Dynamical Coupled-Channel Model of $\pi N$ Scattering in the W $\leq$ 2
  GeV Nucleon Resonance Region
Dynamical Coupled-Channel Model of πN Scattering
in the W ≤ 2 GeV Nucleon Resonance Region∗
(From EBAC, Thomas Jefferson National Accelerator Facility)
B. Juliá-Dı́az,1, 2 T.-S. H. Lee,1, 3 A. Matsuyama,1, 4 and T. Sato1, 5
1 Excited Baryon Analysis Center (EBAC),
Thomas Jefferson National Accelerator Facility, Newport News, VA 22901, USA
2Departament d’Estructura i Constituents de la Matèria,
Universitat de Barcelona, E–08028 Barcelona, Spain
3Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA
4Department of Physics, Shizuoka University, Shizuoka 422-8529, Japan
5Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
Abstract
As a first step to analyze the electromagnetic meson production reactions in the nucleon reso-
nance region, the parameters of the hadronic interactions of a dynamical coupled-channel model,
developed in Physics Reports 439, 193 (2007), are determined by fitting the πN scattering data.
The channels included in the calculations are πN , ηN and ππN which has π∆, ρN , and σN reso-
nant components. The non-resonant meson-baryon interactions of the model are derived from a set
of Lagrangians by using a unitary transformation method. One or two bare excited nucleon states
in each of S, P , D, and F partial waves are included to generate the resonant amplitudes in the
fits. The parameters of the model are first determined by fitting as much as possible the empirical
πN elastic scattering amplitudes of SAID up to 2 GeV. We then refine and confirm the resulting
parameters by directly comparing the predicted differential cross section and target polarization
asymmetry with the original data of the elastic π±p → π±p and charge-exchange π−p → π0n pro-
cesses. The predicted total cross sections of πN reactions and πN → ηN reactions are also in good
agreement with the data. Applications of the constructed model in analyzing the electromagnetic
meson production data as well as the future developments are discussed.
PACS numbers: 13.75.Gx, 13.60.Le, 14.20.Gk
∗ Notice: Authored by Jefferson Science Associates, LLC under U.S. DOE Contract No. DE-AC05-
06OR23177. The U.S. Government retains a non-exclusive, paid-up, irrevocable, world-wide license to
publish or reproduce this manuscript for U.S. Government purposes.
http://arxiv.org/abs/0704.1615v2
I. INTRODUCTION
It is now well recognized that a coupled-channel approach is needed to extract the nucleon
resonance (N∗) parameters from the data of πN and electromagnetic meson production re-
actions. With the recent experimental developments [1, 2], such a theoretical effort is needed
to analyze the very extensive data from Jefferson Laboratory (JLab), Mainz, Bonn, GRAAL,
and Spring-8. To cope with this challenge, a dynamical coupled-channel model (MSL) for
meson-baryon reactions in the nucleon resonance region has been developed recently [3]. In
this paper we report a first-stage determination of the parameters of this model by fitting
the πN scattering data up to invariant mass W = 2 GeV.
The details of the MSL model are given in Ref. [3]. Here we will only briefly recall its
essential features. Similar to the earlier works on meson-exchange models [4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] of pion-nucleon scattering, the
starting point of the MSL model is a set of Lagrangians describing the interactions between
mesons (M =γ, π, η , ρ, ω, σ, . . .) and baryons (B = N,∆, N∗, . . .). By applying a unitary
transformation method [13, 27], an effective Hamiltonian is then derived from the considered
Lagrangian. It can be cast into the following more transparent form
Heff = H0 + ΓV + v22 + hππN , (1)
where H0 =
m2α + ~p
α with mα denoting the mass of particle α, and
ΓV = {
ΓN∗→MB) +
hM∗→ππ}+ {c.c.} , (2)
v22 =
MB,M ′B′
vMB,M ′B′ + vππ , (3)
hππN =
ΓN∗→ππN +
[(vMB,ππN) + (c.c.)] + vππN,ππN . (4)
Here c.c. denotes the complex conjugate of the terms on its left-hand-side. In the above
equations, MB = γN, πN , ηN, π∆, ρN, σN , represent the considered meson-baryon states.
The resonance associated with the bare baryon state N∗ is induced by the vertex interactions
ΓN∗→MB and ΓN∗→ππN . Similarly, the bare meson states M
∗ = ρ, σ can develop into reso-
nances through the vertex interaction hM∗→ππ. Note that the masses M
N∗ and m
M∗ of the
bare states N∗ and M∗ are the parameters of the model which must be determined by fit-
ting the πN and ππ scattering data. They differ from the empirically determined resonance
positions by mass shifts which are due to the coupling of the bare states to the scattering
states. The term v22 contains the non-resonant meson-baryon interaction vMB,M ′B′ and ππ
interaction vππ. The non-resonant interactions involving ππN states are in hππN . All of
these interactions are energy independent, an important feature of the MSL formulation.
We note here that the Hamiltonian defined above does not have a πN ↔ N vertex. By
applying the unitary transformation method, this un-physical process as well as any vertex
interaction A ↔ B + C with a mass relation mA < mB + mC are eliminated from the
considered Hilbert space and their effects are absorbed in the effective interactions v22 and
hππN . This procedure defines the Hamiltonian in terms of physical nucleons and greatly
simplifies the formulation of a unitary reaction model. In particular, the complications
due to the nucleon mass and wavefunction renormalizations do not appear in the resulting
scattering equations. This makes the numerical calculations involving the ππN channel
much more tractable in practice. The details of this approach are discussed in Refs. [13, 27]
as well as in the earlier works on πNN interactions [28].
Starting from the above Hamiltonian, the coupled-channel equations for πN and γN
reactions are then derived by using the standard projection operator technique [29], as given
explicitly in Ref. [3]. The obtained scattering equations satisfy the two-body (πN, ηN , γN)
and three-body (ππN) unitarity conditions. The π∆, ρN and σN resonant components
of the ππN continuum are generated dynamically by the vertex interaction ΓV of Eq. (2).
