On the topological charge of S O(2) gauged Skyrmions in 2 + 1 and 3 + 1 dimensions

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

On

the

topological

charge

of

S O

(

2

)

gauged

Skyrmions

in

2

+

1 and

3

+

1 dimensions

Francisco Navarro-Lérida

a

,

∗

,

Eugen Radu

b

,

c

,

D.H. Tchrakian

c

,

d a_{Dept.deFísicaTeóricaandIPARCOS,CienciasFísicas,UniversidadComplutensedeMadrid,E-28040Madrid,Spain}

b_{CenterforResearchandDevelopmentinMathematicsandApplications,(CIDMA), CampusdeSantiago,3810-183Aveiro,Portugal} c_{SchoolofTheoreticalPhysics,DublinInstituteforAdvancedStudies,10BurlingtonRoad,Dublin4,Ireland}

d_{DepartmentofComputerScience,NUIMaynooth,Maynooth,Ireland}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received11December2018

Receivedinrevisedform5February2019 Accepted11February2019

Availableonline13February2019 Editor:N.Lambert Keywords: Solitons Skyrmions Topologicalcharge Instantons

Thequestionofthedependenceofthetopologicalchargeq ofagaugedSkyrmion,onthegaugefield,is studiedquantitatively.Twoexamples,bothgaugedwith S O(2)arestudiedandcontrasted: i) The O(3) modelin2+1 dimensions,and ii) The O(4)modelin3+1 dimensions.Incase i),wherethe(usual) Chern–Simons(CS) termispresent,thevalue ofq changessign, goingthroughzero.Thisevolutionis trackedbyaparametercharacterisingthesolutionsinthegiventheory.Incase ii),inwhichdimensions noCSdensityisavailable,theevolutionofq isnotobserved.

©2019TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

The topological charge q of gauged Skyrmions presents pecu-liarities that are absent in gauged Higgs solitons. While in the lattercasethisquantityalwaysequalsthewindingnumberofthe Higgsfield,inthecaseofSkyrmionsthevalueof

q in

generalmay departfromthewindingnumber,orthe“baryonnumber”.

The winding number of a Skyrme system is a topological

charge [1],which providesthe staticenergywithalower bound. What we refer to as the topological charge ofa gauged Skyrme system,ismerelythenumberwhichgivesthelowerboundonthe staticenergy. This contrasts with (gauged) Higgs systems where thetopologicalchargeencodedintheHiggs(andgauge)field,also supplies the energy lower bound [2,3]. While in Higgs models this topological charge is the magnetic monopole charge [4], by contrastthemagneticfluxofgauged Skyrmemodelsin space di-mensions D

≥

3 vanishes [5], exceptin D

=

2,1 _{and that onlyin}

thepresenceofChern–Simonsdynamics.

The topological charge for the S O

(2)

gauged O

(4)

(Skyrme) modelin3

+

1 dimensionsisdefinedinRef. [7] andin[8,9],and, forthe

S O

(2)

gauged O

(3)

(planarSkyrme)modelin2

+

1

dimen-*

Correspondingauthor.

E-mailaddress:fnavarro@fis.ucm.es(F. Navarro-Lérida).

1 _{Seeforexample[6].}

sions in Ref. [10]. The topologicalcharges forthe S O

(

D

)

gauged

O

(

D

+

1) sigmamodelon

R

D (D

=

2,3,4)were definedin[11], and the soliton of the S O

(3)

gauged O

(4)

sigma model on

R

3

was first reportedin[5].More detailedstudiesof thelatter soli-tonsweregivenin[12], whilesolitonsofthe S O

(4)

gauged O

(5)

sigmamodelon

R

4 _{werereportedin[13].Themost}

comprehen-sive and recent definitions of the generic topological charges of gaugedSkyrmionsaregiveninAppendix B ofRef. [14].

WiththeexceptionoftheworkinRef. [7],whereattentionwas focusedonthedecayofthebaryon number,2 theworksof[8–11,

5,13] are concerned with theconstruction oftopologically stable solitons.Here,weareconcernedwiththe

S O

(2)

gauged O

(3)

and

O

(4)

sigmamodels in 2

+

1 and 3

+

1 dimensions, respectively. ButunlikeintheworksofRefs. [10,8,9],ourfocusherewillbethe evolutionofthetopologicalcharge,

e.g.,

itspossibledecay.3

In odddimensional spacetimes,where theChern–Simons (CS) termisdefined, one findsthat theeffectof thisCS dynamics re-sults in the mass/energy of the static solitons both to increase andtodecreasewithincreasingglobalcharge(electricchargeand

2 _{Well knownworksinthis directionare,e.g., [16,17],employ instead SU} (2)

gaugingthatdoesnotresultintopologicallystablesolitons.

