Feynman rules & diagram for phi^3 theory

[SkeZa]

I'm reading a course in Introduction to QFT and I'm stuck at a problem.
I'm hoping someone here could point me in the right direction or say if my assumptions are incorrect.

Homework Statement

Derive the Feynman rules and all diagrams at tree-level for \lambda \phi^3 theory using Wick's theorem.
My own questions:
To what n do you take the n-point correlation function \tau(x_1,x_2,...,x_n)?
How does one draw a simple tree-level Feynman diagram depending if you know the interaction term L_i_n_t or H'_I
Is there something I haven't understood yet or is there something I'm forgetting about Feynmans rules and diagrams

Homework Equations

The correlation function:
\tau(x_1,x_2,...,x_n) = \langle0\left\right| T{\phi_i_n(x_1)\phi_i_n(x_2)...\phi_i_n(x_n)exp(-i\int H'_I(t')dt')}\left|\right0\rangle
to all orders in pertubation theory and where
H'_I = \int d^3 x \lambda \phi^3 / 3!

The Attempt at a Solution

I've read (in Peskin & Schroeder) that higher than n = 2 correlation functions have to be solved using brute force (doesn't understand) and thought that it has to be n = 2 otherwise it's beyond the scope of the course.
For that case however, for the first order of pertubation in \lambda, there are an odd number of \phi's and therefore they can't give any contributions due to normal-ordering. Right?
There is, however, an even number of \phi's for the second order in the pertubation.
Do I have to take to the second order of \lambda to get the diagrams?

As for the rules: I've understood that the zeroth order in pertubation just gives the propagator (for n = 2) and that there has to be a 3-way vertex (insted of the 4-way vertex for \lambda \phi^4 theory). Should they also have the same "value" when constructing the M matrix?

The diagrams draw for \lambda \phi^4 theory in Peskin & Schroeder's "An Introduction to Quantum Field Theory" har straight 2 point lines (with some loops), how does one go from these to a tree-level diagram?

I'd also appreciate any tips on books that deal with \phi^3 theory.

 

[weejee]

Srednicki treats phi^3 theory.

 

[LongLiveYorke]

I'm not an expert, but since no one else is answering, I'll throw in my two cents:

For this theory, it's clear what Feynman diagrams will look like. You can draw lines and you can draw three lines meeting at a point. So, we can pretty easily see what diagrams contribute at a particular level of perturbation theory. Of course, the real hard part is coming up with rules for these diagrams, which is your question.

As to your question about the correlation function, I believe that you'll only need to calculate the 2-point function. This is just the propagator for a free theory which comes from the non-perturbative part of the Lagrangian. The fact that the theory has 3-point vertices doesn't mean that we need to calculate a three particle correlation function. It means that what we eventually want to do is to use Wick's theorem to turn longs chains of fields into sums of correlation functions (which is just the idea of Feynman diagrams).

As to your concern about phi^3 giving you an odd number of fields which normal order to 0: Remember, some of the fields will be used to destroy or create initial or final state particles. So, if we have an initial state containing one particle and a final state containing two particles, we can connect these states using the phi^3 interaction term. What we would want to calculate would be

<i| \lambda \phi \phi \phi |f>

with

|i> = \phi^+ |0>
|f> = \phi^+ \phi^+ |0>

(this is just schematic and I haven't included any indices or anything)

But this doesn't vanish because one of the phi's is used to destroy the particle in the initial state and the other two phi's are used to destroy the particles in the final state (or however you want to think about it). So, the odd number of phi's are used to deal with the initial and final state particle wavefunctions. This term ends up not even having a propagator. If we were to calculate a higher order diagram, we would first use up any available phi's to cancel out initial and final states, and then we would group remaining phi's together to form correlation functions (two point propagators), and those would give us loops, etc.

Does this make sense?

 

[LongLiveYorke]

Srednicki treats phi^3 theory.

