# On a conjecture of R. M. Murty and V. K. Murty

Yuchen Ding  (Yuchen Ding) School of Mathematical Science, Yangzhou
University, Yangzhou 225002, People’s Republic of China<EMAIL_ADDRESS>
###### Abstract.

Let $\omega^{*}(n)$ be the number of primes $p$ such that $p-1$ divides $n$.
Recently, R. M. Murty and V. K. Murty proved that

$x(\log\log x)^{3}\ll\sum_{n\leq x}\omega^{*}(n)^{2}\ll x\log x.$

They further conjectured that there is some positive constant $C$ such that

$\sum_{n\leq x}\omega^{*}(n)^{2}\sim Cx\log x$

as $x\rightarrow\infty$. In this short note, we give the correct order of the
sum by showing that

$\sum_{n\leq x}\omega^{*}(n)^{2}\asymp x\log x.$

###### Key words and phrases:

Primes; Bombieri–Vinogradov theorem.

###### 2010 Mathematics Subject Classification:

Primary 11A41.

Let $\omega(n)$ be the number of distinct prime divisors of $n$. In about one
hundred years ago, Hardy and Ramanujan [3] found out that $\omega(n)$ has
normal order $\log\log n$ which means that for all most all integers $n$ we
have $\omega(n)\sim\log\log n$. Later, Turán [6] provided a quite elegantly
simplified proof by establishing

$\sum_{n\leq x}(\omega(n)-\log\log n)^{2}\ll x\log\log x.$

In 1955, Prachar [5] considered a variant arithmetic function of $\omega$. Let
$\omega^{*}(n)$ be the number of primes $p$ such that $p-1$ divides $n$.
Prachar proved that

$\sum_{n\leq x}\omega^{*}(n)=x\log\log x+Bx+O(x/\log x)$

and

$\sum_{n\leq x}\omega^{*}(n)^{2}=O\left(x(\log x)^{2}\right),$

where $B$ is a constant. Motived by Prachar’s work, Erdős and Prachar [2]
proved that the number of pairs of primes $p$ and $q$ so that the least common
multiple $[p-1,q-1]\leq x$ is bounded by $O(x\log\log x)$. Following a remark
of Erdős and Prachar, R. M. Murty and V. K. Murty [4] improved this to $O(x)$.
By this improvement, they reached the nice bounds

$x(\log\log x)^{3}\ll\sum_{n\leq x}\omega^{*}(n)^{2}\ll x\log x.$

With these in hands, R. M. Murty and V. K. Murty conjectured that there is
some positive constant $C$ such that

$\sum_{n\leq x}\omega^{*}(n)^{2}\sim Cx\log x$

as $x\rightarrow\infty$. In this note, the author shall give a slight
improvement of the result due to R. M. Murty and V. K. Murty towards the
correct direction of their conjecture.

######  Theorem 1.

There are two absolute constants $a_{1}$ and $a_{2}$ such that

$a_{1}x\log x\leq\sum_{n\leq x}\omega^{*}(n)^{2}\leq a_{2}x\log x.$

###### Proof.

We only need to prove the lower bound as the upper bound is displayed by R. M.
Murty and V. K. Murty. Throughout our proof, the number $x$ is sufficiently
large. From the paper of R. M. Murty and V. K. Murty [4, equation (4.10)], we
have

$\displaystyle\sum_{n\leq x}\omega^{*}(n)^{2}=x\sum_{d\leq
x}\varphi(d)\bigg{(}\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv 1\\!\\!\\!\
({\rm{mod}}\ d)\end{subarray}}\frac{1}{p-1}\bigg{)}^{2}+O(x).$ (1)

Integrating by parts gives

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv 1\\!\\!\\!\
({\rm{mod}}\
d)\end{subarray}}\frac{1}{p}=\frac{\pi(x;d,1)}{x}+\int_{2}^{x}\frac{\pi(t;d,1)}{t^{2}}dt\geq\int_{x^{3/4}}^{x}\frac{\pi(t;d,1)}{t^{2}}dt,$
(2)

where $\pi(t;d,1)$ is the number of primes $p\equiv 1\ ({\rm{mod}}\ d)$ up to
$t$. Thus, from equations (1) and (2) we obtain

$\displaystyle\sum_{n\leq x}\omega^{*}(n)^{2}\geq x\sum_{d\leq
x^{1/3}}\varphi(d)\left(\int_{x^{3/4}}^{x}\frac{\pi(t;d,1)}{t^{2}}dt\right)^{2}+O(x).$
(3)

For any integer $0\leq j\leq\left\lfloor\frac{\log x}{13\log 2}\right\rfloor$,
let $Q_{j}=2^{j}x^{1/4}$. Then $Q_{j}<x^{1/3}$ for all integers $j$. From a
weak form of the Bombieri–Vinogradov theorem (see for example [1]), we have

$\displaystyle\sum_{Q_{j}<d\leq 2Q_{j}}\max_{y\leq
z}\left|\pi(y;d,1)-\frac{\text{li}~{}y}{\varphi(d)}\right|\ll\frac{z}{(\log
z)^{5}},$

for any $0\leq j\leq\left\lfloor\frac{\log x}{13\log 2}\right\rfloor$ and
$x^{3/4}\leq z\leq x$, where the implied constant is absolute. It follows
immediately that

$\displaystyle\max_{y\leq
z}\left|\pi(y;d,1)-\frac{\text{li}~{}y}{\varphi(d)}\right|<\frac{\text{li}~{}z}{\varphi(d)\log
z}$ (4)

hold for all $Q_{j}<d\leq 2Q_{j}$ but at most $O\left(Q_{j}/(\log
x)^{2}\right)$ exceptions. From equation (4) we have

$\displaystyle\pi(y;d,1)>\frac{\text{li}~{}y}{2\varphi(d)}\quad\left(z/2<y\leq
z\right)$

for all $Q_{j}<d\leq 2Q_{j}$ with at most $O\left(Q_{j}/(\log x)^{2}\right)$
exceptions. A little thought with the dichotomy of $z$ between the interval
$[x^{3/4},x]$ leads to the fact

$\displaystyle\pi(y;d,1)>\frac{\text{li}~{}y}{2\varphi(d)}>\frac{y}{3\varphi(d)\log
y}\quad\left(\forall~{}x^{3/4}\leq y\leq x\right)$ (5)

for all $Q_{j}<d\leq 2Q_{j}$ except for $O\left(Q_{j}/\log x\right)$
exceptions. Let $S_{j}$ be the set of all integers $Q_{j}<d\leq 2Q_{j}$ such
that equation (5) hold. Thus, from the analysis above and equation (3), (5) we
conclude that

$\displaystyle\sum_{n\leq x}\omega^{*}(n)^{2}$
$\displaystyle\geq\frac{x}{9}\sum_{0\leq j\leq\left\lfloor\frac{\log x}{13\log
2}\right\rfloor}\sum_{\begin{subarray}{c}d\in
S_{j}\end{subarray}}\varphi(d)\left(\int_{x^{3/4}}^{x}\frac{1}{\varphi(d)t\log
t}dt\right)^{2}+O(x)$ $\displaystyle\gg x\sum_{0\leq
j\leq\left\lfloor\frac{\log x}{13\log
2}\right\rfloor}\sum_{\begin{subarray}{c}d\in
S_{j}\end{subarray}}\frac{1}{\varphi(d)}\gg x\log x.$

∎

It is worth here mentioning that we have the following corollary

$\sum_{p,q\leq x}\frac{1}{[p-1,q-1]}\asymp\log x$

due to (see [4, page 6, last line])

$\sum_{n\leq x}\omega^{*}(n)^{2}=\sum_{p,q\leq x}\frac{x}{[p-1,q-1]}+O(x).$

## Acknowledgments

The author is supported by the Natural Science Foundation of Jiangsu Province
of China, Grant No. BK20210784, China Postdoctoral Science Foundation, Grant
No. 2022M710121. He is also supported by foundation numbers JSSCBS20211023 and
YZLYJF2020PHD051.

## References

  * [1] H. Davenport, Multiplicative Number Theory, Second edition, Graduate Texts in Mathematics 74, Springer-Verlag, New York, 1980.
  * [2] P. Erdős, K. Prachar, Über die Anzahl der Lösungen von $[p-1,q-1]\leq x$, Monatsh. Math. 59 (1955), 318–319.
  * [3] G. H. Hardy, S. Ramanujan, The normal number of prime factors of a number $n$, Quart. J. Math. 48 (1917), 76–92.
  * [4] R. M. Murty, V. K. Murty, A variant of the Hardy–Ramanujan theorem, Hardy–Ramanujan J. 44 (2021), 32–40.
  * [5] K. Prachar, Über die Anzahl der Teiler einer natürlichen Zahl, welche die Form $p-1$ haben, Monatsh. Math. 59 (1955), 91–97.
  * [6] P. Turán, On a theorem of Hardy and Ramanujan, J. London Math. Soc. 9 (1934), 274–276.


# Storage Rings and Gravitational Waves:
Summary and Outlook

A. Berlin1, M. Brüggen2, O. Buchmueller3, P. Chen4, R. T. D’Agnolo5, R. Deng6,
J. R. Ellis7,×,, S. Ellis5, G. Franchetti8, A. Ivanov9, J. M. Jowett8,
A. P. Kobushkin10, S. Y. Lee11, J. Liske2, K. Oide12, S. Rao2, J. Wenninger13,
M. Wellenzohn9, M. Zanetti14, F. Zimmermann13,×,

×Editors

1New York U., USA; 2U. Hamburg, Germany; 3Imperial College London, UK;
4National Taiwan University, Taiwan; 5IPhT Paris, France; 6SARI Shanghai,
China;
7King’s College London, UK; 8GSI, Germany; 9Vienna U., Austria; 10BITP,
Ukraine;
11Indiana U., USA; 12KEK, Japan; 13CERN, Switzerland; 14U.& INFN Padua, Italy
John.Ellis<EMAIL_ADDRESS>
###### Abstract

We report some highlights from the ARIES APEC workshop on “Storage Rings and
Gravitational Waves” (SRGW2021), held in virtual space from 2 February to 18
March 2021, and sketch a tentative landscape for using accelerators and
associated technologies for the detection or generation of gravitational
waves.

KCL-PH-TH/2021-28, CERN-TH-2021-068

##  1 Introduction

SRGW2021 [1] brought together storage-ring experts, accelerator scientists,
experimental particle physicists, theoretical physicists, astrophysicists, and
members of the gravitational physics community. It surveyed and discussed
various topics including: (1) examples of the sensitivities of storage rings
to tides, earthquakes and other seismic perturbations; (2) earlier and more
recent work on the use of storage rings to detect gravitational waves; (3) the
dependence of the storage ring detection sensitivity on the ring size and
other parameters; (4) possible enhancement strategies such a special optics,
particular magnets and advanced diagnostics; (5) gravitational radiation from
storage ring beams; (6) possible astrophysical and cosmological signals, their
frequencies and amplitudes; (7) gravity research based on accelerator
technologies; and (8) a tentative roadmap for possible future R&D and
measurements.

