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Van Orden, JW Quark-Hadron Duality in Structur PUBLISHED VERSION Isgur, Nathan; Jeschonnek, Sabine; Melnitchouk, Wolodymyr; Van Orden, J. W. Quark-hadron duality in structure functions Physical Review D, 2001; 64(5):054005 © 2001 American Physical Society http://link.aps.org/doi/10.1103/PhysRevD.64.054005 PERMISSIONS PERMISSIONS http://publish.aps.org/authors/transfer -of-copyright-agreement http://publish.aps.org/authors/transfer-of-copyright-agreement http://link.aps.org/doi/10.1103/PhysRevD.62.093023 “The author(s), and in the case of a Work Made For Hire, as defined in the U.S. Copyright“The author(s), Act, 17 and U.S.C. 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”): §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 revision3. The right or modification, to use all or parton the of author(s)’the Article, web including home thepage APS or -employer’sprepared version website without and to makerevision copies or modification, of all or part on of the the author(s)’ Article, including web home the pageAPS- orprepared employer’s version website without and to revisionmake copies or modification, of all or part for of the the author(s)’ Article, including and/or thethe employer’s APS-prepared use versionfor educational without or researchrevision or purposes.” modification, for the author(s)’ and/or the employer’s use for educational or research purposes.” 5th April 2013 http://hdl.handle.net/2440/11173 PHYSICAL REVIEW D, VOLUME 64, 054005 Quark-hadron duality in structure functions Nathan Isgur,1 Sabine Jeschonnek,1 W. Melnitchouk,1,2 and J. W. Van Orden1,3 1Jefferson Lab, 12000 Jefferson Avenue, Newport News, Virginia 23606 2Special Research Centre for the Subatomic Structure of Matter, Adelaide University, Adelaide 5005, Australia 3Department of Physics, Old Dominion University, Norfolk, Virginia 23529 ͑Received 3 April 2001; published 6 August 2001͒ While quark-hadron duality is well established experimentally, the current theoretical understanding of this important phenomenon is quite limited. To expose the essential features of the dynamics behind duality, we use a simple model in which the hadronic spectrum consists of narrow resonances made of valence quarks. We qualitatively reproduce the features of duality as seen in electron scattering data within our model. We show that in order to observe duality, it is essential to use the appropriate scaling variable and scaling function. In addition to its great intrinsic interest in connecting the quark-gluon and hadronic pictures, an understanding of quark-hadron duality could lead to important benefits in extending the applicability of scaling into previously inaccessible regions. DOI: 10.1103/PhysRevD.64.054005 PACS number͑s͒: 12.40.Nn, 12.39.Ki, 13.60.Hb I. INTRODUCTION a free particle during the essential part of its interaction. This A. Background leads to the compellingly simple picture that the electron- nucleon cross section is determined in this kinematic region Duality is a much used and much abused concept. In by free electron-quark scattering, i.e. duality is exact for this some cases it is used to describe an equivalence between process in this kinematic regime. quark- and hadron-based pictures which is trivial; in others For inclusive inelastic electron scattering from a proton in an equivalence which is impossible. In almost all cases, the the scaling region, the cross section is determined by the conceptual framework in which duality is discussed and used convolution of a nonperturbative and currently difficult to is either hopelessly muddled or hopelessly abstract. Never- calculate parton distribution function with an electron-quark theless, the data indicate that some extremely interesting and scattering cross section determined by perturbative QCD. potentially very important ‘‘duality’’ phenomena are occur- For semileptonic decays of heavy quarks, e.g., ¯B!X l¯␯ , ring at low energy. c l one can prove using pQCD that the decay rate is determined We begin by making the trivial observation that any had- by that of the underlying heavy quark, in this case obtained ronic process can be correctly described in terms of quarks ! ¯␯ ͓ ͔ ϩ Ϫ! and gluons, assuming that quantum chromodynamics ͑QCD͒ from the process b cl l 1 .Ine e hadrons,itisthe ϩ Ϫ is the correct theory for strong interactions. While this state- underlying e e !qq¯ process that applies because of ment is obvious, it rarely has practical value, since in most pQCD. However, while duality applies to all of these phe- cases we can neither perform nor interpret a full QCD cal- nomena, we will see that even in these special processes we culation. We will refer to the above statement that any