Comment about the vanishing of the vacuum energy in the Wess–Zumino model

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Physics Letters B

www.elsevier.com/locate/physletb

Comment about the vanishing of the vacuum energy in the Wess–Zumino model

Andrei O. Barvinsky

^a

, Alexander Yu. Kamenshchik

^b,c,∗

, Tereza Vardanyan

^b

aTheoryDepartment,LebedevPhysicsInstitute,LeninskyProspect53,Moscow119991,Russia

bDipartimentodiFisicaeAstronomia,UniversitàdiBolognaandINFN,ViaIrnerio46,40126Bologna,Italy

cL.D.LandauInstituteforTheoreticalPhysicsoftheRussianAcademyofSciences,Kosyginstr.2,119334Moscow,Russia

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

Articlehistory:

Received10April2018 Accepted4May2018 Availableonline17May2018 Editor:M.Cvetiˇc

Keywords:

Supersymmetry Wess–Zuminomodel Vacuumenergy

WecheckthecancellationofthevacuumenergyintheWess–Zuminomodelatthetwo-looporderinthe componentfieldformalismswithandwithoutauxiliaryfields.Weshowthatinbothcasesthevacuum energyisequaltozero.However,intheformalismwheretheauxiliaryfieldsareexcluded,thevanishing ofthevacuumenergyarisesduetothecancellationbetweenthepotentialandkineticenergies,whilein theformalismwiththeauxiliaryfields,bothtermsvanishseparately.

©^{2018PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

Thediscovery ofthe supersymmetry–thesymmetry between bosons andfermionsin theearly seventies [1–3] was one ofthe mostimpressive achievementsofthe modern theoretical physics.

Inspiteofthefactthattheparticles,whichplaytheroleofsuper- partnersoftheknownparticlesbelongingtotheStandardModel, arenotyetobservedexperimentally,theinterestinsupersymmetry andtoitsapplicationsindifferentareasofphysicsandmathemat- icsisstillgrowingandthenumberoforiginalpapersandreviews dedicatedtodifferentaspects ofthe supersymmetryisreallyim- pressive. One of the most studied objects in this context is the Wess–Zuminomodel[4,5] –asimple,butveryrichmodel,where manybasicfeatures ofthesupersymmetrywere observedforthe first time. Amongst these features there are cancellation of the essential part of the ultraviolet divergences [4] and the vanish- ing of the vacuum energy [6]. The fact that the vacuumenergy inexactly supersymmetrictheories vanishes hasa rathergeneral nature. Indeed, the fundamental relation defining the supersym- metricextensionofthePoincaréalgebrais

{

^Qa

, ¯

Q_b

} =

²

γ

_ab^μP_μ

,

(1)

*

Correspondingauthor.

E-mailaddresses:barvin@td.lpi.ru(A.O. Barvinsky),kamenshchik@bo.infn.it (A.Yu. Kamenshchik),tereza.vardanyan@bo.infn.it(T. Vardanyan).

where Q is the generator of the supersymmetry transformation andP μ istheenergy–momentumvector.Thevacuumexpectation valueoftheenergy⁰|^P0|⁰^{vanishesifthe}supersymmetrygener- atorsannihilatethevacuumstate:

Q

|

⁰

 =

⁰

, ¯

Q

|

⁰

 =

⁰

⇒ 

⁰

|

^P0

|

⁰

 =

⁰

.

(2) However, it is interesting also to seehow this cancellationof thevacuumenergyworksattheleveloftheconcretefieldsanddi- agrams.Indeed,inthecaseofabrokensupersymmetryexactcan- cellation of the vacuumenergy, discussed above, doesnot work;

howeveronecanstillhopethatatleasttheultravioletdivergences in the vacuumenergy expressions are cancelled dueto the fact, thatthenumbersofthebosonsandfermionspresentinthemod- elsunderconsiderationareequal.

In fact, already in fifties Pauli [7] suggested that the vacuum (zero-point) energies of all existing fermions and bosons com- pensate each other.This possibilityis based on thefact that the vacuum energy of fermions has a negative sign, whereas that of bosons has a positive one. Later, in a series of papers Zel- dovich [8,9] related a finite part of the vacuum energy to the cosmologicalconstant,howeverratherthaneliminatingthediver- gencesviatheboson–fermioncancellation,hesuggestedthePauli–

Villarsregularisationofalldivergencesbyintroducinganumberof massive regulator fields. Covariant regularisation of all contribu- tionsthenleadstofinitevaluesforboththeenergydensity

ε

and https://doi.org/10.1016/j.physletb.2018.05.008

0370-2693/©^{2018PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

(negative)pressurep correspondingtoacosmologicalconstant,i.e.

relatedbytheequationofstate p= −

ε

.

