Existence of spinning solitons in field theory

0.19 MB·English

by  Mikhail S. Volkov

∗
Existence of spinning solitons in field theory
Mikhail S. Volkov
Laboratoire de Math´ematiques et Physique Th´eorique,
4
Universit´e de Tours, Parc de Grandmont, 37200 Tours, FRANCE
0
0 E-mail: [email protected]
2
n February 1, 2008
a
J

6

Abstract
1
v The first results, both positive and negative, recently obtained in the
0 area of constructing stationary spinning solitons in flat Minkowski space
3 in 3+1 dimensions are discussed.
0
1
0 1 Introduction
4
0
It is well known that in General Relativity there exist classical solutions
/
h describingspinningobjectswithfiniteenergy–Kerr-Newmanblackholes.
t Theyareuniquelycharacterizedbytheirmass,electriccharge,andangu-
-
p lar momentum J. Static and spherically symmetric solutions with J =0
e are relatively easy to obtain, and historically they had been found first,
h whileittookthenmorethan40yearsuntiltheirstationarygeneralizations
v: with J 6=0 were constructed.
i If one descends from curved space to flat Minkowski space, one finds
X
there non-gravitational field theories, as for example the Yang-Mills the-
r ory,describedattheclassical levelbynon-linearpartial differentialequa-
a
tions. Insome cases theseequationsadmit solutions describinglocalized,
globally regular particle-like objects with finite energy – solitons. There
existvarioustypesofsolitonsinfield-theoreticmodels,suchas,forexam-
ple,monopoles[1](theywillbebrieflyreviewedbelow),dyons[2],vortices
[3],sphalerons[4],Skyrmeons[5],knots[6],Q-balls[7],etc. Inmostcases
explicitly known soliton solutions are static and spherically symmetric.
It is then natural to wonder whether one can find for them stationary,
spinning generalizations with J 6= 0, similar to what has been done for
black holes. More generally, one can ask
Do static solitons admit stationary, spinning generalizations ?
It is quite natural to conjecture that the answer is positive. However,
until very recently this intuitive conjecture had neither been supported
∗Talk given at the Seventh Hungarian Relativity Workshop, 10–15 August, 2003,
Sarospatak,Hungary.
1
M.S.Volkov 2
by any explicit examples of spinning solitons, nor had it been frustrated
by any no-go conclusions.
To be precise, by spinning solitons are meant here J 6=0 solutions in
the one-soliton sector. They describe rotational, spinning excitations of
an individual, isolated object. On the other hand, one can also consider
rotating solutions outside the one-soliton sector. These would rather de-
scriberelativeorbitalmotionsinmany-solitonsystems,suchas,forexam-
ple,apairofsolitonandantisoliton rotatingaroundtheircommoncenter
of mass. Another example of systems which could be naturally classified
as orbiting arerotating vortex loops –vortons. Solutionsdescribing such
orbitalrotationsareactuallyknown(see[8]foradiscussion)andwill not
be considered here.
In what follows I shall present the first results on the existence of
spinningsolitonsobtainedrecentlyinourworkwithErikW¨ohnert[8],[9].
Theyaretwo-fold,bothpositiveandnegative. Thepositivestatementis:
•Solitonsintheorieswithrigidsymmetriescanhavespinningexcitations.
Thisissupportedbyanexplicitconstructionofspinningsolutions. Next,
however, comes a no-go result:
• None of the known solitons in gauge field theories with gauge group
SU(2) havespinning generalizations within theaxially symmetric sector.
This fact is quite surprising, as it rules out a large class of spinning soli-
tons, in particular, monopoles, dyons, sphalerons, and vortices.
2 Explicit example of spinning solitons
Let usconsider a theory of a self-interacting scalar field Φ [7],
L=∂ Φ∂µΦ∗−U(|Φ|), (1)
µ
where the potential satisfies thefollowing condition,
2U(φ)
ω2 ≡min <ω2 ≡U′′. (2)
min φ φ2 max
Althoughthisconditioncanbefulfilled,forexample,byapotentialofthe
type U(φ)=aφ6+bφ4+cφ2+d with a6=0, it rules out renormalizable
quarticpotentialswitha=0. Asaresult,themodelunderconsideration
can at best beonly some effective field theory.
