We present an explicit classical dyon solution for the noncommutative version
of the Yang-Mills-Higgs model (in the Prasad-Sommerfield limit) with a tehta
term. We show that the relation between classical electric and magnetic charges
also holds in noncommutative space. Extending the Noether approach to the case
of a noncommutative gauge theory, we analyze the effect of CP violation at the
quantum level, induced both by the theta term and by noncommutativity and we
prove that the Witten effect formula for the dyon charge remains the same as in
ordinary space.Comment: 17 page

Cieri, L.

Schaposnik, F. A.

English

arXiv

We construct a dyon solution for the noncommutative version of the Yang-Mills-Higgs model with a ϑ-term. Extending the Noether method to the case of a noncommutative gauge theory, we analyze the effect of CP violation induced both by the ϑ- term and by noncommutativity proving that theWitten effect formula for the dyon charge remains the same as in ordinary space.Facultad de Ciencias Exacta

Shaposnik, Fidel

El Servicio de Difusión de la Creación Intelectual

Hindawi Publishing Corporation
Research Letters in Physics
Volume 2008, Article ID 890916, 4 pages
doi:10.1155/2008/890916
Research Letter
The Dyon Charge in Noncommutative Gauge Theories
L. Cieri1, 2 and F. A. Schaposnik1, 2
1Departamento de Fı´sica, Universidad Nacional de La Plata, 1900 La Plata, Argentina
2Centro de Fisica y Matemitica de Amrica del Sur (CFIMAS), Avenida Santa Fe 1145, 1069 Buenos Aires, Argentina
Correspondence should be addressed to Fidel Schaposnik, fidel@fisica.unlp.edu.ar
Received 25 September 2007; Accepted 22 November 2007
Recommended by Jacobus Verbaarschot
We construct a dyon solution for the noncommutative version of the Yang-Mills-Higgs model with a ϑ-term. Extending the
Noether method to the case of a noncommutative gauge theory, we analyze the effect of CP violation induced both by the ϑ-
term and by noncommutativity proving that the Witten effect formula for the dyon charge remains the same as in ordinary space.
Copyright © 2008 L. Cieri and F. A. Schaposnik. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
Gauge theories coupled to Higgs scalars exhibit a remarkable
phenomenon called Witten effect [1], related to the ϑ-angle.
Indeed, if one adds to a Yang-Mills-Higgs Lagrangian a ϑ-
term
ΔL = ϑ e
2
32π2
εμναβtr
(
FμνFαβ
)
, (1)
which explicitly violates CP, the electric charge qe of a dyon
is modified. Instead of being quantized, as in the ϑ = 0 case,
one has, for ϑ=0,
qe =
(
n +
ϑ
2π
)
e , n ∈ Z. (2)
This result corresponds to a Julia-Zee dyon [2] with mag-
netic charge m = 4π/e. There are also arguments leading to
the conjecture that other CP violating interactions may also
induce a shift of the dyon charge [1].
CP violation can be induced not just by adding, as in (1),
new interactions to the Yang-Mills-Higgs Lagrangian but by
radically changing the setting of the theory. This is the case
of noncommutative gauge theories (NCGT) where the intro-
duction of noncommutation in space-time coordinates has
shown to affect the behavior under C, P, and T invariances
[3–6]. More specifically, one can prove that when noncom-
mutativity is restricted to space coordinates,
[
xi, xj
] = iθi j ,
[
xi, x0
] = 0, i, j = 1, 2, 3. (3)
As it is well known [4], NCGTs are not charge invariant.
Only if the usual field transformations are accompanied by
a change of sign in θi j , charge invariance is recovered. Hence,
if one takes θi j as a fixed parameter, CP is violated, although
CPT invariance is maintained since parity invariance is not
affected by the introduction of θ and time reversal undergoes
a change that compensates that in C.
It is then natural to pose the question whether the dyon
charge in noncommutative gauge theories receives a contri-
bution from a CP violating effect induced by noncommuta-
tivity even if the ϑ angle vanishes. Moreover, one could also
ask how the addition of the noncommutative version of the
term (1) modifies Q when both θi j =0 and ϑ=0.
We will analyze these questions in the present letter and,
to this end, we first discuss the properties of the dyon in a
noncommutative Yang-Mills-Higgs model with U(2) gauge
symmetry, calculate its charge at the quantum level and also
extend the theory in order to include a ϑ-term. Some of these
issues were briefly discussed in [7] starting from a monopole
solution obtained generalizing Nahm’s equations that de-
scribes BPS solitons [8] (some aspects of dyon solutions were
also considered in [9]). Here, instead, we will extend the
more explicit U(2) monopole solution found in [10, 11] to
the case of a dyon and then establish a noncommutative ver-
sion of the Noether theorem. We then discuss the issue of the
Witten effect in noncommutative space.
We start from the noncommutative U(2) Yang-Mills-
Higgs system
SNCYM = tr
∫
d4x
(
− 1
2
Fμν∗Fμν +DμΦ∗DμΦ
)
. (4)
2 Research Letters in Physics
Gauge fields Aμ = AAμ tA take values in the Lie algebra ofU(2)
with generators tA,A = 0, 1, 2, 3 (t0 = I/2, ta = σa/2, a =
1, 2, 3). Φ = ΦAtA is the Higgs multiplet and we consider
the Prasad-Sommerfield limit [12] in which the symmetry
breaking potential vanishes. Covariant derivatives and field
strength are given by
DμΦ = ∂μΦ− ie[Aμ,Φ]∗,
Fμν = ∂μAν − ∂νAμ − ie[Aμ,Aν]∗,
(5)
where the Moyal product and commutator are defined as
A(x)∗B(x) = exp
(
i
2
θμν∂
μ
x∂
ν
y
)
A(x)B(y)
∣
∣
∣∣
y=x
,
[
A(x),B(x)
]
∗ = A(x)∗B(x)− B(x)∗A(x).
(6)
As in (3) we will take θ0i = 0. This ensures a well-defined
Hamiltonian and unitarity at the quantum level.
To find noncommutative static dyon solutions we will
minimize the energy
E = tr
∫
d3x
(
Ei∗Ei + Bi∗Bi +DiΦ∗DiΦ +D0Φ∗D0Φ
)
,
Ei = −F0i , Bi = −12 εi jkF
jk.
(7)
Since we are working in the BPS limit, the vacuum expecta-
tion value enters as a boundary condition on the Higgs field,
trΦ2vac = v20/2. As in ordinary space, (7) can be written in the
form
E = tr
∫
d3x
((
Ei − sinαDiΦ
)∗(Ei − sinαDiΦ
)
+
(
Bi − cosαDiΦ
)∗(Bi − cosαDiΦ
)
+D0Φ∗D0Φ + 2 sinαEi∗DiΦ
+ 2 cosαBi∗DiΦ
)
,
(8)
leading to a Bogomol’nyi bound on the energy
E ≥ v0 sinαQ + v0 cosαM (9)
with Q and M are the electric and magnetic charges defined
as
Q = 2
v0
tr
∫
d3xEi∗DiΦ, M = 2
v0
tr
∫
d3xBi∗DiΦ.
(10)
The bound is saturated whenever the following BPS equa-
tions hold:
Ei = sinαDiΦ, (11)
Bi = cosαDiΦ, (12)
D0Φ = 0. (13)
