arXiv:1909.11935v2 [nucl-th] 4 Oct 2019arXiv:1909.11935v2 [nucl-th] 4 Oct 2019 The ANL-Osaka...
Transcript of arXiv:1909.11935v2 [nucl-th] 4 Oct 2019arXiv:1909.11935v2 [nucl-th] 4 Oct 2019 The ANL-Osaka...
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The ANL-Osaka Partial-Wave Amplitudes of πN and γN Reactions
H. Kamano1, T.-S. H. Lee2, S.X. Nakamura3, T. Sato11 Research Center for Nuclear Physics, 10-1 Mihogaoka, Ibaraki, Osaka, Japan2Physics Division, Argonne National Laboratory, Argonne, Illinois 60249, USA
3 University of Science and Technology of China, Hefei, 230026, People’s Republic of China
Contact Person: T.-S. H. Lee (email:[email protected]), September 20, 2019(https://www.phy.anl.gov/theory/research/anl-osaka-pwa)
Abstract
The determination of the Argonne National Laboratory-Osaka University (ANL-Osaka)Partial-Wave Amplitudes (PWA) of πN and γN Reactions is reviewed. The predicted PWAare presented on a web page (https://www.phy.anl.gov/theory/research/anl-osaka-pwa). The formulas are given for using the predicted PWA to calculate the cross sectionsof (1) meson-baryon (MB) scattering MB → M ′B′ with MB,M ′B′ = πN, ηN,KΛ, KΣ,(2) two-pion production πN → π∆, ρN, σN → ππN , (3) Meson photoproduction γN →πN, ηN,KΛ, KΣ, (4) Pion electroproduction N(e, e′π)N , (5) inclusive N(e, e′)X . We alsopresent sample results from our fits to the data.
I. INTRODUCTION
The Argonne National Laboratory-Osaka University (ANL-Osaka) collaboration startedin 1996 with a publication [1] of a meson-exchange model for investigating the excitation ofthe ∆ (1232) resonance in πN and γN reactions. The predictions [1, 2] from this model,called the Sato-Lee (SL) model in the community, were found to be consistent with the datafrom MIT-Bates, Mainz, Bonn, and JLab. The results from the SL model strongly suggestedthat a dynamical model based on a Hamiltonian with bare N∗ states and meson-exchangemechanisms can be used to
1. describe the data of πN and γN reactions up to invariant mass W = 2 GeV,
2. extract the masses and widths of nucleon resonances (N∗) from the predicted Partial-Wave Amplitudes (PWA) for investigating the structure of the nucleon and its excitedstates,
3. apply the constructed Hamiltonian with meson and N∗ degrees of freedom to predictthe cross sections of the production of mesons (π, 2π, η,K) from nuclei, which areneeded for analyzing the data of nuclear reactions induced by hadrons, electrons, andneutrinos in the nucleon resonance region.
The formulation of the SL model was then extended to include the higher mass bare N∗
states and the meson-baryon channels which have significant contributions to the πN and
1
γN total cross sections below invariant massW = 2 GeV. Exploratory calculations using thisDynamical Coupled-Channel (DCC) model were carried out by the ANL-Osaka collaborationduring 2004-2006. The analysis of the world data of πN and γN reactions up toW = 2 GeVwas then carried out at the Excited Baryon Analysis Center (EBAC) at JLab during 2006-2012, with 12 publications [3-15]. The ANL-Osaka collaboration continued the analysis toextract [16, 17] 22 nucleon resonances during 2013-2016. The calculations for the analysis in-volved solving coupled-channel scattering equations with 8 channels: γN, πN, ηN,KΛ, KΣ,and ππN which has π∆, σN, ρN resonant components. The parameters of the model weredetermined by performing χ2-fits to the world data of γN, πN → πN, ηN,KΛ, KΣ (about30,000 data points). The ANL-Osaka model was also extended [18] to investigate the elec-tron and neutrino-induced meson production reactions on the nucleon. The computationresources from DOE’s NERSC (about 500,000 hours/year) and Argonne’s LCRC (about300,000 hours/year) had been used to complete this task.
The results presented on https://www.phy.anl.gov/theory/research/anl-osaka-pwa are for the nuclear and hadron physics communities to get access to the predictedpartial-wave amplitudes of γN, πN → πN, ηN,KΛ, KΣ, π∆, σN, ρN . In section II, we de-scribe briefly the ANL-Osaka DCC model and the resulting formulation of πN and γNreactions. The procedures for extracting nucleon resonances are given in section III. Insection IV, we present formula for using the PWA posted on the web page to calculate thecross sections of the considered meson-baryon reactions . The data included in the fits aresummarized in section V. In section VI and Appendix, we present sample results from ourfits to the data.
II. ANL-OSAKA DCC MODEL
The ANL-Osaka DCC Model is based on an effective Hamiltonian of the followingenergy − independent form:
H = H0 + v22 + ΓV + hππN , (1)
where H0 =∑
α
√
m2α + ~p 2
α with mα and ~pα denoting the mass and momentum of particle
α, respectively. The interactions are defined as
v22 =∑
MB,M ′B′
vMB,M ′B′ + vππ,ππ , (2)
ΓV = ∑
N∗
(∑
MB
ΓN∗→MB) +∑
M∗
hM∗→ππ+ c.c , (3)
hππN = vπN,ππN + vγN,ππN+ c.c , (4)
where vMB,M ′B′ is the meson-baryon (MB) interactions, vππ,ππ is the ππ interactions,ΓN∗→MB describes the decay of a bare excited nucleon (N∗) into a MB state, hM∗→ππ
describes the decay of a bare meson (M∗) into a ππ state, and c.c. denotes the complexconjugate of the terms on its left-side. For describing the πN and γN reactions up to in-variant mass W = 2 GeV, we include MB = γN, πN, ηN,KΛ, KΣ, π∆, ρN, σN , about 20bare N∗ states, and M∗ = ρ, σ.
2
A. Hadronic amplitudes
Starting with the Hamiltonian defined by Eqs. (1)-(4), we apply [4] the projection operatormethod to cast the partial-wave amplitudes of the scattering T matrix of the meson-baryon
reaction, M(~k) +B(−~k) →M ′(~k′) +B′(−~k′), into the following form
TM ′B′,MB(k′, k;E) = tM ′B′,MB(k
′, k;E) + tRM ′B′,MB(k′, k;E), (5)
where E is the total energy, k and k′ are the meson-baryon relative momenta in the centerof mass frame, and MB,M ′B′ = πN, ηN, π∆, ρN, σN,KΛ, KΣ are the reaction channelsincluded in the analysis. The notation MB also represents the partial-wave quantum num-bers: [L(sMsB)S]JT ], where J is the total angular momentum, T the total isospin, L theorbital angular momentum, S the total spin which is from the coupling of the meson spinsM and the baryon spin sB.
The direct reaction amplitude tM ′B′,MB(k′, k;E) in Eq. (5) is defined by a set of coupled-
channel equations
tM ′B′,MB(k′, k;E) = VM ′B′,MB(k
′, k;E)
+∑
M ′′B′′
∫
CM′′B′′
k′′2dk′′VM ′B′,M ′′B′′(k′, k′′;E)
×GM ′′B′′(k′′;E)tM ′′B′′,MB(k′′, k;E). (6)
Here CM ′′B′′ is the integration path, which is taken from 0 to ∞ for the physical E; thesummation
∑
MB runs over the orbital angular momentum and total spin indices for allMB channels allowed in a given partial wave; GM ′′B′′(k;E) are the meson-baryon Greenfunctions. Defining Eα(k) = [m2
α + k2]1/2 with mα being the mass of a particle α, themeson-baryon Green functions in the above equations are:
GMB(k;E) =1
E −EM (k)−EB(k) + iǫ, (7)
for the stable πN , ηN , KΛ, and KΣ channels, and
GMB(k;E) =1
E − EM(k)− EB(k)− ΣMB(k;E), (8)
for the unstable π∆, ρN , and σN channels. The self energy ΣMB(k;E) in Eq. (8) is cal-culated from a vertex function defining the decay of the considered unstable particle inthe presence of a spectator π or N with momentum k. For the π∆ and ρN channels, theself-energies are explicitly given by
Σπ∆(k;E) =m∆
E∆(k)
∫
q2dqMπN (q)
[M2πN(q) + k2]1/2
|f∆→πN(q)|2E − Eπ(k)− [M2
πN (q) + k2]1/2 + iǫ, (9)
ΣρN (k;E) =mρ
Eρ(k)
∫
q2dqMππ(q)
[M2ππ(q) + k2]1/2
|fρ→ππ(q)|2E −EN (k)− [M2
ππ(q) + k2]1/2 + iǫ, (10)
where m∆ = 1280 MeV, mρ = 812 MeV, MπN(q) = Eπ(q) + EN (q), and Mππ(q) = Eπ(q) +Eπ(q). The form factors f∆→πN(q) and fρ→ππ(q) are for describing the ∆ → πN and ρ→ ππ
3
decays in the ∆ and ρ rest frames, respectively. They are parametrized as:
f∆→πN(q) = −i (0.98)
[2(mN +mπ)]1/2
(
q
mπ
)
(
1
1 + [q/(358 MeV)]2
)2
, (11)
fρ→ππ(q) =(0.6684)√
mπ
(
q
(461 MeV)
)(
1
1 + [q/(461 MeV)]2
)2
. (12)
The σ self-energy ΣσN (k;E) is calculated from a ππ s-wave scattering model with a vertexfunction g(q) for the σ → ππ decay and a separable interaction v(q′, q) = h0h(q
′)h(q). Theresulting form is
ΣσN (k;E) = 〈gGππg〉(k;E) + τ(k;E)[〈gGππh〉(k;E)]2, (13)
with
τ(k;E) =h0
1− h0〈hGππh〉(k;E), (14)
〈hGππh〉(k;E) =∫
dqq2Mππ(q)
[M2ππ(q) + k2]1/2
× h(q)2
E − EN(k)− [M2ππ(q) + k2]1/2 + iε
, (15)
〈gGππg〉(k;E) =mσ
Eσ(k)
∫
dqq2Mππ(q)
[M2ππ(q) + k2]1/2
× g(q)2
E − EN(k)− [M2ππ(q) + k2]1/2 + iε
, (16)
〈gGππh〉(k;E) =
√
mσ
Eσ(k)
∫
dqq2Mππ(q)
[M2ππ(q) + k2]1/2
× g(q)h(q)
E − EN(k)− [M2ππ(q) + k2]1/2 + iε
. (17)
In the above equations, mσ = 700.0 MeV and the form factors are
g(p) =g0√mπ
1
1 + (cp)2, (18)
h(p) =1
mπ
1
1 + (dp)2. (19)
where g0 = 1.638, h0 = 0.556, c = 1.02 fm, and d = 0.514 fm.The driving terms of Eq. (6) are
VM ′B′,MB(k′, k;E) = vM ′B′,MB(k
′, k) + Z(E)M ′B′,MB(k
′, k;E) , (20)
where vM ′B′,MB(k′, k) are the meson-exchange potentials derived from the tree-diagrams,
as illustrated in Fig. 1, of phenomenological Lagrangians. Within the unitary transforma-tion method used in the derivation, those potentials are energy independent and free ofsingularities. The Lagrangians used in our derivations and the partial-wave expansions ofvM ′B′,MB(k
′, k) are given in Appendices B and C of Ref. [16].
4
The energy-dependent Z(E)M ′B′,MB(k
′, k;E) terms in Eq. (20), as illustrated in Fig. 2, con-tain the moving singularities due to the ππN cuts. The procedures for evaluating the
partial-wave matrix elements of Z(E)M ′B′,MB(k
′, k;E) are explained in detail in the AppendixE of Ref. [4].