Accordingly, the ππN cuts are treated more rigorously than the commonly used quasi-
particle formulation within which these resonant channels are treated as simple two-particle
states with a phenomenological parametrization of their widths. The importance of such a
dynamical treatment of unstable particle channels was well known in earlier studies of πN
scattering [4, 30] and πNN reactions [31].
A complete determination of the parameters of the model Hamiltonian defined by Eqs.(1)-
(4) requires good fits to all of the data of πN and γN reactions up to invariant mass W ≤
about 2 GeV. Obviously, this is a very complex task and can only be accomplished step
by step. Our strategy is as follows. We need to first determine the parameters associated
with the hadronic interaction parts of the Hamiltonian. With the fits to ππ phase shifts in
Ref. [32], the ππ interactions hρ,ππ and hσ,ππ and the corresponding bare masses for ρ and
σ have been determined in an isobar model with vππ = 0. We next proceed in two stages.
The first-stage is to determine the ranges of the parameters of the interactions ΓN∗→MB and
vMB,M ′B′ . This will be achieved by fitting the πN scattering data from performing coupled-
channel calculations which neglect the more complex three-body interaction term hππN . This
simplification greatly reduces the numerical complexity and the number of parameters to be
determined in the fits. This first-stage fit will provide the starting parameters to fit both
the data of πN scattering and πN → ππN reactions. In this second-stage, the parameters
associated with ΓN∗→MB and vMB,M ′B′ will be refined and the parameters of hππN are then
determined. The dynamical coupled-channel calculations for such more extensive fits are
numerically more complex, as explained in Ref. [3].
In this work we report on the results from our first-stage determination of the parameters
of ΓN∗→MB and vMB,M ′B′ of Eqs.(2)-(3) withMB,M
′B′ = πN, ηN, π∆, ρN, σN . We proceed
in two steps. We first locate the range of the model parameters by fitting as much as possible
the empirical πN elastic scattering amplitudes up to W = 2 GeV of SAID [33]. We then
refine and confirm the resulting parameters by directly comparing our predictions with
the original πN scattering data. Our procedures are similar to what have been used in
determining the nucleon-nucleon (NN) potentials [34] from fitting NN scattering data.
The constructed model can describe well almost all of the empirical πN amplitudes in
S, P , D, and F partial waves of SAID [33]. We then show that the predicted differential
cross sections and target polarization asymmetry are in good agreement with the original
data of elastic π±p → π±p and charge-exchange π−p → π0n processes. Furthermore the
predicted total cross sections of the πN reactions and πN → ηN reactions agree well with
the data. Thus the constructed model is at least comparable to, if not better than, all of the
recent πN models [11, 12, 13, 19, 20, 22, 23, 24, 26]. It can be used to perform a first-stage
extraction of the γN → N∗ parameters by analyzing the photo- and electro-production of
single π meson. It has also provided us with a starting point for performing the second-stage
determination of the model parameters by also fitting the data of πN → ππN reactions.
Our efforts in these directions are in progress and will be reported elsewhere.
In Section II, we recall the coupled-channel equations presented in Ref. [3]. The calcula-
MB,M’B’
t MB,M’B’ t MB,M’B’
t MB,M’B’
vMB,M’B’
FIG. 1: Graphical representation of Eqs.(5)-(21).
tions performed in this work are described in Section III. The fitting procedure is described
in Section IV and the results are presented in Section V. In Section VI we give a summary
and discuss future developments.
II. DYNAMICAL COUPLED-CHANNEL EQUATIONS
With the simplification that ππN interaction hππN of Eq. (4) is set to zero, the meson-
baryon (MB) scattering equations derived in Ref. [3] are illustrated in Fig. 1. Explicitly,
they are defined by the following equations
TMB,M ′B′(E) = tMB,M ′B′(E) + t
MB,M ′B′(E) , (5)
where MB = πN, ηN, π∆, ρN, σN . The full amplitudes TπN,πN(E) can be directly used to
calculate πN scattering observables. The non-resonant amplitude tMB,M ′B′(E) in Eq. (5) is
defined by the coupled-channel equations,
tMB,M ′B′(E) = VMB,M ′B′(E) +
M ′′B′′
VMB,M ′′B′′(E) GM ′′B′′(E) tM ′′B′′,M ′B′(E) (6)
VMB,M ′B′(E) = vMB,M ′B′ + Z
MB,M ′B′(E) . (7)
Here the interactions vMB,M ′B′ are derived from the tree-diagrams illustrated in Fig. 2 by
using a unitary transformation method [13, 27]. It is energy independent and free of singu-
larity. On the other hand, Z
MB,M ′B′(E) is induced by the decays of the unstable particles
v v v vs u t c
FIG. 2: Mechanisms for vMB,M ′B′ of Eq. (7): v
s direct s-channel, vu crossed u-channel, vt one-
particle-exchange t-channel, vc contact interactions.
MB,M’B’
FIG. 3: One-particle-exchange interactions Z
π∆,π∆(E), Z
ρN,π∆ and Z
σN,π∆ of Eq. (7).
(∆, ρ, σ) and thus contains moving singularities due to the ππN cuts, as illustrated in
Fig.3. Here we note that if the ππN interaction term hππN of Eq.(4) is included, the driving
term Eq. (7) will have an additional term Z
MB,M ′B′(E) which involves a 3-3 ππN amplitude
tππN,ππN , as given in Ref. [3], and hence is much more difficult to calculate. As explained in
Section I, we neglect this term in this first-stage fit to the πN scattering data.