This theory has the rigid phase symmetry, Φ → eiγΦ, with the asso-
ciated Noether charge
Q=i (Φ˙∗Φ−Φ˙Φ∗)d3x. (3)
Z
If the field does not depend on time, then Derrick’s scaling arguments
apply to rule out finite energy solutions. If Φ˙ 6= 0, however, then the
existence of such solutions is not prohibited. These solutions, called Q-
balls [7], are obtained bygiving to thefield a time-dependent phase,
Φ=eiωtφ(r), (4)
PSfragrepla
ements
M.S.Volkov 3
1
n=0
(cid:30)
0.5
n=2
0
n=1
-0.5
0 2 4 6 8 10 12 14 16 18 20
r
Figure 1: The static Q-ballsolutions.
with real φ(r). The equation of motion for φ reads
2 dU(φ)
φ′′+ φ′+ω2φ= . (5)
r dφ
Asimplequalitativeanalysis[7,8]showsthatifωisrestrictedtotherange
ω2 < ω2 < ω2 , then there are finite energy solutions for which φ(r)
min max
interpolates smoothly between φ(0) 6= 0 and φ(∞) = 0. These Q-balls
form a discrete family labeled by the number n = 0,1,2,... of nodes of
φ(r) [8]; see Fig.1. Since these solutions are spherically symmetric, their
energy-momentumtensor Tµ isdiagonal, andsotheangularmomentum
ν
J = T0d3x (6)
ϕ
Z
vanishes.
One wishes now to spin these solutions up. The idea is to give to the
field a rotating phase [8],
Φ=eiωt−iNϕφ(r,ϑ), (7)
with integerN, wherer,ϑ,ϕarethestandard sphericalcoordinates. Itis
worthnotingthatsuchafieldisneitherstationarynoraxiallysymmetric.
However,itcanbecallednon-manifestlystationaryandaxiallysymmetric
for the following reason1. One notices that the action of the generators
of time translations and axial rotations is equivalent to the action of the
rigid phasesymmetry,
L Φ=iωΦ, L Φ=−iNΦ, (8)
∂t ∂ϕ
where L is the Lie derivative along the vector ξ. This implies that the
ξ
energy-momentum tensor fulfills theconditions
L Tµ =L Tµ=0, (9)
∂t ν ∂ϕ ν
sinceitisinvariantunderthephasesymmetry. Theobservablequantities
are thusindeed stationary and axially symmetric.
1IwouldliketothankBrandonCarterforclarifyingdiscussionsofthisissue
M.S.Volkov 4
PSfragrepla
ements
1.5
N=1
(cid:30)
1
n=0
0.5
n=2
0
-0.5 n=1
-1
0 5 (cid:26) 10 15
Figure 2: The rotating Q-vortex solutions.
Beforeanalyzingthefullaxiallysymmetricproblem,itisinstructiveto
consider its simplified version in which there is the additional symmetry
with respect to translations along thez-axis. The field is then given by
Φ=eiωt−iNϕφ(ρ), (10)
and thefield equation reduces tothe ODEfor φ(ρ),
∂2 1 ∂ N2 dU(φ)
+ − +ω2 φ= . (11)
(cid:18)∂ρ2 ρ∂ρ ρ2 (cid:19) dφ
The solutions are shown in Fig.2, they describe cylindrically symmetric
configurations with a finite energy per unit length – rotating Q-vortices
[8]. TheirchargeQandangularmomentumJ perunitlengtharerelated
as J =NQ.
Returningnowtothefullaxiallysymmetricproblem,thefieldequation
reduces to thefollowing PDE for φ(r,ϑ),
∂2 2 ∂ 1 ∂2 cotϑ ∂ N2 dU(φ)
+ + + − +ω2 φ= . (12)
(cid:18)∂r2 r∂r r2∂ϑ2 r2 ∂ϑ r2sin2ϑ (cid:19) dφ
Tosolvethisequation,thespectraldecomposition hasbeenemployed[8]
∞
φ(r,ϑ)= f (r)PN (cosϑ), (13)
k N+k
Xk=0
wherePN (cosϑ)aretheassociatedLegendrepolynomials. Thisreduces
N+k
the problem to the infinitesystem of coupled ODE’s,
d2 2 d (N +k)(N +k+1)
(cid:18)dr2 + rdr − r2 +ω2(cid:19)fk(r)=Fk(f0,f1,...), (14)
where F stand for the non-linear terms. Truncating this system by set-
k
ting fk = 0 for k ≥ kmax, gives a finite system of kmax coupled ODE’s.
