In the paper [Hong-Shi Zong, Wei-Min Sun, Phys. Lett. B 640 (2006) 196], the
authors claim that our proof of the inconsistency of the ladder approximation
to QCD [Phys. Lett. B 611 (2005) 129] was incorrect. However, their claim is
based on a derivation which contains a rough mathematical mistake, namely the
unjustified change of variables in the divergent (though regularized)
integrals. In this comment I will show this explicitly, so our conclusion that
the ladder approximation to QCD is inconsistent remains, of course, correct.Comment: 2 pages, no figure, no table

Bender

Burden

Cahill

Frank

Gogohia

Hong-Shi Zong

Ivanov

Johnson

Marciano

Maris

Meissner

Roberts

Tandy

Wei-Min Sun

Zong

Physics Letters B

English

arXiv

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A note on failure of the ladder approximation to QCD

AbstractIn this Letter we show that the claim made in [V. Gogohia, Phys. Lett. B 611 (2005) 129] that the ladder approximation to QCD is internally inconsistent is incorrect. The incorrect conclusion in [V. Gogohia, Phys. Lett. B 611 (2005) 129] is based on the incorrect use of a QED-type Ward–Takahashi relation, which does not hold in the ladder approximation to QCD. We give a proof for this fact

Zong, Hong-Shi

Sun, Wei-Min

Elsevier - Publisher Connector 

Physics Letters B 640 (2006) 196–200
www.elsevier.com/locate/physletb
A note on failure of the ladder approximation to QCD
Hong-Shi Zong a,b,c,∗, Wei-Min Sun a,b
a Department of Physics, Nanjing University, Nanjing 210093, China
b Joint Center for Particle, Nuclear Physics and Cosmology, Nanjing 210093, China
c CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China
Received 10 January 2006; received in revised form 12 March 2006; accepted 21 July 2006
Available online 8 August 2006
Editor: J.-P. Blaizot
Abstract
In this Letter we show that the claim made in [V. Gogohia, Phys. Lett. B 611 (2005) 129] that the ladder approximation to QCD is internally
inconsistent is incorrect. The incorrect conclusion in [V. Gogohia, Phys. Lett. B 611 (2005) 129] is based on the incorrect use of a QED-type
Ward–Takahashi relation, which does not hold in the ladder approximation to QCD. We give a proof for this fact.
© 2006 Elsevier B.V.
PACS: 11.15.Tk; 12.38.Lg
Keywords: Ladder approximation to QCD
It is well-known that the full dynamical information on any quantum field theory such as quantum chromodynamics (QCD) is
contained in the corresponding quantum equations of motion, the so-called Dyson–Schwinger (DS) equations. Due to the fact that
the DS equations are an infinite tower of coupled integral equations that relate all of the dressed n-point functions of a quantum field
theory to each other, there is no hope for an exact solutions. Thus, in phenomenological applications, one often proceed by making
certain simplifications and truncations regarding some subset of n-point functions such that the DS equations are reduced to a closed
system of equations which may be solved directly. The simplest of such truncation schemes is the rainbow-ladder approximation.
Over the past few years, considerable progress has been made in the framework of the rainbow-ladder approximation of the DS
approach [1–5], which provides a successful description of various nonperturbative aspects of strong interaction physics [6–20].
Recently, the ladder approximation is examined by Ref. [21] and the author claims that the ladder approximation to QCD is
internally inconsistent. Since the ladder approximation is such a widely used approximation, its internal consistency is an important
issue and deserves careful investigation. We have reexamined the reasoning in [21] and found that this claim is incorrect. The
incorrect conclusion in [21] is based on the incorrect use of a QED-type Ward–Takahashi relation, which does not hold in the
