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Novel colour transparency effect. Scanning the wave function of vector mesons
Physics Letters B 309 (1993) 179-186 North-Holland
PHYSICS LETTERS B
N o v e l colour transparency effect. Scanning the wave function of vector mesons B Z K o p e h o v l c h a,b, j Nemchlck b,c, N N Nlkolaev d and B G Zakharov a a TRIUMF, 4004 Wesbrook Mall, Vancouver, B C V6T 2.43, Canada b Laboratory of Nuclear Problems, Jomt Institute for Nuclear Research, Head Post Office, P 0 Box 79, 101000 Moscow, Russian Federation c Institute of Experimental Physics SA V, Solovjevova 47, CS-04353 Kostce, Slovak Repubhc d L D Landau Institute for Theoretical Physics, GSP-I, 117940, ul Kosygma 2, V-334, Moscow, Russian Federation ReceJved 9 February 1993 Editor Landshoff We demonstrate how the wrtual photoproductJon of vector mesons on nuclei scans the wave function of vector mesons from the large non-perturbat~ve transverse s~ze p ~ Rv down to the small perturbatlve s~ze p ~ 1/V/-~ The mechanism of scanning is based on color transparency and QCD predicted spatml wave function of quark-antlquark fluctuations of virtual photons A rich, energy- and Q2-dependent, pattern of the nuclear shadowing and antlshadowmg is predicted, which can be tested at the European Electron Facihty and SLAC
1. Introduction In this paper we discuss a novel feature o f color transparency (CT) tests o f Q C D the scanning o f the non-perturbatlve hadronic wave functions The virtual photoproductIon o f vector mesons (T, T', J / g , ¢/, p, p', ) is particularly suited for such CT tests At high energy u, the photoproductton can be viewed [ 1,2 ] as a production o f the virtual ~q pair with the coherence length l~ -
2l/ Q2 + m~
(1)
The size pQ o f the produced ~q pair (the ejectile state) is controlled by the virtuahty Q2 o f photons [3 ] The ~q state is projected onto (recombines into) the finalstate vector meson V with the formation (recombination) length /)
If-
mvAm'
(2)
where Am 1s the typical level sphttmg in the quarkom u m We demonstrate, how by changing Q2 and the initial size pQ one can scan the wave function o f vector
mesons starting from the non-perturbative size p ,-~ R v down to the perturbative region o f p ~ 1/Q Besides the scanning radius Po, the formation length If is a second important parameter o f CT physics Changing energy u, one can vary If from If << R~ (quasi-instantaneous formation o f the finalstate hadron) to If > RA (the frozen-size limit), and thus study the dynamical evolution o f the small-sized, perturbatlve q~ pair to the full-sized non-perturbatlve hadron We predict a very rich pattern o f the nuclear shadowing and antlshadowing phenomena, which changes with the scanning radius pQ, i e , with Q2, and with the evolution rate, i e , with energy v It is worthwhile to emphasize that the virtual photoproduct~on exemplifies m a particularly transparent way the principle idea o f CT tests o f Q C D ( 1 ) A small transverse-s~ze component o f the interacting hadrons, 19 << Rh, is selected by the interaction dynamics (2) The interaction cross section a (p) o f the small-sized ejectde is measured by a strength of the final state interaction (FSI) in the target nucleus [4,5] Notice a combination o f the perturbatJve and non-perturbatlve aspects o f Q C D the perturbative production o f the small-sized ejectile is followed by probing its size in the non-perturbatlve, diffractive small-angle scatter-
0370-2693/93/$ 06 00 © 1993-Elsevxer Science Pubhshers B V All rights reserved
179
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16
mg m the nuclear matter Nevertheless, Q C D as a theory o f strong Interactions predicts that the strength o f this diffractive scattenng vanishes as p ~ 0 a ( p ) cx p2 [6,7]
o IO C..)
~05 i1) 2. The scanning mechanism
~oo
The quantum-mechanical descnpUon o f the scannlng goes as follows At lf, lc > Ra the a m p l i t u d e o f the forward photoproductlon on the free nucleon M u and the nuclear transmission coefficient or the nuclear transparency TrA = daA/Adau m the quaslelasttc photoproductlon y*A ~ VA read [8,9]
Mu
=
(3)
Tra=lfd2bT(b) ×
(4)
(Vlo-(p) [y*)2 where T ( b ) = f dzna (b, z) Is the optical thickness o f a nucleus, the nuclear density na (b, z) is normallzed to the nuclear mass number A f d3r nA (r) = A The wave function 17") o f the q~ fluctuations o f the virtual photons was calculated in [3] in the mixed (p, z ) representation, where z is a fractaon o f the photon's light-cone m o m e n t u m , carried by the quark, 0 < z < 1 The most i m p o r t a n t feature o f ]7") ts an exponential decrease at large distances [ 3 ]
(5)
[7") c< e x p ( - e p ) ,
0
i
radius Fig 1 The qualitative pattern of the the Q2-dependent scannlng of the wave functions of the ground state V and the radial excitation V' of the vector meson The scanning distributions a (p)~y, (p) shown by the solid and dashed curve have the scanning radu pQ differing by a factor 3 All wave functions are in arbitrary units product will be sharply peaked at p ~ P2 = 2pQ with the width Ap ~ 2pQ, which leads naturally to the idea of scanning The transition matrix elements (3,4) probe the wave function o f vector mesons at p ,,~ 2pQ, and varying pQ by changing Q2, one can scan the wave function IV) from large to small distances In fig l we demonstrate qualitatively how the scanning works We also show the z-integrated wave functions o f the ground state IV) and o f the radial excitation IV') The nuclear matrix element in (4) can be expanded m the moments ( V[a ( p ) " ]y* ) which are the true QCD observables o f the virtual photoproductlon process These moments probe the wave function IV) at different values of p ~ P2n To leading order in the final state interaction, TrA = 1 - , ~ , v l / d 2 b T ( b )
where e2 = m 2 + z ( 1 - z ) Q
2
(6)
Calculation o f the matrix elements in (3), (4) involves d 2 p d z integration In the non-relatxvlstlc q u a r k o m u m z ~ 1/2, so that the relevant q~ fluctuations have a s~ze P~Pt2-
VI -05
1 2 ~ m 2 + ¼Q2 ~" ~ + Q 2
2,
(8)
where
Xv = (V[a(P)2[Y*)
(Vlo(p)17")
(9)
This expansion works well when 1 - Tra << 1, and is convenient to explain how the scanning proceeds Notice, that t~(p)2[7 *) peaks at p ~ P4 = 4pQ
(7) 3. Scanning the vector mesons: the node effect
W h a t enters ( 3 ), ( 4 ) is a product a ( p ) I;~*) Because o f the CT property, e ( p ) oc p2 at small p [6,7], this 180
Start with the real photoproductlon (Q2 = 0) o f
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the ground-state vector mesons (T, J/g/, po, ) and take the case o f the c h a r m o n i u m In this case PO = 1~me << Rj/v,, P2 is still smaller than Rj/~, but P4 "~ Rj/~, F o r this numerical reason, one finds 27j/~, atot ( J~ g~N), 1 e, the predicted nuclear shadowing will be marginally similar [8,9] to the Vector D o m i n a n c e Model ( V D M ) prediction [10] This holds to a large extent for the T a n d the hght vector mesons as well At larger Q 2 , 0 n e h a s p2,/94 << Rv a n d l v ~ o'(p0) o¢ p~ with the calculable logarithmic corrections [3,11 ], so that the above marginal similarity with the V D M disappears The nuclear transparency will tend to unity from below
8 July 1993
~t O0 . . . . . . . r•l Z r,~
..........
