# Lattice investigations of the chimera baryon spectrum in the $Sp(4)$ gauge
theory

Ed Bennett<EMAIL_ADDRESS>Swansea Academy of Advanced Computing,
Swansea University, Fabian Way, SA1 8EN Swansea, Wales, UK  Deog Ki Hong
<EMAIL_ADDRESS>Department of Physics, Pusan National University, Busan
46241, Korea  Extreme Physics Institute, Pusan National University, Busan
46241, Korea  Ho Hsiao<EMAIL_ADDRESS>Institute of Physics,
National Yang Ming Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 30010,
Taiwan  Jong-Wan Lee<EMAIL_ADDRESS>Particle Theory and Cosmology Group,
Center for Theoretical Physics of the Universe, Institute for Basic Science
(IBS), Daejeon, 34126, Korea  Extreme Physics Institute, Pusan National
University, Busan 46241, Korea  C.-J. David Lin<EMAIL_ADDRESS>Institute of
Physics, National Yang Ming Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu
30010, Taiwan  Center for High Energy Physics, Chung-Yuan Christian
University, Chung-Li 32023, Taiwan  Centre for Theoretical and Computational
Physics, National Yang Ming Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu
30010, Taiwan  Biagio Lucini<EMAIL_ADDRESS>Department of
Mathematics, Faculty of Science and Engineering, Swansea University, Fabian
Way, SA1 8EN Swansea, Wales, UK  Swansea Academy of Advanced Computing,
Swansea University, Fabian Way, SA1 8EN Swansea, Wales, UK  Maurizio Piai
<EMAIL_ADDRESS>Department of Physics, Faculty of Science and
Engineering, Swansea University, Singleton Park, SA2 8PP, Swansea, Wales, UK
Davide Vadacchino<EMAIL_ADDRESS>Centre for Mathematical
Sciences, University of Plymouth, Plymouth, PL4 8AA, United Kingdom

###### Abstract

We report the results of lattice numerical studies of the $Sp(4)$ gauge theory
coupled to fermions (hyperquarks) transforming in the fundamental and two-
index antisymmetric representations of the gauge group. This strongly-coupled
theory is the minimal candidate for the ultraviolet completion of composite
Higgs models that facilitate the mechanism of partial compositeness for
generating the top-quark mass. We measure the spectrum of the low-lying, half-
integer spin, bound states composed of two fundamental and one antisymmetric
hyperquarks, dubbed chimera baryons, in the quenched approximation.

In this first systematic, non-perturbative study, we focus on the three
lightest parity-even chimera-baryon states, in analogy with QCD, denoted as
$\Lambda_{\rm CB}$, $\Sigma_{\rm CB}$ (both with spin $1/2$), and $\Sigma_{\rm
CB}^{\ast}$(with spin $3/2$). The spin-$1/2$ such states are candidates of the
top partners. The extrapolation of our results to the continuum and massless-
hyperquark limit is performed using formulae inspired by QCD heavy-baryon
Wilson chiral perturbation theory. Within the range of hyperquark masses in
our simulations, we find that $\Sigma_{\mathrm{CB}}$ is not heavier than
$\Lambda_{\mathrm{CB}}$.

††preprint: CTPU-PTC-23-49

###### Contents

  1. I Introduction
  2. II Lattice numerical calculations
    1. II.1 Lattice action
    2. II.2 Numerical strategy
    3. II.3 Interpolating operators and correlation functions
  3. III Data analysis and numerical results
    1. III.1 Spin and Parity projection
    2. III.2 Mass hierarchy and hyperquark-mass dependence of chimera baryons
    3. III.3 Mass extrapolations and cross checks
  4. IV Summary and Outlook
  5. A Extracted masses of mesons and chimera baryons
  6. B Smearing techniques
  7. C Details about fitting procedures

##  I Introduction

The discovery of the Higgs boson ATLAS:2012yve ; CMS:2012qbp  has exacerbated
the need for a deeper understanding of the origin of electroweak symmetry
breaking (EWSB). On the one hand, no firm experimental evidence has been found
of violations of the standard model (SM) predictions. On the other hand, a
plethora of considerations, in particular the triviality of the scalar sector
Aizenman:1981zz ; Frohlich:1982tw ; Luscher:1987ay ; Luscher:1987ek ;
Luscher:1988uq ; Bernreuther:1987hv ; Kenna:1992np ; Gockeler:1992zj ;
Wolff:2009ke ; Weisz:2010xx ; Hogervorst:2011zw ; Siefert:2014ela  (and most
likely of the whole Higgs-Yukawa sector Molgaard:2014mqa ; Bulava:2012rb ;
Chu:2018ldw ) implies that the SM cannot be a fundamental theory, but it
rather provides an effective field theory (EFT) description, valid up to some
large, but finite, ultraviolet (UV) cut-off scale, beyond which the SM has to
be completed. The challenge is that any theory serving as the UV completion of
the SM must contain a light scalar state that can be interpreted as the
observed Higgs boson, while also reproducing the observed SM phenomenology, up
to the TeV scale, and down to the current (high) level of precision.

Composite Higgs models (CHMs) Kaplan:1983fs ; Georgi:1984hp ; Dugan:1984hq ,
for example those in Refs. Contino:2003ve ; Agashe:2004rs ; Ferretti:2013kya ;
Ferretti:2016upr ; Cacciapaglia:2019bqz ; Ferretti:2014qta ; Barnard:2013zea ;
Ma:2015gra ; Vecchi:2015fma ; Appelquist:2020bqj —see also the reviews in
Refs. Contino:2010rs ; Panico:2015jxa ; Witzel:2019jbe ; Cacciapaglia:2019vcb
; Cacciapaglia:2020kgq ; Bennett:2023wjw —have been attracting attention in
recent years, because they can naturally accommodate a light Higgs boson. In
these models, a novel strongly coupled sector is introduced, based upon an
asymptotically-free gauge theory coupled to fermions (hyperquarks). At
variance with technicolor models, the SM Higgs boson emerges as one of the
pseudo-Nambu-Goldstone bosons (PNGBs), associated with a global symmetry of
the new strong interaction, to provide a UV completion for the standard model.
The global symmetry is broken both spontaneously (by the condensates forming
dynamically) and explicitly, hence the PNGBs develop a potential due to
(small) symmetry breaking effects. Such effects may arise either within the
strong-coupling sector itself (e.g., hyperquark mass terms) or due to its
coupling to external fields (e.g., couplings to SM fields). As the Higgs
fields are identified with a subset those that describe the PNGBs in the low-
energy EFT description of the theory, EWSB is triggered by the interplay among
different symmetry-breaking effects, along the lines of vacuum alignment
analysis Peskin:1980gc  and radiative EWSB Coleman:1973jx —for recent studies
in the context of CHMs, see for instance Refs. Arkani-Hamed:2002ikv ;
Contino:2003ve ; Agashe:2004rs ; Contino:2010rs ; Golterman:2015zwa ;
Golterman:2017vdj ; Banerjee:2023ipb .

Since CHMs involve strongly coupled dynamics requiring a non-perturbative
treatment, it is natural to rely on lattice calculations for their
investigation. Our collaboration has been performing such calculations for a
particular UV-completion that is built with the $Sp(4)$ gauge theory
containing two flavors ($N_{f}=2$) of Dirac fermions in the fundamental,
$(f)$, representation Barnard:2013zea ; Ferretti:2013kya . We denote these
fundamental hyperquarks by $Q^{i\,a}$, where $a=1,\ldots,4$ is the hypercolor
index and $i=1,2$ the flavor one. Because of the pseudoreality of this
representation of the gauge group, the approximate global symmetry acting on
the $(f)$ hyperquarks is $SU(4)$, which is broken spontaneously to $Sp(4)$
Peskin:1980gc . This results in five PNGBs, four of which can be interpreted
as the SM complex scalar doublet, provided the $SU(2)\times U(1)$ SM gauge
group is chosen as an appropriate subgroup of the components of $SU(4)$. The
$SU(4)/Sp(4)$ coset leads to the minimal CHM amenable to lattice treatment, in
the sense that it gives candidates for the SM Higgs doublet with only one
additional Goldstone mode. Previous publications Bennett:2017kga ;
Bennett:2019jzz ; Bennett:2019cxd  reported on the meson spectra of this
theory obtained from both quenched and dynamical lattice simulations. An
extended study of meson spectra computed in the quenched approximation, for
various $Sp(2N)$ groups, and matter fields transforming in several different
representations of the group, is in preparation Bennett:quenched .

It is possible to extend CHMs to address the flavor problem, or at least its
most challenging aspect: to generate the large mass for the SM top quark,
without spoiling the SM successful description of flavor-changing neutral
current processes and precision electroweak observables. To address this
challenge, the idea of (top) partial compositeness was introduced in Ref.
Kaplan:1991dc  (see also the discussions in Refs. Grossman:1999ra ;
Gherghetta:2000qt ; Lodone:2008yy ; Chacko:2012sy ; Grojean:2013qca ). If one
couples the theory to hyperquarks transforming in two different
representations of the gauge group, and embeds the SM gauge group as an
appropriate subgroup of the global symmetry of the new sector, some of the
bound states formed by hyperquarks in different representations can be
arranged to carry the same quantum numbers as the top quark. Such bound states
can be identified as top partners. The top quark then acquires its mass by
mixing with the top partners. In the $Sp(4)$ gauge theory with $N_{f}=2$ $(f)$
hyperquarks, top partial compositeness can be achieved by adding to the theory
$n_{f}=3$ Dirac fermions in the two-index antisymmetric, $(as)$,
representation of the gauge group Bennett:2022yfa ; Barnard:2013zea . We
denoted the $(as)$ hyperquarks by $\Psi^{k\,ab}$, with $k=1,2,3$ the flavor
index. Because the $(as)$ representation is real, the global symmetry for
three flavors is $SU(6)$, spontaneously broken to $SO(6)$ Peskin:1980gc —see
also the CHMs in Refs. Cacciapaglia:2019ixa ; Cai:2020njb . One can gauge the
$SU(3)$ subgroup of the unbroken $SO(6)$, and identify it with the QCD gauge
group Barnard:2013zea ; Ferretti:2013kya . We call chimera baryons the
hypercolor singlet bound states formed by one $\Psi$ and two $Q$ fields.
Spin-1/2 chimera baryons can then act as candidate top partners. See Refs.
Cossu:2019hse ; DelDebbio:2022qgu ; Bergner:2021ivi ; Ayyar:2019exp ;
Ayyar:2018zuk ; Ayyar:2017qdf  for recent work on candidate top partners in
other gauge theories.

For the purposes of this paper, we consider the strongly coupled theory in
isolation, hence there are no SM fields nor interactions. We present our
measurements of the masses of chimera baryons sourced by the following
operators:

$\displaystyle{\mathcal{O}}_{\rho}^{ijk,5}$ $\displaystyle\equiv$
$\displaystyle
Q^{i\,a}_{\alpha}(C\gamma^{5})_{\alpha\beta}Q^{j\,b}_{\beta}\Omega^{ad}\Omega^{bc}\Psi^{k\,cd}_{\rho}\,,$
(1) $\displaystyle{\mathcal{O}}_{\rho}^{ijk,\mu}$ $\displaystyle\equiv$
$\displaystyle
Q^{i\,a}_{\alpha}(C\gamma^{\mu})_{\alpha\beta}Q^{j\,b}_{\,\beta}\Omega^{ad}\Omega^{bc}\Psi^{k\,cd}_{\rho}\,,$
(2)

where $a,b,c,d$ are hypercolor indices, $\alpha,\beta$, $\rho$ are spinor
indices, $i,j,k$ are flavor indices, $\gamma^{5}$ and $\gamma^{\mu}$ are
$4\times 4$ Dirac matrices, and $C$ is the charge conjugation matrix. The
symplectic matrix, $\Omega$, is defined as

$\Omega\equiv\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&1\\\ -1&0&0&0\\\
0&-1&0&0\\\ \end{array}\right)\,.$ (3)

We restrict our attention to operators for which the $SU(4)$ index is off-
diagonal, $i\not=j$. For mesons this requirement ensures that there is no
disconnected contraction in computing two-point correlation functions
111Pioneering lattice studies of the flavor-singlet meson sector of candidate
completions for CHMs can be found for example in Refs. Arthur:2016ozw ;
Drach:2021uhl ; Bennett:2023rsl .. For chimera baryons it removes from the
calculations diagrams involving $(f)$-type contractions within the initial and
final state. The operator ${\mathcal{O}}^{5}$ annihilates spin-1/2 composite
states. Following an analogy with the $\Lambda$ baryon in QCD, to which we
return later in this section, we denote the lightest state of this type as
$\Lambda_{\mathrm{CB}}$. The operator in Eq. (2), ${\mathcal{O}}^{\mu}$,
overlaps with both spin-1/2 and 3/2 states, and we denote the lightest ones by
$\Sigma_{\mathrm{CB}}$ and $\Sigma^{\ast}_{\mathrm{CB}}$, respectively. Both
$\Lambda_{\mathrm{CB}}$ and $\Sigma_{\mathrm{CB}}$ baryons can be candidate
top partners Gripaios:2009pe ; Banerjee:2022izw . We report the quantum
numbers of the three chimera baryons in Tab. 1, together with some of the
properties of the analogous particle in QCD. Our lattice calculations of the
masses of $\Lambda_{\mathrm{CB}}$, $\Sigma_{\mathrm{CB}}$ and
$\Sigma^{\ast}_{\mathrm{CB}}$ are performed in the quenched approximation. The
determination of these masses is of importance in constructing a viable UV-
complete composite Higgs model with partial compositeness, because it affects
both the mass of the top quark, and direct and indirect new physics searches
for top partners.

Table 1: The chimera baryons of interest in this paper, the interpolating operators that source them, their quantum numbers—spin, $J$, and irreducible representation of the unbroken, global symmetry groups, $Sp(4)$ and $SO(6)$—and the properties of the analogous QCD state—mass in MeV (rounded to unit), strangeness, $S$, isospin, $I$, spin, $J$ ParticleDataGroup:2022pth . See also Refs. Bennett:2022yfa ; Hsiao:2022kxf ; Bennett:2023wjw .  Chimera Baryon |  Interpolating operator |  $J$ |  $Sp(4)$ |  $SO(6)$ |  QCD analogy
---|---|---|---|---|---
$\Lambda_{{\rm CB}}$ |  ${\mathcal{O}}^{5}$ , Eq. (1) |  ${1}/{2}$ |  $5$ |  $6$ |  $\Lambda(1116)$, $S=-1$, $I=0$, $J=1/2$
$\Sigma_{{\rm CB}}$ |  ${\mathcal{O}}^{\mu}$ , Eq. (2) |  ${1}/{2}$ |  $10$ |  $6$ |  $\Sigma(1193)$, $S=-1$, $I=1$, $J=1/2$
$\Sigma^{\ast}_{{\rm CB}}$ |  ${\mathcal{O}}^{\mu}$ , Eq. (2) |  ${3}/{2}$ |  $10$ |  $6$ |  $\Sigma^{\ast}(1379)$, $S=-1$, $I=1$, $J=3/2$

The mechanism by which SM fermions (the top quark in particular) acquire a
mass via their coupling to chimera baryon operators of the strongly coupled
sector is rather different from that provided by the Yukawa couplings in the
standard model, as well as from the coupling to the meson operators adopted in
extended technicolor Dimopoulos:1979es ; Eichten:1979ah  and walking
technicolor Yamawaki:1985zg ; Holdom:1984sk ; Appelquist:1986an  theories. In
particular, the value of the dynamically generated scaling dimension of the
chimera baryon operators enters non-trivially into the estimates of the
resulting SM fermion masses—see for example the discussion in Sections. IV.B
and V.B of Ref. Chacko:2012sy , in the studies reported in Refs. Ayyar:2018glg
; BuarqueFranzosi:2019eee ; DeGrand:2015yna , in Section 2.4.2 of the review
Bennett:2023wjw , and references therein. Measuring these scaling dimensions
is an ambitious task that requires dedicated methodology—see Ref.
Hasenfratz:2023sqa  for recent progress along these lines, but in a different
theory—and that we leave for the future, as it goes far beyond the reach of
the quenched approximation we adopt here.

Yang-Mills theories have a well-defined limit for a large number of colors
tHooft:1973alw . $SU(N_{c})$ theories coupled to a finite number of fermions
give, in the large-$N_{c}$ limit, a good description of important properties
of strong interactions, such as Zweig’s rule, or vector meson dominance
Witten:1979kh . Furthermore, baryons can be realized as solitons, in agreement
with Skyrme’s picture Skyrme:1961vq ; Witten:1983tx . If one naively expects
baryons in $Sp(2N)$ gauge theories to be well-defined in the large-$N$ limit,
yet baryons made of $2N$ fundamental hyperquarks are unstable, decaying into
$N$ mesons, since the totally antisymmetric tensor can be decomposed into
products of symplectic structures, schematically written as

$\epsilon_{a_{1}a_{2}\cdots
a_{2N}}=\Omega_{a_{1}a_{2}}\Omega_{a_{3}a_{4}}\cdots\Omega_{a_{2N-1}a_{2N}}\pm\cdots.$
(4)

One can still have a well-defined limit in two ways. Either one generalizes
the rank-$2$ antisymmetric hyperquark to the antisymmetric rank-$N$
hyperquark, transforming as the Pfaffian of $Sp(2N)$, to form a color-singlet
with $N$ fundamental hyperquarks. As an alternative, one can also consider the
singlet state obtained with one (conjugate) antisymmetric fermion and two
fundamental ones: this state exists for both $SU(N_{c})$ and $Sp(N_{c}=2N)$
theories, the two large $N_{c}$ limits yielding a common, finite mass, and one
can show that in $SU(3)$ this is an ordinary baryon. Because in the following
the two species of fermions have different masses, we can make an analogy for
the heavier, conjugate antisymmetric fermions with the strange quark and for
the fundamental fermions with the up and down quarks, leading to the
aforementioned association of the states of interest in this paper with the
$\Lambda$, $\Sigma$, and $\Sigma^{\ast}$ states in QCD—see for instance Ref.
Corrigan:1979xf  for a discussion within $SU(N_{c})$ gauge theories.

This paper is organized in the following way. In Section II, we describe
lattice field theory basic definitions, such as the simulation algorithm and
the correlation functions that enter our measurements of chimera baryon
masses. Section III describes our data analysis procedure in the extraction of
the chimera-baryon masses. It also details the strategy applied to the
continuum and massless extrapolations of these masses. We then summarize our
findings in Section IV. More technical details are relegated to the
Appendices.

##  II Lattice numerical calculations

Lattice field theory enables to perform first-principle non-perturbative
computations in quantum field theory. Since little is known about chimera
baryon spectra in $Sp(4)$ gauge theories Bennett:2022yfa , we adopt the
quenched approximation, which significantly reduces the demands on computing
resources, while allowing the exploration of parameter space, independent of
the number of fields, $N_{f}$ and $n_{f}$. Based upon experience gained from
quenched calculations of the spectrum of QCD, we envisage that this
approximation gives reasonably accurate results in some of the regions of
parameter space of interest, in which the number of fermions is not too large,
or their mass is not too small. Furthermore, performing this first study in
the quenched approximation facilitates an extensive scan of the space of bare
parameter, to yield benchmarking information for our future computations
involving dynamical hyperquarks.

This section describes the lattice action and provides technical details
necessary to reproduce our calculations. More details, such as the specific
features of our implementation of the heat bath algorithm for $Sp(4)$ gauge
theory and the scale-setting procedure based on the gradient flow, can be
found in Refs. Bennett:2017kga ; Bennett:2019cxd ; Bennett:2021mbw ;
Bennett:2022ftz . We also define the interpolating operators and correlation
functions relevant for this work, in Section II.3. Some additional technical
details can be found in the appendix, in particular pertaining to our use of
smearing.

###  II.1 Lattice action

We discretize the four-dimensional $Sp(4)$ gauge theory on a spatially
isotropic Euclidean lattice. The dynamics of the gauge degrees of freedom is
described by the standard Wilson plaquette action, $S_{g}$, given by

$S_{g}\equiv\beta\sum_{x}\sum_{\mu<\nu}\left(1-\frac{1}{4}\textrm{Re}\,{{\rm
Tr\,}}\,\mathcal{P}_{\mu\nu}\right)\,,$ (5)

where $\beta\equiv 8/g^{2}$ is the bare lattice coupling. The plaquette,
$\mathcal{P}_{\mu\nu}$, is defined as

$\mathcal{P}_{\mu\nu}(x)\equiv
U_{\mu}(x)U_{\nu}(x+\hat{\mu})U^{\dagger}_{\mu}(x+\hat{\nu})U^{\dagger}_{\nu}(x)\,,$
(6)

with the link variable, $U_{\mu}(x)\in Sp(4)$, transforming in the adjoint
representation of the gauge group. The action $S_{g}$ is used in our Monte
Carlo computations to generate gauge-field ensembles.

