BPS Skyrmions as neutron stars

Contents lists available atScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

BPS Skyrmions as neutron stars

C. Adam

^a,∗

, C. Naya

^a

, J. Sanchez-Guillen

^a

, R. Vazquez

^a

, A. Wereszczynski

^b

aDepartamentodeFísicadePartículas,UniversidaddeSantiagodeCompostelaandInstitutoGalegodeFísicadeAltasEnerxias(IGFAE),E-15782Santiagode Compostela,Spain

bInstituteofPhysics,JagiellonianUniversity,Reymonta4,Kraków,Poland

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

Articlehistory:

Received17December2014

Receivedinrevisedform14January2015 Accepted21January2015

Availableonline23January2015 Editor:A.Ringwald

TheBPSSkyrmemodelhasbeendemonstratedalreadytoprovideaphysicallyintriguingandquantita- tivelyreliabledescriptionofnuclearmatter.Indeed,themodelhasboththesymmetriesandtheenergy–

momentum tensor of aperfect fluid, and thus represents a field theoretic realization ofthe “liquid droplet”modelofnuclearmatter.Inaddition,theclassicalsolitonsolutionstogetherwithsomeobvious corrections (spin–isospin quantization, Coulomb energy, proton–neutron mass difference) provide an accuratemodelingofnuclearbindingenergiesforheaviernuclei.Theseresultsleadtotherathernatural proposaltotrytodescribealsoneutronstarsbytheBPSSkyrmemodelcoupledtogravity.Wefindthat theresultingself-gravitatingBPSSkyrmionsprovideexcellentresultsaswellassomenewperspectives for the description ofbulk properties ofneutron stars when the parameter values ofthe model are extractedfromnuclearphysics.Specifically,themaximumpossiblemassofaneutronstarbeforeblack- holeformationsetsinisafewsolarmasses,theprecisevalueofwhichdepends ontheprecisevaluesof themodelparameters,andtheresultingneutronstarradiusisoftheorderof10km.

©^{2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

The calculation of physical observables ofstrongly interacting matteratlowenergies–relevant,e.g.,to nuclearphysics–directly fromQCD isa notoriouslydifficultproblem, whichled tothe in- troductionof low-energyeffectivefield theories(EFTs) asamore tractablealternative.The Skyrmemodelisawell-knownexample ofsucha low-energyEFT.Itwas introducedoriginally bySkyrme [1]asa purelymesonic nonlinearfield theoryforthedescription ofnuclei.Skyrme’sideawasthatnucleonsshouldbedescribed as a kind of “vorticity” in a mesonic “fluid” or, in a more modern language, as topological solitonsof the underlying mesonic non- linear field theory. And, indeed, the Skyrme model is known to possess topological solitons (“Skyrmions”) whose topological in- dex is identified with the baryon number. The original idea of Skyrme gained further support when it was observed that QCD inthelimitofalargenumberofcolors(largeNc)becomes athe- oryofweaklyinteractingmesons(interactionstrength∼^N−c¹)[2].

Suchweaklyinteractingnonlinearfieldtheoriesfrequentlypossess solitonicsolutionswithsolitonmassesproportionaltotheinverse

*

Correspondingauthor.

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

ofthe(weak)coupling,whichinthepresentcaseofthelarge N_c mesonicmodelofQCDareidentifiedwithbaryonsandnuclei,re- coveringtherebytheproposalofSkyrme.

TheSkyrmemodelhasbeenappliedtothedescriptionofnuclei withnotablesuccess,e.g.,inthedescriptionofrotationalexcitation bands of some light nuclei [3,4]. The version ofthe model orig- inally proposed by Skyrme,however,has some drawbacksin the description of physicalnuclei. First of all,Skyrmions withhigher baryon number B have ratherhighbinding energies (i.e.,masses significantly below B times the B=^{1 Skyrmion mass, see, e.g.,} [5]), which is instriking contrast to the low binding energies of physical nuclei. Also,Skyrmions forlarge baryon numbertend to formcrystalsoflower B substructures[6,7],whichisatoddswith theliquid-type behavior ofphysicalheavynuclei. Theseproblems recentlyledtoproposeseveral“nearBPS”Skyrmemodels,thatis, generalizations of the original Skyrme modelwhich are close to BPSmodels[8,9].HerebyaBPSmodelweunderstandafieldthe- orywhichhasbothanenergyboundforstaticfieldconfigurations which isexactlylinearinthe baryoncharge B andsolutions sat- urating the bound forall valuesof B (we shall assume B≥0 in the sequel,i.e., consideronlymatternot antimatter).Theoriginal SkyrmemodelisnotBPS.Ithasalowertopologicalenergybound, butit maybe showneasily that thisbound cannot be saturated.

Specifically,weconsiderthefollowingnearBPSSkyrmemodel[8]

http://dx.doi.org/10.1016/j.physletb.2015.01.027

0370-2693/©^{2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

(forthemomentinflatMinkowskispace;weusethe“mostlymi- nus”metricsignconvention diag(_gμν)= (+,−,−,−)^),

L

=

L0

+

L6

+  (

L2

+

L4

),

(1)

where

L2

= −λ

2tr L_μL^μ

,

L4

= λ

4tr



[

^Lμ

,

L_ν

] 

2

(2) and

L0

= −λ

0U

(

tr U

),

L6

= −λ

6

 

^μνρσtr L_νL_ρL_σ



2

≡ − 

24

π

²

^

²

λ

6BμB^μ

.

