Sp(4) gauge theories on the lattice: Quenched fundamental and antisymmetric fermions

We perform lattice studies of meson mass spectra and decay constants of the $Sp(4)$ gauge theory in the quenched approximation. We consider two species of (Dirac) fermions as matter field content, transforming in the 2-index antisymmetric and the fundamental representation of the gauge group, respectively. All matter fields are formulated as Wilson fermions. We extrapolate to the continuum and massless limits, and compare to each other the results obtained for the two species of mesons. In the case of two fundamental and three antisymmetric fermions, the long-distance dynamics is relevant for composite Higgs models. This is the first lattice study of this class of theories. The global $SU(4) \times SU(6)$ symmetry is broken to the $Sp(4) \times SO(6)$ subgroup, and the condensates align with the explicit mass terms present in the lattice formulation of the theory. The main results of our quenched calculations are that, with fermions in the 2-index antisymmetric representation of the group, the masses squared and decay constant squared of all the mesons we considered are larger than the corresponding quantities for the fundamental representation, by factors that vary between $\sim$1.2 and $\sim$2.7. We also present technical results that will be useful for future lattice investigations of dynamical simulations, of composite chimera baryons, and of the approach to large $N$ in the $Sp(2N)$ theories considered. We briefly discuss their high-temperature behaviour, where symmetry restoration and enhancement are expected.


I. INTRODUCTION
In composite Higgs models (CHMs) [1][2][3], the Higgs fields, responsible for electroweak symmetry breaking, arise as pseudo-Nambu-Goldstone bosons (pNGBs) in a more fundamental theory, hence addressing the little hierarchy problem of generic extensions of the Standard Model (SM) of particle physics.In comparison to the other SM fermions, the top quark has a large mass, making it heavier than the W , the Z, and even the recently discovered Higgs boson [4,5].It is then natural to complete the CHM scenario by postulating that also the top quark has composite nature, at least partially, at the fundamental level.The additional model-building dimension added to this framework by (partial) top compositeness yields a richness of potential implications that has been explored in the literature on the subject in a range of possible directions, and motivates us to study its dynamical origin with nonperturbative techniques.The literature on composite Higgs models is indeed vast (see for instance Refs.), especially in connection with dynamical theories characterised by the SU (4)/Sp(4) coset (see for instance Refs.).
In Ref. [62] (see also Refs.[63,64] and the more recent Refs.[65][66][67][68]), some of us proposed a systematic programme of exploration of the lattice dynamics of Sp(2N ) gauge theories.Our main scientific motivation is the application of the results of such studies to the CHM context.In order to realise also top compositeness, it is necessary to implement on the lattice matter fields with mixed representations.For example, the model discussed in Refs.[12,43] requires that the matter content consists of N f = 2 Dirac fields transforming in the fundamental representation of Sp(2N ), supplemented by n f = 3 Dirac fields transforming in the antisymmetric representation of Sp(2N ).This dynamical system is expected to yield the spontaneous breaking of the SU (4) × SU (6) global symmetry to its Sp(4) × SO (6) subgroup.The introduction of diagonal mass terms for the fermions is compatible (aligned) with the vacuum structure, and provides a degenerate nonvanishing mass for the resulting 5 + 20 pNGBs.The lattice treatment of such a system with multiple dynamical fermion representations is a novel arena for lattice gauge theories, and only recently have calculations of this type been published, in the specific context of theories with SU (4) gauge group [21,27,29,33,38].
In this paper, we take a first step in this direction for Sp(2N ) gauge theories.We consider the Sp(4) gauge theory, and treat the two species of fermions in the quenched approximation; only the gluon dynamics is captured by the lattice numerical study, but the operators used to compute the relevant correlation functions involve both types of matter fields.We compute the mass spectra and decay constants of the mesons built both with fundamental and antisymmetric fermions, and perform their continuum extrapolation.We compare the properties of mesonic observables obtained with the two representations, which, in the dynamical theory, is important for CHM phenomenology.Since very little is known about the Sp(2N ) gauge theories, our quenched study is a first benchmark of these theories and would serve as a starting point for a more extensive and detailed investigation of such models.
We treat the relevant degrees of freedom with a low-energy effective field theory (EFT) that we employ to analyse the numerical data extrapolated to the continuum limit.The EFT proposed in Ref. [62] for the theory with SU (4)/Sp(4) coset is based on the ideas of hidden local symmetry, adapted from Refs.[69][70][71][72][73] (and [74][75][76][77]), and supplemented by some simplifying working assumptions.Here we return to the EFT to improve it and to generalise it to the case of the SU (6)/SO(6) coset.
The paper is organised as follows.In Sec.II, we define the Sp(4) theory with field content we are interested in, by writing both the Lagrangian density of the microscopic continuum theory as well as its low-energy EFT description.We devote Sec.III to describing the lattice action we adopt, the Monte Carlo algorithm we employ, and other important aspects of the lattice study we perform, such as scale setting and topology.In Sec.IV we present our results for the calculation of the masses and the (renormalised) decay constants of the lightest mesons in the quenched approximation.We compare the results for quenched fundamental and antisymmetric fermions.We also discuss in Sec.V a first attempt at matching the results to the low-energy EFT description applicable to pseudoscalar (PS), vector (V) and axial-vector (AV) states.We conclude by summarising and discussing our main findings and by outlining future avenues for investigation in Sec.VI.
The presentation is complemented by a rather generous set of Appendixes, intended to be of use also beyond the specific aims of this paper, for the research programme we are carrying out as a whole.We expose some details and conventions in the treatment of spinors in Appendix A and some technical points about the treatment of massive spin-1 particles in Appendix B. Technical points about the embedding of the SM gauge group in the context of CHMs are highlighted in Appendix C. Appendix D contains some numerical tests of the topological charge history and of its effect on spectral observables, in the illustrative case of a numerical ensemble that has a fine lattice spacing.In Appendix E, besides briefly summarising some properties of QCD light flavoured mesons, we discuss general symmetry properties of the mesons in theories with symmetric cosets, that are important for spectroscopy.We also touch upon possible high-temperature symmetry restoration and enhancement in Appendix E 1.We explicitly write the operators relevant as sources of all the mesons in Appendix F, and in Appendix F 1 we specify the sources of PS, V and AV mesons in the SU (4)/Sp(4) case, by adopting a specific choice of SU (4) generators and normalisations.

II. THE MODEL
In this section, we describe the specific model of interest, borrowing ideas from Refs.[12,43], and we describe the basic properties of the long-distance EFT description(s) we use later.I: Field content of the microscopic theory (Vµ, q, ψ) and of the low-energy EFT describing the pNGBs (Σ6,21, M6,21).Sp( 4) is the gauge group, while SU (4) and SU (6) are the global symmetries.The elementary fields Vµ are gauge bosons, while q and ψ are two-component spinors.Σ6 and Σ21 are composite scalar fields.They capture the long-distance dynamics of operators that are bilinear in q and ψ, the VEVs of which are responsible for the breaking SU (4) → Sp(4) and SU (6) → SO (6), respectively.The mass matrices M6 and M21 are treated as scalar spurions, formally transforming as ∼ 6 ∼ 6 of SU (4), and ∼ 21 of SU (6), respectively.

A. Continuum microscopic theory
The Sp(4) gauge theory we started to study in Ref. [62] has matter content consisting of two Dirac fermions Q i a , where a = 1 , • • • , 4 is the colour index and i = 1, 2 the flavour index, or equivalently four two-component spinors q j a with j = 1 , • • • , 4. Following [12,43], we supplement it by three Dirac fermions Ψ i ab transforming in the antisymmetric 2-index representation of Sp(4), or equivalently by six two-component spinors ψ j ab , with j = 1 , • • • , 6.The field content is summarised in Table I.The Lagrangian density is The covariant derivatives are defined by making use of the transformation properties under the action of an element U of the Sp(4) gauge group-Q → U Q and Ψ → U ΨU T -so that where g is the gauge coupling.
The Lagrangian density possesses a global SU (4) symmetry acting on the fundamental fermions Q and a global SU (6) acting on the antisymmetric-representation fermions Ψ.The mass terms break them to the Sp(4) and SO(6) subgroups, respectively.The unbroken subgroups consist of the transformations that leave invariant the symplectic matrix Ω and the symmetric matrix ω, respectively, that are defined by 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 By rewriting explicitly the fermion contributions to the Lagrangian density in two-component notation as follows (see Appendix A for the list of conventions about spinors) the global symmetries become manifest: Of the 15 generators T A of the global SU (4), and 35 generators t B of SU (6), we denote with A = 1 , • • • , 5 and with B = 1 , • • • , 20 the broken ones, which obey while the unbroken generators with A = 6 , • • • , 15 and with B = 21 , • • • , 35 satisfy As described in Appendix C, the Higgs potential in the SM has a global symmetry with group SU (2) L × SU (2) R ∼ SO (4), which in the present case is a subgroup of the unbroken global Sp(4).The SU (3) c gauge group characterising QCD is a subgroup of the unbroken global SO (6).And finally the generator Y of the hypercharge U (1) Y group is a linear combination of one of the generators of SU (2) R and of the generator of the additional U (1) X unbroken subgroup of SO( 6) that commutes with SU (3) c .

B. The pNGB fields
At low energies, the gauge theory with Sp(4) group is best described by an EFT that contains only the fields corresponding to the pNGBs parametrising the SU (4)×SU (6)  Sp( 4)×SO (6) coset.We define the fields Σ 6 and Σ 21 in terms of the transformation properties of the operators that are responsible for spontaneous symmetry breaking, hence identifying Σ nm

6
∼ Ω ab q n a T Cq m b , Σ nm
Σ 6 transforms as the antisymmetric representation of SU (4), and Σ 21 as the symmetric representation of SU (6).We parameterise them in terms of fields π 6 and π 21 as Σ 6 ≡ e where π 6 ≡ π A 6 T A with A = 1 , • • • , 5 and π 21 ≡ π B 21 t B with B = 1 , • • • , 20 are Hermitian matrix-valued fields, and the generators T A are normalised by the relation Tr T A T B = 1  2 δ AB = Tr t A t B .The decay constants of the pNGBs are denoted by f 6 and f 21 , and the normalisation conventions we adopt correspond to those in which the decay constant in the chiral Lagrangian of QCD is f π 93 MeV.
In order to identify the operators to be included in the Lagrangian density describing the mass-deformed theory, one treats the (diagonal) mass matrices as (nondynamical) spurions M 6 ≡ M Ω and M 21 ≡ −m ω (see Table I).The vacuum expectation value (VEV) of the operators Σ i yields the symmetry breaking pattern SU (4) × SU (6) → Sp(4) × SO(6), aligned with the explicit breaking terms controlled by M 6 and M 21 , and hence in the vacuum of the theory we have π i = 0.
At the leading order in both the derivative expansion and the expansion in small masses, the Lagrangian densities of the EFT describing the dynamics of the pNGBs of both the SU (4)/Sp(4) and SU (6)/SO(6) cosets are given by FIG. 1: The moose diagrams representing the low-energy EFT descriptions.On the left SU (4)6A is gauged, while SU (4)6B is a global symmetry (including the SU (2)L × SU (2)R), and the combination of the nontrivial VEVs of S6 and Σ6 breaks the symmetry to Sp(4), giving mass to all the vector mesons and leaving a set of five light pions.On the right, the same principles are applied to SU (6)21B × SU (6)21A and to its breaking to the SO(6) subgroup.
for i = 6, 21, and with m 6 = M and m 21 = m. 1 The condensates are parameterised by v 6 and v 21 , which have dimension of a mass.In the SU (4)/Sp(4) case Tr Σ 6 Σ † 6 = 4, and in the SU (6)/SO(6) case Tr Σ 21 Σ † 21 = 6.In order to describe the coupling to the Standard Model, one chooses appropriate embeddings for the relevant SU (2) L × SU (2) R and SU (3) c × U (1) X groups and promotes the ordinary derivatives to covariant derivatives.By doing so, the irreducible representations of the unbroken Sp(4) × SO( 6) can be decomposed in representations of the SM groups (see Appendix C).
Starting from the SU (4)/Sp(4) coset, the five pNGBs transform as the fundamental representation of SO(5) ∼ Sp(4).Because SO(4) ∼ SU (2) L × SU (2) R is a natural subgroup of SO(5), one finds the decomposition 5 = 1 + 4, and hence four of the pNGBs are identified with the SM Higgs doublet, while the one additional degree of freedom is a real singlet of SU (2) L × SU (2) R .In the conventions of [50,62], the latter is denoted by π 3 -or π 3  6 if one needs to avoid ambiguity with the set of pNGBs from the SU (6)/SO( 6) coset (see also Appendix C 1).

