11institutetext: Gran Sasso Science Institute, L’Aquila, Italy
22institutetext: INFN Laboratori Nazionali del Gran Sasso, L’Aquila, Italy
33institutetext: INAF - Osservatorio Astronomico d’Abruzzo, Teramo, Italy
44institutetext: University of Naples Federico II, Naples, Italy
55institutetext: INAF - Osservatorio Astronomico di Roma, Monte Porzio Catone
(Roma), Italy 66institutetext: NSF’s NOIRLab, Tucson, AZ 85719, USA

# A study of globular clusters in a lenticular galaxy in Hydra I from deep
HST/ACS photometry

Nandini Hazra 112233 Michele Cantiello 33 Gabriella Raimondo 33 Marco Mirabile
3344 John P. Blakeslee 66 Marica Branchesi 112233 Enzo Brocato 3355

###### Abstract

Aims. We take advantage of exquisitely deep optical imaging data from HST/ACS
in the F475W ($g_{F475W}\ $) and F606W ($V_{F606W}\ $) bands, to study the
properties of the globular cluster (GC) population in the intermediate mass
lenticular galaxy PGC 087327, in the Hydra I galaxy cluster.

Methods. We inspect the photometric (magnitudes and color) and morphometric
(compactness, elongation, etc.) properties of sources lying in an area of
$\sim 19\times 19$ kpc centered on PGC 087327 and compare them with four
neighbouring fields over the same HST/ACS mosaic. This allowed us to identify
a list of GC candidates and to inspect their properties using a background
decontamination method.

Results. Relative to four comparison fields, PGC 087327 shows a robust
overdensity of GCs, $N_{GC}=82\pm 9$. At the estimated magnitude of the
galaxy, this number implies a specific frequency of $S_{N}=1.8\pm 0.7$. In
spite of the short wavelength interval available with the $g_{F475W}\ $and
$V_{F606W}\ $passbands, the color distribution shows a clear bimodality with a
blue peak at $\langle g_{F475W}{-}V_{F606W}\rangle=0.47\pm 0.05$ mag and a red
peak at $\langle g_{F475W}{-}V_{F606W}\rangle=0.62\pm 0.03$ mag. We also
observe the typical steeper slope of the radial distribution of red GCs
relative to blue ones. Thanks to the unique depth of the available data, we
characterize the GC luminosity function (GCLF) well beyond the expected GCLF
turn-over. We find $g^{TOM}_{F475W}=26.54\pm 0.10$ mag and
$V^{TOM}_{F606W}=26.08\pm 0.09$ mag, which after calibration yields a distance
of $D_{GCLF}=56.7\pm 4.3(statistical)\pm 5.2(systematic)$ Mpc.

###### Key Words.:

galaxies: distances and redshifts – galaxies: elliptical and lenticular, cD –
galaxies: star clusters: general – galaxies: individual: PGC 087327

##  1 Introduction

Globular clusters (GC) are dense stellar systems, with typically old ages,
found ubiquitously in massive galaxies and spanning a wide range of magnitudes
(e.g. Brodie & Strader, 2006). The small number of available spectroscopic
studies of extragalactic GCs inferred that the majority of them have ages
comparable to galactic GCs (e.g., Cohen et al., 1998; Beasley & Sharples,
2000; Kuntschner et al., 2002), namely $t\geq 10$ Gyr (Carretta et al., 2000).
In all cases where the population of GCs in a spheroidal galaxy, either
elliptical or lenticular, was observed with an intermediate age component
($t\sim 3-6$ Gyr), this only composed a small fraction of the GC population,
mostly in merger remnants like NGC 3610 or NGC 1316 (Goudfrooij et al., 2001;
Brodie & Strader, 2006; Bassino & Caso, 2017).

Throughout this paper we will focus only on the old GCs component. They are
often some of the most luminous non-transient objects in a galaxy and exhibit
a variety of properties (magnitudes, colors, radial distributions, sizes and
so on) that are used as tracers of galaxy formation and evolution (e.g. Brodie
& Strader, 2006). These old GC systems have been used as distance indicators
since Hanes (1977) due to the fact that they have a nearly universal Gaussian
luminosity function, with a peak at a constant absolute magnitude of
$M_{V}^{TO}\sim-7.6$ mag (Georgiev et al., 2009), known as the turn-over
magnitude. The near-universality of the globular cluster luminosity function
(GCLF) in optical and near-IR bands, has prompted its use as a standard candle
to act as a secondary distance indicator (Ferrarese et al., 2000).

Old GC populations typically exhibit a bimodal color distribution, with a blue
(metal-poor) and a red (metal-rich) peak. This has historically been
attributed to hierarchical formation giving rise to two distinct GC sub-
populations with different peak metallicities (Brodie & Strader, 2006).
Additionally, while the red GC system generally shows a radial profile which
roughly matches with the galaxy field star profile (Harris, 2009), the blue,
more metal-poor GCs, are often observed to be less concentrated close to the
galaxy core and more numerous than the red GCs at larger radii. This seems to
be indicative of an in-situ red GCs population, and of a blue population
acquired through galaxy mergers and tidal events (Forbes et al., 2011).

However, recent works (Yoon et al., 2006; Richtler, 2006; Cantiello et al.,
2007; Cantiello & Blakeslee, 2007, 2012, among others) have shown that the
bimodality could also arise from a continuous metallicity distribution with a
radial gradient, combined with non-linear color-metallicity relations. This
could point to stochastic processes in galaxy formation, without requiring two
major events or mechanisms to generate the two observed sub-populations.

In this work, we take advantage of the exquisite resolution of HST/ACS data,
combined with the extremely deep observations of NGC 3314A/B, to characterize
the GC system in PGC 087327. In particular, we analyze the color and radial
distributions, and the luminosity function of the old GCs in this galaxy.

PGC 087327 is an intermediate-mass galaxy (see Sect. 2) classified as E3/S0,
close in projection to NGC 3314A/B, in the Hydra I cluster. Using the flow-
corrected peculiar velocity from Cosmicflows-3 (Tully et al., 2016) reported
in Table 1, and an $H_{0}\sim 73$ km s-1 Mpc-1(e.g. Blakeslee et al., 2021;
Khetan et al., 2021), we obtain a preliminary estimate of the distance, $D\sim
61$ Mpc. Later in this work, we will use the photometry of GCs to derive a
more refined estimate of this distance.

Table 1: Main properties of PGC 087327

Quantity | Value | Reference
---|---|---
$v_{Helio.}$ | 4114 $km\ s^{-1}$ | Smith et al. (2000)
$v_{Corr.}$ | 4449 $km\ s^{-1}$ | Tully et al. (2016)
Mass | $\sim 10^{10}M_{\odot}$ | This work
Dist.($H_{0}$) | 61 Mpc | Hubble-Lemaitre law & $v_{Corr.}$
$E_{B-V}$ | 0.063 | IRSA
$m_{B}$ |  $15.6$ maga | Bernardi et al. (2002)
$m_{Ks}$ |  $11.7$ maga | 2MASS
$m_{H}$ |  $13.3$ mag | 2MASS
$m_{z}$ |  $13.5$ mag |  $m_{H}$ & $\langle z{-}H\rangle\sim 0.2$ mag
$m_{F606W}$ | 14.7 mag | This work
$m_{F475W}$ | 15.3 mag | This work

Total magnitudes, $m_{band}$, have not been corrected for extinction.
Magnitudes in the $K_{s}$, $H$, $z$ bands as well as those estimated in this
work (see Sect. 2.3) are from fit extrapolation estimates.
Note: $a)$ These magnitudes are in Vegamag, for sake of comparison with the
literature used throughout this work.

This paper is organised as follows: we describe the observational dataset and
target in Section 2, and the procedures to identify GC candidates in Sect. 3.
The analysis of the main properties of the GC sample, along with the
calibrations and results are outlined in Sect. 4, and the conclusions are
summarized in Sect. 5.

##  2 Observational Data

###  2.1 Target properties

The main properties of PGC 087327 are given in Table 1. Figure 1 shows the
color composite image of the galaxy based on the HST data used in this work.

Figure 1: Color composite $2\arcmin\times 1\arcmin$ HST image of PGC 087327.
North is up and east is left

.

We derived an approximate estimate of the galaxy mass using stellar population
models and empirical relations. For the estimate from models we adopted the
colour-mass-luminosity relations of Into & Portinari (2013, see their Tab. 3)
together with the $K_{s}$ magnitude from 2MASS (the magnitude from fit
extrapolation, see Tab. 1), the distance from the Hubble-Lemaitre law and a
range of $V{-}I$ color of 1-1.25 mag (e.g. Tonry et al., 2001). With such
assumptions we evaluate a total mass in the range
$9.85\leq\log(M/M_{\odot})\leq 10.2$, depending on the color used. Using the
empirical mass-luminosity relation from Cappellari (2013, their eq. 2),
combined with the $K_{s}$ magnitude and distance, we obtained
$\log(M/M_{\odot})\sim 10$. From these estimates, PGC 087327 appears to be an
intermediate mass lenticular galaxy.

For the GCLF calibrations (see Sect. 4.4) we need the $B$\- and $z$-band
magnitudes of the galaxy. We adopted the $m_{B}$ estimate from Bernardi et al.
(2002, PGC 087327 is D 135 in their catalog), reported in Table 1. For the
$z$-band magnitude, we used the 2MASS $m_{H}=13.33$ mag111Throughout this
work, we will always consider the ABmag photometric system, unless specified
otherwise. and a $z{-}H\sim 0.2$ color –appropriate for elliptical galaxies,
Lee & Chary (2020)–, thus obtaining $m_{z}=13.53$ mag.

Additionally, we find a $B$-band mass-luminosty ratio of $2.5\leq(M/L)\leq 3$
for this galaxy, which compared to the predictions of Into & Portinari (2013,
see their Fig. 4) indicates stellar population ages older than $\sim 10$ Gyr
for all except the highest metallicities. A field stellar component with
metallicities $[Fe/H]\geq 0$ is ruled out by the measured galaxy color
(computed later in Sect. 2.3, total magnitudes without extinction correction
are in Table 1), which is observed to be slightly bluer (by $\sim 0.1$ mag)
than model predictions for intermediate-age metal rich populations.

###  2.2 HST Data

The data for this analysis was retrieved from the Hubble Legacy Archive222See
https://hla.stsci.edu/, and are part of the gravitational microlensing survey
in the NGC 3314A/B galaxy pair (HST Proposal 9977, PI. D. Bennett).

The observations were carried out in the $F475W$ and $F606W$ passbands
(hereafter also referenced as $g_{F475W}\ $and $V_{F606W}\ $, respectively).
We downloaded the combined images based on the standard HST drizzling and
calibration pipeline.

Table 2 provides information about the observed dataset and the ABmag
zeropoints we adopted. The full frame is $\sim 5\arcmin$ wide, and centered on
the double spiral galaxy NGC 3314A/B. The original ACS resolution is
0.05$\arcsec/pix$ but the mosaic downloaded from the Hubble Legacy Archive has
been drizzled to a lower pixel resolution of $0.04\arcsec/pix$ owing to the
very large number of dithered exposures ($N_{exp}=120$). In order to isolate
the GCs around our target, we chose a cutout of the frame centered on PGC
087327, having an approximate width of $1.1\arcmin\times 1.1\arcmin$
($1600\times 1600$ pixels $\approx 19\times 19$ kpc at the assumed galaxy
distance of 61 Mpc). The galaxy lies at $\sim 1.7\arcmin$ from the overlapping
spirals, and $\sim 10\arcmin$ away from the closest of the two brightest
cluster galaxies (BCGs) in Hydra I, i.e. NGC 3311.

Because of possible residual contamination of GCs belonging to the
neighbouring BCG and spiral galaxies, as well as contamination from other
fore- and back-ground sources, we compare the target frame with a set of
background reference fields from the same HST/ACS mosaic. In particular, we
used four background regions in the vicinity of PGC 087327, chosen to be at
approximately the same distance from NGC 3314A/B, and far away from any
obvious bright source in the field. The coordinates and properties of these
fields are given in Table 3. Figure 2 shows the position of PGC 087327 and the
four background fields.

