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Extending the Kinetic Mass to Higher Orders in $1/m_Q$

Thomas Mannel, Ilija S. Milutin, Rens Verkade, K. Keri Vos

Abstract

Currently, the kinetic mass is defined in terms of the pole mass and operators at order $1/m_Q^2$, which are known to N$^3$LO accuracy in $α_s$. At the same time, the Heavy Quark Expansion (HQE) for inclusive semileptonic decays is known up to and including terms of order $1/m_Q^5$. Therefore, it is desirable to extend the definition of the kinetic mass to higher orders in $1/m_Q$. The original kinetic mass is based on the hadron-mass formula in Heavy Quark Effective Theory (HQET). However, the HQE is formulated in terms of matrix elements defined in full QCD to avoid the appearance of non-local matrix elements. To avoid this, we develop a definition of the kinetic mass rooted in full QCD. Starting from the hadron-mass formula derived from the energy-momentum tensor of full QCD, we define a relation between a general mass and the pole mass. Using a simple cut-off scheme, we compute a generalized kinetic mass at one loop to all powers of $1/m_Q$, which reproduces the well-known results for the kinetic mass up to $1/m_Q^2$. Our approach opens the road to a consistent use of the kinetic mass at higher-orders in the heavy quark expansion.

Extending the Kinetic Mass to Higher Orders in $1/m_Q$

Abstract

Currently, the kinetic mass is defined in terms of the pole mass and operators at order , which are known to NLO accuracy in . At the same time, the Heavy Quark Expansion (HQE) for inclusive semileptonic decays is known up to and including terms of order . Therefore, it is desirable to extend the definition of the kinetic mass to higher orders in . The original kinetic mass is based on the hadron-mass formula in Heavy Quark Effective Theory (HQET). However, the HQE is formulated in terms of matrix elements defined in full QCD to avoid the appearance of non-local matrix elements. To avoid this, we develop a definition of the kinetic mass rooted in full QCD. Starting from the hadron-mass formula derived from the energy-momentum tensor of full QCD, we define a relation between a general mass and the pole mass. Using a simple cut-off scheme, we compute a generalized kinetic mass at one loop to all powers of , which reproduces the well-known results for the kinetic mass up to . Our approach opens the road to a consistent use of the kinetic mass at higher-orders in the heavy quark expansion.
Paper Structure (11 sections, 60 equations, 2 figures)

This paper contains 11 sections, 60 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mass definitions and hadron mass formulae
  3. Quark two-point function and on-shell quark mass
  4. From Hadron Mass to Quark Mass in HQET
  5. Extended kinetic mass from the QCD mass formula
  6. Calculation of the on-shell $\bar{Q}Q$ matrix element
  7. Calculation of the anomaly terms
  8. The extended kinetic mass
  9. Kinetic mass and the HQE
  10. Small Velocity sum rules
  11. Summary and Outlook

Figures (2)

  • Figure 1: The one-loop diagrams contributing to $M_{\bar{Q}Q}$, where (b) and (c) correspond to the external field renormalization. The black square represents the insertion of $\bar{Q}Q$.
  • Figure 2: The Feynman diagram for the on-shell matrix element from the $G_{\mu\nu}^a G^{\mu\nu,a}$ operator, denoted by the black square.