The hyperquarks, constituents of the chimera baryons, are fermions whose
dynamics is described by the Wilson-Dirac lattice action

$S_{f}\equiv
a^{4}\sum_{i=1}^{N_{f}}\sum_{x}\overline{Q^{i}}(x)D^{(f)}_{m}Q^{i}(x)+a^{4}\sum_{j=1}^{n_{f}}\sum_{x}\overline{\Psi^{j}}(x)D^{(as)}_{m}\Psi^{j}(x),$
(7)

where $a$ is the lattice spacing, while $i$ and $j$ are flavor
indices—hypercolor and spinor indexes are understood. Explicitly, we write the
following, with $R=(f)$ for fermions transforming in the fundamental
representation, and $R=(as)$ in the case of the 2-index antisymmetric
representation:

$\displaystyle\overline{\psi^{R\,j}}(x)D^{R}_{m}\psi^{R\,j}(x)$
$\displaystyle\equiv$
$\displaystyle\overline{\psi^{R\,j}}(x)(4/a+m^{R}_{0})\psi^{R\,j}(x)$
$\displaystyle-\frac{1}{2a}\sum_{\mu}\overline{\psi^{R\,j}}(x)\left\\{(1-\gamma_{\mu})U^{R}_{\mu}(x)\psi^{R\,j}(x+\hat{\mu})+(1+\gamma_{\mu})U^{R\,\dagger}_{\mu}(x-\hat{\mu})\psi^{R\,j}(x-\hat{\mu})\right\\}\,,$

with $\psi^{(f)\,j}=Q^{j}$, $\psi^{(as)\,j}=\Psi^{j}$, and
$U_{\mu}^{(f)}=U_{\mu}$. The construction of the antisymmetric gauge link,
$U^{(as)}_{\mu}$, is detailed in Ref. Bennett:2022yfa . The symbol $m_{0}^{R}$
denotes the flavor diagonal (degenerate) bare mass of hyperquarks,
$\psi^{R}_{j}$, transforming in the corresponding representation, $R$, of the
gauge group.

###  II.2 Numerical strategy

For this work we use the open source HiRep code DelDebbio:2008zf , with the
add-ons we developed in the context of earlier publications in order to
implement $Sp(4)$ Bennett:2017kga  —see also the first lattice study of
symplectic gauge group Holland:2003kg  and the recent implementation of
$Sp(2N)$ in the Grid environment Bennett:2023gbe ; Forzano:2023duk ;
Boyle:2015tjk ; Boyle:2016lbp ; Yamaguchi:2022feu . Gauge field ensembles are
generated using one-plus-four combinations of heat bath plus over-relaxation
update algorithms. Two successive configurations in the Markov chain are
separated by twelve such updates of the whole lattice. More details of the
implementation of this procedure can be found in Ref. Bennett:2017kga . Also,
in every Markov chain, the initial $600$ configurations are treated as
thermalization steps and discarded from the measurements of physical
observables. For each ensemble, we generate $200$ configurations. We monitor
the topological charge and its evolution, to ascertain that there is no
evidence of topological freezing. We denote the dimensionless lattice volume
as $N_{t}\times N_{s}^{3}$, where $N_{t}$ and $N_{s}$ are the temporal and
spatial lattice extents, respectively. Periodic boundary conditions are
imposed on gauge fields, in all directions. For hyperquark fields, periodic
and anti-periodic boundary conditions are implemented in spatial and temporal
directions, respectively.

We generate five ensembles with different values of the lattice bare coupling
$\beta$. We summarize in Tab. 2 the defining properties of each ensemble. We
set the scale of dimensionful physical observables by employing the gradient-
flow method Luscher:2010iy ; Luscher:2011bx ; Luscher:2013vga . The procedure
outlined in Ref. BMW:2012hcm  yields the quantity $w_{0}/a$, where $w_{0}$ has
dimension of an inverse mass. This scale-setting exercise was already carried
out and reported in detail in previous publications—see Table II of Ref.
Bennett:2019cxd , as well as the extensive discussions in Ref. Bennett:2022ftz
—hence we borrow results for $w_{0}/a$ from Ref. Bennett:2019cxd . We notice
that, in respect to Eq. (2.3) of Ref. BMW:2012hcm , we use the different
reference value ${\cal W}_{0}=0.35$, rather than $0.30$. The information
presented in Tab. 2 shows that the spread of our choices of the lattice bare
coupling corresponds to a variation of the lattice spacing roughly by a factor
of two, which allows us to perform a first extrapolation of our results
towards the continuum limit. In this work, when a dimensional quantity is
expressed in units of $w_{0}$, the corresponding dimensionless quantity is
denoted with the caret symbol. For instance, $\hat{a}\equiv a/w_{0}$ and
$\hat{m}\equiv w_{0}m$, where $m$ stands for a generic mass. The lattice
parameters being identical, the relevant autocorrelation times can be found in
Table III in Ref. Bennett:2019cxd .

Table 2:  Gauge ensembles generated for the $Sp(4)$ theory. We report the bare coupling $\beta$, the lattice size, $N_{t}\times N^{3}_{s}$, the average plaquette $\left<P\right>$, and the gradient-flow scale $w_{0}/a$. The gradient-flow scales are taken from Ref. Bennett:2019cxd .  Ensemble | $\beta$ | $N_{t}\times N^{3}_{s}$ | $\left<P\right>$ | $w_{0}/a$
---|---|---|---|---
QB1 | $7.62$ | $48\times 24^{3}$ | 0.60192 | 1.448(3)
QB2 | $7.7$ | $60\times 48^{3}$ | 0.608795 | 1.6070(19)
QB3 | $7.85$ | $60\times 48^{3}$ | 0.620381 | 1.944(3)
QB4 | $8.0$ | $60\times 48^{3}$ | 0.630740 | 2.3149(12)
QB5 | $8.2$ | $60\times 48^{3}$ | 0.643228 | 2.8812(21)

###  II.3 Interpolating operators and correlation functions

Following the notation introduced in Ref. Bennett:2022yfa , we denote the
generic structure of the chimera baryon interpolating operators, built out of
two $(f)$ and one $(as)$ hyperquarks, as

$\mathcal{O}_{\rm{CB},\rho}(x)\equiv\left({Q^{i\,a}}_{\alpha}(x)\frac{}{}\Gamma^{1\,\alpha\beta}{Q^{j\,b}}_{\beta}(x)\right)\Omega_{ad}\Omega_{bc}\Gamma^{2\,\rho\sigma}{\Psi^{k\,cd}}_{\sigma}(x)\,,$
(9)

where $\Gamma^{1,2}$ are gamma matrices and $\Omega$ is the $4\times 4$
symplectic matrix defined in Eq. (3), with $a,\,b,\,c,\,d$ being
$Sp(4)$-hypercolor, $i,\,j,\,k$ flavor, and $\alpha,\,\beta,\,\sigma,\,\rho$
spinor indices. Operators given in Eqs. (1) and (2) are special cases of this
generic structure. The Dirac conjugate operator of $\mathcal{O}_{\rm
CB,\rho}(x)$ is

$\overline{\mathcal{O}_{\rm{CB},\rho}}(x)\equiv\overline{\Psi^{k\,cd}}_{\sigma}(x)\Omega_{cb}\Omega_{da}\Gamma^{2\,\sigma\rho}\left({\overline{Q^{j\,b}}_{\beta}}(x)\Gamma^{1\,\beta\alpha}\overline{Q^{i\,a}}_{\alpha}(x)\right)\,.$
(10)

The zero momentum, two-point correlation functions of interest, restricted to
consider only $i\neq j$, are written as

$\displaystyle C_{\rm{CB},\sigma\rho}(t)$ $\displaystyle\equiv$
$\displaystyle\sum_{\vec{x}}\langle\mathcal{O}_{\rm{CB},\sigma}(x)\overline{\mathcal{O}_{\rm{CB},\rho}}(0)\rangle$
(11) $\displaystyle=$
$\displaystyle-\sum_{\vec{x}}\left(\Gamma^{2}{S_{\Psi}^{k\,cd}}_{c^{\prime}d^{\prime}}(x,0)\overline{\Gamma^{2}}\right)_{\sigma\rho}\Omega_{cb}\Omega^{b^{\prime}c^{\prime}}\Omega_{ad}\Omega^{d^{\prime}a^{\prime}}{\rm
Tr\,}\left[\Gamma^{1}S^{b}_{Q\,\,\,b^{\prime}}(x,0)\overline{\Gamma^{1}}S_{Q\,\,\,\,a^{\prime}}^{a}(x,0)\right],$

where $x\equiv(t,\vec{x})$, while
$\overline{\Gamma}\equiv\gamma^{0}\Gamma^{\dagger}\gamma^{0}$. The trace is
taken over the spinor indices. The hyperquark propagators are

$S^{\,i\,a}_{Q\,\,\,\,b\,\alpha\beta}(x,y)=\langle
Q^{i\,a}_{\,\,\,\,\,\alpha}(x)\overline{Q^{i\,b}}_{\beta}(y)\rangle\,,~{}{\textrm{and}}~{}S^{\,k\,ab}_{\Psi\,\,\,\,\,\,\,\,cd\,\alpha\beta}(x,y)=\langle\Psi^{k\,ab}_{\,\,\,\,\,\,\,\,\,\,\alpha}(x)\overline{\Psi^{k\,cd}}_{\beta}(y)\rangle\,.$
(12)

We are interested in operators with
$(\Gamma^{1},\,\Gamma^{2})=(C\gamma^{5},\,\mathbb{1})$ and
$(C\gamma^{\mu},\,\mathbb{1})$. The former overlaps with the $\Lambda_{\rm
CB}$ state, while the latter sources both $\Sigma_{\rm CB}$ and
$\Sigma^{\ast}_{\rm CB}$ baryons. The chimera baryon interpolating operators
in Eq. (9) generally couple to states with both even and odd parity. In order
to facilitate the investigation of the spectrum of $\Lambda_{\rm CB}$,
$\Sigma_{\rm CB}$, and $\Sigma^{\ast}_{\rm CB}$ chimera baryons, which are all
parity-even, we apply appropriate projection operators, as detailed in Section
III.

Our main objective is to study how the mass of the chimera baryons changes in
response to the variation of the hyperquark masses, in particular because it
would be interesting to explore the limits in which $m_{0}^{(f)}$ and
$m_{0}^{(as)}$ approach zero. The methodology we apply to the extraction of
these hadronic masses is described in Section III. We perform our calculations
with several choices of $am_{0}^{(f)}$ and $am_{0}^{(as)}$, on each available
ensemble, and report our results in Appendix A.

For sufficiently light hyperquarks, we expect the square of the pseudoscalar
meson mass to depend linearly on the hyperquark mass. Information on the meson
spectrum hence allows us to perform a combined extrapolation to continuum and
massless-hyperquark limit. As this is a quenched calculation, the results of
the extrapolation towards the massless-hyperquark limit have to be taken with
a grain of salt Sharpe:1992ft ; Bernard:1992mk . Yet, they provide useful
input for future dynamical calculations—see Figs. 17-18 in Ref.
Bennett:2019jzz  for examples of the difference in mesons mass between
quenched and dynamical fermions in the case of the fundamental representation.
We can also monitor the ratio between the masses of pseudoscalar and vector
mesons, as an indicator of the relative size of explicitly breaking of the
global symmetry in the theory.

The meson interpolating operators for $(f)$ and $(as)$ hyperquarks are

${\mathcal{O}}^{(f)}_{{\mathrm{M}}}(x)=\overline{{Q^{i\,a}}}_{\alpha}(x)\Gamma^{\alpha\beta}_{{\mathrm{M}}}{Q^{j\,b}}_{\beta}(x)\,,~{}{\textrm{and}}~{}~{}{\mathcal{O}}^{(as)}_{{\mathrm{M}}}(x)=\overline{{\Psi^{k\,ab}}}_{\alpha}(x)\Gamma_{{\mathrm{M}}}^{\alpha\beta}{\Psi^{m\,cd}}_{\beta}(x)\,,$
(13)

respectively. We can set $\Gamma_{{\mathrm{M}}}=\gamma^{5}$ for the
pseudoscalar, and $\Gamma_{{\mathrm{M}}}=\gamma^{\mu}$ for the vector mesons.
Imposing the restriction $i\neq j$ and $k\neq m$, no disconnected diagrams
contribute to the two-point correlation function, which read

$C_{\rm M}^{(f)}(t)\equiv\sum_{\vec{x}}\langle\mathcal{O}^{(f)}_{\rm
M}(x)\mathcal{O}_{\rm M}^{{(f)}\dagger}(0)\rangle=-\sum_{\vec{x}}{\rm
Tr\,}\left[\gamma^{5}\Gamma_{\rm
M}S^{\,i\,a}_{Q\,\,\,\,a^{\prime}}(x,0)\bar{\Gamma}_{\rm
M}\gamma^{5}S^{\,i\,b\dagger}_{Q\,\,\,\,b^{\prime}}(x,0)\right]\,,$ (14)

for mesons made with $(f)$ hyperquarks, and

$C_{\rm M}^{(as)}(t)\equiv\sum_{\vec{x}}\langle\mathcal{O}^{(as)}_{\rm
M}(x)\mathcal{O}_{\rm M}^{{(as)}\dagger}(0)\rangle=-\sum_{\vec{x}}{\rm
Tr\,}\left[\gamma^{5}\Gamma_{\rm
M}S^{\,k\,ab}_{\Psi\,\,\,\,\,\,a^{\prime}b^{\prime}}(x,0)\bar{\Gamma}_{\rm
M}\gamma^{5}S^{\,m\,cd\dagger}_{\Psi\,\,\,\,\,\,\,\,c^{\prime}d^{\prime}}(x,0)\right]\,,$
(15)

for $(as)$ hyperquarks. The traces are taken over spinor indices. The
propagators of $(f)$ and $(as)$ hyperquarks are given in Eq. (12).

The masses of the mesons are extracted from the large-$t$ behavior of
correlation functions. For convenience, we label the pseudoscalar meson masses
as $m_{\rm PS}$ and $m_{\rm ps}$ and the masses of the vector meson as $m_{\rm
V}$ and $m_{\rm v}$, with upper case subscripts referring to $(f)$ hyperquarks
and lower case one to $(as)$ hyperquarks. It is well known that numerical
results of lattice computations of quantities involving baryons are noisy, and
in this work we resort to modifying the correlation functions and the
propagators used for chimera baryons and mesons, by applying two smearing
techniques: the Wuppertal smearing Gusken:1989qx  for the hyperquark fields
and the APE smearing APE:1987ehd  for the gauge fields. We describe in
Appendix B our implementation of these smearing procedures.

##  III Data analysis and numerical results

In this section, we discuss the strategy of our analysis and report numerical
results for the spectrum of the low-lying chimera baryons. In Section III.1,
we describe how we extract ground-state masses with definite spin and parity
quantum numbers, by applying appropriate spin and parity projections on the
correlation functions. In Section III.2 we report our measurements of the
masses of the pseudoscalar and vector mesons, for both $(f)$ and $(as)$
hyperquarks. We then apply relations inspired by Wilson chiral perturbation
theory to analyze the spectra for various hyperquark masses, and we
extrapolate to the continuum and massless-hyperquark limit. The process and
our results are presented in Section III.3. We employ the Akaike information
criterion (AIC) Akaike:1998zah  to optimize for the best analysis procedure
over various fitting ansatze and different selections of the data points to be
included in this investigation. In addition, we also manually check the
results, to demonstrate the correctness of the automated analysis.

We anticipate here that throughout this work, in the data analysis of
correlation functions, estimates of the statistical errors are obtained via
the bootstrap method. For each measurement we generate $800$ bootstrap
samples. Technical details on the intermediate steps are relegated to the
appendix. In particular, fit results of the ground-state masses are presented
in Appendix A, while the choices of smearing parameters are reported in
Appendix C.

###  III.1 Spin and Parity projection

Correlation functions involving the (chimera) baryon operators in Eqs. (9) and
(10) can be further decomposed into components with different spin and parity
quantum numbers Fucito:1982ip ; Bennett:2022yfa ; Hsiao:2022kxf . We denote by
${\mathcal{O}}^{\mu}_{{\rm CB},\rho}$ the operator with Dirac matrix structure
$(\Gamma^{1},\Gamma^{2})=(C\gamma_{\mu},\mathbb{1})$, with $\mu$ running from
1 to 3. It overlaps with both spin-$1/2$ and $3/2$ states. The corresponding
two-point function with vanishing momentum, $\vec{p}=\vec{0}$, can be written
as

$C_{{\rm
CB},\sigma\rho}^{\mu\nu}(t)\equiv\sum_{\vec{x}}\langle\mathcal{O}^{\mu}_{\textrm{CB}\,,\sigma}(x)\overline{\mathcal{O}^{\nu}_{\textrm{CB}\,,\rho}}(0)\rangle\,.$
(16)

The lightest baryons dominate the large Euclidean-time behaviors of the
spin-1/2 and 3/2 components of $C_{{\rm CB},\sigma\rho}^{\mu\nu}$, and we
identify them with $\Sigma_{\mathrm{CB}}$ and $\Sigma^{\ast}_{\mathrm{CB}}$
(see Section I), respectively. We define the following two correlation
functions:

$C_{\Sigma_{\mathrm{CB}},\sigma\rho}(t)\equiv\left[P^{1/2}_{\mu\nu}C_{{\rm
CB}}^{\mu\nu}(t)\right]_{\sigma\rho}~{}{\textrm{and}}~{}~{}C_{\Sigma^{\ast}_{\mathrm{CB}},\sigma\rho}(t)\equiv\left[P^{3/2}_{\mu\nu}C_{{\rm
CB}}^{\mu\nu}(t)\right]_{\sigma\rho}\,,$ (17)

where the spin projectors UKQCD:1996ssj  are (for $\mu,\,\nu$ = 1, 2, 3)

$P^{1/2}_{\mu\nu}\equiv\frac{1}{3}\gamma^{\mu}\gamma^{\nu}\,~{}{\textrm{and}}~{}~{}P^{3/2}_{\mu\nu}\equiv\delta^{\mu\nu}-\frac{1}{3}\gamma^{\mu}\gamma^{\nu}\,.$
(18)

We define as ${\mathcal{O}}^{5}_{{\rm CB},\rho}$ the operator obtained from
Eq. (9) by considering $(\Gamma^{1},\Gamma^{2})=(C\gamma^{5},\mathbb{1})$.
This operator only overlaps with spin-$1/2$ states, the ground state of which
is the $\Lambda_{\mathrm{CB}}$ introduced in Section I. Therefore, we define

$C_{\Lambda_{\rm
CB},\sigma\rho}(t)\equiv\sum_{\vec{x}}\langle{\mathcal{O}}^{5}_{{\rm
CB},\sigma}(x)\overline{{\mathcal{O}}^{5}_{{\rm CB},\rho}}(0)\rangle\,.$ (19)

For notational simplicity, in the rest of this article we will not write
explicitly the spinor indices, $\sigma$ and $\rho$, in the correlation
functions in Eqs. (17) and (19), but leave them understood. Furthermore, we
use the symbol $C_{\mathrm{CB}}(t)$ to denote generically
$C_{\Lambda_{\mathrm{CB}}}(t)$, $C_{\Sigma_{\mathrm{CB}}}(t)$, or
$C_{\Sigma^{\ast}_{\mathrm{CB}}}(t)$.

The chimera baryon interpolating operators, ${\mathcal{O}}^{5}_{{\rm
CB},\rho}$ and ${\mathcal{O}}^{\mu}_{{\rm CB},\rho}$, couple to both even- and
odd-parity states. At large Euclidean time, due to the use of anti-periodic
boundary conditions in the temporal direction for hyperquark fields, the two-
point correlation function of a chimera baryon, following the convention in
Ref. Montvay:1994cy , behaves asymptotically as

$C_{\rm CB}(t)\xrightarrow{0\ll t\ll
T}P_{+}\left[c_{+}e^{-m^{+}t}-c_{-}e^{-m^{-}(T-t)}\right]+P_{-}\left[c_{-}e^{-m^{-}t}-c_{+}e^{-m^{+}(T-t)}\right]\,,$
(20)

where the parity projectors are

$P_{\pm}\equiv\frac{1\pm\gamma^{0}}{2}\,,$ (21)

while $m^{\pm}$ are the masses of the even- and odd-parity states, and
$c_{\pm}$ the corresponding baryon-to-vacuum matrix elements. We define even-
and odd-parity correlation functions, $C^{+}_{\rm CB}(t)$ and $C^{-}_{\rm
CB}(t)$, by applying the $P_{\pm}$ projectors:

$C^{\pm}_{\rm CB}(t)\equiv P_{\pm}\,C_{\rm CB}(t)\,.$ (22)

For finite but large extent of the temporal lattice, $T$, we therefore find
that the projected correlation functions at large Euclidean time, $0\ll t\ll
T$, behave as

$C^{\pm}_{\rm CB}(t)\longrightarrow
c_{\pm}e^{-m^{\pm}t}-c_{\mp}e^{-m^{\mp}(T-t)}\,.$ (23)

To improve statistics, in the analysis we employ the averaged correlator,

$\overline{C}_{\rm CB}^{\pm}(t)=\frac{C_{\rm CB}^{\pm}(t)-C_{\rm
CB}^{\mp}(T-t)}{2}\,,$ (24)

which exhibits the same asymptotic behavior as in Eq. (23).

