EPJ Web of Conferences Lattice2017  11institutetext: Institute for Theoretical
Physics, UniversitÃt Regensburg, D-93040 Regensburg, Germany 22institutetext:
Department of Theoretical Physics, Tata Institute of Fundamental Research,
Homi Bhabha Road, Mumbai 400005, India

#  $\eta$ and $\eta^{\prime}$ masses and decay constants

Gunnar Bali 1122 Sara Collins 11 Jakob Simeth Speaker, 11<EMAIL_ADDRESS>
###### Abstract

We present preliminary results for the masses and decay constants of the
$\eta$ and $\eta^{\prime}$ mesons using CLS $N_{f}=2+1$ ensembles.

One of the major challenges in these calculations are the large statistical
fluctuations due to disconnected quark loops. We tackle these by employing a
combination of noise reduction techniques which are tuned to minimize the
statistical error at a fixed cost.

On the analysis side we carefully assess excited states contributions by using
a direct fit approach.

##  1 Introduction

The $\eta$ and $\eta^{\prime}$ play an important role in many QCD processes
and are directly connected to the chiral anomaly of QCD. Their electromagnetic
transition form factors are of great phenomenological interest and have been
studied theoretically, e.g., using light cone sum rules Agaev:2014wna .
However, little is known about their wavefunctions and the quark mass
dependence of singlet-octet mixing and first-principles determinations are
needed. On the lattice, the $\eta$ and $\eta^{\prime}$ mesons are difficult to
study. Nonetheless, quite some progress has been made, e.g., in
Kuramashi:1994aj ; Struckmann:2000bt ; McNeile:2000hf ; Lesk:2002gd ;
Christ:2010dd ; Dudek:2011tt ; Gregory:2011sg ; Michael:2013vba ; Bali:2014pva
.

The fact that the $\eta$ and $\eta^{\prime}$ states are not flavour
eigenstates presents a challenge. The mass eigenstates are mixtures of flavour
singlet and octet states, i.e.,

$\mid\eta\rangle=\cos\theta\mid\eta_{8}\rangle-\sin\theta\mid\eta_{1}\rangle\text{,}\qquad\mid\eta^{\prime}\rangle=\sin\theta^{\prime}\mid\eta_{8}\rangle+\cos\theta^{\prime}\mid\eta_{1}\rangle\text{,}$
(1)

where the angles $\theta$ and $\theta^{\prime}$ will in general depend on the
choice of $\mid\eta_{1}\rangle$ and $\mid\eta_{8}\rangle$, e.g., on the scale
and the smearing applied.

This can also be expressed in matrix form,

$\begin{pmatrix}\mathcal{C}_{\eta}&0\\\
0&\mathcal{C}_{\eta^{\prime}}\end{pmatrix}=U\begin{pmatrix}\mathcal{C}_{88}&\mathcal{C}_{81}\\\
\mathcal{C}_{18}&\mathcal{C}_{11}\end{pmatrix}U^{T}\text{,
where}\quad\mathcal{C}_{ij}=\langle P_{i}(x)\mid
P_{j}(y)\rangle\text{,}\quad\text{and}\quad
U(\theta,\theta^{\prime})=\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta^{\prime}&\phantom{-}\cos\theta^{\prime}\end{pmatrix}$ (2)

is a non-unitary rotation matrix to allow for possible gluonic and other
excited states contributions. $\mathcal{C}_{ij}$ with $i,j\in\\{8,1\\}$ are
correlators connecting pseudoscalar octet and singlet operators, respectively:

$P_{8}=\frac{1}{\sqrt{6}}\left(\bar{u}\gamma_{5}u+\bar{d}\gamma_{5}d-2\bar{s}\gamma_{5}s\right)\quad\text{and}\quad
P_{1}=\frac{1}{\sqrt{3}}\left(\bar{u}\gamma_{5}u+\bar{d}\gamma_{5}d+\bar{s}\gamma_{5}s\right)\text{.}$
(3)

When performing the Wick contractions for the matrix elements
$\mathcal{C}_{ij}$, another difficulty emerges: disconnected loops appear
which are inherently noisy and expensive to compute on the lattice. Efficient
solvers for the Dirac equation and noise reduction techniques are mandatory to
obtain decent signals.

In these proceedings we attempt a determination of the masses and leading
distribution amplitudes of the $\eta$ and $\eta^{\prime}$ mesons, refining our
techniques on two of the many existing $N_{f}=2+1$ CLS ensembles Bruno:2014jqa
with non-perturbatively improved Wilson fermions at $\beta=3.4$, corresponding
to a lattice spacing $a\approx 0.0864\,\mathrm{fm}$ (determined using
$\sqrt{8\,t_{0}}=0.415\,\mathrm{fm}$ Bruno:2016plf ). The first of the two,
ensemble U103, is at the $N_{f}=3$ symmetric point with $m_{\pi}=m_{K}\approx
415\,\textrm{MeV}$. The other, H105, is along the same line of constant
average quark mass, with $m_{\pi}\approx 287\,\textrm{MeV}$ and $m_{K}\approx
487\,\textrm{MeV}$.

##  2 Computation of disconnected loops

The entries $\mathcal{C}_{ij}$ of the matrix in Eq. (2) are given by

$\displaystyle\mathcal{C}_{88}=$
$\displaystyle\frac{1}{3}\left(C_{\ell\bar{\ell}}+2C_{s\bar{s}}-2D_{\ell\bar{\ell}}+2D_{\ell\bar{s}}+2D_{s\bar{\ell}}-2D_{s\bar{s}}\right)$
(4) $\displaystyle\mathcal{C}_{18}=\mathcal{C}_{81}=$
$\displaystyle\frac{\sqrt{2}}{3}\left(C_{\ell\bar{\ell}}-C_{s\bar{s}}-2D_{\ell\bar{\ell}}+2D_{\ell\bar{s}}-D_{s\bar{\ell}}+D_{s\bar{s}}\right)$
(5) $\displaystyle\mathcal{C}_{11}=$
$\displaystyle\frac{1}{3}\left(2C_{\ell\bar{\ell}}+C_{s\bar{s}}-4D_{\ell\bar{\ell}}-2D_{\ell\bar{s}}-2D_{s\bar{\ell}}-D_{s\bar{s}}\right)\text{,}$
(6)

and are linear combinations of connected and disconnected pseudoscalar
correlators,

$\displaystyle C^{P}_{f\bar{f}}(\delta
t)=\operatorname{tr}\left(\gamma_{5}S^{-1}_{f}(\delta
t)\gamma_{5}S^{-1}_{f}(0)\right)\text{,}$ (7) $\displaystyle
D^{P}_{f\bar{g}}(\delta t)=\operatorname{tr}\left(\gamma_{5}S^{-1}_{f}(\delta
t)\right)\operatorname{tr}\left(\gamma_{5}S^{-1}_{g}(0)\right)\text{,}$ (8)

with light and strange quark flavours $f,g\in\\{\ell,s\\}$ and $S^{-1}_{f}$
being the quark propagator of flavour $f$. While the connected correlators can
be cheaply computed by exploiting the $\gamma_{5}\text{-hermiticity}$ of the
Dirac operator (we use one point source per configuration), the disconnected
loops $L_{f}(t)=\operatorname{tr}\gamma_{5}S^{-1}_{f}(t)$ must be estimated
stochastically. We do so by inverting the Dirac equation on time-diluted
$\mathbb{Z}_{2}\otimes i\mathbb{Z}_{2}$ sources, where a distance of $4a$
between non-zero time slices was found to be optimal. Additionally, we employ
the hopping parameter expansion to improve the signal-to-noise ratio of the
loops by exploiting $L_{f}(t)=\operatorname{tr}\Gamma
S^{-1}_{f}(t)=\kappa_{f}^{k}\operatorname{tr}\Gamma D^{k}S^{-1}_{f}(t)$, where
$D$ here is the hopping part in the Dirac operator
$S_{f}=\mathds{1}+\kappa_{f}D$ and $k=2$ for $\Gamma=\gamma_{5}$ and $k=4$ for
$\Gamma=\gamma_{\mu}\gamma_{5}$ are the maximum number of applications that
can be used for our action. See Bali:2009hu  and references therein for more
details on these noise reduction techniques.

It turned out that 96 stochastic estimates for each quark flavour and time
dilution are also sufficient to obtain reasonable signals for the axialvector
loop, that is needed for the extraction of the decay constants. When
performing this many solves per configuration, it is crucial to employ a
modern solver. We use a multigrid solver that has been optimized specifically
for the KNL architecture qp3 .

##  3 Extraction of physical states

At the $N_{f}=3$ flavour symmetric point where $S^{-1}_{\ell}=S^{-1}_{s}$ the
elements of the correlator matrix, Eqs. (4) to (6), simplify leading to a
diagonal matrix: In this limit, the $\eta$ is a pure octet state and equal in
mass to the pion, whereas the $\eta^{\prime}$ is entirely a singlet:

$\displaystyle\mathcal{C}_{88}(\delta t)=$
$\displaystyle\mathcal{C}_{\eta}(\delta t)=\mathcal{C}_{\pi}(\delta t),$ (9)
$\displaystyle\mathcal{C}_{11}(\delta t)=$
$\displaystyle\mathcal{C}_{\eta^{\prime}}(\delta t)=\mathcal{C}_{\pi}(\delta
t)-3D(\delta t)\text{.}$ (10)

For sufficiently large times, where excited states can be neglected, both the
pion and the $\eta^{\prime}$ correlators are single-exponentials.
Consequently, the disconnected correlator is a double-exponential

$D(\delta t)=\frac{1}{3}\left(\mathcal{C}_{\pi}(\delta
t)-\mathcal{C}_{\eta^{\prime}}(\delta
t)\right)\to\frac{1}{3}\left(A_{\pi}\exp(-m_{\pi}\delta
t)-A_{\eta^{\prime}}\exp(-m_{\eta^{\prime}}\delta t)\right)\text{,}$ (11)

where $A_{\pi},A_{\eta^{\prime}}>0$. The (lighter) pion mass dominates at
large times over the heavy $\eta^{\prime}$, which leaves only a small window
for the extraction of the $\eta^{\prime}$ state: at very small times excited
states impede the extraction of the ground state while at large time
separations the relative error arising from the disconnected diagrams grows
rapidly and makes resolving the small difference to the pion correlator
impossible.

