Excited J −− meson resonances at the SU(3) flavor point from lattice QCD
JJLAB-THY-20-3291
Excited J −− meson resonances at the SU(3) flavor point from lattice QCD Christopher T. Johnson
1, 2, ∗ and Jozef J. Dudek
1, 2, † (for the Hadron Spectrum Collaboration) Department of Physics, College of William and Mary, Williamsburg, VA 23187, USA Thomas Jefferson National Accelerator Facility, 12000 Jefferson Avenue, Newport News, VA 23606, USA (Dated: December 2, 2020)We present the first calculation within lattice QCD of excited light meson resonances with J PC =1 −− , 2 −− and 3 −− . Working with an exact SU(3) flavor symmetry, for the singlet representationof pseudoscalar-vector scattering, we find two 1 −− resonances, a lighter broad state and a heaviernarrow state, a broad 2 −− resonance decaying in both P – and F –waves, and a narrow 3 −− state.We present connections to experimental ω (cid:63)J , φ (cid:63)J resonances decaying into πρ , KK ∗ , ηω and otherfinal states. I. INTRODUCTION
The lightest vector meson resonances, the ρ , ω and φ ,are benchmark states in our understanding of the quarksubstructure of hadrons [1]. The near degeneracy of the ρ and the ω , and the preference for φ to decay to KK evenwhen πππ has a much larger phase-space, leads to theOZI rule and the u ¯ d, u ¯ u + d ¯ d, s ¯ s assignment for the threestates. That such clear conclusions can be drawn comesin part from the fact that these resonances are rathernarrow (particularly the ω and the φ ) and that they canbe produced in the definitively J P C = 1 −− process of e + e − annihilation, where they appear with essentially nobackground in simple final states like ππ, πππ and KK .In comparison, the spectrum of heavier excited vectormesons is far less clear, with proposed experimental can-didate states being rather poorly understood [2, 3]. Suchstates lie at or above about 1400 MeV, which is well intothe region of coupled-channels, where resonances havemultiple possible decay modes. The PDG consensus is fora ρ (1450) with a large total decay width, and a somewhatnarrower ρ (1700). The isoscalar states are even less welldetermined, with preference for an ω (1420) with a largeuncertainty on the width, and an ω (1650) that is likely tobe broad. A relatively narrow φ (1680) does not appear tohave an obvious partner at higher energy . These assign-ments of isoscalar resonances to the names ω, φ (implyingdominantly hidden-light versus hidden-strange q ¯ q struc-ture) follow from assumptions based upon the OZI ruleapplied to the decay channels in which the resonances areseen (mostly πππ versus KK ( ∗ ) ).Within quark models assuming a minimal q ¯ q struc-ture for mesons, the presence of two 1 −− states in eachflavor channel is quite natural, with there being a firstradial excitation of the lightest vector states having aquark-antiquark pair in a relative S -wave, q ¯ q (cid:2) S (cid:3) , and ∗ [email protected] † [email protected] The next relevant state listed in the PDG is the φ (2170) observedthrough its decay to φf (980) which is too heavy to partner the φ (1680). in addition a D –wave excitation, q ¯ q (cid:2) D (cid:3) . The physi-cal eigenstates can be admixtures of these basis states,although typically the simple model dynamics does notgenerate a large mixing [4]. Within these models, oneof the 1 −− states would come with spin-orbit partners, q ¯ q (cid:2) D , , (cid:3) , leading to an expectation of approximatelydegenerate states with J P C = 1 −− , −− , −− . There areexperimental candidates for 3 −− states in the form of thenarrow resonances ρ (1690) , ω (1670), and φ (1850), butto date there are no clear signals for the corresponding2 −− states [3].Recent support for these longstanding quark modelexpectations comes from lattice QCD calculations of theexcited meson spectrum [5–8]. Lattice QCD is a first-principles numerical approach to QCD in which the quarkand gluon fields are discretized on a periodic grid of finitesize. By sampling gluon field configurations according toa probability distribution fixed by the QCD action, corre-lation functions can be computed, and from these physicalobservables extracted. The simplest calculations of themeson spectrum make use of a large basis of fermion bilin-ear operators in the construction of matrices of correlationfunctions, and diagonalisation of these provides a guideto the excited state spectrum of isovector and isoscalarmesons. Figure 1, taken from Ref. [8], shows the relevantpart of the spectrum from two such calculations, one witha heavier than physical light quark mass such that thepion has a mass ∼
391 MeV (left) and another where thelight and strange quark masses are degenerate leading toan exact SU(3) F symmetry and a lightest pseudoscalarof mass ∼
700 MeV. The observed spectra support thequark model picture described above, provided it is aug-mented with 1 −− hybrid mesons (highlighted with orangeborders) in which a q ¯ q construction in a color octet iscoupled to an excitation of the gluonic field [9]. The lackof significant hidden-light–hidden-strange mixing at thelighter quark mass, and the near degeneracy of the isovec-tor and hidden-light isoscalar states supports the OZI“rule” in which q ¯ q annihilation within a meson, leading toa “disconnected” diagram, is a suppressed process.While relatively simple lattice QCD calculations likethose presented in Figure 1 provide guidance as to whatexcited states we expect to find in QCD, they are clearly a r X i v : . [ h e p - l a t ] D ec FIG. 1. J −− meson spectrum (in MeV units) extracted fromdiagonalization of matrices of fermion bilinear correlationfunctions. Results taken from Ref [8]. Left panel: Isovector(blue) and isoscalar (green/black) mesons with m π ∼
391 MeV.Relative hidden-light and hidden-strange content determinedby size of matrix elements (cid:104) M | ¯ (cid:96) Γ (cid:96) | (cid:105) , (cid:104) M | ¯ s Γ s | (cid:105) . Right Panel:Spectrum in the SU(3) F limit, in octet ( , blue) and singlet( , pink) representations, with m π ∼
700 MeV. incomplete in that they do not resolve that excited statesare in fact resonances which decay rapidly into lighterstable hadrons. In this paper we seek to resolve thisomission.The ρ (cid:63) and ω (cid:63) , φ (cid:63) excited mesons are separated in theirdecay channels by isospin and G -parity. In particular ρ (cid:63) states decay to even numbers of pions ππ, ππππ , while ω (cid:63) states decay to πππ . The separation of isoscalar statesinto ω and φ assignments is not based upon any funda-mental symmetry, but is rather motivated by the OZI rulewhich suggests that φ states prefer to decay to KK ( ∗ ) ,as the initial s ¯ s does not have to annihilate, over forexample πππ where it does . Whether the excited J −− states remain ideally flavor mixed like the lightest ω, φ isa dynamical question, but the lattice calculation shownin the left panel of Figure 1 seems to suggest they do.Regarding the spin-parity structure of decays, we notethat J P = 1 − and 3 − are in the natural parity sequence which means that for example ρ (cid:63) , ρ can decay into pairsof pseudoscalars, while 2 − is in the unnatural parity se-quence preventing ρ decaying into these simplest of finalstates.The state-of-the-art until now for theoretical descriptionof the decays of excited J −− states has been to supplementa q ¯ q bound-state quark model with a q ¯ q pair-creation ver-tex applied in lowest-order perturbation theory. Within OZI does not forbid ω ∗ decays to pairs of strange hadrons whichcan proceed by production of an s ¯ s pair. such a model an estimate for the OZI-allowed decays ofan excited meson M to a pair of lighter mesons, AB , fol-lows from evaluation of the matrix element (cid:104) AB |O q ¯ q | M (cid:105) ,where A and B do not interact with each other, thecalculation of which amounts to computing overlap inte-grals featuring q ¯ q bound-state wavefunctions. The mostsuccessful approach, in the sense of approximately dupli-cating several measured hadron decay rates, is to assumethe q ¯ q pair is produced with the quantum numbers of thevacuum, the “ P -model” [10, 11].Going beyond this to compute directly within QCD, anapproach is available which allows us to access the energydependence of scattering amplitudes , like AB → AB inwhich M appears as a resonance. This method makesuse of the discrete spectrum of QCD in the finite vol-ume defined by the periodic lattice used in lattice QCD.Consideration of field theories in a cubic volume [12–26]provides a relationship between the S -matrix in multiplepartial waves and the finite-volume spectrum such thatthrough lattice QCD spectrum computations we can ob-tain scattering information. By using parameterizationsof scattering amplitudes, resonance information follows ina rigorous way from isolating pole singularities at complexvalues of the scattering energy. Computation of the ρ reso-nance in elastic ππ scattering is now common [27–43], andthe extension into the more complicated coupled-channelsector has been pioneered by the hadspec collaboration,with calculations of scattering systems containing reso-nances resembling the a (980) , f (980) , b (1235) , f (1270)and f (cid:48) (1525), amongst others [44–50].Given the likely complexity of the coupled-channel scat-tering systems housing the physical J −− resonances , wechoose in this first calculation of their properties to workin a simplified version of QCD in which three quark flavorsare degenerate, m u = m d = m s , and where this singlequark mass is tuned to approximately match the valueof the physical strange quark mass. The exact SU(3) F symmetry present in this version of QCD simplifies thescattering systems in which the J −− resonances appear,and the relatively large value of the mass of the lightestpseudoscalar ∼
700 MeV makes decays to three-mesonand higher multiplicity final states kinematically inacces-sible. The spectrum of states obtained at this SU(3) F point when only fermion bilinear operators are used toform correlation functions is shown in the right panel ofFigure 1, where we observe octet ( ) and singlet ( ) exci-tations in good agreement with the q ¯ q picture describedabove. The ideal flavor mixing (states as u ¯ u + d ¯ d, s ¯ s )observed away from the SU(3) F point of course cannot bepresent here as ∼ u ¯ u + d ¯ d − s ¯ s and ∼ u ¯ u + d ¯ d + s ¯ s ,but the near degeneracy of the octet and singlet statesallows for a strong mixing to ideal flavor upon even asmall breaking of the SU(3) F symmetry. In particular the possibility of three-meson decays, the formal-ism for which has only recently been developed and initiallytested [51].
In the first calculation of J −− resonances in latticeQCD presented in this paper, we will focus on the SU(3) F singlet ( ) states, and seek to determine if there are indeedtwo resonances in 1 −− , one in 2 −− and one in 3 −− . Wewill determine the decay widths of these resonances, andexplore how two overlapping resonances might manifestin 1 −− scattering amplitudes. We will initially work in arestricted energy region below the expected location of the1 −− hybrid meson, to avoid the possibility of three-mesondecays becoming relevant. We will find that resonancesare present which appear in the η ω scattering channelwith negligible coupling to other kinematically accessiblechannels – this active scattering channel will be relatedto ω (cid:63) and φ (cid:63) decays to for example πρ and KK ∗ in theSU(3) F -broken case. II. FINITE-VOLUME SPECTRUM
As indicated in the introduction, resonances can bedetermined from the energy dependence of scattering am-plitudes, which are constrained by finite-volume spec-tra computed using lattice QCD. The spectra followfrom diagonalization of matrices of correlation functionswhich were computed on five anisotropic lattices with vol-umes (
L/a s ) × ( T /a t ) = { , , , , } × a s ∼ .