C. EFT: Hidden local symmetry
This subsection is devoted to the treatment of spin-1 composite states.All irreducible representations coming from the SU (4)/Sp(4) theory can be decomposed following the same principles illustrated by the pNGBs, into representations of the groups relevant to SM physics.For example the 10 of SO(5) decomposes as 10 = 4 + 6 of SO(4), so that the composite vector mesons V of the SU (4)/Sp(4) theory (corresponding to the ρ mesons of QCD) decompose into a complex doublet and a complex triplet of SU (2) L × SU (2) R .The axial vectors AV (corresponding to the a 1 mesons in QCD) transform with the same internal quantum numbers as the pNGBs and hence give rise to a complex doublet and a real singlet.In the SU (6)/SO(6) coset, the composite vector mesons V transform as the 15 of SO(6) ∼ SU (4), which decomposes as 15 = 1 + 3 C + 8 of SU (3) c , and the axial-vector mesons AV transform as the 20 of SO (6), which decomposes as 20 We study a reformulation of the low-energy EFT description of the model, that is intended to capture also the behaviour of the lightest vector and axial-vector states, in addition to the pNGBs (as in the chiral Lagrangian).It is based on hidden local symmetry [69][70][71][72][73] (see also [74][75][76][77]) and illustrated by the diagram in Fig. 1.There are well-known limitations to the applicability of this type of EFT treatment, which we will discuss in due time.
We consider the two moose diagrams as completely independent from one another.We follow closely the notation of Ref. [62] in describing the SU (4)/Sp(4) coset, except for the fact that we include only single-trace operators in the Lagrangian density.Because the breaking is due to the condensate of the operator transforming in the 6 of SU (4), we label all the fields of relevance to the low-energy EFT with a subscript, as in S 6 .The scalar fields S 6 transform as a bifundamental of SU (4) 6B × SU (4) 6A , while Σ 6 transform as the antisymmetric representations of SU (4) 6A .Hence the transformation rules are as follows: where U 6A and U 6B are group elements of SU (4) 6A and SU (4) 6B , respectively.The EFT is built by imposing the nonlinear constraints Σ 6 Σ † 6 = 1 4 = S 6 S † 6 , which are solved by parameterising S 6 = e .M 6 = M Ω is a constant matrix, introducing explicit symmetry breaking.One can think of it as a spurion in the antisymmetric representation of SU (4) 6B , so that as a field it would transform according to M * 6 → U 6B M * 6 U T 6B .The 15 real Nambu-Goldstone fields σ 6 = σ A 6 T A and five real π 6 = π A 6 T A are in part gauged into providing the longitudinal components for the 15 gauge bosons of SU (4) 6A , so that only five linear combinations remain in the spectrum as physical pseudoscalars.One then uses Σ i and its derivatives, as well as M i , to build all possible operators allowed by the symmetries, organises them as an expansion in derivatives (momenta p 2 ) and explicit mass terms (M ), and writes a Lagrangian density that includes all such operators up to a given order in the expansion.We also restrict attention to operators that can be written as single traces, as anticipated.
Truncated at the next-to-leading order, the Lagrangian density takes the following form, which we borrow from Ref. [62] 3 : Tr M SΣS T M SΣS T + h.c. .
We omitted, for notational simplicity, the subscript "6" on all fields and all the parameters.We should stress that we made some simplifications, and omitted some operators, as discussed in [62].The covariant derivatives introduce the parameter g V , controlling the coupling of the spin-1 states.They can be written as follows: and The analogue of Eq. ( 16) in the SU (6)/SO(6) case is obtained in the same way.With the conventions outlined above, masses and decay constants are given by the same relations as in Ref. [62], both for the mesons sourced by fundamental and antisymmetric fermion bilinears: The pNGB decay constants obey the following relation: It was observed in Ref. [62] that AV is independent of m as the accidental consequence of the truncations and of the omission of some operators.It was also shown that some of the couplings parameterise the violation of the saturation of the Weinberg sum rules, when truncated at this level-retaining only the lightest excitations sourced by the V and AV operators rather than the whole infinite tower of states.
In both the SU (4)/Sp(4) as well as SU (6)/Sp(6) cosets, truncated at this level the Lagrangian implies that the mass of the pions satisfies a generalised Gell-Mann-Oakes-Renner relation, which reads as follows: which implies a dependence of the condensate on m.We notice the presence of the constant g V , which enters the g VPP coupling between V and two PS states and has an important role in controlling the EFT expansion.

III. LATTICE MODEL
The lattice action and its numerical treatment via Monte Carlo methods are the main topics of this section.Most of the material covered here is based upon well-established processes, and we discussed its application to our programme elsewhere [62,67], hence we summarise it briefly, mostly for the purpose of defining the notation and language we adopt later in the paper.

A. Lattice definitions
In the numerical (lattice) studies, we should adopt a discretised four-dimensional Euclidean-space version of Eq. ( 1).But as we perform our numerical work in the quenched approximation, we only need the pure gauge part of the Lagrangian density, as in pioneering studies of Sp(2N ) Yang-Mills theories in Ref. [78].We employ the standard Wilson action where β = 8/g 2 is the bare lattice coupling and the trace is over colour indices.The elementary plaquette P µν is a path-ordered product of (fundamental) link variables U µ (x), the group elements of Sp(4), and reads as follows: Ensemble β Nt × N   Given the action in Eq. ( 26), we generate the gauge configurations by implementing a heat bath (HB) algorithm with microcanonical overrelaxation updates.Technical details, including the modified Cabbibo-Marinari procedure [79] and the resymplectisation process we adopted, can be found in Refs.[62,65].The HiRep code [80], appropriately adapted to the requirements of this project, is used for the numerical calculations.
The pure Sp(4) Yang-Mills lattice theory at any values of β can in principle be connected smoothly to the continuum, as no evidence of bulk transitions has been found [78].In this study, we work in the regime with β > 7.5.In a previous publication [62], some of us used two values of the coupling (β = 7.62 and β = 8.0), and performed preliminary studies of the meson spectrum with fermions in the fundamental representation, in the quenched limit.In order to carry out the continuum extrapolation, here we extend those studies by including three additional values of the bare lattice coupling, β = 7.7, 7.85, 8.2.The four-dimensional Euclidean lattice has size N t × N 3 s , with N t and N s the temporal and spatial extents, respectively.We impose periodic boundary conditions in all directions for the gauge fields.While for the ensemble at β = 7.62 we reuse the configurations generated on a 48 × 24 3 lattice already employed in the quenched calculations in Ref. [62], for all the other values of the coupling we generate new configurations with 60 × 48 3 lattice points.For each lattice coupling we generate 200 gauge configurations, separated by 12 trajectories5 between adjacent configurations.To ensure thermalisation, we discard the first 600 trajectories.In Table II we summarise the ensembles.In addition to the ensemble name, the lattice coupling and the lattice size, we also present two measured quantities: the average plaquette P and the gradient-flow scale w 0 /a in lattice units.The former is defined by P ≡ Re x µ<ν Tr P µν (x)/(24 × N t × N 3 s ), while the latter will be defined and discussed in the next subsection.The statistical uncertainties are estimated by using a standard bootstrapping technique for resampling, which will also be applied to the rest of this work.

B. Scale setting and topology
In numerical lattice calculations, all dimensional quantities can be written in terms of the lattice spacing a, for example by defining a dimensionless mass as m latt = ma.But in taking the continuum limit, the lattice spacing vanishes, a → 0. Hence, in order to connect the lattice observables to continuum ones, we have to set a common physical scale that allows the comparison.We adopt as our scale-setting method Lüscher's gradient-flow (GF) scheme, using the definition of Wilson flow in Ref. [81] (see also Refs.[82][83][84]).This method is particularly suitable for the purpose of this work, since it relies on theoretically defined quantities that do not require direct experimental input.III.In the plots we show Q evaluated only on the configurations used in the data analysis.
The scale-setting procedure with the GF scheme in Sp(4) theories has been first discussed in Ref. [62], both for the pure Yang-Mills and for the theory with two fundamental Dirac fermions (see also [63,67]).We follow the same procedure throughout this work: we define the flow scale w 0 by W| t=w 2 0 = W 0 [85], where W(t) is the derivative of the action density built from gauge fields at nonzero fictitious flow time t.The reference value W 0 = 0.35 has been chosen to minimise both discretisation and finite-volume effects [62] (though with the caveats discussed in Refs.[86][87][88]).We also choose a four-plaquette clover for the definition of the field-strength tensors [81].The resulting values of the flow scale in lattice units w 0 /a are shown in Table II.
We measure the history of the topological charge Q to monitor the possible emergence of topological freezing, which might affect spectral measurements [89,90] (see also Refs.[91,92]).Since Q ≡ x 1 32π 2 µνρσ Tr {U µν (x)U ρσ (x)} is dominated by ultraviolet (UV) fluctuations when calculated directly on the configurations in ensembles QB1-5, configurations that have been smoothed by the gradient flow are instead used.Q is measured at the point in the flow such that the smoothing radius √ 8t = L/2.In Fig. 2 we present the histories and histograms of Q along the Markov chain for all ensembles in Table II, the latter of which is fitted with the Gaussian fit form n . In Table III we present the results of this fit, and the exponential autocorrelation time τ exp calculated via a fit to the autocorrelation function of Q.In the five ensembles, there is no clear evidence of a freeze-out of the topology; the histograms clearly show sampling from multiple topological sectors, and the distributions are peaked within 1σ of Q = 0.
However, as we move to finer lattice spacing, we observe that the autocorrelation time of the topological charge grows significantly; in the case of QB5, this has grown to around 34 configurations.In this case specifically Q 0 = −4.12 is also marginal compared to σ = 4.81.This effect may be due to the fact that a change of the discrete global quantity Q by local updates becomes disfavoured in the approach to the continuum limit.
To verify that this increasing τ exp and marginal Q 0 do not affect the spectroscopic results we obtain from these ensembles, we generate an additional ensemble QB5 of 2400 trajectories starting from the last configuration in QB5.We repeated the measurements of meson masses and decay constants, and of the topological charge history, on 200 configurations sampled from QB5.While the value of Q 0 differs between the two ensembles, the meson masses and decay constants do not show significant deviations (beyond the statistical fluctuations).We report these tests in detail in Appendix D. This suggests that any systematic effect associated with the long autocorrelation time of the topological charge and the marginal Q 0 on the spectroscopy is comfortably smaller than the statistical error for the ensembles and observables we study, and we use ensembles QB1-5 for the remainder of the analysis.

