PHYSICAL REVIEW C 76, 015205 (2007)

Color transparency in electroproduction of the ρ meson at low energies

B. Z. Kopeliovich,1,2 J. Nemchik,3 and Ivan Schmidt11Departamento de Fısica y Centro de Estudios Subatomicos, Universidad Tecnica Federico Santa Marıa, Valparaıso, Chile

2Joint Institute for Nuclear Research, Dubna, Russia3Institute of Experimental Physics SAS, Watsonova 47, SK-04001 Kosice, Slovakia

(Received 1 March 2007; published 30 July 2007)

The study of color transparency (CT) in elastic electroproduction of vector mesons off nuclei encounters theproblem of the onset of coherence length (CL) effects. The problem of CT-CL separation arises especially atmedium energies, corresponding to the HERMES experiment, when the coherence length is of the order of thenuclear radius RA. Only at asymptotic high energies, corresponding to large CL, lc � RA, can the CT-CL mixingbe eliminated. On the other hand, the net CT effects can be studied in the kinematic range accessible by theexperiment at the Continuous Electron Beam Accelerator Facility’s large acceptance spectrometer (CLAS) at theThomas Jefferson National Accelerator Facility, since in that case the CL is much smaller than the nuclear radius.Using light-cone QCD dipole formalism, we investigate manifestations of CT effects in the electroproduction ofvector mesons. Motivated by expected data from the CLAS experiment, we predict the A and Q2 dependence ofnuclear transparency for ρ0 mesons produced incoherently off nuclei. We demonstrate that in the CLAS kinematicregion, the CL effects are weak enough to keep the photon energy at such values as to obtain maximal photonvirtualities keeping optimal statistics of the data. This has a clear advantage over a standard investigation of netCT effects fixing CL.

DOI: 10.1103/PhysRevC.76.015205 PACS number(s): 13.60.Le, 13.85.Lg, 25.30.Rw, 25.20.Lj

I. INTRODUCTION

One fundamental phenomenon coming from quantumchromodynamics is color transparency (CT), which has beenstudied intensively for more than two decades. It had beenpredicted already in 1981 [1,2] that CT occurred in diffractiveinteraction with nuclei. The CT phenomenon is manifestedas a vanishing interaction cross section σqq(r) ∝ r2 [1]for vanishing hadron (quark configuration) transverse size r .As a result, the nuclear medium is more transparent forsmaller transverse size of the hadron (quark configurations).The history of CT investigation was continued further inRefs. [3,4], where CT was predicted to be manifested alsoin quasielastic high-pT scattering of electrons and hadrons offnuclei with high momentum transfer. Later, CT was predictedto appear also in quasifree charge-exchange scattering [5,6].It was also proposed [7,8] that strong CT effects be searchedfor in the reaction of back-to-back dijet production in coherentdiffractive dissociation on nuclei.

Searches for a CT signal in quasielastic electron scatteringA(e, e′p)A∗ have not been successful so far. The first E18experiment at the Stanford Linear Accelerator Center (SLAC)[9] and later measurements at the Thomas Jefferson NationalAccelerator Facility (JLab) [10,11] failed to detect any signalof CT, which was expected to be a deviation from the Glaubermodel calculations (see the recent overview in Ref. [12]).The energy of recoil protons was too low, and the formationlength was too short in these experiments. A promisingway of detecting a signal of CT at these energies wouldbe a measurement of an asymmetric dependence of nucleartransparency on missed energy [13].

Searches for CT effects in quasielastic proton scattering,A(p, 2p)A∗, were not conclusive either. Although a veryinteresting variation of nuclear transparency was observed

[14], no unambiguous interpretation has been proposed sofar. The observed effect may be related to either the energythreshold for charm production [15] or the interplay betweenhard and soft components of the production mechanism [16](see, however, finite energy corrections in Ref. [6]). One canfind more details in the review in Ref. [17].

Only a few experiments were able to confirm the CTphenomenon. A very strong signal was observed in the PROZAexperiment at Serpukhov [18], in quasifree charge-exchangepion scattering of nuclei, at an energy of 40 GeV. Also, asizable CT effect was detected in the E791 experiment [19] indiffractive coherent dissociation of pions on nuclei.