ladder approximation to QCD. In the following we shall first briefly recall the arguments in [21], and then prove that the QED-type
Ward–Takahashi relation used there does not hold in the ladder approximation to QCD.
Following the notations in [21], the quark DSE under the ladder approximation reads
(1)S−1(p) = S (p)−10 + iΣ(p) = S (p)−10 + g C2 F
∫
d l4
(2π)4
γαS(l)γβDαβ(p − l),
where iΣ(p) is the quark self-energy and CF is the eigenvalue of the quadratic Casimir operator in the fundamental representation
(CF = (N2 − 1)/2N = 4/3, N = 3 for QCD). Here we note that there is a typing error in Eq. (2.1) in Ref. [21] (the sign before
* Corresponding author.
E-mail address: nuclphys@nju.edu.cn (H.-S. Zong).
Open access under CC BY license.0370-2693© 2006 Elsevier B.V.
doi:10.1016/j.physletb.2006.07.065
Open access under CC BY license.
H.-S. Zong, W.-M. Sun / Physics Letters B 640 (2006) 196–200 197g2CF should be positive). S−10 (p) is the free quark propagator
(2)S−10 (p) = −i(/p − m0)
with m0 being the current quark mass, and Dαβ(q) is the gluon propagator in an arbitrary covariant gauge
(3)Dαβ(q) = −i
{[
gαβ − qαqβ
q2
]
d
(
q2, ξ
)+ ξ qαqβ
q2
}
1
q2
,
where ξ is the gauge-fixing parameter. Differentiating both sides of Eq. (1) with respect to pμ, one gets the differential form of the
quark DSE used in [21]
(4)∂μS−1(p) = −iγμ + ∂μiΣ(p) = −iγμ + g2CF
∫
d4l
(2π)4
γα
[
∂μS(l)
]
γβDαβ(p − l).
The quark–gluon vertex DSE under the ladder approximation reads
(5)Γ aμ (p, k) = −iγμT a − g2
∫
d4l
(2π)4
γαT
bS(l)Γ aμ (l, k)S(l − k)γβT bDαβ(p − l),
where a, b are color indices with T a for the standard Gell-Mann SU(3) representation and k is the momentum transfer. Assuming
analyticity of the vertex at zero momentum transfer, one obtains from the above equation
(6)Γ aμ (p,0) = −iγμT a − g2
∫
d4l
(2π)4
γαT
bS(l)Γ aμ (l,0)S(l)γβT bDαβ(p − l).
At this point the author in Ref. [21] argues that in the ladder approximation one should omit the ghost–quark scattering kernel
contained in the Slavnov–Taylor identity of QCD and therefore the Slavnov–Taylor identity reduces to the following QED-type
Ward–Takahashi relation
(7)kμΓ aμ (p, k) = T aS−1(p) − T aS−1(p − k).
From this relation one derives the following Ward–Takahashi identity relating Γ aμ (p,0) and S−1(p)
(8)Γ aμ (p,0) = T a∂μS−1(p).
Here we note that the above Ward–Takahashi identity plays a key role in Ref. [21]’s proof of the “internal inconsistency” of the
ladder approximation to QCD. As we will show below, this Ward–Takahashi identity should not be used because it does not hold
in the ladder approximation to QCD, and thus the proof in Ref. [21] is incorrect.
Using the above Ward–Takahashi identity, one can cast Eq. (6) into the form
(9)∂μS−1(p)T a = −iγμT a + g2T bT aT b
∫
d4l
(2π)4
γα
[
∂μS(l)
]
γβDαβ(p − l),
where we have used the identity ∂μS−1(p) = −S−1(p)[∂μS(p)]S−1(p) in deriving the above equation. Making use of the identity
T bT aT b = (CF − 12CA)T a to cancel the SU(3) generator T a from both sides of the above equation, where CA is the eigenvalue of
the quadratic Casimir operator in the adjoint representation (CA = N = 3 for QCD), one obtains
(10)∂μS−1(p) = −iγμ + g2
(
CF − 12CA
)∫
d4l
(2π)4
γα
[
∂μS(l)
]
γβDαβ(p − l).
Comparing this with Eq. (4) gives
(11)−1
2
g2CA
∫
d4l
(2π)4
γα
[
∂μS(l)
]
γβDαβ(p − l) = 0.
From this result and Eq. (4), one concludes that
(12)∂μiΣ(p) = 0.
The above constraint has only a trivial solution
(13)Σ(p) = mc,
where mc is the constant of the dimensions of mass (constant of integration).
According to the above observation the author of Ref. [21] concludes that in the ladder approximation to QCD the quark
propagator is a free one, apart from a redefinition of the quark mass, i.e. there is no running/dressed quark mass. Based on this