SRGW2021 was held in 5 virtual sessions scheduled between 2 February to 18
March 2021. A total of 115 participants registered for the event. Figure 1
shows the geographical distribution of the participants, along with the time
evolution of their registrations. The workshop schedule, including its
sessions, talks and speakers, is shown in the Appendix.

Figure 1: Geographical distribution of SRGW2021 participants (left) and time
evolution of registrations (right).

##  2 Selected Highlights - Summaries by Presenters

###  2.1 Storage ring sensitivity to tides & large-scale perturbations,
earthquakes, noise - examples from LEP and LHC
J. Wenninger

Tides and other large scale geological deformations like slow ground motion
and earthquakes have been observed routinely with the LHC and the LEP
colliders. As an example, Fig. 2 displays measurements of the seismic noise
power spectrum in the LHC tunnel [2]. Due to the large size of the ring, 26.7
km, even very small strains result in absolute deformations of the
circumference exceeding 1 mm for large tides and large earthquakes occurring
anywhere on the globe (magnitudes of 7 or higher). The beam position monitor
system is currently the best tool to observe such deformations. Changes of the
circumference are particularly well resolved with around 500 measurements
along the LHC circumference. The resolutions on the average circumference
measurement reaches the micrometer scale over time intervals of one hour.
Strains (relative circumference changes) down to around $10^{-11}$ may be
resolved.

Figure 2: Measurements of the seismic noise power spectrum in the LHC tunnel
[2].

###  2.2 Ground Vibration at the SSRF Site
R. Deng

The mechanical stability of key components in an accelerator is one of the
main contributions to the beam orbit stability. A series of measurements and
analyses of ground vibration at the SSRF site have been carried out since
2004. Conventional facilities were located outside the main building so as to
minimize the vibration generated by mechanical noise due to pumps, etc., and a
long-term vibration monitoring system was built to monitor the ground
vibration of SSRF. Recently, in view of the SXFEL and SHINE projects, more
measurements are being performed for the shafts and tunnels about 30m
underground, where the accelerator will be located, and a permanent vibration
monitoring system is also considered. In addition, more investigation and
optimization for the mechanical engineering are underway, along with the
operation of the accelerator, to realize the beam stability goal for SSRF.

###  2.3 Response of a storage-ring beam to a gravitational wave
K. Oide

Summary of the presentation:

  * •

The beam in a storage ring can respond to a gravitational wave (GW), somewhat
similarly to the Weber bar. A helpful reference is the book by Misner, Thorne,
and Wheeler [3].

  * •

The betatron motion of the beam in a ring can respond resonantly to the GW.

  * •

A special beam optics “beam antenna” may enhance the sensitivity. One example
of a 37-km storage ring lattice optimized as a GW antenna is shown in Fig. 3.

  * •

There will be many noise sources to overcome: thermal motion of quadrupoles,
beam fluctuation due to emittance, synchrotron radiation and the acceleration,
etc.

Figure 3: A 37-km storage ring lattice optimized as a GW antenna.

Questions:

  * •

Does the force depend on the choice of the coordinate system? — A combination
of motion of bunches to single out the “quadrupole mode” does not depend on
the choice of the coordinates.

  * •

Are any sources beyond 10 kHz range conceivable in the universe?

  * •

This scheme may be suitable for GWs from the crab pulsar (60 Hz, continuous).
If FCC is used, the necessary betatron tune is 0.02, at which operation is
quite possible.

###  2.4 Detection of millihertz gravitational waves using storage rings
S. Rao, M. Brüggen and J. Liske

Based on our study [4], we showed that a storage ring can be turned into a
gravitational wave (GW) observatory which is sensitive to millihertz GWs from
astrophysical sources. This can be achieved, in principle, by making accurate
timing measurements of a test mass circulating in a storage ring and, after
resolving the noise sources, to observe a unique GW signal which causes the
net travel time of the test mass to deviate from its nominal value (expected
value in the absence of GWs), causing the change in travel time from its
nominal value to oscillate with the periodicity of the GWs, while also being
modulated by the effect of Earth’s rotation. The pros of this detection
technique are that, it potentially allows for an earth-based gravitational
wave observatory that is sensitve to millihertz GWs, which can also complement
the space-based laser interferometer detectors (like LISA, planned to be
launched in 2034), allowing for better triangulation of the position of
millihertz GW sources in the sky ( i.e. improvement of GW source sky-
localization), which is useful for electromagnetic follow-up studies of the GW
event via astronomical telescopes.

The effect of GWs studied in our work is thus along the longitudinal direction
( i.e. along the travel direction of the particle), and our calculations for
this effect (excluding the influence of the Rf system) are parametrically in
agreement with those by R.T. D Agnolo et al. in their works [5, 6], cited with
permission. Therefore, the general relativistic calculations for this GW
effect done from two different reference frames (metric formalism and Riemann
tensor formalism), give effectively the same results. Several previous works
[7, 8, 9, 10, 11] missed this effect, because they only considered the
particular orientation of the GW propagating perpendicular to the ring plane,
where this effect becomes null in the leading order of $h$ (the GW strain
amplitude), and only the effect along the radial direction is maximised, which
has an indirect contribution along the longitudinal direction with leading
order of $h^{2}$, negligible compared to the effect we studied, since $h$ is a
very small number.

The open question is in figuring out a feasible realization of this experiment
concept, and the conclusions of the SRGW2021 workshop (at least for the LHC
and some other current CERN facilities), is that there is a technological
limitation due to the sources of noise. In the presence of an RF system, the
RF can potentially cause a damping of this GW effect (as shown by R.T.
D’Agnolo et al.), or add its own noise. In the absence of an RF system (in a
coasting beam, for instance), there are yet other sources of noise (such as
collisions of the ions with with rest gas molecules etc., pointed out by Fritz
Caspers). Hence, if we can technologically optimise the experiment setup
and/or account for all noise sources, then we can potentially observe the GW
effect using this detection technique. We, the authors of Ref. [4], are
currently working on numerical simulations of the expected GW signal,
accounting for realistic GW waveforms from astrophysical sources, the effect
of Earth’s rotation etc. On another avenue, we are investigating whether
single ion storage rings could be feasible candidates to conduct this
experiment in the future.

###  2.5 Storage rings as detectors for relic gravitational-wave background?
A. N. Ivanov, A. P. Kobushkin, M. Wellenzohn

We argue that storage rings can be used for the detection of low-frequency
gravitational-wave background. We explain the systematic shrinkage of the
machine circumference of the storage ring of the SPring-8, observed by Takao
and Shimada (Proceedings of EPAC 2000, Vienna, 2000, p.1572), by the influence
of the relic gravitational-wave background. We show that the forces, related
to the stiffness of the physical structures, governing the path of the beam,
can be neglected for the analysis of the shrinkage of the machine
circumference caused by the relic gravitational-wave background. We show the
systematic shrinkage of the machine circumference can be explained by a relic
gravitational-wave background even if it is treated as a stochastic system
incoming on the plane of the machine circumference from all quarters of the
Universe. We show that the rate of the systematic shrinkage of the machine
circumference does not depend on the radius of the storage ring and it should
be universal for storage rings with any radius. Details can be found in Ref.
[9].

###  2.6 Gravitational Waves at Particle Storage Rings
R. T. D’Agnolo

#### Introduction

Gravitational Waves (GWs) are an important probe of the Universe today and are
a unique tool to shed light on its history before photon decoupling and Big
Bang Nucleosynthesis [12, 13]. There are several accepted astrophysical
sources of GWs such as white dwarf, neutron star and black hole binaries and a
number of hypothetical cosmological sources including reheating after
inflation and phase transitions in the early universe. The many orders of
magnitude in frequency and amplitude that are still unexplored, strongly
motivate the pursuit of new experimental concepts.

Here we discuss the possible detection of gravitational waves at circular
colliders through measurements of the beam position and revolution period. In
the following we use the LHC as a case study and then generalize to other
storage rings. Most of the discussion is a review of textbook concepts in
general relativity that help us build the right intuition for the LHC.

#### Basics of GW Detection

A gravitational wave can have at least two effects on a storage rings: 1) It
can interact with its electromagnetic fields; 2) it can displace the particles
in the ring in the radial, vertical or longitudinal direction. The first
possibility is discussed in another talk at this workshop (see Sebastian
Ellis’ contribution and [14]). I focus on the second possibility and in
particular on the longitudinal direction, which is the most promising
direction for detection, out of the three.

A priori it might seem hard to disentangle the different effects of a GW and
understand what we can measure. In principle everything in a storage ring
interacts with the gravitational wave at the same time: magnets, tunnel,
protons, electromagnetic fields, and we observers. However if we focus on a
small frequency range, the majority of potential detectors are rigid and do
not appreciably respond to the wave.

Any system in equilibrium displaced by a small amount $\delta$ responds as a
harmonic oscillator

$\displaystyle\ddot{\delta}+\frac{\dot{\delta}}{\tau_{s}}+\omega_{s}^{2}\delta=0\,.$
(1)

In the proper detector frame (described in the next Section) we can
approximate the effect of the wave as

$\displaystyle\ddot{\delta}+\frac{\dot{\delta}}{\tau_{s}}+\omega_{s}^{2}\delta=\omega_{g}^{2}f(\omega_{g},t)\,.$
(2)

Hence any system responds as

$\displaystyle\tilde{\delta}(\omega)=\frac{\omega_{g}^{2}\tilde{f}(\omega_{g},\omega)}{\omega_{s}^{2}-\omega^{2}+i\omega/\tau_{s}}\,,$
(3)

and any object with $\omega_{s}\gg\omega_{g}$ appears as rigid, i.e., it is
not appreciably affected by the GW:

$\displaystyle\tilde{\delta}(\omega)\simeq\frac{\omega_{g}^{2}}{\omega_{s}^{2}}\tilde{f}(\omega_{g},\omega)\ll\tilde{f}(\omega_{g},\omega)\,.$
(4)

For example, protons at the LHC are kept in equilibrium in the longitudinal
direction by the RF system [15]:

$\displaystyle\ddot{\delta}_{l}+\frac{\dot{\delta}_{l}}{\tau_{l}}+\omega_{l}^{2}\delta_{l}=0\,,$
(5)

where $\omega_{l}\simeq 60$ Hz. Protons reaching the RF cavity later than
their nominal revolution period see a larger electric field and are
accelerated more, and the opposite is true for protons arriving earlier than
they should.

If we focus on a GW that changes the protons’ revolution period, we have the
maximal response of the system when the effect is resonant:
$\omega_{g}\simeq\omega_{l}$. Clearly, for GWs of this frequency the magnets
or an observer appear as rigid in the proper detector frame, so we can focus
only on the protons that respond to the wave maximally at this frequency.

