# Singlets in gauge theories with fundamental matter

Ed Bennett<EMAIL_ADDRESS>Swansea Academy of Advanced Computing,
Swansea University (Bay Campus), Fabian Way, SA1 8EN Swansea, Wales, United
Kingdom  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  Department of Physics, Pusan National University, Busan
46241, Korea  Institute for Extreme Physics, Pusan National University, Busan
46241, Korea  Biagio Lucini<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  Axel Maas<EMAIL_ADDRESS>Institute of Physics,
NAWI Graz, University of Graz, Universitätsplatz 5, A-8010 Graz, Austria
Maurizio Piai<EMAIL_ADDRESS>Department of Physics, Faculty of Science
and Engineering, Swansea University (Park Campus), Singleton Park, SA2 8PP
Swansea, Wales, United Kingdom  Fabian Zierler<EMAIL_ADDRESS>Institute of Physics, NAWI Graz, University of Graz, Universitätsplatz 5,
A-8010 Graz, Austria

###### Abstract

We provide the first determination of the mass of the lightest flavor-singlet
pseudoscalar and scalar bound states (mesons), in the $\mathrm{Sp}(4)$ Yang-
Mills theory coupled to two flavors of fundamental fermions, using lattice
methods. This theory has applications both to composite Higgs and strongly-
interacting dark matter scenarios. We find the singlets to have masses
comparable to those of the light flavored states, which might have important
implications for phenomenological models. We focus on regions of parameter
space corresponding to a moderately heavy mass regime for the fermions. We
compare the spectra we computed to existing and new results for
$\mathrm{SU}(2)$ and $\mathrm{SU}(3)$ theories, uncovering an intriguing
degree of commonality. As a by-product, in order to perform the aforementioned
measurements, we implemented and tested, in the context of symplectic lattice
gauge theories, several strategies for the treatment of disconnected-diagram
contributions to two-point correlation functions. These technical advances set
the stage for future studies of the singlet sector in broader portions of
parameter space of this and other lattice theories with a symplectic gauge
group.

###### Contents

  1. I Introduction
  2. II Flavor singlets in symplectic gauge theories
    1. II.1 Microscopic theory and global symmetries
    2. II.2 Light meson spectrum for two fundamental fermions
  3. III Lattice setup
    1. III.1 Interpolating operators and two-point functions
    2. III.2 Variance reduction techniques
  4. IV Results
    1. IV.1 Pseudoscalar singlet in $\mathrm{SU}(2)$ and $\mathrm{Sp}(4)$ with $N_{f}=2$
    2. IV.2 Pseudoscalar singlets in Sp(4) with $N_{f}=1+1$
    3. IV.3 Scalar singlet in Sp(4) with $N_{f}=2$
    4. IV.4 Comparison to SU(3) with $N_{f}=2$
    5. IV.5 Possible phenomenological implications
  5. V Summary
  6. A Constant contributions to the correlators
  7. B Comparison between excited state subtraction and smeared connected diagrams

##  I Introduction

The possible existence of new strongly interacting sectors that extend the
standard model (SM) of particle physics has been the subject of a long-
standing history of theoretical studies. In recent years, this idea has
prominently featured in the context of composite Higgs models (CHMs) in which
the Higgs fields of the SM originate as pseudo-Nambu-Goldstone bosons (PNGBs)
of the underlying theory [1, 2, 3].111The recent literature is vast. See,
e.g., the reviews in Refs. [4, 5, 6], the summary tables in Refs. [7, 8,
9],and the selection of papers in Refs. [10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37,
38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48] and Refs. [49, 50, 51, 52, 53, 54,
55, 56, 57, 58, 59, 60, 61, 62, 63]. A parallel development has led to
strongly coupled gauge theories being considered as the dynamical origin of
hidden sectors in which dark matter consists of strongly interacting massive
particles (SIMPs) [64, 65, 66, 67, 68, 69, 70, 71, 72, 73]. This scenario can
address observational problems such as the ‘core vs. cusp’ [74] and ‘too big
to fail’ [75] ones, and have implications for gravitational wave experiments
[76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93]—see
e.g. Refs. [94, 95, 96, 97, 98, 99], as well as Refs. [100, 101, 102, 103,
104]. Both CHMs and SIMPs give rise to particles, the PNGBs, carrying non-
trivial quantum numbers of non-Abelian global (flavor) symmetries, that
suppress and protect their masses.

In general, strongly coupled theories also yield bound states that are flavor
singlets. Composite models with a strongly coupled origin can in particular
give rise to a light dilaton [105, 106, 107], the PNGB associated with
(approximate) scale invariance. The striking phenomenological implications of
this possibility [108] have also been studied extensively (see e.g. Refs.
[109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120]), and the low
energy effective field theory (EFT) of the dilaton [121, 122] and the other
PNGBs can be combined in the dilaton EFT [123, 124, 125, 126, 127, 128, 129,
130, 131, 132, 133, 134, 135, 136, 137, 138]. Numerical evidence supporting
this possibility has emerged in the context of $\mathrm{SU}(3)$ lattice gauge
theories with special field content [139, 140, 141, 142, 143, 144, 145, 146,
147, 148, 149, 150, 151]. The singlet sector of a strongly coupled theory
contains also pseudoscalar composite states, the phenomenology of which is the
subject of dedicated studies [27, 152, 153, 23, 154, 9]. Their EFT treatment
follows closely that of axion-like particles (ALPs) [155, 156, 157].

The phenomenological consequences of such theories depend crucially on the
mass spectrum of the lightest states, and on the (model dependent) couplings
to the standard model. While an efficient tool in treating the latter is
provided by the EFT methodology, even the construction of such EFTs requires a
good understanding of the lightest portion of the spectrum. Whatever the
original motivation and application envisioned for such new strongly coupled
physical sectors is, they have a plethora of bound states, some of which can
be either stable or long-lived, and potentially light. It is then desirable to
gain a broad non-perturbative understanding of their spectroscopy, for all
bound states, and in the largest possible portions of the parameter space. The
instrument of choice for this endeavor is that of (numerical) lattice gauge
theories. There has been a wide variety of investigations into the spectrum of
many such theories, especially those with $\mathrm{SU}(N)$ group, with matter
in various numbers and representations. Besides the aforementioned work, and
Refs. [158, 159, 160, 161, 143, 144, 162] on $\mathrm{SU}(3)$, see e.g. Refs.
[163, 164, 165, 166, 167, 168, 169, 170, 171] for $\mathrm{SU}(2)$ theories,
Refs. [172, 173, 174, 175] for $\mathrm{SU}(4)$ in multiple representations,
and [176] for $\mathrm{G}_{2}$, as well as the reviews in Refs. [177, 178,
179, 180]

Gauge theories with symplectic group play a special role in all these
contexts, because of the peculiar properties of $\mathrm{Sp}(2N)$ groups and
their representations. For example, the model in Ref. [16] provides the
simplest realization of a CHM that combines it with top partial compositeness
[181], and consists of a $\mathrm{Sp}(2N)$ gauge theory with mixed-
representation fermion content. Likewise, Refs. [64, 66] use it for the
construction of a SIMP-“miracle”. A number of recent lattice studies started
to characterize them [182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192,
193, 194, 195, 196, 197, 198], following the pioneering effort in Ref. [199].
While much such work has focused on the spectroscopy of bound states carrying
flavor, with this paper we report on progress in the singlet sector.

The present study explores the flavor-singlet bound state sector in the
$\mathrm{Sp}(4)$ gauge theory coupled to two fermions in the fundamental
representation, a theory that has gathered substantial interest in the CHM
context, and also provides a minimal realization of the SIMP mechanism [67,
69, 200, 71, 72]. This study is complementary to the available non-singlet
hadron spectrum found in Refs. [190, 197, 189, 187, 186, 184, 185, 182].
Within the general aim of understanding universal features of the low-lying
spectrum across different gauge groups, this is also a step towards
understanding how the approach to the large-$N$ limit of $\mathrm{Sp}(2N)$
gauge theories differs from that of $\mathrm{SU}(N)$ gauge theories,
especially with respect to the axial anomaly and topology. Our results could
in the future help to understand the anomaly-mediated decays of singlets into
SM particles.

We supplement this publication with numerical results obtained in two other
theories. The first is the closely related $\mathrm{SU}(2)$ theory with two
fundamental fermions, for which earlier studies exist [163, 201], and for
which we perform additional, new calculations. In the case of the
$\mathrm{SU}(3)$ theory coupled to fermions in the fundamental representation,
the pseudoscalar singlet has been studied in Refs. [202, 203, 204, 205, 206,
207, 208, 209, 210, 211, 212, 213, 214]. The determination of the mass (and
width) of the lightest scalar singlet has proven to be particularly
challenging and a number of studies in $\mathrm{SU}(3)$ with dynamical
fermions exists both in the context of real-world QCD [215, 216, 217, 218,
219, 220, 221, 222, 223, 224] and more general field content [158, 159, 141,
160, 161, 143, 144]. We borrow results from this extensive literature, for the
purpose of comparing with our own results.

The paper is organized as follows. The pseudo-real nature of the fundamental
representation of $\mathrm{Sp}(4)$—as for $\mathrm{SU}(2)$—leads to symmetry
enhancement by modifying the flavor symmetry and symmetry-breaking pattern.
The structure of the low-lying spectrum is hence different from the more
familiar QCD case. We briefly comment on the most striking such features in
Sect. II, as we define the continuum theory of interest. In Sect. III we
describe the lattice methods that we use to study the flavor-singlet states,
putting some emphasis on the implications for the construction of suitable
operators, in Sect. III.1. The study of correlations functions involving
singlets is affected by notorious difficulties, poor signal-to-noise ratio
featuring prominently among them. This required the adoption of advanced
techniques to obtain a non-zero signal, as we explain in Sect. III.2, and in
more detail in the Appendices.

We present the body of our numerical results in Sect. IV. Section IV.1 is
devoted to the lightest pseudoscalar singlet state, in both the
$\mathrm{Sp}(4)$ and $\mathrm{SU}(2)$ theories, for degenerate masses. We
report on the case of non-degenerate flavor masses for $\mathrm{Sp}(4)$ (which
realizes a scenario relevant to dark matter models) in Sect. IV.2. The scalar
singlet sector, in the degenerate case for the $\mathrm{Sp}(4)$ theory, is the
subject of Sect. IV.3, though we anticipate that, because of the bad signal-
to-noise ratio, only a rough estimate with unclear finite-spacing systematics
can be established at this stage. Finally, we assess our results by a
comparison to the $\mathrm{SU}(3)$ case, and report the results in Sect. IV.4.
Our general conclusion, exposed more critically in Sect. V, is that for the
available range of fermion masses the singlets are indeed light enough to
affect phenomenology and low-energy EFT considerations. We add several
technical appendices, covering further details. We note that some preliminary
results are available in Ref. [225].

##  II Flavor singlets in symplectic gauge theories

We start the presentation by defining explicitly the (continuum) field
theories of interest. We provide both their microscopic definition, in terms
of elementary fields, and their salient long-distance properties, which can be
explained in EFT terms. In doing so, we emphasize the role of the symmetries
of the theory.

