Glueballs and Strings in Sp(2N) Yang-Mills theories
Ed Bennett, Jack Holligan, Deog Ki Hong, Jong-Wan Lee, C.-J. David Lin, Biagio Lucini, Maurizio Piai, Davide Vadacchino
PPrepared for submission to JHEP
PNUTP-20/A06
Glueballs and Strings in Sp (2 N ) Yang-Mills theories
Ed Bennett, Jack Holligan, , Deog Ki Hong, Jong-Wan Lee, , C.-J. David Lin, , Biagio Lucini, , Maurizio Piai, Davide Vadacchino , Swansea Academy of Advanced Computing, Swansea University (Bay Campus), Fabian Way, SA18EN, Swansea, Wales, UK Department of Physics, College of Science, Swansea University (Park Campus), Singleton Park,SA2 8PP, Swansea, Wales, UK The Institute for Computational Cosmology (ICC), Department of Physics, South Road, Durham,DH1 3LE, UK Department of Physics, Pusan National University, Busan 46241, Korea Extreme Physics Institute, Pusan National University, Busan 46241, Korea Institute of Physics, National Chiao-Tung University, 1001 Ta-Hsueh Road, Hsinchu 30010, Tai-wan Centre for High Energy Physics, Chung-Yuan Christian University, Chung-Li 32023, Taiwan Department of Mathematics, College of Science, Swansea University (Bay Campus), Fabian Way,SA1 8EN, Swansea, Wales, UK INFN, Sezione di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy School of Mathematics and Hamilton Mathematics Institute, Trinity College, Dublin 2, Ireland
E-mail: [email protected], [email protected],[email protected], [email protected], [email protected],[email protected], [email protected], [email protected]
Abstract:
Motivated in part by the pseudo-Nambu Goldstone Boson mechanism of elec-troweak symmetry breaking in Composite Higgs Models, in part by dark matter scenarioswith strongly coupled origin, as well as by general theoretical considerations related to thelarge-N extrapolation, we perform lattice studies of the Yang-Mills theories with Sp (2 N ) gauge groups. We measure the string tension and the mass spectrum of glueballs, extractedfrom appropriate 2-point correlation functions of operators organised as irreducible repre-sentations of the octahedral symmetry group. We perform the continuum extrapolationand study the magnitude of finite-size effects, showing that they are negligible in our cal-culation. We present new numerical results for N = 1 , , , , combine them with datapreviously obtained for N = 2 , and extrapolate towards N → ∞ . We confirm explicitlythe expectation that, as already known for N = 1 , also for N = 3 , a confining potentialrising linearly with the distance binds a static quark to its antiquark. We compare our re-sults to the existing literature on other gauge groups, with particular attention devoted tothe large- N limit. We find agreement with the known values of the mass of the ++ , ++ ∗ and ++ glueballs obtained taking the large- N limit in the SU ( N ) groups. In addition, wedetermine for the first time the mass of some heavier glueball states at finite N in Sp (2 N ) and extrapolate the results towards N → + ∞ taking the limit in the latter groups. Sincethe large- N limit of Sp (2 N ) is the same as in SU ( N ) , our results are relevant also for thestudy of QCD-like theories. a r X i v : . [ h e p - l a t ] O c t ontents N limit 23 Sp (2 N )
30B Searching for the bulk phase transition 32C Continuum and infinite volume extrapolations 36
C.1 Finite-size effects (FSEs) 36C.2 Continuum limit extrapolations 37
D A closer look at the Sp (4) data 42E On the inclusion of N = 1 in the large- N extrapolation 46 – 1 – Introduction
Recent years have seen a resurgence of interest in gauge theories based upon symplecticgroups, driven by theoretical as well as phenomenological motivations, related to model-building in the context of physics beyond the Standard Model (SM). In comparison withthe SU ( N ) and (limited to (2 + 1) -dimensional) SO ( N ) cases [1–5], the literature on latticestudies of Yang-Mills theories with Sp (2 N ) gauge groups is limited in its extent, scope andreach (see for instance Ref. [6]). In a recent publication [7] (see also Refs. [8–11]), some ofus announced the intention to carry out a long-term, systematic lattice exploration of thestrong-coupling dynamics of the theories based on Sp (2 N ) gauge symmetry, and proposeda research programme that includes as one of its crucial steps the study of the dynamics ofglueballs and strings in the pure gauge theory.In the same paper [7], we presented our first comprehensive study of the Sp (4) puregauge theory, and computed the spectrum of masses and decay constants of mesons con-sisting of two fundamental (Dirac) fermions, treated in the quenched approximation. In thecontext of Composite Higgs Models (CHMs) [12–14] (see also Refs. [15 ? –54]), that initialstep provided an important source of quantitative information about the underlying dy-namics. In particular, we started to explore and exploit the dynamical origin of low-energyEffective Field Theories (EFT) based upon the SU (4) /Sp (4) coset, which have a prominentrole in the CHM context (see for instance Refs. [55–75]), as well as for related models ofdark matter with strong-coupling origin [76–79]. More recently [80], some of us presentedthe first continuum results of the lattice study of the Sp (4) theory with dynamical Wilsonfermions, hence making the treatment of the dynamics more realistic and useful in the CHMcontext. A first set of exploratory studies of the quenched theory with valence fermions inmultiple representations has been published in Ref. [81].In the present paper (see also Ref. [82, 83]), we take major steps in a complementarydirection, by focusing on the pure gauge theory without matter content, but extending theanalysis to different Sp (2 N ) gauge groups. Our specific objective is to obtain for the Sp (2 N ) Yang-Mills theories in D = 3 + 1 dimensions a comparable level of control over the spectraof strings and glueballs as achieved for the previously studied SU ( N ) and SO ( N ) gaugetheories [1–5]. On a theoretical side, this endeavour will allow us to study the approachtowards the common large- N limit via an alternative sequence of groups in respect to SU ( N ) and SO ( N ) . In turn, this will provide an alternative set of numerical tests for suchconjectural behaviours as those put forwards for example in Refs. [82, 84, 85], as well asallowing comparison to calculations performed within the context of gauge-gravity dualities(see for instance Refs. [86–96]) or with alternative field theoretical methods [97–101]. Inpragmatic terms, we will also set the stage for future studies in quenched theories realisingthe SU (4) /Sp (4) coset, based upon generic Sp (2 N ) groups, extending the results obtainedin [81] for the Sp (4) gauge theory.In our investigation, we adopt a unified approach to the study of Sp (2 N ) gauge theories,by applying the same heat bath (HB) algorithm exploited in Ref. [7] for the Sp (4) theoryto the whole Sp (2 N ) sequence. In addition to reconsidering Sp (2) ∼ SU (2) , which allowsto test our algorithm and procedures by comparing to existing results in the literature and– 2 –o extending the N = 2 results discussed in Ref. [7] with new calculations, we considerthe N > cases. For the latter, with the exception of our study in Ref. [82] (focussingon a discussion of the two lowest-lying glueball states and on a remarkable universalityproperty of their ratio) and Ref. [83] (presenting some preliminary numerical results, furtherdiscussed in the current work), no detailed calculation of the glueballs has been reportedin the literature so far. From an operational perspective, we first compute the effectivestring tension and glueball masses in the large-volume limit for fixed lattice spacing and N = 1 , , , . Then, after taking the continuum limit of the glueball spectrum at eachinvestigated value of N , we perform a critical analysis of the large- N extrapolation andcompare to other results in the literature, as appropriate.The paper is organised as follows. In Section 2 we introduce the basic definitionsand conventions adopted in the lattice calculations. In Section 3 we describe the spectralobservables of interest. In Section 4 we present our numerical results, including also theextrapolations to continuum and large- N limits. Section 5 summarises our conclusions andsuggestions for future further enquiries. We have relegated some important technical detailsto the Appendices. In four Euclidean dimensions, the Sp (2 N ) gauge theory is defined by the following action S YM ≡ − g (cid:90) d x Tr F µν F µν , (2.1)where g is the gauge coupling, the trace is over colour indices, the field-strength tensor F µν ≡ (cid:80) A F Aµν τ A is defined by F Aµν ≡ ∂ µ A Aν − ∂ ν A Aµ + f ABC A Bµ A Cν , (2.2)and the gauge fields are A µ = (cid:80) A A Aµ τ A , with the indices taking the values A, B, C =1 , · · · , N (2 N + 1) , for Sp (2 N ) . The N × N matrices τ A are the generators of thealgebra associated with the Sp (2 N ) group, written in the fundamental representation, andnormalised according to Tr τ A τ B = δ AB . The structure constants of the algebra aredefined as the commutation relations (cid:2) τ A , τ B (cid:3) = i f ABC τ C . (2.3)We regularise the theory on a lattice, in which the continuum coordinates are discre-tised with lattice spacing a . The four dimensional Euclidean hypercubic lattice consists ofsites that are denoted by their position x in the lattice. The sites are connected by linksthat are characterised by the position x and direction µ , where µ, ν = 0 , .., label the fourspace-time coordinates. The elementary variables of the lattice regularised Sp (2 N ) gaugetheory are the link variables , defined as U µ ( x ) ≡ exp (cid:18) i (cid:90) x +ˆ µx d λ µ τ A A Aµ ( λ ) (cid:19) , (2.4)– 3 –ith ˆ µ the unit vector in direction µ . The N × N matrices U µ ( x ) transform accordingto the fundamental representation of the Sp (2 N ) group. Gauge transformations take theform U µ ( x ) → g ( x ) U µ ( x ) g † ( x + ˆ µ ) , with g ( x ) a group element.The simplest gauge invariant operator is the trace of the product of link variablesaround an elementary square of the lattice, P µν ( x ) ≡ U µ ( x ) U ν ( x + ˆ µ ) U † µ ( x + ˆ ν ) U † ν ( x ) . (2.5)The matrices P µν ( x ) are called the elementary plaquette variables or just plaquettes forshort.The Sp (2 N ) lattice gauge theory (LGT) we adopt in this paper is defined by the Wilsonaction , S W ≡ β (cid:88) x (cid:88) µ<ν (cid:18) − N (cid:60) Tr P µν ( x ) (cid:19) . (2.6)In this expression, (cid:60) Tr P µν ( x ) is the real part of the trace of P µν ( x ) . The inverse coupling β is related to g by the request that, when the lattice spacing a → , Eq. (2.6) tends tothe continuum Yang-Mills action in Eq. (2.1), at leading order in a . From this requirement,one finds β = 4 Ng . (2.7)Monte Carlo numerical evaluations of the integrals appearing in the definitions allow usto explore the long-distance regime of the Sp (2 N ) (pure) Yang-Mills theories, capturing non-perturbative phenomena that are not accessible to perturbation theory. For any quantity O ( A µ ) that depends on the gauge fields, the physical observables are estimated as ensembleaverages, which are schematically given by (cid:104)O ( U µ ) (cid:105) ≡ (cid:82) D U µ e − S W O ( U µ ) Z ( β ) , (2.8)where the denominator is Z ( β ) ≡ (cid:90) D U µ e − S W . (2.9)These expressions can be computed numerically by sampling the space of configurations of U µ ( x ) , according to the probability distribution e − S W . This can be achieved by defining aMarkovian process that evolves a particular configuration according to an update algorithm .The algorithm must respect detailed balance and reproduce the correct equilibrium distri-bution. Then, if i labels the M configurations produced sequentially, the ensemble averagecan be obtained as the simple average (cid:104)O(cid:105) = lim M →∞ M M (cid:88) i =1 O i , (2.10)where O i is the value that the observable O ( U µ ) takes on configuration i . The algorithmadopted in this work to produce successive configurations is a combination of local heatbath (HB) and overrelaxation (OR) updates, adapted to Sp (2 N ) from the SU (2 N ) imple-mentation provided in Ref. [102] (see Appendix A for further details). Configurations are– 4 –pdated sequentially, one link at a time, with one HB update followed by four OR updates.An update of all the links on the lattice is called a lattice sweep . Successive configurationsproduced in this manner are correlated; to reduce the effects of autocorrelation, the ensem-ble averages used for physical calculations are restricted by sampling the history in stepsthat are separated by 10 lattice sweeps. Our implementation of the algorithms above isbased on the HiRep code [103], originally designed for the treatment of SU ( N ) theorieswith matter fields in general representations. The lattice size being finite, we impose periodic boundary conditions in all directions.In the continuum, it is known that resulting configurations of gauge fields are characterisedby an integer topological number [104], defined as Q ≡ π (cid:15) µνρσ (cid:90) d x Tr F µν F ρσ . (2.11)The associated susceptibility can be related to the large mass of the η (cid:48) particle [105]. Theconfiguration space is thus divided into sectors, each characterised by an integer value ofthe topological number Q , and separated from each other by potential barriers.Because of the lattice discretisation, the topological charge Q takes nearly integervalues [106–108]. There are many microscopic lattice definitions of the topological chargethat reproduce the same, correct long-distance results in the a → limit. In this work weadopt the definition Q ≡ (cid:88) x q ( x ) , (2.12)with q ( x ) ≡ π (cid:15) µνρσ Tr { U µν ( x ) U ρσ ( x ) } , (2.13)and where x runs upon all lattice sites. Since these definitions make use of the short-distance degrees of freedom, calculations are affected by short-range fluctuations. Theseeffects can be reduced by the use of smoothing operations such as the Gradient (or Wilson)Flow [109], which we will introduce below.As in the continuum, also on the lattice the different topological sectors are separatedby potential barriers. If these barriers are not too steep, in simulations a sufficient number oftunnelling events between sectors will occur, and the resulting measured topological chargewill be Gaussian distributed around zero. However, superselection of topological sectorscan be shown to emerge close to the continuum limit [106, 107]. As a consequence, MonteCarlo update algorithms tend to become trapped inside one of the topological sectors.Hence, close to the continuum limit, the topological charge has a long autocorrelation time.This phenomenon is referred to in the literature as topological freezing . Due to large- N suppression of small-size instantons, which are crucial for changing the topological chargein numerical simulations [110], topological freezing becomes more severe as N increases.We shall discuss implications of this algorithmical trapping more extensively later in thepaper, focusing on the effects of topological freezing on the observables that are of interestto us. HiRep can be downloaded from https://github.com/claudiopica/HiRep. – 5 –o remove ultraviolet fluctuations that would otherwise dominate the signal in theextraction of the topological charge, we employ the Gradient Flow [109, 111] of the Wilsonaction ( i.e. the Wilson flow). The Gradient Flow provides a first-principles approach tothe smoothening of configurations with efficiency comparable to that of the more empiricaland time-honoured cooling methods (see for instance Ref. [112]). Moreover, the evolutionof observables under the Gradient Flow can be determined with numerical procedures thatcan be easily implemented. For this reason, this method has gained a prominent role inlattice studies in recent years.With t the coordinate in an additional fifth dimension (referred to as flow time ) and x a point in four-dimensional space, the Gradient flow B µ ( t , x ) is defined by the followingdifferential equations and boundary conditions: d B µ ( t , x )d t = D ν G νµ ( t , x ) , with B µ ( t = 0 , x ) = A µ ( x ) . (2.14)Here A µ ( x ) is the continuum gauge field, while the covariant derivative is D µ = ∂ µ + [ B µ , · ] ,which yields the field-strength tensor: G µν ≡ [ D µ , D ν ] . (2.15)On the lattice, the Gradient Flow for the action in Eq. (2.6) is defined by ∂V µ ( t , x ) ∂ t ≡ − g (cid:8) ∂ x,µ S flow [ V µ ] (cid:9) V µ ( t , x ) , (2.16)with initial condition V µ ( t = 0 , x ) = U µ ( x ) . Here, S flow is the Wilson plaquette action for V µ . The Gradient Flow describes a diffusion process with time t . At the leading orderin the coupling g , the flow to time t acts on the gauge fields as a Gaussian sphericalsmoothing operation, with root-mean-square radius √ t , the flow time t having the di-mension of a length squared. Furthermore, to all orders in perturbations in g , any gaugeinvariant composite operator constructed from B µ ( t , x ) is renormalised at t > , and thusdirectly encodes physically observable properties. Using a value of the flow time τ such that a (cid:28) √ τ (cid:28) R , where R is a typical hadronic scale, provides four-dimensional smoothedconfigurations V ( τ, x ) that are not affected by ultraviolet fluctuations and still encode thecorrect infrared behaviour of the theory. Non-Abelian Yang-Mills theories confine, and their spectra consist of massive colour-neutralstates called glueballs . If a non-Abelian gauge theory is formulated on a space with one ormore compact directions, wrapping torelon states arise. The validity of the confinementpicture for the specific case of Sp (4) has been confirmed explicitly in the numerical calcula-tions reported in Refs. [6, 7]. The main objectives of this work are to show through latticecalculations that, as one would expect, confinement arises also in Sp (6) and Sp (8) , to mea-sure the resulting glueball mass spectrum, and to determine the large- N limit of the latter.