Gravity and the stability of the Higgs vacuum

0.46 MB·English

by  Philipp Burda

DCPT-15/03
Gravity and the stability of the Higgs vacuum
Philipp Burda1, Ruth Gregory1,2, and Ian G. Moss3
1Centre for Particle Theory, Durham University, South Road, Durham, DH1 3LE, UK
2Perimeter Institute, 31 Caroline St, Waterloo, Ontario N2L 2Y5, Canada
3School of Mathematics and Statistics, Newcastle University, Newcastle Upon Tyne, NE1 7RU, U.K.
(Dated: August 26, 2015)
We discuss the effect of gravitational interactions on the lifetime of the Higgs vacuum where
genericquantumgravitycorrectionsaretakenintoaccount. Usinga“thin-wall”approximation,we
provideaproofofprinciplethatsmallblackholescanactasseedsforvacuumdecay,spontaneously
nucleating a new Higgs phase centered on the black hole with a lifetime measured in millions of
Planck times rather than billions of years. The corresponding parameter space constraints are
extremely stringent however, therefore we also present numerical evidence suggesting that with
thick walls, the parameter space may open up. Implications for collider black holes are discussed.
5
1 PACSnumbers: 04.70.Dy,12.60.Fr,11.15.Tk
0
2
With the discovery of the Higgs boson at the LHC To recall Coleman’s original intuition: the nucleation
g
[1, 2], and the measurement of its mass, it seems we of a bubble costs energy because a wall with energy and
u
A live in interesting times: the running of the coupling of tension is formed as a barrier between the false and true
the Higgs quite possibly means that our vacuum is only vacua, but to counter that, energy is gained by the vol-

5 metastable, and the true Higgs vacuum in fact lies at ume of space inside the bubble having lower energy by
2
large expectation values of the Higgs and negative vac- virtueofhavingtransitionedtothetruevacuum. Abub-

uum energy. Although this metastability at first might bleofjusttherightsizethenoptimisesthisenergypay-off
]
h seemalarming, inorderforthevacuumtodecay, itmust andonceformed,grows. Thepicturewithagravitational
t tunnel through a sizeable energy barrier, and the proba- inhomogeneity is similar; the bubble forms around the
-
p bility for this typically has an exponential factor, (euclidean) black hole but because of the distortion of
e space the payoff between volume inside a bubble and its
h Γ∼Ae−B/(cid:126) (1) surface area is changed, and bubbles form at a smaller
[
radius and hence the ‘cost’ of forming them is lower: the
where B is the action of a solution to the Euclidean field
2 instantonhasasmalleractionandthedecayprocesscan
equations, “the bounce”, which interpolates between the
v be significantly enhanced. Alternatively, in terms of the
7 metastable (false) and true vacua; the prefactor A is de-
original energy argument of Coleman and de Luccia, the
3 terminedfromfluctuationsaroundthebounce. Sincethe
addition of a seed black hole which is eliminated or re-
9 actionB isusuallylarge(wewillset(cid:126)to1fortherestof
duced by the bubble can change the energy balance dra-
4 this discussion) the probability of vacuum decay is very
0 matically.
low. For the decay of a false vacuum, the process was
. Can this process affect the lifetime of the Higgs vac-
1 understood and the probability computed in a series of
uum? We will show that it can, although only if small
0 papers by Coleman, Callan and de Luccia [3–5]. This
5 blackholesnucleatethedecay. Suchblackholescouldre-
‘gold standard’ calculation is now used ubiquitously to
1 sult from gradual evaporation of primordial black holes
estimate decay rates and the half life of a false vacuum
: formedintheearlyuniverse[9];alternatively,ifthereare
v state in field theory, and for the Higgs vacuum predicts
“large” extra dimensions [10, 11] responsible for produc-
i a lifetime well in excess of the age of the universe.
