# On the mixing between flavor singlets in lattice gauge theories coupled to
matter fields in multiple representations

Ed Bennett  0000-0002-1678-6701<EMAIL_ADDRESS>Swansea Academy of
Advanced Computing, Swansea University (Bay Campus), Fabian Way, SA1 8EN
Swansea, Wales, United Kingdom  Niccolò Forzano  0000-0003-0985-8858
<EMAIL_ADDRESS>Department of Physics, Faculty of Science and
Engineering, Swansea University, Singleton Park, SA2 8PP, Swansea, Wales, UK
Deog Ki Hong  0000-0002-3923-4184<EMAIL_ADDRESS>Department of Physics,
Pusan National University, Busan 46241, Korea  Extreme Physics Institute,
Pusan National University, Busan 46241, Korea  Ho Hsiao  0000-0002-8522-5190
<EMAIL_ADDRESS>Institute of Physics, National Yang Ming Chiao
Tung University, 1001 Ta-Hsueh Road, Hsinchu 30010, Taiwan  Jong-Wan Lee
0000-0002-4616-2422<EMAIL_ADDRESS>Particle Theory and Cosmology Group,
Center for Theoretical Physics of the Universe, Institute for Basic Science
(IBS), Daejeon, 34126, Korea  C.-J. David Lin  0000-0003-3743-0840
<EMAIL_ADDRESS>Institute of Physics, National Yang Ming Chiao Tung
University, 1001 Ta-Hsueh Road, Hsinchu 30010, Taiwan  Centre for Theoretical
and Computational Physics, National Yang Ming Chiao Tung University, 1001 Ta-
Hsueh Road, Hsinchu 30010, Taiwan  Centre for High Energy Physics, Chung-Yuan
Christian University, Chung-Li 32023, Taiwan  Biagio Lucini
0000-0001-8974-8266<EMAIL_ADDRESS>Swansea Academy of Advanced
Computing, Swansea University (Bay Campus), Fabian Way, SA1 8EN Swansea,
Wales, United Kingdom  Department of Mathematics, Faculty of Science and
Engineering, Swansea University (Bay Campus), Fabian Way, SA1 8EN Swansea,
Wales, United Kingdom  Maurizio Piai  0000-0002-2251-0111<EMAIL_ADDRESS>Department of Physics, Faculty of Science and Engineering, Swansea University,
Singleton Park, SA2 8PP, Swansea, Wales, UK  Davide Vadacchino
0000-0002-5783-5602<EMAIL_ADDRESS>Centre for Mathematical
Sciences, University of Plymouth, Plymouth, PL4 8AA, United Kingdom  Fabian
Zierler  0000-0002-8670-4054<EMAIL_ADDRESS>Department of
Physics, Faculty of Science and Engineering, Swansea University, Singleton
Park, SA2 8PP, Swansea, Wales, UK

###### Abstract

We provide the first extensive, numerical study of the non-trivial problem of
mixing between flavor-singlet composite states emerging in strongly coupled
lattice field theories with matter field content consisting of fermions
transforming in different representations of the gauge group. The theory of
interest is the minimal candidate for a composite Higgs model that also
accommodates a mechanism for top partial compositeness: the $Sp(4)$ gauge
theory coupled to two (Dirac) fermions transforming as the fundamental and
three as the two-index antisymmetric representation of the gauge group,
respectively. We apply an admixture of APE and Wuppertal smearings, as well as
the generalized eigenvalue problem approach, to two-point functions involving
flavor-singlet mesons, for ensembles having time extent longer than the space
extent. We demonstrate that, in the region of lattice parameter space
accessible to this study, both masses and mixing angles can be measured
effectively, despite the presence of (numerically noisy) contributions from
disconnected diagrams.

††preprint: CTPU-PTC-24-12††preprint: PNUTP-24/A03

###### Contents

  1. I Introduction
  2. II Elements of field theory
  3. III Numerical Strategy
    1. III.1 Decay constants and mixing angles
  4. IV Results
    1. IV.1 Meson masses
    2. IV.2 Mixing angle
  5. V Summary and Outlook

##  I Introduction

A distinguishing feature of a broad class of models of new physics featuring
composite dynamics is the emergence in their spectrum of several scalar
singlets, associated with spontaneously broken, approximate, anomalous Abelian
continuous symmetries Belyaev:2016ftv . In the context of composite Higgs
models (CHMs) Kaplan:1983fs ; Georgi:1984af ; Dugan:1984hq  that also
implement top partial compositeness (TPC) Kaplan:1991dc ,111An incomplete
catalog of such models, in which the Higgs fields of the Standard Model emerge
as pseudo-Nambu-Goldstone bosons (PNGBs) associated with the spontaneous
breaking of approximate global symmetries, and of studies of their
phenomenology, includes for example Refs. Barbieri:2007bh ; Lodone:2008yy ;
Gripaios:2009pe ; Mrazek:2011iu ; Marzocca:2012zn ; Grojean:2013qca ;
Barnard:2013zea ; Cacciapaglia:2014uja ; Ferretti:2014qta ; Arbey:2015exa ;
Cacciapaglia:2015eqa ; vonGersdorff:2015fta ; Feruglio:2016zvt ;
DeGrand:2016pgq ; Fichet:2016xvs ; Galloway:2016fuo ; Agugliaro:2016clv ;
Bizot:2016zyu ; Csaki:2017cep ; Chala:2017sjk ; Golterman:2017vdj ;
Csaki:2017jby ; Alanne:2017rrs ; Alanne:2017ymh ; Sannino:2017utc ;
Alanne:2018wtp ; Bizot:2018tds ; Cai:2018tet ; Agugliaro:2018vsu ;
Cacciapaglia:2018avr ; Gertov:2019yqo ; Ayyar:2019exp ; Cacciapaglia:2019ixa ;
BuarqueFranzosi:2019eee ; Cacciapaglia:2019dsq ; Cacciapaglia:2020vyf ;
Dong:2020eqy ; Cacciapaglia:2021uqh ; Ferretti:2022mpy ; Banerjee:2022izw ;
Cai:2022zqu ; Cacciapaglia:2024wdn . Implementations of similar ideas within
gauge-gravity dualities have been presented for example in Refs.
Contino:2003ve ; Agashe:2004rs ; Agashe:2005dk ; Agashe:2006at ;
Contino:2006qr ; Falkowski:2008fz ; Contino:2010rs ; Contino:2011np  and, more
recently, in Refs. Erdmenger:2020lvq ; Erdmenger:2020flu ; Elander:2020nyd ;
Elander:2021bmt ; Erdmenger:2023hkl ; Elander:2023aow ; Erdmenger:2024dxf , in
the bottom-up approach, and Ref. Elander:2021kxk  in the top-down approach to
holography. the multiplicity of such flavor-singlet states arises because the
short-distance origin of the dynamics involves two separate matter sectors
coupled to the same gauge theory.222This requirement may not apply in $SU(3)$
theories, for which the ordinary baryons may provide the origin of the TPC
fields Vecchi:2015fma — see also the constructions in Refs. Appelquist:2020bqj
; Appelquist:2022qgl , which make use of ideas from Refs. Ma:2015gra ;
BuarqueFranzosi:2018eaj  Examples can be found in the reviews Panico:2015jxa ;
Witzel:2019jbe ; Cacciapaglia:2020kgq , and the tables in Refs.
Ferretti:2013kya ; Ferretti:2016upr ; Cacciapaglia:2019bqz . Closely related
classes of theories are also known to admit an application as new models of
dark matter with strongly coupled origin, for example along the lines of Refs.
Hochberg:2014dra ; Hochberg:2014kqa ; Hochberg:2015vrg ; Hansen:2015yaa ;
Bernal:2017mqb ; Berlin:2018tvf ; Bernal:2019uqr ; Beylin:2019gtw ;
Tsai:2020vpi ; Kondo:2022lgg ; Bernal:2015xba ; Pomper:2024otb ;
Appelquist:2024koa .

