Z. Phys. A 356, 327–338 (1996) ZEITSCHRIFTFUR PHYSIK Ac© Springer-Verlag 1996

Modelling the nucleon wave function from soft and hard processesJ. Bolz?, P. Kroll

Fachbereich Physik, Universitat Wuppertal, D-42097 Wuppertal, Germany (e-mail:bolz,kroll@wpts0.physik.uni-wuppertal.de)

Received: 15 March 1996 / Revised version: 10 May 1996Communicated by W. Weise

Abstract. Current light-cone wave functions for the nucleonare unsatisfactory since they are in conflict with the data ofthe nucleon’s Dirac form factor at large momentum transfer.Therefore, we attempt a determination of a new wave func-tion respecting theoretical ideas on its parameterization andsatisfying the following constraints: It should provide a softoverlap contribution to the proton’s form factor in agreementwith data; it should be consistent with current parameteriza-tions of the valence quark distribution functions and lastlyit should provide an acceptable value for theJ/ψ → NNdecay width. The latter process is calculated within the mod-ified perturbative approach to hard exclusive reactions. Asimultaneous fit to the three sets of data leads to a wavefunction whosex-dependent part, the distribution amplitude,shows the same type of asymmetry as those distribution am-plitudes constrained by QCD sum rules. The asymmetry ishowever much more moderate as in those amplitudes. Ourdistribution amplitude resembles the asymptotic one in shapebut the position of the maximum is somewhat shifted.

PACS: 12.38.Bx; 13.25.Gv; 13.40.Gp; 14.20.Dh

1 Introduction

There is general agreement that the conventional hard scat-tering approach (see [1] and references cited therein) inwhich the collinear approximation is used, gives the cor-rect description of electromagnetic form factors and perhapsof other exclusive processes in the limit of asymptoticallyhigh momentum transfers. This framework relies upon thefactorization of hadronic amplitudes in perturbative, short-distance dominated hard scattering amplitudes and process-independent soft distribution amplitudes (DA). In order tochallenge arguments against the applicability of the hardscattering approach in experimentally accessible regions ofmomentum transfer [2] a modification of this scheme hasbeen proposed by Botts, Li and Sterman [3] in which the

? Supported by the Deutsche Forschungsgemeinschaft

transverse hadronic structure is retained and gluonic radia-tive corrections in form of a Sudakov factor are incorpo-rated. This more refined treatment of exclusive observablesallows to calculate the genuinely perturbative contributionself-consistently in the sense that the bulk of the perturba-tive contribution is accumulated in regions of reasonablysmall values of the strong coupling constantαS . Neverthe-less it turns out that the perturbative contributions to exclu-sive observables are in general too small with perhaps a fewexceptions. In particular in the case of the nucleon’s Diracform factor it is shown in [4] that using end-point concen-trated DAs the perturbative contribution to it can amount toat most 50% of the experimental data. This is a “maximal”result obtained under the assumption of probability one forthe nucleon’s valence Fock state. For a more plausible valueof that probability (e.g. of order 0.1) the perturbative con-tribution is much smaller than the experimental data (orderof 20%). In [5] furtherk⊥-dependent corrections were in-vestigated and it was found that these corrections may leadto even stronger suppressions of the perturbative contribu-tion than found in [4]. Thus, it is likely that the perturbativecontribution to the proton form factor only amounts to atiny fraction of the experimental data. We emphasize thatthis conclusion is in agreement with the criticism againstthe perturbative approach to exclusive reactions raised byIsgur and Llewellyn-Smith [2].

Ostensible agreement between data and calculations car-ried through within the conventional hard scattering ap-proach under use of end-point concentrated DAs like thatone proposed by Chernyak et al. (COZ) [6], can be tracedback to large contributions from the soft end-point regionswhere one of the momentum fractionsxi tends to zero. Inthese regions gluon momenta become small and hence theuse of perturbation theory is unjustified [2].

The smallness of the perturbative contribution is, as webelieve, not a debacle. On the contrary it seems to be fullyconsistent within the entire approach. We remind the readerof the contribution from the overlap of the initial and finalstate (soft) nucleon wave functions. This additional contri-bution, customarily termed the overlap contribution, is usu-ally neglected but it is indeed large, as was shown in [2]for a number of examples and as we are going to demon-

328

strate for a wide class of wave functions. For the end-pointconcentrated wave functions like those based on the COZDA [6], the overlap contributions even exceed the exper-imental data [7] on the Dirac form factor of the nucleon,FN1 , by large amounts. This parallels observations made onthe pion’s electromagnetic form factor recently (see, for in-stance, [8]) according to which end-point concentrated pionwave functions are clearly disfavoured.

The purpose of the present paper is the construction ofthe nucleon’s (valence Fock state) wave function. In accordwith the findings in [4, 5] we demand this wave function toprovide an overlap contribution that completely controls theDirac form factor at momentum transfers around 10 GeV2.Admittedly, this requirement does not suffice to determinethe wave function, further constraints are required. So weuse the available information on the parton distribution func-tions [9] to which the nucleon’s wave function is also re-lated. As a third constraint we employ the decay reactionJ/Ψ → NN . This process is expected to be dominated byperturbative contributions. We are going to calculate the de-cay width for it within the modified perturbative approachof [3] in contrast to previous analyses of theJ/Ψ decay[10, 11, 12, 13]. Employing a parameterization of the wavefunction that complies with theoretical ideas [14, 15], wedetermine the few (actually two) parameters of the wavefunction from a combined fit to the data of the three reac-tions just mentioned. We emphasize that we do not aim at aperfect fit to the data, a number of theoretical uncertaintiesand approximations inherent in our approach would rendersuch an attempt meaningless. The purpose of our analysisis rather to demonstrate the existence of a reasonable wavefunction from which the prominent features of the data canbe reproduced.

The paper is organized as follows: In Sect. 2 we brieflyrecapitulate a few properties of the nucleon’s light-conewave function and we introduce our ansatz for it. Sects. 3,4 and 5 are devoted to the discussions of the overlap con-tributions toFN1 , the parton distribution functions and thedecayJ/ψ → NN , respectively. In Sect. 6 we will presenta new wave function satisfying the constraints discussed inSects. 3, 4 and 5. Finally, Sect. 7 contains our conclusions.An appendix includes a derivation of the nucleonicJ/ψwidth within the modified perturbative approach.

2 The nucleon wave function

Similarly to Sotiropoulos and Sterman [16] we write thevalence Fock state of a proton with positive helicity as (theplane waves are omitted for convenience)

|P,+〉 =1√3!εa1a2a3

∫[dx][d2k⊥]

Ψ123Ma1a2a3

+−+

+ Ψ213Ma1a2a3−++ −(Ψ132 + Ψ231

)Ma1a2a3

++−

(2.1)

where we assume the proton to be moving rapidly in the 3-direction. Hence the ratio of transverse to longitudinal mo-menta of the quarks is small and one may still use a spinorbasis on the light cone. A neutron state is obtained by thereplacementu ↔ d. The integration measures are definedby

[dx] ≡3∏i=1

dxi δ(1−∑i

xi)

[d2k⊥] ≡ 1(16π3)2

3∏i=1

d2k⊥i δ(2)(∑i

k⊥i) . (2.2)

The quarki is characterized by the usual fractionxi of thenucleon’s momentum it carries, by its transverse momentumk⊥i with respect to the nucleon’s momentum as well as byits helicity and color. A three-quark state is then given by

Ma1a2a3λ1λ2λ3

=1√

x1x2x3|ua1;x1, k⊥1, λ1〉

|ua2;x2, k⊥2, λ2〉 | da3;x3, k⊥3, λ3〉. (2.3)

