Resonances of the Nucleon - Part II Dieter Drechsel Institut f¨ ur Kernphysik , Universit¨ at Mainz Bosen Workshop, August 2010
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Resonances of the Nucleon - Part II
Dieter Drechsel
Institut fur Kernphysik , Universitat Mainz
Bosen Workshop, August 2010
Table of Contents
Table of Contents
IntroductionScattering MatrixResonance Model
Resonance ParametersPredictions DMT
Pion PhotoproductionSum Rules
Helicity amplitudesForm Factors
CQM PredictionsConclusions
Pion-Proton Scattering - DMT Model—– DMT fit to complex multipole, - - - nonresonant backgroundO, �: single-energy solutions of PWA (SAID), ♦ bare masses
S11 P11 P33
resultsresults ofof ourour fitsfits toto thethe SAID SAID s.es.e. partial . partial waveswaves
bare resonances
sin l n p n l sis f m SAIDDMT fit
nonresonant background single-energy pw analysis from SAID
G.Y. Chen, S.S. Kamalov et al., PRC 76,035206 (2007)
Argand Plots - DMT Model
—– path of complex multipole as function of W, —– unitary circle+ and +: data, O BW massesDMT DMT ArganArgan diagramsdiagrams
I Only P33(1232) has clear signature to determine BW mass:Re T=0, Im T=1 at WR .I In other cases, model-dependent background must besubtracted.I However, pole position and other parameters uniquely defined.
Contour Plots of Imaginary Part - DMT Model
1200 1400 1600 1800 2000 2200-300-250-200-150-100
-500
Re W
ImW
P33 partial wave
1200 1400 1600 1800 2000 2200-300-250-200-150-100
-500
Re W
ImW
P11 partial wave
1200 1400 1600 1800 2000 2200-300-250-200-150-100
-500
Re W
ImW
D13 partial wave
P331232****1600***1920***
P111440****1710***2100*
D131520****1700***2080**
Pole Positions 4-Star Resonances – DMT Model
97
44
15
weakweak
22 8
weak
3 6
no 4-star
no pole weakp
Spin-dependent forward Compton Scattering
Amplitude g(ν) connected with helicity-dependent absorption crosssection σ3/2(ν)− σ1/2(ν) by dispersion relations:
g(ν) = − e2κ2
8πM2 ν + γdyn0 (ν)ν3 , γdyn
0 (ν) = γ0 + γ0 ν2 + . . .
. anomalous magnetic moment κ (GDH sum rule)πe2κ2
2M2 =∫∞ν0
σ3/2(ν′)−σ1/2(ν
′)
ν′ dν ′ ≡ IGDH
. forward spin polarizabilities (GGT sum rule)leading term γ0, subleading term γ0, etc., summed up: dynamicpolarizability = γdyn
0 (ν), full spin-dependent response of nucleon inforward direction.γ0 = 1
4π2
∫∞ν0
σ1/2(ν′)−σ3/2(ν
′)
ν′3dν ′
γ0 = 14π2
∫∞ν0
σ1/2(ν′)−σ3/2(ν
′)
ν′5dν ′
Re[γdyn0 (ν)] = 1
4π2 P∫∞ν0
σ1/2(ν′)−σ3/2(ν
′)
ν′(ν′2−ν2)dν ′
Im[γdyn0 (ν)] =
σ1/2(ν)−σ3/2(ν)
8πν2
Running Integrals and dynamic polarizability
0
100
200I G
DH
(µb
)
-1
0
1
γ 0 (1
0-4 f
m4 )
0
0.5
1
1.5
0.2 0.4 0.6 0.8
γ– 0 (1
0-4 f
m4 )
Eγ (GeV)
-5
0
5
0 0.2 0.4
Eγ (GeV)
(10-4
fm
4 )
Re[γdyn
0 ]
0.2 0.3 0.4
Eγ (GeV)
Im[γdyn
0 ]
GDH from data analysis ⇒ (210± 6± 14)µb, from sum rule⇒ 204µbγ0 = (−0.90± 0.08± 0.11)10−4fm4
γ0 = (+0.60± 0.07± 0.07)10−6fm4
based on data of the GDH Collaboration and multipole analysis near threshold,see B.Pasquini, P.Pedroni, D.D., PLB (2010)
Note: dynamic model DMT(2001) and dispersion theory HDT(1997) yield best
representation of low-energy data!
Neutral pion photoproduction at threshold
solid line: Hanstein et al. (97)dashed line: Bernard et al. (96)data: MAMI (90,96), SAL (96)
line at −2.2: previous “LET”
FIGURES
144 147 150 153 156 159 162 165 168E / MeV
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
/b
Ref. [7]Ref. [6]this work
FIG. 1. Total cross sections for π0 photoproduction close to threshold with statistical errors
(without systematic error of 5%) as function of incident photon energy (solid squares, this work,
open circles, Ref. [7], open diamonds Ref. [6]).
