The dielectric Skyrme model

Contents lists available atScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

The dielectric Skyrme model

C. Adam

^a,∗

, K. Oles

^b

, A. Wereszczynski

^b

aDepartamentodeFísicadePartículas,UniversidaddeSantiagodeCompostela,InstitutoGalegodeFísicadeAltasEnerxias(IGFAE),E-15782 Santiagode Compostela,Spain

bInstituteofPhysics,JagiellonianUniversity,Lojasiewicza11,Kraków,Poland

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received10May2020

Receivedinrevisedform8June2020 Accepted11June2020

Availableonline17June2020 Editor:A.Ringwald

We consider a version of the Skyrmemodel where both the kineticterm and the Skyrme termare multipliedbyfield-dependentcouplingfunctions.Forsuitablechoices,this“dielectricSkyrmemodel”has staticsolutionssaturatingthepertinenttopologicalboundinthesectorofbaryonnumber(ortopological charge)B= ±^{1 butnotforhigher}|^B|^{.Thisimpliesthathigherchargefield}configurationsareunbound, and looselyboundhigher skyrmionscanbe achievedbysmalldeformations ofthisdielectricSkyrme model.Weprovideasimpleandexplicitexampleforthispossibility.

Further,weshowthatthe|^B|=^{1 BPSsectorcontinuestoexistforcertain}generalizationsofthemodel like,forinstance,afteritscouplingtoaspecificversionoftheBPSSkyrmemodel,i.e.,theadditionofthe sextictermandaparticularpotential.

©^{2020TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

TheSkyrmemodel [1–3] isone particularproposalfora low- energy effective field theory (EFT) of strong-interaction physics [4–6] and,inparticular,forthedescriptionofbaryonsandnuclei.

Itsprimaryfieldsaremesons,whereasbaryonsandnucleiemerge astopologicalsolitons(Skyrmions)[7–10] supportedbythemodel.

The Skyrmemodel incorporatesmany nontrivialfeatures of low- energyQCD(baryonnumberconservation,chiralsymmetryandits breaking,currentalgebraresults,. . . )inacompletelynaturalway.

It also reproduces some quantitative properties of nucleons and severallightnucleiwithreasonablesuccess[11–19].Severalshort- comings,however,impedeits useasa generalandquantitatively preciseEFT ofnuclearphysics.Twomajorproblemsofthemodel arethetoolargebindingenergiesofhigher-chargeSkyrmionsand theabsenceofalpha-particleclustersinsidethem.Physicalnuclei have rather small binding energies (always below 1%) and fre- quentlypossessan alpha-particlesubstructure. TheSkyrmemodel permitsmanygeneralizations,e.g.,theadditionofmoreterms[20]

and the inclusion of further meson fields [21–25], and some of these generalizations allow to significantly alleviate these short- comings[26–34].

Themodelproposed in thepresentletteris mainlymotivated by the first problem (the too large binding energies), because the theory of topological solitons provides simple and system-

*

Correspondingauthor.

E-mailaddress:adam@fpaxp1.usc.es(C. Adam).

aticmethodsto searchformodelswithsmallbindingenergies.A systematicmethod topredict alpha-particle substructuresforthe Skyrmions of a particular model,on the other hand,is currently not known. To answer this question, at present full numerical calculations ofhigher charge solitons are required [34]. Abetter qualitativeunderstandingoftheformationofsubstructureswithin Skyrmionswouldcertainlybedesirable.

TheoriginalversionoftheSkyrmemodelrestrictsthefieldcon- tenttopionsandisgivenbythefollowingLagrangian density

L

=

L2

+

L4

+

L0 (1)

where

L2

=

^c2Tr

∂

_μU

∂

^μU^†

,

L4

=

^c4Tr

( [

^Rμ

,

R_ν

])

^{2 (2)}

are the kinetic (Sigma model or Dirichlet) termand the Skyrme term, respectively. Here, the field U takes values in SU(2), and Rμ = ∂μU U⁻¹ is the right-invariant current. Further, L0 =

−^c0U(^{Tr U}) is a potential term, where a frequent choice is the pionmasspotential Uπ= (¹/2)Tr(I−^U).Finally,thec_i arecou- plingconstants.

Inthesmall-fieldlimit,whichisrelevantforlargedistances,the Dirichlet termquadraticin the pionfield dominates andinduces attractivechannelsbetweenSkyrmions. Thatis tosay,there exist certain relative orientationsbetween individual B=1 Skyrmions (modeling nucleons) such that they attract each other,and they maybearrangedinarrayssuchthat allnearest neighborsattract.

Theformationofboundstates,i.e.,theexistenceofhighercharge

https://doi.org/10.1016/j.physletb.2020.135560

0370-2693/©^{2020TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

Skyrmions with restenergies below the total mass oftheir con- stituentsis,therefore,expected, andthisisindeedwhathappens.

Ifthepionmasstermquadraticinthepionfieldisincluded,then the attractive forces change from power-law to exponential, but boundstatesstill form.Theresultingbindingenergiesare,infact, muchlargerthanthebindingenergiesofphysicalnuclei(see,e.g., [7]).

