# QED Interaction Lagrangian with two different fermions

1. Jun 6, 2013

### lfqm

Let say I want to study electron-proton scattering (without considering proton's quarks, i.e. no QCD), which is the Lagrangian?

I've seen two different answers to this question

First one:
L=$\bar{ψ}$e(i∂-me)$ψ$e+$\bar{ψ}$p(i∂-mp)$ψ$p-$\frac{1}{4}$Fμ$\nu$Fμ$\nu$-e$\bar{ψ}$eγμ$ψ$eAμ+e$\bar{ψ}$pγμ$ψ$pAμ

Second one:
L=$\bar{ψ}$e(i∂-me)$ψ$e+$\bar{ψ}$p(i∂-mp)$ψ$p-$\frac{1}{4}$Fμ$\nu$Fμ$\nu$-g$\bar{ψ}$eγμ$ψ$pAμ-h$\bar{ψ}$pγμ$ψ$eAμ

with g,h some constants... (which constants? )

The difference is obviously the interaction part, both seem reasonable to me... but I've encountered both... So, which one is correct?

Last edited: Jun 6, 2013
2. Jun 6, 2013

### dextercioby

If the proton is assumed without internal structure, one can take the first option. The coupling is really done through the electric charge whose absolute value is the same for both particles. The g and h are probably meant to take into account the proton's internal structure through the so-called form factors.

Oops, i didn't see that the spinors were mixed. Sorry. See Bill's comment.

Last edited: Jun 6, 2013
3. Jun 6, 2013

### Bill_K

Any reference that gives the second form should be run through a shredder. The interaction g ψeγμψpAμ turns a proton into an electron, violating several basic principles including charge conservation. Likewise for the h term.

4. Jun 6, 2013

### VantagePoint72

Agreed with the above. Think about the Feynman diagrams: you have, e.g, an incoming fermion and anti-fermion which annihilate and then the photon decays into a different species of fermion and anti-fermion. That's how you get coupling between the different fermions: mediated by photons, since they both independently couple to the EM field.

5. Jun 6, 2013

### andrien

If you really want to study electron-proton scattering without considering qcd then in process you have to involve the structure of proton.So while drawing the feynman diagram the vertex factor at proton site should be modified from -ieγμ to more general one obeying lorentz invariance,parity and gauge invariance.Most generally it is written as [γμF1(q2)+kqvσμvF2(q2)],apart from a overall factor.It was the basis of rosenbluth formula for electron proton scattering.