# The scaling limit of a quantum field theory

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What happens to a quantum field theory when all physical masses (including those generated dynamically) tend to zero?

(The resulting scaling limit was taken as the starting point of Wilson's hadronic skeleton theory that he extracted from his operator product expansion.)

edited Apr 20, 2015

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The high energy behavior of all quantum field theories (QFTs) is essentially independent of the masses of its fundamental particles, since the latter are small compared to the energy scale. Thus they can be neglected numerically, leaving experimentally accessible scaling laws.

If one tries to do the same in the $n$-point functions of the theory one ends up with a theory having an additional scaling invariance - the dilatation becomes an additional symmetry. Under suitable (apparently weak) additional conditions the resulting field  theory - called the scaling limit  - is even conformally invariant, hence a conformal field theory (CFT).

In the scaling limit there is no notion of bound states or scattering states - the Green's functions have no poles but only branch cuts. Thus a scaling limit is a bootstrap theory, where all bound states are treated on the same footing.

Wilson's hadronic skeleton theory would in today's terms be the scaling limit of QCD, hence should be a conformal field theory with massless mesonic, baryonic, and nucleonic fields, whose scaling dimensions agree with those derived from QCD.

Note that, as for any asymptotically free theory, the renormalization of QCD introduces through dimensional transmutation a physical mass scale $\Lambda_{QCD}\approx 520 MeV$ (not to be confused with any unphysical cutoff used in some renormalization schemes); see, e,g., (2.39) in Fischer. The Gaussian fixed point responsible for asymptotic freedom in the ultraviolet (UV) is given by the infinite mass limit  $\Lambda_{QCD}\to\infty$, while the hadronic skeleton theory is given by the non-Gaussian fixed point obtained in the massless limit $\Lambda_{QCD}\to 0$. The properties of the latter theory are the result of nonperturbative, poorly understood infrared (IR) properties of QCD.

The multitude of local fields is generated (in any scaling limit, not just in hadronic skeleton theory) by taking the local limit of appropriate products of a generating set of fields at infinitesimally close arguments.

After all - infinitely many - local fields are collected, one distinguishes those that are not derivatives of other local fields as primary, calling the others secondary. The primary fields can be ordered by increasing scaling dimension; this places the most relevant fields first.

By truncating the field theory to fields of scaling dimension below some threshold only, one gets an approximation that is (to some extent) numerically tractable via a correspondingly truncated operator product expansion (OPE). The coefficients of this truncated OPE must satisfy very strong restrictions coming from the representation theory of the conformal group. In addition they must satisfy certain inequalities (unitarity constraints) that can be exploited.

No (rigorous) interacting 4D CFT is known at present, but this doesn't mean much as also no interacting 4D massive QFT is known. Thus the properties of the hadronic skeleton theory are still widely unexplored.

Relevant rigorous work is Bostelmann, and in 2D,  Bostelmann et al.  -  Bostelmann et al..

For references about the conformal bootstrap see here, here, and here.

A paper by Witten (1984) called ''Skyrmions and QCD" mentions

although QCD is equivalent to a meson theory [...] it is an extremely complex and unknown theory with an infinite number of elementary fields

and gives more details about how  QCD (at the time unknown to Wilson) is related to this quark-free hadronic theory whose high energy limit would be Wilson' hadronic skeleton theory.

answered Apr 12, 2015 by (14,019 points)
edited May 28, 2015

Isn't QCD asymptotically free so the UV CFT is free as well? (On the physical level of rigor.)

@ArnoldNeumaier: What you are writing is a personal interpretation, this was not at all historically what Wilson meant in his paper in 1969. Wilson meant the high-energy microscopic theory of the strong interaction, which he imagined would be a strongly interacting 4d field theory with noncanonical field dimensions. Asymptotic freedom was not yet discovered in 1969, and people had no idea what the microscopic theory of strong interactions would look like.

At the time, people imagined that the microscopic theory would end up being an exactly scale invariant theory, with the only breaking of scale invariance from the quark mass terms. This expectation wasn't really justified in hindsight. My personal impression is that it came from the Gell-Mann Okubo style analysis of flavor SU(3) breaking, where the form of the mass term considered as a spurion determine the mass-breaking of the SU(3) multiplets very accurately. This made it obvious that QCD was some sort of SU(3) symmetric theory with masses breaking the SU(3) symmetry (this is also true).

But people then (unjustifiably) thought that the masses were the only scales in the theory. At the time, only Gell-Mann and a few others appreciated running coupling and dimensional transmutation, this is before Kallen-Symanzik popularized the RG. So people worked on scale invariant theories, and they imagined that the quark masses were the only dimensional scale in the microscopic theory breaking the scale invariance, and so the remaining theory at zero quark mass would become a nontrivial scale-invariant theory of conformal type, like you describe.

