# Constraining nonstandard neutrino interactions at electron colliders

Jiajun Liao<EMAIL_ADDRESS>School of Physics, Sun Yat-Sen
University, Guangzhou 510275, China  Yu Zhang<EMAIL_ADDRESS>Institutes of
Physical Science and Information Technology, Anhui University, Hefei 230601,
China  School of Physics and Materials Science, Anhui University, Hefei
230601,China

###### Abstract

The presence of nonstandard neutrino interactions (NSI) has a large effect on
the precision measurements at next generation neutrino oscillation
experiments. Other type of experiments are needed to constrain the NSI
parameter space. We study the constraints on NSI with electrons from current
and future $e^{+}e^{-}$ collider experiments including Belle II, STCF and
CEPC. We find that Belle II and STCF will provide competitive and
complementary bounds on electron-type NSI parameters as compared to the
current global analysis, and strong improvements in the constraints on tau-
type NSI. In addition, CEPC alone will impose stringent constraints on the
parameter space of NSI with electrons. We find that the degeneracy between the
left-handed (vector) and right-handed (axial-vector) NSI parameters can be
lifted by combining the data from three different running modes at CEPC, and
the limits on $\epsilon_{ee}^{eL}$ ($\epsilon_{ee}^{eV}$) and
$\epsilon_{ee}^{eR}$ ($\epsilon_{ee}^{eA}$) can reach 0.002 at CEPC even if
both the left and right-handed NSI parameters are present.

##  I Introduction

Neutrino oscillation has been well established after successful measurements
from a variety of neutrino experiments using solar, atmospheric, reactor, and
accelerator neutrinos in the last two decades, and current data from most
neutrino oscillation experiments can be successfully explained in a three
neutrino oscillation framework Tanabashi  _et al._ (2018). Since the
explanation of neutrino oscillations requires the neutrino masses to be
nonzero, the observation of neutrino oscillations provides a clear evidence of
new physics beyond the Standard Model (SM). Next generation neutrino
oscillation experiments,such as DUNE Acciarri  _et al._ (2015), Hyper-
Kamiokande Abe  _et al._ (2018) and JUNO An  _et al._ (2016) will not only do
a high precision measurement of the oscillation parameters in the standard
three neutrino oscillation framework, but also reach the sensitivity to probe
new physics in the neutrino sector. A model-independent way of studying new
physics in neutrino experiments was given in the effective field theory
framework of nonstandard neutrino interactions (NSI); for reviews see Refs.
Ohlsson (2013); Miranda and Nunokawa (2015); Farzan and Tortola (2018). In
this framework, NSI are typically described by dimension-six four-fermion
operators of the form Wolfenstein (1978); Guzzo  _et al._ (1991),

$\displaystyle\mathcal{L}_{\text{NSI}}^{\text{NC}}$
$\displaystyle=-2\sqrt{2}G_{F}\epsilon^{fC}_{\alpha\beta}\left[\overline{\nu_{\alpha}}\gamma^{\rho}P_{L}\nu_{\beta}\right]\left[\bar{f}\gamma_{\rho}P_{C}f\right]\,,$
(1) $\displaystyle\mathcal{L}_{\text{NSI}}^{\text{CC}}$
$\displaystyle=-2\sqrt{2}G_{F}\epsilon^{ff^{\prime}C}_{\alpha\beta}\left[\overline{\nu_{\beta}}\gamma^{\rho}P_{L}\ell_{\alpha}\right]\left[\bar{f^{\prime}}\gamma_{\rho}P_{C}f\right]\,,$
(2)

where $\alpha$, $\beta$ label the lepton flavors ($e,\mu,\tau$), $f$ denotes
the fermion fields ($u,d,e$), and $C$ indicates the chirality ($L,R$),
$\epsilon^{ff^{\prime}C}_{\alpha\beta}$ are dimensionless parameters that
characterize the strength of the new interactions in units of the Fermi
constant $G_{F}$. The neutral current (NC) NSI will affect neutrino
propagation in matter, and the charged current (CC) NSI mainly affect neutrino
production and detection. Both the NC and CC NSI at neutrino oscillation
experiments have been extensively studied in the literature; see the
references in Ref. Dev (2019). Here we focus on NC NSI since the bounds on the
CC NSI are generally much stronger than the NC NSI Biggio  _et al._ (2009).
The presence of NC NSI have a large impact on the determination of the mass
ordering, CP violation and $\theta_{23}$ octant at neutrino oscillation
experiments Gonzalez-Garcia  _et al._ (2001); Miranda  _et al._ (2006); Coloma
_et al._ (2011); Friedland and Shoemaker (2012); Masud  _et al._ (2016); de
Gouvêa and Kelly (2016); Coloma (2016); Liao  _et al._ (2016); Forero and
Huber (2016); Masud and Mehta (2016); Ge and Smirnov (2016); Blennow  _et al._
(2016); Agarwalla  _et al._ (2016); Fukasawa  _et al._ (2017); Deepthi  _et
al._ (2017); Liao  _et al._ (2017). In particular, due to the presence of the
generalized mass ordering degeneracy induced by NSI, the neutrino mass
ordering cannot be resolved by neutrino oscillation experiments alone Coloma
and Schwetz (2016). In order to improve the sensitivity of neutrino
oscillation experiments in the presence of NSI, other types of experiments
such as neutrino scattering or collider experiments are needed to constrain
the NSI parameter space Coloma  _et al._ (2017a).

The constraints on NSI with quarks have been well studied in the literature
Gonzalez-Garcia and Maltoni (2013); Esteban  _et al._ (2018). In particular,
the recent measurements of the coherent neutrino nucleus scattering by the
COHERENT experiment Akimov  _et al._ (2017, 2021) has been used to impose
constraints on NSI with the $u$ and $d$ quarks, and the generalized mass
ordering degeneracy due to NSI with quarks can be excluded by more than
$3\sigma$ with the COHERENT data Coloma  _et al._ (2017b, 2020). In addition,
the parameter space of NSI with quarks can be further constrained by using the
monojet events at LHC Friedland  _et al._ (2012); Choudhury  _et al._ (2018).
However, there are models that predict NSI with only electrons but not with
quarks in the literature; see e.g., Refs. Forero and Huang (2017); Babu  _et
al._ (2020); Liao  _et al._ (2020). Due to degeneracies between the left and
right-handed NSI parameters, constraints on NSI with electrons are still weak
as compared to NSI with quarks.

The Lagrangian of NC NSI with electrons can be written as,

$\displaystyle\mathcal{L}_{\rm{NSI}}^{\rm NC,e}$ $\displaystyle=$
$\displaystyle-2\sqrt{2}G_{F}\epsilon_{\alpha\beta}^{eL}(\bar{\nu}_{\alpha}\gamma^{\mu}P_{L}\nu_{\beta})(\bar{e}\gamma_{\mu}P_{L}e)-2\sqrt{2}G_{F}\epsilon_{\alpha\beta}^{eR}(\bar{\nu}_{\alpha}\gamma^{\mu}P_{L}\nu_{\beta})(\bar{e}\gamma_{\mu}P_{R}e)$
(3) $\displaystyle=$
$\displaystyle-\sqrt{2}G_{F}\epsilon_{\alpha\beta}^{eV}(\bar{\nu}_{\alpha}\gamma^{\mu}P_{L}\nu_{\beta})(\bar{e}\gamma_{\mu}e)+\sqrt{2}G_{F}\epsilon_{\alpha\beta}^{eA}(\bar{\nu}_{\alpha}\gamma^{\mu}P_{L}\nu_{\beta})(\bar{e}\gamma_{\mu}\gamma^{5}e)\,,$
(4)

where

$\epsilon_{\alpha\beta}^{eV}\equiv\epsilon_{\alpha\beta}^{eL}+\epsilon_{\alpha\beta}^{eR}\,,\quad\epsilon_{\alpha\beta}^{eA}\equiv\epsilon_{\alpha\beta}^{eL}-\epsilon_{\alpha\beta}^{eR}\,,$
(5)

with $\epsilon^{eV}_{\alpha\beta}$ ($\epsilon^{eA}_{\alpha\beta}$) describing
the strength of the new vector (axial-vector) interactions between electrons.
NSI with electrons will lead to new contributions to the monophoton process
$e^{+}e^{-}\to\nu\bar{\nu}\gamma$ at electron colliders. An early study of NSI
with electrons using the monophoton events at LEP has been given in Ref.
Berezhiani and Rossi (2002), and constraints on NSI with electrons from
neutrino scattering, solar and reactor neutrino experiments are performed in
Refs. Davidson  _et al._ (2003); Barranco  _et al._ (2006); Bolanos  _et al._
(2009). A combination of the data from LEP, neutrino scattering, solar and
reactor neutrino experiments can be found in Refs. Barranco  _et al._ (2008);
Forero and Guzzo (2011). The $e^{+}e^{-}$ collider experiment LEP can help in
removing degeneracy in the global analysis of constraints on NSI parameters.
Moreover, the current constraints on tau-type NSI with electrons are mainly
coming from LEP. However, due to cancellation between the left-handed (vector)
and right-handed (axial-vector) NSI parameters, the allowed ranges of NSI with
electrons are still large if both the left and right-handed NSI parameters
exist. Considering future $e^{+}e^{-}$ collider experiment will provide much
more data than LEP, we study the constraints on NSI with electrons from
current $e^{+}e^{-}$ collider experiment Belle II Abe  _et al._ (2010), and
the proposed future $e^{+}e^{-}$ collider experiments including the Super Tau
Charm Factory (STCF) Luo  _et al._ (2019) and CEPC Dong  _et al._ (2018) in
this work.

This paper is organized as follows. In Sec. II we describe the signal and
backgrounds of probing NSI at electron colliders. In Sec. III we present the
constraints on NSI at Belle II and STCF that are operated with the center-of-
mass (CM) energy $\sqrt{s}\ll M_{Z}$. In Sec. IV we consider the constraints
on NSI at CEPC that is operated with $\sqrt{s}\geq M_{Z}$. Our conclusions are
drawn in Sec. V.

##  II Signals and backgrounds at electron colliders

The neutrinos can be measured at $e^{+}e^{-}$ colliders via the production of
single photon associated with a neutrino-antineutrino pair,
$e^{+}e^{-}\to\nu\bar{\nu}\gamma$. In the SM, the tree-level Feynman diagrams
for this process are shown in Fig. 1. For muon and tau neutrinos, only the
$Z$-boson exchange diagram contributes; for electron neutrinos, both the
$Z$-boson and $W$-boson exchange diagrams contribute.

Figure 1: The tree-level Feynman diagrams for
$e^{+}e^{-}\to\nu_{\ell}\bar{\nu}_{\ell}\gamma$ process at electron colliders.
The $W$-boson exchange diagrams only contribute to the
$e^{+}e^{-}\rightarrow\nu_{e}\bar{\nu_{e}}\gamma$ process, and the $Z$-boson
exchange diagram can contribute to all neutrino flavors with
$\ell=e,\mu,\tau$.

The cross section for the single photon production from $e^{+}e^{-}$
annihilation, $e^{+}e^{-}\to\nu\bar{\nu}\gamma$, can be approximately
factorized into the process without photon emission,
$e^{+}e^{-}\to\nu\bar{\nu}$, times the improved Altarelli-Parisi radiator
function Nicrosini and Trentadue (1989); Montagna  _et al._ (1995),

$\frac{d^{2}\sigma}{dx_{\gamma}dz_{\gamma}}=H\left(x_{\gamma},z_{\gamma};s\right)\sigma_{0}(s_{\gamma})\,,$
(6)

where the radiator function

$H\left(x_{\gamma},z_{\gamma};s\right)=\frac{\alpha}{\pi}\frac{1}{x_{\gamma}}\left[\frac{1+(1-x_{\gamma})^{2}}{1-z_{\gamma}^{2}}-\frac{x_{\gamma}^{2}}{2}\right]\,.$
(7)

Here, $E_{\gamma}$ is the energy of the initial state radiation (ISR) photon,
$\sqrt{s}$ is the CM energy at the colliders,
$x_{\gamma}=2E_{\gamma}/\sqrt{s}$ is the energy fraction emitted away by ISR,
$z_{\gamma}=\cos\theta_{\gamma}$ with $\theta_{\gamma}$ being the polar angle
of the photon, $s_{\gamma}=(1-x_{\gamma})s$ is the CM energy of the
$\nu\bar{\nu}$ system, and $\sigma_{0}$ is the cross section of the neutrino
pair production without ISR.

