aainstitutetext: Department of Physics and McDonnell Center for the Space
Sciences, Washington University,
St. Louis, MO 63130, USAbbinstitutetext: Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germanyccinstitutetext: Service de
Physique Théorique, Université Libre de Bruxelles, Boulevard du Triomphe,
CP225, 1050 Brussels, Belgiumddinstitutetext: School of Physics, Southeast
University, Nanjing 211189, China

# Searching for $Z^{\prime}$ bosons at the P2 experiment

P. S. Bhupal Dev b Werner Rodejohann c Xun-Jie Xu d,a Yongchao Zhang
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The P2 experiment aims at high-precision measurements of the parity-violating
asymmetry in elastic electron-proton and electron-12C scatterings with
longitudinally polarized electrons. We discuss here the sensitivity of P2 to
new physics mediated by an additional neutral gauge boson $Z^{\prime}$ of a
new $U(1)^{\prime}$ gauge symmetry. If the charge assignment of the
$U(1)^{\prime}$ is chiral, i.e., left- and right-handed fermions have
different charges under $U(1)^{\prime}$, additional parity-violation is
induced directly. On the other hand, if the $U(1)^{\prime}$ has a non-chiral
charge assignment, additional parity-violation can be induced via mass or
kinetic $Z$-$Z^{\prime}$ mixing. By comparing the P2 sensitivity to existing
constraints, we show that in both cases P2 has discovery potential over a wide
range of $Z^{\prime}$ mass. In particular, for chiral models, the P2
experiment can probe gauge couplings at the order of $10^{-5}$ when the
$Z^{\prime}$ boson is light, and heavy $Z^{\prime}$ bosons up to 79 (90) TeV
in the proton (12C) mode. For non-chiral models with mass mixing, the P2
experiment is sensitive to mass mixing angles smaller than roughly $10^{-4}$,
depending on model details and gauge coupling magnitude.

##  1  Introduction

Why parity is violated in elementary particle interactions remains one of the
biggest mysteries in physics. Within the Standard Model (SM) of particle
physics, parity-violation is caused by the weak $Z$ and $W^{\pm}$ bosons,
which couple differently to left- and right-handed fermions. This chiral
charge assignment of the SM fermions leads to various interesting phenomena
and precision tests of the SM Zyla:2020zbs . At the same time, given the fact
that some form of beyond the SM (BSM) physics is expected on general grounds,
an interesting question is whether the new BSM interactions are parity-
conserving or parity-violating. Either way, BSM physics can influence parity-
violating observables in reactions in which it participates. Thus, parity-
violating searches provide an excellent avenue to probe BSM physics; see Ref.
Safronova:2017xyt  for a recent review.

A classic parity-violating observable can be obtained from the scattering of
polarized electrons off unpolarized targets, yielding cross-sections
$\sigma_{R}$ and $\sigma_{L}$ for right- and left-handed electrons,
respectively. The parity-violating left-right asymmetry, defined as

$A^{{\rm PV}}\ =\ \frac{\sigma_{R}-\sigma_{L}}{\sigma_{R}+\sigma_{L}}\,,$ (1)

is then a very useful probe of parity-violation; see Ref. Erler:2014fqa  for a
review. The asymmetry $A^{\rm PV}$ has been (will be) measured with high
precision in low-energy polarized electron scattering processes Souder:2015mlu
, such as Møller scattering $e^{-}e^{-}\to e^{-}e^{-}$ Anthony:2003ub ;
Anthony:2005pm  (Benesch:2014bas ), as well as electron-proton Spayde:1999qg ;
Aniol:2000at ; Aniol:2004hp ; Androic:2011rha ; Androic:2018kni
(Becker:2018ggl ), electron-deuteron Prescott:1978tm ; Prescott:1979dh ;
Ito:2003mr ; Wang:2014bba ; Wang:2014guo ; BalaguerRios:2016ftd , electron-4He
Aniol:2005zf , electron-9Be Heil:1989dz , electron-12C Souder:1990ia
(Becker:2018ggl ) and electron-208Pb Abrahamyan:2012gp  scatterings. In
addition, there are precise measurements of atomic parity violation (APV)
using 133Cs Wood:1997zq ; Bennett:1999pd ; Porsev:2009pr ; Porsev:2010de ,
205Tl Edwards:1995zz ; Vetter:1995vf , 208Pb Meekhof:1993zz , 209Bi
Macpherson:1991opp  and 100,102,104,106Yb Antypas:2018mxf . Parity-violating
asymmetries have also been measured at the high-energy colliders such as LEP,
SLC, Tevatron and LHC Zyla:2020zbs . The interplay of these various parity-
violating measurements with new physics have been discussed, e.g. in Refs.
Marciano:1990dp ; Altarelli:1991ci ; Erler:2003yk ; Kurylov:2003zh ;
Erler:2011iw ; Diener:2011jt ; Davoudiasl:2012qa ; Davoudiasl:2014kua ;
Dev:2018sel ; Long:2018fud ; Carlini:2019ksi ; DAmbrosio:2019tph ;
Arcadi:2019uif ; Ghosh:2019dmi ; Hong:2020dwz ; Sahoo:2021thl .

This paper deals with probing new physics using the measurements of the
parity-asymmetry in elastic electron-proton or electron-12C scatterings in the
proposed P2 experiment at the upcoming Mainz Energy-recovering Superconducting
Accelerator (MESA) facility Becker:2018ggl . The goal of P2, with start of
data-taking expected in 2024, is to measure the parity-violating asymmetries
for polarized electrons scattering off unpolarized protons or 12C nuclei using
a 155 MeV electron beam, where the relative uncertainties $\Delta A^{\rm
PV}/A^{\rm PV}$ are expected to be 1.4% and 0.3% respectively Becker:2018ggl .
Such precise measurements at low momentum transfer provide not only an
important test of the SM, but also a sensitive probe of BSM physics.

One natural scenario of new physics is a (light) $Z^{\prime}$ boson that
couples differently to left- and right-handed SM fermions. The $Z^{\prime}$
boson will mediate new Feynman diagrams for electron scattering off proton or
nucleus (see the bottom two panels in Fig. 1). We specifically examine how new
physics in the form of a new neutral gauge boson $Z^{\prime}$ can be
constrained in the P2 experiment. Depending on the origin of the parity-
violation in the $Z^{\prime}$ couplings, the $U(1)^{\prime}$ models
accommodating the $Z^{\prime}$ gauge boson can be classified into two
categories:

  * •

Chiral theories in which left- and right-handed particles have different
charge assignments under the $U(1)^{\prime}$. They give a direct contribution
to the parity-violating asymmetry. Some anomaly-free $U(1)^{\prime}$ examples
are given in Table 2.

  * •

Vector-like or non-chiral theories in which left- and right-handed particles
have identical charge assignments under the $U(1)^{\prime}$. They give an
indirect contribution to the parity-violating asymmetry if the $Z^{\prime}$
mixes with the SM $Z$ boson. In this paper, we will first consider a generic
$U(1)^{\prime}$ model with either mass mixing $\sin\theta$ or kinetic mixing
$\epsilon$ in the limit of the new gauge coupling $g^{\prime}\to 0$, and then
generalize to the $U(1)_{B}$ and $U(1)_{B-L}$ models (here $B$ and $L$ denote
the baryon and lepton number, respectively), with three benchmark values of
$g^{\prime}/\sin\theta=0.01$, $1$ and $10$, and $\epsilon=0$.

Our results, shown in Figs. 2 to 6, demonstrate that the P2 prospects of the
$U(1)^{\prime}$ models are rather model-dependent. However, even if all
existing constraints are taken into consideration, the P2 experiment can still
probe a wide range of $Z^{\prime}$ masses. For the three chiral models
considered in this paper, the P2 experiment can probe gauge couplings down to
$g^{\prime}\sim 10^{-5}$ when the $Z^{\prime}$ boson is light, as summarized
in Table 4. When the $Z^{\prime}$ boson mass $m_{Z^{\prime}}$ is large, the P2
experiment probes an effective cutoff scale
$\Lambda=m_{Z^{\prime}}/g^{\prime}$, which for $m_{Z^{\prime}}$ can go up to
79 TeV in the $e+p$ mode, and even up to 90 TeV in the $e+^{12}$C mode
(setting the $g^{\prime}$ to be the perturbative limit of $4\pi$), which is
well beyond direct searches at past and current high-energy colliders. For the
non-chiral models, if there is only $Z$-$Z^{\prime}$ mass mixing, in the limit
of $g^{\prime}Q^{\prime}\to 0$, the P2 prospects in the $e+p$ mode have been
precluded by APV measurements, while P2 can probe unexplored mass mixing
angles in the range of $1.1\times 10^{-4}<\sin\theta<0.15$ in $e+^{12}$C
scattering, as shown in Fig. 3 and Table 5. If there is only kinetic mixing,
the $Z^{\prime}$ boson behaves essentially like a dark photon when it is
light, and it is also severely constrained when it is heavy. Such particles
are easily accessible and often searched for. Therefore, the P2 prospects of
the kinetic mixing angle $\epsilon$ have been precluded by existing limits, as
presented in Fig. 4. For illustration, we further apply our analyses to
$U(1)_{B}$ and $U(1)_{B-L}$ models with $g^{\prime}/\sin\theta$ fixed at some
benchmark values. We find that when the gauge coupling $g^{\prime}$ is sizable
compared to the mass mixing angle $\sin\theta$, the P2 sensitivity to
$\sin\theta$ can be significantly improved. Our study also shows that whether
$e+p$ or $e+^{12}$C scattering gives better limits depends largely on the
model.

The rest of the paper is organized as follows: In Section 2 we discuss general
aspects of the parity-violating asymmetry, before discussing various chiral
and non-chiral models that modify the parity-violating asymmetry in Section 3.
A sensitivity study is performed in Section 4: with the procedure for
obtaining the sensitivities given in Section 2, all relevant existing limits
are collected in Section 4.2, and the P2 sensitivities are obtained in Section
4.3. The conclusions are presented in Section 5. More details of the axial-
vector couplings of $Z^{\prime}$ boson to proton and 12C are provided in
Appendix A.

##  2  Parity-violating asymmetry in elastic electron scattering

In this section, we derive the parity-violating asymmetry $A^{{\rm PV}}$
defined in Eq. (1) for elastic electron-proton or electron-nucleus scattering.
In the SM, the leading-order contribution can be computed by evaluating the
first two diagrams in Fig. 1. Parity-violation enters via the $Z$ boson
contribution and its interference with the parity-conserving photon diagram.

