Finite energy electroweak dyon
Kyoungtae Kimm
2
J. H. Yoon
1
Y. M. Cho
0
3
0
Administration Building 3104, Konkuk University
,
Seoul 143701
,
Korea
1
Department of Physics, College of Natural Sciences, Konkuk University
,
Seoul 143701
,
Korea
2
Faculty of Liberal Education, Seoul National University
,
Seoul 151747
,
Korea
3
School of Physics and Astronomy, Seoul National University
,
Seoul 151742
,
Korea
The latest MoEDAL experiment at LHC to detect the electroweak monopole makes the theoretical prediction of the monopole mass an urgent issue. We discuss three different ways to estimate the mass of the electroweak monopole. We first present the dimensional and scaling arguments which indicate the monopole mass to be around 4 to 10 TeV. To justify this we construct finite energy analytic dyon solutions which could be viewed as the regularized ChoMaison dyon, modifying the coupling strength at short distance. Our result demonstrates that a genuine electroweak monopole whose mass scale is much smaller than the grand unification scale can exist, which can actually be detected at the present LHC.

The recent discover of the Higgs particle at LHC and Tevatron
has reconfirmed that the electroweak theory of Weinberg and
Salam provides the true unification of electromagnetic and
weak interactions [13]. Indeed the discovery of the Higgs
particle has been claimed to be the final test of the standard
model. This, however, might be a premature claim. The real
final test should come from the discovery of the electroweak
monopole, because the standard model predicts it [47]. In
fact the existence of the monopole topology in the standard
model tells that the discovery of the monopole must be the
topological test of the standard model.
In this sense it is timely that the latest MoEDAL
detector (The Magnificent Seventh) at LHC is actively
searching for such monopole [811]. To detect the electroweak
monopole experimentally, however, it is important to
estimate the monopole mass in advance. The purpose of this
paper is to estimate the mass of the electroweak monopole.
We show that the monopole mass is expected to be around 4
to 7 TeV.
Ever since Dirac [12] has introduced the concept of the
magnetic monopole, the monopoles have remained a
fascinating subject. The Abelian monopole has been
generalized to the nonAbelian monopoles by Wu and Yang [13
16] who showed that the pure SU (2) gauge theory allows a
pointlike monopole, and by t Hooft and Polyakov [1720]
who have constructed a finite energy monopole solution in
GeorgiGlashow model as a topological soliton. Moreover,
the monopole in grand unification has been constructed by
Dokos and Tomaras [21].
In the interesting case of the electroweak theory of
Weinberg and Salam, however, it has been asserted that there
exists no topological monopole of physical interest [22,23].
The basis for this nonexistence theorem is, of course, that
with the spontaneous symmetry breaking the quotient space
SU (2)U (1)Y /U (1)em allows no nontrivial second
homotopy. This has led many people to believe that there is no
monopole in WeinbergSalam model.
This claim, however, has been shown to be false. If the
electroweak unification of Weinberg and Salam is correct,
the standard model must have a monopole which
generalizes the Dirac monopole. Moreover, it has been shown that
the standard model has a new type of monopole and dyon
solutions [4,5]. This was based on the observation that the
WeinbergSalam model, with the U (1)Y , could be viewed as
a gauged C P1 model in which the (normalized) Higgs
doublet plays the role of the C P1 field. So the WeinbergSalam
model does have exactly the same nontrivial second
homotopy as the GeorgiGlashow model which allows topological
monopoles.
Once this is understood, one could proceed to construct the
desired monopole and dyon solutions in the WeinbergSalam
model. Originally the electroweak monopole and dyon
solutions were obtained by numerical integration. But a
mathematically rigorous existence proof has been established
which endorses the numerical results, and the solutions are
now referred to as ChoMaison monopole and dyon [6,7].
which indicate that the mass of the electroweak monopole
could be around 4 to 10 TeV. In Sect. 4 we discuss the
Abelian decomposition and gauge independent
Abelianization of WeinbergSalam model and GeorgiGlashow model
to help us how to regularize the ChoMaison monopole.
In Sect. 5 we discuss two different methods to regularize
the ChoMaison dyon with the quantum correction which
modifies the coupling constants at short distance, and
construct finite energy dyon solutions which support the
scaling argument. In Sect. 6 we discuss another way to make
the ChoMaison dyon regular, by enlarging the gauge group
SU (2) U (1)Y to SU (2) SU (2)Y . Finally in Sect. 7 we
discuss the physical implications of our results.
2 ChoMaison dyon in WeinbergSalam model:
a review
Before we construct a finite energy dyon solution in the
electroweak theory we must understand how one can obtain the
infinite energy ChoMaison dyon solution first. Let us start
with the Lagrangian which describes (the bosonic sector of)
the WeinbergSalam theory
It should be emphasized that the ChoMaison monopole
is completely different from the electroweak monopole
derived from the Nambu electroweak string. In his
continued search for the stringlike objects in physics, Nambu has
demonstrated the existence of a rotating dumb bell made of
the monopole antimonopole pair connected by the neutral
string of Z boson flux (actually the SU (2) flux) in Weinberg
Salam model [24,25]. Taking advantage of Nambus
pioneering work, others claimed to have discovered another type of
electroweak monopole, simply by making the string infinitely
long and moving the antimonopole to infinity [26]. This
electroweak monopole, however, must carry a fractional
magnetic charge and cannot be isolated with finite energy.
Moreover, this has no spherical symmetry which is manifest
in the ChoMaison monopole [4,5].
The existence of the electroweak monopole makes the
experimental confirmation of the monopole an urgent issue.
