Color transparency after the NE18 and E665 experiments Outllok and perspectives at CEBAF

a r X i v :n u c l -t h /9406005v 1 6 J u n 1994KF A-IKP(Th)-1994-201June 1994

Color transparency

after the NE18and E665experiments:

Outllok and perspectives at CEBAF.?

J.Nemchik 1,3),N.N.Nikolaev 2,3and B.G.Zakharov 21)Institute of Experimental Physics,Slovak Academy of Sciences,Watsonova 47,04353Kosice,Slovak Republik 2)https://www.wendangku.net/doc/f319048265.html,ndau Institute for Theoretical Physics,GSP-1,117940,ul.Kosygina 2,V-334,Moscow,Russia 3)IKP(Theorie),KFA J¨u lich,D-52425J¨u lich,Germany ABSTRACT CEBAF is a high-luminocity factory of virtual photons with variable virtuality Q 2and transverse size.This makes CEBAF,in particular after the energy upgrade to (8-12)GeV,an ideal facility for uncovering new phenomena,and opening new windows,at the interface of the perturbative and nonperturbative QCD.We discuss color transparency as the case for a broad program on electroproduction of vector mesons ρ0,ω0,φ0and their radial excitations ρ′,ω′,φ′at CEBAF.We also comment on the second generation of experiments on color transparency in 4He (e,e ′p )scattering,which are also feasible at CEBAF.In 1994,we can make more reliable projections into future because our understanding of the onset of color transparency has greatly been augmented by two experiments completed in 1993:i)no e?ect of CT was seen in the SLAC NE18experiment on A (e,e ′p )scattering at virtu-

alities of the exchanged photon Q 2～<7GeV 2,

ii)strong signal of CT was observed in the FNAL E665experiment on exclusive ρ0-meson production in deep inelastic scattering in the same range of Q 2.

We discuss the impact of these observations on the CEBAF experimental program.We argue they both are good news,both were anticipated theoretically,and both rule in the correct QCD mechanism of the onset of CT.

?)Presented by NNN at the Workshop on CEBAF at Higher Energies CEBAF,14-16April 1994

1Introduction

The fundamental prediction of QCD is that the quark con?gurations with small transverse size r have small interaction cross section[1],which was dubbed color transparency(CT) [2].Looking for CT is long discussed as the case for the high-luminocity,high-duty cycle, (10-20)GeV electron facility,which is well documented in the ELFE project[3](ELFE= European Laboratory for Electrons).In the meantime,the good news from CEBAF is a possibility of the(8-12)GeV upgrade,which opens exciting possibilities of doing CT physics at CEBAF.

CEBAF is a high-luminocity factory of virtual photons.Higher energy means a higher virtuality Q2of photons,and higher Q2means that smaller sizes are becoming accessible reaching eventually into the perturbative QCD region.Higher energy also means longer lifetime of these small-size states.However,from the very start,we must emphasize that as far as CT physics is concerned,the purely perturbative region lies well beyond the kinemat-ical range of CEBAF experiments on exclusive processes,even after the(8-12)GeV energy upgrade.Testing the purely perturbative QCD to few decimal places is a task of inclusive experiments at superhigh energy facilities like LEP or HERA.Even at LEP and HERA, the predictive power of the purely perturbative QCD rapidly deteriorates when the exclusive processes are considered.The real task of the CEBAF experiments is to uncover new QCD phenomena in exclusive reactions at the interface of the perturbative and nonperturbative QCD.Very conservative conclusion of this overview is that the energy-upgraded CEBAF shall do the job.

Before jumping into conclusions on the feasibility of CT physics at CEBAF,one must recall and critically summarize the results of the two CT experiments completed in1993:?The A(e,e′p)reaction on the D,C,F e and Au targets was studied by the SLAC NE18 collaboration with the negative result:no CT e?ects are seen at Q2≤7GeV2[4].

?The FNAL E665experiment[5]on exclusive production of theρ0mesons in deep inelastic scattering of muons on nuclei produced a solid evidence for CT in precisely the same range of Q2as explored in the NE18experiment.

The early history of CT focused on the quasielastic A(e,e′p)scattering of electrons on nuclei.A number of predictions of precocious CT at low Q2were published(for the review and references see[6]),and the failure to con?rm this precocious CT in the NE18experiment considerably dampened the whole subject of CT.Fortunately,more consistent treatment rather predicted a very slow onset of CT in the A(e,e′p)scattering[7,8].As a matter of fact, the NE18results do perfectly con?rm the correct theory and rule in the mechanism of CT, which is alive and well,and we can joyfully recite Mark Twain’s telegram to the Associated Press:”The reports of my death were an exaggeration”.

The parallel development was a theory of CT in(virtual)photoproduction of vector mesonsγ?N→V N.From the theoretical point of view,this is a much cleaner case,with a well understood shrinkage of the transverse size of the virtual photon with the increase of Q2[9,10].The prediction[9],not a postdiction,of the precocious onset of CT was con?rmed by the FNAL E665experiment[5],which put the CT physics in the right perspective.

