Near the pion-production threshold

PHYSICS · Multi-part · Fermi University Physics · Particle Physics & Quantum Fields · Symmetry, groups and the quark model · Symmetry, groups and the quark model

Particle Physics, Ch Isospin: a symmetry used as bookkeeping

Problem

Near the pion-production threshold, two nucleons can fuse into a deuteron and a single pion:

pp \to d\pi^+, \qquad np \to d\pi^0.

The deuteron

d

is the bound state of a proton and a neutron and carries isospin

I=0

. The pion carries

I=1

, so the isospin of the final state

d\pi

is fixed at

I=1

, whatever the pion's charge. The two-nucleon initial states decompose in isospin as follows (quoted): the

pp

state is purely

I=1

, so it reaches the

I=1

final state with probability

1

; the

np

state is

I=1

with probability

\tfrac12

and

I=0

with probability

\tfrac12

, and only its

I=1

part can reach the

d\pi

channel. ((a)) Using these quoted probabilities, predict the ratio of reaction rates

pp\to d\pi^+ : np\to d\pi^0

. ((b)) In a bubble-chamber exposure under identical beam conditions,

850

examples of

pp\to d\pi^+

and

460

examples of

np\to d\pi^0

are counted. Compute the measured ratio and compare it with the prediction.

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