Using String Theory to Describe Monopole Scattering
We explain how it is possible to describe the scattering of magnetic monopoles using the D-branes of string theory. We use this description to calculate the energy radiated during monopole scattering.
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Відділення фізики і астрономії НАН України
2010
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-561882014-02-14T03:10:14Z Using String Theory to Describe Monopole Scattering Barrett, J.K. Поля та елементарні частинки We explain how it is possible to describe the scattering of magnetic monopoles using the D-branes of string theory. We use this description to calculate the energy radiated during monopole scattering. Пояснено, як можна описати розсiяння магнiтних монополiв, використовуючи D-брани в теорiї струн. Знайдено енергiю, що випромiнюється при розсiюваннi монополiв. 2010 Article Using String Theory to Describe Monopole Scattering / J.K. Barrett // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 473-480. — Бібліогр.: 13 назв. — англ. 2071-0194 PACS 11.25.Uv, 11.25.Wx, 14.80.Hv http://dspace.nbuv.gov.ua/handle/123456789/56188 en Український фізичний журнал Відділення фізики і астрономії НАН України |
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Поля та елементарні частинки Поля та елементарні частинки |
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Поля та елементарні частинки Поля та елементарні частинки Barrett, J.K. Using String Theory to Describe Monopole Scattering Український фізичний журнал |
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We explain how it is possible to describe the scattering of magnetic monopoles using the D-branes of string theory. We use this description to calculate the energy radiated during monopole scattering. |
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Article |
| author |
Barrett, J.K. |
| author_facet |
Barrett, J.K. |
| author_sort |
Barrett, J.K. |
| title |
Using String Theory to Describe Monopole Scattering |
| title_short |
Using String Theory to Describe Monopole Scattering |
| title_full |
Using String Theory to Describe Monopole Scattering |
| title_fullStr |
Using String Theory to Describe Monopole Scattering |
| title_full_unstemmed |
Using String Theory to Describe Monopole Scattering |
| title_sort |
using string theory to describe monopole scattering |
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Відділення фізики і астрономії НАН України |
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2010 |
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Поля та елементарні частинки |
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http://dspace.nbuv.gov.ua/handle/123456789/56188 |
| citation_txt |
Using String Theory to Describe Monopole Scattering / J.K. Barrett // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 473-480. — Бібліогр.: 13 назв. — англ. |
| series |
Український фізичний журнал |
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AT barrettjk usingstringtheorytodescribemonopolescattering |
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2025-07-05T07:25:15Z |
| last_indexed |
2025-07-05T07:25:15Z |
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1836790910178820096 |
| fulltext |
FIELDS AND ELEMENTARY PARTICLES
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 473
USING STRING THEORY TO DESCRIBE MONOPOLE
SCATTERING
J.K. BARRETT
University of Iceland
(Raunvisindastofnun, Taeknigardi, Dunhaga 5, 107 Reykjavik, Iceland; e-mail:
jessica@ raunvis. hi. is )
PACS 11.25.Uv, 11.25.Wx,
14.80.Hv
c©2010
We explain how it is possible to describe the scattering of mag-
netic monopoles using the D-branes of string theory. We use this
description to calculate the energy radiated during monopole scat-
tering.
1. Introduction
String theory is currently the best candidate we have for
a ‘theory of everything’. However there are still many
gaps in our understanding of string theory, and it has
not yet been possible to derive a realistic description of
the world we observe from string theory. But there have
been many ways in which string theory has given us new
perspectives on some of our existing knowledge. We will
consider one such example here; the D-brane realiza-
tion of the ADHMN construction. The ADHMN con-
struction was a mathematical tool which was developed
in the early 1980s to aid the construction of magnetic
monopole solutions. It was realized in the 1990s that,
in string theory, a D1-brane stretched between two D3-
branes embodies the ADHMN construction and gives it
a physical realization.
We describe here some material taken from [1] and [2],
in which we used the D-brane configuration described
above to study magnetic monopoles. Our aim was to
calculate the energy radiated during monopole scatter-
ing. This calculation has already been studied in [3] with
the result
Erad ∼ 1.35mmonv
5
−∞ ,
Erad
Etot
∼ 1.35 v3
−∞ , (1)
where Erad is the energy radiated, Etot is the total en-
ergy in the system, mmon is the mass of each monopole,
and v−∞ is the asymptotic velocity of each monopole.
