1. Introduction
The Yang-Mills equations for the SU(2) gauge field coupled with the Higgs
field suggest many interesting solution types: monopole, dyon, instanton and meron
[2, 3, 5, 7]. Physical applications of the classical Yang-Mills-Higgs theory begin with
exact solutions. The physical properties of monopoles, dyons, instantons and merons are
particularly important. For example, imaginary time solutions of classical theories are
usually interpreted as real time tunneling in the corresponding quantized theory. Classical
Yang-Mills-Higgs theory can be studied independently of exact solutions, of course. This
is an interesting parsuit, because any results may lead to improvements in an integral
formulation of quantum field theory [4].
The Yang-Mills equations are nonlinear differential equations. Exact solutions to
nonlinear field theories are very difficult to find since there exists no general method for
discovering them. The usual approach is to make some guess as to the form of the solution
and insert it into the field equations to see if it solves them. For the Yang-Mills-Higgs
theory there are some known exact solutions, which are found by this approach [1, 8, 9].
In this article, we consider the Yang-Mills equations for the SU(2) gauge field
coupled with two massless Higgs triplets. The exact classical solution of this equations
and corresponding non-Abelin field intensities seem to exhibit the property of cofinement,
which can be found for non-Abelian gauge theories.

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JOURNAL OF SCIENCE OF HNUE
Mathematical and Physical Sci., 2012, Vol. 57, No. 7, pp. 100-105
This paper is available online at
NON-ABELIAN CLASSICAL SOLUTION
OF THE YANG-MILLS-HIGGS THEORY
Nguyen Van Thuan
Faculty of Physics, Hanoi National University of Education
Abstract. In this paper, we investigate an SU(2) gauge field coupled with two
massless Higgs triplets. We obtain a non-Abelian exact classical solution of
corresponding Yang-Mills equations. We also find the energy expression of this
classical solution. Some particular cases of the solution are considered.
Keywords: Non-Abelian gauge fields, Higgs triplets, Yang-Mills equation,
classical solution.
1. Introduction
The Yang-Mills equations for the SU(2) gauge field coupled with the Higgs
field suggest many interesting solution types: monopole, dyon, instanton and meron
[2, 3, 5, 7]. Physical applications of the classical Yang-Mills-Higgs theory begin with
exact solutions. The physical properties of monopoles, dyons, instantons and merons are
particularly important. For example, imaginary time solutions of classical theories are
usually interpreted as real time tunneling in the corresponding quantized theory. Classical
Yang-Mills-Higgs theory can be studied independently of exact solutions, of course. This
is an interesting parsuit, because any results may lead to improvements in an integral
formulation of quantum field theory [4].
The Yang-Mills equations are nonlinear differential equations. Exact solutions to
nonlinear field theories are very difficult to find since there exists no general method for
discovering them. The usual approach is to make some guess as to the form of the solution
and insert it into the field equations to see if it solves them. For the Yang-Mills-Higgs
theory there are some known exact solutions, which are found by this approach [1, 8, 9].
Received July 25, 2012. Accepted September 20, 2012.
Physics Subject Classification: 60 44 01 03.
Contact Nguyen Van Thuan, e-mail address: thuanvatli@yahoo.com
100
Non-Abelian classical solution of the Yang-Mills-Higgs theory
In this article, we consider the Yang-Mills equations for the SU(2) gauge field
coupled with two massless Higgs triplets. The exact classical solution of this equations
and corresponding non-Abelin field intensities seem to exhibit the property of cofinement,
which can be found for non-Abelian gauge theories.
