15 February 2001

Physics Letters B 500 (2001) 53–58www.elsevier.nl/locate/npe

Associative photoproduction of charmed particles near threshold

Michail P. Rekaloa,∗, Egle Tomasi-Gustafssonb

a Middle East Technical University, Physics Department, Ankara 06531, Turkeyb DAPNIA/SPhN, CEA/Saclay, 91191 Gif-sur-Yvette Cedex, France

Received 21 June 2000; received in revised form 4 December 2000; accepted 14 December 2000Editor: W. Haxton

Abstract

We calculate the cross section and the beam asymmetry for exclusive photoproduction of charmed particles near threshold

(γ +p→+c +D0 ), in the framework of an effective Lagrangian model. We discuss the sensitivity of these observables to the

magnetic moment of thec baryon and of the coupling constant in thepcD vertex. We show that exclusive measurementsallow, in principle, to determine the magnetic moments of charmed baryons. 2001 Published by Elsevier Science B.V.

The purpose of this Letter is to study the associa-tive photoproduction of charmed particles on nucleon.Among the simplest reactions of open charm produc-tion,γ +N→ Bc+D, with Bc =c orc, the reac-tion γ +p→+

c +D0 has the lowest threshold. Theexperimental data about charm photoproduction con-cern mainly inclusive production ofD(D∗)-mesons orc(c)-baryons at high energies, very far from thresh-old. The lowest photon energy where charm produc-tion was observed is 20 GeV, at SLAC [1], with anindirect estimation of the corresponding cross section.Up to now no exclusive measurement exists. Photo-production processes involving charmed particles areexperimentally accessible with a photon beam of en-ergy over 10 GeV. The possibility of a systematicstudy of these reactions in the treshold region at fu-ture machines as ELFE [2] or at the Jefferson Lab-oratory (JLab) electron accelerator, after the plannedupgrade [3], makes this problem very actual.

* Permanent address: National Science Center KFTI, 310108Kharkov, Ukraine.

E-mail address: etomasi@cea.fr (E. Tomasi-Gustafsson).

The photon–gluon fusion,γ +g→ c+ c, is consid-ered as the most probable mechanism of inclusive pho-toproduction of charmed particles at high energy. In itsdifferent versions (adding, for example, NLO contri-butions), the inclusive spectra ofD∗- andD-mesons[4] can be satisfactorily reproduced. However thismechanism predictsD/D as well asc/c symme-try, in contradiction with the experimental data [5,6]. Open charm photoproduction can induce, at leastpartly, the observed asymmetry. The standard view ofinclusive charm photoproduction, based on(γ + g)-fusion, with subsequent fragmentation ofc + c intocharmed particles cannot be directly applied to the de-scription of the exclusive reactions, in any kinematicalconditions, and in particular in the threshold region.Also other approaches, based on VDM [7] or QCD [8],which may reproduce the total inclusive cross section,do not give a simple picture of exclusive processes.

We will use a formalism based on an effectiveLagrangian approach (ELA), extrapolating a methodwhich is well known in the domain of the photo-production ofπ - and K-mesons. We will considerthe photoproduction of pions,γ + N → N + π ; of

0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V.PII: S0370-2693(01)00016-8

54 M.P. Rekalo, E. Tomasi-Gustafsson / Physics Letters B 500 (2001) 53–58

strange particles,γ +N→+K; and of open charm,γ + N → Bc + D, as the same class of reactionsγ + N → B + P , whereB is a baryon, with spinJand parityP equal toJ P = 1/2+, (B =N ,,,c ,or c) andP is the corresponding pseudoscalar me-son (P = π , η, K, D). In our knowledge, there is noprincipal restriction on the mass of theP-meson forthe applicability of ELA. Note in this respect, an inter-esting scaling for the masses of pseudoscalar mesons:mK/mπ mD/mK 3.6.

