# Comment on “Consistent hydrodynamics of ferrofluids”

Mark I. Shliomis<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of
Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105,
Israel

In a recent paper [1], Fang presented “a generic and consistent hydrodynamic
theory for ferrofluids” which “is expected to be a cornerstone of ferrofluid
hydrodynamics.” Here is how he formulates the main flaw of existing theory:
“Unfortunately, in the ferrofluid community all previous studies fail to
discriminate between the solvent vorticity ${\bm{\Omega}}_{sol}$ and the
suspension vorticity ${\bm{\Omega}}$.” By his definition, the solvent vortices
are unobserved quantities. To observe (measure) them, it would be necessary to
extract magnetic particles from the ferrofluid, and replace them with holes,
i.e., turn ferrofluid into a kind of Swiss cheese with holes instead of
magnetic grains. Therefore, “${\bm{\Omega}}_{sol}$, the mesoscopic vorticity
due to solvent flow, is not a genuine macroscopic variable and should be
eliminated,” the author writes.

But how to do that? “How to correctly express ${\bm{\Omega}}_{sol}$ in terms
of macroscopically quantities characterizing the suspension?” And Fang found a
“simple but important relation” that relates ${\bm{\Omega}}_{sol}$ to
${\bm{\Omega}}$ and the mean angular velocity of particles ${\bm{\omega}}$ by
a linear relationship

$\displaystyle{\bm{\Omega}}=(1-\phi){\bm{\Omega}}_{sol}+\phi\kern
1.0pt{\bm{\omega}}\,;$ (1)

here $\phi=nV_{p}$ and $n$ are the volume fraction and the number density of
magnetic particles, $V_{p}=\pi d^{3}/6$.

Unfortunately, the relation (1) contradicts to the angular momentum
conservation law and therefore is false. Indeed, when two physical objects (A
and B), rotating with frequencies ${\bm{\omega}}_{A}$ and ${\bm{\omega}}_{B}$,
merge into one (C), then, according to the law of conservation of angular
momentum, the contributions of ${\bm{\omega}}_{A}$ and ${\bm{\omega}}_{B}$ to
${\bm{\omega}}_{C}$ are determined by the ratio of the moments of inertia
$I_{A}/I_{B}$, and not by the relative volume fraction $\phi$, as it is done
in Eq. (1).

To derive the correct relation, one should use the conservation law for the
angular momentum:

$\displaystyle
I_{FF}{\bm{\Omega}}=I_{FF}{\bm{\Omega}}_{0}+I_{MP}({\bm{\omega}}-{\bm{\Omega}}_{0}),$
(2)

where $I_{FF}$ is the moment of inertia of the ferrofluid, and $I_{MP}$ is the
sum of the intrinsic moments of inertia of the magnetic particles contained in
it; ${\bm{\Omega}}$ and ${\bm{\Omega}}_{0}$ are ferrofluid vortices in and
without an external magnetic field. In the absence of a field, the particles
rotate together with the fluid, ${\bm{\omega}}={\bm{\Omega}}_{0}\,$, so
ferrofluid rotates as a whole: ${\bm{\Omega}}={\bm{\Omega}}_{0}\,$. Constant
magnetic field always hampers particle’s rotation, ${\omega<\Omega_{0}}$,
while a rotating magnetic field can provide the inverse inequality.

Dividing Eq. (3) by $I_{FF}$, we arrive at the relation

$\displaystyle{\bm{\Omega}}=\Big{(}1-\frac{I_{MP}}{I_{FF}}\Big{)}{\bm{\Omega}}_{0}+\frac{I_{MP}}{I_{FF}}\kern
1.0pt{\bm{\omega}},$ (3)

that differs from (1) in two ways: by replacing $\phi$ by $I_{MP}/I_{FF}$ and
by replacing the unobserved ${\bm{\Omega}}_{sol}$ to ${\bm{\Omega}}_{0}$. Let
us evaluate these replacements.

The moment of inertia of one magnetic particle is equal
$\rho_{p}V_{p}(d^{2}/10)$, their number in a cylinder of radius $R$ and length
$S$ is $\pi nR^{2}S$, so their total moment of inertia is

$\displaystyle I_{MP}=(\pi/10)\rho_{p}\kern 1.0pt\phi\kern 1.0ptd^{2}R^{2}S.$

The moment of inertia of the ferrofluid filling the cylinder (below $\rho_{s}$
and $\rho_{p}$ are the densities of the solvent and particles) is equal to

$\displaystyle
I_{FF}=(\pi/2)\big{(}\rho_{s}(1-\phi)+\rho_{p}\phi\big{)}R^{4}S\,,$

and the ratio of the above moments of inertia is

$\displaystyle\frac{I_{MP}}{I_{FF}}=C\phi\Big{(}\frac{d}{R}\Big{)}^{2},~{}~{}~{}{\rm
where}~{}~{}~{}C=\frac{\rho_{p}/5}{\rho_{p}\phi+\rho_{s}(1-\phi)}\,.$ (4)

Substituting this relation in Eq. (3), we finally find

$\displaystyle{\bm{\Omega}}=\Big{(}1-C\phi\frac{d^{2}}{R^{2}}\Big{)}{\bm{\Omega}}_{0}+C\phi\frac{d^{2}}{R^{2}}{\bm{\omega}}.$
(5)

For particles of size $d\simeq 20\,{\rm{nm}}$ and vessel radius $R\simeq
2\,{\rm{cm}}$ we have $(d/R)^{2}\simeq 10^{-12}\kern 1.0pt$(!). Since $C$ is
always of the order of unity, Eq. (5) reduces to
${\bm{\Omega}}={\bm{\Omega}}_{0}\,$; this equality holds with high accuracy
for any values of $\phi$. Thus, the uniform rotation of the nanoparticles does
not in itself affect the rotation rate of the ferrofluid; the reasons for this
have been explained in [2]. Nonetheless, the spin of particles may turn on
indirect mechanisms of ferrofluid rotation [2], which are not covered in [1].

