††thanks<EMAIL_ADDRESS>Current address: Department of Physics,
University of Chicago, Chicago, IL 60637, USA††thanks<EMAIL_ADDRESS>
# Bounds to strong gravity from electron-positron annihilations near threshold

Nathaniel Alden  Department of Physics and Astronomy, Dartmouth College, 6127
Wilder Laboratory, Hanover, NH 03755, USA Roberto Onofrio  Department of
Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, NH
03755, USA

###### Abstract

We discuss bounds to strong gravity arising from annihilation of electron and
positrons and the corresponding enhancement of the cross section for the
production of massive particles. Two complementary examples are discussed, the
case of the $e^{+}e^{-}\to W^{+}W^{-}$ processes, with maximum mass and short
lifetime for the particle pairs in the final state, and the case of electron
positron annihilation into heavier charged lepton pairs,
$e^{+}e^{-}\to\mu^{+}\mu^{-}$ and $e^{+}e^{-}\to\tau^{+}\tau^{-}$. These
bounds may be improved by the next generation of electron-positron colliders,
and are complementary to bounds arising from direct graviton searches.

##  I Introduction

In the attempt to solve the hierarchy problem without necessarily invoking
supersymmetric models, a connection between the standard model and gravity at
the Fermi scale is conjectured in several approaches now under scrutiny.
Conjectures regarding a connection between gravitation and neutrinos have been
discussed since several decades (see for instance Dicke ; Brill ), including
efforts to consider weak interactions as the short-range counterpart of
gravitation Hehl1 ; Hehl2 ; Gasperini ; Sivaram , or natural extensions of the
successful mixing between electromagnetic and weak interactions Novello ;
Loskutov . This connection can emerge through several mechanisms proposed more
recently, such as introducing extra dimensions Arkani ; Antoniadis ; Randall ,
four-dimensional models with large number of fermions Dvali ; Calmet , Kerr-
Newman black holes with spin-dragging effects Burinskii1 ; Burinskii2 ;
Burinskii3 , and macroscopic gravity containing a long-range residual of the
Higgs field Dehnen1 ; Dehnen2 ; Consoli1 ; Consoli2 , among many attempts.
Moreover, strong gravity at the attometer scale - with the Newtonian
gravitational constant reaching the strength of the weak interactions - allows
for an interpretation of the Yukawa couplings of fundamental fermions in terms
of relative balance between quantum vacuum and gravitation Onofrio . A
possible mixing between gravity and weak interactions could also give insights
on the chiral structure of the latter Alexander  \- a feature of weak
interactions not shared by electromagnetic and color interactions -
encouraging their embedding into a geometrodynamics program Wheeler .

This multiform theoretical activity has also motivated searches for
gravitational signals at high energy colliders, for instance looking to
deviations from purely quantum electrodynamics processes such as photon pair
production, $e^{+}e^{-}\to\gamma\gamma$, starting from TRISTAN collider Abe .
The search of anomalous signals indicating graviton emission has been
unsuccessful so far, even at the highest energies available at the LHC Bound1
; Bound2 ; Bound3 . Also, precision experiments looking for deviations from
Newtonian gravity at much lower energies, using macroscopic apparata and force
measurements, have not evidenced positive signal, although they have
enormously extended the exclusion region Murata .

Figure 1: Modification of the production cross section due to the presence of
a Coulomb-like potential (a) and a Yukawa-like potential (b) versus the factor
$\beta=v/c$ up to values for which nonrelativistic approximations as the one
in Eq. 3 held. In the Coulomb case the $S$ factor scales as $\beta^{-1}$ at
small $\beta$ for attractive interactions, while it goes to zero for repulsive
interactions in the same limit. The coupling strength used here for both
potentials is $\alpha=1/128$, corresponding to the strength of the
electromagnetic interaction at the $Z^{0}$ pole, as a reference point. The
Yukawa potential is chosen with a mass of the intermediate boson
$m=80\leavevmode\nobreak\ \mathrm{GeV}$. The pattern is similar to the one of
the Coulomb case, except that in this case the $S$ factor saturates below a
certain velocity, with a magnitude depending on the particle mass $M$, here
represented with the cases of $M$=2 TeV (continous line), $M$=1 TeV (dot-
dashed line), and M=0.5 TeV (dashed line), see similar plots in Lattanzi2009 .

In this paper we discuss the possibility to evidence a gravitational
contribution, or at least to provide significant bounds, via precision
measurements of the cross section near the threshold production of particles
and antiparticles. The attention is focused on the cases which appear cleaner
from both the experimental and theoretical standpoints. This leads us to
consider electron-positron annihilation processes as initial states, producing
particle-antiparticle pairs without experiencing color interactions at tree
level. The selected final states are therefore the $W^{+}W^{-}$, with the
advantage of their mass but the drawback of their significantly short
lifetime, and the $\mu^{+}\mu^{-}$ and $\tau^{+}\tau^{-}$, with the advantage
of a better statistics, negligible suppression of the cross section
enhancement due to the lifetime, also utilizing a comparison with the
$e^{+}e^{-}$ final state.

