81. N and ∆ Resonances

1

81. N and ∆ Resonances
Revised August 2019 by V.D. Burkert (Jefferson Lab), V. Crede (Florida State U.), U. Thoma(Bonn U.), L. Tiator (KPH, JGU Mainz) and R.L. Workman (George Washington U.).

81.1 Introduction
The excited states of the nucleon have been studied in a large number of formation and production experiments. Until recently, the Breit-Wigner masses and widths, the pole positions, and theelasticities of the N and ∆ resonances in the Baryon Summary Table came largely from partial-wave
analyses of πN total, elastic, and charge-exchange scattering data. The most comprehensive analyses were carried out by the Karlsruhe-Helsinki (KH80) [1], Carnegie Mellon-Berkeley (CMB80) [2],and George Washington U (GWU) [3] groups. Partial-wave analyses have also been performed onmuch smaller πN reaction data sets to get ηN , KΛ, and KΣ branching fractions (see the Listings
for references). Other branching fractions come from analyses of πN → ππN data.
In recent years, a large amount of data on photoproduction of many final states has been accumulated, and these data are beginning to tell us much about the properties of baryon resonances.
A survey of data on photoproduction can be found in the proceedings of recent conferences [4] andworkshops [5], and in recent reviews [6, 7].

81.2 Naming scheme for baryon resonances
In the past, when nearly all resonance information came from elastic πN scattering, it was common to label resonances with the incoming partial wave L2I,2J , as in ∆(1232)P33 and N (1680)F15 .
However, most recent information has come from γN experiments. Therefore, we have replaced
L2I,2J with the spin-parity J P of the state, as in ∆(1232)3/2+ and N (1680)5/2+ ; this name gives
intrinsic properties of the resonance that are independent of the specific particles and reactions used
to study them. This applies equally to all baryons, including Ξ resonances and charm baryons thatare not produced in formation experiments. We do not, however, attach the mass or spin-parity to
the names of the ground-state (“stable”) baryons N, Λ, Σ, Ξ, Ω, Λc , · · · .

81.3 Using the N and ∆ listings
Tables 81.1 and 81.2 list all the N and ∆ entries in the Baryon Listings and give our evaluationof the overall status and the status channel by channel. Only the established resonances (overall
status 3 or 4 stars) are promoted to the Baryon Summary Table. We long ago omitted from theListings information from old analyses, prior to KH80 and CMB80, which can be found in earlier
editions. A rather complete survey of older results was given in our 1982 edition [8].
As a rule, we award an overall status **** or *** only to those resonances which are derived fromanalyses of data sets that include precision differential cross sections and polarization observables,
and are confirmed by independent analyses. All other signals are given ** or * status. New resultsthat are not accompanied by proper error evaluation are less valuable for evaluating star ratings.
The following criteria are guidelines for future error analysis.
1. Uncertainties in resonance parameters: The publication should have a detailed discussion
on how the uncertainties of parameters were estimated. This requires that the error estimates go
beyond the simple fit error as e.g. given by MINUIT, and the robustness of the results should bedemonstrated.
2. Fit quality: Concrete measures for the fit quality should be provided. The reduced global
2
χ value of the fit, while useful, is insufficient. Other possibilities include quoting variations oflocal χ2 values in kinematic regions where evidence for new resonances, or significantly improved
information on resonance parameters, is claimed.
3. Weight factors in observables: Analyses sometimes use weight factors for certain data sets to
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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81. N and ∆ Resonances