The second term in the right-hand-side of Eq. (5) is the resonant term defined by
tRMB,M ′B′(E) =
Γ̄MB→N∗
(E)[D(E)]i,jΓ̄N∗
→M ′B′(E) , (8)
[D−1(E)]i,j = (E −M0N∗
)δi,j − Σ̄i,j(E) , (9)
where M0N∗ is the bare mass of the resonant state N
∗, and the self-energies are
Σ̄i,j(E) =
→MBGMB(E)Γ̄MB→N∗
(E) . (10)
The dressed vertex interactions in Eq. (8) and Eq. (10) are (defining ΓMB→N∗ = Γ
N∗→MB)
Γ̄MB→N∗(E) = ΓMB→N∗ +
M ′B′
tMB,M ′B′(E)GM ′B′(E)ΓM ′B′→N∗ , (11)
Γ̄N∗→MB(E) = ΓN∗→MB +
M ′B′
ΓN∗→M ′B′GM ′B′(E)tM ′B′,MB(E) . (12)
It is useful to mention here that if there is only one N∗ in the considered partial wave, the
resonant amplitude (Eq. (8)) can be written as
tRMB,M ′B′(E) =
Γ̄MB→N∗
(E)Γ̄N∗
→M ′B′(E)
E − ER(E) + iΓR(E)2
ER(E) = M
N∗ + Re[Σ̄(E)] , (14)
ΓR(E) = −2 Im[Σ̄(E)] , (15)
where,
Σ̄(E) =
ΓN∗→MBGMB(E){
M ′B′
[δMB,M ′B′ + tMB,M ′B′(E)GM ′B′(E)]}ΓM ′B′→N∗(E) .
The form Eq. (13) is similar to the commonly used Breit-Wigner form, but the resonance
position ER(E) and width ΓR(E) are determined by the N
∗ → MB vertex and the non-
resonant amplitude tMB,M ′B′ . This is the consequence of the unitarity condition and is
an important and well known feature of a dynamical approach. Namely, the resonance
amplitude necessarily includes the non-resonant mechanisms. This feature is consistent
with the well developed formal reaction theory [29]. Eq. (16) indicates that it is essential
to understand the non-resonant mechanisms in extracting the bare vertex functions ΓN∗,MB
which contain the information for exploring the N∗ structure. The parameterization used
for ΓN∗,MB will be explained in Section III. We also note here that the energy dependence of
ER(E) and ΓR(E), defined by Eqs (14)-(15), is essential in determining the resonance poles
in the complex E-plane.
The meson-baryon propagators GMB in the above equations are
GMB(k, E) =
E −EM(k)−EB(k) + iǫ
for the stable particle channels MB = πN, ηN , and
GMB(k, E) =
E − EM(k)− EB(k)− ΣMB(k, E)
for the unstable particle channels MB = π∆, ρN, σN . The self-energies [36] in Eq. (18) are
Σπ∆(k, E) =
E∆(k)
MπN(q)
[M2πN(q) + k
2]1/2
|f∆,πN(q)|2
E − Eπ(k)− [(EN(q) + Eπ(q))2 + k2]1/2 + iǫ
ΣρN (k, E) =
Eρ(k)
Mππ(q)
[M2ππ(q) + k
2]1/2
|fρ,ππ(q)|2
E − EN(k)− [(2Eπ(q))2 + k2]1/2 + iǫ
, (20)
ΣσN (k, E) =
Eσ(k)
Mππ(q)
[M2ππ(q) + k
2]1/2
|fσ,ππ(q)|2
E −EN (k)− [(2Eπ(q))2 + k2]1/2 + iǫ
, (21)
where MπN(q) = Eπ(q) + EN (q) and Mππ(q) = 2Eπ(q). The vertex function f∆,πN(q) is
taken from Ref. [13], fρ,ππ(q) and fσ,ππ(q) are from the isobar fits [32] to the ππ phase shifts.
They are also given explicitly in [3].
Here we note that the driving term Z
MB,M ′B′ of Eq. (7) is also determined by the same
vertex functions f∆,πN(q), fρ,ππ(q) and fρ,ππ(q) of Eqs. (19)-(21). This consistency is essential
for the solutions of Eq. (6) to satisfy the unitarity condition.
III. CALCULATIONS
We solve the coupled-channel equations defined by Eqs.(5)-(21) in the partial-wave repre-
sentation. The input of these equations are the partial-wave matrix elements of ΓN∗→MB and
vMB,M ′B′ of Eqs.(2)-(3), with MB,M
′B′ = πN, ηN , π∆, ρN, σN , and Z
MB,M ′B′ of Eq. (7)
with MB,M ′B′ = π∆, ρN, σN . The calculations of these matrix elements have been given
explicitly in the appendices of Ref. [3]. Here we only mention a few points which are needed
for later discussions.
In deriving the non-resonant interactions vMB,M ′B′ of Eq. (7) we consider the tree-
diagrams (Fig. 2) generated from a set of Lagrangians with π, η, σ, ρ, ω, N , and ∆ fields.
The higher mass mesons, such as a0, a1 included in other meson-exchange πN models,
such as the Jülich model [19], are not considered. The employed Lagrangians are ( in the
convention of Bjorken and Drell [37])
LπNN = −
ψ̄Nγµγ5~τψN · ∂µ ~φπ , (22)
LπN∆ = −
~TψN · ∂µ ~φπ , (23)
Lπ∆∆ =
ψ̄∆µγ
νγ5 ~T∆ψ
∆ · ∂ν~φπ , (24)
LηNN = −
ψ̄Nγµγ5ψN∂
µφη . (25)
LρNN = gρNN ψ̄N [γµ −
ν ] ~ρµ ·
ψN , (26)
LρN∆ = −i
νγ5 ~T · [∂µ ~ρν − ∂ν ~ρµ]ψN + [h.c.] , (27)
Lρ∆∆ = gρ∆∆ψ̄∆α[γ
µ − κρ∆∆
σµν∂ν ] ~ρµ · ~T∆ψα∆ , (28)
Lρππ = gρππ[ ~φπ × ∂µ ~φπ] · ~ρµ , (29)
LNNρπ =
gρNN ψ̄Nγµγ5~τψN · ~ρµ × ~φπ , (30)
LNNρρ = −
µν~τψN · ~ρµ × ~ρν . (31)
LωNN = gωNN ψ̄N [γµ −
ν ]ωµψN , (32)
Lωπρ = −
ǫµαλν∂
α ~ρµ∂λ ~φπω
ν , (33)
LσNN = gσNN ψ̄NψNφσ (34)
Lσππ = −
∂µ~φπ∂µ~φπφσ . (35)
To solve the coupled-channel equations, Eq. (6), we need to regularize the matrix elements
of vMB,M ′B′ , illustrated in Fig. 2. Here we follow Ref. [13] in order to use the parameters
determined in the ∆ (1232) region as the starting parameters in our fits. For the vs and
vu terms of Fig. 2, we include at each meson-baryon-baryon vertex a form factor of the
following form
F (~k,Λ) = [~k2/[(~k2 + Λ2)]2 (36)
with ~k being the meson momentum. For the meson-meson-meson vertex of vt of Fig. 2, the
form Eq. (36) is also used with ~k being the momentum of the exchanged meson. For the
contact term vc, we regularize it by F (~k,Λ)F (~k′,Λ′).
