Decays of an exotic 1 −+ hybrid meson resonance in QCD
Antoni J. Woss, Jozef J. Dudek, Robert G. Edwards, Christopher E. Thomas, David J. Wilson
JJLAB-THY-20-3249
Decays of an exotic − + hybrid meson resonance in QCD Antoni J. Woss, ∗ Jozef J. Dudek,
2, 3, † Robert G. Edwards, ‡ Christopher E. Thomas, § and David J. Wilson ¶ (for the Hadron Spectrum Collaboration) DAMTP, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WA, UK Thomas Jefferson National Accelerator Facility, 12000 Jefferson Avenue, Newport News, VA 23606, USA Department of Physics, College of William and Mary, Williamsburg, VA 23187, USA (Dated: 21 September 2020)We present the first determination of the hadronic decays of the lightest exotic J PC = 1 − + resonance in lattice QCD. Working with SU(3) flavor symmetry, where the up, down and strangequark masses approximately match the physical strange-quark mass giving m π ∼
700 MeV, wecompute finite-volume spectra on six lattice volumes which constrain a scattering system featuringeight coupled channels. Analytically continuing the scattering amplitudes into the complex energyplane, we find a pole singularity corresponding to a narrow resonance which shows relativelyweak coupling to the open pseudoscalar–pseudoscalar, vector–pseudoscalar and vector–vector decaychannels, but large couplings to at least one kinematically-closed axial-vector–pseudoscalar channel.Attempting a simple extrapolation of the couplings to physical light-quark mass suggests a broad π resonance decaying dominantly through the b π mode with much smaller decays into f π , ρπ , η (cid:48) π and ηπ . A large total width is potentially in agreement with the experimental π (1564) candidatestate, observed in ηπ , η (cid:48) π , which we suggest may be heavily suppressed decay channels. I. INTRODUCTION
The composition of hadrons has been the subject ofexperimental and theoretical studies for many decades.Historically, the majority of mesons could be understoodin a quark-model picture where they consist of a quark-antiquark pair ( q ¯ q ). There are some notable long-standingexceptions that do not appear to fit into this framework,such as the light scalar mesons, and more recently ithas been challenged by the observation of a number ofunexpected structures in the charm and bottom sectors.In principle mesons can contain constituent combina-tions beyond q ¯ q , but whether QCD allows for such ar-rangements continues to motivate investigations in boththeory and experiment. One particular focus is on hybridmesons in which a quark-antiquark pair is coupled toan excitation of the gluonic field. Such states are an at-tractive target because the additional quantum numberspotentially supplied by the gluonic field allow for J P C combinations not allowed to a q ¯ q system. These exotic J P C = 0 −− , + − , − + , + − . . . serve as a smoking-gunsignature that a novel state has been observed.Suggestions that hybrid mesons are a feature of QCDare longstanding, but until recently predictions of theirproperties came only within models whose connection toQCD is not always clear [1–7]. While dynamical pictureslike the flux-tube model, the bag model, and constituentgluon approaches generally agree that hybrids form part ofthe meson spectrum, some with exotic J P C , they differ indetails. A common feature is that typically a J P C = 1 − + ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] state (labelled π when the state has isospin–1) appearswith a mass somewhere above 1.5 GeV. A particularchallenge has been for these models to provide reliablepredictions for the decay properties of hybrid mesons,which we expect to appear as resonances that can decayinto several final states. Having some advance knowledgeof which final states are more heavily populated in theirdecay is useful to experiments which perform amplitudeanalyses final-state by final-state. A folklore has devel-oped, largely following from models in which the hybriddecay proceeds by the breaking of an oscillating tubeof gluonic flux or through conversion of a constituentgluon to a q ¯ q pair [8–13], where decays featuring only thelightest hadrons are suppressed, such as π → ηπ, η (cid:48) π, ρπ ,while decays which include a more excited hadron areprominent, such as π → b π . Whether these resultsare really a feature of QCD, or reflect the assumptionsbuilt into the flux-tube (a picture whose validity looksincreasingly unlikely [14]) or constituent gluon pictures,has yet to be established.The experimental focus has remained largely on the π , and historically the picture has been quite con-fused [15, 16]. Analyses have mostly considered the ηπ , η (cid:48) π and ρπ → πππ final states which have the lowestpossible multiplicities. Recent data sets of unprecedentedstatistics from COMPASS provide our clearest picture [17]:a broad bump in ηπ peaking near 1400 MeV appears tomatch poorly with another bump in η (cid:48) π peaking near1600 MeV. These results are similar to those observedin earlier experiments which were interpreted as two res-onances, π (1400) and π (1600), with there being somefurther evidence for the heavier resonance in the ρπ finalstate.A recent analysis of the COMPASS data by JPACcomes to a different conclusion [18]: the two bumps in ηπ, η (cid:48) π are actually due to a single resonance decaying a r X i v : . [ h e p - l a t ] S e p into both final states. They proceed by parameterizatingthe production process and the scattering of the coupled-channel ηπ, η (cid:48) π system, respecting unitarity in these twochannels. The scattering t -matrix is constrained for realvalues of the energy using experimental data. When theamplitude is considered for complex values of the energy,a single pole singularity is found which can be interpretedas one resonance with a mass slightly below 1.6 GeVand a width of around 500 MeV. A combined analysis ofCOMPASS and Crystal Barrel data [19] which appearedwhile this paper was in the final stages of preparationfinds a very similar mass, but a slightly smaller width ∼
388 MeV.Currently the GlueX experiment [20, 21] is collectinglarge data sets using photoproduction in which they willsearch for hybrid mesons. Since the higher multiplicityfinal states suggested as preferred by the flux-tube picture,e.g. π → b π → ( ωπ ) π → πππππ , are much harder toanalyze than those investigated in COMPASS, it would beof benefit to have some evidence within QCD that thesechannels are in fact dominant in the decays of hybridmesons. It is to this task that we turn our attention inthis paper, using the technique known as Lattice QCD.Lattice QCD, which offers a first-principles numericalapproach to QCD, has matured to the point where ithas been able to make some fairly definitive statementsabout the excited spectrum of hadrons. In Refs. [22–26],bases of composite operators built from fermion bilinearsand up to three gauge-covariant derivatives were usedto construct matrices of two-point correlation functions.Analyzing the time dependences of these matrices led topredictions for the spectrum of mesons with a wide rangeof J P C . The spectra obtained, for several values of thelight quark mass, show a strong qualitative similarity withthe experimental meson spectrum, but also feature clearindications of exotic J P C states with notably a lightest π . A phenomenology was developed [14] based uponthe observation that this state, along with states having J P C = 0 − + , − + and 1 −− at similar masses, have largematrix elements to be produced by operators of the form ψ Γ t a ψ B a , which has the q ¯ q pair in a color-octet withthe color neutralized by the chromomagnetic field opera-tor, B a . It was proposed that this large overlap signalsthat these states are hybrid mesons, and they systemat-ically appear roughly 1 . − . t -matrix can thenbe obtained through the use of parameterizations whichare constrained at the discrete real values of energy pro-vided by the finite-volume spectra. The t -matrix is thencontinued into the complex energy plane and any polesingularities identified. From these the mass and width ofa resonance can be determined, along with its couplingsto different decay channels, in what can be argued to bethe most rigorous way possible. In the past few yearsthis approach has been used extensively in the study ofelastic scattering, in cases like isospin–1 ππ where the ρ resonance appears [45–61], and in several pioneeringcalculations of coupled-channel scattering [46, 62–68].In this paper we will report on the first calculation ofan exotic J P C = 1 − + meson appearing as a resonancein coupled-channel meson-meson scattering. By workingwith an exact SU(3) flavor symmetry where the u, d quarkmasses are raised to the physical strange quark mass, wewill reduce the effective number of decay channels andmake three-body decays irrelevant.The paper is structured as follows: In Section II wereview the techniques needed to compute finite-volumespectra in lattice QCD and to relate these to scatteringamplitudes. Section III discusses generalities of workingwith an exact SU(3) F symmetry. In Section IV we presentcalculational details and finite-volume spectra relevantfor a 1 − + resonance on six lattice volumes. In Section Vthese spectra are used to constrain a scattering matrix ofeight channels using a range of parameterizations, and inSection VI these parameterizations are analytically con-tinued into the complex energy plane where a resonancepole singularity is found. Section VII interprets the decaycouplings obtained from the residue of the resonance pole,comparing to existing models of hybrid meson decay, andattempts an extrapolation to physical kinematics. Finally,we summarize in Section VIII. Some additional technicalpoints are discussed in appendices, and details of the vari-ous parameterizations used can be found in SupplementalMaterial. II. RESONANCES IN LATTICE QCD
Our approach to determining resonant physics in latticeQCD requires the computation of discrete spectra in thefinite-volume defined by the lattice, and analysis of thesespectra in terms of a scattering matrix using the L¨uschermethod. In this section we will review our approach fordoing this – if further details are required, the field isreviewed in Ref. [69].
A. Finite-volume spectra
In order to constrain the scattering t -matrix over arange of energies, we are required to calculate a largenumber of discrete finite-volume levels sampling the en-ergy region. An approach which has proven to be highlyeffective for the reliable extraction of many excited statesis through the diagonalization of a large matrix of corre-lation functions, C ij ( t ) = (cid:104) |O i ( t ) O † j (0) | (cid:105) . This can beachieved by solving a generalized eigenvalue problem [70–72], with our implementation described in Refs. [23, 73].This approach makes use of orthogonality between en-ergy eigenstates to distinguish contributions of even near-degenerate states, supplying their energies through thetime-dependence of the eigenvalues while the eigenvectorsprovide linear combinations of the basis operators whichserve as the optimal operator, in the variational sense, foreach state.One possible basis of operators, (cid:8) O i (cid:9) , that can be usedto form a matrix of meson correlation functions is builtfrom fermion bilinears featuring gauge-covariant deriva-tives. A large basis can be constructed both with zeromomentum [23] and non-zero momentum [25]. For thedetermination of stable hadrons, such a basis is typicallysufficient and leads to reliable determinations of the mass(or energy with non-zero momentum) and optimized oper-ators which relax to the desired state more rapidly thanany single operator in the basis (see for example Figure 2of Ref. [74] or Figure 3 of Ref. [75]).The reduced rotational symmetry of a cubic latticemeans that meson states are characterized not by integerspin values and parity, but by the irreducible represen-tations ( irreps ) of the octahedral group or the appro-priate little group for non-zero momentum, with theallowed momenta in an L × L × L periodic volume givenby (cid:126)p = πL (cid:0) n x , n y , n z (cid:1) where n i are integers. In general,this means that examination of a particular irrep requiresconsidering multiple J P values, but the group theory de-scribing how spin subduces into irreps [76, 77] and theconstruction of operators in appropriate irreps [23, 25]are well understood. the set of allowed octahedral group rotations and reflections whichleave the momentum vector unchanged When considering energies near and above meson-meson decay thresholds, a basis of only fermion bilinearsis insufficient to capture the complete finite-volume spec-trum, while augmenting this single-meson-like basis witha set of meson-meson-like constructions has proven tobe highly effective [45, 46]. Such operators are built bycombining optimized stable meson operators using appro-priately weighted products. For an M M -like operatorwith overall momentum (cid:126)P in irrep Λ, O Λ µ † M M ( (cid:126)P ) = (cid:88) µ ,µ (cid:88) ˆ p , ˆ p C ([ (cid:126)P ]Λ , µ ; [ (cid:126)p ]Λ , µ ; [ (cid:126)p ]Λ , µ ) × Ω Λ µ † M ( (cid:126)p ) Ω Λ µ † M ( (cid:126)p ) . Here the optimized stable meson operator, Ω Λ i µ i † M i ( (cid:126)p i ), formeson M i with momentum (cid:126)p i , is labelled by the irrep, Λ i ,and the row of that irrep, µ i (analogous to the J z valuefor a spin- J meson in an infinite-volume continuum). Thesum over momentum directions related by allowed cubicrotations is subject to the constraint that (cid:126)p + (cid:126)p = (cid:126)P . Thegeneralised Clebsch-Gordan coefficients, C , are discussedin Ref. [74].Each meson-meson operator can be characterized bythe magnitudes of meson momenta that went into its con-struction, (cid:0) | (cid:126)p | , | (cid:126)p | (cid:1) . This leads to a natural truncationof the basis of operators following from the energy wewould expect if the mesons had no residual interactions, E (2)n . i . = (cid:113) m + | (cid:126)p | + (cid:113) m + | (cid:126)p | . Clearly, as the individual meson momenta increase, thenon-interacting energy increases, and at some point be-comes sufficiently far above the energy region of interestthat we are justified in not including that operator, orany above it, in our basis.Constructing operators which resemble meson-meson-meson systems, relevant in the energy region above three-meson thresholds, can be done by a recursive applicationof the approach described above [68]. However, one sub-tlety that arises here is that intermediate meson-mesonsubsystems may feature resonant behaviour which a singlemeson-meson operator alone will not efficiently capture.In this case, one or more optimized operators can be con-structed for the lowest energy eigenstates in the meson-meson subsystem by diagonalizing a matrix of correlationfunctions formed from a basis of single-meson-like andmeson-meson-like operators. These optimized operatorsare then combined with the remaining optimized stablemeson operator to form three-meson-like operators thatefficiently interpolate the energy eigenstates. Details ofthis type of construction are given in Ref. [68].The inclusion of multi-meson and isospin–0 single-meson operators in our bases naturally leads to Wickcontractions which feature quark-antiquark annihilations;in the context of lattice QCD these appear via t -to- t quarkpropagators. The distillation approach to computing cor-relation functions [78] efficiently handles these, along withthe required source-sink propagators, without the needto make any further approximations or to introduce anystochastic noise. The propagators, which factorize fromthe operator constructions, are extremely general. Theycan be extensively reused in other calculations which re-quire propagation of the same flavor of quarks such thatthe computational cost of obtaining them is spread overmany physics results. B. Scattering amplitudes
Once the finite-volume spectrum has been extractedfrom a variational analysis of a matrix of correlation func-tions it can be used as a constraint on the energy depen-dence of the coupled-channel t -matrix. The relationshipis encoded in the L¨uscher quantization condition [30–44],det (cid:104) + i ρ t (cid:0) + i M (cid:1)(cid:105) = 0 , (1)where the diagonal matrix of phase-space factors, ρ ( E cm ),and M ( E cm , L ) are known functions of essentially kine-matic origin – see Ref. [79] for our conventions. Thematrix space over which the determinant acts is the setof partial-waves subduced into a particular irrep, for allkinematically accessible meson-meson scattering channels.For a given t -matrix, t ( E cm ), the discrete set of solutionsof this equation, [ E cm ( L )] n =1 , ,... , for a fixed value of L isthe finite-volume spectrum in an L × L × L periodic box.A practical approach for reliably finding solutions to thisequation when there are multiple partial waves and/orhadron-hadron scattering channels, which makes use ofan eigenvalue decomposition of a suitable transformationof the matrix under the determinant, was presented inRef. [79].Eq. 1 is capable of describing any number of coupledhadron-hadron channels, but must be supplemented withfurther formalism once three-hadron channels are accessi-ble. Recent progress is reviewed in Refs. [80, 81]. An approach that allows computed finite-volume spec-tra to constrain scattering amplitudes is to propose param-eterizations of t ( E cm ), whose parameters can be varied,with the corresponding finite-volume spectra from solu-tion of Eq. 1 at each iteration compared to the computedspectra [74]. In this way, a χ can be defined which canbe minimized to find the best description of the com-puted lattice QCD spectra (Eq. (9) in Ref. [45]). Use of a K -matrix in the parameterization of the t -matrix ensurescoupled-channel unitarity, and sensitivity to the particu-lar choice of form chosen for K ( E cm ) can be explored byvarying the form [39]. related to the scattering S -matrix via S = + 2 i √ ρ t √ ρ What role the experimentally observed dominance of quasi-two-body isobars plays in these formalisms is not yet known, but itmay lead to considerable simplifications in practice.