The strong point which we wish to make in this review is that after the energy upgrade, CEBAF experiments on exclusive electroproduction of vector mesons can signi?cantly con-tribute to our understanding of the onset of CT.Furthermore,the experiments on produc-tion of the radially excited vector mesons will open an entirely new window not only on the

mechanism of CT,but also on the quark structure and poorly known spectroscopy of radial excitations.In the A(e,e′p)sector,we comment on the potential of experiments on the4He target,in which the onset of CT is sooner than for any other nucleus owing to the small size of the4He nucleus.

In this contribution to the Workshop on CEBAF at Higher Energies we concentrate on the recent experimental and theoretical developments,for the earlier reviews on the subject see[3,11-14].

2CT in exclusive production of vector mesons

2.1CT and dipole cross section

In order to be quantitative,let us set up the theoretical framework,which is the lightcone dipole-cross section representation[15].Mesons can be viewed as color dipoles.The distri-bution of the transverse size r of color dipoles in the meson is given be the qˉq wave function Ψ(z, r),where z is the fraction of meson’s momentum carried by the quark.This mixed(z, r) lightcone representation is custom tailored for description of CT.By the Lorentz-dilation,in the high-energy scattering the dipole size r becomes as good a conserved quantum number as an angular momentum.The fundamental quantity which describes all the scattering processes,is the dipole cross sectionσ(ν,r)for interaction of the color dipole of size r with the target nucleon.Of course,apart from the qˉq Fock state,the lightcone hadrons contain the higher qˉq g....Fock states.The e?ect of gluons in the projectile dipole brings in the dependence ofσ(ν,r)on energyν,which can be related to the gluon structure function G(x,q2)of the target nucleon[16,17].Speci?cally,at small r,the dipole cross section is∝r2 ,

π2

σ(ν,r)=

?The proton structure function F 2(x,Q 2)receives contribution from 1/√

Q 2.

In CT experiments one looks for a weak intranuclear ?nal (and initial)state interactions,which will be the case if the nuclear production amplitude is dominated by the dipole cross section at small r such that σ(ν,r )is much smaller than the free-nucleon cross section.Whether the particular exclusive reaction is selective of such a small r or not,requires the case-by-case study.

2.2How CT is tested in leptoproduction of vector mesons?

The exclusive (elastic)production γ?p →V p (V =ρ0,φ0,J/Ψ,...)is an ideal laboratory for testing CT ideas [9,10,12,21-24].The forward production amplitude equals

M = V |σ(ν,r )|γ? = 1

0dz d 2 r σ(ν,r )Ψ?V (r,z )Ψγ?(r,z )(3)

Here Ψγ?(r,z )is the wave function of the q ˉq

?uctuation of the photon,which at high energy νis formed at a large distance (the coherence length)

l c =2ν/(Q 2+m 2V )(4)

in front of the target nucleon.The most important feature of Ψγ?(r,z )as derived in [15]is an exponential decrease at large size

Ψγ?(r,z )∝exp(?εr ),

(5)where ε2=m 2q +z (1?z )Q 2(6)and m q is the quark mass.Therefore,the amplitude (3)recives the dominant contribution from r ～r S ≈3/ε.In the nonrelativistic quarkonium z ≈1

(Q 2+m 2V )2(8)

M L ≈√

m V M T ∝√m V 1

The scanning radius r S decreases,and the virtual photon shrinks,with increasing Q2. Notice,however,the large numerical factor in Eq.(7),for which the onset of small-size dominance requires very large Q2.Remarkably,this large factor derives precisely from CT property of the dipole cross section.Because of this large scanning radius,the vector meson production probes the gluon structure function G(x,q2)at[23]

q2～0.2(Q2+m2V)(10) The emergence of this very low factorization scale was not noticed in[26].

Because of a large scanning radius,the simple nonrelativistic approximation remains viable in quite a broad range of Q2,and the production amplitude can be calculated using the constituent quark wave functions of vector mesons.For the same reason of large r S, the asymptotic predictions[10]σT∝1/Q8andσL∝1/Q6,can not readily be tested at Q2～<(10?20)GeV2studied sofar.

In the conventional quark model,the radii of theρandπmesons are about identical. Once the radius of theρ0is speci?ed,further predictions[23,27]for theρ0production cross section do not contain any adjustable parameters.They are presented in Fig.3.We use the low-energy dipole cross section of Ref.[15].Shown is the combination of the longitudinal and transverse cross sections as measured by the NMC collaboration.The agreement with the recent NMC data[25]is excellent.This agreement in a broad range of Q2shows we have a good understanding of the dipole cross section from the hadronic scale r～(1-2)f down to the smallest dipole size r～0.3f achieved in the NMC experiment at Q2～20GeV2. Over this range of radii r,the dipole cross section drops by approximately one order in magnitude,and this agreement of the total production rate with experiment is by itself a very important test of CT.Similar calculations[23]give an excellent description of the J/Ψproduction and of the real photoproduction of theρ0at HERA.

