††thanks: Corresponding author

# Probing hidden spin-2 mediator of dark matter with NA64$e$, LDMX, NA64$\mu$
and M3

I. V. Voronchikhin<EMAIL_ADDRESS>Tomsk Polytechnic University, 634050
Tomsk, Russia  D. V. Kirpichnikov<EMAIL_ADDRESS>Institute for Nuclear
Research, 117312 Moscow, Russia

###### Abstract

The connection between Standard Model (SM) particles and dark matter (DM) can
be introduced via hidden spin-2 massive mediator. In the present paper we
consider the simplified benchmark link between charged lepton sector of SM and
DM particles which are considered to be a hidden Dirac fermions from the dark
sector. The regarding couplings are established through the dimension-5
operators involving spin-2 mediator field and the energy-momentum tensors of
both SM and DM sectors. We study in detail the implication of this scenario
for the lepton fixed-target facilities, such as NA64$e$, LDMX, NA64$\mu$ and
M3. In particular, for the specific experiment we discuss in detail the
missing-energy signatures of spin-2 boson production followed by its invisible
decay into stable DM pairs. Moreover, we derive the expected reaches of these
experiments for the projected statistics of the leptons accumulated on the
target. We also discuss the implication of both nuclear and atomic form-factor
parametrizations for the differential spectra of hidden spin-$2$ boson
emission, the total cross-section of its production and the experimental reach
of the fixed-fixed target facilities for probing hidden spin-$2$ DM mediator.

##  I Introduction and framework

The nature of the dark matter (DM) particles remains puzzling for decades. The
indirect evidences for the DM are associated with galaxy rotation velocities,
large scale structures, cosmic microwave background anisotropy, gravitational
lensing, etc. Some extensions of the standard model (SM) imply connection
between SM and DM via an idea of portals. For instance, the dark photon portal
[1, 2, 3, 4] and Higgs boson portal [5, 6, 7, 8]. Such portal scenarios
suggest a systematic probing of DM and also provide a novel specific
experimental signatures. However, recently a scenarios of a massive spin-2
particle as the mediator between DM and SM have been discussed [9, 10, 11, 12,
13] in the context of gravity model with warped extra-dimensions [14, 15, 16,
17, 18, 19, 20, 21]. Furthermore, recently confirmed 4.2$\sigma$ discrepancy
in the anomalous magnetic moment measurement of the muon [22] with respect to
its theoretical prediction [23]:

$\Delta a_{\mu}=a_{\mu}(exp)-a_{\mu}(th)=(251\pm 59)\times 10^{-11},$

can be explained by the one-loop effects induced by a hidden massive spin-2
particle [10, 24]. Let us consider now the couplings between the charged
lepton sector of the SM and the DM particles that can be described by the
benchmark simplified Lagrangian [10] involving dimension-5 operators of the
energy momentum tensor and massive $G_{\mu\nu}$ field:

$\displaystyle\mathcal{L}\supset\frac{c_{\gamma}}{\Lambda}G^{\mu\nu}\left(\frac{1}{4}\eta_{\mu\nu}F_{\lambda\rho}F^{\lambda\rho}+F_{\mu\lambda}F^{\lambda}_{\nu}\right)$
$\displaystyle-\sum_{l}\frac{ic_{l}}{2\Lambda}G^{\mu\nu}\left(\overline{l}\gamma_{\mu}D_{\nu}l-\eta_{\mu\nu}\overline{l}\gamma_{\rho}D^{\rho}l\right)+\frac{c_{\chi}}{\Lambda}G^{\mu\nu}T^{\chi}_{\mu\nu},$
(1)

where $F_{\mu\nu}$ is stress tensor for the photon field, $l$ is the label for
the charged leptons $(l=e,\mu)$, $\Lambda$ is dimensional parameter for spin-2
interactions, $c_{\gamma}$, $c_{l}$, $c_{\chi}$ are the dimensionless coupling
constants for the electromagnetic field, charged leptons of SM and DM sector
respectively; $T^{\chi}_{\mu\nu}$ is the energy momentum tensor of DM
particles. To be more concrete we address DM as hidden Dirac fermion $\chi$
with mass of $m_{\chi}$. In addition, it is worth noting that the coupling
constants in the regarding scenario are assumed to be independent.

In the present paper we show that corresponding setup (1) has a very broad
phenomenological implication and can be examined through the missing energy
signatures in the existent (NA64$e$ [25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36, 37] and NA64$\mu$ [38, 39]) and the projected (LDMX [40, 41, 42, 43]
and M3 [44, 45]) lepton fixed-target experiments. These signatures can be
described by the bremsstrahlung-like reaction of the production of a spin-2
boson by the charged lepton $l^{-}$ impinging on a nucleus $N$. The
corresponding diagrams are shown in Fig. 1. In the present paper we will focus
on mainly invisible channel of dark spin-2 boson decay into DM particles, such
that $\mbox{Br}(G\to\chi\bar{\chi})\simeq 1$. Furthermore, for the $G$-boson
production cross-section calculation we exploit the equivalent photons
approximation, also known as the Weizsacker–Williams (WW) approach that is
typically exploited for the hidden particle yield estimate at both beam dump
and fixed-target experiments. In particular, this approach provides a fairly
reasonable approximation (i. e. at the level of $\lesssim 2\%$) for the exact-
tree-level production cross-sections of both hidden spin-$0$ and spin-$1$
bosons [46, 47, 39].

In addition, we also discuss in detail the impact of both nuclear and atomic
form-factor parametrizations on: (i) the differential spectra of $G$ boson
emission, (ii) the total cross-section of its production (iii) the
experimental reach of the fixed-fixed target facilities for probing hidden
spin-2 boson.

The paper is organized as follows. In Sec. II we derive explicitly the double
differential cross-section of $G$-boson production in WW approach. In Sec. III
we discuss the set of form-factors that are typically used for the calculation
of regarding cross-sections. In this section we also study the impact of
various form-factor parametrizations on the virtual photon flux distribution
from the charged particles. In Sec. IV we provide a description of the missing
energy signatures for the analysis of DM production at lepton fixed-target
experiments, such as NA64$e$, LDMX, NA64$\mu$ and M3. In Sec. V we discuss the
impact of form-factor parametrization on both differential and total cross-
sections of hidden spin-2 boson production. In Sec. VI we obtain the
constraints on the parameter space of spin-2 DM mediator from the NA64$e$,
LDMX, NA64$\mu$ and M3 facilities. In this section we also study implication
of the form-factors for the expected reach of the regarding experiments.

##  II  The production cross section of DM mediator in WW approach

Figure 1: Feynman diagrams describing missing energy signatures.  Figure 2:
Left panel: virtual photon flux for Tsai-Schiff’s $F_{TS}$ (the green solid
line), nuclear Helm’s $F_{H_{nucl}}$ (the blue solid line), atomic Helm’s
$F_{H}$ (the red dashed line) and atomic exponential $F_{E}$ (the orange
dashed line) form-factors as function of the lower limits $t_{min}$ with fixed
upper limit $t_{max}=10\;\text{GeV}^{2}$. Right panel: the relative difference
bewteen benchmark Tsai-Schiff’s flux and other fluxes.

Let us consider the kinematic variables of bremsstrahlung-like process $2\to
3$ in the laboratory frame:

$l^{-}(p)+N(P_{i})\rightarrow l^{-}(p^{\prime})+N(P_{f})+G(k),$ (2)

where $p=(E_{l},\boldsymbol{p})$ is the momentum of incoming charged lepton,
$p^{\prime}=(E^{\prime}_{l},\boldsymbol{p^{\prime}})$ is the momentum of
outgoing lepton, $k=(E_{G},\boldsymbol{k})$ is the momentum of spin-2
mediator, $P_{i}=(M,0)$ and $P_{f}=(P^{0}_{f},\boldsymbol{P}_{f})$ are the
momenta of the initial and outgoing nucleus respectively. We define four-
momentum transfer to nucleus as $P_{i}-P_{f}=q=(q_{0},\boldsymbol{q})$, such
that the photon virtuality takes the form $t=-q^{2}>0$. In order to calculate
the differential cross-section of $G$ boson production in nuclear interaction
one can exploit the Weizsäcker-Williams approximation, implying that the
energy of incoming lepton is much higher than $m_{l}$ and $m_{G}$. In this
case, the incoming charged particle is replaced by its effective photon flux,
such that the phase space of the process $l^{-}(p)N(P_{i})\to
l^{-}(p^{\prime})N(P_{f})G(k)$ is reduced to the Compton-like process
$l^{-}(p)\gamma(q)\to l^{-}(p^{\prime})G(k)$. In particular, one can obtain
the following expression for the double differential cross-section [48, 49,
50]:

