On Lorentz violation in e−+e+→ μ−+μ+ scattering at finite

Citation for this paper:

Souza, P.R.A., Santos, A.F., Ulhoa, S.C. & Khanna, F.C. (2019). On Lorentz

violation in e−+e+→ μ−+μ+ scattering at finite. Physics Letters B, (791), 195-200.

https://doi.org/10.1016/j.physletb.2019.02.033

UVicSPACE: Research & Learning Repository

_____________________________________________________________

Faculty of Science

Faculty Publications

_____________________________________________________________

On Lorentz violation in e

−

_+e

+

_{→ μ}

−

_+μ

+

_{scattering at finite}

P.R.A. Souza, A.F. Santos, S.C. Ulhoa, F.C. Khanna

April 2019

© 2019 The Author(s). Published by Elsevier B.V. This is an open access article

under the CC BY license (

http://creativecommons.org/licenses/by/4.0/

)

This article was originally published at:

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

On

Lorentz

violation

in

e

−

+

e

+

→

μ

−

+

μ

+

scattering

at

finite

temperature

P.R.A. Souza

a

,

A.F. Santos

a,

∗

,

S.C. Ulhoa

b

,

F.C. Khanna

c,1

a_{InstitutodeFísica,UniversidadeFederaldeMatoGrosso,78060-900,Cuiabá,MatoGrosso,Brazil} b_{InternationalCenterofPhysics,InstitutodeFísica,UniversidadedeBrasília,70910-900,Brasília,DF,Brazil} c_{DepartmentofPhysicsandAstronomy,UniversityofVictoria,BCV8P5C2,Canada}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received10October2018 Accepted21February2019 Availableonline26February2019 Editor:A.Ringwald

Keywords: Lorentzviolation Crosssection Finitetemperature

Small violation of Lorentz and CPT symmetries may emerge in models unifying gravity with other forces ofnature.An extensionofthe standardmodel withallpossible termsthatviolate Lorentzand CPTsymmetries are included.Here aCPT-evennon-minimal couplingterm isadded tothe covariant derivative.ThisleadstoanewinteractiontermthatbreakstheLorentzsymmetry.Ourmainobjectiveis tocalculatethecrosssectionforthe

e

−+e+→

μ

−+

μ

+scatteringinordertoinvestigateanyviolation of Lorentz and/orCPT symmetryatfinite temperature. Thermo Field Dynamicsformalism is used to considerfinitetemperatureeffects.

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

1. Introduction

TheStandard Model(SM) is a successfulfield theory that de-scribes fundamental particles and their interactions with high

precision. The SM is a gauge theory with the symmetry group

U

(

1

)

×

SU

(

2

)

×

SU

(

3

)

[1,2].TheSMisafundamentaltheory. How-everitdoesnotincludethetheoryofgravitationinitsframework, thatincludesthreefundamental forcesofnature: the electromag-netic,weakandstrongforces.Thereare severalattemptsto unify allinteractions ofnatureina uniquefundamental theory.Among

various candidates for a unified theory the most famous is the

stringtheory[3].Inadditionthismodeldoesnotexplainina sat-isfactory waysome problemssuch as, the hierarchy problem[4], theneutrinooscillation[5],cosmicparticlesathighenergies[6,7], amongothers. Includingthese issues leads to a physics well be-yondthestandardmodel.

It is anticipated that a fundamental theory would emerge at veryhighenergies(

≈

1019GeV).At sufficientlyhighenergies,the possibilityof small violation of the Lorentz and CPTsymmetries maybe present. Some models,such asstring theory [8], lead to spontaneousbreakingofLorentzsymmetry.Itisinterestingtonote

*

Correspondingauthor.

E-mailaddresses:pablo@fisica.ufmt.br(P.R.A. Souza),

alesandroferreira@fisica.ufmt.br(A.F. Santos),sc.ulhoa@gmail.com(S.C. Ulhoa), khannaf@uvic.ca(F.C. Khanna).

