PANN: Partial-wave analysis of nucleon–nucleon scattering in wide-energy region

Computer Physics Communications 131 (2000) 225–263www.elsevier.nl/locate/cpc

PANN: Partial-wave analysis of nucleon–nucleon scatteringin wide-energy region

Masanori Matsudaa, Junichi Nagatab,∗, Hiro Yoshinoc, Yutaka Yoshinoaa Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima 739-8521, Japan

b Venture Business Laboratory, Hiroshima University, Higashi-Hiroshima 739-8527, Japanc Faculty of Health Sciences, Hiroshima International University, Kurose 724-0695, Japan

Received 29 July 1999; received in revised form 27 January 2000

Abstract

PANN is a partial-wave analysis program which can be used to determine the complex scattering amplitudes of elasticnucleon–nucleon scattering by fitting the experimental data on many kinds of observables. It is able to apply for an analysis inthe wide energy region such as a few MeV∼10 GeV. In PANN, the higher partial-wave amplitudes for the analysis in the lowenergy region (<1 GeV) are given by the one-pion exchange amplitude, and the ones in the high-energy region of several GeVare evaluated by the modified one-boson exchange amplitudes which is equivalent to Veneziano amplitudes. 2000 Publishedby Elsevier Science B.V.

PACS:13.75.C; 13.85

Keywords:Partial-wave analysis; Nucleon–nucleon scattering; Regge amplitudes; Multi-pole; Dibaryon; One-boson-exchange

PROGRAM SUMMARY

Title of program: PANN

Catalogue identifier:ADMN

Program Summary URL:http://cpc.cs.qub.ac.uk/summaries/ADMN

Program obtainable from:Junichi Nagata, [email protected];CPC Program Library, Queen’s University of Belfast, N. Ireland

Licensing provisions:none

Computer: any computer with a Fortran 77 compiler

Programming language used:FORTRAN 77

No. of bits in a word:32

No. of bytes in distributed program, including test data, etc.:68 881

Distribution format: tar gzip file

Libraries and programs needed:none

Nature of physical problemPartial-wave analysis (PWA) is a model independent method to de-termine the scattering amplitudes by means of fitting the experimen-tal data on many kinds of observables. PANN carries out the energy-independent PWA of nucleon–nucleon scattering [1]. The obtainedphase-shifts can be used to construct the potential model at low en-ergies and to determine the coupling constants of the interactions

∗ Corresponding author. Present address: Faculty of Law, Kyusyu International University, Kitakyusyu 805-8512, Japan.

0010-4655/00/$ – see front matter 2000 Published by Elsevier Science B.V.PII: S0010-4655(00)00123-5

226 M. Matsuda et al. / Computer Physics Communications 131 (2000) 225–263

in various models for nucleon–nucleon system at intermediate ener-gies [2]. Unlike the usual PWA program, PANN is extended to makeit available to analyze the nucleon–nucleon scattering data up to afew ten GeV using Veneziano-type amplitudes [3]. The nucleon–nucleon interaction at very short range can be studied by PANN,which are expected to detect some new dynamics.

Method of analysisThe available experimental data are stored in the input file forPANN. Theχ2-minimization is carried out [4] and the free searchedparameters are varied so as to reproduce the experimental data. Aftercompleting theχ2-minimization, the user obtains the best-fit valuesof parameters (phase shifts, mixing parameters and reflection pa-rameters) together with their uncertainties simultaneously. PANNprovides the solution of phase shifts, mixing parameters, reflectionparameters and also the determined helicity amplitudes. In addition,the calculated values of the differential cross section and variousspin observables are provided. New experiments may be proposedby studying the predicted values for spin observables by PANN.

Restrictions on the complexity of the problemThe program is developed to perform the PWAs of the elasticpp

scattering and the elasticnp scattering, where their effects from in-

elastic channels are evaluated by the reflection coefficients ofS-matrix. The inelastic nucleon–nucleon scattering, for instance, one-pion production process can not be treated.

Typical running timeFrom several seconds to several ten minutes depending upon thenumber of experimental data and searched parameters. The com-puter time will increase if one takes smaller step size in the gradientof χ2-space.

References[1] M. Matsuda, J. Nagata, H. Yoshino, K. Harada, S. Ohara, Prog.

Theor. Phys. 93 (1995) 1059;J. Nagata, H. Yoshino, M. Matsuda, Prog. Theor. Phys. 95 (1996)691;H. Yoshino, J. Nagata, Y. Yoshino, M. Matsuda, N. Hiroshige,T. Ueda, Prog. Theor. Phys. 95 (1996) 577.

[2] See, for instance, S. Sawada, T. Ueda, W. Watari, M. Yonezawa,Prog. Theor. Phys. 28 (1962) 28.

[3] M. Kawasaki, Y. Susuki, M. Yonezawa, Prog. Theor. Phys. 47(1972) 589.

[4] Y. Oyanagi, Library of Computer Center of University of Tokyo.

1. Introduction

Partial-wave analysis (PWA) has constantly supplied a motivation for particle-scattering experiments and playedan important role in the determination of hadron scattering amplitudes. Especially for resonance detection thismethod has been fruitful. Recently it has contributed to the study of dibaryon resonance [1–3]. Hereafter we mayexpect these analyses to motivate spin-polarized experiments and be continuously in the forefront of researchconcerning the strong interaction. There will certainly be some problems to be solved involving PWA.

