# Impact of the cosmic neutrino background on long-range force searches

Garv Chauhan  0000-0002-8129-8034<EMAIL_ADDRESS>b Xun-Jie Xu
0000-0003-3181-1386<EMAIL_ADDRESS>Center for Neutrino Physics, Department of
Physics, Virginia Tech, Blacksburg, VA 24061, USAInstitute of High Energy
Physics, Chinese Academy of Sciences, Beijing 100049, China

###### Abstract

Light bosons can mediate long-range forces. We show that light bosonic
mediators interacting with a background medium, in particular, with the cosmic
neutrino background (C$\nu$B), may induce medium-dependent masses which could
effectively screen long-range forces from detection. This leads to profound
implications for long-range force searches in e.g. the Eöt-Wash, MICROSCOPE,
and lunar laser-ranging (LLR) experiments. For instance, we find that when the
coupling of the mediator to neutrinos is above $3\times 10^{-10}$ or $5\times
10^{-13}$, bounds from LLR and experiments employing the Sun as an attractor,
respectively, would be entirely eliminated. Larger values of the coupling can
also substantially alleviate bounds from searches conducted at shorter
distances.

††preprint: September 1, 2024

##  1 Introduction

Among the four fundamental interactions in nature—gravity, electromagnetism,
and the strong and weak interactions—two of them are long-range forces,
observable at macroscopic scales. It is tempting to conceive that new
interactions arising from theories beyond the standard model (SM) might
generate extra long-range forces. If discovered, new long-range forces would
have paradigm-shifting implications for new physics explorations and could
even lead to revolutionary applications in macroscopic-scale technologies.

Long-range forces are mediated by massless or sufficiently light mediators.
Some examples include light scalar or vector bosons in hidden sectors or gauge
extensions Heeck:2014zfa ; Knapen:2017xzo ; Xu:2020qek ; Chauhan:2020mgv ;
Chauhan:2022iuh , axions Hoedl:2011zz ; Mantry:2014zsa ; Okawa:2021fto ,
majorons Chikashige:1980ui , dilatons Taylor:1988nw ; Ellis:1989as ;
Kaplan:2000hh , or even neutrinos Feinberg:1968zz ; Hsu:1992tg ; Dzuba:2017cas
; Orlofsky:2021mmy ; Xu:2021daf ; Coy:2022cpt ; Ghosh:2022nzo . In recent
years, the interest in light, feebly-interacting particles has been
persistently growing (see Refs. Lanfranchi:2020crw ; Antel:2023hkf  for
reviews), while long-range force searches (also known as fifth-force searches
in the literature) play a crucial role in constraining ultra-light particles
if their de Broglie wavelengths can reach macroscopic extent.

Currently, the most sensitive searches are based on torsion-balance tests of
gravity Wagner:2012ui . New forces mediated by light bosons with generic
couplings to SM fermions might cause observable deviations in the probe of the
weak equivalent principle (WEP) or the inverse square law of gravity. At large
distance scales comparable to the Earth’s radius (corresponding to a mediator
mass below $10^{-14}$ eV), the Eöt-Wash experiment has excluded the couplings
with electrons and baryons above $10^{-23}\sim 10^{-24}$. If such a small
coupling arises from high-energy theories (e.g. generated by heavy loops
Xu:2020qek ; Chauhan:2020mgv ), then the extremely small value can be
reinterpreted as a probe of new physics at very high energy scales. Other
experimental probes including lunar laser-ranging (LLR) Williams:1995nq ;
Williams:2004qba ; Merkowitz:2010kka ; Murphy:2013qya ; Viswanathan:2017vob ,
the detection of gravitational waves at LIGO/Virgo TheLIGOScientific:2017qsa ;
Abbott:2020khf ; Croon:2017zcu ; Kopp:2018jom ; Dror:2019uea , and the very
recent MICROSCOPE experiment MICROSCOPE:2022doy  have their respective
advantages in long-range force searches.

Given the remarkably high sensitivity of these experiments to new long-range
forces, it is important to ask: to what extent are these bounds valid and what
factors may significantly alter the bounds?

Taking the example of electromagnetism, the Coulomb potentials of electrons
and nuclei always cancel each other at large scales in electrically neutral
matter, and electromagnetic waves can often be screened by metals.
Consequently, the photon and photon-like particles111By “photon-like”, we mean
that the effective couplings to electrons and nuclei are proportional to their
electric charges. The dark photon and the $L_{\mu}-L_{\tau}$ gauge boson with
loop-induced couplings to electrons and baryons are in this category.  are
unable to cause observable effects in these experiments. For axions and
majorons which are pseudoscalars, there are similar cancellations between
particles of opposite spin directions in unpolarized matter. For light scalar
or vector bosons, in general the forces on individual particles can be added
coherently without cancellation. However, such bosons could still evade the
aforementioned constraints if their propagation in a medium behaves
differently from that in the vacuum.

In this study, we investigate potential medium effects that may exert a
significant influence on the searches for long-range forces. We consider a
light scalar mediator $\phi$ that is generically coupled to all SM fermions
including neutrinos, and show that the propagation of $\phi$ in a medium such
as normal matter or the cosmic neutrino background (C$\nu$B) could be modified
by coherent forward scattering with medium particles, generating a medium-
dependent mass for $\phi$. Such a mass could shorten the interaction range and
alleviate some experimental bounds on long-range forces. We examine this
effect and find that the medium effect caused by normal matter of the Earth or
the Sun is negligible in experiments employing them as the attractor. On the
other hand, the C$\nu$B which is ubiquitous and cannot be evacuated in any
experiments, can cause a considerably strong medium effect due to the
lightness of neutrinos. We show that within a broad range of the neutrino
coupling allowed by cosmology, existing bounds on long-range forces can be
altered by the C$\nu$B very significantly.

This work is organized as follows. In Sec. 2, we briefly review the formalism
for generating long-range forces in the vacuum and elucidate the medium effect
to be taken into account in the formalism. The medium effect can be included
by solving a modified field equation, which is solved in Sec. 3 assuming a
spherically symmetric configuration. Then we inspect to what extent the
experimental bounds on long-range forces can be altered by the C$\nu$B in Sec.
4 and draw our conclusions in Sec. 5. Some minus sign issues are addressed in
Appendix A and a derivation of the medium effect is presented in Appendix B.