– 6 –efore discussing our numerical results, in this section we review the methodology we shalladopt. The methodological material presented in this section is based upon notions thathave been tested and are well established in the literature. Details beyond our expositioncan be found, e.g., in Refs. [113–118], from which we draw heavily in what follows. Let H be a Hamiltonian of the 3-dimensional system of volume L defined on a lattice with L t time slices. Let | n (cid:105) and E n be the eigenstates and eigenvalues of H , i.e., H| n (cid:105) = E n | n (cid:105) . (3.1)The transfer matrix , T ≡ e − a H , (3.2)is the operator that evolves one time slice of the system into the next. Note that in thissection, for simplicity, we reabsorbe β in the definition of H . In terms of T , the partitionfunction in Eq. (2.9) can be expressed as Z = Tr (cid:0) T L t (cid:1) . (3.3)Masses of particle states can be obtained from the large time decay rate of (normalised)two-point correlators of interpolating operators, C ( t ) ≡ (cid:104) Ω | O † (0) O ( t ) | Ω (cid:105)(cid:104) Ω | O † (0) O (0) | Ω (cid:105) = (cid:104) Ω | O † (0)T t/a O (0) | Ω (cid:105)(cid:104) Ω | O † (0) O (0) | Ω (cid:105) , (3.4)where | Ω (cid:105) is the vacuum state, normalised so that | Ω (cid:105) = T | Ω (cid:105) , and O ( t ) is an interpolatingoperator that produces the single-particle state | Ψ (cid:105) by acting on the vacuum, | Ψ (cid:105) = O ( t ) | Ω (cid:105) , (3.5)with (cid:104) Ω | Ψ (cid:105) = 0 . Inserting a complete set of eigenstates of H in Eq. (3.4), we obtain C ( t ) = (cid:88) n | c n | e − E n t , (3.6)where the coefficients c n , given by c n = (cid:104) n | O (0) | Ω (cid:105) (cid:112) (cid:104) Ω | O † (0) O (0) | Ω (cid:105) , (3.7)are called overlaps . If E < E < · · · , then, C ( t ) ∼ | c | e − E t (cid:18) | c | | c | e − ( E − E ) t + · · · (cid:19) ∼ | c | e − E t , t (cid:29) ( E − E ) − . (3.8)Hence E = − lim t →∞ a log C ( t + a ) C ( t ) . (3.9) In our calculations, we set L t = L/a . – 7 –his equation implies that, in principle, E can be obtained by fitting an exponential to thelarge t values of C ( t ) as measured from the lattice. When O ( t ) creates a zero-momentumstate, the energies E i are identified with particle masses m i . In our calculation we willrestrict to this case.Following from Eq. (3.9), we define the effective mass m eff ( t ) as am eff ( t ) ≡ − log C ( t + a ) C ( t ) . (3.10)If a one-particle eigenstate of the Hamiltonian were propagating, m eff ( t ) would be constantwith respect to t with a value equal to the mass of that state. In the presence of otherstates contributing to the correlation function, we expect this effective mass to be an upperbound for the true asymptotic mass at any finite t . In numerical studies, a t min can beidentified such that, for t ≥ t min , only the ground state (or, more precisely, the smallestmass eigenstate with non-zero overlap) contributes to C ( t ) within the statistical precision,and hence am eff ( t ) becomes constant. The plateau value of m eff ( t ) provides an estimate ofthe ground state mass m , which can be extracted by fitting a single exponential to thedata for C ( t ) for t ≥ t min .While this programme is at the basis of standard techniques for extracting massesfrom correlators, its direct implementation is not straightforward, and requires a carefultreatment of numerical data. The first difficulty one encounters stems from the statisti-cal noise affecting the measurements. In fact, while the statistical fluctuations of C ( t ) areroughly independent of t , the magnitude of correlation functions decays exponentially. Thisgives an exponentially-suppressed signal-to-noise ratio which is prohibitively hard to im-prove upon with an increase in the measurement sample size alone. In addition, the value t min of the onset of the single-exponential asymptotic regime is not known a priori; it is amodel-dependent feature, sensitive to the mass spectrum in the given channel and to thechoice of the operator O , as well as to the precision of the numerical calculation. The time t min is extracted from the simulations. Moreover, simple arguments based on asymptoticfreedom show that for a given operator and in a given channel, t min grows exponentially asthe continuum limit is approached.The discussion above highlights the necessity to go to large times to isolate the groundstate, but then the signal-to-noise ratio degrades and this makes it difficult to estimate m in a reliable way. If one could find operators with correlators that provide single exponentialbehaviours, one could perform fits at small times, when the signal is still well visible abovethe noise. Although this ideal situation can not be reproduced in numerical investigations,since the knowledge of operators giving rise to single exponential correlators would only arisefrom an at least partial solution of the theory, one can try to engineer the calculation in sucha way that in each relevant channel t min is as small as possible. For this purpose, at eachvalue of a we construct interpolating operators that maximize the overlaps with the spectralstates of interest. The main idea is to approximate the (unknown) exact eigenfunctions of H with an appropriate linear combination of a set of states {| Ψ i (cid:105)} , chosen on the basis ofsymmetry considerations, as trial wave functions. Then, in the given channel, the mass of– 8 –he lowest lying state above the vacuum can be bound as am ≤ − τ log (cid:26) min {| Ψ (cid:105)} (cid:104) Ψ | T τ | Ψ (cid:105)(cid:104) Ψ | Ψ (cid:105) (cid:27) = am var , (3.11)where Ψ denotes any linear combination of the variational basis Ψ i , subject to the constraint (cid:104) Ψ | Ω (cid:105) = 0 , and τ is a time chosen for minimisation, which is performed across the linearcombinations of our basis operators. This bound is saturated by the lowest-lying eigenstateof the Hamiltonian in the chosen channel, which can be obtained using a complete set ofvariational states {| Ψ i (cid:105)} . Since variational bases used in calculations are necessarily finite,the bound is in general not saturated when the variational method is used in practice.Nevertheless, with a suitably large variational basis, the extracted variational mass m var willeventually be compatible within the statistical errors with m . This framework, referredhenceforth as the variational technique , can be implemented algorithmically in order toextract both the glueball and the torelon spectrum in various channels [117].The success of this approach and the quality of the results obtained with this techniquecrucially depend on the nature of the operators that we include in the variational basis. Forthis reason, particular attention needs to be paid to its construction. We will review in thefollowing two subsections the approach we followed to construct trial states to be used in thevariational calculation, by discussing how gauge invariant states are created on the latticein Sect. 3.2, and how one obtains the irreducible representations of the symmetry group ofthe lattice in which these states must transform in Sect. 3.3. In Sect. 3.4 we will show howto perform the extremisation provided in Eq. (3.11) in an effective way, in order to obtainrobust estimates of m . The effective description of torelon states as closed fluxtubes willbe summarised in Sect. 3.5. The estimates of m will be affected by systematic errors ofdifferent origins, which will be discussed in Sect. 3.6. In this section we explain how to create gauge invariant states out of the vacuum andtheir interpretation in terms of glueball and torelon states. Consider traced path orderedproducts of links, defined in Eq. (2.4), around closed spacelike loops C , U ( C ) = Tr (cid:89) ( x,µ ) ∈C U µ ( x ) , (3.12)where C can be defined as a set of successive displacements, C = [ f , f , · · · , f L ] , (3.13)where each f j is one of the elementary vectors of the lattice { (cid:126)e i } . The sequence f , f , · · · , f L is defined up to cyclic permutations. The closeness of the path C implies that (cid:80) i f i = 0 .A generic gauge invariant operator O such that (cid:104) Ω | O | Ω (cid:105) = 0 can be obtained as a sumof products of operators O C , each defined as O C = U ( C ) − (cid:104) Ω | U ( C ) | Ω (cid:105) (3.14)– 9 –or specified choices of C .Single trace operators create states called glueballs when C is contractible and torelons states when C wraps around a spatial direction of the space-time hypertorus, and is thusnon-contractible. These two classes of states transform in different representations of thecenter of the group and hence do not mix. We will start our analysis from the contractibleloops. Most of our arguments are applicable also to non-contractible loops, which will beanalysed more specifically in Sect. 3.5.Multitrace operators are monomials involving products of at least two of the operatorsin Eq. (3.12). Operators in this class can be used to generate multi-particle states. Someof these states have the same quantum numbers as single particle states we are interestedin, and can thus mix with them. This mixing can result in a systematic error in theextraction of masses of single-particle states. In our calculation, we will neglect mixing ofgenuine glueball states with multi-particle states. The justification for neglecting multi-trace operators resides in the fact that matrix elements involving them go to zero in thelarge- N limit. In the lattice theory, the Poincaré symmetry of the continuum is explicitly broken to thediscrete subgroup of symmetries of the hypercubic lattice and discrete translations by aninteger number of lattice spacings. In particular, on an infinite lattice, for a time slice, thisis the semi-direct product of discretised translations T d and of the point groups of invarianceof the elementary (cubic) cell of the lattice: the Octahedral group O h (see e.g. Ref. [119]).The study of the representations of this symmetry group of the lattice is simplified by thefact that T d is an invariant abelian subgroup. The one-dimensional representations of O h (related to momentum) can thus be studied separately from those of T d .In a finite box of size L , and with periodic boundary conditions, the momentum isquantised in every direction as p n = 2 πn/L . On a lattice its value must also lie in theBrillouin zone containing (cid:126)p = (cid:126) . Operators at fixed momentum can be obtained as Fouriersums of their counterpart in coordinate space, O C ( t, (cid:126)p ) = (cid:88) (cid:126)x e i(cid:126)p · (cid:126)x O C ( (cid:126)x, t ) . (3.15)Zero momentum combinations (to which we restrict ourselves in this study) can be simplyobtained as sums over fixed time slices of operators of the type given in Eq. (3.14), O C ( t, (cid:126)p = (cid:126)
0) = (cid:88) (cid:126)x O C ( (cid:126)x, t ) . (3.16)We now briefly describe the irreducible representations of O h and their relation withthe representations of the Poincaré group. The Octahedral group is the symmetry groupof a cube. This group has elements divided in conjugacy classes. Accordingly, it has inequivalent irreducible representations, labelled by R = A , A , E, T , T , of dimensions , , , , , respectively. The spatial parity P has two eigenstates, which we label byan additional ± sign, depending on whether they remain invariant ( + ) or are reflected (-)– 10 –nder a parity transformation. We will label the states of the lattice theory with R = R P and their mass with m R P . Asterisks will denote excitations of the ground state: A + ∗ willdenote the first excited state of A +1 , A + ∗∗ the second, etc.The states generated from the vacuum by gauge invariant operators U ( C ) will transformin the same representation as the paths on which they were defined according to Eq. (3.12).In general, single trace operators belong to reducible representations of the octahedralgroup. Under the action of an element r of the group, the operators O C transform inrepresentation U ( r ) in the following way U ( r ) O C U − ( r ) = O r C , (3.17)where the law of transformation of C can be inferred from its definition in Eq. (3.13), C (cid:48) = r C = [ rf , rf , · · · , rf L ] . (3.18)The decomposition of U ( r ) in terms of its irreducible components can be obtained from theorthonormality property of characters, supplemented by a choice of orthonormal bases foreach of the irreducible representations R P of O h . For this, the projector method borrowedfrom Ref. [119] has been used.In the continuum limit, we expect the Poincaré symmetry to be recovered. The rela-tionship between the representations of the octahedral group defined above and those of thePoincaré group enables us to decompose the former in their continuous spin components.The representations of the Poincaré group are labelled by the mass m and the quantumnumbers J P C , where J is associated to irreducible representations of the rotation group, P to spatial parity and C to charge conjugation. Owing to the pseudo-reality of the represen-tations of Sp (2 N ) , C is always positive. Hence, we will drop this quantum number fromnow on.If we restrict the elements of the rotation group in a representation J to the discreterotations that lie in O h , we obtain the subduced representation J ↓ O . We report in Tab. 1the subduced representations for the lowest values of J , adapted from Ref. [117]. In O h ,these representations are reducible in terms of A , A , E , T and T . Thus, degeneratestates with the same spin but different polarisation of the continuum spectrum might havea different mass on the lattice. In the continuum limit, nevertheless, the restoration ofcontinuum rotational invariance implies that these states become degenerate. For instance,the E and T representations of the octahedral group contain respectively two and three ofthe five polarisations of spin-2 particles. Hence, corresponding states extracted in the E ± and T ± channels must become degenerate in the continuum limit. The degree of degeneracyof these states at finite lattice spacing will thus provide an important measure of the effectof lattice artefacts. Let us now consider a specific irreducible representation R P and build a generic linearcombination Φ of basis elements O R P at time t , which we denote as Φ( t ) = (cid:88) i v i O R P i ( t ) . (3.19)– 11 – A A E T T Table 1 . Subduced representations R of the continuum rotation group and their componentslabelled with the spin J , up to J = 4 . The 2-point correlation function is (cid:104) Ω | Φ † (0)Φ( t ) | Ω (cid:105) = (cid:88) ij v (cid:63)i v j C ij ( t ) , (3.20)where, in general, C ij ( t ) = (cid:88) a c a(cid:63)i c aj e − m a t , (3.21)with c ai = (cid:104) a | O R P i (0) | Ω (cid:105) . As a result, Eq. (3.10) can be rewritten as am eff ( t ) = − log (cid:80) ij v (cid:63)i v j C ij ( t ) (cid:80) ij v (cid:63)i v j C ij ( t − a ) . (3.22)The matrix C ij ( t ) is positive-definite (see Eq. (3.21)), and its eigenvalues are given by λ a ( t ) = e − m a t . Hence, extracting the spectrum is equivalent to the diagonalisation of C ij ( t ) .Unfortunately, due to statistical fluctuations, eigenvectors and masses of the measured C ij ( t ) do depend on t . In order to resolve this dependency, we seek a solution to thegeneralised eigenvalue problem (cid:88) j C ij ( τ ) v j = λ ( τ, (cid:88) j C ij (0) v j , (3.23)by diagonalising (cid:2) C − (0) C ( τ ) (cid:3) for some τ > . The eigenvectors of (cid:2) C − (0) C ( τ ) (cid:3) provideus with a practical choice ˜Φ i of the optimal operators. The corresponding masses m i canbe obtained from fits of the correlators of the ˜Φ i (which we refer to as ˜ C i ) at t > t , usingthe ansatz ˜ C i ( t ) = 2 | c i | e − m i L t a/ cosh m i (cid:18) t − L t a (cid:19) , (3.24)over ranges of t for which am eff ( t ) = arccosh (cid:32) ˜ C i ( t + a ) − ˜ C i ( t − a )2 ˜ C i ( t ) (cid:33) (3.25)reaches a plateau value. We still denote as am eff the effective mass, although we adopt fromnow on a definition that differs from the one in Eq. (3.10). The reason for the discrepancy,which is visible only away from the large volume limit, is a consequence of adopting periodic– 12 –oundary conditions in time, which allows for both forward and backward propagatingstates. The mass of the ground state, m , is obtained from a fit of the largest eigenvalue λ . The masses of higher energy states can be obtained in the same manner from thediagonal correlators of eigenvectors associated to the other eigenvalues computed in thegeneralised eigenvalue problem.As discussed earlier, a crucial ingredient for an efficient variational calculation is thepreparation of trial states that have an extension comparable to the target glueball state.A priori, we have no information about the physical size characterising glueball states. Todetermine an efficient linear combination, we shall insert in our variational set operatorsobtained from prototypical paths of different sizes and shapes, and also operators obtainedfrom the original basis at each of S iterations of smearing and blocking operations, withthe combination obtained in Ref. [2] (to which we refer the Reader also for the definitionof the operations of blocking and smearing and for specific details on the particular pathsused to define the basis operators). In this way, we obtain a variational basis that finelyscans the propagating states from length scales corresponding to the lattice spacing all theway up to the lattice extent L .From the technical perspective, the only procedural change to the methodology em-ployed for Sp (4) in Ref. [7] (to which we refer for further details) lies in the projection andcooling routines, that had to be adapted to the case of Sp (2 N ) . With M elementary pathsand S smearing steps used for constructing the basis, our variational basis is formed by S × M operators in total, and ˜ C ij ( t ) will accordingly have ( S × M )( S × M + 1) / elements.In our calculations, we perform the maximum number of blocking steps N b allowed by thefinite size, provided by the maximisation of the l.h.s. in the inequality N b ≤ L/a . At eachblocking step we perform smearing steps and cooling steps to reproject on the group.In general, our variational basis contains approximately elements. Torelon states are generated from the vacuum by path ordered products of link variablesalong non-contractible paths, i.e. paths that wrap the periodic lattice along a given direction.These states have the same quantum numbers as physical states in which a wrapping closedloop of glue with fixed length is propagating in the system. We refer to this configurationas a fluxtube . When the fluxtube is long enough, it can be described by an effective stringtheory. This classical effective theory is written in terms of a single physical parameter, thestring tension σ , that governs the energy of the fluctuations. In order to extract it fromthe data with the highest precision, we will make use of effective string theory, as brieflysummarised in this section.Effective string theory is based on approximating the fluxtube as a one dimensionalfluctuating object — a string — with constant energy per unit length. Classically, the mass m and the length L of the fluxtube are proportional, m = σL . (3.26)This classical string description becomes exact in the long string limit L σ → ∞ .