PION PRODUCTION MODEL - CONNECTION BETWEENDYNAMICS AND QUARK MODELS ,

T.-S. H. LEE

Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, U.S,A.

E-mail: lee@anlphy.phy. anl.gov

T. Sato

Department of Physics, Osaka Univemity, Osaka 560-0043, Japan

E-mail: tsato@phys.sci. osaka-u. ac.jp

We discuss the difficulties in testing the hadron models by using the N* parame-

ters extracted from the empirical amplitude analyses of the TN and TN reaction

data. As an alternative or perhaps a more advantageous approach, we present a

Hamiltonian formulation that can relate the pion production dynamics and the

constituent quark models of N* structure. The application of the approach in

investigating the A and N*(SII ) excitations is reviewed. It is found that while the

A excitation can be described satisfactory, the TN scattering in S11 channel can

not be described by the constituent quark models based on either the one-gluon-

exchange or one-meson-exchange mechanisms. A phenomenological quark-quark

potential has been constructed to reproduce the S1l amplitude.

One of the challenging theoretical problems is to understand the hadronstructure within Quantum Chromodynzunics( QCD). There exist two differentapproaches. The most fundamental one is the Lattice QCD calculation. Thesecond one is to develop various QCD-inspired models. While a lot of pro-gresses have been made in this direction, only very limited attention has beenpaid to the need of developing appropriate reaction theories for testing theirpredictions by using the data of TN and yN reactions. In the past few years,we have addressed this question concerning the constituent quark models. Herewe would like to review the progresses we have made [1–3] and discuss futuredirections. The work for N* (S1l ) involves a collaboration with T. Yoshimotoand M. Arima.

We were motivated by the following observation. It is common to comparethe masses and decay widths predicted by the constituent quark models (or anyexisting hadron models) with the data listed by Particle Data Group (PDG). Allcalculations of decay widths have been done [4–9] perturbatively. It has beenfound that such a perturbative calculation can at best describe the generalqualitative trend of the data, but not the quantitative details. It is importantto note here that the PDG’s values are extracted from the empirical 7rN --+ TNand -yN + 7rN amplitudes which contain both resonant and non-resonant com-ponents. In most partial waves, the non-resonant mechanisms are important ;

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The submitted manuscript has been createdby the University of Chicago as Operator ofArgonne National Laboratory (“Argonne”)under Contract No. W-31 -1 09- ENG-38 withthe U.S. Department of Energy. The U.S.Government retains for itself, and others act-ing on its behalf, a paid-up, nonexclusive,

irrevocable worldwide license in said articleto reproduce, prepare derivative works, dis-tribute copies to the public, and perform pub-licly and display publicly, by or on behalf of

~the Government.

one can see thk from the fact that most of the resonances identified by PDG arein fact not visible in ~iV and 7JV cross section data. By the unitarit y condition,therefore the extracted ‘resonance parameters inherently contain non-resonantcontributions. Clearly, except in a region where the non-resonant contributionsare negligibly small, the comparison of the PDG values (or values from otheramplitude analyses) with the decay widths calculated perturbatively from theconstituent ‘quark models could be very misleading. In particular, a perturba-tive calculation of decay widths is obviously not valid for cases in which two

‘., . nearby resonances in the same partial wave can couple with each other throughtheir coupling with the meson-nucleon continuum. Similar precautions ‘mustalso be taken in comparing the predicted masses with the PDG values.

To have a more direct test of constituent quark models, it is necessaryto develop a nonperturbative approach that takes account of the unitaritycondition and can relate the Tiv and -yN reactions directly to the predictedinternal quark wave functions of baryons. We have achieved this by developingan approach within the Hamiltonian formulation. In the following, we willfirst describe the major steps and discuss how our previous work [1] on theA excitation can be interpreted within this framework. We then discuss ourfindings from an investigation of the excitation of the N* (S1l) resonances inTN scattering.

