Tuesday, December 8, 2020

What We Know And Don't Know About Mesons And Baryons


What are mesons?

Mesons are composite particles also known as "hadrons" that are distinguished by their integer total angular momentum J (often called simply "spin") (making them "bosons" which behave consistently with Bose-Einstein statistics which means that there is no restriction on the number of them that occupy the same quantum state).

Spatial extent

Mesons are about the same size (about 1.2 femtometer) as a proton or neutron. 

Mean Lifetime

All known mesons (like all composite particles bound by the strong force, which are collectively called "hadrons", except protons and neutrons which are bound in atoms) are unstable. Those mesons that have not been detected experimentally are expected to be even less stable that those which have been experimentally observed. 

The mesons with the longest mean lifetimes are the charged pions (with an up quark and a down quark, one of which is an antiquark, as valence quarks) and the charged kaons (with an up quark and strange quark, one of which is an antiquark as valence quarks), which have mean lifetimes on the order of 10^-8 seconds. All other known pseudoscalar mesons have mean lifetimes on the order of 10^-11 seconds or less. All other known vector mesons have mean lifetimes on the order of 10^-20 seconds or less. 

Gluons, up quarks, down quarks, strange quarks, charm quarks and bottom quarks are always confined in hadrons, all of which, except for protons and neutrons which are bound in atoms, are unstable. No hadrons with valence quarks other than up quarks or down quarks are stable or even metastable like neutrons. Gluons are theoretically stable and massless, but gluons travel very short distances on the order of one femtometer within the hadrons in which they are confined, at the speed of light, between being emitted and absorbed, so they are very short lived in practice.

Free neutrons have a mean lifetime of about 880 seconds. All hadrons which are fermions rather than bosons (which are called "baryons" and have three valence quarks), other than protons and neutrons, have mean lifetimes on the order of 10^-10 seconds or less. 

Hypothetical pure "glueballs" with no quark components, which are bosons which QCD naively predicts are possible, have never been observed, but would be unstable with similarly short lifetimes to unstable mesons if the existed.

Some fundamental particles in the Standard Model that are found outside of hadrons are also unstable, specifically, there are means lifetimes for the following fundamental particles on the order of the times shown and in the order shown from longest to shortest: muons (2*10^-6 seconds), tau leptons (3*10^-13 seconds), Higgs bosons (2*10^-22 seconds), top quarks (5*10^-25 seconds), W bosons (3*10^-25 seconds) and Z bosons (3*10^-25 seconds). A unit of Planck time (a potential theoretical minimum of time measurement precision equal to the speed of light times the Planck length) is on the order of 10^-43 seconds.

Photons and electrons are stable. Neutrinos oscillate but don't decay. 

Thus, in Nature, we observe only five, free, fundamental particles that have mean lifetimes of 3*10^-6 seconds or more, and two hadrons with lifetimes that long. The two long lived hadrons and the electron, in turn, combine into only a modest number of atomic isotopes which are that long lived, 254 isotopes of 80 elements of which are classified as stable, and there are scores or hundreds more that are unstable but have longer mean lifetimes than 3*10^-6 seconds. (Neutrinos and photons do not become a part of bound composite particles.)


The least massive meson (the neutral pion) has a mass of about 135 MeV. The heaviest experimentally measured meson mass (for an upsilon mesons which is a vector meson made up of a bottom quark and an anti-bottom quark), which is expected to be the heaviest possible simple meson, is about 9,460 MeV. By comparison, a proton is about 938 MeV and a neutron is about 940 MeV.

There are no hadrons (i.e. composite particles bound by the strong force) that are remotely close to the masses of the W boson (about 80,400 MeV), Z boson (about 91,200 MeV), Higgs boson (about 125,000 MeV) or top quark (about 173,000 MeV), all of which are fundamental particles in the Standard Model (with mean lifetimes of 2*10^-22 seconds or less). 

