Ultrahigh-Energy Neutrino-Nucleon Deep-Inelastic Scattering and the Froissart Bound

Abstract

We present a simple formula for the total cross section σ νN of neutral-and charged-current deepinelastic scattering of ultrahigh-energy neutrinos on isoscalar nuclear targets, which is proportional to the structure function F νN 2 (M 2 V /s, M 2 V ), where MV is the intermediate-boson mass and s is the square of the center-of-mass energy. The coefficient in the front of F νN 2 (x, Q 2 ) depends on the asymptotic low-x behavior of F νN 2 . It contains an additional ln s term if F νN 2 scales with a power of ln(1/x). Hence, an asymptotic low-x behavior F νN 2 ∝ ln 2 (1/x), which is frequently assumed in the literature, already leads to a violation of the Froissart bound on σ νN .PACS numbers: 13.15.+g, 13.85.Hd, 25.30.Pt, 95.85.Ry For more than a decade, large experiments have been searching for ultrahigh-energy (UHE) cosmic neutrinos (ν), with energies E ν > 10 6 GeV, using detectors scanning for events in large volumes of water, ice, the Earth's atmosphere, and the lunar regolith [1]. While no clear indication of such an event has yet been reported, experimental bounds on UHE-neutrino fluxes could be established, which, put together, now cover energies way up to 10 12 GeV and start to constrain scenarios of astrophysical interest. Since these limits directly depend on the total cross section σ νN (E ν ) of UHE-neutrino deep-inelastic scattering (DIS) off nucleons (N ), it is an urgent task to provide reliable theoretical predictions for the latter in the asymptotic high-E ν regime, which lies far beyond the one explored by laboratory-based νN DIS experiments and corresponds to asymptotically low values of Bjorken's scaling variable x. This requires extrapolation over several orders of magnitude in E ν , for which various approaches exist [2][3][4][5][6][7][8]. These are based on successful descriptions of the terrestrial data within the framework of perturbative QCD and frequently impose the Froissart bound [9] on σ νN . According to the latter, unitarity and analyticity limit the total cross section of a scattering process not to grow faster with energy than ln 2 s.In this Letter, we derive a general formula for σ νN that is remarkably concise and correctly accounts for the asymptotic high-energy behavior making it perfectly suitable for UHE-neutrino phenomenology. It is proportional to the DIS structure function F νN 2 (x, Q 2 ), which has a well-known representation in terms of parton distribution functions (PDFs) within the parton model (PM) of QCD, with x and the typical energy scale Q appropriately defined in terms of E ν and M V (V = W, Z). To be on the conservative side, we assume for the time being, as in Refs. [2][3][4][5][6][7][8], that the available experimental data on DIS allow for an extrapolation to very high (low) values of E ν (x) using an appropriate parameterization of F νN 2 . If the latter rises too steeply as x → 0, then possible new QCD phenomena, such as gluon saturation or recombination, color glass condensates, multiple pomeron exchanges, etc., are expected to enter the stage as a ...

Connection of the virtualγ*pcross section ofepdeep inelastic scattering to realγ<

“…As a result, a Froissart bounded F γp 2 which behaves asymptotically as ln 2 (1/x) for decreasing x leads to an integrated ep cross section that grows asymptotically as ln 3 E e . This modified Froissart behavior for the integrated cross sections was originally noted in the case of neutrino-proton scattering in [32], where the bound on the integrated cross section is proportional to ln 3 E ν , and was re-emphasized for that case in [31].…”

“…[64] The authors of [32] state incorrectly that the work in [7,56] claims that the νN cross section rises only as ln 2 Eν in the limit of large Eν. Those references actually assume the Froissart saturated form ln 2 (1/x) only for the structure function F γp 2 and the corresponding neutrino structure function F νN 2 , and not for the integrated lowest-order weak νN cross section.…”

Implications of a Froissart bound saturation ofγ*−pdeep inelastic scattering. II. Ultrahigh energy neutrino interactions

“…A discussion of why a strong interaction Froissart bound of ln 2 (1/x) gives rise to a weak interaction νN cross section bound of ln 3 E ν is given in the last paragraphs of Section II B and in [46]. Conversely, a weak cross section νN bound of ln 3 E ν implies a strong cross section hadron-nucleon Froissart bound of ln 2ŝ = ln 2 W 2 .…”

“…This is helpful for assessing the UHE behavior of the total cross section that follows from a given model of the structure functions, as pointed out in [46]. In particular, given the effective cutoff in the Q 2 integration for Q 2 > M 2 W , it shows that when the neutrino energy E ν satisfies the condition E ν /M 2 W ≫ 1, the νN cross section calculated to lowest order in G F will rise asymptotically with neutrino energy as ln 3 E ν for our Froissart-bounded extrapolations of F ν(ν) 2 [52].…”

A phenomenological solution small x to the longitudinal structure function dynamical behavior

“…Since the longitudinal structure function F L contains rather large heavy flavor contributions in the small-x region, therefore the measurement of these observables * boroun@razi.ac.ir have told us about the different scheme used to calculate the heavy quark contribution to the structure function and also the dependence of parton distribution functions (PDFs) on heavy quark masses [5]. For PDFs we need to use the corresponding massless Wilson coefficients up to next-to-next-to leading order (NNLO) [6][7][8][9][10][11][12][13][14], but we determine heavy contributions of longitudinal structure function in leading order and next-to-leading order by using massive Wilson coefficients in the asymptotic region Q 2 ≫m 2 h , where m h is the mass of heavy quark [15][16][17][18][19][20][21]. The dominant small x role is played by gluons, and the basic dynamic quantity is the unintegrated gluon distribution f (x, Q 2 t ), where Q t its transverse momentum.…”