Feynman rules and the tree level cross section of two scalar fields

[MT777]

Homework Statement

Find the Feynman rules in the momentum space for the following interaction lagrangian $\mathcal{L}_{\text{I}} = -(1/2!) \lambda \phi_1^2 \phi_2 + \text{h.c.}$, where $\phi_1$ has charge $-e$ and $\phi_2$ is neutral. Calculate at tree level the efficient cross section from the center of mass of the elastic process $\phi_1 + \phi_2 \rightarrow \phi_1 + \phi_2$.

Relevant Equations

The tree level cross section from the center of mass from two incoming particles $A$ and $B$ is
$$\Bigg(\frac{\text{d} \sigma}{\text{d} \Omega} \Bigg)_{\text{CM}} = \frac{1}{2 E_A 2 E_B} \frac{1}{|v_A - v_B|} \frac{||\textbf{p}||}{4 (2 \pi)^4 \sqrt{S}} |\mathcal{M}|^2$$

Hi there. I'm trying to solve the problem mentioned above, the thing is I'm truly lost and I don't know how to start solving this problem. Sorry if I don't have a concrete attempt at a solution. How do I derive the Feynman rules for this Lagrangian? What I think happens is that in momentum space, for each line there is a Feynman propagator while for each vertex there is a coupling term with a Dirac delta due to conservation of momentum, but how do I draw the corresponding Feynman diagram from the interaction Lagrangian?

For the cross section, who is $\mathcal{M}$ in this case?

 

[Gaussian97]

Homework Statement:: Find the Feynman rules in the momentum space for the following interaction lagrangian $\mathcal{L}_{\text{I}} = -(1/2!) \lambda \phi_1^2 \phi_2 + \text{h.c.}$, where $\phi_1$ has charge $-e$ and $\phi_2$ is neutral. Calculate at tree level the efficient cross section from the center of mass of the elastic process $\phi_1 + \phi_2 \rightarrow \phi_1 + \phi_2$.
Relevant Equations:: The tree level cross section from the center of mass from two incoming particles $A$ and $B$ is
$$\Bigg(\frac{\text{d} \sigma}{\text{d} \Omega} \Bigg)_{\text{CM}} = \frac{1}{2 E_A 2 E_B} \frac{1}{|v_A - v_B|} \frac{||\textbf{p}||}{4 (2 \pi)^4 \sqrt{S}} |\mathcal{M}|^2$$

Hi there. I'm trying to solve the problem mentioned above, the thing is I'm truly lost and I don't know how to start solving this problem. Sorry if I don't have a concrete attempt at a solution. How do I derive the Feynman rules for this Lagrangian? What I think happens is that in momentum space, for each line there is a Feynman propagator while for each vertex there is a coupling term with a Dirac delta due to conservation of momentum, but how do I draw the corresponding Feynman diagram from the interaction Lagrangian?

For the cross section, who is $\mathcal{M}$ in this case?

Ok, first of all, let's make sure you understand the Lagrangian you are given. Can you tell me exactly what you know of each symbol that appears in the Lagrangian? And can you write the whole Lagrangian explicitly?

 

[MT777]

Ok, first of all, let's make sure you understand the Lagrangian you are given. Can you tell me exactly what you know of each symbol that appears in the Lagrangian? And can you write the whole Lagrangian explicitly?

Hi, sorry for the late reply!

As I understand it, the interaction Lagrangian tells us that there are two scalar fields $\phi_1$ and $\phi_2$ with coupling term $\lambda$. Lastly $2!$ is the symmetry factor. Since $\phi_1$ is charged, this field is complex (that's why we include its complex conjugate in $\mathcal{L}_{\text{I}}$), while $\phi_2$ is neutral and thus it's real.

Since $\phi_1$ is squared and $\phi_2$ has exponent 1, at first order there are two lines corresponding to $\phi_1$ and one line for $\phi_2$ for each possible Feynman diagram. However, when using Wick's theorem in order to derive the Feynman rules for this Lagrangian, we see that we need to study at least the second order interaction since at first order there is an odd number of fields.

What I think happens in momentum space, is that each vertex contributes a term of $-i \lambda$, each mediating field contributes a scalar propagator, and each external line contributes a factor of 1. At each vertex we impose momentum conservation and we divide over our symmetry factor. This is what I gather from $\mathcal{L}_{\text{I}}$, however I guess you deduce these rules formally from Wick's theorem in the sense that each contraction leads to a corresponding Feynman diagram.

