Effective photon mass by Super and Lorentz symmetry breaking

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Effective photon mass by Super and Lorentz symmetry

breaking

Luca Bonetti, Luís R. dos Santos Filho, José A. Helayël-Neto, Alessandro

D.A.M. Spallicci

To cite this version:

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Effective

photon

mass

by

Super

and

Lorentz

symmetry

breaking

Luca Bonetti

a,b,c

,

Luís

R. dos Santos Filho

d

,

José

A. Helayël-Neto

d

,

Alessandro D.A.M. Spallicci

a,b,c,

∗

a_{Universitéd’Orléans,ObservatoiredesSciencesdel’UniversenrégionCentre(OSUC),UMS3116,1AruedelaFérollerie,45071Orléans,France} b_{Universitéd’Orléans,CollegiumSciencesetTechniques(COST),PôledePhysique,RuedeChartres,45100Orléans,France}

c_{CentreNationaledelaRechercheScientifique(CNRS),LaboratoiredePhysiqueetChimiedel’Environnementetdel’Espace(LPC2E),UMR7328,CampusCNRS,} 3AAv.delaRechercheScientifique,45071Orléans,France

d_{CentroBrasileirodePesquisasFísicas(CBPF),RuaXavierSigaud150,22290-180Urca,RiodeJaneiro,RJ,Brasil}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received3August2016

Receivedinrevisedform17October2016 Accepted14November2016

Availableonline17November2016 Editor:J.Hisano Keywords: Supersymmetry Lorentzviolation Photons Dispersion

InthecontextofStandardModelExtensions(SMEs),weanalysefourgeneralclassesofSuperSymmetry (SuSy)and LorentzSymmetry(LoSy)breaking,leadingtoobservableimprintsatourenergyscales.The photondispersionrelationsshowanon-MaxwellianbehaviourfortheCPT(Charge-Parity-Timereversal symmetry)oddandevensectors.Thegroupvelocitiesexhibitalsoadirectionaldependencewithrespect tothebreakingbackgroundvector(oddCPT)ortensor(evenCPT).Intheformersector,thegroupvelocity maydecayfollowinganinversesquaredfrequencybehaviour.Thus,weextractamassiveCarroll–Field– JackiwphotontermintheLagrangianandshowthattheeffectivemassisproportionaltothebreaking vectorandmoderatelydependentonthedirectionofobservation.Thebreakingvectorabsolutevalueis estimatedbygroundmeasurementsandleadstoaphotonmassupperlimitof10−19_{eV or2×10}−55_kg,

andtherebytoapotentiallymeasurabledelayatlowradiofrequencies.

©2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

We largely base our understanding of particle physics on the StandardModel(SM).Despitehavingproventobe averyreliable reference,therearestillunsolvedproblems,suchastheHiggs Bo-sonmassoverestimate,theabsenceofacandidateparticleforthe darkuniverse,aswellastheneutrinooscillationsandtheirmass.

Standard Model Extensions (SMEs) tackle these problems. Amongthem,SuperSymmetry (SuSy)[1,2]figures newphysicsat TeVscales[3].Since,inSuSy,BosonicandFermionicparticleseach haveacounterpart,theirmasscontributionscanceleachotherand allowthecorrectexperimentallowmassvaluefortheHiggsBoson. Lorentz Symmetry (LoSy) is assumed in the SM. It emerges

[4–7] that in the context ofBosonic strings, thecondensation of tensor fields is dynamically possible and determines LoSy viola-tion.Thereareopportunitiestotestthelowenergymanifestations ofLoSyviolation, through SMEs[8,9].The effectiveLagrangian is givenby the usual SM Lagrangian corrected by SM operators of anydimensionalitycontractedwithsuitableLorentzbreaking ten-sorial (or simply vectorial) backgroundcoefficients. In thisletter,

*

Correspondingauthorat:Universitéd’Orléans,ObservatoiredesSciencesde l’U-niversenrégionCentre(OSUC),UMS3116,1AruedelaFérollerie,45071Orléans, France.

E-mailaddress:spallicci@cnrs-orleans.fr(A.D.A.M. Spallicci).

we show that photons exhibit a non-Maxwellian behaviour, and possibly manifest dispersion at low frequencies pursued by the newlyoperatinggroundradio observatoriesandfuturespace mis-sions.

