An effective approach to VMD at one loop order and the departures from ideal mixing for vector mesons

Abstract

We examine the mechanisms producing departures from ideal mixing for vector mesons within the context of the Hidden Local Symmetry (HLS) model. We show that kaon loop transitions between the ideal combinations of the ω and φ mesons necessitate a field transformation in order to get the mass eigenstates. It is shown that this transformation is close to a rotation for processes involving, like meson decays, on-shell ω and φ mesons. The HLS model predicts a momentum dependent, slowly varying mixing angle between the ideal states. We examine numerically the consequences of this for radiative and leptonic decays of light mesons. The mean ω − φ mixing angle is found smaller than its ideal value; this is exhibited separately in radiative and in leptonic decays. Effects of nonet symmetry breaking in the vector sector are compared to those produced by the field rotation implied by the HLS model. * Supported by the US Department of Energy under contract DE-AC03-76SF00515The Hidden Local Symmetry (HLS) Model in both its non-anomalous [2] and anomalous (FKTUY) sectors [3] is a powerful tool for analyzing experimental data, by providing a clear framework with the fewest possible number of free parameters. For instance, it allows a 3-parameter description of the I = 1 pion form factor; this gives a statistically optimal description in an energy interval running from threshold to the φ mass. This has been shown by Ref. [4] in analyzing the world data set for e + e − → π + π − annihilation collected in Ref. [5]. The exercise has been repeated as successfully with the data set recently collected by the CMD-2 Collaboration on the VEPP-2M collider at Novosibirsk [6].However, in order to go beyond while staying within the framework defined by the HLS model and its anomalous sector, one needs to define a consistent scheme of symmetry breakings. Without SU(3) breaking, the HLS model cannot successfully describe the kaon form factors; without nonet symmetry breaking in the pseudoscalar (PS) sector, it cannot be used reliably to describe radiative decays of light mesons. The BKY mechanism [7,8] is a consistent way to introduce SU(3) breaking in both the vector (V) and PS sectors. It has been shown recently [9] that the BKY SU(3) breaking in the PS sector is in perfect agreement with all accessible predictions of Chiral Perturbation Theory [10-12] at first order in the breaking parameters. In order to reach this conclusion, the needed ingredients were only the BKY breaking in the PS sector (referred to hereafter as X A breaking), the kinetic energy term of the non-anomalous HLS Lagrangian and the P γγ Lagrangian of Wess, Zumino and Witten [13]. Thus, this part is on secure grounds.When dealing with PS mesons, the question of nonet symmetry breaking (NSB) cannot be avoided, as clear from Refs. [14,15] for instance. It was already introduced in the physics of single photon radiative decays (12 modes) by O'Donnell long ago [16], relying basically only on group theoretical considerations, but outside the context of effective Lagrangians. Th...

The pion form factor within the hidden local symmetry model

“…This can be derived by resumming formally an obvious infinite series of terms, each containing bare propagators and loops (Referred to in [4] as Dyson-Schwinger Summation). This expression can also be obtained by adding an effective piece [20] to the HLS Lagrangian of the form Π ρρ (s)ρ 2 /2, which turns out to modify the vector meson mass term by a s−dependent piece. The (dressed) ρ propagator is derived from this effective Lagrangian at tree level.…”

“…These loops should be subtracted minimally twice (P P ) or three times (V P ) from requiring the corresponding Dispersion integrals [20] to be convergent. Therefore, in the full HLS Model (non-anomalous and anomalous sectors), the subtraction polynomials must be at least second degree in s and we can write :…”

“…Loop effects cannot be avoided in problems where the ρ meson plays a crucial role. These will be considered in the framework of the one-loop order treatment proposed in [20]. Doing this way, one limits the possible couplings by neglecting intermediate states with more than two particles which generate multiparticle loops ; these are expected to produce small effects [4].…”

Chiral behavior of vector meson self-energies

“…In this work, we will not make use of such non-perturbative methods, and restrict ourselves to the one-loop level of perturbation theory to study the quark mass dependence of the vector meson self-energies. For some earlier studies of vector meson selfenergies on the one-loop level, outside the framework of ChPT, we refer to [41][42][43][44].…”

The dipion mass spectrum in annihilation and τ decay: Isospin Symmetry breaking effects from the (ρ, ω, ϕ) mixing

“…It indeed happens that the radiative decays (P V γ and P γγ), which are accounted for by the anomalous sector [21] of the HLS Lagrangian, depend on a large part of the parameters involved in our model and can serve to fix them quite reliably, even by fitting them in isolation [22,23]. If one adds to this data set the leptonic decay information for the ω and φ mesons on the one hand, and two-pion decay information of the φ meson on the other hand, the minimization program becomes numerically well defined.…”