# Charged Quark Stars and Extreme Compact Objects in Regularized 4D Einstein-
Gauss-Bonnet Gravity

Michael Gammon<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1  Robert B. Mann
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of
Waterloo, Waterloo, Ontario, Canada, N2L 3G1  Sarah Rourke
<EMAIL_ADDRESS>Department of Physics, McGill University,
Montreal, Quebec, Canada

###### Abstract

Since the derivation of a well-defined $D\rightarrow 4$ limit for 4
dimensional Einstein Gauss-Bonnet (4DEGB) gravity coupled to a scalar field,
there has been interest in testing it as an alternative to Einstein’s general
theory of relativity. Using the Tolman-Oppenheimer-Volkoff (TOV) equations
modified for charge and 4DEGB gravity, we model the stellar structure of
charged, non-interacting quark stars. We find that increasing the Gauss-Bonnet
coupling constant $\alpha$ or the charge $Q$ both tend to increase the mass-
radius profiles of quark stars described by this theory, allowing a given
central pressure to support larger quark stars in general. We also derive a
generalization of the Buchdahl bound for charged stars in 4DEGB gravity. As in
the uncharged case, we find that quark stars can exist below the general
relativistic Buchdahl bound (BB) and Schwarzschild radius $R=2M$, due to the
lack of a mass gap between black holes and compact stars in the 4DEGB theory.
Even for $\alpha$ well within current observational constraints, we find that
quark star solutions in this theory can describe Extreme Compact Charged
Objects (ECCOs), objects whose radii are smaller than what is allowed by
general relativity.

##  I Introduction

Modified theories of gravity continue to attract attention despite the
empirical success of general relativity (GR). These theories are motivated by
a variety of problems, including addressing issues in modern cosmology
bueno2016 ; sotiriou_2010 ; nojiribook ; clifton_2012 , quantizing gravity
ahmed2017 ; stelle1977 , eliminating singularities
brandenberger1992nonsingular ; BRANDENBERGER_1993 ;
brandenberger1995implementing , and, perhaps most importantly, finding viable
phenomenological competitors against which GR can be tested in the most
stringent manner possible.

Higher curvature theories (or HCTs) are amongst the most popular
modifications. An HCT modifies the assumed linear relationship in GR between
the curvature and the stress-energy, replacing the former with an arbitrary
sum of powers of the curvature tensor (appropriately contracted to two
indices). Such modifications provide us with a foil to further challenge the
empirical success of GR, while also making new testable predictions.

Lovelock theories lovelock1971  have long been at the forefront of this
search, since they possess the distinctive feature of having 2nd order
differential equations of motion. The physical significance of such theories
has been unclear, however, since their higher order terms yield non-trivial
contributions to the equations of motion only in more than four spacetime
dimensions ($D>4$).

Recently this restriction was circumvented hennigar_2020_on ;
Fernandes:2020nbq  in the quadratic case, or what is better known as
“Einstein-Gauss-Bonnet” gravity . The Gauss-Bonnet (GB) contribution to the
gravitational action is

$S_{D}^{GB}=\alpha\int
d^{D}x\sqrt{-g}\left[R^{\mu\nu\rho\tau}R_{\mu\nu\rho\tau}-4R^{\mu\nu}R_{\mu\nu}+R^{2}\right]\equiv\alpha\int
d^{D}x\sqrt{-g}\mathcal{G}$ (1)

(where $R_{\mu\nu\rho\tau}$ is the Riemann curvature tensor), which becomes
the integral of a total derivative in $D=4$, and thus cannot contribute to a
system’s gravitational dynamics in less than five dimensions. For this reason
it is often referred to as a “topological term” having no relevance to
physical problems. Indeed, the Lovelock theorem lovelock1971  ensures that a
$D=4$ dimensional metric theory of gravity must incorporate additional fields
in order to have second order equations of motion and diffeomorphism
invariance.

A few years ago it was noted glavan2020  that several exact solutions to
$D$-dimensional Einstein-Gauss Bonnet gravity have a sensible limit under the
rescaling

$\lim_{D\to 4}(D-4)\alpha\rightarrow\alpha,$ (2)

of the Gauss-Bonnet coupling constant. Using this approach a variety of
4-dimensional metrics can be obtained, including cosmological glavan2020 ;
li2020 ; kobayashi2020 , spherical black hole glavan2020 ; kumar2022 ;
fernandes2020 ; kumar2020 ; kumar2022_2 , collapsing malafarina2020 , star-
like doneva2021 ; charmousis2022 , and radiating ghosh2020  metrics, each
carrying imprints of the quadratic curvature effects of their $D>4$
counterparts. However a number of objections to this approach were
subsequently raised gurses2020 ; ai2020 ; shu2020 , based on the fact that the
existence of a limiting solution does not imply the existence of a well-
defined 4D theory whose field equations have that solution. This shortcoming
was quickly addressed when it was shown that the $D\to 4$ limit in (2) can be
taken in the gravitational action hennigar_2020_on ; Fernandes:2020nbq ,
generalizing an earlier procedure employed in obtaining the $D\to 2$ limit of
GR Mann:1992ar . One can also compactify $D$-dimensional Gauss-Bonnet gravity
on a $(D-4)$-dimensional maximally symmetric space and then use (2) to obtain
a $D=4$ HCT Lu:2020iav . This approach yields the same result (up to trivial
field redefinitions), in addition to terms depending on the curvature of the
maximally symmetric $(D-4)$-dimensional space. Taking this to vanish yields

$\displaystyle S_{4}^{GB}=$ $\displaystyle\alpha\int
d^{4}x\sqrt{-g}\left[\phi\mathcal{G}+4G_{\mu\nu}\nabla^{\mu}\phi\nabla^{\nu}\phi-4(\nabla\phi)^{2}\square\phi+2(\nabla\phi)^{4}\right]$
(3)

where we see that an additional scalar field $\phi$ appears. Surprisingly, the
spherically symmetric black hole solutions to the field equations match those
from the naïve $D\rightarrow 4$ limit of $D>4$ solutions glavan2020 . The
resultant 4D scalar-tensor theory is a particular type of Horndeski theory
horndeski1974 , and solutions to its equations of motion can be obtained
without ever referencing a higher dimensional spacetime hennigar_2020_on .

We are interested here in what is called 4D Einstein-Gauss-Bonnet gravity
(4DEGB), whose action is given by (3) plus the Einstein-Hilbert term:

$\displaystyle S$ $\displaystyle=S^{GR}+S_{4}^{GB}$
$\displaystyle=\frac{1}{8\pi G}\int
d^{4}x\sqrt{-g}\left[R+\alpha\left\\{\phi\mathcal{G}+4G_{\mu\nu}\nabla^{\mu}\phi\nabla^{\nu}\phi-4(\nabla\phi)^{2}\square\phi+2(\nabla\phi)^{4}\right\\}\right]$
(4)

which has been shown to be an interesting phenomenological competitor to GR
Clifton:2020xhc ; charmousis2022 ; Zanoletti:2023ori . Despite much
exploration Fernandes:2022zrq , the role played by these higher curvature
terms in real gravitational dynamics is still not fully understood. One
important arena for testing such theories against standard general relativity
is via observations of compact astrophysical objects like neutron stars. The
correct theory should be able to accurately describe recent gravitational wave
observations of astrophysical objects existing in the mass gap between the
heaviest compact stars and the lightest black holes.

Modern observational astrophysics is rich in findings of compact objects and
as such our understanding of highly dense gravitational objects is rapidly
advancing. However, there is as of yet no strong consensus on their underlying
physics. A number of such objects have been recently observed that are
inconsistent with standard GR and a simple neutron star equation of state. It
was recently shown zhang_2021_unified ; zhang_2021_stellar  that in standard
general relativity, the secondary component of the merger GW190814 could
feasibly be a quark star with an interacting equation of state governed by a
single parameter $\lambda$. This parameterization of strong interaction
effects was inspired by another recent theoretical study showing that non-
strange quark matter (QM) could feasibly be the ground state of baryonic
matter at sufficient density and temperature holdom_2018_quark . Similar
analyses with a different equation of state and/or QM phase miao_2021_bayesian
; lopes_2022_onthe ; oikonomou_2023_colourflavour  found similarly promising
results. This same object was subsequently shown to be well described as a
slowly-rotating neutron star in the 4DEGB theory without resorting to exotic
quark matter EOSs, while also demonstrating that the equilibrium sequence of
neutron stars asymptotically matches the black hole limit, thus closing the
mass gap between NS/black holes of the same radius charmousis2022 . More
recently, some groups have also been interested in modelling ECOs (extreme
compact objects) zhang_2023_rescaling ; gammon_2024 ; Mann:2021mnc ;
Conklin:2021cbc  as well as unusually light compact stars horvath_2023_alight
; oikonomou_2023_colourflavour  (like that in the gamma ray remnant J1731-347,
which is inconsistent with minimum mass calculations of neutron stars
generated by iron cores) as quark stars to explain their unusual properties.

To further illuminate the range of possibilities, we consider in this paper
charged quark star solutions to the 4DEGB theory, as the confining nature of
the strong interaction make quark stars one of the few remaining candidates
for charged stellar objects. Although charged quark star solutions have
previously been considered in the context of 4DEGB pretel_2022 , the upper
limit of the coupling $\alpha$ was taken to be much smaller than that allowed
by current observational constraints Fernandes:2022zrq ; Clifton:2020xhc , and
the relation with the charged 4DEGB Buchdahl bound was not discussed (as the
bound was not derived). In considering values of $\alpha$ closer to presently
allowed bounds, we obtain a number of interesting novel results.

Our most intriguing result (extending similar results found in gammon_2024 )
is that charged quark stars in 4DEGB can be Extreme Compact Charged Objects
(ECCOs), objects whose radii are smaller than that allowed by the Buchdahl
bound and Schwarzschild radius in GR. Indeed, there exist uncharged quark
stars in 4DEGB whose radii are smaller than that of a corresponding black hole
of the same mass in GR111This phenomenon was also shown to be present for
neutron stars charmousis2022 , though the relationship with the Buchdahl bound
was not noted.. Observations of these latter objects, apart from indicating a
new class of astrophysical phenomena Mann:2021mnc , would provide strong
evidence for 4DEGB as a physical theory. These ECOs respect a generalization
of the Buchdahl bound, whose small-radius limit is that of the horizon radius
of the corresponding minimum mass black hole. ECCOs likewise respect a
generalization of the charged Buchdahl bound as we shall demonstrate.

We also find in general that for a given central pressure, charged quark stars
in 4DEGB have a larger mass and radius than their GR counterparts with the
same pressure. This can be attributed to the ‘less attractive’ nature of
gravity in 4DEGB: for $\alpha>0$ the gradient of the effective potential
yields a weaker force than pure GR. We consequently find a larger maximal mass
for a given value of $Q$ in 4DEGB than in general relativity. Similarly,
increasing $Q$ also weakens the overall force gradient compared to the
uncharged case, and leads to larger quark stars relative to their uncharged
counterparts.