Recently, this Pauli–Zeldovich cancellation mechanismfor the divergences of the vacuum energy due to the balance between the contributions of bosons and fermions has attracted growing attention [10–14]. Similar questions were also discussed in pa- pers[15–19].Inparticular,in[14] severalmodelswithinteractions between different particles at the lowest order of the perturba- tion theory were considered. Here a certain subtlety is present.

To provide the cancellationof the vacuumenergy divergences it is necessary to have the balance between the fermion and the boson degrees of freedom not only on shell, but also off shell.

Letustryto bemoreprecise inthe terminology.Speakingabout theonshellorphysicaldegreesoffreedom,one meansthenum- ber of independent Cauchy data, which can be imposed on the fieldunderconsideration.Thisnumberisassociatedalsowiththe numberofdifferentparticlestates.Speakingabouttheoffshellde- greesof freedom,one meansinstead thenumberofindependent fieldcomponents.Forexample,theDiracspinorhasfourcomplex oreight real components.When we impose the first-order Dirac equationthe numberofindependentrealcomponents(or Cauchy data) dwindles andbecomes equalto four, whichcorresponds to a particle andantiparticle, having two helicity states each. Thus, one can say that the number ofdegrees of freedom of a spinor field,interpretedasthenumberofindependentfieldcomponents, doubles when it is off shell. Then the Majorana spinor has two complexcomponents,i.e.fourdegreesoffreedom offshell.When weimposethefirst-orderDiracequationthenumberofdegreesof freedom, interpreted asthe numberof initial datafor thisequa- tion,becomesequaltotwo(seee.g.[20]).

Wecanalsoaddthatwhilethedifferencebetweenthenumber of independent Cauchy data and the number of the field com- ponents for spinors is connected with the fact that they satisfy first-order differential equations, this last fact has in turn deep group-theoretical roots. Indeed, requirement that a field belongs toacertainrepresentationoftheLorentzgroup impliesadefinite formoftheinvariantwaveequationforthisfield.Namely,thebe- longing of the spinorto the (1/2,0)⊕ (⁰,1/2) representation of the Lorentz group impliesthat it subject to the first-order Dirac equation(see,e.g.[21,22]).

As is well known in the Wess–Zumino model one has two fermiondegreesoffreedomoftheMajoranaspinorandtwoboson degrees of freedom associated with the scalar and pseudoscalar fields. Off shell the number of fermion degrees of freedom be- comesequaltofourwhilethe roleoftwoadditionalbosonfields isplayedby twoauxiliary fields,whichare independentoffshell.

If we consider the non-supersymmetric models with the Pauli–

Zeldovich mechanism of cancellation of UV divergences for the vacuumenergyinthepresence ofinteractions, thenthenumbers ofthebosonandfermiondegreesoffreedomshouldalsocoincide notonlyonshell,butoffshellaswell[14].

Thus, it looks like the introduction of the auxiliary fields be- comes unavoidable even for non-supersymmetric models if we wanttomake thePauli–Zeldovichmechanismefficient.Indeed,it wasexplicitlyshownintheseminalpaperby Zumino[6] thatthe potentialvacuumenergy(under thepotentialwe meanthecubic andquarticterms intheaction)intheWess–Zumino modelvan- ishesatthe leveloftwo-loop Feynman diagrams. Then itis easy toshow(andweshalldoitinSect.3ofthepaper)thatthiscan- cellation failsinthe formalism withthe auxiliary fieldsexcluded via the equations of motion.However, explicitcalculations show that in thisformalism the kinetic vacuum energy(under the ki- neticwemeanallthequadratictermsintheaction,includingmass terms)isalsonon-vanishingwhilethesumofbothvanishes.Thus, all the calculations can be coherently done also in the formal-

ism withoutthe auxiliary fields.Hence, thedetailedpresentation andcomparisonoftheresults,obtainedindifferentformalismsfor a simple butvery rich Wess–Zumino model, seems to be rather instructive. We hope that these results could be usefulfor more complicatedmodelsaswell.Besides,whilethegeneralarguments aboutthedifferencebetweenthenumbersoftheonshellandoff shell degrees of freedom for spinors are well known, especially in the context of the supersymmetrictheories (see, e.g. [20,23]), the concrete analysis of the manifestation of thesedifference in thediagram calculationsarenot soelaborated,atleast,up toour knowledge.Thus,italsocanbeofsomeinterest.