The whole Lagrangian would be (if our fields are minimally coupled) the sum of their corresponding kinetic energies and the substraction of their corresponding potential energies, i.e.
$$\mathcal{L} = -\frac{1}{2} \eta^{\mu \nu} \partial_\mu \phi_1 \partial_\nu \phi_1 - \frac{1}{2} \eta^{\mu \nu} \partial_\mu \phi_2 \partial_\nu \phi_2 - V (\phi_1 ) - V (\phi_2 ) + \mathcal{L}_{\text{I}}.$$

For the tree level cross section, we use our Feynman rules to derive our $\mathcal{M}$ matrix for the stated collision, and then we evaluate on the formula I wrote in my question. This is what I thought in how to solve the problem, however I feel that my approach is rather vague and where I'm struggling is to do it formally, mainly using Wick's theorem, but doing it this way is a bit tedious and I was looking for a way to do it more elegant.

 

[Gaussian97]

Ok, let's focus first on the Lagrangian.
First of all, you need to know what the functions $V(\phi)$ are. In this case, you can just assume the fields have no self-interactions, which determines those functions. (Or, alternatively, you can justify that self-interactions would only affect to higher order, and we can neglect them.)
Then, for the kinetic terms, what you have written is not correct, remember one of the fields is complex.
Now for the interaction term, I think it is really important before to try to do any computation that you write it explicitly. Also, make sure you have the correct expression in the first comment.

For your thoughts on how to solve the problem, I think you are basically correct. The correct way to do it is with Wick's theorem or with path integrals. If you are not comfortable doing it that way, I recommend you to keep practising until you have no problem at all. Once you know you could do it with Wick's theorem without a problem, there are ways to find the Feynman rules directly from the Lagrangian and do the computation much faster.

 

[MT777]

Ok, let's focus first on the Lagrangian.
First of all, you need to know what the functions $V(\phi)$ are. In this case, you can just assume the fields have no self-interactions, which determines those functions. (Or, alternatively, you can justify that self-interactions would only affect to higher order, and we can neglect them.)
Then, for the kinetic terms, what you have written is not correct, remember one of the fields is complex.
Now for the interaction term, I think it is really important before to try to do any computation that you write it explicitly. Also, make sure you have the correct expression in the first comment.

For your thoughts on how to solve the problem, I think you are basically correct. The correct way to do it is with Wick's theorem or with path integrals. If you are not comfortable doing it that way, I recommend you to keep practising until you have no problem at all. Once you know you could do it with Wick's theorem without a problem, there are ways to find the Feynman rules directly from the Lagrangian and do the computation much faster.

You're right, I forgot to consider $\phi_1$ is complex when writing down $\mathcal{L}$. With this in mind, writing the potentials and $\mathcal{L}_{\text{I}}$ explicitly in $\mathcal{L}$ gives us

$$\mathcal{L} = -\frac{1}{2} \eta^{\mu \nu} \partial_\mu \phi_1^* \partial_\nu \phi_1 - \frac{1}{2} \eta^{\mu \nu} \partial_\mu \phi_2 \partial_\nu \phi_2 - \frac{1}{2} m_1^2 \phi_1^* \phi_1 - \frac{1}{2} m_2^2 \phi_2^2 - \frac{1}{2!} \lambda \phi_1^2 \phi_2 - \frac{1}{2!} \lambda \phi_1^{*^2} \phi_2$$

As far as I'm aware, all the expressions I wrote in my initial post are correct, the interaction Lagrangian is copied directly from my homework and the scattering cross section is from Peskin's book, I'll check it again, though. Could you share some sources where it's explained how to find the Feynman rules directly from the Lagrangian, please?

 

[Gaussian97]

As far as I'm aware, all the expressions I wrote in my initial post are correct, the interaction Lagrangian is copied directly from my homework and the scattering cross section is from Peskin's book, I'll check it again, though. Could you share some sources where it's explained how to find the Feynman rules directly from the Lagrangian, please?

Ok, just want to make sure, because you are using a strange normalization for the complex field, and also this interaction doesn't conserve charge (which is not a problem, but is strange to see).

Regarding the Feynman rules, the first thing that comes to my mind is Appendix B of Cheng-Li's book.

 

[MT777]

Ok, just want to make sure, because you are using a strange normalization for the complex field, and also this interaction doesn't conserve charge (which is not a problem, but is strange to see).

Regarding the Feynman rules, the first thing that comes to my mind is Appendix B of Cheng-Li's book.

Got it. Thank you very much for your time.