In order to find an explicit dyon solution, we will now
consider an expansion of fields Aμ and Φ in powers of
the noncommutative parameter in θ, thus extending to the
dyon case the approach developed in [10] and [11] where
a purely magnetically charged solution was considered. One
starts from the exact Prasad-Sommerfield solution in ordi-
nary space [12] as giving the zeroth order of an expansion
in powers of θ for the monopole in noncommutative space.
Plugging this expansion into the BPS equations, one obtains
the noncommutative solution order by order in θ.
We take as zeroth-order approximation for the SU(2)
components the Prasad-Sommerfield dyon solution [12],
which we call Φa(0), Aa(0)i , and A
a(0)
0 . Concerning the U(1)
components, absent in the original Julia-Zee dyon, we pro-
pose for simplicity Φ0(0) =A0(0)i =A0(0)0 = 0. To first order in
θ, an ansatz for the gauge potential and Higgs field compo-
nents on U(1), that obey covariance under the SO(3) rota-
tion corresponding to the diagonal subgroup of SO(3)gauge ×
SO(3)space, is
A˜0(1)i = θi jx jA(r) + εi jkθ jkC(r) + xiε jklθ jkxlD(r),
A˜0(1)0 = θi jεi jkxkK(r),
Φ˜
0(1) = θi jεi jkxkB(r),
(14)
where A(r),B(r),C(r),D(r) , and K(r) are radial functions
to be determined. The component on SU(2) to first order
in θ of the Bogomol’nyi equation is not going to be analyze,
because it is pure gauge.
One can easily see that Bogomol’nyi (11) and (13) imply,
to all orders in θ, the relation
A0 = sinαΦ. (15)
We have then proven, using a θ-expansion, a relation that was
proposed as an ansatz in [7] within Nahm’s approach to the
construction of monopole solutions. Now, (15) implies that
the BPS (11) is automatically satisfied and then the study of
the BPS system (11)-(12) is reduced to the analysis of the sole
equation
1
2
εi jkFjk = − cosαDiΦ (16)
which, except for the factor cosα, is nothing but the pure
noncommutative monopole equation. Then, after making
the appropriate rescaling, we have from [10, 11]
A˜(n)i
(
xi
) = cos2n+1αA(n)i
(
cosαxi
)
,
Φ˜
(
xi
)
(n) = cos2nαΦ(n)( cosαxi), n = 1, 2, . . . ,
(17)
where A(n)i and Φ
(n) are the pure monopole solutions and
A˜(n)0 (x
i) is related to them through (15).
One can easily determine the asymptotic behavior of the
proposed solution and then compute the electric and mag-
netic charges of the dyon solution. One finds that the n > 0
terms do not contribute at the surface at infinity so that one
just recovers the n = 0 result, one finds
M = 4π
e
, Q =M tanα. (18)
L. Cieri and F. A. Schaposnik 3
So that one can conclude that n > 0 orders in θ do not con-
tribute to the charges that thus coincide with those in ordi-
nary space. That is, CP violation induced by noncommuta-
tivity does not change, at least at the classical level, the charge
of the dyon, no trace of θi j , appears in Q.
In order to analyze charge quantization at the quantum
level, we use the Noether approach and canonically proceed
as originally done in [1] regarding the unbroken symmetry
which leaves the Higgs vacuum invariant and is associated to
the electric charge. To do this, we will not just consider action
(4) but the one including the noncommutative version of a
ϑ-term,
Sϑ = SNCYM + e
2ϑ
16π2
∫
d4xFμν∗F˜μν. (19)
In this way, we will be able to test possible modifications of
the dyon charge because both the noncommutativity and the
ϑ-term. As in ordinary space, the noncommutative version
of the ϑ-term can be written as a surface term [13] and hence
the equations of motion for action (19) and its dyon solu-
tions remain the same for all values of ϑ.
After symmetry breaking through the condition Φ2vac =
v20/2, the unbroken symmetry is related to rotations Λ in the
direction of Φ leaving the Higgs vacuum invariant. Let us
consider such gauge transformations along the Higgs field
direction
Λ(x) = 1
v0
Φata(x) , (∞) = 1,
δΛΦ = 0, δΛAμ = 1
ev0
DμΦ.
(20)
As first noticed for a scalar theory in [14], in order to apply
the Noether method in the noncommutative case, one has to
take into account ∗-commutators that, once integrated, give
a vanishing contribution. That is, from the well-honnored
Noether formula
δΛ(x)S =
∫
d4x∂μJ
μ
[
Φ(x),Aμ(x)
]
(x) (21)
one can at most infer that
δΛ(x)S = 0 =⇒ (∂μJμ) = tr
(
[O,P]∗ + [B,C]∗∗B + · · ·
)
(22)
for some proper functionals O,P,B,C, . . . since, once inte-
grated over space-time, the right-hand side in (22) vanishes
due to the ∗-product cyclic properties under integration.
Also when integrating (22) over 3 spaces to find the con-
served charge commutators vanish since, as explained above,
θ0i = 0. With all this, we finally have for the conserved
charge, which we call N ,
N = −1
e
Q +
ϑe
8π2
M. (23)
At large distances, exp (2πiN) should implement a 2π rota-
tion about the direction of Φ and hence the identity. Then
the eigenvalues of N have to be quantized in integer units n.
If we call qe and qg , the eigenvalues of the electric and mag-
netic charge operators, one then has
qe =
(
ne +
ϑe2
8π2
qg
)
. (24)
That is, we have obtained for the nonconmmutative dyon the
same formula that holds for the case of ordinary space, (2)
(qg = 4π/e).
In summary, we have constructed an explicit noncom-
mutative dyon solution showing that the relation (18) be-
tween classical electric and magnetic charge also holds
in noncommutative space. Moreover, after extending the
Noether approach to the case of a noncommutative gauge
theory, we have proven that CP violation introduced by the
commutation rule (3) does not change the Witten effect for-
mula; indeed, the dyon’s charge shift is θi j-independent. In
this respect, it should be interesting to consider other type of
noncommutativity and in particular, to investigate the case
of the dyon in the fuzzy sphere along the lines developed in
[15] where monopole solutions were constructed for the case
in which θi j = θrεi jkxk since in that case the coordinate de-
pendence of θi j may introduce definite changes in (24). We
hope to discuss this issue in the future.
ACKNOWLEDGMENTS
F. A. Schaposnik is associated with CICBA. The authors are
grateful to acknowledge G. Giribet and G. Lozano for in-
teresting discussions and comments. This work is partially
supported by CONICET (PIP6160), ANPCyT (PICT 20204),
UNLP, and CICBA grants.
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The Dyon Charge in Noncommutative Gauge Theories