Thepaperhasthefollowingstructure:intheSect.2wedemon- strate atthe two-loop orderthe vanishingof thevacuumenergy intheWess–Zuminomodelinthepresenceoftheauxiliaryfields.

In Sect. 3 we do the same in the formalism with the auxiliary fieldsexcluded,whilethelast sectioncontainsadiscussionwhich briefly mentions a similar derivation in the superfield formalism along withits relationto thePauli–Zeldovichcancellationmecha- nism.

2. Wess–Zuminomodelinthepresenceoftheauxiliaryfields:

calculationofthevacuumenergy

TheLagrangiandensityoftheWess–Zuminomodelis L

=

¹

2

(∂

_μA

)(∂

^μA

) +

¹

2

(∂

_μB

)(∂

^μB

) +

¹

2

¯ψ(

i

ˆ∂ −

^m

)ψ +

¹

2F²

+

¹

2G²

+

^{mF A}

+

^{mG B}

+

^{g F}

(

A²

−

^B2

)

+

^{2gG A B}

−

^g

( ¯ ψ ψ

A

+

ⁱ

¯ψ γ

⁵

ψ

B

),

(3) where A is a scalar field, B is a pseudoscalar field, ψ is a Ma- jorana spinor, F and G are auxiliary fields. The symbol ˆ∂ means ˆ∂ ≡

γ

^μ∂μ. The propagatorsor contractions ofthese fieldsin the momentumrepresentationhavethefollowingform[4,24]

DA A

=

^DB B

=

ⁱ k²

−

^m2

,

D_{A F}

=

^DBG

= −

^im

k²

−

^m2

,

DF F

=

DG G

=

^ik2

k²

−

m²

,

D_{ψ ¯ψ}

=

ⁱ

(ˆ

k

+

^m

)

k²

−

^m2

.

(4)

LetusaddthatincontrasttoDiracspinors,Majoranaspinorsalso haveDψ ψ andD_{¯ψ ¯ψ} propagatorswhicharerelatedtothestandard

“Dirac”propagatorbytheformulae(seee.g.[25]):

D_{ψ ψ}

=

^D_{ψ ¯ψ}C⁻¹

,

D_{¯ψ ¯ψ}

=

C^−1D_{ψ ¯ψ}

,

(5)

whereC^isthechargeconjugationoperator.However,inallthecal- culationsintheWess–Zuminomodelthepresenceofthepropaga- tors(5) isboileddownonlytotheappearanceofsomeadditional combinatorialfactors,whichmultiplythestandard Diracpropaga- tor D_{ψ ¯ψ}.

TheinteractionHamiltoniandensityofthemodel(wealsocall it“potential”)includes5termsandisgivenbytheexpression

H_I

= −

^{g F}

(

A²

−

^B2

) −

^{2gG A B}

+

^g

( ¯ ψ ψ

A

+

ⁱ

¯ψ γ

⁵

ψ

B

).

(6) The vacuumexpectationvalueofthepotential partoftheHamil- tonian(6) isgivenbytheexpression

E_pot

(

x

) =

 [

^d

φ ]

^HI

(

x

)

e^{i S}

 [

d

φ ]

e^{i S}

= 

HI

(

x

)

e^{i SI}





e^{i SI}

 ,

(7)

whereS_I istheinteractionpartofthefullaction S_I

= −



d⁴y H_I

(

y

)

(8)

andthebrackets...^{denotethequantumaveragewithrespectto} the free theory whose action is quadratic in the quantum fields φⁱ=^A,B,F,G,ψ, ¯ψ and incorporates D^{i j}(k) as the momentum spacepropagatorwiththecomponents(4).

