Gravitational corrections to Standard Model vacuum decay
Gino Isidori, Vyacheslav S. Rychkov, Alessandro Strumia, Nikolaos Tetradis
GGravitational corrections toStandard Model vacuum decay
Gino Isidori a,b , Vyacheslav S. Rychkov a , Alessandro Strumia c , Nikolaos Tetradis da Scuola Normale Superiore and INFN, Piazza dei Cavalieri 7, I-56126 Pisa, Italy b INFN, Laboratori Nazionali di Frascati, Via E.Fermi 40, I-00044 Frascati, Italy c Dipartimento di Fisica dell’Universit`a di Pisa and INFN, Italia d Department of Physics, University of Athens, Zographou GR-15784, Athens, Greece
Abstract
We refine and update the metastability constraint on the Standard Model top andHiggs masses, by analytically including gravitational corrections to the vacuum decayrate. Present best-fit ranges of the top and Higgs masses mostly lie in the narrowmetastable region. Furthermore, we show that the SM potential can be fine-tunedin order to be made suitable for inflation. However, SM inflation results in a powerspectrum of cosmological perturbations not consistent with observations.
Assuming that the Standard Model (SM) holds up to some high energy scale close to M Pl = 1 .
22 10 × GeV, present data suggest a light Higgs mass, m h ∼ (115 − We recall vacuum decay within the Standard Model without gravity, and its peculiarities relevant forour later inclusion of gravity. The SM contains one complex scalar doublet H , H = (cid:20) ( h + iη ) / √ χ − (cid:21) , (1)1 a r X i v : . [ h e p - ph ] D ec ith tree-level potential V = m | H | + λ | H | = 12 m h + 14 λh + . . . (2)where the dots stand for terms that vanish when the Goldstones η, χ − are set to zero. With thisnormalization, v = ( G F √ − / = 246 . h is m h = V (cid:48)(cid:48) ( h ) | h = v = 2 λv . As is well known, for h (cid:29) v the quantum corrections to V ( h ) canbe reabsorbed in the running coupling λ (¯ µ ), renormalized at a scale ¯ µ ∼ h . To a good accuracy, V ( h (cid:29) v ) ≈ λ ( h ) h / λ becomes negative for some value of h . For thevalues of m h compatible with data this occurs at scales larger than 10 GeV, suggesting that we cancompute vacuum decay neglecting the quadratic term m h / h ( r ) of the Euclidean equations of motion that depends only on theradial coordinate r ≡ x µ x µ : − ∂ µ ∂ µ h + V (cid:48) ( h ) = − d hdr − r dhdr + V (cid:48) ( h ) = 0 , (3)and satisfies the boundary conditions h (cid:48) (0) = 0 , h ( ∞ ) = v → . (4)We can perform a tree-level computation of the tunnelling rate with a negative λ < µ . In this approximation, the tree-level bounce h ( r ) can be found analyticallyand depends on an arbitrary scale R : h ( r ) = (cid:115) | λ | Rr + R , S [ h ] = 8 π | λ | . (5)At first sight, computing the decay rate among two vacua in the approximation V ( h ) = λh / h ∼ h ∼ M Pl , if the bounce has size R (cid:29) /M Pl : once a tunneling bubble appears, the instability dueto V (cid:48) ( h (0)) (cid:54) = 0 brings h down to the true minimum with unit probability. Formally, by performingthe analytic continuation from Euclidean r = x + t to Minkowskian r = x − t space-time, theevolution inside the bubble is described by eq. (5) with r < h reaches a singularity at r = − R .Indeed our potential is unbounded from below. In general, what happens inside the bubble does notaffect the tunneling rate nor the growth of the bubble: being an O(4)-invariant configuration (i.e. thebounce depends only on r ), its walls expand at the speed of light, so that what happens in the interiorcannot causally affect the exterior.The arbitrary parameter R appears in the expression of the SM bounce h ( r ) because in ourapproximations the tree level SM potential is scale-invariant: at this level, there is an infinite set ofbounces of varying size R , all with the same action S [ h ].Quantum corrections are the dominant effect that breaks scale invariance, and have been computedin [4]. At one-loop order, the tunnelling probability in the universe space-time volume V U is then givenby p = max R [ p ( R )] , p ( R ) = V U R e − S , (6) In the usual case, with a potential with two minima, the bounce can be computed only numerically. The analyticcontinuation can be done by switching r → ir in eq. (3) at r <
