A diquark model for the d*(2380) dibaryon resonance?

Diquark models have been applied with varying degree of success to tetraquark and pentaquark states involving both light and heavy quark degrees of freedom. We discuss the applicability of such models to light quark dibaryons, viewed as three-diquark objects. Highlighting the case of the d*(2380) dibaryon resonance, we demonstrate the inapplicability of diquark models in the light quark sector.


Introduction
The idea that diquarks (D) play a significant role in hadron spectroscopy was raised by Jaffe to explain the 'inverted' SU(3)-flavor symmetry pattern of the lowest 0 + scalar-meson nonet in terms of tetraquarks, each made of a DD pair [1]. Diquarks attracted considerable interest also in trying to understand the structure of the dubious Θ + (1540) pentaquark which in some experiments showed up as a narrow K N resonant state [2,3]. More recently, following the discovery of tetraquark and pentaquark structures in the charmed (c) and bottom (b) quark sectors, diquarks have been used in theoretical studies of the structure and decay patterns of such exotic states; for a recent review see, e.g., Refs. [4,5].
A recent attempt to invoke diquarks to the structure of dibaryons, assuming that six-quark (6q) dibaryons consist dynamically of three diquarks, was made by Shi et al. [6] for the d * (2380) dibaryon resonance shown in Fig. 1. This I(J P )=0(3 + ) fairly narrow resonance, peaked about 80 MeV below the ∆∆ threshold, was observed in several two-pion production channels in pn collisions studied by the WASAat-COSY Collaboration [8]. Its I = 0 isospin assignment follows from balancing isospin in the pn → dπ 0 π 0 production reaction, and J P = 3 + spin-parity follows from the measured deuteron angular distribution. Subsequent measurements of  Fig. 1 The d * (2380) dibaryon resonance seen in the pn → dπ 0 π 0 reaction reported by the WASA-at-COSY Collaboration [7].
pn scattering and analyzing power [9] led to a pn 3 D 3 partialwave Argand diagram that supports the d * (2380) dibaryon resonance interpretation.
A major problem in understanding the structure of the d * (2380), viewed as an L = 0 ∆∆ dibaryon, arises from its relatively small width Γ d * ≈70 MeV, see Fig. 1, which is by far smaller than twice the width of a single ∆ baryon. Considering the reduced decay phase space available on average to a single ∆ bound in d * , its width is lowered from the free-space value of ≈115 MeV to about 80 MeV, so the problem here is how to account for a width reduction from about 160 MeV to Γ d * ≈70 MeV. This problem was considered in three separate approaches, the most recent of which (third one below) is the one we question in this note.
(i) The ∆∆-πN∆ coupled-channel hadronic calculation by Gal and Garcilazo [10,11] finds the d * (2380) resonance at about the right position in between the corresponding thresholds, and with approximately the observed width. The coupled-channel nature of this description is essential for understanding the relatively small width in simple terms [12].
(ii) Six-quark resonating-group-method calculations by Dong et al. [13] conclude that d * (2380) is dominated by a hidden-color ∆ 8 ∆ 8 component, roughly 2:1 with respect to a 'normal' ∆ 1 ∆ 1 component. With color conservation forbidding the decay ∆ 8 → N 1 + π 1 of a color-octet ∆ to colorless hadrons, this leads to a substantial reduction of Γ d * , in good agreement with the observed value. However, the compact nature of the decaying ∆ 1 ∆ 1 component introduces further reduction of the width, thereby resulting in over-suppression of Γ d * [12].
(iii) Assuming that d * (2380) consists of three (6 f , 3 c ) color-flavor S D =1 spin diquarks, Shi et al. [6] argued that the spatial rearrangement involved in transforming three colored diquarks to two color-singlet 3q hadrons, with spin and flavor that identify them with two ∆ baryons, suppresses the ≈160 MeV expected width by a factor of about 0.4. Unfortunately these authors overlooked the rearrangement required also in color-flavor space for a 3D system to become a ∆∆ system. This produces another suppression factor of 1/9, as shown in some detail below, so the resulting width is less than 10 MeV.
Apart from demonstrating explicitly, based on the rough width estimate cited above, why a diquark model is not the right model to describe the d * (2380) dibaryon resonance, the present note also discusses other light-quark dibaryon candidates predicted in this diquark model. It is concluded that diquark models in general are inappropriate for describing light quark dibaryons.

Classification of nonstrange dibaryon candidates
The quark-quark (qq) interaction is particularly strong in the anti-triplet antisymmetric color state3 c [3]. Hence, we limit the discussion to3 c diquarks. For a nonstrange S-wave diquark, requiring antisymmetry in the combined spin-isospincolor space leaves one with just two spin-isospin combinations: S D , I D =0,0 scalar diquarks and S D , I D =1,1 vector diquarks.
Consider first a state consisting of three scalar diquark bosons, antisymmetrized in color space to yield a colorconfined singlet 1 c wavefunction. Bose-Einstein statistics then imposes antisymmetry on the three-diquark space wavefunction. The lowest energy state is then expected to have L = 1, yielding a dibaryon with I = 0, S = 0 and J P = 1 − spin-parity. Within the quark model one estimates the mass of a scalar diquark to be 576 MeV [14]. Assuming the cost of exciting one unit of angular momentum at 150-200 MeV, the mass of such a 3D state would be expected to be between 0 and 50 MeV above the deuteron. Yet, no such isoscalar 1 P 1 resonance state has ever been claimed [8] and the only dibaryon-like P-wave resonances observed are in the isovector 3 P 0 and 3 P 2 − 3 F 2 channels near 2.2 GeV, about 80 MeV above the ∆N threshold [15]. This suggests that by trying to construct a dibaryon from three scalar diquarks one overlooks an important aspect of the dynamics. The most likely culprit is the implicit assumption that in order to satisfy spin-statistics one may ignore the diquarks' substructure and treat them all as elementary bosons. This is a rather dubious presumption, because in a hadron consisting of only light quarks there is a sole dynamical scale -Λ QCD . In the following we limit the discussion of dibaryon candidates to vector 3 c diquarks.
Manipulations with S D , I D =1,1 vector diquarks are a bit more involved. For a symmetric 3D space wavefunction, with orbital angular momentum L = 0 in mind, the spinisospin degrees of freedom have to be considered explicitly in forming together with a 1 c color wavefunction a totally symmetric 3D wavefunction. This is expressed schematically in terms of a product of two antisymmetric components: (1) The (1,1,1) S,I Young tableaux stands for the 84 S,I antisymmetric representation of SU (9) Some of these 3D S, I combinations, specifically 1,1 and 2,2, are spurious in terms of the underlying 6q wavefunctions which are obtained from the following product: where the (3,3) S,I Young tableaux stands for the 50 S,I representation of the standard SU(4)=SU(2) S ⊗ SU(2) I for spin-1/2 and isospin-1/2 quarks. The S, I=3,0 dibaryon candidate in this 6q scheme was calculated to lie more than 150 MeV above the d * (2380) dibaryon resonance [16] which casts doubts on any attempt to ascribe a dominantly hexaquark structure to the observed d * (2380).

