# Probing quark orbital angular momentum at EIC and EicC

Shohini Bhattacharya  RIKEN BNL Research Center, Brookhaven National
Laboratory, Upton, NY 11973, USA  Duxin Zheng  Shandong Institute of Advanced
Technology, Jinan, Shandong, 250100, China  Jian Zhou  School of Physics and
Key Laboratory of Particle Physics and Particle Irradiation (MOE), Shandong
University, QingDao, Shandong, 266237, China

###### Abstract

We propose to extract quark orbital angular momentum (OAM) through exclusive
$\pi^{0}$ production in electron-(longitudinally-polarized) proton collisions.
Our analysis demonstrates that the $\sin 2\phi$ azimuthal angular correlation
between the transverse momentum of the scattered electron and the recoil
proton serves as a sensitive probe of quark OAM. Additionally, we present a
numerical estimate of the asymmetry associated with this correlation for the
kinematics accessible at EIC and EicC. This study aims to pave the way for the
first measurement of quark OAM in relation to the Jaffe-Manohar spin sum rule.

1\. Introduction—The exploration of nucleon spin structure, sparked by the
revelation of the “spin crisis”, has developed into a captivating research
field over the past three decades. A central goal of this field is to
comprehend the nucleon’s spin in terms of contributions from its underlying
partons. Significant progress has been made in deciphering this partonic
content of nucleon spin, particularly in constraining contributions from quark
and gluon spins in the moderate and large $x$ regions through measurements of
parton helicity distributions at accelerator facilities worldwide Adamczyk et
al. (2015); de Florian et al. (2014); Nocera et al. (2014); Abdallah et al.
(2022). The upcoming Electron-Ion Collider in the US and China (EIC and EicC)
Abdul Khalek et al. (2022); Anderle et al. (2021) is expected to play a
crucial role in precisely determining the gluon helicity distribution at small
$x$. While parton helicities represent a significant fraction of nucleon spin,
there remains ample opportunity to investigate the contribution of parton
orbital angular momentum to nucleon spin, constituting another key objective
of the EIC and EicC.

The kinematic orbital angular momentum (OAM) of quarks and gluons is
determined by subtracting their helicity contributions from the total angular
momentum contributions, which can be accessed through hard exclusive processes
Ji (1997a, b). However, extracting Jaffe-Manohar type parton OAM Jaffe and
Manohar (1990), or equivalently canonical OAM, in high-energy scattering
processes poses a significant experimental challenge. Progress in this
direction was limited until a connection between parton OAM and Wigner
distribution functions Belitsky et al. (2004), or equivalently, Generalized
Transverse Momentum-dependent Distributions (GTMDs) Meissner et al. (2009),
was revealed about a decade ago. For the quark case, this connection is given
by Lorce and Pasquini (2011); Hatta (2012); Lorce et al. (2012),

$\displaystyle L^{q}(x,\xi)=-\int
d^{2}k_{\perp}\frac{k_{\perp}^{2}}{M^{2}}F_{1,4}^{q}(x,k_{\perp},\xi,\Delta_{\perp}=0)\,.$
(1)

The quark OAM can be reconstructed by integrating over the $x$-dependent OAM
distribution: $L_{q}=\int_{0}^{1}dxL_{q}(x,\xi=0)$. This relation, coupled
with Eq. (1), thus opens a new avenue to directly access the parton canonical
OAM contribution to the nucleon spin through GTMDs. In recent years,
theoretical efforts have primarily centered on investigating the experimental
signals of the gluon GTMD $F_{1,4}$ Ji et al. (2017); Hatta et al. (2017);
Bhattacharya et al. (2022a, b); Boussarie et al. (2018). In particular,
polarization-dependent diffractive di-jet production has been extensively
studied in this context. Conversely, the exclusive double Drell-Yan process,
the sole known process providing access to quark GTMDs, offers sensitivity to
quark GTMD $F_{1,4}$ only in the ERBL region Bhattacharya et al. (2017). This
poses a challenge when extrapolating the distribution to the forward limit;
see below in Eq. (1).

In this paper, we introduce a novel observable to experimentally detect the
quark GTMD $F_{1,4}$ in the DGLAP region, establishing a direct link to quark
OAM through Eq. (1) and overcoming the difficulty discussed above in the
context of double Drell-Yan. Our proposal involves the exclusive $\pi^{0}$
production process in electron-proton collisions, $ep\rightarrow
e^{\prime}p^{\prime}\pi^{0}$, with a longitudinally polarized proton target.
Our analysis demonstrates that the longitudinal single target-spin asymmetry
results in a $\sin 2(\phi_{l_{\perp}}-\phi_{\Delta_{\perp}})$ azimuthal
angular correlation, where $\phi_{l_{\perp}}$ and $\Delta_{\perp}$ denote the
azimuthal angles of the transverse momentum of the scattered electron and the
recoil proton. This correlation exhibits a direct sensitivity to quark OAM.

The proposed observable stands out as an ideal probe for quark OAM from both
theoretical and practical perspectives. Firstly, the background for this
process remains clean, free from contamination by final-state soft gluon
radiation effects Hatta et al. (2021a, b); Zhang et al. (2020); Tong et al.
(2023a); Gao et al. (2023); Tong et al. (2023b). Additionally, our observable,
akin to the unpolarized cross section, constitutes a twist-3 contribution.
This characteristic enables the maximal enhancement of the asymmetry without
being washed out by the unpolarized cross section. We will present numerical
results for our observable in EIC and EicC kinematics, showcasing its
potential as a promising tool for directly probing quark OAM.

2\. Probing the quark GTMD $F_{1,4}$ in exclusive $\pi^{0}$ production—First,
let us define the kinematics of the process under consideration,

$\displaystyle
e(l)+p(p,\lambda)\longrightarrow\pi^{0}(l_{\pi})+e(l^{\prime})+p(p^{\prime},\lambda^{\prime})\,.$
(2)

The standard kinematic variables are defined as follows:
$Q^{2}=-q^{2}=-(l-l^{\prime})^{2}$, representing the photon’s virtuality; and
the incoming electron’s momentum is parameterized as
$l^{\mu}=(l^{+},l^{-},l_{\perp})=(\tfrac{Q(1-y)}{\sqrt{2}y},\tfrac{Q}{\sqrt{2}y},\tfrac{Q\sqrt{1-y}}{y})$.
Here, $y=p\cdot q/p\cdot l$ represents the usual momentum fraction. The
$\gamma^{*}p$ center-of-mass energy is given by $W^{2}=(p+q)^{2}$. The pion
mass in our calculation is neglected ($l_{\pi}^{2}\approx 0$), simplifying the
analysis. We work in the symmetric frame where the initial state and the final
state proton carry the transverse momenta $p_{\perp}=-\Delta_{\perp}/2$ and
$p_{\perp}^{\prime}=\Delta_{\perp}/2$, respectively. The skewness variable is
given by
$\xi=(p^{+}-p^{\prime+})/(p^{+}+p^{\prime+})=-\Delta^{+}/(2P^{+})=x_{B}/(2-x_{B})$,
where ‘$+$’ stands for the light-cone plus component and $x_{B}=Q^{2}/2p\cdot
q$, and the momentum transfer squared can be expressed as
$t=(p-p^{\prime})^{2}=-\tfrac{4\xi^{2}M^{2}+\Delta_{\perp}^{2}}{1-\xi^{2}}$,
with $M$ being the proton mass.

In the near forward region, specifically when $\Delta_{\perp}\sim 0$, the
exclusive $\pi^{0}$ production process receives its leading power contribution
from the helicity flip Generalized Parton Distribution (GPD) for a
transversely polarized target, as detailed in Ref. Frankfurt et al. (1999).
However, in the scenario of an unpolarized or longitudinally polarized nucleon
target, the leading power contribution emerges at the twist-3 level.
Crucially, the absence of the unpolarized cross section as the leading
contribution is pivotal in making the investigated longitudinal-spin
asymmetry, which shares a twist-3 nature with the unpolarized cross section, a
non-power-suppressed observable. This characteristic positions it as an ideal
tool for probing the quark OAM. In the region where the momentum transfer $t$
is exceedingly small, the exclusive $\pi^{0}$ production process becomes
susceptible to being dominated by the Primakoff process Primakoff (1951);
Gasparian (2016); Liping et al. (2014); Kaskulov and Mosel (2011); Lepage and
Brodsky (1980); Khodjamirian (1999); Jia et al. (2022). This dominance arises
due to the electromagnetic interaction scaling inversely with $t$, while the
hadronic reaction plateaus as $t$ approaches zero. The $\sin 2\phi$ azimuthal
asymmetry, precisely mirroring what we observe in this study, emerges from the
interference between the Primakoff process and the contribution from the gluon
GTMD $F_{1,4}$, as demonstrated in Ref. Bhattacharya et al. (2023). Note also
that the linear polarization of the exchanged photon plays an essential role
in generating asymmetries through such an interference channel Li et al.
(2020); Li and Lu (2022). In this work, we specifically concentrate on the
valence quark region, where $\xi\sim 0.1$, thereby permitting the neglect of
contributions from both the Primakoff process and the gluon-initiated process
Bhattacharya et al. (2023).

We will perform the calculation within the framework of collinear higher-twist
expansion. In the twist analysis, the hard factor
$H(k_{\perp},\Delta_{\perp})$ is expanded in terms of $k_{\perp}/Q$ and
$\Delta_{\perp}/Q$, where $k_{\perp}$ denotes the relative transverse momentum
carried by the exchanged quarks,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!H(k_{\perp},\Delta_{\perp})=H(k_{\perp}=0,\Delta_{\perp}=0)+$
(3) $\displaystyle\\!\\!\\!\\!\\!\frac{\partial
H(k_{\perp},\Delta_{\perp}\\!\\!=0)}{\partial
k_{\perp}^{\mu}}\Big{|}_{k_{\perp}\\!\\!=0}\\!\\!k_{\perp}^{\mu}+\frac{\partial
H(k_{\perp}\\!\\!=0,\Delta_{\perp})}{\partial\Delta_{\perp}^{\mu}}\Big{|}_{\Delta_{\perp}\\!\\!=0}\\!\\!\Delta_{\perp}^{\mu}+...$

The zeroth-order expansion of $k_{\perp}$ and $\Delta_{\perp}$ yields a null
result for both the spin-averaged cross section and the longitudinal
polarization-dependent cross section. Following this expansion, the subsequent
step involves integrating over $k_{\perp}$. Consequently, the scattering
amplitudes are expressed as the convolution of the next-to-leading power of
Eq. (3) with the GPDs or the first $k_{\perp}$-moment of certain GTMDs,
including the $k_{\perp}$-moment of the quark GTMD $F_{1,4}$—in other words,
the quark OAM distribution.

Before discussing the calculation details, let us briefly recapitulate the
definition of the leading twist quark GTMDs for nucleons. GTMDs parameterize
the off-forward transverse momentum dependent quark-quark correlator Meissner
et al. (2009); Lorcé and Pasquini (2013),

$W_{\lambda,\lambda^{\prime}}^{[\Gamma]}=\\!\\!\int\frac{d^{3}z}{2(2\pi)^{3}}\,e^{ik\cdot
z}\,\langle p^{\prime},\lambda^{\prime}|\,\bar{q}(-\tfrac{z}{2})\Gamma
q(\tfrac{z}{2})\,|p,\lambda\rangle\Big{|}_{z^{+}=0}\,,$ (4)

where $\Gamma$ indicates a generic gamma matrix. The Wilson line in Eq. (4) is
suppressed for brevity. Here, we require the parametrization of Eq. (4) in
terms of GTMDs for $\Gamma=\gamma^{+},\gamma^{+}\gamma_{5}$. In the notation
of Meissner et al. (2009), they are expressed as follows:

$\displaystyle W_{\lambda,\lambda^{\prime}}^{[\gamma^{+}]}\\!$
$\displaystyle=$
$\displaystyle\\!\frac{1}{2M}\bar{u}(p^{\prime},\lambda^{\prime})\bigg{[}F_{1,1}+\frac{i\sigma^{i+}}{P^{+}}(k_{\perp}^{i}F_{1,2}+\Delta_{\perp}^{i}F_{1,3})$
(5)
$\displaystyle\\!+\frac{i\sigma^{ij}k_{\perp}^{i}\Delta_{\perp}^{j}}{M^{2}}\,F_{1,4}\bigg{]}u(p,\lambda)\,,$
$\displaystyle W_{\lambda,\lambda^{\prime}}^{[\gamma^{+}\gamma_{5}]}\\!$
$\displaystyle=$
$\displaystyle\\!\frac{1}{2M}\bar{u}(p^{\prime},\lambda^{\prime})\bigg{[}\frac{-i\varepsilon_{\perp}^{ij}k_{\perp}^{i}\Delta_{\perp}^{j}}{M^{2}}G_{1,1}+\frac{i\sigma^{i+}\gamma_{5}k_{\perp}^{i}}{P^{+}}G_{1,2}$
(6)
$\displaystyle\\!+\frac{i\sigma^{i+}\gamma_{5}\Delta_{\perp}^{i}}{P^{+}}G_{1,3}+i\sigma^{+-}\gamma_{5}G_{1,4}\bigg{]}u(p,\lambda)\,,$

where $\varepsilon_{\perp}^{ij}=\varepsilon^{-+ij}$ with
$\varepsilon^{0123}=1$. The arguments of the GTMDs depend on
$(x,\xi,\vec{k}_{\perp},\vec{\Delta}_{\perp},\vec{k}_{\perp}\cdot\vec{\Delta}_{\perp})$
but have been omitted in the above formulas for the sake of notation
convenience. It is interesting to note that, in general, GTMDs are complex-
valued functions Meissner et al. (2008, 2009). In addition to $F_{1,4}$, the
quark GTMD $G_{1,1}$ is particularly intriguing. The real part of $G_{1,1}$
encodes information about the quark’s spin-orbital angular momentum
correlation inside an unpolarized nucleon Meissner et al. (2009); Lorce and
Pasquini (2011). These GTMDs have been explored in various models Meissner et
al. (2008, 2009); Lorce and Pasquini (2011); Kanazawa et al. (2014); Mukherjee
et al. (2014); Hagiwara et al. (2016); Zhou (2016); Courtoy and Miramontes
(2017); Boer and Setyadi (2023, 2021); Hatta and Zhou (2022); Tan and Lu
(2023); Xu et al. (2022); Ojha et al. (2022, 2023).

There are a total of four diagrams contributing to the exclusive $\pi^{0}$
production amplitude. Our explicit calculation has confirmed that the
contributions from all four diagrams vanish at the leading power. To isolate
the twist-3 contribution, we perform an expansion in $k_{\perp}$ and
$\Delta_{\perp}$. In doing so, it is essential to handle the $k_{\perp}$ and
$\Delta_{\perp}$ dependencies from the exchanged quark legs with utmost care.
To address this, we employ a technique known as the special propagator
technique, first introduced in Ref. Qiu (1990). The inclusion of the special
propagator contribution is crucial to ensure electromagnetic gauge invariance.
It is noteworthy that an alternative approach, which also maintains
electromagnetic gauge invariance at twist-3 accuracy, has been developed in
Refs. Anikin et al. (2000); Radyushkin and Weiss (2001).

