Boson and Dirac stars in D ≥ 4 dimensions

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Boson

and

Dirac

stars

in

D

≥

4 dimensions

Jose

Luis Blázquez-Salcedo

a,

∗

,

Christian Knoll

a

,

Eugen Radu

b

a_{InstitutfürPhysik,UniversitätOldenburg,Postfach2503,D-26111Oldenburg,Germany}

b_{DepartamentodeFísicadaUniversidadedeAveiroandCenterforResearchandDevelopmentinMathematicsandApplications(CIDMA),} CampusdeSantiago,3810-183Aveiro,Portugal

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received20February2019 Receivedinrevisedform3April2019 Accepted12April2019

Availableonline17April2019 Editor:M.Cvetiˇc

We present a comparative study of spherically symmetric, localized, particle-like solutions for spin s=0,1/2 and 1 gravitating fields in a D-dimensional,

asymptotically flat

spacetime. These fields are massive, possessing a harmonic time dependence and no self-interaction. Special attention is paid to the mathematical similarities and physical differences between the bosonic and fermionic cases. We find that the generic pattern of solutions is similar for any value of the spin s,

depending only on the

dimension-ality of spacetime, the cases D=4,5 being special.

©2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3_.

1. Introductionandmotivation

Thefirstexplicitrealizationoftheideaofstable,localized bun-dlesofenergyasa modelforparticlescan betracedback tothe work of Kaup [1] and Ruffini andBonazzalo [2], fifty years ago. Theyfoundasymptotically flat, sphericallysymmetricequilibrium solutionsofthe Einstein-scalarfield systeminfourspacetime di-mensions.These bosonstars are macroscopicquantum statesand areonly preventedfromcollapsinggravitationallyby the Heisen-berguncertaintyprinciple[3],[4].However,asshownintherecent work[5],similarconfigurationsexistalsoforamodelwitha grav-itatingmassivespin-onefield, representingProcastars.Analogous solutions(althoughlessknown)werefoundalsoinEinstein-Dirac theory[6],thefermionsbeingtreatedasclassicalfields.

Acomparative studyofallthesecasescanbefoundinthe re-centwork[7],whereithasbeennoticedthat,asclassicalfield the-orysolutions, the existenceof such self-gravitatingenergylumps doesnot distinguishbetweenthefermionic/bosonicnatureofthe fields,possessing avarietyof similarfeatures. Forexample,in all casesthere is a harmonictime dependencein thefields (with a frequency w), together witha confiningmechanism, asprovided byamass

μ

=

0 oftheelementaryquantaofthefield.Moreover, whenignoringthePauli’sexclusionprinciple,the(field frequency-ADMmass)-diagramofthesolutionslookssimilarforbothbosonic andfermionic stars. Also, the existence of these(nontopological)

*

Correspondingauthor.

E-mailaddress:[email protected](J.L. Blázquez-Salcedo).

solitons can be related to the fact that they possess a Noether charge Q , associatedwithaglobalU(1)globalsymmetry.

As withother models,it isof interestto seehow the dimen-sionality D of spacetime affects the properties ofthe solutions.1 Forexample,onewouldliketoknowwhichoftheirpropertiesare peculiartofour-dimensions,andwhichholdmoregenerally.Thus, attheveryleast,suchastudywillleadtoadeeperunderstanding of D

=

4 case. However,relativelylittleisknownaboutthe prop-ertiesofthistypeofconfigurationsinmorethanfourdimensions. WhileD

=

5 spin-zerobosonstarsarediscussedin[9],thecaseof spins

=

1

/

2

,

1 fieldswasnotconsideredatall.

The purpose of this work is to consider a comparative study ofscalar, Proca andDiracparticle-like solutionsin D

≥

4 dimen-sions,lookingforsphericallysymmetricsolutions whichapproach at infinity a Minkowski spacetime background. In this approach, one deals with classical field theory solutions, the quantum ef-fects beingignored. Our results show that, despite the existence of some common features, the cases D

=

4

,

5 are rather special. Perhapsthemoststrikingfeatureisthat theD

>

4 solutions con-figurations do not connect continuously to Minkowskispacetime vacuum, withthe existence of a mass(and Noethercharge) gap. Also,whenignoringthePauli’sexclusionprinciple,theDiracstars sharethepatternofthebosonicconfigurations.