###  II.1 Microscopic theory and global symmetries

The $\mathrm{Sp}(2N)$ gauge theories of interest are characterized by a
Lagrangian density, ${\cal L}$, which in this section we write using a metric
with Lorentzian signature $(+1,-1,-1,-1)$, and takes the form:

$\displaystyle\mathcal{L}=-\frac{1}{2}\Tr\left[G_{\mu\nu}G^{\mu\nu}\right]+\bar{u}\left(i\gamma^{\mu}D_{\mu}-m_{u}\right)u+\bar{d}\left(i\gamma^{\mu}D_{\mu}-m_{d}\right)d\,,$
(1)

where $G_{\mu\nu}\equiv\sum_{A}G_{\mu\nu}^{A}t^{A}$ are the field strength
tensors, and $t^{A}$ are the generators of $\mathrm{Sp}(2N)$, normalized so
that $\Tr T^{A}T^{B}=\frac{1}{2}\delta^{AB}$, while $u$ and $d$ are
4-component Dirac spinors, denoting the two flavors of fermion fields
transforming in the fundamental representation of $\mathrm{Sp}(2N)$. The
Lagrangian is real and Lorentz invariant, as $\bar{u}\equiv
u^{\dagger}\gamma^{0}$, and $\bar{d}\equiv d^{\dagger}\gamma^{0}$. The
covariant derivatives are defined in terms of the gauge fields $A_{\mu}\equiv
A_{\mu}^{A}t^{A}$ as

$\displaystyle D_{\mu}u$ $\displaystyle\equiv$
$\displaystyle\partial_{\mu}u+igA_{\mu}u\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}D_{\mu}d\,\equiv\,\partial_{\mu}d+igA_{\mu}d\,,$
(2)

where $g$ is the gauge coupling. Explicitly, the field-strength tensor is
given by

$\displaystyle G_{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+ig\left[A_{\mu}\,,\,A_{\nu}\right]\,,$
(3)

where $[\cdot,\cdot]$ denotes the commutator.

The fundamental representation of $\mathrm{Sp}(2N)$ is pseudo-real. As a
result, the global symmetry is enhanced: one can show explicitly, by rewriting
Eq. (1) in terms of 2-component fermions,222This somewhat tedious exercise can
be found in all its details in the literature, for instance in Refs. [168,
185] and references therein. that for each Dirac fermion the
$\mathrm{U}(1)_{L}\times\mathrm{U}(1)_{R}$ Abelian global symmetry acting on
its chiral projections is enhanced to a non-Abelian $\mathrm{U}(2)$ global
symmetry. The fermion kinetic terms hence, written in terms of covariant
derivatives, exhibit an enhanced (classical) $\mathrm{U}(4)=U(1)_{A}\times
SU(4)$ global symmetry; we will return to the effect of anomalies in the next
subsection.

The fermion mass terms break explicitly the global symmetry. If the masses are
degenerate, $m_{u}=m_{d}$, as will be the case throughout most of this paper,
then the global symmetry is explicitly broken to $\mathrm{Sp}(4)$. The
bilinear, non-derivative operator appearing in the Lagrangian density as a
mass term is also expected to condense, so that the same symmetry breaking
pattern appears also in spontaneous symmetry breaking effects. For generic
choices of fermion masses, $m_{u}\neq m_{d}$, the approximate global symmetry
is further broken to $\mathrm{Sp}(2)^{2}\sim\mathrm{SU}(2)^{2}$ [197].

###  II.2 Light meson spectrum for two fundamental fermions

We summarize here the main properties of the bound states of interest, guided
by gauge invariance and symmetry arguments, starting from the case in which
the fundamental fermions have degenerate mass, $m_{u}=m_{d}$. We observe that
the group structure has an even number of colors, hence baryons are bosons.
Furthermore, because the matter content consists only of fermions transforming
in the pseudo-real fundamental representation, and as a result the global
symmetry is enhanced, ordinary baryon number is a subgroup of the enhanced
$\mathrm{SU}(4)$, and is unbroken in the vacuum of the theory, and hence
objects that one might be tempted to classify as having different baryon
number (e.g. mesons and diquarks) belong to the same $\mathrm{Sp}(4)$
multiplet.

We restrict the discussion from here onwards to mesons made of two fundamental
fermions. It may be convenient to think of the mesons in terms of their
2-component fermion field content, in order to classify them by their
$\mathrm{Sp}(4)$ transformations, and attribute their $J^{P}$ quantum
numbers.333We define the parity $P$ so that flavor eigenstates are also parity
eigenstates. See Refs. [165, 185, 197] for extended discussions of the
subtleties involving twisting between space and flavor in the definition of
parity. As the 2-component fermions transform as a $4$ of $\mathrm{Sp}(4)$,
the multiplication properties imply that there exist mesons transforming as a
$1$, $5$, and $10$ of $\mathrm{Sp}(4)$.

Starting from the spin-0 states, one expects to find in the spectrum $5$
PNGBs, spanning the $\mathrm{SU}(4)/\mathrm{Sp}(4)$ coset, and transforming as
a $5$ of $\mathrm{Sp}(4)$, to become massless in the $m_{u}=m_{d}\rightarrow
0$ limit.444These are the states playing a role in CHMs; in their EFT
description in terms of weakly coupled fields, four of them are identified
with the components of the SM Higgs doublet, because the
$\mathrm{SU}(2)_{L}\times\mathrm{SU}(2)_{R}$ approximate symmetry of the
electroweak theory is identified with the $SO(4)$ subgroup of
$\mathrm{Sp}(4)$, and hence $5=4\oplus 1$ [16]. The $1$ is an additional
state, that carries no SM gauge quantum numbers, but in the EFT description of
the strong coupling sector it is degenerate with the $4$. Likewise, these
states are the dark matter candidates in SIMP models [64, 65, 197]. These
states have parity partners, generalizing what in QCD literature are usually
denoted as $a_{0}$ particles. Some numerical lattice evidence exists that at
high temperature these two sets of states become degenerate—see Ref. [226] for
$\mathrm{SU}(2)$ and Ref. [227] for $\mathrm{SU}(3)$, both with two flavors of
fundamental fermions—because the $\mathrm{U}(1)_{A}$ symmetry relating them is
restored. But at zero temperature the scalar $5$ is expected to be heavy, the
mass of the particles being of the order of the confinement scale, even in the
$m_{u}=m_{d}\rightarrow 0$ limit.

Classically, one would expect also two singlet scalars to be light: the axion
and the dilaton. Indeed, the classical Lagrangian for $m_{u}=m_{d}=0$ is
invariant also under the action of a $\mathrm{U}(1)_{A}$ symmetry and of
dilatations, the condensates breaking both of them spontaneously, and these
two additional light states can be thought of as the PNGBs associated with
these two Abelian symmetries. Alas, besides being explicitly broken by the
fermion masses, both these symmetries are also anomalous. The
$\mathrm{U}(1)_{A}$ and scale anomalies hence provide masses for the axion-
dilaton system, related to the scale of confinement of the theory. Mixing
effects between these states and other vacuum excitations (e.g. glueballs with
the same $J^{P}$ quantum numbers) are present as well, given that no symmetry
argument can be invoked rigorously to forbid them. The precise determination
of such masses, hence, is non-trivial, and to large extent this paper is about
setting the stage for its future large-scale, high-precision calculation.
Furthermore, in confining theories with large numbers of degrees of freedom,
and when approaching the lower edge of the conformal window, non-perturbative
effects might suppress the mass of the axion and dilaton, respectively; this
is a very active field of research in itself, for a potential phenomenological
role both of this axion and of the dilaton, as we mentioned in the
introduction; the technology we developed and tested for this paper could play
an important future role in either case.

The spin-1 part of the meson spectrum is more rich. In analogy with the case
of QCD, one expects the lightest such states to generalize the $\rho$ mesons;
they transform as a $10$ of $\mathrm{Sp}(4)$ and have $J^{P}=1^{-}$, and their
properties have been studied elsewhere [184]. They have the peculiar property
that they can be sourced by two different interpolating operators, with the
schematic structures $\bar{\psi}\gamma_{\mu}\psi$ and
$\bar{\psi}\sigma_{\mu\nu}\psi$, respectively. In addition, the
generalizations of the $a_{1}$ and $b_{1}$ from QCD transform as a $5$ and a
$10$ of $\mathrm{Sp}(4)$, respectively; these additional states are heavier
than the aforementioned $10$-plet with $J^{P}=1^{-}$—see the discussions in
Refs. [168, 185]. It is worth noticing that some spin-1 singlet mesons of QCD
are actually part of these multiplets, because of the symmetry-enhancement
pattern—noticeably, the particle that corresponds to $\omega$ in QCD.

In the presence of non-degenerate fermion masses, $m_{u}\neq m_{d}$, the
global symmetry breaks further from $\mathrm{Sp}(4)$ down to
$\mathrm{Sp}(2)\times\mathrm{Sp}(2)=\mathrm{SU}(2)\times\mathrm{SU}(2)\sim\mathrm{SO}(4)$.
Consequently, the multiplets decompose with respect to the smaller flavor
symmetry [197]. The $5$-plets split into a $4$-plet and a singlet, whereas the
$10$-plet decomposes into a $6$-plet and a $4$-plet. This implies that an
additional singlet appears in states that would have been a $5$-plet in the
mass-degenerate theory, such as the PNGBs and the axial-vectors. This is the
familiar scenario in QCD: in the presence of a mass difference between up and
down quark, isospin is explicitly broken and the flavor-neutral pion $\pi^{0}$
becomes a singlet, with different mass from the charged $\pi^{\pm}$ states.
The main difference with QCD is that, as the pseudo-reality of the
representation results in additional flavor neutral states, which
microscopically can be written as di-quark states, the multiplet is enlarged.

This differs for mesons in the $10$-plet representation, such as the vector
meson. The $10$ decomposes in a $4$-plet (with the same flavor structure as in
the case of the PNGBs) and a $6$-plet. The latter consists of two states
sourced by ordinary meson operators (in the QCD analogy, they are the
$\rho^{0}$ and the $\omega$ particles with $J^{P}=1^{-}$) and four other
states that are sourced by diquark operators. A further splitting of these
multiplets is possible, for example by gauging a $\mathrm{U}(1)$ subgroup of
the $\mathrm{SO}(4)$ symmetry [197], but we do not consider it here.

##  III Lattice setup

We perform lattice simulations using the standard plaquette action and
standard Wilson fermions [228]. We use the HiRep code [229, 230] extended for
$\mathrm{Sp}(2N)$ gauge theories [231] to generate configurations and to
perform the measurements. In the case of degenerate fermions we use the Hybrid
Monte Carlo (HMC) [232] algorithm and for non-degenerate fermions we use the
rational HMC (RHMC) [233] algorithm. The latter case does not guarantee
positivity of the fermion determinant. In this case we have monitored the
lowest eigenvalue of the Dirac operator which we always found to be positive.
Thus, we do not see any hints of a sign problem for the fermion masses studied
in this work. Results for the non-singlet spectrum for two fundamental
fermions were first reported in Refs.[185, 197]. We give simulation details of
our ensembles in Tabs. 1 and 2.

We perform simulations on Euclidean lattices of size $T\times L^{3}$ and
define the bare inverse gauge coupling as $\beta=8/g^{2}$. We implement
periodic boundary conditions for the gauge fields. For the Dirac fields we
impose periodic boundary conditions in the spatial directions and anti-
periodic boundary conditions in the temporal direction.

