Physics Letters B 740 (2015) 141–146
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Physics Letters B
www.elsevier.com/locate/physletb
A semi-classical analogue of the relation between the chiral and
the gluon QCD condensates
José Emílio F.T. Ribeiro
∗
, Dmitri Antonov
Departamento de Física and CFIF, Instituto Superior Técnico, ULisboa, Av. Rovisco Pais, 1049-001 Lisbon, Portugal
1
a r t i c l e i n f o a b s t r a c t
Article history:
Received
19 August 2014
Received
in revised form 4 October 2014
Accepted
19 November 2014
Available
online 25 November 2014
Editor:
A. Ringwald
We present a semi-classical analogue of the relation between chiral and gluon QCD condensates, in which
a condensation of a massless scalar field is provided by a classical field, instead of a quantum Yang–
Mills
field. For the classical field, we choose a gravitational field of a spherically symmetric object. The
size of such an object becomes then a classical-physics analogue of the QCD vacuum correlation length,
whereas the squared curvature of the gravitational field plays the role of the gluon condensate. Finally,
by iterating the emerging contribution to the trace of the energy–momentum tensor, we prove that the
energy density of the object remains of the same order of magnitude, i.e. no instability of the system
occurs.
© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP
3
.
It is nowadays commonly accepted that the two main non-
perturbative
phenomena in QCD, confinement and chiral symmetry
breaking, are not independent of each other. A clear indication in
favor of this statement stems from the following relation that holds
between the order parameters of these two phenomena, which
are the chiral and the chromo-electric gluon condensates
¯
ψψ
and (gE
a
i
)
2
[1]:
¯
ψψ∝−λ
gE
a
i
2
.
(1)
In this expression, λ is the so-called chromo-electric vacuum cor-
relation
length, while the concrete value of the order-O(1) non-
universal
proportionality coefficient is unimportant. Clearly, this
relation is specific for a stochastic vacuum, such as that of QCD,
while it cannot hold in a vacuum characterized by some constant
chromo-electric field E
a
i
. That is, such a “classical” vacuum cannot
support either of the two phenomena. Nevertheless, it is legiti-
mate
to pose a question of whether other classical fields can lead
to the condensation of quantum fields so as to yield a formula for
the corresponding condensates similar to Eq. (1). In this Letter, we
show that a simple example of such a classical field is provided
by the gravitational field of a spherically symmetric object of a
constant energy density. For our illustrative purposes, it suffices to
calculate the condensate of a scalar field, which is minimally cou-
*
Corresponding author.
E-mail
address: emilioribeiro@tecnico.ulisboa.pt (J.E.F.T. Ribeiro).
1
The former address of the second author.
pled to this gravitational field. In the vanishing-mass limit of this
scalar field, the role of the vacuum correlation length is played by
the size of the object. We will obtain a formula for the scalar-field
condensate through the second order in the curvature R and the
Ricci tensor R
μν
. Within the present analogy, these two quanti-
ties
play the role of classical counterparts of E
a
i
. In order to derive
such a formula, we use the known closed-form expression [2] for
the one-loop effective action of a scalar field in the gravitational
background.
Hence,
let us consider a real-valued massive scalar field φ(x)
interacting with the gravitational field g
μν
(x). The corresponding
Euclidean action has the form
S =
1
2
d
4
x
√
gφ
−2 +m
2
φ,
where
2φ =
1
√
g
∂
μ
√
gg
μν
∂
ν
φ
, and g ≡ det g
μν
.
Integrating over the field φ, one obtains the following effective ac-
tion:
Γ [g
μν
]≡−ln
D φe
−S
=
1
2
ln det
−2 +
m
2
.
In Ref. [2], a closed-form expression for Γ [g
μν
] has been obtained
through the second order in the curvature:
Γ [g
μν
]=
1
2
∞
0
ds
s
d
4
x
√
g
(4π s)
2
http://dx.doi.org/10.1016/j.physletb.2014.11.039
0370-2693/
© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by
SCOAP
3
.