1 _{Thisquestionhasbeenaddressedintheliteraturefor theEinstein(-Maxwell)}

theorymainly,inwhichcaseithasbeenfoundthatwhilethestaticD

≥

5 black holessharethebasicpropertiesofthefour-dimensionalcounterparts,avarietyof newfeaturesappearinthephysicsofhigherdimensionalspinningsolutions[8]. https://doi.org/10.1016/j.physletb.2019.04.035

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

Thepaperisstructuredasfollows:inSection2wepresentthe generalframeworkforbothgeometryandmatterfields.Special at-tentionispaidtotheconstructionofasuitablespin s

=

1

/

2 field Ansatzcompatiblewithasphericallysymmetricspacetimemetric, ataskwhich,toourknowledge,hasnotbeenyetaddressedinthe literature.InSection 3the fieldequationsare solvednumerically, andit isshown howthe dimensionalityof spacetime affectsthe propertiesofthesolutions.WeconcludewithSection4wherethe resultsarecompiled.

2. Theframework

2.1. Thegeneralactionandlineelement

WeconsiderEinstein’sgravityinD-spacetimedimensions min-imally coupled with a spin-s field (with s

=

0

,

1₂

,

1), generically denotedas

U

,thecorrespondingactionbeing

S

=



dDx

√

−

g



1 16

π

GR

+

L

(s)



.

(1)

Extremizingtheaction (1) leads toa systemofcoupled Einstein-matterequationsofmotion

Rαβ

−

1 2R gαβ

=

8

π

G T (s) αβ

,

with T (s) αβ

=

2

√

−

g

δ

L

(s)

δ

gαβ

,

(2)

whilethevariationof(1) w

.

r

.

t.

U

leadstothematterfields equa-tions.

Weareinterestedinhorizonless,nonsingularsolutions describ-ing particle-like, localised configurations with finite energy. Re-strictingforsimplicitytosphericallysymmetricconfigurations,the corresponding spacetime metric is most conveniently studied in Schwarzschild-likecoordinates,with

ds2

=

dr 2 N(r)

+

r 2_d2 D−2

−

N(r)

σ

2

(r)dt

2

,

with N

(r)

≡

1

−

2m

(r)

rD−3

,

(3)

whered



2_D−2denotesthelineelementona

(

D

−

2

)

−

dimensional sphere,whiler andt are theradialandtimecoordinates, respec-tively.This Ansatz introduces two functionsm

(

r

)

and

σ

(

r

)

, with m

(

r

)

being related to the local mass-energy densityup to some D-dependentfactor. Also,its asymptoticvalue,m0,fixes thetotal

ADMmassofthespacetime,

M

=

(D

−

2

)V

D−2

8

π

G m0

,

(4)

(with VD−2 the areaofthe

(

D

−

2

)

-sphere). Anadvantageofthe

above metricformis thatit leads tosimple firstorder equations forthefunctionsm

,

σ

,

m

= −

8

π

G D

−

2r D−2_Tt t

,

σ



=

8

π

G D

−

2 r

σ

N

(T

r r

−

Ttt

).

(5) Thegroundstateofthemodelis

U =

0,togetherwithaflat space-timemetric(i.e. m

=

0,

σ

=

1).

2.2. Themattercontent

Inall threecases,theLagrangian

L

(s) possesses a global U

(

1

)

invariance,underthetransformation

U →

eia

_U

_{,witha being} con-stant.Thisimpliestheexistenceofaconserved4-current, jα_(s);α

=

0.Integratingthetimelike componentofthiscurrentona space-likesliceyieldsaconservedquantity–theNoethercharge:

Q(s)

=

VD−2 ∞



0

dr rD−2

σ

jt_(s)

.

(6)

Upon quantization, Q becomes an integer–the particle number. Also, oneremarks thattheADMmass M and theNoethercharge Q providetheonlyglobalchargesofthesystem.

2.2.1. s

=

0:amassivecomplexscalarfield

We start withthe simplest case of a complex scalar field



withaLagrangiandensity

L

(0)

= −

gαβ

¯

,α



, β

−

μ

2

¯,

(7)

the energy-momentum tensor, the current andthe Klein-Gordon equationbeing T(_α0_β)

= ¯

,α



,β

+ ¯

,β



,α

−

gαβ



1 2g γδ

_{( ¯}

_

,γ



,δ

+ ¯

,δ



,γ

)

+

μ

2

¯



,

(8) jα₍₀₎

= −

i( ¯

∂

α



− ∂

α

¯), ∇

2



−

μ

2



=

0

.

(9) Ascalarfield Ansatzwhichiscompatiblewitha spherically sym-metricgeometryiswrittenintermsofasinglerealfunction

φ (

r

)

, andreads:



= φ(

r)e−i wt

.