12 fm and a t = a s /ξ ∼ (4 . − , withanisotropy ξ ∼ .
5. Details of the generation of thesedynamical three-flavor lattices where m u = m d = m s andwhere the lightest pseudoscalar has mass ∼
700 MeV canbe found in Refs. [52, 53].
Distillation [54] was used to compute correlation func-tions, allowing all relevant Wick contractions to be com-puted including those featuring q ¯ q annihilation, whichare common when SU(3) F singlets are being considered.The rank of the distillation space, the number of timesources and the number of gauge configurations used areprovided in Table I.The spectrum of mesons stable against strong decayin this version of QCD was presented in Ref. [55] andis reproduced in Table II. The dispersion relations (theenergy when at momentum (cid:126)p = πL (cid:126)n ) for the low-lyingmesons which feature in scattering were also computedand found to conform to the relativistic expression, (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 | , (1) L/a s
14 16 18 20 24 N cfgs
397 490 358 477 499 N vecs
48 64 96 128 160 N tsrcs
16 4 4 4 1TABLE I. 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. η η ω ω f f f h h a t m . with an estimate for the anisotropy that accounts forsmall variations observed for different mesons being ξ = 3 . J P are not in general good quantumnumbers, rather we should use the irreducible representa-tions ( irreps ) of the cubic symmetry and of its little group for systems with nonzero momentum. The irreps we willconsider are presented in Table III where we observe the subduction of many J P C values into each irrep (we showonly
J < J P C to subducemore than once into an irrep, an example being 3 −− whichsubduces twice into [110] A – one way to understand thisis in terms of helicity [56] where two linear combinationsof the seven possible helicities of J = 3 end up in thisirrep. The (undesired) presence of positive parities like0 + − , + − in the in-flight irreps is unlikely to pose a prob-lem for our calculation as these J P C quantum numbersare exotic (inaccessible to q ¯ q ), and there is good evidencethat the lightest such resonances appear at much higherenergies than we will consider [8]. The possible in-flightirreps not listed in Table III are excluded because theyinclude subductions of 1 + − which is expected to featureaxial meson resonances – we choose to avoid the complica-tion of simultaneously describing such resonances in thisfirst study.The scattering channels that can contribute to J −− in the energy region where we expect to find resonancesare η ω with threshold 0 . ω f with threshold0 . η ω with threshold 0 . P – and F –waves, while the ω f channel can contribute in S –wave. [000] T − −− −− [000] E − −− [000] T − −− −− [000] A − −− [100] A −− −− + − + − [100] B −− −− + − [100] B −− −− + − [110] A −− −− (cid:0) −− (cid:1) + − (cid:0) + − (cid:1) [111] A −− −− (cid:0) −− (cid:1) + − + − TABLE III. Subductions of J PC into cubic irreps, superscriptsindicate multiple embeddings. Only J < −− η ω (cid:8) P (cid:9) ω f (cid:8) S , D (cid:9) η ω (cid:8) P (cid:9) −− η ω (cid:8) P , F (cid:9) ω f (cid:8) D (cid:9) η ω (cid:8) P , F (cid:9) −− η ω (cid:8) F (cid:9) ω f (cid:8) D (cid:9) η ω (cid:8) F (cid:9) TABLE IV. Meson-meson scattering partial-waves for each J PC – only waves with (cid:96) ≤ We will compute correlation functions using operatorswhich resemble all three of these meson-meson configu-rations, although in practice we will find that ω f and η ω appear to be decoupled from each other, from η ω ,and from resonances. The lowest three-meson channel, η η η , has its threshold at 0 . J −− scattering at least two P –waves arerequired, which will render the channel irrelevant in theenergy region we consider.The construction of meson-meson-like operators whichtransform irreducibly under the relevant symmetries ofthe lattice has been discussed in detail previously (see forexample Refs. [49, 50, 57]), but in short they are built assums of products of definite-momentum operators opti-mized for their overlap onto the relevant scattering meson.The summation runs over possible allowed rotations ofthe momentum of each meson, keeping the total momen-tum fixed. For example, an operator labelled η [100] ω [110] will contribute in the [100] A irrep, and in the limit inwhich the η and ω have no meson-meson interactions,this operator would interpolate an eigenstate with a non-interacting energy of (cid:113) m η + (cid:0) πL (cid:1) + (cid:113) m ω + 2 (cid:0) πL (cid:1) .Interactions will move the actual finite-volume energyaway from this value, and it is ultimately these volume-dependent shifts which allow us to determine the scatter-ing amplitudes.Our approach is to include all meson-meson operatorswhich have a non-interacting energy, as measured in thecenter-of-momentum frame, below roughly a t E cm ∼ . q ¯ q and hybridmeson configurations. With this basis we expect to obtaina set of energy eigenstates which constitute the completefinite-volume spectrum below a t E cm ∼ .
46. The operatorbasis for each irrep is provided in Appendix A.In each irrep, a spectrum is determined by solving ageneralized eigenvalue problem featuring the matrix ofcorrelation functions [6, 58–61]. The resulting eigenval-ues each have a time-dependence controlled dominantlyby the energy of one finite-volume eigenstate, and thecorresponding eigenvectors can be related to the overlapof that state with each operator in the basis. In orderto verify that our finite-volume spectra are not overlysensitive to the specific choice of operator basis, we per-form several diagonalizations, varying which single-mesonoperators we include, and also check that excluding thosemeson-meson operators with the highest non-interacting
FIG. 2. Finite-volume spectra in the [000] T − irrep extractedfrom matrices of correlators built using the operators listedin Table V. States color-coded by their dominant operatoroverlap, as shown in Figure 3. Curves show non-interactingenergies, and when dashed indicate that the correspondingoperator(s) were not included in the basis. energies does not lead to a significantly different low-lyingspectrum. Any such sensitivity (which is rare) is includedas a systematic error on the finite-volume energy.An example set of spectra on the five lattice volumesconsidered is shown in Figure 2 for the case of the [000] T − irrep. The lightest state, present at approximately thesame energy on each volume, can be identified as the sta-ble ω – that it shows essentially no volume dependencesupports the idea that the lattices used are large enoughto avoid significant ‘polarization’ effects, in which a singlemeson can have an effect on itself around the periodicworld. The higher spectra show some large departuresfrom the non-interacting energies (colored curves), andindeed the counting of levels is larger than the numberof non-interacting levels, indicating strong meson-mesoninteractions and likely resonances. The spectra are ob-served to become dense above the ω f threshold, andit is worth examining the overlaps of these finite-volumestates onto the set of operators used.Figure 3 shows the same spectra as Figure 2 with theaddition of histograms that illustrate the size of overlapsonto a subset of the operator basis used . The five orange The normalization is such that for a given operator, the largestoverlap within the complete spectrum of states extracted is giventhe value 1, and all others are expressed relative to this. bars show overlap onto five single-meson operators with J = 1 (to be discussed below), the cyan bar shows a single-meson operator with J = 3, the red bars represent the η ω operators (ordered top-bottom as lowest-highest non-interacting energy), the green bars the ω f operators,and the blue bar the η ω operator. The spectrum hasbeen separated into three panels because it is clear fromthe overlaps that some states have overlap onto only the ω f operators, or onto only the η ω operators, and wenotice that these states are statistically compatible withlying on the non-interacting energy curves. This likelyindicates that the ω f and η ω scattering channels aredecoupled from each other, from η ω , and from anyresonances.Examining the upper panel of Figure 3 we have arather well determined spectrum in which states typi-cally have overlap onto both the single-meson operators(orange, cyan) and the η ω operators, which may betaken as an indication that there are “ q ¯ q -like” resonancespresent which can decay into η ω . The subset of single-meson operators shown are selected for the property that,as discussed in Ref. [9], certain operators can be char-acterized by which q ¯ q constructions they overlap within the non-relativistic limit. The first two orange barsshown represent (cid:0) ρ × D [0] J =0 (cid:1) J =1 and (cid:0) ρ × D [2] J =0 (cid:1) J =1 , whichhave unsuppressed overlap with q ¯ q in a S configura-tion (including radial excitations). The third operator, (cid:0) π × D [2] J =1 (cid:1) J =1 , which features the commutator of twogauge-covariant derivatives, is expected to overlap withhybrid mesons. The fourth operator, (cid:0) a × D [1] J =1 (cid:1) J =1 ,has overlap with both q ¯ q [ S ] and q ¯ q [ D ], while thefifth operator, (cid:0) ρ × D [2] J =2 (cid:1) J =1 , only overlaps with q ¯ q [ D ].We notice that the first excited state, located between a t E cm = 0 .
38 and 0 .
40 always has large overlap with thefirst two orange operators, likely signaling a significant q ¯ q [2 S ] component. On each volume there is a state near a t E cm = 0 .
42 having large overlap onto the fourth andfifth orange operator corresponding to q ¯ q [1 D ]. Thereare no states having large overlap with the third orangeoperator, which matches with our expectation, discussedearlier, that the 1 −− hybrid meson lies at a higher energythan we are considering here . At least one state near to a t E cm = 0 .
44 at each volume has overlap with the cyanoperator, suggesting the presence of a 3 −− resonance.This same procedure of examination of the overlaphistograms has been performed for all computed irrepson all volumes, and in every case it appears that ω f and η ω are decoupled, and we propose to proceed underthe assumption that η ω can be considered as an elasticscattering system. We will seek to describe all finite-volume energy levels that remain when those levels havingoverlap onto ω f or η ω are excluded, as shown in There are finite-volume states at energies larger than we haveplotted with overlap onto the third operator.
Figure 4. In total this amounts to nearly 200 energylevels lying below a t E cm = 0 .