Figure 1:  left: Pseudoscalar connected and disconnected correlators are
shown, both for local sources and sinks ($n_{sm}=0$) and after $n_{sm}=125$
smearing iterations for U103. The correlators are scaled to match such that
they agree at large times. With no smearing excited states are clearly visible
both in the connected and the disconnected correlators. right: The same for
the light-light disconnected correlator of H105. The smeared correlators do
not exhibit significant excited states after $n_{sm}=75$ iterations.

We employ Wuppertal quark smearing Gusken:1989ad  both to the source and the
sink loops to enhance the ground over excited states. A small number of
smearing iterations turned out to be sufficient to suppress excited states and
achieve moderate errors in the final results. In particular for the
disconnected correlators comparably few iterations seem to be sufficient, see
Fig. 1.

The observations above are also valid away from the symmetric point. There,
the correlator matrix, Eq. (2), must first be diagonalized to extract the
physical states. This is usually done by solving the generalized eigenvalue
problem, yielding the physical correlators as eigenvalues. However, we
observed comparably large errors and unstable results when changing the
parameters of the method. Instead, we insert Eqs. (4) – (6) into Eq. (2) and
obtain a system of three linear equations that we solve for the three
independent disconnected correlators,
$D_{\ell\bar{\ell}},D_{\ell\bar{s}},\text{and}D_{s\bar{s}}$, yielding similar
expressions as at the symmetric point, Eq. (11). In the $N_{f}=2+1$ case, the
resulting formulae are more complex and depend on two angles $\theta$ and
$\theta^{\prime}$, two connected correlators for the pion and the $s\bar{s}$,
and the wanted $\eta$ and $\eta^{\prime}$ correlators. Plugging single-
exponentials into the correlators, we end up with a total of ten independent
parameters.

Figure 2:  left: Combined fit to the disconnected correlators of H105, using
information from a previous fit to the connected correlators. We obtain
typical values of $\chi^{2}/\mathrm{d.o.f.}\approx 1$ in correlated fits.
right: Effective masses for the $\eta$ and $\eta^{\prime}$, computed from
their reconstructed correlators, denoted
$a\,m^{\mathrm{eff}}_{\eta^{(\prime)}}$, and with the connected correlators
replaced by their fitted curves ($a\,m^{\mathrm{imp}}_{\eta^{(\prime)}}$),
along with the fitted mass values from the combined fit. Shaded regions
represent the errors of the fitted masses and the fit range of the
disconnected correlators. Connected correlators are fitted in the range
$\delta t/a\in[15,40]$. Excited states arising from the connected correlators
at earlier times are clearly visible.

We perform a combined fit to determine these parameters. The connected
correlators naturally do not depend on the disconnected ones and can be fitted
first, reducing the number of parameters to six. Being able to tune the fit
ranges independently we can fit the connected correlators in a region where we
are sure to be free of excited states contaminations whereas the disconnected
fits can start at smaller times to capture the heavy $\eta^{\prime}$ state
sufficiently well. At later times, the well determined connected correlators
constrain the fit. This approach is similar in spirit to the subtraction
method of Neff:2001zr ; Jansen:2008wv ; Michael:2013vba . Fig. 2 demonstrates
that the fit describes the data points and the fit errors actually decrease at
large times, confirming our expectations.

Note that in Eq. (2) we chose the octet-singlet basis, which is just one
particular choice and we can as well work in any other basis, e.g., in the
flavour basis, where

$\displaystyle\mid\eta\rangle=\frac{1}{\sqrt{2}}\cos\vartheta\mid\bar{u}\gamma_{5}u+\bar{d}\gamma_{5}d\rangle-\sin\vartheta\mid\bar{s}\gamma_{5}s\rangle$
(12)
$\displaystyle\mid\eta^{\prime}\rangle=\frac{1}{\sqrt{2}}\sin\vartheta^{\prime}\mid\bar{u}\gamma_{5}u+\bar{d}\gamma_{5}d\rangle+\cos\vartheta^{\prime}\mid\bar{s}\gamma_{5}s\rangle\text{,}$
(13)

with different angles $\vartheta^{(\prime)}$. In this case, the fit forms look
a bit simpler but the resulting correlators and masses are (numerically)
equal.

To remove any potential constant shifts from our (disconnected) correlators
stemming from an incomplete sampling of the topological sectors, we also fit
to

$-\hat{\partial}_{4}\mathcal{C}(\delta t)\approx\frac{\mathcal{C}(\delta
t-a)-\mathcal{C}(\delta t+a)}{2a}$ (14)

as suggested in Feng:2009ij ; Umeda:2007hy . We found that this reduces the
error and also stabilizes the result with respect to different fit ranges.

In order to further improve statistics in the mass estimates, and as a cross
check on systematic errors we also combine data at finite momentum and fit to
the dispersion relation

$aE(m,p)=\sqrt{a^{2}m^{2}+a^{2}p^{2}}\text{,}$ (15)

where $a^{2}p^{2}=\sum_{\mu}k^{2}_{\mu}\frac{4\pi^{2}}{L_{\mu}^{2}}$ is the
square of the lattice momentum, $L_{\mu}$ is the number of lattice points in
direction $\hat{\mu}$ and $-L_{\mu}/2\leq k_{\mu}<L_{\mu}/2$ are the integer
momentum components.

Figure 3:  left: Fits to the dispersion relation, Eq. (15), combining data at
several momenta. right: Summary plot of recent lattice determinations of the
$\eta$ and $\eta^{\prime}$ masses, cf. Christ:2010dd ; Gregory:2011sg ;
Dudek:2011tt ; Michael:2013vba .

From this fitting approach, we obtain for the U103 ensemble at the flavour
symmetric point,

$m_{\eta}=m_{\pi}=412(2)\,\mathrm{MeV},\quad
m_{\eta^{\prime}}=833(33)\,\mathrm{MeV}\text{,}$ (16)

and away from that point, for H105, at a pion mass of $282\,\mathrm{MeV}$,

$m_{\eta}=504(7)\,\mathrm{MeV},\quad
m_{\eta^{\prime}}=878(35)\,\mathrm{MeV}\text{.}$ (17)

Fig. 3 compares these results with other lattice determinations and the
physical point. It should be stressed that, like in the RQCD 2014 study
Bali:2014pva , the two considered ensembles are on a line of constant average
quark mass and approach the physical point on a different trajectory than
other ($m_{s}\approx\mathrm{const.}$) studies.

##  4 Decay constants

Having extracted masses and the angles, we can construct the correlators

$\mathcal{C}_{A^{i}\eta^{(\prime)}}(\delta t)=\langle 0\mid A^{i}_{\mu}(\delta
t)\mid\eta^{(\prime)}\rangle\text{,}\quad\mathcal{C}_{P^{i}\eta^{(\prime)}}(\delta
t)=\langle 0\mid P^{i}(\delta
t)\mid\eta^{(\prime)}\rangle\rangle\text{,}\quad\mathcal{C}_{\eta^{(\prime)}\eta^{(\prime)}}(\delta
t)=\langle\eta^{(\prime)}(\delta t)\mid\eta^{(\prime)}\rangle$ (18)

where $P^{i}$ labels the octet ($i=8$) and singlet ($i=1$) pseudoscalar local
currents, defined in Eq. (3) and $A^{i}_{\mu}$ similarly defines the
axialvector currents. Using these correlators, one can compute decay constants
which read

$\langle 0\mid
A_{\mu}^{i}\mid\eta^{(\prime)}(p)\rangle=if_{N}^{i}p_{\mu}\text{.}$ (19)

This results in four different effective decay constants at momentum $p=0$,

$f^{i,\mathrm{eff}}_{\eta^{(\prime)}}(\delta
t)=Z_{A}^{i}\frac{\sqrt{2}\mathcal{C}_{A^{i}\eta^{(\prime)}}(\delta
t)}{\sqrt{\mathcal{C}_{\eta^{(\prime)}\eta^{(\prime)}}(\delta
t)m_{\eta^{(\prime)}}}}\exp(m_{\eta^{(\prime)}}\delta t/2)\text{,}$ (20)

which are constant over some range in $\delta t$ where excited states can be
neglected and the correlators have not yet vanished in the noise.

For (partial) $\mathcal{O}(a)$ improvement, we replace

$\mathcal{C}_{A^{i}\eta^{(\prime)}}\to\tilde{\mathcal{C}}_{A^{i}\eta^{(\prime)}}=(1+b_{A}a\,m_{i})(\mathcal{C}_{A^{i}\eta^{(\prime)}}+ac_{A}\partial_{4}\mathcal{C}_{P^{i}\eta^{(\prime)}})\text{,}$
(21)

where $a\,m_{8}=\frac{a}{3}(m_{l}+2\,m_{s})$ and
$a\,m_{1}=\frac{a}{3}(2\,m_{l}+m_{s})$. We take the non-perturbatively
determined improvement coefficients $b_{A}$ from Korcyl:2016ugy , $c_{A}$ from
Bulava:2015bxa  and $Z_{A}^{8}$ from Bulava:2016ktf . For the axial
renormalization factor, the difference between $Z_{A}^{8}$ and $Z_{A}^{1}$ is
of $\mathcal{O}(\alpha_{s}^{2})$ and known perturbatively Constantinou:2016ieh
. For this lattice spacing and action it is at the level of 2 %, which is in
line with preliminary recent non-perturbative estimates Bali:2017jyw . From
these numbers, we estimate the systematic uncertainty from renormalization to
be around 5 %. In this work, we take the non-perturbative octet
renormalization factor together with the perturbative difference at a scale of
$2\,\mathrm{GeV}$.