Using the erroneous initial assumption (1), Fang immediately arrived at the
mistaken equation for the ferrofluid magnetization

$\displaystyle\frac{d{\bf
M}}{dt}=\Big{(}\frac{1}{1-\phi}{\bm{\Omega}}+\frac{\phi}{1-\phi}{\bm{\omega}}\Big{)}\times{\bf
M}-\frac{\bf M-\bf M_{0}}{\tau_{M}}\,,$ (6)

and then to the mistaken expression [3] for the dimensionless rotational
viscosity $\eta_{R}=[\eta(H)-\eta(0)]/\eta(0)$:

$\displaystyle\eta_{R}=\frac{3}{2}\phi\kern 1.0pt\frac{\alpha
L^{2}(\alpha)}{(\alpha-L(\alpha))F(\alpha)}~{}~{}~{}~{}{\rm
where}~{}~{}~{}~{}F(\alpha)=1-\phi\kern 1.0pt\frac{\alpha
L^{2}(\alpha)}{\alpha-L(\alpha)}\,;$ (7)

here $L(\alpha)$ is the Langevin function. Fang compared Eq. (7) with the
result

$\displaystyle\eta_{R}=\frac{3}{2}\phi\kern 1.0pt\frac{\alpha
L^{2}(\alpha)}{\alpha-L(\alpha)},$ (8)

obtained in [4] on the basis of the Martsenyuk-Raikher-Shliomis theory [5]. As
can be seen, these equations differ from each other by the additional function
$F(\alpha)$ in the denominator of Eq. (7). The absence of this function in Eq.
(8), Fang explains by the fact that “Shliomis neglects the difference between
the vorticity of the suspension and the vorticity of the solvent.” This is
true: as shown above, neither this difference nor the solvent vorticity itself
exists.

Assumption (1) is the main, but not the only mistake made in [1]. Fang is
constantly confused about the rotational speeds of particles ${\bm{\omega}}$,
fluid ${\bm{\Omega}}$, and magnetization ${\bm{\omega}}_{M}$. “Should we
identify ${\bm{\omega}}_{M}$ with ${\bm{\omega}}$ or ${\bm{\Omega}}$ or
something else? This answer is far from obvious,”- he writes. Unfortunately,
he forgot about one more frequency: the speed of rotation of the magnetic
field ${\bm{\omega}}_{H}$. The magnetization ${\bf M}$ always follows external
magnetic field ${\bf H}$, so the rotation speeds of both vectors are the same,
${\bm{\omega}}_{M}={\bm{\omega}}_{H}$, and this is the only identity. Further
he writes: “The first magnetization relaxation equation accounting for the
flow effect was proposed by Shliomis in ${\rm{1972}}$ [6]:

$\displaystyle\frac{d{\bf M}}{dt}={\bm{\omega}}\times{\bf M}-\frac{\bf M-\bf
M_{0}}{\tau_{M}}\,.$ (9)

Here, [he continues] ${\tau_{M}}$ is called the Debye’s magnetization
relaxation time, which is often identified with [the Brownian one] $\tau_{B}$.
Hence (?!), the Shliomis–Debye model identifies ${\bm{\omega}}_{M}$ with
${\bm{\omega}}$.” This is a misunderstanding. As mentioned above,
${\bm{\omega}}_{M}$ coincides with the rotation frequency of the field, while
the particle spin ${\bm{\omega}}$ satisfies equation

$\displaystyle I_{MP}\frac{d{\bm{\omega}}}{dt}={\bf M}\times{\bf
H}-6\eta\phi({\bm{\omega}}-{\bm{\Omega}}),$

which reduces – due to the smallness of $I_{MP}$ – to equality

$\displaystyle{\bm{\omega}}={\bm{\Omega}}+\frac{{\bf M}\times{\bf
H}}{6\eta\phi}\,.$ (10)

The latter makes it possible to exclude ${\bm{\omega}}$ from the equations of
fluid motion and magnetization (9), the solution of which determines
${\bm{\Omega}}$ and ${\bf M}$, and thus also ${\bm{\omega}}$ by formula (10).
In a constant field $H$, where always $\omega_{M}=0$, magnetic particles
rotate in a vortex flow, so that $\omega\neq\omega_{M}$. The dependence
${\bm{\omega}}({\bm{\Omega}},{\bf H})$ manifests itself in the
magnetoviscosity, which for $\Omega\tau_{B}\ll 1$ is well described by the
theory [6].

The purpose of including subsection IV B ”Primary dielectric peak of water” in
the article is not clear. This subsection has nothing to do with hydrodynamics
and/or ferrofluids and is completely devoted to the well-known and well-
studied problem of molecular physics about the origin of the Debye absorption
peak in water. The author is simply sharing his thoughts on the subject here.

Summarizing, an attempt to build “a generic and consistent hydrodynamic theory
of ferrofluids” based on relation (1) incompatible with the law of
conservation of angular momentum was doomed to failure, which eventually
happened.

## Notes and references

  * [1] A. Fang, “Consistent hydrodynamics of ferrofluids,” Phys. Fluids 34, 013319 (2022).
  * [2] M. I. Shliomis, “How a rotating magnetic field causes ferrofluid to rotate,” Phys. Rev. Fluids 6, 043701 (2021).
  * [3] The original transcription of Eq. (59) in [1] differs from Eq. (7) because Fang used different notation: $L(\alpha)=m$ and $\alpha=L^{-1}(m)$.
  * [4] M. I. Shliomis, “Ferrohydrodynamics: Testing a third magnetization equation,” Phys. Rev. E 64, 060501(R). (2001).
  * [5] M. A. Martsenyuk, Yu. L. Raikher, and M. I. Shliomis, “Zh. Eksp. Teor. Fiz. 65, 834 (1973) [Sov. Phys. JETP 38, 413 (1974)].
  * [6] M. I. Shliomis, “Effective viscosity of magnetic suspensions,” Zh. Eksp. Teor. Fiz. 61, 2411 (1971) [Sov. Phys. JETP 34, 1291 (1972)].