(a)

(b)

(c)

Figure 1: Illustrative examples of chimera baryon correlation functions, (a),
and effective mass plots, (b) and (c), obtained on the ensemble QB1 with
hyperquark masses $am_{0}^{(f)}=-0.77$ and $am_{0}^{(as)}=-1.05$. (1(a)): the
parity-projected correlators, $C^{\pm}_{\Lambda_{\rm CB}}(t)$. (1(b))
effective masses, $am_{\rm eff}$, extracted from correlation function obtained
with, $C^{\pm}_{\Lambda_{\rm CB}}(t)$, and without parity projection,
$C_{\Lambda_{\rm CB}}(t)$. (1(c)): effective masses, $am_{\rm eff}$, extracted
from correlation functions upon which spin and parity projections are applied,
$C^{+}_{\Sigma_{\rm CB}}(t)$ and $C^{+}_{\Sigma^{\ast}_{\rm CB}}(t)$, or
without any projection, $C_{{\rm CB},\mu\nu}(t)$.

For both even- and odd-parity states, we define the effective masses as

$am^{\pm}_{\rm eff,CB}(t)=\ln{\left[\frac{\overline{C}^{\pm}_{\rm
CB}(t)}{\overline{C}^{\pm}_{\rm CB}(t+1)}\right]}\,,$ (25)

and restrict our attention to ranges of Euclidean time $0\ll t<T/2$. From Eqs.
(23) and (24), one expects that $am^{\pm}_{\rm eff,CB}(t)$, when plotted
against time, will asymptotically display a plateau dominated by either the
even-parity or odd-parity ground states, in $\overline{C}_{\rm CB}^{+}(t)$ and
$\overline{C}_{\rm CB}^{-}(t)$, respectively. By studying and comparing the
resulting effective mass plots, we determine the parity of the lowest-lying
chimera baryon state for each choice of spin and global symmetry quantum
numbers of interest, as listed in Tab. 1. As a cross-check of our results, we
consider also the effective mass computed with unprojected correlation
functions, $C_{\rm CB}$. In analogy with Eq. (25), for $0\ll t<T/2$, we define
it as

$am_{\rm eff,CB}(t)=\ln{\left[\frac{C_{\rm CB}(t)}{C_{\rm
CB}(t+1)}\right]}\,.$ (26)

Given the asymptotic behavior expected in Eq. (20), the value of the plateau
in $am_{\rm eff,CB}(t)$ should appear at a value compatible with the lightest
between $am^{+}_{\rm eff,CB}(t)$ and $am^{-}_{\rm eff,CB}(t)$.

In order to graphically illustrate how projectors affect the effective mass
extraction, we present in Fig. 1(a) the parity-projected correlation
functions, $C^{\pm}_{\Lambda_{\rm CB}}(t)$, obtained from the ensemble QB1
(see Tab. 2) with the bare hyperquark masses in the Wilson-Dirac action set to
$am_{0}^{f}=-0.77$ and $am_{0}^{as}=-1.05$. Notice the logarithm scale on the
vertical axis. The lattice used to generate this ensemble has Euclidean time
extent $T/a=48$. By comparing the slopes with the behavior expected in Eq.
(20), one can infer that the parity-even state is lighter than its parity-odd
partner, and hence that the $\Lambda_{\rm CB}$ chimera baryon (a candidate top
partner) has even parity.

Figure 1(b) shows the effective masses, $am^{\pm}_{{\rm eff},\Lambda_{\rm
CB}}$, extracted with and without applying parity projectors. For the
$\Lambda_{\rm CB}$ state, the plot clearly demonstrates that $m^{+}_{{\rm
eff},\Lambda_{\rm CB}}<m^{-}_{{\rm eff},\Lambda_{\rm CB}}$. Furthermore,
examination of the effective mass extracted from the unprojected correlator,
$am_{{\rm eff},\Lambda_{\rm CB}}$, confirms the hierarchy between the masses
of the two parity eigenstates. It is worthy of notice that in Fig. 1(b) we can
clearly discern the emergence of a plateau for $am^{-}_{{\rm eff},\Lambda_{\rm
CB}}$ at smaller $t/a$. This negative-parity ground state happens to be
substantially heavier, but not parametrically so. It would be interesting to
perform a systematic study of the spectra of this and other heavy baryons, but
doing so would go beyond the purposes of the present study, and requires the
use of dedicated numerically strategies to optimize the signal. We postpone
such a study to the future.

Following the same procedure, applied to the correlation functions involving
the operator ${\cal O}^{\mu}_{{\rm CB},\rho}$, we also demonstrate that
$m^{+}_{{\rm eff},\Sigma_{\rm CB}}<m^{-}_{{\rm eff},\Sigma_{\rm CB}}$, as well
as that $m^{+}_{{\rm eff},\Sigma^{\ast}_{\rm CB}}<m^{-}_{{\rm
eff},\Sigma^{\ast}_{\rm CB}}$. Therefore, it is established that $\Lambda_{\rm
CB}$, $\Sigma_{\rm CB}$, and $\Sigma^{\ast}_{\rm CB}$ are all parity even, and
we only discuss their masses (denoted as $m_{\Lambda_{\rm CB}}$,
$m_{\Sigma_{\rm CB}}$ and $m_{\Sigma^{\ast}_{\rm CB}}$) in the rest of this
paper. These baryon masses are extracted by performing single-exponential fits
of the data for $\overline{C}^{+}_{\rm CB}$ to Eq. (23) in the interval $0\ll
t\leq T/2$. The choice of fit range is guided by the range of the plateau of
the effective mass, and can be optimized by tracking the value of
$\chi^{2}/N_{\rm d.o.f.}$.

Besides parity, we perform also spin projections, as defined in Eq. (17), for
the correlator $C_{{\rm CB}}^{\mu\nu}(t)$. By doing so, we can discriminate
between $\Sigma_{\rm CB}$ and $\Sigma^{\ast}_{\rm CB}$ states. Figure 1(c)
displays the effective masses computed from $C^{\mu\nu,+}_{{\rm CB}}(t)$
measured on ensemble QB1 with spin projections and same hyperquark masses as
in Figs. 1(a) and 1(b). This plot shows the expected hierarchy,
$m_{\Sigma_{\rm CB}}<m_{\Sigma^{\ast}_{\rm CB}}$. Furthermore, we also display
the effective mass obtained from $C_{{\rm CB}}^{\mu\nu}(t)$ with neither spin
nor parity projections. As expected, the plateau value is compatible with that
of the $\Sigma_{\rm CB}$ baryon, but contamination with the heavier states
results in some deterioration of the signal quality.

###  III.2 Mass hierarchy and hyperquark-mass dependence of chimera baryons

One interesting feature we observe is the hierarchy between the ground-state
chimera baryons in the three channels of interest. Figure 2 shows $am_{{\rm
eff},\Lambda_{\rm CB}}(t)$, $am_{{\rm eff},\Sigma_{\rm CB}}(t)$, and $am_{{\rm
eff},\Sigma^{\ast}_{\rm CB}}(t)$ for two representative choices of bare
hyperquark masses, $(am_{0}^{(f)},am_{0}^{(as)})=(-0.6,-0.81)$ and
$(am_{0}^{(f)},am_{0}^{(as)})=(-0.69,-0.81)$, as measured in the ensemble QB4
(see Tab. 2). In the former case, we find convincing evidence that
$\Sigma_{\rm CB}$ is the lightest among these states. In the latter case, the
$(f)$-type bare hyperquark mass is reduced to $am_{0}^{(f)}=-0.69$, and as
shown in the right panel of Fig. 2, $\Lambda_{\rm CB}$ and $\Sigma_{\rm CB}$
become almost degenerate, their masses cannot be discriminated with given
present uncertainties. For all choices we make of bare hyperquark masses, and
in all available ensembles in Tab. 2, $\Sigma^{\ast}_{\rm CB}$ is always the
heaviest amongst the three lowest-lying parity-even baryon states, and
$\Lambda_{\rm CB}$ is never lighter than $\Sigma_{\rm CB}$. More detailed
investigations of the hierarchy in the chimera-baryon masses, in particular
its dependence on the hyperquark masses, will be discussed in this and the
next subsections.

(a)

(b)

Figure 2:  Effective masses of the lightest, even-parity chimera baryons
measured in the ensemble QB4, for two representative choices of hyperquark
masses, $am_{0}^{(as)}=-0.81$ and (2(a)) $am_{0}^{(f)}=-0.6$, or (2(b))
$am_{0}^{(f)}=-0.69$.

In Ref. Bennett:2019cxd , the mass spectrum of the lightest mesons composed of
$(f)$ and $(as)$ hyperquark constituents has been reported, based upon
measurements using the same quenched ensembles as in Tab. 2, while varying the
hyperquark bare masses. For this work, starting from the same choices of bare
masses as in Ref. Bennett:2019cxd , we extend the parameter space into the
regimes of lighter as well as heavier hyperquarks. The inclusion of data
points with smaller hyperquark masses makes the massless extrapolation more
reliable. It also enables access to a wide range of the value of the ratio
between $(f)$ and $(as)$ hyperquark masses. Our aim is to better understand
the interplay between these hyperquark masses and the hierarchy amongst
$m_{\Lambda_{\rm CB}}$, $m_{\Sigma_{\rm CB}}$ and $m_{\Sigma^{\ast}_{\rm
CB}}$. To this purpose, we find it convenient to use the square of the mass of
the pseudoscalar meson as a reference scale, as in Ref. Bennett:2019cxd ,
denoting the masses of the pseudoscalar mesons composed of $(f)$ and $(as)$
hyperquarks by $m_{\rm PS}$ and $m_{\rm ps}$, respectively. For light masses
we then expect $(m_{\rm PS,ps})^{2}\sim m^{(f),(as)}$.

At large Euclidean time, the meson two-point correlation functions in Eqs.
(14) and (15) are expected to behave as

$C_{\rm M}^{R}(t)\xrightarrow{t/a\gg 1}A\left[e^{-m_{\rm M}^{R}t}+e^{-m_{\rm
M}^{R}(T-t)}\right]\,,$ (27)

where $m_{\rm M}^{R}$ is the mass of the ground-state meson, M, composed of
hyperquarks transforming in the representation $R$, and $A$ is the relevant
matrix element. Following Eq. (27), the meson effective mass can be computed
through

$am^{R}_{\rm eff,M}(t)\equiv\cosh^{-1}{\left[\frac{C^{R}_{\rm
M}(t+1)+C^{R}_{\rm M}(t-1)}{2C^{R}_{\rm M}(t)}\right]}\,,$ (28)

in the range of Euclidean time $0<t<T-1$. We then determine the fit interval
for extracting the meson mass by performing a correlated fit of $C^{R}_{\rm
M}(t)$ to Eq. (27), by identifying a suitable plateau in $am^{R}_{\rm
eff,M}(t)$.

In Figs. 3(a), 3(b), and 3(c), we display $\hat{m}_{\Lambda_{\rm CB}}$,
$\hat{m}_{\Sigma_{\rm CB}}$ and $\hat{m}_{\Sigma^{\ast}_{\rm CB}}$ as a
function of $\hat{m}^{2}_{\rm PS}$. For clarity of presentation, the value of
$\hat{m}^{2}_{\rm ps}$ is color-coded. Conversely, in Figs. 3(d), 3(e), and
3(f), the horizontal axis is $\hat{m}^{2}_{\rm ps}$, and the color coding
corresponds to the value of $\hat{m}^{2}_{\rm PS}$. The data points shown in
these six plots are obtained on the five available ensembles listed in Tab. 2,
and are distinguished by the shape of the markers. The meson masses take
values in the range $\hat{m}_{\rm PS}\in[0.28,1.03]$ and $\hat{m}_{\rm
ps}\in[0.35,1.84]$. The plots illustrate how chimera-baryon masses decrease as
either $\hat{m}^{2}_{\rm PS}$ or $\hat{m}^{2}_{\rm ps}$ is reduced,
approaching a non-vanishing limit for $\hat{m}^{2}_{\rm PS}\rightarrow 0$ or
$\hat{m}^{2}_{\rm ps}\rightarrow 0$. To further demonstrate the dependence on
both sources of explicit symmetry breaking (hyperquark masses and lattice
spacing), we show all the data points together in the 3-dimensional plots in
Figs. 3(g)-3(i). These baryon masses measured at different values of
hyperquark masses lie on a surface for each value of $\beta$, and slightly
decrease as we increase $\beta$.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 3:  Masses of chimera baryons as functions of $\hat{m}^{2}_{\rm PS}$
and $\hat{m}^{2}_{\rm ps}$. In panels (3(a)) to (3(c)), the values of
$\hat{m}_{\Lambda_{\rm CB}}$, $\hat{m}_{\Sigma_{\rm CB}}$, and
$\hat{m}_{\Sigma^{\ast}_{\rm CB}}$ are plotted against $\hat{m}^{2}_{\rm PS}$,
while color-coding is used to denote $\hat{m}^{2}_{\rm ps}$. In panels (3(d))
to (3(f)), the horizontal axis is $\hat{m}^{2}_{\rm ps}$, while color-coding
denotes $\hat{m}^{2}_{\rm PS}$. The plot markers represent different
ensembles, characterized by the $\beta$ values indicated in the legend. Panels
(3(g)) to (3(i)) are 3-dimensional plots that show the chimera-baryon masses
as functions of $\hat{m}^{2}_{\rm PS}$ and $\hat{m}^{2}_{\rm ps}$. Different
colors refer to different $\beta$ values, as listed in the legend.

To study how the mass hierarchy depends on the hyperquark masses, we conduct a
thorough exploration across a wide range of $\hat{m}^{2}_{\rm PS}$ and
$\hat{m}^{2}_{\rm ps}$. The left panel of Fig. 4 shows that the ratio between
$m_{\Lambda_{\rm CB}}$ and $m_{\Sigma_{\rm CB}}$ decreases at increasing
$\hat{m}_{\rm ps}$, and tends to unity in the large-$\hat{m}^{2}_{\rm ps}$
regime. The right panel of Fig. 4 exhibits a similar trend with respect to the
variation of $m_{\rm PS}^{2}$. Yet, $m_{\Lambda_{\rm CB}}/m_{\Sigma_{\rm CB}}$
is never consistent with $1$ in the region where the mass of the PS meson is
large. When the $(as)$ hyperquark is heavy (denoted by purple markers in the
right panel of Fig. 4), the ratio between $m_{\Lambda_{\rm CB}}$ and
$m_{\Sigma_{\rm CB}}$ shows a mild dependence on $m_{\rm PS}^{2}$. In this
regime, $m_{\Lambda_{\rm CB}}/m_{\Sigma_{\rm CB}}$ depends primarily on
$\hat{m}^{2}_{\rm ps}$. Within our whole range of hyperquark masses,
$\Lambda_{\rm CB}$ is never lighter than $\Sigma_{\rm CB}$.

Figure 4: Left: ratio between the masses of $\Lambda_{\rm CB}$ and
$\Sigma_{\rm CB}$, plotted against $\hat{m}_{\rm ps}^{2}$, with the value of
$\hat{m}_{\rm PS}^{2}$ color-coded. Right: same ratio plotted against
$\hat{m}_{\rm PS}^{2}$ while color-coding $\hat{m}_{\rm ps}^{2}$. Different
markers denote different $\beta$ values as listed in the legend.

We conduct a similar, systematic investigation of the ratio between
$\hat{m}_{\Sigma_{\rm CB}}$ and $\hat{m}_{\Sigma^{*}_{\rm CB}}$, and the
results are presented in Fig. 5. We find that $\Sigma_{\rm CB}$ is always
lighter than $\Sigma_{\rm CB}^{*}$, their mass gap decreasing as
$\hat{m}^{2}_{\rm PS}$ and $\hat{m}^{2}_{\rm ps}$ increase. This feature can
be interpreted in terms of heavy-hyperquark spin symmetry Isgur:1989vq . As
the hyperquark masses increase, the effects of spin, which account for the
mass difference between $\Sigma_{\rm CB}$ and $\Sigma^{\ast}_{\rm CB}$, are
suppressed.

Figure 5: Left: ratio between the masses of $\Sigma_{\rm CB}$ and
$\Sigma^{\ast}_{\rm CB}$, plotted against $\hat{m}_{\rm ps}^{2}$, with
$\hat{m}_{\rm PS}^{2}$ color-coded. Right: the same ratio plotted against
$\hat{m}_{\rm PS}^{2}$, while color-coding $\hat{m}_{\rm ps}^{2}$. Different
markers denote different $\beta$ values as listed in the legend.

###  III.3 Mass extrapolations and cross checks

We now discuss our extrapolation towards the continuum and the massless-
hyperquark limit. Inspired by baryon chiral perturbation theory for QCD
Jenkins:1990jv ; Bernard:1995dp , and for its lattice realization Beane:2003xv
, we introduce the following ansatz and use it to carry out uncorrelated fits
of our measurements of the chimera-baryon masses in terms of polynomial
functions of $\hat{m}_{\mathrm{PS}}$ and $\hat{m}_{\mathrm{ps}}$, as well as
the lattice spacing, $\hat{a}$,

$\displaystyle\hat{m}_{\textrm{CB}}=\hat{m}_{\rm CB}^{\chi}$ $\displaystyle+$
$\displaystyle F_{2}\hat{m}_{\rm PS}^{2}+A_{2}\hat{m}_{\rm
ps}^{2}+L_{1}\hat{a}$ (29) $\displaystyle+$ $\displaystyle F_{3}\hat{m}_{\rm
PS}^{3}+A_{3}\hat{m}_{\rm ps}^{3}+L_{2F}\hat{m}_{\rm
PS}^{2}\hat{a}+L_{2A}\hat{m}_{\rm ps}^{2}\hat{a}$ $\displaystyle+$
$\displaystyle F_{4}\hat{m}_{\rm PS}^{4}+A_{4}\hat{m}_{\rm
ps}^{4}+C_{4}\hat{m}_{\rm PS}^{2}\hat{m}_{\rm ps}^{2}\,,$

where ${\rm CB}=\Lambda_{\rm CB}$, $\Sigma_{\rm CB}$ or $\Sigma_{\rm
CB}^{\ast}$, and all the hatted dimensionful quantities are expressed in units
of the gradient flow scale, $w_{0}$. Here $\hat{m}_{\rm CB}^{\chi}$ represents
the mass of the chimera baryon in the continuum and massless-hyperquark limit.

As anticipated, the pseudoscalar meson mass squared plays the role of the
hyperquark mass in each representation. We call $F_{j}$ and $A_{j}$ the low
energy constants (LECs) associated with the corrections to $\hat{m}_{\rm CB}$
appearing at the $j$-th power in $\hat{m}_{\rm PS}$ and $\hat{m}_{\rm ps}$,
respectively. As we are limited by the number of available lattice spacings
and by the statistics, we retain terms up to the fourth power in the meson
mass. The coefficient $C_{4}$ controls the cross-term proportional to
$\hat{m}^{2}_{\mathrm{PS}}\hat{m}^{2}_{\mathrm{ps}}$. The $L_{1}$, $L_{2F}$,
and $L_{2A}$ LECs are associated with the finite lattice spacing, $\hat{a}$,
for which we only consider the leading-order, linear in $\hat{a}$, as expected
for Wilson-Dirac fermions. Note that the LECs, $\hat{m}^{\chi}_{\mathrm{CB}}$,
$F_{j}$, $A_{j}$, $L_{j}$ and $C_{4}$, take different values for different
chimera baryons.

Chiral perturbation theory predicts the existence of terms logarithmic in the
hyperquark masses, which we do not include in Eq. (29). These additional terms
have discernible effects only for light enough hyperquark masses (typically in
the regime where the vector meson is more than twice heavier than the
pseudoscalar meson), which is beyond the scope of this study. Furthermore, the
quenched approximation results in diverging terms that in the limit where
$\hat{m}_{\rm PS}$ or $\hat{m}_{\rm ps}$ approaches zero Sharpe:1992ft ;
Bernard:1992mk . Therefore, we only investigate polynomial dependence of
$\hat{m}_{\rm CB}$ on pseudoscalar meson masses in our analysis.

Figure 3 shows clear evidence of a dependence of chimera-baryon masses on
hyperquark masses and lattice spacing in our measurements. The result of a
naive first attempt to fit our whole data set—tabulated in Appendix A—to Eq.
(29) is poor, as indicated by a large value of $\chi^{2}/N_{\rm d.o.f.}$, and
hence we do not report it here. The truncated expansion in Eq. (29) is
expected to be valid only for light enough hyperquarks. To test the
possibility that a portion of our data points lie outside the range of
validity of the expansion, we consider the effect of excluding data points
collected at the largest available masses. To this purpose, we proceed
systematically, according to

  1. 1.

We start by placing cuts,
$\hat{m}_{\mathrm{PS,cut}}=\hat{m}_{\mathrm{ps,cut}}=0.52$, and remove data
points with $\hat{m}_{\rm PS}>\hat{m}_{\mathrm{PS,cut}}$ or $\hat{m}_{\rm
ps}>\hat{m}_{\mathrm{ps,cut}}$, on all the five ensembles in Tab. 2. The
value, 0.52, is chosen such that there remain 13 data points in total. We
verify by inspection that all these 13 measurements satisfy the condition
$am_{\rm PS}<1$ and $am_{\rm ps}<1$. A fit to determine the 11 parameters in
Eq. (29) is then performed.