An observation of the onset of CT in the virtual diffractivephotoproduction of ρ mesons was claimed by the E665Collaboration [20], measuring the Q2 dependence of nucleartransparency defined as

TrincA = σ inc

γ ∗A→VX

Aσγ ∗N→VX(1)

for the diffractive incoherent (quasielasic) production of vectormesons, γ ∗A → VX.1

The CT signal observed by the E665 Collaboration [20] hadbeen predicted in Refs. [21,22] as a rising nuclear transparency(vanishing final state interaction) with increasing hardnessof the reaction Q2. Although electroproduction of vectormesons off nuclei is a very effective tool for the study of CT,available experimental setups do not allow one to reach sucha kinematic region in which a strong CT signal is expected.In these cases, one has to deal only with an onset of this

1An analogous definition of nuclear transparency can also be donefor the coherent or elastic process, γ ∗A → VA.

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B. Z. KOPELIOVICH, J. NEMCHIK, AND IVAN SCHMIDT PHYSICAL REVIEW C 76, 015205 (2007)

phenomenon. Therefore for a detailed study of CT, othereffects which compete with the CT phenomenon of shouldbe considered. This was first done in Refs. [23,24] within theGlauber model. There it was shown that the main competitionto CT effects at moderate energies is the appearance ofthe so-called coherence length (CL) effects, leading to ananalogous Q2 dependence of nuclear transparency. Later, in1997, a multichannel evolution equation for the density matrixwas developed [25], describing the propagation of a hadronicwave packet through nuclear matter. This approach, appliedto the electroproduction of vector mesons off nuclei in thehadronic basis, incorporates both CT and CL effects. It wasalso suggested in this reference to separate the net CT signalby investigating the Q2 dependence of nuclear transparency atvalues of the photon energy ν which keep the CL constant.

An alternative description of CT in the electroproductionof vector mesons can be realized also within the quark-gluonrepresentation [21,22,26–30]. Here a photon of high virtualityQ2 is expected to produce a pair with a small ∼1/Q2 transverseseparation.2 Then CT manifests itself as a vanishing absorptionof the small size colorless qq wave packet during propagationthrough the nucleus. The dynamical evolution of this smallsize qq pair to a normal size vector meson is controlled by thetime scale called formation time. Because of the uncertaintyprinciple, one needs a time interval to resolve different levelsV (the ground state) or V ′ (the next excited state) in the finalstate. In the rest frame of the nucleus, this formation time isLorentz dilated, that is,

tf = 2ν

m2V ′ − m2

V

. (2)

A rigorous quantum-mechanical description of the pair evo-lution was suggested in Ref. [26] based on the nonrelativisticlight-cone (LC) Green’s function technique. The same LCGreen’s function formalism has already been applied to theDrell-Yan production in proton-nucleus and nucleus-nucleusinteractions [31], to nuclear shadowing in deep inelasticscattering [32,33], and to coherent and incoherent electropro-duction of vector mesons off nuclei [28–30].

Another phenomenon known to cause nuclear suppressionis quantum coherence. It results from the destructive interfer-ence of the amplitudes for which the interaction takes place ondifferent bound nucleons. It is characterized by the coherencelength (CL), which is related to the longitudinal momentumtransfer by qc = 1/lc. It also corresponds to the coherencetime (lc = tc), given by

tc = 2ν

Q2 + m2V

. (3)

In the electroproduction of vector mesons off nuclei, oneneeds to disentangle CT (absorption) and CL (shadowing)as the two sources of nuclear suppression. Conventionally,

2In fact, the situation is somewhat more complicated. For veryasymmetric pairs, the q or q carries almost the whole photonmomentum, and then the pair can have a large separation; see, forexample, Ref. [28].

one can associate these effects with final and initial stateinteractions, respectively. Detailed analysis of the CT andCL effects in the electroproduction of light vector mesonsoff nuclei showed [28] that the coherence length is larger orcomparable with the formation one, lc � lf , starting fromthe photoproduction limit up to Q2 ∼ 1–2 GeV2. This doesnot happen in charmonium production, where there is a stronginequality lc < lf , independent of Q2 and ν [29,30], and whichleads to a different scenario of CT-CL mixing in comparisonwith the production of light vector mesons.

