Higher-order effects of asymptotically free field theories

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Abstract
Several experimental consequences of non-Abelian asymptotically free gauge theories are derived, in particular, second-order corrections to the moments of the nonsinglet pieces of the structure functions of deep-inelastic lepton-hadron scattering are calculated. These results are then used to obtain corrections to the parton-model sum rules, as well as to derive some direct functional relationships among the structure functions. It is found that, if asymptotic behavior is assumed, the second-order correction terms involve calculable and uncalculable contributions; however, by considering suitable combinations of structure functions, the latter can be eliminated, thus obtaining some definite predictions. It is also found that some of these correction terms increase as one calculates higher moments of the structure functions. This remarkable fact, which is likely to persist in higher orders, suggests that all the functional relationships that are derived here and elsewhere have questionable validity near threshold; however, away from this region, where higher-order terms become negligible, they provide an important test of these theories. It is also shown that no similar predictions can be made for the singlet pieces of the structure functions since they always involve uncalculable constants. The question whether or not the present range of energies (25 GeV in electroproduction and 150 GeV in neutrino production) is sufficiently high to test these theoretical asymptotic predictions is discussed. By making some reasonable assumptions, it is found that the effective coupling constant in these ranges of energies is ≥1 so that according to these models, the asymptotic region has not yet been achieved. However, by regarding the expansion in terms of the effective coupling constant as an experimentally measurable parameter, it is possible (but not certain) that one obtains measurable theoretical predictions within the range of present energies.