TheSkyrmemodel(1) permitsmanygeneralizations.Firstofall, moregeneral choicesforthe potential apart fromthe pionmass termare possible.Secondly,terms withhigherpowers of deriva- tives may be added. Among these, a particular term of a sixth powerinfirstderivativesissingledout,

L6

= −(

²⁴

π

²

)

²c₆BμB^μ

,

(3)

becausethistermisstillquadraticintimederivativesandleadsto astandardHamiltonian.Here,B^{μ isthebaryoncurrent,}

B^μ

=

¹

24

π

²



^μνρσTr R_νR_ρR_σ

,

B

=



d³xB⁰

,

(4)

whichallowsto calculatethebaryon number(topologicalcharge) B.Thirdly,inadditiontothepions,furtherfieldsmaybeincluded inthemodel.

Giventhis vastlandscape ofpossible generalizations,arbitrar- ily choosing a model inside it and calculating its higher charge Skyrmions does not seem to be an efficient strategy for find- ingmodels withlow bindingenergies. Topologicalsolitonmodels like thegeneralized Skyrmemodels, however,allow to findnon- trivial topological energy bounds [35], [36]. For Skyrme models in Minkowski space-time, they are always exactly linear in the topological charge (baryon number) [28], [37]. Further, so-called Bogomolnyequationscanbefound[35],whichimplythatthecor- respondingboundissaturated.TheseboundsandtheirBogomolny equationsarevaluabletoolsinthesearchforSkyrmemodelswith smallbindingenergies.Inafirststep,certainsubmodels(so-called BPS models) must be identified which possess both a topologi- calbound andnontrivialsolutions(BPSSkyrmions)saturatingthe bound.TheexistenceoftheBPSSkyrmionsimpliesthat thebind- ingenergiesinthemodelareeitherzeroornegative.Modelswith smallbindingenergies canthenbeconstructedbycertain“small”

deformationsoftheBPSsubmodels.

More concretely, a BPS submodel may either support BPS Skyrmions witharbitrarybaryon number.Theirenergies arethen exactlylinearinB andtheresultingbindingenergiesarezero.This isthe caseof theBPS Skyrme model[26], consistingof thesex- tic term (3) and an arbitrary potential, or of the Skyrme model withaninfinitetowerofvectormesons [32],obtainedfroma di- mensional reduction of a higher-dimensional Yang-Mills theory.

TheotherpossibilityisthattheBPSsubmodelhasaBPSSkyrmion onlyin the B= ±^{1 sector.Higher} B solutions are then unstable, andtheresulting“bindingenergies” arenegative. Inother words, theinduced forces betweenthe B=^{1 BPSSkyrmions are always} repulsive. Thisis the caseofthe BPS submodel discovered by D.

Harland[28],andthemodelinvestigatedinthepresentletteralso belongs to this class. We remark that for BPS submodels based onlyonpionfields,eithertheabsenceoftheDirichlettermorits suppression inthe small-field limit (e.g.,by making it effectively higherthansecondorder,orbyenhancingotherterms)isaneces- sarycondition,asfollowsfromtheattractiveforcesinducedbythis term.Any deformationtoarealistic near-BPSmodelmustcorrect thisbehaviorintheregionofsmallpionfields.

Inthepresentletter,wefurtherdeveloptherecentobservation thattheminimalSkyrmemodelwithaveryparticular runningpion decay constant enjoys the BPS property [38]. We generalize this finding to the case of two running coupling constants (the pion decayconstant fπ andtheSkyrmeparametere).Here,theformof

thesefunctionsisarbitrarywhiletheirproductisfixedbytheBPS condition.Thenweshowhowanear-BPSsectorcanbereachedvia abreaking oftheBPScondition. Importantly,alreadythesimplest and most natural BPS breaking is capable of producing arbitrar- ilysmallbindingenergiesandcanbringthecouplingconstantsto their physicalvalues. Finally, we consistently (i.e.,preserving the BPS property) add the usual BPS Skyrme model into the frame- work.

2. ThedielectricSkyrmemodel

WeconsiderthestaticenergyfunctionaloftheminimalSkyrme model(1),

E

=

^Ed2

+

^Ed4

,

(5)

consistingofthekineticterm(Dirichletenergy), E^d₂

=



R³

−

^f2

2 Tr

(

R_iR_i

)

d³x

,

(6)

andthequarticSkyrmeterm E^d₄

=



R³

−

¹

16e² Tr

( [

^Ri

,

R_j

][

^Ri

,

R_j

])

^d3x

.

(7)

Incontrasttothestandardcase,however,weassumethatinstead ofthecouplingconstants c₂ andc₄ nowwehavefield-dependent coupling functions f and e. This is explicitly indicated by the index d (dielectric).In particular, we assume that they are func- tions of the trace of the Skyrme field Tr U , implying that the isospinsymmetry remainsunbroken.Equivalently,usingthestan- dardparametrization(here

τ

arethePaulimatrices)

U

=

^exp

(

i

ξ(

x

) τ  _{· }

n

(

x

)),

(8) oftheSkyrmefieldintermsofaprofilefunctionξ andanisospin unit vector n, the couplingfunctionsonly depend onthe profile,

f=^f(ξ ),e=^e(ξ ).

