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Natural Color Transparency in High Energy (p,pp) Reactions

arXiv:hep-ph/9305317v1 25 May 1993Natural Color Transparency in High Energy (p,pp) ReactionsB.K. JenningsTRIUMF, 4004 Wesbrook Mall, Vancouver, B.C., V6T 2A3, CanadaandG.A.

MillerDepartment of Physics, FM-15, University of Washington, Seattle, WA 98195, USAABSTRACTNew parameter free calculations including a variety of necessary kinematic and dynamiceffects show that the results of BNL (p, 2p) measurements are consistent with the expec-tations of color transparency. (submitted to Phys.

Lett. B)1

The anomalously large transmission of a hadron in the nuclear medium following orpreceding a hard interaction is commonly referred to as color transparency[1, 2]. Thistopic has been actively studied theoretically [3]-[9].

Color transparency depends on theformation of a small-sized wavepacket or point-like configuration PLC in a high momen-tum transfer quasi-elastic reaction. The effects of color screening or color neutrality [10]suppress the interaction between the PLC and the nuclear medium.

At the present ex-perimental energies the PLC expands as it moves through the nucleus and therefore doesinteract. This expansion is treated in Refs.

[3, 8, 9, 11, 12].Color transparency is under active experimental investigation at BNL [13, 14] and atSLAC[15, 16]. The (p, 2p) experiment of Carroll et al[13] finds a transparency T (ratioof nuclear to hydrogen cross section per nucleon after removing the effects of nucleonmotion) with an oscillatory pattern.

T increases as the beam momentum increases from5 to 10 GeV/c but then decreases. The SLAC experiment is unpublished at this time, buta preliminary report [16] found no effect of color transparency (CT) in (e,e’p) reactionsfor Q2 between 1 and 7 GeV2.

However, the stated systematic errors are about ± 10%which are large compared to the size of the predicted CT effects.The purpose of this note is to include all of the known necessary kinematic anddynamic effects: a proper treatment of the longitudinal (parallel to the virtual photon orincident proton momentum) component of the momentum of the detected protons[17, 18];computing the measured experimental observable which involves an integration over thetransverse momentum of the struck proton; including the effects on interference betweenPLC and non-PLC configurations produced in proton-proton elastic scattering [6, 7]; and,a realistic treatment of the baryonic components of the PLC[12]. Thus our aim here isto show that including the three kinematic and dynamic effects mentioned above alongwith the color transparency of ref.

[12] leads to a natural explanation of the existing BNLdata. No free parameters are used in the present work.We start the analysis by recalling the mechanisms proposed to understand the (p,pp)reaction.

The suggestion of Ralston and Pire [7], is that the energy dependence is caused2

by an interference between a hard amplitude which produces a PLC, and a soft one (blob-like configuration BLC) which does not. The Ralston and Pire idea is that the BLC is dueto the Landshoffprocess[19].

Another mechanism with a similar effect is that of Brodskyand de Teramond [6] in which the two-baryon system couples to charmed quarks (thereis a small (6q) and a BLC which is a (6q,c¯c) object) The Brodsky-de Teramond idea ismotivated by the fact that the mass scale of the rapid energy variation in ANN[20] andin the measured transparency matches that of the charm threshold.Both of the mechanisms of [7] and[6] are well motivated, but imprecisely understood.For example, the nature of the Landshoffterm is uncertain. At high energies one expectsthat this component is suppressed by a Sudakov factor [21].

At present energies, theenergy dependence and phase of the term are not well determined, and the size of theobject produced by the Landshoffprocess is not well known. See e.g.

ref. [22].

Never-theless, the Sudakov effects can be expected to a set of configurations with a range ofdifferent sizes. Similarly even if the details of the c¯c production amplitudes are not yetwell- established, threshold effects will naturally lead to mixtures of BLC and PLC.

Thus,it’s natural to discuss high Q2 elastic proton-proton scattering in terms of configurationsof different sizes. Separating the contributing configurations into two, a PLC and a BLCis only a simple first step.The two ideas about the BLC can be used to qualitatively explain the oscillationsobserved in the (p,pp) data, but do not quantitatively reproduce features of the data Ref.

[23, 11] when combined with a proper treatment of the expansion[11]. The agreement isnot quantitative because including the non-zero absorption of the expanding PLC and thenon-complete absorption of the BLC tends to make the two terms similar and weakensthe interference effects.Another relevant effect is to properly account for the correct momentum of the de-tected protons.

