# nLab confinement

Contents

## Surveys, textbooks and lecture notes

#### Differential cohomology

differential cohomology

# Contents

## Idea

Confinement (e.g Espiru 94) is the (expected) phenomenon in Yang-Mills theory/QCD that quark-excitations must form bound states which are neutral under the color charge – such as the hadrons (protons, neutrons) and mesons.

Confinement is a non-perturbative effect (e.g Espiru 94).

## Open problem

While experiment as well as lattice gauge theory-computer simulation clearly show that confinement takes place, a real theoretical understanding is missing (see also at mass gap problem). Here are quotes from some references highlighting the open problem:

• Robert Kutschke, section 3.1 Heavy flavour spectroscopy, in D. Bugg (ed.), Hadron Spectroscopy and the Confinement Problem, Proceedings of a NATO Advanced Study Institute, Plenum Press 1996 (doi:10.1007/978-1-4613-0375-6)

While it is generally believed that QCD is the correct fundamental theory of the strong interactions there are, as yet, no practical means to produce full QCD calculations of hadron masses and their decay widths.

Because of the great importance of the standard model, and the central role it plays in our understanding of particle physics, it is unfortunate that, in one very important respect, we don’t really understand how it works. The problem lies in the sector dealing with the interactions of quarks and gluons, the sector known as Quantum Chromodynamics or QCD. We simply do not know for sure why quarks and gluons, which are the fundamental fields of the theory, don’t show up in the actual spectrum of the theory, as asymptotic particle states. There is wide agreement about what must be happening in high energy particle collisions: the formation of color electric flux tubes among quarks and antiquarks, and the eventual fragmentation of those flux tubes into mesons and baryons, rather than free quarks and gluons. But there is no general agreement about why this is happening, and that limitation exposes our general ignorance about the workings of non-abelian gauge theories in general, and QCD in particular, at large distance scales.

• Brambilla et al.. QCD and strongly coupled gauge theories: challenges and perspectives, Eur Phys J C Part Fields. 2014; 74(10): 2981 (doi:10.1140/epjc/s10052-014-2981-5)

The success of the technique does not remove the challenge of understanding the non-perturbative aspects of the theory. The two aspects are deeply intertwined. The Lagrangian of QCD is written in terms of quark and gluon degrees of freedom which become apparent at large energy but remain hidden inside hadrons in the low-energy regime. This confinement property is related to the increase of $\alpha_s$ at low energy, but it has never been demonstrated analytically.

We have clear indications of the confinement of quarks into hadrons from both experiments and lattice QCD. Computations of the heavy quark–antiquark potential, for example, display a linear behavior in the quark–antiquark distance, which cannot be obtained in pure perturbation theory. Indeed the two main characteristics of QCD: confinement and the appearance of nearly massless pseudoscalar mesons, emergent from the spontaneous breaking of chiral symmetry, are non-perturbative phenomena whose precise understanding continues to be a target of research.

Even in the simpler case of gluodynamics in the absence of quarks, we do not have a precise understanding of how a gap in the spectrum is formed and the glueball spectrum is generated.

$[\cdots]$ the QCD Lagrangian does not by itself explain the data on strongly interacting matter, and it is not clear how the observed bound states, the hadrons, and their properties arise from QCD. Neither confinement nor dynamical chiral symmetry breaking (DCSB) is apparent in QCD’s lagrangian, yet they play a dominant role in determining the observable characteristics of QCD. The physics of strongly interacting matter is governed by emergent phenomena such as these, which can only be elucidated through the use of non-perturbative methods in QCD [4, 5, 6, 7]

Experimentally, there is a large number of facts that lack a detailed qualitative and quantitative explanation. The most spectacular manifestation of our lack of theoretical understanding of QCD is the failure to observe the elementary degrees of freedom, quarks and gluons, as free asymptotic states (color con- finement) and the occurrence, instead, of families of massive mesons and baryons (hadrons) that form approximately linear Regge trajectories in the mass squared. The internal, partonic structure of hadrons, and nucleons in particular, is still largely mysterious. Since protons and neutrons form almost all the visible matter of the Universe, it is of basic importance to explore their static and dynamical properties in terms of quarks and gluons interacting according to QCD dynamics.

