Thesis Projects in Quark-Hadron Duality
My name is Sabine Jeschonnek, and I am a nuclear theorist. My two main research interests are quark-hadron duality and electron scattering from few-body systems. Both topics have a very strong connection to experiments, and I have been involved in making predictions for experimental data, and in planning and proposing experiments at Jefferson Lab.
Contact Info: This quarter, I will be in Columbus more frequently, two or three days per week. Next quarter, I will be teaching at the Lima Campus Monday - Thursday, and I will be in Columbus on Friday. You can reach me by e-mail: jeschonnek.1@osu.edu or by phone (419) 995 8201.
Interested? If you are interested, or simply would like to ask me a few questions, or learn more about duality, please get in touch with me. By the way, if you do not understand some part of what is written below, that's perfectly ok - I will be happy to explain it, and you will learn about it while working on the project.
Perks: Trips to Jefferson Lab in Newport News, Virginia. Many interesting duality experiments are carried out there, see e.g. [1]. There will be ample opportunity to interact with experimentalists and theorists.
What is Quark-Hadron Duality? In the literature, there exist many, slightly varying ``definitions'' and usages of the term duality, and the phenomenon manifests itself experimentally in many different processes. Here is a definition that covers all these cases. First, we need to make an obvious observation: any hadronic process can be correctly described in terms of quarks and gluons. In other words, quantum chromodynamics (QCD) is the correct theory for strong interactions. While this statement is obvious, it has little practical value, as in most cases, we cannot perform a full QCD calculation. E.g., in order to calculate a resonance excitation form factor, one would need to include very many quarks and gluons, and they would all couple strongly. The above statement that any hadronic process can be described by a full QCD calculation is dubbed ``degrees of freedom duality''. There exists a more practical and less obvious version of the first statement: in certain kinematic regions, the average of hadronic observables is described by a perturbative QCD (pQCD) result. This is the statement of duality, and we are going to explain the details in the following. With pQCD result, we indicate the result for the underlying quark process -- for inclusive inelastic electron scattering from a proton, it is free electron-quark scattering; for semileptonic decays, e.g. B --> Xc l \nul, it is the underlying quark decay rate, in this case obtained from the process b --> c l \nul. Now, it is clear that we expect perturbative QCD to describe Nature in a certain kinematic regime, i.e. for very large Q2. In this regime, due to the fact that full QCD is approximated by perturbative QCD, the statement of duality turns into the statement of the ``degrees of freedom duality''. So we have identified one kinematic regime in which even the non-obvious version of duality must hold. We also can identify a kinematic regime for which duality cannot hold: for Q2 --> 0. For duality to hold for the nucleon structure functions in this case, we would need the following: the elastic proton and neutron from factors, which take the value of the nucleon charge for Q2 --> 0, would have to be reproduced by electron scattering off the corresponding u and d quarks. Now, for the proton this can work, as the squares of the charges of two u quarks and one d quark add up to 1. However, for the neutron, the squared quark charges cannot add up to 0, so it is clear that duality in inclusive inelastic electron scattering from a neutron must fail for Q2 --> 0. So now we know that duality has to hold in one kinematic regime and that it has to break down in another kinematic regime. Obviously, a very interesting question is what happens in between these regimes, i.e. how exactly does duality break down, how far does it hold in the regime where it is nontrivial, i.e. for moderate values of Q2, and how accurately does it hold where it holds. In the case of inclusive electron scattering, we have an answer to this question from experiment [2,1], but a theoretical explanation of duality remains elusive. Duality is not only a very interesting phenomenon by itself, it has extremely important applications. As duality connects the resonance region, commonly defined as the region where the final state invariant mass W < 2 GeV, and the deep inelastic region, one may infer information on one from the other. The earliest example discussed was the extraction of the elastic nucleon form factor from the deep inelastic scaling curve [3]. In [4], higher twist contributions were inferred from the resonance data. One of the most exciting and promising applications of duality will be the measurement of the neutron polarization asymmetry A1n at large xBj There are many different theoretical predictions for this quantity, ranging from 0 (unbroken SU(6)) to 1 (pQCD) [5]. The data which are currently available from SLAC have a very large error bar, and do not extend beyond x Bj = 0.5. The experiment is very difficult and the main problem is the lack of cross section at high xBj, which can be reached in the deep inelastic region (W > 2 GeV) only at high Q2. If duality is well understood, one may take data in the resonance region, apply a proper averaging procedure, and thus obtain results for A1n (xBj --> 1) . The large xBj region is much easier to access in the resonance region, as the necessary Q2 values there are much smaller. Obviously, in order to carry out such a procedure reliably, one needs a solid theoretical understanding of the averaging procedure which needs to be applied, and of the systematic error thus incurred.