X ing a hierarchically large Planck scale in our universe,
The Coleman et al. picture of vacuum decay is how-
r then small black holes can be produced at the LHC [12].
a ever very idealized, in that an exactly homogeneous and
Depending on the tension of the bubble wall, which is
isotropic false vacuum decays into a very nearly as sym-
directlyrelatedtoparametersintheHiggspotential,the
metric configuration: a completely spherical bubble of
enhancement of vacuum decay can be large.
true vacuum which expands outwards with uniform ac-
TobrieflyreviewtheHiggspotential,notethatatlarge
celeration. Ineverydayphysicshowever,firstorderphase
values of the Higgs field, we can pick any component φ
transitions are far from clean, and often proceed not via
andapproximatethepotentialusinganeffectivecoupling
some perfect nucleation process but rather by impurities
constant λ ,
acting as sites for the condensation of a new phase. Re- eff
cently in [6] we investigated the impact of gravitational 1
impurities, intheguiseofblackholes, ontheusualCole- V(φ)= 4λeff(φ)φ4. (2)
man de Luccia picture (see also [7, 8] for early investiga-
tions of this issue) and found a significant enhancement The effective coupling is obtained by combining the run-
ofvacuumdecayinawiderangeofcases–theintuitionof ning of λ under the renormalisation group with the low-
an impurity seeding nucleation of a true vacuum bubble energy particle physics parameters. Two-loop calcula-
was entirely borne out by the analysis. tionsoftherunningcoupling[13],includingcontributions
2
V(cid:72)Φ(cid:76) the CDL “thin-wall” description, the tunneling ampli-
1.(cid:180)10(cid:45)20
tudecanbesignificantlyenhancedbyasmallblackhole,
albeit within a small region of parameter space. This
5.(cid:180)10(cid:45)21
provides a ‘proof of principle’ and motivates a numerical
Φ analysis of Higgs instantons, which confirm the presence
0.00005 0.00010 0.00015 Mp ofstronglyenhanceddecayinthepresenceofblackholes.
Instanton
The equations of motion for a wall bounding two dif-
(cid:45)5.(cid:180)10(cid:45)21
ferent regions of spacetime with different cosmological
constants and black hole masses can be expressed in the
(cid:45)1.(cid:180)10(cid:45)20
form R˙2 +V(R) = 0, where R is the bubble radius (as
a function of Euclidean time), and V(R) is an effective
FIG.1. Anillustrationoftheimpactofthequantumgravity
correction (in red), here shown for λ = −0.01,φ = 2M , potential involving the wall tension. For the decay of
∗ ∗ p
and λ =63K. The path of the bounce is sketched. the Higgs vacuum, we assume the standard model has
6
Λ = 0, and write the true vacuum cosmological con-
+
stantasΛ =−3/(cid:96)2,thenthepotentialV dependson(cid:96),
−
fromallofthestandardmodelfields, yieldahigh-energy the black hole masses M and the surface tension of the
±
approximation bubblewall. (See[16]forexplicitformsofthispotential.)
Torecapitulatetheresultsof[6],theactionofageneral
(cid:18) φ (cid:19)2 instanton with a black hole was found to be
λ ≈λ +b ln , (3)
eff ∗ φ A+ A−
∗ B = h − h (5)
4G 4G
where the fit to the results of [13] give parameter ranges 1 (cid:73)
−0.01(cid:46)λ∗ (cid:46)0, 0.1Mp (cid:46)φ∗ (cid:46)Mp and b∼10−4. These +4G dλ{(2R−6GM+)τ˙+−(2R−6GM−)τ˙−}
parametersrangesaremostlyduetoexperimentaluncer-
tainties in the Higgs and top quark mass, however with where R(τ±) is the solution for the bubble wall, and A±h
thecurrentlymeasuredvaluesitseemsthatλeff nearthe are the black hole horizon areas corresponding to M±.
Planck scale is small with a preference towards negative This result includes a careful treatment of the conical
values. deficits which can arise in the Euclidean section when
Of course this discussion assumes no impact from new the periodicity of the bubble solution is not the same
physics between the TeV scale and the Planck scale. At as that of the black hole, and although the specifics of
the very least, quantum gravity effects will have to be computingactionsinvacuumandAdSvaryfromthatof
taken into account. On dimensional grounds, we might dS, the essence of the calculation remains the same as
expect some modifications to the potential of the follow- the presentation in [6], and the result, (5), identical in
ing form [14]: form.
