Quark models of dibaryon resonances in nucleon-nucleon scattering

0.34 MB·English

by  J. L. Ping

Quark models of dibaryon resonances in nucleon-nucleon scattering
J.L. Ping1, H.X. Huang1, H.R. Pang2, Fan Wang3 and C.W. Wong4
1Department of Physics, Nanjing Normal University, Nanjing, 210097, P.R. China
2Department of Physics, Southeast University, Nanjing, 210094, P.R. China
3Department of Physics, Nanjing University, Nanjing, 210093, P.R. China
4Department of Physics and Astronomy, University of California, Los Angeles, CA 90095-1547, USA
Welookfor∆∆andN∆resonancesbycalculatingNN scatteringphaseshiftsoftwointeracting
baryonclustersofquarkswithexplicit couplingtothesedibaryonchannels. Twophenomenological
nonrelativistic chiral quark models giving similar low-energy NN properties are found to give sig-
nificantlydifferentdibaryonresonancestructures. Inthechiralquarkmodel(ChQM),thedibaryon
systemdoesnotresonateintheNN S-waves,inagreementwiththeexperimentalSP07NN partial-
wave scattering amplitudes. In the quark delocalization and color screening model (QDCSM), the
9 S-wave NN resonances disappear when the nucleon size b falls below 0.53 fm. Both quark models
0 give an IJP = 03+ ∆∆ resonance. At b = 0.52fm, the value favored by baryon spectrum, the
0 resonance mass is 2390 (2420) MeV for the ChQM with quadratic (linear) confinement, and 2360
2 MeV for the QDCSM. Accessible from the 3DNN channel, this resonance is a promising candidate
3
n for the known isoscalar ABC structure seen more clearly in the pn→dππ production cross section
a at 2410 MeV in the recent preliminary data reported by the CELSIUS-WASA Collaboration. In
J theisovectordibaryonsector,ourquarkmodelsgiveaboundoralmost bound5S∆∆ statethatcan
2
0 giverisetoa1D2NN resonance. Noneofthequarkmodels usedhasboundN∆ P-statesthatmight
3 generate odd-parity resonances.

] PACSnumbers: 12.39.Jh,14.20.Pt,13.75.Cs
h
t
-
l I. INTRODUCTION nels containing ∆s and pions, and should therefore vary
c
u when more of these channels are explicitly included in
n the coupled-channel calculation. Its effects always in-
Quantum chromodynamics (QCD) is widely accepted
[ cludetheexchangeoftwopions,andarecalledcorrelated
asthefundamentaltheoryofthestronginteraction. Lat-
if the two pions also interact with each other. Modern
2 tice QCD methods have recently been used to study
treatments of correlatedtwo-pionexchange show that in
v low-energy hadronic interactions, including the nucleon-
8 additiontoalong-rangescalar-isoscalarattractiontradi-
nucleon (NN) interaction [1]. However, QCD-inspired
5 tionally associated with scalar exchange, there is also a
quark models are still the main tool for detailed studies
4 strong scalar-isoscalar repulsive core [5], a complication
of the baryon-baryoninteraction.
0 that has not yet been included in the ChQM.
. The phenomenological quark model most commonly
6
used in the study of NN interaction is the nonrelativis- In the quark model, the forces between baryonic clus-
0
8 tic chiral quark model (ChQM) [2, 3, 4]. Nonrelativis- ters of quarks and antiquarks are like molecular forces
0 tic kinematics makes the many-body treatment of the between molecules of atoms built up from the forces be-
: multiquarksystemmanageable,withtheveryconvenient tweentheirconstituents. Thismolecularmodelofnuclear
v
i choice that the light quark mass mq is just one third of forceshasbeenextensivelydevelopedbyusingthequark
X thenucleonmass. Withquarks,oneneedsaconfinement delocalization and color screening model (QDCSM) [6].
r potential to reproduce a distinctive QCD property, and Inthis model, quarksconfined inone baryonare allowed
a a one gluon exchange (OGE) to account for ∆−N mass to delocalize to a nearby baryon and to change the dy-
difference and other details of baryon excitations. The namicsofthebaryon-baryoninteractionthroughareduc-
inclusion of pion exchange will take care of long-range tion of the confinement potential called color screening.
baryon-baryon interactions as well as some features in Thedelocalizationparameterthatappearsisdetermined
baryon structure, and is the consequence of the relative by the dynamics of the interacting quark system, thus
weakness of chiral symmetry breaking. Finally, scalar making it possible for the quark system to reach a more
exchangeis usedto describeanextraintermediate-range favorable configuration through its own dynamics in a
attraction needed in nuclear forces. No other meson ex- larger Hilbert space. The model has been successfully
changes are included. ChQMs with a few chosen and a applied to NN and hyperon-nucleon scatterings. The
few adjusted parameters are able to give a surprisingly importantintermediate-rangeNN attractionisachieved
simple if only semi-quantitative picture of both baryon in this model by the mutual distortion of the interact-
spectra and baryon-baryoninteractions at relatively low ing nucleons, in a way that is very similar to the mutual
energies. It is therefore of some interest to understand distortion of interacting molecules.
some of the limitations of these simple quarks models. The main difference between the ChQM and the QD-
The mostproblematic termin the ChQMis the scalar CSMisthemechanismforintermediate-rangeattraction.
exchange term. It takes into account neglected chan- Recently,weshowedthatboththeChQMcontainingthe
2
σ-mesonandtheQDCSMwithoutitgaveagooddescrip- tions. Ourfirststudyintheresonanceregionwillinclude
tion of the low-energy NN S- and D-wave scattering only∆excitations. Thislimitationofexcitedbaryonde-
phase shifts and the properties of deuteron with almost grees of freedom to ∆s only has often been made in past
thesamequark-modelparameters[7]. Thustheσ-meson studies of nuclear forces. [12]
exchange can effectively be replaced by the quark delo- Weshallusethe sameSalamancaChQM[13]andQD-
calization and color screening mechanism. It is not clear CSMusedpreviouslyforNN channelsonly[7]withaddi-
however if their equivalence persists to higher energies tionalsetsofpotentialparameterstofindoutiftheirsim-
where nucleons overlap more strongly and baryon exci- ilarity persists into the resonance region. We are inter-
tation and multiquark effects become more important. ested in particular in discovering how similar these sim-
Interest in multiquark system has persisted since R. ple quark models are in describing theoretical dibaryon
JaffepredictedtheH-particlein1977[8]. Allquarkmod- resonances originating from ∆∆ or N∆ bound states
els,includingthoseusinglatticeQCDtechniques,predict when the ∆s are treated as stable particles. In other
that in addition to the qq¯mesons and q3 baryons, there words, these dibaryon resonances are theoretical com-
shouldbemultiquarksystems(qq¯)2,q4q¯,q6,quarkgluon pounddibaryonstatesthatareallowedtodecayonlyvia
hybrids qq¯g, q3g, and glueballs [9]. Up to now there the NN channels. Abriefdescriptionofthese twoquark
has been no well established experimental candidate of models of the baryon-baryoninteraction is given in Sec-
these multiquark states [10]. Recently, the CELSIUS- tion II.