Figure 4:  Effective decay constants as defined in Eq. (20) for H105. Shaded
regions indicate fit ranges and errors.

As can be seen in Fig. 4, we face the "window problem" again: The axialvector
loops are even noisier than the pseudoscalar ones so that the signal is lost
very early in Euclidean time. Also, being local at the sink, we need to take
special care of excited states. By fixing the masses to their previously
determined values we can take weighted averages ("one-parameter fits") to each
of the effective decay constants, tuning the ranges individually. The results
are compatible with Bali:2014pva , however, $f_{\eta^{\prime}}^{1}$ is quite a
bit larger, which might be due to insufficient treatment of excited states in
that channel.

It is common to give the decay constants in the flavour basis, which we
recover from the fitted values by taking the linear combinations at the scale
of interest, $\mu=2\,\mathrm{GeV}$,

$f_{\eta^{(\prime)}}^{l}=\sqrt{\frac{1}{3}}f_{\eta^{(\prime)}}^{8}+\sqrt{\frac{2}{3}}f_{\eta^{(\prime)}}^{1},\quad
f_{\eta^{(\prime)}}^{s}=-\sqrt{\frac{2}{3}}f_{\eta^{(\prime)}}^{8}+\sqrt{\frac{1}{3}}f_{\eta^{(\prime)}}^{1}\text{.}$
(22)

Usually, these are parameterized in terms of two constants and two angles,

$\begin{pmatrix}f_{\eta}^{l}&f_{\eta}^{s}\\\
f_{\eta^{\prime}}^{l}&f_{\eta^{\prime}}^{s}\end{pmatrix}=\begin{pmatrix}f_{l}\,\cos\theta_{l}&-f_{s}\,\sin\theta_{s}\\\
f_{l}\,\sin\theta_{l}&f_{s}\,\cos\theta_{s}\end{pmatrix}=U^{T}(-\theta_{l},-\theta_{s})\begin{pmatrix}f^{l}&0\\\
0&f^{s}\end{pmatrix}\text{,}$ (23)

which we extract simply via

$\displaystyle\theta_{l}=$
$\displaystyle\arctan\frac{f_{\eta^{\prime}}^{l}}{f_{\eta}^{l}}\text{,}$
$\displaystyle\quad\theta_{s}=$
$\displaystyle-\arctan\frac{f_{\eta}^{s}}{f_{\eta^{\prime}}^{s}}\text{,}$
$\displaystyle f_{l}=$
$\displaystyle\frac{f_{\eta}^{l}}{2\cos\theta_{l}}+\frac{f_{\eta^{\prime}}^{l}}{2\sin\theta_{l}}\text{,}$
$\displaystyle\quad f_{s}=$
$\displaystyle\frac{f_{\eta^{\prime}}^{s}}{2\cos\theta_{s}}-\frac{f_{\eta}^{s}}{2\sin\theta_{s}}\text{.}$
(24)

Additionally, we monitor unitarity violations by considering the difference of
the angles $\delta\theta=\theta_{l}-\theta_{s}$, which has a non-vanishing
value if there is a gluonic contribution to $f_{\eta^{(\prime)}}^{i}$ and is
related to the low-energy constant $\Lambda_{1}$ Feldmann:1999uf ,

$f_{l}f_{s}\sin(\theta_{l}-\theta_{s})=\frac{\sqrt{2}}{3}f_{\pi}^{2}\Lambda_{1}\text{.}$
(25)

For the two ensembles, we obtain for the decay constants

$\displaystyle\text{U103:}\quad f_{l}/f_{\pi}=$ $\displaystyle 1.20(7),$
$\displaystyle\quad f_{s}/f_{\pi}=$ $\displaystyle
1.27(2)\text{,}\qquad\text{H105:}\quad f_{l}/f_{\pi}=$ $\displaystyle
1.27(7),$ $\displaystyle\quad f_{s}/f_{\pi}=$ $\displaystyle 1.39(5)\text{,}$
(26)

and for the two angles and $\Lambda_{1}$

$\displaystyle\text{U103:}\quad\theta_{l}=$ $\displaystyle 61.3(1.8)\degree,$
$\displaystyle\quad\theta_{s}=$ $\displaystyle 65.4(1.6)\degree,$
$\displaystyle\quad\delta\theta=$ $\displaystyle-4.1(3.4)\degree,$
$\displaystyle\quad\Lambda_{1}=-0.23(17)\text{,}$ (27)
$\displaystyle\text{H105:}\quad\theta_{l}=$ $\displaystyle 59.7(1.6)\degree,$
$\displaystyle\quad\theta_{s}=$ $\displaystyle 66.9(1.3)\degree,$
$\displaystyle\quad\delta\theta=$ $\displaystyle-7.2(2.9)\degree,$
$\displaystyle\quad\Lambda_{1}=-0.47(17)\text{,}$ (28)

Within present statistics, we find $\theta_{l}\approx\theta_{s}$, i.e., we do
not see any strong OZI violation, however, the difference increases at the
H105 ensemble compared to the symmetric point. It will be interesting to see
if this becomes stronger at smaller quark masses.

Compared to phenomenology, the light decay constants have rather large values:

$\displaystyle\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Feldmann:1998vh,Feldmann:1998sh}{\@@citephrase{(}}{\@@citephrase{)}}}}\quad
f_{l}/f_{\pi}=$ $\displaystyle 1.07(2),$ $\displaystyle\quad f_{s}/f_{\pi}=$
$\displaystyle 1.34(6)\text{,}$
$\displaystyle\quad\theta_{l}=\theta_{s}=39.3(1.0)\text{,}$ (29)
$\displaystyle\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Escribano:2005qq}{\@@citephrase{(}}{\@@citephrase{)}}}}\quad
f_{l}/f_{\pi}=$ $\displaystyle 1.09(3),$ $\displaystyle\quad f_{s}/f_{\pi}=$
$\displaystyle 1.66(6)\text{,}$
$\displaystyle\quad\theta_{l}=\theta_{s}=40.7(1.4)\text{.}$ (30)

As already mentioned, this might be due to underestimated excited states, in
particular in the correlator $\mathcal{C}_{A^{i}\eta^{\prime}}^{1}$. Being
local at the sink, the chosen smearing might be insufficient to completely
remove them. It has also been shown Bruno:2016plf  that cutoff effects are
large, e.g., for the combination $\sqrt{8\,t_{0}}f_{\pi K}$ in particular at
this lattice spacing. It remains to be seen how the values change when
approaching the physical point, i.e., at finer lattices and smaller quark
masses.

##  5 Summary

In these proceedings, we employed several noise reduction techniques to obtain
disconnected loops at a precision that allows us to study the
$\eta/\eta^{\prime}$ system on two CLS ensembles.

To extract the physical states, we directly fit to the correlators. This, in
combination with quark smearing enables us to capture the small-t behaviour,
encoding the $\eta^{\prime}$ physics and still have excited states under
control. In a second fit we incorporate even noisier data from the axialvector
channel to determine decay constants directly. The results are encouraging,
however, a detailed analysis of systematic errors still remains to be done.

We plan to do so by analysing more CLS ensembles, following two distinct quark
mass trajectories. The combination of ensembles along the line of constant
average quark mass with ensembles where the strange quark mass is kept fixed,
as well as going towards lighter masses will allow for a controlled chiral
extrapolation. Also finite lattice spacing and volume effects will be
investigated.

##  6 Acknowledgments

This work was supported by the DFG SFB/TRR 55. We thank our colleagues in CLS
[http://wiki-zeuthen.desy.de/CLS/CLS] for the joint effort in the generation
of the gauge field ensembles which form a basis for the here described
computation. The authors gratefully acknowledge the Gauss Centre for
Supercomputing (GCS) for providing computing time through the John von Neumann
Institute for Computing (NIC) on the GCS share of the super-computer JUQUEEN
at JÃlich Supercomputing Centre (JSC). GCS is the alliance of the three
national supercomputing centers HLRS (UniversitÃt Stuttgart), JSC
(Forschungszentrum JÃlich), and LRZ (Bayerische Akademie der Wissenschaften),
funded by the German Federal Ministry of Education and Research (BMBF) and the
German State Ministries for Research of Baden-WÃrttemberg (MWK), Bayern
(StMWFK) and Nordrhein-Westfalen (MIWF).

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# Improving Automatic Endoscopic Stone Recognition Using a Multi-view Fusion
Approach Enhanced with Two-Step Transfer Learning

Francisco Lopez-Tiro1,2 Elias Villalvazo-Avila1 Juan Pablo Betancur-Rengifo1
Ivan Reyes-Amezcua3 Jacques Hubert4 Gilberto Ochoa-Ruiz1 Christian Daul2
1Tecnologico de Monterrey, School of Engineering and Sciences, Mexico
2CRAN UMR 7039, Université de Lorraine and CNRS, Nancy, France
3CINVESTAV, Guadalajara, Mexico
4CHRU Nancy, Service d’urologie de Brabois, Nancy, France

###### Abstract

This contribution presents a deep-learning method for extracting and fusing
image information acquired from different viewpoints, with the aim to produce
more discriminant object features for the identification of the type of kidney
stones seen in endoscopic images. The approach was specifically designed to
mimic the morpho-constitutional analysis to visually classify kidney stones by
jointly using surface and section images of kidney stone fragments. The model
was further improved with a two-step transfer learning approach and by
attention blocks to refine the learned feature maps. Deep feature fusion
strategies improved the results of single view extraction backbone models by
more than 6% in terms of accuracy of the kidney stones classification.