Ensemble | group | $\beta$ | $m_{0}$ | $L$ | $T$ | $n_{\text{conf}}$ | $n_{\text{src}}$ | $I_{\eta^{\prime}}$ | $I_{\pi}$ | $I_{\sigma}$ | $I_{\sigma^{\rm conn.}}$ | $I_{\rho}$ | $\langle P\rangle$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
SU2B1L1M8 | SU(2) | 2.0 | -0.947 | 20 | 32 | 1020 | 300 | - | (10, 16) | (5, 8) | (7, 10) | (10, 16) | 0.56734(2)
SU2B1L1M7 | SU(2) | 2.0 | -0.94 | 14 | 24 | 1851 | 192 | (8, 12) | (8, 12) | - | - | (9, 12) | 0.56516(3)
SU2B1L1M6 | SU(2) | 2.0 | -0.935 | 16 | 32 | 951 | 256 | (7, 11) | (9, 16) | - | - | (9, 16) | 0.563654(28)
SU2B1L1M5 | SU(2) | 2.0 | -0.93 | 14 | 24 | 1481 | 256 | (7, 12) | (8, 12) | - | - | (9, 12) | 0.56245(3)
SU2B1L1M4 | SU(2) | 2.0 | -0.925 | 14 | 24 | 1206 | 192 | (6, 10) | (8, 12) | - | - | (9, 12) | 0.56119(3)
SU2B1L1M3 | SU(2) | 2.0 | -0.92 | 12 | 24 | 2401 | 192 | (6, 9) | (7, 12) | - | (6, 11) | (8, 12) | 0.559983(29)
SU2B1L1M2 | SU(2) | 2.0 | -0.9 | 12 | 24 | 500 | 128 | (6, 9) | (7, 12) | - | - | (8, 12) | 0.55571(6)
SU2B1L1M1 | SU(2) | 2.0 | -0.88 | 10 | 20 | 2582 | 128 | (5, 8) | (8, 10) | - | - | (9, 10) | 0.55225(4)
Sp4B3L1M8 | Sp(4) | 7.2 | -0.799 | 32 | 40 | 451 | 224 | - | (15, 20) | (5, 9) | (11, 19) | (15, 20) | 0.590862(5)
Sp4B3L1M7 | Sp(4) | 7.2 | -0.794 | 28 | 36 | 504 | 288 | (7, 12) | (10, 18) | - | (11, 16) | (11, 18) | 0.590452(7)
Sp4B3L1M6 | Sp(4) | 7.2 | -0.79 | 24 | 36 | 500 | 320 | (7, 12) | (12, 18) | (5, 8) | (10, 16) | (13, 18) | 0.590127(9)
Sp4B3L1M5 | Sp(4) | 7.2 | -0.78 | 24 | 36 | 508 | 384 | (6, 12) | (12, 18) | - | (11, 15) | (13, 18) | 0.589278(8)
Sp4B3L1M4 | Sp(4) | 7.2 | -0.77 | 24 | 36 | 200 | 384 | (6, 11) | (11, 18) | - | (10, 15) | (12, 18) | 0.588460(12)
Sp4B3L1M3 | Sp(4) | 7.2 | -0.76 | 16 | 36 | 200 | 384 | - | (11, 18) | (5, 8) | (9, 14) | (12, 18) | 0.587666(25)
Sp4B1L1M7 | Sp(4) | 6.9 | -0.924 | 24 | 32 | 492 | 320 | - | (9, 16) | (4, 7) | (7, 10) | (10, 16) | 0.56317(2)
Sp4B1L1M6 | Sp(4) | 6.9 | -0.92 | 24 | 32 | 503 | 484 | - | (7, 16) | (4, 9) | (8, 12) | (8, 16) | 0.562075(13)
Sp4B1L2M6 | Sp(4) | 6.9 | -0.92 | 16 | 32 | 176 | 128 | (6, 10) | (9, 16) | (4, 10) | (7, 10) | (9, 16) | 0.56212(5)
Sp4B1L1M5 | Sp(4) | 6.9 | -0.91 | 16 | 32 | 435 | 256 | (6, 11) | (8, 16) | - | (7, 9) | (9, 16) | 0.55935(3)
Sp4B1L2M5 | Sp(4) | 6.9 | -0.91 | 14 | 24 | 513 | 256 | (5, 10) | (8, 12) | (4, 7) | (9, 12) | (9, 12) | 0.55941(3)
Sp4B1L1M4 | Sp(4) | 6.9 | -0.9 | 16 | 32 | 547 | 512 | (6, 10) | (9, 16) | - | (7, 10) | (10, 16) | 0.556921(25)
Sp4B1L2M4 | Sp(4) | 6.9 | -0.9 | 14 | 24 | 942 | 128 | (7, 10) | (8, 12) | (4, 9) | (7, 9) | (9, 12) | 0.556981(26)
Sp4B1L3M4 | Sp(4) | 6.9 | -0.9 | 12 | 24 | 2904 | 128 | (6, 10) | (8, 12) | (4, 8) | (8, 10) | (9, 12) | 0.557009(18)
Sp4B1L2M3 | Sp(4) | 6.9 | -0.89 | 14 | 24 | 461 | 128 | (7, 10) | (8, 12) | (5, 9) | (8, 11) | (9, 12) | 0.55468(4)
Sp4B1L3M3 | Sp(4) | 6.9 | -0.89 | 12 | 24 | 1019 | 320 | (6, 10) | (8, 12) | (3, 6) | (7, 11) | (9, 12) | 0.55467(3)
Sp4B1L2M2 | Sp(4) | 6.9 | -0.87 | 12 | 24 | 1457 | 128 | (7, 11) | (8, 12) | (5, 8) | (8, 10) | (9, 12) | 0.550497(27)
Sp4B1L2M3 | Sp(4) | 6.9 | -0.87 | 10 | 20 | 976 | 128 | (6, 9) | (8, 10) | - | (6, 10) | (8, 10) | 0.55068(5)
Table 1: List of all ensembles with degenerate fermion masses used in this work. We report the number of configurations $n_{\rm conf}$, the number of the stochastic sources used in the approximation of the all-to-all quark propagator $n_{\rm src}$, the intervals for fitting the resulting meson correlators $I_{\rm meson}$ and the average value of the plaquette $\langle P\rangle$. In some cases we were unable to identify a clear plateau in the effective masses and could not determine the singlet masses. In these cases we do not report a fit interval. For the singlet mesons the interval quoted here was used to fit the correlators after subtracting the excited state contributions in the connected pieces and after performing a numerical derivative. Ensemble | $\beta$ | $m_{0}^{1}$ | $m_{0}^{2}$ | $L$ | $T$ | $n_{\text{conf}}$ | $n_{\text{src}}$ | $I_{\eta^{\prime}}$ | $I_{\pi^{0}}$ | $I_{\pi^{\pm}}$ | $I_{\sigma}$ | $I_{\sigma^{\rm conn.}}$ | $I_{\rho}$ | $\langle P\rangle$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Sp4B1L2M4ND1 | 6.9 | -0.9 | -0.89 | 14 | 24 | 300 | 64 | (7, 10) | (7, 10) | (8, 14) | (4, 8) | (8, 14) | (8, 14) | 0.55583(5)
Sp4B1L2M4ND2 | 6.9 | -0.9 | -0.88 | 14 | 24 | 191 | 128 | (7, 10) | (7, 10) | (8, 14) | (4, 8) | (8, 14) | (8, 14) | 0.55474(5)
Sp4B1L2M4ND3 | 6.9 | -0.9 | -0.87 | 14 | 24 | 400 | 128 | (7, 10) | (7, 10) | (8, 14) | (5, 8) | (8, 14) | (8, 14) | 0.55361(4)
Sp4B1L2M4ND4 | 6.9 | -0.9 | -0.85 | 14 | 24 | 300 | 64 | - | - | (8, 14) | (5, 8) | (8, 14) | (8, 14) | 0.55163(4)
Sp4B1L2M4ND5 | 6.9 | -0.9 | -0.8 | 14 | 24 | 400 | 128 | (7, 10) | (7, 10) | (8, 14) | - | (8, 14) | (8, 14) | 0.54735(4)
Sp4B1L2M4ND6 | 6.9 | -0.9 | -0.75 | 12 | 24 | 264 | 64 | (7, 10) | (7, 10) | (8, 12) | - | (8, 12) | (8, 12) | 0.54395(6)
Sp4B1L2M4ND7 | 6.9 | -0.9 | -0.7 | 12 | 24 | 249 | 128 | (7, 10) | (7, 10) | (8, 12) | - | (8, 12) | (8, 12) | 0.54104(6)
Table 2: List of all ensembles with non-degenerate fermion masses used in this
work. We report the number of configurations $n_{\rm conf}$, the number of the
stochastic sources used in the approximation of the all-to-all quark
propagator $n_{\rm src}$, the intervals for fitting the resulting meson
correlators $I_{\rm meson}$ and the average value of the plaquette $\langle
P\rangle$. In some cases we were unable to identify a clear plateau in the
effective masses and could not determine the singlet masses. In these cases we
do not report a fit interval. For the singlet mesons the interval quoted here
was used to fit the correlators after subtracting the excited state
contributions in the connected pieces and after performing a numerical
derivative.

###  III.1 Interpolating operators and two-point functions

We use fermion bilinear operators for sourcing both singlet and non-singlet
mesons. From here onwards, with some abuse of notation, we denote as
$\eta^{\prime}$ and $\sigma$, respectively, the pseudoscalar and scalar,
flavor-singlet states. While the mesonic sectors are enlarged in
$\mathrm{Sp}(2N)$ with fundamental fermions, every non-singlet or singlet
state can still be probed by fermion-antifermion operators, even in the case
of non-degenerate fermions. Furthermore, since fermions are moderately heavy,
we find such operators are sufficient to study the ground states, and for now
do not consider others (such as $\pi\pi$ operators, glueballs, or even
tetraquarks). We use the operators

$\displaystyle O_{1}^{(\Gamma)}(n)$ $\displaystyle=\bar{u}(n)\Gamma d(n)\,,$
$\displaystyle O^{(\Gamma)}_{\pm}(n)$ $\displaystyle=\left(\bar{u}(n)\Gamma
u(n)\pm\bar{d}(n)\Gamma d(n)\right)/\sqrt{2}\,,$ (4)

where $n=(\vec{n},t)$ denote lattice sites. For pseudoscalar mesons
$\Gamma=\gamma_{5}$ and we omit the superscript when its value is clear from
the context. The pseudoscalar operators $O_{-}$ and $O_{1}$ source the pion
$5$-plet, and the operator $O_{+}$ sources the pseudoscalar singlet,
$\eta^{\prime}$. The same pattern persists for the scalar mesons where
$\Gamma=1$, and we use the notation:

$\displaystyle O_{\eta^{\prime}}(n)$
$\displaystyle\equiv\left(\bar{u}(n)\gamma_{5}u(n)+\bar{d}(n)\gamma_{5}d(n)\right)/\sqrt{2}\,,$
$\displaystyle O_{\sigma}(n)$
$\displaystyle\equiv\left(\bar{u}(n)u(n)+\bar{d}(n)d(n)\right)/\sqrt{2}\,.$
(5)

For vector mesons $\Gamma=\gamma_{i}$ and all operators $O_{1}$ and $O_{\pm}$
source states in the vector $10$-plet [185]. In the non-degenerate case, the
flavored multiplet is always probed by $O_{1}$. For the vector mesons both
$O_{-}$ and $O_{+}$ probe the same unflavored multiplet. In the case of the
pseudoscalars and scalars, the $O_{-}$ and $O_{+}$ probe distinct singlets. We
note that the ensembles studied in this work have moderately heavy fermions—in
all cases the vector meson is lighter than twice the pNGB mass. After
performing the required Wick contractions, we arrive at the following two-
point correlation functions.

