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

Nonrelativistic bound states have two low energy scales, a soft scale mu(S) of order ma and an ultrasoft scale mu(u) of order m nu(2) In two-stage run ning, the soft and ultrasoft scales are lowered from nr to me, and then the ultrasoft scale is lowered from m nu to m nu(2). In one-stage running, the two scales are lowered in a correlated way using a subtraction velocity. W e compare these two methods of summing logarithms and show that only the co rrelated running in velocity space of the one-stage method correctly reprod uces the logarithms in nonrelativistic bound states in QED. The argument fo r one-stage running is general, and should apply to any system with correla ted scales. (C) 2000 Elsevier Science B.V. All rights reserved.