1 _{PhysicsDepartment,TheoreticalPhysicsInstitute,UniversityofAlberta}

Edmon-ton,Alberta,Canada.

that a quantum theory ofgravitation may be anticipated to vio-latetheLorentzsymmetry.TheseideasleadtotheStandardModel

Extension (SME) that violates Lorentz and CPT symmetry. Such

models havebeendeveloped[9–11]. The SMEconsistsofmodels ofwellknownphysics oftheSM plusallpossibletermsthat vio-lateLorentz andCPTsymmetry.Inaddition,itisdividedintotwo parts:(i) theminimalversion restrictedtopowercounting renor-malizable operators and(ii) the non-minimal version which also includesoperatorsofhigherdimensions.

ThestructureofSMEisawaytoinvestigatetheLorentz viola-tion.However, there isanotherinteresting wayto investigatethe Lorentz violation that modifies the interaction betweenfermions andphotons,i.e., a newnon-minimal couplingtermaddedto the covariantderivative [12]. Thenon-minimal couplingtermmaybe

CPT-odd or CPT-even that have been considered for various

ap-plications [12–22]. Here a CPT-even non-minimal coupling term will be included to analyze the e−

+

e+

→

μ

−

+

μ

+ scattering, a well-knownquantumelectrodynamicsprocess,atfinite temper-ature.TheThermoFieldDynamics(TFD)formalismwillbeusedto introducetemperatureeffects.

TFD isarealtime finitetemperatureformalism[23–28].It in-cludes the statistical average of an observable

A

expressed asa thermalvacuumexpectationvaluei.e.,



A

= 

0

(β)

|

A|

0

(β)



,where

|

0

(β)



isthethermalvacuum,

β

=

_k1

BT,withT beingthe

temper-atureandkB is theBoltzmannconstant (weusekB

= ¯

h

=

c

=

1). This formalism is composed of two ingredients, the doubling of theHilbertspaceandtheBogoliubovtransformation.Thisdoubling https://doi.org/10.1016/j.physletb.2019.02.033

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

196 P.R.A. Souza et al. / Physics Letters B 791 (2019) 195–200

consists of the Hilbert space composed of the original space, S, andafictitiousspace(tildespace), S.

˜

The mapbetweenthetilde andnon-tildeoperators isdefinedby thetilde(ordual) conjuga-tionrules. Thetemperatureeffectisimplemented inthedoubled Hilbertspaceby a Bogoliubov transformationwhich introduces a rotationofthetildeandnon-tildevariables.

Thispaperisorganizedasfollows.Insection2,an introduction totheTFD formalismisdeveloped.Inthesection 3,the modelis presented.Thetransitionamplitudeandthecrosssectionforthree differentverticesarecalculated.Insection4,someconcluding re-marksarepresented.

2. TFDformalism

TFDisathermalquantumfield theorywithathermalvacuum

|

0

(β)



. It iscomposed by two fundamental ingredients: (1) dou-bling thedegrees of freedom in a Hilbert spaceand (2) the Bo-goliubovtransformation.TheexpandedHilbertspaceisdefinedas ST

=

S

⊗ ˜

S,withS beingthestandardHilbertspaceand

˜

S the fic-titiousHilbertspace.Themap betweenthetilde B

˜

i andnon-tilde

Bioperatorsisdefinedbythefollowingtildeconjugationrules:

(

BiBj

)

∼

= ˜

BiB

˜

j

,

(

c Bi

+

Bj

)

∼

=

c∗B

˜

i

+ ˜

Bj

,

(

B†_i

)

∼

= ˜

Bi †

,

( ˜

Bi

)

∼

= −ξ

Bi

,

(1)

with

ξ

= −

1 forbosons and

ξ

= +

1 forfermions.TheBogoliubov transformation introduces a rotation in the tilde and non-tilde Hilbertspacewith thermaldependence.The Bogoliubov transfor-mationisdifferentforfermionsandbosons.Hereourinterestisin fermions.ThentheBogoliubovtransformationforfermionsis

cp

= u(β)

cp

(β)

+ v(β)˜

c†p

(β),

c†p

= u(β)

c†p

(β)