# Stellar Masses and Star-Formation Rates of Galaxies and AGNs in the eFEDS
GAMA09 Field

Zhibo Yu (喻知博) Department of Astronomy and Astrophysics, 525 Davey Lab, The
Pennsylvania State University, University Park, PA 16802, USA  Email:
<EMAIL_ADDRESS>Institute for Gravitation and the Cosmos, The Pennsylvania
State University, University Park, PA 16802, USA  Fan Zou (邹凡) Department of
Astronomy and Astrophysics, 525 Davey Lab, The Pennsylvania State University,
University Park, PA 16802, USA  Institute for Gravitation and the Cosmos, The
Pennsylvania State University, University Park, PA 16802, USA  William N.
Brandt Department of Astronomy and Astrophysics, 525 Davey Lab, The
Pennsylvania State University, University Park, PA 16802, USA  Institute for
Gravitation and the Cosmos, The Pennsylvania State University, University
Park, PA 16802, USA  Department of Physics, 104 Davey Laboratory, The
Pennsylvania State University, University Park, PA 16802, USA

###### Abstract

The eFEDS is a wide $\approx 140\,\deg^{2}$ field that has extensive
multiwavelength coverage. To improve the utility of the existing data, we use
CIGALE to fit source Spectral Energy Distributions (SEDs) from X-rays to far-
infrared (FIR) mainly to derive stellar masses ($M_{\star}$) and star-
formation rates (SFRs) for normal galaxies and X-ray Active Galactic Nuclei
(AGNs). The catalog consists of 2,057,027 galaxies and 10,373 X-ray AGNs
located in the $\approx 60\deg^{2}$ GAMA09 sub-field. Comparing our
$M_{\star}$ with other available catalogs and our SFRs with FIR-derived SFRs,
we demonstrate the general reliability of our SED-fitting measurements. Our
catalog is publicly available at 10.5281/zenodo.10127224.

##  1 Introduction

The eROSITA Final Equatorial Depth Survey (eFEDS) was the largest
observational investment during the eROSITA performance verification phase.
The entire field, spanning $\approx 140\,\deg^{2}$, was observed to a depth of
$\approx 2.2\,\mathrm{ks}$ by eROSITA (Brunner et al., 2022; Salvato et al.,
2022). eFEDS was constructed to encompass the $\approx 60\deg^{2}$ GAMA09
field (Driver et al., 2022), which has rich multi-wavelength data. Given the
X-ray coverage from eROSITA and the well-cataloged photometric data from UV to
FIR on a $10^{2}$-$\deg^{2}$ scale, eFEDS is useful for both AGN and galaxy
studies. To facilitate future studies, we report $M_{\star}$ and SFRs for
$\approx\,2$ million sources in the eFEDS GAMA09 field primarily utilizing the
GAMA-derived photometric catalog that is based upon KiDS and VIKING imaging
data (Bellstedt et al., 2020). The SED fitting was performed with CIGALE
(Boquien et al., 2019; Yang et al., 2022).

##  2 Data and Methods

Our analysis focuses on the GAMA09 region since the remaining parts of eFEDS
lack sufficient multi-wavelength coverage, especially the NIR coverage
provided by KiDS and VIKING, which are crucial in deriving reliable
$M_{\star}$. We select X-ray AGNs based on the eFEDS X-ray main catalog
(Brunner et al., 2022), and the intrinsic X-ray fluxes are taken from Liu et
al. (2022) who corrected for absorption via X-ray spectral analyses. The rest
of the sources are classified as galaxies. The UV-to-FIR photometry is from
the GAMA-KiDS-VIKING (GKV) catalog compiled in Bellstedt et al. (2020). In
general, the multiwavelength coverage is uniform, reaching $5\sigma$ depths of
24.2 mag and 21.3 mag for $i$-band and $K_{S}$-band, respectively. We also
incorporate the HSC-Wide survey to support the optical coverage, which
uniformly covers the entire GAMA09 region with a $5\sigma$ limiting magnitude
of 26.1 mag in $i$-band (Aihara et al., 2022). The effective filter response
and zero-point calibration across different HSC bands are also corrected. Both
the GKV and HSC-Wide catalogs have accounted for Galactic extinction.
Additionally, we include Herschel FIR photometry from the HELP collaboration
(Shirley et al., 2021). The photometric redshifts (photo-zs) and spectroscopic
redshifts (spec-zs) are taken from the compilations of Salvato et al. (2022)
and Driver et al. (2022) for X-ray AGNs and normal galaxies, respectively. The
photo-zs generally have good quality with typical dispersions of a few percent
and outlier fractions of less than 10%. We also drop sources near bright stars
so that the impact on our overall photometry is minimal. Our final sample
contains 2,057,027 normal galaxies and 10,373 X-ray AGNs (including 9,856 with
primary counterparts and 517 with secondary counterparts based upon the
identifications of Salvato et al., 2022).

We apply the same methods as in Zou et al. (2022) to derive the best-fit
$M_{\star}$ and SFRs. Briefly, CIGALE assumes an energy balance principle and
decomposes a SED into several user-defined components (including AGNs). For
normal galaxies and X-ray AGNs, we use the dense-grid CIGALE parameter
settings for normal galaxies and AGN candidates in Zou et al. (2022),
respectively (see their Tables 4 and 5). We fit the near-UV (NUV) to FIR SEDs
for galaxies and add X-ray photometry for AGNs. To account for systematic
uncertainties, 0.05 mag error is added in quadrature for all bands from NUV to
NIR.