2.3CT in exclusive production on nuclei

Having tested CT property of the dipole cross section in the production on free nucleons, now we turn our attention to the production on nuclei.The qˉq pair produced on the target nucleon,recombines into the observed vector meson with the recombination(formation) length

l f=

ν

Aσp =

12σ(r)T(b) |γ? 2

A d2 bT(b)2+...,(12)

where

T(b)= dzn A(b,z)(13)

is the optical thickness of a nucleus at the impact parameter b and n A (b,z )is the nuclear matter density.In the quasielastic production one sums over all excitations and breakup of the target nucleus.The total cross section of the coherent (elastic)production γ?A →V A ,when the target nucleus remains in the ground state,equals [22]

σcoh (V A )=4

d 2 b V |1?exp ?1dt t =0

d 2 bT (b )2 1?1 V |σ(r )|γ?

.(15)Evidently,the matrix element in the numerator of

(14)is dominated by r ～r F SI =

5Q 2+m 2V ,(16)

which gives an estimate [10,23]

ΣV ≈σ(r F SI )(17)CT and/or weak FSI set in when r F SI ?R V ,i.e.,when ΣV ?σtot (V N ).In this regime of CT,the ΣV is insensitive to the wave function of the vector meson,so that predictions of FSI e?ects are less model independent than predictions for the total production cross section.The large value of r F SI Eq.(16)implies that FSI only slowly vanishes with the increase of Q 2,and this slow onset of CT is driven by the very mechanism of CT.2.4The E665experiment [5]:the decisive proof of CT

Our predictions [9,10]for nuclear e?ects in the coherent and incoherent exclusive ρ0produc-tion are compared with the E665data in Figs.4-7.In Fig.4we show nuclear transparency for the incoherent production.Nuclear attenuation is very strong at small Q 2and gradually decreases with Q 2.The e?ect is particularly dramatic for the heavy nuclei (Ca,P b ),and leaves no doubts the E665observed the onset of CT.The predicted Q 2dependence of nuclear transparency for the forward coherent production on nuclei T (coh )A =(dσ(coh )A /A 2dσN )|t =0is

shown in Fig.5.We predict a rise of T (coh )A with Q 2towards T (coh )A =1for vanishing FSI.

The predicted Q 2dependence of the coherent production cross section relative to the cross section for the carbon nucleus is presented in Fig.6.In the regime of vanishing FSI,

R (CT )coh (A/C )=

12σA 12R ch (A )2,(18)which gives R (CT )coh (Ca/C )=1.56and R (CT )

coh (P b/C )=3.25.Here R ch (A )is the charge

radius of a nucleus.The observed growth of the P b/C ratio with increasing Q 2gives a solid evidence for the onset of CT.

The (approximate)A αparametrization is a convenient short-hand representation of the A -dependence of nuclear cross sections,although the so-de?ned exponent αslightly depends

on the range of the mass number A used in the?t.Then,Eq.(12)predicts

thatα

inc

(Q2)

tends to1from below,as Q2increases.In the limit of vanishing FSI Eqs.(14,18)predict σcoh～A4/3,so thatαcoh(Q2)tends to≈4

1GeV .(19) In the low energy limit of l f?R A we have an instantaneous formation of the?nal-state hadron,which then attenuates with the free-nucleon cross section and CT e?ects are absent.

At high energies l f>R A,the formation of the observed hadron takes place behind the nucleus,and CT becomes possible.For the observation of the fully developed CT one needs

ν～>(3?4)·A1/3GeV.(20) CT e?ects already start showing up,though,if l f～>l int,where the interaction length,or the mean free path,equals

l int=1

σtot(V N) .(21)

Therefore,purely kinematically,CT e?ects are within the reach of CEBAF experiments after the8?12GeV energy upgrade.Notice,that the formation length l f does not depend on the photon’s virtuality Q2.

On the other hand,the coherence length l c tells at which distance from the absorption point the pointlike photon becomes the hadronlike qˉq pair.If l c～>R A,then the whole thickness of the nucleus contributes to attenuation of the qˉq pair.Changing the virtuality of the photon Q2and/or reducing the photon’s energyν,one can make l c?R A.In this case,the incoming photon is absorbed approximately uniformly over the volume of the target nucleus,and attenuation of the produced qˉq pair will take place over half of the total thickness of the nucleus.In the practically interesting cases of Q2?m2V for the light mesons,or in the real and virtual photoproduction of heavy quarkonia J/Ψ,Υwe have l c?l f.If l f～>R A,but l c?R A,then nuclear transparency equals

T A=1

2

σ(ρ)t(b,z)]|γ? 2

l c

=

Q2+m2V

3.2Quantum evolution and energy dependence of FSI.

If l f

The resulting predictions for the energy dependence of nuclear transparency T A for the production of di?erent vector mesons are shown in Figs.9,10.The salient features of T A are [9,21,22]:

?At small Q2,nuclear transparency for theρ0and the J/Ψdecreases with energyν, starting at the value given by Eq.(22)and levelling o?at the value given by Eq.(12).