With the non-resonant amplitudes generated from solving Eq. (6), the resonant ampli-
tude tRMB,M ′B′ Eq. (8) then depends on the bare mass M
N∗ and the bare N
∗ → MB vertex
functions. As discussed in Ref. [3], these bare N∗ parameters can perhaps be taken from
a hadron structure calculation which does not include coupling with meson-baryon contin-
uum states or meson-exchange quark interactions. Unfortunately, such information is not
available to us. We thus use the following parameterization
ΓN∗,MB(LS)(k) =
(2π)3/2
CN∗,MB(LS)
Λ2N∗,MB(LS)
Λ2N∗,MB(LS) + (k − kR)2
(2+L/2) [
.(37)
where L and S are the orbital angular momentum and the total spin of the MB system,
respectively. The above parameterization accounts for the threshold kL dependence and the
right power (2 + L/2) such that the integration for calculating the dressed vertex Eq. (11)-
(12) is finite. Nevertheless as we will discuss in Section V this parameterization could be
too naive.
The partial-wave quantum numbers for the considered channels are listed in Table I. The
numerical methods for handling the moving singularities due to the ππN cuts in Z
MB,M ′B′
(Fig. 3) in solving Eq. (6) are explained in detail in Ref [3]. To get the πN elastic scattering
amplitudes, we can use either the method of contour rotation by solving the equations on
the complex momentum axis k = ke−iθ with θ > 0 or the Spline-function method developed
in Refs. [38, 39] and explained in detail in Ref. [3]. We perform the calculations using these
two very different methods and they agree within less than 1%. When Z
MB,M ′B′ is neglected,
Eq. (6) can be solved by the standard subtraction method since the resonant propagators,
Eqs. (18), for unstable particle channels π∆, ρN , and σN are free of singularity on the real
momentum axis. A code for this simplified case has also been developed to confirm the
results from using the other two methods.
The method of contour rotation becomes difficult at high W since the required rotation
angle θ is very small. The Spline function method has no such limitation and we can perform
calculations at W > 1.9 GeV without any difficulty. Typically, 24 and 32 mesh points
are needed to get convergent solutions of the coupled-channel integral equation (6). Such
mesh points are also needed to get stable integrations in evaluating the dressed resonance
quantities Eqs. (10)-(12).
(LS) of the considered partial waves
πN ηN π∆ σN ρN
S11 (0,
) (0, 1
) (2, 3
) (1, 1
) (0, 1
), (2, 3
S31 (0,
) − (2, 3
) − (0, 1
), (2, 3
P11 (1,
) (1, 1
) (1, 3
) (0, 1
) (1, 1
), (1, 3
P13 (1,
) (1, 1
) (1, 3
),(3, 3
) (2, 1
) (1, 1
),(1, 3
), (3, 3
P31 (1,
) − (1, 3
) − (1, 1
), (1, 3
P33 (1,
) − (1, 3
),(3, 3
) − (1, 1
),(1, 3
), (3, 3
D13 (2,
) (2, 1
) (0, 3
),(2, 3
) (1, 1
) (2, 1
), (0, 3
), (4, 3
D15 (2,
) (2, 1
) (2, 3
) , (4, 3
) (3, 1
) (2, 1
), (2, 3
), (4, 3
D33 (2,
) − (0, 3
),(2, 3
) − (2, 1
), (0, 3
), (2, 3
D35 (2,
) − (2, 3
), (4, 3
) − (2, 1
), (2, 3
), (4, 3
F15 (3,
) (3, 1
) (1, 3
),(3, 3
) (2, 1
) (3, 1
), (1, 3
), (3, 3
F17 (3,
) (3, 1
) (3, 3
),(5, 3
) (4, 1
) (3, 1
), (3, 3
), (5, 1
F35 (3,
) − (1, 3
),(3, 3
) − (3, 1
), (1, 3
), (3, 3
F37 (3,
) − (3, 3
),(5, 3
) - (3, 1
), (3, 3
), (5, 3
TABLE I: The orbital angular momentum (L) and total spin (S)of the partial waves included in
solving the coupled channel Equation (6).
IV. FITTING PROCEDURE
With the specifications given in Section III, the parameters associated with Z
MB,M ′B′ of
Eq. (7) are completely determined from fitting the ππ phase shifts in Refs. [13] and [32]. Thus
the considered model has the following parameters: (a) the coupling constants associated
with the Lagrangians listed in Eqs. (22)-(35), (b) the cutoff Λ for each vertex of vMB,M ′B′
(Fig. 2), (c) the coupling strength CN∗,MB(LS) and range kR and ΛN∗,MB(LS) of the bare
N∗ → MB vertex Eq. (37), and (d) the bare mass M0N∗ of each N∗ state. We determine
these by fitting the πN scattering data.
Our fitting procedure is as follows. We first perform fits to the πN scattering data up to
about 1.4 GeV and including only one bare state, the ∆ (1232) resonance. In these fits, the
starting coupling constant parameters of vMB,M ′B′ are taken from the previous studies of
πN and NN scattering, which are also given in Ref. [3]. Except the πNN coupling constant
fπNN all coupling constants and the cutoff parameters are allowed to vary in the χ
2-fit to
the πN data. The coupled-channel effects can shift the coupling constants greatly from their
starting values. We try to minimize these shifts by allowing the cutoff parameters to vary
in a very wide range 500 MeV < Λ < 2000 MeV. Some signs of coupling constants, which
could not be fixed by the previous works [40], are also allowed to change. We then use the
parameters from these fits at low energies as the starting ones to fit the amplitudes up to
2 GeV by also adjusting the resonance parameters, M0N∗ , CN∗,MB(LS), kR and ΛN∗,MB(LS).
Here we need to specify the number of bare N∗ states in each partial wave. The simplest
approach is to assume that each of 3-star and 4-star resonances listed by the Particle Data
Group [35] is generated from a bare N∗ state of the model Hamiltonian Eq. (1). However,
this choice is perhaps not well justified since the situation of the higher mass N∗’s is not so
clear.
