introduction-weak-interaction-volume-one

5 .4 Form Factors .
Before we may consider form factors in the weak interaction, we mus t
first discuss them with reference to the electromagnetic interaction, in term s
of which they were originally formulated . For a proton, for example, unaffecte d
by strong interactions , the probability or cross-section for elastic electro n
scattering is given by the Dirac formula, which assumes the proton to be a
point with spin + and magnetic moment eV2Mp o
e4 coc2 (A /2) 2
d6 -
dS1 4p o sin 2 -`o sin 2i
.p
2 4 0 r, 2n 2 (1+ .2 tan 2 -2 ) ( . 4.1 )
where M P
is the proton mass, p is its initial three-momentum, and q is th e
o
four-momentum transfer between the electron and the proton during the scatterin g
process . The second term in (5 .4 .1) is due to magnetic scattering, and i s
absent from the Mott scattering cross-section from atomic nuclei . For a real
proton, we must take into account the virtual pairs of hadrons surrounding i t
due to the fact that it takes part in strong interactions, and also th e
anomalous magnetic moment over and above the Dirac prediction of 2 . Thus w e
define two form factors, the electric or charge form factor F, and the magneti c
form factor G . These two form factors are real functions of the four-momentu m
transfer squared, q2 . Thus, for the proto n
F(0) = 1 , (5 .4 .2 )
G(o) = jAp -v 2 .79 n .m . (5 .4 .3 )
where r p denotes the magnetic moment of the proton, and n .m . stands fo r
the units nuclear magnetons 5 . For the neutro n
F(0) = 0 , (5 .4.4 )
G(o) -1 .91 n .m . (5 .4 .5 )
We find that the corrected cross-section has the form
dd
r'p2+(o2/41„2)G2
_ dG
(( l
2 2 2
dSL
+ a2 x tan (A/2) 1 (5 .4 .6 )
\d~/ Mott L1 1 + (q2 /4M - ) / 4r.
(5 .4.6) is known as the Rosenbluth formula (12) . The important feature of it i s
that