other words, if the nucleus
is ”so lucky” to have the excitation level just with the energy
$\Delta\varepsilon_{mn}=T_{\eta}^{\max}/(r+1)$, the excitation of the nucleus
is possible exactly to this ”unique level”. In this case, the kinetic energy
of the nuclear recoil, which inevitably accompanies the excitation of this
particular nuclear level is, according to (64),

$\displaystyle T_{A^{*},\eta}^{,\omega=1}(r)$ $\displaystyle=$
$\displaystyle\dfrac{T_{A,\eta}^{\max}(r,0)}{4}=\Big{[}\dfrac{r}{r+1}\Big{]}^{2}0.27\big{(}2.5+\eta\big{)}^{2}\Big{[}\dfrac{m_{A}}{1\text{~{}GeV}}\Big{]}\text{keV}.$
(66)

From formula (66) one has the ”threshold” value of $r$ (and $m_{\chi}$), at
which this ”unique excitation” takes place, in the following form:

$\displaystyle\dfrac{1}{r}=\sqrt{\dfrac{\Big{[}\dfrac{m_{A}}{1\text{~{}GeV}}\Big{]}}{\Big{[}\dfrac{T_{A^{*},\eta}^{\omega=1}}{1\text{~{}keV}}\Big{]}}}\sqrt{0.27}\big{(}2.5+\eta\big{)}-1\text{~{}~{}or~{}~{}}m_{\chi}^{\min}(T_{A}^{*})=\dfrac{m_{A}}{\dfrac{\sqrt{0.27}(2.5+\eta)}{\sqrt{\Big{[}\dfrac{T_{A^{*},\eta}^{\omega=1}}{1\text{~{}keV}}\Big{]}\Big{[}\dfrac{m_{A}}{1\text{~{}GeV}}\Big{]}^{-1}}}-1}.\quad$
(67)

Vanishing of the denominator in (67) means that the excitation to the nuclear
level with $\Delta\varepsilon_{mn}=T^{\max}_{\eta}\dfrac{1}{r+1}$ accompanied
by the recoil energy of the excited nucleus $T_{A^{*},\eta}^{\omega=1}$ from
(66) is possible only for $m_{\chi}\simeq\infty$. Clearly, this process does
not occur for any $m_{\chi}$.

Figure 4 shows the behavior of the maximal recoil energy of the excited
nucleus, $T_{A^{*},\eta}^{\max}$, as a function of the mass ratio of the DM
particle and the nucleus, $r$, for some typical maximal energies of the DM
particles, $T^{\max}_{\eta}$, and some nuclear excitation energies $\Delta E$.

Figure 4: Theoretical dependence of the maximal recoil energy of the excited
nucleus, $T_{A^{*},\eta}^{\max}$, (expression (64)) on the mass ratio of the
DM particle and the nucleus, $r$, for four ”trial” maximal energies of the DM
particles, $T^{\max}_{\eta}$, and five excitation energies $\Delta E$.

One can see that the maximal recoil energy of excited nuclei,
$T_{A^{*},\eta}^{\max}$, is achieved when the masses of the $\chi$ particle
and the nucleus coincide ($r=1$). As the ratio $r$ increases, the maximal
recoil energy of the excited nucleus rather quickly decreases. The sharp drop
of some curves down to zero occurs when the increasing
$\omega=\dfrac{(r+1)\Delta\varepsilon_{mn}}{T_{\eta}^{\max}}$ from (65)
reaches one. For example, in the top plot, this happens at
$T^{\max}_{\eta}=250$ keV, $\Delta E=50$ keV and $r=4$. There are noticeable
regions of the parameters $r$ and $T^{\max}_{\eta}$ (for example, at
sufficiently high excitation energies of nuclear levels, $\Delta E\geq 200$
keV), where $\omega>1$. In these regions, $T_{A^{*},\eta}^{\max}$ is set equal
to zero, which reflects the impossibility of the nuclear excitation.

Figure 5: Dependence of the maximal recoil energy of the excited nuclei,
$T_{A^{*},\eta}^{\max}(r,\Delta\varepsilon_{mn})$, on $m_{\chi}$ for typical
targets and the DM-flow velocities $\eta=1;2;4$. The value $\eta=1$
corresponds to the Standard Halo Model.

In Fig. 5, the maximal recoil energy of the excited nucleus (64) is given as a
function of the mass of the $\chi$ particle for some dimensionless velocities
of the DM flow $\eta$ and some typical target nuclei (with specific values of
their masses and excitation energies, given in Table 1 below). One can see
that in the case of the Standard Halo Model ($\eta=1$) the inelastic
excitation of the quite heavy nuclei 127I and 131Xe as well as 73Ge (due to
their low excitation energies) is allowed for any $m_{\chi}$. The recoil
energies of these (excited) nuclei reach the MeV scale. For lighter nuclei
(fluorine, gallium and 74Ge), despite the difference in their energy
excitation, the DM particles from the SHM distribution are unable to cause the
inelastic process. With $\eta=2$, the situation changes insignificantly. The
recoil energies for heavy nuclei only increase, and the inelastic channel in
gallium opens at sufficiently large $m_{\chi}$. Finally, in the case of a
strong DM flow ($\eta=4$), the energy of DM particles is quite enough to
excite all the nuclei with noticeable recoil energies, which increase rapidly
with $m_{\chi}$. A ”small exception” is fluorine, where the recoil energy
increase is not so ”rapid” due to the small mass of the fluorine nucleus, and
74Ge, for which the inelastic channel opens with large DM masses
($m_{\chi}\geq 200$ GeV) due to the very high excitation energy (596 keV).

”Conversion” of relation (64) gives $T_{A^{*}}$-dependence of the minimal DM
particle mass, $m_{\chi}^{\min}$, starting from which the excitation of the
nucleus becomes possible together with the recoil energy $T_{A^{*}}$ of the
nucleus. This dependence has the form

$\displaystyle m^{\min}_{\chi}(T_{A^{*}},A,\eta)$ $\displaystyle=$
$\displaystyle\dfrac{m_{A}T_{A}}{\sqrt{2m_{A}T_{A}(x^{\max}_{\eta})^{2}v^{2}_{0}}-(T_{A}+\Delta\varepsilon_{mn})}=$
$\displaystyle=$
$\displaystyle\dfrac{m_{A}}{2(2.5+\eta)\sqrt{0.27}\sqrt{\Big{[}\dfrac{m_{A}}{1\text{~{}GeV}}\Big{]}\Big{[}\dfrac{T_{A^{*}}}{1\text{~{}keV}}\Big{]}^{-1}}-1-\dfrac{\Delta\varepsilon_{mn}}{T_{A^{*}}}}.$

Figure 6: The minimal masses $m_{\chi}^{\min}$ are given (right y-axis) as
functions of the recoil energy, $T_{A^{*}}$, of the (excited) nuclei 131Xe,
127I, and 73Ge for the DM distributions with the maximal flow velocities
$\eta=1$ (green curve), $\eta=2$ (red) $\eta=3$ (purple) and $\eta=4$ (dark
curve). The left vertical y-axis illustrates the behavior of the square form
factor of the nucleus.

In Fig. 6 and Fig. 7 the $m^{\min}_{\chi}$-function (4) is shown for the DM
distributions with maximal velocities $\eta=1$ (green curve), $\eta=2$ (red
curve), $\eta=3$ (purple curve) and $\eta=4$ (dark curve).

Figure 7: Same as in Fig. 6, but for the 19F and 69Ga nuclei.

