Decays of an Exotic 1-+ Hybrid Meson Resonance In

PHYSICAL REVIEW D 103, 054502 (2021)

Decays of an exotic 1 − + hybrid meson resonance in QCD

† ‡ ∥ Antoni J. Woss ,1,* Jozef J. Dudek,2,3, Robert G. Edwards,2, Christopher E. Thomas ,1,§ and David J. Wilson 1,

(for the Hadron Spectrum Collaboration)

1DAMTP, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom 2Thomas Jefferson National Accelerator Facility, 12000 Jefferson Avenue, Newport News, Virginia 23606, USA 3Department of Physics, College of William and Mary, Williamsburg, Virginia 23187, USA

(Received 30 September 2020; accepted 23 December 2020; published 10 March 2021)

We present the first determination of the hadronic decays of the lightest exotic JPC ¼ 1−þ resonance in lattice QCD. Working with SU(3) flavor symmetry, where the up, down and strange-quark masses approximately match the physical strange-quark mass giving mπ ∼ 700 MeV, we compute finite-volume spectra on six lattice volumes which constrain a scattering system featuring eight coupled channels. Analytically continuing the scattering amplitudes into the complex-energy plane, we find a pole singularity corresponding to a narrow resonance which shows relatively weak coupling to the open pseudoscalar– pseudoscalar, vector–pseudoscalar and vector–vector decay channels, 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 π1 resonance decaying dominantly through the b1π 0 mode with much smaller decays into f1π, ρπ, η π and ηπ. A large total width is potentially in agreement 0 with the experimental π1ð1564Þ candidate state observed in ηπ, η π, which we suggest may be heavily suppressed decay channels.

DOI: 10.1103/PhysRevD.103.054502

I. INTRODUCTION theory and experiment. One particular focus is on hybrid mesons in which a quark-antiquark pair is coupled to The composition of hadrons has been the subject of an excitation of the gluonic field. Such states are an experimental and theoretical studies for many decades. attractive target because the additional quantum numbers Historically, the majority of mesons could be understood in JPC a quark-model picture where they consist of a quark- potentially supplied by the gluonic field allow for qq¯ exotic antiquark pair (qq¯). There are some notable long-standing combinations not allowed to a system. These PC 0−− 0þ− 1−þ 2þ−… exceptions that do not appear to fit into this framework, J ¼ ; ; ; serve as a smoking-gun sig- such as the light scalar mesons, and more recently it has nature that a novel state has been observed. been challenged by the observation of a number of Suggestions that hybrid mesons are a feature of QCD are unexpected structures in the charm and bottom sectors. long-standing, but until recently predictions of their proper- In principle mesons can contain constituent combina- ties came only within models whose connection to QCD is – tions beyond qq¯, but whether QCD allows for such not always clear [1 7]. While dynamical pictures like the arrangements continues to motivate investigations in both flux-tube model, the bag model, and constituent gluon approaches generally agree that hybrids form part of the meson spectrum, some with exotic JPC, they differ in *[email protected] PC 1−þ † details. A common feature is that typically a J ¼ [email protected] ‡ state (labeled π1 when the state has isospin-1) appears with [email protected] §[email protected] a mass somewhere above 1.5 GeV. A particular challenge ∥ [email protected] has been for these models to provide reliable predictions for the decay properties of hybrid mesons, which we expect to Published by the American Physical Society under the terms of appear as resonances that can decay into several final states. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to Having some advance knowledge of which final states are the author(s) and the published article’s title, journal citation, more heavily populated in their decay is useful to experi- and DOI. Funded by SCOAP3. ments which perform amplitude analyses final state by final

2470-0010=2021=103(5)=054502(33) 054502-1 Published by the American Physical Society WOSS, DUDEK, EDWARDS, THOMAS, and WILSON PHYS. REV. D 103, 054502 (2021) state. A folklore has developed, largely following from time dependences of these matrices led to predictions for models in which the hybrid decay proceeds by the breaking the spectrum of mesons with a wide range of JPC. The of an oscillating tube of gluonic flux or through conversion spectra obtained, for several values of the light-quark mass, of a constituent gluon to a qq¯ pair [8–13], where decays show a strong qualitative similarity to the experimental featuring only the lightest hadrons are suppressed, meson spectrum, but also feature clear indications of exotic 0 PC such as π1 → ηπ; η π; ρπ, while decays which include a J states with notably a lightest π1. A phenomenology more excited hadron are prominent, such as π1 → b1π. was developed [14] based upon the observation that this Whether these results are really a feature of QCD, or reflect state, along with states having JPC ¼ 0−þ; 2−þ and 1−− at the assumptions built into the flux tube (a picture whose similar masses, have large matrix elements to be produced validity looks increasingly unlikely [14]) or constituent by operators of the form ψ¯ ΓtaψBa, which has the qq¯ pair in gluon pictures, has yet to be established. a color octet with the color neutralized by the chromo- The experimental focus has remained largely on the magnetic field operator, Ba. It was proposed that this large π1, and historically the picture has been quite confused overlap signals that these states are hybrid mesons, and they [15,16]. Analyses have mostly considered the ηπ, η0π and systematically appear roughly 1.3–1.4 GeV above the ρπ → πππ final states which have the lowest possible lightest vector meson, even for quark masses corresponding multiplicities. Recent datasets of unprecedented statistics to charmonium [27,28]. The picture extends into the baryon from COMPASS provide our clearest picture [17]: A broad spectrum [29], where hybrid baryons can be identified, bump in ηπ peaking near 1400 MeV appears to match although in this case exotic quantum numbers are not poorly with another bump in η0π peaking near 1600 MeV. possible. These results are similar to those observed in earlier While these calculations have provided us with the first experiments which were interpreted as two resonances, picture of hybrid hadrons directly connected to QCD, the π1ð1400Þ and π1ð1600Þ, with there being some further picture is clearly incomplete. These excited hadrons are not evidence for the heavier resonance in the ρπ final state. stable particles having a definite mass, rather they are A recent analysis of the COMPASS data by JPAC comes unstable resonances which should appear as enhancements to a different conclusion [18]: The two bumps in ηπ; η0π are in the scattering of lighter stable hadrons, but this was actually due to a single resonance decaying into both final neglected in the calculations. The resonant nature of these states. They proceed by parametrizing the production states has consequences for the spectrum calculated in process and the scattering of the coupled-channel ηπ; η0π lattice QCD, where the important difference with respect to system, respecting unitarity in these two channels. The experiment is the use of a finite spatial volume. scattering t-matrix is constrained for real values of the The discrete spectrum of eigenstates in a finite periodic energy using experimental data. When the amplitude is spatial volume can be related to infinite-volume scattering considered for complex values of the energy, a single pole amplitudes using an approach that is commonly referred to singularity is found which can be interpreted as one as the Lüscher method [30–33], a formalism that has been resonance with a mass slightly below 1.6 GeV and a width extended to systems moving with respect to the lattice, of around 500 MeV. A combined analysis of COMPASS hadrons with nonzero spin and any number of coupled and Crystal Barrel data [19] which appeared while this hadron-hadron channels [34–44]. Obtaining the complete paper was in the final stages of preparation finds a very spectrum of eigenstates requires a larger basis of operators similar mass, but a slightly smaller width ∼388 MeV. than that used in the calculations referred to above [45,46], Currently the GlueX experiment [20,21] is collecting and it has been demonstrated that operators constructed as large datasets using photoproduction in which they will products of meson operators are sufficient. The coupled- search for hybrid mesons. Since the higher multiplicity channel t-matrix can then be obtained through the use of final states suggested as preferred by the flux-tube picture, parametrizations which are constrained at the discrete real e.g., π1 → b1π → ðωπÞπ → πππππ, are much harder to values of energy provided by the finite-volume spectra. The analyze than those investigated in COMPASS, it would be t-matrix is then continued into the complex-energy plane of benefit to have some evidence within QCD that these and any pole singularities identified. From these the mass channels are in fact dominant in the decays of hybrid and width of a resonance can be determined, along with its mesons. It is to this task that we turn our attention in this couplings to different decay channels, in what can be paper, using the technique known as lattice QCD. argued to be the most rigorous way possible. In the past few Lattice QCD, which offers a first-principles numerical years this approach has been used extensively in the study approach to QCD, has matured to the point where it has of elastic scattering, in cases like isospin-1 ππ where the ρ been able to make some fairly definitive statements about resonance appears [45–61], and in several pioneering the excited spectrum of hadrons. In Refs. [22–26], bases of calculations of coupled-channel scattering [46,62–68]. composite operators built from fermion bilinears and up to In this paper we report on the first calculation of an three gauge-covariant derivatives were used to construct exotic JPC ¼ 1−þ meson appearing as a resonance in matrices of two-point correlation functions. Analyzing the coupled-channel meson-meson scattering. By working

054502-2 DECAYS OF AN EXOTIC 1−þ HYBRID MESON … PHYS. REV. D 103, 054502 (2021) with an exact SU(3) flavor symmetry where the u, d quark to reliable determinations of the mass (or energy with masses are raised to the physical strange-quark mass, we nonzero momentum) and optimized operators which relax reduce the effective number of decay channels and make to the desired state more rapidly than any single operator in three-body decays irrelevant. the basis (see for example Fig. 2 of Ref. [75] or Fig. 3 The paper is structured as follows: In Sec. II we review of Ref. [76]). the techniques needed to compute finite-volume spectra in The reduced rotational symmetry of a cubic lattice lattice QCD and to relate these to scattering amplitudes. means that meson states are characterized not by integer Section III discusses the generalities of working with an spin values and parity, but by the irreducible representa- 3 exact SUð ÞF symmetry. In Sec. IV we present calculations (irreps) of the octahedral group or the appropriate tional details and finite-volume spectra relevant for a 1−þ little group1 for nonzero momentum, with the allowed resonance on six lattice volumes. In Sec. V these spectra are momenta in an L × L × L periodic volume given by used to constrain a scattering matrix of eight channels using ⃗ 2π p ¼ L ðnx;ny;nzÞ where ni are integers. In general, this a range of parametrizations, and in Sec. VI these para- means that examination of a particular irrep requires metrizations are analytically continued into the complex- considering multiple JP values, but the group theory energy plane where a resonance pole singularity is found. describing how spin subduces into irreps [77,78] and the Section VII interprets the decay couplings obtained from construction of operators in appropriate irreps [23,25] are the residue of the resonance pole, comparing to existing well understood. models of hybrid meson decay, and attempts an extrapo- When considering energies near and above meson- lation to physical kinematics. Finally, we summarize in meson decay thresholds, a basis of only fermion bilinears Sec. VIII. Some additional technical points are discussed in is insufficient to capture the complete finite-volume spec- the Appendixes, and details of the various parametrizations trum, while augmenting this single-meson-like basis with a used can be found in Supplemental Material [69]. set of meson-meson-like constructions has proven to be highly effective [45,46]. Such operators are built by II. RESONANCES IN LATTICE QCD combining optimized stable meson operators using appro- M M Our approach to determining resonant physics in lattice priately weighted products. For an 1 2-like operator with ⃗ QCD requires the computation of discrete spectra in the overall momentum P in irrep Λ, finite-volume defined by the lattice, and analysis of these X X OΛμ† ⃗ C ⃗Λ μ; ⃗ Λ μ ; ⃗ Λ μ spectra in terms of a scattering matrix using the Lüscher M1M2 ðPÞ¼ ð½P ; ½p1 1; 1 ½p2 2; 2Þ μ μ method. In this section we will review our approach for 1; 2 pˆ 1;pˆ 2 doing this; if further details are required, the field is Λ1μ1† Λ2μ2† × ΩM ðp⃗1ÞΩM ðp⃗2Þ: reviewed in Ref. [70]. 1 2 Λ μ † Here the optimized stable meson operator Ω i i ðp⃗Þ for Mi i A. Finite-volume spectra ⃗ meson Mi with momentum pi, is labeled by the irrep Λi, In order to constrain the scattering t-matrix over a range and the row of that irrep, μi (analogous to the Jz value for a of energies, we are required to calculate a large number of spin-J meson in an infinite-volume continuum). The sum discrete finite-volume levels sampling the energy region. over momentum directions related by allowed cubic rota- An approach which has proven to be highly effective for the ⃗ tions is subject to the constraint that p⃗1 þ p⃗2 ¼ P. The reliable extraction of many excited states is through the generalized Clebsch-Gordan coefficients C are discussed diagonalization of a large matrix of correlation functions, in Ref. [75]. 0 O O† 0 0 CijðtÞ¼h j iðtÞ j ð Þj i. This can be achieved by solv- Each meson-meson operator can be characterized by the ing a generalized eigenvalue problem [71–73], with our magnitudes of meson momenta that went into its con- implementation described in Refs. [23,74]. This approach struction, ðjp⃗1j; jp⃗2jÞ. This leads to a natural truncation of makes use of orthogonality between energy eigenstates to the basis of operators following from the energy we would distinguish contributions of even near-degenerate states, expect if the mesons had no residual interactions, supplying their energies through the time dependence of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the eigenvalues while the eigenvectors provide linear ð2Þ 2 ⃗ 2 2 ⃗ 2 combinations of the basis operators which serve as the En:i: ¼ m1 þjp1j þ m2 þjp2j : optimal operator, in the variational sense, for each state. One possible basis of operators fOig that can be used to Clearly, as the individual meson momenta increase, the form a matrix of meson correlation functions is built from noninteracting energy increases, and at some point fermion bilinears featuring gauge-covariant derivatives. A becomes sufficiently far above the energy region of interest large basis can be constructed both with zero momentum [23] and nonzero momentum [25]. For the determination of 1The set of allowed octahedral group rotations and reflections stable hadrons, such a basis is typically sufficient and leads which leave the momentum vector unchanged.

054502-3 WOSS, DUDEK, EDWARDS, THOMAS, and WILSON PHYS. REV. D 103, 054502 (2021) that we are justified in not including that operator, or any decomposition of a suitable transformation of the matrix above it, in our basis. under the determinant, was presented in Ref. [80]. Constructing operators which resemble meson-meson- Equation (1) is capable of describing any number of meson systems, relevant in the energy region above three- coupled hadron-hadron channels, but must be supplemented meson thresholds, can be done by a recursive application of with further formalism once three-hadron channels are the approach described above [68]. However, one subtlety accessible. Recent progress is reviewed in Refs. [81,82].3 that arises here is that intermediate meson-meson subsys- An approach that allows computed finite-volume spectra tems may feature resonant behavior which a single meson- to constrain scattering amplitudes is to propose paramet- meson operator alone will not efficiently capture. In this rizations of tðEcmÞ, whose parameters can be varied, with case, one or more optimized operators can be constructed the corresponding finite-volume spectra from the solution for the lowest-energy eigenstates in the meson-meson of Eq. (1) at each iteration compared to the computed subsystem by diagonalizing a matrix of correlation func- spectra [75]. In this way, a χ2 can be defined which can tions formed from a basis of single-meson-like and meson- be minimized to find the best description of the computed meson-like operators. These optimized operators are then lattice QCD spectra [Eq. (9) in Ref. [45] ]. Use of a combined with the remaining optimized stable meson K-matrix in the parametrization of the t-matrix ensures operator to form three-meson-like operators that efficiently coupled-channel unitarity, and sensitivity to the particular interpolate the energy eigenstates. Details of this type of choice of form chosen for KðEcmÞ can be explored by construction are given in Ref. [68]. varying the form [39]. The inclusion of multimeson and isospin-0 single-meson This method provides coupled-channel amplitudes con- operators in our bases naturally leads to Wick contractions strained for real values of Ecm, but use of explicit functional which feature quark-antiquark annihilations; in the context forms in the parametrizations means that we can analyti- of lattice QCD these appear via t-to-t quark propagators. cally continue into the complex-energy plane to explore the The distillation approach to computing correlation func- singularity content of the t-matrix. Poles at complex values tions [79] efficiently handles these, along with the required of Ecm can be identified with resonances, with the real and source-sink propagators, without the need to make any imaginary parts of the pole position having an interpreta- further approximations or to introduce any stochastic noise. tion in terms of, respectively, the mass and width of the The propagators, which factorize from the operator con- resonance. Factorizing the residues of elements of t at the structions, are extremely general. They can be extensively pole position leads to decay couplings of the resonance to reused in other calculations which require propagation of the various scattering channels. The statistical uncertainty the same flavor of quarks such that the computational cost originating in the finite number of Monte Carlo samples in of obtaining them is spread over many physics results. the lattice QCD calculation can be propagated through this process, and in addition the scatter over parametrizations B. Scattering amplitudes can be used to estimate a systematic uncertainty from the Once the finite-volume spectrum has been extracted from choice of parametrization. a variational analysis of a matrix of correlation functions it This approach has been applied successfully in several can be used as a constraint on the energy dependence of the recent calculations of coupled-channel scattering, most coupled-channel t-matrix. The relationship is encoded in the notably in a series of papers computing on three lattice Lüscher quantization condition [30–44], volumes with mπ ∼ 391 MeV. In the first calculations [62,63], coupled πK, ηK scattering was investigated. A det½1 þ iρtð1 þ iMÞ ¼ 0; ð1Þ virtual bound state and a broad resonance were found in JP ¼ 0þ, a bound state in 1−, and there was evidence for a where the diagonal matrix of phase-space factors ρðEcmÞ narrow resonance in 2þ, but for all these JP the coupling to and MðEcm;LÞ are known functions of essentially kin- the ηK channel was found to be small in the energy region ematic origin; see Ref. [80] for our conventions. The matrix studied. In Ref. [64], the JP ¼ 0þ coupled πη;KK¯ scatter- space over which the determinant acts is the set of partial ing sector was considered, where an asymmetrical peak in waves subduced into a particular irrep, for all kinematically πη → πη at the KK¯ threshold was found to correspond to a accessible meson-meson scattering channels. For a given 2 resonance pole that could be compared to the experimental t t Ecm P þ þ -matrix ð Þ, the discrete set of solutions of this equation a0ð980Þ. In Ref. [66], the J ¼ 0 and 2 coupled ½EcmðLÞn¼1;2;… for a fixed value of L is the finite-volume ππ;KK;¯ ηη isospin-0 sectors were studied. The scalar spectrum in an L × L × L periodic box. A practical amplitudes show a sharp dip in ππ → ππ at KK¯ threshold approach for reliably finding solutions to this equation that could be associated with a resonance pole related to the when there are multiple partial waves and/or hadron-hadron scattering channels, which makes use of an eigenvalue 3What role the experimentally observed dominance of quasi- pffiffiffi pffiffiffi two-body isobars plays in these formalisms is not yet known, but 2Related to the scattering S-matrix via S ¼ 1 þ 2i ρt ρ. it may lead to considerable simplifications in practice.