$\displaystyle\langle O_{1}(n)\bar{O}_{1}(m)\rangle$
$\displaystyle=-\vbox{\hbox{\includegraphics[scale={0.5},page=1]{contractions.pdf}}}\,,$
(6) $\displaystyle 2\langle O_{\pm}(n)\bar{O}_{\pm}(m)\rangle$
$\displaystyle=-\vbox{\hbox{\includegraphics[scale={0.5},page=2]{contractions.pdf}}}-\vbox{\hbox{\includegraphics[scale={0.5},page=3]{contractions.pdf}}}\pm
2~{}\vbox{\hbox{\includegraphics[scale={0.5},page=5]{contractions.pdf}}}+\vbox{\hbox{\includegraphics[scale={0.5},page=6]{contractions.pdf}}}+\vbox{\hbox{\includegraphics[scale={0.5},page=7]{contractions.pdf}}}\,.$

It can be seen that the singlet mesons only differ from the non-singlets by
the additional disconnected diagrams. In the degenerate limit they cancel
exactly for the $O_{-}$ operators. In order to determine the mesonic spectrum
we need to determine both the connected and disconnected pieces and then fit
the zero momentum correlator,

$\displaystyle C(t)\equiv\sum_{\vec{n}}\langle
O(\vec{n},t)\bar{O}(\vec{0},0)\rangle\,,$ (7)

on a Euclidean time interval $(t_{\rm{min}},t_{\rm{max}})$, where the ground
state dominates, and its energy—and thus the mass—can be extracted. The
different components of $C(t)$ drop off exponentially with their energy
$\propto\exp\left(-E_{n}t\right)$, and thus at sufficiently large $t$ only the
ground state remains, as all other states are exponentially suppressed.
However, we note that an additional constant term can arise, which is the case
for both the $\eta^{\prime}$ and the $\sigma$ meson. In the former case this
can arise due to an insufficient topological sampling of the path integral,
and this constant vanishes in the continuum limit [234, 235]. For the scalar
singlet, $\sigma$, this constant arises due to the vacuum contributions, e.g.
the fermion condensate, and persists in the continuum limit for vanishing
momenta. At large times the correlator $C(t)$ is then given by

$\displaystyle\lim_{t\to\infty}C(t)=a\left(e^{-mt}+e^{-m(T-t)}\right)+\langle
0|O|0\rangle^{2}\,,$ (8)

where the second exponential term is due to the lattice periodicity. While in
the case of the $\eta^{\prime}$ this constant is small compared to the signal
and only affects the correlator at large $t$, this is not the case for the
$\sigma$ meson. In the scalar case, this constant is several orders of
magnitudes larger than the signal and its removal is a significant challenge.

We choose to perform a numerical derivative as proposed in Ref. [236]. The
resulting correlator is then anti-symmetric with respect to the midpoint
$T/2$,

$\displaystyle\tilde{C}(t)\equiv\frac{1}{2}\left(C(t-1)-C(t+1)\right)\xrightarrow{t\rightarrow\infty}a\sinh{(m)}\left(e^{-mt}-e^{-m(T-t)})\right).$
(9)

In order to determine the Euclidean time interval for fitting we use an
effective mass $m_{\rm eff}(t)$ defined by

$\displaystyle\frac{\tilde{C}(t-1)}{\tilde{C}(t)}=\frac{e^{-m_{\rm
eff}(t)\cdot(T-t+1)}\pm e^{-m_{\rm eff}(t)\cdot(t-1)}}{e^{-m_{\rm
eff}(t)\cdot(T-t)}\pm e^{-m_{\rm eff}(t)\cdot t}}\,,$ (10)

where the $+$ is used for periodic correlators and the $-$ sign in case of
anti-periodic correlators with respect to the lattice midpoint $T/2$. We
determine $(t_{\rm min},t_{\rm max})$ by visually inspecting the effective
mass and identifying a plateau at large times $t$. We restrict ourselves to
ensembles where the plateau persists over four or more time slices. We then
perform a fit of a single exponential term to the correlator $\tilde{C}(t)$
for the mesons. In Appendix A we compare this method to computing the
additional constant $\langle 0|O|0\rangle^{2}$ directly, without the use of a
numerical derivative.

For the pseudoscalar sector in the non-degenerate $N_{f}=1+1$ theory both the
$\pi^{0}$ and $\eta^{\prime}$ are pseudoscalar singlets, and the
$\eta^{\prime}$ is not a groundstate. Thus, we need to perform a variational
analysis by computing the correlation matrix of the operators $O_{\pi^{0}}$
and $O_{\eta^{\prime}}$ and solve the resulting generalized eigenvalue problem
(GEVP). In the minimal operator basis of (III.1) and (III.1) the cross-
correlator are diagrammatically given by

$\displaystyle 2\langle O_{+}(n)\bar{O}_{-}(m)\rangle$
$\displaystyle=-\left(\vbox{\hbox{\includegraphics[scale={0.5},page=2]{contractions.pdf}}}-\vbox{\hbox{\includegraphics[scale={0.5},page=3]{contractions.pdf}}}\right)+\left(\vbox{\hbox{\includegraphics[scale={0.5},page=6]{contractions.pdf}}}-\vbox{\hbox{\includegraphics[scale={0.5},page=7]{contractions.pdf}}}\right).$
(11)

In the mass-degenerate limit the cross-correlator vanishes and the
$\eta^{\prime}$ becomes the ground state of the pseudoscalar singlet sector,
whereas the $\pi^{0}$ becomes part of the pNGB multiplet. Note the presence of
connected diagrams in the cross-correlator. This implies that even in the
limit of large fermion masses – which suppresses the disconnected pieces – the
cross-correlator remains large for large mass differences, i.e. a system with
heavy-light properties. Thus, sizeable mixing effects are expected to occur.

For a heavy-light system, a more diagonal basis is obtained by using the
operators $O^{PS}_{A}=\bar{u}\gamma u$ and $O^{PS}_{B}=\bar{d}\gamma d$. The
corresponding cross-correlator vanishes as the heavier fermion mass approaches
infinity and is given by

$\displaystyle\langle O^{PS}_{A}(n)\bar{O}^{PS}_{B}(m)\rangle$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.55},page=5]{contractions.pdf}}}.$
(12)

###  III.2 Variance reduction techniques

In order to obtain the full singlet two-point functions we need to measure
both the connected and disconnected pieces in Eq. (6). The disconnected
diagrams in particular are very noisy, and the signal is already lost at small
to intermediate $t$ where contaminations from excited states are non-
negligible. A direct determination of the ground state mass at large $t$ is
thus not possible. We can circumvent this problem by removing the
contributions of excited states in the singlet correlators manually. This is
straightforward for the connected pieces. There, the signal for the connected
pseudoscalar and vector mesons persists for all time slices $t$ and in the
case of the connected piece of the scalar meson we still have a signal up to
large $t$. We fit the connected piece at large times (see Tabs. 1 and 2 for
our choice of fit intervals) to a single exponential

$\displaystyle C_{{\rm conn}}^{\rm 1exp}(t)$
$\displaystyle=A_{0}\left(e^{-m_{\rm conn}t}+e^{-m_{\rm conn}(T-t)}\right),$
(13)

and replace the full connected piece by the ground state correlator [237],
where $A_{0}$ and $m_{\rm conn}$ are the fit parameters, such that

$\displaystyle C_{\eta^{\prime}}^{\text{1exp}}(t)$ $\displaystyle=C_{\pi,{\rm
conn}}^{\text{1exp}}(t)+C_{\eta^{\prime},{\rm disc.}}(t)\,,$ (14)
$\displaystyle C_{\sigma}^{\text{1exp}}(t)$ $\displaystyle=C_{\sigma,{\rm
conn}}^{\text{1exp}}(t)+C_{\sigma,{\rm disc.}}(t)\,.$ (15)

We find that the excited state contributions in the connected pieces are the
dominant ones, and removing them shows a much earlier onset of a plateau in
the effective masses. The underlying assumption for these correlators is that
the excited state contributions of full and connected pieces are
indistinguishable within data quality as was noted in [209]. This is not
guaranteed a priori. However, we find that the excited state contributions in
the connected pieces are indeed the dominant ones. In Appendix B we show that
our results obtained by subtracting the connected excited state contributions
through a fit at larger times produces the same results as using smeared
operators, for the connected pieces with more overlap with the ground state.

Note, that this technique is not applicable to the non-degenerate case, as the
$\eta^{\prime}$ is no longer a ground state and some relevant information is
actually encoded in the excited states. Thus, we will not apply this technique
there.

The evaluation of disconnected pieces requires all-to-all propagators. We use
$Z_{2}\times Z_{2}$ noisy sources with spin and even-odd dilution [238]. We
typically use $\mathcal{O}(100)$ distinct noise vectors. The connected pieces
are evaluated using stochastic wall sources. Uncertainties are estimated using
the jackknife method.