46. As we will see laterin explicit parameterizations of the relevant scatteringamplitudes, the numbers of levels extracted in each energyregion matches our expectations of there being two 1 −− resonances, one 2 −− resonance, and one 3 −− resonance. FIG. 3. [000] T − spectra as in Figure 2 separated by dominantoverlap onto ¯ ψ Γ ψ, η ω (top panel), ω f (middle panel) or η ω (bottom panel). Histograms show the overlap onto asubset of operators used to build the matrix of correlators.Orange bars (top to bottom): (cid:0) ρ × D [0] J =0 (cid:1) J =1 , (cid:0) ρ × D [2] J =0 (cid:1) J =1 , (cid:0) π × D [2] J =1 (cid:1) J =1 , (cid:0) a × D [1] J =1 (cid:1) J =1 , (cid:0) ρ × D [2] J =2 (cid:1) J =1 , cyan bar: (cid:0) ρ × D [2] J =2 (cid:1) J =3 , red bars: η ω (increasing momentum top tobottom), green bars: ω f , blue bar: η ω . FIG. 4. Energy levels with negligible η ω , ω f overlap, assumed to form part of the η ω scattering system. These levelswill be analysed in terms of elastic η ω scattering, with gray points not used. FIG. 5. ω mass as determining by ‘boosting’ to the cm -framethe lowest energy determined in each of the irreps: [000] T − (black), [100] A (red), [110] A (cyan), [111] A (green). Theorange band indicates the mass used for this state in scatteringanalysis. The growth in the uncertainty on energy levels plottedas a t E cm as the frame-momentum increases can be tracedback to the uncertainty we place on the anisotropy, ξ ,which is accounted for when we ‘boost’ the calculated ener-gies in the moving-frame back to the center-of-momentumframe. This can be seen clearly in Figure 5 where weshow the lowest energy level extracted in the [000] T − ,[100] A , [110] A , and [111] A irreps, which we expectto be the stable ω . We see a consistent mass, but with agrowth in uncertainty as the frame momentum increases.Our hypothesis that the η ω and ω f channels aredecoupled will be tested explicitly later using a limited setof coupled-channel amplitudes, but we note that shouldthe hypothesis be incorrect, it will likely not be possible tofind elastic η ω amplitudes that are capable of describingall the energy levels in Figure 4. We will find that elasticamplitudes are able to describe the spectrum rather well,and we will not find any significant evidence to supportchannel coupling in this system. III. SCATTERING AMPLITUDES
The relationship between the t -matrix describing scat-tering and the finite-volume spectrum in a periodic L × L × L box is encoded in the L¨uscher quantization condition,det (cid:104) + i ρ t (cid:0) + i M (cid:1)(cid:105) = 0 , (2)and an extensive discussion of how the relationship can beimplemented is presented in Ref. [62]. Our approach is tomake use of parameterizations of the energy dependenceof t ( E cm ) and to attempt to describe as much of an ob-tained finite-volume lattice QCD spectrum as possible byvarying the parameters in the parameterization, solvingEqn. 2 for the finite-volume spectrum for each choice ofparameter values, and comparing to the lattice spectrum.An efficient method to solve the above equation, particu-larly applicable in cases of coupled-channels or coupledpartial waves is presented in Ref. [63].An important feature of the above quantization con-dition is that it only has solutions for t -matrices whichsatisfy the unitarity condition that implements the con-servation of probability. A straightforward way to ensurethis is to make use of K -matrices by writing, t − = K − + I , (3)where K ( s = E cm ) is a symmetric real matrix in the spaceof coupled-channels and/or coupled partial waves, andwhere I ( s ) is a diagonal matrix with imaginary parts ofvalue Im I i ( s ) = − ρ i ( s ), where the phase-space , ρ = k √ s .The real part of I i ( s ) can simply be chosen to be zero, inwhich case we speak of using the “naive phase-space”, orwe can make the choice to use the result of placing ρ ( s )in a dispersive integral, leading to what is often called the“Chew-Mandelstam phase-space”. The dispersive integralis once-subtracted, and the location of the subtractioncan be chosen for our convenience – our implementationis described in Appendix B of Ref. [45].We have significant freedom to choose parameterizationforms for K ( s ), and a good approach is to try a rangeof parameterizations, finding as many as possible thatcan describe the finite-volume spectrum. If the resultingamplitude has features that are robust under changesof parameterization, we can be confident that they aretrue features of the actual QCD amplitude. This ap-proach has been used extensively in previous calculationsof elastic and coupled-channel scattering by the hadspec collaboration [34, 44–46, 49, 50].In practice, for scattering in partial waves with non-zero orbital angular momentum, (cid:96) , we extract from the K -matrix the momentum factors needed to get the correctthreshold behavior, writing (cid:2) t − (cid:3) (cid:96)(cid:96) (cid:48) = k ) (cid:96) (cid:2) K − (cid:3) (cid:96)(cid:96) (cid:48) k ) (cid:96) (cid:48) + I , (4)for the case of two coupled partial waves ( (cid:96) , (cid:96) (cid:48) ). Detailsof how dynamically coupled partial waves are handled inthe (cid:96)S basis can be found in Refs. [49, 50, 55].Resonances in a scattering system are associated with pole singularities at complex values of s located on un-physical Riemann sheets (those where the scattering mo-mentum has a negative imaginary part). The real andimaginary parts of the pole position are commonly givenmeanings in terms of the mass and total width of theresonance, √ s = m ± i Γ, where the two signs reflect thefact that these poles always come in complex-conjugatepairs. Couplings to decay channels in each partial wave, c i , can be obtained by factorizing the residue at the poleposition, t ij ( s ) ∼ c i c j s − s . Poles can also lie on the real energy axis below thelowest kinematic threshold – if they appear on the physicalsheet they are associated with stable bound states thatcan appear as asymptotic particles, while if they appearon unphysical sheets they are termed virtual bound states which do not have an associated asymptotic particle.The t -matrix can have other singularities, notably cuts associated with the dynamics of scattering in crossed-channels. These are known as left-hand cuts , and typicallytheir effect on physical scattering is much milder thanthe effects of narrow resonances, and their net effectabove threshold can be modelled by including slowlyvarying polynomial behavior in the K -matrix . They arediscussed further in the current context in Appendix B.We now proceed to present descriptions of the finite-volume spectra introduced in the previous section, begin-ning with the assumed elastic η ω spectra of Figure 4, us-ing parameterizations of elastic scattering in J P C = 1 −− ,2 −− and 3 −− . We will consider several strategies to iso-late the amplitudes, firstly considering those irreps whichonly depend upon scattering with J P C = 2 −− and/or3 −− , then those irreps which depend only upon 1 −− and3 −− , before finally attempting a global description of allthe energy levels. A. J P C = 3 −− from the [000] A − irrep The only partial wave expected to contribute in the[000] A − irrep at the energies we are considering is η ω { F } , and the volume dependence of energy lev-els in the top right panel of Figure 4 appears to be acanonical “avoided level crossing” indicating a narrowresonance near a t E cm ∼ . K -matrix featuring a single pole, K ( s ) = g m − s , and either a Chew-Mandelstam or naivephase-space is capable of describing this spectrum witha χ /N dof = . − = 1 .
31. The resulting amplitude has a But see Refs. [64, 65] for cases where resonances may appear thatare very broad and where the physics of the left-hand cut maybecome relevant. narrow peak and no other features, and the t -matrix hasa pole at a t √ s = 0 . ± i . a t | c η ω | = 0 . K -matrix leads to a negligible change in the qualityof fit, and a consistent resonance pole.From this analysis of a single irrep, it is clear thatthere is a narrow 3 −− resonance – we will delay providingfurther discussion until we report a more precise determi-nation of its pole parameters using a description of moreenergy levels. B. J P C = 2 −− from the [000] E − irrep Assuming negligible J = 4 scattering, the [000] E − irrep spectrum is controlled by the coupled partial waves, η ω { P , F } , requiring a two-dimensional t -matrix, t = (cid:20) t P P t P F t P F t F F (cid:21) . Examining the volume dependence of energy levels inthe second panel of Figure 4, we potentially observe anavoided level crossing with the lowest non-interacting η ω curve, but spread out over a large energy range,which might signal a broad resonance somewhere around a t E cm ∼ .
42. A simple amplitude that proves capable ofdescribing this spectrum is given by the K -matrix, K ( s ) = 1 m − s (cid:20) g P g P g F g P g F g F (cid:21) + (cid:20) γ P P γ P F γ P F γ F F (cid:21) , (5)where when the Chew-Mandelstam phase-space is used(subtracted at s = m ) in Eqn. 4, the 13 energylevels can be described with a χ /N dof = . − = 0 . t -matrix in this case has a pole singularityat a t √ s = 0 . ± i . a t | c η ω { P } | = 0 . a t | c η ω { F } | = 0 . −− and, as we’d ex-pect for a resonance lying only slightly above the relevantdecay threshold, the angular momentum barrier ensuresan F –wave decay coupling that is significantly smallerthan the leading P –wave.A more precise determination of this 2 −− amplitude,and of the 3 −− amplitude discussed previously, can beobtained by simultaneously describing both in a descrip-tion of the 91 finite-volume energy levels in the irreps[000] A − , T − , E − and [100] B , B . C. J P C = 2 −− , −− from the[000] A − , T − , E − , [100] B , B irreps Five irreps, [000] A − , T − , E − , and [100] B , B , areeach sensitive to one or both of the 2 −− and 3 −− scat-tering amplitudes, and together feature 91 energy levelsthat we can use to constrain the energy dependence. Asan example, using a single K -matrix pole for the 3 −− amplitude, K ( s ) = g m − s , and the K -matrix presented inEqn. 5 for the 2 −− amplitude, with Chew-Mandelstam phase-space (subtracted at s = m ) in both cases, weobtain a best-fit description of the finite-volume spectrawith parameters, J = 2 m = 0 . · a − t .
31 0 .
29 0 . − .
37 0 .
31 0 .
19 0 . − . − .
70 0 .
04 0 .
48 0 . − .
231 0 . − . − . − . − . − . − . − .
16 0 . − . − . − .
051 0 .
02 0 . − . g P = 0 . g F = − . · a t γ P P = 0 . · a t γ P F = − · a t γ F F = 143(322) · a t J = 3 (cid:40) m = 0 . · a − t g = 4 . · a t χ /N dof = . − = 1 .
45 . (6)The fit quality is quite reasonable, and the parametersshow no particularly large correlations. We observe thatthe constants γ P P , γ
F F are probably redundant, and laterwe will explore fixing them to zero. The resulting am-plitudes are shown in Figure 6 where the bumps suggestthe presence of a narrow resonance in 3 −− , and a broaderresonance in 2 −− . The dominance of P over F isobvious, and while the resonance bump is still visible inthe off-diagonal element t P F , albeit peaking at a slightlylower energy than in the t P P element, there is no clearpeak in the weak t F F element.These best-fit amplitudes feature 2 −− and 3 −− t -matrixpoles that are compatible with those reported in previoussections in fits to [000] E − , [000] A − alone, but which now FIG. 6. Scattering amplitudes for J PC = 2 −− (Eqn. 5) and3 −− ( K -matrix pole), for the best-fit parameters of Eqn. 6.Points below the abscissa show the positions of the finite-volume energy-levels constraining the amplitudes. have improved statistical uncertainty:2 −− : a t √ s = 0 . ± i . a t c η ω { P } = 0 . e ± iπ . a t c η ω { F } = 0 . e ∓ iπ . −− : a t √ s = 0 . ± i . a t c η ω { F } = 0 . e ± iπ . . We note that the F –wave couplings for the 2 −− and 3 −− resonances are of a very similar size. The 2 −− resonancepole (in the lower half plane) has a ratio of F –wave to P –wave couplings of 0 . e iπ . ≈ − . K -matrix pole in Eqn. 5 to zero, replacing someof the constants with terms of form γ · s , using a secondpole in place of the constants, or by writing K − as amatrix of polynomials. The forms used are listed in Ta-ble IX in Appendix C. The resulting 2 −− amplitudes areshown in Figure 7, and it is quite clear that the amplitudepreviously presented in Figure 6 is representative, andin fact it has among the largest statistical uncertainties The 3 −− amplitudes show completely negligible variation. FIG. 7. Variation of 2 −− amplitude over parameterizationchoice. Solid curves and bands show descriptions of the finite-volume spectra with 1 . < χ /N dof < .