Consider for example, the (e,e’p) reaction in which the virtual photonhas the momentum ⃗q. Suppose the detected proton has momentum ⃗p with ⃗p = ⃗q + ⃗k.For quasi-elastic kinematics ⃗p = ⃗q and ⃗k = ⃗0.

Fermi motion or final state interactions3

can lead to a non-zero value of ⃗k. This is old news.

However the non-zero componentsof ⃗k in the direction of ⃗q (kz) have been shown to give a large numerical effect [17]. Adetailed study of this “Fermi motion” effect was made by Frankfurt et al [18].

The mo-mentum ⃗k is sometimes called the momentum of the struck nucleon (and is that quantityif the plane wave Born approximation is valid ). We also use the term “momentum ofthe struck proton”, but only as an abbreviation.

For the (p,pp) reaction the momentumof the detected protons can also be described in terms of the momentum of the strucknucleon [13]. Indeed, the data are presented in terms of bins of kz, the component of ⃗kparallel to the beam direction.

This is especially important for the (p,pp) reaction be-cause the proton- proton scattering cross section varies approximately as s−10 and heres = 2M2 + 2MELAB −2kzPLAB ≈2MPLAB(1 −kz/M) where M is the proton mass. Ref[18] examined the (p,pp) reaction, including the effects of non-zero values of kz in theircalculation, and obtained qualitative agreement with the 10 GeV data, but not with datataken at the other energies.

Agreement was also obtained by Kopeliovich[24] for someenergies. Neither group included any effects of interference between PLC and BLC.Our calculation of the nuclear (p,pp) cross sections uses the quantum mechanicaltreatment of color transparency developed in ref.

[8, 9, 12]. In that approach the time forthe PLC to expand depends on the masses of the baryonic components of the PLC.

Inearly work [8, 9] only a single average state of mass M1 was used. Then the time for thePLC to expand is2PLAB(M+M1)1(M1−M).

Recently we [12] included the effects of the continuousbaryon spectrum by using measured proton diffractive dissociation and electron deepinelastic scattering data to constrain the baryon masses. Our discussion of the formalismshall be brief and concentrate on the newer kinematic and dynamic aspects.

For detailssee Refs. [9, 11, 12].To be definite, consider the (e,e’p) high momentum transfer process in which a photonof three-momentum ⃗q is absorbed and a nucleon of momentum ⃗p leaves the nucleus.

Asusual, q2 = −Q2. Consider knockout from only a single shell model orbital, denoted byα.

The observable cross sections are computed by making incoherent sums over all the4

occupied orbitals.Let the amplitude be defined as Mα, which can be described in terms of a seriesinvolving multiple wavepacket (PLC) -nucleon scatterings. The first term is the planewave Born term Bα in which the wavepacket undergoes no interactions.

If full colortransparency is obtained, this is the only term to survive. The first correction to theBorn term is the scattering term or second term denoted by STα.

To the stated orderwe have: Mα = Bα + STα. The Born term is given byBα = ⟨⃗p|TH(Q)|α⟩= F(Q2)⟨⃗p −⃗q|α⟩,(1)in which TH(Q2) is the hard scattering operator.

Specific effects of spin are ignored hereand throughout this work.The second term is denoted bySTα = ⟨⃗p|U G TH(Q)|α⟩,(2)where G is the Green’s operator for the emerging small object (PLC). This object is awave packet of which the nucleon is just one component.

The operator U represents theinteraction between the ejected baryon and the nuclear medium. The nuclear interactioncan change the momentum of the ejectile and also excite or de-excite the internal degreesof freedom.Our approach is to treat the Green’s function G as a sum of baryonicpropagators each denoted by m. The eikonal Green’s function for the emerging baryonis:G(Z, Z′) =XmGm(Z, Z′) =Xmθ(Z −Z′)eipm(Z−Z′)2ipm,(3)with ⃗p 2m = ⃗p 2 + M2 −M2m.

Here M is the nucleon mass, Mm the mass of the baryonicstate. Our notation is that p = |⃗p| and pm = | ⃗pm|.

