On the inexistence of self-gravitating solitons in generalised axion electrodynamics

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

On

the

inexistence

of

self-gravitating

solitons

in

generalised

axion

electrodynamics

Carlos A.R. Herdeiro

a

,

b

,

João M.S. Oliveira

b

,

∗

a_{CentrodeAstrofísicaeGravitação- CENTRA,DepartamentodeFísica,InstitutoSuperiorTécnico- IST,UniversidadedeLisboa,AvenidaRoviscoPais1,1049-001,}

Portugal

b_{DepartamentodeMatemáticadaUniversidadedeAveiroandCIDMA,CampusdeSantiago,3810-183Aveiro,Portugal}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received17September2019 Accepted30October2019 Availableonline6November2019 Editor: M.Cvetiˇc

Buildinguponthemethodsusedrecentlyin[1],weestablishtheinexistenceofself-gravitatingsolitonic solutions for both static and strictly stationary asymptotically flat spacetimes in generalised axion electrodynamics. This is anEinstein-Maxwell-axion model, where the axion field θ isnon-minimally coupled to the electromagnetic field.Considering the standard QCD axion coupling, we firstpresent an argument for the absence of static axionic solitons, i.e. localised energy axionic-electromagnetic configurations,yielding aneverywhereregular, horizonless, asymptotically flat,staticspacetime.Then, for generic couplings f(θ ) and g(θ ) (subject to mildassumptions) between the axion field and the electromagnetic field invariants, we show there are still no solitonic solutions, even when dropping the staticityassumptionand merelyrequiringastrictlystationary spacetime,regardlessofthe spatial isometries.

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

1. Introduction

QuantumChromodynamics (QCD) admits a termthat violates the combined CP (Charge conjugation and Parity) discrete sym-metries. Yet, such violationis not observed in any experimental process controlled by thestrong interaction only, which suggests that ifit exists it must be very small. Consequently,the CP vio-latingtermmusthavean unnaturallysmallcoefficient,yielding a fine-tuningproblem.

AningenioussolutiontothisstrongCPproblem wasproposedby PecceiandQuinn[2,3].Theirideawastopromotetheunnaturally smallcoefficientintoadynamicalfieldwhichcouldbedynamically relaxedtozero.Intheoriginalguise,themechanismextendedthe Standard Model with a complex scalar field possessing a global U

(

1

)

symmetryandaMexicanhattypepotential.Thesymmetryis spontaneouslybrokenbelowsome highenergyscale,whereinthe complexscalaracquiresavacuumexpectationvalue(vev),yielding aGoldstonepseudo-scalar –theaxion,

θ (

x

)

–parameterising the degeneracyofthepotentialvacuummanifold.If,moreover,atleast oneofthefermions inthemodelacquiresitsmassvia aYukawa couplingtothecomplexscalar,theaxionacquiresapotential

un-*

Correspondingauthor.

E-mailaddresses:[email protected](C.A.R. Herdeiro),[email protected] (J.M.S. Oliveira).

derachiralanomaly,drivingittoavevthatpreciselycancelsthe CP violating termand, moreover, endowsthe axion witha small mass [4,5]. When later studied in a cosmologicalcontext, it was suggestedthataxionsareinterestingdarkmattercandidates[6–8], seealso[9].Sincethen,thestudyofgravitationaleffectsof axion-likeparticleshavereceivedconsiderableattention.

Inthisworkwe discusswhethersolitonic self-gravitating solu-tions,i.e., everywherenon-singular,asymptotically flatspacetimes without a horizon, are possible ina generalised axion electrody-namicsminimallycoupledtoEinstein’sgravity- Einstein-Maxwell-axion models. Such solitons would describe localised lumps of energy, particle-likesolutions, in afield theory-gravity model.To contextualisethisquestion,it isknownthat theEinstein-Maxwell system admits no static soliton solutions - see e.g. [10]. The sameholdsforstrictlystationary,butnotnecessarilystatic, space-times[11].Similarconclusions stillholdifoneconsiders Einstein-Maxwell-scalarmodels [1] withoutaxion-likecouplings,but allow-ing generic couplings between the scalar field and the Maxwell invariant. Thus, one may ask whether the particle physics moti-vatedaxioncouplingtotheelectromagneticfield[12] couldchange this state of affairs, possibly unveiling another guise for axionic manifestationinNature.

