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On The Difficulty of Computing Higher-Twist Corrections

G. Martinelli, C. T. Sachrajda

TL;DR

The paper investigates whether higher-twist power corrections in QCD can be computed with sufficient precision to meaningfully improve predictions. It centers on renormalon ambiguities in Wilson coefficients and shows that these ambiguities cancel only when the coefficient functions are known to very high perturbative order, or when higher-twist matrix elements are anchored to physical quantities or computed nonperturbatively with careful operator definitions. Through toy models, lattice HQET analyses, and large-$\beta_0$ calculations, it demonstrates that in practice the residual uncertainties are often comparable to, or larger than, the power corrections themselves, making reliable determination of leading power effects very difficult. The results suggest that, with current perturbative knowledge, leading power corrections are not well constrained and that substantial methodological advances are needed for robust higher-twist predictions.

Abstract

We discuss the evaluation of power corrections to hard scattering and decay processes for which an operator product expansion is applicable. The Wilson coefficient of the leading-twist operator is the difference of two perturbative series, each of which has a renormalon ambiguity of the same order as the power corrections themselves, but which cancel in the difference. We stress the necessity of calculating this coefficient function to sufficiently high orders in perturbation theory so as to make the uncertainty of the same order or smaller than the relevant power corrections. We investigate in some simple examples whether this can be achieved. Our conclusion is that in most of the theoretical calculations which include power corrections, the uncertainties are at least comparable to the power corrections themselves, and that it will be a very difficult task to improve the situation.

On The Difficulty of Computing Higher-Twist Corrections

TL;DR

The paper investigates whether higher-twist power corrections in QCD can be computed with sufficient precision to meaningfully improve predictions. It centers on renormalon ambiguities in Wilson coefficients and shows that these ambiguities cancel only when the coefficient functions are known to very high perturbative order, or when higher-twist matrix elements are anchored to physical quantities or computed nonperturbatively with careful operator definitions. Through toy models, lattice HQET analyses, and large- calculations, it demonstrates that in practice the residual uncertainties are often comparable to, or larger than, the power corrections themselves, making reliable determination of leading power effects very difficult. The results suggest that, with current perturbative knowledge, leading power corrections are not well constrained and that substantial methodological advances are needed for robust higher-twist predictions.

Abstract

We discuss the evaluation of power corrections to hard scattering and decay processes for which an operator product expansion is applicable. The Wilson coefficient of the leading-twist operator is the difference of two perturbative series, each of which has a renormalon ambiguity of the same order as the power corrections themselves, but which cancel in the difference. We stress the necessity of calculating this coefficient function to sufficiently high orders in perturbation theory so as to make the uncertainty of the same order or smaller than the relevant power corrections. We investigate in some simple examples whether this can be achieved. Our conclusion is that in most of the theoretical calculations which include power corrections, the uncertainties are at least comparable to the power corrections themselves, and that it will be a very difficult task to improve the situation.

Paper Structure

This paper contains 13 sections, 41 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Power Corrections
  3. Defining The Matrix Elements of $O_2$ Directly in Terms of Physical Quantities
  4. Non-Perturbative Computation of the Matrix Elements of $O_2$
  5. Summary
  6. A Toy Model
  7. The Pole Mass of a Heavy Quark
  8. Evaluation of $\overline{m}$ in Lattice Simulations
  9. Cancellation of Renormalon Ambiguities in the Heavy-Quark Mass
  10. The Gluon Condensate
  11. Evaluation of the Gluon Condensate in Lattice Simulations
  12. The Uncertainty in the Perturbation Series for the D-function
  13. Conclusions

Figures (2)

  • Figure 1: Diagrams which must be evaluated in order to study the heavy quark mass (and hence $\overline{\Lambda}$) in the large-$\beta_0$ limit. The double line represents the propagator of the heavy quark, and the bubbles represent light-quark loops.
  • Figure 2: One-loop bubble graph contributing to the vacuum polarization of the gluon. This graph is the basic ingredient in the evaluation of the large-$\beta_0$ contribution to the quark mass (and other quantities). The solid lines represent light-quark propagators.