$\displaystyle\left.\frac{d\sigma(p+P_{i}\rightarrow
p^{\prime}+P_{f}+k)}{d(pk)d(kP_{i})}\right|_{WW}=$
$\displaystyle=\frac{\alpha\chi}{\pi(p^{\prime}P_{i})}\left.\frac{d\sigma(p+q\rightarrow
k+p^{\prime})}{d(pk)}\right|_{t=t_{min}},$ (3)

where the flux of virtual photon $\chi$ from nucleus is expressed through the
elastic form-factor $F(t)$ as follows:

$\chi=Z^{2}\int\limits^{t_{max}}_{t_{min}}\frac{t-t_{min}}{t^{2}}F^{2}(t)dt,$
(4)

where $Z$ is the atomic number of the nucleus, the explicit expressions for
the form-factors $F(t)$ are discussed below in Section III. The photon flux
for the inelastic form-factor is proportional to $\propto Z$, and thus can be
ignored in the calculation for heavy nuclei $Z\propto\mathcal{O}(100)$. The WW
approach in Eqs. (3) and (4) implies that the virtuality $t$ has its minimum
$t=t_{min}$ when ${\bf q}$ is collinear with ${\bf k}-{\bf p}$. In particular,
the expression for $t_{min}$ is derived below (see, e. g. Eq. (11)). For
ultra-relativistic incident lepton in laboratory frame we have:

$\displaystyle d(kp)d(k\mathcal{P}_{i})\simeq$
$\displaystyle|\boldsymbol{J}(\cos(\theta_{G}),E_{G})|d\cos(\theta_{G})dE_{G}\simeq$
$\displaystyle\simeq
M|\boldsymbol{p}||\boldsymbol{k}|d\cos(\theta_{G})dE_{G},$ (5)

where $\theta_{G}$ is the angle between the initial lepton direction and the
momentum of the produced $G$-boson and
$\boldsymbol{J}(\cos(\theta_{G}),E_{G})$ is the Jacobian of the transformation
from $(kp)$ and $(k\mathcal{P}_{i})$ variables to $\cos\theta_{G}$ and
$E_{G}$. So that by substituting Eq. (5) into Eq. (3), we get the following
expression for the double differential cross-section after some algebraic
simplification:

$\displaystyle\left.\frac{d\sigma(p+P_{i}\rightarrow
p^{\prime}+P_{f}+k)}{dxd\cos(\theta_{G})}\right|_{WW}=$
$\displaystyle\frac{\alpha\chi}{\pi}\frac{E_{l}^{2}x\beta_{G}}{1-x}\left.\frac{d\sigma(p+q\rightarrow
k+p^{\prime})}{d(pk)}\right|_{t=t_{min}},$ (6)

where $\alpha=e^{2}/(4\pi)\simeq 1/137$ is the fine structure constant,
$x=E_{G}/E_{l}$ is the energy fraction of spin-2 mediator that it carries away
and $\beta_{G}=\sqrt{1-m_{G}^{2}/(xE_{l})^{2}}$ is the typical velocity of
$G$-boson. By solving the mass-shell equations for both outgoing electron and
nucleus:

$p^{\prime 2}=(q+p-k)^{2}=m_{l}^{2},\quad P_{f}^{2}=(P_{i}-q)^{2}=M^{2},$ (7)

we get the following auxiliary expressions: $q_{0}=-t/(2M)$ and
$|\boldsymbol{q}|^{2}=t^{2}/(4M^{2})+t$. Furthermore, by taking into account
the small typical energy transferred to the nucleus $|\boldsymbol{q}|/M\ll 1$,
we get both approximate expressions $t\simeq|\boldsymbol{q}|^{2}$ and
$q_{0}\simeq-|\boldsymbol{q}|^{2}/(2M)$ for the photon virtuality and nucleus
energy transfer respectively. The value of $q_{0}$ is fairly small and thus
can be neglected in the calculation, since the typical momentum is
$|\boldsymbol{q}|\lesssim\mathcal{O}(100)\,\mbox{MeV}$ and the mass of nucleus
is of the order of $M\propto\mathcal{O}(100)\,\mbox{GeV}$. On the other hand,
the photon flux $\chi$ is sensitive to the photon virtuality
$t\simeq|\boldsymbol{q}|^{2}$, as long as the screening effects due to the
atomic electrons should be taken into account (see e. g. Sec. III for detail).

Next, let us introduce the Mandelstam variables in the following form:

$s_{2}=(p+q)^{2},\quad u_{2}=(p-k)^{2},\quad t_{2}=(p-p^{\prime})^{2}.$ (8)

It is worth mentioning that Eq. (7) implies,
$q_{0}^{2}-|\boldsymbol{q}|^{2}+u_{2}+2q_{0}V_{0}-2(\boldsymbol{q},\boldsymbol{V})=m_{l}^{2},$
where we define the three-vector as
$\boldsymbol{V}=\boldsymbol{p}-\boldsymbol{k}$. Then for the small energy
transferred to the nucleus $|\boldsymbol{q}|^{2}\ll s_{2},t_{2},u_{2}$, one
obtains:

$t=\frac{(u_{2}-m_{l}^{2})^{2}}{4|\boldsymbol{V}|^{2}\cos^{2}(\theta_{\boldsymbol{q}\boldsymbol{V}})},$
(9)

where $\theta_{\boldsymbol{q}\boldsymbol{V}}$ is the angle between vector
$\boldsymbol{q}$ and $\boldsymbol{V}$. Therefore keeping only leading terms in
$m_{G}^{2}/E_{k}^{2}$, $m_{l}^{2}/E_{l}^{2}$,
$m_{l^{\prime}}^{2}/E_{l^{\prime}}^{2}$ and $\theta_{G}^{2}$, we obtain the
approximate expression for the absolute value of the three vector,
$|\boldsymbol{V}|\simeq E_{l}(1-x)$. Let us introduce the following auxiliary
function:

$U\equiv m_{l}^{2}-u_{2}\simeq
E_{l}^{2}\theta_{G}^{2}x+m_{G}^{2}(1-x)/x+m_{l}^{2}x>0,$ (10)

thus by exploiting Eqs. (9) and (10) one can obtain the expressions for
minimum of the virtuality $t=t_{min}$, implying that
$\theta_{\boldsymbol{q}\boldsymbol{V}}\simeq\pi$, such that the three vector
$\boldsymbol{q}$ is almost collinear with $\boldsymbol{k}-\boldsymbol{p}$:

$t_{min}\simeq|\boldsymbol{q}|^{2}\simeq U^{2}/(4E_{l}^{2}(1-x)^{2}).$ (11)

It is worth noting that the WW approximation for the $2\to 3$ cross–section
implies that $t_{min}$ is the function of both $x$ and $\theta_{G}$ variables,
so that it is fairly accurate approach for the exact tree level cross-section
(for detail, see, e. g. Ref. [39] and references therein). Moreover, we note
that $\theta_{\boldsymbol{q}\boldsymbol{p}}\simeq\pi$ as long as
$\theta_{G}\ll 1$ and ${\bf q}$ is collinear with ${\bf k}-{\bf p}$, therefore
this yields, $(q,k)\simeq|\boldsymbol{q}||\boldsymbol{k}|\simeq Ux/(2(1-x))$,
and we finally get the following expressions for Mandelstam terms expressed
through the both $x$ and $\theta_{G}$ variables:

$u_{2}=m_{l}^{2}-U\lesssim 0,$ (12) $t_{2}=-2(q,k)+t+m_{G}^{2}\simeq-
Ux/(1-x)+m_{G}^{2}\lesssim 0,$ (13)
$s_{2}=2m_{l}^{2}+m_{G}^{2}-t-t_{2}-u_{2}\simeq U/(1-x)+m_{l}^{2}\gtrsim 0.$
(14)

Note that both the energy conservation law,
$q_{0}+E_{l}=E_{l}^{\prime}+E_{G}$, and the condition, $q_{0}/E_{l}\ll 1$,
imply that $E_{l}\simeq E_{G}+E^{\prime}_{l}$, thus for the energy faction
$x=E_{G}/E_{l}$ we get respectively its minimum and maximum values in the
following form $\hat{x}_{min}\simeq m_{G}/E_{l}$ and $\hat{x}_{max}\simeq
1-m_{l}/E_{l}$.