We start with the usual constituent quark model defined by the followingHamiltonian

hB = K + Vccmf+ Vqq. (1)

where K is the kinetic energy, Vconf = Xi<j ffcTij is the usual linear con-finement potential. For the residual qq-interaction Vqq in Eq.(1), we considerboth the usual one-gluon-exchange( OGE) model [10-12] and the recently de-veloped [13, 14] one-meson-exchange (O ME) model(the weak q-exchange is sup-pressed here, but it was included in our investigation [3]). Explicitly, we have

yOGE =‘?9 x <h . Aj > [uz.~jv:(Tij) + SijV; (rij)] , (2)

i<j

vq:ME = E T2 . Tj [~.i . U_jvo=(Tij) + sijvf(~ij)] , (3)Z<j

where the color SU(3) factor is < &. Aj >= – ~, ~i and Ti are respectively thespin and isospin operators, and S’ij is the usual tensor operator. The radialparts of the potentials in Eqs.(2)-(3) are given in Ref. [3]. We only note herethat they are regularized by form factors I’”(q) = A%/(A~ + qz). This isconsistent with the notion that the constituent quarks are not point particles

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This report was prepared as an account of work sponsoredby an agency of the United States Government. Neitherthe United States Government nor any agency thereof, norany of their employees, make any warranty, express orimplied, or assumes any legal liability or responsibility forthe accuracy, completeness, or usefulness of anyinformation, apparatus, product, or process disclosed, orrepresents that its use would not infringe privately ownedrights. Reference herein to any specific commercialproduct, process, or service by trade name, trademark,manufacturer, or otherwise does not necessarily constituteor imply its endorsement, recommendation, or favoring bythe United States Government or any agency thereof. Theviews and opinions of authors expressed herein do notnecessarily state or reflect those of the United StatesGovernment or any agency thereof.

DISCLAIMER

Portions of this document may be illegiblein electronic image products. Images areproduced from the best available originaldocument.

within an effective theory. This regularization of the qq-potential is essentialin obtaining convergent solutions for the bound state problem defined by theHamiltoni&n hB (Eq.(1)). If the potentials are not regularized by form factors,the ground state energy is not bound from below.

It is important to mention here that the considered two models ‘are ratherdifferent mainly due to the flavor-dependent factor Ti. Tj in OME model. Thishas important consequences in predicting the baryon spectra, as discussedin Ref. [13]. It is fair to say that with suitable adjustments and additionalphenomenological parameters, both models can reproduce the general patternof PDG’s baryon spectra. Our objective is to find a way to distinguish themby considering iTN and TN reactions. The situation is similar to atomic andnuclear physics. Only by investigating reactions, the dynamical content of thetheoretical models can be truly tested.

With the Hamiltonian hB (Eq.(1)) defined above, our first step is to solvethe three-quark bound state problem

hBl~) = ~B1~), (4)

where IB) is the baryon wave function with the label B denoting collectivelythe spin-parity Jr and isospin T; mB is the mass eigenvalue. We use thediagonalization method developed in Ref. [15] to solve Eq.(4) by expandingthe baryon wave function as

(5)

where the basis states are appropriately antisymmetrized harmonic oscillatorwavefunctions. The coefficients a~” T “m Eq. (5) and the mass eigenvalues mJ. T

of Eq. (4) are obtained from diagonalizing the matrix

Hi,j = (JTT; ilhBIJmT; j). (6)

In practice diagonalization is performed within a limited number of basis states.Then the solution of Eq.(4) is a function of the oscillator range parameter b.We treat it as a variational parameter and find b by imposing the condition:

i%B *X= - (7)

The basis state is chosen so that the mass eigenvalue mJ9? T does not changeby further extension of the basis states. In practice we include the basis statesup to llhu.

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The next step is to define how an external meson(M) can interact with thebound three-quark systems. Namely, we need to calculate the matrix eIementsfor the transitions from one-baryon states(B) to meson-baryon states(Af’l?’)