WASA Collaboration has reported preliminary results
The NN phase shifts for these quark-quarkpotentials
on the ABC anomaly in the production cross section of
are calculated using a coupled-channel resonating group
the pn→dπ0π0 reaction that suggests the presence of an
formalism [14] that includes ∆∆, N∆ and sometimes
isoscalar JP = 1+ or 3+ subthreshold ∆∆ resonance,
hidden-colorchannelsaswell. Noexplicitpionicchannels
with resonance mass estimated at ∼ 2410 MeV and a
are included as the ∆s are treated as stable particles. In
widthof<100MeV[11]. Therelativelylargebindingen-
this context, a promising dibaryon resonance is taken to
ergyinvolvedgivesanobjectthatismuchclosertothese
be one arising from a bound state below the ∆∆ or N∆
interesting multiquark states than a loosely bound sys-
threshold. The calculated results, including resonance
temlikethedeuteron. Nonrelativisticquarkmodelssuch
masses and widths (FWHMs), are given in section III in
as ChQMs and QDCSMs fitting both N,∆ masses and
those partial waves (PWs) where a theoretical dibaryon
low-energyNN scattering properties can be expected to
resonance appears in at least one quark model. No N∆
give particularly interesting and parameter-free predic-
bound state is found in any isovectorodd-parity state in
tions for such dibaryon multiquark states.
all our quark models.
It thus appears worthwhile to extend our past calcu-
InSect. IV,theseresultsarecomparedtopartial-wave
lation of NN phase shifts to the resonance region near
analysesofNN scatteringamplitudes[15,16],wherethe
the ∆∆ and N∆ thresholds by including these excited
presenceof a dibaryonresonancecauses a rapidcounter-
dibaryon channels in coupled-channel calculations. The
clockwise motion in the Argand diagram. The possibil-
∆ resonance is by far the most important low-energy
ity that the ABC effect in the pn→dππ reaction is an
−
baryon resonance. It dominates even the π p cross sec-
isoscalar NN resonance is also discussed.
tionswhereitsproductionishinderedrelativetothepro-
∗ Section V contains brief concluding remarks on what
duction of isospin 1/2 N resonances by a factor of 2
we have learned about quark dynamics in the NN reso-
from isospin coupling. In dibaryon channels, the ∆∆
nance region.
threshold at 2460 MeV is clearly separated from the
∗
NN (1440) threshold at 2380 MeV, the second most in-
teresting dibaryon threshold in this energy region. ∆∆
bound states are of immediate interest in understanding
II. TWO QUARK MODELS OF
resonance phenomena in this near subthreshold energy BARYON-BARYON INTERACTIONS
∗
region. The inclusion of N (1440) would be of consider-
able interest in a broader study of dibaryon resonances,
A. Chiral quark model
but its inclusion is technically difficult for us because
∗
N (1440)is commonly understood to be a monopole ex-
citation of the nucleon that has a much more compli- TheSalamancaChQMisrepresentativeofchiralquark
catedquarkwavefunction. Incontrast,ourapproximate models. It has also been used to describe both hadron
description of the N → ∆ excitation as a simple spin- spectroscopy and nucleon-nucleon interactions. The
isospin excitation without any change in the radial wave model details can be found in [13]. Only the Hamilto-
function probably captures the essence ofthe physics in- nian and parameters are given here.
volved. ∆ excitations are thus within easy reach of the The ChQM Hamiltonian in the baryon-baryon sector
technologyusedinourpreviouscoupled-channelcalcula- is
3
6 p2
H = m + i −T + VG(r )+Vπ(r )+Vσ(r )+Vρ(r )+VC(r ) , (1)
i c ij ij ij ij ij
2m
i=1(cid:18) i(cid:19) i<j
X X(cid:2) (cid:3)
1 1 π 2 3
VG(r ) = α λ ·λ − 1+ σ ·σ δ(r )− S +VG,LS,
ij 4 s i j r m2 3 i j ij 4m2r3 ij ij
" ij q (cid:18) (cid:19) q ij #
α 1 3
VG,LS = − sλ ·λ [r ×(p −p )]·(σ +σ ),
ij 4 i j8m2r3 ij i j i j
q ij
1 Λ2 Λ3 Λ3
Vπ(r ) = α m Y(m r )− Y(Λr ) σ ·σ + H(m r )− H(Λr ) S τ ·τ ,
ij 3 chΛ2−m2 π π ij m3 ij i j π ij m3 ij ij i j
π (cid:26)(cid:20) π (cid:21) (cid:20) π (cid:21) (cid:27)
4m2 Λ2 Λ g2 m2
Vσ(r ) = −α u m Y(m r )− Y(Λr ) +Vσ,LS, α = ch π ,
ij ch m2 Λ2−m2 σ σ ij m ij ij ch 4π 4m2
π σ (cid:20) σ (cid:21) u
α Λ2 Λ3
Vσ,LS = − ch m3 G(m r )− G(Λr ) [r ×(p −p )]·(σ +σ ),
ij 2m2 Λ2−m2 σ σ ij m3 ij ij i j i j
π σ (cid:20) σ (cid:21)
4m2 Λ2 Λ m2 Λ3
Vρ(r ) = α u m Y(m r )− Y(Λr ) + ρ Y(m r )− Y(Λr ) σ ·σ
ij chv m2 Λ2−m2 ρ ρ ij m ij 6m2 ρ ij m3 ij i j
π ρ ((cid:20) ρ (cid:21) u (cid:20) ρ (cid:21)
1 Λ3 g2 m2
− H(m r )− H(Λr ) S τ ·τ +Vρ,LS, α = chv π ,
2 ρ ij m3 ij ij i j ij chv 4π 4m2
(cid:20) ρ (cid:21) (cid:27) u
3α Λ2 Λ3
Vρ,LS = − chv m3 G(m r )− G(Λr ) [r ×(p −p )]·(σ +σ ) τ ·τ ,
ij m2 Λ2−m2 ρ ρ ij m3 ij ij i j i j i j
π ρ (cid:20) ρ (cid:21)
VC(r ) = −a λ ·λ (r2 +V )+VC,LS,
ij c i j ij 0 ij
1 1 dVc
VC,LS = a λ ·λ [r ×(p −p )]·(σ +σ ), Vc =r2,
ij c i j8m2r dr ij i j i j ij
q ij ij
(σ ·r )(σ ·r ) 1
S = i ij j ij − σ ·σ . (2)
ij r2 3 i j
ij
Here S is quark tensor operator, Y(x), H(x) and G(x) confinementpotential. The modelChQM1 [2] usesa lin-
ij
are standard Yukawa functions [17] , T is the kinetic ear confinement potential instead. The model ChQM2a
c
energy of the center of mass, α is the chiral coupling differs fromChQM2 infitting a differentequilibrium nu-
ch
constant, determined as usual from the π-nucleon cou- cleon size b but for simplicity m ,Λ are allowed to re-
σ
pling constant. An additional ρ meson exchange poten- main at the ChQM2 values. Its deuteronbinding energy
tial Vρ between quarks has been added to give an im- is reduced, but we do not consider the difference to be
provedtreatment of baryon-baryonP-states. Its param- importantinourstudyofdibaryonresonanceproperties.