LoSy violation has been analysed phenomenologically. Stud-iesincludeelectrons,photons, muons,mesons,baryons,neutrinos and Higgs sectors. Limits on the parameters associated withthe breaking of relativistic covariance are set by numerous analyses

[10–12], including with electromagnetic cavities and optical sys-tems [13–19]. Also Fermionic strings have been proposed inthe presenceofLoSyviolation.Indeed,themagneticpropertiesof spin-lessand/orneutralparticleswithanon-minimalcouplingtoaLoSy violation background have been placed in relation to Fermionic matterorgaugeBosons[20–25].

LoSyviolationoccursatlargerenergyscalesthanthose obtain-ableinparticleaccelerators[26–32].Atthoseenergies,SuSyisstill anexactsymmetry,evenifweassumethatitmightbreakatscales closetotheprimordialones.However,LoSyviolationnaturally in-ducesSuSybreaking becausethebackgroundvector (ortensor) – thatimpliestheLoSyviolation–isinfactpartofaSuSymultiplet

[33],Fig. 1.

Thesequenceisassuredbythesupersymmetrisation,intheCPT (Charge-Parity-Timereversalsymmetry)oddsector,oftheCarroll– Field–Jackiw (CFJ)model[34] that emulatesa Chern–Simons[35] http://dx.doi.org/10.1016/j.physletb.2016.11.023

204 L. Bonetti et al. / Physics Letters B 764 (2017) 203–206

Fig. 1. BreakingenergyvaluesandtheLagrangians.AdifferenthierarchyofLoSy,SuSybreakingandGrandUnificationTheories(GUT)doesnotinterferewiththedispersion lawsofthephotonicsectoratlowenergies.

termandincludesabackgroundfield thatbreaks LoSy,underthe pointofviewoftheso-called(active)particletransformations.The latterconsistsoftransformingthepotential Aμ andthefield F μν , whilekeepingthebackgroundvector

V

μ unchanged.Forthe pho-ton sector, when unaffected by the photino contribution,the CFJ Lagrangianreads(ClassI)

LI

= −

1 4F

−

1 2

V

μAνF

˜

μν

_,

₍₁₎

˜

Fμν

=

1 2



μναβ_F αβ

,

(2)

where F

=

F μν Fμν .TheterminEq.(2)couplesthephotontoan external constant fourvector and itviolates parityeven ifgauge symmetryisrespected[34].IftheCFJmodelissupersymmetrised

[36], the vector

V

μ is space-like constant and is given by the gradient ofthe SuSy breakingscalar backgroundfield, presentin themattersupermultiplet.Thedispersion relationyields,denoting kμ

= (

ω

,



k

)

,k2

= (

ω

2

_{− |}

_k

_|

2

₎

_,and

_(V

_μkμ

₎

2

_{= (}

_V

0

_ω

_{− }

_{V · }

_k

₎

2_,

k4

+

V

2k2

− (

V

μkμ

)

2

=

0

.

(3)

IfSuSyholdsandthephotinodegreesoffreedomareintegrated out,weareledtotheeffectivephotonicaction,i.e. theeffectofthe photinoonthephotonpropagation.TheLagrangian(1)isrecastas (ClassII)[33] LI I

= −

1 4F

+

1 4



μνρσ

_V

μAνFρσ

+

1 4H F

+

MμνF μλ_Fνλ

_,

₍₄₎

whereH ,thetensorMμν

= ˜

Mμν

+

1

/

4

η

μν M ,andMμν depend

˜

on thebackgroundFermioniccondensate,originatedbySuSy; Mμν is traceless, M is the trace of Mμν and

η

μν the metric. Thus, the Lagrangian,Eq.(4),intermsoftheirreducibletermsdisplaysas LI I

= −

1 4

(

1

−

H

−

M

)

F

+

1 4



μνρσ

_V

μAνFρσ

+ ˜

MμνFμλFλν

.

(5)

Thecorrespondingdispersionrelationreads

k4

+

V

2

(

1

−

H

−

M

)

2k 2

₋

1

(

1

−

H

−

M

)

2

V

μ_k μ

=

0

.