(10)

Thescalarfieldamplitudesolvestheequation

φ



+



D

−

2 r

+

N N

+

σ



σ



φ



+ (

w 2 N

σ

2

−

μ

2

₎

φ

N

=

0

,

(11)

whiletheEinsteinequationsimply

m

=

8

π

G D

−

2r D−2



Nφ2

+

μ

2

φ

2

+

w 2

_φ

2 N

σ

2



,

σ



=

16

π

G D

−

2r

σ



φ

2

+

w 2

_φ

2 N2

_σ

2



.

(12)

Finally,following[10,11],onecanprovethevirialidentity

∞



0 dr rD−2

σ



(D

−

3

)φ

2

+ (

D

−

1

)

μ

2

φ

2



=

w2 ∞



0 dr

(

2

(D

−

2

)N

−

D

+

3

)

r D−2

_φ

2 N2

_σ

,

(13)

which shows e.g. that a nontrivial scalar field can only be sup-portedifw

=

0.

2.2.2. s

=

1:amassivecomplexvectorfield

For any value of D, the complex Proca field is described by a potential 1-form

A

withthe associated field strength

F =

d

A

(where we denote the corresponding complex conjugates by an overbar,

A

¯

and

F

¯

). The corresponding Lagrangian density, field equations,currentandenergy-momentumtensorare

L

(1)

= −

1 4

F

αβ

F

¯

αβ

₋

1 2

μ

2

_A

α

A

¯

α

,

∇

α

F

αβ

−

μ

2

A

β

=

0

,

jα₍₁₎

=

i 2

¯

F

αβ

_A

β

−

F

αβ

A

¯

β

,

(14) T(_α1_β)

=

1 2

(

F

ασ

F

¯

βγ

+ ¯

F

ασ

F

βγ

)g

σ γ

₋

1 4gαβ

F

σ τ

F

¯

σ τ

+

1 2

μ

2

_A

α

A

¯

β

+ ¯

A

α

A

β

−

gαβ

A

σ

A

¯

σ

.

(15)

Notethat thefield equations implythe Lorentz condition (which fora Proca field is not a gauge choice, buta dynamical require-ment),

∇

α

A

α

=

0.

The 1-form Ansatz compatible witha static, spherically sym-metricgeometrycontainstworealpotentials,F

(

r

)

andG

(

r

)

:

A

=



F

(r)dt

+

iG(r)dr



e−i wt

,

(16)

whichsolvetheequations

F

= (

w2

−

μ

2N

σ

2

)

G w

,

G

+



(D

−

2

)

r

+

N N

+

σ



σ

+

w F N2

_σ

2



G

=

0

,

(17)

thecorrespondingequationsforthemetricfunctionsbeing

m

=

4

π

G D

−

2r D−2



(

F

−

w G)2 2

σ

2

+

μ

2



G2N

+

F 2 N

σ

2



,

σ



=

8

π

G D

−

2

σ

r

μ

2



G2

+

F 2 N2

_σ

2



.

(18)

Inthiscasethesolutionssatisfythefollowingvirialidentity

μ

2 ∞



0 dr rD−2

σ



(D

−

3

)G

2

−

F 2

₍

₂

_(D

₋

₂

_)N

₋

_D

₊

₃

₎

N2

_σ

2



= (

D

−

3

)

∞



0 dr rD−2

(w G

−

F 

₎

2

σ

.

(19)

2.2.3. s

=

1

/

2:massiveDiracfields

It is also of interest to consider a fermionic matter content andlookfor theexistence of particle-likesolutions of the corre-spondinggravitatingsystems.However,while inthebosonic case (s

=

0

,

1)oneexpectsthatsuchaclassicaltreatmentofthe(many particle) systems is consistent, this is not the casefor fermions, which are intrinsically quantum fields. Thus the construction of Dirac stars (although historically motivated by the research on geons[12] and solitonicsolutionswhich canbe usedaseffective modelsofnucleons[13])israthera(pure)mathematicalproblem. Thisis, however,worth to investigate, possessingvarious curious features.