By exploiting FeynCalc package [51, 52] for the Wolfram Mathematica routine
[53], we get matrix element for process $l^{-}\gamma\to l^{-}G$ (see e. g.
Appendix (A)):

$\left|\mathcal{M}_{l^{-}\gamma\rightarrow
Gl^{-}}\right|^{2}=\frac{c_{l}^{2}e^{2}}{\Lambda^{2}}\frac{u_{2}[s_{2}-2m_{l}^{2}][(t_{2}+u_{2})^{2}+(u_{2}-m_{G}^{2})^{2}][4u_{2}(2m_{l}^{2}-s_{2})-m_{G}^{2}t_{2}]}{4t_{2}(u_{2}-m_{l}^{2})^{2}(s_{2}-m_{l}^{2})^{2}}-\\\
-\frac{c_{l}^{2}e^{2}}{\Lambda^{2}}\frac{m_{l}^{2}R(m_{l},m_{G},t_{2},u_{2})}{12t_{2}^{2}(m_{l}^{2}-u_{2})^{2}(s_{2}-m_{l}^{2})^{2}},$
(15)

where $R(m_{l},m_{G},t_{2},u_{2})$ is regular expression for $m_{l}$ and
$m_{G}$ that is given in the Appendix A. Note that in Eq. (15) we set
universal coupling of $G$ boson to both charged lepton and photon, such that
$c_{l}\equiv c_{e}=c_{\mu}=c_{\gamma}$. It implies that the unitarity of the
scenario is not violated at low energies as soon as $m_{G}\to 0$ (see, e. g.
Ref. [10] and references therein for detail). The differential cross section
for process $2\to 2$ is

$\frac{d\sigma_{2\to 2}}{d(pk)}=2\frac{d\sigma_{2\to
2}}{du_{2}}=\frac{1}{8\pi(s_{2}-m_{l}^{2})^{2}}\left|\mathcal{M}_{l^{-}\gamma\rightarrow
Gl^{-}}\right|^{2},$ (16)

where $\left|\mathcal{M}_{l^{-}\gamma\rightarrow Gl^{-}}\right|^{2}$ is
defined by (15). As a result, the double-differential cross section of the
bremsstrahlung-like process $lN\to lNG(\to\chi\overline{\chi})$ takes the
following form:

$\displaystyle\left.\frac{d\sigma(p+P_{i}\rightarrow
p^{\prime}+P_{f}+k)}{dxd\cos(\theta_{G})}\right|_{WW}=$
$\displaystyle=\frac{\alpha\chi}{\pi}\frac{E_{l}^{2}x\beta_{G}}{1-x}\frac{1}{8\pi(s_{2}-m_{l}^{2})^{2}}\left|\mathcal{M}_{l^{-}\gamma\rightarrow
Gl^{-}}\right|^{2}.$ (17)

In order to verify our calculation for the matrix element squared of spin-2
particle production $e^{-}(p)\gamma(q)\to G(k)e^{-}(p^{\prime})$ one can
exploit the crossing symmetry for the well known process of electron positron
pair annihilation $e^{+}(p_{e^{+}})e^{-}(p_{e^{-}})\to
G(p_{G})\gamma(p_{\gamma}^{\prime})$, for which the amplitude squared can be
found in Ref. [10]. In particular, for the massless fermion the regarding
transition element reads as follows [10]:

$\displaystyle\left|\mathcal{M}_{e^{+}e^{-}\rightarrow\gamma G}\right|^{2}=$
$\displaystyle=\frac{c_{e}^{2}e^{2}}{\Lambda^{2}}\frac{\left(\hat{s}_{2}^{2}+2\hat{t}_{2}(\hat{s}_{2}+\hat{t}_{2})-2m_{G}^{2}\hat{t}_{2}+m_{G}^{4}\right)}{4\hat{t}_{2}\hat{s}_{2}(\hat{s}_{2}+\hat{t}_{2}-m_{G}^{2})}\times$
$\displaystyle\times\left(4\hat{t}_{2}(\hat{s}_{2}+\hat{t}_{2})-m_{G}^{2}(\hat{s}_{2}+4\hat{t}_{2})\right),$
(18)

where the Mandelstam variables are defined by:

$\hat{s}_{2}\equiv(p_{e^{-}}+p_{e^{+}})^{2},\quad\hat{t}_{2}\equiv(p_{e^{-}}-p^{\prime}_{G})^{2},\quad\hat{u}_{2}\equiv(p_{e^{-}}-p^{\prime}_{\gamma})^{2}.$
(19)

The crossing-symmetry implies the momentum replacement for the initial process
$e^{+}e^{-}\rightarrow\gamma G$ in the following form:
$p_{e^{+}}\to-p^{\prime}_{e^{-}}$ and $p^{\prime}_{\gamma}\to-p_{\gamma}$. The
regarding Mandelstam variables transform as follows: $\hat{s}_{2}\rightarrow
t_{2}$, $\hat{t}_{2}\rightarrow u_{2}$ and $\hat{u}_{2}\rightarrow s_{2}$,
where the notations for $s_{2},u_{2}$ and $t_{2}$ are introduced (see e. g.
Eq. (8)) in order to match with conventional labels of the authors of Ref.
[50]. Finally this yields the following expression for the matrix element
squared:

$\displaystyle\left|\mathcal{M}_{e^{-}\gamma\rightarrow Ge^{-}}\right|^{2}=$
$\displaystyle=\frac{c_{e}^{2}e^{2}}{\Lambda^{2}}\frac{\left(t_{2}^{2}+2u_{2}(u_{2}+t_{2})-2m_{G}^{2}u_{2}+m_{G}^{4}\right)}{4u_{2}t_{2}(u_{2}+t_{2}-m_{G}^{2})}\times$
$\displaystyle\times\left(4u_{2}(u_{2}+t_{2})-m_{G}^{2}(t_{2}+4u_{2})\right)$
(20)

It is worth noting that Eq. (15) tends to Eq. (20) in the massless lepton
limit as soon as $m_{l}\to 0$.

##  III The virtual photon flux function

Figure 3: Left panel: the differential cross-section of a $G$-boson production
at NA64$e$ experiment as function of the energy fraction for the set of its
mass and the set of form-factors. The set of masses is chosen to be:
$m_{G}=5\,\mbox{MeV}$, $m_{G}=250\,\mbox{MeV}$ and $m_{G}=1\,\mbox{GeV}$. The
cross-section for $F_{TS}$ is shown by the green solid line, the cross-section
for $F_{H_{nucl}}$ is shown by blue solid line, the cross-section for
$F_{H_{nucl}}$ is shown by blue solid line, the cross-section for $F_{H}$ is
shown by the red dashed line, the cross-section for $F_{E}$ is shown by the
orange dashed line. Right panel: the regarding relative differences of the the
cross-sections for various form-factor parametrizations are expressed as
$(\mathcal{O}_{H_{nucl}}-\mathcal{O}_{TS})/\mathcal{O}_{TS}$,
$(\mathcal{O}_{H}-\mathcal{O}_{TS})/\mathcal{O}_{TS}$ and
$(\mathcal{O}_{E}-\mathcal{O}_{TS})/\mathcal{O}_{TS}$ respectively, where
$\mathcal{O}\equiv d\sigma/dx$.

In this section we discuss the impact of different atomic and nuclear form-
factor parametrization $F(t)$ for the photon flux $\chi$ which is given by Eq.
(4). The latter affects the differential and total cross sections of hidden
spin-2 boson production, that will be discussed in Section V.

The nuclear form-factor in the laboratory frame is associated with charge
density of nucleus through the Fourier transformation for both spin-$0$ and
spin-$1/2$ (see e. g. Refs. [54, 55, 50, 46, 47, 39, 49, 56, 55] and
references therein for detail). The atomic form-factor can be represented as
the nuclear form-factor that takes into account the screening of the nucleus
by Coulomb field due to the atomic electrons. Indeed, in the limit $t\to 0$
nuclear form-factor tends to $F_{nucl}(t)\to 1$ in opposite to the atomic
form-factor, which tends to $F_{atom}(t)\to 0$ as $t\to 0$. In addition,
screening charge density of the atomic form-factor can be represented as a
convolution of the nuclear charge density with the specific screening density.
In particular, following L. Schiff [57] to obtain the atomic form-factor one
should multiply the nuclear form-factor by the screening term $t/(t_{a}+t)$.