The second term in the right-hand-side of Eq. (5) is the N∗-excitation amplitude definedby
tRM ′B′,MB(k′, k;E) =
∑
N∗
n,N∗
m
ΓM ′B′,N∗
n(k′;E)[D(E)]n,mΓN∗
m,MB(k;E). (21)
Here the dressed N∗ → MB and MB → N∗ decay vertices are, respectively, defined by
ΓMB,N∗(k;E) = ΓMB,N∗(k) +∑
M ′B′
∫
q2dqtMB,M ′B′(k, q;E)GM ′B′(q, E)ΓM ′B′,N∗(q), (22)
ΓN∗,MB(k;E) = Γ†MB,N∗(k) +
∑
M ′B′
∫
q2dqΓ†M ′B′,N∗(q)GM ′B′(q, E)tM ′B′,MB(q, k;E), (23)
with ΓMB,N∗(k) being the bare N∗ →MB decay vertex [note that ΓN∗,MB(k) = Γ†MB,N∗(k)];
the inverse of the dressed N∗ propagator is defined by
[D−1(E)]n,m = (E −M0N∗
n)δn,m − [ΣN∗(E)]n,m, (24)
where M0N∗ is the mass of the bare N∗ and the N∗ self-energies ΣN∗(E) are given by
[ΣN∗(E)]n,m =∑
MB
∫
CMB
k2dkΓN∗
n,MB(k)GMB(k;E)ΓMB,N∗
m(k;E). (25)
It is emphasized that the N∗ propagator D(E) can have off-diagonal terms. The bare vertexfunctions in Eqs. (22)-(23) are parametrized as
ΓMB(LS),N∗(k) =1
(2π)3/21√mN
CMB(LS),N∗
Λ2MB(LS),N∗
Λ2MB(LS),N∗ + k2
(2+L/2) (k
mπ
)L
, (26)
where L and S denote the orbital angular momentum and spin of theMB state, respectively.This vertex function behaves as kL at k ∼ 0 and k−4 for k → ∞. The coupling constantsCMB(LS),N∗ , the cutoffs ΛMB(LS),N∗ and the bare masses M0
N∗ are the parameters of themodel.
Equations (5)-(25) define the DCC model used in the ANL-Osaka analysis. They areillustrated in Fig. 3. In the absence of theoretical inputs, the DCC model, as well as all
B’B
M M’
B
M
n
n
v v v vs u t c
FIG. 1: Meson-exchange mechanisms.
5
N
π
∆
ρ,σ∆
πZ
(E)
MB,M’B’= +
FIG. 2: Z-diagram mechanisms.
T
= +
= +
=
MB,M’B’t MB,M’B’ t MB,M’B’
R
t MB,M’B’
+
= +
vMB,M’B’
Γ Γ_
FIG. 3: Graphical representation of Eqs. (5)-(25).
hadron reaction models, has parameters that can only be determined phenomenologicallyfrom fitting the data. The meson-exchange interactions vM ′B′,MB depend on the couplingconstants and the cutoffs of form factors that regularize their matrix elements. While thevalues of some of the coupling constants can be estimated from SU(3) and the previousanalysis, we allow most of them to vary in the fits.
B. Electromagnetic amplitudes
With the hadronic amplitudes tM ′B′,MB(k′, k;E) defined in Eq. (6), the partial wave
amplitudes for the γ(~q) +N(−~q) → M ′(~k′) +B′(−~k) reactions are expressed as [4],
TM ′B′,γN(k′, q;E) = tM ′B′,γN(k
′, q;E) + tRM ′B′,γN(k′, q;E), (27)
with
tM ′B′,γN(k′, q;E) = vM ′B′,γN(k
′, q)
+∑
M ′′B′′
∫
p2dptM ′B′,M ′′B′′(k′, p;E)GM ′′B′′(p;E)vM ′′B′′,γN(p, q), (28)
6
tRM ′B′,γN(k′, q;E) =
∑
n,m
ΓM ′B′,N∗
n(k′;E)[D(E)]n,mΓN∗
m,γN(q;E), (29)
where the dressed N∗ → γN vertex is
ΓN∗,γN (q;E) = ΓN∗,γN(q) +∑
M ′B′
∫
p2dpΓ†M ′B′,N∗(p)GM ′B′(p, E)tM ′B′,γN(p, q;E). (30)
By using Eq. (28), the above equation can be written in terms of dressed N∗ →MB vertexΓN∗,M ′B′(p, E) defined by Eq. (23) and
ΓN∗,γN(q;E) = ΓN∗,γN(q) +∑
M ′B′
∫
p2dpΓN∗,M ′B′(p, E)GM ′B′(p, E)vM ′B′,γN (p, q;E).(31)
Here the transition interaction vMB,γN has the tree-diagram mechanisms shown in Fig. 1.Thus the pion-loop contributions (M ′B′ = πN) to the dressed vertex ΓN∗,γN(q;E) definedby Eq. (31) can be illustrated in Fig. 4. The procedures for calculating vMB,γN are detailedin Ref. [16].
For the bare γN → N∗ vertex, we write in the helicity representation as
ΓN∗,γN(q) =1
(2π)3/2
√
mN
EN(q)
√
qRq0GN∗
λ (Q2, q0)δλ,(λγ−λN ), (32)
where qR and q0 are defined by MN∗ = qR+EN (qR) and W = q0+EN (q0), respectively, and
GN∗
λ (Q2, q0) = AN∗
λ (Q2, q0), for transverse photons, (33)
= SN∗
λ (Q2, q0), for longitudinal photons. (34)
The helicity amplitudes AN∗
λ and SN∗
λ in the above equations are related to the multipoleamplitudes EN∗
l± ,MN∗
l± , SN∗
l± of γN → N∗ processes. For the N∗ of spin J = l + 1/2 with lbeing the orbital angular momentum of the γN system, the helicity amplitudes are
AN∗
3/2(Q2, q0) =
√
l(l + 2)
2[−MN∗
l+ (Q2, q0) + EN∗
l+ (Q2, q0)], (35)
AN∗
1/2(Q2, q0) = −1
2[lMN∗
l+ (Q2, q0) + (l + 2)EN∗
l+ (Q2, q0)], (36)
SN∗
1/2(Q2, q0) = SN∗
l+ (Q2, q0), (37)
For N∗ with J = l − 1/2, we have
AN∗
3/2(Q2, q0) = −
√
(l − 1)(l + 1)
2[MN∗
l− (Q2, q0) + EN∗
l− (Q2, q0)], (38)
AN∗
1/2(Q2, q0) =
1
2[(l + 1)MN∗
l− (Q2, q0)− (l − 1)EN∗
l− (Q2, q0)], (39)
SN∗
1/2(Q2, q0) = SN∗
l− (Q2, q0). (40)
The multipole amplitudes are parametrized as
MN∗
l± (Q2, q0) =(
q0mπ
)l(
Λ2N∗,γN +m2
π
Λ2N∗,γN + q20
)(2+l/2)
MN∗
l± (Q2), (41)
EN∗
l± (k)(Q2, q0) =(
q0mπ
)(l±1)(
Λ2N∗,γN +m2
π
Λ2N∗,γN + q20
)[2+(l±1)/2]
EN∗
l± (Q2), (42)
SN∗
l± (k)(Q2, q0) =(
q0mπ
)(l±1)(
Λ2N∗,γN +m2
π
Λ2N∗,γN + q20
)[2+(l±1)/2]
SN∗
l± (Q2), (43)
7
N N∗
π
π, ρω
π
π
N, ∆
γ
FIG. 4: Dressed γN → N∗ vertex defined by Eq. (32).
where the cutoff ΛN∗,γN and the coupling constants Ml±(Q2), El±(Q
2), Sl±(Q2) are deter-
mined in fitting the data. One significant difference between the above parametrization andthe form used in our previous analysis [6] is that the multipole amplitudes, or equivalentlythe helicity amplitudes, for the γN → N∗ processes now have the dependence on the γNrelative momentum q0.
III. EXTRACTIONS OF NUCLEON RESONANCES
We follow the earlier works to define that a nucleon resonance with a complex massMR isan “eigenstate“ of a Hamiltonian: H |ψR
N∗〉 = MR |ψRN∗〉. Then from the spectral expansion
of the Low Equation for reaction amplitude T (E) = H ′ +H ′(E − H)−1H ′, where we havedefined H ′ = H −H0 with H0 being the non-interacting free Hamiltonian, we have
TMB,M ′B′(k0MB, k0M ′B′ ;E →MR) =
〈k0MB|H ′ |ψRN∗〉 〈ψR
N∗|H ′ |k0M ′B′〉E −MR
+ ··, (44)
where k0MB and k0M ′B′ are the on-shell momenta defined by
E = EM (k0MB) + EB(k0MB)
= EM ′(k0M ′B′) + EB′(k0M ′B′) . (45)
Therefore the resonance positions can also be defined as the poles MR of the meson-baryonamplitude TMB,M ′B′(k0MB, k
0M ′B′ ;E) on the complex Riemann E-surface. Because of the
quadratic relation between the energy and momentum variables, each MB channel for agiven E can have a physical (p) sheet characterized by Im(k0MB) > 0 and an unphysical(u) sheet by Im(k0MB) < 0. Like all previous works, we only look for poles close to thephysical region and have large effects on πN scattering observables. All of these poles areon the unphysical sheet of the πN channel, but could be on either (u) or (p) sheets of otherchannels. To find the resonance poles, we analytically continue Eqs. (5)-(25) and Eqs. (28)-(30) to complex E-plane by using the method detailed in Refs. [8, 14]. The main step is tochoose appropriate integration paths CMB of Eq. (6) in solving Eqs. (5)-(25).
8
Explicitly, as energy approaches a resonance position MR in the complex E-plane, thetotal meson-baryon amplitudes can be written as
TMB,M ′B′(kRMB, kRM ′B′ ;E →MR) →
RMB,M ′B′(MR)
E −MR, (46)
with
RMB,M ′B′(MR) = ΓRM ′B′(kRM ′B′ ,MR)Γ
RMB(k
RMB,MR) (47)
where the on-shell momenta kRMB, kRM ′B′ are defined by Eqs. (45) with E =MR. In the actual
calculation, the residues RMB,M ′B′(MR) are extracted numerically by using the well knownmethod based on the Cauchy theorem. It is instructive to mention here that for the partial-wave with only one bare N∗ state and the resonance can be extracted from tRM ′B′,MB(k
′, k;E)of the full amplitude Eq. (5), each factor of the residue defined by Eq. (47) can be relatedto the dressed vertex functions Eqs. (22)-(23) and the self energy Eq. (25):
ΓRM ′B′(kRM ′B′ ,MR) =
ΓM ′B′,N∗(kRM ′B′ ,MR)√
1− Σ′(MR)(48)
ΓRMB(k
RMB,MR) =
ΓN∗,MB(kRMB,MR)
√
1− Σ′(MR)(49)
where Σ′(MR) = [dΣ(E)/dE]E=MR.
For the elastic πN → πN case, it is customary to define
RπN,πN(MR) = ρπN (k0πN)RπN,πN(MR)
= ρπN (k0πN)Γ
RπN(k
RπN ,MR)Γ
RπN(k
RπN ,MR) . (50)
where k0πN is defined by MR = Eπ(k0πN) + EN (k
0πN) and
ρπN (k0πN) = π
k0πNEπ(k0πN)EN (k
0πN)
E. (51)
The helicity amplitudes of γN → N∗ at the resonance pole MR are defined as
A3/2 = C × ΓRγN (q0,MR, λγ = 1, λN = −1/2), (52)
A1/2 = C × ΓRγN (q0,MR, λγ = −1, λN = −1/2) , (53)
where λN and λγ are the helicities of the initial nucleon and photon, respectively, and
C =
√
EN (~q)
mN
1√2K
√
(2JR + 1)(2π)3(2q0)
4π, (54)
where JR is the spin of the resonance state, q0 = |~q| andK = (M2R−m2
N)/(2MR). In practice,we can use the extracted residues RπN,γN(λ) and RπN,πN to calculate the determined helicityamplitudes by
A3/2 = N × RπNγN(3/2), (55)
A1/2 = N × RπNγN(1/2), (56)
N = a×√
√
√
√
konπNK
2π(2JR + 1)MR
mNRπN,πN
, (57)
with a =√
2/3 for the resonance with isospin I=3/2, and a = −√3 for I=1/2 resonance;
the phase is fixed so that −π/2 ≤ arg(N/a) ≤ π/2.
9
IV. CALCULATION OF CROSS SECTIONS
A. Meson-baryon scattering
We follow the convention of Goldberger and Watson to define the meson-baryon scattering
amplitudes. The normalization of states are : < ~k|~k ′
>= δ(~k − ~k′
) for plane wave states,and < φα|φβ >= δα,β for bound states. The T -matrix elements are related to S-matrixelements by
SMB,M ′B′ = δMB,M ′B′ − 2π i δ(EMB − EM ′B′) TMB,M ′B′ (58)
Note that the ”− ” sign in the right side of the above equation is opposite to the ”+ ” signused by the other partial-wave analysis groups such as SAID.
