arXiv:1109.6509v1 [hep-ph] 29 Sep 2011
Bottomonium in a Bethe-Salpeter-equation study
M. Blank∗and A. Krassnigg†
Institut f¨ ur Physik, Karl-Franzens-Universit¨ at Graz, A-8010 Graz, Austria
(Dated: September 30, 2011)
Using a well-established effective interaction in a rainbow-ladder truncation model of QCD, we fix
the remaining model parameter to the bottomonium ground-state spectrum in a covariant Bethe-
Salpeter equation approach and find surprisingly good agreement with the available experimental
data including the 2−−Υ(1D) state. Furthermore, we investigate the consequences of such a fit for
charmonium and light-quark ground states.
PACS numbers: 14.40.-n, 12.38.Lg, 11.10.St
InQCD, mesonsappear asboundstatesof
(anti)quarks and gluons. The bottomonium system, in
particular below the B¯B threshold, is a prototype for
the successful description of a meson using a straight-
forward ¯ qq-picture (for reviews on the subject, see e.g.
the comprehensive compilations of the quarkonium work-
ing group [1, 2]). Such a simple setup is expected to be
generally more realistic and accurate for heavier than for
light quarks. In a much similar way, in a covariant Bethe-
Salpeter-equation (BSE) approach a simple truncation
like the well-established rainbow-ladder (RL) truncation
is expected to be more accurate for heavier quarks and
their bound states. We perform a basic initial test for this
hypothesis by employing a model historically set up to
describe light mesons. In this way, we study ground-state
mesons for spins J = 0,1,2 starting with bottomonium
down to light quarks and check the possibility to arrive
at a reasonable agreement with experiment without fine-
tuning. The latter point is important, since a general
result or trend must be visible before one optimizes or
fine-tunes the available model parameters.
The paper is organized as follows: in Secs. II and
III we review the necessary details of the approach and
the interaction.Section IV deals with the bottomo-
nium ground states, followed by an investigation of the
consequences of our parameter choice for lower quark
masses in Sec. V. We conclude and present an outlook
in Sec. VI. All calculations have been performed using
Landau-gauge QCD in Euclidean momentum space.
II.MESONS IN AN RL MODEL OF QCD
We employ QCD’s Dyson-Schwinger-equations (DSEs)
(see, e.g. [3, 4] for recent reviews) coupled with the
quark-antiquarkBethe-Salpeter equation (BSE). The lat-
ter is the covariant bound-state equation for the study of
mesons in this context [5–7], and analogously one can
use a covariant approach to baryons in both a quark-
diquark picture (e.g. [8–10] and references therein) or a
three-quark setup [11, 12].
Numerical hadron studies such as the present one make
a truncation of this infinite tower of coupled and in gen-
eral nonlinear integral equations necessary. Herein, we
use the so-called rainbow-ladder (RL) truncation, which
is well-established as a tool for modeling hadron physics
in QCD. In particular, it is better-suited as an approxi-
mation to the full set of equations the higher the quark
mass becomes, see e.g., [13–15]. Related concrete results
on the heavy-quark domain and -limit of Coulomb-gauge
QCD have become available recently [16–18].
The RL truncation is simple yet offers the possibility
for sophisticated model studies of QCD within the DSE-
BSE context, since it satisfies the relevant (axial-vector
and vector) Ward-Takahashi identities (see e.g. [19–26]).
Regarding the meson spectrum a generally accurate de-
scription on the basis of a purely phenomenologically ori-
ented model is conceivable. However, to increase the pre-
dictive power of the model it is advisable to reduce the
free parameters in such a model, or more precisely in the
effective model interaction, as much as possible.
The expectation in such a situation is that the de-
scription of light mesons will not be as accurate, which is
a consequence of the fact that additional terms in the
dressed quark-gluon vertex, which are omitted in RL
truncation, cannot be successfully mimicked by a simple
parametrization of the effective interaction such as the
one used here. In the present study however, this is a de-
fect we are willing to accept in order to check the validity
of our assumptions about the heavy-quark domain. Our
restrictions are further justified by meson studies beyond
RL truncation (see, e.g., [27, 28] for relevant references)
that have confirmed effects from correction terms, but at
the same time shown both the numerical complexity of
such investigations as well as the uncertainty of the size
of even further corrections.
In RL truncation the axial-vector Ward-Takahashi
identity dictates that the rainbow-truncated integral-
equation kernel of the quark DSE corresponds to the
ladder-truncated integral-equation kernel of the quark-
antiquark BSE as given below. The identity is crucial
to correctly realize chiral symmetry and its dynamical
breaking in the model calculation from the very be-
FIG. 1. (Color online) The bottomonium ground-state spec-
trum compared to experimental data.