46, while dashedcurves have 1 . < χ /N dof < . of those amplitudes considered. While the bulk of theamplitudes tried have a χ /N dof very close to the value1 .
45 obtained using Eqn. 5, there are three choices thathave somewhat larger values: Eqn. 5 with γ P F fixed tozero ( χ /N dof = 1 . K -matrix built as the sum oftwo poles ( χ /N dof = 1 . K − isparameterized as independent linear polynomials ( a + bs )in each element ( χ /N dof = 1 . t P P peak position, butotherwise only start to show significant deviation fromthe solid curves above the energy region where constraintis provided by the finite-volume spectra.Figure 8 shows the t -matrix pole positions and thecorresponding pole couplings for the parameterizationvariations, indicating a clear consensus that agrees withthe reference amplitude described previously. The am-plitudes considered do have other pole singularities inaddition to the one presented in Figure 8, but they aretypically distant from physical scattering and vary withparameterization choice. A typical example, present inthe reference amplitude, is a pole on the real axis onthe unphysical sheet near a t √ s ∼ .
23 – such a pole ispresent for many of our amplitudes, although its preciseposition varies, always remaining far from physical scat-tering, and as such it remains largely irrelevant to physicalscattering. As one might expect given its inferior analyticproperties, using the naive phase-space in place of theChew-Mandelstam function leads to additional singulari-ties, in particular a physical sheet pole on the real energyaxis at a t √ s ∼ .
24 which is found to have real-valued couplings. In odd- (cid:96) scattering, a true bound-state musthave imaginary couplings, so the presence of this polesignifies a ghost state having negative probability. Such asingularity suggests a flaw in the parameterization, but inpractice the ghost pole is so far from physical scatteringthat it has a negligible impact – we will further discusssuch poles, and their relation to neglected left-hand cuts -0.06-0.04-0.020 0.36 0.38 0.40 0.42 0.44 0.46 0.48 -0.10-0.050.050.10-0.20 -0.15 -0.10 -0.05 0.05 0.10 0.15 0.20
FIG. 8. Top panel: 2 −− t -matrix pole positions for pa-rameterization variations shown in Figure 7 – black pointsshow the fits with 1 . < χ /N dof < .
46 and grey thosewith 1 . < χ /N dof < .
86. Middle panel: the couplings, c η ω { P } (green) and c η ω { F } (sand) obtained from fac-torizing the residue of the t -matrix pole in the lower half-plane– lighter points show those fits with poorer χ . Bottom panel:The magnitude of the ratio of the couplings. later, in the context of 1 −− scattering .The case of an amplitude in which K − is parameter-ized with linear functions features a different pathology:there are poles off the real axis on the physical sheet , albeitfairly deep into the complex plane. Such poles signal abreakdown in causality which comes about because wedo not place analyticity constraints upon our amplitudes.We will later return to further discussion of the 2 −− and 3 −− amplitudes in the context of a global analysis ofall of our finite-volume energies levels, while now we moveto an initial determination of the J P C = 1 −− amplitude. A ghost pole in S -wave is the pathology that causes an amplitudeto fail the ‘sanity check’ of Iritani et.al. [66]. D. J P C = 1 −− , −− from the[000] T − , [100] A , [111] A irreps Irreps [000] T − , [100] A , and [111] A all depend uponboth the 1 −− and the 3 −− amplitude, but not the 2 −− amplitude. Since we have a well-constrained 3 −− ampli-tude from the previous subsection, we choose to initiallyfix this amplitude, and only vary the 1 −− amplitude. Wewill relax this later when we attempt descriptions of ourentire set of finite-volume energy levels.We will not try to include the very deeply-bound stable ω as an explicit pole in our scattering amplitudes, andhence we exclude the lowest energy level in each irrep oneach volume. There are 72 suitable energy levels below a t E cm = 0 .
46 shown in Figure 4, and as discussed in Sec-tion II and shown in Figure 3, the spectra and operatoroverlaps hint at there being two 1 −− resonances present.Narrow resonances are most conveniently parameterizedby including explicit poles in the K -matrix, and as sucha good choice of amplitude to illustrate this case featurestwo poles and a constant, where the constant allows forsome flexibility away from a pure superposition of reso-nances. The Chew-Mandelstam phase-space, subtractedat the lower mass pole ( s = m a ) is used, and the best fitparameters are found to be, m a = 0 . · a − t .
08 0 . − .
33 0 .
191 0 . − .
46 0 . − .
86 0 . − . g a = 1 . m b = 0 . · a − t g b = − . γ = 20 . · a t χ /N dof = . − = 1 .
36 .(7)The description is reasonable, and we note that the param-eter correlations are modest, with the constant γ beingstatistically significant. The resulting amplitude is shownin Figure 9 where we observe a prominent dip with a zerolocated at a t E cm = 0 . t -matrix is examined in thecomplex s -plane, two pole singularities are found close tothe real axis on the unphysical sheet: a t √ s = 0 . ± i . a t c η ω { P } = 0 . e ± iπ . , and a t √ s = 0 . ± i . a t c η ω { P } = 0 . e ± iπ . . FIG. 9. Illustrative two-pole plus constant η ω { P } (1 −− )elastic scattering amplitude (Eqn. 7). Points below the ab-scissa show the positions of the finite-volume energy-levelsconstraining the amplitudes. We interpret these two t -matrix poles as being thesignal for two 1 −− resonances, a lighter broader state,and a heavier narrow state. The zero on the real energyaxis is located close to the second resonance pole .Elastic unitarity is a strong constraint that significantlyrestricts the possible behavior of an amplitude like this,and as seen in Figure 9, there is clearly a non-trivialenergy dependence, one that does not for example simplyconsist of two separated bumps as one might anticipategiven the resonance content. This is one reason why theuse of complex s -plane pole positions is advocated asa rigorous identification of resonances – one could notdescribe this amplitude as a sum of two Breit-Wigners.We now move to explore whether the same finite-volumespectrum can be described by other choices of amplitudeparameterization, and whether the resulting amplitudeshave the same features as just observed. Variations con-sidered include varying the choice for I ( s ), by changingthe subtraction point or by simply using the naive phase-space, and varying what kind of polynomial is addedto the two K -matrix poles. Table VIII in Appendix Clists the variations, and in Figure 10 we show the ampli-tudes obtained using these parameterization variations,all of which prove capable of describing the finite-volumespectra with χ /N dof < . There is guaranteed to be a zero located between s = m a and s = m b whenever a two-pole plus polynomial form isused for an elastic K -matrix. Since t = K IK and K ( s ) = g a m a − s + g b m b − s + γ ( s ), defining P ( s ) = ( m a − s )( m b − s ) K ( s )we have t ( s ) = P ( s )( m a − s )( m b − s )+ I ( s ) P ( s ) and a zero of t ( s ) willappear when P ( s ) = 0. Since P ( m a ) = g a ( m b − m a ) > P ( m b ) = − g b ( m b − m a ) < P ( s ) must cross zero at least oncebetween s = m a and s = m b . If g b is small, it is clear that P ( m b )will have a value close to zero and hence the zero of t ( s ) will beclose to s = m b . FIG. 10. Variation of 1 −− amplitude over parameterizationchoice. Dashed curves show cases with just two poles in the K -matrix and no further freedom. FIG. 11. Elastic scattering phase-shift for 1 −− , variation overparameterization choice. Dashed curves show cases with justtwo poles in the K -matrix and no further freedom. We note that there is very little observed change in theamplitude except at the highest energies, and in particularthe location of the zero in the amplitude appears to bevery stable. We display two examples of allowing toolittle freedom in the amplitude – the dashed curves showparameterizations featuring only two K -matrix poles andno further freedom with either the Chew-Mandelstamphase-space subtracted at the lower pole, or the naivephase-space. We see that they are compatible with theother parameterizations in the region of the resonances,but deviate at high energy. Our conclusion is that somefreedom beyond two poles in the K -matrix is needed tohave the amplitude fall-off at higher energy – adding aconstant seems to be sufficient.Because this process is elastic, we can alternativelydisplay the scattering in terms of an elastic phase shift, δ ( E cm ) , defined by t = ρ e iδ sin δ . The classic signal foran isolated narrow resonance is a rapid rise of δ passingthrough 90 ◦ , with the steepness of the rise correlated withthe smallness of the resonance width. As observed inFigure 11, the phase shift undergoes two such increases,the first with a low slope and the second being much more -0.03-0.02-0.010 0.36 0.38 0.40 0.42 0.44 0.46 0.48 -0.10-0.05 0.05 0.10 0.15 0.20 FIG. 12. Top panel: 1 −− t -matrix pole positions for parame-terization variations shown in Figures 10,11. Bottom panel:the coupling, c η ω { P } obtained from factorizing the residueof the t -matrix poles in the lower half-plane. Gray pointsrepresent the amplitudes having limited freedom shown by thedashed curves in Figures 10,11. rapid. The phase shift passing through 180 ◦ representsthe zero in the amplitude, and the relatively slow approachto 360 ◦ reflects the slow fall off of the amplitude at highenergy.Figure 12 shows the location of the two t -matrix polesingularities and their pole couplings for all the param-eterization variations considered. It is clear that thereis very little scatter, and that robust conclusions can bedrawn about these two resonances appearing in 1 −− . Ad-ditional pole singularities which lie further from physicalscattering are found for some of amplitude variations, inparticular several parameterizations feature an extra un-physical sheet pole, lying slightly above the energy regionthat we have constrained and far into the complex plane.As shown in Figure 13, its position is not well determined,and indeed it is not present in all parameterizations, andas such it appears to be an irrelevant artifact .Another additional pole singularity is present for severalparameterizations, lying on the real energy axis belowthreshold on the physical sheet. Whenever it appears it isfound to have a real-valued coupling, indicating that is is The anticipated 1 −− hybrid meson resonance pole is expectedto lie at a somewhat larger energy, and is unlikely to be wellconstrained without higher-lying energy levels being included inthe analysis. -0.14-0.12-0.10-0.08-0.06-0.04-0.02 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 FIG. 13. As in Figure 12, the 1 −− t -matrix pole positionsfor parameterization variations shown in Figure 10, includinga poorly determined pole at higher energy (red). a ghost . In our illustrative amplitude with two K -matrixpoles and a constant using Chew-Mandelstam phase-space,it is located at a t √ s = 0 . −− , −− , −− partial-waves up to an energy a t E cm ∼ .
