Supersymmetry: Understanding Weinberg's Formula on Page 78

[Eisenhorn]

Greetings everyone,

I have to give a lecture about supersymmetry, so I started reading Weinbergs quantum theory of fields vol 3, which is quite of a task. Sometimes I've trouble with some of his conclusions, and I hope you could help me there.

I really do not understand how he got to the formula on top of page 78 (I've got the paperback version from 2005. there is no number, so I post the page). To write it down, this one here.

$$\left[f(\Phi)\right]_{\theta_L^2} =& \sum_{nm} \left( \theta^T_L \varepsilon \psi_{nL} (x) \right) \left( \theta^T_L \varepsilon \psi_{mL}(x) \right) \frac{\partial^2 f \left( \phi(x)\right)}{\partial\phi_n(x) \partial\phi_m(x)} \\ &+ \sum_n \mathcal{F}_n(x) \frac{\partial f\left( \phi(x)\right)}{\partial \phi_n(x)} \left( \theta^T_L \varepsilon \theta_L \right)$$

I think it has to be some kind of series, but I really can't calculate this. So it would be really great if someone could post the calculation of this formula.

Eisenhorn

 

[blechman]

I don't have W's text in front of me, but this looks like the F-term of a function of a chiral superfield. You compute such a thing simply by taylor expanding the superfield:

$$\Phi(x,\theta)=\phi(x)+\sqrt{2}\theta\psi(x)+\theta^2F(x)+\ldots$$

where I leave out the derivative terms. Remember that the $\theta$ coordinates are Grassman, so the Taylor expansion terminates.

If you are just starting out learing SUSY, may I suggest Weinberg is not the book for you! The canonical text is Wess and Bagger (chapters 3-8). Also there are some great lectures by Philip Argyres at U Cincinnati: http://www.physics.uc.edu/~argyres/661/index.html

These might serve you better. Weinberg is for when you're already a master! If I may ask, what is this lecture for? Are you teaching a class? Or is this a student presentation?

 

[Eisenhorn]

Thank you for your help, but its only half the truth (I've figured it out myself this morning).
You're right, its the F-Term, but its only a tricky way writing it. If you take the $$\Phi$$ you suggested, and construct a polynom out of different $$\Phi_i$$'s and then only take terms of the Order $$\theta_L^2$$ you get my Term. To see this, take only the $$\phi$$-Terms of the polynom but either two of the $$\psi$$'s, or one $$F$$. So all terms are of the order $$\theta_L^2$$. If you now count the possibilities of replacing one of the possibilities $$\phi$$'s with a $$\psi$$ (two times, so you get two $$\theta$$), you get same factor as if you just take the whole polynom in $$\phi$$ and derive in respect to $$\phi$$. Thats the whole trick in there.

You are right, I'm just learning SUSY, but Weinbergs is the best book I could find. In my opinion, all the other books are too brief or just incomplete, including Argyres or Wess and Bagger. And you're right, Weinberg is a hard text, but at least he gives enough motivation to the things he does. I'm only missing some comments here and there. So if you know a script (other than Argyres. I got this one.) somewhere, based on Weinbergs Book with some extra remarks and comments, that would be great.

And yeah, this is a students presentation. 5 weeks to go.

And again, thank you for your help.

Eisenhorn

 

[Haelfix]

Weinberg does a good job going over the algebra of Supersymmetry, writing it out in painful excruciating step by step detail. Its a good exercise for every physicist to see once and awhile and he is my reference book now.

However, you don't really learn how to calculate things fast and efficiently with Weinberg and its hard to learn with. For instance his supergraph and superspace sections is abymsal and completely opaque upon first reading.

I highly suggest any number of alternatives, some of them online

eg hep-th/0108200, hep-th/9612114

Try also D. Bailin & A. Love and Srivastava if you don't like Wess and Bagger

 

[blechman]

Try also D. Bailin & A. Love and Srivastava if you don't like Wess and Bagger

Let me interject a warning about Bailin and Love: I use this book frequently in my research, as it has some very nice and clear explanations. But the editors should be ashamed of themselves: the book is FULL of **BAD** typos - sign errors, incorrect greek letters, incorrect factors of 2, etc. So if you go with their book (and I *DO* like the text) - just be careful about blindly using their equations!