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arXiv:0905.2291v2 [hep-ph] 24 Sep 2009

Probing the gluon self-interaction in light mesons

Christian S. Fischer1,2and Richard Williams1

1Institute for Nuclear Physics, Darmstadt University of Technology,

Schlossgartenstraße 9, 64289 Darmstadt, Germany

2GSI Helmholtzzentrum f¨ ur Schwerionenforschung GmbH, Planckstr. 1 D-64291 Darmstadt, Germany.

(Dated: September 24, 2009)

We investigate masses and decay constants of light mesons from a coupled system of Dyson–

Schwinger and Bethe–Salpeter equations. We explicitly take into account dominant non-Abelian

contributions to the dressed quark-gluon vertex stemming from the gluon self-interaction. We con-

struct the corresponding Bethe-Salpeter kernel that satisfies the axial-vector Ward-Takahashi iden-

tity. Our numerical treatment fully includes all momentum dependencies with all equations solved

completely in the complex plane. This approach goes well beyond the rainbow-ladder approximation

and permits us to investigate the influence of the gluon self-interaction on the properties of mesons.

As a first result we find indications of a nonperturbative cancellation of the gluon self-interaction

contributions and pion cloud effects in the mass of the ρ−meson.

PACS numbers: 11.10.St, 11.30.Rd, 12.38.Lg

Keywords: Dynamical chiral symmetry breaking, light mesons

Introduction

Understanding the details of the light meson spectrum

from underlying QCD is still an intricate and open prob-

lem of considerable interest.

mesons are the (pseudo-)Goldstone bosons of QCD

and as such enjoy an exceptional position amongst the

hadronic states of QCD. Within the framework of Dyson–

Schwinger (DSE) and Bethe-Salpeter (BSE) equations

their Goldstone nature is retrieved in dynamical calcu-

lations provided constraints from chiral symmetry, i.e.

the axial-vector Ward-Takahashi identity (axWTI), are

taken into account, see e.g. [1, 2]. However, apart from

the masses of the (pseudo-)Goldstone bosons all other

properties of light mesons such as masses, decay con-

stants or charge radii are not governed by symmetry but

depend on the details of the strong interaction of their

constituents.This includes effects such as gluon self-

interactions as well as pion cloud corrections [3].

Of course, pseudoscalar

Within the DSE/BSE framework these effects are all

contained in the structure of the nonperturbative quark-

gluon vertex. It is then clear that a simple rainbow-

ladder parametrisation of the quark-gluon interaction

in terms of vector couplings cannot be sufficient to de-

scribe the wealth of phenomena associated with the in-

ternal structure of light mesons, see e.g. [4] and refer-

ences therein. In particular, a long-standing problem of

rainbow-ladder is that it is too attractive, yielding masses

of 800-900 MeV [5] for the axial-vector mesons.

Consequently there have been considerable efforts to

go beyond rainbow-ladder. Here, Abelian corrections to

the quark-gluon vertex have been considered in a number

of works, see e.g. [1, 6, 7, 8, 9]. These are, however, by

far not the dominant contributions to the vertex [10].

Instead, genuinely non-Abelian diagrams including the

gluon self-interaction are most important. Up to now,

these have only been considered on the level of the DSE

for the quark-gluon vertex and the quark propagator [10,

11].

Here we provide for a significant extension of these

efforts.We present the first calculation of meson ob-

servables within a framework where the DSEs for the

quark-gluon vertex and the quark propagator, as well as

the BSE for mesons, have been coupled together without

a trivialization of any momentum dependence. Further-

more it is the first time that corrections from the domi-

nant non-Abelian part of the quark-gluon vertex includ-

ing the gluon self-interaction are tested. We determine

masses and decay constants of light mesons and evaluate

the influence of the gluon self-interaction corrections as

compared to the rainbow-ladder approximation.

Our results are significant for q¯ q bound states and

therefore relevant for the vector mesons. In the axial-

vector and scalar channels, we describe putative q¯ q bound

states which may or may not be realised in nature, see

e.g. [12]. Nevertheless it is satisfactory that the inclu-

sion of gluon self-interaction effects also in these channels

leads to an improved description of light mesons.