The existence of mixing terms in the effective field theory (EFT) description
of such flavor-singlets is ensured by the chiral anomaly, which breaks the
symmetry even when no other explicit symmetry-breaking terms are present. The
phenomenology associated with such scalar singlets is determined by
coefficients in the EFT treatment that have dynamical origin—related to that
of axion-like particles (ALPs) Brivio:2017ije ; Bellazzini:2017neg ;
Bauer:2017ris . This is the case in the dark matter context, as in the
collider phenomenology one Belyaev:2016ftv ; Cai:2015bss ; Belyaev:2015hgo ;
DeGrand:2016pgq ; Cacciapaglia:2017iws ; Cacciapaglia:2019bqz . Hence, gaining
non-perturbative information about them is essential in order to plan and
perform an effective program of experimental searches for new particles in
both visible and dark sectors.

In recent years, extensive numerical investigations have been developed in the
context of extensions to the Standard Model, based on symplectic gauge groups
$SU(2)=Sp(2)$ Hietanen:2014xca ; Detmold:2014kba ; Drach:2015epq ;
Arthur:2016dir ; Arthur:2016ozw ; Pica:2016zst ; Lee:2017uvl ; Drach:2017jsh ;
Drach:2017btk ; Drach:2020wux ; Drach:2021uhl ; Bowes:2023ihh  and $Sp(4)$
Bennett:2017kga ; Lee:2018ztv ; Bennett:2019jzz ; Bennett:2019cxd ;
Lucini:2021xke ; Bennett:2021mbw ; Bennett:2022yfa ; Lee:2022xbp ;
Bennett:2023wjw ; Bennett:2023rsl ; Bennett:2023gbe ; Mason:2023ixv ;
Zierler:2021cfa ; Kulkarni:2022bvh ; Bennett:2023mhh ; Bennett:2023qwx ;
Hsiao:2023nyn . Among the gauge theories with fermions in two distinct
representations, lattice studies have been performed in $SU(2)$
Bergner:2020mwl ; Bergner:2021ivi , as well as in $Sp(4)$ gauge theories
Bennett:2022yfa ; Hsiao:2022kxf ; Bennett:2023mhh ; Hsiao:2023nyn . In a
parallel development, studies of theories with fermions transforming in
multiple representation of $SU(4)$ have also appeared Ayyar:2017qdf ;
Ayyar:2018zuk ; Ayyar:2018ppa ; Ayyar:2018glg ; Cossu:2019hse ; Shamir:2021frg
; Lupo:2021nzv ; DelDebbio:2022qgu ; Hasenfratz:2023sqa . These lattice
studies, motivated by new physics considerations, focus predominantly on the
study of flavored mesons. 333The literature on the $SU(3)$ theory with $N_{\rm
f}=8$ Dirac fermions in the fundamental representation LatKMI:2014xoh ;
Appelquist:2016viq ; LatKMI:2016xxi ; Gasbarro:2017fmi ;
LatticeStrongDynamics:2018hun ; LSD:2023uzj ; LatticeStrongDynamics:2023bqp ;
Ingoldby:2023mtf  or $n_{\rm s}=2$ sextets Fodor:2012ty ; Fodor:2015vwa ;
Fodor:2016pls ; Fodor:2017nlp ; Fodor:2019vmw ; Fodor:2020niv , stands out as
it presents extensive studies of the flavor singlet states. But these studies
focus on the phenomenology of the dilaton, the Goldstone boson associated with
scale invariance, which has different quantum numbers from the PNGBs
associated with Abelian internal symmetries. This lattice subfield is only
beginning to enter its high precision phase, after the necessary exploratory
period. Calculating correlation functions involving singlet mesons requires
specific technology, developed to handle effectively the contributions to
observables coming from disconnected diagrams, which potentially reduce the
signal-to-noise ratio. Only few studies in the context of BSM physics outside
of $SU(3)$ theories exist Drach:2021uhl ; Arthur:2016ozw ; Bennett:2023rsl .
To the best of our knowledge, this paper is the first one to present a
systematic lattice study of the effects of mixing between different flavor
singlets in models of BSM physics. Our aim is to demonstrate the feasibility
of such an endeavor.

In this publication, we present the first results of the calculation of the
mass spectrum of spin-0, flavor singlet states with negative parity, in the
presence of two distinct Abelian PNGBs associated with two $U(1)$ factors. One
of them is expected to be anomalously broken, thus giving an extra
contribution to the mass of one PNGB Belyaev:2016ftv . The lattice ensembles
we study are obtained in the CHM candidate of Ref. Barnard:2013zea : the
$Sp(4)$ gauge theory coupled to $N_{\rm f}=2$ (Dirac) fermions transforming
according to the fundamental, as well as $N_{\rm as}=3$ fermions transforming
as the 2-index antisymmetric representation of the group. We use the ensembles
described in detail in Ref. Bennett:2024cqv , originally produced to study the
spectral density of flavored meson correlators, within the Hansen-Lupo-Tantalo
method Hansen:2019idp ; DelDebbio:2022qgu . Building on the results discussed
in the Appendix of Ref. Bennett:2023rsl , we introduce Wuppertal Gusken:1989ad
; Gusken:1989qx ; Roberts:2012tp ; Alexandrou:1990dq  and APE smearing
APE:1987ehd ; Falcioni:1984ei . We then implement a variation of the
Generalized Eigenvalue Problem (GEVP) to take into account mixing effects
between states that appear in two-point functions involving the two distinct
meson singlets in the theory.

The paper is organized as follows. We very briefly introduce the continuum and
lattice theories of interest in Sects. II and III, respectively. We provide
the minimal amount of detail to make the narrative self-contained, and refer
the Reader to Ref. Bennett:2024cqv  for technical details and in-depth
discussions. Nevertheless, we extensively describe the properties of the
flavor-singlet meson states of interest, with particular reference to decay
constants and mixing angles, as well as our analysis techniques, in Sect.
III.1. Our results are then reported and critically discussed in Sect. IV. A
final summary in Sect. V contains also a brief outlook on future avenues for
research.

##  II Elements of field theory

We study the $Sp(4)$ gauge theory coupled to two Dirac fermions transforming
in the fundamental representation and three Dirac fermions in the
antisymmetric representation of the gauge group. The Lagrangian density, in
Minkowski space, is given by

$\displaystyle\mathcal{L}=-\frac{1}{2}{\rm
Tr}G_{\mu\nu}G^{\mu\nu}+\sum_{I=1}^{2}\bar{Q}^{I}\left(i\gamma^{\mu}D_{\mu}-m^{\rm
f}\right)Q^{I}+\sum_{K=1}^{3}\bar{\Psi}^{k}\left(i\gamma^{\mu}D_{\mu}-m^{\rm
as}\right)\Psi^{k},$ (1)

where $Q^{I}$, with $I=1,\,2$, are the fundamental Dirac fermions and
$\Psi^{k}$, with $k=1,\,2,\,3$, are the antisymmetric Dirac fermions with
their respective degenerate masses $m^{\rm f}$ and $m^{\rm as}$. The field-
strength tensor for the $Sp(4)$ theory is denoted as $G_{\mu\nu}\equiv
G_{\mu\nu}^{a}T^{a}$, with the Hermitian generators normalized so that ${\rm
Tr}T^{a}T^{b}=\frac{1}{2}\delta^{ab}$, for $a,\,b=1,\,\cdots,\,10$.