The quark states are normalized as follows

〈qa′i; x′i, k⊥

′i, λ

′i | qai ;xi, k⊥i, λi〉 =

2xi(2π)3δa′iaiδλ′

iλiδ(x′i − xi)δ(k′⊥i − k⊥i) . (2.4)

Since the 3-component of the orbital angular momentumLzis assumed to be zero the quark helicities sum up to thenucleon’s helicity. As has been demonstrated explicitly in[17] (2.1) is the most general ansatz for theLz = 0 pro-jection of the three-quark nucleon wave function: From thepermutation symmetry between the twou quarks and fromthe requirement that the three quarks have to be coupled inan isospin 1/2 state it follows that there is only one inde-pendent scalar wave function1 which, for convenience, wewrite as [4]

Ψ123(x, k⊥) ≡ Ψ (x1, x2, x3; k⊥1, k⊥2, k⊥3)

=1

8√

3!fN (µF )φ123(x, µF )Ω(x, k⊥) . (2.5)

fN (µF ) plays the role of the nucleon wave function at theorigin of the configuration space and the factorization scaleis denoted byµF . φijk(x, µF ) ≡ φ(xi, xj , xk, µF ) is thenucleon DA conventionally normalized to unity∫

[dx]φ123(x, µF ) = 1 . (2.6)

The DA is commonly expanded in a series of eigenfunctionsφn123(x) of the evolution kernel being linear combinations ofAppell polynomials (see [1]2)

φ123(x, µF ) = φAS(x)

[1 +

∞∑n=1

Bn(µF ) φn123(x)

](2.7)

whereφAS(x) ≡ 120x1x2x3 is the asymptotic (AS) DA [1].Evolution is incorporated by the scale dependences offNand the expansion coefficientsBn :

fN (µF ) = fN (µ0)

(ln(µ0/ΛQCD)ln(µF /ΛQCD)

)2/3β0

,

Bn(µF ) = Bn(µ0)

(ln(µ0/ΛQCD)ln(µF /ΛQCD)

)γn/β0

(2.8)

1 In [17] it has been shown that the entire nucleon state (including theLz /=0 projections) is described by three independent functions

2 Note that theφ2123(x) used here differs from that of [1] by an overall

sign

329

whereβ0 ≡ 11−2/3nf andµ0 denotes the scale of referencecustomarily chosen to be 1 GeV. The exponents ˜γn are thereduced anomalous dimensions. Because they are positivefractional numbers increasing withn [18], higher order termsin (2.7) are gradually suppressed.

Nucleon DAs are frequently utilized in applications ofthe perturbative approach which are constrained by momentsof the DA

φ(n1n2n3)(µ0) =∫

[dx] xn11 x

n22 x

n33 φ123(x, µ0) (2.9)

evaluated by means of QCD sum rules [6]. The few momentsknown only suffice to determine the first five expansion co-efficientsBn. However, since the moments are burdened byerrors theBn, and hence the DA, are not fixed uniquely.In [19], for instance, a set of 45 model DAs has been con-structed where each DA respects the moments of [6] and isstrongly end-point concentrated and asymmetric in thexi. Itis shown in [4], as we already mentioned in the introduction,that the perturbative contributions to the nucleon form factorevaluated with these DAs are too small in comparison withthe data [7]. As we are going to discuss subsequently theDAs constructed by Bergmann and Stefanis [19] also showserious deficiencies in other applications. The value offN isalso determined from QCD sum rules [6]: (5.0± 0.3) · 10−3

GeV2. This value is to be used in conjunction with the COZDA [6] and the DAs of [19].

The transverse momentum dependence of the wave func-tion is contained in the functionΩ which is normalized ac-cording to∫

[d2k⊥]Ω(x, k⊥) = 1 . (2.10)

Throughout we use a simple symmetric Gaussian parame-terization for thek⊥-dependence

Ω(x, k⊥) = (16π2)2 a4

x1x2x3exp

[−a2

3∑i=1

k2⊥i/xi

], (2.11)

which resembles the harmonic oscillator wave function pro-posed in [14]. The ansatz (2.11) keeps our model simpleand appears to be reasonable for a nucleon wave functionΨ123 which is dominantly symmetric. With the ansatz (2.7),(2.11) antisymmetric or mixed symmetric contributions mayonly appear through the DA.

For reasons which will become clear subsequently weonly need the soft part of the wave function, i.e. the full wavefunction with its perturbative tail removed from it [1]. TheGaussiank⊥-dependence is conform with the behaviour ofa soft wave function; the power-like decreasing perturbativetail is removed. The ansatz (2.11) is also supported by recentwork of Chibisov and Zhitnitsky [15] who showed that, onrather general grounds,Ω depends onxi andk⊥i solely inthe combinationk2

⊥i/xi and thatΩ falls off like a Gaussianat largek⊥.3 Equation (2.11) is the simplest way to complywith these requirements. We remark that integratingΩ in

3 The kinematical transverse momentum of the partons is not the sameobject ask⊥ defined through moments as in [15]. However, we will assumethat both are one and the same variable. This assumption corresponds tosumming up soft gluon corrections, i.e. higher twist contributions

(2.10) to infinity instead of to a cut-off scale of orderQintroduces only a small negligible error into the calculation.

According to (2.5) a wave function is defined by a certainDA combined with the Gaussian (2.11) andfN . For the COZwave function which we will subsequently confront to datafor the purpose of comparison, the transverse size parametera, controlling the width of the wave function ink⊥ space,is fixed by requiring a certain value of either the root meansquare (rms) transverse momentum or the probability of thevalence Fock state. We will utilize the COZ wave functionfor three cases:P3q = 1 (a = 0.99 GeV−1, 〈k2

⊥〉1/2 = 272MeV), 〈k2

⊥〉1/2 = 450 MeV (a = 0.60 GeV−1, P3q = 0.13)and 600 MeV (a = 0.45 GeV−1, P3q = 0.04).

3 The overlap contribution

The Dirac form factor of the nucleon can be expressed interms of overlaps of initial and final Fock state wave func-tions [20, 21]. This is an exact representation of the formfactor provided a sum over all Fock states is implied and thefull Fock state wave functions are used. However, one canidentify the overlaps of the hard largek⊥ tails of the wavefunctions with the perturbative contributions [1]. Since thiscontribution is small [4] the form factor is dominated by theoverlaps of the soft parts of the wave functions. The physicalpicture behind the overlap representation is that one singlequark is scattered by the virtual photon with the remain-ing constituents following the struck quark as spectators.The various overlap integrals are dominated by those con-figurations where the struck quark carries almost the entiremomentum of the nucleon. Obviously, with an increasingnumber of partons sharing the nucleon’s momentum it be-comes less likely that one parton carries the full momentumof the nucleon. Therefore, higher Fock state contributionscan be expected to be strongly suppressed at large momen-tum transfer. For wave functions of the type we considerfor the valence Fock state the strong suppresssion of thehigher Fock states can explicitly be seen. The DA of a Fockstate consisting ofng gluons andnq quarks contains to low-est order in the momentum fractions terms proportional tox1x2...xnqx

2nq+1...x

2nq+ng as is supported by power count-

ing arguments given in [22]. Using this asymptotic formof a higher Fock state DA in combination with a Gaussiank⊥-dependence of the type (2.11) one finds the overlap con-tribution to fall off asQ−4(nq+2ng−1).

The valence Fock state provides the most important softcontribution (this is termed the overlap contribution) in theregion of momentum transfer around 10 GeV2. Of course, asrequired by the consistency of the entire picture, the pertur-bative contribution will take control in the limitQ→∞; theoverlap contribution is suppressed by powers ofQ relativeto the perturbative contribution.