#### An Estimate in the Proper Detector Frame

General relativity has a large gauge symmetry. All possible choices of
coordinates

$\displaystyle x^{\mu}\to x^{\mu\prime}(x^{\mu})$ (6)

describe the same physics. We can use this to our advantage by choosing a
reference frame where Newtonian intuition applies. This is known as the proper
detector frame. In this frame: 1) coordinate distances coincide with proper
distances for non-relativistic objects; 2) rigid objects are not appreciably
deformed by the wave; 3) at zero GW frequency $\omega_{g}=0$ we recover flat
space; 4) in the cases of interest to us the GW acts as a Newtonian force.

This is especially convenient in the case under consideration, because we want
to study gravitational waves that do not oscillate appreciably over the size
of the LHC: $L_{\rm LHC}/\lambda_{g}\ll 1$. In this regime we can expand the
metric to second order in $L_{\rm LHC}/\lambda_{g}$. In the proper detector
frame this gives

$\displaystyle
ds^{2}=-dt^{2}\left[1+R_{0i0j}x^{i}x^{j}\right]+2dx^{i}dt\left(\frac{2}{3}R_{0ijk}x^{j}x^{k}\right)+dx^{i}dx^{j}\left(\delta_{ij}-\frac{1}{3}R_{ikjl}x^{k}x^{l}\right)\,,$
(7)

where $R_{\mu\nu\rho\sigma}$ is the Riemann tensor and the spatial coordinates
$x^{i}$ are measured from an origin chosen by the observer (for instance the
center of the LHC ring). For a monochromatic wave of frequency $\omega_{g}$,
neglecting indexes and $\mathcal{O}(1)$ factors describing polarizations, we
have the order of magnitude estimate

$\displaystyle R\sim\omega_{g}^{2}h\cos(\omega_{g}t+\phi)\,,$ (8)

where $h$ is a tiny ($\sim 10^{-20}$) dimensionless number. For a derivation
of the above results, including the metric in the detector frame, we refer to
the first chapter of [16]. We have neglected the Earth’s gravity, which
contributes to the $00$ component of the metric, but does not affect
appreciably the motion of the protons that are kept on a stable circular orbit
by the LHC magnets and RF system.

To compute the effect of the wave described by the metric in Eq. (7) we
imagine an observer fixed at a specific position along the ring. The observer
moves along a geodesic with coordinates $x^{\mu}(\tau)$, where $\tau$ is the
proper time. The protons move along the ring and are in general on a different
geodesic $y^{\mu}(\tau)=x^{\mu}(\tau)+\xi^{\mu}(\tau)$. For small
displacements $\xi^{\mu}$ compared to the wavelength of the GW ($L_{\rm
LHC}/\lambda_{g}\ll 1$) we can use the equation for geodesic deviation [16]

$\displaystyle\frac{D^{2}\xi^{\mu}}{D\tau^{2}}=-R^{\mu}_{\nu\rho\sigma}\xi^{\rho}\frac{dx^{\nu}}{d\tau}\frac{dx^{\sigma}}{d\tau}\,.$
(9)

In the proper detector frame this reduces to

$\displaystyle\frac{d^{2}\xi^{i}}{d\tau^{2}}=-R^{i}_{0j0}\xi^{j}\left(\frac{dt}{d\tau}\right)^{2}+...$
(10)

The terms represented by the ellipses are in the form
$R^{i}_{0jl}\xi^{j}v^{l}$ and $R^{i}_{kjl}\xi^{j}v^{l}v^{k}$. So for highly
relativistic protons, $v\simeq 1$, they are of the same order as
$R^{i}_{0j0}\xi^{j}$ (as is the case at the LHC), but can be neglected for
non-relativistic particles. Since we are interested in an order of magnitude
estimate we approximate all the terms on the right-hand side of the equation
with their parametric value at the LHC for a monochromatic wave propagating in
the plane of the accelerator:
$R^{i}_{0j0}\xi^{j},R^{i}_{0jl}\xi^{j}v^{l},R^{i}_{kjl}\xi^{j}v^{l}v^{k}\simeq\omega_{g}^{2}hL_{\rm
LHC}\cos(\omega_{g}t+\phi)$.

To get the equation of motion for the protons in our reference frame we notice
that for a non-relativistic observer $dt=d\tau+\mathcal{O}(h)$, so

$\displaystyle\frac{d^{2}\xi^{i}}{dt^{2}}=-R^{i}_{0j0}\xi^{j}+...\,,$ (11)

where the $\xi^{i}$ are the coordinates of the protons in a reference frame
moving along the geodesic of the observer and $\frac{d^{2}\xi^{i}}{dt^{2}}$ is
their acceleration. Note that we would have obtained the same result in the
reference frame of the protons $dt=\gamma d\tau+\mathcal{O}(h)$.

The previous equation describes the effect of the wave as a Newtonian force.
To describe correctly the motion of the protons in this frame we just need to
add to the previous equation any other external forces. In particular, if we
are measuring variations over the revolution period of the protons we have

$\displaystyle\frac{d^{2}\delta\xi^{l}}{dt^{2}}+\frac{1}{\tau_{l}}\frac{d\delta\xi^{l}}{dt}+\omega_{l}^{2}\delta\xi^{l}\simeq\omega_{g}^{2}hL_{\rm
LHC}\cos(\omega_{g}t+\phi)\,,$ (12)

where $l$ is the longitudinal direction along the circumference. We could
easily have obtained the same result from dimensional analysis.

#### Possible Avenues towards Detection

If we solve Eq. (12) on resonance to maximize the effect, we obtain

$\displaystyle\frac{\Delta T}{T}\simeq h\omega_{l}\tau_{l}\,.$ (13)

At the LHC we have $\frac{\Delta T}{T}\simeq 10^{-7}$ from the phase
measurement in the RF system [15] and $\omega_{l}\simeq 60$ Hz. We thus get a
sensitivity (without backgrounds)

$\displaystyle h\gtrsim 10^{-13}\left(\frac{2\pi\times 10\;{\rm
Hz}}{\omega_{l}}\right)\left(\frac{10\;{\rm
hours}}{\tau_{l}}\right)\left(\frac{\Delta T/T}{10^{-7}}\right)\,.$ (14)

This is about 7 orders of magnitude larger than what we expect from realistic
GW sources [17]. However, there is in principle a path towards vastly
increasing the sensitivity of a storage ring:

  1. 1.

The effect saturates when $T\simeq\omega_{g}^{-1}$. So we can either increase
$\omega_{l}$ or increase $T$ by about three orders of magnitude compared to
the LHC, where $T\simeq 1/(11\;{\rm kHz})$, increasing the sensitivity on $h$
by the same amount. Increasing $\omega_{l}$ reduces the amplitude of expected
astrophysical signals, so it is better to consider rings with either non-
relativistic protons or sizes even larger than the LHC.

  2. 2.

One can improve the time resolution of the measurement. Using electro-optical
sampling as in other storage rings [18] could improve $\Delta T/T$ by one
order of magnitude.

  3. 3.

Decreasing the energy of the protons also goes in the direction of increasing
$\tau_{l}$. Maybe one can envision operating a storage ring in stable
conditions for days or even weeks.

The three above points indicate future avenues of investigation that might
lead to the detection of GWs at storage rings. Another interesting option is
to make the protons as freely-falling as possible, i.e., $\omega_{l}\to 0$. In
this limit the wave can act coherently on $T$ for a very long time
$1/\omega_{g}\simeq 1/\omega_{l}$. A calculation neglecting the RF system was
presented in another contribution to this workshop (see the talk by Suvrat
Rao). This suggests that in this regime mHz GWs might be detected at a storage
ring. However the problem of keeping the protons in a stable orbit in these
conditions remains open at the time of writing this article.

In conclusion, from the very simple estimates presented here, measuring GWs
using particle motion inside storage rings is quite challenging. However, I
consider the existence of a conceptual path towards GW-level sensitivity, as
outlined in this Section, interesting and deserving of further study.

###  2.7 Gravitational Synchrotron Radiation from Storage Rings
Pisin Chen

In this presentation, we show that relativistic charged particles executing
circular orbital motion, as in a storage ring, can emit gravitational waves
through two channels. One is the gravitational synchrotron radiation (GSR)
emitted directly by the massive particle; the other is ‘resonant conversion’,
i.e., the Gertsenshtein effect, which, in this case, converts the
electromagnetic synchrotron radiation (EMSR) to GWs. It is shown that the
dominant frequency of the direct GSR is its fundamental mode, i.e.,
$\omega_{0}=c/\rho$, where $\rho$ is the radius of the storage ring. In the
case of CERN LHC, $\omega_{0}\sim 10^{4}-10^{5}$ Hz. The dominant frequency of
resonant EMSR-GSR conversion is a factor $\gamma^{3}$ higher, and for LHC it
is around $\omega_{c}=\gamma^{3}\omega_{0}\sim 10^{13}$ Hz, with wavelength at
$\sim\mu$m, which, if detected, would be the first observation of gravitons.
Since the observation point can be located near the storage ring, we find that
the spacetime metric perturbation can be $h_{\rm GSR}\sim n_{b}N\times
10^{-31}\sim 10^{-18}$ for the LHC. More details can be found in Refs. [19,
20].

Open questions and/or planned next steps: The deduced scaling law for the
spacetime perturbation, $h\sim(\gamma^{2}m_{p}^{2}M_{\rm Pl}^{2})(\rho/R)$,
requires further confirmation. Under what conditions can the bunch train be
treated as radiating GSR coherently? Is there a near-field effect in GSR and,
if so, is that more favorable in terms of detection? Would the heavy ion mode
in LHC be preferable for GSR production?

Additional Comments: The author would be most eager to collaborate with
colleagues to push forward an experiment at CERN to generate and detect the
direct GSR and the resonant EMSR-GSR conversion signals.

###  2.8 On Gravitational Synchrotron Radiation
J. M. Jowett

Gravitational synchrotron radiation from the future LEP, LHC and SSC beams
were discussed among a number of people at CERN in the late 1980s, shortly
before LEP started operation. It was realised that these beams would be among
the most powerful terrestrial sources of gravitational radiation, although the
total radiated power would still only be of order $10^{-25}\,{\rm W}$.

This presentation revisited the paper [21] which treated the subject in
parallel to the familiar treatment of incoherent synchrotron radiation. The
numerical results of [21] were reproduced and extended to the beams in the HL-
LHC and FCC-hh.

The question had been raised whether the heavy-ion beams in hadron colliders
could be a more powerful source. It was shown that a Pb nucleus radiates 167
times more power than a proton in the same magnetic field, the same
$Z^{6}/m^{4}$ scaling as for photons. However the total power is smaller
because the number of Pb ions in HL-LHC or FCC-hh is about a factor 1000
smaller.

The relation between these results, based on coherence conditions from [21],
and [19], which considered that all particles in the ring radiate coherently,
requires some clarification.

