Physics Letters B 773 (2017) 172–178
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Renormalized quark–antiquark Hamiltonian induced by a gluon mass
ansatz in heavy-flavor QCD
Stanisław D. Głazek
a,b
, María Gómez-Rocha
c,∗
, Jai More
d
, Kamil Serafin
a
a
Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
b
Department of Physics, Yale University, 217 Prospect Street, New Haven, CT 06520, USA
c
European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*), Villa Tambosi, Strada delle Tabarelle 286, I-38123 Villazzano (Trento), Italy
d
Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
a r t i c l e i n f o a b s t r a c t
Article history:
Received
24 May 2017
Received
in revised form 18 July 2017
Accepted
8 August 2017
Available
online xxxx
Editor:
J.-P. Blaizot
Keywords:
Hamiltonian
QCD
Renormalization
Heavy
quarkonia
Gluon
mass
Symmetry
In response to the growing need for theoretical tools that can be used in QCD to describe and
understand the dynamics of gluons in hadrons in the Minkowski space–time, the renormalization group
procedure for effective particles (RGPEP) is shown in the simplest available context of heavy quarkonia
to exhibit a welcome degree of universality in the first approximation it yields once one assumes
that beyond perturbation theory gluons obtain effective mass. Namely, in the second-order terms, the
Coulomb potential with Breit–Fermi spin couplings in the effective quark–antiquark component of a
heavy quarkonium, is corrected in one-flavor QCD by a spin-independent harmonic oscillator term that
does not depend on the assumed effective gluon mass or the choice of the RGPEP generator. The new
generator we use here is much simpler than the ones used before and has the advantage of being suitable
for studies of the effective gluon dynamics at higher orders than the second and beyond the perturbative
expansion.
© 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
The growing need for understanding dynamics of gluons in
QCD, comprehensively documented in [1], revives interest in
Hamiltonian methods for describing heavy quarkonia in terms of
wave functions in the Fock space of virtual quanta, where glu-
ons
are likely to behave in a relatively simple way because they
are permanently coupled to the massive quarks that move slowly
with respect to each other. Phenomenology and theory of heavy
quarkonia rapidly develops, as is illustrated by many examples
[2–12]. The key feature regarding gluons that requires understand-
ing
is how they acquire the effective mass, so that the hadron
mass spectra [13] do not exhibit any small excitations such as the
atomic spectra do due to massless photons. The gluon-mass gen-
eration
is a subject of research from early on using continuum
Dyson–Schwinger equations [14–16] and it is addressed in lattice
studies [17–20] because its implications for theory of strong inter-
actions
would be broad, including the issue of saturation in dense
gluon systems beyond a single hadron setting [21]. In the canon-
ical
formulation of QCD in the front form (FF) of dynamics in the
*
Corresponding author.
E-mail
address: gomezr@ectstar.eu (M. Gómez-Rocha).
Minkowski space–time [22], the need for understanding implica-
tions
of an effective gluon mass is stressed in general in [23] and
in the context of heavy quarkonia in [24,25]. Theoretical studies
of heavy quarkonia may thus be expected to increasingly focus
on the Hamiltonian dynamics of their gluonic content, cf. [26,27].
This article presents a first step in a program of systematic studies
of dynamics of scale-dependent gluons in heavy quarkonia, start-
ing
with the simplified case of canonical FF formulation of QCD
with quarks of just one heavy flavor and assuming that the effec-
tive
gluons which correspond to momentum scale of quark binding
mechanism in quarkonia have mass.
We
calculate a renormalized, scale-dependent Hamiltonian for
quarkonium constituents in the Fock space using a new formula-
tion
of the renormalization group procedure for effective particles
(RGPEP) in quantum field theory, see below. In this new formula-
tion,
the key RGPEP element called its generator does not depend
on the derivative of the renormalized Hamiltonian with respect
to the scale parameter, in contrast to the earlier versions of the
RGPEP, in which such dependence was involved. The generator de-
pendence
on the derivative of the Hamiltonian made a calculation
of the latter difficult and stalled the development. With the deriva-
tive
issue resolved, the improved method is now prepared for trial
applications and further development. The new generator has been
http://dx.doi.org/10.1016/j.physletb.2017.08.018
0370-2693/
© 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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