With PWA one can satisfy the unitarity condition for scattering amplitudes automatically and take account forspin effects without any assumptions more easily than in total amplitude analyses. On the other hand, some faultsexist. The main fault is the difficulty that arises from the ever-increasing number of participating partial waves asthe energy of the incident particle increases. This causes a doubt on its effectiveness at high energies. Nevertheless,the utility of the PWA ofNN scattering has been demonstrated by some groups in the region of a few GeV/c[1,4–6].

Some ways for dealing with this difficulty have been proposed. They are, for example, the phase-band methodsuggested by Moravcsik [7] and the accelerated-convergence-expansion method proposed by Cutkosky [8]. As iswell known, Cutkosky’s method has had success for PWAs ofπN scattering at a few tens of GeV. Such successby some new methods is expected also inNN scattering. The primitive PANN was developed by Matsuda andWatari [9] for the PWA ofNN scattering in the low energy region. Our expanded PANN described herein isproposed for a treatment of the above-mentioned fault and for a possible extension of the usual PWA to the PWAofNN scattering in the high energy region. Our method of extension might be said to originate in the methodologyproposed by Iwadare et al. [10] in the research of verification of the pion theory of the nuclear force.

The program PANN has been applied to the analyses of Argonne data at 6 GeV/c [11] and 12 GeV/c [12], wherethe modified one-boson exchange (OBE) amplitudes have been included by theK-matrix method. In the programdescribed herein, theS-matrix of the lower partial waves is either the one defined by Matsuda and Watari (MW-representation) [9] or the one by Arndt and Roper (VPI-representation) [13]. The user can choose either of those

M. Matsuda et al. / Computer Physics Communications 131 (2000) 225–263 227

S-matrix representations. The relativistic Coulomb-amplitude is added according to sub-programs developed byArndt.

In the next section, a brief formalism used in PANN program is described. In Section 3 some features of PANNare given and the structure of program is explained in Section 4. We give the input data format and the results oftest runs in Sections 5 and 6, respectively. Our application plan of PANN is given in Section 7. Finally we give anadditional comment in Section 8.

2. Partial-wave analysis of nucleon–nucleon scattering

In the usual PWAs ofNN scattering at low energies, the peripheral part of the scattering amplitude is providedby the one-pion exchange (OPE) amplitude. In the analyses ofNN scattering at intermediate energies, one mightconsider using the one-boson exchange amplitudes instead of the OPE-amplitude according to Iwadare’s way.Iwadare et al. [10] divided the inter-nucleon distance into three regions and estimated the reliability of the nuclearinteraction in each region. And they have shown the validity of the OPE-potential in the outer region, whichhas since been the general view-point of the strong interaction. OBE-contributions are used in the intermediateregion. OBE-amplitudes, however, have at-dependence considerably different from that in the phenomena at highenergies, which is the reason why OBE-amplitudes have not been used for a modified partial-wave analysis. Thisdifference oft-dependence is thought to be due to form-factor-like effects and/or Regge-pole type behavior.

In the PANN program described herein, the modified OBE-amplitudes are used instead of the OPE-amplitude.They have been introduced by Kawasaki et al. [14] in the phenomenology ofpp scattering at high energies as theones equivalent to Veneziano amplitudes.

The OBE-amplitude forNN scattering is modified as follows:

g2

m2− t →g2

m2

(Λ2

Λ2− t/n)n, (1)

wheret is the squared momentum transfer andm the observed mass of the exchanged boson.g, Λ andn are theparameters for a particular boson, which are searched with the phase shifts(δ, η) of low partial waves to determinethe best fit to the experimental data (n is constrained as 106 n6 1 in the search). For the partial wave amplitude,this modification implies the replacement of the Legendre function of the second kind(QJ ):

QJ (x0)→ nΛ2

m2 (x′0− 1)n

(−1)n−1

(n− 1)!Q(n−1)J (x ′0), (2)

x0= 1+ m2

2p2, (3)

x ′0= 1+ nΛ2

2p2 , (4)

whereQ(n)J is thenth derivative ofQJ andp the c.m.s. momentum. This modification makes it possible to do theusual field-theoretical calculation of theNN scattering-amplitude with the multi-pole propagator. We calculate themodified OBE-amplitudes with the well-known exchanged mesons:π , σ , ρ andω [15].

The scattering amplitude is given by the partial wave amplitude and the modified OBE-amplitude as follows:

M =∑`6`m

f`(δ`, η`)+∑

`x>`>`mf`(δ`(OBEC), η`

)+MOBEC(` > `x). (5)

Heref`(δ`, η`) is the contribution from the real and the imaginary phase shifts(δ`, η`) for the orbital angularmomentum . δ` (OBEC) are the real phase shifts obtained from the modified OBE-amplitudes by theK-matrixmethod.


3. Features of program

The features of PANN are as follows:

3.1. Coulomb amplitudes

The non-relativistic Coulomb amplitudes are usually used for PWA. The subroutine for relativistic Coulombamplitudes are included in PANN which was calculated by Lechanoine et al. [16] based on the one-photon exchangeinteraction between nucleons, and developed by Arndt. It is needed for the extension of PWA to relativistic region.

