I am interested in fundamental approaches to QCD as applied to low-energy hadron physics. One such approach is called Lattice QCD which numerically evaluates the path integrals via Monte Carlo methods on a discrete, space-time lattice. It provides the only comprehensive method to solve QCD with controlled systematic errors. The calucaltion involves millions of degrees of freedom which must be manipulated by computers, usually supercomputers. The predictions can be made arbitrarily accurate with increasing computing power. My goal is to provide quantitative predictions of QCD for aspects of hadron structure that have a direct bearing on experiments at JLab and elsewhere, using state-of-the-art technology in the field.
I am also interested in another, complementary approach to QCD: the QCD Sum Rule Method, which reveals the deep connection between the QCD vacuum structure and hadron phenomenology via only a few parameters called vacuum condensates. It is a useful way of getting an unique and valuable insight into a variety of hadronic observables, without doing path integrals. The method is mostly analytical, physically transparent, but with limited, generally-accepted accuracy.
Following is a comprehensive description of the research areas.
The baryon resonances are experimental emblems of QCD (see a postscript figure of some of the observed N*s). The three valence quarks in the proton and its excited states are a manifestation of SU(3); the baryon spectrum with linearly rising Regge trajectories is an indication of a flux-tube-like confining force; and the fine and hyperfine structure of the baryon spectrum reveal details of inter-quark interactions.
We have done exploratory calculations of the low-lying states with spin-parity 1/2- and 3/2- with an O(a^2) improved action. The interpolating fields used to excite the states were proposed, as well as methods for removing the contamination due to positive-parity, ground-state states. The results show that these states can be isolated on the lattice, albeit high statistics are required.
These calculations are being improved in a number of ways: 1) introduce lattice interpolating operators with explicit derivatives to better capture the features of N*s; 2) construct a set of operators corresponding to each of the irreducible representations of the lattice cubic group, enabling us to determine the lightest physical state in each representation. In particular, this will enable the lattice determination of the higher spin states like 5/2; 2) use anisotropic lattices to better resolve the heavier N* states as successfully done in glueball calculations; 3) employ variational methods to isolate states beyond the ground state in each channel. Such methods are essential for determining masses of states of spin greater than 5/2, and radial excitations.
An important new development in the field is the discovery of overlap fermions that preserve exact chiral symmetry even on a finite lattice. In collaboration with Prof. K.F. Liu of University of Kentucky and collaborators, we have succeeded in a numerical implementation of the overlap operator in quenched QCD simulations. The new operator is much more expensive to simulate (about a factor of 100) than traditional operators, but holds great promise as demonstrated in the calculations we have completed so far on small lattices. We are trying to increase the physical volume by incorporating improved actions in the overlap operator. The goal is to increase the physical volume to the level suitable for phenomenology (around 5 fm on a side), while still keeping the CPU demand manageable. The idea of fuzzing is also being studied to accelerate the calculations.
Having gained experience with the spectrum, it is important to extend the study to electromagnetic form factors, which encode rich information about the spatial distributions of quarks inside hadrons. The basic formalism has been established in definitive, first-generation calculations by Leinweber et al. By today's standards, the simulations provided only limited control over systematic and statistical errors, but nonetheless yielded valuable, direct information on the key form factors of octet (1/2+) and decuplet (3/2+) baryons for comparison with experiment.
In collaboration of Prof. Derek Leinweber of University of Adelaide, we plan second-generation calculations of electromagnetic form factors, with highly improved actions featuring good statistics, several lattices of varying size and spacing for taking the continuum limit, more quark masses for better chiral extrapolation, and several values of q^2 up to 2 (GeV/c)^2. Of high priority is a careful calculation of the E2/M1 and C2/M1 ratios in the gamma p --> Delta+ transition to compare with those extracted from data. These ratios are currently of high interest in the community.
Later, the calculations will be extended to negative-parity $N^*$ transitions, including gamma N--> N*(1/2-) and N*(3/2-), for which good-quality data will become available from JLab. Such calculations will provide the first insights into the structure of orbitally excited baryons as predicted by QCD. In particular, the electromagnetic properties of these resonances may be completely different from model expectations if gluons carry a significant fraction of the angular momentum usually attributed to quark degrees of freedom alone.
On the lattice, the polarizabilities can be extracted from current-current correlation functions, or by the external static field method. The latter appears to hold the most promise to extract this quantity. In this method, the interaction of the induced moments with the external fields causes shifts in hadron energy (or mass). For small external fields, polarizabilities manifest as second order effects in the external fields. These ab initio calculations are expected to be instrumental in assessing various models for Compton scattering, such as Dispersion Theory and Chiral Perturbation Theory. Currently various methods for introducing a uniform small magnetic field on a periodic lattice are being considered. It may require large physical volume on the lattice to reduce lattice artifacts. This can be helped by the use of improved actions.
The focus is to provide careful, systematic calculations of N* properties, coupled with a rigorous Monte Carlo-based numerical analysis in order to explore the predictive ability of the approach for hadronic observables, especially N* matrix elements. Such studies help assess the reliability and accuracy of the predictions, and lead to ways for improvement. The improved predictions in turn can lead to better comparisons with data and with other methods. Conventional QCD sum rule studies tend to fall short in the analysis stage where it is limited to only a small portion of the QCD parameter space, and often the uncertainties assigned to the fit parameters have a certain degree of arbitrariness. This has led to some harsh criticism of the approach in the past.