Till recently the experimental effort for the monopole
detection has been on the Dirac monopole [27]. But the
electroweak unification of Maxwells theory requires the
modification of the Dirac monopole, and this modification changes
the Dirac monopole to the ChoMaison monopole. This
means that the monopole which should exist in the real world
is not likely to be the Dirac monopole but the electroweak
monopole.
To detect the electroweak monopole experimentally, it is
important to estimate the mass of the monopole
theoretically. Unfortunately the ChoMaison monopole carries an
infinite energy at the classical level, so that the monopole
mass is not determined. This is because it can be viewed
as a hybrid between the Dirac monopole and the t Hooft
Polyakov monopole, so that it has a U (1)em point singularity
at the center even though the SU (2) part is completely
regular.
A priori there is nothing wrong with this. Classically the
electron has an infinite electric energy but a finite mass. But
for the experimental search for the monopole we need a solid
idea about the monopole mass. In this paper we show how to
predict the mass of the electroweak monopole. Based on the
dimensional argument we first show that the monopole mass
should be of the order of 1/ times the Wboson mass, or
around 10 TeV. To back up this we adopt the scaling argument
to predict the mass to be around 4 TeV. Finally, we show how
the quantum correction could regularize the point singularity
of the ChoMaison dyon, and construct finite energy
electroweak dyon solutions introducing the effective action of
the standard model. Our result suggests that the electroweak
monopole with the mass around 4 to 7 TeV could exist, which
implies that there is a very good chance that the MoEDAL at
the present LHC can detect the electroweak monopole.
The paper is organized as follows. In Sect. 2 we review
the ChoMaison dyon for later purpose. In Sect. 3 we
provide two arguments, the dimensional and scaling arguments,
where is the Higgs doublet, F and G are the gauge
field strengths of SU (2) and U (1)Y with the potentials A
and B, and g and g are the corresponding coupling
constants. Notice that D describes the covariant derivative of
the SU (2) subgroup only. With
1 2
L = 2 ()2 2 D 2 8
Notice that the U (1)Y coupling of makes the theory a gauge
theory of C P1 field [4,5].
From (1) one has the following equations of motion:
D2 = 2 D + D2 + 2 ( D ) ,
Now we choose the following ansatz in the spherical
coordinates (t, r, , ):
Notice that = r. Moreover, A describes the Wu
Yang monopole when A(r ) = f (r ) = 0. So the ansatz is
spherically symmetric. Of course, and B have an
apparent string singularity along the negative zaxis, but this
singularity is a pure gauge artifact which can easily be removed
making the U (1)Y bundle nontrivial. So the above ansatz
describes a most general spherically symmetric ansatz of an
electroweak dyon.
Here we emphasize the importance of the nontrivial
nature of U (1)Y gauge symmetry to make the ansatz
spherically symmetric. Without the extra U (1)Y the Higgs
doublet does not allow a spherically symmetric ansatz. This is
because the spherical symmetry for the gauge field involves
the embedding of the radial isotropy group S O(2) into the
gauge group that requires the Higgs field to be invariant
under the U (1) subgroup of SU (2). This is possible with
a Higgs triplet, but not with a Higgs doublet [28]. In fact,
in the absence of the U (1)Y degrees of freedom, the above
ansatz describes the SU (2) sphaleron which is not
spherically symmetric [2931].
To see this, one might try to remove the string in with
the U (1) subgroup of SU (2). But this U (1) will
necessarily change r and thus violate the spherical symmetry. This
means that there is no SU (2) gauge transformation which
can remove the string in and at the same time keeps the
spherical symmetry intact. The situation changes with the
inclusion of the U (1)Y in the standard model, which
naturally makes a C P1 field [4,5]. This allows the spherical
symmetry for the Higgs doublet.
To understand the physical content of the ansatz we
perform the following gauge transformation on (5):
U = i
and find that in this unitary gauge we have
1 f (r )(sin + sin cos )
A g f (r )(cos sin sin ) .
A(r )t (1 cos )
So introducing the electromagnetic and neutral Z boson
potentials A(em) and Z with the Weinberg angle w
we can express the ansatz (5) in terms of the physical fields
1 i f (r ) ei ( + i sin ),
W = ( A1 + i A2) = g
2 2
e =
This clearly shows that the ansatz is for the electroweak dyon.
The spherically symmetric ansatz reduces the equations
of motion to
2 f 2 1
+ r 2r 2 = 4 ( A B)2 + 2
Obviously this has a trivial solution
A = B = 0,
which describes the point monopole in WeinbergSalam
model
A(em) = 1e (1 cos ).
This monopole has two remarkable features. First, this is
the electroweak generalization of the Dirac monopole, but
not the Dirac monopole. It has the electric charge 4/e, not
2/e [4,5]. Second, this monopole naturally admits a
nontrivial dressing of weak bosons. Indeed, with the nontrivial
dressing, the monopole becomes the ChoMaison dyon.
To see this let us choose the following boundary condition:
A(0) = 0, B(0) = b0,
A() = B() = A0.
Then we can show that the Eq. (10) admits a family of
solutions labeled by the real parameter A0 lying in the range
[47]
g
0 A0 < min e0, 2 0 .
In this case all four functions f (r ), (r ), A(r ), and B(r )
must be positive for r > 0, and A(r )/g2 + B(r )/g 2 and
B(r ) become increasing functions of r . So we have 0 b0
A0. Furthermore, we have B(r ) A(r ) 0 for all range,
and B(r ) must approach A(r ) with an exponential damping.
Notice that, with the experimental fact sin2 w = 0.2312,
(14) can be written as 0 A0 < e0.
With the boundary condition (13) we can integrate (10).
For example, with A = B = 0, we have the ChoMaison
monopole. In general, with A0 = 0, we find the ChoMaison
dyon [4,5].
Near the origin the dyon solution has the following
behavwhere = (g0)2/4 A20, and = (g2 + g 2)0/2.