  2. 2.

After increasing $\hat{m}_{\mathrm{PS,cut}}$ and $\hat{m}_{\mathrm{ps,cut}}$,
independently, in steps of $0.05$, the above selection and fitting procedure
is repeated. At each step, measurements for which $am_{\rm PS}>1$ or $am_{\rm
ps}>1$ are also removed. We stop when $\hat{m}_{\mathrm{PS,cut}}=1.07$ and
$\hat{m}_{\mathrm{ps,cut}}=1.87$, at which point all the data points in
Appendix A have been considered.

Table 3: List of the terms in Eq. (29) that are included in the choices of fit ansatz used in our analysis.  Fit Ansatz |  $\hat{m}_{\rm CB}^{\chi}$ |  $\hat{m}_{\rm PS}^{2}$ |  $\hat{m}_{\rm ps}^{2}$ |  $\hat{m}_{\rm PS}^{3}$ |  $\hat{m}_{\rm ps}^{3}$ |  $\hat{m}_{\rm PS}^{4}$ |  $\hat{m}_{\rm ps}^{4}$ |  $\hat{m}_{\rm PS}^{2}\hat{m}_{\rm ps}^{2}$ |  $\hat{a}$ |  $\hat{m}_{\rm PS}^{2}\hat{a}$ |  $\hat{m}_{\rm ps}^{2}\hat{a}$
---|---|---|---|---|---|---|---|---|---|---|---
M2 |  ✓ |  ✓ |  ✓ |  - |  - |  - |  - |  - |  ✓ |  - |  -
M3 |  ✓ |  ✓ |  ✓ |  ✓ |  ✓ |  - |  - |  - |  ✓ |  ✓ |  ✓
MF4 |  ✓ |  ✓ |  ✓ |  ✓ |  ✓ |  ✓ |  - |  - |  ✓ |  ✓ |  ✓
MA4 |  ✓ |  ✓ |  ✓ |  ✓ |  ✓ |  - |  ✓ |  - |  ✓ |  ✓ |  ✓
MC4 |  ✓ |  ✓ |  ✓ |  ✓ |  ✓ |  - |  - |  ✓ |  ✓ |  ✓ |  ✓

The above procedure results in $158$ distinct data sets, and 158 fitting
analyses, each characterized by an unacceptably large $\chi^{2}/N_{\rm
d.o.f.}$. We interpret this result as evidence that the modelling of our data
set encapsulated by Eq. (29) is too general, so that some of the LECs cannot
be determined by the available data. In view of this, in this article, we do
not report results obtained by fitting our data to Eq. (29). Instead, we
explore a different numerical approach that allows for a variation of the set
of free parameters included in the analysis, besides changing the number of
incorporated measurements.

We summarize in Tab. 3 the five fit ansatze included in our analysis. They are
all based upon Eq. (29), but are obtained by restricting the set of terms used
in the fit, while setting the others to zero. The first fit ansatz, dubbed M2,
includes the polynomial terms in the first line of Eq. (29), i.e.,
$\hat{m}^{\chi}_{\mathrm{CB}}$ and corrections quadratic in pseudoscalar-meson
masses or linear in lattice spacing. In M3, we also incorporate corrections up
to the cubic terms in the pseudoscalar-meson masses, as well as the lattice-
spacing corrections, $\hat{m}_{\textrm{PS}}^{2}\hat{a}$ and
$\hat{m}_{\textrm{ps}}^{2}\hat{a}$. We further include the three highest-order
terms in Eq. (29), one by one, in MF4, MA4, and MC4, corresponding to the
addition of only $F_{4}\hat{m}_{\textrm{PS}}^{4}$,
$A_{4}\hat{m}_{\textrm{ps}}^{4}$, or
$C_{4}\hat{m}_{\textrm{PS}}^{2}\hat{m}_{\textrm{ps}}^{2}$, respectively.

By combining the 5 fit ansatz with the 158 data sets generated by imposing
cuts on the data sets, we are left with $158\times 5=790$ different analysis
procedures. Following the ideas in Ref. Jay:2020jkz , we select the best one
by applying the Akaike information criterion (AIC). For each analysis
procedure, one computes the quantity

${\rm AIC}\equiv\chi^{2}+2k+2N_{\rm cut}\,,$ (30)

where $\chi^{2}$ is the standard chi-square, $k$ is the number of fit
parameters, and $N_{\rm cut}$ is the number of data points removed by the
introduction of the cuts $\hat{m}_{\mathrm{PS,cut}}$ and
$\hat{m}_{\mathrm{ps,cut}}$.

Figure 6: Heat-map plots of $\chi^{2}/N_{\rm d.o.f.}$, $W$ and $\hat{m}_{\rm CB}^{\chi}$ (top to bottom) for the analysis of $\hat{m}_{\Lambda_{\rm CB}}$ using different fit ansatze in Tab. 3. The horizontal and vertical axes are $\hat{m}_{\mathrm{PS,cut}}$ and $\hat{m}_{\mathrm{ps,cut}}$, respectively. | $\Lambda_{\rm CB}$
---|---
Ansatz |  $\hat{m}_{\rm PS,cut}$ |  $\hat{m}_{\rm ps,cut}$ |  $\chi^{2}/N_{\rm d.o.f.}$ |  AIC |  $W$
M2 |  0.77 |  0.87 |  0.44 |  97.79 |  $\sim 10^{-18}$
M3 |  1.07 |  1.87 |  0.97 |  73.48 |  $\sim 10^{-8}$
MF4 |  0.77 |  1.87 |  0.77 |  73.3 |  $\sim 10^{-8}$
MA4 |  0.77 |  1.67 |  0.63 |  70.83 |  $\sim 10^{-7}$
MC4 |  1.07 |  1.87 |  0.72 |  57.47 |  0.47
Table 4: The optimal choices of $\hat{m}_{\rm PS,cut}$ and $\hat{m}_{\rm ps,cut}$ for each fit ansatz in the continuum and massless-hyperquark extrapolation of $\hat{m}_{\Lambda_{\rm CB}}$. Also shown are the corresponding value of $\chi^{2}/N_{\rm d.o.f.}$, AIC and $W$. Figure 7:  Heat-map plots of $\chi^{2}/N_{\rm d.o.f.}$, $W$ and $\hat{m}_{\rm CB}^{\chi}$ (top to bottom) for the analysis of $\hat{m}_{\Sigma_{\rm CB}}$ using different fit ansatze in Tab. 3. The horizontal and vertical axes are $\hat{m}_{\mathrm{PS,cut}}$ and $\hat{m}_{\mathrm{ps,cut}}$, respectively. | $\Sigma_{\rm CB}$
---|---
Ansatz |  $\hat{m}_{\rm PS,cut}$ |  $\hat{m}_{\rm ps,cut}$ |  $\chi^{2}/N_{\rm d.o.f.}$ |  AIC |  $W$
M2 |  0.72 |  0.67 |  0.84 |  116.9 |  $\sim 10^{-13}$
M3 |  0.62 |  1.17 |  0.61 |  109.56 |  $\sim 10^{-10}$
MF4 |  0.62 |  1.17 |  0.62 |  110.47 |  $\sim 10^{-11}$
MA4 |  0.57 |  1.87 |  0.73 |  105.55 |  $\sim 10^{-9}$
MC4 |  0.77 |  1.47 |  0.89 |  88.27 |  0.31
Table 5: The optimal choices of $\hat{m}_{\rm PS,cut}$ and $\hat{m}_{\rm ps,cut}$ for each fit ansatz in the continuum and massless-hyperquark extrapolation of $\hat{m}_{\Sigma_{\rm CB}}$. Also shown are the corresponding value of $\chi^{2}/N_{\rm d.o.f.}$, AIC and $W$. Figure 8:  Heat-map plots of $\chi^{2}/N_{\rm d.o.f.}$, $W$ and $\hat{m}_{\rm CB}^{\chi}$ (top to bottom) for the analysis of $\hat{m}_{\Sigma^{\ast}_{\rm CB}}$ using different fit ansatze in Tab. 3. The horizontal and vertical axes are $\hat{m}_{\mathrm{PS,cut}}$ and $\hat{m}_{\mathrm{ps,cut}}$, respectively.  | $\Sigma^{\ast}_{\rm CB}$
---|---
Ansatz |  $\hat{m}_{\rm PS,cut}$ |  $\hat{m}_{\rm ps,cut}$ |  $\chi^{2}/N_{\rm d.o.f.}$ |  AIC |  $W$
M2 |  0.82 |  0.87 |  1.04 |  113.72 |  $\sim 10^{-5}$
M3 |  0.82 |  1.17 |  1.0 |  104.44 |  0.36
MF4 |  0.82 |  1.17 |  1.01 |  105.51 |  0.12
MA4 |  0.82 |  1.17 |  1.02 |  105.94 |  0.08
MC4 |  0.82 |  1.17 |  1.0 |  104.98 |  0.21
Table 6: The optimal choices of $\hat{m}_{\rm PS,cut}$ and $\hat{m}_{\rm
ps,cut}$ for each fit ansatz in the continuum and massless-hyperquark
extrapolation of $\hat{m}_{\Sigma_{\rm CB}}$. Also shown are the corresponding
value of $\chi^{2}/N_{\rm d.o.f.}$, AIC and $W$.

The corresponding probability weight is expected to be

$W=\frac{1}{\mathcal{N}}\exp\left[{-\frac{1}{2}{\rm AIC}}\right]\,,$ (31)

where $\mathcal{N}$ is a normalization factor that ensures the sum of $W$ over
all 790 analysis procedures equals to one. Maximizing the $W$ over ansatze and
data sets is equivalent to minimizing the AIC. We note that a smaller
$\chi^{2}$ value can normally be obtained by considering more fit parameters,
or by excluding data points that are not well described by the ansatz, e.g.,
points in the region of heavy hyperquark masses in our case. These correspond
to the last two terms on the right-hand side of Eq. (30). They introduce a
penalty by increasing the value of AIC, hence reducing $W$. In Ref.
Jay:2020jkz , the aim was to estimate a measured quantity by averaging over
results from all analysis procedures with their probability weights. The
$\chi^{2}$ therein was augmented to account for prior information. In this
work, we focus on the standard $\chi^{2}$, with the aim of selecting the best
analysis procedure.

Figures 6, 7, and 8 show, in heat-map format, the
$\hat{m}_{\mathrm{PS,cut}}$\- and $\hat{m}_{\mathrm{ps,cut}}$-dependence of
the $\chi^{2}/N_{\rm d.o.f.}$, the probability weight, $W$, in Eq. (31), and
the fitted $\hat{m}_{\rm CB}^{\chi}$, for measurements of the masses of
$\Lambda_{\mathrm{CB}}$, $\Sigma_{\mathrm{CB}}$, and
$\Sigma^{\ast}_{\mathrm{CB}}$, respectively. In each row of a given figure, we
display the results for the five distinct fitting strategies, M2, M3, MF4,
MA4, and MC4, listed in Tab. 3. In all the plots, the horizontal and vertical
axes correspond to $\hat{m}_{\mathrm{PS,cut}}$ and
$\hat{m}_{\mathrm{ps,cut}}$, respectively. The center of each pixel in a heat
map represents a set of cuts
$(\hat{m}_{\mathrm{PS,cut}},\hat{m}_{\mathrm{ps,cut}})$. Notice that changing
the values of $\hat{m}_{\mathrm{PS,cut}}$ and $\hat{m}_{\mathrm{ps,cut}}$ does
not correspond to removing or including data points. Therefore, in each heat
map, the 336 pixels constituting the panel represent only 158 distinct data
sets. This redundancy does not affect the normalization factor, ${\cal N}$, in
Eq. (31).

In Tabs. 4, 5 and 6, we display the optimal choices of
$\hat{m}_{\mathrm{PS,cut}}$ and $\hat{m}_{\mathrm{ps,cut}}$, as well as the
corresponding value of $\chi^{2}/N_{\rm d.o.f}$, AIC and $W$, for all fitting
methods in analyzing data of $\hat{m}_{\Lambda_{\rm CB}}$,
$\hat{m}_{\Sigma_{\rm CB}}$, and $\hat{m}_{\Sigma^{\ast}_{\rm CB}}$,
respectively. From these tables, as well as by inspection of Figs. 6, 7, and
8, we conclude that the best analysis procedure for the continuum and
massless-hyperquark extrapolation of $\hat{m}_{\Lambda_{\rm CB}}$ is the use
of MC4 fit ansatz, with $\hat{m}_{\rm PS,cut}=1.07$ and $\hat{m}_{\rm
ps,cut}=1.87$, while that of $\hat{m}_{\Sigma_{\rm CB}}$ is MC4 with
$\hat{m}_{\rm PS,cut}=0.77$ and $\hat{m}_{\rm ps,cut}=1.47$. Regarding
$\hat{m}_{\Sigma^{\ast}_{\rm CB}}$, we find the optimal procedure to be the M3
ansatz with $\hat{m}_{\rm PS,cut}=0.82$ and $\hat{m}_{\rm ps,cut}=1.17$.

Our results of the estimates of the LECs from the best analysis procedures are
presented in Tab. 7. We notice for example that $F_{i}$ and $A_{i}$ are not
compatible for at least $\Lambda_{\rm CB}$ and $\Sigma_{\rm CB}$ chimera
baryons, confirming the necessity of using separate expansions in
$\hat{m}_{\rm PS}$ and $\hat{m}_{\rm ps}$. Moreover, the $L_{1}$ coefficient
for $\hat{m}_{\Sigma^{\ast}_{\rm CB}}$ is significantly larger than that for
$\hat{m}_{\Lambda_{\rm CB}}$ and $\hat{m}_{\Sigma_{\rm CB}}$, indicating that
lattice artifacts are expected to be more sizeable in this baryon mass, which
is the heaviest of the three.

CB |  Ansatz | $\hat{m}_{\rm CB}^{\chi}$ | $F_{2}$ | $A_{2}$ | $F_{3}$ | $A_{3}$ | $L_{1}$ | $L_{2F}$ | $L_{2A}$ | $C_{4}$
---|---|---|---|---|---|---|---|---|---|---
$\Lambda_{\rm CB}$ |  MC4 | 0.999(27) | 0.709(61) | 0.383(10) | -0.150(34) | -0.092(4) | -0.137(43) | 0.071(69) | 0.004(11) | -0.026(5)
$\Sigma_{\rm CB}$ |  MC4 | 0.841(21) | 0.815(77) | 0.558(13) | -0.252(63) | -0.161(7) | -0.141(33) | 0.187(65) | -0.021(16) | -0.078(7)
$\Sigma^{\ast}_{\rm CB}$ |  M3 | 1.259(34) | 0.360(110) | 0.393(29) | -0.071(93) | -0.129(16) | -0.335(54) | 0.342(83) | 0.004(30) | -
Table 7: Low-energy constants in Eq. (29) for each chimera baryon, as
determined by the best analysis procedure selected from those presented in
Tabs. 4, 5, and 6. The missing coefficients are set to zero, as a result of
the AIC-driven analysis.

To demonstrate the robustness of the AIC-driven analysis, we perform cross
checks by fitting the data obtained by fixing lattice spacing and mass of the
pseudoscalar meson in one of the representations. We first consider fixing the
value of $\hat{m}_{\mathrm{ps}}$ and the lattice spacing. In this case, the
fit function, Eq. (29), reduces to

$\hat{m}_{\textrm{CB}}=\tilde{m}_{\rm
CB}^{\chi}+\tilde{F}_{2}\hat{m}_{\textrm{PS}}^{2}+\tilde{F}_{3}\hat{m}_{\textrm{PS}}^{3}\,.$
(32)

We can then choose our data points at particular fixed
$\hat{m}_{\mathrm{ps}}$, $\hat{a}$, and fit them to Eq. (32) to determine
$\tilde{m}^{\chi}_{\mathrm{CB}}$, $\tilde{F}_{2}$ and $\tilde{F}_{3}$.
Comparing with Eq. (29), it is anticipated that these three parameters depend
on the chosen values of $\hat{m}_{\mathrm{ps}}$ and $\hat{a}$. Nevertheless,
we expect that for small enough values of $\hat{m}_{\mathrm{ps}}$ and
$\hat{a}$, we should see that $\tilde{m}^{\chi}_{\mathrm{CB}}$ approaches
$m^{\chi}_{\mathrm{CB}}$ determined from the global fit discussed above (MC4
for $\hat{m}_{\Lambda_{\rm CB}}$ and $\hat{m}_{\Sigma_{\rm CB}}$, and M3 for
$\hat{m}_{\Sigma^{\ast}_{\rm CB}}$). Similar expectations apply to
$\tilde{F}_{2}$ and $\tilde{F}_{3}$. Also, the
$\hat{m}_{\mathrm{ps}}$-dependence in $\tilde{F}_{2}$ should primarily be
accounted by the cross term,
$C_{4}\hat{m}^{2}_{\mathrm{PS}}\hat{m}^{2}_{\mathrm{ps}}$.

Analogously, Eq. (29), for fixed $\hat{m}_{\rm PS}$ and $\hat{a}$, reduces to

$\hat{m}_{\textrm{CB}}=\tilde{m}_{\rm
CB}^{\chi}+\tilde{A}_{2}\hat{m}_{\textrm{ps}}^{2}+\tilde{A}_{3}\hat{m}_{\textrm{ps}}^{3}\,.$
(33)

Here $\tilde{m}^{\chi}_{\mathrm{CB}}$, $\tilde{A}_{2}$ and $\tilde{A}_{3}$
depend on the chosen values of $\hat{m}_{\mathrm{PS}}$ and $\hat{a}$. They are
expected to approach the appropriate LECs from the global fit when
$\hat{m}_{\mathrm{PS}}$ and $\hat{a}$ are small. These cross checks serve as
examinations of our analyses using the fitting strategies listed in Tab. 3.

Figure 9:  Result of the cross checks based upon Eq. (32). The horizontal axis
denotes the value of $\hat{m}_{\rm ps}$. As indicated in the legend below the
plots, different markers stand for measurements performed on different
ensembles, and the bands represent results of $\hat{m}^{\chi}_{\rm CB}$,
$F_{2}$ and $F_{3}$ from the best global-fit analysis procedures (see Tab. 7).
The height and the width of each band correspond to the size of the
statistical error and the value of $\hat{m}_{\rm ps,cut}$, respectively.
Figure 10:  Result of the cross checks based upon Eq. (33). The horizontal
axis denotes the value of $\hat{m}_{\rm ps}$. As indicated in the legend below
the plots, different markers stand for measurements performed on different
ensembles, and the bands represent results of $\hat{m}^{\chi}_{\rm CB}$,
$A_{2}$ and $A_{3}$ from the best global-fit analysis procedures (see Tab. 7).
The height and the width of each band correspond to the size of the
statistical error and the value of $\hat{m}_{\rm PS,cut}$, respectively.

In Fig. 9, we present results of the fitted $\tilde{m}^{\chi}_{\mathrm{CB}}$,
$\tilde{F}_{2}$ and $\tilde{F}_{3}$ in Eq. (32) for three values of lattice
spacing, corresponding to $\beta=7.62$, 7.7 and 8.0 listed in Tab. 2. As
discussed above, these three parameters should depend upon
$\hat{m}_{\mathrm{ps}}$ and the lattice spacing. The plots in Fig. 9 indicate
that lattice artifacts are small. Yet, notice that most results presented in
this figure are from the coarsest lattice ($\beta=7.62$). This is because the
number of data points in the other two ensembles, when fixing
$\hat{m}_{\mathrm{ps}}$, is small and in many cases does not allow us to carry
out this exercise. This is also the reason why we cannot perform the cross
check on the other two ensembles ($\beta=7.85$ and $8.2$) listed in Tab. 2.

The plots in Fig. 9 demonstrate that $\tilde{m}^{\chi}_{\mathrm{CB}}$ and
$\tilde{F}_{2}$ have non-negligible $\hat{m}_{\mathrm{ps}}$-dependence. The
bands in each plot represents the global fit results, namely $m_{\rm
CB}^{\chi}$, $F_{2}$, and $F_{3}$, respectively, obtained from the best
analysis procedures in Tab. 7. The height and width of each band correspond to
the size of the statistical error and the value of $\hat{m}_{\rm ps,cut}$,
respectively. It can be seen that $\tilde{m}^{\chi}_{\mathrm{CB}}$ is
compatible with the value of $m^{\chi}_{\mathrm{CB}}$ obtained from the best
global-fit analysis procedure for $\hat{m}_{\Lambda_{\rm CB}}$ and
$\hat{m}_{\Sigma_{\rm CB}}$ in the small-$\hat{m}_{\rm ps}$ regime. This is
not the case for $\hat{m}_{\Sigma^{\ast}_{\rm CB}}$, for which we find a
larger value of $L_{1}$ (see Tab. 7), indicating the presence of more
significant lattice artifacts. We further observe that results of
$\tilde{F}_{2}$ for $\hat{m}_{\Lambda_{\rm CB}}$ and $\hat{m}_{\Sigma_{\rm
CB}}$ show non-negligible dependence upon the chosen value of $\hat{m}_{\rm
ps}$, indicating the need to include the cross term, $C_{4}\hat{m}^{2}_{\rm
PS}\hat{m}^{2}_{\rm ps}$, in the global-fit analysis.