Table 2: Properties of the galaxy frame in each band

Property | $g_{F475W}\ $ | $V_{F606W}\ $
---|---|---
Exposure time (s) | 62369.18 | 64800
Zeropoint (AB mag) | 26.071 | 26.507
Encircled Energy (r=0.12”)a | 0.7195 | 0.7145
Extinction $A_{\lambda}$ (mag) | 0.207 | 0.157
$FWHM$ | 0.11″ | 0.12″

Note: $a)$ See Sect. 2.4 for definition and details.

Figure 2: The image of the full frame in $g_{F475W}\ $band with the region of
PGC 087327 highlighted in magenta, and the four regions used as background
frames in black. Each region is $1.1\arcmin\times 1.1\arcmin$. The most
conspicuous object in the centre of this mosaic is NGC 3314A/B.

###  2.3 Modelling the galaxy

In order to improve GC detection in regions of the galaxy with high surface
brightness, we subtracted the galaxy mean profile. We modelled the galaxy with
elliptical isophotes using the Astropy affiliated package photutils (v1.0.2)
in Python.

We first initialized a rough galaxy model using first-guess ellipticity and
position angle parameters using the EllipseGeometry class in photutils. Then
we initialized an object of the Ellipse class with the unmasked galaxy image
data and this geometry, and performed a short and coarse fit using the
fit_image routine within Ellipse. This provided us with a list of isophotes in
the form of an object of the Isophote class, which were used to generate more
refined starting parameters for the final fitting which would be performed on
the masked image.

To obtain the mask, we ran the sewpy wrapper333See
https://github.com/megalut/sewpy for SExtractor (Bertin & Arnouts, 1996),
generating separate photometric catalogs of the extended and compact objects
in the frame. At this stage of the selection, we only masked the brightest
objects ($mag<24/25$ for extended/compact sources, respectively, in both
passbands), and separated extended from compact sources using the SExtractor
CLASS_STAR parameter (CLASS_STAR$>$0.85 for compactness).

The final fit for elliptical isophotes was then performed on the masked image
of PGC 087327 using the fit_image routine. A low-surface brightness feature
(possibly a diffuse foreground galaxy) and the bright star south-east of the
core of PGC 087327, both visible in Figure 1, were also masked out before
running the isophotes fitting. The galaxy modelling run went out to a semi-
major axis of $26.5\arcsec$, where the galaxy surface brightness level reaches
$\approx$26.3/26.0 $mag/arcsec^{2}$ in $g_{F475W}\ $an $V_{F606W}\ $-band,
respectively. The model was synthesized from the list of isophotes generated
by fit_image, using the build_ellipse_model function within Ellipse. Finally,
the residuals were generated by subtracting the model from the original image
of PGC 087327. Figure 3 shows the $g_{F475W}\ $-band image of the galaxy and
the smoothed residuals.

The Isophote class also provides us with information on the total flux within
each fitted elliptical isophote, which is stored in the variable
”$tflux\\_e$”. Using a curve of growth analysis on the total flux within each
isophote, we determine the total apparent magnitude of the galaxy in both
$g_{F475W}\ $and $V_{F606W}\ $. The values of this analysis are quoted in Tab.
1.

We carried out an additional test with Galfit (Peng et al., 2010) in order to
estimate the disk-to-bulge ratio using a combined fit of an elliptical and a
Sersic component to the galaxy. We find that the $B/T$ (bulge-to-total) ratio
is $0.76$, implying an $D/B\approx 0.32$ (disk-to-bulge ratio). This, combined
with the Sersic index of 5.8 from our Galfit run puts PGC 087327 firmly in the
E/S0 category (Baillard, A. et al., 2011).

Figure 3: Left: PGC 087327 $g_{F475W}\ $-band image. Right: galaxy subtracted
residual image. We further subtracted a smooth background map derived from
SExtractor, to improve the visibility of GC candidates near the galaxy center.
The image size is $1.0\arcmin\times 0.85\arcmin$

.

###  2.4 Source detection and photometry

Once we have the residual image, we re-run SExtractor to generate the catalog
of sources in the image cutout. The catalog will be used to identify GCs in
the frame.

The photometry of point-like sources is aperture and extinction corrected as
follows. We obtain the extinction correction from the IRSA Dust query module
of Astropy, which gives us the value of $E_{B-V}$ at the position of our
field(s). The value of $R_{V}$ is adopted from Schlafly & Finkbeiner (2011).
The aperture correction values were obtained from the instrument web-pages,
using the on-line encircled energy plots444See
https://www.stsci.edu/hst/instrumentation/acs/data-analysis/aperture-
corrections, derived by Bohlin (2016), where the encircled energy is the
fraction of flux contained within a certain radius of aperture (Sirianni et
al., 2005). We chose an aperture radius of $r=0.12\arcsec$, which corresponds
approximately to the FWHM of our data (see Tab. 2).

We verified the adopted encircled energy values by obtaining the aperture
correction from the analysis of the most isolated, compact and bright objects
in the field. By inspecting the curve of growth (i.e. the aperture magnitude
at different radii) out to 64 pixels aperture, we obtained aperture correction
values that agree within $\pm$0.04 mag with the ones from the HST/ACS
calibration team. We finally matched the aperture and extinction corrected
catalogs across the $g_{F475W}\ $and $V_{F606W}\ $-bands, using a matching
radius of 3 pixels ($\sim 0.12\arcsec$). This was done in order to select only
the sources that appear in both bands.

The full catalog of GC candidates identified is available Tab.
LABEL:tab:gccat. In the catalog we provide the positions (cols. 2 and 3), the
aperture and extinction corrected magnitudes with errors for the $g_{F475W}$
and $V_{F606W}$ bands (cols. 4-7), the $g_{F475W}\ $-$V_{F606W}\ $color, as
well as the flux radius (half-light radius from SExtractor) $F_{rad}$ and
$FWHM$ in each band (cols. 9-12) for the sample of selected GCs.

The same procedure for the photometry, extinction and aperture corrections,
and cross-matching was also applied to the four background fields.

Table 3: Comparison of PGC 087327 and the background frames. The RA and Dec are the coordinates of the centre of each cutout. $N_{unsel}$ is the total number of sources, and $N_{GC}$ is the number of GC candidates detected in each frame. Frame | RA | Dec | $N_{unsel}$ | $N_{GC}$
---|---|---|---|---
| (deg) | (deg) |  |
PGC 087327 | 159.290 | -27.658 | 804 | 102
Ref 1 | 159.331 | -27.672 | 485 | 20
Ref 2 | 159.316 | -27.657 | 485 | 20
Ref 3 | 159.269 | -27.669 | 369 | 17
Ref 4 | 159.339 | -27.690 | 449 | 20

##  3 Globular Cluster population: sample selection

In this section we describe the procedures adopted to identify GC candidates
from the cross-matched catalog derived in the previous section.

Since GCs at the distance of the Hydra I appear as point-like objects
($1\arcsec$ corresponds to nearly 100 pc), it is reasonable to identify them
by means of their shape, in addition to their photometric properties. We
selected GCs using three criteria based on $i$) compactness, $ii$) color and
$iii$) magnitude.

The same selection procedure is applied to the catalogs of the four background
fields.

###  3.1 Compactness

The selection on compactness is used to separate the compact GC candidates
from the extended sources. We identify compact sources using the SExtractor
$CLASS\\_STAR$ parameter and the concentration index (CI).

For the SExtractor star-galaxy separation parameter $CLASS\\_STAR$, which has
a value close to 1.0 for compact objects, we adopted a threshold value of
$CLASS\\_STAR\geq 0.8$.

The CI is defined as the difference between magnitudes at two different
apertures, and is a further indicator of source compactness (Peng et al.,
2011). After several tests, we set the two aperture diameters at 4 and 8
pixels to calculate $CI=mag_{4pix}-mag_{8pix}$. We then selected all sources
with $0.25\leq CI\leq 0.75$ in both passbands, based on the median and scatter
of the sequence of compact sources in the CI vs magnitude plot.

The selection of compact sources is shown in Figure 4 with black points.

Figure 4: Left: Star-galaxy classifier of all detected sources in the matched
catalog (green), and GC candidates (black). Right: Concentration indices for
all the detected sources in the matched catalog (green) and GC candidates
(black).

###  3.2 Color and magnitudes

The selection based on color enables us to reduce the contamination from fore-
and back-ground compact sources in the field, mostly Milky Way stars and
distant galaxies, respectively.

Figure 5 shows the color-magnitude diagrams of sources detected in PGC 087327
and in the four reference fields. The upper left panel of the figure shows the
color-magnitude diagram of the entire sample of matched sources with no
selection. The panel reveals the presence of a sharp drop in the number of
detected sources for $g_{F475W}\ $-$V_{F606W}\ $color bluer than $\sim$0.3 and
redder than $\sim$0.7 mag, which is not seen in the four background fields,
shown in the other panels of the figure.

Intermediate size galaxies like PGC 087327 typically do not contain a
substantial population of red, metal-rich GCs (Peng et al., 2006). Using an
updated version of the SPoT (e.g. Raimondo et al., 2005) simple stellar
population (SSP) models, we estimate that with $[Fe/H]\leq 0$, and age$\geq
10$ Gyr, the range of color we expect for GCs is conservatively
$0.3\leq\langle g_{F475W}{-}V_{F606W}\rangle\leq 0.7$ mag. We use this color
range from SSP models, combined with the properties observed in the colour-
magnitude diagrams in Fig. 5, to select the GC candidates in our catalog. A
similar range of GC colors was corroborated by examining the YEPS (Chung et
al., 2020) SSP models555The YEPS models can be found on
http://cosmic.yonsei.ac.kr/YEPS.htm. The SPoT models are reported in Tab. 5 in
the Appendix.

After the compactness and color selections over the matched catalogs, we also
applied a cut on the magnitude. As anticipated in Sect. 1, the GCLF has a
universal Gaussian shape, and a width $\sigma^{GCLF}$ which scales with the
galaxy luminosity. Using eq. (5) from Villegas et al. (2010) adopting a total
galaxy magnitude $M_{z,gal}\sim-20.4$ mag (see Tab. 1), we evaluated
$\sigma_{F475W}^{GCLF}=0.93\pm 0.04$ mag. Adopting a preliminary value of
$M_{F475W}^{TO}\sim-7.3$ mag (we will refine this in Sect. 4.4), and a
distance of 61 Mpc, we estimated the expected GCLF peak $m_{F475W}^{TOM}\sim
26.6$ mag. We selected from the matched catalog all sources $\pm
5\sigma_{F475W}^{GCLF}$ around the $m_{F475W}^{TOM}$, using a large interval
in order to avoid introducing bias in our own distance estimate.

A synthesis of the GC candidates identified in PGC 087327 and the background
fields is also given in Tab. 3. In Figure 5 the sources identified as GC
candidates are highlighted in black. The plots and data in Tab. 3 show that
the galaxy hosts a factor of $\sim 5$ more GCs than the background fields.
From Figure 5, we can verify that the number of objects selected through the
compactness criteria with colors bluer than the GCs color, $\langle
g_{F475W}{-}V_{F606W}\rangle<0.3$, have a similar density in the PGC 087327
frame ($N_{red,gal}=35$) as in the background frames (median
$N_{red,bkg}=35\pm 9$). A similar trend is also observed in the population of
objects redder than the GCs color, $\langle g_{F475W}{-}V_{F606W}\rangle>0.7$
($N_{blue,gal}=15$, median $N_{blue,bkg}=14\pm 2$). The last two panels in the
figure also show the colors and magnitudes from the SPoT and YEPS SSP models,
with a magnitude offset factored to match the magnitude range in the plots.

In summary, even though the GC selection is based on a single color and shape
criteria, the plots in Figure 5 reveal a significant over-density of GC
candidates in the galaxy frame compared to the background fields. In the
forthcoming section we will take advantage of that to characterize the GC
population.

Some of the objects in the GC candidates catalog have a larger than average
flux radius, with SExtractror half-light radii $F_{rad}\
\raise-2.0pt\hbox{\hbox to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$>$}\
}2\leavevmode\nobreak\ pix$ (compared to the median $F_{rad}=1.7$ pix),
relatively red colors ($\langle$$g_{F475W}\ $-$V_{F606W}\ $$\rangle$¿0.53 mag)
and magnitude $M_{F606W}\approx-7.1$ mag which make them probable candidates
for ”faint fuzzies”, which have historically been observed with similar
characteristics close to many lenticular galaxies (Brodie & Strader, 2006).

