The renormalization group for correlated scales: one-stage versus two-stage running

Abstract

Nonrelativistic bound states have two low energy scales, a soft scale mu_S of order mv and an ultrasoft scale mu_U of order mv^2. In two-stage running, the soft and ultrasoft scales are lowered from m to mv, and then the ultrasoft scale is lowered from mv to mv^2. In one-stage running, the two scales are lowered in a correlated way using a subtraction velocity. We compare these two methods of summing logarithms and show that only the correlated running in velocity space of the one-stage method correctly reproduces the logarithms in non-relativistic bound states in QED. The argument for one-stage running is general, and should apply to any system with correlated scales.

9 pages, typos fixed, results and conclusions unchanged

References in corpus (6)

• Renormalization group scaling in nonrelativistic QCD
• The infrared behaviour of the static potential in perturbative QCD
• Ultrasoft Effects in Heavy-Quarkonium Physics
• Renormalization group analysis of the QCD quark potential to order v^2
• Order alpha_s^3 ln^2(1/alpha_s) Corrections to Heavy-Quarkonium Creation and Annihilation
• The second-order electron self-energy in hydrogen-like ions

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