Analogously, we conduct a similar cross check by fixing the value of
$\hat{m}_{\rm PS}$, as described by Eq. (33). Results of this process are
displayed in Fig. 10, which show similar features to those we discussed in
commenting on Fig. 9. Since this work is the first exploratory study of the
spectrum of the chimera baryons in the $Sp(4)$ gauge theory, we do not attempt
to estimate systematic errors affecting our results. It has to be emphasized
that the current calculation is performed in the quenched approximation, and
we are interested in the qualitative feature of the spectrum at this stage.
More precise, dynamical computations are deferred to future work.

Figure 11:  Dependence on $\hat{m}^{2}_{\rm PS}$ (left) and $\hat{m}^{2}_{\rm
ps}$ (right) of the mass of three chimera baryons, ${\Lambda_{\rm CB}}$,
${\Sigma_{\rm CB}}$, and ${\Sigma^{\ast}_{\rm CB}}$, in the limit where the
lattice spacing vanishes, while $\hat{m}^{2}_{\rm ps}=0$ (left) and
$\hat{m}^{2}_{\rm PS}=0$ (right). These plots are generated using the best-fit
LECs in Tab. 7, with the bands representing the statistical errors. These
bands straddle in the horizontal direction between zero and the optimal
choices of $\hat{m}^{2}_{\rm PS,cut}$ (left) and $\hat{m}^{2}_{\rm ps,cut}$
(right). Figure 12: Quenched spectrum of the $Sp(4)$ gauge theory in the
continuum and massless-hyperquark limit. The glueball states are labelled
using the $J^{P}$ notation, while PS (ps), V (v), T(t), AV (av), AT (at) and S
(s) denote the pseudoscalar, vector, tensor, axial-vector, axial-tensor and
scalar mesons composed of fundamental (antisymmetric) hyperquarks. The results
of mesons and glueballs are taken from our previous works in Ref.
Bennett:2019cxd ; Bennett:2020qtj . The results for the chimera baryons are
original to this work.

Using the results summarized in Tab. 7, for the LECs denoted as
$\hat{m}^{\chi}_{\rm CB}$, $F_{2,3}$, and $A_{2,3}$, we present the dependence
on $\hat{m}^{2}_{\rm PS}$ and $\hat{m}^{2}_{\rm ps}$ of the chimera-baryon
masses in the continuum limit, in Fig. 11. That is, the plots in this figure
are generated using Eq. (29) with $\hat{a}=0$. The left (right) panel of this
figure shows the evolution of $\hat{m}_{\Lambda_{\rm CB}}$,
$\hat{m}_{\Sigma_{\rm CB}}$, and $\hat{m}_{\Sigma_{\rm CB}}$ as a function of
$\hat{m}_{\rm PS}$ ($\hat{m}_{\rm ps}$) in the limit where $\hat{m}_{\rm
ps}=0$ ($\hat{m}_{\rm PS}=0$). The color bands represent the statistical
errors, and they straddle in the horizontal direction from 0 to the values
$\hat{m}_{\rm PS}=\hat{m}_{\rm PS,cut}$ (left) and $\hat{m}_{\rm
ps}=\hat{m}_{\rm ps,cut}$ (right). The mass hierarchy,

$\hat{m}_{\Sigma_{\rm CB}}\lesssim\hat{m}_{\Lambda_{\rm
CB}}<\hat{m}_{\Sigma^{\ast}_{\rm CB}}\,,$ (34)

emerges in the whole range of hyperquark masses investigated in this work. The
masses $\hat{m}_{\Lambda_{\rm CB}}$ and $\hat{m}_{\Sigma_{\rm CB}}$ become
compatible with one another only in the regime of heavier $(as)$ hyperquarks.
The hierarchy in Eq. (34) can have non-trivial implications in constructing
viable models for top partial compositeness Banerjee:2022izw .

It is interesting to compare the masses of the chimera baryons with those of
other states in the theory, as we do in Fig. 12. Meson and glueball masses are
taken from our previous measurements in the quenched approximation
Bennett:2019cxd ; Bennett:2020qtj  (see also Refs. Bennett:2020hqd ;
Bennett:2022gdz  for related studies). In this figure, mesons denoted by
capital letters are those composed of $(f)$ hyperquarks, while those expressed
by lowercase letters contain $(as)$ hyperquarks only. All the masses presented
in the plot have been extrapolated to the continuum and massless-hyperquark
limit, and are shown in both gradient-flow units (vertical axis on the left-
hand side), as well as in units of the fundamental pseudoscalar meson decay
constant Bennett:2019cxd  (vertical axis on the right-hand side). The height
of the bands represents statistical errors. As shown in the figure, we find
that the masses of the top-partner candidates, $\Lambda_{\rm CB}$ and
$\Sigma_{\rm CB}$, are comparable to those of the $(as)$ vector mesons.

The spectrum of CHMs with top partial compositeness has also been studied
using other methods, such as Schwinger-Dyson equations, Nambu-Jona-Lasinio
models, or in the framework of holography Elander:2021bmt ; Elander:2020nyd ;
Erdmenger:2023hkl ; Erdmenger:2020flu ; Erdmenger:2020lvq ; Abt:2019tas . To
facilitate comparison with these as well as other future studies, and in view
of possible phenomenological applications, we express our final results for
the massless-hyperquark and continuum extrapolations for the chimera baryon
states in units of the mass, $m_{\rm v}$, of the lightest vector meson with
$(as)$-type constituents. We find

$m_{\Lambda_{\rm CB}}/m_{\rm v}=1.234(32)\,,~{}m_{\Sigma_{\rm CB}}/m_{\rm
v}=1.016(25)\,,~{}\rm{and}~{}m_{\Sigma^{*}_{\rm CB}}/m_{\rm v}=1.576(47)\,,$
(35)

where the quoted error is statistical errors without including systematic
effects, for example due to the quenched approximation.

##  IV Summary and Outlook

The strongly interacting $Sp(4)$ gauge theory coupled to $N_{f}=2$
fundamental, $(f)$, and $n_{f}=3$ two-index antisymmetric, $(as)$, Dirac
fermions (hyperquarks) is the minimal model amenable to lattice investigations
that can provide a UV completion of CHMs with top partial compositeness
Barnard:2013zea . Chimera baryons are composite states formed by two $(f)$ and
one $(as)$ hyperquarks, and are sourced by the operators ${\mathcal{O}}^{5}$
in Eq. (1) and ${\mathcal{O}}^{\mu}$ in Eq. (2). The lightest state sourced by
${\mathcal{O}}^{5}$ is the spin-1/2 chimera baryon, $\Lambda_{\mathrm{CB}}$,
while ${\mathcal{O}}^{\mu}$ can source spin-1/2 and -3/2 baryons, and we
denote by $\Sigma_{\mathrm{CB}}$ and $\Sigma_{\mathrm{CB}}^{\ast}$,
respectively, the two lightest states with definite spin. Either
$\Lambda_{\mathrm{CB}}$ or $\Sigma_{\mathrm{CB}}$ are candidate top partners
Banerjee:2022izw .

Because this is the first systematic lattice calculation of the chimera baryon
spectrum in the $Sp(4)$ gauge theory, we perform it in the quenched
approximation, in which the hyperquark determinant in the path integral is set
to a constant—see Ref. Bennett:2022yfa  for pioneering work on the theory with
$N_{f}=2$ $(f)$-type and $n_{f}=3$ $(as)$-type dynamical fermions. Working in
the quenched approximation not only makes the numerical computation
significantly less demanding, but it also allows us to scan a large region of
parameter space and gather useful information for future dynamical
calculations. The interpolating operators, ${\mathcal{O}}^{5}$ and
${\mathcal{O}}^{\mu}$, source states with both even and odd parity. As
discussed in Section III.1, having established that states with even parity
are lighter, we implement projections to the parity-even states in our
analysis. Furthermore, spin projectors are introduced to distinguish between
$\Sigma_{\mathrm{CB}}$ and $\Sigma_{\mathrm{CB}}^{\ast}$ states, that are
sourced by the same operator, ${\mathcal{O}}^{\mu}$.

The main focus of this study is the hyperquark-mass dependence of the chimera
baryon masses. As we use the Wilson-Dirac formulation for hyperquark fields,
we find it convenient to express this dependence in terms of the mass of the
pseudoscalar mesons, which we denote as $\hat{m}_{\rm PS}$ and $\hat{m}_{\rm
ps}$, respectively, for mesons built of $(f)$-type and $(as)$-type
hyperquarks. As is expected, the three chimera baryon masses approach one
another when increasing $\hat{m}_{\rm PS}$ and $\hat{m}_{\rm ps}$. Working
under the assumption that the hyperquark masses are sufficiently light to make
it viable, we use an effective description inspired by baryon chiral EFT
Jenkins:1990jv ; Jenkins:1991ne . We include only polynomial terms in the
continuum and massless-hyperquark extrapolations. In the range of hyperquark
masses probed in this work, we find the mass hierarchy
$m^{\ast}_{\Sigma_{\mathrm{CB}}}>m_{\Lambda_{\mathrm{CB}}}\gtrsim
m_{\Sigma_{\mathrm{CB}}}$. Our measurements show that the ratio
$m_{\Lambda_{\mathrm{CB}}}/m_{\Sigma_{\mathrm{CB}}}$ decreases when $m_{\rm
ps}$ increases, and that, for the heaviest available values of $m_{\rm ps}$,
this ratio is compatible with unity. These findings suggest that the hierarchy
may not hold true in other regimes of hyperquark masses, which warrants
further, more extensive future investigations.

The use of EFT-inspired relations also enables the extrapolation to the
continuum limit, by including effects of lattice-artifact that break the
global symmetries of the system explicitly Beane:2003xv . As explained in
detail in Section III.3, we implement various ansatze for this simultaneous
continuum and massless-hyperquark extrapolation. Leveraging the AIC
Akaike:1998zah ; Jay:2020jkz  to assess the fit quality and performing cross
checks by fixing the pseudoscalar-meson masses in each of the representations,
we find the optimal choice of the fit procedure for the chimera baryons in our
analysis. The evolution of the chimera baryon masses as a function of the
pseudoscalar masses in the continuum limit are displayed in Fig. 11.
Furthermore, in Fig. 12 we display the complete ground state spectrum of the
theory in the limit of vanishing hyperquark masses, displaying together with
chimera baryon results also meson and glueball masses taken from our earlier
studies Bennett:2019cxd ; Bennett:2020qtj .

This investigation of chimera baryon mass spectra sets the stage for lattice
simulations with dynamical matter fields. Understanding the intricate mass
relations between chimera baryons and hyperquarks is pivotal for navigating
the multi-faced lattice parameter space and constructing physically meaningful
models. This commitment to precise and accurate physics is central to our
quest for a deeper understanding of composite Higgs and top partial
compositeness models and their potential implications for particle physics. We
here established a robust analysis framework, that will be crucial in future
high precision studies, particularly when confronting computationally
demanding calculations that require control over the numerous lattice inputs.

###### Acknowledgements.

The work of EB has been supported by the UKRI Science and Technology
Facilities Council (STFC) Research Software Engineering Fellowship
EP/V052489/1, and by the ExaTEPP project EP/X017168/1. The work of DKH was
supported by Basic Science Research Program through the National Research
Foundation of Korea (NRF) funded by the Ministry of Education
(NRF-2017R1D1A1B06033701). The work of JWL was supported in part by the
National Research Foundation of Korea (NRF) grant funded by the Korea
government(MSIT) (NRF-2018R1C1B3001379) and by IBS under the project code,
IBS-R018-D1. The work of DKH and JWL was further supported by the National
Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT)
(2021R1A4A5031460). The work of HH and CJDL is supported by the Taiwanese MoST
grant 109-2112-M-009-006-MY3 and NSTC grant 112-2112-M-A49-021-MY3. The work
of BL and MP has been supported in part by the STFC Consolidated Grants No.
ST/P00055X/1, ST/T000813/1, and ST/X000648/1. BL and MP received funding from
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation program under Grant Agreement No. 813942. The work of
BL is further supported in part by the Royal Society Wolfson Research Merit
Award WM170010 and by the Leverhulme Trust Research Fellowship No.
RF-2020-4619. The work of DV is supported by STFC under Consolidated Grant No.
ST/X000680/1. Numerical simulations have been performed on the Swansea
University SUNBIRD cluster (part of the Supercomputing Wales project) and
AccelerateAI A100 GPU system, on the local HPC clusters in Pusan National
University (PNU) and in National Yang Ming Chiao Tung University (NYCU), and
on the DiRAC Data Intensive service at Leicester. The Swansea University
SUNBIRD system and AccelerateAI are part funded by the European Regional
Development Fund (ERDF) via Welsh Government. The DiRAC Data Intensive service
at Leicester is operated by the University of Leicester IT Services, which
forms part of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The DiRAC Data
Intensive service equipment at Leicester was funded by BEIS capital funding
via STFC capital grants ST/K000373/1 and ST/R002363/1 and STFC DiRAC
Operations grant ST/R001014/1. DiRAC is part of the National e-Infrastructure.
Open Access Statement—For the purpose of open access, the authors have applied
a Creative Commons Attribution (CC BY) license to any Author Accepted
Manuscript version arising. Research Data Access Statement—The data generated
for this manuscript can be downloaded in machine-readable format from Ref.
DATA .