Figure 5: The color-magnitude diagram of all objects detected (green) (top
left), and along with the objects selected by compactness criteria (black) in
the PGC 087327 frame (top centre), and background reference frames (all
others). The GC candidates are the compact objects (black) within the color
interval marked by purple dashed vertical lines, in all panels. The bottom two
panels also show the SPoT (bottom left panel) and YEPS (bottom right panel)
SSP models (with an arbitrary magnitude shift) overlaid on top of the color-
magnitude plots for regions 3 and 4. Different SSP [Fe/H] are shown with
different colors : $-2.5\leq[Fe/H]\leq-1.5$ in orange, $-1.5<[Fe/H]\leq 0$ in
blue, and $0<[Fe/H]\leq 0.5$ in red. The vertical dotted lines in the middle
panel show the color interval adopted for GCs selection.

##  4 Globular Cluster population analysis

###  4.1 Color Distribution

Figure 6: Color histograms of the full matched (left panel), of GC candidates
(centre) and the color distribution of GCs over PGC 087327 corrected for the
background contamination (right). The light blue histograms in the left and
central panels indicate mean color distributions in the background frames, and
the dark blue histograms represent the distribution in the frame of PGC
087327. In the right panel the gaussian fits (from GMM) for the blue (solid
blue line) and red (dot-dashed red line) GC sub-populations are overlaid on
the background-corrected color density histogram.

Despite the small separation in wavelength between the $g_{F475W}\ $and
$V_{F606W}\ $passbands, the color histogram of the selected GCs shows a clear
bimodality. By inspecting the entire population of matched sources on PGC
087327 and in the reference fields, it is hard to identify any evidence of
this feature (Fig. 6, left panel). However, the bimodality emerges when the
color distribution of GC candidates is inspected, and it is especially evident
when we subtract the background obtained from the four reference frames (Fig.
6, middle and right panels).

To study the detailed properties of the color distribution, we generated a
random sample of $\sim$1000 points in the shape of the background-subtracted
histogram of GCs (Fig. 6, right panel), using the $numpy$ routine
random.choice. In order to make this distribution more continuous, each bin in
this histogram of random points was smoothed with a uniform distribution
having a width equal to half of the bin size. The resulting distribution,
exhibiting a dual Gaussian profile, was fitted using Gaussian Mixture Models
(GMM, Muratov & Gnedin, 2010), repeated over 10 iterations. We found that a
bimodal distribution is favored over a unimodal one, and the median blue (red)
peak of the background subtracted density distribution lies at $\langle
g_{F475W}{-}V_{F606W}\rangle=0.47\leavevmode\nobreak\ (0.62)$ mag, with a
width of $0.05\leavevmode\nobreak\ (0.03)$ mag. The resulting fraction of red
GCs is $f_{red}\sim 0.3\pm 0.1$, which agrees with the median value of
$f_{red}\sim 0.18\pm 0.20$ from Peng et al. (2008, their Tab. 3) for galaxies
similar to PGC 087327, with a specific frequency ($S_{N}$) between 1.0 and 2.5
(further discussed in Sect. 4.4).

Due to the lack of existing literature on the bimodality in $g_{F475W}\
$-$V_{F606W}\ $color, we projected the $g_{F475W}\ $-$V_{F606W}\ $peaks to
$V{-}I$ and $g{-}z$ colors, using the SpoT and Yonsei models for the
transformation. Considering an interval of $\langle$$g_{F475W}\ $-$V_{F606W}\
$$\rangle\pm 0.05$ mag around each peak and adopting predictions for ages
$t\geq 10$ Gyr and metallicity $-2.5\leq[Fe/H]\leq 0$, we found that the blue
$g_{F475W}\ $-$V_{F606W}\ $=0.47 mag peak projects into $V{-}I=0.55$ mag and
$g{-}z$=0.93 mag, while the red $g_{F475W}\ $-$V_{F606W}\ $=0.62 mag peak is
projected into $V{-}I$=0.78 mag and $g{-}z$=1.34 mag. We compared the
projected colors for each peak with the values from the review by Brodie &
Strader (2006, their Tab. 1), for galaxies similar to PGC 087327, having type
S0 and $-19.5\leq M_{B}\leq-17.5$ mag. The median colors from this selection
are $\langle V{-}I\rangle=0.51\leavevmode\nobreak\ (0.72)$ mag and $\langle
g{-}z\rangle=0.92\leavevmode\nobreak\ (1.26)$ mag for the blue (red) peak,
which agree with our results within $\sim$0.1 mag. In summary, despite the
narrow wavelength interval available, the GCs color bimodality appears to be
consistent with the results obtained for similar galaxies over a wider
wavelength range.

###  4.2 Radial distribution

The coordinates of the GCs identified in each frame were plotted to understand
the radial distribution of the sources in the frame of PGC 087327 (Fig. 7).
The GCs were separated into two different classes based on their $\langle
g_{F475W}{-}V_{F606W}\rangle$ color: red ($\langle
g_{F475W}{-}V_{F606W}\rangle\geq 0.53$, adopted from GMM fits) and blue
($\langle g_{F475W}{-}V_{F606W}\rangle<0.53$). From Fig. 7 (left panel) and
Fig. 8 we can identify that red, more metal-rich GCs appear clustered near the
centre of the galaxy compared to the blue. The slope of the radial number
density of the red GCs is steeper ($\alpha_{r}=\frac{d\log
N^{red}_{GC}(r)}{d\log R}\approx-1.9$) than the blue
($\alpha_{b}\approx-1.7$). The slopes agree with the values of the slopes for
metal-rich and metal-poor subpopulations quoted by Brodie & Strader (2006). We
also plotted the galaxy surface brightness radial profile (scaled
appropriately), to emphasize its resemblance to the radial density profile of
the red GCs. Around a radius of $\sim 32\arcsec$ we begin to approach the
edges of the galaxy cutout, and to avoid issues due to vignetting we do not
consider this region for our fits (indicated by gray shaded area in Fig. 8).
In Fig. 7 we have plotted the kernel density estimate plots for the red and
blue sub-populations, and we calculated that the ellipticity of the red GCs
(centre frame) is $\epsilon=0.41$, where $\epsilon$ is the ratio of the minor
to the major axis. The $\epsilon$ for red GCs matches that of our model
isophotes in Sect. 2.3. The KDE plot of the blue sub-population, on the other
hand, has a more circular distribution with $\epsilon=0.13$ and a higher
variation at larger radii.

Figure 7: The spatial distribution of the GC candidates divided into red and
blue according to color ($\langle g_{F475W}{-}V_{F606W}\rangle\geq 0.53$ is
red) in the frame of PGC 087327 (left) and the kernel density estimate(KDE)
plot of the blue (centre) and red (right) GC sub-populations. Figure 8: Radial
densities of the full (black triangles), red (red circles) and blue (blue
squares) GC populations in PGC 087327.The radial density profile for the red
(dashed line) and blue (dot-dashed line) GC populations is also shown as a
linear trend. The radial surface brightness profile of the galaxy (scaled
appropriately) is shown with the purple solid line. The shaded gray region
shows the limit of geometric completeness as we approach the margins of our
image cutout.

###  4.3 Luminosity function

The histograms of GCs in the frame of PGC 087327 and the background frames
were inspected to study the luminosity function. As for the color and radial
distribution, we used the four reference fields to define the background level
of contaminant sources to be subtracted from the GC density over PGC 087327.
Since the area of the background fields is the same as the cutout of PGC
087327, to properly subtract the background contamination it is not necessary
to normalize to the area. Figure 9 shows the luminosity function of the
sources detected in PGC 087327 and (the mean) of the reference regions for the
full matched catalogs. The middle panel of the same figure shows the
luminosity functions for the GC candidates. As expected, the GCs on-galaxy
significantly outnumber the counts in the reference fields. We also note a
slight increase of background counts toward fainter magnitudes, as an effect
of the lower efficiency of the adopted selection criteria for sources with
lower S/N. The background subtracted luminosity function of GCs is shown in
Fig. 9 (right panel). In order to use the GCLF for deriving the distance
modulus of PGC 087327, it must be corrected for completeness first.

The completeness function in each band was determined by injecting and then
detecting a total of $\sim 850$ artificial stars on the $g_{F475W}\ $and
$V_{F606W}\ $residual frames. At first, we derived a PSF from the bright,
compact and isolated sources in the image in each band. The brightest and most
isolated compact sources were extracted from the residual image using the
extract_stars routine of Astropy, and these extracted stars were then fed into
the EPSFBuilder routine of the photutils package to generate an effective PSF
of the size $41\times 41$ pixels. The magnitudes of artificial stars to be
injected were obtained by generating a random sample of magnitudes, again
using numpy.random.choice, in the shape of the histogram of all sources
detected in the galaxy frame, limited to the range $23\leq mag\leq 32$. These
stars were injected in the residual frame of PGC 087327 along an equispaced
grid whose position was varied over 50 iterations. A catalog of detected
sources was obtained from a run of SExtractor on this synthesized image,
adopting the same parameters used for generating the GCs catalog. The aperture
correction for each band was then applied on this catalog. A two-fold
selection algorithm was used to match the detected to the injected artificial
stars: a first selection on separation ($\leq 6$ pix), followed by a cut on
the magnitude difference (absolute difference $|m_{injected}-m_{detected}|\leq
0.5$ mag). The ratio of the number of sources thus retrieved vs injected, in
each magnitude bin, gives us the discrete completeness function (Fig. 10).

We fit the completeness function with a modified Fermi function (Alamo-
Martínez et al., 2013, their eq 2), and interpolated it to counteract the
discrete nature of the sampled magnitudes, then applied it to correct the GCLF
in the galaxy frame. We note that the completeness is $90\%$ down to about
$m\approx 29.5$ in both $g_{F475W}\ $and $V_{F606W}\ $, which is fainter than
the 3$\sigma_{GCLF}$ tail of the galaxy $m_{TO}$. In other words, the observed
GCLF is complete and only mildly affected by the completeness, and a
correction $\sim 5\%$ is required for magnitudes $g_{F475W}\ $$=28-29$ mag.
The same analysis was performed on the background frames to determine their
completeness functions individually, and the mean of the completeness-
corrected LFs of GC candidates from the four background frames was used to
determine and subtract the overall background for the GCLF in PGC 087327.

Finally, the background-subtracted GCLF was fitted with a Gaussian in both the
$g_{F475W}\ $and $V_{F606W}\ $bands (Fig. 11) to obtain the peak turn over
apparent magnitude and the width, the results of which are given in Table 4.

Figure 9: Left: The luminosity function (LF) of all detected and matched
objects in the frame of PGC 087327 (in black), and in the background fields
(in green, dashed). Centre: The LF of the globular clusters candidates in the
galaxy frame (in black), and the mean LF of GC candidates over the four
background fields (in green). Right: The LF of the GC candidates in PGC 087327
after subtracting the mean LF from the background fields. This luminosity
function has not yet been corrected for completeness.

Figure 10: The completeness functions in $g_{F475W}\ $(left) and $V_{F606W}\ $(right). Each point represents the ratio of the number of sources detected vs injected, in each magnitude bin, and the green line shows the best fit modified Fermi function. Figure 11: The final GCLF which has been corrected for completeness, and fitted with a Gaussian (green line). Table 4: GCLF parameters Band | $m_{TO}$ | $\sigma_{GCLF}$ | $M_{TO}$ | $m-M$
---|---|---|---|---
$g_{F475W}\ $ | 26.54±0.10 | 0.86±0.10 | -7.29±0.28 | 33.76±0.30
$V_{F606W}\ $ | 26.08±0.09 | 0.76±0.09 | -7.69±0.18 | 33.77±0.20

###  4.4 GCLF distance: calibration and results

In order to derive the GCLF distance to PGC 087327 we adopt the apparent
$m_{TO}$ derived as described in Sect 4.3 combined with $M_{TO}$ calibrations
in $g_{F475W}\ $and $V_{F606W}\ $from the literature, outlined below.