– 13 –t finite length, quantum corrections become relevant. The energy of the fluxtube isobtained as a power expansion in / ( σL ) around the long string limit. In general m = σL (cid:32) ∞ (cid:88) k =1 d k ( σL ) k (cid:33) , (3.27)where the dimensionless coefficients d k , which are in principle calculable, can be determinedby matching the power series to the results of numerical measurements. Universality the-orems allow to fix some of these coefficients on the basis of symmetry arguments. Theformation of the fluxtube can be described as a process of spontaneous breaking of someof the generators of Poincaré symmetry. We omit details, for which we refer the Reader tothe literature [120].The ground state mass m of a torelon wrapping along one direction of extent L is given,in a spacetime of dimension D , by m LO ( L ) = σL − π ( D − L , (3.28)where we included the leading order correction in an expansion in /σL , and m NLO ( L ) = σL − π ( D − L − (cid:18) π ( D − (cid:19) σL , (3.29)which describes m ( L ) up to the next-to-leading-order correction. At this order, one canshow that these predictions are universal , i.e. the coefficients are fixed by Poincaré in-variance and certain geometric dualities. The only physical parameter to consider is thus σ . In general, for the ground state, no deviations with respect to the Nambu-Goto formula m NG ( L ) = σL (cid:114) − ( D − π σL , (3.30)are allowed up the term / ( σ L ) . These results will allow us to compute σ from the massof torelons, keeping under control the effects of working at finite L . There are several sources of systematic errors that affect the computation of glueball andtorelon masses. In this section, we discuss the most relevant ones for our study.As explained in Sect. 3.4, the variational technique depends on our choice of basis ofoperators. A potential source of error is the choice to only include single trace operatorsin our variational set. By doing so, we are neglecting scattering states and multitorelonstates that share the quantum numbers of single glueball states of interest. In the case ofscattering, we deal with states with two or more glueballs. Neglecting the interactions (anapproximation that holds at large N ), these states have masses that are about twice aslarge as the smallest glueball mass. Thus, below this threshold, we can safely neglect theeffect of scattering states. Even above that threshold, scattering states decouple at large N .– 14 –ultitorelon states have a mass that is in general an increasing function of L . Therefore, atlarge enough values of L , they decay quickly in correlators and can hence be neglected aswell. The effect of these states can in principle be controlled by including the correspondingoperators in the variational basis and evaluating their overlaps, as done in Ref. [117].As a consequence of the fact that we are simulating a finite lattice, all our physicalestimates will be affected by finite size effects. These effects have been reported in Ref. [121],where it is shown that they obey the relationship m ( L ) = m (cid:40) be − √ mL mL (cid:41) , (3.31)with m ( L ) and m the masses in volume V = L and at infinite volume, respectively. b isa coefficient that, a priori, depends on the symmetry channel. Under the assumption thatthese corrections are independent of the lattice spacing a , we will be able to compute themat one value of a and use the same prediction for all others. More so, we will be able toneglect them altogether once we find that, at a certain combination of a and L , these effectsare much smaller than the statistical error.Discretisation errors come from the dependence of the masses on the lattice spacing. Atrivial dependence can be inferred from dimensional analysis. The lattice combination ma is dimensionless. Since all masses obtained on the lattice depend on the lattice spacing a inthis way, we consider ratios of dimensionful objects where the trivial dependence simplifiesin the ratio. As a reference scale for the ratio, we use the square root of the string tension √ σ . This choice is motivated by the fact that, thanks to the results discussed in Sect. 3.5,we can measure the string tension more accurately than any other quantity of interest.Hence, the use of the string tension reduces significantly the systematic error due to thescale setting process, which is a necessary step to provide quantities in physical units.Beyond the overall dependence of the mass on the lattice spacing a , we know, by com-puting the naïve continuum limit of the theory described by the lattice action in Eq. (2.6),that the leading corrections to mass ratios start at order a . Therefore, close to the contin-uum limit, for a glueball state R P , we approximate m R P √ σ ( a ) = m R P √ σ (0) (cid:0) c R P σa (cid:1) . (3.32)To conclude this overview of systematic effects, we return to mentioning topologicalfreezing. Near the continuum limit, the Monte Carlo updates tend to get trapped in a sectorat fixed topology. This topological trapping becomes more pronounced at larger N [110].Restricting the gauge theory to a sector at fixed topology generates power-law correctionsin the inverse volume that delay the onset of the large volume regime [122, 123]. Bothlarge- N reduction arguments [124] and the large- N scaling prescription of the θ angle [125]suggest that finite volume corrections due to topological freezing are suppressed at large N .The decreased severity of topological freezing as N increases has been verified explicitly inRef. [126]. In Sect. 4, we will show that topological freezing affects only a small subset ofour calculations. When discussing the relevant ensembles, we shall describe how topologicalfreezing has been accounted for in those specific cases.– 15 – Numerical Results
In this section, we present and discuss our main numerical results. In Sect. 4.1 we performcalibration and validation studies of the underlying algorithm. We also select the values ofthe coupling β at which to compute the masses of torelons and glueballs for the Sp (2 N ) Yang-Mills theories with N = 3 , . In Sect. 4.2, we compute the ground state mass oftorelons of various lengths at fixed lattice spacing. We compare the results to the predictionsdiscussed in Sect. 3.5. We also evaluate finite size effects, alongside exposing our strategy forextracting the string tension using one lattice size, in the asymptotic regime. In Sect. 4.3we report the results of the continuum limit extrapolations of the glueball spectrum for N = 1 , , , , while we cover in detail in Appendix C all pertinent technicalities. Thecontinuum limit values of the masses are then used to extrapolate towards the large- N limit, in Sect. 4.4. We compute the expectation value of the plaquette P , which is defined as (cid:104) P (cid:105) ≡ L N (cid:88) x,µ>ν (cid:60) Tr P µν ( x ) . (4.1)We consider several values of β , and focus attention on N = 3 and N = 4 . Independentensembles are generated for each chosen value of β , with either unit (cold), or random(hot) starting configurations in the Monte Carlo update algorithm. We calculate (cid:104) P (cid:105) each sweeps, record individual measurements of this quantity out of the · sweepsperformed for each β value. By comparing the history of (cid:104) P (cid:105) starting separately with unitand random configurations, we are able to identify and discard the initial transient due tothermalisation. We have verified explicitly that the integrated autocorrelation times areless than . for all values of β . We finally bin and bootstrap the measurements of (cid:104) P (cid:105) .For Sp (2 N ) Yang-Mills theories with N = 3 , , the results are shown in Fig. 1 for latticeswith size ( L/a ) = 16 .Our algorithm is based on a heat bath update of Sp (2) subgroups that when combinedprovide a covering of the whole Sp (2 N ) group (see Appendix A for a detailed explanation).For validation purposes, we obtained alternative, independent estimates for the averageplaquette using the simpler (and slower) Metropolis-Hastings update algorithm. For both N = 3 and N = 4 , and at every relevant value of β , the estimates obtained with thetwo different algorithms are compatible with each other, within one standard deviation.For N = 3 , independent numerical results are also available through Ref. [6], and ourresults are compatible with theirs within one standard deviation, when comparisons arepossible. Finally, the limits of weak ( β → ∞ ) and strong ( β → ) coupling can be controlledanalytically [6]. It is expected that (cid:104) P (cid:105) weak = 1 − ( N + 1)8 β + O (1 /β ) , (4.2)and (cid:104) P (cid:105) strong = βN + O ( β ) , (4.3)– 16 – β/ N . . . . . h P i N = 3 N = 4 Figure 1 . The average plaquette (cid:104) P (cid:105) measured at varying coupling β/ N , for fixed lattice size L = 16 a . The results from the leading order expressions at weak ( β → ∞ ) and strong ( β → )coupling regimes in Eqs. (4.2) and (4.3) are presented by the solid lines for comparison purposes. at weak and strong coupling, respectively. In Fig. 1, we compare the leading terms of theseanalytical predictions to our numerical data in the relevant regime. The combination of allthese tests supports the robustness of the algorithm we are employing.The behaviour of (cid:104) P (cid:105) as a function of β is also used to detect the potential presenceof a bulk phase transition that separates the weak and the strong coupling regimes of thetheory. While the latter is dominated by strong lattice artefacts, the former is relevantto continuum physics. The pseudo inflection point visible in Fig. 1 (for both N = 3 and N = 4 ) is a potential signature of such a phase transition. We study the nature of thischange of regime in Appendix B, where we conclude that our numerical data are compatiblewith a crossover, confirming the findings of Ref. [6] for N = 3 and extending this conclusionto N = 4 .In principle, the absence of a genuine phase transition may allow extraction of physicalobservables by performing an extrapolation to the continuum limit that makes use of genericvalues of β . Nevertheless, by restricting our choices of β to the weak coupling regime wemaintain better control over the approach to the continuum. Our choice of the values of β for the simulations results from a pragmatic compromise aimed at reducing discretisationerrors while deploying the finite amount of available computational resources. In this subsection, we discuss the methodology we use to extract the string tension fromground state mass of torelons of length L for N = 3 , , while also testing the predictions ofSect. 3.5. We first perform an analysis of the L dependence of the mass at a fixed value ofthe lattice coupling, in order to identify the regime in which the string effective description– 17 – p (6) , β = 16 . Sp (8) , β = 26 . L/a am s χ /N d . o . f . am s χ /N d . o . f . . .
82 0 . . . .
72 0 . . . .
52 0 . .
014 0 . .
32 0 . . . .
08 0 . . . .
71 0 . . . .
11 1 . . . . − −
24 0 . . − −
28 0 . . − − Table 2 . Ground state masses of the torelon states of Sp (2 N ) theories for N = 3 and N = 4 , atvarious values of L/a . Masses are obtained from a fit to Eq. (3.24). is applicable. We then extract the string tension from torelon masses measured at oneasymptotic value of L for each choice of β . This procedure allows us to obtain accurately thestring tension as a function of the finite lattice size, using the known functional form of thetorelon mass. We retained thermalised configurations for post-production analysis. Thevariational basis we adopt includes the elementary Wilson line winding around a compactspatial dimension, and averaged over all three spatial directions, alongside its blocked andsmeared improved versions, up to blocking level N b such that in the inequaliy N b ≤ L/a thel.h.s. is maximised. Following Ref. [7], to which we refer for further details, we performedeither one (for the coarsest lattices) or two (for the finest lattices) smearing steps in-betweenone blocking step.For the study of the finite size dependence of the torelon mass, we generated configura-tions with β = 16 . for Sp (6) , and β = 26 . for Sp (8) , on the lattice sizes listed in Tab. 2.These values of β are chosen to be small enough that large physical volumes are reachedwith moderate computing cost, while still remaining within the weak coupling regime. Thevalues of the masses thus obtained, denoted by am s , are reported in Tab. 2 and plotted inFig. 2. In order to extract am s , we performed a maximum likelihood analysis based uponEq. (3.24). The value of the resulting χ /N d.o.f. is usually below or around one; exceptionsto this are mostly restricted to the largest lattice studies in Sp (8) , where am s becomes oforder one and as a consequence the signal decays quickly.We now test the predictions of Sect. 3.5. From Fig. 2, we see that, at the largest valuesof L/a , m s a is an approximately linear, increasing function of the length, both in N = 3 and N = 4 . This behaviour supports the intuitive description of a torelon state as a closedfluxtube with constant energy per unit length. In order to extract the string tension σ , asa first approximation we use Eq. (3.26) applied to the largest value of L/a , treating thefluxtube as a classical string. We call σ cl the resulting string tension. For Sp (6) , we find σ cl a = 0 . at L/a = 28 and for Sp (8) , we obtain σ cl a = 0 . at L/a = 20 .The large- L expansion is expected to be well-behaved when σL (cid:29) . At a given value– 18 – p (6) Sp (8) β = 16 . , L ≥ a β = 26 . , L ≥ aσa χ /N d.o.f σa χ /N d.o.f linear . .
79 0 . . LO . .
74 0 . . NLO . .
89 0 . . NG . .
97 0 . . Table 3 . Measurements of the string tensions σ , based upon applying Eqs. (3.28)-(3.30) and alinear form inspired by Eq. (3.26), to fit the dependence of numerical results of m s a on L/a . of β , the classical string in Eq. (3.26) should hence provide an accurate description of thetorelon when L (cid:29) a for Sp (6) and L (cid:29) a for Sp (8) , the numerical coefficients in thesetwo expressions coming from the condition σL (cid:39) . Corrections to long string behaviour,such as those encoded in Eqs. (3.28)-(3.30), are expected to become important as L/a isdecreased.We show in Fig. 2 our best-fit results of the numerical data, based upon Eqs. (3.28)-(3.30) and a linear form inspired by Eq. (3.26), restricting the fitting region to the range L ≥ a in Sp (6) and L ≥ a in Sp (8) . The results of the fits are also reported inTab. 3. All the values of χ /N d.o.f. are acceptable. Determinations based upon LO, NLOand NG effective string treatments are indistinguishable from one another, but they aredifferent from the classical behaviour represented by the linear approximation. We electedto adopt the NG value as our best determination of the string tension as final result of thispreliminary analysis, and hence we find σ N =3 a = 0 . , β = 16 . , (4.4)for N = 3 , and σ N =4 a = 0 . , β = 26 . , (4.5)for N = 4 .From this analysis, we observe that at the chosen values of β the string picture providesa good description of the torelon mass down to lattice size L = 16 a for Sp (6) and L = 12 a for Sp (8) . These values correspond to L √ σ (cid:39) . for Sp (6 ) and L √ σ (cid:39) . for Sp (8) ,somewhat more generous than the generic, conservative estimates we anticipated. Wehence impose the bound L √ σ ≥ , in order to control the extraction of the string tensionthrough an asymptotic large- L expansion including at the least the LO correction. Thisobservation will be used in the following to extract the string tension at other β values forboth N = 3 and N = 4 , when we will apply the NG expression to torelon masses obtainedat a single size L and test a posteriori that L/a fulfils the condition L √ σ ≥ . In this Section, we report on the the spectrum of glueballs in Sp (2 N ) gauge theories for N = 1 , , , , for each fixed value of N , focusing on the results obtained in the continuum.– 19 – L/a . . . . . . . m s a N = 3 L/a . . . . . . N = 4 linearLONLONG Figure 2 . Masses of the torelons measured on lattices of volume L at fixed lattice coupling β . Wecompare the results obtained by adopting a linear expression (referred to as linear, blue), leadingorder (yellow), next to leading order (green) and Nambu-Goto (red) effective description of thedependence of m s a on L/a . In the left hand panel, we show the results for the Sp (6) theory atcoupling β = 16 . , with fits to the data in the range L/a ≥ . The right hand panel displays theresults for the Sp (8) theory at coupling β = 26 . , with fits to the data in the range L/a ≥ . Fitcurves are displayed outside the fit range in order to expose the short- L deviations of the data fromthe asymptotic string behaviour. See Tab. 3 for the fit results. Calculations of the masses in units of the lattice spacing, finite size effects studies and moretechnical details on the continuum extrapolation can be found in Appendix C.We report the glueball masses in Tab. 4, in units of √ σ (top half of the table) as well asin units of the mass of the E + state (bottom half). The spectra at the various values of N are also presented in Fig. 3, where, together with the lattice quantum numbers, we displaythe continuum quantum numbers of glueball states. The latter have been obtained fromthe decomposition presented in Tab. 1 under the assumption that lighter states correspondto lower continuum spin. For N = 1 , since Sp (2) (cid:39) SU (2) , results for the spectrum arealready present in the literature, see for example Ref. [2]. In this case, the results obtainedin our study are useful for comparison and as a test of our procedure. For SU (2) , Ref. [2]finds the values m A +1 / √ σ = 3 . and m E + / √ σ = 5 . . These values are compatible– 20 – ∞ R P m R P / √ σ m R P / √ σ m R P / √ σ m R P / √ σ m R P / √ σA +1 . . . . . A + ∗ . . . . . A − . . . . . A −∗ . . . . . A +2 . . . . .
0) 8 . A − . . . . .