First, let us consider the elastic atomic form-factor that was studied by Y.
Tsai [49] and L. Schiff [58] in the following form [50]:

$F_{TS}(t)=\frac{t}{(t_{a}+t)}\frac{1}{(1+t/t_{d})},$ (21)

where $\sqrt{t_{a}}=1/R_{a}$ is a momentum transfer associated with nucleus
Coulomb field screening due to the atomic electrons, with $R_{a}$ being a
typical magnitude of the atomic radius $R_{a}=111Z^{-1/3}/m_{e}$,
$\sqrt{t_{d}}=1/R_{n}$ is the typical momentum associated with nuclear radius
$R_{n}$, such that $R_{n}\simeq 1/\sqrt{d}$ and
$d=0.164A^{-2/3}\text{GeV}^{2}$. Since the integration with respect to $t$ in
Eq. (4) is dominated by $t\gtrsim t_{min}$, therefore the magnitude of
$t_{min}$ (see e. g. Eq. (11)) defines the typical form-factor approach to be
considered. In particular, if $t_{min}/t_{a}\ll 1$, then the complete
screening regime takes place, which implies that nucleus transfer momentum is
small and the typical atomic elastic form-factor is much less then unity,
$F_{TS}(t)\simeq t/t_{a}\ll 1$. On the other hand, as soon as
$t_{min}/t_{a}\gg 1$, then no screening regime occurs. In this case the atomic
elastic form-factor is scaled as $F_{TS}(t)\simeq 1/(1+t/t_{d})$ and the
nucleus size effects dominate, it implies also that the typical nucleus
transfer momentum squared is relatively large, $t_{d}\gg t_{a}$. It is worth
noting, given the parameter space of interest $1\,\mbox{MeV}\lesssim
m_{G}\lesssim 1\,\mbox{GeV}$ and $10\,\mbox{GeV}\lesssim E_{l}\lesssim
100\,\mbox{GeV}$ both screening and nucleus size parameters can contribute to
the virtual photon flux and thus to the total yield of the $G$-boson
production.

Next, let us consider the nuclear Helm’s form-factor $F_{H_{nucl}}(t)$ that
corresponds to the inverse Fourier transformation of the nucleus charge
density $\rho(\vec{r})$. The latter can be represented as the convolution of
the spherically uniform charge inside the nucleus and the Gaussian profile
implying better accounting of the the edge of nucleus [59]. Both the nuclear
Helm’s form-factor $F_{H_{nucl}}(t)$ and atomic Helm’s form-factor $F_{H}(t)$
read as follows respectively [60, 61]:

$\displaystyle
F_{H_{nucl}}(t)=\frac{3j_{1}(\sqrt{t}R_{H})}{\sqrt{t}R_{H}}\exp{-s_{H}^{2}t/2},$
(22) $\displaystyle F_{H}(t)=t/(t_{a}+t)F_{H_{nucl}}(t),$ (23)

where $j_{1}(x)$ \- first spherical Bessel function of the first kind, the
effective nuclear radius $R_{H}$ can be parameterized as

$R_{H}=\sqrt{c_{H}^{2}+7/3\pi^{2}a_{H}^{2}-5s_{H}^{2}},$

where $s_{H}=0.9\;\text{fm}$ is the nuclear shell thickness,
$a_{H}=0.52\;\text{fm}$ and $c_{H}=(1.23A^{1/3}-0.6)\;\text{fm}$. It is worth
noting that we also set $F_{H_{nucl}}$ to zero for $t\gtrsim(4.49/R_{H})^{2}$,
it implies that we consider only non-negative values of the Helm’s form-
factor, since it vanishes at $t\simeq(4.49/R_{H})^{2}$ and the dominant
contribution for the photon flux is due to the typical range
$t\lesssim(4.49/R_{H})^{2}$ (for details see e. g. Ref. [61]).

Finlay, let us consider now the exponential atomic form-factor corresponding
Gaussian charge distribution that reads as follows [62]:

$F_{E}(t)=t/(t_{a}+t)\exp\left(-tR^{2}_{exp}/6\right),$ (24)

where the mean radius of nucleus is defined as
$R_{exp}=(0.91A^{1/3}+0.3)\text{fm}$. As was discussed above, the screening
term $t/(t_{a}+t)$ in Eq. (24) is introduced to take into account the
shielding of the nucleus Coulumb field due to the atomic electrons.

Note that the general virtual photon flux (4) depends on both fraction of
energy $x$ and emission angle of spin-2 boson $\theta_{G}$ through the
function of the lower limit $t_{min}(x,\theta_{G})$ (see e. g. Eq. (11) for
detail). So that it is instructive to study the the dependence of $\chi$ upon
$t_{min}$ for various form-factor parametrizations. Intriguingly, that the
integration over $dt$ in the virtual photon flux (4) of the Tsai-Schiff’s
elastic form-factor (21) can be performed in the analytical way through the
elementary functions. The latter simplification can be exploited for the
reducing of the computational time of $\chi_{{}_{TS}}$ integration. As the
result, the virtual photon flux in analytic form for Tsai-Schiff’s form-factor
reads as follows:

$\displaystyle\chi_{{}_{TS}}=\frac{Z^{2}t_{d}^{2}}{(t_{a}-t_{d})^{3}}\Bigl{(}\left[C^{\chi}_{1}+C^{\chi}_{2}t_{min}\right]+$
$\displaystyle+\left[C^{\chi}_{3}+C^{\chi}_{4}t_{min}\right]\ln{\left[\frac{t_{min}+t_{d}}{t_{min}+t_{a}}\right]}\Bigr{)}$
(25)

where the functions $C_{1}^{\chi}$, $C_{2}^{\chi}$, $C_{3}^{\chi}$ and
$C_{4}^{\chi}$ are defined by the following expressions respectively:

$\displaystyle
C^{\chi}_{1}=\Bigl{(}\frac{t_{d}(t_{a}-t_{d})}{t_{d}+t_{max}}+\frac{t_{a}(t_{a}-t_{d})}{t_{a}+t_{max}}-$
(26)
$\displaystyle-2(t_{a}-t_{d})+(t_{a}+t_{d})\ln{\left[\frac{(t_{d}+t_{max})}{(t_{a}+t_{max})}\right]}\Bigr{)},$
(27)
$C^{\chi}_{2}=\left(\frac{t_{a}-t_{d}}{t_{d}+t_{max}}+\frac{t_{a}-t_{d}}{t_{a}+t_{max}}+2\ln{\left[\frac{t_{d}+t_{max}}{t_{a}+t_{max}}\right]}\right),$
(28) $C^{\chi}_{3}=-(t_{a}+t_{d}),\quad C^{\chi}_{4}=-2.$ (29)

The regarding flux for and exponential form-factor can be expressed through
the special function as follows

$\displaystyle\chi_{{}_{E}}=Z^{2}\Bigl{(}\frac{t_{a}+t_{min}}{t+t_{a}}e^{-R_{exp}^{2}t/3}+$
(30)
$\displaystyle+\frac{3+R_{exp}^{2}(t_{a}+t_{min})}{3}e^{R_{exp}^{2}t_{a}/3}\text{Ei}\left[-\frac{R_{exp}^{2}(t+t_{a})}{3}\right]\Bigr{)}\Bigr{|}_{t_{min}}^{t_{max}},$

where $\text{Ei}(x)=\int_{-\infty}^{x}e^{t}/t\;dt$ is the exponential integral
function. Contrary, photon flux with Helm’s form-factor cannot be integrated
over $t$ in analytical way, in what follows we perform the numerical
integration for both $\chi_{{}_{H}}$ and $\chi_{{}_{H_{nucl}}}$. In addition,
the numerical calculations reveal that general photon flux (4) depends weakly
on $t_{max}$ as long as $t_{max}\gg t_{min},t_{a},t_{d}$. On the other hand,
one can show that the typical values of the $t_{min},t_{a}$ and $t_{d}$ don’t
exceed the magnitude of the order of $\lesssim\mathcal{O}(1)\,\mbox{GeV}^{2}$.
In what follows we set $t_{max}\simeq 10\,\mbox{GeV}^{2}$ for the numerical
estimations.

In the left panel of Fig. 2 we show the virtual photon fluxes $\chi$ as a
functions of $t_{min}$ for the lead target material of the NA64$e$ experiment
with atomic number of $Z=82$ and particle number of $A=207$ that parameterize
both typical screening and nuclear radius. In the right panel of Fig. 2 we
choose $\chi_{{}_{TS}}$ as a benchmark photon flux and show the the relative
differences between it and the set of $\chi_{{}_{E}}$, $\chi_{{}_{H}}$ and
$\chi_{{}_{H_{nucl}}}$. For the wide range of the typical momentum squared
$10^{-10}\,\mbox{GeV}^{2}\lesssim t_{min}\lesssim 10^{-3}\,\mbox{GeV}^{2}$ all
the atomic photon fluxes match with a reasonable accuracy at the level of
$\lesssim 5\%$. In order to illustrate the impact of the screening effect we
show both the nuclear and atomic Helm’s photon fluxes in Fig. 2. In
particular, for the screening region $t_{min}\lesssim t_{a}\simeq
10^{-9}\,\mbox{GeV}^{2}$ the nucleus photon flux $\chi_{{}_{H_{nucl}}}$
exceeds the atomic one $\chi_{{}_{H}}$ by $10\%$ approximately. The latter one
is associated also with the smaller slope of $\chi_{{}_{H}}$ in the screening
range $t_{min}\lesssim t_{a}$.

