Singlets in gauge theories with fundamental matter

We provide the first determination of the mass of the lightest flavor-singlet pseudoscalar and scalar bound states (mesons), in the $\rm{Sp}(4)$ 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 $\rm{SU}(2)$ and $\rm{SU}(3)$ 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.

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 SU(N ) group, with matter in various numbers and representations.Besides the aforementioned work, and Refs.[143,144,[158][159][160][161][162] on SU (3), see e.g.Refs.[163][164][165][166][167][168][169][170][171] for SU(2) theories, Refs.[172][173][174][175] for SU (4) in multiple representations, and [176] for 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 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 Sp(2N ) gauge theory with mixedrepresentation 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 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 [67,69,71,72,200].This study is complementary to the available non-singlet hadron spectrum found in Refs.[182, 184-187, 189, 190, 197].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 Sp(2N ) gauge theories differs from that of 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.
The paper is organized as follows.The pseudo-real nature of the fundamental representation of Sp(4)-as for 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 A. 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 B, and in more detail in the Appendices.
We present the body of our numerical results in Sect.IV.Section IV A is devoted to the lightest pseudoscalar singlet state, in both the Sp(4) and SU(2) theories, for degenerate masses.We report on the case of non-degenerate flavor masses for Sp(4) (which realizes a scenario relevant to dark matter models) in Sect.IV B. The scalar singlet sector, in the degenerate case for the Sp(4) theory, is the subject of Sect.IV C, 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 SU(3) case, and report the results in Sect.IV D. 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.

A. Microscopic theory and global symmetries
The Sp(2N ) gauge theories of interest are characterized by a Lagrangian density, L, which in this section we write using a metric with Lorentzian signature (+1, −1, −1, −1), and takes the form: where G µν ≡ A G A µν t A are the field strength tensors, and t A are the generators of Sp(2N ), normalized so that Tr T A T B = 1 2 δ AB , while u and d are 4-component Dirac spinors, denoting the two flavors of fermion fields transforming in the fundamental representation of Sp(2N ).The Lagrangian is real and Lorentz invariant, as ū ≡ u † γ 0 , and d ≡ d † γ 0 .The covariant derivatives are defined in terms of the gauge fields where g is the gauge coupling.Explicitly, the field-strength tensor is given by where [•, •] denotes the commutator.The fundamental representation of 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, 2 that for each Dirac fermion the U(1) L ×U(1) R Abelian global symmetry acting on its chiral projections is enhanced to a non-Abelian U(2) global symmetry.The fermion kinetic terms hence, written in terms of covariant derivatives, exhibit an enhanced (classical) U(4) = U (1) A × 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 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 ̸ = m d , the approximate global symmetry is further broken to Sp(2) 2 ∼ SU(2) 2 [197].

B. 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 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 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 Sp(4) transformations, and attribute their J P quantum numbers. 3As the 2-component fermions transform as a 4 of Sp(4), the multiplication properties imply that there exist mesons transforming as a 1, 5, and 10 of Sp (4).
Starting from the spin-0 states, one expects to find in the spectrum 5 PNGBs, spanning the SU(4)/Sp(4) coset, and transforming as a 5 of Sp(4), to become massless in the m u = m d → 0 limit. 4These 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 SU(2) and Ref. [227] for SU(3), both with two flavors of fundamental fermions-because the 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 → 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 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 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 ρ mesons; they transform as a 10 of 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 ψγ µ ψ and ψσ µν ψ, respectively.In addition, the generalizations of the a 1 and b 1 from QCD transform as a 5 and a 10 of 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 ω in QCD.
In the presence of non-degenerate fermion masses, m u ̸ = m d , the global symmetry breaks further from Sp(4) down to Sp(2) × Sp(2) = SU(2) × SU(2) ∼ 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 π 0 becomes a singlet, with different mass from the charged π ± 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 ρ 0 and the ω 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 U(1) subgroup of the 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 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.I and II.
We perform simulations on Euclidean lattices of size T × L 3 and define the bare inverse gauge coupling as β = 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.