The physical meaning of the asymptotic behavior must be
clear. Obviously , f , and A B represent the Higgs boson,
W boson, and Z boson whose masses are given by MH =
2 = 0, MW = g0/2, and MZ = g2 + g 20/2.
So (16) tells that MH , 1 ( A0/MW )2 MW , and MZ
determine the exponential damping of the Higgs boson,
W boson, and Z boson to their vacuum expectation values
asymptotically. Notice that it is 1 ( A0/MW )2 MW , but
not MW , which determines the exponential damping of the
W boson. This tells that the electric potential of the dyon
slows down the exponential damping of the W boson, which
is reasonable.
where = (3 1)/2. Asymptotically it has the following
behavior:
The dyon has the following electromagnetic charges:
f 2 Adr =
qm = 4e . (17)
Also, the asymptotic condition (16) assures that the dyon
does not carry any neutral charge,
Zm = 0.
r 2( A B )
r=
= 0,
Furthermore, notice that the dyon equation (10) is invariant
under the reflection
A A,
B B.
This means that, for a given magnetic charge, there are always
two dyon solutions which carry opposite electric charges
qe. Clearly the signature of the electric charge of the dyon
is determined by the signature of the boundary value A0.
We can also have the antimonopole or in general
antidyon solution, the charge conjugate state of the dyon, which
has the magnetic charge qm = 4/e with the following
ansatz:
1 1
A = g A(r )t r + g ( f (r ) 1) r r ,
1 1
B = g B(r )t + g (1 cos ),
r = = (sin cos , sin sin , cos ).
Notice that the ansatz is basically the complex conjugation
of the dyon ansatz.
To understand the meaning of the antidyon ansatz notice
that in the unitary gauge
1 f (r )(sin + sin cos )
A g f (r )(cos sin sin ) .
A(r )t + (1 cos )
U = i
This clearly shows that the electric and magnetic charges of
the ansatz (20) are the opposite of the dyon ansatz, which
confirms that the ansatz indeed describes the antidyon.
With the ansatz (20) we have exactly the same Eq. (10)
for the antidyon. This assures that the standard model has
the antidyon as well as the dyon.
The above discussion tells that the W and Z boson part of
the antidyon solution is basically the complex conjugation of
the dyon solution. This, of course, is natural from the physical
point of view. On the other hand there is one minor point to be
clarified here. Since the topological charge of the monopole
is given by the second homotopy defined by r = , one
might expect that r defined by the antidyon ansatz =
must be r. But this is not so, and we have to explain why.
To understand this notice that we can change r to r by
a SU(2) gauge transformation, by the rotation along the
yaxis. With this gauge transformation the ansatz (20) changes
to
, r r,
This tells that the monopole topology defined by r is the
same as that of r.
Since the ChoMaison solution is obtained numerically,
one might like to have a mathematically rigorous existence
proof of the ChoMaison dyon. The existence proof is
nontrivial, because the equation of motion (10) is not the Euler
Lagrange equation of the positive definite energy (26), but
that of the indefinite action
f2 21 (r A)2 f 2 A2
r 2 1
8 (B A)22 + 2r 2
Fortunately the existence proof has been established by Yang
[6,7].
Before we leave this section it is worth to readdress the
important question again: Does the standard model predict
the monopole? Notice that the Dirac monopole in
electrodynamics is optional: It can exist only when U (1)em is
nontrivial, but there is no reason why this has to be so. If so, why
cannot the electroweak monopole be optional?
As we have pointed out, the nontrivial U (1)Y is crucial
for the existence of the monopole in the standard model. So
the question here is why the U (1)Y must be nontrivial. To see
why, notice that in the standard model U (1)em comes from
two U (1), the U (1) subgroup of SU (2) and U (1)Y , and it is
well known that the U (1) subgroup of SU (2) is nontrivial.
Now, to obtain the electroweak monopole we have to make
the linear combination of two monopoles; that of the U (1)
subgroup of SU (2) and U (1)Y . This must be clear from (8).
In this case the mathematical consistency requires the two
potentials A3 and B (and two U (1)) to have the same
structure, in particular the same topology. But we already know
that A3 is nontrivial. So B, and the corresponding U (1)Y ,
has to be nontrivial. In other words, requiring U (1)Y to
be trivial is inconsistent (i.e., in contradiction with the
selfconsistency) in the standard model. This tells that, unlike
Maxwells theory, the U (1)em in the standard model must be
nontrivial. This assures that the standard model must have
the monopole.
But ultimately this question has to be answered by the
experiment. So the discovery of the monopole must be the
topological test of the standard model, which has never been
done before. This is why MoEDAL is so important.
To detect the electroweak monopole experimentally, we have
to have a firm idea on the mass of the monopole.
Unfortunately, at the classical level we cannot estimate the mass of
the ChoMaison monopole, because it has a point singularity
at the center which makes the total energy infinite.
Indeed the ansatz (5) gives the following energy:
3 Mass of the electroweak monopole (26)
+ 41 f 22 + r82 (B A)22 .
The boundary condition (13) guarantees that E1 is finite.
As for E0 we can minimize it with the boundary condition
f (0) = 1, but even with this E0 becomes infinite. Of course
the origin of this infinite energy is obvious, which is
precisely due to the magnetic singularity of B at the origin.
This means that one cannot predict the mass of the dyon.
Physically it remains arbitrary.
To estimate the monopole mass theoretically, we have to
regularize the point singularity of the ChoMaison dyon. One
might try to do that introducing the gravitational interaction,
in which case the mass is fixed by the asymptotic behavior
of the gravitational potential. But the magnetic charge of the
monopole is not likely to change the character of the
singularity, so that asymptotically the leading order of the
gravitational potential becomes that of the ReissnerNordstrom
type [3235]. This implies the gravitational interaction may
not help us to estimate the monopole mass.