The outline of our paper is as follows: In section II we introduce the basic
theory underlying 4DEGB gravity, as well as the non-interacting quark matter
equation of state that we make use of. Following this, a electromagnetic
perfect fluid stress-energy tensor is employed to derive the charged 4DEGB TOV
equations, and current observational constraints on the coupling constant are
briefly discussed. We use section III to derive a generalization of Buchdahl’s
bound for charged stars in 4DEGB gravity to be plotted alongside the solutions
to the modified TOV equations. Section IV outlines the results of our
numerical calculations. We conclude our results with a brief analysis of the
stability of charged 4DEGB quark stars. Section VI summarizes our key findings
and suggests topics for future study.

##  II Theory

###  II.1 4D Einstein-Gauss-Bonnet Gravity

To model the structure of a quark star in 4DEGB gravity, we add to the action
(4) a perfect-fluid term corresponding to the energy of the quark matter and

$S_{\mathrm{Max}}=-\frac{1}{8\pi
G}\int\sqrt{-g}\left({\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}+A_{\mu}J^{\mu}\right)$
(5)

where $J^{\mu}$ is the electromagnetic current of the charged quark matter.
The field equations of 4DEGB are obtained from a straightforward variational
principle applied to this action.

Variation with respect to the scalar $\phi$ yields

$\displaystyle\mathcal{E}_{\phi}=$
$\displaystyle-\mathcal{G}+8G^{\mu\nu}\nabla_{\nu}\nabla_{\mu}\phi+8R^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi-8(\square\phi)^{2}+8(\nabla\phi)^{2}\square\phi+16\nabla^{a}\phi\nabla^{\nu}\phi\nabla_{\nu}\nabla_{\mu}\phi$
(6)
$\displaystyle\qquad+8\nabla_{\nu}\nabla_{\mu}\phi\nabla^{\nu}\nabla^{\mu}\phi$
$\displaystyle=$ $\displaystyle\;0$

while the variation with respect to the metric gives

$\displaystyle\mathcal{E}_{\mu\nu}$ $\displaystyle=\Lambda
g_{\mu\nu}+G_{\mu\nu}+\alpha\left[\phi
H_{\mu\nu}-2R\left[\left(\nabla_{\mu}\phi\right)\left(\nabla_{\nu}\phi\right)+\nabla_{\nu}\nabla_{\mu}\phi\right]+8R_{(\mu}^{\sigma}\nabla_{\nu)}\nabla_{\sigma}\phi+8R_{(\mu}^{\sigma}\left(\nabla_{\nu)}\phi\right)\left(\nabla_{\sigma}\phi\right)\right.$
(7)
$\displaystyle-2G_{\mu\nu}\left[(\nabla\phi)^{2}+2\square\phi\right]-4\left[\left(\nabla_{\mu}\phi\right)\left(\nabla_{\nu}\phi\right)+\nabla_{\nu}\nabla_{\mu}\phi\right]\square\phi-\left[g_{\mu\nu}(\nabla\phi)^{2}-4\left(\nabla_{\mu}\phi\right)\left(\nabla_{\nu}\phi\right)\right](\nabla\phi)^{2}$
$\displaystyle+8\left(\nabla_{(\mu}\phi\right)\left(\nabla_{\nu)}\nabla_{\sigma}\phi\right)\nabla^{\sigma}\phi-4g_{\mu\nu}R^{\sigma\rho}\left[\nabla_{\sigma}\nabla_{\rho}\phi+\left(\nabla_{\sigma}\phi\right)\left(\nabla_{\rho}\phi\right)\right]+2g_{\mu\nu}(\square\phi)^{2}$
$\displaystyle-4g_{\mu\nu}\left(\nabla^{\sigma}\phi\right)\left(\nabla^{\rho}\phi\right)\left(\nabla_{\sigma}\nabla_{\rho}\phi\right)+4\left(\nabla_{\sigma}\nabla_{\nu}\phi\right)\left(\nabla^{\sigma}\nabla_{\mu}\phi\right)$
$\displaystyle\left.-2g_{\mu\nu}\left(\nabla_{\sigma}\nabla_{\rho}\phi\right)\left(\nabla^{\sigma}\nabla^{\rho}\phi\right)+4R_{\mu\nu\sigma\rho}\left[\left(\nabla^{\sigma}\phi\right)\left(\nabla^{\rho}\phi\right)+\nabla^{\rho}\nabla^{\sigma}\phi\right]\right]$
$\displaystyle=\;T_{\mu\nu}$

where $T_{\mu\nu}$ is the stress-energy tensor of the charged quark matter and
the electromagnetic field, and

$\displaystyle
H_{\mu\nu}=2\Big{[}RR_{\mu\nu}-2R_{\mu\alpha\nu\beta}R^{\alpha\beta}+R_{\mu\alpha\beta\sigma}R_{\nu}^{\alpha\beta\sigma}-2R_{\mu\alpha}R_{\nu}^{\alpha}-\frac{1}{4}g_{\mu\nu}\mathcal{G}\Big{]}$
(8)

is the Gauss-Bonnet tensor. These field equations satisfy the following
relationship

$g^{\mu\nu}T_{\mu\nu}=g^{\mu\nu}\mathcal{E}_{\mu\nu}+\frac{\alpha}{2}\mathcal{E}_{\phi}=4\Lambda-R-\frac{\alpha}{2}\mathcal{G}$
(9)

which can act as a useful consistency check to see whether prior solutions
generated via the solution-limit method glavan2020  are even possible
solutions to the theory. For example, using (9) it is easy to verify that the
rotating metrics generated from a Newman-Janis algorithm kumar2020 ; wei2020
are not solutions to the field equations of the scalar-tensor 4DEGB theory.

###  II.2 Charged 4DEGB TOV Equations

Throughout the following we assume a non-interacting quark equation of state

$P(r)=\frac{1}{3}(\rho-4B_{\mathrm{eff}})$ (10)

where $B_{\mathrm{eff}}$ is the MIT bag constant (for which we use a benchmark
value of $60\;\mathrm{MeV}/\mathrm{fm}^{3}$ zhang_2021_stellar ).

The standard Tolman-Oppenheimer-Volkoff (TOV) equations for stellar structure
are well-known in GR. Variation of the action with respect to the gauge
potential $A_{\mu}$, together with (6) and (7) yield the TOV equations for the
charged 4DEGB quark star. To obtain these we begin with a static, spherically
symmetric metric ansatz in natural units ($G=c=1$):

$ds^{2}=-e^{\Phi(r)}c^{2}dt^{2}+e^{\Lambda(r)}dr^{2}+r^{2}d\Omega^{2}.$ (11)

As usual hennigar_2020_on ; gammon_2024 , so long as
$e^{\Phi(r)}=e^{-\Lambda(r)}$ outside the star, the combination
$\mathcal{E}_{0}^{0}-\mathcal{E}_{1}^{1}$ of the field equations can be used
to derive the following equation for the scalar field:

$\left(\phi^{\prime
2}+\phi^{\prime\prime}\right)\left(e^{\Lambda(r)}-\left(r\phi^{\prime}-1\right)^{2}\right)=0.$
(12)

which, apart from the irrelevant $\phi=\ln\left(\frac{r-r_{0}}{l}\right)$
(with $r_{0}$ and $l$ being constants of integration), has the solution

$\phi_{\pm}=\int\frac{1\pm e^{\Lambda(r)/2}}{r}dr$ (13)

where $\phi_{-}$ falls off as as $\frac{1}{r}$ in asymptotically flat
spacetimes. Choosing $\phi=\phi_{-}$ ensures that (6) is automatically
satisfied.

Modelling the charged quark matter by a perfect fluid electromagnetic matter
source, the stress-energy tensor is (setting $G=c=1$)

$T_{\nu}^{\mu}=(P+\rho)u^{\mu}u_{\nu}+P\delta_{\nu}^{\mu}+\frac{1}{4\pi}\left(F^{\mu\alpha}F_{\alpha\nu}-\frac{1}{4}\delta_{\nu}^{\mu}F_{\alpha\beta}F^{\alpha\beta}\right),$
(14)

from which we obtain the equations

$\displaystyle\frac{e^{-2\Lambda(r)}\left(r^{2}e^{\Lambda(r)}\left(1-e^{\Lambda(r)}\right)-r^{3}e^{\Lambda(r)}\Lambda^{\prime}(r)\right)}{r^{4}}$
(15)
$\displaystyle+\alpha\frac{e^{-2\Lambda(r)}\left(-2re^{\Lambda(r)}\Lambda^{\prime}(r)+2r\Lambda^{\prime}(r)-e^{\Lambda(r)}\left(1-e^{\Lambda(r)}\right)-e^{\Lambda(r)}+1\right)}{r^{4}}=-\rho(r)-\frac{E(r)^{2}}{8\pi}$
$\displaystyle\frac{e^{-2\Lambda(r)}\left(r^{3}e^{\Lambda(r)}\Phi^{\prime}(r)+r^{2}e^{\Lambda(r)}\left(1-e^{\Lambda(r)}\right)\right)}{r^{4}}$
$\displaystyle+\alpha\frac{e^{-2\Lambda(r)}\left(2re^{\Lambda(r)}\Phi^{\prime}(r)-2r\Phi^{\prime}(r)-e^{\Lambda(r)}\left(1-e^{\Lambda(r)}\right)-e^{\Lambda(r)}+1\right)}{r^{4}}=P(r)-\frac{E(r)^{2}}{8\pi}$
(16)
$\displaystyle(\rho(r)+P(r))=-\frac{2}{\Phi^{\prime}(r)}\left(\frac{\mathrm{d}P}{\mathrm{~{}d}r}-\frac{q}{4\pi
r^{4}}\frac{\mathrm{d}q}{\mathrm{~{}d}r}\right)$ (17)

in 4DEGB.

Imposing asymptotic flatness implies that $\Phi(\infty)=\Lambda(\infty)=0$,
and regularity at the center of the star implies $\Lambda(0)=0$. Using the
$tt$ field equation it is easy to show that

$e^{-\Lambda(r)}=1+\frac{r^{2}}{2\alpha}\left[1-\sqrt{1+4\alpha\left(\frac{2m(r)}{r^{3}}-\frac{q(r)^{2}}{r^{4}}\right)}\right],$
(18)

in agreement with what was found in pretel_2022 , recalling that
$E(r)^{2}=\frac{q(r)^{2}}{r^{4}}$. With this we arrive at the 4DEGB modified
TOV equations, namely

$\displaystyle\frac{dq}{dr}$ $\displaystyle=4\pi
r^{2}\rho_{e}e^{\frac{\Lambda(r)}{2}}$ (19) $\displaystyle\frac{dm}{dr}$
$\displaystyle=4\pi r^{2}\rho(r)+\frac{q(r)}{r}\frac{dq}{dr}$ (20)
$\displaystyle\frac{dP}{dr}$
$\displaystyle=(P(r)+\rho(r))\frac{\left(r^{3}(\Gamma+8\pi\alpha
P(r)-1)-2\alpha m(r)\right)}{\Gamma
r^{2}\left((\Gamma-1)r^{2}-2\alpha\right)}+\frac{q(r)}{4\pi
r^{4}}\frac{dq}{dr}$ (21)

where
$\Gamma=\sqrt{1+4\alpha\left(\frac{2m(r)}{r^{3}}-\frac{q(r)^{2}}{r^{4}}\right)}$
(matching those found in pretel_2022 ). The vacuum solution is given by
$m(r)=M$ and $q(r)=Q$ where $M$ and $Q$ are constants (the total mass and
charge, respectively), implying that $\Phi=-\Lambda$. Writing
$e^{-\Lambda(r)}=1+2\varphi(r)$, we can compute the gravitational force in
4DEGB due to a spherical body

$\vec{F}=-\frac{d\varphi}{dr}\hat{r}=-\frac{r}{2\alpha}\left(1-\frac{2\alpha
M+r^{3}}{r\sqrt{8\alpha rM-4\alpha Q^{2}+r^{4}}}\right)\hat{r}\,,$ (22)

which is smaller in magnitude than its Newtonian $\alpha=0$ counterpart
($\vec{F}_{N}=-\frac{M}{r^{2}}\hat{r}$) for $\alpha>0$. The force in (22)
vanishes at $r=(\alpha M)^{1/3}$ if $Q=0$, but this is always at a smaller
value of $r$ than the outer horizon $R_{h}=M+\sqrt{M^{2}-Q^{2}-\alpha}$ of the
corresponding black hole. Hence the gravitational force outside of any
spherical body, while weaker than that in GR, is always attractive provided
$\alpha>0$. If $\alpha<0$ then the corresponding gravitational force is more
attractive than in GR. Similarly for nonzero charge, if all other parameters
are held constant, increasing the charge weakens the gravitational attraction,
which only vanishes in regions disallowed by the black hole horizon.