In the limit of vanishing fermion masses, the Lagrangian density is invariant
under the enhanced global internal symmetry with group $U(1)\times U(1)\times
SU(6)\times SU(4)$. This symmetry of the Lagrangian is not reflected in low-
energy physics; it is broken spontaneously by the fermion condensates. The
breaking pattern is governed by the realness of the antisymmetric
representation and the pseudo-realness of the fundamental representation;
$SU(6)$ breaks to its $SO(6)$ subgroup, while $SU(4)$ breaks to $Sp(4)$
Kosower:1984aw . At the same time, the diagonal and degenerate mass terms
induce explicit symmetry breaking, according to the same, aligned symmetry
breaking pattern. For small fermion masses, the spontaneous breaking leads to
20 PNGBs in the antisymmetric sector and 5 PNGBs in the fundamental sector
Kogut:2000ek . In the CHM context, one can choose appropriate embeddings for
the standard-model gauge group so that part of the global symmetry is (weakly)
gauged. For example, in this way the five PNGBs of the fundamental sector can
be identified with the Higgs doublet and one additional singlet
Barnard:2013zea . In this paper we consider the $Sp(4)$ gauge theory in
complete isolation, ignoring the effects due to coupling to other external
sectors.

In theories with only one species of fermions, the global U(1) is anomalous,
and no Goldstone bosons, related to its breaking, appear. In the large-$N$
limit, the associated pseudoscalar flavor-singlet, $\eta^{\prime}$, becomes
the would-be Goldstone mode, as the effects of the axial anomaly are
suppressed in this limit Witten:1978bc ; Witten:1979vv ; Veneziano:1979ec . In
the presence of fermions in two distinct representations, the axial anomaly
can only break one (linear combination) of the two global $U(1)$ symmetry
factors. Hence, an additional PNGB is expected to appear at small fermion
masses, in the pseudoscalar flavor-singlet sector, the lightest mass
eigenstate in this channel. The other pseudoscalar flavor-singlet should
acquire a larger mass through the axial anomaly and show up as an excited
state in the same channel Belyaev:2015hgo .

In this paper, we perform the first measurement of flavor-singlet ground and
excited states in a theory with multiple fermion representations, which we
call ${\eta^{\prime}_{l}}$ and ${\eta^{\prime}_{h}}$444Note, that this
corresponds to the states $a$ and $\eta^{\prime}$ in the notation of Ref.
Belyaev:2016ftv . The theory of interest has $Sp(4)$ gauge group, and explicit
mass terms for the fermions, which are not small. Hence, we expect both the
aforementioned flavor singlets to be comparatively heavy, as the mass terms
contribute together with the anomaly to the explicit symmetry breaking. One
can draw an analogy between this regime and the $\eta-\eta^{\prime}$ system in
the theory of strong nuclear interactions, QCD 555Note, that this analogy has
its limits. In QCD, the mixing between the $\eta$ and the $\eta^{\prime}$ is
introduced by an explicit breaking of the global flavor symmetry. For the
mixed representations system, this is not the case, and both states are
pseudoscalar singlets even without additional symmetry breaking.. In this
case, the $\eta$ meson is a PNGB accommodated within the approximate $SU(3)$
flavor symmetry, while the $\eta^{\prime}$ is the would-be-PNGB associated
with the anomalously broken global $U(1)$. Within QCD, both the mass spectrum
and the mixing between those states have been studied in detail Christ:2010dd
; Dudek:2011tt ; Ottnad:2012fv ; Michael:2013gka ; Bali:2014pva ;
Ottnad:2015hva ; Ottnad:2017bjt ; Bali:2021qem ; CSSMQCDSFUKQCD:2021rvs ;
Bali:2021vsa ; Simeth:2022fuq .

##  III Numerical Strategy

Our numerical analysis is based on ensembles generated using the Wilson
plaquette action for the gauge sector and standard Wilson fermions for both
the fundamental and antisymmetric fermions Wilson:1974sk . We consider
hypercubic lattices with a volume $L^{3}\times T=a^{4}(N_{s}^{3}\times
N_{t})$, where $a$ is the lattice spacing. The ensembles have been generated
using the Grid software library Boyle:2015tjk ; Boyle:2016lbp ;
Yamaguchi:2022feu  which has been extended to $Sp(2N)$ gauge theories
Bennett:2023gbe . We study five ensembles, at three different values of the
bare fundamental fermion mass, $am_{0}^{\rm f}=-0.7,\,-0.71,\,0.72$, while we
keep the antisymmetric fermion mass fixed, $am_{0}^{\rm as}=-1.01$. We
consider only one value of the inverse gauge coupling, $\beta=8/g^{2}=6.5$.
The measurements of the mesonic correlation function are performed using the
HiRep code DelDebbio:2008zf ; HiRepSUN ; HiRepSpN . The configurations have
been converted to the HiRep binary format using the GLU library GLU . We set
the overall scale using the Wilson flow Luscher:2010iy , the lattice
implementation of the gradient flow Luscher:2011bx ; Luscher:2013vga , by
calculating the gradient flow quantity, $w_{0}$ BMW:2012hcm , in units of the
lattice spacing, $a$. For further information on the generation of these
ensembles we refer to Ref. Bennett:2024cqv . Detailed characterization of the
five ensembles is summarized in Tab. 1.

Table 1: Ensembles studied in this paper. For each of the five ensembles, we list the value of the inverse lattice coupling, $\beta$, the masses of the two fermion species, $am_{0}^{\rm as}$ and $am_{0}^{\rm f}$, the extent of the lattice in time, $N_{t}$, and space directions, $N_{s}$, the number of thermalization steps, $N_{\rm therm}$, discarded from the analysis, the number of complete sweeps, $n_{\rm skip}$, discarded between configurations retained in the analysis, the number of configurations constituting the ensemble and used in the analysis, $N_{\rm conf}$, the average plaquette in the ensemble, $\langle P\rangle$, the value of the Wilson flow scale, $w_{0}/a$, the topological autocorrelation time in configuration units, $\tau_{\rm int}^{Q}$, and the average topological charge in the ensemble, $\bar{Q}$. Details, explanations and discussions can be found in Ref. Bennett:2024cqv .  Label | $\beta$ | $am_{0}^{\rm as}$ | $am_{0}^{\rm f}$ | $N_{t}$ | $N_{s}$ | $N_{\rm therm}$ | $n_{\rm skip}$ | $N_{\rm conf}$ | $\langle P\rangle$ | $w_{0}/a$ | $\tau_{\rm int}^{Q}$ | $\bar{Q}$
---|---|---|---|---|---|---|---|---|---|---|---|---
M1 | 6.5 | -1.01 | -0.71 | 48 | 20 | 3006 | 14 | 479 | 0.585172(16) | 2.5200(50) | 6.9(2.4) | 0.38(12)
M2 | 6.5 | -1.01 | -0.71 | 64 | 20 | 1000 | 28 | 698 | 0.585172(12) | 2.5300(40) | 7.1(2.1) | 0.58(14)
M3 | 6.5 | -1.01 | -0.71 | 96 | 20 | 4000 | 26 | 436 | 0.585156(13) | 2.5170(40) | 6.4(3.3) | -0.60(19)
M4 | 6.5 | -1.01 | -0.70 | 64 | 20 | 1000 | 20 | 709 | 0.584228(12) | 2.3557(31) | 10.6(4.8) | -0.31(19)
M5 | 6.5 | -1.01 | -0.72 | 64 | 32 | 3020 | 20 | 295 | 0.5860810(93) | 2.6927(31) | 12.9(8.2) | 0.80(33)