A. 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 η ′ and σ, respectively, the pseudoscalar and scalar, flavor-singlet states.While the mesonic sectors are enlarged in 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 ππ operators, glueballs, or even tetraquarks).We use the operators where n = (⃗ n, t) denote lattice sites.For pseudoscalar mesons Γ = γ 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, η ′ .The same pattern persists for the scalar mesons where Γ = 1, and we use the notation: For vector mesons Γ = γ i and all operators O 1 and O ± 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  I. List of all ensembles with degenerate fermion masses used in this work.We report the number of configurations n conf , the number of the stochastic sources used in the approximation of the all-to-all quark propagator nsrc, the intervals for fitting the resulting meson correlators Imeson and the average value of the plaquette ⟨P ⟩.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 Iρ ⟨P ⟩ Sp4B1L2M4ND1 6.9 -0.9 -0.89 14 II.List of all ensembles with non-degenerate fermion masses used in this work.We report the number of configurations n conf , the number of the stochastic sources used in the approximation of the all-to-all quark propagator nsrc, the intervals for fitting the resulting meson correlators Imeson and the average value of the plaquette ⟨P ⟩.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. functions. .
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, on a Euclidean time interval (t min , t 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 ∝ exp (−E n t), 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 η ′ and the σ 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, σ, 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 lim t→∞ where the second exponential term is due to the lattice periodicity.While in the case of the η ′ this constant is small compared to the signal and only affects the correlator at large t, this is not the case for the σ 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 antisymmetric with respect to the midpoint T /2, In order to determine the Euclidean time interval for fitting we use an effective mass m eff (t) defined by 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 min , t 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 C(t) for the mesons.In Appendix A we compare this method to computing the additional constant ⟨0|O|0⟩ 2 directly, without the use of a numerical derivative.
For the pseudoscalar sector in the non-degenerate N f = 1 + 1 theory both the π 0 and η ′ are pseudoscalar singlets, and the η ′ is not a groundstate.Thus, we need to perform a variational analysis by computing the correlation matrix of the operators O π 0 and O η ′ and solve the resulting generalized eigenvalue problem (GEVP).In the minimal operator basis of ( 4) and ( 5) the cross-correlator are diagrammatically given by In the mass-degenerate limit the cross-correlator vanishes and the η ′ becomes the ground state of the pseudoscalar singlet sector, whereas the π 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 P S A = ūγu and O P S B = dγd.The corresponding cross-correlator vanishes as the heavier fermion mass approaches infinity and is given by

B. 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.I and II for our choice of fit intervals) to a single exponential and replace the full connected piece by the ground state correlator [237], where A 0 and m conn are the fit parameters, such that 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 η ′ 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 × Z 2 noisy sources with spin and even-odd dilution [238].We typically use O(100) distinct noise vectors.The connected pieces are evaluated using stochastic wall sources.Uncertainties are estimated using the jackknife method.

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 Sp(4) theory coupled to two fundamental dynamical fermions, but for degenerate fermions we supplement it with the 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 SU(3) theory.
A. Pseudoscalar singlet in SU(2) and Sp(4) with N f = 2 Our results for the mesons with degenerate fermions are tabulated in Tab.III.All the ensembles satisfy the condition m π L > 6, suggested by the observations in Ref. [184] for Sp(4), and in Ref. [166] for SU (2), that the size of finite volume corrections to the low-lying spectrum for flavored mesons is of the order of 1 ∼ 2% at m π L ≃ 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 π and m ρ at different volumes in the Sp(4) theory with β = 6.9, by varying the bare fermion mass, m 0 .Finite volume corrections to m η ′ 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 π L ≳ 6.We therefore safely neglect finite volume corrections to m η ′ in the following.
In Fig. 1, we present our measurements of the ratios between the mass of the η ′ meson and that of the pseudoscalar non-singlet π, as a function of m π /m ρ .For reference, we indicate the mass of the vector meson ρ by a solid line.In the Sp(4) theory we find that the pseudoscalar singlet is consistently heavier than the non-singlet, over the range of 0.7 ≲ m π /m ρ ≲ 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 η -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  slowly increases as m π decreases in this mass regime, and approaches m ρ /m π for m π /m ρ ≲ 0.75.We do not observe an appreciable difference in the mass ratios obtained with the two different values of β, within the quoted one-sigma error bars.We find a similar trend in the 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 β = 7.2 is approximately 40% smaller than for β = 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 π w 0 and m η ′ 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 ρ and m η ′ , as depicted in the right panel of Fig. 2.

