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Physics Letters B 509 (2001) 231–238www.elsevier.nl/locate/npe

Non-diffractive mechanisms in theφ-meson photoproductionon nucleons

Qiang Zhaoa, B. Saghaib, J.S. Al-Khalilia

a Department of Physics, University of Surrey, GU2 7XH, Guildford, UKb Service de Physique Nucléaire, DSM/DAPNIA, CEA/Saclay, F-91191 Gif-sur-Yvette, France

Received 12 February 2001; received in revised form 15 March 2001; accepted 20 March 2001Editor: W. Haxton

Abstract

We examine the non-diffractive mechanisms in theφ-meson photoproduction from threshold up to a few GeV using aneffective Lagrangian in a constituent quark model. The new data from CLAS at large angles can be consistently accounted forin terms ofs- andu-channel processes. Isotopic effects arising from the reactionsγp → φp andγ n→ φn, are investigatedby comparing the cross sections and polarized beam asymmetries. Our result highlights an experimental means of studyingnon-diffractive mechanisms inφ-meson photoproduction. 2001 Elsevier Science B.V. All rights reserved.

PACS: 12.39.-x; 25.20.Lj; 13.60.LeKeywords: Phenomenological quark model; Photoproduction reactions; Meson production

For a long time, the study ofφ-meson photopro-duction has been concentrated at high energies wherethe diffractive process is the dominant source, and apomeron exchange model based on the Regge phe-nomenology explains the elasticφ production at smallmomentum transfers [1]. In contrast with the high en-ergy reactions, data for the photoproduction of theφ-meson near threshold are still very sparse, and wereavailable only for small momentum transfers [2]. Thenew data from the CLAS collaboration at JLAB [3]cover for the first time momentum transfers above 1.5(GeV/c)2 with 2.66 � W � 2.86 GeV, and provideimportant information about mechanisms leading tonon-diffractive processes at large angles.

E-mail address: [email protected]. (Q. Zhao).

Initiated by the possible existence of strangenessin nucleons, Henley et al. [4] showed that 10–20%of strange quark admixture in the nucleon would re-sult in anss knockout cross section compatible withthe diffractive one near threshold. More recently, ithas been shown by Titov et al. [5–8] using a rela-tivistic harmonic oscillator quark model that an evensmaller fraction ofss of about 5% would produce de-tectable effects in some polarization observables. InRef. [9], Williams studied the effect of an OZI evad-ing φNN interaction by including the Born term withan effectiveφNN coupling. Quite different conclu-sions were drawn from the above approaches, sincethe descriptions of the diffractive process were sig-nificantly model-dependent, and would influence notonly the fraction of a possibless component in thenucleon, but also the OZI evadingφNN coupling.

0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0370-2693(01)00432-4

232 Q. Zhao et al. / Physics Letters B 509 (2001) 231–238

As shown in Ref. [9], the|gφNN | could have a rangeof 0.3–0.8, depending on the model for the diffrac-tive process. Therefore, a reliable description of thediffractive contribution, which determines what scoperemains for other non-diffractive mechanisms, is vi-tal. Near threshold, another question arising from non-diffractive φ-meson production is what the dominantprocess in the large angleφ production might be?In Ref. [10], we showed that OZI suppresseds- andu-channel contributions should be a dominant sourcefor large angleφ production inγp→ φp.

Concerning the two points noted above, we studyhere, within a quark model, the non-diffractiveφ-meson photoproduction in two isotopic channels,γp→φp andγ n→ φn, from threshold to a few GeV of c.m.energy. A pomeron exchange model, which was deter-mined at higher energies, was then extrapolated to thelow energy limit with the same parameter. In this way,we believe the diffractive contribution has been reli-ably evaluated and should be a prerequisite for study ofnon-diffractive mechanisms in both reactions. Pion ex-change was also included but found to be small. More-over, its forward peaking character suggests that someother non-diffractive process is necessary at large an-gles. An effectiveφ–qq interaction was proposed forthes- andu-channelφ-meson production, which willaccount for the large angle non-diffractive contribu-tions up toW ≈ 3 GeV. In the quark model frame-work, the nucleon pole terms (Born term), as well as acomplete set of resonance contributions can be consis-tently included. Our attention will be focused on thelarge angles- andu-channel processes in this work.We do not take into account the strangeness compo-nent, although the effectiveφ–qq coupling might haveincluded effects from an OZI evading process. A com-parison with the new data from the CLAS Collabora-tion should highlight the roles played by thes- andu-channelφ production, and the isotopic study willprovide insight into any non-diffractive mechanism.