##  2 Long-range forces and the medium effect

###  2.1 Long-range forces in the vacuum

Let us briefly review the generation of long-range forces from light bosons,
mainly following the formalism in Refs. Smirnov:2019cae ; Babu:2019iml . We
start by considering the following Lagrangian:

${\cal
L}=\frac{1}{2}\left(\partial\phi\right)^{2}-\frac{1}{2}m_{\phi}^{2}\phi^{2}+\overline{\psi}i\not{\partial}\psi-
m_{\psi}\overline{\psi}\psi-y_{\psi}\phi\overline{\psi}\psi\thinspace,$ (2.1)

where $\phi$ is a light scalar and $\psi$ denotes a generic fermion, with
their respective masses $m_{\phi}$ and $m_{\psi}$. The Lagrangian leads to the
following equations of motion (EOMs):

$\displaystyle i\not{\partial}\psi-(m_{\psi}+y_{\psi}\phi)\psi$
$\displaystyle=0\thinspace,$ (2.2)
$\displaystyle(\partial^{2}+m_{\phi}^{2})\phi+y_{\psi}\overline{\psi}\psi$
$\displaystyle=0\thinspace.$ (2.3)

Eq. (2.2) implies that in the presence of a background $\phi$ field, the mass
of $\psi$ is essentially shifted to $m_{\psi}+y_{\psi}\phi$. For
nonrelativistic $\psi$ particles, this is equivalent to shifting its energy by
$y_{\psi}\phi$. Thus one can introduce an effective potential to account for
the influence of $\phi$ on $\psi$:

$V=y_{\psi}\phi\thinspace.$ (2.4)

When $\phi$ has a nonzero gradient ($\nabla\phi\neq 0$), it causes a force,
$F=y_{\psi}\nabla\phi$, acting on $\psi$ particles. For relativistic $\psi$
particles, the mass shift is not fully equivalent to the energy shift, causing
a reduction of the Yukawa force. We refer to Ref. Smirnov:2022sfo  for
detailed discussions on this issue.

For a bulk of $\psi$ particles statically distributed in space, we replace
$\overline{\psi}\psi$ in Eq. (2.3) with its ensemble average
$\langle\overline{\psi}\psi\rangle\approx n_{\psi}$ where $n_{\psi}$ is the
number density of $\psi$ particles. Since it is static, we can neglect the
temporal derivative in Eq. (2.3) and write it as

$\left[\nabla^{2}-m_{\phi}^{2}\right]\phi=y_{\psi}n_{\psi}\ \text{(non-
relativistic)}.$ (2.5)

Assuming the distribution is spherically symmetric and using spherical
coordinates, we have $\nabla^{2}\phi=\frac{\partial^{2}\phi}{\partial
r^{2}}+2\frac{\partial\phi}{r\partial r}=\frac{u^{\prime\prime}(r)}{r}$ where
$u(r)\equiv r\phi(r)$. This allows us to further rewrite Eq. (2.5) as

$\left[\frac{d^{2}}{dr^{2}}-m_{\phi}^{2}\right]u(r)=ry_{\psi}n_{\psi}(r)\thinspace,$
(2.6)

which can be solved analytically for some simple forms of $n_{\psi}(r)$,
including e.g. constant or piecewisely constant $n_{\psi}$, or the exponential
form $n_{\psi}\propto e^{-r\kappa}$ Smirnov:2019cae . In particular, if
$n_{\psi}=N_{\psi}\delta^{3}(\mathbf{r})$, the solution is
$u(r)=-y_{\psi}N_{\psi}e^{-m_{\phi}r}/(4\pi)$, resulting in the well-known
Yukawa potential:

$V=-\frac{y_{\psi}^{2}N_{\psi}}{4\pi r}e^{-m_{\phi}r}\thinspace.$ (2.7)

The force arising from Eq. (2.7), $F=\nabla V$, is considered long-range if
$m_{\phi}\ll r^{-1}$, in which case the exponential suppression can be
neglected. As is well known, the potential is negative, implying that the
Yukawa force is attractive. For vector interactions, one can derive a similar
potential which is positive, corresponding to a repulsive force.

###  2.2 The medium effect

In a medium, the propagation of the $\phi$ field could be significantly
altered, similar to various known medium effects on electromagnetic waves. For
instance, in plasma, due to interactions with free charged particles, the
photon acquires an effective mass known as the plasmon mass. In cold (non-
ionized) gas such as the air, where electrons and nuclei are bound into atoms
or molecules, the response of atomic or molecular electric dipoles to
electromagnetic waves slightly slows down the propagation of light by a factor
of $1/n$ with $n$ the refraction index of the medium.

For $\phi$ propagating through a medium containing a large number of free
$\psi$ particles, the medium effect is similar to the plasmon mass, giving
rise to the following mass correction for the $\phi$ field:

$m_{\phi}^{2}\to
m_{\phi}^{2}+y_{\psi}^{2}\frac{\tilde{n}_{\psi}}{m_{\psi}}\thinspace,$ (2.8)

where

$\tilde{n}_{\psi}\equiv\int\frac{d^{3}\mathbf{p}}{(2\pi)^{3}}\frac{m_{\psi}}{E_{\mathbf{p}}}f(\mathbf{p})\thinspace,$
(2.9)

with $f(\mathbf{p})$ the phase space distribution function of $\psi$
particles. We refer to $\tilde{n}_{\psi}$ in Eq. (2.9) as the effective number
density, in contrast to the standard number density:

$n_{\psi}\equiv\int\frac{d^{3}\mathbf{p}}{(2\pi)^{3}}f(\mathbf{p})\thinspace.$
(2.10)

Eq. (2.8) can be derived from finite-temperature or finite-density field
theory Babu:2019iml . However, such a derivation involves the underlying
assumption that the medium is in thermal and chemical equilibrium, which
implies that $f(\mathbf{p})$ should take the Fermi-Dirac distribution. In
fact, Eq. (2.8) also holds for general forms of $f(\mathbf{p})$, not
necessarily limited to thermal distributions, since it is a consequence of
coherent forward scattering of $\phi$ with the medium particles. This is again
similar to the photon, for which the plasmon mass and the refraction index are
known to arise from coherent forward scattering. In Appendix B, we rederive
Eq. (2.8) from the theory of coherent forward scattering, following similar
calculations in Refs. Li:2023vpv ; Wu:2024fsf  for photon-dark photon or
photon-axion conversions. This justifies the use of Eq. (2.8) for non-thermal
distributions such as the C$\nu$B.

