nLab instanton in QCD

Idea

In QCD instanton configurations of the gauge field control the experimentally observed vacuum structure of the theory.

One considers QCD on a Minkowski spacetime Wick rotated to a Euclidean $\mathbb{R}^4$ and assumes the field strength to vanish at infinity. This makes the gauge field configurations be equivalently $SU(3)$-principal connections on the 4-sphere $S^4$. The second Chern class of the underlying $SU(3)$-principal bundle is called the Yang-Mills instanton-number of the gauge field configuration. The theoretical/experimentally observed vacuum of QCD is some superposition of gauge fields in various instanton number sectors.

In particular the instanton liquid model in QCD assumes that the vacuum (ground state of QCD) is populated by instanton field configurations of average radius $1/3 fm$ (femtometres) with a density of 1 such instanton per $fm^4$. Some details of this model remain subtle, see notably (Schaefer-Shuryak) for a survey.

This instanton sea model suffers however from infrared divergencies that either require a finite-volume cutoff or appeal to other effects (e.g. Witten 98).

But lattice gauge field theory simulations demonstrate numerically that the QCD vacuum is filled with instantons (e.g. Gruber 13, esp. sections 5 and 7). Whether or not their density and distribution may be or has been theoretically derived from first principles is still being discussed (e.g. Schaefer 02)

The Witten-Veneziano formula (Witten 79, Veneziano 79) relates topological instanton charge to the mass of the $\eta'$-particle in the limiting case that the gauge group is $SU(N)$ for $N \to \infty$. Discussion of this relation for the experimentally observed finite case of $N = 3$ includes (Creutz 11).

Experimental evidence

An instanton vacuum is argued to account for

• formation of the gluon condensate (Mikhail Shifman, Arkady Vainshtein and Vladimir Zakharov?, Nucl. Phys. B 147 (1979) 385).

• spontaneous chiral symmetry breaking (Diakonov 96)

• baryogenesis in the early universe by inducing baryon number non-conservation

Critical discussion of the literature

In the physics literature a style of discussion of instantons is wide-spread where the motivation of the formulas considered is dubious. In these references (for instance Schaefer-Shuryak 96 and Forkel 00 and many more) it is said that

1. a YM field of minimal energy, hence of vanishing field strength is one that is pure gauge;

2. that therefore on a Minkowski spacetime $\mathbb{R}^4$ in temporal gauge and assuming fields are vanishing at infinity, these are given by connection 1-forms on $S^3$ of the form $U^{-1} d U$ for $U \colon S^3 \to G$ a smooth function, and

3. that therefore equivalence classes of such fields are given by homotopy classes of maps $S^3 \to G$, hence by the third homotopy group $\pi_3(G)$.

These points are not really correct, even though the conclusion about the classification of instanton sectors turns out to be right.

The correct statement is that winding-number sectors/instanton sectors are equivalence classes of $G$-principal bundles on the one-point compactification $S^4$ of Minkowski spacetime $\mathbb{R}^4$, and these happen to indeed be given by homotopy classes of maps $S^3 \to G$, with $S^3$ regarded as the “equator” of $S^4$. This is for instance stated explicitly in (Nakahara) and used in many references, such as (Witten 85).

Before we come to this, first notice the problems with the above items from the standard physics reviews:

1. Gauge fields with vanishing field strength are flat connections. These are classified by group homomorphisms from the fundamental group of the given space or spacetime to the gauge group. Hence not every such field configuration is “pure gauge”, this is only the case if the fundamental group is trivial. Of course this is the case on $\mathbb{R}^n$ and $S^n$, so the first item above is correct in these cases. But beware that these authors usually justify the third item above by referring to gauge transformations between gauge fields, which is then again wrong, see the next point.

2. Flat connection on $S^3$ are not classified by homotopy classes of maps $S^3 \to G$. Instead, since $\pi_1(S^3)$ is the trivial group, there is only the trivial such flat connection. This is actually manifest in the formula for such connections which is preferred in these references, which is “$U^{-1} d U$” with $U \colon S^3 \to G$ a gauge transformation. That formula exhibits precisely the trivial gauge connection $d$ as gauge equivalent to $U^{-1} d U$.