198 H.-S. Zong, W.-M. Sun / Physics Letters B 640 (2006) 196–200result, the author further argues that the quark propagator in the ladder approximation in fact equals the zero-mass free propagator
(14)S(p) = i
/p
,
i.e. there is no current quark mass, and the vertex at zero momentum transfer is always trivial
(15)Γ aμ (p,0) = −iγμT a.
From this the author of Ref. [21] concludes that the ladder approximation to QCD is internally inconsistent, and all the results based
on the nontrivial (analytical or numerical) solutions to the quark DSE in the ladder approximation should be reconsidered, and its
use in the whole energy/momentum range should be abandoned.
As was noted earlier, the key point of the above arguments is the use of the Ward–Takahashi identity Γ aμ (p,0) = T a∂μS−1(p),
which provides a constraint between the quark–gluon vertex at zero momentum transfer and the inverse quark propagator. Here we
want to stress that this relation cannot be used because it does not hold in the ladder approximation to QCD. The following is our
proof.
Our starting point is the quark–gluon vertex DSE (5). First let us contract both sides of Eq. (5) with kμ and obtain
(16)kμΓ aμ (p, k) = −i/kT a − g2
∫
d4l
(2π)4
γαT
bS(l)kμΓ
a
μ (l, k)S(l − k)γβT bDαβ(p − l).
Then, if the Ward–Takahashi relation (7) holds, we can substitute kμΓ aμ (l, k) with T aS−1(l) − T aS−1(l − k) and the right-hand
side (RHS) of the above equation becomes
−i/kT a − g2
∫
d4l
(2π)4
γαT
bS(l)
(
T aS−1(l) − T aS−1(l − k))S(l − k)γβT bDαβ(p − l)
= −i/kT a − g2T bT aT b
∫
d4l
(2π)4
γα
(
S(l − k) − S(l))γβDαβ(p − l)
= −i/kT a + g2
(
1
2
CA − CF
)
T a
∫
d4l
(2π)4
γα
(
S(l − k) − S(l))γβDαβ(p − l)
= T aS−10 (p) − T aS−10 (p − k) + T aiΣ(p) − T aiΣ(p − k) +
1
2
g2CAT
a
∫
d4l
(2π)4
γα
(
S(l − k) − S(l))γβDαβ(p − l)
(17)= T aS−1(p) − T aS−1(p − k) + 1
2
g2CAT
a
∫
d4l
(2π)4
γα
(
S(l − k) − S(l))γβDαβ(p − l),
where we have made use of the quark DSE (1). One sees that there appears an extra term
(18)1
2
g2CAT
a
∫
d4l
(2π)4
γα
(
S(l − k) − S(l))γβDαβ(p − l)
on the RHS of (17). If this extra term does not vanish, the RHS of (17) cannot be equal to T aS−1(p)− T aS−1(p − k) and we have
a contradiction, showing that the Ward–Takahashi relation (7) cannot hold in the ladder approximation to QCD. This assertion can
be proved in different ways. One way is to use the weak coupling expansion. In the weak coupling limit, the quark propagator and
the gluon propagator can be expanded in powers of the coupling constant g, with the zeroth order term being the free quark and
gluon propagator. Substituting this expansion into (18) we get a power series expansion for the extra term. In order to prove that the
extra term (18) does not vanish, it is sufficient to prove the leading term of the expansion of (18) does not vanish. It is apparent that
the leading term is obtained by substituting the quark and gluon propagators in (18) by the free ones:
(19)S(l − k) → i
/l − /k − m0 , S(l) →
i
/l − m0 , Dαβ(p − l) →
−igαβ
(p − l)2 ,
where for simplicity we have chosen the Feynman gauge for the gluon propagator. Without loosing generality we may further
assume that the bare quark mass m0 is zero and the leading term of (18) reads
(20)1
2
g2CAT
a
∫
d4l
(2π)4
γα
(
1
/l − /k −
1
/l
)
γα
1
(p − l)2 .
This integral is ultraviolet divergent and can be computed using, for instance, dimensional regularization. In n dimensions, (20)
becomes
(21)1
2
g2
(
μ2
)2− n2 CAT a
∫
dnl
(2π)n
γα
(
1
/l − /k −
1
/l
)
γα
1
(p − l)2 ,
H.-S. Zong, W.-M. Sun / Physics Letters B 640 (2006) 196–200 199where μ is an arbitrary mass scale introduced to make the coupling constant dimensionless. The remaining computation is standard
and we only give the results:
(22)1
2
g2CAT
a(2 − n) i(−π)
n
2
(2π)n