Figure 1: Leading-order processes for $A^{{\rm PV}}$ in the SM (top panels) and in the $Z^{\prime}$ models considered here (bottom panels). $N$ denotes either a proton or a nucleus (${}^{12}{\rm C}$ in the context of P2).  Table 1:  Analytical expressions for the effective couplings of $Z$ and $Z^{\prime}$ bosons to the SM quarks and charged leptons in the SM, chiral $U(1)^{\prime}$ models (from $Q_{f_{L}}\neq Q_{f_{R}}$) and non-chiral $U(1)^{\prime}$ models (from the kinetic mixing $\epsilon B^{\mu\nu}F^{\prime}_{\mu\nu}$ and mass mixing $\delta m^{2}Z^{\mu}Z^{\prime}_{\mu}$). The last two rows are the corresponding analytic formulae of $A^{{\rm PV}}$ for electron scattering off proton or 12C. The superscript “SM” indicates SM contributions. Here $g_{Z}\equiv e/s_{W}c_{W}$, $s_{W}\equiv\sin\theta_{W}$ being the weak mixing angle (and $c_{W}\equiv\cos\theta_{W}$) and we have defined $r_{m}\equiv m_{Z^{\prime}}^{2}/m_{Z}^{2}$. See the main text for details.  models | SM | chiral models | non-chiral models
---|---|---|---
$A^{{\rm PV}}$ caused by | NC | $Q^{\prime}_{e_{L}}\neq Q^{\prime}_{e_{R}}$ | $\epsilon B^{\mu\nu}F^{\prime}_{\mu\nu}$ | $\delta m^{2}Z^{\mu}Z^{\prime}_{\mu}$
$g_{e_{L}}$ | $g_{Z}\left(-\frac{1}{2}+s_{W}^{2}\right)$ | $g_{e_{L}}^{{\rm SM}}$ | $g_{e_{L}}^{{\rm SM}}+\frac{3-2r_{m}-2s_{W}^{2}}{4\left(1-r_{m}\right){}^{2}}g_{Z}s_{W}^{2}\epsilon^{2}$ | $g_{e_{L}}^{{\rm SM}}\cos\theta$
$g_{e_{R}}$ | $g_{Z}s_{W}^{2}$ | $g_{e_{R}}^{{\rm SM}}$ | $g_{e_{R}}^{{\rm SM}}+\frac{2-2r_{m}-s_{W}^{2}}{2\left(1-r_{m}\right){}^{2}}g_{Z}s_{W}^{2}\epsilon^{2}$ | $g_{e_{R}}^{{\rm SM}}\cos\theta$
$g_{u_{L}}$ | $g_{Z}\left(\frac{1}{2}-\frac{2}{3}s_{W}^{2}\right)$ | $g_{u_{L}}^{{\rm SM}}$ | $g_{u_{L}}^{{\rm SM}}-\frac{5-2r_{m}-4s_{W}^{2}}{12\left(1-r_{m}\right){}^{2}}g_{Z}s_{W}^{2}\epsilon^{2}$ | $g_{u_{L}}^{{\rm SM}}\cos\theta$
$g_{u_{R}}$ | $g_{Z}\left(-\frac{2}{3}s_{W}^{2}\right)$ | $g_{u_{R}}^{{\rm SM}}$ | $g_{u_{R}}^{{\rm SM}}-\frac{2-2r_{m}-s_{W}^{2}}{3\left(1-r_{m}\right){}^{2}}g_{Z}s_{W}^{2}\epsilon^{2}$ | $g_{u_{R}}^{{\rm SM}}\cos\theta$
$g_{d_{L}}$ | $g_{Z}\left(-\frac{1}{2}+\frac{1}{3}s_{W}^{2}\right)$ | $g_{d_{L}}^{{\rm SM}}$ | $g_{d_{L}}^{{\rm SM}}+\frac{1+2r_{m}-2s_{W}^{2}}{12\left(1-r_{m}\right){}^{2}}g_{Z}s_{W}^{2}\epsilon^{2}$ | $g_{d_{L}}^{{\rm SM}}\cos\theta$
$g_{d_{R}}$ | $g_{Z}\left(\frac{1}{3}s_{W}^{2}\right)$ | $g_{d_{R}}^{{\rm SM}}$ | $g_{d_{R}}^{{\rm SM}}+\frac{2-2r_{m}-s_{W}^{2}}{6\left(1-r_{m}\right){}^{2}}g_{Z}s_{W}^{2}\epsilon^{2}$ | $g_{d_{R}}^{{\rm SM}}\cos\theta$
$g^{\prime}_{e_{L}}$ | 0 | $Q^{\prime}_{e_{L}}g^{\prime}$ | $\frac{c_{W}^{2}-r_{m}/2}{1-r_{m}}\epsilon g_{Z}s_{W}$ | $g_{e_{L}}^{{\rm SM}}\sin\theta$
$g^{\prime}_{e_{R}}$ | 0 | $Q^{\prime}_{e_{R}}g^{\prime}$ | $\frac{c_{W}^{2}-r_{m}}{1-r_{m}}\epsilon g_{Z}s_{W}$ | $g_{e_{R}}^{{\rm SM}}\sin\theta$
$g^{\prime}_{u_{L}}$ | 0 | $Q^{\prime}_{u_{L}}g^{\prime}$ | $\frac{r_{m}-4c_{W}^{2}}{6\left(1-r_{m}\right)}\epsilon g_{Z}s_{W}$ | $g_{u_{L}}^{{\rm SM}}\sin\theta$
$g^{\prime}_{u_{R}}$ | 0 | $Q^{\prime}_{u_{R}}g^{\prime}$ | $\frac{2(r_{m}-c_{W}^{2})}{3\left(1-r_{m}\right)}\epsilon g_{Z}s_{W}$ | $g_{u_{R}}^{{\rm SM}}\sin\theta$
$g^{\prime}_{d_{L}}$ | 0 | $Q^{\prime}_{d_{L}}g^{\prime}$ | $\frac{r_{m}+2c_{W}^{2}}{6\left(1-r_{m}\right)}\epsilon g_{Z}s_{W}$ | $g_{d_{L}}^{{\rm SM}}\sin\theta$
$g^{\prime}_{d_{R}}$ | 0 | $Q^{\prime}_{d_{R}}g^{\prime}$ | $\frac{c_{W}^{2}-r_{m}}{3\left(1-r_{m}\right)}\epsilon g_{Z}s_{W}$ | $g_{d_{R}}^{{\rm SM}}\sin\theta$
$A^{{\rm PV}}$ ($p$) | Eq. (11) | Eq. (15) | Eq. (26) | Eq. (25)
$A^{{\rm PV}}$ (${}^{12}{\rm C}$) | Eq. (12) | Eq. (15) | Eq. (27) | Eq. (25)

A new neutral gauge boson $Z^{\prime}$ could also contribute to $A^{{\rm PV}}$
either directly or indirectly via mixing with the SM $Z$ boson, depending on
whether the $Z^{\prime}$ couplings to $e_{L}$ and $e_{R}$ are different or
not, respectively. This is illustrated by the diagrams in the bottom two
panels of Fig. 1. In the mixing case, we need to canonicalize kinetic terms or
diagonalize mass terms so that both $Z$ and $Z^{\prime}$ are mass eigenstates,
leading to effective parity-violating couplings of the $Z^{\prime}$. Besides,
in the presence of $Z$-$Z^{\prime}$ mixing, the $Z$ couplings may also deviate
from the SM values. This is discussed in detail in Sec. 3. In general, we can
consider the following Lagrangian, which contains the neutral current (NC)
interactions of the SM $Z$ boson and the most general interactions of a
$Z^{\prime}$ with polarized electrons and a target nucleus $N$:

$\displaystyle{\cal L}$ $\displaystyle\ \supset\ $ $\displaystyle
Z_{\mu}(g_{e_{L}}\overline{e_{L}}\gamma^{\mu}e_{L}+g_{e_{R}}\overline{e_{R}}\gamma^{\mu}e_{R})+Z{}_{\mu}(g_{V}\overline{N}\gamma^{\mu}N+g_{A}\overline{N}\gamma^{\mu}\gamma^{5}N)$
(2)
$\displaystyle\quad+Z^{\prime}_{\mu}(g^{\prime}_{e_{L}}\overline{e_{L}}\gamma^{\mu}e_{L}+g^{\prime}_{e_{R}}\overline{e_{R}}\gamma^{\mu}e_{R})+Z^{\prime}_{\mu}(g^{\prime}_{V}\overline{N}\gamma^{\mu}N+g^{\prime}_{A}\overline{N}\gamma^{\mu}\gamma^{5}N).$

Here $Z$ and $Z^{\prime}$ are mass eigenstates with masses denoted by $m_{Z}$
and $m_{Z^{\prime}}$, $g_{e_{L}}$ and $g_{e_{R}}$ ($g^{\prime}_{e_{L}}$ and
$g^{\prime}_{e_{R}}$) are the effective couplings of $Z$ ($Z^{\prime}$) to
$e_{L}$ and $e_{R}$, $g_{V}$ and $g_{A}$ ($g^{\prime}_{V}$ and
$g^{\prime}_{A}$) are vector and axial-vector couplings of $Z$ ($Z^{\prime}$)
to a nucleus, respectively. The vector couplings can be obtained by simply
adding up the fundamental couplings to quarks in a nucleus with ${\cal N}$
neutrons and ${\cal Z}$ protons as follows:

$\displaystyle g_{V}$ $\displaystyle\ =\ $ $\displaystyle{\cal
Z}\left(g_{u_{L}}+g_{u_{R}}+\frac{1}{2}g_{d_{L}}+\frac{1}{2}g_{d_{R}}\right)+{\cal
N}\left(g_{d_{L}}+g_{d_{R}}+\frac{1}{2}g_{u_{L}}+\frac{1}{2}g_{u_{R}}\right),$
(3) $\displaystyle g^{\prime}_{V}$ $\displaystyle\ =\ $ $\displaystyle{\cal
Z}\left(g^{\prime}_{u_{L}}+g^{\prime}_{u_{R}}+\frac{1}{2}g^{\prime}_{d_{L}}+\frac{1}{2}g^{\prime}_{d_{R}}\right)+{\cal
N}\left(g^{\prime}_{d_{L}}+g^{\prime}_{d_{R}}+\frac{1}{2}g^{\prime}_{u_{L}}+\frac{1}{2}g^{\prime}_{u_{R}}\right).$
(4)

Here the couplings to chiral quarks ($g_{u_{L}}$, $g_{u_{R}}$,
$g^{\prime}_{d_{R}}$, etc.) are defined in a way similar to $g_{e_{L}}$ and
$g_{e_{R}}$ in Eq. (2). In Tab. 1 we list the SM values of these couplings as
well as new physics values which will be derived later in Sec. 3.

As for the axial-vector couplings ($g_{A}$ and $g^{\prime}_{A}$), albeit not
calculable from first principles, their contributions to $A^{{\rm PV}}$ are
suppressed by electron energy over target mass, as we show in Appendix A. If
we therefore ignore the contribution of $g_{A}$ and $g^{\prime}_{A}$, the
amplitude of $e_{L,R}^{-}+N\rightarrow e_{L,R}^{-}+N$ reads

$\displaystyle i{\cal M}_{L,R}$ $\displaystyle\ \propto\ $
$\displaystyle[\overline{u_{3}}\gamma^{\mu}P_{L,R}u_{1}]\left[\overline{u_{4}}\gamma_{\mu}G_{L,R}u_{2}\right]\,,$
(5)

where $\overline{u_{3}}$ ($u_{1}$) and $\overline{u_{4}}$ ($u_{2}$) denote the
final (initial) electron and nucleon states, and

$G_{L,R}\ =\
\frac{-e^{2}}{q^{2}}+\frac{g_{e_{L},e_{R}}g_{V}}{q^{2}-m_{Z}^{2}}+\frac{g^{\prime}_{e_{L},e_{R}}g^{\prime}_{V}}{q^{2}-m_{Z^{\prime}}^{2}}$
(6)

contains contributions from the $t$-channel $\gamma$, $Z$, and $Z^{\prime}$
diagrams, respectively. Applying the standard trace technology, it is
straightforward to obtain

$\frac{|{\cal M}_{L}|^{2}}{|{\cal M}_{R}|^{2}}\ =\
\frac{G_{L}^{2}}{G_{R}^{2}}\,,$ (7)

which, according to Eq. (1), implies

$A^{{\rm PV}}\ =\ \frac{G_{R}^{2}-G_{L}^{2}}{G_{R}^{2}+G_{L}^{2}}\,.$ (8)

Now substituting Eq. (6) in Eq. (8), we obtain

$A^{{\rm PV}}\ \approx\
\frac{\left(g_{e_{L}}-g_{e_{R}}\right)g_{V}}{4\pi\alpha}\frac{Q^{2}}{m_{Z}^{2}}+\frac{\left(g^{\prime}_{e_{L}}-g^{\prime}_{e_{R}}\right)g^{\prime}_{V}}{4\pi\alpha}\frac{1}{1+m_{Z^{\prime}}^{2}/Q^{2}}\,,$
(9)

where $Q^{2}\equiv-q^{2}>0$ and $\alpha\equiv e^{2}/4\pi$ is the fine-
structure constant. In deriving Eq. (9), we have made the following
approximations:

$\frac{Q^{2}}{m_{Z}^{2}}\ \ll\ 1\,,\qquad{\rm
and}\qquad\frac{g^{\prime}_{e}g^{\prime}_{V}}{1+m_{Z^{\prime}}^{2}/Q^{2}}\
\ll\ 1\,,$ (10)

where $g^{\prime}_{e}\sim\max(g^{\prime}_{e_{L}},\ g^{\prime}_{e_{R}})$. The
exact expression of $A^{{\rm PV}}$ is lengthy and not very helpful. Within a
theory containing an additional $Z^{\prime}$ boson, once the six effective
couplings ($g_{e_{L}}$, $g_{e_{R}}$, $g_{V}$, $g^{\prime}_{e_{L}}$,
$g^{\prime}_{e_{R}}$, $g^{\prime}_{V}$) are known, $A^{{\rm PV}}$ can be
obtained using Eq. (9) and confronted with its existing or future constraints,
as exemplified in the following two sections.

We stress that Eq. (9) is obtained from the most general Lagrangian in Eq.
(2). In the limit of $g^{\prime}_{e_{L}}=g^{\prime}_{e_{R}}$,
$g_{V}^{\prime}\to 0$ or $m_{Z^{\prime}}\to\infty$, the BSM contributions are
vanishing, and taking the SM values in Tab. 1 and using Eq. (3), we can easily
reproduce the leading-order SM expressions of the parity-violating asymmetries
$A^{{\rm PV}}$ Becker:2018ggl :

$\displaystyle A_{{\rm SM}}^{{\rm PV}}(e+p)$ $\displaystyle\ \approx\ $
$\displaystyle-\frac{G_{F}Q^{2}\left(1-4s_{W}^{2}\right)}{4\sqrt{2}\pi\alpha}\,,$
(11) $\displaystyle A_{{\rm SM}}^{{\rm PV}}(e+{}^{12}{\rm C})$ $\displaystyle\
\approx\ $ $\displaystyle\frac{3\sqrt{2}G_{F}Q^{2}s_{W}^{2}}{\pi\alpha}\,,$
(12)

for electron-proton scattering (${\cal N}=0$ and ${\cal Z}=1$) and
electron-12C scattering (${\cal N}={\cal Z}=6$), respectively. Here
$G_{F}=\pi\alpha/(\sqrt{2}m_{W}^{2}s_{W}^{2})$ is the Fermi constant.

##  3 New physics effects

This section deals with obtaining the couplings of an additional $Z^{\prime}$
boson with the SM fermions in some representative $U(1)^{\prime}$ models to
study the P2 prospects. We first discuss three chiral $U(1)^{\prime}$ models
with explicit parity-violation, and then turn to parity-conserving
$U(1)^{\prime}$ models with mass or kinetic $Z$-$Z^{\prime}$ mixing.