2.1. TopologicalboundandBogomolnyequations

Tofindthepertinenttopologicalboundwefollowthestandard method based on the three eigenvalues λ²_i of the strain tensor D_{i j}= −¹₂ ^Tr(R_iR_j)[39].Then, thestaticenergycan berewritten as

E^d

=



R³



f²



λ

²₁

+ λ

²₂

+ λ

²₃



+

¹

e²



λ

²₁

λ

²₂

+ λ

²₂

λ

²₃

+ λ

²₃

λ

²₁



d³x

=



R³



(

f

λ

1

±

¹

e

λ

2

λ

3

)

²

+ (

f

λ

2

±

¹

e

λ

3

λ

1

)

²

+ (

f

λ

3

±

¹ e

λ

₁

λ

₂

)

²



d³x

∓

⁶



R³ f

e

λ

₁

λ

₂

λ

₃d³x

≥

⁶



R³ f

e

λ

₁

λ

₂

λ

₃d³x

=

¹²

π

²

f e



|

^B

|,

⁽⁹⁾

whereF istheaveragevalue ofatargetspacefunctionF over thewholeS³ targetspace.Thelaststepfollowsfromthefactthat thebaryondensityB0 isjust

B0

=

¹

2

π

²

^λ

¹

^λ

²

^λ

³

^.

⁽¹⁰⁾

The bound is saturated if and only if the following dielectric Bogomolnyequationshold,

f

λ

1

±

¹

e

λ

2

λ

3

=

0

,

f

λ

2

±

¹

e

λ

1

λ

3

=

0

,

f

λ

3

±

¹

e

λ

1

λ

2

=

0

,

(11) whichafterastraightforwardmanipulationresultin

λ

²₁

= λ

²₂

= λ

²₃

=

^e2

(ξ )

f²

(ξ ) .

(12) Inthestandardcase,where f ande areconstant,theBogomolny equationscannotbesatisfiedfornon-zero B inR³ space.Indeed, thenalleigenvaluesmustbeconstantwhichcontradictsthefinite- nessoftheenergyforsolutionsoftheBogomolnyequations.Here, ontheother hand,the r.h.s.of(12) isafunction ofξ,whichcan tendto0for|^x|→ ∞^{. Ifwe requirethatthe Skyrmefield U ap-} proachestheperturbativevacuumU= I^(i.e.,ξ→^0)for|^x|→ ∞^, thenatleastoneofthetwocouplingfunctions, e or f ,mustap- proachzeroatξ=^{0.Weshallfindthat thesolutionpresentedin} thenextsectionautomaticallyobeysthisrequirement.

2.2. B=^{1 BPSsolution}

It isknown that for R³ base space theBogomolny equations (12) admitatopologicallynontrivialsolutionif

e f

=

¹

2r₀Tr

(I −

^U

)

(13)

where r0>0 (we chose the plus sign). In fact, this result was firstobtainedinthe contextofa BPS submodelconsidered by D.

Harland[28],consistingoftheSkyrmetermandtheparticularpo- tentialU4= (^Tr(I−^U))⁴.ThecorrespondingBogomolnyequations are identical to (12) after the identification (e f)⁴=U4 andhave solitonic solutions in the B= ±1 topological sectors. Skyrmions with higher values of the baryon charge do not obey the Bo- gomolny equations and, therefore, do not saturate the pertinent topologicalbound.

Following[28],thecharge B=^{1 solutionofourmodelcanbe} easily found. We assume a naturalspherical symmetry provided bythehedgehogansatz ξ= ξ(^r),n= (^sinθcosφ,sinθsinφ,cosθ ) insphericalpolarcoordinates (r,θ,φ).Theunit three component isovector^{n can} ^{beexpressedviathe}stereographicprojectionbya complexfieldu



n

=

¹

1

+ |

u

|

²



2

(

u

),

2

(

u

),

1

− |

u

|

²



,

(14)

whereu(θ,φ)=^tanθ₂^eiφ.Thentheeigenvaluesare

λ

²₁

= ξ

r²

, λ

²₂

= λ

²3

=

^sin2

ξ

r²

.

(15)

Theequalityoftheλ²_i leadstoafirstorderODE

ξ

_r

= −

^sin

ξ

r

.

(16)

Thisshouldbecompletedwiththethirdequalityin(12)

ξ

r

= −

¹

r₀

(

1

−

^cos

ξ ).

(17)

Thesetwoequationshaveacommonsolution

ξ =

^{2 arctanr0}

r (18)

whichinterpolates between

π

and0 asr changesfrom 0 toin- finity.ThisfinallyconstitutesaBPSSkyrmionwithunittopological

charge.Weremarkthatsolution(18) alsocoincideswiththesolu- tionfoundin[40],wheretheSkyrmefieldiscoupledtoasecond, non-dynamicalfield.