3.2. New kinds of observables

Recent polarized-spin experiments of nucleon–nucleon scattering include the mixed spin observables of severalusual ones. For instance, the experiments done by SATURNE II provide the following observables [17]. Here wecall them as follows:

SAT1≡A00kk + δA00sk =ALL + δASL, (6)

SAT2≡A00n0+ δA00sk = P + δASL, (7)

SAT3≡K0sk0+ αK0ss0+ βK0kk0=KLS + αKSS + βK0kk0, (8)

SAT4≡N0nkk + βK0kk0=HLLN + βKLL, (9)

SAT5≡N0snk + αK0ss0=HNLS + αKSS, (10)

SAT6≡N0skn + αN0ssn + βN0kkn =HLNS + αHSNS + βKLL, (11)

SAT7≡K0nn0+ αN0nsk + βK0ks0+ γN0knk =HLNS + αHSLN + βLSL + γHNLL, (12)

whereα, β , γ andδ are the coefficients depending on the energy. The observablesAij , Kij andHijk are definedin Appendix. The definition ofAijk`,Kijk` andNijk` should be referred in Ref. [17].

The subroutine PRECES treats these mixed spin observables correctly.

3.3. Higher partial waves

In order to make it possible to perform the PWA in high energy region, the modified OBE amplitudes are usedin PANN instead of the one-pion exchange amplitudes which are included in PWAs in low energy region. We candecrease drastically the number of free-searched parameters in PANN by using modified OBE amplitudes.

3.4. Representation of scattering amplitudes

Currently there are two ways to parameterize the scattering amplitudes of nucleon–nucleon scattering as follows:

3.4.1. MW-representationTheS-matrix with an orbital angular momentumand a total angular momentumJ in the case = J is given

by

S`,J = η`,J exp(2iδ`,J ), (13)

whereη`,J andδ`,J are the reflection parameters and the phase shifts with the orbital angular momentum` and thetotal angular momentumJ .

For coupled state between`= J ± 1 waves,

SJ =( √

1− |ρ|2η−e2iδ− iρ√η−η+ei(δ−+δ+)

iρ√η−η+ei(δ−+δ+)

√1− |ρ|2η+e2iδ+

). (14)


Hereδ+ = δJ+1,J , δ− = δJ−1,J , andη(+)− are the reflection parameters with`= J + 1 (J − 1), respectively.ρ isthe mixing parameter.

3.4.2. VPI-representationThe partial waveS-matrix is given by

S = 1+ iK1− iK (15)

for the singlet state and the uncoupled triplet state.K is represented as

K = tanδ + i tan2ρ. (16)

For the coupled triplet states, theK-matrix is taken as

K =Kr− Kr0

Kr0 Kr+

+ i tan2ρ− tanρ− tanρ+

tanρ− tanρ+ tan2ρ+

, (17)

Kr± = sin(δ+ + δ−)± cos(2ε)sin(δ+ − δ−)cos(δ+ + δ−)+ cos(2ε)sin(δ+ − δ−) , (18)

Kr0= sin(2ε)

cos(δ+ + δ−)+ cos(2ε)cos(δ+ − δ−) . (19)

Hereδ± = δJ±1,J , ρ± = ρJ±1,J andε is the mixing parameter.In PANN, it is possible to perform the PWA by means of either MW- or VPI-representations, and the obtained

solution by VPI-representation is automatically converted to the one by MW-representation.

3.5. Forward amplitudes

The helicity amplitudes which are concerned in nucleon–nucleon scattering are

Φ1= 〈++ |M| ++〉,Φ2= 〈−− |M| ++〉,Φ3= 〈+− |M| +−〉,Φ4= 〈+− |M| −+〉,Φ5= 〈++ |M| +−〉, (20)

where+(−) are the helicity+1/2(−1/2) andM is aM-matrix [18].

Φ1= 12

(Mss + cosθcM00−

√2sinθcM10

),

Φ2= 12

(−Mss + cosθcM00−√

2sinθcM10),

Φ3= 12

[(1+ cosθc)M11+ (1− cosθc)M1−1+

√2sinθcM01

],

Φ4= 12

[(1− cosθc)M11+ (1+ cosθc)M1−1−

√2sinθcM01

],

Φ5= 12

(−sinθcM11+√

2cosθcM01+ sinθcM1−1)

= 12

(−sinθcM00−√

2cosθcM10), (21)

whereθc is the scattering angle in the c.m.s.In the forward direction(θc = 0), only Φ1, Φ2 andΦ3 are not zero. It means that the forward amplitudes are

completely determined by measuring the six kinds of observables atθc = 0.


In our program PANN, the seven observables (σt , σr , 1σT , 1σL, α [19], ReF2 [20] and ReF3 [20]) can betreated as experimental data.

σt = 2√π Im

[Φ1(0)+Φ3(0)

], (22)

σr = π

p2

∑`,J

(2`+ 1)(1− |η`,J |2

), (23)

1σT =−4√π ImΦ2(0), (24)

1σL = 4√π Im

[Φ1(0)−Φ3(0)

], (25)

α =Re[Φ1(0)+Φ3(0)

]/ Im

[Φ1(0)+Φ3(0)

], (26)

ReF2= PL√π

ReΦ2(0), (27)

ReF3= PL√π

Re[Φ1(0)−Φ3(0)

], (28)

wherePL is the momentum in the laboratory system.