The formula for calculating the meson-baryon scattering cross sections given in this sec-
tion are in the center of mass system. For the process M(~k) +B(−~k) → M ′(~k′
) +B′(−~k ′
)the differential cross section can be written as
dσMB→M ′B′
dΩk′=
(4π)2
k2ρM ′B′(k′)ρMB(k)
1
(2jM + 1)(2jB + 1)
∑
mjMmjB
∑
m′
jMm′
jB
| < M ′B′|t(W )|MB > |2 ,
(59)
where the incoming and outgoing momentum k and k′ are defined by the invariant mass W
W = EM(k) + EB(k) = EM ′(k′) + EB′(k′) , (60)
where Eα(k) =√
m2α + ~k 2 with mα being the mass of particle α, and the phase-space factor
is
ρMB(k) = πkEM(k)EB(k)
W. (61)
The scattering amplitude < M ′B′|t(W )|MB > in Eq. (59) can be calculated from thepartial-wave amplitudes tJTL′S′M ′B′,LSMB(k
′, k,W ) as
< M ′B′|t(W )|MB >
=∑
JMJ ,T
∑
L′M ′
L,S′M ′
S
∑
LML,SMS
tJTL′S′M ′B′,‘LSMB(k′, k,W )
×[< TMT |i′Mτ ′Bm′iMm′
τB>< JMJ |L′S ′m′
Lm′S >< S ′m′
S|j′Mj′Bm′jMm′
jB> Y ∗
L′m′
L(k
′
)]
×[< TMT |iMτBmiMmτB >< JMJ |LSmLmS >< SmS|jMjBmjMmjB > YLmL(k)] , (62)
where < jmj|j1j2mj1mj2 > is the Clebsch-Gordon coefficient for the ~j1 + ~j2 = ~j coupling,[(jMmjM ), (iMmiM )] and [(jBmjB), (τBmτB )] are the spin-isospin quantum numbers of mesonsand baryons, respectively; (JMJ)((TMT )) are the total angular momentum (total isospin),(LML) ((SMS)) are the relative orbital angular momentum (total spin) of the consideredtwo-body systems.
By choosing the incoming momentum ~k in the quantization z-component, the totalMB →M ′B′ cross sections are
σtotMB→M ′B′(W ) =
∫
dΩk′dσMB→M ′B′
dΩk′. (63)
10
By optical theorem and the above partial-wave expansion, one can get the πN → X totalcross sections averaged over the initial spins:
σtotπN→X(W ) =
−4π
(2sN + 1)k2∑
J,T,L
(2J + 1) ρπN(k) Im[tJTL 12πN,L 1
2πN(k, k,W )]× [< TMT |1
1
2miπmτN >]2 ,
(64)
where MT = miπ +mτN and sN = 1/2 is the nucleon spin .The ANL-Osaka partial-wave amplitudes tJTL′S′M ′B′,LSMB(k
′, k,W ) can be obtained fromthe following quantities presented on the web page:
< M ′B′|T (W )|MB >= −ρ1/2M ′B′(k′)tJTL′S′M ′B′,LSMB(k′, k,W )ρ
1/2MB(k) , (65)
for MB,M ′B′ = πN, ηN,KΛ, KΣ and W = Wth − 2000 MeV, where Wth is the lower oneof the two threshold energies mM +mB and mM ′ +mB′ . The phase space factors ρMB(k)and ρM ′B′(k′) are defined by Eq. (61).
B. Electro-production of pions
For the pion electroproduction, ( e(pe)+N(pN) → e′(p′e)+π(k)+N(p′N)), the differentialcross section is conventionally written as
dσ
dE ′edΩe′dΩπ
= Γdσv
dΩπ, (66)
where Q2 = −q2, q = pe − p′e = (ωL, qL), W =√
(pN + q)2, and
Γ =αqγL
2π2Q2
E ′e
Ee
1
1− ǫ. (67)
Here, we have defined α = e2/4π = 1/137 and the effective photon energy in the laboratorysystem and ǫ are given as
qγL =W 2 −m2
N
2mN
, (68)
ǫ = [1 +2q2
L
Q2tan2 θe
2]−1, (69)
where θe is the angle between the outgoing and incoming electrons, and mN is the nucleonmass and qL is momentum transfer in the laboratory system.
The differential cross section dσv/dΩπ in Eq. (66) is defined in final πN center of massframe with the following coordinate system:
z = q =q
|q| (70)
y =q × k
|q × k| (71)
x = y × z (72)
11
We then have the following expression:
dσv
dΩπ=
dσTdΩπ
+ ǫdσLdΩπ
+ ǫdσTT
dΩπcos 2φπ +
√
2ǫ(ǫ+ 1)dσLTdΩπ
cos φπ
+he√
2ǫ(1− ǫ)dσLT ′
dΩπsinφπ , (73)
where he is the helicity of the incoming electron, φπ is the pion angle measured from thee− e′ scattering plane of electron, and
dσTdΩπ
=|k||qγ|
∑
spin
F xx + F yy
2, (74)
dσLdΩπ
=|k||qγ|
∑
spin
Q2
|q|2F00 , (75)
dσTT
dΩπ=
|k||qγ|
∑
spin
F xx − F yy
2, (76)
dσLTdΩπ
=|k||qγ|
∑
spin
(−1)
√
√
√
√
Q2
|q|2Re(Fx0) , (77)
dσLT ′
dΩπ=
|k||qγ|
∑
spin
√
√
√
√
Q2
|q|2Im(F x0) . (78)
Here q is the momentum transfer to the initial nucleon and k is the pion momentum in thecenter of mass system of the final πN state:
ω =W 2 −M2
N −Q2
2W, (79)
|q| =√
Q2 + ω2 , (80)
|k| =
√
(W 2 +m2
π −M2N
2W)2 −m2
π , (81)
and
|qγ| =W 2 −M2
N
2W. (82)
Here ω and qγ are the energy transfer and the effective photon energy in the center of masssystem. Integrating pion angles, Eqs. (66) and (73) lead to
dσ
dE ′edΩe′
(Q2,W ) = Γ[σT (Q2,W ) + ǫσL(Q
2,W )]. (83)
where
σT/L =∑
π+,π0
∫
dΩπ
dσT/LdΩπ
(84)
12
In the coordinate system defined by Eqs. (70)-(72), the pion momentum k is on the x− zplane. We thus can define
k = k/|k| = cos θz + sin θx , (85)
where θ is the angle between the outgoing pion and the virtual photon. The quantities F ij
with i, j = x, y, 0 in Eqs. (74)-(78) are defined as
∑
spin
F ij =1
2
∑
msi,msf
< msf |F i|msi >< msf |F j|msi >∗ , (86)
where ms is the z-component of the nucleon spin, and F i is defined by the Chew-Goldberger-Low-Nambu (CGLN) amplitude FCGLN = Fµǫµ.
The CGLN amplitude can be expressed in terms of Pauli operator σ, q, k and the photonpolarization vector ǫµ = (ǫ0, ǫ)
Fµǫµ = −(iσ · ǫ⊥F1 + σ · kσ · q × ǫ⊥F2 + iσ · qk · ǫ⊥F3 + iσ · kk · ǫ⊥F4
+iσ · qq · ǫF5 + iσ · kq · ǫF6) + iσ · kǫ0F7 + iσ · qǫ0F8 , (87)
where ǫ⊥ = q × (ǫ × q). By using Eq. (85) and choosing ǫµ = (ǫ0, ǫ) =(0, 1, 0, 0), (0, 0, 1, 0), (1, 0, 0, 0) to evaluate Eq. (87), we then have
Fx = iσx(F1 − cos θF2 + sin2 θF4 + iσz sin θ(F2 + F3 + cos θF4) , (88)
Fy = iσy(F1 − cos θF2)− sin θF2 , (89)
F0 = iσz(cos θF7 + F8) + iσy sin θF7 . (90)
The amplitudes Fi are calculated from the multipole amplitudes El±,Ml±, Sl± and Ll± ofthe γ∗ +N → πN process :
F1 =∑
l
[P ′l+1(x)El+(Q
2,W ) + P ′l−1(x)El−(Q
2,W ) + lP ′l+1(x)Ml+(Q
2,W )
+(l + 1)P ′l−1(x)Ml−(Q
2,W )] , (91)
F2 =∑
l
[(l + 1)P ′l (x)Ml+(Q
2,W ) + lP ′l (x)Ml−(Q
2,W )] , (92)
F3 =∑
l
[P ′′l+1(x)El+(Q
2,W ) + P ′′l−1(x)El−(Q
2,W )− P ′′l+1(x)Ml+(Q
2,W )
+P ′′l−1(x)Ml−(Q
2,W )] , (93)
F4 =∑
l
[−P ′′l (x)El+(Q
2,W )− P ′′l (x)El−(Q
2,W ) + P ′′l (x)Ml+(Q
2,W )− P ′′l (x)Ml−(Q
2,W )] ,(94)
F5 =∑
l
[(l + 1)P ′l+1(x)Ll+(Q
2,W )− lP ′l−1(x)Ll−(Q
2,W )] , (95)
F6 =∑
l
[−(l + 1)P ′l (x)Ll+(Q
2,W ) + lP ′l (x)Ll−(Q
2,W )] , (96)
F7 =∑
l
[−(l + 1)P ′l (x)Sl+(Q
2,W ) + lP ′l (x)Sl−(Q
2,W )] , (97)
F8 =∑
l
[(l + 1)P ′l+1(x)Sl+(Q
2,W )− lP ′l−1(x)Sl−(Q
2,W )] , (98)
13
where x = q · k, PL(x) is the Legendre function, P ′L(x) = dPL(x)/dx and P ′′
L(x) =d2PL(x)/d
2x. For the photo-production process γN → πN , the differential cross sectionis dσT/dΩπ with Q2 = 0.
The ANL-Osaka multipole amplitudes El±(Q2,W ), Ml±(Q
2,W ), and Ll±(Q2,W ) for
W = 1080− 2000 MeV and Q2 = 0− 3 (GeV/c)2 are presented on the web page.‘
C. Photo-production of mesons
In the center of mass system, the formula for calculating the un-polarized differential crosssection of the photo-production of pseudo-scalar mesons (M = π, η, K) on the nucleon (N),γ(q) +N(−q) →M(k) +N(−k), is
dσ0dΩ
=1
4
∑
msN=±1/2
∑
m′
sN=±1/2
∑
λ=±1
k
q| < m′
sN|FCGLN |msN > |2 (99)
where the Chew-Goldberger-Low-Nambu (CGLN) amplitude is defined as
FCGLN =∑
i=1,4
OiFi(θ,W ) . (100)
Here we have defined W = q +EN(q) = EM(k) +EN (k), where Eα(p) =√
m2α + p2 and mα
is the mass of particle α, and θ is the angle between k and q. The operators in the aboveequation are:
O1 = −iσ · ǫλO2 = −[σ · k][σ · q × ǫλ]
O3 = −i[σ · q][k · ǫλ]O4 = −i[σ · k][k · ǫλ]
where k = k/|k| and q = q/|q| = z, and σ is the standard Pauli operator. The quantizationdirection is chosen to be in the z-direction, and the photon polarization vectors are ǫ± =∓√2[x± iy]. The CGLN amplitudes Fi can be calculated from the multipole amplitudes with
Q2 = 0, El±(W ) and Ml±(W ), by using Eqs. (91)-(94).The ANL-Osaka multipole amplitudes El±(W ) and Ml±(W ) at Q2 = 0 for γN →
πN, ηN,ΛN,ΣN and W = 1080− 2000 MeV are presented on the web page.The formula for using the multipole amplitudes to calculate the polarization observables
can be found in Ref.[15].
