Abstract. From the study of the SU(2) gauge field coupled with two
massless scalar fields, we have found solutions to the corresponding YangMills equations. These classical solutions exhibited a form of confinement
to an SU(2) gauge charge. Energy expressions of the corresponding gauge
and scalar field configurations are obtained.

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JOURNAL OF SCIENCE OF HNUE
Natural Sci., 2010, Vol. 55, No. 6, pp. 14-19
SU(2) GAUGE FIELD COUPLED
WITH TWO MASSLESS SCALAR FIELDS
Nguyen Van Thuan
Hanoi National University of Education
E-mail: thuanvatli@yahoo.com
Abstract. From the study of the SU(2) gauge field coupled with two
massless scalar fields, we have found solutions to the corresponding Yang-
Mills equations. These classical solutions exhibited a form of confinement
to an SU(2) gauge charge. Energy expressions of the corresponding gauge
and scalar field configurations are obtained.
Keywords: gauge field, massless scalar field.
1. Introduction
Non-abenlian gauge theories were invented by Yang-Mills. Solutions to the
classical field equations, in which field functions are c-number (not operators), play
an impotant role in investigating the configurations of the corresponding quantum
field theory. Based on these solutions, applying the semi-classical analysis methods,
important information about the quantum theory can be obtained, which could not
be done with the perturbation theory [1-3]. One important result when studying the
classical Yang-Mills equations is to recognise that the extremum of the functional
in the Euclidian space do not correspond to the zero-unified field, but correspond
to the nontrivial space-time local field configuration known as instanton [4]. In the
quantum theory, instantons describe the tunnel effects of the degenerate vacuum
states. This result brought a new view to the vacuum structure of the Yang-Mills
theory, providing a qualitative explanation of the Quark confinement [5-8].
Researches of classical Yang-Mills equations with external sources are well
documented. With these kind of problems, the most interesting result is to find
some particular solutions that can produce screening effects with external color
charge, similar to the screening effects of the external charge in electrodynamics.
Some studies have found that the Quark confinement may have some connections
with these screening effects [9,10].
14
SU(2) gauge field coupled with two massless scalar fields
In this paper, we investigate the SU(2) gauge field coupled with two massless
scalar fields. In Section 2, we introduce solutions to Yang-Mills equations corre-
sponding to these field configurations. It is argued that these classical solutions
exhibit the property of confinement that has been looked for in non-abelian gauge
theory. In Section 3, energy expressions of the SU(2) gauge field coupled with two
massless scalar fields are identified. Finally, we discuss the results and make con-
clusions in Section 4.
2. Solutions to the Yang-Mills equations
The Lagrangian density of the SU(2) gauge field coupled with two scalar fields
is given by:
L = −1
4
F µνaF aµν +
1
2
(Dµφa)(Dµφ
a) +
1
2
(Dµψa)(Dµψ
a)
− λ
4
(m2
λ
− φ2
)2
− λ
′
4
(m′2
λ′
− ψ2 +W 2
)2
− 1
2
g2
[
φ2ψ2 − (φaψa)
]
, (2.1)
where
F aµν = ∂µW
a
ν − ∂νW aµ + gεabcW bµW cν , (2.2)
Dµφ
a = ∂µφ
a + gεabcW bµφ
c, (2.3)
Dµψ
a = ∂µψ
a + gεabcW bµψ
c. (2.4)
HereW aµ , φ
a and ψa are the non-alelian gauge and scalar potentials, respectively, and
g is the coupling constant. The last term in Equation (2.1) describes the coupling
of φa and ψa. This term vanishes if they are of the same direction in group space.
The model which we consider here is an SU(2) gauge field coupled with two
massless scalar fields. These scalar fields have the same direction in group space.
Therefore the Lagrangian density (1) can be rewritten
L = −1
4
F µνaF aµν +
1
2
(Dµφa)(Dµφ
a) +
1
2
(Dµψa)(Dµψ
a). (2.5)
We are only interested in static solutions so all the time derivatives will be
zero. With this condition the gauge field equations from the Lagrangian density
(2.5) are:
∂iF
µia + gεabcW bi F
µic = gεabc
[
(Dµφb)φc + (Dµψb)ψc
]
, (2.6)
where i = 1, 2, 3 are the space indices.
