mass gap



Quantum field theory


physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics



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 “Millennium Problems”, see Jaffe-Witten, Douglas 04.



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 Millennium Prize Problem, 2004 (pdf)

Further comments are in

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)

See also


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:

many of the essential properties that the theory [[QCD]] is presumed to have, including confinement, dynamical mass generation, and chiral symmetry breaking, are only poorly understood. And apart from the low-lying bound states of heavy quarks, which we believe can be described by a nonrelativistic Schroedinger equation, we are unable to derive from the basic theory even the grossest features of the particle spectrum, or of traditional strong interaction phenomenology

  • 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.

QCD is a perennially problematic theory. Despite its decades of experimental support, the detailed low-energy physics remains beyond our calculational reach. The lattice provides a technique for answering nonperturbative questions, but to date there remain open questions that have not been answered

  • 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 α s\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.

  • Hideo Suganuma, Yuya Nakagawa, Kohei Matsumoto, 1+1 Large N cN_c QCD and its Holographic Dual -Soliton Picture of Baryons in Single-Flavor World, Proceedings of the 14th International Conference on Meson-Nucleon Physics and the Structure of the Nucleon (MENU2016) (arxiv:1610.02074)

Since 1973, quantum chromodynamics (QCD) has been established as the fundamental theory of the strong interaction. Nevertheless, it is very difficult to solve QCD directly in an analytical manner, and many effective models of QCD have been used instead of QCD, but most models cannot be derived from QCD and its connection to QCD is unclear. To analyze nonperturbative QCD, the lattice QCD Monte Carlo simulation has been also used as a first-principle calculation of the strong interaction. However, it has several weak points. For example, the state information (e.g. the wave function) is severely limited, because lattice QCD is based on the path-integral formalism. Also, it is difficult to take the chiral limit, because zero-mass pions require infinite volume lattices. There appears a notorious “sign problem” at finite density.

On the other hand, holographic QCD has a direct connection to QCD, and can be derived from QCD in some limit. In fact, holographic QCD is equivalent to infrared QCD in large N cN_c and strong ‘t Hooft coupling λ\lambda, via gauge/gravity correspondence. Remarkably, holographic QCD is successful to reproduce many hadron phenomenology such as vector meson dominance, the KSRF relation, hidden local symmetry, the GSW model and the Skyrme soliton picture. Unlike lattice QCD simulations, holographic QCD is usually formulated in the chiral limit, and does not have the sign problem at finite density.

  • 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).

  • Christian Drischler, Wick Haxton, Kenneth McElvain, Emanuele Mereghetti, Amy Nicholson, Pavlos Vranas, André Walker-Loud, Towards grounding nuclear physics in QCD (arxiv:1910.07961)

the entirety of the rich field of nuclear physics emerges from QCD: from the forces binding protons and neutrons into the nuclear landscape, to the fusion and fission reactions between nuclei, to the prospective interactions of nuclei with BSM physics, and to the unknown state of matter at the cores of neutron stars.

How does this emergence take place exactly? How is the clustering of quarks into nucleons and alpha particles realized? What are the mechanisms behind collective phenomena in nuclei as strongly correlated many-body systems? How does the extreme fine-tuning required to reproduce nuclear binding energies proceed? – are big open questions in nuclear physics.

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 February 8, 2020 at 10:04:13. See the history of this page for a list of all contributions to it.