6.2 Problems
327
(1 − cos θ )/c
2 , where E is the total initial electron’s energy in the lab system.
State when this approximation is justified.
6.78 A neutral unstable particle decays into π
+ and π
− , each of which has a
momentum 530 MeV/c. The angle between the two pions is 90
◦ . Calculate
the mass of the unstable particle.
6.79 If a particle of mass M decays in flight into m 1 and another m 2 ; m 1 has
momentum p 1 and total energy E 1 , where as m 2 has momentum p 2 and total
energy E 2 . p 1 and p 2 make an angle θ. Show that
E 1 E 2 − p 1 p 2 cos θ = invariant =
1
2
[M
2
− m 1
2
− m 2
2 ]
6.80 The Mandelstam variables s, t, and u are defined for the reaction A + B →
C + D, by
s = (P A + P B )
2
/c
2
, t = (P A − P C )
2
/c
2
, u = (P A − P D )
2
/c
2
where P A , P B , P C , P D are the relevant energy-momentum four vectors. Show
that
s + t + u =
m j
2 ( j = A, B, C, D)
6.81 In Problem 6.80 show for the elastic scattering t = −2 p
2 (1 − cos θ )/c
2 where
p = |p|.p is the center of mass momentum of particle and θ is its scattering
angle in the CMS.
6.82 A neutral pion undergoes radioactive decay into two γ -rays. Obtain the
expression for the laboratory angle between the direction of the γ -rays,
and find the minimum value for the angle when the pion energy is 10 GeV
(m π = 0.14 GeV)
[University of Bristol 1965]
6.83 A bubble chamber event was identified in the reaction
p
−
+ p → π
+
+ π
−
+ ω
0
The total energy available was 2.29 GeV while the kinetic energy of the residual particles was 1.22 GeV. What is the rest energy of ω
0 in MeV?
6.84 A particle of rest mass m 1 and velocity v 1 collides with a particle of mass
m 2 at rest after which the two particles coalesce. Show that the mass M
and velocity v of the composite particle are related by M
2
= m 1
2
+ m 2
2
+
2m 1 m 2 /
1 − v 2 /c 2
6.85 Show that for the decay in flight of a Λ-hyperon into a proton and a pion with
Laboratory momenta P p and P π respectively, the Q value can be calculated
from
Q = (m p
2
+ m π
2
+ 2E p E π − 2P p P π cos θ)
1/2
− (m p + m π )
where θ is the angle between P p and P π in the Laboratory system and E is
the total relativistic energy
[University of Dublin 1967]
6.86 Two particles are moving with relativistic velocities in directions at right
angles, they have momenta p 1 and p 2 and total energies E 1 and E 2 . If they
327
(1 − cos θ )/c
2 , where E is the total initial electron’s energy in the lab system.
State when this approximation is justified.
6.78 A neutral unstable particle decays into π
+ and π
− , each of which has a
momentum 530 MeV/c. The angle between the two pions is 90
◦ . Calculate
the mass of the unstable particle.
6.79 If a particle of mass M decays in flight into m 1 and another m 2 ; m 1 has
momentum p 1 and total energy E 1 , where as m 2 has momentum p 2 and total
energy E 2 . p 1 and p 2 make an angle θ. Show that
E 1 E 2 − p 1 p 2 cos θ = invariant =
1
2
[M
2
− m 1
2
− m 2
2 ]
6.80 The Mandelstam variables s, t, and u are defined for the reaction A + B →
C + D, by
s = (P A + P B )
2
/c
2
, t = (P A − P C )
2
/c
2
, u = (P A − P D )
2
/c
2
where P A , P B , P C , P D are the relevant energy-momentum four vectors. Show
that
s + t + u =
m j
2 ( j = A, B, C, D)
6.81 In Problem 6.80 show for the elastic scattering t = −2 p
2 (1 − cos θ )/c
2 where
p = |p|.p is the center of mass momentum of particle and θ is its scattering
angle in the CMS.
6.82 A neutral pion undergoes radioactive decay into two γ -rays. Obtain the
expression for the laboratory angle between the direction of the γ -rays,
and find the minimum value for the angle when the pion energy is 10 GeV
(m π = 0.14 GeV)
[University of Bristol 1965]
6.83 A bubble chamber event was identified in the reaction
p
−
+ p → π
+
+ π
−
+ ω
0
The total energy available was 2.29 GeV while the kinetic energy of the residual particles was 1.22 GeV. What is the rest energy of ω
0 in MeV?
6.84 A particle of rest mass m 1 and velocity v 1 collides with a particle of mass
m 2 at rest after which the two particles coalesce. Show that the mass M
and velocity v of the composite particle are related by M
2
= m 1
2
+ m 2
2
+
2m 1 m 2 /
1 − v 2 /c 2
6.85 Show that for the decay in flight of a Λ-hyperon into a proton and a pion with
Laboratory momenta P p and P π respectively, the Q value can be calculated
from
Q = (m p
2
+ m π
2
+ 2E p E π − 2P p P π cos θ)
1/2
− (m p + m π )
where θ is the angle between P p and P π in the Laboratory system and E is
the total relativistic energy
[University of Dublin 1967]
6.86 Two particles are moving with relativistic velocities in directions at right
angles, they have momenta p 1 and p 2 and total energies E 1 and E 2 . If they