Solutions of thistruncatedsystem havefiniteenergy whose valuerapidly
M.S.Volkov 5
convergestoalowernon-zerolimit askmax grows. Infact, itturnsoutto
be sufficient to choose kmax =10 to get a reasonable approximation [8].
This gives spinning Q-balls – the first explicit example of stationary,
spinning solitons in Minkowski space in 3+1 dimensions. For each given
value of the charge Q, these solutions are characterized by the winding
number N and also by their parity. The latter is determined by whether
theindexkin(13)takesonlyoddoronlyevenvalues. Thedistributionof
the energy density is strongly non-spherical, and depending on the value
of parity it has the structure of deformed ellipsoids or dumbells oriented
along therotation axis [8]. Theangular momentum is ‘quantised’ as
J =QN, (15)
while the energy increases by about 20% when the winding number N
increasesbyone. Suchabehavioroftheenergysupportstheinterpretation
of solutions with N > 0 as describing spinning excitations of the static
Q-balls.
Perhaps the most important lesson that one can draw from this ex-
plicit exampleofspinningsolitonsisthatstationary rotation ispurefield
systems without gravity is possible. It is then natural to look for spin-
ning solitons also in physically more interesting systems of gauge fields
withspontaneouslybrokensymmetries. Surprisingly,however,theresults
obtained up to now in this direction are all negative.
3 Yang-Mills-Higgs theory
Weshall beconsideringarathergeneral classof gaugefieldtheorieswith
spontaneously broken gauge symmetries [9]. These are the Yang-Mills-
Higgs (YMH) theories with a compact gauge group G defined by theLa-
grangian
1 1 λ
LYMH =− hFµνFµνi+ (DµΦ)†DµΦ− (Φ†Φ−1)2. (16)
4 2 4
Here, F ≡ T Fa = ∂ A −∂ A +[A ,A ] with A ≡ T Aa. The
µν a µν µ ν ν µ µ ν µ a µ
gauge group generators in a r-dimensional representation of G are T =
a
(T ) , where a = 1,2,...,dimG and p,q = 1,2,...r. They satisfy the
a pq
relations [T ,T ] = f T and tr(T T ) = Kδ . The invariant scalar
a b abc c a b ab
productintheLiealgebraisdefinedashABi= 1tr(AB). TheHiggsfield
K
Φ=Φp is a vector in the representation space of G where the generators
T act; this space can be complex or real. D Φ = (∂ +A )Φ is the
a µ µ µ
covariant derivativeof theHiggs field.
Thisfieldtheoryisquitegeneral,andfordifferentchoicesofG itsrep-
resentationsitcoversmostoftheknowngaugemodelsadmittingsolitons.
For example, choosing G=SU(2) and theHiggs field in the adjoint repre-
sentation, in which case the group generators are (T ) = −ǫ , gives
a ik aik
the theory whose solutions are
The ’t Hooft-Polyakov monopoles [1]. For these solutions the
YMH fields can bechosen in theform
xk xp
Aa =0, Aa =ε (1−w(r)), Φp = φ(r), (17)
0 i aikr2 r
M.S.Volkov 6
whereindicesi,k=1,2,3correspondtoCartesiancoordinates. Thisfield
configuration is spherically symmetric in the following sense. For each
Killing vector ξ generating the SO(3) rotation group there exists a Lie-
algebra-valued scalar function W such that the following equations are
ξ
fulfilled [10],
L Aa =δ˜ A , L Φa =δ˜ Φ. (18)
ξ µ Wξ µ ξ Wξ
Here thegauge variations induced by W are defined as
ξ
δ˜ Aa =D W , δ˜ Φ=−W Ψ, (19)
Wξ µ µ ξ Wξ ξ
where D =∂ +[A , ] is thecovariant derivativein theadjoint repre-
µ µ µ
sentation. Conditions (18) express the invariance of the fields under the
combined action of rotations generated by ξ and gauge transformations
generated by W .