eterswillbespecifiedinSect.III. Allothersymbolshave
their usual meanings.
Table I gives the model parameters used. For each Finally, Table I also gives the effective-range(ER) pa-
set of parameters, the nucleon size b that appears in rameters of a 5-parameter ER formula in the NN 3S
1
Eq.(5) is given a pre-determined value. Two of the (1S ) state. The calculation uses with 5 (4) color-singlet
0
parameters (a ,α ) are fitted to ∆−N mass difference channels denoted 5cc (4cc) and defined in the following
c s
(1232−939MeV) and the equilibrium condition for the section. The channels used include 2 NN and 3 ∆∆ (an
nucleonmassatthechosenb. Theabsolutenucleonmass NN, 2 ∆∆, and an N∆) channels. The deuteron bind-
is controlled by a constant term V in the confinement ing energy ε calculated from the two ER parameters of
0 d
potentialthatdoes notaffectthe baryon-baryoninterac- the 5cc calculation is also given in the table. Accord-
tion. In ChQM2 [18], the deuteron binding energy (2.22 ingto[19],thisapproximationoverestimatesthe binding
MeV) is fitted by varying the combination m ,Λ calcu- energy, but by only 0.015 MeV. So the tabulated bind-
σ
lated with the standard two NN coupled channels 3S ing energies are sufficiently accurate for this qualitative
1
and 3D (called 2NNcc in the following). The remaining study. The same ER aproximation for a 2NNcc calcula-
1
parameters m ,m and α are fixed at chosen values. tion gives ε ≈1.86MeV for ChQM2, thus showing that
q π ch d
Thenumeral2inthenameChQM2referstoitsquadratic the 3 ∆∆ channels increase ε by about 1.5 MeV.
d
4
Quak delocalization with color screening is an approxi-
TABLEI:Parametersthatdifferindifferentmodelsaregiven
mate way of including hidden-color (h.c.) effects.
in this table. The dimension of each dimensional parame-
ThecolorscreeningconstantµinEq.(4)isdetermined
terisgivenwithin parenthesesfollowing eachsymbol: b(fm),
ac(MeVfm−2 if quadratic, butMeVfm−1 if linear), V0(fm2) by fitting deuteron properties. The parameters of the
and µ(fm−2). Parameters having the same value for all the QDCSMi(i=1,3)usedherearethoseofSetiof[7],and
quark models discussed in this paper are mq = 313MeV, are given again in Table I. For QDCSM0, µ is an esti-
mπ = 138MeV, Λ/¯hc = 4.2fm−1, and mσ = 675(MeV) matedvaluethathasnotbeenfinetunedtothedeuteron
for ChMQs. The scattering length and effective range calcu- binding energy. These models differ in the equilibrium
lated for each potential are also given: at,rt for the triplet nucleon size b.
state 3S1 and as,rs for the singlet state 1S0, all in fm.
Thedeuteronbindingenergyεd(MeV)iscalculated fromthe
triplet effective-range parameters.
III. RESULTS
ChQM2 ChQM2a QDCSM0 QDCSM1 QDCSM3
(ChQM1)
NN scatteringphaseshiftsarecalculatedforthequark
b 0.518 0.60 0.48 0.518 0.60
models of Table I to energies beyond the ∆∆ or N∆
ac 46.938 12.39 85.60 56.75 18.55
thresholdfordifferentchoiceofcoupledchannels. Wein-
(67.0)
clude channelscontainingoneormore∆s treatedassta-
V0 -1.297 0.255 -1.299 -1.3598 -0.5279
µ 0.30 0.45 1.00 ble particles,andchannelscontaininghidden-color(h.c.)
αs 0.485 0.9955 0.3016 0.485 0.9955 states. The resonating-groupmethod (RGM), described
αch 0.027 0.027 0.027 0.027 0.027 in more details in [14], is used.
(0.0269) Past experience has suggested that reliable estimates
at 4.52 20.8 34.9 5.94 6.03 of resonancemasses canbe made using non-decaying∆s
rt 1.56 2.24 2.27 1.75 1.67 [2]. For the theoretical N∆ resonance d′ (IJP = 20−)
εd 3.35 0.11 0.04 1.75 1.64 at the theoretical mass 2065 MeV (and an N∆ binding
as -170 -2.48 -2.32 -6.90 -5.41 energy of 106 MeV), an increase in the imaginary part
rs 2.17 5.42 4.48 2.63 3.56 of the ∆ resonance energy by 10 − 15 MeV is known
′
to increase the d mass by only a few tenth of an MeV
[21]. Thisresultsuggeststhatthecompleteneglectofthe
B. Quark delocalization, color screening model
imaginarypartΓ/2≈60MeVofthe ∆resonanceenergy
′
will underestimate the d resonance mass by perhaps 2
Themodelanditsextensionwerediscussedindetailin
MeV. If this result holds generally for other resonances
[6,20]. ItsHamiltonianhasthesameformasEq.(1),but
atother binding energies,our estimates of the resonance
used with Vσ =0 and a different confinement potential
masses can be expected to be good to a few MeV.