##  3 Results

To evaluate the reliability of our $M_{\star}$ measurements, we compare our
$M_{\star}$ for normal galaxies with the results from the GAMA collaboration
($M_{\star,\rm ref}$) (Taylor et al., 2011), which only include bright sources
with GAMA spectra. For X-ray AGNs, we refer to the $M_{\star}$ measurements
from Li et al. (2023). The comparisons are shown in the top-left panel of
Figure 1 where $\Delta\log M_{\star}=\log M_{\star}-\log M_{\star,\rm ref}$.
For 59,653 normal galaxies and 2,029 X-ray AGNs, $\sigma_{\rm NMAD}=0.12$ and
$0.23$ with median $\Delta\log M_{\star}=0.02$ and $-0.12$, where $\sigma_{\rm
NMAD}$ is the normalized median absolute deviation (NMAD).111NMAD is defined
as 1.4826 $\times$ median absolute deviation. To assess the SFR measurements,
we compare SFRs based upon our SED-fitting and FIR-based SFRs ($\rm
SFR_{FIR}$) that are derived following the method in Chen et al. (2013). We
also correct for old-star heating following Equation 25 in Zou et al. (2022).
Figure 1 top-right panel shows the comparisons between different SFRs for both
galaxies and X-ray AGNs, where
$\Delta\log\mathrm{SFR}=\log\mathrm{SFR}-\log\mathrm{SFR_{FIR}}$. For 34,610
galaxies and 862 X-ray AGNs with FIR signal-to-noise ratio (SNR) $>5$,
$\sigma_{\rm NMAD}=0.41$ and $0.28$ with median $\Delta\log\mathrm{SFR}=-0.19$
and $0.04$, respectively. These values are generally as good as those in Zou
et al. (2022) where they fit SEDs to three million sources in the $\approx
13.2\deg^{2}$ XMM-SERVS fields. Due to the large area and shallow X-ray depth
of eFEDS, there will be a significant fraction of BL AGNs whose AGN components
generally dominate the NIR emission, causing less reliable $M_{\star}\ $
measurements. Thus, we also plot Broad-Line AGN (BL AGN) candidates in Figure
1, which are selected as having AGN components constituting $>50\%$ of the
total flux density at rest-frame 1 $\mu$m. The impact of a high AGN
contribution on $M_{\star}\ $ is clearly shown by the widely scattered
$\Delta\log M_{\star}$ ($\sigma_{\rm NMAD}=0.66$). Apart from eFEDS, we also
apply the same method for SED-fitting to COSMOS, and the results are
consistent with the COSMOS2020 catalog by Weaver et al. (2022).

We further estimate the nominal depth of our measurements. We define “good
bands” as those with $\rm SNR>5$, and we plot the number of good bands vs.
$i$-band magnitude in Figure 1 bottom panel. The plot indicates that the SED
quality in eFEDS degrades at $i$-mag $\approx 21.5$. Approximately 30% of our
sources are brighter than this magnitude. Among these, 10% of normal galaxies
and 43% of X-ray AGNs have available spec-zs. We only show sources with
$i$-mag $<22$ in Figure 1 top panels, which constitute $\gtrsim 90\%$ of the
sources that we compared. We warn readers to use our catalog at $i$-mag
$\gtrsim 22$ cautiously, as fainter sources normally contain only $\sim 6$
bands primarily from HSC, and it is harder to constrain their properties due
to lack of NIR coverage.

Our catalog is available at 10.5281/zenodo.10127224. We provide the source
$M_{\star}$, SFRs, classifications (X-ray AGNs vs. normal galaxies), and
necessary information that can be helpful to cross-reference GAMA objects
and/or eFEDS X-ray/optical-IR counterparts. We also provide AGN fractional
contributions at rest-frame $5000$ Å, 1 $\mu$m, and integrated $8-1000$ $\mu$m
for X-ray AGNs to help readers identify sources with less-reliable
$M_{\star}$. Readers should also note that our AGN selection is based upon
X-ray detection only, which is generally a good tracer of black-hole accretion
rates (BHAR), but this selection is incomplete. To reach better completeness,
one should combine multiple selection methods such as mid-infrared (MIR; e.g.,
Zou et al., 2022) and radio (e.g., Zhu et al., 2023).

Figure 1: Top-left: Comparison between our $M_{\star}$ and $M_{\star}$ from
existing catalogs in eFEDS. Top-right: Comparison between our SED-based SFRs
and FIR-based SFRs. The dashed lines indicate 0.5 dex deviations. Note that we
only plot sources with $i$-mag $<22$. Bottom: Number of good bands (with $\rm
SNR>5$) vs. $i$-mag. Each blue point represents a source, and the red error
bars represent the median, 25th, and 75th percentile of the number of good
bands in each magnitude bin.

Acknowledgments

We acknowledge support from NSF grant AST-2106990 and Penn State.

##  Appendix A SEDs in COSMOS

The COSMOS field is one of the LSST Deep Drilling Fields (DDFs) and has been
well characterized by many past studies (e.g., Marchesi et al., 2016; Weaver
et al., 2022). Our purpose in working on COSMOS is to consistently measure the
galaxy properties via SED-fitting in the same manner as for other DDFs (e.g.,
Zou et al., 2022). In this work, our methods for COSMOS are the same as those
for eFEDS and Zou et al. (2022). We focus on the 1.27 deg2 UltraVISTA
footprint inside COSMOS because sufficiently deep NIR coverage is critical in
deriving reliable $M_{\star}$. The final catalog consists of 709,087 normal
galaxies and 2,209 X-ray AGNs. The absorption-corrected X-ray data are from
Civano et al. (2016) and Marchesi et al. (2016). The UV-to-MIR data are from
the Farmer catalog in the COSMOS2020 data release (Weaver et al., 2022). To
optimize the magnitude offsets in each band, we apply the correction in Table
3 of Weaver et al. (2022). We also add the 24–500 $\mu$m data from the “super-
deblended” FIR photometric catalog in Jin et al. (2018). The Galactic
extinction is corrected following the methods in Section 2.6 of Zou et al.
(2022). The redshifts for X-ray AGNs and normal galaxies are from Marchesi et
al. (2016) and Weaver et al. (2022), respectively. The quality of the photo-zs
is generally good, with typical dispersions of 1–4% and outlier fractions of a
few percent.

