arXiv:hep-th/0311096v1 11 Nov 2003
Confinement in SU(Nc) Gauge Theory
with a Massive Dilaton
Mohamed Chababa,b ∗
and Latifa Sanhajia
a)LPHEA, Physics Department, Faculty of Science - Semlalia
P.O. Box 2390 Cadi-Ayyad University, Marrakech, Morocco.
b)Centro de Fisica Torica, Departamento de Fisica,
Universidade de Coimbra, 3004-516 Coimbra, Portugal.
Following a recently proposed confinement generating scenario ,
we provide a new string inspired model with a massive dilaton and a
general dilaton-gluon coupling. By solving analytically the equations
of motion, we derive a new class of confining interquark potentials,
which includes most of the QCD motivated potential forms given in
∗Corresponding author: email@example.com
To describe the confinement of quarks and gluons, several low energy effective
models have been proposed. The most popular ones are : Color dielectric
models [1, 2, 3], the constituent models with non relativistic quark and a con-
fining potential , and the dual Landau-Ginzberg Model . Recently, the
extension of gauge field theories by inclusion of dilatonic degrees of freedom
has evoked considerable interest. Particularly, dilatonic Maxwelle and Yang-
Mills theories which, under some assumptions, possess stable and finite energy
solutions . Indeed, in theories with dilaton fields, the topoligical struc-
ture of the vacuum is drastically changed compared to the non dilatonic ones.
It is therefore of great interest to investigate the vacuum solutions induced
by r-dependent dilaton field, through a string inspired effective theory which
may reproduce the main features of QCD, in particular, the quark confine-
ment. Recall that the dilaton is an hypothetical scalar particle appearing in
the spectrum of string theory and Kaluza-Klein type theories . Along with
its pseudo scalar companion, the axion, they are the basis of the discovery F-
theory compactification  and of the derivation of type IIB self duality .
The main features of a dilaton field is its coupling to the gauge fields through
the Maxwell and Yang-Mills kinetic term. In particular, in string theory, the
dilaton field determines the strength of the gauge coupling at tree level of the
effective action. In this context, Dick  observed that a superstring inspired
coupling of a massive dilaton to the 4d SU(Nc) gauge fields provides a phe-
nomenologically interesting interquark potential V (r) with both the Coulomb
and confining phases. The derivation performed in  is phenemenologically
attractive since it provides a new confinement generating mechanism. In this
context, a general formula of a quark-antiquark potential, which is directly
related to the dilaton-gluon coupling function, has been obtained in . The
importance of this formula is manifest since it generalizes the Coulomb and
Dick potentials, and it may be confronted to known descriptions of the con-
finement, particularly, those describing the complex structure of the vacuum
in terms of quarks and gluons condensates. Moreover a generalized version of
Dick model with both a massive and massless dilaton has been proposed in
In this paper, we shall propose a new effective coupling of a massive dilaton
to chromoelectric and chromomagnetic fields subject to the requirement that
the Coulomb problem still admits an analytic solution. Our main interest
concerns the derivation of a new family of confining interquark potentials. As
a by product, we shall set up a theoretical basis to various QCD motivated
1there is a missing factor (q)
term of Eq.(14) should be multiplied by q
1+4δin second term of Eq.(12) and Eq.(13). Also, the second
quark potentials used in the literature. The later would gain in credibility if
they can emerge from low energy effective theories.
The plan of this work is as follows: In section 2, we will develop our model
and derive the main equations of motion. Particularly, emphasis will be put on
the equations of a massive dilaton in the asymptotic regime. The latter should
show the long range behaviour of the solutions, and consequently is connected
with the confining phase. Section 3 will be devoted to the existence of analyt-
ical solutions from which we shall extract a new class of interquark potentials
whose magnitude grows with the separation between the quark and antiquark.
The main features of these potentials will be presented along with their con-
nection to some popular phenomenological ones. Finally our conclusion will
be drawn in section 4.
2 The model
We propose an effective field theory defined by the general Lagrangian:
L(φ,A) = −
2∂µΦ∂µ− V (φ) + Jµ
where the coupling function F(φ) depends on the dilaton field and V (φ) de-
notes the non perturbative scalar potential of φ. Gµνis the field strength in
the language of 4d gauge theory.