054502-4 DECAYS OF AN EXOTIC 1−þ HYBRID MESON … PHYS. REV. D 103, 054502 (2021) experimental f0ð980Þ, while a rapid turn-on of ππ at member cannot easily be associated with either the ω or the threshold was found to be due to a bound state related ϕ, as the experimental ω is believed to have approximate ¯ to the σ=f0ð500Þ. The tensor sector was more straightfor- quark content uu¯ þ dd, while the ϕ is dominantly ss¯. ward, with clear bumps related to two resonances poles, the These correspond to significant admixtures of the octet lighter of which was found to be dominantly coupled to ππ (uu¯ þ dd¯ − 2ss¯) and singlet (uu¯ þ dd¯ þ ss¯). Clearly, when ¯ and the heavier to KK, in line with the experimental SUð3Þ is broken, the flavorless members of ω8 and ω1 1270 0 1525 πω πϕ F f2ð Þ and f2ð Þ. In Ref. [68], coupled , must mix to form the physical eigenstates. ω scattering was considered, with the vector nature of the The notable difference between the pseudoscalar and (which is stable at this quark mass) leading to dynamically vector sectors was explored in lattice QCD in terms of the P 1þ coupled partial waves in J ¼ . A bump was found in qq¯ annihilation,or“disconnected,” contributions to two- πω 3 → πω 3 ð S1Þ ð S1Þ whose origin is a b1-like reso- point correlation functions in Ref. [26]. As can be seen in nance pole. Figs. 4 and 5 of that paper, the vector correlators have Before computing finite-volume spectra and determining extremely small disconnected pieces, both at and away scattering amplitudes relevant for the exotic JPC ¼ 1−þ from the SUð3Þ limit, leading to a lack of hidden-light– channel, we now discuss some of the consequences of F hidden-strange mixing and the ρ and ω mesons being close working with exact SU(3) flavor symmetry. to degenerate. This can be compared to the same quantities III. MESONS WITH EXACT SU(3) in the pseudoscalar sector shown in Figs. 2 and 3 therein. FLAVOR SYMMETRY These observations are related to the Okubo-Zweig- Iizuka (OZI) rule which states that processes where In this paper we will present the first attempt to compute there are no quark lines connecting the initial-state hadrons the properties of a resonance with exotic JPC, the lightest to the final-state hadrons are suppressed. Empirically this π1, which is suspected to be a hybrid meson. As indicated in holds for many JPC, including vectors, where a famous the Introduction, this is a challenging problem owing to the example is the suppression of the otherwise allowed large number of possible decay channels. A significant decay ϕ → πππ which leads to the ss¯ assignment for the simplification would occur if we had an exact SU(3) flavor ϕ. The OZI rule does not seem to apply to the pseudoscalar symmetry, as opposed to the approximate one present in sector. nature, as then many of the apparently independent A major advantage of an exact SU(3) flavor symmetry channels would coalesce into particular representations comes when we consider meson-meson scattering, as of SUð3Þ . In this first calculation, we opt to make this 3 F channels that with broken SUð ÞF were independent and symmetry exact by working with three flavors of light had differing thresholds, like ππ, KK¯ , …, are now equiv- quark all with a mass value tuned to approximately match 8 8 alent, being a single channel, η η . Since the stable scattering the physical strange-quark mass. In this world, the lightest hadrons lie in octets and singlets, the meson-meson products pseudoscalar octet, containing the pion, kaon and η-like 8 ⊗ 8, 8 ⊗ 1 and 1 ⊗ 1 are of interest, with the first of these unflavored member, has a mass around 700 MeV. This 1 ⊕ 8 ⊕ 8 ⊕ 10 ⊕ 10 ⊕ 27 relatively large mass has the additional useful effect of being decomposed into 1 2 .The pushing three-meson thresholds to higher energies such representations 10; 10; 27 lie outside the “conventional” that they become irrelevant in our calculation. sector, requiring at least qqq¯q¯, and are unlikely to be With exact SU(3) flavor symmetry, the “conventional” resonant [67,75,83]. The two octets 81; 82 can be distin- mesons (having flavor quantum numbers accessible to qq¯) guished by their symmetries under the exchange of the flavor lie in octet (8) and singlet (1) representations following of the two hadrons in the product. We follow the conventions ¯ from the decomposition of 3 ⊗ 3. The lightest of these is of Ref. [84],where81 is symmetric and 82 is antisymmetric, the pseudoscalar octet containing degenerate mesons which and we summarize the relevant results in Appendix A.Asan we can associate with the pion, the kaon and something example, using the SU(3) analogs of Clebsch-Gordan close to the η meson. We choose to use the zero-isospin, coefficients in that reference, the flavor structure of the zero-strangeness member of the octet as a label to indicate I ¼ 0, Iz ¼ 0, zero-strangeness members of the two octets in the JPðCÞ, e.g., η8 in this case of 0−ðþÞ. There is also a light the vector-pseudoscalar case can be expressed as pseudoscalar singlet η1, whose sole member is close to the familiar η0.4 The lightest octet of vectors ω8 contains 1 ρ þ − − þ 0 ¯ 0 ¯ 0 0 mesons we identify with the and the K , but its neutral 81 ¼ pffiffiffiffiffi ðK K þ K K − K K − K K Þ 20

4 1 1 The physical η, η0 eigenstates [with broken SUð3Þ ] are þ − − þ 0 0 F − pffiffiffi ðρ π þ ρ π − ρ π Þ − pffiffiffi ω8η8; believed to be admixtures of the octet/singlet basis states with a 5 5 small mixing angle as discussed in Sec. VII. The dependence of 1 þ − − þ 0 ¯ 0 ¯ 0 0 this mixing angle on the light-quark mass was explored using 82 ¼ ðK K − K K − K K þ K K Þ; ð2Þ lattice QCD in Ref. [26]. 2

054502-5 WOSS, DUDEK, EDWARDS, THOMAS, and WILSON PHYS. REV. D 103, 054502 (2021) which makes manifest that 81 is symmetric under the quark masses, retaining an isospin symmetry, we expect η η 3 interchange of the flavor of the two hadrons while 82 is that the 1 can mix with an 1 living in the SUð ÞF singlet, 1 −ð−Þ antisymmetric. the η1, while the kaonic states can mix with 1 kaons In determining what decays are possible, it is important owing to there being no relevant C-parity-like symmetry for to pay attention to the generalization of charge-conjugation mesons with net strangeness. On these grounds it seems symmetry. With exact isospin symmetry it is useful to plausible that the properties of the π1 will change least as 3 consider G-parity and there are natural extensions of this in we move away from the exact SUð ÞF limit. There may be 3 the SUð ÞF case. Because we are at liberty to consider any some mixing with the corresponding states in the 10; 10; 27 member of the target SU(3) multiplet, here we focus on the representations, but this is expected to be negligible given neutral zero-strangeness element where charge-conjugation that there is no evidence for anything beyond rather weak symmetry itself is good and so C-parity is the relevant nonresonant interactions in these multiplets. quantum number to consider. The resulting selection rules The meson-meson scattering channels capable of cou- apply to all members of the multiplet. Details are provided pling to the 1−ðþÞ octet include η1η8, ω8η8, ω8ω8, ω1ω8, in Appendix A and the relevant results are summarized in f8ω8, h8η8, f1η8…. How many of these are kinematically 8 8 1 1 1 Table I where the different symmetries of 1 and 2 are accessible in the decay of a potential lightest 1−ðþÞ apparent. 3 resonance depends upon QCD dynamics which we will When the two scattering mesons are in the same SUð ÞF now explore in a lattice QCD calculation. multiplet, there is the additional constraint of Bose symmetry which requires that the state is symmetric under the interchange of the two mesons, i.e., the overall symmetry IV. LATTICE QCD SPECTRA under the interchange of flavor, spin and spatial position. In Calculations of correlation functions were performed on the pseudoscalar-pseudoscalar case, where there is no spin 3 six anisotropic lattices with volumes ðL=asÞ × ðT=atÞ¼ to be dealt with, we immediately have the restriction that 123 × 96 and f143; 163; 183; 203; 243g × 128. The spatial η8η8 l 8 PðCÞ lþðþÞ with even appear in 1 with J ¼ , while and temporal lattice spacings are as ∼ 0.12 fm and l 8 PðCÞ l−ð−Þ −1 odd appear in 2 with J ¼ . It is therefore not at ¼as=ξ∼ð4.7 GeVÞ respectively, where the anisotropy possible to have an octet 1−ðþÞ resonance decay to η8η8. ξ ∼ 3.5. Gauge fields were generated from a tree-level Slightly more complicated is the case of ω8ω8 where the Symanzik improved gauge action and a Clover fermion spin of the two vectors can combine to total spin S ¼ 0,1,2 action with three degenerate flavors of dynamical quarks (symmetric, antisymmetric, symmetric respectively) which [85,86] tuned to approximately the value of the physical is then coupled to orbital angular momentum l. The strange-quark mass, such that the pion mass is ∼700 MeV. 8 8 1 spin þ space symmetric options (such as ω ω f S0g) On all volumes, exponentially suppressed finite-volume appear in 81, while the spin þ space antisymmetric options and thermal effects remain negligible as mπL ≳ 6 and 8 8 3 ≳ 14 (such as ω ω f S1g) appear in 82. A more complete mπT . discussion of these constraints can be found in Appendix B. Correlation functions were computed using the distil- In this study we will present the result of a calculation of lation framework [79] and we give the rank of the PðCÞ −ðþÞ 8 the J ¼ 1 octet labeled η1. We will choose to focus distillation space Nvecs, number of gauge configurations our later interpretation on the isovector member, the π1, Ncfgs, and time sources Ntsrcs used on each volume in 3 even though with exact SUð ÞF symmetry the properties of Table II. We typically compute all the elements of the the isoscalar member the η1 and the strange members are matrix of correlation functions; however, in a few cases we exactly the same. The reason for this choice is that as we made use of Hermiticity to infer CjiðtÞ from a com- 3 move away from the SUð ÞF limit by reducing the u, d puted CijðtÞ.

TABLE I. C-parity values for the neutral zero-strangeness components of the SU(3) octets and singlets from meson-meson TABLE II. Number of distillation vectors (Nvecs), gauge configurations (Ncfgs) and time sources (Ntsrcs) used in computation products. Ca and Cb denote the C-parity of the neutral zero- strangeness components of the product irreps. We present an of correlation functions on each lattice volume, as described in example for the 1−ð−Þ0−ðþÞ → 1þðCÞ case to illustrate notation. the text. 3 −ð−Þ −ðþÞ þðCÞ ðL=a Þ × ðT=a Þ N N N Fa ⊗ Fb → F C e.g., (1 0 → 1 ) s t vecs cfgs tsrcs 8 8 8 123 96 8 ⊗ 8 → 81 C C (ω η → h C ¼ −) × 48 219 24 a b a b 1 3 8 8 8 14 × 128 64 397 16 8 ⊗ 8 → 82 −C C (ω η → f1 C ¼þ) a b a b 163 × 128 64 529 4 8 ⊗ 8 → 1 ω8η8 → 1 − a b CaCb ( h1 C ¼ ) 183 × 128 96 358 4 8 1 8 3 8a ⊗ 1b → 8 CaCb (ω η → h1 C ¼ −) 20 × 128 128 501 4 1 1 1 243 128 1a ⊗ 1b → 1 CaCb (ω η → h1 C ¼ −) × 160 607 4

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TABLE III. Relevant stable hadron masses, atm. 0.35 η8 0.1478(1) η1 0.2017(11) ω8 0.2154(2) ω1 0.2174(3) 0.30 1 f0 0.2007(18) 8 1 f1 0.3203(6) f1 0.3364(14) 8 1 0.25 h1 0.3272(6) h1 0.3288(17)

0.20 disconnected contribution, while the C ¼ − states are close to degenerate. 0.15 As well as the computations in the rest frame from which the hadron masses in Table III are obtained, matrices of FIG. 1. Spectrum of low-lying octet (red) and singlet (cyan) correlation functions are also computed with nonzero mesons by JPðCÞ obtained using only single-meson operators. ⃗ 2π values of allowed lattice momentum p ¼ L ðnx;ny;nzÞ, Solid boxes show mesons which lie below relevant meson-meson and from these the dispersion relations Eðjp⃗jÞ for the stable thresholds and are thus stable, while hatched boxes show mesons mesons are determined; these are found to be well which lie above threshold and which will require a full finite- described by the expected relativistic form, volume analysis to resolve their resonant nature. Dashed lines show the lowest relevant meson-meson thresholds. 1 2π 2 2 2 ⃗2 ðatEn⃗Þ ¼ðatmÞ þ 2 jnj ; ð3Þ ξ L=as The spectrum of low-lying mesons on these lattices is shown in Fig. 1, obtained as the ground states in variational with the fitted values of anisotropy found for each meson analysis of matrices of correlation functions using a basis of being broadly compatible up to small variations due to 3 fermion-bilinear operators in either SUð ÞF octet or singlet 5 discretization effects. An estimate of the anisotropy with an representations. As we might expect, the pseudoscalar uncertainty that reflects the small variation over different η octet (containing the analogs of the pion, kaon and )is mesons is ξ ¼ 3.486ð43Þ; see Ref. [67] for further details. lightest, with the pseudoscalar singlet (comparable to the Figure 2 illustrates the position of a likely octet 1−ðþÞ η0 ) being somewhat heavier. The octet and singlet vector resonance based upon variational analysis of correlation PC mesons are close to degenerate reflecting that this J has a matrices using only fermion-bilinear constructions, along very small disconnected contribution which distinguishes the singlet from the octet. 1 The singlet scalar meson (f0) is rather light, at a similar mass to the pseudoscalar singlet. As it does not appear in the decays of the 1−ðþÞ resonance we are studying in this paper, we will not discuss it further here. The extracted 8 8 8 0.45 scalar octet meson (f0) mass lies very close to the η η 8 threshold. This indicates that to properly understand the f0, which may be a resonance or a shallow bound state, we would have to include meson-meson operators in our basis. 1 8 Levels corresponding to the tensor mesons (f2;f2) are 0.40 found some way above the η8η8 threshold, strongly suggesting that these states will be resonances capable of decaying into η8η8. PðCÞ þð−Þ 1 8 The axial mesons, the J ¼ 1 h1 and h1, and the 0.35 PðCÞ þðþÞ 1 8 J ¼ 1 f1 and f1, all lie quite far below their relevant decay thresholds, indicating that they are stable. As in the pseudoscalar-vector complex, the C ¼þstates show some octet-singlet splitting owing to a significant 0.30

5More details of the operator construction, and decomposition in terms of connected and disconnected contributions can be found in Ref. [26]. The 163 and 203 volumes used in that FIG. 2. Masses of C ¼þ octet mesons obtained using only reference are supplemented with the other volumes in Table II in single-meson operators (taken from Ref. [26]). Thresholds the current work. relevant for JPðCÞ ¼ 1−ðþÞ are shown.