0, and solving numerically. The qualitative behavior ofthe solution can be understood by noticing that this operation is equivalent to flipping the sign of V : h oscillates aroundthe true minimum, reaching it at r → −∞ , i.e. for asymptotically large times inside the expanding bubble. S = S + ∆ S − loop is the one-loop action: since the bounce is not a static configuration,corrections to both the potential part, as well as to the kinetic part of the action, must be taken intoaccount [4]. To find the bounce configuration that extremizes S , it is enough to evaluate it alongthe family of tree-level bounces, h in eq. (5), and minimize with respect to R . The result is roughly S ≈ π / | λ (¯ µ = 1 /R ) | , i.e. one-loop corrections remove the tree-level ambiguity on the RGE scale ¯ µ by fixing it to be the scale 1 /R of the bounce. Since within the SM λ (¯ µ ) happens to run reaching aminimal value at ¯ µ ∼ − GeV, tunneling is dominated by bounces with this size. A posteriori,this justifies having neglected the SM mass term, that gives a correction ∆ S ∼ ( mR ) (cid:28) We now extend the previous computation taking into account gravity [7]. In our case this is apotentially relevant effect, since gravity breaks scale-invariance and 1 /R is just somewhat smallerthan the Planck scale. One might worry that gravity can have dramatic effects, and that the decayrate starts to depend on the unknown depth V min of the true minimum of the SM potential. This isnot the case. Since the exterior geometry is the flat Minkowski space, the generic argument given inthe non-gravitational case still holds: the bubble is an O(4)-invariant solution and its walls expandsat the speed of light, irrespectively of what happens inside. We recall from [7] the basic formalism needed for a quantitative analysis. We assume an Euclideanspherically symmetric geometry, ds = dr + ρ ( r ) d Ω , where d Ω is the volume element of the unit3-sphere. The Einstein-Higgs action S = (cid:90) d x √ g (cid:20) ( ∂ µ h )( ∂ µ h )2 + V ( h ) − R κ (cid:21) , (8)where κ = 8 πG and G = 1 /M with M Pl = 1 .
22 10 GeV, simplifies to S = 2 π (cid:90) dr (cid:20) ρ ( h (cid:48) V ) + 3 κ ( ρ ρ (cid:48)(cid:48) + ρρ (cid:48) − ρ ) (cid:21) , (9)where (cid:48) denotes d/dr . The equations of motion are h (cid:48)(cid:48) + 3 ρ (cid:48) ρ h (cid:48) = dVdh , ρ (cid:48) = 1 + κ ρ ( h (cid:48) − V ) . (10)We can analytically include the effect of gravity, assuming RM Pl (cid:29)
1, by performing a leading-order expansion in the gravitational coupling κ : h ( r ) = h ( r ) + κh ( r ) + O ( κ ) , ρ ( r ) = r + κρ ( r ) + O ( κ ) . (11)The action is S = S + 6 π κ (cid:90) dr (cid:20) r ρ (cid:18) h (cid:48) V ( h ) (cid:19) + ( rρ (cid:48) + 2 ρ ρ (cid:48) + 2 ρ rρ (cid:48)(cid:48) ) (cid:21) + O ( κ ) . (12) This is what one would na¨ıvely guess from the results of [7] for the bounce action: S with gravity ≈ S without gravity / [1 + R V min /M ] , (7)i.e. the bubble does not exist if the true minimum has a large negative cosmological constant, e.g. V min ∼ − M .However, eq. (7) holds within the thin-wall approximation [7], not applicable when V min is large and negative, and notapplicable to the SM case we are interested in. It is only an observer inside the bubble that experiences a large negative cosmological constant and consequently acontraction down to a big-crunch singularity [7], instead than an expanding bubble. (cid:45) (cid:45) (cid:45) (cid:45) (cid:144) R in GeV v ac uu m d eca yp r ob a b ilit y withoutgravitywithgravity Figure 1:
Probability p ( R ) that the SM vacuum decayed so far for m h = 115 GeV , m t = 174 . , α ( M Z ) = 0 . , due to bounces with size R , without including gravitational effects (dashedcurve [4]) and including gravitational effects (continuous line). The correction is relevant only at /R > ∼ GeV . Uncertainties due to higher-order corrections are not shown.
