Spin-dependent Scattering of Boosted Dark Matter

Wenyu Wang
Faculty of Science, Beijing University of Technology

Lei Wu
Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, 210023, China

Wen-Na Yang
Faculty of Science, Beijing University of Technology

Bin Zhu
School of Physics, Yantai University, Yantai 264005, China

Boosted dark matter is a promising method for probing light dark matter,  with a well-developed computational framework for spin-independent scattering already existing.
The spin-dependent case, on the other hand, lacks a coherent treatment.
We therefore give the first comprehensive derivation of the spin-dependent scattering cross-section for boosted dark matter, finding that certain effects can lead to enhanced experimental sensitivity compared to the conventional contact interaction.
For example, when the transfer momentum is sufficiently large, the time component of the dark matter current contributes significantly to the proton structure factor.
Also, even without a light mediator, we find a residual momentum dependence in the quark-nucleon matching operation which can contribute similarly.
We promote this endeavor by deriving direct limits on sub-GeV spin-dependent scattering of boosted dark matter from terrestrial data.
We find that the exclusion limits from the boosted structure factor differ by as much as six orders of magnitude from those calculated using non-relativistic structure factors.

Valid PACS appear here

Observations in cosmology and astrophysics have supported the presence of dark matter (DM) [Bertone et al., 2005].
Its features, such as mass and interactions are however still unknown.  One of the most promising experimental avenues is to search for the small energy depositions from the DM elastically scattering in sensitive detectors on Earth.
Strict constraints exist on the cross section for DM heavier than 1 GeV.
As the detection limits reach the neutrino floor, we must look for other strategies to explore the surviving parameter space of DM or find some means to detect DM beyond ordinary considerations  [Battaglieri et al., 2017].

A crucial aspect of the theoretical study of DM detection is the elastic scattering process between DM and the nuclei, which determines detection rates.
We can classify the hypothetical DM-nucleon interactions into the spin-independent and spin-dependent cases.
There is already an impressive range of existing constraints on the DM-nucleon cross-section in the MeV-to-GeV mass range, ranging from rare processes involving the emission of photons or ‘Migdal’ electrons from the recoiling atom [Ibe et al., 2018, Dolan et al., 2018, Flambaum et al., 2020, Knapen et al., 2021], and the small flux of boosted dark matter (BDM) arising from interactions between DM and cosmic rays, the sun or mesons [An et al., 2018, Bringmann and Pospelov, 2019, Ema et al., 2019, McKeen and Raj, 2019, Alvey et al., 2019, Dent et al., 2020, Wang et al., 2019, Ge et al., 2020, Su et al., 2020, Xia et al., 2020, Guo et al., 2020, Herrera and Ibarra, 2021].

That said, although there is a well-developed framework for spin-independent scattering of BDM, a thorough treatment for spin-dependent scattering is lacking.

Since WIMPs are heavy, they are estimated to be rotating around our galaxy at a non-relativistic velocity of several hundred kilometers per second, and with a minimum escape velocity $\upsilon_{\rm esc}\sim 544\rm km/s$.
Therefore non-relativistic expansions of the DM-nucleon scattering amplitude in powers of the DM-nucleon relative velocity are appropriate for evaluating the cross-section [Del Nobile, 2021].
Though the non-relativistic expansion or the effective field theory is reliable in dealing with WIMP scattering, the neglect of higher-order terms is problematic in some circumstances. For example, in a novel detection of sub-GeV dark matter, called cosmic ray dark matter
(CRDM)  [Bringmann and Pospelov, 2019, Wang et al., 2019, Dent et al., 2020, Feng et al., 2021], this simple expansion is not reliable.