We determine the ground state energy as well as the energy of the first
excited state for the system of interest by using a variational analysis
Michael:1985ne ; Luscher:1990ck . To this purpose, we measure the (zero-
momentum) correlation matrix for a set of interpolating operators,
$\left\\{O_{i}\right\\}$, which can be expanded in the Hamiltonian eigenstates
as

$\displaystyle
C_{ij}(t-t^{\prime})=\langle\bar{O}_{i}(t)O_{j}(t^{\prime})\rangle=\sum_{n}\frac{1}{2E_{n}}\langle
0|\bar{O}_{i}|n\rangle\langle n|O_{j}|0\rangle e^{-E_{n}(t-t^{\prime})}.$ (2)

The eigenvalues of $C_{ij}$ are obtained by solving the generalized eigenvalue
problem (GEVP) with eigenvalues $\lambda_{n}(t,t_{0})$ and eigenvectors
$v_{n}(t,t_{0})$ given by

$\displaystyle C(t)v_{n}(t,t_{0})=\lambda_{n}(t,t_{0})C(t_{0})v_{n}(t,t_{0}).$
(3)

Assuming that the states created by each of the operators in the variational
basis has a sufficient overlap with the first $M$ eigenstates in the selected
channel, the leading behavior of the $m^{\rm th}$ eigenvalue at large $t$ for
fixed but sufficiently large $t_{0}$ can be written as Blossier:2009kd

$\displaystyle\lambda_{m}(t\to\infty,t_{0})=e^{-E_{m}(t-t_{0})}+\mathcal{O}\left(e^{-(E_{m+1}-E_{m})t}\right).$
(4)

In summary, the solution of the above GEVP enables us to find the operator
that produces states with the maximal overlap with the ground state and the
first excited state, and thus to access their energies.

The two meson operators of interest in the pseudoscalar singlet channel are
coupled, respectively, to $N_{\rm f}$ fundamental fermions and $N_{\rm as}$
antisymmetric fermions, and they are given by the following666We note, that in
principle a contribution from the $J^{P}=0^{-}$ glueball state is also
allowed. However, an investigation of $\eta^{\prime}$-glueball mixing in two-
flavor QCD showed no sizeable contributions from the glueball state
Jiang:2022ffl . Studies of the quenched $Sp(4)$ theory suggest that indeed the
$0^{-}$ glueballs is very heavy with respect to the scale of the vector
mesons—see Fig. 13 of Ref. Bennett:2023mhh , for example. :

$\displaystyle O_{\eta^{\rm as}}(x)$ $\displaystyle=\frac{1}{\sqrt{N_{\rm
as}}}\sum_{k=1}^{N_{\rm as}}\bar{\Psi}^{k}(x)\gamma_{5}\Psi^{k}(x)\,,$ (5)
$\displaystyle O_{\eta^{\rm f}}(x)$ $\displaystyle=\frac{1}{\sqrt{N_{\rm
f}}}\sum_{I=1}^{N_{\rm f}}\bar{Q}^{I}(x)\gamma_{5}Q^{I}(x)\,.$ (6)

Performing the required Wick contractions for the correlation matrix resulting
from Eq. (2), for operators at lattice sites $x$ and $y$, we find the
diagrammatic expression

$\displaystyle C_{11}(x,y)\equiv\langle\bar{O}_{\eta^{\rm as}}(x)O_{\eta^{\rm
as}}(y)\rangle=-\vbox{\hbox{\includegraphics[scale={0.5},page=31]{figures/contractions.pdf}}}+N_{\rm
as}~{}\vbox{\hbox{\includegraphics[scale={0.5},page=33]{figures/contractions.pdf}}},$
(7) $\displaystyle C_{22}(x,y)\equiv\langle\bar{O}_{\eta^{\rm
f}}(x)O_{\eta^{\rm
f}}(y)\rangle=-\vbox{\hbox{\includegraphics[scale={0.5},page=30]{figures/contractions.pdf}}}+N_{\rm
f}~{}\vbox{\hbox{\includegraphics[scale={0.5},page=34]{figures/contractions.pdf}}},$
(8) $\displaystyle C_{12}(x,y)=C_{21}(x,y)\equiv\langle\bar{O}_{\eta^{\rm
f}}(x)O_{\eta^{\rm as}}(y)\rangle=+\sqrt{N_{\rm as}N_{\rm
f}}\vbox{\hbox{\includegraphics[scale={0.5},page=32]{figures/contractions.pdf}}},$
(9)

where dashed lines denote the contraction of two antisymmetric fermion fields
and solid lines denote the contraction of two fundamental fermion fields.
Unlike the singlet-octet basis in QCD for the $\eta$ and $\eta^{\prime}$
mesons, the cross-correlator does not vanish in the limit of vanishing fermion
masses, hence we expect sizeable mixing effects for moderate and light fermion
masses.

The disconnected diagrams in Eqs. (7)-(9) are challenging for lattice
calculations, as they introduce a smaller signal-to-noise ratio than the
connected diagrams—that are also present in non-singlet mesons. Our objective
is to improve the signal by enlarging the variational basis through smearing
techniques. The general principle at work is that one expects the adoption of
larger variational basis in the GEVP analysis to suppress the effects of
excited state contamination at smaller Euclidean times, where the signal-to-
noise ratio is substantially better.

We implement Wuppertal smearing of the fermionic operators Gusken:1989ad ;
Gusken:1989qx ; Roberts:2012tp ; Alexandrou:1990dq , in conjunction with APE
smearing for the gauge fields APE:1987ehd ; Falcioni:1984ei . We follow the
approach used in Ref. Bennett:2023mhh  for the connected diagrams, and apply
the smearing function to point sources. For the disconnected diagrams, we use
spin-diluted stochastic sources Foley:2005ac , with $Z_{2}\times Z_{2}$ noise
Dong:1993pk , and perform the measurements on $n_{\rm src}=64$ stochastic
samples. As pointed out in Ref. Bali:2021qem , it is possible to measure the
disconnected loops at several smearing levels using only one inversion of the
Dirac operator. Doing so comes at the cost of potentially introducing a bias
in the construction of the full correlation function, following the procedure
deployed in Ref. Arthur:2016ozw . The construction of a completely unbiased
estimator would require to perform a separate inversion for every smearing
level, but this would go beyond the purposes of this paper. We apply APE
smearing for every measurement. We choose the smearing parameter as
$\alpha_{\rm APE}=0.4$, with $N_{\rm APE}=50$ smearing steps.

The variational basis is composed of the operators defined in Eq. (5) and Eq.
(6) together with their Wuppertal-smeared versions, with $N^{\rm
smear}=10,\,20,\,\dots,\,80$ smearing steps, and $\epsilon_{\rm f}=0.2$ for
the fundamental fermions, or $\epsilon_{\rm as}=0.12$ for the antisymmetric
fermions (we follow the notation of Ref. Bennett:2023mhh , to which we refer
the reader for technical details). In total, we have a variational basis of 18
operators.

