Implications of a Froissart bound saturation of γ∗\gamma^*-pp deep inelastic scattering. Part II. Ultra-high energy neutrino interactions

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 $E_\nu \sim 10^{16}$-$10^{17}$ 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 $\nu N$ 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 $\ln^3E_\nu$ for leading-order-weak $\nu N$ scattering. The energy dependence of $\nu N$ total cross section measurements consequently has important implications for hadronic interactions at enormous cms (center-of-mass) energies not otherwise accessible.Comment: 15 pages, 6 figures. The paper is now shorter with the new results clearly emphasize

Implications of a Froissart bound saturation of γ∗\gamma^*-pp deep inelastic scattering. Part I. Quark distributions at ultra small xx

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 $x\rightarrow 0$. 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 $E_\nu=10^{17}$ 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 $F_2^{\nu(\bar{\nu})}$ directly in terms of $F_2^{\gamma p}$. We show that the correct distributions nevertheless give results for $F_2^{\nu(\bar{\nu})}$ 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.Comment: 19 pages, 6 figures. The paper is now shorter with the new results clearly emphasize

Connection of the virtual γ∗p\gamma^*p cross section of epep deep inelastic scattering to real γp\gamma p scattering, and the implications for νN\nu N and epep total cross sections

We show that it is possible to fit all of the HERA DIS (deep inelastic scattering) data on $F_2^{\gamma p}$ at small values of Bjorken $x$, including the data at {\em very low} $Q^2$, using a new model for $F_2^{\gamma p}$ which both includes an asymptotic (high energy) part that satisfies a saturated Froissart bound behavior, with a vector-dominance like mass factor in the parameterization, and extends smoothly to $Q^2=0$. We require that the corresponding part of the virtual $\gamma^* p$ cross section match the known asymptotic part of the real $\gamma p$ cross section at $Q^2=0$, a cross section which is determined by strong interactions and asymptotically satisfies a saturated Froissart bound of the form $\alpha +\beta\ln s+\gamma\ln^2s$. Using this model for the asymptotic part of $F_2^{\gamma p}$ plus a known valence contribution, we fit the asymptotic high energy part of the HERA data with $x\le 0.1$ and $W\ge 25$ GeV; the fit is excellent. We find that the mass parameter in the fit lies in the region of the light vector mesons, somewhat above the $\rho$ meson mass, and is compatible with vector dominance. We use this fit to obtain accurate results for the high energy $ep$ and isoscalar $\nu N$ total cross sections. Both cross sections obey an analytic expression of the type $a +b \ln E +c \ln^2 E +d \ln^3 E$ at large energies $E$ of the incident particle, reflecting the fact that the underlying strong interaction parts of the $\gamma^*p$, $Z^*N$ and $W^*N$ cross sections satisfy the saturated Froissart bound. Since approximately 50% of the $\nu N$ center of mass (cms) energy is found in $W$---the cms energy of the strongly interacting intermediate vector boson-nucleon system---a study of ultra-high-energy neutrino-nucleon cross sections would allow us, for the first time, to explore {\em strong interactions at incredibly high energies}.Comment: 17 pages, 9 figures. This version was accepted for publication in Phys. Rev.

An analytic solution to LO coupled DGLAP evolution equations: a new pQCD tool

We have analytically solved the LO pQCD singlet DGLAP equations using Laplace transform techniques. Newly-developed highly accurate numerical inverse Laplace transform algorithms allow us to write fully decoupled solutions for the singlet structure function F_s(x,Q^2)and G(x,Q^2) as F_s(x,Q^2)={\cal F}_s(F_{s0}(x), G_0(x)) and G(x,Q^2)={\cal G}(F_{s0}(x), G_0(x)). Here {\cal F}_s and \cal G are known functions of the initial boundary conditions F_{s0}(x) = F_s(x,Q_0^2) and G_{0}(x) = G(x,Q_0^2), i.e., the chosen starting functions at the virtuality Q_0^2. For both G and F_s, we are able to either devolve or evolve each separately and rapidly, with very high numerical accuracy, a computational fractional precision of O(10^{-9}). Armed with this powerful new tool in the pQCD arsenal, we compare our numerical results from the above equations with the published MSTW2008 and CTEQ6L LO gluon and singlet F_s distributions, starting from their initial values at Q_0^2=1 GeV^2 and 1.69 GeV^2, respectively, using their choices of \alpha_s(Q^2). This allows an important independent check on the accuracies of their evolution codes and therefore the computational accuracies of their published parton distributions. Our method completely decouples the two LO distributions, at the same time guaranteeing that both G and F_s satisfy the singlet coupled DGLAP equations. It also allows one to easily obtain the effects of the starting functions on the evolved gluon and singlet structure functions, as functions of both Q^2 and Q_0^2, being equally accurate in devolution as in evolution. Further, it can also be used for non-singlet distributions, thus giving LO analytic solutions for individual quark and gluon distributions at a given x and Q^2, rather than the numerical solutions of the coupled integral-differential equations on a large, but fixed, two-dimensional grid that are currently available.Comment: 13 pages, 5 figures, typos corrected, references updated and a footnote added; Accepted for publication in Physical Review

Ultrahigh energy neutrino scattering: an update

We update our estimates of charged and neutral current neutrino total cross sections on isoscalar nucleons at ultrahigh energies using a global (x, Q^2) fit, motivated by the Froissart bound, to the F_2 (electron-proton) structure function utilizing the most recent analysis of the complete ZEUS and H1 data sets from HERA I. Using the large Q^2, small Bjorken-x limits of the "wee" parton model, we connect the ultrahigh energy neutrino cross sections directly to the large Q^2, small-x extrapolation of our new fit, which we assume saturates the Froissart bound. We compare both to our previous work, which utilized only the smaller ZEUS data set, as well as to recent results of a calculation using the ZEUS-S based global perturbative QCD parton distributions using the combined HERA I results as input. Our new results substantiate our previous conclusions, again predicting significantly smaller cross sections than those predicted by extrapolating pQCD calculations to neutrino energies above 10^9 GeV.Comment: 8 pages, 1 figure, 3 table

Two historical changes in the narrative of energy forecasts

A collection of 417 energy scenarios was assembled and harmonized to compare what they said about nuclear, fossil and renewable energy thirty years from their publication. Based on data analysis, we divide the recent history of the energy forecasting in three periods. The first is defined by a decline in nuclear optimism, approximately until 1990. The second by a stability of forecasts, approximately until 2005. The third by a rise in the forecasted share of renewable energy sources. We also find that forecasts tend to cohere, that is they have a low dispersion within periods compared to the change across periods