0 20 40 60 80 100 120 140 160 180cms
0 / deg
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
this work fit to the dataDR [20]ChPT [9]
FIG. 2. Photon asymmetry for π0 photoproduction for a photon energy of 159.5 MeV with
statistical errors (without systematic error of 3%) as a function of the polar angle θ (solid line: fit
to the data) in comparison to ChPT [9] (dotted line) and DR [20] (dashed line). With the new value
of the low–energy constant bP the ChPT calculation [26] is in agreement with the experimental
values.
12
data of Schmidt et al. (2001)
compared to Bernard et al. (1996) and
Hanstein et al. (1997)
Differential Cross Section
0( , )p p� � Photoproduction at Threshold
data: Fuchs (Mainz 1996)
Schmidt (Mainz 2001)
Bergstrom (Saskatoon 96)
data: Fuchs, Schmidt (Mainz 1996, 2001)without FSI
with FSI ( )0( ) ( )m m� ��
�
with FSI ( )0( ) ( )m m� ��
�
Im
Re
ChPT (Bernard 1996)
Kamalov et al., DMT(2001)
dotted: without fsi, pionmasses equal
dashed: with fsi, pionmasses equal
solid: with fsi and physicalpion masses
data: Fuchs (1996) andSchmidt (2001)
Threshold Multipoles
Leading multipoles at threshold:{E0+ , P1 = 3E1+ + M1+ −M1−, P2 = 3E1+ −M1+ + M1−, P3 = 2M1+ + M1−}P waves vanish at threshold ⇒ divide by pion momentum, Pi = Pi/q
6 December 2001
Physics Letters B 522 (2001) 27–36www.elsevier.com/locate/npe
π0 Photo- and electroproduction at threshold withina dynamical model
S.S. Kamalova,1, Guan-Yeu Chena, Shin Nan Yanga, D. Drechselb, L. Tiatorb
a Department of Physics, National Taiwan University, Taipei, 10764 Taiwan, ROCb Institut für Kernphysik, Universität Mainz, 55099 Mainz, Germany
Received 10 July 2001; received in revised form 12 September 2001; accepted 15 October 2001Editor: Dr. W. Haxton
Abstract
We show that, within a meson-exchange dynamical model describing most of the existing pion electromagnetic productiondata up to the second resonance region, one is also able to obtain a good agreement with theπ0 photo- and electroproduction datanear threshold. The potentials used in the model are derived from an effective chiral Lagrangian. The only sizable discrepancybetween our results and the data is in theP -wave amplitudeP3 = 2M1+ +M1− where our prediction underestimate the databy about 20%. In the case ofπ0 production, the effects of final state interaction in the threshold region are nearly saturatedby single charge exchange rescattering. This indicates that in ChPT it might be sufficient to carry out the calculation just up toone-loop diagrams for threshold neutral pion production. 2001 Elsevier Science B.V. All rights reserved.
Chiral perturbation theory (ChPT) provides us witha systematic scheme to describe the low energy inter-actions of Goldstone bosons among themselves andwith other hadrons, because it is based on a low en-ergy effective field theory respecting the symmetriesof QCD, in particular, chiral symmetry. There is gen-erally good agreement between the ChPT predictionsand experiments [1]. One case which has been very in-tensively studied isπ0 electromagnetic production ofneutral pions near threshold where very precise mea-surements [2–8] have been performed and the ChPTcalculation to one loopO(p3) (O(p4) in the case ofphotoproduction) has been carried out in the heavybaryon formulation [9,10]. Nice agreement between
E-mail address: [email protected] (S.N. Yang).1 Permanent address: Laboratory of Theoretical Physics, JINR
Dubna, 141980 Moscow region, Russia.
theory and experiment was reached not only for theS-wave multipolesE0+ andL0+ but for theP -waveamplitudes [8,10] as well.
As in ChPT, meson-exchange models also start froman effective chiral Lagrangian. However, they differfrom ChPT in the approach to calculate the scatteringamplitudes. In ChPT, crossing symmetry is maintainedin the perturbative field-theoretic calculation, and theagreement with low energy theorems and the data is tobe expected as long as the series is well convergent. Inmeson-exchange models, the effective Lagrangian isused to construct a potential for use in the scatteringequation. The solutions of the scattering equationwill include rescattering effects to all orders andthereby unitarity is ensured, while crossing symmetryis violated. Such models [11–17] have been able toprovide a good description ofπN scattering lengthsand phase shifts inS-, P -, and D-waves up to600 MeV pion laboratory kinetic energy.