For a finite sample, the correlation functions of pseudoscalar flavor-singlet
mesons (or any other quantity that has the same quantum numbers as the
topological charge density) can acquire an additional, constant contribution
Aoki:2007ka . In order to remove this constant we consider the central
difference approximation to the derivative of the correlation matrix, as
proposed in Ref. Umeda:2007hy  and use of the following redefinition:

$\displaystyle
C_{ij}(t)\to\tilde{C}_{ij}(t)=\frac{C_{ij}(t-1)-C_{ij}(t+1)}{2}\,.$ (10)

Doing so changes the periodicity of the correlation matrix with respect to the
lattice midpoint, $t=T/2$, from periodic to anti-periodic Bennett:2023rsl .

After performing the GEVP analysis, we fit the eigenvalues to an exponentially
decaying function according to Eq. (4). We first visually examine the
effective mass, $m_{\rm eff}(t)$, defined implicitly as

$\displaystyle\frac{\lambda(t-1)}{\lambda(t)}=\frac{e^{-m_{\rm
eff}(t)\cdot(T-t+1)}\pm e^{-m_{\rm eff}(t)\cdot(t-1)}}{e^{-m_{\rm
eff}(t)\cdot(T-t)}\pm e^{-m_{\rm eff}(t)\cdot t}}\,.$ (11)

We solve for the effective mass numerically using a root finding algorithm. It
exhibits a plateau when the eigenvalue is dominated by its leading exponential
term at large Euclidean time. The sign in Eq. (11) is chosen to be positive
for symmetric correlation functions, and negative for antisymmetric ones. We
then perform a fit to the eigenvalue using constrained curve fitting
Lepage:2001ym , utilizing the corrfitter package peter_lepage_2021_5733391 .

Finally, in order to gauge the physical meaning of the results of our
analysis, we use the same ensembles and processes also to perform the
measurement of the mass of the lightest flavored mesons, in the pseudoscalar
and vector channel, for mesons constructed with either species of fermions. We
denote as ${\rm PS}~{}({\rm ps})$ flavored pseudoscalar mesons made of
fermions transforming in the fundamental, ${\rm f}$, (antisymmetric, ${\rm
as}$) representation, and as ${\rm V}~{}({\rm v})$ flavored vector mesons with
the same composition. We refer the reader to Ref. Bennett:2024cqv  for more
details about the operators, and the spectrum of flavored mesons, that are not
central to the results and discussions presented in this paper.

###  III.1 Decay constants and mixing angles

It is interesting to extract the mixing angle between the lightest states
sourced by the operators in Eqs. (5) and (6). For the purposes of this paper,
we assume that the state mixing is given by the mixing of the decay constants.
We call ${\eta^{\prime}_{l}}$ and ${\eta^{\prime}_{h}}$, respectively, the
lightest and next-to-lightest states identified in the GEVP analysis. We
parameterize the non-renormalized matrix elements of axial-vector currents
with the pseudoscalar singlets as follows:

$\displaystyle\begin{pmatrix}F_{\rm
f}^{{\eta^{\prime}_{l}}}p_{\mu}^{{\eta^{\prime}_{l}}}&F_{\rm
as}^{{\eta^{\prime}_{l}}}p_{\mu}^{{\eta^{\prime}_{l}}}\\\ F_{\rm
f}^{{\eta^{\prime}_{h}}}p_{\mu}^{{\eta^{\prime}_{h}}}&F_{\rm
as}^{{\eta^{\prime}_{h}}}p_{\mu}^{{\eta^{\prime}_{h}}}\end{pmatrix}\equiv\begin{pmatrix}\langle
0|\frac{1}{\sqrt{N_{\rm f}}}\sum_{i=1}^{N_{\rm
f}}\bar{Q}_{i}(x)\gamma_{\mu}\gamma_{5}Q_{i}(x)|{\eta^{\prime}_{l}}\rangle&\langle
0|\frac{1}{\sqrt{N_{\rm as}}}\sum_{i=1}^{N_{\rm
as}}\bar{\Psi}_{i}(x)\gamma_{\mu}\gamma_{5}\Psi_{i}(x)|{\eta^{\prime}_{l}}\rangle\\\
\langle 0|\frac{1}{\sqrt{N_{\rm f}}}\sum_{i=1}^{N_{\rm
f}}\bar{Q}_{i}(x)\gamma_{\mu}\gamma_{5}Q_{i}(x)|{\eta^{\prime}_{h}}\rangle&\langle
0|\frac{1}{\sqrt{N_{\rm as}}}\sum_{i=1}^{N_{\rm
as}}\bar{\Psi}_{i}(x)\gamma_{\mu}\gamma_{5}\Psi_{i}(x)|{\eta^{\prime}_{h}}\rangle\end{pmatrix}\,.$
(12)

In this general relation, we define the relevant decay constants computed with
an operator ($op$) and a state ($s$) as $F^{s}_{op}$. By setting $\mu=0$ we
obtain the familiar relation

$\displaystyle\begin{pmatrix}F_{\rm
f}^{{\eta^{\prime}_{l}}}p_{0}^{{\eta^{\prime}_{l}}}&F_{\rm
as}^{{\eta^{\prime}_{l}}}p_{0}^{{\eta^{\prime}_{l}}}\\\ F_{\rm
f}^{{\eta^{\prime}_{h}}}p_{0}^{{\eta^{\prime}_{h}}}&F_{\rm
as}^{{\eta^{\prime}_{h}}}p_{0}^{{\eta^{\prime}_{h}}}\end{pmatrix}=\begin{pmatrix}F_{\rm
f}^{{\eta^{\prime}_{l}}}m_{{\eta^{\prime}_{l}}}&F_{\rm
as}^{{\eta^{\prime}_{l}}}m_{{\eta^{\prime}_{l}}}\\\ F_{\rm
f}^{{\eta^{\prime}_{h}}}m_{{\eta^{\prime}_{h}}}&F_{\rm
as}^{{\eta^{\prime}_{h}}}m_{{\eta^{\prime}_{h}}}\end{pmatrix}.$ (13)

These non-renormalized local matrix elements can be obtained from the
eigenvectors of the GEVP analysis, for a variational basis without Wuppertal
smearing. This matrix is parameterized as follows, in terms of two mixing
angles, and two decay constants Kaiser:1998ds :

$\displaystyle\begin{pmatrix}F_{\rm f}^{{\eta^{\prime}_{l}}}&F_{\rm
as}^{{\eta^{\prime}_{l}}}\\\ F_{\rm f}^{{\eta^{\prime}_{h}}}&F_{\rm
as}^{{\eta^{\prime}_{h}}}\end{pmatrix}\equiv\begin{pmatrix}F_{{\eta^{\prime}_{l}}}\cos{\phi_{\eta^{\prime}_{l}}}&F_{{\eta^{\prime}_{l}}}\sin{\phi_{\eta^{\prime}_{l}}}\\\
-F_{{\eta^{\prime}_{h}}}\sin{\phi_{\eta^{\prime}_{h}}}&F_{{\eta^{\prime}_{h}}}\cos{\phi_{\eta^{\prime}_{h}}}\end{pmatrix}\,,$
(14)

in analogy to the $\eta-\eta^{\prime}$ mixing system in the singlet-octet
basis of QCD.