Depending on the various vector structures, the scattering amplitude can be
organized into three terms,

$\displaystyle{\cal M}_{1}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{g_{s}^{2}ef_{\pi}}{2\sqrt{2}}\frac{(N_{c}^{2}-1)2\xi}{N_{c}^{2}\sqrt{1-\xi^{2}}}\delta_{\lambda\lambda^{\prime}}\frac{\epsilon_{\perp}\times\Delta_{\perp}}{Q^{2}}\left\\{{\cal
F}_{1,1}+{\cal G}_{1,1}\right\\}\,,$ $\displaystyle{\cal M}_{2}\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\frac{g_{s}^{2}ef_{\pi}}{2\sqrt{2}}\frac{(N_{c}^{2}-1)2\xi}{N_{c}^{2}\sqrt{1-\xi^{2}}}\delta_{\lambda,-\lambda^{\prime}}\frac{M\epsilon_{\perp}\cdot
S_{\perp}}{Q^{2}}\left\\{{\cal F}_{1,2}+{\cal G}_{1,2}\right\\}\,,$
$\displaystyle{\cal M}_{4}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{ig_{s}^{2}ef_{\pi}}{2\sqrt{2}}\frac{(N_{c}^{2}-1)2\xi}{N_{c}^{2}\sqrt{1-\xi^{2}}}\lambda\delta_{\lambda\lambda^{\prime}}\frac{\epsilon_{\perp}\cdot\Delta_{\perp}}{Q^{2}}\left\\{{\cal
F}_{1,4}+{\cal G}_{1,4}\right\\}\,,$ (7)

where $f_{\pi}=131$ MeV represents the $\pi^{0}$ decay constant,
$\epsilon_{\perp}$ denotes the virtual photon’s transverse polarization
vector, and $S_{\perp}$ is defined as
$S_{\perp}^{\mu}=(0^{+},0^{-},-i,\lambda)$. ${\cal F}_{i,j}$ and ${\cal
G}_{i,j}$ serve as shorthand notations for complex convolutions involving the
GTMDs $F_{i,j}$, $G_{i,j}$, and the $\pi^{0}$ distribution amplitude (DA)
$\phi_{\pi}(z)$. They are expressed as follows,

$\displaystyle{\cal F}_{1,1}$ $\displaystyle=$
$\displaystyle\int_{-1}^{1}dx\frac{x^{2}\int
d^{2}k_{\perp}F_{1,1}^{u+d}(x,\xi,\Delta_{\perp},k_{\perp})}{(x+\xi-i\epsilon)^{2}(x-\xi+i\epsilon)^{2}}$
(8)
$\displaystyle\times\int_{0}^{1}dz\frac{\phi_{\pi}(z)(1+z^{2}-z)}{z^{2}(1-z)^{2}}\,,$
$\displaystyle{\cal G}_{1,1}$ $\displaystyle=$
$\displaystyle\int_{-1}^{1}dx\int_{0}^{1}dz\frac{\phi_{\pi}(z)(x^{2}+2x^{2}z+\xi^{2})}{z^{2}(x+\xi-i\epsilon)^{2}(x-\xi+i\epsilon)^{2}}$
(9) $\displaystyle\times\int
d^{2}k_{\perp}\frac{k_{\perp}^{2}}{M^{2}}G_{1,1}^{u+d}(x,\xi,\Delta_{\perp},k_{\perp})\,,$
$\displaystyle{\cal F}_{1,2}$ $\displaystyle=$
$\displaystyle\int_{-1}^{1}dxx\frac{\xi(1-\xi^{2})\int
d^{2}k_{\perp}k_{\perp}^{2}F_{1,2}^{u+d}(x,\xi,\Delta_{\perp},k_{\perp})}{M^{2}(x+\xi-i\epsilon)^{2}(x-\xi+i\epsilon)^{2}}$
(10)
$\displaystyle\times\int_{0}^{1}dz\frac{\phi_{\pi}(z)(1+z^{2}-z)}{z^{2}(1-z)^{2}}\,,$
$\displaystyle{\cal G}_{1,2}$ $\displaystyle=$
$\displaystyle\int_{-1}^{1}dx\int_{0}^{1}dz\frac{\phi_{\pi}(z)(x^{2}+2x^{2}z+\xi^{2})(1-\xi^{2})}{z^{2}(x+\xi-i\epsilon)^{2}(x-\xi+i\epsilon)^{2}}$
(11) $\displaystyle\times\int
d^{2}k_{\perp}\frac{k_{\perp}^{2}}{M^{2}}G_{1,2}^{u+d}(x,\xi,\Delta_{\perp},k_{\perp})\,,$
$\displaystyle{\cal F}_{1,4}$ $\displaystyle=$
$\displaystyle\int_{-1}^{1}dx\frac{x\xi\int
d^{2}k_{\perp}k_{\perp}^{2}F_{1,4}^{u+d}(x,\xi,\Delta_{\perp},k_{\perp})}{M^{2}(x+\xi-i\epsilon)^{2}(x-\xi+i\epsilon)^{2}}$
(12)
$\displaystyle\times\int_{0}^{1}dz\frac{\phi_{\pi}(z)(1+z^{2}-z)}{z^{2}(1-z)^{2}}\,,$
$\displaystyle{\cal G}_{1,4}$ $\displaystyle=$
$\displaystyle\int_{-1}^{1}dx\int_{0}^{1}dz\frac{x(4\xi^{2}z+\xi^{2}-2x^{2}z+x^{2})}{z^{2}\xi(x+\xi-i\epsilon)^{2}(x-\xi+i\epsilon)^{2}}\phi_{\pi}(z)$
(13) $\displaystyle\times\int
d^{2}k_{\perp}G_{1,4}^{u+d}(x,\xi,\Delta_{\perp},k_{\perp})\,.$

The superscript $u+d$ indicates the summation of up and down quark
contributions. For example,
$F_{1,1}^{u+d}(x,\xi,\Delta_{\perp},k_{\perp})=\frac{2}{3}F_{1,1}^{u}(x,\xi,\Delta_{\perp},k_{\perp})+\frac{1}{3}F_{1,1}^{d}(x,\xi,\Delta_{\perp},k_{\perp})$.
Here, $z$ represents the longitudinal momentum fraction of $\pi^{0}$ carried
by the quark. The derivation of the above expressions involves the repeated
use of the symmetry property: $\int
dz\frac{z\phi_{\pi}(z)}{z^{2}(1-z)^{2}}=\int
dz\frac{(1-z)\phi_{\pi}(z)}{z^{2}(1-z)^{2}}$.

A few remarks are now in order. First, we obtain the terms ${\cal F}_{1,2}$,
${\cal F}_{1,4}$, ${\cal G}_{1,1}$, and ${\cal G}_{1,2}$ by performing
$k_{\perp}$ expansion, while the $\Delta_{\perp}$ expansion gives rise to the
contributions ${\cal F}_{1,1}$ and ${\cal G}_{1,4}$. Second, the amplitudes
${\cal M}_{1}$, ${\cal M}_{2}$, and ${\cal M}_{4}$ exhibit distinct
$\Delta_{\perp}$-dependent behaviors. Notably, ${\cal M}_{2}$ persists as
$\Delta_{\perp}$ approaches zero, even when averaging over $S_{\perp}$ in the
unpolarized cross section. This persistence is attributed to the helicity flip
mechanism provided by the quark GTMDs $F_{1,2}$ and $G_{1,2}$, akin to what
the gluon GTMD $F_{1,2}$ does Boussarie et al. (2020). In the forward limit,
the GTMD $F_{1,2}$ is related to the Sivers function $f^{\perp}_{1T}$ Meissner
et al. (2009); Boussarie et al. (2020); Boer et al. (2016); Zhou (2014), and
the GTMD $G_{1,2}$ reduces to the worm-gear function $g_{1T}$ Meissner et al.
(2009). The last point to emphasize is that exclusive $\pi^{0}$ production
selects a C-odd exchange. This implies that the hard factors associated with
$F_{1,1}$, $G_{1,1}$, and $G_{1,2}$ must be even functions of $x$, while those
proportional to $F_{1,2}$, $F_{1,4}$, and $G_{1,4}$ must be odd functions of
$x$. This property is explicitly satisfied by our results.

Assembling all the pieces, we derive the following spin-averaged and single
longitudinal polarization-dependent cross section:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\frac{d\sigma}{dtdQ^{2}dx_{B}d\phi}\\!=\\!\frac{(N_{c}^{2}-1)^{2}\alpha_{em}^{2}\alpha_{s}^{2}f_{\pi}^{2}\xi^{3}\Delta_{\perp}^{2}}{2N_{c}^{4}(1-\xi^{2})Q^{10}(1+\xi)}\left[1+\\!(1\\!-\\!y)^{2}\right]$
(14) $\displaystyle\times$ $\displaystyle\\!\\!\\!\left\\{\left[|{\cal
F}_{1,1}+{\cal G}_{1,1}|^{2}+|{\cal F}_{1,4}+{\cal
G}_{1,4}|^{2}+\\!2\frac{M^{2}}{\Delta_{\perp}^{2}}|{\cal F}_{1,2}+{\cal
G}_{1,2}|^{2}\right]\right.\ $ $\displaystyle+\cos(2\phi)a\left[-|{\cal
F}_{1,1}+{\cal G}_{1,1}|^{2}+|{\cal F}_{1,4}+{\cal G}_{1,4}|^{2}\right]$
$\displaystyle+\lambda\sin(2\phi)\,2a\,{\rm Re}\left[\left(i{\cal
F}_{1,4}+i{\cal G}_{1,4}\right)\left({\cal F}^{*}_{1,1}+{\cal
G}^{*}_{1,1}\right)\right]\Big{\\}}$

where $\phi=\phi_{l_{\perp}}-\phi_{\Delta_{\perp}}$ and
$a=\frac{2(1-y)}{1+(1-y)^{2}}$. Eq. (14) stands as the central result of our
paper. As demonstrated, the real part of the quark GTMD $F_{1,4}$, and
consequently, the quark OAM, leaves a distinct signature through an azimuthal
angular correlation of $\sin 2\phi$ in the longitudinal single target-spin
asymmetry. Moreover, a $\cos 2\phi$ azimuthal angular modulation in the spin-
averaged cross section provides an additional handle onto various components
(real and imaginary parts) of the quark GTMDs. Since both unpolarized and
polarized cross sections contribute at twist-3, the magnitudes of the
asymmetries are not power-suppressed. Consequently, we emphasize that
experimentally extracting quark OAM through the $\sin 2\phi$ asymmetry is less
challenging compared to, for instance, the azimuthal asymmetry in diffractive
di-jet production, which constitutes a power correction.

Figure 1: The unpolarized part of the cross section, as given by Eq. (14), is
displayed in the top plot for EIC kinematics with $Q^{2}=10\,\textrm{GeV}^{2}$
and $\sqrt{s_{ep}}=100\,\textrm{GeV}$, as well as for EicC kinematics with
$Q^{2}=3\,\textrm{GeV}^{2}$ and $\sqrt{s_{ep}}=16\,\textrm{GeV}$. The
unpolarized cross section for the EIC case is re-scaled by a factor of 100.
The bottom plot shows the average value of $\langle\sin(2\phi)\rangle$ given
by Eq. (18) in EIC and EicC kinematics. The variable $t$ is integrated over
the range [$-0.5\,\textrm{GeV}^{2}$, $-\frac{4\xi^{2}M^{2}}{1-\xi^{2}}$].

3\. Numerical results—We now present the numerical results for both the
unpolarized cross section and the $\sin 2\phi$ asymmetry. It is noteworthy
that two of the $k_{\perp}$-integrated GTMDs can be linked to the standard
unpolarized GPD and the helicity GPD Meissner et al. (2009),

$\displaystyle\int d^{2}k_{\perp}{\rm
Re}[F_{1,1}(x,\xi,\Delta_{\perp},k_{\perp})]$ $\displaystyle\approx$
$\displaystyle H(x,\xi,\Delta_{\perp})\,,$ (15) $\displaystyle\int
d^{2}k_{\perp}{\rm Re}[G_{1,4}(x,\xi,\Delta_{\perp},k_{\perp})]$
$\displaystyle\approx$ $\displaystyle\tilde{H}(x,\xi,\Delta_{\perp})\,,$ (16)

where we neglect terms proportional to $\xi^{2}$ on the right-hand side of the
aforementioned equations. As previously mentioned, the imaginary part of the
GTMD $F_{1,2}$ is intricately linked to the quark Sivers function in the
forward limit. The Sivers functions for up and down quarks are recognized to
exhibit similar magnitudes but with opposite signs. It is reasonable to
anticipate that the contributions of $F_{1,2}^{u}$ and $F_{1,2}^{d}$ are
largely canceled out. The same rationale applies to the $G_{1,2}$ term, which,
in the forward limit, is associated with the worm-gear function Bhattacharya
et al. (2022c). Thus, in our first attempt at a numerical study, we choose to
neglect contributions from the GTMDs $F_{1,2}$ and $G_{1,2}$, as well as from
$G_{1,1}$, which lacks a GPD or TMD counterpart and remains unconstrained thus
far. Regarding the ${\cal F}_{1,4}$ and ${\cal G}_{1,4}$ terms, we only
consider their pole contributions from their imaginary parts. However, for the
term ${\cal F}_{1,1}$, the magnitude of which is expected to be the largest
among all GTMDs, we include both its imaginary and real parts in the numerical
estimation.

Note that the hard part becomes divergent as $z$ approaches 0 or 1. This
behavior, known as the endpoint singularity, typically signals factorization
breaking. From a phenomenological standpoint, regularization is achievable by
considering the transverse momentum dependence of the pion DA Goloskokov and
Kroll (2005, 2007); Sun et al. (2021). An effective way to introduce
transverse momentum dependence is to modify the upper and lower integration
limits of $z$ to $\int_{\langle p_{\perp}^{2}\rangle/Q^{2}}^{1-\langle
p_{\perp}^{2}\rangle/Q^{2}}dz$, where $\langle p_{\perp}^{2}\rangle$ is the
mean squared transverse momentum of the quark inside the pion, chosen to be
$\langle p_{\perp}^{2}\rangle=0.04\ \textrm{GeV}^{2}$ Goloskokov and Kroll
(2008) in our numerical calculation. For simplicity, we consider the
asymptotic form for the pion’s DA, $\phi_{\pi}(z)=6z(1-z)$.

On the other hand, the discontinuity of the derivative of quark GPDs at the
endpoints $x=\pm\xi$ (as seen in, for example, Refs. Bhattacharya et al.
(2019, 2020)), coupled with the double poles at $x=\pm\xi$, also contributes
to a divergent component in the cross section. To resolve this issue, we shift
the double pole from $\tfrac{1}{(x-\xi+i\epsilon)^{2}}$ to
$\tfrac{1}{(x-\xi-\langle p_{\perp}^{2}\rangle/Q^{2}+i\epsilon)^{2}}$ (and
similarly for the negative $x$ region) following the same rationale.
Essentially, we affirm the necessity of some transverse motion of the partons
within the pion and proton to smear the effects of poles at the endpoints and
induce a shift.

