An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 5.4

Elementary Processes of Quantum Electrodynamics — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · qed-positronium-lifetimes

Problem

(a) Compute the amplitude

\mathcal{M}

for

e^+e^-

annihilation into 2 photons in the extreme nonrelativistic limit (i.e., keep only the term proportional to zero powers of the electron and positron 3-momentum). Use this result, together with our formalism for fermion-antifermion bound states, to compute the rate of annihilation of the

1S

states of positronium into 2 photons. You should find that the spin-1 states of positronium do not annihilate into 2 photons, confirming the symmetry argument of Problem 3.8. For the spin-0 state of positronium, you should find a result proportional to the square of the

1S

wavefunction at the origin. Inserting the value of this wavefunction from nonrelativistic quantum mechanics, you should find

\frac{1}{\tau} = \Gamma = \frac{\alpha^5 m_e}{2} \approx 8.03 \times 10^9 \;\text{sec}^{-1}.

A recent measurement gives

\Gamma = 7.994 \pm .011 \;\text{nsec}^{-1}

; the

0.5\%

discrepancy is accounted for by radiative corrections. (b) Computing the decay rates of higher-

l

positronium states is somewhat more difficult; in the rest of this problem, we will consider the case

l = 1

. First, work out the terms in the

e^+e^- \to 2\gamma

amplitude proportional to one power of the electron and positron 3-momentum. (For simplicity, work in the center-of-mass frame.) Since

\int \frac{d^3p}{(2\pi)^3} p^i \psi(\mathbf{p}) = i \frac{\partial}{\partial x^i} \psi(\mathbf{x})\bigg|_{\mathbf{x}=0},

this piece of the amplitude has overlap with

P

-wave bound states. Show that the

S = 1

, but not the

S = 0

states, can decay to 2 photons. Again, this is a consequence of

C

. (c) To compute the decay rates of these

P

-wave states, we need properly normalized state vectors. Denote the three

P

-state wavefunctions by

\psi_i = x^i f(|\mathbf{x}|), \qquad \text{normalized to } \int d^3x\, \psi_i^*(\mathbf{x})\psi_j(\mathbf{x}) = \delta_{ij},

and their Fourier transforms by

\psi_i(\mathbf{p})

. Show that

|B(\mathbf{k})\rangle = \sqrt{2M} \int \frac{d^3p}{(2\pi)^3} \psi_i(\mathbf{p})\, a^\dagger_{\mathbf{p}+\mathbf{k}/2}\, \Sigma^i\, b^\dagger_{-\mathbf{p}+\mathbf{k}/2} |0\rangle

is a properly normalized bound-state vector if

\Sigma^i

denotes a set of three

2 \times 2

matrices normalized to

\sum_i \text{tr}(\Sigma^{i\dagger} \Sigma^i) = 1.

To build

S = 1

states, we should take each

\Sigma^i

to contain a Pauli sigma matrix. In general, spin-orbit coupling will split the multiplet of

S = 1

L = 1

states according to the total angular momentum

J

. The states of definite

J

are given by

J = 0: \qquad \Sigma^i = \frac{1}{\sqrt{6}} \sigma^i,

J = 1: \qquad \Sigma^i = \frac{1}{2} \epsilon^{ijk} n^j \sigma^k,

J = 2: \qquad \Sigma^i = \frac{1}{\sqrt{3}} h^{ij} \sigma^j,

where

\mathbf{n}

is a polarization vector satisfying

|\mathbf{n}|^2 = 1

and

h^{ij}

is a traceless tensor, for which a typical value might be

h^{12} = 1

and all other components zero. (d) Using the expanded form for the

e^+e^- \to 2\gamma

amplitude derived in part (b) and the explicit form of the

S = 1

L = 1

, definite-

J

positronium states found in part (c), compute, for each

J

, the decay rate of the state into two photons.

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