††thanks: Corresponding author

# Probing Hidden Sectors with a muon beam:
Total and differential cross-sections for vector boson production in muon
bremsstrahlung

D. V. Kirpichnikov<EMAIL_ADDRESS>Institute for Nuclear Research,
117312 Moscow, Russia  H. Sieber<EMAIL_ADDRESS>ETH Zürich,
Institute for Particle Physics and Astrophysics, CH-8093 Zürich, Switzerland
L. Molina Bueno<EMAIL_ADDRESS>ETH Zürich, Institute for Particle
Physics and Astrophysics, CH-8093 Zürich, Switzerland  Instituto de Fisica
Corpuscular (CSIC/UV), Carrer del Catedrátic José Beltrán Martinez, 2, 46980
Paterna, Valencia, Spain  P. Crivelli<EMAIL_ADDRESS>ETH Zürich,
Institute for Particle Physics and Astrophysics, CH-8093 Zürich, Switzerland
M. M. Kirsanov<EMAIL_ADDRESS>Institute for Nuclear Research,
117312 Moscow, Russia

###### Abstract

Vector bosons, such as Dark Photon $A^{\prime}$ or $Z^{\prime}$, can couple to
muons and be produced in the bremsstrahlung reaction
$\mu^{-}+N\rightarrow\mu^{-}+N+A^{\prime}(Z^{\prime})$. Their possible
subsequent invisible decay can be detected in fixed target experiments through
missing energy/momentum signature. In such experiments, not only is the energy
transfer to $A^{\prime}(Z^{\prime})$ important, but also the recoil muon angle
$\psi_{\mu^{\prime}}$. In this paper, we derive the total and the double
differential cross-sections involved in this process using the phase space
Weiszäcker-Williams and improved Weiszäcker-Williams approximations, as well
as using exact-tree-level calculations. As an example, we compare the derived
cross-sections and resulting signal yields in the NA64$\mu$ experiment that
uses a 160 GeV muon beam at the CERN Super Proton Synchrotron accelerator. We
also discuss its impact on the NA64$\mu$ expected sensitivity to explore the
$(g-2)_{\mu}$ anomaly favoured region with a $Z^{\prime}$ boson considering
$10^{12}$ muons accumulated on target.

##  I Introduction

The recently confirmed $4.2\sigma$ discrepancy in the anomalous magnetic
moment measurement of the muon [1] with respect to its theoretical prediction
[2]

$\Delta a_{\mu}\equiv a_{\mu}(exp)-a_{\mu}(th)=(251\pm 59)\cdot 10^{-11}$ (1)

remains one of the long standing puzzles in the Standard Model (SM) of
particle physics. A minimal extension of the SM able to explain this mismatch
consists in the addition of a new weak interaction between the standard matter
and a dark sector. The existence of dark sectors is also strongly motivated as
a framework to explain the origin of Dark Matter (DM) as a thermal freeze-out
relic, a mechanism similar to the one of the Weakly Interacting Massive
Particles (WIMPs), but in a broader and lower mass range [3]. In particular,
the $L_{\mu}-L_{\tau}$ models based on the existence of a new light vector
boson ($Z^{\prime}$) which predominantly couples to the second and third
generation of leptons, are theoretically well motivated as they are an
anomaly-free extension of the gauge sector. The interaction between
$Z^{\prime}$ and the $L_{\mu}-L_{\tau}$ current is given by the following term
[4]:

$L_{Z^{\prime}}=g^{\prime}\left(\bar{\mu}\gamma_{\nu}\mu+\bar{\nu}_{\mu
L}\gamma_{\nu}\nu_{\mu L}-\bar{\tau}\gamma_{\nu}\tau-\bar{\nu}_{\tau
L}\gamma_{\nu}\nu_{\tau L}\right)Z^{\prime\nu},$ (2)

This interaction can give via loop effects the required additional
contribution to explain the discrepancy between the theoretical and
experimental value of the muon magnetic moment [5, 4, 6, 7, 8, 9]:

$\Delta a^{Z^{\prime}}_{\mu}=\frac{g^{\prime
2}}{4\pi^{2}}\int_{0}^{1}dx\frac{x^{2}\left(1-x\right)}{x^{2}+\left(1-x\right)m_{Z^{\prime}}^{2}/m^{2}_{\mu}},$
(3)

for couplings $g^{\prime}\sim 10^{-3}-10^{-5}$ and $Z^{\prime}$ masses in the
MeV-GeV range.

If such a boson exists, it can be produced in the muon nuclear bremsstrahlung
process $\mu^{-}+N\to\mu^{-}+N+Z^{\prime}$ when a high energy muon beam
impinges on a target. In this paper, we focus on the calculation of this
production mechanism but the same results can be directly applied to the
$A^{\prime}$ bremsstrahlung ($A^{\prime}$-strahlung) production from an
initial muon beam. In order to illustrate the impact of our results, we
consider the NA64$\mu$ experiment [10] which aims at exploring dark sector
particles weakly coupled to muons. A similar experiment, $M^{3}$, is also
under consideration at Fermilab [11]. Typically, the $Z’$ is expected to be
produced in the bremsstrahlung reaction of a 160 GeV muon beam impinging on an
electromagnetic calorimeter acting as an active target. A fraction of the
primary beam energy is carried away by the scattered muon, which momentum is
measured by a set of trackers after being deflected by a magnet located
downstream the target. In this work, we derive the cross-sections involved in
the production process and estimate its impact on the signal yield using the
Geant4-based [12] dark matter package DMG4 [13], in which we implemented the
simulation of $Z’$ according to the derived cross-sections. We compare the
results of analytical calculations using phase space approximations such as
Weiszäcker-Williams (WW) and improved Weiszäcker-Williams (IWW) with exact-
tree-level (ETL) calculations. We also derive a new analytical expression to
calculate the photon flux in the WW approximation. Finally, we study the
differential cross sections as a function of the recoil muon and $Z^{\prime}$
angles.

This paper is organised as follows. In section II, we calculate at ETL the
total cross-section for $Z^{\prime}$ production. In section III, we discuss
the differential cross-sections for the angle and energy fraction of the
emitted particle $Z^{\prime}$ for both WW and IWW approximations. In Section
III, we also discuss the zero mass limit $m_{Z^{\prime}}\to 0$ of the
corresponding cross-sections and show that WW and ETL reproduce the
bremsstrahlung spectrum of muons on nucleus. In Section IV, we derive the
differential cross-sections for the angle and energy fraction of the deflected
muon for both WW and IWW approximations. Finally, in section V, we evaluate
the impact of the cross-section calculations on the projected $(g-2)_{\mu}$
sensitivities for NA64μ experiment.

