**Unformatted text preview: **scattering cross section, early experiments relied heavily on the formalism ﬁrst written down in
Llewellyn-Smith (1972). In the case of QE scattering off
free nucleons, the QE differential cross section can be
expressed as
d
G2 M 2
sÀu
ðs À uÞ2
¼ F 2 AÆ
Bþ
C;
(57)
dQ2
M2
8E
M4
where ðÀÞþ refers to (anti)neutrino scattering, GF is the
Fermi coupling constant, Q2 is the squared four-momentum
transfer (Q2 ¼ Àq2 > 0), M is the nucleon mass, m is
the lepton mass, E is the incident neutrino energy, and
s À u ¼ 4ME À Q2 À m2 . The factors A, B, and C are
functions of the familiar vector (F1 and F2 ), axial-vector
(FA ), and pseudoscalar (FP ) form factors of the nucleon
m2 þ Q2
2
2
2
A¼
ð1 þ ÞFA À ð1 À ÞF1 þ ð1 À ÞF2
M2
m2
þ 4F1 F2 À
ðF1 þ F2 Þ2 þ ðFA þ 2FP Þ2
4M 2
2
Q
2
(58)
À
þ 4 FP ;
M2
B¼ Q2
FA ðF1 þ F2 Þ;
M2 12
2
2
C ¼ ðFA þ F1 þ F2 Þ;
4 (59) (60) where ¼ Q2 =4M2 . Much of these equations should be
familiar from Sec. IV. Historically, this formalism was used
to analyze neutrino QE scattering data on deuterium, subject
to minor modiﬁcations for nuclear effects. In this way, experiments studying neutrino QE scattering could in principle
measure the vector, axial-vector, and pseudoscalar form factors given that the weak hadronic current contains all three of
these components. In practice, the pseudoscalar contribution
was typically neglected in the analysis of QE scattering as
it enters the cross section multiplied by m2 =M2 . Using CVC,
the vector form factors could be obtained from electron
scattering, thus leaving the neutrino experiments to measure
the axial-vector form factor of the nucleon. For the axialvector form factor, it was (and still is) customary to assume a
dipole form
gA
;
(61)
FA ðQ2 Þ ¼
2
ð1 þ Q2 =MA Þ2
which depends on two empirical parameters: the value of the
axial-vector form factor at Q2 ¼ 0, gA ¼ FA ð0Þ ¼ 1:2694 Æ
0:0028 (Nakamura, K. et al., 2010), and an ‘‘axial mass’’ MA .
With the vector form factors under control from electron
scattering and gA determined with high precision from nuclear beta decay, measurement of the axial-vector form
factor (and hence MA ) became the focus of the earliest A value of MA ¼ 1:014 Æ 0:014 GeV was obtained from a
recent global ﬁt to the deuterium data in Bodek et al. (2007),
while a consistent value of MA ¼ 0:999 Æ 0:011 GeV was obtained
in Kuzmin et al. (2008) from a ﬁt that additionally includes some of
the early heavy target data.
9 Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 1325 2.5
ANL, PRD 16, 3103 (1977), D GGM, NC A38, 260 (1977), C H8CF3 Br
3 2 BEBC, NP B343, 285 (1990), D 2 MiniBooNE, PRD 81, 092005 (2010), C BNL, PRD 23, 2499 (1981), D NOMAD, EPJ C63, 355 (2009) 2 2 FNAL, PRD 28, 436 (1983), D
NUANCE (M =1.0 GeV)
A 2 Serpukhov, ZP A320, 625 (1985), Al
SKAT, ZP C45, 551 (1990), CF Br
3 1.5 1 µ n 0.5 ( measurements of neutrino QE scattering. Values of MA ra...

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