(6)

ThedispersionlawgivenbyEq.(6)isjustarescalingofEq.(3)

aswe integratedoutthephotinosector. Thebackground parame-tersare very small, beingsuppressedexponentially atthePlanck scale;theyrender thedenominatorinEq.(6)closetounity,

imply-ingsimilarnumericaloutcomesforthetwodispersionsofClasses I and II.

Theevensector[33] assumesthattheBosonicbackground, re-sponsible of LoSy violation, is a background tensor tμν . For the photon sector, if unaffected by the photino contribution, the La-grangianreads(ClassIII)

LI I I

= −

1 4F

−

16tμνF μκ_Fν κ

−

4



tμν

η

μν



F

.

(7)

ThedispersionrelationforClassIII[37]is

ω

2

− (

1

+

ρ

+

σ

)

2

|

k

|

2

=

0

,

(8)

where

ρ

=

1

/

2K α

˜

_{α ,}

σ

=

1

/

2K α

˜

β_Kα

˜

β

−

ρ

2,andK α

˜

β

=

tαβtμν pμ pν

/

|

k

|

2 _{areassociatedtoFermioniccondensates.}

Integratingout thephotino [33],we turntotheLagrangian of ClassIV LI V

= −

1 4F

+

a 2tμνF μ κFνκ

+

b 2tμν

∂

αF αμ

_∂

βFβν

,

(9)

where a is a dimensionless coefficient and b a parameter of di-mensionofmass−2 (herein,c

=

1,unlessotherwisestated).Forthe dispersionrelation,wewritetheEuler–Lagrangeequations,passto Fourierspace andsetto zerothedeterminantof thematrixthat multiplies the Fouriertransformed potential. However, given the complexityofthematrixinthiscaseandthesmallnessofthe ten-sortμν ,we developthe determinantinaseriestruncatedatfirst orderandget[37]

btk4

−

k2

+



3a

+

bk2



tαβk_αkβ

=

0

,

(10)

wheret

=

tμμ .

Fordeterminingthegroupvelocity,wefirstconsider

V

0

=

0 for

ClassI[38,39]andobtain

ω

4

₋



₂

_|

_k

_|

2

_{+ | }

_V

_|

2



_ω

2

_{+ |}

_k

_|

4

_{+ |}

_k

_|

2

_{| }

_V

2

₋



_V



_{· }

_k



2

₌

₀

_.

₍₁₁₎

In[39],theauthorsdonotexploittheconsequencesofthe dis-persionrelationsanddonotconsideraSuSyscenario.Dealingwith Eq.(11),wehaveneglectedthenegativeroots;itturnsoutthatthe two positive rootsdetermine identicalgroupvelocities dw

/

dk up to second order in

V

. For

θ

, the angle between the background vector

V

andk,



weget

vI_g

|

θ_V=π/2

0=0

=

1

−

| 

V

|

2

8

ω

2

(

2

+

cos

Fig. 2. ForClassI,weplotthedelays[s],Eq.(16),fordifferentangles,Eqs.(12),

(13),using| V|=10−19_{eV[40],versusfrequency.Wehavesupposedthesourceto} beatadistanceof4kpc.Thefrequencyrange0.1–1MHzhasbeenchosensince itistargetedbyrecentlyproposedlowradiofrequencyspacedetectors,composed byaswarmofnano-satellites;see[41]andreferencestherein.Thereisafeeble dependenceofthedelaysonθ.Thedelayisofabout50psat1MHzforθ=π/2, Eq.(13),andaroundhalfofthisvalueforθapproachingπ/2,Eq.(12).

for

θ

=

π

/

2. Insteadfor

θ

=

π

/

2, one ofthetwo solutions coin-cideswiththeMaxwellianvalue,whiletheotherisdispersive vI1_g

|

θ_V=π/2 0=0

=

1

,

v I2 g

|

θ=π/2 V0=0

=

1

−

1 2

| 

V

|

2

ω

2

.

(13)

For

V

0

=

0,we suppose that the light propagatesalong the z

axis(k1

=

k2

=

0) whichforconvenienceisalongthelineofsight

ofthesource.Wethenobtain

ω

4

_{− [}

_2k2

3

+

V

12

+

V

22

+

V

32

]

ω

2

+

2

V

0

V

3k3

ω

+

k43

+ (

V

2

1

+

V

22

−

V

02

)

k23

=

0

.