Recently, new HERMES data [34,35] have been graduallyobtained for the diffractive exclusive electroproduction ofρ0 mesons on a nitrogen target. At the beginning, the datawere presented as a dependence of nuclear transparency[Eq. (1)] on coherence length [Eq. (3)]. The data for incoherentρ0 production decrease with lc, as expected from the effectsof initial state interactions. On the other hand, the nucleartransparency for coherent ρ0 production increases with coher-ence length, as expected from the effects of the nuclear formfactor [28]. However, each lc bin of the data contains differentvalues of ν and Q2; i.e., there are different contributions ofboth effects, CT and CL. For this reason, the lc behavior ofnuclear transparency does not allow us to study separately CTand CL effects. Therefore, Refs. [25,28] proposed that CT canbe separately studied eliminating the effect of CL from the dataon the Q2 dependence of nuclear transparency, in a way thatkeeps lc = const. According to this prescription, the HERMESdata [35] were later presented as Q2 dependence of nucleartransparency at different fixed values of lc. Then the rise of TrAwith Q2 represents a signature of CT. The HERMES data [35]are in a good agreement with the predictions coming fromRef. [28]. New HERMES data on neon and krypton targetshould be available soon, which will allow further verificationof the CT predictions from Ref. [28].

Moreover, another investigation of CT has been carriedout by the CLAS Collaboration at JLab [12], studying theincoherent electroproduction of ρ mesons off nuclei at smallphoton energies 2.2 < ν < 4.5 GeV. At such values of ν,the corresponding coherence length lc � RA, where RA isthe nuclear radius. For this reason, CL effects are expectedto be much weaker than CT, and CL-CT mixing does notplay an important role. Therefore the study of vector mesonelectroproduction at low energies represents an alternativeway for investigating a clear signal of CT. Because newdata from the CLAS Collaboration will appear soon, we willpresent here the corresponding predictions for CT. The mainemphasis will be devoted to discussing the advantages in theinvestigation of CT effects in the energy range available atthe the Continuous Electron Beam Accelerator Facility’s largeacceptance spectrometer (CLAS) at JLab in comparison withthe HERMES experiment.

The paper is organized as follows. In Sec. II, we present abrief summary of the light-cone dipole approach to diffractiveelectroproduction of vector mesons. In Sec. III, we calculatethe predictions for the Q2 dependence of nuclear transparency,for various nuclear targets, at the low energies corresponding tothe CLAS experiment at JLab. We analyze also the strength ofCL effects and their influence on the CT signal. We show thatCL-CT mixing at low energies does not spoil the clear onset

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COLOR TRANSPARENCY IN ELECTROPRODUCTION OF . . . PHYSICAL REVIEW C 76, 015205 (2007)

of CT effects and permits one to treat the CLAS experimentalvalues at different small lc � RA, allowing one to processthe data at maximal possible statistics. We discuss the factthat the weak CL effects in the CLAS energy region lead alsoto the possibility of studying the CT phenomenon at muchlarger values of Q2 than those which are presently available,using a prescription to fix CL, in particular for the HERMESexperiment. The results of the paper are summarized anddiscussed in Sec. IV.

II. SHORT REVIEW OF COLORDIPOLE PHENOMENOLOGY

The light-cone (LC) dipole approach for the processγ ∗N → VN was already used in Refs. [26,36] to studythe exclusive photo- and electroproduction of charmonia andin Ref. [28] for the elastic virtual photoproduction of lightvector mesons ρ0 and �0 (for a review, see also Ref. [37]).Therefore, we present only a short review of this approach.Here a diffractive process is treated as elastic scattering of aqq fluctuation of the incident particle. The forward productionamplitude for the process γ ∗N → VN can be represented inthe quantum-mechanical form

Mγ ∗N→VN(s,Q2) = 〈V |σqq(r, s)|γ ∗〉

=1∫0

dα

∫d2r�∗

V (r, α)σqq(r, s)

×�γ ∗ (r, α,Q2), (4)

with the normalization

dσ (γ ∗N → VN)

dt

∣∣∣∣t=0

= |Mγ ∗N→VN(s,Q2)|216π

. (5)

There are three ingredients contributing to the amplitude(4): (i) the dipole cross section σqq(r, s) which depends onthe qq transverse separation r and the c.m. energy squared s,(ii) the LC wave function of the photon, �γ ∗ (r, α,Q2), whichdepends not only on r but also on the photon virtuality Q2 andthe relative share α of the photon momentum carried by thequark, and (iii) the LC wave function �V (r, α) of the vectormeson. Detailed descriptions of these ingredients can be foundin Refs. [28,29].