We thus start the fits including only the bare states which generate the lowest and well-
established N∗ resonance in each partial wave. The second higher mass bare state is then
included when a good fit can not be achieved. We also impose the condition that if the
resulting M0N∗ is too high > 2.5 GeV, we remove such a bare state in the fit. This is due
to the consideration that the interactions due to such a heavy bare N∗ state could be just
the separable representation of some non-resonant mechanisms which should be included
in vMB,M ′B′ . In some partial waves the quality of the fits is not very sensitive to the N
couplings to π∆, ρN , and σN . But the freedom of varying these coupling parameters is
needed to achieve good fits.
It is rather difficult to fit all partial waves simultaneously because the number of resonance
parameters to be determined is very large. We proceed as follows. We first fit only 3 or
4 partial waves which have well established resonant states, and whose amplitudes have an
involved energy dependence. These are the S11, P11, S31 and P33 partial waves. These fits are
aimed at identifying the possible ranges of the parameters associated with vMB,M ′B′ . This
step is most difficult and time consuming. We then gradually extend the fits to include more
partial waves. For some cases, the fits can be reached easily by simply adjusting the bare
N∗ parameters. But it often requires some adjustments of the non-resonance parameters to
obtain new fits. This procedure has to be repeated many times to explore the parameter
space as much as we can. We carry out this very involved numerical task by using the fitting
code MINUIT and the parallel computation facilities at NERSC in US and the Barcelona
Supercomputing Center in Spain.
The most uncertain part of the fitting is to handle the large number of parameters asso-
ciated with the bare N∗ states. Here the use of the empirical partial-wave amplitudes from
SAID is an essential step in the fit. It allows us to locate the ranges of the N∗ parameters
partial-wave by partial-wave for a given set of the parameters for the non-resonant vMB,M ′B,.
Even with this, the information is far from complete for pinning down the N∗ parameters.
Perhaps the N∗ parameters associated with the πN state are reasonably well determined in
this fit to the πN scattering data. The parameters associated with ηN , π∆, ρN and σN
can only be better determined by also fitting to the data of πN → ηN and πN → ππN
reactions. This will be pursued in our second-stage calculations, as discussed in section I.
It is useful to note here that the leading-order effect due to Z(E) of the meson-baryon
interaction Eq. (7) on πN elastic scattering is
δvπN,πN =
MB,M ′B′=π∆,ρN,σN
vπN,MBGMB(E)Z
MB,M ′B′GM ′B′(E)vM ′B′,πN . (38)
We have found by explicit numerical calculations that δvπN,πN is much weaker than vπN,πN
and hence the coupled channel effects due to Z
MB,M ′B′ on πN elastic scattering amplitude
are weak. One example obtained from our model is shown in Table II. Thus we first
perform the fits without including Z(E) term to speed up the computation. We then refine
the parameters by including this term in the fits.
V. RESULTS
As mentioned in section I, we first locate the range of the parameters by fitting the
empirical πN scattering amplitude of SAID [33]. We then check and refine the resulting
parameters by directly comparing our predictions with the original πN scattering data.
Re[tπN,πN ] Re[tπN,πN (Z
(E) = 0)] Im[tπN,πN ] Im[tπN,πN(Z
(E) = 0)]
S11 −0.00481 −0.00557 0.0841 0.0827
P11 0.0937 0.103 0.636 0.640
P13 0.169 0.181 0.275 0.275
D13 0.202 0.194 0.299 0.309
D15 0.117 0.116 0.0179 0.0179
F15 0.290 0.291 0.157 0.155
F17 0.0360 0.0359 0.00293 0.00289
S31 −0.433 −0.437 0.496 0.504
P31 −0.253 −0.230 0.434 0.448
P33 0.0506 0.0306 0.510 0.457
D33 −0.00504 −0.0135 0.106 0.104
D35 0.0551 0.0551 0.0540 0.0537
F35 −0.0214 −0.0229 0.0259 0.0283
F37 0.0625 0.0626 0.00502 0.00512
TABLE II: The effect of Z
MB,M ′B′ on the πN scattering amplitudes tπN,πN from solving Eq. (6)
at W = 1.7 GeV. The normalization is tπN,πN = (e
2iδπN − 1)/(2i), where δπN is the πN scattering
phase shift which could be complex at energies above the π production threshold.
Our fits to the empirical amplitudes of SAID [33] are given in Figs. 4-5 and Figs. 6-7 for
the T = 1/2 and T = 3/2 partial waves, respectively. The resulting parameters are presented
in Appendix I. The parameters associated with the non-resonant interactions, vMB,M ′B′ with
MB,M ′B′ = πN, ηN , π∆, ρN, σN , are given in Table III for the coupling constants of the
starting Lagrangian Eqs.(22)-(35) and Table IV for the cutoffs of the form factors defined
by Eq. (36). The resulting bare N∗ parameters are listed in Tables V-VII
From Figs. 4-7, one can see that the empirical πN amplitudes can be fitted very well.
The most significant discrepancies are in the imaginary part of S31 in Fig.7. The agreement
is also poor for the F17 in Fig.4-5 and D35 in Figs.6-7, but there are rather large errors in
the data. Our parameters are therefore checked by directly comparing our predictions with
the data of differential cross sections dσ/dΩ and target polarization asymmetry P of elastic
π±p → π±p and charge-exchange π−p → π0n processes. Our results (solid red curves) are
shown in Figs.8-12. Clearly, our model is rather consistent with the available data, and are
close to the results (dashed blue curves) calculated from the SAID’s amplitudes. Thus our
model is justified despite the differences with the SAID’s amplitudes seen in Fig.4-7.
It will be important to further refine our parameters by fitting the data of other πN
scattering observables, such as the recoil polarization and double polarization. Hopefully,
such data can be obtained from the new hadron facilities at JPARC in Japan.