(3) HereU: R³× R→^SU(2) istheSkyrmefield, Lμ=^U†∂μU isthe left-invariantMaurer–CartancurrentandU ^{isa potential.The} λn aredimensionful,non-negativecouplingconstants, andBμ isthe topologicalorbaryonnumbercurrentgivingrisetothetopological degree(baryonnumber)B∈ Z^,

B^μ

=

¹

24

π

²



^μνρσtr L_νL_ρL_σ

,

B

=



d³xB⁰

.

(4) L2+L4 is the model originally considered by Skyrme, and the abovegeneralization is essentiallythe mostgeneralmodelwhich isbothPoincaréinvariantandnomorethanquadraticinfirsttime derivatives,such that a standard hamiltonian can be found. This generalizedSkyrmemodelisnearBPSforsufficientlysmallvalues ofthedimensionlessparameter



,becausethesubmodel

L06

=

L0

+

L6 (5)

isBPS. Thatis tosay, the staticenergyfunctional E₀₆[^U] ^{has an} energyboundlinearinB and(infact,infinitelymany)minimizing field configurationssaturating thebound foreach B,[8].Further, thisenergyfunctionalisinvariantundervolume-preservingdiffeo- morphisms (VPDs) on physical space, which are the symmetries of a perfect fluid. The energy–momentum tensor of the model L06 is, in fact, the energy–momentum tensor of a perfect fluid, aswe shall see below. These findings lead to the intriguing hy- pothesis thatthenear-BPSSkyrmemodel(1)mightbethecorrect low-energyEFT for the descriptionof nuclear matter, asthe BPS submodelL06 alreadyprovidesa rathergooddescriptionofsome ofitsstaticproperties.Indeed,theBPSSkyrmemodelallowsfora veryaccurate description of nuclearbinding energies [10,11], es- peciallyforheavy nuclei.Itisthepurposeofthepresentletterto coupletheBPSSkyrme modeltogravity andto usetheresulting self-gravitatingBPSSkyrmionsforthedescriptionofneutronstars.

Weremarkthattherealreadyexistseveralattemptstodescribe neutronstarsusingtheoriginalSkyrmemodel.In[12]thehedge- hogansatz forhigher B wascoupledtogravity butitturned out that – asin the non-gravitating case – higher B hedgehogs are notstable. In [13,14] approximateSkyrmionconfigurations based onrationalmapswereused.Probablythemostpromisingattempt within this context is using Skyrmion crystals [15,16] because Skyrmioncrystalsare thetrueminimizersofthe originalSkyrme modelforlarge B.The crystalstructure is,however,atodds with thefact that,mostlikely, thecoreofneutronstars isina super- fluid phase. Also, full numerical calculations are not possible in thiscasesuchthatcertainassumptionsabouttherightequationof stateofSkyrmecrystalsunder stronggravitationalfieldsmustbe made.Anaccessiblereviewcanbefoundin[17].

2. BPSSkyrmemodelandparametervalues

Conveniently redefining the coupling constants λ6= λ²/(24)² andλ0=

μ

²,thestaticenergyfunctionalofthetheoryis

E₀₆

=



d³x



π

⁴

λ

²B²₀

+ μ

²U

(

tr U

) 

.

(6)

ItsBPSbound E₀₆

≥

²

π

²

λ μ |

^B

| √

U

_S³

,  √

U

_S³

≡

¹ 2

π

²



S³ d

Ω √

U ⁽⁷⁾

(where √

U _S^{3 is the average value of} √

U ^{on the target space} SU(2)∼ S^{3)issaturatedbyinfinitelymanyBPSsolutions[8,18,19],} andthecorrespondingBPSequationis

π

²

λ

B0

± μ ^√

U

=

⁰

.

(8) We now have to determine the valuesof the parameters λ and

μ

to be used in our calculations. The product m≡ λ

μ

has the dimensions of mass (energy; we use units where the speed of light c=^{1). Further, l}≡ (λ/

μ

)^1/3 has the dimensions oflength.

We fit m by requiring that the BPS Skyrmion mass is B times one-fourthof themassof thehelium nucleus, E₀₆=^Bm¯N where

¯

mN=^mHe/4=⁹³¹.75 MeV. We use ^m¯N instead of the nucleon massmN∼940 MeV becausethelatterwillreceivecontributions from(iso)spinexcitationsinaSkyrmemodeldescription,butthese areabsentforhelium.Evenheliumreceivessmall(e.g.,coulombic) contributions in addition to the Skyrmion mass, but the uncer- tainty will be at mosta few MeV. To fix l, we usethe fact that BPS Skyrmions for many potentials (in particular, for the poten- tialsconsideredinthisletter)arecompactonswithastrictlyfinite volume V , andthis volume is the same for all solutions witha given baryon number B and is exactly linear in B. This permits todefineaSkyrmionradius R viaV = (⁴

π

/3)R³.Wenowrequire thatthisradiuscoincideswiththenucleonradiusrN=¹.25 fm for B=^{1,i.e., R}=^rNB^1/3.Wethink thatthefitforthemassparam- eter m isquite accurate, because by far the biggest contribution tothenuclearmassesmustalwayscomefromtheSkyrmionmass.