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PðCÞ 1−ðþÞ 1 8 6 TABLE IV. Multimeson thresholds relevant for J ¼ a modest distance below f1η threshold, are included. shown in Fig. 2. Uncertainties are determined by adding the These operators are presented in Table V, listed by uncertainties on the single-meson masses in quadrature. increasing noninteracting energy. η8η8η8 η1η8 0.3495(11) The only relevant three-meson threshold lies −ðþÞ ω8η8 0.3632(2) slightly below the expected 1 resonance position. ω8ω8 0.4308(3) The lowest noninteracting three-meson energies appear ω1ω8 ð3Þ 0.4324(7) at atE > 0.51. As discussed in Sec. II, resonant excita- 8 8 8 n:i: η η η 0.4434(2) tions in two-meson subsystems may be present and 8η8 f1 0.4681(6) operators that capture these subsystem interactions need 8η8 h1 0.4750(6) to be considered for inclusion. To do this we examine the 1η8 2 1 f1 0.4842(14) “ ” ð þ Þ two-plus-one noninteracting energies atEn:i: , which follow from assuming no residual interaction between the interacting two-meson subsystem and the third meson; with the decay thresholds given in Table IV which follow details are provided in Ref. [68]. The lowest-energy from the masses in Table III. Also shown are the expected combination of three η8 that appears in the T− irrep is octet resonance spectra with other JPðþÞ taken from 1 Ref. [26]. These quantum numbers would contribute if 011 ⊗ 001 ⊗ 001 → 000 − ⊕ … spectra with nonzero overall momentum were to be ½|fflfflfflffl{zfflfflfflffl}A2 |fflfflfflffl{zfflfflfflffl}½ A2 |fflfflfflffl{zfflfflfflffl}½ A2 ½|fflfflfflffl{zfflfflfflffl}T1 : ð5Þ η8 η8 η8 η8 considered, significantly complicating the analysis. For 1 this reason, in this first calculation of the exotic 1−ðþÞ scattering system, we will restrict our attention to the We consider all possible meson-meson subsystems here −ðþÞ spectrum in the overall rest frame, considering the T1 that could feature bound states or resonances. Combining irrep. We will consider the role played by 3−ðþÞ, 4−ðþÞ the first two pseudoscalar octets appearing in Eq. (5) into definite momentum type ½001, we find the only irrep scattering, which in principle contribute in this irrep, later − in the manuscript. combination that yields the T1 irrep is 011 ⊗ 001 ⊗ 001 → 000 − A. Operator bases ð½|fflfflfflffl{zfflfflfflffl}A2 |fflfflfflffl{zfflfflfflffl}½ A2Þ |fflfflfflffl{zfflfflfflffl}½ A2 ½|fflfflfflffl{zfflfflfflffl}T1 − ðþÞ η8 η8 η8 η8 We construct a suitable basis of operators in the T1 1 irrep from a set of single-meson-like operators and a set of − ½001E2 ⊗ ½001A2 → ½000T1 : meson-meson-like operators. A total of 18 fermion bilin- |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} η8η8 η8 η8 ears ψ¯ Γψ are used following Ref. [23], with a spin and 1 spatial structure built from Dirac γ-matrices and gauge- 8 8 covariant derivatives. Gluonic degrees of freedom enter The irrep ½001E2 houses the ω and f1, which we treat as through the gauge-covariant derivatives. For example, one stable scattering particles; any excited finite-volume energy simple 1−ðþÞ bilinear operator constructed using the vector level coupling to η8η8 (in any flavor combination) will lie γ 8 cross product of i and the commutator of two derivatives, above the f1 level, and hence no three-meson-like operators is given by are needed in the basis to study a 1−ðþÞ resonance near atE ∼ 0.46. ðψ¯ ΓψÞ ¼ ϵ ðψγ¯ ψÞB ; ð4Þ −ðþÞ i |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}ijk j k As discussed previously, the T1 irrep also features − 1−−⊗1þ−→1−þ PðCÞ −ðþÞ ðþÞ contributions from J ¼ 3 . Considering the A2 ≤ 4 PðCÞ 3−ðþÞ ↔ ↔ irrep, which for J features only J ¼ sub- 3 where Bk ∝ ϵkpq½Dp;Dq is the chromomagnetic field. In ductions, we can isolate the contribution from the J ¼ practice, when we determine the spectra we vary the partial waves. We will use the finite-volume energy levels number of single-meson operators to establish insensitivity in this irrep to constrain the J ¼ 3 partial waves and show to the details of the choice of operator basis. these are small over the energy range considered here. The − In Table IV, we show the relevant multihadron thresholds operator basis used in the A ðþÞ irrep for each lattice − 2 for two- and three-meson channels that appear in 1 ðþÞ and volume is given in Table VI. that transform in the flavor octet. To ensure all relevant meson-meson operators are included in the operator 6 1 8 In addition, we include an f1η operator corresponding to a basis, we calculate the noninteracting energies for each 1 8 noninteracting level at f1η threshold. A small number of meson- multimeson system by considering all momenta combina- 1 8 meson operators that lie a modest distance above the f1η tions that sum to zero. All meson-meson operators with a threshold were also added to explore the (very mild) sensitivity corresponding noninteracting energy below atEcm ¼ 0.48, to our choice of largest energy.

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−ðþÞ TABLE V. T1 operator basis for each lattice volume. Meson-meson operators are ordered by increasing En:i:. and labeled with the momentum types of the two mesons; different momentum directions are summed over as discussed in Sec. II. The number in braces fNmultg denotes the multiplicity of linearly independent meson-meson operators if this is larger than 1. The maximum number of single- meson operators N is denoted by N × ψ¯ Γψ and various subsets of these were considered to investigate sensitivity to the details of the choice of operator basis.

L=as ¼ 12 L=as ¼ 14 L=as ¼ 16 L=as ¼ 18 L=as ¼ 20 L=as ¼ 24 18 × ψ¯ Γψ 18 × ψ¯ Γψ 18 × ψ¯ Γψ 18 × ψ¯ Γψ 18 × ψ¯ Γψ 18 × ψ¯ Γψ η1 η8 η1 η8 η1 η8 η1 η8 η1 η8 η1 η8 ½001 ½001 ½001 ½001 ½001 ½001 ½001 ½001 ½001 ½001 ½001 ½001 8 η8 ω8 η8 ω8 η8 ω8 η8 ω8 η8 ω8 η8 f1½000 ½000 ½001 ½001 ½001 ½001 ½001 ½001 ½001 ½001 ½001 ½001 ω8 η8 8 η8 8 η8 η1 η8 η1 η8 η1 η8 ½001 ½001 f1½000 ½000 f1½000 ½000 ½011 ½011 ½011 ½011 ½011 ½011 8 η8 8 η8 η1 η8 2 ω8 η8 2 ω8 η8 2 ω8 η8 h1½000 ½000 h1½000 ½000 ½011 ½011 f g ½011 ½011 f g ½011 ½011 f g ½011 ½011 1 η8 1 η8 8 η8 8 η8 ω8 ω8 η1 η8 f1½000 ½000 f1½000 ½000 h1½000 ½000 f1½000 ½000 ½001 ½001 ½111 ½111 ω8 ω8 1 η8 8 η8 8 η8 ω8 η8 ½001 ½001 f1½000 ½000 h1½000 ½000 f1½000 ½000 ½111 ½111 4 ω1 ω8 2 ω8 η8 ω8 ω8 4 ω1 ω8 ω8 ω8 f g ½001 ½001 f g ½011 ½011 ½001 ½001 f g ½001 ½001 ½001 ½001 η1 η8 ω8 ω8 4 ω1 ω8 η1 η8 4 ω1 ω8 ½011 ½011 ½001 ½001 f g ½001 ½001 ½111 ½111 f g ½001 ½001 2 ω8 η8 4 ω1 ω8 1 η8 8 η8 η1 η8 f g ½011 ½011 f g ½001 ½001 f1½000 ½000 h1½000 ½000 ½002 ½002 η1 η8 ω8 η8 8 η8 ½111 ½111 ½111 ½111 f1½000 ½000 ω8 η8 1 η8 ω8 η8 ½111 ½111 f1½000 ½000 ½002 ½002 ω8 η8 8 η8 ½002 ½002 h1½000 ½000 1 η8 f1½000 ½000 η1 η8 ½012 ½012

B. Finite-volume spectra η1η8 operators is a consequence of the substantial dis- − 1 Variational analysis of matrices of T ðþÞ correlation connected contribution to the η . 1 ∼ 0 46 functions on the six volumes leads to the spectrum In an energy region around atEcm . on each presented in Fig. 3. Error bars reflect the statistical volume we find one more energy level than expected on uncertainty and an estimate of the systematic uncertainty the basis of the noninteracting energies, and we begin to from varying the details of the variational analysis (such as observe levels having significant overlaps onto hybridlike operator basis and fit range). For each finite-volume single-meson operator constructions (orange bar). This − eigenstate that will be used to constrain scattering ampli- energy region is where the 1 ðþÞ state proposed to be a tudes, we also show a histogram illustrating the overlap hybrid meson was observed in the analysis using only strength with operators in the basis. single-meson operators discussed earlier. The finite-volume We notice that below atEcm ∼ 0.44, the energy levels lie eigenstates having overlap onto the hybridlike operator are very close to the η1η8 and ω8η8 noninteracting energies, and also observed to have overlap onto meson-meson con- 1 8 8 8 8 8 each level has dominant overlap with just the operator(s) structions, notably η η (dark blue), ω η (red), f1η (cyan) 8 8 corresponding to the particular noninteracting momentum and/or h1η (purple), which might suggest a resonance combination lying nearby (blue and red bars). This tends to coupling to these scattering channels. suggest weak, uncoupled scattering at lower energies. The A level lying very close to the twofold degenerate somewhat larger error bars on levels with large overlap onto ω8 η8 ½011 ½011 noninteracting curve is observed at each volume above L=as ¼ 16 with a characteristic histogram that ω8 η8 −ðþÞ couples strongly to the two ½011 ½011 operators but is TABLE VI. As Table V but showing the A2 operator basis for each lattice volume, with meson-meson operators ordered by decoupled from all other operators. Such behavior would ω8η8 3 increasing En:i:. be expected if the f F3g wave is weak. On the L=a ¼ 18, 20, 24 volumes, a cluster of states 16 18 20 24 s L=as ¼ L=as ¼ L=as ¼ L=as ¼ appears in the energy region of interest close to the lowest 4 × ψ¯ Γψ 4 × ψ¯ Γψ 4 × ψ¯ Γψ 4 × ψ¯ Γψ ω8ω8 (sand) and ω1ω8 (green) noninteracting energies. The ω8 η8 ω8 η8 ω8 η8 ω8 η8 ½011 ½011 ½011 ½011 ½011 ½011 ½011 ½011 histograms for these states, presented at the top of the ω1 ω8 ω1 ω8 ω1 ω8 η1 η8 figure, show that in each case there are five energies which ½001 ½001 ½001 ½001 ½001 ½001 ½111 ½111 1 8 1 8 1 8 have a large overlap with these vector-vector operators, but η 111 η 111 η 111 η 111 ω 001 ω 001 ½ ½ ½ ½ ½ ½ not a large overlap with hybridlike operators. This might be

054502-9 WOSS, DUDEK, EDWARDS, THOMAS, and WILSON PHYS. REV. D 103, 054502 (2021)

0.48

0.46

0.44

0.42

0.40

0.38

0.36

12 16 20 24

−ðþÞ FIG. 3. Finite-volume spectrum in the T1 irrep on six lattice volumes. Points show the extracted energy levels, including uncertainties, from a variational analysis using the operator bases in Table V; black points are included in the subsequent scattering analysis and gray points are not. Some points are slightly displaced horizontally for clarity when near-degenerate energies appear. Curves show meson-meson noninteracting energies, with multiplicities greater than 1 labeled by fng and shown as slightly split curves. Dashed curves correspond to meson-meson operators not included in the basis. Relevant thresholds transcribed from Table IV are shown on the vertical axis. n n O† 0 0 η1η8 ω8η8 Accompanying each energy level is a histogram of the operator-state overlap factors, Zi ¼h j i ð Þj i, for (dark blue), (red), 8 8 1 8 8 8 8 8 1 8 ω ω (sand), ω ω (green), f1η (cyan), h1η (purple) and f1η (brown) meson-meson operators and a sample of ψ¯ Γψ (orange) fermion- bilinear operators. The overlaps are normalized such that the largest value for any given operator across all energy levels is equal to 1. For clarity, the histograms accompanying the cluster of levels on the L=as ¼ 18, 20, 24 volumes are displayed at the top of the figure.

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TABLE VII. Scattering partial waves included in the descrip- 0.50 −ðþÞ tion of T1 finite-volume spectra.

− 1 8 1 1 ðþÞ η η f P1g 8 8 3 ω η f P1g 8 8 3 1 8 1 3 5 0.48 ω ω f P1g, ω ω f P1; P1; P1g 8 8 3 8 8 3 f1η f S1g, h1η f S1g − 1 8 1 3 ðþÞ η η f F3g 8 8 3 0.46 ω η f F3g 1 8 5 ω ω f P3g

0.44 −ðþÞ −ðþÞ −ðþÞ We expect T1 to be dominated by 1 , with 3 , 4−ðþÞ, and still higher J being weak at these energies; these require higher orbital angular momentum l and so are 0.42 suppressed close to threshold in the absence of any dynamical enhancement. There is no evidence from the single-meson operator study in Ref. [26] of a low-lying 3−ðþÞ resonance, and while 4−ðþÞ is nonexotic [it can be ¯ 1 0.36 constructed as the qqð G4Þ state], Ref. [26] suggests that such a state lies at atEcm ∼ 0.58, far above our region of −ðþÞ interest. By computing the A2 spectrum we are able to directly constrain the strength of scattering with JPðCÞ ¼ 12 16 20 24 3−ðþÞ in the energy region of interest. −ðþÞ The first step in analyzing the finite-volume spectrum is FIG. 4. Analogous to Fig. 3 but for the A2 irrep (operator lists shown in Table VI). Note the vertical axis is broken to to establish the basis of relevant meson-meson partial emphasize the relatively low-lying η1η8 and ω8η8 thresholds. waves in the considered energy region which define the matrix space in Eq. (1). The set of meson-meson channels kinematically accessible was presented in the previous taken as a suggestion that a hybrid resonance (if present) section and in Table VII we show the set of partial waves may not be strongly coupled to these vector-vector scatter- we will use. ing channels. A small number of possible partial waves have been Finally, the only states which show any significant excluded from Table VII under the expectation that 1η8 coupling to the f1 (brown) operator lie at rather high they will not contribute significantly. In the 1−ðþÞ sector, energies, suggesting that this channel is probably not 8 8 3 8 8 3 f1η f D1g and h1η f D1g are not included, as the thresholds relevant to any resonance near atEcm ∼ 0.46. −ðþÞ for these channels are very high-lying in our energy region Figure 4 shows the spectrum obtained in the A2 irrep. such that we expect a significant angular momentum sup- It is clear from the histograms, which are dominated in each pression from the D-wave, relative to the leading S-wave, that case by a single meson-meson-like operator, and the will render them practically irrelevant. Similarly, in the proximity of each level to the corresponding noninteracting 1 8 5 vector-vector channels, we exclude ω ω f F1g on the basis curves, that there are only relatively weak interactions. 7 of F-wave angular momentum suppression. There is no sign of any resonant behavior that might be −ðþÞ 8 8 3 In the 3 sector, ω η f F3g is included despite the 3−ðþÞ associated with a low-lying state. large angular momentum barrier. As can be seen in Table V, While a qualitative discussion of the spectra like the there are two independent operators for ω8 η8 and one just presented can suggest possible features of the ½011 ½011 scattering system, a rigorous determination requires an there is a corresponding twofold degenerate noninteracting analysis using the coupled-channel finite-volume formal- energy. In order that there be two solutions of Eq. (1) near ism described in Sec. II from which the t-matrix can be this energy, as observed in our computed spectra and commented on in the previous section, higher ω8η8 partial extracted, and from it properties of any resonance poles. 8 8 3 waves must be considered, so we include the ω η f F3g 8 8 3 wave along with the dominant ω η f P1g. We also include V. SCATTERING AMPLITUDES 1 8 1 1 8 η η f F3g as the η η threshold is relatively low compared −ðþÞ We wish to use the spectra computed in the T1 and with the resonant region, such that the angular momentum −ðþÞ A2 irreps presented in the previous section to determine PðCÞ −ðþÞ 7 8 8 1 5 8 8 5 the matrix describing scattering with J ¼ 1 . Bose symmetry forbids ω ω f P1; P1g and ω ω f F1g.

