Meson-exchange model for π N scattering and γ N →π N reaction

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PHYSICAL REVIEW C NOVEMBER 1996VOLUME 54, NUMBER 5

0556-28

Meson-exchange model forpN scattering andgN˜pN reaction

T. Sato1,2 and T.-S. H. Lee11Physics Division, Argonne National Laboratory, Argonne, Illinois 60439-4843

2Department of Physics, Osaka University, Toyonaka, Osaka 560, Japan~Received 7 June 1996!

An effective Hamiltonian consisting of bareD↔pN, gN vertex interactions and energy-independemeson-exchangepN↔pN,gN transition operators is derived by applying a unitary transformation to a moLagrangian withN,D,p, r, v, andg fields. With appropriate phenomenological form factors and couplconstants forr and D, the model can give a good description ofpN scattering phase shifts up to theDexcitation energy region. It is shown that the best reproduction of the recent LEGS data of the phasymmetry ratios ingp→p0p reactions provides rather restricted constraints on the coupling strengthsGE ofthe electricE2 andGM of the magneticM1 transitions of the bareD↔gN vertex and the less well-determinecoupling constantgvNN of v meson. Within the ranges thatGM51.960.05, GE50.060.025, and7<gvNN<10.5, the predicted differential cross sections and photon-asymmetry ratios are in an overalagreement with the data ofgp→p0p, gp→p1n, andgn→p2p reactions from 180 MeV to theD excitationregion. The predictedM11 andE11 multipole amplitudes are also in good agreement with the empirical valdetermined by the amplitude analyses. The constructed effective Hamiltonian is free of the nucleon renization problem and hence is suitable for nuclear many-body calculations. We have also shown thassumptions made in theK-matrix method, commonly used in extracting empirically thegN→D transitionamplitudes from the data, are consistent with our meson-exchange dynamical model. It is found thhelicity amplitudes calculated from our baregN→D vertex are in good agreement with the predictions of tconstituent quark model. The differences between these bare amplitudes and the dressed amplitudes, wcloser to the empirical values listed by the Particle Data Group, are shown to be due to the nonresonanexchange mechanisms. Within the range 7<gvNN<10.5 of thev meson coupling favored by the data of thphoton-asymmetry ratios ingp→p0p reactions, our values of theE2/M1 ratio for thegN→D transition are~0.061.3!% for the bare vertex and (21.860.9!% for the dressed vertex.@S0556-2813~96!04611-0#

PACS number~s!: 13.75.Gx, 21.45.1v, 24.10.2i, 25.20.Lj

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I. INTRODUCTION

The main objective of investigating photoproduction anelectroproduction of mesons on the nucleon is to study tstructure of the nucleon excited states (N* ). This has beenpursued actively@1# during the period around 1970. With thedevelopments at several electron facilities since 1980, mextensive investigations of theD excitation have been car-ried out both experimentally and theoretically@2#. Apartfrom the need for precise and extensive measurements whwill soon be possible at CEBAF and Mainz, an accuraunderstanding of theN* structure can be obtained only whean appropriate reaction theory is developed to separatereaction mechanisms from the hadron structure in tgN→pN,ppN reactions. The importance of this theoreticaeffort can be understood by recalling many years experienin the development of nuclear physics. For example, theformation about the deformation of12C can be extractedfrom 12C(p,p8)12C* (21,4.44 MeV! inelastic scattering onlywhen a reliable reaction theory@3#, such as the distorted-wave impulse approximation or the coupled-channemethod, is used to calculate the initial and final proton-12Cinteractions. Accordingly, one expects that theN* structurecan be determined only when the interactions in its decchannelsgN, pN, andppN can be calculated from a reli-able reaction theory. It is the objective of this work to address this problem from the point of view of meson exchanmodels. In contrast to approaches based on the dispers

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relations@1# or theK-matrix method@4–7#, our approach isaimed at not only an investigation of theN* structure butalso at the application of the constructed model to a constent calculation ofN* in nuclear many-body systems.

The meson-exchange models have been very successfdescribing nucleon-nucleon interactions@8#, electroweak in-teraction currents@9,10#, meson-meson scattering@11#, andmeson-nucleon scattering@12–14#. It is therefore reasonableto expect that the same success can also be achieved ininvestigation of pion photoproduction and electroproductioThis possibility has, however, not been fully explored. Thdynamical models of pion photoproduction developedRefs. @15–17# did contain the well-established meson exchange mechanisms of pion photoproduction, but phenoenological separable potentials were used to describepN multiple scattering. The improvement made in Ref.@18#suffered from the theoretical inconsistency in defining thmeson exchangepN interaction andpN→gN transition.The model developed in Refs.@19,20# also does not treatmeson exchange completely since a zero-range contact tis introduced to replace the particle-exchange terms of thpN potential. In all of these models, the incomplete treament of the meson-exchange interactions leads to somecertainties in interpreting the parameters characterizinggN→D vertex which is the main interest in testing hadromodels. The formulation developed in Ref.@21# can, in prin-ciple, be used to examine the meson-exchange mechaniin pion photoproduction, but has not been pursued nume

2660 © 1996 The American Physical Society

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54 2661MESON-EXCHANGE MODEL FORpN SCATTERING AND . . .

cally. In this work, we will try to improve the situation byapplying the unitary transformation method developedRef. @22# to derive from a model Lagrangian an effectivHamiltonian for a consistent meson-exchange descriptionboth thepN scattering and pion photoproduction. Furthemore, the constructed model can be directly used to imprand extend thepNN Hamiltonian developed in Ref.@23# toalso describe the electromagneticD excitation in intermedi-ate energy nuclear reactions.

The starting point of constructing a meson-exchanmodel is a model Lagrangian of relativistic quantum fietheory. The form of the Lagrangian is constrained by tobserved symmetries of fundamental interactions, suchLorentz invariance, isospin conservation, chiral symmeand gauge invariance. The most common approach@24# is tofind an appropriate three-dimensional reduction of the ladBethe-Salpeter equation of the considered model LagrangThe meson-exchange potentials are then identified withdriving terms of the resulting three-dimensional scatterequation. The most recent examples are thepN models de-veloped in Refs.@12–14,19#. The extension of this approacto investigate pion photoproduction has also been madRef. @20#.

Alternatively, one can construct a meson-exchange moby deriving an effective Hamiltonian from the consideremodel Lagrangian. Historically, two approaches have bedeveloped. The first one is to use the Tamm-Dancoffproximation@25#. This method leads to an effective Hamitonian which is energy dependent and contains unlinkterms, and hence cannot be easily used in nuclear many-bcalculations. A more tractable approach is to applymethod of unitary transformation which was developedFukuda, Sawada, and Taketani@26# and independently byOkubo @27#. This approach, called the FST-Okubo methohas been very useful in investigating nuclear electromagncurrents@28–30# and relativistic descriptions of nuclear interactions@31–34#. The advantage of this approach is ththe resulting effective Hamiltonian is energy independeand can readily be used in nuclear many-body calculatMotivated by the investigation of thepNN dynamics@23,35#, this method has been extended in Ref.@22# to derivean effective theory involving pion production channels.this work, we adopt this method to develop a dynamicmodel forpN scattering andgN→pN reactions.

It is necessary to explain here how our approach is relato the approach based on chiral perturbation theory~CHPT!@36#. Since chiral symmetry is a well-established dynamicsymmetry of strong interactions, it should be used to costrain our starting Lagrangian. This leads us to assumeour starting Lagrangian is an effective Lagrangian for genating the tree diagrams in CHPT. The parameters are tcompletely determined by the well-established chiral dynaics such as partially conserved axial-vector current~PCAC!and current algebra. Therefore, our model and CHPTidentical in leading orders. The differences come from hthe unitarity is implemented to account for thepN multiplescattering. In the spirit of CHPT, the ‘‘low’’-momentumpions are considered as weakly interacting Goldstone bosand hence their interactions with the nucleon can be treaas perturbations@37#. This amounts to restoring the unitaritperturbatively by calculating loop corrections order by ord

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It is then necessary to include more terms in the effectiLagrangian. A phenomenological procedure is then unavoable to determine the accompanied low-energy constants

In the meson-exchange model, one hopes to describepN multiple scattering in the entire kinematic region including the highly nonperturbativeD excitation region. The es-sential assumption is that thepN multiple scattering is gov-erned by a few-body Schro¨dinger equation with the drivingterms calculated from the starting Lagrangian in a perturbtion expansion in the coupling constants. This can be reized in practice only when the driving terms are regularizeby appropriate phenomenological form factors. Qualitativespeaking, the meson-exchange model is an alternativeCHPT in the kinematic region where perturbative calcultions become very difficult or impossible. Both approachinvolve phenomenological parameters. The success of eapproach depends on whether these parameters can be ipreted theoretically.

In this work we will focus on theD excitation and willlimit our investigation to the energy region where 2p pro-duction is negligibly small. By applying the unitary transformation of Ref.@22# to a model Lagrangian forN, D, p, r,v, andg fields, we have obtained an effective Hamiltoniaconsisting of bareD↔pN,gN vertex interactions andenergy-independentpN↔pN,gN transition operators. ThepN scattering phase shifts@38–40# are used to determine thehadronic part of the constructed effective Hamiltonian whichas only seven parameters for defining the vertices ofmeson-exchangepN potential and theD↔pN transition.The strong vertex functions in thegN→pN transition op-erator are then also fixed. This is a significant improvemeover the previous dynamical models@15–17# in which theemployed separable potentials have no dynamical relatwith the pion photoproduction operator. A consistent dscription of thepN scattering andgN→pN transition iscrucial for separating the reaction mechanisms due to mesexchange nonresonant interactions from the totalgN→Dtransition.

Once the hadronic part of the effective Hamiltoniandetermined, the resulting pion photoproduction amplituhas only three adjustable parameters:GM of magneticM1andGE of electricE2 transitions of the baregN→D vertex,and the less well-determinedvNN coupling constant. Wewill determine these three parameters by consideringmost recent LEGS data@41# of the photon-asymmetry ratiosin gp→p0p reactions. The resulting parameters are thtested against very extensive data in Refs.@42–44#.

It is customary to test hadron models by comparing ttheoretical predictions ofN*→gN transition amplitudeswith the empirical values listed by the Particle Data Grou~PDG! @45#. Since the first systematic calculation@46# basedon the constituent quark model was performed, it has beobserved that the predictedD→gN transition amplitudes@46–50# are significantly smaller than the empirical valuelisted by the PDG@45#. While the problem may be due to thelimitations of the constituent quark model, it is necessaryrecognize that the empirical values of the PDG are obtainby using theK-matrix method@4–7# or dispersion relation@1#. Both approaches contain assumptions about the nonrenant contributions to thegN→D transition and must be jus-tified from a dynamical point of view. Within our dynamica

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2662 54T. SATO AND T.-S. H. LEE

model, we will address this point concerning theK-matrixmethod. This leads us to identify our baregN→D vertexwith the constituent quark model. The dispersion relatiapproach@1,51# is defined in a very different theoreticaframework and therefore is beyond the scope of this invegation.

In Sec. II, we will use a simple model Lagragian to eplain how an effective Hamiltonian can be constructedusing the unitary transformation method of Ref.@22#. Themethod is then applied to realistic Lagrangians to deriveSecs. III and IV an effective Hamiltonian forpN scatteringand pion photoproduction. The equations for calculatingpN scattering andgN→pN amplitudes are also presentethere. The relationships with theK-matrix method are thenestablished. Results and discussions are given in Sec. V.conclusions and discussions of future studies are givenSec. VI.

