Determination of the pole position of the lightest hybrid meson candidate

Mapping states with explicit gluonic degrees of freedom in the light sector is a challenge, and has led to controversies in the past. In particular, the experiments have reported two different hybrid candidates with spin-exotic signature, pi1(1400) and pi1(1600), which couple separately to eta pi and eta' pi. This picture is not compatible with recent Lattice QCD estimates for hybrid states, nor with most phenomenological models. We consider the recent partial wave analysis of the eta(') pi system by the COMPASS collaboration. We fit the extracted intensities and phases with a coupled-channel amplitude that enforces the unitarity and analyticity of the S-matrix. We provide a robust extraction of a single exotic pi1 resonant pole, with mass and width 1564 +- 24 +- 86 MeV and 492 +- 54 +- 102 MeV, which couples to both eta(') pi channels. We find no evidence for a second exotic state. We also provide the resonance parameters of the a2(1320) and a2'(1700).

Introduction.-Explaining the structure of hadrons in terms of quarks and gluons, the fundamental building blocks of Quantum Chromodynamics (QCD), is of key importance to our understanding of strong interactions. The vast majority of observed mesons can be classified as qq bound states, although QCD should have, in principle, a much richer spectrum. Indeed, several experiments have reported resonance candidates that do not fit the valence quark model template [1][2][3][4][5][6][7]. These new experimental results, together with rapid advances in lattice gauge computations, open new fronts in studies of the fundamental aspects of QCD, such as quark confinement and mass generation. Since gluons are the mediators of the strong interaction, QCD dynamics cannot be fully understood without addressing the role of gluons in binding hadrons. The existence of states with explicit excitations of the gluon field, commonly referred to as hybrids, was postulated a long time ago [8][9][10][11][12], and has recently been supported by lattice [13][14][15] and phenomenological QCD studies [16][17][18]. In particular, a state with exotic quantum numbers J P C (I G ) = 1 −+ (1 − ) in the mass range 1.7 − 1.9 GeV is generally expected. The experimental determination of hybrid hadron properties -e.g. their masses, widths, and decay patterns-provides a unique opportunity for a systematic study of low-energy gluon dynamics. This has motivated the COMPASS spectroscopy program [19,20] and the 12 GeV upgrade of Jefferson Lab, with experiments dedicated to hybrid photoproduction at CLAS12 and GlueX [21,22].
The hunt for hybrid mesons is challenging, since the spectrum of particles produced in high energy collisions is dominated by nonexotic resonances. The extraction of exotic signatures requires sophisticated partial-wave amplitude analyses. In the past, inadequate model assumptions and limited statistics resulted in debatable results. The first reported hybrid candidate was the π 1 (1400) in the ηπ final state [23][24][25][26][27]. Another state, the π 1 (1600), was later claimed in the ρπ and η π channels, with different resonance parameters [28,29]. A first coupled-channel analysis of the η ( ) π system from E852 data was not conclusive [30]. The COMPASS experiment confirmed a peak in ρπ and η π at around 1.6 GeV [31,32] and an additional structure in ηπ, at approximately 1.4 GeV [33]. While the π 1 (1600) is close to the expectation for a hybrid, the observation of two nearby 1 −+ hybrids below 2 GeV is surprising. This makes the microscopic interpretation of the π 1 (1400) problematic. Moreover, in the chiral limit, Bose symmetry prevents the decay of a hybrid into ηπ [34]. A tetraquark interpretation of the lighter state might be viable, and would explain why this state has eluded predictions in constituent gluon models. However, this interpretation would lead to the prediction of unobserved doubly charged and doubly strange mesons [35], and is unfavored in the diquark-antidiquark model [36,37]. Establishing whether there exists one or two states in this mass region is thus a stringent test for the available phenomenological frameworks in the nonperturbative regime.
In [38] we analyzed the spectrum of the ηπ D-wave extracted from the COMPASS data. In this Letter, we extend the mass dependent study to the exotic P -wave, and present results of the first coupled-channel analysis of the η ( ) π COMPASS data. We establish that a single exotic π 1 is needed and provide a detailed analysis of its properties. We also determine the resonance parameters of the nonexotic a 2 (1320) and a 2 (1700).