0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0370-2693(01)01241-2
6 December 2001
Physics Letters B 522 (2001) 27–36www.elsevier.com/locate/npe
π0 Photo- and electroproduction at threshold withina dynamical model
S.S. Kamalova,1, Guan-Yeu Chena, Shin Nan Yanga, D. Drechselb, L. Tiatorb
a Department of Physics, National Taiwan University, Taipei, 10764 Taiwan, ROCb Institut für Kernphysik, Universität Mainz, 55099 Mainz, Germany
Received 10 July 2001; received in revised form 12 September 2001; accepted 15 October 2001Editor: Dr. W. Haxton
Abstract
We show that, within a meson-exchange dynamical model describing most of the existing pion electromagnetic productiondata up to the second resonance region, one is also able to obtain a good agreement with theπ0 photo- and electroproduction datanear threshold. The potentials used in the model are derived from an effective chiral Lagrangian. The only sizable discrepancybetween our results and the data is in theP -wave amplitudeP3 = 2M1+ +M1− where our prediction underestimate the databy about 20%. In the case ofπ0 production, the effects of final state interaction in the threshold region are nearly saturatedby single charge exchange rescattering. This indicates that in ChPT it might be sufficient to carry out the calculation just up toone-loop diagrams for threshold neutral pion production. 2001 Elsevier Science B.V. All rights reserved.
Chiral perturbation theory (ChPT) provides us witha systematic scheme to describe the low energy inter-actions of Goldstone bosons among themselves andwith other hadrons, because it is based on a low en-ergy effective field theory respecting the symmetriesof QCD, in particular, chiral symmetry. There is gen-erally good agreement between the ChPT predictionsand experiments [1]. One case which has been very in-tensively studied isπ0 electromagnetic production ofneutral pions near threshold where very precise mea-surements [2–8] have been performed and the ChPTcalculation to one loopO(p3) (O(p4) in the case ofphotoproduction) has been carried out in the heavybaryon formulation [9,10]. Nice agreement between
E-mail address: [email protected] (S.N. Yang).1 Permanent address: Laboratory of Theoretical Physics, JINR
Dubna, 141980 Moscow region, Russia.
theory and experiment was reached not only for theS-wave multipolesE0+ andL0+ but for theP -waveamplitudes [8,10] as well.
As in ChPT, meson-exchange models also start froman effective chiral Lagrangian. However, they differfrom ChPT in the approach to calculate the scatteringamplitudes. In ChPT, crossing symmetry is maintainedin the perturbative field-theoretic calculation, and theagreement with low energy theorems and the data is tobe expected as long as the series is well convergent. Inmeson-exchange models, the effective Lagrangian isused to construct a potential for use in the scatteringequation. The solutions of the scattering equationwill include rescattering effects to all orders andthereby unitarity is ensured, while crossing symmetryis violated. Such models [11–17] have been able toprovide a good description ofπN scattering lengthsand phase shifts inS-, P -, and D-waves up to600 MeV pion laboratory kinetic energy.
0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0370-2693(01)01241-2
S.S. Kamalov et al. / Physics Letters B 522 (2001) 27–36 31
Table 1E0+ (in units 10−3/mπ+ ) at threshold andP1, P2 andP3 (in units 10−3q/m2
π+ ). Contributions of Born terms (Born), vector mesons (ω+ρ),pion rescattering (FSI) and resonances (res.) are shown separately. The predictions of ChPT and recent experimental values are taken fromRef. [10] and Ref. [8]
Born ω+ ρ FSI Res. Tot. ChPT Exp.
E0+ −2.46 0.17 1.06 0.07 −1.16 −1.16 −1.33±0.11
P1 9.12 −0.35 0.15 0.38 9.30 9.14 9.47±0.37
P2 −8.91 0.21 −1.32 −0.13 −10.15 −9.7 −9.46±0.39
P3 0.18 4.61 3.36 1.20 9.35 10.36 11.48±0.41
Fig. 4. Solid curves are our predictions for the energy dependence ofthe photon asymmetryΣ at θπ = 90◦ (upper panel) and its angulardistribution atEγ = 159.5 MeV (lower panel) inγp → π0p.Dashed curves are results obtained with a 15% reduction of theM1−multipole in the model. Experimental data from Ref. [8].
at fixed pion c.m. angleθπ = 90◦. As shown by thesolid curve in Fig. 4, our prediction forΣ(θπ = 90◦)first tends to more negative values before bending overand becoming positive at large photon energies. It wasfound in Ref. [33] that in the threshold region this ob-
servable is very sensitive to theM1− multipole whichstrongly depends on the details of the low energy be-havior of Roper resonance, vector meson and FSI con-tributions. Therefore, a slight modification of one orall of these mechanisms can drastically change thephoton asymmetry. As an illustration, our predictionfor the energy dependence and angular distribution ofΣ(θπ) obtained with a 15% reduction of theM1− mul-tipole, is shown by dashed curves in Fig. 4. This smallmodification of the low energy tail of the Roper reso-nance leads to positive photon asymmetries at all en-ergies!
