The renormalization group for correlated scales: one-stage versus two-stage running
Aneesh V. Manohar, Joan Soto, Iain W. Stewart
Abstract
Nonrelativistic bound states have two low energy scales, a soft scale muS of order mv and an ultrasoft scale muU of order mv2. 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 mv2. 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.
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