To provide a model input for the $x$-dependent quark OAM distribution,
specifically the $k_{\perp}$-moment of $F_{1,4}$, we employ the Wandzura-
Wilczek (WW) approximation Hatta and Yoshida (2012),

$\displaystyle L_{q}(x)\approx
x\\!\int_{x}^{1}\\!\frac{dx^{\prime}}{x^{\prime}}q(x^{\prime})-x\\!\int_{x}^{1}\\!\frac{dx^{\prime}}{x^{\prime
2}}\Delta q(x^{\prime})\,,$ (17)

where $q(x^{\prime})$ and $\Delta q(x^{\prime})$ denote the usual unpolarized
quark PDF and quark helicity distribution, respectively. It is a common
practice to reconstruct the $\xi$-dependence for $L_{q}(x,\xi)$ from its PDF
counterpart $L_{q}(x)$ using the double distribution method Radyushkin (1999,
2000); Goloskokov and Kroll (2008). In this context, we use the JAM (valence)
quark PDFs $q(x)$ and $\Delta q(x)$ as inputs in Eq. (17) from Ref. Cocuzza et
al. (2022). As for its $t$-dependence, we adopt a Gaussian form factor,
represented by $e^{t/\Lambda}$ with $\Lambda=0.5\,\textrm{GeV}^{2}$.
Similarly, we reconstruct the $\xi$ and $t$-dependence of the GPDs
$H(x,\xi,t)$ and $\tilde{H}(x,\xi,t)$ from their PDF counterparts using the
double distribution method.

We now present numerical predictions for the EIC and EicC kinematics. The
$t$-integrated unpolarized cross section is shown as a function of $\xi$ in
the top panel of Fig. 1. The asymmetry, quantified by the average value of
$\sin(2\phi)$ and depicted as a function of $\xi$ in the bottom plot of Fig.
1, is defined as:

$\displaystyle\langle\sin(2\phi)\rangle$ $\displaystyle=$
$\displaystyle\frac{\int\frac{d\Delta\sigma}{d{\cal P.S.}}\sin(2\phi)\ d{\cal
P.S.}}{\int\frac{d\sigma}{d{\cal P.S.}}d{\cal P.S.}}\,,$ (18)

where $d\Delta\sigma=\sigma(\lambda=1)-\sigma(\lambda=-1)$. The unpolarized
cross section exhibits a notable magnitude at EicC energy, whereas it is
relatively small at EIC energy. Additionally, the asymmetries are substantial
for both EIC and EicC kinematics. Examination of the plot suggests an almost
linear rise in asymmetries with increasing $\xi$. Consequently, our numerical
results signify that the azimuthal asymmetry $\sin 2\phi$ in exclusive
$\pi^{0}$ production stands out as a promising avenue for probing the quark
OAM distribution.

4\. Summary—We propose extracting the quark OAM associated with the Jaffe-
Manohar spin sum rule by measuring the azimuthal angular correlation $\sin
2\phi$ in exclusive $\pi^{0}$ production at EIC and EicC. This observable
serves as a clean and sensitive probe of quark OAM for several reasons.
Firstly, the azimuthal asymmetry is not a power correction, as both the
unpolarized and longitudinal polarization-dependent cross sections contribute
at twist-3. Secondly, the produced $\pi^{0}$ transverse momentum
$-\Delta_{\perp}$ remains unaffected by final state QCD radiations. Detecting
$\pi^{0}$ makes it less challenging to experimentally measure
$\Delta_{\perp}$, in contrast to the diffractive di-jet production case where
reconstructing $\Delta_{\perp}$ from the total transverse momentum of the di-
jet system is impossible due to the contamination of final-state soft gluon
radiations. Most importantly, this process enables access to the quark GTMD
$F_{1,4}$ in the DGLAP region for the first time. In contrast, the exclusive
double Drell-Yan process only allows the access of quark GTMD $F_{1,4}$ in the
EBRL region, making it challenging to extrapolate the functional form of
$F_{1,4}$ to the forward limit.

We computed the differential cross section within the collinear higher-twist
expansion framework for both an unpolarized and longitudinally polarized
proton target. Despite the substantial uncertainties associated with the model
inputs, our numerical results reveal a sizable azimuthal asymmetry, which
critically relies on the quark OAM distribution. In the kinematic range
accessible to the EIC and EicC, our observable can be thoroughly investigated,
paving the way for the first experimental extraction of the canonical quark
OAM distribution in the future.

Acknowledgements: The authors would like to express their gratitude to C.
Cocuzza for providing the LHAPDF tables of JAM22 PDFs as referenced in Cocuzza
et al. (2022) and to F. Yuan for insightful discussions. This work has been
supported by the National Natural Science Foundation of China under Grant No.
12175118 (J. Z.), and the National Science Foundation under Contract No.
PHY-1516088. S. B. has been supported by the U.S. Department of Energy under
Contract No. DE-SC0012704, and also by Laboratory Directed Research and
Development (LDRD) funds from Brookhaven Science Associates.

## References

  * Adamczyk et al. (2015) L. Adamczyk et al. (STAR), Phys. Rev. Lett. 115, 092002 (2015), eprint 1405.5134.
  * de Florian et al. (2014) D. de Florian, R. Sassot, M. Stratmann, and W. Vogelsang, Phys. Rev. Lett. 113, 012001 (2014), eprint 1404.4293.
  * Nocera et al. (2014) E. R. Nocera, R. D. Ball, S. Forte, G. Ridolfi, and J. Rojo (NNPDF), Nucl. Phys. B 887, 276 (2014), eprint 1406.5539.
  * Abdallah et al. (2022) M. S. Abdallah et al. (STAR), Phys. Rev. D 105, 092011 (2022), eprint 2110.11020.
  * Abdul Khalek et al. (2022) R. Abdul Khalek et al., Nucl. Phys. A 1026, 122447 (2022), eprint 2103.05419.
  * Anderle et al. (2021) D. P. Anderle et al., Front. Phys. (Beijing) 16, 64701 (2021), eprint 2102.09222.
  * Ji (1997a) X.-D. Ji, Phys. Rev. Lett. 78, 610 (1997a), eprint hep-ph/9603249.
  * Ji (1997b) X.-D. Ji, Phys. Rev. D 55, 7114 (1997b), eprint hep-ph/9609381.
  * Jaffe and Manohar (1990) R. L. Jaffe and A. Manohar, Nucl. Phys. B 337, 509 (1990).
  * Belitsky et al. (2004) A. V. Belitsky, X.-d. Ji, and F. Yuan, Phys. Rev. D 69, 074014 (2004), eprint hep-ph/0307383.
  * Meissner et al. (2009) S. Meissner, A. Metz, and M. Schlegel, JHEP 08, 056 (2009), eprint 0906.5323.
  * Lorce and Pasquini (2011) C. Lorce and B. Pasquini, Phys. Rev. D 84, 014015 (2011), eprint 1106.0139.
  * Hatta (2012) Y. Hatta, Phys. Lett. B 708, 186 (2012), eprint 1111.3547.
  * Lorce et al. (2012) C. Lorce, B. Pasquini, X. Xiong, and F. Yuan, Phys. Rev. D 85, 114006 (2012), eprint 1111.4827.
  * Ji et al. (2017) X. Ji, F. Yuan, and Y. Zhao, Phys. Rev. Lett. 118, 192004 (2017), eprint 1612.02438.
  * Hatta et al. (2017) Y. Hatta, Y. Nakagawa, F. Yuan, Y. Zhao, and B. Xiao, Phys. Rev. D 95, 114032 (2017), eprint 1612.02445.
  * Bhattacharya et al. (2022a) S. Bhattacharya, R. Boussarie, and Y. Hatta, Phys. Rev. Lett. 128, 182002 (2022a), eprint 2201.08709.
  * Bhattacharya et al. (2022b) S. Bhattacharya, A. Metz, V. K. Ojha, J.-Y. Tsai, and J. Zhou, Phys. Lett. B 833, 137383 (2022b), eprint 1802.10550.
  * Boussarie et al. (2018) R. Boussarie, Y. Hatta, B.-W. Xiao, and F. Yuan, Phys. Rev. D 98, 074015 (2018), eprint 1807.08697.
  * Bhattacharya et al. (2017) S. Bhattacharya, A. Metz, and J. Zhou, Phys. Lett. B 771, 396 (2017), [Erratum: Phys.Lett.B 810, 135866 (2020)], eprint 1702.04387.
  * Hatta et al. (2021a) Y. Hatta, B.-W. Xiao, F. Yuan, and J. Zhou, Phys. Rev. D 104, 054037 (2021a), eprint 2106.05307.
  * Hatta et al. (2021b) Y. Hatta, B.-W. Xiao, F. Yuan, and J. Zhou, Phys. Rev. Lett. 126, 142001 (2021b), eprint 2010.10774.
  * Zhang et al. (2020) C. Zhang, Q.-S. Dai, and D. Y. Shao, JHEP 23, 002 (2020), eprint 2211.07071.
  * Tong et al. (2023a) X.-B. Tong, B.-W. Xiao, and Y.-Y. Zhang, Phys. Rev. Lett. 130, 151902 (2023a), eprint 2211.01647.
  * Gao et al. (2023) M.-S. Gao, Z.-B. Kang, D. Y. Shao, J. Terry, and C. Zhang, JHEP 10, 013 (2023), eprint 2306.09317.
  * Tong et al. (2023b) X.-B. Tong, B.-W. Xiao, and Y.-Y. Zhang (2023b), eprint 2310.20662.
  * Frankfurt et al. (1999) L. L. Frankfurt, P. V. Pobylitsa, M. V. Polyakov, and M. Strikman, Phys. Rev. D 60, 014010 (1999), eprint hep-ph/9901429.
  * Primakoff (1951) H. Primakoff, Phys. Rev. 81, 899 (1951).
  * Gasparian (2016) A. H. Gasparian, PoS CD15, 048 (2016).
  * Liping et al. (2014) G. Liping, R. Miskimen, A. Gasparian, J. Goity, W. Rong, and C. Xurong, Nucl. Phys. Rev. 31, 453 (2014).
  * Kaskulov and Mosel (2011) M. M. Kaskulov and U. Mosel, Phys. Rev. C 84, 065206 (2011), eprint 1103.2097.
  * Lepage and Brodsky (1980) G. P. Lepage and S. J. Brodsky, Phys. Rev. D 22, 2157 (1980).
  * Khodjamirian (1999) A. Khodjamirian, Eur. Phys. J. C 6, 477 (1999), eprint hep-ph/9712451.
  * Jia et al. (2022) Y. Jia, Z. Mo, J. Pan, and J.-Y. Zhang (2022), eprint 2207.14171.
  * Bhattacharya et al. (2023) S. Bhattacharya, D. Zheng, and J. Zhou (2023), eprint 2304.05784.
  * Li et al. (2020) C. Li, J. Zhou, and Y.-J. Zhou, Phys. Rev. D 101, 034015 (2020), eprint 1911.00237.
  * Li and Lu (2022) H. Li and Z. Lu, Eur. Phys. J. C 82, 668 (2022), eprint 2111.03840.
  * Lorcé and Pasquini (2013) C. Lorcé and B. Pasquini, JHEP 09, 138 (2013), eprint 1307.4497.
  * Meissner et al. (2008) S. Meissner, A. Metz, M. Schlegel, and K. Goeke, JHEP 08, 038 (2008), eprint 0805.3165.
  * Kanazawa et al. (2014) K. Kanazawa, C. Lorcé, A. Metz, B. Pasquini, and M. Schlegel, Phys. Rev. D 90, 014028 (2014), eprint 1403.5226.
  * Mukherjee et al. (2014) A. Mukherjee, S. Nair, and V. K. Ojha, Phys. Rev. D 90, 014024 (2014), eprint 1403.6233.
  * Hagiwara et al. (2016) Y. Hagiwara, Y. Hatta, and T. Ueda, Phys. Rev. D 94, 094036 (2016), eprint 1609.05773.
  * Zhou (2016) J. Zhou, Phys. Rev. D 94, 114017 (2016), eprint 1611.02397.
  * Courtoy and Miramontes (2017) A. Courtoy and A. S. Miramontes, Phys. Rev. D 95, 014027 (2017), eprint 1611.03375.
  * Boer and Setyadi (2023) D. Boer and C. Setyadi (2023), eprint 2301.07980.
  * Boer and Setyadi (2021) D. Boer and C. Setyadi, Phys. Rev. D 104, 074006 (2021), eprint 2106.15148.
  * Hatta and Zhou (2022) Y. Hatta and J. Zhou, Phys. Rev. Lett. 129, 252002 (2022), eprint 2207.03378.
  * Tan and Lu (2023) C. Tan and Z. Lu (2023), eprint 2301.09081.
  * Xu et al. (2022) S. Xu, C. Mondal, X. Zhao, Y. Li, and J. P. Vary (2022), eprint 2209.08584.
  * Ojha et al. (2022) V. K. Ojha, S. Jana, and T. Maji (2022), eprint 2211.02959.
  * Ojha et al. (2023) V. K. Ojha, S. Jana, and T. Maji (2023), eprint 2309.03917.
  * Qiu (1990) J.-W. Qiu, Phys. Rev. D 42, 30 (1990).
  * Anikin et al. (2000) I. V. Anikin, B. Pire, and O. V. Teryaev, Phys. Rev. D 62, 071501 (2000), eprint hep-ph/0003203.
  * Radyushkin and Weiss (2001) A. V. Radyushkin and C. Weiss, Phys. Rev. D 63, 114012 (2001), eprint hep-ph/0010296.
  * Boussarie et al. (2020) R. Boussarie, Y. Hatta, L. Szymanowski, and S. Wallon, Phys. Rev. Lett. 124, 172501 (2020), eprint 1912.08182.
  * Boer et al. (2016) D. Boer, M. G. Echevarria, P. Mulders, and J. Zhou, Phys. Rev. Lett. 116, 122001 (2016), eprint 1511.03485.
  * Zhou (2014) J. Zhou, Phys. Rev. D 89, 074050 (2014), eprint 1308.5912.
  * Bhattacharya et al. (2022c) S. Bhattacharya, Z.-B. Kang, A. Metz, G. Penn, and D. Pitonyak, Phys. Rev. D 105, 034007 (2022c), eprint 2110.10253.
  * Goloskokov and Kroll (2005) S. V. Goloskokov and P. Kroll, Eur. Phys. J. C 42, 281 (2005), eprint hep-ph/0501242.
  * Goloskokov and Kroll (2007) S. V. Goloskokov and P. Kroll, Eur. Phys. J. C 50, 829 (2007), eprint hep-ph/0611290.
  * Sun et al. (2021) P. Sun, X.-B. Tong, and F. Yuan, Phys. Lett. B 822, 136655 (2021), eprint 2103.12047.
  * Goloskokov and Kroll (2008) S. V. Goloskokov and P. Kroll, Eur. Phys. J. C 53, 367 (2008), eprint 0708.3569.
  * Bhattacharya et al. (2019) S. Bhattacharya, C. Cocuzza, and A. Metz, Phys. Lett. B 788, 453 (2019), eprint 1808.01437.
  * Bhattacharya et al. (2020) S. Bhattacharya, C. Cocuzza, and A. Metz, Phys. Rev. D 102, 054021 (2020), eprint 1903.05721.
  * Hatta and Yoshida (2012) Y. Hatta and S. Yoshida, JHEP 10, 080 (2012), eprint 1207.5332.
  * Radyushkin (1999) A. V. Radyushkin, Phys. Rev. D 59, 014030 (1999), eprint hep-ph/9805342.
  * Radyushkin (2000) A. V. Radyushkin (2000), eprint hep-ph/0101225.
  * Cocuzza et al. (2022) C. Cocuzza, W. Melnitchouk, A. Metz, and N. Sato (Jefferson Lab Angular Momentum (JAM)), Phys. Rev. D 106, L031502 (2022), eprint 2202.03372.