The two-loop or g²-order of the expression (7), obtained by expandingitin S_I∝^g,lookslike

E^2-loop_pot

=

i



HI

(

x

)

SI

 −

i



HI

(

x

)  

SI

,

(9) wherethesecondterm(actuallyvanishingbecausebothHI(x)and SI areoddinquantumfields)subtractsthedisconnecteddiagrams partofE^2-loop_pot .Theexpression(9) correspondstotwotypesofthe Feynmandiagrams: thosewiththe topologyof a“dumbbell” and thosewiththe“nut”topology.Thedumbbelldiagrams–twotad- polesconnectedbythepropagator,



^HI

(

x

)

S_I



^dumbbell

∝

^Djn

(

x

,

x

)

S⁽_jni³⁾



d⁴y D^ik

(

x

,

y

)

S_klm⁽³⁾D^lm

(

y

,

y

),

(10) givethevanishingcontribution(here S_klm⁽³⁾ isathree-vertexcorre- spondingtothecubicinteractionterm(8)).Thisfollowsfromthe factthatthetadpolediagramsintheWess–Zuminomodelarecan- celledduetosupersymmetry.Eventhoughthisfactiswellknown, letusillustrateitonceagainforcompleteness.

In view of the only nonvanishing propagator components (4) thetadpoles

 φ

^iSI

 ∝



d⁴y D^ik

(

x

,

y

)

S_klm⁽³⁾D^lm

(

y

,

y

)

=

^Dik

(

0

)

S_klm⁽³⁾



_d4k

(

2

π )

^4D

lm

(

k

)

(11)

with the fermion leg, φⁱ= ψ ^or φⁱ= ¯ψ^{, vanish} identically. The same is true for the tadpoles with the pseudoscalar field B leg andtheauxiliary fieldG leg.Thetadpolewiththescalarfieldleg, φⁱ=^A,isproportionalattheone-looporderto



^A

(

F A²

−

^{F B2}

+

^{2G A B}

− ¯ψψ

^A

) 

=

^2DA A



D_{A F}

+

^DA F



D_{A A}D_{A F}



D_{B B}

+

^2DA A



D_BG

− (−

¹

)

D_{A A}



Tr D_{ψ ¯ψ}

,

(12)

whereforbrevityweomitthemomentumspaceintegrationmea- sure.Takingintoaccounttheformulae(4) andthefactthat



DA A

=

i I

,



DB B

=

i I



DA F

= −

imI

,



DBG

= −

imI



Tr D_{ψ ¯ψ}

=



_d4k

(

2

π )

⁴

Tr

(ˆ

k

+

^m

)

k²

−

^m2

=

^4imI

,

I

≡

¹

(

2

π )

⁴



d⁴k

k²

−

^m2

,

(13)

weseethat theright-hand side ofEq. (12) vanishes.The tadpole withtheauxiliaryfield F legisproportionalto



F

(

F A²

−

F B²

+

2G A B

− ¯ψψ

A

) 

=

^DF F



(

D_{A A}

−

^DB B

) +

^2DA F



D_{A F}

+

2DA F



DBG

− (−

1

)

DA F



TrD_{ψ ¯ψ}

=

0

.

(14) Thus,consideringtheexpression(9) wetakeintoaccount only the diagrams with a nut topology, i.e. the diagrams, where two verticesareconnectedbythreepropagators.Therearetwotypesof such diagrams,onesincludingfermions,andthose includingonly bosonfields.Thecontributionofthediagramsofthefirsttypeto thevacuumenergyisgivenbytheexpression

E1

(

x

) =



d⁴y



(

g

¯ψψ

A

)(

x

) · (−

ig

¯ψψ

A

)(

y

) +(

ig

¯ψ γ

⁵

ψ

B

)(

x

) · (−

i

(

i

)

g

¯ψ γ

⁵

ψ

B

)(

y

) 

,

(15)

where·^{separatesthefactorsatthepointsx andy,betweenwhich} and only which the chronological contractions are taken in the

“nut”diagram.Thisexpressionreads E1

=

^16g2



d⁴kd⁴p k

· (

^p

+

^k

)

(

k²

−

^m2

)(

p²

−

^m2

)((

p

+

^k

)

²

−

^m2

)

=

^8g2



d⁴kd⁴p

(

p

+

^k

)

²

−

^p2

+

^k2

(

k²

−

^m2

)(

p²

−

^m2

)((

p

+

^k

)

²

−

^m2

)

=

8g²



d⁴kd⁴p 1

(

p²

−

m²

)((

p

+

k

)

²

−

m²

) +

8g²m²



d⁴kd⁴p 1

(

k²

−

m²

)(

p²

−

m²

)((

p

+

k

)

²

−

m²

)

=

^8g2I2

+

^8g2m2K

,

(16)

where K

≡



d⁴kd⁴p 1

(

k²

−

^m2

)(

p²

−

^m2

)((

p

+

^k

)

²

−

^m2

) .