Hadrons including top quarks aren't completely forbidden in the Standard Model (because decay times and hadronization times are probabilistic averages and not fixed values), but require such high energies to produce, and are so unlikely to form even then because of the average time it takes for quarks to hadronize is less than it mean lifetime, and the fact that a single top quark-antiquark pair would normally involve the creation of a pair of particles rapidly moving away from each other, and would be extremely short lived even if they did form again due to the short mean lifetime of a top quark. So, they are usually considered impossible to form for all practical purposes and certainly have never been observed.


The most well understood mesons are well modeled as a single valence quark with a particular color charge, and a single valence anti-quark with the anti-color charge of the same type, bound by gluons carrying the strong force emitted by both of them and by the gluons themselves, in a sea of quarks. 

The valence quarks can be different flavors (i.e. up, down, strange, charm, and bottom/beauty since top quarks are to short lived to form hadrons). If there are the same flavor, they are called "quarkonium". 

These simple mesons can be either spin-0 with odd parity pseudoscalar mesons, or spin-1 with odd parity, which are called vector mesonsIf the valence quarks are of the same flavor they are called quarkonium

In theory there are naively 15 possible simple pseudoscalar meson ground state combinations and 15 possible vector meson ground state combinations. 

So far 16 pseudoscalar mesons and 14 vector mesons of which have been experimentally observed in their ground states, in addition to many excited states of these mesons that have been observed, although they don't line up precisely with the naively expected states. Of the simple mesons that scientists expect exist, only one has not yet been observed, the charmed vector B meson which should have a mass in the ballpark of 6,344 MeV.

This is a lot of meson resonances over a 10 GeV mass range, spanning two orders of magnitude, even before considering myriad excited resonances.

Seven important lighter electrically neutral valence quark pseudoscalar and vector mesons, however, don't neatly fit into these simple states and present as blends of simple states and replace some of the possibilities that would be expected in a truly simple model. 

Three have only up quark and down quark components.

* Pseudoscalar neutral pions, which are a blend of pseudoscalar up quark and down quark quarkonia, producing one type of pseudoscalar meson where two would be naively expected.

* Neutral rho vector mesons and omega vector mesons, which are a blend of vector up quark and down quark quarkonia, producing a different pair of vector mesons than the two vector mesons that would be naively expected, but leaving the total number of vector mesons equal to the expected number.

Four more are electrically neutral pseudoscalar mesons that have a strange quark component. 

* Two are the short and long components of electrically neutral kaons with down quark and strange quark components that occur blended into the one naively expected meson giving it unexpected properties. 

* Two of the others are electrically neutral pseudoscalar mesons with a up quark, down quark and strange quark components: eta mesons and eta prime mesons, producing a pair of mesons where one mesons with only strange quark components would naively be expected. 

Counting the short and long components of the kaon as separate, there is one more pseudoscalar mesons than naively expected overall.

We observe mesons that are scalar mesons (i.e. spin-0 with even parity), or axial vector mesons a.k.a. pseudovector mesons (spin-1 with even parity) whose structure is less well understood. We don't have clear theoretical expectations for how many ground states of these kinds of mesons are possible and the experimental data regarding how many scalar meson resonances and axial vector meson resonances exist in inconclusive because the relevant resonances are not always highly statistically significant or replicated consistently by different experiments, and the structure giving rise to particular well established resonances isn't a matter upon which a consensus has been reached. See review articles at the Particle Data Group here such as this one discussing scalar meson candidates and potential component structures under 2,000 MeV. It notes that:

In the literature, many suggestions are discussed, such as conventional qq¯ mesons, compact (qq)(¯qq¯) structures (tetraquarks) or meson-meson bound states. In addition, one expects a scalar glueball in this mass range. In reality, there can be superpositions of these components, and one often depends on models to determine the dominant one. Although we have seen progress in recent years, this question remains open.

A spin-2 meson (generally limited to excited states) is called a tensor meson.

In addition to spin and parity, two key properties of mesons that have to be determined from experimentally measured constants are the meson's mass (usually expressed in electron-volts with an implied divisor of the speed of light squared) and its resonance width. The mean lifetime for a particle is equal to the reduced Planck's constant (h bar, i.e. ħ) divided by resonance width (usually denoted with the capital Greek letter gamma, i.e. Γ). So, the larger the width of a particle (which can be expressed in electron-volt units used for mass and energy at this scale), the more short lived it will be, on average. Width is a convenient form in which to consider this because overall decay width of a particle is equal to the sum of the width of each of its possible decay channels which can only computed theoretically, one by one.