We hoped to confirm this result using an alternative per-
spective provided by the D-brane configuration.
The layout will be as follows. In Section 2, we will
briefly review some facts about magnetic monopoles, and
we will review the role played by D-branes in string the-
ory in Section 3. In Section 4, we will discuss the D1-D3
brane description of magnetic monopoles, and, in Sec-
tion 5, we will describe our calculations of the energy
radiated during monopole scattering. We will conclude
in Section 6.
2. Magnetic Monopoles
In this section, we review some of the theory concerning
magnetic monopoles. The material from this section can
be found in more details in [4] and [5].
2.1. Dirac monopole
Let us consider Maxwell’s equations of electromag-
netism,
∇ ·E = ρe , ∇×B− ∂E
∂t
= je , (2)
and
∇ ·B = 0 , ∇×E +
∂B
∂t
= 0 . (3)
The electric source terms ρe and je are present in Eqs.
(2), because we have observed electric monopoles in the
nature. On the other hand, magnetic monopoles have
never been observed, and so the corresponding magnetic
equations (3) contain no source terms. However, it is
interesting to think about which consequences would be
J.K. BARRETT
if magnetic monopoles did exist. Equation (3) would
then become
∇ ·B = ρm , ∇×E +
∂B
∂t
= jm . (4)
It was shown by Dirac in the 1930s that the presence of
a single magnetic monopole in the Universe is sufficient
to guarantee the conservation of electric charge in the
form
eg = 2πn , n ∈ Z , (5)
where e is the electric charge, and g is the magnetic
charge. This is a nice explanation of charge quantization,
which makes the existence of magnetic monopoles an
attractive proposition.
2.2. ’t Hooft-Polyakov monopole
Let us move on from the theory of electromagnetism
to the Yang–Mills–Higgs theory. The Yang–Mills–Higgs
action is
SYM =
1
g2
YM
∫
d4x
{
− 1
4
F aµνF
aµν +
+
1
2
DµΦaDµΦa − V (Φ)
}
, (6)
where gYM is the Yang–Mills coupling constant, the field
Fµν is a gauge field, and the field Φ is a scalar field called
the Higgs field. Both Fµν and Φ belong to the adjoint
representation of the gauge group we take to be SU(2).
So, we write
Fµν = F aµνT
a , Φ = ΦaT a , (7)
where T a are the generators of SU(2). The gauge field
is defined in terms of a gauge potential Aµ as follows:
F aµν = ∂µA
a
ν − ∂νAaµ − εabcAbµAcν , (8)
and the covariant derivative is defined to be
DµΦa = ∂µΦa − εabcAbµΦc . (9)
The potential V (Φ) is chosen such that the vacuum ex-
pectation value of Φ is non-zero. This means that the
gauge bosons Aaµ acquire masses, and we say that the
gauge group is broken. We will take
V (Φ) =
λ
4
(
ΦaΦa − v2
)2
, (10)
so that the vacuum expectation value of Φ2 will be v2.
We will look for Yang–Mills–Higgs solutions which
have finite energy. In particular, this means that we
require V (Φ) = 0 on the sphere at spatial infinity, which
we denote S2
∞. Let M be the set of Φa which satisfy
V (Φ) = 0. Then
M = {Φ : ΦaΦa = v2} (11)
which has the topology of a two-sphere. Therefore, the
Higgs field configuration at spatial infinity defines a map
from one two-sphere to another one,
Φ : S2
∞ →MH . (12)
This map has an associated topological quantity called
a winding number, n ∈ Z (it can be thought of as the
number of times the first sphere ‘wraps around’ the sec-
ond).
It turns out that, for a solution with n 6= 0 to have
finite energy, there must be a non-zero gauge field. The
required gauge field has the form of a magnetic field
(where we define Bi = εijkFjk to be the magnetic field)
with magnetic charge given by
g =
4πn
gYM
. (13)
Equation (13) is once again Dirac’s quantization condi-
tion, with gYM = e/2.
