*This preview shows
page 1. Sign up to
view the full content.*

**Unformatted text preview: **he simplest manifestations
of the above formalism, where the reaction is a pure charged
current interaction
l þ eÀ ! l À þ e ðl ¼ or Þ: (9) The corresponding tree-level amplitude can be calculated
from the above expressions. In the case of l þ e (sometimes
known as inverse muon or inverse tau decays) one ﬁnds
GF "
"
MCC ¼ À pﬃﬃﬃ f½l
ð1 À 5 Þl ½e ð1 À 5 Þeg:
2
(10)
TABLE I. Values for the gV (vector), gA (axial), gL (left), and gR
(right) coupling constants for the known fermion ﬁelds.
Fermion
e , ,
e , ,
u, c, t
d, s, b gf
L gf
R gf
V gf
A þ1
2
À 1 þ sin2 W
2
þ 1 À 2 sin2 W
2
3
À 1 þ 1 sin2 W
2
3 0
þsin2 W
À 2 sin2 W
3
þ 1 sin2 W
3 þ1
2
À 1 þ 2sin2 W
2
þ 1 À 4 sin2 W
2
3
À 1 þ 2 sin2 W
2
3 þ1
2
À1
2
þ1
2
À1
2 Joseph A. Formaggio and G. P. Zeller: From eV to EeV: Neutrino cross sections . . . 1310 FIG. 3. Feynman tree-level diagram for charged and neutralcurrent components of e þ eÀ ! e þ eÀ scattering. Here, and in all future cases unless speciﬁed, we assume that
the four-momentum of the intermediate boson is much
2
smaller than its mass (i.e., jq2 j ( MW;Z ) such that propagator
effects can be ignored. In this approximation, the coupling
strength is then dictated primarily by the Fermi constant GF ,
g2
GF ¼ pﬃﬃﬃ 2 ¼ 1:1663 788ð7Þ Â 10À5 GeVÀ2 :
4 2M W (11) By summing over all polarization and spin states, and
integrating over all unobserved momenta, one attains the
differential cross section with respect to the fractional energy
imparted to the outgoing lepton,
dðl e ! e lÞ 2me G2 E
m2 À m2
e
F
¼
1À l
;
(12)
dy
2m e E
where E is the energy of the incident neutrino and me and ml
are the masses of the electron and outgoing lepton,
respectively. The dimensionless inelasticity parameter y reﬂects the kinetic energy of the outgoing lepton, which in this
particular example is y ¼ ½El À ðm2 þ m2 Þ=2me =E . The
e
l
limits of y are such that
0 y ymax ¼ 1 À m2
l
:
2me E þ m2
e (13) Note that in this derivation, we have neglected the contribution from neutrino masses, which in this context is too small
to be observed kinematically. The above cross section has a
threshold energy imposed by the kinematics of the system,
E ! ð m 2 À m 2 Þ =2m e .
e
l
In the case where E ) Ethresh , integration of the above
expression yields a simple expression for the total neutrino
cross section as a function of neutrino energy,
’ 2me G2 E G2 s
F
¼ F;
(14) where s is the center-of-mass energy of the collision. Note
that the neutrino cross section grows linearly with energy.
Because of the different available spin states, the equivalent expression for the inverse lepton decay of antineutrinos,
e þ e ! l þ l
"
" ðl ¼ or Þ; "
l þ e ! l þ e
" ðl ¼ or Þ: (17) In the instance of a pure neutral-current interaction, we are
no longer at liberty to ignore the left- and right-handed
l...

View Full
Document