r"---'----
e
~
~:OBO
d Q
~ _ _ _
C b a
E'-, 0
60
f
£t_.
//
e
/
d c
//
(GeV)
ioo
50 10 5 0
b a
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
100
I000 I0 IV (GeV)
,
100 v
,
iO00 (GeV)
I 00'
Y
>-
Y'
Pb
Pb C
b <
A 2 1 - TrA cx ~aap Q
(10)
~
090'
Z
C (CeV~)
Co:
/#c
E--,
The case o f the radml excitation V' is more interesting Radial excitations have larger radius and larger free nucleon cross section e g , O'tot(g/tN) ,-~ 2 5atot(J/q/N) [8], which classically would suggest a much stronger final state interaction for g/' than for J/g/ The presence o f the node in the V' wave function leads to a rather complex pattern o f shadowing and antishadowmg In the p h o t o p r o d u c t l o n limit, because of P2 '~ R~,, there are rather strong cancellations between the contributions to the a m p h t u d e (3) from p below and above the node (the node effect, see fig 1) F o r this reason, in photoproduction on the free nucleons, one predicts the rat]o of the forward production differential cross sections r(Q 2 = O) = d a ( T N ~ V ' N ) / d a ( T N - * VN)I/= 0 < 1 F o r the c h a r m o n m m the g/'/(J/g/) ratio r ( 0 ) = 0 17 In the l a production, the initial size pQ scales as 1/mq, whereas the radius o f the bb b o u n d states decreases with mq less rapidly F o r this reason for the la the node effect is much weaker and we find the T'/T ratio r ( 0 ) = 0 84 The larger ]s Q2, the smaller size/72 is scanned, a contribution from the regmn above the node becomes negligible, and r (Q2) will increase with Q2 up to r(Q 2 >> m 2) ,-~ 1 (the exact limiting ratio depends on the wave function's at the origin) Since P4 is closer to the node position, the node effect is still stronger in the matrix element (V'la(p)2[7) F o r the g/' one finds (g/'lo'(p)217) < 0, so that Z~, < 0 and TrA(g/') > 1 despite the larger free-nucleon cross section [8,9,11] In the T' productlon the scanning radius c o m p a r e d to the bottom u m r a d m s is relatively smaller than in the char-
// 0 80
100
1000
u (GeV)
r i6o
b
10o 50
......
i8~o
e
~ (GeV)
Fig 2 The predlcted Q2 and u-dependence of the nuclear transparency m the virtual photoproductlon of the heavy quarkoma The quahtatlve pattern isthe same from the hghl to heavy nuclex
m o n m m case, the node effect is weaker and FSI produces the shadowing o f the T' Still, m spite o f atot ( T ' N ) >> atot (TAr), shadowing o f the T p is weaker than shadowing o f the T The Q2-dependent scanning changes the node effect significantly The smaller the scanning radms Pe, the weaker are the cancellations In the g/' case (V'la(p)2]7 *) becomes positive definite, so that the antishadowlng changes to the shadowing, which first nses with Q2, then saturates and Is followed by the onset o f asymptotic decrease (10) In the ~Y productlon the node effect is weaker, still it affects the QE-dependence o f the shadowing making it different from that o f the T The predictions for the heavy q u a r k o n m m photoproduction are shown in figs 2 and 3 Because o f the small size o f heavy q u a r k o n m m , the results are numerically reliable, as they are d o m i n a t e d by the perturbatlve Q C D d o m a i n 4. The two scenarios of scanning the light vector mesons The case o f the light vector mesons is particularly 181
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1 10
10
g.x31 o o
(..)O8 Z F-~ C~06 -<
>.-, [.)
D-, o3 Z •, ~ o a~
Q2 (GeV2) d tO p e 5
~@
c
Cf]
NO4 <
9o
b.~02 O0
0 80
'i
l+Qe/m
e
1o
I
,
10
. . . . . . . .
,
,
,
.
100
(GeV) Fig 4 The predicted Q2_ and v-dependence of the nuclear transparency m the wrtual photoproductlon of the pO-mesons
6-
(.9
function, experamental tests o f which can shed hght on the spectroscopy and identification of the radial excitations o f the light vector mesons
(/2 Z < E-~
(1) The undercompensated free-nucleon amphtude <#la(p)ly) > 0 1 + qZ/m,Z 10
F~g 3 The predicted Q2-dependence of the nuclear transparency m the virtual photoproductlon of the ~u' (the undercompensatlon scenario) and p' (the overcompensatlon scenario) mesons interesting, as similar (anti)shadowing effects occur in the energy and Q2 range accessible at SLAC and at the planned European Electron Facility (EEF) Here at moderate Q2 < m 2 the scanning radius is large, pQ ,-, R e , so that numerically the predicted shadowing o f the pO, fig 4, is not very accurate, although the accuracy increases gradually with Q2 as the scanning radius decreases into the perturbative d o m a i n pQ << R v Nevertheless, since the wave function o f the pO does not have a node, we beheve to describe correctly a smooth transition from the real to virtual photoproductmn In the photoproduction limit we find a strong node effect and strong suppression o f the p h o t o p r o d u c t m n o f the radial excitation p' on nucleons, by more than one order of magnitude c o m p a r e d to the pO production, which broadly agrees with the scanty experimental data [10,12] As here Pt2 ~ Rv, we cannot give a reliable numerical estimate o f how small the p,/pO cross section ratio is Nevertheless, we can describe the two possible scenarios o f scanning the p' wave 182
. . . . . . . . . . . .