Let us note here that the mass correction in Eq. (2.8) is a local rather than
global effect. In a density-varying medium, one should only apply the mass
correction at the EOM level, not the Yukawa potential in Eq. (2.7) which is
obtained by assuming constant $m_{\phi}$. The actual potential should be
determined by solving the modified EOM:

$\left[\nabla^{2}-m_{\phi}^{2}-y_{\psi}^{2}\frac{\tilde{n}_{\psi}}{m_{\psi}}\right]\phi=y_{\psi}\tilde{n}_{\psi}\thinspace.$
(2.11)

Note that comparing to Eq. (2.5), the right-hand side has also been changed.
The right-hand side is changed from $y_{\psi}n_{\psi}$ to
$y_{\psi}\tilde{n}_{\psi}$ because $\langle\overline{\psi}\psi\rangle\approx
n_{\psi}$ is only applicable to nonrelativistic distributions while
$\langle\overline{\psi}\psi\rangle=\tilde{n}_{\psi}$ holds for general cases
including relativistic distributions Smirnov:2022sfo .

By definition, $n_{\psi}$ and $\tilde{n}_{\psi}$ in Eqs. (2.9) and (2.10) do
not include antiparticles. For antiparticles, we define similar quantities,
$n_{\overline{\psi}}$ and $\tilde{n}_{\overline{\psi}}$. When including
antiparticles, there are a few subtle minus signs to be discussed in Appendix
A and the conclusion is that antiparticles contribute positively to Eq. (2.8),
i.e. there is no cancellation between particles and antiparticles in the
medium effect. Therefore, the contribution of antiparticles can be included by

$\tilde{n}_{\psi}\to\tilde{n}_{\psi}+\tilde{n}_{\overline{\psi}}\thinspace.$
(2.12)

In addition, Eq. (2.11) can be straightforwardly generalized to include
multiple fermionic species that are coupled to $\phi$. Altogether, the most
general equation that determines the $\phi$ field in the presence of multiple
species of medium particles together with antiparticles reads:

$\left[\nabla^{2}-\tilde{m}_{\phi}^{2}\right]\phi=\sum_{\psi}y_{\psi}\left(\tilde{n}_{\psi}+\tilde{n}_{\overline{\psi}}\right),$
(2.13)

where

$\tilde{m}_{\phi}^{2}\equiv
m_{\phi}^{2}+\sum_{\psi}y_{\psi}^{2}\frac{\tilde{n}_{\psi}+\tilde{n}_{\overline{\psi}}}{m_{\psi}}\thinspace.$
(2.14)

###  2.3 On the effective number density $\tilde{n}_{\psi}$

In the nonrelativistic limit, the effective number density $\tilde{n}_{\psi}$
defined in Eq. (2.9) simply reduces to $n_{\psi}$ because
$m_{\psi}/E_{\mathbf{p}}\approx 1$. For relativistic distributions,
$\tilde{n}_{\psi}$ is smaller than $n_{\psi}$ and their ratio
$\tilde{n}_{\psi}/n_{\psi}$ is roughly suppressed by $m_{\psi}/\langle
E_{\mathbf{p}}\rangle$ where $\langle E_{\mathbf{p}}\rangle$ denotes the mean
energy of $\psi$ particles.

In general, by computing the integral in Eq. (2.9) for a given
$f(\mathbf{p})$, one obtains

$\tilde{n}_{\psi}\approx\begin{cases}n_{\psi}&(\text{nonrelativistic})\\\ \xi
m_{\psi}n_{\psi}^{2/3}&(\text{relativistic})\end{cases}\thinspace,$ (2.15)

where $\xi$ is an ${\cal O}(1)$ coefficient depending on the shape of
$f(\mathbf{p})$. For the Fermi-Dirac (FD) distribution with zero chemical
potential,

$\xi=\frac{1}{6}\left(\frac{\pi^{2}}{6\zeta(3)}\right)^{2/3}g_{i}^{1/3}\approx
0.2054g_{i}^{1/3}\thinspace,$ (2.16)

with $g_{i}$ the number of internal degrees of freedom: $g_{i}=2$ for
electrons and $g_{i}=1$ for Majorana neutrinos. For the FD distribution
distribution with zero temperature but a finite chemical potential (i.e. the
degenerate limit), $\xi\approx 0.3848g_{i}^{1/3}$. The Maxwell-Boltzmann
distribution would lead to $\xi\approx 0.2331g_{i}^{1/3}$.

Note that the standard cosmic neutrino background (C$\nu$B) with nonzero
neutrino masses is not exactly in the FD distribution (see e.g.
Esteban:2021ozz  for detailed discussions):

$f_{\text{C}\nu\text{B}}(p)=\frac{1}{e^{p/T}+1}\neq\frac{1}{e^{E_{p}/T}+1}\thinspace.$
(2.17)

But for neutrino masses well below $1.9\ \text{K}\approx 1.6\times 10^{-4}\
\text{eV}$ (possible for the lightest neutrino mass eigenstate), the
difference is negligible and one can take $\xi\approx 0.2054g_{i}^{1/3}$.

The neutrino mass squared differences in neutrino oscillations have been well
measured Zyla:2020zbs :

$|\Delta m_{31}^{2}|\approx 2.5\times 10^{-3}\ \text{eV}^{2}\thinspace,\
\Delta m_{21}^{2}\approx 7.4\times 10^{-5}\ \text{eV}^{2}\thinspace,$ (2.18)

which implies that two of the mass eigenstates are heavier than $\sqrt{|\Delta
m_{31}^{2}|}\approx 0.05$ eV or $\sqrt{\Delta m_{21}^{2}}\approx 0.009$ eV.
Therefore, among three neutrino mass eigenstates, at least two of them should
be nonrelativistic today.