According to Drell and Yan [20] we calculate the nucleonmatrix elements of the electromagnetic current in a framewhere the incoming nucleon is rapidly moving in the 3-direction (infinite momentum frame). To leading order inthe nucleon’s momentumP we find from (2.1), combinedwith the ansatz (2.5) for the wave function, the followingexpression for the overlap contribution to the Dirac formfactor

330

FN1 soft(Q2) =

(fN

8√

6

)2 3∑j=1

ej

∫[dx]

[φ2

123(x)

+φ2213(x) + (φ132(x) + φ231(x))2

] ∫[d2k⊥]

Ω(x, k⊥j + (1−xj)q, k⊥i − xiq)Ω(x, k⊥) . (3.1)

This result is obtained from the matrix elements of the so-called good current components (µ = 0, 3). For P → ∞these matrix elements are only fed by such configurationsfor which all constituents of the nucleon move along thesame direction as the nucleon (up to finite transverse mo-menta), i.e. 0≤ xi ≤ 1 for all constituents [23]. Matrixelements of the bad current components (µ = 1, 2) have tobe treated with precaution; they are less reliable. Indeed,these matrix elements are suppressed by 1/P as opposed tothose of the good current components and suffer from manyapproximations made, e.g. off-shell effects, helicity or spinrotations and so on. Therefore, we also refrain from calcu-lating the Pauli form factor,F2, along the same lines asF1since it is controlled by such non-leading (with respect tothe momentumP ) contributions. To be more specific,F2can only be calculated ifLz /= 0 components of the nu-cleon wave function are also included. We also note that theexpression (3.1) somewhat differs from overlap expressionsgiven by Isgur and Llewellyn-Smith [2] who start from cur-rent matrix elements in a Breit frame and boost them to theinfinite momentum frame. In this procedure spin rotationshave to be considered which, in a model dependent way,generateLz /= 0 components in the nucleon wave function.Still the numerical results obtained in [2] are very similar toour ones.

For a completely symmetric wave function the expres-sion (3.1) is proportional to the sum of the quark chargesand thus exactly vanishes in the case of the neutron formfactorFn1 .4 This observation already precludes the AS wavefunction. For this reason we refrain from showing results onF p1soft evaluated with the AS wave function. The integra-tions in (3.1) can easily be performed analytically for wavefunctions of the type we consider consisting of the Gaussian(2.11) and a DA for which the expansion (2.7) is truncatedat some finiten. For such wave functions the overlap con-tribution falls off proportional toQ−8 for Q2 →∞.

In Fig. 1 we show the overlap contribution to the protonform factor evaluated with the COZ wave function5 in com-parison with experimental data [7]. It can be seen that thedata is dramatically exceeded by the overlap contribution.This result is independent on the value of the transverse sizeparameter used; an increase of thek⊥-width only shifts theposition of the maximum value of the overlap contribution tohigherQ2 without considerably reducing its magnitude. TheasymptoticQ−8 behaviour does not set in beforeQ2 ≈ 100GeV2 (since the expansion of (3.1) into a series of 1/Q2

powers converges slowly). Similarly large overlap contribu-tions are obtained from the COZ wave function in the caseof the neutron.

4 The neutron form factor is zero for any wave function at zero momen-tum transfer

5 As the externalQ2 is the only scale present in the process it definesat the same time a natural evolution scale for the DA

0 20 40 60 80

Q2 [GeV2]

0

2

4

6

8

Fig. 1. Overlap contributions to Dirac form factorF p1 using the COZ wave

function. The solid (dashed, dotted) line is evaluated witha = 0.99 (0.60,0.45) GeV−1. Experimental data () are taken from [7]

The COZ wave function is representative for all stronglyend-point concentrated DAs. We checked that all wave func-tions constructed from the Bergmann-Stefanis set of DAs[19] provide similar unphysically strong overlap contribu-tions as the COZ wave function [5]. As was found in [2, 24]anx-independent Gaussian instead of our ansatz (2.11), alsoleads to large overlap contributions toF1 for the COZ DA.These findings give rise to severe objections against DAsconstrained by the QCD sum rule moments of [6].

The overlap contributions are also subject to Sudakovcorrections. An estimate of these corrections on the basisof the Sudakov factor as derived by Botts and Sterman [3](see (A.7)) reveals that the size of the overlap contributionsis somewhat reduced by it. The suppression is stronger forthe end-point concentrated wave functions (but the overlapcontributions still exceed the form factor data dramatically)than for the AS wave function or similar ones. Since thissuppression can be compensated for by readjusting the wavefunction parameters we refrain from taking into account theSudakov corrections to the overlap contributions.

4 Valence quark distribution functions

Deep inelastic lepton-nucleon scattering provides informa-tion on the parton distributions inside nucleons. As is dis-cussed in [14] the parton distribution functionsqN (x) rep-resenting the number of partons of typeq with momentumfraction x inside the nucleon, are determined by the Fockstate wave functions. Each Fock state contributes throughthe modulus squared of its wave function, integrated overtransverse momenta up toQ and over all fractions exceptthose pertaining to partons of typeq. Obviously, the valence

331

Fock state wave function only feeds the valence quark dis-tribution functions,uNV (x) anddNV (x). Since each Fock statecontributes positively the following inequalities hold

upV (x) ≥ 2∫

[dx][d2k⊥] δ(x− x1)

× [Ψ2123 + Ψ2

213 + (Ψ132 + Ψ231)2],

dpV (x) ≥∫

[dx][d2k⊥] δ(x− x3)

× [Ψ2123 + Ψ2

213 + (Ψ132 + Ψ231)2]. (4.1)

In (4.1) the full wave function enters. For reasons obvi-ous from the preceding discussions, we evaluate the valenceFock state contributions toupV anddpV from our soft wavefunctions. On the strength of our experience with the nu-cleon form factor we expect the neglect of the perturbativetails to be admissible. The soft wave functions defined by(2.5), (2.7) – with the expansion truncated at some finite or-der – and (2.11) leads to the behaviourxqNV (x) ∼ (1− x)3

for x→ 1 in fair agreement with the structure function data.This property is related to the asymptotic behaviour of theoverlap contributionFN1 soft ∼ Q−8. The interrelation be-tween a (1− x)p behaviour of the distribution functions forx → 1 and a (1/Q)p+1 behaviour of the overlap contribu-tion at largeQ2 proposed by Drell and Yan [20] is onlyobtained for wave functions factorizing inx andk⊥ (i.e. forΩ not depending on thexi). Asymptotically, if the pertur-bative contribution dominates the form factor, the Drell-Yaninterrelation also holds for wave functions of the type weconsider.

Another interesting property of (4.1) is that any symmet-ric wave function, as e.g. the AS one, provides a value of 2for the ratioupV (x)/dpV (x). This is to be contrasted with thevalue of 5 (atx ≈ 0.6) for that ratio to be seen in currentparameterizations of the distribution functions [9].

In Fig. 2 we compare the valence Fock state contribu-tions, evaluated with the COZ wave function, toupV anddpVwith the Gluck-Reya-Vogt parameterizations [9] at a scaleof 1 GeV2 in the largex region. Forx smaller than 0.6our results well respect the inequalities (4.1). Forx largerthan about 0.6, on the other hand, wave functions normal-ized to large probabilities (or equivalently providing smallvalues of the mean transverse momentum) violate the in-equalities (4.1). For the COZ wave function consistency withthe Gluck-Reya-Vogt parameterizations can only be achievedif P3q is less than about 1% which for our wave functionparameterization would correspond to unrealistically largemean transverse momenta (>∼ 1 GeV). Similar results arefound for the other COZ-like wave functions constructedfrom the Bergmann-Stefanis set of DAs [19]. This observa-tion is to be considered as another serious failure of the COZ-like wave functions. It should be mentioned that Schafer etal. [25] found similar results for the valence quark distribu-tion functions or structure functions, respectively.

