PUBLISHED VERSION Isgur, Nathan; Jeschonnek, Sabine; Melnitchouk, Wolodymyr; Van Orden, J. W. -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

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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 . 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- 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 , assuming that ͑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. These data not only confirmed the existence of the we need to perform some sort of average: the smooth, ana- Bloom-Gilman duality to rather low values of Q2, but also lytical pQCD prediction cannot in general correspond ex- seem to demonstrate that for the proton the equivalence of actly to the generally highly structured hadronic data. For the averaged resonance and scaling structure functions holds low energies this requirement is universally accepted; how- also for each resonance so that duality also exists locally. ever, even in the ‘‘scaling’’ region one must average in prin- ͓ ͔ Here we present a model for the study of quark-hadron ciple. To see this, consider QCD in the large-Nc limit 6 .We can do this because no element of the pQCD results for deep duality in electron scattering that uses only a few basic in- inelastic scattering depends on the number of colors. How- gredients. That is, in addition to requiring that our model be ever, in this limit the hadronic spectrum consists entirely of relativistic, we assume confinement, and assume that it is ͑ infinitely narrow noninteracting resonances ͓7͔, i.e., there are sufficient to consider only valence quarks this latter simpli- only infinitely narrow spikes in the N →ϱ hadronic world. fication being underwritten, as mentioned previously, by the c ͒ Since the quark level calculation still yields a smooth scaling large-Nc limit . In addition, since our model is designed to curve, and the kinematic conditions for being in the scaling explore conceptual issues and not to be compared to data, →ϱ region are unchanged as Nc , we see that we must aver- and since we postpone addressing spin-dependent issues to age even in the scaling region. While in Nature the reso- later work, for simplicity we also take the quarks, electrons nances have fairly broad decay widths, so that the averaging and photons to be scalars. A model with these features will takes place automatically in the data, the large-Nc limit not give a realistic description of any data, but it should shows us that averaging is always required in principle. It is allow us to study the critical questions of when and why thus clearly important to be able to define this averaging duality holds. While this model is extremely simple, we see procedure, e.g., how large the intervals must be and which no impediment to extending it to describe a more realistic resonances have to be included. situation since we find that duality arises from the most basic It is easy to see that this procedure will not be universal, properties of our model. and will certainly not simply be that the resonances one-by- We make several more convenient simplifications. Al- one locally average the pQCD-derived scaling curve: the av- though it is our aim to study duality in electron scattering eraging method will depend on the process and on the target. from the nucleon, i.e., from a three-quark system, as a first Consider, as an illustration of these points, the case of a step we study these issues in what is effectively a one quark spinless quark and antiquark with charges e1 and e2 and system by considering such a quark to be confined to an ¯ infinitely massive antiquark. In