The question of whether a non-diffractive processcan play a role at a few GeV of c.m. energy, is stillan open one. As pointed out by Donnachie and Land-shoff [11], contributions from two-gluon exchangesshould be small at a few GeV, and a pomeron exchangewould be enough. Laget [12] showed that a two-gluonexchange mechanism might start to play a role at large|t| with W ≈ 3 GeV. A relatively large contributionwas found from correlation processes. However, it was

also shown that two-gluon exchange could not accountfor the increase in the cross sections at large angles.A u-channel process, which violated thes-channelhelicity conservation (SCHC), was then employed toexplain the large angle behavior. Interestingly, newlysubmitted results from the CLAS Collaboration for theφ electroproduction at 0.7 �Q2 � 2.2 (GeV/c)2 and2.0�W � 2.6 GeV suggest that some non-diffractivemechanism plays a role at larget [13]. Such resultscannot generally be explained by the SCHC pomeronexchange and the soft two-gluon-exchange model,but strongly imply that some non-perturbative processmight still compete against the progressively more im-portant perturbative QCD processes at a few GeV.To disentangle these mechanisms near threshold, oneshould start with those SCHC violated processes, inparticular, thes- andu-channelφ productions. Theirenergy evolution to a few GeV as well as a measur-able effect arising from their isotopic reaction shouldbe seriously considered.

Our model consists of three processes: (i)s- andu-channelφ production with an effective Lagrangian;(ii) t-channel pomeron exchange; (iii)t-channel pionexchange.

At quark level, theφ–qq coupling is described bythe effective Lagrangian [14,15]:

(1)Leff = ψ

(aγµ + ibσµνq

ν

2mq

)φµmψ,

where the quark fieldψ can beu, d , or s for thelight-quark baryon system, whileφµm represents thevector φ-meson field. The 3-quark baryon systemis described by the nonrelativistic constituent quarkmodel (NRCQM) in theSU(6) ⊗ O(3) symmetrylimit. The vector meson is treated as an elementarypoint-like particle which couples to the constituentquark through the effective interaction. Two parame-ters,a andb, are introduced for the vector and tensorcoupling of theφ–qq in thes- andu-channels.

At tree level, the transition amplitude from the ef-fective Lagrangian can be expressed as the contribu-tions from thes-, u- andt-channel processes:

(2)Mfi =Msf i +Mu

f i +Mtf i.

In γN → φN ,Mtf i vanishes since it is proportional

to the charge of the final stateφ-meson. Introducingintermediate states, thes- andu-channel amplitudes

Q. Zhao et al. / Physics Letters B 509 (2001) 231–238 233

can be written as:

Ms+uf i = iωγ

∑j

〈Nf |Hm|Nj 〉

× 〈Nj | 1

Ei +ωγ −Ejhe|Ni〉

(3)

+ iωγ∑j

〈Nf |he 1

Ei −ωφ −Ej|Nj 〉

× 〈Nj |Hm|Ni〉,with

Hm = −ψ(aγµ + ibσµνq

ν

2mq

)φµmψ

for the quark–meson coupling vertex, and

(4)he =∑l

elrl · εγ(1− α · k

)eik·rl , k = k

ωγ,

wherek andωγ are the three-momentum and energyof the incident photon, respectively.|Nj 〉 representsthe complete set of intermediate states. In the NR-CQM, those low-lying states (n � 2) have been suc-cessfully related to the resonances and can be takeninto account explicitly in the formula. Higher excitedstates can be treated as degenerate in the main quan-tum numbern of the harmonic oscillator basis. A de-tailed description of this approach can be found inRefs. [14] and [15]. It should be noted that resonancesbelonging to quark model representation[70, 48] donot contribute inγp → φp due to the Moorhouseselection rule at the electromagnetic interaction ver-tex [16]. Therefore, eight low-lying resonances willexplicitly appear inγp → φp, while there are 16 inγ n→ φn.

