Hadronic Molecules and the f0(980)/a0(980)
aa r X i v : . [ h e p - ph ] J un Hadronic Molecules and the f ( ) / a ( ) Tanja Branz, Thomas Gutsche and Valery E. Lyubovitskij
Institut für Theoretische Physik, Universität Tübingen,Auf der Morgenstelle 14, D-72076 Tübingen, Germany
Abstract.
We discuss a possible interpretation of the f ( ) and a ( ) mesons as hadronicmolecules - bound states of K and ¯ K mesons. Using a phenomenological Lagrangian approach wecalculate the strong f ( ) → pp and a ( ) → ph as well as the electromagnetic f ( ) → gg and a ( ) → gg decays. The covariant and gauge invariant model, which also allows for finitesize effects of the hadronic molecule, delivers results in good agreement with available data andresults of other theoretical approaches. Keywords: scalar mesons, hadronic molecule, electromagnetic and strong decays
PACS:
INTRODUCTION
The scalar f ( ) and a ( ) mesons have been analyzed in various structure inter-pretations amongst the most important are: quarkonium state, tetraquark configurationand hadronic molecule. In particular, the closeness to the K ¯ K threshold and their nearmass degeneracy, problematic in the q ¯ q picture, give evidence for a hadronic boundstate interpretation. In addition, recent calculations based on QCD sum rules and latticeQCD also support the q ¯ q configuration, where the quarks can either form a compacttetraquark or a loosely bound state of kaons [1, 2].We present a clear and straightforward model which allows for a consistent evaluationof electromagnetic and strong decay properties of the a and f considered as pure K ¯ K bound states [3]. Covariance and gauge invariance are the main features of ourtheoretical framework which additionally considers the spatially extended structure ofthe hadronic molecules with a minimal amount of assumptions. SETUP OF THE MODEL
In this section we focus on the ’supporting pillars’ of our framework. The model is basedon an interaction Lagrangian describing the coupling between f and its K ¯ K constituents L f K ¯ K ( x ) = g f K ¯ K f ( x ) Z dy F ( y ) ¯ K (cid:16) x − y (cid:17) K (cid:16) x + y (cid:17) , (1)with a similar expression for a . Here, L f K ¯ K ( x ) is expressed by the center-of-masscoordinate x and the relative coordinate y. The compositeness condition [4, 5] providesa self-consistent method to fix the coupling g f K ¯ K between the f bound state and itsconstituents. In order to describe the bound state of constituents the field renormalizationconstant Z f is set to zero f = − g f K ¯ K P ′ ( m f ) = , where g f K ¯ K P ( m f ) is the mass operator.The correlation function F ( y ) in (1) allows to account for the finite size of the f asa bound state of K ¯ K . Although the vertex function is related to the shape and size of themeson its explicit form only plays a minor role. The second task of the form factor is theregularization of the kaon loop integral. Here we have chosen a Gaussian form F ( y ) = Z d k ( p ) e − iky e F ( − k ) , e F ( k E ) = exp ( − k E / L ) , where the index E refers to Euclidean momentum space. THE ELECTROMAGNETIC DECAYS
In this section we study the electromagnetic decays of the f ( ) and a ( ) , whichproceed via the charged kaon loop. We derive the form factors in a manifest gauge-invariant way by evaluating the kaon loop integrals and finally deduce the couplings anddecay widths. In the following the radiative decay is discussed for the case of the f , the a is treated in full analogy.First we restrict to the local case which corresponds to a vertex function withlim L → ¥ e F ( − k ) = F ( y ) . As a consequence gauge invarianceof the strong interaction Lagrangian (1) gets lost. Hence, we deal with a modified gauge-invariant form L GIf K ¯ K = g f K ¯ K f ( x ) Z dy F ( y ) × (cid:2) e − ieI ( x + y , x − y ) K + (cid:0) x + y (cid:1) K − (cid:0) x − y (cid:1) + K (cid:0) x + y (cid:1) ¯ K (cid:0) x − y (cid:1)(cid:3) which additionally includes photons via the path integral I ( x , y ) = x R y dz m A m ( z ) [6]. Dia-grammatically, vertices with additional photon lines attached are generated correspond-ing to the graphs of Fig. 1 c) and d). It is important to note that these diagrams only givea minor contribution to the transition amplitude but are required in order to fully restoregauge invariance. Results
Our results for the f → gg decay width are ( m f =0.98 GeV) G f → gg = .