We estimated the absolute $V_{F606W}\ $band turn-over by taking advantage of
existing accurate GCLF analysis by Georgiev et al. (2009) on HST/ACS data of
dwarf galaxies. We use their calibration quoted in the V-band, which gives us
$M_{V}^{TO}=-7.60\pm 0.11$ mag. Using the photometric transformation in Tab.
21 from Sirianni et al. (2005) and inverting the relation given in eq. 12, we
find the $M^{TO}_{F606W}$:

$M_{F606W}^{TO}=M_{V}^{TO}-c_{0}-c_{1}\times(V-I)-c_{2}\times(V-I)^{2}+Zpt_{F606W}(AB)$

The color used in this equation is the $V-I$ color of GCs, which is $\sim 1$
(Harris, 1996; Cantiello et al., 2007), and $Zpt_{F606W}(AB)$ is the zero-
point magnitude of the F606W band (Tab. 2) for the date of our observation,
obtained from the zero-point calculator of the HST/ACS666Can be found at URL :
https://acszeropoints.stsci.edu/. Performing this analysis for the observed
and synthetic (color$>0.4$) coefficients in Sirianni et al. (2005) we find two
individual estimates: $M^{TO}_{F606W,obs}=-7.65\pm 0.14$ mag and
$M^{TO}_{F606W,syn}=-7.73\pm 0.11$ mag. On averaging these two values, we get
$M^{TO}_{F606W}=-7.69\pm 0.18$ mag.

To obtain the $g_{F475W}\ $band $M^{TO}$ we used the ACSVCS results by
Villegas et al. (2010, their Tab. 1) for the Virgo cluster sample. We isolated
a sub-sample of galaxies with $-19\leq M_{B}\leavevmode\nobreak\ (mag)\leq-17$
which corresponds to the magnitude level of PGC 087327 (see Tab. 1). From this
sample, we excluded the galaxies with a number of GCs $N_{GC}<30$, as well as
VCC 1025 (since it belongs to the W’ cloud and has a different distance
modulus). We calculated the median turn-over magnitude of the selected sub-
sample and the $rms$ (derived from the median absolute deviation), which turns
out to be $m^{TO}_{F475W}=23.87\pm 0.25$ mag. Adopting $(m{-}M)_{SBF}=31.09\pm
0.15$ for the Virgo cluster from Blakeslee et al. (2009), we find :

$M^{TO}_{F475W}=m^{TO}_{F475W}-(m{-}M)_{SBF}=-7.22\pm 0.28\>mag$

We calculate the distance modulus $m^{TO}{-}M^{TO}$ in each band, the results
of which are in Tab. 4. Being independent measurements (except the catalogs
being cleaned by matching the $g_{F475W}\ $and $V_{F606W}\ $photometry) we
assume our best distance modulus to PGC 087327 to be the weighted average of
the two values in Table 4: $(m{-}M)_{GCLF}=33.77\pm 0.17$ mag, or $56.7\pm
4.3$ Mpc. This agrees with the distance estimate from the Fundamental Plane
for this galaxy, $(m-M)=33.8\pm 0.5$ mag (Tully et al., 2013).

The systematic error on the calibrations derived above, could be assumed to be
of the same order of the SBF systematic, i.e. $\sim 0.1$ mag (Cantiello et
al., 2018). Lee et al. (2018) claim that the real global systematic for GCLF
is anyway $<0.3$ mag. We adopt a conservative middle-ground estimate of 0.2
mag as our systematic error budget. Hence, including systematic errors we
obtain $D_{GCLF}=56.7\pm 4.3(statistical)\pm 5.2(systematic)$ Mpc.

We estimate the value of the Hubble parameter using the flow corrected radial
velocity for PGC 087327 from Table 4. Combining it with the distance modulus
from this work we find $H_{0}\approx 78.5\pm 6.0(statistical)\pm
7.3(systematic)$ km s-1 Mpc-1.

The distance modulus from this work places PGC 087327 between NGC 3314A
($(m-M)=33.19\pm 0.40$ mag, from Theureau et al., 2007) and NGC 3314B
($(m-M)=34.37\pm 0.15$ mag, from Mould et al., 2000) in projection, consistent
with other distance measurements for the Hydra cluster and lying close to the
upper end of these estimates (for e.g., Blakeslee et al., 2002; Hudson et al.,
2004; Mieske et al., 2005). Our results show that this galaxy lies at a
distance that is $\sim 20\%$ larger than that of the two BGCs in the Hydra I
cluster (Tully et al., 2013), and in a region where we count eight other
brighter galaxies which have upto 20% larger velocity distances than that of
PGC 087327 in the vicinity of $\sim 15\arcmin$777Source: Nasa Extragalatic
Database.

Considering the Hubble-Lemaitre distance to PGC 087327 and the results in
Table 4 we can turn the argument around, and use our results to check whether
the fitted parameters support the universality of the GCLF. Given the fact
that the final distance calculated in our work agrees closely with the
velocity distance (reported in Table 1), combined with the $\sigma_{GCLF}$
values from our analysis that match very well with the expected
$\sigma_{GCLF}=0.93$ mag (see Sect. 3.2), we can conclude that the results
from this work support the universality of the GCLF method.

###  4.5 Specific frequency

We can also estimate the specific frequency (Harris & van den Bergh, 1981) of
the GC system in PGC 087327, which is defined as:

$S_{N}\equiv N_{GC}\times 10^{0.4(M_{V,gal}+15)}$

Using the $m_{F606W,gal}$ obtained from our fit in Sect. 2.3 (quoted in Tab.
1), correcting it for extinction and using Tab. 21 from Sirianni et al. (2005)
we transform $m_{F606W,gal}$ to $m_{V,gal}$ in Vegamag, which is used as
reference for $S_{N}$ estimates.

Combining this apparent magnitude with the distance modulus of PGC 087327
(from Sec. 4.4), we estimate a specific frequency $S_{N}=1.8\pm 0.7$. The
value we estimated is consistent with the observed scatter of $S_{N}$ for
galaxies of similar magnitude (Peng et al., 2008; Harris et al., 2013),
leaning towards the tail of higher GC population density.

##  5 Conclusions

In this work, we benefited from extremely deep images in the HST/ACS
$g_{F475W}\ $and $V_{F606W}\ $bands to characterize the GC population around
PGC 087327. The main results of our study are summarized below:

  * •

Although the population of GCs we find is relatively small, $N_{GC}=82\pm 9$,
and the wavelengths of the $g_{F475W}\ $and $V_{F606W}\ $bands are relatively
close, we find a clear bimodal color distribution in the GC system of this
galaxy;

  * •

The radial distribution of the GCs shows a clustering of the red GCs close to
the core of the galaxy, which falls off rapidly at higher radii
($\alpha_{r}\approx-1.9$) and has a close resemblance to the galaxy surface
brightness profile, whereas the blue GCs are less concentrated at the centre,
taper off more slowly away from the galaxy core ($\alpha_{b}\approx-1.7$) and
appear circularly distributed around the galaxy.

  * •

In spite of the close wavelengths of the $g_{F475W}\ $and $V_{F606W}\ $bands,
and the intermediate mass of PGC 087327, we observe the typical bimodal
characteristics of GC populations that are more pronounced in massive
galaxies. The $g_{F475W}\ $-$V_{F606W}\ $color distribution shows a blue (red)
peak at $\langle g_{F475W}{-}V_{F606W}\rangle=0.47(0.62)$. This further
demonstrates the role of quality of the images on the characterization of GC
populations, since a lower depth and accuracy of images would have smeared out
the color bimodality feature we observe in this work;

  * •

The turn-over magnitude in $g_{F475W}\ $and $V_{F606W}\ $are both $\sim 3$ mag
brighter than the completeness limit ($>29.5$ mag), which to our knowledge is
an unprecedented finding at distances of the order of 60 Mpc;

  * •

We estimate a distance modulus of $(m{-}M)=33.77\pm 0.17$ mag, or $56.7\pm
4.3(statistical)$ Mpc for this galaxy, which places it between NGC 3314A and
NGC 3314B in projection;

  * •

Considering the velocity distance of PGC 087327 and the empirical expectations
for the GCLF peak and width, our analysis for this target supports the
universality of the GCLF method;

  * •

Assuming $\sim 10\%$ systematic error on the galaxy distance, we derive a
Hubble constant value of $H_{0}\approx 78.5\pm 6.0(statistical)\pm
7.3(systematic)$ km s-1 Mpc-1.

###### Acknowledgements.

This work was carried out based on observations made with the NASA/ESA Hubble
Space Telescope, and obtained from the Hubble Legacy Archive, which is a
collaboration between the Space Telescope Science Institute (STScI/NASA), the
Space Telescope European Coordinating Facility (ST-ECF/ESA) and the Canadian
Astronomy Data Centre (CADC/NRC/CSA). The authors made use of
Astropy,888http://www.astropy.org a community-developed core Python package
for Astronomy (Astropy Collaboration et al., 2013, 2018), along with the
databases of HyperLeda (Makarov et al., 2014), the Extragalactic Distance
Database (EDD)999https://edd.ifa.hawaii.edu/ and the NASA/IPAC Extragalactic
Database (NED, which is operated by the Jet Propulsion Laboratory, California
Institute of Technology, under contract with the National Aeronautics and
Space Administration). We also made extensive use of the softwares of
Topcat101010http://www.starlink.ac.uk/topcat/, SExtractor (Bertin & Arnouts,
1996) and Galfit (Peng et al., 2010). MC acknowledges support from MIUR, PRIN
2017 (grant 20179ZF5KS). The authors would also like to acknowledge and thank
the referee for their valuable comments, questions and suggestions.

