Symmetry factors of some diagrams

[CAF123]

I have drawn five connected diagrams that arise in $\phi^4$ theory. I was wondering if the symmetry factors I have for each of them are correct and if I have missed any graphs. I only want to consider the case of $V=3,$ with $J=0,...4$ in turn. (V: number of vertices which i denoted by a cross and J: number of source points denoted by a bold circle)

If I have this case understood I am comfortable i could do other cases.

Thanks!

 

[mfb]

No idea about the symmetry factors, but I think there are diagrams missing. A "cross" where two legs have an additional loop attached (instead of both at one), then something similar to the lower right diagram, but just with one loop in the center and an additional loop at one leg.
Similar modifications are possible for the left diagram.

 

[CAF123]

Hi Mfb

No idea about the symmetry factors,...

Should I write out my explanation for them and see if you agree?

...but I think there are diagrams missing. A "cross" where two legs have an additional loop attached (instead of both at one), then something similar to the lower right diagram, but just with one loop in the center and an additional loop at one leg.
Similar modifications are possible for the left diagram.

I'm having a little difficulty trying to picture what you mean - Is it possible for you to draw what you mean for the latter description?
For

A "cross" where two legs have an additional loop attached (instead of both at one)

do you mean three different crosses each with a loop attached to it (as shown in another attachment in the red box)

 

[vanhees71]

I've also no clue what you mean with your symmetry factors. Please give the interaction Lagrangian (i.e., which convention you are using) and then write down the overall factor. Using
$$\mathcal{L}_I=-\frac{\lambda}{4!} \phi^4$$
for the upper left diagram I get a factor
$$\frac{\lambda^3}{(4!)^3}\cdot \frac{1}{3!}\cdot 12 \cdot 3 \cdot 8 \cdot 3 \cdot 4 \cdot 3=\frac{\lambda^3}{8}.$$

 

[mfb]

Do we have constraints apart from the 3 vertices with 4 lines, and the number of external legs?

I don't think that list is complete (in addition to those already posted). One is there twice, ignore one copy.

 

[CAF123]

Hi vanhees71,

I've also no clue what you mean with your symmetry factors. Please give the interaction Lagrangian (i.e., which convention you are using) and then write down the overall factor. Using
$$\mathcal{L}_I=-\frac{\lambda}{4!} \phi^4$$
for the upper left diagram I get a factor
$$\frac{\lambda^3}{(4!)^3}\cdot \frac{1}{3!}\cdot 12 \cdot 3 \cdot 8 \cdot 3 \cdot 4 \cdot 3=\frac{\lambda^3}{8}.$$

Yes, that is my interaction lagrangian. I think I am getting 1/16 for my factor whereas you're getting 1/8. Just to check, I'm not talking about correlation functions at this point so the bold circles are also allowed to be permuted which might be why answer is off from yours by a factor of 1/2. (maybe)

My reasoning is basically for each loop there (apart from the top one), I can swap permute the prongs at each vertex 2! ways and also swap the propagators. For the top loop, I can swap the prongs and then reverse the propagator direction. This gives an overcounting of 2^3 so far. Then for the permutation of the source points together with reversal of the propagators gives another contribution of 2.

Thanks!

 

[CAF123]

Do we have constraints apart from the 3 vertices with 4 lines, and the number of external legs?

I'm just considering the V=3 case at the moment and take J=0...4 in turn.

I don't think that list is complete (in addition to those already posted). One is there twice, ignore one copy.

View attachment 95422

Ok, I see thanks. My convention for the normalisation of the interaction piece is $-\lambda \phi^4/4!$ so maybe this makes it easier to compare to my symmetry factors.

Thanks!

 

[vanhees71]

Hi vanhees71,

Yes, that is my interaction lagrangian. I think I am getting 1/16 for my factor whereas you're getting 1/8. Just to check, I'm not talking about correlation functions at this point so the bold circles are also allowed to be permuted which might be why answer is off from yours by a factor of 1/2. (maybe)

I see. Before taking the derivatives wrt. the external currents the two-point functions have an additional factor 1/2. So we agree in our counting.

My reasoning is basically for each loop there (apart from the top one), I can swap permute the prongs at each vertex 2! ways and also swap the propagators. For the top loop, I can swap the prongs and then reverse the propagator direction. This gives an overcounting of 2^3 so far. Then for the permutation of the source points together with reversal of the propagators gives another contribution of 2.

Thanks!

I usually start by drawing the elements of the diagram, i.e., vertices and external points, and then I count, how to connect these elements with the corresponding propagator lines. This is kind of a graphical application of Wick's theorem.

 

[CAF123]

I see. Before taking the derivatives wrt. the external currents the two-point functions have an additional factor 1/2. So we agree in our counting.

I usually start by drawing the elements of the diagram, i.e., vertices and external points, and then I count, how to connect these elements with the corresponding propagator lines. This is kind of a graphical application of Wick's theorem.

Ok thanks, good to know we agree. I know sometimes counting these symmetry factors can be a bit of a pain but for some they can see it simply by looking at the diagram. Would it be too much to ask if you could verify if the rest of my symmetry factors I have drawn in my attachment are correct for the rest of the diagrams I am getting for V=3?