Lost: Imaginary term in weak lagrangian

[Inge_k]

Homework Statement

I'm working on the last section of the last chapter (11.9 The Higgs Mechanism) of Griffith's book on Elementary Particles. In Problem 11.25 I'm asked to derive the lagrangian that he gets juts after gauging [Eq. 11.131]. That's the one with a Goldstone boson and a strange imaginary term that he doesn't want there too.
So, he starts with a model lagrangian in which he uses phi = phi1 + i phi2 . Then he finds the deviations of the vacuum state eta = phi1 - mu/lamda and xi = phi2 and my problem is to write the lagrangian in terms of eta and xi

Homework Equations

So I have
phi1 = eta + mu/lambda
phi2 = xi
And I fill this in in the Langrangian I already had

The Attempt at a Solution

Everything works out fine, except that all imaginary terms cancel. In the original langrangian there are two from the covariant derivatives and two from the complex phi. Multiplied with each other they vanish to give -1, multiplied with the other factors they cancel each other out. The term that should be imaginary in the end does appear on my final result, but it's missing a factor -2i .

I've looked everywhere, but I have no idea where to get that from. I'm sorry I'm not very good with Tex so I included a few mathematica screenshots. I hope it's clear.

 

[Jhero]

Hello,

Thank you for sharing your progress on this problem. It seems like you have made some good progress in deriving the Lagrangian in terms of the new variables eta and xi. However, it seems like you are missing a factor of -2i in your final result for the imaginary term.

One possible explanation for this could be that you may have made a mistake in your algebraic manipulations. I would suggest going back through your calculations and checking for any errors or missing steps. It may also be helpful to consult with a classmate or professor for assistance in identifying any mistakes.

Another possibility is that there may be an error in the original Lagrangian or in the textbook's derivation of the Lagrangian. In this case, it would be best to consult with the author or another expert in the field for clarification or to see if there is a typo in the textbook.

I hope this helps and good luck with your calculations!