In the SM, $\sigma_{0}$ is given by

$\displaystyle\sigma_{0}^{\mathrm{SM}}(s)$
$\displaystyle=\sigma_{W}(s)+\sigma_{Z}(s)+\sigma_{W-Z}(s)$ (8)
$\displaystyle=\frac{G_{F}^{2}s}{12\pi}\left[2+\frac{2N_{\nu}\left(g_{L}^{2}+g_{R}^{2}\right)}{\left(1-s/M_{Z}^{2}\right)^{2}+\Gamma_{Z}^{2}/M_{Z}^{2}}+\frac{4g_{L}\left(1-s/M_{Z}^{2}\right)}{\left(1-s/M_{Z}^{2}\right)^{2}+\Gamma_{Z}^{2}/M_{Z}^{2}}\right],$

where $g_{L}=-\frac{1}{2}+\sin^{2}\theta_{W}$, $g_{R}=\sin^{2}\theta_{W}$, and
$N_{\nu}$ is the number of active neutrinos. The three terms in Eq. (8) come
from the contribution of the $W$ boson, the $Z$ boson, and their interference,
respectively. The NSI contribution to the cross section $\sigma_{0}$ can be
written analogously as

$\sigma_{0}^{\mathrm{NSI}}(s)=\frac{G_{F}^{2}}{6\pi}s\left\\{\sum_{\alpha,\beta=e,\mu,\tau}\left[\left((\epsilon_{\alpha\beta}^{eL})^{2}+(\epsilon_{\alpha\beta}^{eR})^{2}\right)-2\left(g_{L}\epsilon_{\alpha\beta}^{eL}+g_{R}\epsilon_{\alpha\beta}^{eR}\right)\frac{M_{Z}^{2}\left(s-M_{Z}^{2}\right)}{\left(s-M_{Z}^{2}\right)^{2}+\left(M_{Z}\Gamma_{Z}\right)^{2}}\right]+2\epsilon_{ee}^{eL}\right\\}.$
(9)

From Eq. (9), we see that there are 12 independent NSI parameters that can be
probed at electron colliders. Also, there is no interference between
$\epsilon_{\alpha\beta}$ with different flavor indices. Therefore, we consider
the NSI parameters with different flavor indices separately. In order to
constrain the NSI parameter space, we perform a variation of the $\epsilon$
pair with the same flavor indices $(\epsilon_{\alpha\beta}^{eL},\
\epsilon_{\alpha\beta}^{eR})$ (or equivalently $(\epsilon_{\alpha\beta}^{eV},\
\epsilon_{\alpha\beta}^{eA})$), setting other NSI parameters to be zero. In
our analysis, we only consider bounds on the flavor-conserving NSI parameters
since the new physics that induces flavor-violating NSI is more likely to be
constrained by lepton flavor violating processes111Note that for the flavor
violating NSI parameters, their contributions to the cross section are twice
of that from $\epsilon^{eL,R}_{\mu\mu,\tau\tau}$, and the constraints on
flavor violating NSI parameters at electron colliders can be easily derived
from those on $\epsilon^{eL,R}_{\mu\mu,\tau\tau}$.. To constrain the electron-
type NSI parameters $\epsilon_{ee}^{eL,R}$, we perform a variation of the
$\epsilon$ pair $(\epsilon_{ee}^{eL},\epsilon_{ee}^{eR})$ at the same time,
setting others to be zero, i.e.,
$\epsilon_{\mu\mu}^{eL,R}=\epsilon_{\tau\tau}^{eL,R}=0$. Similarly, to
constrain the muon-type (tau-type) NSI parameters $\epsilon_{\mu\mu}^{eL,R}$
($\epsilon_{\tau\tau}^{eL,R}$), we take the electron-type and tau-type (muon-
type) NSI parameters to be zero.

When $\sqrt{s}$ is far below the $Z$ resonance ($\sqrt{s}\ll M_{Z}$), the
differential production cross section for the
$e^{+}e^{-}\to\nu\bar{\nu}\gamma$ in the SM can be written as

${d\sigma^{\rm SM}\over
dx_{\gamma}dz_{\gamma}}=H(x_{\gamma},z_{\gamma},s){G_{F}^{2}s_{\gamma}\over
2\pi}C^{\rm SM}\,,$ (10)

where $C^{\rm SM}\equiv g_{L}^{2}+g_{R}^{2}+\frac{2}{3}g_{L}+\frac{1}{3}$.
Here we have integrated over the momenta of the final state neutrinos and
summed all three neutrino flavors. The contribution from NSI can be expressed
as

${d\sigma^{\rm NSI}\over
dx_{\gamma}dz_{\gamma}}=H(x_{\gamma},z_{\gamma},s){G_{F}^{2}s_{\gamma}\over
2\pi}C^{\rm NSI}\,,$ (11)

where $C^{\rm
NSI}\equiv\frac{1}{3}\sum\limits_{\alpha,\beta=e,\mu,\tau}\Big{[}(\epsilon_{\alpha\beta}^{eL})^{2}+(\epsilon_{\alpha\beta}^{eR})^{2}+2(g_{L}\epsilon_{\alpha\beta}^{eL}+g_{R}\epsilon_{\alpha\beta}^{eR})\Big{]}+\frac{2}{3}{\epsilon_{ee}^{eL}}$.
From Eqs. (10) and (11), we see that for $e^{+}e^{-}\to\nu\bar{\nu}\gamma$
production with $\sqrt{s}\ll M_{Z}$,

${\sigma^{\rm NSI}\over\sigma^{\rm SM}}={C^{\rm NSI}\over C^{\rm SM}}\,,$ (12)

The total cross section of the $e^{+}e^{-}\to\nu\bar{\nu}\gamma$ process is
$\sigma=\sigma^{\rm SM}+\sigma^{\rm NSI}$. Now the iso-$\sigma$ contour is a
circle in the $(\epsilon_{\alpha\alpha}^{eL},\ \epsilon_{\alpha\alpha}^{eR})$
plane with the center located at

$(\epsilon_{\alpha\alpha}^{eL},\
\epsilon_{\alpha\alpha}^{eR})=(-g_{L}-\delta_{\alpha e},-g_{R})\,,$ (13)

where $\delta_{\alpha e}=1$ (0) for $\alpha=e$ ($\alpha\neq e$). Similarly,
the center of the iso-$\sigma$ circle in the $(\epsilon_{\alpha\alpha}^{eV},\
\epsilon_{\alpha\alpha}^{eA})$ plane is located at

$(\epsilon_{\alpha\alpha}^{eV},\
\epsilon_{\alpha\alpha}^{eA})=(-g_{V}-\delta_{\alpha e},-g_{A}-\delta_{\alpha
e})\,,$ (14)

where $g_{V}=g_{L}+g_{R}=-\frac{1}{2}+2\sin^{2}\theta_{W}$,
$g_{A}=g_{L}-g_{R}=-\frac{1}{2}$.

For energies above the $Z$ resonance ($\sqrt{s}\geq M_{Z}$), finite distance
effects on the $W$ propagator need to be considered. These effects can be
taken into account by the following substitution Berezhiani and Rossi (2002);
Hirsch  _et al._ (2003):

$\displaystyle\sigma_{W}(s)$
$\displaystyle\rightarrow\sigma_{W}(s)F_{W}\left(s/M_{W}^{2}\right)\,,$ (15)
$\displaystyle\sigma_{W-Z}(s)$
$\displaystyle\rightarrow\sigma_{W-Z}(s)F_{W-Z}\left(s/M_{W}^{2}\right)\,,$

where

$\displaystyle F_{W}(z)$
$\displaystyle=\frac{3}{z^{3}}[-2(z+1)\log(z+1)+z(z+2)]\,,$ (16)
$\displaystyle F_{W-Z}(z)$
$\displaystyle=\frac{3}{z^{3}}\left[(z+1)^{2}\log(z+1)-z\left(\frac{3}{2}z+1\right)\right].$

Using the SM couplings and the contact NSI interaction (3), one can get
Berezhiani and Rossi (2002)

$\displaystyle\sigma_{0}^{\mathrm{SM}}(s)$
$\displaystyle=\frac{N_{\nu}G_{F}^{2}}{6\pi}M_{Z}^{4}\left(g_{L}^{2}+g_{R}^{2}\right)\frac{s}{\left[\left(s-M_{Z}^{2}\right)^{2}+\left(M_{Z}\Gamma_{Z}\right)^{2}\right]}$
(17)
$\displaystyle+\frac{G_{F}^{2}}{\pi}M_{W}^{2}\left\\{\frac{s+2M_{W}^{2}}{2s}\right.-\frac{M_{W}^{2}}{s}\left(\frac{s+M_{W}^{2}}{s}\right)\log\left(\frac{s+M_{W}^{2}}{M_{W}^{2}}\right)$
$\displaystyle-
g_{L}\frac{M_{Z}^{2}\left(s-M_{Z}^{2}\right)}{\left(s-M_{Z}^{2}\right)^{2}+\left(M_{Z}\Gamma_{Z}\right)^{2}}\left.\times\left[\frac{\left(s+M_{W}^{2}\right)^{2}}{s^{2}}\log\left(\frac{s+M_{W}^{2}}{M_{W}^{2}}\right)-\frac{M_{W}^{2}}{s}-\frac{3}{2}\right]\right\\}\,,$

$\displaystyle\sigma_{0}^{\mathrm{NSI}}(s)$
$\displaystyle=\sum_{\alpha,\beta=e,\mu,\tau}\frac{G_{F}^{2}}{6\pi}s\left[\left((\epsilon_{\alpha\beta}^{eL})^{2}+(\epsilon_{\alpha\beta}^{eR})^{2}\right)-2\left(g_{L}\epsilon_{\alpha\beta}^{eL}+g_{R}\epsilon_{\alpha\beta}^{eR}\right)\frac{M_{Z}^{2}\left(s-M_{Z}^{2}\right)}{\left(s-M_{Z}^{2}\right)^{2}+\left(M_{Z}\Gamma_{Z}\right)^{2}}\right]$
(18)
$\displaystyle+\frac{G_{F}^{2}}{\pi}\epsilon_{ee}^{eL}M_{W}^{2}\left[\frac{\left(s+M_{W}^{2}\right)^{2}}{s^{2}}\log\left(\frac{s+M_{W}^{2}}{M_{W}^{2}}\right)-\frac{M_{W}^{2}}{s}-\frac{3}{2}\right]\,.$

According to the coefficient of the NSI parameter, the Eq.(18) can also be
written as

$\displaystyle\sigma_{0}^{\mathrm{NSI}}(s)$
$\displaystyle=\left[\sigma_{0}^{\mathrm{NSI}}(s)\right]_{1}\sum_{\alpha,\beta=e,\mu,\tau}\left((\epsilon_{\alpha\beta}^{eL})^{2}+(\epsilon_{\alpha\beta}^{eR})^{2}\right)$
(19)
$\displaystyle+\left[\sigma_{0}^{\mathrm{NSI}}(s)\right]_{2}\sum_{\alpha,\beta=e,\mu,\tau}\epsilon_{\alpha\beta}^{eL}+\left[\sigma_{0}^{\mathrm{NSI}}(s)\right]_{2}\frac{g_{R}}{g_{L}}\sum_{\alpha,\beta=e,\mu,\tau}\epsilon_{\alpha\beta}^{eR}$
$\displaystyle+\left[\sigma_{0}^{\mathrm{NSI}}(s)\right]_{3}\epsilon_{ee}^{eL}.$

Integrating out the final photon, we can get the total cross section for the
process $e^{+}e^{-}\to\nu\bar{\nu}\gamma$ from NSI at the electron colliders
with $\sqrt{s}\geq M_{Z}$ as

$\sigma^{\mathrm{NSI}}\left(\epsilon_{ee}^{eL},\epsilon_{ee}^{eR}\right)=I_{1}\left(\left(\epsilon_{ee}^{eL}\right)^{2}+\left(\epsilon_{ee}^{eR}\right)^{2}\right)+(I_{2}+I_{3})\epsilon_{ee}^{eL}+I_{2}\frac{g_{R}}{g_{L}}\epsilon_{ee}^{eR}$
(20)

for electron-type NSI parameter, and

$\sigma^{\mathrm{NSI}}\left(\epsilon_{\mu\mu/\tau\tau}^{eL},\epsilon_{\mu\mu/\tau\tau}^{eR}\right)=I_{1}\left(\left(\epsilon_{\mu\mu/\tau\tau}^{eL}\right)^{2}+\left(\epsilon_{\mu\mu/\tau\tau}^{eR}\right)^{2}\right)+I_{2}\epsilon_{\mu\mu/\tau\tau}^{eL}+I_{2}\frac{g_{R}}{g_{L}}\epsilon_{\mu\mu/\tau\tau}^{eR}$
(21)

for muon-type or tau-type NSI parameter, with the definition of

$I_{i}\equiv\int dx\int
dz_{\gamma}H(x_{\gamma},z_{\gamma},s)\left[\sigma_{0}^{\mathrm{NSI}}((s_{\gamma})\right]_{i}.$
(22)

The iso-$\sigma$ contour is also a circle in the NSI parameter plane, of which
the center locates at

$(\epsilon_{ee}^{eL},\ \epsilon_{ee}^{eR})=(-{I_{2}+I_{3}\over
2I_{1}},-{I_{2}g_{R}\over 2I_{1}g_{L}}),$ (23)

or

$(\epsilon_{\mu\mu/\tau\tau}^{eL},\epsilon_{\mu\mu/\tau\tau}^{eR})=(-{I_{2}\over
2I_{1}},-{I_{2}g_{R}\over 2I_{1}g_{L}}).$ (24)

We can see that the coordinates of the circle center is a function of
$\sqrt{s}$.