IV. OF QUENCHED MESONS
In this section, we present the main numerical results of our study.We start by defining the mesonic two-point correlation functions that are computed numerically, and the observables we extract from them, namely the meson masses and decay constants.We provide some technical details about the otherwise standard procedure we follow, in order to clarify how different representations of the gauge group are implemented.Perturbative renormalisation of the decay constants is summarised towards the end of Sec.IV A. We perform continuum extrapolations with the use of Wilson chiral perturbation theory (WχPT) in Sec.IV B. We devote Secs.IV C and IV D to present the numerical results for the mesons made of fermions transforming in the fundamental and 2-index antisymmetric representations, respectively, and conclude with a comparison of the two representations in Sec.IV E. For practical reasons, in this section we specify our results to the theory with N f = 2 fermions on the fundamental representation and n f = 3 on the antisymmetric, though the results of the quenched calculations apply for generic N f and n f .

A. Correlation functions
We extract masses and decay constants of the lightest flavoured spin-0 and spin-1 mesons from the corresponding Euclidean two-point correlation functions of operators O M involving Dirac fermions Q transforming in the fundamental and Ψ in the 2-index antisymmetric representation, as listed in Table IV.In the table, colour and spin indices are implicitly summed over, while the flavour indices i = j (k = m) are chosen.The operators of the form Q i Γ M Q j are gauge invariant and they source the meson states M .Spin and parity J P are determined by the choice of Γ M .The operators built with Γ = γ 5 , γ µ , γ 5 γ µ , with µ =1, 2, 3, correspond to the pseudoscalar (PS), vector (V), and axial-vectors (AV) mesons, respectively.They appeared in the EFT discussion in Sec II C. For all of them, we measure both the masses and the decay constants of the particles that they source.For completeness, we also calculate the correlation functions built with Γ = 1 4 , γ 0 γ µ , γ 5 γ 0 γ µ , which refer to scalar (S), (antisymmetric) tensor (T), and axial tensor (AT), but we extract only the masses of the lightest states sourced by these operators.The operators Label Interpolating operator Meson TABLE IV: Interpolating operators OM built of Dirac fermions on the fundamental Q i a and antisymmetric Ψ k ab .We show explicitly the flavour indices i, j = 1 , 2 and k, m = 1 , 2 , 3, while colour and spinor indices are implicit and summed over.We also show the J P quantum numbers, the corresponding QCD mesons sourced by the analogous operator, and the irreducible representation of the unbroken global Sp(4) × SO(6) spanned by the meson (see also [44]).We indicate in parentheses other nontrivial representations that are obtained with the same operator structure but that we do not study in this paper as they source heavier states.The singlets (1 of both Sp(4) and SO( 6)) are ignored, as we choose to analyse only the operators with i = j or k = m.More details about the symmetries can be found in Appendix E, and the details of a specific choice of basis for the global SU (4) are presented in Appendix F.
Ψ k Γ M Ψ m are defined and classified in the same way, except that we denote them with lowercase letters as ps, v, av, s, t, and at, respectively.In Table IV, we also show the irreducible representation of the unbroken global symmetry Sp(4) × SO(6), as well as the corresponding mesons in QCD, to provide intuitive guidance to the reader.We also recall that because of the (pseudo)real nature of the representations we use, there is no difference between meson and diquark operators.More details about the classification of the mesons and the relation between four-component and two-component spinors can be found in Appendixes E and F. The two-point correlation functions at positive Euclidean time t and vanishing momentum − → p can be written as We extract physical observables from these objects.In most of our calculations we set M = M , with the exception of the extraction of the pseudoscalar decay constant, which involves both O PS and O AV ( O ps and O av in the case of fermions Ψ).The standard procedure requires rewriting C(t) in terms of fermion propagators S aα bβ (x) ≡ Q aα (x)Q bβ (0) (and analogous expressions for the propagators involving Ψ), to yield where the trace is over both spinor indices α, β and gauge indices a, b.
In the simplest case of a point source, the fermion propagator D R (R labeling the fermion representation) is calculated by solving the Dirac equation In order to improve the signal, in our numerical studies throughout this work we use the Z 2 × Z 2 single time slice stochastic wall sources [93] with three different sources considered individually for each configuration, instead of the point sources, on the right-hand side of Eq. (30).In all the spectroscopic measurements using quenched ensembles, we use the (unimproved) Wilson action for the fermions.The corresponding massive Wilson-Dirac operator in the fundamental representation D F is defined by its action on the fermions Q, that takes the form where U µ (x) are the link variables in the fundamental representation of Sp(4), a is the lattice spacing, and μ is the unit vector in the spacelike direction µ.
In order to construct the Dirac operator D AS for fermion fields Ψ ab in the 2-index antisymmetric representation, we follow the prescription in [80].For Sp(2N ), we define an orthonormal basis e , for c < a, and for b = N + a (e (ab) The main difference compared to the case of SU (N ) is that the base e AS is Ω-traceless, satisfying Ω dc e (ab) AS cd = 0.In the Sp(4) case, one can verify that the resulting five nonvanishing matrices satisfy the orthonormalisation condition Tr e AS e (cd) AS = −δ (ab)(cd) , while the matrix e AS vanishes identically.The explicit form of the antisymmetric link variables U AS µ (x) descends from the fundamental link variables U µ (x), as Finally, the Dirac operator for the 2-index antisymmetric representation D AS is obtained by replacing (U µ ) ab by (U AS µ ) (ab)(cd) and Q by Ψ in Eq. ( 31).
Masses and decay constants for the mesons are extracted from the asymptotic behaviour of C M, M (t) at large Euclidean time.We assume it to be dominated by a single mesonic state.If M = M , for all meson interpolating operators we can write where T is the temporal extent of the lattice.In our conventions, the meson states |M are normalised by writing M = M A T A , with T A the generators of the global SU (2N f ) or SU (2n f ) symmetry.The value of the pseudoscalar decay constant in QCD in these conventions would be f PS = f π 93 MeV.We also consider the correlator defined with M = PS and M = AV, for which the large-time behaviour is given by having restricted attention to the components of the AV operator with index µ = 1, 2, 3.
We parameterise the vacuum-to-meson matrix elements for fundamental fermions in such a way that the decay constants f M obey the following relations: where the polarisation vector µ is transverse to the momentum p µ and normalised by * µ µ = 1.(For operators constituted by antisymmetric fermions we replace the fields Q by Ψ.)For spin-1 V and AV mesons we extract both masses and decay constants from Eqs. ( 35) and (37).In the case of the pseudoscalar meson, we determine the masses and decay constants by combining Eqs.(35) with M = PS, Eq. ( 36) and Eq.(37).
The matrix elements at finite lattice spacing a have to be renormalised.For Wilson fermions, the axial and vector currents receive multiplicative (finite) renormalisation.The renormalisation factors Z A and Z V are defined by the relations In this work we determine the renormalisation factors via one-loop perturbative matching, and for Wilson fermions the relevant matching coefficients are written as [94] Z where Γ = γ 5 γ µ for Z A and Γ = γ µ for Z V .The eigenvalues of the quadratic Casimir operators with fermions in the fundamental and antisymmetric representations of Sp(4) are C(F ) = 5/4 and C(AS) = 2, respectively.The matching factors in Eq. ( 39) are computed by one-loop integrals within the continuum MS (modified minimal subtraction) regulalisation scheme.The resulting numerical values are ∆ Σ1 = −12.82,∆ γµ = −7.75 and ∆ γ5γµ = −3.0[62,94].Following the prescription in Ref. [95], in order to improve the convergence of perturbative expansion we replace the bare coupling g by the tadpole improved coupling defined as g2 = g 2 / P .P is the average plaquette value, and this procedure removes large tadpole-induced additive renormalisation arising with Wilson fermions.

B. Continuum extrapolation
Extrapolations to the continuum limit are carried out following the same procedure as in Ref. [67].We borrow the ideas of tree-level Wilson chiral perturbation theory (WχPT), which we truncate at the next-to-leading order (NLO) in the double expansion in fermion mass and lattice spacing [96,97] (see also Ref. [98], as well as [99,100], though written in the context of improvement).Tree-level results for the full theory can be extended to (partially) quenched calculations, since quenching effects only arise from integrals in fermion loops [97].But we cannot a priori determine the range of validity of tree-level WχPT at NLO.On the one hand, if we were too close to the chiral limit, we would need to include loop integrals (the well-known chiral logs).On the other hand, if we were in the heavy mass regime, then we would need to include more higher-order terms.As we will discuss later, most of our data sit somewhere in between these two extrema, and as a consequence we can empirically find appropriate ranges of fermion mass over which tree-level NLO WχPT well describes the numerical data.
We apply the scale-setting procedure discussed in Sec III B, and define the lattice spacing in units of the gradientflow scale as â ≡ a/w 0 .All other dimensional quantities are treated accordingly, so that masses are rescaled as in mM ≡ w 0 m M and decay constants as in fM = w 0 f M .Tree-level NLO WχPT assumes that the decay constant squared f 2,NLO PS is linearly dependent on both m2 PS and â.We extend this assumption to all other observables as well, hence defining the ansatz m2,NLO for decay constants squared and masses squared, respectively.We note that the fermion mass m f appearing in the standard WχPT has been replaced by the pseudoscalar mass squared by using LO χPT results, according to which m2 PS = 2Bm f .The low-energy constant B could in principle be determined via a dedicated study of the fermion mass, but this would go beyond our current aims.The empirical prescription we adopt requires us to identify the largest possible region of lattice data showing evidence of the linear behaviour described above and then fit the data in order to identify the additive contribution proportional to â. Extrapolation to the continuum is obtained by subtracting this contribution from the lattice measurements.