Our model is further checked by examining our predictions of the total cross sections σtot
which can be calculated from the forward elastic scattering amplitudes by using the optical
theorem. The total elastic scattering cross sections σel can be calculated from the predicted
partial wave amplitudes. With the normalization < ~k|~k′ >= δ(~k − ~k′) used in Ref. [3], we
σel(W ) =
T=1/2,3/2
σelT (W ) (39)
1200 1600 2000
1200 1600 2000
1200 1600 2000
-0.15
-0.05
1200 1600 2000
1200 1600 2000
W (MeV)
1200 1600 2000
W (MeV)
1200 1600 2000
W (MeV)
-0.04
-0.02
11 P13
FIG. 4: Real parts of the calculated πN partial wave amplitudes (Eq. (5)) of isospin T = 1/2 are
compared with the energy independent solutions of Ref. [33].
1200 1600 2000
1200 1600 2000
1200 1600 2000
1200 1600 2000
1200 1600 2000
W (MeV)
1200 1600 2000
W (MeV)
1200 1600 2000
W (MeV)
11 P13
FIG. 5: Imaginary parts of the calculated πN partial wave amplitudes (Eq. (5)) of isospin T = 1/2
are compared with the energy independent solutions of Ref. [33].
1200 1600 2000
1200 1600 2000
1200 1600 2000
1200 1600 2000
-0.15
-0.05
1200 1600 2000
W (MeV)
-0.08
-0.06
-0.04
-0.02
1200 1600 2000
W (MeV)
-0.15
-0.05
1200 1600 2000
W (MeV)
31 P33
FIG. 6: Real parts of the calculated πN partial wave amplitudes (Eq. (5)) of isospin T = 3/2 are
compared with the energy independent solutions of Ref. [33].
σelT (W ) =
(4π)2
ρπN (W )
(2J + 1)
|T TJπN(LS),πN(LS)(k, k,W )|2 , (40)
where ρπN(W ) = πkEπ(k)EN (k)/W with k determined by W = Eπ(k) + EN(k) and the
amplitude T TJL′S′(πN),LS(πN)(k, k;W ) is the partial-wave solution of Eq. (5). Similarly, the
total πN → ηN cross sections can be calculated from
σtotπN→ηN =
(4π)2
πN(W )ρ
ηN (W )
(2J + 1)
|T T=1/2,JηN(LS),πN(LS)(k
′, k,W )|2 (41)
where ρηN (W ) = πk
′Eη(k
′)EN(k
′)/W with k′ determined by W = Eη(k
′) + EN (k
′). We
can also calculate the contribution from each of the unstable channels, π∆, ρN , and σN ,
to the total πN → ππN cross sections. For example, we have for the πN → π∆ → ππN
contribution in the center of mass frame
σrecπ∆(W ) =
∫ W−mπ
mN+mπ
E∆(k)
Γπ∆(k, E)/(2π)
|W − Eπ(k)− E∆(k)− Σπ∆(k, E)|2
σπN→π∆(k,W ) (42)
where k is defined by MπN = Eπ(k) +EN(k), EπN(k) = [M
πN + k
2]1/2, Σπ∆(k, E) is defined
in Eq.(19), Γπ∆(k, E) = −2Im(Σπ∆(k, E)), and
σπN→π∆(k,W ) = 4πρπN (k0)ρπ∆(k)
L′S′,LS,J
2J + 1
(2SN + 1)(2Sπ + 1)
|T Jπ∆(L′S′),πN(LS)(k, k0;W )|2
where k0 is defined by W = Eπ(k0) + EN (k0) and ρab(k) = πkEa(k)Eb(k)/W . The am-
plitude T JL′S′(π∆),LS(πN)(k, k0;W ) is the partial-wave solution of Eq.(5). The corresponding
1200 1600 2000
1200 1600 2000
1200 1600 2000
1200 1600 2000
1200 1600 2000
W (MeV)
1200 1600 2000
W (MeV)
1200 1600 2000
W (MeV)
31 P33
FIG. 7: Imaginary parts of the calculated πN partial wave amplitudes (Eq. (5)) of isospin T = 3/2
are compared with the energy independent solutions of Ref. [33].
0 40 80 120 160
θ (cm)
0 40 80 120 160
θ (cm)
p       π
p       π
W=1440 MeV W=1440 MeV
W=1650 MeV W=1650 MeV
W=1800 MeV W=1800 MeV
FIG. 8: Differential cross section for several different center of mass energies. Solid red curve
corresponds to our model while blue dashed lines correspond to the SP06 solution of SAID [33].
All data have been obtained through the SAID online applications. Ref. [33].
0 40 80 120 160
θ (cm)
0 40 80 120 160
θ (cm)
p       π
p       π
W=1235 MeV W=1235 MeV
W=1535 MeV W=1535 MeV
W=1680 MeV W=1680 MeV
FIG. 9: Differential cross section for several different center of mass energies. Similar description
as Fig. 8. All data have been obtained through the SAID online applications. Ref. [33].
0 40 80 120 160
θ (cm)
0 40 80 120 160
θ (cm)
p       π
p       π
W=1230 MeV W=1640 MeV
W=1440 MeV W=1680 MeV
W=1540 MeV W=1800 MeV
FIG. 10: Target polarization asymmetry, P , for several different center of mass energies. Similar
description as Fig. 8. All data have been obtained through the SAID online applications. Ref. [33].
0 40 80 120 160
θ (cm)
0 40 80 120 160
θ (cm)
p       π
p       π
W=1230 MeV W=1640 MeV
W=1440 MeV W=1680 MeV
W=1540 MeV W=1800 MeV
FIG. 11: Target polarization asymmetry, P , for several different center of mass energies. Descrip-
tion as in Fig. 8. All data have been obtained through the SAID online applications. Ref. [33].
expressions for the unstable channels ρN and σN can be obtained from Eqs. (42)-(43) by
changing the channel labels.
