Impact of Gravity on Vacuum Stability

0.41 MB·

by  Vincenzo Branchina

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)<<V(φtop)−V(φfalse).
. enough, as the stability condition of the vacuum is very sen- However,itiscommonlyconsideredasbeingofmoregeneral
1
0 sitive to unknown new physics interactions at high (or very validity.
6 high,sayPlanckian)energyscales[14–17]. GoingbacktotheSM,itisknownthatduetothetoploop
1 Following a pioneering work of Bender and collabora- corrections the Higgs potentialV(φ) bends down for values
:
v tors[18], Coleman and Callan studied the decay of the false of φ > 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)<<V(−ba)−V(−a). Atthesame
polebecomesphysicallyirrelevant(similarlytowhathappens time,byvaryingthedimensionfulparametera(theonlymass
for the Landau pole in QED, where the latter occurs at such scaleofourpotential),andpushingittowardthePlanckscale
ahighenergyscalethatthetheoryhasalreadylostitssignif- andbeyond,wewillbeabletotesttheimpactofstronggrav-
icanceseveralordersofmagnitudesbelowthatscale). Aten- itationalphysicsonthedecayofthefalsevacuum.
ergiesbelowthepolescale,butyetPlanckianortransPlanck- Anticipating from the following analysis, we will see that
ian, the decay probability is non-vanishing and very close to intheb→1limit(thatisthelimitofthelargestpossibledif-
theflatspace-timeresult,i.e. theresultobtainedignoringthe ferenceV(−a)−V(a) for the potential (2)), the decay prob-
presenceofgravity. abilityfromtheMinkowskifalsevacuumtothetruevacuum
These findings are new and totally unexpected, as it was gives exactly the flat space-time result (as if gravity was ab-
thought that in a strong gravity regime the decay of a false sent!), foranyvalueofa, i.e. fromveryweaktoverystrong
MinkoskivacuumtoatrueAdSvacuumisalwaysinhibited. gravitational regimes. For values of b close to (but smaller
Moreover, they are of crucial importance for current studies than)1,thesuppressionofthevacuumdecayshouldoccurfor
and for model building of BSM physics, where we are of- verylargetransplanckianvaluesofa.
teninterestedinconsideringnewphysicsatPlanckianand/or To calculate the bubble nucleation rate Γ we need the
transPlanckianscales. bounce solution to the (Euclidean) equations of motion de-
Analysis–Tostudytheimpactofgravityonfalsevacuum rived by varying the action in Eq.(1). The saddle point ap-
decay, let us consider the (Euclidean) action of a scalar field proximation to the transition rate Γ per unit volume is given
incurvedspace-timetogetherwiththeEinsteinterm: bytheratiobetweentheexponentialofminusthe(Euclidean)
(cid:90) √ (cid:20)1 R (cid:21) action evaluated at the bounce and the action calculated for
S= d4x g gµν∂ φ ∂ φ+V(φ)− , (1) thefalsevacuumsolution[20]:
2 µ ν 16πG
Γ
whereRistheRicciscalar,GtheNewtonconstant,andV(φ) V ∼ e−(Sbounce−Sfalse)≡ e−B. (3)
3
3.0 3.0
2.5 2.5
2.0 2.0
Ρ B
1.5 1.5
r0 B0
1.0 1.0
0.5 0.5
0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
r0 r0
2(cid:76) 2(cid:76)
FIG. 3: ρ/r0 computed for b=0.4 (red online) dotted, b=0.7 FIG.4:B/B0withthesamecodingasinFig.3.
(greenonline)dot-dashed,b=0.85(violetonline)dashedline. The
blacksolidcurvecorrespondstothethinwallprediction.
TheB.T.areinfinite. However,thecomputationofΓonly
involvesthedifferenceBbetweentheactionevaluatedatthe
Following[20], we consider the most general O(4)-
bounceandatthefalsevacuumsolutions(seeEq.(3)). These
symmetric Euclidean metric (r is the radial coordinate along
twosolutionscoincideatinfinity,andthetwoactionsprovide
aradialcurve):
the same divergent contribution, that is then canceled out in
thedifferenceB=S −S .