(17) Thesumofthecontributionsofthediagrams,whichincludeonly thebosonfieldsis

E₂

= −

^ig2



d⁴y



(

F A²

)(

x

)˙(

F A²

)(

y

) + (

^{F B2}

)(

x

) · (

^{F B2}

)(

y

) +

4

(

G A B

)(

x

) · (

G A B

)(

y

) −

4

(

F B²

)(

x

) · (

G A B

)(

y

) 

= −

^ig2



2



D_{F F}D_{A A}D_{A A}

+

⁴



D_{A F}D_{A F}D_{A A}

+

2



DF FDB BDB B

+

4



DG GDA ADB B

+

⁴



D_{A A}D_BGD_BG

−

⁸



D_{A F}D_BGD_BG

.

(18)

Usingthepresentation D_{F F}

=

^DG G

=

ⁱ

+

^im2

k²

−

^{m2 (19)}

and other formulae from Eq. (4), one can reduce the right-hand sideofEq. (18) tothefollowingsum

E₂

= −

^8g2I2

−

^8g2m2K

.

(20) Inother words,thenutdiagrams, includingtwoverticeslinearin auxiliaryfieldsandbilinearinscalar(pseudoscalar)fieldsgiverise totwotypesofdiagramswithsimplescalar-typepropagators:ones with the topology of a nut and those with the topology of the

“eight” diagram, including one 4-vertex and two one-propagator scalar loops– the product of two one-loop integrals.Finally, we seefromEqs. (16) and(20) that

E1

+

E2

=

0

.

(21) Thecorrespondingdiagrams(withoutdetail)werepresentedinthe paperbyZumino[6].

What can one say about the vacuum kinetic energy in the Wess–Zumino model? Oneknows thatin theabsence ofinterac- tionsthequantumfieldscanbeconsidered asthesystemsoffree harmonicoscillators andthat thezero energyofa boson oscilla- torispositivewhilethatofafermionoscillatorisnegative.Aswe have already mentioned in the Introduction, the requirement of thedisappearanceoftheultravioletdivergencesinthevacuumen- ergy ofthe free fieldsis reducedto theequalityof thenumbers ofbosonandfermiondegreesoffreedomandtocertainsumrules [7–12]. Inthe caseofthesupersymmetric modelsall the masses inside of a supemultiplet are equal and the mass sumrules are satisfiedautomatically.Moreover,whentheinteractionisswitched onthemassesbegintheir running,buttheir anomalous massdi- mensionsare such that the running massesofdifferent particles arestill equal.InthecaseoftheWess–Zumino modelinthefor- malismwiththeauxiliaryfieldsthereisonlyonerenormalization constant –the wave functionrenomralization, which isequal for all fields. This impliesthe equality of the running masses. Thus, one cansay thatboth thekinetic vacuumenergyandthe poten- tialvacuumenergyintheWess–Zuminomodelwiththeauxiliary fieldsarecancelledseparately.

3. Wess–Zuminomodelwithoutauxiliaryfieldsandthevacuum energy

TheEuler–Lagrange equationsforthe auxiliary fields F and G are

F

+

^g

(

A²

−

^B2

) =

⁰

,

G

+

^{2g A B}

=

⁰

.

(22)

Substituting the expression for the auxiliary fields taken from Eq. (22) into the Lagrangian of the Wess–Zumino model (3), we obtainanexpressionwhichcontainsonlythe“physical”fields A,B andψ:

L

=

¹

2

(∂

μA

)(∂

^μA

) +

¹

2

(∂

μB

)(∂

^μB

) +

¹

2

¯ψ(

i

ˆ∂ −

^m

)ψ

−

¹

2m²A²

−

¹

2m²B²

−

¹

2g²

(

A²

+

B²

)

²

−

mg A

(

A²

+

B²

)

−

g

( ¯ ψ ψ

A

+

i

¯ψ γ

⁵

ψ

B

).

(23)

NowtheinteractionHamiltoniandensityis HI

=

¹

2g²

(

A²

+

B²

)

²

+

mg A

(

A²

+

B²

) +

g

( ¯ ψ ψ

A

+

i

¯ψ γ

⁵

ψ

B

).