The t-channel diffractive process is accounted forby the pomeron exchange model of Donnachie andLandshoff [1,17,18]. In this model, the pomeron me-diates the long range interaction between two confinedquarks, and behaves rather like aC = +1 isoscalarphoton. We summarize the vertices as follows:

(i) Pomeron–nucleon coupling:

Fµ(t)= 3β0γµf (t),

(5)f (t)= (4M2N − 2.8t)

(4M2N − t)(1− t/0.7)2

,

whereβ0 is the coupling of the pomeron to one lightconstituent quark;f (t) is the isoscalar nucleon elec-

tromagnetic form factor with four-momentum trans-fer t ; the factor 3 comes from the “quark-countingrule”.

(ii) Quark–φ-meson coupling:

(6)Vν(p− 1

2q,p+ 12q

) = fφMφγν,

where fφ = 164.76 MeV is the decay constant ofthe φ-meson inφ → e+e−, which is determined byΓφ→e+e− = 8πα2

e e2Qf

2φ /3Mφ = 1.32 keV [19].

A form factor µ20/(µ

20 + p2) is adopted for the

pomeron–off-shell-quark vertex, whereµ0 = 1.2 GeVis the cut-off energy, andp is the four-momentum ofthe quark. The pomeron trajectory isα(t) = 1 + ε +α′t , with ε = 0.08 andα′ = 0.25 GeV−2.

Theπ0 exchange is introduced via the Lagrangianfor theπNN coupling andφπγ coupling as

(7)LπNN = −igπNN ψγ5(τ · π)ψ,and

(8)Lφπ0γ = eNgφπγ

Mφ

εαβγ δ∂αAβ∂γ φδπ0.

Then the amplitude for theπ0 exchange can be de-rived in the NRCQM. The commonly used couplings,g2πNN/4π = 14, g2

φπγ = 0.143, are adopted. A signexists between the two pion exchange amplitudes forγp → φp andγ n → φn, i.e., gπpp = −gπnn, due tothe isospin symmetry.

In the pion exchange, the only parameterαπ =300 MeV comes from the quark model form factore−(q−k)2/6α2

π given by the spatial integral over thenucleon wavefunctions. Theη meson exchange hasnot been included due to its even smaller contributioncompared to the pion exchange. A recent study [20]showed that thegηNN coupling could be as smallas 1.1, which means thatη exchange can be neglectedsafely inφ-meson production.

A criticism of the application of a NRCQM toW ≈3 GeV is that relativistic effects become important dueto the high momentum transfer between the incomingphoton and the constituent quarks. In principle, oneneeds a relativistic version of the quark model totake into account the time axis. However, a self-consistent relativistic quark model is not available yet.On the other hand, the NRCQM has made impressivesuccess in hadron spectroscopy as well as most photo-excitation helicity amplitudes for baryons [21]. In


our approach, uncertainties arising from NRCQM’sshortcoming can be regarded as being efficiently takeninto account in two ways: (i) The masses as well astotal decay widths of those low-lying resonances comefrom the experimental output. Therefore, one neednot fit the baryon spectroscopy. (ii) A Lorentz boostfactor for each momentum in the spatial integrals isemployed to take into account the Lorentz contractioneffects up toW ≈ 3 GeV. In fact, it shows that energyevolution of thoses- and u-channel terms is veryimportant in relating a pomeron exchange model to theeffective Lagrangian model.