##  3 The spherically symmetric solution

The differential equation (2.11) can be solved analytically if the effective
number density is spherically symmetric with the following form:

$\tilde{n}_{\psi}(r)=\begin{cases}\tilde{n}_{\psi 0}&\text{for }r\leq R\\\
0&\text{for }r>R\end{cases}\thinspace,$ (3.1)

where $\tilde{n}_{\psi 0}$ is a constant and $R$ is the radius of the
spherical distribution. By solving Eq. (2.11) in the $r\leq R$ and $r>R$
regimes and requiring that $\phi$ and $d\phi/dr$ are continuous at $r=R$, we
obtain the solution

$\phi=\begin{cases}\frac{y_{\psi}\tilde{n}_{\psi
0}}{\tilde{m}_{\phi}^{2}}\left[-1+\frac{C_{{\rm
in}}}{r}\sinh(\tilde{m}_{\phi}r)\right]&\text{for }r\leq R\\\\[5.69054pt]
\frac{y_{\psi}\tilde{n}_{\psi 0}}{\tilde{m}_{\phi}^{2}}\cdot\frac{C_{{\rm
out}}}{r}e^{-m_{\phi}(r-R)}&\text{for }r>R\end{cases}\thinspace,$ (3.2)

where $\tilde{m}_{\phi}$ denotes the effective mass within the sphere
($\tilde{m}_{\phi}^{2}=m_{\phi}^{2}+y_{\psi}^{2}\tilde{n}_{\psi 0}/m_{\psi}$)
and

$\displaystyle C_{{\rm in}}$
$\displaystyle=\frac{m_{\phi}R+1}{\tilde{m}_{\phi}\cosh(\tilde{m}_{\phi}R)+m_{\phi}\sinh(\tilde{m}_{\phi}R)}\thinspace,$
(3.3) $\displaystyle C_{{\rm out}}$
$\displaystyle=\frac{\sinh(\tilde{m}_{\phi}R)-\tilde{m}_{\phi}R\cosh(\tilde{m}_{\phi}R)}{m_{\phi}\sinh(\tilde{m}_{\phi}R)+\tilde{m}_{\phi}\cosh(\tilde{m}_{\phi}R)}\thinspace.$
(3.4)

Here we would like to discuss an interesting limit: $m_{\phi}\to 0$, i.e.
$\phi$ is massless in the vacuum but acquires a mass within the sphere due to
the medium. In this limit, Eq. (3.2) reduces to

$\phi=\begin{cases}-\frac{y_{\psi}\tilde{n}_{\psi
0}}{\tilde{m}_{\phi}^{2}}\left[1-\frac{\sinh(\tilde{m}_{\phi}r)}{\tilde{m}_{\phi}r\cosh(\tilde{m}_{\phi}R)}\right]&\text{for
}r\leq R\\\\[5.69054pt] -\frac{y_{\psi}\tilde{n}_{\psi
0}}{\tilde{m}_{\phi}^{2}}\left[1-\frac{\tanh(\tilde{m}_{\phi}R)}{\tilde{m}_{\phi}R}\right]\frac{R}{r}&\text{for
}r>R\end{cases}\thinspace,\ \ (m_{\phi}\to 0)\thinspace.$ (3.5)

Obviously, the $r>R$ part of the solution behaves like an infinitely long-
range force, as can be seen from its $1/r$ dependence. If the medium effect is
weak ($\tilde{m}_{\phi}R\ll 1$), one can further expand it in terms of
$x\equiv\tilde{m}_{\phi}R$:

$\phi=\begin{cases}y_{\psi}\tilde{n}_{\psi
0}\left[\frac{r^{2}-3R^{2}}{6}+\frac{x^{2}\left(r^{2}-5R^{2}\right)^{2}}{120R^{2}}+{\cal
O}(x^{4})\right]&\text{for }r\leq R\\\\[11.38109pt]
-\frac{y_{\psi}\tilde{n}_{\psi 0}R^{3}}{3r}\left[1-\frac{2x^{2}}{5}+{\cal
O}(x^{4})\right]&\text{for }r>R\end{cases}\thinspace,\ \ (m_{\phi}\to
0)\thinspace.$ (3.6)

Here the first terms in squared brackets correspond to known results of the
Coulomb potential, while the second terms represent the leading-order medium
effect.

In the presence of multiple species of medium particles coupled to $\phi$, one
needs to solve the more general differential equation (2.13). Note that due to
the mass correction term in Eq. (2.14), Eq. (2.13) cannot be solved by
linearly adding the solutions in Eq. (3.2) for each species. Unlike Eq. (2.5)
for which the solution scales linearly with respect to the number density,
Eqs. (2.11) and (2.13) exhibit nonlinearity when the medium effect is
significant.

Nevertheless, Eq. (3.2) can still be practically useful under certain
approximations and assumptions. Consider for instance the Earth as a
spherically symmetric object. Assuming that its matter density and chemical
composition are homogeneous, the above solution can be used by replacing
$y_{\psi}\tilde{n}_{\psi 0}\to\sum_{\psi}y_{\psi}(\tilde{n}_{\psi
0}+\tilde{n}_{\overline{\psi}0})$ in Eq. (3.2) and taking
$\tilde{m}_{\phi}^{2}=m_{\phi}^{2}+\sum_{\psi}y_{\psi}^{2}(\tilde{n}_{\psi
0}+\tilde{n}_{\overline{\psi}0})/m_{\psi}$, where the summation $\sum_{\psi}$
goes over all ingredients of the Earth. Furthermore, adding the C$\nu$B to
this problem can also be solved analytically, assuming that the C$\nu$B is
homogeneous and extends infinitely in space. Under this assumption, the medium
effect of C$\nu$B is a constant shift added to $m_{\phi}^{2}$.