| $\beta$ | $m_{0}$ | $L$ | $T$ | $m_{\pi}L$ | $m_{\pi}/m_{\rho}$ | $m_{\pi}$ | $m_{\rho}$ | $m_{\eta^{\prime}}$ | $m_{\sigma}$
---|---|---|---|---|---|---|---|---|---|---
SU(2) | 2.0 | -0.947 | 20 | 32 | 7.47(3) | 0.690(7) | 0.3735(13) | 0.540(5) | - | 0.53(4)
SU(2) | 2.0 | -0.94 | 14 | 24 | 6.40(2) | 0.746(6) | 0.4576(14) | 0.612(4) | 0.67(6) | -
SU(2) | 2.0 | -0.935 | 16 | 32 | 7.91(2) | 0.767(5) | 0.4946(14) | 0.644(4) | 0.60(3) | -
SU(2) | 2.0 | -0.93 | 14 | 24 | 7.491(19) | 0.787(4) | 0.5350(14) | 0.679(3) | 0.65(3) | -
SU(2) | 2.0 | -0.925 | 14 | 24 | 7.999(19) | 0.806(4) | 0.5713(14) | 0.708(3) | 0.634(16) | -
SU(2) | 2.0 | -0.92 | 12 | 24 | 7.323(10) | 0.8210(19) | 0.6102(8) | 0.7432(14) | 0.665(9) | -
SU(2) | 2.0 | -0.9 | 12 | 24 | 8.620(17) | 0.862(3) | 0.7183(14) | 0.832(2) | 0.770(16) | -
SU(2) | 2.0 | -0.88 | 10 | 20 | 8.120(11) | 0.885(2) | 0.8120(11) | 0.9169(19) | 0.842(5) | -
Sp(4) | 7.2 | -0.799 | 32 | 40 | 8.087(16) | 0.668(4) | 0.2527(5) | 0.377(2) | - | 0.36(5)
Sp(4) | 7.2 | -0.794 | 28 | 36 | 8.072(11) | 0.710(2) | 0.2882(4) | 0.4055(11) | 0.397(16) | -
Sp(4) | 7.2 | -0.79 | 24 | 36 | 7.505(19) | 0.742(6) | 0.3127(8) | 0.421(3) | 0.387(13) | 0.56(6)
Sp(4) | 7.2 | -0.78 | 24 | 36 | 8.882(17) | 0.793(4) | 0.3700(7) | 0.466(2) | 0.418(7) | -
Sp(4) | 7.2 | -0.77 | 24 | 36 | 10.16(2) | 0.829(5) | 0.4236(10) | 0.510(3) | 0.456(8) | -
Sp(4) | 7.2 | -0.76 | 16 | 36 | 7.544(17) | 0.850(4) | 0.4715(10) | 0.554(2) | - | 0.64(12)
Sp(4) | 6.9 | -0.924 | 24 | 32 | 8.208(12) | 0.663(2) | 0.3420(5) | 0.5157(17) | - | 0.46(3)
Sp(4) | 6.9 | -0.92 | 24 | 32 | 9.356(12) | 0.7036(17) | 0.3898(5) | 0.5540(12) | - | 0.42(2)
Sp(4) | 6.9 | -0.92 | 16 | 32 | 6.22(2) | 0.696(7) | 0.3889(14) | 0.558(5) | 0.49(3) | 0.45(6)
Sp(4) | 6.9 | -0.91 | 16 | 32 | 7.817(19) | 0.769(5) | 0.4885(12) | 0.634(4) | 0.560(14) | -
Sp(4) | 6.9 | -0.91 | 14 | 24 | 6.86(2) | 0.766(6) | 0.4902(16) | 0.639(5) | 0.541(9) | 0.41(3)
Sp(4) | 6.9 | -0.9 | 16 | 32 | 9.006(13) | 0.815(3) | 0.5629(8) | 0.690(2) | 0.611(9) | -
Sp(4) | 6.9 | -0.9 | 14 | 24 | 7.897(14) | 0.812(3) | 0.5641(10) | 0.694(2) | 0.619(16) | 0.57(4)
Sp(4) | 6.9 | -0.9 | 12 | 24 | 6.796(9) | 0.809(2) | 0.5663(8) | 0.6994(18) | 0.610(6) | 0.55(2)
Sp(4) | 6.9 | -0.89 | 14 | 24 | 8.813(19) | 0.843(4) | 0.6295(13) | 0.746(3) | 0.69(2) | 0.57(9)
Sp(4) | 6.9 | -0.89 | 12 | 24 | 7.581(15) | 0.841(3) | 0.6318(12) | 0.751(3) | 0.661(9) | 0.62(7)
Sp(4) | 6.9 | -0.87 | 12 | 24 | 8.925(10) | 0.878(2) | 0.7437(9) | 0.8468(17) | 0.782(13) | 0.80(15)
Sp(4) | 6.9 | -0.87 | 10 | 20 | 7.470(16) | 0.871(3) | 0.7470(16) | 0.857(3) | 0.764(9) | -
Table 3: The light spectrum of $\mathrm{SU}(2)$ and $\mathrm{Sp}(4)$ with
degenerate fermions: the lightest flavored pseudoscalar, $\pi$, and vector,
$\rho$, and flavor-singlet pseudoscalar, $\eta^{\prime}$, and scalar,
$\sigma$, mesons, measured in different ensembles. All dimensionful quantities
are given in lattice units and the omission of the lattice spacing $a=1$ is
understood. The missing entries for $m_{\eta^{\prime}}$ and/or $m_{\sigma}$
are due to the measurements not fulfilling the fitting criteria discussed in
the main text.

##  IV Results

Here we report the main results of our numerical investigations on the mass
spectrum of flavor-singlet pseudoscalar and scalar mesons, obtained using the
techniques discussed in the previous section. We focus on the $\mathrm{Sp}(4)$
theory coupled to two fundamental dynamical fermions, but for degenerate
fermions we supplement it with the $\mathrm{SU}(2)$ theory with the same
matter content. In the case of the pseudoscalar singlet, we further compare to
the existing literature on lattice results for the $\mathrm{SU}(3)$ theory.

Figure 1: (left panel) Mass ratios $m_{\rm meson}/m_{\pi}$ for pseudoscalar
and vector mesons, including the flavor-singlet pseudoscalar $\eta^{\prime}$,
in the $\mathrm{Sp}(4)$ gauge theory with $N_{f}=2$ Dirac flavors of fermions
in the fundamental representation measured at the two values of the inverse
coupling $\beta=6.9$ and $7.2$. (right panel) The same plot but in the
$\mathrm{SU}(2)$ gauge theory at $\beta=2.0$. The green solid lines
$m_{\rho}/m_{\pi}=1/x$ are displayed for reference.

###  IV.1 Pseudoscalar singlet in $\mathrm{SU}(2)$ and $\mathrm{Sp}(4)$ with
$N_{f}=2$

Our results for the mesons with degenerate fermions are tabulated in Tab. 3.
All the ensembles satisfy the condition $m_{\pi}L>6$, suggested by the
observations in Ref. [184] for $\mathrm{Sp}(4)$, and in Ref. [166] for
$\mathrm{SU}(2)$, that the size of finite volume corrections to the low-lying
spectrum for flavored mesons is of the order of $1\sim 2\%$ at $m_{\pi}L\simeq
6$, and becomes much smaller for the larger volumes, as it is exponentially
suppressed with the volume. This observation is also confirmed by our
measurements of $m_{\pi}$ and $m_{\rho}$ at different volumes in the
$\mathrm{Sp}(4)$ theory with $\beta=6.9$, by varying the bare fermion mass,
$m_{0}$. Finite volume corrections to $m_{\eta^{\prime}}$ are compatible with
the statistical uncertainties and expected to be less than $2\%$, which we
estimated from the most precise results available, for $m_{0}=-0.9$, if
$m_{\pi}L\gtrsim 6$. We therefore safely neglect finite volume corrections to
$m_{\eta^{\prime}}$ in the following.

In Fig. 1, we present our measurements of the ratios between the mass of the
$\eta^{\prime}$ meson and that of the pseudoscalar non-singlet $\pi$, as a
function of $m_{\pi}/m_{\rho}$. For reference, we indicate the mass of the
vector meson $\rho$ by a solid line. In the $\mathrm{Sp}(4)$ theory we find
that the pseudoscalar singlet is consistently heavier than the non-singlet,
over the range of $0.7\lesssim m_{\pi}/m_{\rho}\lesssim 0.9$, but lighter than
the vector mesons. While in the lightest and finest ensembles the hierarchy
between the pseudoscalar singlet and vector mesons is not yet clearly
resolved, the emerging trend is that $m_{\eta^{\prime}}/m_{\pi}$ slowly
increases as $m_{\pi}$ decreases in this mass regime, and approaches
$m_{\rho}/m_{\pi}$ for $m_{\pi}/m_{\rho}\lesssim 0.75$. We do not observe an
appreciable difference in the mass ratios obtained with the two different
values of $\beta$, within the quoted one-sigma error bars. We find a similar
trend in the $\mathrm{SU}(2)$ theory, as shown in the right panel of Fig. 1.
Since in this case only one, fairly coarse lattice is considered, we cannot
comment on the size of finite lattice spacing effects.

The smallness of lattice artifacts in the ratios of meson masses is somewhat
surprising, as the lattice spacing for $\beta=7.2$ is approximately $40\%$
smaller than for $\beta=6.9$ [184]. To assess this point, we present the meson
masses in units of the gradient flow scale $w_{0}$, which defines a common
scale in the continuum theory, and which we use also to compute the
topological charge $Q$—see Appendix A. We borrow the definition and
measurements of the gradient flow scale $w_{0}$ from Ref. [184], and refer the
reader to that publication for details. The left panel of Fig. 2 shows that
both the mass of the pseudoscalar singlet and the vector mesons receives
significant corrections from the finite lattice spacing. By comparing with
Fig. 1, we see that such corrections to $m_{\pi}w_{0}$ and
$m_{\eta^{\prime}}w_{0}$ happen to have the same sign and similar sizes, which
cancel out in the mass ratios. We observe the same pattern for the mass ratio
of $m_{\rho}$ and $m_{\eta^{\prime}}$, as depicted in the right panel of Fig.
2.

Figure 2: (left panel) Same data as in the left panel of Fig. 1, but now
masses are given in units of the gradient flow scale $w_{0}$. The blue solid
line $w_{0}m_{\pi}=x$ is displayed for reference. (right panel) Mass ratio
between of the vector mesons $\rho$ and the scalar singlet $\eta^{\prime}$.

###  IV.2 Pseudoscalar singlets in Sp(4) with $N_{f}=1+1$

$\beta$ | $m_{0}^{(1)}$ | $m_{0}^{(2)}$ | $L$ | $T$ | $m_{\pi^{0}}$ | $m_{\pi^{0}_{c}}$ | $m_{\pi^{\pm}}$ | $m_{\rho^{0}}$ | $m_{\rho^{\pm}}$ | $m_{\eta^{\prime}}$ | $m_{\sigma}$
---|---|---|---|---|---|---|---|---|---|---|---
6.9 | -0.9 | -0.89 | 14 | 24 | 0.60(2) | 0.597(2) | 0.596(2) | 0.723(3) | 0.719(4) | 0.65(4) | 0.45(6)
6.9 | -0.9 | -0.88 | 14 | 24 | 0.63(3) | 0.6261(18) | 0.628(2) | 0.749(2) | 0.745(3) | 0.69(4) | 0.51(8)
6.9 | -0.9 | -0.87 | 14 | 24 | 0.664(17) | 0.6510(14) | 0.6606(17) | 0.7731(19) | 0.775(2) | 0.77(3) | 0.69(14)
6.9 | -0.9 | -0.85 | 14 | 24 | - | 0.6889(19) | 0.709(2) | 0.813(3) | 0.813(4) | - | 0.51(12)
6.9 | -0.9 | -0.8 | 14 | 24 | 0.75(2) | 0.7563(16) | 0.8277(18) | 0.879(2) | 0.922(3) | 0.93(2) | -
6.9 | -0.9 | -0.75 | 12 | 24 | 0.79(3) | 0.803(4) | 0.921(4) | 0.927(6) | 1.005(5) | 1.04(3) | -
6.9 | -0.9 | -0.7 | 12 | 24 | 0.83(3) | 0.833(4) | 0.996(3) | 0.954(5) | 1.073(5) | 1.15(3) | -
Table 4: Results for the light spectrum with non-degenerate fermions: the
flavored pseudoscalar $\pi^{\pm}$ and vector $\rho^{\pm}$, and the flavor-
singlet pseudoscalar $\eta^{\prime}$, $\pi^{0}$, vector $\rho^{0}$ and scalar
$\sigma$ mesons. All dimensionful quantities are given in lattice units
($a=1$). Missing entries for $m_{\eta^{\prime}}$ or $m_{\sigma}$ denote
measurements that do not fulfill the fitting criteria discussed in the main
text. We furthermore report the mass of the flavor-singlet pseudo-Goldstone
$m_{\pi^{0}_{c}}$ in the connected-only approximation.

Figure 3:  Masses of the lightest non-singlet mesons as well as the
pseudoscalar singlet meson in the $\mathrm{Sp}(4)$ theory with non-degenerate
Dirac fermions. We fix the lattice coupling and one of the bare fermion masses
to $\beta=6.9$ and $m_{0}^{(1)}=-0.9$, respectively, while varying the other
bare fermion mass. In the left panel we display our results as a function of
the flavoured pion mass, while in the right panel we show them as a function
of ratio of the PCAC fermion masses in units of the $\pi^{0}$ mass.