3 _{S O(2)gauged Skyrmionsin2+1 dimensionshavebeen studiedextensively}

(seeRefs. [10,6,18–20] andmorerecentlyin[21,22]).TheS O(2)gaugedSkyrmions in3+1 dimensionshavebeenquantitativelystudiedmainlyinRefs. [8,9] (seealso therecentanalyticalresultsinRef. [23]).

https://doi.org/10.1016/j.physletb.2019.02.011

0370-2693/©2019TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

spin). This was clearly demonstrated in Section 4 of Ref. [6] in 2

+

1 dimensions, where the effects of Chern–Simons dynamics were studied also in other 2

+

1 dimensional models. This evo-lutionof themass/energy ofthe staticsolitonswas trackedby a parametercharacterisingthesolutioninthe

given theory,

whichis strictlyduetotheCSdynamics.Inthepresentworkweshowthat thissamedynamicaleffectresultsintheevolutionofthevalueof thetopologicalcharge

q,

changing signandpassing throughzero, in the giventheory. Such a solutioncannot be found in the ab-senceofthe CS term [6].This isthemain resultofthefirst part ofthepresentwork,namelythestudyofthe S O

(2)

gauged O

(3)

model,withemphasisontheissueoftopologicalcharge

q.

(A de-tailedstudyofgeneric S O

(2)

gaugedSkyrmionsisgivenin[15].)

Inevendimensional spacetimeshowever,noCS densityis de-fined. There is nonetheless a class of Skyrme–CS (SCS) densities satisfyingthepropertiesoftheusualCSdensityinalldimensions, including even dimensions. These are introduced in Ref. [14]. It happens that in the present example this quantity vanishes un-dertheimposition ofappropriate symmetries,andintheabsence of any Chern–Simons type dynamics we are limited to studying the modelconsidered in Refs. [8,9]. There,it was found that the mass/energyofthesolitonofthegauged systemincreases mono-tonicallywithincreasing electric(global)charge. Sinceno param-etercharacterisingthesolutionsisavailable,thecouplingstrength of the Maxwell field,

λ

0 in (11) below, was varied in [8]. This

clearlychangesthetheorybeingconsidered,butsinceinanygiven theory the value of both the energy and the electric charge Qe

dependon

λ

0,thisenabled [8] thetrackingoftheenergywith

in-creasingQe.Herebycontrast,weconsiderthedependenceofthe

topologicalchargeon

λ

0 andshowthat

q remains

constant.

This paper is organised as follows. In Section 2 below we present our quantitative results on the S O

(2)

gauged Skyrmion in 2

+

1 dimensions, and in Section 3 our results on the S O

(2)

gaugedSkyrmionin3

+

1 dimensions.InSection4wesummarise ourresultsandpointouttofurtherdevelopments.

2. S O

(

2

)

gaugedSkyrmionsin2

+

1 dimensions

Ever since Schroers’ construction of solitons [10] of the U

(1)

gauged planar Skyrme model, there hasbeen considerable inter-est 4 _{in thisarea. The dynamicsof theAbelian field in Ref. [10]}

wasdescribedbytheMaxwelldensity,subsequentlythelatterwas replacedbytheChern–Simons(CS)density [18–20].Morerecently, theeffectofChern–Simons dynamicsonthe combinedMaxwell– CS O

(3)

sigma model was studied in Ref. [6]. What was found there (in Section 4) is that the mass/energy of the soliton de-creased withincreasing globalcharges – electric charge and an-gularmomentum –insome rangesoftheparameters.Thiseffect isabsent,intheabsenceoftheCStermintheLagrangian,andis strictlyaresultoftheChern–Simonsdynamics.