# Polarized target as a tool for probing the non-standard properties of dark
matter

Arkadiusz Błaut<EMAIL_ADDRESS>Wiesław Sobków
<EMAIL_ADDRESS>Institute of Theoretical Physics, University of
Wrocław, Pl. M. Born 9, PL-50-204 Wrocław, Poland

###### Abstract

Possibility of using the polarized nucleus target for testing the non-standard
properties of fermionic dark matter is considered. It is assumed that the
incoming dark matter scatters on the polarized nuclei in the presence of
vector, axial, scalar and tensor interactions. The scalar and tensor couplings
are assumed to be complex numbers. Various types of measurement settings are
tested depending on the assumptions regarding dark matter polarization and
measurement of the recoil nucleus spin. In particular we address the question
of finding and distinguishing effects of theoretically possible scenarios in
which mixed triple products among various polarization vectors, the momenta of
incoming DM and of recoil nucleus appear in the cross section. Their presence
would indicate the possibility of time reversal symmetry violation in the
polarized dark matter-nucleus scattering. Our considerations are model-
independent and carried out in the non-relativistic dark matter and recoiled
nucleus limit.

###### keywords:

fermionic dark matter, non-standard interactions, polarized target , time
reversal symmetry violation

††journal: Journal of Physics of the Dark Universe

##  1 Introduction

The astronomical and cosmological data indicate the presence of dark matter
(DM), but this evidence is indirect and based on the gravity impact on stars,
galaxies, galaxy clusters, the cosmic microwave background radiation. The
particle nature of DM is still not identified. Let us recall that the Standard
Model(SM) [1, 2, 3, 4, 5] does not explain the concepts of DM and dark energy,
and some fundamental issues such as: the origin of parity violation, the
violation of combined symmetry of charge conjugation and space inversion (CP),
observed baryon asymmetry between matter and antimatter of universe [6]
through a single CP-violating phase of the Cabibbo-Kobayashi-Maskawa quark-
mixing matrix (CKM) [7]. There is also a lack of clarity about the hierarchy
problem (the enormous difference between the weak and Planck scales in the
presence of the Higgs field) and the spontaneous symmetry breaking associated
with the origin of mass. In such a situation it is natural to suppose that SM
is the low-energy limit of a more fundamental theory. This has led to a number
of proposals for non-standard models, but none of them have proved conclusive.
There are many candidates for DM particles predicted by various non-standard
schemes. One of the most intensive and constantly investigated ideas is the
paradigm of weakly interacting massive particles (WIMPs) with masses around
the weak scale [8, 9, 10, 11]. WIMPs can be made of heavy fourth-generation
neutrinos, supersymmetric particles, particles in models with universal extra
dimensions, etc. These may be scalar or fermion particles (Dirac or Majorana
type) [12, 13, 14, 15]. Despite the efforts, there is no direct or indirect
evidence of the existence of the WIMPs to date, but new measurement strategies
have been proposed to detect DM in the laboratory, in accelerators and spaces.
In this situation it makes sense to examine also other theoretically possible
scenarios, such as a light DM (LDM), with masses in the keV to GEV range.
Interesting studies of the use of LDM scattering on electrons can be found in
[16, 17, 18].
The main goal of terrestrial direct-detection experiments is the measurement
of energy spectrum and the direction of the nuclear recoils (or recoil
electrons in the case of LDM) coming from the interactions of DM with ordinary
particles [19, 20, 21, 22, 23]. It must be noted that in these experiments
there is an irreducible background produced by neutrinos, which can mimic the
right signal. The key issue of new experiments will be the possibility of
distinguishing between events from the scattering of DM and events generated
by neutrino scattering in the presence of non-standard interactions [24, 25,
26, 27, 28, 29]. The approach shown in [30] is worth noting, as it assumes
that exotic interactions can communicate with both DM and neutrinos.
Furthermore, neutrino oscillation effects should also be taken into account.
The above ideas involve the un-polarized detection targets.

As an alternative to experiments with un-polarized targets, the idea of using
polarized target (PT) to detect DM is worth to be considered. It should be
stressed that PT has been proposed to probe the flavour composition of
(anti)neutrino beam [31], the neutrino magnetic moments [32, 33], the time
reversal symmetry violation and neutrino nature in the (semi)leptonic
processes [34, 35, 36], and most recently also for searching for DM [37, 38,
39]. A basic advantage of using PT is an opportunity of changing the rate of
weak interaction by inverting the direction of magnetic field. This feature is
very essential in the measurement of low energy neutrinos because the
background level would be precisely controlled in the direct detection DM
experiments [40]. It is important to mention the prototype measurements
confirming the possibility of building the polarized target crystal of
$Gd_{2}SiO_{5}$ (GSO) doped with Cerium (GSO:Ce) [41]. It is also noteworthy
that there are known the methods of producing the polarized gasses such as
helium, argon and xenon [42, 43].

In this paper, we focus on using a polarized target to test the non-standard
properties of fermionic DM in the presence of vector, axial, scalar and tensor
interactions. In particular, we think about the possibility of breaking
symmetry with respect to time inversion for the scenarios with the complex
scalar and tensor couplings of DM. It is worth noting here an interesting
analysis of the possibility of P- and CP-violation in un-polarized DM-nucleus
scattering [44]. We show how the energy and angular (polar-azimuthal)
distributions of recoil nuclei are sensitive to the non-vanishing mixed triple
products among various polarization vectors, the momenta of incoming DM and of
recoil nucleus. In addition, we check whether these observables can be used to
differentiate between the Dirac and the Majorana DM.

##  2 Scattering of fermionic dark matter on polarized target

Figure 1: Kinematics of polarized DM-nucleus scattering. $\hat{v}_{e}$ is the
Earth’s velocity in the galactic rest frame; ${\hat{s}_{1}}$ is the
polarization vector of the nuclear target; $\vec{v}_{\chi}$ is the velocity of
the incoming DM; ${\hat{s}_{2}}$ is the polarization vector of the incoming
DM; $\vec{v}$ is the velocity of the recoil nucleus; ${\hat{s}_{3}}$ is the
polarization vector of the outgoing nucleus; $\alpha$, $\beta$ are the polar
and azimuthal angles of recoil nucleus velocity; $\theta$ and $\phi$ are the
polar and azimuthal angles of incoming DM velocity. The three vectors,
$\left\\{{\hat{s}_{1}},\hat{y},\hat{v}_{e}\right\\}$, form an orthonormal
frame.

We consider the scattering of fermionic DM of the Dirac type on the polarized
nuclear target: $\chi+N\rightarrow\chi+N$. We assume that a DM fermion of mass
$m_{\chi}$ interacts with a spin $1/2$ nucleus of mass $m$ via the four-
fermion point interactions. The initial nucleus is polarized and at rest. For
simplicity, the nucleus is treated a point object (the nuclear effects are
assumed to be contained in the effective couplings). However, it should be
made clear here that a complete analysis of nuclear effects will be necessary
for experiments with a specific target and may be the subject of a new study.

Fig. 1 illustrates the relative orientation of all vectors and explains the
notation for the relevant quantities. The amplitude for this scattering is of
the form:

$\displaystyle M$ $\displaystyle=$ $\displaystyle\left[\overline{u}_{\rm
N^{\prime}}\gamma^{\alpha}(c_{\rm V}+c_{\rm A}\gamma_{5})u_{\rm
N}\right]\left[\overline{u}_{\chi^{\prime}}\gamma_{\alpha}(d_{\rm V}+d_{\rm
A}\gamma_{5})u_{\chi}\right]$ $\displaystyle\mbox{}+\left[c_{\rm S_{\rm
R}}\overline{u}_{\rm N^{\prime}}(1+\gamma_{5})u_{\rm N}+c_{\rm S_{\rm
L}}\overline{u}_{\rm N^{\prime}}(1-\gamma_{5})u_{\rm N}\right]\left[d_{\rm
S_{\rm R}}\overline{u}_{\chi^{\prime}}(1+\gamma_{5})u_{\chi}+d_{\rm S_{\rm
L}}\overline{u}_{\chi^{\prime}}(1-\gamma_{5})u_{\chi}\right]$
$\displaystyle\mbox{}+\frac{1}{2}\left[c_{\rm T_{\rm R}}\overline{u}_{\rm
N^{\prime}}\sigma^{\alpha\beta}(1+\gamma_{5})u_{\rm N}+c_{\rm T_{\rm
L}}\overline{u}_{\rm N^{\prime}}\sigma^{\alpha\beta}(1-\gamma_{5})u_{\rm
N}\right]\left[d_{\rm T_{\rm
R}}(\overline{u}_{\chi^{\prime}}\sigma_{\alpha\beta}(1+\gamma_{5})u_{\chi})+d_{T_{\rm
L}}(\overline{u}_{\chi^{\prime}}\sigma_{\alpha\beta}(1-\gamma_{5})u_{\chi})\right].$

The coupling constants $c_{\rm V}$, $c_{\rm A}$, $d_{\rm V}$, $d_{\rm A}$
corresponding to the vector and axial interactions are real numbers, while the
non-standard $c_{\rm S_{\rm R,L}},c_{\rm T_{\rm R,L}},d_{\rm S_{\rm
R,L}},d_{\rm T_{\rm R,L}}$ couplings, associated to the scalar and tensor
interactions, are the complex numbers denoted as $c_{\rm S_{\rm R}}=|c_{\rm
S_{\rm R}}|e^{i\,\phi_{\rm S_{\rm R}}},d_{\rm S_{\rm R}}=|d_{\rm S_{\rm
R}}|e^{i\,\theta_{\rm S_{\rm R}}}$, etc. Reality of vector and axial coupling
constants follows from the hermiticity of the interaction lagrangian. For the
same reason, we also take into account the relations between the non-standard
complex couplings with left- and right-handed chirality, $d_{\rm S_{\rm L},\rm
T_{\rm L}}=(d_{\rm S_{\rm R},\rm T_{\rm R}})^{*}$, $c_{\rm S_{\rm L},\rm
T_{\rm L}}=(c_{\rm S_{\rm R},\rm T_{\rm R}})^{*}$.

Our main goal to show how the presence of non-standard couplings manifest
themselves in the angular and energy distributions of the recoil nucleus. This
is mainly about the effects of a time reversal violation in the polarized DM-
nucleus scattering. To this end, we calculate the differential cross section,
the differential event rate per unit detector mass and define the function
that purely depends on the target polarization:

$\displaystyle d\sigma$ $\displaystyle=$
$\displaystyle\frac{|M|^{2}}{64\,\pi^{2}\,m_{\chi}^{2}\,m\,v_{\chi}\,|\hat{v}_{\chi}\cdot\hat{v}|}\,\delta\left(v_{\chi}-\bar{v}\right)\,dE_{\rm
R}\,d\beta\,d\cos{\alpha},$ (2) $\displaystyle\frac{d^{2}R}{dE_{\rm
R}\,d\Omega}$ $\displaystyle=$
$\displaystyle\frac{\rho_{\chi}}{m_{\chi}\,m}\int
d^{3}v_{\chi}\,\frac{d^{2}\sigma}{dE_{\rm
R}\,d\Omega}\,v_{\chi}\,f(\vec{v}_{\chi})$ (3) $\displaystyle=$
$\displaystyle\sum\limits_{l=1,2}\frac{\rho_{\chi}}{64\,\pi^{2}\,m_{\chi}^{3}\,m^{2}}\int
d\varphi\int d\cos{\theta}\,\,\frac{\bar{v}^{2}|\,M(E_{\rm
R},\alpha,\beta,\theta,\varphi)|^{2}}{|\hat{v}_{\chi}\cdot\hat{v}|}\,\Theta_{l}\,f(\vec{v}_{\chi}=\bar{v}\,\hat{v}_{\chi}),$
$\displaystyle\frac{d^{2}\Delta R}{dE_{\rm R}d\,\Omega}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{[}\frac{d^{2}R({\hat{s}_{1}})}{dE_{\rm
R}\,d\Omega}-\frac{d^{2}R(-{\hat{s}_{1}})}{dE_{\rm R}\,d\Omega}\bigg{]}.$ (4)

$\rho_{\chi}\simeq 0.4\,$GeV cm-3 is the local DM density;

$f(\vec{v}_{\chi})=\frac{1}{N}e^{-(\vec{v}_{\chi}+\vec{v}_{e})^{2}/v_{0}^{2}}$
(5)

(for $|\vec{v}_{\chi}+\vec{v}_{e}|\leq v_{\rm esc}$) is the DM velocity
distribution in the laboratory frame (Maxwell-Boltzmann distribution truncated
at the galactic escape velocity $v_{\rm esc}=544\,$km s-1); $\vec{v}_{e}=232$
km s-1 is the Earth’s velocity in the galactic rest frame and is assumed to be
constant magnitude; $N$ is the normalized factor, and $\Theta_{1}$,
$\Theta_{2}$ are kinematical factors for the DM distribution, see [38]; $\mu$
is the reduced mass of the DM-nucleus system; $q=mv$ is the nuclear recoil
momentum; $E_{\rm R}=q^{2}/2m$ is the nuclear recoil energy;
$\bar{v}=q/[2\mu(\hat{v}_{\chi}\cdot\hat{v})]$; $d\Omega=d\cos\alpha\,d\beta$
is the infinitesimal solid angle around the nuclear recoil direction;
$|M|^{2}$ is the squared modulus of the scattering amplitude. All results for
the cross sections are expressed in the terms of right chiral couplings for
the scalar and tensor interactions of DM. The pseudoscalar interaction is
omitted because it does not have a qualitative effect on results and
conclusions. In addition, spectral functions $(1/R)(dR/dE_{\rm R})$ and
$(1/|\Delta R|)(d|\Delta R|/dE_{\rm R})$ are used. All plots are made for the
following choice of mass values and coupling constants: $m=m_{\chi}=\,$100
GeV, $c_{\rm V}=-c_{\rm A}=d_{\rm V}=-d_{\rm A}=1/2$, $|c_{S_{\rm R}}|=|c_{\rm
T_{\rm R}}|=|d_{\rm S_{\rm R}}|=|d_{\rm T_{\rm R}}|=1/2$; the overall factor
standing at the detection rate, $\rho_{\chi}/(64\pi^{2}Nm_{\chi}^{3}m^{2})$,
is put to $1$.

##  3 First scenario - polarized target

In this section we inquire how the presence of the vector, axial, tensor and
scalar interactions in the polarized DM-nucleus scattering affects the
spectrum and angular (polar and azimuthal) distributions of nuclear recoils
and whether it allows for the appearance of terms that break the time reversal
symmetry.