II. METHOD OF UNITARY TRANSFORMATION

To explain the unitary transformation method of Ref.@22#@will be referred to as the Sato-Kobayashi-Ohtsubo~SKO!

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method#, it is sufficient to consider a simple system consisting of only neutral pions and fictituouss mesons. The ob-jective is to derive an effective Hamiltonian from the La-grangian density

L~x!5L0~x!1LI~x!, ~2.1!

whereL0(x) is the usual noninteracting Lagrangian, and thinteraction term is taken to be

LI~x!52gsppfp2 ~x!fs~x!. ~2.2!

The Hamiltonian can be derived from Eq.~2.1! by using thestandard method of canonical quantization. In the seconquantization form, we obtain~in the convention of Bjorkinand Drell @52#!

H5H01HI , ~2.3!

with

H05E dkW @Ep~k!a†~kW !a~kW !1Es~k!b†~kW !b~kW !#, ~2.4!

HI5gsppE dkW1dkW2dkW

A~2p!38Ep~k1!Ep~k2!Es~k!$@2a†~kW1!a~kW2!b~kW !d~kW12kW22kW !

1a†~kW1!a†~kW2!b~kW !d~kW11kW22kW !1a~kW1!a~kW2!b~kW !d~kW11kW21kW !#1@ H.c.#%, ~2.5!

es

he

wherea†(kW ) andb†(kW ) are, respectively, the creation operators forp ands particles,Ea(k)5Ama

21k2 is the free en-ergy for the particlea, [email protected].# means taking the Hermit-ian conjugate of the first term in the equation. We furthassume that the mass of thes meson is heavier than two-pion mass, i.e.,ms.2mp .

Because of the intrinsic many-body problem associatwith the starting quantum field theory, it is not possiblesolve exactly the equation of motion for meson-meson sctering defined by the above Hamiltonian. A simplificationobtained by assuming that in the low- and intermediaenergy regions, only ‘‘few-body’’ states are active and mube treated explicitly. The effects due to ‘‘many-body’’ stateare absorbed in effective interaction operators which cancalculated in a perturbation expansion in coupling constanThis few-body approach to field theory was pioneeredAmado and Aaron@53#. In the SKO approach, this isachieved by first decomposing the interaction HamiltoniHI , Eq. ~2.5!, into two parts:

HI5HIP1HI

Q ~2.6!

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HIP5gsppE dkW1dkW2dkW

A~2p!38Ep~k1!Ep~k2!Es~k!

3$a†~kW1!a†~kW2!b~kW !d~kW11kW22kW !1@ H.c.#%,

~2.7!

HIQ5gsppE dkW1dkW2dkW


3$@2a†~kW1!a~kW2!b~kW !d~kW12kW22kW !

1a~kW1!a~kW2!b~kW !d~kW11kW21kW !#1@ H.c.#%.

~2.8!

The elementary processes induced byHIP are illustrated in

the upper half of Fig. 1. Forms.2mp , thes→pp decayandpp→s annihilation are ‘‘real processes’’ and can takeplace in free space. On the other hand, the processp↔ps and vacuum↔pps induced byHI

Q are ‘‘virtualprocesses’’~lower part of Fig. 1!. They cannot occur in freespace because of the energy-momentum conservation. T

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essence of the SKO method is to systematically eliminatevirtual processes from the considered Hamiltonian by usunitary transformations. As a result the effects of ‘‘virtuprocesses’’ are included as effective operators in the traformed Hamiltonian.

The transformed Hamiltonian is defined as

H85UHU†

5U~H01HIP1HI

Q!U†, ~2.9!

whereU5exp(2iS) is a unitary operator defined by a Hemitian operatorS. By expandingU512 iS1••• , thetransformed Hamiltonian can be written as

H85H01HIP1HI

Q1@H0 ,iS#1@HI ,iS#

11

2!†@H0 ,iS#,iS‡1••• ~2.10!

To eliminate from Eq.~2.10! the virtual processes which arof first order in the coupling constantgspp , the SKOmethod imposes the condition that

HIQ1@H0 ,iS#50. ~2.11!

SinceH0 is a diagonal operator in Fock space, Eq.~2.11!clearly implies thatiS must have the same operator structuof HI

Q .To simplify the presentation, we writeHQ as

HIQ5(

nE FnOndkW1dkW2dkW , ~2.12!

whereOn denotes the part containing the creation and anhilation operators, andFn is the rest of thenth term in Eq.~2.8!. In the form of Eq.~2.12!, the solution of Eq.~2.11! canbe written as

iS5(n

i E SnOndkW1dkW2dkW . ~2.13!

FIG. 1. Graphical representation of the interaction HamiltoniaHI

P of Eq. ~2.7! andHIQ of Eq. ~2.8!.

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Our task is to findSn by solving Eq.~2.11!. Considering twoeigenstatesu i & and u f & of the free HamiltonianH0 such that^ f uOnu i &51, Eqs.~2.11!, ~2.12!, and~2.13! then lead to

iSn521

Ef2EiFn . ~2.14!

Note thatEi andEf are the eigenvalues of free HamiltonianH0, and hence the solutionSn is independent of the collisionenergyE of the total HamiltonianH. This is an importantfeature distinguishing our approach from the Tamm-Dancoapproximation. By using the above relation, it is easyverify that the solution of the operator equation~2.11! is

iS5gsppE dkW1dkW2dkW


3S F2a†~kW1!a~kW2!b~kW !d~kW12kW22kW !

2Ep~k1!1Ep~k2!1Es~k!

1a~kW1!a~kW2!b~kW !d~kW11kW21kW !

Ep~k1!1Ep~k2!1Es~k!G1 H.c.D .

~2.15!

By using Eq.~2.11!, Eq. ~2.10! can be written as

H85H01HI8 , ~2.16!

with

HI85HIP1@HI

P ,iS#1 12 @HI

Q ,iS#1 higher-order terms.~2.17!

SinceHIP , HI

Q, andS are all of the first order in the couplingconstantgspp , all processes included in the second and thiterms of theHI8 are of the order ofgspp

2 . But some of themare ‘‘real processes,’’ such as thes-exchangepp interactionand p-exchangepp→ss transition, as illustrated in theupper half of Fig. 2. The other processes are ‘‘virtual processes.’’ An example is the emission of twos mesons by apion illustrated in the lower half of Fig. 2. We thereforerewrite Eq.~2.17! as

ns FIG. 2. Graphical representation of the effective interactioHamiltoniansHI

8P of Eq. ~2.19! andHI8Q of Eq. ~2.20!.


HI85HIP1HI8

P1HI8Q1 (

n>3O~gspp

n !, ~2.18!

where

HI8P5~@HI

P ,iS#1 12 @HI

Q ,iS# !P, ~2.19!

HI8Q5~@HI

P ,iS#1 12 @HI

Q ,iS# !Q. ~2.20!

In the above two definitions,HI8P(HI8

Q) is obtained by evalu-ating the commutators using Eqs.~2.7!, ~2.8!, and~2.15! andkeeping only the real~virtual! processes in the results.

The next step is to perform a second unitary transformtion to eliminateH8Q. In this paper we only consider theffective Hamiltonian up to second order in the coupliconstant, and so we do not need to consider the secondtary transformation. The effective Hamiltonian is then otained by droppingH8Q and higher-order terms in Eq.~2.18!:

Heff5H1HIP1HI8

P , ~2.21!

whereHIP is defined by Eq.~2.7!, andHI8

P can be calculatedfrom Eq. ~2.19! by using the solution Eq.~2.15! for S. Notethat bothHI

Q and iS contain a virtualp↔ps vertex. Theyare included inHI8

P of the effective HamiltonianHeff in acommutator form, as seen in the second term of Eq.~2.19!.Consequently, the effect due to thep↔ps vertex can give ap→ps→p loop correction to the pion mass operator of teffective HamiltonianHeff . Furthermore, one can see by peforming some straightforward derivations that the matrixement ofHeff between ap and aps state vanishes. Thissimplification is the consequence of the SKO method definby Eq. ~2.11! in operator form, and is not due to the choicof a model space in which some virtual states are omittedwe choose the pion mass inH0 as the physical mass, the loocorrection should be dropped fromHI8

P in order to avoiddouble counting. A similar situation will be encounteredthe derivation of thepN effective Hamiltonian. The virtualN↔pN process can give aN→pN→N loop correction tothe nucleon mass operator. The matrix element of the cstructed effective Hamiltonian between anN state and apN state vanishes. In our model we choose the physmasses forN andp in the free Hamiltonian and hence thloop corrections in the effective Hamiltonian are aldropped. This phenomenological procedure saves us ffacing the complicated mass renormalization problemsolving thepN scattering problems.

Because of Eq.~2.14!, Eqs.~2.12! and ~2.13! lead to thesimple relation

^ f u iSu i &521

Ef2Ei^ f uHI

Qu i &, ~2.22!

whereu f & and u i & are two eigenstates ofH0. With the aboverelation, the calculation ofHI8

P , Eq. ~2.19!, becomes

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^ f uHI8Pu i &5(

nH ^ f uHI

Pun&^nuHIQu i &

1

Ei2En

2^ f uHIQun&^nuHI

Pu i &1

En2Ef

1^ f uHIQun&^nuHI

Qu i &1

2 F 1

Ei2En2

1

En2EfG J .

~2.23!

The calculation of the matrix element ofHI8P therefore has a

very simple rule. For a given choice of basis statesu i & andu f &, the allowed intermediate staten is determined by theoperator structure ofOn in Eqs. ~2.12! and ~2.13!. The de-nominator in Eq.~2.23! can easily be written down by usingeigenvalues of the free HamiltonianH0.

Evaluating HIP and HI8

P explicitly within the coupledpp % s space, Eq.~2.21! can be cast into the following morefamiliar form for pp scattering:

Heffpp5H01 f s↔pp1Vpp . ~2.24!

HereH0 is the free Hamiltonian operator forp ands me-sons. The second term describes thes↔pp transition withthe matrix element

^kW1kW2u f pp,sukW &5A2gspp

A~2p!38Ep~k1!Ep~k2!Es~k!.

~2.25!

The pp potentialVpp is obtained by using Eq.~2.23! tocalculate HI8

P between two pp states. For

u i &5ukW i1kW i2&5A12 @akW i1

†akW i2†

#u0& and ^ f u5^kW f1kW f2u

5^0u@akW f1akW f2#A12 the possible intermediate states in Eq.

~2.23! are upps& and upppps& states. Inserting these in-termediate states into Eq.~2.23! and carrying out straightfor-ward operator algebra, we obtain

Vpp5Vpps 1Vpp

t , ~2.26!

with the following the matrix elements between twoppstates:

^kW f1kW f2uVpps ukW i1kW i2&5

gspp2

~2p!31

A2Ep~kf1!

1

A2Ep~kf2!

31

A2Ep~ki1!

1

A2Ep~ki2!

3@Ds~2 !~ki11ki2!1Ds

~2 !~kf11kf2!#,

~2.27!

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^kW f1kW f2uVppt ukW i1kW i2&5

gspp2

~2p!31

A2Ep~kf1!

1

A2Ep~kf2!

31

A2Ep~ki1!

1

A2Ep~ki2!

3@Ds~ki12kf2!1Ds~ki22kf2!

1Ds~ki12kf1!1Ds~ki22kf1!#,

~2.28!

where

Ds~k!51

k22ms2 5D ~1 !~k!1D ~2 !~k!, ~2.29!

with

Ds~6 !~k!5

1

2Es~k!