(2 − n2 )
(n2 )
(n2 − 1)

(n − 1)
((
μ2
(p − k)2
)2− n2
(/p − /k) −
(
μ2
p2
)2− n2
/p
)
.
This is nonzero and therefore (at least in the weak couping limit) our assertion is proved. The above assertion can also be proved
directly by numerical calculations. More specifically, one chooses a suitable model gluon propagator in a specific gauge and use it
as input to numerically solve the quark DSE (Eq. (1)) (more details can be found in Refs. [1–5]). Substituting the obtained quark
propagator and model gluon propagator into (18), one finds that the extra term does not vanish. Here we remark that from the
quark DSE (Eq. (1)) and quark–gluon vertex DSE (Eq. (5)) in the ladder approximation one can obtain nontrivial solutions to the
quark DSE, as was demonstrated in the existing literatures [1–20]. From the above results we conclude that the Ward–Takahashi
relation (7) cannot hold in the ladder approximation to QCD. At this point the readers can see clearly that if one imposes the
Ward–Takahashi relation (7), as was done in Ref. [21], then the extra term (18) must vanish
(23)1
2
g2CAT
a
∫
d4l
(2π)4
γα
(
S(l − k) − S(l))γβDαβ(p − l) ≡ 0.
Expanding the S(l − k) in powers of kμ, one sees that the term linear in kμ gives exactly Eq. (11), which is just the equation used
in Ref. [21] to reach its conclusion (see Eq. (5.4) in that reference).
Here it should be noted that in QCD the quark–gluon vertex function satisfies the Slavnov–Taylor identity which involves the
ghost–quark scattering kernel [22] and the author in Ref. [21] assumes in the ladder approximation the above “reduced” Ward–
Takahashi identity (7) holds. Since this identity does not hold, just as we have proved above, one cannot make use of it to prove
the inconsistency of the ladder approximation to QCD, and the conclusion in Ref. [21] “all the results based on the nontrivial
(analytical or numerical) solutions to the quark DSE in the ladder approximation should be reconsidered, and its use in the whole
energy/momentum range should be abandoned” is incorrect. Here we also note that in the ladder approximation to QCD one does
not have a Ward–Takahashi identity for the quark–gluon vertex (due to SU(3) color structures), but it can be shown that for the
color singlet vector qq¯ bound state vertex the corresponding Ward–Takahashi identity is valid [7].
Acknowledgements
This work was supported in part by the National Natural Science Foundation of China (under Grant Nos. 10575050, 10135030,
10475057) and the Research Fund for the Doctoral Program of Higher Education (under Grant No. 20030284009).
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A note on failure of the ladder approximation to QCD 

Comment to: "A note on failure of the ladder approximation to QCD"
  [Phys. Lett. B 640 (2006) 196 ]

Comment to: "A note on failure of the ladder approximation to QCD" [Phys. Lett. B 640 (2006) 196 ]

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