Description of the data.-We use the mass independent analysis by COMPASS of πp → η ( ) πp, with a 190 GeV pion beam [33]. We focus on the P -and D-wave intensities and their relative phase, in both channels. The published data are integrated over the range of transferred momentum squared −t 1 ∈ [0.1, 1] GeV 2 . However, given the diffractive nature of the reaction, most of the events are produced in the forward direction, near the lower limit in −t 1 . The η ( ) π partial-wave intensities and phase differences are given in 40 MeV mass bins, from threshold up to 3 GeV. Intensities are normalized to the number of observed events corrected by the detector acceptance. The errors quoted are statistical only; systematic uncertainties or correlations were found to be negligible [39]. We thus assume that all data points are independent and normally distributed. As seen in Figs. 4(a) and 5(a) of [33], at the η π mass of 2.04 GeV there is a sharp falloff in the P -wave intensity, and a sudden change by 50 • in the phase difference between the P -and D-wave. The state π 1 (2015) claimed by E852 [40,41] would be too broad to explain such an abrupt behavior and it is difficult to find a reasonable physical explanation. Unfortunately it is not possible to crosscheck this behavior with the ηπ relative phase due to lack of data in the 1.8 − 2.0 GeV region. Moreover, fitting these features of the P -wave drives the position of the a 2 to unphysical values. For these reasons, we fit data up to 2 GeV only.
Enforcing unitarity allows us to properly implement the interference among the various resonances and the background. In principle, one wishes to include all the kinematically allowed channels in a unitary analysis. Recently, COMPASS published the complete 3π partialwave analysis [32], including the exotic 1 −+ wave in the ρπ final state. However, the extraction of the resonance pole in this channel is hindered by the irreducible Deck process [42], which generates a peaking background in the exotic partial wave [43]. Since the Deck mechanism is not fully accounted in the COMPASS amplitude model, we do not include the 3π data in our analysis. As discussed in [38], neglecting additional channels does not affect the pole position, as long as the resonance poles are far away from threshold, which is the case studied here.
The diffractive character of the process entails factor-ization of the πP → η ( ) π reaction, which for fixed t 1 resembles an ordinary helicity amplitude a J iM (s), with i = η ( ) π the final channel index, J the angular momentum, s the invariant mass squared of the η ( ) π system, and M = 0 the Pomeron helicity in the Gottfried-Jackson frame (for more details, see [38]). In what follows, we restrict ourselves to M = ±1, which dominate the reaction. The two components are related by parity and thus we drop the M index. The Pomeron virtuality t 1 is fixed to an effective value t eff = −0.1 GeV 2 , as discussed above.
We parameterize the amplitudes following the coupledchannel N/D formalism [44], where s) the π beam momentum in the η ( ) π rest frame, with λ(a, b, c) being the Källén triangular function. 1 The n J k (s)'s incorporate exchange "forces" in the production process and are smooth functions of s in the physical region. The D J (s) matrix represents the η ( ) π → η ( ) π final state interactions, and contains cuts on the real axis above thresholds (right hand cuts), which are constrained by unitarity.
For the numerator n J k (s), we use an effective expansion in Chebyshev polynomials in the variable ω(s) = s/(s + s 0 ), which for s 0 at the hadronic scale 1 GeV 2 reflects the short range nature of η ( ) π production. A customary parameterization of the denominator is given by [45] where s k is the threshold in channel k and is an effective description of the left hand singularities in the η ( ) π → η ( ) π scattering, which is controlled by the s L parameter fixed at the hadronic scale 1 GeV 2 . Finally, with c J ki = c J ik and d J ki = d J ik , is a standard parameterization for the K-matrix. In our reference model, we consider two K-matrix poles in the D-wave, and one single K-matrix pole in the P -wave; the numerator of each channel and wave is described by a third-order polynomial. We set α = 2 in Eq. (3), which has been effective in describing the single-channel case [38]. The remaining 37 parameters are fitted to data, by performing a χ 2 minimization with MINUIT [46], taking into account the periodicity of the phases. The initialization of the fit is chosen by randomly generating 5 × 10 5 different sets of values for the parameters. The best fit has χ 2 /dof = 162/122 = 1.3, in good agreement with data. In particular, a single K-matrix pole is able to correctly describe the P -wave peaks in the two channels, which are separated by 200 MeV. The shift of the peak in the ηπ spectrum to lower energies originates from the combination between final state interactions and the production process. The uncertainties on the parameters are estimated via the bootstrap method [47,48]: we generate 5 × 10 4 pseudodata sets and refit each one of them. The (co)variance of the parameters provides an estimate of their statistical uncertainties and correlations. The results are shown in Fig. 1, while the values of the fitted parameters and their covariance matrix are provided in the Supplemental Material [49]. The average curve passes the Gaussian test in [50].