###  2.9 Revisiting Gravitational Wave Detection in an SRF Cavity
A. Berlin, R. T. D’Agnolo, S. A. R. Ellis

Since the discovery of gravitational waves (GWs) by the LIGO collaboration
[22], there has been much renewed interest in approaches to their detection
(see, for example, the recent proposals [23, 24, 25, 26, 27, 28, 29, 30]).
Many of these proposals seek to detect low-frequency (sub-10 Hz) GWs, where
standard astrophysical sources are expected aplenty. However, less focus has
been placed on high-frequency GWs, in the kHz$-$MHz range. The reasons for
this include the expectation that no black holes of stellar origin can produce
such high-frequency signals, and that the strength of the signal decreases
significantly at higher frequency, making detection ever more difficult [17,
29]. However, proposals have existed for many years to detect high-frequency
GWs [14, 31, 32, 33], and there has recently been a renewed interest in new
techniques, driven by advances in quantum sensing technology, and by
suggestions of potential sources [34, 28, 29].

In the late 1970s, proposals to search for GWs using superconducting radio-
frequency (SRF) cavities was made by Pegoraro, Picasso and Radicati (PPR) [14,
31] and by Caves [32]. Their idea was that a cavity would be prepared with a
mode split into two nearly-degenerate modes in a symmetric and anti-symmetric
configuration, and one of the two modes would be pumped, while the other would
be kept quiet. Gravitational waves passing through the detector would deform
the cavity walls, thereby transferring power from the pumped mode into the
quiet mode if the GW frequency $\omega_{G}$ was matched to the splitting of
the two cavity modes, $\omega_{G}\simeq|\omega_{a}-\omega_{s}|$, where the
subscripts denote the symmetric ($s$) and anti-symmetric ($a$) modes. On this
basis, prototypes were constructed in the 1980s and early 2000s, culminating
in the MAGO collaboration [33, 35, 36, 37, 38]. The approach was abandoned,
however, due to political decisions prioritizing other GW searches where known
astrophysical sources exist [39].

Due to the renewed theoretical and experimental interest in high-frequency
GWs, and advances in SRF technology, we have revisited the use of carefully-
controlled SRF cavities to detect gravitational waves. Similar to the PPR
approach, the idea would be to design a cavity such that a low-order mode is
split into a symmetric and anti-symmetric mode, to pump one of the two, and
look for a signal in the quiet mode. However, contrary to the PPR approach, we
are interested in detecting GWs through the inverse Gertsenshtein effect (IGE)
[40], whereby GWs convert to photons in a background electromagnetic field.
The reason for considering this signal, rather than the mechanical one of PPR,
is that mechanical vibrations can be a significant noise source at low mode
separations $|\omega_{a}-\omega_{s}|\ll\omega_{a,s}$ [41, 42], as well as
being a transducer of a possible signal. Because of the potential for noise
vibrations to cause significant cavity frequency drift, we propose to use a
precise active feedback system to control the cavity frequency drift to be
smaller than the cavity bandwidth, $\Delta\omega\sim\omega/Q$. This active
feedback would not only damp mechanical noise, but also any mechanically-
induced GW signal, thereby necessitating the study of the IGE signal from GWs.
Recently, the DarkSRF collaboration at FNAL has achieved $\Delta\omega\sim
0.1$ Hz in a cavity operating at $\omega\sim$ GHz, providing the proof of
concept that such control can be achieved with current SRF technology [43].

Employing the approach of searching for the GWs converting the pumped mode
into the quiet mode via the IGE we expect, based on preliminary calculations,
that the strain sensitivity should be on the order of $h\sim 10^{-22}$ for a
thermal noise-limited device operating at $\omega\sim 100$ MHz. The exact
details of the computation will be found in a forthcoming publication [44].

Future directions of this research include establishing an experimental
collaboration to build and operate the SRF cavity, as well as conducting
theoretical work on potential sources at high frequencies. Experimental
activities might be combined with a similar approach to the search for axion
dark matter, which employs similar SRF technology [41, 42].

###  2.10 An atom interferometer to search for ultralight dark matter and
gravitational waves at CERN?
O. Buchmueller and J. R. Ellis

We outline the scientific opportunities of a proposed programme using ground-
and space-based Atom Interferometry experiments to search for ultra-light dark
matter, to explore gravitational waves in the mid-frequency range between the
peak sensitivities of the LISA and LIGO/Virgo/ KAGRA/INDIGO/Einstein
Telescope/Cosmic Explorer experiments, and to probe other frontiers in
fundamental physics. This programme would complement other planned searches
for dark matter, as well as probe mergers involving intermediate-mass black
holes and explore early-universe cosmology.

#### The AION Project

The Atom Interferometer Observatory and Network (AION) is a project to explore
ultra-light dark matter (ULDM) and mid-frequency gravitational waves (GWs)
[25] that has recently received initial funding of about £9.6M from the UK
Quantum Technology for Fundamental Physics Programme and other sources. It is
being executed in national partnership with the UK National Quantum Technology
Hub in Sensors and Timing in Birmingham and in international partnership with
the MAGIS Collaboration and the Fermi National Laboratory in the US [45].

The basic principle of an atom interferometer is illustrated in Fig. 4 [46]. A
laser splits a cloud of atoms into ground and excited states that follow
different trajectories because of the momentum kicks they have undergone.
After following different space-time paths, they exhibit an interference
pattern upon recombination. The same laser beam strikes two or more clouds,
and waves of ULDM or passing GWs can generate differences between their
interference patterns.

Figure 4: The principle of atom interferometry [46]: blue lines are atoms in
the ground state, red dashed lines are atoms in the excited state and the wavy
lines are laser pulses; and a typical interference pattern.

Stage 1 of AION is to build and commission a 10m detector using atom
interferometry, developing existing technology and exploring possible
infrastructure for a 100m that would be Stage 2. In addition to building,
commissioning and exploiting the 100m detector, Stage 2 would also carry out a
design study for a follow-on km-scale detector, which is planned as Stage 3.
AION was selected in 2018 by STFC as a high-priority medium-scale project, and
will work in partnership with MAGIS in the US [45] to form a “LIGO/Virgo-
style” network and collaboration. Stage 1 is to be located in the Oxford
University Physics Department, while Stage 2 could be placed either at a UK
national facility in Boulby or Daresbury, or possibly at CERN. Stage 4 of AION
is a long-term objective to orbit a pair of satellite detectors in medium
Earth orbit called AEDGE [47], which was proposed to ESA in response to its
Voyage2050 call for possible mission concepts. Members of AION took the lead
in proposing AEDGE and developing its science case, bringing together
collaborators from European and Chinese groups including those working on the
MIGA [48], MAGIA [49], ELGAR [27] and ZAIGA [50] projects. Stages 3 and 4 will
probably both require funding on an international level via the EU, ESA, etc.,
and the AION collaboration has already started to build the foundation for
such an international effort.

AION Stage 1 will be built on the basement level of the new purpose-built
building of the Oxford Physics Department, which has 14.7m headroom. The site
has stable concrete construction and space for a temperature-controlled laser
laboratory. In addition to benefiting from this world-class infrastructure,
AION will receive strong engineering support from the nearby STFC Rutherford-
Appleton Laboratory (RAL). The focus during the first 30 months of the project
will be on establishing the prerequisites for the 10m detector, including a
network of ultracold strontium atom sources and laser laboratories in
Birmingham, Cambridge, Imperial College, Oxford and RAL, and training early-
career scientists in the necessary expertise. These laboratories are expected
to be fully operational in late 2021. In parallel, the final design for 10m
detector will be developed, and preparations made for its construction and
subsequent physics exploitation. As already mentioned, this development phase
will be in partnership with the MAGIS experiment in the US, which is to be
installed in a vertical access shaft for the Fermilab neutrino beam. Several
of the other atom interferometer projects, such as the MIGA and ELGAR projects
in Europe and the ZAIGA project in China, are being prepared on similar
timescales.

Figure 5: One possible location of the 100m Stage 2 of AION at CERN is in the
PX46 access shaft to the LHC.

Possible locations for the 100m Stage 2 of AION include a tower at the STFC
Daresbury Laboratory and a shaft in the STFC Boulby underground laboratory. A
third possibility is one of the LHC access shafts, namely PX46, as illustrated
in Fig. 5. Its height (143m) and diameter (10.1m) are ideal for AION, and the
technical infrastructure available at CERN is very attractive. First radiation
studies have also been quite promising, but more work is needed to determine
if PX46 could be a valid option for Stage 2 of AION. We are currently working
with the CERN Physics Beyond Colliders team on a feasibility study, in
particular for the arrangement of shielding that would be required if the LHC
were to suffer a catastrophic beam loss near Point 4 while people are working
in PX46.

#### Science Objectives

The main physics goal of Stage 1 will be the search for ULDM in the form of
some scalar field with a mass around $10^{-15}-10^{-14}$ eV, exploring new
domains of parameter space and complementing other searches, as illustrated in
Fig. 6 [25]. Possible extensions to pseudoscalar and vector dark matter fields
are also under study. The search for ULDM will continue with subsequent stages
of AION, and will be accompanied by growing sensitivities to GWs in the mid-
frequency band between the peak sensitivities of LISA (at $\sim 10^{-2}$ Hz)
and LIGO, Virgo, KAGRA and the proposed Einstein Telescope (ET) (at $\sim
10^{2}$ Hz), as illustrated in Fig. 7 [47]. Prime targets as sources of mid-
frequency GWs include the mergers of black holes, cosmological phase
transitions and cosmic strings.

Figure 6: The sensitivities of various AION stages to ULDM couplings to
electrons (left), photons (middle) and through a Higgs portal (right) [25]
Figure 7: The sensitivities of various AION stages to gravitational wave
strain as functions of frequency, compared with the sensitivities of LIGO,
LISA and ET, indicating also the signals expected for mergers of black holes
of various masses at different redshifts $z$ [47]. Figure 8: The estimated
signal-to-noise ratios attainable with AION 100m for mergers of black holes of
masses $M$ at different redshifts $z$. [25]

High-priority astrophysical questions to be investigated by studies of GWs in
the mid-frequency band include how supermassive black holes in galactic
centres were formed, e.g., by mergers of intermediate-mass black holes. The
capabilities of AION 100m for observing such mergers are shown in Fig. 8.
There will also be synergies with LISA and LIGO/Virgo/KAGRA/ET observations
through observations over long periods that make possible early warnings and
localizations of future mergers. These will also make possible sensitive
probes of fundamental physics, such as the possible mass of the graviton and
violation of Lorentz invariance by GWs [51].

Other topics beyond the Standard Model (SM) of particle physics include the
possible observation of GWs generated during a first-order phase transition
during the early Universe. Among the extensions of the SM that AION could
study are additional $H^{6}/\Lambda^{2}$ interactions and extra $U(1)_{B-L}$
gauge bosons [25]. These scenarios also have prospective collider signatures
that are complementary to those from GWs. AION could also search for evidence
of a network of cosmic strings [52], as has recently been hinted by the
NANOGrav pulsar timing array [53].