Rescaling the various quantities using

$\bar{\rho}=\frac{\rho}{4B_{\mathrm{eff}}}\qquad\bar{p}=\frac{p}{4B_{\mathrm{eff}}}\qquad\bar{\rho}_{e}=\frac{\rho_{e}}{4B_{\mathrm{eff}}}$
(23)

and

$\bar{m}=m\sqrt{4B_{\mathrm{eff}}}\qquad\bar{r}=r\sqrt{4B_{\mathrm{eff}}}\qquad\bar{q}=q\sqrt{4B_{\mathrm{eff}}}\qquad\bar{\alpha}=\alpha\cdot
4B_{\mathrm{eff}},$ (24)

we obtain the dimensionless equations

$\displaystyle\frac{d\bar{q}}{d\bar{r}}$
$\displaystyle=4\pi\bar{r}^{2}\bar{\rho}_{e}e^{\frac{\Lambda(\bar{r})}{2}}$
(25) $\displaystyle\frac{d\bar{m}}{d\bar{r}}$
$\displaystyle=4\pi\bar{r}^{2}\bar{\rho}(\bar{r})+\frac{\bar{q}(\bar{r})}{\bar{r}}\frac{d\bar{q}}{d\bar{r}}$
(26) $\displaystyle\frac{d\bar{P}}{d\bar{r}}$
$\displaystyle=(\bar{P}(\bar{r})+\bar{\rho}(\bar{r}))\frac{\left(\bar{r}^{3}(\Gamma+8\pi\bar{\alpha}\bar{P}(\bar{r})-1)-2\bar{\alpha}\bar{m}(\bar{r})\right)}{\Gamma\bar{r}^{2}\left((\Gamma-1)\bar{r}^{2}-2\bar{\alpha}\right)}+\frac{\bar{q}(\bar{r})}{4\pi\bar{r}^{4}}\frac{d\bar{q}}{d\bar{r}}$
(27)

which may be solved numerically. In the limit $\alpha\to 0$, the above
equations reduce back to the well-known TOV equations for a charged, static,
spherically symmetric gravitating body in GR.

To solve (25), (26) and (27) numerically we impose the boundary conditions

$m(0)=0,\quad q(0)=0,\quad\rho(0)=\rho_{\mathrm{c}},$ (28)

where the star’s surface radius $R$ is defined via $\bar{p}(\bar{R})=0$,
namely the radius at which the pressure goes to 0 (i.e. $p(R)=0$). We
similarly define the total mass of the star to be $M=m(R)$.

For the charge density there are several benchmark models used in the
literature. We shall restrict ourselves to the model in which the charge
density is proportional to energy density (ie. $\rho_{e}=\gamma\rho$ or
$\bar{\rho}_{e}=\gamma\bar{\rho}$ where $0\leq\gamma\leq 1$)
zhang_2021_stellar ; ray2003 ; arbanil2015 . Another popular charge model sets
charge proportional to spatial volume. However, this leads to exotic pressure
profiles that do not decrease monotonically zhang_2021_stellar , and thus are
not consistent with arguments we shall present in section III. As we are
primarily interested in how charged 4DEGB quark stars affect the modified
Buchdahl bound (since many novel results were noticed in the uncharged case
gammon_2024 ), we consider only the model where
$\bar{\rho}_{e}=\gamma\bar{\rho}$. Furthermore, since the interaction with the
Buchdahl bound is the main feature of interest, we choose to present results
with a fixed $Q$ rather than a fixed $\gamma$, with this charge parameter
taking on standard values from the literature ($Q=(0,1,2)\times
10^{20}\mathrm{\;C}$, $\bar{Q}=(0,1.538,3.076)\times 10^{-2}$) for comparison
arbanil2015 ; zhang_2021_stellar . This is done by solving the equations for a
trial value of $\gamma$ and iterating until our desired fixed charge is
attained.

With the above, numerical solutions can thus be obtained by scanning through a
range of values of $\rho_{c}$ and solving for the star’s total mass and
radius.

Before proceeding to solve the TOV equations, we consider the behaviour of the
scalar field $\phi$ in the interior. By continuity with the exterior solution,
the scalar is still described by (13) inside the star charmousis2022 . In
considering the interior behaviour it is instructive to rewrite $q(r)\to
r\tilde{q}(r)$ (which can be done by virtue of equation (19) and the fact that
$\lim_{r\to 0}e^{\frac{\Lambda(r)}{2}}\sim\mathrm{const}+\mathcal{O}(r)$).
Inserting the interior solution (18) into (12) and making the latter
substitution, we find

$\displaystyle\phi^{\prime}(r)$ $\displaystyle=-\sqrt{\frac{m(0)}{2\alpha
r}}-\frac{3m(0)}{4\alpha}+\frac{m(0)\sqrt{r}\left(\alpha\left(\tilde{q}(0)^{2}-2m^{\prime}(0)\right)-5m(0)^{2}\right)}{4\sqrt{2}(\alpha
m(0))^{3/2}}$ (29)
$\displaystyle+\frac{r\left(4\alpha\left(-6m^{\prime}(0)+3\tilde{q}(0)^{2}+2\right)-35m(0)^{2}\right)}{32\alpha^{2}}+\mathcal{O}\left(r^{3/2}\right)$

Furthermore, provided $m(r)$ and $q(r)$ both vanish at least quadratically in
$r$ for small $r$ (which is ensured from (19) and (20) for the boundary
conditions (28)) we find that near the origin

$\displaystyle\lim_{r\to 0}\phi^{\prime}(r)$
$\displaystyle\sim-\sqrt{\frac{\mathcal{M}(0)}{2\alpha}}\sqrt{r}+\frac{r}{4\alpha}+\mathcal{O}\left(r^{3/2}\right)\approx
0$ (30) $\displaystyle\lim_{r\to 0}\phi(r)$
$\displaystyle\sim-\frac{r^{3/2}}{3}\sqrt{\frac{2\mathcal{M}(0)}{\alpha}}+\frac{r^{2}}{8\alpha}+K\approx
K$

(where $\mathcal{M}(r)=r^{-2}m(r)$ and $K$ is a constant) and thus regularity
of the scalar at the origin is ensured.

Finally we note that the effective bag constant $B_{\mathrm{eff}}=60$
MeV/$\mathrm{fm}^{3}$ can be converted to units of inverse length squared
(‘gravitational units’) with the factor $G/c^{4}$, yielding

$B_{\mathrm{eff}}=7.84\times 10^{-5}\;\mathrm{km}^{-2}.$ (31)

###  II.3 Observational Constraints on the 4DEGB Coupling Constant

An investigation of the observational constraints on the coupling $\alpha$
yielded Clifton:2020xhc ; Fernandes:2022zrq

$-10^{-30}\;\textrm{m}^{2}<\alpha<10^{10}\;\textrm{m}^{2}$ (32)

where the lower bound comes from “early universe cosmology and atomic nuclei”
data charmousis2022 , and the upper bound follows from LAGEOS satellite
observations. Regarding the lower bound as negligibly close to zero, the
dimensionless version of (32) reads

$0<\bar{\alpha}\lesssim 3.2.$ (33)

We note that inclusion of preliminary calculations on recent GW data suggest
these constraints could potentially tighten to $0<\alpha\lesssim
10^{7}\;\textrm{m}^{2}$, or alternatively $0<\bar{\alpha}\lesssim 0.0032$.
This would mean that deviations from GR due to 4DEGB would only be detectable
in extreme environments such as in the very early universe or near the surface
of extremely massive objects. Even tighter bounds were assumed in previous
studies of quark stars, where only solutions with $\alpha$ below 6
$\mathrm{km}^{2}$ ($\bar{\alpha}\leq 0.0019$) were considered
banerjee_2021_strange ; banerjee_2021_quark ; pretel_2022 . Adopting such a
tight bound would make compact stars near the upper end of the mass gap an
ideal candidate for investigation the effects of 4DEGB theory.

At this point in time such tighter bounds are not warranted. A proper study of
the effects of gravitational radiation in 4DEGB has yet to be carried out. In
view of this we shall assume the bound (33), which has strong observational
support Clifton:2020xhc ; Fernandes:2022zrq .

##  III Generalization of the Buchdahl Bound for Charged Compact Stars in
4DEGB Gravity

###  III.1 Derivation of a Maximal Mass for Charged Compact Objects in 4DEGB
Gravity

A generalization of the Buchdahl bound has been derived for charged stars in
GR bohmer_2007  and for uncharged objects in 4DEGB theory chakraborty_2020 .
In the following we extend these calculations for the case of a charged star
in 4DEGB gravity.

By virtue of equation (18) it is straightforward to check that

$\frac{d}{dr}\frac{(1-e^{-\Lambda(r)})}{2r^{2}}=\frac{1}{\sqrt{1+4\alpha(\frac{2m(r)}{r^{3}}-\frac{q(r)^{2}}{r^{4}})}}\frac{d}{dr}\left(\frac{m(r)}{r^{3}}-\frac{q(r)^{2}}{2r^{4}}\right)$
(34)

The next step of the argument hinges on the assumption that the quantity
$\frac{(1-e^{-\Lambda(r)})}{r^{2}}$ decreases as radial distance increases
chakraborty_2020 . In the uncharged case this is equivalent to assuming
decreasing mass density as the star’s surface is approached (though we note
that some exotic star models do not have monotonically decreasing density
profiles zhang_2021_stellar ). This assumption also seems reasonable in the
case of a charged sphere chakraborty_2020  and we employ it in what follows:

Utilizing the conservation equation

$(\rho(r)+P(r))=-\frac{2}{\Phi^{\prime}(r)}\left(\frac{\mathrm{d}P}{\mathrm{~{}d}r}-\frac{q}{4\pi
r^{4}}\frac{\mathrm{d}q}{\mathrm{~{}d}r}\right)$ (35)

we can write

$\frac{dP}{dr}=\frac{q}{4\pi
r^{4}}\frac{dq}{dr}-\frac{1}{2}(\rho(r)+P(r))\Phi^{\prime}(r).$ (36)

Recalling the $rr$ component of the field equations, we obtain

$\displaystyle\frac{1}{r^{2}}\left[r\Phi^{\prime}(r)e^{-\Lambda(r)}-\left(1-e^{-\Lambda(r)}\right)\right]$
(37)
$\displaystyle\qquad+\alpha\frac{\left(1-e^{-\Lambda(r)}\right)}{r^{4}}\left[2r\Phi^{\prime}(r)e^{-\Lambda(r)}+\left(1-e^{-\Lambda(r)}\right)\right]=\kappa\left(P(r)-\frac{q(r)^{2}}{8\pi
r^{4}}\right)$

and subsequently get

$\displaystyle\kappa\left(\frac{dP}{dr}+\frac{q(r)^{2}}{2\pi
r^{5}}-\frac{q(r)}{4\pi
r^{4}}\frac{dq}{dr}\right)=\kappa\left(\frac{q(r)^{2}}{2\pi
r^{5}}-\frac{1}{2}(\rho(r)+P(r))\Phi^{\prime}(r)\right)$ (38)

by differentiating the left hand side of (37) using (36).