46. We have notso far made any use of the [110] A irrep which dependsupon the scattering amplitudes of all the above partialwaves. We will now move to consider a global fit of all the η ω energy levels that will confirm the results seen sofar, and lead to reduced statistical uncertainties on someresonance parameters. E. ‘Global fit’ to all η ω energy levels In this section we attempt a description of the full set of192 energy levels shown in black in Figure 4 using param-eterizations of 1 −− , 2 −− and 3 −− scattering amplitudes.To illustrate the approach we select amplitudes where3 −− is described by a single K -matrix pole with Chew-Mandelstam phase-space subtracted at the pole, the 2 −− P , F coupled system is described by a K -matrix poleplus constants in the P P and
P F positions with Chew-Mandelstam phase-space subtracted at the pole, and 1 −− is described by two K -matrix poles plus a constant withChew-Mandelstam phase-space subtracted at the lowerpole. A description of the complete set of energy levelsis found with χ /N dof = 258 . / (192 −
12) = 1 .
43, andthe finite-volume spectrum following from the best-fitamplitude is shown by the orange curves in Figure 15.As we might guess from the relatively small χ , the or-ange curves are in good agreement with the black points.While the spectrum of ‘predicted’ levels can be dense, it al-ways agrees with our expectations of the number of levels FIG. 14. η ω elastic scattering amplitudes obtained bydescribing 192 finite-volume energy levels. arising from non-interacting levels plus resonances, oncemultiple subductions are accounted for – as an example,consider small volumes in the [110] A irrep, where the sixorange curves agrees with an expectation based upon onelow-lying η ω non-interacting level and five resonancecontributions (1 −− a , 1 −− b , 2 −− , and 3 −− subduced twice).We note that in cases where resonances in different J P C overlap, the “avoided level crossing” structure can besomewhat non-trivial.The amplitudes are shown in Figure 14 where they areseen to be compatible with our previous determinationsusing subsets of the spectrum. The t -matrices are foundto feature poles on the unphysical sheet at the followinglocations (and with pole couplings):1 −− ( a ) : a t √ s = 0 . ± i . a t c η ω { P } = 0 . e ± iπ . −− ( b ) : a t √ s = 0 . ± i . a t c η ω { P } = 0 . e ± iπ . −− : a t √ s = 0 . ± i . a t c η ω { P } = 0 . e ± iπ . a t c η ω { F } = 0 . e ∓ iπ . −− : a t √ s = 0 . ± i . a t c η ω { F } = 0 . e ± iπ . . (8)These are compatible with those found previously.Thus far we have not accounted for the effect of the(relatively small) uncertainties on the scattering hadron( η , ω ) masses on the scattering amplitudes, but whenconsidered by varying them by ± σ , there is negligiblechange. The somewhat larger conservative estimate ofthe uncertainty on the anisotropy, ξ = 3 . FIG. 15. Spectra from Figure 4 used to constrain η ω elastic scattering amplitudes. Orange curves show the finite-volumespectrum corresponding to the best-fit amplitudes. For guidance, the purple (1 −− ), green (2 −− ) and cyan (3 −− ) bands show theresonance masses and widths, allowing avoided level crossings to be observed with the non-interacting η ω levels (red dashed). a t E cm . It can also be considered in thecomputation of M in Eqn. 2, where varying by ± σ leadsto small adjustments in the pole positions given above.The largest effects are observed in the real part of the polepositions which can move by amounts comparable withthe statistical error, and in the imaginary part of the 2 −− pole position. In the next section, when we present ourbest estimates for the resonance pole properties we willinclude this source of uncertainty in our error estimates. F. Decoupled η ω and ω f scattering As discussed in Section II, the finite-volume spectrumappears to separate into a spectrum due to the resonating η ω system that we have just considered, and two spectradue to non-resonant systems η ω and ω f . We willconsider these latter sets of energy levels in isolation.For J P C = 1 −− η ω { P } elastic scattering, we usefive energy levels as constraint, three levels in [000] T − (as shown in the bottom panel of Figure 3) and two in[001] A , all of which are compatible with lying on thelowest non-interacting η ω curve.An effective range expansion k cot δ = a + rk + . . . can be used to describe the elastic amplitude. Using onlya scattering length a (cid:54) = 0 , r = 0 and no higher terms in thepolynomial, the five energy levels can be described with a = 4 . × a t with a χ /N dof = 6 . / (5 −
1) = 1 . P –wave scattering length approximation gives a t -matrix pole distribution that is not easily interpreted(three poles evenly spaced around a circle of radius k = a − / ), but allowing also a non-zero effective range,which can generate a realistic pole distribution in thiscase leads to a fit with 100% correlation between theparameters ( a, r ).Alternatively, using a constant K -matrix and theChew-Mandelstam phase-space subtracted at thresh-old, the energy levels can be described witha χ /N dof = 6 . / (5 −
1) = 1 . t -matrix has a ghost pole on the physical sheet at a t √ s = 0 . .
378 (see Appendix B), we can associatethis ghost with our lack of control over the crossed-channelphysics.For J P C = 2 −− , assumed to be only in the η ω { P } partial wave, we use 8 levels from [000] E − ,[000] T − , and [001] B . The F –wave 3 −− ampli-tude is assumed to be negligible. A scatteringlength description finds a = 4 . × a t with a χ /N dof = 12 . / (8 −
1) = 1 .
8, while a constant K -matrixwith Chew-Mandelstam subtracted at threshold has asimilar χ and a ghost pole at a t √ s = 0 . η ω { P , P } arenon-resonant – the elastic phase-shift reaches only ∼ ◦ at the largest energies we consider ( a t E cm ∼ .
46) – andthey appear to have very similar behavior suggesting weakspin-orbit forces in this channel. The large uncertaintieson the scattering parameters are only slightly increased if we include the effect of the uncertainty on the η mass.For J P C = 1 −− ω f , we in principle may have amore significant amplitude, owing to the scattering beingpossible in an S –wave ( S ). In fact a coupled system ofpartial waves { S , D } is required in order to describethe multiplicity of non-interacting energies shown in themiddle panel of Figure 3. We make use of 16 energy levelstaken from [000] T − and [001] A irreps, noting that theyall have rather large statistical uncertainties and are allcompatible with non-interacting ω f energies.An example parameterization uses a diagonal con-stant K -matrix and a Chew-Mandelstam phase-spacesubtracted at threshold. The resulting constants are sta-tistically compatible with zero in a description of theenergy levels with χ /N dof = 21 . / (16 −
2) = 1 .
54. Evenlarger errors are obtained once the uncertainty on the f mass is accounted for, and considering this and variationover parameterizations, the S –wave phase-shift remainscompatible with zero but with an uncertainty that spreadsover at least ± ◦ at a t E cm = 0 . G. Estimating coupled-channel effects
The previous sections indicate that the finite-volumespectra can be well described assuming that the η ω , η ω , and ω f channels are decoupled, with resonancesonly appearing in η ω . Nevertheless we can attempt alimited study of possible channel coupling.Using a set of 52 energy levels in irreps sensitive to J P C = 1 −− and 3 −− , which includes 4 levels having large η ω overlap, we can try to constrain coupled ( η ω , η ω ) J P C = 1 −− amplitudes parameterized with K ( s ) = 1 m a − s (cid:20) ( g a η ω ) g a η ω g a η ω g a η ω g a η ω ( g a η ω ) (cid:21) + 1 m b − s (cid:20) ( g b η ω ) g b η ω g b η ω g b η ω g b η ω ( g b η ω ) (cid:21) + (cid:20) γ η ω ,η ω γ η ω ,η ω γ η ω ,η ω γ η ω ,η ω (cid:21) , (9)and Chew-Mandelstam phase-space subtracted at s = m a .The J P C = 3 −− amplitude is a single K -matrix pole,elastic in η ω with parameters fixed from previous fits.In practice the a –pole being far below η ω thresholdmeans that the parameter g a η ω is basically unconstrainedso we set it equal to zero. γ η ω ,η ω is always left free,and we consider three fits: g b η ω (cid:54) = 0 , γ η ω ,η ω = 0 ,g b η ω = 0 , γ η ω ,η ω (cid:54) = 0 ,g b η ω (cid:54) = 0 , γ η ω ,η ω (cid:54) = 0 , (10)all three of which provide descriptions of the energy levelswith χ /N dof = 1 .
84. The resulting amplitudes are shownin Figure 16 where we see that only the t η ω ,η ω elementis significantly non-zero in each case, and that it broadly6 FIG. 16. Coupled ( η ω , η ω ) 1 −− amplitudes as in Eqns. 9,10. t η ω ,η ω (purple), t η ω ,η ω (blue), t η ω ,η ω (cyan).For comparison the assumed elastic amplitude from Figure 9is shown in grey. The energy levels used to constrain theamplitude are shown below the abscissa, with those havinglarger overlap onto η ω operators shown in cyan. agrees with the previous elastic analysis. The third fitis somewhat optimistic given the small number of η ω dominated energy levels providing constraint, and indeedit is this amplitude that shows the largest difference withrespect to the elastic case, in particular with it havingthe largest shift in the dip position.These t -matrices have pole singularitieson sheets III (Im k η ω < , Im k η ω <
0) and II (Im k η ω < , Im k η ω >
0) that are qualitativelyunchanged compared to the elastic-only assumption,
FIG. 17. J PC = 2 −− t -matrix, as in Figure 6 (grey), withthe addition of the η ω { P } channel. albeit with larger statistical uncertainties. While it is notwell determined, it is possible that the b –pole may havea modest η ω coupling, a t | c η ω | (cid:46) .
04, that does notchange the total width of the heavier resonance becausethere is so little phase-space for the decay.Scattering of η ω in P –wave can also impact J P C = 2 −− , with the F –wave being unlikely to contributesignificantly so close to threshold. We augment the poleplus constant K -matrix of Eqn. 5 with an extra η ω { P } channel, letting the pole coupling to this new channel andthe extra diagonal constant float freely in a description of96 energy levels, 7 of which have large η ω overlap. Thequality of fit, χ /N dof = . − = 1 .
