f0(1370) decay in the Fock-Tani formalism

POSTERS

f₀(1370) decay in the Fock-Tani formalism

Mario L. L. da Silva^I; Daniel T. da Silva^I; Cesar A. Z. Vasconcellos^I; Dimiter Hadjimichef^II

^IInstituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, CEP 91501-970, Porto Alegre, RS, Brazil

^IIDepartamento de Física, Instituto de Física e Matemática, Universidade Federal de Pelotas, Campus Universitário, CEP 96010-900, Pelotas, RS, Brazil

ABSTRACT

We investigate the two-meson decay modes for f₀(1370). In this calculation we consider this resonance as a glueball. The Fock-Tani formalism is introduced to calculate the decay width.

Keywords: Glueballs; Fock-Tani formalism; Meson decay

I. INTRODUCTION

The gluon self-coupling in QCD opens the possibility of existing bound states of pure gauge fields known as glueballs. Even though theoretically acceptable, the question still remains unanswered: do bound states of gluons actually exist? Glueballs are predicted by many models and by lattice calculations. In experiments glueballs are supposed to be produced in gluon-rich environments. The most important reactions to study gluonic degrees of freedom are radiative J/y decays, central productions processes and antiproton-proton annihilation.

Numerous technical difficulties have so far been present in our understanding of their properties in experiments, largely because glueball states can mix strongly with nearby resonances [1],[2].

The best estimate for the masses of glueballs comes from lattice gauge calculations, which in the quenched approximation show [3] that the lightest glueball has J^PC = 0⁺⁺ and that its mass should be in the range 1.45 - 1.75 GeV.

Constituent gluon models have received attention recently, for spectroscopic calculations. For example, a simple potential model, namely the model of Cornwall and Soni [4],[5] has been compared consistently to lattice and experiment [6],[7]. In the present we shall apply the Fock-Tani formalism [8] to glueball decay by defining an effective constituent quark-gluon Hamiltonian. In particular the resonance f₀(1370) shall be considered.

II. THE FOCK-TANI FORMALISM

Now let us to apply the Fock-Tani formalism in the microscopic Hamiltonian to obtain an effective Hamiltonian. In the Fock-Tani formalism we can write the glueball and the meson creation operators in the following form

The indexes a and b are the glueball and meson quantum numbers: a = {space, spin} and b = {space, spin, isospin}. The gluon creation and annihilation a_m operators obey the following commutation relations [a_m,a_n] = 0 and [a_m,] = d_mn. While the quark creation , annihilation q_m, the antiquark creation and annihilation _m operators obey the following anticommutation relations {q_m,q_n} = {_m, _n} = {q_m,_n} = {q_m,} = 0 and {q_m,} = {_m,} = d_mn. In (1) and are the bound-state wave-functions for two-gluons and two-quarks respectively. The composite glueball and meson operators satisfy non-canonical commutation relations

The "ideal particles" which obey canonical relations

This way one can transform the composite state |añ into an ideal state |a), in the glueball case for example we have

where U = exp(tF) and F is the generator of the glueball transformation given by

with

In order to obtain the effective potential one has to use (4) in a set of Heisenberg-like equations for the basic operators g, , a

The simplicity of these equations are not present in the equations for a

The solution for these equation can be found order by order in the wave functions. For zero order one has = a_m, (t) = G_a sint + g_a cost and (t) = G_b cost - g_b sint. In the first order = 0, = 0 and . If we repeat a similar calculation for mesons let us to obtain the following equations solution: , and .

III. THE MICROSCOPIC MODEL

The microscopic model adopted here must contain explicit quark and gluon degrees of freedom, so we obtain a microscopic Hamiltonian of the following form

Where the quark and the gluon fields are respectively [9]

and

We choose this Hamiltonian due to its form that allow to obtain a operators structure of this type q^†

^†q^†

^†aa.

IV. THE FOCK-TANI FORMALISM APPLICATION

Now we are going to apply the Fock-Tani formalism to the microscopic Hamiltonian

which gives rise to an effective interaction H_FT. To find this Hamiltonian we have to calculate the transformed operators for quarks and gluons by a technique known as the equation of motion technique. The resulting H_FT for the glueball decay G ® mm is represented by two diagrams which appear in Fig. (1).

Analyzing these diagrams, of Fig. (1), it is clear that in the first one there is no color conservation. The glueball's wave-function F is written as a product

is the spin contribution, with A_a º {S_a, } , where S_a is the glueball's total spin index and the index of the spin's third component; is the color component given by and the spatial wave-function is

The expectation value of r² gives a relation between the rms radius r₀ and b of the form b = /r₀. The meson wave function Y is similar with parameter b replacing b. To determine the decay rate, we evaluate the matrix element between the states |iñ = |0ñ and |fñ = |0ñ which is of the form

The h_fi decay amplitude can be combined with a relativistic phase space to give the differential decay rate [10]

After several manipulations we obtain the following result

Finally one can write the decay amplitude for the f₀ into two mesons

where

with m_q the u and d quark mass and m_s the mass of the s quark. The decays that are studied are for the following processes f ® pp, f ® K and f ® hh. The parameters used are b = 0.34 GeV, m_q = 0.33, m_q/m_s = 0.6, a_s = 0.6. Experimental data is still uncertain for this resonance. There is a large interval for the full width G = 200 to 500 MeV and the studied decay channels are seen, but still with no estimation.

V. CONCLUSIONS

The Fock-Tani formalism is proven appropriate not only for hadron scattering but for decay. The example decay process f₀(1370) ® pp; K and hh in the Fock-Tani formalism is studied. The same procedure can be used for other f₀(M) and for heavier scalar mesons and compared with similar calculations which include mixtures.

Acknowledgments

The author M.L.L.S. acknowledges support from the Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq and D.T.S. acknowledges support from the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES.

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Received on 23 September, 2006

Publication Dates

• Publication in this collection
  13 Aug 2007
• Date of issue
  July 2007

History

• Received
  23 Sept 2006