The application of ELA to exclusive charm photo-production has the following advantages:

• The transparent physical content of the consideredmechanisms;

• Limited number of parameters with a definite phys-ical meaning: interaction constants and magneticmoments of charmed baryons;

• Theoretical predictions for parameters such as mag-netic moments of charmed baryons can be used inthe numerical calculations;

• The outputs of the model are the absolute valueof the differential cross section and all polarizationobservables, in the near threshold region.

Such approach has been recently used in the stud-ies of different processes involving charm particles, asJ/ψ + π(ρ)→ D + D(D∗) or J/ψ + N → C +D [9–12]. The cross sections for these reactions havebeen calculated in the ELA formalism, considering thepole diagrams (withD,D∗ orC exchanges), in rela-tion with the study of the mechanisms responsible fortheJ/ψ suppression in high energy nuclei collisions.

The present analysis should be considered as a start-ing point in the discussion of the possible mechanismswhich play the most important role in the thresholdregion [13]. The predictive power of such approachshould be useful in planning the possible future mea-surements of cross sections or polarization observ-ables.

We found a large sensitivity of the observables todifferent fundamental parameters: as an example, theabsolute value of the differential cross section de-pends mainly on the interaction constant in the vertexNBcD, while the magnetic moments of the charmedbaryons mostly affect the beam asymmetry. The mag-netic moments of charmed particles can be calculatedin different approaches, from quark models to ChPT+ HQET considerations [14–17]. In this respect the

c is very interesting: the light quarks, here, are in aconfiguration with zero total spin, so thec magneticmoment is due only to the the magnetic moment of theheavyc-quark [15]. There is no experimental infor-mation about the magnetic moments of the charmedbaryons. The standard methods used for usual hyper-ons cannot be applied here, due to the shorter time oflife for Bc. In principle, the method of bending theseparticles in crystals can be used [18]. The processesγ +N→ Bc +D can be considered a parallel and in-dependent way.

According to the lines given by a previous work[13], which we update and complete, we calculate thedifferential cross section and the excitation functionfor the processγ + p→+

c +D0. For this reaction,we show the differential cross section and the beamasymmetry. We discuss, in particular, the sensitivityof different observables to the magnetic moment ofthe+

c .The spin structure for the amplitude of the processes

γ +N→ Bc +D can be written in a general form (inthe CMS of the considered reaction):

F = i σ · ef1 + σ · q σ · k× ef2

+ ie · q σ · kf3 + i σ · qe · qf4,

wheree is the photon polarization three-vector,k andq are the unit vectors along the three-momentum ofγ

andD. The differential cross section is given by:

dσ

dΩ= qk(E1 +m)(E2 +M)B +Asin2ϑ

128π2s

with

A= |f3|2 + |f4|2 + 2Re f2f∗3

+ 2Re(f1 + cosϑf3)f∗4 ,

B = 2(|f1|2 + |f2|2 − 2 cosϑRe f1f

∗2

),

whereE1(E2) andm (M) are the energy and the massof N(Bc), respectively, andϑ is the center of massangle of theD-meson production (with respect to thedirection of the incident photon). In case of a linearlypolarized photon beam, the beam asymmetryΣ canbe determinated as:

Σ = dσ⊥/dΩ − dσ‖/dΩdσ⊥/dΩ + dσ‖/dΩ = −Asin2ϑ

B +Asin2ϑ.

The amplitudesfi , i = 1, . . . ,4, have to be calculatedin the framework of some model. We consider here the

M.P. Rekalo, E. Tomasi-Gustafsson / Physics Letters B 500 (2001) 53–58 55

Fig. 1. Feynman diagrams calculated for the processγ +N→ Bc +D: (a) s-channel, (b)u-channel, (c)t-channel.

pole contributions in thes-, t- andu-channels (Fig. 1),sofi = fi,s + fi,t + fi,u.

