Subatomic Physics

5.3. Mass Measurements 91
Earlier, in Eq. (2.29), we defined the total or invariant mass of a system of
particles. Applying this definition to the two pions and using the notation defined
in Fig. 5.10, the invariant mass m12 of the two pions is
If a magnetic field is applied to the bubble
chamber, the momenta of the two charged
pions can be determined. The energy can
be found from their range (Fig. 3.6) or
their ionization. For every observed pion
pair, the invariant mass m12 can then
be computed from Eq. (5.25). If the reaction
proceeds according to Fig. 5.9(a),
with no correlation between the two pions
and the neutron, they will share energy
and momentum statistically. The
number of pion pairs with a certain invariant
mass, N(m12), can be calculated
in a straightforward way, and the result
is called a phase-space spectrum. (Phase
space will be discussed in Section 10.2.)
It is sketched in Fig. 5.11.
m12 = 1
c 2 [(E1 + E2) 2 − (p 1 + p 2) 2 c 2 ] 1/2 . (5.25)
Figure 5.10: Energies and momenta involved
in the decay of the ρ 0 .
If, on the other hand, the reaction proceeds via the production of a ρ, energy
and momentum conservation demand
The mass of the rho is given by Eq. (1.2) as
or, with Eqs. (5.25) and (5.26), as
Eρ = E1 + E2, p ρ = p 1 + p 2. (5.26)
mρ = 1
c 2 (E2 ρ − p 2 ρc 2 ) 1/2 ;
mρ = m12. (5.27)
If the pions result from the decay of a particle, their invariant mass will be a
constant and will be equal to the mass of the decaying particle. Figure 5.12 shows
an early result, the invariant mass spectrum of pion pairs produced in the reaction
Eq. (5.24) with pions of momentum 1.89 GeV/c. A broad peak at an invariant mass
of 765 MeV/c 2 is unmistakable. The particle giving rise to this peak is called the
rho. Even though it lives only about 6 × 10 −24 sec, its existence is well established
and its mass known.