D. Inclusive N(e, e′) cross sections
For the inclusive process, e(pe) + N(pN ) → e′(p′e) + X , the differential cross section iswritten by using structure functions Wi as
dσ
dE ′edΩe′
(Q2,W ) =α2
4E2e sin
4 θe2
[W2(Q2,W ) cos2
θe2+ 2W1(Q
2,W ) sin2 θe2] (101)
14
The structure functions are defined from the hadron tensor W µν as
W µν =∑
i
∑
f
(2π)3δ4(P + q − P ′)EN (P )
mN< f(P ′)|Jµ
em|N(P ) >< f(P ′)|Jνem|N(P ) >∗
= (−gµν + qµqν
q2)W1(Q
2,W ) +W2(Q
2,W )
m2N
P µP ν, (102)
where P µ = P µ − qµ(P · q)/q2.The structure functions are also expressed by virtual photon cross sections σX
T and σXL as
W1(Q2,W ) =
qγL4π2α
σXT (Q2,W ) (103)
W2(Q2,W ) =
qγL4π2α
Q2
q2L
[σXT (Q2,W ) + σX
L (Q2,W )]. (104)
The inclusive cross section can then be written as
dσ
dE ′edΩe′
(Q2,W ) = Γ[σXT (Q2,W ) + ǫσX
L (Q2,W )]. (105)
where ǫ is defined in Eq. (69). If only the single pion contribution to W1 and W2 is kept inevaluating Eq. (102), σX
T/L will be identical to σT/L defined in Eq. (83).
The ANL-Osaka structure functions W1(Q2,W ) and W2(Q
2,W ) for Q2 = 0−3 (GeV/c)2
and W = 1080− 2000 MeV are presented on the web page.
E. πN → ππN cross sections
In the center of mass frame, the momentum variables of the πN → ππN reaction withinvariant mass W can be specified as
a(~pa) + b(~pb) → c(~pc) + d(~pd) + e(~pe) , (106)
where ~pa = −~pb = ~k with k defined byW = Ea(k)+Eb(k), ~pc+~pd = −~pe = ~k′, and (c+d+e)can be any possible charged states formed from two pions and one nucleon. The total crosssection of the process Eq. (106) can be written as
σrecab→cde =
∫ W−me
mc+md
dσrec
dMcd
dMcd , (107)
with
dσrec
dMcd=ρik2
16π3∫
dΩkcddΩk′kcdk
′
W
1
(2sa + 1)(2sb + 1)
∑
i,f
|√
EcEdEe〈~pc~pd~pe, f |T |~k, i〉|2 ,
(108)
where ρi = π kEa(k)Eb(k)W
,i, f denote all spin (sa, saz) and isospin (ta, taz) quantum numbers,and
∑
i,f means summing over only spin quantum numbers. For a given invariant mass Mcd,
15
~kcd is the relative momentum between c and d in the center of mass of the sub-system (cd). It follows that k′ and kcd are defined by W and Mcd:
Mcd = Ec(kcd) + Ed(kcd) ,
W = Ee(k′) + Ecd(k
′) ,
Ecd(k′) =
√
M2cd + (k′)2 . (109)
The T -matrix elements in the Eq. (108) are of the following form
〈~pc~pd~pe, f |T |~k, i〉 =∑
sRz ,tRz
〈~pc, scz, tcz; ~pd, sdz, tdz|HI |~k′, sRz, tRz〉
W − Ee(k′)− ER(k′)− ΣeR(k′, E)
×〈~k′, sRz, tRz;−~k′, sez, tez|T |~k, saz, taz ;−~k, sbz, tbz〉 , (110)
where R is a bare state which has R → cd decay channel. For the eR = π∆ and eR = ρNchannels, the self-energies are explicitly given by
Σπ∆(k;W ) =m∆
E∆(k)
∫
q2dqMπN (q)
[M2πN(q) + k2]1/2
|f∆→πN(q)|2W − Eπ(k)− [M2
πN(q) + k2]1/2 + iǫ,(111)
ΣρN(k;W ) =mρ
Eρ(k)
∫
q2dqMππ(q)
[M2ππ(q) + k2]1/2
|fρ→ππ(q)|2W − EN(k)− [M2
ππ(q) + k2]1/2 + iǫ, (112)
where m∆ = 1280 MeV, mρ = 812 MeV, MπN(q) = Eπ(q) + EN (q), and Mππ(q) = Eπ(q) +Eπ(q). The form factors f∆→πN(q) and fρ→ππ(q) are for describing the ∆ → πN and ρ→ ππdecays in the ∆ and ρ rest frames, respectively. They are parametrized as:
f∆→πN(q) = −i (0.98)
[2(mN +mπ)]1/2
(
q
mπ
)
(
1
1 + [q/(358 MeV)]2
)2
, (113)
fρ→ππ(q) =(0.6684)√
mπ
(
q
(461 MeV)
)(
1
1 + [q/(461 MeV)]2
)2
. (114)
The σ self-energy ΣσN (k;E) is calculated from a ππ s-wave scattering model with a vertexfunction g(q) for the σ → ππ decay and a separable interaction v(q′, q) = h0h(q
′)h(q). Theresulting form is
ΣσN (k;W ) = 〈gGππg〉(k;W ) + τ(k;E)[〈gGππh〉(k;W )]2, (115)
with
τ(k;W ) =h0
1− h0〈hGππh〉(k;W ), (116)
〈hGππh〉(k;W ) =∫
dqq2Mππ(q)
[M2ππ(q) + k2]1/2
× h(q)2
W −EN (k)− [M2ππ(q) + k2]1/2 + iε
, (117)
〈gGππg〉(k;W ) =mσ
Eσ(k)
∫
dqq2Mππ(q)
[M2ππ(q) + k2]1/2
16
× g(q)2
W −EN (k)− [M2ππ(q) + k2]1/2 + iε
, (118)
〈gGππh〉(k;W ) =
√
mσ
Eσ(k)
∫
dqq2Mππ(q)
[M2ππ(q) + k2]1/2
× g(q)h(q)
W −EN (k)− [M2ππ(q) + k2]1/2 + iε
. (119)
In the above equations, mσ = 700.0 MeV and the form factors are
g(p) =g0√mπ
1
1 + (cp)2, (120)
h(p) =1
mπ
1
1 + (dp)2. (121)
where g0 = 1.638, h0 = 0.556, c = 1.02 fm, and d = 0.514 fm.For any spins and isospins and c.m. momenta ~p and ~p
′
, the MB → M ′B′ T -matrixelements in Eq. (110) are in general defined by
〈~p′, sM ′z, tM ′z;−~p′, sB′z, tB′z|T |~p, sMz, tMz;−~p, sBz, tBz〉=
∑
JM,TTz
∑
L′M ′
L,S′S′
z
∑
LML,SSz
YL′,M ′
L(p′)Y ∗
L,ML(p)
×〈sM ′, sB′ , sM ′z, sB′z|S ′, S ′z〉〈L′, S ′,M ′
L, S′z|J,M〉〈tM ′, tB′ , tM ′z, tB′z|T, Tz〉
×〈sM , sB, sMz, sBz|S, Sz〉〈L, S,ML, Sz|J,M〉〈tM , tB, tMz, tBz|T, Tz〉×tJTL′S′M ′B′,LSMB(p
′, p,W ), (122)
where the matrix elements tJTL′S′M ′B′,LSπN(p′, p,W ) for M ′B′ = π∆, σN, ρN are the PWA
from the ANL-Osaka model.The matrix elements of HI of Eq. (110) describe the decay of a resonance R = ∆, ρ, σ
into a two-particle state cd. It is of the following expression
〈~pc, scz, tcz; ~pd, sdz, tdz|HI |~k′, sRz, tRz〉
= δ(~pc + ~pd − ~k′)
√
√
√
√
Ec(kcd)Ed(kcd)MR
Ec(pc)Ed(pd)ER(k′)〈~kcd, scz, tcz;−~kcd, sdz, tdz|HI |~0, sRz, tRz〉, (123)
with
〈~kcd, scz, tcz;−~kcd, sdz, tdz|HI |~0, sRz, tRz〉=
∑
Lcd,Scd,mcd,Scdz
[〈sc, sd, scz, sdz|Scd, Scdz〉〈Lcd, Scd, mcd, Scdz|sR, sRz〉
×〈tc, td, tcz, tdz|tR, tRz〉YLcd,mcd(kcd)FLR
cd,SR
cd(kcd)]δLcd,L
RcdδScd,S
Rcd. (124)
The vertex functions are
FL∆πN
,S∆πN
(q) = if∆→πN(q) , (125)
FLσππ,S
σππ(q) =
√2g(q) , (126)
FLρππ,S
ρππ(q) = (−1)
√2fρ→ππ(q) , (127)
17
where L∆πN = 1, S∆
πN = 3/2, Lσππ = 0, Sσ
ππ = 0, Lρππ = 1, Sρ
ππ = 1. Here it is noted that thefactor
√2 in Eqs. (126)-(127) comes from the Bose symmetry of pions, and the phase factor
i and (−1) are chosen to be consistent with the non-resonant interactions involving πN∆,σππ and ρππ vertex interactions. The form factors f∆→πN(q) and fρ→ππ(q) have been inEqs. (113)-(114) and g(q) in Eq. (120).
With the above equations, the contribution from πN → π∆ → ππN to the total crosssection σrec
πN→ππN , as defined by Eq. (107)-(108), can be written as
σrecπ∆(W ) =
∫ W−mπ
mN+mπ
dMπNMπN
E∆(k)
Γ∆/(2π)
|W − Eπ(k)− E∆(k)− Σπ∆(k,W )|2 × σπN→π∆ ,(128)
where k and E∆(k) are defined by W and MπN
k =1
2W[(W 2 −M2
πN −m2π)
2 − 4M2πNm
2π]
1/2 , (129)
E∆(k) = [m2∆ + k2]1/2 , (130)
Σπ∆(k,W )is defined in Eq. (111), Γ∆ = −2Im[Σπ∆(k = 0,W )], and
σπN→π∆ =4π
k20
∑
JT,L′S′,LS
2J + 1
(2SN + 1)(2Sπ + 1)|ρ1/2π∆(k)t
JTL′S′π∆,LSπN(k, k0;W )ρ
1/2πN(k0)|2
×〈tπ, tN , tzπ, tzN |T, T z〉2 , (131)
where k0 is defined by W = Eπ(k0) + EN(k0) and ρab(k) = πkEa(k)Eb(k)/W . Similarly,the contributions of πN → ρN → ππN and πN → σN → ππN to the total cross sectionσrecπN→ππN are
σrecaN (W ) =
∫ W−mN
2mπ
dMππMππ
Ea(k)
Γa/(2π)
|W − EN(k)− Ea(k)− ΣaN (k,W )|2 × σπN→aN ,(132)
where a = ρ, σ, k is defined by Mππ and W
k =1
2W[(W 2 −M2
ππ −m2N )
2 − 4M2ππm
2N ]
1/2 , (133)
Ea(k) = [m2a + k2]1/2 , (134)
ΣaN (k,W ) for aN = ρN, σN are is defined in Eqs.(104) and (107), Γa = −2Im[ΣaN (k =0,W )], and
σπN→aN =4π
k20
∑
JT,L′S′,LS
2J + 1
(2SN + 1)(2Sπ + 1)|ρ1/2aN (k)tJTL′S′aN,LSπN(k, k0;W )ρ
1/2πN(k0)|2
×〈tπ, tN , tzπ, tzN |T, T z〉2 . (135)
To perform calculations, we need to have the partial-wave amplitudestJTL′S′M ′B′,LSπN(p, k,W ) for M ′B′ = π∆, ρN, σN . These PWA from ANL-Osaka modelcan be obtained from the web page which present the following :
< π∆|T (W )|πN >= −ρ1/2π∆(p∆)tJTL′S′π∆,LSπN(p∆, k,W )ρ
1/2πN(k) ,
< ρN |T (W )|πN >= −ρ1/2ρN (pρ)tJTL′S′ρN,LSπN(pρ, k,W )ρ
1/2πN(k) ,
< σN |T (W )|πN >= −ρ1/2σN (pσ)tJTL′S′σN,LSπN(pσ, k,W )ρ
1/2πN(k) ,
(136)
18
Here the phase space factors account for the effects due to ∆ → πN , σ → ππ and ρ → ππdecays. Explicitly, we have
ρπ∆(p∆) = πp∆E∆(p∆)Eπ(p∆)
W, (137)
where p∆ and E∆(p∆) are defined by W and the invariant mass MπN in the integrations ofEqs. (107) and (128)
p∆ =1
2W[(W 2 −M2
πN −m2π)
2 − 4M2πNm
2π]
1/2 , (138)
E∆(p∆) = [M2πN + p2∆]
1/2 , (139)
For the calculations of Eqs. (107) and (128), we thus present < π∆|T (W )|πN > in the range0 ≤ p∆ ≤ p∆,max with
p∆,max =1
2W[(W 2 − (mπ +mN)
2 −m2π)
2 − 4(mπ +mN )2m2
π]1/2 . (140)
For the ρN and σN channels, we have
ρ1/2σN (pσ) = π
pσEσ(pσ)EN(pσ)
W, (141)
ρ1/2ρN (pρ) = π
pρEρ(pρ)EN(pρ)
W, (142)
For a = σ and ρ, we have
pa =1
2W[(W 2 −M2
ππ −m2N )
2 − 4M2ππm
2N ]
1/2 , (143)
Ea(pa) = [M2ππ + p2a]
1/2 . (144)
For the calculation of Eqs. (107) and (132), we thus present < ρN |T (W )|πN > and <σN |T (W )|πN > in the range 0 ≤ pa ≤ pa,max with
pa,max =1
2W[(W 2 − (2mπ)
2 −m2N)
2 − 4(2mπ)2m2
N ]1/2 . (145)
The above equations are for the calculations of the πN → ππN through the resonant π∆,σN and ρN channels. There are also weaker contributions from the direct production mech-anisms, as illustrated in Fig. 5, which can be calculated by using the procedures explainedin Ref. [9].