This method provides coupled-channel amplitudes con-strained for real values of E cm , but use of explicit func-tional forms in the parameterizations means that we cananalytically continue into the complex-energy plane toexplore the singularity content of the t -matrix. Poles at complex values of E cm can be identified with reso-nances , with the real and imaginary parts of the poleposition having an interpretation in terms of, respectively,the mass and width of the resonance. Factorizing theresidues of elements of t at the pole position leads todecay couplings of the resonance to the various scatteringchannels. The statistical uncertainty originating in thefinite number of Monte-Carlo samples in the lattice QCDcalculation can be propagated through this process, andin addition the scatter over parameterizations can be usedto estimate a systematic uncertainty from the choice ofparametrization.This approach has been applied successfully in severalrecent calculations of coupled-channel scattering, mostnotably in a series of papers computing on three lat-tice volumes with m π ∼
391 MeV. In the first calcula-tions [62, 63], coupled πK , ηK scattering was investigated.A virtual bound state and a broad resonance were foundin J P = 0 + , a bound state in 1 − , and there was evidencefor a narrow resonance in 2 + , but for all these J P thecoupling to the ηK channel was found to be small inenergy region studied. In Ref. [64], the J P = 0 + cou-pled πη, KK scattering sector was considered, where anasymmetrical peak in πη → πη at the KK threshold wasfound to correspond to a resonance pole that could becompared to the experimental a (980). In Ref. [66], the J P = 0 + and 2 + coupled ππ, KK, ηη isospin-0 sectorswere studied. The scalar amplitudes show a sharp dipin ππ → ππ at KK threshold that could be associatedwith a resonance pole related to the experimental f (980),while a rapid turn on of ππ at threshold was found to bedue to a bound-state related to the σ/f (500). The tensorsector was more straightforward, with clear bumps relatedto two resonances poles, the lighter of which was foundto be dominantly coupled to ππ and the heavier to KK ,in line with the experimental f (1270) and f (cid:48) (1525). InRef. [68], coupled πω , πφ scattering was considered, withthe vector nature of the ω (which is stable at this quarkmass) leading to dynamically coupled partial-waves in J P = 1 + . A bump was found in the πω ( S ) → πω ( S )whose origin is a b -like resonance pole.Before computing finite-volume spectra and determin-ing scattering amplitudes relevant for the exotic J P C =1 − + channel, we now discuss some of the consequences ofworking with exact SU(3) flavor symmetry. III. MESONS WITH EXACT SU ( ) FLAVORSYMMETRY
In this paper we will present the first attempt to com-pute the properties of a resonance with exotic J P C , thelightest π , which is suspected to be a hybrid meson . Asindicated in the introduction, this is a challenging prob-lem owing to the large number of possible decay channels.A significant simplification would occur if we had an exact SU(3) flavor symmetry , as opposed to the approximateone present in nature, as then many of the apparentlyindependent channels would coalesce into particular rep-resentations of SU(3) F . In this first calculation, we opt tomake this symmetry exact by working with three flavors oflight quark all with a mass value tuned to approximatelymatch the physical strange-quark mass. In this world, thelightest pseudoscalar octet, containing the pion, kaon and η -like unflavored member, has a mass around 700 MeV.This relatively large mass has the additional useful effectof pushing three-meson thresholds to higher energies suchthat they become irrelevant in our calculation.With exact SU(3) flavor symmetry, the ‘conventional’mesons (having flavor quantum numbers accessible to q ¯ q ) lie in octet ( ) and singlet ( ) representations fol-lowing from the decomposition of ⊗ ¯ . The lightestof these is the pseudoscalar octet, containing degeneratemesons which we can associate with the pion, the kaonand something close to the η meson. We choose to usethe zero-isospin, zero-strangeness member of the octetas a label to indicate the J P ( C ) , e.g. η in this case of0 − (+) . There is also a light pseudoscalar singlet, η , whosesole member is close to the familiar η (cid:48) . The lightestoctet of vectors, ω , contains mesons we identify withthe ρ and the K ∗ , but its neutral member cannot easilybe associated with either the ω or the φ , as the experi-mental ω is believed to have approximate quark content u ¯ u + d ¯ d , while the φ is dominantly s ¯ s . These correspondto significant admixtures of the octet ( u ¯ u + d ¯ d − s ¯ s ) andsinglet ( u ¯ u + d ¯ d + s ¯ s ). Clearly, when SU(3) F is broken,the flavorless members of ω and ω must mix to formthe physical eigenstates.The notable difference between the pseudoscalar andvector sectors was explored in lattice QCD in terms ofthe q ¯ q annihilation , or ‘ disconnected ’, contributions totwo-point correlation functions in Ref. [26]. As can beseen in Figs. 4 and 5 of that paper, the vector correlatorshave extremely small disconnected pieces, both at andaway from the SU(3) F limit, leading to a lack of hidden-light–hidden-strange mixing and the ρ and ω mesonsbeing close to degenerate. This can be compared tothe same quantities in the pseudoscalar sector shown in The physical eigenstates (with broken SU(3) F ) are believed to beadmixtures of the octet/singlet basis states with a small mixingangle as discussed in Section VII. The dependence of this mixingangle on the light-quark mass was explored using lattice QCD inRef. [26]. Figs. 2 and 3 therein.These observations are related to the Okubo ZweigIizuka (OZI) rule which states that processes where thereare no quark lines connecting the initial-state hadrons tothe final-state hadrons are suppressed. Empirically thisholds for many J P C , including vectors, where a famousexample is the suppression of the otherwise allowed decay φ → πππ which leads to the s ¯ s assignment for the φ .The OZI rule does not seem to apply to the pseudoscalarsector.A major advantage of an exact SU(3) flavor symmetrycomes when we consider meson-meson scattering, as chan-nels that with broken SU(3) F were independent and haddiffering thresholds, like ππ , KK , . . . , are now equivalent,being a single channel, η η . Since the stable scatteringhadrons lie in octets and singlets, the meson-meson prod-ucts ⊗ , ⊗ and ⊗ are of interest, with the first ofthese being decomposed into ⊕ ⊕ ⊕ ⊕ ⊕ .The representations , , lie outside the ‘conven-tional’ sector, requiring at least qq ¯ q ¯ q , and are unlikely tobe resonant [67, 74, 82]. The two octets, , , can bedistinguished by their symmetries under the exchange ofthe flavor of the two hadrons in the product. We followthe conventions of Ref. [83], where is symmetric and is antisymmetric, and we summarize the relevant resultsin Appendix A. As an example, using the SU(3) analoguesof Clebsch-Gordan coefficients in that reference, the flavorstructure of the I = 0, I z = 0, zero-strangeness membersof the two octets in the vector-pseudoscalar case can beexpressed as, = √ (cid:16) K ∗ + K − + K ∗− K + − K ∗ K − K ∗ K (cid:17) − √ (cid:16) ρ + π − + ρ − π + − ρ π (cid:17) − √ ω η , = (cid:16) K ∗ + K − − K ∗− K + − K ∗ K + K ∗ K (cid:17) , (2)which makes manifest that is symmetric under theinterchange of the flavor of the two hadrons while isantisymmetric.In determining what decays are possible, it is impor-tant to pay attention to the generalization of charge-conjugation symmetry. With exact isospin symmetryit is useful to consider G -parity and there are naturalextensions of this in the SU(3) F case. Because we areat liberty to consider any member of the target SU(3)multiplet, here we focus on the neutral zero-strangenesselement where charge-conjugation symmetry itself is goodand so C -parity is the relevant quantum number to con-sider. The resulting selection rules apply to all membersof the multiplet. Details are provided in Appendix A andthe relevant results are summarized in Table I where thedifferent symmetries of and are apparent.When the two scattering mesons are in the same SU(3) F multiplet, there is the additional constraint of Bose sym-metry which requires that the state is symmetric under theinterchange of the two mesons, i.e. the overall symmetryunder the interchange of flavor, spin and spatial position.In the pseudoscalar-pseudoscalar case, where there is no F a ⊗ F b → F C e.g. (1 − ( − ) − (+) → +( C ) ) a ⊗ b → C a C b ( ω η → h C = − ) a ⊗ b → − C a C b ( ω η → f C = +) a ⊗ b → C a C b ( ω η → h C = − ) a ⊗ b → C a C b ( ω η → h C = − ) a ⊗ b → C a C b ( ω η → h C = − )TABLE I. C -parity values for the neutral zero-strangenesscomponents of the SU(3) octets and singlets from meson-mesonproducts. C a and C b denote the C -parity of the neutral zero-strangeness components of the product irreps. We present anexample for the 1 − ( − ) − (+) → +( C ) case to illustrate notation. spin to be dealt with, we immediately have the restrictionthat η η with even (cid:96) appear in with J P ( C ) = (cid:96) +(+) ,while odd (cid:96) appear in with J P ( C ) = (cid:96) − ( − ) . It is there-fore not possible to have an octet 1 − (+) resonance decayto η η . Slightly more complicated is the case of ω ω where the spin of the two vectors can combine to totalspin S = 0 , , (cid:96) . The spin+space symmetric options (such as ω ω { S } ) appear in , while the spin+space antisym-metric options (such as ω ω { S } ) appear in . A morecomplete discussion of these constraints can be found inAppendix B.In this study we will present the result of a calculationof the J P ( C ) = 1 − (+) octet, labelled η . We will chooseto focus our later interpretation on the isovector mem-ber, the π , even though with exact SU(3) F symmetrythe properties of the isoscalar member, the η , and thestrange members are exactly the same. The reason forthis choice is that as we move away from the SU(3) F limitby reducing the u, d quark masses, retaining an isospinsymmetry, we expect that the η can mix with an η livingin the SU(3) F singlet, the η , while the kaonic states canmix with 1 − ( − ) kaons owing to there being no relevant C -parity-like symmetry for mesons with net strangeness.On these grounds it seems plausible that the propertiesof the π will change least as we move away from theexact SU(3) F limit. There may be some mixing withthe corresponding states in the , , representations,but this is expected to be negligible given that there is noevidence for anything beyond rather weak non-resonantinteractions in these multiplets.The meson-meson scattering channels capable of cou-pling to the 1 − (+) octet include η η , ω η , ω ω , ω ω , f ω , h η , f η . . . . How many of these are kinemati-cally accessible in the decay of a potential lightest 1 − (+) resonance depends upon QCD dynamics which we willnow explore in a lattice QCD calculation. IV. LATTICE QCD SPECTRA
Calculations of correlation functions were performed onsix anisotropic lattices with volumes (
L/a s ) × ( T /a t ) =12 ×
96 and { , , , , } × a s ∼ .
12 fmand a t = a s /ξ ∼ (4 . − respectively, where theanisotropy ξ ∼ .
