Scalar Glueball Decay Into Pions In Effective Theory

We examine the mixing between the sigma meson σ and the ”pure” glueball field H and study the decays of the scalar glueball candidate f0(1500) (a linear combination of σ and H) into two and four pions in effective linear sigma model. From recent experimental data on f0(1500) decay to pions we point out that the mixing angle is of order of 0.1. PACS numbers 13.25.Hw 12.28.Lg ∗Email: [email protected], [email protected] 1 Quantum Chromo Dynamics (QCD), as the fundamental theory of the strong interaction predicts the existence of exotic mesons made of gluons. Observations of these gluonium will provide a direct confirmation on the special feature of non-abelian gauge theory. For scalar glueballs the candidates reported in the literature include f0(1500) and f0(1750)[1,2,3]. However, the non-perturbative dynamics of QCD makes it difficult to do the calculation on the glueball mass, decay width and mixing with the sigma meson. Besides Lattice QCD[8], one has considered various phenomenological models, such as the potential models[4], bag models[5], flux tube[6] models and the QCD sum rule[7] in order to study the glueball mass and its two body decays. In this paper we take an effective lagrangian approach and discuss the glueball decay into two as well as four-pion final states. Particularly we will pay attention to the mixing between the glueball and the sigma meson. It is well known that non-linear sigma model describes the dynamical properties of the pions. To construct an effective lagrangian for glueball decay into pions one could simply include the glueball meson into the non-linear sigma model. Under the chiral symmetry, the glueball transforms as a singlet. For instance, interaction responsible for the decay of the scalar glueball into multi-pions is given by HTr∂μΣ ∂Σ (1) where Σ = exp i~τ · ~π/fπ and H the ”pure” glueball field. However, with mass of scalar glueball around 1500 MeV or 1700MeV, the momentum transfer from the decay of the glueball into pions is so large that resonances, such as sigma and rho mesons will show up and play an essential role in the processes of the glueball decay. Thus we use in this paper an effective lagrangian with sigma and rho mesons included explicitly i.e. the linear sigma model. The identification of the sigma particle σ is highly controversial. It is listed in the Particle Data Tables[14] as a very broad meson with mass around 400 − 1200 MeV and full width 600 − 1200 MeV. This particle, needed in the linear sigma model, has been argued to play an important role in nuclear physics[9] and in the study on the chiral phase transition[10]. In recent years, many works are devoted to search for it in the ππ 2 scattering[13]. Regarding its role in the glueball physics, BES[1] and MKIII[2] data on B(J/Ψ → γf0(1500) → γ4π) and B(J/Ψ → γf0(1750) → γ4π) , Crystal Barrel data[3] on p̄p → πf0(1500) → π(2π, K̄K, 2η, ηη, 4π) show f0(1500) decays into 4π dominantly through sigma channel(S partial wave), and also hints it strongly couples to 2σ. In this paper, we consider, for simplicity, the linear sigma model for two flavors. As usual we introduce the field Φ = σ τ 0 2 + i~π ~τ 2 , (2) where τ 0 is unity matrix and ~τ the Pauli matrices with normalization condition Trττ b = 2δ. Under a SUL(2)⊗ SUR(2) chiral transformation, Φ transforms as Φ → LΦR. (3) A renormalizable lagrangian of linear sigma model is given now by LΦ−Φ = Tr{∂μΦ∂Φ} − λ[Tr{ΦΦ} − f 2 π 2 ], (4) where fπ is the vacuum expectation value of the sigma field and λ the self coupling constant. Let’s now add the O ”pure” glueball state H on the lagrangian. Given that the H is made of gluons and singlet under the chiral symmetry, there are only two terms which give rise to the interaction among H, the sigma and pions, LH−Φ = g1fπH [Tr{ΦΦ} − f 2 π 2 ] + g2H [Tr{ΦΦ} − f 2 π 2 ], (5) where two free parameters, g1 and g2 are introduced to describe the strength of couplings of one and two H to the sigma or pions. For the self interaction of glueballs, the lagrangian is given by LH−H = 1 2 ∂μH∂ H + 1 2 m2HH 2 + f3H 3 + f4H , (6) where m2H is the mass of glueball, f3, f4 the self coupling constants. After the chiral symmetry is broken by non-vanishing vacuum expectation value of the sigma field, lagrangian in (5) generates not only the interaction for glueball decay, but also