Z. Phys. A 359, 467–470 (1997) ZEITSCHRIFTFUR PHYSIK Ac© Springer-Verlag 1997

Treatment of the∆ current inelectromagnetic two-nucleon knockout reactions

P. Wilhelm1, H. Arenhovel1, C. Giusti2,3, F.D. Pacati2,3

1 Institut fur Kernphysik, Johannes Gutenberg-Universit¨at, D-55099 Mainz, Germany2 Dipartimento di Fisica Nucleare e Teorica, Universit`a di Pavia3 Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, Italy

Received: 20 June 1997 / Revised version: 4 August 1997Communicated by W. Weise

Abstract. The contribution of the∆(1232) isobar to the elec-tromagnetic current of the two-nucleon system and its rolein (γ,NN) processes is investigated. The difference betweenthe genuine∆-excitation current and that part of the currentconnected to the deexcitation of a preformed∆ in the targetnucleus is stressed. The latter cannot lead to a resonant be-haviour of matrix elements for energies in the∆ region. Thereaction16O(γ, pp)14C, where the∆ contribution is dominantat intermediate energies, is considered. The large variationsfound in the cross sections for different treatments stress theneed for a proper treatment of the∆ current for a clear under-standing of the reaction mechanism of two-nucleon emissionprocesses.

PACS: 25.20.Lj; 14.20.Gk

1 Introduction

The electromagnetically induced two-nucleon knockout servesas a tool to study short-range correlations between two nucle-ons in a nucleus. Thereby, one assumes that the photon inter-acts with the correlated pair through a one-nucleon current.However, nucleon pairs can also be ejected by two-nucleoncurrents which effectively take into account the influence ofsubnuclear degrees of freedom like mesons and isobars. There-fore, in order to estimate this competing effect quantitatively,a reliable treatment of meson exchange as well as isobar cur-rents is necessary before one can draw definite conclusionson the role of short-range correlations. Experimental data oftwo-nucleon photoemission in the∆(1232) region were takenin different laboratories [1, 2, 3, 4], while only one exploratoryexperiment on the (e, e′pp) reaction [5] was completed. How-ever, many new results will become available in the near future,exploiting continuous-wave accelerators and tagged-photonfacilities.

In this note we would like to point out a pitfall which onemay encounter when considering contributions from interme-diate∆ isobars to electromagnetic processes via the effectiveoperator approach. To this end we remind the reader at the two

first-order contributions of the∆ to the effective electromag-netic two-body current of a two-nucleon subsystem showndiagramatically in Fig. 1. The first one (I) describes the∆ ex-citation with subsequent deexcitation by pion exchange whilethe second (II) describes the time interchange of the two steps,i.e., first excitation of a virtual∆ by pion exchange in a NNcollision and subsequent deexcitation by photon absorption.In other words parts I and II correspond to N∆ admixturesin the final and the initial NN subsystem, respectively. Theywould be treated as explicit components of the wave functionin the approach of nuclear isobar configurations (IC) [6, 7].Consequently, it is important to note that in diagram II the∆is always far off-shell being part of the initial state irrespectiveof the energy transferred to the system by the real (or virtual)photon, whereas in diagram I the∆ can become on-shell for asufficiently high energy transfer. This then gives rise to a pro-nounced resonant behaviour of the matrix elements of diagramI when varying the energy transfer in the region between, say,200 and 400 MeV. These features appear automatically in theIC approach, however, in the effective operator approach onlyif one keeps the full∆ propagator in the intermediate state,making the effective two-body operator nonlocal.

For (γ,NN) calculations on complex nuclei, it is necessaryto avoid this nonlocality in order to keep the numerical effortwithin reasonable limits. Therefore, often the∆ propagatoris taken in the simplest static approximation keeping only thebaryon mass differenceM∆ −MN . In this case the two con-

∆γ

( I )

π

N N

N N

( I I )

∆

γ

π

N N

N N

Fig. 1. The∆ contribution to the current of the two-nucleon system:∆-excitation current (I) and∆-deexcitation current (II)

468

tributions of Fig. 1 can be combined into one effective localtwo-nucleon operator. This approximation appears reasonableat low energies but it certainly fails when the transferred en-ergy allows the excitation of an on-shell∆. In the latter caseone might be tempted to replace (M∆ −MN )−1 by a reso-nant energy-dependent∆ propagator. Indeed this procedurehas been followed in the past. However, it leads to a wrongeffective operator, as is outlined in detail in the next section,and results in a strong overestimation of the contribution ofdiagram II. This is shown in section 3, where different treat-ments of the∆ current are compared by means of a calculationfor the16O(γ, pp)14C process within the framework of [8, 9].

