There Are Still Hadrons We Don't Fully Understand (UPDATED November 2, 2021)

In theory, the Standard Model says everything there is to say about particle physics.  

But, in practice, there are many resonances of composite particles made up of quarks and gluons bound by the strong force, called "hadrons" whose structures are still not well understood theoretically. There is no one consensus explanation for most of them. Hadrons come in mesons which are bosons with integer spin, and baryons which a fermions with spin-1/2 + N (where N is a whole number). Quarkonia is a a meson made up of a valence quark and its own valence antiparticle.

A very simple constituent quark model explains almost all of the known pseudoscalar mesons (i.e. spin-0, parity odd) and vector mesons (i.e. spin-1, parity odd) with a valance quark and a valence antiquark. The exceptions are a few blends of light quark quarkonia (the pseudoscalar neutral pion, eta meson, and eta prime meson, and the vector neutral rho meson, and omega meson), or are blends of a neutral light quark pseudoscalar meson and its antiparticle (the long and short types of neutral kaons). All possible pseudoscalar meson ground states, and all but one of the possible vector meson ground states (the charmed vector B meson) have been observed experimentally with enough data to experimentally measure the mass of that type of meson. 

This simple constituent quark model also describes all known three quark baryons, both spin-1/2 and spin-3/2 (which have either three valence quarks, or three valence antiquarks). Of the 74 possible ground state baryons (excluding antiparticles), about 44 have been and about 30 have not been observed experimentally with enough data to experimentally measure the mass of that type of meson. 

All of the baryons of the 30 that have not been observed sufficiently have at least one valence bottom quark and/or at least two valence charm quarks. Ten baryons with a single bottom quark only, and one double charmed baryon (which had no bottom quarks), have been observed. But no triple charmed, or double or triple bottom baryons have been observed, nor have any double charmed baryons with a single valence bottom quark. Almost no particle physicists, however, if any, seriously doubt that the 31 non-exotic hadrons left to be discovered will eventually be discovered with a big enough collider and enough time, or that they can exist, and their expected properties can be well predicted by QCD (or even by far more crude extrapolation from existing data).

Also, all hadrons can form "excited states" at higher masses with different decay chains, that are otherwise identical to the "ground state" resonances with the same valence quarks.

But, the constituent quark model struggles to understand the structure scalar mesons (i,.e. spin-0 parity even) and axial vector a.k.a. isovector mesons (i.e. spin-1 parity even).

It has also been a struggle to identify experimentally observed resonances that correspond to certain structures that are theoretically possible in quantum chromodynamics a.k.a. QCD (the Standard Model theory of the strong force between quarks and gluons). 

These structures include glueballs (which contain only gluons), tetraquarks, pentequarks, hexaquarks, "molecules" made up of ordinary mesons and baryon substructures bound by the residual strong force that also binds atomic nuclei, and hybrid hadrons that bind a glueball and more ordinary quark based hadrons into a bound hadron state. 

Part of the problem in the case of these theoretically possible hadrons is that bosons (i.e. hadrons with integer spin that behave consistent with Bose-Einstein statistics) with the same quantum numbers naturally tend to blend into each other in one or more combinations. In many cases (e.g. up quark quarkonia, down quark quarkonia and strange quark quarkonia) as a result of this blending, a completely pure version of any of the hadrons with those quantum numbers doesn't exist.

It is also possible that some of the theoretically possible heavier or exotic hadrons do not exist because quantitatively the forces that would bind them into a bound hadron are just not quite strong enough to hold the components of the composite particle together with the physical constants of the Standard Model having the values that they do in Nature, much like the hypothetical atomic neutronium state (like a deuterium nucleus bound by the residual strong force, but with two neutrons instead of a proton and a neutron). 

The five pseudoscalar mesons and two vector mesons that are exceptions to the general rule are examples of this, but the problem is pervasive in the cases of scalar mesons, axial vector mesons, glueballs, tetraquarks, hexaquarks, even spin meson molecules, and hybrid hadrons. Just how they end up blending is also not always manifestly obvious and predictable, making it hard to predict in advance the "spectrum" of hadron resonances at various masses that are out there and observed.

So, experimentalists look for resonances with particular masses and quantum numbers, and then phenomenologists try to match those resonances to ordinary ground state pseudoscalar mesons, ground state vector mesons, ground state scalar mesons, ground state axial vector mesons, true tetraquarks, true pentequarks, true hexaquarks, glueballs, hybrid hadrons, hadron molecules, excited state hadrons of ground state hadrons, and blends of hadrons with the same quantum numbers.

The light scalar mesons, discussed below in the quoted introduction to a new high energy physics preprint.

