Parametrizing the line shapes of near-threshold resonances
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rametrizing the Line shapes of Near-Threshold Resonances1 Yu. S. Kalashnikova, R. V. Mizuk, and A. V. Nefediev* National Research Nuclear University MEPhI, Moscow, 115409 Russia *e-mail: [email protected] Abstract—A parametrization is proposed for the line shapes of near-threshold resonances, which is based on the model of coupled channels and can include an arbitrary number of elastic and inelastic channels and bare poles. The proposed parametrization satisfies the requirements imposed by unitarity and analyticity, and is convenient for the data analysis embracing all available experimental information. The model parameters are physically meaningful, and their values can be found using different theoretical schemes. DOI: 10.1134/S1063779617060235
The charmonium state X (3872) which would not be accommodated in the naive quark model was observed in 2003 by the Belle collaboration [1]. Over ten such “supernumerary” charmonia and bottomonia have been discovered to date, and most of them lie close to strong S -wave thresholds that strongly affect their properties. The other characteristic feature of such states is that they have been detected in the channels with the open as well as hidden flavor. Since new high-statistics data are forthcoming in the near future [2–5], consistent approaches to their analysis should be urgently developed. These should involve phenomenologically valid parametrizations consistent with the conditions of analyticity and unitarity, capable of embracing the complete experimental information available for various production channels and decays of the near-threshold resonances. The proposed approach to this problem is based on the model of coupled channels described by the Lippmann–Schwinger equations for the t -matrix. The interaction potential has the matrix form [6]
b = 1, N p
β = 1, N e
v aβ ⎛ v ab ⎜ Vˆ = ⎜ v α b v αβ l in j in l j ⎜ ⎝ λ j (k j ) g jβ(k j )
i = 1, N in
λi(kiin)li g iα (k iin )li 0
⎞ ⎟ ⎟ ⎟ ⎠
a = 1, N p α = 1, N e ,
(1)
where Q and q stand for the heavy and light quarks, denoted by the Greek letters α , β, γ , etc.; and N in inelastic hidden-flavor channels denoted by the Latin letters i , j , k , etc. The potential of the direct elastic interaction, v αβ, is approximately described by the constant matrix, and the transition potentials between the elastic and inelastic channels are assumed in separable forms. The direct interaction in inelastic channels is neglected, whereby the equations acquire simpler forms since inelastic channels decouple from both the elastic channels and bare poles, and their contributions reduce to a universal additive operator in the effective elastic potential. Then, elastic channels can be separated from bare poles, resulting in an additional term in the effective elastic potential that describes the rescattering through the production of quark states. Therefore, the problem reduces to the Lippmann– Schwinger equation for the elastic component t αβ of the complete t -matrix, while its other components are algebraically expressed through t α
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