The Large Coefficient Problem: Can We Make Sense Out of QCD Perturbation Theory?
There is the possibility of an impending crisis looming on the horizon for QCD. The problem is that in many processes, large coefficients arise in the perturbation series expansion leading to serious uncertainties concerning its predictive power. Until recently most of the examples of such a phenomenon occurred in the calculation of decay rates. These were, by and large, either ignored or dismissed using possible scheme-dependence arguments as a way out. However, more recently a calculation of the 3-loop contribution to the total e + e - annihilation cross-section was performed which gave an enormous coefficient of the order of 50 times that of the 2-loop term(1). If correct, this would imply that the 3-loop contribution actually exceeds that of the 2-loop! Thus, from a conservative viewpoint, the validity of the perturbation series expansion as an estimate for the total e + e - cross-section is called into question. Such a cautionary attitude should even be extended to the lowest order parton-model result, Σ (Q i 2 ); (Q i being the charge of the ith quark species). Since this process has played a key role in the development and understanding of QCD and since, in many ways, it is one of the cleanest methods for extracting α 3 (the conventional QCD fine structure constant) the problem can no longer be avoided. Furthermore, there is no reason to doubt (and, in fact, good reasons to believe) that this problem should occur in all physical processes. Coming to grips with it is, of course, not only important for testing QCD but also for extracting fundamental quantities such as α s . Clearly one needs to understand the nature and origin of such large coefficients before one can confidently continue to use perturbative estimates.
KeywordsPerturbation Theory Saddle Point Large Coefficient Perturbation Series Positive Real Axis
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