We compare our $M_{\star}$ with other measurements in the top-left panel of
Figure 2. The $M_{\star}$ values for 703,081 galaxies are compared with those
in the Farmer catalog. The median $\Delta\log M_{\star}=0.07$, and
$\sigma_{\rm NMAD}=0.15$. We compare 2,086 X-ray AGNs with those in Zou et al.
(2019), who also include AGN components when deriving $M_{\star}$. The median
$\Delta\log M_{\star}=0.09$, and $\sigma_{\rm NMAD}=0.21$. We also select BL
AGN candidates with AGN components contributing $>50\%$ of the total flux
density at rest-frame 1 $\mu$m. The result indicates this criterion is
efficient in selecting AGNs with problematic $M_{\star}$ measurements as
almost all sources with $\Delta\log M_{\star}<-0.5$ are selected. The impact
of these BL AGN candidates on COSMOS should be much reduced compared to eFEDS
because of the small area and greater depth of COSMOS. This is supported by
the much-reduced fraction of sources with AGN contribution at rest-frame 1
$\mu$m $>50\%$ in COSMOS (9%) than in eFEDS (32%). In addition, among our
X-ray AGNs, 36 are spectroscopically confirmed BL AGNs by Suh et al. (2020).
The median $\Delta\log M_{\star}$ is $-0.40$, which is consistent with the
findings in Zhuang et al. (2023). In Figure 2 top-right panel, we compare SED-
based SFRs and FIR-based SFRs corrected for old-star heating. For 4,017
galaxies and 225 AGNs with FIR SNR $>5$, $\sigma_{\rm NMAD}=0.23$ and $0.27$
with median $\Delta\log\rm SFR=-0.06$ and $0.05$, respectively. Our
measurements in COSMOS are generally consistent with the previous results.

The nominal depth in COSMOS is also shown in the bottom panel of Figure 2. The
number of good bands in COSMOS is much larger than that in eFEDS at the bright
end, but it dramatically degrades at $i$-mag $\approx 25.5$. Approximately 35%
of our sources are brighter than this magnitude. Among these, 64% of X-ray
AGNs have spec-zs from Marchesi et al. (2016).222The spec-zs for normal
galaxies in the COSMOS2020 data release are not publicly available, so we only
adopt their photo-zs. Since their photo-zs agree well with the spec-zs, the
adoption of photo-zs should not materially affect our conclusions. In the
Figure 2 top panels, we only show sources with $i$-mag $<26$. We warn readers
to use our catalog at $i$-mag $\gtrsim 26$ cautiously as fainter sources
generally have only a handful of bands, so the SED fitting becomes unreliable.

Figure 2: Top-left: Comparison between our $M_{\star}$ and $M_{\star}$ from
existing catalogs in COSMOS. Top-right: Comparison between our SED-based SFRs
and FIR-based SFRs. The dashed lines indicate 0.5 dex deviations. Note that we
only plot sources with $i$-mag $<26$. Bottom: Number of good bands (with $\rm
SNR>5$) vs. $i$-mag. Each blue point represents a source, and the red error
bars represent the median, 25th, and 75th percentile of the number of good
bands in each magnitude bin.

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[c]Rajnandini Mukherjee

# Charm quark mass using a massive nonperturbative renormalisation scheme

Luigi Del Debbio  Felix Erben  Jonathan Flynn  J. Tobias Tsang

###### Abstract

We present a first numerical implementation of a massive nonperturbative
renormalisation scheme, RI/mSMOM, in the study of heavy quarks using the
domain-wall fermion action. In particular, we calculate renormalisation
constants for fermion bilinears at non-vanishing heavy-quark masses and
compare the approach to the continuum of the renormalised charm-quark mass
with that from a mass-independent scheme.

##  1 Introduction

Lattice simulations allow for first principles computations of Standard Model
parameters such as quark masses in the nonperturbative regime of QCD. However,
the bare quantities calculated on the lattice need to be renormalised using
some intermediate renormalisation scheme for them to have a well-defined
continuum limit. There are many such schemes for the nonperturbative
renormalisation (NPR) of bare lattice quantities, such as RI/MOM [1], RI/SMOM
[2] and the Schrödinger functional method [3, 4]. However, these schemes are
implemented in the chiral limit of QCD, and therefore introduce $am$-sized
lattice artefacts when used for renormalising heavy-quark observables. A
massive momentum-subtraction scheme, called RI/mSMOM, prescribing NPR away
from the chiral limit, has been introduced in [5] to possibly ameliorate these
cutoff effects. Designed to have similar properties to the RI/SMOM scheme
(which is already $O(am)$-improved), this massive scheme aims to change
$O(a^{2}m^{2})$-sized lattice artefacts in heavy-quark observables,
potentially leading to smoother continuum extrapolations.

In this study, we present a pilot numerical implementation of the RI/mSMOM (or
massive) scheme on three ensembles with three different lattice spacings,
including a comparison to the RI/SMOM (or massless) scheme. We investigate its
effectiveness in mitigating cutoff effects by studying in particular the
renormalised charm-quark mass.

