Unquenching the meson spectrum:
a model study of exited
Quark models taking into account the dynamical effects of hadronic decay often produce very different predictions for mass shifts in the hadron spectrum. The consequences for meson spectroscopy can be dramatic and completely obscure the underlying confining force. Recent unquenched lattice calculations of mesonic resonances that also include meson-meson interpolators provide a touchstone for such models, despite the present limitations in applicability. On the experimental side, the meson and its several observed radial recurrences are a fertile testing ground for both quark models and lattice computations. Here we apply a unitarised quark model that has been successful in the description of many enigmatic mesons to these vector resonances and the corresponding -wave phase shifts. This work is in progress, with encouraging preliminary results.
14.40.Cs, 13.25.-k, 12.40.Yx, 11.80.Gw
The static quark model, which describes hadrons as pure bound states of
confined quarks and antiquarks, has remained largely unchallenged for about
40 years. Even nowadays most experimentalists still confront the
enhancements in their meson data with the relativised quark model of
Godfrey and Isgur (GI)  in order to arrive at an assignment
or otherwise claim to have found evidence of some exotic state. Now, the
GI model is indeed the most comprehensive calculation of practically all
possible quark-antiquark masses, employing the usual Coulomb-plus-linear
(“funnel”) confining potential. However, the insistence on comparing both
narrow and broad structures in cross sections directly with the infinitely
sharp levels of a manifestly discrete confinement spectrum is clearly a
poor-man’s approach. Yet, models going beyond the static quark model
have been around for almost the same four decades, the pioneering ones
being the Cornell model for charmonium , the Helsinki
model for light pseudoscalars and vectors [3, 4], and
the Nijmegen model for heavy quarkonia  and all
pseudoscalar & vector mesons . Despite the at times huge
mass shifts predicted by these models, for many years the effects of
decay, also called coupled-channel contributions or unitarisation, were
largely ignored. Instead, inspired by perturbative QCD, hadron
spectroscopists made their models more and more sophisticated at the level
of the confining potential, with e.g. spin-orbit splittings and also
relativistic corrections , which are nevertheless quite
insignificant as compared to many of the large mass shifts from unitarisation.
Only since the observation of a growing number of enigmatic mesons, whose masses or observed decays do not seem to fit in the GI and similar static quark models, more authors started to take into account dynamical effects from strong decay and scattering. Parallelly, very recent unquenched lattice computations have shown remarkably large mass shifts due to the inclusion of two-meson interpolators besides the usual quark-antiquark ones, thus confirming the importance of decay for meson spectroscopy.
An appropriate class of mesons to study these issues is and its several radial excitations, together with the corresponding -wave phase shifts, because of the considerable amount of available data, despite being mostly old . Here we shall present preliminary results in the context of the Resonance-Spectrum Expansion (RSE), which is a momentum-space variant of the unitarised model employed in Ref. .
In Sec. 2 meson mass shifts in different quark models that include hadronic decay are compared, also with a recent lattice calculation. Section 3 is devoted to a very brief description of the RSE model as applied to the isovector vector mesons, with some preliminary yet encouraging results. A few conclusions are drawn in Sec. 4.
2 Mass shifts from “unquenching” in models and on the lattice
Quark models that dynamically account for decay are often called “unquenched” [8, 9, 10, 11, 12]. Now, this is actually a very sloppy name, as the term “unquenched” originates in lattice calculations with dynamical instead of static quarks, via a fermion determinant. We shall nevertheless use this inaccurate name when referring to such quark models, because the various approaches are very different. For instance, Refs.  and  evaluate real or complex mass shifts from lowest-order hadronic loops, Ref.  constructs and uses a screened confining potential supposedly resulting from quark loops, while Ref.  includes meson loops to all orders in a fully unitary -matrix formalism. Also the original models of Refs. [2, 5, 6] were truly unitarised. But there are enormous differences as well in the computed mass shifts from unquenching, even among in principle similar models. In Table 1
|[3, 4]||one-loop BT||light||530–780, 320–500|
|[5, 6]||-matrix, -space||, , , , ;||30–350|
|||-matrix, -space||light, intermediate||510–830,|
|||QM, RGM||,||328, 94|
|||RSE, -space||,||260, 410|
|||CC, Lagrangian||,||173, 51|
|||CC, HO WF||charmonium||165–228|
|||RSE, -space||, ;||4–13, 5–93|
we show the corresponding predictions of a number of unquenched quark models
for mesons. Note that the mass shifts in Refs. [5, 6, 13, 15, 19, 20] are
in general complex, in some cases [13, 15, 20]
with huge imaginary parts, corresponding to pole positions in an exactly solved
-matrix. As for the disparate shifts among the various approaches, they are
due to differences in the assumed decay mechanism, included channels, and
possibly drastic approximations. Another crucial point should be to properly
account for the nodal structure of the bare wave functions.
