The Problem.

The issue that I'm talking about is the large-instanton problem of asymptotically-free non-abelian gauge theories. You can read about it in:

Here is basically what it is: for $\mathrm{SU(N)}$ gauge-theories (which admit standard BPST instantons), the dominant contribution of gauge-field configurations to the path integral (i.e. for quark correlators) comes from a single instanton field.

$$ \begin{align} \langle\psi(x_1)\cdots\psi(x_n)\bar\psi(y_1)\cdots\bar\psi(y_m)\rangle&\sim\underset{(1)}{\underbrace{\int d\mathcal{M} \,\exp(-S_{\mathcal{M}})}}\times\\ &\underset{(2)}{\underbrace{\int \mathcal{D}\bar\psi\mathcal{D}\,\exp\left[\bar\psi(\gamma^\mu D(\mathcal{M})_{\mu}+m)\psi\right]\psi(x_1)\cdots\psi(x_n)\bar\psi(y_1)\cdots\bar\psi(y_m)}} \end{align}$$

where $\int d\mathcal{M}$ is an integral over the moduli space for a single $k_{\textrm{winding}=\pm 1}$ (anti-)instanton. The term (1) is an integral over this moduli space, with the gauge-field part of the action. The term (2) is the remaining path integral over $\psi$ and $\bar\psi$, in the background of a single-instanton $\mathcal{M}$. The term (1) looks like:

$$ d\mathcal{M} \,\exp(-S_{\mathcal{M}})\sim \frac{d\rho}{g^8(\mu)}\rho^{-5}\exp\left[-\frac{8\pi^2}{g^2(\mu)}+C_1 \ln(\mu\rho)\right] $$

where there is a lower-bound on $\rho$ set by $1/\mu$. For asymptotically free gauge theories, the limit $\rho\rightarrow 0$ (actually $\rho\rightarrow 1/\mu$, and then $\mu\rightarrow\infty$) is OK in term (1), because

$$g^2\sim 8\pi^2/b\ln(\mu\Lambda)+\mathrm{(subleading corrections)}$$

assuming $b$ is large enough, which for QCD it is ($b=9$ in QCD, so $n(\rho) d\rho \sim \rho^{b-5} d\rho$ is convergent for $\rho\rightarrow 0$).

However we see that the ostensibly unrestricted $\rho\rightarrow \infty$ IR limit is not controlled — term (1) diverges. This is the large-instanton problem, more or less.

Possible Resolutions.

So here are the possible resolutions that one might think of.

  1. The integral $d\mathcal{M}$ necessarily includes the fermionic integral! What I mean is, to evaluate the correlation function, first we must evaluate $(2)$, and then evaluate $(1)$ — they are not separate. What happens if we include this integral? Can we come up with an effective Lagrangian for fermions after integrating over these instantonic configurations? Maybe that will provide an cutoff for large $\rho$.

This is precisely what 't Hooft answered in [1.]. The short answer is, no, the effective Lagrangian for $\psi$ has the same large-$\rho$ issue.

  1. Maybe we were naive in assuming that only single instanton configurations dominate. Maybe we need to also include multi-instanton configurations. When we integrate over their collective radii $d\rho_1\cdots d\rho_n$, maybe we will see a cutoff.

My Questions.

  1. The two papers that I mentioned before [1.][2.] solve this issue in a very strange way - they show that if you include a Higgs field $\mathcal{L}_{\textrm{Higgs}}=-(D\phi)^{\dagger}(D\phi)-\mu^2\left(\phi^{\dagger}\phi-\nu^2\right)^2$, through an interesting method known as constrained instantons, you can show that now the $d\rho$ integrand obtains a contribution $\sim\exp\left[-c_2\rho^2-c_3\rho^4\ln(\nu\rho)\right]$ so it gets rapidly cut off at large $\rho$. But in the real world, we do not have a Higgs field for the asymptotically-free $\textrm{SU(3)}$ sector,i.e. QCD! The real Higgs field is only in a non-trivial representation for the electroweak sector, which at our scales is very weakly coupled and which is not asymptotically free. Maybe this would be fine if we assumed the existence of a super-duper heavy Higgs field for QCD, whose dynamics are basically completely removed from our real world (because of its huge mass).

  2. Many modern papers, such as Dynamical Supperssion of Large Instantons (Munster & Kamp, 2001), say that actually the large instanton sizes are suppressed by multi-instanton interactions. We should not integrate over the moduli space of one instanton, but over many instantons & anti-instantons. So if you look for instanton configurations on the lattice, you will indeed find instantons that have a distribution in $\rho$ that goes like $n(\rho)\sim\exp(-c\rho^2)$ for large $\rho$, i.e. there is no large-instanton problem. If this is indeed the resolution to the large-instanton problem which is relevant to reality, why does anybody even talk about introducing Higgs-fields and using the complicated machinery of constrained instantons?


2 Answers 2


Coleman famously called it an "IR embarassment", not an IR problem, because it takes place in a regime where we cannot perform reliable calculations. The basic answer to your question is that, in general, the fate of large instantons in QCD is not a well defined question.