This decrease is due to the increase of the coherence length l c,discussed in section3.1.?At larger Q2,the trend changes:T A?rst increases with Q2,and then decreases for the same reason of the rise of the coherence length l c.In agreement with Eqs.(25,26),the larger is Q2,the higher is the energyνat which the levelling o?of T A takes place.?For theΥproduction,nuclear transparency T A starts increasing with energy at all values of Q2,in close similarity to the J/Ψproduction at large Q2,and in agreement with our conclusion that nuclear attenuation scales with(Q2+m2V),see section2.5.

For instance,we predict T A(J/Ψ,Q2=100GeV2)≈T A(Υ,Q2=0).

?The radial excitationsΨ′,Υ′have larger size,and larger free-nucleon cross section thereof,than the ground states J/ΨandΥ,respectively.Nonetheless,the radial excitations are predicted to have weaker nuclear attenuation,which is a completely counterintuitive result.

?At last but not the least,for theρ0production we predict very rich pattern of theνand Q2dependence precisely in the kinematical range of CEBAF.CT e?ects are large and can readily be observed at CEBAF.Theρ′production will be treated in section

3.7.

These properties of nuclear transparency can best be understood in terms of the interplay of CT with the node e?ect.

3.3CT and the node e?ect:antishadowing phenomenon.

The wave function of the radial excitation V′(2S)has a node.For this reason,in the V′production amplituide there is the node e?fect-cancellations between the contributions from r below,and above,the node.The productσ(r)Ψγ?(z,r)acts as a distribution,which probes the wave function of the V(1S)and V(2S)states at the scanning radius～r S[9,10,12] and the node e?ect evidently depends on the scanning radius r S,see Fig.2.

If the node e?ect is strong,even the slight variations of r S lead to an anomalously rapid variation of the V′(2S)production amplitude,which must be contrasted to the smooth Q2

and r S dependence of the V(1S)production amplitude.Evidently,the stronger is the node

e?ect and the smaller is the V′(2S)production amplitude,the higher is the sensitivity to the model for the V′(2S)wave function,and in some cases only?rm statement will be the fact of the strong suppression of the V′(2S)production.

For the real photoproduction of theΨ′,the calculations in[21]gaveσ(γN→Ψ′N)/σ(γN→J/ΨN)=0.17,which is in excellent agreement with the NMC result0.20±0.05(stat)±0.07(syst)for this ratio[29].In this case the node e?ect is already rather strong for the fact that the scanning radius r S(Q2=0)is rather close to the J/Ψradius R J/Ψ.For the Υ′,the scanning radius is substantially smaller than RΥ,which is realtively large for the small strong couplingαS(RΥ).For the light mesons,the scanning radius r S is larger and, at small Q2,the node e?ect is much stronger,see section3.7.

Because r F SI is larger than r S,the node e?ect in the strength of FSI given by Eq.(15) becomes stronger.For the J/Ψ,one?nds the overcompensation: V|σ(r)2|γ <0,which leads toΣV<0and to the antishadowing phenomenon T A>1shown in Fig.10.For the Υ′,we?nd the undercompensation: V|σ(r)2|γ >0,which leads toΣV>0and to the shadowing T A<1.None the less,the node e?ect shows up:for theΥ′with its large radius, nuclear attenuation is weaker than for theΥ!With increasing Q2,when r F SI?R V,the node e?ect becomes negligible,and the V(1S)and V′(2S)states will have identical nuclear attenuation,see Fig.10.

3.4The interplay of CT,of the node e?ect and of quantum evo-

lution

Of course,theΨ′has a larger radius and larger free-nucleon cross sectionσtot(Ψ′N)～(2.5-3)σtot(J/ΨN).How come,then,that in the real photoproductionΣΨ′<0and we?nd the antishadowing of the strongly interactingΨ′alongside with shadowing for the J/Ψ?

Although the above derivation of antishadowing was(deceptively)simple,still another look at antishadowing and the variation of energy dependence of nuclear transparency with

Q2is in order[9].Let us consider for simplicity the J/Ψ,Ψ′system.The numerator of the strength of FSIΣV can be expanded in terms of the complete set of intermediate states |i >of charmonium

V|σ(ρ)2|γ? = i V|σ(ρ)|V i V i|σ(ρ)|γ? .(27)

In terms of this expansion,antishadowing of the photoproduction of theΨ′comes from the destructive interference of the two dominant intermediate states:the direct,VDM-like rescattering

γ?→Ψ′→Ψ′(28) and the o?-diagonal rescattering

γ?→J/Ψ→Ψ′(29) (there is a small contribution from other intermediate states too).

Then,for theΨ′production,the strength of FSI is given by

M(γ?N→J/ΨN)

ΣΨ′=σtot(Ψ′N)+M(J/ΨN→Ψ′N)·

Because of the interplay of CT and the node e?ect,we have M(γ?N→J/ΨN)/M(γ?N→Ψ′N)?1.For the same interplay of CT and the node e?ect,there is an overcompensation in the amplitude of the o?-diagonal transition

M(J/ΨN→Ψ′N)<0,(31) and numerically this amplitude is not very small,M(J/ΨN→Ψ′N)～?σtot(J/ΨN). Consequently,the second,negative valued,o?-diagonal term inΣΨ′takes over theσtot(Ψ′N)

(the higher intermediate states also slightly contribute to the antishadowing e?ect).