We construct a dyon solution for the noncommutative version of the Yang-Mills-Higgs model with a ϑ-term. Extending the Noether method to the case of a noncommutative gauge theory, we analyze the effect of CP violation induced both by the ϑ- term and by noncommutativity proving that theWitten effect formula for the dyon charge remains the same as in ordinary space

Schaposnik, Fidel Arturo

Centro de Servicios en Gestión de Información

Hindawi Publishing CorporationResearch Letters in PhysicsVolume 2008, Article ID 890916, 4 pagesdoi:10.1155/2008/890916Research LetterThe Dyon Charge in Noncommutative Gauge TheoriesL. Cieri1, 2 and F. A. Schaposnik1, 21Departamento de Fı´sica, Universidad Nacional de La Plata, 1900 La Plata, Argentina2Centro de Fisica y Matemitica de Amrica del Sur (CFIMAS), Avenida Santa Fe 1145, 1069 Buenos Aires, ArgentinaCorrespondence should be addressed to Fidel Schaposnik, fidel@fisica.unlp.edu.arReceived 25 September 2007; Accepted 22 November 2007Recommended by Jacobus VerbaarschotWe construct a dyon solution for the noncommutative version of the Yang-Mills-Higgs model with a ϑ-term. Extending theNoether method to the case of a noncommutative gauge theory, we analyze the eﬀect of CP violation induced both by the ϑ-term and by noncommutativity proving that the Witten eﬀect formula for the dyon charge remains the same as in ordinary space.Copyright © 2008 L. Cieri and F. A. Schaposnik. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.Gauge theories coupled to Higgs scalars exhibit a remarkablephenomenon called Witten eﬀect [1], related to the ϑ-angle.Indeed, if one adds to a Yang-Mills-Higgs Lagrangian a ϑ-termΔL = ϑ e232π2εμναβtr(FμνFαβ), (1)which explicitly violates CP, the electric charge qe of a dyonis modified. Instead of being quantized, as in the ϑ = 0 case,one has, for ϑ=0,qe =(n +ϑ2π)e , n ∈ Z. (2)This result corresponds to a Julia-Zee dyon [2] with mag-netic charge m = 4π/e. There are also arguments leading tothe conjecture that other CP violating interactions may alsoinduce a shift of the dyon charge [1].CP violation can be induced not just by adding, as in (1),new interactions to the Yang-Mills-Higgs Lagrangian but byradically changing the setting of the theory. This is the caseof noncommutative gauge theories (NCGT) where the intro-duction of noncommutation in space-time coordinates hasshown to aﬀect the behavior under C, P, and T invariances[3–6]. More specifically, one can prove that when noncom-mutativity is restricted to space coordinates,[xi, xj] = iθi j ,[xi, x0] = 0, i, j = 1, 2, 3. (3)As it is well known [4], NCGTs are not charge invariant.Only if the usual field transformations are accompanied bya change of sign in θi j , charge invariance is recovered. Hence,if one takes θi j as a fixed parameter, CP is violated, althoughCPT invariance is maintained since parity invariance is notaﬀected by the introduction of θ and time reversal undergoesa change that compensates that in C.It is then natural to pose the question whether the dyoncharge in noncommutative gauge theories receives a contri-bution from a CP violating eﬀect induced by noncommuta-tivity even if the ϑ angle vanishes. Moreover, one could alsoask how the addition of the noncommutative version of theterm (1) modifies Q when both θi j =0 and ϑ=0.We will analyze these questions in the present letter and,to this end, we first discuss the properties of the dyon in anoncommutative Yang-Mills-Higgs model with U(2) gaugesymmetry, calculate its charge at the quantum level and alsoextend the theory in order to include a ϑ-term. Some of theseissues were briefly discussed in [7] starting from a monopolesolution obtained generalizing Nahm’s equations that de-scribes BPS solitons [8] (some aspects of dyon solutions werealso considered in [9]). Here, instead, we will extend themore explicit U(2) monopole solution found in [10, 11] tothe case of a dyon and then establish a noncommutative ver-sion of the Noether theorem. We then discuss the issue of theWitten eﬀect in noncommutative space.We start from the noncommutative U(2) Yang-Mills-Higgs systemSNCYM = tr∫d4x(− 12Fμν∗Fμν +DμΦ∗DμΦ). (4)2 Research Letters in PhysicsGauge fields Aμ = AAμ tA take values in the Lie algebra ofU(2)with generators tA,A = 0, 1, 2, 3 (t0 = I/2, ta = σa/2, a =1, 2, 3). Φ = ΦAtA is the Higgs multiplet and we considerthe Prasad-Sommerfield limit [12] in which the symmetrybreaking potential vanishes. Covariant derivatives and fieldstrength are given byDμΦ = ∂μΦ− ie[Aμ,Φ]∗,Fμν = ∂μAν − ∂νAμ − ie[Aμ,Aν]∗,(5)where the Moyal product and commutator are defined asA(x)∗B(x) = exp(i2θμν∂μx∂νy)A(x)B(y)∣∣∣∣y=x,[A(x),B(x)]∗ = A(x)∗B(x)− B(x)∗A(x).(6)As in (3) we will take θ0i = 0. This ensures a well-definedHamiltonian and unitarity at the quantum level.To find noncommutative static dyon solutions we willminimize the energyE = tr∫d3x(Ei∗Ei + Bi∗Bi +DiΦ∗DiΦ +D0Φ∗D0Φ),Ei = −F0i , Bi = −12 εi jkFjk.(7)Since we are working in the BPS limit, the vacuum expecta-tion value enters as a boundary condition on the Higgs field,trΦ2vac = v20/2. As in ordinary space, (7) can be written in theformE = tr∫d3x((Ei − sinαDiΦ)∗(Ei − sinαDiΦ)+(Bi − cosαDiΦ)∗(Bi − cosαDiΦ)+D0Φ∗D0Φ + 2 sinαEi∗DiΦ+ 2 cosαBi∗DiΦ),(8)leading to a Bogomol’nyi bound on the energyE ≥ v0 sinαQ + v0 cosαM (9)with Q and M are the electric and magnetic charges definedasQ = 2v0tr∫d3xEi∗DiΦ, M = 2v0tr∫d3xBi∗DiΦ.(10)The bound is saturated whenever the following BPS equa-tions hold:Ei = sinαDiΦ, (11)Bi = cosαDiΦ, (12)D0Φ = 0. (13)In order to find an explicit dyon solution, we will nowconsider an expansion of fields Aμ and Φ in powers ofthe noncommutative parameter in θ, thus extending to thedyon case the approach developed in [10] and [11] wherea purely magnetically charged solution was considered. Onestarts from the exact Prasad-Sommerfield solution in ordi-nary space [12] as giving the zeroth order of an expansionin powers of θ for the monopole in noncommutative space.Plugging this expansion into the BPS equations, one obtainsthe noncommutative solution order by order in θ.We take as zeroth-order approximation for the SU(2)components the Prasad-Sommerfield dyon solution [12],which we call Φa(0), Aa(0)i , and Aa(0)0 . Concerning the U(1)components, absent in the original Julia-Zee dyon, we pro-pose for simplicity Φ0(0) =A0(0)i =A0(0)0 = 0. To first order inθ, an ansatz for the gauge potential and Higgs field compo-nents on U(1), that obey covariance under the SO(3) rota-tion corresponding to the diagonal subgroup of SO(3)gauge ×SO(3)space, isA˜0(1)i = θi jx jA(r) + εi jkθ jkC(r) + xiε jklθ jkxlD(r),A˜0(1)0 = θi jεi jkxkK(r),Φ˜0(1) = θi jεi jkxkB(r),(14)where A(r),B(r),C(r),D(r) , and K(r) are radial functionsto be determined. The component on SU(2) to first orderin θ of the Bogomol’nyi equation is not going to be analyze,because it is pure gauge.One can easily see that Bogomol’nyi (11) and (13) imply,to all orders in θ, the relationA0 = sinαΦ. (15)We have then proven, using a θ-expansion, a relation that wasproposed as an ansatz in [7] within Nahm’s approach to theconstruction of monopole solutions. Now, (15) implies thatthe BPS (11) is automatically satisfied and then the study ofthe BPS system (11)-(12) is reduced to the analysis of the soleequation12εi jkFjk = − cosαDiΦ (16)which, except for the factor cosα, is nothing but the purenoncommutative monopole equation. Then, after makingthe appropriate rescaling, we have from [10, 11]A˜(n)i(xi) = cos2n+1αA(n)i(cosαxi),Φ˜(xi)(n) = cos2nαΦ(n)( cosαxi), n = 1, 2, . . . ,(17)where A(n)i and Φ(n) are the pure monopole