For this work, we choose a variational basis of interpolating operators and
the matrix elements of the currents according to Eqs. (5), (6), and (12). The
difference in mass between mesons made of antisymmetric and fundamental
fermions is expected to be large for the ensembles studied in this paper, on
the basis of existing numerical results for other parts of the spectrum
Bennett:2022yfa . In QCD, the two mixing angles are approximately equal,
$\phi_{\eta^{\prime}_{l}}\approx\phi_{{\eta^{\prime}_{h}}}$, at the physical
point Feldmann:1998vh . It is therefore reasonable to expect, that Eq. (14) is
effectively described by a single mixing angle,
$\phi\equiv\phi_{\eta^{\prime}_{l}}=\phi_{{\eta^{\prime}_{h}}}$, and we elect
to work with this assumption in the following.

A variation of this approach requires to use the pseudoscalar matrix elements.
This alternative approach has also been successfully applied to studying
$\eta-\eta^{\prime}$ mixing in QCD Christ:2010dd ; Dudek:2011tt ;
Ottnad:2012fv ; Michael:2013gka . Again, the matrix elements are parameterized
in terms of two mixing angles, which are expected to be approximately
degenerate:

$\displaystyle\begin{pmatrix}\langle 0|O_{\eta^{\rm
f}}|{\eta^{\prime}_{l}}\rangle&\langle 0|O_{\eta^{\rm
as}}|{\eta^{\prime}_{l}}\rangle\\\ \langle 0|O_{\eta^{\rm
f}}|{\eta^{\prime}_{h}}\rangle&\langle 0|O_{\eta^{\rm
as}}|{\eta^{\prime}_{h}}\rangle\end{pmatrix}=\begin{pmatrix}A_{\rm
f}^{{\eta^{\prime}_{l}}}&A_{\rm as}^{{\eta^{\prime}_{l}}}\\\ A_{\rm
f}^{{\eta^{\prime}_{h}}}&A_{\rm
as}^{{\eta^{\prime}_{h}}}\end{pmatrix}\equiv\begin{pmatrix}A_{{\eta^{\prime}_{l}}}\cos{\phi_{\eta^{\prime}_{l}}}&A_{{\eta^{\prime}_{l}}}\sin{\phi_{\eta^{\prime}_{l}}}\\\
-A_{{\eta^{\prime}_{h}}}\sin{\phi_{\eta^{\prime}_{h}}}&A_{{\eta^{\prime}_{h}}}\cos{\phi_{\eta^{\prime}_{h}}}\end{pmatrix}\,.$
(15)

We can extract the matrix elements of interest from the eigenvectors that
diagonalize the $2\times 2$ matrix of correlation functions defined by the
local operators (i.e., without smearing). Under the assumption that only two
states contribute to the $2\times 2$ correlation matrix, the eigenvectors,
$v_{n}(t,t_{0})$, obtained from the GEVP analysis should be proportional to
time-independent vectors, $u_{n}$ Blossier:2009kd . We use this to determine
the matrix elements $F^{s}_{op}m_{s}$ and $A^{s}_{op}$, by fitting the
effective mixing angle $\phi$ defined as

$\displaystyle-\left(\tan\phi\right)^{2}\equiv\frac{A_{\rm
as}^{{\eta^{\prime}_{l}}}A_{\rm f}^{{\eta^{\prime}_{h}}}}{A_{\rm
as}^{{\eta^{\prime}_{h}}}A_{\rm f}^{{\eta^{\prime}_{l}}}}\,,$ (16)

to a constant value, for a suitable interval $t\in\left[t_{\rm min},t_{\rm
max}\right]$, with $t_{\rm min}>t_{0}$. We use this second procedure to
determine the mixing angles appearing in Eqs. (14) and (15), for every matrix
element.

##  IV Results

Figure 1: Effective masses, $am^{\rm eff}_{{\eta^{\prime}_{l}}}<am^{\rm
eff}_{{\eta^{\prime}_{h}}}$ in units of the lattice spacing, of the ground
state and first excited state in the flavor singlet, pseudoscalar meson sector
of the $Sp(4)$ theory coupled to two Dirac fermions transforming in the
fundamental representation, ${\rm f}$, and three in the antisymmetric, ${\rm
as}$. The effective mass makes use of the eigenvalues extracted in the GEVP
analysis. We display, for comparison, also the masses as extracted by an
exponential fit to the eigenvalues as colored bands. The variational analysis
used three distinct levels of smearing. For some ensembles, there is no clear
plateau for the first excited state. The five panels correspond to the five
available ensembles.

We present in this section our main numerical results. First, we explain our
strategy in applying the GEVP analysis. As discussed in the previous section,
we applied APE smearing to the ensembles, as well as Wuppertal smearing to the
meson operators. A basis of 18 distinct operators was obtained by varying the
number smearing levels applied to the operators in Eqs. (5) and (6). While the
inclusion of further smearing levels in the GEVP improves the determination of
the effective mass, it soon reintroduces a signal-to-noise ratio in the form
of a loss signal at intermediate values of $t$. Hence, we identify by
inspection a subset of smearing levels appropriate to our analysis. By
inspection, we find that the optimal choice for our purpose is to include only
a subset of the operators produced with different $N^{\rm smear}$. We find,
that restricting ourselves to three smearing levels $N^{\rm smear}=0,40,80$ is
a good choice for our ensembles in all channels studied here. We observed that
the stability of the GEVP analysis is improved by using a small value of
$t_{0}=1$. This choice may in principle give rise to unaccounted systematic
effects, that scale with $\mathcal{O}\left(e^{-\Delta Et_{0}}\right)$, where
$\Delta E$ is the energy difference to the next state that is not properly
captured by the chosen variational basis Blossier:2009kd . Yet, for the
purposes of this analysis, the present approach suffices; future, large scale
numerical analysis, aimed at extrapolating the continuum and (hyperquark)
massless limits, will address this limitation.

###  IV.1 Meson masses

In Fig. 1, we show the effective masses of the states, $am^{\rm
eff}_{{\eta^{\prime}_{l}}}<am^{\rm eff}_{{\eta^{\prime}_{h}}}$, expressed in
units of the lattice scale, $a$. The effective masses plots, obtained from the
GEVP analysis, are shown to provide visual guidance in the choice of the
fitting range. In some cases, only an approximate plateau appears within the
statistical errors. This could be attributed to an insufficient stochastic
sampling of the Dirac propagator. In this work we have chosen $n_{\rm src}=64$
stochastic sources to sample the disconnected diagrams in (7)-(9). This choice
introduces an error in the correlation matrix, that is expected to scale as
$\mathcal{O}\left(1/\sqrt{n_{\rm src}}\right)$ Wilcox:1999ab .

As can be seen in Fig. 1, the signal in the effective mass is lost for times
$t$ much smaller than the temporal lattice midpoint $T/2$. Thus, we only fit a
single exponential in the specified fit interval. The choices we made for the
fitting parameters and the associated $\chi^{2}/N_{\rm d.o.f.}$ values are
reported in Tab. 2. The extracted meson masses (expressed in lattice units)
are reported in Tab. 3. We find that the statistical uncertainties obtained
for the flavor-singlet states are up to an order of magnitude larger than
those for the flavored states. For the aforementioned reasons, it is likely
that the systematic errors are also larger than those of the flavored mesons.