the case of scalar quarks con- equal masses bound into a nonrelativistic q1q2 system. The inelastic electron scattering rate calculated at the quark level sidered here, we can therefore describe the system by the in leading twist will then be proportional to e2ϩe2. Since the Klein-Gordon equation. We also select for our confining po- 1 2 2 ជ 2 elastic state will be produced with a rate proportional to tential one which is linear in r, namely V (r)ϭ␣r , where ␣ ϩ 2 is a generalized, relativistic string constant. This choice al- (e1 e2) , it clearly cannot in general be locally dual to the scaling curve ͓8͔. How then is duality realized in this sys- lows us to obtain analytic solutions, without which the re- ជ ជ ͚ iq ri quired numerical work for this study would be daunting. In- tem? Consider the charge operator ieie • : from the ϭͱ ͱ␣ ϩ ϩ 2 ground state it excites even partial wave states with an am- deed, the energy eigenvalues, EN 2 (N 3/2) m , ϩ plitude proportional to e1 e2 and odd ones with an ampli- where m is the mass of interacting quark, can be readily Ϫ tude proportional to e1 e2. Thus the resonances build up a obtained by noting the similarity to the Schro¨dinger equation ␣ ϩ 2ϩ␣ Ϫ 2 cross section of the form 1(e1 e2) 2(e1 e2) for a non-relativistic harmonic oscillator potential: the solu-

054005-2 QUARK-HADRON DUALITY IN STRUCTURE FUNCTIONS PHYSICAL REVIEW D 64 054005 tions for the wave functions are the same as for the non- Scaling in the presence of confining final state interactions relativistic case. was previously investigated in Refs. ͓12–15͔, where similar In Sec. II we construct the structure function out of reso- conclusions are reached. This suggests that scaling may in- nances described by form factors, each of which individually deed be a trivial feature of a large class of simple quantum gives vanishing contributions at large momenta, and show mechanical models. Some sense of how this can occur can be that it both scales and, when suitably averaged, is equal to obtained by considering some of the properties of the rela- the ‘‘free quark’’ result. An analysis in terms of structure tivistic oscillator model used in this paper. In particular, con- function moments is presented in Sec. III. In Sec. IV we sider the properties of the square of the form factors. For a examine the onset of scaling, and the appearance of Bloom- fixed principal quantum number N, the form factor has a ͉ជ͉ ជ 2 ϭ ␤2 ␯ ϭ Ϫ Gilman duality, while in Sec. V we discuss the connection of maximum in q at qN 2 N. Using N EN E0 and EN ϭͱ ␤2 ϩ 2 Bloom-Gilman duality with duality in heavy quark systems. 2 N E0, it can be shown that Finally, in Sec. VI we summarize our results and mention Q2 some possible directions for future research. ␯ ϭ N ͑ ͒ N , 4 2E0 II. QUARK-HADRON DUALITY IN THE SCALING LIMIT 2 ϭ ជ 2 Ϫ␯2 where QN qN N . So the position of the peak in the av- ϭ The differential cross section for inclusive inelastic scat- eraged structure function occurs at uBj m/E0 where uBj ϭ 2 ␯ tering of a ‘‘scalar electron’’ via the exchange of a ‘‘scalar Q /2m is a scaled Bjorken scaling variable uBj ϵ photon’’ is (M/m)xBj which takes into account that as the mass of the →ϱ antiquark M Q¯ , the light constituent quark will carry only ␴ 4 a fraction of order m/E0 of the hadron’s infinite-momentum- d g E f 1 ϭ W, ͑1͒ frame momentum. Furthermore, for fixed qជ the structure ⍀ 2 4 dEf d f 16␲ Ei Q function falls off smoothly for energy transfers away from the peak value. The width of this peak as a function of en- where the scalar coupling constant g carries the dimension of ergy transfer also