19
FIG. 5: The considered vπN,ππN .
20
V. DATA
The parameters of the ANL-Osaka DCC model were determined by performing χ2-fitsto the data of πN, γN → πN, ηN,KΛ, KΣ. The calculations involve solving the coupled-channel Eqs. (5)-(25). The meson-baryon partial-waves included in the calculations arelisted in Table I. The total number of the data included in the fits are about 30,000 datapoints, as listed in table II-V.
Note that the 1940 data points for πN → πN listed in Table II are in fact containinformation of about 30,000 data in the SAID analysis: 14196 of π+p → π+p, 13895 ofπ−p → π−p, and 2877 of π−p → π0n. In addition, the partial-wave amplitudes of πN →ππN determined from the very extensive bubble-chamber data of πN → ππN were alsoused in the earlier analysis of SAID. Thus the determined PWA for πN → πN, π∆, ρN, σNare rather reliable.
On the other hand, the data points for πN → ηN,KΛ, KΣ listed in Table III are muchless, only 294, 941, and 1262, respectively. Thus the determined PWA for these processesneed to be improved by using more extensive data which could be available from experimentsat J-PARC in near future.
In Tables IV and V, we see that the data points for γN → πN are much more thanthose of γN → ηN,KΛ, KΣ. Therefore, the γN → N∗ couplings are mainly determinedby the predicted multipole amplitudes of γN → πN . It will be interesting to include morenew JLab data of γN → KΛ, KΣ in the analysis to further improve the determination ofγN → N∗ couplings which are crucial to test the predictions from LQCD and various hadronmodels.
21
TABLE I: The orbital angular momentum (L) and total spin (S) of each MB channel allowed in
a given partial wave. In the first column, partial waves are denoted with the conventional notation
l2I2J as well as (I,JP ).
l2I2J (I, JP ) (L,S) of the considered partial waves
πN ηN π∆ σN ρN KΛ KΣ
(π∆)1 (π∆)2 (ρN)1 (ρN)2 (ρN)3
S11 (1, 12−) (0, 12) (0, 12) (2, 32 ) – (1, 12) (0, 12 ) (2, 32) – (0, 12) (0, 12)
S31 (3, 12−) (0, 12) – (2, 32 ) – – (0, 12 ) (2, 32) – – (0, 12)
P11 (1, 12+) (1, 12) (1, 12) (1, 32 ) – (0, 12) (1, 12 ) (1, 32) – (1, 12) (1, 12)
P13 (1, 32+) (1, 12) (1, 12) (1, 32 ) (3, 32) (2, 12) (1, 12 ) (1, 32) (3, 32) (1, 12) (1, 12)
P31 (3, 12+) (1, 12) – (1, 32 ) – – (1, 12 ) (1, 32) – – (1, 12)
P33 (3, 32+) (1, 12) – (1, 32 ) (3, 32) – (1, 12 ) (1, 32) (3, 32) – (1, 12)
D13 (1, 32−) (2, 12) (2, 12) (0, 32 ) (2, 32) (1, 12) (2, 12 ) (0, 32) (4, 32) (2, 12) (2, 12)
D15 (1, 52−) (2, 12) (2, 12) (2, 32 ) (4, 32) (3, 12) (2, 12 ) (2, 32) (4, 32) (2, 12) (2, 12)
D33 (3, 32−) (2, 12) – (0, 32 ) (2, 32) – (2, 12 ) (0, 32) (2, 32) – (2, 12)
D35 (3, 52−) (2, 12) – (2, 32 ) (4, 32) – (2, 12 ) (2, 32) (4, 32) – (2, 12)
F15 (1, 52+) (3, 12) (3, 12) (1, 32 ) (3, 32) (2, 12) (3, 12 ) (1, 32) (3, 32) (3, 12) (3, 12)
F17 (1, 72+) (3, 12) (3, 12) (3, 32 ) (5, 32) (4, 12) (3, 12 ) (3, 32) (5, 32) (3, 12) (3, 12)
F35 (3, 52+) (3, 12) – (1, 32 ) (3, 32) – (3, 12 ) (1, 32) (3, 32) – (3, 12)
F37 (3, 72+) (3, 12) – (3, 32 ) (5, 32) – (3, 12 ) (3, 32) (5, 32) – (3, 12)
G17 (1, 72−) (4, 12) (4, 12) (2, 32 ) (4, 32) (3, 12) (4, 12 ) (2, 32) (4, 32) (4, 12) (4, 12)
G19 (1, 92−) (4, 12) (4, 12) (4, 32 ) (6, 32) (5, 12) (4, 12 ) (4, 32) (6, 32) (4, 12) (4, 12)
G37 (3, 72−) (4, 12) – (2, 32 ) (4, 32) – (4, 12 ) (2, 32) (4, 32) – (4, 12)
G39 (3, 92−) (4, 12) – (4, 32 ) (6, 32) – (4, 12 ) (4, 32) (6, 32) – (4, 12)
H19 (1, 92+) (5, 12) (5, 12) (3, 32 ) (5, 32) (4, 12) (5, 12 ) (3, 32) (5, 32) (5, 12) (5, 12)
H39 (3, 92+) (5, 12) (5, 12) (3, 32 ) (5, 32) – (5, 12 ) (3, 32) (5, 32) – (5, 12)
22
TABLE II: Number of the data points of πN → πN amplitudes included in our fits. The data
are from SAID analysis of 14196 data points of π+p → π+p, 13895 of π−p → π−p, and 2877 of
π−p → π0p.
Partial Wave Partial Wave
S11 65×2 S31 65×2
P11 65×2 P31 61×2
P13 61×2 P33 65×2
D13 61×2 D33 55×2
D15 61×2 D35 45×2
F15 48×2 F35 43×2
F17 32×2 F37 44×2
G17 42×2 G37 32×2
G19 28×2 G39 32×2
H19 34×2 H39 31×2
Sum 994 946 1940
TABLE III: Number of data points of hadronic processes included in our fits. Data are from the
compilation of Bonn-Gatchina.
dσ/dΩ P R a sum
π−p → ηp 294 – – – 294
π−p → K0Λ 587 354 – – 941
π−p → K0Σ0 259 90 – – 349
π+p → K+Σ+ 609 304 – – 913
Sum 1749 748 – – 2497
TABLE IV: The number of data points of photoproduction processes included in our fits. The
data are from compilation of Bonn-Gatchina.
dσ/dΩ Σ T P G H E F Ox′ Oz′ Cx′ Cz′ sum
γp → π0p 4414 1866 389 607 75 71 140 – 7 7 – – 7576
γp → π+n 2475 899 661 252 86 128 231 – – – – – 4732
γp → ηp 780 151 50 – – – – – – – – – 981
γp → K+Λ 1320 118 66 1336 – – – – 160 159 66 66 3291
γp → K+Σ0 1280 87 – 95 – – – – – – 94 94 1650
γp → K0Σ+ 276 15 – 72 – – – – – – – – 363
Sum 10545 3136 1166 2362 161 199 371 – 167 166 160 160 18593
23
TABLE V: Observables and number of the data points considered in this coupled-channels analysis.
The data are taken from the database of the INS DAC Services.
Reactions Observables Number of data points
γ‘n’ → π−p dσ/dΩ 2305
Σ 308
T 94
P 88
γ‘n’ → π0n dσ/dΩ 148
Σ 216
Sum 3159
TABLE VI: Number of data points of p(e, e′π)N included in our fits. The data are for about 25
Q2 with W < 2000 MeV.
σT + ǫσL σTT σLT σ′LT sum
p(e, e′π0)p 5830 5830 5830 240 17730
p(e, e′π+)n 2614 2614 2614 566 8408
Sum 8444 8444 8444 806 26138
24
VI. RESULTS
The nucleon resonance properties extracted from ANL-Osaka amplitudes were finalizedin 2016 and published in Ref. [17]. In Table VII, we list the pole positions and the residuesRπN,πN of the extracted resonances. In Ref. [16], the residues for RπN,ηN , RπN,KΛ, andRπN,KΣ are also given. However these results are not as solid as that of RπN,πN becausethe data of πN → ηN,KΛ, KΣ included in the fits are not as accurate as the data ofπN → πN . The residues associated with the unstable channels π∆, ρN and σN are alsonot given because the analytic continuation of the PWA associated with these channels tocomplex E-plane is rather complex and a rigorous way to do this remains to be explored.
The determined helicity amplitudes for the γp → N∗ transition at resonance pole posi-tions, published in Ref. [17], are listed in Table VIII. However the Q2-dependence of γp→ N∗
transitions at resonance pole was not extracted because only the data at few Q2 were in-cluded in the fits.
To illustrate the quality of our fits to the data, we present sample results from our analysisin the Appendix: (A) Total cross sections of πN reactions, (B) Total cross sections of γNreactions, (C) Differential cross sections of πN → πN, ηN, KΛ, KΣ, (D) Differential crosssections of γN → πN, ηN, KΛ, KΣ, (E) Differential cross sections of γ∗N → πN , (F)Inclusive cross sections of p(e, e′).
25
TABLE VII: The extracted nucleon resonance pole mass (MR) and πN elastic residue (RπN,πN ).
MR is listed as (Re(MR),−Im(MR)) in units of MeV, while RπN,πN = |RπN,πN |eiφ is listed as
(|RπN,πN |, φ) in units of MeV for |RπN,πN | and degree for φ. The range of φ is taken to be
−180 ≤ φ < 180. The N∗ resonances for which the asterisk (*) is marked locate in the complex
energy plane slightly off the sheet closest to the physical real energy axis, yet are still expected to
visibly affect the physical observables.
JP (L2I2J) MR RπN,πN
N∗ 1/2−(S11) (1490, 102) (70, −42)
(1652, 71) (45, −74)
1/2+(P11) (1376, 75) (38, −70)
(1741, 139) (15, 80)
3/2+(P13) (1708, 65) (9, −4)
(1765, 160) (30, −105)
3/2−(D13) (1509, 48) (30, −10)
(1702, 148)* (< 1, −161)
5/2−(D15) (1651, 68) (26, −27)
5/2+(F15) (1665, 52) (36, −22)
∆∗ 1/2−(S31) (1597, 69) (21, −111)
(1713, 187) (20, 73)
1/2+(P31) (1857, 145) (11, −118)
3/2+(P33) (1212, 52) (55, −47)
(1733, 162) (16, −108)
3/2−(D33) (1577, 113) (13, −67)
5/2−(D35) (1911, 130) (4, −30)
5/2+(F35) (1767, 88) (11, −61)
7/2+(F37) (1885, 102) (49, −30)
26
TABLE VIII: The determined helicity amplitudes for the γp → N∗ transition at resonance pole
positions. The presented values follow the notation: A1/2,3/2 = A1/2,3/2 × eiφ with φ taken to be in
the range −90 ≤ φ < 90. The units of A1/2,3/2 and φ are 10−3 GeV−1/2 and degree, respectively.
Each resonance is specified by the isospin and spin-parity quantum numbers as well as the real
part of the resonance pole mass.