Meson spectroscopy is the enterprise of trying to determine all possible mesons (including excite state of mesons) which could exist and their properties. It is similar to creating a periodic table and table of isotopes for atoms. This is done with a combination of particle collider experiments and quantum chromodynamics (QCD) calculations. 

Some properties of a hypothetical meson can be determined in a straightforward manner with minimal calculations. All mesons have zero net QCD color charge. The electromagnetic charge, total angular momentum a.k.a. spin, and the parity of a meson, and a list of all types of decays of a meson that are possible in the Standard Model are elementary and trivial to determine for a simple meson from the valence quarks from which it arises and whether the mesons have identical or opposite directions of spin (which is assumed to happen both ways for purposes of determining what is possible). Sometimes, the valence quark structure of a meson clearly isn't something as simple as a quark and antiquark pair and has to be deduced with less settled and consensus interpretations that sometimes produce dozens or even hundreds of papers debating which interpretation is correct.

The key properties determined in the core guts of experimental and theoretical meson spectroscopy primarily consist of the mass and width of the meson, and the properties of its decay channels, which have to be measured or calculated, and depend upon the precise values of experimentally measured Standard Model constants.

The calculations are based mostly* on the QCD equations, the experimentally determined quark masses, and the experimentally determined value of the strong force coupling constant (represented by the lower case Greek letter alpha). 

I say "mostly" because electroweak interactions tweak the predominant effect of the strong force interactions, both directly and by facilitating "loops" of interactions mediated through other particles by means other than the strong force to which the source particle can transform, that have to be considered in a complete and rigorous calculation. 

Arguably hypothetical glueballs (which have no valence quarks and have not be observed in a pure state) and tetraquarks (which have four valence quarks and been observed in a handful of cases and have candidate masses from 2632 MeV to 6900 MeV) and hypothetical hexaquarks are also mesons sometimes known as "exotic mesons". There could also be "exotic mesons" that are "hybrid particles" which contain a valence quark-antiquark pair and one or more gluons in a bound composite particle.

Predicted and observed baryons.

Baryon spectroscopy exists as well, but has posed fewer mysteries.

All of the experimentally observed baryons fit in the 75 naively expected categories and 43 of those combinations have been observed, while 32 have not been observed: 24 of the combinations predicted to exist but not yet observed have at least one bottom quark, 1 has three charm quarks, and 5 have two charm quarks and no bottom quarks. But some baryons with valence bottom quarks have been observed, and some baryons with two charm quarks have been observed. Like mesons, baryons also have excited states, some of which have been observed. 

Every predicted baryon with only up, down and strange valence quarks have been observed, as have all predicted baryons with one valence charm quark. 

No predicted baryons with three charm quarks or two or more bottom quarks have been observed, and it is conceivable that it is impossible for that strong force to bind that much quark mass into a single composite hadron although no QCD calculations to date have reached that conclusion.

All of the baryons not observed experimentally have at least one bottom quark, at least two charm quarks, or both, and hence take very high energy collisions to produce and are produced in smaller numbers than other baryons even at high energies, as a result. Thus, the baryons we haven't yet observed are the ones we would expect not to have observed yet.

The proton at 938 MeV is the least massive baryon. The most massive observed baryon has a mass of about 6,071 MeV (a bottom omega baryon with two valance strange quarks and a valence bottom quark), but it is expected that baryons not yet observed will be more massive. 

Wikipedia notes that the only observed exotic baryons are the pentaquarks P+c(4380) and P+c(4450), discovered in 2015 by the LHCb collaboration.

A hypothetical triple bottom omega baryon, with three valance bottom quarks, if it can exist, would be the heaviest possible baryon and would be expected to have a mass on the order of 15,000 MeV or so. No other  kind baryon with three valance quarks permitted by QCD should have a mass of more than about 11,500 MeV, although some pentaquarks might be more massive.