The possible existence of self-gravitating axionic solitons is discussed here for both static and strictly stationary configura-tions.OuranalysisstartswiththesimplestEinstein-Maxwell-axion

https://doi.org/10.1016/j.physletb.2019.135076

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

model, buta generalisation,considering a non-minimal coupling f

(θ )

between theaxion field andthe Maxwell term, is also dis-cussed,following [1].

Thispaperisorganisedasfollows:insection 2wediscussthe absence ofself-gravitating, staticaxionic solitons,with the usual axionelectrodynamicsdescription.Insection3,weincludea non-minimal coupling f

(θ )

between the axion field and the electro-magnetic field andchange theusual linear axion coupling

κ

θ

to anarbitrarypseudoscalarfunction g

(θ )

,findingthat ano-go the-oremforsolitonsstill holds.Insection 4 wegeneralisethe latter resultfor strictly stationary,but not necessarilystatic, configura-tions.Finally,insection (5),wepresentadiscussionoftheresults andpossiblefuturework.

2. Absenceofstaticaxionicsolitons

Axion electrodynamicsminimally coupledto Einstein’s gravity consists,apartfromthegravitationalaction,oftheusual Maxwell andKlein-Gordontermsalongwithanadditionaltermwhich cou-plesthe electromagneticfield totheaxion field. Itisrepresented bythefollowingaction[12–14]

S

Ax

=

S

E H

+



d4x

√

−

g



−

1 4FμνF μν

₊

κ

θ

4 FμνF

˜

μν

+

1 2

∇

μ

θ

∇

μ

_θ

₋

_U

_{(θ )}



,

(1)

where

S

E H is the Einstein-Hilbert action,

κ

is a constant,

θ

is

the pseudo-scalar axion field and F μν is

˜

the Hodge dual of the Maxwell tensor Fμν

= ∂

μ Aν

− ∂

ν Aμ, F μν

˜

=

12



μναβFαβ, where



μναβ _{isthecontravariantLevi-Civitatensor.Wealsoallowforthe}

existenceofageneralaxionpotentialU

(θ )

.Inthissectionwe con-sider an asymptotically flat,static spacetime withno restrictions onthespatialsymmetries.Thegravitationalpartwillplaynorole inthesubsequentargument.

Theequationsofmotionforthismodelare

∇

μ

(

Fμν

−

κ

θ ˜

Fμν

)

=

0

,

(2)

∇

μF

˜

μν

=

0

,

(3)

2θ =

κ

4FμνF

˜

μν

₋

dU

(θ )

d

θ

,

(4)

where

2

is the covariant d’Alembertian. Since the spacetime is static and without horizons it admits an everywhere timelike Killingvector field k. Thisvector field canbe usedto define the electricandmagneticfields(infact4-(co)vectors)as:

Eμ

≡ −

Fμνkν

,

(5) Bμ

≡ −

1 2

ε

μαβνF αβ_kν

_{= − ˜}

_F μνkν

.

(6)

In Maxwell’s theory, one can rewrite the covariant Maxwell equations interms of E

,

B in a certain canonical form- see e.g. eqs.(38)-(41)in [1].Inaxion electrodynamics,asimilarcanonical formisobtainedifwedefinetwo newfields E andBwhichare relatedtotheoriginalfieldsas

E_μ

≡

Eμ

−

κ

θ

Bμ

,

(7)

B_μ

≡

B_μ

+

κ

θ

E_μ

;

(8)

now,theaxionMaxwellequations (2)-(3) arewrittenas

∇

[μE_ν]

=

0

,

(9)

∇

[μBν]

=

0

,

(10)

∇

μ



Eμ V



=

0

,

(11)

∇

μ



Bμ V



=

0

,

(12)

where V

≡ −

kμkμ

>

0. Due to the absence ofcurrents, the first two equations imply that an electric

ϕ

anda magnetic

ψ

scalar potentialscanbeintroduced,as

Eμ

= ∂

μ

ϕ

,

Bμ

= ∂

μ

ψ .

(13)

The remainder oftheargumentusesthe methodin[1] which was inspired by Heusler’s argument described in [10]. We make use of the following identity: for an arbitrary vector

α

obeying £k

α

= [

k

,

α

]

=

0,itholdsthat [15]:



∂

α

μkνdSμν

=

1 2





∇

μ

α

μkνd



ν

,

(14)

where



isanarbitraryCauchyhypersurfacewithvolumeelement d



ν andboundary

∂

,thelatterwithantisymmetricareaelement dSμν .Specifying thisidentity for

α

μ

₌

_Eμ

_/

_{V andusing the} ax-ionicMaxwellequationsyields



∂

Eμkν

V dSμν

=

0

,

(15)

where we took

∂

to be the surface at spatial infinity (an r

=

∞

2-surface,wherer isthestandardMinkowskiradialcoordinate, whichcanbeusednearinfinityduetoasymptoticflatness).