Goldstone’s theorem  and obtains a generalized Gell-
Mann–Oakes–Renner relation valid for all pseudoscalar
mesons and all current-quark masses [29, 30]. This rela-
tion can also be checked numerically and is satisfied at
the per-mill level in our calculations.
We start out from a model setup defined in Ref. 
and given in detail below, which has since been success-
fully applied to many in particular pseudoscalar- and
vector-meson properties in recent years (see e.g. [4, 32]
for comprehensive bibliographies). Of interest are elec-
tromagnetic hadron properties [25, 33–36], strong hadron
decay widths , valence-quark distributions of pseu-
doscalar mesons [38–40], a study of tensor mesons 
and an exploratory application of this model to the chi-
ral phase transition of QCD at finite temperature . Of
immediate interest are the recent steps to the successful
numerical treatment of heavy quarks in this particular
model setup [32, 43–45].
As the most prominent result, one satisfies
III.QUARK DSE AND MESON BSE
In RL truncation one considers a meson with total
q¯ q momentum P and relative q¯ q momentum q by con-
sistently solving the homogeneous, ladder-truncated q¯ q
Γ(p;P) = −4
χ(q;P) = S(q+)Γ(q;P)S(q−),
G((p − q)2) Df
µν(p − q) γµχ(q;P) γν,
on the one hand, where the semicolon separates four-
vector arguments, and the rainbow-truncated quark DSE
S(p)−1= (iγ · p + mq) + Σ(p),
G((p − q)2) Df
µν(p − q) γµS(q) γν (2)
on the other hand.
The solution of the BSE is the Bethe-Salpeter ampli-
tude (BSA) Γ(q;P) which, combined with two dressed
FIG. 2. (Color online) The charmonium ground-state spec-
trum compared to experimental data.
quark propagators S(q+) and S(q−) gives the “Bethe-
Salpeter wave function” χ(q;P). Note that in the case of
spin J > 0 the BSA carries J open Lorentz indices (for
details see ), which are omitted here for simplicity
together with the BSA’s Dirac and flavor indices. The
free gluon propagator, γνis the bare quark-gluon vertex,
G((p − q)2) is the effective interaction specified in detail
below, and the (anti)quark momenta are q+ = q + ηP
and q− = q − (1 − η)P. η ∈ [0,1] is referred to as the
momentum partitioning parameter and is usually set to
1/2 for systems of equal-mass constitutents, which we do
invariant regularization of the integral, with the regular-
ization scale Λ . Σ(p) denotes the quark self energy
and mq the current-quark mass. The solution for the
quark propagator S(p) requires a renormalization proce-
dure, the details of which can be found together with
the general structure of both the BSE and quark DSE in
For the homogeneous, ladder-truncated q¯ q BSE nu-
merical solution methods have been improved in recent
years  and also cross-checked with the closely related
approach to hadron phenomenology via the correspond-
ing inhomogeneous vertex BSEs, see e.g., [47–49]. Re-
garding the numerical solution of the quark DSE (2) we
note that in order to be able to subsequently and con-
sistently solve the BSE numerically, the propagator must
be known for quark four-momenta whose squares lie in-
side a parabola-shaped region of the complex p2plane
(for a more detailed discussion, see e.g., the appendix
of Ref. ). As a consequence, in particular for heavy
quarks a reliable numerical approach to the quark DSE
is needed and we refer the reader to  for the details
of our particular solution method.
important to note here that the analytical structure of
the quark propagator can place restrictions in terms of
an upper bound on the bound-state masses of mesons
accessible via standard numerical methods (see  for
a detailed discussion and an initial step towards a full
numerical treatment of such a situation). A route dif-
ferent to the one described in  is to extrapolate data
3comes from the color trace, Df
µν(p − q) is the
q=?Λd4q/(2π)4represents a translationally
Furthermore, it is
FIG. 3. (Color online) The light-isovector ground-state spec-
trum compared to experimental data.
obtained from the homogeneous BSE in the accessible
region to the inaccessible point of interest. This is the
approach we use here when necessary, and we give the
details of our procedure in the appendix.
Once RL truncation has been chosen, the integral-
equation kernels of (1) and (2) are essentially character-
ized by an effective interaction G(s), s := (p − q)2. The
parameterization of Ref.  reads
4π γmπ F(s)
This form produces the correct perturbative limit, i.e. it
preserves the one-loop renormalization group behavior
of QCD for solutions of the quark DSE. As given in ,
F(s) = [1 − exp(−s/[4m2
1, Nf = 4, ΛNf=4
= 0.234GeV, and γm = 12/(33 −
2Nf). The motivation for this function, which mimics the
behavior of the product of quark-gluon vertex and gluon
propagator, is mainly phenomenological. While currently
debated on principle grounds (e.g. [52, 53]) the impact
of its particular form in the far IR on meson masses is
expected to be small (see also  for an exploratory
study in this direction).
