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I regularly find that I'll understand where the field content in a particular physics paper comes from, but then a Lagrangian or action or superpotential is stated and I don't know how it's derived. Is there a set of general rules for building a Lagrangian/action/superpotential if you already know the field content of the theory?

Any suggestions of sources that explain how to do so would be very welcome as I'm having little trouble finding much that helps.

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Building an action: If you know the field content (which I assume means you know the gauge group and reps of all the fields) then:

  1. Write down every term that is Lorentz scalar (so combinations like $\partial_\mu A^\mu$, $\bar{\psi}\gamma^\mu \partial_\mu\psi$ allowed but not things like $\vec{n}\cdot\nabla \phi$ where $\vec{n}$ is some random 3-vector). Stop at terms with dimension greater than the spacetime dimension (4 in 4D, so include terms like $\phi^4$ and $\phi\bar{\psi}\psi$ but not higher dimension terms like $\phi \partial_\mu \phi \bar{\psi} \gamma^\mu \psi$).

  2. Cross out terms that are not gauge invariant. This means that gauge fields can only appear through covariant derivatives and field strength tensors, and matter fields must appear in singlet combinations.

  3. Cross out terms that violate any global symmetries you want to impose (though be warned - if these symmetries are anomalous you can't drop these terms consistently). In SUSY theories you need to impose relations between coupling constants as well.

  4. After you've done this you can probably use field redefinitions (orthogonal/unitary rotations in flavour space) to simplify some of the coupling constants. An example would be the standard model where the lepton yukawas can be diagonalised and the quark yukawas can be diagonalised leaving just the physical CKM matrix.

It's really a lot like lego - the fields are the building blocks and symmetries & gauge invariance tell you what you can put together.

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  • $\begingroup$ Nice exlanations, I like this. Supersymmetric Lagrangians can be derived too from Superlagrangians using the superspace formalism. So is there a way to relate the field content you want to have to a corresponding Superlagrangian and to derive from this then the Lagrangian? But doing it as you describe is probably most reasonable and everything else would be like cracking a nut with a sledge hammer (?) ... $\endgroup$
    – Dilaton
    Mar 12, 2013 at 12:07
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    $\begingroup$ @Dilaton I've learned superspace but haven't (yet) used it practically for anything, so don't put too much stock into this: I believe superspace only exists for N=1 SUSY, but I could be wrong. But yes, the basic ingredients are chiral superfields for the matter content and vector superfields for the gauge fields. Using these fields in the superspace formalism automatically gaurantees SUSY. You can write some very general expressions in superspace and obtain the usual Lagrangian by integrating over the fermionic directions. The general strategy of writing down every possible term still holds. $\endgroup$
    – Michael
    Mar 12, 2013 at 12:36

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