becomes constant for large ͉qជ͉. a mass, and the factor multiplying the scalar structure func- Now consider the integral of the structure function tion W corresponds to the Mott cross section. In a model ϱ where the only excited states are infinitely narrow reso- ⌺͑qជ 2͒ϭ ͵ d␯W͑␯,qជ 2͒ nances, W is given entirely by a sum of squares of transition 0 form factors weighted by appropriate kinematic factors, 1 ϭ ͗␺ ͉␳͑Ϫ ជ ͉͒␺ ͗͘␺ ͉␳͑ ជ ͉͒␺ ͘ N ͚ 000 q nlm nlm q 000 , max 1 nlm 4E0EN W ␯ ជ 2͒ϭ ͉ ជ ͉͒2␦ Ϫ Ϫ␯͒ ͑ ͒ ͑ ,q ͚ F0N͑q ͑EN E0 , 2 Nϭ0 4E0EN ͑5͒ where Nϭ2(nϪ1)ϩl with nϭ1,2,3, , and where ␳(qជ ) ជ ϵ ជ Ϫ ជ ជ ជ ••• where q pi p f , the form factor F0N represents a transition ϭ iq x ជ from the ground state to a state characterized by the principal e • . Since the form factor sum for a fixed q peaks about E ϭͱqជ 2ϩE2, we can substitute E →E and then quantum number N, and the sum over states N goes up to the Nmax 0 N Nmax maximum Nmax allowed kinematically. Note that for fixed, sum over the complete set of final states to obtain 2ϵ ជ 2Ϫ␯2 ϭϱ positive Q q , Nmax . 1 1 The excitation form factors can be derived using the re- ⌺͑qជ 2͒Х Х ͑6͒ 4E E 4E q currence relations of the Hermite polynomials. One finds 0 Nmax 0

N for large momentum transfer. Therefore, if we define the 1 ͉qជ͉ ជ ͑ ជ 2͒ϭ Nͩ ͪ ͑Ϫ ជ 2 ␤2͒ ͑ ͒ scaling function as Sϵ͉q͉W, as will be done below, the area F0N q i exp q /4 , 3 ͱN! ͱ2␤ under the scaling function becomes constant at large momen- tum transfer. where ␤ϭ␣1/4. As a precursor to our discussion of duality, Since the scaling function peaks at fixed uBj , smoothly we note that it will be a necessary condition for duality that falls about the peak, and has fixed width and constant area at these form factors ͑or more generally those corresponding to large momentum transfer, the model scales. It is a common some other model potential͒ can represent the pointlike free misconception that the presence of scaling implies that the ͚Nmax͉ ជ ͉2→ final states must become plane waves. In fact, the argument quark. It is in fact the case that Nϭ0 F0N(q) 1as →ϱ above makes it clear that scaling occurs when the structure Nmax , a relation which follows from the completeness of function becomes independent of the final states as in the the confined wave functions. Incidentally, an examination of closure approximation used here. ជ 2 the convergence of this sum as a function of ͉q͉ is sufficient To see duality clearly both experimentally and theoreti- to make the point that reproducing the behavior of a free cally, one needs to go beyond the Bjorken scaling variable ͉ជ͉2 S ϭ␯W quark requires more and more resonances as q increases xBj and the scaling function Bj that goes with it. This ͑details of this will be discussed in a forthcoming publica- is because in deriving Bjorken’s variable and scaling func- tion͒. tion, one not only assumes Q2 to be larger than any mass

054005-3 ISGUR, JESCHONNEK, MELNITCHOUK, AND VAN ORDEN PHYSICAL REVIEW D 64 054005

S 2 ⌫ϭ FIG. 1. The high energy scaling behavior of cq as a function of u for various values of Q . In panel A we have used 100 MeV to give the impression of real resonances even though this large value distorts the scaling curve somewhat; for any width equal to or smaller than this, the distortion is rather innocuous, and for ⌫→0, the structure function approaches the scaling function in Eq. ͑11͒, as shown in panel B. scale in the problem, but also that high Q2 ͑pQCD͒ dynamics The variable u takes values from 0 to a maximal, Q2 depen- controls the interactions. However, duality has its onset in dent value, which can go to infinity. The high energy