+ v(β)˜

cp

(β),

(2)

˜

cp

= u(β)˜

cp

(β)

− v(β)

c†p

(β),

˜

c†p

= u(β)˜

c†p

(β)

− v(β)c

p

(β),

wherecp andc†p arethe annihilationandcreation operators.The factors

u

(β)

and

v

(β)

aregivenas

u

(β)

=

cos

(θ (β))

= (e

−β|κ0|

+

₁

₎

−1

_,

₍₃₎

v

(β)

=

sin

(θ (β))

= (e

β|κ0|

+

₁

₎

−1

_.

Algebraicrulesforthermaloperatorsare

{

cp

(β),

cq†

(β)

} = δ

3

(

p

−

q

),

(4)

{˜

cp

(β),

c

˜

q†

(β)

} = δ

3

(

p

−

q

),

(5)

andotheranti-commutationrelationsarenull.

IntheframeworkoftheTFDformalismthetransitionamplitude foranyQEDprocessisgivenas

ˆ

S

f i

(β)

=



f

, β



ˆ

S



_

i

, β



,

(6)

wherethethermalstatesaredefinedas

|

i

, β

 =

c†_p1

(β,

s1

)

d†p2

(β,

s2

)

|

0

(β)

,

(7)

|

f

, β

 =

c†p3

(β,

s3

)

d

†

p4

(β,

s4

)

|

0

(β)

,

(8)

withsi beingthespin variable(i

=

1

,

2

,

3

,

4)and S-matrix

ˆ

is

de-finedas

ˆ

S

=

∞



n=0

(

−ı)

n n

!



dx1dx2

...

dxn

: [ ˆ

HI

(

x1

) ˆ

HI

(

x2

)... ˆ

HI

(

xn

)

] : ,

(9)

where H

ˆ

I

(

x

)

=

HI

(

x

)

− ˜

HI

(

x

)

is the interaction hamiltonian. Here uptothesecondordertermisconsideredandhastheform

ˆ

S(2)

=

(

−ı)

2 2



d4xd4y

: [ ˆ

HI

(

x1

) ˆ

HI

(

x2

)

] :=

S(2)

− ˜

S(2)

.

(10) Thenthetransitionamplitudebecomes

S

f i

(β)

=



f

, β





S(2)

|i

, β



=

(

−ı)

2 2

!



d4xd4y



f

, β



 : [

L

I

(

x

)

L

I

(

y

)

] :





i

, β



.

(11)

Itisimportanttonotethat,thereisasimilarequationforthetilde part. As the physicalquantities are givenby non-tilde part, only thispartisconsidered.

Using thetransitionamplitude, the crosssection forany scat-teringprocessatfinitetemperatureisconsidered.Thecrosssection isdefinedas d

σ

(β)

d



=

1 64

π

2 1 4s



Spin





S

f i

(β)





2

,

(12)

where

√

s

=

2E

=

EC M andEC Misthecenterofmass(CM)energy. Inadditionanaverageoverthespinoftheincomingparticlesand summingoverthespinofoutgoingparticlesisincluded.

Inthenextsection thetransitionamplitudewillbecalculated. Then the crosssection forthe e−

+

e+

→

μ

−

+

μ

+ scatteringat finitetemperatureiscalculated.

3. Crosssectionofthee−

+

e+

−→

μ

−

+

μ

+scattering

Herethecrosssectionforthee−

+

e+

−→

μ

−

+

μ

+scattering atfinitetemperatureiscalculated.InadditionLorentz-violating ef-fects are included. The Lorentz violation is using a non-minimal couplingtermthatisaddedtothecovariantderivative,i.e.,

D

μ

=

Dμ

+

λ

2Kμνθρ

γ

ν_Fθρ

_,

₍₁₃₎

with Dμ

= ∂

μ

+

ie Aμ.Here

λ

isthecouplingconstantforLorentz violation term. The tensor Kμνθρ belongs to the CPT-even gauge

sector of the SME. It has the same symmetries as that of the

Riemann tensor and it possesses doublenull trace. Thus the in-teractionpartoftheDiracLagrangianbecomes