Yeah, he covers it in chapters 9,10 and lists Feynman rules. If you're just starting out with QFT, I think Srednicki may be a bit awkward in his notation since he includes a lot of factors early on that anticipate renormalization (there are a lot of Z's floating around that may confuse you at first, and he talks about counter terms which you may not need to in your problem)

 

[SkeZa]

For this theory, it's clear what Feynman diagrams will look like. You can draw lines and you can draw three lines meeting at a point. So, we can pretty easily see what diagrams contribute at a particular level of perturbation theory.

These where my thoughts at the beginning. I later found a reference in a book by F. Gross, but he's including the charged \Phi K-G fields as well.
In the book, he tells that the first order pertubation (for 2 point) describes the decay the fields. Second order would then describe scattering (2 \phi fields scatter into one \phi field which then decays in two fields. This could the be a tree-level diagram. What I haven't understood is then how you determine if it's a "s", "t" and/or "u" (Mandelstam variables)- channel tree-level diagram.

Of course, the real hard part is coming up with rules for these diagrams, which is your question.

I'm assuming that they will be similar to the rules for \lambda \phi^4 theory except that you will have a 3-way vertex instead of a 4-way vertex. But I could be wrong.
Would rules like "Divide by a potential symmetry factor" or "Impose 4-momentum conservation at every vertex" for \lambda \phi^4 theory be different for \lambda \phi^3 theory?

As to your concern about phi^3 giving you an odd number of fields which normal order to 0: Remember, some of the fields will be used to destroy or create initial or final state particles. So, if we have an initial state containing one particle and a final state containing two particles, we can connect these states using the phi^3 interaction term.

I haden't considered that. I was under the assumption that they had to act on the ground state of the free theory ( \left|0\right\rangle ).
Thanks.

I've looked into Srednicki's book. His methods are different then what we've talked about in class. This is only an introductory course and Srednicki's methods may be a little too advanced. For example: we haven't discussed path integrals for field theory.

 

[LongLiveYorke]

I'm assuming that they will be similar to the rules for \lambda \phi^4 theory except that you will have a 3-way vertex instead of a 4-way vertex. But I could be wrong.
Would rules like "Divide by a potential symmetry factor" or "Impose 4-momentum conservation at every vertex" for \lambda \phi^4 theory be different for \lambda \phi^3 theory?

I think it you have the rules for phi^4 theory, the rules for phi^3 are nearly identical, and the only difference is the type of diagrams that you are allowed to draw. The initial and final state wave functions will be exactly the same and the propagators will be exactly the same. If you're doing everything at tree level (no loops), it should be easy to go from phi-3 to phi-4. Yes, you'll still have to deal with symmetry factors and such if they apply to a particular diagram.

 

[SkeZa]

That's a relief.
At least I don't have to "reinvent" Faynmans rules.

But I'm still confused as to which channel/channels the diagrams will be.
For actual particles, one gets clues as to which it might be but how does one figure it out for a field \phi

 

[Ol49]

Hej guys !

I'm working on the same problem right now (are you a classmate SkeZa ?)

Anyways, i read in Gross that only the rule 1 for Feynman diagrams is theory-dependent. This can be found both in the appendix of the Gross' book and in his derivation of the \phi^3[\tex] theory (chapter 9.4, introduction to feynman rules, where the \phi^3[\tex] theory is used here as a "simple" example for deriving the rules)<br /> <br /> So the only rule that have to be changed, is that the vertex term is (-i\lambda[\tex]). The problem is that i don't see how to derive this from Wick's theorem...

 

[Davide86]

What I haven't understood is then how you determine if it's a "s", "t" and/or "u" (Mandelstam variables)- channel tree-level diagram.



The three channels represent different ways in which two particles scattering (considering the example presented by Srednicki) can occur: they all contribute to the cross section.
By analizing the non-relativistic and relativistic limits for the Mandelstam variables, you find out that at low energy (non-relativistic limit) the only contribute comes from the channel s.
At higher energy the other two channels also don't vanish.