The New Paper

A new review paper spells out the state of what we do and do not know about meson structure in mesons beyond the simple cases.

The importance of S-matrix unitarity in realistic meson spectroscopy is reviewed, both its historical development and more recent applications. First the effects of imposing S-matrix unitarity on meson resonances is demonstrated in both the elastic and the inelastic case. Then, the static quark model is revisited and its theoretical as well as phenomenological shortcomings are highlighted. 
A detailed account is presented of the mesons in the tables of the Particle Data Group that cannot be explained at all or only poorly in models describing mesons as pure quark-antiquark bound states. 
Next the earliest unitarised and coupled-channel models are revisited, followed by several examples of puzzling meson resonances and their understanding in a modern unitarised framework. Also, recent and fully unquenched lattice descriptions of such mesons are summarised. Finally, attention is paid to production processes, which require an unconventional yet related unitary approach. Proposals for further improvement are discussed.
Eef van Beveren, George Rupp, "Modern meson spectroscopy: the fundamental role of unitarity" arXiv (December 7, 2020) (review to be published in "Progress in Particle and Nuclear Physics" 2021).

The introduction of the paper explains (in language only somewhat more technical than I usually use in blog posts, with citations and internal references omitted, bracketed comments and emphasis mine):
Knowledge of low-energy QCD is encoded in the observable properties of hadrons, that is, mesons and baryons. Most importantly, hadronic mass spectra should provide detailed information on the forces that keep the quarks and/or antiquarks in such systems permanently confined, inhibiting their observation as free particles. However, since QCD is not tractable through perturbative calculations at low energies, owing to a large running coupling in that regime, quark confinement is usually dealt with employing a confining potential in the context of some phenomenological quark model. The shape of this potential is largely empirical, though its short-distance behaviour can be reasonably determined from one-gluon exchange, resulting in a Coulomb-like interaction, usually endowed with an r-dependent coupling constant in order to simulate asymptotic freedom. At large distances, the potential is mostly supposed to grow linearly, on the basis of flux-tube considerations, which have been observed in lattice simulations of string formation for static quarks. The most cited quark model of mesons with such a Coulomb-plus-linear confining potential, sometimes also called “funnel potential”, is due to Godfrey & Isgur (GI), which also accounts for kinematically relativistic effects. The enormous popularity of the model is understandable, in view of its exhaustive description of practically all imaginable qq¯ systems, including those with one or two top quarks, which had not yet even been discovered then. Experimentalists as well as model builders often invoke GI predictions as a touchstone for their observations or results. However, the GI model does not reproduce the excitation spectra of mesons made of light quarks. . . . Its principal shortcoming is the prediction of much too large radial splittings for mesons in the range of roughly 1–2 GeV, resulting in several experimentally observed states that do not fit in the GI level scheme. Also the lowest-lying scalar mesons, below 1 GeV, are not at all reproduced in the GI model. . . . 
Logically, there can be two reasons for the problems of the GI model in the mentioned energy region, namely possible deficiencies of the employed confining potential and/or certain approximations inherent in the model. Let us first consider the employed funnel-type confining potential 
V(conf) = − (αs(r))/r + λr , (1) 