2 Formalism

The effective current operator shown in Fig. 1 is given by

∆ = (I)∆ + (II)

∆ + (1↔ 2). (1)

In the following we restrict ourselves to the dominant magneticdipole N↔∆ transition. For simplicity the hadronic N∆↔NNtransition is described by staticπ-exchange. The inclusionof theρ-exchange is straightforward but not essential for ourpurpose. Under these assumptions, the excitation current reads

(I)∆ (q) = γ τ (1)

N∆ · τ(2)NN

σ(1)N∆ · k σ

(2)NN · k

k2 +m2π

×G∆(√sI ) τ

(1)∆N,3 iσ

(1)∆N × q, (2)

and the deexcitation part

(II)∆ (q) = γ τ (1)

N∆,3 iσ(1)N∆ × q G∆(

√sII ) τ

(1)∆N

· τ (2)NN

σ(1)∆N · k σ

(2)NN · k

k2 +m2π

, (3)

whereq is the photon momentum, andk is the momentumof the exchanged pion. The factorγ collects various couplingconstants,γ = fγN∆fπNNfπN∆/m3

π. The propagator of theresonance is denoted byG∆. It depends on the invariant en-ergy√s of the∆. Neglecting the kinetic energy of the relative

motion of the intermediate N∆ state, one obtains a local ap-proximation toG∆. It reads

G∆(√s) =

1

M∆ −√s− i

2Γ∆(√s), (4)

whereΓ∆ is the energy-dependent decay width of the∆ andM∆ = 1232 MeV its mass.

Clearly,√s can be very different in diagram I and II. In

diagram II, it does not depend on the photon energyEγ , andit is reasonable to approximate it by the nucleon mass√sII = MN . (5)

On the other hand,√sI depends onEγ . It grows withEγ

and for√sI ≈ M∆ the∆ is essentially on-shell. Following

the recent suggestion made in [10] for the choice of√sI , the

calculations presented below use√sI =

√sNN −MN , (6)

where√sNN is the experimentally measured invariant en-

ergy of the two fast outgoing nucleons in anA(γ,NN )A− 2reaction. The energy dependence of the two parts of the∆

current can also be considered from a different point of view.For forward propagating exchanged pions, both parts are re-lated to the process of electromagnetic pion production onone nucleon followed by its reabsorption on the second nu-cleon. Part I is connected to thes-channel contribution of the∆which leads to the well-known resonantM3/2

1+ pion produc-tion multipole, whereas part II is connected to theu-channelcontribution which has only a smooth energy dependence.

Only for low energy transfers, say below 100 MeV, it maybe justified to approximate also

√sI ≈ MN . Then the∆

propagators in the excitation and deexcitation parts are equaland the spin and isospin structure of their sum simplifies due tothe cancellation of terms. To see this, one first has to rewrite (2)and (3) using the following identity for the N↔∆ transitionspin (isospin) operators

σN∆ · a σ∆N · b =23

a · b− i

3σNN · a× b, (7)

wherea andb are two arbitrary vectors. One finds

(I)∆ (q) =

19γ[2τ (2)

NN,3− i(τ (1)NN × τ

(2)NN

)3

](

2ik − k × σ(1)NN

)× q G∆(

√sI )

σ(2)NN · k

k2 +m2π

(8)

and

(II)∆ (q) =

19γ[2τ (2)

NN,3 + i(τ (1)NN × τ

(2)NN

)3

](

2ik + k × σ(1)NN

)× q G∆(

√sII )

σ(2)NN · k

k2 +m2π

. (9)

Then, in the low-energy (le) approximation, i.e. usingG∆(√sI ) = G∆(

√sII ) = (M∆ −MN )−1, one obtains for

the total current

(le)∆ (q) =

29γ i[4τ (2)

NN,3 k +(τ (1)NN × τ

(2)NN

)3

k × σ(1)NN

]×q

1M∆ −MN

σ(2)NN · k

k2 +m2π

. (10)

This form is usually quoted in the literature (see e.g. [11, 12]).It serves as starting point for model calculations in heaviernuclei [12, 13, 14] as well as in few-nucleon reactions [15, 16,17].