Other possible topics in fundamental physics include high-precision
measurements of the Newton constant and the gravitational redshift, probes of
Bell inequalities, the equivalence principle and the constancy of fundamental
parameters, and searching for fundamental (i.e., not environmental)
decoherence. Installing Stage 2 of AION at CERN would add many novel features
to CERN’s programme in Physics Beyond Colliders.

##  3 Final Workshop Discussion
Moderated by J. R. Ellis

The final discussion centred on the following five main topics:

  1. 1.

The proposal by K. Oide (Section 2.3) of GW detection by resonant betatron
oscillations in a storage ring, sensitive in the 10 kHz range;

  2. 2.

A variant of the proposal by S. Rao (Section 2.4) and R. T. D’Agnolo (Section
2.6) of GW detection through the change in revolution period, but using a
“low-energy” coasting ion beam without RF, aiming at sensitivity down to
$10^{-5}$ Hz;

  3. 3.

The proposal by P. Chen (Section 2.7) (discussed also by J. Jowett (Section
2.8) of gravitational synchrotron radiation generated by the beam at
frequencies close to the orbital frequency, $\sim 10^{4}$ Hz for LHC. Section
2.7 also discussed the Gertsenshtein effect, whereby electromagnetic
synchrotron radiation is converted to GWs of much higher frequency;

  4. 4.

The suggestion by S. Ellis and collaborators (Section 2.9) of heterodyne GW
detection using SRF, which could be sensitive up to $\sim 10^{7}$ Hz;

  5. 5.

The possibility suggested by O. Buchmueller and J. Ellis (Section 2.10) of
using an LHC access shaft to house a 100m atom interferometer targeting the 1
to $10^{-2}$ Hz range.

Figure 9: The landscape of the GW frequency spectrum, illustrating the ranges
in which various astrophysical and cosmological sources are expected, and the
sensitivities of some present and planned experiments. Also shown are the
frequency ranges targeted by some of the proposals discussed during this
Workshop.

Figure 9 illustrates the ways in which these suggestions might complement the
frequency coverage of existing, planned and proposed GW experiments. We now
review some of the points that emerged during discussions of these
suggestions.

Proposal 1 is aimed at transverse beam perturbations at frequencies around the
orbital frequency. Key issues include the design of the storage ring, the
suppression of noise sources and the development of beam position monitors
with nanometre precision that do not degrade the beam characteristics through
impedance effects. It was suggested in the Discussion that the CERN AD and
ELENA facilities could be used to study possible noise sources.

Similar issues arise in connection with Proposal 2 for studying longitudinal
perturbations of the beam, which might be able to access part of the gap in
frequencies $\lesssim 10^{-3}$ Hz where LISA loses some sensitivity. However,
there are issues concerning the long-term stability of a beam coasting in the
absence of RF and the development of picosecond time resolution. The
suggestion of using a low-energy ion beam emerged during the Workshop, and
will be explored by CERN accelerator physicists as well as by the authors of
Ref. [4].

Concerning the generation of gravitational synchrotron radiation by a
circulating beam (Proposal 3), there is not yet a consensus on the strength of
this GW source, which is likely to be very small but merits urgent
clarification. It would peak at frequencies above those where the
sensitivities of LIGO and similar laser interferometers are reduced and
astrophysical sources may be weaker.

Proposal 4 for the detection of high-frequency GWs using an SRF cavity could
benefit from synergies with the suggestion [41, 42] to use similar cavities to
search for axions, which is currently being discussed at SLAC and CERN.

Proposal 5 is aimed at the frequency gap between LIGO/Virgo at $\gtrsim 100$
Hz and LISA at $\lesssim 10^{-2}$ Hz. The requisite atom interferometer
technology is being developed by the MAGIS Collaboration in the US as well as
by the AION Collaboration in the UK and other groups in Europe and China. The
AION Collaboration has chosen the Oxford Physics Department as the site for
its precursor 10m experiment, and is currently considering one of the LHC
access shafts for its 100m successor. The possibility of being hosted by
CERN’s excellent infrastructure is very attractive, though other
considerations may affect the choice of site. The radiation protection issue
at CERN is under study, and it would be interesting to extend the LHC seismic
measurements shown in Fig. 2 to the LHC access shaft of interest.

One interesting aspect common to Proposals 4 and 5 is that they could/would be
located close to the LHC beam pipe, where they could benefit from any near-
field enhancement of a gravitational synchrotron radiation signal generated by
the beam. For the same reason, the LHC underground areas might also be
interesting locations for other experiments searching for high-frequency GWs
[29].

We recall in this connection that the classical gravitational perturbation
generated by a relativistic particle source is well understood [54], and can
be applied to bunches of particles in a storage ring, as discussed recently in
[55]. This reference considers various possible detection schemes, including a
monolithic pendulum, and it would be interesting to extend these studies to
other detectors that could be located close to the LHC beams, such as
Proposals 4 and 5 and those reviewed in [29].

Many of the Proposals discussed during this Workshop involve elements of
speculation, are at relatively early stages of development, and require
considerable conceptual and technical development. That said, the prospects of
using a storage ring as a GW detector or source are fascinating, and the
opportunities for using CERN’s technical infrastructure to add GW physics to
its Physics Beyond Colliders programme [56] are too interesting to be ignored.

## Acknowledgements

This work was supported in part by the European Commission under the HORIZON
2020 project ARIES no. 730871. The work of O.B. and J.E. was supported in part
by STFC grant ST/T00679X/1 to the AION project, and the work of J.E. was also
supported in part by STFC grant ST/T000759/1 and by Estonian Research Council
grant MOBTT5. The work of S.R, M.B. and J.L. was was supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy—EXC 2121 “Quantum Universe”—390833306.