C. Quenched spectrum: Fundamental fermions
Reference [62] reported the quenched spectrum of the lightest PS, V, and AV flavoured mesons for two values of the lattice coupling, β = 7.62 and 8.0, with fermions in the fundamental representation.In this section, we extend the exploration of the quenched theory in several directions.First, we consider three more values of the coupling, β = 7.7, 7.85, and 8.2, as mentioned in Sec III A, aiming to perform continuum extrapolations, along the lines described in Sec.IV B. Second, in order to remove potential finite-volume effects, we restrict the bare fermion mass m 0 to ensembles that satisfy the condition m PS L ≥ 7.5, in line with the results of the study with dynamical fermions [67].Only part of the data in [62] meets this restriction, over the range of m 0 ∈ [−0.7, −0.79] at β = 7.62, measured on the lattice with extension 48 × 24 3 -corresponding to the ensemble denoted as QB1 in Table II.For the other values of the lattice coupling we perform new calculations by using lattices with extension 60 × 48 3 .The details of all the ensembles are found in Sec III A and summarised in Table II.For a given ensemble QBi (with i = 1 , • • • , 5) we introduce various choices of bare mass am 0 of the fundamental fermions (see Table IX), and calculate two-point Euclidean correlation functions of pseudoscalar PS, vector V, axialvector AV, scalar S, (antisymmetric) tensor T and axial-tensor AT meson operators, using the interpolating operators in Table IV.We follow the standard procedure described in Sec IV A, and we extract the masses and decay constants from the correlated fit of the data for the correlation functions as in Eq. ( 35).In the case of the pseudoscalar meson, we simultaneously fit the data for the correlators C PS, PS (t) and C PS,AV (t) according to Eqs. (35) and (36).The fitting intervals over the asymptotic (plateau) region at large Euclidean time are chosen to optimise the χ 2 while keeping the interval as large as possible.Such optimised values are shown in the numerical fits presented in Tables IX and X in Appendix G.
Notice that, in the case of AV, AT and S mesons, we are not able to find an acceptable plateau region for several among the lightest choices of fermion masses.This problem appears when approximately reaching the threshold for decay to three pseudoscalars.Similar problems have been observed before in the literature on quenched theories (see for example Refs.[101][102][103]), and may be due to the appearance of two types of new features, both of which are ultimately due to violations of unitarity: polynomial factors correct the exponential behaviour of the large-time correlation functions, and finite-volume effects do not decouple in the infinite-volume limit.We pragmatically decided to ignore measurements showing evidence of these phenomena, and discard them from the analysis.
The resulting values of meson masses and decay constants are presented in Tables XIII, XIV and XVI in Appendix G.In Table XIII we also present the results of m PS L and f PS L. For all the listed measurements the lattice volumes are large enough that the finite-volume effects are expected to be negligible as m PS L > ∼ 7.5, and the low-energy EFT is applicable as f PS L > ∼ 1.6.All fermion masses are large enough that the decay of a V meson into two PS mesons is kinematically forbidden.The resulting values of the masses measured from the correlators involving O V and O T are statistically consistent with each other, in support of theoretical prediction: the V and T operators interpolate the same physical states with J P = 1 − (identified with the ρ meson in the case of real-world QCD).
We perform simultaneous continuum and massless extrapolations by fitting the data for (quenched) meson masses and decay constants to Eqs. (40) and (41).We restrict the range of masses used for the extrapolations to m2 for the PS states, and to ensembles yielding m2 PS 0.6 for all other states, in order to retain the largest possible range of masses within which the data show linear dependence on m2 PS .In Figs. 3 and 4 we show the results of decay constants and masses, with different colours being used to denote ensembles at different β values.In the figures we also present the continuum-extrapolated values (denoted by grey bands the widths of which represent the statistical uncertainties), obtained after subtracting artefacts arising from finite lattice spacing.We find that f 2 PS , m2 V and m2 T are significantly affected by the discretisation of the Euclidian space.And such a long continuum extrapolation can be understood from the fact that we have used the standard Wilson fermions.The size of lattice artefacts in all other quantities is comparable with that of statistical uncertainties.
From the numerical fits we determine the constants appearing in Eqs. ( 40) and ( 41), and we report them in Table V.The numbers in the first and second parentheses are the statistical and systematic uncertainties of the fits, respectively.In the table, we also present the values of χ 2 /N d.o.f .Some large values of χ 2 /N d.o.f indicate that either the uncertainties of the individual data were underestimated or the fit functions are not sufficient to correctly describe the data.Although it would be difficult to fully account for the systematics associated with the continuum extrapolation with limited number of lattice spacings, we estimate the systematic uncertainties in the fits by taking the maximum and minimum values obtained from the set of data excluding the coarsest lattice (the ensemble with β = 7.62) and including or excluding the heaviest measurements.Notice in the table that this process yields large estimates for the systematic uncertainty for those fits that result in a large value of χ 2 /N d.o.f at the minimum.Finally, the resulting values in the massless limit, f 2, χ M and m2, χ M , should be taken with a due level of caution, since the considered masses are still relatively heavy and only the corrections corresponding to the tree-level terms in the chiral expansion are used in the fits.We leave more dedicated studies of the massless extrapolation to our future work with fully dynamical and light fermions.
As seen in Table XV, for each given value of β the ratio mV / fPS is approximately constant over the mass region mV / mPS > ∼ 1.3.From a simple linear extrapolation to the continuum of these constant vector masses in units of fPS , we find that mV / √ 2 fPS = 5.42 (5).A more rigorous, yet compatible, estimate of the massless limit is obtained by taking the extrapolated results in Table V and yields mχ  40) and ( 41), used in the continuum and massless extrapolations of masses and decay constants of mesons in the quenched simulations involving Dirac fermions in the fundamental representation.
The numbers in parentheses represent, respectively, statistical and systematic uncertainties due to the fit.

D. Quenched spectrum: Antisymmetric fermions
We turn now our attention to the quenched spectrum of the lightest flavoured mesons involving the fermions transforming in the antisymmetric representation of Sp(4).We use the same ensembles listed in Table II, but the bare masses m 0 of the fermions are listed in Table XI of Appendix G.As with fundamental fermions, we choose the values of am 0 to satisfy the condition of m ps L ≥ 7.5.In the table, we also present the fitting intervals used for the extraction of the masses and the decay constants of ps, v, av, and s mesons as well as the resulting values of χ 2 /N d.o.f .The results for t and at mesons are shown in Table XII of the same appendix.We apply to the antisymmetric case the same numerical treatment and analysis techniques used for the fundamental fermions.As in the case of the fundamental representation, we could not find an acceptable plateau region for some measurements at the smallest fermion masses, in the cases of v, av and s mesons.
In Appendix G we also present the numerical results of the masses and decay constants of ps, v and av mesons, as well as the masses of s, t and at mesons.See Tables XVII, XVIII and XX.As shown in Table XVII, all the measurements meet the aforementioned condition m ps L ≥ 7.5.In addition, we find that f ps L ≥ 2.3, which supports the applicability of low-energy EFT techniques.Furthermore, the meson masses in units of fps and the ratio fv / fps are presented in Tables XIX and XX.As already seen in the results for fundamental fermions, in all the measurements we find that the results of mt are consistent with those of mv , given the current statistical uncertainties.
We perform the numerical fits of masses and decay constants by using the tree-level NLO WχPT described by Eqs.(40) and (41).In Figs. 5 and 6, we present the fit results denoted by grey bands as well as numerical results of the masses and the decay constants measured at given lattice parameters.For the fits we consider the same ranges of m2 ps taken for the case of fundamental fermions: m2 ps 0.4 and m2 ps 0.6, respectively, for the ps and all other states.Over these mass ranges no significant deviation from linearity of the data in â and m2 ps is visible in our data.Different colours denote different lattice couplings, while the widths of the bands represent the statistical uncertainties of the continuum extrapolations.
The resulting fit values are reported in Table VI.The numbers in the first and second parentheses are the statistical and systematic uncertainties of the fits, respectively.Once more, we estimate the fitting systematics by taking the maximum and minimum values obtained from the set of data excluding the coarsest lattices at β = 7.62 and including or excluding the heaviest measurements.
As in the case of fundamental fermions Q, we find that for each β value the vector masses in units of the pseudoscalar decay constant are almost constant over the range of mv / mps > ∼ 1.3-see Table XIX.After performing a simple linear extrapolation of these constants, we find that mv / √ 2 fps = 4.72(4) in the continuum limit.A more rigorous, yet compatible, estimate is obtained by making use of the extrapolated results in Table VI: we find mχ v / √ 2 f χ ps = 4.80 (12) (4).The resulting value of the ratio is smaller than that for the fundamental fermions by 13%.E. Quenched spectrum: Comparison Fig. 7 shows a visual comparison between the decay constants of the pseudoscalar, vector, and axial-vector mesons, made of fermions transforming in the fundamental representation of Sp(4) (PS, V, AV) and in the 2-index antisymmetric representation (ps, v, av).In order to make the comparison, we plot the continuum-limit results by naively identifying the masses of the pseudoscalars m2 PS = m2 ps as the abscissa.The comparison at finite mass should be taken with some caution, as the symmetry-breaking operators controlling the mass of PS and ps states are distinct, but the massless extrapolations can be compared unambiguously.We repeat the exercise also for the masses of all the mesons, and show the result in Fig. 8.
In all cases we considered, masses and decay constants of bound states made of fermions Ψ transforming in the 2-index antisymmetric representation are considerably larger than those made of fermions Q transforming in the fundamental representation.Focusing on the massless limit, we find that the ratio f 2 av / f 2 AV = 2.7 ± 1.1 is the largest, while m2 s / m2 S = 1.18 ± 0.13 is the smallest, and the other results are distributed in the range between these two values.The hierarchy between the pseudoscalar decay constants is important in the CHM context; we find that f 2 ps / f 2 PS = 1.81 ± 0.04.It is also to be noted that the mass of the vector states v is larger, but not substantially so, in respect to that of the corresponding V mesons, with m2 v / m2 V = 1.46 ± 0.08.How much of the above holds true for the dynamical calculations is not known and is an interesting topic for future studies.It was shown in Ref. [67] that, by comparing quenched and dynamical calculations for mesons in the fundamental representation (performed in comparable ranges of fermion mass), and after both the continuum and massless extrapolations were performed, the discrepancies are not too large: O(25%) for m2 S , O(20%) for f 2 PS , O(10%) for m2 V , and smaller for the other measurements.Whether this is due to the fact that all the calculations in Ref. [67] are performed in a range of fermion masses that are comparatively large or to other reasons-the large-N behaviour of the theory might already be dominating the dynamics of Sp(4) mesons, for example-is not currently known and should be studied in future dedicated investigations.Yet, it is suggestive that no dramatic discrepancy has emerged so far, for all the observables we considered.
We conclude this section by reminding the reader that the calculations performed for this paper, being done with the quenched approximation, are insensitive to the number of fundamental flavours N f and antisymmetric flavours n f and hence apply to other models, beyond the phenomenologically relevant case with N f = 2 and n f = 3.A recent lattice study within the SU (3) gauge theory [104] of the ratio m ρ /f π between the mass of the rho mesons and the decay constant of the pions (corresponding to m V /f PS in this paper) shows no appreciable dependence on the number of flavours N f < ∼ 6-as long as the theory is deep inside the regime in which chiral symmetry breaking occurs.It would be interesting to measure whether this holds true also for other representations, in the dynamical theories.Meanwhile, we find that in our quenched calculation, after taking both the continuum and massless limits, for the fundamental representation we have m2 V / f 2 PS = 59.0 ± 2.2, while for the antisymmetric representation we find  v / f 2 ps = 47.3 ± 2.3. 6The discrepancy reaches beyond the 3σ level, suggesting that this ratio-which enters into the Kawarabayashi-Suzuki-Riazuddin-Fayyazuddin (KSRF) relation M 2 ρ /(g 2 ρππ f 2 π ) = 2 [106,107]-depends on the fermion representation.By comparison, the ratio obtained from the numerical studies with dynamical Dirac fermions in the fundamental representation is m2 V / f 2 P S = 65.4 ± 5.0 [67], which is slightly larger than the result of our quenched calculation.Once more, checking this result (as well as the KSRF relations) in the full dynamical theory with fermions in the antisymmetric representation would be of great interest.In this section, we perform a global fit of the continuum-extrapolated masses and decay constants of PS, V, and AV mesons to the EFT described in Sec.II C. As stated there, the EFT equations are applicable both to mesons constituted of fermions in the fundamental as well as 2-index antisymmetric representations of the Sp(4) gauge group.We also recall from Ref. [62] that several working assumptions have been used to arrive at Eqs. ( 19)- (24).We follow in the analysis the prescription introduced in Ref. [67].We only repeat some of the essential features of the process, while referring the reader to Ref. [67] for details.We focus instead on the results of the global fit.
We start by restricting the data analysed to lie in the mass range over which all the measured masses and decay constants can be extrapolated to the continuum limit using Eqs.(40) and (41).In the case of the fundamental representation, we restrict our measurements to include only QB1FM3−QB1FM6, QB2FM1−QB2FM3, QB3FM4−QB3FM7, QB4FM6−QB4FM8, and QB5FM2−QB5FM3.In the case of antisymmetric representation, we restrict to QB1ASM4−QB1ASM6, QB2ASM3−QB2ASM6, QB3ASM2−QB3ASM4, QB4ASM4−QB4ASM6, and QB5ASM2.As anticipated in Sec IV B, we use the LO χPT result for the pseudoscalar mass, and replace the fermion mass in Eqs. ( 19)-( 24) by m2 PS = 2Bm f .In the mass range considered, this replacement is supported by the numerical data, as m2 PS is found to be approximately linear with the mass of the fermion m0 .Accordingly, we expand the EFT equations and truncate at the linear order in m2 PS .The resulting fit equations have been presented as Eqs.(6.1)-(6.5) in Ref. [67].The ten unknown low-energy constants, denoted as ( f , F , b, c, g V , κ, v1 , v2 , ŷ3 , ŷ4 ), are appropriately redefined by introducing the gradient-flow scale w 0 .
We perform the numerical global fits of the data to the EFTs, via standard χ 2 minimisation, by using 200 bootstrapped samples and a simplified χ 2 function that is built by just summing the individual χ 2 functions for the five independent fit equations.The fit results satisfy the constraints obtained from the unitarity conditions in Eq. (6.8) of Ref. [67].In practice, we guide the fits by an initial minimisation of the full dataset.In Fig. 9 we present the results of the global fit along with the continuum-extrapolated data used for the fits, by further comparing the results originating from fundamental and antisymmetric fermions.In the figure, the fit results are presented by shaded bands, the widths of which represent the statistical uncertainties.The quality of the fits is measured by the fact that χ 2 /N d.o.f ∼ 0.6 at the minimum, although one should remember that correlations have not been taken into consideration in the analysis.The results of continuum and massless extrapolations, displayed in Figs.7 and 8, are in good agreement, even in proximity of the massless limit, with those of this alternative analysis.
As pointed out in Ref. [67], some of the parameters in the EFTs are not well constrained by the global fit of measurements coming from two-point functions only.Hence, we do not report the individual best-fit results, which are affected by flat directions and large correlations.Yet, in the same reference it is observed that some (nontrivial) combinations of the parameters may be determined well.One of the most interesting such quantities is the coupling constant associated with the decay of a vector meson V (or v) into two pseudoscalar mesons PS (ps).These couplings play the same role as the g ρππ in low-energy QCD.The resulting values in the cases of fundamental and antisymmetric fermions are g χ VPP = 4.95(21)(8) and g χ vpp = 3.80(24)( 16), (42) respectively, where the suffix χ denotes the result of simultaneous continuum and massless extrapolations.As shown in Fig. 10, the distributions of this quantity exhibit a regular Gaussian shape, from which we estimate the statistical uncertainty-the numbers in the first parentheses of Eq. ( 42).The numbers in the second parentheses in Eq. ( 42) denote the systematic errors of the fits with similar caveats to those discussed in Sec IV C, that we estimated by taking the maximum and minimum values obtained from the set of data excluding the coarsest ensemble and including or excluding the heaviest measurements.
The EFT analyses performed in this section is affected by several limitations-in particular by the quenched approximation and by the comparatively large fermion masses-and thus one should interpret the results with some caution.Yet, it is interesting to compare the EFT results with phenomenological models and with available measurements obtained with dynamical fermions transforming in the fundamental representation.We first compare the EFT results in Eq. ( 42) with the ones predicted from the KSRF relation, g VPP = m V / √ 2m PS .We find that the left-hand side is smaller than the right-hand side of this relation by about 10% and 23%, for the fundamental and antisymmetric representations, respectively.These discrepancies are larger than the uncertainties associated with the fits, and might indicate that the KSRF relation does not describe the quenched theories accurately, particularly in the case of the antisymmetric representation, although this statement is affected by uncontrolled systematic uncertainties due to the use of the EFT with such large values of g VPP and g vpp , as well as large fermion masses.We also find that for the fundamental representation the quenched value of g χ VPP is smaller by 29% compared to the dynamical value of g χ VPP = 6.0(4)(2) [67], yielding again a discrepancy that is significantly larger than the fit uncertainties.It would be interesting to repeat these tests with dynamical fermions in the antisymmetric representation, and in general to explore more directly the low-mass regimes of all these theories, but these are tasks that we leave for future extensive studies.