The constituent quark model has been strikingly successful, but the nontrivial quark structures of scalar mesons below 1 GeV, f₀(500), f₀(980), and a₀(980)^0(±) (briefly denoted with σ, f₀, and a a₀^0(±), respectively), are not completely classified. Many theoretical hypotheses, such as the tetraquark states, and two-meson bound states, have been proposed for these light scalar mesons but with controversial results. Identifying the correct hypothesis is key to exploring chiral-symmetry-breaking mechanisms of nonperturbative QCD in low-energy region. Therefore, conclusive experimental results are required to interpret these states.

From a recent BESII preprint. A published physics journal article from 2019 tells a similar story:

Mitchell, in the comments, noted an insightful discussion of the topic by Ron Maimon at Physics Stack Exchange (to which I am an occasional mid-tier reputation contributor):

At one point, I decided to make friends with the low-lying spectrum of QCD. By this I do not mean the symmetry numbers (the "quark content"), but the actual dynamics, some insight.

The pions are the sloshing of the up-down condensate, and the other pseudoscalars by extending to strangeness. Their couplings are by soft-particle theorems. The eta-prime is their frustrated friend, weighed down by the instanton fluid. The rho and omega are the gauge fields for flavor SU(2), and A1(1260) gauges the axial SU(2), and they have KaluzaKlein-like echoes at higher energies, these can decay into the appropriate "charged" hadrons with couplings that depend on the flavor symmetry multiplet. The proton and the neutron are the topological defects. That accounts for everything up to 1300 but a few scalars and the b1.

There are scalars starting at around 1300 MeV which are probably some combination of glue-condensate sloshing around and quark-condensate sloshing around, some kind of sound in the vacuum glue. Their mass is large, their lifetime is not that big, they have sharp decay properties.

On the other hand, there is nothing in AdS/QCD which should correspond to the sigma/f0(600), or (what seems to be) its strange counterpart f0(980). While looking around, I found this discussion: http://www.physicsforums.com/showthread.php?t=241073. The literature that it pointed to suggests that the sigma is a very unstable bound state of pions (or, if you like, tetraquarks).

This paper gives strong evidence for an actual pole; another gives a more cursory review. The location of the pole is far away from the real axis, the width is larger than the mass by 20% or so, and the mass is about 400MeV. The authors though are confident that it is real because they tell me that the interpolation the interactions of pions is safe in this region because their goldstone properties dominate the interactions. I want to believe it, but how can you be sure?

I know this particle was controversial. I want to understand what kind of picture this is giving. The dispersion subtraction process is hard for me to visualize in terms of effective fields, and the result is saying that there is an unstable bound state.

Is there a physical picture of the sigma which is more field theoretical, perhaps even just an effective potential for pions? Did anyone who convinced himself of the reality of the sigma have a way of understanding the bound state properties? Is there an analog unstable bound state for other goldstone bosons? Any insight would be welcome.

This produced two answers, the first by Cosmos Zachos and the second by Mitchell Porter:

It is the lowest lying scalar meson, and scalar mesons, sharing quantum numbers with the vacuum, are notoriously hard to study. So, I've watched it entering and dropping off the PDG over decades. Today, it is there as a broad resonance, 

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0

(500)

$f_0(500)$, of about 441 MeV, according to Leutwyler, who should be the most reliable maven of it. The point is that it is wider than it's heavy, as its full width is about 544MeV, a dismal fate that the Higgs escaped!

In effective QCD, it dominates chiral symmetry breaking very analogously to the Higgs' informing EW symmetry breaking: in fact, theoretically, it has served as the conceptual underpinning of the Higgs for more than 40 yrs. Theorists love it more than experimentalists, and here is why:

Introduced in 1960, by Gell-Mann, M.; Lévy, M., "The axial vector current in beta decay", Il Nuovo Cimento 16705–726, doi:10.1007/BF02859738, following a hint by Schwinger, it gives its name to the σ-model introduced there. The effective Largrangian (diffidently called "model" back then) had kinetic terms for the (p,n) light nucleon doublet, the 3 πs and this σ, and a Higgsoid quartic potential 

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$\lambda ({\sigma }^2+{\overrightarrow{\pi }}^2-f_{\pi }^2{)}^2$ where I'm being cavalier with the normalizations of the fields and constants; and, moreover, most crucially, a Yukawa coupling term to the nucleon doublet 

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$g\overline{\psi }(\sigma +i\overrightarrow{\tau }\cdot \overrightarrow{\pi }{\gamma }_5)\psi$.