scaling 2 S the region of low to moderate Q , and there masses and behavior of the appropriately rescaled structure function cq violations of asymptotic freedom do play a role. Bloom and is illustrated in Fig. 1. Gilman used a new, ad hoc scaling variable ␻Ј ͓10͔ in an The structure function has been evaluated using the phe- attempt to deal with this fact. In most contemporary data nomenologically reasonable parameters mϭ0.33 GeV and analyses, the Nachtmann variable ͓16,17͔ is used together ␣1/4ϭ0.4 GeV, though we remind the reader not to compare S with Bj . Nachtmann’s variable contains the target mass as a our results, which might resemble electron scattering from a scale, but neglects quark masses. For our model, the con- B , to nucleon data. To display it in a visually mean- stituent quark mass ͑assumed to arise as a result of sponta- ingful manner, the energy-dependent ␦-function has been neous chiral symmetry breaking͒ is vital at low energy, and a smoothed out by introducing an unphysical Breit-Wigner scaling variable that treats both target and quark masses is shape with an arbitrary but small width ⌫, chosen for pur- desirable. Such a variable was derived more than twenty poses of illustration years ago by Barbieri et al. ͓18͔ to take into account the masses of heavy quarks; we use it here given that after spon- ⌫ f taneous chiral symmetry breaking the nearly massless light ␦͑E ϪE Ϫ␯͒→ , ͑10͒ N 0 2␲ Ϫ Ϫ␯͒2ϩ ⌫ ͒2 quarks have become massive constituent quarks, calling it ͑EN E0 ͑ /2 xcq : 2 ϭ␲ ␲ ϩ ͓ Ϫ ⌫͔ 1 4m where the factor f /„( /2) arctan 2(EN E0)/ … ensures x ϭ ͑ͱ␯2ϩQ2Ϫ␯͒ͩ 1ϩͱ1ϩ ͪ . ͑7͒ that the integral over the ␦-function is identical to that over cq 2M 2 Q the Breit-Wigner shape. The curves in Fig. 1 show that scal- ing sets in rather rapidly. The resonances show up as bumpy The scaling function associated with this variable is given by structures in the low Q2 region ͑which will be discussed in Sec. IV below͒, a trace of which is visible for the Q2 S ϵ͉qជ͉Wϭͱ␯2ϩQ2W. ͑8͒ cq ϭ5 GeV2 curve. By taking the continuum limit for the energy and applying This scaling function and variable were derived for scalar Stirling’s formula, one can obtain an analytic expression for quarks which are free, but have a momentum distribution. the scaling curve, valid in the scaling region, for the transi- The derivation of a new scaling variable and function for tion of the quark from the ground state to the sum of all bound quarks will be published elsewhere. Numerically, this excited states: scaling variable does not differ very much from the one in ͑ ͒ Eq. 7 . Of course all versions of the scaling variable must Ϫ ͒2 1 ͑E0 mu converge to xBj and all versions of the scaling function must S ͑u͒ϭ exp ͩ Ϫ ͪ . ͑11͒ S 2 cq ͱ␲␤ ␤2 converge toward Bj for large enough Q . One can also eas- 4 E0 ily verify that in the limit m→0 one obtains the Nachtmann scaling variable from Eq. ͑7͒. In the following, we use the Of course we still need to verify that this scaling curve as S variable xcq and the scaling function cq . seen in Fig. 1 found by summing over hadrons is the same as We are now ready to look at scaling and duality in our the one which we would obtain from deep inelastic scattering model. Since the target has mass M→ϱ, it is convenient to off the quark, i.e., if we were to switch off the potential in the rescale the scaling variable xcq by a factor M/m: final state. In this case, the tower of hadronic states is re- placed by the free quark continuum. Duality predicts that the M results should be the same in the scaling limit, and by direct uϵ x . ͑9͒ m cq calculation we confirm this.