In this work, we focus on spin-dependent boosted dark matter scattering.
We assume dark matter is a Majorana fermion for the sake of comparison.
This is due to the fact that the Majorana-type DM always occurs in the most popular models of physics beyond the Standard Model, such as those featuring supersymmetry [Martin, 1998] or extra dimensions [Csaki et al., 2005], et. al.
It's worth noting that our calculation is not limited to Majorana DM, which can be easily extended to Dirac fermion or scalar DM.
Previous calculations of spin-dependent WIMP scattering have started from WIMP-nucleon currents and used the nuclear-structure function to convert the result to the nucleus.
However, nuclear structure calculations can be improved with recent advances in nuclear interactions and computing capabilities.
We can thus demonstrate the difference between the WIMP-nuclei scattering and boosted DM scattering on the target.
Furthermore, we find another source of the momentum transfer effect in the contact interaction, where the finite size proton accounts for the momentum dependent behavior.

Computational Framework:
In order to demonstrate the subtlety of BDM spin-dependent scattering, we firstly sketch the ordinary computation of the WIMP spin-dependent cross section.
At low momentum transfer $q$, the Lagrangian for the interaction between DM and quarks can be evaluated using chiral effective field theory [Klos et al., 2013, Bishara et al., 2017].
In the neutron or proton-only case, the differential scattering cross section can be rewritten as

\begin{equation}
\frac{d\sigma^{\mathrm{SD}}}{dq^2}=\frac{\sigma_{\chi N}^{\mathrm{SD}}}{3 \mu_{N}^{2} v^2}
\frac{\pi}{2J+1} S_{A}(q),\label{eq:5_final_diff_cross_section}
\end{equation}

in which $\mu_N$ is the reduced mass of the DM-nucleon system, and $\sigma_{\chi N}$ is the scattering cross section between a DM particle and a single proton or neutron at zero momentum transfer.
$v$ is the WIMP velocity in the rest frame of the detector, and $J$ is the initial ground-state angular momentum of the nucleus.
The total expected non-relativistic spectrum of the detection rate  $dR/dE_{\mathrm{r}}$ is $\frac{dR}{dE_{\mathrm{r}}}= \frac{2\rho_{\chi}}{m_{\chi}} \int
\frac{d\sigma^{\mathrm{SD}}}{dq^2} v f({\bm v}) d^3 v$.
$m_{\chi}$ is the mass of the DM, $\rho_{\chi}$ is the local DM density
and $f({\bm v})$ is the velocity distribution in the rest frame of the detector.
$q = \sqrt{2 E_{\mathrm{r}} m_{\mathrm{N}}}$ is the transfer momentum.
If we assume a  standard isothermal WIMP halo, then $v_0 = 220~\mathrm{km}/\mathrm{s}$,
$v_\mathrm{esc} = 544~\mathrm{km}/\mathrm{s}$,
and Earth velocity $v_\mathrm{E} = 232~\mathrm{km}/\mathrm{s}$.
If we generalize the WIMP into BDM, the Maxwell-Boltzmann distribution is replaced by the incoming DM flux $d\Phi_{\chi}/dE_{\chi}$.

The structure factor $S_A(q)$ plays a crucial role in determining the event rate of spin-dependent scattering process in both relativistic and non-relativistic process,

\begin{equation}
S_A(q)=\frac{1}{4 \pi G_{\rm v}^2}\sum_ {s_f,s_i}\sum_{M_f,M_i}
\left|\left\langle {f}\left|{\mathcal L}^{\rm SD}_\chi\right|{i}\right\rangle\right|^2 \,,
\end{equation}

in which the sum $s_f, s_i = \pm 1/2$ is over Majorana fermion spin projections, and the sum $M_f, M_i$ is over the projections of the total angular momentum of the final and initial states $J_f, J_i$.
We assume the heavy mediator is a vector so that $G_{\rm v}\sim 1/m_{\rm V}^2$.
It is easy to include the scalar mediator scenario.
Usually, the heavy mediator does not lead to a momentum transfer effect in the scattering process.
However, we find there is still a residual momentum transfer effect through the matching procedure from quark to proton.
The structure factor has three contributions: the spatial current, temporal current, and interference component current.
Fortunately, we find that the wave functions of
the spatial current and the temporal current are orthogonal to each other,
making their interference terms vanish.
One can write the structure factor as

