Page 1

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.

and

b)Centro de Fisica Torica, Departamento de Fisica,

Universidade de Coimbra, 3004-516 Coimbra, Portugal.

Abstract

Following a recently proposed confinement generating scenario [6],

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

the literature.

∗Corresponding author: mchabab@ucam.ac.ma

1

Page 2

1Introduction

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 [4], and the dual Landau-Ginzberg Model [5]. 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 [18]. 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 [19]. Along with

its pseudo scalar companion, the axion, they are the basis of the discovery F-

theory compactification [20] and of the derivation of type IIB self duality [21].

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 [6] 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 [6] 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 [7]. 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

[10]1.

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

1+4δin second term of Eq.(12) and Eq.(13). Also, the second

3

4.

2

Page 3

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) = −

1

4F(φ)Ga

µνGµν

a+1

2∂µΦ∂µ− V (φ) + Jµ

aAa

µ

(1)

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

F(Φ) =Φ

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 [22],

we noted in [7] 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

current:

f as in string theory and Kaluza-Klein theories [19];

fin the Cornwall-Soni model parameterizing the glueball-gluon cou-

Φ2. The constant f

Jµ

a= gδ(r)Caνµ

0= ρaνµ

0

(2)

Carepresents the expectation value of the SU(N) generator χafor a normalized

3

Page 4

spinor in Coulomb space. They satisfy the algebraic identity:

N2−1

?

a=1

C2

a=Nc− 1

2Nc

(3)

The equations of motions, inherited from the model (1) and emerging from the

static configuration (2) are given by:

Dµ,F−1(Φ)Gµν?

and

?

= Jν

(4)

∂µ∂µΦ = −∂V (Φ)

∂Φ

−1

4

∂F−1(Φ)

∂Φ

Gµν

aGνµ

a

(5)

By setting G0i

algebra, the simplified expressions:

a= Eiχa= −∇iΦa, we obtain, after some straightforward

dΦa

dr

= r−2F(Φ(r))

?−g

4πCa

?

(6)

∆Φ =∂V

∂Φ−

? α

r4

∂F

∂Φ

(7)

with ? α =

g2

32π2

?Nc−1

2Nc

?

We then derive the important formula of [7, 11],

Φa(r) =−gCa

4π

?

drF(Φ(r))

r2

(8)

which shows that the quark confinement appears if the following condition is

satisfied:

lim

r→∞rF−1(Φ(r)) = finite

(9)

Then, the interquark potential reads as,

U(r) = Φa(r)

?−g

?F(Φ(r))

4πCa

?

= 2? αs

r2

dr

(10)

At this stage, note that the effective charge is defined by,

Qa

eff(r) =

?

gCa

4π

?

F(Φ(r))

4

Page 5

thus the chromo-electronic field takes the usual standard form:

Ea=Qa

eff(r)

r2

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:

F(Φ) =

?

1 − βΦ2

f2

?−n

Then the equation (7) becomes:

∆Φ = m2Φ − 2n

? αs

r4

?

1 −βΦ2

f2

?−(n+1)

βΦ

However since we are

f2

(11)

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

function:

Φ =

f2

β−

?β

f2

? −n

n+1?2nαs

m2

?

1

n+1?1

r

?

4

n+1

1

2

(12)

solves (11) at large r. Therefore, thanks to the master formula (11), we derive

the potential,

Φa(r) = −gCa

4π

?2nαs

m2f2

?−4n

n+1n + 1

3n − 1r(3n−1

n+1)

(13)

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

?1

lecting specific values of n, we may reproduce several popular QCD motivated

3. If moreover, we use the criterion of Seiler [21] which states

1

F(Φ)with

n ∈

3,1

?

. Such class of confining potentials is very attractive. Indeed, by se-

5

Page 6

interquark potentials: Indeed if n = 1, we recover the confining linear term

of Cornell potential [13]. Martin’s potential (V (r) ∼ r0.1)[14] corresponds to

n =11

tial [16], with a long range behaviour scaling as√r, are obtained by setting

n to

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 [15] and Motyka-Zalewski poten-

3

5. Turin potential [17] is recovered for n =

5

9. We see then, that these

3Conclusion

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

f2

?−n. By constraining the

Acknowledgements

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.

6

Page 7

References

[1] K. Johnson, C.B. Thorn, A. Chodos, R.L. Jaffe, V.F. Weisskopf, Phys.

Rev. D 9, 3471 (1974).

[2] R. Friedberg, T.D. Lee, Phys. Rev. D 15, 1964 (1977).

[3] G.E. Brown, M. Rho, Phys. Lett. B 82, 177 (1979).

[4] D. Gromes, Ordinary Hadrons, Lectures given at Yukon Advanced Study

Institute, Published in Yukon ASI, 1 (1984); Z. Phys. C 26, 401 (1984).

[5] J. Hosek, Phys. Rev. D 46, 3645 (1992).

[6] R. Dick, Eur. Phys. J. C 6, 701 (1999); Phys. Lett. B 397, 193 (1999).

[7] M. Chabab, R. Markazi, E. H. Saidi, Eur. Phys. J. C 13, 543 (2000).

[8] J. M. Cornwall, A. Soni, Phys. Rev. D 29, 1424 (1984).

[9] R. Dick, L. P. Fulcher, Eur. Phys. J. C 9, 271 (1999).

[10] M. Slusarczyk, A. Wereszczynsky, Eur. Phys. J. C 28, 151 (2003).

[11] M. Chabab et al., Class. Quantum Gravity 18, 5085 (2001).

[12] E. Seiler, Phys. Rev. D 18, 482 (1978).

[13] E. Eichten et al., Phys. Rev. Lett. 34, 369 (1975).

[14] A. Martin, Phys. Lett. B 100, 511 (1981).

[15] X. Song, H. Lin, Z. Phys. C 34, 223 (1987).

[16] L. Motyka, K. Zalewski, Z. Phys. C 69, 342 (1996).

[17] D. B. Lichtenberg, E. Predazzi, R. Roncaglia, M. Rosso, J.G. Wills, Z.

Phys. C 41, 615 (1989).

[18] M. Cvetic, A.A. Tseytlin, Nucl. Phys. B 416, 137 (1983).

[19] M. Green, J. Schwartz, E. Witten, Superstring Theory, (Cambridge Uni-

versity Press, Cambridge 1987)

[20] C. Vafa, Nucl.Phys. B 469, 403 (1996).

[21] A. Sen, Unification of string dualities, Nucl. Phys. Proc. Suppl. 58, 5

(1997).

7

Page 8

[22] A. Galperin, E. Ivanov, V. Ogievetsky, P.T. Towsend, Class. Quantum

Gravity 1, (1985) 469.

8