From Fig. 6 one can seen that if $\eta=2$ (red curve), the recoil energy
$T_{A}\simeq 50~{}$keV in 131Xe can be caused only by the DM particle with
$m_{\chi}\geq 25$ GeV/$c^{2}$. For the SHM-model ($\eta=1$, green curve), the
same nuclear recoil $T_{A}\simeq 50~{}$keV can only be produced by the DM
particle, with $m_{\chi}\geq 40$ GeV/$c^{2}$. For nuclei in Fig. 6 (with ”a
small exception” of small $T_{A}$), the inelastic channel of interaction can
be open for almost any value of $m_{\chi}$.

In Fig. 7, similar functions are shown for the light nucleus 19F
($\Delta\varepsilon_{mn}=110\,$keV) and the 69Ga nucleus
($\Delta\varepsilon_{mn}=319\,$keV). Comparing with Fig. 6, one can notice
some difference in the behavior of the curves888It is due to a smaller value
of the ratio $\dfrac{m_{A}}{\Delta\varepsilon_{mn}}$ for 19F and 69Ga than for
131Xe and 73Ge, respectively, despite the relatively small difference in their
excitation energies.. For both nuclei the green curve ($\eta=1$) ”lies”
completely at $m_{\chi}=0$ (it is not visible in the plots). This means that
the DM particle with the SHM distribution is unable to excite the 19F and 69Ga
nuclei (together with production of the recoil energy, $T_{A}$, within the
intervals indicated in the plots). With increase of the maximal DM velocity
(purple and black curves), the excitation of 19F becomes possible with
accompanying recoil energy $T_{A}$ in the interval $25\div 575\,$keV ($40\div
320\,$keV) when $\eta=4(3)$, respectively. Note that to achieve the recoil
energy $T_{A^{*}}$ at the edges of these intervals, an infinite initial
kinetic energy is required, which is equivalent to $m_{\chi}\simeq\infty$.
This is seen from the sharp pole peaks in the curves and forbidden regions for
$T_{A}$ outside these peaks both at small and large $T_{A}$. These (in this
case two) poles correspond to the zero denominator in formula (4). In Fig. 6,
the similar pole-like behavior can be noticed for 129I and 131Xe in the region
of very small $T_{A}$.

The excitation of 69Ga (to the lower level) can occur already at $\eta=2$, and
is possible only in some limited intervals of the recoil energy. However,
unlike 19F, these allowed $T_{A}$ intervals are located almost completely at
(very) large $T_{A}$ (see Fig. 7), where the coherent contribution is already
vanished, because the square of the form factor (blue dotted line) has long
been zero. The range of allowed recoil energy, $T_{A^{*}}$, which is
inevitably connected with the excitation of the 19F and 69Ga nuclei (see Fig.
7), is limited, and its width depends on the maximal velocity of the DM
particle. For example, if with the 19F target the 110-keV $\gamma$-ray is
registered together with $T_{A^{*}}\simeq 370\,$keV, one concludes that the DM
particle which caused this $\gamma$-quantum has $m_{\chi}\geq$ 100 GeV$/c^{2}$
and belongs to the DM flow with $\eta\simeq 4$. Similar conclusions can be
made for the 69Ga nucleus.

In Fig. 8, the nuclear factor $R_{A}(T_{A})$ from formula (60) is shown as a
function of $T_{A}$ for some typical target nuclei.

Figure 8: ”Nuclear” ratios $R_{A}(T_{A})$ from (60) as functions of recoil
energy of the nucleus $T_{A}$.

Figure 8 illustrates the effect of ”pure nuclear incoherence”, which is
completely specified by the squared nuclear form factor behavior with
increasing $T_{A}$. The expression $R_{A}(T_{A})$ from (60) has a pole when
$|F(T_{A})|^{2}\simeq 0$. This condition was used to define the maximal recoil
energy $T_{A}^{\max}$ in the figures (x-axis). The nuclear factor
$R_{A}(T_{A})$ allows one to estimate the prospects of the target nucleus
(with specific $m_{A}$ and $\Delta\varepsilon_{mn}$) for registration of the
inelastic excitation by the DM particle.

ANucleus$(Z,N)$ | $Z_{+}$ | $Z_{-}$ | $\Delta Z$ | $N_{+}$ | $N_{-}$ | $\Delta N$ |  $\Delta\epsilon$, keV | $R_{\rm SM}(A)$
---|---|---|---|---|---|---|---|---
${}^{4}\text{He}^{0+}(2,2)$ | 1 | 1 | 0 | 1 | 1 | 0 |  202100+,$21010^{0-}$,218402- | 23.44
${}^{6}\text{Li}^{1+}(3,3)$ | 2 | 1 | 1 | 2 | 1 | 1 |  21863+,35630+,43102+ | 23.44
${}^{12}\text{C}^{0+}(6,6)$ | 3 | 3 | 0 | 3 | 3 | 0 |  44382+,76540+,96413- | 23.44
${}^{19}\text{F}^{\frac{1}{2}+}(9,10)$ | 5 | 4 | 1 | 5 | 5 | 0 |  110${}^{\frac{1}{2}-}$, 197${}^{\frac{5}{2}+}$,1346${}^{\frac{5}{2}-}$,1459${}^{\frac{3}{2}-}$ | 20.69
${}^{23}\text{Na}^{\frac{3}{2}+}(11,12)$ | 6 | 5 | 1 | 7 | 5 | 2 |  440${}^{\frac{5}{2}+}$,2076${}^{\frac{7}{2}+}$,2391${}^{\frac{1}{2}+}$ | 21.18
${}^{35}\text{Cl}^{\frac{3}{2}+}(17,18)$ | 9 | 8 | 1 | 10 | 8 | 2 |  1220${}^{\frac{1}{2}+}$,1763${}^{\frac{5}{2}+}$,2693${}^{\frac{3}{2}+}$ | 21.06
${}^{40}\text{Ar}^{0+}(18,22)$ | 9 | 9 | 0 | 11 | 11 | 0 |  14612+,21210+,25242+ | 19.22
${}^{48}\text{TI}^{0+}(24,24)$ | 12 | 12 | 0 | 12 | 12 | 0 |  9842+,22964+,24212+ | 19.84
${}^{69}\text{Ga}^{\frac{3}{2}-}(31,38)$ | 16 | 15 | 1 | 20 | 18 | 2 |  319${}^{\frac{1}{2}-}$,574${}^{\frac{5}{2}-}$, 872${}^{\frac{3}{2}-}$ ,1029${}^{\frac{1}{2}-}$ | 19.14
${}^{73}\text{Ge}^{\frac{9}{2}+}(32,41)$ | 16 | 16 | 0 | 25 | 16 | 9 |  13.3${}^{\frac{5}{2}+}$,67${}^{\frac{1}{2}-}$, 69${}^{\frac{7}{2}+}$,354${}^{\frac{5}{2}-}$ | 17.00
${}^{74}\text{Ge}^{0+}(32,42)$ | 16 | 16 | 0 | 21 | 21 | 0 |  5962+,12042+, 14830+ | 18.00
${}^{127}\text{I}^{5/2+}(53,74)$ | 27 | 26 | 1 | 39 | 35 | 4 |  58${}^{\frac{7}{2}+}$,203${}^{\frac{3}{2}+}$,375${}^{\frac{1}{2}+}$, 418${}^{\frac{5}{2}+}$ | 17.01
${}^{129}\text{Xe}^{\frac{1}{2}+}(54,75)$ | 27 | 27 | 0 | 38 | 37 | 1 |  40,236,318${}^{\frac{3}{2}+}$,322${}^{\frac{5}{2}+}$,412${}^{\frac{1}{2}+}$ | 17.13
${}^{131}\text{Xe}^{\frac{3}{2}+}(54,77)$ | 27 | 27 | 0 | 40 | 37 | 3 |  80${}^{\frac{1}{2}+}$,164${}^{\frac{11}{2}+}$, 365${}^{\frac{5}{2}+}$, 405${}^{\frac{3}{2}+}$ | 16.72
${}^{133}\text{Cs}^{\frac{7}{2}+}(55,78)$ | 28 | 27 | 1 | 46 | 32 | 6 |  81,161${}^{\frac{5}{2}+}$,384${}^{\frac{3}{2}+}$ 437${}^{\frac{1}{2}+}$,633${}^{\frac{11}{2}+}$ | 16.72
${}^{207}\text{Pb}^{\frac{1}{2}-}(82,125)$ | 41 | 41 | 0 | 63 | 62 | 1 |  570${}^{\frac{5}{2}-}$, 898${}^{\frac{3}{2}-}$1633${}^{\frac{13}{2}+}$, 2340${}^{\frac{7}{2}-}$ | 15.86
Table 1: Characteristics of some target nuclei used. Here $Z=A_{p},\ N=A_{n}$
and $\Delta A_{p}\equiv\Delta Z=Z_{+}-Z_{-}$, $\Delta A_{n}\equiv\Delta
N=N_{+}-N_{-}$. The total nuclear spin is calculated by the formula
$\frac{\Delta Z}{2}+\frac{\Delta N}{2}$. The indices $\pm$ denote the
direction of the nucleon spin with respect to the quantization axis. Values
for $R_{\rm SM}(A)\equiv R_{I}(A)$ from formula (69) are obtained with the SM
coupling constants $\alpha_{p}\simeq 0.02,\ \alpha_{n}=-0.5,\
\delta_{p}=-\delta_{n}\simeq 0.64$ from (4).