A discussion of the ELA applied to charm particles,with broken SU(4) symmetry, can be found in litera-ture [9]. In the framework of this formalism, one ob-tains the following expressions for the matrix element,corresponding to thes-, t- andu-contributions (for thepseudoscalar meson–baryon vertex):

Ms = e gNBcDs −m2

u(p2)γ5(k + p1 +m)

×[QN ε − εk

2mκN

]u(p1),

Mu = e gNBcDu−M2

u(p2)

(Qcε − εk

2MκC

)

× (p2 − k +M)γ5u(p1),

Mt = 2eQDgNBcDt −M2

D

ε · qu(p2)γ5u(p1),

wherek, q , p1 andp2 are the four momenta ofγ ,D, N and c , ε (ε · k = 0) is the four vector ofthe photon polarization,s = (k+p1)

2, u= (k − p2)2,

t = (k − q)2 are the Mandelstam variables,m(QN ),M(Qc) andMD(QD ) are the masses (electric charges)of the nucleon, the charmed baryon and theD-meson,respectively,gNBcD is the coupling constant for thevertex of nucleon-charmed baryon–D-meson interac-tion; κN andκc are the nucleon and charmed baryonanomalous magnetic moment (κN = 1.79(−1.91) forp (n)).

From these formulas we can derive the followingexpressions for the scalar amplitudesfi .

s-channel:

f1,s = e gNBcDW +m

[QN − (W −m) κN

2m

],

f2,s = e gNBcDW +m

[−QN − (W +m) κN

2m

] |q|E2 +M ,

f3,s = f4,s = 0,e2

4π= α 1

137,

whereW = √s, is the total energy, related to theγ -

energy in the laboratory system bys =m2 + 2Eγm.

u-channel:

f1,u = e gNBcDu−M2

×Qc(W −m)

− κc

2M

[t −M2

D + (W −m)(W +m− 2M)],

f2,u = −e gNBcDu−M2

|q|E2 +M

×Qc(W +m)

+ κc

2M

[t −M2

D + (W +m)(W −m+ 2M)]

× W −mW +m,

f3,u = e gNBcDu−M2

|q| W −mW +m

[2Qc + κc W +m

M

],

f4,u = gNBcDeu−M2

(E2 −M)[−2Qc + κc W −m

M

],

56 M.P. Rekalo, E. Tomasi-Gustafsson / Physics Letters B 500 (2001) 53–58

t-channel:

f1,t = f2,t = 0,

f3,t = −2eQDgNBcDt −M2

D

|q|W −mW +m,

f4,u = 2eQDgNBcDt −M2

D

(E2 −M).

The angular distribution and the beam asymmetryfor the processγ + p→+

c +D0, atEγ = 11 GeV,are reported on Fig. 2 as functions ofϑ . The fullline represents the sum of all contributions, the Borns-channel term is given by the dashed line and theBornu-channel term by the dotted line. Thet-channeldiagram does not play any role for this reaction, asQD = 0. Even in the case when the contribution of onediagram is negligible its interference with the otherterms can largely affect the total result: a large effectof the s–u-interference may appear in the differentialcross section (Fig. 2).

The plotted differential cross section is divided byg2NDc

, for which we do not have (theoretical orexperimental) precise indications.

The behavior of theΣ-asymmetry results from thedifference in the spin structure of thes- andu-channelcontributions to the matrix element forγ + p →c + D0. The s-channel diagram (mainly character-ized byS-wavecD-production with a small off-massshell admixture ofP -wave) cannot induce nonzeroΣ-asymmetry, contrary to theu-channel contribution,which contains the majority of the multipole ampli-tudes, even in the threshold region, and is responsiblefor the positive value ofΣ , in the whole angular do-main. However, the final result is essentially driven bythes–u-interference, which is negative, compensatingthe positiveu-channel contribution.

In order to illustrate the sensitivity of this reactionto thec magnetic moments, in Fig. 3 we show, asa function ofµc , the dependence of the asymmetry,taken atϑ = π/2 and of the ratio of the differentialcross section forϑ = π and ϑ = 0 (backward–forward asymmetry). Both these quantities show acharacteristic dependence onµc , in the regionµc 1, being almost independent onµc for µc 1. TheasymmetryΣ(π/2) changes sign, forµc 1.

