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Register a new userCan one improve the Froissart bound?
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Authors
Andre Martin
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We explain why we hope that the Froissart bound can be improved, either qualitatively or, more likely, quantitatively, by making a better use of unitarity, in particular elastic unitarity. In other instances (Gribovs theorem) elastic unitarity played a crucial role. A preliminary requirement for this is to work with an appropriate average of the cross-section, to make the problem well defined. This is possible, without destroying the Lukaszuk--Martin bound.
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We study soft gluon kt-resummation and the relevance of zero momentum gluons for the energy dependence of total hadronic cross-sections. We discuss a model in which consistency of the energy dependence of the cross-section with the limitation of the Froissart bound, is directly related to the behaviour of the strong coupling constant in the infrared region. Our predictions for the asymptotic behaviour are shown to be related to the ansatz that the infrared behaviour of the QCD strong coupling constant follows an inverse power law.
A previously successful model for purely hadronic total cross-sections, based on QCD minijets and soft-gluon resummation, is here applied to the total photoproduction cross section. We find that our model in the gamma p case predicts a rise with energy stronger than in the pp -pbarp case.
We point out that bound states, degenerate in energy but differing in parity, may form in one dimensional quantum systems even if the potential is non-singular in any finite domain. Such potentials are necessarily unbounded from below at infinity and occur in several different contexts, such as in the study of localised states in brane-world scenarios. We describe how to construct large classes of such potentials and give explicit analytic expressions for the degenerate bound states. Some of these bound states occur above the potential maximum while some are below. Various unusual features of the bound states are described and after highlighting those that are ansatz independent, we suggest that it might be possible to observe such parity-paired degenerate bound states in specific mesoscopic systems.
In Part I (in this journal) we argued that the structure function $F_2^{gamma p}(x,Q^2)$ in deep inelastic $ep$ scattering, regarded as a cross section for virtual $gamma^*p$ scattering, has a saturated Froissart-bounded form behaving as $ln^2 (1/x)$ at small $x$. This form provides an excellent fit to the low $x$ HERA data, including the very low $Q^2$ regions, and can be extrapolated reliably to small $x$ using the natural variable $ln(1/x)$. We used our fit to derive quark distributions for values of $x$ down to $x=10^{-14}$. We use those distributions here to evaluate ultra-high energy (UHE) cross sections for neutrino scattering on an isoscalar nucleon, $N=(n+p)/2$, up to laboratory neutrino energies $E_ u sim 10^{16}$-$10^{17}$ GeV where there are now limits on neutrino fluxes. We estimate that these cross sections are accurate to $sim$2% at the highest energies considered, with the major uncertainty coming from the errors in the parameters that were needed to fit $F_2^{gamma p}(x,Q^2)$. We compare our results to recently published neutrino cross sections derived from NLO parton distribution functions, which become much larger at high energies because of the use of power-law extrapolations of quark distributions to small $x$. We argue that our calculation of the UHE $ u N$ cross sections is the best one can make based the existing experimental deep inelastic scattering data. Further, we show that the strong interaction Froissart bound of $ln^2 (1/x)$ on $F_2^{gamma p}$ translates to an exact bound of $ln^3E_ u$ for leading-order-weak $ u N$ scattering. The energy dependence of $ u N$ total cross section measurements consequently has important implications for hadronic interactions at enormous cms (center-of-mass) energies not otherwise accessible.
We argue that the deep inelastic structure function $F_2^{gamma p}(x, Q^2)$, regarded as a cross section for virtual $gamma^*p$ scattering, is hadronic in nature. This implies that its growth is limited by the Froissart bound at high hadronic energies, giving a $ln^2 (1/x)$ bound on $F_2^{gamma p}$ as Bjorken $xrightarrow 0$. The same bound holds for the individual quark distributions. In earlier work, we obtained a very accurate global fit to the combined HERA data on $F_2^{gamma p}$ using a fit function which respects the Froissart bound at small $x$, and is equivalent in its $x$ dependence to the function used successfully to describe all high energy hadronic cross sections, including $gamma p$ scattering. We extrapolate that fit by a factor of $lesssim$3 beyond the HERA region in the natural variable $ln(1/x)$ to the values of $x$ down to $x=10^{-14}$ and use the results to derive the quark distributions needed for the reliable calculation of neutrino cross sections at energies up to $E_ u=10^{17}$ GeV. These distributions do not satisfy the Feynman wee parton assumption, that they all converge toward a common distribution $xq(x,Q^2)$ at small $x$ and large $Q^2$. This was used in some past calculations to express the dominant neutrino structure function $F_2^{ u(bar{ u})}$ directly in terms of $F_2^{gamma p}$. We show that the correct distributions nevertheless give results for $F_2^{ u(bar{ u})}$ which differ only slightly from those obtained assuming that the wee parton limit holds. In two Appendices, we develop simple analytic results for the effects of QCD evolution and operator-product corrections on the distribution functions at small $x$, and show that these effects amount mainly to shifting the values of $ln(1/x)$ in the initial distributions.
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