Figure 2: Dirac DM: plot of $(1/R)(dR/dE_{\rm R})$ as a function of $E_{\rm
R}$ for vector V, axial A and scalar S interactions with $\theta_{\rm S_{\rm
R}}=\mbox{arbitrary}$ (solid black line); plots of $(1/|\Delta R|)(d|\Delta
R|/dE_{\rm R})$ as a function of $E_{\rm R}$ for V, A, S interactions; solid
gray line for $\theta_{\rm S_{\rm R}}=0$; dashed line for $\theta_{\rm S_{\rm
R}}=\pi/4$; dotted line for $\theta_{\rm S_{\rm R}}=\pi/2$.  Figure 3:
Majorana DM: plot of $(1/R)(dR/dE_{\rm R})$ as a function of $E_{\rm R}$ for
vector V, axial A and scalar S interactions with $\theta_{\rm S_{\rm R}}=0$
(solid black line); plots of $(1/|\Delta R|)(d|\Delta R|/dE_{\rm R})$ as a
function of $E_{\rm R}$ for V, A, S interactions; solid gray line for
$\theta_{\rm S_{\rm R}}=0$; dashed line for $\theta_{\rm S_{\rm R}}=\pi/4$;
dotted line for $\theta_{\rm S_{\rm R}}=\pi/2$.  Figure 4: Dirac DM: plot of
$(1/R)(dR/dE_{\rm R})$ as a function of $E_{\rm R}$ for vector V, axial A and
tensor T interactions with $\theta_{\rm S_{\rm R}}=0$ (solid black line);
plots of $(1/|\Delta R|)(d|\Delta R|/dE_{\rm R})$ as a function of $E_{\rm R}$
for V, A, T interactions; solid gray line for $\theta_{\rm S_{\rm R}}=0$;
dashed line for $\theta_{\rm S_{\rm R}}=\pi/4$; dotted line for $\theta_{\rm
S_{\rm R}}=\pi/2$.

In what follows we use the squared amplitudes $|M_{\rm
VA}({\hat{s}_{1}})|^{2}$, $|M_{\rm VA-{S_{R}}}({\hat{s}_{1}})|^{2}$, $|M_{\rm
VA-{T_{R}}}({\hat{s}_{1}})|^{2}$, where

$\displaystyle|M_{\rm VA-{S_{R}}}({\hat{s}_{1}})|^{2}$ $\displaystyle=$
$\displaystyle|M_{\rm VA}({\hat{s}_{1}})|^{2}+|M_{\rm
S_{R}}({\hat{s}_{1}})|^{2}+2\Re[M_{\rm VA-S_{R}}({\hat{s}_{1}})]$ (6)
$\displaystyle|M_{\rm VA-{T_{R}}}({\hat{s}_{1}})|^{2}$ $\displaystyle=$
$\displaystyle|M_{\rm VA}({\hat{s}_{1}})|^{2}+|M_{\rm
T_{R}}({\hat{s}_{1}})|^{2}+2\Re[M_{\rm VA-T_{R}}({\hat{s}_{1}})],$ (7)

and difference of squared amplitudes (for all interactions)

$\Delta
M({\hat{s}_{1}})\equiv\frac{1}{2}\left[|M({\hat{s}_{1}})|^{2}-|M(-{\hat{s}_{1}})|^{2}\right].$
(8)

We refer the reader to the Appendix 1 and Appendix 2 for the explicit formulas
defining amplitudes $|M|^{2}$ and $\Delta M$ in the non-relativistic
approximations of the orders relevant in the analysis.

First we analyse how the normalized spectra depend on the the coupling phases.
From the results given in Appendix 1: Squared modulus of amplitudes for DM
scattering on polarized target we can read that for $\theta_{\rm S_{R}}=\pi/2$
(that is $\Re\left(c_{\rm S_{R}}d_{\rm S_{R}}\right)=0$) the amplitude $\Delta
M_{\rm VA-{S_{R}}}$ reduces to $\Delta M_{\rm VA}$. The leading terms of
$|M_{\rm VA}|^{2}$ and $|M_{\rm VA-{S_{R}}}|^{2}$ are of the order
O$(v^{0},v_{\chi}^{0})$ thus the plots of $(1/R)(dR/(dE_{\rm R}))$ on Fig. 2
do not discern the two cases, whereas for $\theta_{\rm S_{R}}\neq\pi/2$ the
differences are visible.

Second, we ask whether this scenario can help to determine the nature of DM.
It seems that the answer to this question is not affirmative: Fig. 3 displays
only tiny differences between the Dirac and the Majorana DM in the
corresponding cases.

Next, for tensor interactions with $\theta_{\rm T_{R}}=\pi/2$ the amplitude
$\Delta M_{\rm VA-{T_{R}}}$ differs from $\Delta M_{\rm VA}$ by a nontrivial
contribution from $\Delta M_{\rm T_{R}}$ which results in descending behaviour
of $(1/|\Delta R|)(d|\Delta R|/(dE_{\rm R}))$ for small energies. This is
shown on Fig. 4. Similarly to the scalar case the leading term of $|M_{\rm VA-
T_{R}}|^{2}$ is of the order O$(v^{0},v_{\chi}^{0})$ thus the normalized plot
of $(1/R)(dR/(dE_{\rm R}))$ is the same as for $|M_{\rm VA}|^{2}$.

Finally, we analyse the the angular distributions of recoil nucleus on
$(\alpha,\beta)$-maps. Fig. 5 shows contributions of the two terms,
$\vec{v}\cdot{\hat{s}_{1}}$, $\vec{v}_{\chi}\cdot{\hat{s}_{1}}$, to the
detection rate $(d^{2}R)/(dE_{\rm R}\,d\,\Omega)$; the total detection rates
in the discussed scenario are formed by superpositions of these dipol-like
components. Figs. 6 and 7 present $(\alpha,\beta)$-charts of the detection
rates $(d^{2}\Delta R)/(dE_{\rm R}\,d\,\Omega)$ for the scalar and tensor
interactions. The plots reveal a small difference between the ${\rm VA-S_{R}}$
and ${\rm VA-T_{R}}$ scenarios. Moreover the shapes of the corresponding
distributions are sensitive to the coupling phases.

As an additional remark we state the $(\alpha,\beta)$-maps for the Majorana DM
were analyzed as well; compared to the Dirac DM the difference is only in
values (on the level $\sim 60\%$), not in shapes.

Figure 5: Contributions of terms, $\vec{v}\cdot{\hat{s}_{1}}$,
$\vec{v}_{\chi}\cdot{\hat{s}_{1}}$, to $(d^{2}R)/(dE_{\rm R}\,d\,\Omega)$ as a
function of $\alpha$ and $\beta$; incoming DM with $E_{\rm R}=20$ keV. Figure
6: Dirac DM for $E_{\rm R}=20$ keV: plots of $d^{2}\Delta R/(dE_{\rm
R}d\,\Omega)$ as a function of $\alpha$ and $\beta$; left plot for V, A, S
interactions with $\theta_{\rm S_{\rm R}}=\pi/4$; right plot for V, A, T
interactions with $\theta_{\rm S_{\rm R}}=\pi/4$.  Figure 7: Dirac DM for
$E_{\rm R}=20$ keV: plots of $d^{2}\Delta R/(dE_{\rm R}d\,\Omega)$ as a
function of $\alpha$ and $\beta$; left plot for V, A, S interactions with
$\theta_{\rm S_{\rm R}}=\pi/2$; right plot for V, A, T interactions with
$\theta_{\rm S_{\rm R}}=\pi/2$.

It is worth pointing out that the possible effects of the time inversion
symmetry are not visible because all triple mixed products made up of the
momenta of the incoming DM and of the recoil nucleus, and the target
polarization are strongly suppressed. The expected effects can only enter with
the terms of the order O$(v^{2},\;v^{2}_{\chi})$.

##  4 Second scenario - polarized target with polarized recoil nucleus

This section is an extended version of the considerations in the previous one.
In addition, it is assumed that the polarization of the recoil nucleus will be
measured. As a result, various mixed products among the target polarization
vector, the polarization vector of the recoil nucleus, the momenta of incoming
DM and of recoil nucleus appear in the cross section. This indicates a
breaking symmetry with respect to time inversion.

Figure 8: Dirac DM with polarization of recoil nucleus ${\hat{s}_{3}}$ for
$\varphi_{3}=\pi/2$, $\theta_{3}=\pi/2$: plot of $(1/R)(dR/dE_{\rm R})$ as a
function of $E_{\rm R}$ for vector V, axial A and scalar S interactions with
$\theta_{\rm S_{\rm R}}=0$ (solid black line); plots of $(1/|\Delta
R|)(d|\Delta R|/dE_{\rm R})$ as a function of $E_{\rm R}$ for V, A, S
interactions; solid gray line for $\theta_{\rm S_{\rm R}}=0$; dashed line for
$\theta_{\rm S_{\rm R}}=\pi/4$; dotted line for $\theta_{\rm S_{\rm
R}}=\pi/2$.  Figure 9: Dirac DM with polarization of recoil nucleus
${\hat{s}_{3}}$ for $\varphi_{3}=\pi/2$, $\theta_{3}=\pi/2$, $E_{\rm R}=20$
keV: plots of $d^{2}\Delta R/(dE_{\rm R}d\,\Omega)$ as a function of $\alpha$
and $\beta$; top left plot for V, A, S interactions with $\theta_{\rm S_{\rm
R}}=0$; top right plot for V, A, S interactions with $\theta_{\rm S_{\rm
R}}=\pi/4$; bottom left plot for V, A, S interactions with $\theta_{\rm S_{\rm
R}}=\pi/2$; bottom right plot for V, A, T interactions with $\theta_{\rm
S_{\rm R}}=\pi/2$.

From the formulas for $|M_{\rm VA}|^{2}$ and $\Delta M_{\rm VA}$ we see that
in the leading order there is no dependence on the ${\hat{s}_{3}}$, thus the
spectral and angular distributions are insensitive to the polarization recoil
nucleus. With the presence of the scalar interaction the leading order terms
in $|M_{\rm S_{R}}({\hat{s}_{1}},{\hat{s}_{3}})|^{2}$, $\Re(M_{\rm VA-
S_{R}}({\hat{s}_{1}},{\hat{s}_{3}}))$ and $\Delta M_{\rm
S_{R}}({\hat{s}_{1}},{\hat{s}_{3}})$, $\Re(\Delta M_{\rm VA-
S_{R}}({\hat{s}_{1}},{\hat{s}_{3}}))$ have the form,
${\hat{s}_{1}}\cdot{\hat{s}_{3}}$. Therefore unless the recoil nucleus
polarization is orthogonal to the target polarization the normalized spectra
of $R$ and $\Delta R$ will coincide for each ${\hat{s}_{3}}$ orientation. Thus
to observe a potential ${\hat{s}_{3}}$-dependent effects in what follows we
specify the ${\hat{s}_{1}}\cdot{\hat{s}_{3}}=0$ orientation.

As in the first scenario, the normalized spectra are sensitive to the coupling
phases. But now the dependence is modified due to the presence of the new
terms including the polarization of the outgoing nucleus. Fig. 8 and 9 were
plotted for a specific orientation,
${\hat{s}_{3}}=\left(\cos{\varphi_{3}}\sin{\theta_{3}},\sin{\varphi_{3}}\sin{\theta_{3}},\cos{\theta_{3}}\right)$
with $\varphi_{3}=\pi/2$, $\theta_{3}=\pi/2$ (that is
${\hat{s}_{3}}=\hat{y}$). The plots manifest a phase-dependent behaviour and
in addition point to the different patterns for the scalar and tensor
interactions. Interestingly one can observe CP breaking effects directly on
the angular distribution plots in Fig. 9. To see them let us first notice that
the kinematics of the process implies that the terms
$\vec{v}_{\chi}\cdot{\hat{s}_{1}}$ and $\vec{v}\cdot{\hat{s}_{1}}$ in the
amplitudes do not contribute to the differential event rates
$(d^{2}R)/(d\,E_{\rm R}\,d\Omega)$ for the nuclei recoiled in the direction
orthogonal to ${\hat{s}_{1}}$. As a result one can see a “null” circle along
the lines of constant $\beta$ equal to $\pi/2$ and $3/2\,\pi$ in the Fig. 5
and on all $(\alpha,\beta)$-charts of the first scenario. In the present case
this feature is visible on the map with the phase $\theta_{\rm S_{R}}=0$ in
Fig. 9, and is disturbed by the CP breaking effects induced by the mixed
products as is seen on the remaining three maps in Fig. 9 for the non-zero
coupling phases 111Notice that this characteristic can also be destroyed by
the ${\hat{s}_{1}}\cdot{\hat{s}_{3}}$ term but in the discussed case it is
assumed to be zero..

Furthermore, we can ask whether it is possible to find explicit observables
isolating CP breaking terms. In fact, the assumed orientation of
${\hat{s}_{1}}$ and ${\hat{s}_{3}}$ allows us to define quantities

$\displaystyle{\rm D_{(1)}}M_{\rm VA-S_{R}}({\hat{s}_{1}},{\hat{s}_{3}})$
$\displaystyle:=$ $\displaystyle\Delta M_{\rm VA-
S_{R}}({\hat{s}_{1}},{\hat{s}_{3}})-\Delta M_{\rm VA-
S_{R}}({\hat{s}_{1}},-{\hat{s}_{3}})$ (9) $\displaystyle{\rm D_{(1)}}M_{\rm
VA-T_{R}}({\hat{s}_{1}},{\hat{s}_{3}})$ $\displaystyle:=$ $\displaystyle\Delta
M_{\rm VA-T_{R}}({\hat{s}_{1}},{\hat{s}_{3}})-\Delta M_{\rm VA-
T_{R}}({\hat{s}_{1}},-{\hat{s}_{3}})$ (10)

containing exclusively $\hat{v}_{\chi}\cdot({\hat{s}_{1}}\times{\hat{s}_{3}})$
and $\hat{v}\cdot({\hat{s}_{1}}\times{\hat{s}_{3}})$ terms. On the angular
distribution maps each of the mixed products would give contours lying along
lines of constant $\alpha$. Another choice is

$\displaystyle{\rm D_{(2)}}M_{\rm VA-
S_{R}}({\hat{s}_{1}},{\hat{s}_{3}},\vec{v})$ $\displaystyle:=$
$\displaystyle\Delta M_{\rm VA-
S_{R}}({\hat{s}_{1}},{\hat{s}_{3}},\vec{v})-\Delta M_{\rm VA-
S_{R}}({\hat{s}_{1}},{\hat{s}_{3}},-\vec{v}),$ (11) $\displaystyle{\rm
D_{(2)}}M_{\rm VA-T_{R}}({\hat{s}_{1}},{\hat{s}_{3}},\vec{v})$
$\displaystyle:=$ $\displaystyle\Delta M_{\rm VA-
T_{R}}({\hat{s}_{1}},{\hat{s}_{3}},\vec{v})-\Delta M_{\rm VA-
T_{R}}({\hat{s}_{1}},{\hat{s}_{3}},-\vec{v})$

for $\hat{v}\cdot{\hat{s}_{1}}=0$ leaving only the term
$\hat{v}\cdot({\hat{s}_{1}}\times{\hat{s}_{3}})$.

##  5 Third scenario - polarized target with polarized incoming dark matter

In this section, the physical consequences of the assumption that DM is
polarized are discussed. In this case the polarization of the recoil nucleus
is not measured, but the symmetry violation under the time inversion generated
by the triple mixed products among the target polarization vector, the
polarization vector of incoming DM, the momenta of incoming DM and of recoil
nucleus may occur.