### Event rates with the Standard Model weak couplings

The incoherent-to-coherent rate ratio dependence on the nature of the
$\chi$-nucleon interaction is completely determined by the first ”charge-
nuclear” factor in (60), written as follows:

$\displaystyle R_{\rm I}(A)=A\dfrac{Q_{\rm i}(A)}{Q_{\rm
c}(A)}=\dfrac{(A_{p}+A_{n})[A^{p}(\alpha^{2}_{p}+3\delta^{2}_{p})+A^{n}(\alpha^{2}_{n}+3\delta^{2}_{n})]}{(\alpha_{p}A_{p}+\alpha_{n}A_{n})^{2}+(\delta_{p}{\Delta
A_{p}}+\delta_{n}{\Delta A_{n}})^{2}}.$ (69)

In the nonrelativistic approximation, it is determined only by the effective
coupling constants $\alpha_{p,n}$ and $\delta_{p,n}$ from (2), which are
”weighted” with the proton–neutron structure parameters of the nucleus. This
factor $R_{\rm I}(A)$ can either enhance or weaken the ”pure” effect of the
nuclear structure shown in Fig. 8. The SM-values of this factor for a number
of target nuclei are given in Table 1 and Fig. 9.

Figure 9: Nuclear charge factors from formulas (4) and (69) with SM parameters
from (4).

The ”charge-nuclear” factors $R_{\rm SM}(A)$ from (69) give more than an order
of magnitude increase in the ”incoherence effect”. Therefore, the dominance of
the inelastic contribution can begin at twice as low recoil energies, i.e.,
the incoherent nuclear excitation may be accompanied by lower recoil energy
$T_{A}$.

The third, ”cosmic” factor in formula (60), i.e., the ratio
$R_{C}(x_{\min},\eta,\epsilon,T_{A})$ from (60), is always equal to one if
only one DM velocity distribution (for example, SHM) is allowed. If one takes
into account, as presented in formula (60), the two DM distributions
simultaneously (SHM and $\eta>1$), the relation
$R_{C}(x_{\min},\eta,\epsilon,T_{A})$ is equal to one or zero (see lower plot
in Fig. 10) in almost entire range of parameters.

Figure 10: For 73Ge as an example, the coefficient $C_{\rm
i}(x_{\min},\eta,\epsilon,T_{A})$ (upper plot), $C_{\rm
c}(x_{\min},\eta,\epsilon,T_{A})$ (middle plot) from (53), and their ratios
$R_{C}(x_{\min},\eta,\epsilon,T_{A})$ (lower plot) from (60) are given as
functions of the nuclear recoil energy $T_{A}$ for $\eta=1\div 4$,
$T_{A}^{\min}=1\,$keV, $m_{\chi}=75\,$GeV$/c^{2}$ and
$\epsilon=1,\gamma_{1}=\gamma_{\eta}=0.5$. The dashed line shows the squared
73Ge form factor.

The zero value occurs when increasing $T_{A}$ becomes greater than
$T_{A^{*},\eta}^{\max}$ but still smaller than $T_{A,\eta}^{\max}$. In this
case, the numerator of this ratio vanishes (upper plot in Fig. 10) along with
the incoherent contribution. There are also cases, where the first term in the
numerator of $R_{C}(x_{\min},\eta,\epsilon,T_{A})$ corresponding to smaller
velocity $\eta$ vanishes, for example, a little earlier than it happens with
the first term in the denominator. This situation can be seen from the
comparison of the top and middle plots in Fig. 10. As a result, a
characteristic ”dip” occurs in $R_{C}(x_{\min},\eta,\epsilon,T_{A})$, shown in
the bottom plot in Fig. 10. The width of this dip is determined by
$\Delta\epsilon_{mn}$. As $m_{\chi}$ increases, this dip shifts to higher
values of $T_{A}$, where the coherent contribution almost completely
disappears.

The influence of $R_{C}(x_{\min},\eta,\epsilon,T_{A})$ from (60) on the total
incoherent-to-coherent rate ratio (60) for 127I is shown in Fig. 11.

Figure 11: The role of the two-DM-component factor
$R_{C}(x_{\min},\eta,\epsilon,T_{A})$ from (60) is shown for 127I. It
manifests itself in differences of the solid curves $R(A,T_{A})$ marked with
different colors for different $m_{\chi}$, from the ”pure” product
$R_{A}(T_{A})R_{\rm SM}(A)$ (red bold dashed line) only for small
$m_{\chi}\leq 25~{}$GeV$/c^{2}$. The colors of the right-side y-axes
correspond to these masses. The blue thin dotted line shows the square of the
127I form factor $|F_{A}(T_{A})|^{2}$ (left y-axis).

The influence is noticeable only for sufficiently small values of
$m_{\chi}=10\div 35$ GeV$/c^{2}$. For example, if $\eta=2$ (middle plot), the
incoherent process in 127I is impossible for $m_{\chi}=10\,$ GeV$/c^{2}$. The
corresponding (yellow) curve vanishes for all $T_{A}$. The green curve for
$m_{\chi}=25\,$ GeV$/c^{2}$ (and $\eta=2$) is strongly ”suppressed” by the
factor $R_{C}(x_{\min},\eta,\epsilon,T_{A})$. It has a characteristic dip for
middle $T_{A}$ and vanishes earlier with increasing $T_{A}$ than the red
dashed curve of the ”pure” product $R_{A}(T_{A})R_{\rm SM}(A)$. All these
”features” point to the insignificance of the inelastic contribution in the
case of small $m_{\chi}$.

Figure 12: The ”cosmic factors” and $R_{C}(x_{\min},\eta,\epsilon,T_{A})$ as
functions of $T_{A}$ are shown in 73Ge for only one DM distribution with
different parameters $\eta$. The blue dotted line shows $|F_{A}(T_{A})|^{2}$.

