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Challenges of nuclear shadowing in DIS
12 November 1998
Physics Letters B 440 Ž1998. 151–156
Challenges of nuclear shadowing in DIS B.Z. Kopeliovich
a,c
, J. Raufeisen b, A.V. Tarasov
b,c
a
b
Max-Planck Institut fur ¨ Kernphysik, Postfach 103980, 69029 Heidelberg, Germany Institut fur ¨ Theoretische Physik der UniÕersitat, ¨ Philosophenweg 19, 69120 Heidelberg, Germany c Joint Institute for Nuclear Research, Dubna, 141980 Moscow Region, Russia Received 3 July 1998; revised 10 August 1998 Editor: P.V. Landshoff
Abstract Nuclear shadowing in DIS at moderately small x is suppressed by the nuclear formfactor and depends on the effective mass of a hadronic fluctuation of the virtual photon. We propose a solution to the problems Ži. of how to combine a definite transverse size of the fluctuation with a definite effective mass, and Žii. of how to include the nuclear formfactor in the higher multiple scattering terms. Comparison of the numerical results with known approximations shows a substantial difference. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Shadowing in deep-inelastic scattering ŽDIS. off nuclei is a hot topic for the last two decades. In the infinite momentum frame of the nucleus it can be interpreted as a result of parton fusion leading to a diminishing parton density at low Bjorken x w1–4x. A more intuitive picture arises in the rest frame of the nucleus where the same phenomenon looks like nuclear shadowing of hadronic fluctuations of the virtual photon w5–12x. To crystallize the problem and its solution we restrict ourselves in this paper to only quark-antiquark fluctuations of the photon, neglecting those higher Fock components which contain gluons and qq pairs from the sea. The lifetime of the qq fluctuation Žcalled coherence time. is given by 2n tc s 2 Ž 1. Q qM 2 where n is the photon energy, Q 2 its virtuality and M is the effective mass of the qq pair.
Provided that the coherence time is much longer than the nuclear radius, l c 4 R A , the total cross section on a nucleus reads w13x, )
g stot
A
Ž x ,Q 2 .
s 2 d 2 b d 2 r Gg ) Ž Q 2 ,r .
H H
1 = 1 y exp y s Ž r . T Ž b . 2
½
5
1 ' 2 d 2 b 1 y exp y s Ž r . T Ž b . 2
H
½ ¦
;5
.
Ž 2.
Here Gg ) Ž Q 2 ,r . characterizes the probability for the photon to develop a qq fluctuation with transverse separation r. The condition t c 4 R A insures that the r does not vary during propagation through the nucleus ŽLorentz time dilation.. Then the qq pair with a definite transverse separation is an eigenstate
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 0 7 2 - 7
B.Z. KopelioÕich et al.r Physics Letters B 440 (1998) 151–156
152
of the interaction with the eigenvalue of the total cross section s Ž r .. Therefore, one can apply the eikonal expression Ž2. for the interaction with the nucleus. The nuclear thickness function T Ž b . s ` Hy` dz rAŽ b, z . is the integral of nuclear density over longitudinal coordinate z and depends on the impact parameter b. The color dipole cross section s Ž r . introduced in w13x vanishes like r 2 at small r ™ 0 due to color screening. This is the heart of the phenomenon called nowadays color transparency w14,13,15x. For this reason nuclear shadowing in Ž2. is dominated by large size fluctuations corresponding to highly asymmetric sharing of the longitudinal momentum carried by the q and q w16,7,9,12x. This leads to Q 2 scaling of shadowing. Note that the averaging of the whole exponential in Ž2. makes this expression different from the Glauber eikonal approximation where s Ž r . is averaged in the exponent. The difference is known as Gribov’s inelastic corrections w17x. In the case of DIS the Glauber approximation does not make sense, and the whole cross section is due to the inelastic shadowing. For the other case, t c ; R A , one has to take into account the variation of r during the propagation of the qq fluctuation through the nucleus. At present this can only be done for the double scattering term w12x in the expansion of the exponential in Ž2., )
A
)
N
g stot
g stot
f1y
1 ²s 2 Ž r . : 4 ²s Ž r . :
²T : d 2 b FA2 Ž q,b . q . . . ,
H
Ž 3. or in hadronic representation w18x, )
g stot
A
g)N stot
f1y
=
1 )
g 4pstot
N
²T : d 2 b dM X2
H H
d s Ž g ) N ™ XN . dM X2 dt
FA2 Ž q,b . q . . . , ts0
Ž 4.
where M X is the effective mass of the inelastic intermediate state, the mean nuclear thickness and the formfactor read, ²T : s
1 A
Hd
FA Ž q,b . s
2
b T 2 Ž b. ,
1 ²T :
Ž 5.
