Ignore the left and right handed leptonic couplings

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Unformatted text preview: eptonic couplings. As a result, one obtains a more complex expression for the relevant matrix element [for a review, see Adams et al. (2009)] pffiffiffi " MNC ¼ À 2GF f½l  ðg À g 5 Þl Š V A "  ½e  ðgf À gf 5 ÞeŠg: V A (18) We have expressed the strength of the coupling in terms of the vector and axial-vector coupling constants (gV and gA , respectively). An equivalent formulation can be constructed using left- and right-handed couplings pffiffiffi MNC ¼ À 2GF f½g l  ð1 À 5 Þl L" þ g l  ð1 þ 5 Þl Š R"  f½gf e  ð1 À 5 Þe þ gf e  ð1 þ 5 ÞeŠg: L" R" (19) The relation between the coupling constants are dictated by the standard model, pffiffiffiffi g ¼ 0 ; g ¼ ðþ1Þ; L R 2 pffiffiffiffi f pffiffiffiffi f f sin2  Þ; gf ¼ ðÀQf sin2 W Þ; gL ¼ ðI3 À Q W R or, equivalently, pffiffiffiffi g ¼ g þ g ¼ ðþ1Þ; V L R 2 pffiffiffiffi g ¼ g À g ¼ ðþ1Þ; L R A 2 pffiffiffiffi f f f f gV ¼ gL þ gR ¼ ðI3 À 2Qf sin2 W Þ; pffiffiffiffi f gf ¼ gf À gf ¼ ðI3 Þ: L R A f Here I3 and Qf are the weak isospin and electromagnetic charge of the target lepton,  is the relative coupling strength between charged and neutral-current interaction (at tree level,   1), and W is the Weinberg mixing angle. The standard model defines a relation between the electroweak couplings and gauge boson masses MW and MZ (15) has a different dependence on y than its neutrino counterpart, although the matrix elements are equivalent dðe e ! l lÞ 2me G2 E " " F ¼ dy    ðm2 À m2 Þð1 À yÞ e  ð1 À yÞ2 À l : (16) 2me E Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 Upon integration, the total antineutrino cross section is approximately a factor of 3 lower than the neutrino cross section. The suppression comes entirely from helicity considerations. Having just completed a charged current example, we now turn our attention to a pure neutral current exchange, such as witnessed in the reaction sin2 W  1 À 2 MW : 2 MZ (20) In the observable cross section for the neutral-current reactions highlighted above, we find that they are directly sensitive to the left- and right-handed couplings. In the literature, the cross section is often expressed in terms of their vector and axial-vector currents, Joseph A. Formaggio and G. P. Zeller: From eV to EeV: Neutrino cross sections . . . gV  ð2g gf Þ; LV gA  ð2g gf Þ; LA  dðl e ! l eÞ me G2 E F ¼ ðgV þ gA Þ2 þ ðgV À gA Þ2 dy 2  my  ð1 À yÞ2 À ðg2 À g2 Þ e ; V A E  2 dðl e ! l eÞ me GF E " " ¼ ðgV À gA Þ2 þ ðgV þ gA Þ2 dy 2  my  ð1 À yÞ2 À ðg2 À g2 Þ e : V A E Though we have limited ourselves to discussing neutrinolepton scattering, the rules governing the coupling strengths are predetermined by the standard model and can be used to describe neutrino-quark interactions as well. A full list of the different possible coupling s...
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This document was uploaded on 09/28/2013.

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