B. Pseudoscalar singlets in Sp(4) with N
For non-degenerate fermions, the theory contains two flavor-singlet pseudoscalar mesons, the η ′ as well as the flavordiagonal PNGB, π 0 .To understand the effects of (explicit) flavor-symmetry breaking on the low-lying spectrum, we first choose the ensemble for degenerate fermions with β = 6.9 and m 0 = −0.9 and vary the bare mass of one of the Dirac fermion, m (2) 0 ≥ m 0 , for which we effectively increase its mass, while keeping that of the other fixed, m (1) 0 = m 0 .We summarize the numerical results in Tab.IV B. 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 π 0 c .Within the generally larger uncertainties, we find no statistically appreciable difference between m π 0 and m π 0 c , 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 π ± .In the mass-degenerate limit, we recover the mass hierarchy of Sec.IV A, as expected.As we increase one of the fermion masses, we observe a clear separation between flavored mesons and the unflavoured π 0 as well as the unflavoured vector mesons ρ 0 , with the former being heavier than the latter.At the same time, the pseudoscalar singlet η ′ , becomes heavier.This effect leads to an inversion of the mass hierarchy between the vector mesons ρ ± , ρ 0 and the η ′ meson and the η ′ becomes heavier than any other meson considered here. 5 couple of cautionary remarks should be added.First of all, we are in a moderately heavy mass regime, with m π 0 /m ρ 0 ∼ 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 η ′ and the π 0 is approximately twice the mass difference between the π 0 and the π ± 's.This indicates that the mass-differences are driven by the valence fermion masses.
In this regime, the singlet π 0 and the non-singlet ρ 0 approach an approximate one-flavour theory -see e.g.Ref. [170] for lattice results on the low-lying spectrum in the 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 π ± through the relation 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].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 π.For the finer lattice this changes drastically and the scalar singlet σ is usually heavier than the vector meson ρ, though still below the two-π threshold.The green solid line mρ/mπ = 1/x is displayed for reference.
C. Scalar singlet in Sp(4) with N f = 2 In the case of the scalar singlet, σ, 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 Sp(4) on the coarse β = 6.9 lattices, we observe a light σ state, of mass comparable to the mass of the π.This pattern persists over the entire mass range considered.On finer lattices, for β = 7.2, the mass of the σ increases and is heavier than the PNGBs, and comparable to the vector meson, though with much larger statistical errors, and still below the ππ 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 σ 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.