VI. CONCLUSIONS AND OUTLOOK
Composite Higgs and (partial) top compositeness emerge naturally as the low-energy EFT description of gauge theories with fermion matter content in mixed representations of the gauge group.Motivated by this framework, we considered the Sp(4) gauge theory with N f quenched Wilson-Dirac fermions Q transforming in the fundamental representation of Sp(4), as well as n f quenched fermions Ψ in the 2-index antisymmetric representation.While the quenched theory is not expected to reproduce the full dynamics, it provides a useful comparison case for future full dynamical calculations.We generated lattice ensembles consisting of gauge configurations by means of the HB algorithm, modified appropriately the HiRep code [80], considered meson operators O M bilinear in these fermions (see Table IV for explicit definitions of the operators), and measured two-point Euclidean correlation functions of such operators on discrete lattices (and in the quenched approximation).
We hence extracted decay constants f M and masses m M of the flavoured mesons sourced by the operators O M , with M = PS, V, AV, S, T, and AT (and M = ps, v, av, s, t, and at), defined in Table IV.We renormalised the decay constants, expressed all dimensional quantities in terms of the gradient-flow scale w 0 , and-having restricted attention to ensembles for which finite-volume effects can be ignored-applied tree-level WχPT to extrapolate toward the continuum and massless limits the results for mesons constituted of both fermion species.We also performed a first global fit of the continuum results that makes use of the EFT describing the lightest spin-1 states (besides the pseudoscalars).It is constructed by extending with the language of hidden local symmetry the chiral-Lagrangian description of the pNGBs spanning the SU (2N f ) × SU (2n f )/Sp(2N f ) × SO(2n f ) coset.
Our main results for the physical observables in the continuum limit are listed in the tables and plots in Secs.IV C and IV D and graphically illustrated in Sec.IV E (see in particular Figs.7 and 8).They can be summarised as follows.In the quenched approximation, after extrapolation to the massless limit, all dimensional quantities extracted from two-point correlation functions involving operators constituted of Ψ fermions are larger than the corresponding observables involving Q fermions.The two extremes are m2 s / m2 S = 1.18 ± 0.13 and f 2 av / f 2 AV = 2.7 ± 1.1, respectively, with all other ratios between observables in the two sectors falling between these two values.(Of particular interest for model building are the ratios m2 v / m2 V = 1.46 ± 0.08 and f 2 ps / f 2 PS = 1.81 ± 0.04.)The error bars comprise both statistical as well as systematic errors, the latter arising from the continuum and massless extrapolations as discussed in details in Sec IV.Furthermore, we found statistically significant violations of the KSRF relations by the mesons made of antisymmetric fermions, at least in the quenched approximation.(The extraction of the g VPP and g vpp couplings from the global fit of two-point function data collected with large fermion mass to the EFT is affected by unknown systematic effects, and hence this should be taken as a preliminary result.) Despite the physical limitations of the studied quenched theory, this paper opens the way toward addressing a number of interesting questions in future related work, a first class of which is related to the comparison of the quenched calculations to the full dynamical ones, in particular for the case of fermions in the antisymmetric representation.While it was observed elsewhere [67] that the quenched approximation captures remarkably well the dynamics of fundamental fermions (at least for the range of masses hitherto explored), there is no clear reason for this to happen also in the antisymmetric case, for which large-N arguments are less constraining.In order to address this point, one would require to study the dynamical simulations with Ψ fermions, in the phenomenologically relevant low-mass ranges of the dynamical calculations, and also to generalise our approach to Sp(2N ) gauge theories.The reader may be aware of the possibility that, with higher-dimensional representations and large numbers of fermion degrees of freedom, some of the Sp(2N ) theories we are interested in might be close to the edge of the conformal window and behave very differently.(For perturbative studies within the Sp(2N ) class, see for instance Refs.[108][109][110], and references therein.) The extensive line of research outlined in the previous paragraph complements the development of our programme of studies in the context of top compositeness, that as outlined in Ref. [62] requires one to consider the dynamical theory in the presence of mixed representations.This is a novel area of exploration for lattice gauge theories, for which the literature is somewhat limited (see for instance Refs.[21,27,29,33,38]).New fermion bound states, sometimes referred to as chimera baryons, can be sourced by operators that involve gauge-invariant combinations of fermions in mixed representations.(The anomalous dimensions of chimera baryons are discussed for example in [40,61,111].)The study of these states is necessary in the context of top compositeness, as they are interpreted as top partners.
A third group of future research projects can be envisioned to explore the role of higher-dimensional operators, for which the material in the Appendixes of this paper is technically useful.These operators play a role in determining the physics of vacuum (mis)alignment and of electroweak symmetry breaking, as their matrix elements enter the calculation of the potential in the low-energy EFT description.These studies would provide an additional link to phenomenological investigations of composite Higgs models, bringing lattice calculations in close contact with modelbuilding considerations and searches for new physics at the Large Hadron Collider (LHC).
Finally, it would be interesting to investigate the finite temperature behaviour of these theories.As discussed in Appendix E 1, it is important to characterise symmetry restoration and symmetry enhancement that appear at high temperature, generalising what has been studied about QCD to the case of real and pseudoreal representations, for which the group structure of the global symmetries and their breaking is expected to be different.U (2) symmetry.Their components are described by Grassmann variables, satisfying the two nontrivial relations7 and analogous for all other combinations.A Dirac four-component spinor is obtained by joining the LH projection of the Majorana spinor built starting from u and the RH projection of the Majorana spinor corresponding to d, so that and By inspection, one finds that the following relations hold true: and by using the γ µ matrices, one also finds the relations as well as By definition, the transpose of a C-number is trivial, and hence ξAχ = (ξAχ) T ≡ −χ T A T ξ T , for any ξ, χ spinor written in terms of Grassmann variables and A any matrix of C-numbers.This implies the relation which will be useful later.Some algebra shows that the following identity between real numbers holds: where q j = (u , d), and where the U (2) = U (1) × SU (2) global symmetry is now made manifest.This is adopted as the kinetic term of the Dirac spinor Q.
The Lagrangian density for the Dirac spinor Q admits also a mass term.By virtue of the relations C † = − C = CT , and by the Grassmann nature of the spinors, it can be written in terms of the symmetric matrix ω ≡ τ 1 : where with α = α(x).The gauge choice α = −π/m removes π from the Lagrangian density, which then depends only on a massive vector field.
• A 2-index antisymmetric form B µν is coupled to a vector A µ (not to be confused with A µ ), and the Lagrangian density is where The Lagrangian L 1 is invariant under the gauge transformation with the vector α µ = α µ (x).The gauge choice α µ = −A µ /m removes A µ from the Lagrangian density, which then depends only on a massive 2-form field.
The Lagrangian L 1 can also be rewritten, by defining K µν ≡ 1 2m µνρσ H ρσ , in the form Gauge invariance is not manifest in this form.The Lagrangians L 0 and L 1 are equivalent at the level of the path integrals they define [113][114][115][116]. Hence, the use of antisymmetric massive 2-index tensors provides an alternative, equivalent descriptions of massive vectors.In physical terms, there is no difference between these two (or rather, three) formulations.Important differences are introduced by the coupling to matter fields and sources.For example, one can couple fermions to A µ via the new term with Q a Dirac fermion and g the coupling.For the antisymmetric tensor, one may write While L A couples the spin-1 field to the LH component only of Q, in L B the LH and RH projections are coupled to one another, so that while L 0 and L 1 in isolation define the same theory, the addition of L A or L B leaves different global symmetries and different coupled theories.
Appendix C: About Lie groups, algebras and SM embedding Here we summarise some group theory notions relevant for models of composite Higgs and top quark compositeness based on the SU (4)/Sp(4) ⊗ SU (6)/SO(6) coset [12,43].We do not repeat unnecessary details-in particular, our special choice of SU (4) generators can be found elsewhere [50]-but we explicitly show the embedding of the SM gauge group (and fields, when useful).
The SU (4)/Sp(4) coset governs the Higgs sector of the Standard Model.Given the form of Ω in Eq. ( 5), the unbroken subgroup SO(4) ∼ SU (2) L × SU (2) R is the subset of the unbroken global Sp(4) ⊂ SU (4) that is generated by the following elements of the associated algebra: The T L generators satisfy the SU (2) L algebra T i L , T j L = i ijk T k L , and similarly In the vacuum aligned with Ω in Eq. ( 5), this is the natural choice of embedding of the SO( 4) symmetries of the Higgs potential.Following the notation in Refs.[50,62], the matrix of the five pNGB fields parametrising the SU (4)/Sp(4) coset is The real fields π 1 , π 2 , π 4 , and π 5 combine into the Higgs doublet, while π 3 is a SM singlet.
The SU (6)/SO (6) coset is relevant to top compositeness.The choice of n f = 3 Dirac fermions on the 2-index antisymmetric representation of Sp( 4) matches the number of colours in the SU (3) c gauge group of the Standard Model.The natural subgroup SU (3) L × SU (3) R ⊂ SU (6) is generated by with λ B the eight Hermitian Gell-Mann matrices, normalised according to the relation Tr , with the choice of ω in Eq. ( 5), one can verify that ωt B c + t B T c ω = 0, that the structure constants t A c , t B c = if ABC t C c are those of the su(3) c algebra, and that Tr t A c t B c = δ AB is twice the fundamental.The latter property is due to the fact that we are writing the SU (3) c generators as 6 × 6 matrices acting on two-component spinors.We hence identify t B c as the generators of the SU (3) c gauge symmetry of the Standard Model.An additional, independent, unbroken generator of SU (6) is given by which also commutes with the generators of SU (3) c .The generator Y of the hypercharge U (1) Y gauge symmetry of the Standard Model is a linear combination of X and T 3 R (see also Ref. [36] and references therein).