Tocompute the energyofthe B=^{1 solution we mustdecide} howthecouplingfunctionsdependonthetargetspacecoordinate ξ. Thesefunctionsare, however, arbitraryprovided thecondition (13) isfulfilled.Hereweconsiderthefollowingpossibility

e

=

^e0

(

1

−

^cos

ξ )

^α

,

f

=

^f0

(

1

−

^cos

ξ )

^1−α (19) witharealparameter

α

.Further,e₀ and f₀ aredimensionalcon- stantssettingtheenergyscale E0= _e^f0₀ ^{andthelength scaler}0=

1

f0e0. They donothavetocorrespondtophysicalvalues.Thenus- ingtheexplicitformulaforthetargetspaceaverageintegral

f e



=

²

π



π

0

sin²

ξ

d

ξ =

²

π

f₀ e₀



π

0

sin²

ξ(

1

−

^cos

ξ )

^1−2αd

ξ

=

²

·

^41−α

√ π

f₀ e₀

 

₅

2

−

2

α

[

⁴

−

²

α ]

⁽²⁰⁾

finally,weget E^d

=

¹²

π

²²

^·

⁴

1−α

√ π

f₀ e₀

 

₅

2

−

²

α

[

⁴

−

²

α ] .

(21)

Despite the appearance of a zero ine (and, forsome parameter values,asingularityin f )theenergyintegralconvergesfor

α

<⁵₄. This includes threequalitatively very differentcases. Namely, for

α

<1 thecouplingfunction f increaseswithξ,from f(ξ=⁰)=⁰ to f(ξ=

π

)=^{21−α f}0,whichmeansthatithasbiggervalueinside theBPSSkyrmionthaninthevacuum.For

α

=^{1,thefunction f is} just aconstant.Finally,for

α

∈

1,⁵₄

, f decreaseswithξ.There- fore,ithasabiggervalueinthevacuumthaninsidetheSkyrmion (here f(ξ=⁰)= ∞^{).We remarkthatthecase}

α

=^{0 was consid-} eredalreadyin[38],inaslightlydifferentcontext.

TheBPSconstraintonlydeterminestheratiobetweentherun- ningcouplings f ande,therefore,theabovechoiceofthesefunc- tions should be regardedjust asan exampleto get a qualitative pictureoftheallowedpossibilitiesinthedielectricSkyrmemodel.

Indeed,theparticularformofoneofthefunctionscan bechosen arbitrarily, e.g., by fitting to the in-medium dependence of cou- pling constants. Thesefits, however,should always be done after breaking the BPS property (see below), where the vacuum con- stantsapproachtheirphysicalvalues.

The restrictive form of the Bogomolnyequations excludes so- lutions withhigher valuesofthetopologicalcharge forthe same model(thesamedielectricfunctionse and f ).Theargumentspre- sentedin[28] likewiseholdforthedielectric Skyrmemodel.The result is that higher charge Skyrmions are not BPS solitons and haveenergieshigherthan B·^E(B=¹).Thus,theyareenergetically unstabletowardsadecayintoacollectionofseparatedchargeone BPSSkyrmions.Toconclude,therearenostableB>1 Skyrmions.

Obviously,the fact that atleast one ofthe couplingfunctions approacheszeroatthevacuummayhavea significantimpact on the (time-dependence of) non-BPS solutions (solutions ofthe di- electricBPSmodelthatdonotobeytheBPSequations).Forexam- ple, forour choice (19), we findthat theprofile function of any non-BPSsolutionmusttendtothevacuumfasterthan1/rⁿ,where n>1/(4

α

−⁴)(sphericalsymmetryassumed).Forrealisticphysical applications,however,alwaysanear-BPScompletionofthedielec- tricBPSmodelshouldbeconsidered,wherethecouplingfunctions tendtotheirphysicalvaluesinthevacuum(seebelow).Then,the time-dynamicsaswellastheperturbativepropertiesofthemodel closetothevacuumcoincidewiththestandardSkyrmemodel.

2.3. Adielectricnear-BPSSkyrmemodel

Thebehavior(13) ofthedielectricfunctionsisnotphenomeno- logically acceptable closeto the vacuum ξ=^{0. A rather obvious} proposal for a realistic near-BPS model is a deformation which changes(13) toanonzerovalueforξ→^{0 butleavesit}essentially untouched forsufficiently large ξ. Thisproposal has the twofold advantage that, i), it recovers the correct small-field limit and, ii), it should provide small binding energies, because small-field regions only make small contributions to the total energy. For a fullyrealistic modelone probablypreferssmooth dielectric func- tions f(ξ ) and e(ξ ), but for the less ambitious goal of finding estimatesforthebindingenergiesofnear-BPSmodels,continuous functionsaresufficient.Concretely,weshallassumee=^1,whereas f is given by the expression resulting from (13) for ξ∈ [ξ∗,

π

_], butbythe constantvalue f_∗= ^f(ξ_∗) forthenear-vacuumregion ξ∈ [⁰,ξ_∗]^{.Forourspecific examplewe choose}ξ_∗= (

π

/6) which issmallbutnotverysmall.Further,thesizeparameterr0 isirrel- evant for ourenergy considerations, therefore we choose r₀=^1.