What happens with increasing Q2is very simple:The scanning radius r S decreases with Q2and the node e?ect become weaker,the ratio of amplitudes M(γ?N→J/ΨN)/M(γ?N→Ψ′N)decreases with Q2and tends to approximately unity,whereasσtot(Ψ′N)and M(J/ΨN→Ψ′N)do not depend on Q2.As a result,the o?-diagonal contribution in Eq.(30)becomes small,and at large Q2the antishadowing ofΨ′changes to the shadowing.

Similarly,for the J/Ψproduction

M(γ?N→Ψ′N)

ΣJ/Ψ=σtot(J/ΨN)+M(J/ΨN→Ψ′N)·

1

2ν≈

G2A(κ12).(36)

M(γ?N→Ψ′N)

M(γ?N→Ψ′N)

ΣJ/Ψ=σtot(J/ΨN)+M(J/ΨN→Ψ′N)·

Evidently,at low energy such thatκ12R A～>1,i.e.,at l f?R A,the nuclear form factor vanishes G A(κ12)2?1.Only the diagonal contributions toΣV survive,and CT e?ects which come from the o?-diagonal terms inΣV,vanish at low energy.For instance,for the Ψ′one starts with the shadowing T A<1,which with increasing energy and the opening of the o?-diagonal channels,changes to the antishadowing.For the J/Ψandρ0,at small energy there is a competition of CT e?ect which rises with energy,and of the e?ect of growth of the coherence length.The latter takes over at small Q2,whereas at larger Q2the opening of the o?-diagonal transitions leads to a rapid near-threshold rise of nuclear transparency. For theρ0production,nuclear transparency T A is a lively function of energyνand Q2in precisely the kinematical region accessible at CEBAF.

3.5Measuring the J/Ψ-nucleon cross section at CEBAF

The smallness of the o?-diagonal rescattering in the real photoproduction of the J/Ψ,see Eqs.(33),(37),leads to an important prediction[9,22]that nuclear attenuation allows to evaluateσtot(J/ΨN)using the conventional VDM formulas.This suggestion was carried over in an analysis[34]of the data[29,30]on the coherent photoproduction on nuclei with the resultσtot(J/ΨN)～(5-7)mb.The CEBAF experiments will allow measurement of this cross section at low energy,although being very close to the threshold requires a good understanding of the Fermi-smearing e?ects.

For theΨ′photoproduction at low energies,the o?-diagonal transitions(29)are non-negligible even close to the threshold,and for theΨ′the VDM prediction for nuclear shad-owing breaks down completely:T A is always larger than the VDM prediction.

3.6The coherence and formation lengths revisited:Vanishing

nuclear shadowing in inclusive DIS at CEBAF coexists with lots of CT in exlusive production at CEBAF

Above we have repeatedly emphasized that the onset of CT is entirely controlled by the formation length l f,which does not depend on Q2.The onset of CT is quanti?ed by Eqs.(36,37),in which the formation length enters via the nuclear form factor G A(κ12), whereκ12is the longitudinal momentum transfer in the o?-diagonal transition V1N→V2N. At large Q2we have l c?l f and much larger longitudinal momentum transferκEq.(26) in the transitionγ?N→V N.However,becauseκ?κ12,this momentum transfer is approximately the same for all intermediate states.At l c?R A,the corresponding overall phase factor simply drops out from the incoherent production cross section.Neither does this phase factor a?ect T A dramatically in the transient regime of l c～R A,see Eq.(25). In the coherent production on nuclei,the major e?ect of the momentum tranferκis that the nuclear production amplitude acquires the overall factor G A(κ),which signi?cantly suppresses the coherent production amplitude but does not a?ect the nuclear attenuation properties.

As we discussed in section2.2,the vector meson production on the free nucleon probes the gluon distribution in the target proton.Stretching the so-called factorization theorems, one can be tempted to conclude that nuclear attenuation in the production of vector mesons is given by the nuclear shadowing of gluon structure function[26].Indeed,exclusive electro-production of vector mesons is the typical di?raction dissociation of the photonγ?N→XN, and virtual di?ractive transitionsγ?→X→in the nuclear forward Compton scattering

amplitude are precisley the source of nuclear shadowing[15].But,subtle is the nuclear

shadowing!

The mere de?nition of the shadowed nuclear parton distributions is only useful provided that the shadowing term by itself satis?es the conventional evolution equations.To a certain

approximation,this is the case[17].However,there are no theorems on the universality of these shadowing corrections in all hard scattering processes and shadowing corrections

may defy the factorization theorems[16].Here we present simple arguments,essentially of

kinematical origin,why the factorzation theorems must be taken with the grain of salt.