The paper is organized as follows. In Sec. II we discuss the modification to
the cross section for a particle-antiparticle pair near the production
threshold. This is discussed for Yukawa-like potentials, having in mind a
contribution of this form arising from a specific parametrization of strong
gravity, and comparing this enhancement to the one of long-range interactions
also known as Sommerfeld enhancement. In Sec. III we discuss the enhancement
of the cross section in the most massive case for production of colorless
particle-antiparticle pairs, the case of $W^{+}W^{-}$ production,
incorporating its suppression due to the finite lifetime of the $W$s. In Sec.
IV, using the considerations of the previous two sections, we provide bounds
to strong gravity based on measurements of the $W^{+}W^{-}$ production
performed at LEP-II. In Sec. V we provide the complementary example of
production of charged heavier leptonic pairs, for which the comparatively
longer lifetimes are advantageous, at the price of a reduced expected signal
due to the significantly smaller masses, with higher statistics due to the
larger cross sections available at those energies.

These processes, as well as being useful to subtract the effect of the known
Coulomb contribution, could be studied at high precision and high statistics
in future electron-positron machines, as qualitatively discussed in the
conclusions.

##  II cross section enhancement for particle production with static short-
range interactions

The creation process of a particle-antiparticle pair is affected in the early
stage by the presence of interactions, schematized as static, and therefore
non-relativistic, present between the two particles. For instance, in the
simplest case of $e^{+}e^{-}\to e^{+}e^{-}$, one expects Coulomb attraction at
least. The effect of these potentials is increasingly important as one
approaches the production threshold, as the kinetic energy of the produced
pair is low, and therefore the two particles are spending more time nearby
each other.

In order to consider a general form of gravitation allowing for a short-
distance contribution, we introduce the following potential energy between two
pointlike particles of mass $m_{1}$ and $m_{2}$

$V_{G}(r)=-\frac{Gm_{1}m_{2}}{r}\left[1+\alpha_{G}e^{-r/\lambda_{G}}\right],$
(1)

where $\alpha_{G}$ is a dimensionless coupling constant, expressing the
strength of the short-distance component with respect to the known long-
distance, Newtonian, component, and $\lambda_{G}=\hbar/(m_{G}c)$ is its range,
the Compton wavelength of the massive graviton of mass $m_{G}$ expected from
hypothetical strong gravity. The usual long-range Newtonian term is negligible
for individual particles, so we will later on consider just the Yukawa term
when calculating the strong gravity correction.

The presence of a static potential alters the predictions for the cross
sections of annihilation and creation processes, enhancing or suppressing them
depending on the attractive or repulsive character of the interaction. In the
case of long-range interactions such as the Coulomb one, this effect is named
after Sommerfeld Sommerfeld1931  and in diagrammatic terms corresponds to a
ladder diagram in which a virtual particle (photon in the case of Coulomb
interactions) is repeatedly exchanged. The Sommerfeld (also often named
Sommerfeld-Gamow-Sakharov) effect was originally introduced to explain the
shift in the energy distribution of the electrons and positrons emitted
respectively in $\beta^{-}$ and $\beta^{+}$ processes Wu . The Coulomb
repulsion between the positron and the nucleus after $\beta^{+}$ emission
upshifts its energy, the opposite for electrons after $\beta^{-}$ emission.
The effect is velocity selective, and becomes more relevant at low energy.

To quantitatively understand the dependence of the enhancement or suppression
of the cross section due to a more generic class of potentials upon the
relevant quantities, we consider an annihilation process and the effect of the
interaction potential on the dynamics of the creation of particles and
antiparticles. The interaction potential should in general be treated non-
perturbatively, since it will not necessarily be small for small velocities,
inducing a modified wave function $\psi$ with respect to the free particle
wave function $\psi_{0}$, both evaluated at the origin (see, for example,
Arkani_Hamed_2009 ), $S=|\psi(0)|^{2}/|\psi_{0}(0)|^{2}$. These wave functions
are solutions of the Schroedinger equation for a reduced two-body system with
and without the static potential, respectively. A finite interaction region in
which creation processes do not necessarily occur in the origin leads to a
negligible difference in the $S$ factor, as discussed in Nierop2009 . To
evaluate $S$ we need to solve the nonrelativistic Schroedinger equation for
scattering problem in the dominant $s$-wave annihilation. The total energy is
the kinetic energy associated to the relative motions of the two particles,

$-\frac{\hbar^{2}}{M}\frac{d^{2}\psi(r)}{dr^{2}}+V(r)\psi(r)=Mv^{2}\psi(r),$
(2)

with $v$ being their relative velocity. We use throughout the entire paper
units in which $\hbar$ and $c$ appear explicitly, unlike natural units. The
solution to Eq. (2) for the Coulomb interaction yields Lattanzi2009

$S=\frac{\pi\alpha}{\beta}\frac{1}{1-e^{-\pi\alpha/\beta}},$ (3)

where $\alpha$ is the fine structure constant and $\beta=v/c$. This
relationship shows that in the limit of $\beta\to 0$ the enhancement factor
diverges as $S\ \sim\beta^{-1}$.

