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**Unformatted text preview: **st that:
p
n
F1 ðq2 Þ ¼ F1 ðq2 Þ À F1 ðq2 Þ;
p
n
F2 ðq2 Þ ¼ F2 ðq2 Þ À F2 ðq2 Þ:
n;p
n;p
Here F1 and F2 are known in the literature as the electromagnetic Dirac and Pauli form factors of the proton and
neutron, respectively. In the limit of zero-momentum transfer,
the Dirac form factors reduce to the charge of the nucleon,
while the Pauli form factors reduce to the nucleon’s magnetic
moments
1 if proton;
N
F1 ð0Þ ¼ qN ¼
0 if neutron;
8
< p À 1 if proton;
N
F2 ð0Þ ¼ N
: n
if neutron:
N Here qN is the nucleon charge, N is the nuclear magneton,
and p;n are the proton and neutron magnetic form factors.
To ascertain the q2 dependence of these form factors, it is
common to use the Sachs electric and magnetic form factors
N
N
and relate them back to F1 and F2 , 1317 fV ð0Þ F1 ð0Þ ¼ 1;
p À n
À 1 ’ 3:706;
fP ð0Þ F2 ð0Þ ¼
N
fA ð0Þ FA ð0Þ ¼ ÀgA ;
with fA ð0Þ=fV ð0Þ À1:2694 Æ 0:0028, as before
(Nakamura, K. et al., 2010).
The above represents an approach that works quite well
when the ﬁnal states are simple, for example, when one is
dealing with a few-nucleon system with no strong bound
states or when the momentum exchange is very high (see
the next section on quasielastic interactions).
Seminal articles on neutrino (and electron) scattering can
be found in earlier review articles by Donnelly et al. (1974),
Donnelly and Walecka (1975), Donnelly and Peccei (1979),
and Peccei and Donnelly (1979). Peccei and Donnelly equate
the relevant form factors to those measured in ðe; e0 Þ scattering (Drell and Walecka, 1964; de Forest Jr. and Walecka,
1966), removing some of the model dependence and q2
restrictions prevalent in certain techniques. This approach is
not entirely model independent, as certain axial form factors
are not completely accessible via electron scattering. This
technique has been expanded in describing neutrino scattering at much higher energy scales (Amaro et al., 2005, 2007)
with the recent realization that added nuclear effects come
into play (Amaro et al., 2011b). N
N
GN ðq2 Þ ¼ F1 ðq2 Þ À F2 q2 Þ;
E E. Estimating fermi and Gamow-Teller strengths N
N
GN ðq2 Þ ¼ F1 ðq2 Þ þ F2 ðq2 Þ;
M with Àq2 =4MN and
Gp ðq2 Þ ¼ GD ðq2 Þ;
Gn ðq2 Þ ¼ 0;
E
E
p
GD ðq2 Þ;
Gn ðq2 Þ ¼ n GD ðq2 Þ:
Gp ðq2 Þ ¼
M
M
N
N
Here GD ðq2 Þ is a dipole function determined by the charge
radius of the nucleon. Empirically, the dipole term can be
written as
q2 À1
;
(42)
G D ð q2 Þ ¼ 1 À 2
mV
where mV ’ 0:84 MeV.
We now turn our attention to the axial portion of the
current, where the terms FA ðq2 Þ and FP ðq2 Þ play a role. For
FA ðq2 Þ, one also often assumes a dipolelike behavior, but
with a different coupling and axial mass term (mA )
FA ðq2 Þ ¼ ÀgA GA ðq2 Þ; GA ðq2 Þ ¼ 1
:
ð1 À q2 =m2 Þ2
A The Goldberger-Treiman relation allows one to also relate
the pseudoscalar contribution in terms of the axial term as
well; typically
F...

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