We re-examine the role of vector meson dominance in nuclear shadowing at low
Q^2. We find that models which incorporate both vector meson and partonic
mechanisms are consistent with both the magnitude and the Q^2 slope of the
shadowing data.Comment: 7 pages, 2 figures; to appear in Phys. Rev. 

A. Capella

A. Donnachie

A. W. Thomas

A.H. Mueller

B. Badelek

B.Z. Kopeliovich

C. Boros

C.L. Bilchak

E.L. Berger

F.E. Close

G. Ingelman

G. Piller

G. Shaw

G.A. Schuler

G.P. Zeller

J. Kwiecinski

J. Qiu

J.C. Collins

K. Goulianos

L.L. Frankfurt

M. Arneodo

M.R. Adams

N.N. Nikolaev

P. Amaudruz

R.J. Glauber

S. Kovalenko

S. Kumano

S.J. Brodsky

T.H. Bauer

V.N. Gribov

W. Melnitchouk

English

arXiv

We reexamine the role of vector meson dominance in nuclear shadowing at low Q². We find that models incorporating both vector meson and partonic mechanisms are consistent with both the magnitude and the Q² slope of the shadowing data.W. Melnitchouk, A. W. Thoma

Melnitchouk, W.

Thomas, A.

Adelaide Research & Scholarship

PUBLISHED VERSION     Melnitchouk, Wolodymyr; Thomas, Anthony William  Nuclear shadowing at low Q² Physical Review C, 2003; 67(3):038201   © 2003 American Physical Society http://link.aps.org/doi/10.1103/PhysRevC.67.038201                         http://hdl.handle.net/2440/11104 PERMISSIONS http://publish.aps.org/authors/transfer-of-copyright-agreement   “The author(s), and in the case of a Work Made For Hire, as defined in the U.S. Copyright Act, 17 U.S.C. §101, the employer named [below], shall have the following rights (the “Author Rights”): [...] 3. The right to use all or part of the Article, including the APS-prepared version without revision or modification, on the author(s)’ web home page or employer’s website and to make copies of all or part of the Article, including the APS-prepared version without revision or modification, for the author(s)’ and/or the employer’s use for educational or research purposes.”    25th March 2013  PHYSICAL REVIEW C 67, 038201 ~2003!Nuclear shadowing at low Q2W. MelnitchoukJefferson Laboratory, 12000 Jefferson Avenue, Newport News, Virginia 23606A. W. ThomasSpecial Research Centre for the Subatomic Structure of Matter, and Department of Physics and Mathematical Physics,University of Adelaide, Adelaide 5005, Australia~Received 6 August 2002; published 6 March 2003!We reexamine the role of vector meson dominance in nuclear shadowing at low Q2. We find that modelsincorporating both vector meson and partonic mechanisms are consistent with both the magnitude and the Q2slope of the shadowing data.DOI: 10.1103/PhysRevC.67.038201 PACS number~s!: 24.85.1p, 12.40.Vv, 12.40.Nn, 13.60.HbThere has been renewed interest recently in the problemof nuclear shadowing in structure functions at low and inter-mediate Q2. In part, this has been prompted by the analysisof the NuTeV Collaboration @1# of neutrino-nucleus crosssections and subsequent questions about nuclear shadowingcorrections when extracting nucleon quark distributions orelectroweak parameters @2–4#. Indeed, shadowing in neu-trino scattering has received considerably less attention thanin electromagnetic reactions, and currently there are propos-als to utilize high intensity neutrino and antineutrino beamsto perform high statistics measurements of n/n¯ -nucleus crosssections at Fermilab @5#. A pressing need exists, therefore, tounderstand the differences between nuclear shadowing ef-fects in charged lepton and neutrino scattering @6,7#, espe-cially at low Q2.An extensive review of both data and models of nuclearshadowing was given recently by Piller and Weise @8#. Be-fore one can reliably tackle nuclear corrections in neutrinoscattering, however, it is vital to determine the relevant de-grees of freedom responsible for shadowing in charged lep-ton scattering, where data are much more copious. The bestavailable data on nuclear shadowing, including the Q2 de-pendence, are from the New Muon Collaboration ~NMC!