The complex relation between production and scattering amplitudes

Abstract

The unitarity relation ℑm (A) = T * A is derived for a three-body production amplitude A that consists of a complex linear combination of elements of the two-body scattering amplitude T . We conclude that the unitarity relation does not impose a realness condition on the coefficients in the expansion of A in terms of T .

On the existence of a superlight scalar boson

“…[26] and further developed in Refs. [27,28]. This model shows that one must expect non-resonant enhancements in the amplitudes just above pair-creation thresholds, which, in the case of stable hadrons, are accompanied by a clear minimum at threshold, as observed in experiment for the process e + e − → bb, measured and analysed by the BaBar Collaboration [29].…”

Dramatic Implications of Unitarity for Meson Spectroscopy

“…The corresponding production amplitude P is defined [14] in the RSE formalism as a non-resonant, lead term plus its infinite rescattering series via the above RSE T -matrix, i.e., P = resonance around 1.35 GeV, viz. the f 0 ð1370Þ [15].…”

“…Notice that the latter quantum number is related-though unequal-to the relative two-meson angular momentum ', because of total-angular-momentum and parity conservation. Below, we shall discuss the properties of production amplitude (15) for pairs of interacting mesons.…”

“…[14], where the detailed expressions can be found, the Z k are spherical Hankel (1) functions and their real parts spherical Bessel functions. The P k components satisfy [14,15] the extended-unitarity relation Im(P k ) = l T kl P l . Note that this can be rewritten in terms of purely imaginary functionsZ k , so without the inhomogeneous term, but then the real functions iZ k would necessarily include elements of the T -matrix itself and so not be purely kinematical anymore [16].…”

Heavy quarkonia: the beauty and the beasts

“…The situation with ψ(4660) [3] (X(4660) up until 2016 [11]) is related, as we consider [18] it a non-resonant threshold enhancement due to the opening of the Λ cΛc channel (also see Ref. [17]).…”