L

I

D

= −

e



γ

μ



Aμ

+

λ

2Kμνθρ



μν

_

_Fθρ

_,

₍₁₄₎

where



μν

=

2i

[

γ

μ

,

γ

ν

]

is used. The first term describes the

usual QEDvertex andthe second term isa new vertexthat

im-plies violationofLorentz symmetriesduetothe CPT-eventensor. The tensorKμνθρ maybedecomposedintobirefringentand non-birefringentcomponents.Hereourinvestigationisrestrictedtothe non-birefringent components that is represented by a symmetric andtracelessrank-2tensorKμν [29],i.e.,

Kμναβ

=

1 2



gμαKσβ

−

gναKμβ

+

gνβKμα

−

gμβKνα



,

(15)

where Kμν isdefinedbythecontraction Kμν

≡

K ρ μρν .Then the interactionLagrangianbecomes

L

I D

= −

e



γ

μ



Aμ

+ λ



βνKνμ

− 

μνKνβ

κ

βA_μ

,

(16)

with

κ

μ being the 4-momentum of the photon. This interaction Lagrangianimpliesthefollowingvertices:

• →

Vμ₍₀₎

= −

ie

γ

μ

,

(17)

⊗ →

Vμ₍₁₎

= −

i

λ

κ

β



βνKνμ

− 

μνKνβ

Fig. 1. Tree-level Feynman diagrams with different vertices.

The Feynman diagrams that describe this scattering process are giveninFig.1.

Toanalyzethisprocess,considerthecenterofmassframe(CM) suchthat p1

= (

E

,

pi

),

p3

= (

E

,

pi

),

p2

= (

E

,

−

pi

),

p4

= (

E

,

−

pi

),

κ

= (

p1

+

p2

)

= (

√

s

,

0

),

(19)

with p1

,

p2

,

p3 and p4 being the 4-momentum of the electron, positron,muonandanti-muon,respectively.Thenewvertex com-ponentsare

V0₍₁₎

=

0

,

(20)

Vi₍₁₎

=

Vi_+is

+

Vi_+an

+

V₋i (21)

wherethepartassociatedwiththeparity-evenisotropiccoefficient is

Vi_+is

= −ı

√

sK00



0i

,

(22)

theanisotropicparity-evenpartis

Vi_+an

= ı

√

sKij



0j

,

(23)

andtheparity-oddcomponentis

Vi₋

= −ı

√

sKj



ij

.

(24)

Thenthetransitionamplitudeiswrittenas

S

f iλ

(β)

=

1 2



d4xd4y

×



a,b



f

, β



 : (

x

)

Vμ_(a)

(

x

)(

y

)

Vν_(b)

(

y

)

Aμ

(

x

)

Aν

(

y

)

:





i

, β



,

(25)

witha

,

b

=

0

,

1.Consideringthatthewavefunctionofthefermion fieldis

(

x

)

=



dp



cp

(

s

)

u

(

p

,

s

)

e−ıpx

+

d†p

(

s

)

v

(

p

,

s

)

eıpx



,

(26)

with cp and dp being annihilation operators for electrons and positrons,respectivelywithu

(

p

,

s

)

andv

(

p

,

s

)

beingDiracspinors, theneq. (25) becomes

S

f iλ

(β)

=



d4p

(

2

π

)

4



d4xd4ye−ıx(p1−p3)−ıy(p2−p4)

×



a,b



v

(

p2

,

s2

)

V₍μa)u

(

p1

,

s1

)



u

(

p3

,

s3

)

V(νb)v

(

p4

,

s4

)



×



0

(β)



 :

Aμ

(

x

)

Aν

(

y

)

:





0

(β)



,

(27)

where the Bogoliubov transformation and the anti-commutation

relationbetweentheannihilationandcreationoperatorshavebeen used. The photon propagator at finite temperature [24,28,30] is givenas



0

(β)



 :

Aμ

(

x

)

Aν

(

y

)