4) 8 . T +2 . . . . . T − . . . . . E + . . . . . E − . . . . . T +1 . . . . . T − . . . . . ∞ R P m R P /m E + m R P /m E + m R P /m E + m R P /m E + m R P /m E + A +1 . . . . . A + ∗ . . . . . A − . . . . . A −∗ . . . . . A +2 . . . . . A − . . . . . T +2 . . . . . T − . . . . . E + − − − − . E − . . . . . T +1 . . . . . T − . . . . . Table 4 . Calculations of the masses in the continuum limit for each N and each channel, in unitsof √ σ (top) and m E + (bottom). For N = 2 , these values have been computed as weighted meansbetween those in Ref. [7] and those obtained in the present work, see Appendix C.2. In the caseof SU ( N → ∞ ) , we have m/ √ σ = 3 . for the A ++1 channel, . for the A ++ ∗ channeland . for the E ++ channel (data taken from Ref. [2]). As expected, at least for these threechannels, which are the only ones for which we can compare, the masses of Sp ( N → ∞ ) and SU ( N → ∞ ) theories are compatible. within one standard deviation with the values obtained in this work (see Tab. 4). For N = 2 , some of us already obtained first results for the spectrum in Ref. [7]. We combineour new measurements for Sp (4) with our earlier results, and in Tab. 4 we report theweighted averages of the two. The available data sets for Sp (4) are discussed more in detailin Appendix D.A look at Fig. 3 shows that, while specific details depend on N , there are common– 21 – + − + − + + − − + − A +1 A − A +2 A − T +2 E + T − E − T +1 T − m R P √ σ N = 1 N = 2 N = 3 N = 4 Sp ( ∞ ) SU ( ∞ ) Figure 3 . Spectrum of the Sp (2 N ) theory in the continuum limit for N = 1 , , , , and N = ∞ ,in units of √ σ . Continuum quantum numbers are reported at the top. For comparison, we havereported also the masses of the A ++1 , A ++ ∗ and E ++ states for SU ( ∞ ) (borrowed from Ref. [2]).The boxes represent 1 σ statistical errors. patterns across the investigated values of N . As expected, the A +1 channel is consistently thelightest, followed by the ( T +2 , E + ) (degenerate) pair. At a slightly larger mass we find the A − channel and the T − and E − (degenerate) pair. As explained in Sect. 3.3, the degeneracyof these pairs provides evidence that the rotation invariance of the continuum theory isrecovered as a → . The remaining channels, A ± and T ± , are also almost degenerate inpairs and their masses are larger than those of all other states. Since the smallest massesin the A ± and T ± channels are comparable with twice the ground state mass of the lowest-lying A +1 state, numerical results for these masses may be affected by systematic errors dueto mixing with scattering states, as discussed in Sect. 3.6. An indication of this is the factthat the error bars for the masses of those heavier states are visibly larger. Large error barsare also the result of the higher level of noise affecting the extraction of masses of heavierstates.We were able to extract masses of excited states for the A ± channels at all values of N . These masses are reported in Tab. 4 and displayed above the corresponding groundstates in Fig. 3. The error bars of the A + (cid:63) states are comparable to those of the groundstate in the A − channel, while for the A − (cid:63) states they are similar to those found in heavierchannels. – 22 –inally, we note that, where determined in both calculations, corresponding states ob-tained from a recent SU (3) study [127] are in broad agreement with the spectrum resultingfrom our investigation. N limit As shown for instance in Ref. [128], while corresponding quantities in SU ( N ) and Sp (2 N ) Yang-Mills theories converge to a common large- N limit, the /N expansions around thislimit are different: in the case of SU ( N ) , only even powers of /N are present, whilefor Sp (2 N ) the power expansion is genuinely in /N . Following the strategy that hasbeen implemented in the large- N extrapolation of the SU ( N ) glueball masses, we shallinvestigate whether the lowest order correction to the large- N limit is sufficient to describethe large- N glueball spectrum in Sp (2 N ) for all the simulated values of N . Therefore, wefit the finite- N spectrum with the ansatz m R P √ σ ( N ) = m R P √ σ ( ∞ ) + c R P N , (4.6)where c R P is a constant (expected to be of order in a well-behaved expansion) thatdepends on the glueball channel. If the ansatz provides a sufficiently accurate descriptionof the data, m RP √ σ ( ∞ ) is a reliable infinite- N extrapolation of the ratio of the mass in thechannel R P normalised to the square root of the string tension. R P m R P / √ σ ( N = ∞ ) c R P χ /N d.o.f. A +1 . . . A + ∗ . − . .
2) 2 . A − . . . A −∗ . . .
4) 3 . A +2 . − . .
3) 3 . A − . . .
0) 0 . T +2 . . . E + . . . T − . . .
2) 1 . E − . . .
2) 2 . T +1 . . .
6) 1 . T − . . .
6) 0 . Table 5 . Large- N extrapolated masses of the glueball spectrum obtained from a fit of Eq. (4.6).Note that the left hand part of this table is the same as the last column of Tab. 4. For each channel, we perform a separate linear fit to Eq. (4.6) using c R P and m R P / √ σ ( ∞ ) as fittings parameters. The results of the fits are reported in Tab. 5. The fitting range in-cludes all the values of N . From Fig. 4 we see that Eq. (4.6) describes the data well in thisrange of N for the A ± channels, the T ± , E ± degenerate pairs and for the T ± channels.For the A −∗ and for the A +2 channels, the value of χ /N d . o . f . is larger than the criticalvalue at confidence level. For comparison, in Appendix E, the same fits are performed– 23 –roup η (0 + ) χ /N d.o.f. SU ( N ) 5 . . Sp (2 N ) 5 . . Combined . . Table 6 . Resulting values of the universal constant η for the Casimir scaling described in Eq. (4.7)for Sp (2 N ) and SU ( N ) groups. The combined fit to both groups is also reported. for a range N > . Although generally the χ /N d . o . f . are smaller, in this latter case theparameters c R P and m R P / √ σ ( ∞ ) are estimated from three data points only and thus onlyone degree of freedom remains to assess the goodness of the fits. For this reason, we optto present the extrapolations including the (generally still acceptable) N = 1 data points,postponing to future studies that investigate larger N the question of whether N = 1 iscaptured by a simple leading correction with the current precision of the data. For the timebeing, in the absence of any evidence that would suggest otherwise, we assume that indeedthis is the case.For some of the lightest states for which the continuum mass in the large- N limit isavailable in the literature, we can verify that the large- N extrapolation of the Sp (2 N ) and of the SU ( N ) values are compatible. In Fig. 3 the spectrum in the large- N limit isrepresented together with the finite- N one, to allow for such a comparison. Recalling thatcharge conjugation is always positive in Sp (2 N ) , for the sake of comparing correspondingstates, we temporarily reintroduce the corresponding index in the notation for glueballstates for the rest of the current subsection. With the second + superscript identifyingpositive charge conjugation, we borrow the values of the A ++1 , A ++ ∗ and E ++ channelmasses for SU ( ∞ ) from Ref. [2]. Fig. 3 shows that the large- N extrapolations of the A ++1 ,the A ++ ∗ and the E ++ in Sp (2 N ) and SU ( N ) are compatible with each other, as predictedby general large- N arguments.Armed with the results of the mass calculation of the A ++1 , we can provide furthersupport to the conjecture put forward in Ref. [85] that the quantity η in the relationship m ++ σ = η C ( A ) C ( F ) (4.7)is a universal constant depending only on the dimension of space-time. In this equation, C ( F ) and C ( A ) are the quadratic Casimir operators in the fundamental and adjointrepresentations respectively, whose ratio in Sp (2 N ) is given by C ( A ) C ( F ) = 4( N + 1)2 N + 1 . (4.8)After performing the standard identification of the A ++1 with the lowest-lying scalarglueball, we tested this conjecture by performing a fit of Eq. (4.7) to the data, using η asa fitting parameter. The result can be found in Tab. 6 and is represented in the top panelof Fig. 5. The behaviour of η as a function of N is compatible with a constant for both the Sp (2 N ) and the SU ( N ) sequence. Moreover, the values of η extracted in each sequence– 24 –re compatible with each other within one standard deviation, as reported in Tab. 6. Asan additional test of Eq. (4.7), the behaviour of m ++ / ( √ ση ) is represented in Fig. 5 forboth Sp (2 N ) and SU ( N ) groups, along with the ratio of the quadratic Casimir operators.The weighted mean of the values of m ++ / ( √ ση ) obtained in each series is also reported inTab. 6 and represented in Fig. 5. This analysis provides further indications of the validityof the conjectured Casimir scaling, at least within current accuracy and precision.Another remarkable universal property is the independence on the gauge group ofthe ratio between the mass of the tensor glueball and the mass of the scalar glueball.This has been the subject of the investigation reported in Ref. [82], that makes use of themeasurements reported here. We do not repeat the details of that analysis, but refer theinterested Reader to Ref. [82]. – 25 – m R P / √ σ A + ∗ , χ /N d . o . f = 2 . A +1 , χ /N d . o . f = 2 . m R P / √ σ A −∗ , χ /N d . o . f = 3 . A − , χ /N d . o . f = 0 . m R P / √ σ A +2 , χ /N d . o . f = 3 . A − , χ /N d . o . f = 0 . . . . m R P / √ σ T +2 , χ /N d . o . f = 0 . E + , χ /N d . o . f = 0 . . . . m R P / √ σ T − , χ /N d . o . f = 1 . E − , χ /N d . o . f = 2 . . . . . / N m R P / √ σ T +1 , χ /N d . o . f = 1 . T − , χ /N d . o . f = 0 . Figure 4 . Glueball mass in each symmetry channel R P in units of √ σ , as a function of / N .The point corresponding to / N = 0 is the value of m R P / √ σ ( ∞ ) obtained from the best fit ofEq. (4.6) to the numerical measurements reported in this publication. See main text for details. – 26 – η η ( Sp ) = 5 . η ( SU ) = 5 . η ( Sp + SU ) = 5 . . . . . . . /N c . . . . . m ++ σ η Sp ( N c ) SU ( N c ) Figure 5 . Top panel: ratios defining the conjectured universal constant η for both SU ( N c = N ) and Sp ( N c = 2 N ) . Note that the naming convention for the symplectic group has been altered,using the variable N c = 2 N , to better accomodate the data into the plots; fits are also shown for the Sp ( N c ) family, the SU ( N c ) family and the combination of Sp ( N c ) and SU ( N c ) results. Bottom:measured ratios m ++ /σ further divided by the fitted universality constant η plotted as a functionof /N c ; lines are the ratios of the quadratic Casimir operators of the adjoint representation overthe corresponding ones of the fundamental representation as N c varies. (we note that, for the sakeof the visualisation, in this figure we have represented N c as a real number). – 27 – Conclusions
We have performed a numerical study of the low-lying spectrum in Sp (2 N ) Yang-Millsgauge theories. We have considered the lattice theory formulated with N = 1 , , , andwe have measured numerically its torelon and low-lying glueball spectrum as a functionof the lattice coupling β . After estimating finite size effects on the target observables byusing effective-string-theory motivated predictions applied to torelon masses at N = 3 , ,we have extracted the string tension as a function of β and N from the latter quantities.As a byproduct, through this calculation we have explicitly verified the realisation of theconfinement scenario in Sp (6) and Sp (8) Yang-Mills theories, by exposing one of its mosttypical signatures: the presence of stringy states wrapping compact directions. Whilethis is hardly surprising, direct validation of the expected behaviour in these two gaugetheories had never been done before in the literature. We have then extrapolated to thecontinuum limit the measurements of the adimensional ratios between the glueball massesand the square root of the string tension. Finally, we have obtained the large- N limit ofthe glueball spectrum in the Sp (2 N ) sequence of groups through an extrapolation in apower series in /N . For the lowest-lying masses, the leading corrections O (1 /N ) to thelarge- N limit appear to be sufficient to describe the N -dependence down to the smallestvalue N = 1 . We have assessed the size of systematic errors connected with the continuumand the large- N extrapolations and showed that they are negligible at the level of precisionof our data.We have found that, for the states for which the large- N extrapolation in the SU ( N ) sequence has been measured, their masses in the large- N limit agree with the ones wehave extracted taking the same limit in the Sp (2 N ) sequence, as expected. The otherstates we have determined in this calculation extend our knowledge of the continuum large- N spectrum, therefore providing a more complete set of masses to compare to analyticmethods that naturally work at N = ∞ , such as gauge-gravity duality techniques. Throughan analysis of the ratio of the lowest-lying glueball mass squared and the string tension as afunction of N , we have provided further support to the conjecture put forward in Ref. [85],that this ratio is proportional to the ratio of the quadratic Casimir of the adjoint overthat of the fundamental representation of the gauge group. Indeed, we have verified thatthe m ++ /σ ratio normalised with the appropriate ratio of quadratic Casimir operators isconstant within the Sp (2 N ) and the SU ( N ) family, and takes compatible values in the two.Our calculation bounds possible N -dependent corrections to this constant to be less than , the latter being the minimum precision with which we have measured the ratio as afunction of N .In addition to the glueball spectrum at finite N ≤ , our study has also provided apreliminary investigation of the topological charge in Sp (2 N ) gauge theories, in relationto systematics effects in the generation of configurations and in the extraction of spectralmasses. An extended analysis of topological observables and a more thorough analysis oftopological freezing effects at large N is currently in progress, and will be reported elsewhere.We envision a number of possible future avenues for exploration, in order to improveand extend this study. Beside the obvious increase in precision that can be obtained by– 28 –imulating at larger N and smaller lattice spacing (both of which, however, are affectedby increased autocorrelation times near the continuum limit and as N grows), one couldinvestigate the effect of including double-torelon and scattering states in the operator basis,in order to have a better resolution of genuine glueball masses. A study of glueball scatteringwould also provide an extension to the physical reach of our current investigation. Indeed, ascenario in which Sp (2 N ) glueballs may play a central role is gluonic dark matter [129, 130].In order to assess the viability of a dark matter scenario based on Sp (2 N ) glueballs, onewould have to compute the cross section for the decay of the higher-spin glueballs into twoscalar glueballs. This (very challenging) calculation would require a dedicated study ofmulti-point glueball functions. A study of correlators describing glueball scattering wouldbe a natural starting point for such an investigation.Finally, it is worth reminding the reader that the main motivation for our work hasbeen provided by our ongoing investigation of the pseudo-Nambu-Goldstone-Boson mech-anism of electroweak symmetry breaking based on the SU (4) (cid:55)→ Sp (4) global symmetrybreaking pattern in Sp (2 N ) gauge theories with two fundamental Dirac flavours, follow-ing the programme outlined in Ref. [7]. In this context, the obtained glueball masses inYang-Mills provide a reference energy scale to compare with the fermionic matter spectrum,both directly, in the quenched approximation [81], or indirectly, as a proxy for masses ofequivalent states in the full theory, for which calculations performed in QCD suggest thatthe presence of dynamical fermions does not shift significantly the mass of gluonic boundstates (see, e.g., Ref. [131]). The mild N dependence in the gluonic observables provides afirst indication that no large variations will emerge across corresponding relevant physicalobservables evaluated in different Sp (2 N ) gauge theories, as long as the theory is dominatedby gluon dynamics, with small numbers of matter fields. Acknowledgments
A Cabibbo-Marinari updating for Sp (2 N ) The Sp (2 N ) group is the subgroup of SU (2 N ) with elements U satisfying the relationship U Ω U T = Ω , (A.1)where the superscript T indicates the transposition operation and Ω is the symplecticmatrix. The latter can be cast in the form Ω = (cid:32) − (cid:33) , (A.2)with the N × N identity matrix. Eq. (A.1) implies that U has the form U = (cid:32) A B − B ∗ A ∗ (cid:33) , (A.3)with the N × N matrices A and B satisfying the conditions A † A + B † B = and A T B = B T A .As briefly mentioned in Sect. 2, ensembles of Sp (2 N ) configurations distributed ac-cording to Eq. (2.6) are obtained from lattice sweeps of single link heat bath (HB) andoverrelaxation (OR) updates. In our implementation of these algorithms, we have used anadaptation of the Cabibbo-Marinari method [102] to the case of Sp (2 N ) . The Cabibbo-Marinari algorithm updates a group matrix via subsequent updates of SU (2) subgroupscovering the whole target gauge group. The choice of the set of SU (2) subgroups to update– 30 –s crucial. For SU ( N ) , an efficient implementation can be obtained starting from all theCartan generators ( i, j ) having on the i -th diagonal element, − on the j -th diagonalelement (with ≤ i < j ≤ N ) and 0 everywhere else, along with their eigenvectors un-der conjugate action. Each generates an SU (2) subgroup of SU ( N ) . Since Sp (2 N ) is asubgroup of SU (2 N ) , the desired set of subgroups can be obtained from the set found for SU (2 N ) by excluding the SU (2) subgroups that alter the block structure in Eq. (A.3) ofthe Sp (2 N ) matrices. Chosing a larger set improves the decorrelation of the algorithm. Inthis work, we used N subgroups.To better understand how these subgroups are embedded in Sp (2 N ) matrices, we re-formulate the considerations above in the language of group representations. Each choiceof Cartan generators, along with its eigenvectors, exponentiates to a SU (2) subgroup of SU ( N ) . The elements of the matrices of this subgroup are different from a unit matrixonly at the positions { ( i, j ) , ( j, i ) , ( i, i ) , ( j, j ) } . A SU (2) matrix is thus embedded into a SU ( N ) matrix. We denote this embedding with ( i, j ) . The different embeddings ( i, j ) canbe seen as completely reducible representations of SU (2) that are unitarily equivalent to R ⊕ N − ,N − , i.e. to the (1 , embedding, where R is the × irreducible representationof SU (2) . The unitary transformation that maps one embedding into another is the ex-change of rows and columns i and j with and respectively. If [ N ] SU is the fundamentalrepresentation of SU ( N ) , { } the fundamental representation of SU (2) , all the embeddingsabove can be decomposed as [ N ] SU = { } ⊕ ( N − . (A.4)For the Sp (2 N ) case, the allowed SU (2) embeddings must respect the block structureEq. (A.3). These embeddings can be split in three classes that are not unitarily equivalent.The embedding (1 , is unitarily equivalent to the embeddings ( i < N, j < N ) .Embeddings in this class can be denoted by [2 N ] Sp = { } ⊕ { } ⊕ (2 N − . (A.5)Examples are, for N = 3 a b − b ∗ a ∗ a ∗ b ∗ − b a , a b − b ∗ a ∗ a ∗ b ∗ − b a , . . . (A.6)with a, b ∈ C such that | a | + | b | = 1 and a ∗ b − b ∗ a = 0 . There are N ( N − / of thoseembeddings.The embedding (1 , (cid:48) is unitarily equivalent to the embeddings ( i < N, j < N ) (cid:48) .Embeddings in this class can be denoted by [2 N ] (cid:48) Sp = { } (cid:48) ⊕ { } (cid:48) ⊕ (2 N − . (A.7)– 31 –xamples are, for N = 3 a b − a ∗ − b ∗
