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Homework Help: Positronium allowed decays (Peskin)

  1. Jan 14, 2010 #1


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    1. The problem statement, all variables and given/known data
    (This is not homework)
    This refers to question 3.8 in Peskin's QFT

    Using the fact that the electromagnetic interaction term in the Dirac + EM lagrangian is invariant under Parity (P) and Charge conjugation (C), and that spin 0 and spin 1 states are odd and even under exchange of spins,

    show that the
    1) spin 0 positronium ground state (S wavefunction) decays into 2 photons, and that the
    2) spin 1 positronium ground state (S wavefunction) must decay into 3 photons
    3) The above for P, D states.

    2. Relevant equations

    EM coupling
    [tex]\Delta H=\int A_{\mu}j^{\mu}d^{3}x[/tex]

    We know that under parity, [tex]j^{\mu}\rightarrow (j^{0},-j^{1},-j^{2},-j^{3})[/tex]
    We know that under parity, [tex]j^{\mu}\rightarrow -j^{\mu}[/tex]

    3. The attempt at a solution

    By handwaving, we can say that these transitions occur due to conservation of angular momentum as a photon has a spin of 1.

    However, how would these transitions be derived on the basis of C and P symmetries alone?

    One could probably consider the interaction matrix term:

    [tex]\left\langle photons\right|\Delta H\left|positronium\right\rangle[/tex]

    And determine how it transforms under C and P

    The problem I have is in evaluating the P and C eigenvalues of states that contain only photons.

    For a state involving a fermion and antifermion (eg. positronium), and with orbital angular momemtum L, P|state> = (-1)L+1|state>. The extra factor of +1 is due to the anticommutativity of spin 1/2 creation operators.

  2. jcsd
  3. Jan 19, 2010 #2


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