Chiral Extrapolations and Exotic Meson Spectrum

We examine the chiral corrections to exotic meson masses calculated in lattice QCD. In particular, we ask whether the non-linear chiral behavior at small quark masses, which has been found in other hadronic systems, could lead to large corrections to the predictions of exotic meson masses based on linear extrapolations to the chiral limit. We find that our present understanding of exotic meson decay dynamics suggests that open channels may not make a significant contribution to such non-linearities whereas the virtual, closed channels may be important.

1. Introduction. One of the biggest challenges in hadronic physics is to understand the role of gluonic degrees of freedom. Even though there is evidence from high energy experiments that gluons contribute significantly to hadron structure, for example to the momentum and spin sum rules, there is a pressing need for direct observation of gluonic excitations at low energies. In particular, it is expected that glue should manifest itself in the meson spectrum and reflect on the nature of confinement.
In recent years a few candidates for glueballs and hybrid mesons have been reported. For example, there are two major glueball candidates, the f 0 (1500) and f 0 (1710), scalar-isoscalar mesons observed inpp annihilation, in central production as well as in J/ψ (f 0 (1710)) decays [1]. However, as a result of mixing with regular qq states, none of these is expected to be a purely gluonic state. This is a general problem with glueballs -they have regular quantum numbers and therefore cannot be unambiguously identified as purely gluonic excitations. On the other hand, the existence of mesons with exotic quantum numbers (combinations of spin, parity and charge conjugation, J P C , which cannot be attributed to valence quarks alone), would be an explicit proof that gluonic degrees of freedom can indeed play an active role at low energies.
Current estimates based on lattice QCD [2,3], as well as QCD-based models [4], suggest that the lightest exotic hybrid should have J P C = 1 −+ and mass slightly below 2 GeV. On the experimental side, two exotic candidates with these quantum numbers have recently been reported by the E852 BNL collaboration. One, in the ηπ − channel of the reaction π − (18GeV)p → Xp → ηπ − p, occurs at M X ≈ 1400MeV [5], while the other, in the ρ 0 π − and η ′ π − channels, has a mass M X ≈ 1600MeV [6,7].
Although the lighter state could be obscured by final state interaction effects as well as leakage from the strong X = a 2 decay, the heavier one seems to have a fairly clean signal.
In the large N c limit it has been shown that exotic meson widths obey the same N c scaling laws as the regular mesons [8]. The lack of an overwhelming number of exotic resonance candidates may therefore be the result of a small production crosssection in the typical reactions employed in exotic searchese.g., high energy πN or KN scattering. Indeed, low lying exotic mesons are expected to have valence quarks in a spin-1 configuration and their production would therefore be suppressed in peripheral pseudoscalar meson-nucleon scattering. By the same arguments one would expect exotic production to be enhanced in real photon-nucleon scattering [9,10,11]. This is particularly encouraging for the experimental studies of exotic meson photoproduction which have recently been proposed in connection with the JLab 12 GeV energy upgrade [12].
In view of the theoretical importance of the topic and the exciting new experimental possibilities for exotic meson searches it is of prime importance to constrain the theoretical predictions for exotic meson masses -especially from lattice QCD.
One of the important practical questions in this regard is the effect of light quarks on the predicted spectrum. All calculations that have been performed so far involve quarks that are much heavier than the physical u and d quarks and, with the exception of the initial study by the SESAM collaboration [3] (based on full QCD with two degenerate Wilson fermions), are performed in the quenched approximation. This is directly related to the present limitations on computer power. It is well known that, as a consequence of dynamical chiral symmetry breaking, hadron properties are nonanalytic functions of the light quark masses. This non-analyticity can lead to rapid, non-linear variations of masses as the light quark masses approach zero. Whether or not this could make a sizable difference to the lattice QCD predictions for exotic state masses is the question we address.
There has been considerable activity concerning the appropriate chiral extrapolation of lattice results for hadron properties in the past few years, ranging from magnetic moments [13] to charge radii [14] and structure functions [15] as well as masses [16]. The overall conclusion from this work is that for current quark masses in the region above 50-60 MeV, where most lattice results are available, hadron properties are smoothly varying functions of the quark masses -much like a constituent quark model. However, as one goes below this range, so that the corresponding pion Compton wavelength is larger than the source, one finds rapid, non-linear variations which are a direct result of dynamical chiral symmetry breaking. These variations can change the mass of hadrons extracted by naive linear extrapolation by a hundred MeV or more. (Certainly this was the case for the N and ∆ [16], while for the ρ the difference was only a few MeV [17,18].) A variation of this order of magnitude for the predictions of exotic meson masses would clearly be phenomenologically important and our purpose is to check whether this seems likely.