:





0

(β)



=

i



_d4

_κ

(

2

π

)

4e− iκ(x−y)



_

f 0

(

κ

)

− 

f β

(

κ

)



η

μν

,

(28) where



₀f

(

κ

)

=

1

κ

2



1 0 0

−

1



,

(29)

isthezerotemperaturepartofthephotonpropagatorand



_βf

(

κ

)

=

2

π

i

δ(

κ

2

₎

eβ|κ0|

−

₁



1 eβ|κ0|/2 eβ|κ0|/2

−

₁



,

(30)

is the finite temperature part. Using the definition of the four-dimensional delta function and carrying out the

κ

integral, the

matrixelementbecomes

S

f iλ

(β)

=

i

u

2

(β)

− v

2

(β)

2





₀f

(

κ

)

− 

_βf

(

κ

)



S

f iλ

,

(31) with

S

f iλ

=

1

κ

2 1



a,b=0



v

(

p2

,

s2

)

V₍μ_a)u

(

p1

,

s1

)



×



u

(

p3

,

s3

)

V(b)μv

(

p4

,

s4

)



,

(32)

198 P.R.A. Souza et al. / Physics Letters B 791 (2019) 195–200

beingthematrixelementatzerotemperature.Theremainingdelta functionthatexpressesoverallfour-momentumconservationis ig-nored.Usingtherelation

[

v2Vaμu1

][

u1Vbμv2

] =

tr

[

Vμau1u1Vμ_bv2v2

]

(33) andtheeq. (3) forthefunctions

u

(β)

and

v

(β)

,thesquareofthe transitionamplitudeisfoundas



spin

|

S

f iλ

(β)

|

2

=

B

(β)

s2



a,b



c,d

E

μν (a,b)

M(

c,d)μν

,

where

B

(β)

=

tanh4



β

EC M 2



1

+

(

2

π

)

2

_δ

2

₍

_s

₎

(

eβEC M

−

₁

₎

2



.

(34)

Hereonlythephysicalcomponentofthephotonpropagatoris con-sidered

E

μν (a,b)

=

V μ (a)



s1 u

(

p1

,

s1

)

u

(

p1

,

s1

)

Vν(b)



s2 v

(

p2

,

s2

)

v

(

p2

,

s2

)

=

tr



Vμ_(a)

(

/

p1

+

me

)

Vν(b)

(

/

p2

−

me

)



,

(35)

M

μν (a,b)

=

V μ (a)



s3 u

(

p3

,

s3

)

u

(

p3

,

s3

)

Vν(b)



s4 v

(

p3

,

s4

)

v

(

p4

,

s4

)

=

tr



V(c),μ

(

p

/

3

+

mμ

)

V(d)ν

(

/

p4

−

mμ

)



,

(36)

wheretherelations,



s u

(

p

,

s

)

u

(

p

,

s

)

=

p

/

+

m (37)



s v

(

p

,

s

)

v

(

p

,

s

)

=

p

/

−

m

,

(38)

areused.Thepropagatoratfinitetemperatureintroducesproduct ofdeltafunctionswith identicalarguments (34). Thisproblemis avoided by workingwith the regularized form ofdelta-functions andtheirderivatives[31]:

2

π

i

δ

n

(

x

)

=



−

1 x

+

i





n+1

−



−

1 x

−

i





n+1

.

(39)

Thusthedifferentialcrosssectionatfinitetemperatureforthis scatteringis d

σ

λ

(β)

d



=

B

(β)

d

σ

λ d



,

(40) where d

σ

λ d



=

1

(

8

π

)

2_4s3



a,b



c,d

E

μν (a,b)

M(

c,d)μν

,

(41)

isthedifferentialcrosssectionatzerotemperature.Thenthecross sectionatfinitetemperaturehastheform

σ

λ

(β)

=

B

(β)

σ

λ

,

(42) with

σ

λ

=

1 64

π

2 1 4s3



a,b



c,d

E

μν (a,b)



d



M(

c,d)μν

,

(43)

wheretheintegrationisonlyonangularvariablesofscattered par-ticles.