Forexample,restrictingto aDiracfield (s

=

1

/

2), oneremarks thatamodelwithasingle(backreacting)spinorisnotcompatible withasphericallysymmetricspacetime.Thus,asintheD

=

4 case [6], one should consider severalspinors withequal mass

μ

and thesamefrequency w,eachonepossessingaspecificangular de-pendence.Althoughanindividualenergy-momentumtensorisnot sphericallysymmetric,thesumofallcontributionsleadstoaresult whichis compatiblewith the line-element(3). In D-dimensions, suchconfigurationcanbeconstructedwith(atleast)nf

=

2

D−2 2



spinors

[A] ( A

=

1

...

nf),eachonewithaLagrangiandensity

L

[A] (1/2)

= −

i



1 2



{ ˆ/

D [A]

}

[A]

−

[A]D

ˆ/

[A]



+

μ

[A]

[A]



,

(20) thetotalenergy-momentumtensorandtheindividualcurrent be-ing T_α(1_β/2)

=

nf



A=1 T_α[A_β]

,

with T_α[A_β]

= −

i 2



[A]

γ

(αD

ˆ

β)

[A]

−



ˆ

D(α

[A]



γ

β)

[A]



,

jα₍₁[_/A₂]₎

= ¯

[A]

γ

α

[A]

.

(21)

Also,eachspinorsolvestheDiracequation

ˆ/

D [A]

−

μ

[A]

=

0

.

(22)

The Dirac equation in a D

>

4 spherically symmetric back-ground has been extensively studied in the literature, see e.g. Ref. [14].Thushereweshallonlyreviewthebasicsteps,together withthespecialchoiceoftheAnsatzwhichleadstoatotal energy-momentumtensor T_α(1_β/2) whichiscompatiblewiththestaticand spherically symmetric line-element (3). To achieve this aim, we impose the spinors to satisfy a set of conditions, which can be summarizedasfollows.2

i)TheseparableAnsatz. Thefirststepistoassumesome simple butgenericAnsatzforeachfield

[A].Sinceeachindividualspinor satisfies equation (22), we want to make use of the separability oftheangularandtheradial dependenceofeachfield. Hencewe assumethat:

•

theradialdependenceisthesameforallindividualspinors;

•

thetemporaldependenceisoftheformofaphase(similarto thepreviouslyconsidered s

=

0

,

1 fields),andalsocommonto allindividualspinors;

•

theonlydifferencebetweenspinorsisintheangularpart. SuchanAnsatztakestheform

[A]

=

e−iwt

φ

κ

(r)

⊗ 

(κ;A)

,

(23)

where

φ

κ

(

r

)

onlydependsontheradialcoordinate,and



(κ;A) on

theangularcoordinates.

ii) Solutionsoftheangularpart. In the second step, we solve theangularpart ofthedecomposition(23). Sinceeach individual spinorsatisfiestheequation(22),wemakeuseoftheseparability oftheangularandtheradial dependenceofeachfield [15].Also, itisconvenienttochooseaparametrizationofthe

(

D

−

2

)

-sphere with

d



2_D−2

=

d

θ

₁2

+

sin2

θ

1d

φ

21

+

cos2

θ

1d



2D−4

.

(24)

For D

>

5,asimilardecompositionof d



2_D−4 canbe doneby in-troducing further

φ

j and

θ

j coordinates on the

(

D

−

4

)

-sphere. Suchatowercanbeconstructeduntilalltheanglesofthesphere areparametrized(withd



2₀

=

0,d



2₁

=

d

φ

2_D−2

2

,



x

denotingthe ceilingfunction).Thedecompositionmakesthe



D−2

2



commuting azimuthal Killingdirections

φ

j explicit,andwe have

∂

φj



(κ;A)

=

imj



(κ;A) (mjbeingahalf-integer).

Onecanshow thattheangularpartofthespinor



(κ;A) isan

eigenfunction ofa Dirac angularoperator

K

D−2, with

κ

the

cor-respondingangulareigenvalue [14,16]. Workingwiththe vielbein

ω

t

₌

√

_N

_σ

_dt

_,

_ω

r

₌

_√1

Ndr

,

ω

j

₌

_r

_ω

j

D−2 (where j

=

1

,

...,

D

−

2

and

ω

_Dj₋₂ is a vielbein for the

(

D

−

2

)

-sphere), a tower of an-gular operators in the lower dimensional spheres is found, with

[

K

D−2

,

K

D−4

]

=

0,

[∂φ

1

,

K

D−2

]

=

0,etc.

Asa result,withthisparametrization, theangulardependence of the spinors can be explicitly solved, and the general solution isgivenbyacombinationofhypergeometricfunctions.3 _However,

forour purpose,itis enoughto considerthe casewherethe an-gular eigenvalue is minimal (which means that each individual

2 _ForD

₌

_{4,theconstruction ofaspins}

₌

_{1/2 Ansatzcompatiblewitha}

spher-icallysymmetricspacetimeisdiscussedatlengthinRef. [6] (althoughforrather differentconventionsthaninthiswork).