5 J/ψ decay into nucleon-antinucleon

The decay width of the processJ/ψ → NN provides athird constraint on the nucleon wave function. By reasonof the spin and the parity of theJ/ψ meson and of color

0.5 0.6 0.7 0.8 0.9 1

x

0

0.2

x u v

(x)

0.5 0.6 0.7 0.8 0.9 1

x

0

0.05

0.1

0.15

x d v

(x)

Fig. 2. Valence Fock state contributions to the valence quark distributionfunctions of the proton atQ2 = 1 GeV2. The open circles represent theparameterization of [9]. The solid and dashed lines represent the contri-butions of the valence Fock state using a wave function composed of theGaussian (2.11) and either the DA (6.1) or the COZ one (a = 0.60 GeV−1),respectively. The dotted line is obtained from the AS wave function withfN anda as for the wave function (2.11), (6.1)

neutralization the massive charm quarks theJ/ψ is com-posed of, dominantly annihilate through three gluons whichin turn create the light quark pairs necessary for the forma-tion of nucleon and antinucleon (see Fig. 3). There are hintsthat both thecc annihilation and the conversion of the glu-ons intoNN pairs, are under control of perturbative QCD.If, for instance, the decays of charmonium states into lighthadrons are viewed as decays into two or three gluons thewidths can be estimated perturbatively. With acceptable val-ues ofαS one obtains reasonable agreement with experiment[26] although in the case of theJ/ψ that value appears tobe a bit small with regard to the relevant scale provided bythe charm quark mass (see the discussion in [27]). Anotherhint that perturbative QCD is at work inJ/ψ decays is pro-vided by the angular distribution ofNN pairs produced viae+e− → J/ψ → NN . From the data of the DM2 collabora-tion [28] one estimates the fraction ofNN pairs with equalhelicities to amount to (10±3)% of the total number of pairs.This is what is to be expected if the process is dominatedby perturbative QCD: Each of the virtual gluons produces alight, almost massless quark and a corresponding antiquarkwith opposite helicities. Since our nucleon wave functiondoes not embody any non-zero orbital angular momentumcomponent the quark helicities sum up to the nucleon’s he-licity. Hence, nucleon and antinucleon are dominantly pro-duced with opposite helicities. This fact is an example ofthe well-known helicity conservation rule for light hadrons[1, 10]. The small amount ofNN pairs with the wrong he-licity combination observed experimentally, while indicatingthe presence of some soft contributions, can be tolerated.

332

@@@

@@@

@@@

q1

q3

g1

g2

g3

q=2

q=2

1

1

2

2

3

3

x1p+ k1

x01p0 + k0

1

x2p+ k2

x02p0 + k0

2

x3p+ k3

x03p0 + k0

3

q = p+ p0

m

Fig. 3. Decay graphJ/ψ → 3g → 3qq. The momenta ofthe quarks arexip + ki with ki = (0, 0, k⊥i), and those ofthe antiquarks are marked by a prime

One should, however, be aware of these contributions whentheoretical results for theJ/ψ → NN decay are comparedwith experiment.

It is, however, fair to mention that there are several dif-ficulties with the perturbative calculation of exclusive char-monium decays, e.g. the relatively large branching ratiosof ψ′(2s) → pp, ηc → pp and J/ψ → πρ. We will nev-ertheless calculate theJ/ψ → NN decay width within aperturbative approach. It will turn out and this may be re-garded as another argument in favour of the prominent roleof perturbative QCD in charmonium decays, that the samewave function that, on the one hand, leads to a very smallperturbative contribution to the nucleon form factor (and si-multaneously to a large overlap contribution), provides, onthe other hand, a reasonably large value for theJ/ψ → NNdecay width.

We start the calculation of theJ/ψ → NN decay widthfrom an invariant decomposition of the decay helicity am-plitudes (with an appropriately chosen spin quantization axisof theJ/Ψ )

Mλ1λ2λ = u(p1, λ1)

[B (M2

ψ) γµ + C (M2ψ)

(p1 − p2)µ2mN

]v(p2, λ2) εµ(λ) , (5.1)

wherep1 (p2) andλ1 (λ2) are the momentum and the helic-ity of the nucleon (antinucleon) respectively.u andv denotethe nucleon spinors andε the polarization vector of theJ/ψ.In the perturbative approach hadronic helicity conservationforces the invariant functionC to vanish. TheJ/ψ → NNdecay width, therefore, takes the simple form (up to correc-tions of order (mN/Mψ)4 whereMψ is theJ/ψ mass)

Γ (J/ψ → NN ) =Mψ

12π|B |2 . (5.2)

There is a small, almost negligible contribution to the in-variant functionB from the cc annihilation mediated bytwo gluons and a photon. Theggγ contribution toB beingproportional to theggg contribution, amounts to about 1%of the latter. In the following it is understood that theggγcontribution is absorbed into theggg contribution. There isalso a small electromagnetic contribution to the invariantfunctionB from thecc annihilations through a virtual pho-ton. This contribution involves the time-like Dirac and Pauliform factor of the nucleon

Bem =8πα3Mψ

fψ(FN1 (s=M2

ψ) + κNFN2 (s=M2

ψ)). (5.3)

The factor multiplying the form factors is the invariant am-plitude for theJ/ψ decay into a lepton pair (ml = 0). Thisfactor includes theJ/ψ decay constantfψ (being related tothe configuration space wave function at the origin) whichcan be determined from the experimental value of the lep-tonic decay widths [29]. One findsfψ = 409± 14 MeVfrom a leading order calculation. TheαS corrections tofΨare known to be large [30], leading to an increase offΨ byabout 20%. However, there is no assurance that theα2

S cor-rections are not also large and perhaps cancelling partly theαS correction. Therefore we use the leading order value offΨ being aware of this eventual source of uncertainty in ourfinal result. UsingF p1 +κpF

p2 = GpM = 2.5±0.4 GeV4/M4

ψ inagreement with the E760 data [31] andF p2 = 0, we estimatethe electromagnetic contributionBem to theJ/ψ → pp in-variant decay amplitudeB to amount to (14± 4)% of theexperimental value.

Several QCD studies of the decayJ/ψ → 3g → NNhave appeared in the past [10, 11, 12, 13]. A point to criti-cize in these studies which relied on the conventional hardscattering approach, is the treatment of the strong couplingconstantαS . Since, on the average, the virtuality of the in-termediate gluons is roughly 1 GeV2 one would expectαSto be of the order of 0.4 to 0.5 rather than 0.2 to 0.3 whichis usually chosen [11, 12, 13]. SinceαS enters to the sixthpower into the expression for the width a variation ofαSfrom, say, 0.3 to 0.45 would lead to a change by a factor of11 for the width. Thus, a large uncertainty is hidden in thesecalculations preventing any severe test of the DA utilized.

In constrast to previous work [10, 11, 12, 13] we willnot use the collinear approximation but rather use the mod-ified perturbative approach of Sterman et al. [3] in whichtransverse degrees of freedom are retained and Sudakov sup-pression, comprising those gluonic radiative corrections notincluded in the evolution of the wave function, are takeninto account. The calculation of the three-gluon contributionto the invariant functionB is presented in some detail inAppendix A.

An important advantage of the modified perturbative ap-proach is that the strong coupling constant can to be used inone-loop approximation; its singularity, to be reached in theend-point regionsxi → 0, is compensated by the Sudakovfactor. Hence, there is no uncertainty in its use. This is tobe contrasted with the conventional perturbative approachwhere eitherαS is evaluated at anx-independent renormal-ization scale typically chosen to be of the order ofM2

ψ or atscales like (A.4). In the latter case for which, in contrast with

333

the first case, large logs from higher orders of perturbationtheory are avoided,αS has to be frozen in at a certain value(typically 0.5) in order to avoid uncompensatedαS singu-larities in the end-point regions. The modified perturbativeapproach possesses another interesting feature: the soft end-point regions are strongly suppressed. Therefore, the bulk ofthe perturbative contribution comes from regions where theinternal quarks and gluons are far off-shell (order ofM2

ψ).In contrast to the nucleon form factor theJ/ψ → NN am-plitude is not end-point sensitive.6 The suppression of theend-point regions does not, therefore, lead to a substantialreduction of theJ/ψ → NN amplitude.