In the present note, we pursue further consequences of the presence of the Chern–Simons term. The new effect in question istheevolutionofthetopologicalchargeduetothegaugingofthe

O

(3)

sigmamodel.Whilethisisnotstrictlyadynamical effect,it isagainpredicatedonthepresenceofaCStermintheLagrangian. Theprecise mechanismofthisis therole that theCS termplays inestablishing thetopologicallower bound (Belavininequalities). This,andotherdetailswillbeexposedinafutureextendedwork. The Skyrme model on

R

2 is described by the scalar

φ

a

=

(φ

α

,

φ

3

),

|φ

a

_|

2

₌

_{1,andthe S O}

₍₂₎

_{gaugingprescriptionis}5

4 _{SeeRef. [6] andcitationsthere.}

5 _{TheprescriptionforgaugingtheO(D+1)Skyrmescalarφ}a_{,a=1,2,. . . ,D+1,}

onRD_{withgaugegroupS O}

(N),2≤N≤D,issummarisedmostrecentlyin Ap-pendix B ofRef. [14].

Dμ

φ

α

= ∂μ

φ

α

+

Aμ

(

ε

φ)

α

,

(

ε

φ)

α

=

ε

αβ

φ

β

,

(1)

Dμ

φ

3

= ∂μ

φ

3

,

μ

=

i

,

0

,

i

=

1,2

,

(2) Aμ being the

S O

(2)

connection.

The topological charge densityfor the S O

(2)

gauged Skyrme modelis[11],[14],[15]



=



0

+

2

ε

i j

∂

i

[(φ

3

−

v

)

Aj

],

(3)

independently ofwhetherthedynamics isMaxwell orCS. In(3),



0 is the winding numberdensity whose volume integral is the

“baryonnumber”, Ai,

i

=

1,2,arethemagneticcomponentsofthe

Abelianconnection,andv is arealconstant.

What is crucial here is that in the Maxwell gauged case the Belavininequalities,whichmustbesatisfiedbytherequirementof topologicalstability,forcetheconstant

v to

beequaltoone,

v

=

1, andas aconsequence ofthe boundarycondition

φ

3

₍

_∞)

₌

_{1, the}

contribution of the Abelian field to the integral of



disappears. Whatremains istheintegral of



0,namely,the“baryonnumber”.

ThesituationisstarklydifferentwhenthedynamicsoftheAbelian field is described by the Chern–Simonsterm. In that case, it fol-lowsfromtheBogomol’nyianalysis,thattheconstant

v in

(3) can take any value, including v

=

1, as a consequence of which the topologicalcharge willdepart fromthe“baryonnumber”, quanti-tatively.Itisouraimheretotracktheevolutionofthetopological charge,duetothismechanism.

Toillustratethismechanismbriefly,considerthechoiceof po-tential V

≡

V

[φ

a

]

inthe Lagrangian. In the caseof Maxwell dy-namics, this potential is fixed by the constraints of the Belavin inequalitiestobe

VM

=

1 2

λ(1

− φ

3

₎

2

_,

₍₄₎

consistently with the boundary value

φ

3

(

∞)

=

1, when

VM

[φ

3

(

∞)]

=

0.

The corresponding Bogomol’nyi analysisinthe caseofChern– Simonsdynamics,whichwaspresentedin[20] andinAppendixC of[15],leadstothepotential

VCS

=

1 32

λ



_η

κ



2

|φ

α

_|

2

_(φ

3

₋

_v

₎

2

_,

_{λ >}

₀

_.

₍₅₎

(In(4)–(5),

λ

,

η

and

κ

arerealconstants.)Thestrikingpropertyof thepotential(5) isthatthe(finiteenergy)condition

V

CS

[φ

a

(

∞)]

=

0 can be satisfied for any value of the real constant v, since at spatialinfinity

|φ

3

_|

2

_→

₁

_{⇒ |φ}

α

_|

2

_→

₀

_;

_α

₌

_1,2

_.

₍₆₎

Thisispreciselythesituationwherethetopologicalchargewill de-partfromthewindingnumber

n,

namelyfromthe“baryon num-ber”.Inpractice,

v

=

0 isthemostconvenientchoice.

The Chern–Simons–Skyrme (CS–S) model studied inRef. [20], can be augmentedby the Maxwell termwithoutinvalidating the topological inequalities, to yield the Maxwell–CS–S model de-scribedbytheLagrangian

L

= −

1 4F 2 μν

+

κε

λμνAλFμν

−

1 8

τ

|

D[μ

φ

a_D ν]

φ

b

|

2

+

1 2

η

2

_|

_D μ

φ

a

|

2

−

η

4VCS

[φ

a

],

(7) consideredhere.

TheLagrangian(7) isnottheminimalonethatfollowsfromthe Belavin inequalities.It hasbeen further augmented by the quar-tic kinetic Skyrme term with coupling

τ

. The reason for this is

Fig. 1. b∞vs. a∞for vortices with n=1,λ=1.6,η=1,κ=1, andτ=1.

ourrequirementthat after gauge decoupling,the systemsupport Skyrmionswithwindingnumber

n.