4. Structure of program

The program PANN is written in the FORTRAN77 language (or called VS FORTRAN on IBM computers). It iscomposed of the main program and 29 subprograms as follows:• MAIN: Reads the input data for operating PANN and the starting values forχ2-minimizing search, controls

the whole program and prints the obtained results.• EXPERM: Reads the experimental data, arranges them, calculates the associated Legendre polynomials at

each data point and prints the used experimental data.• VPIDTF: Reads the experimental data by the input data format of the VPI (Virginia Polytechnic Institute)-

group.• TANGA: Converts the differential cross section from mb/(GeV/c)2 unit to mb/sr units. One can read the

differential cross section data in either of the units.• TOTAL: Calculates the forward observables.• CROSE: Calculates the other observables. See Appendix.• PHASE: Calculates the partial-wave amplitudes from the supported phase-shift parameters.• SUMSQ: Calculates theχ2 value, which is defined as

χ2=∑ij

[θ thij − nj θex

ij

nj1θexij

]2

+∑j

[1− nj1nj

]2

, (29)

or

χ2=∑ij

[Njθ

thij − θex

ij

1θexij

]2

+∑j

[1−Nj1nj

]2

, (30)

whereθ thij is the theoretical value of an observablei, and θex

ij , 1θexij are its experimental value and error

by the j th experiment, respectively.nj and1nj , are the experimental renormalization parameter and thesystematic error of thej th experiment, which can be searched in theχ2-minimization, if desired.Nj is therenormalization parameter of the calculated observable corresponding to datum with the systematic error1ni ,of the experiment. In this program,Ni can be determined only at the minimum point of theχ2-valley. Theway of using these parameters varies among the different analyses. One can choose either Eq. (29) or (30) inoperating the PANN.


• RENORM: Calculates the parameterNj at theχ2-minimum point.• MATRIX: Calculates theM-matrix from the partial wave amplitudes.• LEGEN: Calculates the Legendre functions of the second kind(QJ ) and theirnth derivativesQ(n)J .• NNOBAM: Calculates the partial-wave OBE-amplitudes ofNN scattering with pseudoscalar boson(π),

scalar boson(σ ) and vector bosons(ρ,ω).• SMOBE: Calculates the OBE-amplitudes for the exchanged bosons (π , σ , ρ, ω) and calculates theirM-

matrix.• MODIAM: Puts the multi-pole parameters(n,Λ) of the OBE-amplitudes into the subprogram NNOBAM.• WATARI: Calculates the parameters in MW-representation.• PHOUT: Calculates Hoshizaki’s parameters [18].• TTOUT: Converts the scattering amplitudes between Arndt’s, Hoshizaki’s and ours. Print the calculated

observables and the helicity amplitudes.• STAPP: Calculates theS-matrix by Stapp’s definition [21]. This subroutine is used only for an analysis of theNN scattering data in the elastic region.• TCAL: Connects the MW-representation and the VPI-representation.• TCALL: Calculates the amplitudes 2iT = S − 1 by using MW-representation.• DTOT & KTOT: Calculates theS-matrix by theK-matrix form.• COULM: Connects this program to the subprogram NNCOUL which was written by R.A. Arndt.• NNCOUL: Calculates the relativistic Coulomb-amplitude forNN scattering.• NRCOUL: Calculates the non-relativistic Coulomb-amplitude forNN scattering.• POWELL & INV: χ2-minimization program which was written by Oyanagi [22] with Powell’s modified

gradient method.• DATAST: Generates an input-data file to be read for starting the 2nd search from the parameters obtained by

the 1st search, which is needed for an analysis of high-energy data in the present status of computer ability. Itmay be unavoidable to do some repeated search and some artificial operations because of the sharp diffractionpattern of the differential cross section.• BLOCK DATA: Gives the values of Planck constant, light velocity and 1 eV= 1.602177× 10−12 erg. In

PANN, all calculations are done in the natural unit and the conversion of calculated results to the usual unit isgiven by the c.g.s. unit.• PRECES: Calculates mixed spin observables measured at Saclay.

5. Input data format

The symbols for parameters and their functions in each read statement are explained in this section. The readingformats are shown in the square brackets. The input-data file must be formatted in the sequence shown in thefollowing:

(1) The 1st read statement:

IWV , ICR, IAMP,NCV, ICOUL [5I2]IWV=0: VPI-representation is used in PANN;=1: MW-representation is used in PANN.

ICR=0: relativistic Coulomb amplitudes are used;=1: non-relativistic Coulomb amplitudes are used.

IAMP=0: execution of a partial wave analysis;=1: calculation of the scattering amplitudes and the observables by using the current phase shifts and/or

boson parameters withoutχ2-minimizing search.


NCV=0: prints the amplitudes of the three groups; VPI, Kyoto Univ. and Hiroshima Univ. If NCV6=0, theyare not printed. So NCV=0 must be used when IAMP=1.ICOUL=0: Coulomb-correction is done;=1: Coulomb amplitude is suppressed;=2: Coulomb correction is excluded.

(2) The 2nd read statement:

EMP,BM(1–4) [5F10.0]EMP: nucleon mass which is taken as 938.272 MeV if defaulted.BM(1): pion mass which is taken as 134.98 MeV if defaulted.BM(2) and BM(3): vector boson masses in MeV.BM(4): scalar boson mass in MeV.

(3) The 3rd read statement:

T,NP,LX , IPHASE, IANG, IOBE,JR, ITEST, IWRITE, IDATA [F10.0, 9I2]T: kinetic energy of incident nucleon in the laboratory system in MeV.NP=1: an analysis ofpp scattering;=2: an analysis ofnp scattering.