In this scenario the p ' case will be similar to the ¢,' case, apart from the possibility o f anomalously strong nuclear enhancement Tra >> 1, which might show strong atomic n u m b e r dependence Indeed, because o f the larger relevant values o f p the cross section tr (p) is larger, and the attenuation factor exp [ - 1 a ( p ) T (b) ] in the nuclear m a t n x element, eq (4), will suppress the large p region, effectively decreasing the scanning radius Pt2 and dlmimshlng the node effect A detailed description o f the atomic number dependence will be presented elsewhere The Q:-scannlng too will follow the ~u'-scenano change from the antlshadowlng to the shadowing with increasing Q2, followed by the saturation and then decrease o f the shadowing according to eq (10) The range o f Q2 at which the major effects should occur corresponds to a change o f the scanning radius Pa by a factor ,,~ 2, i e , to Q2 ,,~ 4mq ,~ m 2
(2) The overcompensated free-nucleon amphtude <#la(P)ly> < 0 This scenario is preferred in the crude oscillator model used in [3 ] and fits the pattern o f the node effect becoming stronger for the lighter flavours In this case P2 ~> R v and the higher moments too will be negative valued ( p ' l a (p)"ly) < 0, so that in the photoproduction h m l t one starts with the conventional shadowing Tra < 1 In the Q2-scanning process the striking effect is
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b n n g m g the free-nucleon amplitude ( V ' l a ( p ) l 9'*) to the exact compensation at certain moderate Qz, when the decreasing P2 intercepts Rv (strictly speaking, because o f the relativistic corrections and different quark hehclty states, the compensation is unlikely to be exact) As a result, one finds a spike in TrA, fig 3 With the further increase o f Q2 one enters the undercompensatIon regime, and the further pattern o f the Q2-scannlng will be essentially the same as in the undercompensatlon or the ~" scenario The above described Q2-dependent scanning o f the wave function o f hght vector mesons offers a unique possibility o f identifying the radial excitation o f the hght vector mesons (for a detailed discussion o f the spectroscopy o f light vector mesons see [12]) The corresponding experiments could easily be performed at SLAC and EEF
8 July 1993
TrA ( u ) ~ Tr(/c << RA)
+ Fch(X)2[TrA(lc > RA) --TrA(lc <
where Fch ( x ) is the charge form factor o f the target nucleus and x = 1/lc = (Q2 + mZ)/2u (For the general idea of the derivation o f (14) and a successful description o f the N M C data [14] o f the photoproduction of the J/~u see ref [9] ) If If < RA, the spatial expansion o f the q~ pair becomes Important It can be described in either the quark basis used above, or in the hadronlc basis, which is a nice demonstration o f the q u a r k - h a d r o n duahty Consider the leading term o f the final state interaction in eq (10) in the hadronlc basis, i e , In terms o f G r l b o v ' s inelastic shadowing [ 15] Inserting a complete set o f the intermediate states, one can write down
If the coherence length lc > RA, then amplitudes of production on different nucleons add up coherently At moderate energy, lc < RA, the production rates on different nucleons at the same impact parameter b a d d up incoherently In the opposite limit o f lc > RA amplitudes o f production on different nucleons add up coherently, and nuclear effects are generally weaker [8,9,11] F o r the heavy quarks we have the strong Inequality If >> lc [ 13 ] The same lnequahty holds for the light vector mesons a t Q2 >> m 2 Consequently, at moderate energy, when l~ < R4, one still has a broad energy range in which If > RA and the transverse size o f the q~ pair is still frozen In this regime the nuclear transparency is given by the simple formula
1/
d2bdznA(b,z)
x (Vla(p)exp[-½a(p)t(b'z)]b'*)2
(13)
l
5. Quantum evolution, coherence and energy dependence of FSI
TrA = ~
(12)
(11)
(vla(p)ly.)2 where t(b, z) = f ~ dZ'nA(b, z') Notice, that compared to the high-energy limit (4), here the attenuation effect in the nuclear matrix element IS weaker The energy dependence o f the nuclear transparency is given by the approximate interpolation formula
In the hadronic basis the antlshadowmg o f the ¢/ comes from the destructive interference o f the direct, VDM-hke rescattering, y* ~ ~l --~ ~ , ,
(14)
and the off-diagonal rescattenng
y*--*J/~--*~'
(15)
(there is a small contribution from other intermediate states too) The reason for the strong cancellation is that the 7 ~ ~u' transition is weak c o m p a r e d to the y ~ J/~, transition, whereas the J/~u ---, ~u' transition has an amplitude o f opposite sign, and smaller, than the ~u' ~ ~" elastic scattering amplitude To the contrary, in the J/~, photoproductlon, both a m p h t u d e s in the off-diagonal transitions like 7 ~ ~t' ---, J / ~ are small, and this explains why one finds a marginal similarity to the V D M in the photoproduction At larger Q2 the node effect is no longer effective in suppressing the y* ---* ~,' transition, the off-diagonal a m p h t u d e s become significant for the J/~u photoproduction too and one finds strong departure from the V D M for the J / ~ too By the nature o f CT experiments, at finite energy a (p)Z in the 1 h s o f e q (13) is the non-local operator the q~ state produced on one nucleon is probed by a 183
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second nucleon a distance Az ~ RA apart As a result, the diagonal and off-dmgonal rescatterlng amplitudes acquire the relative phase [ 15,16 ] AtPEl = Az (m 2 m E)/2u ,.~ A z / l f U p o n the integration over Az only the intermediate states It) for which A~, 1 ~ 1, or
Im 2, - m2vI < 2ulRa
8 July 1993
threshold behavlour o f TrA IS partially due to the kinematical rise o f the threshold energy with Q2 In the b o t t o n i u m the initial size pQ c o m p a r e d to the bottonlum radius IS relatively smaller than m the charmonlum, and Tra (Y') exhibits a monotonous Q2 and v dependence (fig 2) With the rising energy the destructive interference o f transitions (14) and (15) leads to a rapid rise o f the nuclear transparency Tra and the onset o f the antlshadowmg o f the ~,' In the photoproductlon limit At the larger Q2 the smaller p are scanned, and s~mllar nse o f TrA with energy ends up in the shadowing region In the photoproductlon o f the light vector mesons If ~ lc and a more elaborate technique like the effective diffraction operator techmque developed m [ 17 ] ~s called upon This will be the subject of the further investigation The effect on the Q2-dependence IS not slgmficant, though, and the c~ted results for the photoproductlon of the p°-mesons, fig 4, were obtained still assuming lc > If At large Q2, when for all vector mesons If >> l~, the nuclear transparency for pO, j/~/, T exhlblts a similar Qa. and u-dependence
(16)
will contribute to the r h s o f eq (13), so that the strength o f the final state interaction will change rapidly from the near-threshold energy o f If << RA to a higher energy o f If > RA We describe the energydependence in th~s region using the quark-basis pathintegral technique suggested in [3] At If << RA the a m p h t u d e s o f transitions (16) and (17) do not interfere However, since they are of comparable magmtude, for the ~u' the V D M predtctlon for the nuclear shadowing breaks down even near the production threshold TrA is larger than the G l a u b e r model predlctmn F o r the J~ ~, a similar incoherent contribution ~s small at small Q2, rises with Q2 as described above, and the near-threshold value o f Tra rises too In the ~u' case the d o m i n a n t effect o f the Q2-dependent scanning is that the 7" ~,' transition a m p h t u d e increases w~th Q2 relative to the y* ---, J / ~ amphtude, which enhances the diagonal (shadowing) rescatterlng contribution compared to the off-diagonal (antlshadowlng) rescatterlng contribution As a result o f this competition the near-threshold value of TrA (~U') first decreases significantly w~th Q2, followed by the J~ ~,-llke behavlour at larger Q2 (fig 2) F o r both J/~, and g ' the near-
6 Coherence effects and scahng law for FSI
According to eq (10), at large Q2 the FSI effect
scalesasAl/3/Q 2 However, this scahng law, suggested m [18 ], is valid only at the asymptotic energy [ 16] Indeed, at moderate energy, the coherence constralnt (16) hmlts the number o f the interfering states
1!