2. Content
2.1. The exact classical solution of the Yang-Mills equations
The Lagrangian density for the SU(2) gauge fields coupled with two massless
Higgs triplets has the form
L = −1
4
F aµνF
µνa +
1
2
(Dµφ
a)(Dµφa) +
1
2
(Dµψ
a)(Dµψa), (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)
The equations of motion of the SU(2) gauge fields and two massless Higgs triplets
from the Lagrangian density (2.1) are
∂νF aµν = gε
abc
[
F bµνW
νc − (Dµφb)φc − (Dµψb)ψc
]
, (2.5)
∂µ(Dµφ
a) = gεabc(Dµφ
b)W µc, (2.6)
∂µ(Dµψ
a) = gεabc(Dµψ
b)W µc. (2.7)
Assume that the SU(2) gauge fields and two massless Higgs triplets are spherical
symmetry. We use the Wu-Yang ansatz [10]
W ai = εaij
rˆj
gr
[1−K(r)],
W a0 =
rˆa
gr
J(r),
φa =
rˆa
gr
I(r),
ψa =
rˆa
gr
H(r), (2.8)
whereK(r), J(r), I(r) andH(r) are certain functions of the radius r, which satisfy field
equations of motion, and rˆa is the unit radius vector. Inserting this ansatz into the field
101
Nguyen Van Thuan
equations (2.5) - (2.7) yields four coupled nonlinear differential equations
r2
d2K
dr2
= K(K2 +H2 + I2 − J2 − 1),
r2
d2J
dr2
= 2JK2,
r2
d2I
dr2
= 2IK2,
r2
d2H
dr2
= 2HK2. (2.9)
The exact solution of the classical equations of motion of the SU(2) gauge field
coupled with one massless Higgs triplet was found by Singleton [8]. Here we consider
two massless Higgs triplets. The exact solution to the above equations are
K(r) =
Ar
Ar − 1 ,
J(r) =
B
Ar − 1 ,
I(r) =
C
Ar − 1 ,
H(r) =
D
Ar − 1 , (2.10)
where A,B,C, and D are arbitrary constants. The only constraint inposed is that C2 +
D2−B2 = 1 so that the solution of equation (2.10) involves only three arbitrary constants.
Inserting K(r), J(r), I(r) and H(r) into the expressions for the gauge fields and two
massless Higgs triplets of equation (2.8), we see that the gauge fields and two massless
Higgs triplets become infinite at the radius
r = r0 =
1
A
. (2.11)
Using these singular gauge potentials to calculate the non-Abelian electric and
magnetic field intensities, we obtain
Eai = F
a
0i =
1
g
[
(2ABr −B)
r2(Ar − 1)2 rˆ
irˆa − AB
r(Ar − 1)2
(
δia − rˆirˆa
)]
, (2.12)
Bai = −
1
2
εijkF
a
jk =
1
g
[
(1− 2Ar)
r2(Ar − 1)2 rˆ
irˆa +
A
r(Ar − 1)2
(
δia − rˆirˆa
)]
. (2.13)
These non-Abelian field intensities are also infinite at r = r0 =
1
A
. This seems to
exhibit SU(2) gauge charge confinement. An SU(2) gauge charge carring particle, which
102
Non-Abelian classical solution of the Yang-Mills-Higgs theory
enters the region r < r0, is not able to leave this region. As r → ∞, these electric and
magnetic fields fall off like (1/r3), unlike the Prasad-Sommerfield solution, which has a
(1/r2) behavior for large r [6].
There are three particular cases which can be considered. The first case is where the
spatial component of the gauge fields equals zero (W ai = 0). This corresponds to taking
K(r) = 1. In this case equation (2.9) has the solution
K(r) = 1, J(r) =
E
r
, I(r) =
F
r
, H(r) =
G
r
, (2.14)
where E, F , and G are arbitrary constants, which satisfy the codition F 2 +G2 = E2.