LX: the boundary value(`x) of the orbital angular momentum to which the scattering amplitude iscalculated with the partial wave amplitudef` as seen in the expression Eq. (5). If it is defaulted, LX=9 istaken automatically, and if LX>35, LX=35 is taken.IPHASE: the number of fixed parameters which are excluded from the searched parameters, so that it givesthe number of phase-shift parameters and boson parameters which are read in by the 8th read statement.IANG=1: input of the phase-shift parameters in degrees;=0: input of them in radians.

IOBE: specification of the method of calculation of the real phase shiftsδ`(OBEC) by the modified one-boson exchange model, i.e. IOBE=0 forK-matrix calculation and=1 for Born approximation.JR: parameter for grouping the experimental data. It is not needed usually, so should be equal to 0.ITEST: If it is not taken as 0, the finalχ2-value is not only printed out, but also the intermediateχ2-valuesare printed out at each cycle.IWRITE=0: printing the complete information about the obtained result;

=1: a partial print of the obtained result.IDATA: If it is not equal to 0, the subprogram DATAST is called to generate a data file for the next search.

(4) The 4th read statement:

IUDX , IATL , ITOTAL , IOUT, IOUT1,NRM [I3, 5I2]IUDX: total number of experimental groups of the used data.IATL: specification of the unit for the differential cross section, i.e.IATL=0 for mb/sr and IATL=1 for mb/(GeV)2.ITOTAL=1: an analysis with some forward data;

=0: an analysis without forward data.IOUT: =0 to print out the numbered experimental data which are needed for investigation of the M-value of data, and6= 0 for no printout of them. Each term of the 1st summation in Eq. (29) or (30) i.e.[(θi − θei )/1θei ]2 is called as M-value.IOUT1:=0 to print out the used data-file of VPI and6= 0 for no printout of them.NRM: If you want to carry out the renormalization (Ni in Eq. (31)) of the calculated observables atχ2-minimum point, put NRM as6= 0.


(5) The 5th read statement.The 5th, 6th and 7th read statements are for reading the experimental data. In PANN program, the inputdata format is same as the one used in SAID maintained by VPI-group. In this format, the data given bydifferent experiments are divided into the different groups of data:

T,ND,AN(1–4),C(1–5) [F9.3, I4,4F7.3,2(1X,A4),1X,3A4]T: kinetic energy of incident nucleon in the laboratory system for each experimental data group.ND: total number of data provided by each experimental data groups.AN(1): systematic error of the experiments.AN(2–4): coefficients used to calculate the mixed spin observables measured by SATURNE II [17] (forinstance, SAT1, SAT2, etc.).C(2): gives the kind of observables of the experimental data where the four columns are used, that is readby A4 type using four characters. The correspondence between the symbols in PANN and the observablesis as follows:DSG=dσ/d�, P=polarization, R, A, RP, AP= R, A, R′, A′ (Wolfenstein parameters), ALS, ANN, ASS,ASL, ALL, DNN, DSS, DSL, DLS, DLL, KSS, KSL, KLS, KLL, KNN, DSSP, DSLP= ALS , ANN , ASS ,ASL, ALL, DNN , DSS , DSL, DLS , DLL, KSS , KSL, KLS , KLL, KNN , DSS ′ , DSL′ (two spin correlationparameters), SAT1, SAT2, SAT3= SAT1, SAT2, SAT3 (mixed spin observables of two spin correlationparameters), CQKN= Cqkn (observable defined in the c.m.s. [18]), HSNS, HLNL, HSNL, HLNS, HNSS,HLSN, HNLS, HSSN, HSLN, HLLN, HNSL, HNLL= HSNS , HLNL, HSNL, HLNS , HNSS , HLSN ,HNLS ,HSSN ,HSLN ,HLLN ,HNSL,HNLL (three spin correlation parameters), SAT4, SAT5, SAT6, SAT7,SAT8= SAT4, SAT5, SAT6, SAT7, SAT8 (mixed spin observables of three spin correlation parameters),DSN, CRN0, CRU2, CIN1, CIN2, CIN0=DSN , CRN0, CRU2, CIN1, CIN2, CIN0 (constraints), SGTR,SGT, SGTT, SGTL, ALFA, REF2, REF3= σr , σt ,1σT ,1σL, α, ReF2, ReF3 (forward observables).C(1), C(3–5): some remarks for the experiments.


COM(1–18) [18A4]COM: comments for the each experimental data group. References of the experimental data are writtenhere.


ANGL,EXD,DEXD [3(F7.0,F10.0,F7.0),8X]ANGL: scattering angle(θc) in degree unit if IATL=0, or the squared momentum transfer(−t) in (GeV/c)2

unit if IATL =1.EXD, DEXD: experimental value of the observable and its experimental error, respectively.


