where m = electron mass,
and taking into account for
difference in coupling constants, estimate a value for alpha_s
(strong coupling constant).
Bound states of top and anti-top quark :
The interaction between a top quark (t) and a "nearby"
anti-top quark (t-bar) is effectively described by a Coulomb
potential
V(r) = - A / r,
where r is the interqurk distance and A is about 0.1.
The mass of the top quark is measured to be about 174 GeV.
Question
: Consider the lowest-lying tt-bar state
which is bound by this force. Assuming nonrelativistic
dynamics, estimate the size of this state and the
average speed of the quarks in this state.
Question : The total width of the top quark is about 1.5
GeV. Calculate the lifetime of the top quark.
Using the results of the previous question, show that
the t and t-bar do not have enough time to execute
one complete Bohr orbit about each other before
one of them decays, and thus it is very difficult to
produce a bound state.
Question :
Given the top quark lifetime, about how many tt-bar
pairs would have to be produced before one such pair
would be expected to bind ?
Pion Decay :
Pion decay is possible for two different modes,
pi+ -> mu+ nu_mu
and
pi+ -> e+ nu_e.
Question : explain the dramatic difference in the branching
ratio for the two modes.
Hint
Helicity is a well-defined, Lorentz-invariant
quantity for a massless particle. That is, solutions
of the Dirac eq. are pure helicity eigenstates.
Operators P_R = 1/2 (1 + sigma dot p / E),
P_L = 1/2 (1 - sigma dot p / E), acting on
two-component spinors will project out states
of particular helicity from an arbitrary
superposition of positive and negative helicity
states.
However, for a non-zero mass particle, solutions
are not pure helicity engenstates, but mixture of
left-handed (LH) and right-handed (RH) fermions.
(In relativisitic limit, mass can be considered to
be zero, approaching helicity eigenstates.)
Operators P_R = 1/2 (1 + gamma_5), and
P_L = 1/2 (1 - gamma_5) project out
a state of polarization P = +v/c and P =
-v/c.
P can be written as (N_R - N_L) / (N_R +
N_L) where N_R = the number of particles with
RH helicity, N_L = the number of particles with
LH helicity.
Leptons participating in weak interactions are
longitudinally polarised:
P = - v/c for leptons
P = + v/c for anti-leptons
First get the probabilities that the charged lepton
will emerge with the two helicities.
Secondly consider the angular momentum conservation.
Note that pions are pseudo-scalar particles.
Now, take into account for the phase-space factor
p^2 dp / dE0,
where p is the momentum of either lepton in
the cm frame, and E0 = the total energy in the cms.