solutions andA˜(n)0 (xi) is related to them through (15).One can easily determine the asymptotic behavior of theproposed solution and then compute the electric and mag-netic charges of the dyon solution. One finds that the n > 0terms do not contribute at the surface at infinity so that onejust recovers the n = 0 result, one findsM = 4πe, Q =M tanα. (18)L. Cieri and F. A. Schaposnik 3So that one can conclude that n > 0 orders in θ do not con-tribute to the charges that thus coincide with those in ordi-nary space. That is, CP violation induced by noncommuta-tivity does not change, at least at the classical level, the chargeof the dyon, no trace of θi j , appears in Q.In order to analyze charge quantization at the quantumlevel, we use the Noether approach and canonically proceedas originally done in [1] regarding the unbroken symmetrywhich leaves the Higgs vacuum invariant and is associated tothe electric charge. To do this, we will not just consider action(4) but the one including the noncommutative version of aϑ-term,Sϑ = SNCYM + e2ϑ16π2∫d4xFμν∗F˜μν. (19)In this way, we will be able to test possible modifications ofthe dyon charge because both the noncommutativity and theϑ-term. As in ordinary space, the noncommutative versionof the ϑ-term can be written as a surface term [13] and hencethe equations of motion for action (19) and its dyon solu-tions remain the same for all values of ϑ.After symmetry breaking through the condition Φ2vac =v20/2, the unbroken symmetry is related to rotations Λ in thedirection of Φ leaving the Higgs vacuum invariant. Let usconsider such gauge transformations along the Higgs fielddirectionΛ(x) = 1v0Φata(x) , (∞) = 1,δΛΦ = 0, δΛAμ = 1ev0DμΦ.(20)As first noticed for a scalar theory in [14], in order to applythe Noether method in the noncommutative case, one has totake into account ∗-commutators that, once integrated, givea vanishing contribution. That is, from the well-honnoredNoether formulaδΛ(x)S =∫d4x∂μJμ[Φ(x),Aμ(x)](x) (21)one can at most infer thatδΛ(x)S = 0 =⇒ (∂μJμ) = tr([O,P]∗ + [B,C]∗∗B + · · ·)(22)for some proper functionals O,P,B,C, . . . since, once inte-grated over space-time, the right-hand side in (22) vanishesdue to the ∗-product cyclic properties under integration.Also when integrating (22) over 3 spaces to find the con-served charge commutators vanish since, as explained above,θ0i = 0. With all this, we finally have for the conservedcharge, which we call N ,N = −1eQ +ϑe8π2M. (23)At large distances, exp (2πiN) should implement a 2π rota-tion about the direction of Φ and hence the identity. Thenthe eigenvalues of N have to be quantized in integer units n.If we call qe and qg , the eigenvalues of the electric and mag-netic charge operators, one then hasqe =(ne +ϑe28π2qg). (24)That is, we have obtained for the nonconmmutative dyon thesame formula that holds for the case of ordinary space, (2)(qg = 4π/e).In summary, we have constructed an explicit noncom-mutative dyon solution showing that the relation (18) be-tween classical electric and magnetic charge also holdsin noncommutative space. Moreover, after extending theNoether approach to the case of a noncommutative gaugetheory, we have proven that CP violation introduced by thecommutation rule (3) does not change the Witten eﬀect for-mula; indeed, the dyon’s charge shift is θi j-independent. Inthis respect, it should be interesting to consider other type ofnoncommutativity and in particular, to investigate the caseof the dyon in the fuzzy sphere along the lines developed in[15] where monopole solutions were constructed for the casein which θi j = θrεi jkxk since in that case the coordinate de-pendence of θi j may introduce definite changes in (24). Wehope to discuss this issue in the future.ACKNOWLEDGMENTSF. A. Schaposnik is associated with CICBA. The authors aregrateful to acknowledge G. Giribet and G. Lozano for in-teresting discussions and comments. This work is partiallysupported by CONICET (PIP6160), ANPCyT (PICT 20204),UNLP, and CICBA grants.REFERENCES[1] E. Witten, “Dyons of charge eϑ/2Π,” Physics Letters B, vol. 86,no. 3-4, pp. 283–287, 1979.[2] B. Julia and A. Zee, “Poles with both magnetic and elec-tric charges in non-Abelian gauge theory,” Physical Review D,vol. 11, no. 8, pp. 2227–2232, 1975.[3] M. Hayakawa, “Perturbative analysis on infrared aspects ofnoncommutative QED on R4,” Physics Letters B, vol. 478,no. 1–3, pp. 394–400, 2000.[4] M. M. Sheikh-Jabbari, “C, P, and T invariance of noncommu-tative gauge theories,” Physical Review Letters, vol. 84, no. 23,pp. 5265–5268, 2000.[5] P. Aschieri, B. Jurcˇo, P. Schupp, and J. Wess, “Noncummu-tative GUTs, standard model and C,P,T ,” Nuclear Physics B,vol. 651, no. 3, pp. 45–70, 2003.[6] M. Chaichian, K. Nishijima, and A. Tureanu, “Spin-statisticsand CPT theorems in noncommutative field theory,” PhysicsLetters B, vol. 568, no. 1-2, pp. 146–152, 2003.[7] D. J. Gross and N. A. Nekrasov, “Dynamics of strings in non-commutative gauge theory,” Journal of High Energy Physics,vol. 10, no. 4, p. 021, 2000.[8] D. J. Gross and N. A. Nekrasov, “Monopoles and stringsin noncommutative gauge theory,” Journal of High EnergyPhysics, vol. 7, p. 034, 2000.[9] D. Bak and K. Lee, “Elongation of moving noncommutativesolitons,” Physics Letters B, vol. 495, no. 1-2, pp. 231–236, 2000.[10] K. Hashimoto, H. Hata, and S. Moriyama, “Brane configu-ration from monopole solution in non-commutative super4 Research Letters in PhysicsYang-Mills theory,” Journal of High Energy Physics, vol. 12,no. 3, p. 021, 1999.[11] S. Goto and H. Hata, “Noncommutative monopole at the sec-ond order in ϑ,” Physical Review D, vol. 62, no. 8, ArticleID 085022, 10 pages, 2000.[12] M. K. Prasad and C. M. Sommerfield, “Exact classical solutionfor the ’t hooft monopole and the Julia-Zee Dyon,” PhysicalReview Letters, vol. 35, no. 12, pp. 760–762, 1975.[13] K. Furuuchi, “Equivalence of projections as gauge equivalenceon noncommutative space,” Communications in MathematicalPhysics, vol. 217, no. 3, pp. 579–593, 2001.[14] A.Micu andM.M. Sheikh-Jabbari, “NoncommutativeΦ4 the-ory at two loops,” Journal of High Energy Physics, vol. 5, no. 1,p. 025, 2001.[15] E. F. Moreno, “Spherically symmetric monopoles in noncom-mutative space,” Physical Review D, vol. 72, no. 4, ArticleID 045001, 9 pages, 2005.Submit your manuscripts athttp://www.hindawi.comHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014High Energy PhysicsAdvances inThe Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014FluidsJournal of Atomic and  Molecular PhysicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Advances in  Condensed Matter PhysicsOpticsInternational Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014AstronomyAdvances inInternational Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014SuperconductivityHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Statistical MechanicsInternational Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014GravityJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014AstrophysicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Physics Research InternationalHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Solid State PhysicsJournal of Computational  Methods in PhysicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Soft MatterJournal ofHindawi Publishing Corporationhttp://www.hindawi.comAerodynamicsJournal ofVolume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014PhotonicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Journal ofBiophysicsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014ThermodynamicsJournal of