##  II The exact tree level calculation

In this section, we discuss the derivation of the $Z^{\prime}$-boson exact-
tree-level production cross-section, in particular, we follow the notations of
Ref. [14, 15]. Let us consider the kinematic variables of the process

$\mu^{-}(p)+N(P_{i})\rightarrow\mu^{-}(p^{\prime})+N(P_{f})+Z^{\prime}(k),$

where $p=(E_{\mu},{\bf p})$ is the four-momentum of the incoming muon,
$P_{i}=(M,0)$ designates the nucleus four-momentum in the laboratory frame,
and $P_{f}=(P^{0}_{f},{\bf P}_{f})$ is the final state of the nucleus. The
$Z^{\prime}$-boson momentum is $k=(E_{Z^{\prime}},{\bf k})$ and
$p^{\prime}=(E_{\mu^{\prime}},{\bf p}^{\prime})$ is the momentum of the
scattered muon. It is instructive to carry out the calculation in the
geometrical frame where the three-vector ${\bf V=p-k}$ is parallel to $z$-axis
and the three-vector ${\bf k}$ is in the $xz$-plane. We define the four-
momentum transfer to the nucleus as $q=P_{i}-P_{f}$. In that frame the polar
and axial angles of ${\bf q}$ are denoted by $\theta_{q}$ and $\phi_{q}$
respectively. We use minus metric $\eta_{\mu\nu}=\mbox{diag}(+,-,-,-)$, in
contrast to the authors of Ref. [14], which implies that the virtuality of the
photon is $t=-q^{2}=|{\bf q}|^{2}-q_{0}^{2}>0$. After some algebraic
manipulations one obtains the following expressions

$\cos\theta_{q}=-\frac{|{\bf V}|^{2}+|{\bf
q}|^{2}+m_{\mu}^{2}-(E_{\mu}+q_{0}-E_{Z^{\prime}})^{2}}{2|{\bf V}||{\bf q}|},$
(4) $\qquad q_{0}=-\frac{t}{2M},\qquad|{\bf
q}|=\sqrt{\frac{t^{2}}{4M^{2}}+t}.$ (5)

We assume that the nucleus has zero spin [14, 16, 17, 18], as a result the
photon-nucleus vertex is given by

$ieF(t)(P_{i}+P_{f})_{\mu}\equiv ieF(t)\mathcal{P}_{\mu},$ (6)

where the squared elastic form-factor is

$F^{2}(t)\equiv G^{el}_{2}(t)\simeq
Z^{2}\left(\frac{a^{2}t}{1+a^{2}t}\right)^{2}\left(\frac{1}{1+t/d}\right)^{2},$
(7)

with $a=111Z^{-1/3}/m_{e}$ and $d=0.164\,\mbox{GeV}^{2}A^{-2/3}$. The
contribution to the cross-section associated with squared inelastic form-
factor is proportional to $Z$, therefore for heavy nucleus
$Z\sim\mathcal{O}(100)$ that term is negligible. In the case of NA64$\mu$, the
active target is made of lead $(A=207,Z=82)$ being relevant only to the
elastic form-factors giving the following typical momenta transfer associated
with screening effects and nucleus size respectively

$\sqrt{t_{a}}=1/a\simeq 2\cdot 10^{-5}\,\mbox{GeV},$ (8)
$\sqrt{t_{d}}=\sqrt{d}\simeq 6.7\cdot 10^{-2}\,\mbox{GeV}.$ (9)

We define the energy fraction of the $Z^{\prime}$-boson as
$x=E_{Z^{\prime}}/E_{\mu}$ and the angle between ${\bf k}$ and ${\bf p}$ as
$\theta_{Z^{\prime}}$. One can express the differential cross-section in the
following form [14]

$\frac{d\sigma}{dx\,d\cos\theta_{Z^{\prime}}}=\frac{\epsilon^{2}\alpha^{3}|{\bf
k}|E_{\mu}}{|{\bf p}||{\bf
k-p}|}\cdot\int\limits_{t_{min}}^{t_{max}}\frac{dt}{t^{2}}\,G_{2}^{el}(t)\cdot\int\limits_{0}^{2\pi}\frac{d\phi_{q}}{2\pi}\,\frac{|\mathcal{A}^{2\rightarrow
3}_{Z^{\prime}}|^{2}}{8M^{2}},$ (10)

where $\epsilon$ is $Z^{\prime}$ coupling to muons, which is related to
$g^{\prime}$ as follows $\epsilon=g^{\prime}/\sqrt{4\pi\alpha}$, here
$\alpha=1/137$ is fine-structure constant, $t_{\text{min}}$ and
$t_{\text{max}}$ are the values of minimum and maximum momentum transfer
respectively. The quantities for $t_{\text{min}}$ and $t_{\text{max}}$ are
derived explicitly in Ref. [14]. The production amplitude squared for vector
dark boson such as $Z^{\prime}$ is calculated by using FeynCalc tools [19, 20]
for the Wolfram Mathematica package [21]

$\begin{gathered}|\mathcal{A}^{2\rightarrow
3}_{Z^{\prime}}|^{2}=\frac{2}{\tilde{s}^{2}\tilde{u}^{2}}\Bigl{(}+\tilde{s}\tilde{u}\\{\mathcal{P}^{2}[(\tilde{s}+t)^{2}+(\tilde{u}+t)^{2}]-4t[(\mathcal{P}\cdot
p)^{2}+(\mathcal{P}\cdot p^{\prime})^{2}]\\}\\\ \\\
+2m_{\mu}^{2}\\{\mathcal{P}^{2}t(\tilde{s}+\tilde{u})^{2}-4[(\mathcal{P}\cdot
p)\tilde{u}+(\mathcal{P}\cdot
p^{\prime})\tilde{s}]^{2}\\}+m_{Z^{\prime}}^{2}\\{\mathcal{P}^{2}t(\tilde{s}-\tilde{u})^{2}-4[(\mathcal{P}\cdot
p)\tilde{u}+(\mathcal{P}\cdot
p^{\prime})\tilde{s}]^{2}\\}\Bigr{)},\end{gathered}$ (11)

where the Mandelstam variables and relevant dot products are

$\tilde{s}=(p^{\prime}+k)^{2}-m_{\mu}^{2}=2(p^{\prime}\cdot
k)+m_{Z^{\prime}}^{2},\qquad\tilde{u}=(p-k)^{2}-m_{\mu}^{2}=-2(p\cdot
k)+m_{Z^{\prime}}^{2},$ (12) $\mathcal{P}^{2}=4M^{2}+t,\qquad\mathcal{P}\cdot
p=2ME_{\mu}-(\tilde{s}+t)/2,\qquad\mathcal{P}\cdot
p^{\prime}=2M(E_{\mu}-E_{Z^{\prime}})+(\tilde{u}-t)/2.$ (13)

The resulting amplitude squared (11) coincides with the one given in Ref. [14]
for the case considered here, replacing the incident electron with a muon.