00 0 0 1 0 00 0 − b ∗ a ∗ b − a , a b − a ∗ − b ∗ − b ∗ a ∗ b − a , . . . (A.8)with a, b ∈ C such that | a | + | b | = 1 and a ∗ b − b ∗ a = 0 . There are N ( N − / of thoseembeddings.The embedding (1 , N ) is unitarily equivalent to the embeddings ( i, i + N ) . Thesecan be denoted by [2 N ] Sp = { } ⊕ (2 N − . (A.9)Examples are, for N = 3 , a b
00 0 1 0 0 00 0 0 1 0 00 − b ∗ a ∗
00 0 0 0 0 1 , a b − b ∗ a ∗ , . . . (A.10)with a, b ∈ C such that | a | + | b | = 1 and a ∗ b − b ∗ a = 0 . There are N of those embeddings.With our construction, we have identified N embeddings that cover the whole of Sp (2 N ) . A HB iteration on one link consists in updating consecutively each of the em-beddings belonging to classes Eqs. (A.5), (A.7) and (A.9) with the Creutz or Kennedy-Pendleton implementation of the SU (2) HB algorithm. An OR iteration is built in asimilar way.For this work, we performed one HB iteration followed by OR iterations for eachlink variable. Repeating these iterations for all the links of the lattice is a lattice sweep .To prevent the desymplectisation and deunitarisation of the configuration caused by theaccumulation of numerical error, we reprojected each link of the configuration on the groupafter each lattice sweeps with a modified Gram-Schmidt algorithm that preserves the Sp (2 N ) structure [7]. B Searching for the bulk phase transition
A phase transition taking place in the ( d +1) -dimensional classical canonical system definedby Eq. (2.9) is called a bulk phase transition. This transition separates the strong and weakcoupling regimes of the theory, limiting the range of β that is analytically connected to thecontinuum limit. The identification of bulk phase transition points is hence a crucial stepfor extrapolating numerical data to the a → limit in a controlled way.In general terms, a bulk phase transition takes place at values of β for which one ofthe derivatives of Z ( β ) with respect to β is singular (in the L → ∞ limit). For a system– 32 –efined by Eq. (2.9), with the action in Eq. (2.6), the first derivative of ln Z ( β ) correspondsto the expectation value of the average plaquette, (cid:104) P ( β ) (cid:105) ≡ L ∂ ln Z ( β ) ∂β , (B.1)and its second derivative to the plaquette susceptibility, χ p ( β ) ≡ ∂ ln Z ( β ) ∂β = L (cid:2) (cid:104) P ( β ) (cid:105) − (cid:104) P ( β ) (cid:105) (cid:3) . (B.2)As we observed in Section 4.1, (cid:104) P (cid:105) ( β ) shows a pseudo-inflection point at some value β c of the lattice coupling. This pseudo-inflection corresponds to a maximum χ c ( L ) of theplaquette susceptibility. If the latter is associated to the smoothing of a proper phasetransition, we expect χ c ( L ) → ∞ , and β c ( L ) → β c , as L → ∞ . Conversely, if χ c ( L ) staysfinite when L → ∞ , the change from the strong coupling regime to the weak coupling onehappens not through a phase transition, but via a crossover. To study the scaling of the maximum of the plaquette susceptibility with the size ofthe system, we focused our attention on the neighbourhood of the identified pseudocriticalcoupling β c , computing at various values of β near this coupling for L/a = 8 , , , andcollecting measurements of (cid:104) P (cid:105) ( β ) . A total of data points at each investigated valueof β and L were collected, one every five sweeps. The corresponding results for (cid:104) P (cid:105) ( β ) are reported in Tab 7. The Monte Carlo histories of our simulations were searched forany sign of metastability, which would have signalled a first order phase transition, withnegative results. This allows us to exclude the presence of a discontinuous phase transitionfor both N = 3 and N = 4 . The plaquette susceptibility χ p ( β ) was computed at each β .The results are presented in Fig. 6. At each volume, β c and χ c were estimated using themultihistogram method. The obtained values are reported in Tab. 8. No appreciable scalingof the peak values can be detected as L is increased. Thus, from our data we can concludethat no phase transition is present for N = 3 , , with the change of behaviour from strongto weak coupling being described by a crossover. These conclusions are in agreement withthe findings in Ref. [6] for the case N = 3 .Even if a phase transition is excluded, the presence of a crossover can still affect physicalobservables near the change of regime. An example of a similar effect in SU (4) Yang-Mills with a fundamental Wilson action is described in Ref. [118], where the effect of thepresence of a crossover reflects in a dip of the measured scalar glueball masses at thecorresponding values of β . Similar results emerge also in SU (2) with a mixed fundamental-adjoint action [134, 135]. Therefore, to extrapolate lattice observables to the continuumlimit with a simple and controlled dependence in a √ σ , it is still necessary to be in theweak coupling regime. We achieved this by ensuring that our data points were far enoughfrom the inflection points and then verifying that there was no visible signal of bulk phasecontamination in our observables. In principle, higher-order (e.g., third order) phase transitions are also possible. However, the onlyexamples known to us arise strictly in the N → ∞ limit (e.g., Ref. [132, 133]). If a higher-order phasetransition were present, it would be extremely difficult to detect it in our data. At the same time, weexpect its influence on the numerical measurements to mimic a crossover. For this reason, we use here theexpression crossover in a rather loose sense. – 33 – / N (cid:104) P ( β ) (cid:105) , N = 3 (cid:104) P ( β ) (cid:105) , N = 41 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 7 . Expectation values of the plaquette (cid:104) P ( β ) (cid:105) in Sp (2 N ) lattice theories with N = 3 (left)and N = 4 (right) at L = 16 a . N = 3 N = 4 β c χ P max β c χ P max L/a = 8 14 . . . . L/a = 12 14 . . . . L/a = 16 14 . . . . Table 8 . Location of the maximum value of the susceptibility for N = 3 and N = 4 at L/a =8 , , . – 34 – β/ N . . . . . . . χ P N = 3 L/a = 8
L/a = 16 15 20 25 30 β/ N . . . . . . . . . χ P N = 4 L/a = 8
L/a = 16
Figure 6 . The plaquette susceptibility χ p , as defined in Eq. (B.2), as a function of β = 2 N/g , atvolumes L/a = 8 (orange squares),
L/a = 16 (blue triangles) for the Sp (2 N ) lattice gauge theorywith N = 3 (left panel) and N = 4 (right panel). – 35 – Continuum and infinite volume extrapolations
As mentioned in Sect. 3.6, estimates of glueball and torelon masses obtained from the latticeare affected by systematic errors. We focus on the systematic error caused by working on alattice of finite size in Sect. C.1 and on the error caused by the discretisation in Sect. C.2.
C.1 Finite-size effects (FSEs)
The spectrum of a theory defined in a finite box of linear size L with periodic boundaryconditions depends on L . The problem was studied for instance in Ref. [136], and thisdependence was found to be described by Eq. (3.31). The magnitude of the leading finite-size effects decays exponentially as a function of mL , where m is the lightest excitation inthe spectrum.If ma is estimated to a given finite precision, a value L min /a exists such that for L > L min the FSEs on the spectrum are negligible—by which we mean that their size ismuch smaller than the statistical error. For
L > L min , the measured spectral masses canthus be considered as an estimate of the infinite size spectrum at fixed lattice spacing.In the scaling regime, mL is also a constant as a → , and once L min /a is found for avalue of a , the FSEs will remain negligible as a is taken to , provided the physical volumeis kept approximately constant in the process. The precise value of L min /a depends onthe precision of our estimates and on the theory under scrutiny, and must be determinedempirically.To determine L min and obtain an estimate for the spectrum at infinite size for N = 3 , ,we used the ensembles described in Sect. 4.2. For each ensemble, we determined the glueballspectrum and the string tension. The results are reported in Tab. 9 for N = 3 and Tab. 10for N = 4 . In this Appendix we focus on the lightest channel, which is consistently foundto be A +1 and suffers from the largest FSEs. (Exceptions to this rule can be found, butthey can only occur in the small L/a regime, in which we are not interested.)The estimates of am A +1 are presented in Fig. 7, for N = 3 in the top panel, and N = 4 in the bottom panel. From these figures we see that am A +1 rapidly settles on a plateau as L/a is increased. This means that, as expected, FSEs become negligible as L is increased.A rough estimate yields L min /a = 20 for Sp (6) at β = 16 . and for Sp (8) at β = 26 . . Asan a posteriori check, note that m A +1 L min ∼ . for Sp (6) and m A +1 L min ∼ . for Sp (8) .The infinite size spectrum can then be estimated by any one of the results at L > L min .We fitted Eq. (3.31) to the data using b and m ( ∞ ) as fitting parameters. The fitted curvesand the related χ /N d.o.f. are displayed in Fig. 7.From the analysis above, we conclude that FSEs are negligible when L > a , for N =3 , and L > a , for N = 4 . On these lattices, the condition L √ σ ≥ , which identifies thelarge volume regime of torelons, is also fulfilled. Hence, we choose this condition throughoutas an indicator that finite volume effects can be neglected.– 36 – P L/a = 14
L/a = 16
L/a = 18
L/a = 20
L/a = 22
L/a = 24 A +1 . . . . . . A − . . . . . . A +2 . . . . . . A − . . . . . . E + . . . . . . E − . . . . . . T +2 . . . . . . T − . . . . . . T +1 . . . . . . T − . . . . . . Table 9 . Estimates of the masses of glueballs ma in all symmetry channels R P in units of thelattice spacing at β = 16 . , for various L/a and for N = 3 . R P L/a = 8
L/a = 10
L/a = 12
L/a = 14
L/a = 16
L/a = 18
L/a = 20 A +1 . . . . . . . A − . . . . . . . A − . . . . . . . A +2 . . . . . . . E − . . . . . . . E + . . . . . . . T − . . . . . . . T +2 . . . . . . . T − . . . . . . . T +1 . . . . . . . Table 10 . Estimates of the masses of glueballs ma in all symmetry channels R P in units of thelattice spacing at β = 26 . for various L/a and for N = 4 . C.2 Continuum limit extrapolations
The behaviour of the discretisation error was studied in Sect. 3.6. The dimensionless ratio m R P / √ σ behaves, at leading order in a , as m R P √ σ ( a ) = m R P √ σ (cid:0) c R P σa (cid:1) , (C.1)where c R P is a constant that depends on the symmetry channel and on the excitationnumber. The multiplicative term on the right hand side is the continuum limit of the ratio,while the term in σa describes the deviation with respect to this limit for sufficiently small a . The continuum limit of the spectrum of glueballs can be obtained from sets of estimatesthereof obtained at finite lattice spacing with a fit of Eq. (C.1) to the data. This is the– 37 –
12 15 18 21 24 27 30
L/a . . . m A + a ma = 0 . χ /N d . o . f . = 3 . N = 34 6 8 10 12 14 16 18 20 22 L/a . . . . m A + a ma = 0 . χ /N d . o . f . = 1 . N = 4 Figure 7 . Mass of the lightest glueball m A +1 a in units of a , at fixed coupling, as a function of L/a .This corresponds to the A +1 channel for both N = 3 , evaluated at β = 16 . (top panel), and N = 4 ,evaluated at β = 26 . (bottom panel). The solid line is the best fit of Eq. (3.31) to the data. general strategy to extract results of the continuum spectrum from estimates at finite a , ifthe latter are obtained in a regime where Eq. (C.1) is valid.For each N = 1 , , , , ensembles of , thermalised configurations were obtainedat several values of a and L/a and stored for later analysis. The values of
L/a were alwayschosen so that FSEs could be safely neglected. This has been verified a posteriori from themeasured values of m R P L . The glueball and torelon masses were estimated in units of thelattice spacing, as explained in Sect. 3, for each ensemble.While not strictly related to the continuum extrapolation, a comment is in order re-garding the estimation of the uncertainty on m R P a and to guide the Reader in navigatingthe tables of results. To determine t min , defined in Sec. 3.1, the quantity m eff ( t ) is com-puted on all the available range of t/a . If a plateau can be found, the fits of Eq. (3.24)over the range t > t min provide an estimate of m R P a together with the statistical error ofthe fit and the corresponding χ /N d.o.f. . It is often the case, however, that the plateau is– 38 –nly a − a long, that an accurate determination of its preimage in t/a is hindered by thecontamination from larger mass states, or that the mass itself is large. These difficulties indetermining t min lead to a systematic error on m R P a that can be larger than the statisticalerror of the fitting procedure. In such cases, the statistical error of the fit cannot be trustedin describing the fluctuations of m R P a . Hence, we use a safe estimate of the mass and itserror from the envelope of the points at plateau. A value for χ /N d.o.f. cannot be definedand the corresponding entry in the table is left empty. Finally, there are extreme cases forwhich an estimate for the mass simply cannot be found, i.e. a plateau is absent. In thatcase, the corresponding entry in the table is left empty.All our estimates are reported in Tabs. 11-22 for the ensembles with N = 1 , , , .The values of β and L/a are found in the first row of each table; the subsequent rowscorrespond to the symmetry channels, until the last row, that corresponds to the stringtension. For each value of N , these estimates are plotted as a function of σa in Figs. 8-11.In general, we found that the statistical errors and the confidence intervals are of thesame order of magnitude, the latter being slightly larger in the majority of cases. This canbe taken as an indication of the correctness of the method detailed above. In the following,we refer to the uncertainty in the determination of m R P a generically as its “error”.Let us now comment on the features of these finite- a estimates that are common acrossall the values of N . The fact that in every case m R P L > allows us to safely neglect FSEsfor all the ensembles, as anticipated. Moreover, all the estimates satisfy m R P a ≤ , exceptfor the two roughest lattices when N = 1 , corresponding to β = 2 . and β = 2 . .Therefore, we can assert that our choices of β are well calibrated to study the flow tothe continuum limit of the spectrum of these systems. In the glueball sector, the A +1 isconsistently the lightest channel, followed by the ( E ± , T ± ) degenerate pairs. The error ofthe estimates is larger for larger m R P a , as is to be expected on the basis of the discussion inSect. 3.6. The E ± and T ± are degenerate over the whole range of a probed at least at the σ level, with the mass difference being below one standard deviation in most of the cases.This is a non-trivial a posteriori test of the restoration of continuum rotational invarianceand can be taken as a signal that we are in the regime for which Eq. (3.32) is valid.An additional source of systematic error, the effects of which are difficult to account for,is the autocorrelation time of the system, which grows as the continuum limit is approached.Since the topological charge is one of the quantities with the longest autocorrelation time,studying the evolution of this observable yields a conservative estimator of these effects.Particular attention should be paid to cases in which the topology is (nearly-)frozen. Thesecan be detected by analysing the time series of the topological charge. To this end, asubset of configurations were obtained from the N = 3 and N = 4 ensembles, by pick-ing one configuration each . The Gradient flow was then used to smooth each of theconfigurations, and the regularised topological charge was computed using Eq. (2.11) ateach smoothing step. The continuum topological charge is obtained when the regularisedtopological charge reaches a plateau under further smoothing operations. The results ofthis analysis are visible in Fig. 12 and Fig. 13.For both N = 3 and N = 4 , there is a value β min above which the topological chargebarely changes with the Monte Carlo steps that we are able to perform. These ensembles– 39 – = 2 . β = 2 . β = 2 . N = 1 L = 10 a L = 12 a L = 16 am R P a χ /N d . o . f . m R P a χ /N d . o . f . m R P a χ /N d . o . f . A +1 . − . .
77 0 . . A + ∗ . .
41 1 . − . − A − . .
02 1 . − . − A −∗ . − . − . − A +2 . − . .
82 1 . − A − − − − − . − T +2 . − . − . . E + . − . − . . T − . − . .
28 1 . − E − . − . .