The useofcoupledchannelscontainingstable ∆s does
VCON(r )=−a λ ·λ [f (r )+V ], (3)
ij ij c i j ij ij 0 mean that the calculated NN phase shifts do not de-
scribe inelasticities correctly. Thus they cannot be com-
where f (r ) = r2 if quarks i,j, each in the Gaussian
ij ij ij pared quantitatively to experimental phase parameters
single-quarkwavefunctionφ ofEq.(5),areonthesame
α above the pion-production threshold in NN channels
side of the dibaryon, i.e., both centered at S/2 or at
with strong inelasticities. For this reason, the primary
−S/2. Ifquarksi,jareonoppositesidesofthedibaryon,
emphasis of this paper is the extraction of resonance en-
ergies from phase shifts in the resonance region.
fij(rij)= 1 1−e−µri2j . (4) Our theoretical dibaryons are made up of two stable
µ
constituentsbelowtheirbreakupthresholdandarethere-
(cid:16) (cid:17)
forerealresonancesinthemodel. Theyhavefinitewidths
Quark delocalizationin QDCSM is realizedby assum-
that come from the coupling to open NN channels.
ing that the single particle orbital wave function of QD-
CSMasalinearcombinationofleftandrightGaussians,
the single particle orbital wave functions of the ordinary
A. I=0 states
quark cluster model,
ψ (S,ǫ) = (φ (S)+ǫφ (−S))/N(ǫ), Calculational details and results for the IJP = 03+
α α α
ψ (−S,ǫ) = (φ (−S)+ǫφ (S))/N(ǫ), states are given in Table II. The number of chan-
β β β
nels used in the theory is given by N . The theo-
N(ǫ) = 1+ǫ2+2ǫe−S2/4b2; retical pure 7S∆∆ binding energy is nexcthestimated by
3
φα(S) = pπ1b2 3/4e−2b12(rα−S/2)2 daigaegnoenraaltizoirn-gcotohredinHaatmeirletporneisaenntmataitornixwfhoerreththisesatvaetreagine
(cid:18) (cid:19) baryon-baryonseparationistakentobelessthan6fm(in
φβ(−S) = π1b2 3/4e−2b12(rβ+S/2)2. (5) oInrdtehristowkaeye,ptthheepmuaretri7xSd∆i∆meinssifoonunmdatnoagbeeabblyousnmdalbl)y.
(cid:18) (cid:19) 3
5
TABLEII:The∆∆orresonancemassM anddecaywidthΓ, TABLEIII:The∆∆orresonancemassM anddecaywidthΓ,
in MeV, in five quark models for the IJP =03+ states. The in MeV, in five quark models for the IJP =01+ states. The
channels included are one channel or 1c (7S∆∆ only), two channels included are 1c (3S∆∆ only), 2cc (1c +3SNN), and
3 1 1
coupled channels or 2cc (1c +3DNN), 4cc (2cc +7,3D∆∆), 5cc (2cc +3DNN + 3,7D∆∆). The pure 3S∆∆ bound-state
3 3 1 1 1
and 10 coupled channels as described in the text. The pure mass for ChQM1 is 2274 MeV [2].
7S3∆∆ bound state mass for ChQM1 is 2456 MeV [2]. Nch ChQM2 ChQM2a QDCSM0 QDCSM1 QDCSM3
Nch ChQM2 ChQM2a QDCSM0 QDCSM1 QDCSM3 M M M M M Γ
M Γ M Γ M Γ M Γ M Γ 1c 2366 2344 2317 2206 2115 -
1c 2425 - 2430 - 2413 - 2365 - 2276 - 2cc nra nr nr nr 2408 74
2cc 2428 17 2433 10 2416 20 2368 20 2278 19 5cc nr nr nr nr 2393 70
4cc 2413 14 2424 9 2400 14 2357 14 2273 17 aNoresonanceinthesecoupledchannels.
10cc 2393 14 - - - - - -
′
10cc 2353 17 - - - - - -
′′
10cc 2351 21 - - - - - - Thecalculated3DNN phaseshifts areshowninFig.1.
3
Theydiffernoticeablybetweentwoquarkmodelsofvery
similar parameters that differ only in the replacement of
35−190MeV,withtheChQM2mass60MeVlowerthan the scalarexchangeinChQM2bythe QDCSmechanism
theChQM1massobtainedby[2]. Couplingtothe3DNN inQDCSM1. TheChQM2phaseshiftsagreebetterwith
3
channelcausesthis bound state to changeinto anelastic experiment than the QDCSM1 phase shifts for 100 <
resonance where the phase shift, shown in Fig. 1, rises E < 400MeV. The situation is different in the 3SNN
cm 1
through π/2 at a resonance mass that has been shifted state shown in Fig. 2 where the phase shifts from these
up by 3 MeV. The same small mass increase caused by two quark models agree, as already pointed out by [7].
thecouplingtotheNN continuumisseeninallJP states Dibaryon resonance masses can be quite sensitive to
studied here. The result shows that the mass shift is al- short-range dynamics. For example, the ChQM2 reso-
ways dominated by the NN scattering states below the nance mass can be forced down to the lower QDCSM1
pure bound state mass rather than those above it. valuebyartificiallyincreasingtheattractiveconfinement
The table next shows that on coupling to the two interactioninvolvingh.c. configurationsthatappearonly
7,3D∆∆ channels above the pure 7S∆∆ bound state, the for overlapping baryons. The change can be made in
3 3
resonance is pushed down in mass, as expected. The ef- two different ways: Either (1) increase the color confine-
fect is not large, however,being 15 MeV in ChQM2 and ment interaction strength among h.c. channels, and also
11 MeV in QDCSM1, which has the same αs. that between color-singlet channels and h.c. channels
Both the pure 7S∆∆ bound state and the associated (a model denoted 10cc’) by an overall factor of 1.3; or
3
3DNN resonancearelowerinmassinQDCSM1,byabout (2) increase the color confinement interaction between
3
60 MeV, than in ChQM2. Since the QDCSMs contain color-singlet channels and h.c. channels only (model
h.c. effects, we must first determine how much h.c chan- 10cc”) by an overall factor 1.55. Fig. 1(b) shows that
nels contribute to this mass difference. This study is the resulting 3DNN phase shifts change noticeably only
3
done with ten channels (or case 10cc). Besides the four for E > 350MeV. In other words, only these “high-
cm
baryon channels shown in Table II, they include the fol- energy” NN phase shifts are sensitive to changes in
lowing six h.c. channels: four 3D channels of 2∆ 2∆ , short-range dynamics when shielded by the strong cen-
3 8 8
4N 4N , 4N 2N , 2N 2N , the 7S (4N 4N ) and the trifugalbarrierintheNN D-waves. Hencethesechanges
8 8 8 8 8 8 3 8 8
7D (4N 4N ). Here the baryon symbol is used only to intheconfinementinteractions,artificialasthey are,are
3 8 8
denotetheisospinsothat2∆ meanstheT,S =3/2,1/2 not excluded by the experimental phase shifts.