• V. A. Petrov, Asymptotic Regimes of Hadron Scattering in QCD (arXiv:1901.02628)

This is a commonplace that so far we do not have a full-fledged theory of interaction of hadrons, derived from the first principles of QCD and having a regular way of calculating of hadronic amplitudes, especially at high energies and small momentum transfers. The problem is related to a more general problem that QCD Lagrangian would yield colour confinement with massive colourless physical states (hadrons).

## Potential solutions

### Via Calorons

It has been argued that, after Wick rotation, confinement may be derived from the behaviour of instantons (Schaefer-Shuryak 96, section III D), or rather their positive temperature-incarnations as calorons, Greensite 11, section 8.5:

it is natural to wonder if confinement could be derived from some semiclassical treatment of Yang–Mills theory based on the instanton solutions of non-abelian gauge theories. The standard instantons, introduced by Belavin et al. (40), do not seem to work; their field strengths fall off too rapidly to produce the desired magnetic disorder in the vacuum.

In recent years, however, it has been realized that instanton solutions at finite temperature, known as calorons, might do the job. These caloron solutions were introduced independently by Kraan and van Baal (41, 42) and Lee and Lu (43) (KvBLL), and they have the remarkable property of containing monopole constituents which may, depending on the type of caloron, be widely separated.

$[...]$

The caloron idea is probably the most promising current version of monopole confinement in pure non-abelian gauge theories, but it is basically (in certain gauges) a superposition of monopoles with spherically symmetric abelian fields, and this leads to the same questions raised in connection with monopole Coulomb gases.

For the moment see at AdS/QCD correspondence.

### In $\mathcal{N}=2$ super Yang-Mills theory via Seiberg-Witten theory

While confinement in plain Yang-Mills theory is still waiting for mathematical formalization and proof (see Jaffe-Witten), there is a variant of Yang-Mills theory with more symmetry, namely supersymmetry, where the phenomenon has been giving a decent argument, namely in N=2 D=4 super Yang-Mills theory (Seiberg-Witten 94).

Also a strategy for a proof for N=1 D=4 super Yang-Mills theory has been proposed, see below.

### In $\mathcal{N} = 1$ super Yang-Mills theory via M-theory on $G_2$-manifolds

An idea for a strategy towards a proof of confinement in N=1 D=4 super Yang-Mills theory via two different but conjecturally equivalent realizations as M-theory on G2-manifolds has been given in Atiyah-Witten 01, section 6, review is in Acharya-Gukov 04, section 5.3.

The idea here is to consider a KK-compactification of M-theory on fibers which are G2-manifolds that locally around a special point are of the form

$X_{1,\Gamma} \;\coloneqq\; \big( S^3 / \Gamma \big) \times Cone\big(S^3\big) \phantom{AA} \text{or} \phantom{AA} X_{2,\Gamma} \;\coloneqq\; S^3 \times Cone\big(S^3/\Gamma\big)$

where

• $\Gamma$ is a finite subgroup of SU(2) that acts canonically by left-multiplication on $S^3 \simeq$ SU(2);

• $Cone(\cdots)$ denotes the metric cone construction.

This means that $X_{1,\Gamma}$ is a smooth manifold, but $X_{2,\Gamma}$, as soon as $\Gamma$ is not the trivial group, $\Gamma \neq 1$, is an orbifold with an ADE singularity.

Now the lore of M-theory on G2-manifolds predicts that KK-compactification

1. on $X_{1,\Gamma}$ yields a 4d theory without massless fields (since there are no massless modes on the covering space $S^3$ of $X_{1,\Gamma}$)

2. on the ADE-singularity $X_{2,\Gamma}$ yields non-abelian Yang-Mills theory in 4d coupled to chiral fermions.

So in the first case a mass gap is manifest, while non-abelian gauge theory is not visible, while in the second case it is the other way around.