where the constant parameter λ is the so-called string tension and αs(r) a configuration-space parametrisation of the running strong coupling. Ignoring for the moment the r-dependence of αs, we see that V(conf) is independent of (quark) mass and therefore also flavour-independent, in accordance with the QCD Lagrangian. Consequently, the mass spectrum of a Schr¨odinger or related relativistic equation with such a potential will inexorably be mass-dependent, that is, radial splittings will increase according as the quark mass decreases. This had already been realised by the Cornell group when developing their potential model of charmonium, in which the strength κ of the Coulombic part was fitted to the few then known charmonium observables (quote): 
“The recent discovery of the µ +µ− enhancement Υ probably implies the existence of another QQ¯ family, where Q is a quark carrying a new flavor and having a mass of 4–5 GeV. The variation of the spectrum with quark mass mQ is very sensitive to the form of V(r), and present indications are that our ansatz (1.1) may not pass this test.” [ed. In other words, these experimenters modeling mesons made up of a charm quark and an anti-charm quark resolved differences between their theory and experiment by accurately predicting the existence of the soon to be discovered bottom quark.] 
Two years later, with more data available, the same authors adjusted their model parameters so as to try to accommodate charmonium as well as bottomonium states, both in the static and the coupled channel version of the model, though resulting in a clearly too heavy 3 3(S(1)) state in the latter case. Also, one of the authors used a slightly smaller value of κ when applying the coupled-channel model only to bottomonium states, by indeed arguing on the basis of a running coupling in the Coulombic part. . . . An empirical way to account for a running strong coupling αs(r) in the context of the static quark model simultaneously applied to charmonium and bottomonium states was [developed] . . . .
Now, the mechanism that allows to reproduce the approximate equal radial spacings in charmonium and bottomonium for the funnel potential, despite the quark-mass independence of its confining part, is a very delicate balance between the Coulombic behaviour and running coupling at short distances, as well as the linear rising at larger distances. However, this is impossible to sustain for light quarkonia, resulting in considerably increased spacings in the energy region of 1–2 GeV, in conflict with experiment. A model proposed by us almost four decades ago, does reproduce the observed radial spacings for light, charm, and bottom quarks, being based on a flavour-dependent harmonic-oscillator confining potential, while also accounting for non-perturbative coupled-channel effects in a manifestly unitary S-matrix formalism. . . . 
The other possible reason why the GI and related quark models often fail in light-meson spectroscopy is the usual neglect of unitarisation effects. By this we mean that, as most mesons are resonances and not stable or quasi-stable qq¯, they must strictly speaking be described as poles in some unitary S-matrix, and not as bound states in a static potential. This becomes all the more true if one realises that many mesonic resonances are broad or very broad, some of which having widths of the same order of magnitude as the observed radial spacings. Picking just one typical example from the PDG  Meson Summary Tables, we see that the mass difference between the ground-state ss¯ tensor meson f'(2) (1525) and its first radial excitation f(2)(1950) is about 420 MeV, while the full width of the latter resonance is (464 ± 24) MeV. So it is clear that a reliable determination of radial splittings that originate exclusively in the underlying confining potential demands to account for unitarisation effects, which inevitably will give rise to both the observed decay widths and real mass shifts hidden in the spectrum. Note that the size of such a mass shift may very well depend on the specific radial (or angular) quantum number and/or the vicinity of some decay threshold.