5. Gauge fields were generated from atree-level Symanzik improved gauge action and a Cloverfermion action with three degenerate flavors of dynamicalquarks [84, 85], tuned to approximately the value of thephysical strange quark mass, such that the pion mass is ∼
700 MeV. On all volumes, exponentially-suppressedfinite-volume and thermal effects remain negligible as m π L (cid:38) m π T (cid:38) distilla-tion framework [78] and we give the rank of the distillationspace, N vecs , number of gauge configurations, N cfgs , andtime-sources, N tsrcs , used on each volume in Table II.We typically compute all the elements of the matrix ofcorrelation functions; however, in a few cases we madeuse of hermiticity to infer C ji ( t ) from a computed C ij ( t ).The spectrum of low-lying mesons on these latticesis shown in Figure 1, obtained as the ground states invariational analysis of matrices of correlation functionsusing a basis of fermion bilinear operators in either SU(3) F octet or singlet representations . As we might expect,the pseudoscalar octet (containing the analogues of thepion, kaon and η ) is lightest, with the pseudoscalar singlet(comparable to the η (cid:48) ) being somewhat heavier. The octetand singlet vector mesons are close to degenerate reflectingthat this J P C has a very small disconnected contributionwhich distinguishes the singlet from the octet.The singlet scalar meson ( f ) is rather light, at a similarmass to the pseudoscalar singlet. As it does not appear inthe decays of the 1 − (+) resonance we are studying in thispaper, we will not discuss it further here. The extractedscalar octet meson ( f ) mass lies very close to the η η ( L/a s ) × ( T /a t ) N vecs N cfgs N tsrcs ×
96 48 219 2414 ×
128 64 397 1616 ×
128 64 529 418 ×
128 96 358 420 ×
128 128 501 424 ×
128 160 607 4TABLE II. Number of distillation vectors ( N vecs ), gauge config-urations ( N cfgs ) and time-sources ( N tsrcs ) used in computationof correlation functions on each lattice volume, as describedin the text. More details of the operator construction, and decomposition interms of connected and disconnected contributions can be foundin Ref. [26]. The 16 and 20 volumes used in that reference aresupplemented with the other volumes in Table II in the currentwork. FIG. 1. Spectrum of low-lying octet (red) and singlet (cyan)mesons by J P ( C ) obtained using only single-meson operators.Solid boxes show mesons which lie below relevant meson-meson thresholds and are thus stable, while hatched boxesshow mesons which lie above threshold and which will requirea full finite-volume analysis to resolve their resonant nature.Dashed lines show the lowest relevant meson-meson thresholds. η η ω ω f f f h h a t m . threshold. This indicates that to properly understand the f , which may be a resonance or a shallow bound state,we would have to include meson-meson operators in ourbasis. Levels corresponding to the tensor mesons ( f , f )are found some way above the η η threshold, stronglysuggesting that these states will be resonances capable ofdecaying into η η .The axial mesons, the J P ( C ) = 1 +( − ) h and h , andthe J P ( C ) = 1 +(+) f and f , all lie quite far below theirrelevant decay thresholds, indicating that they are stable.As in the pseudoscalar-vector complex, the C = + statesshow some octet-singlet splitting owing to a significantdisconnected contribution, while the C = − states areclose to degenerate.As well as the computations in the rest frame fromwhich the hadron masses in Table III are obtained,matrices of correlation functions are also computedwith non-zero values of allowed lattice momentum, (cid:126)p = πL ( n x , n y , n z ), and from these the dispersion rela-tions, E ( | (cid:126)p | ), for the stable mesons determined – theseare found to be well described by the expected relativisticform, (cid:0) a t E (cid:126)n (cid:1) = (cid:0) a t m (cid:1) + 1 ξ (cid:18) πL/a s (cid:19) | (cid:126)n | , (3)with the fitted values of anisotropy found for each mesonbeing broadly compatible up to small variations due to discretization effects. An estimate of the anisotropy withan uncertainty that reflects the small variation over dif-ferent mesons is ξ = 3 . − (+) resonance based upon variational analysis of correlationmatrices using only fermion-bilinear constructions, alongwith the decay thresholds given in Table IV which fol-low from the masses in Table III. Also shown are theexpected octet resonance spectra with other J P (+) takenfrom Ref. [26]. These quantum numbers would contributeif spectra with non-zero overall momentum were to beconsidered, significantly complicating the analysis. Forthis reason, in this first calculation of the exotic 1 − (+) scattering system, we will restrict our attention to thespectrum in the overall rest-frame, considering the T − (+)1 irrep. We will consider the role played by 3 − (+) , 4 − (+) scat-tering, which in principle contribute in this irrep, later inthe manuscript. FIG. 2. Masses of C = + octet mesons obtained using onlysingle-meson operators (taken from Ref. [26]). Thresholdsrelevant for J P ( C ) = 1 − (+) are shown. η η ω η ω ω ω ω η η η f η h η f η J P ( C ) = 1 − (+) shown in Fig. 2. Uncertainties are determined by adding theuncertainties on the single-meson masses in quadrature. A. Operator Bases
We construct a suitable basis of operators in the T − (+)1 irrep from a set of single-meson-like operators and a setof meson-meson-like operators. A total of 18 fermionbilinears, ¯ ψ Γ ψ , are used following Ref. [23], with a spinand spatial structure built from Dirac γ -matrices andgauge-covariant derivatives. Gluonic degrees of freedomenter through the gauge-covariant derivatives. For exam-ple, one simple 1 − (+) bilinear operator, constructed usingthe vector cross product of γ i and the commutator of twoderivatives, is given by,( ¯ ψ Γ ψ ) i = (cid:15) ijk ( ¯ ψγ j ψ ) B k (cid:124) (cid:123)(cid:122) (cid:125) −− ⊗ + − → − + , (4)where B k ∝ (cid:15) kpq [ ←→ D p , ←→ D q ] is the chromomagnetic field.In practice, when we determine the spectra we vary thenumber of single-meson operators to establish insensitivityto the details of the choice of operator basis.In Table IV, we show the relevant multi-hadron thresh-olds for two- and three-meson channels that appear in1 − (+) and that transform in the flavor octet. To ensureall relevant meson-meson operators are included in theoperator basis, we calculate the non-interacting energiesfor each multi-meson system by considering all momentacombinations that sum to zero. All meson-meson opera-tors with a corresponding non-interacting energy below a t E cm = 0 .
48, a modest distance below f η threshold,are included. These operators are presented in Table V,listed by increasing non-interacting energy.The only relevant three-meson threshold, η η η , liesslightly below the expected 1 − (+) resonance position. Thelowest non-interacting three-meson energies appear at a t E (3)n.i. > .
51. As discussed in Sec. II, resonant excita-tions in two-meson subsystems may feature and operatorsthat capture these subsystem interactions need to beconsidered for inclusion. To do this we examine the ‘two-plus-one’ non-interacting energies, a t E (2+1)n.i. , which followfrom assuming no residual interaction between the inter-acting two-meson subsystem and the third meson – detailsare provided in Ref. [68]. The lowest-energy combinationof three η that appears in the T − irrep is[011] A (cid:124) (cid:123)(cid:122) (cid:125) η ⊗ [001] A (cid:124) (cid:123)(cid:122) (cid:125) η ⊗ [001] A (cid:124) (cid:123)(cid:122) (cid:125) η → [000] T − (cid:124) (cid:123)(cid:122) (cid:125) η ⊕ . . . . (5)We consider all possible meson-meson subsystems herethat could feature bound states or resonances. Combining In addition, we include an f η operator corresponding to anon-interacting level at f η threshold. A small number ofmeson-meson operators that lie a modest distance above the f η threshold were also added to explore the (very mild) sensitivityto our choice of largest energy. the first two pseudoscalar octets appearing in Eq. 5 intodefinite momentum type [001], we find the only irrepcombination that yields the T − irrep is, (cid:16) [011] A (cid:124) (cid:123)(cid:122) (cid:125) η ⊗ [001] A (cid:124) (cid:123)(cid:122) (cid:125) η (cid:17) ⊗ [001] A (cid:124) (cid:123)(cid:122) (cid:125) η → [000] T − (cid:124) (cid:123)(cid:122) (cid:125) η [001] E (cid:124) (cid:123)(cid:122) (cid:125) η η ⊗ [001] A (cid:124) (cid:123)(cid:122) (cid:125) η → [000] T − (cid:124) (cid:123)(cid:122) (cid:125) η . The irrep [001] E houses the ω and f , which we treatas stable scattering particles – any excited finite-volumeenergy level coupling to η η (in any flavor combination)will lie above the f level, and hence no three-meson-like operators are needed in the basis to study a 1 − (+) resonance near a t E ∼ . T − (+)1 irrep also featurescontributions from J P ( C ) = 3 − (+) . Considering the A − (+)2 irrep, which for J ≤ J P ( C ) = 3 − (+) sub-ductions, we can isolate the contribution from the J = 3partial-waves. We will use the finite-volume energy levelsin this irrep to constrain the J = 3 partial-waves andshow these are small over the energy range consideredhere. The operator basis used in the A − (+)2 irrep for eachlattice volume is given in Table VI. B. Finite-volume spectra
Variational analysis of matrices of T − (+)1 correlationfunctions on the six volumes leads to the spectrum pre-sented in Figure 3. Errorbars reflect the statistical un-certainty and an estimate of the systematic uncertaintyfrom varying the details of the variational analysis (suchas operator basis and fit range). For each finite-volumeeigenstate that will be used to constrain scattering ampli-tudes, we also show a histogram illustrating the overlapstrength with operators in the basis.We notice that below a t E cm ∼ .
44, the energy levels lievery close to the η η and ω η non-interacting energies,and each level has dominant overlap with just the oper-ator(s) corresponding to the particular non-interactingmomentum combination lying nearby (blue and red bars).This tends to suggest weak, uncoupled scattering at lowerenergies. The somewhat larger errorbars on levels withlarge overlap onto η η operators is a consequence of thesubstantial disconnected contribution to the η .In an energy region around a t E cm ∼ .
46 on each vol-ume we find one more energy level than expected onthe basis of the non-interacting energies, and we beginto observe levels having significant overlaps onto hybrid-like single-meson operator constructions (orange bar).This energy region is where the 1 − (+) state proposed tobe a hybrid meson was observed in the analysis using FIG. 3. Finite-volume spectrum in the T − (+)1 irrep on six lattice volumes. Points show the extracted energy levels, includinguncertainties, from a variational analysis using the operator bases in Table V; black points are included in the subsequentscattering analysis and grey points are not. Some points are slightly displaced horizontally for clarity when near-degenerateenergies appear. Curves show meson-meson non-interacting energies, with multiplicities greater than one labelled by { n } and shown as slightly split curves. Dashed curves correspond to meson-meson operators not included in the basis. Relevantthresholds transcribed from Table IV are shown on the vertical axis. Accompanying each energy level is a histogram of theoperator-state overlap factors, Z n i = (cid:104) n |O † i (0) | (cid:105) , for η η (dark blue), ω η (red), ω ω (sand), ω ω (green), f η (cyan), h η (purple) and f η (brown) meson-meson operators and a sample of ¯ ψ Γ ψ (orange) fermion-bilinear operators. The overlapsare normalized such that the largest value for any given operator across all energy levels is equal to one. For clarity, thehistograms accompanying the cluster of levels on the L/a s = 18 , ,
24 volumes are displayed at the top of the figure. L/a s = 12 L/a s = 14 L/a s = 16 L/a s = 18 L/a s = 20 L/a s = 2418 × ¯ ψ Γ ψ × ¯ ψ Γ ψ × ¯ ψ Γ ψ × ¯ ψ Γ ψ × ¯ ψ Γ ψ × ¯ ψ Γ ψη [001] η [001] η [001] η [001] η [001] η [001] η [001] η [001] η [001] η [001] η [001] η [001] f η [000] ω [001] η [001] ω [001] η [001] ω [001] η [001] ω [001] η [001] ω [001] η [001] ω [001] η [001] f η [000] f η [000] η [011] η [011] η [011] η [011] η [011] η [011] h η [000] h η [000] η [011] η [011] { } ω [011] η [011] { } ω [011] η [011] { } ω [011] η [011] f η [000] f η [000] h η [000] f η [000] ω [001] ω [001] η [111] η [111] ω [001] ω [001] f η [000] h η [000] f η [000] ω [111] η [111] { } ω [001] ω [001] { } ω [011] η [011] ω [001] ω [001] { } ω [001] ω [001] ω [001] ω [001] η [011] η [011] ω [001] ω [001] { } ω [001] ω [001] η [111] η [111] { } ω [001] ω [001] { } ω [011] η [011] { } ω [001] ω [001] f η [000] h η [000] η [002] η [002] η [111] η [111] ω [111] η [111] f η [000] ω [111] η [111] f η [000] ω [002] η [002] ω [002] η [002] h η [000] f η [000] η [012] η [012] TABLE V. T − (+)1 operator basis for each lattice volume. Meson-meson operators are ordered by increasing E n.i. and labelledwith the momentum types of the two mesons; different momentum directions are summed over as discussed in Section II. Thenumber in braces, { N mult } , denotes the multiplicity of linearly-independent meson-meson operators if this is larger than one.The maximum number of single-meson operators, N , is denoted by N × ¯ ψ Γ ψ and various subsets of these were considered toinvestigate sensitivity to the details of the choice of operator basis. L/a s = 16 L/a s = 18 L/a s = 20 L/a s = 244 × ¯ ψ Γ ψ × ¯ ψ Γ ψ × ¯ ψ Γ ψ × ¯ ψ Γ ψω [011] η [011] ω [011] η [011] ω [011] η [011] ω [011] η [011] ω [001] ω [001] ω [001] ω [001] ω [001] ω [001] η [111] η [111] η [111] η [111] η [111] η [111] ω [001] ω [001] TABLE VI. As Table V but showing the A − (+)2 operator basisfor each lattice volume, with meson-meson operators orderedby increasing E n.i. . only single-meson operators discussed earlier. The finite-volume eigenstates having overlap onto the hybrid-likeoperator are also observed to have overlap onto meson-meson constructions, notably η η (dark blue), ω η (red), f η (cyan) and/or h η (purple), which might suggest aresonance coupling to these scattering channels.A level lying very close to the two-fold degenerate ω [011] η [011] non-interacting curve is observed at each vol-ume above L/a s = 16 with a characteristic histogram thatcouples strongly to the two ω [011] η [011] operators but isdecoupled from all other operators. Such behavior wouldbe expected if the ω η { F } wave is weak.On the L/a s = 18 , ,
24 volumes, a cluster of statesappears in the energy region of interest close to the lowest ω ω (sand) and ω ω (green) non-interacting energies.The histograms for these states, presented at the top of thefigure, show that in each case there are five energies whichhave large overlap with these vector-vector operators,but not large overlap with hybrid-like operators. Thismight be taken as a suggestion that a hybrid resonance(if present) may not be strongly coupled to these vector-vector scattering channels.Finally, the only states which show any significant cou-pling to the f η (brown) operator lie at rather highenergies, suggesting that this channel is probably notrelevant to any resonance near a t E cm ∼ . A − (+)2 irrep.It is clear from the histograms, which are dominated ineach case by a single meson-meson-like operator, andthe proximity of each level to the corresponding non-interacting curves, that there are only relatively weakinteractions. There is no sign of any resonant behaviourthat might be associated with a low-lying 3 − (+) state.While a qualitative discussion of the spectra like theone just presented can suggest possible features of thescattering system, a rigorous determination requires ananalysis using the coupled-channel finite-volume formal-ism described in Section II from which the t -matrix canbe extracted, and from it properties of any resonancepoles. FIG. 4. Analogous to Figure 3 but for the A − (+)2 irrep (operatorlists shown in Table VI). Note the vertical axis is broken toemphasise the relatively low-lying η η and ω η thresholds. V. SCATTERING AMPLITUDES
We wish to use the spectra computed in the T − (+)1 and A − (+)2 irreps, presented in the previous section, todetermine the matrix describing scattering with J P ( C ) =1 − (+) . We expect T − (+)1 to be dominated by 1 − (+) , with3 − (+) , 4 − (+) , and still higher J being weak at these energies– these require higher orbital angular momentum (cid:96) and soare suppressed close to threshold in the absence of anydynamical enhancement. There is no evidence from thesingle-meson operator study in Ref. [26] of a low-lying3 − (+) resonance, and while 4 − (+) is non-exotic (it can beconstructed as the q ¯ q ( G ) state), Ref. [26] suggests thatsuch a state lies at a t E cm ∼ .