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##  Appendix A Tables

Table 5: SPoT models z | Age | $M_{V}$ |  $V$-$g_{F475W}\ $ |  $V$-$V_{F606W}\ $ |  $g_{F475W}\ $-$V_{F606W}\ $
---|---|---|---|---|---
| (GYr) | (Vegamag) | (Vegamag) | (Vegamag) | (Vegamag)
0.0001 | 2 | -12.8572 | -0.2743 | 0.1329 | 0.4072
0.0001 | 3 | -12.5237 | -0.3051 | 0.144 | 0.4491
0.0001 | 4 | -12.282 | -0.3301 | 0.1563 | 0.4864
0.0001 | 6 | -11.9145 | -0.3521 | 0.1623 | 0.5144
0.0001 | 7 | -11.79 | -0.3609 | 0.1662 | 0.5271
0.0001 | 8 | -11.674 | -0.3645 | 0.1684 | 0.5329
0.0001 | 10 | -11.4846 | -0.3741 | 0.1748 | 0.5489
0.0001 | 12 | -11.3264 | -0.3856 | 0.1805 | 0.5661
0.0001 | 14 | -11.173 | -0.3947 | 0.1833 | 0.578
0.001 | 2 | -12.8318 | -0.3423 | 0.1602 | 0.5025
0.001 | 3 | -12.4435 | -0.3725 | 0.1707 | 0.5432
0.001 | 4 | -12.1836 | -0.4004 | 0.1843 | 0.5847
0.001 | 6 | -11.7924 | -0.4258 | 0.1947 | 0.6205
0.001 | 7 | -11.6625 | -0.4341 | 0.1979 | 0.632
0.001 | 8 | -11.5549 | -0.4412 | 0.2009 | 0.6421
0.001 | 10 | -11.363 | -0.4468 | 0.2034 | 0.6502
0.001 | 12 | -11.2115 | -0.4433 | 0.2032 | 0.6465
0.001 | 14 | -11.0623 | -0.4394 | 0.2031 | 0.6425
0.004 | 2 | -12.638 | -0.3996 | 0.1844 | 0.584
0.004 | 3 | -12.2876 | -0.4364 | 0.1998 | 0.6362
0.004 | 4 | -11.9106 | -0.4526 | 0.2087 | 0.6613
0.004 | 6 | -11.5842 | -0.4719 | 0.2139 | 0.6858
0.004 | 7 | -11.4589 | -0.4823 | 0.2184 | 0.7007
0.004 | 8 | -11.3427 | -0.4908 | 0.2221 | 0.7129
0.004 | 10 | -11.1549 | -0.5015 | 0.2263 | 0.7278
0.004 | 12 | -10.9981 | -0.5044 | 0.2286 | 0.733
0.004 | 14 | -10.8679 | -0.5069 | 0.2301 | 0.737
0.008 | 2 | -12.445 | -0.4307 | 0.1973 | 0.628
0.008 | 3 | -12.1208 | -0.4667 | 0.2114 | 0.6781
0.008 | 4 | -11.8348 | -0.4827 | 0.2176 | 0.7003
0.008 | 6 | -11.4408 | -0.5024 | 0.225 | 0.7274
0.008 | 7 | -11.2877 | -0.5094 | 0.2278 | 0.7372
0.008 | 8 | -11.1762 | -0.5182 | 0.2313 | 0.7495
0.008 | 10 | -10.982 | -0.5307 | 0.2364 | 0.7671
0.008 | 12 | -10.8242 | -0.5397 | 0.24 | 0.7797
0.008 | 14 | -10.6995 | -0.5473 | 0.2436 | 0.7909
0.02 | 2 | -12.1286 | -0.4662 | 0.2079 | 0.6741
0.02 | 3 | -11.7954 | -0.4984 | 0.2208 | 0.7192
0.02 | 4 | -11.5515 | -0.5195 | 0.2295 | 0.749
0.02 | 6 | -11.1586 | -0.5425 | 0.2389 | 0.7814
0.02 | 7 | -10.9995 | -0.5499 | 0.2423 | 0.7922
0.02 | 8 | -10.8795 | -0.5589 | 0.2461 | 0.805
0.02 | 10 | -10.6701 | -0.572 | 0.2517 | 0.8237
0.02 | 12 | -10.5088 | -0.5843 | 0.2569 | 0.8412
0.02 | 14 | -10.3779 | -0.5954 | 0.2616 | 0.857
0.04 | 2 | -11.8901 | -0.5012 | 0.2224 | 0.7236
0.04 | 3 | -11.5415 | -0.535 | 0.236 | 0.771
0.04 | 4 | -11.2996 | -0.5539 | 0.2441 | 0.798
0.04 | 6 | -10.9501 | -0.5799 | 0.2553 | 0.8352
0.04 | 7 | -10.8114 | -0.5933 | 0.2611 | 0.8544
0.04 | 8 | -10.67 | -0.6002 | 0.2643 | 0.8645
0.04 | 10 | -10.4599 | -0.617 | 0.2718 | 0.8888
0.04 | 12 | -10.2896 | -0.6303 | 0.2778 | 0.9081
0.04 | 14 | -10.1477 | -0.6416 | 0.2828 | 0.9244
Table 6: GC catalog Num | RA (J2000) | Dec (J2000) | $g_{F475W}$ | $\Delta g_{F475W}$ | $V_{F606W}$ | $\Delta V_{F606W}$ |  $g_{F475W}\ $-$V_{F606W}\ $ | $F_{rad,F475W}$ | $FWHM_{F475W}$ | $F_{rad,F606W}$ | $FWHM_{F606W}$
---|---|---|---|---|---|---|---|---|---|---|---
| [deg] | [deg] | [mag] | [mag] | [mag] | [mag] | [mag] | [pix] | [pix] | [pix] | [pix]
1 | 159.28939 | -27.66667 | 28.412 | 0.062 | 27.75 | 0.028 | 0.662 | 1.898 | 3.67 | 1.991 | 3.54
2 | 159.294 | -27.66458 | 27.278 | 0.022 | 26.893 | 0.013 | 0.385 | 1.935 | 3.31 | 1.854 | 3.28
3 | 159.29552 | -27.6633 | 26.68 | 0.052 | 26.234 | 0.022 | 0.446 | 1.798 | 3.47 | 1.982 | 3.5
4 | 159.28733 | -27.66329 | 26.449 | 0.01 | 25.871 | 0.006 | 0.578 | 1.727 | 2.95 | 1.727 | 2.87
5 | 159.28472 | -27.66319 | 27.623 | 0.008 | 27.081 | 0.004 | 0.542 | 1.618 | 2.83 | 1.709 | 2.78
6 | 159.29647 | -27.66299 | 27.037 | 0.03 | 26.598 | 0.015 | 0.439 | 1.626 | 2.8 | 1.62 | 2.83
7 | 159.29048 | -27.66266 | 26.412 | 0.018 | 25.997 | 0.01 | 0.415 | 1.737 | 2.71 | 1.743 | 2.9
8 | 159.28771 | -27.66255 | 26.682 | 0.01 | 26.292 | 0.006 | 0.39 | 1.723 | 2.87 | 1.702 | 2.9
9 | 159.28956 | -27.66221 | 26.13 | 0.013 | 25.664 | 0.008 | 0.466 | 2.252 | 3.56 | 2.104 | 3.58
10 | 159.29129 | -27.66182 | 28.629 | 0.008 | 28.139 | 0.005 | 0.49 | 1.862 | 2.94 | 1.795 | 2.96
11 | 159.28278 | -27.66107 | 26.782 | 0.01 | 26.421 | 0.006 | 0.361 | 1.87 | 3.53 | 1.952 | 3.24
12 | 159.29126 | -27.66104 | 26.317 | 0.01 | 25.837 | 0.006 | 0.48 | 1.665 | 2.86 | 1.746 | 3
13 | 159.28722 | -27.66098 | 26.653 | 0.01 | 26.102 | 0.006 | 0.551 | 1.663 | 2.72 | 1.847 | 2.95
14 | 159.29716 | -27.66081 | 25.586 | 0.005 | 25.108 | 0.003 | 0.478 | 1.734 | 2.92 | 1.795 | 2.95
15 | 159.29071 | -27.6607 | 25.76 | 0.006 | 25.329 | 0.004 | 0.431 | 1.716 | 2.73 | 1.718 | 2.85
16 | 159.28341 | -27.65999 | 25.823 | 0.004 | 25.398 | 0.002 | 0.425 | 1.775 | 3.04 | 1.744 | 2.91
17 | 159.29349 | -27.65992 | 26.654 | 0.014 | 26.185 | 0.009 | 0.469 | 1.566 | 2.85 | 1.652 | 2.94
18 | 159.29071 | -27.65988 | 24.668 | 0.002 | 24.22 | 0.002 | 0.448 | 1.766 | 2.9 | 1.794 | 2.97
19 | 159.28904 | -27.65975 | 27.808 | 0.042 | 27.467 | 0.03 | 0.341 | 1.637 | 3.49 | 1.619 | 2.98
20 | 159.29409 | -27.65969 | 27.084 | 0.02 | 26.568 | 0.012 | 0.516 | 1.46 | 2.69 | 1.601 | 2.81
21 | 159.28797 | -27.65952 | 26.467 | 0.011 | 26.046 | 0.007 | 0.421 | 1.735 | 2.93 | 1.795 | 3.23
22 | 159.29009 | -27.6595 | 25.81 | 0.008 | 25.3 | 0.005 | 0.51 | 1.689 | 2.69 | 1.702 | 2.97
23 | 159.2909 | -27.65927 | 26.673 | 0.019 | 26.07 | 0.012 | 0.603 | 1.542 | 2.91 | 1.601 | 2.93
24 | 159.29113 | -27.65919 | 26.825 | 0.023 | 26.311 | 0.017 | 0.514 | 1.321 | 2.72 | 1.821 | 4.03
25 | 159.28978 | -27.65895 | 27.965 | 0.009 | 27.497 | 0.007 | 0.468 | 1.595 | 2.76 | 1.68 | 2.85
26 | 159.2912 | -27.65889 | 25.811 | 0.022 | 25.315 | 0.016 | 0.496 | 1.687 | 2.88 | 1.989 | 3.93
27 | 159.28517 | -27.65886 | 26.575 | 0.003 | 26.06 | 0.002 | 0.515 | 1.758 | 2.87 | 1.786 | 2.93
28 | 159.28785 | -27.65868 | 25.409 | 0.021 | 24.919 | 0.013 | 0.49 | 1.628 | 2.93 | 1.629 | 2.87
29 | 159.28734 | -27.65863 | 27.039 | 0.004 | 26.488 | 0.003 | 0.551 | 1.806 | 3.04 | 1.728 | 2.95
30 | 159.29143 | -27.65848 | 25.503 | 0.024 | 25.023 | 0.02 | 0.48 | 1.85 | 2.86 | 1.853 | 2.92
31 | 159.2913 | -27.65848 | 26.418 | 0.079 | 25.991 | 0.055 | 0.427 | 1.967 | 3.65 | 1.849 | 3.61
32 | 159.29075 | -27.65843 | 27.657 | 0.036 | 27.021 | 0.026 | 0.636 | 1.609 | 2.79 | 1.593 | 2.93
33 | 159.2922 | -27.65836 | 26.634 | 0.007 | 26.027 | 0.005 | 0.607 | 1.723 | 2.77 | 1.73 | 2.92
34 | 159.29242 | -27.65832 | 25.322 | 0.013 | 24.715 | 0.01 | 0.607 | 1.745 | 2.8 | 1.68 | 2.88
35111These objects (#35 and #46) are the ”faint fuzzies” candidates. | 159.2914 | -27.6583 | 27.226 | 0.06 | 26.598 | 0.042 | 0.628 | 2.017 | 5.74 | 2.003 | 4.9
36 | 159.28811 | -27.65829 | 26.071 | 0.019 | 25.617 | 0.013 | 0.454 | 1.656 | 2.87 | 1.684 | 2.96
37 | 159.2915 | -27.65823 | 27.292 | 0.012 | 26.649 | 0.009 | 0.643 | 1.746 | 2.97 | 2.04 | 3.11
38 | 159.29142 | -27.65821 | 26.757 | 0.017 | 26.21 | 0.012 | 0.547 | 1.774 | 2.79 | 1.858 | 2.94
39 | 159.28193 | -27.6582 | 25.509 | 0.047 | 25.025 | 0.019 | 0.484 | 1.675 | 4.27 | 1.929 | 3.83
40 | 159.2905 | -27.6582 | 25.874 | 0.055 | 25.212 | 0.043 | 0.662 | 1.476 | 2.78 | 1.492 | 2.89
41 | 159.29438 | -27.65815 | 26.672 | 0.014 | 26.125 | 0.008 | 0.547 | 2.225 | 3.72 | 2.346 | 3.86
42 | 159.28869 | -27.65812 | 26.581 | 0.038 | 25.953 | 0.025 | 0.628 | 1.516 | 2.84 | 1.571 | 2.86
43 | 159.29263 | -27.6581 | 27.207 | 0.012 | 26.572 | 0.008 | 0.635 | 1.525 | 2.86 | 1.637 | 2.88
44 | 159.28965 | -27.65807 | 26.017 | 0.022 | 25.518 | 0.016 | 0.499 | 1.602 | 2.85 | 1.733 | 3.17
45 | 159.28919 | -27.65801 | 25.964 | 0.051 | 25.345 | 0.041 | 0.619 | 1.609 | 3.08 | 1.532 | 2.93
46111These objects (#35 and #46) are the ”faint fuzzies” candidates. | 159.28808 | -27.658 | 27.163 | 0.041 | 26.67 | 0.029 | 0.493 | 2.037 | 3.84 | 2.389 | 3.58
47 | 159.29098 | -27.65796 | 27.528 | 0.011 | 27.029 | 0.008 | 0.499 | 1.906 | 2.97 | 1.956 | 2.97
48 | 159.29256 | -27.65791 | 24.911 | 0.014 | 24.323 | 0.009 | 0.588 | 1.906 | 3.84 | 1.852 | 3.33
49 | 159.29104 | -27.65789 | 26.197 | 0.017 | 25.577 | 0.014 | 0.62 | 1.775 | 2.85 | 1.929 | 2.96
50 | 159.28788 | -27.65784 | 25.455 | 0.009 | 24.996 | 0.006 | 0.459 | 1.633 | 2.66 | 1.738 | 2.96
51 | 159.28901 | -27.65783 | 25.881 | 0.003 | 25.382 | 0.002 | 0.499 | 1.914 | 3.15 | 1.957 | 3.23
52 | 159.28814 | -27.65778 | 23.951 | 0.016 | 23.481 | 0.01 | 0.47 | 1.715 | 2.77 | 1.721 | 2.89
53 | 159.28799 | -27.65773 | 27.925 | 0.025 | 27.61 | 0.016 | 0.315 | 1.495 | 2.67 | 1.651 | 2.87
54 | 159.28792 | -27.65765 | 26.412 | 0.041 | 25.767 | 0.033 | 0.645 | 1.409 | 2.73 | 1.421 | 2.72
55 | 159.28923 | -27.65756 | 26.968 | 0.045 | 26.344 | 0.034 | 0.624 | 2.044 | 3.48 | 2.09 | 3.93
56 | 159.28932 | -27.65756 | 27.535 | 0.025 | 27.131 | 0.02 | 0.404 | 2.004 | 3.06 | 1.637 | 2.99
57 | 159.28942 | -27.65749 | 26.813 | 0.004 | 26.244 | 0.003 | 0.569 | 1.755 | 2.86 | 1.737 | 2.99
58 | 159.29139 | -27.65747 | 26.091 | 0.039 | 25.555 | 0.032 | 0.536 | 1.375 | 2.76 | 1.294 | 2.67
59 | 159.29152 | -27.65744 | 24.057 | 0.018 | 23.601 | 0.013 | 0.456 | 1.727 | 2.76 | 1.653 | 2.86
60 | 159.29113 | -27.65738 | 26.964 | 0.01 | 26.537 | 0.009 | 0.427 | 1.825 | 3.11 | 1.767 | 3.16
61 | 159.28876 | -27.65738 | 26.216 | 0.014 | 25.688 | 0.01 | 0.528 | 1.633 | 2.74 | 1.58 | 2.81
62 | 159.29132 | -27.6573 | 25.449 | 0.055 | 25.075 | 0.04 | 0.374 | 1.522 | 2.8 | 1.45 | 2.71
63 | 159.2879 | -27.65729 | 25.895 | 0.044 | 25.333 | 0.028 | 0.562 | 1.444 | 2.71 | 1.464 | 2.7
64 | 159.28911 | -27.65726 | 27.452 | 0.028 | 26.886 | 0.02 | 0.566 | 1.941 | 3.05 | 2.058 | 3.43
65 | 159.29074 | -27.65725 | 27.573 | 0.012 | 26.931 | 0.01 | 0.642 | 1.615 | 2.83 | 1.909 | 3.56
66 | 159.29998 | -27.65715 | 26.481 | 0.004 | 25.86 | 0.002 | 0.621 | 1.684 | 2.76 | 1.718 | 2.83
67 | 159.29036 | -27.65712 | 25.575 | 0.027 | 25.052 | 0.019 | 0.523 | 1.373 | 2.58 | 1.451 | 2.66
68 | 159.2899 | -27.65706 | 25.463 | 0.032 | 25.003 | 0.027 | 0.46 | 1.439 | 2.6 | 1.484 | 2.7
69 | 159.29694 | -27.65697 | 26.454 | 0.022 | 25.833 | 0.011 | 0.621 | 1.739 | 2.91 | 1.773 | 2.94
70 | 159.28768 | -27.65692 | 26.653 | 0.007 | 26.236 | 0.005 | 0.417 | 1.871 | 3 | 1.72 | 2.84
71 | 159.29066 | -27.6568 | 27.308 | 0.038 | 26.718 | 0.025 | 0.59 | 1.522 | 2.84 | 1.587 | 2.95
72 | 159.29024 | -27.65671 | 25.877 | 0.032 | 25.313 | 0.025 | 0.564 | 2.193 | 4.96 | 2.043 | 5.07
73 | 159.29169 | -27.65669 | 27.239 | 0.021 | 26.604 | 0.015 | 0.635 | 1.706 | 2.97 | 1.691 | 2.91
74 | 159.28918 | -27.65666 | 27.037 | 0.029 | 26.573 | 0.025 | 0.464 | 1.669 | 3.01 | 1.543 | 2.88
75 | 159.28757 | -27.65636 | 26.828 | 0.002 | 26.386 | 0.002 | 0.442 | 1.653 | 2.87 | 1.734 | 2.81
76 | 159.28955 | -27.65627 | 26.952 | 0.053 | 26.599 | 0.029 | 0.353 | 1.636 | 3.22 | 1.888 | 3.28
77 | 159.29346 | -27.65625 | 24.853 | 0.076 | 24.42 | 0.047 | 0.433 | 1.837 | 3.61 | 1.83 | 3.53
78 | 159.28982 | -27.65617 | 28.507 | 0.04 | 28.028 | 0.025 | 0.479 | 1.828 | 2.95 | 2.319 | 3.58
79 | 159.29111 | -27.65616 | 27.586 | 0.01 | 26.961 | 0.007 | 0.625 | 1.67 | 2.85 | 1.812 | 3.12
80 | 159.29182 | -27.65609 | 26.127 | 0.004 | 25.719 | 0.003 | 0.408 | 1.75 | 2.87 | 1.862 | 3.06
81 | 159.28261 | -27.64962 | 25.307 | 0.001 | 24.864 | 0.001 | 0.443 | 1.813 | 2.87 | 1.902 | 2.93
82 | 159.28092 | -27.6502 | 24.554 | 0.009 | 24.151 | 0.005 | 0.403 | 1.623 | 2.95 | 1.888 | 3.15
83 | 159.29015 | -27.6545 | 26.695 | 0.003 | 26.241 | 0.002 | 0.454 | 2.004 | 3.12 | 2.046 | 3.15
84 | 159.29275 | -27.65174 | 25.165 | 0.018 | 24.716 | 0.01 | 0.449 | 1.678 | 2.76 | 1.858 | 3.15
85 | 159.2868 | -27.65346 | 27.105 | 0.003 | 26.646 | 0.002 | 0.459 | 1.738 | 2.81 | 1.752 | 2.84
86 | 159.28333 | -27.65349 | 25.44 | 0.002 | 25.001 | 0.001 | 0.439 | 1.746 | 2.78 | 1.741 | 2.81
87 | 159.28723 | -27.65379 | 24.799 | 0.007 | 24.399 | 0.004 | 0.4 | 1.615 | 2.76 | 1.709 | 2.82
88 | 159.28677 | -27.65439 | 26.412 | 0.022 | 25.934 | 0.012 | 0.478 | 2.139 | 3.79 | 1.753 | 2.66
89 | 159.29047 | -27.65437 | 27.603 | 0.068 | 27 | 0.032 | 0.603 | 2.065 | 3.54 | 1.841 | 3.82
90 | 159.29032 | -27.65482 | 24.244 | 0.001 | 23.774 | 0.001 | 0.47 | 1.696 | 2.82 | 1.724 | 2.93
91 | 159.2888 | -27.65454 | 25.853 | 0.006 | 25.371 | 0.003 | 0.482 | 1.587 | 2.73 | 1.689 | 2.89
92 | 159.29654 | -27.6513 | 26.89 | 0.015 | 26.398 | 0.008 | 0.492 | 1.754 | 2.91 | 1.713 | 2.98
93 | 159.2899 | -27.65487 | 25.801 | 0.006 | 25.366 | 0.004 | 0.435 | 1.911 | 3.1 | 1.913 | 3.16
94 | 159.29005 | -27.65557 | 25.891 | 0.007 | 25.367 | 0.004 | 0.524 | 1.725 | 2.85 | 1.761 | 2.94
95 | 159.28814 | -27.6556 | 25.851 | 0.006 | 25.248 | 0.004 | 0.603 | 1.639 | 2.87 | 1.814 | 3.02
96 | 159.28987 | -27.6557 | 27.162 | 0.024 | 26.538 | 0.014 | 0.624 | 1.743 | 2.72 | 1.554 | 2.86
97 | 159.29037 | -27.65583 | 26.216 | 0.01 | 25.62 | 0.006 | 0.596 | 1.665 | 2.86 | 1.683 | 2.98
98 | 159.29296 | -27.65223 | 24.58 | 0.002 | 24.181 | 0.001 | 0.399 | 1.88 | 2.91 | 1.924 | 3.06
99 | 159.28485 | -27.6556 | 24.649 | 0.001 | 24.119 | 0.001 | 0.53 | 1.559 | 2.63 | 1.588 | 2.56
100 | 159.29129 | -27.65171 | 26.722 | 0.013 | 26.263 | 0.007 | 0.459 | 1.847 | 3.12 | 2.027 | 3.66
101 | 159.29226 | -27.652 | 26.458 | 0.01 | 26.041 | 0.006 | 0.417 | 1.651 | 2.65 | 1.713 | 2.89
102 | 159.28552 | -27.65198 | 26.045 | 0.005 | 25.638 | 0.003 | 0.407 | 1.954 | 3.37 | 1.865 | 2.94
|  |  |  |  |  |  |  |  |  |  |