In the case of a generic Yukawa potential with parameters $\alpha$ and
$\lambda=\hbar/(mc)$, with $m$ the mass of the intermediate boson

$V(r)=-\frac{\alpha}{r}e^{-r/\lambda},$ (4)

the Schroedinger equation must be solved numerically. This has been performed
using the formulation developed in Iengo2009 , obtaining the dependence upon
velocity of the particle pair as in Fig. 1, for a Coulomb-like potential (left
plot) and for a Yukawa potential (right plot).

Increasing the mass of the particle pairs leads to an interesting resonant
behavior, previously discussed Nierop2009 ; Iengo2009 , as shown in Fig. 2.
This behavior can be semiqualitatively understood by considering the limit of
small distance, $r/\lambda<<1$, in which the Yukawa interaction resembles a
long-range interaction such as ordinary Newtonian gravitation or the Coulomb
interaction. Indeed, in this limit the exponential in the potential in Eq. (4)
can be Taylor-expanded to get $V(r)\approx-\alpha/r+\alpha/\lambda$. For small
velocities, the potential term dominates over the total energy term,
$Mv^{2}\ll\alpha/\lambda$, and the Schrodinger equation appears as identical
to the one for the bound states of a hydrogen atom. We therefore expect
resonant peaks at discrete values of particle masses, for a more detailed
derivation of this behavior, see Lattanzi2009 . This feature is retained also
when the analysis is numerically extended to the full Yukawa potential, as
shown in Fig. 2, although with quantitative differences. The resonant behavior
is evident especially at low relative velocities, and have been previously
discussed in the context of WIMP dark matter detection Lattanzi2009 ;
Slatyer2010 .

Figure 2: Enhancement of the cross section for a Yukawa potential of coupling
strength $\alpha=1/128$ and mass of the intermediate boson $m$= 90 GeV,
plotted versus the mass of the created pair of particles $M$, for various
values of the particles relative velocity. Resonances appear for masses $M$
above a certain threshold, more pronounced at low velocities.

Note that the mass scale of all known particles is below the resonant regime
for an interaction on the order of the weak interaction, so the enhancement
increases monotonically with increasing particle mass. Therefore for both
types of attractive potential, a larger particle mass leads to a larger
enhancement. This makes the analysis of production of the most massive
particles a convenient choice, on top of the natural gain due to the presence
of the mass in the gravitational potential. To avoid complications due to
color interaction effects, we focus on $W^{+}W^{-}$ production. However, in
this case we need to incorporate the corrections due to the finite lifetime of
the $W$ boson, which are particularly relevant very near the production
threshold, when the time spent by the two bosons within their Compton
wavelength becomes comparable or longer than their lifetime, as we discuss in
the next section.

##  III Production of unstable particles

We now consider the annihilation process of the $e^{+}e^{-}$ pair into a pair
of unstable particles and antiparticles, having in mind the concrete case of
the $e^{+}e^{-}\to W^{+}W^{-}$ process. Indeed, the $W$ bosons have the
largest mass for charged particles available so far, considering that top-
antitop production is not yet available in electron-positron colliders,
therefore they maximize the expected gravitational signal. The choice of
colorless final states helps to reduce the effect of QCD final state
interconnection effects such as color reconnection and Bose-Einstein
correlations. However, as a consequence of their large mass and therefore of
the available phase space for decays, $W$ bosons have a short lifetime. This
is affecting the physics around the production threshold, and therefore we
expect suppression of the $S$ factor at small relative velocity. The
incorporation of the lifetime of final states has been the subject of several
studies, starting from Fadin1988 ; Fadin1989  in which it has been discussed
as an effect protecting the top-antitop production from non-perturbative QCD,
potentially leading to toponium bound states, a topic still of current
interest. The analysis was then extended to the case of W-pair production,
including the discussion of higher order effects Bardin1993a ; Bardin1993b ;
Fadin1993 ; Fadin1995D ; Fadin1995B .

