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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.