Several forms of the function F(Φ) appeared in different theoretical frame-
works: F(Φ) = e−kΦ
pling [8, 9]. As to Dick model, F(Φ) is given by F(Φ) = k+f2
is a characteristic scale of the strength of the dilaton/glueball-gluon. By using
the formal analogy between the Dick problem and the Eguchi-Hansen one ,
we noted in  that f is similar to the 4dN = 2 Fayet-Illioupoulos coupling in
the Eguchi-Hansen model. It may be interpreted as the breaking scale of the
U(1) symmetry rotating the dilaton field.
Now, to analyze the problem of the Coulomb gauge theory augmented with
dilatonic degrees of freedom in (1), we proceed as follows: first, we consider
a point like static Coulomb source which is defined in the rest frame by the
f as in string theory and Kaluza-Klein theories ;
fin the Cornwall-Soni model parameterizing the glueball-gluon cou-
Φ2. The constant f
Carepresents the expectation value of the SU(N) generator χafor a normalized
spinor in Coulomb space. They satisfy the algebraic identity:
The equations of motions, inherited from the model (1) and emerging from the
static configuration (2) are given by:
∂µ∂µΦ = −∂V (Φ)
By setting G0i
algebra, the simplified expressions:
a= Eiχa= −∇iΦa, we obtain, after some straightforward
with ? α =
We then derive the important formula of [7, 11],
which shows that the quark confinement appears if the following condition is
r→∞rF−1(Φ(r)) = finite
Then, the interquark potential reads as,
U(r) = Φa(r)
= 2? αs
At this stage, note that the effective charge is defined by,
thus the chromo-electronic field takes the usual standard form:
Therefore, it is the running of the effective charge that makes the potential
stronger than the Coulomb potential. Indeed if the effective charge did not
run, we recover the Coulomb spectrum.
To solve the equations of motion (6) and (7), we need to fix two of the four
unknown quantities Φ(r), F(Φ), V (Φ) and Φa(r) in our model. We set V (Φ)
to V (Φ) =1
2m2Φ and we introduce a new coupling function:
1 − βΦ2
Then the equation (7) becomes:
∆Φ = m2Φ − 2n
However since we are
This equation is very difficult to solve analytically.
usually interested by the large distance behaviour of the dilaton field and its
impact on the Coulomb problem, an analytical solution of (11) in the asymp-
totic regime is very satisfactory. Indeed, it is easily shown that the following
solves (11) at large r. Therefore, thanks to the master formula (11), we derive
Φa(r) = −gCa
n+1n + 1
3n − 1r(3n−1
By imposing the condition (9), we obtain a family of confining interquark po-
tentials if n ≥1
that the magnitude of confining potentials, can not grow more rapidly than
linear, then the values of n are constrained to the range n ≤ 1. Therefore
the confinement in our model (1) appears for the coupling function
lecting specific values of n, we may reproduce several popular QCD motivated
3. If moreover, we use the criterion of Seiler  which states
. Such class of confining potentials is very attractive. Indeed, by se-
interquark potentials: Indeed if n = 1, we recover the confining linear term
of Cornell potential . Martin’s potential (V (r) ∼ r0.1) corresponds to
tial , with a long range behaviour scaling as√r, are obtained by setting
phenomenological potentials, which gained credibility only through their con-
frontation to the hadron spectrum, can now have a theoretical basis since they
can be derived from the low energy effective theory.
29, while Song-Lin interquark potential  and Motyka-Zalewski poten-
5. Turin potential  is recovered for n =
9. We see then, that these
In this paper we have found a family of electric solutions corresponding to
a string inspired effective gauge theory with a massive dilaton varying with
r and a new coupling function F(Φ) =
values of n by both the Seiler criterion and by condition of Eq.(9) we have
shown the existence of a class of confining interquark potentials. The lat-
ter are phenomenologically interesting since they reproduce, through selecting
specific values of n, several QCD motivated potentials which successfully de-
scribe meson and baryon spectra. Clearly these popular potentials would gain
in credibility since they emerge from an low energy effective theory, and at the
same time fit well the hadron spectrum.
1 − βΦ2
?−n. By constraining the
One of the authors (M.C) is deeply grateful to the Centro de Fisica Teorica for
its warm hospitality in Coimbra. He wishes to thank prof.J. da Providencia
for the valuable discussions and comments.
This work is supported by the convention CNRST-Morocco/GRICES-Porugal,
grant 681.02/ CNR and by the PROSTARS III program D16/04.
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