To analyse the system (7) numerically, we subject it to az-imuthalsymmetry

φ

α

=

sin f

(

r

)

nα

,

φ

3

=

cos f

(

r

) ,

nα

=



cos nθ sin nθ



,

(8) Ai

=



a

(

r

)

−

n r



(

ε

ˆ

x

)

i

,

A0

=

b

(

r

) ,

(9)

where

θ

is theazimuthal angle andn is the winding numberof theSkyrmescalar.

WehaveanalysedthereducedonedimensionalLagrangian de-scended from (7) subject to (8), (9), numerically. Also, subject to the above Ansatz, the volume integral of



(as given by (3)), namelythetopologicalcharge,canbereadilyevaluatedas6

q

= −

1 8

π





d2x

=

1 2

(

n

+

a∞

) ,

(10) where

a

∞

=

a

(

∞)

.

We see from relation (10) that the value of the topological charge

q depends

onthesignandmagnitudeof

a

_∞.Inparticular, when

a

∞ isnegative (a_∞

<

0) thevalue of

q will

be diminished, resultingintheannihilationof

q when a

∞

= −

n. Thismechanism enablesthetrackingoftheevolutionofthetopologicalcharge by meansoftheparameter

a

∞,characterisingthesolutionsina

given

theory/model.

Thespectrumoftheasymptoticvalue

a

∞ ofmagneticfunction

a

(

r

)

resultsfromthespectrumoftheasymptoticvalue

b

∞ ofthe electricfunction b

(

r

),

asseen in Fig.1. Thismechanismwas en-counteredinRef. [6],andtechnicallyit resultsfromthefact that theequation ofmotionof

b

(

r

)

can besolved forarbitraryvalues oftheasymptoticquantity

b

∞.

This is a property of theories featuring a Chern–Simons (CS) density,whichcombinesboththemagneticandtheelectricfields. In2

+

1 dimensions,inparticular,itiswellknownthattheelectric fluxis actuallyproportional to the

vortex number,

inother words the(topological)magneticflux.Thusthemechanisminquestionis strictlypredicatedbythepresenceofaCSterm,

κ

=

0.

Wehaveconstructedsuchsolutionswith

κ

=

0,therelationof theasymptoticvalue oftheelectricfunction,

b

∞,tothemagnetic

6 _{SubjecttotheAnsatz(8)–(9),theelectricchargeQ}

eisQe=8π κ(n−a∞),while thetotalmass/energyE andangularmomentum J arecomputedasintegralsofthe

T00andTi0componentsoftheenergymomentumtensor[6].

counterpart,

a

∞,beingdisplayedinFig.1.Weknowalreadyfrom ourresultsinRef. [6] thatduetothiseffect,thedependenceofthe mass/energy E on theglobalchargesisnon-standard, namelythe dependenceontheelectriccharge Qe andtheangularmomentum

J displayed respectively(leftandrightpanels)inFig.2.

Here,we analysefurthertherelationofthetopologicalcharge

q and the energy E. In Fig. 3 (left panel) we represent the en-ergyperunit windingnumber

n versus

thetopologicalchargefor three valuesof

n for

thesame parameters asinFig. 1. Forsmall valuesof thecouplingstrength

τ

ofthe (quartic)Skyrmekinetic term, the minimum of the energy occurs at values of the topo-logicalchargearoundthewindingnumber.However,whenhigher valuesof

τ

are considered,theminimumoftheenergyoccursat values clearly different fromthe winding number.This is shown inFig.3(rightpanel),wheretheminimumofthecurve for

n

=

1,

τ

=

10 islocated at q

≈

0.556.Concerning the stability of these solutions, one wouldbe tempted to state that most stable solu-tionwouldcorrespondtothat withleastenergy,whichaccording toFig.3(rightpanel)doesnotpossessintegertopologicalcharge, ingeneral.

From Fig. 3. (right panel), we see that in the absence of the Skyrmeterm(

τ

=

0) theenergeticallyfavouredconfigurationsare those with topological charge equal to the winding number. But when the Skyrme term is introduced, with increasing values of thecouplingstrength

τ

,itappearsthattheenergeticallyfavoured solutions have topological charge progressively smaller than the winding number, the latter being the topological charge of the Skyrmioninthegaugedecouplinglimit.