The general expression for the $W$-pair production cross section from the
$e^{+}e^{-}$ initial state with off-shell $W$ bosons is given by Muta1986

$\displaystyle\sigma(s)$ $\displaystyle=$ $\displaystyle
B_{f_{1}{\bar{f}_{2}}}B_{f_{3}{\bar{f}_{4}}}\int_{0}^{s}ds_{1}\rho(s_{1})\times$
(5)
$\displaystyle\int_{0}^{(\sqrt{s}-\sqrt{s_{1}})^{2}}ds_{2}\rho(s_{2})\sigma(s,s_{1},s_{2}),$

where $s_{1}$ and $s_{2}$ are the Mandelstam variables for each of the two
outcoming $W$ bosons, $\sigma(s,s_{1},s_{2})$ the corresponding cross section
fully detailed in Muta1986 , $B_{f_{1}{\bar{f}_{2}}}$ and
$B_{f_{3}{\bar{f}_{4}}}$ are the branching ratios for the decay of the $W$s
into fermion pairs $f_{1}{\bar{f}_{2}}$ and $f_{3}{\bar{f}_{4}}$,
$\rho(s_{i})$ are the Breit-Wigner form factors

$\rho(s_{i})=\frac{1}{\pi}\frac{\sqrt{s_{i}}\leavevmode\nobreak\
\Gamma_{W}(s_{i})}{(s_{i}-M_{W}^{2})^{2}+M_{W}^{2}\Gamma_{W}^{2}(s_{i})},$ (6)

and $\Gamma_{W}(s_{i})$ is the (running) full width of the $W$ boson,
$\Gamma_{W}(s_{i})=3g^{2}\sqrt{s_{i}}/(16\pi)$. This width accounts for the
radiative effects due to the $W$ boson decays Fadin1995D , and with respect to
Muta1986  has been updated since the third generation with the top quark does
not contribute. It remains to calculate the effect of interactions between the
particles on the cross section for $W^{+}W^{-}$ $s$-wave production, also
including the effect of strong gravity

$\displaystyle S$ $\displaystyle=$ $\displaystyle
1+\frac{\alpha_{C}}{\beta}\delta_{C}+\frac{\alpha_{Z}}{\beta}\delta_{Z}+\frac{\alpha_{H}}{\beta}\delta_{H}+\frac{\alpha_{g}}{\beta}\delta_{g}$
(7) $\displaystyle=$ $\displaystyle
1+\sum_{int}\frac{\alpha_{int}}{\beta}\delta_{int},$ (8)

where $\beta=4pc/\sqrt{s}$, and

$p=\frac{1}{2c\sqrt{s}}[(s-s_{1}-s_{2})^{2}-4s_{1}s_{2}]^{1/2},$ (9)

for off-shell $W$ bosons. The terms $\delta_{C}$, $\delta_{Z}$ and
$\delta_{H}$ are the expected contributions from standard model interactions,
the Sommerfeld one due to photon, and the ones due to $Z^{0}$ and Higgs
exchange between the W pair, respectively. We have also introduced the
dimensionless coupling constant $\alpha_{g}$

$\alpha_{g}\equiv{\frac{Gm_{1}m_{2}}{\hbar c}}{\alpha_{G}},$ (10)

which allows for a more direct comparison with the other contributions. In the
case of $W$ bosons ($m_{1}=m_{2}=M_{W}$) we have $\alpha_{g}=3.82\times
10^{-35}\alpha_{G}$, the largest apart from the one pertaining to a $t\bar{t}$
pair. Due to the static character of the various potentials, no interference
effects are possible. However, in assessing bounds to strong gravity it is
crucial to notice that the largest $\alpha_{g}$ meaningfully bounded should be
smaller than the next-to-leading contribution to $\alpha_{C}$ in the
perturbative expansion of the Coulomb contribution and other sources of signal
such as the initial state radiation Bardin1993a ; Fadin1995B , expected to be
negligible at the threshold, and amounting to about 0.1 $\%$ in its vicinity
Note .

We now evaluate the quantity $\sigma(s,s_{1},s_{2})$ in Eq. (5) defined as
$\sigma(s,s_{1},s_{2})=\sigma_{0}(s,s_{1},s_{2})(1+\delta(R,I))$ Fadin1995D ,
where

$\delta(R,I)=\frac{\alpha_{C}}{\pi}\delta_{ISR}S\simeq\frac{\alpha_{C}}{\pi}\delta_{ISR}+\sum_{int}\frac{\alpha_{int}}{\beta}\delta_{int},$
(11)

with the Born cross section for off-shell $W$ bosons
$\sigma_{0}(s,s_{1},s_{2})$ given explicitly in Muta1986 . In the second line,
we use the fact that hard initial state radiation decouples from the static
potential contributions to the cross section Fadin1995D . The last
relationship in Eq. (11) holds as long as the corrections to the cross section
due to all interactions are perturbative.

Figure 3: Integrated enhancement of the cross section for a short-range
gravitational interaction between off-shell $W$ bosons with coupling strength
equal to the electromagnetic interaction, versus $\sqrt{s}$. The cases of a
massless graviton (therefore contributing identically to a photon) (b), and
massive gravitons of mass $m_{G}$ equal to 10 GeV (c), 100 GeV (d), and 1 TeV
(e) are shown, as well as the hypothetical case of stable $W$ bosons (a). The
maximum in the enhancement is progressively shifted away from the W-production
threshold, and quickly decreasing in amplitude when $m_{G}$ becomes of the
same order of magnitude of $M_{W}$. The vertical dashed line indicates the on-
shell threshold for W-pair production.