\begin{eqnarray}
S_A(q)= S^0_A(q)+S_T(q),
\end{eqnarray}

where $S^0_A(q)$ denotes the contribution from the spatial current couplings and $S_T(q)$ is the time component contributions.
Evaluating the Lagrangian density between the initial and final state, the spatial contribution is

\begin{equation}
\left\langle{f}\left| {\mathcal L}^{\rm SD}_\chi \right|{i}\right\rangle = -\frac{G_{\rm v}}{\sqrt{2}}
\int d^3{\bm r} \, e^{-i{\bm{q}} \cdot {\bm r}} \, \overline{\chi}_f
{\bm \gamma} \gamma_5 \chi_i \, {\bm J}^A_{i}({\bm r}) \,,
\end{equation}

in which  $e^{-i{\bm {q}} \cdot {\bm r}} \, \overline{\chi}_f {\bm \gamma}
\gamma^5 \chi_i =\left\langle {\chi_f}\left|{\bm j}({\bm r})\right|{\chi_i}\right\rangle $
is the matrix element of the  current of the DM and
${\bm J}^{A}_{i}({\bm r})=\sum_q A_q \overline \psi_q{\bm \gamma}\gamma_5\psi_q$ denotes the hadronic current [Klos et al., 2013].
Therefore, the structure factor $S^0_A(q)$ can be decomposed as a sum over multipoles $L$ with reduced matrix elements of the longitudinal ${\mathcal L}^5_L$, transverse electric ${\mathcal T}^{\mathrm{el}5}_L$, and transverse magnetic ${\mathcal T}^{\mathrm{mag}5}_L$ projections of the axial-vector currents

\begin{equation}
\begin{aligned}
S^0_A(q) &= \sum_{L \geqslant 0}\left|\langle {J_f}\!|\!|{\mathcal L}_L^5|\!|\!
&+\sum_{L \geqslant 1} \left(\left|\langle{ J_f}\!|\!|
{\mathcal T}_L^{\mathrm{e l}5}|\!|\!{J_i}\rangle\right|^2
+\left|\langle{ J_f}\!|\!|{\mathcal T}_L^{\mathrm{mag}5}|\!|\!
{ J_i}\rangle\right|^2\right) \,.\label{nrs0}
\end{aligned}
\end{equation}

For fast-moving DM, the projections of the axial-vector currents will not be changed but become dependent on the momentum of the incident DM particle

\begin{eqnarray}
&& S^0_A(q,p_i,p_f)=\nonumber\\
&& \frac{1}{2}\biggl[\sum_{L\geqslant0}\frac{1}{E_f E_i}\left(2p_{f}^{3}p_{i}^{3}
+p_{f}\centerdot p_{i}+m_\chi^{2}\right)\left|\langle J_{f}||{\mathcal L}_L^5||J_{i}\rangle\right|^{2}\,\nonumber \\
&&+\sum_{L\geqslant1}\frac{1}{E_f E_i}\left(p_{f}^{1}p_{i}^{1}+p_{f}^{2}p_{i}^{2}+p_{f}\centerdot
p_{i}+m_\chi^{2}\right)\,\nonumber \\
&&\left(\left|\langle J_{f}||{\mathcal T}_L^{\mathrm{e l}5}||J_{i}\rangle\right|^{2}+\left|\langle J_{f}||{\mathcal T}_L^{\mathrm{mag}5}||J_{i}\rangle\right|^{2}\right)\biggl]\,.
\end{eqnarray}

We can also get the time component contribution to the structure factor at the nucleon level[A detailed derivation can be found in the supplementary material.]