To make the energy of the ChoMaison monopole finite,
notice that the origin of the infinite energy is the first term
1/g 2 in E0 in (26). A simple way to make this term finite
is to introduce a UVcutoff which removes this divergence.
This type of cutoff could naturally come from the quantum
correction of the coupling constants. In fact, since the
quantum correction changes g to the running coupling g , E0 can
become finite if g diverges at short distance.
We will discuss how such quantum correction could take
place later, but before doing that we present two arguments,
the dimensional argument and the scaling argument, which
could give us a rough estimate of the monopole mass.
3.1 Dimensional argument
To have the order estimate of the monopole mass it is
important to realize that, roughly speaking, the monopole mass
comes from the Higgs mechanism which generates the mass
to the Wboson. This can easily be seen for the t Hooft
Polyakov monopole in the GeorgiGlashow model:
LGG = 41 F2 21 (D ) 2 4
is the Higgs triplet. Here the monopole ansatz is
where C represents the WuYang monopole potential [13,
14,36]. Notice that the Wboson part of the monopole is given
by the WuYang potential, except for the overall amplitude
f .
With this we clearly have
D 2 = ()2 + g22 f 2(C)2.
So, when the Higgs field has a nonvanishing vacuum
expectation value, C acquires a mass (with f 1). This, of
course, is the Higgs mechanism which generates the
Wboson mass. The only difference is that here the Wboson
is expressed by the WuYang potential and the Higgs
coupling becomes magnetic (C contains the extra factor 1/g).
Similar mechanism works for the WeinbergSalam model.
Here again A (with A = B = 0) of the ansatz (5) is identical
to (28), and we have
D 2 = D 2  D 2
( D i g2 B)2 = 41 g2 f 2(C)2,
1 1 2
D2 = 2 ()2 + 2 D 2
= 21 ()2 + 81 g22 f 2(C)2.
This (with f 1) tells that the electroweak monopole
acquires mass through the Higgs mechanism which
generates mass to the Wboson.
Once this is understood, we can use the dimensional
argument to predict the monopole energy. Since the monopole
mass term in the Lagrangian contributes to the monopole
energy in the classical solution we may expect
This implies that the monopole mass should be about 1/
times bigger than the electroweak scale, around 10 TeV. But
this is an order estimate. Now we have to know how to
estimate C .
3.2 Scaling argument
We can use the Derrick scaling argument to estimate the
constant C in (31), assuming the existence of a finite energy
monopole solution. If a finite energy monopole does exist, the
action principle tells that it should be stable under the
rescaling of its field configuration. So consider such a monopole
configuration and let
K A =
K B =
With the ansatz (5) we have (with A = B = 0)
1
Notice that K B makes the monopole energy infinite.
Now, consider the spatial scale transformation
Under this we have
K 1 K , V 3V .
With this we have the following requirement for the stable
monopole configuration:
From this we can estimate the finite value of K B .
Now, for the ChoMaison monopole we have (with MW
80.4 GeV, MH 125 GeV, and sin2 w = 0.2312)
From this we estimate the energy of the monopole to be
This strongly endorses the dimensional argument. In
particular, this tells that the electroweak monopole of mass around
a few TeV could be possible.
The important question now is to show how the
quantum correction could actually make the energy of the Cho
Maison monopole finite. To do that we have to understand the
structure of the electroweak theory, in particular the Abelian
decomposition of the electroweak theory. So we review the
gauge independent Abelian decomposition of the standard
model first.
4 Abelian decomposition of the electroweak theory Consider the YangMills theory
The best way to make the Abelian decomposition is to
introduce a unit SU (2) triplet n which selects the Abelian
direction at each spacetime point, and impose the isometry on the
This, with (37), tells that
gauge potential which determines the restricted potential A
[3639]
1
A A = An g n n = An + C,
Notice that the restricted potential is precisely the
connection which leaves n invariant under parallel transport. The
restricted potential is called a Cho connection or a Cho
DuanGe (CDG) connection [4045].
With this we obtain the gauge independent Abelian
decomposition of the SU (2) gauge potential adding the
valence potential W which was excluded by the isometry
[3639]
A = A + W, (n W = 0).
The Abelian decomposition has recently been referred to as
Cho (also ChoDuanGe or ChoFaddeevNiemi)
decomposition [4045].
Under the infinitesimal gauge transformation
1 1
A = g n , A = g D ,
This tells that A by itself describes an SU (2) connection
which enjoys the full SU (2) gauge degrees of freedom.
Furthermore the valence potential W forms a gauge covariant
vector field under the gauge transformation. But what is really
remarkable is that the decomposition is gauge independent.
Once n is chosen, the decomposition follows automatically,
regardless of the choice of gauge.
Notice that A has a dual structure,
Moreover, H always admits the potential because it
satisfies the Bianchi identity. In fact, replacing n with a C P1 field
(with n = ) we have
H = C C = 2gi ( ),
C = 2gi = gi .
F = F + D W D W + gW W ,
so that the YangMills Lagrangian is expressed as
LY M = 41 F2 41 (D W D W)2
g2 F (W W ) g42 (W W )2.
This shows that the YangMills theory can be viewed as
a restricted gauge theory made of the restricted potential,
which has the valence gluons as its source [3639].
An important advantage of the decomposition (43) is that
it can actually Abelianize (or more precisely dualize) the
nonAbelian gauge theory gauge independently [3639]. To
see this let(n 1, n 2, n ) be a righthanded orthonormal basis
of SU (2) space and let
With this we have
D W = W1 g( A + C )W2 n1
+ W2 + g( A + C )W1 n 2,
so that with
Of course C is determined uniquely up to the U (1) gauge
freedom which leaves n invariant. To understand the meaning
of C , notice that with n = r we have
This is nothing but the Abelian monopole potential, and the
corresponding nonAbelian monopole potential is given by
the WuYang monopole potential C [1316]. This justifies
us to call A and C the electric and magnetic potential.