either increase or reduce their impact on the results. This has been particularly important whenpolarization observables are involved, which often are sensitive to resonance amplitudes throughinterferences, but usually have much poorer statistics than differential cross section data. To
evaluate sensitivities, the resulting resonance parameters should be checked against variations of
the specific weight factors.
Table 81.1: The status of the N resonances
and their decays. Sub-threshold decay modes areomitted. Only resonances with an overall statusof ∗∗∗ or ∗∗∗∗ are included in the main Baryon
Summary Table.
Status as seen in
Particle J P
overall N γ N π ∆π N σ N η ΛK ΣK N ρ N ω N η0N
1/2+ ∗∗∗∗N (1440) 1/2+ ∗∗∗∗ ∗∗∗∗ ∗∗∗∗ ∗∗∗∗ ∗∗∗N (1520) 3/2− ∗∗∗∗ ∗∗∗∗ ∗∗∗∗ ∗∗∗∗ ∗∗ ∗∗∗∗N (1535) 1/2− ∗∗∗∗ ∗∗∗∗ ∗∗∗∗ ∗∗∗ ∗∗∗∗∗N (1650) 1/2− ∗∗∗∗ ∗∗∗∗ ∗∗∗∗ ∗∗∗ ∗∗∗∗∗ ∗N (1675) 5/2− ∗∗∗∗ ∗∗∗∗ ∗∗∗∗ ∗∗∗∗ ∗∗∗ ∗∗∗N (1680) 5/2+ ∗∗∗∗ ∗∗∗∗ ∗∗∗∗ ∗∗∗∗ ∗∗∗ ∗∗∗N (1700) 3/2− ∗∗∗∗∗∗∗∗ ∗∗∗ ∗∗∗N (1710) 1/2+ ∗∗∗∗ ∗∗∗∗ ∗∗∗∗ ∗∗∗∗ ∗∗∗∗∗N (1720) 3/2+ ∗∗∗∗ ∗∗∗∗ ∗∗∗∗ ∗∗∗ ∗∗∗∗∗∗ ∗∗∗N (1860) 5/2+ ∗∗∗∗∗∗∗N (1875) 3/2− ∗∗∗
∗∗
∗∗
∗
∗∗ ∗
∗
∗
∗
∗
N (1880) 1/2+ ∗∗∗
∗∗
∗
∗∗
∗
∗
∗∗
∗∗
∗∗
N (1895) 1/2− ∗∗∗∗ ∗∗∗∗ ∗∗∗∗∗∗∗ ∗∗∗∗ ∗∗∗∗∗∗N (1900) 3/2+ ∗∗∗∗ ∗∗∗∗ ∗∗
∗∗
∗
∗
∗∗
∗∗
∗
∗∗
N (1990) 7/2+ ∗∗
∗∗
∗∗
∗
∗
∗
N (2000) 5/2+ ∗∗
∗∗
∗
∗∗
∗
∗
∗
N (2040) 3/2+ ∗∗N (2060) 5/2− ∗∗∗∗∗∗ ∗∗
∗
∗
∗
∗
∗
∗
∗
N (2100) 1/2+ ∗∗∗∗∗∗∗∗ ∗∗
∗∗ ∗
∗
∗
∗
∗∗
N (2120) 3/2− ∗∗∗∗∗∗ ∗∗
∗∗
∗∗
∗∗
∗
∗
∗
N (2190) 7/2− ∗∗∗∗ ∗∗∗∗ ∗∗∗∗ ∗∗∗∗ ∗∗ ∗∗∗∗∗∗N (2220) 9/2+ ∗∗∗∗ ∗∗
∗∗∗∗
∗
∗
∗
N (2250) 9/2− ∗∗∗∗ ∗∗
∗∗∗∗
∗
∗
∗
N (2300) 1/2+ ∗∗∗∗N (2570) 5/2− ∗∗∗∗N (2600) 11/2− ∗∗∗∗∗∗N (2700) 13/2+ ∗∗∗∗∗ ∗ ∗∗
Existence is certain.
∗∗∗
Existence is very likely.
∗∗
Evidence of existence is fair.
∗
Evidence of existence is poor.

Claims of evidence for new baryon states must be based on a sufficiently complete set of partial
waves in the fit. The robustness of signals must be demonstrated, e.g. by examining the effect ofhigher partial waves in the fit.