For a generic potential $V_{int}(r)$, the corresponding correction is given by
Fadin1995D

$\displaystyle\alpha_{int}\delta_{int}$ $\displaystyle=$
$\displaystyle-\frac{2}{\hbar}\int_{0}^{\infty}dr\;V_{int}(r)e^{-p_{1}r/\hbar}\times$
(12) $\displaystyle\\{\sin[(p+p_{2})r/\hbar]+\sin[(p-p_{2})r/\hbar]\\},$

where $p_{1}$, $p_{2}$ are defined by

$\displaystyle p_{1}$ $\displaystyle=$
$\displaystyle\left[\frac{M_{W}}{2}\left(\sqrt{E^{2}+\Gamma_{W}^{2}}-E\right)\right]^{1/2},$
(13) $\displaystyle p_{2}$ $\displaystyle=$
$\displaystyle\left[\frac{M_{W}}{2}\left(\sqrt{E^{2}+\Gamma_{W}^{2}}+E\right)\right]^{1/2},$
(14)

and $E=(s-4M_{W}^{2}c^{4})/(4M_{W}c^{2})$.

By rewriting the strong gravity potential as

$\displaystyle V_{g}$ $\displaystyle=-\hbar
c\frac{\alpha_{g}}{r}e^{-r/\lambda_{G}},$ (15)

we can solve exactly Eq. (12)

$\displaystyle\alpha_{g}\delta_{g}=-2\int_{0}^{\infty}dr\frac{1}{r}e^{-(p_{1}+\hbar/\lambda_{G})r}\times$
$\displaystyle\\{\sin[(p+p_{2})r/\hbar]+\sin[((p-p_{2})r/\hbar\\}=$ (16)
$\displaystyle
2\left[\mathrm{arctan}\left(\frac{p+p_{2}}{p_{1}+m_{G}c^{2}}\right)+\mathrm{arctan}\left(\frac{p-p_{2}}{p_{1}+m_{G}c^{2}}\right)\right],$

which also holds for the other Yukawa-like interaction of the $Z_{0}$ and the
Higgs. Long-range interactions such as the Coulomb one are just recovered in
the $\lambda_{G}^{-1}\to 0$ limit and appropriate coupling strength Fadin1995D
.

The enhancement factor for several different Yukawa potentials, all with
$\alpha_{g}=1/128$ and various $m_{G}$ is plotted as a function of $\sqrt{s}$
in Fig. 3. Due to the off-shellness of the $W$ bosons, the enhancement is non
negligible at threshold and even below it, without the expected divergent
behavior expected at threshold for on-shell particles. Relative to the case of
infinite lifetime (black curve) the finite lifetime suppresses the
enhancement, as do values of $m_{G}$ approaching or larger than $M_{W}$,
thereby enforcing an effective limitation for searching strong gravity with
this method.

##  IV Bounds from W-pair production at LEP-II

We now describe possible bounds to strong gravity from the measurements on
$e^{+}e^{-}\to W^{+}W^{-}$ performed at LEP-II in the
$161.3\;\mathrm{GeV}\leq\sqrt{s}\leq 206.6\;\mathrm{GeV}$ range. The lower
energy values are closer to threshold and will be more affected by all the
possible sources of static interaction between the two Ws. However the cross
section is small in the same region, so statistical errors, for a given
integrated luminosity, are large. Vice versa, at higher energies the
statistical error is smaller, but the expected enhancement is also smaller. We
then need to assess a range of optimal value of $\sqrt{s}$ for precision
measurements, in which one is maximizing the sensitivity to the putative
strong gravity signal.