##  2 Nonperturbative renormalisation

Let us consider the Green’s function of a quark bilinear operator
$\mathcal{O}_{\Gamma}=\overline{\psi}_{f}\Gamma\psi_{f^{\prime}}$ between two
external off-shell quark lines in a fixed (Landau) gauge, given by (all
quantities in Euclidean space)

$\displaystyle
G_{\Gamma}(p_{3},p_{2})=\langle\psi_{f}(p_{3})\mathcal{O}_{\Gamma}(q)\overline{\psi}_{f^{\prime}}(p_{2})\rangle,$
(1)

where $\psi_{f}$ and $\psi_{f^{\prime}}$ are quark fields of different
flavours, and $q=p_{2}-p_{3}$ (see figure 1 for conventions used). We are
interested in scalar ($\Gamma=\mathbb{1}$), pseudoscalar
($\Gamma=i\gamma_{5}$), vector ($\Gamma=\gamma^{\mu}$) and axial vector
($\Gamma=\gamma^{\mu}\gamma_{5}$) bilinears. The quark propagator is defined
as

$\displaystyle
S_{f}(p)=\langle\psi_{f}(p)\overline{\psi}_{f}(p)\rangle=\frac{1}{i\cancel{p}+m_{f}},$
(2)

and the amputated Green’s function is obtained by amputating each leg with the
inverse quark propagator of the corresponding flavour

$\displaystyle\Lambda_{\Gamma}(p_{2},p_{3})=S^{-1}_{f}(p_{3})G_{\Gamma}(p_{3},p_{2})S^{-1}_{f^{\prime}}(p_{2}).$
(3)

In our study, we consider the flavour diagonal case
$\psi_{f}=\psi_{f^{\prime}}=c$, so we drop the flavour subscripts from here
on. The propagators are also related to the amputated Green’s functions of the
vector and axial vector operators via the Ward-Takahashi identities

$\displaystyle q_{\mu}\Lambda_{V}^{\mu}(p_{2},p_{3})$
$\displaystyle=iS^{-1}(p_{3})-iS^{-1}(p_{2}),$ (4) $\displaystyle
q_{\mu}\Lambda_{A}^{\mu}(p_{2},p_{3})$
$\displaystyle=-2m\Lambda_{P}(p_{2},p_{3})+i\gamma_{5}S^{-1}(p_{2})+S^{-1}(p_{3})i\gamma_{5}.$
(5)

Renormalised and bare quantities are related via renormalisation constants $Z$
as

$\displaystyle\psi_{R}$ $\displaystyle=Z_{q}^{1/2}\psi,\quad
m_{R}=Z_{m}m,\quad\mathcal{O}_{\Gamma,R}=Z_{\Gamma}\mathcal{O}_{\Gamma},$ (6)
$\displaystyle\implies S_{R}(p)$
$\displaystyle=Z_{q}S(p),\quad\Lambda_{\Gamma,R}(p_{2},p_{3})=\frac{Z_{\Gamma}}{Z_{q}}\Lambda_{\Gamma}(p_{2},p_{3}).$
(7)

Renormalised quantities are denoted with a subscript $R$, while bare
quantities are without a subscript. The renormalisation conditions in the
RI/mSMOM scheme are

$\displaystyle Z_{q}:\quad$
$\displaystyle\left.\lim_{m_{R}\rightarrow\overline{m}}\frac{1}{12p^{2}}\operatorname{Tr}\left[-iS_{R}(p)^{-1}\cancel{p}\right]\right|_{p^{2}=\mu^{2}}=1,$
(8) $\displaystyle Z_{m}:\quad$
$\displaystyle\lim_{m_{R}\rightarrow\overline{m}}\frac{1}{12m_{R}}\left\\{\left.\operatorname{Tr}\left[S_{R}(p)^{-1}\right]\right|_{p^{2}=\mu^{2}}+\left.\frac{1}{2}\operatorname{Tr}\left[\left(iq\cdot\Lambda_{\mathrm{A},R}\right)\gamma_{5}\right]\right|_{\mathrm{sym}}\right\\}=1,$
(9) $\displaystyle Z_{V}:\quad$
$\displaystyle\lim_{m_{R}\rightarrow\overline{m}}\frac{1}{12q^{2}}\operatorname{Tr}\left[\left.\left(q\cdot\Lambda_{\mathrm{V},R}\right)\cancel{q}\right]\right|_{\mathrm{sym}}=1,$
(10) $\displaystyle Z_{A}:\quad$
$\displaystyle\left.\lim_{m_{R}\rightarrow\overline{m}}\frac{1}{12q^{2}}\operatorname{Tr}\left[\left(q\cdot\Lambda_{\mathrm{A},R}\,+\,2m_{R}\Lambda_{P,R}\right)\gamma_{5}\cancel{q}\right]\right|_{\mathrm{sym}}=1,$
(11) $\displaystyle Z_{P}:\quad$
$\displaystyle\left.\lim_{m_{R}\rightarrow\overline{m}}\frac{1}{12i}\operatorname{Tr}\left[\Lambda_{\mathrm{P},R}\gamma_{5}\right]\right|_{\mathrm{sym}}=1,$
(12) $\displaystyle Z_{S}:\quad$
$\displaystyle\left.\lim_{m_{R}\rightarrow\overline{m}}\left\\{\frac{1}{12}\operatorname{Tr}\left[\Lambda_{\mathrm{S},R}\right]+\,\frac{1}{6q^{2}}\operatorname{Tr}\left[2m_{R}\Lambda_{P,R}\gamma_{5}\cancel{q}\right]\right\\}\right|_{\mathrm{sym}}=1.$
(13)

The subscript ‘sym’ denotes the symmetric momentum configuration
$q^{2}=p_{2}^{2}=p_{3}^{3}=\mu^{2}$, where $\mu$ is the renormalisation scale.
These kinematics are the same as those in the massless scheme, RI/SMOM. The
massive scheme requires the introduction of another scale, $\overline{m}$, a
renormalised mass at which the renormalisation conditions are imposed. The
massless scheme is recovered in the limit $\overline{m}\to 0$.