8
color-octet state. The table shows that these six h.c. The resonance widths given in Table II are FWHMs.
channels lower the ChQM2 resonance mass by 20 MeV. They are quite small, and agree with one another.
Assuming that the QDCSMs accountadequatelyfor h.c. Results for the IJP = 01+ state are shown in Ta-
contributions, we see that the QDCSM1 dibaryon reso- ble III and Fig. 2. The theoretical pure 3S∆∆ state is
1
nancemassisnowlowerthanthatinChQM2byonly36 bound by 100−350 MeV, around twice the 7S∆∆ bind-
3
MeV. ing energy. The coupling to the 3SNN channel has an
1
Itisinterestingthatthepure7S∆∆boundstatemasses unexpectedly large effect, pushing up the lowestof these
3
in the ChQM2 and ChQM2a of different nucleon size b bound3S∆∆ masses,by293MeVinQDCSM3,sothatit
1
differ by only 5 MeV. In contrast, both bound state and becomes a resonance at 2408 MeV. This very large mass
resonancemasseschangeby90MeVinthetwoQDCSMs shift is caused by the presence of a lower-mass state,
withthesamedifferenceinthenucleonsizeb. Thisshows the deuteron, in the admixed 3SNN channel. Admix-
1
that the QDCS mechanism depends sensitively on the ingthree additionalchannelswithno lowerbound states
nucleon size. The possibility of a 7S∆∆ bound state giv- pushes the resonance mass down a little to give for the
3
ing rise to a subthreshold resonance has been suggested 5cc treatment the mass shift
previously [22], but the resonancemasses for the present
quark models are higher. ∆M ≡M (5cc)−M(1c)=278MeV. (6)
R
6

200 200
sc. in ChQM2
(a) 2cc. in ChQM2 (a) 2cc. in ChQM2
5cc. in ChQM2
sc. in QDCSM1 100 2cc. in QDCSM1
deg.) 100 3D 2ScPc0. 7in QDCSM1 g.) 5ScPc0. 7in QDCSM1
s ( 3 de 0
Phase shift 2000 (b) ChQM2 411c00ccc .cc.’_1.3 Phase shifts ( -110000 3S1 (b) ChQM2 511c44ccc.cc.’_1.3
14cc’’_1.55
10cc’’_1.55
100
0
3
3
D
3 S
1
0
-100
0 100 200 300 400 500 600 0 100 200 300 400 500 600

Ec.m .(MeV) Ec.m .(MeV)
FIG.1: (a)3D3NN phaseshiftscalculatedforasinglechannel FIG.2: (a)3S1NN phaseshiftscalculatedwithtwoorfivecou-
(sc) and two coupled channels (2cc) (3DNN + 7S∆∆) in two pledcolor-singletchannelsdefinedinTableIIIintwodifferent
3 3
different quark models (ChQM2 and QDCSM1) as functions quark models ChQM2 and QDCSM1. (b) 3S1NN phase shifts
of the c.m. kinetic energy Ecm = W −2MN, where W is calculated in the ChQM2 using different numbers of coupled
the total c.m. energy and MN is the nucleon mass. The channels, as described in more detail in thetext.
experimental phase shifts of the partial-wave solution SP07
[16] are also shown. (b) NN 3D3 phase shifts calculated in
the ChQM2 using different numbers of coupled channels, as mass for ChQM2s as b increases from 0.52 fm to 0.60 fm
described in more detail in thetext. is only 20% that of QDCSMs. Furthermore, b can be
increasedto only0.645fm, for the confinementpotential
strengtha turnsnegativeabovethatvalueandhenceno
c
Thepure3S1∆∆ boundstatemassappears100MeVor ChQM2s can be constructed. At b = 0.645fm, the pure
more higher in the other four quark models. The addi- 3S∆∆ bound state mass is 2332 MeV, which is too large
1
tional large mass increase caused by the coupling to the for the system to resonate on coupling to NN channels.
3S1NN channel without or with the additional channels Hence ChQM2s have no 3S1NN resonance.
then pushes the state above the ∆∆ threshold. Then no The pure 3S∆∆ bound state mass for ChQM1 is 2274
1
resonance appears.
MeV, 92 MeV below the ChQM2 mass. If its b depen-
The pure 3S1∆∆ bound state mass inChQM1 is higher dence is the same as ChQM2s, the decrease in bound
by 160 MeV than the QDCSM3 mass. So the 3S1NN state mass is insufficient, by about 50 MeV, to induce a
resonance also does not appear in ChQM1. We shall resonance.
findthatQDCSM3modelhasanunusuallyrichdibaryon Turning now to the NN phase shifts, Fig. 2 shows
spectrum arising from an unusually strong attraction in that ChQM2 and QDCSM1 give quite similar results,
∆∆ channels. with the 5cc treatment giving much more attraction in
ItisclearfromTableIIIthatthestrong∆∆attraction bothquarkmodels,especiallyforE <150MeV.There
cm
in QDCSM3 is caused by the large nucleon size b used is fair agreement with experiment for E < 150MeV,
cm
there. As b decreases, the ∆∆ attraction also decreases. but all theoretical phase shifts become increasingly too
This sensitivity to b is not seen in ChQMs, thus showing attractive at higher energies.
that it is caused by the QDCS mechanism. Fig. 2(b) shows the 14-channel 3SNN phase shifts
1
Infact,the3SNN resonancedisappearssomewherebe- calculated in ChQM2 with the addition of nine h.c.