054502-11 WOSS, DUDEK, EDWARDS, THOMAS, and WILSON PHYS. REV. D 103, 054502 (2021) barrier may not sufficiently suppress contributions from We now seek to use the 61 energy levels shown in black this higher partial wave in the energy region of interest. in Figs. 3 and 4 to constrain parametrizations of the 8 8 3 1 8 1 3 5 Other possible F-waves, ω ω f F3g, ω ω f F3; F3; F3g t-matrix in the partial-wave basis presented in Table VII only generate additional solutions to Eq. (1) at somewhat by solving Eq. (1). Solutions of Eq. (1) are only possible for higher energies and have relatively high-lying thresholds t-matrix parametrizations which satisfy multichannel uni- for which we expect the angular momentum suppression to tarity. The simplest way to implement that constraint is to be significant. In practice we will find that all the 3−ðþÞ make use of the K-matrix, writing partial waves we consider are modest over the energy range considered, with direct constraints coming from the com- −1 1 −1 1 −ðþÞ ½t ðsÞ 0 0 ¼ ½K ðsÞ 0 0 A lSJa;l S Jb 2 l lSJa;l S Jb l0 puted 2 spectra. ð kaÞ ð2kbÞ The 4−ðþÞ sector is populated only by partial waves that δll0 δ 0 I s ; are F-wave or higher, all of which we assume to be small þ SS abð Þ enough as to be negligible, and none of which generate additional solutions of Eq. (1) in the energy region where K is a symmetric matrix taking real values on the considered. real energy axis and IðsÞ is a diagonal matrix satisfying 1−ðþÞ One partial wave with is excluded on dynamical ImIabðsÞ¼−ρaðsÞ above the threshold for channel a. The 1 8 3 grounds: f1η f S1g is observed to be completely decoupled simplest choice is to set IðsÞ¼−iρðsÞ, but other options from the other scattering channels when operator overlaps may have better analytic properties below threshold and (as presented in Fig. 3) are examined. This leads to a natural away from the real energy axis; for example, the Chew- choice of energy cutoff at atEcm ¼ 0.48, a modest distance Mandelstam prescription for which our implementation is 1 8 below the f1η threshold, and we only use energies with no described in Ref. [64]. The K-matrix is block diagonal in J, 1 8 significant dependence on the f1η -like operator. The levels reflecting the fact that total angular momentum is a good to be used in constraining amplitudes are shown in black in quantum number in infinite volume and only “mixes” in a Figs. 3 and 4. finite volume, through the matrix M, due to the reduced The contribution of the three vector-vector partial waves symmetry of the lattice. 1 8 1 3 5 ω ω f P1; P1; P1g, which differ only in the total coupled The presence in the spectrum of an additional level ∼ 0 46 intrinsic spin of the two vector mesons, to Eq. (1) requires around atEcm . and the lack of significant energy some care. In the ½000T− irrep that we are considering, shifts at lower energies hints at a likely narrow resonance in 1 ∼ 0 46 Eq. (1) is invariant under the interchange of any of these the energy region around atEcm . . This is also − partial waves, and it follows that the corresponding rows consistent with the exotic 1 ðþÞ octet level seen in and columns of the t-matrix cannot be uniquely determined Fig. 2. The large overlap with axial-vector–pseudoscalar (see also Appendix C). There is reason, from an approxi- meson-meson operators seen in Fig. 3 suggests significant mate extension of Bose symmetry, to expect that only coupling to these channels, whose thresholds lie just above 1 8 3 amplitudes featuring ω ω f P1g could be significant while the anticipated resonant region. 1 8 1 5 An efficient way to parametrize coupled-channel scatter- those with ω ω f P1; P1g will be very small. The Wick contractions for diagrams featuring these channels differ ing when a narrow resonance appears is to use a K-matrix only from those featuring ω8ω8 by the presence of the featuring an explicit pole. For the case of a single channel 1 this form of parametrization is closely related to the conven- disconnected contribution to the ω , but this contribution is tional Breit-Wigner and for coupled channels it is related to a very small (reflected in the near degeneracy of ω1; ω8). In ω1 ω8 multichannel Breit-Wigner, sometimes referred to as a Flatt´e practice we expect the and to have almost identical amplitude in the two-channel case [87,88]. The K-matrix ω8ω8 1 5 spatial wave functions, and since f P1; P1g are can also be straightforwardly augmented by the addition of a forbidden by Bose symmetry, we anticipate that the polynomial matrix in s, which in the simplest case can just be ω1ω8 corresponding amplitudes will be heavily suppressed. a constant matrix, that allows additional freedom beyond a In fact we will observe that all vector-vector amplitudes are pure resonance interpretation. This is crucial to test the found to be very small over the energy range considered. robustness of scattering amplitudes and allow more flexible η8η8η8 While the three-meson channel becomes kine- forms, as, for example, a pure pole parametrization exhibits matically accessible at the upper end of the energy region the phenomenon of “trapped” levels, where a single energy we are considering, we do not include such partial waves. level is forced to appear between every pair of noninteracting − To couple to JPðCÞ ¼ 1 ðþÞ, this channel requires at least energies; see Appendix D. two P-waves, and since our expected resonance lies barely In addition to varying the form of the K-matrix, the 8 8 8 above the η η η threshold, the angular momentum sup- choice of IðsÞ may also be varied. The Chew-Mandelstam pression implied is expected to render the partial waves prescription improves the analytic continuation below irrelevant. thresholds, which is particularly useful here where, as

054502-12 DECAYS OF AN EXOTIC 1−þ HYBRID MESON … PHYS. REV. D 103, 054502 (2021) discussed above, the axial-vector–pseudoscalar thresholds lie above the resonant region. In this study, we will consider a variety of parametriza- 0.48 tions, finding the best description of the finite-volume spectrum for each choice, ultimately leading to compatible results for the amplitudes and their resonant content. As we 0.46 are only using rest-frame energy levels to determine the large coupled-channel scattering system (see Sec. IV), we have less constraint than in previous calculations of simpler 0.44 systems where in-flight spectra were computed [46,62–68]. However, the use of six volumes appears to provide enough 0.42 information to isolate most of the important features. −ðþÞ The A2 spectra allow us to determine the J ¼ 3 amplitudes which provide a “background” contribution 12 16 20 24 −ðþÞ to the T1 spectra. As discussed in Sec. IV, there is no FIG. 5. As Fig. 4 but including, as orange bands, the energy sign of any resonant behavior associated with a 3−ðþÞ state levels calculated from the amplitude in Eq. (6). The thickness of in this energy region and the histograms in Fig. 4 suggest a the bands reflects the statistical uncertainty. The dashed curves ω8η8 η1η8 totally decoupled system. A reasonable form of paramet- show the noninteracting energy levels for (red), (blue) ω1ω8 rization, capable of successfully describing the finite- and (green). volume spectra, is a diagonal constant K-matrix, 2 3 γ 8 8 3 000 2 3 ω ω f P1g γ 1 8 1 00 6 7 η η f F3g 0 γ 1 8 1 00 6 ω ω f P1g 7 6 7 K s 6 7: 4 0 γ 8 8 3 0 5 VVð Þ¼ K3ðsÞ¼ ω η f F3g ; ð6Þ 4 00γ 1 8 3 0 5 ω ω f P1g 00γ 1 8 5 ω ω f P3g 000γ 1 8 5 ω ω f P1g where the Chew-Mandelstam prescription with subtraction 1−ðþÞ at thresholds was used for IðsÞ. The resulting fit describes For the remaining four channels, motivated by the − likely presence of a narrow resonance, we parametrize the the A ðþÞ finite-volume spectra with a χ2=N 2 d:o:f: ¼ amplitudes using a “pole plus constant” form, 2.53=ð8 − 3Þ¼0.51, as shown in Fig. 5. Other parametrizations give a compatible set of amplitudes and quality of 2 3 −ðþÞ γ 1 8 1 000 fit. The 3 amplitudes are modest over the entire energy η η f P1g 1 8 8 8 6 7 range, with the η η and ω η being mildly repulsive, and T 0 γ 8 8 3 00 gg 6 ω η f P1g 7 ω1ω8 0 48 K ðsÞ¼ þ 6 7; the being mildly attractive; at atEcm ¼ . the VV m2 − s 4 00005 decoupled phase shifts reach only −13ð3Þ°, −6ð1Þ° and 0000 5(4)° respectively.

A. An illustrative t-matrix parametrization where We now consider the eight coupled-channel 1−ðþÞ −ðþÞ g gη1η8 1 ;gω8η8 3 ;g 8η8 3 ;g 8η8 3 ; 7 scattering system that features in T1 . We will illustrate ¼ð f P1g f P1g f1 f S1g h1 f S1gÞ ð Þ the scattering analysis using a single choice of amplitude parametrization, and later explore variations in that choice. The properties of the illustrative amplitude choice are so that all four channels are coupled to the resonance as motivated by the observations of the finite-volume spectra motivated by the histograms in Fig. 3. We also add a made in Sec. IV. The four vector-vector channels appear to constant term in the lowest two channels as the corre- be decoupled for all considered energies and show no sponding thresholds lie very low relative to the resonant significant energy shifts, so in this parametrization we region, and the close proximity of the energy levels with the make the decoupling manifest, parametrizing the ampli- noninteracting energies low down in the spectra suggested tudes with a diagonal K-matrix of constants,8 a region of nonresonant behavior (see the discussion in Sec. IV). We use the Chew-Mandelstam prescription for I 2 8 ðsÞ subtracting at the K-matrix pole mass (s ¼ m ). The In some of the parametrizations we will consider, vector- 1−ðþÞ vector and non-vector-vector channels are allowed to couple to eight-channel K-matrix appears combined with the each other though their coupling to the pole term. three-channel 3−ðþÞ K-matrix as given in Eq. (6),

054502-13 WOSS, DUDEK, EDWARDS, THOMAS, and WILSON PHYS. REV. D 103, 054502 (2021) 2 3 K s 00 degenerate noninteracting energy. As discussed in 6 VVð Þ 7 6 7 Appendix C, using only rest-frame energies does not K s 0 K s 0 ; 8 − 1 8 ð Þ¼4 VVð Þ 5 ð Þ uniquely constrain the three 1 ω ω amplitudes and there 00K3ðsÞ is a freedom to permute these channels within the t-matrix. We therefore consider the envelope of these three ampli- in the finite-volume spectrum condition, Eq. (1).We tudes, as determined from the minimization, as our best 2 ω1ω8 X minimize the χ by varying the 11 parameters in KVVðsÞ estimate for the size of the f P1g amplitudes. This is and KVV, with the parameters in K3 fixed according to the shown in Fig. 7(f) where we see that they are weak over the −ðþÞ entire range, consistent with the observations made in fit to the A2 lattice spectra. The resulting description of − Sec. IV. It is important to note that, as shown in the T ðþÞ spectra gives a very reasonable χ2=N 1 d:o:f: ¼ Appendix C, energy spectra obtained in moving-frame 43.6= 53 − 11 1.04 shown in Fig. 6. ð Þ¼ irreps modify the boundary conditions of the quantization Plotting the resulting t-matrix elements as ρ ρ t 2 a bj abj condition and do distinguish the contributions of the shown in Fig. 7, we can make a number of qualitative and 1 3 5 f P1; P1; P1g partial waves. As discussed in Sec. IV, quantitative observations. 1 8 8 8 8 8 8 8 we do not include moving-frame energy spectra owing to The diagonal amplitudes for the η η , ω η , f η , h η 1 1 the appearance of the relatively low-lying positive-parity channels are shown in Fig. 7(a) where a clear bumplike resonances, as parities mix at nonzero momentum, and this enhancement can be seen in the η1η8 and ω8η8 channels at would significantly complicate the analysis. a Ecm ∼ 0.46, close to the mass obtained using only single- t For this particular parametrization, we also examine the meson operators (see Fig. 2). We observe a sharp turn-on of – 8η8 8η8 effects of varying the stable hadron masses and anisotropy the axial-vector pseudoscalar channels (f1 , h1 )at within their respective uncertainties, as given in Sec. IV,to threshold, allowed for S-wave amplitudes. The associated – get an estimate of some of the systematic uncertainties on off-diagonal amplitudes are plotted in Figs. 7(b) 7(d). the amplitudes. We adopt a conservative approach where Here, we see again a bumplike enhancement in the χ2 1 8 8 8 we repeat the minimization procedure using the extremal η η → ω η amplitude at atEcm ∼ 0.46, with the other values mi → mi þ δmi and ξ → ξ − δξ, and vice versa. off-diagonal amplitudes being mostly small with the These combinations yield the largest deviations in the 8η8 → 8η8 exception of the f1 h1 amplitude which shows a noninteracting energies, and therefore the largest shifts in modest rise from threshold. the energy differences between the computed energy levels The four decoupled vector-vector channels are presented and the noninteracting values; these ultimately constrain in Figs. 7(e) and 7(f). We observe that the single ω8ω8 3 the scattering parameters. We anticipate that these combi- amplitude, in the P1 partial wave, is weak across the entire nations will therefore result in the largest changes in the energy range, consistent with our observations from the scattering parameters and so yield a conservative estimate 1 8 finite-volume spectra in Sec IV. For the ω ω amplitudes, of the systematic uncertainties on the parameters from P − 1 3 5 we require four partial waves, three J ¼ 1 ð P1; P1; P1Þ uncertainties in the hadron masses and anisotropy. − 5 P − and one 3 ð P3Þ, in order to obtain the correct number of For the J ¼ 3 amplitudes, we find that varying the finite-volume energies at the corresponding fourfold anisotropy yields the largest systematic uncertainties. The rather weak interactions in this system lead to small shifts in energies from their noninteracting values, as seen in 0.48 Fig. 5, which receive significant adjustment as the anisotropy is varied.9 The quality of fits under these χ2 0.46 systematic variations also became rather poor: =Nd:o:f: ¼ 2 26 → δ ξ → ξ − δξ χ2 . for mi mi þ mi and , and =Nd:o:f: ¼ 4.82 for mi → mi − δmi and ξ → ξ þ δξ, reflecting that 0.44 even small discretization effects can be visible in weakly interacting systems where the energy levels have been determined with high statistical precision. Nevertheless, we 0.42 find all JP ¼ 3− amplitudes remain small over the entire energy region considered. P − 0.40 Regarding the J ¼ 1 amplitudes, having fixed the (newly determined) JP ¼ 3− parameters, we find the effects of varying the masses and anisotropy are much 0.38 12 16 20 24 9This effect was observed previously in ρπ isospin-2 scattering −ðþÞ FIG. 6. As Fig. 5 but for the T1 irrep using the illustrative where the very small interactions meant that the systematic amplitude described in Eq. (8). uncertainties dominated over the statistical ones [67].