The Quark-Gluon Vertex DSE

The fundamental dynamical quantity in the DSE/BSE

approach to hadrons is the quark-gluon vertex, whose

specification will ultimately determine the truncation

scheme. We therefore begin with a discussion of its DSE

depicted in Eq. (1) and Eq. (2) of Fig. 1.

the detailed investigation of the quark-gluon vertex in

Refs. [10, 13] we approximate the full DSE (1) with the

(nonperturbative) one-loop structure of (2).

first ‘non-Abelian’ loop-diagram in (2) subsumes the first

two diagrams in the full DSE to first order in a skele-

ton expansion of the four-point functions. The two-loop

diagram in the full DSE (1) is neglected. This approx-

imation is well justified in the large and small momen-

tum r` egime [10, 11] and is assumed to be tractable for

Following

Here the

Page 2

2

=

−

+1

2

++1

6

(1)

=

+Nc

2

−

2

Nc

+

π

(2)

FIG. 1: The truncation employed for the quark-gluon vertex.

All internal propagators are dressed, with wiggly lines indi-

cating gluons, straight lines quarks and dashed lines mesons.

White-filled circles indicate bound-state amplitudes whilst

black-filled represent vertex dressings. Note that the last di-

agram is also proportional to 1/Nc.

intermediate momenta.

tributions are split into the non-resonant second loop-

diagram in (2) and a third diagram containing effects

due to hadron backreactions. These are dominated by

the lightest hadrons, i.e. pseudoscalar mesons.

Note that the non-Abelian and non-resonant Abelian

diagrams are associated with colour factors Nc/2 and

−2/Nc, respectively. The diagram including the pion ex-

change is also proportional to 1/Ncdue to the implicit

1/√Nc-dependence of the two pion amplitudes. Thus,

both Abelian diagrams are suppressed by a factor of N2

as compared to the non-Abelian one, a fact that is also ev-

idenced through direct numerical calculation [10]. Con-

sequently, these diagrams are by far not the leading ones

in the vertex-DSE. Moreover, as concerns meson masses,

the Abelian diagrams are generally attractive. The ef-

fects of these diagrams on the mass of the ρ-meson have

been estimated to be about 30 MeV due to non-resonant

Abelian diagrams in the quark-gluon vertex [8] with an-

other 90 MeV due to pion cloud effects [3].

In this letter we concentrate on the leading non-

Abelian diagram in Eq. (2) and explore its effects on me-

son observables as compared to pure rainbow-ladder ap-

proximations. In order to keep our calculation tractable

we employ the well-established strategy of absorbing all

internal vertex dressings into effective dressing functions

for the two internal gluon propagators. The resulting

DSE for the quark-gluon vertex Γµ(p1,p2) with quark

momenta p1and p2and gluon momentum p3reads

The remaining ‘Abelian’ con-

c

Γµ(p1,p2) = Z1Fγµ+

?−iNc

2

g2Z2

1FZ1

?

×

(3)

×

?

q

?

γνS(q)γρΓ3g

σθµ(k1,k2)Dνσ(k1)Dρθ(k2)

?

with

and Γ3gthe bare three-gluon vertex.

quark-gluon vertex Γµis given by a combination of twelve

?

q≡

?

d4q

(2π)4, the renormalization factors Z1F,Z1

In general, the

independent tensors built up of the quark momenta pµ

pµ

dressed quark and gluon propagators are given by

1,

2and γµ; for a detailed discussion see Ref. [10]. The

S(p) = [−ip ? A(p2) + B(p2)]−1,

Dµν(k) =

(4)

?

δµν−kµkν

k2

?Z(k2)

k2

,(5)

where Z is the gluon dressing. The quark dressing func-

tions A, B are determined from the DSE for the quark

propagator given diagrammatically in Fig. 2. With bare

quark mass m and renormalization factor Z2it reads

S−1(p) = Z2[−ip ?

+ g2CFZ1F

+ m]

(6)

?

q

γµS(q)Γν(q,k)Dµν(k).