It is worth noting, that there are also form-factor models that consider the
charge density as the sum of Gaussian, Fourier-Bessel functions and Klein-
Nystrand’s charge density [63, 64, 65, 66, 67]. In addition, the Fermi
distribution for the nucleus charge is also discussed in the literature [60,
66]. We note that these nucleus form-factor parametrizations are beyond the
scope of the present paper, even though we would expect that they provide a
similar results for the flux shown in Fig. 2.

##  IV Missing energy signal

In this section we discuss the setups for the fixed target experiments such as
NA64e (SPS, CERN) [25, 26, 27, 28, 29, 30, 31, 68, 32, 33, 34, 35, 36, 37],
LDMX (Fermilab) [40, 41, 42, 43], NA64$\mu$ (SPS, CERN) [38, 39] and M3
(Fermilab) [44, 45], that can potentially probe invisible decay of
$G\to\chi\overline{\chi}$ in the associated charged lepton missing energy
process $lN\to lNG(\to\chi\overline{\chi})$, where the label $l=(e,\mu)$
denotes either electron or muon beam and $N$ denotes the nucleus of the
target. Given the benchmark coupling Eq. (1), the spin-2 Dark Matter mediator
$G$ can decay through the different channels. In particular, as soon as
$m_{G}\gtrsim 2m_{l}$ the visible decay $G\to l^{+}l^{-}$ is allowed with a
specific decay width [9]

$\displaystyle\Gamma(G\to
l^{+}l^{-})=\frac{c_{l}^{2}m_{G}}{160\pi}\left(\frac{m_{G}}{\Lambda}\right)^{2}\times$
$\displaystyle\times\left(1-4m_{l}^{2}/m_{G}^{2}\right)^{3/2}\left(1+8m_{l}^{2}/(3m_{G}^{2})\right),$
(31)

where $m_{l}$ is the mass of the charged lepton. The Lagrangian (1) also
implies that for $m_{G}\gtrsim 2m_{\chi}$ the invisible decay into fermion DM
pair $G\to\chi\bar{\chi}$ is kinetically allowed with a decay width

$\displaystyle\Gamma(G\to\chi\bar{\chi})=\frac{c_{\chi}^{2}m_{G}}{160\pi}\left(\frac{m_{G}}{\Lambda}\right)^{2}\times$
$\displaystyle\times\left(1-4m_{\chi}^{2}/m_{G}^{2}\right)^{3/2}\left(1+8m_{\chi}^{2}/(3m_{G}^{2})\right),$
(32)

where $m_{\chi}$ is the mass of the hidden Dirac DM fermion. In this paper we
focus on the processes of the invisible channel of $G$-boson decay into pair
of hidden dark fermions, $G\to\chi\bar{\chi}$ with
$\mbox{Br}(G\to\chi\bar{\chi})\simeq 1$ for the sufficiently light DM
particles, $m_{G}\gtrsim 2m_{\chi}$. It implies that decay widths obey the
condition $\Gamma_{G\to\chi\bar{\chi}}\gg\Gamma_{G\to l^{+}l^{-}}$ and
therefore the coupling constants are chosen to be $c_{\chi}\gg c_{l}$ for the
parameter space of interest $1\,\mbox{MeV}\lesssim m_{G}\lesssim
1\,\mbox{GeV}$. As the result, this benchmark conditions imply the rapid decay
of spin-2 DM mediator to $\chi\bar{\chi}$ pair after its production in the
process $lN\to lNG$.

Figure 4: Same as in Fig. 3 but for the LDMX experiment.

Let us estimate $N_{G}$ the number of $G$ produced by the lepton beam at fixed
target as follows

$N_{G}\simeq\mbox{LOT}\cdot\frac{\rho
N_{A}}{A}L_{T}\int\limits^{x_{max}}_{x_{min}}dx\frac{d\sigma_{2\to
3}(E_{l})}{dx}\mbox{Br}(G\to\chi\bar{\chi})\,,$ (33)

where LOT is number of leptons accumulated on target, $\rho$ is target
density, $N_{A}$ is Avogadro’s number, $A$ is atomic weight number, $L_{T}$ is
the effective interaction length of the lepton in the target, $d\sigma_{2\to
3}/dx$ is the differential cross-section of the lepton missing energy process
$lN\to lNG(\to\chi\bar{\chi})$, $E_{l}$ is the initial energy of lepton beam,
$x_{min}$ and $x_{max}$ are the minimal and maximal fraction of missing energy
respectively for the regarding experimental setup, $x\equiv E_{miss}/E_{l}$,
where $E_{miss}\equiv E_{G}$. The $x_{min}\lesssim x\lesssim x_{max}$ cuts are
determined by specific fixed–target facility. In what follows, we describe
below the input benchmark parameters of the lepton fixed-target experiments.

Figure 5: Same as in Fig. 3 but for the NA64$\mu$ experiment.

###  IV.1 NA64$e$

The spin-2 mediator of DM can be produced in the reaction of ultra-
relativistiс electrons of $E_{e}\simeq 100\,\mbox{GeV}$ scattering off the
nuclei of an active target $eN\to eNG$ followed by rapid $G\to\chi\bar{\chi}$
decay into DM particles. The fraction of the primary electron energy
$E_{miss}=xE_{e}$ can be carried away by $\chi\bar{\chi}$ pair, that passes
the NA64$e$ detector without energy deposition. The remaining part of the beam
energy fraction, $E_{e}^{rec}\simeq(1-x)E_{e}$, can be deposited in the
electromagnetic calorimeter (ECAL) of NA64$e$ by the recoil electrons. So
that, the production of the hidden spin-2 boson can be observed as an excess
of events with a single electromagnetic shower of energy $E^{rec}_{e}$ above
the predicted background [26]. In this paper we carry out an estimate that
implies the localization of that electromagnetic shower in the first radiation
length of the lead target. As a result one can set $L_{T}\simeq X_{0}$ in Eq.
(33), where $X_{0}\simeq 0.56\,\mbox{cm}$ is the typical radiation length of
the electron in the lead target. In addition, the candidate event is required
to have a missing energy in the range $E^{rec}_{e}\lesssim 0.5E_{e}\simeq
50\,\mbox{GeV}$, that leads to the specific energy fraction cut $x_{min}\simeq
0.5$ for the NA64$e$ facility in Eq. (33).

Moreover we note that the electromagnetic calorimeter of the NA64$e$ serves as
the active target for the incident electron beam and contains $6\times 6$
Shahalyk-type modules which are made of the both plastic scintillator (Sc) and
lead (Pb, $\rho\simeq 11.34\,\mbox{g cm}^{-3}$, $A=207\,\mbox{g mole}^{-1}$,
$Z=82$) plates. It is worth mentioning that the production of $G$-boson inside
the scintillator plate is subdominant due to its smaller density,
$\rho(\mbox{Sc})\ll\rho(\mbox{Pb})$, and larger radiation length,
$X_{0}(\mbox{Sc})\gg X_{0}(\mbox{Pb})$. In what follows in the numerical
calculations we neglect the contribution of Sc to the production rate of the
spin-2 mediator.

To conclude this subsection we note that NA64$e$ uses the electron beam from
the $H4$ beam line at the CERN SPS. The regarding beam intensity is estimated
to be of the order of $\simeq 10^{7}$ electrons per spill of $\simeq
4.6\,\mbox{seconds}$, however the number of good spills per day is expected to
be $\simeq 4000$. As a result, about $\simeq 120$ days are needed to collect
$\mbox{EOT}\simeq 5\times 10^{12}$ at the $H4$ beam line for the projected
statistics of the NA64$e$. In this work we also perform the analysis of the
sensitivity of NA64$e$ to probe DM for the $\mbox{EOT}\simeq 3.22\times
10^{11}$ (see e. g. Ref. [32] for detail). The regarding statistics has been
already collected by NA64$e$ during previous experimental runs in 2016-2021.

###  IV.2 LDMX

The Light Dark Matter Experiment (LDMX) is the projected electron fixed-target
facility at Fermilab, that aims probing the relic DM particles in the mass
range between $1\,\mbox{MeV}$ and $1\,\mbox{GeV}$. It can be considered as a
facility that is complimentary to NA64$e$ experiment due to its unique
electron missing momentum technique [69]. It is remarkable that the missing-
energy cuts and active veto system of both NA64$e$ and LDMX experiments
provide a significant background suppression at the level of $\lesssim
10^{-12}$.