3. S O

(

2

)

gaugedSkyrmionsin3

+

1 dimensions

ThestationarysolutionsofthissystemwerestudiedinRefs. [8] and[9].In[8],itwasseenthatthemassoftheelectricallycharged Skyrmionwaslargerthanthatoftheelectricallyneutralone,while in[9] itwas shownthat theseSkyrmions spin. Thatthemass of theelectricallychargedSkyrmionisalwayslargerthanthe electri-callyneutralonewasseenin[8] byobservingthattheseenergies both depend on the Maxwell coupling

λ

0 (in (11) below), thus

plottingboththeseenergies

versus

λ

0 illustratesthisfact,asseen

inFig.4(leftpanel).

That the mass increases monotonically with increasing elec-triccharge seenin[8] isnot surprising, sincewe knowfromour workinRef. [6] thattoreversethistrendcanbeachievedonlyby theintroductionofChern–Simons(CS)dynamics,and,in3

+

1 di-mensionsno (usual) CS densityisdefined. While thereis aclass of densities introduced in Ref. [14] which share the properties ofChern–Simons that canbe definedineven dimensional space-times,theparticularonethatapplieshere,definedintermsofthe Maxwellfieldandthe O

(6)

Skyrmescalar,actuallyvanisheswhen subjectedtoaxialsymmetry.Hence,intheabsenceofanyChern– Simonsdynamics,wearebacktothestudyinRefs. [8,9].

Whatwe havedonehere, additionallytotrackingtheincrease ofmasswithincreasingelectricchargeandangularmomentumin Refs. [8,9],istostudytheeffectoftheelectromagneticfieldonthe topologicalcharge.7

Thestudied

S O

(2)

gaugedSkyrmionsystemisdescribedbythe Lagrangian

L

= −

1 4

λ

0

|

Fμν| 2

₊

1 2

λ

1

|

Dμ

φ

a

_|

2

−

1 4

λ

2

|

D[μ

φ

a_D ν]

φ

b

|

2

+ λ

3V

(φ

a

),

(11)

7 _{Strictlyspeakingthisisthedeformationofthetopologicalchargebythegauge}

Fig. 2. Leftpanel: EnergyE vs.electricchargeQeforvorticeswithn=1,λ=1.6,

η

=1,

τ

=1,andseveralvaluesof

κ

.Rightpanel: EnergyE vs.angularmomentum J for

vorticeswithn=1,λ=1.6,

η

=1,

τ

=1,andseveralvaluesof

κ

.

Fig. 3. Leftpanel: EnergyE vs.topologicalchargeq forvorticeswithn=1,2,3,λ=1.6,

η

=1,

κ

=1,and

τ

=1.Rightpanel: Sameforvorticeswithn=1,λ=1.6,

η

=1,

κ=1,andseveralvaluesof

τ

.

Fig. 4. Leftpanel: TheenergyE andtheelectricchargeQeareshownvs.thecouplingconstantλ0forSO(2)gaugedSkyrmionsin3+1 dimensions,withn=1,λ1= λ2= λ3=1,andseveralvaluesofb∞.Rightpanel: Therelativedifferenceq= (qG−qW)/(qG+qW)oftheindividualcontributionstothetopologicalchargeq=qG+qW=1 is

where

φ

a

= (φ

α

,

φ

A

)

;

α

=

1,2

;

A

=

3,4,with

|φ

a

_|

2

₌

_1,Skyrme

scalars of the O

(4)

sigma model, with V

=

1

− φ

4 the usual Skyrmepotential.

λ

i

(

i

=

0,1,2,3),in (11) are realnumbers

pa-rameterisingthecouplingconstantsofthemodel.Also,thegauging prescriptionsfor

φ

ais

Dμ

φ

α

= ∂μ

φ

α

+

Aμ

(

ε

φ)

α

,

Dμ

φ

A

= ∂μ

φ

A

,

α

=

1,2

,

A

=

3,4. (12) In the ungauged case, the topological charge density of the Skyrmionsis



0

=



i jk



abcd

∂

i

φ

a

∂

j

φ

b

∂

k

φ

c

φ

d

.

(13)

The natural generalization of this expression in the presence of gauge fields is found by replacing partial derivatives with gauge derivatives,



G

=



i jk



abcdDi

φ

aDj

φ

bDk

φ

c

φ

d

,

(14)

which,however,doesnotlead toaconservedcharge.