Table 2: Fit parameters used in the extraction of the meson masses, reported in Tab. 3. We report the fitting ranges, $I$, the number of exponential terms, $N_{\rm exp}$, used in the fit, and the values of $\chi^{2}/N_{\rm d.o.f.}$ for every fit. Label | $I_{\eta^{\prime}_{l}}$ | $I_{\eta^{\prime}_{h}}$ | $I_{\rm{PS}}$ | $I_{\rm{ps}}$ | $I_{\rm{V}}$ | $I_{\rm{v}}$ | $N_{\rm exp}$ | $\chi^{2}/N_{\rm d.o.f.}$ | $\chi^{2}/N_{\rm d.o.f.}$ | $\chi^{2}/N_{\rm d.o.f.}$ | $\chi^{2}/N_{\rm d.o.f.}$ | $\chi^{2}/N_{\rm d.o.f.}$ | $\chi^{2}/N_{\rm d.o.f.}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  | $\eta^{\prime}_{l}$ | $\eta^{\prime}_{h}$ | ${\rm PS}$ | ${\rm ps}$ | ${\rm V}$ | ${\rm v}$
M1 | (9,15) | (5,11) | (7,14) | (9,14) | (7,11) | (7,14) | 1 | 2.7 | 2.4 | 3.8 | 2.9 | 0.9 | 1.5
M2 | (9,13) | (7,11) | (8,13) | (8,15) | (8,13) | (8,16) | 1 | 1.8 | 1.9 | 2.7 | 1.7 | 2.3 | 1.8
M3 | (9,14) | (7,11) | (9,16) | (10,13) | (11,20) | (10,13) | 1 | 1.4 | 1.4 | 1.1 | 1.9 | 1.3 | 1.8
M4 | (7,13) | (6,10) | (8,18) | (7,16) | (8,16) | (7,13) | 1 | 2.2 | 2.0 | 1.3 | 2.0 | 1.3 | 3.4
M5 | (8,14) | (5,9) | (9,16) | (7,13) | (12,16) | (8,13) | 1 | 1.4 | 1.6 | 1.4 | 2.2 | 1.5 | 1.9
Table 3: Meson masses extracted from large Euclidean-time behavior of the eigenvalues, $\lambda(t)$, within the GEVP analysis. We report the ground states and first excited states in the flavor-singlet pseudoscalar channel as well as the ground states for flavored pseudoscalar and vector mesons made of either species of fermions, and for all five ensembles, which we characterize by the lattice coupling, $\beta$, its number of sites in the temporal, $N_{t}$, and spatial, $N_{s}$, directions, and the bare masses of the fermions in lattice units, $am_{0}^{\rm f}$ and $am_{0}^{\rm as}$. Label | $\beta$ | $N_{t}$ | $N_{l}$ | $am_{0}^{\rm f}$ | $am_{0}^{\rm as}$ | $am_{\eta^{\prime}_{l}}$ | $am_{\eta^{\prime}_{h}}$ | $am_{\rm PS}$ | $am_{\rm ps}$ | $am_{\rm V}$ | $am_{\rm v}$
---|---|---|---|---|---|---|---|---|---|---|---
M1 | 6.5 | 48 | 20 | -0.71 | -1.01 | 0.3769(96) | 0.6334(59) | 0.3639(14) | 0.6001(11) | 0.4030(33) | 0.6452(18)
M2 | 6.5 | 64 | 20 | -0.71 | -1.01 | 0.3867(68) | 0.619(13) | 0.3648(13) | 0.59856(82) | 0.4038(17) | 0.6421(15)
M3 | 6.5 | 96 | 20 | -0.71 | -1.01 | 0.3826(67) | 0.588(12) | 0.3652(16) | 0.59940(79) | 0.4040(18) | 0.6467(21)
M4 | 6.5 | 64 | 20 | -0.7 | -1.01 | 0.4381(33) | 0.6433(88) | 0.4067(13) | 0.62426(85) | 0.4476(17) | 0.6742(13)
M5 | 6.5 | 64 | 32 | -0.72 | -1.01 | 0.3591(53) | 0.637(26) | 0.31076(68) | 0.57718(85) | 0.3518(12) | 0.6223(15)

Figure 2: Masses, $am$, of the lightest flavor-singlet and flavored
pseudoscalar mesons, and flavored vector mesons, constituted of either species
of fermions. Top left panel: measurements of the masses in lattice units for
the three ensembles (M1 - M3) that have common values of $am_{0}^{\rm
as}=-1.01$ and $am_{0}^{\rm f}=0.71$. The masses as shown for the three
available choices of spatial lattice extent $N_{t}$. Top right panel:
measurements of the masses in lattice units for the three available choices of
bare fundamental fermion mass. A small horizontal offset has been introduced,
for visual clarity, in the ensembles M1 - M3. Bottom left panel: measurement
of the masses in lattice units for all ensembles with $N_{t}=64$ (M2, M4, and
M5), plotted against the ratio of the mass of the flavored pseudoscalar and
vector mesons constituted of fermions in the fundamental representation.
Bottom right panel: same as bottom left, but the masses of the mesons are
expressed in gradient flow units.

In Fig. 2, we show all the low-lying pseudoscalar and vector meson masses.
They are displayed as a function of $N_{t}$ in the left-hand panel and as a
function of $am_{0}^{\rm f}$ in the right-hand panel. On the same plots, we
display the masses of the corresponding states in the flavored channels.
Overall, the ground states and first excited state masses in the flavor-
singlet channels are found to lie, within uncertainties, in the range
determined by the masses corresponding flavored channels. We find larger
uncertainties for the $\eta^{\prime}$ state, hence the mass hierarchy is not
fully determined. For the lightest ensemble, M5, the flavor-singlet
pseudoscalar states have mass compatible to the flavored vector mesons.

The general trends exhibited by our results suggest that the effect of the
disconnected diagrams is suppressed by the heavy fermion masses present in
these ensembles. Thus, the spectrum resembles that of $N_{\rm f}=2$ single-
representation theories at moderately-heavy to heavy fermion mass
Bennett:2023rsl . A suppression of the disconnected diagrams also implies a
suppression of the ${\eta^{\prime}_{l}}-{\eta^{\prime}_{h}}$ mixing effects,
according to Eq. (9). Thus, we expect that the associated mixing angle is
small and that the state $a$ is mostly dominated by the fundamental fermionic
contribution, whereas the mass of the $\eta^{\prime}$ is mostly determined by
the antisymmetric fermion masses. We will return to the determination of the
mixing angle in the next subsection.

Because the ensembles M1-M3 share the same lattice parameters, with only the
temporal lattice extent, $N_{t}$, distinguishing them, we can compare across
the three to ascertain the contribution of $N_{t}$ to systematic effects. We
find that for both the singlet ground states, ${\eta^{\prime}_{l}}$, and the
first excited state, ${\eta^{\prime}_{h}}$, measurements with different
$N_{t}$ yield compatible results. This may not be the case for further excited
states Bennett:2024cqv , but these considerations go beyond the purposes of
the present study.

###  IV.2 Mixing angle

Figure 3: Effective mixing angles, $\phi$, between the lightest pseudo-scalar
singlets, for all ensembles M1-M5, based on the pseudoscalar singlet matrix
elements defined in Eq. (15). We fit the effective mixing angle in the range
from a minimum value $t=t_{0}+1$ until a maximum value $t=t_{\rm max}$.
Conventionally, we choose the latter to coincide with the smallest $t$ value
at which the relative error of the ground-state effective mass is larger than
50%. We find good signal-to-noise ratios in all ensembles, even for the
comparatively large value of $t_{0}=5$. We find no clear evidence of mass
dependence of the mixing angle.