Another example is the Yukawa force between two celestial bodies. Assuming
that each celestial body is a spherically symmetric object with a homogeneous
matter density and the two bodies are well separated, one can first solve
their respective $\phi$ field equations:

$\displaystyle\left[\nabla^{2}-\tilde{m}_{\phi 1}^{2}\right]\phi_{1}$
$\displaystyle=y_{\psi}\tilde{n}_{\psi 1}\thinspace,$ (3.7)
$\displaystyle\left[\nabla^{2}-\tilde{m}_{\phi 2}^{2}\right]\phi_{2}$
$\displaystyle=y_{\psi}\tilde{n}_{\psi 2}\thinspace,$ (3.8)

where $\phi_{i}$ ($i=1$, 2) denotes the field strength generated by the $i$-th
celestial body and $\tilde{m}_{\phi
i}^{2}=m_{\phi}^{2}+y_{\psi}^{2}\tilde{n}_{\psi i}/m_{\psi}$. Here for
simplicity we assume that they all consist of a single species of $\psi$
particles. Summing the two solutions $\phi_{1}$ and $\phi_{2}$ together cannot
generate an exact solution of the combined system, but
$\phi=\phi_{1}+\phi_{2}$ satisfies

$\left[\nabla^{2}-\tilde{m}_{\phi}^{2}\right]\phi=y_{\psi}\tilde{n}_{\psi
1}+y_{\psi}\tilde{n}_{\psi 2}+y_{\psi}^{2}\frac{\tilde{n}_{\psi
2}\phi_{1}+\tilde{n}_{\psi 1}\phi_{2}}{m_{\psi}}\thinspace,$ (3.9)

where $\tilde{m}_{\phi}^{2}=m_{\phi}^{2}+y_{\psi}^{2}(\tilde{n}_{\psi
1}+\tilde{n}_{\psi 2})/m_{\psi}$ and the last term in Eq. (3.9) comes from
cross terms in $\tilde{m}_{\phi}^{2}\phi$. Compared to
$y_{\psi}\tilde{n}_{\psi 1}$ or $y_{\psi}\tilde{n}_{\psi 2}$, the last term is
suppressed by a factor of $y_{\psi}\phi_{i}/m_{\psi}$ which is $\ll 1$ as long
as the mass shift of $\psi$ [see the discussion above Eq. (2.4)] is small.
Hence $\phi=\phi_{1}+\phi_{2}$ can be viewed as an approximate solution of the
field equation for the two-body system.

##  4 Altering the experimental bounds on long-range forces

A variety of experiments measuring gravitational effects or testing
gravitational laws can also be utilized to search for extra long-range forces,
as has been conducted by a number of experimental groups Hoskins:1985tn ;
Schlamminger:2007ht ; Adelberger:2009zz ; Wagner:2012ui ; Yang:2012zzb ;
Tan:2016vwu ; Tan:2020vpf ; fifth-force ; Lee:2020zjt . In recent years, the
detection of gravitational waves from black hole or neutron star binary
mergers also offers an important avenue for probing long-range forces
Croon:2017zcu ; Kopp:2018jom ; Dror:2019uea .

We focus our analyses on experimental bounds derived from measuring
gravitational effects of the most well-understood celestial bodies nearby,
namely the Sun, Earth, and Moon. With these celestial bodies as attractors,
new long-range forces can be severely constrained by measuring possible
acceleration differences between two test bodies composed of different
materials (e.g. Be, Ti, Al, Pt), known as the test of the weak equivalence
principle (WEP). Such a test has been conducted by the Princeton Roll:1964rd
and Moscow 1972JETP  groups utilizing the Sun as the attractor and
subsequently by the Eöt-Wash group Schlamminger:2007ht ; Wagner:2012ui  using
the Earth as the attractor. In addition to the WEP test, there is another
interesting type of experimental probe—the lunar laser-ranging (LLR)
experiments Williams:1995nq ; Williams:2004qba ; Merkowitz:2010kka ;
Murphy:2013qya ; Viswanathan:2017vob  which are capable to measure anomalous
precession of the lunar orbit and hence sensitive to new forces that modify
the inverse square law. Very recently, the MICROSCOPE experiment
MICROSCOPE:2022doy  deployed test masses orbiting the Earth in a drag-free
satellite and achieved an unprecedented precision of measuring WEP violation.

In the literature, these measurements have been used to use to set constraints
on the standard Yukawa potential in Eq. (2.7). Typically for $m_{\phi}$
ranging from $10^{-14}$ eV (corresponding to the length scale of the Earth
radius) to $10^{-18}$ eV (corresponding to the length scale of Earth-Sun
distance), the experimental bounds on $y_{\psi}$ for $\psi\in\\{e^{-},\ p,\
n\\}$ vary from $10^{-22}$ to $10^{-24}$—see the top left panel of Fig. 1.

Let us first estimate the medium effect caused by $\phi$ interacting with
electrons in the medium, which leads to a mass correction of the order of

$y_{e}\left(\frac{n_{e}}{m_{e}}\right)^{1/2}\sim 10^{-22}\
\text{eV}\cdot\left(\frac{y_{e}}{10^{-24}}\right)\cdot\left(\frac{\rho}{5\
\text{g}/\text{cm}^{3}}\right)^{1/2},$ (4.1)

where $\rho$ denotes the matter density and $5\ \text{g}/\text{cm}^{3}$ is the
average density of the Earth. In Eq. (4.1) we have assumed $n_{e}=n_{p}\approx
n_{n}$ so that $n_{e}\approx 0.5\rho/m_{p}$. Obviously this medium effect is
negligibly small for the experiments considered here.

Next, we estimate the medium effect caused by the C$\nu$B which is ubiquitous
in the entire universe and according to the standard cosmological model has
the number density $56\ \nu/{\rm cm}^{3}+56\ \overline{\nu}/{\rm cm}^{3}$ for
Dirac neutrinos or $112\ \nu/{\rm cm}^{3}$ for Majorana neutrinos. The medium
effect due to the C$\nu$B can be substantially enhanced by the smallness of
neutrino masses. Besides, the Yukawa coupling $y_{\nu}$ for very light $\phi$
can be much greater than $y_{e}$ as known bounds on $y_{\nu}$ are much less
restrictive than those on $y_{e}$. Theoretically, $y_{\nu}\gg y_{e}$ can be
well motivated from models that introduce light mediators via the neutrino
portal Xu:2020qek ; Chauhan:2020mgv ; Chauhan:2022iuh .

Figure 1: The impact of the C$\nu$B medium effect on long-range force
searches. The top left panel shows the experimental bounds on $y_{e}$ without
the C$\nu$B medium effect. The subsequent panels demonstrate the variation of
these bounds due to the C$\nu$B medium effect, with the dashed and solid lines
representing the bounds before and after including such an effect,
respectively. The $x$-axis on the top of each panel indicates the wavelength
scale $\lambda\equiv 1/m_{\phi}$.

