Recent advances in experimental high-energy physics have provided a wealth of data on fundamental interactions. It is important that the data be completely analyzed within a coherent theoretical framework extending beyond leading-order perturbation theory. The insight gained should prove invaluable in planning experiments for the future, and, in the process, discovering unexpected "new physics."

This work is part of a long-term effort by several individuals to incorporate advances of theoretical calculations into experimental data analysis to address two fundamental issues: 1) the structure of hadrons in the Quantum Chromodynamic-based parton model; and 2) the resulting precision comparisons with new data.

The large renormalization-scale dependence of next-to-leading-order (NLO) heavy quark hadroproduction cross sections[1] and the underestimate of experimental data present a serious challenge for QCD. We resum large initial and final state logarithms, via flavor excitation Feynman graphs and heavy quark PDFs combined in a consistent manner, to extend and improve present NLO calculations. This theoretical analysis will have implications for experiments performed at the Tevatron, at whose energy scale the "heavy" quarks behave more like partons.

We note that the difference between the variable-flavor scheme (VFS) calculation[2,3] and the fixed-flavor scheme (FFS) calculation[4] suggests higher order contributions yet to be included may be important. Combining these two next-to-leading-order calculations in a consistent fashion (with the additional mass factorizations required) will allow us to make predictions based upon a three-order result that combines the best attributes of both calculations.

Such a result would be theoretically significant since there are very few processes are computed to three orders. More importantly, it would be experimentally significant as the increased luminosity at HERA allows the study of heavy quark production in detail. The combination should provide an important test of perturbative QCD when compared with the results from HERA.

This three-order calculation is in progress, and merges the expertise of the LRSN[4] and ACOT[2,3] groups. The intermediate calculations necessary have been completed, and are in the process of being checked. Preliminary results of this calculation were presented at the 1995 Moriond conference[5].

Essential to QCD applications of the operator product expansion, etc., is a knowledge of those operators that mix with gauge-invariant operators. A standard theorem asserts that the renormalization matrix is triangular: Gauge-invariant operators have "alien" gauge-variant operators among their counterterms, but, with a suitably chosen basis, the necessary alien operators have only themselves as counterterms. Moreover, the alien operators are supposed to vanish in physical matrix elements. A recent calculation by Hamberg and van Neerven apparently contradicts these results. By explicit calculations with the energy-momentum tensor, we show that the problems arise because of subtle infra-red singularities that appear when gluonic matrix elements are taken on-shell at zero momentum transfer.

In particular, we study the one-loop renormalization of the
covariant gluon operator which arises in the operator product
expansion of the non-local product of electromagnetic currents in
processes involving hadrons. The inverse Mellin transform of the
anomalous dimension of this operator is the Altarelli-Parisi gluon
splitting kernel, and is used to evolve parton distribution functions
to higher values of Q^{2}.

This work relies heavily on the BRST symmetry of the Yang-Mills Lagrangian density. We use the invariance of certain operators under the BRST transformation to isolate the violation, at zero momentum transfer, of some theorems in renormalization. This symmetry is also critical in the selection of a basis of operators which mixes with the gluon operator under renormalization.

An application of the techniques developed in the one-loop case to the two-loop calculation is work in progress. Eventually, we should like to attempt the three-loop calculation since this is needed at present to complete the two-loop corrections to the proton structure functions in the small-x range now accessible at HERA, for example.

[2] "Leptoproduction of Heavy Quarks. 1. General Formalism and Kinematics of Charged Current and Neutral Current Production Processes", M.A.G. Aivazis, F.I. Olness (Southern Methodist U.), W.-K. Tung (Michigan State U.). MSU-HEP-93-15, Oct 1993. 27pp. Published in Phys.Rev.D50:3085-3101,1994. e-Print Archive: hep-ph/9312318

[3] "Leptoproduction of Heavy Quarks. 2. A Unified QCD Formulation of Charged and Neutral Current Processes from Fixed Target to Collider Energies", M.A.G. Aivazis (Southern Methodist U.), J.C. Collins (Penn State U.), F.I. Olness (Southern Methodist U.), W.-K. Tung (Michigan State U.). SMU-HEP-93-17, Oct 1993. 29pp. Published in Phys.Rev. D50:3102-3118,1994. e-Print Archive: hep-ph/9312319

[4] "Complete O(alpha_s) Corrections to Heavy Flavor Structure Functions in Electroproduction" E. Laenen, S.T. Riemersma, J. Smith, and W.L. van Neerven, Nucl. Phys. B392, 162 (1993); "O(alpha_s) Corrections to Heavy Flavor Inclusive Distributions in Electroproduction",ibid., 229 (1993).

[5] "Leptoproduction of Heavy Quarks", P. Agrawal (Michigan State U.), F.I. Olness, S.T. Riemersma (Southern Methodist U.), W.-K. Tung (Michigan State U.). SMU-HEP-95-04, May 1995. 7pp. To be published in the proceedings of 30th Rencontres de Moriond: QCD and High Energy Hadronic Interactions, Meribel les Allues, France, 19-25 Mar 1995. e-Print Archive: hep-ph/9507295

[6] "Heavy Quark Hadroproduction: Resumming Large Logarithms via Heavy Quark PDFs", F.I. Olness, R.J. Scalise, and W.-K. Tung, SMU Preprint (in preparation)

[7] "Heavy Quark Production in Deep Inelastic Scattering at HERA", P. Agrawal, F.I. Olness, S.T. Riemersma, R.J. Scalise, and W.-K. Tung. SMU preprint (in preparation)

[8]
"Renormalization of Composite Operators in Yang-Mills Theories
Using a General Covariant Gauge" with
John
C. Collins, Physical Review D **50** (1994) p4117-36.

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