The contribution of the intermediate state X to the nuclear Compton scattering ampli-

tude contains the excitationγ?N→XN on one nucleon and the de-excitation XN→γ?N on another nucleon.In both transitions there is a longitidunal momentum transferκ

Eq.(26).Consequently,the corresponding contribution to nuclear shadowing in the struc-

ture function will enter with the suppression factor G A(κ)2.The onset of nuclear shadowing requires l c～>R A,so that the larger is Q2,the higher energy is required for the onset of nucler shadowing.In the opposite to that,nuclear attenuation and CT e?ects in the nu-

clear electroproduction of vector mesons only requires l f～>l int,R A,and this condition does not depend on Q2and does not require l c～>R A.For instance,there will be no nuclear shadowing in the inclusive electroproduction on nuclei in the kinematical range of the CEBAF experiments,but lots of CT e?ects in the exclusive production of vector mesons on nuclei at CEBAF.We wish to emphasize this simple,but important,point in view of the opposite claims made to this e?ect by Frankfurt and Strikman at this Workshop.

3.7Anomalous electroproduction of radial excitationsρ′,φ′:CE-

BAF’s new window at CT

For the light vector mesons,at small Q2the scanning radius r S～R V,and there is an exciting,and most likely,possibility of the overcompensation already in the free-nucleon production amplitude:M= V′|σ(r)|γ? <0.Theρ′production on nuclei is indispensable for testing the node e?ect and Q2dependence of the scanning radius r S,because nuclear attenuation gives still another handle on the scanning radius[9,13,27].For the sake of simplicity,we discuss the quasielastic(incoherent)ρ′production on nuclei assuming that l f～>R A.Extension to lower energies and to the coherent production is straightforward and the prediction[13,27]of the anomalous Q2and A dependence persists in these cases too.

The A-dependence of the node e?ect comes from the nuclear attenuation exp[?1

2σ(r)T(b) |γ? .The possibility of

the A-dependent node e?ect e?ect in hadronic di?raction production on nuclei hA→h?A was pointed out in[35].

Firstly,consider the Q2dependence of theρ′/ρ0ratio on the free nucleon.Increasing Q2 and decreasing the scanning radius r S,one will bring theρ′production on the free nucleon to the exact node e?ect,and theρ′/ρ0ratio takes the minimum value at a certain?nite Q2, see Fig.11.Because of the r-dependence of the attenuation factor,in the nuclear amplitude the node e?ect will be incomplete.Consequently,as a function of Q2,nuclear transparency T A will have a spike T A?1at a?nite value of Q2[9].

Secondly,consider theρ′production on nuclei at a?xed value of Q2such that the free nucleon amplitude is still in the overcompensation regime.Increasing A and enhancing the importance of the attenuation factor exp[?1

to the nearly exact compensation regime.Therefore,theρ′/ρ0production ratio,as well as nuclear transparency for theρ′production,will decrease with A and take a minimum value at a certain?nite A.With the further increase of A,the undercompensation regime takes over,and we encounter very counterintuitive situation:nuclear transparency for theρ′is larger for heavier,more strongly absorbing nuclei!This situation is illustrated in Fig.12a and must be contrasted with a smooth and uneventful decrease of transparency for theρ0 production on heavy nuclei.

With the further increase of Q2one enters the pure undercompensation regime for all the targets.Nuclear undoing of the node e?ect enhances M A and nuclear transparency T A, whereas the overall attenuation factor exp[?1

ergy,and numerically is much stronger than for theΨ′.Nuclear attenuation of theφ′(1680) is weaker than for theφ0starting already at low energy(Fig.14).Because of the large mass and small radius of theφ0,nuclear transparency for theφ0production only weakly depends on Q2at CEBAF,but for theφ′(1680)production the predicted Q2dependence is quite strong(Fig.15)and can easily be measured at CEBAF.Here the larger Q2and smaller scanning radius r S predict increasing attenuation of theφ′(1680).However,for the lead target we expect the onset of increasing nuclear transparency at already moderate Q2～>1GeV2.

Few more comments about the possibilities of CEBAF are worth while.Because of the strong suppression of theρ′/ρ0andφ′/φ0production ratio by the CT and node e?ects,the high luminosity of CEBAF is absolutely crucial for high-statistics experiments on theρ′,φ′production.Notice,that the most interesting anomalies in the A and Q2dependence take place near the minimum of theρ′production cross section.Furthermore,the observation of theρ′production requires detection of its4-pion decays,and here one can take advantage of the CLAS multiparticle spectrometer available at CEBAF.

4FSI and nuclear transparency in A(e,e′p)scattering 4.1Multiple-scattering expansion for the nuclear spectral func-

tion

We are interested in A(e,e′p)scattering at large Q2>(1-2)GeV2,when the struck proton has the kinetic energy T kin≈Q2/2m p～>(0.5-1)GeV.Such a proton has the free nucleon total cross sectionσtot(pN)≈40mb.The corresponding mean free path in the nuclear medium l int～1.5f is short and implies strong FSI and strong nuclear attenuation of struck protons.At large Q2this FSI is expected to vanish by virtue of CT.One needs?rst a reliable formalism for description of FSI of the struck proton,and the coupled-channel generalization [7,8,36,37]of the Glauber’s multiple scattering theory[38]provides the necessary framework.