(14)

Wenow set

V

3

=

0, that is,the light propagatesorthogonally

to the background vector. Further, for

V

spacelike and 4

V

₀2k2₃

/

| 

V|

4

_

_{1,wegettwogroupvelocities,oneofwhichisdispersive}

vI1_g

|

_V3=0

=

1

−

V

2 0

| 

V

|

2

,

v I2 g

|

V3=0



α



1

−

1 2

| 

V

|

2

ω

2



.

(15) The solution vI1

g

|

V3=0 is always subluminal for

V

spacelike.

The solution vI2 g

|

V3=0 assumes

ω

 | 

V|

. Since

α

=

1

+

V

2 0

/

| 

V|

2, vI2_g

|

_V3=0 is superluminalfor

√

2

ω

>

| 

V|(

1

+ | 

V|

2

/

V

₀2

)

1/2_.Further,

thevalueof

α

isnotLorentz–Poincaréinvariant.Superluminal be-haviourisavoidedassumingforbothsolutions

V

0

=

0.

Ifdealingonlywithanull

V

0 andwithdispersivegroup

veloc-ities,forasourceatdistance



,thetime delayoftwo photonsat differentfrequencies,AandB,isgivenby(inSIunits)



tCFJ

=



| 

V

|

2 2ch

¯

2



1

ω

2 A

−

1

ω

2 B



x

,

(16)

where x takes the values

(

2

+

cos2

_{θ )/}

_{4, for Eq. (12), and 1 for}

Eqs.(13),(15).Thedelays,Eq.(16),areplottedinFig. 2.Comparing withthedeBroglie–Proca(dBP)delay



tdBP

=



m2 γc3 2h

_¯

2



1

ω

2 A

−

1

ω

2 B



,

(17)

weconcludethatthebackgroundvectorinducesaneffectivemass tothephoton,mγ ,ofvalue

m_γ

=

| 

V

|

c2x

.

(18)

Equation (18)is gauge-invariant, butnot Lorentz–Poincaré in-variant.Nevertheless,thereisa subset ofLorentz–Poincaré trans-formation that leave the value of Eq.(18) unchanged. Under the assumption of

V

0

=

0 and thus

| 

V|

constant,the value ofmγ is

constant whenthe originofthe referenceis translatedalong the lineofsightoftheobserverto thesourceand/or underthe rota-tiongroup SO(3). Themassappears asthepoleofthetransverse componentofthephotonpropagator[39].

ClassII, justa rescaling of ClassI, implies identicalsolutions, differingbyanumericalfactoronly.

The group velocities of Classes III and IV show no sign of dispersion; they are slightly smaller than c – as light travelling throughmatter,butsufferfromanisotropytoalargerdegreethan inClassesIandII.Indeed,theisotropyislostduetothetensorial natureoftheLoSYandSuSybreakingperturbation.Thefeebleness ofthecorrectionsisduetothecoefficient

T

beingproportionalto thepowersofthetensortμν components, of10−19eV magnitude

[37]

vI I I_g ,I V

=

1

−

T



t1sin2

θ

cos2

ϕ

+

t2sin2

θ

sin2

ϕ

+

t3cos2

θ



,

(19)

where

θ

and

ϕ

aretheazimuthalandplanaranglesof



k with re-specttotheaxesrespectively.

Havingseenamassive-likephoton behaviourinthe group ve-locitiesoftheoddsector,werewriteEq.(1)intermsofthe poten-tialstoletamassive-liketermemerge

L

=

1 2



∇φ + ˙

A



2

−

1 2



∇ × 

A



2

+

V

0



A

·



∇ × 

A



−

2

∇φ ·





V

× 

A



− 

V

·





A

× ˙

A



.

Sincethe

φ

fieldappears onlythroughits gradient,inthe ab-senceof

φ

timederivatives andtherebyof dynamics,

∇φ

actsas an auxiliary field andcanbe integratedout fromtheLagrangian. Defining

χ

= ∇φ + ˙

A

−

2

V × 

A,weget L

=

1 2

χ

2

₋

₂



_V



_{× }

_A



2

_{+ }

_V

_·





_A

_{× ˙}

_A



₋

1 2



∇ × 

A



2

+

V

0



A

·



∇ × 

A



.