45, is reasonable, andthe resulting t -matrix elements are shown in Figure 17.Clearly the additional channel has only a weak effect. The t -matrix has a pole (on sheets III and II ) that is in a lo-cation compatible with previous estimates. Similarly the η ω couplings in P – and F –waves are not significantlychanged. There is a pole coupling to η ω that whilesmall, a t | c η ω | ∼ . η ω F –wave coupling.It proves to be the case that the large statistical uncer-tainties on the ω f energy levels prevent any meaningfulattempt at coupled-channel ( η ω , ω f ) analysis. As suchwhile we cannot rule out non-zero couplings to ω f forour resonances, such an outcome seems unlikely given ourability to describe the a huge number of finite-volumeenergy levels using a set of decoupled amplitudes.7 IV. RESONANCE INTERPRETATION
In the previous section we presented descriptions of η ω scattering with J P C = 1 −− , −− and 3 −− findingseveral resonances appearing as poles in the t -matrix. Wechoose to set the lattice scale using the decuplet Ω–baryonmass computed on these lattices, finding a − t = 4655 MeV.Our best estimates of the resonance pole properties, withuncertainties which reflect the variations seen in the pre-vious section are,1 −− , ω a : √ s = 1772(7) ± i (cid:12)(cid:12) c η ω (cid:12)(cid:12) = 656(37) MeV1 −− , ω b : √ s = 1969 +5 − ± i (cid:12)(cid:12) c η ω (cid:12)(cid:12) = 242(93) MeV2 −− , ω : √ s = 1975(10) ± i (cid:12)(cid:12) c η ω ( P ) (cid:12)(cid:12) = 815(30) MeV (cid:12)(cid:12) c η ω ( F ) (cid:12)(cid:12) = 275(37) MeV c η ω ( F ) c η ω ( P ) = − . −− , ω : √ s = 2021(8) ± i +2 − MeV (cid:12)(cid:12) c η ω (cid:12)(cid:12) = 298 +23 − MeV . (11)The 1 −− amplitude features two resonances: a lighterstate with a larger width, and a heavier narrow statewhich has a tight dip and a zero of the amplitude asso-ciated with it. A common parameterization approach inelastic scattering is the effective range expansion , in which k (cid:96) +1 cot δ (cid:96) is expanded as a polynomial in k , truncatedat some finite order, with the polynomial coefficients be-ing free parameters, the first two of which are knownas the scattering length and the effective range . Thejustification for the use of such a series is that it is ex-pected to converge for energies inside a circle centeredat threshold which just touches the left-hand cut, thenearest unconsidered singularity. Even with only twoterms such a parameterization is capable of describing asingle resonance. It is simple to see that our extractedamplitude as shown in e.g. Figure 9 cannot be describedby an effective range expansion, owing to the presence ofa zero in the amplitude, which would require k cot δ todiverge at some positive value of k , which cannot happenfor any finite order polynomial. This appears to presentsomething of a paradox if one takes the view that theleft-hand cut represents the “potential” due to particleexchanges in the crossed channels that act to bind thescattering hadrons into a resonance – such a potentialcannot generate the observed zero. Indeed our findingof a lighter broad resonance and a heavier narrow reso-nance looks quite unnatural in a potential picture where for realistic potential shapes with a centrifugal barrier,one expects the lighter state to have to tunnel through alarger distance than the heavier state in order to decay.The way out of this is to recognize that relativisticscattering systems have more freedom than those drivenby non-relativistic potentials. This can be illustrated byexpressing t ( s ) as a ratio of functions, t ( s ) = N ( s ) /D ( s ),where the numerator houses the left-hand cut, and thedenominator has the unitarity cut. In the case of poten-tial scattering, N ( s ) serves as the potential, and then D ( s ) is uniquely determined from N ( s ) by evaluating adispersive integral. Relativistic scattering differs from thisin that there is the freedom to add an arbitrary numberof poles to D ( s ), known as “CDD poles” [67]. In elasticscattering, these poles will generate zeros in t ( s ) at realvalues of s , and nearby t -matrix poles at complex valuesof s . Although not a unique interpretation, they are oftenassociated with the idea that the underlying theory (QCDin our case) features particles that would be stable were itnot for the presence of pairs of lighter hadrons into whichthey can decay. This of course matches quite closely withthe quark-model picture of q ¯ q mesons that become stableas the quark mass increases.Considering the set of resonances as a whole supportsan interpretation, bolstered by the overlaps discussed inSection II, of the lighter 1 −− state as being dominantly q ¯ q [2 S ], and the remaining three states as being q ¯ q [1 D J ]with only small spin-orbit splittings. Which η ω partialwaves are accessible appears to play a role in setting thestate decays widths: the 3 −− resonance decays only in F –wave and is narrow, while 2 −− also has a P –wave decay,and is significantly broader. There is not any obviousexplanation for why the lighter 1 −− has a much largerwidth than the rather narrow heavier 1 −− state.The leading method for predicting meson decays priorto this calculation was the P –model. When its assumedform for the q ¯ q creation vertex is used with harmonic os-cillator wavefunctions for the bound q ¯ q mesons, simple ex-pressions follow for ratios of decay amplitudes of q ¯ q [1 D J ]mesons to pseudoscalar-vector pairs [10] (where we areneglecting the effect of the small mass differences betweenthe decaying mesons). For the F –wave decays of the 3 −− and 2 −− states, we have g F (3 −− ) g F (2 −− ) = (cid:113) ≈ .
20, and if forcomparison we use the ratio of pole couplings presentedin Eqn. 11, we obtain ∼ . P –wave decays of the 1 −− [1 D ]and 2 −− , the model predicts g P (1 −− ) g P (2 −− ) = √ ≈ .
75, while,assuming the b –pole is the q ¯ q [ D ] state, Eqn. 11 suggests ∼ . P –model provides an expression for the 2 −− F/P amplitude ratio, one that depends only on the ratio of thedecay momentum, k , to the harmonic oscillator parameter, β : g F (2 −− ) g P (2 −− ) = − (cid:113) (cid:16) kβ (cid:17) (cid:18) − (cid:16) kβ (cid:17) (cid:19) − . To describe physical light and strange-quark mesons, it8is usual to choose β = 400 MeV, but since the quarks inour study are somewhat heavier than physical quarks,we might expect the wavefunctions of the mesons tobe smaller, and β to be larger. When the P –modelis applied to charmonium, with still heavier quarks, β = 500 MeV is typical [68]. For our 2 −− resonance, k ≈
504 MeV, such that taking 400 < β <
500 MeV, theequation above predicts an
F/P ratio between − .
13 and − .
22, which while the sign agrees, is somewhat smallerin magnitude than our lattice QCD result of − . A. Estimating J −− meson properties at thephysical u, d quark mass While we have only computed for a single unphysicallyheavy quark mass, we can attempt to extrapolate conse-quences at the physical quark mass. This will necessarilybe a crude estimate in which we will need to imposeadditional phenomenological constraints not following di-rectly from our calculation. We begin by expressing theSU(3) F representations in terms of more familiar mesonstates – the SU(3) F singlet can be decomposed [55, 69]into states labeled by isospin and strangeness as, = √ (cid:16) K + K ∗− + K − K ∗ + − K K ∗ − K K ∗ + π + ρ − + π − ρ + − π ρ − η ω (cid:17) , where we use the PDG naming scheme, except for η , ω by which we mean the neutral flavorless element of thepseudoscalar or vector octet. It is generally accepted thatwith physical mass quarks, the η meson is very close tobeing η with only a small admixture of η , while the ω and φ are nearly ideally flavor mixed, ω = (cid:113) ω + √ ω φ = √ ω − (cid:113) ω . Similar mixing appears in lattice QCD calculations atlarger than physical light quark masses, as can be seenin Figure 1, and indeed the ideal flavor mixing appearsto be present for excited ω (cid:63)J , φ (cid:63)J states also. This mixingposes a challenge for us if we wish to estimate decays ofthese states, as we have only computed the SU(3) F singletcomponent and not the octet. The octet, which for C = − decays to pseudoscalar-vector in the representation(see Ref. [55]), has decomposition, = √ (cid:16) K + K ∗− + K − K ∗ + − K K ∗ − K K ∗ (cid:17) − √ (cid:0) π + ρ − + π − ρ + − π ρ − η ω (cid:17) , and since we would like to have decays to ηω and ηφ ,we also require the process → ⊗ , which has atrivial decomposition = η ω . We will assume that wecan neglect the small admixture of η in the η as a firstapproximation. While we have only computed the singlet decays, wecan relate the octet decays to these if we implement theOZI rule in a way consistent with the assumed ideal flavormixing. We define a notation where g represents the → ⊗ decay coupling, g represents the → ⊗ decay coupling and h represents the → ⊗ decaycoupling. A first condition follows from imposing thatthe decay φ (cid:63) → πρ must be zero for exact OZI – theamplitude for this process is proportional to √ √ g + (cid:16) − (cid:113) (cid:17)(cid:16) − √ (cid:17) g , where the factors √ , − (cid:113) are the combination of singletand octet required to produce the ideally flavor mixed s ¯ sφ (cid:63) . It follows that exact OZI implies g = − √ g . (12)We can establish the accuracy of this relation by comput-ing scattering in the SU(3) F octet representation, whichwill be done in the near future.A second condition following from OZI can be obtainedby insisting that there is zero amplitude for the decay φ (cid:63) → ηω , which follows since every possible diagram forthis process is disconnected. The amplitude is propor-tional to √ (cid:16) − √ (cid:17) √ g + (cid:16) − (cid:113) (cid:17)(cid:16) − √ (cid:17) √ g + (cid:16) − (cid:113) (cid:17)(cid:113) h , where the rightmost factors of √ , (cid:113) are the combina-tions of singlet and octet required to produce the ideallyflavor mixed √ (cid:0) u ¯ u + d ¯ d (cid:1) ω in the decay. Using Eqn. 12,this amplitude is only zero if h = − √ g , (13)and again the accuracy of this expression will be testedin future calculations.Making use of the two OZI conditions, we can writeexpressions for decays of ω (cid:63)J , φ (cid:63)J mesons into pseudoscalar-vector final states solely in terms of the computed singletcoupling g : g (cid:0) φ (cid:63) → KK ∗ (cid:1) = √ √ g g (cid:0) φ (cid:63) → ηφ (cid:1) = g g (cid:0) ω (cid:63) → πρ (cid:1) = √ g g (cid:0) ω (cid:63) → KK ∗ (cid:1) = √ g g (cid:0) ω (cid:63) → ηω (cid:1) = − g , where these couplings represent a single charge state. Infact, if we take the OZI relations seriously, they also allowus to use the singlet coupling to predict some decays of theisoscalar members of the octet, the ρ (cid:63)J mesons, where for9these we can use the decomposition of the I = 1 , I z = +1member of the octet, − (cid:113) (cid:16) K + K ∗ + K K ∗ + (cid:17) + √ π + ω + √ η ρ + , so that g (cid:0) ρ (cid:63) → πω (cid:1) = − √ g g (cid:0) ρ (cid:63) → KK ∗ (cid:1) = √ √ g . Using Γ = g ρM for the partial width of a meson ofmass M into a final state with coupling g , we can obtainΓ (cid:0) ω (cid:63) → πρ (cid:1) = 3 ρM (cid:0) g (cid:1) Γ (cid:0) ω (cid:63) → KK ∗ (cid:1) = 4 ρM (cid:0) g (cid:1) Γ (cid:0) ω (cid:63) → ηω (cid:1) = 1 ρM (cid:0) g (cid:1) Γ (cid:0) φ (cid:63) → KK ∗ (cid:1) = 4 ρM (cid:0) g (cid:1) Γ (cid:0) φ (cid:63) → ηφ (cid:1) = 1 ρM (cid:0) g (cid:1) Γ (cid:0) ρ (cid:63) → πω (cid:1) = 1 ρM (cid:0) g (cid:1) Γ (cid:0) ρ (cid:63) → KK ∗ (cid:1) = 2 ρM (cid:0) g (cid:1) , where the leftmost integers count the final charge states,and where KK ∗ is a shorthand for a sum over all thepossible pseudoscalar-vector kaonic final states.Clearly this combination of SU(3) F symmetry and im-position of exact OZI implies there are many relationshipsthat should hold for the experimental states, but unfor-tunately the lack of a clear experimental picture makesthe relationships rather hard to test. Perhaps the sim-plest is the prediction that, to the extent that an ω (cid:63) isdegenerate with the corresponding ρ (cid:63) , the decay widthof the former into ρπ should be three times larger thanthe decay width of the latter into πω . For the experimen-tal ρ (1690), according to the PDG, the partial widthinto πω is ∼ ω (1670) into πρ is not known, the total widthof this state, 168(10) MeV, provides an upper limit, sothe relation might hold provided that decays other than πρ are significant. For the vector states, the analysis ofDonnachie and Clegg [2] suggests ω (cid:63) (1440) Γ πρ ∼