Thatistosay,we choosededielectricSkyrmemodelwiththedi- electricfunctions

e

=

¹

,

f

=



₁

−

^cos

ξ . . . ξ ∈ [ξ

∗

, π ]

f_∗

≡

¹

−

^cos

ξ

_∗

=

¹

−

^√₂³

. . . ξ ∈ [

⁰

, ξ

_∗

] , ξ

_∗

≡ π

6

.

(22)

Forξ∈ [ξ∗,

π

]^,thecontribution E^> totheenergyisgivenby the accordinglyrestrictedBPSbound,

E^>

=

^E>BPS

=

¹²

π

²

_

1

−

^cos

ξ 

>

=

²⁴

π



π

π 6

d

ξ

sin²

ξ(

1

−

^cos

ξ )

= π (

10

π +

¹

+

³

√

3

) 

¹¹⁸

.

162

.

(23) Forξ∈ [⁰,ξ_∗]^,thecontributionto theBPS boundissimply given bytheaccordinglyrestrictedSkyrme-Faddeevbound,

E^<_BPS

=

²⁴

π

f_∗

π



6

0

d

ξ

sin²

ξ = π

f_∗

(

2

π ₋

3

√

3

) 

⁰

.

4576

.

(24)

ForthecontributionE^<tothetrueenergy,weshould,inprinciple, find the hedgehog solution of the minimal Skyrme model (with couplingconstantse=^{1 and f}= ^f∗)intheregionξ∈ [⁰,ξ_∗]^with the corresponding boundary conditions. But if we only want to findanupperboundforthebindingenergy,thenanupperbound E^<_b >E^< issufficient.Suchupperboundscanbefoundbyinsert- ing certain trial functions instead of the true hedgehog solution intotheenergyfunctional.Onefirstpossibilityistousethesame BPSsolution(18), butitturnsout thatthisisalousyapproxima- tion,becauseithasthewronglarger behavior.Thesolutionofthe minimal Skyrme modelbehaves like r⁻² for large r, so the sim- plestpossibletrialfunctionwiththisbehavioris

ξ

^<

(

r

) = ξ

∗r²_∗

r²

. . .

r

>

r_∗

,

r_∗

=

¹

tan₁₂^π

=

¹ 2

− √

3

.

(25) Here,r_∗ istheradiuswheretheBPSsolution(18) takesthevalue ξ_∗,such thattheBPSsolutionforr≤^r∗ andthetrialfunction ξ^<

forr>r_∗ togetherdefineacontinuousfunction.Itsfirstderivative isno longercontinuous (has a finitejumpatr_∗), butthisissuf- ficientlyregular forourenergyestimates.Thenumericalvaluesof ourparametersare

f_∗



0

.

1340

,

r_∗



3

.

7320

,

C

≡ ξ

_∗r²_∗



7

.

2928

.

(26) Fortheupperenergyboundwefinallyget

E^<_b

=

E^<_b,2

+

E^<_b,4

≡

4

π



∞ r_∗

drr²

(

Eb,2

+

Eb,4

)

(27)

where Eb,2

=

^f_∗²



4C²

r⁶

+

^2sin2

(

C

/

r²

)

r²



(28)

Eb,4

=

^8C2 r⁸ sin² C

r²

+

^sin4

(

C

/

r²

)

r⁴

.

(29)

Performingtheintegrationsnumerically,thisleadsto

E^<_b,2

=

⁴

π

f_∗²

·

²

.

0203

=

⁰

.

4559

,

(30) E^<_b,4

=

⁴

π _·

0

.

2359

=

⁰

.

2359

,

(31) E^<_b

=

^E<_b,2

+

^E_b^<_,4

=

⁰

.

6918

.

(32) For therelative binding energies we, therefore, getthe following upperbound,



E

(

B

)

E

(

B

) ≡

^{B E}

(

1

) −

^E

(

B

)

E

(

B

) ≤

^E<b

−

^E<_BPS

E^>_BPS

+

^E_BPS^<



⁰

.

235

118

.

6

=

⁰

.

00198

,

(33) where E(B)is the energyofa skyrmion withbaryon number B.

Inotherwords,relativebindingenergiesforthedielectricSkyrme model defined by the dielectric functions of Eq. (22) must al- waysbebelow0.2%.Thisexampledemonstratesthatthedielectric Skyrmemodelnotonlyallowstofindsubmodelswithsmallbind- ing energies.It provides,infact,an extremelysimple andnatural mechanismtoconstructsuchmodelswitharbitrarilysmallbinding energies.

Weremarkthatthesamemechanismholdsforanyotherchoice oftherunningfunctions,alsothosewhere f takesasingularvalue atξ=^{0.Thisisimportant,sinceaparticularformof f (and,con-} sequently,e)shouldnotonlyreproduceitsvacuumvalue fπ (after theBPSbreaking) butalso,atleastqualitatively, theexperimental in-mediumdependence.

