Asymptotics of the S-matrix and unitarisation
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ESSION “QUANTUM FIELD THEORY AND THEORY OF ELEMENTARY PARTICLES”
Asymptotics of the SMatrix and Unitarisation1 O. V. Selyugina, b and J.R. Cudellb a
Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Russia b Institut de Physique, Bât. B5a, Université de Liège, Belgium
Abstract—The manner in which the elastic scattering amplitude obeys unitarity, how it enters the circle of unitarity, and what its asymptotic limit is, remains a problem for models which include terms that rise fast with s. We have checked that the features of cross sections which come from unitarisation are present for most unitarisation schemes, e.g. those that saturate the profile function or those that describe multiple exchanges via an analytic formula. We have also obtained a scheme which interpolates between different classes of the unitarisation and found corresponding nonlinear equations. Considering different forms of energy depen dence of the scattering amplitude, and a variety of unitarisation schemes, we show that, in order to reproduce the data, the fits choose an amplitude that corresponds to an asymptotic value S = 0. DOI: 10.1134/S1063779610060213 1
The asymptotic properties of the Smatrix are con nected to the unitarity condition SS+ ≤ 1 [1]. In the impact parameter representation after unitarisation the total cross section is ∞
∫
σ tot ( s ) = 4πIm bG ( χ ( s, b ) ) db,
(1)
s ∞, it leads to S = eiχ 0: the large imaginary part of the scattering amplitude damps the size of Smatrix down to zero. Another value can however be envisaged. Indeed, a unitarisation scheme was proposed [4], which leads to a different asymptotic value: + iχ ( s, b )/2 S ( s, b ) = 1 1 – iχ ( s, b )/2
0
where G(χ(s, b)) is the unitarised amplitude built from the Born term χ(s, b). There exists two main classes of unitarisation [2]: the standard one leads to the Black Disk Limit (BDL), and includes the standard eikonal representation, while the Full Unitarity Circle (FCU) class, which includes the standard Umatrix, does not lead asymptotically to the BDL, but rather to a scatter ing amplitude which is mostly elastic. In the absence of inelastic processes, the conserva tion of probability in the scattering process requires that each phase shift δl or, in the impact parameter representation, that the Born term χ(s, b) be real. In the standard representation the Smatrix has an expo nential form S(s, b) = eiχ(s, b), where, at low energy, the phase χ(s, b) is taken as purely real. Obviously, one then has |S(s, b) |2 = 1. When one thought that total cross sections at high energies decreased, it was believed [3] that the scattering amplitude tended to its Born term and that the asymptotic value of the Smatrix was S(s ∞) = 1. Later, it was realised that total cross sections grew with s. If χ ∞ the value of the Smatrix will oscil late. In this case the asymptotic value of Smatrix is indefinite. However, as the phase χ(s, b) develops a positive imaginary part, χ(s) = iImχ(s) + Reχ(s), we obtain the inequality SS+
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