240 MeV ρ (cid:63) (1463) Γ πω ∼ −
78 MeV , which is in reasonable agreement with a factor of three,while ω (cid:63) (1606) Γ πρ ∼
84 MeV ρ (cid:63) (1730) Γ πω ∼ , is less obviously compatible.We will follow the approach laid out in Ref. [55] toextrapolate our couplings to the physical light quark mass. We interpret the magnitude of the pole couplings | c η ω | asbeing suitable for use as g , and make the simple-mindedassumption that there is no dependence on the light-quarkmass apart from the scaling of the angular-momentumbarrier in a decay with orbital angular momentum (cid:96) , g = (cid:12)(cid:12)(cid:12)(cid:12) k phys ( M phys ) k ( M ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:96) (cid:12)(cid:12) c η ω (cid:12)(cid:12) . This approach breaks SU(3) F symmetry only through themasses of the decay hadrons, and requires us to know therelevant resonance masses for physical light quark masses, M phys , which we will take from the PDG when known,or will estimate when not known.For J P C = 3 −− , using the experimental masses of ρ (1690), ω (1667) and φ (1854) we predictΓ (cid:0) ρ → πω, KK ∗ (cid:1) = 22 , (cid:0) ω → πρ, KK ∗ , ηω (cid:1) = 62 , , (cid:0) φ → KK ∗ , ηφ (cid:1) = 20 , , and we will not quote errors for fear of implying a levelof certainty that surely is not present in such a crudeextrapolation. There is limited scope for comparison toexperiment owing to there being few measured branchingratios. The summed ω estimated partial widths are atleast below the measured total width ∼ φ partial widths compared to 87(25)MeV, and in that case there may be a significant contri-bution from φ → KK . The ρ does have some measuredpartial widths: Γ πω ∼ KKπ ∼ KK ∗ as a sub-process.The P –model has been used to predict decays ofthese states [10, 11]. It has φ decays to KK ∗ , ηφ thatare in good agreement with our estimates, and in addi-tion predicts larger rates to KK and K ∗ K ∗ . The modelpredictions for ω and ρ are also in reasonable agreementwith our estimates, with the ρ also having significantrates to ππ and ρρ . To get access to these additional de-cay modes in the current framework we need to calculateSU(3) F octet scattering.For J P C = 2 −− there are no experimental candidatestates, and as such we will proceed assuming masses thatare approximately equal to the corresponding ρ , ω and φ states. In this case there are both P –wave and F –wavedecays and the total partial width for each channel is anincoherent sum of the two. We predictΓ (cid:0) ρ → πω, KK ∗ (cid:1) = 125 ,
36 MeVΓ (cid:0) ω → πρ, KK ∗ , ηω (cid:1) = 365 , ,
17 MeVΓ (cid:0) φ → KK ∗ , ηφ (cid:1) = 148 ,
44 MeV , which suggests that the ω is likely to have quite a largetotal width, particularly once decays to final states otherthan pseudoscalar-vector are added in. The ρ and ω might be narrower, particularly given that the largest0phase-space modes ππ and KK are not accessible to a2 − resonance.The P –model has φ partial widths that are in goodagreement with our estimates, while the ω and ρ comeout lower in the model. The model predicts a very large ρ → a π rate that leads to a rather large total width forthis state.For J P C = 1 −− we have the problem of associatingour two resonances, the lighter broad state a , and theheavier narrow state b , with the physical states. Thesimplest assumption is that in each flavor channel, thelighter state is purely a and the heavier state purely b ,with no evolution in a possible basis-state mixing anglewith change in light-quark mass. With this assignmentwe predict Γ (cid:0) ρ a → πω, KK ∗ (cid:1) = 133 , (cid:0) ω a → πρ, KK ∗ , ηω (cid:1) = 384 , , (cid:0) φ a → KK ∗ , ηφ (cid:1) = 154 ,
25 MeV , and we can say little more than that these summed partialwidths do not over saturate the experimental total widthsof the ρ (1450), ω (1420) and φ (1680). For the heavierstate we predictΓ (cid:0) ρ b → πω, KK ∗ (cid:1) = 9 , (cid:0) ω b → πρ, KK ∗ , ηω (cid:1) = 25 , , (cid:0) φ b → KK ∗ , ηφ (cid:1) = 13 , , which appears to suggest that unless the other alloweddecays of the ρ (1700), ω (1650) and a hypothetical φ (1900)provide large partial widths, these states should be muchnarrower than they seem to be in experiment. We donot have a good explanation of this observation, althoughsome degree of basis-state mixing of a , b into the physicalstates might share the decays more evenly and give riseto two moderately broad states.The P –model, assuming the lighter state is pure q ¯ q [2 S ] has somewhat larger decay rates for the φ state,and rates for the ω and ρ states that are in reasonableagreement with our estimates. Assuming the heavier stateis pure q ¯ q [1 D ], the model predicts decays for the hypo-thetical φ (cid:63) that are much larger than our estimates, andalso has a huge ∼
500 MeV branch into K K . A similarpattern is observed for the ω (cid:63) and ρ (cid:63) states, indicatingquite poor agreement with our estimates. V. SUMMARY
In this paper we have reported on a first lattice QCDstudy of excited mesons with J −− quantum numbers,computing in a version of QCD having exact SU(3) F sym-metry, and focussing on the singlet representation. Wefound that the 1 −− , 2 −− and 3 −− partial waves at lowenergies have only a single strongly-interacting channel ofpseudoscalar-vector scattering, η ω , with other kinemat-ically open channels being decoupled and weakly inter-acting. Constraining scattering amplitudes using nearly200 energy levels across five lattice volumes, we found aunique picture featuring four resonances.A single, isolated narrow resonance with 3 −− ap-pears to match with the well-known experimental states( ρ , ω , φ ). A first computation within lattice QCD of2 −− amplitudes, which appear as dynamically coupled { P , F } partial waves yields a much broader resonancefor which there is no experimental evidence to date. Therather novel 1 −− partial wave amplitude features a lighterbroad resonance and a heavier narrow resonance. A tightdip and a zero in the amplitude appears on the real en-ergy axis, very close to the heavier resonance pole. Wesummarize our amplitudes and our best estimates of theresonance poles in Figure 18.In a natural extension of the work reported on in thispaper, our next calculation will consider the SU(3) F octetsystem on the same lattices. This will require a first FIG. 18. Upper panel: η ω scattering amplitudes as pre-sented in Section III E. Lower panel: Our best estimate forresonance poles from Eqn. 11, including variation of amplitudeparameterization, scattering meson masses and anisotropy inthe error estimates. Scale set to MeV units using the Ω-baryonmass. −− is of partic-ular interest given the suggestion that the next resonanceabove those we have extracted is expected to be a hybridmeson. The challenge here is the need to implement a sofar underdeveloped extension of finite-volume three-bodyformalism in which two- and three- meson sectors arecoupled, but we expect to see progress in this directionin the near future.Another interesting expansion of scope of the currentstudy would consider the process in which a vector cur-rent (describing the virtual photon in e + e − annihilation)produces the η ω system with J P C = 1 −− . If the quarkmodel picture of the two vector resonances is correct, we’dexpect the q ¯ q (cid:2) D (cid:3) state to contribute very little (it haszero wavefunction at the origin, and only appears throughthe suppressed second derivative), while the q ¯ q (cid:2) S (cid:3) state could be significant. Such a calculation would bea first step towards a first-principles QCD based phe-nomenology to be used to describe resonance productionin e + e − , a process of primary importance at experimentslike BES III.Developing an understanding of the excited J −− reso-nances in QCD is timely, as we expect a huge new experi-mental data set in photoproduction from the GlueX exper-iment using which we can obtain better constraint on theproperties of these states. It remains to be seen whetheran extension of the method presented in Ref. [70, 71] forproduction of the ρ resonance can be practically appliedto the current case in order to describe the pion-exchangecontribution to photoproduction of excited J −− mesons.The calculation presented in this paper is the first steptowards a QCD-based theoretical understanding of themysterious excited J −− resonances. ACKNOWLEDGMENTS
We thank our colleagues within the Hadron Spec-trum Collaboration. Special thanks to Antoni Wossand Christopher Thomas for their assistance computingcorrelation functions with computer resources at Cam-bridge, and Robert Edwards for his support with usingthe
Redstar software system. JJD and CTJ acknowledgesupport from the U.S. Department of Energy contract DE-SC0018416 at William & Mary, and contract DE-AC05-06OR23177, under which Jefferson Science Associates,LLC, manages and operates Jefferson Lab.The software codes
Chroma [72] and
QUDA
Appendix A: Operator basis
Details of the fermion-bilinear operators, the methodfor obtaining optimized “single-meson” operators, andthe construction of meson-meson operators can be foundin Refs. [6, 34, 45, 49, 50, 56, 57]. The operator basisused in the current calculation is presented in Tables V,VI, VII where the meson-meson operators are listed inorder of increasing non-interacting energy. Those cases inwhich more than one construction appears with the samenon-interacting energy are indicated by the multiplicity , { N } . Appendix B: Left-hand cut singularities
The complete scattering amplitude for η ω → η ω , T ( s, t ), has properties which follow from crossing symme-try, the simplest of which is that unitarity should apply notjust in the s -channel ( η ω → η ω ) but also in the (sym-metric) u -channel, and in the t -channel ( η η → ω ω ).The impact of the required discontinuities across theunitarity branch cut in Mandelstam t and u when theamplitude is projected into s -channel partial waves is togenerate left-hand cuts , i.e. branch cuts which typicallylie on the real axis to the left of the s -channel thresholdin the complex s -plane.While the discontinuity across these cuts requires knowl- edge of the scattering dynamics, the position of thesecuts is simply a function of the masses of the scatteringhadrons. Unitarity in the u -channel implies a cut run-ning along the real s axis from −∞ to (cid:0) m ( ω ) − m ( η ) (cid:1) ,while unitarity in the t -channel provides a cut along theentire negative real s axis, and a circular cut of radius s = m ( ω ) − m ( η ) .In the current case there are additional cuts due to thefact that stable mesons appear as bound-state poles inthe crossed channels. ω appears in the u -channel, andgenerates an extra “short-cut” from s = ( m ( ω ) − m ( η ) ) m ( ω ) to s = 2 (cid:0) m ( ω ) + m ( η ) (cid:1) − m ( ω ) . f appears in the t -channel and generates a cut running from −∞ to s = (cid:16)(cid:113) m ( η ) − m ( f ) + (cid:113) m ( ω ) − m ( f ) (cid:17) . Usingthe hadron masses in Table II, we find that the rightmostextent of the left-hand cut lies at a t √ s = 0 .