The projected LDMX experiment would employ the target, the silicon tracker and
both the hadron and electromagnetic calorimeter. The lost of the primary
electron beam energy can be associated with $G$-boson emission in the thin
upstream target of the LDMX. The regarding missing momentum of the electron
can be measured by the silicon tracker system, electromagnetic and hadron
calorimeter which are located downstream. The missing energy cut of the recoil
electron is set to be $E^{rec}_{e}\lesssim 0.3E_{e}$, that implies
$x_{min}=0.7$ in Eq. (33). We perform the analysis of the LDMX sensitivity for
aluminium target (Al) $(\rho=2.7\,\mbox{g cm}^{-3},A=27\,\mbox{g
mole}^{-1},Z=13)$ with a thickness of $L_{T}\simeq 0.4X_{0}\simeq
3.56\,\mbox{cm}$, where $X_{0}=8.9\leavevmode\nobreak\ \mbox{cm}$ is a
radiation length of the electron in the aluminium. The energy of the beam is
chosen to be $E_{e}\simeq 16\,\mbox{GeV}$ and the projected statistics
corresponds to $\mbox{EOT}\simeq 10^{16}$ for the final phase of experimental
running after 2027 (see e. g. Ref. [43] and references therein for detail).

###  IV.3 NA64$\mu$

Figure 6: Same as in Fig. 3 but for the M3 experiment.  Figure 7: Left panel:
total cross section of $G$-boson production as function of its mass for the
set of experiments associated with benchmark Tsai-Shiff’s form-factor. We set
here $c_{l}/\Lambda=1\,\mbox{GeV}^{-1}$ and integrate the cross-section over
specific experimental cut range, $x_{min}\lesssim x\lesssim x_{max}$ (see text
for detail). Right panel: the relative difference is expressed as
$(\mathcal{O}_{H}-\mathcal{O}_{TS})/\mathcal{O}_{TS}$ (right upper) and
$(\mathcal{O}_{E}-\mathcal{O}_{TS})/\mathcal{O}_{TS}$ (right bottom) for
atomic Helm’s and the exponential form-factors respectively, where
$\mathcal{O}=\sigma_{tot}$. The green line corresponds to the NA64$e$
facility, the violet line is associated with the NA64$\mu$ experiment, the red
line corresponds to the M3 facility, the blue line is associated with the LDMX
experiment.

The NA64$\mu$ experiment is the fixed-target facility at the CERN SPS that
searches for the dark sector particles in the muon beam missing momentum mode
$\mu N\to\mu NG$. It can be considered as a complementary experiment to
NA64$e$. For the sensitivity estimation of NA64$\mu$ we set the energy of the
muon beam to be $E_{\mu}\simeq 160\,\mbox{GeV}$ and choose $\mbox{MOT}\simeq
5\times 10^{13}$ as the projected statistics for the muons accumulated on
target. The NA64$\mu$ experiment employs the lead Shashlyk-type
electromagnetic calorimeter that serves as a target with a thickness of
$L_{T}\simeq 40X_{0}\simeq 22.5\,\mbox{cm}$. We note that one can neglect the
muon stopping loss in the lead target of $\simeq 22.5\,\mbox{cm}$ for the
ultra-relativistic muons of $E_{\mu}\simeq 160\,\mbox{GeV}$ due to small
energy attenuation in the lead $\langle dE_{\mu}/dx\rangle\simeq 12.7\times
10^{-3}\,\mbox{GeV}/\mbox{cm}$ (see e. g. Ref. [70] for detail).

In order to measure the momentum of incident and outgoing muon, the NA64$\mu$
facility employs two magnet spectrometers. We choose the typical cut for the
outgoing muon as $E_{\mu}^{rec}\lesssim 0.5E_{\mu}\simeq 80\,\mbox{GeV}$, that
corresponds to $x_{min}=0.5$ in Eq. (33). We note that for NA64$\mu$ facility
approximately $\simeq 120$ days are needed to accumulate statistics of
$\mbox{MOT}\simeq 5\times 10^{13}$ relative to $\mbox{EOT}\simeq 5\times
10^{12}$ for NA64$e$ facility. This can be explained by the increased
intensity of muon beam line at M3 that is higher by factor of $\simeq 10$ than
the intensity of electron beam at H4.

###  IV.4 M3

The muon missing momentum experiment (M3) at Fermilab is the projected fixed
target facility that aims probing dark sector particles by employing the muon-
specific missing energy signature $\mu N\to\mu NG(\to\chi\bar{\chi})$. It
utilizes the muon beam of $E_{\mu}\simeq 15\,\mbox{GeV}$ impinging on the
tungsten target (W) ($\rho=19.3\,\mbox{g cm}^{-3},A=184\,\mbox{g
mole}^{-1},Z=74$) of the typical thickness of $L_{T}\simeq 50X_{0}\simeq
17.5\,\mbox{cm}$, where $X_{0}\simeq 0.35\leavevmode\nobreak\ \mbox{cm}$ is
the radiation length of electron in the tungsten. The regarding facility also
exploits a downstream detector to veto the Standard model background. It aims
to collect $\mbox{MOT}\simeq 10^{3}$ during $\simeq 3$ months of the
experimental running. The cut on the missing momentum of muon is chosen to be
$E^{rec}_{\mu}\lesssim 9\,\mbox{GeV}$, this yields the lower limit on the
energy fraction $x_{min}\simeq 0.4$ in Eq. (33).

To conclude this subsection we note that for muons of $E_{\mu}\simeq
15\,\mbox{GeV}$ the muon stopping loss in the tungsten [45] is estimated to be
of the order of $530\,\mbox{MeV}$ through the target medium of $50X_{0}$. This
allows one to neglect $\langle dE_{\mu}/dx\rangle$ in the numerical
calculation of the yield of $G$-boson at M3. As a result that justifies the
exploiting of Eq. (33) for the estimate of $N_{G}$ with $L_{T}\simeq 50X_{0}$
at M3 facility.

##  V The differential and total сross sections

In this section we study both the single-differential and total cross sections
in the WW approximation for the set of the form-factors discussed in Section
III. The results are presented for the benchmark parameters of the lepton
fixed target experiments which can potentially probe the invisible signatures
associated with a lepton missing energy in bremsstrahlung-like process $lN\to
lNG(\to\chi\overline{\chi})$. The total $\sigma_{tot}$ cross-sections are
obtained by the numerical integration of the double-differential cross section
(6) for the specific experimental cuts. In the left panels of Figs. 3, 4, 5,
and 6 we show the differential cross-sections of $G$-boson production as
function of energy fraction $x=E_{G}/E_{l}$ for the NA64$e$, LDMX, NA64$\mu$
and M3 experiments respectively and for the set of form-factors discussed in
Sec. III. These cross-sections are also presented for the specific set of
$G$-boson masses, $m_{G}=5\,\mbox{MeV}$, $m_{G}=250\,\mbox{MeV}$ and
$m_{G}=1\,\mbox{GeV}$.

Let us describe now the typical properties and the general kinematics of
$G$-boson production by the charged lepton beams impinging on the solid
target. It is worth mentioning that both NA64$e$ and LDMX electron cross-
sections have a peak in the region $x\lesssim 1$ for the relatively heavy
masses $m_{G}\gtrsim 250\,\mbox{MeV}$, it means that the signal of $G$-boson
production is strongly forward peak, such that the dominant part of the
initial beam energy transfers to the hidden spin-2 boson. On the other hand,
for the relatively light masses $m_{G}\gtrsim 5\,\mbox{MeV}$ the sharp forward
peak is mitigated and the regarding differential cross-section is in the soft
bremsstrahlung-like regime. It implies that the cross-section peaks in the
infra-red (IR) region $x=E_{G}/E_{e}\ll 1$. However, for the higher masses
$m_{G}\gtrsim 250\,\mbox{MeV}$ the IR peak is smeared as long as the energy
fraction is small.

Remarkably that both NA64$\mu$ and M3 muon cross-sections are in the soft
breamsstrahlung-like regime $d\sigma_{\mu}/dx\propto\mathcal{O}(1/x)$ as long
as $x\ll 1$ for all masses in the range $1\,\mbox{MeV}\lesssim m_{G}\lesssim
1\,\mbox{GeV}$, since the mass of spin-2 boson is comparable to the muon mass
in that region, $m_{G}\propto\mathcal{O}(m_{\mu})$.

Let us describe now the impact of various form-factor parametrizations on the
shape of the differential cross-sections. In order to compare these cross-
sections we choose Tsai-Shiff’s form-factor as a benchmark one, since the
latter is exploited widely in the calculations of WW cross-section for both
beam dump and fixed target experiments (see e. g. Refs. [70, 50, 45, 39] and
references therein for detail). In what follows in the right panels of Figs.
3, 4, 5, and 6 we show the relative differences between $(d\sigma/dx)_{TS}$
and the set of $(d\sigma/dx)_{H_{nucl}}$, $(d\sigma/dx)_{H}$ and
$(d\sigma/dx)_{E}$ respectively for the specific experiments and the specific
masses of $G$-boson. We recall again for clarity, that the subscriptions of
$F_{TS}(t)$, $F_{H}(t)$ and $F_{E}(t)$ are related to the atomic elastic form-
factors with screening term, while the label of $F_{H_{nucl}}(t)$ is
associated with the nuclear elastic form-factor that doesn’t take into account
the screening effect.