The second case is where the time component of the gauge fields equals zero
(W a0 = 0). This corresponds to taking J(r) = 0, which implies B = 0 in equation
(2.10). The above codition C2 + D2 − B2 = 1 yields C2 + D2 = 1. Therefore we can
write C = sinθ,D = cosθ where θ is an arbitrary constant. In this case the solution
becomes
K(r) =
Ar
Ar − 1 , I(r) =
sinθ
Ar − 1 , H(r) =
cosθ
Ar − 1 . (2.15)
The last case is where there are no two Higgs triplets. This corresponds to I(r) =
0, H(r) = 0, which implyC = 0, D = 0 in equation (2.10). The coditionC2+D2−B2 =
1 now requires that B = ±i. The solution becomes
K(r) =
Ar
Ar − 1 , J(r) = ±
i
Ar − 1 . (2.16)
2.2. Energy of the SU(2) gauge fields and two massless Higgs triplets
The energy of the SU(2) gauge fields and two massless Higgs triplets of our
exact classical solution 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. (2.17)
The energy of the fields is
E =
∫
T 00d3x. (2.18)
From equations (2,17), (2.18) and using ansatz (2.8) we have
E =
4π
g2
∫ ∞
ra
[(dK
dr
)2
+
(K2 − 1)2
2r2
+
J2K2
r2
+
(
r
dJ
dr
− J
)2
2r2
+
+
I2K2
r2
+
(
r
dI
dr
− I
)2
2r2
+
H2K2
r2
+
(
r
dH
dr
−H
)2
2r2
]
dr. (2.19)
103
Nguyen Van Thuan
Notice that the integral has been cut off from below at an arbitrary distance ra,
which must be large than r0. This procedure is done to avoid the singularities at r = 0
and r = r0, since integrating through r = 0, r0 would give an infinite field energy. This
is similar to the Coulomb potential of point electric charge, which yields an infinite field
energy when integreted down to zero. Inserting K(r), J(r), I(r), and H(r) of equation
(2.10) into equation (2.19) we obtain
E =
2π
g2
(B2 + C2 +D2 + 1)
(2Ara − 1)
ra(Ara − 1)3 . (2.20)
From the codition C2 +D2 − B2 = 1, equation (2.20) can be rewitten as
E =
4π
g2
(C2 +D2)
(2Ara − 1)
ra(Ara − 1)3 . (2.21)
In the case where there is only one massless Higgs triplet (i.e., C = 0, or D = 0),
equation (2.21) becomes
E =
4πC2
g2
(2Ara − 1)
ra(Ara − 1)3 for the case D = 0, (2.22)
or
E =
4πD2
g2
(2Ara − 1)
ra(Ara − 1)3 for the case C = 0, (2.23)
For the pure gauge field case (C2 = 0, D2 = 0, B2 = −1), the energy of equation
(2.20) becomes zero. This together with the requirement that theW a0 components of this
solution are pure imaginary raises doubts about the physical importance of this particular
case. If we want to discard the zero energy pure gauge case, then it is necessary for the
Lagrangian density to always include the Higgs fields.
3. Conclusion
Considering the SU(2) gauge field coupled with two massless Higgs triplets, we
have found the exact classical solution of the corresponding Yang-Mills equations. We
also obtain non-Abelian electric and magnetic field intensities of this solution. The exact
classical solution and field intensities have singularity at r0 =
1
A
. It can be seen that
a particle, which caries an SU(2) gauge charge, becomes confined if it crosses into the
region r < r0 =
1
A
. Thus the solution exhibits the property of the SU(2) gauge charge
confinement. We investigated three particular cases. First the spatial component of the
104
Non-Abelian classical solution of the Yang-Mills-Higgs theory
gauge fields equals zero, leaving only two massless Higgs triplets and the time component
of the gauge field. Second the time component of the gauge fields is zero, leaving only
two massless Higgs triplets and the spatial component of the gauge field. And third, there
is the pure gauge solution, where two massless Higgs triplets are absent, leaving only the
time and spatial components of the gauge fields. In pure gauge field case, energy of field
equals zero. It follows that the Lagrangian density of the fields always include the Higgs
fields so that the zero energy gauge case can be discarded.
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