In Table 1, the ChPT predictions and the experi-mental values extracted from recent TAPS polarizationmeasurements [8] are listed for comparison. Our pre-dictions are in good overall agreement with the ChPTpredictions [10] and the TAPS results. However, thereis a 15–20% difference inP3 which leads to an under-estimation of our result for the photon asymmetry, asshown in Fig. 4. Note that, in contrast to our model,P3 is essentially determined by a low energy constantin ChPT.
Pion electroproduction provides us with informa-tion on theQ2 = −k2 dependence of the transverseE0+ and longitudinalL0+ multipoles in the thresh-old region. The “cusp” effects in theL0+ multipoleis taken into account in a similar way as in the case ofE0+,
ReLγπ0
0+ = ReLγπ0
0+ (IS)
(6)− aπNωcReLγπ+
0+ (IS)
√1− ω2
ω2c
,
where all the multipoles are functions of total c.m.energyE and virtual photon four-momentum squa-
I Status of 2001: DMT underestimates data for P3 by 20%.I one-loop is good approximation.
Photon asymmetry – new experiment
Asymmetries
Asymmetry Comparison
(deg)cmθ0 20 40 60 80 100 120 140 160 180
Σ
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
11.5 MeV± = 157.4 γAsymmetry for E
This WorkDMT 2001ChPTSchmidt - TAPS
11.5 MeV± = 157.4 γAsymmetry for E
David Hornidge (Mount Allison University) Mainz 10 March 2010 40 / 48
Photon Asymmetry – Models
0 30 60 90 120 150 180Θ@degD
-0.1
-0.05
0
0.05
0.1
0.15
0.2Photon Asymmetry Predictions
HBChPTDRHDTDMT
Multipoles HBChPT
130 135 140 145 150pion c.m. energy @MeVD
-2
-1
0
1
2
3
Γ+p ® Π0+p, Multipoles @10-3
�mΠD
Re@M1-DRe@M1+DRe@E1+DIm@E0+DRe@E0+D
Heavy Baryon ChPT (Bernard et al., PLB 1996):S and P wave multipoles in the threshold region.
Dashed line: Photon asymmetry experiment near pion c.m. energy 146 MeV.
Low Energy Constants
Photon asymmetry Σ ∼| P3 |2 − | P2 |2
135 140 145 150pion c.m. energy @MeVD
0
2
4
6
Γ+p ® Π0+p, P wave combinations @10-3
�mΠD
-P2P1P3 c.t.P3
Heavy Baryon ChPT (Bernard et al., PLB 1996): P1 and P2 mostly Born terms
P3: essentially LEC due to ∆(1232) excitation + t-channel ω exchange.
Pion electroproduction
0 10 20 30 400.0
0.5
1.0
1.5
2.0
2.5σ 0
(θ=
90°)
[μb
/sr]
ΔW [MeV]
0
5
0
5
0
5 ChPT
0
5
0
5
0
5Maid
0
5
0
5
0
5
DMT
0 10 20 30 40−1.5
−1.0
−0.5
0.0
0.5
σ TT(θ
=90
°) [
μb/s
r]
ΔW [MeV]
5
0
5
0
5
ChPT
5
0
5
0
5
Maid
5
0
5
0
5
DMT
0 10 20 30 40−0.60
−0.50
−0.40
−0.30
−0.20
−0.10
−0.00
σ TL(
θ=90
°) [
μb/s
r]
ΔW [MeV]
0
0
0
0
0
0
0
ChPT
0
0
0
0
0
0
0
Maid
0
0
0
0
0
0
0
DMT
0 10 20 30 40−1.0%
0.0%
1.0%
2.0%
3.0%
4.0%
5.0%
AT
L’(θ
=90
°)
ΔW [MeV]
ChPTMaidDMT
Separated cross sections σ0, σLT ,σTT and beam helicity asymmetry A′LT at
θ∗π = 90◦ and Q2 = 0.05GeV2. Solid line: HBChPT (Bernard, 1996), dashed
line: MAID, dashed-dotted line: DMT (Kamalov, 2001). Data: Weis (2007)
FFR sum rule
Connects anomalous magnetic moment κp with dispersion integralover photoproduction amplitude A1 in limit ν → 0,mπ → 0.
Re A(p π0),disp1 (ν, tthr) =
2
πP
∫ ∞
νthr
dν ′ν ′ Im A
(p π0)1 (ν ′, tthr)
ν ′2 − ν2
= κp + ∆p(ν, tthr)isoscalar (top)/ isovector (bottom)
integrands
B. Pasquini et al.: The Fubini-Furlan-Rossetti sum rule revisited 285
Fig. 2. The correction to the FFR sum rule, ∆N (ν, tthr), asdefined by eq. (36) and obtained from the dispersion integralfor proton (full line) and neutron (dashed line).