###  3.1 Chiral $U(1)^{\prime}$ models

First let us consider new $U(1)^{\prime}$ gauge symmetries under which $e_{L}$
and $e_{R}$ have different charges. We refer to such models as chiral
$U(1)^{\prime}$ models. Anomaly cancellation usually requires the existence of
three right-handed neutrinos $\nu_{R}$ (for simplicity we have neglected the
generation indices $i=1,\,2,\,3$), and implies the following charge
assignments of chiral fermions Oda:2015gna ; Campos:2017dgc :

$\displaystyle Q_{u_{R}}^{\prime}$ $\displaystyle\ =\ x,$ $\displaystyle
Q_{d_{R}}^{\prime}$ $\displaystyle\ =\ y,$ $\displaystyle
Q_{\nu_{R}}^{\prime}$ $\displaystyle\ =\ -x-2y,$ $\displaystyle
Q_{e_{R}}^{\prime}$ $\displaystyle\ =\ -2x-y,$ $\displaystyle
Q^{\prime}_{Q_{L}}$ $\displaystyle\ =\ \frac{1}{2}(x+y),$ $\displaystyle
Q_{L_{L}}^{\prime}$ $\displaystyle\ =\ -\frac{3}{2}(x+y)\,.$ (13)

Here $Q_{L}=(u_{L},\;d_{L})^{\sf T}$ and $L_{L}=(\nu_{L},\;e_{L})^{\sf T}$ are
respectively the left-handed quark and lepton doublets, and $Q_{f}^{\prime}$
is the $U(1)^{\prime}$ charge of the chiral fermion $f$. The charges $x$ and
$y$ can be arbitrary integers or fractional numbers. However, from the model
building perspective, simple integers or fractions are preferred in order to
construct simple (low-dimensional) gauge-invariant operators. For simplicity,
we will assume that the right-handed neutrinos are heavier than
$m_{Z^{\prime}}/2$, such that the decay channel
$Z^{\prime}\to\nu_{R}\bar{\nu}_{R}$ is kinematically forbidden and the
$\nu_{R}$ do not have any effect on the P2 sensitivities.

The simplest example is perhaps the $U(1)^{\prime}_{R}$ model given in Tab. 2.
In this model, all right-handed fermions are charged under
$U(1)^{\prime}_{R}$, and none of the left-handed fermions are, i.e.
$(x,y)=(1,-1)$ in Eq. (13); see e.g. Refs. Alikhanov:2019drg ; Dutta:2019fxn ;
Jana:2019mez . One might also consider the opposite case where all left-handed
fermions are charged and none of the right-handed fermions are. Unfortunately
for such assignment it is impossible to achieve chiral anomaly cancellation.
If, for instance, $e_{R}$ is not charged under the chiral $U(1)^{\prime}$,
which corresponds to $x/y=-1/2$ in Eq. (13), then the only solution is where
the other right-handed fermions are charged for anomaly cancellation. This is
the $U(1)^{\prime}_{L}$ case shown in Tab. 2. In addition to
$U(1)^{\prime}_{R}$ and $U(1)^{\prime}_{L}$, we choose another solution with
$(x,y)=(0,1)$, where right-handed up-quarks do not couple of $Z^{\prime}$, and
refer to it as $U(1)_{X}^{\prime}$ in Tab. 2. Obviously there are many more
possibilities. Our choice of the three models summarized in Tab. 2 is
motivated by the fact that they are typical for different corners of the
parameter space to be explored at P2. For instance, the $U(1)_{R}$ model will
not contribute to the asymmetry in electron-12C scattering (cf. Eq. (17)
below).

Table 2: Quantum numbers of fermions in the three chiral $U(1)^{\prime}$ examples considered here.  model | $(x,y)$ |  $U(1)^{\prime}$ charge assignment
---|---|---
$(\nu_{L},e_{L})$ | $(\nu_{R},e_{R})$ | $(u_{L},d_{L})$ | $(u_{R},d_{R})$
$-\frac{3}{2}(x+y)$ | $-(x+2y,\ 2x+y)$ | $\frac{1}{2}(x+y)$ | $(x,\ y)$
$U(1)_{L}^{\prime}$ | $(-\frac{2}{3},\frac{4}{3})$ | $-1$ | $(-2,0)$ | $\frac{1}{3}$ | $(-\frac{2}{3},\frac{4}{3})$
$U(1)_{R}^{\prime}$ | $(1,-1)$ | $0$ | $(1,-1)$ | $0$ | $(1,-1)$
$U(1)_{X}^{\prime}$ | $(0,1)$ | $-\frac{3}{2}$ | $(-2,-1)$ | $\frac{1}{2}$ | $(0,1)$

Computing $A^{{\rm PV}}$ for chiral $U(1)^{\prime}$ models is now
straightforward. In the absence of $Z$-$Z^{\prime}$ mixing, the $Z$ couplings
are not modified and the $Z^{\prime}$ couplings in Eq. (2) are given by

$g^{\prime}_{f}\ =\ g^{\prime}Q^{\prime}_{f}\,,$ (14)

where $g^{\prime}$ is the fundamental gauge coupling of $U(1)^{\prime}$. Using
Eq. (9), we obtain

$\Delta A^{{\rm PV}}\ \equiv\ A^{{\rm PV}}-A_{{\rm SM}}^{{\rm PV}}\ \approx\
\frac{g^{\prime
2}Q^{\prime}_{N}Q^{\prime}_{LR}}{4\pi\alpha}\frac{1}{1+m_{Z^{\prime}}^{2}/Q^{2}}\,,$
(15)

where $Q^{\prime}_{LR}\equiv Q^{\prime}_{e_{L}}-Q^{\prime}_{e_{R}}$, and
$Q^{\prime}_{N}$ is either the weak charge $Q^{\prime}_{p}$ of the proton or
the weak charge $Q^{\prime}_{{}^{12}{\text{C}}}$ of 12C, with

$\displaystyle Q^{\prime}_{p}$ $\displaystyle\ \equiv\
Q^{\prime}_{u_{L}}+Q^{\prime}_{u_{R}}+\frac{1}{2}\left(Q^{\prime}_{d_{L}}+Q^{\prime}_{d_{R}}\right)\,,$
(16) $\displaystyle Q^{\prime}_{{}^{12}{\rm C}}$ $\displaystyle\ \equiv\
9\left(Q^{\prime}_{u_{L}}+Q^{\prime}_{u_{R}}+Q^{\prime}_{d_{L}}+Q^{\prime}_{d_{R}}\right)\,.$
(17)

From Tab. 2, we note that in the $U(1)^{\prime}_{R}$ model,
$Q^{\prime}_{{}^{12}{\rm C}}$ vanishes, which implies that electron-12C
scattering has no sensitivity to $Z^{\prime}$ in this model.

Here we comment on the potential effect of incoherence in $e+{}^{12}{\rm C}$
scattering and the validity of summing up the charges in Eq. (17). In the P2
experiment, $Q\approx 93\ {\rm MeV}$ is comparable to the inverse of the
${}^{12}{\rm C}$ nucleus radius and there is a considerable amount of loss of
coherence, which is usually taken into account by including a form factor
depending on the nucleon distributions. Due to the small difference between
neutron and proton distributions, two types of form factors are often
considered, namely weak and charge form factors, denoted respectively by
$F_{W}(Q^{2})$ and $F_{{\rm Ch}}(Q^{2})$. The form factor $F_{\rm Ch}(Q^{2})$
depends on the proton distribution in the nucleus, while $F_{W}(Q^{2})$ is
determined mainly by the neutron distribution, as the coupling of $Z$ to
neutrons is significantly larger than the coupling of $Z$ to protons. In
presence of the couplings of $Z^{\prime}$ to protons and neutrons, the cross
sections $\sigma_{L,\,R}$ depend on the combination of $F_{W}(Q^{2})-F_{{\rm
Ch}}(Q^{2})$. Using the Helm analytic approximation for form factors
Helm:1956zz , we find that the two form factors $F_{W,\,{\rm Ch}}(Q^{2})$ are
both around $0.75$ with percent-level uncertainties for ${}^{12}{\rm C}$ at
$Q\approx 93$ MeV. This implies that with the approximation of
$F_{W}(Q^{2})\simeq F_{\rm Ch}(Q^{2})$ the effect of incoherence cancels out
for $A^{\rm PV}$, which justifies Eq. (17). It should be noted that this
justification does not depend on the details of $U(1)^{\prime}$ models and
applies also to all the non-chiral $U(1)^{\prime}$ models discussed below.

###  3.2 Non-chiral $U(1)^{\prime}$ models and mixing-induced couplings

There are plenty of non-chiral $U(1)^{\prime}$ models with parity-conserving
charge assignments. For instance, the $U(1)_{B-L}$ model Davidson:1978pm ;
Mohapatra:1980qe ; Buchmuller:1991ce , which assigns all leptons (both left-
and right-handed) a charge of $-1$ and all quarks a charge of $\frac{1}{3}$,
is among the most extensively studied models in the literature. Although the
non-chiral charge assignments do not cause any additional source of asymmetry
in polarized electron scattering, constraints on non-chiral $U(1)^{\prime}$
models are possible in the presence of $Z$-$Z^{\prime}$ mixing, as illustrated
by the lower right diagram in Fig. 1.

Let us consider that the $Z$-$Z^{\prime}$ mixing arises from the following
terms:

${\cal L}\ \supset\ -\frac{\epsilon}{2}B^{\mu\nu}F^{\prime}_{\mu\nu}+\delta
m^{2}\hat{Z}^{\mu}\hat{Z}^{\prime}_{\mu}\,,$ (18)

where $B^{\mu\nu}$ and $F^{\prime}_{\mu\nu}$ are the field strength tensors of
hypercharge $U(1)_{Y}$ and $U(1)^{\prime}$, respectively. The second term is a
mass-mixing term which exists if the symmetry breaking of $U(1)^{\prime}$ is
not decoupled from the electroweak symmetry breaking. We define a mass mixing
parameter $\theta$ by

$\tan\theta\ \equiv\ \frac{\delta m^{2}}{m_{Z^{\prime}}^{2}-m_{Z}^{2}}\,,$
(19)

where $m_{Z}^{2}$ is the $Z$ boson mass. Note that in the presence of Eq.
(18), the kinetic terms are not canonical and the mass matrix is not diagonal.
One needs to perform necessary transformations and redefine $Z$ and
$Z^{\prime}$ as the physical mass eigenstates. To avoid potential confusion,
we have denoted the original states as $\hat{Z}$ and $\hat{Z}^{\prime}$ in Eq.
(18). Following the notation in Ref. Lindner:2018kjo , the couplings in the
physical basis can be written as

$\displaystyle{\cal L}$ $\displaystyle\ \supset\ Z_{\mu}\left[J_{{\rm
NC}}^{\mu}\cos\beta-J_{X}^{\mu}\sin\beta\right]+Z^{\prime}_{\mu}\left[J_{{\rm
NC}}^{\mu}\sin\beta+J_{X}^{\mu}\cos\beta\right]\,,$ (20)

where the effective $Z$-$Z^{\prime}$ mixing angle is

$\displaystyle\tan\beta\ \approx\ \tan\theta+\frac{\epsilon
s_{W}}{r_{m}-1}+{\cal O}(\theta^{2},\ \epsilon^{2})\,,$ (21)

where we have defined the mass ratio

$\displaystyle r_{m}\ \equiv\ \frac{m_{Z^{\prime}}^{2}}{m_{Z}^{2}}\,.$ (22)

Eq. (21) incorporates contributions from the mass mixing $\theta$ and the
kinetic mixing $\epsilon$. The SM NC $J_{{\rm NC}}^{\mu}$ and the
$Z^{\prime}$-induced neutral current $J_{X}^{\mu}$ in Eq. (21) are defined
respectively as

$\displaystyle J_{{\rm NC}}^{\mu}$ $\displaystyle\ =\ \sum_{f}g_{f}^{{\rm
SM}}\overline{f}\gamma^{\mu}f\,,$ (23) $\displaystyle J_{X}^{\mu}$
$\displaystyle\ =\
\sum_{f}\frac{\overline{f}\gamma^{\mu}f}{\sqrt{1-\epsilon^{2}}}\left[g^{\prime}Q_{f}^{\prime}-g_{Z}\epsilon
s_{W}Y_{f}\right]\,,$ (24)

with $g_{f}^{{\rm SM}}$ the SM gauge couplings for the chiral fermions $f$
collected in the second column of Tab. 1. In Eq. (24),
$g_{Z}\equiv{g}/{c_{W}}={\sqrt{4\pi\alpha}}/{s_{W}c_{W}}$ is a SM coupling,
and $Y_{f}$ is the SM $U(1)_{Y}$ charge for the fermions.

From Eqs. (20)-(24) it is straightforward to derive the effective couplings of
$Z$ and $Z^{\prime}$ to fermions; these are summarized in Tab. 1. Note that
for $Z$ couplings we have expanded the results to order $\epsilon^{2}$ because
the $Z$ diagram with one vertex modified by $\epsilon^{2}$ and the
$Z^{\prime}$ diagram with two vertices proportional to $\epsilon$ have
comparable contributions to the parity-violating asymmetry.

Now using the effective couplings and Eq. (9), we can compute the mixing-
induced contribution to the parity-violating asymmetry, $\Delta A^{{\rm
PV}}\equiv A^{{\rm PV}}-A^{{\rm PV}}_{\rm SM}$, for non-chiral $U(1)^{\prime}$
models. The result, in the presence of only mass-mixing, is

$\frac{\Delta A^{{\rm PV}}}{A_{{\rm SM}}^{{\rm PV}}}\ =\
\sin^{2}{\theta}\left(\frac{m_{Z}^{2}}{m_{Z^{\prime}}^{2}+Q^{2}}-1\right).$
(25)

For only kinetic mixing, the result is

$\displaystyle\frac{\Delta A^{{\rm PV}}}{A_{{\rm SM}}^{{\rm PV}}}(e+p)$
$\displaystyle\ =\ $
$\displaystyle\frac{m_{Z^{\prime}}^{2}(1-r_{m})\left(1-4s_{W}^{2}\right)+Q^{2}\left(4r_{m}s_{W}^{2}+2r_{m}-3\right)}{(1-r_{m})^{2}\left(1-4s_{W}^{2}\right)\left(m_{Z^{\prime}}^{2}+Q^{2}\right)}s_{W}^{2}\epsilon^{2}\,,$
(26) $\displaystyle\frac{\Delta A^{{\rm PV}}}{A_{{\rm SM}}^{{\rm
PV}}}(e+{}^{12}{\rm C})$ $\displaystyle\ =\ $
$\displaystyle\frac{m_{Z^{\prime}}^{2}s_{W}^{2}(1-r_{m})+Q^{2}\left(1-r_{m}-r_{m}s_{W}^{2}\right)}{(1-r_{m})^{2}\left(m_{Z^{\prime}}^{2}+Q^{2}\right)}\epsilon^{2}\,,$
(27)

for electron-proton and electron-12C scattering, respectively.