2.3. BPS limit
The BPS limit of the theory described in the previous
section is the limit λ→ 0, where the limit is taken while
maintaining the boundary condition on the Higgs field,
Φ2 → v2 as r →∞ . (14)
Consider a static configuration with the electric field,
Ei = F0i, set to zero. In the BPS limit, it can be
shown that the energy of such a configuration with a
given magnetic charge g is minimized if and only if the
configuration satisfies the Bogomol’nyi equation
Ba = DΦa . (15)
The energy of the configuration is then given by
E =
|vg|
g2
YM
. (16)
474 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
USING STRING THEORY
2.4. ADHMN construction
The ADHMN construction is a technique which can be
used to obtain magnetic monopole solutions of Eq. (15).
Instead of working with the fields Aµ and Φ, we work
with Nahm data which consist of the matrices Ti. There
is a one-to-one map between Aµ and Φ and the Ti which
is called the Nahm transformation,
Aµ ,Φ ←→ Ti . (17)
To be more precise, Ti are n × n anti-Hermitian matri-
ces, where n is the winding number of the corresponding
monopole solution. They depend on the real parameter
ξ ∈ [0, 2] and satisfy the criteria
1. Nahm’s equations
dTi
dξ
=
i
2
εijk[Tj , Tk] . (18)
2. Ti have simple poles at ξ = 0 and ξ = 2.
3. The matrix residues at the poles form an irre-
ducible n-dimensional representation of SU(2).
Having obtained Nahm data which satisfy the above cri-
teria, we can use the Nahm transformation to obtain the
fields Aµ and Φ of the corresponding monopole solution.
3. D-Branes in String Theory
In this section, we will review some of the basic facts
concerning D-branes in string theory. See [6, 7] and [8]
for more details.
It was realized in the mid-1990s that string theory is
not only a theory of strings; to obtain a consistent the-
ory, we must include higher-dimensional objects which
are called D-branes. The definition of a Dp-brane is
a surface with p spacelike dimensions and one timelike
dimension, on which the ends of open strings are con-
strained to lie.
D-branes have an alternative description as soliton
(i.e., static and energy-minimizing) solutions of ten-
dimensional supergravity which is the low energy limit
of string theory.
In the spectrum of open string theory, there is a mass-
less vector which has the U(1) gauge symmetry. We can
promote this to the U(N) gauge symmetry by endowing
the ends of the open string with non-dynamical charges
(these are called Chan–Paton factors), as in the diagram
in Fig. 1, where a, b = 1, . . . N .
The number of massless vectors is now N2, because
there are N choices for the charge of each end of the
a b
Fig. 1. A fundamental string with Chan–Paton factors at the ends
D−Brane
D−Brane
Fig. 2. Parallel D-branes with strings stretching between them
string. These correspond to the gauge bosons of the
U(N) theory. Let us consider Chan–Paton factors in the
presence of D-branes. Suppose we have N parallel D-
branes, as in Fig. 2.
Then we can imagine the Chan–Paton factors at each
end of the string might correspond, in some sense, to the
D-brane on which the end of the string lies. (Note that
we are discussing the oriented string theory, in which a
string which begins on brane 1 and ends on brane 2 is
distinct from a string which begins on brane 2 and ends
on brane 1). So, for N parallel D-branes, the gauge the-
ory is U(N). When the D-branes are separated, as in Fig.
2, the strings stretching between D-branes are massive,
and the gauge group is U(N) broken down to U(1)N (as
in Section 2 2.2, where the breaking of the gauge group
by the Higgs vacuum expectation value leads to gauge
bosons acquiring masses). When the D-branes are coin-
cident, then all the open string vector states are massless,
and the gauge group is the unbroken U(N).
We will denote the gauge potential, which is confined
to the world volume of the D-brane since it is an open
string field, by Aµ with the corresponding field strength
Fµν , where µ, ν = 0, . . . , p denote D-brane directions.
There are other fields which exist on the brane’s surface
we will denote by ΦI , I = p + 1, . . . , 9 (the existence
of these fields can be shown using T -duality - see one of
the references on D-branes mentioned above for details).
The quantities ΦI also belong to the adjoint representa-
tion of the gauge group, and they roughly correspond to
the position of the D-brane(s) in the dimensions which
are transverse to theD-brane’s surface (recall that string
theory contains ten dimensions in total). We can label
x0, . . . , xp as the brane’s directions. Then XI = α′ΦI
tell us about the position of the brane(s) in the xI -
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 475
J.K. BARRETT
D−Brane
String
Fig. 3. A BIon spike solution
directions (more precisely, in the case of N branes, the
N eigenvalues of ΦI represent the positions! of the N
branes in the xI direction).