It is worth mentioning that given the mass $m_{G}$ all the atomic form-factor
cross-sections for the NA64$e$ experiment match with a reasonable accuracy at
the level of $2\%-10\%$ as soon as $0.2\lesssim x\lesssim 1$. However, the
sizable deviation (i. e. at the level of $\propto\mathcal{O}(2)$) between the
nuclear and atomic cross-sections appears only for the small mass region long
as $m_{G}\lesssim 5\,\mbox{MeV}$. These effects are shown in the right panel
of Fig. 3. The regarding impact of the form-factor parametrization on the
differential cross-section shape is shown in the right panel of Fig. 4 for the
LDMX electron fixed-target experiment. In addition we note that the shapes of
the cross-sections can be varied for $x\ll 1$ and $m_{G}\lesssim
1\,\mbox{GeV}$, however this energy fraction region doesn’t provide a sizeable
contribution to the total cross-section of heavy masses.

For the muon beam cross-sections of the NA64$\mu$ experiment the impact of the
different form-factor parametrization can be estimated at the level of
$\lesssim 2\%-8\%$ for energy fraction in the range $0.2\lesssim x\lesssim 1$.
The latter is shown in the right panel of Fig. 5. One can see also from Fig. 6
that the shape of the differential cross-section for the M3 experiment can be
varied significantly due to the form-factors as long as $x\ll 1$ and
$m_{G}\lesssim 1\,\mbox{GeV}$.

Let us study now the specific shape of the total cross-sections of $G$-boson
production. In the left panel of Fig. 7 we show the regarding cross-sections
for the NA64$e$, LDMX, NA64$\mu$ and M3 facilities, which are calculated for
the benchmark Tsai-Shiff’s atomic elastic form-factor. One can see from Fig. 7
that both NA64$e$ and NA64$\mu$ cross-sections are calculated for the lead
(Pb) target ($Z\simeq 82$), nevertheless the the total electron beam cross-
section $\sigma^{tot}_{e}$ is generally larger by factor of $\simeq 5$ than
the muon beam cross-section $\sigma^{tot}_{\mu}$, even though they have a
comparable energies of the impinging beams $E_{e}\simeq
E_{\mu}\simeq\mathcal{O}(100)\,\mbox{GeV}$. That implies the advantage of
using the electron beam instead of muon beam at the CERN SPS facility in the
mass range $1\,\mbox{MeV}\lesssim m_{G}\lesssim 1\,\mbox{GeV}$. However, if
one compares the LDMX electron cross-section for the aluminium (Al) target
($Z\simeq 13$) with the M3 cross-section for the muon beam impinging on the
tungsten ($W$) target ($Z\simeq 74$), one can conclude that the regarding
advantage of using the electron beam is compensated by the nucleus charge
suppression $\simeq 13/74$, even though both LDMX and M3 experiments have a
comparable beam energies, $E_{\mu}\simeq E_{e}\simeq 15\,\mbox{GeV}$. In
addition we note from the NA64$\mu$ and M3 cross-sections that the greater
energy of the muon beam implies greater rate of the $G$-boson production,
$\sigma^{tot}_{\mu}(160\,\mbox{GeV})\gtrsim\sigma^{tot}_{\mu}(16\,\mbox{GeV})$.
To conclude this section, we specify briefly the impact of the form-factor
parametrization on the total cross-section. In the right panel of Fig. 7 the
relative differences of the total cross-sections are shown for the set of
atomic form-factors. In the small mass region $m_{G}\lesssim 100\,\mbox{MeV}$
these differences are relatively small, i. e. at the level of $\lesssim 2\%$.
However, the regarding discrepancies can be as large as $\gtrsim 8\%$ for the
heavy mass region $m_{G}\lesssim 1\,\mbox{GeV}$.

##  VI The experimental limits

In this section we study current and expected reach of the lepton fixed-target
experiments. The limit on the coupling $c_{l}/\Lambda$ can be obtained as
follows. By exploiting both Eq. (33) for the number of produced $G$-bosons and
the result on its production cross-section with specific experimental cuts,
one can require that the number of signal events is $N_{G}\gtrsim 2.3$. As a
result, this yields $90\%C.L.$ exclusion limit on the coupling constant
$c_{l}/\Lambda$ for the background free case and null result of the fixed-
target facilities. In the left panel of Fig. 8 we show by the dashed lines the
projected sensitivities of NA64$e$, LDMX, NA64$\mu$ and M3 fixed target
facilities.

It is worth mentioning that the expected reach of the LDMX is rather strong,
i. e. at the level of $c_{l}/\Lambda\gtrsim 10^{-5}\,\mbox{GeV}^{-1}$, even
though the typical cross-section of the $G$-boson production is fairly small
(see e. g. the blue line in the left panel of Fig. 7). The enhanced expected
reach of the LDMX is associated with fairly large number of the electrons that
will be accumulated on target. In particular, by the final phase of data
taking LDMX plans to accumulate $\mbox{EOT}\simeq 10^{16}$, as we discussed it
above in Sec. IV.2.

In addition, from the projected sensitivities of both NA64$\mu$ and M3 muon
experiments shown in Fig. 8, one can conclude that compared to the M3 option
with $16\,\mbox{GeV}$ beam muons and $\mbox{MOT}\simeq 10^{13}$, the higher
energy muons of NA64$\mu$ (e. g. $E_{\mu}\simeq 160\,\mbox{GeV}$) with
$\mbox{MOT}\simeq 5\times 10^{13}$ allow examining a wider region in the
parameter space of the spin-2 boson scenario. Nevertheless, the ultimate
projected bounds of the NA64$e$ experiment for $\mbox{EOT}\simeq 5\times
10^{12}$ (see e. g. the green dashed line in the left panel of Fig. 8 at the
level of $c_{l}/\Lambda\gtrsim 10^{-4}\,\mbox{GeV}^{-1}$), can be ruled out
even by the M3 facility at the level of $c_{l}/\Lambda\gtrsim 7\times
10^{-5}\,\mbox{GeV}^{-1}$ (see e. g. the red dashed line in the left panel of
Fig. 8). In the left panel of Fig. 8, we show by the solid green line the
excluded limit of the NA64$e$ experiment (at the level of
$c_{l}/\Lambda\gtrsim 7\times 10^{-4}\,\mbox{GeV}^{-1}$) for the current
accumulated statistics of $\mbox{EOT}\simeq 3.22\times 10^{11}$. In
particular, from the analysis of the NA64$e$ data collected during 2016-2021
runs no signal events were found for the background free case (see e. g. Ref.
[32]), as we discussed it in Sec. IV.1.

It is worth mentioning that authors of Ref. [10] have been already considered
the bounds on the light spin-2 mediator from $e^{+}e^{-}\to\gamma+E_{miss}$ at
BaBar experiment [71]. The analysis was carried out for mainly invisible decay
mode $\mbox{Br}(G\to\chi\bar{\chi})\simeq 1$. The regarding exclusion limit
yields the magnitude of the coupling constant at the level of
$c_{e}/\Lambda\gtrsim 2\times 10^{-4}\,\mbox{GeV}^{-1}$ for the masses in the
range $1\,\mbox{MeV}\lesssim m_{G}\lesssim 1\,\mbox{GeV}$. These bounds are
shown by the solid black line in the left panel of Fig. 8. One can conclude
that it rules out the current experimental constraints of the NA64$e$ facility
for $\mbox{EOT}\simeq 3.22\times 10^{11}$.

To conclude this section, we discuss briefly the impact of the form-factor
parametrization on the experimental limits $c_{l}/\Lambda$ for the fixed
target facilities. In the right panel of Fig. 8 the relative differences of
the coupling constants are shown for the set of atomic form-factors. In the
small mass region $m_{G}\lesssim 100\,\mbox{MeV}$ these differences are
relatively small, i. e. at the level of $\lesssim 2\%$. However, the regarding
discrepancies can be as large as $\gtrsim 8\%$ for the heavy mass region
$m_{G}\lesssim 1\,\mbox{GeV}$.