When both mass and kinetic mixing are present, one can add their contributions
linearly as long as both $\epsilon$ and $\theta$ are perturbatively small. In
the above results, we have neglected the contribution of the fundamental gauge
coupling $g^{\prime}$ in Eq. (24). Although for non-chiral $U(1)^{\prime}$
models this part does not account for parity violation, sizable values of
$g^{\prime}$ do affect the P2 sensitivity by changing the effective couplings
to nucleons if $g^{\prime}$ is comparable to $\theta$ or $\epsilon$. This
effect can be readily included by adding a correction of
$g^{\prime}Q_{f}^{\prime}$ to the mixing-induced couplings as long as both are
perturbatively small. In this case, the P2 sensitivity to $\epsilon$ and
$\theta$ becomes model-dependent. We select two non-chiral $U(1)^{\prime}$
models when the model-dependent details are required, $U(1)_{B-L}$ as
aforementioned, and $U(1)_{B}$ which gauges the baryon number only Foot:1989ts
. For these models with sizable $g^{\prime}$ and $\theta$, the new physics
contributions can be formulated as

$\frac{\Delta A^{{\rm PV}}}{A_{{\rm SM}}^{{\rm PV}}}\ =\
\sin^{2}\theta\left[\frac{m_{Z}^{2}(1+R)}{m_{Z^{\prime}}^{2}+Q^{2}}-1\right],$
(28)

where $R$ takes the following expressions for the $U(1)_{B}$ model:

$\displaystyle R_{e+p}[U(1)_{B}]$ $\displaystyle\ =\ $
$\displaystyle\frac{12g^{\prime}}{g_{Z}\sin\theta(1-4s_{W}^{2})}\,,$ (29)
$\displaystyle R_{e+^{12}\text{C}}[U(1)_{B}]$ $\displaystyle\ =\ $
$\displaystyle-\frac{6g^{\prime}}{g_{Z}\sin\theta s_{W}^{2}}\,.$ (30)

For $U(1)_{B-L}$, we simply need to multiply the $R$ factors in Eqs. (29) and
(30) by a factor of $-1/3$, i.e.

$\displaystyle R[U(1)_{B-L}]\ =\ -\frac{1}{3}R[U(1)_{B}]\,.$ (31)

##  4 P2 sensitivity study

In this section, we derive the P2 sensitivity for the $U(1)$ models discussed
in the previous section.

###  4.1 Method

The P2 experiment will measure polarized electron scattering with an electron
beam of 155 MeV Kempf:2020bfx . The momentum transfer $Q^{2}$ in this process
can be determined by

$Q^{2}\ \approx\ 4E_{i}E_{f}\sin(\theta_{f}/2)\,,$ (32)

where $\theta_{f}$ is the scattering angle of the outgoing electron with
respect to the incoming one, and $E_{f}\approx E_{i}\approx 155$ MeV are the
final/initial electron energies. The angular acceptance of the detector is

$\theta_{f}\ \in\ \left[\overline{\theta}_{f}-\frac{1}{2}\delta\theta_{f},\
\overline{\theta}_{f}+\frac{1}{2}\delta\theta_{f}\right],\quad\text{with}\quad\overline{\theta}_{f}=35^{\circ},\
\delta\theta_{f}=20^{\circ}\,.$ (33)

In our analysis we take $Q^{2}\approx(93\ {\rm MeV})^{2}$ evaluated from the
central value $\overline{\theta}_{f}$. For $e+p$ scattering, the expected
value of $A^{{\rm PV}}$ is $39.94\times 10^{-9}$ and the P2 experiment will be
able to measure it with an uncertainty of $\Delta A^{{\rm PV}}=0.56\times
10^{-9}$, which corresponds to a relative uncertainty of

$\frac{\Delta A^{{\rm PV}}}{A^{{\rm PV}}}\ \approx\ 1.4\%\,,\ \ (e+p\text{
scattering}).$ (34)

This would imply a relative uncertainty of $0.14\%$ for $s_{W}^{2}$. For
$e+{}^{12}$C scattering, the achievable uncertainty is

$\frac{\Delta A^{{\rm PV}}}{A^{{\rm PV}}}\ \approx\ 0.3\%\,,\ \ (e+^{12}{\rm
C}\text{ scattering}).$ (35)

Since $A^{{\rm PV}}(^{12}{\rm C})$ is proportional to $s_{W}^{2}$ rather than
$1-4s_{W}^{2}$, the relative uncertainty for $s_{W}^{2}$ is also $0.3\%$.

With the experimental setup described above and the new physics contributions
formulated in Eqs. (15) and (25) to (27), it is straightforward to study the
sensitivity of P2 to $Z^{\prime}$ models. By requiring that the new physics
contributions do not exceed

$\displaystyle\frac{\Delta A^{{\rm PV}}}{A^{{\rm PV}}}\ =\
\begin{cases}\sqrt{2.71}\times 1.4\%\ =\ 2.30\%&(\text{for $e+p$
scattering})\,,\\\ \sqrt{2.71}\times 0.3\%\ =\ 0.49\%&(\text{for $e+^{12}$C
scattering})\,,\end{cases}$ (36)

where the factor $\sqrt{2.71}$ converts 1$\sigma$ to 90% confidence level
(C.L.), we obtain the P2 sensitivity curves in Figs. 2 to 6 for the variety of
$U(1)^{\prime}$ models under study.

As for the value of the weak mixing angle $\theta_{W}$ used in our numerical
calculations, it is precisely measured at the electroweak scale from $Z$ pole
observables, the $W$ boson mass and a variety of NC processes. In the global
fitting of high-energy data, it is determined to be $s^{2}_{W}=0.23122\pm
0.00003$ Zyla:2020zbs , which can be run down via the SM renormalization group
equations to lower energies Erler:2004in ; Erler:2017knj  and compared to the
direct measurements of the weak mixing angle at that low-energy scales
Kumar:2013yoa . In the numerical calculations, we have fixed the value of
$s_{W}^{2}$ to its expected central low-$q^{2}$ value of $0.230\pm 0.00003$.
As shown in Eq. (11), in the SM the parity-violating asymmetry $A^{\rm
PV}_{\rm SM}$ in electron-proton scattering is proportional to the factor of
$(1-4s_{W}^{2})$, therefore the uncertainty of $A^{\rm PV}_{\rm SM}$ due to
the error bars $\Delta s_{W}^{2}$ of $s_{W}^{2}$ goes like

$\displaystyle\Delta A_{\rm SM}^{\rm PV}(e+p)\ \propto\ \frac{-4\Delta
s_{W}^{2}}{1-4s_{W}^{2}}\,.$ (37)

For scattering with 12C, the SM $A^{\rm PV}$ depends linearly on the weak
mixing angle, i.e.

$\displaystyle\Delta A_{\rm SM}^{\rm PV}(e+^{12}{\rm\\!C})\ \propto\
\frac{\Delta s_{W}^{2}}{s_{W}^{2}}\,.$ (38)

In the case of $Z$-$Z^{\prime}$ mixing, the new physics contributions to
$A^{\rm PV}$ are also subject to the weak mixing angle uncertainties, as
implied in Eqs. (21) to (24). They enter the result in the form of
$\epsilon\,\Delta s_{W}^{2}$ or $\sin\theta\,\Delta s_{W}^{2}$. In light of
the small experimental uncertainties of $s_{W}^{2}$, we neglect this higher-
order effect in this paper.

Since the measurable $A^{\rm PV}$ is a single number, there is an unavoidable
degeneracy between new physics and a variation of $s_{W}^{2}$. This could,
however, be resolved by performing the P2 measurements not only of the total
asymmetry but of the angular distribution of the cross-sections over the whole
range of scattering angle $\theta_{f}$ (see Eq. (33)). In this way one could
perform measurements at different $Q^{2}$, see Eq. (32), which could for
appropriate $Z^{\prime}$ masses break the degeneracy of the Weinberg angle
with new physics contributions to the asymmetry. Of course, measuring at two
$Q^{2}$ with the same amount of available beam-time will cost precision. In
case interesting departures from the expectation are found during the run-time
of the experiment, a trade-off between precision and new physics sensitivity
could be a possibility. We note that a measurement at large scattering angle
is foreseen by P2, in order to measure corrections from the proton structure
to the measurement Becker:2018ggl .

###  4.2 Current experimental limits

Before discussing our sensitivity results for P2, we briefly review the
existing bounds on $Z^{\prime}$ in each of the $U(1)^{\prime}$ models
considered in Sec. 3, following the compilations in, e.g. Refs. Essig:2013lka
; Ilten:2018crw ; Bauer:2018onh ; Berryman:2019dme ; Fabbrichesi:2020wbt ;
Lanfranchi:2020crw . Since the P2 experiment is based on electron-nucleon
scattering, bounds on electron and quark couplings should be taken into
account. In addition, for specific models, bounds on neutrino or muon
couplings can also be used to constrain $g^{\prime}$, $\epsilon$, and
$\theta$. In addition, gauge invariance links electron scattering to neutrino
scattering.

$\bullet$ Atomic parity violation:

Precise low-energy measurements of APV have been performed using atomic 133Cs
Wood:1997zq ; Bennett:1999pd ; Porsev:2009pr ; Porsev:2010de , 205Tl
Edwards:1995zz ; Vetter:1995vf , 208Pb Meekhof:1993zz , 209Bi
Macpherson:1991opp  and 100,102,104,106Yb Antypas:2018mxf , and the most
precise ones are the 6$S_{1/2}$ $-$ 7$S_{1/2}$ nuclear transition in 133Cs. To
lowest order, the nuclear weak charge measured in APV is given by
$Q_{W}=-{\cal N}_{\rm Cs}+{\cal Z}_{\rm Cs}(1-4s_{W}^{2})$, with ${\cal
Z}_{\rm Cs}=55$ and ${\cal N}_{\rm Cs}=78$ being the proton and neutron
numbers in 133Cs respectively. Combining calculation of the nuclear spin-
independent parity-violating electric dipole transition amplitude and the most
accurate experimental measurements Marciano:1982mm ; Marciano:1983ss ;
Marciano:1990dp ; Sahoo:2021thl , we have the following SM predictions and
experimental values of $Q_{W}(^{133}{\rm Cs})$:

$\displaystyle Q_{W}^{\rm SM}(^{133}{\rm Cs})$ $\displaystyle\ =\ $
$\displaystyle-73.23(1)\,,\qquad Q_{W}^{\rm exp}(^{133}{\rm Cs})\ =\
-73.71(35)\,,$ (39)

which corresponds to an accuracy of $0.47\%$ ($0.77\%$) at the $1\sigma$ (90%)
C.L., and can be used to set limits on any new physics contributions to APV
Davoudiasl:2012ag ; Safronova:2017xyt ; Davoudiasl:2014kua ; Abdullah:2018ykz
. In presence of the $Z^{\prime}$ boson, the 6$S_{1/2}$ $-$ 7$S_{1/2}$ nuclear
transition can be viewed as an electron-nucleus scattering in the Cesium atom,
and the calculation methods for proton and 12C can be applied directly to the
case of 133Cs, for instance Eqs. (15), (25) and (28). For concreteness, we set
the energy scale of 133Cs experiments to be $Q=30$ MeV Abdullah:2018ykz , and
the resultant limits are shown in Figs. 2 to 6 as the pink curves.

$\bullet$ Beam dumps:

Beam dump experiments such as E137 Bjorken:1988as , E141 Riordan:1987aw , E774
Bross:1989mp  and Orsay Davier:1989wz  have searched for particles that could
be produced in electron scattering off nuclei in fixed targets and their
subsequent decay to visible final states. In these experiments, the production
process is electron bremsstrahlung and the detection relies on
$Z^{\prime}\rightarrow e^{+}e^{-}$ decay. Therefore the searches are sensitive
to the electron coupling of a low-mass $Z^{\prime}$ in the range of 1 to 100
MeV. We use the package DARKCAST Ilten:2018crw  to recast and combine the
bounds from these experiments.

There was recently a search of $eN\to eNX$ at NA64, with the $X$ boson
decaying invisibly Andreev:2021xpu . The null result at this experiment (which
is not yet incorporated in DARKCAST) sets an upper bound on the effective
coupling of $X$ to electron, which can go up to the order of $10^{-5}$ for a
MeV-scale $X$ boson mass and up to $10^{-2}$ if $X$ is at the GeV-scale,
depending on the Lorentz structure of the couplings. When the NA64 limits in
Ref. Andreev:2021xpu  are cast onto the chiral and non-chiral $U(1)^{\prime}$
models in this paper, the corresponding limits are expected to be the same
order as that of the electron $g-2$ limits or up to one order of magnitude
better (cf. Fig. 4 in Ref. Andreev:2021xpu ), and are thus not shown in Figs.
2 to 6.