1
tween QDCSM1 and QDCSM3. The critical value bcrit channels: eight 3(S,D) channels of 2∆ 2∆ , 4N 4N ,
1 8 8 8 8
below which the resonance disappears can be estimated 4N 2N , 2N 2N , and 7D 4N 4N . Their inclusion
8 8 8 8 1 8 8
undertheassumptionthatthemassshift∆M causedby causes the phase shift to become only a little more at-
the coupling to the NN channels is the same for all pa- tractive.
rameter sets. The critical point then appears when the The figure also shows the phase shifts obtained af-
bare3S∆∆ bound-state mass is 2186MeV.Interpolation ter the two arbitrary increases of the color confinement
1
from the bound-state masses shown in Table III gives strength involving h.c. channels made previously for the
bcrit ≈0.53fm. 3DNN system. The phase shifts are now noticeably dif-
3
Incontrast,thedecreaseofthepure3S∆∆ boundstate ferent from each other. Both are considerably more at-
1
7
TABLEIV:Thedibaryonorresonancemassanddecaywidth,
all in MeV, in four quark models for I =1 states. The chan-
nels included in the JP = 0+ state are 1c (1S∆∆ only), 2cc 100
0
(1c +1SNN), and 4cc (2cc +5D∆∆+ 5DN∆). The channels
0 0 0
included in the JP = 2+ state are 1c (5S2N∆ only), and 2cc g.) 24cccc.. iinn CChhQQMM22
(1c +1D2NN). The bound-state masses for ChQM1 are 2304 de 50 2cc. in QDCSM1
MeV for 1S0∆∆ and 2171 MeV for 5S2N∆ [2]. s ( 4cc. in QDCSM1
Nch ChQM2 ChQM2a QDCSM0 QDCSM1 QDCSM3 hift SP07
M M M M M Γ e s 0
JP = 0+ : as
h
1c 2395 2390 2335 2231 2148 - P
2cc nra nr nr nr 2448 106 -50
1
4cc nr nr nr nr 2433 128 S
0
JP = 2+ :
1c ubb ub ub ub 2167 - -100
0 100 200 300 400 500 600
2cc nr nr nr nr 2168 4
Ec.m .(MeV)
aNoresonanceinthesecoupledchannels.
bUnbound.
FIG. 3: NN 1S0 phase shifts calculated with two or four
coupled color-singlet channels in two different quark models
tractive than those for case 14cc, but still not attractive ChQM2 and QDCSM1.
enough at high energies for a resonance to appear below
the∆∆threshold. Theselargechangesinthelow-energy
phaseshiftsareinconsistentwithexperiment,thusshow- nucleonsizebelowwhichtheresonancedisappearsinthe
ingthattheseadditionalh.c. effectscannowbeexcluded. QDCSM is found to be bcrit ≈0.56fm.
Two other NN partialwavesmerit a shortdiscussion. Figure 3 shows the non-resonant behavior of the
The5cc3D1NN phaseshifts,liketheir3S1NN partners,are coupled-channel1S0NN phaseshifts forChQM2 andQD-
nonresonant except for QDCSM3. All theoretical phase CSM1. These phase shifts become much more attractive
shifts agree well with experiment, to around 200 MeV. than the experimental values from SP07 as the scatter-
The quality of the theoretical 3S1NN and 3D1NN phase ing energy increases into the resonance region,the effect
shifts shows that both quark models give good descrip- being more than twice as strong as the similar behavior
tions of the longer range part of the effective isoscalar inthe 3SNN state. Allthe quarkmodelsstudiedheredo
1
central potential. The isoscalar 5S2∆∆ is Pauli forbid- not give enough short-range repulsion in these S-states.
den, while the pure3D2∆∆ state is unbound. As a result, In the JP = 2+ state, the pure 5SN∆ bound state
the 2cc 3DNN phase shifts are nonresonant. The QD- 2
2 appears at roughly the same mass straddling the N∆
CSM values agree quite well with experiment, while the
threshold in all six quark models used. (Model differ-
ChQMs agree less well.
ences in the N∆ S-state masses are much smaller for
the high intrinsic spin state than for the low intrinsic
spin states, just like the model differences in the ∆∆ S-
B. I=1 states states.) The pure 5SN∆ mass is pushed up only a little
2
by coupling to the 1DNN continuum. It remains bound
2
Table IV summarizes the results for two isovector only in QDCSM3. The pure 5S2N∆ binding energy for
states with possible resonances. The JP = 0+ (1SNN) ChQM1 is only 0.14 MeV. So the state is unlikely to re-
0
state is qualitatively similar to the isoscalar JP = 1+ main below the N∆ threshold after coupling to the NN
(3SNN) state since they are mostly different spin states channel.
1
ofthesamedibaryonpairsinthesamerelativeorbitalan- The pure N∆ state is unbound in ChQM2. (It be-
gularmomenta. Thepure1S∆∆boundstateinQDCSM3 comes bound if the attractive cental part of the scalar
0
is pushed up from its unperturbed energy of 2148 MeV potentialVσ showninEq.(2)isincreasedinstrengthby
by 300 MeV on coupling to the 1SNN channel by the amultiplicativefactor1.7.) The calculatednon-resonant
0
presence of a lower-mass state, the well-known slightly NN phase shifts are shown in Fig. 4 for ChQM2 and
unbound 1SNN state. The perturbed mass is still small QDCSM1. The prominentcusp is a thresholdor Wigner
0
enoughfor the system to resonatebelow the ∆∆ thresh- cusp with its maximum located right at the N∆ thresh-
old. old. The phase shift above the threshold is the phase of
Intheremainingfivequarkmodels,thepure∆∆mass S11, where the subscript1 denotes the NN channel. For
is80−250MeVhigher. Ineachcase,thestrongcoupling comparison, the resonant phase shifts for QDCSM3 are
to the NN channel pushes the state well into the ∆∆ also given.
continuum, thus preventingaresonancefrommaterializ- Even though a resonance appears only in one quark
ing. Following the procedure used for 3S∆∆, the critical model in our limited theoretical treatment, the masses
1
8
180
2cc. in ChQM2
) 160
g. 2cc. in QDCSM1 40
s (de 112400 2ScPc0. 7in QDCSM3 g.) CChhQQMM22 QQDDCCSSMM11 SSPP0077 33PPCLS
hift 100 (de 20 ChQM2 QDCSM1 SP07 3PT
Phase s 6800 se shifts 0
40 1 a
h
20 D2 P
0 -20
-20
0 50 100 15E0 (M2e00V) 250 300 350
c.m.
-40
0 50 100 150 200
FIG. 4: NN 1D2 phase shifts calculated with two coupled Ec m (MeV)
color-singletchannelsinthreedifferentquarkmodelsChQM2,
QDCSM1 and QDCSM3.
