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**Unformatted text preview: **lectron at rest
with four-momentum pe . See text for details. d
1
jM2 j
¼
dq2 16 ½s À ðme þ m Þ2 ½s À ðme À m Þ2 (1) which, in the context of very small neutrino masses,
simpliﬁes to
d
1
jM2 j
¼
:
2
16 ðs À m2 Þ2
dq
e (2) Here M is the matrix element associated with our particular
interaction (Fig. 2). In the laboratory frame, it is always
possible to express the cross section in alternative ways by
making use of the appropriate Jacobian. For example, to
determine the cross section as a function of the muon’s
scattering angle , the Jacobian is given by
dq2
~~
¼ 2jp jjk j;
d cos (3) while the Jacobian written in terms of the fraction of the
neutrino energy imparted to the outgoing lepton energy (y) is
given by
dq2
¼ 2me E :
dy (4) Pending on what one is interested in studying, the differential cross sections can be recast to highlight a particular
dependence or behavior.
B. Formalism: Matrix elements The full description of the interaction is encoded within the
matrix element M. The standard model readily provides a
prescription to describe neutrino interactions via the leptonic
charged current and neutral current in the weak interaction
Lagrangian. Within the framework of the standard model, a
variety of neutrino interactions are readily described
(Weinberg, 1967). These interactions all fall within the context of the general gauge theory of SUð2ÞL Â Uð1ÞY . This
divides the types of possible interactions for neutrinos into
three broad categories. The ﬁrst is mediated by the exchange
of a charged W boson, otherwise known as a charged current
(CC) exchange. The leptonic charged weak current j is
W
given by the form
Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 (6) Here LðRÞ and lLðRÞ correspond to the left (right) neutral
and charged leptonic ﬁelds, while g , gf , and gf represent
L
L
R
the fermion left- and right-handed couplings (for a list of
these values, see Table I). Though the charged leptonic ﬁelds
are of a deﬁnite mass eigenstate, this is not necessarily so for
the neutrino ﬁelds, giving rise to the well-known phenomena
of neutrino oscillations.
Historically, the neutrino-lepton charged current and
neutral-current interactions have been used to study the nature of the weak force in great detail. We now return to the
case of calculating the charged and neutral-current reactions.
These previously deﬁned components enter directly into the
Lagrangian via their coupling to the heavy gauge bosons,
W Æ and Z0 ,
g
y
LCC ¼ À pﬃﬃﬃ ðj W þ j;y W Þ;
W
W
22 (7) g
j Z :
2 cosW Z (8) LNC ¼ À Here W and Z represent the heavy gauge boson ﬁeld, g is
the coupling constant while W is the weak mixing angle. It is
possible to represent these exchanges with the use of
Feynman diagrams, as is shown in Fig. 3. Using this formalism, it is possible to articulate all neutrino interactions
(’t Hooft, 1971) within this simple framework.
We begin by looking at one of t...

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