1-
.....
z
r,..)
--
Fe
z o~
6-
\ a
t.--,
4
b
e
,
\\_ '
1 l+qZ/mZ
d 001
b lo
J/*
\
10 1 + Q a / m ~
Fig 5 Test of the scaling law (10) in the virtual photoproducUon of the J/~u and pO mesons The right box shows the pO production at fixed energy The dashed straight line corresponds to the 1/(Q 2 + m 2) dependence The left box shows the pO production at asymptotic energy The dotted straight hne corresponds to the 1/ (Q2 + rn2j/~,) dependence 184
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Neff To a crude approximation, m this case the effective a t t e n u a t m n wdl be controlled by not a small size Pt2 of the initml q~ pair, but rather by the least possible size prom of the wave packet, which can be constructed on a truncated basis of Neff conspiring states [17] (Above we have already encountered a strong effect of the coherence on the energy-dependence of Trn at fixed Q2 ) Only in the very high energy limit, when Neff is not b o u n d e d from above, pmm ~ PQ The constraint (16) is less stringent for the heavy quarkonIa, as the level splitting Am << 2mq, and is more noticeable for the light vector mesons, as here the mE-splitting between higher excitations, which enters the coherence condition (16), rises rapidly with the mass In fig 5 we present our results for the Q2 dependence of 1 - T r n ( p °) at fixed energy u it is much weaker than ~ 1 / ( Q 2 + rn~), which is an appropriate variable in view of eq (7) At asymptotic energy the scaling law ( l l ) works for all vector mesons, fig 5 A somewhat late onset of the scahng law ( l l ) for the p0 can be understood in terms of the overcompensation scenario in the pO, p, photoproduction, as one needs a relatively larger Q2 to make the node effect negligible
7. Conclusions We have shown that QCD observables of CT experiments correspond to scanning the non-perturbative hadronxc wave functions with the Qa-dependent scanning radius pQ The strength of the final state interaction in the exclusive virtual photoproductIon of vector mesons is shown to depend strongly on the nodal structure of wave functions At large energy and Q2 we predict a universal pattern of shadowing for all the photoproduced vector mesons The node effect in the virtual photoproduction can be used to identify the radial excitation states The predicted Q2 and energy dependence of virtual photoproduction on nuclei can easily be tested at SLAC and the planned European Electron Facility The dedicated experiments on virtual photoproduction of vector mesons deserve special attention, since theoretical predictions of CT signal are much more reliable than in (e,e'p) or (p,p'p) reactions (for a recent review see [19])
8 July 1993
Acknowledgement B Z K thanks Theory Division of T R I U M F for the hospitality
References [1] L D Landau and IYa Pomeranchuk, ZhETF 24 (1953) 505, E L Feinberg and I Ya Pomeranchuk, Doklady AN SSSR 93 (1953) 439, I Ya Pomeranchuk, Doklady AN SSSR 96 (1954) 265, 481, E L Felnberg and I Ya Pomeranchuk, Nuovo Clm Suppl 4 (1956) 652 [ 2 ] V N Grlbov, Sov Phys JETP 29 (1969) 483, 30 (1970) 709 [3] N N NikolaevandB G Zakharov, Z Phys C49 (1991) 607, C 53 (1992) 331 [4] A H Mueller, in Proceedings of the XVII Rencontre de Monond (Les Arcs, France), ed J Tran Thanh Van (Editions Frontleres, Glf-sur-Yvette, 1982) p 13 [5 ] S J Brodsky, in Proceedings of the XIII International Symposium on Multiparticle dynamics (Volendam, The Netherlands), eds E W Klttel, W Metzger and A Stergion (World Scientific, Singapore, 1982) p 963 [6] AB Zamolodchlkov, B Z Kopehovlch and LI Lapldus, JETP Lett 33 (1981) 595, 612 [7] G Bertsch, S J Brodsky, A S Goldhaber and J R Gunion, Phys Rev Lett 47 (1981)267 [ 8 ] B Z Kopehovich and B G Zakharov, Phys Rev D 44 (1991) 3466 [9] O Benhar, B Z Kopehovlch, Ch Manotti, N N Nikolaev and B G Zakharov, Phys Rev Lett 69 (1992) 1156 [10] T H Bauer et al, Rev Mod Phys 50 (1978) 261 [ 11 ] N N Nlkolaev, Quantum mechanics of color transparency, Comments on Nuclear and Particle Physics (1992), in press, Color transparency facts and fancy, IJMPE Reports on Nuclear Physics (1992), in press [ 12] A Donnachle and H Mlrzale, Z Phys C 33 (1987) 407, A Donnachie and A B Clegg, Z Phys C 34 (1987)257, C40 (1980) 313, C 42 (1989) 663, C 45 (1990) 677 [13] SJ BrodskyandA H Mueller, Phys Lett B206 (1988) 685 [ 14] NMC Collab, C Manotti, Invited Talk at Europhyslcs High-energy and lepton-photon Conference (Geneva, August 1991 ), to be published in the Proceedings [15]VN Grlbov, Sov Phys JETP 29 (1969) 483, 30 (1970) 709 [ 16 ] V A Karmanov and L A Kondratyuk, JETP Lett 18 (1973) 266
185
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[17] N N Nlkolaev, A Szczurek, J Speth, J Wambach, B G Zakharov and B R Zoller, Juhch preprmt KFA° IKP (Th)- 1992-16 (1992), submitted to Nucl Phys A, and paper in preparation [ 1 8 ] J P Raison and B Plre, Phys Rev Lett 65 (1990) 2343
186
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[ 19 ] N N Nlkolaev, High energy nuclear reacUons m QCD color transparency aspects, Lecture course at RCNP Kakuchl School on Spin physics at intermediate energies (16-19 November 1992), Osaka University, to be pubhshed in the Proceedings, RCNP preprlnt 051 (December 1992)
PHYSICS LETTERS B
N o v e l colour transparency effect. Scanning the wave function of vector mesons B Z K o p e h o v l c h a,b, j Nemchlck b,c, N N Nlkolaev d and B G Zakharov a a TRIUMF, 4004 Wesbrook Mall, Vancouver, B C V6T 2.43, Canada b Laboratory of Nuclear Problems, Jomt Institute for Nuclear Research, Head Post Office, P 0 Box 79, 101000 Moscow, Russian Federation c Institute of Experimental Physics SA V, Solovjevova 47, CS-04353 Kostce, Slovak Repubhc d L D Landau Institute for Theoretical Physics, GSP-I, 117940, ul Kosygma 2, V-334, Moscow, Russian Federation ReceJved 9 February 1993 Editor Landshoff We demonstrate how the wrtual photoproductJon of vector mesons on nuclei scans the wave function of vector mesons from the large non-perturbat~ve transverse s~ze p ~ Rv down to the small perturbatlve s~ze p ~ 1/V/-~ The mechanism of scanning is based on color transparency and QCD predicted spatml wave function of quark-antlquark fluctuations of virtual photons A rich, energy- and Q2-dependent, pattern of the nuclear shadowing and antlshadowmg is predicted, which can be tested at the European Electron Facihty and SLAC