##  III The approximations for the $Z^{\prime}$ emission cross-sections

To calculate the differential $Z^{\prime}$ emission cross-sections in the
muon-nuclei interactions we can use the so called Weiszäcker-Williams
approximation by assuming that the energy of the initial particle is much
higher than $m_{\mu}$ and $m_{Z^{\prime}}$. In this case, the flux of
generated virtual photons can be considered as a plane wave and be
approximated by a real photon. This approximation allows thus to reduce the
phase space of a $\mu(p)+N(\mathcal{P}_{i})\to
Z^{\prime}(k)+\mu(p^{\prime})+N(\mathcal{P}_{f})$ process to a Compton-like
$\mu(p)+\gamma(q)\to Z^{\prime}(k)+\mu(p^{\prime})$ process [22, 15, 6]. In
particular, following the procedure described in Appendix D of [22], which
generalizes the classical WW one photon exchange process, we write down the
following expression

$\frac{d\sigma(p+\mathcal{P}_{i}\to
k+p^{\prime}+\mathcal{P}_{f})}{d(pk)d(k\mathcal{P}_{i})}\Big{|}_{WW}=\frac{\alpha\chi}{\pi(p^{\prime}\mathcal{P}_{i})}\cdot\frac{d\sigma(p+q\to
k+p^{\prime})}{d(pk)}\Big{|}_{t=t_{min}}.$ (14)

For ultra-relativistic incident muons we take into account that in the
laboratory frame $d(k\mathcal{P}_{i})=MdE_{Z^{\prime}}$,
$(p^{\prime}\mathcal{P}_{i})=ME_{\mu^{\prime}}$, $d(pk)\simeq-|{\bf p}||{\bf
k}|d\cos\theta_{Z^{\prime}}$, $|{\bf
k}|=\sqrt{E_{Z^{\prime}}^{2}-m_{Z^{\prime}}^{2}}$ and $E_{\mu^{\prime}}\simeq
E_{\mu}-E_{Z^{\prime}}$. The WW approach implies that the virtuality $t$ has
its minimum $t_{min}$ when ${\bf q}$ is collinear with ${\bf k}-{\bf p}$. For
the sake of simplicity we denote the cross-section on the left side of Eq.(14)
as $d\sigma_{2\to 3}$ and in the right side as $d\sigma_{2\to 2}$. We obtain
the following expression for the double differential cross-section:

$\frac{d\sigma_{2\to
3}}{dx\,d\cos\theta_{Z^{\prime}}}\Big{|}_{WW}\simeq\frac{\alpha\chi}{\pi(1-x)}\cdot
E_{\mu}^{2}x\beta_{Z^{\prime}}\cdot\frac{d\sigma_{2\to
2}}{d(pk)}\Big{|}_{t=t_{min}},$ (15)

where $\beta_{Z^{\prime}}=\sqrt{1-m_{Z^{\prime}}^{2}/(x^{2}E_{\mu}^{2})}$. It
is worth mentioning that the authors of Ref. [23] made a typo in the
$\beta_{Z^{\prime}}$ definition. We take into account the $x$ dependence in
$\beta_{Z^{\prime}}$ as in Ref. [14], which plays an important role for $x\ll
1$. The effective photon flux $\chi$ in the WW approach is defined by

$\chi^{WW}=\int\limits_{t_{min}}^{t_{max}}dt\frac{t-t_{min}}{t^{2}}F^{2}(t),$
(16)

where $t_{min}\approx U^{2}(x,\theta_{Z^{\prime}})/(4E_{\mu}^{2}(1-x)^{2})$
and $t_{max}=m_{Z^{\prime}}^{2}+m_{\mu}^{2}$ are the minimum and maximum
squared momentum transfer to the nucleus. The expression for $t_{min}$ is
derived below (for details, see e. g. Eq. (21) and its discussion) and the
function $U(x,\theta_{Z^{\prime}})$ is defined by

$U\equiv 2(E_{Z^{\prime}}E_{\mu}-|{\bf k}||{\bf
p}|\cos\theta_{Z^{\prime}})-m_{Z^{\prime}}^{2}\simeq
E_{\mu}^{2}\theta^{2}_{Z^{\prime}}x+m_{Z^{\prime}}^{2}(1-x)/x+m_{\mu}^{2}x.$
(17)

In Eq. (17) we keep only leading terms in
$m_{Z^{\prime}}^{2}/E_{Z^{\prime}}^{2}$, $m_{\mu}^{2}/E_{\mu^{\prime}}^{2}$
and $\theta_{Z^{\prime}}^{2}$. Considering the subsequent identities for the
momentum squared

$M^{2}=(\mathcal{P}_{i}-q)^{2},\qquad
q^{2}=(\mathcal{P}_{i}-\mathcal{P}_{f})^{2}=2Mq_{0},$ (18)

implies that the expression for the typical energy transferred to the nucleus
can be expressed as:

$q_{0}\simeq-|{\bf q}|^{2}/(2M).$ (19)

This value is negligible and will be ignored in the calculation below. One can
easily obtain the expression $|{\bf k}-{\bf p}|\simeq E_{\mu}(1-x)$ by taking
into account the approximation discussed above. The next step is to consider
the kinematic identity for the four-momentum squared of the outgoing muon:

$(p^{\prime})^{2}=(q-k+p)^{2}\equiv m_{\mu}^{2}.$ (20)

For $|{\bf q}|^{2}\ll U$ and $|{\bf q}|^{2}\ll|{\bf k}-{\bf p}|^{2}$ Eq. (20)
implies the relation between the absolute value of nucleus momentum transfer
and other kinematic variables

$|{\bf q}|\simeq U/(2|{\bf k}-{\bf p}|)\simeq U/(2E_{\mu}(1-x)).$ (21)

Note that in Eq. (21) we take into account that the vectors ${\bf q}$ and
${\bf k}-{\bf p}$ are collinear. Therefore, for WW approximation we get the
expression for the minimum nucleus momentum transfer, $t_{min}\simeq|{\bf
q}|^{2}$.