Current Status of the Project: In previous work on duality, the experimental data were analyzed in terms of the operator product expansion (OPE) [3,4]. There, it was observed that at moderate Q2, the higher twist corrections to the lower moments of the structure function are small. The higher twist corrections arise due to initial and final state interactions of the quarks and gluons. Hence, the average value of the structure function at moderate $Q^2$ is not very different from its value in the scaling region. While true, this statement is merely a rephrasing in the language of the operator product expansion of the experimentally observed fact that the resonance curve averages to the scaling curve. However, the operator product expansion does not explain why a certain correction is small or why there are cancellations: the expansion coefficients which determine this behavior are not predicted. The confirmation of these coefficients will eventually come from a numerical solution of QCD on the lattice, but an understanding of the physical mechanism that leads to the small values of the expansion coefficient will almost certainly only be found in the framework of a model. In [6],we have presented a simple, quantum-mechanical model in which we were able to qualitatively reproduce the features of Bloom-Gilman duality. The model assumptions we made are the most basic ones possible: we assumed relativistic, confined, scalar valence quarks and treated the hadrons in the infinitely narrow resonance approximation. To further simplify the situation, we did not consider a three quark ``nucleon'' target, but a target made up by an infinitely heavy anti-quark and a light quark. The work presented in [6] does not attempt to quantitatively describe any data, but to give qualitative insight into the physics of duality. A very nice feature of our model is that all the results can be worked out analytically. This is due to the use of a linear confining potential, which leads to a relativistic version of the harmonic oscillator. This means also that comparisons between relativistic and non-relativistic treatments are particularly easy. Our first published results dealt with an all scalar case, i.e. not only with scalar ``quarks'', but also with scalar ``photons'' and ``electrons''. In our current paper [7], we have extended our model to include proper electrons and photons, and only the quarks remain spinless. If you would like to work on duality, you would also collaborate with Wally Van Orden (Jefferson Lab & Old Dominion University) and probably with Dick Furnstahl from Ohio State.
Some Projects to get started: Here are a two possible projects for starting out in quark hadron duality. Besides working on one of these projects, you will also have the opportunity to read and discuss the relevant articles on the subject in the first couple of months.
Model dependence: Obviously, by picking a specific scalar confining potential, we made a particular choice. Actually, most model calculations which address duality or the related, but somewhat more limited, issue of scaling in the presence of final state interactions, make use of harmonic oscillator potentials. While the harmonic oscillator has many clear advantages, which make it a natural choice for a first study, it also has an unrealistic short-range behavior. In order to study model dependence, we intend to use a variety of model potentials. Obvious choices are square and spherical wells because the solutions are easily obtained, and a Coulomb-type potential because of its more realistic short range behavior. One important question we wish to address is the following feature of duality: the scaling curve obtained from summing over all bound states should agree with the scaling curve obtained when placing the final particle in a free, plane-wave state. Paris and Pandharipande [8] found a 30 % difference in their numerical calculation, while we could show analytically that in our model, the scaling curves agree exactly. This is an important issue which we would like to investigate further by using different confining potentials. While we do expect duality to hold independent of the particular potential, we have no general proof yet. It is reasonable to expect that calculations with different potentials will give us more insight into the general realization of duality, leading towards a general proof of scaling and duality. Studying duality in the case of quarks confined in a Coulomb-like potential should be a good starting point for a thesis project.
Developing and testing averaging methods: The key point in exploiting duality to learn about deep inelastic quantities by measuring in the resonance region is employing the correct averaging procedure, and knowing its systematic error. In our model, we basically generate ``data points'', and by trying out various averaging methods, we can construct scaling curves from the model results in the resonance region. Then, as we know the scaling results for the model, we can compare the true scaling result with the result obtained in the averaging. This should lead to robust averaging methods with a well-known error-bar. We will repeat this investigation of averaging methods for all new reaction types and targets investigated in our model, as these methods and their reliability will exhibit a dependence on the process.
References:
[1] I. Niculescu et al., Phys. Rev. Lett. 85, 1182 (2000); 85, 1186 (2000); R. Ent, C.E. Keppel and I. Niculescu, Phys. Rev. D 62, 073008 (2000).
[2] E.D. Bloom and F.J. Gilman, Phys. Rev. Lett. 25, 1140 (1970); Phys. Rev. D 4, 2901 (1971).
[3] A. DeRujula, H. Georgi, and H. D. Politzer, Ann. Phys. (N.Y.) 103 315 (1977).
[4] X. Ji and P. Unrau, Phys. Rev. D 52 72 (1995).
[5] ``Valence Quark Spin Distribution Functions'', N. Isgur, Phys. Rev. D 59 034013 (1999).
[6] ``Quark-hadron duality in structure functions'', N. Isgur, S. Jeschonnek, W. Melnitchouk, and J. W. Van Orden, Phys. Rev. D 64, 054005 (2001).
[7] ``Quark-Hadron Duality in a Relativistic, Confining Model'', S. Jeschonnek and J. W. Van Orden, in preparation.
[8] ``Scaling of space and timelike response of confined relativistic particles'', M. W. Paris and V. R. Pandharipande, nucl-th/0105076.