We have taken into account that many terms in the expansion vanish either because the integrand isa total derivative (e.g. the negative power 1 /κ in eq. (8) is just apparent) or thanks to the equationsof motion. Indeed h does not appear in eq. (12) because we are functionally expanding around theextremum h of the non-gravitational action, so that the first functional derivatives vanish thanks tothe equations of motion. So, we only need to compute ρ : its equation of motion is ρ (cid:48) = 16 r (cid:18) h (cid:48) − V ( h ) (cid:19) . (13)Inserting it into eq. (12) completes the computation of gravitational corrections to leading-order in κ . We notice that the first term in eq. (12), which is linear in ρ , contributes − ρ . This happens because S must havean extremum at c = 1 under the variation ρ ( r ) → cρ ( r ). The discussion is so far general, and bychoosing toy potentials we verified that eq. (12) agrees with the full numerical result. Going to the SM case, using the analytic expression of eq. (5) for the bounce h , we can perform allintegrations analytically finding S = 8 π | λ | + ∆ S − loop + ∆ S gravity , ∆ S gravity = 256 π RM Pl λ ) (14)where ∆ S gravity is the gravitational correction and ∆ S − loop the one-loop correction, given in eq. (3.3)of [4]. Eq. (6) gives the tunneling probability p ( R ).Fig. 1 shows an example of the relevance of gravitational corrections. We checked that the leading-order approximation agrees with the result of a full numerical computation: eq. (14) correctly approx-imates the action of the true bounce, and the true bounce h ( r ) is correctly approximated by h ( r )with the value of R that minimizes S . Here we comment about the comparison between the analytic result in eq. (14) and the full numerical computation. nstability110 120 130 140 150165170175180 Higgs mass (cid:32) m h (cid:32) in GeV t op m a ss (cid:32) m t (cid:32) i n G e V StabilityMeta (cid:45) stability
Figure 2:
Metastability region of the Standard Model vacuum in the ( m h , m t ) plane, for α s ( m Z ) =0 . (solid curves). Dashed and dot-dashed curves are obtained for α s ( m Z ) = 0 . ± . . Theshaded half-ellipses indicates the experimental range for m t and m h at and confidence level.Sub-leading effects could shift the bounds by ± in m t . Fig. 2 shows the regions in the ( m h , m t ) plane where the SM vacuum is stable, meta-stable or toounstable. Gravitational corrections only induce a minor shift on the ‘instability’ border, less relevantthan present experimental and theoretical (higher-order) uncertainties. The ellipses truncated at m h = 115 GeV are the best-fit values for the top and Higgs masses, from our up-to-date global fit ofprecision data, that includes the latest direct measurement of the top mass, m t = (170 . ± .
8) GeV [8].Present data and computations indicate that we do not live in the unstable region (such that the SMcan be valid up to the Planck scale), but increased accuracy is needed to determine if we live in thestable or in the small meta-stable region.Adding to the SM action possible dimension-6 non-renormalizable operators suppressed by thePlanck scale would give similar corrections to the bounce action. In particular, adding to the SMLagrangian the operators∆ L = 1 M (cid:18) − ξM R | H | + c | H |
3! + c | H | | D µ H | (cid:19) , (15)where ξ and c , are unknown dimensionless coefficients, gives the following correction∆ S (cid:48) gravity = 8 π M Pl Rλ ) (cid:18) πξ + c | λ | + 4 c (cid:19) , (16)which can be comparable to the model-independent gravitational effect computed in eq. (14). With a typical potential this is a straightforward procedure: the bounce is determined numerically as a compromisebetween classical solutions which under-shoot and over-shoot the true bounce at large r . With a potential close to h ,finding the bounce numerically is more involved: with this potential classical solutions necessarily go to zero at large r ;however, they generically oscillate to zero as 1 /r giving a divergent action. The special feature of h ( r ) is that it vanishesas 1 /r giving a finite action. The true bounce should maintain this behavior. In practice, this is achieved imposing avanishing difference between h ( r ) and the numerical bounce. The advantage of our analytic approximation based onthe set of candidate bounces h ( r ) with different values of R is that ill-behaved never enter the computation. (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Higgs vev h in Planck units V (cid:144) i n P l a n c kun it s n ee d e d Ε (cid:61) V (cid:144) (cid:72) . M P l (cid:76) SM Higgs potential m t (cid:61)
170 GeV m h (cid:187) m t (cid:61)
171 GeV m h (cid:187) m t (cid:61)
172 GeV m h (cid:187) Figure 3:
Examples of fine-tuned SM potentials that might allow inflation. The right handed axisshows the value of the slow-roll parameter ε that would give the observed amount of anisotropies. The values of the coefficients ξ and c , change under field redefinitions and only their linearcombination entering (16) is physical. Indeed, under H → H (1 + a | H | /M ) we have δc = 24 λa,δc = 6 a and δξ = 0; this transformation can be used to set c →