58, far above our regionof interest. By computing the A − (+)2 spectrum we areable to directly constrain the strength of scattering with J P ( C ) = 3 − (+) in the energy region of interest.The first step in analysing the finite-volume spectrumis to establish the basis of relevant meson-meson partial-waves in the considered energy region which define thematrix space in Eq. 1. The set of meson-meson channelskinematically accessible was presented in the previoussection and in Table VII we show the set of partial waveswe will use.2 − (+) η η (cid:8) P (cid:9) ω η (cid:8) P (cid:9) ω ω (cid:8) P (cid:9) , ω ω (cid:8) P , P , P (cid:9) f η (cid:8) S (cid:9) , h η (cid:8) S (cid:9) − (+) η η { F } ω η { F } ω ω { P } TABLE VII. Scattering partial waves included in the descrip-tion of T − (+)1 finite-volume spectra. A small number of possible partial waves have beenexcluded from Table VII under the expectation thatthey will not contribute significantly. In the 1 − (+) sec-tor, f η (cid:8) D (cid:9) and h η (cid:8) D (cid:9) are not included, as thethresholds for these channels are very high-lying in ourenergy region such that we expect a significant angularmomentum suppression from the D -wave, relative to theleading S -wave, that will render them practically irrele-vant. Similarly, in the vector-vector channels, we exclude ω ω (cid:8) F (cid:9) on the basis of F -wave angular momentumsuppression. In the 3 − (+) sector, ω η (cid:8) F (cid:9) is included despite thelarge angular momentum barrier. As can be seen in Ta-ble V, there are two independent operators for ω [011] η [011] and there is a corresponding two-fold degenerate non-interacting energy. In order that there be two solutionsof Eq. 1 near this energy, as observed in our computedspectra and commented on in the previous section, higher ω η partial-waves must be considered, so we include the ω η (cid:8) F (cid:9) wave along with the dominant ω η (cid:8) P (cid:9) . Wealso include η η (cid:8) F (cid:9) as the η η threshold is relativelylow compared with the resonant region, such that theangular momentum barrier may not sufficiently suppresscontributions from this higher partial-wave in the energyregion of interest. Other possible F -waves, ω ω (cid:8) F (cid:9) , ω ω (cid:8) F , F , F (cid:9) only generate additional solutions toEq. 1 at somewhat higher energies and have relativelyhigh-lying thresholds for which we expect the angularmomentum suppression to be significant. In practice wewill find that all the 3 − (+) partial waves we consider aremodest over the energy range considered, with directconstraints coming from the computed A − (+)2 spectra.The 4 − (+) sector is populated only by partial-wavesthat are F -wave or higher, all of which we assume tobe small enough as to be negligible, and none of whichgenerate additional solutions of Eq. 1 in the energy regionconsidered.One partial wave with 1 − (+) is excluded on dynamicalgrounds: f η (cid:8) S (cid:9) is observed to be completely decou-pled from the other scattering channels when operatoroverlaps (as presented in Figure 3) are examined. This Bose symmetry forbids ω ω (cid:8) P , P (cid:9) and ω ω (cid:8) F (cid:9) . leads to a natural choice of energy cutoff at a t E cm = 0 . f η threshold, and weonly use energies with no significant dependence on the f η -like operator. The levels to be used in constrainingamplitudes are shown in black in Figs. 3 and 4.The contribution of the three vector-vector partial-waves, ω ω (cid:8) P , P , P (cid:9) , which differ only in the totalcoupled intrinsic spin of the two vector mesons, to Eq. 1requires some care. In the [000] T − irrep that we areconsidering, Eq. 1 is invariant under the interchange ofany of these partial-waves, and it follows that the cor-responding rows and columns of the t -matrix cannot beuniquely determined (see also Appendix C). There is rea-son, from an approximate extension of Bose symmetry, toexpect that only amplitudes featuring ω ω (cid:8) P } couldbe significant while those with ω ω (cid:8) P , P } will bevery small. The Wick contractions for diagrams featuringthese channels differ only from those featuring ω ω bythe presence of the disconnected contribution to the ω ,but this contribution is very small (reflected in the neardegeneracy of ω , ω ). In practice we expect the ω and ω to have almost identical spatial wavefunctions, andsince ω ω (cid:8) P , P } are forbidden by Bose symmetry, weanticipate that the corresponding ω ω amplitudes willbe heavily suppressed. In fact we will observe that all vector-vector amplitudes are found to be very small overthe energy range considered.While the three-meson channel η η η becomes kine-matically accessible at the upper end of the energy regionwe are considering, we do not include such partial waves.To couple to J P ( C ) = 1 − (+) , this channel requires at least two P -waves, and since our expected resonance lies barelyabove the η η η threshold, the angular momentum sup-pression implied is expected to render the partial wavesirrelevant.We now seek to use the 61 energy levels shown in blackin Figures 3 and 4 to constrain parameterizations of the t -matrix in the partial-wave basis presented in Table VIIby solving Eq. 1. Solutions of Eq. 1 are only possible for t -matrix parameterizations which satisfy multi-channelunitarity. The simplest way to implement that constraintis to make use of the K -matrix, writing, (cid:2) t − ( s ) (cid:3) (cid:96)SJa,(cid:96) (cid:48) S (cid:48) Jb = k a ) (cid:96) (cid:2) K − ( s ) (cid:3) (cid:96)SJa,(cid:96) (cid:48) S (cid:48) Jb k b ) (cid:96) (cid:48) + δ (cid:96)(cid:96) (cid:48) δ SS (cid:48) I ab ( s ) , where K is a symmetric matrix taking real values on thereal energy axis and I ( s ) is a diagonal matrix satisfyingIm I ab ( s ) = − ρ a ( s ) above the threshold for channel a .The simplest choice is to set I ( s ) = − i ρ ( s ), but otheroptions may have better analytic properties below thresh-old and away from the real energy axis; for example,the Chew-Mandelstam prescription for which our imple-mentation is described in Ref. [64]. The K -matrix isblock-diagonal in J , reflecting the fact that total angularmomentum is a good quantum number in infinite volumeand only ‘mixes’ in a finite volume, through the matrix M , due to the reduced symmetry of the lattice.3The presence in the spectrum of an additional levelaround a t E cm ∼ .
46 and the lack of significant energyshifts at lower energies hints at a likely narrow resonancein the energy region around a t E cm ∼ .
46. This is alsoconsistent with the exotic 1 − (+) octet level seen in Figure 2.The large overlap with axial-vector pseudoscalar meson-meson operators seen in Figure 3 suggests significantcoupling to these channels, whose thresholds lie just abovethe anticipated resonant region.An efficient way to parameterize coupled-channel scat-tering when a narrow resonance appears is to use a K -matrix featuring an explicit pole. For the case ofa single channel this form of parameterisation is closelyrelated to the conventional Breit-Wigner and for coupledchannels it is related to a multi-channel Breit-Wigner,sometimes referred to as a Flatt´e amplitude in the two-channel case [86, 87]. The K -matrix can also be straight-forwardly augmented by the addition of a polynomialmatrix in s , which in the simplest case can just be aconstant matrix, that allows additional freedom beyonda pure resonance interpretation. This is crucial to testthe robustness of scattering amplitudes and allow moreflexible forms, as, for example, a pure pole parameterisa-tion exhibits the phenomenon of “trapped” levels, wherea single energy level is forced to appear between everypair of non-interacting energies – see Appendix D.In addition to varying the form of the K -matrix, thechoice of I ( s ) may also be varied. The Chew-Mandelstamprescription improves the analytic continuation belowthresholds, which is particularly useful here where, asdiscussed above, the axial-vector–pseudoscalar thresholdslie above the resonant region.In this study, we will consider a variety of parameteri-sations, finding the best description of the finite-volumespectrum for each choice, ultimately leading to compati-ble results for the amplitudes and their resonant content.As we are only using rest-frame energy levels to deter-mine the large coupled-channel scattering system (seeSection IV), we have less constraint than in previous cal-culations of simpler systems where in-flight spectra werecomputed [46, 62–68]. However, the use of six volumesappears to provide enough information to isolate most ofthe important features.The A − (+)2 spectra allow us to determine the J = 3amplitudes which provide a ‘background’ contribution tothe T − (+)1 spectra. As discussed in Section IV, there isno sign of any resonant behaviour associated with a 3 − (+) state in this energy region and the histograms in Figure 4suggest a totally decoupled system. A reasonable form ofparameterisation, capable of successfully describing thefinite-volume spectra, is a diagonal constant K -matrix, K ( s ) = γ η η { F } γ ω η { F }
00 0 γ ω ω { P } , (6) FIG. 5. As Figure 4 but including, as orange bands, the energylevels calculated from the amplitude in Eq. 6. The thicknessof the bands reflects the statistical uncertainty. The dashedcurves show the non-interacting energy levels for ω η (red), η η (blue) and ω ω (green). where the Chew-Mandelstam prescription with subtrac-tion at thresholds was used for I ( s ). The resultingfit describes the A − (+)2 finite-volume spectra with a χ /N dof = 2 . / (8 −
3) = 0 .
51, as shown in Figure 5.Other parameterisations give a compatible set of am-plitudes and quality of fit. The 3 − (+) amplitudes aremodest over the entire energy range, with the η η and ω η being mildly repulsive, and the ω ω being mildlyattractive – at a t E cm = 0 .
48 the decoupled phase shiftsreach only − ◦ , − ◦ and 5(4) ◦ respectively. A. An illustrative t -matrix parameterization We now consider the eight coupled-channel 1 − (+) scat-tering system that features in T − (+)1 . We will illustrate thescattering analysis using a single choice of amplitude pa-rameterization, and later explore variations in that choice.The properties of the illustrative amplitude choice aremotivated by the observations of the finite-volume spec-tra made in Section IV. The four vector-vector channelsappear to be decoupled for all considered energies andshow no significant energy shifts, so in this parameteri-zation we make the decoupling manifest, parameterizingthe amplitudes with a diagonal K -matrix of constants, K V V ( s ) = γ ω ω { P } γ ω ω { P } γ ω ω { P }
00 0 0 γ ω ω { P } . In some of the parameterizations we will consider, vector-vectorand non vector-vector channels are allowed to couple to eachother though their coupling to the pole term. − (+) channels, motivated by thelikely presence of a narrow resonance, we parameterizethe amplitudes using a ‘pole plus constant’ form, K (cid:8)(cid:8) V V ( s ) = g g T m − s + γ η η { P } γ ω η { P } , where g = (cid:0) g η η { P } , g ω η { P } , g f η { S } , g h η { S } (cid:1) , (7)so that all four channels are coupled to the resonanceas motivated by the histograms in Figure 3. We alsoadd a constant term in the lowest two channels as thecorresponding thresholds lie very low relative to the reso-nant region, and the close proximity of the energy levelswith the non-interacting energies low down in the spec-tra suggested a region of non-resonant behavior (see thediscussion in Section IV). We use the Chew-Mandelstamprescription for I ( s ) subtracting at the K -matrix polemass ( s = m ). The eight-channel 1 − (+) K -matrix ap-pears combined with the three-channel 3 − (+) K -matrix asgiven in Eq. 6, K ( s ) = K V V ( s ) 0 00 K (cid:8)(cid:8) V V ( s ) 00 0 K ( s ) , (8)in the finite-volume spectrum condition, Eq. 1. We min-imise the χ by varying the 11 parameters in K V V ( s )and K (cid:8)(cid:8) V V , with the parameters in K fixed accordingto the fit to the A − (+)2 lattice spectra. The resulting de-scription of the T − (+)1 spectra gives a very reasonable χ /N dof = 43 . / (53 −
11) = 1 .
04, shown in Figure 6.
FIG. 6. As Figure 5 but for the T − (+)1 irrep using the illustrativeamplitude described in Eq. 8. Plotting the resulting t -matrix elements as ρ a ρ b | t ab | ,shown in Figure 7, we can make a number of qualitativeand quantitative observations.The diagonal amplitudes for the η η , ω η , f η , h η channels are shown in Figure 7 (a) where a clear bump-likeenhancement can be seen in the η η and ω η channelsat a t E cm ∼ .
46, close to the mass obtained using onlysingle-meson operators (see Figure 2). We observe asharp turn-on of the axial-vector–pseudoscalar channels( f η , h η ) at threshold, allowed for S -wave amplitudes.The associated off-diagonal amplitudes are plotted inpanels (b), (c), (d) of Figure 7. Here, we see again abump-like enhancement in the η η → ω η amplitudeat a t E cm ∼ .
46, with the other off-diagonal amplitudesbeing mostly small with the exception of the f η → h η amplitude which shows a modest rise from threshold.The four decoupled vector-vector channels are presentedin panels (e) and (f) of Figure 7. We observe that thesingle ω ω amplitude, in the P partial-wave, is weakacross the entire energy range, consistent with our ob-servations from the finite-volume spectra in Sec IV. Forthe ω ω amplitudes, we require four partial-waves, three J P = 1 − ( P , P , P ) and one 3 − ( P ), in order toobtain the correct number of finite-volume energies atthe corresponding four-fold degenerate non-interactingenergy. As discussed in Appendix C, using only rest-frame energies does not uniquely constrain the three 1 − ω ω amplitudes and there is a freedom to permute thesechannels within the t -matrix. We therefore consider theenvelope of these three amplitudes, as determined fromthe minimisation, as our best estimate for the size ofthe ω ω { X P } amplitudes. This is shown in Figure 7panel (f) where we see that they are weak over the entirerange, consistent with the observations made in Sec IV.It is important to note that, as shown in Appendix C,energy spectra obtained in moving-frame irreps modifythe boundary conditions of the quantisation condition and do distinguish the contributions of the { P , P , P } partial-waves. As discussed in Sec IV, we do not includemoving-frame energy spectra owing to the appearanceof the relatively low-lying positive-parity resonances, asparities mix at non-zero momentum, and this would sig-nificantly complicate the analysis.For this particular parameterisation, we also exam-ine the effects of varying the stable hadron masses andanisotropy within their respective uncertainties, as givenin Sec. IV, to get an estimate of the systematic uncertain-ties on the amplitudes. As a very conservative approach,we repeat the χ -minimisation procedure using the ex-tremal values m i → m i + δm i and ξ → ξ − δξ , and vice-versa, in the evaluations of the finite-volume functionsand momenta. These combinations yield the largest devi-ations in the non-interacting energies, and therefore thelargest energy shifts away from the non-interacting valueswhich ultimately constrain the scattering parameters.For the J P = 3 − amplitudes, we find that varyingthe anisotropy yields the largest systematic uncertain-ties. The rather weak interactions in this system lead5 FIG. 7. (a):
Diagonal t -matrix elements, plotted as ρ a ρ b | t ab | , for the illustrative amplitude presented in Eq. 8 for nonvector-vector channels: η η (cid:8) P (cid:9) , ω η (cid:8) P (cid:9) , f η (cid:8) S (cid:9) and h η (cid:8) S (cid:9) . Shaded bands reflect statistical uncertainties on thescattering parameters. (b), (c), (d): As above, but for the off-diagonal amplitudes between the four channels displayed above. (e), (f ):
Diagonal vector-vector amplitudes: ω ω (cid:8) P (cid:9) , ω ω (cid:8) P , P , P (cid:9) . As discussed in the text, the ω ω partial-wavesare indistinguishable and are combined in a single plot, labelled ω ω { X P } , the shaded band reflecting an envelope over thestatistical uncertainties for each partial-wave. . The quality of fits un-der these systematic variations also became rather poor: χ /N dof = 2 . , for m i → m i + δm i and ξ → ξ − δξ ,and χ /N dof = 4 .