1. Introduction In this paper we discuss a novel feature o f color transparency (CT) tests o f Q C D the scanning o f the non-perturbatlve hadronic wave functions The virtual photoproductIon o f vector mesons (T, T', J / g , ¢/, p, p', ) is particularly suited for such CT tests At high energy u, the photoproductton can be viewed [ 1,2 ] as a production o f the virtual ~q pair with the coherence length l~ -
2l/ Q2 + m~
(1)
The size pQ o f the produced ~q pair (the ejectile state) is controlled by the virtuahty Q2 o f photons [3 ] The ~q state is projected onto (recombines into) the finalstate vector meson V with the formation (recombination) length /)
If-
mvAm'
(2)
where Am 1s the typical level sphttmg in the quarkom u m We demonstrate, how by changing Q2 and the initial size pQ one can scan the wave function o f vector
mesons starting from the non-perturbative size p ,-~ R v down to the perturbative region o f p ~ 1/Q Besides the scanning radius Po, the formation length If is a second important parameter o f CT physics Changing energy u, one can vary If from If << R~ (quasi-instantaneous formation o f the finalstate hadron) to If > RA (the frozen-size limit), and thus study the dynamical evolution o f the small-sized, perturbatlve q~ pair to the full-sized non-perturbatlve hadron We predict a very rich pattern o f the nuclear shadowing and antlshadowing phenomena, which changes with the scanning radius pQ, i e , with Q2, and with the evolution rate, i e , with energy v It is worthwhile to emphasize that the virtual photoproduct~on exemplifies m a particularly transparent way the principle idea o f CT tests o f Q C D ( 1 ) A small transverse-s~ze component o f the interacting hadrons, 19 << Rh, is selected by the interaction dynamics (2) The interaction cross section a (p) o f the small-sized ejectde is measured by a strength of the final state interaction (FSI) in the target nucleus [4,5] Notice a combination o f the perturbatJve and non-perturbatlve aspects o f Q C D the perturbative production o f the small-sized ejectile is followed by probing its size in the non-perturbatlve, diffractive small-angle scatter-
0370-2693/93/$ 06 00 © 1993-Elsevxer Science Pubhshers B V All rights reserved
179
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16
mg m the nuclear matter Nevertheless, Q C D as a theory o f strong Interactions predicts that the strength o f this diffractive scattenng vanishes as p ~ 0 a ( p ) cx p2 [6,7]
o IO C..)
~05 i1) 2. The scanning mechanism
~oo
The quantum-mechanical descnpUon o f the scannlng goes as follows At lf, lc > Ra the a m p l i t u d e o f the forward photoproductlon on the free nucleon M u and the nuclear transmission coefficient or the nuclear transparency TrA = daA/Adau m the quaslelasttc photoproductlon y*A ~ VA read [8,9]
Mu
=
(3)
Tra=lfd2bT(b) ×
(4)
(Vlo-(p) [y*)2 where T ( b ) = f dzna (b, z) Is the optical thickness o f a nucleus, the nuclear density na (b, z) is normallzed to the nuclear mass number A f d3r nA (r) = A The wave function 17") o f the q~ fluctuations o f the virtual photons was calculated in [3] in the mixed (p, z ) representation, where z is a fractaon o f the photon's light-cone m o m e n t u m , carried by the quark, 0 < z < 1 The most i m p o r t a n t feature o f ]7") ts an exponential decrease at large distances [ 3 ]
(5)
[7") c< e x p ( - e p ) ,
0
i
radius Fig 1 The qualitative pattern of the the Q2-dependent scannlng of the wave functions of the ground state V and the radial excitation V' of the vector meson The scanning distributions a (p)~y, (p) shown by the solid and dashed curve have the scanning radu pQ differing by a factor 3 All wave functions are in arbitrary units product will be sharply peaked at p ~ P2 = 2pQ with the width Ap ~ 2pQ, which leads naturally to the idea of scanning The transition matrix elements (3,4) probe the wave function o f vector mesons at p ,,~ 2pQ, and varying pQ by changing Q2, one can scan the wave function IV) from large to small distances In fig l we demonstrate qualitatively how the scanning works We also show the z-integrated wave functions o f the ground state IV) and o f the radial excitation IV') The nuclear matrix element in (4) can be expanded m the moments ( V[a ( p ) " ]y* ) which are the true QCD observables o f the virtual photoproductlon process These moments probe the wave function IV) at different values of p ~ P2n To leading order in the final state interaction, TrA = 1 - , ~ , v l / d 2 b T ( b )
where e2 = m 2 + z ( 1 - z ) Q
2
(6)
Calculation o f the matrix elements in (3), (4) involves d 2 p d z integration In the non-relatxvlstlc q u a r k o m u m z ~ 1/2, so that the relevant q~ fluctuations have a s~ze P~Pt2-
VI -05
1 2 ~ m 2 + ¼Q2 ~" ~ + Q 2
2,
(8)
where
Xv = (V[a(P)2[Y*)
(Vlo(p)17")
(9)
This expansion works well when 1 - Tra << 1, and is convenient to explain how the scanning proceeds Notice, that t~(p)2[7 *) peaks at p ~ P4 = 4pQ
(7) 3. Scanning the vector mesons: the node effect
W h a t enters ( 3 ), ( 4 ) is a product a ( p ) I;~*) Because o f the CT property, e ( p ) oc p2 at small p [6,7], this 180
Start with the real photoproductlon (Q2 = 0) o f
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the ground-state vector mesons (T, J/g/, po, ) and take the case o f the c h a r m o n i u m In this case PO = 1~me << Rj/v,, P2 is still smaller than Rj/~, but P4 "~ Rj/~, F o r this numerical reason, one finds 27j/~, atot ( J~ g~N), 1 e, the predicted nuclear shadowing will be marginally similar [8,9] to the Vector D o m i n a n c e Model ( V D M ) prediction [10] This holds to a large extent for the T a n d the hght vector mesons as well At larger Q 2 , 0 n e h a s p2,/94 << Rv a n d l v ~ o'(p0) o¢ p~ with the calculable logarithmic corrections [3,11 ], so that the above marginal similarity with the V D M disappears The nuclear transparency will tend to unity from below
8 July 1993
~t O0 . . . . . . . r•l Z r,~
..........
r"---'----
e
~
~:OBO
d Q
~ _ _ _
C b a
E'-, 0
60
f
£t_.