Notice that in the LC formalism, the photon andmeson wave functions contain also higher Fock states|qq〉, |qqG〉, |qq2G〉, etc. The effect of these higher Fockstates is implicitly incorporated into the energy dependenceof the dipole cross section σqq(r, s), as is given in Eq. (4).

Since our main interest is the electroproduction of lightvector mesons, we explicitly consider the nonperturbativeinteraction effects between the q and q, which can be includedin the LC wave function of the photon, �γ ∗ (r, α,Q2). For thispurpose, we use the corresponding phenomenology based onthe LC Green’s function approach, developed in Refs. [26,38].Green’s function Gqq(z1, r1; z2, r2) describes the propagationof an interacting qq pair between points with longitudinalcoordinates z1 and z2 and with initial and final transverse

separations r1 and r2. This Green’s function satisfies thetwo-dimensional Schrodinger equation

id

dz2Gqq(z1, r1; z2, r2) =

{ε2 − r2

2να(1 − α)+ Vqq(z2, r2, α)

}

×Gqq(z1, r1; z2, r2). (6)

The real part of the LC potential Vqq(z2, r2, α) in Eq. (6)is responsible for the interaction between the q and q in thevacuum. For the sake of simplicity, we use the oscillator formof the potential,

ReVqq(z2, r2, α) = a4(α)r22

2να(1 − α), (7)

which leads to a Gaussian r dependence of the LC wavefunction of the meson ground state. The shape of the functiona(α) can be found in Ref. [38].

In this case, Eq. (6) has an analytical solution leading to anexplicit form of the harmonic oscillator Green’s function [39],

Gqq(z1, r1; z2, r2) = a2(α)

2πisin(ωz)exp

{ia2(α)

sin(ωz)

× [(r2

1 + r22

)cos(ω z) − 2 r1 · r2

]}

× exp

[− iε2z

2να(1 − α)

], (8)

where z = z2 − z1 and

ω = a2(α)

ν α(1 − α). (9)

The boundary condition is Gqq(z1, r1; z2, r2)|z2=z1 = δ2(r1 −r2).

Assuming s-channel helicity conservation, the forwardproduction amplitude for the process γ ∗N → VN, separatelyfor transverse (T ) and longitudinal (L) photons and vectormesons, reads

MTγ ∗N→VN(s,Q2)|t=0 = NCZq

√2αem

∫d2rσqq(r, s)

×1∫0

dα{m2

q�0(ε, r, λ)�TV (r, α)

− [α2 + (1 − α)2] �1(ε, r, λ)

· ∇r�TV (r, α)

}, (10)

MLγ ∗N→VN(s,Q2)|t=0

= 4NCZq

√2αemmV Q

∫d2rσqq(r, s)

×1∫0

dαα2(1 − α)2

×�0(ε, r, λ)�LV (r, α), (11)

with the normalization given by Eq. (5) for both T and L

polarizations. The functions �0,1 in Eqs. (10) and (11) includenonperturbative interaction effects between q and q, and �V

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B. Z. KOPELIOVICH, J. NEMCHIK, AND IVAN SCHMIDT PHYSICAL REVIEW C 76, 015205 (2007)

represents the vector meson wave function, whose explicitform will be taken from Ref. [28]. The real part of theamplitude is included according to the prescription describedin Ref. [28].

Usually the data are presented in the form of the productioncross section σ = σT + ε′σL, at fixed photon polarization ε′.Here the transverse and longitudinal cross sections, integratedover t , read

σT,L(γ ∗N → VN) = |MT ,L|216πBγ ∗N

, (12)

where Bγ ∗N is the t-slope parameter in the differential crosssection for reaction γ ∗p → Vp, Eq. (5).