For nonrelativistic cosmic relic neutrinos ($m_{\nu}\gg 1.6\times 10^{-4}$
eV), the mass correction can be estimated as follows:

$y_{\nu}\left(\frac{n_{\nu}}{m_{\nu}}\right)^{1/2}\sim 10^{-16}\
\text{eV}\cdot\left(\frac{y_{\nu}}{10^{-10}}\right)\cdot\left(\frac{n_{\nu}}{56/\text{cm}^{3}}\right)^{1/2}\left(\frac{0.1\text{eV}}{m_{\nu}}\right)^{1/2}.$
(4.2)

For relativistic neutrinos, according to Eq. (2.15), the mass correction
should be independent of $m_{\nu}$:

$y_{\nu}\left(\frac{\tilde{n}_{\nu}}{m_{\nu}}\right)^{1/2}\sim 10^{-15}\
\text{eV}\cdot\left(\frac{y_{\nu}}{10^{-10}}\right)\cdot\left(\frac{n_{\nu}}{56/\text{cm}^{3}}\right)^{1/3}.$
(4.3)

The mass corrections of $10^{-15}$ or $10^{-16}$ eV in Eqs. (4.2) and (4.3)
are comparable to the inverse of the geometrical sizes of the celestial bodies
being considered. Hence the above estimates imply that the C$\nu$B medium
effect can be considerably large in the aforementioned experiments.

Indeed, as is shown in Fig. 1, for a rather broad range of $y_{\nu}$, the
C$\nu$B may significantly alter some of the experimental bounds on long-range
forces. In this figure, we select three representative experiments, the Moscow
experiment using the Sun as the attractor (green curves), the Eöt-Wash
experiment using the Earth as the attractor (blue), and LLR which measures the
varying Earth-Moon distance (orange). The experimental bounds without
including medium effects are taken from Ref. Wagner:2012ui . We recast the
bounds into those presented in Fig. 1 with significant C$\nu$B medium effects
included. Since the C$\nu$B is a homogeneous distribution, the main effect is
the mass correction in Eq. (2.14). The C$\nu$B could also contribute to the
source term on the right-hand side of Eq. (2.13), but due to the homogeneity
this contribution only causes an constant shift of $\phi$, corresponding to
adding a constant to the effective potential. Hence it can be omitted.

In Fig. 1, we gradually increase the coupling $y_{\nu}$ from $0$ to $10^{-6}$
to show the impact of the C$\nu$B medium effect on long-range force searches,
assuming relativistic C$\nu$B. For nonrelativistic C$\nu$B, the results are
similar and the difference can be absorbed by rescaling $y_{\nu}$ as follows:

$y_{\nu}\to 18.5\times\left(\frac{m_{\nu}}{0.1\
\text{eV}}\right)^{1/2}y_{\nu}\thinspace.$ (4.4)

In the upper right panel, we set $y_{\nu}=10^{-13}$ which according to Eq.
(4.3) leads to a mass correction of $10^{-18}$ eV, corresponding to the
inverse of the Earth-Sun distance. Consequently, the green curve in the top
right panel is significantly altered and becomes a weaker bound. It is
interesting to note that the LLR bound (orange curve) in this panel becomes
more constraining for $m_{\phi}\lesssim 10^{-18}$ eV. This is because the LLR
bound is essentially a constraint on deviations from the inverse square law.
Without the medium effect, sufficiently light mediators cannot be effectively
constrained by LLR since the new force in the massless limit also follows the
inverse square law. With the medium effect, the effective mass is dominated by
the medium-induced mass if the vacuum mass $m_{\phi}$ is small. So LLR sets a
bound on $y_{e}$ independent of $m_{\phi}$ in this regime.

Further increasing $y_{\nu}$, some bounds can be substantially alleviated or
even entirely eliminated. For example, for $y_{\nu}\gtrsim 5\times 10^{-13}$
and $3\times 10^{-10}$, the green and orange bounds would disappear
respectively, as the medium effect renders the forces short-range with respect
to the relevant distances. One may ask to what extent these long-range force
bounds can be alleviated by continuing increasing $y_{\nu}$. In fact, when
$y_{\nu}$ is too large, the $\phi$ field would thermalize via neutrino
scattering in the early universe, modifying the BBN observables and the
cosmological effective number of relativistic neutrino species, $N_{{\rm
eff}}$. According to Refs. Huang:2017egl ; Li:2023puz , for light scalars
predominantly coupled to neutrinos, the cosmological upper bound on $y_{\nu}$
is about $10^{-5}$. Therefore, we stop increasing $y_{\nu}$ at $10^{-6}$ in
Fig. 1.

From Fig. 1, we conclude that the C$\nu$B medium effect on long-range force
searches can fully eliminate the bounds derived from LLR and the Sun, and
alleviate the bound from the Earth by about one order of magnitude.

##  5 Conclusions

Long-range force searches generally set the most constraining bounds on ultra-
light bosons that interact with normal matter. In this work, we assume that
such bosons are also coupled to neutrinos and investigate whether the presence
of cosmic neutrino background (C$\nu$B) could have an impact on long-range
force search experiments which are soaked in this ubiquitous background. We
find that the medium effect of C$\nu$B can significantly shorten the
interaction ranges of long-range forces, due to coherent forwarding scattering
of the force mediator with neutrinos. Consequently, the experimental bounds on
long-range forces can be altered by the C$\nu$B with sizable couplings of the
mediator to neutrinos.

Specifically, for experiments sensitive to new forces with interaction ranges
longer than $\sim 10^{8}$ km (the Sun-Earth distance, corresponding to a
mediator mass of $\sim 10^{-18}$ eV) or $\sim 10^{5}$ km (the Earth-Moon
distance), the C$\nu$B could effectively screen the forces from detection with
the neutrino coupling greater than $5\times 10^{-13}$ or $3\times 10^{-10}$,
respectively, causing such experimental bounds to be entirely eliminated. For
experiments utilizing the Earth as an attractor, the bounds can be
substantially alleviated by about one order of magnitude with cosmologically
allowed values of the neutrino coupling.