054005-4 QUARK-HADRON DUALITY IN STRUCTURE FUNCTIONS PHYSICAL REVIEW D 64 054005

III. MOMENTS OF STRUCTURE FUNCTIONS Bloom-Gilman duality relates structure functions at low and high Q2 averaged over appropriate intervals of the had- ronic mass W. As a quantitative measure of this feature of the data, one conventionally examines the Q2 dependence of moments of structure functions. The moments offer the cleanest connection with the operator product expansion of QCD, and provide a natural connection between duality in the high- and low-Q2 regions. By considering the moments, we also remove artifacts introduced through the smoothing procedure described above for the structure function itself. S 2 The moments of the structure function cq(u,Q ) are de- fined as 2 FIG. 2. Some moments M n as a function of Q . umax ͑ 2͒ϭ ͵ nϪ2S ͑ 2͒ ͑ ͒ M n Q duu cq u,Q , 12 rule is not applicable here because the scalar current which 0 couples to our quark is not conserved. Nonetheless, the mo- where u corresponds to the maximum value of u which is ments in Fig. 2 do serve to demonstrate that scaling is a max natural consequence of our model, and illustrate the relative kinematically accessible at a given Q2. Evaluating the mo- onset of scaling for different moments. ments of the structure function ͓Eq. ͑8͔͒ explicitly one has ͑provided the kinematics allow us to access all excited states͒, IV. ONSET OF SCALING AND BLOOM-GILMAN DUALITY ϱ r nϪ1 ͑ 2͒ϭͩ ͪ ͑ͱ␯2 ϩ 2Ϫ␯ ͒nϪ1 After studying the scaling behavior of the structure func- M n Q ͚ N Q N 2 2m Nϭ0 tions in our model at high Q and the moments over a range of 4-momentum transfers, we now study the structure func- E 2 ϫ 0 ͉ ͑ͱ␯2 ϩ 2͉͒2 ͑ ͒ tions at low Q . There, resonances are visibly dominant over F0N N Q , 13 EN a wide range in the scaling variable, not only in the large Nc limit, but also in Nature. Here, we consider a target where ␯ ϭ Ϫ ϭ ϩͱ ϩ 2 2 where N EN E0 and r 1 1 4m /Q . The higher only one quark carries all the charge of the system, so there moments, i.e. nϭ4,6,..., tend to emphasize the resonance is no forced breakdown of duality at Q2ϭ0 GeV2 of the region, as for fixed Q2, the resonances are found at larger type noted earlier for the neutron. Still, one cannot expect values of the scaling variable. The elastic contribution to the that the perturbative QCD result will describe even averaged moments is hadronic observables well at very low Q2: these are after all strong interactions. nϪ1 r Ϫ If local duality holds, one might expect the resonance M elastic͑Q2͒ϭͩ ͪ QnϪ1͉F ͑Q2͉͒2ϭun 1͉F ͑Q2͉͒2, n 2m 00 0 00 ‘‘spikes’’ to oscillate around the scaling curve and to average ͑14͒ to it, once Q2 is large enough. ͑We remind the reader that while scaling in deep-inelastic electron scattering from the 2 2 where u0(Q ) is the position in u of the ground state. Note nucleon is known from experiment to set in by Q elastic 2 2 2 that M n (Q ) becomes independent of n in the limit Q ϳ2 GeV , the target considered here corresponds to an in- → 2 0, approaching the value 1/4E0 and that the inelastic con- finitely heavy ‘‘meson’’ composed of scalar quarks interact- tributions to the moments vanish for vanishing Q2. ing with a scalar current, so one should not expect numeri- ϭ ͒ In Fig. 2 we show the n 2, 4, 6 and 8 moments M n as a cally realistic results, only qualitative ones. Figure 3 shows 2 S function of Q . While all moments appear qualitatively simi- the onset of scaling for the structure function cq as a func- lar, as would be expected the higher moments become inde- tion of u,asQ2 varies from 0.5 to 2 GeV2. As in Fig. 1, for pendent of Q2 only at larger values of Q2. The lowest mo- each of the resonances ͑excluding the elastic peak͒ the en- 2 ␦ ment M 2 is very close to its asymptotic value at Q ergy -function has been smoothed out using the Breit- ϭ 2 ⌫ϭ 5 GeV , while the highest shown moment, M 8, is still Wigner method with a width 100 MeV. The elastic peak slightly increasing at the highest Q2 shown. This is qualita- is displayed with the arbitrarily chosen width ⌫ϭ30 MeV. tively consistent with the expectation from the operator prod- With increasing Q2, each of the resonances moves out to- uct expansion discussed in Ref. ͓19͔, where it was argued ward higher u, as dictated by kinematics. At Q2ϭ0, the