##  VII Conclusion

In the present paper we have discussed in detail the probing of the massive
spin-2 boson $G$ trough its invisible decay into pair of DM particles $\chi$
with $\mbox{Br}(G\to\chi\bar{\chi})\simeq 1$. The regarding scenario implies
that the $G$-boson serves as a mediator between charged leptons of SM and DM
sector. The benchmark simplified coupling of this model involves the
dimension-5 operators of massive $G_{\mu\nu}$ field and the energy momentum
tensors of both SM and DM particles. We have studied explicitly the missing
energy signatures for the projected and existing lepton fixed-target
experiments, such as NA64$e$, LDMX, NA64$\mu$ and M3. Namely, by exploiting
the WW approach, we have calculated $G$-boson production cross-section in the
process $lN\to lNG$ followed by its invisible decay into DM particles
$G\to\chi\bar{\chi}$ for the specific fixed-target experiments. We have
calculated the expected reach for the regarding experiments, implying the
background free case and null result for DM detection. Moreover, we have also
discussed in detail the impact of both nuclear and atomic form-factor
parametrizations on (i) the differential spectra of $G$ boson emission, (ii)
the total cross-section of its production (iii) the experimental reach of the
fixed-fixed target facilities for probing hidden spin-2 boson. It was found
that the total yield of spin-2 boson production and the regarding experimental
reach can be affected by the form-factor at the level of $\simeq 8\%-10\%$ for
the large mass region $m\lesssim 1\,\mbox{GeV}$. For the light masses of
$G$-boson, $m_{G}\lesssim 100\,\mbox{MeV}$, the implication of the form-
factors for the yield and experimental reach is estimated at the level of
$\lesssim 2\%$.

###### Acknowledgements.

We would like to thank P. Crivelli, S. Demidov, S. Gninenko, D. Gorbunov, M.
Kirsanov, N. Krasnikov, V. Lyubovitskij, L. Molina Bueno, A. Pukhov, H. Sieber
and A. Zhevlakov for very helpful discussions and correspondences. The work of
D. V. K on description of the dark matter missing energy signatures of NA64$e$
and regarding exclusion limits for spin-2 DM mediator is supported by the
Russian Science Foundation RSF grant 21-12-00379.

Figure 8: Left panel: $90\,\%\,C.L.$ limits on $c_{l}/\Lambda$ coupling
constant for the fixed-target and BaBar experiment [10] as a function of
$G$-boson mass $m_{G}$. For all expected reaches, we imply that
$\mbox{Br}(G\to\chi\bar{\chi})\simeq 1$ and $m_{G}\gtrsim 2m_{\chi}$. In
addition, in the left panel all curves for the fixed-target facilities imply
the Tsai-Shiff’s form-factor in the cross-section. The green dashed line is
the projected sensitivity of the NA64$e$ experiment for $\mbox{EOT}\simeq
5\times 10^{12}$, the red dashed line is the expected reach of the M3 facility
for $\mbox{MOT}\simeq 10^{13}$, the violet dashed line is the projected
sensitivity of the NA64$\mu$ experiment for $\mbox{MOT}\simeq 5\times 10^{13}$
and the blue dashed line is the expected reach of the LDMX facility for
$\mbox{EOT}\simeq 10^{16}$. The green solid line represents the excluded at
$90\,\%\,C.L.$ bound of the NA64$e$ experiment for $\mbox{EOT}\simeq
3.22\times 10^{11}$. Right panel: the relative difference is expressed as
$(\mathcal{O}_{H}-\mathcal{O}_{TS})/\mathcal{O}_{TS}$ (right upper) and
$(\mathcal{O}_{E}-\mathcal{O}_{TS})/\mathcal{O}_{TS}$ (right bottom) for
atomic Helm’s and the exponential form-factors respectively, where
$\mathcal{O}=c_{l}/\Lambda$.

##  Appendix A Matrix element for process $l^{-}\gamma\rightarrow Gl^{-}$

In this section, we collect an expressions for the Feynman diagrams [9, 72]
associated with spin-2 boson production and the matrix element squared for
process $l^{-}\gamma\rightarrow Gl^{-}$. In particular, we use the expression
for polarization sum of spin-2 mediator:

$\sum\limits_{s}e_{\mu\nu}(k,s)e_{\alpha\beta}(k,s)=1/2(P_{\alpha\mu}P_{\beta\nu}+P_{\beta\mu}P_{\alpha\nu}-2/3P_{\mu\nu}P_{\alpha\beta}),$
(34)

where $P_{\mu\nu}=g_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{m_{G}^{2}}$. The vertex for
the outgoing $G$-boson in case of incoming $l(p_{1})$ and outgoing $l(p_{2})$
leptons is

$\begin{gathered}\includegraphics[width=108.405pt,valign={c}]{Graphics/6.fTofG.pdf}\end{gathered}=\begin{aligned}
-\frac{ic_{l}}{4\Lambda}\left\\{(p_{2}+p_{1})^{\mu}\gamma^{\nu}+(p_{2}+p_{1})^{\nu}\gamma^{\mu}-\right.\\\
\left.-2g^{\mu\nu}(\not{p_{2}}+\not{p_{1}}-2m_{l})\right\\}.\end{aligned}$
(35)

For the incoming $V^{\alpha}(k_{1})$ and outgoing $V^{\beta}(k_{2})$ massless
vector bosons one has

$\begin{gathered}\includegraphics[width=108.405pt,valign={c}]{Graphics/7.vTovG.pdf}\end{gathered}=\begin{aligned}
-\frac{ic_{\gamma}}{\Lambda}\left\\{\eta^{\alpha\beta}k_{1}^{\mu}k_{2}^{\nu}+\eta^{\mu\alpha}((k_{1},k_{2})\eta^{\nu\beta}-k_{1}^{\beta}k_{2}^{\nu})\right.-\\\
-\left.\eta^{\mu\beta}k_{1}^{\nu}k_{2}^{\alpha}+1/2\;\eta^{\mu\nu}(k_{1}^{\beta}k_{2}^{\alpha}-(k_{1},k_{2})\eta^{\alpha\beta})+(\mu\longleftrightarrow\nu)\right\\}.\end{aligned}$
(36)

Finally the vertex for 4-point interaction is

$\begin{gathered}\includegraphics[width=108.405pt,valign={c}]{Graphics/8.4vToG.pdf}\end{gathered}=\begin{aligned}
\frac{c_{l}e}{2}\left(\eta^{\mu\alpha}\gamma^{\nu}+\eta^{\nu\alpha}\gamma^{\mu}\right).\end{aligned}$
(37)

As a result, by using FeynCalc package [51, 52] for the Wolfram Mathematica
[53], we get matrix element squared for process $l^{-}\gamma\to l^{-}G$:

$\left|\mathcal{M}_{l^{-}\gamma\rightarrow
Gl^{-}}\right|^{2}=\frac{c_{l}^{2}e^{2}}{\Lambda^{2}}\frac{u_{2}[s_{2}-2m_{l}^{2}][(t_{2}+u_{2})^{2}+(u_{2}-m_{G}^{2})^{2}][4u_{2}(2m_{l}^{2}-s_{2})-m_{G}^{2}t_{2}]}{4t_{2}(u_{2}-m_{l}^{2})^{2}(s_{2}-m_{l}^{2})^{2}}-\\\
-\frac{c_{l}^{2}e^{2}}{\Lambda^{2}}\frac{m_{l}^{2}R(m_{l},m_{G},t_{2},u_{2})}{12t_{2}^{2}(m_{l}^{2}-u_{2})^{2}(s_{2}-m_{l}^{2})^{2}},$
(38)

where $R(m_{l},m_{G},t,u)$ is regular expression for $m_{l}$ and $m_{G}$:

$R(m_{l},m_{G},t,u)=24m_{l}^{10}t+24m_{l}^{8}[m_{G}^{4}+5m_{G}^{2}t-6tu]+\\\
+2m_{l}^{6}[24m_{G}^{6}+6m_{G}^{4}(15t-8u)-3m_{G}^{2}t(33t+92u)+2t(13t^{2}+18tu+90u^{2})]+\\\
+2m_{l}^{4}[12m_{G}^{8}+12m_{G}^{6}(5t-6u)+m_{G}^{4}(72u^{2}-113t^{2}-288tu)+\\\
+4m_{G}t(t+3u)(13t+42u)-t(7t^{3}+64t^{2}u+144tu^{2}+240u^{3})]+\\\
+m_{l}^{2}[12m_{G}^{8}(3t-4u)-3m_{G}^{6}(29t^{2}+80tu-48u^{2})+m_{G}^{4}(70t^{3}+544t^{2}u+696tu^{2}-96u^{3})-\\\
-m_{G}^{2}t(27t^{3}+364t^{2}u+852tu^{2}+912u^{3})+2t(t^{4}+46t^{3}u+92t^{2}u^{2}+216tu^{3}+180u^{4})]+\\\
+[-3m_{G}^{8}(3t^{2}+16tu-8u^{2})+m_{G}^{6}(15t^{3}+128t^{2}u+168tu^{2}-48u^{3})-\\\
-m_{G}^{4}(9t^{4}+122t^{3}u+362t^{2}u^{2}+384tu^{3}-24u^{4})+\\\
+m_{G}^{2}t(3t^{4}+68t^{3}u+240t^{2}u^{2}+552tu^{3}+408u^{4})-\\\
-2tu(t+u)(7t^{3}+24t^{2}u+72tu^{2}+72u^{3})],$ (39)

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