`
Hy`dz r Ž b, z . e A
iqz
,
Ž 6.
with longitudinal momentum transfer q s 1rt c given by Ž1.. In the case of Ž3. the uncertain fluctuation mass is fixed at M 2 s Q 2 , q s 2 m N x. Two expressions Ž3. and Ž4. are related since the integrated forward diffractive dissociation cross section g ) N ™ XN equals to ² s 2 :r16p . There are two problems remaining which are under discussion: Ø How the nuclear formfactor can be included in the higher order scattering terms which are of great importance for heavy nuclei? For instance, the shadowing term in Ž3., Ž4. for lead is of the order of one at low x, so the need of the higher order terms is obvious. Ø Even for the double scattering term in Ž3. it is still unclear which argument should enter the formfactor. Indeed, the effective mass of the qq fluctuation needed for the coherence time in Ž1. cannot be defined in the quark representation with a definite qq separation. On the other hand, Eq. Ž4. exhibits an explicit dependence on M X and the longitudinal momentum transfer is known. However, unknown in this case is the absorptive cross section of the intermediate state X. We suggest a solution of both problems in the next section. The goal of this paper is restricted to the study of the difference between the predictions of the correct quantum-mechanical treatment of nuclear shadowing and known approximations. We do it on an example of the valence qq part of the photon and neglect the higher Fock components containing gluons and sea quarks, which may be important if to compare with data especially at very low x. Nuclear anti-shadowing effect is omitted as well, since we believe it is beyond the shadowing dynamics Že.g. bound nucleon swelling.. Numerical results and a comparison with the standard approach are presented in Section 3.
B.Z. KopelioÕich et al.r Physics Letters B 440 (1998) 151–156
153
2. The Green function of a qq pair in nuclear medium We start with the generalizing of Eq. Ž2. for the case l c F R A , )
g stot
A
1
)
Ž x ,Q 2 . s Hd 2 b H d a stotg A Ž x ,Q 2 ;b, a . , 0
Ž 7. where g) A stot Ž x ,Q 2 ;b, a . sT Ž b.
Hd
Fig. 1. A cartoon for the shadowing Žnegative. term in Ž8.. The Green function W Ž r 2 , z 2 ;r 1 , z 1 . results from the summation over different paths of the qq pair propagation through the nucleus.
2
2
r Cg ) Ž r , a . s Ž r .
`
y 2 Re
Hy`dz
1
rA Ž b, z 1 .
`
=
Hz dz
2
rA Ž b, z 2 . A Ž z 1 , z 2 , a . .
Ž 8.
1
The first term in r.h.s. of Ž8. corresponds to the second, lowest order in s Ž r .T Ž b ., term in expansion of the exponential in Ž2.. The shadowing terms are contained in the second term in Ž8.. Cg ) Ž r, a . is the Žnon-normalized. wave function of the qq fluctuation of the virtual photon, where a is the fraction of the light-cone momentum of the photon carried by the quark. An explicit expression of transverse and longitudinally polarized photons can be found in w19,9x. The function AŽ z 1 , z 2 , a . in Ž8. reads Žsee also in w20x., AŽ z1 , z 2 , a . 1 s d 2 r 1 d 2 r 2 Cg )) Ž r 2 , a . W Ž r 2 , z 2 ;r 1 , z 1 . 4 = Cg ) Ž r 1 , a . s Ž r 2 . s Ž r 1 . e i q mi n Ž z 2yz 1 . , Ž 9 . with Q 2a Ž 1 y a . q m 2q qmin s . Ž 10 . 2 na Ž 1 y a . This is the fraction of the longitudinal momentum transfer q s 1rt c s qmin q D q, which originates from longitudinal motion. Another part D q s k T2rw2 na Ž1 y a .x which is due to the transverse motion of the qq with momentum k T is included in W Ž r 2 , z 2 ;r 1 , z 1 . Žsee below.. The second Žshadowing. term in Ž8. is illustrated in Fig. 1. At the point z 1 the photon diffractively produces the qq pair Žg ) N ™ qqN . with transverse separation r 1. The pair propagates through the nu-
H
cleus along arbitrarily curved trajectories Žshould be summed over. and arrives at the point z 2 with a separation r 2 . The initial and the final separations are controlled by the distribution amplitude Cg ) Ž r .. While passing the nucleus the qq pair interacts with bound nucleons via the cross section s Ž r . which depends on the local separation r. The function W Ž r 2 , z 2 ;r 1 , z 1 . describing the propagation of the pair from z 1 to z 2 also includes that part of the phase shift between the initial and the final photons, which is due to transverse motion of the quarks, while longitudinal motion is already included in Ž10. via the exponential. Thus, Eq. Ž8. does not suffer from either of the two problems of the approximations Ž3. - Ž4.. The longitudinal momentum transfer is known and all the multiple interactions are included. The propagation function W Ž r 2 , z 2 ;r 1 , z 1 . in Ž9. satisfies the equation w20x, i
E W Ž r 2 , z 2 ;r 1 , z 1 . E z2 sy
D Ž r2 . 2 na Ž 1 y a . i
W Ž r 2 , z 2 ;r 1 , z 1 .