For NSI searches with the monophoton signature at electron colliders, the
background can be classified into two categories: the irreducible background
and the reducible background. The irreducible background has the final state
containing one photon and two neutrinos arising from the SM, which has been
discussed above. The reducible background comes from a photon produced in the
final state together with several other visible particles which are however
not detected due to limitations of the detector acceptance. Since the
reducible background strongly depends on the detector performance, we discuss
it later in details for each experiment. To evaluate the sensitivity to NSI at
electron collider experiments, we define a $\chi^{2}$ as the function of the
two NSI parameters, i.e.,

$\chi^{2}(\epsilon_{\alpha\beta}^{eL},\ \epsilon_{\alpha\beta}^{eR})\equiv
S^{2}(\epsilon_{\alpha\beta}^{eL},\ \epsilon_{\alpha\beta}^{eR})/(B_{\rm
ir}+B_{\rm re}),$ (25)

where $S$, $B_{\rm ir}$ and $B_{\rm re}$ are the number of events in the
signal, irreducible and reducible background, respectively.

##  III $e^{+}e^{-}$ colliders operated with $\sqrt{s}\ll M_{Z}$

We first consider constraints on NSI with electrons at $e^{+}e^{-}$ colliders
that are operated with the CM energies $\sqrt{s}\ll M_{Z}$, such as Belle II
and STCF. Here we present the detailed analyis of these two experiments below.

###  III.1 Belle II

At Belle II, photons and electrons can be detected in the Electromagnetic
Calorimeter (ECL), which covers a polar angle region of $(12.4-155.1)^{\circ}$
and has inefficient gaps between the endcaps and the barrel for polar angles
between $(31.4-32.2)^{\circ}$ and $(128.7-130.7)^{\circ}$ in the lab frame
Altmannshofer  _et al._ (2019). For monophoton signature at Belle II, the
reducible background comes from two major parts: one is mainly due to the lack
of polar angle coverage of the ECL near the beam directions, which is referred
to as the “bBG”; the other one is mainly because of the gaps between the three
segments in the ECL detector, which is referred to as the “gBG”. The bBG
arises from the electromagnetic processes $e^{+}e^{-}\to\gamma+\not{X}$,
mainly containing $e^{+}e^{-}\to\gamma\not{\gamma}(\not{\gamma})$ and
$e^{+}e^{-}\to\gamma\not{e}^{+}\not{e}^{-}$, in which except the detected
photon all the other final state particles are emitted along the beam
directions with $\theta>155.1^{\circ}$ or $\theta<12.4^{\circ}$ in the lab
frame. At Belle II, we adopt the detector cuts for the final detected photon
(hereafter the “basic cuts”): $12.4^{\circ}<\theta_{\gamma}<155.1^{\circ}$ in
the lab frame.

For the Belle II detector, which is asymmetric, the maximum energy of the
monophoton events in the bBG in the CM frame, $E_{\gamma}^{m}$, is given by
Liang  _et al._ (2020); Zhang  _et al._ (2021) (if not exceeding $\sqrt{s}/2$)

$E_{\gamma}^{m}(\theta_{\gamma})=\frac{\sqrt{s}(A\cos\theta_{1}-\sin\theta_{1})}{A(\cos\theta_{1}-\cos\theta_{\gamma})-(\sin\theta_{\gamma}+\sin\theta_{1})},$
(26)

where all angles are given in the CM frame, and
$A=(\sin\theta_{1}-\sin\theta_{2})/(\cos\theta_{1}-\cos\theta_{2})$, with
$\theta_{1}$ and $\theta_{2}$ being the polar angles corresponding to the
edges of the ECL detector. In order to remove the above bBG, we use the
detector cut

$E_{\gamma}>E_{\gamma}^{m}$ (27)

for the final monophoton (hereafter the “bBG cut”).

Since the gaps in the ECL are significantly away from the beam direction, in
the gBG, the monophoton energy can be quite large in the central
$\theta_{\gamma}$ region. The gBG simulations have been presented by Ref.
Altmannshofer  _et al._ (2019) in searching for an invisibly decaying dark
photon. In order to optimize the detection efficiency for different masses of
the dark photon, they design two different sets of detector cuts: the “low-
mass cut” and “high-mass cut”. By integrating the differential cross section
in Eq. (10) in the phase space regions under these two different detector
cuts, and assuming photon detection efficiency as 95% Altmannshofer  _et al._
(2019), with 50 ab-1 integrated luminosity about 2280 (15230) irreducible BG
events with the “low-mass cut” (“high-mass cut”) can be reached at Belle
IILiang  _et al._ (2020). For the reducible background, it is found that about
300 (25000) gBG events survived the “low-mass cut” (“high-mass cut”) with 20
fb-1 integrated luminosity Altmannshofer  _et al._ (2019).

With the “low-mass cut” (“high-mass cut”) at 50 ab-1 Belle II, the numbers of
event from irreducible and reducible background are $B_{\rm ir}=2280\ (15230)$
and $B_{\rm re}=7.5\times 10^{5}\ (6.25\times 10^{7})$. From Eqs. (12) and
(25), we can get $\chi^{2}=\delta^{2}B^{2}_{\rm ir}/(B_{\rm ir}+B_{\rm re})$,
where $\delta\equiv{C^{\rm NSI}/C^{\rm SM}}$. By solving
$\chi^{2}(\epsilon_{\alpha\alpha}^{eL/V},\
\epsilon_{\alpha\alpha}^{eR/A})-\chi^{2}(0,0)=4.61$ for each specific pair, we
present 90% C.L. allowed regions at Belle II with the “low-mass cut” and
“high-mass cut” in Fig. 2 for electron-type and Fig. 3 for muon/tau-type NSI
parameters, respectively. Notice that, the constraints on muon-type NSI
parameters from electron colliders are the same as tau-type NSI parameters.
The allowed region under the “low-mass cut” lies between two concentric black
circles, and under the “high-mass cut” in the red circle. We can see that the
centers of the circles are all located at the $(\epsilon_{\alpha\alpha}^{eL},\
\epsilon_{\alpha\alpha}^{eR})=(-g_{L}-\delta_{\alpha e},-g_{R})$ and
$(\epsilon_{\alpha\alpha}^{eV},\
\epsilon_{\alpha\alpha}^{eA})=(-g_{V}-\delta_{\alpha e},-g_{A}-\delta_{\alpha
e})$, which agree with Eqs. (13) and (14).

We also show together the $(\epsilon_{ee/\mu\mu}^{eL},\
\epsilon_{ee/\mu\mu}^{eR})$ allowed regions at 90% C.L. from the global
analysis with data of the LEP, CHARM II, LSND, and reactor neutrino
experiments Barranco  _et al._ (2008), which are shown in the shaded gray
regions. The constraints on the parameters $(\epsilon_{\tau\tau}^{eL},\
\epsilon_{\tau\tau}^{eR})$ only come from the LEP data Barranco  _et al._
(2008), and are shown in dashed gray lines. The allowed regions from the
global analysis for $(\epsilon_{\alpha\alpha}^{eV},\
\epsilon_{\alpha\alpha}^{eA})$ are translated from the results of
$(\epsilon_{\alpha\alpha}^{eL},\ \epsilon_{\alpha\alpha}^{eR})$ in Ref.
Barranco  _et al._ (2008) by using the relations in Eq. (5). We can see that
the sensitivity under the “low-mass cut” is better than the one under the
“high-mass cut”. Using either “low-mass cut” or “high-mass cut”, the 50 ab-1
Belle II will be not competitive with current global analysis
$(\epsilon_{ee,\mu\mu}^{eL},\ \epsilon_{ee,\mu\mu}^{eR})$ and
$(\epsilon_{ee,\mu\mu,\tau\tau}^{eV},\ \epsilon_{ee,\mu\mu,\tau\tau}^{eA})$,
but can be complementary with existing LEP data for
$(\epsilon_{\tau\tau}^{eL},\ \epsilon_{\tau\tau}^{eR})$.

In order to compare with other experiments where detailed simulations with gBG
are not available, we present the limits with gBG ignored for illustration. We
use the “bBG cut” defined above to remove the reducible background events. At
this time, if gBG is not taken into account, the remain background events
survived the “bBG cut” come from irreducible backgrounds. One can get
$\chi^{2}=\delta^{2}\sigma^{SM}L$, where $\sigma^{SM}$ is the total cross
section by integrating Eq. (10) with the “bBG cut”. The 90% C.L. allowed
regions analyzed with the “bBG cut” are shown in Figs. 2 and 3, which lie
between the two concentric blue circles. One can find that the constraints is
obvious stronger than the one when gBG is considered under the “low-mass cut”.
Under “bBG cut”, the 50 ab-1 Belle II with gBG ignored can provide a good
complement with current global to constrain the NSI parameter.

Figure 2: The allowed 90% C.L. region for electron-type neutrino NSI in the
planes of $(\epsilon_{ee}^{eL},\ \epsilon_{ee}^{eR})$ (left panel) and
$(\epsilon_{ee}^{eV},\ \epsilon_{ee}^{eA})$ (right panel) at L = 50 ab-1 Belle
II with “low-mass cut” (black lines), “high-mass cut” (red lines) and “bBG
cuts” (blue lines), respectively. The allowed 90% C.L. regions arising from
the global analysis of the LEP, CHARM II, LSND, and reactor data Barranco  _et
al._ (2008), are shown in the shaded gray regions.

Figure 3: The allowed 90% C.L. region for muon-type and tau-type neutrino NSI
in the planes of $(\epsilon_{\mu\mu,\tau\tau}^{eL},\
\epsilon_{\mu\mu,\tau\tau}^{eR})$ (left panel) and
$(\epsilon_{\mu\mu,\tau\tau}^{eV},\ \epsilon_{\mu\mu,\tau\tau}^{eA})$ (right
panel) at L = 50 ab-1 Belle II with “low-mass” cut (black lines), “high-mass”
cut (red lines) and “bBG cuts” (blue lines), respectively. The allowed 90%
C.L. regions for muon-type arising from the global analysis of the LEP, CHARM
II, LSND, and reactor data Barranco  _et al._ (2008), are shown in the shaded
gray regions, and the allowed 95% C.L. regions for tau-type NSI arising from
the LEP data are shown in dashed gray lines.

###  III.2 STCF

The proposed STCF Luo  _et al._ (2019) in China is a symmetric double ring
electron-positron collider. It is the next generation tau charm facility and
successor of the BESIII experiment, and designed to have CM energy ranging
from 2 to 7 GeV. Due to the similarity of STCF and BESIII, we present a
preliminary projection limit for STCF, assuming the same detector performance
with BESIII. For the final photon, we adopt the cuts: $E_{\gamma}>25$ MeV in
the barrel ($|z_{\gamma}|<0.8$) or $E_{\gamma}>50$ MeV in the end-caps
($0.86<|z_{\gamma}|<0.92$), following the cuts used by the BESIII
Collaboration Ablikim  _et al._ (2017), which are defined as the “basic cuts”
hereafter.