##  1 Introduction

The formation of kidney stones that cannot freely pass through the urinary
tract is a major public health issue [2, 15, 13]. In industrialized countries,
it has been reported that at least 10% of the population suffers from a kidney
stone episode once in their lifetime. In the United States alone, the risk of
relapse of the same type of kidney stone has increased by up to 40% [24, 27].
The formation of kidney stones is caused by different factors such as diet,
low fluid intake, and a sedentary lifestyle [25, 6]. However, there are other
unavoidable factors such as age, genetic inheritance, and chronic diseases
that increase the risk of forming kidney stones [12]. Therefore, methods for
identifying the different types of kidney stones are crucial for the
prescription of appropriate treatments and to reduce the risk of relapses [14,
12]. In order to carry out this identification in clinical practice, different
procedures have been developed, such as the Morpho-Constitutional Analysis
(MCA), and Endoscopic Stone Recognition (ESR) [5, 11].

MCA is commonly accepted as the standard procedure for determining the
different types of kidney stones (up to 21 different types and sub-types
including pure and mixed compositions are recognized during the MCA) [3]. MCA
consists of a double laboratory analysis of kidney stone fragments extracted
from the urinary tract during an ureteroscopy [7].

First, a biologist performs a visual inspection of the kidney stone which is
observed with a magnifying glass. This inspection aims to describe kidney
stones in terms of colors, textures, and morphology [3]. This visual analysis
is done both for the surface view (the external part of the kidney stone
fragment), and for a cross-section of the kidney stone fragment (the internal
stone part may consist of several layers surrounding a nucleus). Then, the
kidney stones are ground up and the resulting powder is used to perform a
biochemical analysis using a Fourier Transform Infrared Spectroscopy (FTIR)
[16]. The FTIR provides a detailed description of the chemical composition of
the kidney stone. Finally, the MCA analysis returns the type of kidney stone
through a detailed report of the biochemical and morphological characteristics
of both views of the kidney stone. However, MCA has some major drawbacks: the
results are often available only after several weeks, and it is difficult to
have a specialized team in each hospital to perform MCA.

Therefore, urologists have proposed, as a possible alternative, the Endoscopic
Stone Recognition (ESR) procedure in which the most common kidney stones are
visually identified on the video displayed on a screen during the ureteroscopy
itself [9]. However, this visual analysis of the surface and section views
requires a great deal of expertise due to the high similarities between
classes. Only a limited number of specialists have this expertise. In
addition, this technique is more operator dependent and subjective than MCA.
Therefore, new approaches based on deep-learning (DL) methods have been
proposed to automate and speed-up the kidney stone identification. Such
automated recognition can potentially assists urologists for a real-time
decision-making during an ureteroscopy.

(a)

(b)

Figure 1: Examples of ex-vivo kidney stone images acquired with (a) a CCD
camera and (b) an endoscope. SEC and SUR stand for section and surface views,
respectively. Table 1: Description of the two ex-vivo datasets.

Dataset A. M. DAUDON et al. [3] |  | Dataset B. J. EL-BEZE et al. [8]
---|---|---
Subtype | Main component (Key) | Surface | Section | Total |  | Subtype | Main component (Key) | Surface | Section | Total
Ia | Whewellite (WW) | 50 | 74 | 124 |  | Ia | Whewellite (WW) | 62 | 25 | 87
IVa1 | Carbapatite (CAR) | 18 | 18 | 36 |  | IIa | Weddelite (WD) | 13 | 12 | 25
IVa2 | Carbapatite (CAR2) | 36 | 18 | 54 |  | IIIa | Uric Acid (UA) | 58 | 50 | 108
IVc | Struvite (STR) | 25 | 19 | 44 |  | IVc | Struvite (STR) | 43 | 24 | 67
IVd | Brushite (BRU) | 43 | 17 | 60 |  | IVd | Brushite (BRU) | 23 | 4 | 27
Va | Cystine (CYS) | 37 | 11 | 48 |  | Va | Cystine (CYS) | 47 | 48 | 95
Number of images dataset A | 209 | 157 | 366 |  | Number of images dataset B | 246 | 163 | 409

This paper has two contributions: i) it proposes a novel DL-model for fusing
information included in endoscopic images of the two views (surface and
section) of a kidney stone fragment with the aim to increase the
discrimination performance and, ii) it shows how a multi-branch model can be
trained using a two-step transfer learning (TL) approach in order to improve
the model generalization capabilities.

This paper is organized as follows. Section 2 reviews the literature on
automated ESR and introduces the key concepts used in this work, namely multi-
view fusion and two-step TL. Section 3 describes the construction of the
dataset, details the two-step TL setup, and presents the pre-training stage of
the multi-view model. Section 4 compares the results obtained with the
proposed model in several configurations, with that of other models given in
previous works. Finally, section 5 discusses future research directions.

##  2 State-of-the-art

Different DL approaches for an automated classification of kidney stones
demonstrated encouraging results [17, 23]. However, DL-models require large
data amounts to yield accurate results [22, 21]. In ureteroscopy, it is
difficult to collect such large datasets. A solution to this issue lies in
methods such as TL and fine-tuning from other distributions (ImageNet) as a
weight initialization technique [18]. Such techniques also enable to avoid
training from scratch. However, for an automated endoscopic stone recognition
(aESR), these initialization techniques are not useful, since the distribution
of ImageNet and endoscopic (ureteroscopic) images are substantially different.
Thus, customized TL methods that initialize useful weights closer to the
target domain are required.

Furthermore, most models performing aESR were trained on surface or section
images taken separately [9, 10, 22]. However, the visual inspection in MCA (by
biologists) and ESR (by urologists) is based on both views by jointly
exploiting information from fragment surfaces and sections [3, 5, 4]. So far,
the DL-models in the literature did not use together surface and section
information to improve the classification efficiency. Multi-View (MV)
classification is exploited in this contribution to combine the features
observed in the two fragment-type views.

The aim of this paper is to show that an MV-model outperforms models without
an elaborated fusion strategy. MV is performed by fusing features (of shallow
models) or feature maps (for DL-models) determined for various images with the
aim to learn more complete representations and to obtain more effective
classifiers [28, 26]. Contrary to a MV-approach, previous works for aESR were
based on a DL-model, trained three times (only with section data, only for
surface data, and for surface and section data gathered in the same class).
This contribution leverages recent advances in DL-based models that combine
information from multiple viewpoints and improve the results using domain
adaptation techniques.

##  3 Materials and Methods

###  3.1 Datasets

Two kidney stone datasets were used in our experiments [3, 8]. According to
the dataset, the images were acquired either with standard CCD cameras or with
an ureteroscope (i.e., an endoscope). These datasets are described below.

Dataset A, [3]. This ex-vivo dataset of 366 CCD camera images (see, Figure
1(a), left column Table 1) is split in 209 surface and 157 section images, and
contains six different stone types sorted by sub-types denoted by WW
(Whewellite, sub-type Ia), CAR (Carbapatite, IVa), CAR2 (Carbapatite, IVa2),
STR (Struvite, IVc), BRU (Brushite, IVd), and CYS (Cystine, Va). The stone
fragment images were acquired with a digital camera under controlled lighting
conditions and with a uniform background. The dimensions of the images in
dataset A are 2848$\times$4288 pixels.

Dataset B, [8]. The endoscopic dataset consists of 409 images (see Figure
1(b), right column Table 1). This dataset includes 246 surface and 163 section
images. Dataset B involves the same classes as dataset A, except that the
Carbapatite fragments (sub-types IVa1, and IVa2) are replaced by the Weddelite
(sub-type IIa) and Uric Acid (IIIa) classes. The images of dataset B were
captured with an endoscope by placing the kidney stone fragments in an
environment simulating in a quite realistic way in-vivo conditions (for more
details, see [8]). The dimensions of the images in dataset B are
576$\times$768 pixels.

Automatic kidney stone classification is usually not performed on full images
due to the limited size of the datasets. Therefore, as in previous works [18],
patches of 256$\times$256 pixels were extracted from the original images to
increase the size of the training dataset (for more details, see [17]).
However, it should also be mentioned whether the disadvantages have been taken
into account, some of which are (i) loss of context since by cropping small
regions of an image, contextual and spatial information may be lost, (ii) when
patching an image, certain features may be present in multiple patches. On the
other hand, one of the advantages is that by using patches it is possible to
train machine learning models (which are complicated with few samples) and
ensure an increase in the number of samples, and balance between classes.

A total of 12,000 patches were generated for each dataset which is organized
into six classes as follows: For dataset A (WW, STR, CYS, BRU, CAR, CAR2) and
dataset B (WW, WD,UA, STR, BRU, CYS). For each data set, $80\%$ of the patches
(9600 patches) are used for the training and validation steps, while the
remaining $20\%$ of the patches (2400 patches) act as test data (200 patch-
images for each class). Patches of the same image contribute either only to
the training/validation data or solely to the test data. The patches were also
“whitened” using the mean $m_{i}$ and standard deviation $\sigma_{i}$ of the
color values $I_{i}$ in each channel [17].

###  3.2 Proposed approach

Figure 2: Proposed multiview-fusion model assisted by two-step transfer
learning for aESR.

Several approaches [17] have demonstrated the ability of DL-based models to
recognize in single views (SUR or SEC) different types of kidney stones with
high performance. However, in most cases, they have been trained by fine-
tuning with a totally different distribution than kidney stones, or worse,
they have been trained from scratch with the endoscopic images for individual
views. On the other hand, although in the work [17], it was observed that
combining features (color and texture) from both views of an endoscopic kidney
stone image (surface and section) produces more useful vectors for
classification using shallow machine learning methods, or training surface and
section patches together in a DL-based model produce more discriminative
features compared to models trained with surface or section images, so far no
elaborated technique was exploited to combine the surface and section
information.