This interaction term, like the rest of the O(4)~ SU(2)xSU(2) invariant action, is also SU(2)xSU(2) invariant. When this group is broken down to the (strong!) isospin SU(2) spontaneously (Nambu's Nobel prize), the σ shifts by 

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$f_{\pi }$, and crucially the 3 Axial vector current combinations are now realized nonlinearly, i.e. 𝐴𝜇

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=

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+

$\overrightarrow{A_{\mu }}=f_p{\mathrm{\partial }}_{\mu }\overrightarrow{\pi }+$ bilinear terms... so they pump Goldstone pions into and out of the degenerate vacua while the isospin vector currents remain bilinear, so isospin is still unbroken.

Of course, from the Higgsoid potential, e.g. quartic, the quadratic term of the σ picks up a mass proportional to 

𝑓

𝜋

=93𝑀𝑒𝑉

$f_{\pi }=93MeV$ and the square root of the mystery effective quartic coupling of order one, which one never specified, as this is the strong interactions, after all... So, one expects a few hundreds of MeVs, which is what you get... Note it is, of course, heavier than the pions and kaons, which are pseudogoldstons, but not the ρ, the otherwise lightest "real" hadron.

The magic serving as the prototype of the EW standard model SSB is that now the masses of the nucleons come out of the above Yukawa term, 

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$g\overline{\psi }(f_{\pi }+{\sigma }^{\prime }+i\overrightarrow{\tau }\cdot \overrightarrow{\pi }{\gamma }_5)\psi$ to be about 

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$g{f}_{\pi }$, about a GeV, schematically: 𝑔

$\mathrm{<b>g</b>}$ is a strong nucleon-pion coupling larger than 10.

Today, this dynamical symmetry breaking is all described by quark condensates and calculated on lattice QCD, but the simplicity and elegance of the model in parsing out the logic is unbeatable. Since the σ can be there symmetry-wise, it would be odd if QCD did not manage to conjure an avatar for it at some level or other, but, in practice, it is a resonance from hell---or nuclear/hadronic physics.

And:

The literature that it pointed to suggests that the sigma is a very unstable bound state of pions (or, if you like, tetraquarks).

This has reminded me of a really interesting paper by Shifman and Vainshtein - http://arxiv.org/abs/hep-ph/0501200 - which talks about an exact symmetry between pions and diquarks in two-color QCD. They speculate that this symmetry should have an analogue in three-color QCD. I'm wondering if the sigma might be a sort of diquark-diquark correlation enhanced by the Shifman-Vainshtein symmetry.

I'd also look at Hilmar Forkel's work for AdS/QCD insight.

Untangling the Hadron Spectrum

Untangling the resonances in terms of underlying QCD structures is further complicated by the fact that the range of masses of the different possible hadrons and hadron molecules (other than plain old atoms consisting of protons and neutrons bound by the residual strong force) is quite narrow.

The least massive hadron which is also the least massive meson is the neutral pion with a mass of 134.9766(6) MeV. The most massive ground state meson without at least one bottom quark (the J/Psi) has a mass of 3,096.92(1) MeV. The most massive ground state pseudoscalar meson, the Upsilon (which is a spin-1 meson with a bottom quark and a bottom anti-quark), which is a form of bottomonium, has a measured mass of 9,460.30(26) MeV. The heaviest observed meson with a well measured mass, which is believed to be an excited form of bottomonium, has a mass of about 11,020 MeV. 

The least massive baryon is the proton with a mass of 938.272 0813(58) MeV. No measured ground state baryon mass without at least one bottom quark or three charm quarks as a valence quark is more massive than 3,621.2(7) MeV  (which is the double charmed Xi baryon). The heaviest observed ground state baryon with a well measured mass, the bottom omega baryon, has a mass of 6,046(2) MeV. No excited state three valence quark baryons with masses in excess of 5,955 MeV have been observed. The triple bottom omega baryon should have the highest possible baryon mass, which should be about 19,000 MeV.

In this mass range there are nearly 500 different observed hadron resonances, of which about 80% have well understood simple structures: 51 ground state pseudoscalar mesons (treating distinguishable antimesons as separate), 154 ground state baryons, about 200 excited states of these mesons, about 70 scalar and axial vector meson resonances (including both ground states and excited states), a dozen or so meson resonances not measured well enough to establish all of their quantum numbers with confidence, and a dozen or two resonances that are probably tetraquarks, pentequarks, or hexaquarks with hadron molecules with the same number of valence quarks. This is a new resonance, on average, every 30 MeV, but heavily crowded into the low end of this range but not quite the bottom of this range.

Exotic Hadrons

Exotic hadrons could be more massive.

The heaviest observed tetraquark, the X(6900), first reported in the year 2020, with two valence charm quarks and two valence charm antiquarks has a measured mass of about 6,900 MeV.

The heaviest observed pentaquark out of about half a dozen (including sub-discovery class claimed observations) types seen, the charmonium-pentaquark P_c(4457)⁺ has a mass of about 4,457 MeV first reported in the year 2019.

