RevModPhys.84.1307

Pseudoscalar contribution in terms of the axial term

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Unformatted text preview: P ðq2 Þ ¼ 2 2M N FA ðq2 Þ; À q2 m2  where m is the pion mass. In the limit that the momentum exchange is small (such as in neutron decay or inverse beta decay), the form factors reduce to the constants defined previously in this section Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 For very small momentum transfers, the relevant impact of these various form factors take a back seat to the individual final states accessible to the system. Under this scheme, it is customary to divide into two general groupings: the Fermi transitions [associated with fV ð0Þ] and the Gamow-Teller transitions [associated with fA ð0Þ]. In doing so, the cross section can be rewritten as  d G2 jV j2 FðZf ; Ee ÞEe pe ’ F ud fF ðq2 ÞjMF j2 d cos 2  2 Þ 1 jM j2 þ interference terms ; þ fGT ðq 3 GT (43) where jMF j2 ¼ 2 A X X 1 hJf ;Mf j Æ ðkÞeiqÁrk jJi ;Mi i ; 2Ji þ 1 Mf ;Mi k¼1 (44) jMGT j2 ¼ 1 2J i þ 1 2 A X X hJf ;Mf j Æ ðkÞðkÞeiqÁrk jJi ;Mi i : Â M ;M k¼1 f i (45) We note that we have altered our notation slightly to denote explicit summation over individual accessible nuclear states. Equations (44) and (45) show explicitly the summation across Joseph A. Formaggio and G. P. Zeller: From eV to EeV: Neutrino cross sections . . . 1318 both initial (jJi ; Mi i) and final (jJf ; Mf i) spin states. In general, the terms associated with the Fermi transitions fF ðq2 Þ and the Gamow-Teller transitions fGT ðq2 Þ are nontrivial combinations of the various form factors described previously [see also Kuramoto et al., (1990)]. However, as one approaches zero momentum, we can immediately connect the relevant Fermi and Gamow-Teller amplitudes directly to decay M ¼ fV ð0Þ2 jMF j2 þ fA ð0Þ2 1jMGT j2 ; 3 jMF j2 ¼ (46) 2 A X X 1 h Jf ; M f j Æ ðkÞjJi ; Mi i ; 2Ji þ 1 Mf ;Mi k ¼1 (47) 2 A X X 1 hJf ;Mf j Æ ðkÞðkÞjJi ;Mi i ; jMGT j2 ¼ 2Ji þ 1 M ;M k¼1 f i (48) and 23 ln2 : (49) G2 jVud j2 m5 M e F Hence, in the most simplistic model, the total charged current cross section can be calculated directly from evaluating the appropriate decay reaction and correcting for the spin of the system Ft ¼ ¼ 2J f þ 1 22 ln2 : pe Ee FðEe ; Zf Þ 5 2J i þ 1 me F t (50) Further information on the relevant coefficients can also be obtained by studying muon capture on the nucleus of interest (Luyten, Rood, and Tolhoek, 1963; Nguyen, 1975; Ricci and Truhlik, 2010), or by imposing sum rules on the total strength of the interaction.6 Another extremely powerful technique in helping discern the contributions to the neutrino cross section, particularly for Gamow-Teller transitions, has been through ðp; nÞ scattering. Unlike its decay counterpart, ðp; nÞ scattering does not suffer from being limited to a particular momentum band; in principle, a wider band is accessible via this channel. Since the processes involved for ðp; nÞ scattering are essentially the same as those for the weak interaction in general, one can obtain an...
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