From the formulas $|M({\hat{s}_{1}},{\hat{s}_{2}})|^{2}$ and $\Delta
M({\hat{s}_{1}},{\hat{s}_{2}})$ for the vector, scalar and tensor interactions
we see that the dominating contribution comes from the scalar product
${\hat{s}_{1}}\cdot{\hat{s}_{2}}$. This term on the angular distribution maps
would produce contours along lines of constant $\alpha$ (reflecting behaviour
of the integral kernel used in 3); the overall scale would depend on a mutual
orientation between ${\hat{s}_{1}}$ and ${\hat{s}_{2}}$. At this point it is
worth to note that the presence of DM polarization makes the detection rates
larger by the factor of the order of $\sim 10^{3}$ as compared to the scenario
without polarization. This also is true in the absence of scalar and tensor
interactions.

Figure 10:  Contributions of terms, $\vec{v}\cdot{\hat{s}_{2}}$,
$\vec{v}\cdot({\hat{s}_{1}}\times{\hat{s}_{2}})$, to $(d^{2}R)/(dE_{\rm
R}\,d\,\Omega)$ as a function of $\alpha$ and $\beta$; incoming DM with
$\kappa=\pi/3$, $\zeta=\pi/2$, $E_{\rm R}=20$ keV.

Theoretically one can even choose an orientation of the polarized target to
eliminate this predominant term to observe higher order effects. Fig. 10 shows
the two contributions, one from P-breaking term $\vec{v}\cdot{\hat{s}_{2}}$,
and second, from P and CP-breaking term
$\vec{v}\cdot({\hat{s}_{1}}\times{\hat{s}_{2}})$. The DM polarization is there
parameterized as
${\hat{s}_{2}}=\left(\cos{\zeta}\sin{\kappa},\sin{\zeta}\sin{\kappa},\cos{\kappa}\right)$;
on both maps $\kappa=\pi/3$, $\zeta=\pi/2$ and ${\hat{s}_{1}}=(1,0,0)$.
Similarly to Fig. 5 the present maps have dipolar characteristics, but now
with respect to different axes whose orientation depends on ${\hat{s}_{2}}$.
The analogous (not presented) contributions coming from the terms,
$\vec{v}_{\chi}\cdot{\hat{s}_{2}}$ and
$\vec{v}_{\chi}\cdot({\hat{s}_{1}}\times{\hat{s}_{2}})$, have comparable
shapes.

##  6 Polarized target with polarized incoming dark matter and polarized
recoil nucleus

The most general and at the same time quite exotic case assumes that apart
from the polarized target, the incoming DM can be polarized and the
polarization of the recoil nucleus is measured. This means that the effects of
time inversion symmetry violation can be caused by triple mixed products built
only from the polarization vectors themselves.

In the present scenario the leading terms of
$|M({\hat{s}_{1}},{\hat{s}_{2}},{\hat{s}_{3}})|^{2}$ are of the order
O$(v_{\chi}^{0},v^{0})$; they read:

$\displaystyle|M_{\rm VA-
S_{R}}({\hat{s}_{1}},{\hat{s}_{2}},{\hat{s}_{3}})|^{2}$ $\displaystyle=$
$\displaystyle 2m^{2}\left(4|c_{\rm S_{R}}|{}^{2}|d_{\rm
S_{R}}|{}^{2}+2\hat{s}_{1}\cdot(\hat{s}_{2}\times\hat{s}_{3})\Im\left(c_{\rm
S_{R}}d_{\rm S_{R}}\right)-2\hat{s}_{2}\cdot\hat{s}_{3}\Re\left(c_{\rm
S_{R}}d_{\rm S_{R}}\right)\right.$ (12) $\displaystyle\left.+2\Re\left(c_{\rm
S_{R}}d_{\rm S_{R}}\right)-\hat{s}_{1}\cdot\hat{s}_{2}\left(2\Re\left(c_{\rm
S_{R}}d_{\rm S_{R}}\right)+1\right)+1\right)m_{\chi}^{2}$
$\displaystyle|M_{\rm VA-
T_{R}}({\hat{s}_{1}},{\hat{s}_{2}},{\hat{s}_{3}})|^{2}$ $\displaystyle=$
$\displaystyle 2m^{2}\left(4|c_{\rm
T_{R}}|{}^{2}\left(-2\hat{s}_{1}\cdot\hat{s}_{2}+2\hat{s}_{2}\cdot\hat{s}_{3}+3\right)|d_{\rm
T_{R}}|{}^{2}+2\hat{s}_{1}\cdot\hat{s}_{2}\times\hat{s}_{3}\Im\left(c_{\rm
T_{R}}d_{\rm T_{R}}\right)\right.$ (13)
$\displaystyle\left.-2\hat{s}_{2}\cdot\hat{s}_{3}\Re\left(c_{\rm T_{R}}d_{\rm
T_{R}}\right)-6\Re\left(c_{\rm T_{R}}d_{\rm
T_{R}}\right)+\hat{s}_{1}\cdot\hat{s}_{2}\left(6\Re\left(c_{\rm T_{R}}d_{\rm
T_{R}}\right)-1\right)+1\right)m_{\chi}^{2}$

The above formulae assume that the polarization of the recoil nucleus is
perpendicular to the polarization of the target. We can construct two new
observables to rule out all the double products $\Delta_{\rm S}:=|M_{\rm VA-
S_{R}}|^{2}({\hat{s}_{1}},{\hat{s}_{2}},{\hat{s}_{3}})+|M_{\rm VA-
S_{R}}|^{2}(-{\hat{s}_{1}},{\hat{s}_{2}},-{\hat{s}_{3}})$, and $\Delta_{\rm
T}$ in the analogous way; in the assumed order they read:

$\displaystyle\Delta_{\rm S}$ $\displaystyle=$ $\displaystyle
4m^{2}\left(4|c_{\rm S_{R}}|{}^{2}|d_{\rm
S_{R}}|{}^{2}+2\hat{s}_{1}\cdot(\hat{s}_{2}\times\hat{s}_{3})\Im\left(c_{\rm
S_{R}}d_{\rm S_{R}}\right)+2\Re\left(c_{\rm S_{R}}d_{\rm
S_{R}}\right)+1\right)m_{\chi}^{2}$ (14) $\displaystyle\Delta_{\rm T}$
$\displaystyle=$ $\displaystyle 4m^{2}\left(12|c_{\rm T_{R}}|{}^{2}|d_{\rm
T_{R}}|{}^{2}+2\hat{s}_{1}\cdot(\hat{s}_{2}\times\hat{s}_{3})\Im\left(c_{\rm
T_{R}}d_{\rm T_{R}}\right)-6\Re\left(c_{\rm T_{R}}d_{\rm
T_{R}}\right)+1\right)m_{\chi}^{2}$

We see that the detection rate depends now on the orientation between
${\hat{s}_{1}}\times{\hat{s}_{3}}$ and DM polarization ${\hat{s}_{2}}$. Any
change of the number of events that appears during rigid rotations of the pair
${\hat{s}_{1}}$, ${\hat{s}_{3}}$ would signal time reversal violation (i.e.
presence of $\hat{s}_{1}\cdot(\hat{s}_{2}\times\hat{s}_{3})\Im\left(c_{\rm
S_{R}}d_{\rm
S_{R}}\right)\sim{\hat{s}_{1}}\cdot({\hat{s}_{2}}\times{\hat{s}_{3}})\,\sin{(\theta_{\rm
S_{\rm R}}})$ term). Moreover, rigid rotations of ${\hat{s}_{1}}$,
${\hat{s}_{3}}$ in principle would allow for determination of the DM
polarization direction. From theoretical point of view the scenario analyzed
is rather peculiar because it primarily assumes a mere presence of DM
polarization. In addition, an experiment would require a control over
polarization of the recoil nucleus.

##  7 Conclusions

We have analyzed the theoretically possible scenarios for the fermionic dark
matter interacting with the target of polarized nuclei in the presence of non-
standard vector, axial, scalar and tensor interactions. We have specified to
the two kinds of measurements: normalized spectra and angular distribution of
the recoil nucleus. We have considered four experiments with polarized target.
In the first we assumed that the incoming DM is unpolarized and the spin of
the recoiled nucleus is not measured. In the second option we allow for the
recoiled spin measurement. The third scenario permits a hypothetical
polarization of DM; in this case the recoiled nucleus polarization is not
registered. In the fourth experiment we assume both, DM polarization and the
recoiled nucleus spin detection.

The analysis presented in the paper shows that each scenario offers new ways
to study a nature of DM and its interaction with SM particles. It is reflected
in the dependence of the detection rates on the coupling constants and
orientations of the spin polarizations. This feature is seen on both, spectral
diagrams and angular distribution maps. The parameter space of the considered
models is rather large and to make general conclusions a complete analysis
would be required which goes beyond the cases considered. However even in the
subset of presented models we can observe some regularities in the normalized
spectra as is seen in Figs. (2, 3, 4, 8). On the other hand we can also notice
an unexpected behaviour of the spectrum at low energies for tensor
interactions (see Fig. 4). Next, we would like to point out an important role
that the angular distributions may play in studies of DM nature. The analysed
examples have shown that $(\alpha,\beta)$-maps could be useful in testing time
reversal symmetry breaking effects, see Figs. (9, 10). In particular one can
define observables which isolate CP-breaking interaction terms, Eqs. (11, 14).
In the more exotic scenarios with polarized DM one can, in principle, put
forward a measurement scheme determining DM spin orientation, see Sec. 5.

The great problem is the lack of any experiments with a polarized target, so
this and previous works of other authors have been carried out in the hope
that they will encourage research groups to take action in this direction.

##  Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgment

This research did not receive any specific grant from funding agencies in the
public, commercial, or not-for-profit sectors.

## Appendix 1: Squared modulus of amplitudes for DM scattering on polarized
target

${\hat{s}_{1}}\neq 0,\;{\hat{s}_{2}}=0,\;{\hat{s}_{3}}=0$ case, terms to the
order O$(v,v_{\chi})$:

$\displaystyle|M_{\rm VA}({\hat{s}_{1}})|^{2}$ $\displaystyle=$ $\displaystyle
4m^{2}m_{\chi}^{2}\left(\hat{v}_{\chi}\cdot\hat{s}_{1}v_{\chi}+1\right)$
$\displaystyle|M_{\rm S_{R}}({\hat{s}_{1}})|^{2}$ $\displaystyle=$
$\displaystyle 16m^{2}\left|c_{\rm S_{R}}\right|{}^{2}\left|d_{\rm
S_{R}}\right|{}^{2}m_{\chi}^{2}$ $\displaystyle|M_{\rm
T_{R}}({\hat{s}_{1}})|^{2}$ $\displaystyle=$ $\displaystyle-16m\left|c_{\rm
T_{R}}\right|{}^{2}\left|d_{\rm
T_{R}}\right|{}^{2}m_{\chi}\left(m\left(m_{\chi}\left(4\hat{v}_{\chi}\cdot\hat{s}_{1}v_{\chi}-3\right)-2mv\hat{v}\cdot\hat{s}_{1}\right)-mv\hat{v}\cdot\hat{s}_{1}m_{\chi}\right)$
$\displaystyle 2\Re[M_{\rm VA-S_{R}}({\hat{s}_{1}})]$ $\displaystyle=$
$\displaystyle 4m\Re\left(c_{\rm S_{R}}d_{\rm
S_{R}}\right)m_{\chi}\left(2m\left(-mv\hat{v}\cdot\hat{s}_{1}+m_{\chi}+\hat{v}_{\chi}\cdot\hat{s}_{1}m_{\chi}v_{\chi}\right)-mv\hat{v}\cdot\hat{s}_{1}m_{\chi}\right)$
$\displaystyle 2\Re(M_{\rm VA-T_{R}}({\hat{s}_{1}}))$ $\displaystyle=$
$\displaystyle 8m\Re\left(c_{\rm T_{R}}d_{\rm
T_{R}}\right)m_{\chi}\left(m\left(m_{\chi}\left(\hat{v}_{\chi}\cdot\hat{s}_{1}v_{\chi}-3\right)-mv\hat{v}\cdot\hat{s}_{1}\right)-mv\hat{v}\cdot\hat{s}_{1}m_{\chi}\right)$
$\displaystyle|M_{\rm VA}^{\rm Maj}({\hat{s}_{1}})|^{2}$ $\displaystyle=$
$\displaystyle
4m\,m_{\chi}\left(mv\hat{v}\cdot\hat{s}_{1}m_{\chi}+m\left(m_{\chi}\left(2\hat{v}_{\chi}\cdot\hat{s}_{1}v_{\chi}+3\right)-mv\hat{v}\cdot\hat{s}_{1}\right)\right)$
$\displaystyle|M_{\rm S_{R}}^{\rm Maj}({\hat{s}_{1}})|^{2}$ $\displaystyle=$
$\displaystyle 64m^{2}\left|c_{\rm S_{R}}\right|{}^{2}\left|d_{\rm
S_{R}}\right|{}^{2}m_{\chi}^{2}$ $\displaystyle 2\Re(M_{\rm VA-S_{R}}^{\rm
Maj}({\hat{s}_{1}}))$ $\displaystyle=$
$\displaystyle-16m^{3}v\hat{v}\cdot\hat{s}_{1}\Re\left(c_{\rm S_{R}}d_{\rm
S_{R}}\right)m_{\chi}$

${\hat{s}_{1}}\neq 0,\;{\hat{s}_{2}}=0,\;{\hat{s}_{3}}\neq 0$ case, terms to
the order O$(v,v_{\chi})$:

$\displaystyle|M_{\rm VA}({\hat{s}_{1}},{\hat{s}_{3}})|^{2}$ $\displaystyle=$
$\displaystyle
2m\,m_{\chi}\left(m\left(-mv\hat{v}\cdot\hat{s}_{3}+m_{\chi}+\hat{v}_{\chi}\cdot\hat{s}_{1}m_{\chi}v_{\chi}+\hat{v}_{\chi}\cdot\hat{s}_{3}m_{\chi}v_{\chi}\right)-mv\hat{v}\cdot\hat{s}_{3}m_{\chi}\right)$
$\displaystyle|M_{\rm S_{R}}({\hat{s}_{1}},{\hat{s}_{3}})|^{2}$
$\displaystyle=$ $\displaystyle 8m^{2}\left|c_{\rm
S_{R}}\right|{}^{2}\left|d_{\rm
S_{R}}\right|{}^{2}\left(\hat{s}_{1}\cdot\hat{s}_{3}+1\right)m_{\chi}^{2}$
$\displaystyle|M_{\rm T_{R}}({\hat{s}_{1}},{\hat{s}_{3}})|^{2}$
$\displaystyle=$ $\displaystyle-8m\left|c_{\rm T_{R}}\right|{}^{2}\left|d_{\rm
T_{R}}\right|{}^{2}m_{\chi}\left(mv\left(\hat{v}\cdot\hat{s}_{1}-3\hat{v}\cdot\hat{s}_{3}\right)m_{\chi}+m\left(2mv\left(\hat{v}\cdot\hat{s}_{1}-\hat{v}\cdot\hat{s}_{3}\right)\right.\right.$
$\displaystyle\quad\left.\left.+m_{\chi}\left(\hat{s}_{1}\cdot\hat{s}_{3}-4\hat{v}_{\chi}\cdot\hat{s}_{1}v_{\chi}+4\hat{v}_{\chi}\cdot\hat{s}_{3}v_{\chi}-3\right)\right)\right)$
$\displaystyle 2\Re(M_{\rm VA-S_{R}}({\hat{s}_{1}},{\hat{s}_{3}}))$
$\displaystyle=$ $\displaystyle 2m\,m_{\chi}\left(\Im\left(c_{\rm S_{R}}d_{\rm
S_{R}}\right)\left(2m\hat{v}_{\chi}\cdot(\hat{s}_{1}\times\hat{s}_{3})m_{\chi}v_{\chi}-mv\hat{v}\cdot(\hat{s}_{1}\times\hat{s}_{3})\left(2m+m_{\chi}\right)\right)\right.$
$\displaystyle\quad\left.-\Re\left(c_{\rm S_{R}}d_{\rm
S_{R}}\right)\left(mv\left(\hat{v}\cdot\hat{s}_{1}+\hat{v}\cdot\hat{s}_{3}\right)m_{\chi}-2m\left(m_{\chi}\left(\hat{s}_{1}\cdot\hat{s}_{3}+\hat{v}_{\chi}\cdot\hat{s}_{1}v_{\chi}+\hat{v}_{\chi}\cdot\hat{s}_{3}v_{\chi}+1\right)\right.\right.\right.$
$\displaystyle\quad\left.\left.\left.-mv\left(\hat{v}\cdot\hat{s}_{1}+\hat{v}\cdot\hat{s}_{3}\right)\right)\right)\right)$
$\displaystyle 2\Re(M_{\rm VA-T_{R}}({\hat{s}_{1}},{\hat{s}_{3}}))$
$\displaystyle=$ $\displaystyle 4m\,m_{\chi}\left(m\Im\left(c_{\rm
T_{R}}d_{\rm
T_{R}}\right)\left(\hat{v}_{\chi}\cdot(\hat{s}_{1}\times\hat{s}_{3})m_{\chi}v_{\chi}-mv\hat{v}\cdot(\hat{s}_{1}\times\hat{s}_{3})\right)\right.$
$\displaystyle\quad\left.-\Re\left(c_{\rm T_{R}}d_{\rm
T_{R}}\right)\left(mv\left(\hat{v}\cdot\hat{s}_{1}-3\hat{v}\cdot\hat{s}_{3}\right)m_{\chi}+m\left(mv\left(\hat{v}\cdot\hat{s}_{1}-3\hat{v}\cdot\hat{s}_{3}\right)\right.\right.\right.$
$\displaystyle\quad\left.\left.\left.-m_{\chi}\left(\hat{s}_{1}\cdot\hat{s}_{3}+\hat{v}_{\chi}\cdot\hat{s}_{1}v_{\chi}-3\hat{v}_{\chi}\cdot\hat{s}_{3}v_{\chi}-3\right)\right)\right)\right)$

${\hat{s}_{1}}\neq 0,\;{\hat{s}_{2}}\neq 0,\;{\hat{s}_{3}}=0$ case, terms to
the order O$(v,v_{\chi})$:

$\displaystyle|M_{\rm VA}({\hat{s}_{1}},{\hat{s}_{2}})|^{2}$ $\displaystyle=$
$\displaystyle-4m^{2}m_{\chi}^{2}\left(\hat{s}_{1}\cdot\hat{s}_{2}-\hat{v}_{\chi}\cdot\hat{s}_{1}v_{\chi}+\hat{v}_{\chi}\cdot\hat{s}_{2}v_{\chi}-1\right)$
$\displaystyle|M_{\rm S_{R}}({\hat{s}_{1}},{\hat{s}_{2}})|^{2}$
$\displaystyle=$ $\displaystyle 16m^{2}\left|c_{\rm
S_{R}}\right|{}^{2}\left|d_{\rm
S_{R}}\right|{}^{2}m_{\chi}\left(mv\hat{v}\cdot\hat{s}_{2}+m_{\chi}\right)$
$\displaystyle|M_{\rm T_{R}}({\hat{s}_{1}},{\hat{s}_{2}})|^{2}$
$\displaystyle=$ $\displaystyle 16m\left|c_{\rm
T_{R}}\right|{}^{2}\left|d_{\rm
T_{R}}\right|{}^{2}m_{\chi}\left(mv\left(\hat{v}\cdot\hat{s}_{1}-2\hat{v}\cdot\hat{s}_{2}\right)m_{\chi}\right.$
$\displaystyle\quad\left.+m\left(m_{\chi}\left(-2\hat{s}_{1}\cdot\hat{s}_{2}-4\hat{v}_{\chi}\cdot\hat{s}_{1}v_{\chi}+4\hat{v}_{\chi}\cdot\hat{s}_{2}v_{\chi}+3\right)-mv\left(\hat{v}\cdot\hat{s}_{2}-2\hat{v}\cdot\hat{s}_{1}\right)\right)\right)$
$\displaystyle 2\Re(M_{\rm VA-S_{R}}({\hat{s}_{1}},{\hat{s}_{2}}))$
$\displaystyle=$ $\displaystyle
4m\,m_{\chi}\left(-mv\hat{v}\cdot(\hat{s}_{1}\times\hat{s}_{2})\Im\left(c_{\rm
S_{R}}d_{\rm S_{R}}\right)\left(2m+m_{\chi}\right)-\Re\left(c_{\rm
S_{R}}d_{\rm
S_{R}}\right)\left(mv\left(\hat{v}\cdot\hat{s}_{1}-\hat{v}\cdot\hat{s}_{2}\right)m_{\chi}\right.\right.$
$\displaystyle\quad\left.\left.+2m\left(mv\left(\hat{v}\cdot\hat{s}_{1}-\hat{v}\cdot\hat{s}_{2}\right)+m_{\chi}\left(\hat{s}_{1}\cdot\hat{s}_{2}-\hat{v}_{\chi}\cdot\hat{s}_{1}v_{\chi}+\hat{v}_{\chi}\cdot\hat{s}_{2}v_{\chi}-1\right)\right)\right)\right)$
$\displaystyle 2\Re(M_{\rm VA-T_{R}}({\hat{s}_{1}},{\hat{s}_{2}}))$
$\displaystyle=$ $\displaystyle-8m\,m_{\chi}\left(\Im\left(c_{\rm T_{R}}d_{\rm
T_{R}}\right)\left(mv\hat{v}\cdot(\hat{s}_{1}\times\hat{s}_{2})\left(m+m_{\chi}\right)-4m\hat{v}_{\chi}\cdot(\hat{s}_{1}\times\hat{s}_{2})m_{\chi}v_{\chi}\right)\right.$
$\displaystyle\quad\left.-\Re\left(c_{\rm T_{R}}d_{\rm
T_{R}}\right)\left(mv\left(\hat{v}\cdot\hat{s}_{2}-\hat{v}\cdot\hat{s}_{1}\right)m_{\chi}+m\left(mv\left(\hat{v}\cdot\hat{s}_{2}-\hat{v}\cdot\hat{s}_{1}\right)\right.\right.\right.$
$\displaystyle\quad\left.\left.\left.+m_{\chi}\left(3\hat{s}_{1}\cdot\hat{s}_{2}+\hat{v}_{\chi}\cdot\hat{s}_{1}v_{\chi}-\hat{v}_{\chi}\cdot\hat{s}_{2}v_{\chi}-3\right)\right)\right)\right)$

${\hat{s}_{1}}\neq 0,\;{\hat{s}_{2}}\neq 0,\;{\hat{s}_{3}}\neq 0$ case, terms
to the order O$(v^{0},v_{\chi}^{0})$:

$\displaystyle|M_{\rm VA}({\hat{s}_{1}},{\hat{s}_{2}},{\hat{s}_{3}})|^{2}$
$\displaystyle=$
$\displaystyle-2m^{2}\left(\hat{s}_{1}\cdot\hat{s}_{2}-1\right)m_{\chi}^{2}$
$\displaystyle|M_{\rm S_{R}}({\hat{s}_{1}},{\hat{s}_{2}},{\hat{s}_{3}})|^{2}$
$\displaystyle=$ $\displaystyle 8m^{2}\left|c_{\rm
S_{R}}\right|{}^{2}\left|d_{\rm
S_{R}}\right|{}^{2}\left(\hat{s}_{1}\cdot\hat{s}_{3}+1\right)m_{\chi}^{2}$
$\displaystyle|M_{\rm T_{R}}({\hat{s}_{1}},{\hat{s}_{2}},{\hat{s}_{3}})|^{2}$
$\displaystyle=$ $\displaystyle 8m^{2}\left|c_{\rm
T_{R}}\right|{}^{2}\left|d_{\rm
T_{R}}\right|{}^{2}\left(-2\hat{s}_{1}\cdot\hat{s}_{2}-\hat{s}_{1}\cdot\hat{s}_{3}+2\hat{s}_{2}\cdot\hat{s}_{3}+3\right)m_{\chi}^{2}$
$\displaystyle 2\Re(M_{\rm VA-
S_{R}}({\hat{s}_{1}},{\hat{s}_{2}},{\hat{s}_{3}}))$ $\displaystyle=$
$\displaystyle
4m^{2}\left(\hat{s}_{1}\cdot(\hat{s}_{2}\times\hat{s}_{3})\Im\left(c_{\rm
S_{R}}d_{\rm
S_{R}}\right)-\left(\hat{s}_{1}\cdot\hat{s}_{2}-\hat{s}_{1}\cdot\hat{s}_{3}+\hat{s}_{2}\cdot\hat{s}_{3}-1\right)\Re\left(c_{\rm
S_{R}}d_{\rm S_{R}}\right)\right)m_{\chi}^{2}$ $\displaystyle 2\Re(M_{\rm VA-
T_{R}}({\hat{s}_{1}},{\hat{s}_{2}},{\hat{s}_{3}}))$ $\displaystyle=$
$\displaystyle
4m^{2}\left(\hat{s}_{1}\cdot(\hat{s}_{2}\times\hat{s}_{3})\Im\left(c_{\rm
T_{R}}d_{\rm
T_{R}}\right)-\left(-3\hat{s}_{1}\cdot\hat{s}_{2}-\hat{s}_{1}\cdot\hat{s}_{3}+\hat{s}_{2}\cdot\hat{s}_{3}+3\right)\Re\left(c_{\rm
T_{R}}d_{\rm T_{R}}\right)\right)m_{\chi}^{2}$

## Appendix 2: Difference of squared modulus of amplitudes for DM scattering
on polarized target

${\hat{s}_{1}}\neq 0,\;{\hat{s}_{2}}=0,\;{\hat{s}_{3}}=0$ case, terms to the
order O$(v,v_{\chi})$:

$\displaystyle\Delta M_{\rm VA}({\hat{s}_{1}})$ $\displaystyle=$
$\displaystyle 4m^{2}\hat{v}_{\chi}\cdot\hat{s}_{1}m_{\chi}^{2}v_{\chi}$
$\displaystyle\Delta M_{\rm S_{R}}({\hat{s}_{1}})$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle\Delta M_{\rm T_{R}}({\hat{s}_{1}})$
$\displaystyle=$ $\displaystyle 16m\left|c_{\rm
T_{R}}\right|{}^{2}\left|d_{\rm
T_{R}}\right|{}^{2}m_{\chi}\left(2v\hat{v}\cdot\hat{s}_{1}m^{2}+v\hat{v}\cdot\hat{s}_{1}m_{\chi}m-4\hat{v}_{\chi}\cdot\hat{s}_{1}m_{\chi}v_{\chi}m\right)$
$\displaystyle 2\Re(\Delta M_{\rm VA-S_{R}}({\hat{s}_{1}}))$ $\displaystyle=$
$\displaystyle-4m\Re\left(c_{\rm S_{R}}d_{\rm
S_{R}}\right)m_{\chi}\left(2v\hat{v}\cdot\hat{s}_{1}m^{2}+v\hat{v}\cdot\hat{s}_{1}m_{\chi}m-2\hat{v}_{\chi}\cdot\hat{s}_{1}m_{\chi}v_{\chi}m\right)$
$\displaystyle 2\Re(\Delta M_{\rm VA-T_{R}}({\hat{s}_{1}}))$ $\displaystyle=$
$\displaystyle 8m\Re\left(c_{\rm T_{R}}d_{\rm
T_{R}}\right)m_{\chi}\left(-v\hat{v}\cdot\hat{s}_{1}m^{2}-v\hat{v}\cdot\hat{s}_{1}m_{\chi}m+\hat{v}_{\chi}\cdot\hat{s}_{1}m_{\chi}v_{\chi}m\right)$
$\displaystyle\Delta M_{\rm VA}^{\rm Maj}({\hat{s}_{1}})$ $\displaystyle=$
$\displaystyle
4m\,m_{\chi}\left(-v\hat{v}\cdot\hat{s}_{1}m^{2}+v\hat{v}\cdot\hat{s}_{1}m_{\chi}m+2\hat{v}_{\chi}\cdot\hat{s}_{1}m_{\chi}v_{\chi}m\right)$
$\displaystyle\Delta M_{\rm VA}^{\rm Maj}({\hat{s}_{1}})$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle 2\Re(\Delta M_{\rm VA-S_{R}}^{\rm
Maj}({\hat{s}_{1}}))$ $\displaystyle=$
$\displaystyle-16m^{3}v\hat{v}\cdot\hat{s}_{1}\Re\left(c_{\rm S_{R}}d_{\rm
S_{R}}\right)m_{\chi}$

${\hat{s}_{1}}\neq 0,\;{\hat{s}_{2}}=0,\;{\hat{s}_{3}}\neq 0$ case, terms to
the order O$(v,v_{\chi})$:

$\displaystyle\Delta M_{\rm VA}({\hat{s}_{1}},{\hat{s}_{3}})$ $\displaystyle=$
$\displaystyle 2m^{2}\hat{v}_{\chi}\cdot\hat{s}_{1}m_{\chi}^{2}v_{\chi}$
$\displaystyle\Delta M_{\rm SR}({\hat{s}_{1}},{\hat{s}_{3}})$ $\displaystyle=$
$\displaystyle 8m^{2}\left|c_{\rm S_{R}}\right|{}^{2}\left|d_{\rm
S_{R}}\right|{}^{2}\hat{s}_{1}\cdot\hat{s}_{3}m_{\chi}^{2}$
$\displaystyle\Delta M_{\rm T_{R}}({\hat{s}_{1}},{\hat{s}_{3}})$
$\displaystyle=$ $\displaystyle-8m\left|c_{\rm T_{R}}\right|{}^{2}\left|d_{\rm
T_{R}}\right|{}^{2}m_{\chi}\left(mv\hat{v}\cdot\hat{s}_{1}m_{\chi}+m\left(2mv\hat{v}\cdot\hat{s}_{1}+m_{\chi}\left(\hat{s}_{1}\cdot\hat{s}_{3}-4\hat{v}_{\chi}\cdot\hat{s}_{1}v_{\chi}\right)\right)\right)$
$\displaystyle 2\Re(\Delta M_{\rm VA-S_{R}}({\hat{s}_{1}},{\hat{s}_{3}}))$
$\displaystyle=$ $\displaystyle-2m\,m_{\chi}\left(\Im\left(c_{\rm S_{R}}d_{\rm
S_{R}}\right)\left(mv\hat{v}\cdot(\hat{s}_{1}\times\hat{s}_{3})\left(2m+m_{\chi}\right)-2m\hat{v}_{\chi}\cdot(\hat{s}_{1}\times\hat{s}_{3})m_{\chi}v_{\chi}\right)\right.$
$\displaystyle\quad\left.+\Re\left(c_{\rm S_{R}}d_{\rm
S_{R}}\right)\left(mv\hat{v}\cdot\hat{s}_{1}m_{\chi}+2m\left(mv\hat{v}\cdot\hat{s}_{1}-m_{\chi}\left(\hat{s}_{1}\cdot\hat{s}_{3}+\hat{v}_{\chi}\cdot\hat{s}_{1}v_{\chi}\right)\right)\right)\right)$
$\displaystyle 2\Re(\Delta M_{\rm VA-T_{R}}({\hat{s}_{1}},{\hat{s}_{3}}))$
$\displaystyle=$ $\displaystyle-4m\,m_{\chi}\left(m\Im\left(c_{\rm
T_{R}}d_{\rm
T_{R}}\right)\left(mv\hat{v}\cdot(\hat{s}_{1}\times\hat{s}_{3})-\hat{v}_{\chi}\cdot(\hat{s}_{1}\times\hat{s}_{3})m_{\chi}v_{\chi}\right)\right.$
$\displaystyle\quad\left.+\Re\left(c_{\rm T_{R}}d_{\rm
T_{R}}\right)\left(mv\hat{v}\cdot\hat{s}_{1}m_{\chi}+m\left(mv\hat{v}\cdot\hat{s}_{1}-m_{\chi}\left(\hat{s}_{1}\cdot\hat{s}_{3}+\hat{v}_{\chi}\cdot\hat{s}_{1}v_{\chi}\right)\right)\right)\right)$