In the range of the CLAS measurements, the value|t| = 2 (GeV/c)2 corresponds to a scattering angleof θ ≈ 90◦ in the c.m. system. For larger valuesof |t|, the cross section will reflect features from anon-diffractive mechanism, which in our model is de-scribed by thes- andu-channelφ-meson production.The energy evolution as well as the large angle crosssections provide a direct constraint on the parame-ters in our model. A numerical fit of the old data [2]at Eγ = 2.0 GeV and the new ones [3] at 3.6 GeVgives a = 0.241± 0.105 andb′ = −0.458± 0.091,which are consistent with previous work [10]. Qual-itatively, the ratio of parametera for the φ- andω-meson (see Ref. [22]) can be related to the ratiogφNN/gωNN , namely,gφNN/gωNN = a(φ)/a(ω). InRef. [22],a(ω)= −2.5 accounted for the differentialand total cross sections reasonably. In Ref. [23], thebest valuea(ω) = −2.72 was derived. It shows thata(φ)/a(ω) = −0.096∼ −0.087 covers a range veryclose to the value determined bySU(3) symmetry, i.e.,gφNN/gωNN = − tan3.7◦ = −0.065, where the angle3.7◦ is the deviation from the idealω–φ mixing [19].This feature is strongly related to the effective quark–vector-meson coupling and quark model phenomenol-ogy which perhaps need to be seriously considered infuture investigation. In this work, we just treat the cou-plings as parameters and leave them determined bythe data. The signs of the parameters reflect the rela-tive phases between the pomeron exchange terms andthes- andu-channel transition amplitudes. We assumethat the quark–photon vertices and quark–φ-mesonvertices in both the pomeron exchange ands- andu-channel processes have the same signs, even thoughthe quark flavors are different. Then we leave the rela-tive phases determined by the signs of the parameters.The sign for pion exchange is fixed by Eqs. (7) and (8).

Fig. 1. Differential cross section forγp → φp at Eγ = 3.6 GeV.The dot-dashed, dashed, and solid curves denote the pion exchange,pomeron plus pion, and full model calculations, respectively, whilethe dotted curve represents full model calculation excluding theu-channel contribution. Data come from [2] (dot), [3] (square), and[25] (diamond).

In Fig. 1, the differential cross section is calcu-lated at Eγ = 3.6 GeV for γp → φp. The dot-dashed and dotted curves denote the results for ex-clusive pion exchange and pion plus pomeron ex-change, respectively. Clearly, the pomeron exchangeis the dominant mechanism at small momentum trans-fers. It can be seen that above|t| = 2 (GeV/c)2,the pomeron plus pion exchange cannot reproducethe flattened feature of the cross section. With thes- and u-channel contributions taken into account,the full model calculation is presented by the solidcurve. It is also found that theu-channel has a rel-atively stronger contribution to the cross sectionsabove the resonance energy region. Meanwhile, theu-channel nucleon pole term is dominant over otheru-channel contributions. This feature is in agreementwith the findings of Ref. [12]. The dotted curve de-notes the result excluding theu-channel from con-tributing. It should be noted that thes- andu-channelcontributions might be slightly over-estimated sincethe small two-gluon-exchange contributions are over-looked here.


Next, we show that an isotopicφ-meson photopro-duction on the neutron will be able to provide us withinformation about the large angleφ-meson productionmechanism.

Theφ–qq coupling inγ n→ φn can be described inthe same way as inγp→ φp. But the isospin degreesof freedom distinguish between proton and neutronvia differentg-factors defined for the meson–baryoncouplings [15]. Significant changes occur due to thedisappearance of the electro-interaction in the nucleonpole terms. The anomalous magnetic moment of theneutron will result in phase change effects in thes-andu-channel amplitudes. The nucleon pole terms inγ n→ φn can be written as

Msn(T )= gAµn

b′

2mq

Mn

Pi · k e−(q2+k2)/6α2

(9)

× 〈χf |[(εφ × q) · (εγ × k)

+ iσ · (εφ × q)× (εγ × k)]|χi〉,

for the transverseφ production in thes-channel, and

Msn(L)= −iagtvµn

Mφ

|q|(W +Mn)

2Pi · k e−(q2+k2)/6α2

(10)× 〈χf |σ · (εγ × k)|χi〉,for the longitudinalφ production. The correspondingu-channel amplitudes are

Mun (T )= gAµn

b′

2mq

Mn

Pf · k e−(q2+k2)/6α2

(11)

× 〈χf |[(εφ × q) · (εγ × k)