The main objective is to obtain a complete set of QCD sum rules for N* masses, and to investigate their predictive ability via the Monte Carlo analysis. One important issue is a careful determination of the pole residues (also called current couplings) which describe the ability of the interpolating field to create and annihilate the hadron in question. These couplings are crucial for matrix element calculations as they enter as normalization. They were not well determined in traditional calculations. An improved analysis for 3/2+ decuplet baryons has been completed. Such studies are already paying dividends. The couplings were used to check a cubic scaling law between baryon couplings and masses. The results show a significant reduction in the scaling constant and some possible deviations from the cubic law. Also they were used in obtaining better understanding of the magnetic moment of the Omega particle. A complete re-analysis is being done for the 1/2+, 1/2- and 3/2- channels. New, general QCD sum rules have been derived using general interpolating fields. Comparisons with the old ones in special cases reveal many differences. A numerical analysis is under progress.
Based on the newly-determined current couplings, the first systematic calculation of magnetic moments of the entire family of 3/2+ baryons using the external field method has also been carried out. Using realistic estimates of the QCD input parameters, the uncertainties on the magnetic moments are found relatively large and they can be attributed mostly to a poorly-known vacuum susceptibility. It is shown that the accuracy can be improved to the 20% level, provided the uncertainties in the QCD input parameters can be determined to the 10% level. The computed magnetic moments are consistent with existing data.
The next step is to re-examine the 1/2+ baryon magnetic moments using the general interpolating fields. The advantage of using general interpolating fields is that more and better sum rules can be discovered. The goal is to use the excellent experimental data available for these magnetic moments and better sum rules to extract improved determination of the vacuum susceptibilities, which can be then used in calculating transition amplitudes. There is little information on these amplitudes from QCD. The OPE side of a complete set of QCD sum rules for the entire family of transitions gamma N --> N(3/2+) have been derived, and the results are being checked. The calculation, which involves thousands of Dirac gamma matrix reductions, is aided by a computer algebra software called REDUCE. The spectral representation requires careful treatment because of the transitions to excited states. They are not exponentially suppressed relative to the ground state. The solution is to use a combination of multiple sum rules to eliminate these contaminations, or find sum rules that have such contributions minimized. These sum rules are being checked. Numerical analysis is also under progress.
QCD has SU(2) isospin symmetry if the up and down quarks are identical. The fact that $m_u\neq m_d$ in nature causes a small breaking in this symmetry, reflected in small, subtle and interesting effects in physical quantities, such as n-p or $\pi^0-\pi^{\pm}$ mass differences, non-universality of the $\pi N N$ couplings, $\rho-\omega$ or $\pi-\eta$ mixings. Of the two sources of isospin symmetry breaking: electromagnetic interactions and strong interactions, the PI is only concerned with its strong interaction effects. The QCD sum rules method is well-suited to the study of isospin symmetry breakings, since the contributions of the up and down quarks can be explicitly kept separate in each step of the calculation. One can dial their contributions in various ways to see the effects on observables. The role of the strange quark can also be studied in this way. The PI is interested in isospin symmetry breaking effects in the $\pi N N$, $K \Lambda N$, $K \Sigma N$ systems. The results will have impact on the study of $NN$ potential, the three-body force, and other processes involving $\pi N$ and $KN$ couplings.
This work provides the link between the two non-perturbative approaches. A direct comparison of hadron correlation functions calculated via the Operator Product Expansion (OPE) with those from lattice QCD can test the realm of validity of the OPE in the nonperturbative sector and our understanding of quantum field theory. The Wilson coefficients of the OPE would be calculated via lattice perturbation theory and the two-point functions would be calculated on a fine lattice to provide sufficient information within the ``radius of convergence'' of the OPE. Vacuum expectation values (VEV) of normal ordered operators could be determined from OPE fits to the lattice correlation functions. In turn, these VEV's could be compared with direct lattice calculations of the operators. In addition, essential information on the importance of direct instanton contributions to the OPE could be obtained from such an investigation. A resolution of these issues would provide the final judgment on the method of QCD sum rules and a deeper understanding of non-perturburtive nature of QCD. This a long-term goal in collaboration with Prof. Derek Leinweber of University of Adelaide, Australia.
Meson photoproduction from the nucleon involves rich dynamics of hadron structure: such as the coupling at the strong vertex, the electromagnetic and transition form factors. It provides one of the cleanest ways of extracting N* properties experimentally, and is a major part of the physics program at JLab. Most of our knowledge on N* comes from such reactions. This is the field that led to my PhD degree. I'm still being asked occasionally by experimentalists to provide theoretical curves for comparison with their data.
An important theoretical issue in interpreting the data is a consistent treatment of hadron-hadron interactions beyond tree-level. In this aspect, the emphasis of the nuclear theory group at GW is on a consistent description of hadronic and electromagnetic reactions in a field-theoretical framework that can properly incorporate unitarity, Lorentz covariance and gauge invariance. Such a theoretical framework requires a coupled-channels approach leading to a set of Bethe-Salpeter integral equations whose solutions can only be achieved by numerical methods. In this area, I participate in discussions and contribute in computational expertise.