$am_{0}^{(f)}$ | $am_{0}^{(as)}$ | $am_{\rm PS}$ | $m_{\rm PS}/m_{\rm V}$ | $am_{\rm ps}$ | $m_{\rm ps}/m_{\rm v}$ | $am_{\Lambda_{\rm CB}}$ | $am_{\Sigma_{\rm CB}}$ | $am_{\Sigma^{\star}_{\rm CB}}$
---|---|---|---|---|---|---|---|---
-0.7 | -0.9 | 0.55051(64) | 0.8814(14) | 0.96083(76) | 0.93732(66) | 1.2251(28) | 1.2047(26) | 1.2539(33)
-0.73 | -0.9 | 0.48083(65) | 0.8420(18) | 1.1705(31) | 1.1498(29) | 1.2023(38)
-0.75 | -0.9 | 0.43051(65) | 0.8047(24) | 1.1343(36) | 1.1129(32) | 1.1680(45)
-0.77 | -0.9 | 0.37585(69) | 0.742(11) | 1.0984(46) | 1.0757(41) | 1.1331(51)
-0.79 | -0.9 | 0.31357(79) | 0.6801(35) | 1.0633(49) | 1.0427(47) | 1.0918(76)
-0.7 | -0.95 | 0.55051(64) | 0.8814(14) | 0.86018(70) | 0.91745(72) | 1.1759(27) | 1.1510(26) | 1.2069(32)
-0.73 | -0.95 | 0.48083(65) | 0.8420(18) | 1.1208(29) | 1.0947(27) | 1.1541(39)
-0.75 | -0.95 | 0.43051(65) | 0.8047(24) | 1.0841(33) | 1.0570(29) | 1.1197(45)
-0.77 | -0.95 | 0.37585(69) | 0.742(11) | 1.0470(38) | 1.0195(37) | 1.0845(52)
-0.79 | -0.95 | 0.31357(79) | 0.6801(35) | 1.0093(51) | 0.9838(46) | 1.0483(66)
-0.7 | -1.0 | 0.55051(64) | 0.8814(14) | 0.74972(70) | 0.88675(95) | 1.1264(30) | 1.0940(26) | 1.1599(34)
-0.73 | -1.0 | 0.48083(65) | 0.8420(18) | 1.0689(28) | 1.0363(26) | 1.1049(38)
-0.75 | -1.0 | 0.43051(65) | 0.8047(24) | 1.0314(31) | 0.9977(29) | 1.0704(41)
-0.77 | -1.0 | 0.37585(69) | 0.742(11) | 0.9935(36) | 0.9591(36) | 1.0352(47)
-0.79 | -1.0 | 0.31357(79) | 0.6801(35) | 0.9551(47) | 0.9220(44) | 0.9996(61)
-0.7 | -1.05 | 0.55051(64) | 0.8814(14) | 0.62516(68) | 0.8347(19) | 1.0689(28) | 1.0311(24) | 1.1048(34)
-0.73 | -1.05 | 0.48083(65) | 0.8420(18) | 1.0109(31) | 0.9718(25) | 1.0518(38)
-0.75 | -1.05 | 0.43051(65) | 0.8047(24) | 0.9720(35) | 0.9318(30) | 1.0168(42)
-0.77 | -1.05 | 0.37585(69) | 0.742(11) | 0.9325(45) | 0.8914(34) | 0.9817(47)
-0.79 | -1.05 | 0.31357(79) | 0.6801(35) | 0.8935(60) | 0.8517(43) | 0.9472(61)
-0.7 | -1.1 | 0.55051(64) | 0.8814(14) | 0.47769(75) | 0.7428(27) | 1.0070(30) | 0.9624(24) | 1.0521(31)
-0.73 | -1.1 | 0.48083(65) | 0.8420(18) | 0.9495(32) | 0.9011(25) | 1.0002(34)
-0.75 | -1.1 | 0.43051(65) | 0.8047(24) | 0.9105(33) | 0.8593(27) | 0.9655(34)
-0.77 | -1.1 | 0.37585(69) | 0.742(11) | 0.8708(37) | 0.8165(29) | 0.9309(37)
-0.79 | -1.1 | 0.31357(79) | 0.6801(35) | 0.8295(48) | 0.7724(35) | 0.8966(43)
-0.7 | -1.15 | 0.55051(64) | 0.8814(14) | 0.27757(81) | 0.5289(31) | 0.9399(34) | 0.8842(29) | 0.9908(44)
-0.73 | -1.15 | 0.48083(65) | 0.8420(18) | 0.8804(37) | 0.8190(30) | 0.9314(71)
-0.75 | -1.15 | 0.43051(65) | 0.8047(24) | 0.8400(39) | 0.7737(31) | 0.8953(80)
-0.77 | -1.15 | 0.37585(69) | 0.742(11) | 0.7989(46) | 0.7266(34) | 0.8689(46)
-0.79 | -1.15 | 0.31357(79) | 0.6801(35) | 0.7564(56) | 0.6765(40) | 0.8275(62)
-0.77 | -1.12 | 0.37585(69) | 0.742(11) | 0.40811(66) | 0.6845(26) | 0.8420(37) | 0.7821(30) | 0.9004(50)
-0.78 | -1.12 | 0.34593(72) | 0.7200(43) | 0.8214(41) | 0.7594(33) | 0.8829(53)
-0.79 | -1.12 | 0.31357(79) | 0.6801(35) | 0.8003(47) | 0.7362(35) | 0.8655(60)
-0.77 | -1.14 | 0.37585(69) | 0.742(11) | 0.32632(76) | 0.5944(38) | 0.8133(41) | 0.7461(31) | 0.8763(50)
-0.78 | -1.14 | 0.34593(72) | 0.7200(43) | 0.7926(46) | 0.7223(33) | 0.8582(52)
-0.79 | -1.14 | 0.31357(79) | 0.6801(35) | 0.7713(52) | 0.6972(38) | 0.8401(57)
-0.7 | -1.15 | 0.55051(64) | 0.8814(14) | 0.27757(81) | 0.5289(31) | 0.9399(34) | 0.8842(29) | 0.9908(44)
-0.73 | -1.15 | 0.48083(65) | 0.8420(18) | 0.8804(37) | 0.8190(30) | 0.9314(71)
-0.75 | -1.15 | 0.43051(65) | 0.8047(24) | 0.8400(39) | 0.7737(31) | 0.8953(80)
-0.77 | -1.15 | 0.37585(69) | 0.742(11) | 0.7989(46) | 0.7266(34) | 0.8689(46)
-0.79 | -1.15 | 0.31357(79) | 0.6801(35) | 0.7564(56) | 0.6765(40) | 0.8275(62)
Table 8:  Numerical values of the bare masses, $am_{0}^{(f)}$ and $am_{0}^{(as)}$, used in the measurements on the ensemble QB1, that has gradient flow scale $w_{0}/a=1.448(3)$. For each set of bare masses, we present the pseudoscalar meson mass, $am_{\rm PS}$ and $am_{\rm ps}$, the mass ratio between pseudoscalar and vector mesons in both representations, $m_{\rm PS}/m_{\rm V}$ and $m_{\rm ps}/m_{\rm v}$, and the chimera-baryon masses $am_{\Lambda_{\rm CB}}$, $am_{\Sigma_{\rm CB}}$ and $am_{\Sigma^{\ast}_{\rm CB}}$.  $am_{0}^{(f)}$ | $am_{0}^{(as)}$ | $am_{\rm PS}$ | $m_{\rm PS}/m_{\rm V}$ | $am_{\rm ps}$ | $m_{\rm ps}/m_{\rm v}$ | $am_{\Lambda_{\rm CB}}$ | $am_{\Sigma_{\rm CB}}$ | $am_{\Sigma^{\star}_{\rm CB}}$
---|---|---|---|---|---|---|---|---
-0.72 | -0.91 | 0.41055(54) | 0.8151(16) | 0.85430(49) | 0.92741(77) | 1.0350(29) | 1.0154(25) | 1.0656(42)
-0.74 | -0.91 | 0.35509(61) | 0.7649(27) | 0.9964(34) | 0.9761(28) | 1.0299(50)
-0.77 | -0.91 | 0.25517(63) | 0.6240(41) | 0.9400(58) | 0.9161(43) | 0.9726(76)
-0.72 | -0.92 | 0.41055(54) | 0.8151(16) | 0.83312(49) | 0.92381(87) | 1.0243(29) | 1.0034(25) | 1.0555(41)
-0.74 | -0.92 | 0.35509(61) | 0.7649(27) | 0.9855(34) | 0.9644(28) | 1.0197(49)
-0.77 | -0.92 | 0.25517(63) | 0.6240(41) | 0.9290(57) | 0.9042(43) | 0.9625(74)
-0.7 | -0.89 | 0.46136(51) | 0.8541(12) | 0.89541(64) | 0.93561(91) | 1.0912(38) | 1.0745(27) | 1.1190(37)
-0.72 | -0.89 | 0.41055(54) | 0.8151(16) | 1.0522(46) | 1.0363(29) | 1.0833(42)
-0.74 | -0.89 | 0.35509(61) | 0.7649(27) | 1.0166(42) | 0.9981(37) | 1.0468(50)
-0.77 | -0.89 | 0.25517(63) | 0.6240(41) | 0.9622(74) | 0.9407(61) | 0.9867(82)
-0.7 | -0.93 | 0.46136(51) | 0.8541(12) | 0.81140(63) | 0.9182(11) | 1.0494(36) | 1.0302(26) | 1.0788(37)
-0.72 | -0.93 | 0.41055(54) | 0.8151(16) | 1.0135(32) | 0.9916(24) | 1.0429(41)
-0.74 | -0.93 | 0.35509(61) | 0.7649(27) | 0.9748(36) | 0.9526(28) | 1.0063(49)
-0.76 | -0.93 | 0.29175(62) | 0.6873(44) | 0.9362(45) | 0.9129(36) | 0.9677(63)
-0.77 | -0.93 | 0.25517(63) | 0.6240(41) | 0.9185(56) | 0.8923(41) | 0.9462(78)
-0.76 | -0.97 | 0.29175(62) | 0.6873(44) | 0.72046(63) | 0.8930(14) | 0.8906(43) | 0.8626(32) | 0.9254(62)
-0.76 | -1.01 | 0.29175(62) | 0.6873(44) | 0.62049(64) | 0.8545(17) | 0.8397(41) | 0.8077(30) | 0.8810(61)
-0.72 | -1.09 | 0.41055(54) | 0.8151(16) | 0.36948(65) | 0.6720(49) | 0.8171(33) | 0.7688(22) | 0.8615(44)
-0.74 | -1.09 | 0.35509(61) | 0.7649(27) | 0.7750(35) | 0.7237(24) | 0.8250(53)
-0.76 | -1.09 | 0.29175(62) | 0.6873(44) | 0.7322(45) | 0.6768(29) | 0.7898(69)
-0.72 | -1.1 | 0.41055(54) | 0.8151(16) | 0.32806(68) | 0.6275(56) | 0.8026(34) | 0.7518(28) | 0.8495(48)
-0.74 | -1.1 | 0.35509(61) | 0.7649(27) | 0.7603(38) | 0.7061(30) | 0.8130(57)
-0.76 | -1.1 | 0.29175(62) | 0.6873(44) | 0.7175(47) | 0.6573(27) | 0.7780(78)
-0.77 | -1.1 | 0.25517(63) | 0.6240(41) | 0.6964(38) | 0.6323(30) | 0.7628(53)
-0.72 | -1.11 | 0.41055(54) | 0.8151(16) | 0.28138(69) | 0.5674(75) | 0.7883(38) | 0.7351(35) | 0.8352(55)
-0.74 | -1.11 | 0.35509(61) | 0.7649(27) | 0.7458(42) | 0.6858(26) | 0.8040(52)
-0.76 | -1.11 | 0.29175(62) | 0.6873(44) | 0.7027(50) | 0.6369(28) | 0.7707(63)
-0.77 | -1.11 | 0.25517(63) | 0.6240(41) | 0.6808(40) | 0.6111(29) | 0.7506(57)
-0.72 | -1.12 | 0.41055(54) | 0.8151(16) | 0.22629(62) | 0.4959(61) | 0.7737(43) | 0.7140(26) | 0.8269(50)
-0.74 | -1.12 | 0.35509(61) | 0.7649(27) | 0.7309(49) | 0.6655(27) | 0.7904(56)
-0.76 | -1.12 | 0.29175(62) | 0.6873(44) | 0.6875(57) | 0.6152(30) | 0.7549(64)
-0.77 | -1.12 | 0.25517(63) | 0.6240(41) | 0.6645(43) | 0.5885(30) | 0.7382(68)
Table 9:  Numerical values of the bare masses, $am_{0}^{(f)}$ and $am_{0}^{(as)}$, used in the measurements on the ensemble QB2, that has gradient flow scale $w_{0}/a=1.6070(19)$. For each set of bare masses, we present the pseudoscalar meson mass, $am_{\rm PS}$ and $am_{\rm ps}$, the mass ratio between pseudoscalar and vector mesons in both representations, $m_{\rm PS}/m_{\rm V}$ and $m_{\rm ps}/m_{\rm v}$, and the chimera-baryon masses $am_{\Lambda_{\rm CB}}$, $am_{\Sigma_{\rm CB}}$ and $am_{\Sigma^{\ast}_{\rm CB}}$.  $am_{0}^{(f)}$ | $am_{0}^{(as)}$ | $am_{\rm PS}$ | $m_{\rm PS}/m_{\rm V}$ | $am_{\rm ps}$ | $m_{\rm ps}/m_{\rm v}$ | $am_{\Lambda_{\rm CB}}$ | $am_{\Sigma_{\rm CB}}$ | $am_{\Sigma^{\star}_{\rm CB}}$
---|---|---|---|---|---|---|---|---
-0.66 | -0.85 | 0.41447(68) | 0.8644(16) | 0.83372(63) | 0.94174(74) | 0.9889(30) | 0.9742(22) | 1.0151(21)
-0.68 | -0.85 | 0.36182(67) | 0.8229(21) | 0.9465(30) | 0.9335(19) | 0.9789(24)
-0.7 | -0.85 | 0.30299(59) | 0.7545(24) | 0.9071(41) | 0.8938(22) | 0.9449(35)
-0.72 | -0.85 | 0.23505(56) | 0.6485(35) | 0.8641(58) | 0.8538(31) | 0.9103(55)
-0.66 | -0.87 | 0.41447(68) | 0.8644(16) | 0.79115(62) | 0.93379(83) | 0.9682(28) | 0.9509(16) | 0.9959(24)
-0.68 | -0.87 | 0.36182(67) | 0.8229(21) | 0.9275(33) | 0.9107(18) | 0.9599(27)
-0.7 | -0.87 | 0.30299(59) | 0.7545(24) | 0.8854(40) | 0.8701(21) | 0.9243(34)
-0.72 | -0.87 | 0.23505(56) | 0.6485(35) | 0.8421(57) | 0.8295(31) | 0.8897(54)
-0.66 | -0.9 | 0.41447(68) | 0.8644(16) | 0.72421(62) | 0.9187(10) | 0.9335(25) | 0.9146(16) | 0.9626(20)
-0.68 | -0.9 | 0.36182(67) | 0.8229(21) | 0.8923(29) | 0.8738(18) | 0.9259(23)
-0.7 | -0.9 | 0.30299(59) | 0.7545(24) | 0.8497(37) | 0.8326(21) | 0.8895(27)
-0.72 | -0.9 | 0.23505(56) | 0.6485(35) | 0.8079(55) | 0.7917(29) | 0.8528(39)
-0.66 | -0.93 | 0.41447(68) | 0.8644(16) | 0.65283(65) | 0.8979(12) | 0.9008(27) | 0.8772(16) | 0.9312(25)
-0.68 | -0.93 | 0.36182(67) | 0.8229(21) | 0.8594(31) | 0.8356(18) | 0.8949(27)
-0.7 | -0.93 | 0.30299(59) | 0.7545(24) | 0.8165(38) | 0.7934(20) | 0.8590(34)
-0.72 | -0.93 | 0.23505(56) | 0.6485(35) | 0.7719(52) | 0.7510(30) | 0.8244(50)
-0.72 | -0.97 | 0.23505(56) | 0.6485(35) | 0.54933(59) | 0.8575(14) | 0.7203(54) | 0.6921(27) | 0.7770(52)
-0.73 | -0.97 | 0.19352(66) | 0.5635(45) | 0.6953(68) | 0.6705(37) | 0.7610(72)
-0.72 | -0.99 | 0.23505(56) | 0.6485(35) | 0.49152(59) | 0.8256(17) | 0.6926(53) | 0.6599(26) | 0.7516(53)
-0.73 | -0.99 | 0.19352(66) | 0.5635(45) | 0.6667(69) | 0.6376(35) | 0.7350(72)
-0.72 | -1.01 | 0.23505(56) | 0.6485(35) | 0.42835(59) | 0.7811(23) | 0.6634(53) | 0.6253(26) | 0.7247(57)
-0.73 | -1.01 | 0.19352(66) | 0.5635(45) | 0.6366(72) | 0.6021(32) | 0.7073(79)
-0.72 | -1.03 | 0.23505(56) | 0.6485(35) | 0.35724(59) | 0.7152(29) | 0.6326(57) | 0.5879(26) | 0.6962(61)
-0.73 | -1.03 | 0.19352(66) | 0.5635(45) | 0.6051(75) | 0.5634(30) | 0.6774(89)
-0.7 | -1.04 | 0.30299(59) | 0.7545(24) | 0.31740(54) | 0.6741(25) | 0.6652(37) | 0.6183(21) | 0.7192(34)
-0.71 | -1.04 | 0.27064(52) | 0.7087(28) | 0.6409(44) | 0.5938(23) | 0.6999(36)
-0.72 | -1.04 | 0.23505(56) | 0.6485(35) | 0.6155(54) | 0.5689(25) | 0.6808(42)
-0.7 | -1.05 | 0.30299(59) | 0.7545(24) | 0.27275(51) | 0.6150(30) | 0.6498(39) | 0.5986(23) | 0.7042(41)
-0.71 | -1.05 | 0.27064(52) | 0.7087(28) | 0.6254(47) | 0.5735(25) | 0.6847(44)
-0.72 | -1.05 | 0.23505(56) | 0.6485(35) | 0.6000(54) | 0.5481(27) | 0.6653(53)
-0.7 | -1.06 | 0.30299(59) | 0.7545(24) | 0.22070(52) | 0.5309(37) | 0.6341(43) | 0.5767(24) | 0.6900(47)
-0.71 | -1.06 | 0.27064(52) | 0.7087(28) | 0.6089(49) | 0.5506(25) | 0.6704(52)
-0.72 | -1.06 | 0.23505(56) | 0.6485(35) | 0.5821(60) | 0.5237(26) | 0.6508(63)
Table 10:  Numerical values of the bare masses, $am_{0}^{(f)}$ and $am_{0}^{(as)}$, used in the measurements on the ensemble QB3, that has gradient flow scale $w_{0}/a=1.944(3)$. For each set of bare masses, we present the pseudoscalar meson mass, $am_{\rm PS}$ and $am_{\rm ps}$, the mass ratio between pseudoscalar and vector mesons in both representations, $m_{\rm PS}/m_{\rm V}$ and $m_{\rm ps}/m_{\rm v}$, and the chimera-baryon masses $am_{\Lambda_{\rm CB}}$, $am_{\Sigma_{\rm CB}}$ and $am_{\Sigma^{\ast}_{\rm CB}}$.  $am_{0}^{(f)}$ | $am_{0}^{(as)}$ | $am_{\rm PS}$ | $m_{\rm PS}/m_{\rm V}$ | $am_{\rm ps}$ | $m_{\rm ps}/m_{\rm v}$ | $am_{\Lambda_{\rm CB}}$ | $am_{\Sigma_{\rm CB}}$ | $am_{\Sigma^{\star}_{\rm CB}}$
---|---|---|---|---|---|---|---|---
-0.6 | -0.81 | 0.44096(43) | 0.9122(14) | 0.79148(43) | 0.95387(47) | 0.9475(18) | 0.9379(16) | 0.9643(18)
-0.62 | -0.81 | 0.39299(44) | 0.8899(19) | 0.9071(21) | 0.8981(17) | 0.9270(19)
-0.64 | -0.81 | 0.34160(44) | 0.8518(24) | 0.8655(25) | 0.8580(19) | 0.8892(21)
-0.66 | -0.81 | 0.28512(43) | 0.7896(23) | 0.8245(28) | 0.8182(21) | 0.8512(26)
-0.68 | -0.81 | 0.21852(47) | 0.6825(38) | 0.7817(43) | 0.7788(28) | 0.8153(36)
-0.69 | -0.81 | 0.17780(57) | 0.5922(49) | 0.7651(45) | 0.7609(35) | 0.7988(48)
-0.7 | -0.81 | 0.12461(75) | 0.4505(96) | 0.7486(68) | 0.7409(54) | 0.7818(90)
-0.62 | -0.82 | 0.39299(44) | 0.8899(19) | 0.77020(47) | 0.95130(52) | 0.8963(21) | 0.8867(17) | 0.9165(19)
-0.64 | -0.82 | 0.34160(44) | 0.8518(24) | 0.8547(25) | 0.8464(19) | 0.8786(21)
-0.68 | -0.82 | 0.21852(47) | 0.6825(38) | 0.7704(45) | 0.7680(30) | 0.8054(34)
-0.7 | -0.82 | 0.12461(75) | 0.4505(96) | 0.7370(67) | 0.7287(53) | 0.7713(91)
-0.62 | -0.84 | 0.39299(44) | 0.8899(19) | 0.72632(46) | 0.94418(57) | 0.8743(21) | 0.8633(17) | 0.8950(19)
-0.64 | -0.84 | 0.34160(44) | 0.8518(24) | 0.8326(24) | 0.8227(18) | 0.8570(21)
-0.68 | -0.84 | 0.21852(47) | 0.6825(38) | 0.7475(41) | 0.7433(28) | 0.7833(34)
-0.7 | -0.84 | 0.12461(75) | 0.4505(96) | 0.7132(63) | 0.7036(51) | 0.7498(89)
-0.64 | -0.91 | 0.34160(44) | 0.8518(24) | 0.55722(50) | 0.9003(11) | 0.7493(22) | 0.7324(17) | 0.7750(26)
-0.66 | -0.91 | 0.28512(43) | 0.7896(23) | 0.7064(25) | 0.6903(20) | 0.7371(25)
-0.64 | -0.93 | 0.34160(44) | 0.8518(24) | 0.50263(48) | 0.8779(13) | 0.7234(21) | 0.7039(17) | 0.7504(27)
-0.66 | -0.93 | 0.28512(43) | 0.7896(23) | 0.6804(25) | 0.6610(20) | 0.7122(24)
-0.64 | -0.95 | 0.34160(44) | 0.8518(24) | 0.44385(51) | 0.8466(17) | 0.6949(22) | 0.6738(17) | 0.7237(28)
-0.66 | -0.95 | 0.28512(43) | 0.7896(23) | 0.6525(26) | 0.6298(19) | 0.6865(26)
-0.66 | -0.99 | 0.28512(43) | 0.7896(23) | 0.30571(47) | 0.7232(25) | 0.5948(23) | 0.5603(18) | 0.6347(28)
-0.68 | -0.99 | 0.21852(47) | 0.6825(38) | 0.5478(31) | 0.5117(24) | 0.5965(34)
-0.66 | -1.01 | 0.28512(43) | 0.7896(23) | 0.21414(42) | 0.5735(32) | 0.5627(26) | 0.5198(20) | 0.6063(31)
-0.68 | -1.01 | 0.21852(47) | 0.6825(38) | 0.5148(35) | 0.4677(24) | 0.5688(40)
-0.66 | -1.015 | 0.28512(43) | 0.7896(23) | 0.18561(39) | 0.5161(38) | 0.5536(30) | 0.5099(21) | 0.5993(34)
-0.67 | -1.015 | 0.25347(60) | 0.7468(39) | 0.5295(34) | 0.4836(22) | 0.5814(45)
-0.68 | -1.015 | 0.21852(47) | 0.6825(38) | 0.5056(41) | 0.4567(25) | 0.5623(40)
-0.69 | -1.015 | 0.17780(57) | 0.5922(49) | 0.4826(56) | 0.4290(30) | 0.5475(60)
Table 11:  Numerical values of the bare masses, $am_{0}^{(f)}$ and $am_{0}^{(as)}$, used in the measurements on the ensemble QB4, that has gradient flow scale $w_{0}/a=2.3149(12)$. For each set of bare masses, we present the pseudoscalar meson mass, $am_{\rm PS}$ and $am_{\rm ps}$, the mass ratio between pseudoscalar and vector mesons in both representations, $m_{\rm PS}/m_{\rm V}$ and $m_{\rm ps}/m_{\rm v}$, and the chimera-baryon masses $am_{\Lambda_{\rm CB}}$, $am_{\Sigma_{\rm CB}}$ and $am_{\Sigma^{\ast}_{\rm CB}}$.  $am_{0}^{(f)}$ | $am_{0}^{(as)}$ | $am_{\rm PS}$ | $m_{\rm PS}/m_{\rm V}$ | $am_{\rm ps}$ | $m_{\rm ps}/m_{\rm v}$ | $am_{\Lambda_{\rm CB}}$ | $am_{\Sigma_{\rm CB}}$ | $am_{\Sigma^{\star}_{\rm CB}}$
---|---|---|---|---|---|---|---|---
-0.62 | -0.95 | 0.25077(68) | 0.8158(29) | 0.22264(47) | 0.6751(34) | 0.4901(28) | 0.4600(17) | 0.5205(31)
-0.64 | -0.95 | 0.18239(82) | 0.6997(57) | 0.4371(37) | 0.4061(21) | 0.4778(40)
-0.646 | -0.95 | 0.15759(89) | 0.6397(76) | 0.4203(45) | 0.3897(24) | 0.4657(44)
-0.62 | -0.956 | 0.25077(68) | 0.8158(29) | 0.19303(45) | 0.6200(42) | 0.4792(29) | 0.4469(18) | 0.5111(34)
-0.64 | -0.956 | 0.18239(82) | 0.6997(57) | 0.4251(40) | 0.3918(22) | 0.4682(43)
-0.646 | -0.956 | 0.15759(89) | 0.6397(76) | 0.4078(47) | 0.3749(25) | 0.4560(49)
-0.62 | -0.961 | 0.25077(68) | 0.8158(29) | 0.16488(48) | 0.5568(52) | 0.4693(37) | 0.4350(20) | 0.5033(35)
-0.64 | -0.961 | 0.18239(82) | 0.6997(57) | 0.4145(52) | 0.3781(25) | 0.4601(46)
-0.646 | -0.961 | 0.15759(89) | 0.6397(76) | 0.3971(61) | 0.3605(29) | 0.4481(52)
Table 12:  Numerical values of the bare masses, $am_{0}^{(f)}$ and
$am_{0}^{(as)}$, used in the measurements on the ensemble QB5, that has
gradient flow scale $w_{0}/a=2.8812(21)$. For each set of bare masses, we
present the pseudoscalar meson mass, $am_{\rm PS}$ and $am_{\rm ps}$, the mass
ratio between pseudoscalar and vector mesons in both representations, $m_{\rm
PS}/m_{\rm V}$ and $m_{\rm ps}/m_{\rm v}$, and the chimera-baryon masses
$am_{\Lambda_{\rm CB}}$, $am_{\Sigma_{\rm CB}}$ and $am_{\Sigma^{\ast}_{\rm
CB}}$.

##  Appendix A Extracted masses of mesons and chimera baryons

In this appendix, we provide a comprehensive summary of the choices of bare
hyperquark masses we used in the measurements we made, along with the
resulting masses of composite states, all expressed in lattice units. Our
dataset includes the masses of pseudoscalar mesons composed of $(f)$ and
$(as)$ hyperquarks, which we denote as $m_{\rm PS}$ and $m_{\rm ps}$,
respectively, as well as the masses of the chimera baryons $\Lambda_{\rm CB}$,
$\Sigma_{\rm CB}$, and $\Sigma^{\ast}_{\rm CB}$. We also report the mass
ratios between pseudoscalar and vector mesons, which serves as an indicator of
the amount of explicit breaking of the global symmetry of the theory. We have
organized the data into tables based on the ensemble in which the measurements
were performed, individual ensembles differing both by the lattice coupling
$\beta$ and the volume (see Tab. 2). Specifically, Tab. 8–12 correspond to
measurements on ensembles QB1–QB5, respectively. All numbers presented here
are available in machine-readable format in the data release associated with
this work DATA . Further technical details, such as smearing parameters,
fitting ranges applied to the mass extraction, and values of the resulting
$\chi^{2}/N_{\rm d.o.f.}$, are presented in Appendix C.

##  Appendix B Smearing techniques

Wuppertal smearing Gusken:1989qx  and APE smearing APE:1987ehd  are well-
developed lattice techniques, which are normally applied simultaneously, to
improve the overlap of a ground state. The former amounts to a modification of
the operators used to define the 2-point functions, in particular on the
position of the hyperquarks constituting mesons and chimera baryons. The
latter consists of a smoothening of the gauge configurations, that removes
short-distance fluctuations.