Onthe other hand,thefitforthe lengthparameter l is probably lessprecise.Firstly, although thecompactonradius isquitenatu- ral, thereareadditionaldefinitions forradii(diversecharge radii) which could be used. For compactons these charge radii are al- wayssmallerthanthecompactonradius,indicatingthatthelatter could be slightly bigger than thenucleon radius. Secondly,going beyondtheBPSsubmodelbyincluding,e.g.,theDirichlettermL2, the effectsofthe pioncloud willtendto increase theradius, in- dicating that the compacton radius R without pion cloud could besomewhat smaller.Tosummarize, althoughoursimplefitforl certainlyprovidesareasonablevalue,thetruebestfitvaluecould easily deviate about20% or 30% in either direction. Determining thistruevalue, however,requiresthe knowledgeofthe complete low-energy EFT withall terms (also the non-BPS ones) included, whichisbeyondthescopeofthisletter.

Concretely,weshallconsiderthepionmasspotentialUπ=¹− cosξ andthepionmasspotential squaredU4=Uπ with² aquartic behavior nearthevacuum(hereU=^exp(ξⁿ· 

τ

),ⁿ²=^{1 and}

τ

are thePaulimatrices)withenergies,compactonradiiandfitvalues Uπ

:

^E06

=

⁶⁴

√

2

π

15 B

λ μ ,

R

= √

2

 λ

B

μ



¹₃

⇒

^m

=

⁴⁹

.

15 MeV

,

l

=

⁰

.

884 fm (9)

U_π²

:

^E06

=

²

π

²B

λ μ ,

R

=



3

π

B 2



¹₃

 λ μ



¹₃

⇒

^m

=

⁴⁷

.

20 MeV

,

l

=

⁰

.

746 fm (10)

These expressions for the energies andcompacton radii may be calculateddirectlyfromthepotentials,see[20] (knowledgeofthe Skyrmionsolutionsisnotrequired).

3. BPSSkyrmionscoupledtogravity

The actionof the BPS Skyrmemodel ina generalmetric gρσ (hereg=det gρσ ),

S₀₆

=



d⁴x

|

^g

|

¹²



−λ

²

π

⁴

|

^g

|

^−1gρσB^ρB^σ

− μ

²U



,

(11)

leadstotheenergy–momentumtensor(B^{ρ isdefinedinEq.(4))} T^ρσ

= −

²

|

^g

|

⁻¹²

δ

δ

g_ρσ S₀₆

=

²

λ

²

π

⁴

|

^g

|

⁻¹B^ρB^σ

− 

λ

²

π

⁴

_|

g

|

^−1gπ ωB^πB^ω

− μ

²U



g^ρσ

,

(12)

which is the energy–momentum tensor of a perfect fluid (the perfect-fluidpropertyofthetermL6alone,aswellasitscoupling togravity,havealreadybeendiscussedin[21]),

T^ρσ

= (

p

+ ρ )

u^ρu^σ

−

pg^ρσ (13) wherethefour-velocityuρ ,energydensity

ρ

andpressurep are u^ρ

=

B^ρ

/ 

g_{σ π}B^σB^π (14)

ρ _{= λ}

²

π

⁴

_|

g

|

^−1gρσB^ρB^σ

+ μ

²U

p

= λ

²

π

⁴

_|

g

|

^−1gρσB^ρB^σ

− μ

²U

.

(15) Inthestaticcase,andforadiagonalmetric(whichissufficientfor ourpurposes)wehaveuρ= (

g⁰⁰,0,0,0)and

T⁰⁰

= ρ

g⁰⁰

,

T^{i j}

= −

^{pgi j}

.

(16) Intheflatspacecase,e.g., thisimpliesthat thepressuremustbe constant(zero forBPSsolutions, nonzerofornon-BPS staticsolu- tions[20]),asaconsequenceofenergy–momentumconservation,

D_ρT^ρσ

→ ∂

iT^{i j}

= δ

^{i j}

∂

ip

=

0

,

(17) whereas

ρ

will bea nontrivialfunction ofthespacecoordinates.

In general,

ρ

andp willbequitearbitraryfunctionsofthespace–

time coordinates, so theredoes not exista universal equation of state(EoS)p=^p(

ρ

)whichwouldbevalidforallsolutions.

WenowwanttocoupletheBPS Skyrmemodelto gravityand solvetheresultingEinsteinequationsforastatic,sphericallysym- metricmetricwhichinstandardSchwarzschildcoordinatesreads ds²

=

^A

(

r

)

dt²

−

^B

(

r

)

dr²

−

^r2



d

θ

²

+

^sin2

θ

d

φ

²



.

(18)

Forusthefollowingobservationiscrucial.Theaboveansatzforthe metrictogetherwiththeaxiallysymmetricansatzfortheSkyrme fieldwithbaryonnumberB

ξ = ξ(

^r

),

n

 = (

^sin

θ

cos B

φ,

sin

θ

sin B

φ,

cos

θ )

(19) leads to a baryon density B^{0, energy density}

ρ

and pressure p whicharefunctionsofr only.Theansatzis,thus,compatiblewith theEinsteinequations

G_ρσ

= κ

²

2 T_ρσ (20)

(here Gρσ is the Einstein tensor and

κ

² = ¹⁶

π

G = ⁶.654· 10⁻⁴¹fm MeV⁻¹) andthestaticEuler–Lagrangeequationsforthe Skyrmefield,andreducestheseequationstoasystemofordinary differential equations (ODEs) in the variable r for the three un- knownfunctionsA(r),B(r)andξ(r).Beforepresentingthissystem

of ODEs and the results of a numerical integration, we want to makesomecomments.