3. Inclusionofthesextictermandpotential 3.1. DielectricBPSSkyrmemodel

If generalizations of the Skyrme model at most quadratic in time derivatives are considered, which still lead to a standard Hamiltonian, thentwopossibletermsmaybe added.Namely, the sextictermwithstaticenergy E6=^g2

π

²B²₀,andanon-derivative term, i.e., a potential E₀≡U^{. The two terms together form the} so-calledBPSSkyrmemodel

E_{B P S}

=

^E6

+

^E0 (34)

Here g isapositiveconstant.

This model is interesting because some of its properties co- incide with several relevant features of nuclear matter (atomic nuclei).Firstofall,itisaBPStheorywherethecorrespondingBo- gomolnyequationhassolitonicsolutionsinany topologicalsector.

Thus,stableBPSsolitonswitharbitraryvaluesofthebaryoncharge exist.Asaconsequence,themodelprovideszerobindingenergies attheclassicallevel.Ratherrealisticphysicalbindingenergiescan beobtainedalreadywithinthemodel,ifsomenaturalclassicaland quantumcorrectionsareincluded[27].Secondly,thestaticenergy

functionalenjoysalargesymmetrygroup,i.e.,thevolumepreserv- ingdiffeomorphisms.ThismeansthattheenergyofaBPSsoliton doesnot depend on its shape and is constant provided the vol- umeremains unchanged. Infact, thissymmetry isthe symmetry ofaliquidifthesurfaceenergyisnegligible.Furthermore,theBPS SkyrmeLagrangiandescribesaperfectfluid.Thisreproducesatthe fieldtheoreticleveltheliquiddropmodel.

Letusnowpromotetheconstant g toa functionofthetarget spacevariable,Tr U .Wedonotconsideranydielectricfunctionfor E0,asanyfunctionaldependencemayincorporatedintothepoten- tial.Theresultingdielectricversion oftheBPSSkyrmemodelstill possesses a non-empty self-dual sector. To see this, we compute thecorrespondingtopologicalbound

E^d_{B P S}

=



R³



g² 4

π

²

^λ

2

1

λ

²₂

λ

²₃

+

U



d³x

=



R³



_g

2

π ^λ

¹

^λ

²

^λ

³

^±

√

U



d³x

∓

¹

π



R³ g

√

U

λ

1

λ

2

λ

3d³x

≥

¹

π



R³ g

√

U

λ

1

λ

2

λ

3d³x

=

²

π ^

g

√

U



|

^B

|.

⁽³⁵⁾

The bound is saturated if and only if the following Bogomolny equationisobeyed

g

2

π ^λ

¹

^λ

²

^λ

³

^±

√

U

=

0

.

(36)

ThiscanbetransformedintotheBogomolnyequationofthestan- dard(non-dielectric)BPSSkyrmemodelbyintroducinganewpo- tential ˜U =^g2U^{.Inotherwords,thedielectricfunctioncanalways} beincorporatedintothepotential.Asaconsequence,theSDiff in- varianceofthe BPSsolutions survives.To conclude,the dielectric generalizationoftheBPSSkyrmemodeldoesnotchangethemain qualitative properties of the model. In realistic applications, the functionsg andU^{should,ofcourse,be}constrainedbyexperimen- taldata.

3.2. B=^{1 BPSsolutionofthe}generalizeddielectricSkyrmemodel

Owing to the large freedom in the (dielectric) BPS Skyrme model,itispossibletofindsuchfunctionsg andUthattheBogo- molny equationshares somesolutions withtheBogomolny equa- tionsforthedielectricminimalSkyrmemodel(11).

Intheunitchargesector,aSkyrmionagaincanbeobtainedby thehedgehogansatz.Thus,eq. (36) leadsto

1 2

π

g

(ξ )

sin²

ξ ξ

r

r²

= − √

U ⁽³⁷⁾

Assumingthatthesolutionshouldbeoftheform(18) wecanfind that

√

U

g

=

¹

2

π

r³₀

(

1

−

cos

ξ )

³

≡

¹

16

π

r³₀

(

Tr

(I −

U

))

³ (38) Thus,wehaveproventhatthegeneralizeddielectricSkyrmemodel

E^d

=

^Ed2

+

^Ed4

+

^Ed6

+

^E0 (39)

has a BPS unit charge solution provided the dielectric functions andthepotentialobeyrelations(13) and(38).

Notethattheadditionoftheminimal(dielectric)Skyrmemodel breaks the SDiff symmetry explicitly.Furthermore, higher charge Skyrmions are again unstable towards decay into separated BPS Skyrmions.

4. Possibleapplicationsandconclusions

A first, general observation is that there exists a rather large freedom in the construction of near-BPS Skyrme models, where solitons form bound states with small binding energies. The di- electricSkyrmemodelsproposedinthepresentletterconstitutea newandinterestingpossibilityforthisphenomenon,whichshould befurther explored.Inparticular,theinclusionofthesexticterm (or the BPS Skyrme model) into the full BPS submodel is inter- esting,because thispartofthe completemodelgivestheleading behaviorinthehighdensity(pressure)regime[41].

