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**Unformatted text preview: **sizing our current theoretical
and experimental understanding of the processes involved.
Though it may be tempting to interpret these delineations as
hard and absolute, they are only approximate in nature, meant
as a guide for the reader.
II. A SIMPLE CASE: NEUTRINO-LEPTON SCATTERING
A. Formalism: Kinematics We begin with the simplest of neutrino interactions,
neutrino-lepton scattering. As a purely leptonic interaction, neutrino-lepton scattering allows us to establish the formalism and terminology used through the paper, without introducing some of the complexity that often accompanies
neutrino-nuclear scattering. The general form of the twobody scattering process is governed by the dynamics of the
process encoded in the matrix elements and the phase space
available in the interaction. Figure 2 shows the tree-level
diagram of a neutrino-lepton charged current interaction,
known as inverse muon decay. A muon neutrino with fourmomentum p (aligned along the z direction) scatters in this
example with an electron with four-momentum pe , which is
at rest in the laboratory frame. This produces an outgoing
muon with four-momentum k and a scattered electron neutrino with four-momentum ke . In the laboratory frame, the
components of these quantities can be written as
~
p ¼ ðE ; p Þ;
pe ¼ ðme ; 0Þ; ~
k ¼ ðE ; k Þ;
~
ke ¼ ðEe ; ke Þ: Here we use the convention of the zeroth component corresponding to the energy portion of the energy-momentum
vector, with the usual energy-momentum relation E2 ¼
i
~
jkj2 þ m2 . From these four-vector quantities, it is often useful
i
i
to construct new variables which are invariant under Lorentz
transformations:
s ¼ ð p þ pe Þ 2 ðcenter of mass energyÞ; ð4-momentum transferÞ;
Q ¼ À q ¼ ð p À k Þ 2
pe Á q
y¼
ðinelasticityÞ:
pe Á p
2 2 In the case of two-body collisions between an incoming
neutrino and a (stationary) target lepton, the cross section is
given in general by (ℏ ¼ c ¼ 1) (Berestetskii, Lifshitz, and
Pitaevski, 1974), FIG. 1 (color online). Representative example of various neutrino sources across decades of energy. The electroweak cross section for
e eÀ ! e eÀ scattering on free electrons as a function of neutrino energy (for a massless neutrino) is shown for comparison. The peak at
"
"
1016 eV is due to the W À resonance, which we discuss in greater detail in Sec. VII.
Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 Joseph A. Formaggio and G. P. Zeller: From eV to EeV: Neutrino cross sections . . . j ¼ 2
W X 1309 L; lL :
" (5) ¼e;; The second type of interaction, known as the neutralcurrent (NC) exchange, is similar in character to the charged
current case. The leptonic neutral-current term, j , describes
Z
the exchange of the neutral boson, Z0 ,
X
"
j ¼ 2
g L L þ gf lL lL
L"
Z
L ¼e;; "
þ gf lR lR :
R
FIG. 2. Diagram of two-body scattering between an incoming
muon neutrino with four-momentum p and an e...

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