The above analysis tells that A retains all essential
topological characteristics of the original nonAbelian potential.
First, n defines 2(S2) which describes the nonAbelian
monopoles. Second, it characterizes the Hopf invariant
3(S2) 3(S3) which describes the topologically
distinct vacua [4648]. Moreover, it provides the gauge
independent separation of the monopole field from the generic
nonAbelian gauge potential.
With the decomposition (43), we have
we can express the Lagrangian explicitly in terms of the dual
potential A and the complex vector field W,
LY M = 41 F2 21 D W D W2
+i gF WW + g42 (WW WW)2,
where F = F + H and D = + i gA. This shows
that we can indeed Abelianize the nonAbelian theory with
our decomposition.
Notice that in the Abelian formalism the Abelian potential
A has the extra magnetic potential C . In other words, it
is given by the sum of the electric and magnetic potentials
A + C . Clearly C represents the topological degrees of
the nonAbelian symmetry which does not show up in the
naive Abelianization that one obtains by fixing the gauge
[3639].
Furthermore, this Abelianization is gauge independent,
because here we have never fixed the gauge to obtain this
Abelian formalism. So one might ask how the nonAbelian
gauge symmetry is realized in this Abelian formalism. To
discuss this let
Certainly the Lagrangian (52) is invariant under the active
(classical) gauge transformation (45) described by
But it has another gauge invariance, the invariance under the
following passive (quantum) gauge transformation:
Clearly this passive gauge transformation assures the desired
nonAbelian gauge symmetry for the Abelian formalism.
This tells that the Abelian theory not only retains the original
gauge symmetry, but actually has an enlarged (both active
and passive) gauge symmetries.
The reason for this extra (quantum) gauge symmetry is
that the Abelian decomposition automatically puts the theory
in the background field formalism which doubles the gauge
symmetry [49]. This is because in this decomposition we
can view the restricted and valence potentials as the classical
and quantum potentials, so that we have freedom to assign the
gauge symmetry either to the classical field or to the quantum
field. This is why we have the extra gauge symmetry.
The Abelian decomposition has played a crucial role
in QCD to demonstrate the Abelian dominance and the
monopole condensation in color confinement [5058]. This
is because it separates not only the Abelian potential but also
the monopole potential gauge independently.
Now, consider the GeorgiGlashow model (27). With
we have the Abelian decomposition,
LGG = 21 ()2 g22 2(W)2 4
41 F2 41 (D W D W)2
g2 F (W W ) g42 (W W )2.
With this we can Abelianize it gauge independently,
LGG = 21 ()2 g22W2 4
41 F2 21 D W D W2 + i gF WW
+ g42 (WW WW)2.
This clearly shows that the theory can be viewed as a
(nontrivial) Abelian gauge theory which has a charged vector field
as a source.
The Abelianized Lagrangian looks very much like the
GeorgiGlashow Lagrangian written in the unitary gauge.
But we emphasize that this is the gauge independent
Abelianization which has the full (quantum) SU (2) gauge symmetry.
Obviously we can apply the same Abelian decomposition
to the WeinbergSalam theory
L = 21 ()2 22 D 2 8 (2 02)2
we can Abelianize it gauge independently
L = 21 ()2 8 2 02 2
41 F(em)2 41 Z 2 g42 2W2
21 (D(em)W D(em)W) + i e g (ZW Z W)2
g
+ g42 (WW WW)2,
where D(em) = + i e A(em). Again we emphasize that this
is not the WeinbergSalam Lagrangian in the unitary gauge.
This is the gauge independent Abelianization which has the
extra quantum (passive) nonAbelian gauge degrees of
freedom. This can easily be understood comparing (60) with (8).
Certainly (60) is gauge independent, while (8) applies to the
unitary gauge.
This provides us important piece of information. In the
absence of the electromagnetic interaction (i.e., with A(em) =
W = 0) the WeinbergSalam model describes a
spontaneously broken U (1)Z gauge theory,
L = 21 ()2 8 2 02 2
which is nothing but the GinzburgLandau theory of
superconductivity. Furthermore, here MH and MZ corresponds
to the coherence length (of the Higgs field) and the
penetration length (of the magnetic field made of Z field). So,
when MH > MZ (or MH < MZ ), the theory describes a
type II (or type I) superconductivity, which is well known to
admit the AbrikosovNielsenOlesen vortex solution. This
confirms the existence of Nambus string in WeinbergSalam
model. What Nambu showed was that he could make the
string finite by attaching the fractionally charged monopole
antimonopole pair to this string [24,25].
5 Comparison with JuliaZee dyon
The ChoMaison dyon looks very much like the wellknown
JuliaZee dyon in the GeorgiGlashow model. Both can be
viewed as the WuYang monopole dressed by the weak
boson(s). However, there is a crucial difference. The the
JuliaZee dyon is completely regular and has a finite energy,
while the ChoMaison dyon has a point singularity at the
center which makes the energy infinite.
So, to regularize the ChoMaison dyon it is important to
understand the difference between the two dyons. To do that
notice that, in the absence of the Z boson, (61) reduces to
L = 21 ()2 8 2 02 2
41 F(em)2 21 D(em)W D(em)W2
+i e F(em)WW + g42 (WW WW)2.
This should be compared with (58), which shows that the
two theories have exactly the same type of interaction in
the absence of the Z boson, if we identify F in (58) with
F(em) in (63). The only difference is the coupling strengths of
the W boson quartic selfinteraction and Higgs interaction of
W boson (responsible for the Higgs mechanism). This
difference, of course, originates from the fact that the Weinberg
Salam model has two gauge coupling constants, while the
GeorgiGlashow model has only one.