Figure 1: Choice of kinematics. The dashed bubble represents the operator
insertion and higher order corrections, $p_{2}$ and $p_{3}$ are the momenta of
the external off-shell quark lines.

These conditions are defined such that the vector and axial vector Ward-
Takahashi identities (4) and (5) are satisfied by the renormalised quantities,
and the renormalisation constants preserve the relations

$\displaystyle Z_{V}=Z_{A}=1,\quad Z_{P}=1/Z_{m},\quad Z_{S}=Z_{P},$ (14)

as in the continuum $\overline{\text{MS}}$ scheme.

##  3 Computational details

Given the framework outlined above, the bare quantities needed from lattice
simulations are the Green’s functions for the bilinear operators
$G_{\Gamma}(p_{2},p_{3})$ for $\Gamma=S,P,V,A$, and the corresponding external
leg propagators $S(p_{2})$ and $S(p_{3})$. Using these, we can compute the
amputated Green’s functions and impose the renormalisation conditions to
calculate the $Z$-factors in the RI/mSMOM scheme.

We use three RBC/UKQCD ensembles with unphysical pion masses using $N_{f}=2+1$
flavours of domain wall fermions (DWF). These ensembles, called C1, M1 and F1S
[6, 7, 8, 9, 10], have inverse lattice spacings $a^{-1}\approx 1.78,2.38,$ and
$2.79\,\mathrm{GeV}$ respectively. We compute bilinear Green’s functions and
external leg propagators for various bare quark masses $am_{0}$ including the
light-quark mass and heavy-quark masses around the charm-quark mass. This is
done for various external momenta $ap_{\mu}$ corresponding to physical momenta
in the range $2\,\mathrm{GeV}\lesssim\sqrt{p^{2}}\lesssim 4\,\mathrm{GeV}$.

The key point of the RI/mSMOM scheme is the imposition of the renormalisation
conditions at some finite value of a renormalised mass $\overline{m}$. We
generate connected pseudoscalar-pseudoscalar two-point correlation functions
for the various choices of simulated bare quark masses. We denote their
respective ground-state masses $\eta_{h}(am)$. Interpolations in the values
for $\eta_{h}$ to a common value on all ensembles can then used to fix
$\overline{m}$.

Furthermore, in the DWF setup the bare mass requires additive renormalisation
that depends on the domain wall parameters, in addition to multiplicative
renormalisation. The renormalised mass is thus
$m_{R}=Z_{m}m=Z_{m}(m_{0}+m_{\text{res}})$, where $m_{0}$ is the bare mass
which is an input to the simulation, and the residual mass $m_{\text{res}}$ is
measured for each choice of bare mass. In the discussion that follows, $m$
will always denote the multiplicatively renormalised bare mass.

##  4 Methodology

In this section, we describe the procedure used to implement the RI/mSMOM
scheme to the study of the charm-quark mass. As we repeat this calculation on
multiple ensembles, we need a lattice-independent reference quantity that
allows us to compare the results across different lattice spacings. The
$\eta_{h}$ meson mass in physical units enters as this reference quantity and
we use it to set the various scales involved in this procedure.

We start by using the bilinear Green’s functions and external leg propagators
to calculate the renormalisation constants $Z(a\mu,am)$ in the massive scheme
over a range of renormalisation scales $a\mu$ and bare masses $am$, on each
ensemble.

Next, we compute the bare charm-quark mass $m_{c}$ on each ensemble. This is
done by interpolating to the bare mass that corresponds to the physical
$\eta_{c}$ mass as listed in the PDG [11] as

$\displaystyle
m_{c}=a^{-1}\cdot(am_{c})=a^{-1}\cdot\big{(}am(M_{\eta_{h}}=M_{\eta_{c}})\big{)}.$
(15)

Having computed the bare charm-quark mass, we renormalise it using $Z_{m}$ as
calculated using eqn (9). For this we choose the renormalisation scale
$a\mu=a\cdot(2$ GeV$)$ (keeping in mind the Rome-Southampton window
$\Lambda_{\text{QCD}}^{2}\ll\mu^{2}\ll(\pi/a)^{2}$). Additionally, we
introduce a physical scale by choosing a value of the $\eta_{h}$ mass, denoted
by $M^{\star}$. On each ensemble, we can then interpolate to the corresponding
bare mass $am^{\star}$ given by

$\displaystyle am^{\star}=am(M_{\eta_{h}}=M^{\star}).$ (16)

This choice of a physical $\eta_{h}$ mass connects the choice of bare masses
across the different lattice spacings and allows us to subsequently take a
continuum limit to obtain the renormalised charm-quark mass

$\displaystyle m_{c,R}(\mu,M^{\star})=\lim_{a\to 0}Z_{m}(a\mu,am^{\star})\cdot
a^{-1}\cdot(am_{c}).$ (17)

The renormalised mass $\overline{m}$ is calculated by extrapolating the
renormalised bare mass $m^{\star}$

$\displaystyle\overline{m}(\mu,M^{\star})=\lim_{a\to
0}Z_{m}(a\mu,am^{\star})\cdot a^{-1}\cdot(am^{\star}).$ (18)

The choice of the physical scale $M^{\star}$ therefore gives us a handle on
the scheme-defining renormalised mass $\overline{m}$. We perform this
continuum extrapolation for different choices of $M^{\star}$ — corresponding
to different $\overline{m}$ — to study the absorption of cutoff effects in the
final results, and compare it to the massless scheme. In order to make this
comparision, the final step of the procedure is to match the results for each
$\overline{m}$ to a scheme $S$ in the continuum (for example,
$\overline{\text{MS}}$) such that the dependence on $\overline{m}$ is removed

$\displaystyle
m_{c,R}^{S}(\mu)=R_{m}^{S\leftarrow\text{mSMOM}}\left(\mu,\overline{m}\right)m_{c,R}(\mu,\overline{m}).$
(19)

It is expected that the renormalised charm-quark mass from the different
choices of $\overline{m}$ should coincide with each other and with results
from RI/SMOM in this definition.