IQ,NQ,LQ,JQ,QEL [4(2I1,2I2,1X,F9.0)]IQ=0 and 1 mean for QEL to be the real phase shiftδ and the imaginary oneη (or ρ in the case of the VPI-representation), respectively. In this case, NQ=1 for the spin-singlet state and NQ=3 for the spin-tripletstate. LQ and JQ are the orbital and the total angular momentum of phase shift QEL, respectively. IQ=2means QEL is the boson–nucleon coupling parameterg. In this case, JQ=1 (gP ) for the pseudoscalarboson, JQ=2 (gV ) and 3(fV ) for the vector bosons and JQ=4 (gS) for the scalar boson. NQ and LQhave no meanings. IQ=3 means that (1.0+ QEL ) is renormalization parametersni multiplied by theexperimental data and their errors in Eq. (29). In this case, (NQ, LQ) shows the kind of observables, andJQ its data group. IQ=4 means that QEL is the auxiliary mass parameter A of the boson in GeV unit, whichmust be equal to the observed boson mass BM if you want the unmodified OBE-amplitude. IQ=5 means


that QEL is the pole-multiplicity of the modified OBE-amplitude, which must be taken as 1 if you wantthe unmodified OBE-amplitude. Here for IQ=4 and 5, NQ, LQ and JQ have the same meanings as thoseof the case IQ=2, respectively. The number of QEL parameters has to be equal to IPHASE in the 3rd readstatement. The values of QEL given here are fixed in theχ2-minimizing search.


NFIRST,LANG,NOTE,LOS,NCUT,DELTA,ERROR, ISTART [I3,2I2, I1, I4,2F8.0,1X, I1]NFIRST: the total number of varied parameters inχ2-minimizing.LANG: specification of the unit of phase-shift parameters inputted in the next read statement, i.e. LANG=0for radians and=1 for degrees.NOTE: the number of the comment cards which are read in by the 11th read statement.LOS: If LOS6=0, the theoretical values of observables calculated by the obtained solution are not printedout.NCUT: the upper limit of the iteration ofχ2-minimizing search in the subroutine POWELL.DELTA: this gives the step size in the gradient ofχ2-space, i.e. (DMAX–DMIN)/DELTA. If it is leftout, DELTA=500.0 is taken. Here DMAX and DMIN are the maximum and the minimum values of thesearched domain of parameters, respectively and read in by the next read statement.ERROR: this gives the uncertainty C ofχ2-value atχ2-minimum point where C= ERROR∗(Ne −Np),Ne is the total number of experimental data, andNp the total number of searched parameters. If it is leftout, it is taken as 0.0001.ISTART=0: an execution of theχ2-minimizing search and no outputs of the calculated obervables forconnecting with the other program of X–Y plotting;

=1: a searching and plotting;=2: no searching and no plotting;=3: no searching and plotting.


IR,NR,LD,JD,DEL,DMAX ,DMIN [3(2I1,2I2,F10.0,2F4.0)]IR, NR, LD and JD have the same means as IQ, NQ, LQ and JQ in the 8th read statement, respectively.DEL: starting values of the varied parameter.DMAX, DMIN: the maximum and the minimum values of the searched domain of DEL. If they are notgiven, they are taken as (180.,−180.) for the phase-shift parameters in degrees, (10.,−10.) for the boson–nucleon coupling parameters, (10., 0.) for the auxiliary mass parameters, (10., 1.) for the pole-multiplicityparameters and (0.3,−0.3) for the experimental renormalization parameters. The total number of DELparameters has to be equal to NFIRST in the 9th read statement.


COMMENT [9A8]You can write some comment statements about your analysis in the number of lines specified by theparameter NOTE in the 9th read statement.

6. Results of test runs

In Fig. 1 the calculated values of dσ/d�, ASL, ANN andALL by the obtained solutions in PANN are shownwith experimental data, which are also outputed by PANN.


Fig. 1. The experimental data of elasticpp scattering atPL = 6 GeV/c, and their predictions by the PWA solution with PANN.

7. Application plans of PANN

See Refs. [5,23–25] for the detailed applications of PANN to the analyses ofNN scattering. The followingapplications are prepared in future by dividing the energy region to three parts as,

7.1. TL = 10 MeV∼ 800 MeV

There are plentiful data forNN scattering in this energy region. Therefore we can determine the pion–nucleoncoupling constants precisely, that is now important for a study of the hadron dynamics. The determination by usingPANN is now under progress.


7.2. TL = 0.5∼ 2.7 GeV

Recently COSY provided a plentiful and precise dσ/d� data in this energy region [26]. PANN can provide thebest solutions of phase-shifts and reflection parameters by using their data, the energy dependence of which woulddetermine the spin-parity, mass and width of a dibaryon [5,23,24].

7.3. TL = 3∼ 11 GeV

From our successive studies ofNN scattering by PANN, the catastrophic energy-dependence of the spin-orbitinteraction was found in this energy region [25]. This result was obtained by analyzing the data in wide energyregion (TL = 1∼ 11 GeV). It is very interesting to make clear the origin of this catastrophe.

8. Additional comment

In order to obtain a reasonable solution of PWA by using this program PANN, the user needs not only substantialdatabase to do it, but also enough knowedge of nuclear physics. One could not automatically obtain any reasonablesolutions. As posted up in SAID presented by R.A. Arndt, one’s profanity should be always noticed.

Acknowledgements

One of the authors (M.M.) is grateful to Professor R.A. Arndt and Professor L.D. Roper for valuable discussionsand suggestions. One of the authors (J.N.) would like to thank to Prof. J.J. de Swart and Dr. Th.A. Rijken forvaluable discussions during his stay at University of Nijmegen. He (J.N) was supported by the postdoctoralfellowship for research abroad of the Japan Society for the Promotion of Science.