What remains to be specified in both the vertex and

the quark DSE is the effective dressing of the gluon prop-

agator. Here we use the momentum dependent ansatz [5]

Z(q2) =4π

g2

πD

ω2q4e−q2/ω2,(7)

with two parameters D and ω which provide for the scale

and strength of the effective gluon interaction. Natu-

rally, such an ansatz provides only a first step towards

a full calculation of the non-Abelian diagram including

input from the DSEs for the three-gluon vertex and the

gluon propagator. Given that the numerical treatment

of the coupled system of vertex-DSE, quark-DSE and the

Bethe-Salpeter equation for light mesons is somewhat in-

volved we defer such a calculation for future work. Nev-

ertheless we believe that the ansatz (7) is sufficient to

provide for reliable qualitative results as concerns the ef-

fects due to the non-Abelian diagram onto meson prop-

erties. In particular it is not sensitive to the question of

scaling vs. decoupling [14] in the deep infrared, p < 50

MeV: both scaling and decoupling lead to a combination

of three-gluon vertex and gluon propagatordressings that

is vanishing in the infrared in qualitative agreement with

the ansatz (7). In addition, quantitative effects in the

interaction below p < 50 MeV are not expected to affect

observables in the flavour non-singlet sector since the dy-

namical mass of the quark, M ≈ 350 MeV, suppresses all

physics on scales p ≪ M (see however [15]). Finally, the

ansatz (7) is not sensitive to details of the Slavnov-Taylor

identity (STI) for the three-gluon vertex: exactly those

longitudinal parts of the vertex that are constrained from

??

?

??

?

FIG. 2: The DSE for the fully dressed quark propagator.

Page 3

3

the STI are projected out in any Landau gauge calcula-

tion by the attached transverse gluon propagators.

As concerns our numerical treatment we solve the

coupled system of quark-gluon vertex DSE and quark

DSE for complex Euclidean momenta p1and p2with the

‘shell-method’ described in the second work of Ref. [13].

Through judicious choice of momentum routing in both

the quark DSE and vertex DSE, this can be accomplished

without unconstrained analytic continuation of the gluon

propagator and three-gluon vertex. The BSE is solved for

complex momenta by standard methods [5, 17].

The Bethe-Salpeter equation

The Bethe-Salpeter amplitude Γ(p;P) ≡ Γ(µ)(p;P) for a

bound state of mass M is calculated through

[Γ(p;P)]tu= λ

?

k

Krs

tu(p,k;P)[S(k+)Γ(k;P)S(k−)]sr,

(8)

with k±= k ± P/2 and eigenvalue λ(P2= −M2) = 1.

The amplitude can be decomposed into at most eight

Lorentz and Dirac structures constrained by transforma-

tion properties under CPT [16]. It is well-known that

one may construct a Bethe-Salpeter kernel Krs

ing the axWTI by means of a functional derivative of

the quark self-energy [1, 2, 17, 18]. Applying this cutting

procedure to the quark DSE specified by Eq. (6) with the

quark-gluon vertex of Eq. (4) yields the Bethe-Salpeter

equation portrayed in Fig. 3 [18].

tusatisfy-

=++

FIG. 3: The axWTI preserving BSE corresponding to our

vertex truncation. All propagators are dressed, with wiggly

and straight lines showing gluons and quarks respectively.

Normalisation

Since we solve the homogeneous Bethe-Salpeter equation,

the correct normalisation of the amplitude is achieved

through an auxiliary condition [19], given pictorially in

Fig. 4. This involves evaluating three-loop integrals over

nonperturbative quantities, which we tackle with the aid

of standard Monte-Carlo techniques. Alternatively, us-

ing the eigenvalue λ in (8) and the conjugate amplitude

Γ(k,−P) = CΓ(−k,−P)C−1one may use the equivalent

normalisation condition [20]

?dln(λ)

dP2

?−1

= tr

?

k

3Γ(k,−P)S(k+)Γ(k,P)S(k−).

(9)

This requires significantly less numerical effort and can

be simply applied to all truncations of the BSE.

We will see that the diagrams of Fig. 4 beyond the

impulse approximation give large corrections of the order

of 30%. This is important for observables calculated from

the amplitudes such as leptonic decay constants [17]

Results

Table I details the results of our truncation scheme, in-

cluding gluon self-interaction effects in all twelve ten-

sor structures of the quark-gluon vertex, compared to

the rainbow-ladder with only vector-vector interactions.

The model parameters ω and D were tuned such that for

the latter we obtain reasonable pion observables, and the

quark mass is fixed at 5 MeV. We do not fit observables

for our beyond rainbow-ladder truncation scheme since

the inclusion of additional resonant and non-resonant

corrections would require further parameter tuning.