$\sqrt{s}$ (GeV) |  $\sigma_{\mathrm{exp}}$ (pb) | $S_{C}$ | $S_{Z}$ | $S_{H}$ | $S_{g}$ | $\xi$
---|---|---|---|---|---|---
161.3 | 3.69 $\pm$ 0.45 | 1.042 | 1.049 | 1.236 | 1.144 | 0.89
172.1 | 12.0 $\pm$ 0.7 | 1.029 | 1.051 | 1.253 | 1.115 | 1.48
182.7 | 15.92 $\pm$ 0.34 | 1.023 | 1.051 | 1.258 | 1.094 | 3.31
188.6 | 16.05 $\pm$ 0.21 | 1.021 | 1.050 | 1.259 | 1.087 | 4.99
191.6 | 16.42 $\pm$ 0.47 | 1.020 | 1.050 | 1.260 | 1.084 | 2.21
Table 1: Analysis of the LEP-II measurements on the $e^{+}e^{-}\to W^{+}W^{-}$
channel close to its production threshold. In the first column the $\sqrt{s}$
closest to threshold appear, followed by the measured cross section for
$W$-pair production, including the error bars, from the analysis of the LEP
Electroweak Working Group LEP200 . The columns from third to sixth contain the
$S$ factor for each interaction to which the W pair is sensitive in the
Standard Model. The final column contains the values for
$\xi\equiv(\bar{S}_{g}-1)\left({\Delta\sigma}/{\sigma}\right)^{-1}$, using the
measured $\Delta\sigma$ and $\sigma$ from the second column. The gravitational
component of the factor $\bar{S}_{g}$ is evaluated as described in the text,
using the parameters $\alpha_{g}=1/30$ and $m_{G}=$ 90 GeV. Figure 4: Contour
plot for the exclusion of strong gravity in the $\alpha_{G}-\lambda_{G}$
plane, both in log10 scale, from the measured cross section of $e^{+}e^{-}\to
W^{+}W^{-}$ at LEP-II. The black square is the location of the hypothetical
gravitoweak unification, at $\lambda_{G}=8\times 10^{-19}$ m and
$\alpha_{G}=1.2\times 10^{33}$, at which gravitation has the same strength as
charged weak interactions. The Standard Model cross section used here includes
the contributions from photon exchange, $Z^{0}$ exchange and Higgs boson
exchange. We then calculate the ratio of the cross section, including a
hypothetical strong gravity interaction, to the Standard Model cross section.
Contours delineate the exclusion regions based on the LEP-200 measurements at
$\sqrt{s}=188.6$ GeV. The maximum meaningful value on $\alpha_{G}$ is imposed
by the validity of the perturbation expansion, and corresponds to a
dimensionless coupling parameter $\alpha_{g}\sim 0.25$, therefore
$\alpha_{G}\sim 6.54\times 10^{33}$, i.e. 33.82 in the log10 scale.

In terms of measurements of the cross section, we need to take into account
the uncertainty on the cross section for $W^{+}W^{-}$ pair production. To
date, there has been no sign of tension between measured and predicted cross
section. Therefore the goal is to place bounds on strong gravity parameter
space by calculating the $S$ factor for a range of strong gravity potentials,
and to then rule out regions of parameter space for which S is too large to
agree with observations. Assuming agreement between experiments and the
Standard Model prediction, we can obtain confidence contours on $S$ factor
using the relative error on the cross section measurement at a given
$\sqrt{s}$, the ratio $\Delta\sigma/\sigma$.

We use the cross section measurements taken at LEP-II LEP200 . The energy
measured closest to threshold is $\sqrt{s}=161.3$ GeV. However, we expect to
obtain better constraints on strong gravity parameter space using measurements
of the cross section taken at higher values of $\sqrt{s}$. After combining the
data from the ALEPH, DELPHI, L3, and OPAL collaborations, we collect Table I,
containing the center of mass energy, the measured cross section for the
production of $W^{+}W^{-}$, the contributions to the enhancement due to each
exchanged boson expected in the standard model, plus an example of strong
gravity contribution to $S$. We also include the quantity
$\xi\equiv(\bar{S}_{g}-1)\left({\Delta\sigma}/{\sigma}\right)^{-1}$, where
$\bar{S}_{g}$ is the integrated $S$ factor, given by
$\bar{S}_{g}=(\sigma_{SM}+\sigma_{g})/\sigma_{SM}$, where
$\sigma_{SM}=\sigma_{0}+\sigma_{c}+\sigma_{z}+\sigma_{h}$, with each quantity
being evaluated according to the integration as in Eq. (5). The statistical
bounds are determined by the product value of $\xi$, with a high value of
$\xi$ implying greater statistical significance of the measurement to
constrain strong gravity parameter space.

The largest value of $\xi$ in Table I occurs at $\sqrt{s}=188.6$ GeV. We
therefore fix $\sqrt{s}$ at this value, and vary $\alpha_{G}$ and
$\lambda_{G}$ for a constant $\bar{S}_{g}$ to obtain a plot of the
$W^{+}W^{-}$ pair production $S$ factor in the strong gravity parameter space
(see figure 4). At this energy, the percent uncertainty on the cross section
as measured at LEP is given by
$0.21\;{\mathrm{pb}}/16.05\;{\mathrm{pb}}\approx 1.31\%$. We use this relative
error to assign statistical significance to the contours in Fig. 4, with
darker colors corresponding to higher statistical significance in terms of
standard deviations.

As a benchmark, if gravity is embedded into a more extended structure
including weak interactions, such that weak interactions are interpreted as
its quantum and relativistic manifestation at short distances, with the
relationship Onofrio

$\frac{G_{F}}{\sqrt{2}}=\left(\frac{\hbar}{c}\right)^{2}\tilde{G}_{N},$ (17)

where $\tilde{G}_{N}$ should be a renormalized Newtonian gravitational
constant at the attometer scale, we expect this to occur in the
$\lambda_{G}-\alpha_{G}$ plane at $\mathrm{log}_{10}\alpha_{G}=33.08$ and
Yukawa range $8\times 10^{-19}\mathrm{m}$. This possibility is not yet ruled
out, see black square in Fig. 4, but it is within an order of magnitude of the
1 $\sigma$ confidence level contour.