The action for a single D-brane in a flat background
(i.e. in a space with a flat metric, and all other closed
string fields set to zero) is the Born–Infeld action
S ∼ −
∫
dp+1x
(
− det
(
ηµν + (α′)2∂µΦI∂νΦI +
+ α′Fµν
))1/2
, (19)
where the integral is taken over the D-brane directions.
The non-Abelian extension of this action, at least up to
order (α′)2, is given by
S ∼ −
∫
dp+1x STr
(
−det
(
ηµν +
+ (α′)2DµΦIQ−1
IJDνΦJ + α′Fµν
))1/2
, (20)
where Dµ denotes the covariant derivative and
QIJ = δIJ + iα′[ΦI ,ΦK ]EKJ . (21)
In (20), STr denotes a symmetrized trace, which indi-
cates that we should symmetrize over all orderings of
Fµν , DµΦI , and [ΦI ,ΦJ ], when we take the trace (this
avoids ordering ambiguities in calculating the determi-
nant of a matrix, whose entries are non-Abelian objects).
4. Magnetic Monopoles as Solutions of the
Born–Infeld Action
It has been known for some time that the end of a fun-
damental string attached to a D-brane acts as a source
for the electric field on the brane, and the end of a D-
string (i.e. a D1-brane) attached to a D-brane acts as a
source for the magnetic field. It was shown in [9] that the
Born–Infeld action for a D-brane has a solution which
corresponds to a fundamental string attached to the D-
brane. (called the BIon spike solution, because the ten-
sion of the string pulls the brane into the form of a spike
– see Fig. 3).
The case of a D-string attached to a D3-brane was
studied in [10] and [11], and we shall review calculations
from these papers in more details here. In particular,
there are two ways of studying this object – we can use
the Born–Infeld action for the D3-brane or the Born–
Infeld action for the D-string.
Consider the Born–Infeld action for a D3-brane with
the magnetic field on the brane, Bi, excited, and with
a single transverse field, Φ, excited. If we look for a
static solution which minimizes the energy (i.e. a soliton
solution), we find that the solution obeys the usual BPS
equations for a magnetic monopole
Bi = DiΦ . (22)
The simplest solution is
Φ(r) =
n
2r
, B(r) = ∓ n
2r3
r , (23)
where r is the radial coordinate in the D-brane’s world
volume. Note that this solution for Φ(r) indicates that
the D3-brane has been pulled into an infinitely long spike
in the direction corresponding to the field Φ. It can
be shown that this solution represents n semiinfinite D-
strings attached to the D3-brane at the origin.
On the other hand, this configuration can also be stud-
ied using the non-Abelian Born–Infeld action for n D-
strings. Exciting three transverse scalars in the action,
say Φ1, Φ2, and Φ3, and looking for a soliton solution,
lead to the BPS equations
∂σΦi +
1
2i
εijk[Φj ,Φk] = 0 , (24)
where σ is the D-strings’ spatial direction. Note that
Eqs. (24) are identical to Nahm’s equations (18) from
the ADHMN construction of a magnetic monopole from
Section 2 2.4. The solution corresponding to n semiinfi-
nite D-strings ‘funnelling out’ into a D3-brane is
Φi = ±α
i
2σ
, (25)
where αi are an n×n representation of the SU(2) algebra.