31st May, 2024

3

81. N and ∆ Resonances
Table 81.2: The status of the ∆ resonances
and their decays. Sub-threshold decay modes areomitted. Only resonances with an overall statusof ∗∗∗ or ∗∗∗∗ are included in the main Baryon
Summary Table.
Status as seen in
Particle J P
overall N γ N π ∆π ΣK N ρ ∆η+∆(1232) 3/2∗∗∗∗ ∗∗∗∗ ∗∗∗∗∆(1600) 3/2+ ∗∗∗∗ ∗∗∗∗ ∗∗∗ ∗∗∗∗∆(1620) 1/2− ∗∗∗∗ ∗∗∗∗ ∗∗∗∗ ∗∗∗∗∆(1700) 3/2− ∗∗∗∗ ∗∗∗∗ ∗∗∗∗ ∗∗∗∗ ∗∗+∆(1750) 1/2∗∗∗∗∆(1900) 1/2− ∗∗∗∗∗∗ ∗∗∗ ∗∗∗ ∗∆(1905) 5/2+ ∗∗∗∗ ∗∗∗∗ ∗∗∗∗ ∗∗∗∗∗∗∆(1910) 1/2+ ∗∗∗∗ ∗∗∗ ∗∗∗∗ ∗∗∗∗∗+∆(1920) 3/2∗∗∗∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗∗∆(1930) 5/2− ∗∗∗
∗
∗∗∗ ∗
∗
∆(1940) 3/2− ∗∗∗∗∗∗∗∆(1950) 7/2+ ∗∗∗∗ ∗∗∗∗ ∗∗∗∗ ∗∗∗∗∗∆(2000) 5/2+ ∗∗∗∗∗∗∗∆(2150) 1/2− ∗∗∆(2200) 7/2− ∗∗∗∗∗∗ ∗∗∗∗∗ ∗∗∆(2300) 9/2+ ∗∗∗∗∆(2350) 5/2− ∗∗∆(2390) 7/2+ ∗∗−∆(2400) 9/2∗∗∗∗∗∗∆(2420) 11/2+ ∗∗∗∗ ∗∗∗∗∗∆(2750) 13/2− ∗∗∗∗+∆(2950) 15/2 ∗∗∗∗∗ ∗ ∗∗ Existence is certain.
∗∗∗
Existence is very likely.
∗∗
Evidence of existence is fair.
∗
Evidence of existence is poor.

81.4 Properties of resonances
Resonances are defined by poles of the S-matrix, whether in scattering, production or decay
matrix elements. These are poles in the complex plane in s, as discussed in the new review on√
Resonances. As is traditional, we quote here the pole positions in the complex energy w = s
plane. Crucially, the position of the pole of the S-matrix is independent of the process, and the
production and decay properties factorize. This is the rationale for listing the pole position first foreach resonance. These key properties of the S-matrix pole are in contrast to other quantities related
to resonance phenomena, such as Breit-Wigner parameters or any K-matrix pole. Breit-Wigner
parameters depend on the formalism used, such as angular-momentum barrier factors, or cut-off
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81. N and ∆ Resonances

4parameters, and the assumed or modeled background. However, the accurate determination ofpole parameters from the analysis of data on the real energy axis is not necessarily simple, or evenstraightforward. It requires the implementation of the correct analytic structure of the relevant(often coupled) channels.
In principle, there are two ways to extract pole parameters from experimental data: (i) analytic
continuation of theoretical single- or multi-channel models into the complex energy plane or (ii)local expansions of the partial-wave T-matrix amplitudes in the complex energy plane in the vicinityof a pole.
At present, poles are usually extracted using the first method [9–14], but considerable effort has
been put into the development of alternate approaches, such as the speed plot [15], time delay [16],N/D method [17], regularization procedure [18], or Padè approximation [19].
Methods of the second type are based on the idea to use first or higher-order derivatives inenergy to reduce the importance of, or totally eliminate, the background contribution. One either
has to model the background contribution and introduce model dependence, or one is faced withnumerical derivatives of single-energy data. In both cases, one reaches almost unsurmountabledifficulties.
An alternate way to extract pole parameters from partial waves has been proposed by introducing a Laurent+Pietarinen (L+P) expansion [20–22]
M nX
max
X
αj − xj − W nResip
T (W ) =
+
cjn (
) ,
W
−
W
α
+
xj − Wiji=1
j=1 n=0NXp(81.1)

where T (W ) is a given partial wave amplitude, Wi and Resi are the N complex pole positions andresidues. The background is parameterized with M Pietarinen functions, where αj are positive
range parameters and xj are real or complex branch points; cjn are real expansion coefficients.
The main idea of this procedure is to find the simplest analytic function, with well-definedpoles and cuts, regardless of whether they are generated by a theoretical model or some energyindependent procedure. Instead of searching for the function which reproduces the input amplitudes
over the complete complex energy plane, on all Riemann sheets, a representation is searched only
in a limited complex energy range, near the real axis, which is defined by the radius of convergenceof the Laurent decomposition, and which contains the input amplitudes. All details are found in
Ref. [21]. Applications of the method can be found in [20–27].