The simple replacement of the low energy propagator(M∆ −MN )−1 in (10) by the resonantG∆ of (4) in orderto obtain an operator which is more appropriate for studyingphoton absorption at higher energies is, however, in no wayjustified. This procedure implies a strong overestimation ofthe deexcitation part at higher energies since it introduces aresonant behaviour also there. Nevertheless, it appears in somepapers (see e.g. [18, 19])1. To make this point clearer, it is use-ful to consider the isospin matrix structure of the excitation anddeexcitation current. Table 1 summarizes the relevant isospinmatrix elements. Note that transitions betweennp pairs withisospinT = 1 are always forbidden for an isovector current.According to Table 1, the above replacement leads to wrongconclusions, in particular when the absorption onpp pairs isstudied, or in cases where the absorption onnp pairs with

1 After completion of this work we have been informed by J. Ryckebuschthat for the numerical calculations of [19] a different operator for the∆ currenthas been used instead of the one written in [19]

469

Table 1. Isospin matrix elements (terms in square brackets in (8) and (9))of the excitation (I) and deexcitation part (II) of the∆ current for varioustransitions

Transition (I)∆ (II)

∆

np(T = 0)→ np(T = 1) −4 0np(T = 1)→ np(T = 0) 0 −4pp/nn→ pp/nn ±2 ±2

isospinT = 1 is expected to be relevant. In (γ, np) reactions,only those contributions which proceed via absorption onnppairs withT = 0 (like the quasi deuteron mechanism for exam-ple) remain unaffected with respect to the correct prescription,as the contribution of the operator (9) vanishes.

3 Results

In this section we investigate the dependence of (γ, pp) crosssections on the different treatments of the∆ current withinthe theoretical model of [9]. Although different channels canbe considered in the model, we here have chosen the (γ, pp)channel since there the∆ current is dominant at intermediateenergies.

Exclusive cross sections of the16O(γ, pp)14C knockoutreaction have been calculated for transitions to the groundstate and low-lying discrete excited states of14C. In the modelthe final state|Jπ〉 of the residual nucleus is obtained fromthe removal of a nucleon pair coupled toJ . The correlatedwave function of the pair is calculated with the single-particlestates of [20] and the Jastrow-type correlation function of [21].The final-state interaction is taken into account by means of theoptical potential of [22], describing the interaction of each oneof the two protons with the residual nucleus. More details ofthe model and of the theoretical ingredients of the calculationare given in [9].

The differential cross sections of the16O(γ, pp)14C(g.s.)reaction atEγ = 150 MeV and 300 MeV are shown in Fig. 2in a coplanar and symmetrical kinematics as a function of theangleγ between the photon and one of the symmetrically out-going protons. Three different treatments of the∆ current arecompared. The solid curves refer to the correct form, i.e., use ofthe resonant∆ propagator in the excitation current only (partI of Fig. 1). Adopting the resonant propagator in both excita-

Fig. 2. The differential cross section of the16O(γ, pp)14C(g.s.) reaction incoplanar and symmetrical kinematics as a function ofγ using different treat-ments of the∆ current:∆(RN) solid, ∆(RR) dotted, and∆(NN) dashedcurves

tion and deexcitation currents (part I and II) leads to the dottedcurves. The dashed curves refer to the low energy approxima-tion of (10). In the following, these three currents are referredto as∆(RN), ∆(RR), and∆(NN), respectively, indicatingthe use of either the resonant (R) or energy-independent non-resonant (N) propagator in part I and II. Already at 150 MeV,∆(RN) and∆(RR) lead to peak cross sections which differby nearly a factor two. The difference grows with the photonenergy and at 300 MeV the peak cross sections differ by morethan a factor five. As one would expect,∆(NN) underesti-mates and∆(RR) overestimates the cross section. One notesalso a different shape of the angular distributions for∆(RN)on the one hand and∆(RR) or∆(NN) on the other hand. At300 MeV the minimum is less pronounced for the solid curve.This is a consequence of the fact that∆(RN) has a differentoperator structure with respect to its spin and angular momen-tum dependence, since its parts I and II do not enter with equalweight as in∆(RR) and∆(NN). For the same reason onemay expect qualitatively different predictions for polarizationobservables.