Appendix. Observables for nucleon–nucleon scattering

In this appendix we summarize various expressions of observables for the elastic nucleon–nucleon scattering.In this list, the meaning of spin-correlation parameters is clear by the bracket symbol(i, j→ i ′, j ′) originally dueto Thomas [27] where four charactersi, j , i ′ andj ′ denote the spin directions of the beam, target, scattered andrecoil nucleons, respectively, andL, N andS indicate the longitudinal, the normal to the scattering plane and theN ×L directions, 0 implying the non-observation of spin-states.

dσ/dt = (0,0;0,0)= 12

[|Φ1|2+ |Φ2|2+ |Φ3|2+ |Φ4|2+ 4|Φ5|2], (31)

P = (0,N;0,0)= (N,0;0,0)= Im[(Φ1+Φ2+Φ3−Φ4)

∗Φ5]/(dσ/dt), (32)

D = (N,0;N,0)= {Re[(Φ∗1Φ∗3 −Φ∗2Φ∗4)

]+ 2|Φ5|2}/(dσ/dt), (33)

R = (S,0;S,0)= {−Re

[Φ∗5(Φ1−Φ2+Φ3+Φ4)

]sinθS +Re(Φ∗1Φ3+Φ∗2Φ4)cosθS

}/(dσ/dt), (34)

R′ = (S,0;L,0)= {−Re

[Φ∗5(Φ1−Φ2+Φ3+Φ4)

]cosθS −Re(Φ∗1Φ3+Φ∗2Φ4)sinθS

}/(dσ/dt), (35)

A= (L,0;S,0)= {Re

[Φ∗5(Φ1−Φ2+Φ3+Φ4)

]cosθS + 1

2

(|Φ1|2− |Φ2|2+ |Φ3|2− |Φ4|2)sinθS

}/(dσ/dt), (36)


A′ = (L,0;L,0)= {−Re

[Φ∗5(Φ1−Φ2+Φ3+Φ4)

]sinθS + 1

2

(|Φ1|2− |Φ2|2+ |Φ3|2− |Φ4|2)cosθS

}/(dσ/dt), (37)

ANN = (N,N;0,0)=Re[(Φ∗1Φ2−Φ∗3Φ4)+ 2|Φ5|2

]/(dσ/dt), (38)

ASS = (S,S;0,0)=Re[(Φ∗1Φ2+Φ∗3Φ4)

]/(dσ/dt), (39)

ASL = (S,L;0,0)=Re[(Φ1+Φ2−Φ3+Φ4)

∗Φ5]/(dσ/dt), (40)

ALL = (L,L;0,0)= 12

[−|Φ1|2+ |Φ2|2+ |Φ3|2+ |Φ4|2]/(dσ/dt), (41)

DNN = (0,N;0,N)={Re[(Φ∗1Φ3−Φ∗2Φ4)

]+ 2|Φ5|2}/(dσ/dt), (42)

DSS = (0, S;0, S)= {−sinθR Re

[(Φ1−Φ2+Φ3+Φ4)

∗Φ5]− cosθR Re[Φ∗1Φ3+Φ∗2Φ4]

}/(dσ/dt), (43)

DSL = (0, S;0,L)= {−sinθR Re[Φ∗1Φ3+Φ∗2Φ4] + cosθR Re

[(Φ1−Φ2+Φ3+Φ4)

∗Φ5]}/(dσ/dt), (44)

DLS = (0,L;0, S)= { 1

2 sinθR[|Φ1|2− |Φ2|2+ |Φ3|2− |Φ4|2

]− cosθR Re[(Φ1−Φ2+Φ3+Φ4)

∗Φ5]}/(dσ/dt), (45)

DLL = (0,L;0,L)= {−sinθR

[(Φ1−Φ2+Φ3+Φ4)

∗Φ5]− 1

2 cosθR Re[|Φ1|2− |Φ2|2+ |Φ3|2− |Φ4|2

]}/(dσ/dt), (46)

KNN = (N,0;0,N)={−Re

[(Φ∗1Φ4−Φ∗2Φ3)

]+ 2|Φ5|2}/(dσ/dt), (47)

KSS = (S,0;0, S)= {−sinθR Re

[(Φ1−Φ2−Φ3−Φ4)

∗Φ5]− cosθR Re

[Φ∗1Φ4+Φ∗2Φ4

]}/(dσ/dt), (48)

KSL = (S,0;0,L)= {−sinθR Re

[Φ∗1Φ4+Φ∗2Φ3

]− cosθR Re[(Φ1−Φ2−Φ3−Φ4)

∗Φ5]}/(dσ/dt), (49)

DLS = (L,0;0, S)= {−1

2 sinθR[Φ∗1Φ4+Φ∗2Φ3

]− cosθR[|Φ1|2− |Φ2|2+ |Φ3|2− |Φ4|2

]}/(dσ/dt), (50)

KLL = (L,0;0,L)= {sinθR Re

[(Φ1−Φ2−Φ3−Φ4)

∗Φ5]+ 1

2 cosθR(|Φ1|2− |Φ2|2− |Φ3|2+ |Φ4|2

)}/(dσ/dt), (51)

HSNS = (S,N;0, S)= {− sinθR Im[Φ∗1Φ2+Φ∗3Φ4] + cosθR Im

[(Φ1−Φ2−Φ3−Φ4)

∗Φ5]}/(dσ/dt), (52)