61

k07Es~k!. ~2.30!

This completes the illustration of the SKO method in driving an effective Hamiltonian from a model Lagrangianrelativistic quantum field theory. The extension of tmethod to consider more realistic Lagrangians is straightward and will not be further detailed. In the following setions, we will simply write down the starting Lagrangianand the resulting effective Hamiltonians up to second orin the coupling constants forpN scattering and thegN→pN reaction.

III. p-N SCATTERING

We start with the following commonly assumed@17# La-grangian forN,D,p, andr fields:

L~x!5L0~x!1LI~x!, ~3.1!

whereL0(x) is the usual noninteracting Lagrangian, and tinteraction is taken to be

LI~x!5LpNN~x!1LpND~x!1LrNN~x!1Lrpp~x!,~3.2!

with ~in the convention of Bjorkin and Drell@52#!

LpNN~x!52f pNN

mpcN~x!g5gmtWcN~x!]m

•fW p~x!,

~3.3!

LpND5f pND

mpcD

m~x!TW cN~x!•]mfW p~x!1@ H.c.#, ~3.4!

LrNN~x!5grNNcN~x!tW

2•FgmfW r

m~x!2kr

2mNsmn]nfW r

m~x!G3cN~x!, ~3.5!

Lrpp~x!5grpp~fW p3]mfW p!•fW rm . ~3.6!

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HereTW is aN→D isospin transition operator defined by threduced matrix element@54# ^ 3

2uuTuu 12&52^ 12uuT†uu

32&52. By

using the standard canonical quantization, a Hamiltonianbe derived from the above Lagrangian except the termvolving theD field. The difficulty of quantizing theD field iswell known, as discussed, for example, in Refs.@55# and@56#. As part of our phenomenology, we take the simpleprescription by imposing the anticommutation relation

$DpW ,DpW 8†

%5d~pW 2pW 8!, ~3.7!

whereDpW (DpW†! is the annihilation~creation! operator for a

D state. This choice then leads@55# to the D propagatorgiven later in Eq.~3.18!. The alternative approaches proposed in Ref.@56# will not be considered.

Following the procedure described in Sec. II, the next sis to decompose the resulting Hamiltonian into aHI

P for‘‘physical processes’’ and aHI

Q for ‘‘virtual processes.’’From Eqs.~3.3!–~3.6!, it is clear that the real processesthis case areD↔pN andr↔pp transitions which can takeplace in free space~becausemD.mN1mp andmr.2mp).The virtual processes areN↔pN, N↔rN, N↔pD, andp↔pr transitions. These virtual processes can be elimnated by introducing a unitary transformation operatorSwhich can be determined by using the similar methodobtaining the solution Eqs.~2.13! and ~2.14!. Here, we ofcourse encounter a much more involved task to accountthe Dirac spin structure, isospin, and also the antiparticomponents ofN andD. To see the main steps, we presentthe Appendix an explicit derivation of the potential duetheLpNN term, Eq.~3.3!.

For practical applications, it is sufficient to present oresults in the coupledpN% D subspace in which thepNscattering problem will be solved. The resulting effectivHamiltonian then takes the form

HeffpN5H01GD↔pN1vpN , ~3.8!

whereH0 is the free Hamiltonian forp, N, andD. NotethatGD↔pN @Figs. 3~a! and 3~b!# is the only vertex interac-tion in the constructed effective Hamiltonian. Our modeltherefore distinctively different from the previous mesoexchangepN models@12–14,19# which all involve a barenucleon stateN0 and aN0↔pN vertex.

ThepN potentialvpN in Eq. ~3.8! is found to be

vpN5vND1vNE1vr1vDD1vDE

, ~3.9!

wherevND is the direct nucleon pole term@Fig. 3~c!#, vNE thenucleon-exchange term@Fig. 3~d!#, vr the r-exchange term@Fig. 3~e!#, vDD

the interaction due to the anti-D component

of the D propagation@Fig. 3~f!#, and vDEthe D-exchange

term @Fig. 3~g!#. To simplify the presentation, we will onlygive the matrix element ofvpN in the pN center-of-mass


FIG. 3. Graphical representation of the inter-actions of the effective Hamiltonian, Eq.~3.8!, inthe coupledpN% D space.

m

frame. The initial and final four-momentakm,k8m for pionsandpm,p8m for the nucleons in Fig. 3 are therefore defineas

km5„Ep~k!,kW…, ~3.10!

pm5„EN~k!,2kW…,

dk8m5„Ep~k8!,kW…,

p8m5„EN~k8!,2kW….

In terms of these variables, the matrix element of each terof Eq. ~3.9! between twopN states can be written as

^kW8i 8,ms8mt8uvaukW i ,ms ,mt&51

~2p!31

A2Ep~k8!A mN

EN~k8!

1

A2Ep~k!A mN

EN~k!.u2kW8,ms8,mt8

I a~kW8i 8,kW i !u2kW ,ms ,mt, ~3.11!

where upW ,ms ,mtis the Dirac spinor,ms and mt are the

nucleon spin and isospin quantum numbers, andi and i 8 arethe pion isospin components. The interaction mechanisare contained in the functionsI a(kW8i 8,kW i ). After performinglengthy derivations, we find that these functions can be wrten in the concise forms

I ND~kW8i 8,kW i !5S f pNN

mpD 2t i 8g5k” 8 1

2 @SN~p1k!1SN~p81k8!#

3t ig5k” , ~3.12!

I NE~kW8i 8,kW i !5S f pNN

mpD 2t ig5k” 12 @SN~p2k8!1SN~p82k!#

3t i 8g5k” 8, ~3.13!

I r~kW8i 8,kW i !5igrNNgrpp

4e i i 8ktkH Fgm2

kr

2mNismn~p2p8!nG

3@Drml~p2p8!~k1k8!l#

1@~p2p8!↔~k82k!#J , ~3.14!

ms

it-

IDD~kW8i 8,kW i !5S f pND

mpD 2Ti 8† km8

12 @SD

mn~p1k!1SDmn~p81k8!

2SD~1 !mn~p1k!2SD

~1 !mn~p81k8!#Tikn ,

~3.15!

IDE~kW8i 8,kW i !5S f pND

mpD 2Ti†km

12 @SD

mn~p2k8!

1SDmn~p82k!#Ti 8kn8 . ~3.16!

The propagators in the above equations are defined as

SN~p!51

p”2mN, ~3.17!

SDmn~p!5

1

3~p”2mD!F2S 2gmn1

pmpn

mD2 D 1

gmgn2gngm

2

2pmgn2pngm

mDG , ~3.18!

ly

x

-e

ed


Drmn~p!52

gmn2pmpn/mr2

p22mr2 . ~3.19!

In Eq. ~3.15!, we also have introduced a propagator

SD~1 !mn~p!5

mD

ED~p!

vpmvp

n

p02ED~p!, ~3.20!

wherevpm is the Rarita-Schwinger spinor~as explicitly de-

fined in Ref.@17#!. In theD rest frame, this propagator reduces to the simple form

S~1 !i j ~p! ——→pW→0

11g0

6~3d i j2s is j !

1

p02mD

~3.21!

for i , j51,2,3. The other elements involving time components vanish in this special frame,S(1)m05S(1)0n50. Theappearance of this propagator in Eq.~3.15! is to remove thepN→D→pN mechanism which can be generated by thvertex interactionGD↔pN of the effective Hamiltonian Eq.~3.8!. This comes about naturally in our derivations.

We note that the above expressions are remarkbly simto those derived from using Feynman rules. The only diffeences are in the propagators of the intermediate particThese propagators are evaluated by using the momenta oexternal particles which are restricted on their mass shell,defined in Eq.~3.10!. For the off-energy-shell dynamics@EN(k)1Ep(k)ÞEN(k8)1Ep(k8)#, these propagators canhave two possible forms, depending on which set of extermomenta is used. The propagators in Eqs.~3.12!–~3.16! arethe average of these two possible forms of propagators. Mdetails can be seen in the Appendix where the derivationvN5vND1vNE is given explicitly.

In the pN center-of-mass frame, theD in the vertex in-teraction GD↔pN is at rest. In this particular frame, theRarita-Schwinger spinors reduce to a simple form such tthe matrix element of the vertex interactionGD↔pN takes thefamiliar form

^DuGD↔pNukW i &52f pND

mp

i

A~2p!31

A2Ep~k!

3AEN~k!1mN

2EN~k!SW •kWTi . ~3.22!

HereSW is aN→D transition spin operator. It is defined bythe same reduced matrix element as the transition isosoperatorT.

Because of the absence of apN↔N vertex in the effec-tive Hamiltonian, Eq.~3.8!, it is straightforward to derive thepN scattering equations in the coupledpN% D space. Thederivation procedure is similar to that given in Ref.@23# forthe more complicatedpNN problem. The essential idea is toapply the standard projection operator technique of nuclreaction theory@3#. The resulting scattering amplitude can bcast into the form

TpN~E!5tpN~E!1GD→pN~E!GD~E!GpN→D~E!. ~3.23!

-

-

e

ilarr-les.f theas

nal

oreof

hat

pin

eare

The first term is the nonresonant amplitude determined onby thepN potential:

tpN~E!5vpN1vpNGpN~E!tpN~E!, ~3.24!

with

GpN~E!5PpN

E2EN~k!2Ep~k!1 i e, ~3.25!

wherePpN is the projection operator for thepN subspace.The second term of Eq.~3.23! is the resonant term deter-mined by the dressedD propagator and the dressed vertefunctions. They are defined by

GD~E!5GD0 ~E!1GD

0 ~E!SD~E!GD~E!

5PD

E2mD2SD~E!, ~3.26!

with

GD0 ~E!5

PD

E2mD~3.27!

and

GpN→D~E!5GpN→D@11GpN~E!tpN~E!#, ~3.28!

GD→pN~E!5@11GpN~E!tpN~E!#GD→pN , ~3.29!

wherePD is the projection operator for theD state, and theD self-energy is defined by

SD~E!5GpN→DGpN~E!GD→pN~E!. ~3.30!

Equations~3.23!–~3.30! are illustrated in Fig. 4. These equations are solved in partial-wave representation. To find thsolution for the integral equation~3.24!, it is necessary toregularize thepN potential by introducing a form factor foreach vertex in Eqs.~3.12!–~3.16!. In this work, we choose

FIG. 4. Graphical representation of scattering equations definby Eqs.~3.23!–~3.30!.

E

ns


@ I ND1I NE#~kW8i 8,kW i !→@ I ND1I NE#~kW8i 8,kW i !

3FpNN~kW8!FpNN~kW !, ~3.31!

I r~kW8i 8,kW i !→I r~kW8i 8,kW i !FrNN~kW2kW8!Frpp~kW2kW8!,~3.32!

with

FpNN~kW !5S LpNN2

LpNN2 1kW2

D 2, ~3.33!

FrNN~kW2kW8!5Frpp~kW2kW8!, ~3.34!

5S Lr2

Lr21~kW2kW8!2

D 2.~3.35!

For pND vertex with an external pion momentumkW , wechoose

FpND~kW !5S LpND2

LpND2 1kW2

D 2. ~3.36!

We have also tried other parametrizations of form factobut they do not give better fits to thepN scattering phaseshifts.

IV. PION PHOTOPRODUCTION

To proceed, we need to first extend the Lagrangian,~3.1!, to includev meson coupling which is known@4,17# toplay an important role in pion photoproduction. We choothe following rather conventional form~with kv;0)

LvNN5gvNNcN~x!Fgmfvm~x!2

kv

2mNsmn]nfv

m~x!GcN~x!.