Once the parameters are determined, the amplitudes can be analytically continued to complex values of s. The D J (s) matrix in Eq. (2) can be continued underneath the unitarity cut into the closest unphysical Riemann sheet. A pole s P in the amplitude appears when the determinant of D J (s P ) vanishes. Poles close to the real axis influence the physical region and can be identified as res-onances, whereas further singularities are likely to be artifacts of the specific model with no direct physical interpretation. For any practical parameterization, especially in a coupled-channel problem, it is not possible to specify a priori the number of poles. Appearance of spurious poles far from the physical region is thus unavoidable. It is however possible to isolate the physical poles by testing their stability against different parameterizations and data resampling. We select the resonance poles in the m ∈ [1, 2] GeV and Γ ∈ [0, 1] GeV region, where customarily m = Re √ s P and Γ = −2 Im √ s P . We find two poles in the D-wave, identified as the a 2 (1320) and a 2 (1700), and a single pole in the P -wave, which we call π 1 . The pole positions are shown in Fig. 2, and the resonance parameters in Table I. To estimate the statistical significance of the π 1 pole, we perform fits using a pure background model for the P -wave, i.e. setting g P,1 η ( ) π = 0 in Eq. (4). The best solution having no poles in our reference region has a χ 2 almost 50 times larger, which rejects the possibility for the P -wave peaks to be generated by nonresonant production. We also considered solutions having two isolated P -wave poles in the reference region, which would correspond to the scenario discussed in the PDG [51]. The χ 2 for this case is equivalent to the single pole solution. One of the poles is compatible with the previous determination, while the second is unstable, i.e. it can appear in a large region of the s-plane depending on the initial values of the fit parameters. Moreover, the behavior of the ηπ phase required by the fit is rather peculiar. A 180 • jump (due to a zero in the amplitude) Mass (GeV) Systematic uncertainties.-To assess the systematic uncertainties of our results, we vary the parameters and functional forms which were kept fixed in the previous fits. We can separate these in two categories: i) variations which affect the smooth expansion of the numerator n J k (s) in Eq. (1), and ii) variations which drive the behavior of the imaginary part of the denominator in Eq. (2). As for the latter, we study if the specific form we chose for the function ρN (s ) biases the determination of the poles. We vary the value of s L in the reference model within [0.8, 1.8] GeV 2 , and change the value of α. Both variations alter the shape of the dispersive integral in Eq. (2), but the fit quality is unaffected. The pole positions change roughly within 2σ, as one can see in Fig. 2. The variation of α provides the largest source of systematics.
We also test a different functional form inspired by the exchange of a particle in the cross channel, where Q J (z s ) is the second kind Legendre function, and z s = 1 + 2s s L /λ(s , m 2 η ( ) , m 2 π ), with s L = 1 GeV 2 . Asymptotically it behaves as s −α , has a left hand cut starting at s = 0, a short cut between (s − m η ( ) ) 2 and (s + m η ( ) ) 2 , and an incomplete circular cut. Despite differences in the analytic structure, it gives similar results to our reference model. We also varied α ∈ [1,2], obtaining results similar to the model in Eq. (3) with the same value of α. As for the numerator n J k (s), we varied the effective value of the Pomeron virtuality to t eff = −0.5 GeV 2 and increased the order of the polynomial expansion by one unit. None of these cause important changes in pole locations. Our final estimate for the uncertainties is reported in Table I,  while the detailed summary is given in the Supplemental Material [49].
Conclusions.-We performed the first coupledchannel analysis of the P -and D-waves in the η ( ) π system measured at COMPASS [33]. We used an amplitude parameterization constrained by unitarity and analyticity. We find two poles in the D-wave, which we identify as the a 2 (1320) and the a 2 (1700), with resonance parameters consistent with the single-channel analysis [38]. In P -wave, we find a single exotic π 1 in the region constrained by data. This determination is compatible with the existence of a single isovector hybrid meson with quantum numbers J P C = 1 −+ , as suggested by lattice QCD [13][14][15]. Its mass and width are determined to be 1564 ± 24 ± 86 MeV and 492 ± 54 ± 102 MeV, respectively. The statistical uncertainties are estimated via the bootstrap technique, while the systematics due to model dependence are assessed by varying parameters and functional forms that are not directly constrained by unitarity. We find no evidence for a second pole that could be identified with another π 1 resonance. Solutions with two poles are possible, but do not improve the fit quality and, when present, the position of the second pole is unstable against different starting values of the fit. It is worth not-ing that the two-pole solutions have a peculiar behavior of the ηπ phase in the 2 GeV mass region, where no data exist. New data from GlueX and CLAS12 experiments at Jefferson Lab in this and higher mass region will be valuable to test this behavior.
Acknowledgments.-We would like to thank the COM-PASS collaboration for useful comments. AP thanks the Mainz Institute for Theoretical Physics (MITP) for its kind hospitality while this work was being completed. This work was supported by the U.  IV: Summary of systematic studies. For each systematic variation, 5 × 10 4 bootstrapped pseudodatasets are produced, and the average is shown here. For each parameter varied, we consider the maximum deviation of the pole position from the one in the reference fit. If that is compatible with the statistical uncertainty, we neglect the effect. If larger, we assign a systematic uncertainty to it, and eventually add in quadrature all the systematic uncertainties.