This bounce action feeds directly into the exponent in
φ4 φ4 λ φ6 λ φ8 (1), and following Callan and Coleman [4], we estimate
V(φ)=λeff(φ) 4 +(δλ)bsm 4 + 66M2+ 88M4+... (4) the prefactor by taking a factor of (B/2π)1/2 for the sin-
p p gletime-translationalzeromodeoftheinstantonbutuse
the light crossing time of the black hole, (GM )−1, as a
where (δλ) includes possible corrections to the run- +
bsm rough estimate of the remaining determinant of fluctua-
ning coupling from physics beyond the standard model,
tions giving
and the polynomial terms arise with new physics identi-
fied with the Planck scale. If the coefficients λ6 etc. are (cid:18) B (cid:19)1/2
similar in magnitude, then the small size of λeff at the ΓD ≈ 2πGM e−B. (6)
Planck scale means that the interesting physics occurs +
where the potential is determined predominantly by λeff Typically, the CDL action is of order O(103−6) for the
andλ6. Infigure1weillustratetheeffectofthesecorrec- Higgs potentials, leading to a huge exponential suppres-
tions on the standard model potential with λ∗ = −0.01. sion of the decay rate, and to the conclusion that gravi-
tational tunneling is irrelevant.
Quantum tunnelling in a corrected potential such as However, the effect of a black hole, (5), on the tun-
thishasbeenlookedatbyBranchinaetal.[15],whotake neling action can be very significant for low tension bub-
λ ∼−0.1,wherethepotentialbarrieroccursatφ(cid:28)M , ble walls and small mass black holes. As the seed black
∗ p
and take a negative λ = −2. They argue the existence hole mass M is switched on, the instanton action drops
6 +
ofagreatlyenhancedtunnellingrate, however, theirdis- rapidly, and the bubble initially nucleates by removing
cussion entirely neglected gravitational back-reaction of the black hole. However, as the seed black hole mass
the instanton on the geometry. continues to increase, a critical mass M is reached at
C
Inordertoexploretheimpactofagravitationalimpu- which the potential V(R) has a single point at which
ritywewillextendthemethodofColemananddeLuccia V = V(cid:48) = 0, and there exists a static bubble wall so-
(CDL) [5], to include a black hole. We find that within lution. In this case, an unstable static bubble nucleates
3
whichwilleitherrecollapseorexpandwithroughlyequal 104
probability. As the seed black hole mass increases fur- Λ6(cid:61)0
ther, thenucleatedbubblenowhasablackholeremnant Λ6(cid:61)1x105
inthebubbleinterior,withtheactionnowrisingwithin- 103 Λ6(cid:61)2x105
creasing seed mass. The quantitative values of this crit- Λ6(cid:61)2.4x105
thinwall
ical mass, and the maximal suppression of the bounce
action at M depend on the wall tension parameter σ, B102
C
and the true vacuum energy, however, unless the com-
bination σ(cid:96) is Planck scale, this suppression is several
orders of magnitude at M , thus changing the exponen- 10
C
tial factor in (6) from an irrelevant 106 to a potentially
extremely relevant 100−2.
Whether or not this enhancement is relevant depends 1
1 10 102 103 104 105
on its magnitude relative to other physical decay pro-
SeedMassM(cid:43)(cid:144)Mp
cesses, specifically, black hole evaporation. The key
indicator is therefore the branching ratio of the static
FIG. 2. The action for a bounce solution. Each plot corre-
tunneling decay rate to the Hawking evaporation rate, sponds to a different value of λ in the Higgs potential (4),
6
ΓH ≈3.6×10−4(G2M+3)−1 [17]: with λ∗ =−0.01 and b=1.0×10−4. The largest value of λ6
is within the range of the thin wall approximation, and the
Γ /Γ ≈44(M2/M2)B1/2e−B. (7) thin wall result is shown for comparison.
D H + p
For our thin wall instantons, there is indeed a range
of M (small, though still above the Planck mass), for
+ The solutions are obtained using a shooting technique,
which we have very strong enhancement of bubble tun-
varying the value of the scalar field at the black hole
neling.
horizon and aiming for φ → 0 as r → 0. In Ref. [6], it
The main wrinkle in this argument is that the condi-
was shown that the action is given by the area terms in
tionforthethinwallapproximationrequiresthattheen-
(5), as in the thin wall case. The resulting values of the
ergyatthepotentialminimumissmallerthanthepoten-
actionforaselectionofHiggsmodelsisshowninfigure2.
tialbarrierheight,andscanningthroughparameterspace
Note that the semi-classical bubble nucleation argument
wefindthatrequiringathinwallisveryconstraining: the
only applies when the action B >1.