The effects of the quark mass terms disappear at high energy as a power of the energy-scale over the mass, so if you imagine scattering at higher energies than the mass, and you recover a scale-invariant theory quickly.

Nowadays, we know that this QCD is not scale invariant even at zero quark masses, the breaking of scale invariance is in the running coupling, which gives a slow variation with scale. The true high energy limit of QCD is a free theory, and the scaling corrections are logarithmic running, so that there is a slow dependence of the QCD coupling on energy, which produces a dimensional scale of $\Lambda$ in all hadronic cross sections, with the only input generating $\Lambda$ a dimensionless coupling defined at some tiny dimensional length scale. But this insight was a few years in the future when Wilson wrote his paper.

Wilson defines the hadronic skeleton theory as the limit where all quark masses are set to zero, because he imagined that when you set the quark masses to zero, there is no more energy scale, and you recover the true microscopic strong-interaction theory. He also assumed this theory is a crazy strong-coupling bootstrap-style theory of the kind you talk about, so he defined the OPE to deal with this putative theory nonperturbatively. The thing that was radical about using the OPE is that it isn't really a bootstrap in the sense of 1960s physics (although now it is called the conformal bootstrap), it is introducing local fields and defining an algebra of multiplication on these, which means it is really field theory. It is just a new kind of field theory which does not start from free field theory, rather it starts from arbitrary conformal fields. This new kind of field theory was taken up by Polyakov, who also called it a "conformal bootstrap", partly for political reasons, but partly because it doesn't involve perturbative analysis, and it determines critical exponents. A real bootstrap would not admit any local fields at all, like in string theory. This is not a bootstrap in the latter strong sense.

The actual high-energy fixed point of QCD is just zero coupling QCD, or free gluons and quarks, where the OPE dimensions are integers, and the OPE coefficients can be computed perturbatively. But because Wilson imagined something else at high energies, and he gave the name "Hadron skeleton theory" to this UV fixed point. It's good that he did, because this analysis is useful for 2d conformal fields, and other situations where you actually have a nontrivial scale-invariant fixed point, as compared to the relatively boring case of asymptotic freedom.

Given a $\Lambda_{QCD}$ and massless quarks, you can take the limit $\Lambda\rightarrow 0$, but this is nothing interesting, it's the same as going to the ultraviolet fixed point. This limit gives free theory of QCD with g=0, eight free photons and 6 triplets of quarks, each infinitesimally charged under a different combination of the eight free gluons, the charge going to zero when $\Lambda=0$. As $\Lambda$ gets smaller relative to any fixed energy scale, the coupling at that fixed scale gets weaker, by the running. This limit is not the nontrivial Hadronic fixed point you want.

What you are talking about is the other limit, the limit of low-energy theory, the infrared theory. Here you take $\Lambda \rightarrow\infty$ at fixed energy scale, which, if you keep, say, the pion mass fixed, would indeed be the limit of zero quark mass. You could not take this limit and keep the Hadron mass fixed, because the Hadron mass blows up as $\Lambda\rightarrow\infty$.

An alternative way to say the same thing is to take the pion mass to zero and look at pion-interactions at long distances. This theory in principle could be a nontrivial scaling theory of pion-like objects and other massless hadrons with branch cuts and no distance scale. That's what happens in Banks-Zaks theories.

But unfortunately, in the case of QCD and massless quarks, the infrared limit is sigma-model type and also boring--- it's just free massless pions. The pion interactions are sigma-model type, as the pions are flavor symmetry gauge bosons with sigma-model interactions, and these interactions in the effective pion theory are non-renormalizable and go to zero quickly at long-wavelengths. The result is free field theory in both the infrared (pions), and ultraviolet (quarks/gluons).

I agree that the nontrivial theories that you are talking about is sort of what Wilson meant, but he wasn't clairvoyant. In the 1960s, the two regimes of perturbative QCD at high energies and infrared QCD with the effective sigma-model of Chiral perturbation theory were not yet completely defined and distinguished well, and Wilson meant a putative non-free ultraviolet fixed point of QCD. The real fixed point is free gauge theory, so that part of his paper is obsolete (but still important). It inspires modern work on the nonfree infrared fixed points, and Wilson did imagine such things, but they aren't relevant to real QCD.

@ArnoldNeumaier; I agree he didn't mean an asymptotically free theory of quarks and gluons, he thought something completely different was going on. He meant an unknown strongly interacting field theory, with nontrivial noncanonical dimensions for the fields, which he imagined was the ultraviolet limit of the strong interactions. But he was wrong in this expectation, the ultraviolet limit of QCD is not what Wilson imagined in 1969. As we know today, it is free.