Firstly, in flat Minkowski space the same axially symmetric ansatz (19) (but referring to spherical polar coordinates in that case) was used in the calculations of nuclear binding energies in [10]. As said, the resulting binding energies are very accu- rate for heavier nuclei, but, nevertheless, once additional terms (like, e.g., the Dirichlet term



E2) are taken into account, there are strong arguments indicating that the axially symmetric BPS Skyrmions are not the adequate ones (they do not minimize E₂ amongallBPSSkyrmions)[22].Animprovedcalculationusingthe true minimizersof E2 andtakingthecontribution of E2 intoac- count shouldlead toevenbetter resultsforthebinding energies.

Herewejustwanttoemphasizethatinthecaseofself-gravitating BPS Skyrmions theaxially symmetric ansatz leading to a spheri- cally symmetricmetric,energydensityandpressureis thecorrect one,essentiallybecausegravitystraightensoutalldeviationsfrom sphericalsymmetry.Secondly,inthesubspaceofsphericallysym- metric solutions we maydefine a kindofEoS p=^p(

ρ

),because both

ρ

andp arefunctionsofr.Wefindnumericallythatasimple powerlaw

p

=

^a

ρ

^b (21)

reproduces thisEoS witha highprecision. Here, however, a and b arenot universalconstants.Instead,theydependonthebaryon number B. In particular, for“small” baryon number(small com- pared, e.g., tothe solar baryon number B ), where the effectof gravity maybe neglected, the constant a vanishes, limB→⁰a=⁰ (the pressure is zero like in the case without gravity). If we treated the gravitational coupling

κ

as a parameter which may vary then,of course,it wouldalso holdthat limκ→⁰a=^0.(Here wedefine B asB =^M /m¯N=¹.116·^1060MeV/931.75 MeV= 1.198·^{1057,sostrictly speakingB} isnotthenumberofbaryons inthesun,butthenumberofbaryons(neutrons)inaneutronstar withthesamenon-gravitationalmassasthesun.)

4. Numericalresults

We findit convenientto introducethe newtarget spacevari- able h= (¹/2)(1−^cosξ )=^sin2ξ₂ ^{inwhatfollows,withh}∈ [⁰,1] andUπ=^{2h.ThesystemofODEsresultingfromtheEinsteinequa-} tions maybe broughtintotheformofasystemoftwoequations forh andB,plusathirdequationwhichdeterminesA intermsof h andB.Explicitly,theseequationsread(^≡ ∂r)

1 r

B^ B

= −

¹

r²

(

B

−

¹

) + κ

²

2 B

ρ

(22)

r

(

Bp

)

^

=

¹

2

(

1

−

^B

)

B

( ρ +

^3p

) + κ

²

4 r²B²

( ρ −

^p

)

p (23) A^

A

=

¹

r

(

B

−

¹

) + κ

²

2 rBp (24)

where

ρ

andp fortheaxiallysymmetricansatz(hr≡ ∂rh)read

ρ ₌

^4B

2

λ

²

Br⁴ h

(

1

−

^h

)

h²_r

+ μ

²U

(

h

),

p

= ρ ₋

2

μ

²U

(

h

).

(25) Weintegratethesystem(22),(23)numericallyviaashootingfrom the center. That is to say, we impose the boundary conditions h(r=⁰)=1 (anti-vacuum value) and B(r=⁰)=^{1 (the amount} of matterenclosed atr=^{0 is zero}⇒ ^{flatspacemetric). We are} left with one free parameter, h₂, in the expansion about r=^0, h(r)∼¹− (¹/2)h2r²+ . . . ^or, equivalently,with

ρ

0≡

ρ

(r=⁰)= B²λ²h³₂+

μ

²U(1).Wethenintegratefromr=^{0 uptoapointr}=^R (compactonradius)whereh(R)=^{0 (thevacuum).Itfollowseasily}

Fig. 1. a) Neutronstarmassasafunctionofbaryonnumber,bothinsolarunits.Symboldot(redonline):potentialUπ.Symbolsquare(blueonline):potentialUπ².b) Neutron starmassasafunctionoftheneutronstarradius.PotentialUπ:symboldot(redonline).PotentialUπ²:symbolsquare(blueonline).Maximummassvaluesareindicatedby circles.ThestraightlineistheSchwarzschildmass.

fromEq.(23)that,foranon-singularmetricfunctionB,p atr=^R must obey p^(R)=^{0 which leads to a condition on h}r(R), con- cretely

4B²

λ

²

B

(

R

)

R⁴h²_r

(

R

) − μ

²Uh

(

0

) =

⁰

.

(26) Inthenumericalintegration,thefreeparametervalue

ρ

0isvaried untilthisconditionismet.Formalsolutionswhichdonotobeythis conditionproducemetricfunctionsB whicharesingularatr=^R.

In particular, such a metric function cannot be joined smoothly to theSchwarzschild solution in empty space(for r≥^{R) andis,} therefore,physicallyunacceptable.

We findthe following behavior in thenumerical integrations.