had- must invoke an averaging procedure to identify the hadronic ronic process can be described by a full QCD calculation as results with the quark-gluon predictions. ‘‘degrees of freedom duality’’; if one could perform and in- An important application of quark-hadron duality is QCD terpret the calculations, it would not matter at all which set sum rules; see, e.g., Ref. ͓2͔. There, properties of the ground of states—hadronic states or quark and gluon states—was state are calculated by matching results obtained in the par- used. ton picture, using the operator product expansion, and in the On the other hand, there are rare cases where the average hadron picture, using dispersion relations, in a certain kine- of hadronic observables is described by a perturbative QCD matic region where the ground state is dominant. The region ͑PQCD͒ calculation. We reserve the use of the term ‘‘dual- of applicability of duality in QCD sum rules was studied ity’’ to describe these rare correspondences, in contrast to the extensively in various channels and structure functions, and trivial ‘‘degrees of freedom duality’’ described above. In has been studied more recently in quantum mechanical mod- these rare cases, a quark-gluon calculation leads to a very els ͓3͔. simple description of some phenomenon even though this In addition to its need of an averaging procedure, it is phenomenon ‘‘materializes’’ in the form of hadrons. Deep easy to see that the pQCD picture of inelastic electron scat- inelastic scattering is the prototypical example, and the one tering must fail for Q2!0. For duality to hold for the on which we focus here. These rare examples are all charac- nucleon structure functions in this case, the elastic electric terized by a special choice of kinematic conditions which proton and neutron form factors, which take the value of the serve to expose the ‘‘bare’’ quarks and gluons of the QCD nucleon charges for Q2!0, would have to be reproduced by Lagrangian. In the case of deep inelastic scattering, the kine- electron scattering off the corresponding u and d quarks. This matics are such that the struck quark receives so much en- is possible for the proton since the squares of the charges of ergy over such a small space-time region that it behaves like two u quarks and one d quark add up to 1 ͓4͔. However, for 0556-2821/2001/64͑5͒/054005͑8͒/$20.0064 054005-1 ©2001 The American Physical Society ISGUR, JESCHONNEK, MELNITCHOUK, AND VAN ORDEN PHYSICAL REVIEW D 64 054005 ϩ␣ ϩ 2ϩ the neutron, the squared quark charges cannot add up to 0, so 3(e1 e2) •••, and one can see by explicit calculations it is clear that local duality in inclusive inelastic electron in models that ͑up to phase space factors͒ the cross terms in scattering from a neutron must fail for Q2!0. Also, we this sum will cancel to give a cross section proportional to know that duality must fail for polarized structure functions 2ϩ 2 e1 e2 once averaged over nearby even and odd parity reso- 2 at low Q , as the Ellis-Jaffe ͑EJ͒ sum rule and the nances. It is clear that such target- and process-dependence is ͑ ͒ Gerassimov-Drell-Hearn GDH sum rule, which can be worthy of study. However, in this paper we will restrict our- ␯ 2 2 written as integrals over g1( ,Q ) at different Q , are nega- selves to a model with e ϭ0 so that local duality might ͑ 2ϭ ͑ 2 tive GDH sum rule for Q 0) and positive EJ sum rule at apply even at low Q2 ͓9͔. 2 2͒ ͓ ͔ Q of several GeV , respectively 5 . The question of the validity of low energy duality, i.e., Thus duality in inelastic electron scattering has to hold in duality in electron scattering at finite beam energies in in- the scaling regime and must in general break down at low elastic electron scattering after suitable averaging, is as old energy. Obviously, a very interesting question is what hap- as the first inclusive electron scattering experiments them- pens in between these regimes, i.e., how does duality break selves. It began with the seminal paper of Bloom and Gilman down? This paper answers this question, which is not only ͓ ͔ 10 , which made the observation that the inclusive F2 struc- interesting in itself, but also crucial for practical, quantitative 2 applications of duality. ture function in the resonance region at low Q generally oscillates about and averages to a global scaling curve which 2 B. Introducing local averaging and our model describes high Q data. More recently, interest in Bloom- Gilman duality was revived with the collection of high pre- We begin by discussing the issue of averaging. If duality cision data on the F2 structure function from Jefferson Lab is relevant at all at low energy, then it is quite obvious that ͓11͔.
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