RevModPhys.84.1307

P and an electron at rest with four momentum pe see

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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 Š½s À ðme À m Þ2 Š (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  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 l LðRÞ 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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This document was uploaded on 09/28/2013.

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