(20)

The Euler–Lagrangian equation for

χ

is disregarded since

χ

=

0. The term



V × 

A



2 is expanded as



V

2

δ

kn

−

V

k

V

n

×

AkAn

:=

Mkn



V



AkAn,where Mkn is asymmetric diagonalisable

matrix,thanks to a suitable matrix ofthe S O

(

3

)

rotationgroup. Performing such a changein Eq.(20),the termunderdiscussion changesinto

˜

AiM

˜

i jA

˜

j

=



V

2

˜

A2₂

+

_

V

2A

˜

2₃

,

(21)

therebyshowingamassive-likephotontermasinthedeBroglie– ProcaLagrangian.

The quest fora photon withnon-vanishing mass is definitely notnew.The firstattemptscanbe tracedback todeBrogliewho conceived an upper limit of 10−53 _{kg, and achieved a}

compre-hensive formulation of the photon [42],also thanks to the rein-terpretation of the work of his doctorate student Proca. To the Lagrangian of Maxwell’s electromagnetism, they added a gauge breaking term proportional to the square of the photon mass. A laboratoryCoulomb’slawtestdetermined themassupperlimit of 2

×

10−50 _{kg [43]. In the solar wind, Ryutov found 10}−52 _kg

206 L. Bonetti et al. / Physics Letters B 764 (2017) 203–206

putintoquestion [47].1 _{Thelowestvalue foranymassisdictated}

by Heisenberg’s principle m

≥ ¯

h

/

tc2, andgives 1

.

3

×

10−69 kg, where



t isthesupposedageoftheUniverse.

In this letter, we have focused on SuSy and LoSy breaking andderived theensuing dispersionrelations andgroupvelocities for four types of Lagrangians. All group velocities show a non-Maxwellianbehaviour,intheangulardependenceandthroughsub orsuperluminalspeeds.Superluminalbehaviourisexclusivetothe oddCPTsector, andmayoccur onlyifthetimecomponentofthe perturbing vector isnon-null.Further, in theodd CPTsector, the effectivemassshowsadispersion,proportionalto1

/

ω

2_,asindBP

formalism.Conversely,tothedBPphoton,wheremassisimposed abinitio, the CFJ photon acquires a mass through a mechanism, namely from LoSy violation through the background vector. The otherdifferenceslieinthelackofLorentz–Poincaréinvarianceand intheangulardependenceoftheCFJphotonmass.

The delays are more important at lower frequencies andthe opening of the 0.1–100 MHz window would be of importance

[41].Elsewhere,wehaveanalysedthepolarisationandevincedthe transversalandlongitudinal(massive)modes[37].

From the rotation of the plane of polarisation from distant galaxies,orfromtheCosmicMicrowaveBackground(CMB), ithas beenassessedthat

|

V

μ

|

<

10−34eV[12,34,48].Thisresultis com-parabletotheHeisenberg masslimitvalue atthe ageofthe uni-verse.Alessstringent,butinteresting, limitof10−19_{eV [40] has}

beenset through laboratory basedexperiments involving electric dipole moments of charged leptons or the inter-particle poten-tialbetweenFermionsandtheassociatedcorrectionstothe spec-trumoftheHydrogenatom.Theselatterestimatesimply,Eq.(18), a massupperlimitof10−55_kg.

The detection of the CFJ massive photon can be pursued by other means, e.g., through analysisof Ampère’s law in the solar wind [47].Incidentally, the oddandeven CPTsectorscan be ex-perimentallyseparable[12].

What is the role of a massive photon for SMEs? String the-oryhashintedtomassivegravitonsandphotons[5,6],whileProca electrodynamicswas investigatedinthecontext ofLoSyviolation, butoutsideaSuSyscenario[20].However,ifLoSytakesplaceina supersymmetricscenario,thephotonmass maybe naturally gen-eratedfromSuSybreakingcondensates[33,36].Wepointoutthat theemergenceofamassivephotonispertinentalsotootherSME formulations.

LBandADAMS acknowledge CBPFforhospitality, whileLRdSF andJAHNaregratefultoCNPq-Brasilforfinancialsupport.All au-thorsthanktherefereeforstimulatingcomments.

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