The final ingredient to obtain the differential cross-section is the
$\sigma_{2\rightarrow 2}$ calculation. The differential cross-section in the
Lorentz invariant notations [23] is:

$\frac{d\sigma_{2\to 2}}{d(pk)}=\frac{d\sigma_{2\to
2}}{d(pp^{\prime})}=\frac{2\pi\alpha^{2}\epsilon^{2}}{\tilde{s}}\cdot|\mathcal{A}_{2\to
2}^{Z^{\prime}}|^{2},$ (22)

where [14]

$|\mathcal{A}_{2\to
2}^{Z^{\prime}}|^{2}=-2\frac{\tilde{s}}{\tilde{u}}-2\frac{\tilde{u}}{\tilde{s}}+4(m_{Z^{\prime}}^{2}+2m_{\mu}^{2})\left[\left(\frac{\tilde{s}+\tilde{u}}{\tilde{s}\tilde{u}}\right)^{2}m_{\mu}^{2}-\frac{t_{2}}{\tilde{s}\tilde{u}}\right].$
(23)

where the Mandelstam variables are defined as follows

$\begin{gathered}\tilde{u}=(p-k)^{2}-m_{\mu}^{2},\\\ \\\
\tilde{s}=(p^{\prime}+k)^{2}-m_{\mu}^{2},\\\ \\\
t_{2}=(p-p^{\prime})^{2}.\end{gathered}$ (24)

We note that authors of Ref. [23] dropped terms in (23) which are associated
with a squared incident particle mass (the electron mass term in Ref. [23]).
As explained in [14], dropping these terms will lead to an underestimation of
the experiment sensitivity region. Moreover, when the incident particle is a
muon, one can not drop the corresponding terms [14] since the underestimation
of the available parameter space will be significant as soon as
$m_{Z^{\prime}}\lesssim m_{\mu}$. Finally by using the relation for the scalar
product $(qk)\simeq Ux/(2(1-x))$ and the approximate expression
$\tilde{s}+t_{2}+\tilde{u}\simeq m_{Z^{\prime}}^{2}$ one can easily get

$\tilde{u}\simeq-U,\quad\tilde{s}\simeq U/(1-x),\qquad t_{2}\simeq-
xU/(1-x)+m_{Z^{\prime}}^{2}.$ (25)

We use Eqs. (25) below in Sections III.1 and III.2 to derive the double
differential cross-section in both WW and IWW approach.

###  III.1 Cross-section in WW approximation

The photon flux can be integrated numerically according to Eq. (16). However,
it is instructive and less computational demanding to develop an analytical
expression for this equation by using the specific form of elastic form-
factor, Eq. (7). In this case, one can get the following analytical expression
for the effective photon flux

$\chi=Z^{2}[\tilde{\chi}(t_{max})-\tilde{\chi}(t_{min})],$ (26)

where

$\tilde{\chi}(t)=\frac{t_{d}^{2}}{(t_{a}-t_{d})^{3}}\Bigl{[}\frac{(t_{a}-t_{d})(t_{a}+t_{min})}{t+t_{a}}+\frac{(t_{a}-t_{d})(t_{d}+t_{min})}{t+t_{d}}+(t_{a}+t_{d}+2t_{min})\log\left(\frac{t+t_{d}}{t+t_{a}}\right)\Bigr{]},$
(27)

which is valid for both WW and IWW approaches. Here $t_{a}$ and $t_{d}$ are
defined by Eq. (8) and Eq. (9) respectively. However, in this subsection we
specify the photon flux for the case of double differential WW cross-section,
that simply implies that $t_{min}\approx
U^{2}(x,\theta_{Z^{\prime}})/(4E_{\mu}^{2}(1-x)^{2})$. In particular, one has
the following expression for the WW cross-section of $Z^{\prime}$ boson

$\left(\frac{d\sigma_{2\to
3}}{dx\,d\cos\theta_{Z^{\prime}}}\right)_{WW}=2\epsilon^{2}\alpha^{3}\sqrt{x^{2}-\frac{m_{Z^{\prime}}^{2}}{E_{\mu}^{2}}}E^{2}_{\mu}(1-x)\frac{\chi^{WW}}{\tilde{u}^{2}}|\mathcal{A}^{Z^{\prime}}_{2\to
2}|^{2}.$ (28)

By substituting Eqs. (17) and (25) into Eq. (23), the amplitude squared can be
rewritten as [14]

$|\mathcal{A}^{Z^{\prime}}_{2\to
2}|^{2}=2\frac{2-2x+x^{2}}{1-x}+4(m_{Z^{\prime}}^{2}+2m_{\mu}^{2})\frac{\tilde{u}x+m_{Z^{\prime}}^{2}(1-x)+m_{\mu}^{2}x^{2}}{\tilde{u}^{2}}.$
(29)

Now let us specify the phase space for the produced $Z^{\prime}$. In
particular, Eq. (28) implies that $x_{min}\equiv
m_{Z^{\prime}}/E_{\mu}\lesssim x$. The nuclear transfer energy is given by Eq.
(19), so that it can be neglected for the large $Z^{\prime}$ energy fraction
bound $x\lesssim 1$. Therefore one can immediately get the following upper
bound on $x\lesssim x_{max}\equiv 1-m_{\mu}/E_{\mu}$.

###  III.2 Calculations using IWW approximation

In the so-called improved WW approach the dependence of $t_{min}$ on $x$ and
$\theta_{Z^{\prime}}$ in the flux derivation (Eq. (16)) is omitted to simplify
calculations, which is important in Monte-Carlo (MC) simulations, such that
$t^{IWW}_{min}\simeq m_{Z^{\prime}}^{4}/(4E_{\mu}^{2})$. This means however
that the IWW approach is less accurate [14].

We compute the differential cross-section using the IWW approximation to
compare with the approach developed in the previous section. The integration
over $\theta_{Z^{\prime}}$ in this case can be performed analytically (see,
e.g. Eq. (30) in Ref. [14]). In particular, one has

$\left(\frac{d\sigma_{2\to
3}}{dx}\right)_{IWW}=2\epsilon^{2}\alpha^{3}\chi^{IWW}\sqrt{x^{2}-\frac{m_{Z^{\prime}}^{2}}{E_{\mu}^{2}}}\left[\frac{m_{\mu}^{2}x(-2+2x+x^{2})-2(3-3x+x^{2})\tilde{u}}{3x\tilde{u}^{2}}\right]\Bigr{|}^{\tilde{u}_{max}}_{\tilde{u}_{min}}$
(30)

where $\tilde{u}_{max}=-m_{Z^{\prime}}^{2}\frac{1-x}{x}-m_{\mu}^{2}x$ and
$\tilde{u}_{min}=-xE_{\mu}^{2}(\theta^{max}_{Z^{\prime}})^{2}-m_{Z^{\prime}}^{2}\frac{1-x}{x}-m_{\mu}^{2}x$.
In contrast to Ref. [14], we keep $\tilde{u}_{min}$ finite to take into
account the effects associated with the maximal angle of emitted $Z^{\prime}$.
As discussed before, in the IWW approximation $\chi$ is independent of $x$ and
$\theta_{Z^{\prime}}$. The resulting IWW cross-section has a sharp peak at
$x\simeq 1$, as illustrated in Fig. 1. The numerical calculations reveal that
this peak is smeared in WW approach due to the fact that $\chi$ is a function
of $x$ and $\theta_{Z^{\prime}}$ (see the green line in the right plot of Fig.
1).