Particle JP (L2I2J) A1/2 φ A3/2 φ
N(1490)1/2−(S11) 160 8 - -
N(1652)1/2−(S11) 36 −28 - -
N(1376)1/2+(P11) −40 −8 - -
N(1741)1/2+(P11) −47 −24 - -
N(1708)3/2+(P13) 131 7 −33 12
N(1765)3/2+(P13) 123 −11 −71 3
N(1509)3/2−(D13) −28 < 1 102 4
N(1703)3/2−(D13) 13 50 31 −71
N(1651)5/2−(D15) 8 19 49 −12
N(1665)5/2+(F15) −44 −11 60 −2
∆(1597)1/2−(S31) 105 1 - -
∆(1713)1/2−(S31) 40 13 - -
∆(1857)1/2+(P31) −1 −78 - -
∆(1212)3/2+(P33) −134 −16 −257 −3
∆(1733)3/2+(P33) −48 63 −94 74
∆(1577)3/2−(D33) 128 19 119 46
∆(1911)5/2−(D35) 48 −22 11 −36
∆(1767)5/2+(F35) 38 −7 −24 −80
∆(1885)7/2+(F37) −69 −14 −83 2
27
TABLE IX: The isovector and isoscalar helicity amplitudes for γN → N∗ are defined as AT=1λ =
(Apλ −An
λ), and AT=0λ = (Ap
λ +Anλ), where Ap
λ and Anλ are the helicity amplitudes of γp → N∗ and
γn → N∗, respectively. See the caption of Table VIII for the notation of the table.
Particle JP (L2I2J) AT=11/2 φ AT=0
1/2 φ AT=13/2 φ AT=0
3/2 φ
N(1490)1/2−(S11) 136 11 26 −10 - - - -
N(1652)1/2−(S11) 19 −29 18 −28 - - - -
N(1376)1/2+(P11) −68 −13 28 −21 - - - -
N(1741)1/2+(P11) −120 −11 75 −3 - - - -
N(1708)3/2+(P13) 95 7 36 8 −9 68 −29 −2
N(1765)3/2+(P13) 78 −10 45 −14 −55 4 −16 −2
N(1509)3/2−(D13) 7 −2 −35 −1 106 4 −5 8
N(1703)3/2−(D13) 20 −28 −22 −63 54 −61 −24 −48
N(1651)5/2−(D15) 42 4 −34 1 44 −8 6 −37
N(1665)5/2+(F15) −39 −11 −5 −8 58 −3 2 18
28
Acknowledgments
This work is supported by the U.S. Department of Energy, Office of Science, Office ofNuclear Physics, Contract No. DE-AC02-06CH11357.
Appendix A: Total Cross sections of πN reactions
0
50
100
150
200
250
1000 1200 1400 1600 1800 2000
π+p --> X
π-p --> X
σtot (
mb)
W (MeV)
0
5
10
15
20
25
30
1000 1200 1400 1600 1800 2000
π+p --> π+π+n + π+π0p
σtot (
mb)
W (MeV)
0
5
10
15
20
25
30
1000 1200 1400 1600 1800 2000
π-p --> π0π0n + π0π-p + π+π-n
σtot (
mb)
W (MeV)
FIG. 6: σtot of π±p → X,ππN
29
0
50
100
150
200
250
1000 1200 1400 1600 1800 2000
π+p --> π+p
σtot (
mb)
W (MeV)
0
0.2
0.4
0.6
0.8
1
1000 1200 1400 1600 1800 2000
π+p --> K+Σ+
σtot (
mb)
W (MeV)
0
10
20
30
40
50
1000 1200 1400 1600 1800 2000
π-p --> π-p
σtot (
mb)
W (MeV)
0
10
20
30
40
50
1000 1200 1400 1600 1800 2000
π-p --> π0n
σtot (
mb)
W (MeV)
0
1
2
3
4
5
1000 1200 1400 1600 1800 2000
π-p --> ηn
σtot (
mb)
W (MeV)
0
0.2
0.4
0.6
0.8
1
1000 1200 1400 1600 1800 2000
π-p --> K0Λ
σtot (
mb)
W (MeV)
0
0.1
0.2
0.3
0.4
0.5
1000 1200 1400 1600 1800 2000
π-p --> K0Σ0
σtot (
mb)
W (MeV)
0
0.1
0.2
0.3
0.4
0.5
1000 1200 1400 1600 1800 2000
π-p --> K+Σ-
σtot (
mb)
W (MeV)
FIG. 7: σtot of π±p → πN, ηN,KΛ,KΣ
30
0
5
10
15
20
25
1000 1200 1400 1600 1800 2000
σtot (
mb)
W (MeV)
π+p --> π+π+n + π+π0p
π+p --> π∆ --> π+π+n + π+π0p
0
5
10
15
20
25
30
1000 1200 1400 1600 1800 2000
σtot (
mb)
W (MeV)
π-p --> π0π0n + π0π-p + π+π-n
π-p --> π∆ --> π0π0n + π0π-p + π+π-n
0
5
10
15
20
25
1000 1200 1400 1600 1800 2000
σtot (
mb)
W (MeV)
π+p --> π+π+n + π+π0p
π+p --> ρ+p --> π+π0p
0
5
10
15
20
25
30
1000 1200 1400 1600 1800 2000
σtot (
mb)
W (MeV)
π-p --> π0π0n + π0π-p + π+π-n
π-p --> ρN --> π0π-p + π+π-n
0
5
10
15
20
25
30
1000 1200 1400 1600 1800 2000
σtot (
mb)
W (MeV)
π-p --> π0π0n + π0π-p + π+π-n
π-p --> σn --> π0π0n + π+π-n
FIG. 8: Total cross sections for π±p → (MB)R → ππN (dashed curves) are compared with π±p →ππN(solid curves). Top row: (MB)R = π∆; middle row: (MB)R = ρN ; bottom row:(MB)R =
σN .
31
Appendix B: Total Cross sections of γN reactions
1200 1400 1600 1800 2000W (MeV)
0
100
200
300
400
500
600
σ (µ
b)γ p X
γ p π0 p
γ p π+ n
0
5
10
15
20
0 200 400 600 800 1000 1200 1400 1600 1800
γp --> η+p
σtot (
µb)
Eγ (MeV)
0
0.5
1
1.5
2
2.5
3
0 200 400 600 800 1000 1200 1400 1600 1800
γp --> K+Λ
σtot (
µb)
Eγ (MeV)
0
0.5
1
1.5
2
2.5
3
0 200 400 600 800 1000 1200 1400 1600 1800
γp --> K+Σ0
σtot (
µb)
Eγ (MeV)
0
0.2
0.4
0.6
0.8
1
0 200 400 600 800 1000 1200 1400 1600 1800
γp --> K0Σ+
σtot (
µb)
Eγ (MeV)
FIG. 9: Top :γp → X,π0p, π+n, dashed curves are our previous analysis; Lower four figures:
γp → ηp,KΛ,K+Σ0,K0Σ+.
32
Appendix C: Differential cross sections of πN reactions
1. πN → πN
0
20
40
dσ/d
Ω (
mb/
sr)
0
1.5
3
0
5
10
0
15
dσ/d
Ω (
mb/
sr)
0
2
0
4
0
4
dσ/d
Ω (
mb/
sr)
0
10
0
2
0
8
dσ/d
Ω (
mb/
sr)
0
8
0
0.5
0 90θ (deg.)
0
10
dσ/d
Ω (
mb/
sr)
0 90 180θ (deg.)
0 90θ (deg.)
0
8
0 90 180θ (deg.)
0 90θ (deg.)
0
0.5
0 90 180θ (deg.)
1153 MeV 1235 MeV 1153 MeV 1235 MeV 1158 MeV 1232 MeV
1449 MeV1320 MeV1443 MeV1320 MeV1443 MeV1320 MeV
1535 MeV 1643 MeV 1535 MeV 1531 MeV 1630 MeV
1755 MeV 1898 MeV1730 MeV
1641 MeV
1847 MeV1753 MeV1847 MeV
1958 MeV 2057 MeV 1959 MeV 2047 MeV 1974 MeV 2066 MeV
π+ p π+
p π− p π−
p π− p π0
n
FIG. 10: Differential cross section for πN → πN . The red solid curves are the current results while
the blue dashed curves are from our previous analysis of 2007.
33
-1
0
1
P
-1
0
1
-1
0
1
-1
0
1
P
-1
0
1
-1
0
1
-1
0
1
P
-1
0
1
-1
0
1
-1
0
1
P
90 180θ (deg.)
-1
0
1
90 180θ (deg.)
-1
0
1
0 90 180θ (deg.)
-1
0
1
P
0 90 180θ (deg.)
-1
0
1
0 90θ (deg.)
-1
0
1
0 90 180θ (deg.)
1153 MeV 1235 MeV 1151 MeV 1234 MeV 1160 MeV 1214 MeV
1445 MeV1320 MeV
1443 MeV1326 MeV1445 MeV1318 MeV
1532 MeV 1631 MeV
1558 MeV 1543 MeV 1657 MeV
1764 MeV
1839 MeV1764 MeV
1635 MeV
1856 MeV1767 MeV
1844 MeV
1934 MeV
1934 MeV 1978 MeV 2056 MeV
π+ p π+
p π− p π−
p π− p π0
n
FIG. 11: The target polarization P of πN → πN . The red solid curves are the current results
while the blue dashed curves are from our previous analysis of 2007.
34
2. πN → ηN
0
0.2
dσ/d
Ω (
µb/s
r)
0 90θ (deg.)
0
0.2
dσ/d
Ω (
µb/s
r)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
0
0.2
0.4dσ
/dΩ
(µb
/sr) 1488 MeV 1492 MeV 1499 MeV 1507 MeV 1512 MeV 1516 MeV
1587 MeV1577 MeV1547 MeV1535 MeV1525 MeV
1523 MeV
1609 MeV 1674 MeV
1699 MeV 1805 MeV 1832 MeV 1898 MeV1730 MeV 1948 MeV 1988 MeV
FIG. 12: Differential cross sections of π−p → ηn. The red solid curves are the current results while
the blue dashed curves are from our previous analysis of 2007.
3. πN → KΛ
0
100
200
dσ/d
Ω (
µb/s
r)
0
100
dσ/d
Ω (
µb/s
r)
0
100
dσ/d
Ω (
µb/s
r)
0 90θ (deg.)
0
100
dσ/d
Ω (
µb/s
r)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
1633 MeV 1661 MeV 1670 MeV 1676 MeV 1678 MeV 1681 MeV
1701 MeV1698 MeV1694 MeV1689 MeV1687 MeV1686 MeV
1724 MeV 1743 MeV 1758 MeV 1825 MeV 1847 MeV
1909 MeV 2059 MeV2027 MeV
1792 MeV
1999 MeV1966 MeV1938 MeV
1684 MeV
1707 MeV
1879 MeV
2104 MeV
FIG. 13: Differential cross sections of π−p → K0Λ0. The red solid curves are the current results
while the blue dashed curves are from our previous analysis of 2007.
4. πN → KΣ
35
-1
0
1
P
-1
0
1
P
90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
0 90θ (deg.)
-1
0
1
P
0 90θ (deg.)
0 90 180θ (deg.)
1633 MeV 1661 MeV 1683 MeV 1694 MeV 1724 MeV 1758 MeV
1966 MeV1938 MeV1909 MeV1879 MeV1847 MeV1825 MeV
2027 MeV 2059 MeV
2104 MeV 0
1792 MeV
1999 MeV
FIG. 14: Differential cross sections of π−p → K0Λ0. The red solid curves are the current results
while the blue dashed curves are from our previous analysis of 2007.
0
50
100
dσ/d
Ω (
µb/s
r)
0
100
dσ/d
Ω (
µb/s
r)
90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
0 90θ (deg.)
0
100
dσ/d
Ω (
µb/s
r)
0 90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
1732 MeV 1764 MeV 1783 MeV 1791 MeV 1813 MeV 1822 MeV
0
1970 MeV1939 MeV1926 MeV1891 MeV1870 MeV
2031 MeV 2059 MeV 2074 MeV 2106 MeV
1845 MeV
1985 MeV 2019 MeV
0
80
dσ/d
Ω (
µb/s
r)
0 90θ (deg.)