What would give the intended classification is if one considered connections of the form $U^{-1} d U$ only modulo those gauge transformations $h \colon S^3 \to G$ which can be smoothly extended to maps $D^4 \to G$. This is what these references seem to be implicitly assuming. But this is not the gauge transformation law for connections… this is instead the gauge transformation law for transition functions of $G$-principal bundles.

And this brings us to the correct mathematical interpretation of instantons, as in (Nakahara) and similar:

We are looking for gauge field configurations on $\mathbb{R}^4$ that have field strength vanishing at infinity. This means we are looking for $G$-principal connections on the one-point compactification $S^4$ of $\mathbb{R}^4$ such that the curvature 2-form vanishes at one “pole”.

Now by the discussion at principal bundle, $G$-principal bundles can be constructed for instance by cocycles in Cech cohomology of $S^4$ with coefficients in $G$. In general this means to specify $G$-valued transitions functions on overlaps of a good open cover of $S^4$. But a standard argument shows that for spheres the following covering is already sufficient: we cover the $n$-sphere $S^n$ by two hemi-$n$-spheres that overlap a bit at the equator, which is hence of the form $S^{n-1} \times [-\epsilon, \epsilon]$. A Cech cocycle on such a cover is then simply given by one single $G$-valed transition function $S^{n-1} \to G$. This is known as the clutching construction.

Moreover, a gauge transformation of such a Cech cocycle is given by multiplying with the restriction of $G$-valued functions on the two hemi-$n$-spheres $\simeq D^n$ to the equator. This just means that those transition functions $S^3 \to G$ are gauge-trivial which may be extended smoothly to funtions $D^4 \to G$.

In conclusion, the desired classification of instanton solutions by homotopy classes of maps $U \colon S^3 \to G$ is precisely the clutching construction of $G$-principal bundles.

Finally, in order to add the principal connections to the picture, we think of one of the two semi-$n$-spheres to be the neighbourhood of infinity and hence, since the field strength is demanded to be vanishing at infinity, take the local gauge field to be the vanishing Lie algebra valued differential 1-form there. But then the rules for Cech cohomology with coefficients in the groupoid of Lie algebra valued 1-forms, and using the above clutching construction, it follows that on the equator overlap the gauge field has to be $U^{-1} d U \in \Omega^1( S^3 \times (-\epsilon, \epsilon) )$. This we may then extend in any desired way to a (non-flat) gauge connection over the remaining semi-$n$-sphere $D^4$, hence over the actual spacetime.

Notice that this is precisely the argument which for $G = U(1)$ the circle group and for $n = 2$ is known as the argument of Dirac charge quantization (which is also often misrepresented in the physics literature…)

For more see at Yang-Mills instanton – SU(2)-instantons from the correct maths to the traditional physics story.

Related concepts

• interacting vacuum

• fiber bundles in physics

• non-perturbative effect

• Yang-Mills instanton

  • theta vacuum

  • BPST-instanton

• quantum tunneling

• at finite temperature: caloron

  • caloron correspondence

• CP violation

• baryogenesis

• vortex atoms – Similarity to concepts in modern particle physics

References

General

(See also the general references at instanton.)

Textbook account:

• Robert E. Marshak, Section 10.3 of: Conceptual Foundations of Modern Particle Physics, World Scientific 1993 (doi:10.1142/1767)

Physics-style surveys include the introductory lecture notes

• Marcos Marino, Instantons and Large $N$ – An introduction to non-perturbative methods in QFT (pdf)

• Hilmar Forkel, A Primer on Instantons in QCD (arXiv:hep-ph/0009136)

• Edward Shuryak, Physics of QCD Instantons, Proceedings of the International School of Subnuclear Physics Erice, Sicily, Italy, August – September 2002 (web)

and the fairly detailed account (with lots of pointers to the literature)

• Thomas Schaefer, Edward Shuryak, Instantons in QCD, Rev. Mod. Phys.70:323-426,1998 (arXiv:hep-ph/9610451)

  section III D: relation to confinement

• Dmitri Diakonov, Instantons at work (arXiv:hep-ph/0212026)

See also the survey in

• Marcus Hutter, Instantons in QCD: Theory and Application of the Instanton Liquid Model (arXiv:hep-ph/0107098)

On possible experimental signatures at the LHC:

• Simone Amoroso, Deepak Kar, Matthias Schott, How to discover QCD Instantons at the LHC (arXiv:2012.09120)

Detailed argument for the theta-vacuum structure from chiral symmetry breaking is offered in

• Curtis Callan, R.F. Dashen, David Gross, The Structure of the Gauge Theory Vacuum, Phys.Lett. 63B (1976) 334-340 (spire)

• G. Morchio, Franco Strocchi, Chiral symmetry breaking and theta vacuum structure in QCD, Annals Phys.324:2236-2254, 2009 (arXiv:0907.2522)

A review in view of the asymptotic series nature of the Feynman perturbation series is in

• Igor Suslov, section 4.5 of Divergent perturbation series, Zh.Eksp.Teor.Fiz. 127 (2005) 1350; J. Exp. Theor. Phys. 100 (2005) 1188 (arXiv:hep-ph/0510142)

A survey with emphasis on the strong CP problem is in

• Roberto Peccei, The Strong CP Problem and Axions, Lect.Notes Phys.741:3-17,2008 (arXiv:hep-ph/0607268)

A more mathematically precise account which identifies instanton solutions explicitly equivalence classes of $G$-principal bundles is around Example 9.12 (p.320) and Section 10.5.5 (p.360) of

• Nakahara, Geometry Topology and Physics

This perspective is for instance also the one used in

• Edward Witten, Global gravitational anomalies, Comm. Math. Phys. Volume 100, Number 2 (1985), 197-229. (Euclid)

Discussion in the context of chiral symmetry breaking includes

• Dmitri Diakonov, Chiral Symmetry Breaking by Instantons (arXiv:hep-ph/9602375)

Discussion in the context of confinement includes

• Y. M. Cho, Dimensional Transmutation by Monopole Condensation in QCD (arXiv:1206.6936)

Discussion in the context of the quark-gluon plasma includes

• Edward Shuryak, Nonperturbative QCD and Quark-Gluon Plasma, lectures at Trieste, 2001 (pdf)

• Nikolai Kochelev, Ultra-light Glueballs in Quark-Gluon Plasma (arXiv:1501.07002)

Discussion of the infrared problem of the QCD instanton vacuum model includes

• Edward Witten, Theta Dependence In The Large N Limit Of Four-Dimensional Gauge Theories, Phys.Rev.Lett.81:2862-2865,1998 (arXiv:hep-th/9807109)

Discussion of instantons in lattice gauge theory includes

• Guy Moore, section 7 of Informal lectures on lattice gauge theory, 2003 (pdf)

• Michael Creutz, Anomalies, gauge field topology, and the lattice, Annals of Physics 326 (2011) 911–925 (pdf)

• Florian Gruber, Topology in dynamical Lattice QCD simulations, 2013 (web, pdf)

and via the homotopy theory of bundle gerbes:

• Jing-Yuan Chen, Instanton Density Operator in Lattice QCD from Higher Category Theory [arXiv:2406.06673]

Discussion of the $\eta'$-mass from instantons includes

• Edward Witten, Nucl. Phys. B 156 (1979) 269.

• Gabriele Veneziano, Nucl. Phys. B 159 (1979) 213

• Thomas Schaefer, QCD and the eta prime Mass: Instantons or Confinement?, Phys.Rev. D67 (2003) 074502 (arXiv:hep-lat/0211035)

• A renewed look at $\eta'$ in medium, (arXiv:1203.6740)

Relation to the scanning map and configuration spaces of points in:

• Michael Atiyah, John David Stuart Jones, Topological aspects of Yang-Mills theory, Comm. Math. Phys. Volume 61, Number 2 (1978), 97-118 (arXiv:1103904210)

Relation to skyrmions, calorons, monopoles

The construction of Skyrmions from instantons is due to

• Michael Atiyah, Nicholar S. Manton?, Skyrmions from instantons, Phys. Lett. B, 222(3):438–442, 1989 (doi:10.1016/0370-2693(89)90340-7)

The relation between skyrmions, instantons, calorons, solitons and monopoles is usefully reviewed and further developed in

• Josh Cork, Calorons, symmetry, and the soliton trinity, PhD thesis, University of Leeds 2018 (web)

• Josh Cork, Skyrmions from calorons, J. High Energ. Phys. (2018) 2018: 137 (arXiv:1810.04143)