6.2 Problems 327
(1−cosθ)/c^2 , whereEis the total initial electron’s energy in the lab system.
State when this approximation is justified.6.78 A neutral unstable particle decays intoπ+ andπ−, 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 massMdecays in flight intom 1 and anotherm 2 ;m 1 has
momentump 1 and total energyE 1 , where asm 2 has momentump 2 and total
energyE 2 .p 1 andp 2 make an angleθ. Show that
E 1 E 2 −p 1 p 2 cosθ=invariant=^12 [M^2 −m 12 −m 22 ]
6.80 The Mandelstam variabless,t, anduare defined for the reactionA+B→
C+D,by
s=(PA+PB)^2 /c^2 ,t=(PA−PC)^2 /c^2 ,u=(PA−PD)^2 /c^2
wherePA,PB,PC,PDare the relevant energy-momentum four vectors. Show
that
s+t+u=
∑
mj^2 (j=A,B,C,D)6.81 In Problem 6.80 show for the elastic scatteringt=− 2 p^2 (1−cosθ)/c^2 where
p=|p|.pis the center of mass momentum of particle andθis its scattering
angle in the CMS.
6.82 A neutral pion undergoes radioactive decay into twoγ-rays. Obtain the
expression for the laboratory angle between the direction of theγ-rays,
and find the minimum value for the angle when the pion energy is 10 GeV
(mπ= 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 resid-
ual particles was 1.22 GeV. What is the rest energy ofω^0 in MeV?
6.84 A particle of rest massm 1 and velocityv 1 collides with a particle of mass
m 2 at rest after which the two particles coalesce. Show that the massM
and velocityvof the composite particle are related byM^2 =m 12 +m 22 +
2 m 1 m 2 /
√
1 −v^2 /c^26.85 Show that for the decay in flight of aΛ-hyperon into a proton and a pion with
Laboratory momentaPpandPπrespectively, theQvalue can be calculated
from
Q=(mp^2 +mπ^2 + 2 EpEπ− 2 PpPπcosθ)^1 /^2 −(mp+mπ)
whereθis the angle betweenPpandPπin the Laboratory system andEis
the total relativistic energy
[University of Dublin 1967]
6.86 Two particles are moving with relativistic velocities in directions at right
angles, they have momentap 1 andp 2 and total energiesE 1 andE 2 .Ifthey