64 1 . − T +1 − − . − . − T − − − − − . − σ s a σ s a σ s a . − . − . − Table 11 . Estimates of the glueball and the torelon masses for N = 1 , in units of the latticespacing a , on lattices of linear size L and lattice spacing determined by the inverse coupling β . Theerror in brackets are discussed in the main text. are topologically frozen. From visual inspection of the figures we estimate that β min ( N = 3) (cid:39) . , β min ( N = 4) (cid:39) . . (C.2)Given that topological freezing affects only our two largest values of N , where the system-atic effects it induces on measurements of masses are expected to become less severe (asdiscussed in Sect. 3.6), we included the estimates obtained from these frozen ensembles inthe extrapolation to the continuum limit.A related potential source of systematic error lies in the length of the initial thermali-sation. Our earlier estimates of the continuum spectrum, especially for N = 3 and N = 4 ,presented a visible dip in the calculated masses for the smallest values of σa . This urgedus to perform an overall check of the invariance of the final result under the prolongation ofthe simulation trajectory. In Fig. 14, we show results of m R P / √ σ at finite lattice spacing asa function of the initial thermalisation time. The fact that these estimates are largely inde-pendent of this initial thermalisation time suggests that the Markov chains from which thefinal averages are obtained are long enough for the system to be at statistical equilibrium.Let us now discuss the continuum extrapolations of the ratios m R P / √ σ for given valuesof a . These ratios can be easily formed for each ensemble from the estimates in Tabs. 11-22.At each value of N , fits of Eq. (C.1) using m R P / √ σ in the continuum and c R P as fittingparameters were performed for each symmetry channel. These linear fits are plotted as solidlines in Figs. 8-11, where the corresponding values of the χ /N d.o.f. are also reported. Notethat because of the way that the error on these measurements was evaluated, the χ /N d.o.f. are slightly underestimated. These extrapolations are discussed in Sect. 4.3.– 40 – = 2 . β = 2 . β = 2 . N = 1 L = 20 a L = 24 a L = 26 am R P a χ /N d . o . f . m R P a χ /N d . o . f . m R P a χ /N d . o . f . A +1 . .
36 0 . .
16 0 . . A + ∗ . − . − . − A − . − . .
74 0 . − A −∗ . − . − . − A +2 . − . − . . A − . − . − . − T +2 . .
78 0 . − . − E + . − . .
58 0 . . T − . − . − . − E − . − . − . − T +1 . − . − . − T − . − . − . − σ s a σ s a σ s a . − . − . − Table 12 . Estimates of glueball masses and string tensions for N = 1 , in units of the lattice spacing a , on lattices of linear size L and lattice spacing determined by the inverse coupling β . The errorin brackets are discussed in the main text. β = 2 . N = 1 L = 32 am R P a χ /N d . o . f . A +1 . . A + ∗ . − A − . . A −∗ . − A +2 . − A − . − T +2 . − E + . . T − . − E − . − T +1 . − T − . − σ s a . − Table 13 . Estimates of glueball masses and string tension for N = 1 , in units of the lattice spacing a , on lattices of linear size L and lattice spacing determined by the inverse coupling β . The errorin brackets are discussed in the main text. – 41 – = 7 . β = 7 . β = 7 . N = 2 L = 16 a L = 16 a L = 18 am R P a χ /N d . o . f . m R P a χ /N d . o . f . m R P a χ /N d . o . f . A +1 . − . .
62 0 . . A + ∗ − − . − . − A − . − . − . − A −∗ . − . − . − A +2 . − . − . − A − − − . − . − T +2 − − . − . . E + . − . .
86 0 . − T − − − . − . − E − . − . .
58 1 . − T +1 − − . − . − T − . − . .
56 1 . − σ s a σ s a σ s a . − . − . − Table 14 . Estimates of glueball masses and string tensions for N = 2 , in units of the lattice spacing a , on lattices of linear size L and lattice spacing determined by the inverse coupling β . The errorin brackets are discussed in the main text. β = 8 . β = 8 . β = 8 . N = 2 L = 20 a L = 26 a L = 32 am R P a χ /N d . o . f . m R P a χ /N d . o . f . m R P a χ /N d . o . f . A +1 . .
42 0 . .
31 0 . . A + ∗ . − . − . − A − . − . − . − A −∗ . − . − . − A +2 . − . − . − A − . − . − . − T +2 . − . − . − E + . − . .
31 0 . − T − . − . − . − E − . − . − . − T +1 . − . − . − T − . − . − . − σ s a σ s a σ s a . − . − . − Table 15 . Estimates of glueball masses and string tensions for N = 2 , in units of the lattice spacing a , on lattices of linear size L and lattice spacing determined by the inverse coupling β . The errorin brackets are discussed in the main text. D A closer look at the Sp (4) data In Sect. 3 we presented the spectrum of Sp (2 N ) theories in the continuum and large- N limits. Two sets of ensembles are available for N = 2 . The one obtained for Ref. [7] (oldensembles) and another, independent one, obtained for the present work (new ensembles).– 42 – = 15 . β = 15 . β = 15 . N = 3 L = 12 a L = 12 a L = 12 am R P a χ /N d . o . f . m R P a χ /N d . o . f . m R P a χ /N d . o . f . A +1 . .
18 0 . .
07 0 . . A + ∗ . .
99 1 . − . . A − . .
79 1 . .
92 1 . − A −∗ . − . − . . A +2 . − . .
51 1 . . A − . − . − . − T +2 . − . .
05 1 . . E + . .
37 1 . .
65 1 . . T − . − . − . − E − . − . − . − T +1 . − . − . − T − . − . − . − σ s a σ s a σ s a . − . − . − Table 16 . Estimates of glueball masses and string tensions for N = 3 , in units of the lattice spacing a , on lattices of linear size L and lattice spacing determined by the inverse coupling β . The errorin brackets are discussed in the main text. β = 15 . β = 16 . β = 16 . N = 3 L = 14 a L = 16 a L = 20 am R P a χ /N d . o . f . m R P a χ /N d . o . f . m R P a χ /N d . o . f . A +1 . − . .
25 0 . . A + ∗ . − . − . − A − . − . .
49 0 . . A −∗ . − . − . . A +2 . − . .
97 1 . − A − . − . − . − T +2 . .
49 0 . .
14 0 . − E + . .
88 0 . .
98 0 . − T − . − . − . . E − . − . .
94 1 . − T +1 . − . − . − T − . − . − . − σ s a σ s a σ s a . − . − . − Table 17 . Estimates of glueball masses and string tensions for N = 3 , in units of the lattice spacing a , on lattices of linear size L and lattice spacing determined by the inverse coupling β . The errorin brackets are discussed in the main text. The estimates shown in Tab. 4, in the column N = 2 , are the weighted averages of thecontinuum limits obtained from the new and old ensembles. In this Appendix, we presentseparately the two analysis for the new and old ensembles for N = 2 .For the new ensembles, the continuum and large- N extrapolated estimates can be found– 43 – = 16 . β = 16 . β = 16 . N = 3 L = 20 a L = 28 a L = 24 am R P a χ /N d . o . f . m R P a χ /N d . o . f . m R P a χ /N d . o . f . A +1 . .
11 0 . − . − A + ∗ . − . − . − A − . .
62 0 . − . − A −∗ . − . − . − A +2 . − . − . − A − . − . − . − T +2 . − . .
92 0 . − E + . − . − . . T − . − . − . − E − . .
33 0 . − . . T +1 . − . − . − T − . − . − . − σ s a σ s a σ s a . − . − . − Table 18 . Estimates of glueball masses and string tensions for N = 3 , in units of the lattice spacing a , on lattices of linear size L and lattice spacing determined by the inverse coupling β . The errorin brackets are discussed in the main text. β = 17 . N = 3 L = 28 am R P a χ /N d . o . f . A +1 . . A + ∗ . − A − . − A −∗ . − A +2 . . A − . − T +2 . . E + . . T − . − E − . − T +1 . − T − . − σ s a . − Table 19 . Estimates of glueball masses and string tension for N = 3 , in units of the lattice spacing a , on lattices of linear size L and lattice spacing determined by the inverse coupling β . The errorin brackets are discussed in the main text. in Tab. 23, both in units of √ σ (top) and m E + (bottom). The former extrapolated values,together with the corresponding large- N extrapolated results, are displayed, in units of √ σ ,in Fig. 9, and the large- N extrapolation is shown in Fig. 16.The old ensembles have been reanalysed following the approach used in this work—see– 44 – = 26 . β = 26 . β = 26 . N = 4 L = 14 a L = 14 a L = 14 am R P a χ /N d . o . f . m R P a χ /N d . o . f . m R P a χ /N d . o . f . A +1 . .
88 0 . . . . A + ∗ . − . .
08 1 . . A − . − . .
06 1 . − A −∗ . − . − . . A +2 . − . .
97 1 . − A − . .
56 1 . − . . T +2 . − . .
23 1 . . E + . − . . . . T − . − . − . − E − . − . − . . T +1 . − . − . − T − . − . − . − σ s a σ s a σ s a . − . − . − Table 20 . Estimates of glueball masses and string tensions for N = 4 , in units of the lattice spacing a , on lattices of linear size L and lattice spacing determined by the inverse coupling β . The errorin brackets are discussed in the main text. β = 27 . β = 27 . β = 27 . N = 4 L = 16 a L = 16 a L = 16 am R P a χ /N d . o . f . m R P a χ /N d . o . f . m R P a χ /N d . o . f . A +1 . .
08 0 . − . . A + ∗ . − . − . − A − . .
53 1 . .
16 0 . − A −∗ . − . − . − A +2 . − . − . − A − . − . − . − T +2 . .
08 0 . .
98 0 . . E + . − . .
94 0 . − T − . − . − . − E − . − . − . . T +1 . − . .
36 1 . − T − . − . − . − σ s a σ s a σ s a . − . − . − Table 21 . Estimates of glueball masses and string tensions for N = 4 , in units of the lattice spacing a , on lattices of linear size L and lattice spacing determined by the inverse coupling β . The errorin brackets are discussed in the main text. Sect. C.2. The results are displayed in Fig. 17. The continuum and large- N extrapolatedvalues can be found in Tab. 24, both in units of √ σ (top) and m E + (bottom), and aredisplayed, in units of √ σ , in Figs. 18 and 19, respectively.As expected, the estimates of the spectrum obtained from the two ensembles are statis-– 45 – = 27 . β = 27 . β = 28 . N = 4 L = 18 a L = 20 a L = 22 am R P a χ /N d . o . f . m R P a χ /N d . o . f . m R P a χ /N d . o . f . A +1 . − . − . − A + ∗ . − . − . − A − . − . .
79 0 . − A −∗ . − . − . − A +2 . − . . . − A − . − . − . − T +2 . .
84 0 . − . . E + . .
41 0 . − . . T − . − . .
97 0 . . E − . − . .
94 0 . − T +1 . − . − . − T − . − . − . − σ s a σ s a σ s a . − . − . − Table 22 . Estimates of glueball masses and string tensions for N = 4 , in units of the lattice spacing a , on lattices of linear size L and lattice spacing determined by the inverse coupling β . The errorin brackets are discussed in the main text. tically compatible. This justifies taking weighted averages as our best values for N = 2 . Wenote that, while the new ensemble provides extrapolations with good values of χ /N d.o.f. ,in the old ensemble higher values of the reduced χ are present. This hints toward slightlydifferent systematics between the old and new simulations. This could explain the higher χ /N d.o.f. for some extrapolations presented in the analysis in Sect. 3. The broad compati-bility of the data, nevertheless, suggests that the effect is not dominant. E On the inclusion of N = 1 in the large- N extrapolation In Sect. 3, the large- N extrapolation of the spectrum has been provided including the value N = 1 for all channels. For a handful of channels, this gives a value of χ /N d.o.f. above 2,indicative of a lower statistical significance of the extrapolation. This may suggest that, forthese specific channels, the N = 1 value is not captured by the large- N expansion. Indeed,in previous studies of SU ( N ) gauge theories (e.g., Ref. [2]), the value of the χ /N d.o.f. hasbeen used as an indication of the reliability of the truncation of the large- N series at agiven order for capturing results at some finite value of N . In our work, excluding thedata for N = 1 generally improves the value of the χ /N d.o.f. . However, this leaves onlythree points for the extrapolation, and hence creates a larger systematic bias on the latter.Likewise, adding a higher order correction will decrease the number of degrees of freedomand hence introduce more noise. Being faced with the necessity to make a choice, we haveopted to systematically include N = 1 in all large- N extrapolations. This means thatwe interpret larger value of the χ /N d.o.f. as results of fluctuations in the data or of some– 46 – = 2 ∞ R P m R P / √ σ χ /N d . o . f . m R P / √ σ χ /N d . o . f . A + ∗ . .
04 5 . . A +1 . .
69 3 . . A −∗ . .
09 6 . . A − . .
22 4 . . A +2 . .
89 7 . . A − . .
4) 0 .
18 8 . . E + . .
71 4 . . E − . .
56 6 . . T +2 . .
82 4 . . T − . .
37 6 . . T +1 . .
43 8 . . T − . . . . N = 2 ∞ R P m R P /m E + χ /N d . o . f . m R P /m E + χ /N d . o . f . A + ∗ . .
03 1 . . A +1 . .
44 0 . . A −∗ . .
34 1 . . A − . .
18 1 . . A +2 . .
17 1 . . A − . .
13 1 . . E − . .
45 1 . . T +2 . .
13 1 . . T − . .
43 1 . . T +1 . .
14 1 . . T − . .
01 1 . . Table 23 . In the left column, estimates of the spectrum at N = 2 , in units of √ σa and m E + .These are obtained from the data generated for this work (new ensembles). In the right column,the extrapolation to N = ∞ obtained from fits of Eq. (4.6) to the data using the data in the leftcolumn for N = 2 and the same data as before for N = 1 , , . unknown systematics, rather than as stemming from the fact that N = 1 is not describedby the expansion. The question is left open by this study. For completeness, we compare inTab. 25 our results for the extrapolations with N = 1 systematically included and excluded.Most of the results are compatible at the two sigma level.– 47 – = 2 ∞ m R P √ σ χ /N d . o . f . m R P √ σ χ /N d . o . f . A + ∗ . .
64 6 . . A +1 . .
27 3 . . A −∗ . .
63 7 . . A − . .
37 5 . . A +2 . .
35 8 . . A − . .
67 8 . . E + . .
87 4 . . E − . .
14 6 . . T +2 . .
63 4 . . T − . .
41 6 . . T +1 . .
13 8 . . T − . .
03 8 . . N = 2 ∞ m R P /m E + χ /N d . o . f . m R P /m E + χ /N d . o . f . A + ∗ . .
16 1 . . A +1 . .
62 0 . . A −∗ . .
89 1 . . A − . .
82 1 . . A +2 . .
72 1 . . A − . . . . E − . .
73 1 . . T +2 . .
57 1 . . T − . .
66 1 . . T +1 . .
71 1 . . T − . .
51 1 . . Table 24 . Calculations of the masses in the continuum limit for N = 2 and each channel, in unitsof √ σa and m E + , using only a reanalysis of the N = 2 data from Ref. [7] (old ensembles), asexplained in the text. – 48 – m R P / √ σ A + ∗ , χ /N d . o . f . = 0 . A +1 , χ /N d . o . f . = 1 . m R P / √ σ A −∗ , χ /N d . o . f . = 0 . A − , χ /N d . o . f . = 0 . m R P / √ σ A +2 , χ /N d . o . f . = 0 . A − , χ /N d . o . f . = 0 . m R P / √ σ T +2 , χ /N d . o . f . = 0 . E + , χ /N d . o . f . = 0 . m R P / √ σ T − , χ /N d . o . f . = 0 . E − , χ /N d . o . f . = 0 . .
00 0 .
05 0 .
10 0 . σa m R P / √ σ T +1 , χ /N d . o . f . = 0 . T − , χ /N d . o . f . = 8 × − Figure 8 . Glueball mass in each symmetry channel R P of the Sp (2 N ) theory with N = 1 , in unitsof √ σ , as a function of σa . For each symmetry channel R P , the value at σa = 0 is the continuumlimit, obtained from a best fit of Eq. (C.1) to the data. The best fits lines are represented as solidlines. – 49 – m R P / √ σ A + ∗ , χ /N d . o . f . = 0 . A +1 , χ /N d . o . f . = 0 . m R P / √ σ A −∗ , χ /N d . o . f . = 0 . A − , χ /N d . o . f . = 0 . m R P / √ σ A +2 , χ /N d . o . f . = 0 . A − , χ /N d . o . f . = 0 . . . . m R P / √ σ T +2 , χ /N d . o . f . = 0 . E + , χ /N d . o . f . = 0 . m R P / √ σ T − , χ /N d . o . f . = 1 . E − , χ /N d . o . f . = 0 . .
00 0 .
02 0 .
