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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

I.INTRODUCTION

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

∗martina.blank@uni-graz.at

†andreas.krassnigg@uni-graz.at

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-

Page 2

2

????????

????????

????????

????????

????????

????????

????????

????????

Ηb?1S?

??1S?

Χb0?1P?

Χb1?1P?

hb?1P?

Χb2?1P?

??1D?

0??

1??

0??

1??

1??

2??

2??

2??

JPC

9.4

9.6

9.8

10.0

10.2

M?GeV?

FIG. 1. (Color online) The bottomonium ground-state spec-

trum compared to experimental data.

ginning.

Goldstone’s theorem [23] 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. [31]

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 [37], valence-quark distributions of pseu-

doscalar mesons [38–40], a study of tensor mesons [41]

and an exploratory application of this model to the chi-

ral phase transition of QCD at finite temperature [42]. 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

BSE

Γ(p;P) = −4

χ(q;P) = S(q+)Γ(q;P)S(q−),

3

?Λ

q

G((p − q)2) Df

µν(p − q) γµχ(q;P) γν,

(1)

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),

Σ(p) =4

3

q

?Λ

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

????????

????????

????????

????????

????????

????????

????????

????????

Ηc?1S?

J?Ψ?1S?

Χc0?1P?

Χc1?1P?

hc?1P?

Χc2?1P?

0??

1??

0??

1??

1??

2??

2??

2??

JPC

3.0

3.2

3.4

3.6

3.8

M?GeV?

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 [41]), which are omitted here for simplicity

together with the BSA’s Dirac and flavor indices. The

factor

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

as well.

?Λ

invariant regularization of the integral, with the regular-

ization scale Λ [29]. 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

[29, 31].

For the homogeneous, ladder-truncated q¯ q BSE nu-

merical solution methods have been improved in recent

years [46] 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. [41]). As a consequence, in particular for heavy

quarks a reliable numerical approach to the quark DSE

is needed and we refer the reader to [50] 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 [51] 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 [51] is to extrapolate data

4

3comes from the color trace, Df

µν(p − q) is the

q=?Λd4q/(2π)4represents a translationally

Furthermore, it is

Page 3

3

????????

????????

????????

????????

????????

????????

????????

????????

Π0

Ρ?770?0

a0?980?0

a1?1260?0

b1?1235?0

a2?1320?0

Π2?1670?0

Ρ2?1940?

0??

1??

0??

1??

1??

2??

2??

2??

JPC

0.0

0.5

1.0

1.5

2.0

2.5

3.0

M?GeV?

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. [31] reads

G(s)

s

=4π2D

ω6

s e−s/ω2+

4π γmπ F(s)

1/2ln[τ+(1+s/Λ2

QCD)2]. (3)

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 [31],

F(s) = [1 − exp(−s/[4m2

1, Nf = 4, ΛNf=4

QCD

= 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 [54] for an exploratory

study in this direction).

In [31], 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

[31] 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−

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?

?

?

?

?

?

?

?

?

??

???????

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?

?

?

?

?

?

?

?

?

?

?

?

?

??

???????

?

?

?

?

?

?

?

?

?

?

?

?

?

??

????????

????????

????????

????????

????????

Η

Η??958?

Ω?782?

Φ?1020?

f0?600?

f0?980?

f1?1285?

f1?1420?

h1?1170?

h1?1595?

f2?1270?

f2?1430?

Η2?1645?

Η2?1870?

Ω2?1975?

0??

1??

0??

1??

1??

2??

2??

2??

JPC

0.0

0.5

1.0

1.5

2.0

2.5

3.0

M?GeV?

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.

IV.BOTTOMONIUM

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.

While

Page 4

4

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.

JPC

0−+

9405

2928

637

248

1−−

9488

3111

980(3)

730(120)

0++

9831

3321

904

405(1)

1++

9878

3437

1+−

9873

3421

2++

9927

3582

2−+

2−−

¯bb

10184(8)

3818(8)

1852(58)

1465(136)

10188(8)

3818(9)

1541(247)

1673(817)

¯ cc

¯ ss

¯ nn

1293(299)

861(292)

1244(15)

886(92)

1560(81)

1480(149)

TABLE II. Calculated pseudoscalar and vector meson decay

constants in MeV (rounded) compared to experimental data

[55], where available.

JPC

0−+

708

1−−

687

715

448

416

237+4

227

276+61

221

¯bb

Experiment

−

¯ cc

399

361

173

Experiment

¯ ss

−3

Experiment

¯ nn isovector

Experiment

−

108

131

−113

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

(anti)quarks.

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 [56] 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 bottomonium.

For the strange and light quark cases we present our re-

sults compared to isoscalar and isovector states as listed

by the PDG [58]. 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

[32]; for tensor mesons the situation is better [41]. 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

Wehavereportedastudyofthebottomo-

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

Page 5

5

(anti)quarks.

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.

ACKNOWLEDGMENTS

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.

Appendix: Technicalities

As mentioned in Sec. III and described in detail in

the appendix of Ref. [41] 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 [41] 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 [46]). As discussed in detail

in [49], 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

function,

λ(P2)

1 − λ(P2)=

r

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-

nation

λ(P2)

becomes linear in P2, which was used for

the extrapolations done in [49]. However, more reliable

results can be obtained if the corrections are taken into

account. In this work, we assume corrections of polyno-

mial form,

1−λ(P2)

corrections =

N

?

i=1

?P2?ici, (A.2)

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-

tween the average and the largest and smallest value,

respectively.

λ(P2)

1−λ(P2)in

√−P2= M. Again, the average of

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