In [1], we discussed the solution to Nahm’s equations
which describes two D-strings stretched between two D3-
branes. First, we define a new string coordinate ξ =
476 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
USING STRING THEORY
-40
ξ
1.61.20.80.4
f 1
20
10
0
-10
-20
-30
0
-10
-20
-30
-40
ξ
1.61.20.80.4
f 2
20
10
f 3
20
10
0
-10
-20
-30
-40
ξ
1.61.20.80.4
Fig. 4. Graphs of f1(ξ, k), f2(ξ, k), and f3(ξ, k) with k = 0.999999999
2σ/L, where L is the distance between the branes in the
σ coordinate. Then the distance between the branes in
the ξ coordinate is 2. We make the ansatz
Φi =
2
L
fi(ξ, t)σi, no summation over i, (26)
where σi are the Pauli matrices. Then Nahm’s equations
(24) reduce to
f ′1 − f2f3 = 0 , f ′2 − f3f1 = 0 , f ′3 − f1f2 = 0 , (27)
where ′ denotes the differentiation with respect to ξ. The
appropriate solution to these equations which were first
derived in [12] are
f1(ξ, k) =
−K(k)
sn(K(k)ξ, k)
, (28)
f2(ξ, k) =
−K(k)dn(K(k)ξ, k)
sn(K(k)ξ, k)
, (29)
f3(ξ, k) =
−K(k)cn(K(k)ξ, k)
sn(K(k)ξ, k)
, (30)
where K(k) is the complete elliptic integral of the first
kind and sn(ξ, k), cn(ξ, k), and dn(ξ, k) are the Jacobian
elliptic functions; and k is a parameter of the solutions
with 0 ≤ k < 1. Note that the fi all have poles at ξ = 0
and ξ = 2, which correspond to the D-strings ‘funnelling
out’ into D3-branes.
Let us consider the effect of the parameter k on solu-
tions (28)–(30). The graphs in Fig. 4 are of the fi with
k = 0.999999999.
In the limit k → 1, we can approximate as follows:
f1(ξ, k) ∼ K(k) , f2(ξ, k) ∼ f3(ξ, k) ∼ 0 , (31)
where the approximation is valid, except near the poles
at ξ = 0 and ξ = 2. Recall from Section 3 that the
eigenvalues of Φ1 are the positions of the D-strings in
the x1-direction. Equation (31) then implies that the
positions of the D-strings in the x1-direction are ±K(k).
So we see that the value of the parameter k tells us
about the position of the D-strings – the closer k is to
1, the further apart the D-strings are along the x1-axis.
The configuration with k = 0 is also of interest, since
f1(ξ, k = 0) = f2(ξ, k = 0). So this configuration is
axially symmetric in the x1-x2 plane; it corresponds to
a known two-monopole solution which has the shape of
a ‘doughnut’.
5. Energy Radiated During Monopole
Scattering
5.1. Description of monopole scattering
Let us consider taking a static D-string configuration
with k close to 1, and giving the D-strings a nudge, so
that they are moving slowly toward each other. The
energy of the D-strings is very small because we have
taken the initial configuration to be (28)–(30), which
minimizes the potential energy, and the kinetic energy
is small because the velocity is small. Therefore, as this
configuration evolves in time, it must always be close
to the static solutions – it does not have enough en-
ergy to move far from them. So we can approximate the
D-strings’ motion by allowing the configuration only to
depend on time through k(t) in (28)–(30); at any point
in time, the configuration still has the form of the static
solution with small velocity. This is Manton’s moduli
space approximation of [13] – the motion is described
by a geodesic in the moduli space, whose coordinate is
k. Since the D-strings are moving toward one another
initially, we must have k̇ < 0. As k → 0, the D-strings
continue moving toward one another, until they are co-
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 477
J.K. BARRETT
incident at k = 0. At this point, f1 and f2 swap roles
(we have shown numerically that this is the case). So the
D-strings have scattered at 90◦ – it is known that two
monopoles scatter at 90◦, so this was to be expected.
In order to calculate the energy radiated during scat-
tering, we need to go beyond the moduli space approx-
imation described above. We write the full solution to
the equations of motion as follows:
ϕi(ξ, t) = fi(ξ, k(t)) + εi(ξ, t) , (32)
where the fi are the static solutions which are the zero
modes, and the εi are perturbations which are small
for the reasons described above – they are the non-zero
modes. We showed in [2] that the zero modes and the
non-zero modes decouple after the scattering, so that
the energy can no longer be transferred between them.
The energy in non-zero modes after the scattering is the
energy that has been radiated during the scattering.
5.2. Numerical solution of the equations of
motion
In order to be able to solve the equations of motion nu-
merically, we will take the limit α′ → 0 which is the
low energy limit of string theory. There are two ways
to take this limit, as we discuss in [1] in detail. The
distance between the D3-branes is given by
L = α′v , (33)
where, in the D3-brane description, v is the expectation
value of the field Φ. So when we take the limit α′ → 0,
we can either keep v fixed, with L → 0, or we can keep
L fixed, with v → ∞. We will choose the second limit
here, because it leads to a simpler low-energy action. A
consequence of this is that the mass of the monopole/D-
string, which is given by
mmon =
v
gs
, (34)
is infinite. To keep our calculations finite, we will calcu-
late the ratio of the energy radiated to the total energy
(the total energy is conserved).
