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which already is violated for E > 2 x 108 GeV. From this they deduce that at yet higher energies, where the
right-hand side of Eq. (3) increases, while the left is constant (at O(GF)), previously neglected terms that
are higher order in the weak coupling g, in particular, g6 or g8 terms, must become important. They go on
to suggest that this signals a breakdown in perturbation theory in the weak coupling, g. This is a striking
implication indeed, especially given the small size of the cross section in Eq. (4).
There is another, and we believe more natural, interpretation of the equality when E > 2 x 10' GeV. First,
we observe that the forward elastic cross section receives two qualitatively different and quantum mechanically
incoherent contributions. The first of these describes the coherent elastic scattering of the entire nucleon through
weak vector boson exchange, which begins at tree level, that is, at GF in the cross section. The second is the
contribution of high-Q2 virtual states that results from the incoherent scattering of partons. The latter, not
the former, is related independently by the optical theorem to the inelastic cross section on the right-hand side
of Eq. (3), and will saturate that inequality identically at order GF, regardless of its size, just as at order G'
the forward cross section is identically equal to the corresponding contribution from the square of the real part,
which has been neglected on the right of Eq. (3).
That being said, we may still ask whether the dominance of the partonic part of the cross section, higher-
order by g2 compared to the elastic part, might not be a sign of large contributions from yet higher orders in
the weak coupling. Integrating the factorized form Eq. (1), over x and Q2, however, shows that at very high
energy the square of the total cross section behaves as GF times [g2(S/M .)^]2. This is to be compared to
G2 on the left-hand side of Eq. (3). The factor g2 is the default size of a higher-order electroweak correction.
The factor (S/M&)A is due to the large number of partons of size I/Mw at x Mi/S. For higher orders in
g2 to contribute at a similar level, they would have to come accompanied by a similar large counting factor.
At the leading power in 1/Mw, which is given by Eq. (1), this cannot happen, simply because q(x) and q(x)
already count the partons. It would still be possible if more partons are involved in the hard scattering, but this
involves going to higher twist, that is, to explicit suppression by additional powers of 1/Mw , which would have
to be compensated for by higher-twist multi-parton matrix elements. While such contributions are not very
well-known even at low momentum transfers, there is no experimental indication of such large scales implicit
within the nucleon.
The forgoing arguments, of course, assume that the unaided QCD extrapolations described above are equal to
the task of so many orders of magnitude. We have shown above the self-consistency of these extrapolations, and
that they do not, by themselves, lead to problems with unitarity, or give evidence of a breakdown in perturbation
theory in the weak coupling . The very fact of the self-consistency of the QCD extrapolations shows that
ultra high energy neutrinos offer an exploration of the strong interactions, as well as of cosmic dynamics, into
unprecedented length scales.
We thank Duane Dicus, Stefan Kretzer, and Jamal Jalilian-Marian for discussions. This work was supported in
part by NSF Grants No. PHY-9802403 (M.H.R..), PHY-0098527 (G.S.), DOE Contracts DE-FG02-95ER40906
and DE-FG03-93ER40792 (I.S.). M.S. thanks SUNY Stony Brook and RIKEN, and G.S. and M.S. thank
Brookhaven National Laboratory for hospitality and support. W.V. is grateful to RIKEN, Brookhaven National
Laboratory and the U.S. Department of Energy (contract number DE-AC02-98CH10886) for providing the
facilities essential for the completion of his work.
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Hall Reno, M.; Sarcevic, I. N.; Sterman, G.; Stratmann, M. & Vogelsang, W. ULTRAHIGH ENERGY NEUTRINOS, SMALL X AND UNITARITY., article, June 30, 2001; Upton, New York. (digital.library.unt.edu/ark:/67531/metadc742880/m1/4/: accessed January 22, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.