In general, the role of the factor $R_{C}(x_{\min},\eta,\epsilon,T_{A})$,
which is usually equal to one (see Fig. 10 and 11), disappears as $m_{\chi}$
and $\eta$ increase999The cases, where it is not true (too small $m_{\chi}$
and/or too large $\Delta\epsilon_{mn}$), require more careful analysis.. This
means that the kinetic energy of the incident DM particle becomes sufficient
to excite the nucleus and to produce simultaneously large enough nuclear
recoil, $T_{A}$. Therefore, the inelastic event rate101010In the
nonrelativistic SM-weak interaction approximation. appears as a permanent
component of the DM interaction, in contrast to the elastic one, which quickly
disappears with increasing $T_{A}$.

The ”peculiar” $T_{A}$-dependence of the rate ratios (60) shown in Fig. 11
illustrate in a different way (than in Fig. 8) the interplay between the
elastic and inelastic event rates (or their ”transition dynamics” with
increasing $T_{A}$). The interplay is given for the specific nuclei ($m_{A}$
and $\Delta\epsilon_{mn}$) and the DM masses as a function of the DM kinetic
energy $T^{\max}_{\eta}$.

In other words, one can argue that the ratio between differential incoherent
and coherent event rates (60), $R_{I}(A,T_{A})$, coincides with the ratio
between the differential incoherent and coherent $\chi A$ scattering cross
sections (62) provided conditions (63) and (64) are satisfied

$T_{A}^{\min}\leq T_{A}\leq T_{A^{*},\eta}^{\max}(r,\Delta\varepsilon_{mn}).$

For fixed $\eta$, $m_{A}$, $\Delta\varepsilon_{mn}$ and $T_{A}$ it restricts
the allowed DM masses $m_{\chi}$ (see Fig. 6 and Fig. 7).

The case of two different DM distributions (29), despite some versatility,
looks quite exotic with existence of just these two DM distributions, and
looks quite complicated for numerical study with the ”pecular” curves shown in
Fig. 11.

Therefore, in what follows, we consider only one DM distribution with a set of
maximal dimensionless velocities $\eta=1,2,4$ and weight $\gamma_{\eta}=1$.
Then the ”cosmic factors” (58) become identical in size and differ only in the
”length of $T_{A}$”

$C_{\rm
c/i}(x_{\min},\eta,\epsilon,T_{A})=C(x_{\min},\eta,\epsilon)\Theta(T_{A/A^{*},\eta}^{\max}-T_{A}),$

since always $T_{A,\eta}^{\max}\geq T_{A^{*},\eta}^{\max}$ (see Fig. 12). Here
one has the notation

$\displaystyle C(x_{\min},\eta,\epsilon)$ $\displaystyle\equiv$ $\displaystyle
S(x_{\min},\eta,\epsilon)-S({2.5}+\eta,\eta,\epsilon).$ (70)

Then the total (coherent plus incoherent) differential event rate (53) takes
the form

$\displaystyle\frac{dR(T_{A})_{\text{}}}{R_{0}^{w}dT_{A}}$ $\displaystyle=$
$\displaystyle C(x_{\min},\eta,\epsilon)\dfrac{m_{A}}{Am_{\chi}}[Q_{\rm
c}(A)\Phi_{\rm c}(T_{A})\Theta(T_{A,\eta}^{\max}\\!-\\!T_{A})+Q_{\rm
i}(A)\Phi_{\rm i}(T_{A})\Theta(T_{A^{*},\eta}^{\max}\\!-\\!T_{A})]\qquad$ (72)
$\displaystyle=$ $\displaystyle
C(x_{\min},\eta,\epsilon)\dfrac{m_{A}}{Am_{\chi}}Q_{\rm c}(A)\Phi_{\rm
c}(T_{A})\Theta(T_{A,\eta}^{\max}\\!-\\!T_{A})[1+{R(A,T_{A},T_{A}^{\min})}],\qquad$
$\displaystyle\quad\text{where}\quad
R_{0}^{w}\equiv\dfrac{G^{2}_{F}}{4\pi}\dfrac{\rho_{\chi}N_{\rm
Av}}{v_{0}}\cong 10^{3}\dfrac{\rm events}{\rm~{}kg~{}keV~{}year}.$

Relations (60) and (4) are also simplified (because $T_{A,\eta}^{\max}\geq
T_{A^{*},\eta}^{\max}$) as follows:

$\displaystyle R(A,T_{A},T_{A}^{\min})$ $\displaystyle=$
$\displaystyle{\dfrac{dR_{\text{i}}}{dT_{A}}}/{\dfrac{dR_{\text{c}}}{dT_{A}}}=\dfrac{Q_{\rm
i}(A)}{Q_{\rm
c}(A)}\dfrac{1-|F(T_{A})|^{2}}{|F(T_{A})|^{2}}\Theta(T_{A}-T_{A}^{\min})\Theta(T_{A^{*},\eta}^{\max}-T_{A}),\qquad$
(73) $\displaystyle R(A,T_{A}^{\min})$ $\displaystyle\equiv$
$\displaystyle\dfrac{R_{\text{i}}}{R_{\text{c}}}(T_{A}^{\min})=\dfrac{Q_{i}(A)}{Q_{c}(A)}\Theta(T_{A}-T_{A}^{\min})\dfrac{\int_{T_{A}^{\min}}^{T_{A^{*},\eta}^{\max}}dT_{A}[1-|F(T_{A})|^{2}]}{\int_{T_{A}^{\min}}^{T_{A,\eta}^{\max}}dT_{A}|F(T_{A})|^{2}}.$

Figure 13: For the 73Ge nucleus, the dash-dotted line shows the ratio
$R(A,T_{A},T_{A}^{\min}=1\,$keV) from (73) as a function of $T_{A}$ (right
y-axis) for three velocities $\eta$ (plots from top to bottom) and five
$m_{\chi}$ (marked in different colors). The solid thin line (left y-axis)
shows the differential event rate from (72), which corresponds to $m_{\chi}$
(of the same color). The $\chi$-nucleon interaction is defined by the non-
relativistic SM coupling constants from (4). Figure 14: Same as in Fig. 13,
but for the 127I nucleus.

The total, integrated over $T_{A}$, event rate as a function of the energy
threshold $T_{A}^{\min}$, as well as of $m_{A}$, $\Delta\varepsilon_{mn}$
together with ”cosmic parameters”, $m_{\chi},\eta,\gamma_{\eta},\epsilon$,
takes the form

$\displaystyle
R_{\text{i+c}}(T_{A}^{\min})=C(x_{\min},\eta,\epsilon)\Big{[}Q_{i}(A)\int_{T_{A}^{\min}}^{T_{A^{*},\eta}^{\max}}dT_{A}\Phi_{\rm
i}(T_{A})+Q_{c}(A)\int_{T_{A}^{\min}}^{T_{A,\eta}^{\max}}dT_{A}\Phi_{\rm
c}(T_{A})\Big{]}.\qquad$ (74)

Expressions (72), (73) and (74) are the main formulas for further
consideration.

Figure 15: Same as in Fig. 13, but for the 133Cs nucleus.

In Fig. 13–15, the dash-dotted line shows the ratio $R(A,T_{A},T_{A}^{\min})$
given by (73) as a function of $T_{A}$ for some values of $\eta$ and
$m_{\chi}$ (highlighted by color) when the $\chi A$ interaction is defined by
the SM coupling constants in (4). The thin solid lines illustrate the
$T_{A}$-behavior of the (total) differential event rates
$\dfrac{dR(T_{A})_{\text{}}}{dT_{A}}$ from (72) corresponding to the
$R(A,T_{A},T_{A}^{\min})$-curves (shown in the same color).