82 for m i → m i − δm i and ξ → ξ + δξ ,which likely indicates that this approach is overly con-servative. Nevertheless, we find all J P = 3 − amplitudesremain small over the entire energy region considered.Regarding the J P = 1 − amplitudes, having fixed the(newly determined) J P = 3 − parameters, we find theeffects of varying the masses and anisotropy are muchsmaller relative to those for J P = 3 − , as expected in amore strongly interacting system. There are some mod-est variations in the amplitudes, but these are broadlywithin the statistical uncertainties and certainly withinthe differences we will see in the subsequent variation inthe parameterisation. For example, we find the peak ofthe bump-like enhancements in the η η and ω η ampli-tudes are consistent in height and only slightly displacedin energy (higher or lower depending upon the sign ofthe systematic variations). This will be reflected in theposition of a pole singularity of the t -matrix which variesat a level comparable to the statistical uncertainty.A larger source of uncertainty arises when we considervarying the form of parameterisation, to which we nowturn. B. Parameterization variations
In order to determine the extent to which the amplituderesults presented in Figure 7 are a unique description ofthe scattering system, we try a number of parameteriza-tions, attempting to describe the finite-volume spectrumwith each choice. Variations in the K -matrix includeallowing energy-dependence in the numerator of the pole-term, and changes in the polynomial matrix added to thepole. The prescription used for I ( s ) is also adjusted, whilemaintaining coupled-channel unitarity in all parameteri-zations. We retain 27 parameterizations which are ableto describe the finite-volume spectra with χ /N dof ≤ . η η { P } and ω η { P } , shown in Figure 8, we see abump-like enhancement around a t E cm ∼ .
46 for the ma-jority of parameterisations, but we note that it is possibleto describe our finite-volume spectra without seeing sucha clear bump. We will revisit this observation when weexamine the pole singularities of the t -matrix and the This effect was observed previously in ρπ isospin-2 scatteringwhere the very small interactions meant that the systematicuncertainties dominated over the statistical ones [67]. A full description of each of these parameterisations is providedin the Supplemental Material.
FIG. 8. Diagonal t -matrix elements, plotted as ρ a ρ b | t ab | ,for each parameterisation successfully describing the finite-volume spectra as discussed in the text, for non vector-vectorchannels: η η (cid:8) P (cid:9) , ω η (cid:8) P (cid:9) , f η (cid:8) S (cid:9) and h η (cid:8) S (cid:9) .Shaded bands reflect statistical uncertainties on the illustrativeamplitude shown in Figure 7. FIG. 9. As Figure 8 but for off-diagonal η η { P } → ω η { P } , f η { S } , h η { S } amplitudes. FIG. 10. As Figure 9 but for ω η { P } → f η { S } , h η { S } amplitudes. FIG. 11. As Figure 9 but for the f η { S } → h η { S } amplitude. FIG. 12. As Figure 8 but for vector-vector channels: ω ω (cid:8) P (cid:9) , ω ω (cid:8) X P (cid:9) amplitudes. corresponding couplings. For the remaining two diago-nal amplitudes in channels f η { S } and h η { S } , weobserve that the relatively sharp turn-on at thresholdis a quite general feature, with only the magnitude ofthe effect varying somewhat. That there should be someparameterization dependence here should not come as toomuch of a surprise given the relatively small number offinite-volume energy levels constraining the amplitudesabove the axial-vector–pseudoscalar thresholds.The off-diagonal amplitude, η η { P } → ω η { P } ,shown in Figure 9, typically features a bump-like en-hancement around a t E cm ∼ .
46, but as for the diagonalentries, it is possible to describe the spectra without sucha bump and indeed without any coupling between thesetwo channels. The remaining off-diagonal amplitudes re-main modest under parameterization variation and areshown in Figures 9 – 11.The vector-vector amplitudes shown in Figure 12 havethe same qualitative behavior as in the illustrative ex-ample presented previously. The small bump around a t E cm ∼ .
46 for ω ω { P } → ω ω { P } on a smallnumber of parameterisations reflects allowing freedom forthis channel to couple to the K -matrix pole – it is observedto be a very weak effect and is statistically compatiblewith zero.Collectively, these parameterisation variations tell usthat the limited number of rest-frame energy levels withwhich we are constraining the large number of coupledchannels is not sufficient to completely uniquely determinethe t -matrix. Nevertheless, behavior consistent with asingle resonant enhancement can typically be seen in the η η { P } and ω η { P } amplitudes. We will find thateven those parameterisations that do not appear to showsignificant enhancement in either η η { P } or ω η { P } still feature a nearby resonance. The rapid turn-on of theaxial-vector–pseudoscalar amplitudes will prove to be dueto a large coupling of this resonance to one or both ofthese channels.In order to demonstrate the presence of a resonance, wewill now examine the amplitudes presented in this sectionat complex values of s = E cm where a pole singularity isexpected to feature. VI. RESONANCE POLE SINGULARITIES
At each meson-meson threshold, unitarity necessitatesa branch-point singularity and the corresponding branch-cut divides the complex s -plane into two Riemann sheets.For the system we are considering, there are six relevantkinematic thresholds and hence a total of 64 Riemannsheets. The physical sheet, the sheet on which physicalscattering occurs just above the real energy axis, is identi-fied by all scattering momenta having positive imaginaryparts, i.e. Im( k ( a ) cm ) > a ). Sheets withother sign combinations of the imaginary component ofmomenta are called unphysical , and it is on these sheetswhere pole singularities corresponding to resonances areallowed to live as complex-conjugate pairs away from thereal axis.In each energy region between thresholds, the unphys-ical sheet closest to the region of physical scattering,has Im( k ( a ) cm ) < k ( a ) cm ) > proximal sheet,and a nearby pole singularity on the proximal sheet willhave a significant impact on physical scattering.For brevity, sheets are labelled as an ordered list ofsix signs, where the order reflects increasing thresholdenergies ( η η , ω η , ω ω , ω ω , f η , h η ), and thesign reflects the imaginary component of momenta forthat channel. For example [+ + + + ++] represents thephysical sheet, and [ − − + + ++] represents the proximalsheet for scattering above the ω η threshold, but belowthe ω ω threshold.The position of pole singularities can be related toconventional pictures of meson states. Poles on the realaxis below the lowest threshold on the physical sheetcorrespond to stable bound states , while poles in thatlocation on unphysical sheets are virtual bound states that do not appear as asymptotic particles. Poles offthe real axis on unphysical sheets are associated with resonances , and it is common to interpret the real andimaginary components of the pole position s in terms ofthe mass m R and width Γ R , via √ s = m R ± i Γ R . Nearthe pole, the t -matrix takes the form, t (cid:96)SJa,(cid:96) (cid:48) S (cid:48) Jb ∼ c (cid:96)SJa c (cid:96) (cid:48) S (cid:48) Jb s − s where the factorized residues give access to c (cid:96)SJa , whichare interpreted as complex-valued resonance couplings forthe channel a in partial-wave S +1 (cid:96) J .The amplitudes presented in Sec. V suggest a likelyresonance with real energy a t √ s ∼ .
46, in which casethe proximal sheet is [ − − − − ++]. Indeed, for every Causality forbids poles on the physical sheet off the real axis, andany amplitudes featuring such singularities close enough to thereal axis to have a non-negligible effect should be discarded asunphysical. -0.02-0.01 0.43 0.44 0.45 0.46 0.47 FIG. 13. Pole singularities on the proximal sheet for all successful parameterisations as described in the text. Error bars reflectthe statistical uncertainties on the pole position for each parameterisation. parameterisation which successfully describes the finite-volume spectrum, we find a complex-conjugate pair ofpoles on the proximal sheet whose real energy is in theneighborhood of the anticipated mass and which has onlya small imaginary energy . For the illustrative amplitudegiven by Eq. 8, the poles on the proximal sheet lie at a t √ s [ −−−− ++] = 0 . ± i . , (9)where the uncertainty is statistical. Based upon the varia-tion of scattering hadron masses and anisotropy describedin Sec. V, an additional conservative systematic errorcould be added of similar size to the statistical error.For each of the parameterisations found to successfullydescribe the finite-volume spectrum, we show in Figure 13the proximal sheet pole position situated in the lowerhalf-plane. In every case, the pole is found with a smallimaginary component and hence is very close to the regionof physical scattering, strongly influencing the amplitudesat real energies. As expected there are also ‘mirror poles’distributed across some of the remaining unphysical Rie-mann sheets, but these have a negligible effect on physicalscattering by virtue of lying further away.While it is clear that a nearby pole is required to de-scribe the finite-volume spectra, the channel couplingswhich come from the factorized residues of this pole arenot uniquely determined across different parameteriza-tions. We find that the couplings of the pole to the η η { P } and ω η { P } channels are small relative toa large value of the coupling to h η { S } and in somecases a large value of the coupling to f η { S } .Focusing on the axial-vector–pseudoscalar channels, weisolate two classes of results across our range of parameter-ization forms, one in which the coupling to f η { S } is For a few parameterisations, the pole is found to lie on the realaxis below f η threshold. These parameterizations are thosewhich decouple the resonance from the η η , ω η , ω ω and ω ω channels, such that the pole describes a stable bound statein a coupled f η , h η system. large, of comparable size to a large coupling to h η { S } ,and a second in which the coupling to f η { S } is small.The couplings for these two classes are shown in the topand middle panels of Fig. 14. Their sizes are governedlargely by the corresponding g -parameters in the numera-tor of the pole term in the K -matrix, as given in Eq. 7.For a range of parameterisations that allow both of these g -parameters to freely vary, we find that the ratio of thecorresponding couplings is of order one, with both foundto be significantly non-zero – these are shown in the toppanel of Fig. 14.We also find a number of parameterisations wherethe f η { S } coupling is negligibly small whilst the h η { S } coupling remains large, and parameterizationsin which the coupling of the resonance to f η { S } isset to be exactly zero are also capable of describing thefinite-volume spectra. This class of results are shown inthe middle panel of Fig. 14.Parameterizations in which the coupling of the reso-nance to the h η { S } channel is fixed to zero are foundto be incapable of describing well the finite-volume spec-tra. They either have a poor χ , or predict additionalfinite-volume energy levels that lie very close to our energycutoff, levels for which there is no evidence in the latticecalculation.The ambiguity in the relative size of the f η { S } and h η { S } couplings can be explained in terms ofthere being only a small gap between the relevant kine-matic thresholds. These two channels both have the samepartial-wave structure ( S ), so from the point of viewof the finite-volume functions M in Eq. 1 they differonly in the mass difference between f and h . If the f and h masses were degenerate, then the quantisationcondition would be invariant under permutations of the t -matrix elements in these two channels, analogous tothe indistinguishable vector-vector amplitudes we discussin Appendix C. It follows that we are only able to dis-tinguish these channels by the mass splitting of the twoaxial-vector octets, and we explore the degree to whichthe finite-volume spectra are sensitive to different reso-nance couplings in a toy-model in Appendix E. In this9 -0.050.05 0.05 0.10 0.15 0.20 0.25 0.45-0.050.05 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45-0.050.05 0.05 FIG. 14. Couplings corresponding to the pole singularities shown in Fig. 13 as described in the text. Error bars reflect thestatistical uncertainties on each coupling for each parameterisation. Shaded bars show ranges and upper limits on the couplingsdescribed in the text.
