Begin by looking at one of the simplest

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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 Š½e  ð1 À 5 ÞeŠg: 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 Š=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...
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