− iσ · (εφ × q)× (εγ × k)]|χi〉,

for the transverseφ production, and

Mun (L)= iagtvµn

Mφ

|q|(W +Mn)

2W

Mn

Pf · k e−(q2+k2)/6α2

(12)× 〈χf |σ · (εγ × k)|χi〉,for the longitudinalφ production. In the above equa-tions, µn = −1/3mq is the neutron’s magnetic mo-ment, andmq = 330 MeV is the constituent quarkmass;Mn andMφ are the neutron andφ-meson, re-spectively;k and q are the momenta of the incom-ing photon and outgoing meson, respectively, whileεγ andεφ are the polarization vectors of the photonand meson. Two parameters,a andb ≡ b′ + a, denote

the vector and tensor coupling of theφ–qq interaction,and are determined inγp→ φp.

An interesting feature related to the gauge invari-ance condition and arising from the longitudinalφproduction terms is that the separate calculation of thes- andu-channel nucleon pole terms will result in di-vergence at threshold|q| → 0. To get rid of such aproblem, we need to add thes- andu-channel termstogether. Notice that|k| = ωγ in the real photon reac-tion, we obtainPi · k = |k|W . Thus,Ms+u

n (L) can bewritten as

Ms+un (L)

= iagtvµnMφ

|q|(W +Mn)

2We−(q2+k2)/6α2

×[− W

ωγ (Ei +ωγ )+ Mn

ωγ (Ef + |q|cosθ)

]

× 〈χf |σ · (εγ × k)|χi〉= −iagtvµn

Mφ

|q|(W +Mn)

2We−(q2+k2)/6α2

× 1

Pf · k[ |q|2Ef +Mn

+ |q|cosθ

]

(13)× 〈χf |σ · (εγ × k)|χi〉,where gtv = 3 is derived in the quark model. Inthe last equation, the factor|q| in the denomina-tor will be cancelled by a corresponding one inthe square-bracket, and the divergence at threshold(|q| → 0) is avoided. Meanwhile, theu-channel prop-agator(Pf · k)−1 partly explains why theu-channelplays an important role in the isoscalar vector meson(ω, φ) photoproduction.

Using the parameters derived inγp → φp, thecross sections for bothγp → φp andγ n → φn arecalculated atEγ = 2.0 GeV (Fig. 2). An obviousfeature is that the large angle cross sections aresignificantly smaller forγ n→ φn than forγp→ φp.Meanwhile, a relatively stronger backward peaking isfound from theu-channel nucleon pole term. We alsopresent the results without theu-channel contributionsin Fig. 2 (see the dotted curves). Comparing the dashedcurves (pomeron plus pion exchange) to the dottedones, we find that theu-channel contributions playa dominant role in both reactions. In another word,thes-channel resonance contributions are significantlysmaller than theu-channel contributions in theφ-meson photoproduction. This feature, which has not


Fig. 2. Differential cross section forγp → φp andγ n→ φn atEγ = 2.0 GeV. The dot-dashed, dashed, and solid curves denote thes- andu-channel, pomeron plus pion, and full model calculations, respectively, while the dotted curve represents full model calculation excluding theu-channel contribution. Data come from Ref. [2].

been seen in theω-meson photoproduction, mightmake it difficult to filter signals for individuals-channel resonances in theφ photoproductions. Thisresult might be regarded as a negative result in thecontext of searching for “missing resonances” invarious reaction channels, however it has a positiveside in that the forward angle kinematics might be anideal region for studying the strangeness componentin nucleons. The dot-dashed curves in Fig. 2 denoteresults for thes- andu-channel processes.

In Fig. 3, the isotopic effects of these two reactionsare shown for the polarized beam asymmetryΣ atEγ = 2.0 GeV. Here,Σ is defined as

(14)Σ = σ‖ − σ⊥σ‖ + σ⊥

,

whereσ‖ andσ⊥ denote the cross sections forφ →K+K− when the decay plane is parallel or perpen-dicular to the photon polarization vector. The dashedcurves represent results for the pomeron plus pionexchange, which deviate from+1 due to the pres-ence of theunnatural parity pion exchange. With thes- and u-channel contributions, the full model cal-culations are denoted by the solid curves. Explic-itly, the large angle asymmetry is strongly influencedby the presence of thes- and u-channel processes,while the forward angles are not sensitive to them.

