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in the form of model-independent differential cross sections. Neutrino deep inelastic scattering has long been used to
validate the standard model and probe nucleon structure.
Over the years, experiments have measured cross sections,
electroweak parameters, coupling constants, nucleon structure functions, and scaling variables using such processes. In
deep inelastic scattering (Fig. 27), the neutrino scatters off a
quark in the nucleon via the exchange of a virtual W or Z
boson producing a lepton and a hadronic system in the ﬁnal
state.11 Both CC and NC processes are possible VI. HIGH-ENERGY CROSS SECTIONS: E $ 20–500 GeV Up to now, we have largely discussed neutrino scattering
from composite entities such as nucleons or nuclei. Given
enough energy, the neutrino can actually begin to resolve the
Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 N ! À X; N ! þ X;
" (80) N ! X; N ! X:
"
" (81) Here we restrict ourselves to the case of scattering, as an
example, though e and DIS interactions are also possible.
Following the formalism introduced in Sec. II, DIS processes can be completely described in terms of three dimensionless kinematic invariants. The ﬁrst two, the inelasticity
ðyÞ and the four-momentum transfer (Q2 ¼ Àq2 ), have
already been deﬁned. We now deﬁne the Bjorken scaling
variable x,
x¼ Q2
2pe Á q ðBjorken scaling variableÞ: (82) The Bjorken scaling variable plays a prominent role in
deep inelastic neutrino scattering, where the target can carry a
portion of the incoming energy momentum of the struck
nucleus.
11 Quarks cannot be individually detected; they quickly recombine
and thus appear as a hadronic shower. Joseph A. Formaggio and G. P. Zeller: From eV to EeV: Neutrino cross sections . . . 1334 On a practical level, these Lorentz-invariant parameters
cannot be readily determined from four-vectors, but they can
be reconstructed using readily measured observables in a
given experiment,
x¼ Q2
Q2
¼
;
2M 2ME y where the sum is over all quark species. The struck quark
carries a fraction x of the nucleon’s momentum, such that xq
"
(xq) is the probability of ﬁnding the quark (antiquark) with a
given momentum fraction. These probabilities are known as
parton distribution functions (PDFs). In this way, F2 ðx; Q2 Þ
measures the sum of the quark and antiquark PDFs in the
nucleon, while xF3 ðx; Q2 Þ measures their difference and is
therefore sensitive to the valence quark PDFs. The third
structure function 2xF1 ðx; Q2 Þ is commonly related to
F2 ðx; Q2 Þ via a longitudinal structure function, RL ðx; Q2 Þ, (83) y ¼ Ehad =E ; (84) Q2 ¼ Àm2 þ 2E ðE À p cos Þ;
(85)
F2 ðx; Q2 Þ ¼ where E is the incident neutrino energy, MN is the nucleon
mass, ¼ Ehad is the energy of the hadronic system, and E ,
p , and cos are the energy, momentum, and scattering
angle of the outgoing muon in the laboratory frame. In the
case of NC scattering, th...

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