Terms beyond the first order in Uare included by exponentiation, which is an excellent approximation for our applications[25].The work of ref. [9] involves a simple model in which the operator U gives only one5

excited state (m=1) when acting on a nucleon. In that modelSTα = −F(Q2)Zd2BdZρ(B, Z) e−ipZ(2π)3/2Z Z−∞dZ′e−ip(Z′−Z)σeff2 (Z, Z′)eiqZ′⟨⃗B, Z′|α⟩, (4)where the ρ(B, Z) is the nuclear density, andσeff(Z, Z′) ≡σ1 −ei(p−p1)(Z′−Z),(5)and σ is the proton-nucleon total cross section.

This is eq (32) of ref. [9].Now we can exhibit the importance of the momentum (kz) of the struck nucleon.

Wetake p = q + kz for components parallel to the direction of ⃗q. Change the integrationvariable Z′ to D via Z′ = Z −D.

ThenSTα = −F(Q2)Zd2BdZσρ(B, Z) e−ikzZ(2π)3/2Z ∞0dDeikzD 1 −e−i(p−p1)D⟨⃗B, Z′|α⟩. (6)The effect of kz appears in two places.

The e−ikzZ factor is the same as in standardBorn or Distorted Wave Born calculations (DWBA or Glauber optical model). The eikzDfactor involves a modification of the color transparency physics.

To see this note thatthe real part of STα dominates the numerics. Then cos(kzD) −cos ((kz −p + p1)D) ≈D22 (p −p1) (p −p1 −2kz) for small D. Since p > p1 a positive value of kz reduces thisterm and increases the transparency for any given value of p. This latter effect doesnot occur in the color transparency of ref.

[3], but would occur in models in which thebaryon-nucleon interaction is treated as a finite-dimensional matrix[24, 26].The above paragraph is meant as a simple explanation of a numerical effect that issurprisingly large. The computations of (p, pp) reactions are more involved since there isone incident proton wave function and two outgoing ones.

Furthermore the amplitudesfor the production of both the PLC and BLC must be taken into account. Howeverthe qualitative effect of including non-zero values of kz (here z is the direction of thebeam proton) is similar to the electron scattering case.

Note also that our more realisticcalculations [12] replace the single mass M1 by an appropriate distribution of masses.The next step is to discuss the observables measured in ref. [13].Let the four-momentum of the target proton be denoted as (M,⃗k).

The transparency T is defined asa ratio T = dσ/dσB with6

dσ =Z kbkadkzZ Zdkxdky(dσA/dt)(s)(7)anddσB =Z kbkadkzZ Zdkxdky(dσB/dt)(s)(8)in which the superscript A denotes the nuclear cross section divided by the numberof target protons and the subscript B denotes the same quantity but computed in theplane wave Born approximation. In the experiment the integrations over the transversemomenta kx, ky were limited to about 250 MeV/c.

This corresponds to almost all of theprobability so we integrate over all kx, ky in our calculations. The integration over thetransverse components of ⃗k reduces the Glauber DWBA result for T by about 30%.

Thedata of ref. [13] are presented in terms of T for each (ka, kb).We now present the results.

The calculations for an Al target (three beam momentaand four bins of kz) are shown in Fig 1. The solid curves show the effect of CT usingthe complete power law form of the distribution of baryon masses as in ref.

[12] anddashed curves show the results of using Ref. [9] with a value of M1 = 1550MeV .

Thisvalue gives small enough color transparency at low Q2 so as to be consistent with theNE-18 data. The Ralston-Pire parameterization of dσ/dt for the free protons is used herealong with their separation of the BLC and PLC terms.

Details of the implementationof this are to be found in ref [11]. The use of the corresponding Brodsky-de Teramondmodel for the PLC-BLC interference would lead to similar results for the energy rangewe consider here [11, 12].

The data are from ref. [13].

The target-dependent and targetindependent uncertainties in the normalization of T are about 10% and 25% respectively[13]. We multiply the central values of the Al, Cu, and Pb data points by a factor of 0.75,and those of Li and C by 0.85.

This is consistent with the published errors. Each data“point” represents a bin of kz, represented as a horizontal line.

The integration of Eqs. (7),(8) over the small bins 0.1 or 0.2 GeV/c of ref.

[13] causes negligible differences withsimply using the central values. The use of a distribution of masses starting at M + mπincreases the computed dσ/dσB at the lowest beam momentum.

In either case, including7

the effects of color transparency gives good agreement with the data.Fig. 2, shows a summary of the different calculations for 27Al.

Here as in Figs. 1 and3 the solid curve represents the full color transparency calculation, the dashed curves areobtained by neglecting the effects of the PLC-BLC interference, and the dotted curvesrepresent the use of the standard optical model or “Glauber” calculation.