$\bullet$ Collider searches:

New $Z^{\prime}$ gauge bosons could be probed at colliders in several
different ways. For low-mass $Z^{\prime}$ at $e^{+}e^{-}$ colliders, the
mono-$\gamma$ channel $e^{+}e^{-}\rightarrow\gamma Z^{\prime}$, followed by
decays $Z^{\prime}\rightarrow e^{+}e^{-}$ or $\mu^{+}\mu^{-}$ or into
invisible states, is usually considered as the most restrictive one. Searches
in these channels have been performed at BaBar Lees:2014xha ; Lees:2017lec ,
NA48/2 Batley:2015lha , KLOE Babusci:2012cr  and LEP Fox:2011fx . At hadron
colliders, the Drell-Yan process $pp\rightarrow
Z^{\prime}\rightarrow\mu^{+}\mu^{-}$ or dijet, searched for by LHCb
Aaij:2019bvg , ATLAS Aad:2019fac ; ATLAS:2019bov , and CMS CMS:2019tbu ;
Sirunyan:2019vgj , produces the leading constraints on heavy $Z^{\prime}$
above the $Z$ pole. Besides, the LEP data can also be used to constrain four-
lepton effective operators generated by heavy $Z^{\prime}$ Abdallah:2005ph .
We adopt the ATLAS/CMS dijet limit from Fig. 3 in Ref. Bagnaschi:2019djj .
Other collider bounds are obtained using the DARKCAST package Ilten:2018crw .
The LEP bounds in Figs. 2 and 6 (absent in Figs. 3 to 5 due to insignificance)
are derived by fitting four-lepton effective operators to precision
measurements of $e^{+}e^{-}\rightarrow\ell^{+}\ell^{-}$ at LEP Abdallah:2005ph
.111The LEP limits on flavored $Z^{\prime}$ bosons can be found e.g. in Ref.
Chun:2018ibr .

$\bullet$ Neutrino scattering:

Elastic neutrino scattering data collected by CHARM II Vilain:1993kd ;
Vilain:1994qy , LSND Auerbach:2001wg , Borexino Bellini:2011rx , TEXONO
Deniz:2009mu  and COHERENT Akimov:2017ade  are sensitive to a low-mass
$Z^{\prime}$ as it could mediate a $t$-channel process in $\nu+e^{-}$ or
$\nu+N$ scattering enhanced by the low mass and low momentum transfer.
According to studies in Refs. Bilmis:2015lja ; Farzan:2018gtr ;
Lindner:2018kjo ; Abdullah:2018ykz , we select two of the most restrictive
sets of data, CHARM II and TEXONO, and perform an independent data
fitting222Code is available from https://github.com/xunjiexu/Dark_Z. for the
various models considered in this work. The neutrino scattering bounds apply
to most of the models except for $U(1)_{R}$ and $U(1)_{B}$, in which left-
handed neutrinos $\nu_{L}$ are not charged under the new $U(1)$. However, in
the presence of kinetic/mass mixing, these models could still be constrained
by neutrino scattering.

$\bullet$ Lepton anomalous magnetic moment:

The muon and electron anomalous magnetic moments are sensitive to generic
neutral bosons coupled to them. The $Z^{\prime}$ contribution to
$a_{\ell}\equiv(g-2)/2$ can be evaluated by Leveille:1977rc :

$\Delta a_{\ell}\ =\
\frac{1}{8\pi^{2}}(g^{\prime}_{\ell}\epsilon_{Z^{\prime}})^{2}\int_{0}^{1}\frac{2x^{2}(1-x)}{(1-x)\left(1-x\epsilon_{Z^{\prime}}^{2}\right)+x\epsilon_{Z^{\prime}}^{2}}dx\,,$
(40)

where $\epsilon_{Z^{\prime}}\equiv m_{\ell}/m_{Z^{\prime}}$ and $\ell=\mu$ or
$e$. The muon $g-2$ has a long-standing $3.7\sigma$ discrepancy between the
experimental and the calculated SM values Aoyama:2020ynm , i.e.

$\displaystyle\Delta a_{\mu}\ \equiv\ a_{\mu}^{{\rm exp}}-a_{\mu}^{{\rm SM}}\
=\ (27.9\pm 7.6)\times 10^{-10}\,.$ (41)

The electron $g-2$ had earlier seen a $2.4\sigma$ discrepancy in the opposite
direction Davoudiasl:2018fbb , based on the comparison between the SM
prediction Aoyama:2017uqe  and a measurement of the fine-structure constant
using Cesium Parker:2018vye :

$\displaystyle\Delta a_{e}^{\rm Cs}\ \equiv\ a_{e}^{{\rm exp\
(Cs)}}-a_{e}^{{\rm SM}}\ =\ (-8.7\pm 3.6)\times 10^{-13}\,.$ (42)

However, a recent measurement of the fine-structure constant using Rubidium
Morel:2020dww  has pushed the discrepancy to a mere $1.6\sigma$ in the same
direction as $(g-2)_{\mu}$:

$\displaystyle\Delta a_{e}^{\rm Rb}\ \equiv\ a_{e}^{{\rm exp\
(Rb)}}-a_{e}^{{\rm SM}}\ =\ (4.8\pm 3.0)\times 10^{-13}\,.$ (43)

Although this weakens the possibility of any new physics contribution to
$\Delta a_{e}$, the Cs and Rb measurements of $\alpha$ now disagree by more
than $5\sigma$, which makes this whole issue quite murky. Here we take
conservative bounds from $(g-2)_{\mu}$ and $(g-2)_{e}$ by requiring that
$\Delta a_{\ell}$ does not exceed the experimental best-fit values given by
Eqs. (41) and (42) by 5$\sigma$. These bounds are presented in Figs. 2 to 6 as
green curves. Note that a $Z^{\prime}$ that couples to electrons and muons
with the same strength, as the case for the models studied here, can not
explain the muon $g-2$ discrepancy Lindner:2016bgg .

$\bullet$ Electroweak precision tests:
Electroweak precision tests can also constrain $Z^{\prime}$ parameters. In
Fig. 4 we present a constraint on $\epsilon$ taken from Ref. Curtin:2014cca .
In principle, similar constraints are present for all the $Z^{\prime}$ models,
but need to be evaluated model-by-model, which is beyond the scope of our
paper.

$\bullet$ Loop-level meson and $Z$ decay limits
The $Z^{\prime}$ boson in the $U(1)^{\prime}$ model could induce 1-loop level
flavor-changing neutral current (FCNC) meson decays such as $B\rightarrow
K+Z^{\prime}$ and $K\rightarrow\pi+Z^{\prime}$. The corresponding partial
decay widths are not only directly relevant to the mass $m_{Z^{\prime}}$,
gauge coupling $g^{\prime}$ (and also $\sin\theta$), but also depend subtly on
the charges of the SM fermions under the $U(1)^{\prime}$. For the $U(1)_{B-L}$
case, all the SM mass and interaction terms are invariant under the
$U(1)_{B-L}$ gauge symmetry, and the widths $\Gamma(K\to\pi Z^{\prime})$ and
$\Gamma(B\to KZ^{\prime})$ can be found e.g. in Refs. Pospelov:2008zw ;
Davoudiasl:2014kua , which go to zero in the $m_{Z^{\prime}}\to 0$ limit, as
expected from angular momentum conservation. As for the other $U(1)^{\prime}$
and $Z-Z^{\prime}$ mixing models, the SM fermion couplings are explicitly
chiral and so the fermion mass terms are not invariant under the
$U(1)^{\prime}$ gauge transformations. This leads to loop-level FCNC decays
enhanced by the ratio $M^{2}/m_{Z^{\prime}}^{2}$ for a very light
$Z^{\prime}$, with $M$ being the corresponding energy scale (typically the
electroweak scale) for these decays Dror:2017ehi ; Dror:2017nsg ; Dror:2018wfl
. Similarly, one can also have the 1-loop level decay $\Upsilon\to\gamma
Z^{\prime}$ which is enhanced by $m^{2}_{b}/m^{2}_{Z^{\prime}}$ when
$Z^{\prime}$ is light Carone:1994aa ; Fayet:2006sp ; Fayet:2007ua . Thus the
searches of $\Upsilon\to\gamma+{\rm inv.}$ delAmoSanchez:2010ac  and
$\Upsilon\to\gamma+X$ with $X\to\mu^{+}\mu^{-}$ Love:2008aa  can be used to
set limits on these $U(1)^{\prime}$ models. If kinematically allowed, there is
also the loop-level exotic $Z$ decay $Z\to\gamma Z^{\prime}$. The searches of
single photon from $Z$ decay Acciarri:1997im ; Abreu:1996pa  and
$Z\to\gamma\ell^{+}\ell^{-}$ Adriani:1992zm ; Acton:1991dq ; Adeva:1991dw  at
the LEP can be used to constrain the $Z^{\prime}$ bosons. However, these FCNC
decay constraints become important only for sub-MeV scale $Z^{\prime}$ bosons,
and are not relevant for our P2 sensitivity analysis. Moreover, the
$M^{2}/m_{Z}^{\prime 2}$-enhanced bounds cannot be arbitrarily strong in the
$m_{Z^{\prime}}\to 0$ limit and one has to take the full momentum dependence
into account in loop calculations.

$\bullet$ Astrophysical and cosmological limits:
A sufficiently light $Z^{\prime}$ can also be constrained by astrophysical
limits such as from supernova 1987A Rrapaj:2015wgs ; Chang:2016ntp  and
cosmological limits such as from the big bang nucleosynthesis (BBN)
Fradette:2014sza ; Escudero:2019gzq . However, these limits mostly lie outside
the window of parameter space considered in Figs. 2 to 6. The BBN limits are
significant only for $Z^{\prime}$ masses below 10 MeV, and the supernova
limits apply additionally only for small couplings, see e.g. Ref.
Knapen:2017xzo . Moreover, the BBN bound depends on the branching ratio of
$Z^{\prime}\rightarrow\nu\overline{\nu}$ and $Z^{\prime}\rightarrow
e^{-}e^{+}$. Thus these astrophysical and cosmological limits are not relevant
to the P2 prospects, and thus are not shown in Figs. 2 to 6.

###  4.3 P2 sensitivities

In this subsection, we derive the P2 sensitivities for the models considered
in Sec. 3 and compare them with the current experimental constraints discussed
above.

####  4.3.1 Chiral models

Figure 2: Sensitivities of the P2 experiment in $e+p$ (solid black) and
$e+^{12}$C (dashed black) scatterings to the gauge coupling $g^{\prime}$ in
three chiral $U(1)^{\prime}$ models given in Tab. 2. For simplicity we have
assumed here that there is no mass or kinetic $Z$-$Z^{\prime}$ mixing. The
shaded regions are excluded by APV measurements Wood:1997zq ; Bennett:1999pd ;
Porsev:2009pr ; Porsev:2010de , beam-dump experiments Bjorken:1988as ;
Riordan:1987aw ; Bross:1989mp ; Davier:1989wz , electron and muon $g-2$
Zyla:2020zbs ; Davoudiasl:2018fbb , BaBar Lees:2014xha ; Lees:2017lec , NA48/2
Batley:2015lha , KLOE Babusci:2012cr , LHCb Aaij:2019bvg , LEP
$e^{+}e^{-}\to\ell^{+}\ell^{-}$ data Abdallah:2005ph  and the ATLAS and CMS
dijet data Bagnaschi:2019djj . Note that, due to the cancellation of quantum
numbers in Eq. (17), the P2 prospect in the $e+^{12}$C channel is absent for
the $U(1)^{\prime}_{R}$ model (middle panel). See Sections 4.2 and 4.3 for
more details.

The P2 sensitivities for the three chiral $U(1)^{\prime}$ models given in
Table 2 are shown in Fig. 2, where the $e+p$ sensitivities are shown as solid
black lines, while the $e+^{12}$C sensitivities are in dashed black lines. All
existing relevant limits discussed in Section 4.2 are presented in Fig. 2 as
the shaded regions. Here we have assumed that there is no mass or kinetic
$Z-Z^{\prime}$ mixing. It is clear from Fig. 2 that the limits for the three
chiral $U(1)^{\prime}$ models are rather similar: at low masses
$m_{Z^{\prime}}\lesssim 100$ GeV, the strongest limits are mainly from APV
measurements (see below), beam-dump experiments, and from BaBar, NA48/2 and
KLOE and LHCb data. For the models $U(1)_{L}^{\prime}$ and $U(1)_{X}^{\prime}$
the limits from $\nu+e$ scattering are also important, while for
$U(1)^{\prime}_{R}$ the $Z^{\prime}$ boson does not couple directly to
neutrinos (see Table 2), and as a result these are no neutrino scattering
limits for the $U(1)^{\prime}_{R}$ model. Suppressed by the charged lepton
masses (see Eq. (40)), the electron and muon $g-2$ limits are relatively
weaker for these chiral models (and also for all other models in this paper).
When $Z^{\prime}$ is heavier than the electroweak scale, the coupling
$g^{\prime}$ is constrained by the direct searches of $Z^{\prime}$ in the LHC
dijet data and the limits from the LEP $e^{+}e^{-}\to\ell^{+}\ell^{-}$ data.