Our results can also be used in the studies of neutrino oscillations with
long-range forces—see e.g. Joshipura:2003jh ; Grifols:2003gy ; Gonzalez-
Garcia:2006vic ; Heeck:2010pg ; Bustamante:2018mzu ; Smirnov:2019cae ;
Babu:2019iml ; Singh:2023nek . We leave a dedicated study on this subject for
future work.

###### Acknowledgements.

We thank Xuheng Luo for early discussions that inspired this work. The work of
GC is supported by the U.S. Department of Energy under the award number DE-
SC0020250 and DE-SC0020262. The work of XJX is supported in part by the
National Natural Science Foundation of China under grant No. 12141501 and also
by the CAS Project for Young Scientists in Basic Research (YSBR-099).

##  Appendix A Some minus signs

In this appendix, we shall clarify a few minus signs important to our
analysis.

###  A.1 Minus signs responsible for repulsive and attractive forces

If the scalar mediator $\phi$ is replaced by a vector mediator $\phi^{\mu}$,
the long-range force between two $\psi$ particles becomes repulsive instead of
attractive. This change arises essentially from the differences in their
kinetic and mass terms. The mass term of a vector boson is positive, implying
a sign flip in the mass term:

$-\frac{1}{2}m_{\phi}^{2}\phi^{2}\ \to\
+\frac{1}{2}m_{\phi}^{2}\phi^{\mu}\phi_{\mu}\thinspace.$ (A.1)

On the other hand, the kinetic terms also differ by a minus sign:

$\frac{1}{2}\left(\partial\phi\right)^{2}=-\frac{1}{2}\phi\partial^{2}\phi\
\to\
-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}=\frac{1}{2}\phi^{\mu}\partial^{2}\phi_{\mu}\thinspace,$
(A.2)

where $F^{\mu\nu}\equiv\partial^{\mu}\phi^{\nu}-\partial^{\nu}\phi^{\mu}$.
Consequently, the EOM in Eq. (2.3) changes to

$-(\partial^{2}+m_{\phi}^{2})\phi^{\mu}+y_{\psi}\overline{\psi}\gamma^{\mu}\psi=0\thinspace.$
(A.3)

Due to the minus sign in Eq. (A.3), the force becomes repulsive between two
$\psi$ particles. One may wonder if the sign of $y_{\psi}$ in the Lagrangian
(2.1) is relevant. This sign is unimportant because the effective potential
and the mass correction due to the medium effect are all proportional to
$y_{\psi}^{2}$. So flipping the sign of $y_{\psi}$ has no physical
consequences.

###  A.2 When do antiparticle and particle contributions cancel?

Antiparticles may introduce some extra minus signs to the calculations. In
Eqs. (2.5), (2.8) and (2.11), only particles (no antiparticles) are included.
In the presence of antiparticles, one may be concerned with possible
cancellations between antiparticles and particles.

Let us first inspect the role of antiparticles in the ensemble averages
$\langle\overline{\psi}\psi\rangle$ and
$\langle\overline{\psi}\gamma^{\mu}\psi\rangle$, which determine the scalar
and vector field strengths of $\phi$ ($\phi^{\mu}$), respectively. Under
charge conjugation (the $C$ transformation), we have

$\displaystyle\langle\overline{\psi}\gamma^{\mu}\psi\rangle$
$\displaystyle\overset{C}{\to}-\langle\overline{\psi}\gamma^{\mu}\psi\rangle\thinspace,$
(A.4) $\displaystyle\langle\overline{\psi}\psi\rangle$
$\displaystyle\overset{C}{\to}\langle\overline{\psi}\psi\rangle\thinspace,$
(A.5)

which implies that $\langle\overline{\psi}\gamma^{\mu}\psi\rangle$ and
$\langle\overline{\psi}\psi\rangle$ should be antisymmetric and symmetric
under the particle-antiparticle interchange:

$\displaystyle\langle\overline{\psi}\gamma^{\mu}\psi\rangle$
$\displaystyle=\int\frac{d^{3}\mathbf{p}}{(2\pi)^{3}}\frac{p^{\mu}}{E_{\mathbf{p}}}\left[f_{+}(\mathbf{p})-f_{-}(\mathbf{p})\right],$
(A.6) $\displaystyle\langle\overline{\psi}\psi\rangle$
$\displaystyle=\int\frac{d^{3}\mathbf{p}}{(2\pi)^{3}}\frac{m_{\psi}}{E_{\mathbf{p}}}\left[f_{+}(\mathbf{p})+f_{-}(\mathbf{p})\right],$
(A.7)

where $f_{+}$ and $f_{-}$ denote the phase space distribution functions of
$\psi$ particles and antiparticles, respectively. Note that here
$\frac{d^{3}\mathbf{p}}{(2\pi)^{3}}\frac{1}{E_{\mathbf{p}}}$ is Lorentz
invariant.

Eqs. (A.6) and (A.7) are obtained simply from the argument of the $C$
symmetry. They can also be obtained by explicitly computing the ensemble
averages (i.e. performing Wick contractions of creation and annihilation
operators) using the following identities:

$\displaystyle\overline{u}\gamma^{\mu}u$ $\displaystyle=2p^{\mu}\thinspace,\ \
\overline{v}\gamma^{\mu}v=2p^{\mu}\thinspace,$ (A.8)
$\displaystyle\overline{u}u$ $\displaystyle=2m_{\psi}\thinspace,\ \
\overline{v}v=-2m_{\psi}\thinspace,$ (A.9)

where $u$ and $v$ denote the particle and antiparticle solutions of the Dirac
equation in momentum space, i.e. $(\not{p}-m_{\psi})u=0$ and
$(\not{p}+m_{\psi})v=0$. Note that the $u\leftrightarrow v$ interchange causes
a minus sign for the scalar case in Eq. (A.9), as opposed to Eq. (A.5) or
(A.7) where the scalar product is symmetric under the particle-antiparticle
interchange. This minus sign is canceled by interchanging anti-commuting
fermionic creation and annihilation operators. For the vector case, the same
reason accounts for the anti-symmetry in Eq. (A.6) when deriving it from Eq.
(A.8).