The generalization of the LC dipole approach to nucleartargets is straightforward. We focus in the present paper onthe diffractive incoherent (quasielastic) production of vectormesons off nuclei, γ ∗A → VX. Since the t slope of thedifferential quasielastic cross section is the same as on anucleon target, instead of the integrated cross sections onecan also use the forward differential cross sections, Eq. (5), towrite

TrincA = 1

A

∣∣∣∣Mγ ∗A→VX(s,Q2)

Mγ ∗N→VN(s,Q2)

∣∣∣∣2

. (13)

Besides the energy dependence of the photoproductionreaction on a nucleon, the higher Fock states of virtual photoncontaining gluons lead also to gluon shadowing (GS) as faras nuclear effects are concerned. Detailed description andcalculation of GS for vector meson production off nucleiis presented in Refs. [28,40]. However, in the CLAS andHERMES kinematic range studied in the present paper, GSis negligible and therefore not included in our calculations.

The propagation of an interacting qq pair in a nuclearmedium is also described by the Green’s function satisfying theevolution Eq. (6). However, the potential in this case acquiresan imaginary part which represents absorption in the medium,that is,

ImVqq(z2, r, α) = −σqq(r, s)

2ρA(b, z2), (14)

where ρA(b, z) is the nuclear density function defined at givenimpact parameter b and longitudinal coordinate z.

The analytical solution of Eq. (6) is only known for theharmonic oscillator potential V (r) ∝ r2. Therefore we use thedipole cross section approximation

σqq(r, s) = C(s)r2, (15)

with the energy-dependent factor C(s) adjusted by the proce-dure described in Ref. [28]. Then the solution of Eq. (6) has thesame form as Eq. (8), except that one should replace ω ⇒ �,where

� =√

a4(α) − iρA(b, z)να(1 − α)C(s)

να(1 − α). (16)

The evolution equation (6) was recently solved numericallyfor the first time in Ref. [33], with the potential Vqq(z2, r2, α)

containing the imaginary part [Eq. (14)] and with the realisticdipole cross section given in Ref. [38].

As is discussed in Ref. [28], the value of lc can distinguishdifferent regimes of vector meson production. We presenthere only the general case in which there are no restrictionsfor either lc or lf . The corresponding incoherent nuclearproduction amplitude squared is represented as a sum of twoterms [41],

|Mγ ∗A→VX(s,Q2)|2 =∫

d2b

∞∫−∞

dzρA(b, z)|F1(b, z)

−F2(b, z)|2. (17)

The first term,

F1(b, z) =∫ 1

0dα

∫d2r1d

2r2�∗V (r2, α)G(z′, r2; z, r1)

× σqq(r1, s)�γ ∗ (r1, α)|z′→∞, (18)

corresponds to the short-lc limit, lc → 0, but where lf ∼ RA.The second term F2(b, z) in Eq. (17) corresponds to

the situation in which the incident photon produces a qq

pair diffractively and coherently at the point z1, prior to anincoherent quasielastic scattering at point z. Correspondingly,this term has the form

F2(b, z) = 1

2

z∫−∞

dz1ρA(b, z1)

1∫0

dα

∫d2r1d

2r2d2r�∗

V (r2, α)

×G(z′ → ∞, r2; z, r)σqq(r, s)G(z, r; z1, r1)

× σqq(r1, s)�γ ∗ (r1, α). (19)

III. SIGNAL OF CT EXPECTED AT LOW ENERGIES

Exclusive incoherent electroproduction of vector mesonsoff nuclei has been shown in Refs. [22,28] to be a veryeffective tool for the investigation of CT. Increasing the photonvirtuality Q2 has the effect of squeezing the produced qq wavepacket. Such a small colorless system propagates through thenucleus with little attenuation, provided that the energy issufficiently high (lf � RA), so the fluctuations of the qq

separation are “frozen” during propagation. Consequently, arise of the nuclear transparency Trinc

A (Q2) with Q2 should givea signal for CT. Indeed, such a rise was observed in the E665experiment [20] at Fermilab for exclusive production of ρ0

mesons off nuclei, and this has been claimed as a manifestationof CT.