(ds)2=(dr)2+ρ(r)2(dΩ )2 (4) bounce false
3 Itisworthnotingthat,duetotheparticulardependenceon
the coupling g of the potential V(φ) in (2), it is possible to
where dΩ is the element of distance on a unit three-sphere
3
re-expressEqs.(5)and(6)intermsoftherescaledquantities
andρ(r)theradiusofcurvatureofeachthree-sphere.
rˆ=gr,ρˆ =gρ andVˆ =V/g2,sothatnoexplicitdependence
Withthemetric(4),theEinsteinequationG =−κT re-
rr rr
ducestothefollowingequationforρ(r)(the (cid:48)denotesderiva- on g is left. Accordingly, the dependence on g of the action
canbefactoredout: S=Sˆ/g2andB=Bˆ/g2.
tionwithrespecttor):
TocalculatethetransitionrateΓfordifferentpotentialsof
κ (cid:18)1 (cid:19) the kind given in Eq.(2), i.e. for different values of the pa-
ρ(cid:48)2=1+ ρ2 φ(cid:48)2−V(φ) , (5)
3 2 rametersaandb,weneedfirsttosolvethesystemofcoupled
differentialequations(5)and(6)withtheboundaryconditions
whereκ≡8πG≡8π/M2,andM isthePlanckscale.Eq.(5) givenabove.
P P
iscoupledtothescalarfieldequationderivedfrom(1), Wedothatwiththehelpofanumericalshootingprocedure.
The typical bounces are decreasing profiles φ (r), such that
b
φ(cid:48)(cid:48)+3ρ(cid:48)φ(cid:48)= dV . (6) φb(r→∞)→−a,andcanbeparametrizedbytheirsizer,the
ρ dφ value of the radial coordinate where the height of the profile
isdecreasedofonehalf. Forlargevaluesofr,i.e. forr>>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<<r, i.e. inside
constantsolutionfortheprofile,φ (r)=−a.
false thebubbleoftruevacuum,thepresenceofanegativeenergy
In order to compute Γ, we have to select the classical so-
densitycreatesadistortionofthissolution.
lutionsofEqs.(5)and(6)thatfulfilltheboundaryconditions
Bydefiningρ asthevalueofthecurvatureatr=r,thatis
φ(+∞)=−a, φ(cid:48)(0)=0 and ρ(0)=0.Inourcase,theaction
ρ ≡ρ(r),asimplewayofdisplayingourfindingsistocom-
inEq.(1)canbewrittenas:
pare the results obtained for B and ρ with the corresponding
(cid:90) ∞ (cid:34) (cid:18)1 (cid:19) quantitiesderivedintheflatspace-timecase,thatwillbeindi-
S= 2π2 dr ρ3 φ(cid:48)2+V(φ) + catedasB0 andr0 respectively(needlesstosay, inthislatter
2
0 caseρ(r)coincideswithr).
(cid:35) Inthethinwallregime,thatforourpotentialisthecaseof
3(cid:16) (cid:17)
ρ2ρ(cid:48)(cid:48)+ρρ(cid:48)2−ρ = smallvaluesoftheparameterb,itwasshownin[20]that:
κ ρ=r (cid:2)1−(r /(2Λ))2(cid:3)−1 and B=B (cid:104)1−(r /(2Λ))2(cid:105)−2,
(cid:90) ∞ (cid:18) 3ρ(cid:19) 0 0 0 0
4π2 dr ρ3V(φ)− +B.T., (7) i.e.ρandBhaveapoleatr /2=Λ≡(cid:112)3/(8πG|V(a)|.The
0 κ 0
corresponding curves are reported in Figs.1, 2, 3, 4 as solid
wheretherighthandsideisobtainedbymakinguseofEqs.(5) lineshavinganasymptoteatr /(2Λ)=1.
