Introduction to Elementary Particle Physics
Physics 622, Fall 2002
First midterm exam. Due Monday, 11/11/02, 11:00 AM EST in class. It may be faxed
to 215-898 8512.
(1) Calculate the spin-averaged center of mass differential and total cross sections for
e+e−→π+π−to order α2. In particular, show that the cross section is proportional
to β3, where βis the pion center of mass velocity, and display the angular dependence.
Compare your result with the differential cross section for e+e−→µ+µ−computed in
class. Neglect the electron mass and strong interaction effects.
(2) The interaction Lagrangian for the coupling of the top quark tto decay into a bottom
quark band a W+, a massive charged spin-1 particle, is
L=gW †
µ¯
bγµ(1 −γ5)t+ H.C,
where g is a coupling constant.
(a) Calculate the differential decay rate dΓ/d cos θfor t→bW+in the trest frame, where
θis the angle between the tspin direction and the bmomentum. Sum over the band W+
spins. Neglect mbbut keep MW.
(b) Calculate the fraction of all tdecays that are into a longitudinal W. This should be
done for the total decay rate, not the differential rate.
Note: there was a typo in the posted notes for the sum over spin vectors for a massive
spin-1 particle. The correct formula is:
X
λ
µ(~p, λ)ν(~p, λ)∗=−gµν +pµpν
M2.
(3) Consider the SU (2)R×SU (2)Lσmodel, with quark doublets qLand qR, and four real
fields σand πi. They transform as
qL→ei~
βL·~τ/2qL
qR→ei~
βR·~τ/2qR
M→ei~
βR·~τ/2M e−i~
βL·~τ/2,
where Mis the 2×2 matrix σI +i~π ·~τ .
(a) Write down the most general renormalizable SU (2)R×SU (2)Linvariant interaction
and mass terms. They should be expressed in terms of the matrix M.