@9–11#. We shall concentrate on a model based on a two-phase picture of nuclear shadowing @12–14#, similar to thatpioneered by Kwiecinski and Badelek @15–17#, which wepublished just before the release of the final NMC data @11#.For clarity we briefly review this model.At high virtuality, the interaction of a photon with anucleus can be efficiently parametrized through a partonicmechanism, involving diffractive scattering through thedouble and triple Pomeron @18#. For Q2*2 GeV2, the con-tribution to the nuclear structure function F2A ~per nucleon!from this mechanism can be written asd (P)F2A~x ,Q2!5 1AEyminAdy f P/A~y !F2P~xP ,Q2!, ~1!where f P/A(y) is the Pomeron (P) flux, and F2P is the effec-tive Pomeron structure function @19#. The variable y5x(11M X2 /Q2) is the light-cone momentum fraction carried by0556-2813/2003/67~3!/038201~3!/$20.00 67 0382the Pomeron (M X is the mass of the diffractive hadronicdebris! and xP5x/y is the momentum fraction of thePomeron carried by the struck quark. The dependence of F2Pon Q2 at large Q2, in the region where perturbative QCD canbe applied, arises from radiative corrections to the partondistributions in the Pomeron @17,20#, which leads to a weak,logarithmic, Q2 dependence for the shadowing correctiond (P)F2A. Alone, the P contribution to shadowing would givea structure function ratio F2A/F2D that would be almost flat forQ2*2 GeV2 @21#.On the other hand, the description of shadowing at lowQ2 requires a higher-twist mechanism, such as vector mesondominance ~VMD!, which can map smoothly onto the pho-toproduction limit at Q250. The VMD model is empiricallybased on the observation that some aspects of the interactionof photons with hadronic systems resemble purely hadronicinteractions @22,23#. In QCD language this is understood interms of the coupling of the photon to a correlated qq¯ pairwith low invariant mass, which may be approximated as avirtual vector meson. One can then estimate the amount ofshadowing in terms of the multiple scattering of the vectormeson using Glauber theory @24#. The corresponding VMDcorrection to F2A isd (V)F2A~x ,Q2!5 1AQ2p (VM V4 dsVAf V2 ~Q21M V2 !2, ~2!where dsVA is the shadowing correction to the vectormeson-nucleus cross section, f V is the photon-vector mesoncoupling strength @22#, and M V is the vector meson mass. Inpractice, only the lowest mass vector mesons (V5r0,v ,f)are important at low Q2. ~Inclusion of higher mass states,including continuum contributions, leads to so-called gener-alized vector meson dominance models @25#.! The vectormeson propagators in Eq. ~2! lead to a strong Q2 dependenceof d (V)F2A at low Q2, which peaks at Q2;1 GeV2, althoughone should note that the nucleon structure function itself alsovaries rapidly with Q2 in this region. For Q2→0 and fixed x,d (V)F2A disappears because of the vanishing of the total F2A.Furthermore, since this is a higher twist effect, shadowing in©2003 The American Physical Society01-1BRIEF REPORTS PHYSICAL REVIEW C 67, 038201 ~2003!the VMD model dies off quite rapidly between Q2;1 and10 GeV2, so that for Q2*10 GeV2 it is almost negligible—leaving only the diffractive partonic term, d (P)F2A.The accuracy of the model can be tested by looking fordeviations from the logarithmic Q2 dependence of shadow-ing at low and intermediate Q2. Actually, a detailed analysisof the Q2 dependence of the NMC data, as well as the lower-Q2 Fermilab E665 data @26#, was performed in Refs. @13,14#for various nuclei from A52 to A5208 ~viz., for D, Li, Be,C, Al, Ca, Fe, Sn, Xe, and Pb!. Ratios of F2A/F2D were cal-culated @13,14# for a range of x(1025&x&0.1) andQ2 (0.03&Q2&100 GeV2). Subsequent to these analyses,high precision data on the Q2 dependence of Sn/C structurefunction ratios were published @11#, which provided the firstdetailed evidence concerning the Q2 dependence of nuclearshadowing.In Fig. 1 we show the calculated ratio R(Sn/C)[F2Sn/F2Cas a function of Q2 for x50.0125 ~solid curve! and x50.045 ~dashed!, compared with the NMC data @11#. Theoverall agreement between the model and the data is clearlyexcellent. In particular, the observed Q2 dependence of theratios is certainly compatible with that indicated by the NMCdata. At large Q2 (Q2*10 GeV2), the Q2 dependence isvery weak, as expected from a partonic, leading-twist mecha-nism @14#—see also Refs. @27–31#. In the smallest x bins,however, the Q2 values reach down to Q2’1 GeV2. Thedata on the C/D and Ca/D ratios analyzed in Ref. @14# ateven smaller x(x*0.0003) extend down to Q2’0.05 GeV2. This region is clearly inaccessible to anymodel involving only a partonic mechanism, and it is essen-tial to invoke a nonscaling mechanism here, such as vectormeson dominance. One should also note that, even thoughthe shadowing corrections may depend strongly on Q2, be-cause the nucleon structure function itself is rapidly varyingat low Q2, the Q2 dependence of the ratio will not be asstrong as in the absolute structure functions. In any case, thefact that the two-phase model @14# describes the NMC dataover such a wide range of Q2 gives one added confidence inextending this model to neutrino scattering @6#.FIG. 1. Q2 variation of the Sn/C structure function ratio in themodel of Ref. @14# for x50.0125 ~solid! and x50.045 ~dashed!.The data are from NMC @11#, with statistical and systematic errorsadded in quadrature.03820To illustrate the Q2 dependence of R over the full range ofx covered in the NMC experiment, Arneodo et al. @11# pa-rametrized the Sn/C ratio as R(Sn/C)5a1b ln Q2, and ex-tracted the logarithmic slopes b5dR/d ln Q2 as a function ofx. As illustrated in Fig. 2, the NMC find that the slopes arepositive and differ significantly from zero for 0.01,x,0.05, indicating that the amount of shadowing decreaseswith increasing Q2 @11#. The logarithmic slope b is found todecrease from ’0.04 at the smallest x value to zero at x*0.06. The result of the model calculation @14# is perfectlyconsistent with the NMC data over the full range of x cov-ered, as Fig. 2 demonstrates @see also Fig. 3~b! of Ref. @14##.In particular, the P-exchange mechanism alone, modified byapplying a factor Q2/(Q21Q02) @16,32# to ensure thatd (P)F2A→0 as Q2→0, is clearly insufficient @21# to describethe logarithmic slope in Q2 at low x, whereas the addition ofa VMD component does allow one to describe the data quitewell ~the shaded region indicates an estimate of the uncer-tainty in the model calculation!.In summary, the results of this analysis demonstrate that acombination of VMD at low Q2 to describe the transition tothe photoproduction region, with parton recombination, pa-rametrized via P-exchange, at high Q2 allows one to accu-rately describe shadowing in electromagnetic nuclear struc-ture functions over a large range of Q2. As well asconfirming that higher-twist effects are numerically impor-tant at intermediate Q2;1 –4 GeV 2, our findings also sug-gest that the two-phase model can serve as an excellent basison which to reliably tackle the question of shadowing inneutrino reactions.We thank M. Arneodo, A. Bru¨ll, and M. Szleper for pro-viding the NMC data. This work was supported by the Aus-tralian Research Council, and the U.S. Department of EnergyContract No. DE-AC05-84ER40150, under which the South-eastern Universities Research Association ~SURA! operatesthe Thomas Jefferson National Accelerator Facility ~Jeffer-son Lab!.FIG. 2. Logarithmic slope b in Q2 of the NMC Sn/C ratio as afunction of x @11#, compared with the nuclear shadowing model ofRef. @14#. The statistical and systematic errors are added in quadra-ture.1-2BRIEF REPORTS PHYSICAL REVIEW C 67, 038201 ~2003!@1# G.P. Zeller et al., NuTeV Collaboration, Phys. Rev. Lett. 88,091802 ~2002!; hep-ex/0207052.@2# G.A. Miller and A.W. 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Nuclear shadowing at low Q^2

Nuclear shadowing at low Q^2

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