Interferences between the pomeron exchanges- andu-channel processes can be seen by excluding thepion exchanges (see the dot-dashed curves). It showsthat asymmetries produced by thes- and u-channelprocesses at forward angles are negligible. Since thepion exchange becomes very small at large angles, weconclude that the large angle asymmetry is determinedby thes- andu-channel processes and reflects the iso-topic effects. The role played by thes-channel reso-nances in the two reactions are presented by exclud-ing theu-channel contributions. As shown by the dot-ted curves, the interferences from thes-channel reso-nances are much weaker than that from theu-channel.However, they are still an important non-diffractivesource at large angles.

It should be noted that no isotopic effects can beseen if only the pomeron and pion exchange contributeto the cross section. This is because the transitionamplitude of the pomeron is purely imaginary, whilethat of pion exchange is purely real. InΣ , the signarising from thegπNN will disappear, which is whythe dashed curves in Fig. 3 are the same. We alsopoint out that our results for theΣ are quite similar tofindings of Ref. [24] at small angles, but very differentat large angles. This is because only the nucleonpole terms for thes- and u-channel processes wereincluded in Ref. [24].


Fig. 3. Polarized beam symmetry for the proton and neutron reactions atEγ = 2.0 GeV. The dashed, and solid curves denote the pomeron pluspion, and full model calculations, respectively, while the dotted curve represents full model calculation excluding theu-channel contribution.The dot-dashed curves denote full model calculation excluding the pion exchange.

Fig. 4. Total cross section forγp → φp and γ n → φn. Thesolid (dotted) and dashed (dot-dashed) curves denote the fullmodel calculation and exclusives- and u-channel cross sectionfor the proton (neutron) reaction, respectively. Data come fromRefs. [26–31]. See text for curve notations.

To show our model can be smoothly extended toa few GeV, we present the total cross sections inFig. 4 for both isotopic channels. The solid and dottedcurve denote the full calculations for the proton andneutron reaction, respectively, while the dashed and

dot-dashed curve denote the exclusive calculationsof the s- and u-channel contributions for these tworeactions, respectively. Although significant differenceexists between the exclusives- andu-channel isotopicreactions, the total cross section is not sensitive to suchan effect due to the dominance of pomeron exchange.This feature explains why such a mechanism has in thepast been neglected.

In summary, we studied the non-diffractive mecha-nisms in theφ-meson photoproductions using a quarkmodel with an effective Lagrangian in two isotopicchannels. The diffractive process is accounted for by apomeron exchange model. The pion exchange is alsoincluded and found to be small. The newly publisheddata from CLAS provides a good test of our modeland highlights the mechanisms of non-diffractiveφproduction through the directs-channel and crossingu-channel processes. The result shows that, up to afew GeV, these two channels might still play a roleat large angles, although their cross sections becomesmall. Isotopic effects arising from the proton and neu-tron reaction provide a means of study thes- andu-channel processes in experiment. The measurementof the polarized beam asymmetry at large angles canprovide detectable effects between these two isotopicreactions.

Concerning the search for signals ofss componentin the nucleon, the forward angle kinematics might be


selective if the findings of Refs. [6,7] are true, sinceat forward angles thes- and u-channel only play anegligible role. Certainly, since a possible strangenesscontent has not been explicitly included in this model,the effectiveφ–qq coupling cannot distinguish be-tween an OZI evadingφNN coupling and a strange-ness component in the nucleon. In future study, a morecomplex approach including the possible strangenesscomponent in the nucleon will be explored. To disen-tangle all the possible non-diffractive mechanisms inφ-meson photoproduction, a measurement of the iso-topic reactions covering the full angle range would bealso required.

Acknowledgements

Useful comments from Z.-P. Li are gratefully ac-knowledged. Q.Z. thanks M. Guidal and J.-M. Lagetfor valuable communications. We thank J.-P. Didelez,and E. Hourany for their interest in this work. TheCLAS data from G. Audit are gratefully acknowl-edged.

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