3 _{Theseangularsolutionsarethespinormonopoleharmonicsfora}

(D−2)-sphere [17];also,theyarerelatedwiththesolutionsoftheangularpartin[18].

spinor carries the minimally allowed angular momentum value). Theanalysisofthesolutionsshowsthatthishappenswhen

κ

= ±

(D

−

2

)

2 and m1

= ±

1

2

.

(25)

Apartfromtheglobalsignof

κ

,whichdeterminesthe“spirality”of theconfiguration,in practice,thesolution onthe

(

D

−

2

)

-sphere possesses

D−₂2



free sign combinations (



x



denoting the floor function).

iii) Combinationoftheangularpart. Hence, the third step isto select a proper combination of all these different angular parts ofthe spinors,making useof allthese freesigns. Onecan easily provethat, ifonechooses nf

=

2

D−2 2



spinors, theangularparts canbe combined insuch awaythat almost all thenon-diagonal components ofthe total energy-momentumtensorare zero [16]. As a result, the total combination of fields carries zero angular momentum.Theonlynon-diagonalcomponentofthetotal energy-momentumtensor thatcannot be settozerojustby making use oftheangulardependenceofthespinorsisthe

(

t

,

r

)

-component.

iv) Vanishingradialcurrent. Oneremarks that Tt

r

=

0 isrelated tothe existenceof anon-vanishing radial currentforthegeneric Ansatz(23).Nonetheless,inordertosetittozeroitisenoughto imposeoneconditionontheradialpartofthespinorsAnsatz.For simplicity,itisconvenienttofixtherepresentationoftheradial

γ

matricesatthisstage,4

γ

t

=



0 1 1 0



,

γ

r

=



0

−

1 1 0



, φ

κ

=



φ

1

φ

2



,

(26)

with

φ

1,

φ

2 twocomplexfunctionsdependingonlyonr.Thenthe

vanishingoftheradialcurrent, j₍r₁[A_/]₂₎

=

0,fortheAnsatz(23) im-pliesthat

|φ

1

|

2

= |φ

2

|

2.Onecanchoose

φ

1

= ˆφ , φ

2

=

eiν

ˆφ ,

(27)

with

ˆφ

acomplexfunction and

ν

a realvaluedfunction.5 This is enough to assure that the

(

t

,

r

)

-component of the total energy-momentumtensorvanishes.

Withall theserequirements,thetotal energy-momentum ten-sorisdiagonalandcompatiblewiththestaticandspherically sym-metriclineelement(3).

Tosimplifythenumericaltreatmentoftheproblemitis conve-nienttodefine

ˆφ

ei2ν

=

_g

−

_{i f}

_,

₍₂₈₎

theequationsforthenewmatterfunctionsbeing

f

+



N N

+

2

σ



σ

+

2

(D

−

2

)

r

+

2

(D

−

2

)

r

√

N



1 4f

+ (

√

μ

N

−

w N

σ

)g

=

0

,

(29) g

+



N N

+

2

σ



σ

+

2

(D

−

2

)

r

−

2

(

D

−

2

)

r

√

N



1 4g

+ (

√

μ

N

+

w N

σ

)

f

=

0

.

(30)

Also,thecorrespondingequationsforthemetricfunctionsare

4 _{Therepresentationoftheangularγmatricesisthatemployedin[18].} 5 _{However,letuscommentherethatinthefieldequations,thephaseof} ˆφ_plays

norole,andthusonecanchoosewithoutlossofgenerality

ˆφ

tobearealfunction).

m

=

64

π

Gr D−2 D

−

2 w(f2

₊

_g2

₎

√

σ

N

,

σ



σ

=

64

π

G D

−

2 r N



−

(D

−

2

)

f g r

+

2w

(

f2

₊

_g2

₎

σ

√

N

+

μ

(

f 2

₊

_g2

₎



.

(31) Note that in theabove equations (29)-(31) wehave fixed the “spirality”to

+

1,theonlycasediscussedinthiswork.Nonetheless, all thequalitativepropertiesoftheDiracstars thatwepresentin thenextSectionarenotaffectedbythischoice(forinstance,other properties of the Dirac field on staticand spherically symmetric configurations, such as the quasinormal modes, are not qualita-tivelyaffectedbythechangeofspirality[18]).