L. Cieri

F. A. Schaposnik

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Research Letters in Physics
Volume 2008, Article ID 890916, 4 pages
doi:10.1155/2008/890916
Research Letter
The Dyon Charge in Noncommutative Gauge Theories
L. Cieri1, 2 and F. A. Schaposnik1, 2
1Departamento de Fı´sica, Universidad Nacional de La Plata, 1900 La Plata, Argentina
2Centro de Fisica y Matemitica de Amrica del Sur (CFIMAS), Avenida Santa Fe 1145, 1069 Buenos Aires, Argentina
Correspondence should be addressed to Fidel Schaposnik, fidel@fisica.unlp.edu.ar
Received 25 September 2007; Accepted 22 November 2007
Recommended by Jacobus Verbaarschot
We construct a dyon solution for the noncommutative version of the Yang-Mills-Higgs model with a ϑ-term. Extending the
Noether method to the case of a noncommutative gauge theory, we analyze the eﬀect of CP violation induced both by the ϑ-
term and by noncommutativity proving that the Witten eﬀect formula for the dyon charge remains the same as in ordinary space.
Copyright © 2008 L. Cieri and F. A. Schaposnik. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
Gauge theories coupled to Higgs scalars exhibit a remarkable
phenomenon called Witten eﬀect [1], related to the ϑ-angle.
Indeed, if one adds to a Yang-Mills-Higgs Lagrangian a ϑ-
term
ΔL = ϑ e
2
32π2
εμναβtr
(
FμνFαβ
)
, (1)
which explicitly violates CP, the electric charge qe of a dyon
is modified. Instead of being quantized, as in the ϑ = 0 case,
one has, for ϑ=0,
qe =
(
n +
ϑ
2π
)
e , n ∈ Z. (2)
This result corresponds to a Julia-Zee dyon [2] with mag-
netic charge m = 4π/e. There are also arguments leading to
the conjecture that other CP violating interactions may also
induce a shift of the dyon charge [1].
CP violation can be induced not just by adding, as in (1),
new interactions to the Yang-Mills-Higgs Lagrangian but by
radically changing the setting of the theory. This is the case
of noncommutative gauge theories (NCGT) where the intro-
duction of noncommutation in space-time coordinates has
shown to aﬀect the behavior under C, P, and T invariances
[3–6]. More specifically, one can prove that when noncom-
mutativity is restricted to space coordinates,
[
xi, xj
] = iθi j ,
[
xi, x0
] = 0, i, j = 1, 2, 3. (3)
As it is well known [4], NCGTs are not charge invariant.
Only if the usual field transformations are accompanied by
a change of sign in θi j , charge invariance is recovered. Hence,
if one takes θi j as a fixed parameter, CP is violated, although
CPT invariance is maintained since parity invariance is not
aﬀected by the introduction of θ and time reversal undergoes
a change that compensates that in C.
It is then natural to pose the question whether the dyon
charge in noncommutative gauge theories receives a contri-
bution from a CP violating eﬀect induced by noncommuta-
tivity even if the ϑ angle vanishes. Moreover, one could also
ask how the addition of the noncommutative version of the
term (1) modifies Q when both θi j =0 and ϑ=0.
We will analyze these questions in the present letter and,
to this end, we first discuss the properties of the dyon in a
noncommutative Yang-Mills-Higgs model with U(2) gauge
symmetry, calculate its charge at the quantum level and also
extend the theory in order to include a ϑ-term. Some of these
issues were briefly discussed in [7] starting from a monopole
solution obtained generalizing Nahm’s equations that de-
scribes BPS solitons [8] (some aspects of dyon solutions were
also considered in [9]). Here, instead, we will extend the
more explicit U(2) monopole solution found in [10, 11] to
the case of a dyon and then establish a noncommutative ver-
sion of the Noether theorem. We then discuss the issue of the
Witten eﬀect in noncommutative space.
We start from the noncommutative U(2) Yang-Mills-
Higgs system
SNCYM = tr
∫
d4x
(
− 1
2
Fμν∗Fμν +DμΦ∗DμΦ
)
. (4)
2 Research Letters in Physics
Gauge fields Aμ = AAμ tA take values in the Lie algebra ofU(2)
with generators tA,A = 0, 1, 2, 3 (t0 = I/2, ta = σa/2, a =
1, 2, 3). Φ = ΦAtA is the Higgs multiplet and we consider
the Prasad-Sommerfield limit [12] in which the symmetry
breaking potential vanishes. Covariant derivatives and field
strength are given by
DμΦ = ∂μΦ− ie[Aμ,Φ]∗,
Fμν = ∂μAν − ∂νAμ − ie[Aμ,Aν]∗,
(5)
where the Moyal product and commutator are defined as
A(x)∗B(x) = exp
(
i
2
θμν∂
μ
x∂
ν
y
)
A(x)B(y)
∣
∣
∣∣
y=x
,
[
A(x),B(x)
]
∗ = A(x)∗B(x)− B(x)∗A(x).
(6)
As in (3) we will take θ0i = 0. This ensures a well-defined
Hamiltonian and unitarity at the quantum level.
To find noncommutative static dyon solutions we will
minimize the energy
E = tr
∫
d3x
(
Ei∗Ei + Bi∗Bi +DiΦ∗DiΦ +D0Φ∗D0Φ
)
,
Ei = −F0i , Bi = −12 εi jkF
jk.
(7)
Since we are working in the BPS limit, the vacuum expecta-
tion value enters as a boundary condition on the Higgs field,
trΦ2vac = v20/2. As in ordinary space, (7) can be written in the
form
E = tr
∫
d3x
((
Ei − sinαDiΦ
)∗(Ei − sinαDiΦ
)
+
(
Bi − cosαDiΦ
)∗(Bi − cosαDiΦ
)
+D0Φ∗D0Φ + 2 sinαEi∗DiΦ
+ 2 cosαBi∗DiΦ
)
,
(8)
leading to a Bogomol’nyi bound on the energy
E ≥ v0 sinαQ + v0 cosαM (9)
with Q and M are the electric and magnetic charges defined
as
Q = 2
v0
tr
∫
d3xEi∗DiΦ, M = 2
v0
tr
∫
d3xBi∗DiΦ.
(10)
The bound is saturated whenever the following BPS equa-
tions hold:
Ei = sinαDiΦ, (11)
Bi = cosαDiΦ, (12)
D0Φ = 0. (13)
In order to find an explicit dyon solution, we will now
consider an expansion of fields Aμ and Φ in powers of
the noncommutative parameter in θ, thus extending to the
dyon case the approach developed in [10] and [11] where
a purely magnetically charged solution was considered. One
starts from the exact Prasad-Sommerfield solution in ordi-
nary space [12] as giving the zeroth order of an expansion
in powers of θ for the monopole in noncommutative space.
Plugging this expansion into the BPS equations, one obtains
the noncommutative solution order by order in θ.
We take as zeroth-order approximation for the SU(2)
components the Prasad-Sommerfield dyon solution [12],
which we call Φa(0), Aa(0)i , and A
a(0)
0 . Concerning the U(1)
components, absent in the original Julia-Zee dyon, we pro-
pose for simplicity Φ0(0) =A0(0)i =A0(0)0 = 0. To first order in
θ, an ansatz for the gauge potential and Higgs field compo-
nents on U(1), that obey covariance under the SO(3) rota-
tion corresponding to the diagonal subgroup of SO(3)gauge ×
SO(3)space, is
A˜0(1)i = θi jx jA(r) + εi jkθ jkC(r) + xiε jklθ jkxlD(r),
A˜0(1)0 = θi jεi jkxkK(r),
Φ˜
0(1) = θi jεi jkxkB(r),
(14)
where A(r),B(r),C(r),D(r) , and K(r) are radial functions
to be determined. The component on SU(2) to first order
in θ of the Bogomol’nyi equation is not going to be analyze,
because it is pure gauge.
One can easily see that Bogomol’nyi (11) and (13) imply,
to all orders in θ, the relation
A0 = sinαΦ. (15)
We have then proven, using a θ-expansion, a relation that was
proposed as an ansatz in [7] within Nahm’s approach to the
construction of monopole solutions. Now, (15) implies that
the BPS (11) is automatically satisfied and then the study of
the BPS system (11)-(12) is reduced to the analysis of the sole
equation
1
2
εi jkFjk = − cosαDiΦ (16)