These lattercurves fall far below the data, but including the effects of CT leads to a reasonablereproduction of the data.Fig. 3 shows the A-dependence of the data of ref.[13].

These are taken for a bin ofkz ranging from -0.2 GeV/c to 0.1 GeV/c. Here the use of Eqs.

(7) and (8) does matter.To see this compare the 27Al data of this figure with the previous figures. The Glauberstandard optical model leads to results (dotted curves) that again fall well below thedata, this time for each target nucleus.

The solid curves, which show our full calculation,are in excellent agreement with all of the data, except for the 12 GeV Al data point. Thedashed curves show the results obtained without the BLC-PLC interference Ralston-Pireeffect, so we see that the latter helps to account for the energy dependence, even thoughit is not a very large numerical effect.It is necessary to comment on the single particle nuclear shell model wave functionsused here.

Harmonic oscillator (¯hω = 41MeV/A1/3 ) wave functions are used for lightnuclei (Li and C). The other nuclei are treated in the Hartree-Fock (HF) approximationwith the SGII interaction of Ref.

[27]. (HF wave functions are used in obtaining theresults of Figs.

1,2.) If the oscillator frequency is chosen appropriately, the HF wavefunctions are well approximated by a single harmonic oscillator wave function.

Thuswe find the most important effect of using the HF wavefunctions is to shift the valueof ¯hω from 41MeV/A1/3 to one that more precisely represents the nuclear mean squareradius. The effect of using HF wave functions with the correct exponential dependenceat long distance on dσ/dσB is largest, a 20% increase independent of energy, for theGlauber optical model calculations for the Pb target.

But the effect is negligible for theCT calculations. See also ref.

[28].8

The main effect of the color transparency is to provide an increase in the predictedmagnitude of the cross sections. One might wonder if some combination of reasonableeffects applied to the Glauber optical model calculations yield enhancements that leadto reproducing the data without including color transparency effects.

One possibilityis to to claim [29] that the experimental resolution allows the inclusion of all nuclearexcited states in the measurement. In that case, the optical model wave functions wouldbe computed using the proton-proton reaction cross section σr instead of the total crosssection σ as we have done.However, σr/σ ≈0.85 [30] and has only a small energydependence, for the energy range of interest here.

Calculations show that including thiseffect provides only 10-20 % enhancements. Including the effects of nucleon-nucleon shortrange correlations leads to a 20% increase for 12C [28], but is a much smaller effect forheavier nuclei.

Thus, in the absence of CT effects, we see no possibility to elevate thethe DWBA cross sections to the levels observed by the experiment.We believe the BNL experiment and the present calculations calibrate the size of colortransparency effects. Thus we determine the energies for measurable effects.

Our calcu-lations provide a good guide to the (e,e’p) experiments. The SLAC Ne-18 experiment isset for very small values of kz, so the previous results of ref [12] stand unchanged (exceptpossibly for the effects of using improved nuclear structure information for heavy targets).Since the predicted transparency is not large in the range of Q2 between 1 and 7 GeV2, weexpect that our earlier calculations of color transparency effects for the power-law form ofg(M2X) will not be ruled out by the final results of the SLAC experiment[15, 16].

Further-more, our CT calculations predict enhancements for Q2 between 7 and 15 GeV2. Thus ahigher energy experiment which examines the kz dependence should observe measurableeffects.Thus, we summarize: If CT effects are included, the qualitative features of the data ofref [13] can be reproduced in a very natural way.

No adjustment of parameters is needed.If the new BNL (p,pp) color transparency experiment [14] confirms the central values ofthe older results, one can be confident that color transparency has been discovered.9

Acknowledgment: One of the authors (BKJ) thanks the Natural Sciences and En-gineering Research Council of Canada for financial support. The other (GAM) thanksthe DOE for partial support.

Discussions with W.R. Greenberg and M. Strikman areappreciated. We thank G. Bertsch for making the HF program of ref.

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Figure CaptionsFigure 1: Effect of using distributed baryonic masses. Solid uses distributed masses.Dashed- M1 =1550 MeV.

The data in all figures are those of Ref. [13].Figure 2.

Full Al data. Solid- full calculation.

Dashed- with out the Ralston-Pireeffect. Dotted -Glauber.Figure 3.

A dependence of transparency. Solid- full calculation.

Dashed- with outthe Ralston-Pire effect. Dotted -Glauber.13

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