Finally, let us inspect the role of antiparticles in the mass correction
(2.8). The medium-induced mass correction for the scalar case arises from the
evaluation of the following ensemble average:

$\Delta m_{\phi}^{2}\phi^{2}\propto
y_{\psi}^{2}\langle\overline{\psi}\phi\psi\overline{\psi}\phi\psi\rangle\thinspace,$
(A.10)

which is obviously a $C$-even quantity according to Eq. (A.5). Hence
antiparticles contribute positively to Eq. (2.8), which after including such a
contribution should be

$m_{\phi}^{2}\to
m_{\phi}^{2}+y_{\psi}^{2}\frac{\tilde{n}_{\psi}+\tilde{n}_{\overline{\psi}}}{m_{\psi}}\thinspace.$
(A.11)

As for the vector case, although $\overline{\psi}\gamma^{\mu}\psi$ is $C$-odd,
the medium-induced mass correction
$\langle\overline{\psi}\gamma\cdot\phi\psi\overline{\psi}\gamma\cdot\phi\psi\rangle$
is $C$-even. Hence the antiparticle contribution is also positive, which is a
known conclusion for the plasmon mass in $e^{+}$$e^{-}$ plasma.

##  Appendix B Derivation of the medium effect from coherent forward
scattering

As mentioned in Sec. 2.2, the medium effect given by Eqs. (2.8) and (2.9),
known from the finite-temperature field theory, is also valid for non-thermal
distributions, because it fundamentally arises from coherent forward
scattering of $\phi$ with medium particles. In this appendix, we rederive this
medium effect using the theory of coherent forward scattering—similar
calculations for photon-dark photon or photon-axion conversions can be found
in Refs. Li:2023vpv ; Wu:2024fsf .

In coherent forward scattering, the medium particles do not change their
states after such scattering, implying that the amplitude of scattering with
each medium particle can be added coherently, as is illustrated by Fig. 2. The
combined effect corresponds to the mass correction in Eq. (2.8).

Figure 2: A schematic picture illustrating coherent forward scattering of
$\phi$ with medium particles. The momenta of both $\phi$ and $\psi$ remain
unchanged after the scattering. As a consequence, all scattering amplitudes
can be added coherently. The combined medium effect slows down the propagation
of the $\phi$ field.

Let us first compute the scattering of $\phi$ with a single $\psi$ particle,
for which the amplitude reads

$i{\cal M}=y_{\psi}^{2}\overline{u}(p)\left[\frac{i}{(p+k)\cdot\gamma-
m_{\psi}}+\frac{i}{(p-k)\cdot\gamma-m_{\psi}}\right]u(p)\thinspace,$ (B.1)

where $u$ and $\overline{u}$ denote the initial and final fermion states; $p$
and $k$ denote the momenta of the fermion and the scalar, respectively. The
second term in square brackets accounts for the interchange of the two Yukawa
vertices. Applying on-shell conditions for the fermion, $p^{2}=m_{\psi}^{2}$
and $\not{p}u=m_{\psi}u$, we obtain

$i{\cal M}=iy_{\psi}^{2}\overline{u}\left[\frac{4k^{2}m_{\psi}-4(p\cdot
k)(k\cdot\gamma)}{k^{4}-4(k\cdot p)^{2}}\right]u\thinspace.$ (B.2)

Next, we consider such scattering in a medium containing a large number of
$\psi$ particles, which form a background state denoted by
$|\text{bkg}\rangle$. In this case, we need to evaluate the following matrix
element:

$\langle{\rm bkg}|{\overline{\psi}}\phi{\psi}{\overline{\psi}}\phi{\psi}|{\rm
bkg}\rangle=\bcontraction{\langle}{\rm
bkg}{|}{\overline{\psi}}{}\bcontraction{\langle{\rm
bkg}|{\overline{\psi}}\phi}{\psi}{}{\overline{\psi}}{}\bcontraction{\langle{\rm
bkg}|{\overline{\psi}}\phi{\psi}{\overline{\psi}}\phi}{\psi}{|}{\rm
bkg}\langle{\rm
bkg}|{\overline{\psi}}\phi{\psi}{\overline{\psi}}\phi{\psi}|{\rm
bkg}\rangle+\text{vertex interchange}\,,$ (B.3)

where the first and third Wick contractions correspond to $\overline{u}$ and
$u$ in Eq. (B.1), and the second contraction corresponds to the fermion
propagator in Eq. (B.1), respectively. The “vertex interchange” term
corresponds to the second term in Eq. (B.1). Therefore, Eq. (B.2) implies

$\displaystyle\langle{\rm
bkg}|\overline{\psi}\phi\psi\overline{\psi}\phi\psi|{\rm bkg}\rangle$
$\displaystyle=\phi^{2}\langle\text{bkg}|\overline{\psi}\thinspace\frac{4k^{2}m_{\psi}-4(p\cdot
k)(k\cdot\gamma)}{k^{4}-4(k\cdot p)^{2}}\thinspace\psi|\text{bkg}\rangle$
$\displaystyle=\phi^{2}\int\frac{d^{3}\mathbf{p}}{(2\pi)^{3}}\frac{f_{+}(\mathbf{p})}{E_{\mathbf{p}}}\frac{4k^{2}m_{\psi}^{2}-4(p\cdot
k)(k_{\mu}p^{\mu})}{k^{4}-4(k\cdot p)^{2}}\thinspace,$ (B.4)

where in the second step we have used Eqs. (A.6) and (A.7). For simplicity, we
assume that the background does not contain antiparticles. The antiparticle
contribution can be readily included by performing charge conjugation on the
final result.

For applications in generating effective potentials, one can take the limit
$k^{2}\to 0$. Then Eq. (B.4) implies

$y_{\psi}^{2}\langle{\rm
bkg}|\overline{\psi}\phi\psi\overline{\psi}\phi\psi|{\rm
bkg}\rangle=y_{\psi}^{2}\phi^{2}\int\frac{d^{3}\mathbf{p}}{(2\pi)^{3}}\frac{f_{+}(\mathbf{p})}{E_{\mathbf{p}}}=y_{\psi}^{2}\phi^{2}\frac{\tilde{n}_{\psi}}{m_{\psi}}\thinspace,$
(B.5)

which is exactly the mass correction in Eq. (2.8).

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