Figure 16: Total, integrated over $T_{A}$, event rate (74) in 73Ge as a
function of the DM mass $m_{\chi}$ and other ”cosmic parameters”,
$\eta,\gamma_{\eta},\epsilon$. In contrast to the SM for $R_{0}$ from (72)
here one has $R_{0}=1.0$. The coherent and incoherent contributions to the
total rate are also shown. Figure 17: Same as in Fig. 16, but for the 127I
nucleus.

From these plots one can see the following main features. For small masses
$m_{\chi}$ and small velocities $\eta$, the inelastic interaction is
impossible, and the dash-dotted curves of the inelastic-to-elastic rate ratio
”lie at zero”. With $m_{\chi}$ increasing by an order of magnitude, the total
rate decreases by about an order of magnitude. The sharp termination of the
rate curves is due to the restrictions on the velocity $\eta$ for fixed
$m_{\chi}$. The double kink in these curves with increasing $T_{A}$ results
from the vanishing of the incoherent contribution to the total rate and
subsequent vanishing of the coherent contribution with the large enough
$T_{A}$. Over the entire interval of $T_{A}$, the total differential rates
decrease by no more than one order of magnitude. Moreover, with increasing
$T_{A}$, the purely coherent rate falls off much faster than its sum with the
incoherent rate.

Approximately in the middle of the $T_{A}$ intervals depicted in Fig. 13–15,
the incoherent rate becomes comparable with the coherent one, and as $T_{A}$
increases, it begins to dominate. For example, if the ”measured” recoil energy
in the 133Cs nucleus becomes $T_{A}\geq 60$ keV (see Fig. 15), one can
conclude that this signal is due to the incoherent interaction. In fact, this
signal should also be accompanied by a $\gamma$-quantum with the energy of 81
keV.

Furthermore, the total (coherent plus incoherent) expected event rate in 73Ge
and 127I, given by expression (74) is shown in Fig. 16–17 as a function of
$m_{\chi}$. It is seen that the maximal rate is expected, as a rule, for DM
masses about or a bit smaller than $m_{\chi}\cong 50~{}$GeV$/c^{2}$.

Note that in Fig. 13–15 the absolute value of the rate (for illustrative
purposes) is determined by the intensity of the weak lepton–nucleon
interaction of the Standard Model, although it is already quite clear that the
interaction of DM particles with matter is noticeably weaker. Furthermore,
since the results of the ”classical” direct DM search experiments Smith:1990kw
; Vergados:1996hs ; Bertone:2004pz ; Spooner:2007zh ; Bednyakov:2008gv ;
Saab:2012th ; Baudis:2012ig ; Cushman:2013zza ; Livio:2014gda ;
Gelmini:2015zpa ; Bernabei:2022ath ; Bernabei:2020mon ; Bednyakov:2022dmc are
interpreted in terms of spin-independent and spin-dependent cross sections of
the DM particle interaction with nucleons, let us discuss these two types of
$\chi A$ interaction in more detail below.

Figure 18: For 73Ge and 127I, the ratio
$R^{\text{iso}}_{\text{Scalar}}(A,T_{A})$ of the incoherent-to-coherent rate
(right y-axis) from (76) and the differential rate (72) (left y-axis) are
given as a function of the nuclear recoil energy, $T_{A}$, in the case of
isoscalar scalar $\chi A$ interaction ($\alpha_{p}=\alpha_{n}=0.5$).

### Spin-independent event rates

In the nonrelativistic approximation Bednyakov:2018mjd ; Bednyakov:2019dbl ;
Bednyakov:2021ppn , the scalar (in terminology of the direct DM search the
spin-independent) interaction has the same form as the purely vector one, when
in the scalar products (13) only constants $\alpha_{p}$ and $\alpha_{n}$
remain nonzero Bednyakov:2022dmc . Then ”Scalar” ratio (60) of the
differential incoherent-to-coherent rate can be written as follows:

$\displaystyle R_{\text{Scalar}}(A,T_{A})$ $\displaystyle=$ $\displaystyle
R_{A}(T_{A})A\dfrac{A_{p}\alpha_{p}^{2}+A_{n}\alpha_{n}^{2}}{(A_{p}\alpha_{p}+A_{n}\alpha_{n})^{2}}\Theta(T_{A}-T_{A}^{\min})\Theta(T_{A^{*},\eta}^{\max}(r,\Delta\varepsilon_{mn})-T_{A}).\qquad$
(75)

In the isoscalar case (when $\alpha_{p}=\alpha_{n}$), the dependence on these
constants cancels out completely, and this formula behaves like the ”pure
nuclear ratio” $R_{A}(T_{A})$, i.e., with $R_{I}(A)=1$

$\displaystyle
R^{\text{iso}}_{\text{Scalar}}(A,T_{A})=R_{A}(T_{A})\Theta(T_{A}-T_{A}^{\min})\Theta(T_{A^{*},\eta}^{\max}(r,\Delta\varepsilon_{mn})-T_{A}).\qquad$
(76)

If one of these parameters is significantly smaller than the other, for
example, as in the SM, where $\alpha_{p}\ll\alpha_{n}$ (or, for simplicity,
$\alpha_{p}\simeq 0$), one has the relation

$R^{\alpha_{p}=0}_{\text{Scalar}}(A,T_{A})=R_{A}(T_{A})\dfrac{A_{p}+A_{n}}{A_{n}}=\dfrac{1-F^{2}_{A}(T_{A})}{A_{n}F^{2}_{A}(T_{A})},$

which is also independent of $\alpha_{n}$ and differs from $R_{A}(T_{A})$ only
by about a factor of 2.

Figure 19: Nuclear charge factors from formulas (4) and (69) with parameters
$\alpha_{p}=-\alpha_{n}=0.5$.

In Fig. 18, isoscalar ratios (76) are shown for 73Ge and 127I along with
differential total (coherent plus incoherent) rates from (72) calculated with
equal scalar couplings $a_{p}=a_{n}=0.5$. For this scalar $\chi A$ interaction
the inelastic rate does not noticeably contribute to the total differential
event rate. The ratio $R^{\text{iso}}_{\text{Scalar}}(A,T_{A})$ is less than
one almost everywhere.

In the anti-isoscalar case, where $\alpha_{n}=-\alpha_{p}$, as in the SM, one
has

$R^{\text{SM}}_{\text{Scalar}}(A,T_{A})=R_{A}(T_{A})(A_{p}+A_{n})\dfrac{A_{p}\alpha_{p}^{2}+A_{n}\alpha_{n}^{2}}{(A_{p}\alpha_{p}-A_{n}\alpha_{n})^{2}}=R_{A}(T_{A})\Big{(}\dfrac{p+1}{p-1}\Big{)}^{2},\text{~{}~{}where~{}~{}}p=\dfrac{A_{n}}{A_{p}}.$

Figure 20: The ratio $R^{\text{SM}}_{\text{Scalar}}(A,T_{A})$ of the
incoherent-to-coherent rate from (76) (right y-axis) and the differential
rates (72) (left y-axis) for 73Ge and 127I as a function of the nuclear recoil
energy $T_{A}$ in the case of anti-isosalar scalar $\chi A$ interaction
($\alpha_{p}=-\alpha_{n}=0.5$).