In calculations involving point sources, a hyperquark propagator, $S_{R}$,
involving fermions transforming in the $R$ representation of the gauge group,
is obtained by solving the Dirac equation

$D^{R}_{a\alpha,b\beta}(x,y)S^{\,b\beta}_{R\,\,\,\,c\gamma}(y)=\delta_{x,0}\delta_{\alpha\gamma}\delta_{ac}\,,$
(36)

where $D^{R}_{a\alpha,b\beta}$ is the Wilson-Dirac operator in Eq. (II.1).
Smearing of the source is obtained by replacing the spatial delta function,
$\delta_{x,0}$, by a new source, $q^{(n)}(x)$, constructed with the Wuppertal
smearing function, which is defined recursively by the relation

$q^{(n+1)}(x)=\frac{1}{1+6\varepsilon}\left[q^{(n)}(x)+\varepsilon\sum_{\mu=\pm
1}^{\pm 3}U_{\mu}(x)q^{(n)}(x+\hat{\mu})\right]\,,$ (37)

with $n=0\,,\cdots\,,N_{\text{W}}$. The initial condition is
$q^{(0)}(x)=\delta_{x,0}$ refers to a point source. The tunable parameters,
$\varepsilon$ and $N_{\text{W}}$, are referred to as step size and number of
iterations, respectively. The smearing of the sink is obtained by replacing
$q^{(0)}$ with a hyperquark propagator $S_{R}$.

We supplement Wuppertal smearing by smoothening gauge links with APE smearing.
The smearing function is

$U^{(n+1)}_{\mu}(x)=P\left\\{(1-\alpha)U^{(n)}_{\mu}(x)+\frac{\alpha}{6}S^{(n)}_{\mu}(x)\right\\}\,,$
(38)

where $S_{\mu}$ denotes the staple operator, defined as

$S_{\mu}(x)=\sum_{\pm\nu\neq\mu}U_{\nu}(x)U_{\mu}(x+\hat{\nu})U^{\dagger}_{\nu}(x+\hat{\mu}).$
(39)

The iteration number, $n=0\,,\cdots\,,N_{\text{APE}}$, and step size,
$\alpha$, are tunable parameters. Because of the summation over neighboring
gauge links in Eq. (38), the smeared gauge links should be projected back to
the group. A project $P$ is provided by the re-symplectisation algorithm in
the HB calculations, which inherits the numerical stability of the Gram-
Schmidt process. It takes advantage of the symplectic structure of $Sp(2N)$:
once the first $N$ columns, $col_{j}$, with $j=1,\,\cdots,\,N$, of the
$Sp(2N)$ matrix are given, the remaining ones are given by

$col_{j+N}=-\Omega\,col^{\ast}_{j}\,$ (40)

Having normalized the first column, the $(N+1)$-th one is obtained
algebraically. The second column is computed by orthonormalizing the first and
the $(N+1)$-th. By repeating the process for every column, one arrives at a
complete $Sp(2N)$ matrix.

In previous work Bennett:2017kga ; Bennett:2019jzz ; Bennett:2019cxd , we used
stochastic wall sources Boyle:2008rh  for the meson calculations. To improve
the signal of chimera baryons, in this study, we apply Wuppertal and APE
smearing simultaneously to obtain $(f)$ and $(as)$ hyperquark propagators.
Wuppertal smearing step sizes are chosen differently for $(f)$ and $(as)$
hyperquark propagators, and we denote them as $\epsilon^{(f)}$ and
$\epsilon^{(as)}$, respectively. We fix the number of iterations at the
source, and select individually the number of iterations at the sink that
display the optimal plateau for each meson and each chimera baryon. Our
choices of the Wuppertal smearing parameters are presented in Appendix C. The
APE smearing parameters are fixed in all the calculations to be $\alpha=0.4$
and $N_{\rm APE}=50$.

The same techniques are also applied to our study on the spectrum of $Sp(4)$
gauge theory with $n_{f}=3$ antisymmetric fermions Lee:2022m , particularly
for the calculation of the first excited state of the vector and tensor
mesons. Additionally, these techniques have been utilized as a cross-
verification in the excited state subtraction method applied to the
computation of a singlet meson—see Appendix B of Ref. Bennett:2023rsl .

##  Appendix C Details about fitting procedures

In this appendix, we tabulate numerical information relevant to the mass
extractions for mesons and chimera baryons. In Tab. 13, we list values of the
smearing parameters of Wuppertal smearing, the fitting interval for the mass
extraction of mesons made of $(f)$ hyperquark constituents and the
corresponding $\chi^{2}/N_{\rm d.o.f.}$ of the fits. Similar information for
mesons made of $(as)$ hyperquarks is presented in Tab. 14. For chimera
baryons, we provide the relevant details separately for each ensemble in Tabs.
15–19, where the number of iterations of Wuppertal smearing at the source and
sink, the fitting intervals imposed to fit the correlation functions, as well
as the resulting $\chi^{2}/N_{\rm d.o.f.}$ are displayed. While we set the
number of iterations to be the same for both $(f)$ and $(as)$ hyperquarks, we
set the step size differently for each type of hyperquark. All parameters
presented are also available in machine-readable format in the data release
associated with this publication DATA .