2. Chiral corrections from pion loops. Our investigation of the possible nonlinearity of the extrapolation of exotic meson masses to the chiral limit, as a function of quark mass, will follow closely the earlier investigations for the N and ∆ baryons and the ρ meson. The essential point is that rapid non-linear variations can only arise from coupling to those pion-hadron channels which give rise to the leading (LNA) or next-to-leading non-analytic (NLNA) behavior of the exotic particle's self-energy. In a decay such as E → πH, this means that H must be degenerate or nearly degenerate with the exotic meson E. In addition, the relevant coupling constant should be reasonably large. For example, in the case of the ρ meson the relevant channels are ωπ (LNA) and ππ (NLNA) [17] and the extrapolation of the ρ meson mass was carried out using: where σ ij are the self-energy contributions from the channels ij = ωπ or ππ.
Our first step in studying the extrapolation of exotic meson masses is therefore to look at the channels to which the exotics can couple which involve a pion and to check what is known about the corresponding coupling constants. The matrix element describing a decay of the J P C = 1 −+ , isovector, exotic mesons with mass m X , is given by: where k(m X ) = |k| is the break-up momentum of the AB meson pair, from a decay of a state of mass m X , produced with angular momentum L and spin S, with L + S = 1. In terms of the couplings, g LS , the partial decay widths are given by

Decay channel wave
These couplings have been calculated in Refs. [19,20], in two models based on the flux tube picture of gluonic excitations but assuming different qq production mechanisms.
In general the models agree on predicting sizable couplings to the so called "S+P" channels, where one of the two mesons in the final state has quarks in relative Swave and the other in relative P-wave, e.g., πb 1 . In Table 1 we summarize the results of these calculations, listing only the decay channels containing a pion (S-wave quarks). Since the overall normalization of the decay matrix elements in these models is somewhat arbitrary, we have rescaled the original predictions given in Ref. [20] to match the total width ( 170 MeV) and mass (m X = 1.6 GeV) of the exotic ρπ state found by the E852 experiment [6].  tically different when it comes to predicting ratios of branching ratios. In general, PSS predicts that higher partial waves should be strongly suppressed.
An alternative approach was presented in Ref. [11]. There, it was assumed that the decay of the observed exotic is dominated by just two channels : ρπ -the one in which it has been seen -and the rest, say b 1 π (in an S-wave). The couplings obtained this way are shown in Table II.
The shift in the mass of the exotic meson associated with the pion loop, Σ, can be calculated in second order perturbation theory (from Σ = P ′ |V GV |P ′ / P ′ |P ) and is given by: with k(M) = λ(M, m A , m B ). The leading non-analytic (LNA) behavior of this selfenergy is obtained by extracting the piece, independent of the ultra-violet cut-off (or form factor, g L (k)/g L (k(m X ))), which exhibits the strongest non-analytic variation as a function of the quark mass as one approaches the chiral limit. (This is the term with the lowest odd power of m π or the lowest power of m π multiplying ln m π .) Setting m B = m π the form of the LNA behavior depends on the relation between m X − m A and m π and can be easily extracted analytically from Eq. (5) in two limiting cases.
Consider first the limit m X − m A << m π , corresponding to an off-shell transition between the exotic meson and a nearly degenerate meson plus pion. In this case the self energy contribution reduces to and the LNA behavior is given by and k π ≈ 0. In the case m X − m A >> m π , corresponding to a physical decay process to two light mesons (one of them being the pion), one obtains, leading to Even though, in the present case, neither of these is the exact, four-dimensional, pion loop contribution, they should give a reliable guide as to the non-linearity that one may expect in the extrapolation of lattice results to small quark mass.
