No radial excitations in low energy QCD. I. Diquarks and classification of mesons

Abstract

We propose a new schematic model for mesons in which the building blocks are quarks and flavor-antisymmetric diquarks. The outcome is a new classification of the entire meson spectrum into quark–antiquark and diquark–antidiquark states which does not give rise to a radial quantum number: all mesons which have so far been believed to be radially excited are orbitally excited diquark–antidiquark states; similarly, there are no radially excited baryons. Further, mesons that were previously viewed as “exotic” are no longer exotic as they are now naturally integrated into the classification as diquark–antidiquark states. The classification also leads to the introduction of isorons (iso-hadrons), which are analogs of atomic isotopes, and their magic quantum numbers, which are analogs of the magic numbers of the nuclear shell model. The magic quantum numbers of isorons match the quantum numbers expected for low-lying glueballs in lattice QCD. We observe that interquark forces in mesons behave substantially differently from those in baryons: qualitatively, they are color–magnetic in mesons but color–electrostatic in baryons. We comment on potential models and the hydrogen atom. The implications of our results for confinement, asymptotic freedom, and a new set of relations between two fundamental properties of hadrons—their size and their energy—are discussed in our companion paper (Eur. Phys. J. C (2013). doi:10.1140/epjc/110052-013-2299-8).

Appendices

Appendix A: Nonet by nonet discussion

In this appendix we provide a multiplet by multiplet discussion of the classification.

1.1 A.1 Light mesons

J ^PC=0⁻⁺

We have two 0⁻⁺ nonets. Available assignments are an S-wave of $q\bar{q}$ and a P-wave of $\mathcal{Q}_{2} \bar{\mathcal{Q}}_{2}$. The orbital excitation rule tells us to assign the lower-lying nonet to the S-wave and the second nonet to the P-wave. Other 0⁻⁺ are isorons.

We took the η(1475) to be the heavier isosinglet in the second nonet, leaving out the η(1405). Our choice is due to the fact that the heavier isosinglet in any nonet should couple to kaons, and the η(1475) couples to kaons more strongly than η(1405) (see “Note on η(1405)” in [13]).

Note that the second nonet was previously taken to consist of radially excited mesons [13].

J ^PC=0⁺⁺

The lightest scalar nonet is $\mathcal{Q}_{1} \bar{\mathcal{Q}}_{1}$ with L=S=0; an assignment of these mesons to four–quark states was suggested by Jaffe in 1976 [16]; see also [84].

The next nonet is the quark model’s $q\bar{q}$ P-wave. The choice of isoscalar that would complete this nonet has always been ambiguous [85–87]. Following [87], we choose the f ₀(1710). The other f ₀ mesons are isorons.

The third (partial) nonet has masses around 2 GeV, so by the orbital excitation rule it should be either a D-wave or an F-wave; the only option is a $\mathcal{Q}_{2} \bar{\mathcal{Q}}_{2}$ D-wave.

J ^PC=1⁻⁻

There are three complete or close-to-complete 1⁻⁻ nonets, and two incomplete nonets which consist of only the isospin triplet (the ρ). Available assignments are ³ S ₁ or ³ D ₁ of $q\bar{q}$, ¹ P ₁ of $\mathcal{Q}_{1} \bar{\mathcal{Q}}_{1}$, and ¹ P ₁ or ⁵ P ₁ or ⁵ F ₁ of $\mathcal{Q}_{2} \bar{\mathcal{Q}}_{2}$. By the orbital excitation rule, the lowest-lying nonet, with masses less than 1 GeV, is an S-wave so we assign it to ³ S ₁ of $q\bar{q}$.