HNSS = (N,S;0, S)= {sinθR Im[Φ∗1Φ2−Φ∗3Φ4] − cosθR Im

[(Φ1−Φ2−Φ3−Φ4)

∗Φ5]}/(dσ/dt), (53)

HNLS = (N,L;0, S)= {sinθR Im

[(Φ1−Φ2+Φ3+Φ4)

∗Φ5]− cosθR Im[Φ∗1Φ4+Φ∗2Φ3]

}/(dσ/dt), (54)

HNLS = (L,N;0, S)= {sinθR Im

[(Φ1−Φ2−Φ3−Φ4)

∗Φ5]− cosθR Im[Φ∗1Φ3+Φ∗2Φ4]

}/(dσ/dt), (55)

HLNL = (L,N;0,L)= {− sinθR Im[Φ∗1Φ3+Φ∗2Φ4] − cosθR Im

[(Φ1−Φ2−Φ3−Φ4)

∗Φ5]}/(dσ/dt), (56)

HSNL = (S,N;0,L)= {sinθR Im

[(Φ1−Φ2−Φ3−Φ4)

∗Φ5]+ cosθR Im[Φ∗1Φ2+Φ∗3Φ4]

}/(dσ/dt), (57)


HSSN = (S,S;0,N)= Im[(Φ1−Φ2−Φ3−Φ4)

∗Φ5]/(dσ/dt), (58)

HLSN = (L,S;0,N)=− Im[Φ∗1Φ3−Φ∗2Φ4

]/(dσ/dt), (59)

HSLN = (S,L;0,N)= Im[Φ∗1Φ4−Φ∗2Φ3

]/(dσ/dt), (60)

whereθS andθR are the scattering and recoil angle in the laboratory system, respectively.

References

[1] N. Hoshizaki, Prog. Theor. Phys. 57 (1977) 335; Prog. Theor. Phys. 67 (1977) 1099.[2] R.A. Arndt, quoted in Phys. Rep. 64 (1980) 47.[3] A. Yokosawa, Phys. Rep. 64 (1980) 47;

J. Nagata, H. Yoshino, M. Matsuda, Prog. Theor. Phys. 95 (1996) 691.[4] R.A. Arndt, C.H. Oh, I.I. Strakovsky, R.L. Workman, F. Dohrman, Phys. Rev. C 56 (1997) 3005.[5] M. Matsuda, Nucl. Phys. A 631 (1998) 436c.[6] J. Bystricky, C. Lechanoine-LeLuc, F. Lehar, Eur. Phys. J. C 4 (1998) 607.[7] M.J. Moravcsik, Phys. Rev. 177 (1969) 2587.[8] R.E. Cutkosky, Phys. Rev. 174 (1968) 1859.[9] M. Matsuda, W. Watari, Lett. Nuovo Cimento 6 (1973) 23.

[10] J. Iwadare, D. Otsuki, R. Tamagaki, W. Watari, Suppl. Prog. Theor. Phys. 3 (1956) 32.[11] M. Matsuda, H. Suemitsu, W. Watari, M. Yonezawa, Prog. Theor. Phys. 62 (1979) 1436; 64 (1980) 1344;

M. Matsuda, H. Suemitsu, M. Yonezawa, Phys. Rev. D 33 (1986) 2563.[12] M. Matsuda, H. Suemitsu, W. Watari, M. Yonezawa, Prog. Theor. Phys. 66 (1981) 1102.[13] R.A. Arndt, L.D. Roper, Phys. Rev. D 25 (1982) 2011.[14] M. Kawasaki, Y. Susuki, M. Yonezawa, Prog. Theor. Phys. 47 (1972) 589.[15] S. Sawada, T. Ueda, W. Watari, M. Yonezawa, Prog. Theor. Phys. 28 (1962) 991.[16] C. Lechanoine, F. Lehar, F. Perrot, P. Winternitz, Nuvo Cimento A 56 (1980) 201.[17] C.D. Lac et al., Nucl. Phys. B 297 (1988) 653; Nucl. Phys. B 315 (1989) 284; Nucl. Phys. B 321 (1989) 269; Nucl. Phys. B 321 (1988)

284.[18] N. Hoshizaki, Suppl. Prog. Theor. Phys. 60 (1968) 107.[19] W. Grein, Nucl. Phys. B 131 (1977) 255.[20] W. Grein, Kroll, Nucl. Phys. B 137 (1978) 173.[21] H.P. Stapp, T.J. Ypsilantis, N. Metropolis, Phys. Rev. 105 (1957) 302.[22] Y. Oyanagi, Library of Computer Center of University of Tokyo;

M.J.D. Powell, Comput. J. 7 (1965) 303.[23] J. Nagata, H. Yoshino, M. Matsuda, Prog. Theor. Phys. 95 (1996) 691.[24] M. Matsuda, H. Yoshino, J. Nagata, Y. Yoshino, N. Hiroshige, T. Ueda, Prog. Theor. Phys. 95 (1996) 577.[25] M. Matsuda, J. Nagata, H. Yoshino, K. Harada, S. Ohara, Prog. Theor. Phys. 93 (1995) 1059.[26] D. Albers et al., Phys. Rev. Lett. 78 (1997) 1652.[27] G.H. Thomas, Ph.D. thesis, University of California, LA (1969).


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PANN: Partial-wave analysis of nucleon–nucleon scattering in wide-energy region

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