For STCF, which are symmetric, the maximum energy of the monophoton events in
the bBG in the CM frame, $E_{\gamma}^{m}$, is given by Liu  _et al._ (2019)

$E_{\gamma}^{m}(\theta_{\gamma})=\sqrt{s}\left(\frac{\sin\theta_{\gamma}}{\sin\theta_{b}}\right)^{-1},$
(28)

where $\theta_{b}$ is the polar angle corresponding to the edge of the
detector. We will collectively refer to the “basic cuts” and cut
$E_{\gamma}>E_{\gamma}^{m}$ as the “advanced cuts” hereafter. After
considering all the boundary of the subdetectors, we take
$\cos\theta_{b}=0.95$ at the STCF Liu and Zhang (2019).

Using the $\chi^{2}$ defined in Eq. (25) and the relation in Eq. (12), we can
get $\chi^{2}=\delta^{2}\sigma^{SM}L$ with the “advanced cuts” cut at $L=$ 30
ab-1 STCF. We present 90% C.L. allowed regions by solving
$\chi^{2}(\epsilon_{\alpha\alpha}^{eL/V},\
\epsilon_{\alpha\alpha}^{eR/A})-\chi^{2}(0,0)=4.61$ for each specific pair
with $\alpha=e,\ \mu,\ \tau$, in Fig. 4 for electron-type and Fig. 5 for
muon/tau-type NSI parameters, respectively. There we consider three typical
running energies $\sqrt{s}=2,\ 4,\ 7$ GeV for STCF.

Allowed region at 90% C.L. from each STCF experiment is between the two
concentric circles. The centers of the circles are independent of the running
energy of STCF, and are also the same as Belle II. With same integrated
luminosity, better constraints can be obtained with higher running energy at
STCF due to the larger cross section for neutrino produciton. These plots
clearly indicate that future STCF can be complementary with current global
analysis in constraining $(\epsilon_{ee}^{eL},\ \epsilon_{ee}^{eR})$ and
$(\epsilon_{ee}^{eV},\ \epsilon_{ee}^{eA})$ and with existing LEP data for
tau-type NSI parameter limits, while it will not be competitive for muon-type
NSI parameters.

In Table 1, by varying only one parameter at a time and fixing the remaining
parameters to zero, i.e., by solving
$\chi^{2}(\epsilon_{\alpha\alpha}^{e})-\chi^{2}(0)=2.71$, we obtain the
constraints on each NSI parameter on the neighborhood of the SM point
$\epsilon_{\alpha\alpha}^{e}=0$ at $L=$ 30 ab-1 STCF with $\sqrt{s}=2,\ 4,\ 7$
GeV and $L=$ 50 ab-1 Belle II without considering gBG, respectively. One can
see how future STCF and Belle II experiments lead to an improvement in the
constraints for eletron-type $\epsilon_{ee}^{eL,R}$ and tau-type
$\epsilon_{\tau\tau}^{eL,R}$ NSI parameters. The previous constraints on
$\epsilon_{ee}^{eL,R}$ coming from global analysis of the LEP, CHARM II, LSND
and reactor data, and $\epsilon_{\tau\tau}^{eL,R}$ coming from LEP data can be
superseded. However, there is no improvement in the constraints for the
parameters $\epsilon_{\mu\mu}^{eL,R}$ at STCF.

Figure 4: The allowed 90% C.L. region for electron-type neutrino NSI in the
planes of $(\epsilon_{ee}^{eL},\ \epsilon_{ee}^{eR})$ (left panel) and
$(\epsilon_{ee}^{eV},\ \epsilon_{ee}^{eA})$ (right panel) at future L = 30
ab-1 STCF running with $\sqrt{s}=$2 (blue lines), 4 (black lines) and 7 (red
lines) GeV, respectively. The allowed 90% C.L. regions arising from the global
analysis of the LEP, CHARM II, LSND, and reactor data Barranco  _et al._
(2008), are shown in the shaded gray regions.

Figure 5: The allowed 90% C.L. region for muon-type and tau-type neutrino NSI in the planes of $(\epsilon_{\mu\mu,\tau\tau}^{eL},\ \epsilon_{\mu\mu,\tau\tau}^{eR})$ (left panel) and $(\epsilon_{\mu\mu,\tau\tau}^{eV},\ \epsilon_{\mu\mu,\tau\tau}^{eA})$ (right panel) at future L = 30 ab-1 STCF running with $\sqrt{s}=$2 (blue lines), 4 (black lines) and 7 (red lines) GeV, respectively. The allowed 90% C.L. regions for muon-type arising from the global analysis of the LEP, CHARM II, LSND, and reactor data Barranco  _et al._ (2008), are shown in the shaded gray regions, and the allowed 95% C.L. regions for tau-type NSI arising from the LEP data are shown in dashed gray lines. | STCF-2 | STCF-4 | STCF-7 | Belle II | Previous Limit
---|---|---|---|---|---
|  $L=30$ ab-1 |  $L=30$ ab-1 |  $L=30$ ab-1 |  $L=50$ ab-1 | 90% Allowed Barranco  _et al._ (2008)
$\epsilon_{ee}^{eL}$ | [-0.067,0.061] | [-0.033,0.031] | [-0.018,0.018] | [-0.0091,0.0089] | [-0.03,0.08]
$\epsilon_{ee}^{eR}$ | [-0.60,0.15] | [-0.163,0.087] | [-0.070,0.053] | [-0.031,0.028] | [0.004,0.15]
$\epsilon_{ee}^{eV}$ | [-0.103,0.093] | [-0.050,0.048] | [-0.028,0.027] | [-0.014,0.014] | –
$\epsilon_{ee}^{eA}$ | [-0.24,0.16] | [-0.103,0.085] | [-0.056,0.050] | [-0.027,0.026] | –
$\epsilon_{\mu\mu/\tau\tau}^{eL}$ | [-0.13,0.69] | [-0.073,0.101] | [-0.044,0.052] | [-0.023,0.025] | [-0.03,0.03]/[-0.5,0.2]
$\epsilon_{\mu\mu/\tau\tau}^{eR}$ | [-0.60,0.15] | [-0.163,0.087] | [-0.070,0.053] | [-0.031,0.028 ] | [-0.03,0.03]/[-0.3,0.4]
$\epsilon_{\mu\mu/\tau\tau}^{eV}$ | [-0.38,0.49] | [-0.25,0.36] | [-0.18,0.29] | [-0.12,0.23] | –
$\epsilon_{\mu\mu/\tau\tau}^{eA}$ | [-0.16,0.24] | [-0.085,0.103] | [-0.050,0.056] | [-0.026,0.027] | –
Table 1: Constraints on NSI parameters by varying only one parameter at a time
at STCF and Belle II with gBG ignored. For comparison, the previous reported
results from the global analysis with the LEP, CHARM II, LSND, and reactor
data are shown in the last column.

##  IV $e^{+}e^{-}$ colliders operated with $\sqrt{s}\geq M_{Z}$

In this section, we present the sensitivity on the strength of NSI at
$e^{+}e^{-}$ colliders operated with $\sqrt{s}\geq M_{Z}$. The results with
the monophoton searches at LEP have been shown in Refs. Berezhiani and Rossi
(2002); Barranco  _et al._ (2008); Forero and Guzzo (2011). Here, we will
focus on the future projected CEPC Dong  _et al._ (2018). Three different
running modes have been proposed for CEPC, including the Higgs factory mode
for the $e^{+}e^{-}\rightarrow ZH$ production with a total luminosity of $\sim
5.6\ \mathrm{ab}^{-1}$ for seven years running at $\sqrt{s}=240\ \mathrm{GeV}$
, the $Z$ factory mode for the $e^{+}e^{-}\rightarrow Z$ production with a
total luminosity of $\sim 16\ \mathrm{ab}^{-1}$ for two years running at
$\sqrt{s}=91.2\ \mathrm{GeV}$ , and the $W^{+}W^{-}$ threshold scan mode for
the $e^{+}e^{-}\rightarrow W^{+}W^{-}$ production with a total luminosity of
$\sim 2.6\ \mathrm{ab}^{-1}$ for one year running at $\sqrt{s}\sim$ $158-172\
\mathrm{GeV}$. 222We take $\sqrt{s}=160$ GeV for the $W^{+}W^{-}$ threshold
scan mode throughout our analysis. For the monophoton signature at CEPC, we
use the cuts for the final detected photon (hereafter the “basic cuts”)
following the CEPC CDR Dong  _et al._ (2018): $|z_{\gamma}|<0.99$ and
$E_{\gamma}>0.1$ GeV. Similar as the symmetric STCF, we also apply the cuts
(28) as “advanced cuts” with $\cos\theta_{b}=0.99$ to remove the reducible
background events Liu  _et al._ (2019).

Integrating out the final photon with the “advanced cuts” for the differential
cross section of Eq. (6) with the radiator function and the neutrino pair
production cross section from SM of Eq. (17) or from NSI of Eq. (18), we can
get the corresponding total cross section. Using the $\chi^{2}$ defined in Eq.
(25), and solving $\chi^{2}(\epsilon_{\alpha\alpha}^{eL/V},\
\epsilon_{\alpha\alpha}^{eR/A})-\chi^{2}(0,0)=4.61$, the 90% C.L. allowed
regions for each specific pair with $\alpha=e,\ \mu,\ \tau$ are shown in Figs.
(6-9), which lie between the two concentric circles lablled black for
$\sqrt{s}=240\ \mathrm{GeV}$, red for $\sqrt{s}=160\ \mathrm{GeV}$ and blue
for $\sqrt{s}=91.2\ \mathrm{GeV}$, respectively. We also show the allowed
regions by combining the data from the three different running modes with
planned luminosity, which are enclosed by the blue curves in right panels of
Figs. (6-9). As we see from Figs. (6-9), after combining the data from three
different modes, the allowed regions for all NSI parameters can be severely
constrained as compared to the global analysis.

These strong constraints on the NSI parameter space can be understood as
follows. Unlike the STCF and Belle II with $\sqrt{s}\ll M_{Z}$, the centers of
the iso-$\sigma$ circles with different running energy no longer stay the same
but become a function of $\sqrt{s}$. By integrating out the final photon with
the “advanced cuts from Eqs. (23) and (24), we can get the coordinates of the
circle center for each pair of NSI parameter $(\epsilon_{\alpha\alpha}^{eL},\
\epsilon_{\alpha\alpha}^{eR})$ or $(\epsilon_{\alpha\alpha}^{eA},\
\epsilon_{\alpha\alpha}^{eV})$ as the function of $\sqrt{s}$, which are shown
in Fig. 10.

Since we simulate the data assuming the SM, the point (0,0) will always be
included in the allowed regions. Also, the allowed regions from each running
mode will lie between two concentric circles. From Fig. (6-9), we see that the
direction from the SM point (0,0) to the circle center with $\sqrt{s}=240\
\mathrm{GeV}$ is almost parallel to that with $\sqrt{s}=160\ \mathrm{GeV}$,
and the allowed regions with $\sqrt{s}=160\ \mathrm{GeV}$ can be superseded by
$\sqrt{s}=240\ \mathrm{GeV}$ in the neighborhood of the SM point (0,0). For
electron-type NSI, the direction from the SM point (0,0) to the circle center
with $\sqrt{s}=91.2\ \mathrm{GeV}$ is approximately perpendicular to that with
the other two running mode. Therefore, the allowed parameter space of
electron-type NSI are severely constrained by combining the data from three
running modes. Even if both $\epsilon_{ee}^{eL}$ and $\epsilon_{ee}^{eR}$ are
present, the allowed ranges for $|\epsilon_{ee}^{eL}|$ or
$|\epsilon_{ee}^{eR}|$ can be constrained to be smaller than 0.002. As for
muon-type or tau-type NSI, the direction from the SM point (0,0) to the circle
center with $\sqrt{s}=91.2\ \mathrm{GeV}$ is approximately opposite to that
with the other two running modes. Hence, the allowed regions will become long
ellipses after combining the data from three running modes. If both
$\epsilon_{\mu\mu,\tau\tau}^{eL}$ and $\epsilon_{\mu\mu,\tau\tau}^{eR}$ are
present, the allowed ranges of muon-type and tau-type NSI are about one order
of magnitude weaker than that of electron-type NSI.