Now let usconsider the contribution of each vertex given in eqs. (22), (23) and (24) intheultra-relativisticlimit. Inthislimit

assume me

=

mμ

=

0, then the electronic and muonic

contribu-tionsbecome

E

μν (a,b)

=

tr



Vμ_(a)

/

p1Vν(b)

/

p2



,

(44)

M

μν (a,b)

=

tr



Vμ_(c)

/

p3Vν(d)

/

p4



.

(45)

It is important to note that, the non-null components are those witha

=

b.Whena

=

b thereareanoddnumberofDiracmatrices andtheir trace iszero.They are alsonullwhen

μ

=

0 or

ν

=

0, since thisimpliesthat V0₍₁₎

=

0 and then,

E

₍0i_ab)

= E

i0_(ab)

= M

0i_(ab)

=

M

i0

(ab)

=

0.Thereforethenon-zerocomponentsare

E

ij (0,0)

=

tr

[

V i (0)

/

p1Vj₍₀₎p

/

2

],

(46)

E

ij (1,1)

=

tr

[

V i (1)

/

p1Vj₍₁₎p

/

2

],

(47)

M

ij (0,0)

=

tr

[

V i (0)

/

p4Vj₍0)

/

p3

],

(48)

M

ij (1,1)

=

tr

[

Vi(1)

/

p4Vj(1)p

/

3

].

(49) Usingtheseresultsthecrosssectionbecomes

σ

λ

=

1 64

π

2 1 4s3

E

i j (0,0)



d



M(

0,0)i j

+ E

₍i j_0,0)



d



M(

1,1)i j

+ E

i j (1,1)



d



M(

0,0)i j

+ E

₍i j_1,1)



d



M(

1,1)i j

.

(50)

3.1. Isotropicparity-evencontribution

In thiscasethevertexisgivenby eq. (22) and thenthe non-zerocomponentsofeqs. (46)-(49) are

E

ij (0,0)

=

2e 2

₍

_s

_δ

ij

₋

_4pi_pj

_),

₍₅₁₎

E

ij (1,1)

=

8

λ

2_sK2 00pipj

.

(52)

M

ij (0,0)

=

2e2

(

s

δ

ij

−

4pipj

),

(53)

M

ij (1,1)

=

8

λ

2_sK2 00pipj

.

(54)

Thentheintegralsineq. (50) become

E

ij (0,0)



d



M(

0,0)ij

=

16

π

3 4s 2_e4

_,

₍₅₅₎

E

ij (0,0)



d



M(

1,1)ij

= E

ij_(1,1)



d



M(

0,0)ij

=

16

π

3 2s 3_e2

_λ

2_K2 00

.

(56)

Here the term

E

₍i j_1,1)



d



M

(1,1)i j is ignored, since it is of the fourthorderinLorentz-violatingparameter.Thusthecrosssection atfinitetemperature(uptosecond orderinLorentz-violating pa-rameter)is

σ

i +λ

(β)

=

B

(β)

σ

QED



1

+



λ

√

sK00 e



2



,

(57)

with

σ

QED

=

64

π

s2e4

/

3 and

B(β)

is defined in eq. (34). When temperature effects go to zero

B(β)

→

1, the resultis the same asin[19].

3.2.Anisotropicparity-evencontribution

Herethevertexisgivenineq. (23).Thenelectronicfactorsare givenas

E

ij (0,0)

=

2e 2

₍

_s

_δ

ij

₋

_4pi_pj

_),

₍₅₈₎

E

ij (1,1)

=

8

λ

2_sKik_Kj l_p lpk

,

(59)

withcomponentsofthetensor

M

μν_(a,b) beingsameif p ischanged top.Thentheintegralsineq. (50) are

E

ij (0,0)



d



M(

0,0)ij

=

16

π

3 4s 2_e4

_,

₍₆₀₎

E

ij (0,0)



d



M(

1,1)ij

=

16

π

3 s 2_e2

_λ

2



s

|

K

|

2

−

4

(

piKij

)

2



,

(61)

E(

1,1)ij



d



M(

0,0)ij

=

16

π

3 4e 2_s2

_λ

2

₍

_pi_K ij

)

2 (62)

where

|

K

|

2

=

KijKij.Thereforethecrosssection atfinite tempera-tureis

σ

_+λa

(β)

=

B

(β)

σ

Q E D

1

+



λ

e



2

 √

s|K

|/

2



2

+ (

piKij

)

2



.