Usually, to exploit the information from SUR and SEC images, the patches of
the two views of a fragment are simply seen as instances of the same class.
Although such methods fuse both views information and more data available for
the training, the way in which image features are extracted and combined is
far from being optimal, as it does not emulate how the visual inspection of
MCA/ESR is performed. To make matters worse, mixing the features in this way
does not always improve the classification results. As can be observed in the
MIX column of Table 4 (values marked by the * symbol), in some cases fusing
features from SUR and SEC patches does not produce better feature maps, as
this information combination is not optimal and hinders the model performance
[18].

In order to exploit the best features of both views, the proposed DL-model
(see Figure 2) combines the information in a systematic way using a fusion
strategy based on a multi-view scheme, introducing attention mechanisms to
further filter out unnecessary features maps of our CNN-model. Moreover,
instead of training from the scratch the individual branches, we assist the
model training with a two-step TL approach as a method of initializing weights
from a similar distribution (CCD-camera images) to the endoscopic images.

Figure 3: Two-step Transfer Learning. Model A was initialized with the weights
of a ResNet50 network pre-trained with ImageNet, and fine-tuned with Dataset
A. Next, Model B starts with the weights learned from Model A and is finally
fine-tuned with Dataset B. The objective is to pre-train dataset B with
weights similar to its distribution (kidney stone images).

###  3.3 Two-step Transfer Learning

There are different ways to acquire knowledge for a DL model (e.g. training
from scratch). However, in applications where data scarcity is a constraint,
techniques such as Two-step Transfer Learning (Figure 3) are useful as a pre-
training or initialization method for a specific domain (dataset) [18].

Two-step Transfer Learning learns weights from different distributions
approaching the final domain/target (endoscopic dataset). First, during the
HeTL (HeTL stands for heterogeneous TL), the pre-training is performed with a
general domain (ImageNet). The model weights are updated during a HoTL
(homogeneous TL) using a domain whose data distribution is the closest to that
of the target (domain adaptation process). In the kidney stone application,
the pre-training on ImageNet improves the generalization capabilities of the
DL-model and the CCD camera images of ex-vivo fragments are used as a first
fine-tuning. This fine-tuning is finalized using a part of the target dataset
B (fragment images acquired with endoscopes), the remaining patches of dataset
B being used for the validation and testing steps. More specifically, during
the HeTL-step, the large ImageNet dataset is used to transfer knowledge into a
ResNet50 network which is fine-tuned by the smaller kidney stone image set
acquired under controlled acquisition conditions (dataset A) as shown on the
left part of Figure 2. Then, fine-tuning is achieved for each branch (i.e.
individual model for each view) during the HoTL-step. This final tuning
exploits dataset B which is composed of endoscopic images close to dataset A,
but with higher variability in terms of image contrast, noise, and resolution,
emulating thus the illumination and scene conditions actually encountered in
ureteroscopy when patient data are acquired with an endoscope. The second-step
TL is performed for each of the views (SUR/SEC) by obtaining two independent
models trained with dataset B of endoscopic images for their respective views
(for more details, see [18]). As described below, a MV-model, assisted by the
second TL-step, is used to combine the SUR and SEC views into a mixed model
(MIX).

Table 2: Comparison of the performance of various kidney stone identification
methods. The value of the accuracy over all classes was determined with
dataset B for all methods.

View | Dataset | TL Step | Accuracy | Precision | Recall | F1-Score | Training details
---|---|---|---|---|---|---|---
| B | – | 0.702$\pm$0.012 | 0.718$\pm$0.010 | 0.702$\pm$0.012 | 0.701$\pm$0.008 | Trained with endoscopic images
SUR | A | 1 | 0.649$\pm$0.050 | 0.655$\pm$0.039 | 0.649$\pm$0.050 | 0.642$\pm$0.046 | Fine-tuning with ImageNet weights
| B | 2 | 0.832$\pm$0.012 | 0.845$\pm$0.012 | 0.832$\pm$0.012 | 0.829$\pm$0.012 | Fine-tuning with microscopic images weights
| B | – | 0.738$\pm$0.022 | 0.772$\pm$0.015 | 0.738$\pm$0.022 | 0.722$\pm$0.023 | Trained with endoscopic images
SEC | A | 1 | 0.824$\pm$0.022 | 0.834$\pm$0.020 | 0.824$\pm$0.022 | 0.820$\pm$0.023 | Fine-tuning with ImageNet weights
| B | 2 | 0.904$\pm$0.048 | 0.915$\pm$0.037 | 0.904$\pm$0.048 | 0.903$\pm$0.050 | Fine-tuning with microscopic images weights
| B | – | 0.760$\pm$0.024 | 0.773$\pm$0.029 | 0.760$\pm$0.024 | 0.752$\pm$0.024 | Fine-tuning with ImageNet weights
MIX | A | 1 | 0.800$\pm$0.013 | 0.809$\pm$0.013 | 0.800$\pm$0.013 | 0.797$\pm$0.013 | Fine-tuning with ImageNet weights
| B | 2 | 0.856$\pm$0.001 | 0.868$\pm$0.002 | 0.856$\pm$0.001 | 0.854$\pm$0.001 | Fine-tuning with microscopic images weights

Table 3: Comparison of different fusion strategies (concatenation and pooling) applying attention in different blocks (last and second last).  Fusion method | Accuracy | Precision | Recall | F1-score | Attention block
---|---|---|---|---|---
Concatenation | 91.00$\pm$3.03 | 92.00$\pm$3.16 | 91.00$\pm$3.03 | 91.00$\pm$3.03 | Without attention
Pooling | 90.40$\pm$3.29 | 90.80$\pm$3.11 | 90.40$\pm$3.29 | 90.40$\pm$3.29 | Without attention
Pooling | 88.33$\pm$6.43 | 88,33$\pm$5.51 | 88.00$\pm$6.08 | 88.00$\pm$6.08 | Last block
Pooling | 91.25$\pm$0.50 | 91.75$\pm$0.50 | 91.25$\pm$0.50 | 91.25$\pm$0.50 | Last and second last block

###  3.4 Multi-view model

Once the two SUR and SEC models are trained through the previous two-step TL,
the feature extraction layers of this single-view network are frozen to ensure
that each branch of the multi-view model extracts the same features and that
any variation in performance depends on the non-frozen layers (merge and full
connection layers). These frozen layers are connected to a fusion layer, which
is responsible for mixing the information of the two views.

In this work, the two late-fusion methods proposed for kidney stone
classification [28] were exploited (concatenation and max-pooling). The first
fusion method (concatenation) concatenates the feature vectors obtained from
each view and merges the resulting representation through a fully connected
layer. On the other hand, in the second fusion method (max-pooling), feature
vectors are stacked and max-pooling is applied to them. Three configurations
were used to implement max-pooling. The first corresponds to a model without
attention mechanisms. The second consists of one layer of attention (last
block). Finally, the third consists of two-layer of attention (the last, and
second last block, arranged as shown in Figure 2). Lastly, the output of the
late-fusion layer is connected to the remaining part of the MV-model, which
merely consists of the classifier. The full proposed model is shown in Figure
2.

##  4 Results and Discussion

###  4.1 Two-step Transfer Learning results

(a)

(b)

(c)

Figure 4: UMAP [20] dimensionality feature reduction (a) SUR, (b) SEC, and (c)
MIX patch sets.

Three experiments (SUR model, SEC model, MIX model) were carried out to assess
the performance of the two-step TL approach applied to the patch data
described in Section 3.1. In the first and second experiments, the two-step TL
approach described in Section 3.3 was used to predict kidney stone types in
endoscopic images for individual SUR and SEC views. Then, in the third
experiment, the SUR and SEC datasets are mixed into a single dataset denoted
as MIX. The model is trained by taking into account the patches of both views
during training. The results obtained are shown in Table 2.

In order to describe the results, in this work we will use the Accuracy
metric, since the testing set is balanced and all the metrics (Precision,
Recall and F1-Score) follow the same trend (as shown in Table 2).

SUR model. The Two-step Transfer Learning (fine-tuning with microscopic images
weights) results for the SUR view patch dataset is 83.2$\pm$01.2 (in $\%$
measured with accuracy). The performance of the Two-Step based model
outperforms training from scratch with endoscopic images (70.2$\pm$01.2), and
also greatly improves on the performance obtained by doing a single step of
Transfer Learning with ImageNet (64.9$\pm$05.00%). Although the latter result
would be expected to be superior to training from scratch (with endoscopic
images), we assume that the ImageNet weights for the SUR view are not useful
for this task. Which gives us a slight hint that weight initialization is not
always useful.

SEC model. The results obtained in the SEC view (90.4$\pm$04.8) are promising
using two-step Transfer Learning, and suggest better performance than the SUR
view (83.2$\pm$01.2), probably due to the extraction of more discriminant
features. In comparison, training a SEC model from scratch with endoscopic
images (73.8$\pm$02.2), the performance improved significantly. On the other
hand, pre-training with ImageNet (82.4$\pm$02.2) shows an increase in accuracy
quite useful for this task.

MIX model. In order to measure the performance of a model that considers both
views during training, MIX model was evaluated. An overall yield of
85.6$\pm$0.10 was obtained by applying two-step TL to the MIX assembly. The
results are good, with respect to training from scratch with endoscopic images
(76.0$\pm$02.4) or fine-tuning with ImageNet (80.0$\pm$01.3). However, it
would be expected that combining the information during training would yield
better features than individual views (SUR 83.2$\pm$01.2 and SEC
90.4$\pm$04.8). These results for MIX suggest that combining information
during training does not always generate good results. To deal with this
problem, it is proposed to use the models trained on SUR and SEC datasets
implemented with two-step Transfer Learning were combined using the MV-model
described in Section 3.4.