Addition of the $tt$ and $rr$ equations yields

$(\rho(r)+P(r))=\frac{1}{\kappa}\left(\frac{1}{r}e^{-\Lambda(r)}\left(\Lambda(r)^{\prime}+\Phi(r)^{\prime}\right)+\frac{2\alpha}{r^{3}}e^{-\Lambda(r)}\left(1-e^{-\Lambda(r)}\right)\left(\Lambda(r)^{\prime}+\Phi(r)^{\prime}\right)\right)$
(39)

after division by $r^{2}$. This can be substituted into (38) to obtain

$\displaystyle\kappa\left(\frac{dP}{dr}+\frac{q(r)^{2}}{2\pi
r^{5}}-\frac{q(r)}{4\pi r^{4}}\frac{dq}{dr}\right)$
$\displaystyle=\kappa\frac{q(r)^{2}}{2\pi
r^{5}}-\frac{1}{2}\left(\frac{1}{r}e^{-\Lambda(r)}+\frac{2\alpha}{r^{3}}e^{-\Lambda(r)}\left(1-e^{-\Lambda(r)}\right)\right)\left(\Lambda(r)^{\prime}+\Phi(r)^{\prime}\right)\Phi^{\prime}(r).$
(40)

Making use of equation (37) we find

$\displaystyle\frac{d}{dr}\left(\frac{1}{r^{2}}\left[r\Phi^{\prime}(r)e^{-\Lambda(r)}-\left(1-e^{-\Lambda(r)}\right)\right]+\alpha\frac{\left(1-e^{-\Lambda(r)}\right)}{r^{4}}\left[2r\Phi^{\prime}(r)e^{-\Lambda(r)}+\left(1-e^{-\Lambda(r)}\right)\right]\right)$
(41) $\displaystyle=\kappa\frac{q(r)^{2}}{2\pi
r^{5}}-\frac{1}{2}\left(\frac{1}{r}e^{-\Lambda(r)}+\frac{2\alpha}{r^{3}}e^{-\Lambda(r)}\left(1-e^{-\Lambda(r)}\right)\right)\left(\Lambda(r)^{\prime}+\Phi(r)^{\prime}\right)\Phi^{\prime}(r).$

If we define $\beta(r)=\frac{1-e^{-\Lambda(r)}}{r^{2}}$ the above can be
rewritten as

$\displaystyle\frac{d}{dr}\left(\frac{\Phi^{\prime}(r)e^{-\Lambda(r)}}{r}\right)(1+2\alpha\beta(r))+Z(r)=\kappa\frac{q(r)^{2}}{2\pi
r^{5}}-\frac{\left(\Phi^{\prime}(r)+\Lambda^{\prime}(r)\right)}{2}\frac{\Phi^{\prime}(r)e^{-\Lambda(r)}}{r}(1+2\alpha\beta(r))$
(42)

where
$Z(r)=\left(\frac{2\alpha\Phi^{\prime}(r)e^{-\Lambda(r)}}{r}-1+2\alpha\beta(r)\right)\beta^{\prime}(r)$.
Equivalently,

$\displaystyle
2e^{-(\Phi(r)+\Lambda(r))/2}\frac{d}{dr}\left[\frac{e^{-\Lambda(r)/2}}{r}\frac{de^{\Phi(r)/2}}{dr}\right](1+2\alpha\beta(r))-\kappa\frac{q(r)^{2}}{2\pi
r^{5}}$
$\displaystyle\qquad\qquad=\left(1-\frac{2\alpha\Phi^{\prime}(r)e^{-\Lambda(r)}}{r}-2\alpha\beta(r)\right)\beta^{\prime}(r)$
(43)

which can be shown using the identity

$\left(\frac{d}{dr}\left(\frac{\Phi^{\prime}(r)e^{-\Lambda(r)}}{r}\right)+\frac{1}{2}(\frac{\Phi^{\prime}(r)e^{-\Lambda(r)}}{r})(\Phi^{\prime}(r)+\Lambda^{\prime}(r))\right)=2e^{-(\Phi(r)+\Lambda(r))/2}\frac{d}{dr}\left[\frac{e^{-\Lambda(r)/2}}{r}\frac{de^{\Phi(r)/2}}{dr}\right].$
(44)

To proceed further we assume the factors $(1+2\alpha\beta(r))$ and
$\left(1-\frac{2\alpha\Phi^{\prime}(r)e^{-\Lambda(r)}}{r}-2\alpha\beta(r)\right)$
are both non-negative, which is valid for sufficiently small, positive
$\alpha$. The right hand side of (43) is then negative due to the
monotonically decreasing nature of $\beta(r)$. We can thus write

$2e^{-(\Phi(r)+\Lambda(r))/2}\frac{d}{dr}\left[\frac{e^{-\Lambda(r)/2}}{r}\frac{de^{\Phi(r)/2}}{dr}\right](1+2\alpha\beta(r))-\kappa\frac{q(r)^{2}}{2\pi
r^{5}}\leq 0$ (45)

which upon integration yields

$e^{\Phi(r^{\prime\prime})/2}\leq\kappa\int^{r^{\prime\prime}}dr^{\prime}r^{\prime}e^{\Lambda(r^{\prime})/2}\int^{r^{\prime}}dre^{(\Phi(r)+\Lambda(r))/2}\frac{q(r)^{2}}{4\pi
r^{5}}(1+2\alpha\beta(r))^{-1}\;.$ (46)

Making for convenience the following definitions

$\displaystyle\mathcal{Q}$ $\displaystyle\equiv
e^{(\Phi(r)+\Lambda(r))/2}\frac{q(r)^{2}}{4\pi r^{5}}(1+2\alpha\beta(r))^{-1}$
(47) $\displaystyle\eta(r^{\prime\prime})$
$\displaystyle\equiv\int^{r^{\prime\prime}}e^{\Lambda(r^{\prime})/2}r^{\prime}\int^{r^{\prime}}\mathcal{Q}(r)drdr^{\prime}$
$\displaystyle\psi$ $\displaystyle\equiv
e^{\Phi(r^{\prime\prime})/2}-\eta(r^{\prime\prime})\qquad\xi\equiv\int^{r^{\prime\prime}}r^{\prime}e^{\Lambda(r^{\prime})/2}dr^{\prime}$

it is straightforward to show that

$\displaystyle\frac{d\psi}{d\xi}=\frac{1}{r^{\prime\prime}}e^{-\Lambda(r^{\prime\prime})/2}\frac{d}{dr^{\prime\prime}}e^{\Phi(r^{\prime\prime})/2}-\int^{r^{\prime\prime}}dr\mathcal{Q}(r^{\prime}),$
(48)
$\displaystyle\frac{d^{2}\psi}{d\xi^{2}}=\frac{1}{r^{\prime\prime}}e^{-\Lambda(r^{\prime\prime})/2}\frac{d}{dr}\left[\frac{1}{r^{\prime\prime}}e^{-\Lambda(r^{\prime\prime})/2}\frac{d}{dr^{\prime\prime}}e^{\Phi(r^{\prime\prime})/2}\right]-\frac{1}{r^{\prime\prime}}e^{-\Lambda(r^{\prime\prime})/2}\mathcal{Q}(r^{\prime\prime})\leq
0$ (49)

where integration is from the center ($r^{\prime\prime}>r^{\prime}>r$), or
alternatively

$\frac{d^{2}\psi}{d\xi^{2}}re^{\Lambda(r)/2}\leq 0$ (50)

which follows from (45). Since $e^{\Lambda(r)/2}>0$ and $r>0$ this is
equivalent to

$\frac{d^{2}\psi}{d\xi^{2}}\leq 0.$ (51)

Trivially integrating and applying the mean value theorem bohmer_2007  we see
that

$\frac{d\psi}{d\xi}\leq\frac{\psi(\xi)-\psi(0)}{\xi-0}\leq\frac{\psi(\xi)}{\xi}$
(52)

since at the center of the star $\xi=0$ and $\psi(0)>0$ since
$e^{\Phi(r)/2}>0$ inside the star.

Putting everything together we find

$\displaystyle\left(\frac{1}{r}e^{-\Lambda(r^{\prime\prime})/2}\frac{d}{dr}e^{\Phi(r^{\prime\prime})/2}-\int^{r^{\prime\prime}}dr^{\prime}\mathcal{Q}(r^{\prime})\right)\left(\int^{r^{\prime\prime}}dr^{\prime}r^{\prime}e^{\Lambda(r^{\prime})/2}\right)$
$\displaystyle\qquad\qquad\qquad\leq
e^{\Phi(r^{\prime\prime})/2}-\int^{r^{\prime\prime}}r^{\prime}e^{\Lambda_{\mathrm{int}}/2}\int^{r^{\prime}}\mathcal{Q}(r)drdr^{\prime}.$
(53)

Recalling that
$\beta(r)=\frac{1-e^{-\Lambda(r)}}{r^{2}}=-\frac{1}{2\alpha}\left[1-\sqrt{1+\frac{\kappa\alpha}{2\pi}\left(\frac{2m(r)}{r^{3}}-\frac{q(r)^{2}}{r^{4}}\right)}\right]$
such that

$e^{-\Lambda(r)}=1-r^{2}\beta(r),$ (54)

and that $\beta(r)\geq\beta(r^{\prime})\geq\beta(r^{\prime\prime})$ (more
primes corresponding to further from the center), we obtain

$\displaystyle\xi\equiv\int^{r^{\prime\prime}}dr^{\prime}r^{\prime}e^{\Lambda(r^{\prime})/2}=\int^{r^{\prime\prime}}dr^{\prime}r^{\prime}\frac{1}{\sqrt{1-r^{\prime
2}\beta(r^{\prime})}}$
$\displaystyle\geq\int^{r^{\prime\prime}}dr^{\prime}r^{\prime}\frac{1}{\sqrt{1-r^{\prime
2}\beta(r^{\prime\prime})}}$ (55)
$\displaystyle=\frac{1}{\beta(r^{\prime\prime})}\left(1-\sqrt{1-r^{\prime\prime
2}\beta(r^{\prime\prime})}\right)$

as a bound on $\xi$.