In all cases the non-Abelian corrections are seen to

have only a small impact on the mass of the pion, though

the decay constant is enhanced from 94 MeV to about

110 MeV. This shift is comparable in size to the negative

one of the order of 10 MeV [3] induced by pion cloud

effects, i.e. the third loop-diagram in (2). We also inves-

tigated the impact of the Abelian, second loop-diagram

in (2) without use of the real-axis approximation made

in [8]. We confirm that the change in the pion mass is

negligible for their preferred value of G = 0.5 and find a

small reduction of ∼ 2 MeV in the decay constant. Once

all corrections are combined, we expect significant can-

cellations such that the mass and decay constant of the

pion are close to the experimental value.

For the rho meson, including the gluon self-interaction

enhances its mass by ∼ 120 MeV compared to pure

rainbow-ladder, with the decay constant increased by

∼ 20 MeV. This is an intriguing result since it has

long been suspected that corrections beyond rainbow-

ladder cancel amongst themselves in the vector channel

[1, 6]. Indeed, the resonant and non-resonant contribu-

tions from the Abelian diagrams in (2) are known to pro-

vide reductions of the rho mass of ∼ 90 MeV and 30 MeV

respectively [3, 8]. Similar cancellations happen for the

decay constant. We therefore see for the first time strong

evidence of this nonperturbative cancellation mechanism.

This is one of the main results of this letter.

? = 2

∂

∂P2tr

"

+

++

#

FIG. 4:

Dashed lines represent quark propagators that are kept fixed

under action of the derivative.

Normalisation of the Bethe-Salpeter amplitude.

Page 4

4

ModelωDmπ

fπ

ˆfπ

mρ

fρ

ˆfρ

mσ

ma1

mb1

R-L

BTR

R-L

BTR

R-L

BTR

0.5016

138

138

138

141

136

142

94

111

95

112

92

110

– 758

881

763

887

746

873

154

176

154

176

149

173

– 645

884

676

886

675

796

926

1055

937

1040

917

1006

912

972

895

946

858

902

(127)

–

(129)

–

(128)

(181)

–

(181)

–

(177)

0.48 20

0.4525

Experiment [22]13892.4– 776156–400–120012301230

TABLE I: Masses and decay constants for a variety of mesons, calculated using rainbow-ladder (R-L) and our beyond-the-

rainbow (BTR) truncation. Decay constants are determined from the full normalisation condition, except those indicated by a

caret where only the leading term of the impulse approximation is used. Masses and decay constants are given in MeV.

Fig. 5 shows the rho mass as a function of the pion

mass for our truncation beyond-the-rainbow (BTR) com-

pared to a chiral extrapolation based on (corrected) lat-

tice data, Ref. [21]. Due to the discussion above we

expect the explicit inclusion of the resonant and non-

resonant Abelian contributions in our BTR-scheme to

move our results close to the lattice data.

For completeness, we also report the masses of the

scalar and axial-vectors in Table I. In all cases we see

an enhancement compared to rainbow-ladder.

Conclusions

We presented an exploratory study of light mesons using

a sophisticated truncation of the Bethe-Salpeter equa-

tions beyond rainbow-ladder, in which we consider the

gluon self-interaction contributions to the quark-gluon

vertex. Close to the chiral limit we obtain masses for

the rho meson of ∼ 900 MeV, consistent with extrapo-

lations from quenched lattice simulations. There is evi-

dence that the subsequent inclusion of pion cloud effects

and non-resonant contributions to the quark-gluon inter-

action brings the rho mass back to its physical value thus

supporting a long suspected nonperturbative cancellation

mechanism. Our truncation provides for a well-founded

setup to further explore the details of the nonperturba-

tive quark-gluon interaction and gluon self-interaction ef-

0.00.20.4

0.6

0.8

1.0

mπ [GeV]

0.7

0.8

0.9

1

1.1

1.2

1.3

mρ [GeV]

BTR

CP-PACS + Adelaide

FIG. 5: Rho mass as a function of the pion mass compared

to a chiral extrapolation based on (corrected) lattice data,

Ref. [21].

fects in mesons. In this respect it is complementary to

the one very recently suggested in Ref. [23].

Acknowledgements

We thank Pete Watson for discussions. This work was

supported by the Helmholtz Young Investigator Grant

VH-NG-332 and the Helmholtz International Center for

FAIR within the LOEWE program of the State of Hesse.

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