Comparing the solid and dashed black lines in the upper panel of Fig. 2, we
see that the $e+^{12}$C sensitivity for the $U(1)^{\prime}_{L}$ model is more
stringent than that from $e+p$ scattering by a factor of 1.4. This is mainly
due to the enhancement of the effective coupling of $Z^{\prime}$ to the 12C
nucleus with respect to the proton. In particular, following Eqs. (16) and
(17), we have

$\displaystyle Q^{\prime}_{p}\ =\ \frac{1}{2}\,,\quad Q^{\prime}_{{}^{12}{\rm
C}}\ =\ 12\,\quad\text{for $U(1)^{\prime}_{L}$}\,.$ (44)

Taking into account the two different factors for $\Delta A^{\rm PV}/A^{\rm
PV}_{\rm SM}$ in Eq. (36), and the SM predictions of $A^{\rm PV}_{\rm SM}$ for
proton and 12C in Eqs. (11) and (12), one gets the enhancement factor of 1.4.
Similarly, as a result of

$\displaystyle Q^{\prime}_{p}\ =\ \frac{5}{4}\,,\quad Q^{\prime}_{{}^{12}{\rm
C}}\ =\ 18\,\quad\text{for $U(1)^{\prime}_{X}$}\,,$ (45)

the $e+p$ and $e+^{12}$C sensitivities for the $U(1)^{\prime}_{X}$ model are
almost the same, as can be seen in the bottom panel of Fig. 2. For the
$U(1)^{\prime}_{R}$ model, due to the accidental cancellation of

$\displaystyle
Q^{\prime}_{u_{L}}+Q^{\prime}_{u_{R}}+Q^{\prime}_{d_{L}}+Q^{\prime}_{d_{R}}\
=\ 0\,,$ (46)

we find that $Q^{\prime}_{{}^{12}{\rm C}}=0$, and there is no $e+^{12}$C
sensitivity for the $U(1)^{\prime}_{R}$ model.

Table 3: P2 sensitivities for the chiral and non-chiral models shown in Figs.
2 to 6 in the limits of $m_{Z^{\prime}}\to 0$ and $m_{Z^{\prime}}\to\infty$.
“$U(1)$ ($\sin\theta$, $\epsilon$)” refers to the generic $U(1)^{\prime}$
model with only mass mixing $\sin\theta$ or kinetic mixing $\epsilon$. The
numbers in the parentheses for the $U(1)_{B}$ and $U(1)_{B-L}$ models are the
values of $g^{\prime}/\sin\theta$. Note that for the $U(1)^{\prime}_{R}$ model
there is no sensitivity in the $e+^{12}$C scattering mode due to an accidental
cancellation (cf. Eq. (46)).

models | $m_{Z^{\prime}}\to 0$ | $m_{Z^{\prime}}\to\infty$
---|---|---
$e+p$ |  $e+^{12}$C | $e+p$ |  $e+^{12}$C
$U(1)_{L}^{\prime}$ | $g^{\prime}<1.7\times 10^{-5}$ | $g^{\prime}<1.2\times 10^{-5}$ |  $\frac{m_{Z^{\prime}}}{g^{\prime}}>5.9$ TeV |  $\frac{m_{Z^{\prime}}}{g^{\prime}}>8.3$ TeV
$U(1)_{R}^{\prime}$ | $g^{\prime}<1.7\times 10^{-5}$ | $-$ |  $\frac{m_{Z^{\prime}}}{g^{\prime}}>5.6$ TeV | $-$
$U(1)_{X}^{\prime}$ | $g^{\prime}<1.5\times 10^{-5}$ | $g^{\prime}<1.5\times 10^{-5}$ |  $\frac{m_{Z^{\prime}}}{g^{\prime}}>7.7$ TeV |  $\frac{m_{Z^{\prime}}}{g^{\prime}}>7.7$ TeV
$U(1)^{\prime}$ ($\sin\theta$) | $\sin\theta<1.7\times 10^{-4}$ | $\sin\theta<7.3\times 10^{-5}$ | $\sin\theta<0.13$ | $\sin\theta<0.07$
$U(1)^{\prime}$ ($\epsilon$) | $\epsilon<0.05$ | $\epsilon<0.07$ |  $\frac{m_{Z^{\prime}}}{\epsilon}>350$ GeV |  $\frac{m_{Z^{\prime}}}{\epsilon}>600$ GeV
$U(1)_{B}$ (0.01) | $\sin\theta<9\times 10^{-5}$ | $\sin\theta<9\times 10^{-5}$ | $\sin\theta<0.13$ | $\sin\theta<0.07$
$U(1)_{B}$ (1) | $\sin\theta<1.1\times 10^{-5}$ | $\sin\theta<1.2\times 10^{-5}$ | $\sin\theta<0.13$ | $\sin\theta<0.07$
$U(1)_{B}$ (10) | $\sin\theta<3.7\times 10^{-6}$ | $\sin\theta<4.1\times 10^{-6}$ | $\sin\theta<0.13$ | $\sin\theta<0.07$
$U(1)_{B-L}$ (0.01) | $\sin\theta<2.9\times 10^{-4}$ | $\sin\theta<6.9\times 10^{-5}$ | $\sin\theta<0.13$ | $\sin\theta<0.07$
$U(1)_{B-L}$ (1) | $\sin\theta<2\times 10^{-5}$ | $\sin\theta<2\times 10^{-5}$ | $\sin\theta<0.13$ | $\sin\theta<0.07$
$U(1)_{B-L}$ (10) | $\sin\theta<6\times 10^{-6}$ | $\sin\theta<6.5\times 10^{-6}$ | $\sin\theta<0.13$ | $\sin\theta<0.07$

Table 4:  Ranges of $m_{Z^{\prime}}$ and $g^{\prime}$ where P2 will be able to improve current limits for chiral $U(1)^{\prime}$ models. This table is a summary of Fig. 2. The columns of $U(1)^{\prime}_{L}$ in the $e+p$ mode are void because P2 does not have sensitivity beyond existing APV limits.  models | mode |  $m_{Z^{\prime}}$ ranges |  $g^{\prime}$ ranges
---|---|---|---
$U(1)_{L}^{\prime}$ | $e+p$ | $-$ | $-$
$e+^{12}$C |  | [100 MeV, 200 MeV]
---
[70 GeV, 600 GeV]
[5 TeV, 90 TeV]
| $[2.0\times 10^{-5},\,3.3\times 10^{-5}]$
---
$[0.01,\,0.09]$
$[0.70,\,4\pi]$
$U(1)_{R}^{\prime}$ | $e+p$ |  | [20 MeV, 200 MeV]
---
[70 GeV, 600 GeV]
[5 TeV, 70 TeV]
| $[1.7\times 10^{-5},\,4.0\times 10^{-5}]$
---
$[0.012,\,0.1]$
$[0.9,\,4\pi]$
$U(1)_{X}^{\prime}$ | $e+p$ |  | [93 MeV, 430 MeV]
---
[70 GeV, 700 GeV]
[5 TeV, 79 TeV]
| $[2.0\times 10^{-5},\,7.5\times 10^{-5}]$
---
$[0.01,\,0.11]$
$[0.8,\,4\pi]$
$e+^{12}$C |  | [93 MeV, 430 MeV]
---
[70 GeV, 700 GeV]
[5 TeV, 79 TeV]
| $[2.0\times 10^{-5},\,7.5\times 10^{-5}]$
---
$[0.01,\,0.11]$
$[0.8,\,4\pi]$

For masses $m_{Z^{\prime}}\lesssim 100$ MeV, the P2 sensitivities are almost
independent of the $Z^{\prime}$ mass, which correspond to the plateau in the
three panels of Fig. 2. This is also true for the other models considered here
(see Figs. 3 to 6). In the high $m_{Z^{\prime}}$ limit, i.e.
$m_{Z^{\prime}}\to\infty$, we can safely neglect the momentum transfer
$Q^{2}$, and the P2 experiment can probe an effective ultraviolet (UV) cutoff
scale of $\Lambda=m_{Z^{\prime}}/g^{\prime}$ at the few-TeV scale. All the P2
sensitivities in the limits of $m_{Z^{\prime}}\to 0$ and
$m_{Z^{\prime}}\to\infty$ for the three chiral models are collected in Table
3.

The low-energy nuclear transition in 133Cs is essentially equivalent to
electron-nucleus scattering, thus for all our cases the properties of the APV
limits are very similar to those for the P2 prospects. Comparing the 133Cs
atom in the most precise APV measurement with 12C in P2, the differences lie
mainly in the following factors: the nuclear weak charges, the effective
couplings of $Z^{\prime}$ to nuclei and the energy scale $Q$. Taking the
$U(1)^{\prime}_{L}$ model as an explicit example, the coupling of $Z^{\prime}$
to 133Cs is

$\displaystyle Q^{\prime}_{{}^{133}{\rm Cs}}\ =\
\frac{1}{2}\left(266\,Q^{\prime}_{u_{L}}+266\,Q^{\prime}_{u_{R}}+211\,Q^{\prime}_{d_{L}}+211\,Q^{\prime}_{d_{R}}\right)\
=\ \frac{263}{2}\,.$ (47)

Following Eq. (15), the resulting parity violation for the atom $X$ with
respect to the SM value is proportional to

$\displaystyle\frac{\Delta A^{\rm PV}(X)}{A_{\rm SM}^{\rm PV}}\ \propto\
Q^{-2}\frac{Q^{\prime}_{X}}{Q_{W}(X)}\frac{Q^{2}}{Q^{2}+m_{Z^{\prime}}^{2}}\,.$
(48)

In the limit of $m_{Z^{\prime}}\ll Q$, we can neglect the last factor in the
equation above, and compare the APV sensitivities in 133Cs and the P2
prospects using 12C:

$\displaystyle\frac{\Delta A^{\rm PV}(^{133}{\rm Cs})/A_{\rm SM}^{\rm
PV}(^{133}{\rm Cs})}{\Delta A^{\rm PV}(^{12}{\rm C})/A_{\rm SM}^{\rm
PV}(^{12}{\rm C})}(m_{Z^{\prime}}\ll Q)$ $\displaystyle\ =\ $
$\displaystyle\left(\frac{Q_{\rm P2}}{Q_{\rm
APV}}\right)^{2}\frac{Q^{\prime}_{{}^{133}{\rm Cs}}/Q_{W}(^{133}{\rm
Cs})}{Q^{\prime}_{{}^{12}{\rm C}}/Q_{W}(^{12}{\rm C})}$ (49) $\displaystyle\
\simeq\ $ $\displaystyle\left(\frac{93\;{\rm MeV}}{30\;{\rm
MeV}}\right)^{2}\left(\frac{-1.78}{-2.17}\right).$

Taking into account the difference of the APV accuracy (0.77%) and the P2
precision in 12C (0.49%), Eq. (54) implies that for the $U(1)^{\prime}$ model
at low energies $m_{Z^{\prime}}\ll Q$, the APV limit on $g^{\prime}$ is more
stringent than the P2 prospect using 12C by a factor of

$\displaystyle\left(\frac{0.49\%}{0.77\%}\right)^{1/2}\left(\frac{93\;{\rm
MeV}}{30\;{\rm MeV}}\right)\left(\frac{-1.78}{-2.17}\right)^{1/2}\ \simeq\
2.24\,.$ (50)

In the limit of $m_{Z^{\prime}}\gg Q$, the last factor in Eq. (48) can be
simplified to $Q^{2}/m_{Z^{\prime}}$ and any momentum dependence cancels. In
this case the comparison of APV and P2 with the target 12C is determined only
by the coupling factors $Q^{\prime}_{N}$ and $Q_{W}$, i.e.

$\displaystyle\frac{\Delta A^{\rm PV}(^{133}{\rm Cs})/A_{\rm SM}^{\rm
PV}(^{133}{\rm Cs})}{\Delta A^{\rm PV}(^{12}{\rm C})/A_{\rm SM}^{\rm
PV}(^{12}{\rm C})}(m_{Z^{\prime}}\gg Q)$ $\displaystyle\ =\ $
$\displaystyle\frac{Q^{\prime}_{{}^{133}{\rm Cs}}/Q_{W}(^{133}{\rm
Cs})}{Q^{\prime}_{{}^{12}{\rm C}}/Q_{W}(^{12}{\rm C})}\ \simeq\
\left(\frac{-1.78}{-2.17}\right),$ (51)

and the APV limit on $g^{\prime}$ is weaker than the P2 prospect with 12C by a
factor of

$\displaystyle\left(\frac{0.49\%}{0.77\%}\right)^{1/2}\left(\frac{-1.78}{-2.17}\right)^{1/2}\
\simeq\ 0.73\,,$ (52)

as shown in the upper panel of Fig. 2. In a similar way, one can compare the
APV limits with the P2 sensitivities for the proton target for the
$U(1)^{\prime}$ models. It turns out that for the $U(1)^{\prime}_{L}$ model
the APV constraint is almost the same as that for the P2 prospect with the
proton target in the limit of large $m_{Z^{\prime}}$. For the
$U(1)^{\prime}_{R}$ model, the APV limit is slightly weaker than the P2
sensitivity with protons for all values of $m_{Z^{\prime}}$. For the
$U(1)^{\prime}_{X}$ model the APV limits exclude the P2 prospects at the low
mass range of $m_{Z^{\prime}}$, and are slightly weaker than the P2
sensitivities for both proton and 12C when the $Z^{\prime}$ mass is large.

The P2 sensitivities in the lower mass range have also been precluded
partially by the beam-dump experiments, NA62, Na48/2 and KLOE experiments, and
in the high mass range partially by the LHC dijet data. However, even if all
these constraints are taken into consideration, there is still some parameter
space in the $m_{Z^{\prime}}$-$g^{\prime}$ plane that can be probed at the P2
experiment, which is collected in Table 4. Note that the P2 prospects for the
$U(1)^{\prime}_{L}$ model in the proton mode have been precluded by the APV
limits. Depending on the $U(1)^{\prime}$ model, $Z^{\prime}$ boson masses can
be probed down to roughly 20 MeV, and the coupling $g^{\prime}$ can be probed
down to the order of $10^{-5}$. In the high $Z^{\prime}$ mass end, if the
gauge coupling is fixed at the perturbative limit of $4\pi$, the P2 experiment
can reach $Z^{\prime}$ masses up to 70 TeV and 79 TeV for the chiral
$U(1)^{\prime}_{R}$ and $U(1)^{\prime}_{X}$ models respectively in $e+p$
scattering, and up to 90 TeV and 79 TeV for the $U(1)^{\prime}_{L}$ and
$U(1)^{\prime}_{X}$ models in $e+^{12}$C mode. The P2 sensitivities of heavy
$Z^{\prime}$ bosons are largely complementary to the direct searches at the
LHC and future higher energy colliders.