0
and(6),andtheboundarytermsB.T.appearafterintegrating Ournumericalapproachallowstochecktheaccuracyofthe
bypartsρ2ρ(cid:48)(cid:48)inthesecondlineofEq. (7). thinwallpredictions,andbeforemovingtothecasesofpartic-
4
ularinteresttous,wecalculateρ/r andB/B forsomesmall stressed above, this is expected to hold even out of the thin
0 0
values of b (thin wall regime). To facilitate the comparison wallregime,andmodelsofBSMphysicsandconclusionson
with[20], we present our results by plotting ρ/r and B/B thevacuumstabilityanalysisareoftenbasedonthisexpecta-
0 0
against r /(2Λ), that is an increasing function of a for each tion[23–25].
0
fixed value of b. The parameter a gives the location of the However, we have seen that, when the energy density dif-
minima, and increasing values of a toward the Planck scale ference between the two vacua of the potential increases (so
(andbeyond)markthetransitionfromtheweaktothestrong that the conditions of[20] are no longer verified), the gravi-
gravitationalregime. TheresultsareplottedinFigs.1and2. tationalcontributiontothetransitionrateΓbecomessmaller
Notethattherescalingtothehattedquantitiesdiscussedabove and smaller. Moreover, when this difference becomes suffi-
shows that all variables on the x and y axes of the figures do cientlylarge,thatforourmodeloccursforb→1,thequenc-
notdependonthecouplingg. ing of the vacuum decay does not occur even in very strong
For the smallest value considered for b, b = 0.001, the gravityregimes!
pointspracticallysitonthethinwallcurve(blueonlinesolid Asmentionedbefore, theseresultsareverysurprisingand
curve),andarenotreportedinFigs.1and2. Atb=0.01(red unexpected, and also very important for phenomenological
onlinecircles)westarttoobserveasmalldeviationfromthe applications. In fact, extrapolating from[20], it was thought
thinwallcurve,thatbecomesmoresizableforb=0.13(green thatwheneverweenterinastronggravityregime, i.e. when
onlinesquares). Figs.1and2showthatforsmallvaluesofb, weapproach(andpossiblygobeyond)thePlanckscale,there
when the two minima of the potential are almost degenerate shouldalwaysbeastrongsuppressionandevenavanishingof
(thin wall regime), starting from r0/(2Λ)∼0.5 the effect of thefalsevacuumdecayprobability,resultinginastabilization
gravitygrowsrapidlyforincreasingvaluesofr0/(2Λ). ofthefalsevacuum.Onthecontrary,weseethatforpotentials
We now consider larger values of b, thus leaving the thin wherethedifferenceinthedepthoftheminimaissufficiently
wall regime. Some of the curves for ρ/r0 and B/B0 versus large,thissuppression(ifstillpresent)ispushedtoveryhigh
r0/(2Λ)resultingfromournumericalanalysis,namelythose regimes,even(much)abovethePlanckscale,sothatthepres-
obtained for b=0.4,0.7,0.85, are reported in Figs.3 and 4. enceofapoleofB/B becomesphysicallyirrelevant1.
0
These figures clearly show that for increasing values of b, Moreover,thefactthatintheseconditionsthetransitionrate
i.e. when the difference in height between the two minima fromthefalsetothetruevacuumissoclosetotheflatspace-
increases,bothcurvesB/B0andρ/r0getflattened. timeresultisofcrucialimportanceforphenomenologicalap-
Moreover, for larger and larger values of b (b→1), both plications. In fact, the case that we have studied is relevant
curves get closer and closer to the orizontal lines B/B0 =1 for the stability analysis of the electroweak vacuum, also in
andρ/r0=1. WehavenotaddedothercurvesinFigs.3and4 connection with present searches for BSM physics, where a
astheywouldunnecessarilyclutterthesefigures.Forthesame stabilityanalysisandaclearunderstandingoftheroleplayed
reason,wehavealsolimitedtherangeofvaluesofr0/(2Λ)to bygravityisgreatlyneeded.