Some comments:

  1. What is the large instanton problem? We are attempting to apply the semi-classical approximation to Yang-Mills theory and QCD. We find a non-trivial saddle point, the instanton, with action $S=8\pi^2/g^2$. Because of classical scale invariance instantons come in all sizes, so the saddle point is labeled by a collective coordinate (moduli) $\rho$ that we have to integrate over. This seems like a problem, because the measure on the moduli space (again, just by scale invariance) is $d\rho/\rho^5$. Here, Gaussian fluctuations around the saddle come to the rescue, because they convert $g^2$ to $g^2(\rho)\sim b\log(\rho\Lambda)$. Now the integral is convergent in the UV (good), but diverges in the IR. We should not really be surprised, because the expansion parameter in the semi-classical expansion is $S\gg 1$. However, YM and QCD do not have any dimensionless parameters that could determine the value of $S$.

  2. Remarks about Higgsing etc refer to a slightly different question: Are there QCD-like theories (or scenarios in which QCD is coupled to external fields) in which the semi-classical expansion is rigorous, and the fate of large instantons can be resolved?

  3. The answer is yes, there are many theories of this type: 1) QCD at large temperature. Debye screening in the quark gluon plasma acts like a colored Higgs field, and the topological susceptibility at $T\gg\Lambda_{QCD}$ is calculable. 2) QCD at large baryon density. Again, Debye screening in a dense quark liquid acts like a Higgs vev, see, for example, here. 3) Supersymmetric extensions of QCD or YM with colored Higgs fields, most famously ${\cal N}=2$ SUSY YM, studied by Seiberg and Witten. 4) QCD compactified on suitable manifolds, for example QCD on a circle $R^3\times S^1$ (this is basically finite temperature QCD, except that one may modify boundary conditions on the circle), or on a torus.

  4. Some of these scenarios are interesting beyond their immediate regime of applicability. For example, in SUSY theories we encounter cases where the semi-classical calculation can be performed for large Higgs vev or on a small circle, and then SUSY ensures that the result is correct for any Higgs vev or circle size (that is in a limit in which the semi-classical calculation is sensitive to large instantons).

  5. Lattice QCD is a rigorous approach to QCD in the strongly coupled regime. In lattice QCD we can identify small instantons and study their distribution. We cannot identify large instantons, because large instantons have weak fields and small action, so they cannot be distinguished from ordinary perturbative fluctuations. Only the total topological charge $Q$ of a configuration can be measured. We cannot determine how $Q$ breaks down into the number of instantons and anti-instantons, $Q=N_+-N_-$.

  6. Progress has been made in analyzing the (resurgent) semi-classical expansion. For some generic observable $O$ $$ O = (a_0+a_1g^2 + a_2 g^4 + \ldots ) \\ + (b_0 + b_1 g^2 + \ldots) \exp(-8\pi^2/g^2) \\ + (c_0 + c_1 g^2 + \ldots ) \exp(-16\pi^2/g^2) + \ldots $$ which is a sum over the 0-instanton, 1-instanton, etc, sector. It was shown that ambiguities in summing the perturbative series $a_i$ are canceled by ambiguities in $c_i$, and ambiguities in summing $b_i$ are related to ambiguities in $d_i$, etc. This is known as the resurgence program. However, this program still requires an external scale (a compactification scale, for example) to define the coupling $g$. In QCD at zero temperature and without external scales the dependence on the coupling is fixed by RG invariance, and there is no expansion parameter.

  7. Ideas about resolving the large instanton problem by multi-instanton effects are model-dependent statement about what might happen in the strong coupling regime. It is obviously true that there is a relation between large instantons and the instanton-anti-instanton interaction. If instantons are very large, then they also overlap strongly. However, the resurgence program makes it clear that these are separate issues. There are systems where large instantons are removed by introducing a scale, and the remaining instanton-anti-instanton problem is solved by resurgence.


In "Instantons in QCD" by E. Shuryak and T. Schafer, this issue is answered in section III.C.4. The answer is basically that we have ignored quantum fluctuations, whose effects go into $g(\mu)$ which we have assumed scales according to the one-loop perturbative result, which is obviously not valid in the strong-coupling region $\mu\rightarrow 0$. The one-loop weak-coupling formula we had used was

$$g(\mu)\sim 8\pi^2/ b\ln(\mu\Lambda) \tag{1}$$

When measured numerically on the lattice, the coupling actually begins to level off at lower $\mu$, in contrast with the divergence predicted by eq (1). By loosely plugging in this property to the equation for the single-instanton measure (or equivalently, the single-instanton size distribution $n(\rho)$), we see that $n(\rho)$ could converge, and in fact it does. To get an idea, if the coupling were to saturate a constant value $g(\mu ) = g_*$ (it definitely does not, since QCD has no IR fixed point), then $n(\rho)\sim \rho^{-5}$, which is totally convergent for $\rho\rightarrow \infty$.

One might question whether the semi-classical expansion in terms of increasing number of instantons is even valid for $g(\mu\rightarrow 0 )$. Recall that the validity of this expansion is not reliant on whether $g(\mu)$ is small or large, but rather if $S_0=8\pi^2/g^2(\mu)$ is large - the statistical weight is given by $\exp (-S)$. Nonperturbative estimations of $g(\mu)$, which admittedly are scheme dependent, show that the quantity $8\pi^2/g^2(\mu)$ remains large as $\mu\rightarrow 0$.


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