Fig. 3. The integrands of the dispersion integrals (see RHS ofeq. (36)) for the isoscalar (top) and isovector (bottom) combi-nations of the amplitudes A1. The full curves are obtained forν = νthr, the dashed curves for ν = 0.
from fig. 4; in that case the S-wave provides the lion’sshare of the amplitude in the region of ν . 160 MeV.
Figures 5 and 6 compare the predictions of HBChPTand DR for ∆N (ν, tthr) as functions of ν in the range 0 ≤ν ≤ 200 MeV, for the proton and the neutron, respectively.As we have seen before, all the predictions agree quite well
0
0.5
1
1.5
2
2.5
3
0 20 40 60 80 100 120 140 160 180 200ν (MeV)
∆ p + κ
p
Fig. 4. The values of κp + ∆p(ν, tthr) obtained from the dis-persion integral of eq. (36) with a multipole decomposition ofImA1. Solid line: full result for ImA1 evaluated with MAID03.Dashed line: results for the dispersion integral with only theS-wave contribution to ImA1, dotted line: P -wave contributiononly, dash-dotted line: sum of D- and F -wave contributions.
in the threshold region (upper panels) , which includes thecusp effect due to the opening of the charged-pion channel.
There is also a reasonable agreement with the datapoints obtained by first inserting the experimental valuesof the multipoles (ref. [17], see table 2) in eq. (22) andthen subtracting the pole terms. As an example, the “ex-perimental” threshold value has the following multipoledecomposition:
κp +∆p(νthr, tthr) = 2.06 (S) + 0 (P1)
+0.26 (P2)− 0.19 (P3) + 0.03 (D) = 2.16 . (37)
The result is clearly dominated by the S-wave. However,the small total P -wave term comes about by a delicatecancellation among the P -waves, which leads to a rela-tively large error bar for the FFR correction ∆N if calcu-lated from the real part of the amplitude A1. As we haveseen in figs. 3 and 4, the situation is quite different inthe dispersive approach. In this case, the correction ∆N
is essentially determined by ImM1+ in the region of the∆(1232) and a somewhat smaller contribution of ImE0+
in the threshold region. Both contributions are well undercontrol and additive, and therefore the dispersive evalua-tion of ∆N should be quite stable.
Outside of the threshold region (lower panels), we ob-serve 3 principal differences between HBChPT and thedispersive approach:
(I) The rise of ∆N for ν & 170 MeV is, of course, due tothe ∆(1232)-resonance. It cannot be described by the“static” LECs of appendix B, but will require a dynam-ical description of the resonance degrees of freedom as,
multipole contributions to
dispersion integral
0
0.5
1
1.5
2
2.5
3
0 20 40 60 80 100 120 140 160 180 200ν (MeV)
Δ p +
κ p
Relativistic Amplitudes A1 − A4
amplitudes A1 − A4 nearthreshold
238 The European Physical Journal A
012345678
0 0.1 0.2 0.3
A1 (
GeV
-2 )
-40
-30
-20
-10
0
10
0 0.1 0.2 0.3
A2 (
GeV
-4 )
-25
-20
-15
-10
-5
0
0 0.1 0.2 0.3ν (GeV)
A3 (
GeV
-3 )
010203040506070
0 0.1 0.2 0.3ν (GeV)
A4 (
GeV
-3 )
Fig. 4. The real parts of the amplitudes A(pπ0)i for the reaction
γp→ π0p as a function of ν and at t = tthr. Solid lines: disper-sive contributions according to eqs. (12) and (13) as obtainedwith the imaginary amplitudes from MAID05 containing par-tial waves up to Lmax = 3. Short-dashed lines: S-wave contri-butions; dotted lines: P -wave contributions; long-dashed lines:sum of D- and F -wave contributions. The dash-dotted linesstarting at pion threshold show the real parts taken directlyfrom MAID05.
-2
0
2
4
6
8
0 0.1 0.2 0.3
A1 (+
) ( G
eV-2
)
-40
-20
0
20
0 0.1 0.2 0.3
A2 (+
) ( G
eV-4
)
-20
0
20
0 0.1 0.2 0.3ν (GeV)
A3 (+
) ( G
eV-3
)
-25
0
25
50
75
0 0.1 0.2 0.3ν (GeV)
A4 (+
) ( G
eV-3
)
Fig. 5. The real parts of the amplitudes A(+)i as a function of
ν. The dispersive contributions according to eqs. (12) and (13)as evaluated with the imaginary amplitudes from MAID05 forthe following values of t: M2
π (solid lines), tthr (dashed lines),−4M2
π (dash-dotted lines), and −10M2π (dotted lines).