As in Sects. 3 and 4 and before we turn to the deter-mination of a new nucleon wave function, we test the ASand the COZ wave functions. Evaluating the three-gluoncontributionB3g from (A.2) and leaving aside the incon-sistencies with the form factor and the valence quark dis-tribution functions we find acceptable values for the de-cay width (ΛQCD = 220 MeV): Using the AS wave func-tion (fN = 6.64 · 10−3GeV2, a = 0.75 GeV−1) we findΓ3g(J/ψ → pp) = 0.09 keV and for the COZ wave func-tion (a = 0.6 GeV−1) 0.23 keV. Smaller (higher) valuesfor the width are obtained if the transverse size parametera is decreased (increased). For comparison the experimen-tal value for theJ/ψ → pp decay width is 0.188± 0.014keV [29]. Similar results for the width are obtained withthe wave functions constructed from the Bergmann-StefanisDAs [19]. The only exception is the Gari-Stefanis DA [32]which provides a very small value forΓ3g of the order of afew eV.

6 A model for the nucleon wave function

We have demonstated that the COZ-like wave functions aswell as the AS one do not describe the three processes dis-cussed in the preceding sections in a satisfactory manner.For this reason we will now try to determine a new wavefunction from a fit to the proton form factor data [7] (using(3.1)), the valence quark distribution functions of Gluck etal. [9] (using (4.1)) as well as theJ/ψ → pp decay width[29] (using (A.2)). We start from the ansatz (2.1), (2.5) andassume the Gaussiank⊥-dependence (2.11) again. An im-portant question is, how many terms one has to allow in theexpansion (2.7). It would be obvious to truncate the expan-sion atn = 5 again. As explorative fits immediately reveal,five terms in (2.7) provide too much freedom. For severalexpansion coefficients, in particular forB5, one always ob-tains very small values. The corresponding terms in (2.7)can be neglected without worsening the fits noticeably. In-deed it suffices to consider only the first order expansionterms,φ1

123 andφ2123, and even that still implies unnecessary

freedom in case both the coefficients,B1 andB2 are treatedas free parameters. As it turns out ultimately, the simple DA

φ123(x) = φAS(x)

[1 +

34φ1

123(x) +14φ2

123(x)

]6 This is due to the fact that in the collinear approximation the propagator

denominators of the hard scattering amplitude vanish only linearly in theend-point regions whereas e.g. in the case of the nucleon form factor theyvanish quadratically for some of thexi

= φAS(x)12

[1 + 3x1] (6.1)

meets all requirements. The only free parameters left overin this case, namelyfN and a, are determined by a fit tothe data for the three processes mentioned above. The fitprovides the following values for the two parameters:fN =6.64·10−3 GeV2, a = 0.75 GeV−1. The fitted wave functionimplies a value of 0.17 for the probability of the valenceFock state and a value of 411 MeV for the rms transversemomentum. Both values appear to be reasonable. We stressthat a larger flexibility in the DA, i.e. allowing for more freeparameters to be adjusted in the fit, does not improve the fitsubstantially. We also remark that the DA is not uniquelydetermined by the data. Another solution of similar qualityexists for which the DA contains the expansion termsφ2

123,φ3

123 andφ4123. Although the mathematical expression of that

DA looks rather complicated the DA itself is similar to (6.1)in shape and magnitude.

Our value forfN is about 30% larger than that obtainedfrom QCD sum rules [6]. In lattice QCD, on the other hand,the following values forfN are found: (2.9±0.6)·10−3 GeV2

[33] and 6.6·10−3 GeV2 [34]. Thus, within a factor of about2 all values agree with each other. With regards to the largesystematic uncertainties in the various approaches the spreadof the fN values cannot be considered as a contradiction.

The proton decay offers another check of our wave func-tion. Calculating from it the three-quark annihilation matrixelement of the proton (termedα in [35]) along the samelines as in [35], we find for it a value of 0.012 GeV3 whichis about three times smaller than the value quoted in [35].Our value is almost identical to a recent result from latticeQCD [34] and rather close to many other results [36]. Thesource of the difference between our result and that one ofBrodsky et al. [35] chiefly lies in the fact that we give upthe idea of calculating the proton form factor perturbatively.

The DA (6.1), displayed in Fig. 4, possesses interest-ing features. It is much less asymmetric and less end-pointconcentrated than the COZ-like DAs which exhibit threepronounced maxima and regions where the DAs acquirenegative values. Our DA rather resembles the AS one inshape but with the position of the only maximum shiftedto x1 = 0.44, x2 = x3 = 0.28. Thus, as the COZ DA butto a lesser amount, our DA possesses the property that, onthe average, au-quark in the proton carries a larger fractionof the proton’s momentum than thed-quark. Related to theshift of the maximum’s position is the asymmetry in the firstorder moments of our DA:

〈x1〉FIT =8

21, 〈x2〉FIT = 〈x3〉FIT =

1342, (6.2)

which values are to be contrasted with the QCD sum rulevalues [6]

〈x1〉COZ = 0.54− 0.62, 〈x2〉COZ = 0.18− 0.20,

〈x3〉COZ = 0.20− 0.25. (6.3)

The moments (6.2) are consistent with those obtained byMartinelli and Sachrajda from lattice QCD [33]. For a dis-cussion of the various approximations made in the QCD sumrule analysis of [6, 22] and their implications see [37].

The DA (6.1) or, when combined with the totally sym-metric Gaussian (2.11), the wave function, can be decom-

334

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

10

1

2

3

4

123(x)

x3x1

Fig. 4. The DA (6.1) as a function ofx1 andx3

posed into a symmetric part and a part of mixed symmetryunder permutations (we adopt the notations of [2])

φS(x) = −√

6φAS(x) , φρ(x) = −32

(x1 − x2)φAS(x) ,

φλ(x) =

√3

2(1− 3x3)φAS(x) .

These functions are related to the56 and70, L=0 rep-resentations of the permutation group on three objects. Rep-resentations with non-zero orbital angular momentumL donot contribute to (6.1). The symmetric56 part is dominantsince the ratio of probabilitiesPS3q/P3q is 28/29 = 0.9655.The strength of the70, L=0 admixture is about the sameas found from equal-time wave function analysis [38]. Thisobservation provides some justification of the symmetricansatz (2.11) for thek⊥-dependencea posteriori.

The results for the valence quark distribution functionsobtained from the fit are shown in Fig. 2. The valenceFock state contribution toxdV (x) comes out comparativelylarge leaving hardly room for contributions from higher Fockstates forx>∼ 0.6. The effect of the asymmetric part of our

DA provided by the eigenfunctionsφ1123 and φ2

123 is clearlyvisible in Fig. 2: It pushes upuV at largex and diminishesdV at the same time, thus producing a ratiouV : dV of about5:1.