~4.1!

rs,

q.

se

Following the approach of Ref.@17#, the pion photoproduc-tion mechanisms are defined by the hadronic Lagrangiadefined by Eqs.~3.1! and ~4.1! and the electromagnetic in-teraction Lagrangians

Lem5LgNN1Lgpp1LgpNN1Lgrp1Lgvp1LgND ,~4.2!

with

LgNN52ecN~x!F eAW ~x!2k

2mNsmn@]nAm~x!#GcN~x!,

~4.3!

LgpNN52e fpNNmp

cN~x!g5AW ~x!@tW3fW p~x!#3cN~x!,

~4.4!

Lgpp5e@]mfW p~x!3fW p~x!#3Am~x!, ~4.5!

Lrpg5grpg

mpeabgd@]aAb~x!#fW p~x!•@]gfW r

d~x!#, ~4.6!

Lvpg5gvpg

mpeabgd@]aAb~x!#fp

3 ~x!@]gfvd ~x!#, ~4.7!

LgND5 iecDmT3GmnA

[email protected].#. ~4.8!

Here e5(11t3)/2, k5(kp1kn)/21(kp2kn)t3/2. ThegND coupling in Eq.~4.8! is

Gmn5~GMKmnM 1GEKmn

E !, ~4.9!

as defined in Eqs.~2.10b! and~2.10c! of Ref. @17#. Its matrixelement between anN with momentump and aD with mo-mentumpD can be written explicitly as

^D~pD!uGmnuN~p!&5~GM2GE!F 3

~mD1mN!22q2mD1mN

2mNemnabP

aqbG1GEig

5F 6

@~mD1mN!22q2#@~mD2mN!22q2#

mD1mN

mNemlabP

aqbengdl pD

gqdG , ~4.10!

e
with P5(p1pD)/2 andpD5p1q.By applying the usual canonical quantization procedu
we can obtain from the above Lagrangians an electromnetic interaction HamiltonianHem. In this work, we willtreat the electromagnetic field as an external classical fiand hence the electromagnetic interactionHem can be ne-glected in constructing the unitary transformation operaS. The effective Hamiltonian for describing pion photopr

re,ag-

eld,

toro-

duction is therefore a simple extension of the effectivHamiltonian of the form of Eq.~2.21!:

Heff→Heff1Heffem

5H01HIP1HI8

P1Heffem, ~4.11!

with

les

-

n-

o


Heffem5Hem1@Hem,iS#. ~4.12!

By evaluatingHeffem in the coupledD % pN% gN subspace, we

obtain an extension of Eq.~3.8!:

Heffp →Heff

gp5H01GD↔pN1vpN1GD↔gN1vpg ,~4.13!

where GD↔pN and vpN have already been given in Eq~3.11!–~3.22!. The resonant and nonresonant electromnetic interactions are, respectively, described byGD↔gN andvpg , and are illustrated in Fig. 5. We again omit the detaof the derivation of these two terms, and simply present

FIG. 5. Graphical representation of the effective interactiGD↔gN andvgN of Eq. ~4.13!.

s.ag-

ilsour

results in the center-of-mass frame. The momenta variabqm for the photon,pm for the initial nucleon,km for the pion,andp8m for the final nucleon, in Fig. 5, are therefore

qm5~q,qW !,

pm5„EN~q!,2qW …,

km5„Ep~k!,kW…,

p8m5„EN~k!,2kW…. ~4.14!

In terms of these variables, expression~4.10! becomes verysimple. The matrix element of theGD↔gN vertex can then beexpressed in terms of the spin and isospinN→D transition-operatorsS andT introduced in Sec. III. At the resonanceenergy,mD5EN(q)1q, Eq. ~4.10! leads to

^DuGgN→DuqW l&52e

~2p!3/2AEN~q!1mN

2EN~q!

31

A2q3mD

2mN~mD1mN!T3@ iGMSW 3qW •eWl

1GE~SW •eWlsW •qW 1SW •qW sW •eWl!#, ~4.15!

whereeWl is the photon polarization vector. The matrix element of the nonresonant interactionvpg can be written as

^kW i ,ms8mt8uvpguqW l,msmt&

51

~2p!31

A2Ep~k!

1

A2qA mN

EN~k!A mN

EN~q!

3u2kWms8mt8F(a I a

pg~kW i ,qW l!Gu2kWmsmt. ~4.16!

The nonresonant pion photoproduction mechanisms are cotained in I N

pg for the direct nucleon terms@Figs. 5~c!, 5~d!,and 5~e!# , Ip

pg for the pion pole term@Fig. 5~f!#, I r,vpg for the

vector meson-exchange@Fig. 5~g!#, andIDD,DEpg for the direct

and exchangeD terms @Figs. 5~h! and 5~i!#. Explicitly, wehave

ns

I Npg~kW i ,qW l!5

e fpNNmp

F i t ig5k”SN~p81k!S ee” ~l!1 ik

2mNsmne~l!mqnD1 i S ee” ~l!1 i

k

2mNsmne~l!mqnD

3SN~p2k!t ig5k”2e i j 3t jg

5e” ~l!G , ~4.17!

Ippg~kW i ,qW l!52

e fpNNmp

e i j 3t jg5~p” 82p” !@~k1p2p8!•e~l!Dp~p2p8!#, ~4.18!


I rpg~kW i ,qW l!5

grNNgrpg

mp

t i2 Fgm2

ikr

2mNsmh~p2p8!hG3Dr

mn~p2p8!eabgnea~l!qb~p2p8!g,

Ivpg~kW i ,qW l!5

gvNNgvpg

mpd i ,3Fgm2

ikv

2mNsmh~p2p8!hG3Dv

mn~p2p8!eabgnea~l!qb~p2p8!g ~4.19!

IDEpg~kW i ,qW l!5

e fpND

mpT3†Gmd

† ed~l!SDmn~p2k!Tikn , ~4.20!

IDDpg ~kW i ,qW l!52

e fpND

mpTi†km@SD

mn~p81k!2S~1 !mn~p81k!#T3Gnded~l!. ~4.21!

in

d

n

Here we observe again that the above expressions aresimilar to the results derived by using Feynman rules. Hoever, they have an important feature that the time compnents of the momenta in the propagators and strong intetion vertices are evaluated by using the external momentathe finalpN state. This is the consequence of applying tunitary transformation method defined in Eq.~4.12!. In ad-dition to including nonresonantD terms@Figs. 5~h! and 5~i!#,this is another feature which makes our model different frothe model developed in Ref.@17#.

It is straightforward to derive from the effective Hamiltonian, Eq.~4.13!, the t matrix of pion photoproduction:

Tgp~E!5tgp~E!1GD→pN~E!GD~E!GgN→D~E!,~4.22!

where the nonresonant amplitude is defined by

tgp~E!5vgp1tpN~E!GpN~E!vgp . ~4.23!

The dressedgN↔D is defined by

GgN→D~E!5GgN→D1GpN→D~E!GpN~E!vgp .~4.24!

In the above equations,GD ,GD↔pN ,GpN, andtpN have beendefined in Sec. III. The standard partial-wave decompositiis used to obtain the multipole amplitudes fromTgp for thegN→pN reaction and from GD↔pN for the dressedD↔pN vertex. Equations~4.22!–~4.24! are illustrated inFig. 6.

TheK-matrix formulation of thegN→pN reaction is of-ten used@4–7# in the analysis of data. Within our formulation, this can be obtained by replacing thepN free GreenfunctionGpN , Eq. ~3.25!, by

GpN~E!→GpNP ~E!5P

PpN

E2EN~k!2Ep~k!, ~4.25!

whereP means taking the principal-value part of the propgator. If this replacement is used in the calculations of Eq~3.23!–~3.30!, all scattering quantities will be real numbersTheseK-matrix quantities are defined by exactly the samEqs.~3.23!–~3.30! with the changes

veryw-o-rac-of

he

m

-

on

-

a-s..e

GpN→GpNP ,

TpN→KpN ,

tpN→kpN ,

GD↔pN→GD↔pNk ,

GD→GDP5

PD

E2mD2SDk ~E!

, ~4.26!

with

SDk ~E!5GpN→DGpN

P ~E!GD→pNk ~E!. ~4.27!

Note that theD self-energySDk is now a real number, and the

propagatorGDP has a pole atE5MR5mD1SD

k (MR).The correspondingK matrix for pion photoproduction can

be obtained from Eqs.~4.22!–~4.24! by the same replace-ment, Eq.~4.25!:

Kgp~E!5kgp~E!1GD→pNk ~E!GD

P~E!GgN→Dk ~E!,

~4.28!

with

kgp5vgp1kpN~E!GpNP ~E!vgp , ~4.29!

GgN→Dk ~E!5GgN→D1GpN→D

k ~E!GpNP ~E!vgp .

~4.30!

For the on-shell matrix elements @E5EN(k0)1Ep(k0)5q1EN(q)#, it is straightforward to find the fol-lowing relation in each partial wave:

Tgp~k0 ,q!5@12 iprTpN~k0 ,k0!#Kgp~k0 ,q!.~4.31!

For investigating the hadron structure, we are interestedthe gN↔D vertex. As seen in Eqs.~4.24! and ~4.30!, thedressed vertices in thet matrix and in theK matrix are dif-ferent. In thet-matrix formulation GgN,D , Eq. ~4.24!, is acomplex quantity, while in theK-matrix formulationGgN,Dk , Eq. ~4.30!, is a real function. Consequently, we nee

to be careful about the meaning of theE2/M1 ratio of thedressedgN↔D vertex. The clearest definition seems to be i

-

.

f

-

l

-y

d

f


theK matrix formulation because as the energy approachthe resonance positionMR5mD1SD

k (MR), Eq. ~4.28! is re-duced to

Kgp~E!5A

E2MR1B, ~4.32!

with

A5GD→pNk ~E!GgN→D

k ~E!, ~4.33!

B5kgp~E!. ~4.34!

The separable form of the residueA of theK matrix leads toan interesting result that the ratio between theE1 andM1multipole amplitudes of the dressedgND vertex can be di-rectly calculated from the residues of the corresponding mtipole amplitudes of thegN→pN reaction. The reason isthat both amplitudes have the same strong interactdressed vertex in theP33 channel, and hence the ratio between the residues does not depend on it. Explicitly, we ha

REM5F A~E11!

A~M11!GgN→pN

5GD→pNk ~P33!GgN→D

k ~E11!

GD→pNk ~P33!GgN→D

k ~M11!

5GgN→D~E11!

GgN→D~M11!. ~4.35!

The above relation is the basis of the model-independanalysis of Ref.@6#. We will discuss this issue in the nexsection.