In Table 2, we present the 90% C.L. constraints on only one parameter at a
time by fixing the remaining parameters to zero at CEPC. One can find that
with three different running modes, CEPC can lead to a great improvement in
constraining each NSI parameter compared to the previous global analysis of
the LEP, CHARM II, LSND and reactor data Barranco  _et al._ (2008). Except for
$\epsilon_{\mu\mu,\tau\tau}^{eV}$, one can constrain all NSI parameters to be
smaller than 0.002 by combining the data from three running modes.

Figure 6: The allowed 90% C.L. region for electron-type neutrino NSI in the
planes of $(\epsilon_{ee}^{eL},\ \epsilon_{ee}^{eR})$ at future CEPC with 5.6
ab-1 data of $\sqrt{s}=240$ GeV (Black), with 2.6 ab-1 data of $\sqrt{s}=160$
GeV (Red), and with 16 ab-1 data of $\sqrt{s}=91.2$ GeV (Blue), respectively.
The allowed 90% C.L. regions arising from the global analysis of the LEP,
CHARM, LSND, and reactor data Barranco  _et al._ (2008), are shown in the
shaded gray regions. With all the data collected in all three running modes,
the combined result is labelled by green and shown in the right panel.

Figure 7: The allowed 90% C.L. region for electron-type neutrino NSI in the
planes of $(\epsilon_{ee}^{eV},\ \epsilon_{ee}^{eA})$ at future CEPC with 5.6
ab-1 data of $\sqrt{s}=240$ GeV (Black), with 2.6 ab-1 data of $\sqrt{s}=160$
GeV (Red), and with 16 ab-1 data of $\sqrt{s}=91.2$ GeV (Blue), respectively.
The allowed 90% C.L. regions arising from the global analysis of the LEP,
CHARM II, LSND, and reactor data Barranco  _et al._ (2008), are shown in the
shaded gray regions. With all the data collected in all three running modes,
the combined result is labelled by green and shown in the right panel.

Figure 8:  The allowed 90% C.L. region for muon-type and tau-type neutrino NSI
in the planes of $(\epsilon_{\mu\mu,\tau\tau}^{eL},\
\epsilon_{\mu\mu,\tau\tau}^{eR})$at future CEPC with 5.6 ab-1 data of
$\sqrt{s}=240$ GeV (Black), with 2.6 ab-1 data of $\sqrt{s}=160$ GeV (Red),
and with 16 ab-1 data of $\sqrt{s}=91.2$ GeV (Blue), respectively. The allowed
90% C.L. regions for muon-type arising from the global analysis of the LEP,
CHARM II, LSND, and reactor data Barranco  _et al._ (2008), are shown in the
shaded gray regions, and the allowed 90% C.L. regions for tau-type NSI arising
from the LEP data are shown in dashed gray lines. With all the data collected
in all three running modes, the combined result is labelled by green and shown
in the right panel.

Figure 9:  The allowed 90% C.L. region for muon-type and tau-type neutrino NSI in the planes of $(\epsilon_{\mu\mu,\tau\tau}^{eV},\ \epsilon_{\mu\mu,\tau\tau}^{eA})$at future CEPC with 5.6 ab-1 data of $\sqrt{s}=240$ GeV (Black), with 2.6 ab-1 data of $\sqrt{s}=160$ GeV (Red), and with 16 ab-1 data of $\sqrt{s}=91.2$ GeV (Blue), respectively. The allowed 90% C.L. regions for muon-type arising from the global analysis of the LEP, CHARM II, LSND, and reactor data Barranco  _et al._ (2008), are shown in the shaded gray regions, and the allowed 90% C.L. regions for tau-type arising from the LEP data are shown in dashed gray lines. With all the data collected in all three running modes, the combined result is labelled by green and shown in the right panel. | CEPC-91.2 | CEPC-160 | CEPC-240 | CEPC-combined | Previous Limit
---|---|---|---|---|---
|  $L=16$ ab-1 |  $L=2.6$ ab-1 |  $L=5.6$ ab-1 |  $L=24.2$ ab-1 | 90% Allowed Barranco  _et al._ (2008)
$\epsilon_{ee}^{eL}$ | [-0.0037,0.0037] | [-0.0036,0.0035] | [-0.0010,0.0010] | [-0.00095,0.00095] | [-0.03,0.08]
$\epsilon_{ee}^{eR}$ | [-0.0017,0.0017] | [-0.014,0.015] | [-0.0065,0.0070] | [-0.0017,0.0017] | [0.004,0.15]
$\epsilon_{ee}^{eV}$ | [-0.0024,0.0023] | [-0.0094,0.0093] | [-0.0024,0.0024] | [-0.0017,0.0017] | –
$\epsilon_{ee}^{eA}$ | [-0.0065,0.0066] | [-0.0057,0.0057] | [-0.0018,0.0018] | [-0.0017,0.0017] | –
$\epsilon_{\mu\mu/\tau\tau}^{eL}$ | [-0.0014,0.0014] | [-0.012,0.012] | [-0.0055,0.0053] | [-0.0013,0.0013] | [-0.03,0.03]/[-0.5,0.3]
$\epsilon_{\mu\mu/\tau\tau}^{eR}$ | [-0.0017,0.0017] | [-0.014,0.015] | [-0.0065,0.0070] | [-0.0017,0.0017] | [-0.03,0.03]/[-0.3,0.4]
$\epsilon_{\mu\mu/\tau\tau}^{eV}$ | [-0.013,0.015] | [-0.194,0.074] | [-0.085,0.033] | [-0.013,0.014] | –
$\epsilon_{\mu\mu/\tau\tau}^{eA}$ | [-0.0015,0.0015] | [-0.013,0.013] | [-0.0061,0.0060] | [-0.0015,0.0015] | –
Table 2: Constraints of NSI parameters by varing only one parameter at a time
at CEPC. For comparasion, the previous reported results from the global
analysis with the data of the LEP, CHARM II, LSND, and reactor pieces are
shown in the last coloumn.

Figure 10: The coordinates of the circle centers for each pair of NSI
parameter $(\epsilon_{\alpha\alpha}^{eL},\ \epsilon_{\alpha\alpha}^{eR})$
(Left) and $(\epsilon_{\alpha\alpha}^{eV},\ \epsilon_{\alpha\alpha}^{eA})$
(Right) as a function of $\sqrt{s}$ at CEPC with the “advanced cuts”.

##  V Conclusions

Neutrino oscillation experiments will be affected by the presence of NSI, and
it is natural to seek complementary constraints on NSI from other experiments.
In this work, we investigate the constraints on NC NSI with electrons at
current $e^{+}e^{-}$ collider experiment Belle II, and the proposed future
$e^{+}e^{-}$ collider experiments STCF and CEPC. The allowed regions in the
$(\epsilon_{\alpha\alpha}^{eL},\ \epsilon_{\alpha\alpha}^{eR})$ (or
$(\epsilon_{\alpha\alpha}^{eV},\ \epsilon_{\alpha\alpha}^{eA})$) plane from
these $e^{+}e^{-}$ collider experiments are shown in Figs. (2-9). The
constraints on one NSI parameter at a time are listed in Table. 1 and 2.

For $e^{+}e^{-}$ collider experiments with CM energy $\sqrt{s}\ll M_{Z}$, we
find that the centers of the iso-$\sigma$ circles are independent of the
operating energy. Both Belle II and STCF can provide competitive and
complementary bounds on electron-type NSI parameters as compared to current
global analysis, and strong improvements in the constraints on tau-type NSI.
Belle II with gBG ignored will also yield better constraints than STCF due to
the larger cross section for neutrino production. However, due to the
cancellations between the left-handed (vector) and right-handed (axial-vector)
NSI parameters, the allowed ranges on the NSI parameters from Belle II and
STCF will be still large if multiple NSI parameters are present.

For $e^{+}e^{-}$ collider experiments with $\sqrt{s}\geq M_{Z}$, the centers
of the iso-$\sigma$ circles are dependent on $\sqrt{s}$. We find that future
CEPC experiment with three CM energies $\sqrt{s}=240,160,91.2$ GeV will break
the degeneracy between the left-handed (vector) and right-handed (axial-
vector) NSI parameters after combining the data from the three running modes,
and the allowed parameter space of NSI with electrons will be severely
constrained even if both the left and right-handed NSI parameters are present.
By combining the data from the three running modes, we find that the allowed
ranges for $|\epsilon_{ee}^{eL}|$ or $|\epsilon_{ee}^{eR}|$ can be smaller
than 0.002, and the allowed ranges for muon-type and tau-type NSI are about
one order of magnitude weaker than electron-type NSI if both the left and
right-handed NSI parameters are present.

###### Acknowledgements.

JL is supported by the National Natural Science Foundation of China under
Grant No. 11905299, Guangdong Basic and Applied Basic Research Foundation
under Grant No. 2020A1515011479, the Fundamental Research Funds for the
Central Universities, and the Sun Yat-Sen University Science Foundation. YZ is
supported by National Science Foundation of China under Grant No. 11805001 and
No. 11935001.

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# Status of Birkhoff’s theorem in polymerized semiclassical regime of Loop
Quantum Gravity

Luca Cafaro<EMAIL_ADDRESS>Jerzy Lewandowski<EMAIL_ADDRESS>
The collapse of a spherically symmetric ball of dust has been intensively
studied in Loop Quantum Gravity (LQG). From a quantum theory, it is possible
to recover a semiclassical regime through a polymerization procedure.
In this setting, general solutions to the polymerized Einstein field equations
(PEFE) will be discussed both for the interior and the exterior of the dust
cloud. Exterior solutions are particularly interesting since they may lead to
a semiclassical version of the Birkhoff’s theorem. It is seen that if time
independence of the vacuum is imposed, there exists a unique class of
solutions depending on two parameters. Nevertheless, the possibility of more
intricate time dependent solutions is not ruled out completely.
Ultimately, these results will be compared to a model of spherical collapse
obtained independently from the Einstein equations.

##  I. Introduction

Quantum gravity models lead to semiclassical corrections to spacetime
geometry. Those spacetimes are still described by a metric tensor with
Lorentzian signature, but this tensor contains new expressions proportional to
positive powers of Planck’s constant. Often (though not always) these new
terms smooth out the spacetime singularities predicted by the classical
theory. In the following work we restrict ourselves to semiclassical
spacetimes predicted by LQG models. The first family of semiclassical
spacetimes, which has been very well studied and described in the literature,
consists of spatially homogeneous isotropic spacetimes filled with matter.
They either describe the collapse of a dust ball or the expansion of the
universe. A second family includes spherically symmetric vacuum as well and it
will be the kingpin of our paper. From the point of view of Einstein’s
classical theory, all the spherically symmetric vacuum spacetimes are
characterized by Birkhoff’s theorem as a one-parameter family of spacetimes
given by the Schwarzschild metric tensor. This raises the question of how this
theorem generalizes (in possibly modified form) into the semiclassical theory.
Semiclassical spherically symmetric vacuum spacetimes are obtained in various
ways. The direct method involves finding an exact quantum state in which the
quantum observables have minimal quantum uncertainties, followed by analyzing
the expectation values of the quantum operators composing the metric tensor.
Alternatively, it is possible to consider suitably modified Einstein’s
equation and look for solutions. A third method relies on the Oppenheimer-
Snyder (OS) approach. That is to say, the classical collapsing dust ball is
replaced with a semiclassical version, and then the static, spherically
symmetric metric tensor outside is computed, determined by the known matching
conditions. The last two approaches will be the main focus of this paper.
At first we will look at semiclassical Einstein equations. The aim is to find
the most general solution describing a collapsing ball of homogeneous dust
embedded in a vacuum. Interior and exterior of the ”star” will be treated
separately.
Lastly, said solution could be compared to spacetime metrics obtained by the
OS approach. In contrast to the classical OS models, their semiclassical
counterparts provide three different branches of spherically symmetric vacuum,
depending on the parameter values $k=-1,0,1$ (open, flat, and closed universe,
respectively). In the case of $k=0$, the obtained family of metric tensors
outside the dust ball is parameterized by one parameter, the mass. When $k\neq
0$ a second parameter, related to the initial radius of the ball, appears. Its
absolute value is the same in both cases, however it is positive if $k=-1$ and
negative if $k=1$.
Do these three metric satisfy the semiclassical Einstein’s equations? Do these
equations admit any other vacuum spherically symmetric solutions? Do all the
solutions have static asymptotically flat regions? Finding answers to these
questions is the subject of the current paper.