###  4.2 Multiview-fusion model

The individual SUR and SEC models trained with two-step Transfer Learning are
used as two individual branches for the fusion scheme. In this work, two
fusion methods were implemented: max-pooling and concatenation. For max-
pooling, three variants were implemented (without attention, with attention in
the last block, and with attention in the last two blocks). The results of
these experiments are gathered in Table 3 and discussed below.

Concatenation. For a simple fusion technique that does not require attention
mechanisms, promising results were obtained. For this technique, a performance
of 91.00$\pm$3.03 was obtained in the inference of surface or section patches.
With respect to the MIX model of section (4.1) implemented with two-step
Transfer Learning , it can be observed that a significant improvement of up to
6% accuracy.

Pooling without attention. The performance obtained with max-pooling without
attention (90.40$\pm$3.29, max-pooling slightly below concatenation) as well
as concatenation (91.00$\pm$3.03) maintains a similar accuracy. This suggests
that both techniques of fusing information in an organized manner and assisted
by two-step Transfer Learning, are a viable alternative for the classification
of renal calculi in endoscopic images, outperforming the MIX model by a good
margin (up to 6%).

Pooling with attention (last block). Under the premise of extracting better
features from the individual SUR and SEC models, an attention mechanism was
applied to the last block of the fusion scheme. The results for pooling with
attention in the last block (88.33$\pm$6.43), remained slightly below
concatenation and max-pooling, and are still superior to MIX.

Pooling with attention (last and second last block). In order to improve the
performance of the max-pooling results, attention was applied to the last two
blocks of the model (as shown in Figure 2). In addition, the results show an
improvement over the other experiments with max-pooling and concatenation. The
max-pooling with attention (last and second last block) approach is the model
selected for comparison with the state of the art that will be described
below.

###  4.3 Comparison with the state-of-the-art

In order to compare the results obtained in this work for SEC, SUR and MIX
models, the works [19, 9, 1, 18] were reimplemented and trained with the
datasets described in Sec. 3.1. The evaluation of the models was performed
with the same testset.

Regarding the SUR view, the performance obtained in this contribution using
two-step Transfer Learning exceeds the performance obtained by related works.
Qualitatively, we can observe in Figure 4(a), that the features extracted and
plotted by UMAP remain dispersed among elements of the same classes. However,
the qualitative performance is good, surpassing the results of the state of
the art which follows the same trend.

On the other hand, in the SEC view, the model presented in this work shows
better results than those described in previous works, and maintains a
superior performance. In Figure 4(b), it can be observed that the features
extracted by the SEC view are more discriminative with respect to the SUR
view. Also, it can be seen that elements of the same class are grouped
correctly, and that the extra-class distance is large, supporting the
quantitative results of Table 2.

Finally, the fusion scheme has been shown to be efficient combining both views
(SUR/SEC) in a ”mixed” model. In addition the proposed model can maintain the
performance, contrary to the models of state-of-the-art that do not have an
organized way of combining information (marked by the * symbol). The latter
shows that combining the SUR and SEC information of stones in a single class
leads to a performance decrease. Qualitatively (Figure 4(c)), the MIX view
presents compact clusters for all classes; however, the inter-class distance
is desired to increase. Applying attention techniques at deeper levels could
improve these features, or implement them to the concatenation model.

Table 4: Comparison of the performance of various aESR DL-methods. The
classification accuracy (in percentage) overall classes was determined with
test dataset B for all methods.

Method | SUR | SEC | MIX
---|---|---|---
Martinez, et al. [19] | 56.2$\pm$23.3 | 46.6$\pm$12.8 | *52.7$\pm$18.9
Estrade, et al. [9] | 73.7$\pm$17.9 | 78.8$\pm$10.6 | *70.1$\pm$22.3
Black, et al. [1] | 73.5$\pm$19.0 | 76.2$\pm$18.5 |  *80.1$\pm$13.8
Lopez-Tiro, et al. [17] | 81.0$\pm$03.0 | 88.0$\pm$02.3 | *85.0$\pm$03.0
This contribution | 83.2$\pm$01.2 | 90.4$\pm$04.8 | *91.2$\pm$0.50

##  5 Conclusion and future work

This contribution shows that, by mixing information from two views, it is
possible to train more accurate models to identify kidney stones acquired with
endoscopes. Thus, AI technology can be an interesting solution for assisting
urologists. However, these contributions used a very limited dataset in terms
of class number and patch samples. The learning approaches on few samples must
be improved to cope with the small amount of training data, and especially to
increase the class separability when more kidney stone types have to be
identified.

## Acknowledgements

The authors wish to acknowledge the Mexican Council for Science and Technology
(CONACYT) for the support in terms of postgraduate scholarships in this
project, and the Data Science Hub at Tecnologico de Monterrey for their
support on this project. This work has been supported by Azure Sponsorship
credits granted by Microsoft’s AI for Good Research Lab through the AI for
Health program. The project was also supported by the French-Mexican ANUIES
CONAHCYT Ecos Nord grant 322537.

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# The Challenge to MOND from ultra faint dwarf galaxies

Mohammadtaher Safarzadeh & Abraham Loeb  Center for Astrophysics — Harvard &
Smithsonian, 60 Garden Street, Cambridge, MA;
<EMAIL_ADDRESS>
###### Abstract

Modified Newtonian Dynamics (MOND) at low acceleration has been astonishingly
powerful at explaining the rotation curve of galaxies and the relation between
the baryonic content of the galaxies and their observed circular velocity,
known as the Baryonic Tully-Fisher Relationship (BTFR). It is known that MOND
fails at explaining the observed velocity dispersion of the ultra-faint dwarf
galaxies (UFDs) with the justification that UFDs are more prone to tidal
disruption in MOND compared to cold dark matter model. We show that: (i) the
ratio of tidal to internal acceleration in UFDs is extremely low, (ii) there
is no correlation between the deviation of UFDs from MOND’s prediction as a
function of tidal susceptibility, and (iii) recent constraints from Gaia
proper motion analysis on the orbital parameters of the UFDs exacerbates the
challenge to MOND. In particular, Gaia data indicates that Ursa Major I is
experiencing a recent infall into the Milky Way’s halo, and its inconsistency
with MOND at 7-$\sigma$ level can not be attributed to being an early infall
satellite. Moreover, the new data from Gaia DR2 shows Willman I to have the
least eccentric orbit of all UFDs, and its deviation from MOND at 4-$\sigma$
level can not be attributed to a highly eccentric orbit as previously
suggested. Finally, given that Tuc III is the only UFD observed to show tidal
features, Reticulum II and Segue I are two other UFDs that potentially
challenge MOND as they have comparable galactocentric distances to Tuc III
while showing no tidal features.

###### Subject headings:

Modified Gravity, Ultra Faint Dwarf Galaxies

##  1\. Introduction

Despite all the efforts from multiple theoretical and experimental directions,
the nature of dark matter remains unknown (see Profumo et al., 2019, for a
recent review). One important clue to its nature could come from the fact that
it should manifest itself on Galactic scales as is successfully described by
Modified Newtonian Dynamics (MOND; Milgrom, 1983). MOND has specific
predictions for systems with acceleration below $a_{0}=1.2\times 10^{-8}\rm
cm~{}s^{-2}=3.8~{}\rm pc/Myr^{2}$ such that the acceleration becomes
$g=\sqrt{g_{N}a_{0}}$, where $g_{N}$ is the Newton’s acceleration.
Intriguingly, the value of $a_{0}$ is related to the Hubble constant by
$a_{0}\approx cH_{0}/6$ where $c$ is the speed of light.

Two successes of MOND on galactic scales are: i) explaining the observed steep
relation with low scatter between baryonic mass in galaxies and their circular
velocity known as Baryonic Tully-Fisher Relationship (BTRF; Tully & Fisher,
1977), and ii) explaining the rotation curve of spiral galaxies (Rubin et al.,
1980). and the. While in the standard $\Lambda\rm CDM$ cosmology fine-tuning
of baryonic physics is required to recover the BTRF (McGaugh et al., 2000;
Abadi et al., 2003; Governato et al., 2007; Bouché et al., 2010; Vogelsberger
et al., 2013; Chan et al., 2015; Sales et al., 2017), MOND can reproduce the
observed relationship with no fine-tuning as it requires no dark matter in its
formulation (McGaugh, 2005). Likewise, the modified dynamics have been
astonishigly successful at reproducing the observed rotation curve of the
galaxies (Sanders & McGaugh, 2002).

In the limit of isolated systems, MOND’s prediction for the 1 D velocity
dispersion of a dispersion supported spherical galaxies for $g_{in}\ll a_{0}$
is given by:

$\sigma_{iso}\approx\left(\frac{4}{81}a_{0}GM_{*}\right)^{1/4},$ (1)

where $M_{*}$ is the stellar mass of the galaxy, and the internal
acceleration, $g_{in}$, is given by:

$g_{in}\approx\frac{3\sigma_{iso}^{2}}{r_{1/2}},$ (2)

where $r_{1/2}$ is the half light radius. However, if the galaxy is a
satellite of the Milky Way in the regime of $g_{in}<g_{ex}<a_{0}$, where
$g_{ex}$ is the gravitational acceleration of the MW, given by:

$g_{ex}\approx\frac{V_{MW}^{2}}{D_{GC}},$ (3)

MOND’s predicted dispersion due to the external field effect (EFE; McGaugh &
Milgrom, 2013a) is given by:

$\sigma_{efe}\approx\left(\frac{a_{0}GM_{*}}{3g_{ex}r_{1/2}}\right)^{1/2},$
(4)

where $V_{MW}$ is the circular velocity of the MW, for which we adopt a
constant value of $200~{}\,{\rm km\,s}^{-1}$ for all the calculations in this
work (Reid et al., 2014). $D_{GC}$ is the galactocentric distance of the
satellite from the MW. We note that the EFE of M31 on MW’s satellites is
irrelevant since the internal acceleration of the satellites are stronger than
M31’s external acceleration.