The sensitivity of the absolute cross section toµc appears explicitely at threshold, where only the

Fig. 2. Differential cross section and beam asymmetry as functionsof theD-meson CMS angle,ϑ : full calculation (solid line), Borns-channel (dashed line), and Bornu-channel (dotted line).

amplitudef1 is present and can be written as:

f1 1− W −m2MD

(µc − 1+ κp M

m

)

∝ 1+ 0.46µc.

The excitation functions for all the consideredreactions, with magnetic moments as reported inTable 1, are shown in Fig. 4:γ + p → +

c + D0

(thick solid line),γ + p→++c +D− (dashed line),

γ + p→+c +D0 (dotted line),γ + n→+

c +D−(dot-dashed line),γ +n→+

c +D− (solid line),γ +n→0

c +D0 (thick dot-dashed line). The calculation

for the reactionγ +p→+c +D0, using the magnetic

M.P. Rekalo, E. Tomasi-Gustafsson / Physics Letters B 500 (2001) 53–58 57

moment suggested by [16] (µc = 0.37[muN]) isreported as a thick dashed line.

These calculations reproduce satisfactorily the mea-surements at high energy (on proton target) but overes-timate the measurement at the lowest energy, from [1].The interest of such a comparison with the experimen-tal is to give an estimation of the upper limit of theconstantgNBcD , here taken as unity.

In the present model, the cross section for theprocessγ + n→ 0

c + D0 is larger when comparedwith the other processes, due to the fact that themagnetic moments ofn and0

c contribute coherentlyin thes- andu-channels.

Fig. 3. Asymmetry calculated atϑ = π/2 (top) and ratio of thecross section forϑ = π andϑ = 0 (bottom) as functions of thecmagnetic moment.

Note that the possibleT -odd polarization observ-ables have to be identically zero, in the framework ofthe considered model, in any kinematical conditions,for any reactionγ +N→ B+

c +D.In conclusion, we have calculated the differential

cross section and the beam asymmetry for the exclu-sive photoproduction of charmed particles near thresh-old, γ +N→ Bc +D. We have studied the effects ofthe magnetic moments of charmed baryons and of thecoupling constant for theNBcD vertex and discussed

Fig. 4. Sample of existing experimental data from [1] (circle), [19](square), [20] (triangles), [21] (reversed triangle), [5] (open circle),[22] (open square). The different lines show the calculations for thedifferent reactions (see text and Table 1), assuming the interactionconstantgNBcD = 1.

Table 1Table of reactions and constants. The values ofµc,c are taken from [14]

Reaction Threshold [GeV] µc,c [µN ] κc [µN ] MBc [GeV] MD [GeV]

γ +p→+c +D0 8.7064 1.86 0.86 2.2849 1.8646

γ +p→++c +D− 9.4856 1.86 −0.14 2.4528 1.8693

γ +p→+c +D0 9.4677 0 −1 2.4536 1.8646

γ + n→+c +D− 8.7139 1.86 0.86 2.2849 1.8693

γ + n→+c +D− 9.4750 0 −1 2.4536 1.8693

γ + n→0c +D0 9.4469 −1.86 −1.86 2.4522 1.8646

58 M.P. Rekalo, E. Tomasi-Gustafsson / Physics Letters B 500 (2001) 53–58

the possibility to determine these quantities, which arenot known experimentally. We have shown that the ab-solute value of the cross section depends mainly on theinteraction constant in the vertexNBcD. On the otherhand, the angular dependence and even the sign of thebeam asymmetry is affected by the value of the mag-netic moment of the charmed baryon.

Other diagrams, such asD∗-exchange int-channeland c exchange inu-channel, can, in principle,contribute. The size of the last contribution is driven bythe transition magnetic moment ofc–c conversion.On the other hand, the reaction threshold for charmedparticle production is sufficiently high to exclude thecontribution of nucleonic resonances ins-channel.

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