A chiral covariant approach to \(\varvec{\rho \rho }\) scattering
Eur. Phys. J. C
A chiral covariant approach to ρρ scattering
D. Gülmez 2
U.-G. Meißner 1 2
J. A. Oller 0
0 Departamento de Física, Universidad de Murcia , 30071 Murcia , Spain
1 Institute for Advanced Simulation, Institut für Kernphysik and Jülich Center for Hadron Physics, Forschungszentrum Jülich , 52425 Jülich , Germany
2 Helmholtz-Institut für Strahlenund Kernphysik and Bethe Center for Theoretical Physics, Universität Bonn , 53115 Bonn , Germany
We analyze vector meson-vector meson scattering in a unitarized chiral theory based on a chiral covariant framework restricted to ρρ intermediate states. We show that a pole assigned to the scalar meson f0(1370) can be dynamically generated from the ρρ interaction, while this is not the case for the tensor meson f2(1270) as found in earlier work. We show that the generation of the tensor state is untenable due to the extreme non-relativistic kinematics used before. We further consider the effects arising from the coupling of channels with different orbital angular momenta which are also important. We suggest to use the formalism outlined here to obtain more reliable results for the dynamical generation of resonances in the vector-vector interaction.
1 Introduction
It is now commonly accepted that some hadron resonances
are generated by strong non-perturbative hadron–hadron
interactions. Arguably the most famous example is the
(1405), which arises from the coupled-channel dynamics
of the strangeness S = −1 ground -state octet meson–baryon
channels in the vicinity of the π and K − p thresholds [1].
This resonance also has the outstanding feature of being
actually the combination of two near poles, the so-called two-pole
nature of the (1405). In a field-theoretic sense, one should
consider this state as two particles. This fact was predicted
theoretically [2,3] and later unveiled experimentally [4] (see
also the discussion in Ref. [5]). Another example is the scalar
meson f0(980) close to the K¯ K threshold, which is often
considered to arise due to the strong S-wave interactions in
the π π –K¯ K system with isospin zero [6–8]. A new twist was
given to this field in Ref. [9] where the S-wave vector–vector
(ρρ) interactions were investigated and it was found that due
to the strong binding in certain channels, the f2(1270) and
the f0(1370) mesons could be explained as ρρ bound states.
This approach offered also an explanation why the tensor
state f2 is lighter than the scalar one f0, as the leading order
attraction in the corresponding ρρ channel is stronger. This
work was followed up by extensions to SU(
3
) [10], to account
for radiative decays [11,12], and much other work; see e.g.
the short review in Ref. [13].
These findings concerning the f2(1270) are certainly
surprising and at odds with well-known features of the strong
interactions. It is a text-book result that the f2(1270) fits
very well within a nearly ideally mixed P-wave qq¯ nonet
comprising as well the a2(1320), f2(1525) and Ks∗(1430)
resonances [14–17]. Values for this mixing angle can be
obtained from either the linear or quadratic mass relations as
in Ref. [17]. Non-relativistic quark model calculations [18],
as well as with relativistic corrections [19], predict that the
coupling of the tensor mesons to γ γ should be
predominantly through helicity two by an E 1 transition. This
simple qq¯ picture for the tensor f2(1270) resonance has been
recently validated by the analyses performed in Ref. [20] of
the high-statistics Belle data [21–25] on γ γ → π π in both
the neutral and the charged pion channels. Another point
of importance in support of the qq¯ nature of the f2(1270)
is Regge theory, since this resonance lies in a parallel
linear exchange-degenerate Regge trajectory with a
“universal” slope parameter of around 1 GeV [26–30]. Masses and
widths of the first resonances with increasing spin lying on
this Regge trajectory (ρ, f2, ρ3, f4) are nicely predicted [31]
by the dual-hadronic model of Lovelace–Shapiro–Veneziano
[32–34].
One should stress that the results of Ref. [9] were obtained
based on extreme non-relativistic kinematics, pi2/m2ρ 0,
with p the rho-meson three-momentum and mρ the
vectormeson mass. This approximation, however, leads to some
severe simplifications:
• Due to the assumed threshold kinematics, the full ρ
propagator was reduced to its scalar form, thus enabling the
use of techniques already familiar from the pion–pion
interaction [8]. This was applied when considering the
iteration of the interactions in the Bethe–Salpeter
equation.
• Based on the same argument, the algebra involving the
spin and the isospin projectors of the two vector-meson
states could considerably be simplified.
However, as √sth = 2mρ = 1540 MeV, the lighter of the
bound states obtained in Ref. [9] is already quite far away
from the 2ρ threshold, with a nominal three-momentum of
modulus larger than a 60% of the ρ mass. It is therefore
legitimate to question the assumptions there involved. I (...truncated)