0123456789
0 0.1 0.2 0.3
A1 (
GeV
-2 )
-40
-20
0
0 0.1 0.2 0.3
A2 (
GeV
-4 )
-20
-10
0
0 0.1 0.2 0.3ν (GeV)
A3 (
GeV
-3 )
0
20
40
60
80
100
0 0.1 0.2 0.3ν (GeV)
A4 (
GeV
-3 )
Fig. 6. The real parts of the amplitudes A(pπ0)i for the reac-
tion γp → π0p as a function of ν and at t = tthr. Solid lines:dispersive contributions according to eqs. (12) and (13) as eval-uated with the imaginary amplitudes from MAID05 contain-ing partial waves up to Lmax = 3. Dashed lines: same resultscalculated with SAID. The dash-dotted lines are obtained byadding the vector-meson contributions of eq. (46) to the MAIDresult. The data points near threshold are derived from the ex-perimental values of ref. [27] for the S and P waves plus theMAID correction for the D-waves.
persive parts of the isospin amplitudes as a function of νfor a series of t values in the range M 2
π > t > −10M2π . As
shown in fig. 5, the t-dependence develops quite a regularpattern, with the cusp moving to larger ν values with in-creasing values of t. At the same time the physical thresh-old (see the line θ = 180◦ in fig. 1) moves to smaller valuesof ν down to the minimum at ν = νthr, from whereon itincreases again (see the line θ = 0 in fig. 1).
The comparison with the experiment [27] in fig. 6shows that the dispersion integral by itself misses thethreshold data for the amplitudes A2–A4. If we add thet-channel ρ and ω poles according to MAID05, we obtainan almost perfect agreement for A1, A2 and A4. The ap-parent discrepancy between theory and experiment for A3
is an open question, which will be discussed later.
After these tests we feel safe to expand the amplitudesabout the unphysical point at ν = 0 and t = M 2
π whereall the involved particles are on their mass shell. For thispurpose we proceed as in our previous work [1] by castingthe first amplitude in the form
Adisp1 (ν, t) = A1(ν, t)−Apole
1 (ν, t)
=egπN2M2
N
(κ+∆1(ν, t)) , (49)
240 The European Physical Journal A
The numbers in table 4 should be compared to anexpansion of the loop plus counter terms in covariantBChPT. Such a calculation has been performed in ref. [7]by evaluating the third-order loop corrections and sup-plementing them by a fourth-order polynomial. Since thefourth-order loop corrections are large, the resulting powerseries is only indicative of the expected LECs [7]. However,the coefficients compare favorably with the LECs obtainedfrom the earlier HBChPT calculations [28]. Including forconsistency an additional factor e on the RHS, we obtainfrom eq. (12) of ref. [7] the following coefficients from thefourth-order polynomial contribution: δ1
00 = 0, δ120 = 0.53,
δ102 = 3.40, δ2
00 = −6.33, δ310 = −2.58, and δ4
00 = 22.4.These numbers are in qualitative agreement with our re-sults in table 4. The differences are due to
I) the power series expansion of the loop corrections,which has to be added to the polynomial,
II) the effects of higher partial waves, particularly withregard to the t-dependence given by δ02, and
III) possible s-channel resonances above 2.2GeV and t-channel exchange of heavier objects, not included inthe dispersive approach.
As we have shown, the dispersion calculation includingthe vector meson poles is able to reproduce the experimen-tal threshold data except for the crossing-odd amplitudeA3. In order to pin down the origin of this discrepancy,we have checked the integrands of the dispersion integralsby repeating our calculation with the SAID [29] analysis.Whereas the real parts of the MAID and SAID multipolesdiffer in some cases, particularly at the higher energies, theimaginary parts generally agree well for W . 2.2GeV.As shown in fig. 6 the dispersive contributions of thetwo models turn out to be quite similar. Whereas theexperimental threshold values for A2 and A4 can be de-scribed after adding the vector meson pole contributions,no such remedy exists for the crossing-odd amplitude A3
for which both SAID and MAID are off by a factor of2. We have also checked the high-energy contributionsby varying the onset of the asymptotic tail in the range1.8GeV 6 W 6 2.5GeV and its shape from a simple1/W -dependence to various Regge prescriptions. In thisway we can modify the threshold amplitudes Athr
1 , Athr2 ,
and Athr4 by at most 10% in the π0p channel. However,
the asymptotic contribution to Athr3 reaches at most 1%
because of the better convergence for this crossing-oddamplitude.