The quantity measured in the ideal A(e,e′p)scattering is the spectral function S(E m, p m) as a function of the missing energy E m and missing momentum p m=(p m,z, p⊥),for the kinematics see Fig.16.The discussion greatly simpli?es if the measured cross section is integrated over the su?ciently broad range of the missing energy E m,when the closure can be applied[7,8,36].We assume this is the case.If the plane-wave impulse approximation (PWIA)were applicable,then one would have have measured the single-particle momentum distribution n F( p m)([39],for the recent review see[40]),which is related to the one-body nuclear density matrixρ1( r, r′)as

dσA∝n F( p m)=1 Array

Z dE m S(E m, p m)=

d r′d rρ1( r, r′)exp[i p m( r′? r)]·exp t( b,max(z,z′))ξ( ?)

·exp ?12(1+iαpN)σtot(pN)t( b′,z′) ,(39)

where r=( b,z), r′=( b′,z′), ?= r? r′,αpN denotes the ratio of the real to imaginary parts of the forward proton-nucleon scattering amplitude and

ξ( ?)= d2 q dσel(pN)

2σtot(pN)αpN n A(b,z)(z?z′)]the spectral function S(k⊥

,

k z)is probed at a shifted

value of the longitudinal momentum with

k z?p m,z=?p m,z～1

Z dE m dp m,z S(E m,p m,z,p⊥)=∞ ν=0W(ν)n(ν)( p⊥).(42) Here the p⊥-distribution in theν-fold rescattering n(ν)( p⊥)equals

n(ν)( p⊥)= d2 s Bνs2 n F( p⊥? s)(43) where B denotes the di?raction slope for elastic pN scattering,dσel(pN)/dt∝exp(?B|t|),

and

W(ν)=1

ν!

(44)

gives the probability of havingνelastic rescatterings.

?Thirdly,FSI introduces the attenuation e?https://www.wendangku.net/doc/f319048265.html,ly,whereas in the PWIA one has d3 p m n F(p m)=1,with allowance for FSI

T A=

1

A dzd2 b n A( b,z)exp ?σin(pN)t( b,z) <1.(45)

In the completely integrated nuclear spectral function(45),nuclear attenuation is given byσin(pN)[7].This result is self-obvious:elastic rescatterings only de?ect,but do not absorb,the struck proton,and the e?ect of de?ection is not relevant for the full4πacceptance.

?On the other hand,the forward peak of f A(p⊥)at p⊥=0is dominated by W(0),which is also given by Eq.(45)but withσin(pN)substituted byσtot(pN).In this case elastic rescatterings also contribute to the observed attenuation.

The further discussion will be centered on:(1)how CT a?ects the integrated T A and forward(the in-parallel kinematics)W(0)nuclear transparency;(2)why the onset of CT in A(e,e′p)scattering is so slow;(3)the theoretical interpretation of the nonobservation of CT in the NE18experiment;(4)the discussion of feasibility of CT studies at CEBAF.

4.2FSI e?ect dominates at large transverse missing momenta Measuring the spectral function at large missing momenta p m is of great interest,because large p m are expected to give a direct handle on the short-range correlations(SRC)of nucleons,which is widely discussed for30years since seminal works by Gottfried and Sri-vastava[43].Our important?nding[36]is that in the transverse kinematics,the FSI e?ect completely takes over the SRC e?ect.

The global e?ects of rescatterings are summarized in Table1.Here we present,for dif-ferent values of Q2,the fractions P(ν)=W(ν)/T A of theν-fold rescatterings,the nuclear transparency W(0)for quasifree knockout in parallel kinematics p⊥=0,the total trans-parency T A and the average number of secondary rescatterings ν .The Q2dependence of1?W(0)is reminiscent of the energy dependence ofσtot(pN),which is nearly?at at the kinetic energy T kin≈Q2/2m p～>0.5GeV[42].The Q2dependence of1?T A repeats the energy dependence ofσin(pN),which rises rapidly up to T kin～1.5GeV and then ap-proximately levels o?[42].The Q2dependence of the di?erence of the total and forward transparency T A?W(0)and of the multiplicity of secondary rescatterings ν repeats the energy dependence of the elastic cross sectionσel(pN)=σtot(pN)?σtot(pN),which steadily decreases with T kin[42].The di?erence between W(0)and T A is a convenient measure of the strength of rescatterings,and in the CEBAF range of Q2this di?erence is very large.

For our numerical estimates of the p⊥-distribution we use a simple,yet realistic,param-eterization of the SPMD[44]

n F( k)∝exp ?5k2F +?0exp ?5k2F ,(46)

where?=0.03and the value of the Fermi-momentum k F has been taken from Ref.[39]: k F(C)=221MeV/c and k F(P b)=265MeV/c.This parameterization is consistent with the results from the y-scaling analysis[45].The steeply decreasing?rst term corresponds to the mean-?eld component of the SPMD,while the second term describes the SRC tail. In Fig.17we present our predictions for f A( p⊥)/T A for the12C(e,e′p)and208P b(e,e′p) reactions at Q2=1,2,4(GeV/c)2,respectively.The FSI contribution to f A( p⊥)starts to dominate over the PWIA component P(0)n F(p⊥)already at p⊥～>350MeV/c for carbon and p⊥～>300MeV/c for lead,which is precisely the region thought of being dominated by the SRC tail in the single-particle momentum distribution.