25 keV ( L = ) and G f → gg = .
29 keV (local) , which are in good agreement with experimental data: IGURE 1.
Diagrams contributing to the electromagnetic f → gg decay.Reference [7] [8] [9] [10] G ( f → gg ) [keV] 0 . + . − . . + . + . − . − . . ± . ± .
09 0 . ± . ± . For the a we used m a =0.9847 GeV and obtain G a → gg = .
20 keV ( L = ) and G a → gg = .
23 keV (local) , lying within the quoted range of present data G a gg = . ± . q ¯ q , q ¯ q ) overlap(Tab. 1). Therefore, at this stage, the radiative decay cannot be used to determine thestructure content of the a and f . However, the K ¯ K molecular configuration is sufficientto describe the electromagnetic decay.For the f → gg decay properties, finite size effects play a minor role when bothphotons are on-shell. In contrast, the form factor depends strongly on the size parameter FIGURE 2.
The form factor Q F f gg ∗ ( Q ) in dependence on Q for the local (LC) and nonlocal case(NC). For the latter the form factor is given for different size parameters 0.7, 1.0 and 1.3 GeV. ABLE 1. f → gg decay width: comparison with q ¯ q , q ¯ q , hadronicapproaches.Reference [12] [13] [14] [15] [16]Meson structure q ¯ q q ¯ q q ¯ q hadronic hadronic G ( f → gg ) [keV] 0.33 0.31 0.27 0.2 0.22 ± L in case of virtual photons. In Fig. 2 we indicate the form factor F f gg ∗ ( Q ) with onereal and one virtual photon with Euclidean momentum squared − Q . To demonstratethe sensitivity of this form factor on finite-size effects we plot the results both for thelocal case and for the nonlocal vertex function with different size parameters L =0.7, 1.0and 1.3 GeV. In summary, an experimental determination of F f gg ∗ ( Q ) might pose apossibility to identify the underlying structure of the f / a . THE STRONG DECAYS
We also studied the strong f → pp and a → ph decays within our framework.Both decays proceed via the diagrams generated by the contact coupling of pionsand kaons [Fig.3(a)] and K ∗ meson exchange [Fig. 3(b)]. Within this model, the K ∗ meson is described by antisymmetric tensor fields. As was stressed in Ref. [17], thepropagators S VK ∗ ; mn , ab ( x ) and S WK ∗ ; mn , ab differ by the contact term contained in thetensorial propagator S WK ∗ ; mn , ab ( x ) = S VK ∗ ; mn , ab ( x ) + im K ∗ [ g ma g nb − g mb g na ] d ( x ) . Usingthis identity one can show that the contribution of the diagram Fig. 3(b) in tensorialrepresentation is given by the sum of the graph of Fig. 3(b) in vectorial representationplus a graph, which is diagrammatically described by Fig. 3(a), but has opposite sign. Inaddition we obtain a term resulting effectively from the difference of two graphs of thetype Fig. 3(a), but with different numerators in the expression. Numerically it is foundthat in the last term these two contributions almost compensate each other. f KK K ∗ p q q ππ (b) f KK ππ p q q (a) FIGURE 3.
Diagrams contributing to the strong f → pp decay. Results
The experimental results for the dominant decay processes f → pp and a → ph stillhave large errors. Our results are compatible with the data as indicated in the followingtable: ata G ( f → pp ) [MeV]PDG (2007) (total width) 40 − . + . + . − . − . Analysis [18] 64 ± L =1 GeV) Data G ( a → ph ) [MeV]PDG (2007) (total width) 50 − ± ± ± L =1 GeV) SUMMARY
The scalar f and a mesons were studied in a hadronic molecule model which is fullycovariant and gauge invariant. Additionally, the finite size of the hadronic molecule aretaken into consideration which leads to the only free parameter of the model being thecut-off L . Our results are in rather good agreement with experimental data.We also showed that for the electromagnetic form factor finite size effects becomeessential in the case of virtual photons. A more precise measurement of the decay widthand in particular an experimental determination of the form factor could help to constrainthe K ¯ K content of the f and a . ACKNOWLEDGMENTS
This work was supported by the DFG under contracts FA67/31-1 and GRK683. This work is also part ofthe EU Integrated Infrastructure Initiative Hadronphysics project under contract number RII3-CT-2004-506078 and president grant of Russia “Scientific Schools” No. 5103.2006.2.
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