Dispersive calculation of complex Regge trajectories for the lightest f2 resonances and the K⁎(

Abstract

We apply a recently developed dispersive formalism to calculate the Regge trajectories of the f2(1270) and f ′ 2 (1525) mesons. Trajectories are calculated, not fitted to a family of resonances. Assuming that these spin-2 resonances can be treated in the elastic approximation the only input are the pole position and residue of the resonances. In both cases, the predicted Regge trajectories are almost real and linear, with slopes in agreement with the universal value of order 1 GeV −2 . I. INTRODUCTIONThere is growing evidence for the existence of non-ordinary hadrons that do not follow the quark model, i.e. the quark-antiquark-meson or three-quark-baryon classification. Meson Regge trajectories relate resonance spins J to the square of their masses and for ordinary mesons they are approximately linear. The functional form of a Regge trajectory depends on the underlying dynamics and, for example, the linear trajectory for mesons is consistent with the quark model as it can be explained in terms of a rotating relativistic flux tube that connects the quark with the antiquark. Regge trajectories associated with non-ordinary mesons do not, however, have to be linear. The nonordinary nature of the lightest scalar meson, the f 0 (500) also referred to as the σ, together with a few other scalars, has been postulated long ago [3]. In the context of the Regge classification, in a recent study of the meson spectrum in [1] it was concluded that the σ meson does not belong to the same set of trajectories that many ordinary mesons do. In [2], it was concluded that the σ can be omitted from the fits to linear (J, M 2 ) trajectories because of its large width. The reason is that its width was taken as measure of the uncertainty on its mass and it was found that, when fitting trajectory parameters, its contribution to the overall χ 2 was insignificant.In a recent work [4] we developed a formalism based on dispersion relations that, instead of fitting a specific, e.g. linear, form to spins and masses of various resonances, enables us to calculate the trajectory using as input the position and the residue of a complex resonance pole in a scattering amplitude. When the method was applied to the ρ(770) resonance, which appears as a pole in the elastic P -wave ππ scattering, the resulting trajectory was found to be, to a good approximation, linear. The resulting slope and intercept are in a good agreement with phenomenological Regge fits. The slope, which is slightly less than 1 GeV −2 , is expected to be universal for all ordinary trajectories. It is worth noting that in this approach the resonance width is, as it should be, related to the imaginary part of the trajectory and not a source of an uncertainty. The σ meson also appears as a pole in the ππ S-wave scattering. The position and residue of the pole has recently been accurately determined in [5] using rigorous dispersive formalisms. When the same method was applied to the σ meson, however, we found quite a different trajectory. It has a significantly larger imaginary part and ...

The non-ordinary Regge behavior of the $$K^*_0(800)$$ K 0 ∗ ( 800 ) or $$\kappa $$ κ -meson versus the ordinary $$K^*_0(1430)$$ K 0 ∗ ( 1430 )

“…Let us remark that in [2] it was shown that considering three subtractions lead to almost indistinguishable results. Therefore linear trajectories are not imposed a priori and, actually, a non-linear behavior was found for the f 0 (500) resonance [1].…”

“…Following [1,2], let us briefly recall the notation and the derivation of the dispersion relations that determine the Regge trajectory and residue of an elastic resonance just from its pole parameters. The partial wave expansion of the kaonpion scattering amplitude T (s, t) is…”

“…(5) it then follows that c = α (log(α s 0 ) − 1) and that P(s) can only be a constant. Therefore, we arrive at the following equations [1,2,6,7] describing the Regge trajectory of a resonance pole when it dominates a partial wave as in Eq. (4):…”

“…Let us remark that, due to the analytic properties of amplitudes in the complex plane, in certain cases Regge trajectories can be calculated from the properties of just one resonance [1,2], instead of fitted to several resonances assuming that a straight line should describe them. This approach relies on dispersion relations and unitarity constraints for Regge trajectories and residues [3][4][5][6][7] and is more fundamental and predictive than a pure straight line fit.…”

“…This allows for a clear identification of wide resonances, instead of using, incorrectly, the width as a source of uncertainty in the fits. The method has been described and applied recently in [1,2]. On the one hand, four ordinary linear Regge trajectories were found from the ρ(770), f 2 (1270), f 2 (1525) and K * (892) resonance poles.…”