~, Af,,B(k) = (B’; M’lfifif(k)p),r+ (8)

w-here B is a bound state wave function generated from the above structurecalculation, and H,vr(k) is an appropriate operator describing how a meson Mwith a momentum k is emitted or absorbed by constituent quarks. Followingthe previous works [4–8], we assume that 17M(k) for M = n is a one-bodyoperator which can be derived directly from the nonrelativistic limit of the

~ii t 5-#kPup for the q H rq transition. To beFeynman amplitude - ~= p ~consistent with the nonrelativistic treatment of constituent quarks, we keeponly the terms up to the order of p/mq. In coordinate space, the resultingq + m + q transition operator is

where a denotes the z-component of pion isospin and pi (p:) is the derivativeoperator acting on the initial (final) baryon wave function; k and WXare themomentum and energy of pion, respectively.

We fix the rqq coupling constant by assuming that the ~NN vertex func-tion can be calculated from Eqs. (8)-(9) using the (0s)3 nucleon wavefunction.

We then find that ~ = $ w, where MN is the observed mass of the nu-

cleon and we use the empirical value g~NJJ14x = 14. The same j.qq COUPl@

constant is also used to evaluate the qq potential Eq. (3) of OME model. Ac-cordingly, we also introduce a form factor F(k) in Eq. (9) to account for theeffect due to the finite size of constituent quarks.

For photon coupling, we calculate the B + l?’~ by assuming the commonlyused ~ – qq vertex interaction. The resulting form is

~,~,~(k) = (13’; ~lHT(k)IB),~t (lo)

with

H,(k) = “ ~ ~eik”rie - [zcrzx k – (pi +p~)] .~&,=, ‘hq

(11)

By using the (0s)3 wave functions for N and A and setting mq N 300 MeVto evaluate Eqs. (10)-(11), one finds that the nucleon magnetic moments can

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be reproduced well and the resulting helicity amplitudes for the A -+ N--ytransition are A312 N –160 x 10-3 (GeV)–1/2(A1i2 = A3/2/fi). These valuesof heliclty amplitudes, which are about 40 ‘Yo lower than the values listed byPDG but are cIose to various quark model predictions [8, 12], are consistent

with our interpretation in Ref. [1] that the constituent quark model predictionscan only be compared with the bare values extracted from the data by removingthe on- and off-shell effects due to non-resonant interactions in a dynamicalapproach based on the Hamiltonian formulation.

With the vertex functions defined above, we can then define a hadronicHamiltonian for investigating nN and TN reactions. It has the following form

H= HO+H1, (12)

where the free Hamiltonian can be written in a second-quantization form

Ho=~ J@&~(p)bb(p)b~(p)+~/dk~itr(k)ah(k)~,w(~)-(13)B lvi

Here, b~(b~) and ah (a&f) are the creation (annihilation) operators for thebaryons and mesons respectively, and EB (p) = (m%+ p2)1/2, WM (k) = (WLL+kz)l/z It is important to note here that the baryon mass m~ in Eq.(13)is generated dynamically from solving the three-quark bound state problemdefined by Eqs.(4)-(7). Their values could be significantly different from reso-nance positions listed by PDG. We use the experimental value for the mesonmass m~.

The interaction term in Eq. (12) is written in terms of the vertex functionsdefined by Eqs.(8) and (10)

BBIA4J [

HI= ~ dpdp’dk (B’lH&flB; M)b~, (p’) b~(p)a~ (k) + h.c.] . (14)

The above interaction Hamiltonian is similar to that of the dynamical modeldeveloped in Ref. [1], except that the B’ -+ BJ4 transition amplitudes arenow determined by the predicted quark wavefunctions. As discussed in Ref.[I], it is a non-trivial many-body problem to calculate TN and TN reactionswith the use of HI. To obtain a manageable reaction theory, we follow Refs.[1, 16] and apply the unitary transformation up to the second order in HI toderive an effective Hamiltonian. The essence of the unitary transformationmethod applied in Ref. [1] is to absorb the unphysical transition B + M“B’with m~ < m~) + m ~~ into non-resonant potentials. The resulting effectiveHamiltonian then takes the following form

Heff=Ho+r+r~+?J, (15)