At this point it is in order to draw the reader’s attention toa little difficulty: The evolution behaviour of our soft con-tributions to the valence quark distribution functions doesnot exactly match with that of the phenomenological distri-bution functions. This entails violations of the inequalities(4.1) forQ2 1 GeV2. The imperfect evolution behaviourappears as a consequence of several approximations madein our approach. Thus, for instance, we consider only thesoft part of the wave function and yet extend the upper limitof the k⊥ integration to infinity (numerically this is of littleimportance for a Gaussian like (2.11) ). We also ignored apossible evolution of the transverse size parameter (see [14]where the pion case is discussed). With respect to our ob-

0 5 10 15 20 25 30

Q2 [GeV2]

0

0.5

1

Q4 F

1p (Q2 )

[GeV

4 ]

0 5 10 15 20 25 30

Q2 [GeV2]

0

0.2

0.4

-Q4 F

1n (Q2 )

[GeV

4 ]

Fig. 5. Overlap contribution to the Dirac form factor of the proton (top)and the neutron (bottom) evaluated from the wave function (2.11), (6.1).The dashed line in the upper figure is obtained from the AS wave functionwith fN anda as for the wave function (2.11), (6.1). Data () are takenfrom [7, 39]

335

jective of a more qualitative understanding of the nucleon’sform factors and distribution functions rather than a perfectquantitative description we tolerate that minor drawback.

In Fig. 5 we show the results for the overlap contribu-tions to the proton and neutron form factors in comparisonwith the data [7, 39]. While the data onF p1 is input to thefit the results for the neutron form factor are predictions. Itcan be seen that our wave function provides overlap con-tributions which exhibit a broad maximum near 15 GeV2.The asymmetric part of the DA (6.1) is solely responsiblefor the neutron form factor (pushing it down from zero toa negative value) and pushes up the overlap contribution tothe proton form factor (see the difference between the solidand dashed lines in Fig. 5). ForQ2 smaller than about 8GeV2 the fit is somewhat below the data. We are contentwith that result because, as we already mentioned, our goalis not a perfect fit but rather to demonstrate the existenceof a wave function that provides an overlap contribution tothe nucleon’s form factor of the right magnitude and that isconsistent with the constraints from the other two processeswe consider and thus implicitly gives an explanation for thesmallness of the perturbative contribution to the nucleon’sform factor. An improved fit in agreement with the formfactor data over a wide range ofQ2 can likely be achievedwith minor modifications of the Gaussian (2.11) at finitetransverse momentum. Since such modifications, an exam-ple of which has been given by Zhitnitsky [40] for the caseof the pion wave function, require more free parameters wepersist in the simple Gaussian (6.1). ForQ2<∼ 8 GeV2 thepresence of higher Fock states is to be expected (rememberP3q = 0.17).

Our wave function provides the value

Γ3g(J/ψ → NN ) = 0.117 keV (6.4)

for the three gluon contribution to the nucleonicJ/ψ width.If comparing the result with experiment [29] one has to beaware of the electromagnetic contribution and the spin effect.The electromagnetic contribution toB (see (5.3)) amountsto about 15% (see Sect. 5). Since, however, the phase of theexperimental time-like form factor is unknown we are notin the position to addB3g andBem coherently at present.7

Thus, we can only say that, at best, the electromagnetic con-tribution may increase our prediction by 30%. Consideringalso thatNN pairs with equal helicity contribute about 10%to the total width, we regard our result for the three-gluoncontribution as consistent with experiment.

Other sources of uncertainties in our calculation of theJ/ψ decay widths are introduced by the value offΨ chosen(see the discussion in Sect. 5) and byΛQCD. SinceαS entersto the sixth power inΓ3g the result (6.4) is rather sensitiveto the value ofΛQCD employed. A mild increase ofΛQCDby 10% enlargesΓ3g by about 30% without changing theresults shown in Figs. 2 and 5 noticably; only the evolutionbehaviour of the nucleon wave function in the calculationof the overlap contribution is slightly modified. Thus, weconclude that our wave function (2.11), (6.1) provides a rea-sonably large three gluon contribution to theJ/ψ → pp de-cay width. In contrast to previous calculations of this width

7 In a common perturbative analysis of both the proton form factor andtheJ/Ψ decay, the relative phase between the two contributions would befixed

carried through in collinear approximation, our averageαS(being 0.43) is consistent with the available scale in theJ/ψdecay which is provided by thec-quark mass. We also notethat our perturbative calculation is self-consistent. The bulkof the contribution, i.e. more than 50%, is accumulated inregions whereα3

S < 0.473.

At this point a remark concerning thenn decay channelis in order. Since the three-gluon contribution is flavour-blind any difference between the decay widths intopp andnn must be due to the electromagnetic contribution. Fromexperiment it is known that the widths forJ/ψ → pp andJ/ψ → nn agree within the experimental errors [41] andthat the time-like form factors for the proton and the neu-tron, to which theBem are directly proportional, are approx-imately equal in modulus ats = 5.4 GeV2 [42]. Since boththe contributions,B3g and Bem, are in general complexnumbers with non-trivial phases the only conclusion to bedrawn at present is that the relative phase betweenB3g andBem is the same (up to an eventual sign) for the proton andthe neutron channel.

The generalization to the decayΥ → NN is a straight-forward task. For bottomium systems the application of ourapproach is even better justified than for charmonia becauseof the larger gluon virtualities. Using a value of 710 MeVfor the decay constantfΥ of theΥ meson which, in a simi-lar manner asfψ, is determined from the leptonic width ofthe Υ [29], we find Γ3g(Υ → NN ) = 1.3 · 10−2 eV. Theelectromagnetic contribution toΥ → NN decays is negli-gible as an estimate in analogy to that one in theJ/Ψ caseshows. Up to now there is only an experimental upper limitof about 1 eV for this decay width [29] which is safely metby our result. The effectiveαS is 0.26 in theΥ case.

At the end of this section we want to comment on otherforms of the nucleon’s wave function to be found in theliterature. For instance,x-independent parameterizations ofΩ are used sometimes [2, 24]. Although such factorizingforms of the wave functions are in conflict with rotationalinvariance and with the arguments given in [15], the numer-ical results, say, for the overlap contribution to the nucleonform factor obtained with them do not differ much fromour results. In other cases the wave function is regarded asa constituent wave function. Consequently the parametersare adjusted so that the wave function is normalized to unityand that the charge radius of the nucleon is reproduced. Con-stituent wave functions also provide large overlap contribu-tions which, however, decrease more rapidly with increasingmomentum transfer than our results obtained from a wavefunction describing an object that is smaller than the nu-cleon. The introduction of form factors for the constituentquarks mediating the transition to current quarks, may im-prove the large momentum transfer behaviour of the formfactor (see, for instance, [43]). Another possibility to con-struct a wave function is to start from an equal time wavefunction (e. g. that of a harmonic oscillator) in the nucleon’srest frame and transform it to the light cone under the as-sumption that a Melosh transform for free, non-interactingquarks can be applied (see, for instance, [44]). While this isa very interesting approach attempting to construct a unifiedpicture of the non-relativistic quark model and light-cone

336

physics, the resulting wave function presented in [44] doesnot pass our tests against data.

Occasionally DAs are used which possess a multiplica-tive factor exp[−a2∑

im2q/xi] (see, for instance, [14, 44])

where the parametermq is to be interpreted as a constituentquark mass. Although the infinite series (2.7) can, in prin-ciple, accommodate such a mass exponential, the truncatedexpansion may not reproduce it sufficiently accurate. It mayperhaps be better to consider the mass exponential explicitly.To get an idea about the importance of that mass exponentialwe modify the DA (6.1) by it and repeat our calculations.It turns out that theJ/ψ decay width as well as the distri-bution functions are mildly affected by the mass exponen-tial while the overlap contribution is reduced by about 20%in the momentum transfer region between 5 and 15 GeV2

but decreases somewhat faster with increasingQ2 than theoverlap contribution obtained from the original DA (6.1).The reduction around 10 GeV2 can be compensated for bya 10% increase offN .

Finally, one may think of additional powers of transversemomenta multiplying the Gaussian (2.11) (see, for instance,[44]). Despite of this and many other possible complicationswe stick to our simple ansatz (2.1), (2.5) and (2.11) because,as we have shown, it is flexible enough to account for theavailable data with a sufficient degree of accuracy.