Similarly we can consider the term

$\displaystyle\int^{r^{\prime\prime}}\mathcal{Q}(r^{\prime})dr$
$\displaystyle=\kappa\int^{r^{\prime\prime}}e^{(\Phi(r^{\prime})+\Lambda(r^{\prime}))/2}\frac{q(r^{\prime})^{2}}{4\pi
r^{\prime 5}}(1+2\alpha\beta(r^{\prime}))^{-1}dr$ (56)
$\displaystyle=\frac{\kappa}{4\pi}\int^{r^{\prime\prime}}\frac{e^{\Phi(r^{\prime})/2}}{\sqrt{1-r^{\prime
2}\beta(r^{\prime})}\left(1+2\alpha\beta(r^{\prime})\right)}\frac{q(r^{\prime})^{2}}{r^{\prime
5}}dr^{\prime}.$

Combining our previous assumption on $\beta$ with the following assumption
regarding the behaviour of the charge:

$e^{\Phi_{\mathrm{int}}(r)/2}\frac{q(r)^{2}}{r^{5}}\geq
e^{\Phi_{\mathrm{int}}(r^{\prime})/2}\frac{q(r^{\prime})^{2}}{r^{\prime
5}}\geq
e^{\Phi_{\mathrm{int}}(r^{\prime\prime})/2}\frac{q(r^{\prime\prime})^{2}}{r^{\prime\prime
5}}$ (57)

the above integral can be bounded as follows:

$\displaystyle\frac{\kappa}{4\pi}\int^{r^{\prime\prime}}\frac{e^{\Phi(r^{\prime})/2}}{\sqrt{1-r^{\prime
2}\beta(r^{\prime})}\left(1+2\alpha\beta(r^{\prime})\right)}\frac{q(r^{\prime})^{2}}{r^{\prime
5}}dr^{\prime}$
$\displaystyle\geq\frac{\kappa}{4\pi}\frac{e^{\Phi_{\mathrm{int}}(r^{\prime\prime})/2}}{\left(1+2\alpha\beta(0)\right)}\frac{q(r^{\prime\prime})^{2}}{r^{\prime\prime
5}}\int^{r^{\prime\prime}}\frac{dr^{\prime}}{\sqrt{1-r^{\prime
2}\beta(r^{\prime\prime})}}$ (58)
$\displaystyle=\frac{\kappa}{4\pi}\frac{e^{\Phi_{\mathrm{int}}(r^{\prime\prime})/2}}{\left(1+2\alpha\beta(0)\right)}\frac{q(r^{\prime\prime})^{2}}{r^{\prime\prime
5}}\frac{\mathrm{arcsin}\left(\sqrt{\beta(r^{\prime\prime})}r^{\prime\prime}\right)}{\sqrt{\beta(r^{\prime\prime})}}.$

Finally we can consider the most complicated integral term:

$\displaystyle\int^{r^{\prime\prime}}r^{\prime}e^{\Lambda_{\mathrm{int}}/2}\int^{r^{\prime}}\mathcal{Q}(r)drdr^{\prime}=\frac{\kappa}{4\pi}\int^{r^{\prime\prime}}r^{\prime}e^{\Lambda_{\mathrm{int}}/2}\int^{r^{\prime}}\frac{e^{\Phi_{\mathrm{int}}/2}}{\sqrt{1-r^{2}\beta(r)}\left(1+2\alpha\beta(r)\right)}\frac{q(r)^{2}}{r^{5}}drdr^{\prime}$
(59)
$\displaystyle\geq\frac{\kappa}{4\pi}\left(1+2\alpha\beta(0)\right)^{-1}\int^{r^{\prime\prime}}r^{\prime}e^{(\Phi_{\mathrm{int}}(r^{\prime})+\Lambda_{\mathrm{int}}(r^{\prime}))/2}\frac{q(r^{\prime})^{2}}{r^{\prime
5}}\frac{\mathrm{arcsin}\left(\sqrt{\beta(r^{\prime})}r^{\prime}\right)}{\sqrt{\beta(r^{\prime})}}dr^{\prime}$
$\displaystyle\geq\frac{\kappa}{4\pi}\left(1+2\alpha\beta(0)\right)^{-1}e^{\Phi_{\mathrm{int}}(r^{\prime\prime})/2}\frac{q(r^{\prime\prime})^{2}}{r^{\prime\prime
5}}\int^{r^{\prime\prime}}r^{\prime 2}\frac{1}{\sqrt{1-r^{\prime
2}\beta(r^{\prime})}}\frac{\mathrm{arcsin}\left(\sqrt{\beta(r^{\prime})}r^{\prime}\right)}{\sqrt{\beta(r^{\prime})}r^{\prime}}dr^{\prime}.$

Since the function $\mathrm{arcsin}(x)/x$ monotonically increases on the
interval $x\in[0,1]$, the integral is minimal when the smallest $\beta$ is
chosen. Thus

$\displaystyle\int^{r^{\prime\prime}}r^{\prime}e^{\Lambda_{\mathrm{int}}/2}\int^{r^{\prime}}\mathcal{Q}(r)drdr^{\prime}$
(60)
$\displaystyle\geq\frac{\kappa}{4\pi}\left(1+2\alpha\beta(0)\right)^{-1}e^{\Phi_{\mathrm{int}}(r^{\prime\prime})/2}\frac{q(r^{\prime\prime})^{2}}{r^{\prime\prime
5}}\int^{r^{\prime\prime}}r^{\prime 2}\frac{1}{\sqrt{1-r^{\prime
2}\beta(r^{\prime\prime})}}\frac{\mathrm{arcsin}\left(\sqrt{\beta(r^{\prime\prime})}r^{\prime}\right)}{\sqrt{\beta(r^{\prime\prime})}r^{\prime}}dr^{\prime}$
$\displaystyle=\frac{\kappa}{4\pi}\left(1+2\alpha\beta(0)\right)^{-1}e^{\Phi_{\mathrm{int}}(r^{\prime\prime})/2}\frac{q(r^{\prime\prime})^{2}}{r^{\prime\prime
5}}\frac{1}{\beta(r^{\prime\prime})}\left(r^{\prime\prime}-\sqrt{\frac{1-\beta(r^{\prime\prime})r^{\prime\prime
2}}{\beta(r^{\prime\prime})}}\mathrm{arcsin}(\sqrt{\beta(r^{\prime\prime})}r^{\prime\prime})\right).$

Inserting these results into (III.1) yields

$\displaystyle\left(e^{(\Phi(r^{\prime\prime})-\Lambda(r^{\prime\prime}))/2}\frac{\Phi^{\prime}(r^{\prime\prime})}{2r}-\frac{\kappa}{4\pi}\frac{e^{\Phi(r^{\prime\prime})/2}}{\left(1+2\alpha\beta(0)\right)}\frac{q(r^{\prime\prime})^{2}}{r^{\prime\prime
5}}\frac{\mathrm{arcsin}\left(\sqrt{\beta(r^{\prime\prime})}r^{\prime\prime}\right)}{\sqrt{\beta(r^{\prime\prime})}}\right)\left(\frac{\left(1-\sqrt{1-r^{\prime\prime
2}\beta(r^{\prime\prime})}\right)}{\beta(r^{\prime\prime})}\right)$ (61)
$\displaystyle\qquad\leq
e^{\Phi(r^{\prime\prime})/2}-\frac{\kappa}{4\pi}\left(1+2\alpha\beta(0)\right)^{-1}e^{\Phi(r^{\prime\prime})/2}\frac{q(r^{\prime\prime})^{2}}{r^{\prime\prime
5}}\frac{1}{\beta(r^{\prime\prime})}\left(r^{\prime\prime}-\sqrt{\frac{1-\beta(r^{\prime\prime})r^{\prime\prime
2}}{\beta(r^{\prime\prime})}}\mathrm{arcsin}(\sqrt{\beta(r^{\prime\prime})}r^{\prime\prime})\right)$

and cancelling a factor of $e^{\Phi(r^{\prime\prime})/2}$ leaves

$\displaystyle\left(\frac{\sqrt{1-r^{\prime\prime
2}\beta(r^{\prime\prime})}}{r^{\prime\prime}}\frac{\Phi^{\prime}(r^{\prime\prime})}{2}-\frac{\kappa}{4\pi}\frac{1}{\left(1+2\alpha\beta(0)\right)}\frac{q(r^{\prime\prime})^{2}}{r^{\prime\prime
5}}\frac{\mathrm{arcsin}\left(\sqrt{\beta(r^{\prime\prime})}r^{\prime\prime}\right)}{\sqrt{\beta(r^{\prime\prime})}}\right)\left(\frac{\left(1-\sqrt{1-r^{\prime\prime
2}\beta(r^{\prime\prime})}\right)}{\beta(r^{\prime\prime})}\right)$ (62)
$\displaystyle\qquad\qquad\leq
1-\frac{\kappa}{4\pi}\left(1+2\alpha\beta(0)\right)^{-1}\frac{q(r^{\prime\prime})^{2}}{r^{\prime\prime
5}}\frac{1}{\beta(r^{\prime\prime})}\left(r^{\prime\prime}-\sqrt{\frac{1-\beta(r^{\prime\prime})r^{\prime\prime
2}}{\beta(r^{\prime\prime})}}\mathrm{arcsin}(\sqrt{\beta(r^{\prime\prime})}r^{\prime\prime})\right)$

which depends explicitly on $\Phi^{\prime}(r^{\prime\prime})$. This can be
determined from the $rr$ field equation (16) to give

$\Phi^{\prime}(r)=\frac{\alpha+e^{2\Lambda(r)}\left(\alpha-8\pi
r^{4}P(r)+q(r)^{2}-r^{2}\right)+e^{\Lambda(r)}\left(r^{2}-2\alpha\right)}{2\alpha
r-re^{\Lambda(r)}\left(2\alpha+r^{2}\right)}.$ (63)

Making this substitution and simplifying, we find the desired inequality

$\displaystyle\frac{\left(\sqrt{1-R^{2}\beta(R)}-1\right)\left(q(R)^{2}+R^{4}\beta(R)(-1+\alpha\beta(R))\right)}{2R^{4}\beta(R)\sqrt{1-R^{2}\beta(R)}(1+2\alpha\beta(R))}$
(64) $\displaystyle\qquad\qquad\qquad\leq
1+\frac{2q(r)^{2}}{R^{4}\beta(R)}\left(\frac{\mathrm{arcsin}(\sqrt{\beta(R)}R)}{\sqrt{\beta(R)}R}-1\right)$

which is the generalized Buchdahl bound for charged spheres in 4DEGB gravity.

In the $q\to 0$ limit we recover the following inequality chakraborty_2020

$\sqrt{1-\beta(R)R^{2}}\left(1+\alpha\beta(R)\right)>\frac{1}{3}\left(1-\alpha\beta(R)\right)$
(65)

for uncharged objects in 4DEGB.

Alternatively, in the small $\alpha$ limit $\beta_{\mathrm{\alpha\to
0}}(r)=\frac{2m(r)}{r^{3}}-\frac{q(r)^{2}}{r^{4}}$, and consequently

$\frac{\left(\sqrt{1-r^{2}\beta(r)}-1\right)\left(q(r)^{2}-rm(r)\right)}{r^{4}\sqrt{1-r^{2}\beta(r)}}\leq\beta(r)+\frac{2q(r)^{2}}{r^{4}}\left(\frac{\sin^{-1}\left(\sqrt{r^{2}\beta(r)}\right)}{\sqrt{r^{2}\beta(r)}}-1\right)$
(66)

which matches222At the time of publication bohmer_2007  has a factor of 2
missing from their final expression which Dr. Harko has kindly confirmed to be
a typo. We note that in bohmer_2007  $\alpha(r)=\beta_{\mathrm{\alpha\to
0}}(r)\frac{r^{3}}{2m(r)}$ in our notation.  the final inequality from
bohmer_2007 .