0. Under the Weyl transformationof the metric g µν → g µν (1 + a | H | /M ) we have δξ = a/ π , δc = 12 aλ , δc = a ; this transformationcan be used to set ξ →
0. Both these field redefinitions leave ∆ S (cid:48) gravity invariant.To estimate the magnitude of ∆ S (cid:48) gravity we can thus restrict the attention to only one of thethree operators in ∆ L (we choose the | H | term), and estimate its coupling using na¨ıve dimensionalanalysis. At one loop, graviton exchanges generate the | H | operator with c ∼ g /π as well as the λ | H | operator with coefficient λ ∼ g /π . Here g s is an unknown coefficient which determines ifquantum gravity is weakly or strongly coupled, with strong coupling corresponding to g s ∼ π . Onemight therefore argue that c ∼ λπ , which implies ∆ S (cid:48) gravity ∼ ∆ S gravity . For m t ≈
173 GeV and m h ≈
130 GeV (i.e. within the experimentally allowed region) both the quarticHiggs coupling λ and its β -function happen to vanish, at some RGE scale around M Pl . Is this just acoincidence, or this boundary condition carries some message? Some speculations about this fact havebeen presented in [9]. Here we explore a different aspect, namely a possible connection with inflation.The quasi-vanishing of both λ and β ( λ ) allows to have a quasi-flat Higgs potential at h ∼ M Pl ,suitable for inflation. Indeed, we can approximate the RGE running of λ as λ ( µ ∼ h ) (cid:39) λ min + γ (4 π ) ln µh (17)around the special value h where λ reaches its minimal value λ min . The constant γ is related to β ( β ( λ )) and has the numerical value γ ≈ . We do not distinguish between | H † D µ H | and | H | | D µ H | since these operators coincide on the configurations H = ( h/ √ ,
0) we are interested in.
6M potential V (cid:39) λ ( h ) h / h = h ∗ ≡ h e − / if λ min = γ/ π , such that the slow-rollparameters ε ≡ M π (cid:18) V (cid:48) V (cid:19) , η ≡ M π V (cid:48)(cid:48) V vanish, allowing for inflation.The lack of convincing natural models for inflation might indicate that it happens when scalarfields, fluctuating along some vast ‘landscape’ potential generically unsuitable for inflation, encountera small portion of the potential which accidentally is flat enough. This is what might happen withinthe SM. This potential is illustrated in fig. 3, where we do not show the uncertainty due to higher-order corrections, which effectively amounts to a ± m t . Can this SM potentialbe responsible for inflation and the generation of anisotropies δρ/ρ ? The answer is: not both. Thebasic problem is that the requirement of having enough e -folds of inflation, N = 2 √ π (cid:90) dh/M Pl √ ε ≈ , (18)can be met with a small enough ε , but this conflicts with the requirement that quantum fluctuationof the Higgs inflaton should also generate the observed power spectrum of anisotropies, δρ/ρ ∼ − ,i.e. Vε ≈ (0 . M Pl ) . (19)Indeed the height V of the SM potential in its flat region is predicted and cannot be arbitrarily adjustedto be as low as needed. This result can be understood by either doing explicit computations with theapproximated potential λ ( h ) h /
4, or by looking at the sample SM potentials plotted in fig. 3. For atop mass within the observed range, the plateau is at values of h and V / which are are somewhatbelow the Planck scale, but δρ/ρ at N ≈
60 comes out larger than the observed value. Successfulinflation and successful generation of anisotropies would be obtained if for some unknown reason thepotential would remain flat from h ∼ h ∗ up to h ∼ M Pl . In this paper we have refined and updated the metastability constraint on the Higgs mass, assumingthe validity of the Standard Model up to the highest possible energy scale, Λ ≈ M Pl . In particular, wehave taken into account gravitational corrections, which were neglected in previous analyses. Thesecorrections turn out to be small and calculable in the phenomenologically interesting region of m h close to its experimental lower bound. The updated constraints in the ( m h , m t ) plane are reported infig. 2. Among all possible values, the Higgs mass seems to lie in the narrow region which allows theSM to be a consistent theory up to very high energy scales, with a perturbative coupling and a stableor sufficient long-lived vacuum. Fig. 4 illustrates the constraints on the Higgs mass as function of Λ,and shows that the (meta)stability constraints do not depend on Λ when it is around the Planck scale.We have also shown that the SM potential can be fine-tuned in order to be made suitable forinflation. However, the resulting power spectrum of anisotropies is larger than the observed one. Acknowledgements
We thank Paolo Creminelli and Enrico Trincherini for useful discussions.
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