(63) 3.3.Parity-oddcontribution

Tocalculateparity-oddcontributionsthevertexgivenineq. (24) isconsidered.Then

E

ij (0,0)

=

2e 2

₍

_s

_δ

ij

₋

_4pi_pj

_),

₍₆₄₎

E

ij (1,1)

=

8

λ

2_s

_

ikl

_

jm n_K kKlpmpn

,

(65)

andthe muons contributions are obtained in a similar way. The relevantintegralsare

E

ij (0,0)



d



M(

0,0)ij

=

16

π

3 4s 2_e4

_,

₍₆₆₎

E

ij (0,0)



d



M(

1,1)ij

=

16

π

3 e 2

_λ

2



s|K

|

2

+

4

(

p

·

K

)

2



,

(67)

E(

1,1)ij



d



M(

0,0)ij

=

16

π

3 e 2

_λ

2



s

|

K

|

2

−

8

(

p

·

K

)

2



,

(68) with

(

p

·

K

)

=

pi_K

j.Thenthecrosssectionis

σ

−

(β)

=

B

(β)

σ

Q E D

1

+



λ

2e



2



3s

|

K

| −

4

(

|

p

||

K

|

cos

(θ ))

2



,

(69)

where

θ

istheanglebetweentheparticlebeamandthefieldK. Theresultsobtainedaregeneralandshowthatthetemperature effectsmodify the crosssection of thescatteringprocess for any chosenvertex.Inthelimitofzerotemperaturethestandardresult forthe QED modified by Lorentz-violatingparameters are recov-ered,inallcases.Theseresultsalsoindicatethat thetemperature effectsmayimproveconstraintsonLorentz-violatingparameter.

4. Conclusion

TheSMEisaframeworktostudyLorentzandCPTviolationthat includestheSM,generalrelativityandall possibletermsthat vio-latetheLorentzandCPTsymmetries.Anotherinterestingwayisto modifythe interactionvertexbetweenfermionsandphotons,i.e., a newnon-minimal couplingtermaddedtothecovariant deriva-tive. Here a Lorentz violating CPT-even term is chosen to study thee−

+

e+

→

μ

−

+

μ

+scatteringatfinitetemperature.Thisnew couplinghasmassdimensionequalto

−

1,whichleads toa non-renormalizabletheory atpower counting.Howeverinthepresent casethisdoesnotposeanyproblemsinceourinterest isin ana-lyzingthetree-levelscatteringprocess.Thetemperatureeffectsare introducedusingtheTFDformalism.Threedifferentverticeswhich introducetheLorentzviolationareconsidered.Thenthecross sec-tionatfinitetemperatureiscalculated.Ourresultsshow thatthe temperatureeffectsmodifythecrosssection.Thennewconstraints

on Lorentz-violatingparametermay beimposed by the

tempera-tureeffects.Inaddition astrophysicalprocessesmaybe studiedif thetemperatureisveryhigh.

Acknowledgements

This work by A.F.S. is supported by CNPq project 308611/

2017-9. References

[1]T. Kioshita, Quantum Eletrodynamics, Adv. Ser. Dir. High Energy Phys., vol. 7, Word Scientific, 1990.

[2]M.E. Peskin, D.V. Schroeder, An Introduction to Quantum Field Theory, West-view, 1995.

[3]E. Witten, Nucl. Phys. B 443 (1995) 85.

[4]N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 429 (1998) 263. [5]V. Barger, D. Marfatia, K.L. Whisnant, The Physics of Neutrinos, Princeton

Uni-versity Press, ISBN 0-691-12853-7, 2012.