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0.5 (a) 0.4 0.3 0.2 0.1

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(e) 0.2 0.1

0.43 0.44 0.45 0.46 0.47 0.48 (f) 0.2 0.1 0.43 0.44 0.45 0.46 0.47 0.48

2 FIG. 7. (a) Diagonal t-matrix elements plotted as ρaρbjtabj , for the illustrative amplitude presented in Eq. (8) for non-vector-vector 1 8 1 8 8 3 8 8 3 8 8 3 channels: η η f P1g, ω η f P1g, f1η f S1g and h1η f S1g. Shaded bands reflect statistical uncertainties on the scattering parameters. (b)–(d) As above, but for the off-diagonal amplitudes between the four channels displayed above. (e),(f) Diagonal vector-vector 8 8 3 1 8 1 3 5 1 8 amplitudes: ω ω f P1g, ω ω f P1; P1; P1g. As discussed in the text, the ω ω partial waves are indistinguishable and are combined in 1 8 X a single plot labeled ω ω f P1g, the shaded band reflecting an envelope over the statistical uncertainties for each partial wave. smaller relative to those for JP ¼ 3−, as expected in a more we will see in the subsequent variation in the parametriza- strongly interacting system. There are some modest variation. For example, we find the peak of the bumplike tions in the amplitudes, but these are broadly within the enhancements in the η1η8 and ω8η8 amplitudes is consistent statistical uncertainties and certainly within the differences in height and only slightly displaced in energy (higher or

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lower depending upon the sign of the systematic varia- 1.0 tions). This will be reflected in the position of a pole 0.8 singularity of the t-matrix which varies at a level compa- 0.6 rable to the statistical uncertainty. 0.4 A larger source of uncertainty arises when we consider 0.2 varying the form of parametrization, to which we now turn. 0.43 0.44 0.45 0.46 0.47 0.48 1.0 B. Parametrization variations 0.8 0.6 In order to determine the extent to which the amplitude 0.4 results presented in Fig. 7 are a unique description of the 0.2 scattering system, we try a number of parametrizations, 0.43 0.44 0.45 0.46 0.47 0.48 attempting to describe the finite-volume spectrum with each choice. Variations in the K-matrix include allowing 0.2 energy dependence in the numerator of the pole term, and 0.43 0.44 0.45 0.46 0.47 0.48 changes in the polynomial matrix added to the pole. The 0.4 prescription used for IðsÞ is also adjusted, while maintain- 0.2 ing coupled-channel unitarity in all parametrizations. We 0.43 0.44 0.45 0.46 0.47 0.48 retain 27 parametrizations10 which are able to describe the χ2 ≤ 1 25 ρ ρ 2 finite-volume spectra with =Nd:o:f: . , showing the FIG. 8. Diagonal t-matrix elements plotted as a bjtabj , for resulting amplitudes in Figs. 8–12. each parametrization successfully describing the finite-volume spectra as discussed in the text, for non-vector-vector channels: For the diagonal amplitudes in the lowest two channels 1 8 1 8 8 3 8 8 3 8 8 3 1 8 1 8 8 3 η η f P1g, ω η f P1g, f1η f S1g and h1η f S1g. Shaded bands η η f P1g and ω η f P1g shown in Fig. 8, we see a reflect statistical uncertainties on the illustrative amplitude shown bumplike enhancement around a Ecm ∼ 0.46 for the major- t in Fig. 7. ity of parametrizations, but we note that it is possible to describe our finite-volume spectra without seeing such a clear bump. We will revisit this observation when we examine the pole singularities of the t-matrix and the corresponding couplings. For the remaining two diagonal 8 8 3 8 8 3 0.3 amplitudes in channels f1η f S1g and h1η f S1g,we observe that the relatively sharp turn-on at threshold is a 0.2 quite general feature, with only the magnitude of the effect 0.1 varying somewhat. That there should be some parametri- 0.43 0.44 0.45 0.46 0.47 0.48 zation dependence here should not come as too much of a 0.1 surprise given the relatively small number of finite-volume energy levels constraining the amplitudes above the axial- 0.43 0.44 0.45 0.46 0.47 0.48 vector–pseudoscalar thresholds. 0.1 η1η8 1 → ω8η8 3 The off-diagonal amplitude f P1g f P1g 0.43 0.44 0.45 0.46 0.47 0.48 shown in Fig. 9, typically features a bumplike enhancement ∼ 0 46 1 8 1 8 8 3 around atEcm . , but as for the diagonal entries, it is FIG. 9. As Fig. 8 but for off-diagonal η η f P1g → ω η f P1g, 8 8 3 8 8 3 possible to describe the spectra without such a bump and f1η f S1g, h1η f S1g amplitudes. indeed without any coupling between these two channels. The remaining off-diagonal amplitudes remain modest under parametrization variation and are shown in Figs. 9–11. The vector-vector amplitudes shown in Fig. 12 have the same qualitative behavior as in the illustrative example 0.1 presented previously. The small bump around atEcm ∼ 0.46 8 8 3 8 8 3 0.0 for ω ω f P1g → ω ω f P1g on a small number of para- 0.43 0.44 0.45 0.46 0.47 0.48 metrizations reflects allowing freedom for this channel to 0.2 couple to the K-matrix pole; it is observed to be a very 0.1 weak effect and is statistically compatible with zero. 0.0 0.43 0.44 0.45 0.46 0.47 0.48

10 8 8 3 8 8 3 8 8 3 A full description of each of these parametrizations is FIG. 10. As Fig. 9 but for ω η f P1g → f1η f S1g, h1η f S1g provided in the Supplemental Material [69]. amplitudes.

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In each energy region between thresholds, the unphys- 0.2 ical sheet closest to the region of physical scattering, 0.1 ðaÞ has ImðkcmÞ < 0 for all kinematically open channels and ðaÞ 0.43 0.44 0.45 0.46 0.47 0.48 ImðkcmÞ > 0 for all kinematically closed channels. For convenience we will refer to this as the proximal sheet, and 8η8 3 → 8η8 3 FIG. 11. As Fig. 9 but for the f1 f S1g h1 f S1g ampli- a nearby pole singularity on the proximal sheet will have a tude. significant impact on physical scattering. For brevity, sheets are labeled as an ordered list of six signs, where the order reflects increasing threshold energies 1 8 8 8 8 8 1 8 8 8 8 8 (η η , ω η , ω ω , ω ω , f1η , h1η ), and the sign reflects 0.2 the imaginary component of momenta for that channel. For example ½þþþþþþ represents the physical sheet, and 0.0 −− 0.43 0.44 0.45 0.46 0.47 0.48 ½ þþþþ represents the proximal sheet for scattering ω8η8 ω8ω8 0.2 above the threshold, but below the threshold. The position of pole singularities can be related to conventional pictures of meson states. Poles on the real 0.0 0.43 0.44 0.45 0.46 0.47 0.48 axis below the lowest threshold on the physical sheet correspond to stable bound states, while poles in that ω8ω8 3 FIG. 12. As Fig. 8 but for vector-vector channels: f P1g, location on unphysical sheets are virtual bound states that ω1ω8 X f P1g amplitudes. do not appear as asymptotic particles. Poles off the real axis on unphysical sheets11 are associated with resonances, Collectively, these parametrization variations tell us that and it is common to interpret the real and imaginary the limited number of rest-frame energy levels with which components of the poleffiffiffiffiffi position s0 in terms of the mass p i we are constraining the large number of coupled channels mR and width ΓR, via s0 ¼ mR 2 ΓR. Near the pole, the is not sufficient to completely uniquely determine the t-matrix takes the form, t-matrix. Nevertheless, behavior consistent with a single resonant enhancement can typically be seen in the cl cl0 0 1 8 1 8 8 3 ∼ SJa S Jb η η f P1g and ω η f P1g amplitudes. We will find that tlSJa;l0S0Jb s0 − s even those parametrizations that do not appear to show 1 8 1 8 8 3 significant enhancement in either η η f P1g or ω η f P1g still feature a nearby resonance. The rapid turn-on of the where the factorized residues give access to clSJa, which axial-vector–pseudoscalar amplitudes will prove to be due are interpreted as complex-valued resonance couplings for 2Sþ1 to a large coupling of this resonance to one or both of these the channel a in partial wave lJ. channels. The amplitudes presentedp inffiffiffi Sec. V suggest a likely In order to demonstrate the presence of a resonance, we resonance with real energy at s ∼ 0.46, in which case the will now examine the amplitudes presented in this section proximal sheet is ½−−−−þþ. Indeed, for every para- 2 at complex values of s ¼ Ecm where a pole singularity is metrization which successfully describes the finite-volume expected to feature. spectrum, we find a complex-conjugate pair of poles on the proximal sheet whose real energy is in the neighborhood of VI. RESONANCE POLE SINGULARITIES the anticipated mass and which has only a small imaginary energy.12 For the illustrative amplitude given by Eq. (8), the At each meson-meson threshold, unitarity necessitates a poles on the proximal sheet lie at 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 relevant pffiffiffiffiffi i a s0 ¼ 0.4609ð12Þ 0.0036ð15Þ; ð9Þ kinematic thresholds and hence a total of 64 Riemann t ½−−−−þþ 2 sheets. The physical sheet, the sheet on which physical scattering occurs just above the real energy axis, is 11Causality forbids poles on the physical sheet off the real axis, identified by all scattering momenta having positive imagi- and any amplitudes featuring such singularities close enough to ðaÞ nary parts, i.e., ImðkcmÞ > 0 for all channels, (a). Sheets the real axis to have a non-negligible effect should be discarded with other sign combinations of the imaginary component as unphysical. 12For a few parametrizations, the pole is found to lie on the real of momenta are called unphysical, and it is on these sheets 8 8 axis below the f1η threshold. These parametrizations are those where pole singularities corresponding to resonances are which decouple the resonance from the η1η8, ω8η8, ω8ω8 and allowed to live as complex-conjugate pairs away from the ω1ω8 channels, such that the pole describes a stable bound state 8 8 8 8 real axis. in a coupled f1η , h1η system.

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FIG. 13. Pole singularities on the proximal sheet for all successful parametrizations as described in the text. Error bars reflect the statistical uncertainties on the pole position for each parametrization. where the uncertainty is statistical. Based upon the varia- zero are also capable of describing the finite-volume spectra. tion of scattering hadron masses and anisotropy described This class of results is shown in the middle panel of Fig. 14. in Sec. V, an additional conservative systematic error could Parametrizations in which the coupling of the resonance 8 8 3 be added of similar size to the statistical error. to the h1η f S1g channel is fixed to zero are found to be For each of the parametrizations found to successfully incapable of describing well the finite-volume spectra. describe the finite-volume spectrum, we show in Fig. 13 the They either have a poor χ2, or predict additional finite- proximal sheet pole position situated in the lower half- volume energy levels that lie very close to our energy plane. In every case, the pole is found with a small cutoff, levels for which there is no evidence in the lattice imaginary component and hence is very close to the region calculation. 8 8 3 of physical scattering, strongly influencing the amplitudes The ambiguity in the relative size of the f1η f S1g and “ ” 8 8 3 at real energies. As expected there are also mirror poles h1η f S1g couplings can be explained in terms of there distributed across some of the remaining unphysical being only a small gap between the relevant kinematic Riemann sheets, but these have a negligible effect on thresholds. These two channels both have the same partial- 3 physical scattering by virtue of lying farther away. wave structure ( S1), so from the point of view of the finite- While it is clear that a nearby pole is required to describe volume functions M in Eq. (1) they differ only in the mass the finite-volume spectra, the channel couplings which 8 8 8 8 difference between f1 and h1. If the f1 and h1 masses were come from the factorized residues of this pole are not degenerate, then the quantization condition would be uniquely determined across different parametrizations. We 1 8 1 invariant under permutations of the t-matrix elements in find that the couplings of the pole to the η η f P1g and 8 8 3 these two channels, analogous to the indistinguishable ω η f P1g channels are small relative to a large value of the vector-vector amplitudes we discuss in Appendix C.It 8 8 3 coupling to h1η f S1g and in some cases a large value of follows that we are only able to distinguish these channels 8 8 3 the coupling to f1η f S1g. by the mass splitting of the two axial-vector octets, and we Focusing on the axial-vector–pseudoscalar channels, we explore the degree to which the finite-volume spectra are isolate two classes of results across our range of para- sensitive to different resonance couplings in a toy model in 8 8 3 metrization forms, one in which the coupling to f1η f S1g Appendix E. In this model, the scattering system is 8 8 3 8 8 3 is large, of comparable size to a large coupling to simplified to a two-channel (f1η f S1g, h1η f S1g) case 8 8 3 h1η f S1g, and a second in which the coupling to with a bound-state pole lying below both thresholds. We 8 8 3 f1η f S1g is small. The couplings for these two classes find the finite-volume spectra in the rest frame constrain are shown in the top and middle panels of Fig. 14. Their very well the sum of the squared couplings, but offer sizes are governed largely by the corresponding g param- relatively little constraint on the ratio of the coupling eters in the numerator of the pole term in the K-matrix, as strengths. An energy level that is sensitive to the ratio lies given in Eq. (7). For a range of parametrizations that allow between the two thresholds, but because the thresholds are both of these g parameters to freely vary, we find that the so close together, this lever arm is not large. ratio of the corresponding couplings is of order one, with In summary, while we can confidently state that the 8 8 3 both found to be significantly nonzero; these are shown in h1η f S1g coupling is large, the constraints from the finite- 8 8 3 the top panel of Fig. 14. volume spectra can allow the f1η f S1g coupling to be as We also find a number of parametrizations where the small as zero. 8 8 3 8 8 3 1 8 1 f1η f S1g coupling is negligibly small while the h1η f S1g Examining the η η f P1g coupling in Fig. 14, we find 8 8 3 coupling remains large, and parametrizations in which the this to be small compared with h1η f S1g. There is a clear 8 8 3 coupling of the resonance to f1η f S1g is set to be exactly preference for a value close to 0.04, but there are

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0.05

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-0.05

FIG. 14. Couplings corresponding to the pole singularities shown in Fig. 13 as described in the text. Error bars reflect the statistical uncertainties on each coupling for each parametrization. Shaded bars show ranges and upper limits on the couplings described in the 8 8 3 text. Top: couplings to non-vector-vector channels for parametrizations where the f1η f S1g coupling was found to be significantly 8 8 3 nonzero. Middle: as top but for parametrizations where the f1η f S1g coupling was found to be zero or fixed to be identically zero. Bottom: couplings to vector-vector channels, ω8ω8; ω1ω8.

1 8 X parametrizations capable of describing the finite-volume The upper limit for jatcω ω f P1gj reflects the preferred zero spectra in which this coupling is set to be zero. The value of this coupling, while the other couplings show 8 8 3 coupling to ω η f P1g shows a very similar behavior. evidence that they scatter around some nonzero value; see Finally, for the vector-vector channels, we find the Fig. 14. These ranges and upper limit are shown by the 8 8 3 ω ω f P1g coupling shows signs of being small but shaded bars in the figure. Similarly, a best estimate of the nonzero on some parametrizations, but again the finite- pole position is given by volume spectra can be equally well described with this 1 8 X pffiffiffiffiffi i coupling set to zero. The ω ω f P1g couplings are 0 4606 26 0 0039 39 at s0 ¼ . ð Þ2 . ð Þ: ð11Þ negligibly small on every parametrization and again we find perfectly reasonable descriptions of the spectra when The small total width of the resonance, despite the large these are set to exactly zero. 8 8 coupling to h η , is explained by there being no phase Given this discussion, we summarize the behavior of the 1 space for this subthreshold decay. couplings in Fig. 14 with the following best estimates, The results presented in this section describe a very which we suggest are a conservative reflection of allowed − narrow exotic 1 ðþÞ resonance that appears in a version of ranges or limits taking into account statistical uncertainties QCD where the u, d quarks are as heavy as the physical s and parametrization variations, quark. We will now discuss an interpretation of these results, aiming to provide a description of the π1 resonance ja cη1η8 1 j¼0 → 0.055; t f P1g at the physical light-quark mass. 8 8 3 0 → 0 060 jatcω η f P1gj¼ . ; VII. INTERPRETATION 8 8 3 0 → 0 020 jatcω ω f P1gj¼ . ; In this section we will discuss what can be learned from ja cω1ω8 X j ≲ 0.020; t f P1g the observation of a JPðCÞ ¼ 1−ðþÞ resonance at the SU(3) a c 8η8 3 0 → 0.21; j t f1 f S1gj¼ flavor point as presented above. As discussed in Sec. III,we choose to focus our interpretation on the isovector member jatc 8η8 3 j¼0.21 → 0.41: ð10Þ h1 f S1g of the SU(3) octet, the π1. We will attempt to infer possible