Making a second useoftheidentity (14) but now with

α

μ

₌

ϕ

Eμ

/

V andonceagainusingtheaxionicequations,weobtain

1 2



 EμE_μ V k ν_d



ν

=



∂

ϕ

E μ_kν V dSμν

=

ϕ

∞



∂ Eμkν V dSμν

=

0

,

(16)

where

ϕ

_∞ isthevalueoftheelectricpotentialatr

= ∞

whichis constant,andthelastequalityused (15).

The sameargument canbe used for B and B by replacing

ϕ

by

ψ

,obtaining



 BμB_μ V k ν_d



ν

=

0

.

(17)

Wecannowexpand

(

E

,

B

)

intermsof

(

E

,

B

)

,via (7)-(8) toobtain theidentities:



 EμEμ V k ν_d



ν

−





κ

θ

E μ_B μ V k ν_d



ν

=

0

,

(18)



 BμBμ V k ν_d



ν

+





κ

θ

E μ_B μ V k ν_d



ν

=

0

.

(19)

Addingupthelasttwoequationsyields



 EμEμ

+

BμBμ V k ν_d



ν

=

0

.

(20)

From their definitions (5)-(6), kμ Eμ

=

0

=

kμ Bμ. Thus, these fieldsarenevertimelike.Itfollowsthatboth Eμ Eμ andBμ Bμ are

alwaysnon-negative.Consequently,theonlywayforeq. (20) tobe verified is if both fields vanish for every Cauchy surface



and, consequently,forthewholespacetime.Thisresultisindependent ofthepotentialU

(θ )

.Withvanishingelectromagneticfields,allwe haveleftisthepossibilityofself-gravitatingaxion(scalar)solitons. Howeverithasbeenshownthat therearenoscalarfieldsolitons aslongasthedominantenergyconditionisobeyedandthestrong energyconditionisviolated,whichisthecaseforscalarfieldswith apositivepotential(see[1,15,16]).Therefore,theonlypossible so-lutionforsuchpotentialsisMinkowskispacetime.

Asafinal remarkinthissection,the maindifferencebetween theresulthereinandtheone forEinstein-Maxwelltheory isthat insteadofestablishingthatthenormsofboth E andB vanish,we canonlyestablishthatthesumofthesenormsmustvanish.Since boththesenorms arepositive definite,however,thefinal conclu-sion is that each must vanish, recovering the result of Einstein-Maxwelltheory.

3. Generalisedaxionelectrodynamics

Theresultofsection (2)canbestraightforwardlyextendedtoa modelofgeneralisedaxion electrodynamicsminimallycoupledto Einstein’sgravity

S

A

=

S

E H

+



d4x

√

−

g



−

f

(θ )

4 FμνF μν

₊

g

(θ )

4 FμνF

˜

μν

+

1 2

∇

μ

θ

∇

μ

_θ

₋

_U

_{(θ )}



,

(21)

whichintroduces thearbitraryfunctions f

(θ )

andg

(θ )

ofthe ax-ionfield. The function g

(θ )

is a pseudoscalarfunction and f

(θ )

is a non-minimal coupling between the axion and the standard Maxwellterm,asdiscussedin[1] motivatedby therecentresults ofscalarisationinEinstein-Maxwell-scalarmodels [17].

Inorder to recover Einstein-Maxwellwhen there is no axion, we assume that f

(

0

)

=

1.It is alsoassumed that both functions donot diverge inour spacetime.1 The equationsof motionare a simplegeneralisationofthepreviousones(2)-(4) andread

∇

μ

(

f Fμν

−

gF

˜

μν

)

=

0

,

(22)

∇

μF

˜

μν

=

0

,

(23)

2

2

_θ

₌

1 4 dg d

θ

FμνF

˜

μν

₋

1 4 df d

θ

FμνF μν

₋

dU

(θ )

d

θ

.

