Effective field theory for ultrasoft momenta in NRQCD and NRQED

Abstract

We propose an effective field theory for heavy quark-antiquark bound states once the soft gluons have been integrated out. We also give new results for the matching between QCD (QED) and NRQCD (NRQED).

Heavy quarkonium and nonrelativistic effective field theories

“…A detailed description of the procedure can be found in [3,9,7,13]. Here we just state the main results and briefly discuss how they are obtained.…”

“…The last years have witnessed important progress in the theoretical understanding of heavy-quark-antiquark systems near threshold through the use of effective field theories [1][2][3][4][5][6][7] (see also [8][9][10] for related work in QED). The key point relies on the fact that, since the quark velocity v is a small quantity, v ≪ 1, a hierarchy of widely separated scales: m ≫ mv ≫ mv 2 ... is produced in these systems, where m is the heavy quark mass (hard scale), mv the soft scale and mv 2 the ultrasoft scale.…”

“…The Lagrangian of NRQCD can be organised in powers of 1/m. After integrating out the soft scale in NRQCD, potential NRQCD (pNRQCD) is obtained [3]. The Lagrangian of pNRQCD is organised in powers of 1/m and the relative coordinate r (multipole expansion).…”

Top quark production near threshold

“…In the simplest case, the instantaneous interactions reduce to what is commonly called the 'heavy quark potential'. This theory is appropriately termed potential non-relativistic QCD (PNRQCD) [12]. Although both are integrated out together, it is necessary to distinguish potential gluons (which are off mass shell) and soft gluons (which are on-shell) to obtain an homogeneous (single scale) expansion near threshold [5].…”

The muonic hydrogen Lamb shift and the proton radius

“…In our approach we combine the use of Heavy Baryon Effective Theory (HBET) [5,6], Non-Relativistic QED (NRQED) [7] and, specially, potential NRQED (pN-RQED) [8][9][10]. Partial results following this approach can be found in [11][12][13] (see [14] for a review).…”