# What does Gribov's last paper tell about coloured states?

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In the first days of July 1997, after a long driving effort, crossing all of Europe to come to a meeting in Peñiscola, Vladimir Gribov fell fatally sick and he passed away one month later. His paper "QCD at large and short distances" was uploaded posthumously to arXiv:hep-ph/9708424 and, in edited version, to arXiv:hep-ph/9807224 one year later. Still a remaining write-up, "The theory of quark confinement", was collected, edited and uploaded later to arXiv:hep-ph/9902279.

I guess that I am asking for some general review of these articles, but I am particularly intrigued by the conclusion as it is exposed in the introduction to the first one (bold emphasis is mine).

"The conditions for axial current conservation of ﬂavour non-singlet currents (in the limit of zero bare quark masses) require that eight Goldstone bosons (the pseudo-scalar octet) have to be regarded as elementary objects with couplings deﬁned by Ward identities."

"... the ﬂavour singlet pseudoscalar η′ is a "normal bound state of qq¯ without a point-like structure"

How serious is this elementary status? For instance, in order to do calculations of electroweak interaction, should the bosons in the octet be considered as point particles, say in a Lagrangian? Does it imply that the QCD string, at low energy at least, is point-like?

And, does it happen the same for the colored state having two quarks? (What I mean here is the following: the color-neutral states have been produced from the decomposition $3 \times \bar 3 = 8 + 1$ of SU(3) flavour, joining a quark and an antiquark. Similarly, we could use QCD to join two quarks, and then SU(3) flavour should decompose as $3 \times 3 = 6 + 3$) Is the flavour sextet "pointlike"? And the triplet then, is it still a "normal bound state"?

I expect the argument does not depend of the number of flavours, so the same mechanism for SU(5) flavour should produce a point-like color-neutral 24 and, if the answer to the previous question is yes, a point-like colored 15. Is it so?

Let me note that most work on diquarks concludes that only the antisymmetric flavour triplet, not the sextet, is bound by a QCD string --e.g., measured as $\sqrt 3/2$ times the meson string here in PRL 97, 222002 (2006)--.

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I don't understand this question: there are no "coloured states" in a confining theory. The paragraph you quote seems completely standard textbook material about pions being Goldstone bosons, and the $\eta'$ getting a mass from the anomaly (i.e. the U(1) problem). Other than that, you are just flagging two old papers for attention.

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Representation theory is hardly the issue. All physical states in QCD are colour singlets, why are we discussing coloured states? I'd like to see some connection to current mainstream research, potentially something I am not familiar with.

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I am sorry, what did X mean when he said Y is not a physics question. Those labels you singled out have no technical meaning, at least not specified in the question. Once again, the sextet you mention is not a physical state, so you are not asking a well-defined physics question. I still don't see how this contains a precise answerable question.

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@Moshe let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/1718/discussion-between-arivero-and-moshe) if you wish

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On the general topic of Gribov's work, such as the pion-adjusted quark Greens functions, I think it points the way to an interpretation of QCD in terms of the bootstrap, just as some recent work on N=8 sugra S-matrix is portrayed as revival of the bootstrap. At Physics.SE, Ron Maimon has been the great bootstrap missionary, and he even recommends a specific book by Gribov http://books.google.com.au/books?id=gS7we7IjvKQC ... Also see hep-ph/0106200 for related commentary, e.g. the argument that SQCD behavior supports Gribov's ideas about origin of confinement.

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Please note that I have included the initial narrative not only for sentimental reasons, but also to stress that the work was never communicated in public. It is possible that the point-likeness arguments are just remains of previous speculations.

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If the chiral flavor $SU(3)$ symmetry were exact to start with, one could define $j_\mu(x^\alpha)$ currents transforming as the adjoint (octet) which would be strictly local fields. Once the symmetry would get broken, the very same currents which don't display any internal structure could act on the vacuum – which isn't invariant under the symmetry anymore – and create the Goldstone bosons. In this construction, they're not constructed out of anything more elementary; they seem to be elementary fields.

However, this statement is really vacuous physically because all the complications about the possible (and, in the real world, real) internal structure of the Goldstone bosons would be reflected in the interactions of the Goldstone bosons with each other and with other things. Nothing prevents these interactions from becoming very complicated and non-polynomial right at the QCD scale and above it (and even slightly beneath it). And indeed, this is what happens.

So these Goldstone bosons may look like elementary fields in some description but of course, you won't be able to find any renormalizable theory in which they would be elementary point-like bosons with polynomial interactions. Instead, all objects in QCD are made out of quarks and gluons and this fact makes it pretty obvious that the "non-locality" i.e. apparent size of internal structure of all these particles is comparable with the QCD scale i.e. the proton radius. The flavor currents formally depend on $q$ and $\bar q$ fields at the same point only. However, that's just the leading approximation and there are loop corrections suppressed by the strong coupling constant. Because the strong coupling runs and is really equal to $O(1)$ at the QCD scale, these corrections really allow non-local interactions between the pions at distances comparable to the QCD distance scale.

The Goldstone theorem's description doesn't allow you to simplify any physics or derive any shocking insights about the QCD dynamics; in particular, the QCD string is as complicated as we always knew it to be.

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answered Nov 3, 2011 by (10,278 points)
It seems that Gribov was arguing beyond Goldstone theorem, introducing a "supercritical" phase where a bound particle "falls into the centre" and showing that this was the case for the octet but not for the singlet.

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Well, maybe he was, @arivero, but what's more important on this forum is that the claim is incorrect. If you want to discuss incorrect things that were argued by the people in the past, you should go to a History of Physics Stack Exchange.

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