A view of flavour physics in 2021

Riccardo Barbieri

Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy

Based on a view of current flavour physics and motivated by the hierarchy
problem and by the pattern of quark masses and mixings, I describe a picture
of flavour physics that should give rise in a not too distant future to
observable deviations from the SM in Higgs compositeness and/or in B-decays
with violations of Lepton Flavour Universality, as hinted by current data, or
perhaps even in supersymmetry, depending on the specific realisation.

A contribution to:

The special volume of Acta Physica Polonica B to commemorate

Martinus Veltman

##  1 A master

I met Martinus Veltman for the first time in 1970 when, together with Ettore
Remiddi, we went to visit him in Utrecht. We were looking for help to solve
some problems we had in the use of Schoonschip to perform analytic
calculations of higher order QED processes. Two years later Veltman came to
the SNS in Pisa to give lectures on field theory, which I attended, that
became the basis for the CERN yellow report in 1973 under the title of
Diagrammar with Gerard ’t Hooft as coauthor. All this is recalled by Ettore in
an article also to appear on this volume, with particular reference to the
various tools - Schoonschip, dimensional regularisation, the largest time
equation - introduced by Veltman and crucial to our work throughout the
seventies in QED and QCD. Since then I have met Veltman many other times,
especially, but not only, at CERN.

As far as I can tell, the subject of flavour, considered here, has not been
particularly important in all the work of Veltman, except for a few early
papers that he wrote in the sixties. Given the key pioneering role that
Veltman played in the ElectroWeak precision tests, however, one may find a
connection with his work in the fact that I underline the importance of
precision measurements in flavour. True enough, Veltman always liked precise
calculations, whereas here I am often bound to semi-quantitative
considerations based on a ”picture” rather than on a defined theory. Based on
this picture, however, I argue that the experimental flavour program in the
next decade or so might help to design a complete theory of flavour, thus
allowing for precise calculations in this area as well.

##  2 Introduction and motivations

To the extent that the level of my perception has been constant in time, I
feel pretty sure to say that this is the relatively most uncertain time that I
have seen in the last fifty years of Particle Physics (PP). To a large extent,
although not exclusively, this is due to the contrast between the impressive
and steadily increasing experimental success of the Standard Model (SM) since
its complete formulation in the early seventies and, on the opposite side, the
yet unsolved ”structural” problems of the SM itself. The sharpest of these
problems, in my view, is the unpredictability of the masses of all its massive
particles, 15=17-2 (although with a number of independent observables
correlated by $m_{W},m_{Z}$, and also, via loops, by $m_{t}$ and $m_{H}$).
Among them there is the Higgs mass, $m_{H}$, with the specific property of
being sensitive to any UV scale of physics to which the Higgs boson couples:
the ”hierarchy problem”.

Where could some light come from? Perhaps a theoretical breakthrough in Beyond
the SM - not that one hasn’t already tried, sometimes with great ideas, none
of which, however, confirmed by experiments so far - or in the very
foundations of the SM, like Field Theory or even Quantum Mechanics. A possible
new finding in astrophysics or cosmology with some connection with PP. In this
area the fundamental nature of issues like Dark Matter or the origin of the
Baryon Asymmetry in the Universe is not disputable. The open question,
however, is whether or not their solution will have an impact on the
structural problems of the SM. Or, finally, an experimental deviation from the
SM in the form of a new particle or of a discrepancy with the SM in some
precision Low Energy observables.

Figure 1: Left: The ElectroWeak observable parameters $\epsilon_{1,3}$, taken
from Ref. [1]. Right: The relative deviations from the SM in $M_{12}(B_{d})$,
$h_{d}$, and $M_{12}(B_{s})$, $h_{s}$, taken from Ref. [2].

