ImpactofGravityonVacuumStability Vincenzo Branchina1,2, Emanuele Messina1,2, and D. Zappala`2 1 DepartmentofPhysicsandAstronomy,UniversityofCatania,ViaSantaSofia64,95123Catania,Italyand 2 INFN, Sezione di Catania, Via Santa Sofia 64, 95123 Catania, Italy (Dated:May19,2016) Inapioneeringpaperontheroleofgravityonfalsevacuumdecay,ColemanandDeLucciashowedthata stronggravitationalfieldcanstabilizethefalsevacuum,suppressingtheformationoftruevacuumbubbles.This resultisobtainedforthecasewhentheenergydensitydifferencebetweenthetwovacuaissmall,thesocalled thinwallregime,butisconsideredofmoregeneralvalidity. Hereweshowthatwhenthisconditiondoesnot hold,however,astronggravitationalfield(Planckianphysics)doesnotnecessarilyinduceatotalsuppression oftruevacuumbubblenucleation. Contrarytocommonexpectationsthen,gravitationalphysicsatthePlanck scaledoesnotstabilizethefalsevacuum.Theseresultsareofcrucialimportanceforthestabilityanalysisofthe 6 electroweakvacuumandforsearchesofnewphysicsbeyondtheStandardModel. 1 0 PACSnumbers:14.80.Bn,11.27.+d,04.62.+v 2 y Introduction–ThesearchfornewphysicsbeyondtheStan- largethebubblesize. Inacurvedbackground,thingsaredif- a dardModel(BSM)isoneofthemostimportantandchalleng- ferent. Inthefollowingweconcentrateonthecaseofafalse M ingquestionsofpresentexperimentalandtheoreticalphysics. vacuum with vanishing energy density, Minkowski vacuum, 8 ThefirstrunofLHCdidnotshowanysignofnewphysics[1]. andatruevacuumwithnegativeenergydensity,anti-deSitter 1 Thesecondrun,reportinglocalexcessofsignalinthedipho- (AdS) vacuum, as this case is very relevant for applications, ton channel, has generated a certain expectation and enthu- inparticularforthestabilityanalysisoftheEWvacuum. ] h siasm in the community[2]. Whether or not this signal will The transition from a false Minkowski vacuum to a true p be confirmed by future work, clues are also needed from the AdS vacuum was studied in[20], where it was shown that, - theoretical side. In this respect, crucial for our understand- when the size of the Schwarzschild radius of the bubble of p e ingandconstructionofBSMtheoriesisthestabilityanalysis truevacuumismuchsmallerthanitssize,i.e. whenthegrav- h oftheelectroweak(EW)vacuum[3–9]. Thediscoveryofthe itational effects are weak, the probability of materialization [ Higgsbosonboostednewinterestonthisquestion,makingit of such a bubble is close to the flat space-time result, while, 2 oneofthehottesttopicintheoreticalparticlephysics[10–17]. whentheSchwarzschildradiusbecomescomparablewiththe v At first, it seemed that the knowledge of the Higgs and top bubblesize,i.e. inastronggravitationalregime,thepresence 3 masses, MH and Mt, was sufficient to determine the stability ofgravitystabilizesthefalsevacuum, preventingthemateri- 6 condition of the SM vacuum, that is to determine whether it alization of a true vacuum bubble. This result was obtained 9 isametastablestate(falsevacuum)orastableone(truevac- under the thin wall condition mentioned above, namely for 6 0 uum).Itwaslaterrealized,however,thatthisknowledgeisnot the case when V(φfalse)−V(φtrue)< φ(1) =v, where v∼246 GeV is the location of the min Xi vacuuminaflatspace-timebackground[19]. Later,Coleman EWminimum,andforthepresentexperimentalvaluesofMH andDeLucciaextendedthisanalysistoincludetheimpactof and M, namely M ∼ 125.09 GeV and M ∼ 173.34 GeV r t H t a gravity[20]. The physical mechanism that triggers the false [21,22], developsasecondminimum, muchdeeperthanthe vacuum decay is quite simple: due to quantum fluctuations, EW one and at a much larger value of the field, φ(2) >> v. min thereisafiniteprobabilitythatabubbleoftruevacuummate- Clearly, the conditions under which the Coleman-De Luccia rializesintheseaoffalsevacuum. result is derived are not fulfilled. As said above, however, it Coleman and collaborators worked within the so called isstillexpectedthatinthepresenceofPlanckianphysicsthe “thin wall” approximation, that applies when the two min- bubble nucleation rate vanishes[23–25]. In more technical ima of the potential, φ and φ , are such that the en- words,itisexpectedthatthebouncesolutiontotheeuclidean false true ergy density differenceV(φ )−V(φ ) is much smaller equationofmotion(fromwhichthesaddlepointapproxima- false true than the height of the potential barrier V(φ )−V(φ ), tionforthebubblenucleationrateisobtained)disappears. top false whereV(φ ) is the maximum of the potential between the Motivated by the above phenomenological considerations, top twominimaatφ andφ . Theyfoundthattheprobabil- inthisLetterwefocusourattentiononthecasethatisrelevant false true ity of materialization of a true vacuum bubble decreases for fortheapplicationstotheanalysisoftheSMvacuumdecay, increasingvaluesofitssize. Inaflatspace-timebackground, namelythedecayofafalsevacuumwithzeroenergydensity this probability turns out to be always finite, no matter how toatruevacuumwithnegativeenergydensity,i.e. thedecay 2 6 2.5 (cid:230) (cid:224) (cid:230) (cid:224) 5 2.0 (cid:230)(cid:230)(cid:224)(cid:224)(cid:224) (cid:230) (cid:224) 4 (cid:230) (cid:224) (cid:224) Ρ 1.5 (cid:230) (cid:224) B r0 1.0 (cid:230)(cid:230)(cid:230)(cid:224)(cid:230) (cid:230)(cid:224) (cid:230)(cid:224)(cid:224) (cid:224)(cid:230)(cid:224)(cid:224)(cid:230)(cid:230)(cid:224)(cid:230)(cid:230)(cid:230)(cid:230)(cid:224)(cid:224)(cid:230)(cid:230)(cid:224)(cid:230)(cid:230)(cid:230) B0 23 (cid:230) (cid:224) 0.5 1 (cid:230)(cid:230)(cid:230)(cid:224)(cid:230) (cid:230)(cid:224) (cid:230)(cid:224)(cid:224) (cid:224)(cid:230)(cid:224)(cid:224)(cid:230)(cid:230)(cid:224)(cid:230)(cid:230)(cid:230)(cid:230)(cid:224)(cid:224)(cid:230)(cid:230)(cid:224)(cid:230)(cid:230)(cid:230) 0.0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 r0 r0 2(cid:76) 2(cid:76) FIG. 1: ρ/r0 computed for b=0.01 (red online) circles, and for FIG.2: B/B0computedforthesamevaluesoftheparametersused b=0.13(greenonline)squares,plottedtogetherwiththethinwall inFig.1. prediction,(blueonline)solidline. thepotential[26]: fromaMinkowskifalsevacuumstatetoanAdStruevacuum. g2(cid:20) 4b (cid:21) Tostudytheimpactofgravityonthenucleationrateofabub- V(φ)= (φ2−a2)2+ (aφ3−3a3φ−2a4) . (2) 4 3 ble of true vacuum, we consider a potential[26] that allows toexploreinaunifiedframeworkallpossiblecases,fromthe When 0 < b < 1, V(φ) has two non-degenerate minima at thinwallregimetothecasewhenthedifferenceinheightbe- φ =±a, φ =a being the absolute minimum, and a poten- tweenthetwominimaofthepotentialislarge. Forthislatter tial barrier that separates the two minima, with maximum at case,wefindthatthenucleationrateofatruevacuumbubble φ =−ba. Notethatforb=0thisisjustthedoublewellpo- hasavalueclosetotheresultobtainedinflatspace-timeeven tentialwithtwodegenerateminimaatφ=±a,whileforb=1 in a strong gravity regime, when the typical mass scales of theminimumatφ =−abecomesaninflectionpointandthe thepotential(seeEq.(2)below)becomecomparablewith(or barrierdisappears. evenlargerthan)thePlanckscale. This potential is perfectly suited for our scopes. By vary- In fact, we will see that in this case no vanishing of the ing the dimensionless parameter b, we control the difference bubble materialization rate is observed at these high energy inheight,V(−a)−V(a), betweenthefalseandthetruevac- scales. On the contrary, we find that the suppression of the uum. Thecomparisonbetweenthisdifferenceandtheheight Minkowski false vacuum decay (if still present) is pushed to ofthepotentialbarrier,V(−ba)−V(−a),allowstodetermine veryhighenergyregimes, evenmuchhigherthanthePlanck whetherornotweareinthethinwallregime,thelatteroccur- scale, so that the possible presence of a Coleman-De Luccia ingwhenV(−a)−V(a)<>r, Thefalsevacuumsolutiontothesecoupledequationsconsists thesolutionfortheradiusofcurvatureρ(r)approachestheflat