For non-degenerate fermions, the theory contains two flavor-singlet
pseudoscalar mesons, the $\eta^{\prime}$ as well as the flavor-diagonal PNGB,
$\pi^{0}$. To understand the effects of (explicit) flavor-symmetry breaking on
the low-lying spectrum, we first choose the ensemble for degenerate fermions
with $\beta=6.9$ and $m_{0}=-0.9$ and vary the bare mass of one of the Dirac
fermion, $m_{0}^{(2)}\geq m_{0}$, for which we effectively increase its mass,
while keeping that of the other fixed, $m_{0}^{(1)}=m_{0}$. We summarize the
numerical results in Tab. 4. In the table, we also present the mass of the
flavor-singlet PNGB obtained by computing only the connected diagrams after
dropping the last three terms in Eq. 6, which we denote by $m_{\pi_{c}^{0}}$.
Within the generally larger uncertainties, we find no statistically
appreciable difference between $m_{\pi^{0}}$ and $m_{\pi_{c}^{0}}$, which
supports the connected-only approximation considered in Ref. [197].

In the left panel of Fig. 3, we show the meson masses, as a function of the
flavoured pNGBs mass $m_{\pi^{\pm}}$. In the mass-degenerate limit, we recover
the mass hierarchy of Sec.IV.1, as expected. As we increase one of the fermion
masses, we observe a clear separation between flavored mesons and the
unflavoured $\pi^{0}$ as well as the unflavoured vector mesons $\rho^{0}$,
with the former being heavier than the latter. At the same time, the
pseudoscalar singlet $\eta^{\prime}$, becomes heavier. This effect leads to an
inversion of the mass hierarchy between the vector mesons $\rho^{\pm}$,
$\rho^{0}$ and the $\eta^{\prime}$ meson and the $\eta^{\prime}$ becomes
heavier than any other meson considered here. 555This is qualitatively
different from the preliminary results in [225], where the mixing between the
pseudoscalar singlets was neglected and the excited state subtraction of Sec.
III.2 was incorrectly applied to the non-degenerate theory.

A couple of cautionary remarks should be added. First of all, we are in a
moderately heavy mass regime, with $m_{\pi^{0}}/m_{\rho^{0}}\sim 0.85$.
Secondly, some meson masses for heavy ensembles sit close to the lattice cut-
off and thus could be affected by significant lattice artifacts.

For the heaviest fermion masses, the mass difference between the
$\eta^{\prime}$ and the $\pi^{0}$ is approximately twice the mass difference
between the $\pi^{0}$ and the $\pi^{\pm}$’s. This indicates that the mass-
differences are driven by the valence fermion masses.

In this regime, the singlet $\pi^{0}$ and the non-singlet $\rho^{0}$ approach
an approximate one-flavour theory — see e.g. Ref. [170] for lattice results on
the low-lying spectrum in the $\mathrm{SU}(2)$ gauge theory with one Dirac
fermion.

In the right panel of Fig. 3 we plot the meson masses as a function of the
quark mass ratio, defined through the partially conserved axial current (PCAC)
relation. We identify the average quark mass in the flavored pion $\pi^{\pm}$
through the relation

$\displaystyle m^{\rm PCAC}_{\rm
avg.}=\lim_{t\to\infty}\frac{1}{2}\frac{\partial_{t}C_{\gamma_{0}\gamma_{5},\gamma_{5}}(t)}{C_{\gamma_{5}}(t)}=\lim_{t\to\infty}\frac{1}{2}\frac{\partial_{t}\int
d^{3}\vec{x}\langle\left(\bar{u}(\vec{x},t)\gamma_{0}\gamma_{5}d(\vec{x},t)\right)^{\dagger}\bar{u}(0)\gamma_{5}d(0)\rangle}{\int
d^{3}\vec{x}\langle\left(\bar{u}(\vec{x},t)\gamma_{5}d(\vec{x},t)\right)^{\dagger}\bar{u}(0)\gamma_{5}d(0)\rangle}.$
(16)

The unrenormalized PCAC quark mass ratio $m_{d}/m_{u}$ for non-degenerate
fermions is extracted by performing an additional measurement of the PCAC
average mass at degeneracy. For a detailed discussion of the PCAC relation and
different ways to calculate it, we refer to Ref. [239].

###  IV.3 Scalar singlet in Sp(4) with $N_{f}=2$

Figure 4: Mass of the $\sigma$ meson with degenerate fermions in the
$\mathrm{Sp}(4)$ theory. We find signs of finite spacing effects even when
considering ratios of hadron masses. On the coarser lattice the scalar singlet
appears to be quite light, in some cases even lighter than the $\pi$. For the
finer lattice this changes drastically and the scalar singlet $\sigma$ is
usually heavier than the vector meson $\rho$, though still below the two-$\pi$
threshold. The green solid line $m_{\rho}/m_{\pi}=1/x$ is displayed for
reference.

In the case of the scalar singlet, $\sigma$, the signal is consistently worse
than for the other states discussed so far. Furthermore, we see signs of
finite spacing effects, shown in Fig. 4. For $\mathrm{Sp}(4)$ on the coarse
$\beta=6.9$ lattices, we observe a light $\sigma$ state, of mass comparable to
the mass of the $\pi$. This pattern persists over the entire mass range
considered. On finer lattices, for $\beta=7.2$, the mass of the $\sigma$
increases and is heavier than the PNGBs, and comparable to the vector meson,
though with much larger statistical errors, and still below the $\pi\pi$
threshold.

These results suggest the existence of larger finite-spacing effects that
affect the mass of the scalar singlet. Yet, some caution should be used,
because, due to the large noise in our signal, the mass is extracted from much
shorter times on the finer lattices, and may therefore also be more severely
affected by excited-state contamination, and possibly other systematics.
Nevertheless, even for the finer lattice, the $\sigma$ state is lighter than
its non-singlet counterpart, suggesting that further studies are still needed.
The scalar singlet might be a stable light meson at moderately heavy fermion
masses, and thus phenomenologically relevant.

###  IV.4 Comparison to SU(3) with $N_{f}=2$

In Fig. 5 we show a compilation of the available data published on the
pseudoscalar singlet for the $\mathrm{SU}(3)$ theory with $N_{f}=2$ (upper
panels) as well a comparison of our results for $\mathrm{Sp}(4)$ and
$\mathrm{SU}(2)$ to the available data for $\mathrm{SU}(3)$ (lower panels). In
some cases, the measurement has been performed using different methods in the
analysis or different operators have been used to study the same mesons (e.g.
the mass of the $\eta^{\prime}$ has been obtained from pure gluonic operators
as well as the usual fermionic operators, or in the case of twisted mass
fermions the non-singlet mesons include isospin breaking effects) and sets of
results are available. In such cases, we have chosen the results that are
closest to the standard determination of directly fitting the correlator of a
pure fermionic operator. When this was not possible we quote the largest and
smallest values of $m_{i}\pm\Delta m_{i}$ of all measurements $i$ and
symmetrize the uncertainties. The data depicted in Fig. 5 has been taken from
the UKQCD collaborations (denoted by UKQCD1 [202, 203] and UKQCD2 [207]); the
SESAM/T$\chi$L collaboration [204, 205]; the CP-PACS collaboration [206]; the
RBC collaboration using domain-wall fermions [210]; the CLQCD collaboration
using Wilson clover fermions on anisotropic lattices [211]; the ETMC
collaboration (denoted by ETMC1 [208, 209] and ETMC2 [212, 213]); and from the
analysis of $\eta^{\prime}$-glueball mixing (denoted by Beijing [214]).

In all but the very lightest ensemble (and one obvious outlier at heavy
fermion mass) the vector meson, $\rho$, is found to be heavier than the
pseudoscalar singlet, $\eta^{\prime}$. The authors of Ref. [213] point out
that in the lightest ensemble the $\rho$ particle might be unusually light due
to the small number of energy levels below the inelastic threshold in the
determination of the $\pi\pi$ phase shift. It is lighter than their
extrapolation to the physical point at which $m_{\rho}=786(20)$ and even
lighter than their extrapolation to the chiral limit. The mass dependence of
the $\eta^{\prime}$ meson was found to be flat and an extrapolation in Ref.
[212] to the physical point gave $m_{\eta^{\prime}}=772(18)\rm{MeV}$. This is
in contrast to SM QCD where the $\eta^{\prime}$ is significantly heavier—the
current PDG lists $m_{\eta^{\prime}}^{\rm{PDG}}=957.78(6)\rm{MeV}$ [240],
which is in agreement with recent $\mathrm{SU}(3)$, $N_{f}=2+1$ lattice
results of $m_{\eta^{\prime}}=929.9\left(\begin{subarray}{c}47.5\\\
21.0\end{subarray}\right)$ [241]. This suggests it is the contribution of the
s-quark that leads to the heavier mass. This can be understood in a quark
model of the pseudoscalar singlet mesons based on approximate
$\mathrm{SU}(3)_{F}$ flavor symmetry [242, 243, 244] which was applied to
early lattice results in [203].

The bottom line of this brief survey is that in the regime of moderately large
fermion masses the pattern of ground state masses observed so far in $SU(3)$
is quite similar, both qualitatively and quantitatively, to our findings in
the $\mathrm{Sp}(4)$ case as can be seen in the lower panels of Fig. 5. The
gauge group and modified chiral structure do not seem to have a very strong
impact on mass of the $\eta^{\prime}$.

Figure 5: Comparison to the available lattice data in $\mathrm{SU}(3)$ with
two fundamental fermions. The upper panels depict all available lattice
results in $\mathrm{SU}(3)$. In the upper left panel the green markers denote
the vector meson $\rho$ and the orange ones the pseudoscalar singlet
$\eta^{\prime}$. The different marker shapes denote the different
collaborations. We see that the $\eta^{\prime}$ is lighter than the vector
mesons almost everywhere. In the upper right panel we directly plot the ratio
$m_{\eta^{\prime}}/m_{\rho}$. In the lower panels we compare the
$\mathrm{SU}(3)$ results to our $\mathrm{Sp}(4)$ and $\mathrm{SU}(2)$ data. In
the lower left panel, we show the ratio $m_{\eta^{\prime}}/m_{\pi}$ as a
function of $m_{\pi}/m_{\rho}$ for values of $m_{\pi}/m_{\rho}\approx 0.7$ and
larger. In the lower right plot we compare the different results of the ratio
$m_{\eta^{\prime}}/m_{\rho}$.

###  IV.5 Possible phenomenological implications

Our results provide evidence that the singlet sector, computed for moderately
large fermion masses in the $\mathrm{Sp}(4)$ theory, is not dissimilar from
what is observed in the $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$ theories coupled
to two fundamental fermions. In particular, both pseudoscalar and scalar
singlets are light enough to be stable against decay into Goldstone bosons,
over a fermion-mass range within which also the flavored vector mesons would
not decay. We now present a few examples of potential implications for
phenomenological models for which these theories can be invoked to yield a
short-distance completion.

Firstly, because the flavor singlets are not much heavier than the flavored
mesons, if a model of this type were used as part of a hidden valley scenario,
or a new dark sector, these states would then only decay via a mediator
mechanism into standard model particle, but not strongly, and would be long-
lived. Their lifetime and branching fractions would be determined by the
detailed structure of the coupling to the standard-model fields. They are
unlikely to be long-lived enough to play a significant role in a model
explaining current dark matter density, yet they can easily appear as long-
lived particles in experiments [245, 246, 247, 248], and hence the existence
of a new dark sector containing this theory is experimentally testable.