81.5 Photoproduction
A new approach to the nucleon excitation spectrum is provided by dedicated facilities at the
Universities of Bonn, Grenoble, and Mainz, and at the national laboratories Jefferson Lab in the US
and SPring-8 in Japan. High-precision cross sections and polarization observables for the photoproduction of pseudoscalar mesons provide a data set that is approaching a “complete experiment,” onethat fully constrains the four complex amplitudes describing the spin-structure of the reaction [28].
A large number of photoproduction reactions has been studied.
In pseudoscalar meson photoproduction, the four independent helicity amplitudes can be expressed in terms of the four CGLN [29] amplitudes allowed by Lorentz and gauge invariance. These
amplitudes can be expanded in a series of electric and magnetic multipoles. Except for J = 1/2,
one electric and one magnetic multipole contributes to each J P combination.
For a given state, these two amplitudes determine the resonance photo-decay helicity amplitudes
A1/2 and A3/2 . As described below, this resonance extraction has been carried out either assuming
a Breit-Wigner resonance or at the pole.
If a Breit-Wigner parametrization is used, the N γ partial width, Γγ , is given in terms of the
31st May, 2024

81. N and ∆ Resonances5

helicity amplitudes A1/2 and A3/2 by
Γγ =

2


2mN
kBW
|A1/2 |2 + |A3/2 |2 .
π (2J + 1)mBW(81.2)

Here mN and mBW are the nucleon and resonance masses, J is the resonance spin, and kBW is thephoton c.m. decay momentum. Most earlier analyses have provided these real quantities A1/2 andA3/2 .
More recent studies have quoted related complex quantities, evaluated at the T-matrix pole.
These complex helicity amplitudes, Ã1/2 and Ã3/2 , can be cast onto the formvuu π(2J + 1)wpole Res(Th (γN → N b))pÃh = t2
mN kpole

Res(T (N b → N b))(81.3)

22
2
where the residues (Res) are evaluated at the pole position, wpole , and kpole= (wpole−m2N )2 /4wpole[30]. For Breit-Wigner amplitudes, wpole = mBW and Ãh = Ah . Similar relations for the photo andelectro couplings at the pole position can be found in [31, 32].
The determination of eight real numbers from four complex amplitudes (with one overall phaseundetermined) requires at least seven independent measurements. At least one further measurement
is required to resolve discrete ambiguities that result from the fact that data are proportional tosquared amplitudes. Photon beams and nucleon targets can be polarized (with linear or circular~ can bepolarization P⊥ , P and T~ , respectively); the recoil polarization of the outgoing baryon R
measured. The experiments can be divided into three classes: (1) the beam and target are polarized(BT); (2) the beam is polarized and the recoil baryon polarization is measured (BR); (3) the target
is polarized and the recoil polarization is measured (TR). Different sign conventions are used in theliterature, as summarized in [33].
One of the best studied reactions is γp → ΛK + . Published data include differential crosssections, the beam asymmetry Σ, the target asymmetry T , the recoil polarization P , and the BR
double-polarization variables Cx0 , Cz 0 , Ox0 , and Oz 0 . For the photoproduction of pions and etas, off
proton and neutron targets, differential cross sections, single- and double-polarization asymmetries
have been measured, mainly for pions.

81.6 Electroproduction
Electroproduction of mesons provides information on the internal structure of resonances. The
helicity amplitudes are functions of the (squared) momentum transfer Q2 = −(e − e0 )2 , wheree and e0 are the 4-momenta of the incident and scattered electron, and a third amplitude, S1/2 ,measures the resonance response to the longitudinal component of the virtual photon. Most data
stem from the reactions e− p → e− nπ + and e− p → e− pπ 0 but also the reactions e− p → e− pη,e− p → e− pπ + π − , and e− p → e− Λ(Σ 0 )K + have been studied. The data and their interpretation
are reviewed in Refs. [34, 35].
The transition to the ∆(1232)3/2+ is often quantified in terms of the magnetic dipole transition
moment M1+ (or the magnetic transition form factor G∗M,Ash (Q2 )) [36], and the electric and scalar
quadrupole transition moments E1+ and S1+ . Figure 81.1 shows the strength of the p → ∆+transition plotted versus the photon virtuality Q2 . At Q2 = 0, M1+ dominates the resonancetransition strength. The two amplitudes E1+ and S1+ imply a quadrupole deformation of thetransition to the lowest excited state. The magnitude of REM = E1+ /M1+ remains nearly constant,while the magnitude of RSM = S1+ /M1+ increases rapidly up to 25% at the highest Q2 value.
Figure 81.2 shows the transverse and scalar helicity amplitudes for the N (1440)1/2+ , N (1520)3/2− ,and N (1535)1/2− resonances from JLab [34]. Similar results have been achieved at Mainz [35]. For
31st May, 2024