ξ
Inserting (17) to the YMH equations obtained by varying the La-
grangian (16) gives a coupled system of ODE’s,
r2w′′ = (w2+r2φ2−1)w,
(r2φ′)′ = 2w2φ+λr2(φ2−1)φ. (20)
These equations admit solutions w(r), φ(r) smoothly interpolating be-
tween the asymptotic values w(0) = 1 and w(∞) = 0 and φ(0) = 0 and
φ(∞)=1. Thesearethe’tHooft-Polyakovmonopoles. Onecanvisualize
themasextendedparticlescontainingaheavycorefilledwiththemassive
non-linear YMH fields, while only one massless component of the Yang-
Mills field extends outside the core giving rise at large distances to the
Colombian magneticfieldwith unitmagneticcharge. Theenergydensity
is O(1) in thecore, while asymptotically it is O(r−4),such that thetotal
energy is finite.
The ’t Hooft-Polyakov monopoles have vanishing angular momentum
–sincetheyaresphericallysymmetric. Onecangeneralizethesesolutions
toincludealsoanelectricfield,whichgiveselectricallychargedmonopoles
– dyons [2]. However, it is unknown whether one can further generalize
thesesolutionstoincludealsoanangularmomentumJ 6=0. Weshallnow
showthatthisisnotpossiblewithinthestationaryandaxiallysymmetric
sector, at least for G=SU(2).
4 Non-existence of spinning solitons in
the Yang-Mills-Higgs theory
Let (A0,Φ0) be a static, spherically symmetric soliton solution of the
µ
YMH theory (16) for some choice of gauge group G and its representa-
tion. Let us consider stationary, axially symmetric on-shell deformations
(A ,Φ) of this solution. They satisfy the symmetry conditions (18) with
µ
the Killing vector ξ being either stationary, ξ =∂ , or axial, ξ =∂ . Let
t ϕ
W and W be thecorresponding parameters of the compensating gauge
t ϕ
transformations in theright hand side of (18).
Thedeformationsaresupposedtobeeverywheresmoothandvanishing
in theasymptotic region,
a =A −A0 →0, φ=Φ−Φ0 →0, as r→∞, (21)
µ µ µ
M.S.Volkov 7
whichinsuresthatdeformedconfigurationsbelongtothesametopological
sector as the original one. However, for finite values of r deformations
are not supposed to be small, as long as the total energy of the axial
configuration (A ,Φ) is finite.
µ
To prove the absence of spinning solitons the strategy is as follows.
First, one notices [12] that for axially symmetric fields there exists the
following remarkable surface integral representation of the angular mo-
mentum:
J = T0d3x=− h(A −W )Fk0idS . (22)
ϕ ϕ ϕ k
Z I
Herethesurfaceintegrationisperformedoveraclosedtwo-surfaceexpand-
ingtospatialinfinity. Thecalculationoftheintegralisthenfacilitatedby
the fact that near infinity deformations a and φ are small, and so they
µ
can be described perturbatively. One then carries out a linear perturba-
tion analysis in the asymptotic region in order to decide whether there
existperturbationsgivinganon-zerocontributiontothesurfaceintegral.
It is worth emphasizing that results obtained in this way are non-
perturbative, since deviations from the static background are not sup-
posedtobesmalleverywhere. Thisallows onetodrawconclusionsabout
theexistenceof spinningsolitons for arbitrary valuesof J, without being
restricted to the slow rotation limit [11].
The key role in the programme outlined above certainly belongs to
the surface integral representation (22) of the angular momentum. It
is therefore important to understand where it comes from. It is worth
noting that, normally, theangular momentum,being theNoether charge
associated with the rigid rotational symmetry, is given by a volume and
not surface integral. A surface integral representation for the Noether
charge associated with a rigid symmetry can exist if only there is also
a local symmetry in the problem, and this local symmetry contains the
rigidsymmetryasaparticularcase. TheNoethercurrentinthiscasehas
the total divergence structure typical for theories with local symmetries.
Thisimplies thatthevolumeintegralforNoether’scharge canbefurther
transformed to a surface integral.
AgoodexampleofsuchasituationcanbefoundinGeneralRelativity,
where asymptotic Poincar´e symmetries in asymptotically flat spacetime
can be viewed as a particular case of general diffeomorphisms. As a re-
sult, theassociated Noethercurrentshavethetotal divergencestructure,
and the corresponding Noether charges – mass, momentum and angular
momentum – are given by theADMsurface integrals.