rangeofλ forwhichthisoccursisverysmall,andoccurs
6
forlargevaluesoftheparameterλ (cid:38)103−105,depend- Computing the branching ratio now with these “thick
6
wall” solutions gives figure 3. Although black holes pro-
ing on λ . On the other hand, computing the branching
∗
duced in the early universe start out with relatively high
ratio, (7), for these models shows that tunneling does
masses, their temperature is nonetheless above that of
indeed dominate. Thus, while our pseudo-analytic dis-
themicrowavebackground,andtheyevaporatedowninto
cussion is limited in the sense of parameter space, it has
therangeplottedinfigure3. Atthispoint,themasshits
provided a proof of principle that black holes could po-
arangeinwhichvacuumdecayismoreprobable, i.e.the
tentially seed vacuum decay.
tunneling half life becomes smaller than the (instanta-
In order to decide whether this effect is restricted to
neous)Hawkinglifetimeoftheblackhole. Notethatthis
a niche of parameter space, or is potentially relevant,
rangeiswellabovethePlanckmass,wherewehavesome
a full exploration of instantons outside of the thin wall
confidence in the validity of the vacuum decay calcula-
approximation is necessary. Motivated by our thin wall
tion. Given that this evaporation timescale is ∼ 10−28s
results,inwhichtheenhancedtunnelingtakesplacewith
for a 105M mass black hole, it is clear that once a pri-
the static instanton (as M > M , which is typically p
+ C
mordial black hole nears the end of its life cycle, it will
less than the Planck mass), we have made a preliminary
seed vacuum decay in these models. Hence with these
numerical investigation of static instantons, taking λ =
∗
Higgs potentials, the presence of any primordial black
−0.01, and b = 10−4 as representative values for the
holes will eventually trigger a catastrophic phase tran-
Higgs potential.
sition from our standard model vacuum thus ruling out
Static bounce solutions to the Einstein-scalar equa-
potentials with parameters in these ranges.
tionswithrotationalsymmetryonablackholeAdSback-
groundcanbefoundusingasphericallysymmetricmetric Since our results show that it is precisely for small
ansatz black holes that the risk of seeded tunneling is great-
est, a natural question is what happens with collider
ds2 =f(r)e2δ(r)dτ2+ dr2 +r2(dθ2+sin2θdϕ2), (8) black holes. These can be produced if the fundamental
f(r) (higher dimensional) Planck scale is near the TeV scale
[12]. Theseblackholeshavefeaturesinheritedfromtheir
where
higherdimensionalnature,andwhiletherearenoknown
2Gµ(r) exact solutions, evaporation rates have been computed
f =1− . (9) assumingahigherdimensionalMyers-Perrysolution[18],
r
4
106 relatively insensitive to the number of extra dimensions
105 ΛΛΛΛ6666(cid:61)(cid:61)(cid:61)(cid:61)0122.xx411x0015505 (otrhaetiohnorrizaotne oafrebalsacAk∝hoMles(ni+s2e)n/(hna+n1c))edw,hinerepaasrtthbeeceavuapse-
thinwall of the increased Hawking temperature, T ∝ M−1/(n+1),
104
and in part because of grey-body factors. The branch-
(cid:71)D 103 ing ratio tends to be suppressed with extra dimensions,
(cid:71)H making collider black holes less risky for vacuum decay,
howeverblackholesproducedbyparticlecollisionscould
102
still cause vacuum decay in certain regions of parameter
10 space. Fortunately, we have some reassurance about the
safetyoftheLHCfromthefactthatcosmicraycollisions
1 have occurred at energies higher than those reached at
1 10 102 103
the collider [20]!