There has been only one claimed observation of a hexaquark (in the year 2014) and that with less than discovery class statistical significance, with a mass of about 2,380 MeV.

Heptaquarks are theoretically possible in QCD but none have ever been claimed to be observed. The lightest should be about 2500 MeV and the most stable one should have three strange quarks and two anti-strange quarks, in addition to two first generation quarks.

The heaviest conceivable ground state hadron with six or fewer valence quarks that do not include top quarks should be a mass of less than 35,000 MeV, and excited versions of it shouldn't be that much more massive.

Top quark hadrons would theoretically be vastly more massive (on the order of 173,000 MeV for top quark mesons and baryons with just one valence top quark, 350,000 MeV for toponium, and 520,000 MeV for a triple top omega baryon), but the top quark lifetime is so short that it almost never lasts long enough to hadronize, even in circumstances where there is enough mass-energy in an interaction to form a top quark hadron.

By comparison the energy scale of the highest energy runs of the Large Hadron Collider (the most powerful manmade particle collider ever) is about 14,000,000 MeV (which only produces significant numbers of hadrons and fundamental particles much less massive than this peak energy, however). Still, we can be fairly confident that there are not a lot of very massive hadrons that have gone undiscovered simply because we haven't had powerful enough colliders.

The Fundamental Particle Spectrum Compared

In contrast, the fundamental Standard Model particles have a much wider mass range, although only one fundamental Standard Model particle that exists outside of hadrons in normal circumstances has a mass that overlaps the hadron spectrum.

Eight massive Standard Model fundamental particles are less massive than the least massive hadron: the neutrinos (on the order of 0.000 000 000 1 to 0.000 000 072 0 MeV), the up quark (2.5 MeV), the down quark (4.88 MeV), the strange quark (93.44 MeV), the electron (0.511 MeV) and the muon (105.66 MeV) are all less massive than the least massive hadron. In addition nine Standard Model fundamental bosons: photon and eight kinds of gluons, and the hypothetical graviton, all have exactly zero rest mass.

Five Standard Model fundamental particles are more massive than the most massive hadron: the W+ boson and the W- boson (80,379 MeV), the Z boson (91,187.6 MeV), the Higgs boson (125,250 MeV), and the top quark (176,760 MeV).

Only three Standard Model fundamental particles have masses that overlap with the hadron mass spectrum: the charm quark (1272.3 MeV), the bottom (a.k.a. beauty) quark (4,198 MeV), and the charged tau lepton (1776.86 MeV). But, of course, up, down, strange, charm and bottom quarks are always observed either confined in a hadron, or blurred in a mish-mash within an extremely high energy quark-gluon plasma containing large numbers of quarks and gluons which are part of the plasma as a whole (rather than being truly "free") but are not part of specific hadrons.

Since the lightest five quarks are always hadronized, only one fundamental Standard Model particle, the charged tau lepton, has a mass overlapping the hadron mass spectrum. But this isn't even a particularly crowded part of the hadron spectum. It is close in mass (i.e. ± 75 MeV or so) to some easily distinguished electromagnetically neutral scalar mesons (which are electromagnetically neutral) and excited electromagnetically neutral baryons (e.g. some excited neutron states), and to a few charged hadrons including some excited eta mesons (which are charged) and several kinds of charged excited baryons. 

Thus, only on the order of 1% of electromagnetically charged resonances in the hadron spectrum are fairly close in mass to the charged tau lepton mass.

The Atomic And Molecular Mass Spectrum Compared

Periodic table atomic isotope nuclei also overlap only slightly in mass with hadrons. The heaviest measured baryon mass is about 6.5 daltons, the heaviest measured meson mass is about 11.83 daltons, and the heaviest theoretically possible ordinary two or three valence quark hadron is about 20.4 daltons.

The elements in the first two rows of the period table, with their average masses averaged over all isotopes weighted by frequency in daltons, rounded to three significant digits, are:

1. Hydrogen 1.01

2. Helium 4.00

3. Lithium 6.94 

4. Beryllium 9.01

5. Boron 10.81

6. Carbon 12.01

7. Nitrogen 14.01

8. Oxygen 16.00

9. Fluorine 19.00

10. Neon 20.18

More specifically, for example, Lithium-6 (an isotope of the third element in the periodic table) has a mass of 6.02 daltons, and Lithium-7 has a mass of 7.02 daltons.

For the most part, only common isotopes of hydrogen and helium overlap the hadron mass spectrum, although common isotopes of of elements up to neon overlap the very high end of this range for ordinary hadrons that is theoretically possible.

Some very light molecules (e.g. ordinary hydrogen gas H2) also overlap this mass range.