0
80
dσ/d
Ω (
µb/s
r)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
1694 MeV 1724 MeV 1758 MeV 1792 MeV 1825 MeV 1847 MeV
2059 MeV2027 MeV1999 MeV1966 MeV1938 MeV1909 MeV
1879 MeV
2104 MeV
FIG. 15: Differential cross sections of π+p → K+Σ+(upper) and π−p → K0Σ0(lower). The red
solid curves are the current results while the blue dashed curves are from our previous analysis of
2007.
36
-1
0
1
P
-1
0
1
P
-1
0
1
P
90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
0 90θ (deg.)
-1
0
1
P
0 90θ (deg.)
0 90 180θ (deg.)
1729 MeV 1732 MeV 1757 MeV 1764 MeV 1783 MeV 1789 MeV
2019 MeV
1891 MeV
1870 MeV1845 MeV1822 MeV1813 MeV
1934 MeV 1939 MeV 1954 MeV 1985 MeV
2106 MeV2059 MeV
1970 MeV 2031 MeV
2074 MeV
1791 MeV
1909 MeV 1926 MeV
0
-1
0
1
P
0 90θ (deg.)
-1
0
1
P
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
1694 MeV 1724 MeV 1758 MeV 1792 MeV
1825 MeV
1847 MeV
2059 MeV2027 MeV1999 MeV1966 MeV1938 MeV1909 MeV
1879 MeV
2104 MeV
FIG. 16: Polarization P of π+p → K+Σ+(upper) and π−p → K0Σ0(lower). The red solid curves
are the current results while the blue dashed curves are from our previous analysis of 2007.
37
Appendix D: Differential Cross sections of γN → πN, ηN,KΛ,KΣ
1. γp → π0p
0
12dσ
/dΩ
(µb
/sr)
0
24
dσ/d
Ω (
µb/s
r)
0
12
dσ/d
Ω (
µb/s
r)
0
6
dσ/d
Ω (
µb/s
r)
0
6
dσ/d
Ω (
µb/s
r)
0 90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
0 90θ (deg.)
0
6
dσ/d
Ω (
µb/s
r)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
1110 MeV 1120 MeV 1130 MeV 1140 MeV 1150 MeV 1160 MeV
1386 MeV
1249 MeV1234 MeV1217 MeV1209 MeV1180 MeV
1305 MeV 1325 MeV 1340 MeV 1375 MeV
1427 MeV1402 MeV
1497 MeV
1358 MeV 1390 MeV
1414 MeV
1170 MeV
1265 MeV 1283 MeV
1439 MeV 1450 MeV 1462 MeV 1474 MeV
1490 MeV1484 MeV 1506 MeV 1517 MeV 1527 MeV 1537 MeV
1589 MeV1579 MeV1569 MeV 1599 MeV 1608 MeV1558 MeV1548 MeV
0
4
dσ/d
Ω (
µb/s
r)
0
4
dσ/d
Ω (
µb/s
r)
0
2
dσ/d
Ω (
µb/s
r)
0
2
dσ/d
Ω (
µb/s
r)
0
2
dσ/d
Ω (
µb/s
r)
0 90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
0 90θ (deg.)
0
2
dσ/d
Ω (
µb/s
r)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
1617 MeV 1627 MeV 1636 MeV 1646 MeV 1654 MeV 1663 MeV
1790 MeV
1720 MeV1709 MeV1698 MeV1690 MeV1682 MeV
1750 MeV 1756 MeV 1763 MeV 1783 MeV
1835 MeV1813 MeV
1895 MeV
1777 MeV 1803 MeV
1827 MeV
1672 MeV
1730 MeV 1737 MeV
1842 MeV 1855 MeV 1860 MeV 1869 MeV
1887 MeV1882 MeV 1908 MeV 1920 MeV 1933 MeV 1942 MeV
2053 MeV2029 MeV2006 MeV 2075 MeV 2097 MeV1983 MeV1959 MeV
FIG. 17: Differential cross sections of γp → π0p. The red solid curves are the current results while
the blue dashed curves are from our previous analysis.
38
-1
0
1
Σ
-1
0
1Σ
-1
0
1
Σ
-1
0
1
Σ
-1
0
1
Σ
0 90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
0 90θ (deg.)
-1
0
1
Σ
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
1140 MeV 1149 MeV 1157 MeV 1165 MeV 1173 MeV 1182 MeV
1292 MeV
1232 MeV1227 MeV1219 MeV1209 MeV1200 MeV
1255 MeV 1262 MeV 1270 MeV 1284 MeV
1366 MeV1335 MeV
1459 MeV
1277 MeV 1313 MeV
1349 MeV
1190 MeV
1240 MeV 1247 MeV
1384 MeV 1402 MeV 1416 MeV 1426 MeV
1449 MeV1433 MeV 1465 MeV 1483 MeV 1497 MeV 1504 MeV
1559 MeV1544 MeV1528 MeV 1573 MeV 1580 MeV1524 MeV1513 MeV
-1
0
1
Σ
-1
0
1
Σ
-1
0
1
Σ
-1
0
1
Σ
-1
0
1
Σ
0 90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
0 90θ (deg.)
-1
0
1
Σ
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
1588 MeV 1600 MeV 1603 MeV 1620 MeV 1632 MeV 1638 MeV
1756 MeV
1691 MeV1678 MeV1673 MeV1659 MeV1656 MeV
1714 MeV 1724 MeV 1732 MeV 1750 MeV
1802 MeV1771 MeV
1858 MeV
1740 MeV 1768 MeV
1787 MeV
1646 MeV
1696 MeV 1708 MeV
1816 MeV 1819 MeV 1831 MeV 1836 MeV
1853 MeV1845 MeV 1872 MeV 1885 MeV 1898 MeV 1903 MeV
1967 MeV1951 MeV1935 MeV 1982 MeV 1998 MeV1919 MeV1910 MeV
FIG. 18: The photon asymmetries Σ of γp → π0p. The red solid curves are the current results
while the blue dashed curves are from our previous analysis.39
2. γp → π+n
0
16dσ
/dΩ
(µb
/sr)
0
16
dσ/d
Ω (
µb/s
r)
0
16
dσ/d
Ω (
µb/s
r)
0
16
dσ/d
Ω (
µb/s
r)
0
16
dσ/d
Ω (
µb/s
r)
0 90 180θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
0 90θ (deg.)
0
16
dσ/d
Ω (
µb/s
r)
0 90 180θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
1104 MeV 1112 MeV 1121 MeV 1129 MeV 1137 MeV 1145 MeV
1351 MeV
1240 MeV1225 MeV1209 MeV1194 MeV1178 MeV
1284 MeV 1299 MeV 1313 MeV 1337 MeV
1405 MeV1378 MeV
1491 MeV
1323 MeV 1364 MeV
1392 MeV
1162 MeV
1255 MeV 1270 MeV
1418 MeV 1432 MeV 1445 MeV 1458 MeV
1483 MeV1470 MeV 1497 MeV 1508 MeV 1513 MeV 1521 MeV
1558 MeV1543 MeV1537 MeV 1575 MeV 1588 MeV1533 MeV1528 MeV
0
12
dσ/d
Ω (
µb/s
r)
0
6
dσ/d
Ω (
µb/s
r)
0 90θ (deg.)
0
2
dσ/d
Ω (
µb/s
r)
0 90θ (deg.)
90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
1603 MeV 1617 MeV 1646 MeV 1674 MeV 1702 MeV 1730 MeV
2075 MeV
1885 MeV1860 MeV1835 MeV1809 MeV1783 MeV
1959 MeV 1982 MeV 2006 MeV 2052 MeV
0
2029 MeV 2097 MeV
1756 MeV
1910 MeV 1934 MeV
FIG. 19: Differential cross sections of γp → π+n. The red solid curves are the current results while
the blue dashed curves are from our previous analysis.
40
-1
0
1
Σ
-1
0
1Σ
-1
0
1
Σ
-1
0
1
Σ
-1
0
1
Σ
0 90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
0 90θ (deg.)
-1
0
1
Σ
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
1131 MeV 1140 MeV 1149 MeV 1157 MeV 1165 MeV 1173 MeV
1262 MeV
1201 MeV1194 MeV1190 MeV1186 MeV1182 MeV
1225 MeV 1232 MeV 1240 MeV 1255 MeV
1292 MeV1277 MeV
1416 MeV
1247 MeV 1270 MeV
1284 MeV
1178 MeV
1209 MeV 1218 MeV
1313 MeV 1335 MeV 1349 MeV 1362 MeV
1403 MeV1383 MeV 1430 MeV 1449 MeV 1468 MeV 1474 MeV
1543 MeV1520 MeV1513 MeV 1572 MeV 1594 MeV1504 MeV1481 MeV
-1
0
1
Σ
90 180θ (deg.)
0 90θ (deg.)
-1
0
1
Σ
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
1603 MeV 1633 MeV 1660 MeV 1688 MeV 1716 MeV 1770 MeV
2041 MeV1994 MeV1947 MeV1898 MeV1873 MeV
1822 MeV
2086 MeV
0
FIG. 20: The photon asymmetries Σ of γp → π+n. The red solid curves are the current results
while the blue dashed curves are from our previous analysis.
41
3. γp → ηp
0
2dσ
/dΩ
(µb
/sr)
0
2
dσ/d
Ω (
µb/s
r)
0
2
dσ/d
Ω (
µb/s
r)
0
1.2
dσ/d
Ω (
µb/s
r)
0
0.8
dσ/d
Ω (
µb/s
r)
0 90 180θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
0 90θ (deg.)
0
0.5
dσ/d
Ω (
µb/s
r)
0 90 180θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
1490 MeV 1491 MeV 1493 MeV 1496 MeV 1499 MeV 1501 MeV
1573 MeV
1528 MeV1524 MeV1517 MeV1513 MeV1509 MeV
1540 MeV 1543 MeV 1553 MeV 1564 MeV
1597 MeV1588 MeV
1641 MeV
1558 MeV 1583 MeV
1593 MeV
1506 MeV
1534 MeV 1537 MeV
1603 MeV 1613 MeV 1617 MeV 1623 MeV
1636 MeV1632 MeV 1646 MeV 1649 MeV 1659 MeV 1662 MeV
1694 MeV1688 MeV1685 MeV 1702 MeV 1705 MeV1675 MeV1668 MeV
0
0.5
dσ/d
Ω (
µb/s
r)
0
0.5
dσ/d
Ω (
µb/s
r)
90 180θ (deg.)
0 90θ (deg.)
0
0.5
dσ/d
Ω (
µb/s
r)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
0 90θ (deg.)
1710 MeV 1716 MeV 1721 MeV 1729 MeV 1739 MeV 1756 MeV
2098 MeV
1911 MeV1886 MeV1860 MeV1835 MeV1810 MeV
1983 MeV 2005 MeV 2028 MeV 2076 MeV 02053 MeV
1783 MeV
1935 MeV 1959 MeV
FIG. 21: Differential cross sections of γp → ηp. The red solid curves are the current results while
the blue dashed curves are from our previous analysis.
42
-1
0
1
Σ
0 90θ (deg.)
-1
0
1Σ
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
1508 MeV 1547 MeV 1585 MeV 1622 MeV 1657 MeV 1689 MeV
1841 MeV1813 MeV1786 MeV1758 MeV1725 MeV
1692 MeV
1867 MeV 1896 MeV
-1
0
1
T
90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
0 90θ (deg.)
-1
0
1
T
0 90θ (deg.)
0 90 180θ (deg.)
1492 MeV 1505 MeV 1522 MeV 1537 MeV 1555 MeV 1578 MeV
1719 MeV1673 MeV1630 MeV
1600 MeV
0
FIG. 22: Σ and T of γp → ηp. The red solid curves are the current results while the blue dashed
curves are from our previous analysis.
43
4. γp → K+Λ
0
0.5dσ
/dΩ
(µb
/sr)
0
0.5
dσ/d
Ω (
µb/s
r)
0
0.5
dσ/d
Ω (
µb/s
r)
0
0.5
dσ/d
Ω (
µb/s
r)
0
0.5
dσ/d
Ω (
µb/s
r)
0 90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
0 90θ (deg.)