04 0 . σa . . . m R P / √ σ T +1 , χ /N d . o . f . = 0 . T − , χ /N d . o . f . = 1 . Figure 9 . Glueball mass in each symmetry channel R P of the Sp (2 N ) theory with N = 2 , in unitsof √ σ , as a function of σa . For each symmetry channel R P , the value at σa = 0 is the continuumlimit, obtained from a best fit of Eq. (C.1) to the data. The best fits lines are represented as solidlines. – 50 – m R P / √ σ A + ∗ , χ /N d . o . f . = 0 . A +1 , χ /N d . o . f . = 0 . m R P / √ σ A −∗ , χ /N d . o . f . = 0 . A − , χ /N d . o . f . = 0 . . . . m R P / √ σ A +2 , χ /N d . o . f . = 1 . A − , χ /N d . o . f . = 0 . m R P / √ σ T +2 , χ /N d . o . f . = 0 . E + , χ /N d . o . f . = 0 . m R P / √ σ T − , χ /N d . o . f . = 0 . E − , χ /N d . o . f . = 0 . .
00 0 .
02 0 .
04 0 . σa m R P / √ σ T +1 , χ /N d . o . f . = 1 . T − , χ /N d . o . f . = 1 . Figure 10 . Glueball mass in each symmetry channel R P of the Sp (2 N ) theory with N = 3 , in unitsof √ σ , as a function of σa . For each symmetry channel R P , the value at σa = 0 is the continuumlimit, obtained from a best fit of Eq. (C.1) to the data. The best fits lines are represented as solidlines. – 51 – m R P / √ σ A + ∗ , χ /N d . o . f . = 0 . A +1 , χ /N d . o . f . = 1 . m R P / √ σ A −∗ , χ /N d . o . f . = 0 . A − , χ /N d . o . f . = 0 . m R P / √ σ A +2 , χ /N d . o . f . = 0 . A − , χ /N d . o . f . = 0 . . . . m R P / √ σ T +2 , χ /N d . o . f . = 0 . E + , χ /N d . o . f . = 1 . m R P / √ σ T − , χ /N d . o . f . = 0 . E − , χ /N d . o . f . = 0 . .
00 0 .
02 0 .
04 0 . σa m R P / √ σ T +1 , χ /N d . o . f . = 0 . T − , χ /N d . o . f . = 0 . Figure 11 . Glueball mass in each symmetry channel R P of the Sp (2 N ) theory with N = 4 , in unitsof √ σ , as a function of σa . For each symmetry channel R P , the value at σa = 0 is the continuumlimit, obtained from a best fit of Eq. (C.1) to the data. The best fits lines are represented as solidlines. – 52 – Q β = 15 . L = 12 a -5.00.05.0 Q β = 15 . L = 14 a -5.00.05.0 Q β = 16 . L = 16 a -5.00.05.0 Q β = 16 . L = 20 a Q β = 17 . L = 28 a Count
Figure 12 . Histories and statistical distributions (histograms)of the topological charge defined inEq. (2.11) for the ensembles obtained at N = 3 . The configurations of each ensemble are smoothedwith the gradient flow defined in Eq. (2.16). The frequency of sampling in running time is describedin the text. – 53 – Q β = 26 . L = 14 a -5.00.05.0 Q β = 26 . L = 14 a -5.00.05.0 Q β = 27 . L = 16 a -10.00.010.0 Q β = 27 . L = 18 a -2.50.02.5 Q β = 27 . L = 20 a Q β = 28 . L = 22 a Count
Figure 13 . Histories and statistical distributions (histograms) of the topological charge as definedin Eq. (2.11) for the ensembles obtained at N = 4 . The configurations of each ensemble aresmoothed with the gradient flow defined in Eq. (2.16). The frequency of sampling in running timeis described in the text. – 54 – . . . m A + / √ σ Sp (2), β = 2 . . . . m A + / √ σ Sp (4), β = 8 . . . . m A + / √ σ Sp (6), β = 16 . Sp (6), β = 17 . . . . m A + / √ σ Sp (8), β = 27 . Sp (8), β = 28 . Figure 14 . Mass of the A +1 glueball, in units of √ σ , measured for different lengths of the initialthermalization. Each data point displays the running average over subsequent configurations,with the last one in the series having the sequential index corresponding to the abscissa. – 55 – + − + − + + − − + − A +1 A − A +2 A − T +2 E + T − E − T +1 T − m R P √ σ N = 1 N = 2 N = 3 N = 4 Sp ( ∞ ) SU ( ∞ ) Figure 15 . Spectrum of the Sp (2 N ) theory in the continuum limit for N = 1 , , , and N = ∞ ,in units of √ σ . For N = 2 , only the numerical measurements reported in this publication were used(new ensembles). For ease of comparison, we have reported also the masses of the A ++1 and E ++ channels for SU ( ∞ ) (borrowed from [2]). R P m R P / √ σ c R P χ /N d.o.f. m R P / √ σ c R P χ /N d.o.f. A +1 . . .
38 2 . . .
0) 0 . A + ∗ . − . .
2) 2 .
91 4 . . .
0) 0 . A − . . .
63 4 . . .
5) 0 . A −∗ . . .
4) 3 . . .
1) 12 . .
7) 0 . A +2 . − . .
3) 3 . . .
2) 10 . .
3) 0 . A − . . .
0) 0 . . .
7) 8 . .
5) 0 . T +2 . . .
65 4 . . .
2) 1 . E + . . .
72 4 . . .
1) 0 . T − . . .
2) 1 .
97 5 . . .
1) 0 . E − . . .
2) 2 .
03 5 . . .
7) 0 . T +1 . . .
6) 1 .
15 7 . .
0) 5 . .
0) 1 . T − . . .
6) 0 .
02 8 . .
3) 1 . .
3) 0 . Table 25 . Large- N extrapolated masses of the glueball spectrum obtained from a fit of Eq. (4.6),in the case in which the estimates at N = 1 are included (left) or excluded (right). Note that theleft hand part of this table is the same as the last column of Tab. 4 and the same as Tab. 5. – 56 – m R P / √ σ A + ∗ , χ /N d . o . f = 0 . A +1 , χ /N d . o . f = 1 . m R P / √ σ A −∗ , χ /N d . o . f = 1 . A − , χ /N d . o . f = 0 . m R P / √ σ A +2 , χ /N d . o . f = 1 . A − , χ /N d . o . f = 0 . . . . m R P / √ σ T +2 , χ /N d . o . f = 1 . E + , χ /N d . o . f = 0 . m R P / √ σ T − , χ /N d . o . f = 0 . E − , χ /N d . o . f = 0 . . . . . / N m R P / √ σ T +1 , χ /N d . o . f = 0 . T − , χ /N d . o . f = 0 . Figure 16 . Glueball mass in each symmetry channel R P of the Sp (2 N ) theory, in units of √ σa ,as a function of / N . For N = 2 only the numerical measurements reported in this publicationwere used (new ensembles). The point corresponding to / N = 0 is the value of m R P / √ σ ( ∞ ) obtained from the best fit of Eq. (4.6) to the data. See main text for details. – 57 – m R P / √ σ A + ∗ , χ /N d . o . f . = 0 . A +1 , χ /N d . o . f . = 0 . m R P / √ σ A −∗ , χ /N d . o . f . = 0 . A − , χ /N d . o . f . = 1 . m R P / √ σ A +2 , χ /N d . o . f . = 0 . A − , χ /N d . o . f . = 0 . m R P / √ σ T +2 , χ /N d . o . f . = 1 . E + , χ /N d . o . f . = 0 . m R P / √ σ T − , χ /N d . o . f . = 0 . E − , χ /N d . o . f . = 3 . .
00 0 .
02 0 .
04 0 . σa . . . m R P / √ σ T +1 , χ /N d . o . f . = 2 . T − , χ /N d . o . f . = 1 . Figure 17 . Glueball mass in each symmetry channel R P of the Sp (2 N ) theory with N = 2 , inunits of √ σ , as a function of σa . Only the data produced for Ref. [7] were used (old ensembles).For each symmetry channel R P , the value at σa = 0 is the continuum limit, obtained from a bestfit of Eq. (C.1) to the data. The best fits lines are represented as solid lines. The extrapolatedresults are reported in Tab. 24 – 58 – + − + − + + − − + − A +1 A − A +2 A − T +2 E + T − E − T +1 T − m R P √ σ N = 1 N = 2 N = 3 N = 4 Sp ( ∞ ) SU ( ∞ ) Figure 18 . Spectrum of the Sp (2 N ) theory in the continuum limit for N = 1 , , , and N = ∞ ,in units of √ σ from the data collected for Ref. [7]. For ease of comparison, we have reported alsothe masses of the A ++1 and E ++ channels for SU ( ∞ ) (borrowed from [2]). – 59 – m R P / √ σ A + ∗ , χ /N d . o . f = 3 . A +1 , χ /N d . o . f = 1 . m R P / √ σ A −∗ , χ /N d . o . f = 3 . A − , χ /N d . o . f = 1 . m R P / √ σ A +2 , χ /N d . o . f = 3 . A − , χ /N d . o . f = 1 . . . . m R P / √ σ T +2 , χ /N d . o . f = 0 . E + , χ /N d . o . f = 0 . . . . m R P / √ σ T − , χ /N d . o . f = 1 . E − , χ /N d . o . f = 1 . . . . . / N m R P / √ σ T +1 , χ /N d . o . f = 1 . T − , χ /N d . o . f = 0 . Figure 19 . Glueball mass in each symmetry channel R P in units of √ σ , as a function of / N .For N = 2 only the data created for Ref. [7] were used. The point corresponding to / N = 0 isthe value of m R P / √ σ ( ∞ ) obtained from the best fit of Eq. (4.6) to the data. – 60 – eferences [1] B. Lucini and M. Panero, SU(N) gauge theories at large N , Phys. Rept. (2013) 93[ ].[2] B. Lucini, M. Teper and U. Wenger,
Glueballs and k-strings in SU(N) gauge theories:Calculations with improved operators , JHEP (2004) 012 [ hep-lat/0404008 ].[3] A. Athenodorou, R. Lau and M. Teper, On the weak N -dependence of SO(N) and SU(N)gauge theories in 2+1 dimensions , Phys. Lett.
B749 (2015) 448 [ ].[4] R. Lau and M. Teper,
SO(N) gauge theories in 2 + 1 dimensions: glueball spectra andconfinement , JHEP (2017) 022 [ ].[5] M. Teper, SO(4), SO(3) and SU(2) gauge theories in 2+1 dimensions: comparing glueballspectra and string tensions , .[6] K. Holland, M. Pepe and U. J. Wiese, The Deconfinement phase transition of Sp(2) andSp(3) Yang-Mills theories in (2+1)-dimensions and (3+1)-dimensions , Nucl. Phys.
B694 (2004) 35 [ hep-lat/0312022 ].[7] E. Bennett, D. K. Hong, J.-W. Lee, C. J. D. Lin, B. Lucini, M. Piai et al.,
Sp(4) gaugetheory on the lattice: towards SU(4)/Sp(4) composite Higgs (and beyond) , JHEP (2018)185 [ ].[8] E. Bennett, D. K. Hong, J.-W. Lee, C. J. D. Lin, B. Lucini, M. Piai et al., Higgscompositeness in Sp(2N) gauge theories — Resymplecticisation, scale setting and topology , EPJ Web Conf. (2018) 08012 [ ].[9] E. Bennett, D. K. Hong, J.-W. Lee, C. J. D. Lin, B. Lucini, M. Piai et al.,
Higgscompositeness in Sp(2N) gauge theories – Determining the low-energy constants with latticecalculations , EPJ Web Conf. (2018) 08011 [ ].[10] E. Bennett, D. K. Hong, J.-W. Lee, C. J. D. Lin, B. Lucini, M. Piai et al.,
Higgscompositeness in Sp(2N) gauge theories — The pure gauge model , EPJ Web Conf. (2018) 08013 [ ].[11] J.-W. Lee, E. Bennett, D. K. Hong, C. J. D. Lin, B. Lucini, M. Piai et al.,
Progress in thelattice simulations of Sp(2 N ) gauge theories , PoS
LATTICE2018 (2018) 192[ ].[12] D. B. Kaplan and H. Georgi,
SU(2) x U(1) Breaking by Vacuum Misalignment , Phys. Lett. (1984) 183.[13] H. Georgi and D. B. Kaplan,
Composite Higgs and Custodial SU(2) , Phys. Lett. (1984) 216.[14] M. J. Dugan, H. Georgi and D. B. Kaplan,
Anatomy of a Composite Higgs Model , Nucl.Phys.
B254 (1985) 299.[15] K. Agashe, R. Contino and A. Pomarol,
The Minimal composite Higgs model , Nucl. Phys.
B719 (2005) 165 [ hep-ph/0412089 ].[16] R. Contino, L. Da Rold and A. Pomarol,
Light custodians in natural composite Higgsmodels , Phys. Rev.
D75 (2007) 055014 [ hep-ph/0612048 ].[17] R. Barbieri, B. Bellazzini, V. S. Rychkov and A. Varagnolo,
The Higgs boson from anextended symmetry , Phys. Rev.
D76 (2007) 115008 [ ]. – 61 –
18] P. Lodone,
Vector-like quarks in a ’composite’ Higgs model , JHEP (2008) 029[ ].[19] D. Marzocca, M. Serone and J. Shu, General Composite Higgs Models , JHEP (2012) 013[ ].[20] G. Ferretti and D. Karateev, Fermionic UV completions of Composite Higgs models , JHEP (2014) 077 [ ].[21] G. Cacciapaglia and F. Sannino, Fundamental Composite (Goldstone) Higgs Dynamics , JHEP (2014) 111 [ ].[22] A. Arbey, G. Cacciapaglia, H. Cai, A. Deandrea, S. Le Corre and F. Sannino, FundamentalComposite Electroweak Dynamics: Status at the LHC , Phys. Rev.
D95 (2017) 015028[ ].[23] L. Vecchi,
A dangerous irrelevant UV-completion of the composite Higgs , JHEP (2017)094 [ ].[24] G. Panico and A. Wulzer, The Composite Nambu-Goldstone Higgs , Lect. Notes Phys. (2016) pp.1 [ ].[25] G. Ferretti,
Gauge theories of Partial Compositeness: Scenarios for Run-II of the LHC , JHEP (2016) 107 [ ].[26] A. Agugliaro, O. Antipin, D. Becciolini, S. De Curtis and M. Redi, UV complete compositeHiggs models , Phys. Rev.
D95 (2017) 035019 [ ].[27] T. Alanne, D. Buarque Franzosi and M. T. Frandsen,
A partially composite GoldstoneHiggs , Phys. Rev.
D96 (2017) 095012 [ ].[28] F. Feruglio, B. Gavela, K. Kanshin, P. A. N. Machado, S. Rigolin and S. Saa,
The minimallinear sigma model for the Goldstone Higgs , JHEP (2016) 038 [ ].[29] S. Fichet, G. von Gersdorff, E. Pontón and R. Rosenfeld, The Excitation of the GlobalSymmetry-Breaking Vacuum in Composite Higgs Models , JHEP (2016) 158[ ].[30] J. Galloway, A. L. Kagan and A. Martin, A UV complete partially composite-pNGB Higgs , Phys. Rev.
D95 (2017) 035038 [ ].[31] T. Alanne, D. Buarque Franzosi, M. T. Frandsen, M. L. A. Kristensen, A. Meroni andM. Rosenlyst,
Partially composite Higgs models: Phenomenology and RG analysis , JHEP (2018) 051 [ ].[32] C. Csaki, T. Ma and J. Shu, Maximally Symmetric Composite Higgs Models , Phys. Rev.Lett. (2017) 131803 [ ].[33] M. Chala, G. Durieux, C. Grojean, L. de Lima and O. Matsedonskyi,
Minimally extendedSILH , JHEP (2017) 088 [ ].[34] C. Csáki, T. Ma and J. Shu, Trigonometric Parity for Composite Higgs Models , Phys. Rev.Lett. (2018) 231801 [ ].[35] V. Ayyar, T. Degrand, D. C. Hackett, W. I. Jay, E. T. Neil, Y. Shamir et al.,
Baryonspectrum of SU(4) composite Higgs theory with two distinct fermion representations , Phys.Rev.
D97 (2018) 114505 [ ]. – 62 –
36] V. Ayyar, T. DeGrand, D. C. Hackett, W. I. Jay, E. T. Neil, Y. Shamir et al.,
Finite-temperature phase structure of SU(4) gauge theory with multiple fermionrepresentations , Phys. Rev.
D97 (2018) 114502 [ ].[37] C. Cai, G. Cacciapaglia and H.-H. Zhang,
Vacuum alignment in a composite 2HDM , JHEP (2019) 130 [ ].[38] A. Agugliaro, G. Cacciapaglia, A. Deandrea and S. De Curtis, Vacuum misalignment andpattern of scalar masses in the SU(5)/SO(5) composite Higgs model , JHEP (2019) 089[ ].[39] V. Ayyar, T. DeGrand, D. C. Hackett, W. I. Jay, E. T. Neil, Y. Shamir et al., Partialcompositeness and baryon matrix elements on the lattice , Phys. Rev.