|  |  | PS | V
---|---|---|---|---
Ensemble | $am_{0}^{(f)}$ | $\epsilon^{(f)}$ | $N_{\rm W}^{\rm source}$ | $N_{\rm W}^{\rm sink}$ | I | $\chi^{2}/N_{\rm d.o.f.}$ | $N_{\rm W}^{\rm source}$ | $N_{\rm W}^{\rm sink}$ | I | $\chi^{2}/N_{\rm d.o.f.}$
QB1 | -0.7 | 0.05 | 100 | 0 | [13 24] | 0.88 | 100 | 0 | [15 23] | 0.85
-0.73 | 0.05 | 100 | 0 | [13 24] | 1.01 | 100 | 0 | [15 23] | 0.78
-0.75 | 0.05 | 100 | 0 | [13 24] | 1.1 | 100 | 0 | [15 23] | 0.7
-0.77 | 0.18 | 60 | 0 | [14 24] | 1.17 | 60 | 20 | [15 23] | 0.31
-0.78 | 0.18 | 60 | 0 | [14 24] | 1.21 | 60 | 40 | [10 22] | 0.48
-0.79 | 0.18 | 60 | 0 | [14 24] | 1.25 | 60 | 40 | [ 8 22] | 0.85
QB2 | -0.7 | 0.18 | 50 | 0 | [13 29] | 0.69 | 50 | 0 | [12 29] | 1.6
-0.72 | 0.18 | 50 | 0 | [13 29] | 0.62 | 50 | 0 | [12 25] | 1.53
-0.74 | 0.18 | 50 | 0 | [19 29] | 0.53 | 50 | 0 | [12 20] | 1.0
-0.76 | 0.18 | 50 | 0 | [19 29] | 0.44 | 50 | 0 | [12 20] | 0.85
-0.77 | 0.18 | 50 | 0 | [19 29] | 0.38 | 50 | 0 | [12 20] | 1.6
QB3 | -0.66 | 0.18 | 50 | 0 | [19 26] | 0.24 | 50 | 0 | [21 30] | 0.68
-0.68 | 0.18 | 50 | 0 | [19 26] | 0.36 | 50 | 0 | [21 30] | 0.77
-0.7 | 0.18 | 50 | 0 | [19 30] | 0.72 | 50 | 0 | [14 26] | 0.43
-0.71 | 0.18 | 50 | 0 | [18 30] | 0.55 | 50 | 0 | [14 26] | 0.5
-0.72 | 0.18 | 50 | 0 | [20 30] | 0.53 | 50 | 0 | [14 26] | 0.62
-0.73 | 0.18 | 50 | 0 | [22 30] | 0.45 | 50 | 0 | [14 29] | 1.05
QB4 | -0.6 | 0.18 | 100 | 0 | [20 29] | 1.14 | 100 | 0 | [19 29] | 0.97
-0.62 | 0.18 | 60 | 40 | [18 30] | 1.1 | 60 | 20 | [18 28] | 0.84
-0.64 | 0.18 | 60 | 40 | [18 30] | 1.1 | 60 | 10 | [18 28] | 0.67
-0.66 | 0.18 | 50 | 0 | [15 28] | 1.92 | 50 | 0 | [18 29] | 0.99
-0.68 | 0.18 | 50 | 0 | [15 28] | 1.47 | 50 | 0 | [18 29] | 0.4
-0.69 | 0.18 | 50 | 0 | [15 28] | 1.04 | 50 | 0 | [18 29] | 0.42
-0.7 | 0.18 | 50 | 0 | [15 28] | 0.96 | 50 | 0 | [18 27] | 0.91
QB5 | -0.62 | 0.2 | 50 | 0 | [25 30] | 0.19 | 50 | 0 | [25 30] | 0.11
-0.64 | 0.2 | 50 | 0 | [25 30] | 0.3 | 50 | 0 | [25 30] | 0.57
-0.646 | 0.2 | 50 | 0 | [25 30] | 0.32 | 50 | 20 | [23 30] | 0.75
Table 13:  Technical details about the computation of the masses of the pseudoscalar and vector mesons constituted by hyperquarks in $(f)$ representation. For Wuppertal smearing, we denote the step size by $\epsilon^{(f)}$, the number of iterations at the source by $N_{\rm W}^{\rm source}$ and the number of iterations at the sink by $N_{\rm W}^{\rm sink}$. The APE smearing parameters $\alpha$ and $N_{\rm APE}$ are fixed to $0.4$ and $50$, respectively, in all the calculations. For each choice of bare mass, $am_{0}^{(f)}$, and each meson, we show the fitting intervals as Euclidean time $I=[t_{i},t_{f}]$, between the initial time $t_{i}$ and the final time $t_{f}$. We perform a correlated fit with standard $\chi^{2}$-minimization to the function in Eq. (27). We report the values of $\chi^{2}$ normalized by the number degrees of freedom, $\chi^{2}/N_{\rm d.o.f.}$.  |  |  | ps | v
---|---|---|---|---
Ensemble | $am_{0}^{(as)}$ | $\epsilon^{(as)}$ | $N_{\rm W}^{\rm source}$ | $N_{\rm W}^{\rm sink}$ | I | $\chi^{2}/N_{\rm d.o.f.}$ | $N_{\rm W}^{\rm source}$ | $N_{\rm W}^{\rm sink}$ | I | $\chi^{2}/N_{\rm d.o.f.}$
QB1 | -0.8 | 0.01 | 100 | 0 | [ 8 24] | 0.53 | 100 | 0 | [ 8 24] | 0.6
-0.9 | 0.01 | 100 | 20 | [15 23] | 0.28 | 100 | 20 | [ 8 24] | 0.41
-0.95 | 0.01 | 100 | 0 | [14 23] | 0.42 | 100 | 0 | [10 24] | 0.57
-1.0 | 0.01 | 100 | 0 | [14 23] | 0.55 | 100 | 0 | [10 24] | 0.75
-1.05 | 0.01 | 100 | 0 | [10 23] | 0.41 | 100 | 0 | [15 24] | 0.52
-1.1 | 0.08 | 100 | 0 | [12 24] | 0.53 | 100 | 0 | [12 24] | 1.28
-1.12 | 0.18 | 60 | 0 | [12 24] | 0.58 | 60 | 0 | [ 9 20] | 1.2
-1.14 | 0.18 | 60 | 0 | [12 24] | 0.76 | 60 | 0 | [ 9 20] | 0.61
-1.15 | 0.1 | 100 | 0 | [10 24] | 0.76 | 100 | 0 | [ 7 18] | 0.48
QB2 | -0.89 | 0.02 | 50 | 0 | [23 29] | 0.68 | 50 | 0 | [17 27] | 1.15
-0.91 | 0.02 | 50 | 0 | [17 29] | 1.29 | 50 | 0 | [14 25] | 1.35
-0.92 | 0.02 | 50 | 0 | [17 29] | 1.28 | 50 | 0 | [17 27] | 1.4
-0.93 | 0.02 | 50 | 0 | [23 29] | 0.63 | 50 | 0 | [16 25] | 1.16
-0.97 | 0.02 | 50 | 0 | [23 29] | 0.52 | 50 | 0 | [16 25] | 0.75
-1.01 | 0.02 | 50 | 0 | [23 29] | 0.51 | 50 | 0 | [16 25] | 0.47
-1.09 | 0.12 | 50 | 0 | [23 29] | 0.89 | 50 | 0 | [16 25] | 0.39
-1.1 | 0.12 | 50 | 0 | [23 29] | 0.65 | 50 | 0 | [16 25] | 0.43
-1.11 | 0.14 | 50 | 0 | [18 29] | 0.5 | 50 | 0 | [12 24] | 0.42
-1.12 | 0.2 | 50 | 0 | [18 29] | 1.2 | 50 | 0 | [12 21] | 0.51
QB3 | -0.85 | 0.01 | 50 | 0 | [21 29] | 0.29 | 50 | 0 | [19 29] | 0.41
-0.87 | 0.01 | 50 | 0 | [21 29] | 0.33 | 50 | 0 | [19 29] | 0.43
-0.9 | 0.01 | 50 | 0 | [21 29] | 0.42 | 50 | 0 | [19 28] | 0.5
-0.93 | 0.01 | 50 | 0 | [21 29] | 0.56 | 50 | 0 | [19 29] | 0.54
-0.97 | 0.01 | 50 | 0 | [16 25] | 0.28 | 50 | 0 | [15 26] | 0.68
-0.99 | 0.01 | 50 | 0 | [16 25] | 0.24 | 50 | 0 | [15 26] | 0.82
-1.01 | 0.01 | 50 | 0 | [16 25] | 0.24 | 50 | 0 | [15 26] | 0.9
-1.03 | 0.01 | 50 | 0 | [15 25] | 0.25 | 50 | 0 | [14 26] | 0.76
-1.04 | 0.16 | 50 | 0 | [15 25] | 0.35 | 50 | 0 | [12 21] | 0.64
-1.05 | 0.18 | 50 | 0 | [15 25] | 0.37 | 50 | 0 | [12 21] | 0.69
-1.06 | 0.2 | 50 | 0 | [15 29] | 0.86 | 50 | 0 | [12 23] | 0.85
QB4 | -0.81 | 0.01 | 60 | 0 | [18 30] | 0.91 | 60 | 0 | [19 30] | 1.45
-0.82 | 0.01 | 60 | 0 | [19 30] | 0.82 | 60 | 0 | [19 30] | 1.3
-0.84 | 0.01 | 60 | 0 | [18 30] | 0.69 | 60 | 0 | [19 30] | 1.2
-0.91 | 0.01 | 60 | 0 | [18 30] | 0.62 | 60 | 0 | [19 30] | 0.95
-0.93 | 0.01 | 60 | 0 | [18 30] | 0.7 | 60 | 0 | [19 30] | 0.98
-0.95 | 0.01 | 60 | 0 | [18 30] | 0.84 | 60 | 0 | [19 30] | 1.06
-0.99 | 0.16 | 50 | 0 | [17 30] | 1.42 | 50 | 0 | [16 30] | 0.77
-1.01 | 0.16 | 50 | 0 | [14 30] | 1.42 | 50 | 0 | [12 28] | 1.97
-1.015 | 0.2 | 50 | 20 | [11 30] | 0.85 | 50 | 40 | [11 25] | 1.85
QB5 | -0.95 | 0.2 | 50 | 0 | [20 30] | 0.68 | 50 | 0 | [17 29] | 0.97
-0.956 | 0.2 | 50 | 0 | [18 30] | 1.02 | 50 | 0 | [17 29] | 1.06
-0.961 | 0.2 | 50 | 0 | [18 30] | 1.04 | 50 | 0 | [17 29] | 1.05
Table 14:  Technical details about the computation of the masses of the pseudoscalar and vector mesons constituted by hyperquarks in $(as)$ representation. For Wuppertal smearing, we denote the step size by $\epsilon^{(as)}$, the number of iterations at the source by $N_{\rm W}^{\rm source}$ and the number of iterations at the sink by $N_{\rm W}^{\rm sink}$. The APE smearing parameters $\alpha$ and $N_{\rm APE}$ are fixed to $0.4$ and $50$, respectively, in all the calculations. For each choice of bare mass, $am_{0}^{(as)}$, and each meson, we show the fitting intervals as Euclidean time $I=[t_{i},t_{f}]$, between the initial time $t_{i}$ and the final time $t_{f}$. We perform a correlated fit with standard $\chi^{2}$-minimization to the function in Eq. (27). We report the values of $\chi^{2}$ normalized by the number degrees of freedom, $\chi^{2}/N_{\rm d.o.f.}$.  |  |  |  | $\Lambda_{\rm CB}$ | $\Sigma_{\rm CB}$ | $\Sigma^{\ast}_{\rm CB}$
---|---|---|---|---|---|---
$am_{0}^{(f)}$ | $am_{0}^{(as)}$ | $\epsilon^{(f)}$ | $\epsilon^{(as)}$ | $N_{\rm W}^{\rm source}$ | $N_{\rm W}^{\rm sink}$ | I | $\chi^{2}/N_{\rm d.o.f.}$ | $N_{\rm W}^{\rm source}$ | $N_{\rm W}^{\rm sink}$ | I | $\chi^{2}/N_{\rm d.o.f.}$ | $N_{\rm W}^{\rm source}$ | $N_{\rm W}^{\rm sink}$ | I | $\chi^{2}/N_{\rm d.o.f.}$
-0.7 | -0.8 | 0.05 | 0.01 | 100 | 10 | [12 24] | 0.41 | 100 | 0 | [12 24] | 0.32 | 100 | 0 | [12 24] | 0.62
-0.73 | -0.8 | 0.05 | 0.01 | 100 | 0 | [12 24] | 0.48 | 100 | 0 | [12 24] | 0.31 | 100 | 0 | [12 24] | 0.58
-0.75 | -0.8 | 0.05 | 0.01 | 100 | 0 | [12 24] | 0.54 | 100 | 0 | [14 24] | 0.22 | 100 | 0 | [12 24] | 0.49
-0.77 | -0.8 | 0.05 | 0.01 | 100 | 0 | [12 20] | 0.88 | 100 | 0 | [14 22] | 0.36 | 100 | 0 | [14 20] | 0.21
-0.79 | -0.8 | 0.05 | 0.01 | 100 | 0 | [12 24] | 0.83 | 100 | 0 | [14 24] | 0.51 | 100 | 0 | [12 24] | 0.34
-0.7 | -0.9 | 0.05 | 0.01 | 100 | 0 | [12 24] | 0.32 | 100 | 0 | [14 24] | 0.2 | 100 | 0 | [12 24] | 0.65
-0.73 | -0.9 | 0.08 | 0.01 | 100 | 0 | [12 24] | 0.31 | 100 | 0 | [14 24] | 0.23 | 100 | 0 | [12 24] | 0.6
-0.75 | -0.9 | 0.08 | 0.01 | 100 | 0 | [12 24] | 0.36 | 100 | 0 | [14 24] | 0.25 | 100 | 0 | [12 24] | 0.52
-0.77 | -0.9 | 0.08 | 0.01 | 100 | 0 | [12 24] | 0.49 | 100 | 0 | [14 24] | 0.53 | 100 | 0 | [12 24] | 0.42
-0.79 | -0.9 | 0.08 | 0.01 | 100 | 0 | [10 24] | 0.67 | 100 | 0 | [12 24] | 0.69 | 100 | 0 | [14 24] | 0.35
-0.7 | -0.95 | 0.05 | 0.01 | 100 | 0 | [12 24] | 0.3 | 100 | 0 | [14 24] | 0.21 | 100 | 0 | [12 24] | 0.63
-0.73 | -0.95 | 0.2 | 0.01 | 100 | 0 | [10 24] | 0.46 | 100 | 0 | [12 24] | 0.46 | 100 | 0 | [12 24] | 0.58
-0.75 | -0.95 | 0.2 | 0.01 | 100 | 0 | [10 24] | 0.57 | 100 | 0 | [12 24] | 0.5 | 100 | 0 | [12 24] | 0.54
-0.77 | -0.95 | 0.2 | 0.01 | 100 | 0 | [10 24] | 0.7 | 100 | 0 | [12 24] | 0.65 | 100 | 0 | [12 24] | 0.48
-0.79 | -0.95 | 0.2 | 0.01 | 100 | 0 | [10 24] | 0.79 | 100 | 0 | [12 24] | 0.97 | 100 | 0 | [12 24] | 0.46
-0.7 | -1.0 | 0.05 | 0.01 | 100 | 0 | [10 20] | 0.6 | 100 | 0 | [12 22] | 0.48 | 100 | 0 | [11 20] | 0.9
-0.73 | -1.0 | 0.1 | 0.01 | 100 | 0 | [10 24] | 0.39 | 100 | 0 | [12 24] | 0.44 | 100 | 0 | [12 24] | 0.53
-0.75 | -1.0 | 0.1 | 0.01 | 100 | 0 | [10 24] | 0.46 | 100 | 0 | [12 24] | 0.57 | 100 | 0 | [12 24] | 0.45
-0.77 | -1.0 | 0.1 | 0.01 | 100 | 0 | [10 24] | 0.59 | 100 | 0 | [12 24] | 0.79 | 100 | 0 | [12 24] | 0.36
-0.79 | -1.0 | 0.1 | 0.01 | 100 | 0 | [10 24] | 0.76 | 100 | 0 | [12 24] | 1.14 | 100 | 0 | [12 24] | 0.37
-0.7 | -1.05 | 0.05 | 0.01 | 100 | 0 | [12 24] | 0.59 | 100 | 0 | [12 24] | 0.41 | 100 | 0 | [12 24] | 0.5
-0.73 | -1.05 | 0.08 | 0.01 | 100 | 0 | [12 24] | 0.46 | 100 | 0 | [12 24] | 0.45 | 100 | 0 | [12 24] | 0.44
-0.75 | -1.05 | 0.08 | 0.01 | 100 | 0 | [12 24] | 0.48 | 100 | 0 | [12 24] | 0.61 | 100 | 0 | [12 24] | 0.36
-0.77 | -1.05 | 0.08 | 0.01 | 100 | 0 | [12 24] | 0.64 | 100 | 0 | [12 24] | 0.81 | 100 | 0 | [12 24] | 0.3
-0.79 | -1.05 | 0.08 | 0.01 | 100 | 0 | [12 24] | 0.96 | 100 | 0 | [12 24] | 1.11 | 100 | 0 | [12 24] | 0.37
-0.7 | -1.1 | 0.05 | 0.01 | 100 | 0 | [10 20] | 1.09 | 100 | 0 | [12 22] | 0.53 | 100 | 0 | [10 20] | 0.77
-0.73 | -1.1 | 0.05 | 0.08 | 100 | 0 | [10 20] | 1.03 | 100 | 0 | [12 22] | 0.53 | 100 | 0 | [10 20] | 0.73
-0.75 | -1.1 | 0.05 | 0.08 | 100 | 0 | [10 20] | 0.96 | 100 | 0 | [12 22] | 0.51 | 100 | 0 | [10 20] | 0.73
-0.77 | -1.1 | 0.05 | 0.08 | 100 | 0 | [10 20] | 0.89 | 100 | 0 | [12 22] | 0.51 | 100 | 0 | [10 20] | 0.76
-0.79 | -1.1 | 0.05 | 0.08 | 100 | 0 | [10 20] | 0.86 | 100 | 0 | [12 22] | 0.59 | 100 | 0 | [10 20] | 0.7
-0.77 | -1.12 | 0.18 | 0.18 | 60 | 0 | [10 22] | 1.23 | 60 | 0 | [12 24] | 0.51 | 60 | 0 | [12 20] | 0.49
-0.78 | -1.12 | 0.18 | 0.18 | 60 | 0 | [10 22] | 1.16 | 60 | 0 | [12 24] | 0.62 | 60 | 0 | [12 20] | 0.48
-0.79 | -1.12 | 0.18 | 0.18 | 60 | 0 | [10 22] | 1.1 | 60 | 0 | [12 24] | 0.66 | 60 | 0 | [12 20] | 0.45
-0.77 | -1.14 | 0.18 | 0.18 | 60 | 0 | [10 22] | 1.09 | 60 | 0 | [12 24] | 0.65 | 60 | 0 | [11 20] | 0.52
-0.78 | -1.14 | 0.18 | 0.18 | 60 | 0 | [10 22] | 1.02 | 60 | 0 | [12 24] | 0.73 | 60 | 0 | [11 20] | 0.46
-0.79 | -1.14 | 0.18 | 0.18 | 60 | 0 | [10 20] | 1.18 | 60 | 0 | [12 22] | 0.73 | 60 | 0 | [11 20] | 0.42
-0.7 | -1.15 | 0.05 | 0.1 | 100 | 0 | [10 20] | 1.32 | 100 | 0 | [12 22] | 0.55 | 100 | 0 | [11 20] | 0.86
-0.73 | -1.15 | 0.05 | 0.1 | 100 | 0 | [10 20] | 1.12 | 100 | 0 | [12 22] | 0.63 | 100 | 0 | [14 20] | 0.15
-0.75 | -1.15 | 0.05 | 0.1 | 100 | 0 | [10 20] | 0.96 | 100 | 0 | [12 22] | 0.71 | 100 | 0 | [14 20] | 0.17
-0.77 | -1.15 | 0.05 | 0.1 | 100 | 0 | [10 20] | 0.85 | 100 | 0 | [12 22] | 0.79 | 100 | 0 | [10 20] | 0.75
-0.79 | -1.15 | 0.05 | 0.1 | 100 | 0 | [10 20] | 0.77 | 100 | 0 | [12 22] | 0.89 | 100 | 0 | [11 20] | 0.48
Table 15:  Technical details about the computation of the masses of $\Lambda_{\rm CB}$, $\Sigma_{\rm CB}$, and $\Sigma_{\rm CB}^{\ast}$, with bare masses $am_{0}^{(f)}$ and $am_{0}^{(as)}$, on ensemble QB1. For Wuppertal smearing parameters, we represent the number of iterations at the source as $N_{\rm W}^{\rm source}$ and at the sink as $N_{\rm W}^{\rm sink}$. For each chimera baryon, we select the number of sink iterations to present the optimal plateau, considering both its length and error size. APE smearing parameters $\alpha$ and $N_{\rm APE}$ are $0.4$ and $50$, respectively, for all the calculations. For each set of bare masses and each chimera baryon, we report the fitting intervals as the Euclidean time $I=[t_{i},t_{f}]$, between the initial time $t_{i}$ and the final time $t_{f}$. We perform a correlated fit with the standard $\chi^{2}$-minimization to the function Eq. (24). We report the values of $\chi^{2}$ normalized by the degrees of freedom, $\chi^{2}/N_{\rm d.o.f.}$.  |  |  |  | $\Lambda_{\rm CB}$ | $\Sigma_{\rm CB}$ | $\Sigma^{\ast}_{\rm CB}$
---|---|---|---|---|---|---
$am_{0}^{(f)}$ | $am_{0}^{(as)}$ | $\epsilon^{(f)}$ | $\epsilon^{(as)}$ | $N_{\rm W}^{\rm source}$ | $N_{\rm W}^{\rm sink}$ | I | $\chi^{2}/N_{\rm d.o.f.}$ | $N_{\rm W}^{\rm source}$ | $N_{\rm W}^{\rm sink}$ | I | $\chi^{2}/N_{\rm d.o.f.}$ | $N_{\rm W}^{\rm source}$ | $N_{\rm W}^{\rm sink}$ | I | $\chi^{2}/N_{\rm d.o.f.}$
-0.7 | -0.89 | 0.18 | 0.02 | 50 | 0 | [17 29] | 0.49 | 50 | 0 | [17 28] | 0.48 | 50 | 0 | [17 25] | 0.21
-0.72 | -0.89 | 0.18 | 0.02 | 50 | 0 | [17 29] | 0.63 | 50 | 0 | [17 28] | 0.25 | 50 | 0 | [17 25] | 0.25
-0.74 | -0.89 | 0.18 | 0.02 | 50 | 0 | [15 29] | 1.14 | 50 | 0 | [17 28] | 0.2 | 50 | 0 | [17 25] | 0.29
-0.77 | -0.89 | 0.18 | 0.02 | 50 | 0 | [15 29] | 1.01 | 50 | 0 | [17 28] | 0.77 | 50 | 0 | [17 25] | 0.25
-0.72 | -0.91 | 0.18 | 0.02 | 50 | 0 | [14 27] | 0.62 | 50 | 0 | [14 30] | 0.49 | 50 | 0 | [16 25] | 0.44
-0.74 | -0.91 | 0.18 | 0.02 | 50 | 0 | [14 27] | 0.7 | 50 | 0 | [14 28] | 0.35 | 50 | 0 | [16 25] | 0.53
-0.77 | -0.91 | 0.18 | 0.02 | 50 | 0 | [14 27] | 0.71 | 50 | 0 | [14 28] | 0.76 | 50 | 0 | [16 25] | 0.46
-0.72 | -0.92 | 0.18 | 0.02 | 50 | 0 | [14 27] | 0.61 | 50 | 0 | [14 28] | 0.44 | 50 | 0 | [16 25] | 0.45
-0.74 | -0.92 | 0.18 | 0.02 | 50 | 0 | [14 27] | 0.73 | 50 | 0 | [14 28] | 0.32 | 50 | 0 | [16 25] | 0.54
-0.77 | -0.92 | 0.18 | 0.02 | 50 | 0 | [14 27] | 0.88 | 50 | 0 | [14 28] | 0.69 | 50 | 0 | [16 25] | 0.47
-0.7 | -0.93 | 0.18 | 0.02 | 50 | 0 | [17 28] | 0.45 | 50 | 0 | [17 29] | 0.55 | 50 | 0 | [17 26] | 0.21
-0.72 | -0.93 | 0.18 | 0.02 | 50 | 0 | [14 25] | 0.59 | 50 | 0 | [14 27] | 0.33 | 50 | 0 | [17 26] | 0.25
-0.74 | -0.93 | 0.18 | 0.02 | 50 | 0 | [14 25] | 0.68 | 50 | 0 | [14 27] | 0.34 | 50 | 0 | [17 26] | 0.3
-0.76 | -0.93 | 0.18 | 0.02 | 50 | 0 | [14 26] | 0.73 | 50 | 0 | [14 26] | 0.36 | 50 | 0 | [17 24] | 0.36
-0.77 | -0.93 | 0.18 | 0.02 | 50 | 0 | [14 25] | 0.9 | 50 | 0 | [14 27] | 0.5 | 50 | 0 | [17 24] | 0.29
-0.76 | -0.97 | 0.18 | 0.02 | 50 | 0 | [14 26] | 0.68 | 50 | 0 | [14 26] | 0.36 | 50 | 0 | [17 24] | 0.39
-0.76 | -1.01 | 0.18 | 0.02 | 50 | 0 | [13 26] | 0.67 | 50 | 0 | [14 26] | 0.35 | 50 | 0 | [17 25] | 0.44
-0.72 | -1.09 | 0.18 | 0.12 | 50 | 0 | [12 22] | 0.71 | 50 | 0 | [14 24] | 0.61 | 50 | 0 | [16 25] | 0.33
-0.74 | -1.09 | 0.18 | 0.12 | 50 | 0 | [12 22] | 0.54 | 50 | 0 | [14 24] | 0.68 | 50 | 0 | [16 25] | 0.35
-0.76 | -1.09 | 0.18 | 0.12 | 50 | 0 | [12 22] | 0.4 | 50 | 0 | [14 24] | 0.75 | 50 | 0 | [16 25] | 0.41
-0.72 | -1.1 | 0.18 | 0.12 | 50 | 0 | [12 23] | 0.69 | 50 | 0 | [17 24] | 0.77 | 50 | 0 | [16 26] | 0.42
-0.74 | -1.1 | 0.18 | 0.12 | 50 | 0 | [12 23] | 0.51 | 50 | 0 | [17 24] | 0.68 | 50 | 0 | [16 26] | 0.44
-0.76 | -1.1 | 0.18 | 0.12 | 50 | 0 | [12 23] | 0.37 | 50 | 0 | [14 26] | 0.6 | 50 | 0 | [16 25] | 0.4
-0.77 | -1.1 | 0.2 | 0.18 | 50 | 0 | [ 9 22] | 0.37 | 50 | 0 | [14 28] | 0.88 | 50 | 0 | [12 23] | 0.69
-0.72 | -1.11 | 0.18 | 0.14 | 50 | 0 | [12 23] | 0.68 | 50 | 0 | [18 25] | 0.78 | 50 | 0 | [16 24] | 0.32
-0.74 | -1.11 | 0.18 | 0.14 | 50 | 0 | [12 23] | 0.5 | 50 | 0 | [14 26] | 0.69 | 50 | 0 | [14 23] | 0.63
-0.76 | -1.11 | 0.18 | 0.14 | 50 | 0 | [12 23] | 0.38 | 50 | 0 | [14 26] | 0.71 | 50 | 0 | [14 23] | 0.61
-0.77 | -1.11 | 0.2 | 0.18 | 50 | 0 | [ 9 22] | 0.42 | 50 | 0 | [14 28] | 0.94 | 50 | 0 | [12 23] | 0.71
-0.72 | -1.12 | 0.18 | 0.2 | 50 | 0 | [12 23] | 0.6 | 50 | 0 | [14 26] | 1.04 | 50 | 0 | [12 23] | 0.56
-0.74 | -1.12 | 0.18 | 0.2 | 50 | 0 | [12 23] | 0.43 | 50 | 0 | [14 26] | 0.9 | 50 | 0 | [12 23] | 0.55
-0.76 | -1.12 | 0.18 | 0.2 | 50 | 0 | [12 23] | 0.36 | 50 | 0 | [14 26] | 1.05 | 50 | 0 | [12 23] | 0.64
-0.77 | -1.12 | 0.2 | 0.18 | 50 | 0 | [ 9 22] | 0.45 | 50 | 0 | [14 28] | 1.08 | 50 | 0 | [12 23] | 0.7
Table 16:  Technical details about the computation of the masses of $\Lambda_{\rm CB}$, $\Sigma_{\rm CB}$, and $\Sigma_{\rm CB}^{\ast}$, with bare masses $am_{0}^{(f)}$ and $am_{0}^{(as)}$, on ensemble QB2. For Wuppertal smearing parameters, we represent the number of iterations at the source as $N_{\rm W}^{\rm source}$ and at the sink as $N_{\rm W}^{\rm sink}$. For each chimera baryon, we select the number of sink iterations to present the optimal plateau, considering both its length and error size. APE smearing parameters $\alpha$ and $N_{\rm APE}$ are $0.4$ and $50$, respectively, for all the calculations. For each set of bare masses and each chimera baryon, we report the fitting intervals as the Euclidean time $I=[t_{i},t_{f}]$, between the initial time $t_{i}$ and the final time $t_{f}$. We perform a correlated fit with the standard $\chi^{2}$-minimization to the function Eq. (24). We report the values of $\chi^{2}$ normalized by the degrees of freedom, $\chi^{2}/N_{\rm d.o.f.}$.  |  |  |  | $\Lambda_{\rm CB}$ | $\Sigma_{\rm CB}$ | $\Sigma^{\ast}_{\rm CB}$
---|---|---|---|---|---|---
$am_{0}^{(f)}$ | $am_{0}^{(as)}$ | $\epsilon^{(f)}$ | $\epsilon^{(as)}$ | $N_{\rm W}^{\rm source}$ | $N_{\rm W}^{\rm sink}$ | I | $\chi^{2}/N_{\rm d.o.f.}$ | $N_{\rm W}^{\rm source}$ | $N_{\rm W}^{\rm sink}$ | I | $\chi^{2}/N_{\rm d.o.f.}$ | $N_{\rm W}^{\rm source}$ | $N_{\rm W}^{\rm sink}$ | I | $\chi^{2}/N_{\rm d.o.f.}$
-0.66 | -0.85 | 0.18 | 0.01 | 50 | 0 | [19 27] | 1.01 | 50 | 0 | [19 29] | 1.5 | 50 | 0 | [16 30] | 0.72
-0.68 | -0.85 | 0.18 | 0.01 | 50 | 0 | [17 30] | 0.92 | 50 | 0 | [16 30] | 1.04 | 50 | 0 | [16 28] | 0.71
-0.7 | -0.85 | 0.18 | 0.01 | 50 | 0 | [17 25] | 0.32 | 50 | 0 | [16 27] | 0.93 | 50 | 0 | [18 28] | 0.55
-0.72 | -0.85 | 0.18 | 0.01 | 50 | 0 | [17 25] | 0.3 | 50 | 0 | [16 27] | 0.66 | 50 | 0 | [18 28] | 0.54
-0.66 | -0.87 | 0.18 | 0.01 | 50 | 0 | [17 25] | 0.76 | 50 | 0 | [16 27] | 1.4 | 50 | 0 | [18 28] | 0.55
-0.68 | -0.87 | 0.18 | 0.01 | 50 | 0 | [17 25] | 0.57 | 50 | 0 | [16 27] | 1.26 | 50 | 0 | [18 28] | 0.53
-0.7 | -0.87 | 0.18 | 0.01 | 50 | 0 | [17 25] | 0.33 | 50 | 0 | [16 27] | 0.97 | 50 | 0 | [18 28] | 0.53
-0.72 | -0.87 | 0.18 | 0.01 | 50 | 0 | [17 25] | 0.31 | 50 | 0 | [16 27] | 0.68 | 50 | 0 | [18 28] | 0.5
-0.66 | -0.9 | 0.18 | 0.01 | 50 | 0 | [17 27] | 1.0 | 50 | 0 | [16 29] | 1.31 | 50 | 0 | [16 30] | 0.78
-0.68 | -0.9 | 0.18 | 0.01 | 50 | 0 | [17 27] | 0.9 | 50 | 0 | [16 29] | 1.19 | 50 | 0 | [16 30] | 0.96
-0.7 | -0.9 | 0.18 | 0.01 | 50 | 0 | [17 27] | 0.99 | 50 | 0 | [16 29] | 0.9 | 50 | 0 | [16 30] | 1.1
-0.72 | -0.9 | 0.18 | 0.01 | 50 | 0 | [17 25] | 0.34 | 50 | 0 | [16 25] | 0.45 | 50 | 0 | [16 26] | 0.61
-0.66 | -0.93 | 0.18 | 0.01 | 50 | 0 | [17 25] | 0.88 | 50 | 0 | [16 27] | 1.5 | 50 | 0 | [18 28] | 0.5
-0.68 | -0.93 | 0.18 | 0.01 | 50 | 0 | [17 25] | 0.68 | 50 | 0 | [16 27] | 1.36 | 50 | 0 | [18 28] | 0.45
-0.7 | -0.93 | 0.18 | 0.01 | 50 | 0 | [17 25] | 0.42 | 50 | 0 | [16 27] | 1.07 | 50 | 0 | [18 28] | 0.41
-0.72 | -0.93 | 0.18 | 0.01 | 50 | 0 | [17 25] | 0.39 | 50 | 0 | [16 27] | 0.71 | 50 | 0 | [18 28] | 0.36
-0.72 | -0.97 | 0.18 | 0.01 | 50 | 0 | [17 25] | 0.48 | 50 | 0 | [16 27] | 0.67 | 50 | 0 | [18 28] | 0.27
-0.73 | -0.97 | 0.18 | 0.01 | 50 | 0 | [17 25] | 0.59 | 50 | 0 | [16 27] | 0.6 | 50 | 0 | [18 28] | 0.35
-0.72 | -0.99 | 0.18 | 0.01 | 50 | 0 | [17 25] | 0.52 | 50 | 0 | [16 27] | 0.64 | 50 | 0 | [18 28] | 0.25
-0.73 | -0.99 | 0.18 | 0.01 | 50 | 0 | [17 25] | 0.65 | 50 | 0 | [16 27] | 0.55 | 50 | 0 | [18 28] | 0.31
-0.72 | -1.01 | 0.18 | 0.01 | 50 | 0 | [17 25] | 0.56 | 50 | 0 | [16 27] | 0.6 | 50 | 0 | [18 28] | 0.27
-0.73 | -1.01 | 0.18 | 0.01 | 50 | 0 | [17 25] | 0.68 | 50 | 0 | [16 27] | 0.52 | 50 | 0 | [18 28] | 0.3
-0.72 | -1.03 | 0.18 | 0.01 | 50 | 0 | [17 25] | 0.61 | 50 | 0 | [16 27] | 0.59 | 50 | 0 | [18 28] | 0.34
-0.73 | -1.03 | 0.18 | 0.01 | 50 | 0 | [17 25] | 0.67 | 50 | 0 | [16 27] | 0.55 | 50 | 0 | [18 28] | 0.32
-0.7 | -1.04 | 0.18 | 0.16 | 50 | 0 | [17 25] | 0.79 | 50 | 0 | [17 27] | 0.96 | 50 | 0 | [15 27] | 0.52
-0.71 | -1.04 | 0.18 | 0.16 | 50 | 0 | [17 25] | 0.71 | 50 | 0 | [17 27] | 0.81 | 50 | 0 | [15 27] | 0.4
-0.72 | -1.04 | 0.18 | 0.16 | 50 | 0 | [17 25] | 0.79 | 50 | 0 | [17 27] | 0.67 | 50 | 0 | [15 27] | 0.33
-0.7 | -1.05 | 0.18 | 0.18 | 50 | 0 | [16 25] | 0.76 | 50 | 0 | [17 27] | 1.0 | 50 | 0 | [16 27] | 0.54
-0.71 | -1.05 | 0.18 | 0.18 | 50 | 0 | [16 25] | 0.66 | 50 | 0 | [17 27] | 0.88 | 50 | 0 | [16 27] | 0.42
-0.72 | -1.05 | 0.18 | 0.18 | 50 | 0 | [16 25] | 0.68 | 50 | 0 | [17 27] | 0.78 | 50 | 0 | [16 27] | 0.37
-0.7 | -1.06 | 0.18 | 0.2 | 50 | 0 | [16 30] | 0.74 | 50 | 0 | [16 30] | 1.05 | 50 | 0 | [16 28] | 0.45