Weakly coupling the SM gauge group
In this Appendix, we perform a technical exercise.We compute the (divergent) contributions to the effective potential due to the gauging of the relevant SM subgroups of the global SU (4) × SU (6) symmetry, and discuss their effects on the potential of the pNGBs.The purpose of this exercise is to show explicitly how by gauging part of the global symmetry one breaks it.We also identify the decomposition of the representations according to the unbroken subgroup.
We adopt the external field method and borrow the regulated Coleman-Weinberg potential V 1 from Ref. [117], computed by assuming that a hard momentum cutoff Λ is applied to the one-loop integrals.With our conventions we write where in the trace STr fermions have negative weight and where c i are scheme-dependent coefficients.The matrix M 2 is obtained as follows: consider L i in Eq. ( 13), gauge the relevant subgroups, by promoting the derivatives to covariant derivatives, and compute the mass matrices of all the fields, as a function of the (background, external) scalar fields.When applied to the SU (4)/Sp(4) part of the theory (and for M = 0), this procedure involves only loops of gauge bosons and yields a quadratically divergent contribution to the mass of four of the pNGBs-labeled π 1 , π 2 , π 4 and π 5 in Eq. (C3): where g L is the coupling associated with the SU (2) L group with generators in Eqs.(C1), while g R is the coupling associated with the U (1) R subgroup generated by T 3 R from Eqs. (C2).The four masses are exactly degenerate, and the mass of π 3 does not receive a correction, as it is associated with a generator that commutes with SU (2) L × SU (2) R , and is hence left unbroken by the weak gauging of the SM gauge group-in practice, the mass of π 3 arises for M = 0 due to the explicit breaking of the global symmetry of the Lagrangian.
When applied to the 20 pNGBs that describe the SU (6)/SO(6) coset, the loops involve the SU (3) c gauge bosons, with the embedding chosen in this Appendix, and strength g S , as well as the U (1) X gauge boson generated by Eq. (C5), with strength g X .We find that the mass of 12 pNGBs-transforming as 6 C of SU (3) c -receive the quadraticallydivergent contribution and the other eight, which form the adjoint of SU (3) c , receive the mass correction The complex 6 C of SU (3) c has nontrivial U (1) X charge, while the eight real components of the adjoint representation of SU (3) c have vanishing U (1) X charge.All 20 pNGBs receive also a degenerate, explicit contribution to their mass, which is controlled by m.  [118,119], their approximate masses, and quantum numbers.Charge conjugation C refers to the neutral component in multiplets including electrically charge particles, and G ≡ C(−1) I for the whole multiplet.We show also representative hadronic decay modes.The three-body decay of η violates isospin, and hence G-parity, and yields Γ/M ∼ 10 −5 .
the theory at low temperatures and describe patterns of symmetry restoration and symmetry enhancement at high temperatures.We keep the discussion as general as possible but occasionally exemplify our observations with the specific case of the Sp(4) gauge theory with N f = 2 and n f = 3.Given a Lie group G and its subgroup H, a generic element g of the associated Lie algebra G can be decomposed as g = h + k, with h ∈ H an element of the algebra associated with H and k ∈ K an element of the complement of H in G.The coset space G/H is said to be symmetric if, for all possible choices of h and k, the following properties are true: These properties define in an unambiguous way an unbroken, multiplicative Z 2 symmetry, which we can call G-parity, which is compatible with the Lie algebra, and upon which k is assigned G-parity −, while h is assigned G-parity +.Three classes of cosets are commonly considered in the CHM context (see for instance Table I in Ref. [36]).They all emerge from gauge theories at the microscopic level.
• SU (N f ) L ×SU (N f ) R /SU (N f ) V cosets are, for example, realised in SU (N ) gauge theories with N f fundamental Dirac fermions.
• SU (2N f )/SO(2N f ) cosets are, for example, realised in Sp(2N ) gauge theories with N f Dirac fermions transforming in the 2-index antisymmetric representation.
All these cosets are symmetric, and the resulting G-parity is a symmetry of the theories.It allows selection rules for scattering and decay processes to be established.We now discuss each of these possibilities in some detail, with emphasis on the properties of the mesons associated with the theories they emerge from.We begin by reviewing the case of the SU (3) gauge theory with N f = 2 light flavours.It describes the light mesons in QCD.The associated coset is Much of what one learns from this theory is applicable to the other symmetric cosets listed above, with modifications that will be discussed later.In Table VIII, we report some information about light QCD mesons with S = C = B = 0, taken from the Particle Data Group [118].We found Refs.[124][125][126][127][128][129][130][131], and Appendix B in Ref. [132] particularly useful for the discussion that follows.
It is conventional to denote the states of QCD by the quantum numbers I G J P C , where I is the isospin (the representation of the unbroken SU (2) V ) and J the spin.The assignment of G-parity for the isotriplets coincides with the traditional G-parity: it is related to charge-conjugation C of the neutral component in an isomultiplet, and the isospin I of the isomultiplet, by the relation G ≡ C(−1) I , hence providing a link between the internal symmetry described above and a space-time symmetry.A second subtle link between internal and space-time symmetries involves the notion of spatial parity P ; the SU (2) L and SU (2) R symmetries act, respectively, on the LH and RH projections of the spinors, while the unbroken subgroup is the (vectorial) symmetric combination of the two.
We start the discussion from the isotriplets I = 1.The pions π and axial vectors a 1 are, respectively, the lightest spin-0 and spin-1 states and are associated with the broken generators of the global symmetry, so that they have G = −; the vector mesons ρ are associated with the unbroken group SU (2) V , and have G = +.As a result, the ρ decays to two π's, while the a 1 decays to three π's (or also one π and one ρ).G-parity is a useful practical tool: while ρ and a 1 particles both transform in the adjoint representation of the unbroken SU (2) V (isospin)-the unbroken subgroup and the coset have the same dimension-they are distinguished unambiguously by the different assignments of G, and hence they decay in different ways.
The global symmetries naturally extend to The additional unbroken vectorial U (1) B is the baryon number, and all mesons have vanishing U (1) B charge.The broken, anomalous, global, axial U (1) A symmetry plays an interesting role in relation to parity P .The axial U (1) A partners of the pions π, named a 0 , have the same G-parity but opposite P .If the U (1) A were exact, π and a 0 would be degenerate.
In the J = 1, I = 1 sector, the role played by the U (1) A symmetry is more subtle.The ρ and a 1 mesons are sourced by bilinear operators V and AV, in which spinor indices are contracted on the γ µ and γ µ γ 5 matrices, respectively.Such operators involve either two LH or two RH chiral spinors: the action of U (1) A leaves them both invariant, as they are two independent singlets of U (1) A .But there is an important complication, as massive vectors in four dimensions can equivalently be described by 2-index antisymmetric tensors (see Appendix B and references therein).Two additional sources of spin-1 states T and AT are obtained by contracting the spinor indices on σ µν and σ µν γ 5 , respectively.The two operators T and AT couple the LH and RH chiral projections of the fermions, in a way that is similar to the J = 0 isotriplet operators PS and S, that source the π and a 0 particles.They form a doublet of the U (1) A ∼ O(2) symmetry.
Because of symmetry breaking, the operator T (built with σ µν ) has the same quantum numbers I G J P C = 1 + 1 −− as the source V (built from γ µ ).The lightest and next-to-lightest states in this channel (ρ and ρ(1450) in Table VIII) can approximately be thought of as resulting from the mixing of two states that have different SU (2) L × SU (2) R transformation properties and are sourced by different operators V and T. The U (1) A partner of the combination of ρ and ρ(1450) sourced by T is denoted by b 1 (see again Table VIII), and sourced by AT.
Because we are also comparing with real-world QCD, we should notice that the isosinglet I = 0 sector is complicated by the fact that real-world light mesons are better explained by a model in which one includes 2 + 1 quarks, including the heavier strange quark s.The isosinglet mesons include an additional tower of states, due to mixing with the (ss) singlet.In Table VIII, this results in the doubling of states with I = 0 in respect to the I = 1 case, as we chose to retain pairs of mesons with identical quantum numbers.The G-parity assignment of each isosinglet state is the opposite of that of the corresponding isotriplet with the same J P C .Notice in the table that the decay of the η to 3π violates G-parity: it yields a suppressed rate Γ, with Γ/M ∼ 10 −5 , and originates from explicit breaking of isospin in real-world QCD, in which, for instance, up and down quarks are not degenerate in mass.
Let us see how these considerations have to be modified for enlarged cosets.(Both the SU (2N f )/Sp(2N f ) and enhanced because of the (pseudo)real nature of the underlying fermion representations.)The unbroken baryon number U (1) B is a subset of the unbroken part of these two cosets rather than commuting with it.Diquark operators hence source mesons, that complete the representation of SU (N f ) V into full representations of the unbroken Sp(2N f ) or SO(2N f ).The unbroken group and the coset have in these cases different dimension, so that representations with different G-parity have different dimensionality as well (see Table IV for instance), rendering G-parity redundant, at least as a way to distinguish among them.
We conclude by discussing explicitly the general form of the operators to be used as sources.We start from a two-component spinor χ transforming in the fundamental representation of the global SU (2N f ) symmetry group.A spin-0 local operator J 0 takes the schematic form χ T Cχ.The product of two fundamental decomposes as into the 2-index antisymmetric and symmetric representations of SU (2N f ), respectively.Furthermore, depending on whether the unbroken subgroup is Sp(2N f ) or SO(2N f ), either the antisymmetric or the symmetric combination is reducible and further decomposes into the unbroken subgroup by projecting one element along the elementary symplectic matrix Ω or the symmetric ω, respectively.Excitations sourced by this operator correspond to the f 0 of QCD-like theories, while those along the complement correspond to the PS flavoured states (the π of QCD).The representation that is irreducible, instead, would coincide with the adjoint of the unbroken subgroup, except that it vanishes because of Fermi statistics-unless one considers nonlocal operators.
Along the same lines, the spin-1 local operator J 1 schematically reads χ † σµ χ, and the decomposition in SU (2N f ) takes the form After decomposition into the representations of the unbroken subgroup, the adjoint splits into its antisymmetric and symmetric parts, and hence one ultimately finds the same decomposition as in the spin-0 case.This property descends from the (pseudo)real nature of Sp(2N f ) and SO(2N f ), that do not distinguish the fundamental representation 2N f from its conjugate.It is useful to notice that the four operators V, T, AV, and AT states source only one state that is a singlet.This is different from the , where each of the four operators sources a singlet of the unbroken group: the three additional J = 1 states are part of the irreducible representations sourced by V, T, and AT.Summarising for the SU (2N f )/Sp(2N f ) and SU (2N f )/SO(2N f ) cases: pseudoscalar PS and axial-vector AV multiplets have the same degeneracies, as do the flavoured scalar S-the U (1) A partners of the PS states.The vector states V belong to a different representation of the unbroken group common also to the antisymmetric tensor T, as well as to its U (1) A partner AT.In the unbroken Sp(2N f ) case, the V, T and AT mesons span a complete symmetric representation of Sp(2N f ), with PS, S, and AV on the (antisymmetric) complement.The reverse is true in the case of SO(2N f ) (see again Table IV).In both cases, the singlet sector is simpler: it consists of two spin-0 states forming a U (1) A doublet and of one isolated spin-1 singlet state.