s Ž r 2 . rA Ž b, z 2 . W Ž r 2 , z 2 ;r 1 , z1 . , Ž 11 . 2 with the boundary condition W Ž r 2 , z 1;r 1 , z 1 . s d Ž r 2 y r 1 .. The Laplacian DŽ r 2 . acts on the coordinate r 2 . The full derivation of Ž11. will be given elsewhere w21x. Here we only notice that it looks natural like Schrodinger equation with the kinetic term ¨ Drw2 na Ž1 y a .x which takes care of the varying effective mass of the qq pair and provides a proper phase shift, and z 2 plays the role of the time. The y
B.Z. KopelioÕich et al.r Physics Letters B 440 (1998) 151–156
154
imaginary part of the optical potential describes the absorptive process. In the ‘‘frozen’’ limit n ™ ` the kinetic term in Ž11. can be neglected and W Ž r 2 , z 2 ;r 1 , z 1 . z2 1 s d Ž r 2 y r 1 . exp y s Ž r 2 . dz rA Ž b, z . . 2 z1
H
Ž 12 . When this expression is substituted into Ž8. - Ž9. and with qmin ™ 0 one arrives at result Ž2. with 2 Gg ) Ž Q 2 ,r . s H01 d a Cg ) Ž r , a . . We can also recover the approximation Ž3. - Ž4. if one neglects the absorption of the qq pair in the medium. Then W becomes the Green function of a free motion, W Ž r 2 , z 2 ;r 1 , z 1 . 1 s 2p
H
s ™0
d 2 k exp ik Ž r 2 y r 1 . q
ik 2 Ž z 2 y z 1 . 2 na Ž 1 y a .
,
Ž 13 . where k is the transverse momentum of the quark. With this expression the shadowing term in Ž8. reproduces the second term in Ž4.. Indeed, the amplitude of the photon diffractive dissociation in the plane wave approximation reads, 1 fd d Ž k . s d 2 r Cg ) Ž r , a . s Ž r . e i k r . Ž 14 . 2 Therefore, Ž9. can be represented as, AŽ z1 , z 2 , a . 1 2 s d 2 k fd d Ž k . 2p Q 2a Ž 1 y a . q m 2q q k 2 = exp i Ž z 2 y z 1 . 2 na Ž 1 y a .
H
H
Ž 15 . 2
s Ž m2q q k 2 .ra Ž1 y a .
Taking into account that M is the effective mass squared of the qq pair and substituting Ž15. to Ž8. we arrive at Eq. Ž4.. 3. Numerical results We calculate nuclear shadowing for calcium and lead from the above displayed equations. As was mentioned in the Introduction, only the valence qq-
part of the photon is taken into account, but the higher Fock components containing gluons and sea quarks are neglected, as well as the effect of antishadowing. Therefore, we do not compare our results with data, but only to the standard approach Ž3. - Ž4.. We do the same calculations again, using the free Green function Ž13.. This makes it possible to disentangle between the influence of higher scattering terms and the formfactor. We approximate the cross section by the dipole form s Ž r . s Cr 2 , C f 3, which is a good approximation at r ) 0.2 y 0.3 fm w22x. However, we calculated the proton structure function F2 Ž x,Q 2 . perturbatively Žwe fixed the quark masses at m q s 0.3 GeV, m s s 0.45 GeV and m c s 1.5 GeV. what leads to an additional logarithmic r-dependence at small r. This is important since results in the double-log Q 2 dependence of F2 . Nuclear shadowing, however, is dominated by soft fluctuations with large separation w12x, therefore, the dipole form of the cross section is sufficiently accurate. We use a uniform density for all nuclei, rA s 0.16 fmy3 , what is sufficient for our purpose, comparison with the standard approach calculated under the same assumption. Within these approximations it is possible to solve Ž11. analytically. The solution is the harmonic oscillator Green function with a complex frequency w22x, W Ž r 2 , z 2 ;r 1 , z 1 . a s 2p sinh Ž vD z .
½
=exp y y
a 2
Ž r 22 q r12 . coth Ž vD z .
2 r2 P r1 sinh Ž vD z .
5
,
Ž 16 .
where D z s z 2 y z1 C rA v2si na Ž 1 y a . a 2 s yi C rA na Ž 1 y a . .
Ž 17 .
This formal solution properly accounts for all multiple scatterings and finite lifetime of hadronic fluctuations of the photon, as well as for fluctuations of the transverse separation of the qq pair.
B.Z. KopelioÕich et al.r Physics Letters B 440 (1998) 151–156
The results of calculations are shown in Fig. 2. The dashed curves show predictions of Ž3. which we call standard approach. The mean values of s 2 and s are calculated using the same qq distribution functions w19,9x as in Ž9. and the intermediate state
155
mass is fixed at M 2 s Q 2 . At low x - 0.01 shadowing saturates because q s 2 m N x < 1rR A . The thin solid curve also corresponds to a double scattering approximation, i.e. absorption Žthe second term in Ž11.. is omitted. However, the formfactor is treated
Fig. 2. Nuclear shadowing for calcium and lead. The dotted curve is calculated in the standard approach Ž3.. The thin solid curve corresponds to the double scattering approximation with the free Green function, Ž13., and the thick solid curve shows the full calculation, Ž16..