7 Summary and conclusions

In this paper we have investigated the soft light-cone wavefunction of the nucleon. For the DAs we use the expansion interms of the eigenfunctions of the evolution equation trun-cated at some finite order. For the transverse momentumdependence of the wave function we use a specific formwhich is supported by results obtained in [14, 15]. On thestrength of rather general arguments thek⊥-dependence ofthe wave function appears in the formk2

⊥i/xi with a Gaus-sian fall-off at largek⊥. Studying soft overlap contributionsto the Dirac form factorFN1 of the nucleon and valenceFock state contributions to the quark distribution functionsof the nucleon at largex we found that wave functions con-structed on the basis of QCD sum rules in general lead totoo large predictions. These results resemble similar consid-erations in the case of the pion [8]. In the pion case there isstrong evidence that the DA is close to the asymptotic one,whereas in the case at hand asymmetries between the protonand neutron form factor as well as structure function dataprevent the use of the asymptotic DA.

We take these observations as a motivation to model anew wave function which consistently describes the valenceFock state contributions to the quark distribution functionsand the Dirac form factorFN1 by its overlap contribution.In order to demonstrate the consistency of our approach wemention that the perturbative contribution to the proton formfactor calculated along the same lines as in [4] using ourfitted wave function is indeed negligible within theQ2 rangeof the experimental data: It amounts to values forQ4FN1between 0.0014 GeV4 at Q2 = 8 GeV2 and−0.0024 GeV4

atQ2 = 32 GeV2.In addition we require that our wave function also leads

to a proper prediction for theJ/ψ → NN decay width

within the modified perturbative approach. The perturbativecalculation is self-consistent in that case and, in contrast tothe case of the nucleon form factor, the perturbative con-tribution is in fair agreement with the experimental resulton the decay width. It should be noted that there is stillsome residual uncertainty in the perturbative contributionfrom the imperfect knowledge of the strong coupling con-stantαS and theJ/ψ decay constant. At any rate we havebeen able to rectify the treatment ofαS by avoiding somefixed scale prescriptions in contrast to previous calculations.In our calculation the effective value ofαS is 0.43 for ourwave function atΛQCD = 220 MeV instead of about 0.3 asin previous calculations.

The wave function which we determine from the com-bined fit to the three sets of data consists of a Gaussiank⊥-dependence and a very simple DA which bears resem-blance to the asymptotic DA in shape but with the positionof the only maximum shifted somewhat. Like the COZ-typeDAs our DA possesses the interesting property that, on theaverage, au-quark in the proton carries a larger fraction ofthe proton’s momentum than thed-quark. Our wave func-tion defined by (2.1), (2.5), (2.11) and (6.1), has only twofree parameters to be adjusted. With more complicated wavefunctions, containing more free parameters, the fit can cer-tainly be improved. However, with regard to our aim ofdemonstrating the existence of a soft nucleon wave functionwhich complies with theoretical ideas and from which theprominent features of the data can be reproduced, we refrainfrom introducing such complications.

We would like to thank N.G. Stefanis for useful discussions and S.J. Brod-sky for valuable comments.

A Appendix: Perturbative calculationof the J/ψ → NN decay amplitude

As in previous perturbative calculations [10–13] theJ/ψmeson will be treated as a non-relativisticcc system withv2/c2 corrections neglected. According to [45] we write theJ/ψ state in a covariant fashion

| J/ψ; q, λ 〉 =δab√

3

fψ

2√

6

(q/ +Mψ)ε/(λ)√2

, (A.1)

wherea and b are color indices. Within the modified per-turbative approach the three-gluon contributionB3g to theJ/ψ decay intoNN is of the form

B3g =fψ

2√

6

∫[dx][dx′]

∫d2b1

(4π)2

d2b3

(4π)2TH (x, x′, b)

× exp[−S(x, x′,Mψ)][Ψ123(x, b)Ψ123(x

′, b)

+12

(Ψ123(x, b) + Ψ321(x, b)

)× (

Ψ123(x′, b) + Ψ321(x

′, b)) ]. (A.2)

This convolution of wave functions and a hard scatteringamplitudeTH can formally be derived by using the meth-ods described in detail by Botts and Sterman [3]. Thebi,canonically conjugated to the transverse momentak⊥i, arethe quark separations in the transverse configuration space.

337

b1 and b3 correspond to the locations of quarks 1 and 3 inthe transverse plane relative to quark 2 andb2 = b1 − b3.Ψijk represents the Fourier transform of the wave functionΨijk.

The hard scattering amplitude, to be calculated fromFeynman graphs of the type shown in Fig. 3, reads

TH (x, x′, k⊥, k′⊥) =3∏i=1

αS(ti)g2i − (k⊥i+k′⊥i)

2 + iε

× 5120√

6/27π3M5ψ (x1x

′3 + x3x

′1)

[q21 + (k⊥1+k′⊥1)2] [ q2

3 + (k⊥3+k′⊥3)2], (A.3)

where

q2i = [xi (1− x′i) + (1− xi)x

′i]M

2ψ/2 ,

g2i = xix

′iM

2ψ. (A.4)

Note thatTH depends on sums of transverse momenta,K i ≡k⊥i + k′⊥i, and because of the constraint

∑i K i = 0 only

two of theK i, sayK 1 andK 3, are independent. Hence, theFourier transformed hard amplitude

TH (x, x′, b1, b3) =∫

d2K 1

(2π)2

d2K 3

(2π)2

TH (x, x′,K ) exp[−i K 1·b1 − i K 3·b3] , (A.5)

depends only on the vectorial distancesb1 andb3. In physi-cal terms this means that theNN pairs emerge with identicaltransverse separation configurations from the decay becauseeach gluon produces a quark-antiquark pair at the same lo-cation in the transverse configuration plane which thereafterdo not interact. We emphasize that in contrast to the case ofthe nucleon form factor [4] this circumstance is not due toapproximations concerning thek⊥-dependences ofTH butinstead a direct consequence of the decay kinematics.

Inserting (A.3) into (A.5), one finds for the hard scatter-ing amplitude inb space

TH (x, x′, b1, b3) = −256027

fψM5ψ

(x1x′3 + x3x

′1)

[q21 + g2

1][ q23 + g2

3]

× 1(2π)3

3∏i=1

(παS(ti))∫

d2b0iπ2

H(1)0 (g2b0)

×[

iπ2

H(1)0 (g1 |b1 + b0| )− K0(q1 |b1 + b0| )

]×[

iπ2

H(1)0 (g3 |b3 + b0| )− K0(q3 |b3 + b0| )

]. (A.6)

The auxiliary variableb0 in (A.6) serves as a Lagrange mul-tiplier to the constraint

∑K i = 0. Inserting (A.6) into the

expression (A.2) for the invariant functionB3g we see that anine dimensional numerical integration is to be performed.8

Although this is a rather involved technical task it can becarried through with sufficient accuracy if some care is putinto it. Since the virtualities of the gluons are timelike,THincludes complex-valued Hankel functions H(1)

0 which are

8 Taking into account relativistic corrections to theJ/ψ wave function,i.e. its transverse momentum dependence, one would have to perform a14 dimensional numerical integration which is impossible with present daycomputers to a sufficient degree of accuracy

related to the usual modified Bessel functions K0, appearingfor space-like propagators, by analytic continuation in themomentum transfer variable which in our case is theJ/ψmass. Thus,B3g has a non-trivial phase as for instance thetime-like form factors [46].

