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.


The mass gap problem

A survey and problem description in mathematics/mathematical physics:

Early report on the progress (essentially none):

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

Further comments:

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)

Problem description in terms of probabilistic lattice gauge theory:

See also

In the nuclear physics literature

In the nuclear physics/QCD-literature the mass gap problem is (and has long been) known as the confinement problem. Explicit mentioning of the CMI’s “mass gap” Millennium Problem in nuclear physics/phenomenological discussion of confinement:

The open problem of confinement

While experiment as well as lattice gauge theory-computer simulation clearly show that confinement takes place, a real theoretical understanding has been missing (though AdS-QCD is now on a good track). This is the confinement problem. The same problem from the point of view of mathematics is called the mass gap Millennium Problem. A related problem is the flavor problem.

The following is a list of quotes highlighting the open problem of confinement:

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 partticle spectrum, or of traditional strong interaction phenomenology

There are theoretical attempts to connect the fundamental theory of QCD with the very successful meson picture at low energy. The Skyrme model is an example. In other attempts, one tries to derive the NN interaction more or less directly from QCD. At present, the predictions are more of a qualitative kind. For quantitative results, the one-pion and two-pion contributions have to be added by hand, as they do not emerge naturally out of QCD-inspired models. Knowing that π\pi and 2π2\pi are the most important parts of the nuclear force, this defect of present quark model calculations is serious.

  • 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.

  • Barry R. Holstein, A Brief Introduction to Chiral Perturbation Theory, Czech. J. Phys. 50S4:9-23, 2000 (arXiv:hep-ph/9911449)

the holy grail sought by particle/nuclear knights has been to verify the correctness of the “ultimate” theory of strong interactions – quantum chromodynamics (QCD).

The theory is, of course, deceptively simple on the surface. [...][...] So why are we still not satisfied? While at the very largest energies, asymptotic freedom allows the use of perturbative techniques, for those who are interested in making contact with low energy experimental findings there exist at least three fundamental difficulties:

i) QCD is written in terms of the “wrong” degrees of freedom – quarks and gluons – while low energy experiments are performed with hadronic bound states;

ii) the theory is non-linear due to gluon self-interactions;

iii) the theory is one of strong coupling so that perturbative methods are not practical

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.

  • Mike Guidry, Gauge Field Theories: An Introduction with Applications, Wiley 2008 (ISBN:978-3-527-61736-4)

    Section 13.1.9:

The holy grail of QCD is the proof that a color SU(3) gauge theory confines in the non-perturbative regime.

This is not difficult to show for lattices with large spacing; unfortunately, such a demonstration does not constitute a proof of QCD confinement: to do that we must also demonstrate that the same theory that confines at large lattice spacing (strong coupling) has a continuum limit (weak coupling) that is consistent with the asymptotically free short distance behavior of QCD.

QCD is a challenging theory. Its most interesting aspects, namely the confinement of color and the chiral symmetry breaking, have defied all analytical approaches. While there are now many data accumulated from the lattice gauge theory, the methodology falls well short of giving us insights on how one may understand these phenomena analytically, nor does it give us a systematic way of obtaining a low energy theory of QCD below the confinement scale.

Nuclear physics is one of the oldest branches of high energy physics, yet remains one of more difficult. Despite the fact that we know the underlying fundamental theory, i.e. QCD, we are still unable to predict, reliably and analytically, behavior of nuclei or even a single proton. The problem is of course that one must understand the strong-coupling regime of QCD, which by and large remains inaccessible except by large-scale lattice simulations. Traditionally, this sets nuclear physics apart from the rest of high energy physics in many aspects. However, recent developments in the so-called gauge/gravity duality began to solve certain strongly coupled field theories,possibly including QCD or its close relatives, allowing the two communities to merge with each other.

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.

The problem with a derivation of nuclear forces from QCD is that this theory is non-perturbative in the low-energy regime characteristic of nuclear physics, which makes direct solutions very difficult. Therefore, during the first round of new attempts, QCD-inspired quark models became popular. The positive aspect of these models is that they try to explain hadron structure and hadron-hadron interactions on an equal footing and, indeed, some of the gross features of the nucleon-nucleon interaction are explained successfully.

However, on a critical note, it must be pointed out that these quark-based approaches are nothing but another set of models and, thus, do not represent fundamental progress. For the purpose of describing hadron-hadron interactions, one may equally well stay with the simpler and much more quantitative meson models.

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.

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.

[][\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]

  • 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).

One of the long-standing problems in QCD is to reproduce profound nuclear physics. The strong coupling nature of QCD prevents us from solving it analytically, and even numerical simulations have a limitation such as the volume of the atomic nucleus versus the lattice size. It is quite important to bridge the particle physics and the nuclear physics, by solving QCD to derive typical fundamental notions of the nuclear physics, such as the magic numbers, the nuclear binding energy and the nuclear shell model.