This tells that, in spite of the fact that the ChoMaison
dyon has infinite energy, it is not very different from the Julia
Zee dyon. To amplify this point notice that the spherically
symmetric ansatz of the JuliaZee dyon
1 1
A = g A(r )t r g r r
can be written in the Abelian formalism as
i f (r ) ei ( + i sin ),
W = g 2
With the ansatz we have the following equation for the
dyon:
+ r2 2 rf 22 = 2 2
f
This should be compared to the equation of motion (10) for
the ChoMaison dyon. They are not very different.
With the boundary condition
A(0) = 0,
A() = A0, (67)
one can integrate (66) and obtain the JuliaZee dyon which
has a finite energy. Notice that the boundary condition
A(0) = 0 and f (0) = 1 is crucial to make the solutions
regular at the origin. This confirms that the JuliaZee dyon is
nothing but the Abelian monopole regularized by and W,
where the charged vector field adds an extra electric charge
to the monopole. Again it must be clear from (66) that, for
a given magnetic charge, there are always two dyons with
opposite electric charges.
Moreover, for the monopole (and antimonopole)
solution with A = 0, the equation reduces to the
following BogomolnyiPrasadSommerfield equation in the limit
= 0:
g1r 2 ( f 2 1) = 0, f g f = 0. (68)
This has the analytic solution
= 0 coth(g0r ) e1r , f = sinhg(g0r0r ) ,
which describes the PrasadSommerfield monopole [20].
Of course, the ChoMaison dyon has a nontrivial dressing
of the Z boson which is absent in the JuliaZee dyon. But
notice that the Z boson plays no role in the ChoMaison
monopole. This confirms that the ChoMaison monopole and
the t HooftPolyakov monopole are not so different, so that
the ChoMaison monopole could be modified to have finite
energy.
For the antidyon we can have the following ansatz:
1 1
A = g A(r )t r g r r
r = (sin cos , sin sin , cos ),
1 1
A = g A(r )t + g (1 cos ).
This ansatz looks different from the popular ansatz described
by = (r ) r, but we can easily show that they are gauge
equivalent. With this we have exactly the same equation,
Eq. (66), for the antidyon, which assures that the theory has
both dyon and antidyon.
6 Ultraviolet regularization of ChoMaison dyon
Since the ChoMaison dyon is the only dyon in the standard
model, it is impossible to regularize it within the model.
However, the WeinbergSalam model is the bare theory which
should change to the effective theory after the quantum
correction, and the real electroweak dyon must be the solution
of such theory. So we may hope that the quantum correction
could regularize the ChoMaison dyon.
The importance of the quantum correction in classical
solutions is best understood in QCD. The bare QCD
Lagrangian has no confinement, so that the classical
solutions of the bare QCD can never describe the quarkonium or
hadronic bound states. Only the effective theory can.
To see how the quantum modification could make the
energy of the ChoMaison monopole finite, notice that after
the quantum correction the coupling constants change to the
scale dependent running couplings. So, if this quantum
correction makes 1/g 2 in E0 in (26) vanishing in the short
distance limit, the ChoMaison monopole could have finite
energy.
To do that consider the following effective Lagrangian
which has the noncanonical kinetic term for the U (1)Y gauge
field:
Leff = D2 2
where ( 2) is a positive dimensionless function of the
Higgs doublet which approaches 1 asymptotically. Clearly
modifies the permittivity of the U (1)Y gauge field, but the
effective action still retains the SU (2) U (1)Y gauge
symmetry. Moreover, when 1 asymptotically, the effective
action reproduces the standard model.
This type of effective theory which has the field
dependent permittivity naturally appears in the nonlinear
electrodynamics and higherdimensional unified theory, and has
been studied intensively in cosmology to explain the
latetime accelerated expansion [5963].
From (72) we have the equations for and B
2 = D 2 + 2 (2 02) + 21
= d / d2. This changes the dyon equation (10) to
2 f 2 1
+ r 2r 2 = 4 ( A B)2 + 2 (2 02)
f f 2r 2 1 f = g42 2 A2
2 2 f 2 g2
A + r A r 2 A = 4 2( A B),
This tells that effectively changes the U (1)Y gauge coupling
g to the running coupling g = g / . This is because
with the rescaling of B to B/g , g changes to g / . So,
by making g infinite (requiring vanishing) at the origin,
we can regularize the ChoMaison monopole.
From the equations of motion we find that we need the
following condition near the origin to make the monopole
energy finite:
, n > 4 + 23
Fig. 1 The finite energy electroweak dyon solution obtained from the
effective Lagrangian (72). The solid line represents the finite energy
dyon and dotted line represents the ChoMaison dyon, where Z =
A B and we have chosen f (0) = 1 and A() = MW /2
Fig. 2 The running coupling g of U (1)Y gauge field induced by the
effective Lagrangian (72)
near the origin, and we have the finite energy dyon solution
shown in Fig. 1. It is really remarkable that the regularized
solutions look very much like the ChoMaison solutions,
except that for the finite energy dyon solution Z (0) becomes
0. This confirms that the ultraviolet regularization of the Cho
Maison monopole is indeed possible.
As expected with n = 8 the running coupling g becomes
divergent at the origin, which makes the energy contribution
from the U (1)Y gauge field finite. The scale dependence of
the running coupling is shown in Fig. 2. With A = B = 0
we can estimate the monopole energy to be
This tells that the estimate of the monopole energy based on
the scaling argument is reliable. The finite energy monopole
solution is shown in Fig. 3.