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where Ho is defined in Eq. (13). The vertex I’t contains only the physicaldecay process B + i\f’B’ with mB > m,b~, + m& ,

r$= ~/

dkdpdp’ (B’; M’lH~q,lB)bL, (p’)ajf~ (k’)b~ (P)MBB’

x 8(mB – (~Bt + mMt)), (16)

where 6(z) = 1(0) for z > O(X < O). The non-resonant MB + M’B’ two-bodyinteractions are defined by

IJ= z/ dkdk’dpdp’(B’; M’lfilB; M)a~, (k’)a,u(k)b~, (p’) b~(p). (17)A4M’BB’

The construction of the matrix elements of the nonresonant potential O isexplained in Ref. [1]. Typically, it consists of the energy-forbidden s-channelterms such as TN + N -i nN and rA + A G ~A, and particle-exchangeterms such as the usual crossed nucleon and A terms and the p- and w-exchangeterms.

By using the standard projection operator method [1], it is straightforwardto derive from the effective Hamiltonian Eq. (15) a calculational framework forrN and -yN reactions leading to various final meson-nucleon final states. Theresulting transition operator can be written as

T.,p = fcio + XL’, ,[D-’(E)], Jy?,p. (18)

i,~

Here CY,/? denote the meson-baryon states such as TN, TN, qN and nA. N:are mass eigenstates of Eq. (4). The first term in Eq. (18) is the non-resonantamplitude determined only by the non-resonant interaction v

tcr,p = Va,p + E va#; (W-yp, (19)7

with

[G:(E)]’1 = ~ – EFJ,(p) –w~, (k) +ze. (20)

The second term in Eq. (18) is the resonant term determined by the dressedN* propagator and the dressed vertex functions:

[~(~)],,j= @ – my )&j - Zj(E), (21)

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.

.

“2C0 250 300 350 400 450

.?7[MeV]

(a)

0.8

0.6

Q0.4

0.2

p(ymp 85”

G*.O 02S —,-

I 12c0 250 300 350 400 450

f, [MeV]

(b)

Figure 1: The recent Mainz data[19] of differential cross section and photon asymmetry X

of the p(-y, X“)p reaction are compared with the predictions from the SL model. GE is thestrengthof the El+ amplitude of the A + TN transition.

In Eq. (21), the N* self-energy is defined by

(24)

-r

We now note that the above coupled equations relate the full scatteringamplitude Ta,@ Eq. (18) nonperturbatively to the quark wavefunctions calcu-lated from the considered constituent quark model. In particular, the decaywidths, as listed by PDG, correspond to the dressed vertex rN.,7 defined byEq.(22), and hence they are expected to be different dynamically from thebare vertex functions l?~.,7 which are calculated from using Eqs.(8)-(11). Itis also clear that the accuracy of the non-resonant interaction V*,P, which de-termine the nonresonant amplitude ta,p via Eq. (19), plays an important rolein identifying the resonant amplitude of Eq. (18) from the data. It is thereforealso essential to have it calculated from the same quark model. This can beachieved in our approach through the use of unitary transformation method,as discussed above(the developed method can be straightforwardly extended toaccount for the interactions due to heavy mesons). Without such a consistenttreatment of both the resonant and non-resonant interactions, it is difficult todraw conclusions.

We now turn to discussing our findings. First we consider the A statein F’33 channel. For this investigation, we only have one N* (= A) state andtwo channels a, B = TN, TN. Eqs. (18)-(24) are then reduced to the set ofequations developed in Ref. [1](called SL model from nowon). The accuracyof the SL model are illustrated in Figs.1-2. We see that the predictions are

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,.

0.2 W,~=l.171GeV 0.2 W,m=l.232GeV

<d 0 0

E I-0.2 -0!2 -

‘1

o 20 40 60 0 20 40 60

ep 9P

Figure 2: The MIT-Bates data[20] of asymmetry TT.L of the p(e, e’x”) reaction at Q2 = 0.126(GeV/c)2 are compared with the predictions by the SL model.

in good agreement with the recent data from Mainz [19] and MIT-Bates [20].It is therefore sufficient to adjust the parameters characterizing the the qqpotentials Eqs.(3) or (4) and the constituent quark form factor F(k) for thevertex interaction Eq. (9) to fit the bare A parameters of the SL model. This isa rather restricted procedure since there are only about 4 parameters for eachof the considered OGE and OME models.