V. RESULTS AND DISCUSSIONS

Our first task is to determine the parameters of the effetive pN Hamiltonian derived in Sec. II. Apart from theknown pNN coupling constantfpNN

2 /4p50.08, the modelhas seven parameters: the coupling consta(gr

25grNNgrpp , kr , fpND) of Eqs.~3.12!–~3.16!, the cutoffparameters (LpNN ,LpND ,Lr) of the form factors~3.34!–

FIG. 6. Graphical representation of pion production amplituddefined by Eqs.~4.22!–~4.24!.

es

ul-

ion-ve

entt

c-

nts

~3.37!, andmD of the D bare mass. These parameters aredetermined by fitting thepN phase shifts. Without includinginelastic channels, our scattering equations Eqs.~3.23!–~3.30! are valid rigorously only in the energy region wherethepN scattering is purely elastic. We therefore first take aconservative approach to only fit the data in the energy region below TL5250 MeV pion laboratory energy. Thismodel, called model L, is sufficient for investigating pionphotoproduction up to 400 MeV photon laboratory energyOur results are displayed in Fig. 7. We see that within theuncertainties of the phase shift data@38–40# the model cangive a good account of alls andp partial waves except theP13 channel atTL.120 MeV. We have found that this dif-ficulty cannot be removed by trying various form factorsother than those given in Eqs.~3.34!–~3.37!, and followingthe previous works@12,13# to include the exchange of a fic-titious scalars meson. To see the origin of this problem, weshow in Fig. 8 the contributions from each mechanism oFig. 3 to the on-shell matrix elements of thepN potential.Clearly, the fit to the phase shift data involves delicate cancellations between different mechanisms. It is possible toimprove the fit toP13 by weakening ther exchange or theD exchange. But this change will destroy the good fits to alother partial waves. Fortunately, thepN scattering effect dueto theP13 channel is weak in determining the pion photopro-duction cross sections. We therefore will not pursue the solution of this problem here. Perhaps this can be solved onlwhen ther exchange is replaced by the two-pion exchangeconsidered in Ref.@14#. To be consistent, the coupling withtwo-pion channels, such aspD and rN, must also be in-cluded. These two possible improvements can be achieveby extending the unitary transformation method introducedin Secs. II–IV to second order in the coupling constants.

Let us now examine in more detail theP33 channel whichis most relevant to our later investigation of theD excitation

es

FIG. 7. ThepN phase shifts calculated from thepN model L ofTable I are compared with the empirical values of the analyses oRef. @39# ~open squares! and Ref.@38# ~solid squares!. TL is thepion laboratory energy.

t

t

a

e

e,

r-s-hishe

of

ive

es

i-

,

ed


in pion photoproduction. As seen in Eq.~3.23!, the resonantpart of theP33 amplitude is determined by the dressed propgatorGD of Eq. ~3.26! and the dressed vertexGD↔pN of Eqs.~3.28! and~3.29!. Clearly, theD resonance peak ofpN scat-tering can be obtained only when the model can generaD self-energy such that the real part o@E2mD2SD(E)#→0 as thepN-invariant massW ap-proaches the resonance eneryW5MR51236 MeV. Ourmodel has this desired property, as illustrated in the uphalf of Fig. 9. Another important feature in theP33 channelis that thepN potential generates the dressedD↔pN ver-tex, as defined in Eqs.~3.28! and~3.29!. We have found thatthis renormalization effect modifies greatly theD↔pN formfactor in the low-momentum region. To see this, we castbare vertex@Eq. ~3.22! including the form factorFpND(k)defined by Eq.~3.36!# and the dressed vertex@Eq. ~3.28!#into the forms


mp

i

A~2p!31

A2mp

Fbare~k!SW •kWTi ,

~5.1!


mp

i

A~2p!31

A2mp

Fdressed~k!SW •kWTi ,

~5.2!

with the normalizationuFdressed(0)u5Fbare(0)51. We findthat the dressed coupling constantf pND is 1.3 of the barecoupling constantfpND . The dressed form factorFpND(k)falls off faster than the bare form factorFpND(k) in momen-tum space, as seen in the lower half of Fig. 9. This methat the nonresonantpN interaction has extended theD ex-citation region to a larger distance in coordinate space.

FIG. 8. The on-shell matrix elements ofpN potentials definedby Eqs.~3.11!–~3.16!. TL is the pion laboratory energy. The notations areND :vND, NE :vNE, r:vr , DD :vDD

, DE :vDE, and Tot is the

sum.

a-

e af

per

he

ns

A significant difference between our approach and thpreviouspN models@12–14,19# is in the treatment of theP11 channel. By employing the unitary transformation, thpN↔N vertex does not appear in the effective HamiltonianEq. ~3.8!, and hence our formulation ofpN scattering isstraightforward. It does not require the nucleon mass renomalization. It is natural to ask whether our approach posesses the well-established nucleon-pole dynamics. Tquestion can be answered by examining Fig. 10 in which tpN phase shifts and the scatteringt-matrix elements calcu-lated from the nucleon pole termvND only @Fig. 3~c!# and thefull potential are compared. We see in the the upper halfFig. 10 thatpN phase shifts due to the nucleon-pole term~dotted curve! are repulsive as expected. The fit to theP11data is due to a delicate cancellation between the repulsnucleon-pole term and the attraction coming mainly fromr- andD-exchange terms~see theP11 case in Fig. 8!. In thelower half of Fig. 10, we see that as the energy approachthe threshold,W5mp1mN , the nucleon pole term~dottedcurve! apparently dominates the interaction. If we analytcally continue to the nucleon pole position,k05 ikN@mN5EN( ikN)1Ep( ikN)#, the scattering amplitude will bedetermined by the nucleon pole term, i.e.

-

FIG. 9. The upper half shows the mass of theD state defined byEq. ~3.30!.mD51299.07 MeV is the bare mass of model L of TableI, MR51236 is the experimental resonance position, andW is thepN-invariant mass. The lower half shows the bare and dressD→pN form factors defined by Eqs.~5.1! and ~5.2! withf pND51.3f pND . k is the pion momentum in theD rest frame.

he

d

e

n


t(k0 ,k0 ,W→mN);vND;1/(W2mN).The parameters of the constructed model are listed in

first row ~model L! of Table I. The calculated scatterinlengths are presented in the first column of Table II. Theyall in good agreement with the data. If we assume the uversality of r coupling, we then havegrNN5grpp56.2which is close to that determined in Refs.@12,13#. The fit isalso sensitive to ther tensor coupling constantkr . Ourvalue is close to that of Ref.@12#, but is much smaller than6.6 used in Ref.@13#.

We now turn to presenting our results of pion photoprduction. With thepNN,pDN, andrNN vertices defined bythe parameters given in Table I, the considered pion pho

FIG. 10. ThepN phase shifts (d) and scattering amplitude@ t(k0 ,k0 ,W)# with thepN-invariant massW5EN(k0)1Ep(k0) intheP11 channel. The solid curves are from the full calculation. Tdotted curves are from the calculation including only the nuclepole potentialvND of Fig. 3~c!.

thegareni-

o-

to-

production mechanisms~Fig. 5! still have unknown param-eters associated with the vector meson-exchange and tgN↔D vertex. Following the previous approach@17#, weassume that the photon-meson coupling constantsgrpg andgvpg can be determined from the partial decay widths listeby the Particle Data Group@45#. For thev meson, we furtherassume that the tensor couplingkvNN50 and thevNN formfactor is identical to therNN form factor given in Table I.The coupling constantgvNN is not well determined in theliterature. We will treat it as a free parameter, although thquark model valuegvNN5(3grNN)/2 seems to be a reason-able guess. Thus, our investigation of pion photoproductiohas only three adjustable parameters:GM andGE of the bareD↔gN vertex, and the coupling constantgvNN of the v

FIG. 11. The photon-asymmetry ratiosRg5ds i /ds' ofgp→p0p at 90° .Eg is the photon energy in the laboratory frame.The results in the upper half are from usingGM51.85, GE50.025, and three values ofgvNN . The results inthe lower half are from usingGM51.85, gvNN510.5, and fourvalues ofGE . The data are from Refs.@41,42#.

heon

TABLE I. The parameters of thepN models. The units are 1/F for cutoff parametersLa and MeV for thebareD mass. Model L and model H are obtained respectively from fittingpN phase shifts up to 250 MeVand 400 MeV.

Model f pNN2 /4p LpNN grNNgrpp kr Lr f pND LpND mD

Model L 0.08 3.2551 38.4329 1.825 6.2305 2.049 3.29 1299.07Model H 0.08 3.7447 39.0499 2.2176 7.5569 2.115 3.381 1318.52

Form factorFpNN~k!5S LpNN2

LpNN2 1k2D 2, Fr~q!5S Lr

2

Lr21qW 2

D 2, FpND~k!5S LLND2

LpND2 1k2D 2

he

ly

of

nce

l,-

se

-


exchange. We have, however, some ideas about the rangthese parameters. If we assume that bare vertex interacGD↔gN can be identified with the constituent quark mod@46,48,47,49,50#, thenuGE /GMu;0 since the one-gluon exchange interaction gives negligibleD-state components inN andD. We also expect that thev coupling should be closeto the quark model prediction,gvNN53grNN/2;9, if the rcoupling from ourpN model~Table I! is used. It is thereforereasonable to only consider the regiongvNN<15 anduGE /GMu<0.1.

Since thev-exchange mechanism@Fig. 5~g!# does notproduce charged pions directly~only throughpN charge ex-change!, the ranges ofgvNN , GM , andGE can be mostsensitively determined by considering the data ofp0 photo-production. In the considered region thatuGE /GMu<0.1 andgvNN< 15, we have found that the magnitudes of thep0

differential cross sections depend mainly onGM . The valuesof gvNN andGE can be narrowed down by considering spobservables. In this work, we make use of the recent LE@41# data of photon-asymmetry ratiosRg5ds i /ds' ofgp→p0p reaction. We have found that the slope ofRg(E)at a fixed pion angle is sensitive to the value ofgvNN . Thisis illustrated in the upper half of Fig. 11 for the caseGM51.85 andGE50.025. A smallergvNN yields a steeperslope. The data clearly favorgvNN;10.5. In the lower halfof Fig. 11, we see that the magnitude, not the slope, ofRg issignificantly changed by varying the value ofGE from20.1 to 10.1. The data are consistent wit20.025<GE<0.025, whileGE510.025 seems to give abetter fit. Results similar to Fig. 11 can be obtained by usa higher valueGM51.95. In this case, a smaller value ogvNN57 is needed to maintain the same fit to the magnituof the differential cross section as well as the slope ofRg . But the best value ofGE to reproduce the magnitude oRg is 20.025 instead of10.025 for thegvNN510.5 case. Infact, we have observed a strong correlation between thelowed values ofGM and gvNN . In all cases, the allowedvalue of GE is consistent with20.025<GE<10.025.Therefore, the acceptable values of (GM ,gvNN) are on thecurve between theGE520.025 andGE510.025 lines inFig. 12. To determine the precise value ofREM5GE /GM ,which measures the deformation of theD, we clearly need topin down thev meson coupling constantgvNN . In Table III,we list the determined values ofGM , GE , andgvNN alongwith all other parameters used in our calculationsgN→pN reactions.

The predictions from using the parameters lying on tcurve betweenGE510.025 andGE520.025 lines of Fig.

TABLE II. The calculatedpN scattering lengths~in unit offermis! are compared with the values determined in Ref.@40#.

Model L Model H Koch-Pietarinen

S11 0.1588 0.1737 0.17360.003S31 20.1191 20.1198 20.10160.004P11 20.0976 20.0864 20.08160.002P31 20.0509 20.0478 20.04560.002P13 20.0363 20.0383 20.03060.002P33 0.2523 0.2797 0.21460.002

es oftionel-

inGS

of

h

ingfdethef

al-

of

he

12 are also in good agreement with thep0 data at otherangles and thep1 data. These are shown in Fig. 13 forp0

production and Fig. 14 forp1 production. These results areobtained from using the parameters defined by tinteraction points of the GE560.025 lines inFig. 12: (gvNN ,GM ,GE)5(10.5,1.85,10.025), and(7.0,1.95,20.025). Both sets of parameters yield equalgood agreement with thep0 data~Fig. 13!. Forp1 produc-tion, the predictions are in good agreements with the datathe photon-asymmetry ratiosRg , but underestimate the dif-ferential cross sections by about 10% at most energies. Sithev exchange has a small contribution to thep1 produc-tion ~only through charge-exchangepN final state interac-tion!, the only way to resolve this difficulty within our modelis to increase the value ofGM . But this will lead to anoverestimate of thep0 cross section from Bonn.