81. N and ∆ Resonances

1

REM (%)

G*M,Ash/3GD

6

0.8

RSM (%)

0.6

0.4

2
1
0
-1
-2
-3
-4
-5
-6
-7
0

-5

-10
-15
-20

0.2

-25
-30
-35

10

-1

1

Q2 (GeV2)

10

-1

1

2

2

Q (GeV )Figure 81.1: Left: The magnetic transition form factor for the γ ∗ p → ∆+ (1232) transition versus
the photon virtuality Q2 . Right: The electric and scalar quadrupole ratios REM and RSM . The
different symbols are results from different experiments at JLab (squares, diamonds, circle) and
MAMI (triangle, cross). The boxes near the horizontal axis indicate model uncertainties of thesquares. Curves to guide the eyes.
the states N (1440)1/2+ and N (1520)3/2− , helicity amplitudes and π∆ and ρp decays were determined at JLab in an analysis of π + π − p electroproduction [37]. The data show distinctly different
Q2 dependencies that indicate different internal structures.
The N (1520)3/2− helicity amplitudes reveal the dominance of its three-quark nature: the A3/2
amplitude is large at the photon point and decreases rapidly ∼ Q−5 with increasing Q2 ; A1/2 issmall at the photon point, increases rapidly with Q2 and then falls off with ∼ Q−3 . Quantitative
agreement with the data is, however, achieved only when meson cloud effects are included.
At high Q2 , both amplitudes for N (1440)1/2+ are qualitatively described by light front quark
models [38]: at short distances the resonance behaves as expected from a radial excitation of thenucleon. On the other hand, A1/2 changes sign at about 0.6 GeV2 . This remarkable behavior
has not been observed before for any nucleon form factor or transition amplitude. Obviously, an
important change in the structure occurs when the resonance is probed as a function of Q2 .
The Q2 dependence of A1/2 of the N (1535)1/2− resonance exhibits the expected Q−3 dependence, except for small Q2 values where meson cloud effects set in.
Figure 81.3 shows the transverse and scalar amplitudes for three states in the 3rd nucleon
resonance region, the ∆(1620)1/2− , the N (1675)5/2− and N (1680)5/2+ . The latter two states
have nearly degenerate masses and are parity partners. In the quark model picture, the transverse
amplitudes for N (1675)5/2− on the proton are suppressed due to the Moorhouse selection rule,
allowing for a quantitative evaluation of the meson-baryon contributions. The data show significant
meson-baryon strength in the A1/2 amplitude even at quite high Q2 , while A3/2 drops much fasterwith Q2 . N (1680)5/2+ shows qualitatively the features predicted in constituent quark models, adominant A3/2 at the real photon point that drops rapidly with increasing Q2 , while A1/2 becomes
the dominant contribution at high Q2 , indicating a switch of the helicity structure in the resonancetransition at short distances.

31st May, 2024

81. N and ∆ Resonances
60

-1/2

)

80

S1/2 (10 GeV

60
40

40

-3

-3

A1/2 (10 GeV

-1/2

)

7

20
0
-20
-40

20
0
-20

-60
-40

-80
0

1

2

3

4
2

0

1

2

2

2

)

)

-1/2

-1/2

S1/2(10 GeV

A3/2(10 GeV

-50

-3

-3

)
-1/2

4

Q (GeV )

150

-25

-3

A1/2(10 GeV

03
2

Q (GeV )

100

-75

0
-20
-40

50
-60

-100
0
0

2

-80

4
2

0

2

2

2

2

4
2

2

-1/2

)

Q (GeV )

120
100

20
10

-3

-3

0
2

Q (GeV )S1/2 (10 GeV

-1/2

)

Q (GeV )A1/2 (10 GeV

4

80
60

0
-10
-20

40

-30

20

-40
0
0

2

4

6
2

0

1

2

2

3

4
2

Q (GeV )2

Q (GeV )Figure 81.2: Transverse and scalar (longitudinal) helicity amplitudes for γp → N (1440)1/2+ (top),
γp → N (1520)3/2− (center), and γp → N (1535)1/2− (bottom) as extracted from the JLab/CLAS
data in nπ + production (full circles), MAMI/A1 data in pπ 0 production (full down triangle), in
pπ + π − (open triangles), and combined single and double pion production (open squares). The solid
triangle is the PDG 2014 value at Q2 = 0. The open boxes are the model uncertainties of the fullcircles.