Theconserved,gaugeinvariantgeneralizationof(13) isthesum oftwoterms8



=



0

+ ∂

i



i

,

with



i

=

3



i jk

ε

A BAj

∂

k

φ

A

φ

B

,

(15)

whichcanbewritten alternativelyinamanifestlygaugeinvariant form



=



G

+

W

,

with W

=

3 2



i jkFi j

(

ε

A B_A j

φ

BDk

φ

A

).

(16)

Assuch,thetopologicalcharge

q consists

ofthesumoftwoterms, encodingthecontributionof



G and

W in

(16)

q

= −

1

2

π

2





d3x

=

qG

+

qW

,

(17)

with

q

G the

bare charge

(asfoundbyintegratingdirectlythe

den-sity(14)),and

q

W a

compensating contribution

(from(16)).

More-over,asseenfrom(15) thetopologicalchargeisstillaninteger,as longasthere are no singularitiesandthe surface termsfrom



i

vanish.

Imposing axial symmetry on the Maxwell connection Aμ

=

(

Ai

,

A

z

.

A0

),

wehavetheAnsatz Ai

= −



a

+

n

ρ



(

ε

ˆ

x

)

i

,

Az

=

0

,

A0

= −

b

,

(18) where

a

=

a

(

ρ

,

z

)

,

b

=

b

(

ρ

,

z

),

and,

ρ

2

_{= |}

_x i

|

2,

i

=

1,2.

Thesolutionsconstructedaretypifiedbytheasymptoticvalues

Ai

= −



a_∞

+

n

ρ



(

ε

x

ˆ

)

i

,

A0

= −

b∞

+



Q

ρ

2

₊

_z2

,

(19)

where Q is the electric charge and b_∞ is a free constant char-acterisingthe givensolution (the electrostatic potential).Also, in contrastwiththesolutionsin2

+

1 dimensionsconstructedinthe previous Section, there isno freedom in the asymptoticvalue of themagneticpotential, witha_∞

= −

n for all solutions.Again we areusingthenotation

a

∞

=

a

(

∞)

and

b

∞

=

b

(

∞)

.

ThecorrespondingAnsatzforthe

O

(4)

scalar

φ

a is

φ

α

=

R nα

,

φ

3

=

S

,

φ

4

=

T

,

(20) 8 _{ThisdefinitionisthatusedearlierinRefs. [7] and[8].}

where R

,

S and

T are functions of

ρ

and z, subject to the con-straint

R

2

₊

_S2

₊

_T2

₌

_1.Also,

_{nα is}

_{theunitvectorinthesubplane}

(

x1

_,

_x

2

_),

_{withvorticity}

_n.

Subject to symmetry (18)–(20), thetopological charge density (15) (or(16))reducestothetwodimensionaldensity



=

1

ρ



2nR

R

∂

_[ρS

∂

z]T

+

S

∂

_[ρT

∂

z]R

+

T

∂

_[ρR

∂

z]S

−

S

∂

_[ρa

∂

z]T

−

T

∂

[ρa

∂

z]S

−

2(a

+

n

)∂

_[ρS

∂

z]T



,

(21) which,asexpected,isa

curl.

Inprinciple,thevalueoftheintegral of(21) couldbedifferentfromtheintegralof



0,sinceitencodes

a contribution ofthe gauge potential. However, in contrast with the 2

+

1 dimensional system endowed with Chern–Simons dy-namics,9studiedinthepreviousSection,thiscontributionvanishes forregularconfigurations,andtheintegral of(21) isstill equalto the baryon number, q

=

n, for any value ofthe varying the cou-pling constants, in particular for any

λ

0. At the same time, the

contributionofthegaugeinvariantterms

q

G and

q

W tothe

topo-logicalchargeismodeldependent,asconfirmedbyournumerical results.10

Thedependenceofthemass/energy

E and

oftheelectriccharge

Qe of the spinning Skyrmions on

λ

0 is displayed in Fig. 4 (left

panel). As expected, forfixed coupling constants

λ

i andwinding

number

n,

E increases with increasing electrostatic potential b_∞

forallvaluesof

λ

0.Inthegaugedecoupling limit

λ

0

→ ∞

,these

aretheuncharged,spinningSkyrmionscharacterisedby

b

∞,found inRef. [26].

Thedifferencebetweenthe‘bare’charge

q

G andthe ‘compensat-ing’ one

q

W isdisplayedinFig.4(rightpanel)asafunctionof

λ

0.

Weseethattheinfluenceofthegaugefieldistocausethis quan-titytohavepositive,zeroandnegativevaluesindistincttheories. For large

λ

0, the contribution of qW is negligible, with qG

→

q.