Finally,thevirialidentitysatisfiedbythesolutionsreads

∞



0 dr rD−2

σ



(D

−

2

)

2f g r

+

(

2

(

D

−

1

)

+ (

D

+

1

)(N

−

1

))

2

√

N

(

g f 

₋

_{f g}

₎



=

∞



0 dr rD−2

σ



w(f2

+

g2

)

2N

√

N

(

2

(D

−

1

)

+ (

3D

−

5

)(N

−

1

))

+ (

D

−

1

)

μ

(

f2

−

g2

)



.

(32) 3. Theresults

We are interested in particle-like solutions of the eqs. (11), (12),(17), (18) and(29),(31), witha topologicallytrivial,smooth geometry and a regular matter distribution. Thus, asr

→

0, one imposes N

(

r

)

→

1 (with m

(

r

)

∼

O

(

rD−1

₎

_{), while}

_σ

₍

_r

₎

_→

_σ

0

>

0.

The requirement of finite mass and asymptotic flatness imposes m

(

r

)

→

m0 (i.e. N

(

r

)

→

1) and

σ

(

r

)

→

1 asr

→ ∞

.Also,all

mat-ter functions vanish are infinity, while their behaviour near the origin is more complicated, with f

(

r

)

and G

(

r

)

vanishing there, while

φ (

r

)

, g

(

r

)

and F

(

r

)

satisfy Neumann boundary conditions. For any s, an approximate form of the solutions can be system-aticallyconstructed inboth regions,nearthe originandforlarge values ofr. Forexample,the near-origin expansion contains two freeparameters,oneofthembeing

σ

(

0

)

,andtheotheronebeing

φ (

0

)

, g

(

0

)

orF

(

0

)

,whilethematter fieldsdecayexponentiallyin thefarfield.

The solutionsthatsmoothly interpolatebetweenthese asymp-totics are constructed numerically. The results reported in this work are found in units with

μ

=

1, 4

π

G

=

1 (thus we use a scaled radial coordinate r

→

r

/

μ

(together with m

→

m

/

μ

D−3_,

w

→

w

μ

and,fors

=

1

/

2, f

→

f

√

μ

,g

→

g

√

μ

)whilethefactor of4

π

G isabsorbedintheexpressionofthematterfunctions).The equationsare solvedby usingastandard Runge-KuttaODEsolver andimplementing ashootingmethodin termsofthenear-origin essentialparameters,integratingtowardsr

→ ∞

.

For a given spin-s model, the only input parameters are the number D of spacetimedimensions andthe value w of the fre-quency. Thena (presumably infinite)setof solutionsisfound for some rangeof w,asindexedbythenumberofnodesofthe mat-terfunctions. Note that,however,only fundamental solutions are reportedinthiswork.

Wehaveconstructedinasystematicwayscalar,ProcaandDirac stars in D

=

4

,

5 and 6 dimensions;partial (lessaccurate)results werealsofoundfor D

=

7,8.TheprofileoftypicalD

=

5 configu-rationswiththesameratiow

/

μ

areshowninFig.1,togetherwith

Fig. 1. Topleftpanel: theenergydensityisshownfora(typical)solutionofeachspin-s modelinD

=

5 dimensions,allwiththesamefrequencytoparticlemassratio,

w/μ=0.98.Toprightpanel: thematterfunctionsprofilesforthesamesolutions.Bottompanels: themetricfunctionsareshownforthesamesolutions.

thecorrespondingenergy-densitydistribution.Thisplotappearsto be generic, a(qualitatively) similar picture beingfound for other valuesofw (orevenforD

=

5).

Remarkably,thes

=

0

,

1

/

2

,

1 solutionsexhibita certaindegree ofuniversality,thepatterndependingonthenumberofthe space-timedimensions.Theirbasic(qualitative)featuresaredisplayedin Fig. 2 ina frequency-mass diagram, which is the main resultof thiswork,whilequantitativeresultsareshowninFig.3.

Thestudyoftheseplotsindicatestheexistenceofanumberof basicpropertieswhichholdforboth bosonic andfermionic solu-tionsandcanbesummarizedasfollows.

•

ForanyD, a(continuous)familyofs

=

0

,

1

/

2

,

1 solutions ex-istsforalimitedrangeoffrequenciesonly,wmin

<

w

<

μ

,the minimal value of the ratio w

/

μ

decreasing with the space-time dimension. After reaching the minimal frequency, the M

(

w

)

-curve backbends into a second branch; moreover, fur-ther branches and backbendings are found. We conjecture that, foranyD, the M

(

w

)

-curvebecomes aspiral, which ap-proaches at its centre a critical solution, a result rigorously establishedsofaronlyfortheD

=

4

,

s

=

0 case[21],[22].