Other possible observable effects in this context could arise because the
singlets can enhance interaction cross-sections, as virtual particles,
affecting processes even below production threshold. They can therefore play a
relevant role for dark matter self-interactions [249]. Their effect could even
affect form factors relevant to direct detection experiments [250]. Depending
on details, they could also offer a possibility to create indirect detection
signatures in cases of high dark matter densities. Finally, both singlets can
serve, together or individually, as a Higgs portal, removing the need for an
independent messenger.

In the alternative context of composite Higgs scenarios, in which the PNGBs
provide the longitudinal components of the $W$-bosons and $Z$ boson, as well
giving rise to the experimentally observed Higgs boson, the pseudoscalar
singlet can become a surprisingly strong limiting factor [251]. As its
signature is possibly similar to that of the pseudoscalar Higgs in the minimal
supersymmetric standard model, or in classes of two-Higgs doublet models,
strong exclusion limits already exist, both for a pseudoscalar Higgs heavier
and lighter than the standard model Higgs. To avoid these bounds requires to
open up substantially the mass gap between the scalar and pseudoscalar
singlets, but in our measurements we always observe the opposite hierarchy.

##  V Summary

We have presented the results of the first dedicated lattice study of flavor
singlet meson states in the $\mathrm{Sp}(4)$ theory coupled to two (Wilson-
Dirac) fundamental, dynamical fermions. We have computed the masses of the
lightest pseudoscalar and scalar singlets in a portion of parameter space in
which the fundamental fermion are moderately heavy. We have considered both
the case of degenerate and of non-degenerate masses for the fermions. The
continuum limit of this theory, in the range of parameters explored, is of
interest because it provides the ultraviolet completion of several proposals
for new physics extensions of the standard model, in the contexts of composite
Higgs models and strongly interacting dark matter. In order to perform this
study, we implemented in our analysis state-of-the-art techniques to account
for the contribution of disconnected diagrams to correlation functions
involving flavor singlets.

We observe that the qualitative (and to large extent even the quantitative)
features of the mass spectrum we find in this $\mathrm{Sp}(4)$ theory are
similar to those of $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$ theories with the
same field content, in comparable ranges of parameter space. More
specifically, the mass range of the singlet states, in particular of the
lightest pseudoscalar, is comparable to the masses of the lightest flavored
mesons, at least for our choices of fermion masses. This remains true also in
the mass-non-degenerate case.

Our findings suggest that the singlet sector cannot be neglected in
phenomenological studies of models that have their dynamical, short-distance
origin in this theory. However, notwithstanding the technical implementation
of several techniques to enhance the signal-to-noise ratio in our
measurements, and the comparatively large statistics provided by our numerical
ensembles, we have also found that the observables are affected by large
lattice artifacts, especially in the case of the scalar singlet. While we have
noticed that taking certain ratios of masses reduces drastically the size of
such effects, if phenomenological considerations require precision
measurements for the mass spectrum, then this would provide strong incentive
to further improve this study, in particular in order to better understand the
approach to the continuum limit.

On more general and abstract theoretical ground, the similarity of our main
results with the $\mathrm{SU}(N)$ cases strongly suggests that the altered
chiral structure and gauge group has limited impact on the underlying
dynamics. On the one hand, this might be expected in a gauge theory with small
number of moderately heavy fermions. On the other hand, though, by extending
this kind of analysis to different $N$ and/or further gauge groups we envision
to be able to gain quantitative understanding the relevance of gauge dynamics
for hadron dynamics beyond group-theoretical, and thus non-dynamical, aspects.

###### Acknowledgements.

We are grateful to S. Kulkarni for helpful discussions and to the authors of
[252, 194] for providing access to the smearing code in HiRep prior to
publication. FZ is supported by the Austrian Science Fund research teams grant
STRONG-DM (FG1) and acknowledges travel support for part of this work by the
city of Graz. The work of EB is 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 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 BL and MP has been supported in
part by the STFC Consolidated Grants No. ST/P00055X/1 and 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 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 HH is supported by the
Taiwanese MoST grant 109-2112-M-009-006-MY3. The computations have been
partially performed on the HPC cluster of the University of Graz and on the
Vienna Scientific Cluster (VSC4), and partially on the DiRAC Data Intensive
service at Leicester. The DiRAC Data Intensive service at Leicester is
operated by the University of Leicester IT Services, and 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) licence to any Author Accepted Manuscript version
arising. Research Data Access Statement—The data generated for this manuscript
can be downloaded from Ref. [253], and the software used to analyze and
present it is similarly available from Ref. [254].

##  Appendix A Constant contributions to the correlators

In Sect. III.1 we noted the occurrence of constant terms in the propagators of
both the pseudoscalar singlet $\eta^{\prime}$ meson and the scalar singlet
$\sigma$ meson. This makes it difficult to determine when the excited states
in the meson correlator are sufficiently suppressed and a fit can be
performed. As shown in Eqs. (6) and (8), we can circumvent this issue either
by direct calculation of $\langle 0|O|0\rangle$, or by performing a numerical
derivative. Once we determine the interval $[t_{i},t_{f}]$ where only the
ground state contributes, we can also fit the correlator to an exponentially
decaying term plus a constant. In Fig. 6 we give an example of the correlator
for the flavor singlet, $\eta^{\prime}$, and the flavored mesons, $\pi$. The
flavor-singlet correlator shows a constant term at large Euclidean times while
such a contribution is absent for the $\pi$ meson. This is expected to occur
for the disconnected pieces in a finite volume and at finite statistics if the
topological sampling is insufficient [234, 212]. Then, the constant takes the
form

$\displaystyle\mathrm{const}\propto\frac{1}{V}\left(\frac{Q^{2}}{V}-\chi_{t}-\frac{c_{4}}{2\chi_{t}V}\right)+\mathcal{O}(V^{-3})+\mathcal{O}(e^{-m_{\pi}|x|}),$
(17)

where $V$ denotes the spatial volume of the lattice, $\chi_{t}$ is the
topological susceptibility and $c_{4}$ is a coefficient from a saddle-point
expansion. Algorithms that provide an ergodic exploration of topological
sectors in Yang-Mills theories and are compatible with our prescription for
the boundary conditions have recently been introduced (see, e.g., [255, 256,
257]). However, they are generally computationally costly, and their
adaptation to our model is outside the scope of this work. Hence, in our
study, we explore other strategies, based on analyses that account for
topological freezing. Therefore, we tried to test the relation in Eq. (17) by
taking our ensemble with the largest statistics (corresponding to the bare
parameters $m_{0}=-0.90$, $\beta=6.9$ on a $24\times 12^{3}$ lattice),
measuring the topological charge $Q$ using the same approach as in Ref. [184],
and smoothening the gauge fields using the gradient flow. We then partition
our full statistics into sets of configurations with equal topological
charge666In practice, the topological charge is not strictly an integer on a
finite lattice in the employed approach as described for instance in [184] and
references therein. We thus round $Q$ to the closest integer. and compute the
correlator for the pseudoscalar singlet, $\eta^{\prime}$, at fixed $Q$. We
depict examples of the correlators for some values of $Q$ with sufficient
statistics in Fig. 6. The constant term arising is never statistically
different for any pair of $Q$’s present in this ensemble. However, we see a
slight trend towards a larger constant for larger $|Q|$ as expected from Eq.
(17).

Figure 6: (left) Correlator of the pseudoscalar non-singlet $\pi$ and the
pseudoscalar singlet $\eta^{\prime}$. For visual clarity we multiplied the
$\pi$ correlator by a constant factor of $2$. At large times the singlet
correlator shows a constant term, while this is absent for the non-singlet
case. (right) Correlators of the pseudoscalar singlet for fixed values of the
topological charge $Q$. The constant term shows signs of a dependence on $Q$.
While the constant is not significantly different for any two examples shown
here, the constant appears to be increasing with $|Q|$.

In order to test the robustness of our subtraction choice, we report here the
mass of the pseudoscalar singlet $\eta^{\prime}$ for various techniques. We
remind the reader that the results reported in Sec. IV are based on
correlators where the connected part is modelled by a single sum of
exponentials as in Eq. (8), taking lattice periodicity into account and
removing the constant by a numerical derivative. In Tab. 5 we compare this to
four alternative methods:777We also applied an entirely different method
designed for situations with large statistical noise [258]. Also, this
approach gave results consistent with those shown in the main part of the
paper for both singlets. (i) Direct calculation and subtraction of $\langle
0|O_{\eta^{\prime}}|0\rangle$, (ii) Ignoring the constant and restricting the
fit to early time-slices, (iii) Performing a three-parameter fit of the
decaying exponential plus a term modelling the constant,888This procedure
gives the numerical value of the constant as a byproduct. We subtract the
constant from the correlator and _a posteriori_ check that the resulting
effective mass shows a plateau. (iv) Removing the constant using a numerical
derivative but without any modelling of the connected part.

Whenever we obtain a signal without an explicit modelling of the connected
pieces our results agree within errors. The removal of excited state
contamination (as used in methods (i), (ii) and (iii)) leads to masses that
are generally slightly lighter. The same pattern has been observed in
$\mathrm{SU}(3)$ [212]. We note that the removal of excited state
contaminations should not be confused with the removal of the constant
contribution to the correlator, as discussed earlier. The explicit calculation
of the constant $\langle 0|O_{\eta^{\prime}}|0\rangle$ in Eq. (8) does not
quantitatively capture the constant in the correlator. The results are almost
indistinguishable from not taking the constant into account. For some
ensembles (e.g. $\mathrm{Sp}(4)$ with $\beta=7.2$) these methods appear to
underestimate the meson mass. This is a result of combining the modelling of
the connected piece with an insufficient subtraction of the constant. Due to
the absence of connected excited states in the correlator, the effective mass
is increasing at small $t$, while for large $t$ the constant leads to a
decrease of the effective masses. This can lead to the formation of an
apparent plateau in the effective mass and thus to a possible underestimation
of the meson mass. Overall, we conclude that methods (ii) and (iii) do not
appear sufficiently reliable. Modelling the constant as an additional fit
parameter did not lead to any significant improvements. In most cases we
cannot extract a reliable signal. In the few cases where this is possible the
constant term is quantitatively small and this method agrees with the others
while providing no improvement at the cost of an additional fit parameter.

We conclude that the method used throughout the main part of this work has
proven to be the most reliable approach among the options considered here. Its
results are always consistent with forgoing the explicit removal of subtracted
states, and the removal of the additional constant through taking the
derivative avoids any further estimations of the topological constant terms at
the expense of a shorter plateau in the effective masses and thus, a smaller
interval for fitting the correlator.