Although the difficulty in reproducing thep1 data in Fig.14~b! could be an indication of the deficiency of our modethe possibility of a largerp0 cross section has been suggested by threep0 data atu5120° from Ref. @44# @Fig.13~b!#. To fit these three data points, we need to increaGM from 1.85 to 2.0 for the case ofgvNN 5 10.5 and from1.95 to 2.1 for the case ofgvNN 5 7. The results from thesetwo changes inGM are, respectively, the solid and dotted

FIG. 12. The region of the parameters (gvNN ,GM ,GE) for de-scribing the data ofgN→pN reactions. See text for the explanations.

TABLE III. The parameters for thegN→pN interactions de-fined in Eqs.~4.16!–~4.20! and Fig. 5.

fpNN ,LpNN – Table If pND ,LpND – Table IgrNN5AgrNNgrpp – Table ILrNN5Lr ,kr – Table Igrpg 0.1027e Ref. @45#gvpg 0.3247e Ref. @45#gvNN 7 – 10.5 See textLvNN5Lr – Table Ikv50 – See textGM(0) 1.85 – 2.0 See textGE(0) 6 0.025 See text

er-isin-lestic

e

nds..e

tion

el

ees

-


curves in Figs. 15 and 16. The predicted photon-asymmratios@Figs. 15~a! and 16~a!# are still in good agreement withthe data. The agreement with thep1 data @Fig. 16~b!# isclearly improved. But the calculatedp0 differential crosssections@Fig. 15~b!# overestimate the Bonn data@42,43# byabout 15%. Clearly, the disagreement between thep0 data atu5120° @Fig. 15~b!# from Refs. @42,43# and @44# must beresolved by new measurements.

To further reveal the dynamical content of our model,compare in Fig. 17 our predictions of angular distributiowith the data compiled in Ref.@43# for gp→p0p @Fig.17~a!#, gp→p1n @Fig. 17~b!#, andgn→p2p @Fig. 17~c!#reactions. All results calculated from using the paramelying on the curve between theGE510.025 andGE520.025 lines in Fig. 12 are very close to the socurves which are from using (gvNN ,GM ,GE)5(10.5,1.85,0.025). Again, we see that the charged pproduction cross sections are underestimated. If a laGM52.0 is used in this calculation, we obtain the dottcurves which are in a better agreement with the chargeddata, but overestimate thep0 data by about 15% at reso

FIG. 13. The photon-asymmetry ratios~a! and differential crosssections~b! for thegp→p0p reaction at four angles in the centeof-mass frame. The solid and dotted curves are, respectively, fthe calculation using the parameters (gvNN ,GM ,GE)5(10.5,1.85,10.025), and (7.0,1.95,20.025). The data are fromRefs.@41,42#. Eg is the photon energy in the laboratory frame.

etry

wens

ters

lid

ionrgeredpion-

nance peaks. In all cases, the theoretical predictions undestimate the data at 380 MeV and higher energies. Thisexpected, since the constructed model does not includeelastic channels which should start to play a significant roat energies above about 350 MeV. For example, the inelaproduction mechanismgN→pD→pN should exist since itis known that thepN scattering at this higher energy can bdescribed only when the coupling with thepD channel isincluded. To investigate this effect, it is necessary to extethe derivation of effective Hamiltonians presented in SecIII and IV to include thepD as well as other two-pion states

We now focus on the theoretical interpretations of thD↔gN vertex. The values ofGM andGE determined abovecharacterize the bareD↔gN vertex which can only be iden-tified with hadron models with no coupling with thepN orother hadronic reaction channels. One possible interpretais to compare the determinedGM andGE with the predic-tions of the most well-developed constituent quark mod@46,48,47,49,50#. To explore this possibility, it is necessaryto first discuss the quantities in our model which can bcompared with the results from empirical amplitude analys@6,7,57#. For investigating theD mechanism, we need toonly consider thegN→pN multipole amplitudesM11 andE11 with a P33 final pN state and the dressed vertex func

r-rom

FIG. 14. Same as Fig. 13, except for thegp→p1n reaction.The data are from Ref.@42#.

-

e

-re

o-.s

l

of

,


tion GgN,D . These can be computed from Eqs.~4.22!–~4.24!or Eqs. ~4.28!–~4.30! by performing the standard partial-wave decomposition~see, for example, the Appendix of Ref@17#!. We will discuss these quantities using the results caculated from setting (gvNN ,GM ,GE)5(10.5,1.85,0.025)~solid curves in Figs. 13, 14, and 17!.

The predicted amplitudesM11 andE11 are compared inFig. 18 with the results from the empirical amplitude analyses@7,57#. We see in the upper part of Fig. 18 that the predictedM11 amplitudes are in good agreement with empiricavalues. In the lower half, we show that both theE11 ampli-tudes calculated from usingGE510.025~solid curves! andGE520.025 ~dotted curves! are within the uncertainties ofthe amplitude analyses. This is consistent with our analyusing LEGS data, as seen in the lower part of Fig. 11. Tuncertainties of the empirical values of theE11 amplitudeare due to the lack of complete data of spin observableMore experimental efforts are clearly needed to pin down tvalue ofGE which is needed to test models of hadron struture.

FIG. 15. The photon-asymmetry ratios~a! and differential crosssections~b! for thegp→p0p reaction at four angles. The solid anddotted curves are, respectively, from the calculations usithe parameters (gvNN ,GM ,GE)5(10.5,2.0,10.025), and(7.0,2.10,20.025). The data are from Refs.@41,42#. Eg is the pho-ton energy in the laboratory frame.

.l-

--l

sishe

s.hec-

The dressedD↔gN vertex, defined by Eq.~4.24!, is acomplex number. By making the usual partial-wave decomposition, its magneticM1 and electricE2 components can bewritten asG(a)5uG(a)ueif(a) with a5M11 ,E11 . The pre-dicted dressed vertex functionsG(a) are the solid curves inFig. 19. We see that their magnitudesuG(a)u are very differ-ent from the corresponding values~dotted curves! of the bareD↔gN vertex. The differences are due to the very largecontribution of the nonresonant mechanism described by thsecond term of Eq.~4.24!. Our results indicate that an accu-rate reaction theory calculation of the nonresonant pion photoproduction mechanisms is needed to determine the baD↔gN vertex from the pion photoproduction data. This re-quires a dynamical treatment of the nonresonant pion photproduction mechanisms, as we have done in this workWithin the meson-exchange formulation presented in thiwork, the determinedGM andGE of the bareD↔gN vertexcan be compared with the predictions from a hadron modewhich does not include the coupling with thepN ‘‘reaction’’channel~both pion and nucleon are on their mass shell!.

We now turn to investigate theK-matrix method whichhas been the basis of the empirical amplitude analysesRefs.@6,7#. In Ref. @6#, it was shown that if the backgroundterm is assumed to be a slowly varying function of energy

ng

FIG. 16. Same as Fig. 15, except for thegp→p1n reaction.The data are from Ref.@42#.


FIG. 17. Differential cross sections ofgp→p0p ~a!, gp→p1n ~b!, andgn→p2p, ~c! reactions. The solid~dotted! curves are calculatedby usingGM51.85 (2.0). Both calculations using the same (gvNN ,GE)5(10.5,10.025). The data are from Ref.@43#. Eg is the photonenergy in the laboratory frame.

x

the ratioREM between theE11 and M11 of the gN→Dtransition at the resonant energyW5MR can then be ex-tracted ‘‘model independently’’ from the data of thgN→pN reaction. The only limitation is the accuracy of themployedpN amplitudes and theM11 andE11 multipoleamplitudes of thegN→pN reaction. By using Eqs.~4.32!–~4.35!, we can examine whether theK-matrix method of Ref.@6# is consistent with our dynamical model. In Fig. 20, wdisplay our predictions of the energy dependence of the tK matrix ~solid curve!, the contribution from the resonanterm ~dotted curve! which has a pole atW51236 MeV, and

ee

eotalt

the nonresonant contributionB ~dashed curve!. Clearly, theenergy dependence of the nonresonant termB is rather weak.The assumption made in the empirical analysis of Ref.@6# isfairly consistent with our dynamical model.

By using Eq.~4.33!, we can calculate the residueA of theK matrix from the dressed vertexGgN→D

k defined by Eq.~4.30!. The results~solid curve! for theM11 transitions arecompared with that calculated from using the bare verteGgN→D in the the upper half of Fig. 21. Similar to the resultsin Fig. 19 in thet-matrix formulation, we see the large non-resonant mechanisms in dressing thegN→D vertex. The

leldre-

-ric-ese-delantcal-nsic-

the

isthely,

he


correspondingE2/M1 ratiosREM are compared in the lowerhalf of Fig. 21. The nonresonant mechanisms changeratio by a factor of about 2 at resonance energyW51236MeV.

In Table IV, we list the predictedE11 andM11 ampli-tudes of theD↔gN vertex evaluated at the resonance eergy W51236 MeV. The parameters (gvNN ,GM ,GE) 5~10.5, 1.85, 0025! and~7.0, 1.95,20.025! from the fits to thedata~Figs. 13 and 14! are used in these calculations. We sethat our average valueREM 5 (21.8 6 0.9!% is not toodifferent from the average value (21.07 6 0.37!% of theempirical analysis@6#. Since the assumption made in Ref.@6#is consistent with our model as discussed above, the diffence perhaps mainly comes from the experimental uncertaties of the multipole amplitudes employed in the analysis

FIG. 18. The predicted multipole amplitudesM11 andE11 inthe total isospinI53/2 channel are compared with the empiricavalues of Refs.@38# ~solid squares! and @57# ~open squares!. Theparameters used in this calculation areGM51.85,gvNN510.5, withGE50.025~solid curve! and20.025~dotted curve!. Eg is the pho-ton energy in the laboratory frame.

FIG. 19. TheM11 andE11 multipole amplitudes of the dressedvertexGgN→D defined by Eq.~4.24! and the bare vertexGgN→D arecompared. The dressed vertex is a complex function writtenG(a)5uG(a)ueif(a) with a5M11 ,E11 . W is the gN-invariantmass.

the

n-

e

er-in-of

Ref. @6#. The differences between our predicted multipoamplitudes and the empirical values shown in Fig. 18 coualso be responsible to this discrepancy. To compare oursults with the values listed by the Particle Data Group~PDG!@45#, we calculate the helicity amplitudes by

A3/25A32

@E112M11#,

A1/252 12 @3E111M11#.

The results at the resonance energyW51236 MeV are listedin Table V. The predictions from two constituent quark models @48,47# are also listed for comparison. We notice that oubare values are close to the constituent quark model predtions @47,48#, and the dressed values are close to the valuof the PDG@45#. This suggests that our bare vertex can bidentified with the constituent quark model. The longstanding discrepancy between the constituent quark mopredictions and the PDG values is due to the nonresonmeson-exchange production mechanisms which must beculated from a dynamical approach. Similar consideratiomust be taken in comparing the PDG values with the predtions of higher massN* resonances from hadron models.