The conclusion is as follows:

In this review we aimed at making the case for a systematic treatment of meson spectroscopy based on the quark model for qq¯ states only, yet imposing the requirements of S-matrix unitarity. Thus, in Sec. 1 we started with a brief introduction to mainstream quark models of mesons using a Coulomb-plus linear confining potential, and mentioned the inevitable problem with radial spacings in the spectra of especially mesons made of light and strange quarks. In Sec. 2 we employed a simple unitary single channel model for the S-matrix in order to show discrepancies that may arise when using standard Breit-Wigner parametrisations, in particular when applied to very broad resonances not far above the lowest threshold, like in the case of the light scalar mesons f0(500) and K? 0 (700). Section 3 was devoted to a detailed discussion of static quark models, in which the dynamical effects of strong decay or virtual meson loops on the spectra are ignored. The shortcomings of the relativised meson model of Godfrey and Isgur were illustrated with many examples from particularly light-meson spectra. Furthermore, two fully relativistic static quark models were reviewed as well and shown to have similar or even worse problems. In Sec. 4 we briefly reviewed the very disparate predictions for meson mass shifts, some of them really huge, due to unitarisation or coupled channels in a series of old and more recent models, discussing their differences. Section 5 treated a simple unitarised model in momentum space, called Resonance Spectrum Expansion (RSE) and inspired by the unitarised quark-meson model in coordinate space developed by the Nijmegen group. Its predictive power was demonstrated by successfully describing e.g. the K? 0 (700) resonance, the charmed scalar meson D? s0 (2317), the charmed J P = 1+ mesons, and — in a multichannel extension of the model — even the whole light scalar meson nonet. Furthermore, the most general RSE model, applicable to systems with various quark antiquark channels coupled to an arbitrary number of meson-meson channels, was shown to be exactly solvable, both algebraically and analytically, owing to the separability of the effective meson-meson interaction and the employed string-breaking mechanism. In Sec. 6 the latter general RSE model was used to analyse again the charmed J P = 1+ mesons, thus allowing to dynamically produce the physical states as orthogonal mixtures of the 3P1 and 1P1 quark-antiquark components. This gives rise to two quasi-bound states in the continuum and two strongly shifted states, thus reproducing the observed disparate pattern of masses and widths with remarkable accuracy. Moreover, the same full RSE model as well as its multichannel coordinate space version were employed to describe the axialvector charmonium state χc1(3872), modelling it as a unitarised 2 3P1 charmonium state. The resulting pole trajectories, wave function, and electromagnetic transitions support our interpretation of this very enigmatic meson. Section 7 was devoted to several recent lattice calculations of controversial meson resonances that include meson-meson interpolating fields in the simulations, in order to allow for the computation of phase shifts and extract the corresponding resonance or bound-state pole positions. The results for χc1(3872), D? s0 (2317), the charmed J P = 1+ mesons, and the light scalars largely confirm our description of these mesons. In Sec. 8 we presented a formalism for production processes that is strongly related to our RSE model and satisfies the extended-unitary condition =m(P) = T ? P, with T the RSE T-matrix. The general expression features a purely kinematical, non-resonant real lead 45 term, plus a combination of two-body T-matrix elements with also kinematical yet complex coefficients. Fully empirical applications of the formalism to the controversial vector charmonium states ψ(4260) and ψ(4660) as well as the established Υ(10580) bottomonium state allowed to fit all three resonance-like structures as non-resonant threshold enhancements. 

Let us repeat that we did not aspire to carry out a comprehensive review of general meson spectroscopy. Therefore, several alternative descriptions of mesons with very interesting results, like e.g. unitarised chiral models,  the generalised Nambu–Jona-Lasinio model, or the quark-level linear σ model, have not been dealt with here at all. Nevertheless, we believe that these approaches have a more restricted applicability to meson spectroscopy, being usually limited to specific resonances or ground states only. In order to be able to infer information on the confining potential, it is necessary to be able to calculate radially excited states without introducing new parameters. We have also not paid attention to truly exotic meson candidates, as e.g. the charmed charmonium-like and bottomonium-like states Z ± c (3900), Z ± c (4430), Zb(10610), and Zb(10650). . . . Nevertheless, in this context it is worthwhile to mention a very recent paper by the COMPASS Collaboration, in which for the first time a triangle-singularity model is fitted directly to partial-wave data, viz. for the controversial a(1)(1420) state reported by COMPASS itself five years ago. The conclusion of this fit is that including the triangle singularity allows for a better fit to the data with fewer parameters, so that after all there is no need to introduce the new a(1)(1420) resonance. This result may have far-reaching consequences for exotic spectroscopy, in view of the increasing number of observed enhancements in the data that cannot be accommodated as qq¯ mesons. Clearly, all such controversial states will have to be refitted in a similar fashion. 

To conclude, we recall the mentioned email exchange with a co-spokesperson of the E791 Collaboration about the need for easy formulae to fit the data and a related discussion at the LHCb workshop ‘Multibody decays of D and B mesons”, in Rio de Janeiro, 2015. The latter meeting focused on alternatives to the usual Breit-Wigner (BW) and Flatt´e parametrisations that guarantee multichannel unitarity, even in the case of overlapping broad resonances. In that spirit, we proposed our general RSE formalism, yet with the bare HO energies replaced by a few to-be-fitted real energies and possibly also the Bessel and Hankel-1 functions by more flexible expressions, thus allowing much more accurate fits to the data. Apart from thus guaranteeing manifest multichannel unitarity, the usual two BW parameters for each resonance could then be replaced by only one real energy. Finally, a similar generalisation of our production formalism should also be possible. 

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