In Fig. 3 the differential cross section in coplanar and sym-metrical kinematics at zero recoil momentum of the residualnucleus is plotted as a function of the photon energy. Thiscorresponds to the region in Fig. 2 where the cross sectionis maximal. Fig. 3 clearly shows the strong overestimation ofthe cross section when∆(RR) is used. This certainly wouldaffect any analysis of experimental (γ, pp) data in the∆ regionin view of the role of short-range correlations. The low-energyapproximation∆(NN) can of course not predict a resonancepeak. The dash-dotted curve has been calculated with∆(RR)as the dotted one, but the choice in (6) has been replaced bythe energy assignment

√sI = Eγ +MN used, e.g., in [19]. It

shifts the resonance position towards lower energies and thusleads to a further overestimation of the cross section for ener-gies below 260 MeV. The effect of this choice of

√sI on the

angular distribution has already been discussed in Ref. [9] forthe (e, e′pp) reaction.

Calculations for the transition to the excited 1+ and tothe first excited 2+ state of14C have also been performed.Qualitatively, the results agree with the former ones. However,the size of the overestimation varies. This again can be tracedback to the different operator structure of∆(RN) and∆(RR).It becomes visible when the quantum numbers of the active

Fig. 3. The differential cross section of the16O(γ, pp)14C(g.s.) reaction incoplanar and symmetrical kinematics at zero recoil momentum as a functionof the photon energy. Line convention as in Fig. 2. Thedash-dotted curve

has been calculated with∆(RR), but using the energy assignments1/2I =

Eγ +MN as in [19]

470

nucleon pair change and cannot simply be simulated even byan energy-dependent rescaling factor.

Similar results are obtained in the (e, e′pp) reaction, wherethe presence of the longitudinal contribution, due to correla-tions, reduces the effects of the different treatments of the∆current.

4 Summary and outlook

In this paper we have discussed the treatment of the∆ iso-bar current in electromagnetic two-nucleon knockout reac-tions based on an effective two-nucleon operator. It has beenemphasized that this current consists of a∆ excitation part anda∆ deexcitation part corresponding to a∆ admixture in theinitial state. These two parts are related to thes- andu-channelcontributions of the∆ to electromagnetic pion production on anucleon. Only the excitation part of the current has a resonantenergy dependence which can result in a characteristic energydependence of observables.

This different energy dependence of excitation and deex-citation parts has to be taken care of when going to higherenergy transfers. In particular, it does not allow to replacethe propagator (M∆ −MN )−1 of the simplest static approx-imation to the current (which is justified in the low-energyregion) by a resonant propagator. The error is two-fold: first,the deexcitation part would get a wrong energy dependenceand, secondly, one would loose parts of the excitation currentwhich have been cancelled by terms in the deexcitation cur-rent. In view of the fact that this prescription has been usedin the literature, we have performed an explicit calculation forthe 16O(γ, pp)14C reaction. It shows that the simple replace-ment may lead to an overestimation of the cross section (inthe preferred kinematic region around zero recoil momentumof the residual nucleus) by more than a factor five. Such avariation of the∆ contribution is definitely too large whenone wants to use two-proton knockout as a tool to study shortrange correlations in the nucleus.

Finally, we would like to point out that we are well awarethat the above proposed treatment of the∆ current is still toosimplistic in two respects and should be improved in the future.First, the use of the local approximation in (4) for the∆ prop-agator which ignores the kinetic energy in the intermediateN∆ state tends in general to overestimate the∆ contribution.This has been shown in the past for various hadronic and elec-

tromagnetic few-body reactions (see, e.g. [23, 24]). As a sideremark, in the framework of nucleon isobar configurations thekinetic energy is treated properly [7]. Second, the use of astatic NN↔N∆ transition potential in part I of Fig. 1 ignoresthe∆-N mass difference and thus overestimates the range ofthe effective operator.

P.W. thanks the Dipartimento di Fisica Nucleare e Teorica of the Universityof Pavia for the hospitality during part of this work. H.A. and P.W. weresupported by the Deutsche Forschungsgemeinschaft (SFB 201).

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