####  4.3.2 Non-chiral models

Figure 3: Sensitivity of the P2 experiment to the generic $U(1)^{\prime}$
model with only mass mixing $\sin\theta$ and in the limit of
$g^{\prime}Q^{\prime}=0$. The notation is the same as in Fig. 2.  Figure 4:
Sensitivity of the P2 experiment to the generic $U(1)^{\prime}$ model with
only kinetic mixing $\epsilon$ and in the limit of $g^{\prime}Q^{\prime}=0$.
The notation is the same as in Fig. 2.

For the generic non-chiral $U(1)^{\prime}$ model with $Z$-$Z^{\prime}$ mass
mixing, the P2 sensitivities and corresponding existing limits are shown in
Fig. 3 for the simplest case of $g^{\prime}Q^{\prime}=0$. This corresponds to
the cases of negligibly small couplings ($g^{\prime}/\sin\theta\ll 1$) or no
SM fermions carrying $U(1)^{\prime}$ charges (known as the secluded
$Z^{\prime}$ model). The limits from LEP $e^{+}e^{-}\to\ell^{+}\ell^{-}$ data
are very weak and thus are not shown in Fig. 3. The P2 sensitivity for this
model is given in Eq. (25). As in Fig. 2, for $m_{Z^{\prime}}\lesssim$ 100
MeV, the P2 sensitivities are almost independent of $m_{Z^{\prime}}$. In the
mass range $100\;{\rm MeV}\lesssim m_{Z^{\prime}}\lesssim m_{Z}$, the factor
in the parentheses of Eq. (25) can be simplified to be

$\displaystyle\frac{m_{Z}^{2}}{m_{Z^{\prime}}^{2}+Q^{2}}-1\ \simeq\
\frac{m_{Z}^{2}}{m_{Z^{\prime}}^{2}}\,.$ (53)

Therefore, in this mass range P2 excludes an effective cutoff scale
$\Lambda=m_{Z^{\prime}}/\sin\theta$, which is 560 GeV and 1.1 TeV for the
$e+p$ and $e+^{12}$C scattering, respectively. The P2 sensitivity vanishes
when the $Z^{\prime}$ mass is close to $m_{Z}$. When $m_{Z^{\prime}}>m_{Z}$,
the factor in Eq. (53) changes its sign, and approaches the value $-1$ in the
limit of $m_{Z^{\prime}}\gg m_{Z}$. As a result, the P2 sensitivities are
constant in the heavy $Z^{\prime}$ limits, which are respectively
$\sin\theta<0.13$ and $0.07$ for $e+p$ and $e+^{12}$C scattering.

Table 5:  Similar to Tab. 4 except for non-chiral $U(1)^{\prime}$ models, summarized from Figs. 3 to 6. The columns of $U(1)^{\prime}$ ($\sin\theta$) in the $e+p$ mode and $U(1)^{\prime}$ ($\epsilon$) for all models are void because P2 does not have better sensitivities than the existing limits.  models | mode |  $m_{Z^{\prime}}$ ranges |  $\sin\theta$ or $\epsilon$ ranges
---|---|---|---
$U(1)^{\prime}$ ($\sin\theta$) | $e+p$ | $-$ | $-$
$e+^{12}$C |  | [110 MeV, 200 MeV]
---
($>70$ GeV)
| $[1.1\times 10^{-4},\,1.9\times 10^{-4}]$
---
$[0.07,\,0.15]$
$U(1)^{\prime}$ ($\epsilon$) | $e+p$ | $-$ | $-$
$e+^{12}$C | $-$ | $-$
$U(1)_{B}$ (0.01) | $e+p$ |  | [120 MeV, 200 MeV]
---
[70 GeV, 100 GeV]
| $[1.4\times 10^{-4},\,2.3\times 10^{-4}]$
---
$[0.08,\,0.20]$
$e+^{12}$C |  | [120 MeV, 200 MeV]
---
($>70$ GeV)
| $[1.4\times 10^{-4},\,2.3\times 10^{-4}]$
---
$[0.07,\,0.13]$
$U(1)_{B}$ (1) | $e+p$ |  | [70 MeV, 10 GeV]
---
[70 GeV, 500 GeV]
| $[1.2\times 10^{-5},\,1.2\times 10^{-3}]$
---
$[8\times 10^{-3},\,0.06]$
$e+^{12}$C |  | [83 MeV, 10 GeV]
---
[70 GeV, 650 GeV]
($>2.3$ TeV)
| $[1.4\times 10^{-5},\,1.3\times 10^{-3}]$
---
$[9\times 10^{-3},\,0.055]$
$[0.067,\,0.07]$
$U(1)_{B}$ (10) | $e+p$ | [70 MeV, 305 GeV] | $[4.0\times 10^{-6},\,0.012]$
$e+^{12}$C |  | [83 MeV, 305 GeV]
---
($>5$ TeV)
| $[4.2\times 10^{-6},\,0.012]$
---
$[0.067,\,0.07]$
$U(1)_{B-L}$ (0.01) | $e+p$ | [90 GeV, 110 GeV] | $[0.17,\,0.19]$
$e+^{12}$C |  | [110 MeV, 560 MeV]
---
($>70$ GeV)
| $[1\times 10^{-4},\,5.0\times 10^{-4}]$
---
$[0.07,\,0.2]$
$U(1)_{B-L}$ (1) | $e+p$ |  | [80 MeV, 600 MeV]
---
[70 GeV, 620 GeV]
| $[2.4\times 10^{-5},\,1.2\times 10^{-4}]$
---
$[0.013,\,0.1]$
$e+^{12}$C |  | [80 MeV, 600 MeV]
---
[70 GeV, 260 GeV]
($>400$ GeV)
| $[2.4\times 10^{-5},\,1.2\times 10^{-4}]$
---
$[0.013,\,0.18]$
$U(1)_{B-L}$ (10) | $e+p$ |  | [70 MeV, 200 MeV]
---
[70 GeV, 420 GeV]
| $[7.0\times 10^{-6},\,1.4\times 10^{-5}]$
---
$[4.3\times 10^{-3},\,0.028]$
$e+^{12}$C |  | [83 MeV, 600 MeV]
---
[70 GeV, 420 GeV]
| $[9.0\times 10^{-6},\,1.5\times 10^{-5}]$
---
$[4.5\times 10^{-3},\,0.032]$

Similar to the chiral models, the APV limits in the generic $U(1)^{\prime}$
model have the same features as those for the P2 prospects. In the limit of
$m_{Z^{\prime}}\ll Q$, Eq. (25) implies that the new physics contributions to
APV is mostly determined by the energy scale $Q$, i.e.

$\displaystyle\frac{\Delta A^{\rm PV}(^{133}{\rm Cs})/A_{\rm SM}^{\rm
PV}(^{133}{\rm Cs})}{\Delta A^{\rm PV}(^{12}{\rm C})/A_{\rm SM}^{\rm
PV}(^{12}{\rm C})}(m_{Z^{\prime}}\ll Q)$ $\displaystyle\ =\ $
$\displaystyle\left(\frac{93\;{\rm MeV}}{30\;{\rm MeV}}\right)^{2},$ (54)

and the resulting APV limit is stronger than the P2 prospect in $e+^{12}$C
scattering by a factor of

$\displaystyle\left(\frac{0.49\%}{0.77\%}\right)^{1/2}\left(\frac{93\;{\rm
MeV}}{30\;{\rm MeV}}\right)\ \simeq\ 2.47\,.$ (55)

In the limit of $m_{Z^{\prime}}\gg Q$, comparison of the APV limits and the P2
prospects will be only dictated by the accuracies, i.e. the APV limit is
weaker than the P2 sensitivity for $e+^{12}$C by a factor of
$\sqrt{0.49\%/0.77\%}\simeq 0.80$. As shown in Fig. 3, the APV constraints
have excluded P2 sensitivities for $e+p$ scattering. The limit from $\nu+e$
scattering has the same feature in the high $Z^{\prime}$ mass range, and can
go to masses beyond the ones from direct LHC searches, as shown in Fig. 3.
However, the P2 experiment can exceed the neutrino scattering limits. The
resultant P2 ranges of $m_{Z^{\prime}}$ and $\sin\theta$ are collected in
Table 5.

The P2 sensitivities for the most generic $U(1)^{\prime}$ model with a kinetic
mixing of $Z^{\prime}$ with the SM $Z$ boson and with
$g^{\prime}Q^{\prime}=0$, as well as the corresponding existing limits, are
presented in Fig. 4. As shown in this figure, the P2 sensitivities for this
case are not as competitive as others. This is because $Z^{\prime}$ with a
sizable kinetic mixing is more photon-like when $m_{Z^{\prime}}$ decreases. In
particular, based on the couplings in Tab. 1, in the limit of
$m_{Z^{\prime}}\rightarrow 0$,

$\displaystyle g^{\prime}_{e_{L}}-g^{\prime}_{e_{R}}\ \propto\
\frac{m_{Z^{\prime}}^{2}}{m_{Z}^{2}}\ \to\ 0\,.$ (56)

In fact, all the $g^{\prime}_{f}$ couplings are proportional to the electric
charge of $f$ in this limit, which implies that $Z^{\prime}$ would not mediate
parity-violating processes. In this case, $\epsilon$ is mainly constrained by
the modification of $Z$ couplings. As a result of the enhancement of $\Delta
A^{\rm PV}$ due to the $(1-r_{m})$ factor in the denominator of Eqs. (26) and
(27), there are dips at the $Z$ mass. However, the P2 sensitivities at the
high $Z^{\prime}$ mass range are still precluded by the electroweak precision
data.

Figure 5: Sensitivity of the P2 experiment to the chiral $U(1)_{B}$ model with
mass mixing $\sin\theta$ and the three values of $g^{\prime}/\sin\theta=0.01$
(upper), $1$ (middle) and $10$ (lower). The notations are the same as in Fig.
2.

The P2 sensitivities for the $U(1)_{B}$ model with $Z$-$Z^{\prime}$ mass
mixing and three values of $g^{\prime}/\sin\theta=0.01$, $1$ and $10$ are
shown respectively in the upper, middle and lower panels of Fig. 5. The master
formula for $\Delta A^{\rm PV}$ is given in Eq. (28). For the three benchmark
values, the $R$ ratio is respectively 1.96, 196 and 1960 for $e+p$ scattering,
and $-0.34$, $-34$ and $-340$ for $e+^{12}$C scattering. Generally speaking,
when the $Z^{\prime}$ mass is small such that $m_{Z}^{2}\gtrsim
m_{Z^{\prime}}^{2}+Q^{2}$, the P2 sensitivities of $\sin\theta$ can be
significantly improved by a factor of $\sqrt{1+R}$. For sufficiently large
$m_{Z^{\prime}}$, i.e.

$\displaystyle m_{Z^{\prime}}\ =\ \sqrt{1+R}\,m_{Z}\,,$ (57)

the two terms in Eq. (28) cancel each other for all three values of
$g^{\prime}$ in $e+p$ scattering, which turns out to happen respectively at

$\displaystyle m_{Z^{\prime}}=1.72m_{Z}\;\;[157\;{\rm GeV}]\,,\quad
14.1m_{Z^{\prime}}\;\;[1.28\;{\rm TeV}]\,,\quad
44.3m_{Z^{\prime}}\;\;[4.03\;{\rm TeV}]\,,$ (58)

as indicated by the solid black peaks in Fig. 5. For the case of
$g^{\prime}/\sin\theta=0.01$ in $e+^{12}$C scattering, the cancellation
happens at

$\displaystyle m_{Z^{\prime}}\ =\ 0.81m_{Z}\;\;[74\;{\rm GeV}]\,,$ (59)

as shown by the peak of the dashed black line in the upper panel of Fig. 5.
For the two cases of $g^{\prime}/\sin\theta=1$ and $10$ in $e+^{12}$C
scattering, the factor $R$ is below $-1$ and there is no cancellation between
the two terms in Eq. (28). Therefore there are no peaks for the dashed curves
in the middle and lower panels of Fig. 5. In the limit of
$m_{Z^{\prime}}\gg\sqrt{1+R}\,m_{Z}$, the first term in the bracket of Eq.
(28) can be neglected, and the P2 sensitivity to $\sin\theta$ approaches a
constant value, which is the same as in Fig. 3. For the case of
$g^{\prime}/\sin\theta=0.01$, the current limits on the mass mixing angle
$\sin\theta$ in the upper panel of Fig. 5 are almost the same as in Fig. 3,
while the limits for the cases of $g/\sin\theta=1$ and 10 with larger gauge
couplings tend to be more stringent, as presented in the middle and lower
panels of Fig. 5. In light of all the limits, the ranges of $m_{Z^{\prime}}$
and $\sin\theta$ that can be probed at P2 for the $U(1)_{B}$ model are
collected in Table 5.

Figure 6: Similar to Fig. 5 but for the $U(1)_{B-L}$ model.