156
B.Z. KopelioÕich et al.r Physics Letters B 440 (1998) 151–156
properly, i.e. the kinetic term in Ž11. taking into account the relative transverse motion of the qq pair, correctly reproduces the phase shift. The difference between the curves is substantial. The thin solid curve does not show saturation even at x s 0.001. The next step is to do the full calculations and study importance of the higher order rescattering terms in Ž11.. The results are shown by the thick solid curves. Higher order scattering brings another substantial deviation Žespecially for lead. from the standard approach. At very low x the curves saturate at the level given by Ž2..
4. Conclusions and outlook We suggest a solution for the problem of nuclear shadowing in DIS with correct quantum-mechanical treatment of multiple interaction of the virtual photon fluctuations and of the nuclear formfactor. We perform numerical calculations for qq fluctuations of the photon and find a significant difference with known approximations. Realistic calculations to be compared with data on nuclear shadowing should incorporate the higher Fock components which include gluons. The same path integral technique can be applied in this case. The x-dependence of the dipole cross section s Ž r, x . Žcorrelated with r w23x. should be taken into account. One should also include the effect of anti-shadowing, although it is only a few percent. A realistic form for the nuclear density should be used Žthis can be done replacing rAŽ b, z . by a multistep function like in w22x.. We are going to settle these problems in a forthcoming paper.
Acknowledgements We are grateful for stimulating discussions to Jorg ¨ Hufner and Gerry Garvey who read the paper and ¨ made many useful comments.
The work of J.R. and A.V.T was supported by the Gesellschaft fur ¨ Schwerionenforschung, GSI, grant ¨ T, and B.K. was partially supported by HD HUF European Network: Hadronic Physics with Electromagnetic Probes, No FMRX CT96-0008, and by INTAS grant No 93-0239ext. J.R and A.V.T. greatly acknowledge the hospitality at the MPI.
References w1x O.V. Kancheli, Sov. Phys. JETP Lett. 18 Ž1973. 274. w2x L.V. Gribov, E.M. Levin, M.G. Ryskin, Phys. Rept. 100 Ž1983. 1. w3x A.H. Mueller, J. Qiu, Nucl. Phys. B 268 Ž1986. 427. w4x J. Qiu, Nucl. Phys. B 291 Ž1987. 746. w5x L.D. Landau, I.Ya. Pomeranchuk, Dokl. Akad. Nauk SSSR 92 Ž1953. 535; ibid 735. In English see in L.D. Landau The Collected Papers of L.D. Landau ŽPergamon Press, New York, 1965. w6x T.H. Bauer, R.D. Spital, D.R. Yennie, F.M. Pipkin, Rev. Mod. Phys. 50 Ž1978. 261. w7x L.L. Frankfurt, M.I. Strikman, Phys. Rept. 160 Ž1988. 235. w8x S.J. Brodsky, H.J. Lu, Phys. Rev. Lett. 64 Ž1990. 1342. w9x N.N. Nikolaev, B.G. Zakharov, Z. Phys. C 49 Ž1991. 607. w10x W. Melnitchouk, A.W. Thomas, Phys. Lett. B 317 Ž1993. 437. w11x G. Piller, W. Ratzka, W. Weise, Z. Phys. A 352 Ž1995. 427. w12x B.Z. Kopeliovich, B. Povh, Phys. Lett. B 367 Ž1996. 329; Z. Phys. A 356 Ž1997. 467. w13x Al.B. Zamolodchikov, B.Z. Kopeliovich, L.I. Lapidus, Sov. Phys. JETP Lett. 33 Ž1981. 612. w14x S.J. Brodsky, A. Mueller, Phys. Lett. B 206 Ž1988. 685. w15x G. Bertsch, S.J. Brodsky, A.S. Goldhaber, J.F. Gunion, Phys. Rev. Lett. 47 Ž1981. 297. w16x J.D. Bjorken, J. Kogut, Phys. Rev. D 8 Ž1973. 1341. w17x V.N. Gribov, Sov. Phys. JETP 57 Ž1969. 1306. w18x V. Karmanov, L. Kondratyuk, JETP Lett. 18 Ž1973. 451. w19x S. Gevorkyan, A.M. Kotsinian, V.M. Jaloian, Phys. Lett. 212B Ž1988. 251. w20x B.G. Zakharov,Light-cone path integral approach to the LPM effect, MPI-H-V44-1997 Žunpublished.. w21x J. Raufeisen, A.V. Tarasov, paper in preparation. w22x B.Z. Kopeliovich, B.G. Zakharov, Phys. Rev. D 44 Ž1991. 3466. w23x B.Z. Kopeliovich, B. Povh, hep-phr9806284.