\begin{eqnarray}
\frac{1}{4E_{i}E_{f}}\frac{1}{4E_i^{\prime} E_f^{\prime}}\frac{1}{2\pi}\biggl[ 4\left(2E_{i}E_{f}-\frac{q^{2}}{2}-2m_{\chi}^{2}\right)\nonumber\\
&&\times\left(2E_{i}^{\prime}E_{f}^{\prime}-\frac{q^{2}}{2}-2m_{N}^{2}\right) \, \nonumber\\
&& \hspace{-7mm}+4\frac{m_{N}}{q^2+m_{\pi}^2} \left(2E_{i}E_{f}-\frac{q^{2}}{2}-2m_{\chi}^{2}\right)
\left(E_{i}^{\prime}-E_{f}^{\prime}\right) m_{N} q_{0}\nonumber\\
&& \hspace{-7mm}+\left(\frac{m_{N}}{q^2+m_{\pi}^2} \right)^{2}\left(2E_{i}E_{f}-\frac{q^{2}}{2}-2m_{\chi}^{2}\right) \frac{q^{2}}{2}
q_0^2 \biggl] \,.
\end{eqnarray}

We can make some general comments about the boosted structure factor before concluding this section.
The spatial and time components of the structure factor for semi-relativistic kinematics are not a function of $q$ only, but instead the incoming dark matter momentum $p_i$ or kinetic energy $T_{\chi}$.
To obtain an effective structure factor, we need to integrate out the phase space of incoming momentum

\begin{equation}
S_{\mathrm{eff}}(q)=\int dT_{\chi} \frac{d\Phi_{\chi}}{dT_{\chi}}S(T_{\chi},q)\,.
\end{equation}

This is equivalent to the conventional Maxwell-Boltzmann velocity integral when the incoming DM momentum reduces to the non-relativistic regime.

To get some feeling for this modification, we choose a specific collision $p_{i}=(E_{i},0,0,p_{i})\,,~p_{f}=(E_{f},p_{f},0,0)\,,~
to obtain the structure factor

\begin{eqnarray}
S_A^0(q)&=&\frac{4m_{\chi}^{2}+q^{2}}{4m_{\chi}^{2}+2q^{2}}\sum_{L\geqslant0}\left|\langle J_{f}||{\mathcal L}_L^5||J_{i}\rangle\right|^{2}\,\\
\sum_{L \geqslant 1}\left|\langle J_{f}||({\mathcal T}_L^{\mathrm{el}5}+{\mathcal T}_L^{\mathrm{mag}5} )||J_{i}\rangle\right|^{2}\,,\nonumber
\label{eqss0}\\
&& \hspace{-5mm}\times\left[q^{4}-\frac{1}{2}\frac{q^{6}}{q^{2}+m_{\pi}^{2}}
+\left(\frac{1}{q^{2}+m_{\pi}^{2}}\right)^{2} \frac{q^{8}}{16}\right].\nonumber
\end{eqnarray}

The differences between non-relativistic and boosted structure factors are indicated in Fig. <ref> where we can see that the spatial contribution of the new structure factor almost equals the WIMP structure factor in case of heavy DM.

When the DM is significantly lighter, however, the spatial contribution of
the new structure factor is about one-half of the WIMP structure factor.
In general, BDM gives a suppression factor to the original WIMP factor.
One can easily check that the suppression can be substantially greater in other recoil directions.
Another aspect to consider is that the time component contribution can enhance the proton structure factor to a considerably higher value when $q$ is sufficiently large.
Note that the time component contribution dominates the proton structure factor only in the large $u$ region, which is beyond the ordinary detection region of the recoil energy ($u\ll 1$) in typical DM detectors.