which, except for the factor cosα, is nothing but the pure
noncommutative monopole equation. Then, after making
the appropriate rescaling, we have from [10, 11]
A˜(n)i
(
xi
) = cos2n+1αA(n)i
(
cosαxi
)
,
Φ˜
(
xi
)
(n) = cos2nαΦ(n)( cosαxi), n = 1, 2, . . . ,
(17)
where A(n)i and Φ
(n) are the pure monopole solutions and
A˜(n)0 (x
i) is related to them through (15).
One can easily determine the asymptotic behavior of the
proposed solution and then compute the electric and mag-
netic charges of the dyon solution. One finds that the n > 0
terms do not contribute at the surface at infinity so that one
just recovers the n = 0 result, one finds
M = 4π
e
, Q =M tanα. (18)
L. Cieri and F. A. Schaposnik 3
So that one can conclude that n > 0 orders in θ do not con-
tribute to the charges that thus coincide with those in ordi-
nary space. That is, CP violation induced by noncommuta-
tivity does not change, at least at the classical level, the charge
of the dyon, no trace of θi j , appears in Q.
In order to analyze charge quantization at the quantum
level, we use the Noether approach and canonically proceed
as originally done in [1] regarding the unbroken symmetry
which leaves the Higgs vacuum invariant and is associated to
the electric charge. To do this, we will not just consider action
(4) but the one including the noncommutative version of a
ϑ-term,
Sϑ = SNCYM + e
2ϑ
16π2
∫
d4xFμν∗F˜μν. (19)
In this way, we will be able to test possible modifications of
the dyon charge because both the noncommutativity and the
ϑ-term. As in ordinary space, the noncommutative version
of the ϑ-term can be written as a surface term [13] and hence
the equations of motion for action (19) and its dyon solu-
tions remain the same for all values of ϑ.
After symmetry breaking through the condition Φ2vac =
v20/2, the unbroken symmetry is related to rotations Λ in the
direction of Φ leaving the Higgs vacuum invariant. Let us
consider such gauge transformations along the Higgs field
direction
Λ(x) = 1
v0
Φata(x) , (∞) = 1,
δΛΦ = 0, δΛAμ = 1
ev0
DμΦ.
(20)
As first noticed for a scalar theory in [14], in order to apply
the Noether method in the noncommutative case, one has to
take into account ∗-commutators that, once integrated, give
a vanishing contribution. That is, from the well-honnored
Noether formula
δΛ(x)S =
∫
d4x∂μJ
μ
[
Φ(x),Aμ(x)
]
(x) (21)
one can at most infer that
δΛ(x)S = 0 =⇒ (∂μJμ) = tr
(
[O,P]∗ + [B,C]∗∗B + · · ·
)
(22)
for some proper functionals O,P,B,C, . . . since, once inte-
grated over space-time, the right-hand side in (22) vanishes
due to the ∗-product cyclic properties under integration.
Also when integrating (22) over 3 spaces to find the con-
served charge commutators vanish since, as explained above,
θ0i = 0. With all this, we finally have for the conserved
charge, which we call N ,
N = −1
e
Q +
ϑe
8π2
M. (23)
At large distances, exp (2πiN) should implement a 2π rota-
tion about the direction of Φ and hence the identity. Then
the eigenvalues of N have to be quantized in integer units n.
If we call qe and qg , the eigenvalues of the electric and mag-
netic charge operators, one then has
qe =
(
ne +
ϑe2
8π2
qg
)
. (24)
That is, we have obtained for the nonconmmutative dyon the
same formula that holds for the case of ordinary space, (2)
(qg = 4π/e).
In summary, we have constructed an explicit noncom-
mutative dyon solution showing that the relation (18) be-
tween classical electric and magnetic charge also holds
in noncommutative space. Moreover, after extending the
Noether approach to the case of a noncommutative gauge
theory, we have proven that CP violation introduced by the
commutation rule (3) does not change the Witten eﬀect for-
mula; indeed, the dyon’s charge shift is θi j-independent. In
this respect, it should be interesting to consider other type of
noncommutativity and in particular, to investigate the case
of the dyon in the fuzzy sphere along the lines developed in
[15] where monopole solutions were constructed for the case
in which θi j = θrεi jkxk since in that case the coordinate de-
pendence of θi j may introduce definite changes in (24). We
hope to discuss this issue in the future.
ACKNOWLEDGMENTS
F. A. Schaposnik is associated with CICBA. The authors are
grateful to acknowledge G. Giribet and G. Lozano for in-
teresting discussions and comments. This work is partially
supported by CONICET (PIP6160), ANPCyT (PICT 20204),
UNLP, and CICBA grants.
REFERENCES
[1] E. Witten, “Dyons of charge eϑ/2Π,” Physics Letters B, vol. 86,
no. 3-4, pp. 283–287, 1979.
[2] B. Julia and A. Zee, “Poles with both magnetic and elec-
tric charges in non-Abelian gauge theory,” Physical Review D,
vol. 11, no. 8, pp. 2227–2232, 1975.
[3] M. Hayakawa, “Perturbative analysis on infrared aspects of
noncommutative QED on R4,” Physics Letters B, vol. 478,
no. 1–3, pp. 394–400, 2000.
[4] M. M. Sheikh-Jabbari, “C, P, and T invariance of noncommu-
tative gauge theories,” Physical Review Letters, vol. 84, no. 23,
pp. 5265–5268, 2000.
[5] P. Aschieri, B. Jurcˇo, P. Schupp, and J. Wess, “Noncummu-
tative GUTs, standard model and C,P,T ,” Nuclear Physics B,
vol. 651, no. 3, pp. 45–70, 2003.
[6] M. Chaichian, K. Nishijima, and A. Tureanu, “Spin-statistics
and CPT theorems in noncommutative field theory,” Physics
Letters B, vol. 568, no. 1-2, pp. 146–152, 2003.
[7] D. J. Gross and N. A. Nekrasov, “Dynamics of strings in non-
commutative gauge theory,” Journal of High Energy Physics,
vol. 10, no. 4, p. 021, 2000.
[8] D. J. Gross and N. A. Nekrasov, “Monopoles and strings
in noncommutative gauge theory,” Journal of High Energy
Physics, vol. 7, p. 034, 2000.
[9] D. Bak and K. Lee, “Elongation of moving noncommutative
solitons,” Physics Letters B, vol. 495, no. 1-2, pp. 231–236, 2000.
[10] K. Hashimoto, H. Hata, and S. Moriyama, “Brane configu-
ration from monopole solution in non-commutative super
4 Research Letters in Physics
Yang-Mills theory,” Journal of High Energy Physics, vol. 12,
no. 3, p. 021, 1999.
[11] S. Goto and H. Hata, “Noncommutative monopole at the sec-
ond order in ϑ,” Physical Review D, vol. 62, no. 8, Article
ID 085022, 10 pages, 2000.
[12] M. K. Prasad and C. M. Sommerfield, “Exact classical solution
for the ’t hooft monopole and the Julia-Zee Dyon,” Physical
Review Letters, vol. 35, no. 12, pp. 760–762, 1975.
[13] K. Furuuchi, “Equivalence of projections as gauge equivalence
on noncommutative space,” Communications in Mathematical
Physics, vol. 217, no. 3, pp. 579–593, 2001.
[14] A.Micu andM.M. Sheikh-Jabbari, “NoncommutativeΦ4 the-
ory at two loops,” Journal of High Energy Physics, vol. 5, no. 1,
p. 025, 2001.
[15] E. F. Moreno, “Spherically symmetric monopoles in noncom-
mutative space,” Physical Review D, vol. 72, no. 4, Article
ID 045001, 9 pages, 2005.
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We construct a dyon solution for the noncommutative version of the Yang-Mills-Higgs model with a θ-term. Extending the Noether method to the case of a noncommutative gauge theory, we analyze the effect of CP violation induced both by the θ- term and by noncommutativity proving that the Witten effect formula for the dyon charge remains the same as in ordinary space