[6]L.J. Watson, D.J. Mortlock, A.H. Jaffe, Mon. Not. R. Astron. Soc. 418 (2011) 206, arXiv:1010 .0911.

[7]W. Bernreuther, M. Suzuki, Rev. Mod. Phys. 63 (1991) 313. [8]V.A. Kostelecky, S. Samuel, Phys. Rev. D 39 (1989) 683;

V.A. Kostelecky, R. Potting, Nucl. Phys. B 359 (1991) 545. [9]V.A. Kostelecky, R. Potting, Phys. Rev. D 51 (1995) 3923.

[10]D. Colladay, V.A. Kostelecky, Phys. Rev. D 55 (1997) 6760, Phys. Rev. D 58 (1998) 116002.

[11]V.A. Kostelecky, Phys. Rev. D 69 (2004) 105009.

[12]H. Belich, T. Costa-Soares, M.M. Ferreira Jr., J.A. Helayël-Neto, Eur. Phys. J. C 41 (2005) 421.

[13]H. Belich, T. Costa-Soares, M.M. Ferreira Jr., J.A. Helayël-Neto, M.T.D. Orlando, Phys. Lett. B 639 (2006) 678.

[14]H. Belich, L.P. Colatto, T. Costa-Soares, J.A. Helayël-Neto, M.T.D. Orlando, Eur. Phys. J. C 62 (2009) 425.

[15]H. Belich, T. Costa-Soares, M.M. Ferreira Jr., J.A. Helayël-Neto, F.M.O. Moucherek, Phys. Rev. D 74 (2006) 065009.

[16]H. Belich, M.M. Ferreira Jr., E.O. Silva, M.T.D. Orlando, Phys. Rev. D 83 (2011) 125025.

[17]B. Charneski, M. Gomes, R.V. Maluf, A.J. da Silva, Phys. Rev. D 86 (2012) 045003. [18]G.P. de Brito, J.T.G. Junior, D. Kroff, P.C. Malta, C. Marques, Phys. Rev. D 94

(2016) 056005.

[19]R. Casana, M.M. Ferreira, R.V. Maluf, F.E.P. dos Santos, Phys. Rev. D 86 (2012) 125033.

[20]R. Casana, M.M. Ferreira Jr, E.O. Silva, E. Passos, F.E.P. dos Santos, Phys. Rev. D 87 (2013) 047701.

[21]R. Casana, M.M. Ferreira Jr., R.V. Maluf, F.E.P. dos Santos, Phys. Lett. B 726 (2013) 815.

[22]J.B. Araujo, R. Casana, M.M. Ferreira Jr., Phys. Rev. D 92 (2015) 025049. [23]Y. Takahashi, H. Umezawa, Collect. Phenom. 2 (1975) 55, Int. J. Mod. Phys. B 10

(1996) 1755.

[24]Y. Takahashi, H. Umezawa, H. Matsumoto, Thermofield Dynamics and Con-densed States, North-Holland, Amsterdam, 1982.

[25]H. Umezawa, Advanced Field Theory: Micro, Macro and Thermal Physics, AIP, New York, 1993.

200 P.R.A. Souza et al. / Physics Letters B 791 (2019) 195–200

[26]A.E. Santana, F.C. Khanna, Phys. Lett. A 203 (1995) 68.

[27]A.E. Santana, F.C. Khanna, H. Chu, C. Chang, Ann. Phys. 249 (1996) 481. [28]F.C. Khanna, A.P.C. Malbouisson, J.M.C. Malboiusson, A.E. Santana, Thermal

Quantum Field Theory: Algebraic Aspects and Applications, World Scientific, Singapore, 2009.

[29]V.A. Kostelecky, M. Mewes, Phys. Rev. D 80 (2009) 015020. [30]A.F. Santos, F.C. Khanna, Int. J. Mod. Phys. A 31 (2016) 1650122.

[31]Ch.G. van Weert, An Introduction to Real- and Imaginary-Time Thermal Field Theory, Lect. Notes Stat. Field Theory, 2001.