Forsufficientlysmallbaryonnumber B,thereexistspreciselyone

“initial value”

ρ

0 which obeys(26),i.e., one unique neutronstar solution.Forlargervaluesof B inacertainintervalB∈ [B_∗,Bmax], there exist two values for

ρ

0 leading to solutions fulfilling con- dition (26). Interestingly, this is exactly like in the case of the Tolman–Oppenheimer–Volkoff (TOV) calculation where the neu- tronsare described by a free relativistic fermi gas(see e.g. [23], Chapter11.4,p. 321).AsintheTOVcase,weassumethatthelower value

ρ

0 corresponds tothe stablesolution.Finally,for B>B_max solutions obeyingcondition (26)no longer exist.In other words, physicallyacceptablestaticsolutions(neutronstars)withB>Bmax donotexist. Instead,fieldconfigurations withsuch alarge B are unstable,indicatingthecollapsetoablackhole.

Theneutronstarsolutionfoundintheintervalr∈ [⁰,R]^isthen smoothlyjoinedto thevacuumsolutionforr≥^{R.Thatis tosay,} h(r)=^{0 forr}≥^R,andB(r)= (¹−^{2G M}_r )⁻¹,fromwhichthephys- icalmass M oftheneutronstar(withthegravitationalmassloss takenintoaccount)maybereadoff.

Oneofthemostimportantresultsis,ofcourse,thevalue Bmax andthecorrespondingmaximalneutronstarmassesM andradiiR forthetwopotentialsUπ andU_{π we}^{2 consider.Itisconvenientto} measureB insolarunitsn≡ (^B/B )(equivalently,n= (^Bm¯N/M ), i.e.,thenon-gravitationalmassofthebaryon numberB Skyrmion insolarmassunits).Then,usingthefitvalues(9)and(10),respec- tively,wefindforthemaximumvalues

Uπ

:

ⁿmax

=

⁵

.

005

,

M_max

=

³

.

734M

,

Rmax

=

¹⁸

.

458 km

,

(27)

U_π²

:

ⁿmax

=

³

.

271

,

Mmax

=

²

.

4388M

,

Rmax

=

¹⁶

.

801 km

.

(28)

We remark that neutron star masses up to about M∼^2M are firmlyestablished,whereas thereare indicationsformassesup to about2.5M ,seee.g. [24,25]foranoverviewofrecentmeasure- ments. The results of our calculationsare, therefore,in excellent agreementwiththeseobservations,indicatingthatourmodelpro- vides a very good description of the bulk properties of nuclear matter also inthe presence of thegravitational interaction.Con- cerningtheradii,weremarkthattheobservationalresultsareless precise.Besides,R isthegeometricradiuswhichleadstoaneutron starsurface area of4

π

R²,whereas whencomparingto measure- mentssometimesotherradiiaremoreappropriate,liketheproper distancefromtheorigintothesurface,R¯=R

0 dr√

B(r),orthera- diationradius R^∗=^R√

B(R).Both R and¯ R^∗ aresomewhatbigger than R becauseB(r)≥^{1.Inanycase,alsoourvaluesfortheradii} areintheexpectedrangeofaboutR∼^{10–20 km.Assaidalready,} ourfitfortheunitofmassm isquiteprecise(determinedby the nuclearmass^m¯N),buttheunitoflengthl islessso,thereforeitis interesting tostudythesensitivityofboth Mmax and Rmax under achangeofthelengthscale,l→^l=

α

l.Wefindnumericallythat both M_max and R_max approximately change by a factorof

α

^(3/2)

underthisrescaling.

Finally,weshowourmainnumericalresultsinFigs. 1–4.Con- cretely,inFig. 1aweplottheneutronstarmassasafunctionofthe non-gravitationalSkyrmionmass,bothinsolarunits.Wefindthat for the extremal case M_max the gravitationalmass loss is about 25%.InFig. 1bweplotM againstthe(geometric)neutronstarra- dius R. We findthat even in theextremal casethe neutron star radiusisaboutafactoroftwo abovetheSchwarzschildradius.In Fig. 2weshowtheequationofstatefordifferentvaluesofB (con- cretelyforn=^{1 inFig. 2a,andforn}max inFig. 2b) togetherwith thefitfunction p=^a

ρ

^bforappropriatevaluesofa,b.

InFig. 3,weplotthemetricfunction B(r)forseveralvaluesof the baryon number n=^B/B close to its maximum value n_max. We find for both potentials that the maximum value which B takes forn=ⁿmax is about Bmax∼².7. It is interesting to com- parethisfindingwiththeanalogousresultfortheSkyrmioncrystal of Ref. [16].There the authors calculated the minimumvalue of B⁻¹(whichwascalled S inthatpaper)fordifferentsolutionsand always found that Smin>0.4, which translates into Bmax<2.5.

So theBmax we findforthe maximummass caseisslightlybig- ger (i.e., the induced self-gravitation slightly stronger), but still

Fig. 2. Symbol plus (+, red online): potentialUπ. Symbol cross (×, green online): potentialUπ². Dotted lines: corresponding fit functions.

Fig. 3. The metric function B(r), for different solutions close to the maximum mass solution.

quite similar tothe resultof [16]. The positionof the maximum ofB(r)isquiteclosetotheneutron starsurface forthepotential Uπ ,whereasitisshiftedtowardsthecenterforU_{π .}² Thisisrelated tothe factthat, forU_{π ,}^{2 theenergy densityismore}concentrated aboutthecenter(seeFig. 4).