15
Nguyen Van Thuan
For the massless scalar fields the equations are:
∂i(D
iφa) + gεabcW bµ(D
µφc) = 0, (2.7)
and
∂i(D
iψa) + gεabcW bµ(D
µψc) = 0. (2.8)
Further assuming that the gauge field and two scalar fields are radial, we use
the Actor ansatz [3]
W ai = εaij
rˆj
gr
[
1−K(r)], (2.9)
W a0 = 0, (2.10)
φa =
rˆa
gr
J(r), (2.11)
ψa =
rˆa
gr
H(r), (2.12)
where K(r), J(r), and H(r) are funtions of the radius r, rˆ is unit vector codirectional
with ~r. Inserting this ansatz into Equations (2.6)-(2.8) produces three coupled
nonlinear differential equations
r2K” = K(K2 +H2 + J2 − 1), (2.13)
r2J” = 2JK2, (2.14)
r2H” = 2HK2. (2.15)
The solutions to the above equations are
K(r) =
Cr
Cr − 1 , (2.16)
J(r) =
A
Cr − 1 , (2.17)
H(r) =
B
Cr − 1 , (2.18)
where A,B, and C are constants. The only constraint imposed is that A2 + B2 =
1, so that Equations (2.16)-(2.18) involve only two arbitrary constants. Inserting
K(r), J(r), andH(r) into the expressions for the gauge and scalar fields of Equations
(2.9)-(2.12), we can see that gauge field and both the scalar fields become infinite
at the radius
r0 =
1
C
. (2.19)
16
SU(2) gauge field coupled with two massless scalar fields
Further, using these singular gauge potentials to calculate the ”electric” and
”magnetic” field (Eia = F
i0
a and B
i
a = −
1
2
εijkF jka , respectively) it is seen that these
fields are also infinite at r0 = 1/C. Therefore a particle, which carries an SU(2)
gauge charge, becomes permanently confined if it crosses into the region r < r0. The
non-abelian gauge potentials and both the scalar fields of Equations (2.9)-(2.12) also
become singular at r = 0, which is true as well for the Coulomb potential of a point
charge in classical eletromagnetism. The singularity of all these solutions at r = 0 is
of the same character in that they all imply a δ-function point charge sitting at the
origin (where the charge of our non-abelian model is SU(2) gauge charge). Just as
the singularity at the origin can be taken to be a point source of SU(2) charge, so
the singularity at r = 1/C can be taken to be a spherical shell of SU(2) charge. This
shell structure is a feature of our solution, and it points to a possible connection
with the various models of SU(2) gauge charge bound states.
There are two particular cases which can be considered. The first case is for
K(r) = 0, solutions to Equations (2.14)-(2.15) are:
J(r) = D, H(r) = E, (2.20)
where D, and E are arbitrary constants. With K(r) = 0, SU(2) gauge poten-
tial W ai = εaij
rˆj
gr
, this is the potential of a point monopole with value of 1/g.
The second case is for K(r) = ±1, in this case the solutions to Equations (2.14)-
(2.15) are
J(r) =
G
r
, H(r) = i
G
r
, (2.21)
where G is arbitrary constant. In this case two scalar fields are only different to one
imaginary unit.
3. Energy expressions
The energy of the gauge and scalar field configurations can be obtained by
taking the volume integral of the time-time component of the energy-momentum
tensor
T µν = F µρaF νaρ + (D
µφa)(Dνφa) + (Dµψa)(Dνψa) + gµνL, (3.1)
hence
E =
∫
T 00d3x. (3.2)
17
Nguyen Van Thuan
is the energy of the fields, which is given by
E =
4pi
g2
∫ ∞
rc
[
K
′2 +
(K − 1)2
2r2
+
+
J2K2
r2
+
(rJ ′ − J)2
2r2
+
H2K2
r2
+
(rH ′ −H)2
2r2
]
dr. (3.3)
Notice that the integral has been cut off from below at an arbitrary distance
rc, which must be larger than r0. This was done to avoid the singularities at r = 0
and r = r0, since integrating through r = 0 and r = r0 would give an infinite field
energy in the same way that the Coulomb potential of a point electric charge yield
an infinite field energy when integrated down to zero. Inserting K(r), J(r) and H(r)
into Equation (3.3) we obtain
E =
2pi
g2
(A2 +B2 + 1)
(2Crc − 1)
rc(Crc − 1)3 . (3.4)
For the pure gauge case (A2 = 0, B2 = 0) energy expression is
Eg =
2pi
g2
(2Crc − 1)
rc(Crc − 1)3 . (3.5)
Finally, just like the Prasad-Sommerfield solution [11], our solution can be
seen to carry a topological magnetic charge when the electromagnetic field is em-
bedded into the SU(2) via ′t Hooft generalized, gauge invariant, electromagnetic
field strength tensor [12].
4. Discussion and conclusion
Studying the SU(2) gauge field coupled with two massless scalar fields, we
have found solutions to the corresponding Yang-Mills equations. These classical
solutions exhibited a form of confinement. Any particle that caries an SU(2) gauge
charge and enters the region r < r0 = 1/C would no longer be able to leave this
region. This is analogous to what happens with the Schwarzschild solution in general
relativity, where once a particle passes the event horizon it is permanently confined.
If our solution is responsible for the confinement mechanism in non-abelian field
theories then it is necessary for the Lagrangian density to always include scalar
fields in order for an acceptable solution to exist. Under these assumptions scalar
fields become crucial to the confinement mechanism. Still it is encouraging that at
the classical level an analytical solution, which seems to exhibit confinement, can
be found for a non-abelia gauge theory. Two particular cases of the solutions are
18
SU(2) gauge field coupled with two massless scalar fields
considered. In the case function K(r) = 0, the space component of the gauge field
have the form of the potential of a point monopole with the value of 1/g. In other
cases, if function K(r) = ±1, then two massless scalar fields are only different to
one imaginary unit. We have also found the energy expressions of the the SU(2)
gauge field coupled with two massless scalar fields.
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