The second nonet is about 0.5 GeV heavier, so it is a P-wave of either $\mathcal{Q}_{1} \bar{\mathcal{Q}}_{1}$ or $\mathcal{Q}_{2} \bar {\mathcal{Q}}_{2}$. We assign it to ¹ P ₁ of $\mathcal{Q}_{1} \bar {\mathcal{Q}}_{1}$, though this choice is rather arbitrary—this nonet could be a mixture of $\mathcal{Q}_{1} \bar{\mathcal{Q}}_{1}$ and $\mathcal{Q}_{2} \bar{\mathcal{Q}}_{2}$.

The next nonet has only the ρ(1570), which appeared in the PDG for the first time in 2008. It is slightly heavy for a P-wave by the orbital excitation rule, but we still assign it to the P-wave of $\mathcal{Q}_{2} \bar{\mathcal{Q}}_{2}$ because there is a more suitable nonet for the available D-wave assignment; since it is heavy relative to other P-waves, we choose the ⁵ P ₁ rather than the ¹ P ₁ assignment because it is plausible that higher S may mean higher mass (also see Eq. (26)).

The next nonet, which is about 1 GeV higher than the lightest nonet, is a D-wave by the orbital excitation rule and we assign it to ³ D ₁ of $q\bar{q}$. Another isovector is at the mass range of F-waves, and we assign it to $\mathcal{Q}_{2} \bar{\mathcal{Q}}_{2}$.

Note that the second 1⁻⁻ nonet was previously taken to consist of radially excited mesons [13].

J ^PC=1⁺⁺

There are two nonets. Available assignments are a P-wave of $q\bar{q}$ and a D-wave of $\mathcal{Q}_{2} \bar{\mathcal{Q}}_{2}$. Using the orbital excitation rule, we assign the lighter nonet to a P-wave and the second nonet to a D-wave.

J ^PC=1⁻⁺

There are no complete nonets here. However, from Table 2 we know that a 1⁻⁺ nonet should appear as $\mathcal{Q}_{2} \bar{\mathcal{Q}}_{2}$ in a P-wave. We classify the π ₁(1600) and K(1630) as members of this nonet even though it is a bit heavy for a P-wave (we could have taken the π ₁(1400), but we opted to make the nonet consist of mesons whose masses are closer together); the π ₁(1400) is an isoron. Note that it has been argued [67, 88] that if the 1⁻⁺ pion is a four-quark state, then it should be part of a large flavor multiplet, i.e. larger than a nonet. Such a multiplet has not been observed, and in our model it is not expected to be—we expect only nonets in the light flavor sector (Sect. 2.1). See [13, 68, 89–91] for more on the 1⁻⁺ pions.

J ^PC=2⁻⁺

There are three nonets, and there are three available assignments: a ³ P ₂ of $\mathcal{Q}_{2} \bar{\mathcal{Q}}_{2}$, a ¹ D ₂ of $q\bar {q}$, and a ³ F ₂ of $\mathcal{Q}_{2} \bar{\mathcal{Q}}_{2}$. We assign the lightest nonet to the P-wave (even though it is a bit heavy for a P-wave), the next one to the D-wave, and the last one to the F-wave. Note that the second nonet has so far only the isovector π ₂(1880), which in fact entered the PDG only in 2008; if it were not for its appearance, we would have assigned the lightest nonet to the D-wave based on the orbital excitation rule.

J ^PC=2⁻⁻

There are two 2⁻ kaons here. We assign the lighter to a P-wave (though it is a bit heavy based on the orbital excitation rule) and the heavier to a D-wave.

J ^PC=2⁺⁺

There are three almost complete nonets. The lightest and heaviest are both $q\bar{q}$, while the middle one is a $\mathcal{Q}_{1} \bar {\mathcal{Q}}_{1}$ in a D-wave. The other two have only a single isoscalar in each; they are $\mathcal{Q}_{2} \bar {\mathcal{Q}}_{2}$ D-waves. The $\mathcal{Q}_{2} \bar{\mathcal{Q}}_{2}$ isoscalars and the isoscalars in the middle nonet, all D-waves, could mix.