###  III.2 Relation to the Black Hole Horizon

For a given $\alpha$ the radius of a black hole is
$R=M+\sqrt{M^{2}-Q^{2}-\alpha}$. Since the minimum mass
$M_{\mathrm{min}}=\sqrt{Q^{2}+\alpha}$, we obtain
$R_{\mathrm{Mmin}}=\sqrt{Q^{2}+\alpha}$. Substituting this into the Buchdahl
bound (64) we find that it is automatically satisfied as an equality; in other
words the Buchdahl bound intersects the minimum mass point of the black hole
horizon. The $M/R$ curves for nonzero $\alpha$ also smoothly join this point
in the limit of large central pressure, as can be seen in figures 1 and 2.

##  IV Results

(a)  (b)  (c)  (d)

(e)  (f)

Figure 1: Mass vs. radius/mass vs. central density curves for charged 4DEGB
quark stars for three different fixed charges. The (blue, orange, green)
curves correspond to charges $\bar{Q}=(0,1.538,3.076)\times 10^{-2}$
respectively, with black dots corresponding to local maximum mass points (when
present). Note that the orange and green curves begin at a nonzero value of
$\overline{R}$. The grey solid and dashed lines are the uncharged GR
Schwarzschild and Buchdahl bounds, respectively. The coloured dashed lines are
the 4DEGB Buchdahl bounds for the three different charges, with the colours
corresponding to the associated $M/R$ curve. The black curves are the 4DEGB
black hole horizons, with charges matching the Buchdahl/$MR$ curves that
intersect them.

(a)  (b)  (c)  (d)

(e)  (f)

Figure 2: Mass vs. radius/mass vs. central density curves for charged 4DEGB
quark stars for three different fixed charges. The (blue, orange, green)
curves correspond to charges $\bar{Q}=(0,1.538,3.076)\times 10^{-2}$
respectively, with black dots corresponding to local maximum mass points (when
present). Note that the orange and green curves begin at a nonzero value of
$\overline{R}$ and almost perfectly overlap with the blue curve at larger
$\alpha$. The grey solid and dashed lines are the uncharged GR Schwarzschild
and Buchdahl bounds, respectively. The coloured dashed lines are the 4DEGB
Buchdahl bounds for the three different charges, with the colours
corresponding to the associated $M/R$ curve. The black curves are the 4DEGB
black hole horizons, with charges matching the Buchdahl/$MR$ curves which
intersect them.

The mass/radius ($M/R$) and mass/central density ($M/\rho_{c}$) curves for
charged 4DEGB quark stars are presented in figures 1 and 2. As expected from
previous work gammon_2024 ; zhang_2021_stellar , we observe the general trends
that for a larger $\alpha$ and/or $Q$, the mass/radius profiles of the quark
stars increase in size. For a fixed nonzero charge the stars have a minimum
size, below which the gravitational attraction cannot overcome the self-
repulsion of the charge. This is most evident in the right-hand lower panels
of figure 2. As $\alpha$ gets large the $M/R$ curves converge for all values
of fixed $Q$ (and similarly with the $M/\rho_{c}$ curves) with a slight
divergence near $\bar{\rho_{c}}=1$ in the $M/\rho_{c}$ plots to account for
the structures mentioned above.

As in gammon_2024 , even for small, nonzero $\alpha$ we find curves for the
Buchdahl bound that intersect the black hole horizon at the minimum mass
point. A new feature, evidently not previously noted, is that this same
interaction between the black hole horizon and Buchdahl bound can be observed
in GR when $Q\neq 0$. This is evident in the orange and green dashed curves in
the upper left panel of figure 1. The difference is that for nonzero $\alpha$
the $M/R$ curves approach this intersection point smoothly, whereas in GR the
curves turn away from the horizon/BB and thus never meet.

On the solution curves we have indicated maximum mass points (when this point
doesn’t occur at the intersection with the horizon) with a black dot - in
general relativity these maximum mass points indicate a transition point from
stability against radial perturbations to instability for uncharged stars. In
4DEGB it’s not clear whether this coincidence holds, and offers an interesting
avenue for future research.

In the uncharged case a criticality condition was previously noted gammon_2024
wherein for $\bar{\alpha}\geq\frac{3}{4\pi}$ a critical central pressure was
present - for central pressures below this critical value, the pressure
function diverges with no real roots to define a star’s surface. Above this
critical pressure we find numerical solutions which lie at or beyond the black
hole horizon, indicating a lack of stable physical stars in this critical
regime. Deriving an analogous bound for the charged case is considerably more
complicated, however numerically we find that for $Q>0$ (and hence
$\gamma>0$), such criticality can take place for
$\bar{\alpha}<\frac{3}{4\pi}$. Because of this lack of an analytic bound for
the charged case, we do not present numerical solutions in which such
behaviour is manifest. With this, we note that we have considered values of
$\bar{\alpha}\leq 0.1$. For larger values of $\bar{\alpha}$ ‘critical central
pressure’ behaviour emerges soon after for the charged cases. Examination of
these cases requires a more detailed stability analysis, which we leave for
future investigation.

Most interestingly we note, similar to the uncharged case gammon_2024 , that
the stars described by this theory can be ECCOs (Extreme Compact Charged
Objects): charged stellar objects that are smaller (more compact) than the
uncharged Buchdahl bound (and sometimes even the Schwarzschild radius) of
general relativity. For large enough $\alpha$ the associated $M/R$ curves do
not reach their maximum mass until they intersect the black hole horizon -
these are particularly interesting candidates for stable ECCOs since in pure
uncharged GR we only see stability against radial perturbations when
$dM/d\rho>0$. However, a net charge offsets the stability point from this
maximum mass point zhang_2021_stellar  \- it is likely that the modifications
to gravity will have a similar offsetting effect which should be investigated
in future studies before declaring these solutions stable against
perturbations.

##  V Stability

We now briefly consider the stability of stars respecting the generalized
Buchdahl bound. In general relativity a necessary but insufficient condition
for an uncharged compact star to be stable against radial perturbations is
$dM/d\rho_{c}>0$ zhang_2021_stellar ; glendenningbook ; arbanil2015 ,
corresponding to the part of the solution curve before a maximal mass point is
reached. In Einstein-Maxwell theory a net charge has been shown to offset the
stability point from the maximum mass point in either direction
zhang_2021_stellar . In a similar vein, when the coupling to higher curvature
gravity is nonzero, it is not obvious whether this coincidence of stability
and the maximum mass point will hold for uncharged stars. While we leave a
thorough analysis of the fundamental radial oscillation modes for future work
(where stability against radial oscillations can be ensured), the speed of
sound and effective adiabatic index inside the star can still be discussed.

In the interior of a stable star, the sound speed $c_{s}$ must never exceed
the speed of light $c$. The non-interacting quark equation of state (10) has a
constant subluminal sound speed of $c_{s}=\frac{c}{\sqrt{3}}$, and thus the
causality condition is always satisfied. Similarly, the effective adiabatic
index

$\gamma_{\mathrm{eff}}\equiv\left(1+\frac{\rho}{P}\right)\left(\frac{dP}{d\rho}\right)_{S}$
(67)

is often used as another indicator of stability as it is seen as a bridge
between “the relativistic structure of a spherical static object and the
equation of state of the interior fluid” moustakidis2017_stability . The
subscript $S$ in the above equation indicates that we consider the sound speed
at a constant specific entropy. In principle a critical value for
$\langle\gamma_{\mathrm{eff}}\rangle$ exists, below which configurations are
unstable against radial perturbations. In standard general relativity this
critical value can be written chandrasekhar1964 ; moustakidis2017_stability
$\gamma_{cr}=\frac{4}{3}+\frac{19}{42}\beta$, where $\beta=2M/R=R_{S}/R$ is
the compactness parameter. If $\beta\to 0$ the well-known classical Newtonian
limit is recovered as expected
($\langle\gamma_{\mathrm{eff}}\rangle\geq\frac{4}{3}$).

An equivalent bound has not yet been derived for the 4DEGB theory. Despite
this, it is common practice to plot the adiabatic index of the star relative
to the Newtonian critical value banerjee_2021_quark ; banerjee_2021_strange ;
hansraj_2020_isotropic ; singh_2022_anisotropic . Since we restrict ourselves
to a simple, non-interacting equation of state (10) it is straightforward to
show that $\gamma_{\mathrm{eff}}=\frac{1}{3}(4+\frac{1}{P(r)})$ and hence the
Newtonian bound of 4/3 is always satisfied for non-negative pressure.

##  VI Summary

In this paper we have investigated the stellar structure of strongly
interacting quark stars in the 4D Einstein Gauss-Bonnet theory of gravity for
different combinations of the charge $Q$ and 4DEGB coupling constant $\alpha$.
In accord with the lack of a mass gap in the 4DEGB theory charmousis2022 , we
find that even for small $\alpha$ the quark star solutions asymptotically
approach the 4DEGB black hole horizon radius, and thus have solutions with
smaller radii than the GR Buchdahl/Schwarzschild limits. In general, larger
$Q$ and $\alpha$ tend to increase the mass-radius profile of quark stars, with
large $\alpha$ suppressing the differences between different charges. These
findings are generally consistent with what was found in the regime of weak
coupling to the 4DEGB theory pretel_2022 .

We have found many additional features in the unexplored regions of parameter
space, the most striking of which is that 4DEGB charged quark stars can exist
with radii not only smaller than the general relativistic Buchdahl bound, but
also smaller than the $2M$ Schwarzschild radius. Such Extreme Compact Charged
Objects represent a possible new state of matter. A full analysis of the
stability of such objects would be an interesting topic for future study.

## Acknowledgements

This work was supported in part by the Natural Sciences and Engineering
Research Council of Canada. We would like to thank Dr. Tiberiu Harko for
useful correspondence regarding the existing literature.

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††thanks: Now at: GSI Helmholtzzentrum für Schwerionenforschung, 64291
Darmstadt, Germany

# Quantum mechanics of radiofrequency-driven coherent beam oscillations
in storage rings

J. Slim  III. Physikalisches Institut B, RWTH Aachen University, 52056 Aachen,
Germany  N.N. Nikolaev  L.D. Landau Institute for Theoretical Physics, 142432
Chernogolovka, Russia  Moscow Institute for Physics and Technology, 141700
Dolgoprudny, Russia  F. Rathmann  Institut für Kernphysik, Forschungszentrum
Jülich, 52425 Jülich, Germany  A. Wirzba  Institut für Kernphysik,
Forschungszentrum Jülich, 52425 Jülich, Germany  Institute for Advanced
Simulation, Forschungszentrum Jülich, 52425 Jülich, Germany

###### Abstract

Precision searches for the electric dipole moment of protons in storage ring
experiments call for beam-position monitoring in the picometer range. We
present the relevant fully quantum mechanical derivation of radiofrequency-
driven collective oscillations of a beam in a storage ring.

