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
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 .
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.  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  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 .
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,
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. . 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
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.
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 . 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  with an-
other 90 MeV due to pion cloud effects .
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-
Γµ(p1,p2) = Z1Fγµ+
and Γ3gthe bare three-gluon vertex.
quark-gluon vertex Γµis given by a combination of twelve
(2π)4, the renormalization factors Z1F,Z1
In general, the
independent tensors built up of the quark momenta pµ
dressed quark and gluon propagators are given by
2and γµ; for a detailed discussion see Ref. . The
S(p) = [−ip ? A(p2) + B(p2)]−1,
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 ?
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 
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  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 ). 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.
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. .
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
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 . 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 .
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.
Since we solve the homogeneous Bethe-Salpeter equation,
the correct normalisation of the amplitude is achieved
through an auxiliary condition , 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 
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 
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  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 . 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
Dashed lines represent quark propagators that are kept fixed
under action of the derivative.
Normalisation of the Bethe-Salpeter amplitude.
Experiment 13892.4– 776 156– 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. . 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.
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-
CP-PACS + Adelaide
FIG. 5: Rho mass as a function of the pion mass compared
to a chiral extrapolation based on (corrected) lattice data,
fects in mesons. In this respect it is complementary to
the one very recently suggested in Ref. .
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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