The uncertainty of the current situation motivates the widest possible
approach. This is indeed what is happening, as far as I can tell. Here, with a
time horizon of a decade or so or, most likely, before the operation of the
next High Energy collider, I focus my attention to the potential of the
precision programme in flavour physics, based on a picture of the origin of
charged fermion masses that I shall describe. The underlying assumption is
that there is new physics in the MultiTeV energy region. The motivation for
such an assumption is both generally exploratory and suggested by the still
pending hierarchy problem, mentioned above. Depending on the specific
realisation of the flavour picture, observable deviations from the SM might
emerge in Higgs compositeness and/or in B-decays with violations of Lepton
Flavour Universality, as hinted by current data, or perhaps even in
supersymmetry.

As a specific motivation of this programme I find useful the two Figures 1,
taken from Refs. [1, 2], where one can compare the current status of some key
precision observables in ElectroWeak Physics, the parameters $\epsilon_{1,3}$
[3, 4], and of the relative deviations from the SM, $h_{d}$ and $h_{s}$, in
$\Delta B_{d}=2$ and $\Delta B_{s}=2$ transitions. As one sees, the current
relative precision, respectively sensitive to ”ElectroWeak loops” and to
”Flavour Changing Neutral Current loops”, is in both cases at the level of
$10\div 20\%$. Taking into account experimental, theoretical and parametric
uncertainties, at a future facility like FCC-ee the foreseen relative
precision on $\epsilon_{1,3}$ might reach the $2\div 4\%$ level [5]. A similar
level of relative precision could be reached on $\Delta B_{d}=2$ and $\Delta
B_{s}=2$ by an intense flavour physics programme in the next 10-15 years, as
shown in [2].

Figure 2: Lower bounds on the scale associated with 4-fermion, $\Delta F=2$
operators in different Lorenz configurations, as defined in [9]:
$\mathcal{O}_{1}=(\bar{q}_{Li}\gamma_{\mu}q_{Lj})^{2}$;
$\mathcal{O}_{2,3,4,5}$ contain admixtures of left and right fermion fields.
Courtesy of Luca Silvestrini.

An apparent obstacle to the programme just outlined seems well illustrated in
Fig. 2, where one shows the lower bound on the scales associated with
4-fermion, dimension-6 operators which violate flavour by two units in
different Lorenz configurations. The implications of these bounds are
apparently clear. Flavour is determined by very high energy physics; any new
MultiTeV physics, if existent at all, is flavour independent; the hierarchy
problem is not related to flavour physics. The flavour picture outlined in the
next Section provides an alternative view. Broadly speaking it is inspired by
old ideas, like Extended Technicolour, or less old ones like Extra-dimensions
with branes at different positions [6], or by more recents attempts, as, e.g.,
in Ref.s [7, 8].

##  3 A 3 Scale Flavour Picture

Based on the hierarchies of charged fermion masses and quark mixing angles, I
assume that the three different families are distinguished by new interactions
at (at least) three different scales, $\Lambda_{3}<\Lambda_{2}<\Lambda_{1}$,
from where their masses originate. Denoting collectively by $f_{i}$ the
$SU_{3,2,1}$ fermion multiplets of the i-th generation,
$f_{i}=(q,u,d,l,e)_{i}$ in standard notation, the pattern of masses and
mixings is supposed to arise, at least in part, from the fact that at
$\Lambda_{i}$ no fermion $f_{j}$ is involved with $j<i$. This is summarised in
Fig. 3, where I also assume ”nearest-neighbour” interactions, as implied,
e.g., by the existence of an accidental discrete symmetry, $f_{3}\rightarrow-
f_{3}$ at $\Lambda_{1}$. The presence of more than one family at
$\Lambda_{2,1}$ is what gives rise to mixings. I ignore two issues: the
hierarchy of masses inside one generation, which may be due to the existence
of more than three scales, and the origin of the mixing in the lepton sector,
which is far from hierarchical and may be attributed to the scale(s) and to
the peculiar structure of the right-handed neutrino mass matrix, of different
origin than the mass matrices of the charged fermions.

Figure 3: A scheme of the 3 Scale Flavour Picture as described in the text

The dominant manifestations at low energy, $E<\Lambda_{3}$, of this picture
will be:

  * •

The effects of higher dimensional operators scaled by $\Lambda_{3}$ and only
involving the third generation in the interaction basis;

  * •

An accidental $U(2)^{n}$ symmetry, under which the $f_{3}$ are singlets and
$f_{1,2}$ transform as doublets [10], progressively broken at $\Lambda_{2,1}$,
which controls the Yukawa couplings and the consequent involvement of the
first two generations, in the mass basis, at the scale $\Lambda_{3}$.

I expect that specific realisations of this flavour picture will be relevant
in presence of other new physics in the MultiTeV with consequently different
specific manifestations. In the following Sections I consider in turn the
cases of Composite Higgs, of B-decays with Lepton Flavour Violation (if
confirmed by data), which may in fact be coupled with Higgs compositeness
itself, or even the case of supersymmetry. A central question in all of these
cases is how low can the scale $\Lambda_{3}$ be consistently with current
constraints. Unless otherwise stated, I shall ignore possible low energy
effects of the interactions at $\Lambda_{2,1}$ (except for their contribution
to the Yukawa couplings) assuming that they do not lead to more stringent
constraints than the interactions at $\Lambda_{3}$ itself.

###  3.1 Remaining uncertainties

To be able to determine in a concrete way the bounds on the scale
$\Lambda_{3}$ in the different situations, one needs the diagonalisation
matrices of the different Yukawa couplings

$Y^{U}=(U^{L})^{+}Y^{U}_{diag}U^{R},\quad
Y^{D}=(D^{L})^{+}Y^{D}_{diag}D^{R},\quad Y^{E}=(E^{L})^{+}Y^{E}_{diag}E^{R}.$
(1)

The picture of Fig. 3 implies a hierarchical structure of $Y^{U,D,E}$ and the
vanishing of their $Y_{13},Y_{31}$ elements. In turn, up to small relative
errors controlled by the precise knowledge of $Y^{U,D,E}_{diag}$, every
diagonalisation matrix in eq. (1), collectively called $\mathcal{U}$, reduces
to the product of two subsequent unitary transformations in the 12 and 23
sectors, $\mathcal{U}=\mathcal{U}(12)\mathcal{U}(23)$.

Taking advantage of $V_{CKM}=U^{L}(D^{L})^{+}$, on can show [10] that $U^{L}$
and $D^{L}$ are determined up to one physical phase and their 23 elements,
constrained by

$V_{cb}=U_{23}^{L}-D_{23}^{L}.$ (2)

This last ambiguity can be fixed by invoking an accidental discrete symmetry
at $\Lambda_{2}$, $d_{3}\rightarrow-d_{3}$ or $u_{3}\rightarrow-u_{3}$, giving
respectively $D_{23}^{L}=0$ or $U_{23}^{L}=0$.

The right handed diagonalisation matrices, in particular $U^{R},D^{R}$, remain
unknown. They can leave the third generation almost fully decoupled from the
first two, i.e.
$\mathcal{U}^{R}(23)\approx\mathcal{D}^{R}(23)\approx\mathbb{1}$, by
demanding, again at $\Lambda_{2}$, that $q_{3}\rightarrow-q_{3}$ be an
invariance. All of this is effectively equivalent, in the quark sector, to a
”minimally violated $U(2)^{3}$” flavour symmetry [10].

##  4 Flavour in Composite Higgs

In this Section I assume, as customary in the literature, that a new strong
interaction with a confinement scale $m_{*}$ and a strong coupling
$1<g_{*}<4\pi$ gives rise, after spontaneous symmetry breaking, to the Higgs,
$H$, as a Pseudo-Nambu-Goldstone-Boson (PNGB) [11]. The standard fermions do
not feel directly this new strong interaction, but, to get a mass, they are
connected at the scales $\Lambda_{3,2,1}$, as specified above, with a
composite operator of the strong sector, $\mathcal{O}_{H}$, with
$<0|\mathcal{O}_{H}|H>\neq 0$. The strongest of these couplings occurs at
$\Lambda_{3}$

$\mathcal{L}^{top}_{Y}(\Lambda_{3})=\frac{x_{L}x_{R}}{\Lambda_{3}^{d_{H}-1}}\bar{q}_{3}\mathcal{O}_{H}u_{3},$
(3)

and gives rise to the top Yukawa coupling $y_{t}$ at the compositeness scale
$m_{*}$, after $\mathcal{O}_{H}\rightarrow g_{*}m_{*}^{d_{H}-1}H$,

$y_{t}=g_{*}x_{L}x_{R}(\frac{m_{*}}{\Lambda_{3}})^{d_{H}-1}.$ (4)

Here $d_{H}$ is the anomalous dimension of the operator $\mathcal{O}_{H}$ and
$x_{L,R}\leq 1$ are dimensionless parameters, universally associated with
$q_{3},u_{3}$, which measure their respective ”level of compositeness”. Since
$y_{t}\approx 1$ and $d_{H}\lesssim 2$ [12], eq. (4) suggests that
$\Lambda_{3}$ be identified with the compositeness scale $m_{*}$. The question
to ask is which are the main effects in flavour physics that one expects from
this picture at low energy with respect to $\Lambda_{3}\approx m^{*}$.

$\begin{array}[]{c|c|c|c|c}Observable&Operator&\Lambda/TeV(pres)&\Lambda/TeV(fut)&m_{*}/\Lambda\\\
\hline\cr\epsilon_{K}&\mathcal{Q}_{1}^{sd}=(\bar{s}_{L}\gamma_{\mu}d_{L})^{2}&7&13&x_{t}\\\
\hline\cr\Delta
M_{B_{d}}&\mathcal{Q}_{1}^{bd}=(\bar{b}_{L}\gamma_{\mu}d_{L})^{2}&8&18&x_{t}\\\
\hline\cr\Delta
M_{B_{s}}&\mathcal{Q}_{1}^{bs}=(\bar{b}_{L}\gamma_{\mu}s_{L})^{2}&9&20&x_{t}\\\
\hline\cr\Delta
M_{D},p/q&\mathcal{Q}_{1}^{cu}=(\bar{c}_{L}\gamma_{\mu}u_{L})^{2}&3&10&x_{t}\\\
\hline\cr\hline\cr b\rightarrow s\leavevmode\nobreak\
\bar{l}l&(\bar{s}_{L}\gamma_{\mu}b_{L})H^{+}i\mathcal{D}_{\mu}H&4.5&12&\sqrt{g^{*}x_{t}}\\\
\hline\cr s\rightarrow d\leavevmode\nobreak\
\bar{l}l(\bar{q}q)&(\bar{d}_{L}\gamma_{\mu}s_{L})H^{+}i\mathcal{D}_{\mu}H&1.7&5&\sqrt{g^{*}x_{t}}\\\
\hline\cr\hline\cr neutronEDM\leavevmode\nobreak\ (*)&\approx
m_{t}(\bar{t}_{L}\sigma_{\mu\nu}T^{a}t_{R})g_{S}G^{\mu\nu}_{a}&\approx
5.5&16&g^{*}/4\pi\\\ \hline\cr\hline\cr electronEDM\leavevmode\nobreak\
(*)&\approx m_{t}(\bar{t}_{L}\sigma_{\mu\nu}t_{R})eF^{\mu\nu}&\approx
50&?&\sqrt{g^{*}x_{t}}/(4\pi)\\\ \hline\cr\hline\cr\end{array}$ Table 1:
Estimated lower bounds, present and future, on the scale associated with
representative operators, from different observables. See text.

As well known and manifest from Fig. 2 , the strongest constraint from $\Delta
F=2$ transitions without a flavour structure comes from $\Delta S=2$,
4-fermion, LR interactions. To keep this constraint under control, I assume
the symmetry $q_{3}\rightarrow-q_{3}$ at $\Lambda_{2}$, which decouples in
particular $d_{3}$ from $d_{1,2}$, but also $u_{3}$ from $u_{1,2}$. This
leaves, upon use of eq. (4),

$\mathcal{L}^{(4f)}=\frac{y_{t}^{2}x_{t}^{2}}{m_{*}^{2}}(\bar{q}_{3}\gamma_{\mu}q_{3})^{2},\quad\quad\frac{y_{t}}{g^{*}}<x_{t}=\frac{x_{L}}{x_{R}}<\frac{g^{*}}{y_{t}}$
(5)

as the only possible source of $\Delta F=2$ transitions in the mass basis,
controlled by the mixing matrices $U^{L}$ and $D^{L}$111For the purpose of the
estimates of interest here, I neglect a possible cancellation between the
operator in eq. (5) and $(\bar{q}_{3}\gamma_{\mu}\sigma_{a}q_{3})^{2}$..

