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Connection of the virtual $γ^*p$ cross section of $ep$ deep inelastic scattering to real $γp$ scattering, and the implications for $νN$ and $ep$ total cross sections

Martin M. Block, Loyal Durand, Phuoc Ha

TL;DR

The paper develops a Froissart-bounded, two-component model for the proton structure function $F_2^{\gamma p}$ at small $x$, incorporating a high-energy asymptotic term and a valence contribution. It rigorously connects the virtual photon-proton cross section to the real- photon cross section at $Q^2=0$ by using a dispersion-inspired, dispersion-compatible parametrization and matching to the Block-Halzen Froissart-bound $\\

Abstract

We show that it is possible to fit all of the HERA DIS (deep inelastic scattering) data on $F_2^{γp}$ at small values of Bjorken $x$, including the data at {\em very low} $Q^2$, using a new model for $F_2^{γp}$ 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 $γ^* p$ cross section match the known asymptotic part of the real $γp$ 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 $α+β\ln s+γ\ln^2s$. Using this model for the asymptotic part of $F_2^{γp}$ 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 $ρ$ meson mass, and is compatible with vector dominance. We use this fit to obtain accurate results for the high energy $ep$ and isoscalar $νN$ 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 $γ^*p$, $Z^*N$ and $W^*N$ cross sections satisfy the saturated Froissart bound. Since approximately 50% of the $νN$ 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}.

Connection of the virtual $γ^*p$ cross section of $ep$ deep inelastic scattering to real $γp$ scattering, and the implications for $νN$ and $ep$ total cross sections

TL;DR

The paper develops a Froissart-bounded, two-component model for the proton structure function at small , incorporating a high-energy asymptotic term and a valence contribution. It rigorously connects the virtual photon-proton cross section to the real- photon cross section at by using a dispersion-inspired, dispersion-compatible parametrization and matching to the Block-Halzen Froissart-bound $\\

Abstract

We show that it is possible to fit all of the HERA DIS (deep inelastic scattering) data on at small values of Bjorken , including the data at {\em very low} , using a new model for 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 . We require that the corresponding part of the virtual cross section match the known asymptotic part of the real cross section at , a cross section which is determined by strong interactions and asymptotically satisfies a saturated Froissart bound of the form . Using this model for the asymptotic part of plus a known valence contribution, we fit the asymptotic high energy part of the HERA data with and 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 meson mass, and is compatible with vector dominance. We use this fit to obtain accurate results for the high energy and isoscalar total cross sections. Both cross sections obey an analytic expression of the type at large energies of the incident particle, reflecting the fact that the underlying strong interaction parts of the , and cross sections satisfy the saturated Froissart bound. Since approximately 50% of the center of mass (cms) energy is found in ---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}.

Paper Structure

This paper contains 14 sections, 27 equations, 9 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Continuation of the virtual $\gamma^*\,p$ cross section to $Q^2=0$
  3. Definition of $\gamma^*\,p$ cross section
  4. Extrapolation of $F_2^{\gamma p}$ and the $\gamma^*\,p$ cross section to $Q^2=0$
  5. A Froissart-bounded fit to the $\gamma p$ and HERA $\gamma^*p$ data
  6. The Block-Halzen fit to the $\gamma p$ cross section
  7. A new Froissart-bounded fit to the HERA data at high energies
  8. Applications
  9. $\gamma^* p$ cross sections from the fit
  10. $e^{\pm}p$ neutral current cross sections
  11. Neutrino-nucleon cross sections
  12. Numerical evaluation of the CC and NC $\nu N$ cross sections
  13. Implications of high energy weak $\nu N$ scattering for strong hadronic interactions
  14. Summary and conclusions

Figures (9)

  • Figure 1: Plots of the fitted proton structure function, $F_2^{\gamma p}(x,Q^2)$ versus Bjorken $x$ for virtualities, bottom to top, $Q^2$=0.15, 0.25, 0.65, 3.5, 4.5, 6.5, 10, 15, 22, 35, 70, 250, and 1200 GeV$^2$.
  • Figure 2: Plots of our fit to $F_2^{\gamma p}$ as a function of $Q^2$ at fixed $x$, compared to the corresponding HERA data HERAcombined. Top panel, top to bottom: $x=0.0002$ (red), 0.0005 (black), 0.0032 (green), 0.008 (blue), 0.02 (orange), 0.08 (purple); Bottom panel, top to bottom: $x=0.00013$ (red), 0.00032 (black), 0.0005 (green), 0.0008 (blue), 0.002 (orange), 0.05 (purple).
  • Figure 3: Plots of the fitted proton structure function $F_2^{\gamma p}(W,Q^2)$ versus $W$ for representative values of $Q^2$. The vertical lines indicate the cutoff used in the fit, $W\ge25$ GeV. In the upper panel, top to bottom: $Q^2=0.15$ (red), $0.25$ (black), $0.65$ (green), $3.5$ (blue), $4.5$ (orange), and $6.5$ (purple) GeV$^2$ In the lower panel: top to bottom: $Q^2=35$ (red), $90$ (black), $120$ (green), $250$ (blue), $500$ (orange), and $1200$ (purple) GeV$^2$. The curves in this panel extend in $W$ only to the minimum value allowed by the condition $x\le0.1$. All data at the specified values of $Q^2$ are shown.
  • Figure 4: Plots of the $\gamma^* p$ cross section $\sigma^{\gamma^*p}$, in $\mu$b. vs. $W$, the cms energy of the $\gamma^*p$ system. The upper curve is for $Q^2$ = 0 to 10 GeV$^2$; the lower curve is for $Q^2=60$ to 1000 GeV$^2$. Notice the very different scales of the vertical axes (cross sections) of the two curves. The circles which are plotted for Q$^2\ne 0$ are $\gamma^*p$ cross section data from HERA DIS (deep inelastic scattering) that satisfy the cuts $W\ge 25$ GeV and $x\le 0.1$; the $W$ cut for the curves is indicated by the thick dot-dot-dashed vertical line in the upper plot and the thick dot-dot-dashed boundary in the lower plot . The plotted cross section curves of $\sigma^{\gamma^*p}$ for $Q^2>0$ are the sum of the asymptotic cross section plus the valence cross section. For $Q^2=0$, the curve is the sum of the asymptotic DIS cross section plus the rapidly decreasing Regge-like term used in the Block-Halzen fit to real $\gamma p$ data BHgamma-p; the data for $W>2$ GeV are shown as (blue) triangles.
  • Figure 5: Plots of $e^ \pm p$ NC cross section for $Q^2>1$ GeV$^2$, in cm$^2$, vs. $E_e$, the laboratory neutrino energy, in GeV. The points are the numerical calculations of Table \ref{['table:epCS']}.
  • ...and 4 more figures