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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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