## References

  * ARIES WP6 Workshop: Storage Rings and Gravitational Waves “SRGW2021” () [2 February to 18 March 2021] ARIES WP6 Workshop: Storage Rings and Gravitational Waves “SRGW2021” (2 February to 18 March 2021).  https://indico.cern.ch/event/982987/.  Accessed: 3 April 2021.
  * Collette et al. [2010] C. Collette, K. Artoos, M. Guinchard, and C. Hauviller.  Seismic response of linear accelerators.  _Phys. Rev. ST Accel. Beams_ , 13:072801, 2010.  doi: 10.1103/PhysRevSTAB.13.072801.
  * Misner et al. [1973] Charles William Misner, Kip S Thorne, and John Archibald Wheeler.  _Gravitation; 1st ed._ Freeman, San Francisco, CA, 1973.  URL http://cds.cern.ch/record/100593.
  * Rao et al. [2020] Suvrat Rao, Marcus Brüggen, and Jochen Liske.  Detection of gravitational waves in circular particle accelerators.  _Phys. Rev. D_ , 102(12):122006, 2020.  doi: 10.1103/PhysRevD.102.122006.
  * [5] R.T. D’Agnolo.  Storage rings as antennas for Gravitational waves?, EuCARD-2 XBEAM Strategy Workshop, Valencia, Spain, February 2017.  https://indico.cern.ch/event/587477/contributions/2464930/.  Accessed: 3 April 2021.
  * [6] R.T. D’Agnolo.  Update on Theoretical Effects of Gravitational Waves on Storage Rings, ARIES APEC Workshop: Storage Rings and Gravitational Waves ”SRGW2021” in virtual space, Feb. – Mar. 2021 (this workshop).  https://indico.cern.ch/event/982987/contributions/4197058/.  Accessed: 3 April 2021.
  * Dong [2002] D. Dong.  Study of Charged Particle Storage Ring as Detector of Gravitational Waves.  _eConf_ , C0211115:030, 2002.
  * [8] Daniel Zer-Zion.  On the detection of gravitational waves through their interaction with particles in storage rings, CERN-EP-98-063.
  * [9] A. N. Ivanov, A. P. Kobushkin, and M. Wellenzohn.  Storage rings as detectors for relic gravitational-wave background ?  arXiv:gr-qc/0210091.
  * [10] J. W. van Holten.  Cyclotron motion in a gravitational wave background.  arXiv:gr-qc/9906117.
  * Faber and Suda [2018] Manfried Faber and Martin Suda.  Influence of gravitational waves on circular moving particles.  _J. Mod. Phys._ , 9(4):651–668, 2018.  doi: 10.4236/jmp.2018.94045.
  * Schutz [1999] Bernard F. Schutz.  Gravitational wave astronomy.  _Class. Quant. Grav._ , 16:A131–A156, 1999.  doi: 10.1088/0264-9381/16/12A/307.
  * [13] Curt Cutler and Kip S. Thorne.  An Overview of gravitational wave sources.  In _16th International Conference on General Relativity and Gravitation (GR16)_.  doi: 10.1142/9789812776556˙0004.
  * Pegoraro et al. [1978a] F. Pegoraro, E. Picasso, and L. A. Radicati.  On the Operation of a Tunable Electromagnetic Detector for Gravitational Waves.  _J. Phys. A_ , 11:1949–1962, 1978a.  doi: 10.1088/0305-4470/11/10/013.
  * P. Baudrenghien, T. Mastoridis [2012] P. Baudrenghien, T. Mastoridis.  Performances and Future Plans of the LHC RF.  In _Proceedings of the Advanced Beam Dynamics Workshop on High-Intensity and High-Brightness Hadron Beams, Beijing, China, September 17-21, 2012_ , 2012.
  * Maggiore [2007] Michele Maggiore.  _Gravitational Waves. Vol. 1: Theory and Experiments_.  Oxford Master Series in Physics. Oxford University Press, 2007.  ISBN 978-0-19-857074-5, 978-0-19-852074-0.
  * Maggiore [2000] Michele Maggiore.  Gravitational wave experiments and early universe cosmology.  _Phys. Rept._ , 331:283–367, 2000.  doi: 10.1016/S0370-1573(99)00102-7.
  * Casalbuoni et al. [2008] S. Casalbuoni, H. Schlarb, B. Schmidt, P. Schmuser, B. Steffen, and A. Winter.  Numerical studies on the electro-optic detection of femtosecond electron bunches.  _Phys. Rev. ST Accel. Beams_ , 11:072802, 2008.  doi: 10.1103/PhysRevSTAB.11.072802.
  * Chen [1994] Pisin Chen.  Resonant photon-graviton conversion in EM fields: from earth to heaven.  In _1st International Conference on Phenomenology of Unification: from Present to Future_ , SLAC-PUB-6666, 1994.
  * Chen [1995] Pisin Chen.  Resonant photon - graviton conversion and cosmic microwave background fluctuations.  _Phys. Rev. Lett._ , 74:634–637, 1995.  doi: 10.1103/PhysRevLett.74.634.  [Erratum: Phys.Rev.Lett. 74, 3091 (1995)].
  * Diambrini Palazzi and Fargion [1987] G. Diambrini Palazzi and Daniele Fargion.  On Gravitational Radiation Emitted by Particles Circulating in High-energy Accelerators.  _Phys. Lett. B_ , 197:302–306, 1987.  doi: 10.1016/0370-2693(87)90388-1.
  * Abbott et al. [2016] B. P. Abbott et al.  Observation of Gravitational Waves from a Binary Black Hole Merger.  _Phys. Rev. Lett._ , 116(6):061102, 2016.  doi: 10.1103/PhysRevLett.116.061102.
  * Graham et al. [2013] Peter W. Graham, Jason M. Hogan, Mark A. Kasevich, and Surjeet Rajendran.  A New Method for Gravitational Wave Detection with Atomic Sensors.  _Phys. Rev. Lett._ , 110:171102, 2013.  doi: 10.1103/PhysRevLett.110.171102.
  * El-Neaj et al. [2020a] Yousef Abou El-Neaj et al.  AEDGE: Atomic Experiment for Dark Matter and Gravity Exploration in Space.  _EPJ Quant. Technol._ , 7:6, 2020a.  doi: 10.1140/epjqt/s40507-020-0080-0.
  * Badurina et al. [2020] L. Badurina et al.  AION: An Atom Interferometer Observatory and Network.  _JCAP_ , 05:011, 2020.  doi: 10.1088/1475-7516/2020/05/011.
  * Dimopoulos et al. [2008] Savas Dimopoulos, Peter W. Graham, Jason M. Hogan, Mark A. Kasevich, and Surjeet Rajendran.  An Atomic Gravitational Wave Interferometric Sensor (AGIS).  _Phys. Rev._ , D78:122002, 2008.  doi: 10.1103/PhysRevD.78.122002.
  * Canuel et al. [2020] B. Canuel et al.  ELGAR—a European Laboratory for Gravitation and Atom-interferometric Research.  _Class. Quant. Grav._ , 37(22):225017, 2020.  doi: 10.1088/1361-6382/aba80e.
  * [28] Nancy Aggarwal, George P. Winstone, Mae Teo, Masha Baryakhtar, Shane L. Larson, Vicky Kalogera, and Andrew A. Geraci.  Searching for new physics with a levitated-sensor-based gravitational-wave detector.  arXiv:2010.13157.
  * [29] Aggarwal, N. and others.  Challenges and Opportunities of Gravitational Wave Searches at MHz to GHz Frequencies.  arXiv:2011.12414.
  * [30] Nicolas Herman, André Füzfa, Sébastien Clesse, and Léonard Lehoucq.  Detecting Planetary-mass Primordial Black Holes with Resonant Electromagnetic Gravitational Wave Detectors.  arXiv:2012.12189.
  * Pegoraro et al. [1978b] F. Pegoraro, L. A. Radicati, P. Bernard, and E. Picasso.  Electromagnetic Detector for Gravitational Waves.  _Phys. Lett. A_ , 68:165–168, 1978b.  doi: 10.1016/0375-9601(78)90792-2.
  * Caves [1979] C. M. Caves.  Microwave Cavity Gravitational Radiation Detectors.  _Phys. Lett. B_ , 80:323–326, 1979.  doi: 10.1016/0370-2693(79)90227-2.
  * Reece et al. [1984] C. E. Reece, P. J. Reiner, and A. C. Melissinos.  Observation of $4\times 10^{-17}$ cm Harmonic Displacement Using A 10-GHz Superconducting Parametric Converter.  _Phys. Lett. A_ , 104:341–344, 1984.  doi: 10.1016/0375-9601(84)90811-9.
  * Caprini and Figueroa [2018] Chiara Caprini and Daniel G. Figueroa.  Cosmological Backgrounds of Gravitational Waves.  _Class. Quant. Grav._ , 35(16):163001, 2018.  doi: 10.1088/1361-6382/aac608.
  * [35] P. Bernard, A. Chincarini, G. Gemme, R. Parodi, and E. Picasso.  A Detector of gravitational waves based on coupled microwave cavities.  arXiv:gr-qc/020304.
  * [36] R. Ballantini, P. Bernard, E. Chiaveri, A. Chincarini, G. Gemme, R. Losito, R. Parodi, and E. Picasso.  A detector of high frequency gravitational waves based on coupled microwave cavities.  _Class. Quant. Grav._ , pages 3505–3522.  doi: 10.1088/0264-9381/20/15/316.
  * Bernard et al. [2001] P. Bernard, G. Gemme, R. Parodi, and E. Picasso.  A Detector of small harmonic displacements based on two coupled microwave cavities.  _Rev. Sci. Instrum._ , 72:2428–2437, 2001.  doi: 10.1063/1.1366636.
  * [38] R. Ballantini et al.  Microwave apparatus for gravitational waves observation.  arXiv:gr-qc/0502054.
  * [39] Personal communication from G. Gemme.
  * Gertsenshtein [1961] M. Gertsenshtein.  Wave resonance of light and gravitational waves.  _JETP_ , 41:113–114, July 1961.
  * Berlin et al. [2020] Asher Berlin, Raffaele Tito D’Agnolo, Sebastian A. R. Ellis, Christopher Nantista, Jeffrey Neilson, Philip Schuster, Sami Tantawi, Natalia Toro, and Kevin Zhou.  Axion Dark Matter Detection by Superconducting Resonant Frequency Conversion.  _JHEP_ , 07(07):088, 2020.  doi: 10.1007/JHEP07(2020)088.
  * [42] Asher Berlin, Raffaele Tito D’Agnolo, Sebastian A. R. Ellis, and Kevin Zhou.  Heterodyne Broadband Detection of Axion Dark Matter.  arXiv:2007.15656.
  * Grassellino [2020] Anna Grassellino.  SRF-based dark matter search: Experiment.  2020\.  URL https://indico.physics.lbl.gov/event/939/contributions/4371/attachments/2162/2915/DarkSRF-Aspen-2.pdf.
  * [44] Asher Berlin, Raffaele Tito D’Agnolo, and Sebastian A. R. Ellis.  To appear.
  * [45] Peter W. Graham, Jason M. Hogan, Mark A. Kasevich, Surjeet Rajendran, and Roger W. Romani.  Mid-band gravitational wave detection with precision atomic sensors.  arXiv:1711.02225.
  * Arvanitaki et al. [2018] Asimina Arvanitaki, Peter W. Graham, Jason M. Hogan, Surjeet Rajendran, and Ken Van Tilburg.  Search for light scalar dark matter with atomic gravitational wave detectors.  _Phys. Rev._ , D97(7):075020, 2018.  doi: 10.1103/PhysRevD.97.075020.
  * El-Neaj et al. [2020b] Yousef Abou El-Neaj et al.  Aedge: Atomic experiment for dark matter and gravity exploration in space.  _EPJ Quantum Technology_ , 7(1):6, 2020b.  doi: 10.1140/epjqt/s40507-020-0080-0.  URL https://doi.org/10.1140/epjqt/s40507-020-0080-0.
  * Canuel et al. [2018] B. Canuel et al.  Exploring gravity with the MIGA large scale atom interferometer.  _Sci. Rep._ , 8(1):14064, 2018.  doi: 10.1038/s41598-018-32165-z.
  * [49] G. T. Tino et al.  Ultracold atoms and precision measurements.  http://coldatoms.lens.unifi.it.
  * Zhan et al. [2019] Ming-Sheng Zhan et al.  ZAIGA: Zhaoshan Long-baseline Atom Interferometer Gravitation Antenna.  _Int. J. Mod. Phys._ , D28:1940005, 2019.  doi: 10.1142/S0218271819400054.
  * Ellis and Vaskonen [2020] John Ellis and Ville Vaskonen.  Probes of gravitational waves with atom interferometers.  _Phys. Rev. D_ , 101(12):124013, 2020.  doi: 10.1103/PhysRevD.101.124013.
  * Ellis and Lewicki [2021] John Ellis and Marek Lewicki.  Cosmic String Interpretation of NANOGrav Pulsar Timing Data.  _Phys. Rev. Lett._ , 126(4):041304, 2021.  doi: 10.1103/PhysRevLett.126.041304.
  * Arzoumanian et al. [2020] Zaven Arzoumanian et al.  The NANOGrav 12.5 yr Data Set: Search for an Isotropic Stochastic Gravitational-wave Background.  _Astrophys. J. Lett._ , 905(2):L34, 2020.  doi: 10.3847/2041-8213/abd401.
  * Aichelburg and Sexl [1970] P. C. Aichelburg and R. U. Sexl.  The gravitational field of a massless particle.  _Lett. Nuovo Cim._ , 4S1:1316–1318, 1970.  doi: 10.1007/BF02753774.
  * [55] Felix Spengler, Dennis Rätzel, and Daniel Braun.  Perspectives of measuring gravitational effects of laser light and particle beams.  arXiv:2104.09209.
  * Alemany et al. [arXiv:1902.00260] R. Alemany et al.  Summary Report of Physics Beyond Colliders at CERN.  arXiv:1902.00260.

##  Appendix A Workshop Programme and Schedule

As coordinators of ARIES WP6 and its task 6.6, Giuliano Franchetti, Marco
Zanetti and Frank Zimmermann jointly co-chaired this workshop. The SRGW2021
programme had been established with the help of an International Scientific
Programme Committee consisting of William A. Barletta (MIT), Pisin Chen (NTU),
Raffaele Tito D’Agnolo (IPhT), Raffaele Flaminio (LAPP), Giuliano Franchetti
(GSI), Shyh-Yuan Lee (Indiana U.), Katsunobu Oide (CERN & KEK), Qing Qin (ESRF
& U. Peking) Jörg Wenninger (CERN), Marco Zanetti (U. Padova) and Frank
Zimmermann (CERN). The detailed SRGW2021 workshop schedule with sessions,
presentations and speakers is presented in Table 1.