Symmetry restoration and enhancement
At high temperatures, the fermion condensates melt, leading to restoration of the global symmetries.Both the non-Abelian SU (2N f ) (or SU (N f ) × SU (N f )) global symmetry, as well as the Abelian U (1) A symmetry are restored [120] (see also [121][122][123] for progress on N f = 2 lattice QCD).As a consequence of the former, one might find that the states sourced by V and AV operators become degenerate (the ρ and a 1 in the QCD-like case).Because of the latter, U (1) A multiplets should become degenerate, for example the states sourced by PS and S operators (the π and a 0 mesons in the QCD-like case).See for instance Ref. [50] and references therein.
Recent studies have emerged suggesting that, because the thermal bath reduces the space-time symmetries, the global internal symmetry is further enhanced, with the emergence of a new chiral-spin symmetry that combines with the global symmetries.We refer the reader to Refs.[124][125][126][127][128][129][130] for this research field, in which the specific case of the QCD-like, SU (3) theory with N f = 2 is discussed in great detail, and numerical evidence of the emergence of a SU (4) global symmetry is exposed, in the channels with spin J > 0. In the following, we limit ourselves to producing a summary of what would be the (testable) expectations in the three main cosets of interest to CHMs, if the corresponding symmetry-restoration and symmetry-enhancement patterns were to be confirmed.
• The SU (N f ) L × SU (N f ) R /SU (N f ) V cosets can emerge, at T = 0, from theories with complex representations, for example the SU (N ) gauge theories with N f fundamental Dirac fermions.These cosets are accompanied by the anomalous U (1) A = U (1) L × U (1) R /U (1) B Abelian coset.At high temperatures, the global U (N f ) L × U (N f ) R symmetry is restored and enhanced to U (2N f ) = U (1) A × SU (2N f ).For example, in the N f = 2 case that approximates QCD, in the J = 1 sector, four of the I = 1 states (the two lightest ρ, the a 1 , and the b 1 ) and four of the I = 0 states (the two ω/φ, the f 1 /f 1 , and the h 1 /h 1 ) have been measured to become degenerate, which would be compatible with forming a complete 16-dimensional adjoint representation of the U (1) × SU (4) enhanced global symmetry group [124][125][126][127][128][129][130][131].In the J = 0 sector, this symmetry is not manifest: the π and a 0 combine with the f 0 /f 0 and the η/η to form 2N 2 f degenerate states, the adjoint of the symmetry group U (N f ) L × U (N f ) R .(It would require an additional 2N 2 f components, two copies of the adjoint of U (N f ), to complete the adjoint of U (2N f ).) • The SU (2N f )/Sp(2N f ) cosets can emerge, at T = 0, from theories with pseudoreal representations, for example the Sp(2N ) gauge theories with N f Dirac fundamental fermions.In addition, the anomalous U (1) A is also spontaneously broken.What would have been the unbroken U (1) B associated with the baryon number in the case of complex representations is now a subgroup of the nonanomalous Sp(2N f ).Going to a high temperature, the symmetry is expected to be first restored and then enhanced to Sp(4N f ).For example, in the N f = 2 case, in the J = 1 sector of the spectrum, the ten V and ten T mesons (which include both the correspondent of the ρ and ω/φ of QCD), the five AV mesons (corresponding to the a 1 ), the ten AT mesons (which include both states corresponding to the b 1 and h 1 /h 1 of QCD) and the singlet vector state (corresponding to the f 1 /f 1 ) form a complete, 36-dimensional adjoint representation of Sp (8).The J = 0 sector is not expected to show high-T symmetry enhancement.PS flavoured states combine with the singlet to form the antisymmetric representation (in the QCD analogy, they correspond to the π and the f 0 /f 0 ), and their U (1) A partners (the a 0 and the η/η in QCD) combine to form the complex, antisymmetric 2-index representation of SU (2N f ), which has dimension 2 × N f (2N f − 1).(It would require finding an adjoint representation of SU (2N f ) of dimension 4N 2 f − 1, to yield a total of 8N 2 f − 2N f − 1, which is the 2-index antisymmetric of Sp(4N f ).In the N f = 2 case, the antisymmetric of Sp(8) decomposes as 27 = 15 + 6 C in terms of SU (4), and the 15 is missing.) • The SU (2n f )/SO(2n f ) cosets can emerge, at T = 0, from theories with real representations, for example the Sp(2N ) gauge theories with n f Dirac fermions transforming as the 2-index antisymmetric representation.At high temperatures, the restoration of the symmetry is expected to be followed by its enhancement to a global SO(4n f ).For example, if n f = 3, in the spin-1 flavoured sector the 15 v, 15 t and 15 at mesons sourced by the operators in Table IV will be degenerate with the 20 av mesons.In addition, a flavour-singlet vector will also be degenerate, and together will yield the 66-dimensional antisymmetric (adjoint) representation of SO (12), with spin J = 1.In the spin J = 0 sector, the 20 ps and 20 s mesons, together with the 1 + 1 flavor singlets form the 21 C representation of SU (6).(It would require an additional 35, the adjoint of SU (6), to make the 2-index symmetric traceless 77-dimensional representation of SO (12).) Appendix F: Bilinear operators as sources We collect in this Appendix technical clarifications about gauge-invariant operators O M , written in terms of the four-component fermions Q and Ψ, to be used as sources in the lattice calculations of the spectrum of composite states.We consider the Sp(2N ) gauge theory, without specifying N .When possible, we also write our expression in a form that applies to Sp(2N f ) and SO(2n f ) groups with general N f fundamental and n f antisymmetric Dirac fermions.For concreteness, in Appendix F 1 we explicitly identify the irreducible representations of the unbroken global Sp(4) group of relevance in the context of CHMs, as well as their SU (2) L × SU (2) R decompositions.
In the case of N f Dirac spinors transforming in the fundamental representation of the gauge group Sp(2N ), the fermion bilinear operators are written as in terms of N f × N f block matrices, built from the N f Dirac fermions and, following the same lines leading to Eqs. (A12), their conjugate Dirac fermion as Notice a difference, with respect to the definition leading to Eq. (A12), in how we define the conjugate spinor: the factor of −γ 5 Ω ab is introduced, in order to make the decomposition in LH and RH chiral components of Q and Q C take the same form.We stress that Q i C is physically equivalent to Q i , and hence one can identify the N f Dirac fermions with Q i .
We write explicitly the form of the general 2N f × 2N f matrices built as bilinears in spinors, both in two-component and four-component notation, which read and The symplectic matrix Ω 2N f is defined as a 2N f ×2N f antisymmetric matrix such that (Ω 2N f ) 2 = −1 2N f .Because of the contraction with Ω ab , J 0 is antisymmetric.This bilinear condenses, and hence the unbroken subgroup is Sp(2N f ).The decomposition of J 0 , J µ 1 and J µν We highlighted here the fact that the symmetric part of J 0 vanishes identically.Notice also that the singlets are antisymmetric.Both the J 1 and J 1 decompose into symmetric and antisymmetric parts, the latter expected to be related to heavier states.For the SU (2N f )/Sp(2N f ) coset the operators in Table IV are identified as follows: with the O S and O AT operators being the U (1) A conjugates of O PS and O T , respectively.With matter content including n f Dirac fermions in the antisymmetric representation of Sp(2N ) (for N > 1) we introduce the analogous j 0,1,1 operators built from the fermions We conventionally align the vacuum with the matrix ω 2n f , the 2n f × 2n f symmetric matrix such that (ω 2n f ) 2 = 1 2n f , generalising Eq. ( 5).We decompose j 0,1,1 in irreducible representations as 2n We highlighted the fact that the antisymmetric part of j 0 vanishes.In this case, the singlets are symmetric matrices.We write these operators in n f × n f blocks as matrix representations of the global SU (2n f ) symmetry: and In the SU (2n f )/SO(2n f ) case, the operators in Table IV are identified as follows: As explained also in Appendix E, operators J 0 that source spin-0 states, and operators J 1 sourcing spin-1 states can be classified in terms of the original global SU (2N f ) (enlarged) symmetry, according to which the former transform and the latter as Only once the SU (2N f ) is broken to its Sp(2N f ) group, J 0 and J 1 decompose in the same set of irreducible representations of the subgroup.Similarly, operators J 1 couple LH to RH components of the Dirac fields, and hence while they source particles with the same spin as J 1 , they transform under SU (2N f ) in the same way as J 0 .Once the symmetry is broken, mixing between the particles sourced by J 1 and J 1 will ensue, but in general these are different sources for different SU (2N f ) particles.The same considerations apply to the j 0,1,1 operators, and their decompositions in irreducible representations of the SO(2n f ) subgroup.
One can now explicitly decompose the matrices J 0,1,1 (and j 0,1,1 ) into a given basis of the U (N f ) (and U (n f )), by making a choice of generators T A (and t B ) of the group.We devote the next subsection to showing the result of this process in the SU (4)/Sp(4) case, for some of the interesting operators.
The pions in Eq. (C3) are sourced by the antisymmetric O PS = J (A) 0 operators.We can add to the Lagrangian density in Eq. ( 16) the following source term: where Σ = e 2iπ f Ω is the antisymmetric matrix defined in Eq. (12).We can expand for small π/f and make use of the antisymmetry in flavour space of J 0 .We make explicit use of the generators as written in Eq. (B.4) of Ref. [50].Looking at the decomposition of the 5 according to SO(4) ∼ SU (2) L ×SU (2) R ⊂ Sp(4), we find that the SO(4)-singlet π 3 is sourced by the operator O PS,3 ≡ i δL π /δπ 3 that reads where we made abundant use of Eqs.(A13).The 4 of SO(4) are sourced by the following operators O PS,A ≡ i δL π /δπ A given by9 The additional operator given by is aligned along Ω, in the internal space.It is the operator that develops a nontrivial VEV.It sources the Sp(4) singlet, which has the same role as the f 0 (500) of QCD (see Table VIII).Notice that these 6 operators put together transform as the antisymmetric representation of the global SU (4), before the decomposition in Sp( 4) irreducible representations as 6 = 1 + 5. Six additional operators O S,i can be obtained by replacing 1 4 → iγ 5 from the operators O PS,i for i = 0 , • • • , 5. Such operators are related to the former by the (anomalous) global U (1) A ∼ O(2) symmetry.(In QCD, the U (1) A partners of the π and f 0 (500) particles are, respectively, the a 0 and η/η particles.) The sources of the spin-1 fields are given in terms of the operators in J µ 1 .In Sec.II C we introduced the fields A µ = 15 A=1 A A µ T i (we drop here the subscript 6), with T A the Hermitian generators of SU (4) normalised as Tr T A T B = 1 2 δ AB .As in Ref. [50], it is convenient to label the broken generators with A = 1 , • • • , 5, and the unbroken ones with A = 6 , • • • , 15.We hence add to the Lagrangian density the following source term: Starting from the AV sources, we find that O µ AV,A ≡ δL A δA A µ are given by the following where we used relations such as Similar expressions hold for the ten operators sourcing the V mesons.We adopt the same basis as in Ref. [50], and within these conventions we find the following: The SO(5) adjoint 10 decomposes as 4 + 6 of SO(4): the fundamental 4 is sourced by O µ V,7 , O µ V,8 , O µ V,11 and O µ V,13 , with the adjoint 6 sourced by the other six operators.Again, for completeness, O µ V,8 , O µ V,13 and O µ V,14 would be the unbroken generators in the SU (2) L × SU (2) R /SU (2) V coset, corresponding to the ρ mesons in QCD, in which case O µ V,15 would be associated with baryon number and would source the ω/φ in QCD.We can introduce an additional vector field A 0 µ , to complete the adjoint SU (4) to the adjoint of the whole U (1) × SU (4), by adding the generator T 0 = 1 2 √ 2 1 4 .We hence identify the additional operator One can recognise this operator to be aligned with the generator of the anomalous U (1) A .It sources the equivalent of the f 1 (1285) of QCD (see Table VIII).These 16 operators do not have a U (1) A partner.But a second set of sources of spin-1 states T is built from J (S) 1 states, with the σ µν tensor.Their U (1) A partners (corresponding to the more exotic states b 1 and h 1 /h 1 mesons of QCD) are sourced by operators AT involving the tensor σ µν γ 5 (see also [55]).We do not show this explicitly, as it requires performing again the same exercise illustrated in the previous pages.Sourcing even more exotic states such as π 1 may require using derivatives and nonlocal operators [131], which is beyond our current purposes.