054502-19 WOSS, DUDEK, EDWARDS, THOMAS, and WILSON PHYS. REV. D 103, 054502 (2021) properties of this resonance at the physical light-quark mass We can explore some aspects of this observation by by performing a crude extrapolation, making use of the considering generic properties of correlation functions JPAC/COMPASS candidate state mass [18] to set the having a hybrid meson interpolator at the source and a relevant decay phase spaces. We will compare our results meson-meson-like operator at the sink, following argu- to existing predictions for hybrid meson decay properties ments along the lines of those given by Lipkin [91], which made in models. were later placed in a limited field-theoretic framework by In order to present results in physical units, we must set the “field symmetrization selection rules” [92]. For the 3 the lattice scale using a physically measured quantity, an decay of an SUð ÞF octet into either an octet-octet pair or an approach which is necessarily ambiguous, particularly given octet-singlet pair, the possible Wick contractions are shown that we are far from the physical u, d masses. As in previous in Fig. 15. In the case of decays of a 1−ðþÞ octet to a pair of publications, we choose to use the Ω-baryon mass as a identical octet mesons, if the spin þ space configuration of quantity which should not have a strong dependence on the the meson-meson pair is antisymmetric, from Bose sym- u, d quark masses. Calculated on the L=as ¼ 16 lattice, we metry the flavor configuration must be antisymmetric, but find atmΩ ¼ 0.3593ð7Þ [89], so that using the experimental this would have the wrong C-parity as discussed in Sec. III mass 1672.45(29) MeV [90], we obtain an inverse temporal and Appendix B, and the correlation function is therefore −1 lattice spacing at ¼ 4655 MeV. This scale setting yields zero. Examples of such decays that are not allowed include 8 8 1 8 8 1 5 stable hadron masses of η η f P1g and ω ω f P1; P1g. A nontrivial implication in the SU(3) limit is for octet-singlet meson pairs. For η8 688 1 1 8 1 3 5 mð Þ¼ ð Þ MeV; example, in principle all of ω ω f P1; P1; P1g can have 1 η8 mðη Þ¼939ð5Þ MeV; a nonzero coupling to the 1, but the fact that the disconne- ω1 ω8 1003 1 cted contributions to the are very small (see Sec. III) mð Þ¼ ð Þ MeV; renders the diagram D small, leaving only diagram C.As mðω1Þ¼1012ð1Þ MeV; the spatial qq¯ wave function of the ω1 is expected to be very 8 8 1491 3 similar to that of the ω , we can anticipate that the mðf1Þ¼ ð Þ MeV; 1 5 antisymmetric combinations P1 and P1 from diagram C 8 1523 3 3 mðh1Þ¼ ð Þ MeV: will be small, while the symmetric combination P1 needs 1 8 3 8 8 3 not be suppressed. That the ω ω f P1g and ω ω f P1g η8 The 1 resonance pole described in the previous section couplings prove to be small appears to be due to dynamics when expressed in physical units has a mass mR ¼ that go beyond simple symmetry arguments. 2144 12 Γ 21 21 1 8 1 ð Þ MeV, and a width R ¼ ð Þ MeV, and the In the case of η η f P1g, if the spatial qq¯ wave functions couplings to meson-meson channels are of the η1 and η8 were the same, diagram C would be zero 1 owing to the antisymmetry of P1. In our calculation the jcη1η8 1 j¼0 → 256 MeV; f P1g optimized single-meson operators are constructed using the 8 8 3 0 → 279 same fermion-bilinear basis for both the octet and singlet, jcω η f P1gj¼ MeV; and we find that essentially the same optimal linear jcω8ω8 3 j¼0 → 93 MeV; 1 8 f P1g superposition is present for the η and the η , suggesting

1 8 X ≲ 93 jcω ω f P1gj MeV;

c 8 8 3 0 → 978 MeV; j f1η f S1gj¼

c 8η8 3 978 → 1909 MeV; j h1 f S1gj¼ where we have given an upper bound on the magnitude of 1 8 X ω ω f P1g to acknowledge the preferred value of zero coupling to this channel. These results can be viewed in the context of past predictions for the decays of hybrid mesons made within models. In both flux-tube breaking pictures and bag models, decays to meson pairs in which one meson has qq¯ in a P-wave and the other has qq¯ in an S-wave are enhanced over cases where both mesons have qq¯ in an S-wave [2–4,13]. In this particular case, that would suggest 8 8 8 8 1 8 8 8 8 8 1 8 dominance of f1η ;h1η over η η ; ω η ; ω ω ; ω ω , which appears to be borne out in the couplings found in FIG. 15. Wick contraction topologies for 8 → 8 ⊗ 8 (left) and our QCD calculation. 8 → 8 ⊗ 1 (right).

054502-20 DECAYS OF AN EXOTIC 1−þ HYBRID MESON … PHYS. REV. D 103, 054502 (2021) that they have similar spatial wave functions. However, phys phys l k ðmR Þ C jcjphys ¼ jcj: ð12Þ even if diagram is heavily suppressed, there remains kðm Þ diagram D which can be significant in this case owing to R the large disconnected contribution to the η1 (which generates the mass splitting between the η8 and the η1). This approach is motivated by observations made in lattice calculations of the decays of b1 → ωπ dominantly in S-wave [68], ρ to ππ in P-wave [46], K to Kπ in P-wave A. Flavor decomposition of the SU(3) amplitudes 0 ¯ [93] and f2;f2 decays to ππ and KK in D-wave [66], which This is the first determination of the couplings of an appear to show quark-mass independence when treated this exotic JPC resonance to its decay channels within a first- way. For example in the b1 case, the coupling computed in principles approach to QCD, but of course it has been done [68] at mπ ∼ 391 MeV is jcj¼564ð114Þ MeV, in good with u, d quarks that are much heavier than those in nature. agreement with the coupling jcjphys ¼ 556ð17Þ MeV In order to predict how this resonance would appear extracted from the experimental b1 decay width. In the experimentally, we have to make a large extrapolation P-wave ρ decay, an explicit factor of k is required for the down to the physical light-quark mass. We will attempt this scaling to work, as presented in Ref. [46]. In addition, as in a crude way, by assuming that the pole couplings are shown in Fig. 4 of Ref. [93], the K coupling scaled in this quark-mass independent except for a factor of the angular way is approximately constant for four different light-quark l momentum barrier k evaluated at the resonance mass. To masses corresponding to mπ ¼ 239 to 391 MeV, even when obtain this factor, and to determine the relevant phase the K is a shallow bound state, and is in agreement with 0 space, we require the mass of the π1 at the physical light- the experimentally measured coupling. Scaling the f2;f2 quark mass. Given that we do not have a calculation of this, D-wave couplings computed at mπ ∼ 391 MeV in [66] we use the experimental candidate mass 1564 MeV found gives, in comparison to values extracted from the Particle in the JPAC analysis of COMPASS data [18], and we also Data Group (PDG) review [90], consider a window of masses between 1500 and 1700 MeV. Scaled PDG In order to extrapolate to the physical light-quark mass, → ππ þ9 we need to break the SU(3) flavor symmetry present in our jcðf2 Þj 488(28) 453−4 , ¯ calculation. We will retain isospin symmetry. Because the jcðf2 → KKÞj 139(27) 132(7), 0 → ππ neutral flavorless mesons can now become admixtures of jcðf2 Þj 103(32) 33(4), 0 → ¯ octet and singlet, we will have to introduce mixing angles, jcðf2 KKÞj 321(50) 389(12), which we will take from phenomenological descriptions of which is quite a reasonable agreement given the large experimental data. We will first break up the SU(3) octets extrapolation in quark mass.13 into their component states, making use of the SU(3) Using the couplings scaled to the physical quark mass, Clebsch-Gordan coefficients provided in Ref. [84].As we can estimate partial widths for decay into kinematically an example, for the decays of the πþ, the I 1;I 1 1 ¼ z ¼þ open channels using the approach presented in the PDG member of the octet, into a vector-pseudoscalar pair we review [90] where the real part of the pole position is used would have the combination, to determine the phase space in

phys 2 1 1 jc j phys pffiffiffi ðπþρ0 − π0ρþÞþpffiffiffi ðKþK¯ 0 − K¯ 0KþÞ; ΓðR → iÞ¼ i · ρ ðm Þ: ð13Þ 3 6 phys i R mR Summing up all nonzero partial widths, we can obtain an such that the relevant couplings would be estimate for the total width.14 We will consider each η1η8 rffiffiffi rffiffiffi constrained decay channel in turn, beginning with . 2 1 1 8 1 (i) η η f P1g:For1 ⊗ 8 → 8, we have only, trivially, π → πρ 8 8 π → ¯ 8 8 jcð 1 Þj ¼ jcω η j; jcð 1 KK Þj ¼ jcω η j; þ 3 3 η1π , where η1 is the only member of the SU(3) 8 8 1 singlet. Because η η is forbidden in P1 by Bose pffiffiffi η πþ where the additional factor of 2 reflects the desire to sum symmetry, no components of the form 8 can appear, where η8 is the flavorless, neutral member of over all final state charge combinations when the decay rate is calculated. It is these separated couplings which we attempt to 13An additional quark-model “form factor” as part of the extrapolate to the physical light-quark mass, by making the scaling is advocated by Burns and Close [94]. 14Note that doing this at the SU(3) point using the couplings in simple-minded assumption that each coupling is indepen- Eq. (14) gives a total width in the range 0 → 45 MeV, in dent of the light-quark mass after appropriately rescaling reasonable agreement with our best estimate for the width from the angular momentum barrier, the pole position, 21(21) MeV.

054502-21 WOSS, DUDEK, EDWARDS, THOMAS, and WILSON PHYS. REV. D 103, 054502 (2021)

the octet. The η8 and η1 are related to the physical η The ω8, ω1 mixing to give ω, ϕ is well known to 0 and η states via a mixing angle θP, be very different from the pseudoscalar case, with 1 ¯ the ω being dominantly pffiffi ðuu¯ þ ddÞ and ϕ domi- η θ θ η 2 8 cos P sin P ¯ ¼ ; ð14Þ nantly ss. Using the same conventions as Eq. (14) 0 0 η1 − sin θP cos θP η with η → ω, η → ϕ, this “ideal” mixing would correspond to a mixing angle of θ ≈−54.7°. A θ V where phenomenological estimates for P place it mixing angle of θV ∼−52° extracted from a model close to −10° [90,95–97]. The couplings of the π1 to fit describing experimental vector to pseudoscalar ηπ and η0π then follow, radiative transitions [96] is in good agreement with this (see also [90]). It follows that the ωρ, ϕρ jcðπ1 → ηπÞj ¼ jcη1η8 sin θPj; 3 couplings for P1 are π → η0π 1 8 θ jcð 1 Þj ¼ jcη η cos Pj: rffiffiffi 1 3 jcðπ1 → ωρf P1gÞj ¼ 2 jcω8ω8 3 j cos θ The relatively small mixing angle and the lack of 5 f P1g V η8η8 coupling to Bose-forbidden suggests that the η0π coupling should be around 6 times larger than − jcω1ω8 3 j sin θV ; the coupling to ηπ, independent of the particular f P1g 15 rffiffiffi value of cη1η8 . 1 3 π → ϕρ 2 8 8 3 θ 8 8 3 jcð 1 f P1gÞj ¼ jcω ω f P1gj sin V (ii) ω η P1 : For the vector-pseudoscalar channel, the 5 f g relevant flavor embedding is 8 ⊗ 8 → 82, and the

1 8 3 θ components are þjcω ω f P1gj cos V : 1 1 pffiffiffi ðπþρ0 − π0ρþÞþpffiffiffi ðKþK¯ 0 − K¯ 0KþÞ; These expressions are consistent with the expect- 3 6 ations of the OZI rule: If the disconnected diagram D and the corresponding couplings accounting for a in Fig. 15 vanishes and ω,ϕ mixingqffiffi is ideal, 3 8 sum over charge states to be done in the partial width π →ϕρ 0 1 8 3 8 8 3 cð 1 f P1gÞ¼ and cω ω f P1g ¼ 5cω ω f P1g. calculation are rffiffiffi The coupling to kaons is 2 rffiffiffiffiffi jcðπ1 → ρπÞj ¼ jcω8η8 j; 3 3 ¯ 3 jcðπ1 → K K f P1gÞj ¼ 2 jcω8ω8 3 j: rffiffiffi 10 f P1g 1 π → ¯ 8 8 jcð 1 K KÞj ¼ 3jcω η j: 1 5 8 8 8 8 3 1 8 X 8 8 1 8 For P1 and P1, ω ω is forbidden by Bose (iii) ω ω f P1g, ω ω f P1g: The ω ω and ω ω vec- symmetry and the only contribution comes from the tor-vector channels must be considered together. 1 8 8 8 1 ω ω . The corresponding couplings are therefore, Unlike the η η channel forbidden in P1, the non- 8 8 3 trivial spin coupling in ω ω means that the P1 is in 1 5 jcðπ1 → ωρf P1; P1gÞj ¼ jcω1ω8 1 5 sin θ j; a totally symmetric configuration and thus not f P1; P1g V 1 5 forbidden; see Appendix B. This means the corre- jcðπ1 → ϕρf P1; P1gÞj ¼ jcω1ω8 1 5 cos θVj: 8 8 1 8 3 f P1; P1g sponding components for ω ω and ω ω in P1 8 8 both feature ρω and ρϕ.For8 ⊗ 8 → 81, the ω ω These couplings are expected to be very small components are because only the disconnected diagram contributes rffiffiffiffiffi to these decays. 3 1 þ ¯ 0 ¯ 0 þ þ þ 8 8 3 8 8 3 8 8 8 8 − ðK K þ K K Þþpffiffiffi ðρ ω8 þ ω8ρ Þ η η ω ω η 10 5 (iv) f1 f S1g, h1 f S1g: Similar to , f1 em- rffiffiffiffiffi rffiffiffi beds in 81 and decomposes into 3 1 −2 þ ¯ 0 2 ω ρþ rffiffiffiffiffi ¼ 10K K þ 5 8 ; 3 1 − þ ¯ 0 ¯ 0 þ ffiffiffi þη πþ ðK1AK þ K1AK Þþp ða1 8 þðf1Þ8 Þ; 1 8 þ 10 5 and trivially the only component of ω ω is ω1ρ . where we see the neutral, flavorless members of the 15 Allowing a range of −10°to−20° suggests an η0π coupling þðþÞ pseudoscalar and 1 octets, the η8 and ðf1Þ8, and 3 to 6 times the ηπ coupling, in good agreement with a ratio of 1þðþÞ 3.0(3) suggested in the very recent analysis of COMPASS and the strange members of the octet, K1A.We 1 8 Crystal Barrel data [19]. have not included the f1η channel in the scattering

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TABLE VIII. Thresholds, couplings and partial widths for each channel kinematically open at mR ¼ 1564 MeV. Couplings are derived as discussed in the text and partial widths are determined according to the definition given in Eq. (13). For both couplings and partial widths we present a range calculated from the corresponding SU(3) couplings, while those shown as an upper bound have a preferred value of zero.

phys Γ Thr./MeV jci j=MeV i=MeV ηπ 688 0 → 43 0 → 1 ρπ 910 0 → 203 0 → 20 η0π 1098 0 → 173 0 → 12 b1π 1375 799 → 1559 139 → 529 KK¯ 1386 0 → 87 0 → 2 f1ð1285Þπ 1425 0 → 363 0 → 24 1 ρωf P1g 1552 ≲19 ≲0.03 3 ρωf P1g 1552 ≲32 ≲0.09 5 ρωf P1g 1552 ≲19 ≲0.03 f1ð1420Þπ 1560 0 → 245 0 → 2 P Γ Γ 139 → 590 ¼ i i ¼

rffiffiffi calculation, given that this was largely decoupled in 2 jcðπ1 → b1πÞj ¼ jc 8η8 j: our observations of the finite-volume spectra in 3 h1 Sec. IV, and we therefore assume here that the 1 8 1 1270 f1η coupling is zero. The physical axial-vector kaons, the K ð Þ and K1ð1400Þ, are not eigenstates of charge con- The mixing of ðf1Þ8 and ðf1Þ1 to form the jugation and can be considered to be admixtures of physical states f1ð1285Þ and f1ð1420Þ can be þðþÞ the K1A from the 1 octet and the K1B from the determined from the radiative decays of the þð−Þ 1 octet. This mixing, in terms of an angle θK, can f1 1285 to γρ and γϕ, which suggests a mixing ð Þ be defined through angle of θA ∼−34°, following the formalism presented in [98], using the PDG averages [90], and K1B cosθK −sinθK K1ð1270Þ using the same conventions as Eq. (14) with ¼ ; ð15Þ 0 θ θ 1400 η → f1ð1285Þ, η → f1ð1420Þ (see also Ref. [99]). K1A sin K cos K K1ð Þ The corresponding couplings in decays involving the nonstrange 1þðþÞ mesons are which is consistent with the conventions in Ref. [100]. There is not clear consensus on the θ ∼45 1 value of K, but it could be as large as °. In ffiffiffi jcðπ1 → a1ηÞj ¼ p jc 8η8 cos θPj; practice there is only dependence on this mixing 5 f1 ¯ angle if the decay to the K1ð1270ÞK channel is open; 1 π 0 ffiffiffi this requires the 1 to have a mass above 1747 MeV, jcðπ1 → a1η Þj ¼ p jc 8η8 sin θPj; 5 f1 significantly heavier than the JPAC/COMPASS 1 candidate. ffiffiffi jcðπ1 → f1ð1285ÞπÞj ¼ p jc 8η8 cos θAj; 5 f1 π 1 B. Partial widths for a 1ð1564Þ ffiffiffi jcðπ1 → f1ð1420ÞπÞj ¼ p jc 8η8 sin θAj: 5 f1 Combining the flavor decompositions in the previous section with the scaling given by Eq. (12) we obtain the 16 couplings for a 1564 MeV π1 presented in Table VIII. Using these couplings, we populate Table VIII with partial – 8η8 The other axial-vector pseudoscalar channel h1 widths determined using Eq. (13). We assume that the 8 embeds in 2 and has components, subsequent decays of unstable isobars (e.g., ρ, b1) factorize from the initial π1 decays given in the table. 1 1 ffiffiffi þ ¯ 0 − ¯ 0 þ ffiffiffi þπ0 − 0πþ p ðK1BK K1BK Þþp ðb1 b1 Þ; 16 3 3 6 3 For the ωρf P1g and ϕρf P1g momentum scaling, where there is a linear combination of two SU(3) couplings, we evaluate the momentum at the SU(3) point with m1 ¼ m2 ¼ mω8 in both 1þð−Þ 8 1 where K1B are the strange members of the cases as the mass difference between the ω and ω is negligibly octet. The coupling to b1π is then small and it simplifies the resulting algebra.