31st May, 2024

81. N and ∆ Resonances

-1/2

)

0

60

S1/2 (10 GeV

50

-3

-3

A1/2 (10 GeV

-1/2

)

8

40
30

-10
-20
-30
-40

20

-50

10

-60

0
0

1

2

0

2

1

2

2

2

2

-1/2

20

10

10

-2.5
-5

0
0

-7.5

-10
0

2

4
2

0

2

2

2

)

150

4
2

Q (GeV )-3

S1/2 (10 GeV

-20

2

-1/2

)
-1/2
-3

)
-1/2

0
2

Q (GeV )A3/2 (10 GeV

-10

-3

A1/2 (10 GeV

42

Q (GeV )
0

0

-3

-3

S1/2 (10 GeV

-1/2

)

)

Q (GeV )A3/2 (10 GeV

20-3

A1/2 (10 GeV

-1/2

)

Q (GeV )

100

0

-10

50

-30

-20
-40

0
-30
0

2

4
2

0

2

2

4
2

Q (GeV )02

Q (GeV )

2

4
2

2

Q (GeV )Figure 81.3: Transverse and scalar helicity amplitudes for γp → ∆(1620)1/2− (top), γp →N (1675)5/2− (center), and γp → N (1680)5/2+ (bottom) as extracted from the JLab/CLAS datain nπ + production (full circles), pπ + π − (open triangles), combined single and double pion production (open square). The solid triangle is the 2014 PDG value at Q2 = 0. The open boxes are themodel uncertainties of the full circles. The curves are to guide the eye.

31st May, 2024

981. N and ∆ Resonances

81.7 Partial wave analyses
Several PWA groups are now actively involved in the analysis of the new data. The GWU group
maintains a nearly complete database covering reactions from πN and KN elastic scattering to
γN → N π, N η, and N η 0 . It is presently the only group determining πN elastic amplitudes fromscattering data in sliced energy bins. Given the high-precision of photoproduction data already orsoon to be collected, the spectrum of N and ∆ resonances will in the near future be better known.
Fits to the data are performed by various groups with the aim to understand the reaction
dynamics and to identify N and ∆ resonances. For practical reasons, approximations have to
be made. We mention several analyses here: (1) The Mainz unitary isobar model [39] focuseson the correct treatment of the low-energy domain. Resonances are added to the unitary amplitude as a sum of Breit-Wigner amplitudes. This model also obtains resonance transition form
factors and helicity amplitudes from electroproduction [35]. (2) For N π electroproduction, the Yerevan/JLab group uses both the unitary isobar model and the dispersion relation approach developedin [38]. A phenomenological model was developed to extract resonance couplings and partial decay
widths from exclusive π + π − p electroproduction [37]. (3) Multichannel analyses using K-matrix
parameterizations derive background terms from a chiral Lagrangian - providing a microscopicaldescription of the background - (Giessen [40, 41]) or from phenomenology (KSU [42, 43], BonnGatchina [44]). (4.) Several groups (EBAC-Jlab [45,46], ANL-Osaka [47], Dubna-Mainz-Taipeh [48],Bonn-Jülich [49–51], Valencia [52]) use dynamical reaction models, driven by chiral Lagrangians,
which take dispersive parts of intermediate states into account. Several other groups have madeimportant contributions. The Giessen group pioneered multichannel analyses of large data setson pion- and photo-induced reactions [40, 41]. The Bonn-Gatchina group included recent highstatistics data and reported systematic searches for new baryon resonances in all relevant partial
waves. A summary of their results can be found in [44].
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Springer Verlag.
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Toronto, ed. N. Isgur.
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June, 13 - 17, 2011, published in eConf.
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31st May, 2024