(24) Wecannowcalculatethepotentialvacuumenergycorresponding to the potential (24). Obviously, the contribution of the Yukawa termswill coincide withthat calculatedin the precedingsection andwillbe givenbytheformula(16).Then wehavethe“eight”- diagramtypecontributionofthequarticscalar-pseudoscalarinter- action

E3

=

^g2 2

 (

A²

+

B²

)

²



=

³ 2g²



D_{A A}

·



D_{A A}

+

³ 2g²



D_{B B}

·



D_{B B}

+

g²



DA A

·



DB B

= −

4g²I²

.

(25)

The contributionsofthetriplescalar–preudoscalarinteractions havetheformofthe“nut”diagramintegral

E4

= −

im²g²



d⁴y



A³

(

x

) ·

A³

(

y

) +

A B²

(

x

) ·

A B²

(

y

) 

= −

^im2g2

(

6



D_{A A}D_{A A}D_{A A}

+

²



D_{A A}D_{B B}D_{B B}

)

= −

8m²g²K

.

(26)

Thus,weseethatthetotalcontributionofthepotentialtermsinto thevacuumenergyis

E1

+

^E3

+

^E4

=

^4g2I2

=

⁰

.

(27) Where was the balance between thefermion andboson con- tributions lost? To answer this question let us try to calculate explicitlythe vacuumkinetic energyterms.Thisterms willcome fromtheexpression

E_kin

=

¹ 2

 ∂

μA

∂

μA

+

^m2A2



e^{i SI}

c

+

¹ 2

 ∂

_μB

∂

_μB

+

^m2B2



e^{i SI}

c

+

¹ 2

¯ψ

i

γ

i

∂

i

+

^m

 ψ

e^{i SI}

c

,

(28)

wherei=¹,2,3 and

∂

μA

∂

μA

= (∂

0A

)

²

+ (∂

iA

)

²

ofcourse denotesLorentz non-invariantcombinationcontributing totheenergydensityofthescalarfield A.Thesubscript“c”means that onlyconnecteddiagramsshouldbe considered,disconnected contributionsbeingsubtractedsimilarlytoEq. (9).

The two-loop contribution of the quartic interaction into the kineticenergyofthescalarfieldis

E₅

=



d⁴y



1

2

∂

_μA

∂

_μA

+

¹ 2m²A²

(

x

)

·



−

ⁱ

2g²

(

A²

+

B²

)

²

(

y

) 

c

= −

^4g2I

·



d⁴kk_μk_μ

+

^m2

(

k²

−

^m2

)

²

.

(29)

The corresponding contribution of the quartic interaction to the kineticenergyofthepseudoscalarfieldisthesame.

The contributionofthe triplescalar (pseudoscalar)interaction to thekineticenergyofthescalarfield, whichisobtainedbyex- panding e^{i SI} toquadratic orderinmg

d⁴yA(A²+^B2)(y),isthe following“nut”diagram

E₆

= −

¹ 2m²g²



d⁴y d⁴z



₁

2

∂

_μA

∂

_μA

+

¹ 2m²A²

(

x

)

· 

A³

(

y

) ·

^A3

(

z

) +

^{A B2}

(

y

) ·

^{A B2}

(

z

)



c

= −

^10m2g2



d⁴kd⁴p k_μk_μ

+

^m2

(

k²

−

^m2

)

²

(

p²

−

^m2

)((

k

+

^p

)

²

−

^m2

) .

(30) Thesign·^{againseparatesgroupsoffactorsatdifferentspacetime} points,thechronologicalcontractionsinsideeachgroupbeingdis- regardedlikein(15).

Asimilarcontributionofthetriplescalarinteractiontotheki- neticvacuumenergyofthepseudscalarfieldisdifferent:

E₇

= −

¹ 2m²g²



d⁴y d⁴z



₁

2

∂

μB

∂

μB

+

¹ 2m²B²

(

x

)

·

^{A B2}

(

y

) ·

^{A B2}

(

z

)

c

= −

2m²g²



d⁴kd⁴p k_μk_μ

+

^m2

(

k²

−

m²

)

²

(

p²

−

m²

)((

k

+

p

)

²

−

m²

) .

