aa r X i v : . [ h e p - ph ] N ov RADIATIVE DECAYS OF HADRONICMOLECULES
A.V. Nefediev
Institute of Theoretical and Experimental Physics,117218, B.Cheremushkinskaya 25, Moscow, Russia
Abstract
It is argued that radiative decays of scalars a /f (980) can serve as adecisive tool in establishing the nature of the latter. In particular, pre-dictions for the widths of the radiative decays S → γV ( S = a /f (980), V = ω/ρ/γ ) are given in the framework of the molecule model of thescalars. Finite–range corrections are discussed in detail for the two-gammadecays of hadronic molecules, with a special attention payed to the inter-play of various scales involved in the problem and to the gauge invarianceof the amplitude. The results are applied to the two-photon decay of the f (980), and the existing experimental data on this decay are argued tosupport the molecule assignment for the scalar f (980). The problem of the structure of light scalar mesons is of a fundamental im-portance for understanding the properties of the entire scalar sector, that is,the sector of states with the quantum numbers of the vacuum, including purelygluonic excitations. In particular, the identification of the a (980) and f (980)mesons, together with the experimental studies of the lightest scalars ( σ and κ ), will allow one to establish the structure of multiplets of scalars and to findthe signature of the scalar glueball in the spectrum of physical states. Thereare several models for the a (980) and f (980). The latter can be consideredas P quark–antiquark states [1] strongly coupled to the mesonic continuumand thus strongly distorted with the unitarisation process. However, due to theproximity of the K ¯ K threshold, it is natural to assume a considerable admix-ture of the four–quark component in the wave functions of these mesons, eitheras a compact four–quark with hidden strangeness [2, 3], or as a K ¯ K molecule.These might be t-channel exchanges to be responsible for the formation of sucha molecule [4, 5, 6, 7]. It is therefore important to establish a test which wouldallow one to distinguish between these models and thus to reveal the actualnature of these scalars (in particular, efficient methods to discriminate betweenthe molecule and compact states are strongly needed — for the recent progress1a) (b) (c)Figure 1: Diagrams contributing to the scalar decay amplitude. see [8]). Since years, radiative decays of the φ (1020), φ → γS , have been consid-ered as such an experimental tool [9]. Indeed, these decays point to a large K ¯ K component in the scalars wave function [10, 11, 12]. Still a number of shortcom-ings of this approach should be mentioned. First of all, the radiative decays ofthe φ do not allow one to probe the nonstrange component of the scalars, andthe contribution of the quark loops is strongly suppressed as compared to thecontribution of the meson loops. Finally, the phasespace available in the finalstate of these decays is limited to a large extend. In the meantime, anotherclass of radiative decays involving scalars is known — the radiative decays ofthe scalars themselves: S → γV , where the vector in the final state is eithermassive ( ρ or ω ) or massless, that is one deals with a two–photon decay in thelatter case. Whatever model of scalars is used, gauge invariance imposes strongconstraints on the decay amplitude: iW µν = M ( a, b )[ P µV P νγ − g µν ( P V P γ )] , a = m V m , b = m S m , (1)where m is the kaon mass and P V,γ are the four–momenta of the vector particles.Below we shall evaluate the widths of the radiative decays involving the scalars a /f (980) in the molecule assignment for the latter.First of all, it is important to notice that there are three scales in the problemunder consideration, which are (i) the binding force scale β ≃ m ρ ≈
800 MeV,(ii) the kaon mass m , and (iii) the binding energy ε , and the hierarchy of thesescales is ε ≪ m . β . The last inequality suggests that it is natural to startfrom the point-like limit of β → ∞ and to include finite–range corrections (inthe form a 1 /β expansion) afterwards. Thus we stick to the point-like limit. Thefirst ingredient one needs to know is the coupling of the loosely bound moleculestate to the K ¯ K pair, which reads [13]: g S π = 32 m √ mε ≈ .
