At one point, I decided to make friends with the low-lying spectrum of QCD. By this I do not mean the symmetry numbers (the "quark content"), but the actual dynamics, some insight.

The pions are the sloshing of the up-down condensate, and the other pseudoscalars by extending to strangeness. Their couplings are by soft-particle theorems. The eta-prime is their frustrated friend, weighed down by the instanton fluid. The rho and omega are the gauge fields for flavor SU(2), and A1(1260) gauges the axial SU(2), and they have KaluzaKlein-like echoes at higher energies, these can decay into the appropriate "charged" hadrons with couplings that depend on the flavor symmetry multiplet. The proton and the neutron are the topological defects. That accounts for everything up to 1300 but a few scalars and the b1.

There are scalars starting at around 1300 MeV which are probably some combination of glue-condensate sloshing around and quark-condensate sloshing around, some kind of sound in the vacuum glue. Their mass is large, their lifetime is not that big, they have sharp decay properties.

On the other hand, there is nothing in AdS/QCD which should correspond to the sigma/f0(600), or (what seems to be) its strange counterpart f0(980). While looking around, I found this discussion: http://www.physicsforums.com/showthread.php?t=241073. The literature that it pointed to suggests that the sigma is a very unstable bound state of pions (or, if you like, tetraquarks).

This paper gives strong evidence for an actual pole; another gives a more cursory review. The location of the pole is far away from the real axis, the width is larger than the mass by 20% or so, and the mass is about 400MeV. The authors though are confident that it is real because they tell me that the interpolation the interactions of pions is safe in this region because their goldstone properties dominate the interactions. I want to believe it, but how can you be sure?

I know this particle was controversial. I want to understand what kind of picture this is giving. The dispersion subtraction process is hard for me to visualize in terms of effective fields, and the result is saying that there is an unstable bound state.

Is there a physical picture of the sigma which is more field theoretical, perhaps even just an effective potential for pions? Did anyone who convinced himself of the reality of the sigma have a way of understanding the bound state properties? Is there an analog unstable bound state for other goldstone bosons? Any insight would be welcome.

  • $\begingroup$ and also i don't think this particle exists, they probably made it up so they can get more government grant money. $\endgroup$
    – Timtam
    Aug 13, 2011 at 12:56
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    $\begingroup$ @Timatam: you're not the only person to have this idea. This particle was delisted for a long time, meaning people concluded that it was just a made up particle, just a bump in the cross section due to pion interactions. But the experimentalists shouted that it was really there, and people who do extrapolations of experimental data to extract particles now say that have a pole. I also used to think it was made up, this is why I am confused. $\endgroup$
    – Ron Maimon
    Aug 13, 2011 at 17:05

2 Answers 2


It is the lowest lying scalar meson, and scalar mesons, sharing quantum numbers with the vacuum, are notoriously hard to study. So, I've watched it entering and dropping off the PDG over decades. Today, it is there as a broad resonance, $f_0 (500)$, of about 441 MeV, according to Leutwyler, who should be the most reliable maven of it. The point is that it is wider than it's heavy, as its full width is about 544MeV, a dismal fate that the Higgs escaped!

In effective QCD, it dominates chiral symmetry breaking very analogously to the Higgs' informing EW symmetry breaking: in fact, theoretically, it has served as the conceptual underpinning of the Higgs for more than 40 yrs. Theorists love it more than experimentalists, and here is why:

Introduced in 1960, by Gell-Mann, M.; Lévy, M., "The axial vector current in beta decay", Il Nuovo Cimento 16705–726, doi:10.1007/BF02859738, following a hint by Schwinger, it gives its name to the σ-model introduced there. The effective Largrangian (diffidently called "model" back then) had kinetic terms for the (p,n) light nucleon doublet, the 3 πs and this σ, and a Higgsoid quartic potential $\lambda (\sigma^2+\vec{\pi}^2 -f_\pi^2)^2$ where I'm being cavalier with the normalizations of the fields and constants; and, moreover, most crucially, a Yukawa coupling term to the nucleon doublet $g\overline{\psi}(\sigma +i\vec{\tau}\cdot\vec{\pi} \gamma_5)\psi$.

This interaction term, like the rest of the O(4)~ SU(2)xSU(2) invariant action, is also SU(2)xSU(2) invariant. When this group is broken down to the (strong!) isospin SU(2) spontaneously (Nambu's Nobel prize), the σ shifts by $f_\pi$, and crucially the 3 Axial vector current combinations are now realized nonlinearly, i.e. $\vec{A_\mu} = f_p \partial_\mu \vec{\pi} +$ bilinear terms... so they pump Goldstone pions into and out of the degenerate vacua while the isospin vector currents remain bilinear, so isospin is still unbroken.

Of course, from the Higgsoid potential, e.g. quartic, the quadratic term of the σ picks up a mass proportional to $f_\pi=93 MeV$ and the square root of the mystery effective quartic coupling of order one, which one never specified, as this is the strong interactions, after all... So, one expects a few hundreds of MeVs, which is what you get... Note it is, of course, heavier than the pions and kaons, which are pseudogoldstons, but not the ρ, the otherwise lightest "real" hadron.

The magic serving as the prototype of the EW standard model SSB is that now the masses of the nucleons come out of the above Yukawa term, $g\overline{\psi}( f_\pi +\sigma' +i\vec{\tau}\cdot\vec{\pi} \gamma_5)\psi$ to be about $g f_\pi$, about a GeV, schematically: $g$ is a strong nucleon-pion coupling larger than 10.

Today, this dynamical symmetry breaking is all described by quark condensates and calculated on lattice QCD, but the simplicity and elegance of the model in parsing out the logic is unbeatable. Since the σ can be there symmetry-wise, it would be odd if QCD did not manage to conjure an avatar for it at some level or other, but, in practice, it is a resonance from hell---or nuclear/hadronic physics.

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    $\begingroup$ What a nice answer! $\endgroup$
    – rob
    Mar 3, 2016 at 1:22

The literature that it pointed to suggests that the sigma is a very unstable bound state of pions (or, if you like, tetraquarks).

This has reminded me of a really interesting paper by Shifman and Vainshtein - http://arxiv.org/abs/hep-ph/0501200 - which talks about an exact symmetry between pions and diquarks in two-color QCD. They speculate that this symmetry should have an analogue in three-color QCD. I'm wondering if the sigma might be a sort of diquark-diquark correlation enhanced by the Shifman-Vainshtein symmetry.

I'd also look at Hilmar Forkel's work for AdS/QCD insight.


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