FIG. 5: The central, spin-orbit and tensor components of
the 3PNN phase shifts calculated with three coupled color-
J
involved are sufficiently close to one another and to the singletchannels(containingNN,N∆and∆∆)intwodiffer-
N∆thresholdsothattheydescribesimilardynamicalsit- ent quark models ChQM2 and QDCSM1.
uations to within the uncertainties of the models. More-
over, the large ∆ width when included would cause the
state to straddle the N∆ threshold for all these quark takesgoodcareofthetensorcomponent,butbothquark
models. We therefore consider a 5SN∆ resonance near models give too weak spin-orbit components and too
2
the N∆ threshold to be possible in all these quark mod- repulsive central components, especially for QDCSM1
els. [2, 6].
Infact,inelasticArgandlooping(whichweshalldefine As is well known [2], the problem with the spin-orbit
inSect.IV)hasbeenobtainedbyEntem,Fernandezand component comes about because the included OPE po-
Valcarce [23] in the 1D and possibly also 3F systems tential though clearly needed to generate a NN tensor
2 3
foraChQMhavingα =0.4977,onlyalittle largerthan force also contributes to the ∆−N mass difference. The
s
the value 0.485 used in ChQM1 or ChQM2. The crucial color coupling constant αs needed to account for this
featureintheirtreatmentisthe explicitinclusionofNN mass difference is then reduced to 0.3-1.0from the value
inelasticities by giving decay widths to the ∆s appear- 1.7inquarkmodels withoutpionexchange. This weaker
ing in the coupled-channel treatment. There exist quite αs gives in turn a weaker baryon-baryon spin-orbit po-
extensivePWsolutionsofbothNN [16]andπd[24]scat- tential from OGE. The additional spin-orbit contribu-
tering amplitudes in these and other isovector dibaryon tion from the scalar exchange used in the ChQMs is not
systems. They could yield interesting information con- enough to compensate for the deficit. Resolution of this
cerning quark dynamics in this resonance region. problem would require significant modifications to the
WedonotfindanyresonanceattributabletoanN∆or quark models used.
∆∆boundstateinanyofthequarkmodelsinthefollow- A simple way to study the spin-orbit problem in the
ing four isovector states: (a) the 3P ,3F states (with present limited objective of looking for dibaryon reso-
0,1 3
each state calculated using three coupled color-singlet nanceswithoutoverhaulingthesequarkmodelsistojust
channels of the same quantum numbers for NN, N∆ modify a term or add terms to the quark-quark inter-
and∆∆constituents,respectively),and(b)theJP =2− action phenomenologically. A number of related quark
state (using the four color-singlet channels 3P of NN, models are thus generated: a modified ChQM2m model,
2
N∆, ∆∆, and 7P2∆∆). where the one gluon exchange strength αs of the spin-
It is worthwhile to show in Fig. 5 the difference be- orbit potential VG,LS has been increased 5 times, and
ij
tween ChQM2 and QDCSM1 in the well-known decom- a number of ChQM2ρ(f) models with the additional ρ
position into central, spin-orbit and tensor components meson exchange potential Vρ displayed in Eq. (1). Here
of the 3PNN phase shifts:
J α
chv
f = (8)
3P = 13P + 13P + 53P , α
C 9 0 3 1 9 2 chv0
3PLS = −613P0− 143P1+ 1523P2, is the multiplicative increase of the ρ-quark-quark cou-
3P = − 5 3P + 5 3P − 5 3P . (7) pling strength above the usual and customary value
T 36 0 24 1 72 2 α = 0.021, corresponding to the coupling constant
chv0
We see that the inclusion of one pion exchange (OPE) g = 2.351 used in [17]. Since the effects on these
chv0
9
forces.

40 3 Turning now to our experimental knowledge of pos-
) ChQM2 (11) SP07 3PC sible resonances in NN P-states, we recall that in the
eg. ChQM2 (11) SP07 3PLS PWanalysisFA91[15],resonancepolesarefoundforthe
s (d 20 ChQM2 (11) SP07 PT isovectorodd-parity NN states 3P2,3F2 and 3F3. These
hift empirical resonance-like solutions reproduce the empiri-
s calArgandloopingsofthePWsolutions,butmanystud-
se 0 ies in the past [27] have left unresolved the question of
a
h whether these Argand loopings represent new dibaryon
P
resonances. Thedifficultycentersaroundtheobservation
-20
ofBrayshaw[28]thatwhendecaychannelswiththreeor
more final particles are present, Argand looping can ap-
pearinmodelsknowntohavenoresonancebecausetheir
-40
0 50 100 150 200 S-matrixhasnopole. Brayshawhasgivenanexplicitdy-
Ec m (MeV) namicalexampleforthecaseofthepp1D2 statecoupled
toN∆andπ+dchannels. Thestrongenergydependence
that causes the Argandlooping in his model comes from
FIG.6: Thecentral,spin-orbitandtensorcomponentsofthe
a logarithmic rescattering singularity in the N∆↔π+d
3PNN phase shifts calculated with only a single uncoupled
J transition amplitude near the N∆ threshold. The phys-
NN channelin thequark model ChQM2ρ(11).
ical situation this singularity describes is the oscillation
or exchange of a nucleon between the decaying ∆ and
the second or spectator nucleon with which it forms the
P-wave NN phase shifts of the coupling to N∆ and
bounddeuterond. InBrayshaw’smodel,thereisnonew
∆∆ channels are quite small, we use only a single color- 1Dpp resonance near the N∆ threshold.
singletNN channelto calculate the NN phaseshifts for 2
these modified models. Fig. 6 shows that the resulting
central, spin-orbit and tensor components of the 3PNN To complete our very brief review of resonance condi-
J
phase shifts for the quarkmodelChQM2ρ(11)givequite tions, we shouldmention that recent studies of πN reso-
good agreement with the experimental SP07 values. By nances [29, 30] have shown that the speed test can track
diagonalizingthe Hamiltonian matrix in the appropriate the positionsofresonancepoles,if present,morereliably
N∆ channel, we find that none of the modified quark thanthetimedelaycriterion. (Thespeedtestdetermines
models described in this paragraph has a pure N∆ P- theresonanceenergyfromthemaximumspeedofArgand
wave bound state. loopingasafunctionoftheon-shellkineticenergy,while
The difficulty of forming P-wave bound states can the time delay test locates it by maximizing the positive
readily be appreciated in the attractive square-well po- time delay of the scattered wave packet relative to the
tentialmodel. ItiswellknownthattobindaP-state,its free wave packet.) A different kind of complication can
attractivepotentialdepthmustbefourtimesthatneeded appearindynamicalmodelsthatalreadyhaveresonance
to bind an S-state [25]. To see how far we are from N∆ poles, namely that the speed test can fail because the
P-wave bound states for our quark models, we increase Argand looping does not have a solution with maximum
the strengthofthe attractivecentralscalarpotential Vσ speed [29]. This is an extension to another dynamical
inChQM2byamultiplicativefactorf . The3PN∆ state model (one having two coupled nonrelativistic two-body
s 2
then becomes bound at fs >∼ 3.1, with a binding energy channels) of another old observation of Brayshaw that
of6 (22)MeVwhen f =3.2(3.4). The potentials inthe when relativistic many-body channels are present, true
s
other 3PN∆ states are less attractive. resonance poles can appear without any Argand looping
J
A similar situation holds for ChQM2ρ(11): Its 3PN∆ [31].