Faced with these discrepancies, one is led to look at unitarised lattice results, preferably on and its radial recurrences, which we will study here with the RSE formalism. Unfortunately, no such calculations have been published so far. Nevertheless, a recent paper  on the related vector meson and the associated -wave phase shifts provides very useful information. Not only were the mass and extrapolated width reasonably well reproduced, but also a prediction, albeit approximate, was made for the first radial excitation , finding a mass of GeV. Now, the main surprise about the latter number is not so much its relative closeness to the experimental value and the 250-MeV gap with e.g. the “quenched” GI  prediction. Rather, being about 300 MeV lower than the value found by the same lattice group in another unquenched calculation  yet with no two-meson interpolators included, it showcases the potentially dramatic effects of unitarisation on meson spectra. This is an excellent incentive to study the spectrum and -wave phases in detail.
3 RSE modelling of recurrences and -wave scattering
The experimental status of radial excitations was reviewed minutely in Ref. . Suffice it here to stress the clearly biased handling of a frequently reported resonance by the Particle Data Group (PDG), by lumping some of its observations under , instead of creating a separate entry in the meson listings. The PDG also bluntly omits a reference to a relatively recent phase-shift anaysis  that concludes to be the most important excitation in order to fit the data. Moreover, there is the well-established excited resonance (see PDG  summary table), now also confirmed on the lattice . This lends further evidence to the existence of , as predicted long ago in the model of Ref. .
The general expressions for the RSE off-energy-shell -matrix and
corresponding on-shell -matrix have been given in several
papers (see e.g. Ref. ). In the present case of -wave
scattering, the quantum numbers of the system are
, which couples to the quark-antiquark state
in the spectroscopic channels and .
In the meson-meson sector, we only consider channels allowed by total angular
momentum , isospin , parity , and when possible G-parity . The
included combinations from the lowest-lying meson nonets  are:
PP, VP, VV, VS, AP, and AV, where P stands for , V for
, S for , and A for or .
This choice of meson-meson channels is motivated by the
observed two- and multi-particle decays of the recurrences up to
, which include several intermediate states
containing resonances from the referred nonets. For instance, the PDG lists
 under the decays of the modes
, , , , , and
is probably dominated by the
 scalar resonance. By the same token, the
decays of will most likely include important contributions
from modes as , , etc.. For consistency of
our calculation, we generally include complete nonets in the allowed decays,
and not just individual modes observed in experiment. The only exception is the
important PP mode, because no complete nonet of
radially excited pseudoscalar mesons has been observed so far .
The resulting 26 channels are given in Table 2.
|VP||, , ,||1|
|AP||, , , , ,||0|
|AV||, , , ,||0|
With the few available parameters , a good fit to the -wave phase shifts is only possible up to about 1.2 GeV, whereabove the phases rise a bit too fast, though their qualitative behaviour can be reproduced. Improvements may require more flexibility in the transition potential, by allowing different decay radii for the various classes of two-meson channels, and/or allowing for complex-mass resonances in the final states . The present fit yields a reasonable pole, viz. at MeV, while there are two poles in the range 1.2–1.5 GeV, compatible with both and .
Meson spectroscopists are slowly starting to leave the stone-age behind, by realising that effects from strong decay can be of the same order as the bare level splittings themselves. Enormous obstacles lie on the road ahead, demanding more theoretical work, improved lattice calculations, and much better experimental analyses. The excited spectrum provides an excellent laboratory for such efforts. To make life even harder, several bumps in meson production processes  may just be non-resonant threshold enhancements (see talk by E. van Beveren ).
- thanks: Talk by G. Rupp at Workshop “Excited QCD 2016”, Costa da Caparica, Portugal, March 6–12, 2016.
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