The comparison between the differential cross-section as a function of $x$
calculated using the three approaches for different $Z^{\prime}$ masses is
shown in Fig. 1. The muon energy has been set to 160 GeV as expected in
NA64$\mu$ experiment. We note that the typical angle of $Z^{\prime}$ emission
does not depend on its mass as it scales as $\theta_{Z^{\prime}}\sim
m_{\mu}/E_{\mu}$. This can be seen in Fig. 3, left plot. Thus, the
differential cross-sections (see e.g. Eqs. (30), (32) and (10)) are
numerically integrated using respectively adaptive and Monte-Carlo procedures
[24] up to a maximum $Z^{\prime}$ angle $\theta^{max}_{Z^{\prime}}\sim 0.1$.

The values obtained for the differential cross-section are different depending
on the mass because of the dynamic conditions. In the right plots of Fig. 1,
the relative difference between the approximations with respect to the the
exact-tree-level calculation is shown. The differences between WW and the
exact calculation are below 2$\%$ in the mass region between $10$ MeV and $1$
GeV. Deviations from exact cross-sections are significant in the case of IWW
approximation, especially in the extreme fractional energy regions, due to the
flux simplification.

Figure 1: (Left) Differential cross-section as a function of $x$ for IWW (blue
dashed line), WW (green line) and ETL (red dotted line) for different
$Z^{\prime}$ masses. The single- and double-differential cross-sections (see
e.g. Eqs. (30), (32) and (10)) are numerically integrated using respectively
adaptive and Monte-Carlo procedures [24]. (Right) Relative error between WW
and IWW approximations as a function of $x$ defined as
$(\mathcal{O}_{approx}-\mathcal{O}_{exact})/\mathcal{O}_{exact}$ (similar
approach than the one used in [14]).

In Fig. 2 the evolution of the total cross-section, calculated with the
different approaches, is presented as a function of the initial muon energy
for different $Z^{\prime}$ masses. The WW approximation reproduces the ETL
cross-section with a relative difference below 2$\%$ for the whole mass range.
For illustration, the energy region interesting for the NA64$\mu$ experiment
is highlighted in green. In this region, the relative differences between IWW
and ETL (Sec. II) are smaller than in the low energy region, where they tend
to diverge, when the muon energy approaches the $Z^{\prime}$ mass.

Figure 2: (Left) Total cross-section as a function of the beam energy for IWW
(blue dashed line), WW (green line) expressions and ETL (red dotted line)
integrated over $x,\ \theta$. (Right) Relative error between WW, IWW
approximations and ETL as a function of the beam energy expressed as
$(\mathcal{O}_{approx}-\mathcal{O}_{exact})/\mathcal{O}_{exact}$. The typical
NA64$\mu$ beam energy range is shown as a green band.

###  III.3 Zero mass limit for WW approach

An independent cross-check to validate our calculations, is to consider the
case of massless $Z^{\prime}$, $(m_{Z^{\prime}}\rightarrow 0)$, since this
should reproduce the muon nuclear bremsstrahlung $\mu N\to\mu N\gamma$ cross-
section [25]. We have used a simplified analytical expression of the flux for
these calculations valid for the parameter space of interest $t_{max}\gg
t_{d}\gg t_{min}$ and $t_{d}\gg t_{a}$ with logarithmic accuracy:

$\frac{\chi}{Z^{2}}\simeq\log\left(\frac{t_{d}}{t_{min}+t_{a}}\right)-2.$ (31)

We note that this expression is no longer valid as $t_{min}\simeq t_{d}$,
which implies large masses $\mathcal{O}(1)$ GeV, low energies $E_{\mu}\lesssim
10$ GeV and $x\simeq 1$. However, for the interesting NA64$\mu$ parameter
space, one can use the approximated formula for the calculation of the
analytical differential cross-section $(d\sigma/dx)_{WW}$.

In particular, the numerical calculation reveals that for
$m_{A^{\prime}}\lesssim 500$ MeV and $E_{\mu}\simeq 150-160$ GeV the
discrepancy between the cross-sections calculated with (31) and (26) is well
below $1\%$. We derive the approximate analytical formula for $Z^{\prime}$
production cross-section using WW approximation for this particular case
considering only an elastic nuclear form-factor. The first step is to
integrate out $\theta_{Z^{\prime}}$ in Eq. (29). We obtain

$\left(\frac{d\sigma}{dx}\right)_{WW}=\epsilon^{2}\alpha^{3}\sqrt{x^{2}-m_{Z^{\prime}}^{2}/E_{\mu}^{2}}\,\,\frac{(1-x)}{x}\int\limits^{\tilde{u}_{max}}_{\tilde{u}_{min}}d\tilde{u}\frac{|\mathcal{A}^{Z^{\prime}}_{2\to
2}|^{2}}{\tilde{u}^{2}}\chi^{WW}.$ (32)

Finally, we get

$\left(\frac{d\sigma}{dx}\right)_{WW}=\epsilon^{2}\alpha^{3}Z^{2}\sqrt{x^{2}-m_{Z^{\prime}}^{2}/E_{\mu}^{2}}\,\,\frac{(1-x)}{x}\times\left\\{2\frac{2-2x+x^{2}}{1-x}I_{2}+4(m_{Z^{\prime}}^{2}+2m_{\mu}^{2})\left[xI_{3}+(m_{Z^{\prime}}^{2}(1-x)+m_{\mu}^{2}x^{2})I_{4}\right]\right\\}.$
(33)

where the auxiliary integrals are

$I_{n}=\int\limits^{\tilde{u}_{max}}_{\tilde{u}_{min}}d\tilde{u}\,\frac{1}{\tilde{u}^{n}}\log\left[\frac{\beta_{d}^{2}}{\tilde{u}^{2}+\beta_{a}^{2}}\right],\quad\mbox{for}\quad
n=2,3,4.$ (34)

In this expression $\beta_{d}^{2}=4E_{\mu}^{2}(1-x)^{2}t_{d}/e^{2}$, being
$e\simeq 2.71828$ the Euler’s number,
$\beta_{a}^{2}=4E_{\mu}^{2}(1-x)^{2}t_{a}$ and
$\tilde{u}_{min}=-x(\theta^{max}_{Z^{\prime}})^{2}E_{\mu}^{2}-m_{Z^{\prime}}^{2}\frac{1-x}{x}-m_{\mu}^{2}x$.
The resulting integrals for different $n$ values are:

$I_{2}=\left[-\frac{1}{\tilde{u}}\log\left(\frac{\beta_{d}^{2}}{\beta_{a}^{2}+\tilde{u}^{2}}\right)-\frac{2}{\beta_{a}}\arctan\left(\frac{\tilde{u}}{\beta_{a}}\right)\right]\Bigg{|}^{\tilde{u}_{max}}_{\tilde{u}_{min}},$
(35)
$I_{3}=\left[-\frac{1}{2\tilde{u}^{2}}\log\left(\frac{\beta_{d}^{2}}{\beta_{a}^{2}+\tilde{u}^{2}}\right)+\frac{1}{2\beta_{a}^{2}}\log\left(\frac{\beta_{a}^{2}+\tilde{u}^{2}}{\tilde{u}^{2}}\right)\right]\Bigg{|}^{\tilde{u}_{max}}_{\tilde{u}_{min}},$
(36)
$I_{4}=\left[-\frac{1}{3\tilde{u}^{3}}\log\left(\frac{\beta_{d}^{2}}{\beta_{a}^{2}+\tilde{u}^{2}}\right)+\frac{2}{3\tilde{u}\beta_{a}^{2}}+\frac{2}{3\beta_{a}^{3}}\arctan\left(\frac{\tilde{u}}{\beta_{a}}\right)\right]\Bigg{|}^{\tilde{u}_{max}}_{\tilde{u}_{min}}.$
(37)

In Fig. 4 the differential cross-sections for the three approaches in the low
$Z^{\prime}$ mass region are compared to the cross section obtained for the
QED muon bremsstrahlung process (purple dotted line). The shape of the
distributions is identical, and the relative differences of the WW and ETL
distributions with respect to the muon nuclear bremsstrahlung are below 2$\%$.

Figure 3: Differential cross-sections in the WW approach as a function of
$\theta$ (Left) and $\psi$ (Right), peaking respectively around
$\theta_{Z^{\prime}}\simeq m_{\mu}/E_{\mu}$ and $\psi_{\mu^{\prime}}\simeq
m_{Z^{\prime}}/E_{\mu}$.

Figure 4: (Left) Differential cross-sections as a function of $Z^{\prime}$
energy fraction $x$ calculated at ETL and in WW and IWW approaches for
$m_{Z^{\prime}}=0$ and $\epsilon=1$. Muon nuclear bremsstrahlung $\mu N\to\mu
N\gamma$ cross-section [26] is shown as a purple dotted line. One can see that
as soon as $m_{Z^{\prime}}\to 0$ the shape of the muon bremsstrahlung $\mu
N\to\mu N\gamma$ is reproduced by $\mu N\to\mu NZ^{\prime}$. (Right) Relative
error defined as
$(\mathcal{O}_{m=0}-\mathcal{O}_{brem.})/\mathcal{O}_{brem.}$. The angle of
$Z^{\prime}$ emission is integrated in the range
$\theta_{Z^{\prime}}\in[0,0.1]$.

##  IV Differential cross-section for the scattered muon

For the successful detection of $Z^{\prime}$ in fixed target experiments such
as NA64$\mu$, it is crucial to tag the scattered muon after $Z^{\prime}$
emission. In this section, we focus on calculating the differential cross
section as a function of the scattered muon angle $\psi_{\mu^{\prime}}$ to
evaluate its possible impact in the expected signal yield. In the WW
approximation, the differential cross-section for the process
$\mu(p)+N(\mathcal{P}_{i})\to
Z^{\prime}(k)+\mu(p^{\prime})+N(\mathcal{P}_{f})$ is:

$\frac{d\sigma}{d\cos\psi_{\mu^{\prime}}dy}=\frac{\alpha\chi}{\pi}\frac{E_{\mu}^{2}y\beta_{\mu^{\prime}}}{(1-y)}\times\frac{d\sigma_{2\to
2}}{d(pp^{\prime})}\Big{|}_{t=t_{min}},$ (38)

where $y=E_{\mu}^{\prime}/E_{\mu}$ and
$\beta_{\mu^{\prime}}=\sqrt{1-m_{\mu}^{2}/(E_{\mu}^{{}^{\prime}2})}$ are the
fraction energy and the velocity of the outgoing muon respectively. One can
obtain the following differential cross-section of the Compton-like process
$e\gamma\to eZ^{\prime}$, which is written in Lorentz-invariant form

$\frac{d\sigma_{2\to
2}}{d(pp^{\prime})}\Big{|}_{t=t_{min}}=\frac{4\pi\alpha^{2}\epsilon^{2}}{\tilde{s}^{2}}\times\left(-\frac{\tilde{s}}{\tilde{u}}-\frac{\tilde{u}}{\tilde{s}}+2(m_{Z^{\prime}}^{2}+2m_{\mu}^{2})\left[\left(\frac{\tilde{s}+\tilde{u}}{\tilde{s}\tilde{u}}\right)^{2}m_{\mu}^{2}-\frac{t_{2}}{\tilde{s}\tilde{u}}\right]\right).$
(39)

The WW approach implies that the cross-section (39) is calculated for the
minimum value of the virtuality $t_{min}$, i. e. when ${\bf q}$ is collinear
with ${\bf p}^{\prime}-{\bf p}$. The later condition in association with the
relation $k^{2}\equiv m_{Z^{\prime}}^{2}=(q-(p^{\prime}-p))^{2}$ yields

$2|{\bf q}||{\bf p}^{\prime}-{\bf p}|+t_{2}\simeq m_{Z^{\prime}}^{2}.$ (40)

One can write down the relation $|{\bf p}^{\prime}-{\bf p}|\simeq
E_{\mu}(1-y)$, which is valid as soon as $m_{\mu}\ll E_{\mu},E_{\mu^{\prime}}$
and $\psi_{\mu^{\prime}}\ll 1$. Therefore it yields

$t_{min}=|{\bf q}|^{2},\qquad|{\bf
q}|=\frac{m_{Z^{\prime}}^{2}-t_{2}}{2E_{\mu}(1-y)}.$ (41)

We also use below the following notation for the auxiliary Mandelstam variable
$\tilde{t}=m_{Z^{\prime}}^{2}-t_{2}$. The Mandelstam variables $\tilde{s}$,
$\tilde{u}$ and $t_{2}$ can be written then as

$\begin{gathered}\tilde{s}=(p^{\prime}+k)^{2}-m_{\mu}^{2}=\tilde{t}/(1-y),\\\
\\\ \tilde{u}=(p^{\prime}-q)^{2}-m_{\mu}^{2}\simeq-2(p^{\prime}q)\simeq 2|{\bf
p}^{\prime}||{\bf q}|\cos\theta_{p^{\prime}q}\simeq-2|{\bf p}^{\prime}||{\bf
q}|\simeq-y\,\tilde{t}/(1-y),\\\ \\\
t_{2}=-[E_{\mu}^{2}\psi^{2}_{\mu^{\prime}}y+m_{\mu}^{2}(1-y)/y+m_{\mu}^{2}y]+m_{\mu}^{2}.\end{gathered}$
(42)