(31) ThecontributionoftheYukawainteractiontothekinetic vacuum energyofthescalarfieldis

E8

= −

¹ 2g²



d⁴y d⁴z



₁

2

∂

_μA

∂

_μA

+

¹ 2m²A²

(

x

)

· ¯ψψ

^A

(

y

) · ¯ψψ

^A

(

z

)

c

=

^4g2



d⁴kd⁴p

(

k_μk_μ

+

^m2

) (

p

· (

^p

+

^k

) +

^m2

) (

k²

−

^m2

)

²

(

p²

−

^m2

)((

k

+

^p

)

²

−

^m2

)

=

4g²I

·



d⁴kk_μk_μ

+

m²

(

k²

−

^m2

)

²

−

^2g2



d⁴kd⁴p k_μk_μ

+

^m2

(

k²

−

^m2

)(

p²

−

^m2

)((

k

+

^p

)

²

−

^m2

) +

^6g2m2



d⁴kd⁴p k_μk_μ

+

^m2

(

k²

−

^m2

)

²

(

p²

−

^m2

)((

k

+

^p

)

²

−

^m2

) .

(32) ThecontributionoftheYukawainteractiontothekinetic vacuum energyofthepseudoscalarfieldisinstead

E₉

= −

¹ 2g²



d⁴y d⁴z



₁

2

∂

_μB

∂

_μB

+

¹ 2m²B²

(

x

)

·

ⁱ

¯ψ γ

⁵

ψ

B

(

y

) ·

ⁱ

¯ψ γ

⁵

ψ

B

(

z

)

c

=

^4g2



d⁴kd⁴p

(

k_μk_μ

+

m²

) · (

p

· (

p

+

k

) −

m²

) (

k²

−

^m2

)

²

(

p²

−

^m2

)((

k

+

^p

)

²

−

^m2

)

=

^4g2I

·



d⁴kk_μk_μ

+

^m2

(

k²

−

^m2

)

²

−

^2g2



d⁴kd⁴p k_μk_μ

+

^m2

(

k²

−

^m2

)(

p²

−

^m2

)((

k

+

^p

)

²

−

^m2

)

−

2g²m²



d⁴kd⁴p k_μk_μ

+

m²

(

k²

−

m²

)

²

(

p²

−

m²

)((

k

+

p

)

²

−

m²

) .

(33) Finally,thecontributionoftheYukawainteractionwithscalarand pseudoscalarfieldsto thekinetic vacuumenergyoftheMajorana spinoris

E10

= −

¹ 2g²



d⁴y d⁴z

₁

2

¯ψ(

i

γ

i

∂

i

+

m

)ψ (

x

) ·  ¯ ψ ψ

A

(

y

)

· ¯ψψ

^A

(

z

) +

ⁱ

¯ψ γ

⁵

ψ

B

·

ⁱ

¯ψ γ

⁵

ψ

B



c

=

^4g2



d⁴kd⁴pTr

[( γ

ik_i

+

^m

)(ˆ

k

+

^m

)(

^p

ˆ + ˆ

^k

)(ˆ

k

+

^m

)]

(

k²

−

^m2

)

²

(

p²

−

^m2

)((

k

+

^p

)

²

−

^m2

)

=

8g²



d⁴kd⁴p kiki

+

^m2

(

k²

−

m²

)(

p²

−

m²

)((

k

+

p

)

²

−

m²

) +

^8g2



d⁴kd⁴p m²

(

k²

−

^m2

)(

p²

−

^m2

)((

k

+

^p

)

²

−

^m2

) +

16g²



d⁴kd⁴p m²

(

kiki

+

m²

)

(

k²

−

m²

)

²

(

p²

−

m²

)((

k

+

p

)

²

−

m²

) .

(34)

Now, using the equality for Lorentz non-invariant combination kμkμ=^k20+^k2_i^,

2k_ik_i

−

^kμk_μ

=

^k2

≡

^kμk^μ

,

(35) onecanobtain

E_kin

=

^2E5

+

^E6

+

^E7

+

^E8

+

^E9

+

^E10

= −

^4g2



d⁴kd⁴p 1

(

p²

−

^m2

)((

k

+

^p

)

²

−

^m2

= −

^4g2I2

.

(36)

Assembling Eqs. (36) and (27) we see that, even though sepa- rately the kinetic andpotential vacuum energies in the on-shell formalismofWess–Zuminomodelarenonvanishing,theirsumstill equalszeroasitshouldbe.