(24)

Althoughthe

θ

equationcanbeconsiderablymoredifficultdueto thearbitrarycouplings,definingnowthefieldsEandBas

E

=

f E

−

g B

,

(25)

B

=

f B

+

g E

,

(26)

itfollowsthatthesenewfieldsrespecttheexactsameequationsas (9)-(12). Consequently,we followtheexact sameprocedureasin section2toobtainthecorrespondingrelationsto (18)-(19),which nowread



 fE μ_E μ V k ν_d



ν

−



 gE μ_B μ V k ν_d



ν

=

0

,

(27)



 fB μ_B μ V k ν_d



ν

+



 gE μ_B μ V k ν_d



ν

=

0

.

(28)

1 _{ThisassumptionisconsideredastheapplicationoftheStokestheoremwould}

includeconstantcontributionsduetodivergencesinthespacetime.Therefore,our approachisnotvalidfordivergingcoefficientfunctions.

Addingtheseequationsnowyields



 f E μ_E μ

+

BμBμ V k ν_d



ν

=

0

.

(29)

As both Eμ Eμ and Bμ Bμ are non-negative, this identity implies asimilar resultto theoneobtainedin[1] forthetheory withno axions(g

=

0):the fieldsmust vanishandthere are no solitonic solutions aslong asthecoupling f

(θ )

doesnot changesign. We can see that the mainreason forthisresult to be similar to the onewithg

=

0 isbecauseg,ascomplicatedafunctionasitmight be, does not contribute to the argument due to its contribution disappearingwhenweaddequations(27) and(28).

4. Absenceofstrictlystationaryaxionicsolitons

Sofarwe haveconsidered staticspacetimes.The methodused aboveallowed ustoruleout staticsolitonswithoutrequiringany spatial isometry (see [18] for other approaches to establish the absence ofstaticsolitons). Now we wish to consider strictly sta-tionary, butnotnecessarily static, axionicsolitonswiththe more generalmodel (21). This accountsnow forpossibly rotating soli-tons, aslongasrotationdoesnotcreateergo-regions, sincestrict stationaritymeansthat thereexistsaneverywhere timelikeKilling vector field.Followinga proceduresimilar to[1] where weusea Lichnerowicztypeargument,seee.g. [11],weshallalsoestablisha no-gotheoremforsolitons.InthiscasetheEinsteinequationsplay animportantroleintheargument.

TheEinsteinequationsforthismodelare

Rμν

=

f

( )



F_μαFνα

−

1 4gμνF 2



+ ∂

μ

∂

ν

+

gμνU

( ) .

(30) Theaxionictermispurelytopologicalsoitdoesnotcontribute to the Einsteinequations.Using the timelikeKilling vector field,we definethetwistvector

ω

μ as

ω

μ

=

1

2

ε

μναβ_k

ν

∇

αkβ

;

(31)

thisvectorobeys

∇

μ



ω

μ V2



=

0

.

(32)

The Maxwell equations (9)-(12), with the primed fields defined by (25)-(26) aregeneralisedforastrictlystationaryspacetimeas:

∇

[μE_ν]

=

0

,

(33)

∇

[μB_ν]

=

0

,

(34)

∇

μ



Eμ V



=

2 V2

ω

μB μ

_,

₍₃₅₎

∇

μ



Bμ V



= −

2 V2

ω

μE μ

_.

₍₃₆₎

Itcanbeshownthat

∇

[μ

ω

ν]

=

1₂

ε

μναβk[αRβ]γkγ

,

(37)

sothatusingtheEinsteinequations(30) relatesthecurlof

ω

with thePoyntingvector:

∇

[μ

ω

ν]

=

f B_[μE_ν]

.

(38)

Onecanfreelyaddvanishingtermssuchas

−

g B_{[μ Bν]}andg E_{[μ Eν]} torewritetherighthandsideintwodifferentways

f B_[μE_ν]

=

B_[μE_ν]

=

B_[μE_ν]

.

(39)

We choosethe expressionwith B and E asthesetwo fieldsare theoneswhichwecanrewriteaspotentials

ψ

and

φ

respectively, cf. (13). Thismeans that equation (38) impliesthe followingtwo identities

∇

[μ



ω

ν]

− ψ

E_ν]



=

0

,

(40)

∇

[μ



ω

ν]

+ φ

B_ν]



=

0

,

(41)

whichinturnimplytheexistence oftwonewpotentialsUB and UE

∇

μUE

=

ω

μ

− ψ

Eμ

,

(42)

∇

μUB

=

ω

μ

+ φ

Bμ

.