To estimate the bounds on $\Lambda_{3}\approx m^{*}$ from the various $\Delta
F=2$ transitions is now a straightforward matter, starting from the bounds in
Fig. 2 and using $U^{L}$ and $D^{L}$ as determined in the previous Section. At
least for concreteness, I use $U^{L}_{23}=0$ or $D^{L}_{23}=0$ when
considering the bounds from $\Delta S=2,1$ and $\Delta B=2,1$ or $\Delta C=2$
respectively. As recalled in Section 3.1, this leaves one single phase free in
$U^{L}$ and $D^{L}$, which however does not affect the $\Delta S=2$ nor the
$\Delta C=2$ transitions, where the distinction between CP-conserving and CP-
violating effects is most crucial.

Figure 4: Left: Estimate of the currently excluded regions for $U^{L}_{23}=0$.
Right: Estimate of the currently excluded regions and of the foreseen
sensitivity (dotted lines) for $D^{L}_{23}=0$. The lines associated with the
electron and the neutron EDMs assume maximal phases. Everywhere $x_{t}=1$.

Along similar lines one can extract bounds on $\Lambda_{3}\approx m^{*}$ from
$\Delta F=1$ transitions and from the neutron and electron Electric Dipole
Moments (EDM), although with an increasing level of uncertainty and model
dependence both in $\Delta F=1$ transitions and even more so in the EDMs.
Table 1 summarises an estimate of these bounds, including a future projection,
from different observables. In the third and fourth columns, denoted
$\Lambda/TeV$, I set $g^{*}=x_{t}=1$ and I ignore a strong loop factor
$(g^{*}/4\pi)^{2}$ in the EMDs. The final column gives the relation between
$\Lambda$, thus defined, and $m^{*}\approx\Lambda_{3}$. An overall
illustration of these bounds is given in Fig. 4, both for the case
$U^{L}_{23}=0$ or $D^{L}_{23}=0$222The bound from CP violation in $\Delta C=2$
does not yet include the recent LHCb results [13].

In this Section I have neglected direct significant couplings of the leptons
to the composite sector, i.e. to the scale $\Lambda_{3}$; an assumption to be
changed in the following Section.

##  5 Anomalies in B-decays

If the third generation leptons are involved as well with a significant
strength in the new interactions at $\Lambda_{3}$, violations of Lepton
Flavour Universality become of clear interest, in particular due to the
theoretical cleanness of observables sensitive to them. This unavoidably
brings the attention to the persistent, though not yet decisive, observations
of anomalies in semi-leptonic B-decays, especially since, at first sight, they
occur with a pattern that looks quite compatible with the picture described
above:

  * •

A roughly $10\%$ deviation from a tree level SM process in the charged channel
$b\rightarrow c\leavevmode\nobreak\ l\nu$, quantified in the ratio [14, 15,
16]

$R_{D^{(*)}}=\frac{BR(B\rightarrow D^{(*)}\tau\nu)}{BR(B\rightarrow
D^{(*)}l\nu,l=\mu,e)},$ (6)

where, in the numerator, only one second generation particle is involved (and
three in the denominator, neglecting corrections to the electron component);

  * •

A roughly $10\%$ deviation from a loop induced SM process in the neutral
channel $b\rightarrow s\leavevmode\nobreak\ ll$, quantified in the ratio [17,
18]

$R_{K^{(*)}}=\frac{BR(B\rightarrow K^{(*)}\mu\mu)}{BR(B\rightarrow
K^{(*)}ee)},$ (7)

where, in the numerator, three second generation particles are involved. In
fact the anomaly in $R_{K^{(*)}}$ appears consistently reinforced by the
measurements of the angular distributions in $B\rightarrow K^{*}\mu\mu$ and of
$BR(B_{s}\rightarrow\mu\mu)$ [19, 20].

An independent good reason to pay attention to these anomalies is that the
experimental error on the key observables is expected to be reduced by a large
factor in the foreseen programme in the next decade or so [21, 22].

In view of all this, interesting questions to ask are: 1. Which interaction at
$\Lambda_{3}$ can be responsible for these anomalies, assumed to persist? 2.
Can this interaction be in some way related to Higgs Compositeness, which, due
to the hierarchy problem, motivates a new scale in the MultiTeV?

Figure 5: Isolines of the charged current anomaly ($\Delta R_{D}$, red solid
lines) and of the neutral current anomaly ($\Delta C^{\mu}_{9}=2.1\Delta
R_{K},x_{e}=|E^{L}_{23}/V_{cb}|$, blue dashed lines) from the sole interaction
at $\Lambda_{3}$, see text. Taken from Ref. [26].

To the first question a candidate interaction is

$\mathcal{L}_{U}=g_{U}U_{\mu}^{a}(\bar{q}_{3}^{a}\gamma_{\mu}l_{3})$ (8)

where $U_{\mu}^{a}$ is a vector leptoquark of mass $M_{U}$, transforming as
$(\mathbb{3},\mathbb{1})_{2/3}$ under the SM gauge group[23, 24, 25]. After
going to the mass basis, as in the previous Section, this interaction is
indeed capable of producing both the charged and the neutral current anomaly.
In the parameter space $(M_{eff}=M_{U}/g_{U},D^{L}_{23},E^{L}_{23})$ the size
of the observables $R_{D^{(*)}}$ and $R_{K^{(*)}}$ is shown in Fig. 5, taken
from Ref. [26]. This is within reach of the expected future experimental
precision [21, 22]. Especially $R_{D^{(*)}}$, however, is below the current
preferred value. To get to this value a contribution from the interactions of
the leptoquark at the scale $\Lambda_{2}$ is needed, as illustrated, e.g., in
Ref. [27, 26].

The second question can help to address the unavoidable issue of UV-completing
the vector leptoquark interaction in eq. (8). In Higgs Compositeness it is
commonly assumed that the new strong interaction has a global symmetry, within
which the SM vector interactions are ”weakly gauged”. In turn, always
mimicking QCD, this leads to the existence of $\rho$-like composite vectors
with a mass close to $m^{*}$, as defined in the previous Section. Back to the
leptoquark I think that it is completely natural to view it as part of
composite vectors transforming as the adjoint of an $SU(4)$ global symmetry,
within which the standard strong interactions are gauged. Remember that the
standard quarks and leptons are spectators with respect to the new strong
interaction, but, to get their coupling to the composite Higgs boson, they
have to find suitable degrees of freedom to communicate with the composite
structure itself, like, e.g., composite fermions to mix with [25, 28, 29,
30]333An apparently alternative viewpoint consists in embedding the leptoquark
in a gauged $SU(4)$ group commuting with $SU(3)\times SU(2)\times U(1)$, and
with a suitable breaking pattern to generate the standard gauge interactions
at low energies [31, 32, 33, 34, 8]..

The global $SU(4)$ symmetry, with the same algebraic structure as the familiar
Pati-Salam $SU(4)$ group [35], cannot be universally coupled to the three
different families since in this case, to cope with $BR(K_{L}\rightarrow\mu
e)<4.7\cdot 10^{-12}$, $SU(4)$ would have to be broken down to $SU(3)$ at a
much higher scale than $\Lambda_{3}\approx m^{*}$ in the MultiTeV. Here again
the appearance of more than one scale distinguishing the different families
may come into play.

The number of constraints on this interpretation of the leptoquark in flavour
or even in ElectroWeak precision physics is definitely challenging.
Constraints appear as well in high-$p_{T}$ physics, like in leptoquark pair
production or, even more so, in $\tau\bar{\tau}$ production [36, 37]. To see
if these constraints can all be met, a useful step may be provided by the
model proposed in Ref. [8], in particular its interpretation in 5D [38], which
incorporates the three different scales for the different families, or a
suitable modification of it. Even within obvious uncertainties, the
interpretation of the anomalies as a first sign of Higgs compositeness, with
flavour somehow implemented as described in Section 3, appears consistent with
all observations today provided $D_{23}\approx 0$ (See Fig. 4) and
$M_{U}\gtrsim 3\div 4\leavevmode\nobreak\ TeV,g_{U}\gtrsim 2\div 3$.

##  6 Supersymmetry

To address the hierarchy problem, supersymmetry in the MultiTeV remains the
prominent if not the only weakly coupled alternative to Higgs Compositeness.
In this case, however, there are reasons to think that precision in flavour
physics may not be as effective in searching for indirect signals as in the
case of a strongly interacting theory, even if one blocks the possibility of
highly degenerate superpartners in family space as in the case of gauge-
mediated supersymmetry breaking.

Figure 6: Left: relative correction to $\epsilon_{K}$, enhanced by a factor
100, from the gluino exchange box-diagram with left-handed sbottom only.
Right: relative correction to $\epsilon_{K}$ from the gluino exchange box-
diagram with left-handed and right-handed sbottoms and $D^{R}_{23}=V_{cb}$.

Suppose that one does not assume any flavour structure at all, as in the case
that leads to the bounds in Fig. 2. Even if these bounds are rescaled down by
a factor $(g^{2}/4\pi)$, where $g$ stands for any of the SM gauge couplings,
as necessary from a gaugino-mediated $\Delta F=2$ box diagram, many of them
are left well above the MultiTeV. On the contrary, by assuming ”minimally
broken $U(2)^{3}"$, the same rescaling, this time applied to the bounds in
Table 1, would lead to scales, i.e. the superpartner masses, normally well
inside the region already explored at LHC. This is confirmed by Fig. 6 left,
taken from Ref. [39], which shows the relative deviation from the SM, enhanced
by a factor $100$, in the $\Delta S=2$ parameter $\epsilon_{K}$

$R\equiv\frac{\epsilon_{K}}{\epsilon_{K}^{SM}}-1,$ (9)

coming from the gluino exchange box diagrams, as function of the gluino and
the left-handed sbottom masses, $M_{\tilde{g}}$ and $m_{\tilde{Q}_{3}}$. The
current bound is $R\lesssim 0.2$.

As recalled in Section 4, a key feature here is the suppression of the LR
4-fermion $\Delta S=2$ interaction, obtained by demanding invariance under
$q_{3}\rightarrow-q_{3}$ at $\Lambda_{2}$ or $D^{R}_{23}=0$ (See Section 3.1).
If this requirement is abandoned, the same gluino box diagrams with
$D^{R}_{23}=V_{cb}$ and with both left and right sbottom exchanges, taken
degenerate, contribute to $R$ approximately as in Fig. 6 right, adapted from
Ref. [39]. Note that $R$ scales as $R\approx(D^{R}_{23}/V_{cb})^{2}$, thus
leaving room for a possible effect in $\epsilon_{K}$ even for superpartner
masses exceeding the current LHC bounds.

##  7 Conclusions

The 3 Scale Flavour Picture (3SFP) presented in Section 3 provides an
alternative to the view that the scale of flavour is far beyond the MultiTeV,
as such decoupled from the physics that could explain the hierarchy problem. I
admit that the construction of an actual precise theory incorporating this
3SFP is not a trivial task, hence the use of the word ”picture”. Nevertheless
Higgs Compositeness in the MultiTeV or the confirmation of LFV in B-decays
seem to require some version of the 3SFP. Furthermore, to motivate the
relatively low scale required to explain the current LFV signals, if
confirmed, their coupling to Higgs Compositeness, though not necessary, is
nevertheless highly suggestive, perhaps along the lines outlined in Section 5
with a key role of the $SU(4)$ symmetry. The strongly reassuring fact is that
in any case precision flavour measurements in the next 10-15 years can decide
on these issues [21, 22]. In the positive case for the 3SFP, the number of
different observables that would show a deviation from the SM should offer
enough tools for the construction of a fully complete theory of flavour.

The role of flavour precision tests in a weakly coupled theory, like
supersymmetry, appears relatively less bright, with signals that could emerge,
under suitable conditions, in $\epsilon_{K}$ or in the electron EDM.

## Acknowledgments

I am indebted to Gino Isidori for his collaborations on this subject and for
the many discussions on most the issues of relevance to this paper.

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