Physics Letters B 440 Ž1998. 151–156
Challenges of nuclear shadowing in DIS B.Z. Kopeliovich
a,c
, J. Raufeisen b, A.V. Tarasov
b,c
a
b
Max-Planck Institut fur ¨ Kernphysik, Postfach 103980, 69029 Heidelberg, Germany Institut fur ¨ Theoretische Physik der UniÕersitat, ¨ Philosophenweg 19, 69120 Heidelberg, Germany c Joint Institute for Nuclear Research, Dubna, 141980 Moscow Region, Russia Received 3 July 1998; revised 10 August 1998 Editor: P.V. Landshoff
Abstract Nuclear shadowing in DIS at moderately small x is suppressed by the nuclear formfactor and depends on the effective mass of a hadronic fluctuation of the virtual photon. We propose a solution to the problems Ži. of how to combine a definite transverse size of the fluctuation with a definite effective mass, and Žii. of how to include the nuclear formfactor in the higher multiple scattering terms. Comparison of the numerical results with known approximations shows a substantial difference. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Shadowing in deep-inelastic scattering ŽDIS. off nuclei is a hot topic for the last two decades. In the infinite momentum frame of the nucleus it can be interpreted as a result of parton fusion leading to a diminishing parton density at low Bjorken x w1–4x. A more intuitive picture arises in the rest frame of the nucleus where the same phenomenon looks like nuclear shadowing of hadronic fluctuations of the virtual photon w5–12x. To crystallize the problem and its solution we restrict ourselves in this paper to only quark-antiquark fluctuations of the photon, neglecting those higher Fock components which contain gluons and qq pairs from the sea. The lifetime of the qq fluctuation Žcalled coherence time. is given by 2n tc s 2 Ž 1. Q qM 2 where n is the photon energy, Q 2 its virtuality and M is the effective mass of the qq pair.
Provided that the coherence time is much longer than the nuclear radius, l c 4 R A , the total cross section on a nucleus reads w13x, )
g stot
A
Ž x ,Q 2 .
s 2 d 2 b d 2 r Gg ) Ž Q 2 ,r .
H H
1 = 1 y exp y s Ž r . T Ž b . 2
½
5
1 ' 2 d 2 b 1 y exp y s Ž r . T Ž b . 2
H
½ ¦
;5
.
Ž 2.
Here Gg ) Ž Q 2 ,r . characterizes the probability for the photon to develop a qq fluctuation with transverse separation r. The condition t c 4 R A insures that the r does not vary during propagation through the nucleus ŽLorentz time dilation.. Then the qq pair with a definite transverse separation is an eigenstate
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 0 7 2 - 7
B.Z. KopelioÕich et al.r Physics Letters B 440 (1998) 151–156
152
of the interaction with the eigenvalue of the total cross section s Ž r .. Therefore, one can apply the eikonal expression Ž2. for the interaction with the nucleus. The nuclear thickness function T Ž b . s ` Hy` dz rAŽ b, z . is the integral of nuclear density over longitudinal coordinate z and depends on the impact parameter b. The color dipole cross section s Ž r . introduced in w13x vanishes like r 2 at small r ™ 0 due to color screening. This is the heart of the phenomenon called nowadays color transparency w14,13,15x. For this reason nuclear shadowing in Ž2. is dominated by large size fluctuations corresponding to highly asymmetric sharing of the longitudinal momentum carried by the q and q w16,7,9,12x. This leads to Q 2 scaling of shadowing. Note that the averaging of the whole exponential in Ž2. makes this expression different from the Glauber eikonal approximation where s Ž r . is averaged in the exponent. The difference is known as Gribov’s inelastic corrections w17x. In the case of DIS the Glauber approximation does not make sense, and the whole cross section is due to the inelastic shadowing. For the other case, t c ; R A , one has to take into account the variation of r during the propagation of the qq fluctuation through the nucleus. At present this can only be done for the double scattering term w12x in the expansion of the exponential in Ž2., )
A
)
N
g stot
g stot
f1y
1 ²s 2 Ž r . : 4 ²s Ž r . :
²T : d 2 b FA2 Ž q,b . q . . . ,
H
Ž 3. or in hadronic representation w18x, )
g stot
A
g)N stot
f1y
=
1 )
g 4pstot
N
²T : d 2 b dM X2
H H
d s Ž g ) N ™ XN . dM X2 dt
FA2 Ž q,b . q . . . , ts0
Ž 4.
where M X is the effective mass of the inelastic intermediate state, the mean nuclear thickness and the formfactor read, ²T : s
1 A
Hd
FA Ž q,b . s
2
b T 2 Ž b. ,
1 ²T :
Ž 5.