We have also directly estimated the high-energy tailwith the parameters given by the Regge models of the1970s [20,21], which were constructed to fit the data inthe 5–20GeV region. Since these models include the ex-change of axial vector mesons and Regge cuts correspond-ing to many-particle exchange, they also contribute tothe crossing-odd amplitude A3. However, the predictedstrength of the high-energy contribution to Athr
3 turns outto be even smaller and of the order of 10−3 only. Onthe phenomenological level, possible candidates for axialvector meson exchange (JPC = 1++) are the a1 (1260)with quantum numbers IG = 1− and the f1 (1285) withIG = 0+. The a1 has the same isospin and G parity as
Table 5. The multipole decomposition of the dispersive partof the experimental threshold amplitudes, constructed fromthe data of ref. [27] and the MAID05 value for the D-statecontributions. The errors include statistical and systematic er-rors from the experiment and model errors due to uncertain-ties from the pion-nucleon coupling and from the D-waves, alladded in quadrature.
E0+ E1+ M1+ M1− D TotalA1 4.75 0.08 2.02 −1.94 0.08 5.00± 0.25A2 0 −0.34 −29.44 0 5.13 −24.7± 5.9A3 2.36 −0.65 −18.67 14.34 −0.65 −3.3± 1.7A4 2.36 0.04 40.55 14.34 0.04 57.3± 1.9
the pion, and therefore contributes only for charged-pionproduction. The f1, on the other hand, has positive G par-ity and a branching ratio of about 5% for the γρ0 chan-nel. It is therefore a good candidate for ρ-meson photo-production and, together with the a1, also for the Reggetail of the helicity-dependent inclusive cross-sections asmeasured by the Gerasimov-Drell-Hearn experiment [30].However, there appear to be no clear candidates to solvethe A3 puzzle by a t-channel pole term.
Let us finally discuss the multipole content of the rela-tivistic amplitudes and the related error bars in the thresh-old region. As in our previous analysis we subtract thenucleon pole terms, which may vary within about 2% de-pending on the value chosen for the pion-nucleon couplingconstant gπN . Since the pole term constitutes about 85%of the total threshold amplitudes Athr
2 and Athr3 , the choice
of gπN leads to an error of about 12% in the remaining dis-persive amplitudes. For Athr
1 and Athr4 , however, the pole
contributions are small and therefore the model error ofgπN can be neglected with regard to the dispersive ampli-tudes. In table 5 the threshold amplitudes are constructedfrom the experimental values of the S- and P -wave mul-tipoles and the MAID05 value for the D-waves. The errorgiven in the table is obtained by adding the experimen-tal errors for the threshold multipoles. We recall at thispoint that here and in the following all values refer to thedispersive contributions only.
As is evident from table 5, Athr1 is dominated by the
S-wave, because the magnetic contributions cancel nearlycompletely. The large S-wave contribution originates fromthe FFR current [1] and rescattering corrections, whichare of course included in BChPT by the chirally invariantpion-nucleon coupling and the pion loops leading to thepronounced cusp effect at the nπ+ threshold. These ef-fects are nicely reproduced by the dispersion integral, andthe small t-channel pole contribution is well within the ex-perimental error bars. More precisely, the vector mesonsproduce large effects of about equal size for both mag-netic multipoles, but the discussed cancellation betweenM1+ and M1− leads to a reduction by a large factor.
Quite different physical information is sampled byAthr
2 . Due to the structure of the associated four-vectorMµ
2 , the spin J = 1/2 multipoles E0+ and M1− do notappear in this amplitude. It is dominated by M1+ butalso receives surprisingly large contribution from the D-
TABLE: Contribution of multipoles tothreshold amplitudes for data (2006).Only A1 dominated by S wavesA2, A3, A4 essentially P waves
⇒ large contribution of LECS
A3 puzzle: Experimental amplitudes A1, A2, A4 can be described by MAID orSAID multipoles plus reasonable ω exchange.
Vector mesons do not contribute to A3. Experimental value can not be
reproduced, although dispersion integral for (crossing-odd) A3 converges well.
E.M. Multipoles, Helicity Amplitudes, Form Factors
I Experimental information contained in multipoles MI`±(W ,Q2).
Note: `± and I are quantum numbers of πN final state, M = {E , M, S} refer
to electric, magnetic, and Coulomb excitation.
I Definition of reduced multipoles:
MI`±(Q2) = 1
cI
√(2J+1)πqπ,RWRΓR
κR,cMΓπ/ΓRImMI ,res
`± (W = WR ,Q2)
I Different sets of form factors are linear combinations of M:
(i) HELICITY AMPLITUDES {A1/2(Q2), A3/2(Q
2), S1/2(Q2)}
(ii) SACHS FFs {GE (Q2), GM(Q2), GC (Q2)}(iii) DIRAC FFs {F1(Q
2), F2(Q2), F3(Q
2)}Note: Sachs & Dirac transition form factors not uniquely defined.
Resonances with J = 1/2 have only 2 form factors.