With the available ensembles we measured the pseudoscalar matrix elements of
Eq. (15), from which we extracted the effective mixing angle, $\phi$,
determined according to Eq. (16), which we display in Fig. 3 as a function of
the time $t$. A similar analysis for the axial-vector currents, in Eq. (12),
did not yield a significant signal, hence we did not determine the
pseudoscalar decay constants. This behavior is similar from what has been
found in QCD, where the flavor-singlet axial-vector matrix elements are
affected by poor signal-to-noise ratios Ottnad:2017bjt .

We stress that the result for the effective mixing angle displayed in Fig. 3
have been obtained with a simplified analysis that only involves local
sources. The simpler GEVP is analyzed with a larger value of $t_{0}=5$, as it
is in this regime where the systematic error on the determination of the
matrix element should be smaller, see Ref. Blossier:2009kd . We leave to
future, high-precision studies the task of measuring the relevant components
of the correlation matrix itself Bali:2021qem  to determine simultaneously the
mixing angle and the decay constants.

We extract the mixing angle, $\phi$, via a constant fit to the effective
mixing angle. We choose the fitting interval to start at $t=t_{0}+1$ and end
at the first $t=t_{\rm max}$ for which the relative uncertainty in the
effective mass of the flavor-singlet pseudoscalar ground state exceeds $50\%$.
We report the extracted mixing angles in Tab. 4. We find a small mixing angle
in all available ensembles. We do not find clear evidence of mass dependence
in the mixing angles in the ensembles studied here. All measurements indicate
a mixing angle of roughly $6^{\circ}$. These results further support earlier
conclusions: because of the large value of the fermion masses, the mixing
effects between the flavor-singlet pseudoscalar states ${\eta^{\prime}_{l}}$
and ${\eta^{\prime}_{h}}$ are strongly suppressed.

Table 4: Results for the pseudoscalar singlet mixing angle, $\phi$, expressed in degrees, extracted from the flavor-singlet pseudoscalar matrix elements, using correlation functions with point-like source and sink (no Wuppertal smearing). The quoted uncertainties have been obtained using jackknife resampling. Label | $~{}~{}~{}~{}\beta~{}~{}~{}~{}$ | $~{}~{}~{}~{}N_{t}~{}~{}~{}~{}$ | $~{}~{}~{}~{}N_{s}~{}~{}~{}~{}$ | $~{}~{}~{}~{}\phi/{}^{\circ}~{}~{}~{}~{}$
---|---|---|---|---
M1 | 6.5 | 48 | 20 | 6.15(83)
M2 | 6.5 | 64 | 20 | 6.07(63)
M3 | 6.5 | 96 | 20 | 6.16(66)
M4 | 6.5 | 64 | 20 | 7.44(58)
M5 | 6.5 | 64 | 32 | 6.61(54)

##  V Summary and Outlook

In this paper, we have presented the first numerical lattice study of the mass
spectrum of flavor-singlet pseudoscalar bound states in an $Sp(4)$ gauge
theory with fermions in multiple representations. The channels of interest
result from the mixing effects between the Abelian PNGBs associated with the
two global $U(1)$ factors in the approximate symmetry of the system. These
Abelian symmetries are broken explicitly both by the masses of the fermions,
and by the axial anomaly. The motivation for this study is that, in the
context of composite Higgs models, the phenomenology of such flavor singlets
deserves special consideration Belyaev:2016ftv , and requires detailed
knowledge of non-perturbative properties of the theory in its strongly coupled
regime.

Our measurements have been performed on five available ensembles, generated in
a lattice field theory that is of interest in its own terms, as it has been
proposed as a short-distance completion to the minimal CHM that also
implements top partial compositeness Barnard:2013zea .

This type of spectroscopy measurement is challenging because it requires
performing explicit calculations of disconnected diagrams contributing to the
two-point functions of interest, which introduce high noise level in the
numerical analysis. These measurements were obtained with a combination of APE
and Wuppertal smearing, as well as with an implementation of the GEVP
analysis.

The results presented above provide the first determination–and a
demonstration of feasibility–of the masses of the ground and first excited
state in the flavor-singlet pseudoscalar channel of this theory, as well as of
their associated mixing angle.

Some limitations of the analysis descend from the fact that all the ensembles
have the same value of the lattice coupling, $\beta$, so that a continuum-
limit extrapolation is beyond the reach of this analysis. Furthermore, the
masses for the two species of fermions belong to a regime in which the theory
is far from the massless limit. As a result, the masses of neither the flavor-
singlet nor the flavored pseudoscalar states in the theory are particularly
light, and a massless extrapolation of the results is also beyond current
reach.

The natural next step for future studies would be to deploy the numerical
strategy we developed and tested in a large-scale investigation that would
allow performing continuum and massless extrapolations. This endeavor would
allow to better assess the effect on the spectrum and mixing angle of the
axial anomaly, and connect this study to ongoing model-building and collider
phenomenology programs. Furthermore, having demonstrated the viability of the
numerical techniques, it would be interesting to understand how the results
depend also on other intrinsic parameters of the theory, such as the numbers,
$N_{\rm f}$ and $N_{\rm as}$, of fermions transforming on the fundamental and
antisymmetric representation of the group. An interesting connection to more
formal field-theory research topics would involve changing the group, within
the $Sp(2N)$ class of theories, to explore the approach to the large-$N$
limit. We plan to exploit all of these opportunities for further research in
the future.

###### Acknowledgements.

We would like to thank Giacomo Cacciapaglia, Gabriele Ferretti, Thomas Flacke,
Anna Hasenfratz, Chulwoo Jung, and Sarada Rajeev, for useful discussions
during the ”PNU Workshop on Composite Higgs: Lattice study and all”, at
Haeundae, Busan, in February 2024, where preliminary results of this study
were presented. The work of EB and BL is supported in part by the EPSRC
ExCALIBUR programme ExaTEPP (project EP/X017168/1). The work of EB, BL, MP,
and FZ has been supported by the STFC Consolidated Grant No. ST/X000648. The
work of EB has also been supported by the UKRI Science and Technology
Facilities Council (STFC) Research Software Engineering Fellowship
EP/V052489/1. The work of NF has been supported by the STFC Consolidated Grant
No. ST/X508834/1. 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 DKH was further
supported by the National Research Foundation of Korea (NRF) grant funded by
the Korea government (MSIT) (2021R1A4A5031460). The work of JWL is supported
by IBS under the project code, IBS-R018-D1. 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 CJDL is also supported by Grants No.
112-2639-M-002-006-ASP and No. 113-2119-M-007-013-. The work of BL and MP has
been further supported in part by the STFC Consolidated Grant No.
ST/T000813/1. 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 DV is supported by STFC under
Consolidated Grant No. ST/X000680/1. Numerical simulations have been performed
on the DiRAC Extreme Scaling service at the University of Edinburgh, and on
the DiRAC Data Intensive service at Leicester. The DiRAC Extreme Scaling
service is operated by the Edinburgh Parallel Computing Centre on behalf of
the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by
BEIS capital funding via STFC capital grant ST/R00238X/1 and STFC DiRAC
Operations grant ST/R001006/1. DiRAC is part of the National e-Infrastructure
Research Data Access Statement—The data generated for this manuscript can be
downloaded from Ref. data_release  and the analysis code from Ref
analysis_release . Open Access Statement—For the purpose of open access, the
authors have applied a Creative Commons Attribution (CC BY) licence to any
Author Accepted Manuscript version arising.

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