##  II. Classical theory

As it is common in literature, the classical theory will be described through
the Hamiltonian formalism and Ashtekar variables. The starting point is the
most general spherically symmetric spacetime which can be expressed in the
following way [1]

$ds^{2}=-Nd\tau^{2}+\frac{(E^{\varphi})^{2}}{E^{x}}(dx+N^{x}d\tau)^{2}+E^{x}d\Omega^{2}\,.$
(1)

The functions $N=N(\tau,x)$ and $N^{x}=N^{x}(\tau,x)$ are respectively the
Lapse and the x-component of the Shift vector. The other two variables,
$E^{x}=E^{x}(\tau,x)$ and $E^{\varphi}=E^{\varphi}(\tau,x)$ are related to the
densitised triads $E^{a}_{i}=\sqrt{q}e^{a}_{i}$ through the relations:

$\displaystyle E^{x}_{1}=$ $\displaystyle E^{x}\sin{\theta}$ (2a)
$\displaystyle E^{\theta}_{2}=$ $\displaystyle E^{\varphi}\sin{\theta}$ (2b)
$\displaystyle E^{\varphi}_{3}=$ $\displaystyle E^{\varphi}\,.$ (2c)

The dynamical variables $E^{x}$ and $E^{\varphi}$ are conjugated to the
Ashtekar-Barbero connection $A^{i}_{a}=\omega_{a}^{i}+\gamma K_{a}^{i}$, given
that $\omega_{a}^{i}$ is the spin connection, $K_{a}^{i}=K_{a}^{\;b}e_{b}^{i}$
the extrinsic curvature and $\gamma$ the Barbero-Immirzi parmeter.
Renaming the component of the extrinsic curvature as:

$\displaystyle\gamma K^{1}_{x}=$ $\displaystyle K_{x}$ (3a)
$\displaystyle\gamma K^{2}_{\theta}=$ $\displaystyle K_{\varphi}$ (3b)
$\displaystyle\gamma K^{3}_{\varphi}=$ $\displaystyle
K_{\varphi}\sin{\theta}\,,$ (3c)

leads to two couples of conjugated variables $(K_{x},E^{x})$ and
$(K_{\varphi},E^{\varphi})$.

Since the main interest of this paper will be focused on LTB and
Schwarzschild-like metrics, the line element in (1) can be Gauge fixed.
Firstly, the dust field is taken as the time parameter $\mathcal{T}=\tau$
(Dust Gauge). It follows that $N=1$ is needed in order to preserve the Gauge
choice in time [2]. Such fixing implies that the gravitational Hamiltonian
constraint becomes the physical Hamiltonian (density) $H_{phys}=H^{(g)}=-4\pi
p_{\mathcal{T}}$, where $p_{\mathcal{T}}$ is the conjugate momentum to the
dust field. It follows that the dust density is related to the Hamiltonian by
$\rho=\frac{p_{\mathcal{T}}}{\sqrt{E^{x}}E^{\varphi}}=-\frac{H_{phys}}{4\pi\sqrt{E^{x}}E^{\varphi}}$
(see [3] for details).
Additionally, the Areal Gauge will be imposed too, $E^{x}=x^{2}$.
For convenience, $E^{\varphi}$ will be rewrtten as

$E^{\varphi}=\pm\frac{x}{\sqrt{1+\epsilon(\tau,x)}}\,.$ (4)

All in all the metric becomes

$ds^{2}=-d\tau^{2}+\frac{1}{1+\epsilon(\tau,x)}(dx+N^{x}d\tau)^{2}+x^{2}d\Omega^{2}\,.$
(5)

The coordinates adopted in (5) are known as generalised Painlevé-Gullstrand
(PG) coordinates $(\tau,x,\theta,\varphi)$.
All those models with $\epsilon\neq 0$ are known in literature as ”non-
marginally bound” [4, 5, 6] and will be the main interest of this paper.
Now, the dynamic of the system is all encoded in one dynamical couple
$(K_{\varphi},E^{\varphi})$, whose Poisson bracket reads
$\\{K_{\varphi}(y_{1}),E^{\varphi}(y_{2})\\}=\gamma G\delta(y_{1}-y_{2})$.
From now on the unit system will be set such that $G=1$.

From this point, the LTB model in comoving coordinates $(T,R,\theta,\varphi)$
is recovered by choosing $\tau=T$, $x=\xi(T,R)$ and
$N^{x}=-\partial_{T}\xi(T,R)$

$ds^{2}=-dT^{2}+\frac{[\partial_{R}\xi(T,R)]^{2}}{1+E(R)}dR^{2}+\xi(t,R)^{2}d\Omega^{2}\,,$
(6)

with $\epsilon(\tau,x)=E(R)$ (this justifies the choice in (4)).
It is useful to mention that the Friedmann-Lemaître-Robertson-Walker (FLRW)
metric is a particular case of the LTB model.
By choosing $\xi(T,R)=a(T)\chi_{k}(R)$ and $E(R)=-k\chi_{k}^{2}(R)$, one gets
to

$ds^{2}=-dT^{2}+a^{2}dR^{2}+a^{2}\chi_{k}^{2}d\Omega^{2}\,.$ (7)

The function $a(T)$ is the scale factor obeying the Friedmann equation,
whereas the function $\chi_{k}(R)$ is defined as

$\chi_{k}(R)=\begin{cases}\frac{1}{\sqrt{k}}\sin{(\sqrt{k}R)},&\text{if}\quad
k=1\\\ R&\text{if}\quad k=0\;.\\\
\frac{1}{\sqrt{|k|}}\sinh{(\sqrt{|k|}R)}&\text{if}\quad k=-1\end{cases}$ (8)

with $R\in[0,\pi]$.

On the other hand, the metric in (5) describes the Schwarzschild solution as
well [7]. Such a metric is indeed recovered by setting
$E^{x}=(E^{\varphi})^{2}=x^{2}$ ($\epsilon=0$), $N=1$ and
$N^{x}=\sqrt{\frac{2M}{x}}$. Then, changing coordinates such that $dx=dr$ and
$d\tau=dt+dr\sqrt{\frac{2M}{r}}\frac{1}{1-\frac{2M}{r}}$ , the Schwarzschild
line element is obtained.
It will be shown later that one could actually retrieve any metric of the form

$ds^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}d\Omega^{2}\,,$ (9)

when semiclassical corrections are considered.

##  III. Semiclassical collapse of a dust ball

The problem to be addressed is the collapse of a finite-size dust ball,
characterised by a metric provided by (6), embedded in a vacuum described by
(9) using quantum correction coming from LQG. The whole spacetime can be
described by a single coordinate system using the PG coordinates as in (5).

The quantum theory of such a system relies on a 1-dimensional graph as it has
been described in [8, 9, 10, 11]. A laconic review follows.
The couple $(K_{\varphi},E^{\varphi})$ acts on each vertex $x_{j}$, whereas
$(K_{x},E^{x})$ acts on each edge. Given the Gauge fixing that has been
performed at the beginning, only the variables on $\varphi$ will play a role.
In the $E^{\varphi}$ representation, one could define a state
$\ket{E^{\varphi}_{j}}_{j}$ that spans a Hilbert space associated to the
vertex $x_{j}$, i.e. $H_{j}$. The full Hilbert space is the tensor product
$H=\otimes_{j}H_{j}$. The scalar product on $H$ is provided by
$\leftindex[I]_{i}{\bra{\tilde{E}^{\varphi}_{i}}\ket{E^{\varphi}_{j}}}_{j}=\delta_{ij}\delta_{\tilde{E}^{\varphi}_{i}E^{\varphi}_{j}}$.
It is natural to associate an operator to the triad component
$\hat{E^{\varphi}_{j}}$, while the extrinsic curvature requires a point
holonomy operator $\hat{U}_{j}=e^{i\bar{\mu}_{j}\hat{K}_{\varphi,j}}$. The
scheme adopted here is the $\bar{\mu}$ scheme of Loop Quantum Cosmology [12,
13], that is $\bar{\mu}_{j}=\frac{\sqrt{\Delta}}{x_{j}}$, with $\Delta$ the
smallest non-zero eigenvalue of the area operator. The action of those
operators is given by

$\displaystyle\hat{E}^{\varphi}_{j}\ket{E^{\varphi}_{j}}_{j}=$ $\displaystyle
E^{\varphi}_{j}\ket{E^{\varphi}_{j}}_{j}$ (10a)
$\displaystyle\hat{U}_{j}\ket{E^{\varphi}_{j}}_{j}=$
$\displaystyle\ket{E^{\varphi}_{j}+\bar{\mu}_{j}}_{j}\,.$ (10b)

Since the inverse of $E^{\varphi}$ appears in the Hamiltonian, one is forced
to introduce corrections from the inverse triad operator. Operationally, given
a state $\ket{E^{\varphi}_{j}}_{j}$, the eigenvalue of the operator
$\hat{\nicefrac{{1}}{{E^{\varphi}_{j}}}}$ is taken to be zero whenever the
eigenvalue of $\hat{E}^{\varphi}_{j}$ is zero. In all other cases the
eigenvalue of the inverse triad operator is
$\nicefrac{{1}}{{E^{\varphi}_{j}}}$.
The key point to recover a semiclassical theory is the introduction of the
polymerization
$K_{\varphi,j}\rightarrow\frac{\hat{U}_{j}-\hat{U}_{j}^{\dagger}}{2i\bar{\mu}_{j}}=\widehat{\frac{\sin{\bar{\mu}_{j}K_{\varphi,j}}}{\bar{\mu}_{j}}}$.
Notice that in the classical limit $(\bar{\mu}_{j}\rightarrow 0)$, one has
$\frac{\sin{\bar{\mu}_{j}K_{\varphi,j}}}{\bar{\mu}_{j}}\rightarrow
K_{\varphi,j}$.
All in all, the semiclassical Hamiltonian is recovered by writing its
corresponding operators in terms of the classical phase space variables and by
reintroducing the continuum limit.
By taking the Posisson brackets of $E^{\varphi}$ and $K_{\varphi}$ with the
new Hamiltonian, one recovers the Polymerized Einstein field equations (PEFE).
Along with the PEFE, one can write the expressions for the density and the
radial component of the shift. All together they read [3, 14, 8, 15]

$\displaystyle\dot{E}^{\varphi}=-\frac{x^{2}}{\gamma\sqrt{\Delta}}\partial_{x}\left(\frac{E^{\varphi}}{x}\right)\sin{\beta}\cos{\beta}$
(11a) $\displaystyle\dot{K}_{\varphi}=\frac{\gamma
x}{2(E^{\varphi})^{2}}-\frac{\gamma}{2x}-\frac{\partial_{x}\left(x^{3}\sin^{2}{\beta}\right)}{2\gamma\Delta
x}$ (11b) $\displaystyle\rho=\frac{1}{8\pi
xE^{\varphi}}\left[\frac{E^{\varphi}}{\gamma^{2}\Delta
x}\partial_{x}\left(x^{3}\sin^{2}{\beta}\right)+\frac{x}{E^{\varphi}}+\frac{E^{\varphi}}{x}-2\partial_{x}\left(\frac{x^{2}}{E^{\varphi}}\right)\right]$
(11c) $\displaystyle
N^{x}=-\frac{x}{\gamma\sqrt{\Delta}}\sin{\beta}\cos{\beta}\,,$ (11d)

with $\beta\coloneqq\frac{\sqrt{\Delta}}{x}K_{\varphi}$ and $\dot{()}$ the
derivative with respect to time.
Alternatively, one could substitute (11b) with a combination of (11b) and
(11c). Namely

$\dot{\beta}=-4\pi\gamma\Delta\rho+\frac{\gamma\sqrt{\Delta}}{E^{\varphi}}\left[\frac{1}{E^{\varphi}}-\frac{1}{x}\partial_{x}\left(\frac{x^{2}}{E^{\varphi}}\right)\right]\,.$
(12)

As it was pointed out in [5], the sign of $N^{x}$ requires a careful analysis
in order to avoid discontinuities of the metric. As it will be clarified
later, collapsing solutions need the RHS of (11d) to be positive. This implies
that the product $\sin{\beta}\cos{\beta}$ has to be negative. On the other
hand, the expansion requires $\sin{\beta}$ and $\cos{\beta}$ to be equal in
sign. The best choice relies on choosing $\sin{\beta}>0$ (that is
$\beta\in(0,\pi)$) and
$\cos{\beta}\stackMath\mathrel{\stackinset{c}{-0.25ex}{c}{1.45ex}{<}{\raisebox{-0.5pt}{$\eqslantgtr$}}}0$
(see figure 1).
By keeping this fact in mind, from this point on, the focus will be drawn on
collapsing solutions only ($\cos{\beta}<0$). Anytime that the bounce will be
mentioned it is implied that the $\cos{\beta}$ swaps sign.
Given the equations for the metric, it is suitable to study separately the
interior and the exterior of the dust ball.