`
Hy`dz r Ž b, z . e A
iqz
,
Ž 6.
with longitudinal momentum transfer q s 1rt c given by Ž1.. In the case of Ž3. the uncertain fluctuation mass is fixed at M 2 s Q 2 , q s 2 m N x. Two expressions Ž3. and Ž4. are related since the integrated forward diffractive dissociation cross section g ) N ™ XN equals to ² s 2 :r16p . There are two problems remaining which are under discussion: Ø How the nuclear formfactor can be included in the higher order scattering terms which are of great importance for heavy nuclei? For instance, the shadowing term in Ž3., Ž4. for lead is of the order of one at low x, so the need of the higher order terms is obvious. Ø Even for the double scattering term in Ž3. it is still unclear which argument should enter the formfactor. Indeed, the effective mass of the qq fluctuation needed for the coherence time in Ž1. cannot be defined in the quark representation with a definite qq separation. On the other hand, Eq. Ž4. exhibits an explicit dependence on M X and the longitudinal momentum transfer is known. However, unknown in this case is the absorptive cross section of the intermediate state X. We suggest a solution of both problems in the next section. The goal of this paper is restricted to the study of the difference between the predictions of the correct quantum-mechanical treatment of nuclear shadowing and known approximations. We do it on an example of the valence qq part of the photon and neglect the higher Fock components containing gluons and sea quarks, which may be important if to compare with data especially at very low x. Nuclear anti-shadowing effect is omitted as well, since we believe it is beyond the shadowing dynamics Že.g. bound nucleon swelling.. Numerical results and a comparison with the standard approach are presented in Section 3.
B.Z. KopelioÕich et al.r Physics Letters B 440 (1998) 151–156
153
2. The Green function of a qq pair in nuclear medium We start with the generalizing of Eq. Ž2. for the case l c F R A , )
g stot
A
1
)
Ž x ,Q 2 . s Hd 2 b H d a stotg A Ž x ,Q 2 ;b, a . , 0
Ž 7. where g) A stot Ž x ,Q 2 ;b, a . sT Ž b.
Hd
Fig. 1. A cartoon for the shadowing Žnegative. term in Ž8.. The Green function W Ž r 2 , z 2 ;r 1 , z 1 . results from the summation over different paths of the qq pair propagation through the nucleus.
2
2
r Cg ) Ž r , a . s Ž r .
`
y 2 Re
Hy`dz
1
rA Ž b, z 1 .
`
=
Hz dz
2
rA Ž b, z 2 . A Ž z 1 , z 2 , a . .
Ž 8.
1
The first term in r.h.s. of Ž8. corresponds to the second, lowest order in s Ž r .T Ž b ., term in expansion of the exponential in Ž2.. The shadowing terms are contained in the second term in Ž8.. Cg ) Ž r, a . is the Žnon-normalized. wave function of the qq fluctuation of the virtual photon, where a is the fraction of the light-cone momentum of the photon carried by the quark. An explicit expression of transverse and longitudinally polarized photons can be found in w19,9x. The function AŽ z 1 , z 2 , a . in Ž8. reads Žsee also in w20x., AŽ z1 , z 2 , a . 1 s d 2 r 1 d 2 r 2 Cg )) Ž r 2 , a . W Ž r 2 , z 2 ;r 1 , z 1 . 4 = Cg ) Ž r 1 , a . s Ž r 2 . s Ž r 1 . e i q mi n Ž z 2yz 1 . , Ž 9 . with Q 2a Ž 1 y a . q m 2q qmin s . Ž 10 . 2 na Ž 1 y a . This is the fraction of the longitudinal momentum transfer q s 1rt c s qmin q D q, which originates from longitudinal motion. Another part D q s k T2rw2 na Ž1 y a .x which is due to the transverse motion of the qq with momentum k T is included in W Ž r 2 , z 2 ;r 1 , z 1 . Žsee below.. The second Žshadowing. term in Ž8. is illustrated in Fig. 1. At the point z 1 the photon diffractively produces the qq pair Žg ) N ™ qqN . with transverse separation r 1. The pair propagates through the nu-
H
cleus along arbitrarily curved trajectories Žshould be summed over. and arrives at the point z 2 with a separation r 2 . The initial and the final separations are controlled by the distribution amplitude Cg ) Ž r .. While passing the nucleus the qq pair interacts with bound nucleons via the cross section s Ž r . which depends on the local separation r. The function W Ž r 2 , z 2 ;r 1 , z 1 . describing the propagation of the pair from z 1 to z 2 also includes that part of the phase shift between the initial and the final photons, which is due to transverse motion of the quarks, while longitudinal motion is already included in Ž10. via the exponential. Thus, Eq. Ž8. does not suffer from either of the two problems of the approximations Ž3. - Ž4.. The longitudinal momentum transfer is known and all the multiple interactions are included. The propagation function W Ž r 2 , z 2 ;r 1 , z 1 . in Ž9. satisfies the equation w20x, i
E W Ž r 2 , z 2 ;r 1 , z 1 . E z2 sy
D Ž r2 . 2 na Ž 1 y a . i
W Ž r 2 , z 2 ;r 1 , z 1 .