In , together with the current-quark mass mq, the
parameters ω and D were fitted to pion observables and
the chiral condensate. In this way, this effective coupling
provided the correct amount of dynamical chiral symme-
try breaking as well as quark confinement via the absence
of a Lehmann representation for the dressed quark prop-
agator. Also, from the results of the fitting procedure in
 it was apparent that for a fixed current-quark mass
one can obtain a good description of light pseudoscalar
and vector meson masses and decay constants by keep-
ing the product D × ω = 0.372 GeV3fixed and varying
ω in the range [0.3,0.5] GeV. In particular, these observ-
ables were independent of ω, which thus defines a one-
parameter model. Later, it was found that such a negli-
gible dependence on ω is the characteristic of a ground
state, while exctiations—both radial and orbital—in gen-
eral depend strongly on ω [30, 32]. As a result, it is pos-
sible to fix also ω to phenomenology, which we do in the
t])]/s, mt= 0.5 GeV, τ = e2−
FIG. 4. (Color online) The light-isoscalar ground-state spec-
trum compared to experimental data. The red symbols cor-
respond to the n¯ n-states of our ideally mixed setup, the blue
symbols denote pure s¯ s states.
present study, albeit without fine-tuning the parameters
any further, which is both besides the point as well as
beyond the scope of this work.
With D × ω fixed to the original value of 0.372 GeV3
and taking into account the trend visible in earlier com-
putations of parts of the bottomonium spectrum e.g.,
in [41, 49], we varied ω > 0.5 GeV and found good
agreement with all experimentally known bottomonium
ground states at ω = 0.61 GeV via a least-squares fit of
the masses of the states with JP= 0−, 0+, 1−, 1+, and
2+. The masses for the states with JP= 2−are thus pre-
dictions of the model. The results of our calculations are
summarized in the first row of Tab. I and compared to
the available experimental data in Fig. 1. Our numerical
uncertainties are smaller than the sizes of the symbols in
the figure for all cases.
These results for the masses show that already with-
out fine-tuning we have achieved agreement on the level
of 3 per-mill for all states considered, which is rather re-
markable. In addition to the excellent overall agreement
we also achieve reasonable agreement with e.g., the ex-
perimental value of 69 MeV for the hyperfine splitting
between the 0−+and 1−−ground states, which is repro-
duced to 20%.
Furthermore we computed the leptonic decay constants
of the pseudoscalar and vector bottomonium states and
collect our results together with a comparison to exper-
imental numbers (where available) in Tab. II.
the experimental value for fηbis yet unknown, we arrive
within 4% of the experimental value for fΥ. This latter
observation is remarkable in particular, since the compu-
tation of the leptonic decay constants goes beyond spec-
troscopy in that it involves also the BSAs of the states
under investigation. This means that the structure of the
meson as described by the BSA is captured reasonably
by our setup as well.
TABLE I. Calculated meson masses in MeV (rounded) for the four quark-mass values with extrapolation uncertainties given in
brackets where applicable. A comparison to experimental data is given in Figs. 1 - 4. Note that our results for ¯ nn correspond
to both light isovector and isoscalar states.
TABLE II. Calculated pseudoscalar and vector meson decay
constants in MeV (rounded) compared to experimental data
, where available.
¯ nn isovector
V.CHARMONIUM AND LIGHT MESONS
While the main point of interest in the present work
is the compatibility of our ansatz with the ground state
masses and decay constants of the bottomonium system,
it is natural to ask how the same parameter set per-
forms for charmonium as well as strange and light quark
masses. We thus present the corresponding results in the
remaining rows of Tabs. I and II and compare to avail-
able experimental data in Fig. 2 for charmonium and
Figs. 3 and 4 for states made out of light and strange
It is important to note here that in our present trunca-
tion there is no flavor mixing, i.e., all states are thus ide-
ally mixed and consequently a priori can be expected to
correspond to experimental states only in the appropriate
cases. Note also that we work in the isospin-symmetric
limit. In our tables we therefore list our results for pure
¯ ss and ¯ nn states, where in the usual notation n labels
light quark flavors. While one could apply simple flavor-
mixing rules to our results  we do not attempt this
here to maintain the clarity of our results as well as the
simplicity of both the model and the intention of our
work. It is not aimed at a perfect description of light
meson masses; in fact, such an outcome cannot be ex-
pected of our present study, since RL truncation over-
simplifies the structure of the quark-gluon vertex for light
and strange quarks (and apparently to some degree also
for charm quarks). However, it appears that a set of
results such as the present one can in the future be rec-
onciled with experiment upon proper inclusion of correc-
tions beyond RL truncation as they have been explored
in the recent literature, e.g., [57, 58].