An obvious signal of rescattering is the multinucleon emission(MNE).Rescatterings are not imperative for MNE,but rescatterings leading to p⊥～>k F are necessarily followed by MNE.The contribution of rescatterings to MNE is characterized by the average number of secondary rescatterings ν(p⊥) ,which is very large(Table1).Evidently,the strength of MNE coming from the rescattering mechanism must be the same in the longitudinal and transverse cross sections,which is a strong prediction.Presently,it can not be tested for the lack of the experimental data on large transverse missing momenta p⊥～>k F taken at Q2～>(1-2)GeV2,which is the domain of the forthcoming CEBAF experiments.This FSI driven MNE contributes signi?cantly to the spectral function at large missing energy E m. Namely,the contribution?E m to the missing energy E m from the kinetic energy of the recoil nucleons can be estimated as

?E m≈ s25m p+p2⊥

?E m

dE m∝exp

8πσel(pN)

2Bm p≈

?1

16π2 d2 kφd(p m,z, p⊥? k)exp

p max d2 p⊥φd( p⊥)2(51)

The attenuation e?ect T d<1comes from the interference of the undistorted and rescattering terms in(50).If R2d p2max?1,but Bp2max?1,which is the case for the NE18,then

1?T d～σtot(pn)

which is about twice the Glauber shadowing e?ect in theσtot(Nd).Our prediction for T d is shown in Fig.18and the agreement with the NE18data[4]is good.

At still larger values of p⊥,the spectral function will be entirely dominated by square of the rescattering term in(50),cf.Eqs.(42,43),

σel(pn)

f d(p m,z,p⊥)～

D(NE18)dE m dp m,z dp⊥S P W IA(E m,p m,z+?p m,z,p⊥)(54)

In order to eliminate the spurious e?ect in T A(NE18)due to the FSI generated asymmetry of the nuclear spectral function,we strongly advocate including the e?ect of the shift(41) in the calculation of the PWIA cross section.It was not included in the NE18analysis.The e?ect of the shift(41)on T A depends on the p m,z-acceptance.It vanishes for the wide p m,z-acceptance.For the narrow acceptance centered at p m,z=p?,the longitudinal asymmetry e?ect can be evaluated as

(?p m,z+p?)2?(p?)2

T A(?p m,z)

2·

protons.Since the di?raction slope B rises with the proton energy,the higher is Q2the larger is the fraction of the elastically rescattered struck protons included in T A(NE18). In Fig.18we present our predictions[38]for T A,W(0)and T A(NE18)in which the p⊥integration for heavy nuclei is extended up to p max=250MeV/c as relevant to the NE18 situation[4].We?nd very good quantitative agreement with the NE18data.The principle conclusion from this comparison is that there is no large signal of CT in the A(e,e′p) scattering at Q2≤7GeV2.

For the12C target we also show the e?ect of CT which is still small even at the largest Q2of the NE18experiment(Fig.18b),which must be contrasted with large CT e?ect in the E665experiment.What makes the onset of CT in exclusive vector meson production and(e,e′p)scattering so much di?erent?

5The onset of CT in A(e,e′p)scattering

5.1The ejectile state has a large size

The(e,e′p)scattering can be viewed as an absorption of the virtual photon by the target proton,which leads to the formation of the ejectile state|E ,which then is projected onto the observed?nal state proton.In the A(e,e′p)scattering,this ejectile state propagates in the nuclear medium before evolving into the observed proton.The strength of FSI depends on what is the transverse size and the interaction cross section of the ejectile state.

In the exclusive production of vector mesons,the wave function of the ejectile state equals[9,21]

ΨE(r)∝σ(r)Ψγ?(r)(56) Because of CT,in this case the ejectile wave functionΨE(r)has a hole at r=0,but at large Q2the decrease of the wave function of the photon(5)takes over andΨE(r)has a small size～r S.Notice,that in terms of the ejectile state|E the strength of FSI can be rewritten asΣV= V|σ(r)|E / V|E .

The wave function of the ejectile state in the quasielastic scattering of electrons is much simpler.It can be found in any textbook in quantum mechanics and/or nuclear/particle physics.Indeed,the charge form factor G em(Q)of the qˉq”proton”can be written as(here the r-plane is normal to the momentum transfer Q):

G em(Q)= p|E = dzd2 rΨ?p(z,r)ΨE( r,z)= dzd2 rΨ?p(z,r)exp i

Qz Ψp( r,z).(58)

2

Because|ΨE( r,z)|2=|Ψp( r,z)|2,the ejectile wave packet has the transverse size identical to the size of the proton[46,13].This result is readily generalized to the three-quark proton

and to the relativistic lightcone formalism.The often made statement that the ejectile has √

a small size∝1/