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Power Counting and βFunction in NRQCD

H. W. Griesshammer

TL;DR

This work extends velocity power counting to NRQCD by identifying soft, potential, and ultrasoft regimes and deriving rescaling, vertex, and loop rules that preserve power counting under dimensional regularisation. It demonstrates that NRQCD reproduces the QCD $eta$ function at one loop, independent of gauge choice, via contributions from soft and ultrasoft modes and the Coulombic vacuum polarisation, while proving ultrasoft-quark decoupling and establishing a formal link to threshold expansion. The diagrammatic rules simplify identifying scale-less graphs and ensure gauge-invariant running, highlighting the soft regime’s essential role in the infrared limit of QCD and providing a robust framework for effective field theory analyses of heavy-quark systems. The results reinforce NRQCD as a consistent low-energy limit of QCD and lay groundwork for systematic higher-order and bound-state investigations in a gauge-invariant, dimensionally regularised setting.

Abstract

A computation of the NRQCD $β$ function both in the Lorentz gauge family and in the Coulomb gauge to one loop order endorses a velocity power counting scheme for dimensionally regularised NRQCD. In addition to the ultrasoft scale represented by bremsstrahlung gluons and the potential scale with Coulomb gluons and on-shell quarks, a soft régime is identified in which energies and momenta are of order $Mv$, gluons are on shell and the quark propagator becomes static. The instantaneous gluon propagator has a non-zero vacuum polarisation only because of contributions from this régime, irrespective of the gauge chosen. Rules are derived which allow one to read up from a given graph whether it is zero because of the homogene{\ia}ty of dimensional regularisation. They also apply to threshold expansion and are used to prove that ultrasoft quarks with energy and momentum of order $Mv^2$ decouple from the theory.

Power Counting and βFunction in NRQCD

TL;DR

This work extends velocity power counting to NRQCD by identifying soft, potential, and ultrasoft regimes and deriving rescaling, vertex, and loop rules that preserve power counting under dimensional regularisation. It demonstrates that NRQCD reproduces the QCD function at one loop, independent of gauge choice, via contributions from soft and ultrasoft modes and the Coulombic vacuum polarisation, while proving ultrasoft-quark decoupling and establishing a formal link to threshold expansion. The diagrammatic rules simplify identifying scale-less graphs and ensure gauge-invariant running, highlighting the soft regime’s essential role in the infrared limit of QCD and providing a robust framework for effective field theory analyses of heavy-quark systems. The results reinforce NRQCD as a consistent low-energy limit of QCD and lay groundwork for systematic higher-order and bound-state investigations in a gauge-invariant, dimensionally regularised setting.

Abstract

A computation of the NRQCD function both in the Lorentz gauge family and in the Coulomb gauge to one loop order endorses a velocity power counting scheme for dimensionally regularised NRQCD. In addition to the ultrasoft scale represented by bremsstrahlung gluons and the potential scale with Coulomb gluons and on-shell quarks, a soft régime is identified in which energies and momenta are of order , gluons are on shell and the quark propagator becomes static. The instantaneous gluon propagator has a non-zero vacuum polarisation only because of contributions from this régime, irrespective of the gauge chosen. Rules are derived which allow one to read up from a given graph whether it is zero because of the homogene{\ia}ty of dimensional regularisation. They also apply to threshold expansion and are used to prove that ultrasoft quarks with energy and momentum of order decouple from the theory.

Paper Structure

This paper contains 19 sections, 73 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Velocity Power Counting
  3. Régimes of NRQCD
  4. Rescaling Rules and Propagators
  5. Vertex Rules
  6. Loop Rules
  7. Gauge Invariance
  8. The NRQCD $\beta$ Function
  9. NRQCD and Threshold Expansion: Potential Quark Self-Energy
  10. Some Diagrammatic Rules
  11. Theorems from Diagrammatic Rules
  12. The Quark Self-Energy
  13. The Vacuum Polarisation
  14. The Vertex Correction and NRQCD $\beta$ Function
  15. The Coulomb Gauge $\beta$ Function
  16. ...and 4 more sections

Figures (10)

  • Figure 1: Power counting with soft loops. The loops in the second and third diagram obtain an inverse power of $v$, the last diagram of $v^2$ in addition to the power counting following from the vertex rules.
  • Figure 2: The first rule as proven step by step in the text. (a) Simple case; (b) a more complicated sub-diagram; (c,d) generalisations. The overlay of double and zigzag line stands for any number of arbitrary soft particles, the triple line for any number of potential or ultrasoft particles, all entering at the same vertex. The blob represents vertex and propagator dressings. No time ordering is implied in the way the diagrams are drawn.
  • Figure 3: The second rule in its bare (a) and dressed (b) version; (c) the generalisation analogous to Fig. \ref{['rule1']} (c) fails. Conventions as in Fig. \ref{['rule1']}.
  • Figure 4: (a) Soft-to-ultrasoft vertices are cut off in a third rule; (b) an example. Conventions as in Fig. \ref{['rule1']}.
  • Figure 5: The fourth rule in its (a) bare and (b) extended version, where the rule in Fig. \ref{['rule3']} (a) was used for attaching ultrasoft gluons to the soft gluon. Conventions as in Fig. \ref{['rule1']}.
  • ...and 5 more figures