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I was wondering if anyone could help guide me in finding the Feynman rules for a real pseudoscalar field ($\phi$) interacting with the electromagnetic field $(F^{\mu\nu})$.

The (effective) interaction Lagrangian is: $$ \phi \epsilon^{\alpha\mu\beta\nu} F_{\mu\alpha} F_{\nu\beta} $$

In my textbooks, the interaction with a complex scalar field is taken into account by imposing gauge symmetry and using the covariant derivative. Can I use a similar method for real pseudoscalar fields?

I'm hoping to evaluate the amplitude for an incoming pseudoscalar decaying to 2 photons. Much like the decay of a neutral pion to 2 photons, but using an effective Lagrangian instead of evaluating the triangle diagram.

I'm still fairly new to quantum field theory so apologies if this is ill posed or you need more information.

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    $\begingroup$ physics.stackexchange.com/questions/44202/… $\endgroup$
    – Siva
    Dec 21, 2013 at 19:35
  • $\begingroup$ @Pont does it make any sense to couple a real scalar to a $U(1)$ gauge field? I mean, a gauge transformation would imply a complex phase, but this is forbidden since the gauge transformed field should be real. I am not 100% sure but I think real scalars cannot couple to photons $\endgroup$
    – Yossarian
    Jun 14, 2015 at 23:16

1 Answer 1

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The S-matrix is given by the equation $$S=\sum_{n=0}^{\infty}{i^n\over n!}\int\prod_{i=1}^n d^4 x_i T\prod_{i=1}^n \mathcal{L}_{int}(x_j)=\sum_{n=0}^{\infty}S^{(n)}\;,$$ where $T$ stands for the time ordered product. Substitute $\mathcal{L}_{int}=\phi \epsilon^{\alpha\mu\beta\nu} F_{\mu\alpha} F_{\nu\beta}$, so that $$\sum_{n=0}^{\infty}S^{(n)}=\sum_{n=0}^{\infty}{i^n\over n!}\int\prod_{i=1}^n d^4 x_i T\prod_{i=1}^n \phi (x_j)\epsilon^{\alpha\mu\beta\nu}(x_j) F_{\mu\alpha}(x_j) F_{\nu\beta}(x_j)$$ For example, for $n=2$, one has $$S^{(2)}={i^2\over 2}\int d^4 xd^4 x^{'} T \phi (x)\epsilon^{\alpha\mu\beta\nu}(x) F_{\mu\alpha}(x) F_{\nu\beta}(x)T\phi (x^{'})\epsilon^{\alpha\mu\beta\nu}(x^{'}) F_{\mu\alpha}(x^{'}) F_{\nu\beta}(x^{'})$$ Perform a Wick expansion of this term, and then follow the rules below:

  1. Each integration coordinate $x_j$ is represented by a point (sometimes called a vertex);

  2. A bosonic propagator is represented by a wiggly line connecting two points;

  3. A fermionic propagator is represented by a solid line connecting two points;

  4. A bosonic field $A_\mu(x_i)$ is represented by a wiggly line attached to the point $x_i$;

  5. A scalar field $\phi(x_i)$ is represented by a solid line attached to the point $x_i$ with an arrow toward the point;

  6. A scalar field $\bar\phi(x_i)$ is represented by a solid line attached to the point $x_i$ with an arrow from the point.

These rules lead one to the Feynman diagrams.

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