IntheYMHtheoryunderconsiderationtherelationbetweentherigid
rotationalsymmetryandthelocalgaugesymmetryisprovidedbythecon-
ditions(18). Referringto[9]fordetails,theideaofhowthiscomesabout
is as follows. The Noether current associated with the symmetry gener-
ated by a Killing vector ξ acting on a field system with the Lagrangian
L(uB,∂ uB) is
µ
∂L
Θµ= δ uB−ξµL. (23)
ξ ∂(∂ uB) ξ
XB µ
In our case uB =(A ,Φ,Φ†) and thefield variations are given by
µ
δ A =(L −δ˜ )A , δ Φ=(L −δ˜ )Φ. (24)
ξ µ ξ W µ ξ ξ W
M.S.Volkov 8
TheyconsistoftheLiederivativesalongξandalsoofthepuregaugevari-
ations generated by W =ξαA (the gauge variations should be included
α
to (24) in order to make the Noether current gauge-invariant). Inserting
(24) to (23) gives
Θµ =ξνTµ, (25)
ξ ν
where Tµ is the metrical energy-momentum tensor of the YMH system
ν
(16). The corresponding Noethercharge is given by thevolume integral
Θ = ξνT0d3x. (26)
ξ ν
Z
So far nothing new has been obtained, since this is just the standard
expressionfortheconservedNoetherchargeassociatedwiththespacetime
symmetrygenerated byξ. Choosing ξ=∂ orξ=∂ givestheconserved
t ϕ
energy or angular momentum.
Let us now impose the symmetry conditions (18). Using these, one
can eliminate theLie derivatives from the variations (24), which gives
δξAµ=(δ˜Wξ−δ˜W)Aµ≡δ˜ΨξAµ, δξΦ=(δ˜Wξ−δ˜W)Φ≡δ˜ΨξΦ, (27)
with Ψ = W −ξαA . As a result, the field variations for symmetric
ξ ξ α
fields are puregauge variations ! Inserting (27) to (23) gives
Θµ =−∂ hΨ Fαµi−ξµL. (28)
ξ α ξ
If ξ =∂ , thesecond term on the right does not give contribution to the
ϕ
Noether charge,
J = Θ0ξd3x= ∂khΨϕFk0id3x= h(Aϕ−Wϕ)F0kidSk. (29)
Z Z I
Thisexplainstheappearanceofthesurfaceintegralrepresentationforthe
angular momentum.
To analyze the expression obtained, it is convenient to pass to the
gauge where the fields do not explicitly depend on t,ϕ, so that W =
t
W = 0. The existence of such a gauge is entailed by the fact that the
ϕ
twoKillingvectors∂ and∂ commute. Sincethedeviationsa =A −A0
t ϕ µ µ µ
from thestaticJ =0backgroundaresmallintheasymptoticregion,one
can linearize the integrand in (29) with respect to them. This gives
J = haϕF0k+Aϕ(−Dka0+[A0,ak])idSk. (30)
I
The problem therefore reduces to studying linear perturbations modes
in the asymptotic region that might give a non-zero contribution to this
integral.
LinearizingtheYMHequationsaroundthestaticbackground(A0,Φ0)
µ
with respect to a and φ gives
µ
D Dσa − D D aσ+2[F ,aσ]−M aaT
σ µ µ σ µσ ab µ b
1
= {φ†T D Φ−(D Φ)†T φ+Φ†T D φ−(D φ)†T Φ}T ,
2 a µ µ a a µ µ a a
D Dσφ + D aσΦ+2a DσΦ =−λ (Φ†Φ−1)φ+(Φ†φ+φ†Φ)Φ ,
σ σ σ
n o
M.S.Volkov 9
where the superscript ‘0’ has been omitted and all quantities denoted by
capital letters relate from now on to thestatic background.
The long range behavior of solutions of this system is determined by
eigenvalues of themass matrix M = 1Φ†(T T +T T )Φ. If all eigen-
ab 2 a b b a
valuesarepositive,thenallfieldsa aremassiveandtendtozeroasymp-
µ
totically fast for large r. The integral in (30) vanishes then. This is the
case, for example, for G=SU(2) with a doublet Higgs field, in which case
thestaticsolitons aresphalerons[4]. Theconclusion thereforeisthatthe
SU(2) sphalerons do not admit spinning generalizations within the sta-
tionary,axially symmetricsector. The sameconclusion can bemadealso
in thecase of vortices [9].
