[Paper Review] Ultrahigh energy neutrinos, small x and unitarity

Mary Hall Reno, Ina Sarčević+3 more|arXiv (Cornell University)|Oct 17, 2001

Particle physics theoretical and experimental studies8 citations

TL;DR

This paper examines ultrahigh energy neutrino-nucleon cross sections using perturbative QCD extrapolations to small parton momentum fractions $x$, showing that existing QCD-based predictions remain self-consistent and unitarity-preserving up to $10^{12}$ GeV. It argues that apparent unitarity violations in earlier studies arise from misapplying the optical theorem and that higher-order weak coupling corrections do not signal a breakdown of perturbation theory.

ABSTRACT

The ultrahigh energy cross section for neutrino interactions with nucleons is reviewed, and unitarity constraints are discussed. We argue that existing QCD extrapolations are self-consistent, and do not imply a breakdown of the perturbative expansion in the weak coupling.

Motivation & Objective

• To assess the validity of perturbative QCD extrapolations for neutrino-nucleon cross sections at ultrahigh energies ($E \gtrsim 10^7$ GeV).
• To resolve apparent unitarity violations reported in prior work by clarifying the role of the optical theorem in relating elastic and inelastic cross sections.
• To determine whether higher-order weak coupling corrections ($g^6$, $g^8$) are required to preserve unitarity, as claimed by Dicus et al.
• To evaluate whether small-$x$ parton distribution functions and saturation effects affect the cross section at extreme energies.

Proposed method

• Uses the factorized expression for the charged current neutrino-nucleon cross section, $d^2\sigma/dxdQ^2 = G_F^2 / \pi \cdot (M_W^2 / (Q^2 + M_W^2))^2 \cdot [q(x,Q) + (1-y)^2 \bar{q}(x,Q)]$, to compute cross sections at high energies.
• Applies DGLAP evolution to extrapolate parton distribution functions (PDFs) to small $x \sim 10^{-8}$, using power-law forms $xg(x,Q) \sim x^{-\lambda}$ with $\lambda \approx 0.5$.
• Compares QCD-based PDF extrapolations with BFKL-type evolution and experimental data from HERA and D0 to validate small-$x$ behavior.
• Applies the optical theorem to relate the forward elastic amplitude to the total inelastic cross section, clarifying that only incoherent partonic scattering contributes to the inelastic rate.
• Analyzes the scaling behavior of the total cross section as $\sigma \sim G_F^2 \cdot [g^2 (S/M_W^2)^\lambda]^2$, showing that higher-order corrections require large counting factors to compete.
• Demonstrates that self-consistent QCD extrapolations do not violate unitarity and do not necessitate new physics or breakdown of perturbation theory.

Experimental results

Research questions

• RQ1 Does the perturbative QCD extrapolation of parton distribution functions to $x \sim 10^{-8}$ remain valid and self-consistent at ultrahigh energies?
• RQ2 Why do earlier studies claim a violation of unitarity at $E \gtrsim 2 \times 10^8$ GeV, and is this a real breakdown or a misapplication of the optical theorem?
• RQ3 Can higher-order weak coupling corrections ($g^6$, $g^8$) be neglected at $E \sim 10^{12}$ GeV, or do they become dominant?
• RQ4 Is the dominance of partonic inelastic scattering at high energies a sign of non-perturbative effects in the weak interaction?
• RQ5 Do saturation effects in small-$x$ PDF evolution significantly alter the neutrino cross section at $E \sim 10^{12}$ GeV?

Key findings

• The charged current neutrino-nucleon cross section scales as $\sigma = 5.5 \times 10^{-36} (E/\text{GeV})^{0.36}$ cm$^2$ for $10^7$ GeV $< E < 10^{12}$ GeV using CTEQ4 PDFs.
• QCD extrapolations of PDFs to $x \sim 10^{-8}$ using DGLAP or BFKL evolution are self-consistent and do not violate unitarity.
• The apparent unitarity violation reported by Dicus et al. arises from incorrectly equating the elastic $G_F^2$ amplitude with the inelastic $G_F^4$ cross section; the two are incoherent and not directly comparable.
• Higher-order weak coupling corrections do not become dominant because they lack the necessary large counting factors to compensate for the $g^2$ suppression.
• The total cross section grows as $[g^2 (S/M_W^2)^\lambda]^2$, and no new physics or breakdown of perturbation theory is required to preserve unitarity.
• There is no experimental or theoretical evidence for large-scale structures within the nucleon that would enhance higher-twist or multi-parton contributions at high energies.