We find a different behavior for the scalar singlet. The constant term is not
related to an insufficient sampling of all topological sectors but arises due
to the vacuum quantum numbers of the scalar singlet. In addition, the
modelling of the connected pieces is less important, since the non-singlet
state appears generally heavier than the singlet states and the connected
pieces show a stronger exponential decay. In this case the direct estimation
of the constant term $\langle 0|O_{\sigma}|0\rangle$ in Eq. (8) appears to be
quantitatively reliable. Still, in some cases the modelling of the connected
pieces can extend the plateau in the effective mass to lower timeslices $t$.
Since the constant is several order of magnitude larger than the actual signal
of the $\sigma$ state, a direct modelling of it as a fit parameter is
infeasible and the constant can also not be ignored in the analysis. In Tab. 6
we compare the approach used in the main part of this paper to: (i) both a
numerical derivative and a direct calculation and subsequent subtraction of
the vacuum term $\langle 0|O_{\sigma}|0\rangle$, (ii) only direct subtraction
of the vacuum terms as in [215], (iii) a numerical derivative without an
explicit subtraction of excited states in the connected pieces and without a
direct subtraction of $\langle 0|O_{\sigma}|0\rangle$.

| $\beta$ | $m_{0}$ | $L$ | $T$ | $m_{\eta^{\prime}}$ | $m_{\eta^{\prime}}^{\rm(i)}$ | $m_{\eta^{\prime}}^{\rm(ii)}$ | $m_{\eta^{\prime}}^{\rm(iii)}$ | $m_{\eta^{\prime}}^{\rm(iv)}$
---|---|---|---|---|---|---|---|---|---
SU(2) | 2.0 | -0.947 | 20 | 32 | - | - | - | - | -
SU(2) | 2.0 | -0.94 | 14 | 24 | 0.67(6) | 0.699(14) | 0.572(14) | - | 0.67(6)
SU(2) | 2.0 | -0.935 | 16 | 32 | 0.60(3) | 0.67(3) | 0.61(5) | 0.60(3) | -
SU(2) | 2.0 | -0.93 | 14 | 24 | 0.65(3) | - | - | - | 0.68(5)
SU(2) | 2.0 | -0.925 | 14 | 24 | 0.634(16) | - | - | - | 0.63(8)
SU(2) | 2.0 | -0.92 | 12 | 24 | 0.665(9) | - | - | - | 0.66(3)
SU(2) | 2.0 | -0.9 | 12 | 24 | 0.770(16) | - | - | - | 0.79(8)
SU(2) | 2.0 | -0.88 | 10 | 20 | 0.842(5) | 0.855(14) | 0.855(14) | - | -
Sp(4) | 7.2 | -0.799 | 32 | 40 | - | 0.37(2) | 0.37(2) | - | 0.57(6)
Sp(4) | 7.2 | -0.794 | 28 | 36 | 0.397(16) | 0.368(12) | 0.368(12) | - | 0.47(7)
Sp(4) | 7.2 | -0.79 | 24 | 36 | 0.387(13) | - | - | - | 0.36(6)
Sp(4) | 7.2 | -0.78 | 24 | 36 | 0.418(7) | 0.43(2) | 0.45(2) | - | 0.43(5)
Sp(4) | 7.2 | -0.77 | 24 | 36 | 0.456(8) | 0.450(6) | 0.450(6) | 0.459(7) | -
Sp(4) | 7.2 | -0.76 | 16 | 36 | - | 0.511(13) | 0.512(13) | - | 0.59(3)
Sp(4) | 6.9 | -0.924 | 24 | 32 | - | - | - | - | 0.60(8)
Sp(4) | 6.9 | -0.92 | 24 | 32 | - | 0.51(4) | 0.52(4) | 0.486(16) | 0.40(6)
Sp(4) | 6.9 | -0.92 | 16 | 32 | 0.49(3) | 0.46(2) | 0.46(2) | 0.50(2) | 0.45(13)
Sp(4) | 6.9 | -0.91 | 16 | 32 | 0.560(14) | 0.59(4) | 0.59(4) | 0.560(13) | 0.59(4)
Sp(4) | 6.9 | -0.91 | 14 | 24 | 0.541(9) | 0.58(3) | 0.58(3) | - | -
Sp(4) | 6.9 | -0.9 | 16 | 32 | 0.611(9) | - | - | - | 0.63(3)
Sp(4) | 6.9 | -0.9 | 14 | 24 | 0.619(16) | 0.614(12) | 0.615(12) | 0.620(9) | 0.63(3)
Sp(4) | 6.9 | -0.9 | 12 | 24 | 0.610(6) | 0.620(15) | 0.620(15) | 0.612(5) | -
Sp(4) | 6.9 | -0.89 | 14 | 24 | 0.69(2) | 0.680(16) | 0.681(16) | 0.69(2) | 0.72(4)
Sp(4) | 6.9 | -0.89 | 12 | 24 | 0.661(9) | 0.660(5) | 0.660(5) | 0.660(10) | -
Sp(4) | 6.9 | -0.87 | 12 | 24 | 0.782(13) | 0.80(4) | 0.80(4) | - | 0.80(2)
Sp(4) | 6.9 | -0.87 | 10 | 20 | 0.764(9) | 0.763(6) | 0.763(6) | - | -
Table 5: Determination of the masses of the pseudoscalar singlets using different techniques for removing the constant term in the correlator. We compare the method used in the main part of this work to: (i) Direct calculation and subtraction of $\langle 0|O_{\eta^{\prime}}|0\rangle$; (ii) Ignoring the constant and restricting the fit to early time-slices; (iii) Performing a three-parameter fit of the decaying exponential plus a term modelling the constant; and (iv) Removing the constant using a numerical derivative but without any modelling of the connected part. | $\beta$ | $m_{0}$ | $L$ | $T$ | $m_{\sigma}$ | $m_{\sigma}^{(i)}$ | $m_{\sigma}^{\rm(ii)}$ | $m_{\sigma}^{\rm(iii)}$
---|---|---|---|---|---|---|---|---
SU(2) | 2.0 | -0.947 | 20 | 32 | 0.53(4) | 0.53(3) | 0.58(5) | 0.54(4)
SU(2) | 2.0 | -0.94 | 14 | 24 | - | 0.64(5) | 0.61(4) | 0.64(5)
SU(2) | 2.0 | -0.935 | 16 | 32 | - | 0.47(10) | 0.55(8) | 0.47(10)
SU(2) | 2.0 | -0.93 | 14 | 24 | - | - | 0.62(9) | -
SU(2) | 2.0 | -0.925 | 14 | 24 | - | - | - | -
SU(2) | 2.0 | -0.92 | 12 | 24 | - | 0.71(7) | 0.75(13) | 0.72(7)
SU(2) | 2.0 | -0.9 | 12 | 24 | - | - | - | -
SU(2) | 2.0 | -0.88 | 10 | 20 | - | - | - | -
Sp(4) | 7.2 | -0.799 | 32 | 40 | 0.36(5) | 0.35(8) | 0.41(4) | 0.38(8)
Sp(4) | 7.2 | -0.794 | 28 | 36 | - | 0.55(8) | - | 0.55(8)
Sp(4) | 7.2 | -0.79 | 24 | 36 | 0.56(6) | 0.56(6) | 0.48(7) | 0.65(12)
Sp(4) | 7.2 | -0.78 | 24 | 36 | - | - | 0.55(6) | -
Sp(4) | 7.2 | -0.77 | 24 | 36 | - | - | - | -
Sp(4) | 7.2 | -0.76 | 16 | 36 | 0.64(12) | 0.64(7) | - | -
Sp(4) | 6.9 | -0.924 | 24 | 32 | 0.46(3) | 0.46(3) | 0.45(3) | 0.48(7)
Sp(4) | 6.9 | -0.92 | 24 | 32 | 0.42(2) | 0.43(3) | 0.45(3) | -
Sp(4) | 6.9 | -0.92 | 16 | 32 | 0.45(6) | 0.40(8) | 0.42(7) | 0.37(11)
Sp(4) | 6.9 | -0.91 | 16 | 32 | - | - | - | 0.71(8)
Sp(4) | 6.9 | -0.91 | 14 | 24 | 0.41(3) | 0.41(3) | - | -
Sp(4) | 6.9 | -0.9 | 16 | 32 | - | - | - | -
Sp(4) | 6.9 | -0.9 | 14 | 24 | 0.57(4) | 0.56(4) | 0.48(5) | 0.51(10)
Sp(4) | 6.9 | -0.9 | 12 | 24 | 0.55(2) | 0.56(4) | - | 0.57(3)
Sp(4) | 6.9 | -0.89 | 14 | 24 | 0.57(9) | 0.56(9) | 0.61(9) | 0.59(9)
Sp(4) | 6.9 | -0.89 | 12 | 24 | 0.62(7) | - | 0.64(4) | -
Sp(4) | 6.9 | -0.87 | 12 | 24 | 0.80(15) | 0.76(11) | - | 0.72(7)
Sp(4) | 6.9 | -0.87 | 10 | 20 | - | 0.70(6) | - | -
Table 6: Results for the masses of the scalar singlet $\sigma$ using our
standard approach of a numerical derivative as well as (i) both a numerical
derivative and a direct calculation of the vacuum term $\langle
0|O_{\sigma}|0\rangle$, (ii) only direct calculation of the vacuum term, and
(iii) a numerical derivative without an explicit subtraction of excited states
in the connected piece when possible.

##  Appendix B Comparison between excited state subtraction and smeared
connected diagrams

In Sect. III.2 we noted that the signal of the singlet mesons can be extended
to smaller time separations, $t$, if we replace its connected contribution by
approximating it with a single exponential term having the energy of the non-
singlet meson. This removes all the excited state contaminations of the
connected piece.

A similar effect can be obtained by using smearing techniques on the connected
piece. This can increase the overlap of the source operator with the ground
state of the non-singlet and reduce the contribution of excited states.
Recently, this approach has been implemented, tested, and shown to work in
$\mathrm{Sp}(4)$ gauge theories [194]. These developments allow us to compare
our excited-state subtraction technique.999We thank the authors of [194, 252]
for performing smeared measurements on a set of our configurations for
comparison prior to publication.

In order to compare the two techniques we need to apply smearing to only the
connected piece. However, the use of smearing techniques leads to an overall
change of normalization. Applying Wuppertal smearing [259] with $N_{1}$ steps
at the source and $N_{2}$ steps at the sink leads to an asymptotic correlator
of the form

$\displaystyle
C_{N_{1},N_{2}}(t\to\infty)=\alpha_{N_{1}}\alpha_{N_{2}}e^{-m_{\rm conn}t}\,,$
(18)

where the normalization of unsmeared point sources is recovered for the choice
of the parameters $\alpha_{N_{1}}=\alpha_{N_{2}}=\alpha_{0}$. We consider two
sets of correlators with the smearing steps $(N_{1},N_{2})=(N,0)$ and
$(N_{1},N_{2})=(N,N)$, to restore the normalization as the point source. We
define a new correlator

$\displaystyle C^{\rm smeared}_{\rm
conn.}(t)\equiv\frac{C_{N,0}(t)^{2}}{C_{N,N}(t)}\,,$ (19)

by squaring the connected correlator with $N$ steps of source smearing and no
sink smearing and divide that by the connected correlator with an equal amount
of smearing steps at both the source and the sink. This correlator has the
same large-$t$ behavior and the same normalization as a non-smeared one. From
this we then construct the full correlator of the singlet meson. In Fig. 7 we
compare the full singlet correlator obtained from Eq. (19) using Wuppertal
smearing with $N=60$ smearing steps, to the correlator obtained using the
single-exponential modelling and subtraction of the connected piece. We see
that the subtracted correlator and the smeared correlator agree remarkably
well in the interesting, plateau region.

Figure 7: Comparison between excited state subtraction (orange squares),
obtained by modelling the connected part of the correlator as a single
exponential term, with smeared operators (green pentagon), on a preliminary
set of configurations, for a single ensemble. For reference, we also plot the
correlator without excited state subtraction and without smearing (blue
circles).

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