What is the Lagrangian of interaction of photon and spin zero scalar?

[ndung200790]

What is the Lagrangian of interaction of photon and spin zero charge scalar?The vertex of photon and spin 1/2 charge fermion is proportional with e multiplied vertor gamma matrix,but I do not know what is the vertex of photon and charge scalar.I hear that a vertex is proportional with polynomial of vector momenta(in general case) but I do not understand why.

 

[The_Duck]

You want to read about "scalar electrodynamics." There are two types of vertices. One looks like

$ie A_\mu \phi \partial^\mu \phi$.

This vertex has one photon line and two scalar lines and is proportional to the momentum of one of the scalar lines. The momentum has to be there to contract with the Lorentz index $\mu$ on the photon line.

The other vertex looks like

$e^2 A_\mu A^\mu \phi^2$.

This has two photon lines and two scalar lines and no momenta.

 

[buoyant]

you can express full hamiltonian by using gauge-covariant
this is described in problem 9.1 of peskin and shroeder's book.

 

[tom.stoer]

The gauge invariant expression uses

$D_\mu \phi = (\partial_\mu - ieA_\mu) \phi$

and

$\mathcal{L}_\text{int} = \frac{1}{2}(D_\mu \phi)^\ast (D_\mu \phi)$

as interaction term.