Question

The Lagrangian for three interacting real fields $\phi_1, \phi_2, \phi_3$ is $$ e=\frac{1}{2}\left(\partial_\mu \phi_i\right)^2-\frac{1}{2} \mu^2 \phi_i^2-\frac{1}{4} \lambda\left(\phi_i^2\right)^2 $$ with $\mu^2<0$ and $\lambda>0$, and where a summation of $\phi_i^2$ over $i$ is implied. Show that it describes a massive field of mass $\sqrt{-2 \mu^2}$ and two massless Goldstone bosons.

    The Lagrangian for three interacting real fields $\phi_1, \phi_2, \phi_3$ is
$$
e=\frac{1}{2}\left(\partial_\mu \phi_i\right)^2-\frac{1}{2} \mu^2 \phi_i^2-\frac{1}{4} \lambda\left(\phi_i^2\right)^2
$$
with $\mu^2<0$ and $\lambda>0$, and where a summation of $\phi_i^2$ over $i$ is implied. Show that it describes a massive field of mass $\sqrt{-2 \mu^2}$ and two massless Goldstone bosons.

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Quarks and leptons: introductory course in modern particle physics

Quarks and leptons: introductory course in modern particle physics

Francis Halzen, Alan… 1st Edition

Chapter 14, Problem 12

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Step 1

** The given Lagrangian is: $$ e = \frac{1}{2} (\partial_\mu \phi_i)^2 - \frac{1}{2} \mu^2 \phi_i^2 - \frac{1}{4} \lambda (\phi_i^2)^2 $$ where the summation over repeated indices $i$ from 1 to 3 is implied (Einstein summation convention). This Lagrangian is Show more…

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The Lagrangian for three interacting real fields $\phi_1, \phi_2, \phi_3$ is $$ e=\frac{1}{2}\left(\partial_\mu \phi_i\right)^2-\frac{1}{2} \mu^2 \phi_i^2-\frac{1}{4} \lambda\left(\phi_i^2\right)^2 $$ with $\mu^2<0$ and $\lambda>0$, and where a summation of $\phi_i^2$ over $i$ is implied. Show that it describes a massive field of mass $\sqrt{-2 \mu^2}$ and two massless Goldstone bosons.

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