2
(5PN∆) state is unbound. It becomes bound only when
2
its attractive central scalar potential strength has been In view of all these complications, we shall take the
increased by a multiplicative factor fs ≈ 3.0 (2.6). It tentative but conservative position that a promising
is thus clear that N∆ P-wave bound states would ap- dibaryonresonanceisoneinvolvingatleastone∆orN∗
pear only for quark models with substantially stronger baryonthatisaboundstatebelowthedibaryonbreakup
attraction than those studied here. threshold in the absence of a centrifugal potential when
Concerning the missing P-wave attraction in both these baryons are treated as stable particles. In prac-
ChQM2 andQDCSM1,one cannotjust use the isoscalar tice,theonlyexcitedbaryonweareabletodescribewith
scalarδmesonofmass980MeVthatappearsintheBonn somedegreeofconfidenceisthe∆. Inthelimitedcontext
potentials [26] because of the cutoff mass Λ ≈ 830MeV of our quark models, we consider the 1DNN structure a
2
used in our chiral quark models. It is not a trivial prob- promising dibaryon resonance, at least for some of our
lem to reconcile these two classes of models for nuclear quarkmodels,butnottheNN P-waveArgandloopings.
10
shows rapid counterclockwise motion on the unitarity
TABLEV:Mass,decaywidths(bothinMeV)andbranching
circle. This mathematical behavior describes a physi-
ratiooftheoreticalbaryonresonancesinfivequarkmodelsand
cal picture where the NN system resonates or continues
comparison with partial-wave analyses of experimental data
to “sound”due to its partialtrappinginto the resonance
from SP07 [16] and FA91[15]. MR for ChQM2a is estimated
region, namely the closed channel containing one or two
from the 4cc valueand is shown within parentheses.
NN 1S0 3S1 1D2 3D3 ∆s. Eachoftheseelasticresonanceshasafinitebutvery
3D1 small elastic (or NN) width.
∆∆ 1S0 3S1 7S3 The resonance properties shown in the table must be
5D0 3,7D1 3,7D3 corrected for the width of a decaying ∆, leading to the
N∆ 5D0 5S2 appearance of pionic channels. Inelasticities cause the
reduction |S|<1 and the restriction of the Argand plot
ChQM2:
to the interior of the unitarity circle. We shall avoid
MR 2393
ΓNN 14 Brayshaw’stwo complicationsfrommany-body channels
Γinel 136 by considering only channels where for stable ∆s, the
BNN 0.09 dibaryonsysteminourquarkmodeltreatmentisabound
ChQM2a: or almost bound two-body state. This restriction al-
MR (2404) lows us to take the standard position that for these spe-
ΓNN 9 cial states, rapid counterclockwise Argand looping is an
Γinel 149 acceptable signal of an inelastic resonance [27, 32]. In
BNN 0.06 physicalterms,leakageofthetrappedsystemintopionic
QDCSM0:
channels reduces the effect “heard” in the NN channel
MR 2400
but does not eliminate it altogether. From the perspec-
ΓNN 14
tive of the quark models used here, the I =1 odd-parity
Γinel 144
BNN 0.09 NN Argand structures are not promising candidates for
QDCSM1: dibaryon resonances.
MR 2357 Weshallestimateinelasticitiesonlyatthecrudestlevel
ΓNN 14 of branching ratios, with
Γinel 96
Γ
BNN 0.13 B = NN, (10)
αcrit 0.60 0.57 NN Γ
s
QDCSM3:
where Γ = Γ +Γ depends on the inelastic width
MR 2433 2393 2168 2273 Γ caused NbyN decainyeilng ∆s. Close to the breakup
ΓNN 130 70 4 17 inel
threshold where the ∆s are almost on-shell, the in-
Γinel 190 136 117 33
elastic width can be related approximately to the ∆
BNN 0.41 0.34 0.03 0.34
SP07: width Γf∆ = 120MeV in free space by only accounting
MR none none? 2148a ? for the reduction in phase space available to a decay-
Γ 118a ing bound ∆ whose mass has been reduced to roughly
BNN 0.29 0.26b Mb∆ ≈ Mf∆ −0.5B, where B is the binding energy of
aPolepositionofFA91. the dibaryon. Then [33]
bAtW =2400MeV.
k2ℓρ(M )
Γ (M ) ≈ Γ b b∆ , (11)
b∆ b∆ f∆k2ℓρ(M )
f f∆
IV. DISCUSSION
where k is the pion momentum in the rest frame of the
decaying ∆, ℓ=1 is the pion angular momentum, and
The theoretical resonance properties calculated in the
last section are summarized in Table V. The first line kE E
π N
foreachdibaryontype givesthe dominantPW.The PW ρ(M)=π (12)
M
responsible for the resonance trapping in the theory is
shown in bold type. The experimental information used isthe two-bodydecayphasespaceatmassM wheneach
for the comparison is the PW solution SP07 [16] of NN decayproducthasc.m. energyEi, i=π,N.Weshalluse
scattering data. The four states are arrangedin order of this crude estimate indiscriminately even far below the
increasing relative orbital angular momentum ℓ . breakup threshold, but the harm done is not great be-
NN
Each of our calculated resonances is an elastic res- cause most promising resonances are near the threshold.
onance where the scattering phase shift rises sharply Ifeachdecaying∆inthe dibaryonhasaBreit-Wigner
through π/2. The Argand plot of its complex PW am- (BW) distribution of width Γb∆, the total mass of two
plitude decaying ∆s can be shown to have a BW distribution
with width 2Γ . Hence
b∆
S−1
T = (9) Γ ≈ n Γ (M ), (13)
2i inel b∆ b∆ b∆