(43)

Using thesepotentialsand theidentity (32), the following diver-genceidentityisobtained

∇

μWμ

=

4

ω

μ

ω

μ V2

−

E_μ Eμ

+

B_μBμ V

,

(44) where Wμ

=

2

(

UE

+

UB

)

ω

μ V2

−

ψ

Bμ

+ φ

Eμ V

.

(45)

On the other hand, the contraction of the Einstein equations (30) withtheKillingfieldyields

2 VRμνk

μ_kν

₌

_fEμEμ

+

BμBμ

V

−

2U

(θ ) .

(46)

Thefirsttermonthe righthandside canbe slightlyreshaped by notingthat f

(

Eμ Eμ

+

Bμ Bμ

)

maybewrittenas

f

(

EμEμ

+

BμBμ

)

= (

f Eμ

−

g Bμ

)

Eμ

+ (

f Bμ

+

g Eμ

)

Bμ

=

E_μEμ

+

B_μBμ

.

(47)

Thenaddingequations(44) and(46) yields

2 VRμνk

μ_kν

₋

4

ω

μ

ω

μ

V2

= −∇

μW

μ

₋

_2U

_{(θ ) .}

₍₄₈₎ The final step of the argument consists on taking the Komar massintegralonaCauchysurface



[19]:

M

= −







2Rμνkμkν V

−

4

ω

μ

_ω

μ V2



kαd



α

,

(49)

which,via(48),reads

M

=







∇

μWμ

+

2U



kαd



α

.

(50)

As£kW

=

0,theidentity(14) canbe usedtowrite thefirstterm

intheintegralas





∇

μWμkαd



α

=

2



∂ WμkαdS_μα

.

(51)

The surface

∂

is the 2-surface at infinity andall the terms in W μ decay, asymptotically,faster than r−2, so that (51) vanishes. Thus (50) becomes M

=

2



 U kαd



α

= −

2



 U V d

 ,

(52)

asd



α

=

kαd



.Consequently,aslongasthepotentialU

(θ )

is pos-itive,theonlycontributiontotheKomarmassM willbenegative. Then, by the positive mass theorem,2 _M

₌

_{0 and the only}

solu-tionisflatspacetime.Therefore,noaxionicsolitonsarepossiblein strictly stationaryspacetimes,againregardlessofthespatial sym-metries.

5. Conclusion

In thispaper we haveassessed the possibleexistence of self-gravitating solitons in axion electrodynamics and generalisations thereof.Weestablishedthatthepresenceofaxionsandtheir cou-pling to the electromagneticfield doesnot change theresults of (in)existence ofEinstein-Maxwellsolitons instaticorstrictly sta-tionaryspacetime [1].Thisholds evenwhenconsidering amodel with rather generic couplings between the axion field and the electromagneticinvariants,and,inparticularallowinganarbitrary coefficientfunction g

(θ )

intheaxiontermF

· ˜

F .

A possiblegeneralisationwould be toconsider acoupling be-tweentheelectromagneticfieldandadifferentscalarfield (rather than the axion).However, withoutany kindofcouplingbetween these two scalar fields, the result will likely remain unchanged. One interesting future work route would be then to generalize this model to allow for two different scalar fields, coupled to eachother,andtotheelectromagneticfieldthroughthecouplings f and g. An example of a model that corresponds to this type of framework is the Einstein-Maxwell-dilaton-Axion model [20], where the coupling f

(

ϕ

)

=

e−αϕ depends onthe dilaton field

ϕ

(

α

isaconstant)and g

(θ )

=

κ

θ

hastheusualdependenceonthe axion field

θ

. These two fields also include a coupling between them,possibly allowing forthe existenceofscalar solitonsinthe model.

Acknowledgements

We would like to thank E. Radu for discussions. J.O. is sup-portedby theFCTgrant PD/BD/128184/2016.Thiswork hasbeen supported by FCT (Portugal) through: the IF programme, grant PTDC/FIS-OUT/28407/2017, the strategic project UID/MAT/04106/ 2019 (CIDMA) and the CENTRA strategic project UID/FIS/00099/ 2013. We also acknowledge support from the European Union’s Horizon2020researchandinnovation(RISE)programmes H2020-MSCA-RISE-2015GrantNo.StronGrHEP-690904and H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740. The authors would like to acknowledgenetworkingsupportbytheCOSTActionCA16104. References

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