The $U(1)_{B-L}$ model with $Z$-$Z^{\prime}$ mass mixing is quite similar to
the $U(1)_{B}$ model, and the corresponding P2 sensitivities and current
limits are presented in Fig. 6, with the three benchmark values of
$g^{\prime}/\sin\theta=0.01$, $1$ and $10$ respectively in the upper, middle
and lower panels. The key difference between the $U(1)_{B}$ and $U(1)_{B-L}$
models is the $R$ factor relation in Eq. (31). As a result, for $e+p$
scattering in the $U(1)_{B-L}$ model, we have a disappearing P2 sensitivity
only at

$\displaystyle m_{Z^{\prime}}\ =\ 0.57m_{Z}\;\;[51\;{\rm GeV}]$ (60)

for $g^{\prime}/\sin\theta=0.01$, while for larger $g^{\prime}=1$ and $10$ the
factor $R$ is below $-1$ and there is no peak. Regarding $e+^{12}$C
scattering, P2 looses sensitivity at the following $Z^{\prime}$ masses:

$\displaystyle m_{Z^{\prime}}\ =\ 1.06m_{Z}\;\;[96\;{\rm GeV}]\,,\quad
3.57m_{Z^{\prime}}\;\;[325\;{\rm GeV}]\,,\quad
10.9m_{Z^{\prime}}\;\;[990\;{\rm GeV}]$ (61)

for the three values of $g^{\prime}/\sin\theta=0.01$, $1$ and $10$,
respectively. The P2 sensitivities of $m_{Z^{\prime}}$ and $\sin\theta$ are
also summarized in Table 5.

For the $U(1)_{B}$ and $U(1)_{B-L}$ models, the couplings of the $Z^{\prime}$
boson to proton and 12C receive contributions from the direct coupling and the
$Z$-$Z^{\prime}$ mixing, which are proportional to $g^{\prime}$ and
$\sin\theta$, respectively. When the gauge coupling $g^{\prime}$ is
sufficiently large, the couplings of $Z^{\prime}$ to proton and 12C will be
dominated by the direct coupling $g^{\prime}$, which is different from the
pure $Z$-$Z^{\prime}$ mass mixing case in Fig. 3. On the other hand, for
sufficiently large $g^{\prime}$ and light $Z^{\prime}$, the new physics
contribution to the asymmetry will be proportional to the $R$ factor, i.e.
$\Delta A^{\rm PV}\propto R$ in Eq. (28). Taking into account the different
factors for proton and 12C in Eq. (36), for the cases of
$g^{\prime}/\sin\theta=0.01$, 1 and 10 for $U(1)_{B}$ and the cases of
$g^{\prime}/\sin\theta=1$ and 10 for $U(1)_{B-L}$, the P2 sensitivities for
low $m_{Z^{\prime}}$ are accidentally roughly the same for proton and 12C, as
shown in Figs. 5 and 6.

##  5 Conclusion

With longitudinally polarized electron scattering off proton and 12C, the P2
experiment can perform a measurement of the parity-violating asymmetry $A^{\rm
PV}$ at the unprecedented precision of below per cent level, which would
result in a precision of $3.3\times 10^{-4}$ for the weak mixing angle $s_{W}$
at sub-GeV scales. Such high-precision measurements have excellent
sensitivities to potential BSM physics, as we have illustrated in this paper,
taking as a case study arguably the most straightforward scenario: new
$Z^{\prime}$ bosons. We assumed chiral and non-chiral $U(1)^{\prime}$ models,
for the latter case assuming kinetic or mass $Z$-$Z^{\prime}$ mixing. For the
chiral case, where the $Z^{\prime}$ couplings to left- and right-handed
fermions are different, we adopt three $U(1)^{\prime}$ models, i.e. the
$U(1)^{\prime}_{L}$, $U(1)^{\prime}_{R}$ and $U(1)^{\prime}_{X}$ models given
in Table 2. For the non-chiral models, we first consider a generic
$U(1)^{\prime}$ model with either mass ($\sin\theta$) or kinetic ($\epsilon$)
mixing of $Z^{\prime}$ with the $Z$ boson and setting $g^{\prime}$ to zero,
and then generalize to the $U(1)_{B}$ and $U(1)_{B-L}$ models with three
different benchmark values of $g^{\prime}/\sin\theta=0.01$, $1$ and $10$.

For all chiral and non-chiral models, the leading-order new physics
contributions are dominated by the interference terms of the BSM diagrams with
the SM terms, i.e. terms proportional to the gauge couplings $g^{\prime 2}$,
the mass mixing angle $\sin^{2}\theta$ or the kinetic mixing parameter
$\epsilon^{2}$. The P2 sensitivities for these $U(1)^{\prime}$ models, as well
as the current limits, are presented in Figs. 2 to 6. Let us summarize our
main results from these figures:

  * •

It is a general feature that when the $Z^{\prime}$ mass is smaller than the
energy scale $Q\simeq 93$ MeV of the P2 experiment, i.e.
$m_{Z^{\prime}}\lesssim Q$, the P2 sensitivities will be independent of the
$Z^{\prime}$ mass, as shown in the second and third columns of Table 3.
However, the P2 prospects in the low $Z^{\prime}$ mass range are mostly
precluded by the beam-dump experiments, and some are also limited by APV
measurements, neutrino scattering data, electron and muon $g-2$, and
electroweak precision data.

  * •

In the large $Z^{\prime}$ mass limit, for the chiral $U(1)^{\prime}$ models
and the kinetic mixing case of non-chiral models, the P2 sensitivities are
proportional to $g^{\prime}/m_{Z^{\prime}}$ or $\epsilon/m_{Z^{\prime}}$. In
other words, the P2 experiment can probe effectively a UV cutoff scale
$\Lambda=m_{Z^{\prime}}/g^{\prime}$ or $\Lambda=m_{Z^{\prime}}/\epsilon$. For
the chiral models, the P2 experiment has sensitivities to a $Z^{\prime}$ mass
up to 79 TeV in the proton mode and up to 90 TeV in 12C mode, assuming the
perturbative gauge coupling limit of $g^{\prime}=4\pi$, as shown in Table 4.
For the case of mass mixing $\sin\theta$ in non-chiral models, when the
$Z^{\prime}$ boson is very heavy, say
$m_{Z^{\prime}}\gtrsim\sqrt{1+R}\,m_{Z}$, the first term in the bracket of Eq.
(25) or (28) will be highly suppressed, and the P2 prospects for $\sin\theta$
will be independent of the $Z^{\prime}$ mass, which is very different from the
cases above. The P2 sensitivities in the limit of heavy $Z^{\prime}$ bosons
are collected in the fourth and fifth columns of Table 3.

One can see that for most of the cases in this paper, i.e. the three chiral
models, generic $U(1)^{\prime}$ with mass mixing ($\sin\theta$), the
$U(1)_{B}$ model, and the $U(1)_{B-L}$ model with $g^{\prime}/\sin\theta=0.01$
and $1$, the P2 experiment can probe a high UV scale $\Lambda$ or a mass
mixing angle $\sin\theta$ that is currently not constrained and even goes
beyond the direct search limits from LHC, in particular for $e+^{12}$C
scattering for most of the models. It is promising that a superheavy
$Z^{\prime}$ boson can be directly searched for at future 100 TeV colliders
Arkani-Hamed:2015vfh ; Golling:2016gvc , which is largely complementary to the
prospects at the high-precision P2 experiment. The P2 prospects for a heavy
$Z^{\prime}$ boson in the kinetic mixing $U(1)^{\prime}$ model is precluded by
electroweak precision data, while the $U(1)_{B-L}$ model with a large gauge
coupling $g^{\prime}/\sin\theta$ is excluded by neutrino scattering data for a
heavy $Z^{\prime}$ boson.

  * •

One may wonder whether the P2 experiment running with a ${}^{12}{\rm C}$
target could have significantly improved sensitivity to new physics compared
to that with a proton target. From Figs. 2 to 6, we can see that this is
rather model-dependent, subject to the quantum numbers of SM fermions under
the $U(1)^{\prime}$ gauge group for the chiral $U(1)^{\prime}$ model, and
depending on the gauge coupling $g^{\prime}$ for the non-chiral models. For
the $U(1)^{\prime}_{R}$ model, there is no sensitivity with the 12C target,
which is due to an accidental cancellation of the quantum numbers in Eq. (17).
For the $U(1)^{\prime}_{L}$ model, the 12C target can improve the sensitivity
from the proton target by a factor of 1.4, whereas for the $U(1)^{\prime}_{X}$
model the prospects at proton and 12C targets are roughly the same. For non-
chiral models, the ${}^{12}{\rm C}$ target does exhibit better sensitivity
when $g^{\prime}$ is small; see Figs. 3, 5 and 6. In the limit of
$g^{\prime}=0$, as shown in Figs. 3, this improvement can reach a factor of 2.
In any case, the comparison of electron-proton and electron-12C scatterings
can obviously help in distinguishing models. Comparing further with the parity
asymmetry in polarized electron-electron (Møller) scattering provides further
handle on identifying the underlying physics Benesch:2014bas .

  * •

The nuclear 6$S_{1/2}$ $-$ 7$S_{1/2}$ transition in 133Cs provides stringent
limits on the chiral and non-chiral models which are comparable to the P2
prospects. Depending on model details, the P2 experiment can achieve better
sensitivities than the APV measurements, in particular if the $Z^{\prime}$
boson is relatively heavy.

Taking into account all relevant existing constraints, we find that P2 can
probe a broad range of $m_{Z^{\prime}}$ in the $U(1)^{\prime}$ models
considered in this paper, which are collected in Table 4 for chiral models and
Table 5 for non-chiral models.

One should note that the specific $U(1)^{\prime}$ models considered here are
only for illustration purposes, and the analysis in this paper can be easily
generalized to other $U(1)^{\prime}$ models, by taking different choices of
the quantum numbers $x$ and $y$ in Eq. (13), or to flavor-sensitive scenarios
such as the $L_{\mu}-L_{\tau}$ model.

###### Acknowledgements.

We thank Niklaus Berger, Jeff Dror, Bhaskar Dutta and Krishna Kumar for
helpful discussions. The work of B.D. is supported in part by the US
Department of Energy under Grant No. DE-SC0017987, by the Neutrino Theory
Network Program, and by a Fermilab Intensity Frontier Fellowship. X.J.X. is
supported by the “Probing dark matter with neutrinos” ULB-ARC convention and
by the F.R.S./FNRS under the Excellence of Science (EoS) project No. 30820817
- be.h “The $H$ boson gateway to physics beyond the Standard Model”. Y.Z. is
partially supported by “the Fundamental Research Funds for the Central
Universities”.

##  Appendix A The contribution of the axial-vector coupling

In this appendix we show that the effect of $g_{A}$ (and $g^{\prime}_{A}$) is
suppressed by a factor of $E_{e}/m_{N}$ (which in P2 is about 0.16 for
electron-proton scattering, and 0.013 for electron-12C scattering) compared to
the leading-order contribution to $A^{{\rm PV}}$ from $g_{V}$ (and
$g_{V}^{\prime}$). To evaluate the contributions of $g_{A}$ and
$g^{\prime}_{A}$, we first write down the full amplitude:

$\displaystyle i{\cal M}_{L,R}$ $\displaystyle\ \propto\ $
$\displaystyle[\overline{u_{3}}\gamma^{\mu}P_{L,R}u_{1}][\overline{u_{4}}\gamma_{\mu}u_{2}]\frac{-e^{2}}{q^{2}}$
(62)
$\displaystyle+[\overline{u_{3}}\gamma^{\mu}P_{L,R}g_{L,R}u_{1}][\overline{u_{4}}\gamma_{\mu}(g_{V}+g_{A}\gamma^{5})u_{2}]\frac{1}{q^{2}-m_{Z}^{2}}$
$\displaystyle+[\overline{u_{3}}\gamma^{\mu}P_{L,R}g^{\prime}_{L,R}u_{1}][\overline{u_{4}}\gamma_{\mu}(g^{\prime}_{V}+g^{\prime}_{A}\gamma^{5})u_{2}]\frac{1}{q^{2}-m_{Z^{\prime}}^{2}}\,.$

Hence Eq. (5) is modified to

$\displaystyle i{\cal M}_{L,R}$ $\displaystyle\ \propto\ $
$\displaystyle[\overline{u_{3}}\gamma^{\mu}P_{L,R}u_{1}]\left[\overline{u_{4}}\gamma_{\mu}(G_{L,R}+G_{A}\gamma^{5})u_{2}\right]\,,$
(63)

where

$G_{A}\ =\
\frac{g_{L,R}g_{A}}{q^{2}-m_{Z}^{2}}+\frac{g^{\prime}_{L,R}g^{\prime}_{A}}{q^{2}-m_{Z^{\prime}}^{2}}\,.$
(64)

Applying the standard trace technology, we obtain:

$\displaystyle|{\cal M}_{L,R}|^{2}$ $\displaystyle\ \propto\ $ $\displaystyle
G_{L,R}^{2}\left(2-2y-4r_{2}y-r_{1}^{2}+4r_{2}^{2}y^{2}\right)-8G_{L,R}G_{A}\left(1-r_{2}y\right)r_{2}y$
(65)
$\displaystyle+G_{A}^{2}\left(2+2y-4r_{2}y+r_{1}^{2}+4r_{2}^{2}y^{2}\right),$

where $r_{1}\equiv m_{e}/E_{e}\approx 3\times 10^{-3}$, $r_{2}\equiv
E_{e}/m_{N}\approx 0.16$ (for $e+p$ scattering) or 0.013 (for
$e+{}^{12}\text{C}$ scattering), and $y\equiv-q^{2}/4E_{e}^{2}\approx 0.09$.
The second term in Eq. (65) proportional to $G_{A}$ implies that the axial-
vector contribution is suppressed by $r_{2}y$. The third term proportional to
$G_{A}^{2}$, though not suppressed, cancels in $|{\cal M}_{R}|^{2}-|{\cal
M}_{L}|^{2}$.

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