D99 (2019) 094502[ ].[40] G. Cacciapaglia, S. Vatani, T. Ma and Y. Wu,
Towards a fundamental safe theory ofcomposite Higgs and Dark Matter , .[41] O. Witzel, Review on Composite Higgs Models , PoS
LATTICE2018 (2019) 006[ ].[42] G. Cacciapaglia, G. Ferretti, T. Flacke and H. Serôdio,
Light scalars in composite Higgsmodels , Front.in Phys. (2019) 22 [ ].[43] V. Ayyar, M. F. Golterman, D. C. Hackett, W. Jay, E. T. Neil, Y. Shamir et al., RadiativeContribution to the Composite-Higgs Potential in a Two-Representation Lattice Model , Phys. Rev.
D99 (2019) 094504 [ ].[44] G. Cossu, L. Del Debbio, M. Panero and D. Preti,
Strong dynamics with matter in multiplerepresentations:
SU(4) gauge theory with fundamental and sextet fermions , Eur. Phys. J.
C79 (2019) 638 [ ].[45] G. Cacciapaglia, H. Cai, A. Deandrea and A. Kushwaha,
Composite Higgs and Dark MatterModel in SU(6)/SO(6) , JHEP (2019) 035 [ ].[46] D. Buarque Franzosi and G. Ferretti, Anomalous dimensions of potential top-partners , SciPost Phys. (2019) 027 [ ].[47] C. Grojean, O. Matsedonskyi and G. Panico, Light top partners and precision physics , JHEP (2013) 160 [ ].[48] T. DeGrand, M. Golterman, E. T. Neil and Y. Shamir, One-loop Chiral PerturbationTheory with two fermion representations , Phys. Rev. D (2016) 025020 [ ].[49] V. Ayyar, T. DeGrand, M. Golterman, D. C. Hackett, W. I. Jay, E. T. Neil et al., Spectroscopy of SU(4) composite Higgs theory with two distinct fermion representations , Phys. Rev. D (2018) 074505 [ ].[50] C.-S. Guan, T. Ma and J. Shu, Left-right symmetric composite Higgs model , Phys. Rev. D (2020) 035032 [ ].[51] B. S. Kim, D. K. Hong and J.-W. Lee,
Into the conformal window: Multirepresentationgauge theories , Phys. Rev. D (2020) 056008 [ ].[52] G. Cacciapaglia, C. Pica and F. Sannino,
Fundamental Composite Dynamics: A Review , .[53] G. Cacciapaglia, S. Vatani and C. Zhang, The Techni-Pati-Salam Composite Higgs , . – 63 –
54] J. Erdmenger, N. Evans, W. Porod and K. S. Rigatos,
Gauge/gravity dynamics forcomposite Higgs models and the top mass , .[55] E. Katz, A. E. Nelson and D. G. E. Walker, The Intermediate Higgs , JHEP (2005) 074[ hep-ph/0504252 ].[56] B. Gripaios, A. Pomarol, F. Riva and J. Serra, Beyond the Minimal Composite HiggsModel , JHEP (2009) 070 [ ].[57] J. Barnard, T. Gherghetta and T. S. Ray, UV descriptions of composite Higgs modelswithout elementary scalars , JHEP (2014) 002 [ ].[58] R. Lewis, C. Pica and F. Sannino, Light Asymmetric Dark Matter on the Lattice: SU(2)Technicolor with Two Fundamental Flavors , Phys. Rev.
D85 (2012) 014504 [ ].[59] A. Hietanen, R. Lewis, C. Pica and F. Sannino,
Fundamental Composite Higgs Dynamicson the Lattice: SU(2) with Two Flavors , JHEP (2014) 116 [ ].[60] R. Arthur, V. Drach, M. Hansen, A. Hietanen, C. Pica and F. Sannino, SU(2) gauge theorywith two fundamental flavors: A minimal template for model building , Phys. Rev.
D94 (2016) 094507 [ ].[61] R. Arthur, V. Drach, A. Hietanen, C. Pica and F. Sannino, SU (2) Gauge Theory with TwoFundamental Flavours: Scalar and Pseudoscalar Spectrum , .[62] C. Pica, V. Drach, M. Hansen and F. Sannino, Composite Higgs Dynamics on the Lattice , EPJ Web Conf. (2017) 10005 [ ].[63] W. Detmold, M. McCullough and A. Pochinsky,
Dark nuclei. II. Nuclear spectroscopy intwo-color QCD , Phys. Rev.
D90 (2014) 114506 [ ].[64] J.-W. Lee, B. Lucini and M. Piai,
Symmetry restoration at high-temperature in two-colorand two-flavor lattice gauge theories , JHEP (2017) 036 [ ].[65] G. Cacciapaglia, H. Cai, A. Deandrea, T. Flacke, S. J. Lee and A. Parolini, Compositescalars at the LHC: the Higgs, the Sextet and the Octet , JHEP (2015) 201 [ ].[66] N. Bizot, M. Frigerio, M. Knecht and J.-L. Kneur, Nonperturbative analysis of the spectrumof meson resonances in an ultraviolet-complete composite-Higgs model , Phys. Rev.
D95 (2017) 075006 [ ].[67] D. K. Hong,
Very light dilaton and naturally light Higgs boson , JHEP (2018) 102[ ].[68] M. Golterman and Y. Shamir, Effective potential in ultraviolet completions for compositeHiggs models , Phys. Rev.
D97 (2018) 095005 [ ].[69] V. Drach, T. Janowski and C. Pica,
Update on SU(2) gauge theory with NF = 2fundamental flavours , EPJ Web Conf. (2018) 08020 [ ].[70] F. Sannino, P. Stangl, D. M. Straub and A. E. Thomsen,
Flavor Physics and FlavorAnomalies in Minimal Fundamental Partial Compositeness , Phys. Rev.
D97 (2018) 115046[ ].[71] T. Alanne, N. Bizot, G. Cacciapaglia and F. Sannino,
Classification of NLO operators forcomposite Higgs models , Phys. Rev.
D97 (2018) 075028 [ ].[72] N. Bizot, G. Cacciapaglia and T. Flacke,
Common exotic decays of top partners , JHEP (2018) 065 [ ]. – 64 –
73] D. Buarque Franzosi, G. Cacciapaglia and A. Deandrea,
Sigma-assisted natural compositeHiggs , .[74] H. Gertov, A. E. Nelson, A. Perko and D. G. E. Walker, Lattice-Friendly Gauge Completionof a Composite Higgs with Top Partners , JHEP (2019) 181 [ ].[75] G. Cacciapaglia, S. Vatani and C. Zhang, Composite Higgs Meets Planck Scale: PartialCompositeness from Partial Unification , .[76] Y. Hochberg, E. Kuflik, T. Volansky and J. G. Wacker, Mechanism for Thermal Relic DarkMatter of Strongly Interacting Massive Particles , Phys. Rev. Lett. (2014) 171301[ ].[77] Y. Hochberg, E. Kuflik, H. Murayama, T. Volansky and J. G. Wacker,
Model for ThermalRelic Dark Matter of Strongly Interacting Massive Particles , Phys. Rev. Lett. (2015)021301 [ ].[78] A. Berlin, N. Blinov, S. Gori, P. Schuster and N. Toro,
Cosmology and Accelerator Tests ofStrongly Interacting Dark Matter , Phys. Rev. D (2018) 055033 [ ].[79] Y.-D. Tsai, R. McGehee and H. Murayama, Resonant Self-Interacting Dark Matter fromDark QCD , .[80] E. Bennett, D. K. Hong, J.-W. Lee, C.-J. D. Lin, B. Lucini, M. Piai et al., Sp(4) gaugetheories on the lattice: N f = 2 dynamical fundamental fermions , JHEP (2019) 053[ ].[81] E. Bennett, D. K. Hong, J.-W. Lee, C.-J. D. Lin, B. Lucini, M. Mesiti et al., Sp (4) gaugetheories on the lattice: quenched fundamental and antisymmetric fermions , Phys. Rev. D (2020) 074516 [ ].[82] E. Bennett, J. Holligan, D. K. Hong, J.-W. Lee, C.-J. D. Lin, B. Lucini et al.,
On the colourdependence of tensor and scalar glueball masses in Yang-Mills theories , .[83] J. Holligan, E. Bennett, D. K. Hong, J.-W. Lee, C.-J. D. Lin, B. Lucini et al., Sp (2 N ) Yang-Mills towards large N , in ,2019, .[84] A. Athenodorou, E. Bennett, G. Bergner, D. Elander, C. J. D. Lin, B. Lucini et al., Largemass hierarchies from strongly-coupled dynamics , JHEP (2016) 114 [ ].[85] D. K. Hong, J.-W. Lee, B. Lucini, M. Piai and D. Vadacchino, Casimir scaling andYang–Mills glueballs , Phys. Lett.
B775 (2017) 89 [ ].[86] R. C. Brower, S. D. Mathur and C.-I. Tan,
Glueball spectrum for QCD from AdSsupergravity duality , Nucl. Phys.
B587 (2000) 249 [ hep-th/0003115 ].[87] C.-K. Wen and H.-X. Yang,
QCD(4) glueball masses from AdS(6) black hole description , Mod. Phys. Lett.
A20 (2005) 997 [ hep-th/0404152 ].[88] S. Kuperstein and J. Sonnenschein,
Non-critical, near extremal AdS(6) background as aholographic laboratory of four dimensional YM theory , JHEP (2004) 026[ hep-th/0411009 ].[89] D. Elander, A. F. Faedo, C. Hoyos, D. Mateos and M. Piai, Multiscale confining dynamicsfrom holographic RG flows , JHEP (2014) 003 [ ].[90] D. Elander and M. Piai, Glueballs on the Baryonic Branch of Klebanov-Strassler:dimensional deconstruction and a light scalar particle , JHEP (2017) 003 [ ]. – 65 –
91] D. Elander and M. Piai,
Calculable mass hierarchies and a light dilaton from gravity duals , Phys. Lett.
B772 (2017) 110 [ ].[92] D. Elander, M. Piai and J. Roughley,
Holographic glueballs from the circle reduction ofRomans supergravity , JHEP (2019) 101 [ ].[93] W. Mueck and M. Prisco, Glueball scattering amplitudes from holography , JHEP (2004)037 [ hep-th/0402068 ].[94] R. Apreda, D. E. Crooks, N. J. Evans and M. Petrini, Confinement, glueballs and stringsfrom deformed AdS , JHEP (2004) 065 [ hep-th/0308006 ].[95] D. Elander, A. F. Faedo, D. Mateos, D. Pravos and J. G. Subils, Mass spectrum of gapped,non-confining theories with multi-scale dynamics , JHEP (2019) 175 [ ].[96] D. Elander, M. Piai and J. Roughley, Probing the holographic dilaton , JHEP (2020) 177[ ].[97] R. G. Leigh, D. Minic and A. Yelnikov, Solving pure QCD in 2+1 dimensions , Phys. Rev.Lett. (2006) 222001 [ hep-th/0512111 ].[98] R. G. Leigh, D. Minic and A. Yelnikov, On the Glueball Spectrum of Pure Yang-MillsTheory in 2+1 Dimensions , Phys. Rev. D (2007) 065018 [ hep-th/0604060 ].[99] M. Bochicchio, Glueball and meson spectrum in large-N massless QCD , .[100] M. Bochicchio, An asymptotic solution of Large-N QCD, for the glueball and mesonspectrum and the collinear S-matrix , AIP Conf. Proc. (2016) 030004.[101] M. Bochicchio,
Glueball and Meson Spectrum in Large-N QCD , Few Body Syst. (2016)455.[102] N. Cabibbo and E. Marinari, A New Method for Updating SU(N) Matrices in ComputerSimulations of Gauge Theories , Phys. Lett. (1982) 387.[103] L. Del Debbio, A. Patella and C. Pica,
Higher representations on the lattice: Numericalsimulations. SU(2) with adjoint fermions , Phys. Rev.
D81 (2010) 094503 [ ].[104] A. A. Belavin, A. M. Polyakov, A. S. Schwartz and Yu. S. Tyupkin,
PseudoparticleSolutions of the Yang-Mills Equations , Phys. Lett. (1975) 85.[105] P. Di Vecchia, F. Nicodemi, R. Pettorino and G. Veneziano,
Large n, Chiral Approach toPseudoscalar Masses, Mixings and Decays , Nucl. Phys.
B181 (1981) 318.[106] M. Luscher,
Topology of Lattice Gauge Fields , Commun. Math. Phys. (1982) 39.[107] A. Phillips and D. Stone, Lattice Gauge Fields, Principal Bundles and the Calculation ofTopological Charge , Commun. Math. Phys. (1986) 599.[108] B. Alles, M. D’Elia, A. Di Giacomo and R. Kirchner,
A Critical comparison of differentdefinitions of topological charge on the lattice , Phys. Rev.
D58 (1998) 114506[ hep-lat/9711026 ].[109] M. Lüscher,
Properties and uses of the Wilson flow in lattice QCD , JHEP (2010) 071[ ].[110] B. Lucini, M. Teper and U. Wenger, Topology of SU(N) gauge theories at T = 0 and T= T(c) , Nucl. Phys.
B715 (2005) 461 [ hep-lat/0401028 ].[111] M. Luscher and P. Weisz,
Perturbative analysis of the gradient flow in non-abelian gaugetheories , JHEP (2011) 051 [ ]. – 66 – Comparison of the gradient flow with cooling in SU (3) pure gaugetheory , Phys. Rev.
D89 (2014) 105005 [ ].[113] B. Berg and A. Billoire,
Glueball Spectroscopy in Four-Dimensional SU(3) Lattice GaugeTheory. 1. , Nucl. Phys.
B221 (1983) 109.[114] C. J. Morningstar and M. J. Peardon,
Efficient glueball simulations on anisotropic lattices , Phys. Rev.
D56 (1997) 4043 [ hep-lat/9704011 ].[115] C. J. Morningstar and M. J. Peardon,
The Glueball spectrum from an anisotropic latticestudy , Phys. Rev.
D60 (1999) 034509 [ hep-lat/9901004 ].[116] J. Hoek, M. Teper and J. Waterhouse,
Topological Fluctuations and Susceptibility in SU(3)Lattice Gauge Theory , Nucl. Phys.
B288 (1987) 589.[117] B. Lucini, A. Rago and E. Rinaldi,
Glueball masses in the large N limit , JHEP (2010)119 [ ].[118] B. Lucini and M. Teper, SU(N) gauge theories in four-dimensions: Exploring the approachto N = infinity , JHEP (2001) 050 [ hep-lat/0103027 ].[119] M. Hamermesh, Group theory and its application to physical problems . Addison-Wesley,Reading, MA, 1962.[120] O. Aharony and E. Karzbrun,
On the effective action of confining strings , JHEP (2009)012 [ ].[121] M. Luscher, K. Symanzik and P. Weisz, Anomalies of the Free Loop Wave Equation in theWKB Approximation , Nucl. Phys.
B173 (1980) 365.[122] R. Brower, S. Chandrasekharan, J. W. Negele and U. J. Wiese,
QCD at fixed topology , Phys. Lett.
B560 (2003) 64 [ hep-lat/0302005 ].[123] S. Aoki, H. Fukaya, S. Hashimoto and T. Onogi,
Finite volume QCD at fixed topologicalcharge , Phys. Rev. D (2007) 054508 [ ].[124] T. Eguchi and H. Kawai, Reduction of Dynamical Degrees of Freedom in the Large N GaugeTheory , Phys. Rev. Lett. (1982) 1063.[125] E. Witten, Theta dependence in the large N limit of four-dimensional gauge theories , Phys.Rev. Lett. (1998) 2862 [ hep-th/9807109 ].[126] A. Amato, G. Bali and B. Lucini, Topology and glueballs in SU (7) Yang-Mills with openboundary conditions , PoS
LATTICE2015 (2016) 292 [ ].[127] A. Athenodorou and M. Teper,
The glueball spectrum of SU(3) gauge theory in 3+1dimension , .[128] C. Lovelace, Universality at Large N , Nucl. Phys. B (1982) 333.[129] L. Okun,
THETONS , JETP Lett. (1980) 144.[130] L. Okun, THETA PARTICLES , Nucl. Phys. B (1980) 1.[131] E. Gregory, A. Irving, B. Lucini, C. McNeile, A. Rago, C. Richards et al.,
Towards theglueball spectrum from unquenched lattice QCD , JHEP (2012) 170 [ ].[132] D. Gross and E. Witten, Possible Third Order Phase Transition in the Large N LatticeGauge Theory , Phys. Rev. D (1980) 446. – 67 – N = Infinity Phase Transition in a Class of Exactly Soluble Model LatticeGauge Theories , Phys. Lett. B (1980) 403.[134] L. Caneschi, I. Halliday and A. Schwimmer, The Phase Structure of Mixed Lattice GaugeTheories , Nucl. Phys. B (1982) 409.[135] B. Lucini, A. Patella, A. Rago and E. Rinaldi,
Infrared conformality and bulk critical points:SU(2) with heavy adjoint quarks , JHEP (2013) 106 [ ].[136] M. Luscher, Volume Dependence of the Energy Spectrum in Massive Quantum FieldTheories. 1. Stable Particle States , Commun. Math. Phys. (1986) 177.(1986) 177.