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Exploring the Invisible Renormalon: Renormalization of the Heavy-Quark Kinetic Energy

Matthias Neubert

TL;DR

The paper investigates whether the heavy-quark kinetic-energy operator in HQET mixes with the identity under ultraviolet renormalization and whether UV renormalons affect its matrix elements. It first shows, using the virial theorem, that one-loop mixing is forbidden when the regulator preserves Lorentz invariance, explaining the observed absence of a renormalon in this channel within such schemes. It then demonstrates that at two loops the mixing is nonzero, computing a quadratically divergent additive renormalization $Z_{D^2\to\hat{1}}$ via dispersion-regulated two-loop diagrams and linking it to the chromo-electric operator; this implies a UV renormalon at $u=1$ and regulator-dependent coefficients. The findings establish that $\lambda_1^H$ is not a physical parameter without a nonperturbative subtraction, clarify the role of regularization schemes (Lorentz-invariant vs lattice), and highlight the need for careful scheme matching in phenomenology and lattice calculations. Overall, the work resolves the earlier puzzle of the invisible renormalon and clarifies the regulatory dependence and subtraction required for the heavy-quark kinetic energy parameter.

Abstract

Using the virial theorem of the heavy-quark effective theory, we show that the mixing of the operator for the heavy-quark kinetic energy with the identity operator is forbidden at the one-loop order by Lorentz invariance. This explains why such a mixing was not observed in several one-loop calculations using regularization schemes with a Lorentz-invariant UV regulator, and why no UV renormalon singularity was found in the matrix elements of the kinetic operator in the bubble approximation (the ``invisible renormalon''). On the other hand, we show that the mixing is not protected in general by any symmetry, and it indeed occurs at the two-loop order. This implies that the parameter $λ_1^H$ of the heavy-quark effective theory is not directly a physical quantity, but requires a non-perturbative subtraction.

Exploring the Invisible Renormalon: Renormalization of the Heavy-Quark Kinetic Energy

TL;DR

The paper investigates whether the heavy-quark kinetic-energy operator in HQET mixes with the identity under ultraviolet renormalization and whether UV renormalons affect its matrix elements. It first shows, using the virial theorem, that one-loop mixing is forbidden when the regulator preserves Lorentz invariance, explaining the observed absence of a renormalon in this channel within such schemes. It then demonstrates that at two loops the mixing is nonzero, computing a quadratically divergent additive renormalization via dispersion-regulated two-loop diagrams and linking it to the chromo-electric operator; this implies a UV renormalon at and regulator-dependent coefficients. The findings establish that is not a physical parameter without a nonperturbative subtraction, clarify the role of regularization schemes (Lorentz-invariant vs lattice), and highlight the need for careful scheme matching in phenomenology and lattice calculations. Overall, the work resolves the earlier puzzle of the invisible renormalon and clarifies the regulatory dependence and subtraction required for the heavy-quark kinetic energy parameter.

Abstract

Using the virial theorem of the heavy-quark effective theory, we show that the mixing of the operator for the heavy-quark kinetic energy with the identity operator is forbidden at the one-loop order by Lorentz invariance. This explains why such a mixing was not observed in several one-loop calculations using regularization schemes with a Lorentz-invariant UV regulator, and why no UV renormalon singularity was found in the matrix elements of the kinetic operator in the bubble approximation (the ``invisible renormalon''). On the other hand, we show that the mixing is not protected in general by any symmetry, and it indeed occurs at the two-loop order. This implies that the parameter of the heavy-quark effective theory is not directly a physical quantity, but requires a non-perturbative subtraction.

Paper Structure

This paper contains 3 sections, 28 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Operator Mixing
  3. Conclusions

Figures (3)

  • Figure 1: One-loop diagrams that could contribute to the mixing of the kinetic energy with the identity operator. The kinetic operator is represented by a grey square; heavy-quark propagators are drawn as double lines.
  • Figure 2: One-loop diagrams that could contribute to the mixing of the chromo-electric operator with lower-dimensional operators. The velocity-changing chromo-electric operator is represented by a white square.
  • Figure 3: Two-loop diagrams that could contribute to the mixing of the chromo-electric operator with lower-dimensional operators. Not shown are two copies of the first two diagrams with the "outer" gluon attached to the other heavy-quark line.