However, the effect of coherence length [23,24] leads alsoto a rise of Trinc

A (Q2) with Q2, so it can imitate the CT effects.Both effects work in the same direction, and this produces theproblem of CT-CL separation, which was solved in Refs. [25,28], which presented a simple prescription for the eliminationof CL effects from the data on the Q2 dependence of nucleartransparency. One should bin the data in a way that keepslc = const. It means that ν and Q2 should be correlated,

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COLOR TRANSPARENCY IN ELECTROPRODUCTION OF . . . PHYSICAL REVIEW C 76, 015205 (2007)

that is,

ν = 1

2lc(Q2 + m2

V

). (20)

In this case, any rise with Q2 of the nuclear transparency ratiois a clear signal of CT [25,28].

In the present paper, we investigate a manifestation of CTeffects in the production of vector mesons at low energies,corresponding to CLAS experiment at JLab. Motivated byexpected new data from the CLAS collaboration we con-centrate on the production of ρ0 vector mesons. Besidesthis experimental motivation, there is also theoretical interest,because the coherence and formation effects are much morevisible for the light than for the heavy vector mesons, as wasdiscussed and analyzed in Refs. [28–30].

We start, however, with a discussion about the firsttheoretical analysis of CT phenomenon, which is related tothe E665 experiment [20]. The corresponding predictions [22]were based on the assumption that the energy correspondingto the E665 experiment is high enough, so that lf , lc � RA.Correspondingly, one can neglect a variation of CL with Q2

and use a simplified formula for the ratio TrincA [22]. However,

this is true only at Q2 � 3–5 GeV2, as was shown later inRef. [28]. It means that fluctuations of the size of the qq

pair become important at larger Q2 � 5 GeV2; therefore,at such Q2 values, one should use the general expression(17) for the calculation of nuclear transparency. Thus theobserved variation of Trinc

A (Q2) is a net manifestation ofCT only at smaller Q2. Although the model predictions,within the color dipole approach [22], agreed with the E665data, small corrections for CL effects, done at larger Q2 of5 < Q2 < 10 GeV2, in Ref. [28], did not spoil this agreement.

On the other hand, CT effects are also under investigation atlower energies in both the HERMES experiment at HERA andthe CLAS experiment at JLab, measuring the same process ofincoherent electroproduction of ρ mesons. Because CL effectsare also important, one should use the LC Green’s functionformalism as a very effective tool for such a study, because thenboth CT and CL effects are naturally incorporated. Becauseof strong CL-CT mixing, one should eliminate the effect ofCL according to the prescription mentioned above [see Eq.(20)]. The corresponding predictions for CT [28] are in a goodagreement with the HERMES data [35] on the Q2 dependenceof nuclear transparency at different fixed values of lc. Expectednew HERMES data on neon and krypton targets should allowone to verify further the CT predictions from Ref. [28].

Following the relation in Eq. (20), CT effects are foundto be much stronger at low (HERMES, CLAS) than at high(E665) energies [28]. At high energies, the CL lc is long.Consequently, the formation length is long, too, lf � lc � RA,and nuclear transparency rises with Q2 only because the meantransverse size of the qq photon fluctuations decreases. It isnot so at lower energies, when lc � RA. Then, according tothe prescription in Eq. (20), the photon energy rises with Q2,and consequently the formation length, Eq. (2), rises as well.Thus, these two effects add up, leading to a steeper growth ofTrinc

A (Q2) for short lc.The prescription (20) for studying net signals of CT fixing

the values of lc has one hidden disadvantage. Increasing Q2

requires that one needs to increase also the photon energy,and this can be done only up to the kinematic limit givenby the HERMES experiment, νmax = 24 GeV. In turn, thisleads then to a restriction for the maximal values of the photonvirtualities Q2

max. The situation gets a bit more complicatedbecause an effort to obtain high statistics data leads to a ν-Q2

correlation and then to stronger restrictions on Q2max at different

fixed values of lc. For example, at 〈lc〉 = 1.35 fm, the valueof Q2

max = 3.5 GeV2; and at 〈lc〉 = 2.45 fm, the value isQ2

max = 1.5 GeV2, as follows from the HERMES experiment[35,42].

Therefore, we present in Fig. 1 the Q2 dependence of thenuclear transparency Trinc

A for the exclusive electroproductionof ρ mesons on a 207Pb target at different fixed lc values.The top and bottom dotted lines represent the predictions atfixed lc = 1.0 and 5.0 fm, respectively. These two values limitapproximately the range of lc values corresponding to a mean〈ν〉 = 13 GeV and 0.5 < Q2 < 5.0 GeV2, which follow fromthe HERMES kinematics [42]. Because lc � RPb, one canexpect a strong onset of CL effects, as one can see in Fig. 1 asthe difference between the top and bottom dotted lines.