In the WW approach the effective photon flux $\chi$ in (38) is defined in (16)
with $t_{min}$ to be calculated using (41) and
$t_{max}=m_{Z^{\prime}}^{2}+m_{\mu}^{2}$. Finally, the differential cross-
section as a function of the muon fractional energy $y$ and the recoil muon
angle $\psi_{\mu^{\prime}}$ (Eq. (38), can be written as follows

$\frac{d\sigma}{d\cos\psi_{\mu^{\prime}}dy}=8\alpha^{3}\epsilon^{2}\chi^{WW}E_{\mu}^{2}\sqrt{y^{2}-m_{\mu}^{2}/E_{\mu}^{2}}\cdot\frac{(1-y)}{\tilde{t}^{2}}\cdot\left[\frac{1}{2y}+\frac{y}{2}+\left(m_{Z^{\prime}}^{2}+2m_{\mu}^{2}\right)\frac{(1-y)^{2}}{\tilde{t}^{2}y}\left(m_{\mu}^{2}\frac{(1-y)^{2}}{y}+m_{Z^{\prime}}^{2}-\tilde{t}\right)\right].$
(43)

The next step is to define our integration limits. One can see from Eq. (43)
that the minimal muon fractional energy after emission is $y_{min}\simeq
m_{\mu}/E_{\mu}$, when almost all energy of the initial muon is transferred to
$Z^{\prime}$. On the other hand, when the muon transfers close to zero energy
to $Z^{\prime}$ we have $y_{max}\simeq 1-m_{Z^{\prime}}/E_{\mu}$.

Fig. 5 and 6 show the results of the numerical integration of the double-
differential cross-section with respect to the muon fractional energy and the
recoil angle $\psi_{\mu^{\prime}}$ for different $Z^{\prime}$ masses in WW and
IWW approximations. The difference between the two approximations becomes
significant for large recoil muon angles and for large $y$ values due to the
flux simplification used in IWW approximation described in the previous
section. The behaviour of the differential cross-section as a function of $y$,
reproduces what we observe as a function of $x$ taking into account that
$y=1-x$ (see Fig. 1).

Figure 5: (Left) Differential cross-section as a function of
$\psi_{\mu^{\prime}}$ for IWW (blue dashed line) and WW (green line) for
different $Z^{\prime}$ masses. (Right) Relative error between IWW and WW
defined as $(\mathcal{O}_{IWW}-\mathcal{O}_{WW})/\mathcal{O}_{WW}$.

Figure 6: (Left) Differential cross-section as a function of $y$ for IWW (blue
dashed line) and WW (green line) for different $Z^{\prime}$ masses. (Right)
Relative error between IWW and WW defined as
$(\mathcal{O}_{IWW}-\mathcal{O}_{WW})/\mathcal{O}_{WW}$. The angle of muon
deflection is integrated in the range $\psi\in[0,0.1]$.

The total cross-sections in the Weiszäcker-Williams approximation for both
sets of variables, $(x,\theta_{Z^{\prime}})$ and $(y,\psi_{\mu^{\prime}})$
have been calculated as an independent cross-check. The results are compared
in Fig. 7 as a function of the beam energy. One can see from Fig. 3 that the
typical angle of $Z^{\prime}$ emission is constant, it depends only on the
energy of incoming muon $E_{\mu}\simeq 160$ GeV as $\theta_{Z^{\prime}}\simeq
m_{\mu}/E_{\mu}$. This implies that the total cross-section calculated for
$Z^{\prime}$ depends weakly on the angle cut $\theta^{max}_{Z^{\prime}}$, as
long as $\theta^{max}_{Z^{\prime}}\gg m_{\mu}/E_{\mu}\sim 4\times 10^{-4}$. On
the other hand, the typical muon deflection angle is a function of the
$Z^{\prime}$ mass, scaling as $\psi_{\mu^{\prime}}\simeq
m_{Z^{\prime}}/E_{\mu}$ for a fixed incoming muon energy. Thus, the total
cross-section calculated for the deflected muon is sensitive to the maximum
angle if $\psi^{max}_{\mu^{\prime}}\gtrsim m_{Z^{\prime}}/E_{\mu}$. For this
reason, in this case we integrate over the full parameter space of the muon
and $Z^{\prime}$ without restricting the outgoing particles angles. In
particular, for Fig. 7 the integration ranges are
$m_{Z^{\prime}}/E_{\mu}<x<1-m_{\mu}/E_{\mu}$ and $0<\theta_{Z^{\prime}}<\pi$
and $m_{\mu}/E_{\mu}<y<1-m_{Z^{\prime}}/E_{\mu}$ and
$0<\psi_{\mu^{\prime}}<\pi$ for $Z^{\prime}$ and the muon respectively. The
relative error of both cross-sections with respect to the exact cross-section
integrated over $x$ and $\theta$ for all the masses considered here is below
2$\%$ for energies above 10 GeV.
The total cross-section derived integrating the muon fractional energy $y$ and
the muon recoil angle $\psi_{\mu^{\prime}}$ is shown in Fig. 8 in IWW and WW
approximations for both cases $\psi^{max}_{\mu^{\prime}}=0.1$ and
$\psi^{max}_{\mu^{\prime}}=\pi$. The result is compared to the exact tree-
level cross-section integrated over $x$ and $\theta_{Z^{\prime}}$. Differences
with respect to the exact values are negligible in the case of WW. However, in
the low energy region, especially for higher masses, the differences become
significant. In order not to restrict regions the scattered muon cross-section
has been integrated in a large recoil muon angle range
($\psi^{max}_{\mu^{\prime}}=\pi$) (see magenta and orange dotted lines in Fig.
8). The result is compatible with the one shown in Fig. 2.

Figure 7: (Left) Total cross-section as a function of the beam energy for WW
expressions integrated over $y,\ \psi$ (blue dashed line), $x,\ \theta$ (green
line) and ETL integrated over $x,\ \theta$ (red dotted line). (Right) Relative
error between WW approximations and ETL as a function of the beam energy
expressed as $(\mathcal{O}_{WW}-\mathcal{O}_{exact})/\mathcal{O}_{exact}$. The
typical NA64$\mu$ beam energy range is shown in green. The angles are
integrated in the ranges $\theta\in[0,\pi]$ and $\psi\in[0,\pi]$, the
benchmark mixing strength is taken to be $\epsilon=10^{-4}$.