By definition, satellites with luminosity less than $10^{5}L_{\odot}$ are
considered to be ultra-faint dwarf galaxies (UFDs). In the standard
$\rm\Lambda CDM$ cosmology, UFDs are dark matter-dominated systems with
quenched star formation history (Simon & Geha, 2007; Simon, 2019). New UFDs
were recently discovered in the Dark Energy Survey (Bechtol et al., 2015;
Koposov et al., 2015a). As relics from the early universe, UFDs are testbeds
for examining various theories from the standard collisionless dark matter
(Nadler et al., 2021), to fuzzy dark matter (Safarzadeh & Spergel, 2020;
Burkert, 2020; Hayashi et al., 2021) and MOND (McGaugh & Wolf, 2010).

A notable example of the success of MOND for the velocity dispersion of the
Milky Way’s satellites is MOND’s prediction for the ultra diffuse galaxy
Crater 2 (McGaugh, 2016) which has been proved by the observation (Caldwell et
al., 2017). This is an important prediction for MOND since Crater 2 is free of
current tidal interference (though it may have suffered in the past). However,
it has been shown that MOND’s prediction for the velocity dispersion of the
UFDs does not match the observed values, but the discrepancy has been
attributed to tidal susceptibility (McGaugh & Wolf, 2010). In this work, we
revisit this claim in light of the recently discovered UFDs. We argue that
tidal susceptibility does not seem to justify the deviation of MOND’s
predictions from the observed velocity dispersion of UFDs.

The structure of this paper is as follows: In §2 we present our method of
approximating tidal susceptibility. We investigate which UFDs are within the
regime that MOND’s EFE. We point out a few UFDs that challenge MOND and
present the caveats around those systems. In §3, we summarize our results and
discuss the most promising avenues towards understanding the nature of dark
matter.

##  2\. DATA

The data on the velocity dispersion and half-light radius of the UFDs are
primarily collected from the compilation of Wolf et al. (2010). We use the
estimated galactocentric distances of the UFDs from the recent proper motion
analysis of the Gaia DR2 (Fritz et al., 2018; Simon, 2018). The stellar mass
of the UFDs is from the compilation of McConnachie (2012). For the UFDs that
are not presented in Wolf et al. (2010) we have collected the data from the
following references: Grus II and Tucana IV from Simon et al. (2020), Eridanus
II from Li et al. (2017) and its stellar mass from Gallart et al. (2021) and
Bechtol et al. (2015). We obtain the data for Horologium I (Hor I) from
Koposov et al. (2015b) and its stellar mass from Bechtol et al. (2015) which
is consistent with the data reported in Jerjen et al. (2018). Hor I is close
to the Magellanic Clouds, and whether this is a satellite of the Large
Magellanic Clouds (LMC) is discussed in Nagasawa et al. (2018), and we
elaborate on in the next section. The data for Triangulum II is obtained from
Kirby et al. (2017), and for Segue 2 from Kirby et al. (2013a). The data for
Reticulum II is obtained from Simon et al. (2015). For Bootes I we adopt a
velocity dispersion of 2.4 $\,{\rm km\,s}^{-1}$ which comes from the majority
cold component of stellar population (Koposov et al., 2011). As Leo T is a
gas-rich satellite with total HI mass of $4\times 10^{5}~{}{\rm\,M_{\odot}}$
(Adams & Oosterloo, 2018) we consider a total baryonic mass of
$\log(M_{b}/M_{\odot})=5.75$ for this system. In cases where there is no
estimate of the stellar mass of the system, we use the relationship between
the metallicity of the system and its stellar mass derived from dwarf
satellites of the MW (Kirby et al., 2013b).

##  3\. Method

To begin with, we display the predicted velocity dispersion of MOND for the
UFDs as compared to the observed line of sight estimates in Figure 1. We have
divided the satellites into two groups, one in which the $g_{in}<g_{ex}$ as
shown in red points, and those that internal acceleration is larger than
$g_{ex}$ as shown in green points. The squares show the observational
estimates, the longer cap error bars are showing the 1-$\sigma$ (1 standard
deviation) error, and the longer and thinner caps showing the 3-$\sigma$ error
bars.

Figure 1.— The predicted velocity dispersion of the UFDs and dwarf spheroidal
galaxies from MOND are shown in red circles. The observed line of sight values
(blue squares) of velocity dispersion of the UFDs and dwarf spheroidal
galaxies are shown as blue squares with the longer cap error bars are showing
1-$\sigma$ (1 std) error, and the longer and thinner caps showing the
3-$\sigma$ error. We have divided the sample into two groups: those with
$g_{in}/g_{ex}<1$ shown with red points, and those with $g_{in}>g_{ext}$ shown
with green points. For the former category, we use the $\sigma_{efe}$ of MOND,
and for the latter systems, we use the isolated estimate prediction,
$\sigma_{iso}$, of MOND. The errorbars for MOND predictions come from assuming
the $V_{\rm MW}$ a range between 170 $\,{\rm km\,s}^{-1}$ to 230 $\,{\rm
km\,s}^{-1}$.

To investigate whether tidal susceptibility can explain the deviation of
MOND’s prediction and the observed velocity dispersion of the UFDs, we
calculate the ratio of the tidal acceleration and the internal acceleration of
a satellite. The tidal accelerations are estimated by:

$g_{t}=g_{ex}\frac{2r_{1/2}}{D_{GC}}.$ (5)

The yellow hexagons in Figure 2 show the satellites where $0.15\leq
g_{in}/g_{ex}\leq 1$, and in the red points show the system where
$g_{in}/g_{ex}\leq 0.15$. The reason for this separation is that the EFE
prediction is only valid when $g_{in}\ll g_{ex}$, and its validity when
$g_{in}\approx g_{ex}$ is unkonwn (McGaugh & Milgrom, 2013a). Therefore, we
show the results once for all the sample with $g_{in}/g_{ex}<1$, and another
time for when $g_{in}/g_{ex}<0.15$, a regime in which EFE is shown to give
consistent results with the observations. The value of 0.15 is chosen based on
McGaugh (2016) EFE model’s success in predicting the correct velocity
dispersion for Crater II, which has a ratio of $g_{in}/g_{ex}\approx 0.12$.

The red points in Figure 2 imply that if tidal susceptibility is the reason
for the discrepancy between the predictions of MOND and the observed
$\sigma_{los}$ of the UFDs, we have to see a positive correlation between the
ratio of the tidal over internal acceleration and the deviation of MOND from
the observed velocity dispersion. Does such a correlation exist? With low
number statistics, one can either argue for or against a positive correlation.
However, the fact that Willman I still deviates from the MOND’s prediction at
four $\sigma$ remains a challenge to MOND. Moreover, Hor I with
$g_{in}/g_{ex}\approx 0.3$ stands at 4 $\sigma$ from the EFE prediction while
having the lowest of the $g_{t}/g_{in}$ in our sample.

Figure 2.— Deviation of the predicted MOND velocity dispersion of the UFDs and
their observed value as a function of the $\log(g_{t}/g_{in})$ for systems in
which $g_{in}<g_{ex}<a_{0}$. _Yellow hexagons:_ for systems with
$g_{in}<g_{ex}<1$. _Red points:_ for systems with $g_{in}/g_{ex}<0.15$. Tuc
III is shown with a vertical line since, at the moment, only an upper limit
exists for its velocity dispersion, $\sigma_{los}<1.5\,{\rm km\,s}^{-1}$ at
95% confidence. Therefore, the true ratio of tidal over the internal
acceleration of Tuc III could be any value smaller than $10^{-1.5}$.  Figure
3.— The distribution of MW satellites with $g_{in}/g_{ex}<0.15$ in
galactocentric distance vs. the ratio of tidal to internal acceleration. Tuc
III is shown with a triangle since we only know the lower limit on
$g_{t}/g_{in}$ for Tuc III. The fact that tidal features are observed for Tuc
III but not for Ret II or Seg I is a concerning fact. However, this could be
justified given the highly eccentric orbit of Tuc III with a pericenter
approach of $3^{+1}_{-1}$ kpc.

But the most stringent constraint comes from Ret II due to its proximity to
the Sun for the following reason: Tucana III (Tuc III) is the first UFD
observed to show a velocity gradient of $8.0\pm 0.4\,{\rm km\,s}^{-1}\rm
deg^{-1}$, suggesting the galaxy is ongoing tidal disruption (Li et al.,
2018). Simon et al. (2017) estimated the $\sigma_{los}$ to be less than 1.5
$\,{\rm km\,s}^{-1}$ at 95% C.I. With a half-light radius of about 34 pc and a
distance of about 25 kpc, Tuc III is our nearest neighbor UFD with
$g_{in}/g_{ex}\approx 0.04$. Given the old stellar population of the UFDs, one
can adopt a value for the stellar mass-to-light ratio of $M_{*}/L_{V}\approx
2M_{\odot}/L_{\odot}$ assuming standard initial mass function (Simon, 2019).
This leads to an estimate of the stellar mass of Tuc III to be $M_{*}\approx
1500{\rm\,M_{\odot}}$. The same estimate yields a stellar mass of about 5,000
${\rm\,M_{\odot}}$ for Ret II. With $\sigma_{los}=1.5\,{\rm km\,s}^{-1}$ for
Tuc III (which is the estimated lower limit for this system), the ratio of the
tidal and internal acceleration for Tuc III is $\log(g_{t}/g_{in})=-1.5$. We
show this with a vertical line in the bottom panel of Figure 2. This value is
comparable to the ratio of tidal over internal acceleration for Ret II.