However,

q

G decreaseswith

λ

0,with

q

W dominatingforsmall

λ

0.

Wenotethatforverysmallvaluesof

λ

0,whenthedifference

be-tween

q

G and

q

W isvanishing(orevenbecomes negative,asseen

fromFig.4(rightpanel)),theenergyoftheSkyrmion(asseen in Fig.4(leftpanel)),isfinite.

4. Summary

Wehavestudiedtwo

S O

(2)

gaugedSkyrmesystems,the O

(3)

and

O

(4)

sigmamodelsin2

+

1 and3

+

1 dimensions,respectively. Inthe firstcase, thesystemcan beendowedwithChern–Simons dynamics,whileinthesecondcase,not.Themainmessagehereis tohighlightthecontrastwhenChern–Simonsdynamicsispresent orabsent.

WhenChern–Simonsdynamicsispresent,tworelated phenom-enaareobserved. i) Themassofthesolitoncanbothincreaseand decrease with increasing global charge (electric or spin), and ii) Thetopologicalcharge canevolvethroughpositivetozeroto neg-ativevalues,forvarioussolutionscharacterisedby

a

∞,the asymp-totic value of the magnetic field defining the topological charge. Thisevolutioniscontingentonthedependenceof

b

∞,the asymp-totic value ofthe electric field, thisrelation beingdependent on

9 _{In2+1 dimensions,a}

∞ wasrelatedtob∞,whichprovidedalevertopass fromonesolutiontoanother,resultingintheevolutionofq,whilea∞= −n in

3+1 dimensions.

10 _{Differentfromthepreviouswork[8,9],thesigma-modelconstraint R}2_+S2₊ T2_{=1 isimposedherebyusingthe Lagrange multipliermethod, as explained} e.g. in[24,25].Thesolutionswith b∞=0 arerotating[9],possessinganangular momentum J=4πλ0Qe.Themass/energy E andtheangular momentum J are

computedfromthecomponentsT00and Ti0,respectively,oftheenergy

theintertwining oftheelectricandmagneticfieldsinthe defini-tionoftheChern–Simons density.Thisisclearlydemonstratedin the2

+

1 dimensionalsystemstudiedhere, forthe S O

(2)

gauged Skyrmion,in a modelwith both Maxwell andChern–Simons dy-namics,andbothwithandwithoutaquartickineticSkyrmeterm. Intheparticularmodelstudiedhere,theenergeticallyfavoured configurationsappeartooccurforsmallervaluesofthetopological charge,withprogressivelystrongercoupling

τ

oftheSkyrmeterm. Indeed,when

τ

=

0 the minimumenergyconfigurationscoincide withthewindingnumber

n (see

Fig.3),whileforverylargevalues of

τ

theenergeticallyfavouredconfigurationtendstothe

n

/2 (see

quantitativedetailsin[15]).

In the 3

+

1 dimensional system studied, where there is no Chern–Simons dynamics,these phenomena are absent. The mass ofthesolitonincreasesmonotonicallywithincreasing theelectric charge,andthetopologicalchargeisfixed.Moreover,changingthe theory by varying a couplingconstant doesnot lead tochanging valuesofthetopologicalcharge. Wesuggest that thephenomena ofnon-standard mass-electriccharge/spin11andevolving topolog-icalchargeobservedinthe2

+

1 dimensionalexampleareabsent in the 3

+

1 dimensionalmodel here, because of the absence of Chern–Simonsdynamicsinthelattercase.

The challenge is to supply an example of a gauged Skyrme model in 3

+

1 dimensions, where a Chern–Simons term is de-fined. Such examples are considered in Ref. [14]. In the case of

S O

(2)

gauging considered in this work, such a Skyrme–CS term was considered, butit turned out that thatterm vanished under symmetry imposition. The challenge thus is to consider gauging witha higher (appropriate) gauge group, butthis isa subject of anotherinvestigation.

Acknowledgements

WeareverygratefultoV. A.Rubakovforvaluableand penetrat-ing comments. E.R. gratefullyacknowledges the support of DIAS. The work of E.R.was also partiallysupported by the Portuguese nationalfundingagencyforscience,researchandtechnology(FCT), within the Center for Research and Development in Mathemat-ics and Applications (CIDMA), project UID/MAT/04106/2019, and alsobytheH2020-MSCA-RISE-2015GrantNo.StronGrHEP-690904,

11 _{Thisphenomenonisobservedalsoin(gauged)Higgstheories,wheretheChern–}

Simons (CS) dynamics is supplied by Higgs–CS [14] (HCS) terms, reported in Refs. [27,28].

theH2020-MSCA-RISE-2017GrantNo.FunFiCO-777740,andbythe CIDMAprojectUID/MAT/04106/2013.Thenumericalcomputations in Section3 wereperformedatthe BLAFIScluster inAveiro Uni-versity.