The predicted σtot (solid curves) along with the resulting total elastic scattering cross
sections σel compared with the data of π+p reaction are shown in Fig. 13. Clearly, the
model can account for the data very well within the experimental errors. Here only the
T = 3/2 partial waves are relevant. Equally good agreement with the data for π−p reaction
are shown in the left side of Fig. 14. In the right side, we show how the contributions from
each channel add up to get the total cross sections. The comparison of the contribution
from ηN channel with the data is shown in Fig. 15. It is possible to improve the fit to this
data by adjusting N∗ → ηN parameters. But this can be done correctly only when the
differential cross section data of πN → ηN are included in the fit. This is beyond the scope
of this work and will be pursed in our second-stage calculations.
The contributions from π∆, ρN and σN intermediate states to the π−p → ππN total
cross sections calculated from our model can be seen in the right side of Fig. 14. These
predictions remain to be verified by the future experiments. The existing πN → ππN data
are not sufficient for extracting model independently the contributions from each unstable
channel.
The results shown in Figs. 13-15 indicate that our parameters are consistent with the
total cross section data.
We now discuss the parameters presented in Appendix A. It is rather difficult to compare
the resulting non-resonant coupling constants listed in Table III with the values from other
works, since the coupling strengths are also determined by the cutoff parameters listed in
0 40 80 120 160
θ (cm)
0 40 80 120 160
θ (cm)
p       π
p       π
W=1230 MeV
W=1640 MeV
W=1440 MeV W=1680 MeV
W=1540 MeV
W=1800 MeV
FIG. 12: Target polarization asymmetry, P , for several different center of mass energies. Descrip-
tion as in Fig. 8. All data have been obtained through the SAID online applications. Ref. [33].
Table IV. Perhaps it is possible to narrow their differences by using a different parameter-
ization of the form factors. However, the fit is a rather time consuming process and hence
no attempt is made in this work to try other forms of form factors.
In Table V, we see that all of the bare masses are higher than the PDG’s resonance
positions. This can be understood from the expression Eq. (14) for the partial waves with
only one N∗ since one finds in general that Re[Σ̄(E)] < 0. For the S11, P11, P33 and D13
partial waves, two bare N∗ states are mixed by their interactions, as can be seen in Eq. (10).
Thus the relation between their bare masses and the resonance positions identified by PDG
is much more complex.
As we mentioned above, the fit to πN elastic scattering is can not determine well the
bare N∗ → π∆, ρN, σN parameters. Thus the results for these unstable particle channels
listed in Tables III-VII must be refined by fitting the πN → ππN data.
VI. SUMMARY AND FUTURE DEVELOPMENTS
Within the formulation developed in Ref. [3], we have constructed a dynamical coupled-
channel model of πN scattering by fitting the πN scattering data. The parameters of the
model are first determined by fitting as much as possible the empirical πN elastic scattering
amplitudes of SAID up to 2 GeV. We then refine and confirm the resulting parameters by
directly comparing the predicted differential cross section and target polarization asymmetry
with the original data of the elastic π±p→ π±p and charge-exchange π−p→ π0n processes.
1200 1400 1600 1800 2000
W (MeV)
FIG. 13: The predicted total cross sections of the π+p → X (solid curve)and π+p → π+p (dashed
curve) reactions are compared with the data. Squares and triangles are the corresponding data
from Ref. [35].
The predicted total cross sections of πN reactions and are also in good agreement with the
data. The model thus can be used as a starting point for analyzing the very extensive data
of electromagnetic π production reactions.
The predicted total cross sections of πN → ηN reactions are also in fair agreement with
the data. However, the parameters associated with the ηN channel need to be refined to
also fit the differential cross section data of πN → ηN before the model can be used to
analyze the data of electromagnetic η production reactions.
The main shortcoming of this work is that the ππN interaction term hππN of Eq.(4) is
not included in the calculations. As derived in Ref. [3], the effects due to this interaction
can be included by adding a term Z
MB,M ′B′(E), which contains the ππN → ππN scattering
amplitude, to the driving term VMB,M ′B′(E) of Eq.(6). Our effort in this direction is in
progress along with the development of a more complete determination of the parameters
of the model by fitting both the data of πN elastic scattering and πN → ππN reactions.
This is also essential to pin down the parameters of the interactions associated with the
π∆, ρN and σN states. Only when this second-stage is completed, we then can perform
dynamical coupled-channel analysis of the very extensive and complex data of photo- and
electro-production of two pions. This is an essential step to probe the W > about 1.7 GeV
resonance region where the information on N∗ is very limited and uncertain.
Finally, a necessary next step is to extract the resonance poles and the associated residues
from the predicted πN amplitudes. This is being pursued and will be published else-
where [46].
Acknowledgments
We would like to thank M. Paris for his assistance in using the parallel processors at
NERSC and A. Parreño for her help and encouragement to use the BSC. This work is sup-
ported by the U.S. Department of Energy, Office of Nuclear Physics Division, under contract
1200 1400 1600 1800
W (MeV)
1200 1400 1600 1800 2000
W (MeV)
πN, ηN
πN, ηN, π∆
πN, ηN, pD, σN
πN, ηN, π∆, σN, ρN
FIG. 14: Left: The predicted total cross sections of the π−p → X (solid curve) and π−p →
π−p + π0n (dashed curve) reactions are compared with the data. Open squares are the data on
π−p → X from Ref. [35], open triangles are obtained by adding the π−p → π−p and π−p → π0n
data obtained from Ref. [35] and SAID database [41] respectively. Right: Show how the predicted
contributions from each channel are added up to the predicted total cross sections of the π−p → X.
1400 1600 1800 2000
W (MeV)
FIG. 15: The predicted total cross sections of πp → ηp reaction are compared with the data [42, 43].
No. DE-AC02-06CH11357, and Contract No. DE-AC05-060R23177 under which Jefferson
Science Associates operates Jefferson Lab, and by the Japan Society for the Promotion of Sci-
ence, Grant-in-Aid for Scientific Research(c) 15540275. This work is also partially supported
by Grant No. FIS2005-03142 from MEC (Spain) and FEDER and European Hadron Physics
Project RII3-CT-2004-506078. The computations were performed at NERSC (LBNL) and
Barcelona Supercomputing Center (BSC/CNS) (Spain). The authors thankfully acknowl-
edges the computer resources, technical expertise and assistance provided by the Barcelona
Supercomputing Center - Centro Nacional de Supercomputacion (Spain).