Table 1: SRGW2021 workshop sessions, session chairs, talks, and speakers Session 1 | Introduction to Gravitational Waves and | Pisin Chen, NTU (Chair)
---|---|---
2/2/21 | their Effects |
| Welcome and Introduction | Frank Zimmermann, CERN
| A Brief History of Gravitational Waves | Jorge Cervantes Cota, ININ, Mexico
| Expected Sources of Gravitational Waves | Bangalore Sathyaprakash, Pennsylvania
|  | State U., USA, & Cardiff U., UK
| Response of storage-ring beam to gravitational | Katsunobu Oide, KEK & CERN
| waves - preliminary considerations |
| Discussion |
Session 2 | Measurements and sensitivity | Shyh-Yuan Lee, Indiana U. (Chair)
2/18/21 |  |
| Measurement approach and sensitivities of | Raffaele Flaminio, IN2P3 LAPP
| detectors like LIGO and VIRGO |
| Storage ring sensitivity to tides & large-scale | Jörg Wenninger, CERN
| perturbations \- examples from LEP & LHC |
| Discussion |
Session 3 | Proposals and Schemes | Jörg Wenninger, CERN (Chair)
3/4/21 |  |
| Detection of gravitational waves in circular | Suvrat Rao, Hamburg U.
| particle accelerators — a proposal for the LHC |
| Storage rings as detectors for relic | Andrei Ivanov, Vienna U.
| gravitational-wave background? |
| Update on Theoretical Effects of Gravitational | Raffaele Tito D’Agnolo, IPhT
| Waves on Storage Rings |
| Discussion |
Session 4 | GW generation & detection | Frank Zimmermann, CERN (Chair)
3/11/21 |  |
| Radiofrequency cavities & gravitational | Sebastian Ellis, IPhT
| wave signals? |
| Using storage rings as a GW source | Pisin Chen, NTU
| GSR, some history revisited, and FCC-hh | John Jowett, GSI
| Use of atom-interferometry for possible GW | Oliver Buchmueller, ICL
| detection & other gravity experiments | & John Ellis, KCL
| Discussion |
Session 5 | Summary and Outlook | Giuliano Franchetti, GSI (Chair)
3/18/21 |  |
| Ground Vibration at SSRF Site | Rongbing Deng, SARI Shanghai
|  Contributed: What Governs Flow of Energy? | Peter Cameron, BNL (ret.)
| Questions on Grav. Impedance Matching |
| Concluding Discussion | Moderated by John Ellis, KCL


# Towards a fairer reimbursement system for burn patients using cost-sensitive
classification

Chimdimma Noelyn Onah*
Richard Allmendinger
Julia Handl  University of Manchester, UKOxford RoadManchesterUnited Kingdom
<EMAIL_ADDRESS>and  Ken W. Dunn  University Hospital
South Manchester, UKSouthmoor RoadWythenshaweUnited Kingdom
<EMAIL_ADDRESS>
(2021)

###### Abstract.

The adoption of the Prospective Payment System (PPS) in the UK National Health
Service (NHS) has led to the creation of patient groups called Health Resource
Groups (HRG). HRGs aim to identify groups of clinically similar patients that
share similar resource usage for reimbursement purposes. These groups are
predominantly identified based on expert advice, with homogeneity checked
using the length of stay (LOS). However, for complex patients such as those
encountered in burn care, LOS is not a perfect proxy of resource usage,
leading to incomplete homogeneity checks. To improve homogeneity in resource
usage and severity, we propose a data-driven model and the inclusion of
patient-level costing. We investigate whether a data-driven approach that
considers additional measures of resource usage can lead to a more
comprehensive model. In particular, a cost-sensitive decision tree model is
adopted to identify features of importance and rules that allow for a focused
segmentation on resource usage (LOS and patient-level cost) and clinical
similarity (severity of burn). The proposed approach identified groups with
increased homogeneity compared to the current HRG groups, allowing for a more
equitable reimbursement of hospital care costs if adopted.

Health Resource Groups, Casemix, Patient Segmentation, XAI in Healthcare,
Interpretable Healthcare

††copyright: acmcopyright††journalyear: 2021††doi:
10.1145/nnnnn.n††conference:  Proceedings of DSHealth ’21: KDD Workshop on
Applied Data Science for Healthcare; August 14–18, 2021; DSHealth,
Virtual††booktitle:  Proceedings of DSHealth ’21: KDD Workshop on Applied Data
Science for Healthcare, August 14–18, 2021, DSHealth, Virtual††price:
15.00††isbn: 978-1-4503-XXXX-X/18/06††ccs: Applied computing Health
informatics

##  1\. Introduction

The National Health Service (NHS) serves a broad UK population with varied
demographic and medical histories. An acceleration in medical advances has led
to a greater scope of treating chronic illness (Naylor et al., 2015). In turn,
this burden the NHS with rising costs and with limitation of access to funds,
these have fuelled the popularity and the adoption of Prospective Payment
System (PPS) over retrospective systems. PPS is a reimbursement method that
allows for the predetermination of a fixed amount to be allocated to health
providers for the treatment of patients with a specific diagnosis. The overall
aim is to ensure an equitable and fair distribution of funds. It also acts as
an instrument to understand the organisational activity, such as the type of
patient cared for and treatment delivered. The system adopted in the UK is
called Health Resource Groups (HRGs). HRGs are groupings of clinically similar
patients which use comparable healthcare resources. Although these groups are
generated using patient-level data, the grouping rules are typically generated
by transcribing expert advice into if-else rules meant to capture differing
patient severity and length of stay (LOS).

These HRGs should be clinically meaningful groups of diagnoses and
interventions that consume similar levels of NHS resources. However, the aim
of having homogeneous groups in terms of resource usage might not be met due
to: (i) The methodology’s dependence on expert advice, which carries the risk
of ignoring less established factors that account for specific patient
subgroups’ complexity; (ii) the validation of identified groups using LOS
only, an imperfect indicator of resource usage (Street et al., 2012); (iii)
the determination of payment rate as an average of the cost of each HRG. This
is, in essence, a generalisation that will exacerbate the effect of any non-
homogeneous groups. Non-homogeneity in the resource usage of the identified
groups may lead to a significant disadvantage to small and highly specialised
services that, by necessity, operate at very high expenditure. Burn services
are one such service – they are small, rely on specialist equipment and
intervention, deal with various case complexity and stay open regardless of
the number of patients admitted, with a minimum staff number constantly on the
rota (NHS England, 2018).

We investigate if the current burn HRG can be improved while remaining
interpretable, by using a cost-sensitive decision tree model. Our approach
aims to create a fairer system by identifying more homogeneous groups of burn
patients, thus ensuring a consistent cost reimbursement for the provision of
similar patient care. To validate the methodology, we use data of actual burn
patients. We show that the proposed approach can identify more homogeneous
patient groups, enabling a more equitable and fair reimbursement.

##  2\. COHORT AND DATA DESCRIPTION

The current burn care pathway is designed to treat paediatrics separately from
adults due to evidence of young age as a significant complicator of burn
injury (National Burn Care Review Committee, 2001). Further evidence was found
in (Onah et al., 2019) that different factors and resource requirements
characterise adult and young burn patients. Thus, any developed approach
should be customised for adult and child patients. This paper focuses on
identifying homogeneous groups of young (¡16 years old) burn patients only, by
using anonymised patient-level data from all burn services in England and
Wales.

The information of patients that attended a burn service is collected by
clinicians and nurses, from the first contact with a burn service to
rehabilitation and any late reconstruction procedure (Stylianou et al., 2015).
The dataset excludes burn injury patients who died before reaching a burn
service and those with minor injuries cared for in the community or hospitals
that are not specialist burn services. The data covers 2003 to 2019,
comprising just over 18,000 young patients and was extracted from the
international Burn Injury Database (iBID) (International Burn Injury Database,
2014). An internal iBID patient-level costing methodology (Duncan and Dunn.,
2020) is applied to generate the cost of the burn patients.

##  3\. ANALYTICAL PIPELINE

In healthcare, adopted models are potentially responsible for human life and,
thus, there is a need for high confidence from clinicians on the results of a
model. Therefore, it is imperative that adopted models are explainable. An
explainable model, such as a decision tree, allows result tracing, increases
transparency and facilitates model improvement (Pawar et al., 2020). For
instance, offering a clear explanation of why a specific burn patient is
placed in a higher severity group than another would increase the trust level
of clinicians. Thus, we propose a cost-sensitive decision tree model, trained
and tested to evaluate model performance on unseen data.

A decision tree classifier was chosen as it allows the identification of rules
generated for classification. To increase homogeneity, our paper introduces a
key feature: the allocation of misclassification costs between classes. We
chose a cost-sensitive decision tree that ensures that even with a
misclassification, the predicted class is in close proximity (by cost, LOS and
severity) to the expected patient class. This model enables model penalisation
as the distance of the class predicted compared to the actual class increases
(Domingos, 1999). This distance is identified by ranking classes by the
average value of LOS, cost and total burn surface area (TBSA). The classifier
was implemented with the mlr (Bischl et al., 2016) and rpart (Therneau and
Atkinson., 2019) packages in R, with a customised cost measure.

In summary, the analytical pipeline adopted to build this proposed model
includes data preprocessing, identifying target features and variables of
importance, and building the final model.

###  3.1. Data Preprocessing

With the need to compare our adopted model results to the current HRG groups,
access to the HRG labels for each patient is needed. However, due to non-
access to the Health Episode Statistics (HES) dataset typically used to create
HRGs and thus the inability to directly merge the iBID dataset to the HES, it
is essential to recreate the HRG burn methodology using the iBID dataset. This
was done in close collaboration with the NHS National Casemix office. The
first step was to replace all missing values with 0. The assumption is that
fields are left empty when the value is 0, no or not applicable (also adopted
by the current HRG methodology) (NHS Digital, 2020). The next step was the
identification of iBID variables or a proxy to variables used in the HES
dataset. The HRG replication was then done using if-else rules created by the
Casemix team and adapted by an NHS Casemix expert to suit the iBID dataset. We
expect 13 classes to be generated for the young burn patient subgroup, ranked
from most severe to least severe.

The next data preprocessing step involves removing all cases grouped into the
unclassifiable group, as these cases have no TBSA or depth of burn recorded
for any of the 27 potential burn sites (parts of the body), which is a pre-
requisite (as stated in the burn HRG methodology) for reimbursement as a burn
case. Cases with LOS greater than 360 and cost greater than £1,000,000 are
considered outliers by the iBID data administrators and thus removed. The data
preprocessing also includes removing irrelevant variables (administrative
variables, variables with high missing values, variables with unstructured
data, variables with one unique value and variables which are duplicates of
another).

###  3.2. Identification of Target Features and Variables of Importance

The proposed model is a decision tree (DT) classifier, a supervised model that
requires ‘true’ labels. Thus, the target classes (true labels) were generated
using three critical factors in burn care – LOS, cost and TBSA. These factors
were identified from previous analysis and the literature and represent the
aim of HRGs to create clinically meaningful groups (burn severity) that have
similar resource usage (LOS and cost of treatment). These factors were first
log-transformed to reduce skew, followed by k-means cluster analysis to
segment each factor into 13 target classes. The identified classes were ranked
from most severe/costly to least severe/costly (to allow for adequate
comparison with the current HRGs). With these, three cost-sensitive DT models
(LOS, TBSA and cost DT) were built using the ranked classes to identify the
variables of importance from each model. The target class ranking is necessary
to enable comparison with the ranked HRG and the incorporation of a cost-
sensitive model.

###  3.3. Final Decision Tree (DT) Model

Here, we describe the final model for comparison with the current HRG and
extraction of classification rules. The 13 target classes for this model are
derived using k-means cluster analysis. The feature space for the cluster
analysis is computed by calculating the average rank across all LOS, TBSA and
cost DT target classes as identified using the approach described in Section
3.2. Adopting the mean rank of each case allows for the identification of
classes that reflect the average severity, cost and length of hospital stay.