Different fixed lc values determine the different maximalphoton virtualities Q2

max allowed for the experimental analysisof the CT phenomenon. To reach the maximal possible valuesof Q2, one should fix lc at its minimal possible values, keepingreasonable statistics of the data. Such a situation is depicted bythe top dotted line in Fig. 1, and allows one to study the onsetof CT effects up to Q2

max = 5.0 GeV2 within the HERMESkinematics. The higher fixed values of lc mean a strongerrestriction for Q2

max, and consequently a weaker onset of CTeffects. For example, at the second fixed value of lc = 5.0 fm,

one can study CT effects only up to Q2max = 1.3 GeV2,

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Q2 (GeV2)

Tr A

207Pb

lc = 1 fmlc = 5 fm

HERMES

FIG. 1. Q2 dependence of the nuclear transparency TrincA for

exclusive electroproduction of ρ mesons on 207Pb target. The dottedlines represent the CL fixed at lc = 1.0 and 5.0 fm. The solid linesrepresent predictions at the fixed mean photon energy 〈ν〉 = 13 GeVcorresponding to HERMES experiment.

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B. Z. KOPELIOVICH, J. NEMCHIK, AND IVAN SCHMIDT PHYSICAL REVIEW C 76, 015205 (2007)

although the model calculations shown by the bottom dottedline in Fig. 1 formally reach the value Q2

max = 5.0 GeV2.As a demonstration of the importance of CL effects in the

HERMES kinematic range, we also present, by the solid linein Fig. 1, predictions for nuclear transparency at fixed meanphoton energy 〈ν〉 = 13 GeV. This corresponds to a change ofCL from 5.0 to 1.0 fm. Because both CL and CT effects affectthe rise of TrA with Q2, the net observable effect is much morevisible than that when the effect of CL is eliminated by theprescription (20) (compare each dotted line with the solid onein Fig. 1).

Now we switch to the low-energy region correspondingto that of the CLAS Collaboration at JLab. Because the newdata will appear soon, we provide predictions for the expectedonset of CT effects for different nuclei. The model calculationsfor nuclear transparency as a function of Q2 are depicted inFig. 2 for the nuclear targets 12C, 56Fe, and 207Pb. Accordingto the CLAS kinematics [12], the values of CL change fromlc = 0.9 to 0.5 fm when the photon virtuality rises from Q2 =0.9 to 2.5 GeV2. Eliminating the CL effects by fixing lc = 0.9and 0.5 fm, one can obtain the predictions shown in Fig. 2 bythe difference between the top and bottom dotted lines withrespect to each solid line, for different nuclear targets. Onecan see that in the CLAS kinematic range, the CL effects arerather weak, much weaker than the observed CT signal. Thiscomes from the rather low photon energies for values of CL,lc � RA. A weak onset of CL effects can be seen in Fig. 2as the difference between the top and bottom dotted lines, foreach nuclear target.

Therefore in contrast with the HERMES kinematics, onecan study the variation of nuclear transparency with Q2 at

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3

Q2 (GeV2)

Tr A

12C

56Fe

207Pb

FIG. 2. Q2 dependence of the nuclear transparency TrincA for

exclusive electroproduction of ρ mesons on nuclear targets 12C, 56Fe,and 207Pb (from top to bottom). The top and bottom dotted line withrespect to each solid line represents predictions fixing CL at lc = 0.9and 0.5 fm, respectively. The solid lines represent predictions at thefixed mean photon energy 〈ν〉 = 3.5 GeV corresponding to the CLASexperiment at JLab.

approximately fixed photon energy, ν = 3.5–3.6 GeV, usingmaximal statistics of the data. Such a situation is depicted inFig. 2 by the solid lines, for different nuclear targets. It allowsone to reach much higher values of Q2 than those reached inthe HERMES experiment, using the CL-elimination proceduredescribed above.