//
e
/
d c
//
(GeV)
ioo
50 10 5 0
b a
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
100
I000 I0 IV (GeV)
,
100 v
,
iO00 (GeV)
I 00'
Y
>-
Y'
Pb
Pb C
b <
A 2 1 - TrA cx ~aap Q
(10)
~
090'
Z
C (CeV~)
Co:
/#c
E--,
The case o f the radml excitation V' is more interesting Radial excitations have larger radius and larger free nucleon cross section e g , O'tot(g/tN) ,-~ 2 5atot(J/q/N) [8], which classically would suggest a much stronger final state interaction for g/' than for J/g/ The presence o f the node in the V' wave function leads to a rather complex pattern o f shadowing and antishadowmg In the p h o t o p r o d u c t l o n limit, because of P2 '~ R~,, there are rather strong cancellations between the contributions to the a m p h t u d e (3) from p below and above the node (the node effect, see fig 1) F o r this reason, in photoproduction on the free nucleons, one predicts the rat]o of the forward production differential cross sections r(Q 2 = O) = d a ( T N ~ V ' N ) / d a ( T N - * VN)I/= 0 < 1 F o r the c h a r m o n m m the g/'/(J/g/) ratio r ( 0 ) = 0 17 In the l a production, the initial size pQ scales as 1/mq, whereas the radius o f the bb b o u n d states decreases with mq less rapidly F o r this reason for the la the node effect is much weaker and we find the T'/T ratio r ( 0 ) = 0 84 The larger ]s Q2, the smaller size/72 is scanned, a contribution from the regmn above the node becomes negligible, and r (Q2) will increase with Q2 up to r(Q 2 >> m 2) ,-~ 1 (the exact limiting ratio depends on the wave function's at the origin) Since P4 is closer to the node position, the node effect is still stronger in the matrix element (V'la(p)2[7) F o r the g/' one finds (g/'lo'(p)217) < 0, so that Z~, < 0 and TrA(g/') > 1 despite the larger free-nucleon cross section [8,9,11] In the T' productlon the scanning radius c o m p a r e d to the bottom u m r a d m s is relatively smaller than in the char-
// 0 80
100
1000
u (GeV)
r i6o
b
10o 50
......
i8~o
e
~ (GeV)
Fig 2 The predlcted Q2 and u-dependence of the nuclear transparency m the virtual photoproductlon of the heavy quarkoma The quahtatlve pattern isthe same from the hghl to heavy nuclex
m o n m m case, the node effect is weaker and FSI produces the shadowing o f the T' Still, m spite o f atot ( T ' N ) >> atot (TAr), shadowing o f the T p is weaker than shadowing o f the T The Q2-dependent scanning changes the node effect significantly The smaller the scanning radms Pe, the weaker are the cancellations In the g/' case (V'la(p)2]7 *) becomes positive definite, so that the antishadowlng changes to the shadowing, which first nses with Q2, then saturates and Is followed by the onset o f asymptotic decrease (10) In the ~Y productlon the node effect is weaker, still it affects the QE-dependence o f the shadowing making it different from that o f the T The predictions for the heavy q u a r k o n m m photoproduction are shown in figs 2 and 3 Because o f the small size o f heavy q u a r k o n m m , the results are numerically reliable, as they are d o m i n a t e d by the perturbatlve Q C D d o m a i n 4. The two scenarios of scanning the light vector mesons The case o f the light vector mesons is particularly 181
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1 10
10
g.x31 o o
(..)O8 Z F-~ C~06 -<
>.-, [.)
D-, o3 Z •, ~ o a~
Q2 (GeV2) d tO p e 5
~@
c
Cf]
NO4 <
9o
b.~02 O0
0 80
'i
l+Qe/m
e
1o
I
,
10
. . . . . . . .
,
,
,
.
100
(GeV) Fig 4 The predicted Q2_ and v-dependence of the nuclear transparency m the wrtual photoproductlon of the pO-mesons
6-
(.9
function, experamental tests o f which can shed hght on the spectroscopy and identification of the radial excitations o f the light vector mesons
(/2 Z < E-~
(1) The undercompensated free-nucleon amphtude <#la(p)ly) > 0 1 + qZ/m,Z 10
F~g 3 The predicted Q2-dependence of the nuclear transparency m the virtual photoproductlon of the ~u' (the undercompensatlon scenario) and p' (the overcompensatlon scenario) mesons interesting, as similar (anti)shadowing effects occur in the energy and Q2 range accessible at SLAC and at the planned European Electron Facility (EEF) Here at moderate Q2 < m 2 the scanning radius is large, pQ ,-, R e , so that numerically the predicted shadowing o f the pO, fig 4, is not very accurate, although the accuracy increases gradually with Q2 as the scanning radius decreases into the perturbative d o m a i n pQ << R v Nevertheless, since the wave function o f the pO does not have a node, we beheve to describe correctly a smooth transition from the real to virtual photoproductmn In the photoproduction limit we find a strong node effect and strong suppression o f the p h o t o p r o d u c t m n o f the radial excitation p' on nucleons, by more than one order of magnitude c o m p a r e d to the pO production, which broadly agrees with the scanty experimental data [10,12] As here Pt2 ~ Rv, we cannot give a reliable numerical estimate o f how small the p,/pO cross section ratio is Nevertheless, we can describe the two possible scenarios o f scanning the p' wave 182
. . . . . . . . . . . .