The independent variables are those identified as important from the LOS, TBSA
and cost DT. The dataset is then split into a train and test set. These
subsets are oversampled independently by duplicating cases in the minority
classes. The oversampling is needed to ensure the adopted model has enough
samples to learn the decision boundary effectively.

The model used is a cost-sensitive DT model, trained and tested to evaluate
model performance on unseen data. The cost-sensitive DT model is used to
increase model penalisation as the distance of the class predicted compared to
the actual class increases.

##  4\. Results

This section presents the results of the data-driven model generated using a
cost-sensitive DT. We first evaluate the engineered target classes generated
to ensure it reflects homogeneity in severity and resource usage. Figure 1
shows a scatterplot of LOS, TBSA, and Cost DT model ranked target classes,
where class 1 depicts least severe/costly, and class 13 depicts most
severe/costly, against that of the Final DT target classes (also ranked). The
plotted Final DT target classes reflects, as expected, homogeneity in severity
and resource usage. It shows that the target classes have a decreasing LOS,
cost, and TBSA as its severity (reflected by the Final DT ranked classes)
decreases.

Figure 1. Spread of ranked LOS, Cost and TBSA model target classes - each dot
represents a patient’s LOS, cost and TBSA group and the colour represents the
Final DT group.

Shows the distribution of LOS/TBSA/Cost DT by the final DT target classes

Comparing the HRGs with the generated target classes, we show histograms of
intra-group variance (in terms of cost, TBSA and LOS) of the DT and HRG model
to evaluate how much homogeneity can be found. Figure 2 illustrates a lower
intra-group variance in the DT groups than to the HRG groups for the train
data. In particular, we found that, on average, the cost intra-group variance
(0.4) for DT groups is three times lower than the HRG groups cost intra-group
variance (1.5). This pattern is seen for the LOS intra-group variance - 0.8
for DT groups and 1.7 for HRG groups. As expected, a lower difference when we
compare the average severity (TBSA) intra-group variance of DT groups at 0.5
and that of the HRG groups at 0.6. Evaluating these intra-group variances on
the train data is needed to ensure the engineered classes meet the homogeneity
criteria. These evaluations indicate that the new target classes provide
higher intra-group similarity in terms of the three critical factors than the
HRGs. Thus, these target classes are adopted.

Figure 2. DT and HRG intra-group variance - LOS, Cost and TBSA - on train
data. The value on each legend is the average Cost, LOS and TBSA intra-group
variance for the DT and HRG groups.

This shows the cluster variance measured on LOS, TBSA and cost.

Figure 3. Cost matrix representing the penalty used in the cost-sensitive DT
to ensure any misclassifications are close to the true class.

This shows the cost of misclassification

With these promising results, it is imperative to understand the rules that
allow the classification of cases into each engineered class and evaluate the
model’s performance on unseen cases. This is done by identifying the feature
space, which are the variables of importance identified from the TBSA, LOS and
Cost DT classifiers. Then the creation of a cost matrix for penalisation. The
cost matrix is shown in Figure 3, with the rows indicating true and the
columns predicted class labels. The diagonal zero values indicate a cost of 0
for accurate prediction, with an increase in cost as predicted class deviation
from the true class increases. This is added to the Final DT model pipeline
with a train and test split to evaluate performance.

Figure 4. Confusion matrix representing the impact of penalisation on the DT
model, using the test data, this shows that all misclassifications are at most
3 neighbours away from the true class.

This shows the cost of misclassification

The confusion matrix shown in Figure 4 reveals good classification of unseen
cases (test data). The confusion matrix shows that most of the cases are
predicted as the precise class. Where the classification is wrong, the
misclassified class is in close proximity to the true class. Except for class
8, with most cases predicted as class 9 and some cases misclassified three
classes away. This higher misclassification of class 8 suggests the likely
need to reduce the total number of classes in the young patient subgroup. The
model identifies class 8 and 9 as both made up of cases with TBSA ¿=5 but
differentiated by the LOS. Class 8 are those with LOS ¡ 2.5 while class 9 are
patients with LOS ¿= 2.5 (see Figure 7).

Figure 5. DT and HRG groups by LOS, Cost and TBSA on test data. Ordered in
decreasing order of injury severity.

The image shows a boxplot of critical factor distribution in the HRG and DT
groups.

Figure 6. DT and HRG intra-group variance - LOS, Cost and TBSA - on test data.
The value on each legend is the average LOS, Cost and TBSA intra-group
variance for the DT and HRG groups.

This shows the predicted DT versus HRG intra-group variance measured on LOS,
TBSA and cost.

Figure 5 illustrates the difference in the homogeneity of the HRG groups and
the DT groups using a boxplot to show the distribution of TBSA, LOS and cost
in each group. The groups are ordered by severity/resource usage, with class
13 representing HRG and DT groups with the most severe cases and high resource
usage. Class 1 represents the groups with the least severe and least costly
cases in both the HRG and DT groups. In both, as expected, we see a gradual
reduction in these three factors as severity and resource usage decreases.
Notably, this is more apparent in the DT groups, which also have shorter
boxplots indicating higher homogeneity across DT groups. Comparing the cluster
variance (using LOS, Cost and TBSA) of the predicted groups to the HRG groups
of the same cases, we confirm this higher homogeneity. As seen in Figure 6 on
average, the DT groups have a lower intra-group variance compared to the HRG
groups.

Exploring the classification rules as shown in Figure 7, the adopted model
does well in producing an explainable result. The classification tree
illustrates the criteria for adding a case to a particular group. Class 1’s
(the least severe/costly group) case membership are those with LOS ¡ 1 and
TBSA ¡ 1.6 while class 13 (the most severe/costly group) are made up of cases
with LOS ¿=11 and TBSA ¿= 19. We can also identify the three most important
features as LOS, TBSA, and total theatre visits from the classification tree.

Figure 7. Classification tree showing explainable classification rules
generated from the cost-sensitive DT model.

This is a view of the decision tree

##  5\. Conclusion and Future Work

The adopted model creates a classification rule that is easy to explain and
less complex than the current HRG methodology. This meets the HRG requirement
that groups are clinically relevant to allow for organisational activities.
Additional, our adopted model meets the requirements of group homogeneity in
terms of resource usage and severity. The resultant groups from the adopted DT
model evidenced increased homogeneity across the three critical factors –
length of hospital stay, cost to the hospital and severity of burn, compared
to the current HRG. Thus, our identified groups are better suited for cost
reimbursement.

Our future work includes the customisation of the cost matrix to reflect a
non-linear and non-monotonic penalty (e.g. actual monetary cost) for
misclassification and applying this analytical pipeline to adult burn
patients.

## References

  * (1)
  * Naylor et al. (2015) Chris Naylor, Candace Imison, Rachael Addicott, David Buck, Nick Goodwin, Tony Harrison, Shilpa Ross, Lara Sonola, Yang Tian, and Natasha Curry. 2015\.  _Transforming our health care system: ten priorities for commissioners_.  Technical Report. The King’s Fund.  www.expertpatients.co.uk/
  * Street et al. (2012) Andrew Street, Conrad Kobel, Thomas Renaud, and Josselin Thuilliez. 2012.  How Well Do Diagnosis-Related Groups Explain Variations In Costs Or Length Of Stay Among Patients And Across Hospitals? Methods For Analysing Routine Patient Data.  _Health Economics_ 21 (8 2012), 6–18.  https://doi.org/10.1002/hec.2837
  * NHS England (2018) NHS England. 2018\.  _NHS England Emergency Preparedness, Resilience and Response Concept of Operations for managing Mass Casualties_.  Technical Report. NHS England.  www.england.nhs.uk/ourwork/eprr
  * National Burn Care Review Committee (2001) National Burn Care Review Committee. 2001\.  _National Burn Care Review Committee Report Standards and Strategy for Burn Care: A Review Of Burn Care In The British Isles_.  Technical Report. National Burn Care Review Committee.  https://www.britishburnassociation.org/national-burn-care-review/
  * Onah et al. (2019) Chimdimma Noelyn Onah, Richard Allmendinger, Julia Handl, Paraskevas Yiapanis, and Ken W. Dunn. 2019\.  A Clustering-Based Patient Grouper for Burn Care.  In _Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)_. Vol. 11872 LNCS. Springer, Switzerland, 123–131.  https://doi.org/10.1007/978-3-030-33617-2{_}14
  * Stylianou et al. (2015) Neophytos Stylianou, Iain Buchan, and Ken W Dunn. 2015\.  A review of the international Burn Injury Database (iBID) for England and Wales: descriptive analysis of burn injuries 2003–2011.  _BMJ Open_ 5, 2 (2 2015), 1–10.  http://bmjopen.bmj.com/content/5/2/e006184.abstract
  * International Burn Injury Database (2014) International Burn Injury Database. 2014\.  _International Burn Injury Database Annual Report ( 2014 / 15 )_.  Technical Report. International Burn Injury Database.  http://www.ibidb.org/
  * Duncan and Dunn. (2020) Robert T Duncan, and Ken W Dunn. 2020\.  Burn service costing using a mixed model methodology.  _Burns: journal of the International Society for Burn Injuries_ 46, 3 (3 2020), 520–530.  https://doi:10.1016/j.burns.2020.02.010
  * Pawar et al. (2020) Urja Pawar, Donna O’Shea, Susan Rea, and Ruairi O’Reilly. 2020\.  Explainable AI in Healthcare. In _2020 International Conference on Cyber Situational Awareness, Data Analytics and Assessment, Cyber SA 2020_. Institute of Electrical and Electronics Engineers Inc., Dublin, 1–2.  https://doi.org/10.1109/CyberSA49311.2020.9139655
  * Domingos (1999) Pedro Domingos. 1999\.  MetaCost. In _Proceedings of the fifth ACM SIGKDD international conference on Knowledge discovery and data mining - KDD ’99_. ACM Press, New York, New York, USA, 155–164.  https://doi.org/10.1145/312129.312220
  * Bischl et al. (2016) Bernd Bisch, Michel Lang, Lars Kotthoff, Julia Schiffner, Jakob Richter, Erich Studerus, Giuseppe Casalicchio and Zachary M. Jonesr. 2016.  mlr: Machine Learning in R.  _Journal of Machine Learning Research_ 17, 1–5.  https://jmlr.org/papers/v17/15-066.html
  * Therneau and Atkinson. (2019) Bernd Bisch, Terry Therneau and Beth Atkinson. 2019.  rpart: Recursive Partitioning and Regression Trees.  https://CRAN.R-project.org/package=rpart
  * NHS Digital (2020) NHS Digital. 2020\.  _Chapter Summaries: HRG4+ 2020/21 Local Payment Grouper Version 2 (COVID-19)_.  Technical Report July. NHS Digital. 70–81 pages.  https://doi.org/10.4324/9781315187747-2