The results we have presented so far are based onpN model determined in a fit to thepN phase shifts only upto 250 MeV. It is interesting to see the extent to which thmodel can be extended to a higher-energy region whereinelastic processes are still not dominant. More important

l

as

FIG. 20. The predictedK matrix defined by Eq.~4.28!. Thesolid curves are the full calculations. The dotted curves are from tresonant term. The dashed curves~denoted asB) are the contribu-tions from the nonresonant termkgp defined by Eq.~4.29!.W is thegN-invariant mass.

-ey

ios

e-

itheel

e-

n-

tals

ec-


we would like to examine whether the extended model cyield significantly differentpN off-shell dynamics whichperhaps can help resolve the difficulty in reproducing thmagnitudes ofp1 cross section@see Fig. 14~b!#. To explorethese possibilities, apN model is constructed by fitting thephase shifts data up to 400 MeV. This model is called modH to distingush it from model L from the fit up to only 250MeV. The resulting parameters are also listed in Table I. Tphase shifts calculated from these two models are compain Fig. 22. We see that model H~dotted curves! clearly givesa much better fit to the data in the entire considered eneregion. But it is not as accurate as model L in describing t

FIG. 21. TheM11 residues A of theK matrix @Eq. ~4.32!#calculated from the dressedD→gN defined by Eq.~4.30! and thebare vertex are compared in the upper half. Their correspondratiosREM5E11 /M11 are compared in the lower half.W is thegN-invariant mass.

TABLE IV. The magneticM11 and electricE11 amplitudes ofD→gN transition atW51236 MeV.REM5E11 /M11 . The ampli-tudes are in units of 1023 ~GeV!21/2. The numbers in the upper andlower rows for each case are respectively from usin(gvNN ,GM ,GE) 5 ~10.5, 1.85,1 0.025! and ~7.0, 1.95,20.025!.

M11 E11 REM Average

GD→gN ~bare! 175 22.28 21.3%~0.06 1.3!%

184 12.28 11.2%

GD→gN ~dressed! 257 26.9 –2.7%

~21.86 0.9!%258 22.26 20.9%

an

e

el

hered

rgyhe

the crucialP33 channel in the low-energy region. To accurately fit theP33 in the entire energy region and to resolvthe difficulty in theP13 channel, additional mechanisms mabe needed.

ThegN→pN results calculated from using the models Hand L are compared in Fig. 23. The photon-asymmetry rat@Figs. 23~a! and 23~c!# are equally well described by bothmodels. They yield, however, significant differences in dscribing the differential cross sections. In Fig. 23~b!, we seethat model H gives a much better description of thep0 dif-ferential cross sections in the high-energy region, butslightly overestimates the cross sections at low energies. Tp1 differential cross sections are better described by modH, as seen in Fig. 23~d!. But the difficulty in reproducing themagnitude is not removed entirely.

The results in Fig. 23 suggest that our predictions do dpend to some extent on the accuracy of the constructedpNmodel in describing thepN phase shifts. A natural next stepis to extend the present model to include the inelastic cha

ing

g

TABLE V. Helicity amplitudes of theD→gN transition atW51236 MeV are compared with the values from Particle DaGroup ~PDG! @45# and the predictions of constituent quark modeof Refs. @47,48#. The amplitudes are in units of 1023 ~GeV!21/2.The numbers in the upper and lower rows for each case are, resptively, from using (gvNN ,GM ,GE) 5 ~10.5, 1.85,10.025! and~7.0,1.95,20.025!.

A PDG Dressed Bare Ref.@48# Ref. @47#

A3/2 22576 8 2228 2153 2157 21862225 2158

A1/2 21416 5 2118 284 291 21082126 296

FIG. 22. ThepN phase shifts calculated from model L~solidcurves! and model H~dotted curves! are compared. The data arefrom Refs.@38# ~solid squares! and@39# ~open squares!. TL is pionenergy in the laboratory frame.


FIG. 23. Photon-asymmetry ratios and the differential cross sections for thegp→p0p ~a! and~b! andgp→p1n ~c! and~d! reactions.The solid~dotted! curves are from calculations usingpN model L ~model H!. The parameters are (gvNN ,GM ,GE)5(10.5,1.85,10.025).The data are from Refs.@41,42#. Eg is the photon energy in the laboratory frame.

o-

-tn-

tialrall

an-

nels to obtain an accurate fit up to 400 MeV. This extensiothen will introduce inelastic pion photoproduction mechanisms, such as thegN→pD→pN process, which may beneeded to resolve the difficulty in getting an accurate dscription of both thep0 andp1 processes. Such a coupledchannels approach must also include the effect due toexcitations of higher massN* nucleon resonances. This musbe pursued in order to make progress in using the forthcoing data from CEBAF to test hadron models.

VI. CONCLUSIONS AND FUTURE STUDIES

We have applied the unitary transformation method firproposed in Ref.@22# to derive from a model Lagrangianwith N,D,p, r, v, and g fields an effective Hamiltonianconsisting of bareD↔pN, gN vertices and an energy-independent meson-exchangepN potential ~Fig. 3! andgN→pN transition operator~Fig. 5.!. With the parameterslisted in Table I for the strong form factors and the bare ma

n-

e--thetm-

st

ss

of theD, the model can give a good description ofpN scat-tering phase shifts up to theD excitation energy region. Theonly adjustable parameters in the resulting pion photoprduction amplitude are the coupling strengthsGE of the elec-tric E2 andGM of the magneticM1 transitions of the bareD↔gN vertex and the less well-determined coupling constantgvNN of the v meson. We have shown that the besreproduction of the recent LEGS data of the photoasymmetry ratios of thegp→p0p reaction depends sensi-tively on these three parameters and yieldsGM51.960.05and GE50.060.025 within the range 7<gvNN<10.5.Within these ranges of parameters, the predicted differencross sections and photon-asymmetry ratios are in an ovegood agreement with the data ofgp→p0p, gp→p1p, andgn→p2p reactions from 180 MeV to theD excitation re-gion. The model, however, underestimates thegN→pNcross section at energies above theD region. This is expectedsince the constructed model does not include inelastic chnels, such aspD, rN channels, which should start to play a

eheerfnd

ofa

fe-

n-


significant role at energies above about 350 MeV. Includinthese channels could also be needed to resolve the difficin fitting P13 pN phase shifts~Fig. 22!. The constructedeffective Hamiltonian is free of the nucleon renormalizatioproblem and hence is suitable for nuclear many-body calclations.

We have also analyzed theK-matrix method which iscommonly used to extract empirically thegN→D transitionamplitudes from thegN→pN data. It is found that the as-sumptions made in theK-matrix method@6# are consistentwith our meson-exchange dynamical model. Our averavalue of theE2/M1 ratioREM 5 (21.86 0.9!% is close to(21.076 0.7!% of Ref. @6#. The helicity amplitudes calcu-lated from our baregN→D vertex are in good agreemenwith the predictions of the constituent quark models~TableIV !. The differences between these bare amplitudes andempirical values extracted from the data by using thK-matrix method are shown to be due to the nonresonmeson-exchange mechanisms. This suggests that thevertex interactions in our effective Hamiltonian can be idetified with hadron models in which thepN andppN ‘‘re-action’’ channels~bothp andN are on their mass shell! areexcluded in the calculation of theN* excitation. Unfortu-nately we are not able to pin down theE2/M1 ratio of thebareg→D vertex by considering the existing data of photonasymmetry ratios and differential cross sections. More pcise data of other spin observables are needed to mprogress. This will be pursued when the data becomes avable, along with the extension of our approach to investigapion electroproduction.

The unitary transformation method developed here canextended to higher-energy regions for investigating highmassN* resonances. To proceed, we need to perform tunitary transformation up to second order in the couplinconstants to account for the 2p production channels. Theresulting scattering equations will be defined in a largcoupled channel spaceN* % pN% gN% ppN. This researchprogram can be carried out in practice since the numerimethods for solving such a Faddeev-type coupled-channequations~because of the presence of the three-bodyppNunitary cut! have been well developed@53#. Our effort in thisdirection will be published elsewhere.

ACKNOWLEDGMENTS

We would like to thank Andy Sandorfi for his very stimulating discussions and providing experimental data. One

gulty

nu-

ge

t

theeantbaren-

-re-akeail-te

beerheg

er

calels

-of

the authors~T. S.! would like to thank the Theory Group ofPhysics Division at Argonne National Laboratory for thhospitality and discussions. This work is supported by tU.S. Department of Energy, Nuclear Physics Division, undContract No. W-31-109-ENG-38, and by Grant-in-Aid oScientific Research, the Ministry of Education, Science aCulture, Japan under Contract No. 07640405.

APPENDIX: DERIVATION OF pN POTENTIAL

To see how the Feynman-amplitude-like expressionsEqs. ~3.12!–~3.16! are obtained in our approach, we givedetailed derivation ofpN potential from the familiar La-grangian

L5F2f pNN

mpc~x!g5g

mtWc~x!•]mfW ~x!

1f pND

mpcW D

m~x!TW c~x!•]mfW [email protected].#, ~A1!

where c(x), cDm(x), and f(x) are, respectively, the field

operators forN, D, andp, TW is theD→N isospin transitionoperator, [email protected].# means taking the Hermitian conjugate othe first term. By applying the canonical quantization procdure~see Sec. III about the problem concerning theD field!,we can derive a Hamiltonian from Eq.~A1!. The resultingHamiltonian can be written as

H5H01HI , ~A2!

with

HI5HIP1HI

Q , ~A3!

whereHP(HQ) describes processes which can~cannot! takeplace in free space. Explicitly, we can write in second quatization form

tual

HIP5(

aE 1

~2p!3/21

A2Ep~k!dpWdpW 8dkW H i f pND

mpA mN

EN~p!A mD

ED~p8!vpW

m8TaupWkmDpW8

†bpWakWad~pW 1kW2pW 8!

2 if pND

mpA mN

EN~p8!A mD

ED~p!upW 8Ta

†vpWmkmbpW8

†DpWakW ,a

† d~pW 2kW2pW 8!, ~A4!

whereak,a† , bpW

†, andDpW† are, respectively, the creation operators forp, N, andD states,a is the pion isospin index, andu and

vm are, respectively, the spinors of Dirac and Rarita-Schwinger fields. Clearly,HP describes theD↔pN real processes@similar to Figs. 1~a! and 1~b! with the changes→D# which can take place in free space. On the other hand, the virprocesses@similar to Figs. 1~c!–1~f!# are due to the Hamiltonian

e


HIQ5(

aE 1

~2p!3/21

A2Ep~k!dpWdpW 8dkW

3F i f pNN

mpA m

EN~p!A m

EN~p8!$upW 8g5t

ak”upWbpW8†bpW@2d~pW 82pW 2kW !akW ,a1d~pW 82pW 1kW !akW ,a

†#

1upW 8g5tak”vpWbpW8

†dpW†@2d~pW 81pW 2kW !akW ,a1d~pW 81pW 1kW !akW ,a

†#1 vpW 8g5k”t

aupWdpW 8bpW

3@2d~2pW 82pW 2kW !akW ,a1d~2pW 82pW 1kW !akW ,a†

#1 vpW 8g5k”tavpWdpW 8dpW

†@2d~2pW 81pW 2kW !akW ,a1d~2pW 81pW 1kW !akW ,a

†#%

2 if pND

mpA m

EN~p8!A mD

ED~p8!$vpW 8

mupWkmDpW 8

†bpWd~pW 82pW 1kW !akW ,a

†1vpW 8

m•vpWkmDpW 8

†dpW

1d~pW 81pW 1kW !ak,a† %

1 if pND

mpA m

EN~p8!A mD

ED~p!$upW 8vpW

mkmbpW 8

†DpWd~pW 82pW 2kW !akW ,a1 vpW 8vpW

mkmdpW 8DpWd~2pW 82pW 2kW !akW ,a%G . ~A5!

