# Where do pions go in the spontaneous symmetry breaking of the linear sigma model?

I have a few questions to figure out Peskin 4.3 problem which is Linear sigma model about the interactions of pions at low energy. This model consist of N scalar fields governed by the Hamiltonian ($i = 1,2,3....,N)$ $$H=\int d^3x (\frac{1}{2}|\partial_{\mu}\Phi^i|^2+V(\Phi^2))$$ $$V(\Phi^2)=\frac{1}{2}m^2(\Phi^i)^2+\frac{\lambda}{4}(\Phi^i)^4$$ I find the min/max of the potential energy $V(\Phi^2)$, by assing negative mass $$\frac{\partial V}{\partial\Phi}=m^2\Phi+\lambda(\Phi^3)=0$$ $$m^2=-\mu^2$$ $$\Phi(-\mu^2+\lambda\Phi^2)=0$$ $$\Phi=\frac{\mu}{\sqrt{\lambda}}=v$$ It is the vacuum expectation value of the field and since it's not zero, it causes spontaneous symmetry breaking. In part b, Peskin assign N scalar field as $$\Phi^i(x)=\pi^i(x)$$ $$\Phi^N(x)=v+\sigma(x)$$ v is a constant $\pi^i$pion field and $i=1,2,3...,N-1$.

Shifted potential, after some algebra, is like below then $$V(\Phi^2)=\mu^2\sigma^2+\frac{-\mu^4}{4\lambda}+\frac{\lambda}{4}\pi^4+\frac{\lambda}{2}\pi^2\sigma^2+\sqrt{\lambda}\mu\sigma\pi^2+\sqrt{\lambda}\mu\sigma^3+\frac{\lambda}{4}\sigma^4$$ Here we see $\sigma$ field has a mass term. My question is, what is this field which gain mass and why pion fields become massless suddenly? First I assigned mass to pion fields , $-\mu$, but at the end I get massless pion fields and a field $\sigma$ with $\sqrt{2}\mu$ mass. Is it somehow about pions are the pseudo-Nambu-Goldstone bosons of spontaneously broken chiral symmetry as written wikipedia ?

• Is there an explicit question here somewhere? – ACuriousMind Feb 7 '15 at 20:26
• what is this field which gain mass suddenly? And my assumption at the end is it true ? I do not think it is so hard to understand that. But I will edit to be more clear – aQuestion Feb 7 '15 at 20:29
• The massive field σ is a celebrated resonance in QCD; in the Higgs multiplet, it corresponds to the massive Higgs particle. The massless pions are Goldstone bosons, as in the article you are quoting. This is precisely the SSB mechanism the text you are quoting is illustrating. If you can follow the math, you are there already. – Cosmas Zachos Jan 10 '17 at 22:33