N ⇒ P33(1232)
Helicity Amplitudes Form Factors
magnetic (M1) magnetic (M1) , , electric (E2) & Coulomb (C2)electric (E2) & Coulomb (C2)N N ΔΔ t iti f f tt iti f f tN N --> > ΔΔ transition form factorstransition form factors
from MAID analysisfrom MAID analysisyy
Solid lines: global fit, “data points”: single-Q2 fit (MAID analysis)
p ⇒ P33(1232) – Ratios
EMR = E1+/M1+ CMR = S1+/M1+
solid lines: global MAID fit, data: single-Q2 fit
p ⇒ P11(1440)
empirical e.m. transition FFs for empirical e.m. transition FFs for mp .m. FF fmp .m. FF fN N --> > NN**(1440)(1440) excitationexcitation
F1
F2F2data : CLASanalysis : MAID
Data: CLAS, MAID analysisRelativistic Dirac (F1) and Pauli (F2) form factorsL. Tiator and M. Vanderhaeghen, PLB 672 (2009)
p ⇒ D13(1520)
A1/2
S1/2D (1520)13
A3/2
p ⇒ S11(1535)
S1/2S (1535)11
A1/2
Data: CLAS, M JLab single-Q2
—– MAID global, O MAID single-Q2
P33(1232) Multipoles: BW vs. Pole Position
Multipoles Ratios
MAID analysis
Multipoles: BW vs. Pole Position
p ⇒ P11(1440) p ⇒ S11(1535)
MAID analysis
Proton Helicity Amplitudes CQM vs. MAID
CQM MAID2007
N∗, ∆ `± A1/2 A3/2 S1/2 ζ A1/2 A3/2 S1/2
P33(1232) 1+ -109 -188 0 +1 -140 -265 17.5
P11(1440) 1− 26 – 28 −1 -61 – 4.2
D13(1520) 2− -25 134 -96 +1 -27 161 -64
S11(1535) 0+ 181 – 68 +1 66 – -2.0
S31(1620) 0+ 68 – 68 −1 66 – 16
S11(1650) 0+ 0 – 0 +1 33 – -3.5
D15(1675) 2+ 0 0 0 +1 15 22 0
F15(1680) 3− 0 79 -42 −1 -25 134 -44
D33(1700) 2− 101 100 94 −1 226 210 0
P13(1720) 1+ 118 -39 36 −1 73 -11 -53
F35(1905) 3− -14 -61 0 −1 18 -28 0
P31(1910) 1− 17 – -32 +1 18 – 0
F37(1950) 3+ -37 -48 0 +1 -94 -121 0
Neutron Helicity Amplitudes CQM vs. MAID
CQM MAID2007
N∗ `± A1/2 A3/2 S1/2 ζ A1/2 A3/2 S1/2
P11(1440) 1− −17 – 0 −1 54 – −41
D13(1520) 2− −43 −134 96 +1 −77 −154 14
S11(1535) 0+ −132 – −68 +1 −51 – 28
S11(1650) 0+ 28 – 0 +1 9 – 10
D15(1675) 2+ −39 −55 0 +1 −62 −84 0
F15(1680) 3− 37 0 0 −1 28 −38 0
P13(1720) 1+ −34 0 0 −1 −3 −31 0
Total Cross Sections from Helicities
1.2 1.4 1.6 1.8 2 2.2W@GeVD
-50
0
50
100
150
200
250ΣTT@ΜbD - 13 proton resonances MAID
backgroundF37H1950LD33H1700LD13H1520LP33H1232Ltotal
σT – MAID vs CQM
1.2 1.4 1.6 1.8 2 2.2W@GeVD
0
100
200
300
400
500
MAID vs. CQM, ΣT@ΜbD
CQM neutron ??CQM proton ??CQM neutronMAID neutronCQM protonMAID proton
σTT – MAID vs CQM
1.2 1.4 1.6 1.8 2 2.2W@GeVD
-50
0
50
100
150
200
250MAID vs. CQM, ΣTT@ΜbD
CQM neutron ??CQM proton ??CQM neutronMAID neutronCQM protonMAID proton
σLT – MAID vs CQM
1.2 1.4 1.6 1.8 2 2.2W@GeVD
-10
0
10
20MAID vs. CQM, ΣLT@ΜbD
CQM neutron ??CQM proton ??CQM neutronMAID neutronCQM protonMAID proton
σL – MAID vs CQM
1.2 1.4 1.6 1.8 2 2.2W@GeVD
0
10
20
MAID vs. CQM, ΣL@ΜbD
CQM neutron ??CQM proton ??CQM neutronMAID neutronCQM protonMAID proton
Conclusions
I Spatial extension and excitation spectrum of a particle relatedaspects of internal degrees of freedom
I Threshold and resonance phenomena connected by analyticityof S matrix
I Sum rules give explicit relations between low-energyphenomena and resonance structure
I New data for photon asymmetry may solve “A3” puzzle
I Extend the experiment of Weis et al to larger Q2?
I More and better data/data analysis for neutron needed.
I Have we really understood the longitudinal current in pionelectroproduction above the P33(1232)?
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