In this scenario the p ' case will be similar to the ¢,' case, apart from the possibility o f anomalously strong nuclear enhancement Tra >> 1, which might show strong atomic n u m b e r dependence Indeed, because o f the larger relevant values o f p the cross section tr (p) is larger, and the attenuation factor exp [ - 1 a ( p ) T (b) ] in the nuclear m a t n x element, eq (4), will suppress the large p region, effectively decreasing the scanning radius Pt2 and dlmimshlng the node effect A detailed description o f the atomic number dependence will be presented elsewhere The Q:-scannlng too will follow the ~u'-scenano change from the antlshadowlng to the shadowing with increasing Q2, followed by the saturation and then decrease o f the shadowing according to eq (10) The range o f Q2 at which the major effects should occur corresponds to a change o f the scanning radius Pa by a factor ,,~ 2, i e , to Q2 ,,~ 4mq ,~ m 2
(2) The overcompensated free-nucleon amphtude <#la(P)ly> < 0 This scenario is preferred in the crude oscillator model used in [3 ] and fits the pattern o f the node effect becoming stronger for the lighter flavours In this case P2 ~> R v and the higher moments too will be negative valued ( p ' l a (p)"ly) < 0, so that in the photoproduction h m l t one starts with the conventional shadowing Tra < 1 In the Q2-scanning process the striking effect is
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b n n g m g the free-nucleon amplitude ( V ' l a ( p ) l 9'*) to the exact compensation at certain moderate Qz, when the decreasing P2 intercepts Rv (strictly speaking, because o f the relativistic corrections and different quark hehclty states, the compensation is unlikely to be exact) As a result, one finds a spike in TrA, fig 3 With the further increase o f Q2 one enters the undercompensatIon regime, and the further pattern o f the Q2-scannlng will be essentially the same as in the undercompensatlon or the ~" scenario The above described Q2-dependent scanning o f the wave function o f hght vector mesons offers a unique possibility o f identifying the radial excitation o f the hght vector mesons (for a detailed discussion o f the spectroscopy o f light vector mesons see [12]) The corresponding experiments could easily be performed at SLAC and EEF
8 July 1993
TrA ( u ) ~ Tr(/c << RA)
+ Fch(X)2[TrA(lc > RA) --TrA(lc <
where Fch ( x ) is the charge form factor o f the target nucleus and x = 1/lc = (Q2 + mZ)/2u (For the general idea of the derivation o f (14) and a successful description o f the N M C data [14] o f the photoproduction of the J/~u see ref [9] ) If If < RA, the spatial expansion o f the q~ pair becomes Important It can be described in either the quark basis used above, or in the hadronlc basis, which is a nice demonstration o f the q u a r k - h a d r o n duahty Consider the leading term o f the final state interaction in eq (10) in the hadronlc basis, i e , In terms o f G r l b o v ' s inelastic shadowing [ 15] Inserting a complete set o f the intermediate states, one can write down
If the coherence length lc > RA, then amplitudes of production on different nucleons add up coherently At moderate energy, lc < RA, the production rates on different nucleons at the same impact parameter b a d d up incoherently In the opposite limit o f lc > RA amplitudes o f production on different nucleons add up coherently, and nuclear effects are generally weaker [8,9,11] F o r the heavy quarks we have the strong Inequality If >> lc [ 13 ] The same lnequahty holds for the light vector mesons a t Q2 >> m 2 Consequently, at moderate energy, when l~ < R4, one still has a broad energy range in which If > RA and the transverse size o f the q~ pair is still frozen In this regime the nuclear transparency is given by the simple formula
1/
d2bdznA(b,z)
x (Vla(p)exp[-½a(p)t(b'z)]b'*)2
(13)
l
5. Quantum evolution, coherence and energy dependence of FSI
TrA = ~
(12)
(11)
(vla(p)ly.)2 where t(b, z) = f ~ dZ'nA(b, z') Notice, that compared to the high-energy limit (4), here the attenuation effect in the nuclear matrix element IS weaker The energy dependence o f the nuclear transparency is given by the approximate interpolation formula
In the hadronic basis the antlshadowmg o f the ¢/ comes from the destructive interference o f the direct, VDM-hke rescattering, y* ~ ~l --~ ~ , ,
(14)
and the off-diagonal rescattenng
y*--*J/~--*~'
(15)
(there is a small contribution from other intermediate states too) The reason for the strong cancellation is that the 7 ~ ~u' transition is weak c o m p a r e d to the y ~ J/~, transition, whereas the J/~u ---, ~u' transition has an amplitude o f opposite sign, and smaller, than the ~u' ~ ~" elastic scattering amplitude To the contrary, in the J/~, photoproductlon, both a m p h t u d e s in the off-diagonal transitions like 7 ~ ~t' ---, J / ~ are small, and this explains why one finds a marginal similarity to the V D M in the photoproduction At larger Q2 the node effect is no longer effective in suppressing the y* ---* ~,' transition, the off-diagonal a m p h t u d e s become significant for the J/~u photoproduction too and one finds strong departure from the V D M for the J / ~ too By the nature o f CT experiments, at finite energy a (p)Z in the 1 h s o f e q (13) is the non-local operator the q~ state produced on one nucleon is probed by a 183
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second nucleon a distance Az ~ RA apart As a result, the diagonal and off-dmgonal rescatterlng amplitudes acquire the relative phase [ 15,16 ] AtPEl = Az (m 2 m E)/2u ,.~ A z / l f U p o n the integration over Az only the intermediate states It) for which A~, 1 ~ 1, or
Im 2, - m2vI < 2ulRa
8 July 1993
threshold behavlour o f TrA IS partially due to the kinematical rise o f the threshold energy with Q2 In the b o t t o n i u m the initial size pQ c o m p a r e d to the bottonlum radius IS relatively smaller than m the charmonlum, and Tra (Y') exhibits a monotonous Q2 and v dependence (fig 2) With the rising energy the destructive interference o f transitions (14) and (15) leads to a rapid rise o f the nuclear transparency Tra and the onset o f the antlshadowmg o f the ~,' In the photoproductlon limit At the larger Q2 the smaller p are scanned, and s~mllar nse o f TrA with energy ends up in the shadowing region In the photoproductlon o f the light vector mesons If ~ lc and a more elaborate technique like the effective diffraction operator techmque developed m [ 17 ] ~s called upon This will be the subject of the further investigation The effect on the Q2-dependence IS not slgmficant, though, and the c~ted results for the photoproductlon of the p°-mesons, fig 4, were obtained still assuming lc > If At large Q2, when for all vector mesons If >> l~, the nuclear transparency for pO, j/~/, T exhlblts a similar Qa. and u-dependence
(16)
will contribute to the r h s o f eq (13), so that the strength o f the final state interaction will change rapidly from the near-threshold energy o f If << RA to a higher energy o f If > RA We describe the energydependence in th~s region using the quark-basis pathintegral technique suggested in [3] At If << RA the a m p h t u d e s o f transitions (16) and (17) do not interfere However, since they are of comparable magmtude, for the ~u' the V D M predtctlon for the nuclear shadowing breaks down even near the production threshold TrA is larger than the G l a u b e r model predlctmn F o r the J~ ~, a similar incoherent contribution ~s small at small Q2, rises with Q2 as described above, and the near-threshold value o f Tra rises too In the ~u' case the d o m i n a n t effect o f the Q2-dependent scanning is that the 7" ~,' transition a m p h t u d e increases w~th Q2 relative to the y* ---, J / ~ amphtude, which enhances the diagonal (shadowing) rescatterlng contribution compared to the off-diagonal (antlshadowlng) rescatterlng contribution As a result o f this competition the near-threshold value of TrA (~U') first decreases significantly w~th Q2, followed by the J~ ~,-llke behavlour at larger Q2 (fig 2) F o r both J/~, and g ' the near-
6 Coherence effects and scahng law for FSI
According to eq (10), at large Q2 the FSI effect
scalesasAl/3/Q 2 However, this scahng law, suggested m [18 ], is valid only at the asymptotic energy [ 16] Indeed, at moderate energy, the coherence constralnt (16) hmlts the number o f the interfering states
1!