In the case of ’t Hooft-Polyakov monopoles and Julia-Zee dyons the
situationisslightlymorecomplicated,sincethemassmatrixhasonezero
eigenvalue. This gives rise to a long range massless component of the
gauge field. In this case one has to solve the perturbation equations in
ordertodeterminetheasymptoticbehaviorofthemostgeneralstationary
and axially symmetric perturbations a , φ. The corresponding general
µ
solution was found in [9]. Inserting thissolution to (30) gives
J =0 (31)
since the asymptotic inverse power-law falloff of the perturbations turns
out to be too fast to support a nonzero value of the integral. This shows
that the ’t Hooft-Polyakov monopoles and Julia-Zee dyons do not admit
spinning generalizations within thestationary, axially symmetric sector.
We are therefore bound to conclude that
None of the known gauge field theory solitons with gauge group SU(2) –
monopoles, dyons, sphalerons, vortices – admit spinning generalizations
within thestationary, axially symmetric sector.
Of course, thisdoesnot yet eliminate completely spinningSU(2)soli-
tons, butonlyrestricts theirexistence. Atthesame time,thisrestriction
israthersevere,sinceitimpliesthatspinningcounterpartsfortheknown
solitons,ifexistatall,arenotaxiallysymmetric. Suchanoption,however,
seemstoberatherimplausible. Theonlypossibilityofaxiallysymmetric,
spinning solitons with gauge group SU(2) that is still left unexplored is
related toa non-manifest symmetry,in analogy with theQ-balls.
As we have seen, in theories with a rigid phase invariance the spin-
ningispossiblefornon-manifestlyaxisymmetricfieldscontainingrotating
phases. If theinvarianceis local, then therotating phases can be gauged
away,inwhichcasethefieldsaremanifestlyindependentoft,ϕ. However,
aswehaveseen,thespinningisthenimpossible. Atthesametime,there
exist field systems with both local and global symmetries. In such sys-
temsallcomplexfieldscouldbegivenrotatingphases,butnotallofthese
phaseswould beremovablebylocal gaugetransformations. Forexample,
this would be the case if the dimension of a complex representation of
the gauge group is larger than the dimension of the group itself. In this
case the number of independent gauge parameters would be insufficient
togaugeawayallthephasesofthecomplexHiggsfield. Anon-manifestly
axially symmetric gauge field would be then invariant with respect to a
combined action oftheaxial symmetryplusalocal gauge symmetry,and
M.S.Volkov 10
plus an additional global symmetry that is not a particular case of the
gaugesymmetry. Apossibilityofhavingspinningsolitonsinsuchsystems
remains open.
Another possibility of constructing spinning solitons could be related
to higher gauge groups. The pattern of the symmetry breaking and the
number of massless gauge fields that can contribute to the angular mo-
mentumsurfaceintegraldependverymuchonthegroup. Asaresult,the
possibility of havingmanifestly axially symmetricrotating solitons is not
excluded for higher gauge groups.
References
[1] G.’tHooft,Nucl.Phys.B79,276(1974);A.M.Polyakov,JETPLett.
20, 430 (1974).
[2] B. Julia, A.Zee, Phys. Rev. D11, 2227 (1975).
[3] H.B. Nielsen and P. Olesen, Nucl. Phys. B61, 45 (1973).
[4] F.R. Klinkhamer and N.S.Manton, Phys. Rev. D30, 2212 (1984).
[5] T.H.R. Skyrme, Nucl. Phys. 31, 556 (1962).
[6] L.D. Faddeev and A.J. Niemi, Nature 387, 58 (1997).
[7] S. Coleman, Nucl. Phys. B262, 263 (1985).
[8] M.S. Volkovand E. W¨ohnert,Phys. Rev. D66, 085003 (2002).
[9] M.S. Volkovand E. W¨ohnert,Phys. Rev. D67, 105006 (2003).
[10] P. Forgacs and N.S. Manton, Comm. Math. Phys. 72, 15 (1980).
[11] M.Heusler,N.StraumannandM.S.Volkov,Phys.Rev.D58,105021
(1998).
[12] J.J. VanderBij andE. Radu,Int. J. Mod. Phys. A17,1477 (2002).