But if there were an argument that M-theory on G2-manifolds is in fact equivalent for compactification both on $X_{1,\Gamma}$ and on $X_{2,\Gamma}$. To the extent that this is true, it looks like an argument that could demonstrate confinement in non-abelian 4d gauge theory.

This approach is suggested in Atiyah-Witten 01, pages 84-85. An argument that this equivalence is indeed the case is then provided in sections 6.1-6.4, based on an argument in Atiyah-Maldacena-Vafa 00

## References

### In Yang-Mills theory

#### General

Textbook accounts include

• Jeff Greensite, An Introduction to the Confinement Problem, Lecture Notes in Physics, Volume 821, 2011 (doi:10.1007/978-3-642-14382-3)

• Robert Iengo, section 9.1 of Quantum Field Theory (pdf)

• D. Bugg (ed.), Hadron Spectroscopy and the Confinement Problem, Proceedings of a NATO Advanced Study Institute, Plenum Press 1996 (doi:10.1007/978-1-4613-0375-6)

Introductions and surveys include

Discussion includes

A formulation of confinement as an open problem of mathematical physics, together with many references, is in

Other technical reviews include

#### Via monopole condensation

An original suggestion that confinement in Yang-Mills theory may be understood via monopole condensation as a dual Meissner effect? is due to

• Gerard 't Hooft, in Proceed.of the Europ.Phys.Soc. 1975, ed.by A.Zichichi (Editrice Compositori, Bologna, 1976), p.1225.

• S. Mandelstam, Phys.Rep. 23C (1976) 145;

(That this is indeed the case has not yet been demonstarted for plain Yang-Mills theory, but it was later shown for N=2 D=4 super Yang-Mills theory in (Seiberg-Witten 94). What this does or does not imply for the case of QCD is discussed in (Yung 00) ).

The relation to QCD instantons/monopoles in the QCD vacuum is discussed in

and analogously (at positive temperature) relation to calorons:

• Greensite 11, section 8.5

• P. Gerhold, E.-M. Ilgenfritz, M. Müller-Preussker, An $SU(2)$ KvBLL caloron gas model and confinement, Nucl.Phys.B760:1-37, 2007 (arXiv:hep-ph/0607315)

• Rasmus Larsen, Edward Shuryak, Classical interactions of the instanton-dyons with antidyons, Nucl. Phys. A 950, 110 (2016) (arXiv:1408.6563)

• Rasmus Larsen, Edward Shuryak, Interacting Ensemble of the Instanton-dyons and Deconfinement Phase Transition in the SU(2) Gauge Theory, Phys. Rev. D 92, 094022, 2015 (arXiv:1504.03341)

For further developments see

• Dimensional Transmutation by Monopole Condensation in QCD (arXiv:1206.6936)

### Interacting field vacuum

Discussion of confinement as a result of the interacting vacuum includes

• Johann Rafelski, Vacuum structure – An Essay, in pages 1-29 of H. Fried, Berndt Müller (eds.) Vacuum Structure in Intense Fields, Plenum Press 1990 (GBooks)

### In super-Yang-Mills theory

Confinement in N=2 D=4 super Yang-Mills theory by a version of the monopole condensation of (t Hooft 75, Mandelstam 76) was demonstrated in

Reviews with discussion of the impact on confinement in plain YM include

• Alexei Yung, What Do We Learn about Confinement from the Seiberg-Witten Theory (arXiv:hep-th/0005088)

Discussion in the context of the AdS-CFT correspondence is in

• David Berman, Maulik K. Parikh, Confinement and the AdS/CFT Correspondence, Phys.Lett. B483 (2000) 271-276 (arXiv:hep-th/0002031)

• Henrique Boschi Filho, AdS/QCD and confinement, Seminar at the Workshop on Strongly Coupled QCD: The confinement problem, November 2011 (pdf)

### In M-theory on $G_2$-manifolds

An idea for how to demonstrate confinement in models of M-theory on G2-manifolds is given in

based on