${\hat{s}_{1}}\neq 0,\;{\hat{s}_{2}}\neq 0,\;{\hat{s}_{3}}=0$ case, terms to
the order O$(v,v_{\chi})$:

$\displaystyle\Delta M_{\rm VA}({\hat{s}}_{1},{\hat{s}}_{2})$ $\displaystyle=$
$\displaystyle-4m^{2}m_{\chi}^{2}\left(\hat{s}_{1}\cdot\hat{s}_{2}-\hat{v}_{\chi}\cdot\hat{s}_{1}v_{\chi}\right)$
$\displaystyle\Delta M_{\rm SR}({\hat{s}_{1}},{\hat{s}_{2}})$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle\Delta M_{\rm
T_{R}}({\hat{s}_{1}},{\hat{s}_{2}})$ $\displaystyle=$ $\displaystyle
16m\left|c_{\rm T_{R}}\right|{}^{2}\left|d_{\rm
T_{R}}\right|{}^{2}m_{\chi}\left(mv\hat{v}\cdot\hat{s}_{1}m_{\chi}+2m\left(mv\hat{v}\cdot\hat{s}_{1}-m_{\chi}\left(\hat{s}_{1}\cdot\hat{s}_{2}+2\hat{v}_{\chi}\cdot\hat{s}_{1}v_{\chi}\right)\right)\right)$
$\displaystyle 2\Re(\Delta M_{\rm VA-S_{R}}({\hat{s}_{1}},{\hat{s}_{2}}))$
$\displaystyle=$
$\displaystyle-4m\,m_{\chi}\left(mv\hat{v}\cdot(\hat{s}_{1}\times\hat{s}_{2})\Im\left(c_{\rm
S_{R}}d_{\rm S_{R}}\right)\left(2m+m_{\chi}\right)+\Re\left(c_{\rm
S_{R}}d_{\rm
S_{R}}\right)\left(mv\hat{v}\cdot\hat{s}_{1}m_{\chi}\right.\right.$
$\displaystyle\quad\left.\left.+2m\left(mv\hat{v}\cdot\hat{s}_{1}+\hat{s}_{1}\cdot\hat{s}_{2}m_{\chi}-\hat{v}_{\chi}\cdot\hat{s}_{1}m_{\chi}v_{\chi}\right)\right)\right)$
$\displaystyle 2\Re(\Delta M_{\rm VA-T_{R}}({\hat{s}_{1}},{\hat{s}_{2}}))$
$\displaystyle=$ $\displaystyle-8m\,m_{\chi}\left(\Im\left(c_{\rm T_{R}}d_{\rm
T_{R}}\right)\left(mv\hat{v}\cdot(\hat{s}_{1}\times\hat{s}_{2})\left(m+m_{\chi}\right)-4m\hat{v}_{\chi}\cdot(\hat{s}_{1}\times\hat{s}_{2})m_{\chi}v_{\chi}\right)\right.$
$\displaystyle\quad\left.+\Re\left(c_{\rm T_{R}}d_{\rm
T_{R}}\right)\left(mv\hat{v}\cdot\hat{s}_{1}m_{\chi}+m\left(mv\hat{v}\cdot\hat{s}_{1}-m_{\chi}\left(3\hat{s}_{1}\cdot\hat{s}_{2}+\hat{v}_{\chi}\cdot\hat{s}_{1}v_{\chi}\right)\right)\right)\right)$

${\hat{s}_{1}}\neq 0,\;{\hat{s}_{2}}\neq 0,\;{\hat{s}_{3}}\neq 0$ case, terms
to the order O$(v,v_{\chi})$:

$\displaystyle\Delta M_{\rm VA}({\hat{s}_{1}},{\hat{s}_{2}},{\hat{s}_{3}})$
$\displaystyle=$ $\displaystyle-2m^{2}\hat{s}_{1}\cdot\hat{s}_{2}m_{\chi}^{2}$
$\displaystyle\Delta M_{\rm S_{R}}({\hat{s}_{1}},{\hat{s}_{2}},{\hat{s}_{3}})$
$\displaystyle=$ $\displaystyle 8m^{2}m_{\chi}^{2}\left|c_{\rm
S_{R}}\right|{}^{2}\left|d_{\rm
S_{R}}\right|{}^{2}\hat{s}_{1}\cdot\hat{s}_{3}$ $\displaystyle\Delta M_{\rm
T_{R}}({\hat{s}_{1}},{\hat{s}_{2}},{\hat{s}_{3}})$ $\displaystyle=$
$\displaystyle-8m^{2}\left|c_{\rm T_{R}}\right|{}^{2}\left|d_{\rm
T_{R}}\right|{}^{2}\left(2\hat{s}_{1}\cdot\hat{s}_{2}+\hat{s}_{1}\cdot\hat{s}_{3}\right)m_{\chi}^{2}$
$\displaystyle 2\Re(\Delta M_{\rm VA-
S_{R}}({\hat{s}_{1}},{\hat{s}_{2}},{\hat{s}_{3}}))$ $\displaystyle=$
$\displaystyle
4m^{2}\left(\hat{s}_{1}\cdot(\hat{s}_{2}\times\hat{s}_{3})\Im\left(c_{\rm
S_{R}}d_{\rm
S_{R}}\right)-\left(\hat{s}_{1}\cdot\hat{s}_{2}-\hat{s}_{1}\cdot\hat{s}_{3}\right)\Re\left(c_{\rm
S_{R}}d_{\rm S_{R}}\right)\right)m_{\chi}^{2}$ $\displaystyle 2\Re(\Delta
M_{\rm VA-T_{R}}({\hat{s}_{1}},{\hat{s}_{2}},{\hat{s}_{3}}))$ $\displaystyle=$
$\displaystyle
4m^{2}\left(\hat{s}_{1}\cdot(\hat{s}_{2}\times\hat{s}_{3})\Im\left(c_{\rm
T_{R}}d_{\rm
T_{R}}\right)+\left(3\hat{s}_{1}\cdot\hat{s}_{2}+\hat{s}_{1}\cdot\hat{s}_{3}\right)\Re\left(c_{\rm
T_{R}}d_{\rm T_{R}}\right)\right)m_{\chi}^{2}$

## References

  * [1] S. L. Glashow, Partial-symmetries of weak interactions, Nucl. Phys. 22 (1961) 579.
  * [2] S. Weinberg, A Model of Leptons, Phys. Rev. Lett. 19 (1967) 1264.
  * [3] A. Salam, in Elementary Particle Theory (Almquist and Wiksells, Stockholm, 1969).
  * [4] R. P. Feynman, M. Gell-Mann, Theory of the Fermi Interaction, Phys. Rev. 109 (1958) 193.
  * [5] E. C. G. Sudarshan, R. E. Marshak, Chirality Invariance and the Universal Fermi Interaction, Phys. Rev. 109 (1958) 1860.
  * [6] A. Riotto, M. Trodden, Recent Progress in Baryogenesis, Annu. Rev. Nucl. Part. Sci. 49 (1999) 35.
  * [7] M. Kobayashi, T. Maskawa, CP-Violation in the Renormalizable Theory of Weak Interaction, Prog. Theor. Phys. 49 (1973) 652.
  * [8] G. Bertone, D. Hooper, J. Silk, Particle dark matter: Evidence, candidates and constraints, Phys. Rept. 405 (2005) 279 [hep-ph/0404175].
  * [9] G. Jungman, M. Kamionkowski, K. Griest, Supersymmetric dark matter, Phys. Rept. 267 (1996) 195 [hep-ph/9506380].
  * [10] L. Bergstrom, Non-Baryonic Dark Matter - Observational Evidence and Detection Methods, Rept. Prog. Phys. 63 (2000) 793 [hep-ph/0002126].
  * [11] G. D’Amico , M. Kamionkowski, K. Sigurdson, Dark Matter Astrophysics, [https://doi.org/10.48550/arXiv.0907.1912].
  * [12] T. Appelquist, H.-C. Cheng, B.A. Dobrescu, Bounds on universal extra dimensions, Phys. Rev. D 64 (2001) 035002 [hep-ph/0012100].
  * [13] H.-C. Cheng, J.L. Feng, K.T. Matchev, Kaluza-Klein Dark Matter, Phys. Rev. Lett. 89 (2002) 211301 [hep-ph/0207125].
  * [14] G. Servant, T.M.P. Tait, Elastic Scattering and Direct Detection of Kaluza-Klein Dark Matter, New J. Phys. 4 (2002) 99 [hep-ph/0209262].
  * [15] D. Hooper, S. Profumo, Dark Matter and Collider Phenomenology of Universal Extra Dimensions, Phys. Rept. 453 (2007) 29 [hep-ph/0701197].
  * [16] G.D. Starkman, D.N. Spergel, Proposed New Technique for Detecting Supersymmetric Dark Matter, Phys. Rev. Lett. 74 (1995) 2623.
  * [17] R. Essig, J. Mardon, T. Volansky, Direct Detection of Sub-GeV Dark Matter, Phys. Rev. D 85 (2012) 076007 [hep-ph/ 1108.5383].
  * [18] P.W. Graham, D.E. Kaplan, S. Rajendran, M.T. Walters, Semiconductor Probes of Light Dark Matter, Phys. Dark Univ. 1 (2012) 32 [hep-ph/1203.2531].
  * [19] M.W. Goodman, E. Witten, Detectability of certain dark-matter candidates, Phys. Rev. D 31 (1985) 3059.
  * [20] I. Wasserman, Possibility of detecting heavy neutral fermions in the Galaxy, Phys. Rev. D 33 (1986) 2071.
  * [21] D.N. Spergel, Motion of the Earth and the detection of weakly interacting massive particles, Phys. Rev. D 37 (1988) 1353.
  * [22] P. Gondolo, Recoil momentum spectrum in directional dark matter detector, Phys. Rev. D 66 (2002) 103513 [hep-ph/0209110].
  * [23] M. Lisanti, J.G. Wacker, Disentangling Dark Matter Dynamics with Directional Detection, Phys. Rev. D 81 (2010) 096005 [hep-ph/0911.1997].
  * [24] R. Harnik, J. Kopp, P.A.N. Machado, Exploring nu Signals in Dark Matter Detectors,JCAP 07 (2012) 026 [arXiv:1202.6073].
  * [25] M. Pospelov, J. Pradler, Elastic scattering signals of solar neutrinos with enhanced baryonic currents, Phys. Rev. D 85 (2012) 113016 [arXiv:1203.0545].
  * [26] D.G. Cerdeño et al., Physics from solar neutrinos in dark matter direct detection experiments, JHEP 05 (2016) 118 [Erratum ibid. 09 (2016) 048] [arXiv:1604.01025].
  * [27] J.B. Dent, B. Dutta, J.L. Newstead, L.E. Strigari, Dark matter, light mediators and the neutrino floor, Phys. Rev. D 95 (2017) 051701 [arXiv:1607.01468].
  * [28] J.B. Dent, B. Dutta, J.L. Newstead, L.E. Strigari, Effective field theory treatment of the neutrino background in direct dark matter detection experiments, Phys. Rev. D 93 (2016) 075018 [arXiv:1602.05300].
  * [29] P. Grothaus, M. Fairbairn, J. Monroe, Directional dark matter detection beyond the neutrino bound, Phys. Rev. D 90 (2014) 055018 [arXiv:1406.5047].
  * [30] E. Bertuzzo, F. F. Deppisch, S. Kulkarni, Y. F. Perez Gonzalez, R. Zukanovich Funchal, Dark matter and exotic neutrino interactions in direct detection searches, JHEP 04 (2017) 073.
  * [31] P. Minkowski, M. Passera, Elastic Scattering of Neutrinos off Polarized Electrons, Phys. Lett. B 541 (2002) 151.
  * [32] T. I. Rashba, V. B. Semikoz, Neutrino scattering on polarized electron target as a test of neutrino magnetic moment, Phys. Lett. B 479 (2000) 218.
  * [33] J. Bernabeu J. Papavassiliou, M. Passera, Dynamical zero in $\overline{\nu}_{e}e^{-}$ scattering and the neutrino magnetic moment Phys. Lett. B 613 (2005) 162.
  * [34] S. Ciechanowicz, W Sobków, M Misiaszek, Scattering of neutrinos on a polarized electron target as a test for new physics beyond the standard model, Phys. Rev. D 71 (2005) 093006.
  * [35] W. Sobków, A. Błaut, On possibility of time reversal symmetry violation in neutrino elastic scattering on polarized electron target, Eur. Phys. J. C 78 (2018) 197.
  * [36] A. Błaut, W. Sobków, Neutrino elastic scattering on polarized electrons as a tool for probing the neutrino nature, Eur. Phys. J. C 80 (2020) 261.
  * [37] C.-T. Chiang, M. Kamionkowski, G. Z. Krnjaic, Dark Matter Detection with Polarized Detectors, Phys. Dark Univ. 1 (2012) 109 [1202.1807].
  * [38] R. Catena, K. Fridell, V Zema, Direct detection of fermionic and vector dark matter with polarised targets, JCAP 11 (2018) 018 [1810.01515].
  * [39] T. Franarin, M. Fairbairn, Reducing the solar neutrino background in dark matter searches using polarized helium-3, Phys. Rev. D94 (2016) 053004 [1605.08727].
  * [40] M. Misiaszek, S Ciechanowicz, W Sobków, The polarized electron target as a new solar-neutrino detector, Nucl. Phys. B 734 (2006) 203.
  * [41] B. Babussinov et al., An active electron polarized scintillating GSO target for neutrino physics, Nucl. Instrum. and Meth. A 694 (2012) 335.
  * [42] M. A. Bouchiat, T. R. Carver, C. M. Varnum, Nuclear Polarization in $He^{3}$ Gas Induced by Optical Pumping and Dipolar Exchange, Phys. Rev. Lett. 5 (1960) 373.
  * [43] T. G. Walker, W. Happer, Spin-exchange optical pumping of noble-gas nuclei, Rev. Mod. Phys. 69 (1997) 629.
  * [44] R. Catena, J. Hagel, C. Yaguna, Probing P- and CP-violation in dark matter interactions, JCAP 05 (2021) 016 [2007.01262].