Note that the above equation includes an antinucleon spinorv, which is included to maintain the relativistic feature of thstarting quantum field theory.

To proceed, we need to derive the unitary transformation operatorS. By the procedures outlined in Sec. II,S is related toHIQ . Hence, the actual task of deriving thepN potential is to evaluate Eq.~2.23! from theHI

P andHIQ defined above. Let us

first focus on the first fourpNN coupling terms of Eq.~A5!. We need to consider

u i &5bpW†akWa† u0&,

u f &5bp8† akWa8

† u0&.

The allowed intermediate states in Eq.~2.23! are un&5bpW n† u0&, bpWm

†ak8a8† aka

† u0& for the first term, and

un&5d2pW n

†bpW 8†bpW†akWa†akW8a8† u0&, d2pWm

†bpW 8†bpW†u0& for the other three terms involving the antinucleon componentv. Substituting

these intermediate states into Eq.~2.23! and performing straightforward operator algebra, we then obtain

^kW8a8,pW 8uHIPukWa,pW &5

~ f pNN /mp!2

~2p!31

A2Ep~k8!A mN

EN~p8!upW 8F(

i51

4

M ~ i !GA mN

EN~p!

1

A2Ep~k!upW , ~A6!

where

M ~1!5mN

EN~pn!g5k” 8ta8upW nupW n8g5k”ta

1

2F 1

EN~p!2EN~pn!1Ep~k!2

1

EN~pn!2EN~p8!2Ep~k8!G ,with pW n5kW1pW 5kW81pW 8,

M ~2!5mN

EN~pm!g5k”taupWmupWmg5k” 8ta8

1

2 F 1

EN~p!2EN~pm!2Ep~k8!2

1

EN~pm!2EN~p8!1EN~k!G ,with pWm5pW 2kW85pW 82kW , and

M ~3!52mN

EN~pn!g5k” 8ta8v2pW n

v2pW ng5k”ta

1

2 F 1

2EN~p8!2EN~pn!2Ep~k8!2

1

EN~p!1EN~pn!1Ep~k!G ,M ~4!52

mN

EN~pm!g5k”tav2pWm

v2pWmg5k” 8ta8

1

2F 1

2EN~p8!2EN~pm!1Ep~k!2

1

EN~p!1EN~pm!2Ep~k8!G .By using the properties that

mN

EN~p!upW upW5

1

2EN~p!@mN1g0EN~p!2gW •pW #,

~A7!mN

EN~p!vpW vpW5

1

2EN~p!@2mN1g0EN~p!2gW •pW #,

analuated


one can easily show that, for an arbitraryp0,

mN

EN~pn!upW nupW n

1

p02EN~pN!1

mN

EN~pN!v2pW n

v2pW n

1

p01EN~pN!5

1

2EN~pn!F ~mN2gW •pW n!S 1

p02EN~pn!2

1

p01EN~pn!D

1g0EN~pn!S 1

p02EN~pn!1

1

p01EN~pn!D G

51

p022EN

2 ~pn!@mN2gW •pW n1g0p0#5

p” n1mN

pn22mN

2 51

p” n2mN,

~A8!

wherepn5(p0 ,pW n).By using Eq.~A8!, we can combine various propagators in Eq.~A6! to obtain

(i51

4

M ~ i !5g5k” 8ta8

1

2F 1

~p”1k” !2mN1

1

~p”1k” 8!2mNGg5k”ta1g5k”ta

1

2F 1

~p”2k” 8!2mN1

1

~p” 82k” !2mNGg5k” 8ta8,

wherep5„EN(p),pW … andk5„Ep(k),kW…. The above result looks remarkably simple. It resembles very much the usual Feynmamplitudes, except that the intermediate nucleon propagator is the average of two Dirac propagators for the momenta evusing the incoming or outgoingpN momentum variables.

The evaluation of theD terms is much more involved, but yields a similar form as given in Eqs.~3.15! and~3.16!. Similarderivations can also be carried out to define thepN interactions, Eq.~3.14!, due to ther meson.

y

s.

.

@1# See review by A. Donnachie, inHigh Energy Physics, editedby E. Burhop~Academic Press, New York, 1972!, Vol. 5, p. 1.

@2# See review by T.-S. H. Lee, inBaryon ’92, edited by MosheGai ~World Scientific, Singapore, 1993!, p. 99.

@3# Herman Feshbach,Theoretical Nuclear Physics, Nuclear Reactions~Wiley, New York, 1992!.

@4# M. G. Olsson and E. T. Osypowski, Nucl. Phys.B87, 399~1975!; Phys. Rev. D17, 174 ~1978!.

@5# R. M. Davidson, N. C. Mukhopadhyay, and R. S. WittmaPhys. Rev. D43, 71 ~1991!.

@6# R. M. Davidson and N. C. Mukhopadhyay, Phys. Rev. D42,20 ~1990!.

@7# Z. Li, R. A. Arndt, L. D. Roper, and R. L. Workman, PhysRev. C47, 2759~1993!.

@8# R. Machleidt, inAdvances in Nuclear Physics, edited by J.W.Negele and E. Vogt~Plenum, New York, 1989!, Chap. 2, Vol.19.

@9# D. O. Riska and G. E. Brown, Phys. Lett.32B, 193 ~1970!.@10# M. Chemtob and M. Rho, Nucl. Phys.A163, 1 ~1971!.@11# D. Lohse, J. W. Durso, K. Holinde, and J. Speth, Nucl. Ph

A516, 513 ~1990!.@12# B. C. Pearce and B. Jennings, Nucl. Phys.A528, 655 ~1991!.@13# C. T. Hung, S. N. Yang, and T.-S. H. Lee, J. Phys. G20, 1531

~1994!.@14# C. Schutz, J. W. Durso, K. Holinde, and J. Speth, Phys. Rev

49, 2671~1994!.@15# H. Tanabe and K. Ohta, Phys. Rev. C31, 1876~1985!.@16# S. N. Yang, J. Phys. G11, L205 ~1985!.@17# S. Nozawa, B. Blankleider, and T.-S. H. Lee, Nucl. Phy

A513, 459 ~1990!.@18# T.-S. H. Lee and B. C. Pearce, Nucl. Phys.A530, 532 ~1991!.@19# F. Gross and Y. Surya, Phys. Rev. C47, 703 ~1993!.@20# Y. Surya and F. Gross, Phys. Rev. C53, 2422~1996!.

-

n,

.

ys.

. C

s.

@21# M. Araki and I. R. Afnan, Phys. Rev.36, 250 ~1987!; C. vanAntwerpen and I. Afnan,ibid. C 52, 554 ~1995!.

@22# T. Sato, M. Doi, N. Odagawa, and H. Ohtsubo, Few-BodSyst. Suppl.5, 254 ~1992!; T. Sato, M. Kobayashi, and H.Ohtsubo~unpublished!.

@23# T.-S. H. Lee and A. Matsuyama, Phys. Rev. C32, 516~1986!;36, 1459~1987!.

@24# As reviewed by A. Klein and T.-S. H. Lee, Phys. Rev. D10,4308 ~1974!.

@25# I. Tamm, J. Phys.~Moscow! 9, 449 ~1945!; S. M. Dancoff,Phys. Rev.78, 382 ~1950!.

@26# N. Fukuda, K. Sawada, and M. Taketani, Prog. Theor. Phy12, 156 ~1954!.

@27# S. Okubo, Prog. Theor. Phys.12, 603 ~1953!.@28# K. Ohta and M. Wakamatsu, Prog. Theor. Phys.55, 131

~1976!.@29# M. Gari and H. Hyuga, Z. Phys. A277, 291 ~1976!.@30# T. Sato, M. Kobayashi, and H. Ohtsubo, Prog. Theor. Phys.68,

840 ~1982!.@31# W. Glockle and L. Muller, Phys. Rev. C23, 1183~1981!.@32# L. Muller, Nucl. Phys.A360, 331 ~1981!@33# T. Sato, K. Tamura, T. Niwa, and H. Ohtsubo, J. Phys. G17,

303 ~1991!.@34# K. Tamura, T. Niwa, T. Sato, and H. Ohtsubo, Nucl. Phys

A536, 597 ~1992!.@35# H. Gacilazo and T. Mizutani,pNN System~World Scientific,

Singapore, 1990!.@36# V. Bernard, N. Kaiser, and U. Meissner, Nucl. Phys.B383,

442~1992!; V. Bernard, N. Kaiser, T.-S. H. Lee, and U. Meiss-ner, Phys. Rev. Lett.70, 387 ~1993!; Phys. Rep.246, 315~1994!.

@37# S. Weinberg, Physica A96, 327 ~1979!.@38# R. A. Arndt, J. M. Ford, and L. D. Roper, Phys. Rev. D32,

1085 ~1985!.

.

t

y,


@39# G. Hohler, F. Kaiser, R. Koch, and E. Pietarinen,Handbook ofPion-Nucleon Scattering, Physics Data Vol. 12-1~Karlsruhe,1979!.

@40# R. Koch and E. Pietarinen, Nucl. Phys.A336, 331 ~1980!.@41# LEGS Collaboration, A. M. Sandorfiet al., Few Body Syst.

Suppl.7, 317~1994!; LEGS Data Release No. L7a8.0,~1994!;A.M. Sandorfi~private communication!.

@42# H. Genzelet al., Z. Phys. A268, 43 ~1974!.@43# D. Menze, W. Pfeil, and R. Wilcke, ZAED Compilation of

Pion Photoproduction Data, University of Bonn, 1977.@44# P. Douganet al., Z. Phys. A276, 155 ~1976!.@45# Review of particle properties, L. Montanetet al., Phys. Rev. D

50, 1173~1994!, p. 1710.@46# N. Isgur and G. Karl, Phys. Rev. D19, 2653 ~1979!; R. Ko-

niuk and N. Isgur,ibid. 21, 1868~1990!.@47# S. Capstick, Phys. Rev. D46, 2864~1992!.

@48# R. Bijker, F. Iachello, and A. Leviatan, Ann. Phys.~N.Y.! 236,69 ~1994!.

@49# F. E. Close and Z. Li, Phys. Rev. D42, 2194~1990!.@50# S. Capstick and B. D. Keister, Phys. Rev. D51, 3598~1995!.@51# O. Hanstein, D. Drechsel, and L. Tiator, Mainz report, 1996@52# J. D. Bjorken and S. D. Drell,Relativistic Quantum Mechanics

~McGraw-Hill, New York, 1964!.@53# As reviewed by R. D. Amado and R. Aaron, inModern Three-

Hadron Physics, edited by A. W. Thomas, Topics in CurrenPhysics~Springer-Verlag, Berlin, 1977!.

@54# We use Edmond’s convention ^J8M 8uTkquJM&5(21)2k^J8M 8uJkMq&/A2J811^J8iTkiJ&.

@55# David Lurie, Particles and Fields~Wiley, New York, 1968!.@56# M. Benmerrouche, R. M. Davidson, and N. C. Mukhopadhya

Phys. Rev. C39, 2339~1989!.@57# F. A. Berends and A. Donnachie, Nucl. Phys.B84, 342~1975!.

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Meson-exchange model for π N scattering and γ N →π N reaction