elas- that the effective expansion parameter in the twist expansion tic peak is the only allowed state and contributes about 44% 2 ϳn/Q , so that for higher moments n the higher twist terms of the asymptotic value of M 2. It remains rather prominent 2 2ϭ 2 survive to larger values of Q . for Q 0.5 GeV , though most of M 2 is by this point built Unfortunately, these moments do not have such useful up of excited states, and it becomes negligible for Q2 interpretations here as they do in deep inelastic scattering. у2.0 GeV2. Remarkably, the curves at lower Q2 do tend to For example, the analog of the Gross-Llewellyn Smith sum oscillate ͑at least qualitatively͒ around the scaling curve, as

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S S FIG. 3. Onset of scaling for the structure function cq as a FIG. 4. Onset of scaling for the structure function Bj as a 2ϭ 2 ͑ ͒ 2ϭ 2 2ϭ 2 ͑ ͒ 2ϭ 2 function of u for Q 0.5 GeV solid curve , Q 1GeV function of uBj for Q 0.5 GeV solid curve , Q 1GeV ͑short-dashed curve͒,2 GeV2 ͑long-dashed curve͒ and 5 GeV2 ͑short-dashed curve͒,2 GeV2 ͑long-dashed curve͒ and 5 GeV2 ͑dotted curve͒. Although off-scale, the elastic peak at Q2 ͑dotted curve͒. ϭ0.5 GeV2 accounts for about 22% of the area under the scaling curve. that interacts with the current and not the light quark as in our model.͒ At the free quark level, the decay of Q* at rest is observed in proton data. Note that these curves are at fixed ␾ will produce the with a single sharp kinetic energy T free Q2, but sweep over all ␯. In a typical low energy experiment, and corresponding Q recoil velocity vជ . ͑We use the standard ␯ will also be limited; in such circumstances these curves ជ ␾ still apply, but they are cut off at the minimum value of u that variables T free and v, but others, like the recoil momen- ͒ is kinematically allowed. For another perspective on these tum, could be chosen. In reality, since the heavy quarks are bound into , ␾ will ͑in the narrow resonance approxi- curves, note that ͉qជ͉2ϭQ2ϩ␯2, so for fixed Q2,as␯ is mation͒ emerge from the decay at rest of the initial meson’s increased so that more and more highly excited states are ¯ created, the struck quark is being hit harder and harder. ground state (Q*q)0 with any of the sharp kinetic energies S ¯ → ¯ ϩ␾ In contrast, the structure function Bj when plotted as a allowed by the processes (Q*q)0 (Qq)n as determined function of the scaled Bjorken variable uBj shows very poor by the strong interaction spectra of these two mesonic sys- 2 tems. Since in the heavy quark limit m ¯ Ϫm ¯ duality between its low- and high-Q behaviors, as seen in (Q*q)n (Qq)n Fig. 4. One of the reasons for this failure is that x and S Ӎm Ϫm , m ¯ Ӎm , and m ¯ Ӎm , the had- Bj Bj Q* Q (Q*q)n Q* (Qq)n Q know nothing about the constituent quark mass, while low ronic spectral lines are guaranteed to cluster around T free , energy free quark scattering certainly does, so the corre- →ϱ and to coincide with it exactly as mQ . Moreover, since sponding pQCD cross section calculated neglecting the ӷ⌳ mQ* ,mQ QCD , one can show using an analog of the op- quark mass is simply wrong at low energy. erator product expansion ͓20͔ that the strong interactions can be neglected in calculating the total decay rate ͑i.e., the V. DUALITY IN SEMILEPTONIC DECAYS heavy quarks Q* and Q are so heavy that the decay proceeds OF HEAVY QUARKS as though the quarks were free͒. Thus the sum of the strengths of the spectral lines clustering around T is the ͑ ͒ free We have seen that low-energy Bloom-Gilman duality is free quark strength: there is perfect low energy duality as displayed by our model in terms of the appropriate low- m* ,m →ϱ. energy variable u and described some of the physics behind Q Q What is now especially interesting is to unravel this dual- this duality ͑completeness of the bound state wave functions ity to understand how the required ‘‘conspiracy’’ of spectral to expand a plane wave and an approximate closure based on line strengths arises physically. Because the heavy quark is the required expansion states being in a narrow band of ␯ ͒ so massive, if it would as a free particle recoil with a velocity relative to those that are kinematically allowed . To obtain a ជ deeper understanding of the physics behind low energy