299 and isdue to either of the stable exchanges, ω , f .For the process η ω → η ω , the nearest left-handcut is at a t √ s = 0 . f exchange in the t -channel, and for ω f → ω f is at a t √ s = 0 .
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20 ¯ ψ Γ ψ ×
20 ¯ ψ Γ ψ ×
20 ¯ ψ Γ ψ ×
20 ¯ ψ Γ ψ × η [100] ω [100] η [100] ω [100] η [100] ω [100] η [100] ω [100] η [100] ω [100] η [110] ω [110] { } η [110] ω [110] { } η [110] ω [110] { } η [111] ω [111] ω [000] f [000] ω [000] f [000] ω [000] f [000] ω [000] f [000] ω [000] f [000] ω [100] f [100] { } ω [100] f [100] { } ω [100] f [100] { } η [100] ω [100] η [100] ω [100] η [100] ω [100] [ ] E − ¯ ψ Γ ψ ×
12 ¯ ψ Γ ψ ×
12 ¯ ψ Γ ψ ×
12 ¯ ψ Γ ψ ×
12 ¯ ψ Γ ψ × η [100] ω [100] η [100] ω [100] η [100] ω [100] η [100] ω [100] η [100] ω [100] η [110] ω [110] { } η [110] ω [110] { } η [110] ω [110] { } η [111] ω [111] η [100] ω [100] η [100] ω [100] η [100] ω [100] [ ] T − ¯ ψ Γ ψ ×
18 ¯ ψ Γ ψ ×
18 ¯ ψ Γ ψ ×
18 ¯ ψ Γ ψ ×
18 ¯ ψ Γ ψ × η [100] ω [100] η [100] ω [100] η [100] ω [100] η [100] ω [100] η [100] ω [100] η [110] ω [110] { } η [110] ω [110] { } η [110] ω [110] { } η [111] ω [111] { } ω [100] f [100] ω [100] f [100] ω [100] f [100] η [100] ω [100] η [100] ω [100] η [100] ω [100] [ ] A − ¯ ψ Γ ψ × ψ Γ ψ × ψ Γ ψ × ψ Γ ψ × ψ Γ ψ × η [110] ω [110] η [110] ω [110] η [110] ω [110] TABLE V. Operators used to compute matrices of correlations functions in rest-frame irreps. ¯ ψ Γ ψ × N indicates the maximumnumber of “single-meson” operators included in the basis.[29] S. Aoki et al. (CS), Phys. Rev. D84 , 094505 (2011),arXiv:1106.5365 [hep-lat].[30] C. B. Lang, D. Mohler, S. Prelovsek, and M. Vid-mar, Phys. Rev.
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D93 , 094506 (2016),arXiv:1602.05122 [hep-ph]. [ ] A ¯ ψ Γ ψ ×
18 ¯ ψ Γ ψ ×
18 ¯ ψ Γ ψ ×
18 ¯ ψ Γ ψ ×
18 ¯ ψ Γ ψ × η [100] ω [110] η [100] ω [110] η [100] ω [110] η [100] ω [110] η [100] ω [110] η [110] ω [100] η [110] ω [100] η [110] ω [100] η [110] ω [100] η [110] ω [100] η [110] ω [111] η [110] ω [111] η [110] ω [111] η [111] ω [110] η [111] ω [110] η [110] ω [210] η [210] ω [110] ω [100] f [000] ω [100] f [000] ω [100] f [000] ω [100] f [000] ω [100] f [000] ω [000] f [100] ω [000] f [100] ω [000] f [100] ω [000] f [100] ω [000] f [100] ω [110] f [100] { } ω [100] f [110] { } η [100] ω [110] η [110] ω [100] [ ] B ¯ ψ Γ ψ × ψ Γ ψ × ψ Γ ψ × ψ Γ ψ × ψ Γ ψ × η [100] ω [110] η [100] ω [110] η [100] ω [110] η [100] ω [110] η [100] ω [110] η [110] ω [100] η [110] ω [100] η [110] ω [100] η [110] ω [100] η [110] ω [100] η [110] ω [111] { } η [110] ω [111] { } η [110] ω [111] { } η [111] ω [110] { } η [111] ω [110] { } η [111] ω [110] { } η [110] ω [210] η [210] ω [110] ω [110] f [100] { } ω [100] f [110] { } η [100] ω [110] η [110] ω [100] [ ] B ¯ ψ Γ ψ × ψ Γ ψ × ψ Γ ψ × ψ Γ ψ × ψ Γ ψ × η [100] ω [110] { } η [100] ω [110] { } η [100] ω [110] { } η [100] ω [110] { } η [100] ω [110] { } η [110] ω [100] { } η [110] ω [100] { } η [110] ω [100] { } η [110] ω [100] { } η [110] ω [100] { } η [110] ω [111] η [110] ω [111] η [110] ω [111] η [111] ω [110] η [111] ω [110] η [110] ω [210] { } ω [110] f [100] ω [100] f [110] η [100] ω [110] { } η [110] ω [100] { } TABLE VI. As Table V for irreps with (cid:126)P = [100].[47] G. Moir, M. Peardon, S. M. Ryan, C. E. Thomas, andD. J. Wilson, JHEP , 011 (2016), arXiv:1607.07093[hep-lat].[48] R. A. Briceno, J. J. Dudek, R. G. Edwards, and D. J.Wilson, Phys. Rev. D97 , 054513 (2018), arXiv:1708.06667[hep-lat].[49] A. Woss, C. E. Thomas, J. J. Dudek, R. G. Edwards,and D. J. Wilson, JHEP , 043 (2018), arXiv:1802.05580[hep-lat].[50] A. J. Woss, C. E. Thomas, J. J. Dudek, R. G. Ed-wards, and D. J. Wilson, Phys. Rev. D100 , 054506 (2019), arXiv:1904.04136 [hep-lat].[51] M. T. Hansen and S. R. Sharpe, Ann. Rev. Nucl. Part.Sci. , 65 (2019), arXiv:1901.00483 [hep-lat].[52] R. G. Edwards, B. Joo, and H.-W. Lin, Phys. Rev. D78 ,054501 (2008), arXiv:0803.3960 [hep-lat].[53] H.-W. Lin et al. (Hadron Spectrum), Phys. Rev.
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13 ¯ ψ Γ ψ ×
21 ¯ ψ Γ ψ ×
13 ¯ ψ Γ ψ ×
21 ¯ ψ Γ ψ × η [100] ω [100] η [100] ω [100] η [100] ω [100] η [100] ω [100] η [100] ω [100] η [100] ω [111] η [100] ω [111] η [100] ω [111] η [100] ω [111] η [100] ω [111] η [110] ω [110] { } η [110] ω [110] { } η [110] ω [110] { } η [110] ω [110] { } η [111] ω [100] η [111] ω [100] η [111] ω [100] η [111] ω [100] η [100] ω [210] η [100] ω [210] η [100] ω [210] η [110] ω [200] η [200] ω [110] η [210] ω [200] ω [110] f [000] ω [110] f [000] ω [110] f [000] ω [110] f [000] ω [110] f [000] ω [000] f [110] ω [000] f [110] ω [000] f [110] ω [000] f [110] ω [100] f [100] { } ω [100] f [100] { } ω [100] f [100] { } ω [100] f [100] { } η [100] ω [100] η [100] ω [100] η [100] ω [100] η [100] ω [100] [ ] A ¯ ψ Γ ψ ×
21 ¯ ψ Γ ψ ×
21 ¯ ψ Γ ψ ×
21 ¯ ψ Γ ψ ×
21 ¯ ψ Γ ψ × η [100] ω [110] η [100] ω [110] η [100] ω [110] η [100] ω [110] η [100] ω [110] η [110] ω [100] η [110] ω [100] η [110] ω [100] η [110] ω [100] η [110] ω [100] η [100] ω [211] η [100] ω [211] η [200] ω [111] η [100] ω [211] η [110] ω [210] { } η [210] ω [110] { } η [211] ω [100] ω [111] f [000] ω [111] f [000] ω [111] f [000] ω [111] f [000] ω [111] f [000] ω [110] f [100] { } ω [110] f [100] { } ω [110] f [100] { } ω [110] f [100] { } ω [110] f [100] { } ω [000] f [111] ω [000] f [111] ω [000] f [111] ω [000] f [111] ω [000] f [111] ω [100] f [110] { } ω [100] f [110] { } ω [100] f [110] { } ω [100] f [110] { } ω [100] f [110] { } η [100] ω [110] η [100] ω [110] η [100] ω [110] η [100] ω [110] η [100] ω [110] η [110] ω [100] η [110] ω [100] η [110] ω [100] η [110] ω [100] η [110] ω [100] TABLE VII. As Table V for irreps with (cid:126)P = [110] , [111].Parameterization Phase-space N pars χ /N dof K = g a m a − s + g b m b − s naive 4 CM(pole a ) 4 K = g a m a − s + g b m b − s + γ naive 5 1.38CM(pole a ) 5 1.36CM(pole b ) 5 1.36 K = g a m a − s + g b m b − s + γ s naive 5 1.37CM(pole a ) 5 1.35CM(pole b ) 5 1.35 K = g a m a − s + g b m b − s + γ s CM(pole a ) 5 1.35TABLE VIII. 1 −− amplitude parameterizations, as plotted inFigures 10, 11, 12, 13. χ /N dof values in italics indicate theamplitudes shown by dashed curves. [55] A. J. Woss, J. J. Dudek, R. G. Edwards, C. E. Thomas,and D. J. Wilson, (2020), arXiv:2009.10034 [hep-lat].[56] C. E. Thomas, R. G. Edwards, and J. J. Dudek, Phys.Rev. D85 , 014507 (2012), arXiv:1107.1930 [hep-lat].[57] J. J. Dudek, R. G. Edwards, and C. E. Thomas, Phys.Rev.
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