In Fig.3, we show that the A +- XN form factor calculated from usingEqs. (8)-(9) agree to a large extent with the phenomenological bare form fac-tor(dashed curve) of SL model. The predicted mass(via Eq.(4)) is also requiredto be the bare mass mA = 1300 MeV of SL model.

Once the parameters of the considered constituent quark model are de-termined, we then can generate the mass m~ and quark wavefunctions for allpossible partial waves by solving Eqs. (4)-(7). The relevant vertex functionscan then be calculated by using Eqs, (8)-(11) and the TN and 7N reaction am-plitudes can be predicted by solving Eqs. (18)-(24). The comparison with thedata will tell us whether the considered constituent quark model is correct. Wehave explored this by investigating the TN scattering in Sll partial wave. Thecalculation was done by considering two N* and three channels TN, qN andrA. The predicted ZN phase shifts are shown in Fig.4a. We see that neitherOGE(solid curve) or OME(dashed curve) models can describe the data(opencircles) in the entire energy region.

As an attempt to improve the fit to the S1l amplitude, we have also ex-plored the mixture of OGE and OME models. It turns out that such a hybridmodel also fails, mainly due to the very disruptive tensor component of theOME model in determining the phases of wave functions.

8

...

02 4 6 8 10

k [fro-i]

Figure 3: The A + ~N vertex function(solid curve) determined within the OGE model is

compared with the phenomenological form(dashed curve) of the SL model[l].

We therefore turn to investigating a purely phenomenological model. Byanalyzing an analytical model [3], we have found that the data of A excitationand S1x rN scattering seem to favor a tensor term due to one-gluon-exchangeand a spin-spin interaction due to one-meson-exchange. This has guided us toexplore many phenomenological models. For example, we have found that theXN S1l amplitudes can be much better described by the following phenomeno-logical model

The results are shown in Fig.4b. We see that the general feature of the datacan now be reproduced. The remaining discrepancies perhaps can be removedif we refine various form factors in the qq potential and N* + TN, qN, XAvertex functions(Eq.(9)).

In conclusion, we have developed a Hamiltonian formulation for relatingthe dynamics of ZN and -yN reactions to the constituent quark models. Theapproch has been applied to investigate the A and N* (SI 1) resonances. It isfound that while the A excitation can be described satisfactory, the TN scat-tering in S1l channel can not be described by the constituent quark modelsbased on either the one-gluon-exchange( OGE) or one-meson-exchange( OME)mechanisms. The data seem to favor the spin-spin interaction due to one-meson-exchange and the tensor interaction due to one-gluon-exchange. A phe-nomenological quark-quark potential has been constructed to reproduce theSll amplitude.

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1000 1200 1400 1600 1800 1000 1200 1400 1600 1800

W.m [MeV] W,- [MeV]

(a) (b)

Figure 4: The predicted TN phase shifts are compared with the data(open circles):(a)

from OGE(solid curve) and OME(dashed curve) models, (b) from phenomenological model

Eq. (25).

To end, we would like to point out that the multi-channels multi-resonancesparameterization of ZN reaction amplitudes, first developed by Cutkosky andhis collaborators [17] and has been revived recently [18], can be derived [21]from Eqs. (18)-(24). The bare parameters associated with this phenomenolog-ical model can have dynamical interpretations within our Hamiltonian formu-lation. Consequently, it may be more advantageous to apply this parametri-zation to analyze the forthcoming data. The extracted bare N* parameterscan be used as the data for exploring the constituent quark interactions usingthe procedures detailed in Ref. [3] and outlined in this contribution.

Acknowledgments

Thk work was supported U .S. DOE Nuclear Physics Division Contract No. W-31-109-EPJG-38 and JSPS Grant-in-Aid for Scientific Research (C) 12640273.

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