du- v, then this velocity would be changed only negligibly by the ality, it is instructive to compare and contrast duality in elec- strong interaction since in the heavy quark limit it carries off tron scattering with that in heavy quark decays. We will be- a negligible kinetic energy, but a momentum much larger ⌳ gin by carefully examining duality in heavy-light systems, than QCD . In the rest frame of the recoiling meson, this where it is exact in the heavy quark limit even down to zero configuration requires that the two constituents have a rela- ជ ជ recoil, and where the mechanisms behind this exact duality tive momentum q which grows with v. Thus the strong in- are very clear. teraction dynamics is identical to that of our model in which Duality in heavy quark systems is easily understood intu- the relative momentum qជ is supplied by the scattered elec- ¯ ӷ⌳ tron. Moreover, in this case, with duality exact at all ener- itively. Consider a Q*q system where mQ* QCD , and imagine that Q* can decay to Q by emitting a scalar particle gies, we can reconstruct exactly how it arises. What one sees ␾ Q*→Qϩ␾. ͑Note that in this case it is the heavy quark is remarkably simple ͓21,22͔. At low vជ corresponding to low

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ជ ¯ → ¯ ϩ␾ VI. SUMMARY AND OUTLOOK q, only the ground state process (Q*q)0 (Qq)0 oc- curs. Since the masses and matrix elements for the transi- We have presented a simple, quantum-mechanical model ¯ → ¯ ϩ␾ → ϩ␾ ͑ tions (Q*q)0 (Qq)0 and Q* Q are identical the in which we were able to reproduce the features of Bloom- elastic form factor goes identically to unity as qជ →0), the Gilman duality qualitatively. The model assumptions we hadronic and quark spectral lines and strengths are also iden- made are the most basic ones possible: we assumed relativ- tical and duality is valid at ͉qជ͉2ϭ0! Next consider duality at istic, confined, valence scalar quarks and treated the hadrons a different kinematic point ͑which one might reach by choos- in the infinitely narrow resonance approximation. To sim- plify the situation further, we did not consider a three quark ing a smaller ␾ mass͒ where vជ and therefore qជ have in- ‘‘nucleon’’ target, but a target made up by an infinitely heavy creased. The elastic form factor will fall, so its spectral line antiquark and a light quark. The present work does not at- ͑which is still found at exactly the new value of T in the free tempt to quantitatively describe any data, but to give quali- ͒ ជ heavy quark limit will carry less strength. However, once q tative insight into the physics of duality. ¯ differs from zero, excited states (Qq)n can be created, and Our work complements previous work on duality, where indeed are created with a strength that exactly compensates the experimental data were analyzed in terms of the operator for the loss of elastic rate. These excited state spectral lines product expansion ͓19,25͔. There, it was observed that at 2 also coincide with T free and duality is once again exact. moderate Q , the higher twist corrections to the lower mo- Indeed, no matter how large ͉qជ͉2 becomes, all of the excited ments of the structure function are small. The higher twist states produce spectral lines at T free with strengths that sum corrections arise due to initial and final state interactions of to that of the free quark spectral line. the quarks and gluons. Hence the average value of the struc- Heavy quark theory also allows one to go beyond the ture function at moderate Q2 is not very different from its heavy quark limit to the case of quarks of finite mass. In this value in the scaling region. While true, this statement is case, of course, one finds that duality violation occurs, but merely a rephrasing in the language of the operator product ⌳ that it is formally suppressed by two powers of QCD /mQ expansion of the experimentally observed fact that the reso- ͓ ͔ nance curve averages to the scaling curve. However, the op- 20,23 , with the spectral lines now clustered about T free but not coinciding with it. A remarkable feature of this duality erator product expansion does not explain why a certain cor- violation is that the spectral line strengths differ from those rection is small or why there are cancellations: the expansion of the heavy