Regarding our charmonium results we observe an over-
all pattern of agreement with experimental data although
some deficiencies are visible compared to the bottomo-
nium case.Most notably the scalar and axial-vector
masses are underestimated while the hyperfine splitting
is overestimated. Our results for the pseudoscalar and
vector leptonic decay constants are off 11 and 8% of the
experimental numbers, which is about twice as much as
For the strange and light quark cases we present our re-
sults compared to isoscalar and isovector states as listed
by the PDG . Note that our results for ¯ nn as listed
in Tab. I correspond to both light isovector and isoscalar
states. Due to the setup of the present study, our obser-
vations here can reasonably only be of a general nature.
For the isovector case one observes a well-known pattern
from the literature, namely that pseudoscalar and vector
meson masses compare well to experiment, while scalar
and axial-vector states are substantially underestimated
; for tensor mesons the situation is better . Also
for the decay constants, our model provides a reasonable
description of the data, even in its present form.
For the isoscalar case, the situation is naturally more
complex. Still, the overall impression is the same as de-
scribed for the isovector case above except for the pseu-
doscalar mesons, whose flavor composition cannot be de-
scribed in RL truncation.
VI.CONCLUSIONS AND OUTLOOK
nium ground-state meson masses using and adjusting
a well-established rainbow-ladder truncated effective-
interaction setup of the Bethe-Salpeter-equation. Our
goal was to provide—without fine-tuning—an initial test
of the potential such an approach holds to describe exper-
imental data in a reasonable fashion. Our results show
that such a description is indeed possible and beyond a
surprisingly good match in bottomonium also provides a
reasonable description of charmonium and a consistent
picture for meson masses containing light and strange
This is an encouraging first step towards a comprehen-
sive study of mesons in this approach. Further steps will
involve fine-tuning but will also have to include appro-
priate corrections beyond RL truncation to provide the
mechanism for a reconciliation of the deficiencies appar-
ent in our present results for lower quark masses. To-
gether with an anchor of the model in the heavy-quark
domain such as the one exemplified here this will lead to
a successful model description of hadrons.
We acknowledge helpful conversations with D. Hor-
vati´ c, V. Mader, C. Popovici, and R. Williams. This work
was supported by the Austrian Science Fund FWF under
project no. P20496-N16, and was performed in associa-
tion with and supported in part by the FWF doctoral
program no. W1203-N08.
As mentioned in Sec. III and described in detail in
the appendix of Ref.  the Euclidean-space treatment
of the meson BSE demands considerable conceptual and
numerical care. In particular, for the reasons given in
Sec. III one in some cases of higher-lying meson masses
in our present approach has to resort to extrapolation
techniques. In  a straight-forward method was used,
which we have since improved upon; our refined method
is reported in the following.
In the standard approach, the homogeneous BSE is
solved as an eigenvalue equation, where the on-shell point
(and thus the mass of the meson in question) is reached
if the eigenvalue λ(P2= M2) = 1 (for a detailed discus-
sion of BSE eigenvalues, see ). As discussed in detail
in , this eigenvalue and its dependence on the total-
momentum squared is deeply connected to the bound-
state poles that appear in the corresponding four-point
1 − λ(P2)=
P2+ M2+ corrections ,(A.1)
where r denotes the residue at the pole corresponding to
a particle of mass M.
If the corrections in Eq. (A.1) are neglected, the combi-
becomes linear in P2, which was used for
the extrapolations done in . However, more reliable
results can be obtained if the corrections are taken into
account. In this work, we assume corrections of polyno-
with constants ci. These constants ci, the residue r as
well as the resulting masses M are obtained in a straight-
forward fit to calculated values of the function
the range of P2that can be accessed directly. In order to
estimate the uncertainty of the extrapolation, the poly-
nomial order N of the corrections is varied from N = 1
to 6, and our final result is computed from the arith-
metic mean of this sample; the error bars are given by
the spread of the largest and smallest value.
A similar method is used fort the extrapolation of the
decay constants f of the light and strange vector states.
In these cases, polynomials of degree 3, 4, and 5 have
been fitted to the available values of f(√−P2) ×√−P2
and extrapolated to
the three resulting values is quoted as final result, and
the uncertainties are estimated from the differences be-
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