Top:
Couplings to non vector-vector channels for parameterisations where the f η { S } coupling wasfound to be significantly non-zero. Middle:
As top but for parameterisations where the f η { S } coupling was found to bezero or fixed to be identically zero. Bottom:
Couplings to vector-vector channels, ω ω , ω ω . model, the scattering system is simplified to a two-channel( f η { S } , h η { S } ) case with a bound-state pole ly-ing below both thresholds. We find the finite-volumespectra in the rest-frame constrain very well the sum ofthe squared couplings, but offer relatively little constrainton the ratio of the coupling strengths. An energy levelthat is sensitive to the ratio lies between the two thresh-olds, but because the thresholds are so close together, thislever-arm is not large.In summary, while we can confidently state that the h η { S } coupling is large, the constraints from the finite-volume spectra can allow the f η { S } coupling to beas small as zero.Examining the η η { P } coupling in Fig. 14, we findthis to be small compared with h η { S } . There is aclear preference for a value close to 0.04, but there areparameterizations capable of describing the finite-volumespectra in which this coupling is set to be zero. Thecoupling to ω η { P } shows a very similar behavior.Finally, for the vector-vector channels, we find the ω ω { P } coupling shows signs of being small but non-zero on some parameterisations, but again the finite-volume spectra can be equally well described with thiscoupling set to zero. The ω ω { X P } couplings are negli-gibly small on every parameterisation and again we findperfectly reasonable descriptions of the spectra when theseare set to exactly zero.Given this discussion, we summarize the behavior ofthe couplings in Figure 14 with the following best esti- mates, which we suggest are a conservative reflection ofallowed ranges or limits taking into account statisticaluncertainties and parameterization variations, | a t c η η { P } | = 0 → . | a t c ω η { P } | = 0 → . | a t c ω ω { P } | = 0 → . | a t c ω ω { X P } | (cid:46) . | a t c f η { S } | = 0 → . | a t c h η { S } | = 0 . → . . (10)The upper limit for | a t c ω ω { X P } | reflects the preferredzero value of this coupling, while the other couplings showevidence that they scatter around some non-zero value –see Figure 14. These ranges and upper limit are shown bythe shaded bars in the figure. Similarly, a best estimateof the pole position is given by a t √ s = 0 . ± i . . (11)The small total width of the resonance, despite the largecoupling to h η , is explained by there being no phasespace for this sub-threshold decay.The results presented in this section describe a verynarrow exotic 1 − (+) resonance that appears in a version ofQCD where the u, d quarks are as heavy as the physical s quark. We will now discuss an interpretation of theseresults, aiming to provide a description of the π resonanceat the physical light-quark mass.0 VII. INTERPRETATION
In this section we will discuss what can be learnedfrom the observation of a J P ( C ) = 1 − (+) resonance at theSU(3) flavor point as presented above. As discussed inSection III, we choose to focus our interpretation on theisovector member of the SU(3) octet, the π . We will at-tempt to infer possible properties of this resonance at thephysical light-quark mass by performing a crude extrap-olation, making use of the JPAC/COMPASS candidatestate mass [18] to set the relevant decay phase-spaces. Wewill compare our results to existing predictions for hybridmeson decay properties made in models.In order to present results in physical units, we mustset the lattice scale using a physically measured quantity,an approach which is necessarily ambiguous, particularlygiven that we are far from the physical u, d masses. As inprevious publications, we choose to use the Ω-baryon massas a quantity which should not have a strong dependenceon the u, d quark masses. Calculated on the L/a s = 16lattice, we find a t m Ω = 0 . . a − t = 4655 MeV. Thisscale setting yields stable hadron masses of m ( η ) = 688(1) MeV m ( η ) = 939(5) MeV m ( ω ) = 1003(1) MeV m ( ω ) = 1012(1) MeV m ( f ) = 1491(3) MeV m ( h ) = 1523(3) MeV . The η resonance pole described in the previous sec-tion when expressed in physical units has a mass, m R = 2144(12) MeV, and a width, Γ R = 21(21) MeV,and the couplings to meson-meson channels are (cid:12)(cid:12) c η η { P } (cid:12)(cid:12) = 0 →
256 MeV (cid:12)(cid:12) c ω η { P } (cid:12)(cid:12) = 0 →
279 MeV (cid:12)(cid:12) c ω ω { P } (cid:12)(cid:12) = 0 →
93 MeV (cid:12)(cid:12) c ω ω { X P } (cid:12)(cid:12) (cid:46)
93 MeV (cid:12)(cid:12) c f η { S } (cid:12)(cid:12) = 0 →
978 MeV (cid:12)(cid:12) c h η { S } (cid:12)(cid:12) = 978 → , where we have given an upper bound on the magnitudeof ω ω { X P } to acknowledge the preferred value of zerocoupling to this channel.These results can be viewed in the context of past pre-dictions for the decays of hybrid mesons made withinmodels. In both flux-tube breaking pictures and bag-models, decays to meson pairs in which one meson has q ¯ q in a P -wave and the other has q ¯ q in an S -wave areenhanced over cases where both mesons have q ¯ q in an S -wave [2–4, 13]. In this particular case, that would sug-gest dominance of f η , h η over η η , ω η , ω ω , ω ω , which appears to be borne out in the couplings found inour QCD calculation.We can explore some aspects of this observation byconsidering generic properties of correlation functionshaving a hybrid meson interpolator at the source and ameson-meson-like operator at the sink, following argu-ments along the lines of those given by Lipkin [90], whichwere later placed in a limited field-theoretic framework bythe “Field Symmetrization Selection Rules” (FSSR) [91].For the decay of an SU(3) F octet into either an octet-octetpair or an octet-singlet pair, the possible Wick contrac-tions are shown in Figure 15. In the case of decays ofa 1 − (+) octet to a pair of identical octet mesons, if thespin+space configuration of the meson-meson pair is anti-symmetric, from Bose symmetry the flavor configurationmust be antisymmetric, but this would have the wrong C -parity as discussed in Section III and Appendix B, andthe correlation function is therefore zero. Examples ofsuch decays that are not allowed include η η { P } and ω ω { P , P } . A non-trivial implication in the SU(3)limit is for octet-singlet meson pairs. For example, inprinciple all of ω ω { P , P , P } can have a non-zerocoupling to the η , but the fact that the disconnectedcontributions to the ω are very small (see Section III)renders the diagram D small, leaving only diagram C . Asthe spatial q ¯ q wavefunction of the ω is expected to bevery similar to that of the ω , we can anticipate that theantisymmetric combinations P and P from diagram C will be small, while the symmetric combination P neednot be suppressed. That the ω ω { P } and ω ω { P } couplings prove to be small appears to be due to dynamicsthat go beyond simple symmetry arguments.In the case of η η { P } , if the spatial q ¯ q wavefunctionsof the η and η were the same, diagram C would be zeroowing to the antisymmetry of P . In our calculation theoptimized single-meson operators are constructed usingthe same fermion bilinear basis for both the octet andsinglet, and we find that essentially the same optimallinear superposition is present for the η and the η ,suggesting that they have similar spatial wavefunctions.However, even if diagram C is heavily suppressed, thereremains diagram D which can be significant in this caseowing to the large disconnected contribution to the η (which generates the mass splitting between the η andthe η ). FIG. 15. Wick contraction topologies for → ⊗ (left) and → ⊗ (right). A. Flavor decomposition of the SU(3) amplitudes
This is the first determination of the couplings of anexotic J P C resonance to its decay channels within a firstprinciples approach to QCD, but of course it has beendone with u, d quarks that are much heavier than thosein nature. In order to predict how this resonance wouldappear experimentally, we have to make a large extrap-olation down to the physical light-quark mass. We willattempt this in a crude way, by assuming that the polecouplings are quark-mass independent except for a factorof the angular momentum barrier, k (cid:96) , evaluated at theresonance mass. To obtain this factor, and to determinethe relevant phase-space, we require the mass of the π at the physical light-quark mass. Given that we do nothave a calculation of this, we use the experimental can-didate mass, 1564 MeV, found in the JPAC analysis ofCOMPASS data [18], and we also consider a window ofmasses between 1500 MeV and 1700 MeV.In order to extrapolate to the physical light-quark mass,we need to break the SU(3) flavor symmetry present in ourcalculation. We will retain isospin symmetry. Because theneutral flavorless mesons can now become admixtures ofoctet and singlet, we will have to introduce mixing angles,which we will take from phenomenological descriptions ofexperimental data. We will first break up the SU(3) octetsinto their component states, making use of the SU(3)Clebsch-Gordan coefficients provided in Ref. [83]. As anexample, for the decays of the π +1 , the I = 1 , I z = +1member of the octet, into a vector-pseudoscalar pair wewould have the combination, √ (cid:0) π + ρ − π ρ + (cid:1) + √ (cid:0) K + K ∗ − K K ∗ + (cid:1) , such that the relevant couplings would be, (cid:12)(cid:12) c ( π → πρ ) (cid:12)(cid:12) = (cid:113) (cid:12)(cid:12) c ω η (cid:12)(cid:12) , (cid:12)(cid:12) c ( π → KK ∗ ) (cid:12)(cid:12) = (cid:113) (cid:12)(cid:12) c ω η (cid:12)(cid:12) , where the additional factor of √ (cid:12)(cid:12) c (cid:12)(cid:12) phys = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k phys ( m phys R ) k ( m R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:96) (cid:12)(cid:12) c (cid:12)(cid:12) . (12)This approach is motivated by observations made inlattice calculations of the decays of b → ωπ domi-nantly in S -wave [68], ρ to ππ in P -wave [46], K ∗ to Kπ in P -wave [92] and f , f (cid:48) decays to ππ and KK in D -wave [66], which appear to show quark-mass indepen-dence when treated this way. For example in the b case, the coupling computed in [68] at m π ∼
391 MeV is | c | = 564(114) MeV, in good agreement with the coupling | c | phys = 556(17) MeV extracted from the experimen-tal b decay width. In the P -wave ρ decay, an explicitfactor of k is required for the scaling to work, as pre-sented in Ref. [46]. In addition, as shown in Fig. 4 ofRef. [92], the K ∗ coupling scaled in this way is approx-imately constant for four different light-quark massescorresponding to m π = 239 MeV to 391 MeV, even whenthe K ∗ is a shallow bound state, and is in agreement withthe experimentally-measured coupling. Scaling the f , f (cid:48) D -wave couplings computed at m π ∼
391 MeV in [66]gives, in comparison to PDG-extracted values,scaled PDG (cid:12)(cid:12) c ( f → ππ ) (cid:12)(cid:12) +9 − (cid:12)(cid:12) c ( f → KK ) (cid:12)(cid:12) (cid:12)(cid:12) c ( f (cid:48) → ππ ) (cid:12)(cid:12) (cid:12)(cid:12) c ( f (cid:48) → KK ) (cid:12)(cid:12) .Using the couplings scaled to the physical quark mass,we can estimate partial widths for decay into kinematicallyopen channels using the approach presented in the PDGreview [89] where the real part of the pole position is usedto determine the phase-space inΓ( R → i ) = (cid:12)(cid:12) c phys i (cid:12)(cid:12) m phys R · ρ i ( m phys R ) . (13)Summing up all non-zero partial widths, we can obtainan estimate for the total width . We will consider eachconstrained decay channel in turn, beginning with η η . • η η { P } : For ⊗ → , we have only, trivially, η π + , where η is the only member of the SU(3) singlet.Because η η is forbidden in P by Bose symmetry, nocomponents of the form η π + can appear, where η is theflavorless, neutral member of the octet. The η and η are related to the physical η and η (cid:48) states via a mixingangle θ P , (cid:18) η η (cid:19) = (cid:18) cos θ P sin θ P − sin θ P cos θ P (cid:19) (cid:18) ηη (cid:48) (cid:19) , (14)where phenomenological estimates for θ P place it close to − ◦ [89, 94–96]. The couplings of the π to ηπ and η (cid:48) π then follow, (cid:12)(cid:12) c ( π → ηπ ) (cid:12)(cid:12) = (cid:12)(cid:12) c η η sin θ P (cid:12)(cid:12) , (cid:12)(cid:12) c ( π → η (cid:48) π ) (cid:12)(cid:12) = (cid:12)(cid:12) c η η cos θ P (cid:12)(cid:12) . An additional quark-model ‘form-factor’ as part of the scaling isadvocated by Close and Burns [93]. Note that doing this at the SU(3) point using the couplings inEq. 14 gives a total width in the range 0 →
45 MeV, in reasonableagreement with our best estimate for the width from the poleposition, 21(21) MeV. η η suggests that the η (cid:48) π couplingshould be around six times larger than the coupling to ηπ , independent of the particular value of c η η . • ω η { P } : For the vector-pseudoscalar channel, therelevant flavor embedding is ⊗ → , and the compo-nents are √ (cid:0) π + ρ − π ρ + (cid:1) + √ (cid:0) K + K ∗ − K K ∗ + (cid:1) , and the corresponding couplings, accounting for a sumover charge states to be done in the partial width calcula-tion are (cid:12)(cid:12) c ( π → ρπ ) (cid:12)(cid:12) = (cid:113) (cid:12)(cid:12) c ω η (cid:12)(cid:12) , (cid:12)(cid:12) c ( π → K ∗ K ) (cid:12)(cid:12) = (cid:113) (cid:12)(cid:12) c ω η (cid:12)(cid:12) . • ω ω { P } , ω ω { X P } : The ω ω and ω ω vector-vector channels must be considered together. Unlike the η η channel forbidden in P , the non-trivial spin cou-pling in ω ω means that the P is in a totally symmetricconfiguration and thus not forbidden – see Appendix B.This means the corresponding components for ω ω and ω ω in P both feature ρω and ρφ . For ⊗ → ,the ω ω components are − (cid:113) (cid:0) K ∗ + K ∗ + K ∗ K ∗ + (cid:1) + √ (cid:0) ρ + ω + ω ρ + (cid:1) = − (cid:113) K ∗ + K ∗ + 2 (cid:113) ω ρ + , and trivially the only component of ω ω is ω ρ + .The ω , ω mixing to give ω , φ is well known to bevery different to the pseudoscalar case, with the ω beingdominantly √ (cid:0) u ¯ u + d ¯ d (cid:1) and φ dominantly s ¯ s . Usingthe same conventions as Eq. (14) with η → ω , η (cid:48) → φ ,this ‘ideal’ mixing would correspond to a mixing angleof θ V ≈ − . ◦ . A mixing angle of θ V ∼ − ◦ , extractedfrom a model fit describing experimental vector to pseu-doscalar radiative transitions [95], is in good agreementwith this (see also [89]). It follows that the ωρ , φρ cou-plings for P are, (cid:12)(cid:12) c ( π → ωρ { P } ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) (cid:113) (cid:12)(cid:12) c ω ω { P } (cid:12)(cid:12) cos θ V − (cid:12)(cid:12) c ω ω { P } (cid:12)(cid:12) sin θ V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c ( π → φρ { P } ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) (cid:113) (cid:12)(cid:12) c ω ω { P } (cid:12)(cid:12) sin θ V + (cid:12)(cid:12) c ω ω { P } (cid:12)(cid:12) cos θ V (cid:12)(cid:12)(cid:12)(cid:12) . Allowing a range − ◦ to − ◦ suggests an η (cid:48) π coupling threeto six times the ηπ coupling, in good agreement with a ratio of3 . These expressions are consistent with the expectations ofthe OZI rule: if the disconnected diagram, D in Fig. 15,vanishes and ω , φ mixing is ideal, c ( π → φρ { P } ) = 0and c ω ω { P } = (cid:113) c ω ω { P } . The coupling to kaonsis (cid:12)(cid:12) c ( π → K ∗ K ∗ { P } ) (cid:12)(cid:12) = 2 (cid:113) (cid:12)(cid:12) c ω ω { P } (cid:12)(cid:12) . For P and P , ω ω is forbidden by Bose symmetryand the only contribution comes from the ω ω . Thecorresponding couplings are