P640 – Fall 2003 – Subatomic Physics I
Problem Set #3
Due Wednesday, 8 Oct (postponed from Oct. 1).
1. (Lorentz Invariance) Show that d3 p~/2E = d3 p~ 0 /2E 0 is invariant under Lorentz boosts Λ.
2. (Phase space, Halzen & Martin 4.2) Consider a process a + b → c + d. A differential cross
section can be written
|M|2
dLips,
dσ =
F
where dLips is the Lorentz invariant phase space factor and the incident flux in the laboratory
is
F = |~va |(2Ea )(2Eb ).
Show that
dLips =
√
1 pf
√ dΩ, F = 4pi s,
2
4π 4 s
and hence that the differential cross section is
dσ
1 pf
=
|M|2 ,
dΩ
64π 2 s pi
where dΩ is the element of solid angle about p~c , s = (Ea + Eb )2 , |~
pa | = |~
pb | = pi and
|~
pc | = |~
pd | = p f .
3. (Spinless Scattering, Halzen & Martin 4.3) For high-energy “spinless” electron-muon scattering, show that
dσ
α2 3 + cos θ 2
(CM ) =
,
dΩ
4s 1 − cos θ
where θ is the scattering angle in the center of mass frame and α = e2 /4π. Plot as a function
of cos θ.
4. (Phase space ranges, Dalitz Plots)
(a) Consider the process a + b → c + d + b0 with masses ma , mb ... At fixed beam momentum
pL , find the physical region for the variable mcd = scd = (pc + pd )2 , i.e., find the range
of mcd where scattering is kinematically allowed.
(b) Find the range of energies allowed for the ν̄ in the decay n → pe− ν̄e . Put in numbers.
(c) Sketch the allowed region/boundary (with numbers) of Dalitz plots for the decays ω →
3π and η → π + π − γ.
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5. (More phase space, matrix element practice) We will show later that the spin-averaged matrix
element for the decay π → µ− ν̄µ is
|M|2 = 4G2F fπ2 m2µ (p · k)
where GF is Fermi’s constant, fπ is the pion “form factor”, and p and k are the four-vectors
of the µ and ν̄ respectively.
Use this show that the decay rate for pions is
m2µ
GF 2
Γ=
fπ mπ m2µ 1 − 2
8π
mπ
!
.
Look up the actual lifetime and estimate the value of fπ .
6. (Projection Operators, Dirac Equation) Show that the operator Λ± = (± 6 p + m)/2m is a
projection operator that projects over positive and negative energy states or else projects out
the ± frequencies of the plane wave solutions of the Dirac equation.
(a) First verify that the operator Λ± = (± 6 p + m)/(2m) is a projection operator on the
Dirac wavefunctions, i.e., show Λ± Λ± = Λ± .
(b) Then show that if Ψ = aue−ip·x + bveip·x , then Λ+ Ψ = aue−ip·x and Λ− Ψ = bveip·x
projects over the states.
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