Taking the limit α′ → 0, v → ∞ of the Born–Infeld
action results in the action
SYM ∼
∞∫
−∞
dt
2∫
0
dξ
(
ϕ̇2
1 + ϕ̇2
2 + ϕ̇2
3 − (ϕ′1 − ϕ2ϕ3)2 −
− (ϕ′2 − ϕ3ϕ1)2 − (ϕ′3 − ϕ1ϕ2)2
)
. (35)
The energy of a static solution is minimized when
ϕ′1−ϕ2ϕ3 = 0 , ϕ′2−ϕ3ϕ1 = 0 , ϕ′3−ϕ1ϕ2 = 0 , (36)
which are identical in form to Eqs. (27) which were
derived from the full Born–Infeld action. The equations
of motion are
ϕ̈1 − ϕ′′1 + ϕ1(ϕ2
2 + ϕ2
3) = 0 , (37)
ϕ̈2 − ϕ′′2 + ϕ2(ϕ2
3 + ϕ2
1) = 0 , (38)
ϕ̈3 − ϕ′′3 + ϕ3(ϕ2
1 + ϕ2
2) = 0 . (39)
We solved the equations of motion numerically using
the RK4 method. Our initial configuration was a static
solution of the form (28)–(30) with k = 0.9999999999,
so that the D-strings were a long way apart initially. We
ran our numerical programs twice, once with vinit = 0.05
and once with vinit = 0.1.
5.3. Calculating the Energy Radiated
5.3.1. The energy in the ϕi
To begin with, we calculated the energy in the full solu-
tions, the ϕi. The total energy is conserved, so we can
use the order of the discrepancies in the total energies
to get an idea of the numerical inaccuracy in our results.
We found that, for vinit = 0.05, the inaccuracy in the to-
tal energy was around 10−8, and, for vinit = 0.1, it was
around 10−9.
The potential energy density is given by
P.E. density =
mmon
2
(
(ϕ′1 − ϕ2ϕ3)2 + (ϕ′2 − ϕ1ϕ3)2 +
+ (ϕ′3 − ϕ1ϕ2)2
)
, (40)
where mmon is the mass of a monopole, which we can
neglect since we will always deal with ratios of energies.
The equations for the zero modes fi show that the poten-
tial energy is zero. Therefore, the only contribution to
the potential energy in the ϕi comes from the non-zero
modes εi. Figures 5 and 6 show the logarithmic plots of
the potential energy in the ϕi. In both cases, the po-
tential energy peaks at the point of scattering, around
t = 200 for vinit = 0.05, and t = 100 for vinit = 0.1,
and then falls off again. Note that the final potential en-
ergy is around the level of numerical inaccuracy in both
cases, which suggests that no energy has been radiated.
We will confirm this result by calculating the energy in
the non-zero modes in the next section.
478 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
USING STRING THEORY
PE
10
-5
10
-6
time
10
-7
10
-8
600
10
-9
10
-10
400
10
-11
2000
Fig. 5. Logarithmic plot of the potential energy in the ϕi with
vinit = 0.05
time
300250
PE
10
-4
200
10
-5
10
-6
150
10
-7
10
-8
10
-9
100
10
-11
0
10
-10
50
10
-12
Fig. 6. Logarithmic plot of the potential energy in the ϕi with
vinit = 0.1
5.3.2. The energy in εi
We separated the zero modes from the non-zero modes
in our numerical solutions using a technique which de-
pended on approximation (31) for k close to 1 (see [2]
for the details of our method). When the D-strings are
far apart, the kinetic energy density for εi is given by
K.E. density =
1
2
(ε̇21 + ε̇22 + ε̇23) , (41)
and the potential energy density is
P.E. density =
1
2
(ε′21 + ε′22 + ε′23 )+
+
1
ξ2
(
ε21 + ε22 + ε23 + ε1ε2 + ε2ε3 + ε3ε1
)
+
time
700600500400
KE
5x10
-8
10
-8
5x10
-9
10
-1
10
-2
10
-3
10
-4
10
-5
time
10
-6
10
-7
700
10
-8
600500400
PE
Fig. 7. Logarithmic plots of the kinetic and potential energies in
εi with vinit = 0.05
+
1
ξ
(
ε1(ε′2 + ε′3) + ε2(ε′3 + ε′1) + ε3(ε′1 + ε′2)
)
, (42)
where we have neglected all terms of order ε3 and higher
in the potential energy density (42).