quark limit in ways that tend to compensate for coefficients which determine this behavior are not predicted. the duality-violating phase space effects from the spread of The confirmation of these coefficients will eventually come from a numerical solution of QCD on the lattice, but an spectral lines around T . An additional source of duality free understanding of the physical mechanism that leads to the violation is that some of the high mass resonances that are small values of the expansion coefficient will almost cer- required for exact duality are kinematically forbidden, since Ϫ tainly only be found in the framework of a model like ours. for finite heavy quark masses mQ* mQ is finite. For example, one clear lesson from our study of duality is From this discussion it is clear that the strong interaction that the commonly made sharp distinction between the dynamics of heavy-light decays is the same as that of scat- ‘‘resonance region,’’ corresponding to an invariant mass W tering a probe off of the Q of a Qq¯ system ͓24͔: what is Ͻ2 GeV for scattering from a proton, and the deep inelastic relevant is that the system must in each case respond to a region, where WϾ2 GeV, is completely artificial. relative momentum kick qជ . Needless to say, one must still Finally, we remind the reader that our model, with all the carefully organize the kinematics to expose duality: in a de- charge on a single quark, with scalar currents, and with no cay to a fixed mass ␾ only a single magnitude ͉qជ͉2 is pro- spin degrees of freedom, leaves much to be done in model- duced at the quark level, while in electron scattering a large building. The next step is to use more realistic currents. ͉ជ͉2 ␯ While making the calculations more complicated, coupling range of q and is produced by a given electron beam. to the conserved quark current will allow one to study the Q2 Given these connections, it is relevant to note that in ad- evolution of the Gross-Llewellyn Smith and momentum sum dition to the obvious conceptual relevance of heavy-light 1 rules. To use a spin- target will also be a useful step for- systems, model studies indicate that in these systems heavy 2 ward, but it may require foregoing the great advantages of quark behavior continues to hold qualitatively even for mQ ϳ the analytic solutions of the Klein-Gordon equation. As we m. These models are, as one might expect, similar to ours, have emphasized, the local duality seen here cannot be ex- which displays the same clustering of spectral lines, the same pected for more complicated targets and processes, and pur- tendency for excited state spectral lines to compensate for suing this issue is also clearly very important ͓9͔. Here we- ជ 2 the fall with ͉q͉ of lighter states, and the same sources of have taken a first small step which nevertheless has been duality violation such as kinematically forbidden states and enough to strongly suggest that for these more realistic mod- mismatches between the mass of the recoiling hadrons and els and more general processes there will be a generalization the struck quark. We believe that these elements of the dy- of local averaging—a theoretically well-defined procedure namics are clearly in operation, and that we have understood through our model that the qualitative applicability of duality for integrating over regions of xcq—which will also display for real systems should indeed extend all of the way down to low energy duality. If so, we will not only have understood zero recoil as seen in Nature. quark-hadron duality, we will also have opened the door to

054005-7 ISGUR, JESCHONNEK, MELNITCHOUK, AND VAN ORDEN PHYSICAL REVIEW D 64 054005 extending studies of a variety of structure functions into pre- cussions, and N.N. Nikolaev and S. Simula for pointing out viously unreachable kinematic regimes. several references. This work was supported in part by DOE Contract No. DE-AC05-84ER40150 under which the South- ACKNOWLEDGMENTS eastern Universities Research Association ͑SURA͒ operates We thank C. Carlson, F. Close, R. Ent, J. Goity, C. Kep- the Thomas Jefferson National Accelerator Facility ͑Jeffer- pel, R. Lebed, I. Niculescu and S. Liuti for stimulating dis- son Lab͒, and by the Australian Research Council.

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