The Sudakov factor exp[−S] in (A.2) takes into accountthose gluonic radiative corrections not accounted for in theQCD evolution of the wave function as well as the renormal-ization group transformation from the factorization scaleµFto the renormalization scalesti at which the hard amplitudeTH is evaluated. The Sudakov exponent reads

S(x, x′,Mψ) =3∑i=1

[s(xi, b,Mψ) + s(x′i, b,Mψ)

+4β

loglog(ti/ΛQCD)

log(1/biΛQCD)

], (A.7)

whereβ ≡ 11−2/3nf . The functions(ξ, b,Mψ), originallyderived by Botts and Sterman [3] and later on slightly im-proved, can be found in [5, 47].Mψ appears in the functions since it provides the large scale in the process of interest.

The quantitiesbi are infrared cut-off parameters, nat-urally related to, but not uniquely determined by the mu-tual separations of the three quarks [48]. Following [4] wechosebi = b = maxb1, b2, b3. With this “MAX” prescrip-tion the hard scattering amplitude is unencumbered byαSsingularities in the soft end-point regions. As a consequenceof the regularizing power of the “MAX” prescription, theperturbative contribution saturates in the sense that the re-sults become insensitive to the inclusion of the soft regions.A saturation as strong as possible is a prerequisite for theself-consistency of the perturbative approach. The infraredcut-off b marks the interface betweeen the non-perturbativesoft gluons, which are implicitly accounted for in the nu-cleon wave function, and the contributions from soft gluons,incorporated in a perturbative way in the Sudakov factor.Obviously, the infrared cut-off serves at the same time asthe gliding factorization scaleµF to be used in the evolu-tion of the wave function.

The renormalization scalesti are defined in analogy tothe case of electromagnetic form factors [3, 4] as the max-imum scale of either the longitudinal momentum or the in-verse transverse separation associated with each of the glu-ons

t1 = max(q1, g1, 1/b3) , t2 = max(g2, 1/b2) ,

t3 = max(q3, g3, 1/b1) . (A.8)

The above assignment ofb-scales is not compelling. Re-arrangements in theb-scales, however, induce only slightchanges in the numerical results.

References

1. G.P. Lepage, S.J. Brodsky: Phys. Rev. D22 (1980) 2157.2. N. Isgur, C.H. Llewellyn-Smith: Nucl. Phys. B317 (1989) 526.3. J. Botts, G. Sterman: Nucl. Phys. B325 (1989) 62; H.-N. Li, G. Ster-

man: Nucl. Phys. B381 (1992) 129.4. J. Bolz, R. Jakob, P. Kroll, M. Bergmann, N.G. Stefanis: Z. Phys. C66

(1995) 267; Phys. Lett. B342 (1995) 345.

338

5. J. Bolz: Ph.D. thesis, WUB - DIS 95-10, University of Wuppertal,1995; work in preparation.

6. V.L. Chernyak, A.A. Ogloblin, I.R. Zhitnitsky: Z. Phys. C42 (1989)569.

7. A.F. Sill et al. : Phys. Rev. D48 (1993) 29.8. R. Jakob, P. Kroll, M. Raulfs: J. Phys. G22 (1996) 45.9. M. Gluck, E. Reya, A. Vogt: Z. Phys. C67 (1995) 433.

10. S.J. Brodsky, G.P. Lepage: Phys. Rev. D24 (1981) 2848.11. V.L. Chernyak, A.A. Ogloblin, I.R. Zhitnitsky: Z. Phys. C42 (1989)

583.12. N.G. Stefanis, M. Bergmann: Phys. Rev. D47 (1993) 3685.13. P.H. Damgaard, K. Tsokos, E.L. Berger: Nucl. Phys. B259 (1985) 285.14. G.P. Lepage, S.J. Brodsky, T. Huang, P.B. Mackenzie: Banff Summer

Institute, Particles and Fields 2, A.Z. Capri, A.N. Kamal (eds.) (1983),p. 83; S.J. Brodsky, T. Huang, G.P. Lepage: ibid., p. 143.

15. B. Chibisov, A.R. Zhitnitsky: Phys. Rev. D52 (1995) 5273.16. M.G. Sotiropoulos, G. Sterman: Nucl. Phys. B425 (1994) 489.17. Z. Dziembowski: Phys. Rev. D37 (1988) 768.18. M. Peskin: Phys. Lett. B88 (1979) 128.19. N.G. Stefanis, M. Bergmann: Phys. Rev. D48 (1993) 2990; Phys. Lett.

B325 (1994) 183.20. S.D. Drell, T.-M. Yan: Phys. Rev. Lett.24 (1970) 181.21. G.B. West: Phys. Rev. Lett.24 (1970) 1206.22. V.L. Chernyak, A.R. Zhitnitsky: Phys. Rep.112 (1984) 173.23. S.D. Drell, D.J. Levy, T.M. Yan: Phys. Rev. 187 (1969) 2159.24. C.E. Carlson, F. Gross: Phys. Rev. D36 (1987) 2060.25. A. Schafer, L. Mankiewicz, Z. Dziembowski: Phys. Lett. B233 (1989)

217.26. M.L. Mangano, A. Petrelli: Phys. Lett. B352 (1995) 445.27. M. Consoli, J.M. Field: Phys. Rev. D49 (1994) 1293.28. D. Pallin et al. (DM2 collaboration): Nucl. Phys. B292 (1987) 653.29. Particle Data Group: Review of Particle Properties, Phys. Rev. D50

(1994) 1173.30. R. Barbieri, R. Kogerler, Z. Kunszt, R. Gatto: Nucl. Phys. B105 (1976)

125.

31. T. A. Armstrong et al. (E760 collaboration): Phys. Rev. Lett.70 (1993)1212.

32. M. Gari, N.G. Stefanis: Phys. Lett. B175 (1986) 462.33. G. Martinelli, C.T. Sachrajda: Phys. Lett. B217 (1989) 319.34. K.C. Bowler et al. : Nucl. Phys. B296 (1988) 431.35. S.J. Brodsky, J. Ellis, J.S. Hagelin, C.T. Sachrajda: Nucl. Phys. B238

(1984) 561.36. P. Langacker: Preprint Penn University UPR-0263T, published in pro-

ceedings of the workshop “Inner space/ outer space”, Batavia, IL, May2-5, 1984, edts. E.W. Kolb, M.S. Turner, D. Lindley, K.A. Olive andD. Seckel, University of Chicago Press (1986).

37. A. Duncan, S. Pernice, E. Schnapka: “Pion wave function and trunca-tion sensitivity of QCD sum rules”, hep-ph/9602247.

38. N. Isgur, G. Karl, R. Koniuk: Phys. Rev. Lett.41 (1978) 1269; ibid.45 (1980) 1738(E).

39. A. Lung et al. : Phys. Rev. Lett.70 (1993) 718.40. A.R. Zhitnitsky: Preprint SMU Dallas, SMU-HEP-94-01 (1994), hep-

ph/9402280.41. A. Antonelli et al. (FENICE): Phys. Lett. B301 (1993) 317.42. A. Antonelli et al. (FENICE): Phys. Lett. B313 (1993) 283.43. F. Cardelli, E. Pace, G. Salme, S. Simula: Phys. Lett. B357 (1995)

267.44. Z. Dziembowski: in Quark Cluster Dynamics, Lecture Notes in

Physics, Vol. 417, edited by K. Goeke, P. Kroll and H. R. Petry(Springer Verlag, Berlin, Heidelberg (1992); Z. Dziembowski,C. J. Martoff, P.Zyla: Phys. Rev. D50 (1994) 5613.

45. J.H. Kuhn, J. Kaplan, E.G.O. Safiani: Nucl. Phys. B157 (1979) 125.46. T. Gousset, B. Pire: Phys. Rev. D51 (1995) 15.47. M. Dahm, R. Jakob, P. Kroll: Z. Phys. C68 (1995) 595.48. J.C. Collins, D.E. Soper: Nucl. Phys. B193 (1981) 381.

This article was processed by the authors using the LaTEX style file pljour2from Springer-Verlag.