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Harald Grießhammer

Threshold Expansion and Dimensionally Regularised Non-Relativistic Quantum Chromo Dynamics

Abstract

A Lagrangean and a set of Feynman rules are presented for non-relativistic quantum field theories with manifest power counting in the heavy particle velocity v . A regime is identified in which energies and momenta are of order Mv . It is neither identical to the ultrasoft regime corresponding to radiative processes with energies and momenta of order Mv^2 , nor to the potential regime with on shell heavy particles and Coulomb binding. In this soft regime, massless particles are on shell, and heavy particle propagators become static. Examples show that it contributes to one- and two-loop corrections of scattering and production amplitudes near threshold. Hence, non-relativistic quantum field theory agrees with the results of threshold expansion. A simple example also demonstrates the power of dimensional regularisation in non-relativistic quantum field theory.

Introduction

Velocity power counting in Non-Relativistic Quantum Field Theories (Caswell and Lepage, Braaten et al.), especially in non-relativistic Quantum Electrodynamics and non-relativistic Quantum Chromodynamics (NRQCD), and identification of the relevant energy and momentum regimes has proven more difficult than previously believed. In a recent article, Beneke and Smirnov pointed out that the velocity rescaling rules proposed by Luke and Manohar, and Grinstein and Rothstein, and united by Luke and Savage, do not reproduce the correct behaviour of the two gluon exchange contribution to Coulomb scattering between non-relativistic particles near threshold. This has cast some doubt whether NRQCD, especially in its dimensionally regularised version following Luke and Savage, can be formulated using a self-consistent low energy Lagrangean. The aim of this letter is to demonstrate that a Lagrangean establishing explicit velocity power counting exists, and to show that this Lagrangean reproduces the results obtained by Beneke and Smirnov.

This letter is confined to outlining the ideas to resolve the puzzle, postponing more formal arguments, calculations and derivations to a future, longer publication which will also deal with gauge theories and exemplary calculations. It is organised as follows: In section 2, the relevant regimes of NRQFT are identified. A simple example demonstrates the usefulness of dimensional regularisation in enabling explicit velocity power counting. Section 3 proposes the rescaling rules necessary for a Lagrangean with manifest velocity power counting. The Feynman rules are given. Simple examples in Section 4 establish further the necessity of the new, soft regime introduced in Section 3. Summary and outlook conclude the letter.

Conclusions and Outlook

The objective of this letter was a simple presentation of the ideas behind explicit power counting in dimensionally regularised NRQFT. The identification of three different regimes of scale for on-shell particles in NRQFT leads in a natural way to the existence of a new quark field and a new gluon field in the soft scaling regime . Neither of the five fields in the three regimes should be thought of as physical particles. Rather, they represent the true quark and gluon in the respective regimes. A Lagrangean for non-relativistic quantum field theory has been proposed which leads to the correct behaviour of scattering and production amplitudes. It establishes explicit velocity power counting which is preserved to all orders in perturbation theory. The reason for the existence of such a Lagrangean, once dimensional regularisation is chosen to complete the theory, was elaborated upon in a simple example: the non-commutativity of the expansion in small parameters with dimensionally regularised integrals.

Due to the similarity between the calculation of the examples in the work presented here and in the paper by Beneke and Smirnov, one may get the impression that the Lagrangean presented is only a simple re-formulation of the threshold expansion. Partially, this is true, and a future publication will indeed show the equivalence of the two approaches to all orders in the threshold and coupling expansion. A list of other topics to be addressed there contains: the straightforward generalisation to NRQCD; a proof whether the particle content outlined above is not only fully consistent but complete, i.e. that no new fields (e.g. an ultrasoft quark) or exceptional regimes arise; an investigation of the influence of soft quarks and gluons on bound state calculations in NRQED and NRQCD; a full list of the various couplings between the different regimes and an exploitation of their relevance for physical processes. The formal reason why double counting between different regimes and especially between soft and ultrasoft gluons does not occur, a derivation of the way soft quarks couple to external sources, and the role of soft gluons in Compton scattering deserve further attention, too.

I would like to stress that the diagrammatic threshold expansion derived here allows for a more automatic and intuitive approach and makes it easier to determine the order in to which a certain graph contributes. On the other hand, the NRQFT Lagrangean can easily be applied to bound state problems. As the threshold expansion of Beneke and Smirnov starts in a relativistic setting, it may formally be harder to treat bound states there. Indeed, I believe that even if one may not be able to prove the conjectures of the one starting from the other, both approaches will profit from each other in the wedlock of NRQFT and threshold expansion.