6.2 Problems 329
6.97 Assume the decayK^0 →π++π−. Calculate the mass of the primary particle
if the momentum of each of the secondary particles is 3× 108 eV and the
angle between the tracks is 70◦
[University of Durham 1960]
6.98 Neutral pions of fixed energy decay in flight into twoγ-rays. Show that the
velocity of pion is given by
β=(Emax−Emin)/(Emax+Emin)
whereEis theγ-ray energy in the laboratory
6.99 In Problem 6.98 show that the rest mass energy ofπ^0 is given bymc^2 =
2(EmaxEmin)^1 /^26.100 In Problem 6.98 show that the energy distribution ofγ-rays in the laboratory
is uniform under the assumption thatγ-rays are emitted isotropically in the
rest system ofπ^0
6.101 In Problem 6.98 show that the angular distribution ofγ-rays in the laboratory
is given by
I(θ)= 1 / 4 πγ^2 (1−βcosθ)^2
6.102 In Problem 6.98 show that the locus of the tip of the momentum vector is an
ellipse
6.103 In Problem 6.98 show that in a given decay the angleφbetween twoγ-rays
is given by
sin(φ/2)=mc^2 /2(E 1 E 2 )^1 /^2
6.104 In Problem 6.98 show that the minimum angle between the twoγ-rays is
given by
φmin= 2 mc^2 /Eπ
6.105 In Problem 6.98 find an expression for the disparityD(the ratio of energies)
of theγ-rays and show thatD>3 in half the decays andD>7 in one
quarter of them
6.106 In the interactionπ−+p→ K∗(890)+Y 0 ∗(1,800) at pion momentum
10 GeV/c ,K∗is produced at an angleθin the lab system. Calculate the
maximum valueθm,givenmπ= 0 .140 GeV/c^2 andmp= 0 .940 GeV/c^2
6.107 A particle of massm 1 travelling with a velocityv=βccollides elastically
with the particlem 2 at rest. The scattering angles ofm 1 in the LS and CMS
areθandθ∗. Show that
(a)γc=(γ+ν)/(1+ 2 γν+ν^2 )^1 /^2
(b)γ∗=(γ+ 1 /γ)/
√
(1+ 2 γ/ν+ 1 /ν^2 )
(c) tanθ=sinθ∗/γc(cosθ∗+βc/β∗)
(d) tanθ∗=sinθ/γc(cosθ−βc/β∗)
where βc is the CMS velocity, β∗c is the velocity of m 1 in CMS,
γc=(1−βc^2 )−^1 /^2 ,γ∗=(1−β∗^2 )−^1 /^2 ,ν=m 2 /m 1