From Table 1 and Fig. 19 one can see that for the lightest nuclei there is no
coherent contribution in this case, since the number of protons coincides with
the number of neutrons and $p=1$. For fluorine, sodium, and chlorine, the
elastic contribution is strongly suppressed, and the inelastic charge factor
is orders of magnitude higher than the elastic one. The heavier mass of the
target nucleus, the greater the value of $p$, and the difference between
$R^{\text{SM}}_{\text{scalar}}(A,T_{A})$ and $R_{A}(T_{A})$ presented in Fig.
18 is smaller. Figure 20 illustrates the anti-isocalar scalar $\chi A$
interaction by means of $R^{\text{SM}}_{\text{Scalar}}(A,T_{A})$ and
differential event rate. The total, integrated over $T_{A}$, event rate as a
function of $m_{\chi}$ in approximation (74) is shown in Fig. 21 for the
scalar (with $\alpha_{n}=-\alpha_{p}$) interaction with (simplification)
$R_{0}=1.0$. The coherent and incoherent contributions to the total rate are
also given in Fig. 21.

Figure 21: Integrated event rate in the 73Ge and 127I nuclei as a function of
the mass $m_{\chi}$ and other ”cosmic parameters” in the case of anti-
isoscalar scalar $\chi A$ interaction ($\alpha_{p}=-\alpha_{n}=0.5$). The
coherent and incoherent contributions to the total rate are also shown.

From Fig. 18 and 20 one can see the ”main trend”. There is a smooth change of
the ”content” in the measurable event rate as $T_{A}$ increases, i.e., the
elastic contribution to the rate is being smoothly replaced by the inelastic
(incoherent) contribution. The total (elastic+inelastic) expected event rate
decreases, but not significantly (less than by an order of magnitude). It can
be believed that the number of expected events will not be too small to
completely escape the registration, despite these events change their origin.
From Fig. 21 one can also see that the inelastic (incoherent) scalar
contribution to the total integrated rate dominates over the elastic
(coherent) one with increasing $m_{\chi}$ ($m_{\chi}>30\div 50\,$GeV$/c^{2}$).

Therefore, if a device is configured to register only the elastic scattering
events, it rather quickly begins to see nothing, since if $T_{A}$ increases,
the elastic processes quickly decrease. At the same time, one has an increase
of the expected inelastic events, which the device is not able of see.

Figure 22: The ratio $R^{\text{SM}}_{\text{Axial}}(A,T_{A})$ of the total
incoherent-to-coherent rate from (77) and the differential rates (72) are
given for 73Ge and 127I as functions of the nuclear recoil energy $T_{A}$ for
the SM-like axial SM interaction when $\delta_{n}=-\delta_{p}$ (the dependence
on these parameters survives only in the rates).

The ability of the device to detect interaction of the DM particles with
sufficiently large mass also falls sharply. It turns out that the ”wanted”
$\chi A$ interaction could be possible, but no way to detect it.

### Spin-dependent event rates

Let us turn to the axial-vector $\chi A$ interaction, which is called spin-
dependent in the terminology of the direct DM search. In this case, the vector
coupling constants $\alpha_{p}=\alpha_{n}=0$ in formula (69), and relation
(60) becomes

$\displaystyle
R^{\text{}}_{\text{Axial}}(A,T_{A})=R_{A}(T_{A})\dfrac{3A(A^{p}\delta^{2}_{p}+A^{n}\delta^{2}_{n})}{(\delta_{p}\Delta
A_{p}+\delta_{n}\Delta
A_{n})^{2}}\Theta(T_{A}-T_{A}^{\min})\Theta(T_{A^{*},\eta}^{\max}(r,\Delta\varepsilon_{mn})-T_{A}).\quad$
(77)

One can see that for nuclei with zero spin (more precisely, when $\Delta
A_{p/n}=0$) the coherent contribution to the total cross section (rate as
well) from the purely axial-vector interaction is absent, and this relation
loses its meaning.

Figure 23: Integrated total event rate in 73Ge and 127I as a function of the
mass $m_{\chi}$ and other ”cosmic parameters” in the case of SM-like axial-
vector $\chi A$ interaction ($\delta_{n}=-\delta_{p}$). The coherent and
incoherent contributions to the total rate are also shown.

Since in the SM the axial effective constants of the proton and neutron are
related, $\delta_{n}=-\delta_{p}$, the formula

$R^{\text{SM}}_{\text{Axial}}(A,T_{A})=\dfrac{3R_{A}(T_{A})A^{2}}{(\Delta
A_{p}-\Delta A_{n})^{2}}$

does not depend on the axial coupling constant $\delta_{p}$. This ratio is not
suppressed by $A$, but, on the contrary, it is directly proportional to $A$,
provided $\Delta A_{p}\neq\Delta A_{n}$ (see Fig. 22). Otherwise, relation
(77) loses its meaning again. The total (coherent+incoherent), integrated over
$T_{A}$, event rate as a function of $m_{\chi}$ for 73Ge and 127I with
approximation (74) is shown in Fig. 23 for SM-like axial-vector interaction
($\delta_{n}=-\delta_{p}$), where $R_{0}=1.0$ is taken for simplicity. One can
see from Fig. 22 and Fig. 23 that the coherent contribution to the total event
rate is either very small or completely absent in this case of $\chi A$
interaction.

Figure 24: The ratio $R^{\text{spin}}_{\text{axial}}(A,T_{A})$ from (77)
(right y-axis) and differential rate (72) (left y-axis) are presented for 73Ge
and 127I as a function of the nuclear recoil energy $T_{A}$ in the case of
isoscalar axial-vector interaction, when $\delta_{n}=\delta_{p}$, (dependence
on these parameters is preserved only in the rates).

When $\delta_{n}=\delta_{p}$, relation (77) also loses any dependence on the
axial couplings and takes the form

$\displaystyle R^{\text{spin}}_{\text{Axial}}(A,T_{A})$ $\displaystyle=$
$\displaystyle\dfrac{3R_{A}(T_{A})A^{2}}{(\Delta A_{p}+\Delta A_{n})^{2}},$

where $\Delta A\equiv\Delta A_{p}+\Delta A_{n}$ plays the role of the total
spin of nucleus $A$. This situation can be considered as traditional spin-
dependent interaction of (dark matter) $\chi$ particles with the target
nuclei, since the coherent cross section is proportional to the square of the
nuclear spin. In Fig. 24, the set of plots relevant to this kind of $\chi A$
interaction is given. The total (coherent+incoherent), integrated over
$T_{A}$, event rate as a function of $m_{\chi}$ for 73Ge and 127I nuclei (see
(74)) is shown in Fig. 25 for the iso-scalar ($\delta_{n}=\delta_{p}$) axial-
vector $\chi A$ interaction with $R_{0}=1.0$ taken for simplicity.

Figure 25: Counting rate integrated over $T_{A}$ in 73Ge and 127I as a
function of the mass $m_{\chi}$ and other ”cosmic parameters” in the case of
isoscalar ($\delta_{n}=\delta_{p}$) axial (spin-dependent) $\chi A$
interaction. The coherent and incoherent contributions to the total count rate
are also shown.

From Fig. 22 and Fig. 24 the ”general trend” in the balance between the
coherence and incoherence becomes ”obvious”. Indeed, if $\chi$ particles
interact with nucleons via axial-vector current only, it is not possible to
detect these particles by means of their coherent (elastic) interaction with
zero-spin target nuclei, simply because the interaction does not exist. The
only way to discover $\chi$ particles is registration of their inelastic
(incoherent) interaction with nuclei. Moreover, even if the target consists of
nuclei with a nonzero spin, the coherent (elastic, proportional to the square
of the nuclear spin) contribution to the expected total rate is ”not visible”
above the dominant ”background” of the incoherent (inelastic, proportional to
the atomic mass $A$) contribution almost in the entire range of recoil energy
$T_{A}$.