###### pacs:

13.40.Em, 11.30.Er, 29.20.Dh, 29.27.Hj

Introduction One of grand challenges in particle physics is the search for
physics beyond the Standard Model (BSM). At the forefront of the high
precision frontier is the proposed search for the electric dipole moment (EDM)
of protons stored in an all-electric frozen-spin storage rings with a
sensitivity of
$d_{p}\sim${10}^{-29}\text{\,}\mathrm{e}\,\mathrm{c}\mathrm{m}$$ that is some
15 orders of magnitude smaller than the magnetic dipole moment of the proton
Anastassopoulos  _et al._ (2016); Abusaif  _et al._ (2021); Omarov  _et al._
(2020). The primary motivation is that the experimental observation of a
permanent EDM of any subatomic particle implies the explicit violation of time
reflection (T) and parity (P) symmetries, and therefore, according to the CPT
theorem, also involves the violation of CP in the flavor-preserving channel.
The presence of the latter could shed light on the mystery of the anomalously
large baryon asymmetry in the Universe, which vastly exceeds the expectations
within the Standard Models of particle physics and cosmology Bernreuther
(2002); Bödeker and Buchmüller (2021).

Proper control of systematic effects encountered in the search for such a
minuscule EDM requires concurrent measurements of the spin rotations of beams
propagating in opposite directions in an all-electric ring. To achieve such an
ambitious goal, one of the crucial tasks is to control the difference of the
vertical positions of the two beams along the orbit in the machine with an
accuracy of about $5\text{\,}\mathrm{pm}$ Abusaif  _et al._ (2021). One may
wonder whether such enormously demanding accuracy is not prohibited by the
Heisenberg uncertainty principle. Moreover, this accuracy is also in the range
of the vertical displacement of the beams due to Earth’s gravitational force.
It should be noted that up to now such tiny gravitational effects were never
considered in the design and construction of accelerators, but gravity causes
an observable background in the searches for the proton EDM Silenko and
Teryaev (2007); Orlov  _et al._ (2012); Obukhov  _et al._ (2016); Nikolaev
_et al._ (2019).

Recently, the first direct measurement of the amplitude of collective
oscillations in the micrometer range of an intense beam of deuterons in a
storage ring, excited by an intentionally mismatched radiofrequency (RF) Wien
filter (WF) Slim  _et al._ (2016) at the Cooler Synchrotron COSY of
Forschungszentrum Jülich has been reported Slim  _et al._ (2021). The achieved
accuracy is about a factor of 10 larger than the amplitude of the zero-point
betatron oscillation of single particles. The latter is
$\approx$41\text{\,}\mathrm{nm}$$ (see (Slim  _et al._ , 2021, Eq. (9))).

The approach to the quantum ground state and the possibility of detecting
displacements of macroscopic bodies near the quantum limit are the subject of
intense theoretical and experimental efforts Schreppler  _et al._ (2014);
Abbott  _et al._ (2009); Murch  _et al._ (2008); Biercuk  _et al._ (2010);
Rugar  _et al._ (2004). A notable example is the detection of gravitational
waves with interferometric detectors using kilogram-scale mirrors Abbott  _et
al._ (2016). The case of ultra-small coherent oscillations of particle bunches
of rarefied gas confined in the focusing fields of storage rings is
complementary to, and different from, the above examples. The possibility of
detecting small amplitudes arises from the fact that the signal of coherent
oscillations is generated by $\approx${10}^{9}$$ particles contained in the
bunch.

The subject of the present paper is the transition from the description of
classical mechanics suitable for micron amplitudes in the COSY experiment Slim
_et al._ (2021) to an obviously deep quantum regime of picometer amplitudes in
the proposed ultimate proton EDM experiment. Our main conclusion is that the
functional form of the coherent beam oscillation amplitude does note change
during the transition from the classical to the quantum regime. It follows
that considerations of the Heisenberg quantum uncertainty do not preclude sub-
picometer accuracy in the determination of the coherent beam oscillation
amplitude – the challenge lies in the sensitivity of the beam position
monitors (BPM) Haciömeroglu  _et al._ (2019). We also address the effects of
intrabeam scattering and interactions with the residual gas on the coherent
excitations of the beam. We illustrate the origin of the picometer domain by
the example of the derivation of the vertical beam displacement by the Earth’s
gravity in terms of the vertical betatron frequency.

Vertical splitting of counter-propagating beams due to Earth’s gravity As an
introduction into the subject, we explain how picometer-scale beam
displacements emerge in the EDM experiment in storage rings. The attraction of
the Earth on the beam particles can not be switched off and must be
compensated for by the focusing electromagnetic fields: either radial magnetic
fields in the case of focusing by magnetic quadrupoles Silenko and Teryaev
(2007); Omarov  _et al._ (2020), or vertical electric fields in the case of
electric focusing Obukhov  _et al._ (2016); Orlov  _et al._ (2012);
Anastassopoulos  _et al._ (2016); Abusaif  _et al._ (2021). The force of the
gravitational attraction is given by

$\bm{F}_{g}={\frac{2\gamma^{2}-1}{\gamma}}\,m\bm{g}_{\oplus}\,,$ (1)

where $\bm{g}_{\oplus}$ is the acceleration of free fall at the Earth’s
surface, $\gamma=1/\sqrt{1-(v/c)^{2}}$ is the Lorentz factor of the stored
particles in the ring having velocity $v$, and $m$ is the rest mass of each
particle. The spring constant $\langle k\rangle$ of a confining oscillator
potential can be related to the angular velocity $\omega_{y}$ of the vertical
betatron oscillation as follows,

$\langle k\rangle\approx\gamma m\omega_{y}^{2}\,.$ (2)

Then gravity causes a downward displacement of the beam, which is given by

$\Delta
y\approx\frac{(2\gamma^{2}-1)\,|g_{\oplus}|}{\gamma^{2}\nu_{y}^{2}\,\omega_{\text{rev}}^{2}}\,,$
(3)

where $\nu_{y}=\omega_{y}/\omega_{\text{rev}}$ is the vertical betatron tune
and $\omega_{\text{rev}}$ is the beam orbital angular velocity.

Of particular interest is the all-electric frozen-spin ring for counter-
rotating beams of protons with kinetic energy
$T_{p}=$233\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$. The often discussed
lattice for such a ring with focusing by electric quadrupoles anticipates a
circumference of $500\text{\,}\mathrm{m}$ with
$\omega_{\text{rev}}=$2.26\text{\times}{10}^{6}\text{\,}{\mathrm{s}}^{-1}$$
and vertical betatron tune $\nu_{y}=0.45$ Anastassopoulos  _et al._ (2016);
Abusaif  _et al._ (2021). The coherent vertical displacement of the beams
induced by the Earth’s gravity is therefore

$\Delta y_{E}\approx$13\text{\,}\mathrm{pm}$\,.$ (4)

In this scenario, both counter-rotating beams will have an identical
gravitational displacement. As for the spin rotations of the protons, the
impact of the gravity-compensating electric field focusing in the vertical
direction would correspond to a rotation of their magnetic dipole moments in a
radial magnetic field of opposite sign for clockwise and counter-clockwise
propagating beams, respectively. This would distinguish such a signal from a
real EDM signal, which will be identical for both beams. However small the
displacement may be, the false EDM effect due to the gravity-compensating
electric field is even larger than the EDM effect expected for
$d_{p}=${10}^{-29}\text{\,}\mathrm{e}\,\mathrm{c}\mathrm{m}$$ Orlov  _et al._
(2012); Obukhov  _et al._ (2016); Abusaif  _et al._ (2021).

A recent proposal of a hybrid electric ring with magnetic focusing Omarov  _et
al._ (2020) assumes a circumference of $800\text{\,}\mathrm{m}$ with
$\omega_{\text{rev}}=$1.4\text{\times}{10}^{6}\text{\,}{\mathrm{s}}^{-1}$$ and
a vertical betatron tune $\nu_{y}=2.3$, giving a coherent gravitational
vertical displacement of

$\Delta y_{B}\approx$1.3\text{\,}\mathrm{pm}$\,,$ (5)

and an average beam splitting of $2.6\text{\,}\mathrm{pm}$. Note that the
magnetic quadrupoles exert forces of opposite sign on counter-rotating beams,
resulting in vertically non-identical orbits of the two beams.

Classical mechanics of RF-driven beam oscillations The treatment of the
classical limit provides the necessary background and elucidates the
subsequent transition to the deep quantum regime. Here we follow the
discussion of Ref. Slim  _et al._ (2021) and extend it to include the effects
of intrabeam scattering and interaction with the residual gas. In the
experiment described in Ref. Slim  _et al._ (2021), the beam was
stroboscopically excited with a mismatched RF Wien filter once per turn. In
this way, a vertical Lorentz force

$F_{y}(n)=F_{y}\cos(n\,\omega_{\text{WF}}T)\,,$ (6)

is exerted on the stored particle, where $n$ is the number of turns,
$\omega_{\text{WF}}$ denotes the angular velocity of the RF in the Wien filter
and $T=2\pi/\omega_{\text{rev}}$ is the beam revolution period. When the
Lorentz force vanishes, the beam performs idle vertical (and horizontal)
betatron oscillations

$y(t)=y(0)\sqrt{\frac{\beta_{y}(t)}{\beta_{y}(0)}}\cos\left[\psi_{y}(t)\right]\,,$
(7)

where $\beta_{y}(t)$ is the vertical betatron amplitude function. The betatron
phase advance $\psi_{y}(t)$ satisfies
$\psi_{y}(t+T)-\psi_{y}(t)=\omega_{y}T=2\pi\nu_{y}$. The change at turn $n$ of
the vertical velocity of the stored particle accumulated during the time
interval $\Delta t=\ell/v_{z}$ spent by the particle per turn inside the Wien
filter of length $\ell$ is given by

$\Delta v_{y}(n)=\frac{F_{y}(n)\Delta t}{\gamma
m}=-\zeta\omega_{y}\cos(n\,\omega_{\text{WF}}T)\,,$ (8)

where again $\gamma$ and $m$ are the Lorentz factor and the mass of the
particle, respectively. The change $\Delta y$ of the vertical position $y$ in
the Wien filter can be neglected.