(Color online) Structure factors $S_{p}(u)$ for ${\rm Xe}^{131}$ as a function of $u=q^2b^2/2$, the harmonic-oscillator lengths are $b=2.2905 \rm{fm}$.
$m_\chi=1 \mev$ (upper panel) and $m_\chi=100 \mev$ (lower panel)
at the 1b current level.
The black solid lines show the structure (WIMP) factor of the WIMP DM case.
The red solid lines show the new structure (BDM) factor obtained by considering both the time component and the transfer of momentum contribution of the axial current in the case of boosted scattering between the DM and the nucleus.
The pink dotted lines  are the spatial component contribution and the blue dotted lines are the time component contribution.

In addition, we derive the structure factor for the four-fermion contact interaction.
As a result, it is logical to disregard the momentum dependence i.e. $F_{\mathrm{DM}}=1$.
In spin-dependent scattering, however, the matching between quarks and nucleons gives residual momentum dependence via pion exchange $b_1=m_N a_1/(m_{\pi}^2+q^2)$, which is the coupling of the nucleon matrix.
Such an effect does not come from the light mediator exchange, but rather the finite size effect of protons and neutrons.

Benchmark Model: Cosmic-Ray Boosted Dark Matter:
In the CRDM scenario, DM is boosted by energetic galactic cosmic rays, and it subsequently becomes a fast-moving particle which is one component of  cosmic rays.
Following scattering in detectors, new limits on the DM-nucleon scattering cross section below 1 GeV can be obtained.
CRs transfer kinetic energy to the static DM particle, making it form an energetic flux.
The DM flux in this situation resembles the neutrino flux scattering from outer space, allowing neutrino detectors, such as MiniBooNE et. al.  to give constraints on the CRDM parameter space.
This relativistic DM flux can be obtained via the collision rate of CRs with DM per unit  kinetic energy of CRs ($T_i$) and DM ($T_\chi$) in a differential volume $dV$

\begin{equation}
\frac{d^{2} \Gamma_{\mathrm{CR}_{\mathrm{i}} \rightarrow \chi}}{d T_{i} d T_{\chi}}
=\frac{\rho_{\chi}}{m_{\chi}} \frac{d \sigma_{\chi i}}{d T_{\chi}}
\frac{d \Phi_{i}^{\rm{LIS}}}{d T_{i}} d V,\
\end{equation}

where the flux is taken in the local interstellar (LIS) population
of the CRs  [Della Torre et al., 2016], and $i$ stands for the
specific species of the cosmic rays.
Integrating this over the relevant volume and CR energies yields a boosted DM flux

\begin{equation}
\begin{aligned}
\frac{d \Phi_{\chi}}{d T_{\chi}}&=\int_{\Omega} \frac{d \Omega}{4\pi d^2} \int_{T_{i}^{\min }} d T_{i} \frac{d^{2} \Gamma_{\mathrm{CR}_{i} \rightarrow \chi}}{d T_{i} d T_{\chi}}\\
&=D_{\mathrm{eff}} \frac{\rho_{\chi}}{m_{\chi}} \sum_{i} \int_{T_{i}^{\min }} d T_{i} \frac{d \sigma_{\chi i}}{d T_{\chi}} \frac{d \Phi_{i}^{\mathrm{LIS}}}{d T_{i}}\,.
\end{aligned}\label{eqn:flux}
\end{equation}

Since incoming proton CRs are highly relativistic, the structure factor reduces to $1$.
When the CRDM particle travels from the upper atmosphere to the detector, the scattering with dense matter attenuates the flux to zero, which explains why CRDM searches are sometimes blind to large cross sections: large scattering cross-sections generally give a large CRDM flux, however this also leads to a significant attenuation of the flux.
The degradation of the energy of the CRDM component can be expressed via

\begin{equation}
\frac{d T_{\chi}}{d x}=-\sum_{N} n_{N} \int_0^{T_r^{\rm max}}
\frac{d \sigma}{d T_{r}} T_{r} d T_{r}.
\end{equation}

Here, $T_r$ refers to the energy loss by a CRDM particle in a collision with a nucleus $N$.
${d \sigma}/d T_{r}$ is the differential cross section of DM scattering on dense matter.