12 GeV , (2)where m = 495 MeV and the molecule binding energy is taken to be ε = 10 MeV.In addition, the φK ¯ K and the V K ¯ K coupling constants can be evaluated using2able 1: The widths (in keV) of the radiative decays involving scalars; θ is the(small) φ − ω mixing angle. Quark–antiquark Molecule Data (PDG) φ → γa .
37 sin θ . ± . φ → γf (¯ nn ) /f (¯ ss ) 0 .
04 sin θ/ .
18 0.6 0 . ± . a → γγ ∼ . ± . f → γγ ∼ . +0 . − . a γω/ρ f (¯ nn ) γρ/ω f (¯ ss ) γρ/ω /
31 sin θ φ and the ρππ constant under the assumption of the SU(3)invariance. One arrives then at g φ = 4 . g V = 2 .
13. It is straightforwardthen to arrive at the predictions of the point-like model for the radiative decaysinvolving scalars. We give these predictions in tab.1. For illustrative purposesand for future references, let us quote the formula for the two-photon decaywidth of a point-like scalar (see fig.1 for the diagrams contributing to this decay):Γ( S → γγ ) = 12 (cid:16) απ (cid:17) √ mε (cid:18) mm S (cid:19) "(cid:18) mm S (cid:19) arcsin (cid:18) m S m (cid:19) − , (3)where m S = 2 m − ε .Notice that another approach to two–photon decays of molecules is knownin the literature, namely the approach based on the formula Γ( S → γγ ) = πα m | Ψ(0) | which is written in analogy with that for the positronium two–photondecay. Although this approach appears quite successful in QED, it has to failin hadronic physics. First of all, the w.f. of the kaon molecule is simply notknown, so one has to rely on models. Moreover, since Ψ(0) is very sensitiveto the details of the bound–state formation, the predictions of this approachmay vary drastically (the predictions found in the literature vary by an orderof magnitude, from 0.6 keV in [14] to 6 keV in [15]). Furthermore, any attemptto evaluate corrections to this leading term results in either gauge invarianceor energy conservation law breaking. Indeed, the decay amplitude in this ap-proach is usually given as an overlap integral between the molecule w.f. and theamplitude of the process K + K − → γγ : W ∝ Z d k (2 π ) ψ ( ~k ) h W ( K + ( ~k ) K − ( − ~k ) → γγ ) i . (4)If the amplitude W ( K + K − → γγ ) is taken off-shell, then it obviously fails to begauge–invariant. On the contrary, for the on–shell gauge–invariant amplitude3 ( K + K − → γγ ), kaons carry the energy √ k + m , rather than m S /
2, so thatenergy conservation law is violated.The last, but not the least, argument against using the Ψ(0)-based approachto hadronic processes is that the hierarchy of scales has to be different, namely ε ≪ β ≪ m , in order to validate the given formula with Ψ(0).Finally, we estimate the finite–range corrections to the point-like predictionsquoted in tab.1 — we are interested in the potentially large corrections of order m /β . If the vector in the final state is massive, then the photon in the finalstate is soft ( ω ≪ β ), and the kinematics of the loop becomes nonrelativistic[7, 19, 13]. Inclusion of the finite-range effects amounts to the substitution g S → Γ ( ~k ), with a suitable form of the vertex Γ ( ~k ). Gauge invariance requiresthen that the extra momentum dependence coming from the vertex argumentshould be gauged, that is Γ ( ~k ) → Γ ( ~k + e ~A ) ≈ Γ ( ~k ) + e ~A ∂Γ ( ~k ) ∂~k + O (cid:18) ω β (cid:19) , (5)and the derivative term gives rise to an extra contact diagram with the photonemission from the scalar vertex. Gauge invariance is therefore preserved to order O ( ω/β ). Notice however that one should be extremely careful when treating theloop integrals entering the decay amplitude. Indeed, although the full amplitudeis finite, every individual integral is divergent. If a cut-off is introduced then tomake them finite, gauge invariance may be badly broken and, as a result, wrongconclusions may be deduced (see, for example, [16] and explanations in [17]).Finally one arrives at the conclusion that no corrections of order m /β appearfor the point-like predictions [7, 19, 13]. The same conclusion holds for thetwo–photon decays of scalars, though it is not straightforward to arrive