Thus, in the case of axial-vector $\chi A$ interaction only, the elastic
scattering channel, which is traditionally used with nonzero spin target
nuclei in the spin-dependent DM search, is doomed to see nothing. This is due
to the fact that all ”signal-forming” $\chi A$ interactions occur via the
inelastic channel, to which traditional DM detectors are, as a rule,
insensitive. The detector usually aimed at the direct DM search is constructed
to have maximal sensitivity in the low energy recoil region (near the
threshold). It is unable to register the high-energy (compared with the
threshold) radiation from the nuclear deexcitation caused by the inelastic
$\chi A$ interaction and has also bad sensitivity at high $T_{A}$. Again, it
looks quite possible that the ”wanted” (spin-dependent) DM interaction can
have a noticeable intensity, but a typical DM detector simply cannot detect
it.

The dominance of the inelastic over the elastic spin-dependent interaction of
DM particles with 129,131Xe nuclei was pointed out in Baudis:2013bba , where
the inelastic channel was determined by the excitation of one low-lying level,
and the expected detectable (scintillation) signal was the sum of the recoil
energy and the energy of photons from the nuclear deexcitation.

Indeed, if the transition to the excited state of the nucleus is possible and
took place, i.e., the $\chi A$ inelastic interaction occurred111111This
“classical” inelastic approach should not be mixed up with “inelasticity”
caused by the transition of the incident $\chi_{1}$ particle (of dark matter)
to a more massive $\chi_{2}$ particle (also from the dark sector). The nucleus
is interacting coherently Giudice:2017zke ; XENON:2020fgj ; Zurowski:2020dxe ;
Baryakhtar:2020rwy ; Feng:2021hyz ; Filimonova:2022pkj ; Bell:2022yxn ;
Aboubrahim:2022lwb ., the nucleus should eventually return to its original
(ground) state. This deexcitation of the nucleus should be accompanied by some
release of energy, for example, in the form of $\gamma$ radiation. The
detection of these $\gamma$-quanta was proposed in 1975 Donnelly:1975ze . The
energy spectrum of these photons is determined by the structure of nuclear
excitation levels and is fixed for each nucleus. These photons can produce a
detectable signal, which will be correlated with the target irradiation time
if the source of the external $\chi$ particles is, for example, an
accelerator. According to the figures given above (see, for example, Fig. 24),
the prospects for registration of a noticeable number of events with $\gamma$
quanta from incoherent $\chi A$ interaction do not seem futile, provided that
the $\chi A$ interaction itself has detectable intensity for modern detectors.

Recall Bednyakov:2018mjd ; Bednyakov:2021ppn ; Bednyakov:2021bty ;
Bednyakov:2022dmc that for each target nucleus these photons will be
characterized by three important parameters. First, their energy is often
noticeably higher than the nuclear recoil energy ($T_{\text{${}^{127}$I}}\leq$
20 keV in Fig. 24). Second, emission of photons with the typical nucleus
deexitation energies will be shifted in time (relative to the beginning of the
excitation) by the deexitation time typical of the excited nuclear level.
Third, the counting rate of these $\gamma$ quanta will be proportional to the
ratio of expected incoherent-to-coherent number of events,
$N_{\text{inc}}/N_{\text{coh}}$, where $N_{\text{inc/coh}}=\int
dE_{\nu}\Phi(E_{\nu})\int_{dT_{A}^{\text{min}}}^{dT_{A}^{\text{max}}}dT_{A}\dfrac{d\sigma_{\text{inc/coh}}}{dT_{A}}\varepsilon(T_{A}),$
and $\varepsilon(T_{A})$ is the detector efficiency.

Let us discuss the accuracy of the inelastic $\chi A$ cross section in the
approach Bednyakov:2018mjd ; Bednyakov:2019dbl ; Bednyakov:2021ppn ;
Bednyakov:2022dmc . This (combined) inelastic cross section is the upper
boundary for contributions to the total measured $\chi A$ cross section from
the totality of all allowed inelastic subprocesses (in the given kinematic
region). This upper limit only follows from the probability conservation rule:
the sum of the probabilities of all inelastic processes and the probability of
the elastic process is equal to unity. The degree of “saturation” of this
combined inelastic cross section by individual inelastic channels
(contributions from transitions to different allowed levels) depends on the
structure of excitation levels of a particular nucleus and on the incident
particle energy. Clearly, this saturation cannot be 100% reproduced
numerically.

Inelastic cross sections for the $\nu(\bar{\nu})A$ scattering due to weak
neutral currents were calculated within modern nuclear models for specific
excitation levels of particular nuclei, for example, in Divari:2012zz ;
Divari:2012cj ; Lykasov:2007iy . Recently, similar calculations have been
generalized to the neutral DM particle inelastic scattering off nuclei
Sahu:2020kwh ; Sahu:2020cfq ; Dutta:2022tav . In Sahu:2020kwh ; Sahu:2020cfq ,
detailed calculations were performed for the elastic and inelastic event rates
in 73Ge, 127I, 133Cs, 133Xe as well as in the 23Na and 40Ar nuclei. It was
shown that incoherent (inelastic) WIMP–nucleus processes can be noticeably
enhanced. Therefore, the authors of Sahu:2020kwh ; Sahu:2020cfq confirmed
importance of the inelastic channel at high recoil energies first indicated in
Bednyakov:2018mjd ; Bednyakov:2019dbl ; Bednyakov:2021ppn .

In Dutta:2022tav , inelastic cross sections for scattering of neutrinos and DM
particles off the 40Ar, 133Cs and 127I nuclei were calculated within the
nuclear shell model (NSM). The (total) inelastic $\nu A$ cross section
Dutta:2022tav at rather low neutrino energies ($E_{\nu}<$ 20 MeV) was in
agreement with other calculations (including ones from Bednyakov:2018mjd ),
but at $E_{\nu}\simeq$ 40 MeV the cross sections were about an order of
magnitude smaller than the relevant upper limits from Bednyakov:2018mjd .
Furthermore, from Fig. 12 of Dutta:2022tav , one can notice that the ”simple”
Helm nuclear form factor $|F_{\rm H}(T_{A})|^{2}$ used here and in
Bednyakov:2018mjd ; Bednyakov:2019dbl ; Bednyakov:2021ppn coincides with high
accuracy with the more sophisticated form factor $|F_{\rm NSM}(T_{A})|^{2}$
from Dutta:2022tav , provided both $|F_{\rm NSM/H}(T_{A})|^{2}\geq 0.01$.

Finally let us stress the following. The (individual) inelastic cross sections
(and rates) calculated within modern nuclear models are always smaller than
the ones obtained here within the formalism Bednyakov:2018mjd ;
Bednyakov:2019dbl ; Bednyakov:2021ppn . It looks reasonable, because the upper
bound for all possible channels is always larger than a sum of some set of
individual channels. The calculations of inelastic cross sections (and rates)
within advanced nuclear models do not allow the coherence-to-incoherence
transition to be quantitatively controlled. If the ”coherency condition” is
satisfied ”with high precision”, one can reliably use the ”coherent” formulas,
otherwise one can reliably use the ”incoherent” formulas. Nobody knows what to
do if the ”coherency condition” is satisfied ”with bad precision”. There is no
any ”coherency condition” in the approach Bednyakov:2018mjd ;
Bednyakov:2019dbl ; Bednyakov:2021ppn , since the coherence-to-incoherence
transition is automatic.