Effectively, we can find the CRDM flux at the depth $z$ from the flux at the upper atmosphere via

\begin{equation}
{\frac{d\Phi_{\chi}}{dT_{\chi}^z} = \left(\frac{dT_{\chi}}{dT_{\chi}^z}\right)
\frac{d\Phi_{\chi}}{dT_{\chi}}
= \frac{4 m_{\chi}^2e^{z/\ell}}{\left(
2m_{\chi}+{T_{\chi}^z}  -{T_{\chi}^z} e^{z/\ell}
\right)^{2}}  \frac{d\Phi_{\chi}}{dT_{\chi}}}\,,
\end{equation}

where ${d\Phi_{\chi}}/{dT_{\chi}}$ needs to be evaluated at

\begin{equation}
\label{eq:TztoT}
2m_{\chi} T_{\chi}^ze^{z/\ell}\left(
2m_{\chi} \!+\! {T_{\chi}^z}  \!-\! {T_{\chi}^z} e^{z/\ell}
\right)^{-1}.
\end{equation}

Here $\ell$ denotes the  mean free path of the DM particles, which can be calculated using the scattering cross section and the density of ordinary matter on Earth,

\begin{equation}
\ell^{-1} \equiv \sum_N n_N \int_0^{T_\chi^{\max}} dT_\chi \frac{d\sigma_{\chi N}}{d T_\chi},
\label{freepath}
\end{equation}

where $T_\chi^{\max}$ is the maximal recoil energy of a DM particle.
Note that our definition of the mean free path $\ell$ is different from that of Ref. [Bringmann and Pospelov, 2019] which uses a simpler one.

CRDM particles can transfer the energy to a target nucleus inside the detector, triggering detection events, just as with ordinary DM direct detection.
Therefore there is a natural bridge to reinterpret existing data in the CRDM context.
The equivalence between their event rate gives rise to a constraint on the critical values of mass and coupling.
For WIMP DM, it can read from experiment directly, while for CRDM

\begin{equation}
R =\int_{T_1}^{T_2} {d E_{\rm r}}\frac{1}{m_{\mathrm{T}}} \int_{T_{\chi}^{z,\min }}^{\infty}
d T_{\chi}^z \frac{d \Phi_{\chi}}{d T_{\chi}^z} \frac{d \sigma_{\chi T}}{d E_{\rm r}},
\label{eq:CRDMrate}
\end{equation}

where $T_1$ and $T_2$ are the analysis window for the detectors, and the DM differential flux can be regarded as a modification of the velocity distribution $f({\bm v})$.
The differential event rate is composed of the flux and differential cross section, and the flux $d\Phi/dT_{\chi}^z$ is evaluated at the detector after considering attenuation processes.
$d\sigma_{\chi T}/dE_{\rm r}$ is the differential
cross section of DM-nucleus elastic scattering. One can easily
check that the dominant component $S_A^0$ of the structure factor
can be written as a function of the energy $T_\chi^z$ of the incident DM

\begin{eqnarray}
S_A^0&=& \frac{4(T_\chi^z+m_\chi)^2-q^2}{4(T_\chi^z+m_\chi)^2}
\sum_{L \geqslant 0}\left|\langle {J_f}\!|\!|{\mathcal L}_L^5|\!|\!
&&\hspace{-8mm}+\frac{q^2+4m_\chi^2}{4(T_\chi^z+m_\chi)^2}\sum_{L \geqslant 1} \left(\left|\langle{ J_f}\!|\!|
{\mathcal T}_L^{\mathrm{e l}5}|\!|\!{J_i}\rangle\right|^2
+\left|\langle{ J_f}\!|\!|{\mathcal T}_L^{\mathrm{mag}5}|\!|\!
{ J_i}\rangle\right|^2\right) \,.\nonumber
\end{eqnarray}

Integrating the event rate, we can obtain a relationship between experimental data and theoretical models.
In our numerical study, the package DarkSUSY [Bringmann et al., 2018] is used for simulating CRDM detection.