therefore, (cid:12)(cid:12) c ( π → ωρ { P , P } ) (cid:12)(cid:12) = (cid:12)(cid:12) c ω ω { P , P } sin θ V (cid:12)(cid:12)(cid:12)(cid:12) c ( π → φρ { P , P } ) (cid:12)(cid:12) = (cid:12)(cid:12) c ω ω { P , P } cos θ V (cid:12)(cid:12) . These couplings are expected to be very small because onlythe disconnected diagram contributes to these decays. • f η { S } , h η { S } : Similar to ω ω , f η embedsin and decomposes into, − (cid:113) (cid:0) K +1 A K + K A K + (cid:1) + √ (cid:0) a +1 η + ( f ) π + (cid:1) , where we see the neutral, flavorless members of the pseu-doscalar and 1 +(+) octets, the η and ( f ) , and thestrange members of the 1 +(+) octet, K A . We have notincluded the f η channel in the scattering calculation,given that this was largely decoupled in our observationsof the finite-volume spectra in Sec. IV, and we thereforeassume here that the f η coupling is zero.The mixing of ( f ) and ( f ) to form the physicalstates f (1285) and f (1420) can be determined from theradiative decays of the f (1285) to γρ and γφ , whichsuggests a mixing angle of θ A ∼ − ◦ , following the for-malism presented in [97], using the PDG averages [89], andusing the same conventions as Eq. (14) with η → f (1285), η (cid:48) → f (1420) (see also Ref. [98]). The corresponding cou-plings in decays involving the non-strange 1 +(+) mesonsare (cid:12)(cid:12) c ( π → a η ) (cid:12)(cid:12) = √ (cid:12)(cid:12) c f η cos θ P (cid:12)(cid:12)(cid:12)(cid:12) c ( π → a η (cid:48) ) (cid:12)(cid:12) = √ (cid:12)(cid:12) c f η sin θ P (cid:12)(cid:12)(cid:12)(cid:12) c ( π → f (1285) π ) (cid:12)(cid:12) = √ (cid:12)(cid:12) c f η cos θ A (cid:12)(cid:12)(cid:12)(cid:12) c ( π → f (1420) π ) (cid:12)(cid:12) = √ (cid:12)(cid:12) c f η sin θ A (cid:12)(cid:12) . The other axial-vector–pseudoscalar channel, h η , em-beds in and has components, √ (cid:0) K +1 B K − K B K + (cid:1) + √ (cid:0) b +1 π − b π + (cid:1) , where K B are the strange members of the 1 +( − ) octet.The coupling to b π is then (cid:12)(cid:12) c ( π → b π ) (cid:12)(cid:12) = (cid:113) (cid:12)(cid:12) c h η (cid:12)(cid:12) . The physical axial-vector kaons, the K (1270) and K (1400), are not eigenstates of charge-conjugation and3can be considered to be admixtures of the K A from the1 +(+) octet and the K B from the 1 +( − ) octet. This mixing,in terms of an angle θ K , can be defined through (cid:18) K B K A (cid:19) = (cid:18) cos θ K − sin θ K sin θ K cos θ K (cid:19) (cid:18) K (1270) K (1400) (cid:19) , (15)which is consistent with the conventions in Ref. [99].There is not a clear consensus on the value of θ K , but itcould be as large as ∼ ◦ . In practice there is onlydependence on this mixing angle if the decay to the K (1270) K channel is open – this requires the π tohave a mass above 1747 MeV, significantly heavier thanthe JPAC/COMPASS candidate. B. Partial widths for a π (1564) Combining the flavor decompositions in the previoussection with the scaling given by Eq. 12 we obtain thecouplings for a 1564 MeV π presented in Table VIII. Using these couplings, we populate Table VIII with partialwidths determined using Eq. 13. We assume that thesubsequent decays of unstable isobars (e.g. ρ , b ) factorizefrom the initial π decays given in the table.It is clear that the dominant decay mode is b π , withthe next largest channels, η (cid:48) π, ρπ and f (1285) π being thr./MeV (cid:12)(cid:12) c phys i (cid:12)(cid:12) /MeV Γ i /MeV ηπ
688 0 →
43 0 → ρπ
910 0 →
203 0 → η (cid:48) π →
173 0 → b π → → K ∗ K →
87 0 → f (1285) π →
363 0 → ρω { P } (cid:46) (cid:46) . ρω { P } (cid:46) (cid:46) . ρω { P } (cid:46) (cid:46) . f (1420) π →
245 0 →
2Γ = (cid:80) i Γ i = 139 → m R = 1564 MeV. Couplingsare derived as discussed in the text and partial widths aredetermined according to the definition given in Eq. 13. Forboth couplings and partial widths we present a range calculatedfrom the corresponding SU(3) couplings, while those shownas an upper bound have a preferred value of zero. For the ωρ { P } and φρ { P } momentum scaling, where thereis a linear combination of two SU(3) couplings, we evaluate themomentum at the SU(3) point with m = m = m ω in bothcases as the mass difference between the ω and ω is negligiblysmall and it simplifies the resulting algebra. significantly smaller. Despite the larger phase space, thepartial width into ηπ is approximately ten times smallerthan η (cid:48) π , independent of the coupling and dependingonly on the mixing angle and phase space. Only onekaonic decay mode is kinematically accessible, K ∗ K , witha very small partial width. Decays to ρω are negligible.Summing all partial widths we obtain an estimate for thetotal width in the range 139 to 590 MeV which includes thevalue 492(47)(102) MeV found in the JPAC/COMPASSanalysis . If our extrapolation is accurate, it suggeststhat the observation of the π in ηπ and η (cid:48) π is throughdecays which are very far from being the dominant decaymodes.It is possible that this estimate of the total decay widthmay be missing contributions from channels which areclosed at the SU(3) point, whose couplings we have notdetermined, but which become open at physical kine-matics. Examples might include f π (although this is a D -wave decay with relatively little phase-space, so a largewidth is unlikely), or η (1295) π (a P -wave decay with avery small phase-space). Any truly multibody decays tothree or more mesons, i.e. those not proceeding througha resonant isobar, are also not included in this estimate,but the conventional wisdom is that such decays are notlarge.Figure 16 shows the partial widths for each channel inTable VIII as a function of the physical resonance mass, m phys R , allowed to vary in the range 1500 − π resonance, with the exception of the f (1420) π channelwhich becomes kinematically open in this energy range.The only prior estimate of decay rates for a π ob-tained using lattice QCD was the calculation presentedin Ref. [100] which used a rather different approach tothe one followed in this paper. By tuning the value ofthe light-quark mass in a two-flavor calculation (with-out strange quarks), the authors were able to make themass of the π be approximately equal to the sum ofthe masses of the π and the b . They argued that thetime-dependence of a single two-point function having a1 − (+) single-meson operator at the source and a b π -likeoperator at the sink can be used to infer a transition rate.The method makes a number of assumptions that havenot yet been validated, but their result for pion massesnear 500 MeV does suggest a large coupling. They alsofound a somewhat smaller coupling to f π .We can also compare our result extrapolated to phys-ical kinematics with the predictions of models. Modelsbased upon breaking of the flux-tube [4, 13] do not allowdecays to identical mesons, but these are typically pre-vented by Bose symmetry anyway. The ability of thesemodels to predict decays involving the η or η (cid:48) is somewhatquestionable given that no disconnected contributions areconsidered. Within these models, the quark spin coupling and the somewhat smaller value ∼
388 MeV found in the veryrecent analysis of COMPASS and Crystal Barrel data [19]. FIG. 16. Partial widths as a function of the π pole mass. The bands reflect the coupling ranges given in Table VIII. The totalwidth, obtained by summing the partial widths, is shown by the grey band. factorizes from the spatial matrix element such that ρπ decays are only allowed to the extent that the spatial q ¯ q wavefunctions of the π and the ρ differ. This differ-ence is quite hard to estimate in quark models where thevery light pseudo-Goldstone boson π is typically not welldescribed.If this model picture of the coupling being sensitive tothe difference between the π and ρ radial wavefunctionis correct, our simple extrapolation of the ρπ couplingmay lead to an under estimate. We can use the chargeradius as a guide to the wavefunction size, and at theSU(3) flavor symmetric point these radii were computedin Ref. [75]: (cid:104) r (cid:105) / π = 0.47(6) fm, (cid:104) r (cid:105) / ρ = 0.55(5) fm.These sizes are not that different, as one might expectgiven the heaviness of the quarks, but we expect thedifference to grow as the light-quark mass reduces. Oursimple extrapolation of the ρπ coupling would not capturethis change, and hence our ρπ partial width might be anunder-estimate.The flux-tube breaking models have larger couplings toaxial-vector–pseudoscalar channels like b π and f π thanto, for example, ρπ , but these couplings are still muchsmaller than the ones we are predicting. Bag models showsimilar decay systematics [2, 3]. VIII. SUMMARY
Prior lattice QCD calculations which treated excitedhadrons as stable particles indicated the presence of exotichybrid mesons in the spectrum, but until now the onlytheoretical information on the decay properties of thesestates came from models whose connection to QCD isnot always clear. In this paper we presented the firstdetermination of the lightest J P ( C ) = 1 − (+) resonance within lattice QCD. The resonance was observed in arigorous way as a pole singularity in a coupled-channelscattering amplitude obtained using constraints providedby the discrete spectrum of eigenstates of QCD in six different finite volumes. These spectra were extractedfrom matrices of correlation functions computed in latticeQCD using a large basis of operators.In order to make this first calculation practical we optedto work with quark masses such that m u = m d = m s ,with the quark mass selected to approximately matchthe physical strange-quark mass. The resulting SU(3) F symmetry leads to a simplified set of decay channels, andthe relatively heavy quark mass means that only meson-meson decays are kinematically accessible in the energyregion of interest.The computed lattice QCD spectra are described byan eight-channel flavor-octet 1 − (+) scattering system inwhich a narrow resonance appears, lying slightly be-low the opening of axial-vector–pseudoscalar decay chan-nels, but well above pseudoscalar–pseudoscalar, vector–pseudoscalar and vector–vector decay thresholds. Theresonance pole shows relatively weak couplings to theopen channels, hence the narrow width, but large cou-plings to at least one kinematically-closed axial-vector–pseudoscalar channel.A simple-minded approach was used to predict decayproperties of a π resonance with physical light-quarkmass from these results. We extrapolated the determinedcouplings, assuming their only adjustment is in the an-gular momentum barrier (an approach that has provenreasonably successful when applied to previous latticeQCD determinations of vector, axial-vector and tensormesons). This suggests a potentially broad π resonance,the bulk of whose decay goes into the b π mode.Comparing to the experimental π (1564) candidatestate found by the JPAC/COMPASS analysis [18], ourpredicted range of total width is compatible with theirwidth taken from the resonance pole position. We notethat the ηπ , η (cid:48) π modes in which the resonance is observedexperimentally are relatively rare decays in our picture.Although the b π decay mode is somewhat challengingexperimentally, ending up in five pions through b → ωπ ,these results suggest that it is a promising channel tosearch in.5 ACKNOWLEDGMENTS
We thank our colleagues within the Hadron SpectrumCollaboration. AJW, CET and DJW acknowledge sup-port from the U.K. Science and Technology FacilitiesCouncil (STFC) [grant number ST/P000681/1]. JJDacknowledges support from the U.S. Department of En-ergy contract DE-SC0018416. JJD, RGE and CET ac-knowledge support from the U.S. Department of Energycontract DE-AC05-06OR23177, under which JeffersonScience Associates, LLC, manages and operates JeffersonLab. DJW acknowledges support from a Royal SocietyUniversity Research Fellowship. CET and JJD acknowl-edge support from the Munich Institute for Astro- and Par-ticle Physics (MIAPP), which is funded by the DeutscheForschungsgemeinschaft (DFG, German Research Foun-dation) under Germany’s Excellence Strategy – EXC-2094– 390783311, while attending a program. CET also ac-knowledges CERN TH for hospitality and support duringa visit.The software codes
Chroma [101] and
QUDA
Appendix A: SU ( ) Clebsch-Gordan Coefficients
Unlike in SU(2), where the product of two representa-tions of definite isospin decomposes into a sum of isospinseach of which appears only once, in SU(3) a representa-tion can appear more than once in a product. A relevantexample is ⊗ = ⊕ ⊕ ⊕ ⊕ ⊕ where weobserve two octet embeddings, and .Following the conventions given in Ref. [83], the SU(3)Clebsch-Gordan coefficients, C ( . . . ), for ⊗ → , arerespectively symmetric, antisymmetric under exchangingthe hadrons in the product or conjugating the hadrons inthe product, C (cid:18) i ν ν ν (cid:19) = ξ ( i ) C (cid:18) i ν ν ν (cid:19) , C (cid:18) i ν ν ν (cid:19) = ξ ( i ) C (cid:18) i − ν − ν − ν (cid:19) , with ξ (1) = ξ (1) = 1 and ξ (2) = ξ (2) = −
1, and using = . Here a particular member of the octet is labelled byits isospin I , hypercharge Y , and z -component of isospin I z , in ν = ( I, Y, I z ), and for mesons the hypercharge issimply equal to the strangeness, Y = S .It is useful at this point to write out the non-zeroSU(3) Clebsch-Gordan coefficients for the two embeddingsexplicitly. As we are at liberty to work with any memberof the target octet, we choose ν = (0 , , a and b in order to distinguish them.Applying the rules given in Ref. [83], we have for thesymmetric combination, | ; 0 , , (cid:105) = √ (cid:16) (cid:12)(cid:12) a ; , , (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K ∗ + (cid:12)(cid:12) b ; , -1 , - (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K − + (cid:12)(cid:12) a ; , -1 , - (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K ∗− (cid:12)(cid:12) b ; , , (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K + − (cid:12)(cid:12) a ; , , - (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K ∗ (cid:12)(cid:12) b ; , -1 , (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K − (cid:12)(cid:12) a ; , -1 , (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K ∗ (cid:12)(cid:12) b ; , , - (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K (cid:17) − √ (cid:16) | a ; 1 , , (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) ρ + | b ; 1 , , -1 (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) π − + | a ; 1 , , -1 (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) ρ − | b ; 1 , , (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) π + − | a ; 1 , , (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) ρ | b ; 1 , , (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) π (cid:17) − √ | a ; 0 , , (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) ω | b ; 0 , , (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) η , while for the antisymmetric combination, | ; 0 , , (cid:105) = (cid:16) (cid:12)(cid:12) a ; , , (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K ∗ + (cid:12)(cid:12) b ; , -1 , - (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K − − (cid:12)(cid:12) a ; , -1 , - (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K ∗− (cid:12)(cid:12) b ; , , (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K + − (cid:12)(cid:12) a ; , , - (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K ∗ (cid:12)(cid:12) b ; , -1 , (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K + (cid:12)(cid:12) a ; , -1 , (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K ∗ (cid:12)(cid:12) b ; , , - (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K (cid:17) , where we have provided the PDG notation for vector and pseudoscalar