SeedMassM(cid:43)(cid:144)Mp
To sum up: We have shown that the Coleman de Luc-
FIG. 3. The branching ratio of the false vacuum nucleation cia result for the lifetime of our universe in Higgs poten-
ratetotheHawkingevaporationrateasafunctionoftheseed tialswithmetastabilityseemscruciallydependentonthe
mass for a selection of Higgs models. Each plot corresponds absence of inhomogeneities - the presence of primordial
a different value of λ6 in (4), with λ∗ =−0.01. black holes can dramatically reduce the barrier to vac-
uumdecay,andseednucleationtoauniversewithavery
different “standard model”. Such a conclusion of course
depends on the existence of said small black holes – by
with emission cross sections appropriate to a braneworld
nomeansacertainty–andadetailednumericalstudyof
scenario [19].
parameter space. However, these results are suggestive
Black hole seeded tunneling is now a more involved that the issue of metastability of our universe may not
process, as it should involve a bubble forming around be as simple as initially was thought.
the higher dimensional black hole triggered by the Higgs
fieldtransitioningonthebrane,thebubblethenexpand-
ing out to fill the extra dimensions before finally becom- ACKNOWLEDGMENTS
ing effectively four-dimensional and seeding true decay
of our universe. While this process is beyond the reach We would like to thank Ben Withers for collabora-
of the analytic approximations we have used here, we tion in the early stages of this project, and Yiota Kanti
canestimatetheeffectbymodellingtheinstantonwitha for comments on the manuscript. PB is supported by
higherdimensionalcounterpartofthesolutionsdescribed an EPSRC International Doctoral Scholarship, RG and
above. In this case, the form of the potential V(R) for IGMaresupportedinpartbySTFC(ConsolidatedGrant
thebubblemotionismodified, butofaremarkablysimi- ST/L000407/1). RG is also supported by the Wolfson
lar form, essentially replacing R→Rn+1, where n is the Foundation and Royal Society, and Perimeter Institute
number of extra dimensions. Assuming the static bub- for Theoretical Physics. Research at Perimeter Institute
ble we can then calculate the horizon radius and area: is supported by the Government of Canada through In-
the action will be the difference in seed and remnant dustry Canada and by the Province of Ontario through
black hole horizon areas. It turns out this calculation is the Ministry of Research and Innovation.
[1] G.Aadet al.(ATLASCollaboration),Phys.Lett.B710, [10] N. Arkani-Hamed, S. Dimopoulos, and G. Dvali,
49 (2012), arXiv:1202.1408 [hep-ex]. Phys.Lett. B429, 263 (1998), arXiv:hep-ph/9803315
[2] S. Chatrchyan et al. (CMS Collaboration), Phys.Lett. [hep-ph].
B710, 26 (2012), arXiv:1202.1488 [hep-ex]. [11] L. Randall and R. Sundrum, Phys.Rev.Lett. 83, 3370
[3] S. R. Coleman, Phys.Rev. D15, 2929 (1977). (1999), arXiv:hep-ph/9905221 [hep-ph].
[4] J.Callan,CurtisG.andS.R.Coleman,Phys.Rev.D16, [12] S.DimopoulosandG.L.Landsberg,Phys.Rev.Lett.87,
1762 (1977). 161602 (2001), arXiv:hep-ph/0106295 [hep-ph].
[5] S. R. Coleman and F. De Luccia, Phys.Rev. D21, 3305 [13] G. Degrassi, S. Di Vita, J. Elias-Miro, J. R. Es-
(1980). pinosa, G. F. Giudice, et al., JHEP 1208, 098 (2012),
[6] R. Gregory, I. G. Moss, and B. Withers, JHEP 1403, arXiv:1205.6497 [hep-ph].
081 (2014), arXiv:1401.0017 [hep-th]. [14] I. G. Moss, (2014), arXiv:1409.2108 [hep-th].
[7] W. A. Hiscock, Phys. Rev. D35, 1161 (1987). [15] V. Branchina and E. Messina, Phys.Rev.Lett. 111,
[8] V. A. Berezin, V. A. Kuzmin, and I. I. Tkachev, Phys. 241801 (2013), arXiv:1307.5193 [hep-ph].
Rev. D43, 3112 (1991). [16] M. Sasaki and D.-h. Yeom, JHEP 12, 155 (2014),
[9] B. J. Carr and S. Hawking, Mon.Not.Roy.Astron.Soc. arXiv:1404.1565 [hep-th].
168, 399 (1974). [17] D. N. Page, Phys.Rev. D13, 198 (1976).
5
[18] R.C.MyersandM.Perry,AnnalsPhys.172,304(1986). 229 (2015), arXiv:1402.3952 [hep-th].
[19] P. Kanti and E. Winstanley, Fundam.Theor.Phys. 178, [20] P. Hut and M. Rees, Nature 302, 508 (1983).