Smallclassical binding energies require that theBPS property in the B=^{1 sector is weakly broken. Thiscan be achievedby a} deformationoftheconstraintsonthedielectricfunctionsandthe potential (13), (38). For anydeformation to a physically relevant near-BPS modelwe should impose thatclose to theperturbative vacuum(ξ=^{0)thedielectric functionstendto thenon-zerovac-} uumvalues f(ξ=⁰)= fπ ande(ξ=⁰)=^e0.Buttheseconditions onlyaffectthesmall-fieldregionswhichproviderathersmallcon- tributions to the total energyof a soliton. Inother words, ifwe choosedeformationssuch thatthecouplingfunctionsremain(al- most)unchangedforlargefield values,thentheresultingbinding energiesareexpectedtobesmall.Wedemonstratedinaconcrete andsimpleexamplethatthisisindeedthecase,andtheresulting bindingenergiescan bemadeextremelysmall.Thatistosay,the mostnaturaldeformationsofthedielectricBPSSkyrmemodel,i.e., those which just recover the phenomenologically correct small- field limit, automatically provide very low binding energies, by construction.

Arelatedquestionconcernspossiblephysicalinterpretationsor justificationsforthe(deformed)couplingfunctions.Fromaneffec- tive field theory point ofview, they simplycorrespond to higher order terms inthe field expansion which maybe taken into ac- count. They are not forbidden because they respect the relevant symmetries.Further,afterthedeformation,theyalsoreproducethe correctsmall-fieldlimit.Anotherpossibilityistointerpretthedi- electricfunctions asin-medium coupling constants, inparticular, f asanin-mediumpiondecayconstant.Infact,owingtothearbi- trarinessofoneofthecouplingfunctionsintheminimaldielectric Skyrmemodel, f maybechosentoqualitativelyreproducethein- medium behavior of thepiondecayconstant. Thesameconcerns thedielectric function g and especiallythepotential U^{,which in} thesmallfield-limitinvacuumshould tendtothepionmasspo- tentialm²Tr(I−^U).

The possible relation between (near)-BPS structures and in- mediumpropertiesofSkyrmions(in-mediumSkyrmemodels,see, e.g., [42–45])indicated aboveis aninteresting observationwhich deservesamoreprofoundinvestigation.

Weremarkthatthecondition(38) alsoallowstoaddthemodel E4+U4 of[28].

Declarationofcompetinginterest

Theauthorsdeclarethattheyhavenoknowncompetingfinan- cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.

Acknowledgements

TheauthorsacknowledgefinancialsupportfromtheMinistryof Education,CultureandSports,Spain(GrantNo.FPA2017-83814-P), theXuntadeGalicia(Grant No.INCITE09.296.035PRandConselle- riadeEducacion),theSpanishConsolider-Ingenio2010Programme CPAN(CSD2007-00042),MariadeMaetzuUnitofExcellenceMDM- 2016-0692,andFEDER.

References

[1]T.H.R.Skyrme,Anonlinearfieldtheory,Proc.R.Soc.Lond.260(1961)127.

[2]T.H.R.Skyrme,Aunifieldfieldtheoryofmesonsandbaryons,Nucl.Phys.31 (1962)556.

[3]T.H.R.Skyrme,KinksandtheDiracequation,J.Math.Phys.12(1971)1735.

[4]G.t’Hooft,Aplanardiagramtheoryforstronginteractions,Nucl.Phys.B72 (1974)461.

[5]E.Witten,Baryonsinthe1/N expansion,Nucl.Phys.B160(1979)57.

[6]E.Witten,Currentalgebra,baryons,andquarkconfinement,Nucl.Phys.B223 (1983)433.

[7]N.Manton,P.Sutcliffe,TopologicalSolitons,CambridgeUniversityPress,Cam- bridge,UK,2007.

[8]H.Weigel,Chiralsolitonmodelsforbaryons,Lect.NotesPhys.743(2008)1.

[9]M.Rho,I.Zahed(Eds.),TheMultifacetedSkyrmion,2edition,WorldScientific, Singapore,2016.

[10]Y.-L.Ma,M.Rho,EffectiveFieldTheoriesforNucleiandCompact-StarMatter, WorldScientific,Singapore,2019.

[11]G.S.Adkins,C.R.Nappi,E.Witten,StaticpropertiesofnucleonsintheSkyrme model,Nucl.Phys.B228(1983)552.

[12]E.Braaten,L.Carson,ThedeuteronasasolitonintheSkyrmemodel,Phys.Rev.

Lett.56(1986)1897.

[13]L.Carson,B=3 nucleiasquantizedmultiskyrmions,Phys.Rev.Lett.66(1991) 1406.

[14]T.S.Walhout,Multiskyrmionsasnuclei,Nucl.Phys.A531(1991)596.

[15]O.V.Manko, N.S.Manton, S.W.Wood,Light nucleias quantizedSkyrmions, Phys.Rev.C76(2007)055203.