His expections (like everyone else's in 1969 except perhaps Bjorken, Symanzik, and Khriplovich) turned out to be wrong, which is why I asked you to update your review to allow people to know the background of 1969 physics he was working in. This is something people should know about, to be able to get the right lessons from the paper about OPE and strongly interacting theories, not the wrong lesson, which is that QCD has a theory of this sort in the ultraviolet or infrared limit.

Because he didn't know there was an asymptotically free regime, he imagined that the confining theory was a hadronic theory with all mass scales collapsed to zero, so that you would get all the Regge trajectory mass scales going to zero, all the hadron masses going to zero, and there would be some conformally invariant strongly interacting theory. I know what he thought would happen, it's exactly what you're saying. It's a great original important idea. Except it's not true, that's not what happens. It's wrong in nature for QCD. We know better today, in hindsight. There is an additional scale in QCD, $\Lambda$, and this scale is not quark masses, and in the limit that $\Lambda$ goes to zero, you recover the true ultraviolet fied point, which is free quarks and gluons, and when $\Lambda$ goes to infinity you get the theory of free pions, and there is no nontrivial interacting theory at either limit.

The gluon and quark fields close under OPE with the composite operators formed by multiplication and derivatives. These operators are "composite", but don't have non-canonical dimensions in the deep ultraviolet, rather they have a free-field OPE corrected perturbatively. with the corrections to free OPE going to zero logarithmically at short distances. This is not what Wilson imagined, he imagined a high-energy theory with noncanonical scaling dimensions and nontrivial composite OPE not like in free field theory, not like in perturbative corrections to field theory, i.e not like in QCD. Like in the Wilson-Fisher fixed point in 3d, which he describes next.

While it is possible that the dynamics of long range QCD can be described dynamically using a completely different long-range approach from QCD, such an approach is not found by "taking the masses to zero", as Wilson imagined. It is necessarily a more sophisticated idea, like finding a string dual to QCD. This involves other limits than simply scaling to short distances, it is perhaps a limit of large N, and if you want it to be scale invariant theory of hadrons in the ultraviolet, you perhaps need to start from a different theory as the initial approximation, perhaps a supersymmetric one. This is a subject of active research. Field theories of the Wilson type, with nontrivial non-canonical field scaling dimensions, are also important, but it is not clear you have this type of theory sitting in any variant of QCD in any limit. The case where you do have such a theory in the infrared in a QCD-like theory is in Banks-Zaks theories, which is why Banks-Zaks theory was such an important breakthrough--- it is a realization of a nontrivial Wilsonian field theory by embedding in an asymptotically free theory.

@ArnoldNeumaier: This is a consequence of asymptotic freedom. The limit $\Lambda\rightarrow 0$ means the limit $g(\mu)\rightarrow 0$ for any fixed $\mu$. I don't know how to give a more detailed argument, this is both the beginning and the end. This is the ultraviolet limit, and the local fields appropriate to this regime are quark and gluon fields, the Hadrons become infinite sized bags, and fall out of the local field description altogether.

I know the intuition you have (and Wilson), it is that there is some magic mass-scale in the strong interactions which, when set to zero, produces a conformally invariant fixed point using local hadronic fields. it's just not true in the usual case in QCD, just varying quark masses, or setting quark masses to zero and varying $\Lambda$ because QCD is asymptotically free in the ultraviolet and trivial in the infrared, so any field theory conformal scaling limit is always a free-field limit.

It is true that you get nontrivial conformal points in other theories (probably not realized in nature in particle physics) which are either asymptotically conformal-nonfree (like N=4 gauge theory) or infrared nontrivial (like Banks-Zaks theories in 4d or the Ising model Wilson-Fisher fixed point in 3d).

@ArnoldNeumaier: The comment above is incorrect about two points--- it is incorrect about Wilson's thinking, and it is incorrect regarding the effect of quark mass perturbation on a conformal theory.

First, regarding the quark mass perturbation--- the perturbations of quark mass might intuitively seem to be a convergent series which you can sum and so on, but this is not all that happens. When you start with a theory with a large number of massless quarks and an infrared fixed point, and you slowly make the quarks more massive (and differently massive), at first, you simply break the fixed point into multiplets of SU(N_c)xSU(N_c), with small breakings, but eventually, at some point as a fraction of the quarks get massive enough keeping the others light, you must introduce a chiral quark condensate for the massless quarks. The turning on of the condensate is not exactly a perturbative effect, it turns the effective potential over and shifts the vacuum to a new position, and the spectrum of the theory has a radical change starting at this point--- it is a phase transition. Describing this phase transition is important, but it would require a careful study not just of a "massless hadronic skeleton theory" (by this I mean what you interpret, namely the large N_f,N_c with N_f/N_c at some conformal limit), but the exact N_f/N_c combination which is precisely at the transition point to forming a condensate, so you know how the condensate turns on as you perturb away from this point, and with what strength. The result is not a simple summation of mass-term perturbations, as you are imagining, because of the phase transition.

