Understanding interaction terms in the Lagrangian Given a Lagrangian, how does one understand the dynamics by just observing the interaction terms? specifically, can I determine all the possible or at least some of the possible scattering processes, just by observing the interaction terms in the lagrangian?.
Particularly, Let's say, $$L_{int} = g\bar{\psi}\psi\phi + \lambda \phi^4.$$ In this case, how do I figure out the possible interactions?
I can see that, there will be self-interactions of $\phi$ field, and the "Yukawa term", $g\bar{\psi}\psi\phi$, can be used to model, Nucleon - Nucleon scattering, how do we know that such a term can model Nucleon scattering, and are there other non-trivial interactions?
How does one figure out that, just from the Lagrangian?
 A: It's important to look at what field creates/annihilates what. A real scalar field has both $a, a^\dagger$ and can therefore both create/annihilate particles. A complex scalar field comes in 2 versions-$\phi\sim a+b^\dagger$ and $\phi^\dagger\sim a^\dagger+b$ where $a,b$ refer to particle, antiparticle. It is clear that $\phi$ annihilates a particle but creates an antiparticle, whereas $\phi^\dagger$ does the opposite. Similarly, for fermions, we have $\psi\sim a+b^\dagger$, $\bar{\psi}\sim a^\dagger+b$. It is clear that $\psi$ annihilates a fermion and creates an antifermion, and $\bar{\psi}$ does the opposite. Then, in a Yukawa like interaction, we see that given $\bar{\psi}\psi\phi$; we can either

*

*Annihilate antifermion with $\bar{\psi}$ and annihilate fermion with $\psi$ at a vertex, and annihilate/create a scalar $\phi$(with a complex scalar, there's a distinction between what's annihilated and created, but not here).


*Annihilate fermion with $\psi$ and create fermion with $\bar{\psi}$ and create or annihilate a scalar $\phi$. You can read these as 'fermion and scalar annihilate to produce fermion' or as 'fermion is annihilated to create a fermion-scalar pair' respectively.
and so on.
Some details are sketched below.

*

*The scattering amplitudes are constructed from correlation functions using standard techniques. In particular, we have the Dyson formula where we do a time ordered integral over a term that looks like $e^{-i\int dt H(t)}$, where $H$ is the corresponding hamiltonian that captures the interactions.


*In perturbation theory, the Hamiltonian contains some small coupling constant (say, $g$ or $\lambda$) which allows us to expand the exponential term above in a power series in these couplings and fields. For example, schematically, $\langle T\{e^{-i\int dt\int \lambda\phi^4 d^3x}\}\rangle\approx \langle 1-\lambda\int\phi^4dtd^3x+\lambda^2(\cdot)+...\rangle$. So in perturbation theory, the problem reduces one to that of finding n-point functions of these fields.


*These correlators are evaluated by Wick's theorem. The question now is, what correlations are we computing i.e. what are the initial and final states. The correlator simply measures the probability amplitude for a given state to go into another one in the presence of interactions(given by the field content/operator inside the $\langle\cdot\rangle$).


*Remember that the fields inside the correlators are 'interaction picture' fields and have an expansion in terms of the creation and annihilation operators, generically denoted $a_p^\dagger, a_p$. Now, an external state could be something like a one particle state made of a scalar, such as $|p\rangle=a^\dagger_p|0\rangle$, or a 2 particle state made out of, say, a fermion and an anti-fermion $|p_f, p'_{af}\rangle=a^\dagger_pb^\dagger_{p'}|0\rangle$(spin labels suppressed). When calculating a correlation function between such states, these external state creation/annihilation operators can contract with the
relevant fields and Wick's theorem tells us how to do these contractions.


*The key part then becomes identifying what external states are we looking at. The minimal building block of these contractions-the lowest possible number of fields you require to get a nonzero Wick contraction-is diagrammatically interpreted as a vertex. Then, studying scatterings amounts to understanding the allowed vertices(and the propagators connecting them, which come from the sandwiched contracting with each other instead of external states). For example, in $\phi^4$ theory; you cannot have a correlation function involving 3 external states and say 4 powers of $\phi$, as something is sure to go uncontracted and the result will be zero. 2 external states is just the 2 point function-the propagator-which we already know. If we had 4 external states however-then each external state could be contracted with one of the $\phi$'s inside the correlator and one would say a scattering has occurred. If this is a $1\to 3$ scattering or $2\to 2$ is a different question that depends on specifying precisely the external states and their time stamps. So, the 'vertex' here is a 4 point function-it means that the lower order process involves 4 particles.


*In Yukawa theory, you have the interaction $g\bar{\psi}\psi\phi$. This field has 2 sets of fermion-antifermion creation/annihilation operators, and one set of scalar creation/annihilation operators. This means the external states should be comprised of atleast 2 fermions/antifermions and one scalar to have anything that's nonzero. Depending on your choice of initial states, such a vertex can describe a process like $fermion+antifermion\to scalar$, or $(anti)fermion+scalar\to(anti)fermion$. Again, what process actually happens requires you to specify what the precise external states are.