Ret II and Tuc III have similar Galactocentric distances, implying that if we
see tidal features in Tuc III, we would also expect to see tidal features in
Ret II as well, but no such features have been observed. However, one can
argue that the orbital motion of the two satellites is different: based on
Gaia DR2, Simon (2018) estimates a pericenter and apocenter distance for Tuc
III to be $3^{+1}_{-1}$ kpc and $49^{+6}_{-5}$ kpc with an orbital period of
0.7 Gyr, and the corresponding values for Ret II are derived from being
$29^{+14}_{-6}$ kpc for pericenter approach and $91^{+91}_{-39}$ kpc for
apocenter approach with an orbital period of 1.6 Gyr. These numbers agree with
the results reported in Fritz et al. (2018). The inferred orbital motion based
on Gaia DR2 suggests that despite their current similar galactocentric
distance, Tuc III has been more prone to tidal disruption as opposed to Ret
II. However, these orbital motion estimates are based on cold dark matter
potential for the MW and whether the same orbital dynamics would be inferred
in MOND remains to be verified with detailed orbital motion simulation.
Moreover, we note that although Ret II is an elongated UFD with an ellipticity
of about $\epsilon=0.6$, there have been no signs of tidal features within our
field of view (Mutlu-Pakdil et al., 2018).

Figure 3 shows the distribution of the satellites in the galactocentric
distance and $\log(g_{t}/g_{i})$ plane. Other than Ret II, Seg I also has a
comparable galactocentric distance to Tuc III. The pericenter and apocenter
approach for Segue I (Seg I) is estimated to be $20^{+4}_{-5}$ kpc and
$61^{+34}_{-18}$ kpc with an orbital period of about 1.1 Gyr. This makes Seg I
more similar to Tuc III than Ret II. However, Simon et al. (2011) does not
find evidence suggesting Seg I being tidally disrupted.

Willman I stands 4 $\sigma$ away from the MOND’s prediction. This satellite
has a pericenter and apocenter approach of $44^{+15}_{-19}$ and
$53^{+57}_{-13}$ kpc respectively, which yields this UFD the lowest orbital
eccentricity of $e\approx 0.2$. While the deviation of Willman I was
identified in McGaugh & Wolf (2010), this was attributed to speculating a
highly eccentric orbit for Willman I while the new data from Gaia DR2 actually
suggests Willman I to be actually on the least eccentric orbit of all the UFDs
(Simon, 2018).

Similar to Willman I, Hor I also stands 4 $\sigma$. away from MOND’s
prediction; however, with the ratio of $g_{in}/g_{ex}\approx 0.3$, it is not
clear whether the isolated or the EFE regime is adequate for predicting its
internal dynamics. Furthermore, the close proximity of Hor I to the LMC
(Nagasawa et al., 2018) might further complicate the expected dynamics of this
satellite, although $g_{in}/g_{ex}$ with respect to the Magellanic clouds
remains less than 1 for this satellite.

Perhaps the most deviant of all the satellites is U Maj I standing about 7
$\sigma$ from MOND’s prediction. This deviation was attributed to the early
infall time of this satellite in (McGaugh & Wolf, 2010). While early analysis
of the infall times of the satellites categorized this satellite as indeed one
with an early infall time of about 8 Gyr However (Rocha et al., 2012), recent
Gaia DR2 suggests that this satellite is actually entering the MW’s halo
recently in the past $1.5^{+5.1}_{-1.6}$ Gyr (Fillingham et al., 2019).

So far, all calculations have been done based on the current distance of the
satellites from the Galactic center. However, we know that the satellites have
experienced a pericenter approach, and if their orbit is highly eccentric,
tidal acceleration at pericenter passage is important. This is shown in Figure
4. We obtain the estimated pericenter approach from Fritz et al. (2018)
assuming two different halo mass for the MW’s halo. The obtained estimates
from the low virial mass calculation of Fritz et al. (2018) agrees with the
results in Simon (2018). We note that the agreement is not perfect, and some
deviations in the estimated orbital parameters are present. If the existence
of a correlation between the MOND’s deviation and tidal effects is the desired
feature to look for, then the data shows that such a correlation disappears if
we focus on satellites with low ratios of tidal to internal acceleration even
at pericenter passage. We note that the data point for U Maj I indicates its
estimated pericenter approach while the satellite has not reached its
pericenter distance. Tucana II and Seg I deviate at 3 and 4 $\sigma$ level
from MOND’s prediction despite having a low tidal to internal acceleration at
pericenter approach. Coma Berenices stands at 5-$\sigma$ deviation from MOND’s
prediction while having a rather large pericenter approach of 43 kpcs.
However, since this satellite is an early infall system, its deviation could
be justified by being in the tidal field of its host for about a Hubble time.
Moreover, despite its high ratio of tidal to internal acceleration, Hercules
does not deviate from MOND’s predictions as much as is expected from Draco, U
Min, and Sextans.

Figure 4.— Same as in Figure 2 but calculated at the distance of pericenter
approach. Top (bottom) panel shows the derived pericenter distance assuming a
MW’s potential with a virial mass of $0.8\times 10^{12}$ ($1.6\times
10^{12}$)${\rm\,M_{\odot}}$. The correlation between deviations from MOND’s
prediction disappears in the limit of low tidal to internal acceleration
ratio, which can be a challenging result for MOND to account for. Hercules
stands out as a satellite with rather high tidal to internal acceleration
ratio at its pericenter passage but not deviating as much as Draco, U Mi, and
Sextans would suggest.

##  4\. Discussion

The latest Gaia DR 2 data shows a lack of strong correlation between tidal
susceptibility and deviation from MOND’s predictions and observed line of
sight velocity dispersion of the UFDs, as presented in Figure 2. While the
data challenges MOND, more data is needed. In this work, we solely focused on
tidal features as a proxy for deviation from MOND. We did not investigate
correlations with other estimators such as ellipticity of the UFD as projected
on the sky, or adiabaticity (which is a measure of the number of internal
orbits of stars over the number of orbits of the satellite around its host).
Ellipticity, adiabaticity, and other estimators have been used in McGaugh &
Wolf (2010) and shown to be a proxy to tidal features and we do not
investigate these in this work.

Could we be biased in inferring the intrinsic velocity dispersion of the UFDs?
We know that binary stars can inflate the estimated 1 D velocity dispersion
specifically for systems in which the total number of the stars is $\leq 10$.
For example, a binary star did inflate the velocity dispersion for Triangulum
II (Kirby et al., 2017). However, for well-studied systems such as seg I,
binaries contribute to the velocity dispersion at 10% level (Simon et al.,
2011). In the meantime, a detailed analysis of Draco and U Min shows that the
contribution of binaries to the observed velocity dispersion can be
significant (Spencer et al., 2018). While single epoch observations of stars
are vulnerable to binaries, future multi-epoch campaigns will say the final
word on the issue of binaries.

It is also worth mentioning the remarkable prediction of MOND for M31 dwarf
satellites (McGaugh & Milgrom, 2013b, c). Moreover, the original prediction of
MOND for Cetus and Tucana of 8.2 and 5.3 $\pm 0.4\,{\rm km\,s}^{-1}$,
respectively, was not in agreement with the observations at the time
(estimated values to be about $17.2\pm 2$ and $15.8^{+4.1}_{-3.1}~{}\,{\rm
km\,s}^{-1}$ for Cetus and Tucana, respectively). However, recent observations
point to a much better agreement with MOND’s prediction with Tucana having a
velocity dispersion of $6.2^{+1.6}_{-1.3}~{}\,{\rm km\,s}^{-1}$ (Taibi et al.,
2020). This can be taken as MOND will survive with time as better data is
gathered for the UFDs.

The issue pertaining to MOND’s predictions is not limited to the line of sight
velocity dispersion. For example, it is known that dynamical friction is
stronger in MOND (shorter timescale for a satellite) compared to dark matter
models (Nipoti et al., 2008) which makes Fornax and its globular cluster a
serious problem for MOND. However, Fornax is a problem for the standard model
of $\rm\Lambda CDM$ too, and a core-like structure for its dark matter halo
can allow for the dynamics of globular clusters observed in Fornax. We note
that other circular orbits outside the tidal radius of Fornax can justify the
dynamics of globular clusters in Fornax, even in MOND (Angus & Diaferio,
2009).

What do the seemingly simple modification to Newtonian gravity and its success
at the galactic scale teach us about the nature of dark matter? Why does MOND
have shortcomings at larger scales, such as in clusters of galaxies? And based
on the results we analyzed in this work, apparently shortcoming in the scale
of UFDs? Is this shortcoming in satellites scale an issue with EFE modeling?
Attempts to successfully merge these apparently different behaviors of dark
matter under one umbrella have been limited (see for example the superfluid
interpretation of the dark matter; Berezhiani & Khoury, 2015; Khoury, 2015),
and the emergent gravity paradigm (Verlinde, 2017), but perhaps such avenues
hold the key to understand the nature of dark matter.

We are hopeful that future data obtained by the Vera C. Rubin Observatory,
which will find all satellites of the MW within its virial radius down to V
band absolute magnitude fainter than $M_{V}=-4$, discover enough UFDs to have
a statistically significant sample for testing MOND (Tollerud et al., 2008).

We are thankful to the referee for their constructive comments. We are
thankful to Mordehai Milgrom, Stacy McGaugh, and Josh Simon for their
insightful comments. This work was supported in part by the Dean’s Competitive
Fund for Promising Scholarship at the Faculty of Arts & Sciences of Harvard
University.

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