##  5 Results

Figure 2: ($\mu=2$ GeV) Top row: $\eta_{h}$ meson mass $M_{\eta_{h}}$ vs bare
mass $am$. $M_{\eta_{c}}^{\text{PDG}}$ corresponds to $am_{c}$ and choice of
physical scale $M^{\star}$ corresponds to $am^{\star}$ on each lattice. Middle
row: renormalised charm mass $Z_{m}m_{c}$ interpolated to $am^{\star}$. Bottom
row: renormalised mass $Z_{m}m$ interpolated to $am^{\star}$.

(a) Continuum extrapolation of the renormalised charm mass in the massless
scheme and in the massive scheme for various choices of $M^{\star}$.

(b) Continuum extrapolation of the renormalised mass for various choices of
$M^{\star}$, corresponding to the scheme-defining renormalised mass
$\overline{m}$.

Figure 3: Continuum extrapolations of $Z_{m}m_{c}$ and $Z_{m}m^{\star}$ using
a quadratic ansatz in $a^{2}$. Renormalised charm mass from different schemes
and choices of $M^{\star}$ should converge after all quantities are matched to
a common scheme. Errorbars include statistical uncertainties and the error in
lattice spacing only.

The methodology described above is implemented using three ensembles, and the
preliminary results are shown in figures 2 and 3. Figure 2 shows the procedure
of referencing the $\eta_{h}$ meson mass measurements to calculate the bare
charm mass on each lattice, and implementing the NPR procedure at a chosen
physical scale $M^{\star}$. The top panel shows the interpolation performed
for computing the bare charm mass in lattice units $am_{c}$ using the physical
value of $M_{\eta_{c}}$. Additionally, the scale $M^{\star}$ is introduced,
and again an interpolation is performed to find the corresponding bare mass in
lattice units $am^{\star}$. Having applied these common scales across all
lattice spacings, we can compute the renormalised charm mass using
$Z_{m}(a\mu,am^{\star})$, as shown in the middle panel. The same
renormalisation constant is also used for renormalising the bare mass in the
bottom panel.

Having computed the renormalised charm and bare masses on all lattices for
some choice of $M^{\star}$, we proceed to perform continuum extrapolations, as
prescribed in equations (17) and (18), to obtain the continuum quantities
$m_{c,R}(\mu,M^{\star})$ and $\overline{m}(\mu,M^{\star})$. This procedure is
shown in figures 3(a) and 3(b) respectively. This is the extrapolation where
we expect absorption of higher order cutoff effects with the use of the
massive scheme.

In figure 3(a), we see that higher order cutoff effects are evidently present
in the use of both the massive and massless schemes. We observe an alleviation
of $\mathcal{O}(a^{4})$ effects with the use of the massive scheme, however
this is not yet quantified using a variation in fit ansatze, or additional
lattice spacings. The cutoff effects in these continuum extrapolations are
further exacerbated by the small errors on the datapoints, which currently
only include statistical uncertainties and the error in lattice spacing, and
we plan to improve upon these with a careful study of systematic effects that
arise in the several successive steps of interpolations in the procedure. The
inclusion of these systematic errors in the massive scheme may also allow
renormalised quantities to be canditates for a linear fit in $a^{2}$ for the
continuum extrapolation, for some choices of $M^{\star}$.

##  6 Conclusions and future directions

We have performed a preliminary numerical implementation of a massive
momentum-subtraction scheme and used the renormalised charm-quark mass as a
test case to study its effect on higher-order lattice artefacts.
Qualitatively, employing this massive scheme, RI/mSMOM, helps reduce the
severity of $O(a^{4})$ cutoff effects in the continuum extrapolation, when
compared to the same extrapolation using the massless scheme, RI/SMOM. In
order to further investigate and quantify this claim, this study needs to be
extended to include systematic uncertainties and fit variations.

It is also of interest to do a systematic study of the dependence of this
change in lattice artefacts with the choice of $\overline{m}$ (in practice,
the choice of $M^{\star}$), as it could be expected that there would be an
identifiable ‘window’ of choices that are most effective in absorbing cutoff
effects.

Note that the final step of matching to a common scheme still remains to be
implemented, making use of one-loop perturbative calculations for $Z_{m}$ at
finite quark mass (already computed in the original work [5]). This step will
also allow us to verify the convergence of all the schemes, and to produce a
value of the renormalised charm-quark mass in a universal continuum scheme
such as $\overline{\text{MS}}$ which can be directly compared to other
computations.

If successful, this massive scheme could be used in the study of other
bilinear operators, and the RI/mSMOM prescription could potentially be
extended for four-quark operators as well.

##  7 Acknowledgements

We thank Christopher Sachrajda, Andreas Jüttner, Peter Boyle, and Matteo Di
Carlo for several useful discussions and inputs to this project. This work
used the DiRAC Extreme Scaling service at the University of Edinburgh,
operated by the Edinburgh Parallel Computing Centre on behalf of the STFC
DiRAC HPC Facility (www.dirac.ac.uk) under project codes dp008 and dp207 using
Grid and Hadrons. This equipment was funded by BEIS capital funding via STFC
capital grant ST/R00238X/1 and STFC DiRAC Operations grant ST/R001006/1. DiRAC
is part of the National e-Infrastructure. R.M. is supported by the
Presidential Scholarship from the University of Southampton. F.E. has received
funding from the European Union’s Horizon Europe research and innovation
programme under the Marie Skłodowska-Curie grant agreement No 101106913.

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