054502-23 WOSS, DUDEK, EDWARDS, THOMAS, and WILSON PHYS. REV. D 103, 054502 (2021)

600

400

200

1500 1550 1600 1650

FIG. 16. Partial widths as a function of the π1 pole mass. The bands reflect the coupling ranges given in Table VIII. The total width obtained by summing the partial widths is shown by the gray band.

It is clear that the dominant decay mode is b1π, with the The only prior estimate of decay rates for a π1 obtained 0 next largest channels η π; ρπ and f1ð1285Þπ being signifi- using lattice QCD was the calculation presented in Ref. [101] cantly smaller. Despite the larger phase space, the partial which used a rather different approach to the one followed in width into ηπ is approximately 10 times smaller than η0π, this paper. By tuning the value of the light-quark mass in a independent of the coupling and depending only on the two-flavor calculation (without strange quarks), the authors mixing angle and phase space. Only one kaonic decay were able to make the mass of the π1 be approximately equal ¯ mode is kinematically accessible, KK, with a very small to the sum of the masses of the π and the b1.Theyarguedthat partial width. Decays to ρω are negligible. Summing all the time dependence of a single two-point function having a −ðþÞ partial widths we obtain an estimate for the total width in 1 single-meson operator at the source and a b1π-like the range 139 to 590 MeV which includes the value operator at the sink can be used to infer a transition rate. The 492(47)(102) MeV found in the JPAC/COMPASS analy- method makes a number of assumptions that have not yet sis.17 If our extrapolation is accurate, it suggests that the been validated, but their result for pion masses near 500 MeV 0 observation of the π1 in ηπ and η π is through decays which does suggest a large coupling. They also found a somewhat are very far from being the dominant decay modes. smaller coupling to f1π. It is possible that this estimate of the total decay width We can also compare our result extrapolated to physical may be missing contributions from channels which are kinematics with the predictions of models. Models based closed at the SU(3) point, whose couplings we have not upon breaking of the flux tube [4,13] do not allow decays to determined, but which become open at physical kinematics. identical mesons, but these are typically prevented by Bose Examples might include f2π (although this is a D-wave symmetry anyway. The ability of these models to predict decay with relatively little phase space, so a large width is decays involving the η or η0 is somewhat questionable given unlikely), or ηð1295Þπ (a P-wave decay with a very small that no disconnected contributions are considered. Within phase space). Any truly multibody decays to three or more these models, the quark spin coupling factorizes from the mesons, i.e., those not proceeding through a resonant spatial matrix element such that ρπ decays are only allowed isobar, are also not included in this estimate, but the to the extent that the spatial qq¯ wave functions of the π and conventional wisdom is that such decays are not large. the ρ differ. This difference is quite hard to estimate in Figure 16 shows the partial widths for each channel in quark models where the very light pseudo-Goldstone boson Table VIII as a function of the physical resonance mass π is typically not well described. mphys allowed to vary in the range 1500–1700 MeV. If this model picture of the coupling being sensitive to R π ρ We observe only a modest dependence upon the mass of the difference between the and radial wave function is correct, our simple extrapolation of the ρπ coupling may the π1 resonance, with the exception of the f1ð1420Þπ channel which becomes kinematically open in this energy lead to an underestimate. We can use the charge radius as a range. guide to the wave function size, and at the SU(3) flavor symmetric point these radii were computed in Ref. [76]: 2 1=2 2 1=2 hr iπ ¼ 0.47ð6Þ fm, hr iρ ¼ 0.55ð5Þ fm. These sizes 17And the somewhat smaller value ∼388 MeV found in the are not that different, as one might expect given the very recent analysis of COMPASS and Crystal Barrel data [19]. heaviness of the quarks, but we expect the difference to

054502-24 DECAYS OF AN EXOTIC 1−þ HYBRID MESON … PHYS. REV. D 103, 054502 (2021) grow as the light-quark mass reduces. Our simple extrapo- decay mode is somewhat challenging experimentally, lation of the ρπ coupling would not capture this change, ending up in five pions through b1 → ωπ, these results and hence our ρπ partial width might be an underestimate. suggest that it is a promising channel to search in. The flux-tube breaking models have larger couplings to axial-vector–pseudoscalar channels like b1π and f1π than ACKNOWLEDGMENTS to, for example, ρπ, but these couplings are still much smaller than the ones we are predicting. Bag models show We thank our colleagues within the Hadron Spectrum similar decay systematics [2,3]. Collaboration. A. J. W., C. E. T. and D. J. W. acknowledge support from the U.K. Science and Technology Facilities Council (STFC) (Grant No. ST/P000681/1). J. J. D. VIII. SUMMARY acknowledges support from the U.S. Department of Prior lattice QCD calculations which treated excited Energy Contract No. DE-SC0018416. J. J. D., R. G. E. hadrons as stable particles indicated the presence of exotic and C. E. T. acknowledge support from the U.S. hybrid mesons in the spectrum, but until now the only Department of Energy Contract No. DE-AC05- theoretical information on the decay properties of these 06OR23177, under which Jefferson Science Associates, states came from models whose connection to QCD is not LLC, manages and operates Jefferson Lab. D. J. W. always clear. In this paper we presented the first determi- acknowledges support from a Royal Society University nation of the lightest JPðCÞ ¼ 1−ðþÞ resonance within lattice Research Fellowship. C. E. T. and J. J. D. acknowledge QCD. The resonance was observed in a rigorous way as a support from the Munich Institute for Astro- and pole singularity in a coupled-channel scattering amplitude Particle Physics, which is funded by the Deutsche obtained using constraints provided by the discrete spectrum Forschungsgemeinschaft (German Research Foundation) of eigenstates of QCD in six different finite volumes. These under Germany’s Excellence Strategy EXC-2094– spectra were extracted from matrices of correlation functions 390783311, while attending a program. C. E. T. also computed in lattice QCD using a large basis of operators. acknowledges CERN TH for hospitality and support during In order to make this first calculation practical we opted to a visit. The software codes CHROMA [102] and QUDA work with quark masses such that mu ¼ md ¼ ms, with the [103–105] were used. The authors acknowledge support quark mass selected to approximately match the physical from the U.S. Department of Energy, Office of Science, 3 strange-quark mass. The resulting SUð ÞF symmetry leads to Office of Advanced Scientific Computing Research and a simplified set of decay channels, and the relatively heavy Office of Nuclear Physics, Scientific Discovery through quark mass means that only meson-meson decays are Advanced Computing (SciDAC) program. Also acknowl- kinematically accessible in the energy region of interest. edged is support from the Exascale Computing Project The computed lattice QCD spectra are described No. 17-SC-20-SC, a collaborative effort of the U.S. by an eight-channel flavor-octet 1−ðþÞ scattering system Department of Energy Office of Science and the in which a narrow resonance appears, lying slightly National Nuclear Security Administration. This work below the opening of axial-vector–pseudoscalar decay was performed using the Cambridge Service for Data channels, but well above pseudoscalar–pseudoscalar, Driven Discovery (CSD3) operated by the University of vector–pseudoscalar and vector–vector decay thresholds. Cambridge Research Computing Service provided by Dell The resonance pole shows relatively weak couplings to EMC and Intel using Tier-2 funding from the Engineering the open channels, hence the narrow width, but large and Physical Sciences Research Council (Capital Grant couplings to at least one kinematically closed axial- No. EP/P020259/1), and Distributed Research utilising vector–pseudoscalar channel. Advanced Computing (DiRAC) funding from STFC. The A simple-minded approach was used to predict decay DiRAC component of CSD3 was funded by Department for properties of a π1 resonance with physical light-quark mass Business, Energy and Industrial Strategy (BEIS) capital from these results. We extrapolated the determined cou- funding via STFC Capital Grants No. ST/P002307/1 and plings, assuming their only adjustment is in the angular No. ST/R002452/1 and STFC operations Grant No. ST/ momentum barrier (an approach that has proven reasonably R00689X/1. DiRAC is part of the National e-Infrastructure. successful when applied to previous lattice QCD determi- This work was also performed on clusters at Jefferson Lab nations of vector, axial-vector and tensor mesons). This under the USQCD Collaboration and the LQCD American suggests a potentially broad π1 resonance, the bulk of Recovery and Reinvestment Act (ARRA) Project. This whose decay goes into the b1π mode. research was supported in part under an ALCC grant, and Comparing to the experimental π1ð1564Þ candidate state used resources of the Oak Ridge Leadership Computing found by the JPAC/COMPASS analysis [18], our predicted Facility at the Oak Ridge National Laboratory, which is range of total width is compatible with their width taken supported by the Office of Science of the U.S. Department of from the resonance pole position. We note that the ηπ, η0π Energy under Contract No. DE-AC05-00OR22725. This modes in which the resonance is observed experimentally research used resources of the National Energy Research are relatively rare decays in our picture. Although the b1π Scientific Computing Center (NERSC), a Department of

054502-25 WOSS, DUDEK, EDWARDS, THOMAS, and WILSON PHYS. REV. D 103, 054502 (2021)

Energy Office of Science User Facility supported by Following the conventions given in Ref. [84], the SU(3) the Office of Science of the U.S. Department of Energy Clebsch-Gordan coefficients Cð…Þ for 8 ⊗ 8 → 81; 82 are under Contract No. DE-AC02-05CH11231. The authors respectively symmetric, antisymmetric under exchanging acknowledge the Texas Advanced Computing Center the hadrons in the product or conjugating the hadrons in the (TACC) at The University of Texas at Austin for providing product, HPC resources. Gauge configurations were generated using resources awarded from the U.S. Department of Energy 888 888 C i ξ C i Innovative and Novel Computational Impact on Theory and ¼ 1ðiÞ ; ν1 ν2 ν ν2 ν1 ν Experiment (INCITE) program at the Oak Ridge Leadership 888 888 Computing Facility, the NERSC, the National Science i i C ¼ ξ3ðiÞ C ; Foundation (NSF) Teragrid at the TACC and the ν1 ν2 ν −ν1 −ν2 −ν Pittsburgh Supercomputer Center, as well as at the Cambridge Service for Data Driven Discovery (CSD3) with ξ1ð1Þ¼ξ3ð1Þ¼1 and ξ1ð2Þ¼ξ3ð2Þ¼−1, and ¯ and Jefferson Lab. using 8 ¼ 8. Here a particular member of the octet is labeled by its isospin I, hypercharge Y, and z-component of isospin I ,inν ¼ðI;Y;I Þ, and for mesons the hypercharge APPENDIX A: SU(3) CLEBSCH-GORDAN z z is simply equal to the strangeness, Y ¼ S. COEFFICIENTS It is useful at this point to write out the nonzero SU(3) Unlike in SU(2), where the product of two representa- Clebsch-Gordan coefficients for the two embeddings tions of definite isospin decomposes into a sum of isospins explicitly. As we are at liberty to work with any member each of which appears only once, in SU(3) a representation of the target octet, we choose ν ¼ð0; 0; 0Þ. We label the can appear more than once in a product. A relevant multiplied octets 8a and 8b in order to distinguish them. example is 8 ⊗ 8 ¼ 1 ⊕ 81 ⊕ 82 ⊕ 10 ⊕ 10 ⊕ 27 where Applying the rules given in Ref. [84], we have for the we observe two octet embeddings, 81 and 82. symmetric 81 combination, j81;0;0;0i E E E E E E E E 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ffiffiffiffiffi ¼p 8a; ;1; 8b; ;−1;− þ 8a; ;−1;− 8b; ;1; − 8a; ;1;− 8b; ;−1; − 8a; ;−1; 8b; ;1;− 20 |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}2 2 |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}2 2 |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}2 2 |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}2 2 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}2 2 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}2 2 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}2 2 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}2 2 − − Kþ K K Kþ K0 K¯ 0 K¯ 0 K0 E E E 1 1 ffiffiffi ffiffiffi −p 8a;1;0;1 8b;1;0;−1 þ 8a;1;0;−1 j8b;1;0;1i−j8a;1;0;0ij8b;1;0;0i −p j8a;0;0;0ij8b;0;0;0i; 5 |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}|fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl}|fflfflfflfflfflffl{zfflfflfflfflfflffl} 5|fflfflfflfflfflffl{zfflfflfflfflfflffl}|fflfflfflfflfflffl{zfflfflfflfflfflffl} πþ ρ0 π0 ω8 η8 ρþ π− ρ− while for the antisymmetric 82 combination, j82;0;0;0i E E E E E E E E 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ¼ 8a; ;1; 8b; ;−1;− − 8a; ;−1;− 8b; ;1; − 8a; ;1;− 8b; ;−1; þ 8a; ;−1; 8b; ;1;− ; 2 |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}2 2 |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}2 2 |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}2 2 |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}2 2 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}2 2 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}2 2 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}2 2 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}2 2 − − Kþ K K Kþ K0 K¯ 0 K¯ 0 K0 where we have provided the PDG notation for vector and C ¼þ1 for η8 and C ¼ −1 for ω8. There are SU(3) analogs pseudoscalar mesons as an example, as was done in Eq. (2). of G-parity where the rotation is between the u, s or d, s ˆ ˆ Defining G in the usual way as C followed by a rotation quarks rather than the u, d quarks. When SU(3) is broken by π about the y-component of isospin Rˆ , it is straightfor- these are no longer good quantum numbers whereas ward to show [106] that, G-parity is still good as long as there is isospin symmetry. Acting with Cˆ or Gˆ on the decompositions above gives

ˆ Y=2þIz Cj8; I;Y;Izi¼Cð−1Þ j8; I;−Y;−Izi; − ˆ ˆ ˆ I Iz Cj81;0; 0; 0i¼Gj81;0; 0; 0i¼þC C j81;0; 0; 0i; Rj8; I;Y;Izi¼ð−1Þ j8; I;Y;−Izi; a b ˆ ˆ ˆ Y=2þI Cj82;0; 0; 0i¼Gj82;0; 0; 0i¼−C C j82;0; 0; 0i; Gj8; I;Y;Izi¼Cð−1Þ j8; I;−Y;Izi; a b where C is the intrinsic charge-conjugation quantum where Ca and Cb are the intrinsic charge-conjugation number of the neutral element of the octet, for example, quantum numbers of the neutral element of the octets

054502-26 DECAYS OF AN EXOTIC 1−þ HYBRID MESON … PHYS. REV. D 103, 054502 (2021)

8 8 8 8 a and b. Therefore, 1 and 2 have isoscalar members In the case of 8 ⊗ 8 → 1, the SU(3) Clebsch-Gordan which are eigenstates of charge conjugation with opposite coefficients are symmetric under interchange; explicitly the values of C. construction is j1;0;0;0i E E E E E E 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ffiffiffi ¼ p 8a; ;1; 8b; ;−1;− þ 8a; ;−1;− 8b; ;1; − 8a; ;1;− 8b; ;−1; −j8a; ;−1; ij8b; ;1;− i 2 2 |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}2 2 |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}2 2 |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}2 2 |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}2 2 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}2 2 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}2 2 |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}2 2 |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}2 2 − − Kþ K K Kþ K0 K¯ 0 K¯ 0 K0 1 1 þ pffiffiffiðj8 ;1;0;1ij8 ;1;0;−1iþj8 ;1;0;−1ij8 ;1;0;1i−j8 ;1;0;0ij8 ;1;0;0iÞ− pffiffiffij8 ;0;0;0ij8 ;0;0;0i; 2 2 |fflfflfflfflfflffl{zfflfflfflfflfflffl}a |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}b |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}a |fflfflfflfflfflffl{zfflfflfflfflfflffl}b |fflfflfflfflfflffl{zfflfflfflfflfflffl}a |fflfflfflfflfflffl{zfflfflfflfflfflffl}b 2 2|fflfflfflfflfflffl{zfflfflfflfflfflffl}a |fflfflfflfflfflffl{zfflfflfflfflfflffl}b − − ρþ π ρ πþ ρ0 π0 ω8 η8

ˆ ˆ and Cj1;0; 0; 0i¼Gj1;0; 0; 0i¼CaCbj1;0; 0; 0i. above—only even l partial waves are permitted in 81, For the cases of 8 ⊗ 1 → 8 and 1 ⊗ 1 → 1, the Clebsch- while only odd l appear for 82. In the case of S ¼ 1,byan Gordan coefficients are trivial, analogous argument, only partial waves of odd l are 8 8 permitted in 81 and even l in 82. Hence ω ω is forbidden 8;0 0 0 8 ;0 0 0 1 ;0 0 0 1 5 PðCÞ −ðþÞ j ; ; i¼j a ; ; ij b ; ; i; in P1 and P1 decays of a J ¼ 1 octet resonance, 3 while it is allowed in P1. Table IX summarizes the Bose- j1;0; 0; 0i¼j1a;0; 0; 0ij1b;0; 0; 0i; allowed partial-wave content of 81 and 82 for identical and obviously C ¼ CaCb. vector meson octets.

