# Question about the linear sigma-model

Suppose the linear sigma-model lagrangian: $$L = \bar{N}(i\gamma_{\mu}\partial^{\mu}-g_A \phi)N + |\partial_{\mu}\phi|^{2} - V(|\phi|) - c\sigma ,$$ where $$N = \begin{pmatrix} p \\ n\end{pmatrix}, \quad \phi = i\tau_{a}\gamma_{5}\pi_{a}+\sigma, \quad |\phi|^{2} = \pi_{a}^{2}+\sigma^{2}, \quad |\partial_{\mu}\phi|^{2} = (\partial_{\mu}\sigma)^{2} + (\partial_{\mu}\pi_{a})^{2},$$ $p,n$ are proton and neutron field respectively, $\pi_{a}$ is pion triplet, while $\sigma$ is so-called sigma-particle. Historically it was introduced by Gell-Mann and Levy in 1960. By this theory, initial global internal symmetry of theory is spontaneously broken, $$SU_{L}(2)\times SU_{R}(2) \to SU_{\text{isospin}}(2),$$ nucleons obtain masses in invariant way, while pions (being Goldstone bosons) obtain masses because of explicit breaking by $c\sigma$ term.

The question is: what was experimental origin of proposing $SU_{L}(2)\times SU_{R}(2)$ symmetry? I don't understand this clearly from cited article.

You might consider 241162 and 102575. This is actually a history of science question, on a half-century old landmark in the intellectual history of 20th century physics, whose importance cannot be overstated. Perhaps it belongs to another site.

For a quartic potential, following Schwinger, the σ mass may be sent to infinity, thereby also introducing the nonlinear σ model, as the authors do in section 6.

Your question of the "experimental origin" of contemplating the chiral symmetry of two flavors (before quarks or an inkling of the underlying dynamics, or effective lagrangians, which this represents, of course!) misreads the significance of the paper: it is not a phenomenological fit to data: it is a creative grand synthesis conjecture of several physical facts and ideas coming from all over the place.

The stated purpose of the paper is to explain/rationalize the baffling Goldberger-Treiman formula of the time and other low energy theorems of hadronic physics in terms of PCAC, all without an inkling of QCD or quarks. G-M & L adapted this simple model to illustrate how the 3 vector currents of isospin was "almost" conserved (CVC), but the other 3 current you could make out of the Dirac spinors of the nucleons, the Axial ones, were "partially conserved" (PCAC): their divergences cannot vanish, since then the pions would not decay, but they "nearly vanish", as Feynman discovered by fiddling with weak axial currents, i.e. these are proportional to the pion fields, the hallmark of spontaneous chiral symmetry breaking.

Specifically, for the vectors denoting isospin indices, $$\vec{A}_\mu \sim \sigma \partial_\mu \vec{\pi} + g_A \bar{N} \gamma_\mu \gamma_5 \frac{\vec{\tau}}{2} N ~,$$ and shifting σ ⟶fπ + σ' would yield a leading term of $f_\pi \partial \vec{\pi}$ for the current, so $$\langle 0| A^a_\mu(x) | \pi^b(p)\rangle \sim \delta^{ab}~ f_\pi ~ p_\mu ~e^{-ip\cdot x},$$ whose divergence would then go like $m_\pi^2$ and so vanish for vanishing pion masses. "Nearly"...

This, then would embolden good theorists to theorize about the limit of vanishing pion masses, today called the "chiral limit", and perturb around it, what is now called "chiral perturbation theory" in the explicit breaking masses...

Conserved Vs and As would then be trivially unscrambled to the standard current algebra of $SU(2)_L\times SU(2)_R$ that you mention, V-A=L, V+A=R, with all Rs commuting with all Ls. Feynman and G-M and others had already deciphered the L, V-A nature of the weak interactions, pion decay, etc...

Now for the Goldberger-Treiman relation, the piece de resistance of that paper. By positing a form of the Axial current related to the nucleon mass (=fπ gA !) you could relate that to pion decay ($f_\pi$), the weak decay constant and the pion-nucleon Yukawa coupling ($g_A$).

At the time, the confluence of weak and strong and current algebraic quantities into a tight relationship appeared almost miraculous, and a model that rationalized them all in the context of elegant global spontaneous symmetry breaking a godsent! Nowadays, they are mere footnotes in a QFT text, like M Schwartz's, for instance. You might read up in Georgi's classic text.

Supplementary pedantic edit on chiral perturbation: It struck me this might well offer a gratuitous teachable moment on the explicit breaking term , which you appear to appreciate, anyway, but some of us never tire of it when teaching the course... While the rest of the model is invariant under the 3 isospin and the 3 axial transformations (admire the concerted subtle invariance of the first, fermion, term under the axials and hence $SU(2)_L \times SU(2)_R$) the - term is not under the axials: it shifts by $\propto c \vec{\theta}_A\cdot \vec{\pi}$.

To lowest order in c, then, this extra perturbation shifts <σ> from $f_\pi$ to $f_\pi (1-\frac{c}{8\lambda f_\pi^3}+...)$ and so the mass of π from 0 to $m_\pi^2 \sim c/2f_\pi$ (Dashen's theorem). So taking the 4-divergence of the axial current $\vec{A}_\mu(x)$ on shell produces the $m_\pi^2$ of PCAC. In contemporary QCD language, $c=m_q \Lambda^3/f_\pi$, where Λ is a chiral condensate and $m_q$ is some average of GellMann-Oakes-Renner quark masses.

A final aside, just like theirs. On p708, their "note added in proof" introduces the Cabbibo angle, arcsin $\epsilon/\sqrt{1-\epsilon^2}$, three years before Cabbibo's paper (which references this one): the fine print behind the other constants of the G-T relation mumbled about above. They relate the coupling strengths of the strange and non-strange hadronic currents to the weak leptonic one, as the sides of a right triangle.

• Thank You for Your question! But now I don't understand how to formulate the PCAC relation. It has the form $$\tag 1 \langle \pi^{b}(p)| J^{\mu}_{a}(x)|0\rangle \sim \delta^{b}_{a}p_{\mu}f_{\pi}e^{-ipx}$$ But what is $J^{\mu,5}_{a}$? In the linear $\sigma$-model it has the form $A_{\mu}$ from Your answer, but what did people mean when writing $(1)$ before the linear $\sigma$-model? Jan 20, 2017 at 16:12
• You are plumbing the depths of my historical knowledge... I thought (1) started with the σ-model, even though eqn (5) , p 709, insinuates MGM helped RPF in discovering it by fiddling... The axial part including nucleons, however, is the decisive piece of the prior Feynman-GellMann V-A weak interactions' effective theory. As I indicated, the σ-model paper bears rereading half a dozen times.... Jan 20, 2017 at 16:29
• so, if I understand You correctly, You say that PCAC was rather the guess of Gell-Mann and Levy? Jan 20, 2017 at 19:36
• I certainly believe so! but I am open to revisions, if I get to see more/older stuff. I am certain that PCAC got taken seriously, if not invented, in that miraculous GM-L paper.... Jan 20, 2017 at 19:45
• Well, this is the problem with history... Few people care or know about G-T today... it's an epiphenomenon of QCD. PCAC, on the other hand, proved to be the key and foundation of SSB, in retrospect. But maybe you should read up on A Pais's Inward Bound if you wished to get too historical... I can't claim I know all details... Maybe Nambu's Nobel speech.... Jan 20, 2017 at 22:49