0<r /(2Λ)<1.5. Infact, theColeman-DeLucciapole(of
0
the thin wall case) occurs at r0/(2Λ)=1, and Figs.3 and4 b gr0a g2B0
clearlyshowthatthispointisnotapoleforlargevaluesofb. 0.001 2121.32 4.44132×1010
Finally, for b→1 these curves reach the horizontal lines 0.01 212.113 4.44094×107
B/B0 =1 and ρ/r0 =1 for the whole range of values 0< 0.13 16.2379 19921.0
r /(2Λ)<∞. Thislatterresultisimmediatelyfoundbycon- 0.4 3.93098 276.288
0
sidering the bounce profile function φ (r), that connects the 0.7 2.73633 70.9009
b
center of the bounce, φ (0), with the false vacuum state at 0.85 2.51328 22.2602
b
r→∞, φ (∞)=−a, and seeking for the numerical solution
b
ofEq.(6). Forsmallvaluesofb,φ (0)isclosetoa,thetrue
b TABLEI:Setofvaluesofgr aandg2B computedatdifferentb.
vacuum,whileforincreasingvaluesofb,φ (0)decreasesto- 0 0
b
wardφ (0)∼−a. Inthelimitb→1,φ (0)reachesthislatter
b b
For the reader’s convenience, in Table I some values of
value,sothat
gr a and g2B for different values of b are reported. These
0 0
limφ (r)=−a (8) areusefulparameterstoreconstructphysicalquantitiesrelevat
b
b→1 totheanalysisthatwehavepresentedandtobetterappreciate
the results reported in the figures (see below). These quan-
inthewholerangeofr. Inthislimit,Eq.(5)becomesρ(cid:48)(r)=
tities are obtained in the flat space-time case, and are both
0,andwerecovertheflatspace-timeresult.
g-independent(ascanbeseenbyresortingtothehattedvari-
These are the central results of the present Letter. In fact,
thelessonfrom[20](seeFigs.1and2)isthatwhenwemove
from weak to strong gravity regime, the impact of gravity
becomes enormous, to the point that, for the critical value
1In[27]argumentsaregiveninfavouroftheexistenceofsuchapole.How-
r /(2Λ) = 1, B diverges and the corresponding rate Γ for
0 ever,inthecasesofinterestconsideredinthepresentpaper,thispolewould
the materialization of a bubble of true vacuum vanishes. As beinaregionwherethetheoryisnolongervalid.
5
ables introduced before) and a-independent (they are dimen- problemandforsearchesofnewphysicsbeyondtheStandard
sionlessquantities,andintheabsenceofgravitythatbringsin Model. They represent a real progress in understanding the
thedimensionfulNewtonconstantG, theycannotdependon impactofgravityonthestabilityofthevacuum. Inthepast,
theonlydimensionfulvariablea). Therefore,theyareunivo- despiteattemptstostudythisquestion[28],theroleofgravity
callydeterminedoncebisgiven. beyondthethinwallcasewaspoorlyunderstood.
In particular, the values of gr a in Table I are useful to
0
establishtherelationbetweenr /(2Λ),thatappearsinthex-
0
axisofallfigures,andtheratioa/M (whereM isthePlanck
P P
mass)foreachvalueofb,thatisforeachsinglecurveinthe
[1] J. Olsen, CMS physics results from Run 2 presented on Dec.
figures. In fact, by recalling the definition of Λ, one obtains
√
15th,2015,https://indico.cern.ch/event/442432/;CMSCollab-
thefollowingrelationr /(2Λ)=(2/3) 2πb(gr a)(a/M ).
0 0 P oration,CMSPASEXO-15-004,https://cds.cern.ch/record/
Then,forinstance,thepointr /(2Λ)=1correspondsforb=
0 2114808/files/EXO-15-004-pas.pdf.
0.001toa∼MP/100,andforb=0.85toa∼MP/4(i.e. we [2] M.Kado,ATLASphysicsresultsfromRun2presentedonDec.
arealreadyinasronggravityregime). 15th, 2015, https://indico.cern.ch/event/442432/; ATLASCol-
If we now look at the curves in Fig.4, these examples laboration,ATLAS-CONF-2015-081,https://atlas.web.cern.ch/
clearly illustrate the central results of our work. In fact, we Atlas/GROUPS/PHYSICS/CONFNOTES/ATLAS-CONF-
2015-081/.
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