4. Conclusions

In thispaper we havecompared the two-loop cancellation of the vacuum energy in the Wess–Zumino model in different for- malisms– inthecomponentfield formalismwithauxiliary fields andinthecomponentformalismwithoutauxiliaryfields.Itisalso well-knownthatzerovalueofthevacuumenergydirectlyfollows withoutexplicitcalculationsfromthesuperfield formalismandis basedonthegeneralpropertiesofthesuperfieldFeynmangraphs.

Without going into details we just briefly remind the superfield mechanismresponsibleforthat[26–28].

This mechanism holds separately for all superfield vacuum Feynman diagrams – any vacuum diagram is equal to zero. The pointisthatintheabsenceofexternallines,adiagramundercon- sideration dependsonly on thedifferences ofthe internal Grass- mann variables θα . Thus, the number of Grassmannintegrations is larger than the number of independent Grassmann variables andtheresultequals zerobecauseoftheintegration ruleforθα , namely 

dθ^α =^{0 (see [29]). Thus, when we work in terms of} thesuperfields, relativelysimplealgebraicmanipulationsallowus easily to attain the vanishing of the total vacuum energy along withadrasticdecreaseoftheultravioletdivergencesinthesuper- symmetricmodels. Inparticular, intheWess–Zumino modelit is necessarytointroduce onlyonecounter-term –thewavefunction renormalization constant (see e.g. the review [28] andthe refer- encestherein).

Explicit calculations in the component formalism with auxil- iary fields are still not so complicated. The calculations in the formalismwheretheauxiliaryfieldsareexcludedbymeansofthe equationsofmotionaremorecumbersomeandwehavepresented them in some detail in the section 3 of the present paper. We have seen that inthis casethe vacuumenergy isagain equal to zero.Howeveritsvanishingisaresultofthecancelationbetween thenon-zerocontributionstothevacuumenergy,representingits potential and kinetic parts. Note,that in the formalism with the explicitpresenceoftheauxiliaryfieldsthesetwoterms(potential andkinetic)areequaltozeroseparately.

Why did we undertake these calculations in spiteof the fact that the vanishing of the vacuumenergy in the supersymmetric models is well established many years ago? There are two rea- sonsforthat.First,itisinteresting tostudyononemoreexample an involvedquestion concerningtherole oftheauxiliaryfieldsin the supersymmetric theories. To illustrate once again this prob- lem,weshallcitesomesentencesfromthereviewbySohnius[23]

(p. 94):“Thereare severalwaystounderstandwhyon–andoff- shellrepresentationsaredifferentandwhyauxiliaryfieldsappear insupersymmetrictheories.Attherootoftheproblemliesthedif- ferencebetweenthevectorspacefortheoff-shellrepresentations,

fields over xμ and that for on-shell representations, the Cauchy dataforthefieldsequations.Inbothcasesthefermions=^bosons rulemusthold.Fordifferentspins,however,thereisadifferentre- lationshipbetweenCauchydataandfields.While,e.g.,arealscalar field A(x)describesoneneutralscalarparticle,theDiracfieldψ(x) haseightrealcomponents,butdescribesonlythefourstatesofa charged spin – ¹₂ particle. Thus, in going from the fields to the states,wehavelostsomedimensionsofourrepresentationspace, butdifferentlysofordifferentspins.Supersymmetricmodelswith their strict fermions= ^{bosons rulemust somehowwiggleout of} this,andtheydosobymeansofauxi!liaryfieldswhoseoff-shell degreesoffreedomdisappearcompletelyon-shell”.

Ontheotherhand,inrecentyearstherehasbeenagrowingin- terestinthePauli–Zeldovich mechanism[7–9] of thecancellation ofthevacuumenergydivergencesinthenon-supersymmetricthe- orieswithequalnumberoffermionandbosondegreesoffreedom [10–14].Inthecorrespondingmodelssomerelationsbetweencou- plingconstantsandmassesshouldbeimposed,butonecannotuse thesuperfieldformalism.Thus,adetailedknowledgeofthemech- anism ofthecancellations betweendifferentcontributions tothe vacuumenergycanbeofinterestinthiscontext.

Acknowledgements

This work was supported by the RFBR grant No.17-02-00651.

We are grateful to F. Bastianelli, I.V. Mishakov, A.A. Starobinsky, O.V. Teryaev, A. Tronconi, A.A. Tseytlin andG. Venturifor useful discussions.

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