According to Eq. (7), the stroboscopic signal of the betatron motion observed
at any point in the ring follows the harmonic law as a function of $nT$ with
angular velocity $\omega_{y}$, and we invoke the familiar description of the
oscillatory motion in terms of the complex variable $z=y-iv_{y}/\omega_{y}$.
The one-particle master equation for the buildup of RF-driven oscillations
directly downstream of the Wien filter is

$z(n)=z(n-1)\exp(i\omega_{y}T)-\frac{i}{\omega_{y}}\Delta v_{y}(n)\,.$ (9)

Subject to the initial condition $z(0)=0$, it has the generic solution

$z(n)=-\frac{i}{\omega_{y}}\exp(i\omega_{y}nT)\sum_{k=1}^{n}\Delta
v_{y}(k)\exp(-i\omega_{y}kT)\,,$ (10)

which yields for the stroboscopic force of Eq. (8)

$\begin{split}z(n)=&\frac{i\zeta}{2}\cdot\frac{\exp(in\omega_{y}T)-\exp(in\,\omega_{\text{WF}}T)}{\exp[i(\omega_{y}-\omega_{\text{WF}})T]-1}\\\
&+\\{\omega_{\text{WF}}\to-\omega_{\text{WF}}\\}\,.\end{split}$ (11)

A similar analytic result holds also for generic AC dipole-driven betatron
oscillations, applied in a completely different context of machine
diagnostics, described in Ref. Miyamoto  _et al._ (2008) (see also references
therein). The amplitude of the RF Wien filter Fourier component of the beam
oscillation, $y_{\text{WF}}(n)=-\xi_{y}\cos(n\,\omega_{\text{WF}}T)$, is given
by

$\xi_{y}=\frac{\zeta}{2}\cdot\frac{\sin(2\pi\nu_{y})}{\cos(2\pi\nu_{\text{WF}})-\cos(2\pi\nu_{y})}\,,$
(12)

where the WF tune $\nu_{\text{WF}}=\omega_{\text{WF}}/\omega_{\text{rev}}$.
Note the resonance at $\nu_{\text{WF}}=\nu_{y}$.111In fact, the real part of
$z(n)$, as given in Eq. (11), simply reads
$y(n)=-\zeta_{y}\bigl{\\{}\cos(n\omega_{\text{WF}}T)-\cos(n\omega_{y}T)\bigr{\\}}+\frac{\zeta}{2}\sin(n\omega_{y}T)$.
Thus on resonance, we obtain
$y^{\text{res}}(n)=-\frac{\zeta}{2}(n-1)\sin(n\omega_{y}T)$. The amplitude in
Eq. (12) is independent of the betatron idle motion of individual particles
and is shared by all particles in the bunch. It can be filtered out and
measured by Fourier analysis of the BPM response with lock-in amplifiers.

The Heisenberg uncertainty limit $Q$ for the betatron oscillation amplitude
$\xi_{y}$ is related to the zero-point oscillator energy
$\frac{1}{2}\hbar\omega_{y}$ through

$Q^{2}=\frac{\hbar}{m\gamma\omega_{y}}\,\,.$ (13)

Under the conditions of the experiment, this gives

$Q=\frac{82}{\sqrt{\gamma\nu_{y}}}\,{\rm nm}=$41\text{\,}\mathrm{nm}$\,,$ (14)

while the smallest value of the measured oscillation amplitude was
$\xi_{y}^{\text{min}}=(1.08\pm 0.52)\,$\mathrm{\SIUnitSymbolMicro m}$$ Slim
_et al._ (2021).

One can extend the above considerations to include the impact of the
interaction with the residual gas (RG). The scattering off the residual gas,
followed by the vertical velocity kick $v_{y}^{\text{RG}}$ within the ring
acceptance angle, $\theta_{\text{acc}}$, is said to occur during the turn
$n_{\text{RG}}$. According to Eq. (10), the corresponding contribution to the
betatron motion is

$z^{\text{RG}}(n)=-\frac{i}{\omega_{y}}\
v_{y}^{\text{RG}}\exp\left[i\omega_{y}(n-n_{\text{RG}})T)+\psi_{y}^{\text{RG}}\right]\,,$
(15)

where $\psi_{y}^{\text{RG}}$ is the betatron phase advance from the scattering
point to the WF. Evidently, scattering from the residual gas does not
contribute at all to the RF-driven oscillations with angular velocity
$\omega_{\text{WF}}$. The same is true, of course, for intrabeam scattering.
However, losses due to elastic scattering beyond the ring acceptance angle and
the absorption due to electromagnetic and hadronic interactions must be taken
into account by the obvious substitution in Eq. (11),

$\omega_{y}\to\omega_{y}+\frac{i}{\tau}\,,$ (16)

where $\tau$ is the lifetime of the beam which amounts to
$\approx$1500\text{\,}\mathrm{s}$$ (see (Slim  _et al._ , 2021, Fig. 14)). The
beam attenuation during one revolution is entirely negligible and the net
effect on the oscillation amplitude is the well-known attenuation factor in
Eq. (12),

$\xi_{y}\to\xi_{y}\exp\left(-\frac{nT}{\tau}\right)\,.$ (17)

Quantum mechanics of RF Wien filter-driven oscillations The above treatment of
the proximity to the quantum limit by classical mechanics can be justified a
posteriori, since the observed amplitudes are much larger than the amplitude
of the zero-point quantum oscillations, given in Eq. (14). Such a posteriori
comparison of two amplitudes could have been the other way around if the
perturbation of the beam had been much smaller than the zero-point quantum
amplitude. Then one has to resort to the time-dependent Schrödinger equation

$i\frac{d}{dt}\Psi(t)=\left\\{H_{0}+V(t)\right\\}\Psi(t)\,,$ (18)

where $H_{0}$ is the time-independent potential of the harmonic oscillator,
while the perturbative potential reads

$\begin{split}V(t)&=-F_{y}\cdot y\cdot\cos(\omega_{\text{WF}}t)\\\
&=-\frac{F_{y}}{\sqrt{2m\gamma\omega_{y}}}(a^{\dagger}+a)\cos(\omega_{\text{WF}}t)\,,\end{split}$
(19)

where $a^{\dagger}$ and $a$ are the harmonic oscillator creation and
annihilation operators. This perturbation stroboscopically acts for very short
time intervals, $\Delta t\ll T$, once per turn at time $t_{n}=nT$.

Let $\Psi(-;n)$ be the wave function before, and $\Psi(+;n)$ directly behind
the WF. The impact of the WF potential leads to the following discontinuity of
the wave function

$i\\{\Psi(+;n)-\Psi(-;n)\\}=V(nT)\,\Delta t\,\Psi(-;n)\,,$ (20)

which gives

$\begin{split}&\Psi(+;n)=\\\ &\quad\left\\{1+i\frac{F_{y}\Delta
t}{\sqrt{2m\gamma\omega_{y}}}\cos(n\omega_{\text{WF}}T)(a^{\dagger}+a)\right\\}\Psi(-;n)\,.\end{split}$
(21)

The rest of the turn proceeds in the time-independent harmonic potential. The
creation and annihilation operators change the energy of the state by
$\pm\omega_{y}$. Taking this into account, we obtain the master equation

$\begin{split}&\Psi(-;n)=\\\ &\mbox{}\quad\left\\{1+i\frac{F_{y}\Delta
t}{\sqrt{2m\gamma\omega_{y}}}\cos(n\omega_{\text{WF}}T)\left(a^{\dagger}e^{-i\omega_{y}T}+ae^{i\omega_{y}T}\right)\right\\}\\\
&\mbox{}\quad\times\Psi(-;n-1)e^{-i\omega_{\text{in}}T}\,.\end{split}$ (22)

Here $\omega_{\text{in}}$ is the energy of the initial state
$\Psi(+;0)=|\text{in}\rangle$.

A perturbative solution of this master equation proceeds as follows. The beam
passes the Wien filter at times $t_{k}=kT$ with $k=1,2,\ldots,n$, and the
transition from the initial state $|\text{in}\rangle$ to the perturbation
components $a^{\dagger}|\text{in}\rangle$ and $a|\text{in}\rangle$ can take
place during any pass $k$. Consequently, in the former case, the transition
amplitude acquires $\cos(k\omega_{\text{WF}}T)$ from the Wien filter potential
and the phase factor $\exp(-i(n-k)\omega_{y}T)$ from the subsequent evolution
in the confining potential. In the latter case the transition amplitude
acquires $\cos(k\omega_{\text{WF}}T)$ from the Wien filter potential and the
phase factor $\exp(i(n-k)\omega_{y}T)$ from the subsequent evolution in the
confining potential. Here $\exp(i(n-k)\omega_{y}T)$ is the complex conjugate
to $\exp(-i(n-k)\omega_{y}T)$, and upon summation of all transitions we obtain

$\begin{split}|\Psi(+;n)\rangle&=\left\\{1+i\frac{F_{y}\Delta t}{\sqrt{2\gamma
m\omega_{y}}}\left(w(n)a^{\dagger}+w^{*}(n)a\right)\right\\}|\text{in}\rangle\,,\end{split}$
(23)

where

$\begin{split}w(n)&=\sum_{k=1}^{n}\cos(k\omega_{\text{WF}}T)\exp\\{-i(n-k)\omega_{y}T\\}\\\
&=\frac{1}{2}\cdot\Bigl{[}\frac{\exp(-in\omega_{y}T)-\exp(-in\omega_{\text{WF}}T)}{\exp(-i(\omega_{y}-\omega_{\text{WF}})T)-1}\\\
&\qquad+\\{\omega_{\text{WF}}\to-\omega_{\text{WF}}\\}\Bigr{]}\,.\end{split}$
(24)

Now we are in the position to evaluate the driven oscillation amplitude. To
linear order in $\Delta t$ we get

$\begin{split}y(n)&=\frac{1}{\sqrt{2\gamma
m\omega_{y}}}\left\langle\Psi^{*}(+;n)\left|(a^{\dagger}+a)\right|\Psi(+;n)\right\rangle\\\
&=i\frac{F_{y}\Delta t}{2\gamma
m\omega_{y}}\left\langle\text{in}\left|\left[\left(a^{\dagger}+a\right),\left(w(n)a^{\dagger}+w^{*}(n)a\right)\right]\right|\text{in}\right\rangle\\\
&=-i\frac{F_{y}\Delta t}{2\gamma
m\omega_{y}}\left(w^{*}(n)-w(n)\right)\left\langle\text{in}\left|[a,a^{\dagger}]\right|\text{in}\right\rangle\\\
&=-i\frac{F_{y}\Delta t}{2\gamma
m\omega_{y}}\left(w^{*}(n)-w(n)\right)\,.\end{split}$ (25)

This expression is exactly the same as for $y(n)={\rm Re}\,z(n)$ from Eq.
(11), with $\zeta$ replaced by the ratio $\frac{-F_{y}\Delta t}{\gamma
m\omega_{y}}$, cf. Eq. (8).

The appearance of the commutator $[a,a^{\dagger}]=1$ in Eq. (25) is crucial.
It makes the driven coherent oscillation amplitude universal for all particles
in the beam, regardless of their individual quantum state. This finding is
tantamount to the independence of the classical driven oscillation amplitude
from the amplitude and phase of the inherent betatron motion Slim  _et al._
(2021). The amplitude of the RF Wien filter-driven oscillations of the bunch
is identical to the amplitude of each individual particle in the bunch.
Moreover, quantum mechanics and classical mechanics give absolutely identical
results for the amplitude of driven oscillations. One may consider this as an
exemplary case of Ehrenfest’s theorem.

Summary and Conclusions We have presented a quantum mechanical description of
the excitation of coherent betatron oscillations by radiofrequency
electromagnetic fields. Remarkably, one and the same formula [Eq. (12)] covers
the whole range of amplitudes from large classical ones to well below the one-
particle quantum limit. Neither scattering from the residual gas nor intrabeam
scattering contribute to these coherent betatron oscillations.

We conclude that in principle the amplitude of coherent oscillations of the
center of mass of a particle bunch in a storage ring can be measured with an
accuracy of better than one picometer within the framework of the Heisenberg
uncertainty principle. Our analysis may be applied to other problems involving
pulsed excitation of quantum oscillators.

###### Acknowledgements.

This work has been performed in the framework of the JEDI collaboration, and
is financially supported by an ERC Advanced-Grant (srEDM # 694390) of the
European Union and by the Russian Fund for Basic Research (Grant No.
18-02-40092 MEGA).

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