QB1FM1
-0.7 14-24 1.2 13-24 1.1 9-13 0.  IX: Technical details about PS, V, AS, and S lattice correlation functions.For each ensemble and each choice of bare mass am0, we show the fitting intervals of the Euclidean time I fit = [ti, t f ] between the minimum time ti and maximum time t f retained in the single-exponential fit to the measured correlators of mesons made of fundamental Dirac fermions.We carry out a correlated fit via standard χ 2 -minimisation.We report the values of χ 2 (normalised by the number of degrees of freedom) at the minima.In the case of AV and S states, we leave blank some entries for which the numerical data do not exhibit a plateau in the effective mass plots, because of numerical noise.
fermions, labeled as QBiASMj, we presented the results in Table XI for the cases of ps, v, av and s mesons, and in Table XII for t and at mesons.
In Table XIII we present the extracted values of PS masses and decay constants, as well as S masses.In the table we also present the results of m PS L and f PS L. Similarly, in Table XIV we show the results of the masses and decay constants of V and AV mesons.For presentation purposes, we find it useful to show also the meson masses in units of the PS decay constant, as well as the ratios of fV / fPS , in Table XV.In Table XVI we report the masses of T and AT states, both in units of the lattice spacing a and of the PS decay constant fPS .Analogous results for the masses and decay constants of ps, s, v, av, t, and at mesons composed of antisymmetric fermions are presented in Tables XVII-XX.
QB1ASM1 -1.05 14-24 1.2 13-24 0.5 7-12 0.1 10-15 0.5 QB1ASM2 -1.08 12-24 1.9 12-24 0.5 7-10 0.1 7-13 1.8 QB1ASM3 -1.1 12-24 1.1 12-24 1.0 7-12 1.1 8-13 0.7 QB1ASM4 -1.12 12-24 0.8 12-20 0.  XI: Technical details pertaining the measurements of the correlation functions of operators built with 2-index antisymmetric fermions.For each ensemble and each choice of bare mass am0, we show the fitting intervals of the Euclidean time I fit = [ti, t f ] between the minimum time ti and maximum time t f retained in the single-exponential fit to the measured correlators of mesons made of antisymmetric Dirac fermions.We carry out a correlated fit via standard χ 2 -minimisation.We report the values of χ 2 (normalised by the number of the degrees of freedom) at the minima.In the case of the v, av and s states, we leave blank some entries for which the numerical data do not exhibit a plateau in the effective mass plots, due to numerical noise.

FIG. 2 :
FIG.2: Topological charge histories (left panels), and histograms (right panels), for the five ensembles QB1-QB5, listed from top to bottom.Fitted parameters are given in TableIII.In the plots we show Q evaluated only on the configurations used in the data analysis.
AS (with the multi-index (ab) running over ordered pairs with 1 ≤ a < b ≤ 2N ) for the appropriate vector space of 2N × 2N antisymmetric matrices.The N (2N − 1) − 1 such matrices have the following nonvanishing entries.For b = N + a and 2 ≤ a ≤ N (e (ab) AS ) c,N +c ≡ −(e (ab)

FIG. 3 :
FIG.3: Decay constants squared of PS, V, and AV mesons (constituted of fermions transforming in the fundamental representation), as a function of the PS meson mass squared m2PS , for β = 7.62 (blue), 7.7 (purple), 7.85 (green), 8.0 (red), and 8.2 (brown).All quantities are expressed in units of the gradient-flow scale w0.The results of the continuum and massless extrapolations are represented by the grey bands.

FIG. 4 :
FIG. 4: Masses squared of V, T, AV, AT and S mesons (constituted of fermions in the fundamental representation), as a function of the PS meson mass squared m2 PS , for β = 7.62 (blue), 7.7 (purple), 7.85 (green), 8.0 (red), and 8.2 (brown).All quantities are expressed in units of the gradient-flow scale w0.The results of the continuum and massless extrapolations are represented by the grey bands.

FIG. 5 :
FIG.5: Decay constants squared of ps, v, and av mesons (constituted of fermions in the antisymmetric representation), as a function of the ps meson mass squared m2 ps , for β = 7.62 (blue), 7.7 (purple), 7.85 (green), 8.0 (red), and 8.2 (brown).All quantities are expressed in units of the gradient-flow scale w0.The results of the continuum and massless extrapolations are represented by the grey bands.

FIG. 6 :
FIG.6: Masses squared of v, t, av, at and s mesons (constituted of fermions transforming in the antisymmetric representation), as a function of the ps meson mass squared m2 ps , for β = 7.62 (blue), 7.7 (purple), 7.85 (green), 8.0 (red), and 8.2 (brown).All quantities are expressed in units of the gradient-flow scale w0.The results of the continuum and massless extrapolations are represented by the grey bands.

FIG. 7 :
FIG.7: Comparison of the decay constant squared (in the continuum limit) of the mesons as a function of the pseudoscalar meson mass squared, in units of w0, for fermion constituents transforming in the fundamental (blue) or 2-index antisymmetric (red) representation.

FIG. 8 :
FIG.8: Comparison of the mass squared (in the continuum limit) of the mesons as a function of the pseudoscalar meson mass squared, in units of w0, for fermion constituents transforming in the fundamental (blue) or 2-index antisymmetric (red) representation. m2

FIG. 9 :
FIG.9: Decay constants and masses in the continuum limit after subtracting lattice artefacts due to the finite lattice spacing.The global fit results are denoted by blue solid bands for the mesons constituted of fundamental fermions Q, and red bands for the ones constituted of antisymmetric fermions Ψ.The width of the bands indicates the statistical errors.

FIG. 10 :
FIG.10: Histogram distribution of the gV P P (left panel) and gvpp (right panel) couplings, obtained from the quenched calculation by applying the global fit strategy discussed in the text.In the former, gV P P denotes the coupling between mesons composed of fermions in the fundamental.In the latter case the constituent fermions transform in the 2-index antisymmetric representation.
) Again, O s and O at operators are the U (1) A conjugates of O ps and O t , respectively.

TABLE II :
List of ensembles used for quenched calculations.For each ensemble, we report the bare coupling β, the lattice size Nt × N 3 s , the average plaquette P and the gradient-flow scale w0/a = 1/â.

TABLE III :
Fitted parameters from topological charge histories; see also Fig.2.The autocorrelation time τexp is expressed in units of consecutive configurations.

TABLE V :
Results of the fit of the coefficients in Eqs. (

TABLE VI
(41)sults of the fit of the coefficients in Eqs.(40)and(41), used in the continuum and massless extrapolations of masses and decay constants of mesons in the quenched simulations involving Dirac fermions transforming in the 2-index antisymmetric representation.In parentheses we show statistical and systematic errors, respectively.

TABLE XIX :
Some useful ratios of (quenched) masses and decay constants of mesons made of Dirac fermions transforming in the antisymmetric representation.In parentheses we report the statistical uncertainties.

TABLE XX :
Masses of t and at states, in units of a and fps, for each of the ensembles and bare masses am0.The Dirac fermions transform in the 2-index antisymmetric representation.In parentheses we report the statistical uncertainties.