Scattering amplitudes at multi TeV energies

Abstract

We show that a generalized Regge behaviour, $$ImF(s,t)\simeq \Phi (t)(\log s/\hat{s}{)}^{\nu (t)}(s/\hat{s}{)}^{{\alpha }_P(t)},\mathrm{for}|t|<|t_0|,s\to \infty$$ where $\Phi (t)\simeq e^{bt}$, ${\alpha }_P(t)\simeq {\alpha }_P(0)+{\alpha }_P^{\prime }(0)t$, and $t_0$ is the first zero of ${\alpha }_P(t)$, ${\alpha }_P(t_0)=0$, implies that the corresponding cross section is bounded by $${\sigma }_{\mathrm{tot}}(s)<(Const.)\times \log s/\hat{s}.$$ This growth, however, is not sufficient to fit the experimental cross sections. If, instead of this, we assume saturation of the improved Froissart bound, i.e., a behaviour $$ImF(s,0)\simeq A(s/\hat{s}){\log }^2\frac{s}{s_1{\log }^{7/2}s/s_2},$$ a good fit is obtained to $\pi \pi$, $\pi N$, $KN$ and $NN$ cross sections from c.m. kinetic energy $E_{\mathrm{kin}}\simeq 1$ GeV to 30 TeV (producing a cross section of $108\pm 6$ mb at LHC energy). This suggests that the Regge-type behaviour only holds for values of the momentum transfer very near zero.

Plain TeX. Dedicated to Prof. Yuri Simonov in his 70th birthday