•

The behaviour of the solutions as w

→

μ

depends on the number of spacetime dimensions. For D

=

4, the matter field(s) becomes very diluted and the solutions trivialize in that limit, with M

→

0, the maximal mass value being at-tainedatsomeintermediatefrequency.

ThepictureforD

=

5 isdifferentandamassgapisfound be-tweenthe

U =

0 vacuumMinkowskigroundstateandtheset

Fig. 2. Thegeneric M(w)-diagram(with M theADMmassand w thefield fre-quency)isshownforthes=0,1/2,1 familiesofsolutionsinD≥4 dimensions.

ofsolutionswith

U =

0.Moreover,thelimitingconfigurations witha frequency w arbitrarilyclose to

μ

exhibit a different patternascomparedtothefourdimensionalcase.Thatis, al-though the matter fields spread andtend to zero, while the geometry becomes arbitrarily close to that of flat spacetime, themassremainsfiniteandnonzero(beingalsothemaximal allowedvalue).

Fig. 3. The ADM mass M vs. field frequency w is shown for scalar (s=0), Proca (s=1) and Dirac(s=1/2)stars in D=4,5,6 dimensions.

TheexistenceofamassgapisapropertyfoundalsoforD

≥

6 solutions.Also,againthesolutionsdonottrivializeasw

→

μ

. However, in this case their mass appears to diverge in that limit.

•

Asimilar picture is found whenconsidering instead a Q

(

w

)

diagram. In particular,one finds that M

>

μ

Q for all D

>

4 solutions,with(atthelevelofnumericalaccuracy) M

→

μ

Q as w

→

μ

. The picture for D

=

4 ismore complicated, with M

<

μ

Q for some range of frequencies ranging between

μ

andsomecriticalvalue[3],[4]. 3.1. Fermions:thesingleparticlecondition

Inthediscussionabove,nodistinctionhasbeenmadebetween the bosonic andfermionic solutions. However, although we have treated the Dirac equation classically, the fermionic nature of a spin s

=

1

/

2 field would manifest atthe level of theoccupation number.Thus,foreachfield

[A],atmostasingleparticleshould exist,inaccordancetoPauli’sexclusionprinciple.

The single particle condition, Q

=

1 is imposed by usingthe followingscalingsymmetryoftheEinstein-Diracsystem6:

6 _{Notethat Q correspondstotheNoetherchargeforasinglespinor,thetotal}

chargebeingQtot=AQ=2

D−2 2  Q . r

→ λ¯

r, w

→

w

¯

λ

,

μ

→

¯

μ

λ

,

m

→ λ

D−3_m,

_¯

_f

_→

_√

¯

f

λ

,

g

→

¯

g

√

λ

,

while M

→ λ

D−3M,

¯

Q

→ λ

D−2Q

¯

,

(with

λ

>

0 arbitraryand

σ

invariant).Onecan easily verifythat thistransformationdoesnotaffecttheequationsofmotion; how-ever, it changes the model, since it leads to a different field mass

μ

.Then,foraninitialsolutionwithagivenQ ,thecondition

¯

Q

=

1 isimposedbytaking

λ

=

QD1−2,whichaccordinglychanges the corresponding values for the ADM mass andthe field’s fre-quencyandmass.

The resulting M

(

μ

)

-diagram is shown in Fig. 4 for Einstein-DiracsolutionsinD

=

4

,

5 and6dimensions(notethattherewe dropthe overlineforboth M and

μ

). Again,one noticesa differ-ent behaviour in each spacetime dimension. In four dimensions, the (nf

=

2 particle)Einstein-Dirac solutionsexistfora familyof modelswithboth

μ

andM rangingfromzerotoamaximalvalue oforderone[7].ThesituationisdifferentinD

=

5,inwhichcase theminimalvalueof

μ

andM isnonzero(andnf

=

2 again).This feature is preserved by D

=

6 solutions (with nf

=

4 particles); however, no upper bounds for

μ

and M appear to exist in that case.

In the inset of Fig. 4, we zoom a part of the D

=

5 curve, revealing a structure of peaks.This behaviour is generic for any dimension D, andit is related withthe inspiral behaviour com-mented in theprevious section. As aresult, one can seethat for fixed values the dimension D and the scaled field mass

μ

(i.e.

Fig. 4. TheADMmassisshownvs.fieldmassμforDiracstarsinD

=

4,5 and6 dimensions.Thesingleparticlecondition,Q=1 forafermionicfield,isimposed here.Notethatthe D

=

6 curvecontinuestolargevalues,presumablydiverging. TheinsetfocusesontheD=5 case.

forfixed models), it is possible to have situations with a single solution (only one value of M), a discrete set of solutions (with differentvaluesofM),ornosolution.