Dibaryon masses and rearrangement factors
We focus now on the I = 0 L = 0 S = 3 3D state identified in Ref. [6] with the I = 0 J P = 3 + d * (2380) dibaryon resonance. Its mass value was reproduced there by using an effective diquark mass plus color-electric and color-spin interaction matrix elements deduced from applying scalar and vector diquark models in the charmed sector, above 2 GeV. The applicability of these diquark mass and interaction parameters to the light-quark sector is questionable. Nevertheless accepting this, perhaps fortuitous reproduction of the d * (2380) mass, we ask where the I = 0 J P = 1 + deuteronlike and the I = 1 J P = 0 + virtual-like N N states are located in this 3D model. Identifying these states with the I = 0 S = 1 and the I = 1 S = 0 states of the 84 S,I SU(9) representation discussed in the previous section, we evaluate their masses using the same D mass and DD interaction parameters used by Shi et al. [6] to evaluate the location of d * (2380). Details are given here in the Appendix. The deuteron-like state d is found then 263 MeV below the d * (2380), about 245 MeV above the physical deuteron, with the virtual-like state v further 53 MeV down below d. However, no resonance feature in the corresponding I = 0 J P = 1 + and I = 1 J P = 0 + N N partial-wave phase shifts up to at least E cm =2.4 GeV has ever been observed without any doubt [18]. Next we evaluate the rearrangement factors involved in transforming the 3D model I = 0 L = 0 S = 3 state to a ∆∆ I = 0 J P = 3 + d * (2380). Since the S = 3 Pauli spin configuration is fully stretched in both 3D and ∆∆ bases, the spin rearrangement factor is simply 1. This is not the case for isospin and for color. Starting with isospin, we write schematically the 3D model couplings in the form where the isospin structure of the I 3 = 1 third diquark is spelled out explicitly in terms of its quark component isospins i 3 = i 4 = 1 2 . We now recouple isospins, so that the quark isospin i 3 joins the diquark isospin I 1 = 1 to form a ∆ baryon isospin I 13 = 3 2 , and similarly the quark isospin i 4 joins the diquark isospin I 2 = 1 to form another ∆ isospin I 24 = 3 2 , viz.
Recoupling in color space is done by generalizing from SU(2)-isospin to SU(3)-color. The corresponding unitary operator matrix element is given by [20]: where the notation 'dim' stands for the dimension (=3) of the marked SU(3) c representations. The combined recoupling coefficient in both isospin and color spaces is given by a product of the values noted in Eqs. (7) and (8) which amounts to −1/3. It enters quadratically in the evaluation of the d * (2380) decay width to nucleons and pions via a ∆∆ hadronic doorway state, hence the width suppression factor 1/9 overlooked in Ref. [6].
In a similar way, rearrangement factors for d and v to go into the corresponding N N doorway states can also be evaluated, yielding somewhat smaller values of less than 0.1. This means that the widths involved in decays of such hypothetical dibaryons should be in the range of tens of MeV at most. Therefore, if the d and v 3D dibaryon states exist, they should have been already observed in N N scattering experiments.

Discussion and summary
In this brief note we discussed the applicability of3 c diquark models to light-quark nonstrange dibaryons, following a suggestion made by Shi et al. [6] that the observed d * (2380) dibaryon is dominantly of a 3D structure. A useful test of any dibaryon model is provided by the extent to which it describes well the low lying hadronic spectrum. In this respect, we found that the 3D I = 0 J P = 1 + deuteron-like and the I = 1 J P = 0 + virtual-like states in the particular diquark model suggested by these authors are located some 200-250 MeV above the physical deuteron, where no hint of irregularities in the corresponding N N phase-shift analyses exist. This demonstrates that diquark models are not physically appropriate models for binding six quarks into a dibaryon. Hadronic sizes that are relevant for binding together two baryons, particularly through pion exchange, are of order 1-2 fm and are considerably larger than the subfermi sizes expected for deeply bound 3D structures. This results in extremely small 6q admixtures in the deuteron, see e.g. Ref. [21] for a recent discussion.
As for the d * (2380) dibaryon specifically, which is observed through decay modes involving nucleons and pions that are consistent with a size of 1-2 fm [12], we noted that if it were dominated by a 3D structure, its decay width would have been suppressed by at least an isospin-color recoupling factor of 1/9 with respect to the initial ∆∆ hadronic estimate of 160 MeV width. We conclude that assigning a 3D struc-