Calculating Feynman Rules for Effective Electroweak Chiral Lagrangian

[CoolPhysics5]

Hi all,

I'm trying to calculate the Feynman Rules for the effective electroweak chiral Lagrangian. For example, this is the first term in the Lagrangian:

\begin{eqnarray}
\mathcal{L}=\frac{v^2}{4}\text{Tr}(D_{\mu}U D^{\mu}U^{\dagger})
\end{eqnarray}

where

\begin{eqnarray}
U=\text{exp}(\frac{i\pi \cdot \tau}{v})
\end{eqnarray}

where $\tau$ are the Pauli matrices and $\pi$ are the Goldstone bosons. See,for example equations 1 and 2 in:

http://arxiv.org/PS_cache/hep-ph/pdf/0201/0201098v1.pdf"

My questions are:

1. Does anyone know of a reference in which the Feynman rules for the electroweak Lagrangian are given?

2. Does anyone know how to get the Feynman Rules out of this Lagrangian? It seems that this is a little complicated because of the presence of the trace.

3. Also, do I have to know explicitly what the $\pi$ matrix is? I would assume so, as I seem to be struggling otherwise. I don't know what it is though. For example for chiral perturbation theory in QCD Wikipedia has the following article:

http://en.wikipedia.org/wiki/Chiral_perturbation_theory

in which it gives an explicit representation for the matrix U. But what is the equivalent in the electroweak theory?

Hope I've explained myself clearly. I'm not quite sure if these are the right questions to ask...

Anyhow, any advice or help would be very much appreciated! If anyone knows of any good online resources e.g. papers/lecture notes which might help that would be great too!

Thank you very much.

 

[fzero]

Hi all,

I'm trying to calculate the Feynman Rules for the effective electroweak chiral Lagrangian. For example, this is the first term in the Lagrangian:

\begin{eqnarray}
\mathcal{L}=\frac{v^2}{4}\text{Tr}(D_{\mu}U D^{\mu}U^{\dagger})
\end{eqnarray}

where

\begin{eqnarray}
U=\text{exp}(\frac{i\pi \cdot \tau}{v})
\end{eqnarray}

where $\tau$ are the Pauli matrices and $\pi$ are the Goldstone bosons. See,for example equations 1 and 2 in:

http://arxiv.org/PS_cache/hep-ph/pdf/0201/0201098v1.pdf"

My questions are:

1. Does anyone know of a reference in which the Feynman rules for the electroweak Lagrangian are given?

2. Does anyone know how to get the Feynman Rules out of this Lagrangian? It seems that this is a little complicated because of the presence of the trace.

Each term in the expansion of [itex]\mathcal{L}[/itex] is of the form

[itex] \text{Tr}\left( \tau^{a_1} \cdots \tau^{a_n} \right) \left( \pi_{a_3}\cdots \pi_{a_n} D_{\mu}\pi_{a_1} D^{\mu} \pi_{a_2} \right).[/itex]

The first term gives the propagator, while the other terms give vertices with an even number of external legs and two insertions of momenta. The Pauli matrix traces can be worked out at each order by repeatedly using the identity

[itex] \tau^a \tau^b =\delta^{ab} I +i\epsilon^{abc}\tau^c. [/itex]

The vertices will carry a factor of the form [itex]C_{a_1\cdots a_n} k_{a_1} k_{a_2}[/itex], where the [itex]C[/itex]s are related to the tensor expressions for the traces, along with the numerical factors and powers of [itex]v[/itex]. When you build amplitudes from diagrams, you have to sum over the different ways to insert the momentum factors.

3. Also, do I have to know explicitly what the $\pi$ matrix is? I would assume so, as I seem to be struggling otherwise. I don't know what it is though. For example for chiral perturbation theory in QCD Wikipedia has the following article:

http://en.wikipedia.org/wiki/Chiral_perturbation_theory

in which it gives an explicit representation for the matrix U. But what is the equivalent in the electroweak theory?

As it says in the paper that you referred to, the [itex]\pi_a[/itex] are the would-be Goldstone bosons of EWSB. These are the 3 components of the complex Higgs doublet that are absorbed into the gauge bosons giving them the 3rd component needed to form a massive vector. There are more details in Farhi and Susskind "Technicolor" Phys.Rept. 74 (1981) 277 (a scan of the preprint version is available at http://www-lib.kek.jp/cgi-bin/kiss_prepri.v8?KN=198104309&TI=&AU=&AF=&CL=&RP=&YR= ).

 

[daschaich]

Thanks for that explanation, fzero. I'm also working on a project involving the electroweak chiral lagrangian, so I appreciate this information.

CoolPhysics5, Appelquist and Wu write down Feynman rules for triple gauge vertices from the EWchiL. It might be an instructive exercise to reproduce them if you are so inclined.