mesons as an example, as was done in Eq. 2.Defining ˆ G in the usual way as ˆ C followed by a ro-tation by π about the y -component of isospin, ˆ R , it isstraightforward to show [105] that,ˆ C | ; I, Y, I z (cid:105) = C ( − Y/ I z | ; I, − Y, − I z (cid:105) ˆ R | ; I, Y, I z (cid:105) = ( − I − I z | ; I, Y, − I z (cid:105) ˆ G | ; I, Y, I z (cid:105) = C ( − Y/ I | ; I, − Y, I z (cid:105) , where C is the intrinsic charge-conjugation quantum num-ber of the neutral element of the octet, for example, C = +1 for η and C = − ω . There are SU(3) ana-logues of G -parity where the rotation is between the u, s or d, s quarks rather than the u, d quarks. When SU(3) is broken these are no longer good quantum numberswhereas G -parity is still good as long as there is isospinsymmetry.Acting with ˆ C or ˆ G on the decompositions above gives,ˆ C | ; 0 , , (cid:105) = ˆ G | ; 0 , , (cid:105) = + C a C b | ; 0 , , (cid:105) ˆ C | ; 0 , , (cid:105) = ˆ G | ; 0 , , (cid:105) = − C a C b | ; 0 , , (cid:105) . where C a and C b are the intrinsic charge-conjugationquantum numbers of the neutral element of the octets a and b . Therefore, and have isoscalar memberswhich are eigenstates of charge-conjugation with oppositevalues of C .In the case of ⊗ → , the SU(3) Clebsch-Gordancoefficients are symmetric under interchange – explicitlythe construction is,7 | ; 0 , , (cid:105) = √ (cid:16) (cid:12)(cid:12) a ; , , (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K ∗ + (cid:12)(cid:12) b ; , -1 , - (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K − + (cid:12)(cid:12) a ; , -1 , - (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K ∗− (cid:12)(cid:12) b ; , , (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K + − (cid:12)(cid:12) a ; , , - (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K ∗ (cid:12)(cid:12) b ; , -1 , (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K − (cid:12)(cid:12) a ; , -1 , (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K ∗ (cid:12)(cid:12) b ; , , - (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) K (cid:17) + √ (cid:16) | a ; 1 , , (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) ρ + | b ; 1 , , -1 (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) π − + | a ; 1 , , -1 (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) ρ − | b ; 1 , , (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) π + − | a ; 1 , , (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) ρ | b ; 1 , , (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) π (cid:17) − √ | a ; 0 , , (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) ω | b ; 0 , , (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) η , and ˆ C | ; 0 , , (cid:105) = ˆ G | ; 0 , , (cid:105) = C a C b | ; 0 , , (cid:105) .For the cases of ⊗ → and ⊗ → , the Clebsch-Gordan coefficients are trivial, | ; 0 , , (cid:105) = | a ; 0 , , (cid:105) | b ; 0 , , (cid:105)| ; 0 , , (cid:105) = | a ; 0 , , (cid:105) | b ; 0 , , (cid:105) , and obviously C = C a C b . Appendix B: SU ( ) Bose symmetry
A practical consequence of Bose symmetry is the elimi-nation of certain partial-wave configurations in the scatter-ing of identical mesons. A familiar example assuming onlyisospin symmetry is that ππ scattering with isospin=1is only in odd partial waves, while isospin=0,2 are onlyin even partial waves. The SU (3) Clebsch-Gordan coeffi-cients discussed in Appendix A have definite symmetryunder the exchange of the two scattering hadrons, andthis makes the application of Bose symmetry straight-forward when we need to combine two identical mesonmultiplets.Consider first identical pseudoscalar meson octets – thetotal spin S is zero and the spin wavefunction is triviallysymmetric. To ensure overall symmetry under exchangewe require the product of flavor and spatial wavefunctionsto be overall symmetric, meaning they are either bothsymmetric or both antisymmetric. In Appendix A weshowed that and are symmetric and antisymmetricin flavor respectively, so we deduce that only partial wavesof even (cid:96) are permitted in and odd (cid:96) in . It followsthat, for example, η η appears with even (cid:96) in with J P ( C ) = (cid:96) +(+) and odd (cid:96) in with J P ( C ) = (cid:96) − ( − ) . Aconsequence is that η η is forbidden in decays of an J P ( C ) = 1 − (+) octet resonance.For identical vector meson octets, the symmetry of thespin wavefunction depends on the total spin S : symmetricfor S = 0 , S = 1. It follows thatfor S = 0 ,
2, the product of flavor and spatial wavefunc-tions must be totally symmetric, so either they are bothsymmetric or both antisymmetric, similar to the caseabove – only even (cid:96) partial waves are permitted in ,while only odd (cid:96) appear for . In the case of S = 1, byan analogous argument, only partial waves of odd (cid:96) arepermitted in and even (cid:96) in . Hence ω ω is forbidden ( C = +) ( C = − ) S , D , G , . . . P , F , . . . P , , , F , , , . . . S , D , , , G , , , . . . S , D ... , G ... , . . . P , , , F ... , . . . TABLE IX. Bose-allowed partial-wave content of multiplets and from a product of two identical vector meson octets, a ⊗ a , for (cid:96) ≤ in P and P decays of an J P ( C ) = 1 − (+) octet reso-nance, while it is allowed in P . Table IX summarisesthe Bose-allowed partial-wave content of and foridentical vector meson octets. Appendix C: Indistinguishable vector-vector P -waves in T − In this appendix we show that the quantization con-dition, Eq. 1, when subduced into the T − irrep at restcannot uniquely constrain the ω ω (cid:8) P , P , P (cid:9) am-plitudes owing to a residual S permutation symmetryon these channels, i.e. the corresponding scattering pa-rameters in the t -matrix can be freely interchanged whileleaving the determinant invariant. We also show that thesame permutation symmetry is not present for systemswith overall non-zero momentum, so including energylevels obtained in such irreps would provide a uniqueconstraint for each of these partial waves.Recalling the form of the quantization condition,det (cid:96)SJma (cid:2) + i ρ t (cid:0) + i M (cid:1)(cid:3) = 0 , we note that the finite-volume nature of the problemresides in the matrix M whose components are definedexplicitly in App. A of Ref. [79]. M is trivially diagonalin hadron channel and intrinsic spin, leading to it beingdiagonal in ω ω (cid:8) P , P , P (cid:9) channels. The reasonthat these channels cannot be distinguished at overallzero momentum is that the diagonal entries of M in eachof ω ω (cid:8) P , P , P (cid:9) are equal . It is also diagonal in the ω ω (cid:8) P , P (cid:9) subspace at rest. (cid:82) d Ω Y ∗ m (cid:96) Y ∗ ¯ (cid:96)m (cid:96) Y m (cid:48) (cid:96) , it is clear that only ¯ (cid:96) ≤ Z (cid:126) (cid:96)/ ∈ Z ,m/ ∈ Z = 0 , Z (cid:126) = 0 , only ¯ (cid:96) = 0 , m (cid:96) = 0 survives. The elements of M thusreduce to the rather simple form, M (cid:0) S +1 P , m ; S (cid:48) +1 P , m (cid:48) (cid:1) = δ S,S (cid:48) δ m,m (cid:48) πk c (cid:126) , ( k ; L ) , and it follows that the rest-frame M does not distinguishbetween the ω ω (cid:8) P , P , P (cid:9) channels. The resultof this is that permutations of the ω ω (cid:8) P , P , P (cid:9) channels will leave the determinant in Eq. 1 invariant.These partial waves become distinguishable if we con-sider the system at overall non-zero momentum. Followinga similar derivation to the zero momentum case, owingto Z (cid:126)P being non-zero in general, we find that elementsof M are spin dependent. For example, in the case that (cid:126)P = [00 n ], the m = +1, m (cid:48) = +1 elements are given by, M (cid:0) P , +1; P , +1 (cid:1) = πk c (cid:126)P , ( k ; L ) − √ πk c (cid:126)P , ( k ; L ) M (cid:0) P , +1; P , +1 (cid:1) = πk c (cid:126)P , ( k ; L ) + √ πk c (cid:126)P , ( k ; L ) M (cid:0) P , +1; P , +1 (cid:1) = πk c (cid:126)P , ( k ; L ) − √ πk c (cid:126)P , ( k ; L ) , where we observe that the coefficients of the c (cid:126)P , termdistinguishes the different spin configurations. Appendix D: “Trapped” levels for factorizedK-matrix poles
A parameterization in common use to describe a singlecoupled-channel resonance with angular momentum J assumes a factorized pole in the K -matrix and the simplephase space ( I a ( s ) = − iρ a ( s )) in the construction of the t -matrix, t = (cid:102) K (cid:0) − i ρ (cid:102) K (cid:1) − , (cid:2) (cid:101) K ( s ) (cid:3) (cid:96)SJa,(cid:96) (cid:48) S (cid:48) Jb = (2 k a ) (cid:96) g (cid:96)SJa g (cid:96) (cid:48) S (cid:48) Jb m − s (2 k b ) (cid:96) (cid:48) . Here we will show that this particular form can leadto the phenomenon of “trapped” levels in finite-volumespectra, a situation where there is guaranteed to be ex-actly one finite-volume energy level lying between everyneighboring non-interacting energy. In particular, we willpresent a proof of how trapped levels emerge in coupledmeson-meson scattering in S and { P , P , P } -wavein the rest frame irreps, as relevant for this study. Thiseffect is not a general feature of the finite-volume method– for example, upon adding a matrix of polynomials in s tothe K -matrix above (as we commonly do) the guaranteeis removed. The L¨uscher quantisation condition, Eq. 1, can berewritten in terms of the K -matrix defined above yieldingthe convenient form,det[ − ρ (cid:102) K M ] = 0 , where the determinant is taken over the N -dimensionalspace of hadron-hadron channels and partial waves.When (cid:102) K is factorized as above, the matrix ρ (cid:102) K M isof the form a b T for all energies, where a ( s ) and b ( s ) are(energy dependent) vectors, and hence of rank one. Ithas one non-zero eigenvalue, µ ( s ) = b T a , with eigenvec-tor, v = a , and N − µ i ( s ) = 0 for i = 1 , . . . , N −
1, whose eigenvectors span the hyperplaneorthogonal to a . It immediately follows that − ρ (cid:102) K M has exactly one eigenvalue capable of taking a zero value, λ ( s ) = 1 − b T a – all other eigenvalues λ i ( s ) = 1 for i = 1, . . . , N −
1. The finite-volume spectrum is thereforegiven by the solutions to λ ( s ) = 0.For ease of illustration, consider the case of severalcoupled meson-meson channels, each in a single partial-wave. The nontrivial eigenvalue λ ( s ) takes the form, λ ( s ) = 1 − √ s ( m − s ) (cid:88) a (2 k a ) (cid:96) g a k a M a , (D1)where M a are the elements of the diagonal in channel-space M . Recalling the definition of these presented inRef. [79], for S - and P -waves in the rest-frame, M a ( s ) = √ π k a L Z (cid:126) (cid:104) (cid:0) k a L π (cid:1) (cid:105) , independent of the intrinsic spin of the system. Theonly differences between the objects M a ( s ) for differentchannels come from the momenta k a . It is thereforeinstructive to examine the functional form of − (2 k a ( s )) (cid:96) k a ( s ) M a ( s ) , (D2)that appears for each channel in Eq. D1. We now investi-gate the consequences of this for S -wave scattering beforeconsidering P -wave scattering. S -wave scattering In Fig. 17 we plot Eq. D2 for the f η and h η S channels. These functions are real above threshold, andshow monotonic decrease between divergences at eachnon-interacting energy. The finite-volume spectrum inthis case is given by the solutions of, √ s ( s − m ) = − g k ( s ) M ( s ) − g k ( s ) M ( s ) , (D3)where the RHS of this expression is just a weighted sumof the expressions plotted in Fig. 17. The effect of chang-ing the values of g and g simply moves the point ofinflection of the RHS in each region between neighboringnon-interacting energies. As the LHS is a monotonically9 FIG. 17. Energy dependence of − k ( s ) M ( s ) and − k ( s ) M ( s ) for a lattice of spatial extent L/a s = 24. Both functions arepurely real across this energy region. Dashed vertical lines indicate the location of non-interacting energies. FIG. 18.
Top:
The function − g k ( s ) M ( s ) − g k ( s ) M ( s ), for L/a s = 24 and g = g = 1, plotted in black, and the function √ s ( s − m ) with m = 0 . Bottom:
Same as the top plot but with a zoomedin vertical scale. increasing function for √ s > √ m , this will intersect theRHS exactly once in each energy region between non-interacting energies. This results in what we refer to as“trapped” levels. We see exactly this in Fig. 18, whereabove f η threshold we a single solution in each regionas described. P -wave scattering For P -wave scattering, Eq. D2 has an extra factor ofa smooth real function, 4 k a ( s ), compared to the S -wavecase. This is positive above threshold, negative below andhas a zero exactly at threshold, and this zero is the reasonfor there being no non-interacting level at threshold in P -wave. The argument that led to “trapped” levels in S -wave applies here too.It is interesting to revisit the indistinguishability of { P , P , P } in vector-vector scattering in the contextof a factorized pole K -matrix. If we consider this systemwhich has only a single open channel but three partial-waves, then the single non-trivial eigenvalue which haszeros at the finite-volume energy levels is, λ ( s ) = 1 − k M√ s ( m − s ) (cid:0) g + g + g (cid:1) , as the momenta k ( s ) and the function M ( s ) are identicalin each of these partial-waves.Naively, we would expect to find only a single rootbetween neighboring non-interacting energies; however,this would overlook the fact that the multiplicity of eachof these non-interacting energies is in fact three , andso we should find three roots associated with each non-interacting energy (these roots are not necessarily triplydegenerate as we will see).This can be seen most easily by treating each partialwave as an independent hadron-hadron scattering channelby perturbing the scattering vector meson mass slightly ineach partial wave. In P we take m ω → m ω − (cid:15) and in P m ω → m ω + (cid:15) , so that the perturbed finite-volumeenergy levels are roots of, (cid:101) λ ( s ) =1 − √ s ( m − s ) (cid:0) g k − (cid:15) M − (cid:15) + g k M + g k (cid:15) M (cid:15) (cid:1) , where the subscript ± (cid:15) means that the vector meson masshas been perturbed by ± (cid:15) . The previously triply degen-erate non-interacting energies are now split by order (cid:15) .However, there are trapped roots between these perturbednon-interacting energies which forces at least two of theroots to lie within (cid:15) of the unperturbed non-interactingenergy. In the limit (cid:15) →
0, we find (cid:101) λ ( s ) → λ ( s ) with at0least two roots positioned exactly at the non-interactingenergy. The third root is free to vary in position betweenthese two roots and the next non-interacting energy, its lo-cation depending on the value of g + g + g ; it is exactly atthe non-interacting energy if and only if g = g = g = 0,in which case the roots are triply degenerate. Appendix E: Sensitivity to f η { S } , h η { S } couplings In this appendix, we will examine the sensitivity offinite-volume spectra to the relative size of the f η { S } and h η { S } couplings. In Sec. VI, we found the ratioof these couplings to be poorly determined, while the sumof the squared couplings was well determined. We willinvestigate this effect using a simplified two-channel toymodel where the t -matrix is given by, t ab ( s ) = g a g b m − s + g I ( s ) + g I ( s ) . The mass parameter, m = 0 .
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12 16 20 24 12 16 20 24 12 16 20 24 12 16 20 24 12 16 20 240.460.470.480.490.50
FIG. 19. Finite-volume spectra for the toy model as described in the text for the T − irrep at rest and the A irreps in flight. a ≡ g + g = 1 is kept fixed, while the ratio, b ≡ g /g , is varied.
12 16 20 24 12 16 20 24 12 16 20 24 12 16 20 24 12 16 20 240.460.470.480.490.50
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