The graphs in Figs. 7 and 8 show the potential and ki-
netic energies calculated for vinit = 0.05 and vinit = 0.1,
respectively. In Fig. 7, for vinit = 0.05, the energy in
εi after the scattering, in the region where the approxi-
mation is valid, is ∼ 10−8, which agrees with the results
discussed in Section 5.3. In Fig. 8, for vinit = 0.1, the
energy radiated is ∼ 10−7, which also agrees with the
results of Section 5.3.
6. Conclusions
We have reviewed how string theory can provide a physi-
cal description of the ADHMN construction of magnetic
monopoles. We used the string theory description to
calculate the energy radiated during the monopole scat-
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 479
J.K. BARRETT
KE
10
-7
8x10
-8
6x10
-8
4x10
-8
time
300280260240220200
time
300280260240
PE
220
10
-3
10
-5
10
-4
200
10
-7
10
-6
Fig. 8. Logarithmic plots of the kinetic and potential energies in
εi with vinit = 0.1
tering. The results we obtained were
Erad
Einit
tot
∼ 10−8 for v−∞ = 0.05 ,
Erad
Einit
tot
∼ 10−7 for v−∞ = 0.1 . (43)
Since our results are approximately of the same order
as the numerical inaccuracy in our calculations, they are
consistent with those there being no energy radiated dur-
ing the monopole scattering.
Our results contrast with those of [3], whose prediction
(1) gives
Erad
Etot
∼ 10−4 for v−∞ = 0.05 ,
Erad
Etot
∼ 10−3 for v−∞ = 0.1 . (44)
In the future, we should seek to understand this discrep-
ancy.
Among the other forms of support, the author is
especially grateful to the Austrian Academy of Sci-
ences which supported her travel expenses to Ukraine
in the framework of the collaboration with the Na-
tional Academy of Sciences of the Ukraine. The au-
thor also could have not succeeded in pursuing this pro-
gram for many years without the collaborations with
Prof. W. Kummer and Prof. M. Kreuzer.
1. J.K. Barrett and P. Bowcock, arXiv:hep-th/0402163.
2. J.K. Barrett and P. Bowcock, arXiv:hep-th/0512211.
3. N.S. Manton and T.M. Samols, Phys. Lett. B 215, 559
(1988).
4. J.A. Harvey, arXiv:hep-th/9603086.
5. P.M. Sutcliffe, Int. J. Mod. Phys. A 12, 4663 (1997).
6. C.V. Johnson, D-Branes (Cambridge Univ. Press, Cam-
bridge, 2003).
7. J. Polchinski, String Theory. Vol. 2: Superstring Theory
and Beyond (Cambridge Univ. Press, Cambridge, 1998).
8. J. Polchinski, String Theory. Vol. 1: An Introduction to
the Bosonic String (Cambridge Univ. Press, Cambridge,
1998).
9. C.G. Callan and J.M. Maldacena, Nucl. Phys. B 513,
198 (1998).
10. N.R. Constable, R.C. Myers, and O. Tafjord, Phys. Rev.
D 61, 106009 (2000).
11. N.R. Constable, R.C. Myers, and O. Tafjord, JHEP
0106, 023 (2001).
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Rev. D 26, 854 (1982).
13. N.S. Manton, Phys. Lett. B 110, 54 (1982).
Received 28.05.09
ЗАСТОСУВАННЯ ТЕОРIЇ СТРУН ДЛЯ ОПИСУ
РОЗСIЯННЯ МОНОПОЛIВ
Дж.К. Барретт
Р е з ю м е
Пояснено, як можна описати розсiяння магнiтних монополiв,
використовуючи D-брани в теорiї струн. Знайдено енергiю, що
випромiнюється при розсiюваннi монополiв.
480 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
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