##  5 Conclusion

Recall that the formalism Bednyakov:2018mjd ; Bednyakov:2019dbl ;
Bednyakov:2021ppn proposed to describe the neutrino–nuclear interaction, was
generalized in Bednyakov:2022dmc to the case of non-relativistic121212The
relativistic version of the description is given in Bednyakov:2023bbg . weak
interaction of a massive neutral particle ($\chi$ lepton) with a nucleus as a
composite system. This interaction was parameterized via effective coupling
constants which determine the probability amplitudes by means of the scalar
products of the lepton and nucleon currents. Regardless of the specific
nuclear models, explicit use of the complete set of the nuclear quantum states
allowed one Bednyakov:2018mjd ; Bednyakov:2019dbl ; Bednyakov:2021ppn to
obtain the unified description of the elastic (coherent) and inelastic
(incoherent) scattering of the $\chi$ lepton off nucleus $A$. The behavior of
the elastic and inelastic $\chi A$ cross sections is determined by the nucleon
form factors, $|F_{p/n}(\bm{q})|^{2}$, averaged over the initial nuclear
states. As the nuclear recoil energy $T_{A}$ increases, one has a controlled
transition from the dominance of the elastic interaction to the dominance of
the inelastic one in the nonrelativistic scattering of the massive $\chi$
particle off the nuclei. In other words, as $T_{A}$ increases, the registrable
events will change their ”origin”. If a DM detector is configured to register
only elastic scattering events, it begins to lose the ability to ”see”
anything, because the elastic processes rather quickly decrease with
increasing $T_{A}$. At the same time, the expected number of inelastic events
increases, but the detector is unable to register them, because it cannot see
the deexcitation photons, or their energies are beyond the detector
capabilities, etc.

Another case is also possible. If the nuclear recoil energy produced after
$\chi A$ scattering is below the threshold of the detector,
$T_{A}<T_{A}^{\min}$, the elastic signal cannot be detected. With this
”invisible” $T_{A}$, the only evidence of the $\chi A$ interaction could be
from the nuclear deexitation radiation, i.e., the inelastic signal, although
its intensity can be an order of magnitude smaller than the intensity of the
elastic one. In general, when only the nuclear recoil energy, $T_{A}$, can be
registered, it is impossible to understand whether the elastic or the
inelastic interaction took place. When the inelastic signal ”falls” into the
registration area of the elastic one, its origin is completely unclear.

In the most critical way, this ”phenomenon” can manifest itself in the direct
DM search experiments, the results of which are usually interpreted in terms
of the spin-independent and spin-dependent cross sections of the galactic DM
particle interaction with nucleons.

In this paper, the main statements of the previous one Bednyakov:2022dmc are
transformed into numerical predictions for the direct DM search experiments.
For the first time in the formalism Bednyakov:2018mjd ; Bednyakov:2019dbl ;
Bednyakov:2021ppn ; Bednyakov:2022dmc the effect of nonzero nuclear excitation
energy was explicitly taken into account in calculation of the event rates
caused by inelastic $\chi A$ interactions. The resulting rigid kinematic
correlation between this excitation energy and the allowed recoil energy of
the excited nucleus appreciably limits the possibility of detecting inelastic
events with some set of target nucleus.

In addition to the Standard Halo Model (SHM) of the DM particle distribution
near the Earth, the role of other DM distributions with higher maximal speeds
of DM particles was considered. There are several mechanisms (proposed, for
example, in Feng:2021hyz ; Bardhan:2022ywd ; CDEX:2022fig ; Xia:2022tid ;
Granelli:2022ysi ; Wang:2021jic ), which allow higher DM-particle velocities
at a level of $|\bm{v}|/c\simeq 10^{-2}\div 10^{-1}$.

Three variants of the DM particle interaction with nucleons are considered
Bednyakov:2022dmc . The first is the nonrelativistic weak interaction of the
Standard Model with some suppression parameter $R_{0}\simeq 10^{-(3\div
4)}R^{w}_{0}$ (see formula (72)). The plots for the differential and total
expected event rates are shown in Figs. 13–17. The incoherent (inelastic)
channel does not look much more promising than the usual coherent (elastic)
one. Roughly speaking, in the SM-like $\chi A$ interaction they almost
coincide. This can also be seen for the event rates integrated over $T_{A}$ in
Figs. 16-17.

The second is the spin-independent, or scalar DM-particle interaction. When
the scalar interaction is the same for the proton and the neutron (iso-
scalar), the above-mentioned smooth change of the total rate ”filling” with
increasing $T_{A}$ is also visible in Fig. 18. Nevertheless, the incoherent
contribution does not look dominant here. However, the anti-isoscalar spin-
independent $\chi A$ interaction, where the proton and neutron couplings
differ only by the sign, looks more interesting. Here (see Fig. 20) the smooth
change of the total rate ”filling” with the growth of $T_{A}$ is visible very
clear. The inelastic contribution completely dominates at $T_{A}\geq 50\div
60\,$keV. This dominance, especially with increasing $m_{\chi}$, is also
clearly seen for the total event rates in Fig. 21.

The third is the spin-dependent, or axial-vector DM particle interaction with
nucleons. In this case, any detection of the DM particles via the coherent
(elastic) interaction with zero-spin nuclei is not possible. The only way to
observe this interaction is through the inelastic (incoherent) channel.
Moreover, even if the nucleus has the nonzero spin, the coherent (proportional
to the square of the nuclear spin) contribution to the measurable rate is
hardly visible above the incoherent (proportional to the atomic mass of the
nucleus $A$) contribution for almost all $T_{A}$ (see Fig. 22 and 24). The
coherent contribution to the total (coherent plus incoherent and integrated
over $T_{A}$) rate is not visible and is negligible (see Fig. 25). Therefore,
in the case of the pure axial-vector $\chi A$ interaction, a detector
traditionally focused on recording the elastic spin-dependent DM signal is
”doomed” to see nothing, since all potentially ”signal-forming” interactions
go through the inelastic channel, to which this detector is, as a rule,
insensitive.

One meets the situation where the “wanted” DM interaction may well have a
rather noticeable (potentially detectable) intensity, but the instrument used
to search for the DM is unable to detect it. Furthermore, since the nature of
the DM particle interaction with nucleons is beyond the Standard Model and is
not yet experimentally determined, the other combinations of $\chi$–nucleon
couplings (in addition to discussed above), which can cause the ”interesting
balance” between the elastic and inelastic rates, could be quite interesting
for the direct DM search.

Therefore, one should plan experiments aimed at direct detection of dark
matter particles in a way, where one can measure two signals. The first is the
nuclear recoil energy $T_{A}$ (traditional, elastic), and the second is the
$\gamma$-quantum with the nuclear deexcitation energy $\Delta\epsilon_{mn}$
accompanied by the (excited-nucleus) recoil energy $T_{A^{*}}$, which, as a
rule, is higher than the elastic $T_{A}$. This two-fold experiment will give
complete information about the $\chi A$ interaction, if it occurs.

Finally recall, that for reliable registration of dark matter particles from
our Galaxy by means of direct detection one needs the positive signature
Bednyakov:2015uoa ; Bednyakov:2020njj , which nowadays is only the annual
modulation of the observed signal Freese:1987wu . In the context of this
paper, both elastic (coherent) and inelastic (incoherent) contributions to the
modulation signal are very important Sahu:2020kwh ; Sahu:2020cfq .

The author is grateful to D.V. Naumov, E.A. Yakushev, N.A. Russakovich and
I.V. Titkova for fruitfull discussions and important remarks.

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