Apart from decreasing the statistical error with larger number of events in
future measurements at high luminosity $e^{+}e^{-}$ colliders, such as the
ones currently under design for the detailed study of the Higgs boson, one
possibility to pinpoint a strong gravity contribution consists in studying
also the process $e^{+}e^{-}\to ZZ$, in which the Coulomb contribution is
absent. The data already available on this process from the measurements at
LEP-II are not directly competitive with the $W^{+}W^{-}$ channel due to the
intrinsically lower cross section, generating statistical errors generally
larger by one order of magnitude Jadach ; LEP200 .

Figure 5: Ratio between the total cross section for the
$e^{+}e^{-}\to\mu^{+}\mu^{-}$ process and the corresponding Born cross section
versus $\beta$. The same curve also characterizes the
$e^{+}e^{-}\to\tau^{+}\tau^{-}$ process, indicating that the lifetime of the
final states do not influence the dynamics even at the lowest values of
$\beta$ shown. For comparison, the behavior of the same quantity for the
$e^{+}e^{-}\to W^{+}W^{-}$ process is shown, with nearly complete suppression
of the cross section enhancement.

Figure 6: Contour plot for the exclusion of strong gravity in the
$\alpha_{G}-\lambda_{G}$ plane, both in log10 scale, from possible
measurements of the $S$ factor in $e^{+}e^{-}\to\mu^{+}\mu^{-}$ (left) and in
$e^{+}e^{-}\to\tau^{+}\tau^{-}$ (right). In both cases the small, but finite
width is incorporated. The difference in sensitivity compared to $W^{+}W^{-}$
pair production is due to the smaller mass of the final state leptons, which
suppresses the gravitational coupling constant $\alpha_{g}$, with respect to
the W pair case, by a factor $5.8\times 10^{5}$ for the muon pairs, and
$2\times 10^{3}$ for the $\tau$ pairs. The false color scale on the right side
of each plot indicates the magnitude of the expected gravitational
contribution to the $S$ factor for each projected exclusion region.

##  V Strong gravity in heavy lepton pair production

This analysis can be naturally extended to the case of
$e^{+}e^{-}\to\mu^{+}\mu^{-}$. Existing electron-positron annihilation
experiments have studied the related cross section into muon pairs either with
low statistics near the threshold Borgia ; Balakin ; Alles , or more precisely
(order of 1$\%$) at higher energies Akhmetshin ; Kurdadze ; Ambrosino ;
Achasov . Precision measurements of this process are still crucial to better
understand the hadronic contribution to the anomalous magnetic moment of the
muon. Moreover, if a muon collider is successfully built, it will be possible
to achieve similarly near-threshold cross section measurements as LEP.
Moreover, the feasibility of muon colliders requires schemes with low
transverse emittance beams, and in this context experiments with positron
beams slightly above the muon pair production threshold are crucial
Amapane2020 .

The total cross section for $e^{+}e^{-}\to\mu^{+}\mu^{-}$ including leading
order radiative and initial state radiation corrections has been calculated
Bystritskiy2005

$\sigma_{ISR}(s)=\left(1+\frac{2\alpha}{\pi}\Delta_{ISR}\right)\sigma_{B}(s),$
(18)

where $\alpha$ is the electromagnetic coupling constant $e^{2}/(\hbar c)$
evaluated at the scale $s$, $\sigma_{B}$ the Born cross section, and

$\displaystyle\Delta_{ISR}$ $\displaystyle=$ $\displaystyle(\ell_{e}-1)\times$
(19)
$\displaystyle\left[-\frac{1-3\beta+\beta^{3}}{\beta(3-\beta^{2})}L_{\beta}-\frac{4}{3}+2\ln\left(\frac{2\beta}{1+\beta}\right)\right]+$
$\displaystyle\frac{3}{4}\ell_{e}-1+\frac{\pi^{2}}{6},$

where $L_{\beta}=\ln[(1+\beta)/(1-\beta)]$ and
$\ell_{e}=\ln(s/m_{e}^{2}c^{4})$. In the limit of small $\beta$ Eq. 19 becomes

$\Delta_{ISR}^{(0)}\simeq
2(\ell_{e}-1)\left[\ln(2\beta)-1\right]+\frac{3}{4}\ell_{e}+\frac{\pi^{2}}{6}-1.$
(20)

In this limit, radiative and initial state radiation contributions are
quenched, as the static part of the corrections prevails (see Smith ; Voloshin
for a detailed discussion) and therefore we need to interpolate Eq. 18 with
the following replacement,
$\Delta_{ISR}^{\mu^{+}\mu^{-}}\to\Delta_{ISR}^{\mu^{+}\mu^{-}}-\Delta_{ISR}^{(0)}$,
and to incorporate the correction $S$ as a multiplicative factor $1+S$ in Eq.
18.