4. Furtherremarks

The main purpose ofthis work was to provide a preliminary investigationofa specialtype ofsolitonicsolutionsof gravitating mattersystemsina number D

≥

4 ofspacetimedimensions. The (massive)matter fieldscorrespond tobosons (spin 0,1) orDirac fermions (spin 1

/

2), respectively, and possess an harmonic time dependence.Noquantum effectswere included,both bosonsand fermionsbeing treatedasclassical fields.7 _{The simplestsolutions}

ofthistypearethewell-known D

=

4,s

=

0 bosonstars[1],[2]. Ourresultsshowthat,foranyD,theexistenceofthese particle-likesolutions,doesnotdistinguishbetweenthefermionic/bosonic natureofthefield,withtheexistenceofageneralpatternfixedby the numberof spacetime dimensions. Moreover, while the cases D

=

4 and D

=

5 are special,the D

>

5 solutionsappeartoshare thesame(qualitative)picture.Perhapsthemostcuriousfeature re-vealedbythisstudyistheexistence,forD

>

4,ofamassgap,the set ofspin-s configurations beingnot continuously connected to Minkowskispacetimevacuum.

Among thenumerousavenues thatone maypursue following thestudyherewementionafew.First,itwouldbeinterestingto clarifythestabilityofD

>

4 solutions,anissuewhichhasbeen ex-tensivelystudiedforD

=

4 (seee.g. [3],[4] fors

=

0,[5],[20] for s

=

1, and[6] for the s

=

1

/

2 case).Since, asnoticed above, the higherdimensionalconfigurationshave M

>

μ

Q ,theypossessan excessenergyandthusweexpectthemtobeunstableagainst fis-sion(thecases

=

1

/

2 beingmoresubtle,duetothesingleparticle condition).Abetterunderstandingofthelimitingbehaviourofthe solutions as w

→

μ

andtowards the centre of the M

(

w

)

-spiral isanotherimportantopen question.Inthescalar fieldcase(with D

=

4), an explanationofthisbehaviour isprovided in[21],[22] (seealsotheexplanationinRef. [9] forthebehaviour oftheD

=

5 bosonsstars(s

=

0)asw

→

μ

).

Furthermore,one may inquire if, similar to other field theory models [27], thesespin-s solitons possess generalizationswith a

7 _{Oneremarksthatsuchatreatmentcouldbeinadequateforafermionicmatter}

content,seee.g. Ref. [19] foranearlydiscussionofthisissue.Assuch,incontrast tobosonstars,theconstructionofDiracstarsisratherapurelymathematicaltask.

horizon at their centre. In four dimensions, no such (spherically symmetric)solutionsexist,asshownin[23] forspin0,in[24] for spin1andin[25],[26] foraDiracfield.Weexpectasimilarresult toholdalsointhehigherdimensionalcase.

As furtherpossible physicalapplications,we mentionfirstthe possibility of studying thistype of solutions in an anti-de Sitter spacetime background. Such configurations are of interest in the contextoftheAdS/CFTcorrespondence,somepartialresultsbeing already reported in the literature (see e.g. [28], [29]). Moreover, thesolutionsinthisworkcouldalsobeofinterestinscenarios in-volving large extra-dimensions [30] andbraneworld models [31]. Finally, it would be interesting to investigate this type of con-figurations within the framework ofthe large-D limitof General Relativity[32].

Acknowledgements

JLBS and CK would like to acknowledge support by the DFG Research Training Group 1620 ModelsofGravity. JLBS would like toacknowledgesupport fromthe DFGprojectBL1553. Thework of E.R. was supported by Fundação para a Ciência e a Tecnolo-gia (FCT), within project UID/MAT/04106/2019 (CIDMA), and by national funds (OE),through FCT, I.P., inthe scope of the frame-work contract foreseen in the numbers 4,5 and6 of the article 23, of the Decree-Law 57/2016, of August 29, changed by Law 57/2017,ofJuly19.E.R.alsoacknowledgesupportbytheFCTgrant PTDC/FIS-OUT/28407/2017. This work hasfurther beensupported by the European Union’s Horizon 2020 research and innovation (RISE)programmesH2020-MSCA-RISE-2015GrantNo. StronGrHEP-690904 and H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740. The authors would also like to acknowledge networking support bytheCOSTActionCA16104GWverse.

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