Cieri, Leandro

Schaposnik, Fidel A.

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Hindawi Publishing CorporationResearch Letters in PhysicsVolume 2008, Article ID 890916, 4 pagesdoi:10.1155/2008/890916Research LetterThe Dyon Charge in Noncommutative Gauge TheoriesL. Cieri1, 2 and F. A. Schaposnik1, 21 Departamento de Fı́sica, Universidad Nacional de La Plata, 1900 La Plata, Argentina2 Centro de Fisica y Matemitica de Amrica del Sur (CFIMAS), Avenida Santa Fe 1145, 1069 Buenos Aires, ArgentinaCorrespondence should be addressed to Fidel Schaposnik, fidel@fisica.unlp.edu.arReceived 25 September 2007; Accepted 22 November 2007Recommended by Jacobus VerbaarschotWe construct a dyon solution for the noncommutative version of the Yang-Mills-Higgs model with a ϑ-term. Extending theNoether method to the case of a noncommutative gauge theory, we analyze the effect of CP violation induced both by the ϑ-term and by noncommutativity proving that the Witten effect formula for the dyon charge remains the same as in ordinary space.Copyright © 2008 L. Cieri and F. A. Schaposnik. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.Gauge theories coupled to Higgs scalars exhibit a remarkablephenomenon called Witten effect [1], related to the ϑ-angle.Indeed, if one adds to a Yang-Mills-Higgs Lagrangian a ϑ-termΔL = ϑ e232π2εμναβtr(FμνFαβ), (1)which explicitly violates CP, the electric charge qe of a dyonis modified. Instead of being quantized, as in the ϑ = 0 case,one has, for ϑ=0,qe =(n +ϑ2π)e , n ∈ Z. (2)This result corresponds to a Julia-Zee dyon [2] with mag-netic charge m = 4π/e. There are also arguments leading tothe conjecture that other CP violating interactions may alsoinduce a shift of the dyon charge [1].CP violation can be induced not just by adding, as in (1),new interactions to the Yang-Mills-Higgs Lagrangian but byradically changing the setting of the theory. This is the caseof noncommutative gauge theories (NCGT) where the intro-duction of noncommutation in space-time coordinates hasshown to affect the behavior under C, P, and T invariances[3–6]. More specifically, one can prove that when noncom-mutativity is restricted to space coordinates,[xi, xj] = iθi j ,[xi, x0] = 0, i, j = 1, 2, 3. (3)As it is well known [4], NCGTs are not charge invariant.Only if the usual field transformations are accompanied bya change of sign in θi j , charge invariance is recovered. Hence,if one takes θi j as a fixed parameter, CP is violated, althoughCPT invariance is maintained since parity invariance is notaffected by the introduction of θ and time reversal undergoesa change that compensates that in C.It is then natural to pose the question whether the dyoncharge in noncommutative gauge theories receives a contri-bution from a CP violating effect induced by noncommuta-tivity even if the ϑ angle vanishes. Moreover, one could alsoask how the addition of the noncommutative version of theterm (1) modifies Q when both θi j =0 and ϑ=0.We will analyze these questions in the present letter and,to this end, we first discuss the properties of the dyon in anoncommutative Yang-Mills-Higgs model with U(2) gaugesymmetry, calculate its charge at the quantum level and alsoextend the theory in order to include a ϑ-term. Some of theseissues were briefly discussed in [7] starting from a monopolesolution obtained generalizing Nahm’s equations that de-scribes BPS solitons [8] (some aspects of dyon solutions werealso considered in [9]). Here, instead, we will extend themore explicit U(2) monopole solution found in [10, 11] tothe case of a dyon and then establish a noncommutative ver-sion of the Noether theorem. We then discuss the issue of theWitten effect in noncommutative space.We start from the noncommutative U(2) Yang-Mills-Higgs systemSNCYM = tr∫d4x(− 12Fμν∗Fμν + DμΦ∗DμΦ). (4)2 Research Letters in PhysicsGauge fields Aμ = AAμ tA take values in the Lie algebra of U(2)with generators tA,A = 0, 1, 2, 3 (t0 = I/2, ta = σa/2, a =1, 2, 3). Φ = ΦAtA is the Higgs multiplet and we considerthe Prasad-Sommerfield limit [12] in which the symmetrybreaking potential vanishes. Covariant derivatives and fieldstrength are given byDμΦ = ∂μΦ− ie[Aμ,Φ]∗,Fμν = ∂μAν − ∂νAμ − ie[Aμ,Aν]∗,(5)where the Moyal product and commutator are defined asA(x)∗B(x) = exp(i2θμν∂μx∂νy)A(x)B(y)∣∣∣∣y=x,[A(x),B(x)]∗ = A(x)∗B(x)− B(x)∗A(x).(6)As in (3) we will take θ0i = 0. This ensures a well-definedHamiltonian and unitarity at the quantum level.To find noncommutative static dyon solutions we willminimize the energyE = tr∫d3x(Ei∗Ei + Bi∗Bi + DiΦ∗DiΦ + D0Φ∗D0Φ),Ei = −F0i , Bi = −12 εi jkFjk.(7)Since we are working in the BPS limit, the vacuum expecta-tion value enters as a boundary condition on the Higgs field,trΦ2vac = v20/2. As in ordinary space, (7) can be written in theformE = tr∫d3x((Ei − sinαDiΦ)∗(Ei − sinαDiΦ)+(Bi − cosαDiΦ)∗(Bi − cosαDiΦ)+ D0Φ∗D0Φ + 2 sinαEi∗DiΦ+ 2 cosαBi∗DiΦ),(8)leading to a Bogomol’nyi bound on the energyE ≥ v0 sinαQ + v0 cosαM (9)with Q and M are the electric and magnetic charges definedasQ = 2v0tr∫d3xEi∗DiΦ, M = 2v0tr∫d3xBi∗DiΦ.(10)The bound is saturated whenever the following BPS equa-tions hold:Ei = sinαDiΦ, (11)Bi = cosαDiΦ, (12)D0Φ = 0. (13)In order to find an explicit dyon solution, we will nowconsider an expansion of fields Aμ and Φ in powers ofthe noncommutative parameter in θ, thus extending to thedyon case the approach developed in [10] and [11] wherea purely magnetically charged solution was considered. Onestarts from the exact Prasad-Sommerfield solution in ordi-nary space [12] as giving the zeroth order of an expansionin powers of θ for the monopole in noncommutative space.Plugging this expansion into the BPS equations, one obtainsthe noncommutative solution order by order in θ.We take as zeroth-order approximation for the SU(2)components the Prasad-Sommerfield dyon solution [12],which we call Φa(0), Aa(0)i , and Aa(0)0 . Concerning the U(1)components, absent in the original Julia-Zee dyon, we pro-pose for simplicity Φ0(0) =A0(0)i =A0(0)0 = 0. To first order inθ, an ansatz for the gauge potential and Higgs field compo-nents on U(1), that obey covariance under the SO(3) rota-tion corresponding to the diagonal subgroup of SO(3)gauge ×SO(3)space, isÃ0(1)i = θi jx jA(r) + εi jkθ jkC(r) + xiε jklθ jkxlD(r),Ã0(1)0 = θi jεi jkxkK(r),Φ̃0(1) = θi jεi jkxkB(r),(14)where A(r),B(r),C(r),D(r) , and K(r) are radial functionsto be determined. The component on SU(2) to first orderin θ of the Bogomol’nyi equation is not going to be analyze,because it is pure gauge.One can easily see that Bogomol’nyi (11) and (13) imply,to all orders in θ, the relationA0 = sinαΦ. (15)We have then proven, using a θ-expansion, a relation that wasproposed as an ansatz in [7] within Nahm’s approach to theconstruction of monopole solutions. Now, (15) implies thatthe BPS (11) is automatically satisfied and then the study ofthe BPS system (11)-(12) is reduced to the analysis of the soleequation12εi jkFjk = − cosαDiΦ (16)which, except for the factor cosα, is nothing but the purenoncommutative monopole equation. Then, after makingthe appropriate rescaling, we have from [10, 11]Ã(n)i(xi) = cos2n+1αA(n)i(cosαxi),Φ̃(xi)(n) = cos2nαΦ(n)( cosαxi), n = 1, 2, . . . ,(17)where A(n)i and Φ(n) are the pure monopole solutions andÃ(n)0 (xi) is related to them through (15).One can easily determine the asymptotic behavior of theproposed solution and then compute the electric and mag-netic charges of the dyon solution. One finds that the n > 0terms do not contribute