Second, regarding Wilson's Hadronic Skeleton Theory: Wilson did not mean a limit of 6 quark masses zero, he meant 3 quark masses zero (but even 6 quark masses zero might not be enough to get rid of the chiral condensate).  Wilson did not mean a conformal window of infrared physics in a theoretical modification of QCD, he didn't know about QCD. He imagined that the quark masses were the only mass scales in the otherwise conformal strongly interacting theory and that the theory actually became a conformal theory in the ultraviolet, so that if you were to look at scales much larger than 1GeV, you would flow to a nontrivial interacting fixed point which looked like a Banks-Zaks theory, with nontrivial dimensions and so on.

This was a common assumption in the strong interaction community in 1969, that the quark masses are the only mass scale, and the theory doesn't depend on scale otherwise.  I'll write a separate review regarding the historical inaccuracies in the paper. They are not Wilson's fault, he was doing correct theory given the context of the time and the assumptions he was working with.

Wilson simply doesn't mean a putative infrared conformal fixed point (or conformal window) in a theory with a large number of light flavors, this is only the most charitable correct interpretation of his ideas today. There's nothing wrong with making a charitable interpretation of ideas, it is the right way to read literature and assign credit, but it's not good for placing historical material in context, and it confuses students.

Wilson believed the strong interactions were a scale-invariant nontrivial field theory in the ultraviolet. All study of such theories owe a lot to Wilson's mistaken thinking. But it is ultimately not true, and the violation of this expectation was exactly why asymptotic freedom was such a revolutionary surprise.

If you interpret Wilson's "hadronic skeleton theory" as the conformal limit of Banks-Zaks type, it wouldn't involve a chiral condensate, so that none of the identifiable Hadrons would appear until you broke the chiral symmetry. All the hadrons would be SU(3)xSU(3) multiplets, rather than SU(3) multiplets. It's not the high-energy limit of taking the Hadrons we know to be massless by parameter modification.

In reality, the strong interactions are not described by a non-free conformal theory anywhere outside of artificial mathematical limits, where you add more light quark flavors, of which 6 might or might not be enough to approach any sort of conformal window. But Wilson only know about 3 quarks, and had no appreciation of QCD gauge coupling running, let alone two-loop running, as this was 1969 and nobody knew there was a gauge theory there.

It is not at all clear that QCD with 6 massless flavors has a significant "conformal window", that's a numerical claim of this paper. This needs to be checked with either simulations or calculations which are reliable, I agree that the situation is murky. I don't know if it's there, or what it looks like. It might be there, but then only because the chiral condensate scale might shrink to small enough size compared to $\Lambda$ that you get some near-conformal region. There is presumably still a chiral condensate at 6 massless quarks, but I'm not sure, it would require a simulation. I guess the claim is that the magnitude of the chiral condensate decreases far enough so that there is an order of magnitude or two where the theory starts to resemble a putative conformal point. It's an interesting question.

Whether this is true or not, it is a significantly different claim than Wilson's. I agree with you that these types of infrared conformal theories are the closest thing to a sensible modern interpretation of the picture that Wilson intended. I just was pointing out that in 1969, Wilson believed that high energy QCD, as it is in nature, was really going to turn out like this, not asymptotically free, and this expectation simply turned out to be wrong.

@ArnoldNeumaier: Nobody is disputing that large N QCD is equivalent to a meson theory, and that the Baryons in this limit are solitons. The only point I'm making is that Wilson in 1969 didn't yet know about QCD, and he imagined only the pure Hadron theory and an ultraviolet fixed point, so he made mistakes in how to take the limit, he thought the Hadron theory is the limit of quark-masses going to zero, and then hadron masses he imagined would collapse to zero. This is not the only limit you need to take, you need to also take the right N_f/N_c combination in the large N limit to be right on the cusp of forming a Chiral condensate, and the interpretation of this theory is quite different than just taking the observed meson masses to zero. I said this already, and this dispute is very minor anyway.

The short distance limit of the meson theory to which QCD is equivalent is different from the short distance limit of QCD itself, as one cannot get in any limit free quarks from mesons. But it should be simpler than the meson theory itself.

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