References

[1] N.S.Manton,P.Sutcliffe,https://doi.org/10.1017/CBO9780511617034. [2]J.Arafune,P.G.O.Freund,C.J.Goebel,J.Math.Phys.16(1975)433.

[3]T.Tchrakian,in:V.G.Gurzadyan,etal.(Eds.),FromIntegrableModelstoGauge Theories,pp. 283–291,arXiv:hep-th/0204040.

[4]T.Tchrakian,J.Phys.A44(2011)343001,arXiv:1009.3790 [hep-th].

[5]K.Arthur,D.H.Tchrakian,Phys.Lett.B378(1996)187,arXiv:hep-th/9601053.

[6]F.Navarro-Lérida,E.Radu,D.H.Tchrakian,Phys.Rev.D95 (8)(2017)085016, arXiv:1612.05835 [hep-th].

[7]C.G.CallanJr.,E.Witten,Nucl.Phys.B239(1984)161.

[8]B.M.A.G.Piette,D.H.Tchrakian,Phys.Rev.D62(2000)025020,arXiv:hep-th/ 9709189.

[9]E.Radu,D.H.Tchrakian,Phys.Lett.B632(2006)109,arXiv:hep-th/0509014.

[10]B.J.Schroers,Phys.Lett.B356(1995)291,arXiv:hep-th/9506004.

[11]D.H.Tchrakian,Lett.Math.Phys.40(1997)191.

[12]Y.Brihaye,B.Kleihaus,D.H.Tchrakian,J.Math.Phys.40(1999)1136,arXiv: hep-th/9805059.

[13]Y.Brihaye,D.H.Tchrakian,Nonlinearity11(1998)891–912.

[14]D.H.Tchrakian,J.Phys.A48 (37)(2015)375401,arXiv:1505.05344 [hep-th].

[15]F.Navarro-Lérida,D.H.Tchrakian,Phys.Rev.D99(2019)045007,arXiv:1812. 03147 [hep-th].

[16]E.D’Hoker,E.Farhi,Nucl.Phys.B241(1984)109.

[17]V.A.Rubakov,Nucl.Phys.B256(1985)509.

[18]P.K.Ghosh,S.K.Ghosh,Phys.Lett.B366(1996)199,arXiv:hep-th/9507015.

[19]K.Kimm,K.-M.Lee,T.Lee,Phys.Rev.D53(1996)4436,arXiv:hep-th/9510141.

[20]K.Arthur,D.H.Tchrakian,Y.-s.Yang,Phys.Rev.D54(1996)5245.

[21] A.Samoilenka,Y.Shnir,Phys.Rev.D95 (4)(2017)045002,https://doi.org/10. 1103/PhysRevD.95.045002,arXiv:1610.01300 [hep-th].

[22] A.Samoilenka,Y.Shnir,Phys.Rev.D93 (6)(2016)065018,https://doi.org/10. 1103/PhysRevD.93.065018,arXiv:1512.06280 [hep-th].

[23]F.Canfora,M. Lagos,S.H.Oh,J. Oliva, A.Vera, Phys. Rev.D98 (8) (2018) 085003,arXiv:1809.10386 [hep-th].

[24]R.Rajaraman,SolitonsandInstantons:AnIntroductiontoSolitonsand Instan-tonsinQuantumFieldTheory,North-HollandPublishingCompany,1982.

[25]E.Radu,M.S.Volkov,Phys.Rep.468(2008)101,arXiv:0804.1357 [hep-th].

[26]R.A.Battye,S.Krusch,P.M.Sutcliffe,Phys.Lett.B626(2005)120,arXiv:hep-th/ 0507279.

[27]F.Navarro-Lérida,E.Radu,D.H.Tchrakian,Int.J.Mod.Phys.A29 (26)(2014) 1450149,arXiv:1311.3950 [hep-th].

[28]F.Navarro-Lérida,D.H.Tchrakian,Int.J.Mod.Phys.A30 (15)(2015)1550079, arXiv:1412.4654 [hep-th].