### Interior

The interior is distinguished by $\rho\neq 0$. Moreover, it is assumed that
the dust is homogeneous, $\partial_{x}\rho=0$.
The starting point is to plug the ansatz (4) into (11c) and solve it for
$\sin{\beta}$. It results in

$\sin^{2}{\beta}=\frac{8\pi\gamma^{2}\Delta}{3}\rho+\frac{\gamma^{2}\Delta}{x^{2}}\epsilon\,.$
(13)

Notice that, since the dust is collapsing, $\cos{\beta}$ is imposed to be
negative.
From this result, equation (11a) can be morphed into

$\dot{\epsilon}=\epsilon^{\prime}\sqrt{\frac{8\pi}{3}\rho
x^{2}+\epsilon}\sqrt{1-\frac{\rho}{\rho_{c}}-\frac{3}{8\pi\rho_{c}}\frac{\epsilon}{x^{2}}}\,,$
(14)

where $\rho_{c}\coloneqq\frac{3}{8\pi\gamma^{2}\Delta}$.

In LTB coordinates $(T,R)$, one has $x=\xi(T,R)$ and $\epsilon(\tau,x)=E(R)$.
It is easy to verify that
$N^{x}=\frac{\dot{\epsilon}}{\epsilon^{\prime}}=-\partial_{T}\xi$. Therefore
(14) can be rewritten as

$\frac{\partial_{T}\xi}{\xi}=-\sqrt{\frac{8\pi}{3}\rho+\frac{E}{\xi^{2}}}\sqrt{1-\frac{\rho}{\rho_{c}}-\frac{3}{8\pi\rho_{c}}\frac{E}{\xi^{2}}}\,,$
(15)

which is general for any LTB model. The ”-” sign in the RHS of (15) represents
a collapse. This has been made possible by choosing $\sin{\beta}\cos{\beta}<0$
as it was stated before.
A simplification occurs by imposing $x=\xi(T,R)=a(T)\chi_{k}(R)$. Given that,
in (T, R) coordinates, the time dependence disappears from $E(R)$, it is
expected that $\epsilon(\tau,x)$ is a generic function of the form
$\epsilon=h(\frac{x}{a})$. Therefore, equation (14) becomes

$\left(\frac{\dot{a}}{a}\right)=-\sqrt{\frac{8\pi}{3}\rho+\frac{h}{x^{2}}}\sqrt{1-\frac{\rho}{\rho_{c}}-\frac{3}{8\pi\rho_{c}}\frac{h}{x^{2}}}\,.$
(16)

The scale factor $a$ is a function solely of the time parameter $\tau=T$,
which means that the LHS is independent of x. In order to cancel the x
dependence from the RHS, it is needed to impose $h\propto x^{2}$. The most
general form of h is then $h=e_{0}\frac{x^{2}}{a^{2}}$, with $e_{0}$ a
constant. The collapse is then described by the following Friedmann equation

$\left(\frac{\dot{a}}{a}\right)=-\sqrt{\frac{8\pi}{3}\rho+\frac{e_{0}}{a^{2}}}\sqrt{1-\frac{\rho}{\rho_{c}}-\frac{3}{8\pi\rho_{c}}\frac{e_{0}}{a^{2}}}\,.$
(17)

In $(T,R)$ coordinates, one has $E(R)=e_{0}\chi_{k}^{2}(R)$ and the LTB line
element is

$ds^{2}=-dT^{2}+\frac{a^{2}(\chi_{k}^{\prime})^{2}}{1+e_{0}\chi_{k}^{2}}dR^{2}+a^{2}\chi_{k}^{2}d\Omega^{2}\,.$
(18)

Without loss of generality, it is possible to write $e_{0}$ as
$e_{0}=-k\tilde{e}_{0}$, with $k=0,\pm 1$ and $\tilde{e}_{0}>0$. Introducing a
new variable $\hat{R}=\sqrt{\tilde{e}_{0}}\chi_{k}$, one has

$ds^{2}=-dT^{2}+\frac{1}{\tilde{e}_{0}}\frac{a^{2}}{1-k\hat{R}}d\hat{R}^{2}+\frac{1}{\tilde{e}_{0}}a^{2}\hat{R}d\Omega^{2}\,.$
(19)

Finally, the new constant can be reabsorbed in the scale factor so as to have
the FLRW metric in reduced-circumference coordinates. The line element (18)
can be therefore rewritten as the classical one in (7) provided that $a(T)$
now obeys the modified Friedmann equation (17).
As a consistency check, one can verify that $\rho\propto a^{-3}$ as it is
expected for dust. By differentiating equation (13) with respect to $\tau$ and
comparing it with (12), one is led to

$\dot{\rho}=\frac{3}{x}\frac{\dot{\epsilon}}{\epsilon^{\prime}}\rho\,.$ (20)

Provided that $\epsilon=e_{0}\frac{x^{2}}{a^{2}}$, one finds
$\rho=\frac{\mathcal{C}}{a^{3}}$, with $\mathcal{C}=const$.

### Exterior

#### Time independent exterior

Exterior solutions to the PEFE are found by imposing $\rho=0$.
At first, one can look for time independent solutions, i.e.
$\dot{E}^{\varphi}=0$. It follows that it is either $E^{\varphi}=Ax$
($A=const\neq 0$), $\sin{\beta}=0$ or $\cos{\beta}=0$. Nevertheless, the
second solution is identical to the first with $A=1$, whereas the third runs
into inconsistencies. For this reasons the only interesting solution is
$E^{\varphi}=Ax$.
Notice that the latter is recovered by imposing $\epsilon=B=\frac{1}{A^{2}}-1$
in (4).
From inserting this last condition in (11b), it is found $\dot{K}_{\varphi}=0$
which is consistent with the time independence of the metric.
The exterior analogue of (13) is

$\sin^{2}{\beta}=\frac{\gamma^{2}\Delta}{x^{2}}B+\frac{C}{x^{3}}\,$ (21)

with $C$ an integration constant. Its value is recovered by looking at the
Schwarzschild limit ($\Delta\rightarrow 0$), which corresponds to
$C=2\gamma^{2}\Delta M$.
The same value for $C$ can also be recovered from a suitably modified ADM
mass. Classically it is defined as [10]:

$M=\frac{1}{2}\sqrt{E^{x}}(1+\frac{K_{\varphi}^{2}}{\gamma^{2}})-\frac{\sqrt{E^{x}}[(E^{x})^{\prime}]^{2}}{8(E^{\varphi})^{2}}\,.$
(22)

If one performs the substitution
$K_{\varphi}\rightarrow\frac{\sin{\bar{\mu}K_{\varphi}}}{\bar{\mu}}$ (with
$\bar{\mu}=\frac{\sqrt{\Delta}}{x}$), then a quick calculation shows that
$M=\frac{C}{2\gamma^{2}\Delta}$, which is the value shown above.
It must be noticed that since $\sin^{2}{\beta}\geq 0$, one has $B>0$ in order
to satisfy the disequality at every $x$. This is equivalent to say that
$A^{2}<1$, hence $-1<A<1$.
On the other hand, it is still possible to pick $B<0$ but restrict the
solutions at $x$ lower than a certain $x_{max}$.
The complementary bound, $\sin{\beta}\leq 1$ sets the existence of a minimal
$x$ which is nothing but the bouncing radius $x_{b}=\left(2\gamma^{2}\Delta
M\right)^{\nicefrac{{1}}{{3}}}+\frac{B}{6M}\left(2\gamma^{2}\Delta
M\right)^{\nicefrac{{2}}{{3}}}+O(\Delta^{\nicefrac{{4}}{{3}}})$.
As it was mentioned at the beginning of this section, the $\sin{\beta}$ is
kept positive while $\cos{\beta}$ is negative in the collapsing phase and
positive during the expansion. This is portrayed in figure 1.

Figure 1:  Display of the different values of $\beta$ throughout the
spacetime. At the bouncing point $\sin^{2}{\beta}=1$, therefore
$\cos{\beta}=0$. At spatial infinity, $\sin^{2}{\beta}=0$, then
$\cos{\beta}=\pm 1$. Since $\sin{\beta}$ is kept positive, the collapsing
solution is provided by $\cos{\beta}<0$. The opposite happens for the
expansion in the post-bounce phase.
Notice that the very same result could have been achieved by imposing
$\sin{\beta}<0$. In this setting $\cos{\beta}<0$ would represent the expansion
whereas $\cos{\beta}>0$ the collapse.

Substituting (21) into (11d) leads to

$\left(N^{x}\right)^{2}=\frac{2M}{x}-\frac{\alpha}{x^{2}}\left(\frac{M}{x}+\frac{B}{2}\right)^{2}+B\,,$
(23)

where $\alpha$ has been defined as $\alpha=4\gamma^{2}\Delta$.
This last equation determines the metric completely. However, it is more
convenient to change coordinate so as to have a line element like (9). This is
obtained by setting $dx=dr$ and $d\tau=\frac{1}{A}dt+\frac{N^{x}}{f(r)}dr$,
with

$f(r)=\frac{1}{A^{2}}-(N^{x})^{2}=1-\frac{2M}{r}+\frac{\alpha}{r^{2}}\left(\frac{M}{r}+\frac{B}{2}\right)^{2}\,.$
(24)

In the case where $A=\pm 1$, then $B=0$.
The conclusion is that the family of solutions to the equations (11) (the
semiclassical modification of Einstein’s equations), such that $\epsilon={\rm
const}$, is mapped by a coordinate transformation into the family of metric
tensors (9) defined by (24) for all the values of the constants $M$ and $B$. A
catch is that, according to equation (21), the radial coordinate $x=r$ is to
be restricted either to $r_{b}\leq r$ for $B\geq 0$ or $r_{b}\leq
r\leq\frac{2M}{|B|}$ for $B<0$. However, the resulting metric tensor (24) is
analytic and there is no obstruction for considering it for the other values
of $r$ as long as $f$ is finite, or extending the spacetime even more by using
the Eddington-Finkelstein coordinates.

#### Time dependent exterior

A more general solution for the exterior is found by restoring the time
dependence of $E^{\varphi}$. This time it is possible to impose (4), as for
the interior, and follow the same procedure. One finds:

$\displaystyle\sin^{2}{\beta}=\frac{\gamma^{2}\Delta}{x^{2}}\epsilon+\frac{2\gamma^{2}\Delta
M}{x^{3}}$ (25)
$\displaystyle\dot{\epsilon}=\epsilon^{\prime}\sqrt{\epsilon+\frac{2M}{x}}\sqrt{1-\gamma^{2}\Delta\left(\frac{\epsilon}{x^{2}}+\frac{2M}{x^{3}}\right)}\,.$
(26)

Another useful equation follows from (12):

$\dot{\beta}=-\frac{\gamma\sqrt{\Delta}}{2x}\epsilon^{\prime}\,.$ (27)

At first, one could check singularly the cases $\epsilon=\epsilon(x)$ and
$\epsilon=\epsilon(\tau)$.

  * •

$\epsilon=\epsilon(x)$:
From equation (25), it appears that $\beta$ too is a function solely of x,
implying $\dot{\beta}=0$. However, this is in contrast with equation (27).
Therefore, $\epsilon=\epsilon(x)$ is ruled out and the square roots appearing
in (26) are non-zero. It follows that the same formula can be inverted to have
an expression for $\epsilon^{\prime}$.

  * •

$\epsilon=\epsilon(\tau)$:
Since $\epsilon^{\prime}=0$, it is straightforward to see, from equation (26),
that $\dot{\epsilon}=0$. Therefore, this possibility too can be ruled out.

From this very short analysis, it is to be concluded that either
$\epsilon=const$ (retrieving the time independent case discussed above), or
$\epsilon=\epsilon(\tau,x)$.
However, the time dependence of the metric components does not exclude the
existence of a timelike Killing vector field.