[16]R.A.Battye,N.S.Manton,P.M.Sutcliffe,S.W.Wood,Lightnucleiofevenmass numberintheSkyrmemodel,Phys.Rev.C80(2009)034323.

[17]P.H.C.Lau,N.S.Manton,Statesofcarbon-12intheSkyrmemodel,Phys.Rev.

Lett.113(2014)232503.

[18]C.J.Halcrow,VibrationalquantisationoftheB=7 Skyrmion,Nucl.Phys.B904 (2016)106.

[19]C.J.Halcrow,C.King,N.S.Manton,Adynamicalα-clustermodelofO-16,Phys.

Rev.C95(2017)031303.

[20]A.Jackson,A.D.Jackson,A.S.Goldhaber,G.E.Brown,L.C.Castillejo,Amodified Skyrmion,Phys.Lett.B154(1985)101.

[21]G.S.Adkins,C.R.Nappi,Stabilizationofchiralsolitonsviavectormesons,Phys.

Lett.B137(1984)251.

[22]G.S.Adkins,RhomesonsintheSkyrmemodel,Phys.Rev.D33(1986)193.

[23]U.G.Meissner,I.Zahed,Skyrmionsinthepresenceofvectormesons,Phys.Rev.

Lett.56(1986)1035.

[24]B.Schwesinger,H. Weigel,Vectormesonsversushigherordertermsinthe Skyrmemodelapproachtobaryonresonances,Nucl.Phys.A465(1987)733.

[25]M.Bando,T.Kugo,K.Yamawaki,Nonlinearrealizationandhiddenlocalsym- metries,Phys.Rep.164(1988)217.

[26]C.Adam, J.Sánchez-Guillén, A.Wereszczynski, A Skyrme-typeproposal for baryonicmatter,Phys.Lett.B691(2010)105,arXiv:1001.4544.

[27]C.Adam,C.Naya,J.Sánchez-Guillén,A.Wereszczynski, Bogomol’nyi-Prasad- SommerfieldSkyrmemodelandnuclearbindingenergies,Phys.Rev.Lett.111 (2013)232501,arXiv:1309.0820.

[28]D.Harland,Topologicalenergybounds forthe Skyrmeand Faddeevmodels withmassivepions,Phys.Lett.B728(2014)518,arXiv:1311.2403.

[29]M.Gillard,D.Harland,M.Speight,Skyrmionswithlowbindingenergies,Nucl.

Phys.B895(2015)272,arXiv:1501.05455.

[30]S.B.Gudnason,B.Zhang,N.Ma,GeneralizedSkyrmemodelwiththeloosely boundpotential,Phys.Rev.D94(2016)125004,arXiv:1609.01591.

[31]S.B.Gudnason,ExploringthegeneralizedlooselyboundSkyrmemodel,Phys.

Rev.D98(2018)096018,arXiv:1805.10898.

[32]P.Sutcliffe,Skyrmions,instantonsandholography,J.HighEnergyPhys.1008 (2010)019,arXiv:1003.0023.

[33]C.Naya,P.Sutcliffe,Skyrmionsinmodelswithpionsandrhomesons,J.High EnergyPhys.1805(2018)174,arXiv:1803.06098.

[34]C.Naya,P.Sutcliffe,Skyrmionsandclusteringinlightnuclei,Phys.Rev.Lett.

121(2018)232002,arXiv:1811.02064.

[35]E.B.Bogomolnyi, Stabilityofclassicalsolutions,Sov.J. Nucl.Phys.24(1976) 449.

[36]L.D.Faddeev,Somecommentsonthemany-dimensionalsolitons,Lett.Math.

Phys.1(1976)289.

[37]C.Adam,A.Wereszczynski,TopologicalenergyboundsingeneralizedSkyrme models,Phys.Rev.D89(2014)065010.

[38]C.Naya,K.Oles,Backgroundfieldsandself-dualSkyrmions,arXiv:2004.07069.

[39]N.Manton,GeometryofSkyrmions,Commun.Math.Phys.111(1987)469.

[40]L.A.Ferreira,Exactself-dualityinamodifiedSkyrmemodel,J. HighEnergy Phys.1707(2017)039,arXiv:1705.01824.

[41]C.Adam,M.Haberichter,A.Wereszczynski,Skyrmemodelsandnuclearmatter equationofstate,Phys.Rev.C92(2015)055807.

[42]U.-G.Meissner,Bosonexchangephenomenology:afirststepfromnucleonsto nuclei,Nucl.Phys.A503(1989)801.

[43]U.-G.Meissner,V.Bernard,TheNambu-Jona-Lasiniomodel:applications and limitationsofastrongcouplingtheory,CommentsNucl.Part.Phys.19(1989) 67.

[44]G.E.Brown,M.Rho, ScalingeffectiveLagrangiansinadensemedium, Phys.

Rev.Lett.66(1991)2720.

[45]Y.-L.Ma,M.Rho,RecentprogressondensenuclearmatterinSkyrmionap- proaches,Sci.China,Phys.Mech.Astron.60(2017)032001,arXiv:1612.06600.