Taking into account the lifetime of the muon, we plot the enhancement factor
with respect to the Born cross section, $\sigma_{tot}/\sigma_{B}$, versus
$\beta$ in Fig. 5. For comparison, we also show the case of the Ws, in which
the lifetime has a significant impact. The long lifetime of the muons of
$2.2\mu$s, corresponding to a linewidth $\Gamma_{\mu}=3\times 10^{-10}$ eV,
does not play a role in suppressing the effect of the static potentials,
allowing for gains of order $10^{2}-10^{3}$ in a range of $\beta$ between
$10^{-5}$ and $10^{-4}$. Analogous considerations can be applied to the case
of $e^{+}e^{-}\to\tau^{+}\tau^{-}$, with the linewidth of the $\tau$ being
$2.27\times 10^{-4}$ eV, with an enhancement indistinguishable from the one of
the muons, but with the benefit of a larger mass.

We plot a projection of the exclusion region in Fig. 6, where in the absence
of actual precision measurements close to threshold we map a possible outcome
of the resulting gravitational $S$ factor into the corresponding bounds in the
$\alpha_{G}-\lambda_{G}$ plane. Both these cases do not seem competitive with
the bounds available from the W pair production, but a dedicated analysis is
in order since the larger cross sections at these lower energies can result in
smaller statistical errors near the production threshold. In addition, the
comparison of the cross section enhancement dependence on $\sqrt{s}$ for the
three processes $e^{+}e^{-}\to e^{+}e^{-}$, $e^{+}e^{-}\to\mu^{+}\mu^{-}$,
$e^{+}e^{-}\to\tau^{+}\tau^{-}$, allows to better disentangle the
contributions of the Coulomb interaction from putative strong gravity. For
instance, the cross section for the production of lepton pairs increases
linearly with $\beta$ near the threshold, and at threshold is inversely
proportional to the mass of the produced lepton,
$\sigma_{0}\simeq\alpha^{2}\beta/s$. Due to the inverse dependence of the
Coulomb Sommerfeld factor on $\beta$, we expect a finite cross section exactly
at threshold $\sigma(m_{\ell})\simeq\alpha^{2}/m_{\ell}^{2}$, followed by a
monotonic decrease at higher energies. The strong gravity contribution does
not follow the same pattern as the related $S$ factor saturates as discussed
in Fig. 1. The related cross section is therefore expected to reach a maximum
above the threshold and, due to the dependence of the gravitational potential
on the square of the mass of the produced leptons, is rather insensitive to
the lepton mass. We therefore expect local peaks located at approximately the
same distance, in $\sqrt{s}$, from the threshold, regardless of the production
of electron, muon and tau pairs. This shows the advantages of a comparative
analysis of the various cases.

##  VI Conclusions

We have shown that measurements of the $W^{+}W^{-}$ production cross section
near threshold can be used to constrain the parameter space for strong
gravity, a sort of microscopic counterpart to the Cavendish experiment to
study macroscopic gravitation. We have also briefly discussed the potential
for analogous bounds using production of heavier leptons close to threshold.
The analysis can be refined in the presence of concrete integrated
luminosities for various intermediate and high energy electron-positron
colliders, using optimization approaches as the one discussed in Stirling .

More generally, accurate measurements of the cross section can be used to
detect Beyond-Standard-Model (BSM) interactions, since any interaction leads
to a Sommerfeld-like effect on the cross section close to the production
threshold. A systematic subtraction of the Coulomb contribution to the
Sommerfeld factor can be achieved by measuring the $e^{+}e^{-}\to ZZ$ cross
section in the case of the $W^{+}W^{-}$ channel, and of the $e^{+}e^{-}\to
e^{+}e^{-}$ cross section in the case of the heavier lepton production
channels. This can be also relevant to evidence the presence of toponium
formation Fuks , as well as to understand possible tensions in measurements of
the $W$ boson mass Grzadkowski ; Aaltonen2013 ; Aaltonen2022 .

The bounds on strong gravity presented here are about three orders of
magnitude weaker than those presented in Murata  using LHC data. However,
those techniques rely on the presence of a graviton being produced in a
process, while the results described above are valid for any model which can
be described effectively by a Yukawa potential, for instance in approaches
where spacetime itself has a foamy structure at the Planck scale in which
gravitons are not necessarily required Hu . Furthermore, the methodology we
propose has systematic and statistical errors quite different from the ones
for direct graviton search, ensuring independent checks in case positive
signals are evidenced in the latter scenario.

These considerations provide further motivation to accurately measure the
cross section of fundamental particles near threshold. This would be also
feasible either modifying existing machines, such as DA$\Phi$NE and VEPP-2000,
or in the first stage of development and calibration of future muon colliders,
together with the development of efficient detection of low $\beta$ particles.

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