0
0.5
dσ/d
Ω (
µb/s
r)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
1625 MeV 1635 MeV 1645 MeV 1655 MeV 1665 MeV 1675 MeV
1815 MeV
1735 MeV1725 MeV1715 MeV1705 MeV1695 MeV
1765 MeV 1775 MeV 1785 MeV 1805 MeV
1855 MeV1835 MeV
1925 MeV
1795 MeV 1825 MeV
1845 MeV
1685 MeV
1745 MeV 1755 MeV
1865 MeV 1875 MeV 1885 MeV 1895 MeV
1915 MeV1905 MeV 1935 MeV 1945 MeV 1965 MeV 1975 MeV
2005 MeV1995 MeV1985 MeV 2015 MeV 2025 MeV 2035 MeV 2055 MeV
FIG. 23: Differential cross sections of γp → K+Λ. The red solid curves are the current results
while the blue dashed curves are from our previous analysis.
44
-1
0
1
P
-1
0
1P
-1
0
1
P
90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
0 90 180θ (deg.)
-1
0
1
P
1625 MeV 1635 MeV 1645 MeV 1649 MeV 1655 MeV 1660 MeV
1715 MeV1705 MeV1695 MeV1685 MeV1675 MeV
1665 MeV
1725 MeV 1735 MeV
1743 MeV 1848 MeV 1898 MeV
1947 MeV
0
1796 MeV
2086 MeV
2041 MeV1994 MeV
-1
0
1
Σ
-1
0
1
Σ
90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
0 90θ (deg.)
-1
0
1
Σ
0 90 180θ (deg.)
1649 MeV 1676 MeV 1702 MeV 1728 MeV 1754 MeV 1781 MeV
1883 MeV1859 MeV1833 MeV
1808 MeV
1906 MeV 1947 MeV
2109 MeV2086 MeV
1994 MeV 2041 MeV
0
-1
0
1
T
90θ (deg.)
90θ (deg.)
0 90 180θ (deg.)
0 90θ (deg.)
-1
0
1
T
0 90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
1649 MeV 1676 MeV 1702 MeV 1728 MeV 1754 MeV 1781 MeV
1883 MeV1859 MeV1833 MeV
1808 MeV
1906 MeV 00
FIG. 24: Observables P , Σ, T of of γp → K+Λ. The red solid curves are the current results while
the blue dashed curves are from our previous analysis.
45
5. γp → KΣ
0
0.1dσ
/dΩ
(µb
/sr)
90 180θ (deg.)
0 90θ (deg.)
0
0.1
dσ/d
Ω (
µb/s
r)
0 90θ (deg.)
0 90θ (deg.)
90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
1716 MeV 1770 MeV 1796 MeV 1822 MeV 1873 MeV 1898 MeV
2109 MeV2086 MeV2064 MeV2018 MeV1994 MeV1970 MeV 0
0
1922 MeV
FIG. 25: Differential cross sections of γp → K0Σ+. The red solid curves are the current results
while the blue dashed curves are from our previous analysis.
46
0
0.2
dσ/d
Ω (
µb/s
r)
0
0.4
dσ/d
Ω (
µb/s
r)
0
0.4
dσ/d
Ω (
µb/s
r)
0
0.4
dσ/d
Ω (
µb/s
r)
0
0.4
dσ/d
Ω (
µb/s
r)
0 90θ (deg.)
0 90θ (deg.)
0 90 180θ (deg.)
0 90θ (deg.)
0
0.4
dσ/d
Ω (
µb/s
r)
0 90θ (deg.)
0 90θ (deg.)
0 90θ (deg.)
1695 MeV 1705 MeV 1715 MeV 1725 MeV 1735 MeV 1745 MeV
1885 MeV
1805 MeV1795 MeV1785 MeV1775 MeV1765 MeV
1835 MeV 1845 MeV 1855 MeV 1875 MeV
1925 MeV1905 MeV
2005 MeV
1865 MeV 1895 MeV
1915 MeV
1755 MeV
1815 MeV 1825 MeV
1935 MeV 1945 MeV 1965 MeV 1975 MeV
1995 MeV1985 MeV 2015 MeV 2025 MeV 2035 MeV 2045 MeV
2075 MeV2065 MeV2055 MeV 2085 MeV 2095 MeV 2105 MeV 2115 MeV
FIG. 26: Differential cross sections of γp → K+Σ0. The red solid curves are the current results
while the blue dashed curves are from our previous analysis.
47
Appendix E: Differential Cross sections of γ∗p → πN reactions
1. γ∗p → π0p
0
10
20
301100 1120 1140 1160 1180
1200
0
10
20
1220 1240
1260 1280 1300 1320
0
4
1340 1360 1380 1400
1420 1440
0
4 1460
1480 1500 1520
1540 1560
0
4
-1 0 1cos θπ*
1580
0 1cos θπ*
1600
0 1cos θπ*
1620
0 1cos θπ*
1640
0 1cos θπ*
1660
0 1cos θπ*
1680
FIG. 27: Differential cross section (σT + ǫσL) for γ∗p → π0p at Q2 = 0.4 (GeV/c)2.
48
0
1
2
3
1100 1120 1140 1160 1180
1200
0
1
2
1220 1240
1260 1280 1300 1320
0
1
21340
0 1
cos θπ*
1360
0 1
cos θπ*
1380
0 1
cos θπ*
1400
0 1
cos θπ*
1420
0 1
cos θπ*
1440
0
1
2
-1 0 1
cos θπ*
1460
FIG. 28: Differential cross section (σT + ǫσL) for γ∗p → π0p at Q2 = 1.76 (GeV/c)2.
0
0.5
1
1100 1130 1150 1170 1190
1210
0
0.5
1230
1250 1270
0 1
cos θπ*
1290
0 1
cos θπ*
1310
0 1
cos θπ*
1330
0
0.5
-1 0 1
cos θπ*
1350
0 1
cos θπ*
1370
0 1
cos θπ*
1390
FIG. 29: Differential cross section (σT + ǫσL) for γ∗p → π0p at Q2 = 3.00 (GeV/c)2.
49
2. γ∗p → π+n
0
10
201110 1130 1150 1170 1190
1210
0
10
1230 1250 1270 1290 1310 1330
0
5
101350 1370 1390 1410 1430
0 1cos θπ*
1450
0
5
10
-1 0 1cos θπ*
1470
0 1cos θπ*
1490
0 1cos θπ*
1510
0 1cos θπ*
1530
0 1cos θπ*
1550
FIG. 30: Differential cross section (σT + ǫσL) for γ∗p → π+n at Q2 = 0.4 (GeV/c)2.
50
0
1
2
3
1150 1170 1190
1210 1230 1250
0
1
21270 1290 1310 1330 1350 1370
0
1
21390 1410 1430 1450
1470 1490
0
1
2
1510 1530
1550 1570 1590 1610
0
1
21620 1630 1650 1660 1670 1700
0
1
2
-1 0 1
cos θπ*
1740
0 1
cos θπ*
1780
0 1
cos θπ*
1830
0 1
cos θπ*
1890
0 1
cos θπ*
1950
0 1
cos θπ*
2010
FIG. 31: Differential cross section (σT + ǫσL) for γ∗p → π+n at Q2 = 1.76 (GeV/c)2.
51
0
0.5
1
1150 1170 1190
1210 1230 1250
0
0.5
1270 1290 1310 1330 1350 1370
0
0.5
1 1390 1410 1430
1450 1470 1490
0
0.5
1
1510 1530 1550
0 1
cos θπ*
1570
0 1
cos θπ*
1590
0 1
cos θπ*
1610
0
0.5
-1 0 1
cos θπ*
1630
0 1
cos θπ*
1650
0 1
cos θπ*
1670
FIG. 32: Differential cross section (σT + ǫσL) for γ∗p → π+n at Q2 = 2.91 (GeV/c)2.
52
Appendix F: Inclusive cross section of p(e, e′)
0
1
2
3
1.2 1.4 1.6 1.8 2
Q2=0.21-0.40 (GeV/c)2
d σ/
d Ω
d E
’ (µb
/sr
GeV
)
W (GeV)
Ee=3.118 GeVθe’=12.45o
0
0.1
0.2
1.2 1.4 1.6 1.8 2
Ee=5.498 GeVd σ/
d Ω
d E
’ (µb
/sr
GeV
)
W (GeV)
θe’=12.97o
Q2=0.96-1.3 (GeV/c)2
0
0.01
0.02
1.2 1.4 1.6 1.8 2
Ee=5.498 GeVd σ/
d Ω
d E
’ (µb
/sr
GeV
)
W (GeV)
θe’=20.45o
Q2=1.98-2.69 (GeV/c)2
FIG. 33: Differential cross sections of p(e, e′)X at Q2 ∼ 0.3, 1.1, 2.3 (GeV/c)2. Dashed curves are
from the contributions of p(e, e′π)N .
53
[1] Meson-exchange model of πN scattering and γN → πN reaction
T. Sato, T.-S. H. Lee, Phys. Rev. C 34, 2660 (1996)
[2] Dynamical study of ∆ excitation in N(e, e′π) reaction
T. Sato, T.-S. H. Lee, Phys. Rev. C 63, 055201 (2001)
[3] Extraction and Interpretation ofγN → ∆ form Factors within a Dynamical Model
B. Julia-Diaz, T.-S.H. Lee, T. Sato, L.C. Smith, Phys. Rev. C 75, 015205 (2007)
[4] Dynamical coupled-channel model of meson production reaction in the nucleon resonance
region
A. Matsuyama, T. Sato, T.-S. H. Lee, Phys. Rept 439, 193 (2007)
[5] Dynamical coupled-channel model of πN scattering in the W < 2 GeV nucleon resonance
region
B. Julia-Diaz, T.-S. H. Lee, A. Matsuyama, T. Sato, Phys. Rev. C 76, 065201 (2007)
[6] Dynamical coupled-channel effects on pion photoproduction
B. Julia-Diaz, T.-S. H. Lee, A. Matsuyama, T. Sato, L.C. Smith, Phys. Rev. C 77, 045201
(2008)
[7] Dynamical coupled-channel study of π−p → ηn process
J. Durand, B. Julia-Diaz, T.-S. H. Lee, B. Saghai, T. Sato, Phys. Rev. C 78, 025204 (2008)
[8] Extraction of resonances from meson-nucleon reaction
N. Suzuki, T. Sato, T.-S. H. Lee, Phys. Rev. C 79, 025205 (2009)
[9] Dynamical coupled-channels study of πN → ππN reactions
H. Kamano, B. Julia-Diaz, T.-S. H. Lee, A. Matsuyama, T. Sato, Phys. Rev. C 79, 025206
(2009)
[10] Dynamical coupled-channels analysis of 1H(e, e′π)N reactions
B. Julia-Diaz, H. Kamano, T.-S. H. Lee, A. Matsuyama, T. Sato, N. Suzuki Phys. Rev. C 80,
025207 (2009)
[11] Double and single pion photoproduction within a dynamical coupled-channels model
H. Kamano, B. Julia-Diaz, T.-S.H. Lee, A. Matsuyama, T. Sato, Phys. Rev. C 80, 065203
(2009)
[12] Disentangling the dynamical origin of P11 nucleon resonances
N. Suzuki, B. Julia-Diaz, H. Kamano, T.-S. H. Lee, A. Matsuyama, T. Sato, Phys. Rev. Lett.
104, 042302 (2010)
[13] Extraction of P11 resonances from πN data
H. Kamano, S.X. Nakamura, T.-S.H. Lee, T. Sato, Phys. Rev. C 81, 065207 (2010)
[14] Extraction of electromagnetic transition form factors for nucleon resonances within a dynam-
ical coupled-channels model
N. Suzuki, T. Sato, T.-S.H. Lee, Phys. Rev. C 82, 045206 (2010)
[15] Determining pseudoscalar meson photo-production amplitudes from complete experiments
M. Sandorfi, S. Hoblit, H. Kamano, T.-S. H. Lee, J. Phys. G 38, 053001 (2011)
[16] Nucleon resonances within a dynamical coupled-channel model of πN and γN reactions
H. Kamano, S.X. Nakamura, T.-S.H. Lee, T. Sato, Phys. Rev. C 88, 035209 (2013)
[17] Isospin decomposition of γ∗N → N∗ transitions within a dynamical coupled-channels model
H. Kamano, S.X. Nakamura, T.S.H. Lee, T. Sato, Phys. Rev. C 94, 015201 (2016)
[18] Dynamical coupled-channels model for neutrino-induced meson productions in resonance re-
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S.X. Nakamura, H. Kamano, T. Sato, Phys. Rev. D 92, 074024 (2015)
55