s Ž r 2 . rA Ž b, z 2 . W Ž r 2 , z 2 ;r 1 , z1 . , Ž 11 . 2 with the boundary condition W Ž r 2 , z 1;r 1 , z 1 . s d Ž r 2 y r 1 .. The Laplacian DŽ r 2 . acts on the coordinate r 2 . The full derivation of Ž11. will be given elsewhere w21x. Here we only notice that it looks natural like Schrodinger equation with the kinetic term ¨ Drw2 na Ž1 y a .x which takes care of the varying effective mass of the qq pair and provides a proper phase shift, and z 2 plays the role of the time. The y
B.Z. KopelioÕich et al.r Physics Letters B 440 (1998) 151–156
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imaginary part of the optical potential describes the absorptive process. In the ‘‘frozen’’ limit n ™ ` the kinetic term in Ž11. can be neglected and W Ž r 2 , z 2 ;r 1 , z 1 . z2 1 s d Ž r 2 y r 1 . exp y s Ž r 2 . dz rA Ž b, z . . 2 z1
H
Ž 12 . When this expression is substituted into Ž8. - Ž9. and with qmin ™ 0 one arrives at result Ž2. with 2 Gg ) Ž Q 2 ,r . s H01 d a Cg ) Ž r , a . . We can also recover the approximation Ž3. - Ž4. if one neglects the absorption of the qq pair in the medium. Then W becomes the Green function of a free motion, W Ž r 2 , z 2 ;r 1 , z 1 . 1 s 2p
H
s ™0
d 2 k exp ik Ž r 2 y r 1 . q
ik 2 Ž z 2 y z 1 . 2 na Ž 1 y a .
,
Ž 13 . where k is the transverse momentum of the quark. With this expression the shadowing term in Ž8. reproduces the second term in Ž4.. Indeed, the amplitude of the photon diffractive dissociation in the plane wave approximation reads, 1 fd d Ž k . s d 2 r Cg ) Ž r , a . s Ž r . e i k r . Ž 14 . 2 Therefore, Ž9. can be represented as, AŽ z1 , z 2 , a . 1 2 s d 2 k fd d Ž k . 2p Q 2a Ž 1 y a . q m 2q q k 2 = exp i Ž z 2 y z 1 . 2 na Ž 1 y a .
H
H
Ž 15 . 2
s Ž m2q q k 2 .ra Ž1 y a .
Taking into account that M is the effective mass squared of the qq pair and substituting Ž15. to Ž8. we arrive at Eq. Ž4.. 3. Numerical results We calculate nuclear shadowing for calcium and lead from the above displayed equations. As was mentioned in the Introduction, only the valence qq-
part of the photon is taken into account, but the higher Fock components containing gluons and sea quarks are neglected, as well as the effect of antishadowing. Therefore, we do not compare our results with data, but only to the standard approach Ž3. - Ž4.. We do the same calculations again, using the free Green function Ž13.. This makes it possible to disentangle between the influence of higher scattering terms and the formfactor. We approximate the cross section by the dipole form s Ž r . s Cr 2 , C f 3, which is a good approximation at r ) 0.2 y 0.3 fm w22x. However, we calculated the proton structure function F2 Ž x,Q 2 . perturbatively Žwe fixed the quark masses at m q s 0.3 GeV, m s s 0.45 GeV and m c s 1.5 GeV. what leads to an additional logarithmic r-dependence at small r. This is important since results in the double-log Q 2 dependence of F2 . Nuclear shadowing, however, is dominated by soft fluctuations with large separation w12x, therefore, the dipole form of the cross section is sufficiently accurate. We use a uniform density for all nuclei, rA s 0.16 fmy3 , what is sufficient for our purpose, comparison with the standard approach calculated under the same assumption. Within these approximations it is possible to solve Ž11. analytically. The solution is the harmonic oscillator Green function with a complex frequency w22x, W Ž r 2 , z 2 ;r 1 , z 1 . a s 2p sinh Ž vD z .
½
=exp y y
a 2
Ž r 22 q r12 . coth Ž vD z .
2 r2 P r1 sinh Ž vD z .
5
,
Ž 16 .
where D z s z 2 y z1 C rA v2si na Ž 1 y a . a 2 s yi C rA na Ž 1 y a . .
Ž 17 .
This formal solution properly accounts for all multiple scatterings and finite lifetime of hadronic fluctuations of the photon, as well as for fluctuations of the transverse separation of the qq pair.
B.Z. KopelioÕich et al.r Physics Letters B 440 (1998) 151–156
The results of calculations are shown in Fig. 2. The dashed curves show predictions of Ž3. which we call standard approach. The mean values of s 2 and s are calculated using the same qq distribution functions w19,9x as in Ž9. and the intermediate state
155
mass is fixed at M 2 s Q 2 . At low x - 0.01 shadowing saturates because q s 2 m N x < 1rR A . The thin solid curve also corresponds to a double scattering approximation, i.e. absorption Žthe second term in Ž11.. is omitted. However, the formfactor is treated
Fig. 2. Nuclear shadowing for calcium and lead. The dotted curve is calculated in the standard approach Ž3.. The thin solid curve corresponds to the double scattering approximation with the free Green function, Ž13., and the thick solid curve shows the full calculation, Ž16..
