Abstract
Assuming that the pion-pion scattering amplitude and its absorptive part are analytic inside an ellipse in $t$- plane with foci $t=0$, $u=0$ and right extremity $t=4 m_{\pi}^2 +\epsilon $, ($\epsilon > 0$), except for cuts prescribed by Mandelstam representation for $t\geq 4 m_{\pi}^2$, $u\geq 4 m_{\pi}^2$ , and bounded by $s^N$ on the boundary of this domain, we prove that for $s\rightarrow \infty$, \sigma_{inel} (s) > \frac{Const}{s^{5/2} }\exp {[-\frac{\sqrt{s}}{4} (N+5/2) \ln {s} ]}.
Highlights
It is well known that if there is no inelasticity, the scattering amplitude must be zero
Assuming that the pion-pion scattering amplitude and its absorptive part are analytic inside an ellipse in the complex t plane with foci t 1⁄4 0, u 1⁄4 0 and right extremity t 1⁄4 4m2π þ ε, (ε > 0)—except for cuts prescribed by the Mandelstam representation for t ≥ 4m2π, u ≥ 4pmffi 2π, and bounded by sN on the boundary of this domain—we prove that for s const s5=2 exp
Since we shall use the dominance of the nearest singularities for large angular momenta, we state at the same time the assumption he needs and our assumption
Summary
It is well known that if there is no inelasticity, the scattering amplitude must be zero. If we use the standard Mandelstam variables s, t, u and choose units such that the pion mass mπ 1⁄4 1, we need fixed-energy analyticity in an ellipse with foci at t 1⁄4 0 and u 1⁄4 0 and right extremity at t 1⁄4 4 þ ε, minus the obvious cuts t ≥ 4, u ≥ 4 for the amplitude, and t ≥ 4 þ 64= ðs − 16Þ, u ≥ 4 þ 64=ðs − 16Þ for the absorptive part (see Fig. 1). We seek a lower bound on its imaginary part Imfl, for which we need a bound on the discontinuity of the absorptive part which is nothing but the Mandelstam double spectral function This is what was missing in the work of Dragt [3].
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