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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

, Kamil Seraﬁn

Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland

Department of Physics, Yale University, 217 Prospect Street, New Haven, CT 06520, USA

European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*), Villa Tambosi, Strada delle Tabarelle 286, I-38123 Villazzano (Trento), Italy

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.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

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.

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/

SCOAP

S.D. Głazek et al. / Physics Letters B 773 (2017) 172–178 173

already veriﬁed to work beyond perturbative expansion in simple

theories [28,29] and it has passed the test of producing asymptotic

freedom in the renormalized FF Hamiltonian of QCD [30]. Here we

present the new-generator calculation of a renormalized Hamilto-

nian

in one-ﬂavor theory including terms up to second order in

powers of the strong-interaction coupling constant at the scale of

heavy quarkonia.

The

renormalized Hamiltonian at a suitable scale is subse-

quently

applied to formulate the eigenvalue problem for a quarko-

nium

in the Fock space basis that is constructed using the cre-

ation

operators for effective particles of the corresponding size.

We make a drastically simplifying assumption that the non-Abelian

and non-perturbative effects due to components with more effec-

tive

gluons than one, can be mimicked by a gluon-mass ansatz

introduced in the dynamics of the component with only one ef-

fective

gluon and a quark–antiquark pair. Namely, the components

with more than one effective gluon are dropped at the price of in-

troducing

the ansatz. We then express the quark–antiquark–gluon

component in terms of the quark–antiquark component and cal-

culate

the effective Hamiltonian in the quark–antiquark sector. As

a result, we establish the universality of effective interaction one

obtains in the quark–antiquark component: the new RGPEP gen-

erator

produces the same Coulomb term with Breit–Fermi spin-

dependent

factors and the same spin-independent harmonic os-

cillator

term that were previously obtained with the old genera-

tor [31,32].

The oscillator term is independent of the value of the

mass ansatz.

The

new effective term respects rotational symmetry in the

quark–antiquark relative three-momentum space, in which the

eigenvalue problem for quarkonium two-body component resem-

bles

a non-relativistic Schrödinger equation with a potential, ex-

cept

that the eigenvalue is the quarkonium mass squared instead

of its energy. The effective eigenvalue equation is invariant with re-

spect

to the FF kinematical Lorentz transformations. We also note

that the relative momentum variables we use for heavy quarko-

nia

resemble the variables used in the light-front holographic

approach to the phenomenology of light hadrons, based on the

AdS/QCD duality ideas [33]. The holographic potential is also of the

harmonic oscillator form, but its frequency is different, which can

be associated with much smaller mass of light quarks than 

QCD

while the heavy quark mass is much greater.

Section 2 e

xplains the preliminary nature of the gluon mass

ansatz that is used to ﬁnesse the effective quark–antiquark inter-

action.

Section 3 presents the RGPEP in application to one-ﬂavor

QCD. The renormalized eigenvalue equation for the entire quarko-

nium

state and the effective Hamiltonian for its quark–antiquark

component are introduced in Section 4. Section 5 shows how

the effective harmonic oscillator potential emerges in the non-

relativistic

limit, using holographic quark–antiquark relative mo-

mentum

variables. Section 6 concludes the article with comments

on how our results prepare ground for renormalized Hamiltonian

studies of gluon dynamics in heavy quarkonia.

2. The initial need for gluon mass ansatz

The diﬃculty of describing bound states in the Fock space is

that one needs to deal with an a priori inﬁnite number of compo-

nents.

In the case of quarkonium, the bound state

|ψ=|Q

Q +|Q

QG+|Q

QGG+... , (1)

is built from canonical quanta of quark and gluon ﬁelds. To deal

with this issue, the RGPEP uses the concept of effective particles.

They are characterized by an effective size s and are related to the

bare, point-like particles of canonical theory by means of an oper-

ator

transformation. The idea is that for description of observables

characterized by the momentum scale λ = 1/s, one can use the

renormalized Hamiltonian H

, so that the number of relevant Fock

components in Eq. (1) is suﬃciently small for carrying out com-

putations,

except that Q ,

Q and G are replaced by Q

and

, respectively. Thus, inﬁnitely many Fock components for effec-

tive

particles can be neglected. The bound-state problem becomes

so greatly simpliﬁed that one can attempt to seek solutions to the

eigenvalue equation numerically.

The

above strategy may work when the RGPEP equation for H

is solved exactly. When one uses expansions of H

in a series of

powers of effective coupling constant, non-perturbative effects in

itself are not included. The eigenvalue problem for such H

still couples the Fock components made of effective particles in

signiﬁcant ways. Initially, one cannot be certain that the Fock com-

ponents

with more than one effective gluon can be dropped from

the eigenvalue problem with no consequence. Since power count-

ing

allows a mass term for gluons, perturbative RGPEP calculations

imply a need for a gluon mass counterterm in the canonical Hamil-

tonian

of one-ﬂavor QCD, whose ﬁnite part is not known, and the

phenomenology of hadrons appears to exclude massless effective

gluons, one is in need to assume that neglecting components with

more than one effective gluon of size s in heavy quarkonia of char-

acteristic

physical size s may only be admissible if one allows for

appearance of the effective mass term for the gluon in the Q

component, presumably produced by the non-Abelian interaction

effects descendant from higher components, absent in Abelian the-

ories.

the variety of terms that are conceivable in renormalized

Hamiltonians using FF power counting [23], rotational symmetry

is not explicitly respected. Any ansatz one introduces to hypothet-

ically

account for the omitted terms, must satisfy the condition

that the resulting effective eigenvalue problem produces the mass

spectrum that exhibits degeneracy implied by rotational symmetry.

Making an ansatz for the gluon mass term, rather than any other

potentially allowed term, turns out to lead to the effective oscilla-

tor

correction to the Coulomb terms that satisﬁes the rotational-

symmetry

condition already at the level of effective Hamiltonian

itself, exhibiting the symmetry in relative momentum variables. It

is unlikely that ansatz terms for quantities other than mass can

easily produce such a universal result. Moreover, the mass ansatz

turns out to possess the unique feature that the oscillator potential

it leads to is independent of the actual value of the ansatz mass,

provided it is not too small.

When

the RGPEP calculation of renormalized Hamiltonian H

extended to higher orders than second, and when the unknowns

of the effective gluon mass are relegated to the Fock sectors with

more than one gluon, the preliminary gluon-mass ansatz in the dy-

namics

of Q

component will be replaced by true QCD terms.

However, since the complexity of renormalized Hamiltonian opera-

tors

requires approximations, and since their spectra are calculated

numerically making further simpliﬁcations, a purely mathemati-

cal

approach to identiﬁcation of all important terms may turn out

hard to execute. Fortunately, the spectra of Hamiltonians for heavy

quarkonia in the Fock space representation with effective gluons

can also be studied in comparison with experimental data. As-

suming

that the canonical theory one starts from is right, one can

use the data in identifying scale dependence of various terms al-

lowed

by the power counting. The mass ansatz seems to be the

simplest admissible term to falsify. The universality of the quarko-

nium

effective Hamiltonian described here prepares the ground

for required theoretical studies in parallel with quarkonium phe-

nomenology.

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