Contents

# Contents

## Idea

The mass gap problem is an open conceptual problem in the quantization of Yang-Mills theory, closely related to what in the phenomenology of quantum chromodynamics is called the confinement of quarks, hence the existence of ordinary hadronic, in particular baryonic matter.

The Lagrangian for Yang-Mills theory coupled to fermion fields (such as for QCD) makes manifest the existence of mass-less quarks and gluons. Indeed, at very high temperature QCD is thought to exhibit a quark-gluon plasma well described by these degrees of freedoms.

But at comparatively smaller temperature it is observed in experiment as well as in lattice QCD computer experiment that QCD exhibits confinement, meaning that the low energy states of the theory are not free massless quarks and gluons anymore, but that these form bound states in the form of hadrons (including, notably, protons and neutrons, and hence all the ordinary baryonic matter of the observable universe).

The existence of massive hadron-bound states in low-energy QCD is thus well-established phenomenologically, but it is as yet lacking a conceptual theoretical explanation.

The mass gap problem is the problem in mathematical physics to demonstrate theoretically (i.e. not just by computer simulation) the existence of this mass gap/confinement-phenomenon in QCD and in Yang-Mills theory coupled to fermion fields in general.

This issue is well and widely known in the particle physics-community, see for instance Kutschke 96, Section 3.1, INFN 15. It gained more attention among the mathematics/mathematical physics-community when the Clay Mathematics Institute declared the problem to be one in a list of “Millenium Problems”, see Jaffe-Witten, Douglas 04.

## References

### Mathematics

A survey and problem description in mathematics/mathematical physics is in

and a report on the progress (essentially none) is in

• Michael Douglas, Report on the Status of the Yang-Mills Millenium Prize Problem, 2004 (pdf)

Notes reviewing more technical details of the problem are in

• Jay Yablon, The Origins of QCD Confinement in Yang-Mills Gauge Theory, January 2008 (pdf)

### Phenomenology

In the application of QCD to particle physics phenomenology, the mass gap problem is incarnated in the open problem of demonstrating confinement of quarks at low energies to colour-neutral bound states, hence to hadrons (in particular baryons, hence the problem of existence of ordinary matter). There does exist some qualitative understanding as well as computer simulations in lattice QCD, but there remain fundamental issues with deriving basic properties of hadrons such as their mass (e.g. Kutschke 96, Section 3.1) and spin (the “proton spin crisis”).

This is widely and well known, but particle physics does not quite share the mathematician’s culture of succinctly highlighting open problems. Here are some sources that make this explicit:

• 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).

Of course various partial approaches exist, notably computer-experiment in lattice QCD. (Such computer-checks of the mass-gap problem are analogous to computer checks of the Riemann hypothesis, see there) High-accuracy computation of hadron-masses in lattice QCD-simulations are reported here:

• S. Durr, Z. Fodor, J. Frison, C. Hoelbling, R. Hoffmann, S.D. Katz, S. Krieg, T. Kurth, L. Lellouch, T. Lippert, K.K. Szabo, G. Vulvert,

Ab-initio Determination of Light Hadron Masses,

Science 322:1224-1227,2008 (arXiv:0906.3599)

• Zoltan Fodor, Christian Hoelbling, section V of Light Hadron Masses from Lattice QCD, Rev. Mod. Phys. 84, 449, (arXiv:1203.4789)

• S. Aoki et. al. Review of lattice results concerning low-energy particle physics (arXiv:1607.00299)

(…)

Last revised on January 10, 2019 at 02:21:13. See the history of this page for a list of all contributions to it.