APPENDIX A: PARAMETERS FROM THE FITS
Parameter SL Model
f2πNN/(4π) 0.08 0.08
mσ (MeV) 500.1 −
fπN∆ 2.2061 2.0490
fηNN 3.8892 −
gρNN 8.7214 6.1994
κρ 2.654 1.8250
gωNN 8.0997 10.5
κω 1.0200 0.0
gσNN 6.8147 −
gρππ 4. 6.1994
fπ∆∆ 1.0000 −
fρN∆ 7.516 −
gσππ 2.353 −
gωπρ 6.955 −
gρ∆∆ 3.3016 −
kρ∆∆ 2.0000 −
TABLE III: The parameters associated with the Lagrangians Eqs.(22)-(35). The results are from
fitting the empirical πN partial-wave amplitudes [33] of a given total isospin T = 1/2 or 3/2. The
parameters from the SL model of Ref. [13] are also listed.
Parameter (MeV) SL model (MeV)
ΛπNN 809.05 642.18
ΛπN∆ 829.17 648.18
ΛρNN 1086.7 1229.1
Λρππ 1093.2 1229.1
ΛωNN 1523.18 −
ΛηNN 623.56 −
ΛσNN 781.16 −
ΛρN∆ 1200.0 −
Λπ∆∆ 600.00 −
Λσππ 1200.0 −
Λωπρ 600.00 −
Λρ∆∆ 600.00 −
TABLE IV: Cut-offs of the form factors, Eq. (36), of the non-resonant interaction vMB,M ′B′ . The
results are from fitting the empirical πN partial-wave amplitudes [33] of a given total isospin
T = 1/2 or 3/2. The parameters from the SL model of Ref. [13] are also listed.
LTJ PDG’s Mass( MeV) M1 (MeV) M2 (MeV)
S11 1535; 1655 1800. 1880.
S31 1630 1850.
P11 1440; 1710 1763 2037
P13 1720 1711
P31 1910 1900.3
P33 1232; 1600 1391 1602.
D13 1520; 1700 1899.1 1988.
D15 1675 1898
D33 1700 1976
D35 1960 −
F15 1685 2187
F35 1890 2162
F37 1930 2137.8
TABLE V: The masses of the nucleon excited states included in the fits. (second and third
columns). The first column contains the masses of the nucleon resonances given by PDG [35].
πN ηN π∆ σN ρN
S11 (1) 7.0488 9.1000 −1.8526 −2.7945 2.0280 .02736
S11 (2) 9.8244 .60000 .04470 1.1394 −9.5179 −3.0144
S31 5.275002 − −6.17463 − −4.2989 5.63817
P11 (1) 3.91172 2.62103 −9.90545 −7.1617 −5.1570 3.45590
P11 (2) 9.9978 3.6611 −6.9517 8.62949 −2.9550 −0.9448
P13 3.2702 −.99924 −9.9888 −5.0384 1.0147 −.00343 1.9999 −.08142
P31 6.80277 − 2.11764 − 9.91459 0.15340
P33 (1) 1.31883 − 2.03713 9.53769 − −.3175 1.0358 0.76619
P33 (2) 1.3125 − 1.0783 1.52438 − 2.0118 −1.2490 0.37930
D13 (1) .44527 −.0174 −1.9505 .97755 −.481855 1.1325 −.31396 .17900
D13 (2) .46477 .35700 9.9191 3.8752 −5.4994 .28916 9.6284 −.14089
D15 .31191 −.09594 4.7920 .01988 −.45517 −.17888 1.248 −.10105
D33 .9446 − 3.9993 3.9965 − .16237 3.948 −.85580
F15 .06223 0.0000 1.0395 .00454 1.5269 −1.0353 1.6065 −.0258
F35 .173934 − −2.96090 −1.09339 − −.07581 8.0339 −.06114
F37 0.25378 − −0.3156 −0.0226 − .100 .100 .100
TABLE VI: The coupling constants CN∗,JTLS;MB of Eq. (37) with MB = πN, ηN, π∆, σN, ρN
for each of the resonances. When there are more than one value for π∆ and ρN channels, they
correspond to the possible quantum numbers (LS) listed in Table 2.
πN ηN π∆ σN ρN
S11 (1) 1676.4 598.97 554.04 801.03 1999.8 1893.6
S11 (2) 533.48 500.02 1999.1 1849.5 796.83 500.00
S31 2000.00 − 500.00 − 500.031 500.00
P11 (1) 1203.62 1654.85 729.0 1793.0 621.998 1698.90
P11 (2) 646.86 897.84 501.26 1161.20 500.06 922.280
P13 1374.0 500.23 500.00 500.770 640.50 500.00 500.10 1645.2
P31 828.765 − 1999.9 − 1998.8 2000.6
P33 (1) 880.715 − 507.29 501.73 − 606.78 1043.4 528.37
P33 (2) 746.205 − 846.37 780.96 − 584.98 500.240 1369.7
D13 (1) 1658. 1918.2 976.36 1034.5 1315.8 599.79 1615.1 1499.50
D13 (2) 1094.0 678.41 1960.0 660.02 1317.0 550.14 597.57 1408.7
D15 1584.7 1554.0 500.77 820.17 507.07 735.40 749.41 937.53
D33 806.005 − 1359.38 608.090 − 1514.98 1998.99 956.61
F15 1641.6 655.87 1899.5 522.68 500.93 500.76 500.0 1060.9
F35 1035.28 − 1227.999 586.79 − 1514.3 593.84 1506.0
F37 1049.04 − 1180.2 1031.81 − 600.02 600.00 600.02
TABLE VII: The range parameter ΛN∗,JTLS;MB (in unit of (MeV/c)) of Eq. (37) with MB =
πN, ηN, π∆, σN, ρN for each of the resonances. When there are more than one value for π∆ and
ρN channels, they correspond to the possible quantum numbers (LS) listed in Table 2.
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http://arxiv.org/abs/nucl-th/0611101
http://gwdac.phys.gwu.edu
	Introduction
	Dynamical Coupled-channel equations
	Calculations
	Fitting Procedure
	Results
	Summary and Future Developments
	Acknowledgments
	Parameters from the fits
	References