##  IV. PEFE and Oppenheimer-Snyder model

The goal of this section is to glue together the cloud of collapsing
semiclassical dust and a spherically symmetric, static exterior of the form
(9). In order to do so, it is possible to match the intrinsic metrics and
extrinsic curvatures at the surface of the ball. This will determine uniquely
the time/radial component of the exterior metric, $f(r)$. That result was also
obtained in [16]. Next, we will compare the derived function $f$ with the
exterior solutions (24).
The semiclassical dust ball is described by (7) obeying the two Friedmann
equations

$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}=$
$\displaystyle\left(\frac{8\pi}{3}\rho-\frac{k}{a^{2}}\right)\left(1-\frac{\rho}{\rho_{c}}+\frac{3}{8\pi\rho_{c}}\frac{k}{a^{2}}\right)$
(28a)
$\displaystyle\begin{split}\left(\frac{\ddot{a}}{a}\right)\;=&-\frac{4\pi}{3}\rho\left(1-\frac{\rho}{\rho_{c}}+\frac{3}{8\pi\rho_{c}}\frac{k}{a^{2}}\right)+\\\
&+\left(\frac{8\pi}{3}\rho-\frac{k}{a^{2}}\right)\left(\frac{3}{2}\frac{\rho}{\rho_{c}}-\frac{3}{8\pi\rho_{c}}\frac{k}{a^{2}}\right)\,,\end{split}$
(28b)

where the second one is the derivative of the first. The density $\rho$ for
dust reads $\rho=\frac{\mathcal{C}}{a^{3}}$.
As mentioned at the beginning, this metric is junctioned to an exterior of the
kind (9), by applying the Israel condition so as to match the two extrinsic
curvatures [17, 18]. In the end, one finds

$f(r)=1-\frac{2M}{r}+\frac{\alpha}{r^{2}}\left(\frac{M}{r}-\frac{k\chi^{2}_{k,0}}{2}\right)^{2}\,.$
(29)

It is easy to verify that this last function is exactly the one recovered from
the PEFE (24) provided that $B=-k\chi^{2}_{k,0}$. Notice that we do not
require in this section, that the exterior metric tensor satisfies any
equations. The function (29) is determined by the junction conditions and by
an assumption that the vector field $\partial_{t}$ is a Killing vector. This
is a direct generalisation of the model studied in [19, 20, 16].
The junction of the two line elements leads to
$M=\frac{4\pi}{3}\rho_{0}r^{3}_{0}=\frac{4\pi}{3}\mathcal{C}\chi_{k,0}^{3}$,
where the subscript ”0” has been used to label the variables evaluated on the
surface of the ball.
The horizons of such a metric are provided by real zeros of (29). In the end
one has

$\displaystyle r_{-}$ $\displaystyle=\left(\frac{\alpha
M}{2}\right)^{\nicefrac{{1}}{{3}}}+\frac{1-2k\chi_{k,0}^{2}}{6M}\left(\frac{\alpha
M}{2}\right)^{\nicefrac{{2}}{{3}}}+\frac{(1-k\chi_{k,0}^{2})^{2}}{24M}\alpha+O(\alpha^{\nicefrac{{4}}{{3}}})$
(30a) $\displaystyle r_{+}$
$\displaystyle=2M-\frac{(1-k\chi_{k,0}^{2})^{2}}{8M}\alpha+O(\alpha^{\nicefrac{{4}}{{3}}})\,.$
(30b)

The exact expression of $r_{+}$ contains a square root whose positivity of the
argument is assured when $M\leq M_{-}$ or $M\geq M_{+}$. The lower and upper
bound of $M$ are functions of $k\chi_{k,0}^{2}$ which read

$M_{\pm}=\frac{\sqrt{\alpha}}{6\sqrt{6}}\sqrt{64-96k\chi_{k,0}^{2}+30k^{2}\chi_{k,0}^{4}+k^{3}\chi_{k,0}^{6}\pm\left(16-16k\chi_{k,0}^{2}+k^{2}\chi_{k,0}^{4}\right)^{\nicefrac{{3}}{{2}}}}\,.$
(31)

It is easy to verify that if $k=0$, then $M_{-}=0$ and
$M_{+}=\frac{4}{3\sqrt{3}}\sqrt{\alpha}$ (consistent with the result found in
[16]). Their behaviour for $k=\pm 1$ will be discussed separately below.

The bouncing radius too can be extracted from (29) by using the geodesic
equation. For a massless dust particle, one has $\dot{r}=1-f(r)$. The turning
point occurs when $\dot{r}=0$, i.e.
$r_{b}=\left(\frac{\alpha
M}{2}\right)^{\nicefrac{{1}}{{3}}}-\frac{k\chi_{k,0}^{2}}{6M}\left(\frac{\alpha
M}{2}\right)^{\nicefrac{{2}}{{3}}}+O(\Delta^{\nicefrac{{4}}{{3}}})$ (same
value found using the PEFE with $B=-k\chi^{2}_{k,0}$).

### General properties of the Oppenheimer-Snyder model wth $k\neq 0$

As it has already been discussed, the interior metric (7) is an exact solution
to the PEFE with $\epsilon(\tau,x)=E(R)=-k\chi_{k}^{2}(R)$.
The exterior (29) instead requires a more detailed analysis. Hence the cases
of $k=\pm 1$ will be studied separately ($k=0$ can be found in [16]).

#### k=1

Let the ball be motionless at $T=T_{0}$, implying
$a_{0}=a(T=T_{0})=\frac{8\pi}{3}\mathcal{C}$ (value for which the RHS of (28a)
vanishes). It can be checked that $\ddot{a}<0$, so the ball starts its
collapse. Let $M$ be the mass of the ball and $\chi_{1,0}$ its initial radius.
Finally, let $\rho_{0}\ll\rho_{c}$.
The complete set of initial condition is then

$\displaystyle a_{0}=$
$\displaystyle\frac{2M}{\chi^{3}_{1,0}}\,,\quad\dot{a}_{0}=0\,,\quad\ddot{a}_{0}<0$
(32a) $\displaystyle r_{0}=$ $\displaystyle
a_{0}\chi_{1,0}=\frac{2M}{\chi^{2}_{1,0}}$ (32b) $\displaystyle\rho_{0}=$
$\displaystyle\frac{3}{32\pi}\frac{\chi^{6}_{1,0}}{M^{2}}\,.$ (32c)

Notice that the constant $\mathcal{C}$ has been replaced by $M$ through the
relation given at the beginning.

The exterior solution is of the form (24) but the constant $B=-\chi_{1,0}^{2}$
is now negative. Nevertheless, it still satisfies the PEFE as long as
$\sin^{2}{\beta}=\gamma^{2}\Delta\left(-\frac{k\chi_{1,0}^{2}}{r^{2}}+\frac{2M}{r^{3}}\right)\geq
0$ (21). This leads to $r\leq r_{max}=\frac{2M}{\chi^{2}_{k,0}}$, which is
exactly $r_{0}$.
In other words, at the starting point $T=T_{0}$, the ball of dust occupies all
the region contained in $r_{0}$ and it satisfies the interior PEFE (iPEFE). At
a subsequent time, the ball occupies a spherical subregion of radius
$r_{1}<r_{0}$. It implies that the metric restrained by $r_{1}$ is solution of
the iPEFE whereas the ePEFE are satisfied between $r_{1}$ and $r_{0}$.
The metric outside $r_{0}$ is not solution to the Einstein equations.

From equations (28a) and (28b), one can study the evolution of the system. It
turns out that when $\rho$ approaches $\rho_{c}$ (i.e. when $a$ decreases),
$\ddot{a}$ changes sign and the ball re-expand as soon as the other zero of
(28a) is touched, $\rho=\rho_{c}+\frac{3}{8\pi}\frac{1}{a^{2}_{b}}$. The value
of $a_{b}$ is recovered by solving the third order equation obtained by
replacing $\rho\propto a^{-3}$.
The expansion of the dust cloud will stop when reaching once again $a=a_{0}$,
where $\dot{a}=0$ and $\ddot{a}<0$. At this point it will recollapse as it did
at the beginning [21].

Horizons are formed when $M<M_{-}$ and $M>M_{+}$. From their expressions, one
finds the profile in figure 2.

Figure 2:  Profiles of $M_{+}^{2}$ and $M_{-}^{2}$ as functions of
$\chi_{1,0}$. If $k=1$, $\chi_{1,0}$ is a $\sin$ and therefore goes from 0 to
1 ($R\in[0,\pi]$).

Given that $M_{-}^{2}<0$, the only condition to have a horizon is $M>M_{+}$.
Surprisingly, the actual value of the critical mass is dictated solely by its
initial radius and $M_{+}=0$ if $\chi_{k,0}=1$. On the other hand, if
$\chi_{k,0}=0$ the critical mass is the same as in $k=0$. This is due to the
fact that $M_{+}$ is a function of the product $k\chi_{k,0}^{2}$.

#### k=-1

When $k$ is negative, the RHS of (28a) cannot vanish for any value of $a$ such
that $\rho\ll\rho_{c}$ and there exists only one zero of $\dot{a}$ (as in
$k=0$). The ball will theeqrefore contract for a while but eventually it is
bound to re-expand to infinity.
Unlike the $k=1$ case though. This is everywhere a solution to the PEFE (both
interior and exterior).

For what concerns the critical mass, this scenario is pretty similar to the
previous one for $M_{-}$. Once again $M_{-}^{2}<0$ for every $\chi_{-1,0}$, so
a horizon can form if $M>M_{+}$. Nevertheless, this time there is no value
$\chi_{k,0}$ for which the critical mass is zero (see figure 3).

Figure 3:  Profiles of $M_{+}^{2}$ and $M_{-}^{2}$ as functions of
$\chi_{-1,0}$. This time, $\chi_{-1,0}$ is a $\sinh$ and therefore goes from 0
to $\infty$.

##  V. Discussion

The semiclassical modification of Einstein’s equations has been studied both
in the presence of homogeneous dust (interior) and in the vacuum (exterior).
In the first case, one gets an LTB model whose collapse is dictated by a
generalised Friedmann equation [5, 22]. This scenario predicts a bounce once
the density of dust reaches a certain value. A further generalization to this
model occurs when inhomogeneity of the dust is imposed [22].
On the other hand, when no dust is present, assuming that the function
$\epsilon$ (4) is constant, we determined a two-parameter class of static
solutions. Such metrics are dependent on two free parameters, $M$ and $B$.
These line elements are suitably modified Schwarzschild solutions which
coincide with Schwarzschild in the classical limit ($\Delta\rightarrow 0$).
Nevertheless, if the function $\epsilon$ is not constant, it is subject to the
partial differential equation in (26). It has been discussed that solutions to
said equation are either constant or depending on both time and radial
coordinate. If the latter case can be (somehow) ruled out, then there exists a
unique solution which is time independent and Schwarzschild-like.
The static exterior metric is clearly asymptotically flat. It is conceivable,
that if $\epsilon$ is not a constant, the situation might change drastically.
We do not exclude the existence of a temporal Killing field in this case as
well. Nonetheless, it must be taken into account that all of these results
strongly depend on the choice of quantization [23, 21], the polymerization
scheme [24] and inverse triad corrections in the quantum theory.

This analysis can be compared to a model of spherically symmetric collapse
obtained in a completely different way. It is indeed possible to match a cloud
of homogeneous dust (described by a Friedmann metric) to a general line
element of the form (9).
It turns out that this model is an exact solution to the PEFE when $\rho\neq
0$. In the vacuum instead, it satisfies globally the PEFE when $k=0,-1$ and
just locally when $k=1$.
The latter may suggest that the generalized PG coordinates simply fail if
$k=1$. This assumption is backed up by the fact that in Schwarzschild-AdS
metric the situation is quite similar. The non diagonal piece of the metric in
PG coordinates reads $\sqrt{\frac{2M}{r}+\frac{\Lambda r^{2}}{3}}$ (with
$\Lambda<0$). As a consequence, the metric is not well defined if $r$ big
enough.
On the other hand, it must not be excluded that by correcting the quantum
theory, one can extend the $k=1$ case to a global solution without changing
coordinates.

##  VI. Acknowledgements

Authors would like to thank Ed Wilson-Ewing and Cong Zhang for numerous
discussions. The research presented here was funded by the National Science
Centre, Poland as part of the OPUS 24 grant number 2022/47/B/ST2/027.

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