at the surface at infinity so that onejust recovers the n = 0 result, one findsM = 4πe, Q =M tanα. (18)L. Cieri and F. A. Schaposnik 3So that one can conclude that n > 0 orders in θ do not con-tribute to the charges that thus coincide with those in ordi-nary space. That is, CP violation induced by noncommuta-tivity does not change, at least at the classical level, the chargeof the dyon, no trace of θi j , appears in Q.In order to analyze charge quantization at the quantumlevel, we use the Noether approach and canonically proceedas originally done in [1] regarding the unbroken symmetrywhich leaves the Higgs vacuum invariant and is associated tothe electric charge. To do this, we will not just consider action(4) but the one including the noncommutative version of aϑ-term,Sϑ = SNCYM + e2ϑ16π2∫d4xFμν∗F̃μν. (19)In this way, we will be able to test possible modifications ofthe dyon charge because both the noncommutativity and theϑ-term. As in ordinary space, the noncommutative versionof the ϑ-term can be written as a surface term [13] and hencethe equations of motion for action (19) and its dyon solu-tions remain the same for all values of ϑ.After symmetry breaking through the condition Φ2vac =v20/2, the unbroken symmetry is related to rotations Λ in thedirection of Φ leaving the Higgs vacuum invariant. Let usconsider such gauge transformations along the Higgs fielddirectionΛ(x) = 1v0Φataε(x) , ε(∞) = 1,δΛΦ = 0, δΛAμ = 1ev0DμΦ.(20)As first noticed for a scalar theory in [14], in order to applythe Noether method in the noncommutative case, one has totake into account ∗-commutators that, once integrated, givea vanishing contribution. That is, from the well-honnoredNoether formulaδΛ(x)S =∫d4x∂μJμ[Φ(x),Aμ(x)]ε(x) (21)one can at most infer thatδΛ(x)S = 0 =⇒ (∂μJμ) = tr([O,P]∗ + [B,C]∗∗B + · · ·)(22)for some proper functionals O,P,B,C, . . . since, once inte-grated over space-time, the right-hand side in (22) vanishesdue to the ∗-product cyclic properties under integration.Also when integrating (22) over 3 spaces to find the con-served charge commutators vanish since, as explained above,θ0i = 0. With all this, we finally have for the conservedcharge, which we call N ,N = −1eQ +ϑe8π2M. (23)At large distances, exp (2πiN) should implement a 2π rota-tion about the direction of Φ and hence the identity. Thenthe eigenvalues of N have to be quantized in integer units n.If we call qe and qg , the eigenvalues of the electric and mag-netic charge operators, one then hasqe =(ne +ϑe28π2qg). (24)That is, we have obtained for the nonconmmutative dyon thesame formula that holds for the case of ordinary space, (2)(qg = 4π/e).In summary, we have constructed an explicit noncom-mutative dyon solution showing that the relation (18) be-tween classical electric and magnetic charge also holdsin noncommutative space. Moreover, after extending theNoether approach to the case of a noncommutative gaugetheory, we have proven that CP violation introduced by thecommutation rule (3) does not change the Witten effect for-mula; indeed, the dyon’s charge shift is θi j-independent. Inthis respect, it should be interesting to consider other type ofnoncommutativity and in particular, to investigate the caseof the dyon in the fuzzy sphere along the lines developed in[15] where monopole solutions were constructed for the casein which θi j = θrεi jkxk since in that case the coordinate de-pendence of θi j may introduce definite changes in (24). Wehope to discuss this issue in the future.ACKNOWLEDGMENTSF. A. Schaposnik is associated with CICBA. The authors aregrateful to acknowledge G. Giribet and G. Lozano for in-teresting discussions and comments. This work is partiallysupported by CONICET (PIP6160), ANPCyT (PICT 20204),UNLP, and CICBA grants.REFERENCES[1] E. Witten, “Dyons of charge eϑ/2Π,” Physics Letters B, vol. 86,no. 3-4, pp. 283–287, 1979.[2] B. Julia and A. Zee, “Poles with both magnetic and elec-tric charges in non-Abelian gauge theory,” Physical Review D,vol. 11, no. 8, pp. 2227–2232, 1975.[3] M. Hayakawa, “Perturbative analysis on infrared aspects ofnoncommutative QED on R4,” Physics Letters B, vol. 478,no. 1–3, pp. 394–400, 2000.[4] M. M. Sheikh-Jabbari, “C, P, and T invariance of noncommu-tative gauge theories,” Physical Review Letters, vol. 84, no. 23,pp. 5265–5268, 2000.[5] P. Aschieri, B. Jurčo, P. Schupp, and J. Wess, “Noncummu-tative GUTs, standard model and C,P,T ,” Nuclear Physics B,vol. 651, no. 3, pp. 45–70, 2003.[6] M. Chaichian, K. Nishijima, and A. Tureanu, “Spin-statisticsand CPT theorems in noncommutative field theory,” PhysicsLetters B, vol. 568, no. 1-2, pp. 146–152, 2003.[7] D. J. Gross and N. A. Nekrasov, “Dynamics of strings in non-commutative gauge theory,” Journal of High Energy Physics,vol. 10, no. 4, p. 021, 2000.[8] D. J. Gross and N. A. Nekrasov, “Monopoles and stringsin noncommutative gauge theory,” Journal of High EnergyPhysics, vol. 7, p. 034, 2000.[9] D. Bak and K. Lee, “Elongation of moving noncommutativesolitons,” Physics Letters B, vol. 495, no. 1-2, pp. 231–236, 2000.[10] K. Hashimoto, H. Hata, and S. Moriyama, “Brane configu-ration from monopole solution in non-commutative super4 Research Letters in PhysicsYang-Mills theory,” Journal of High Energy Physics, vol. 12,no. 3, p. 021, 1999.[11] S. Goto and H. Hata, “Noncommutative monopole at the sec-ond order in ϑ,” Physical Review D, vol. 62, no. 8, ArticleID 085022, 10 pages, 2000.[12] M. K. Prasad and C. M. Sommerfield, “Exact classical solutionfor the ’t hooft monopole and the Julia-Zee Dyon,” PhysicalReview Letters, vol. 35, no. 12, pp. 760–762, 1975.[13] K. Furuuchi, “Equivalence of projections as gauge equivalenceon noncommutative space,” Communications in MathematicalPhysics, vol. 217, no. 3, pp. 579–593, 2001.[14] A. Micu and M. M. Sheikh-Jabbari, “NoncommutativeΦ4 the-ory at two loops,” Journal of High Energy Physics, vol. 5, no. 1,p. 025, 2001.[15] E. F. Moreno, “Spherically symmetric monopoles in noncom-mutative space,” Physical Review D, vol. 72, no. 4, ArticleID 045001, 9 pages, 2005.Submit your manuscripts athttp://www.hindawi.comHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014High Energy PhysicsAdvances inThe Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014FluidsJournal of Atomic and  Molecular PhysicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Advances in  Condensed Matter PhysicsOpticsInternational Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014AstronomyAdvances inInternational Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014SuperconductivityHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Statistical MechanicsInternational Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014GravityJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014AstrophysicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Physics Research InternationalHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Solid State PhysicsJournal of Computational  Methods in PhysicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Soft MatterJournal ofHindawi Publishing Corporationhttp://www.hindawi.comAerodynamicsJournal ofVolume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014PhotonicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Journal ofBiophysicsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014ThermodynamicsJournal of

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The dyon charge in noncommutative gauge theories

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