1-
.....
z
r,..)
--
Fe
z o~
6-
\ a
t.--,
4
b
e
,
\\_ '
1 l+qZ/mZ
d 001
b lo
J/*
\
10 1 + Q a / m ~
Fig 5 Test of the scaling law (10) in the virtual photoproducUon of the J/~u and pO mesons The right box shows the pO production at fixed energy The dashed straight line corresponds to the 1/(Q 2 + m 2) dependence The left box shows the pO production at asymptotic energy The dotted straight hne corresponds to the 1/ (Q2 + rn2j/~,) dependence 184
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Neff To a crude approximation, m this case the effective a t t e n u a t m n wdl be controlled by not a small size Pt2 of the initml q~ pair, but rather by the least possible size prom of the wave packet, which can be constructed on a truncated basis of Neff conspiring states [17] (Above we have already encountered a strong effect of the coherence on the energy-dependence of Trn at fixed Q2 ) Only in the very high energy limit, when Neff is not b o u n d e d from above, pmm ~ PQ The constraint (16) is less stringent for the heavy quarkonIa, as the level splitting Am << 2mq, and is more noticeable for the light vector mesons, as here the mE-splitting between higher excitations, which enters the coherence condition (16), rises rapidly with the mass In fig 5 we present our results for the Q2 dependence of 1 - T r n ( p °) at fixed energy u it is much weaker than ~ 1 / ( Q 2 + rn~), which is an appropriate variable in view of eq (7) At asymptotic energy the scaling law ( l l ) works for all vector mesons, fig 5 A somewhat late onset of the scahng law ( l l ) for the p0 can be understood in terms of the overcompensation scenario in the pO, p, photoproduction, as one needs a relatively larger Q2 to make the node effect negligible
7. Conclusions We have shown that QCD observables of CT experiments correspond to scanning the non-perturbative hadronxc wave functions with the Qa-dependent scanning radius pQ The strength of the final state interaction in the exclusive virtual photoproductIon of vector mesons is shown to depend strongly on the nodal structure of wave functions At large energy and Q2 we predict a universal pattern of shadowing for all the photoproduced vector mesons The node effect in the virtual photoproduction can be used to identify the radial excitation states The predicted Q2 and energy dependence of virtual photoproduction on nuclei can easily be tested at SLAC and the planned European Electron Facility The dedicated experiments on virtual photoproduction of vector mesons deserve special attention, since theoretical predictions of CT signal are much more reliable than in (e,e'p) or (p,p'p) reactions (for a recent review see [19])
8 July 1993
Acknowledgement B Z K thanks Theory Division of T R I U M F for the hospitality
References [1] L D Landau and IYa Pomeranchuk, ZhETF 24 (1953) 505, E L Feinberg and I Ya Pomeranchuk, Doklady AN SSSR 93 (1953) 439, I Ya Pomeranchuk, Doklady AN SSSR 96 (1954) 265, 481, E L Felnberg and I Ya Pomeranchuk, Nuovo Clm Suppl 4 (1956) 652 [ 2 ] V N Grlbov, Sov Phys JETP 29 (1969) 483, 30 (1970) 709 [3] N N NikolaevandB G Zakharov, Z Phys C49 (1991) 607, C 53 (1992) 331 [4] A H Mueller, in Proceedings of the XVII Rencontre de Monond (Les Arcs, France), ed J Tran Thanh Van (Editions Frontleres, Glf-sur-Yvette, 1982) p 13 [5 ] S J Brodsky, in Proceedings of the XIII International Symposium on Multiparticle dynamics (Volendam, The Netherlands), eds E W Klttel, W Metzger and A Stergion (World Scientific, Singapore, 1982) p 963 [6] AB Zamolodchlkov, B Z Kopehovlch and LI Lapldus, JETP Lett 33 (1981) 595, 612 [7] G Bertsch, S J Brodsky, A S Goldhaber and J R Gunion, Phys Rev Lett 47 (1981)267 [ 8 ] B Z Kopehovich and B G Zakharov, Phys Rev D 44 (1991) 3466 [9] O Benhar, B Z Kopehovlch, Ch Manotti, N N Nikolaev and B G Zakharov, Phys Rev Lett 69 (1992) 1156 [10] T H Bauer et al, Rev Mod Phys 50 (1978) 261 [ 11 ] N N Nlkolaev, Quantum mechanics of color transparency, Comments on Nuclear and Particle Physics (1992), in press, Color transparency facts and fancy, IJMPE Reports on Nuclear Physics (1992), in press [ 12] A Donnachle and H Mlrzale, Z Phys C 33 (1987) 407, A Donnachie and A B Clegg, Z Phys C 34 (1987)257, C40 (1980) 313, C 42 (1989) 663, C 45 (1990) 677 [13] SJ BrodskyandA H Mueller, Phys Lett B206 (1988) 685 [ 14] NMC Collab, C Manotti, Invited Talk at Europhyslcs High-energy and lepton-photon Conference (Geneva, August 1991 ), to be published in the Proceedings [15]VN Grlbov, Sov Phys JETP 29 (1969) 483, 30 (1970) 709 [ 16 ] V A Karmanov and L A Kondratyuk, JETP Lett 18 (1973) 266
185
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[17] N N Nlkolaev, A Szczurek, J Speth, J Wambach, B G Zakharov and B R Zoller, Juhch preprmt KFA° IKP (Th)- 1992-16 (1992), submitted to Nucl Phys A, and paper in preparation [ 1 8 ] J P Raison and B Plre, Phys Rev Lett 65 (1990) 2343
186
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[ 19 ] N N Nlkolaev, High energy nuclear reacUons m QCD color transparency aspects, Lecture course at RCNP Kakuchl School on Spin physics at intermediate energies (16-19 November 1992), Osaka University, to be pubhshed in the Proceedings, RCNP preprlnt 051 (December 1992)