As a consequence of a small onset of CL effects, thecorresponding maximal CLAS value of photon virtuality usedfor studying CT effects Q2

max ∼ 2.5–3.0 GeV2 does notdiffer much from the HERMES value Q2

max ∼ 3.5–4.0 GeV2,although the mean photon energy is four times lower thanthat in the HERMES experiment. It means that the CLASkinematic range leads to a more effective investigation ofCT phenomenon than when we are forced to use the CL-elimination procedure, i.e., keeping lc = constant at differentvalues of Q2, which is typical for the HERMES experiment.

Such an advantage in the investigation of CT effects in theCLAS experiment should be maintained also after its futureupgrade to a beam energy of 12 GeV [43]. As a consequence ofhigher photon energies, greater values of the photon virtualitiesQ2 will be reached, keeping the condition of a weak onset ofCL effects lc � RA for heavy nuclear targets. Thus it allowsone to study a stronger onset of CT effects by investigatingnuclear transparency at larger Q2 values.

IV. SUMMARY AND CONCLUSIONS

Electroproduction of vector mesons off nuclei is a veryeffective tool for studying the interplay between coherence(shadowing) and formation (color transparency) effects. Inthe present paper, we investigated how those effects manifestthemselves in the production of vector mesons off nuclei,at different energies corresponding to those applied in theE665, HERMES, and CLAS experiments. We used, fromRef. [28], a rigorous quantum-mechanical approach basedon the light-cone QCD Green’s function formalism, whichnaturally incorporates these interference effects. Motivated byexpected new data from the CLAS Collaboration, we studiedthe onset of CT effects at low energies, predicting a risingnuclear transparency as a function of Q2.

As the first step, we discussed CT effects at high energiescorresponding to those in the E665 experiment. It was alreadyshown in Ref. [28] that the high energy limit (lf ∼ lc � RA)can be applied only at small and medium values of photonvirtualities, Q2 � 3–5 GeV2. Here the expressions for nuclearproduction cross sections are sufficiently simplified. Thisso-called frozen approximation includes only CT, becausethere are no fluctuations of the transverse size of the qq pair.At larger values of Q2 (� 5 GeV2), both the CL lc � RA

and the onset of CL effects start to be important. The frozenapproximation can no longer be applied to the study of asignal of CT effects. Therefore, one should use the generalexpression in Eq. (17), which incorporates, in addition, the CLeffects and thereby interpolates between the previously knownlow and high energy limits for the incoherent production ofvector mesons.

In the incoherent electroproduction of vector mesons atlow and medium energies, the onset of coherence effects

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(shadowing) can mimic the expected signal of CT. Both effects,CT and CL, work in the same direction. To single out theformation effect, the experimental data should be binned in lcand Q2 [28]. The observation of a rising nuclear transparencyas a function of Q2 for fixed lc would signal CT. Such aprocedure for the elimination of CL effects must be applied atmedium energies, corresponding to the HERMES kinematicrange, when lc � RA. We showed that a Q2 variation of CLaffects a strong onset of CL effects, as one can see in Fig. 1 asthe difference between the top and bottom dotted lines.

At still lower photon energies, corresponding to those in theCLAS experiment, both lc and lf are short enough comparedto the mean spacing of the bound nucleons. Consequently, theCL effects are rather weak and, in fact, much weaker than theobserved CT effect (see the difference between the top andbottom dotted lines for each nuclear target in Fig. 2). Thisprovides an advantage in the investigation of the onset of CTwhen one can study the rise of nuclear transparency with Q2 atfixed photon energies, processing the data at maximal statistics

(see the solid lines in Fig. 2). We showed that such a procedureallows one to reach much higher values of Q2 effective forstudying the CT phenomenon, than those used in the HERMESexperiment applying the CL-elimination prescription of fixinglc [see Eq. (20)].

In conclusion, the exploratory study of CT effects atdifferent energies in the electroproduction of light vectormesons off nuclei opens new possibilities for the investigationof different manifestations of CL effects. It gives a good basisfor the further effective studies of the onset of CT effects,after an upgrade of the CLAS experiment at JLab to a 12 GeVenergy electron beam.

ACKNOWLEDGMENTS

This work was supported in part by Fondecyt (Chile) GrantNo. 1050519, by DFG (Germany) Grant No. PI182/3-1, andby the Slovak Funding Agency Grant No. 2/7058/27.

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