InFig. 4,weplottheenergydensitiesforseveralvaluesofthe baryonnumbercloseton_max.Wefindthat,especiallyforthepo- tentialU_{π ,}^{2 theenergydensityisquitesharply}concentratedabout the center. This may look surprising at first sight, but is simply related to the shape of the potential Uπ ,² which is quite peaked aboutthe anti-vacuum(h=^{1). Indeed,theBPS equation (8) just} statesthatthebaryondensityisproportionaltothesquarerootof thepotential,so peakedpotentialsleadtopeakedbaryon density (and energy density) profiles already in the case without grav- ity.It is perhaps more instructiveto compare the central energy densityofthe casewithout gravityto the central energydensity for n_max. The central energy density for the case without grav- ity does not depend on the baryon number B and is given by

ρ

BPS(r=⁰)=²

μ

²U(^h=¹). Using the parameter values (9), we

find for Uπ :

ρ

BPS(r=⁰)=⁴(m/l³)=^{285 MeV fm−3. The central} energy density for n_max is, therefore, about 2.7 times the non- gravitationalenergydensity

ρ

BPS(r=⁰),see Fig. 4a.Similarly, we get for Uπ :²

ρ

BPS(r=⁰)=⁸(m/l³)=^{909 MeV fm−3. In this case,} thecentralenergydensityfornmaxisjustabout2.2timesthenon- gravitationalenergydensity

ρ

BPS(r=0),seeFig. 4b.Theseresults in bothcasesindicate aratherhighstiffness oftheeffective(on- shell)EoSofstronglyself-gravitatingBPSSkyrmions,i.e.,anuclear matter which isonly weakly compressiblein stronggravitational fields. Thisresult, again, compares quite well with the Skyrmion crystalresultsofRef.[16],whereacompressionofthecentralen- ergydensitybynotmorethanafactorofthreeisobservedforall solutions.

5. Discussion

WeusedtheBPSSkyrmemodel(6)forthedescriptionofneu- tronstarsandfoundthatbysimplyfittingthetwomodelparam- eters to the nucleon mass and radius we already get very rea-

Fig. 4. The energy densityρ(r), for different solutions close to the maximum mass solution.

sonableresultsfortheresultingneutronstarmassesandradii.In particular, forthe maximum possible neutron star mass we find M_max=².44M orM_max=³.73M ,respectively,forthetwopo- tentialsconsidered. Thiscomparesextremelywellwiththeobser- vationalconstraintMmax∼².5M .Wetakethis,togetherwiththe perfectfluidbehavior ofthemodel,asafurtherverystrongindica- tionthat,indeed,theBPSSkyrmemodelprovidesthemostimpor- tantcontributiontothestaticbulkpropertiesofnuclearmatter.In astrict sense, ourresultsare notyet final predictionsofneutron starproperties,becausegenuinepredictionsrequiretheknowledge ofthefullnear-BPSSkyrmemodel(1)togetherwiththevaluesof allitscouplingconstants,whichshouldfollowfromanapplication to nuclear physics and the corresponding detailed fit to nuclear data.The fullnear-BPS Skyrmemodel mayalsolead toa further improvementinthe descriptionofneutronstars,inthefollowing sense. Even if the additional (standard Skyrme) terms are quite unimportant inthe bulk, this is not trueat the surface, because atthesurface the Skyrmefield is closeto its vacuumvalue, and theterm L6 approachesthe vacuummuch faster than the stan- dardSkyrmemodelterms.The standard Skyrmemodelisknown toprefercrystallinestructuresforlargeB,socrystallinestructures (“neutronstarcrust”)canbeexpectedatthesurfaceofaneutron stardescribed by the near-BPS Skyrme model,whereas the bulk andcore remain in a fluid phase. But precisely this structure is expectedincurrentmodelsofneutronstars(see,e.g.,[26]).

When compared withother, more traditional methods of nu- clearphysics, the advantage of the (near-)BPS Skyrme model at thismomentisnot so muchits abilityto makequantitative pre- dictions – although this, too, should change with more detailed investigationsandwithadvanced numericalmethods, assisted by a rigorousanalytical control which followsfrom theintegrability propertiesoftheBPSmodel.Afterall,themethodsandmodelsof nuclear physics are well developed and lead to very precise de- scriptions of nucleiand nuclear matter. However, a drawback of manymodelsofnuclearphysicsisthattheyaretailor-madetode- scriberatherspecific physical phenomena, thereforeit isdifficult to usethem for extrapolations to new phenomena or parameter values where they have not been employed before. We think it isone ofthe outstandingfeatures oftheBPS Skyrmemodel that itcapturesageneric propertyof(bulk)nuclearmatterandallows, therefore,forfar-reachingextrapolations.Concretely,inthepresent letterwe extrapolatedfrom B=^{1 (whichprovidedtheparameter} fitvalues)toB∼^{1057(theneutronstar)andfroma}nonrelativistic toahighlyrelativisticregime,withveryaccurateresults.Inother

words, the (near) BPS Skyrme model provides a unified descrip- tionof nuclearmatter, reachingfromnucleons andatomicnuclei toneutronstars.