oftheflatspace-timemetric, ρ(cid:48) (r)=1, togetherwiththe false space-time solution, ρ(cid:48)(r)=1, while for r<>M ! Finally, P [6] M.Sher,Phys.Lett.B317(1993)159. withthehelpofgr aandg2B inTableI,onecanderivethe 0 0 [7] G.Altarelli,G.Isidori,Phys.Lett.B337(1994)141. valuesofρ andBfromthevariouscurvesinthefigures. [8] J.A.Casas,J.R.Espinosa,M.Quiro´s,Phys.Lett.B342(1995) Before ending this Letter, we have to say some words on 171;Phys.Lett.B382(1996)374. previous work[28] on the role of gravity on the EW vac- [9] G.Isidori,G.Ridolfi,A.Strumia,Nucl.Phys.B609(2001)387. uumdecay,whereaperturbativeexpansionofφ (r)andρ(r) [10] L.N.Mihaila,J.SalomonandM.Steinhauser,Phys.Rev.Lett. b 108 (2012) 151602; K. Chetyrkin and M. Zoller, JHEP 06 in powers of κ around the flat space-time solution was at- (2012)033;F.Bezrukov,M.Yu.Kalmykov,B.A.Kniehl,M. tempted. Actually, this analytical method seems to have a Shaposhnikov,JHEP1210(2012)140. seriousdrawback,astheboundaryconditions(seeabove),es- [11] J.Elias-Miro,J.R.Espinosa,G.F.Giudice,G.Isidori,A.Riotto, sentialfortheclassicalsolutiontobeabounceandinturnfor A.Strumia,Phys.Lett.B709(2012)222. ΓinEq.(3)tobethebubblenucleationrate,arenotrespected [12] G.Degrassi,S.DiVita,J.Elias-Miro,J.R.Espinosa,G.F.Giu- alreadyatfirstorderinκ. Therefore,theoutputofthisanaly- dice,G.Isidori,A.Strumia,JHEP1208(2012)098. siscannotbetrustedandafortioricannotbeusedforcompar- [13] D.Buttazzo,G.Degrassi,P.P.Giardino,G.F.Giudice,F.Sala, A.Salvio,A.Strumia,JHEP1312(2013)089. ison. Adetailedstudyofthisissuegoesbeyondthescopeof [14] V.Branchina,E.Messina,Phys.Rev.Lett.111(2013)241801. thepresentpaperandwillbepresentedelsewhere. [15] V.Branchina,E.Messina,A.Platania,JHEP1409(2014)182. Conclusions – We have studied the decay of a false [16] V. Branchina, E. Messina, M. Sher, Phys. Rev. D91 (2015) MinkowskivacuumtoatrueAdSvacuumbytakingintoac- 013003. counttheimpactofgravity. [17] V.Branchina,E.Messina,“StabilityandUVcompletionofthe When the energy density difference between the false and StandardModel”,arXiv:1507.08812. the true vacuum is large compared to the height of the po- [18] T.Banks,C.M.Bender,T.T.Wu,Phys.Rev.D8(1973)3346; Phys.Rev.D8(1973)3366. tential barrier (a condition that in the model discussed in the [19] S.Coleman,Phys.Rev.D15(1977)2929;C.G.Callan,S.Cole- presentLetterisobtainedforb→1inthepotentialofEq.(2)), man,Phys.Rev.D16(1977)1762. wefindthattheeffectofgravityonthebubbleproductionrate [20] S.Coleman,F.DeLuccia,Phys.Rev.D21(1980)3305. practically disappears even in the strong gravity regime, and [21] G. Aad et al. (ATLAS Collaboration, CMS Collaboration), theflatspace-timeresultisrecovered. Phys.Rev.Lett.114(2015)19. From a phenomenological point of view, the case that we [22] ATLASandCDFandCMSandD0Collaborations,“Firstcom- havestudiedisofparticularinterest,asitisclosetowhathap- binationofTevatronandLHCmeasurementsofthetop-quark mass”,arXiv:1403.4427[hep-ex]. pensinthestabilityanalysisoftheelectroweakvacuum,and [23] J.R.Espinosa,J.F.Fortin,M.Tre´panier,“ConsistencyofScalar several models considered for physics beyond the Standard PotentialsfromQuantumdeSitterSpace”,arXiv:1508.05343. Model require a stability analysis where the role played by [24] L.DiLuzio,G.Isidori,G.Ridolfi,Phys.Lett.B753(2016)150. gravityneedstobecarefullyunderstood. [25] J. R.Espinosa, “Implications ofthetop(and Higgs)massfor Therefore,thenewresultspresentedinthisworkareofthe vacuumstability”,arXiv:1512.01222. greatest importance for present studies on the SM stability [26] E.J.Weinberg,“ClassicalsolutionsinQuantumFieldTheory”, 6 CambridgeUniversityPress(2012). [28] G. Isidori, V. S. Rychkov, A. Strumia, N. Tetradis, Phys.Rev. [27] A. Aguirre, T. Banks, M. Johnson, JHEP 0608 (2006) 065. D77(2008)025034 arXiv:hep-th/0603107.

Impact of Gravity on Vacuum Stability

Similar Impact of Gravity on Vacuum Stability