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B.Z. KopelioÕich et al.r Physics Letters B 440 (1998) 151–156
properly, i.e. the kinetic term in Ž11. taking into account the relative transverse motion of the qq pair, correctly reproduces the phase shift. The difference between the curves is substantial. The thin solid curve does not show saturation even at x s 0.001. The next step is to do the full calculations and study importance of the higher order rescattering terms in Ž11.. The results are shown by the thick solid curves. Higher order scattering brings another substantial deviation Žespecially for lead. from the standard approach. At very low x the curves saturate at the level given by Ž2..
4. Conclusions and outlook We suggest a solution for the problem of nuclear shadowing in DIS with correct quantum-mechanical treatment of multiple interaction of the virtual photon fluctuations and of the nuclear formfactor. We perform numerical calculations for qq fluctuations of the photon and find a significant difference with known approximations. Realistic calculations to be compared with data on nuclear shadowing should incorporate the higher Fock components which include gluons. The same path integral technique can be applied in this case. The x-dependence of the dipole cross section s Ž r, x . Žcorrelated with r w23x. should be taken into account. One should also include the effect of anti-shadowing, although it is only a few percent. A realistic form for the nuclear density should be used Žthis can be done replacing rAŽ b, z . by a multistep function like in w22x.. We are going to settle these problems in a forthcoming paper.
Acknowledgements We are grateful for stimulating discussions to Jorg ¨ Hufner and Gerry Garvey who read the paper and ¨ made many useful comments.
The work of J.R. and A.V.T was supported by the Gesellschaft fur ¨ Schwerionenforschung, GSI, grant ¨ T, and B.K. was partially supported by HD HUF European Network: Hadronic Physics with Electromagnetic Probes, No FMRX CT96-0008, and by INTAS grant No 93-0239ext. J.R and A.V.T. greatly acknowledge the hospitality at the MPI.
References w1x O.V. Kancheli, Sov. Phys. JETP Lett. 18 Ž1973. 274. w2x L.V. Gribov, E.M. Levin, M.G. Ryskin, Phys. Rept. 100 Ž1983. 1. w3x A.H. Mueller, J. Qiu, Nucl. Phys. B 268 Ž1986. 427. w4x J. Qiu, Nucl. Phys. B 291 Ž1987. 746. w5x L.D. Landau, I.Ya. Pomeranchuk, Dokl. Akad. Nauk SSSR 92 Ž1953. 535; ibid 735. In English see in L.D. Landau The Collected Papers of L.D. Landau ŽPergamon Press, New York, 1965. w6x T.H. Bauer, R.D. Spital, D.R. Yennie, F.M. Pipkin, Rev. Mod. Phys. 50 Ž1978. 261. w7x L.L. Frankfurt, M.I. Strikman, Phys. Rept. 160 Ž1988. 235. w8x S.J. Brodsky, H.J. Lu, Phys. Rev. Lett. 64 Ž1990. 1342. w9x N.N. Nikolaev, B.G. Zakharov, Z. Phys. C 49 Ž1991. 607. w10x W. Melnitchouk, A.W. Thomas, Phys. Lett. B 317 Ž1993. 437. w11x G. Piller, W. Ratzka, W. Weise, Z. Phys. A 352 Ž1995. 427. w12x B.Z. Kopeliovich, B. Povh, Phys. Lett. B 367 Ž1996. 329; Z. Phys. A 356 Ž1997. 467. w13x Al.B. Zamolodchikov, B.Z. Kopeliovich, L.I. Lapidus, Sov. Phys. JETP Lett. 33 Ž1981. 612. w14x S.J. Brodsky, A. Mueller, Phys. Lett. B 206 Ž1988. 685. w15x G. Bertsch, S.J. Brodsky, A.S. Goldhaber, J.F. Gunion, Phys. Rev. Lett. 47 Ž1981. 297. w16x J.D. Bjorken, J. Kogut, Phys. Rev. D 8 Ž1973. 1341. w17x V.N. Gribov, Sov. Phys. JETP 57 Ž1969. 1306. w18x V. Karmanov, L. Kondratyuk, JETP Lett. 18 Ž1973. 451. w19x S. Gevorkyan, A.M. Kotsinian, V.M. Jaloian, Phys. Lett. 212B Ž1988. 251. w20x B.G. Zakharov,Light-cone path integral approach to the LPM effect, MPI-H-V44-1997 Žunpublished.. w21x J. Raufeisen, A.V. Tarasov, paper in preparation. w22x B.Z. Kopeliovich, B.G. Zakharov, Phys. Rev. D 44 Ž1991. 3466. w23x B.Z. Kopeliovich, B. Povh, hep-phr9806284.