There are two particular (related) results of our calculations which are somewhat different from most traditional nuclear physicscalculationsofneutronstarsusingtheTOVequations(22), (23), although they are completely compatible with all observa- tional data.In thetraditional approach, themetric function B(r), the energydensity

ρ

(r) andthe pressure p(r) are considered as independent field variables, so the two TOV equations(22), (23) must be closed by a third equation. For this, usually a univer- sal algebraic equation of state (EoS) p=^p(

ρ

) resultingfrom the thermodynamiclimitofanucleareffectivefield theory(EFT)(like QuantumHadronDynamics(QHD)[27])isassumed.Inourmodel, on the other hand, we find that already the EFT itself is of the perfect-fluid type defining its own energy density and pressure, both of whichdepend on the metric inan explicitfashion. It is, therefore,notpossibletodefineauniversal,algebraicoff-shellEoS, andthe trueoff-shellEoSrelating

ρ

and p isa complicatedand metric-dependent differential equation. We remark that our off- shellEoSsharesomefeatureswiththe“quasi-local”EoSexplicitly dependingonthegeometry(e.g.,metricorcurvature),whichwere introducedin[28]forthedescriptionofanisotropicstarsandfur- therstudiedin[29]and, inrelationwithneutronstars,in[30].It turnsoutthatinstarswithanisotropicmattersuchquasi-localEoS are evenrequiredforconsistency[28].Inourcase, itis stillpos- sibleto find(numerically)an on-shell algebraic EoSforsolutions

ρ

(r)andp(r),butthison-shellEoSisnolongeruniversalandde- pends on the neutron starmass or baryon number B.This does not mean thatthe EoSof nuclearmatter dependson thesample size.TheEoSfortheBPSSkyrmemodelwithoutgravityisalways thesame,p=^{0 at}equilibrium(nuclearsaturation),forarbitrary B.

The B dependence ofthe on-shellEoSforself-gravitatingnuclear matter in the BPS Skyrme model is exclusively a consequence of self-gravitation. Due to the nonlinearity of gravity, the effects of self-gravitationarestrongerforlarger B (largerneutronstarmass) and the effective on-shell EoS, therefore, gets stiffer. Concretely, we foundaneffectiveon-shellEoSofthetype p=^a(B)

ρ

^b(B),see Eq.(21),wherea(B)increaseswithincreasing B whereasb(B)de- creases.

Thisincreasing stiffness hasa particularphysical effectinthe casesweconsidered,namelyaneutronstarradius R whichgrows withtheneutronstarmassM, i.e., (dM/dR)>0 (except forstars veryclosetotheirmaximummassinthecaseofthepotentialUπ ,

seeFig. 1).Thisbehaviorisatvariancewiththeresultsfoundfor solutionsoftheTOVequationsformany(fixed,universal)nuclear physics EoS, where the neutron star radius is either essentially constantfora rangeofneutronstarmassesorevenshrinks with increasingmass[24].Thereasonforthisbehavioristhatforafixed EoS the increasing strength of self-gravitation for larger masses maycollapsethestartomuchhigherdensitiesand,forsofterEoS, eventosmallersizes.OnlysufficientlystiffuniversalEoSarecom- patiblewith(dM/dR)>0.We remarkthatone particularcaseof an EoS which is sufficientlystiff to support (dM/dR)>0 for al- mostall values of M is precisely given by the Skyrmecrystal of Ref. [16]. The M(R) curve found there is, in fact, quite similar to the one we find for the pion mass potential Uπ , see Fig. 1b.

Inthe BPS Skyrmemodel,the squeezing effectofnonlinear self- gravitationis balanced by the increasing stiffness of theon-shell EoS. We emphasize that, at present, (dM/dR)>0 is compatible withobservationsandthattheobservationaldataarenotyetsuf- ficientlyprecisetosettlethisquestion.If(dM/dR)>0 finallyturns outtobetrue,thiseitherrulesoutalargeclassofEoSwhichare wellmotivatedfromnuclearphysics,becauseonlyverystiff fixed EoSare compatiblewith(dM/dR)>0. Orit mayindicatethat in the traditional derivation of the EoS from an EFT like QHD one hastogobeyondmeanfieldtheory,suchthatbackreactioneffects ofgravityon theEoSmaybe takeninto account,atleastfornu- clear matter in sufficiently strong gravitational fields. A detailed discussionoftheseissueswillbegivenelsewhere.Inanycase,the qualitative results we found for the EoS within the BPS Skyrme modelalso pointtowards possible improvementsofthe standard nuclear physics approach to neutron stars in strong gravitational fields.

There are many ways in which the present investigation can bedeepenedandextended.Oneobviouspossibilityistouseaddi- tionalpotentialsandtostudyhowthe shapesofthesepotentials influencethepropertiesoftheresultingneutronstars,e.g.,which maximal massescan be reachedandforwhich potentialsthere- lation(dM/dR)>0 remains true.Anotherinterestingresearch di- rection is related to rotating neutron stars and to neutron stars inmagneticfields.Inprinciple,boththesetasksarerenderedfea- sible by the fact that it is known how to rotate Skyrmions (for a recent discussion see,e.g., [31]) and what is the correct, QCD induced couplingofSkyrmionsto theelectromagneticinteraction [32].Still,theresultingsystemsarenolongersphericallysymmet- ric,soafullsystemofPDEshastobesolvednumericallyinthese cases.Afurtherstepintheanalysiswouldbetousethefullnear- BPS Skyrme modelas a basis for the calculation of neutronstar solutionsandproperties,buthere,inafirststep,thedetailedap- plicationofthe near-BPSSkyrmemodelwithoutgravity tonuclei andnuclearmatterisrequired.AstheBPSandintegrability prop- erties are no longer available in thiscase, full three-dimensional numericalcalculationswillbenecessary.