Limits on the spin-dependent cross section from Xenon1T data.
The red area shows the exclusion results using the boosted structure factor derived in this work; the dark gray contour shows the results from the non-relativistic structure factor.
The difference between the two is evidently significant, especially for light DM.
For comparison, the limits from the direct detection experiments CDMS light [Agnese et al., 2018], PICO60 [Amole et al., 2017] and PICASSO [Aartsen et al., 2020], Borexino
[Alimonti et al., 2002] as well as from delayed-coincidence searches in near-surface detectors by Collar [Collar, 2018] are also shown in the plot..

The numerical results of our calculation are shown in the Fig. <ref>, from which we may conclude the following.

* The upper limits for the two instances considered are the same in the $m_{\chi}\gsim 0.1\rm GeV$ regime because the BDM structure factor recovers the WIMP structure factor when the DM mass is heavier than $1\rm GeV$.
We can see that the exclusion regions are distinct below $0.1\rm GeV$, where the BDM limits become much weaker due to the suppression factor added in Eq. (<ref>).
Notably, BDM limits can be as much as six orders of magnitude weaker for light DM.

* It's worth noting that the detection of the recoil energy $E_{\rm r}$ necessitates a very tiny $u$, yet the transfer momentum $q$ is sufficient.
The maximum $E_{\rm r}$ (40 KeV), for example, indicates that $q$ is at about $100$ MeV.
These results reveal that the new spatial contribution to the structure factor is critical in the evaluation of the spin-dependent scattering between DM and nuclei.

* Another point to note is that the exclusion regions are similar whether or not there is a light mediator between DM and the target nuclei. This can be seen in the WIMP structure factor, although the boost effects on the other hand somewhat cancel it out.
The contact interaction leads to a constant cross section, and the exclusion region is horizontal (as shown
in the results of Borexino [Bringmann and Pospelov, 2019]).
Our results indicate two distinct regions of momentum dependence: the shape of the lower limits is comparable to those from the WIMP structure factor but with an overall difference of two orders of magnitude when $m_\chi \gsim 40\rm MeV$; below 40 MeV, the exclusion region is modified to a considerably larger value canceling the momentum dependence.

The value $m_{\chi}$ dominates the suppression factor when $m_{\chi}\gsim 40\rm MeV$, according to numerical results, whereas $T_{\chi}^z$ dominates in the $m_{\chi}\lsim 40\rm MeV$ region.
This is owing to the fact that the typical transfer momentum $q$ is around 4$\sim$100 MeV.
This means the confined quarks in the nucleon can cause the physics of DM detection to differ significantly from the ordinary non-relativistic scenario.
Equivalently, the finite nuclear size effects lead to momentum dependence in the scattering.

The ongoing search for dark matter is of critical importance to modern physics.
In this letter, we provide the first comprehensive treatment of spin-dependent scattering of boosted dark matter (BDM), a type of dark matter where the usual non-relativistic approximations are not reliable.

We found that when the DM is light, the spatial contribution of the boosted structure factor can be much smaller than the WIMP structure factor, while the time component contribution can boost the proton structure factor when the momentum transfer is large enough.
We also discovered that, even in the absence of a mediator between dark matter and nucleus target, finite nuclear size effects lead to a residual momentum transfer effect.
The complexity in the calculations of the novel boosted structure factor arises because it not only depends on the transfer of momentum $q$, but also the incoming DM momentum.

This new structure factor was applied to the CRDM scenario, providing novel
insight into light DM detection.
In particular, we showed that the exclusion limits can differ by as much as six
orders of magnitude from those calculated using the ordinary non-relativistic
paradigm. Our findings would give conceivable hints on the future
search for the light dark matter.

We thank Nick Houston for useful discussions on the manuscript.
This work was supported by the Natural Science Foundation of China
under grant number
11775012 and 11805161.

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