APPENDIX B: SU(3) BOSE SYMMETRY APPENDIX C: INDISTINGUISHABLE − A practical consequence of Bose symmetry is the elimi- VECTOR-VECTOR P-WAVES IN T1 nation of certain partial-wave configurations in the scatter- In this Appendix we show that the quantization condition ing of identical mesons. A familiar example assuming only − Eq. (1) when subduced into the T1 irrep at rest cannot isospin symmetry is that ππ scattering with isospin ¼ 1 is 1 8 1 3 5 uniquely constrain the ω ω f P1; P1; P1g amplitudes only in odd partial waves, while isospin = 0,2 are only in owing to a residual S3 permutation symmetry on these even partial waves. The SU(3) Clebsch-Gordan coefficients channels; i.e., the corresponding scattering parameters in discussed in Appendix A have definite symmetry under the the t-matrix can be freely interchanged while leaving the exchange of the two scattering hadrons, and this makes the determinant invariant. We also show that the same permu- application of Bose symmetry straightforward when we tation symmetry is not present for systems with overall need to combine two identical meson multiplets. nonzero momentum, so including energy levels obtained in Consider first identical pseudoscalar meson octets—the such irreps would provide a unique constraint for each of total spin S is zero and the spin wave function is trivially these partial waves. symmetric. To ensure overall symmetry under exchange we Recalling the form of the quantization condition, require the product of flavor and spatial wave functions to be overall symmetric, meaning they are either both sym- detl ½1 þ iρtð1 þ iMÞ ¼ 0; metric or both antisymmetric. In Appendix A we showed SJma that 81 and 82 are symmetric and antisymmetric in flavor we note that the finite-volume nature of the problem resides respectively, so we deduce that only partial waves of even l in the matrix M whose components are defined explicitly are permitted in 81 and odd l in 82. It follows that, for in Appendix A of Ref. [80]. M is trivially diagonal in the 8 8 PðCÞ þðþÞ example, η η appears with even l in 81 with J ¼ l PðCÞ −ð−Þ and odd l in 82 with J ¼ l . A consequence is that η8η8 is forbidden in decays of a JPðCÞ ¼ 1−ðþÞ octet TABLE IX. Bose-allowed partial-wave content of multiplets 81 and 82 from a product of two identical vector meson octets resonance. 8 ⊗ 8 for l ≤ 4. For identical vector meson octets, the symmetry of the a a spin wave function depends on the total spin S: symmetric 81 ðC ¼þÞ 82 ðC ¼ −Þ for S ¼ 0, 2 and antisymmetric for S ¼ 1. It follows that 1 1 1 1 1 S0; D2; G4; … P1; F3; … for S ¼ 0, 2, the product of flavor and spatial wave 3 3 3 3 3 P0;1;2; F2;3;4; … S1; D1;2;3; G3;4;5; … functions must be totally symmetric, so either they are 5 5 5 5 5 S2; D0…4; G2…6; … P1 2 3; F1…5; … both symmetric or both antisymmetric, similar to the case ; ;

054502-27 WOSS, DUDEK, EDWARDS, THOMAS, and WILSON PHYS. REV. D 103, 054502 (2021) hadron channel and intrinsic spin, leading to it being t ¼ K˜ ð1 − iρK˜ Þ−1; 1 8 1 3 5 18 diagonal in ω ω P1; P1; P1 channels. The reason f g ˜ l glSJagl0S0Jb l0 ½KðsÞl l0 0 ¼ð2k Þ ð2k Þ : that these channels cannot be distinguished at overall zero SJa; S Jb a m2 − s b momentum is that the diagonal entries of M in each of 1 8 1 3 5 ω ω f P1; P1; P1g are equal. Here we will show that this particular form can lead to From theR product of spherical harmonics in Eq. (A1) of the phenomenon of “trapped” levels in finite-volume ¯ Ref. [80], dΩY1 Y ¯ Y1 0 , it is clear that only l ≤ 2 spectra, a situation where there is guaranteed to be exactly ml lm¯ l ml contribute, and from the symmetries of the Lüscher zeta one finite-volume energy level lying between every neigh- functions at zero momentum, boring noninteracting energy. In particular, we will present a proof of how trapped levels emerge in coupled meson- 3 1 3 5 meson scattering in S1 and f P1; P1; P1g-wave in the rest 0⃗ 0 0⃗ 0 Zl∉2Z;m∉4Z ¼ ;Z20 ¼ ; frame irreps, as relevant for this study. This effect is not a general feature of the finite-volume method; for example, ¯ upon adding a matrix of polynomials in s to the K-matrix only l 0; m¯ l 0 survives. The elements of M thus ¼ ¼ above (as we commonly do) the guarantee is removed. reduce to the rather simple form, The Lüscher quantization condition Eq. (1) can be rewritten in terms of the K-matrix defined above yielding 4π the convenient form, 2Sþ1 2S0þ1 0 0⃗ 2 Mð P1;m; P1;mÞ¼δ 0 δ 0 c ðk ; LÞ; S;S m;m k 0;0 det½1 − ρK˜ M¼0; and it follows that the rest frame M does not distinguish 1 8 1 3 5 where the determinant is taken over the N-dimensional between the ω ω P1; P1; P1 channels. The result of f g space of hadron-hadron channels and partial waves. ω1ω8 1 3 5 this is that permutations of the f P1; P1; P1g chan- When K˜ is factorized as above, the matrix ρK˜ M is of the nels will leave the determinant in Eq. (1) invariant. form abT for all energies, where aðsÞ and bðsÞ are (energy- These partial waves become distinguishable if we con- dependent) vectors, and hence of rank one. It has one sider the system at overall nonzero momentum. Following a T ⃗ nonzero eigenvalue μ0ðsÞ¼b a, with eigenvector v0 ¼ a, similar derivation to the zero momentum case, owing to ZP 20 and N − 1 zero eigenvalues μ ðsÞ¼0 for i ¼ 1; …;N− 1, being nonzero in general, we find that elements of M are i whose eigenvectors span the hyperplane orthogonal to a. ⃗ 00 spin dependent. For example, in the case that P ¼½ n, It immediately follows that 1 − ρK˜ M has exactly 1 0 1 the m ¼þ , m ¼þ elements are given by one eigenvalue capable of taking a zero value, T λ0ðsÞ¼1 − b a—all other eigenvalues λiðsÞ¼1 for 4π 1 4π i ¼ 1, …, N − 1. The finite-volume spectrum is therefore M 1 1;1 1 P⃗ 2; − ffiffiffi P⃗ 2; ð P1;þ P1;þ Þ¼ c0 0ðk LÞ p 3 c2 0ðk LÞ; λ 0 k ; 5 k ; given by the solutions to 0ðsÞ¼ . 4π 1 4π For ease of illustration, consider the case of several 3 3 P⃗ 2 P⃗ 2 Mð P1;þ1; P1;þ1Þ¼ c ðk ;LÞþ pffiffiffi c ðk ;LÞ; coupled meson-meson channels, each in a single partial k 0;0 2 5 k3 2;0 wave. The nontrivial eigenvalue λ0ðsÞ takes the form, 4π 1 4π M 5 1;5 1 P⃗ 2; − ffiffiffi P⃗ 2; ð P1;þ P1;þ Þ¼ c0;0ðk LÞ p 3 c2;0ðk LÞ; 2 X k 10 5 k λ 1 − ffiffiffi 2 2l 2 M 0ðsÞ¼ p 2 − ð kaÞ gaka a; ðD1Þ sðm sÞ a P⃗ where we observe that the coefficients of the c2;0 term where Ma are the elements of the diagonal in channel distinguish the different spin configurations. space M. Recalling the definition of these presented in Ref. [80], for S- and P-waves in the rest frame, APPENDIX D: TRAPPED LEVELS FOR 2 1 2 ffiffiffi 0⃗ kaL FACTORIZED K-MATRIX POLES MaðsÞ¼p Z00 1; ; π kaL 2π A parametrization in common use to describe a single coupled-channel resonance with angular momentum J independent of the intrinsic spin of the system. The only assumes a factorized pole in the K-matrix and the simple differences between the objects MaðsÞ for different chan- phase space [IaðsÞ¼−iρaðsÞ] in the construction of the nels come from the momenta ka. It is therefore instructive t-matrix, to examine the functional form of

2l 18 1 8 5 5 − 2k s k s M s D2 It is also diagonal in the ω ω f P1; P3g subspace at rest. ð að ÞÞ að Þ að ÞðÞ

054502-28 DECAYS OF AN EXOTIC 1−þ HYBRID MESON … PHYS. REV. D 103, 054502 (2021) that appears for each channel in Eq. (D1).Wenow 2. P-wave scattering investigate the consequences of this for S-wave scattering For P-wave scattering, Eq. (D2) has an extra factor of a before considering P-wave scattering. 2 smooth real function 4kaðsÞ compared to the S-wave case. This is positive above threshold, negative below and has a 1. S-wave scattering zero exactly at threshold, and this zero is the reason for 8 8 8 8 3 In Fig. 17 we plot Eq. (D2) for the f1η and h1η S1 there being no noninteracting level at threshold in the channels. These functions are real above threshold, and P-wave. The argument that led to “trapped” levels in the show monotonic decrease between divergences at each S-wave applies here too. noninteracting energy. The finite-volume spectrum in this It is interesting to revisit the indistinguishability of 1 3 5 case is given by the solutions of f P1; P1; P1g in vector-vector scattering in the context of a factorized pole K-matrix. If we consider this system 1 ffiffiffi p 2 2 2 which has only a single open channel but three partial sðs − m Þ¼−g k1ðsÞM1ðsÞ − g k2ðsÞM2ðsÞ; ðD3Þ 2 1 2 waves, then the single nontrivial eigenvalue which has zeros at the finite-volume energy levels is where the rhs of this expression is just a weighted sum of the expressions plotted in Fig. 17. The effect of changing 3 8k M 2 2 2 the values of g1 and g2 simply moves the point of inflection λ0ðsÞ¼1 − pffiffiffi ðg1 þ g2 þ g3Þ; sðm2 − sÞ of the rhs in each region between neighboring noninteracting energies. Asffiffiffi the lhs is a monotonically increasing M p 1 as the momenta kðsÞ and the function ðsÞ are identical in pffiffi function for s > 3 m, this will intersect the rhs exactly each of these partial waves. once in each energy region between noninteracting ener- Naively, we would expect to find only a single root gies. This results in what we refer to as “trapped” levels. We between neighboring noninteracting energies; however, 8 8 see exactly this in Fig. 18, where above f1η threshold we this would overlook the fact that the multiplicity of each see a single solution in each region as described. of these noninteracting energies is in fact 3, and so we

0 0.46 0.47 0.480.49 0.50 0.51 0.52 0.53

FIG. 17. Energy dependence of −k1ðsÞM1ðsÞ and −k2ðsÞM2ðsÞ for a lattice of spatial extent L=as ¼ 24. Both functions are purely real across this energy region. Dashed vertical lines indicate the location of noninteracting energies.

0 0.46 0.47 0.480.49 0.50 0.51 0.52 0.53

− 2 M − 2 M 24 1 FIG.ffiffiffi 18. Top: the function g1k1ðsÞ 1ðsÞ g2k2ðsÞ 2ðsÞ, for L=as ¼ and g1 ¼ g2 ¼ , plotted in black, and the function 1 p 2 2 sðs − m Þ with m ¼ 0.461057 plotted in gray. The points of intersection (circles) correspond to the finite-volume energy levels. Dashed vertical lines indicate the location of noninteracting energies. Bottom: same as the top plot but with an enlarged vertical scale.

054502-29 WOSS, DUDEK, EDWARDS, THOMAS, and WILSON PHYS. REV. D 103, 054502 (2021)

8 8 3 should find three roots associated with each noninteracting and h1η f S1g couplings. In Sec. VI, we found the ratio of energy (these roots are not necessarily triply degenerate as these couplings to be poorly determined, while the sum of we will see). the squared couplings was well determined. We will This can be seen most easily by treating each partial investigate this effect using a simplified two-channel toy wave as an independent hadron-hadron scattering channel model where the t-matrix is given by by perturbing the scattering vector meson mass slightly in 1 → 8 − ϵ g g each partial wave. In P1 we take mω8 mω and in t s a b : 5 abð Þ¼ 2 − 2 2 P1 mω8 → mω8 þ ϵ, so that the perturbed finite-volume m s þ g1I1ðsÞþg2I2ðsÞ energy levels are roots of The mass parameter m ¼ 0.46 is chosen to be below the 8 8 8 3 ˜ 2 3 2 3 2 3 f1η f S1g threshold, a value which is comparable to the λ0ðsÞ¼1−pffiffiffi ðg1k−ϵM−ϵ þg2k Mþg3k ϵM ϵÞ; sðm2 −sÞ þ þ pole mass found in Sec. VI. By choosing the Chew- Mandelstam phase space, with real part such that where the subscript ϵ means that the vector meson mass I s m2 0, we ensure that when varying the cou- að ¼ Þ¼ pffiffiffi has been perturbed by ϵ. The previously triply degenerate plings the bound-state pole remains at s ¼ m. This noninteracting energies are now split by order ϵ. However, enables us to test the dependence of the finite-volume there are trapped roots between these perturbed noninter- spectra on the magnitude and ratio of the couplings for a acting energies which forces at least two of the roots to lie fixed pole position. within ϵ of the unperturbed noninteracting energy. In the In Fig. 19, we present finite-volume spectra obtained by ˜ limit ϵ → 0, we find λ0ðsÞ → λ0ðsÞ with at least two roots solving Eq. (1) for several values of the ratio b ≡ g1=g2 for 2 2 positioned exactly at the noninteracting energy. The third a fixed magnitude, a ≡ g1 þ g2 ¼ 1. Shown are the rest- − root is free to vary in position between these two roots and frame T1 irrep considered in this paper, and also the the next noninteracting energy, its location depending on moving-frame A1 irreps. It is clear that the subthreshold 2 2 2 the value of g1 þ g2 þ g3; it is exactly at the noninteracting level, while volume dependent, is quite insensitive to the energy if and only if g1 ¼ g2 ¼ g3 ¼ 0, in which case the coupling ratio in all irreps. The level lying between the − roots are triply degenerate. thresholds in the T1 irrep is sensitive to the ratio, but that is of limited use because the thresholds are rather close 8η8 3 together, split only by the mass difference between the APPENDIX E: SENSITIVITY TO f 1 f S1g, 8 8 8η8 3 f1 and h1. h1 f S1g COUPLINGS On the contrary, for a fixed ratio g1=g2 ¼ 1, it is clear In this Appendix, we will examine the sensitivity of from Fig. 20 that the subthreshold level is rather sensitive to 8 8 3 finite-volume spectra to the relative size of the f1η f S1g the sum of the squared couplings a, with a smaller value

0.50

0.49

0.48

0.47

0.46

12 16 20 24 12 16 20 24 12 16 20 24 12 16 20 24 12 16 20 24

− FIG. 19. Finite-volume spectra for the toy model as described in the text for the T1 irrep at rest and the A1 irreps in flight. 2 2 2 a ≡ g1 þ g2 ¼ 1 is kept fixed, while the ratio b ≡ g1=g2 is varied.

054502-30 DECAYS OF AN EXOTIC 1−þ HYBRID MESON … PHYS. REV. D 103, 054502 (2021)

0.50

0.49

0.48

0.47

0.46

12 16 20 24 12 16 20 24 12 16 20 24 12 16 20 24 12 16 20 24

FIG. 20. As Fig. 19 but for a fixed ratio b ¼ 1 and varying magnitude a. leading to an energy level much closer to the value of It appears that to have a well-determined ratio of m and with less volume dependence. The level between couplings in this case, we need greater statistical precision − the thresholds in the T1 irrep is somewhat less sensitive on the finite-volume energies and/or additional constraint to a. from several energy levels in moving frames.

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