Figure 8: (Left) Total cross-section as a function of the beam energy for IWW
(blue/magenta dashed line), WW (green/orange line) expressions and ETL (red
dotted line) integrated in the ranges $0<\psi_{\mu}<0.1$ (blue and green) and
$0<\psi_{\mu}<\pi$ (magenta and orange). (Right) Relative error between WW,
IWW approximations and ETL as a function of the beam energy expressed as
$(\mathcal{O}_{WW}-\mathcal{O}_{exact})/\mathcal{O}_{exact}$. The typical
NA64$\mu$ beam energy range is shown in green. The benchmark mixing strength
is taken to be $\epsilon=10^{-4}$.

##  V Projected sensitivities to the mixing strength

The sensitivity of the experiment to $Z^{\prime}$ is calculated according to
[10]. In particular, the expected number of $Z^{\prime}$,
$N_{Z^{\prime}}^{(\bar{\nu}\nu)}$ produced through muon bremsstrahlung and
decaying invisibly to SM neutrinos, $Z^{\prime}\rightarrow\bar{\nu}\nu$, is
given by:

$\begin{split}N_{Z^{\prime}}^{(\bar{\nu}\nu)}&=N_{MOT}\cdot\frac{\rho\mathcal{N}_{A}}{A}\cdot\sum_{i}\Delta
L_{i}\cdot\sigma^{Z^{\prime}}_{tot}(E_{\mu}^{i})\cdot
Br(Z^{\prime}\rightarrow\bar{\nu}\nu)\\\ \end{split}$ (44)

where $A$ is the atomic weight, $\mathcal{N}_{A}$ is the Avogadro’s number,
$N_{MOT}$ is the number of muons on target, $\rho$ is the target density,
$E_{\mu}^{i}$ is the muon energy at the $i$th step in the target, $\Delta
L_{i}$ is the step length of the muon path and $\sigma^{Z^{\prime}}_{tot}$ tot
is the total cross-section of the $Z^{\prime}$ production,
$Br(Z^{\prime}\rightarrow\bar{\nu}\nu)$ corresponds to the branching ratio of
$Z^{\prime}$ decaying invisibly to SM neutrinos [4].
In the NA64$\mu$ facility it is assumed to utilize two, upstream and
downstream, magnetic spectrometers allowing for precise measurements of
momenta for incident and recoiled muons, respectively [4]. The muon missing
energy signal of the reaction $\mu N\to\mu
NZ^{\prime},\,Z^{\prime}\to\bar{\nu}\nu$ is defined by a scattered muon energy
cut $E_{\mu^{\prime}}\lesssim 80$ GeV.
The projected sensitivities are then calculated at 90% C.L. for the number of
signal events, i.e. it is required that $N_{Z^{\prime}}^{(\bar{\nu}\nu)}>2.3$
events, assuming zero background. Fig. 9 shows the projected sensitivities for
both IWW and WW approaches in the plane $(m_{Z^{\prime}},\ \epsilon)$ for
$10^{12}$ 160 GeV muons on target (MOTs), together with the values of
$\epsilon$ necessary for the explanation of the $(g-2)_{\mu}$ anomaly. Those
are obtained separately through numerical integration of Eq. (44) with GSL
[24] and with robust MC simulations using the DMG4 simulation package [13]. It
can be seen that in the low mass region, $m_{Z^{\prime}}\lesssim 10$ MeV, the
relative error between IWW and WW does not exceed $10\%$, whereas it reaches
about $40\%$ at masses $m_{Z^{\prime}}\sim\mathcal{O}(1\ \text{GeV})$. It is
also worth noting that more conservative sensitivity lines are obtained with
the realistic MC simulations, as a result of simulating the full muon physics
within the active target, with about $\lesssim 10\%$ relative error with
respect to the numerically integrated results.

Figure 9: (Left) Projected sensitivity in the $(m_{Z^{\prime}},\
\epsilon=g^{\prime}/\sqrt{4\pi\alpha})$ phase space obtained through both MC
simulations with DMG4 package [13] (blue lines) and GSL [24] numerical
integration (red lines) for $Z^{\prime}$. The limits are calculated at 90%
C.L. for $10^{12}$ MOT and 160 GeV. Also shown is the $(g-2)_{\mu}$ favored
band within $\pm 2\sigma$. (Right) Relative error between IWW and WW defined
as $(\mathcal{O}_{IWW}-\mathcal{O}_{WW})/\mathcal{O}_{WW}$. Is also shown the
average relative error between MC simulations and numerical integration,
defined as
$(\mathcal{O}_{DMG4}-\mathcal{O}_{numerical})/\mathcal{O}_{numerical}$.

##  VI Summary

In this work, we have derived, based on the work of [14], the differential and
total cross-sections for dark vector boson production in fixed target
experiments through muon bremsstrahlung. We have shown that the commonly used
improved Weiszäcker-Williams approximation differs significantly from the
exact-tree-level calculations. On the other hand, the WW approach reproduces
well the cross-section at a level of $<5\%$ in the high-energy beam regime,
such as the one of NA64$\mu$ experiment. However, for very low beam energies,
where $E_{\mu}\sim m_{Z^{\prime}}$, the collinear regime is no longer valid
and the phase-space approximations fail. We have also calculated the $Z’$
double-differential cross-section as a function of new variables, namely the
scattered muon fractional energy and recoil angle, of particular importance
for Monte Carlo simulations and estimates in missing momentum experiments.
Although the ETL for those variables was not calculated, we have cross-checked
our results against both WW approximation and ETL calculations as a function
of the emitted vector boson variables and found that the newly derived WW
cross section reproduces the total cross-section with a good accuracy
($\lesssim 1\%)$. Additionally, we developed an analytical expression of the
photon flux in WW approximation to reduce computational time due to numerical
integration. Finally, our calculations were used to derive possible projected
sensitivities in missing momentum experiments with both numerical integration
and a full realistic Geant4-based MC simulation. It was found that in the high
mass region, IWW calculations differ from WW ones, and thus ETL, by as much as
40$\%$, over-estimating the sensitivity of muon beam fixed target experiments.
Our results demonstrate the potential of these experiments to explore a broad
coupling and mass parameter space region of dark vector bosons such as
$Z^{\prime}(A^{\prime})$, including the very interesting muon $(g-2)$
unexplored region.

## Acknowledgments

We acknowledge the members of the NA64 collaboration for fruitful discussions,
in particular, S. N. Gninenko, N. V. Krasnikov and E. Depero. The work of P.
Crivelli, L. Molina Bueno and H. Sieber is supported by ETH Zürich and SNSF
Grant No. 169133, 186181, 186158 and 197346 (Switzerland). The work of D. V.
Kirpichnikov on MC simulation of $Z^{\prime}$ emission is supported by the
Russian Science Foundation RSF grant 21-12-00379.

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