# Why interacting-kinetic terms can't exist? (Can they?)

It seems that the Lagrangian of QED describing electrons and muons cannot include terms like that: $$\overline{\psi_{(e)}}i\not\!\partial\psi_{(\mu)}$$ where $$\psi_{(e)}$$ and $$\psi_{(\mu)}$$ are 4-component dirac spinors describing the electron and the muon respectively.

I don't understand why this is illicite, as it seems to me that this term is Lorentz-invariant (both spinors transform in the same way), and we could make it gauge invariant. I'm missing something.

Thank you very much,

Anthony.

• – Cosmas Zachos Sep 27 at 18:48
• Thank you Cosmas Zachos, I wouldn't have found it. So it's basically due to the conservation of lepton number. – Anthony Sep 27 at 19:19
• (and it is very consistent with the exercise I was dealing with, since by making this term gauge-invariant by the gauge prescription, we make the decay $\mu^- \rightarrow e^- \gamma$ possible because of the consecutive new interacting term in the Lagrangian, hence the lepton number violation) – Anthony Sep 27 at 19:28
• Careful, as always (below), e and μ are defined by the respective mass terms, so the $\mu \to e\gamma$ amp at the tree level is gone, as part of the minimal coupling prescription. (It is thought to possibly exist at the loop level with enormous suppression.) – Cosmas Zachos Sep 27 at 20:02

One assumes you also have the conventional diagonal kinetic and unequal-mass terms: the definition of $$\psi_e$$ and $$\psi_\mu$$ follows from the propagating mass eigenstates. If the masses were the same, you'd diagonalize the kinetic term which includes your off-diagonal proposal and its h.c., consistently with the (identity) mass matrix in the space of these two states.
If the masses are not the same, as in our world, diagonalizing the kinetic term would then throw the mass matrix off-kilter, and would introduce mixing terms at the mass level, $$\epsilon ~\overline \psi_e \psi_\mu$$ + h.c. Moreover, you'd then have to adjust your normalizations of the canonical fields in the diagonal kinetic term, to ensure the term is the identity matrix! This adjustment would further alter the newly off-diagonal, but still symmetric, mass term.