What is the Fermion's mass in this Lagrangian?

AI Thread Summary
The discussion focuses on determining the fermion mass within a specific Lagrangian framework that involves spontaneous symmetry breaking. The vacuum state is identified by minimizing the potential, leading to a vacuum expectation value of the scalar field, represented as v. The mass of the fermion, denoted as mψ, is derived from the terms in the Lagrangian, particularly focusing on the coupling to the scalar fields h and π. The resulting fermion mass is expressed as gv, with additional couplings involving h and π also identified. The analysis highlights the relationship between the scalar field dynamics and the fermion mass generation in this theoretical context.
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Homework Statement
.
Relevant Equations
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We have a Lagrangian of the form:
$$

\mathcal{L} = \overline{\psi} i \gamma_{\mu} \partial^{\mu} \psi - g \left( \overline{\psi}_L \psi_R \phi + \overline{\psi}_R \psi_L \phi^* \right) + \mathcal{L}_{\phi} - V(|\phi|^2)

$$
Essentially, what we are studying is spontaneous symmetry breaking. First, we must find the minimum of $$V(|\phi|^2)$$ to determine the vacuum state. We obtain:
$$

\langle \phi \rangle = v = \sqrt{\frac{m^2}{\lambda}}

$$
Now, let's perform the following expansion:
$$

\phi = (v + h(r, t)) e^{-\frac{i \pi(r, t)}{f}}

$$
Now, the question arises: How do we find the mass of the "new particles," ##\pi## and ##h##? This part is straightforward. However, the challenge lies in determining the fermion mass, denoted as ##m_{\psi}##, and its coupling to ##\pi## and ##h##.

I assume that the only terms that matter in answering this question are:

$$

\overline{\psi} i \gamma_{\mu} \partial^{\mu} \psi - g \left( \overline{\psi}_L \psi_R \phi + \overline{\psi}_R \psi_L \phi^* \right)

$$
Now, let's expand this term as follows:
$$

\overline{\psi} i \gamma_{\mu} \partial^{\mu} \psi - g \left( \overline{\psi}_L \psi_R \left( (v + h) e^{i \frac{\pi}{f}} \right) + \overline{\psi}_R \psi_L \left( (v + h) e^{-i \frac{\pi}{f}} \right) \right)

$$
The challenge here is to determine the fermion mass. My idea is to write a Lagrangian equivalent to the Dirac Lagrangian, where the constant ##c## that should appear in the Lagrangian, i.e., ##c \overline{\psi} \psi##, represents the mass. However, I can't find such a term in the Lagrangian we have. To proceed, I first rewrite ##\psi_{L,R}## in terms of ##\psi## itself, resulting in:

$$

- g (v+h) \overline{\psi} \left( \cos\left(\frac{\pi}{f}\right) + i \gamma^5 \sin\left(\frac{\pi}{f}\right) \right) \psi

$$

Next, I expand the trigonometric expressions to obtain:

$$

g (v+h) \overline{\psi} \left( 1 - \frac{1}{2} \left(\frac{\pi}{f}\right)^2 + i \gamma^5 \frac{\pi}{f} \right) \psi

$$
This expansion results in terms such as:

$$

- g v \overline{\psi} \psi - g h \overline{\psi} \psi - \frac{i g v \gamma^5}{f} \overline{\psi} \pi \psi + \frac{g v}{2 f^2} \overline{\psi} \pi \pi \psi + O(\ldots)

$$

So, the fermion mass would be ##g v##, the coupling ##h \psi \overline{\psi}## would be ##g##, and the ##\overline{\psi} \pi \psi## coupling would be ##\frac{i g v \gamma^5}{f}##?
 
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