J ^PC=3⁻⁻

There is one complete nonet, which is the D-wave of $q\bar{q}$. There are also two heavier isovectors with the same J ^PC. Of those, the lighter one is below the baryon–antibaryon threshold, so may be $\mathcal {Q}_{1} \bar{\mathcal{Q}}_{1}$ in an F-wave. The second is above this threshold and therefore is unlikely to have $\mathcal{Q}_{1}$ as a constituent (see decay properties, p. 10); therefore, we assign it to $\mathcal{Q}_{2} \bar {\mathcal{Q}}_{2}$ as an F-wave.

J ^PC=4⁺⁺

There is one complete nonet in this sector, a $q\bar{q}$ in an F-wave. An f ₄(2300) should be an F-wave by the orbital excitation rule, but there are no available assignments, so it is an isoron.

J ^PC=5⁻⁻

The 5⁻⁻ nonet could be a G-wave $q\bar{q}$ or an F-wave $\mathcal {Q}_{2} \bar{\mathcal{Q}}_{2}$, or a $\mathcal{Q}_{1} \bar{\mathcal{Q}}_{1}$ with even higher L. By the orbital excitation rule, it should be an F-wave, so we assign it to $\mathcal{Q}_{2} \bar{\mathcal {Q}}_{2}$. However, it could be a G-wave as classified in the PDG.

J ^PC=6⁺⁺

The 6⁺⁺ has the mass range appropriate for an F-wave or at most a G-wave. The lowest L available for this J ^PC is a G-wave of $\mathcal{Q}_{2} \bar{\mathcal{Q}}_{2}$, which is our assignment. However, it could also be the H-wave as classified in the PDG.

1.2 A.2 Charmed and bottom mesons

J ^PC=0⁻⁺

There is one complete multiplet and one partial multiplet. Note that the J ^PC of the bottom mesons in the first multiplet have not been determined experimentally. As is standard, we assign them to S-wave of $q\bar{q}$.

J ^PC=0⁺⁺

Recent suggestions (see [29, 30] for reviews) that $D_{s0}^{*}(2317)$ may be a tetraquark support the possibility that it completes the $\mathcal{Q}_{1} \bar{\mathcal{Q}}_{1}$ nonet rather than the $q\bar{q}$ nonet.

J ^PC=1⁻⁻

Note that since its first appearance in the 1970s, the ψ(2S) was assigned to be a radial excitation [92–95]. Until now, this assignment does not seem to have ever been questioned or challenged and is even part of the particle’s name. In our paper, the ψ(2S) is a diquark–antidiquark P-wave (L=1).

J ^PC=1⁺⁺

There are two multiplets, one complete and one incomplete. Recent suggestions (see [29, 30] for reviews) that $D_{s1}^{*}(2460)$ may be a tetraquark support our classification to $\mathcal{Q}_{2} \bar{\mathcal{Q}}_{2}$ rather than $q\bar{q}$.

We classify the X(3872) as a member of the $\mathcal{Q}_{2} \bar {\mathcal{Q}}_{2}$ as well. The J ^PC=1⁺⁺ assignment for this particle is favored [96] but 2⁻⁺ is also possible [97]; see also [31, 98, 99]. Its isospin has not been determined yet; we listed it only under I=0 in the table, but its decays indicate that it must mix with I=1. See [31, 100–102].

Note that we include the new bottom mesons B ₁ and B _s1; Table 14.3 of the PDG does not include them.

J ^PC=2⁺⁺

There is one complete multiplet and one partial one. We include the new $B_{2}^{*}$ and $B_{s2}^{*}$ (which do not appear in Table 14.3 of the PDG).

Appendix B: “Exotic” and outcast mesons

We list in Table 5 all the mesons that appear in the 2008 PDG particle listing but are not classified in Tables 14.2 and 14.3 there.¹⁰ In our model, all these mesons are no longer exotic or outcast but are part of the model and classified in our tables.

Table 5 “Exotic” and Outcast Mesons in the 2008 PDG