Holographic QCD is an analytic method to approach these problems in the strong coupling limit and at a large N cN_c. The nuclear matrix model is a many-body quantum effective mechanics for multiple baryons, derived by the AdS/CFT correspondence applied to QCD.

  • Yonggoo Heo, C. Kobdaj, M.F.M. Lutz, Constraints from a large-N cN_c analysis on meson-baryon interactions at chiral order Q 3Q^3, Phys. Rev. D 100, 094035 (2019) (arXiv:1908.11816)

Still after many decades of vigorous studies the outstanding challenge of modern physics is to establish a rigorous link of QCD to low-energy hadron physics as it is observed in the many experimental cross section measurements.

  • 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.

More than 98% of visible mass is contained within nuclei. In first approximation, their atomic weights are simply the sum of the masses of all the neutrons and protons (nucleons) they contain. Each nucleon has a mass m N1m_N \sim 1 GeV, i.e. approximately 2000-times the electron mass. The Higgs boson produces the latter, but what produces the masses of the neutron and proton? This is the question posed above, which is pivotal to the development of modern physics: how can science explain the emergence of hadronic mass (EHM)? [][\cdots]

Modern science is thus encumbered with the fundamental problem of gluon and quark confinement; and confinement is crucial because it ensures absolute stability of the proton. [][\cdots] Without confinement,our Universe cannot exist.

As the 21st Century began, the Clay Mathematics Institute established seven Millennium Prize Problems. Each represents one of the toughest challenges in mathematics. The set contains the problem of confinement; and presenting a sound solution will win its discoverer 1,000,000 bucks. Even with such motivation, today, almost fifty years after the discovery of quarks, no rigorous solution has been found. Confinement and EHM are inextricably linked. Consequently, as science plans for the next thirty years, solving the problem of EHM has become a grand challenge. [][\cdots]

In trying to match QCD with Nature, one confronts the many complexities of strong, nonlinear dynamics in relativistic quantum field theory, e.g. the loss of particle number conservation, the frame and scale dependence of the explanations and interpretations of observable processes, and the evolving character of the relevant degrees-of-freedom. Electroweak theory and phenomena are essentially perturbative; hence, possess little of this complexity. Science has never before encountered an interaction such as that at work in QCD. Understanding this interaction, explaining everything of which it is capable, can potentially change the way we look at the Universe.

The confinement of quarks is one of the enduring mysteries of modern physics. [][\ldots] In spite of many decades of research, physically relevant quantum gauge theories have not yet been constructed in a rigorous mathematical sense. [[ non-perturbatively, that is ]] [][\ldots] Taking this program to its completion is one of the Clay millennium problems. [][\ldots] Perhaps the most important example is four-dimensional SU(3)-lattice gauge theory. If one can show that this theory has a mass gap at all values of the coupling strength, that would explain why particles known as glue-balls in the theory of strong interactions have mass. All such questions remain open.

The second big open question is the problem of quark confinement. Quarks are the constituents of various elementary particles, such as protons and neutrons. It is an enduring mystery why quarks are never observed freely in nature. The problem of quark confinement has received enormous attention in the physics literature, but the current consensus seems to be that a satisfactory theoretical explanation does not exist.

QCD, the theory of the strong interactions, involves quarks interacting with non-Abelian gluon fields. This theory has many features that are difficult to impossible to see in conventional diagrammatic perturbation theory. This includes quark confinement, mass generation, and chiral symmetry breaking.

The origin of the proton mass, and with it the basic mass-scale for all nuclear physics, is one of the most profound puzzles in Nature.

Although QCD is defined by a seemingly simple Lagrangian, it specifies a problem that has defied solution for more than forty years. The key challenges in modern nuclear and high-energy physics are to reveal the observable content of strong QCD and, ultimately, therefrom derive the properties of nuclei.

In spite of the important progress of Euclidean lattice gauge theory, a basic understanding of the mechanism of color confinement and its relation to chiral symmetry breaking in QCD has remained an unsolved problem.

Recent developments based on superconformal quantum mechanics in light-front quantization and its holographic embedding on a higher dimensional gravity theory (gauge/gravity correspondence) have led to new analytic insights into the structure of hadrons and their dynamics.


Computer lattice QFT

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)

  • Yonggoo Heo, C. Kobdaj, M.F.M. Lutz, Constraints from a large-N cN_c analysis on meson-baryon interactions at chiral order Q 3Q^3, Phys. Rev. D 100, 094035 (2019) (arXiv:1908.11816)

Still after many decades of vigorous studies the outstanding challenge of modern physics is to establish a rigorous link of QCD to low-energy hadron physics as it is observed in the many experimental cross section measurements.

Rigorous lattice QFT

But there is also an approach via rigorous analytic lattice gauge theory:


Last revised on March 29, 2021 at 01:25:13. See the history of this page for a list of all contributions to it.