D. Comparison to SU(3) with
In Fig. 5 we show a compilation of the available data published on the pseudoscalar singlet for the SU(3) theory with N f = 2 (upper panels) as well a comparison of our results for Sp(4) and SU(2) to the available data for 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 η ′ 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 ± ∆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/TχL 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 η ′ -glueball mixing (denoted by Beijing [214]).
In all but the very lightest ensemble (and one obvious outlier at heavy fermion mass) the vector meson, ρ, is found 0.2 0.4 0.6 0.8 1  4)(β = 6.9)Sp(4)(β = 7.2) FIG. 5. Comparison to the available lattice data in SU(3) with two fundamental fermions.The upper panels depict all available lattice results in SU(3).In the upper left panel the green markers denote the vector meson ρ and the orange ones the pseudoscalar singlet η ′ .The different marker shapes denote the different collaborations.We see that the η ′ is lighter than the vector mesons almost everywhere.In the upper right panel we directly plot the ratio m η ′ /mρ.In the lower panels we compare the SU(3) results to our Sp(4) and SU(2) data.In the lower left panel, we show the ratio m η ′ /mπ as a function of mπ/mρ for values of mπ/mρ ≈ 0.7 and larger.In the lower right plot we compare the different results of the ratio m η ′ /mρ. to be heavier than the pseudoscalar singlet, η ′ .The authors of Ref. [213] point out that in the lightest ensemble the ρ particle might be unusually light due to the small number of energy levels below the inelastic threshold in the determination of the ππ phase shift.It is lighter than their extrapolation to the physical point at which m ρ = 786 (20) and even lighter than their extrapolation to the chiral limit.The mass dependence of the η ′ meson was found to be flat and an extrapolation in Ref. [212] to the physical point gave m η ′ = 772 (18)MeV.This is in contrast to SM QCD where the η ′ is significantly heavier-the current PDG lists m PDG η ′ = 957.78(6)MeV[240], which is in agreement with recent SU(3), N f = 2 + 1 lattice results of m η ′ = 929.9( 47. 5  21.0 ) [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 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 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 η ′ .

E. Possible phenomenological implications
Our results provide evidence that the singlet sector, computed for moderately large fermion masses in the Sp(4) theory, is not dissimilar from what is observed in the SU(2) and 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 longlived 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 crosssections, 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 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 Sp(4) theory are similar to those of SU(2) and 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 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.singlet, η ′ , 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. (A1).
In order to test the robustness of our subtraction choice, we report here the mass of the pseudoscalar singlet η ′ 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.V we compare this to four alternative methods:7 (i) Direct calculation and subtraction of ⟨0|O η ′ |0⟩, (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,8 (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 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 ⟨0|O η ′ |0⟩ 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.Sp(4) with β = 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.
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 ⟨0|O σ |0⟩ 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 σ 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.VI 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 ⟨0|O σ |0⟩, (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 ⟨0|O σ |0⟩.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).

2 E
singlets in symplectic gauge theories A. Microscopic theory and global symmetries B. Light meson spectrum for two fundamental fermions III.Lattice setup A. Interpolating operators and two-point functions B. Variance reduction techniques IV.Results A. Pseudoscalar singlet in SU(2) and Sp(4) with N f = 2 B. Pseudoscalar singlets in Sp(4) with N f = 1 + 1 C. Scalar singlet in Sp(4) with N f = 2 D. Comparison to SU(3) with N f =

FIG. 1 . 2 FIG. 2 .
FIG. 1. (left panel) Mass ratios mmeson/mπ for pseudoscalar and vector mesons, including the flavor-singlet pseudoscalar η ′ , in the 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 β = 6.9 and 7.2.(right panel) The same plot but in the SU(2) gauge theory at β = 2.0.The green solid lines mρ/mπ = 1/x are displayed for reference.

FIG. 3 .
FIG.3.Masses of the lightest non-singlet mesons as well as the pseudoscalar singlet meson in the Sp(4) theory with nondegenerate Dirac fermions.We fix the lattice coupling and one of the bare fermion masses to β = 6.9 and m

FIG. 4 .
FIG.4.Mass of the σ meson with degenerate fermions in the 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 π.For the finer lattice this changes drastically and the scalar singlet σ is usually heavier than the vector meson ρ, though still below the two-π threshold.The green solid line mρ/mπ = 1/x is displayed for reference.

32 ×
FIG. 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).
, β = 6.9, SP (4), m q = −0.9,nsrc= 128C η (t), |Q| = 1, N cfg = 488 C η (t), |Q| = 2, N cfg = 439 C η (t), |Q| = 3, N cfg = 418 C η (t), |Q| = 4, N cfg = 378FIG.6. (left) Correlator of the pseudoscalar non-singlet π and the pseudoscalar singlet η ′ .For visual clarity we multiplied the π 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|.