There is another way to regularize the ChoMaison
monopole. Suppose we have the following ultraviolet
modiFig. 3 The finite energy electroweak monopole solution obtained from
the effective Lagrangian (79). The solid line (red) represents the
regularized monopole and the dotted (blue) line represents the ChoMaison
monopole
fication of (59) from the quantum correction:
where , , are the scale dependent parameters which
vanish asymptotically (and modify the theory only at short
distance). The justification of these counterterms is clear.
The existence of the monopole could affect the W boson
magnetic moment and induce the term. The and terms
could come from the coupling constant and mass
renormalizations of the W boson.
Thus we have the modified WeinbergSalam Lagrangian
L = 21 ()2 8 2 02 2 41 F(em)2 41 Z 2
1 g
2 (D(em)W D(em)W)+i e g (ZW Z W)
Of course, this modification is supposed to hold only in the
short distance, so that asymptotically , , should
vanish to make sure that L reduces to the standard model. But
we will treat them as constants, partly because it is difficult
to make them scale dependent, but mainly because
asymptotically the boundary condition automatically makes them
irrelevant and assures the solution to converge to the Cho
Maison solution.
To understand the physical meaning of (79) notice that in
the absence of the Z boson the above Lagrangian reduces
to the GeorgiGlashow Lagrangian where the W boson has
an extra anomalous magnetic moment when (1 + ) =
e2/g2 and (1 + ) = 4e2/g2, if we identify the coupling
constant g in the GeorgiGlashow model with the
electromagnetic coupling constant e. Moreover, the ansatz (5) can
be written as
1 1
A(r ) + g 2 B(r ) t r e r r,
This shows that, for the monopole (i.e., for A = B = 0), the
ansatz becomes formally identical to (64) if W is rescaled
by a factor g/e. This tells that, as far as the monopole solution
is concerned, in the absence of the Z boson the Weinberg
Salam model and GeorgiGlashow model are not so
different.
With (79) the energy of the dyon is given by
E = E0 + E1,
2 dr g2
E0 = g2 0 r 2 g 2 + 1 2(1 + ) f 2 + (1 + ) f 4
Notice that E1 remains finite with the modification, and
plays no role to make the monopole energy finite.
To make E0 finite we must have
1 g2 1 g2
1 + = f (0)2 e2 , 1 + = f (0)4 e2 ,
so that the constants and are fixed by f (0). With this the
equation of motion is given by
= 41 ( A B)2 + 2 2 02 ,
f
Fig. 4 The finite energy electroweak dyon solution obtained from the
modified Lagrangian (79). The solid line represents the finite energy
dyon and dotted line represents the ChoMaison dyon
The solution has the following behavior near the origin:
Notice that all four deltas are positive (as far as (1 + ) > 0),
so that the four functions are well behaved at the origin.
If we assume = = 0 we have f (0) = g/e, and we
can integrate (83) with the boundary condition
A(0) = 0, B(0) = b0,
A() = B() = A0.
The finite energy dyon solution is shown in Fig. 4. It should
be emphasized that the solution is an approximate solution
which is supposed to be valid only near the origin, because
the constants , , are supposed to vanish
asymptotically. But notice that asymptotically the solution
automatically approaches the ChoMaison solution even without
making them vanish, because we have the same boundary
condition at the infinity. Again it is remarkable that the finite
energy solution looks very similar to the ChoMaison
solution.
Of course, we can still integrate (83) with arbitrary f (0)
and have a finite energy solution. The monopole energy for
f (0) = 1 and f (0) = g/e (with = = 0) are given by
E ( f (0) = 1)
Fig. 5 The energy dependence of the electroweak monopole on f (0)
In general the energy of a dyon depends on f (0), but must be
of the order of (4/e2)MW . The energy dependence of the
monopole on f (0) is shown in Fig. 5. This strongly supports
our prediction of the monopole mass based on the scaling
argument.
As we have emphasized, in the absence of the Z boson
(79) reduces to the GeorgiGlashow theory with
e2 4e2
= 0, 1 + = g2 , 1 + = g2 .
In this case (83) reduces to the following Bogomolnyi
PrasadSommerfield equation in the limit = 0 [20]:
f 2 1
This has the analytic monopole solution
1 g0r
= 0 coth(e0r ) er , f = sinh(e0r ) ,
whose energy is given by the Bogomolnyi bound
From this we can confidently say that the mass of the
electroweak monopole could be around 4 to 10 TeV.
This confirms that we can regularize the ChoMaison
dyon with a simple modification of the coupling strengths
of the existing interactions, which could be caused by the
quantum correction. This provides a most economic way to
make the energy of the dyon finite without introducing a new
interaction in the standard model.
7 Embedding U (1)Y to SU (2)Y
Another way to regularize the ChoMaison dyon, of course,
is to enlarge U (1)Y and embed it to another SU(2). This
type of generalization of the standard model could naturally
arise in the leftright symmetric grand unification models,
in particular in the SO(10) grand unification, although this
generalization may be too simple to be realistic.
To construct the desired solutions we introduce a
hypercharged vector field X and a Higgs field , and we
generalize the Lagrangian (59) adding the following Lagrangian:
L = 21 D X D X2 + i g G X X
+ 41 g 2(X X X X)2
21 ( )2 g 2 2X2 4
where D = + i g B. To understand the meaning of it
let us introduce a hypercharge SU (2) gauge field B and a
scalar triplet , and consider the SU (2)Y GeorgiGlashow
model
Now we can have the Abelian decomposition of this
Lagrangian, with = n, and have (identifying B and
X as the Abelian and valence parts)
This clearly shows that Lagrangian (91) describes nothing
but the embedding of the hypercharge U (1) to an SU (2)
GeorgiGlashow model.
Now for a static spherically symmetric ansatz we choose
(5) and let
X = gi h(r2) ei ( + i sin ).