Acknowledgement

The authors acknowledge financial support fromthe Ministry ofEducation,CultureandSports,Spain(grantFPA2011-22776),the XuntadeGalicia(grantINCITE09.296.035PRandConselleriadeEd- ucacion), the Spanish Consolider-Ingenio 2010 Programme CPAN

(CSD2007-00042), and FEDER. CN thanks the Ministry of Educa- tion, Culture and Sports, Spain for financial support (grant FPU AP2010-5772).Further,AWwassupportedbypolishNCN(National Science Center)grantDEC-2011/01/B/ST2/00464(2012–2014).JSG thanksM.A.Perez-Garciafordiscussions.

References

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

T.H.R.Skyrme,Nucl.Phys.31(1962)556;

T.H.R.Skyrme,J.Math.Phys.12(1971)1735.

[2]G.t’Hooft,Nucl.Phys.B72(1974)461;

E.Witten,Nucl.Phys.B160(1979)57;

E.Witten,Nucl.Phys.B223(1983)433.

[3]R.A.Battye, N.S.Manton,P.M. Sutcliffe, S.W.Wood,Phys. Rev.C 80(2009) 034323.

[4]P.H.C.Lau,N.S.Manton,e-Print,arXiv:1408.6680[nucl-th].

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

[6]M.Kugler,S.Shtrikman,Phys.Lett.B208(1988)491.

[7]L.Castillejo,P.S.J.Jones,A.D.Jackson,J.J.M.Verbaarschot,A.Jackson,Nucl.Phys.

A501(1989)801.

[8]C.Adam,J.Sanchez-Guillen,A.Wereszczynski,Phys.Lett.B691(2010)105;

C.Adam,J.Sanchez-Guillen,A.Wereszczynski,Phys.Rev.D82(2010)085015.

[9]P.Sutcliffe,J.HighEnergyPhys.1008(2010)019;

P.Sutcliffe,J.HighEnergyPhys.1104(2011)045.

[10]C.Adam,C.Naya,J.Sanchez-Guillen,A.Wereszczynski,Phys. Rev.Lett.111 (2013)232501;

C.Adam,C.Naya,J.Sanchez-Guillen,A.Wereszczynski,Phys.Rev.C88(2013) 054313.

[11]E.Bonenfant,L.Marleau,Phys.Rev.D82(2010)054023;

E.Bonenfant,L.Harbour,L.Marleau,Phys.Rev.D85(2012)114045;

M.-O.Beaudoin,L.Marleau,Nucl.Phys.B883(2014)328.

[12]P.Bizon,T.Chmaj,Phys.Lett.B297(1992)55.

[13]B.M.A.G.Piette,G.I.Probert,Phys.Rev.D75(2007)125023.

[14]S.G.Nelmes,B.M.A.G.Piette,Phys.Rev.D84(2011)085017.

[15]T.S.Walhout,Nucl.Phys.A484(1988)397;

T.S.Walhout,Nucl.Phys.A519(1990)816.

[16]S.G.Nelmes,B.M.A.G.Piette,Phys.Rev.D85(2012)123004.

[17] S.G.Nelmes,Skyrmionstars,Durhamtheses,DurhamUniversity,2012,avail- ableonlineat:http://etheses.dur.ac.uk/5258/.

[18]C.Adam,C.D.Fosco,J.M.Queiruga,J.Sanchez-Guillen,A.Wereszczynski,J.Phys.

A46(2013)135401.

[19]J.M.Speight,J.Phys.A43(2010)405201.

[20]C.Adam,C.Naya,J.M.Speight,J.Sanchez-Guillen,A.Wereszczynski,Phys.Rev.

D90(2014)045003.

[21]R.Slobodeanu,Int.J.Geom.MethodsMod.Phys.08(2011)1763.

[22]J.M.Speight,arXiv:1406.0739.

[23]S.Weinberg,GravitationandCosmology,Wiley,NewYork,1972.

[24]J.M.Lattimer,Annu.Rev.Nucl.Part.Sci.62(2012)485;

A.W.Steiner,J.M.Lattimer,E.F.Brown,Astrophys.J.765(2013)L5;

K.Hebeler,J.M.Lattimer,C.J.Pethick,A.Schwenk,Astrophys.J.773(2013)11;

J.M.Lattimer,A.W.Steiner,Astrophys.J.784(2014)123.

[25]F.Özel,G.Baym,T.Güver,Phys.Rev.D82(2010)101301;

F.Ozel,D.Psaltis,R.Narayan,A.SantosVillarreal,Astrophys.J.757(2012)55;

T.Guver,F.Ozel,Astrophys.J.765(2013)L1.

[26]H.Horowitz,J.Piekarewicz,Phys.Rev.Lett.86(2001)5647;

J.Piekarewicz,J.Phys.Conf.Ser.492(2014)012008;

C.J.Horowitz, M.A.Perez-Garcia, D.K. Berry,J. Piekarewicz,Phys. Rev.C72 (2005)035801.

[27]J.D.Walecka,Ann.Phys.83(1974)491;

B.D.Serot,J.D.Walecka,Int.J.Mod.Phys.E6(1997)515.

[28]C.Cattoen,T.Faber,M.Visser,Class.QuantumGravity22(2005)4189;

M.Visser,PoSBHGRS(2008)001.

[29]D.Horvat,S.Ilijic,A.Marunovic,Class.QuantumGravity28(2011)025009.

[30]H.Silva,C.Macedo,E.Berti,L.Crispino,arXiv:1411.6286.

[31]R.A.Battye,M.Haberichter,S.Krusch,e-Print,arXiv:1407.3264[hep-th].

[32]C.G.Callan,E.Witten,Nucl.Phys.B239(1984)161.