at thisconclusion and one needs to develop a selfconsistent gauge-invariant approachto this decay. It was suggested in [18] to use an effective kaon interactionLagrangian (for the neutral-particle exchange, generalisation to the charged-particle exchange being trivial) written to order 1 /β : L int = 12 λ ( ϕ † ϕ ) + λ β (cid:2) ∂ µ ( ϕ † ϕ ) (cid:3) , (6)with the coupling constants λ , being of the same order of magnitude. ThisLagrangian is subject to renormalisation to order 1 /β — see [18] for the de-tails. In the renormalised theory, the kaon propagator and the photon–emissionvertex, which are the dressed quantities, built as solutions of the correspondingfield theoretical equations, read: S ( p ) = Zp − m , v µ ( p, q ) = Z − (2 p − q ) µ + . . . , (7)4here the ellipsis denotes terms which do not contribute to the decays underconsideration. The renormalisation constants for the kaon propagator and forthe photon emission vertex coincide due to gauge invariance. The kaon mass m is the renormalised physical mass. Finally, the most important ingredient— the scalar vertex beyond the point-like limit — comes as a solution of thehomogenious Bethe–Salpeter equation [18]: Γ ( p, P ) = Z − g S (cid:18) λ λ p ( p − P ) β (cid:19) . (8)This vertex is to be normalised [20], which gives for the scalar coupling theformula g S π = 32 m √ mε (cid:18) λ λ m β (cid:19) , (9)which coincides with the point-like result (2) as β → ∞ . With the scalar vertex,the dressed kaon propagator, and the dressed photon emission vertex in handwe are in a position to evaluate the width of the scalar two-photon decay up tothe order 1 /β . The amplitude of the process is given by the set of diagramsformally coinciding with those for the point-like vertex, depicted at fig.1. Notice,however, an important difference: all ingredients are dressed now, and this isa necessary condition to preserve gauge invariance beyond the point-like limit.The only quantity which should not be dressed is the KKγγ vertex in the thirddiagram. Indeed, the scalar vertex Γ obeys the Bethe–Salpeter equation andthus absorbs all dressing diagrams.In view of the fact that we deal with an explicitly gauge–invariant amplitude,we use the trick suggested in [21] and, in order to extract the amplitude, weread-off the coefficient at the structure q ν q µ in the transition matrix element iW = M ( P )[ q ν q µ − g µν ( q q )] ǫ ∗ µ ǫ ∗ ν , P = q + q , (10)which is a particular case of (1) adapted for the two–photon case. Then M ( m S ) = M (0) ( m S ) + λ λ m β M (1) ( m S ) , (11)and, by an explicit calculation, one can find that M (1) ( m S ) = 0. Therefore, nolarge corrections of order m /β appear for the point-like result (3).We conclude therefore, that finite–range effects give only moderate correc-tions to the point-like predictions (of order 10 ÷
20% in the amplitude), pro-vided they are included in a self-consistent and gauge-invariant way [7, 13, 18].We refer to the point-like results presented in tab.1 as to the molecule modelpredictions for the radiative decays involving scalars. For the sake of com-parison, we quote in tab.1 the results of calculations in the quark–antiquark5ssignment for the scalars, which can be obtained with the help of the resultsof [22, 23]. From tab.1 one can conclude that experimental data are well de-scribed in the molecule assignment for the scalars (a recent result by Belle[24] Γ( f (980) → γγ ) = 0 . +0 . − . (stat) +0 . − . (syst) keV gives an even bettercoincidence with the point-like prediction). Furthermore, predictions for theradiative decays of scalars with massive vectors in the final state demonstrate aclear hierarchy, depending on the assignment prescribed to the scalar mesons.This makes these decays an extremely promising tool in establishing the natureof the a /f (980). This work was supported by the Federal Agency for Atomic Energy of Rus-sian Federation and by grants NSh-843.2006.2, DFG-436 RUS 113/820/0-1(R),RFFI-05-02-04012-NNIOa, and PTDC/FIS/70843/2006-Fisica.
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