With the spherically symmetric ansatz the equations of
motion are reduced to
f f 2r 2 1 f = g42 2 A2 f,
2 f 2 1
+ r 2r 2 = 4 ( A B)2 + 2
2 2 f 2 g2
A + r A r 2 A = 4 2( A B),
h h2r 2 1 h = (g 2 2 B2)h,
+ r2 2rh22 = 2 m2
Fig. 6 The SU (2) SU (2) monopole solution with MH /MW = 1.56,
MX = 10 MW , and = 0
E = EW + E X ,
0 dr f2 + ( f 22r21)2
+ f 2 A2 + g22 (r )2 + g42 f 22 +
E X = g 2
where 0 = m2/, MX = g 0, and C1 and C2 are
constants of the order 1. The boundary conditions for a regular
field configuration can be chosen as
f (0) = h(0) = 1,
f () = h() = 0,
A() = A0, B() = B0,
Notice that this guarantees the analyticity of the solution
everywhere, including the origin.
With the boundary condition (97) one may try to find the
desired solution. From the physical point of view one could
assume MX MW , where MX is an intermediate scale
which lies somewhere between the grand unification scale
and the electroweak scale. Now, let A = B = 0 for
simplicity. Then (95) decouples to describe two independent systems
so that the monopole solution has two cores, the one with the
size O(1/MW ) and the other with the size O(1/MX ). With
MX = 10MW we obtain the solution shown in Fig. 6 in the
limit = 0 and MH /MW = 1.56.
In this limit we find C1 = 1.53 and C2 = 1, so that the
energy of the solution is given by
110.17 MX .
Clearly the solution describes the ChoMaison monopole
whose singularity is regularized by a PrasadSommerfield
monopole of the size O(1/MX ).
Notice that, even though the energy of the monopole is
fixed by the intermediate scale, the size of the monopole is
determined by the electroweak scale. Furthermore from the
outside the monopole looks exactly the same as the Cho
Maison monopole. Only the inner core is regularized by the
hypercharged vector field.
8 Conclusions
In this paper we have discussed three ways to estimate
the mass of the electroweak monopole, the dimensional
argument, the scaling argument, and the ultraviolet
regularization of the ChoMaison monopole. As importantly, we
have shown that the standard model has the antidyon as
well as the dyon solution, so that they can be produced in
pairs.
It has generally been believed that the finite energy
monopole could exist only at the grand unification scale [21].
But our result tells that the genuine electroweak monopole of
mass around 4 to 10 TeV could exist. This strongly implies
that there is an excellent chance that MoEDAL could actually
detect such monopole in the near future, because the 14 TeV
LHC upgrade now reaches the monopoleantimonopole
pair production threshold. But of course, if the mass of the
monopole exceeds the LHC threshold 7 TeV, we may have
to look for the monopole from cosmic ray with the cosmic
MoEDAL.
The importance of the electroweak monopole is that it is
the electroweak generalization of the Dirac monopole, and
that it is the only realistic monopole which can be produced
and detected. A remarkable aspect of this monopole is that
mathematically it can be viewed as a hybrid between the
Dirac monopole and the t HooftPolyakov monopole.
However, there are two crucial differences. First, the
magnetic charge of the electroweak monopole is two times
bigger than that of the Dirac monopole, so that it satisfies the
Schwinger quantization condition qm = 4 n/e. This is
because the electroweak generalization requires us to embed
U (1)em to the U(1) subgroup of SU(2), which has the period
of 4 . So the magnetic charge of the electroweak monopole
has the unit 4/e.
Of course, the finite energy dyon solutions we discussed
above are not the solutions of the bare standard model.
Nevertheless they tell us how the ChoMaison dyon could
be regularized and how the regularized electroweak dyon
would look like. From the physical point of view there is no
doubt that the finite energy solutions should be interpreted
as the regularized ChoMaison dyons whose mass and size
are fixed by the electroweak scale.
We emphasize that, unlike Diracs monopole, which can
exist only when U (1)em becomes nontrivial, the electroweak
monopole must exist in the standard model. So, if the standard
model is correct, we must have the monopole. In this sense,
the experimental discovery of the electroweak monopole
should be viewed as the final topological test of the standard
model.
At this point it is worth mentioning other closely related
topological solutions of the standard model, in particular the
sphaleron which could induce the baryon number violation
[2931,64,65]. As we have already remarked the sphaleron
and our monopole share the same topology. In this sense the
discovery of the sphaleron could provide another topological
test of the standard model. On the other hand the sphaleron
has an intrinsic instability because it is a saddle point solution,
while our monopole has the topological stability. This makes
it easier to find the monopole.
But they have similar feature. The energy of the sphaleron
is estimated to be about the same as the monopole mass,
Es 3.9 4/g2 MW 7.25 TeV [64,65]. This, of course
is not an accident. Clearly our dimensional argument on the
monopole mass equally applies to the sphaleron, so that they
must have similar energy.
Another related topological object is the electroweak
skyrmion [6668]. But this soliton becomes possible only in
the modified standard model which has an extra term
motivated by the renormalization, so that it becomes less relevant.
Clearly the electroweak monopole invites more difficult
questions. How can we justify the perturbative expansion
and the renormalization in the presence of the monopole?
What are the new physical processes which can be induced
by the monopole? Most importantly, how can we construct
the quantum field theory of the monopole?
Moreover, the existence of the finite energy electroweak
monopole should have important physical implications. In
particular, it could have important implications in cosmology,
because it can be produced after the inflation. The physical
implications of the monopole will be discussed in a separate
paper [69].
Acknowledgments The work is supported in part by the National
Research Foundation (2012002134) of the Ministry of Science and
Technology and by Konkuk University.
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