In order to estimate the full self energy contribution, we need to know the off-shell dependence of the coupling constant. In the models of Ref. [11,19,20] the momentum dependence of the couplings arises from the Fourier transform of quark wave function overlaps and is typically of a gaussian form, with the scale parameter expected to be in the range β ≈ 0.2 − 1 GeV. To be consistent with the approximation of a heavy source, corresponding to Eqs. (6) and (8), we need to set k 2 (m X ) ≈ 0 or k 2 (m X ) ≈ (m X − m A ) 2 respectively. Thus the momentum dependence of the form factor near k 2 = k 2 π is not expected to significantly renormalize the value at k 2 = k 2 (m X ) and in Eqs. (6) and (8)  possibility in mind, using Eq. (7) we find that for the maximal strength P -wave decays with couplings of the order of 20, with both Σ and m π in GeV, and all couplings of order 1 are irrelevant. For the D-waves with coupling as large as 300 (IKP model, D-wave in the b 1 π channel) we obtain, from Eq.( 7) or Σ ≈ +0.4m 6 π ln m π from Eq. (9). In the case of the b 1 π or f 1 π decay channels the second approximation is probably more accurate.
To be more precise we also check how the full formula for the energy shift compares with the simpler expression for a heavy source given above. In Fig. 1 the dashed line shows Σ as a function of the pion mass, calculated including the three largest P wave open channels from Table 1 (ρ(1450)π, η u (1295)π and ρπ) using the PSS (larger) couplings. The magnitude of Σ is determined by the scale parameter chosen as β = 400 MeV in Fig. 1(a) and β = 700 MeV in Fig 1(b). For small β the dominant contribution comes from the lightest open channel. This is because there is a large mismatch between k(m X ) and k ∼ β which leads to enhancement from and lower bounds, decreasing for higher partial waves, is consistent with Eqs. (11) and (12).
These are small shifts as compared, for example, to the case of the pion contribution to the nucleon mass, where the LNA term is of order −5.6m 3 π . To be more specific, the πNN vertex, written in the notation of Eq. (2), becomes Eq. (7) then gives, However, just as in the case of the nucleon, one expects the largest LNA behavior to come from virtual transitions to nearly degenerate states; in this case from transitions between an isovector exotic and a pion plus an isoscalar exotic meson. The couplings in Tables 1 and 2 only account for real decays and therefore cannot be used to estimate such transitions. To the best of our knowledge there are no microscopic calculations of such matrix elements, however, these can in principle be derived from PCAC. In the soft pion limit one has, If one assumes the flux tube does not affect the axial charge, then in the static limit the valence quark contribution to the lhs is given by 2m X δ ab ǫ * (λ ′ ) · [ ǫ(λ) × q]. After comparing with Eq. (2) this yields The lack of any contribution from the flux-tube makes this coupling identical to the one for ordinary mesons, e.g. the ρωπ coupling (with m X = m ρ ∼ m ω ). The phenomenological value for the g ρωπ , in our notation, is g ρωπ /m ρ = 8π/3g ρωπ with g ρωπ ∼ 15 GeV −1 , so that the simple quark model overestimates the coupling by approximately 50%. The flux-tube may contribute to the axial current if it couples to the spin of the quarks. In particular, since it is expected that the ground state corresponds to a flux-tube in a P -wave with respect to the valence QQ pair, the overlap of the spin and orbital wave functions may modify the numerical factor in Eq. (17). However, it is not expected to alter the 8π/3m X /f π enhancement arising from the soft pion emission. Thus, from Eqs. (7) and (17) we estimate The magnitude of this correction is almost as large as that found in the nucleon case. The contribution to Σ from the virtual transitions calculated using the expression in Eq. (6), is shown in Fig. 1 with the dotted line. Its magnitude is strongly dependent on the cutoff parameter β, and can be as large as O(100MeV) for β = 700 MeV. It also has significant variation as a function of m π , as expected from Eq. (18). The total self energy shift arising from the real and virtual corrections is shown in Fig. 1 with the solid line. This simple analysis of virtual corrections also applies to the ρ(1600)π channel which, depending on the relative phase, could enhance the overall LNA behavior by a factor of two. Finally if the other light exotics J P C = 0 +− , 2 +− are not too far from the J P C = 1 −+ , they could significantly enhance the d-wave (∝ m 5 π ) behavior (cf. Fig.2). 3. Conclusions. We have explored the self-energy corrections to the mass of the