Top mass determination, Higgs inflation, and vacuum stability

The possibility that new physics beyond the Standard Model (SM) appears only at the Planck scale $M_P$ is often considered. However, it is usually argued that new physics interactions at $M_P$ do not affect the SM stability phase diagram, so the latter is obtained neglecting these terms. According to this diagram, for the current experimental values of the top and Higgs masses, our universe lives in a metastable state (with very long lifetime), near the edge of stability. Contrary to these expectations, however, we show that the stability phase diagram strongly depends on new physics and that, despite claims to the contrary, a more precise determination of the top (as well as of the Higgs) mass will not allow to discriminate between stability, metastability or criticality of the electroweak vacuum. At the same time, we show that the conditions needed for the realization of Higgs inflation scenarios (all obtained neglecting new physics) are too sensitive to the presence of new interactions at $M_P$. Therefore, Higgs inflation scenarios require very severe fine tunings that cast serious doubts on these models.

the experimental point is considered by some as the most important message from the data on the Higgs boson [25]. The Higgs inflation scenario of [32], in particular, strongly relies on the realization of the conditions λ(M P ) ∼ 0 and β(λ(M P )) ∼ 0.
Then, given this phase diagram, it is expected that, with the help of more refined measurements, we should be able to see whether the experimental point sits on the border between stability and metastability or is located inside one of these two regions. More precisely, as the dominant uncertainty comes from the value of the top mass, the expectation is that a more precise determination of M t will finally allow to discriminate between a stable or metastable (or critical) EW vacuum [33], [34].
The above analysis, however, presents some delicate aspects. For the central values M t ∼ 173. 34 GeV and M H ∼ 125.7 GeV, the Higgs potential V ef f (φ) at M P is negative (unstable), V ef f (φ = M P ) < 0, and for φ > M P it continues to go down for a long while, developing the new minimum at φ min much larger than M P , φ min ∼ 10 30 GeV. Usually, this is not viewed as a serious drawback [20]. In fact, it is argued that V ef f (φ) is eventually stabilized by new physics interactions present at the Planck scale, that should bring the new minimum around M P . At the same time, it is also argued that the new physics terms should not affect the EW vacuum lifetime [20,21], so the latter is computed by considering the unmodified potential V ef f (φ), i.e. neglecting the presence of new physics.
It has been recently shown, however, that the EW vacuum lifetime can be strongly affected by new physics [35]. By carring further on this analysis, in the present work we show that the expectation that better measurements of the top mass will allow to discriminate between a stable, a metastable or a critical EW vacuum [33,34] is not fulfilled. Even very precision measurements of the top mass M t (as well as of M H ) cannot decide of the EW vacuum stability condition. As we will see, the phase diagram of fig.1 where λ ef f (φ) depends on φ essentially as the running quartic coupling λ(µ) depends on the running scale µ. V ef f (φ) is the renormalization group improved (RGI) Higgs potential, and for λ ef f (φ) we have the corresponding one-loop, two-loops or three-loops expressions.
In the following we consider the up to date Next-to-Next-to-Leading-Order (NNLO) results [24,[36][37][38]. We are now interested in studying what happens when new physics interactions at the Planck scale are taken into account. Following [35], we study the impact of new physics by adding to the potential two higher order operators, φ 6 and φ 8 . With the inclusion of these terms, the classical potential V (φ) = λ 4 φ 4 becomes (np is for new physics) with λ 6 and λ 8 dimensionless coupling constants. Running the RG equations for all of the SM parameters, including λ 6 and λ 8 , we get the where λ 6 (φ) and λ 8 (φ), as λ ef f (φ), are the RG improved couplings, and ξ(φ) comes from the anomalous dimension of φ. For the purposes of this work, however, it is sufficient to keep only the tree level corrections coming from the new operators, although the effect of the running can be easily taken into account.
The potential V new ef f (φ) modified by the presence of the new physics interactions is then obtained from Eq. (1), with λ ef f (φ) replaced by and the stability line is given by those values of φ such that (5)  In the present section we set up the tools for the determination of the stability line, with and without new physics interactions taken into account. In Section 4, where the stability phase diagrams are studied, we will make use of this analysis. In the next section, we move to the metastability case, i.e. we consider values of M H and M t such that the second minimum . In these cases, the EW minimum is a false vacuum, and we need the tools to determine the instability line, i.e. the boarder between the region where the EW vacuum lifetime τ is larger than the age of the universe T U , and the region where τ is shorter than T U , the instability region.

EW vacuum lifetime. Metastability
The standard analysis of the metastability case is performed by computing the EW vacuum lifetime τ with the help of the Higgs potential V ef f (φ) that is obtained by considering SM interactions only [20][21][22][23][24]. As already noted in the Introduction, this is related to the expectation that new physics interactions should not affect τ .
Referring to [20,23,24,35,39,40] for details, we recall here that for a given potential V (φ), the general procedure to obtain the tunnelling time τ is to look first for the bounce solution (tree level) to the euclidean equation of motion [41], and to compute then the quantum fluctuations on the top of it [42]. For the Higgs potential V (φ) = λφ 4 /4, once the running of the quartic coupling is taken into account, this amounts to the following minimization where T (µ) is Together with Eq.(2), Eq. (6)  From Eqs. (6) and (7), the condition τ = T U is immediately translated into the condition min µ log 10 T (µ) = 0 . GeV (black dot in fig.1), the EW vacuum is metastable and τ is much greater than T U . If we now keep M H fixed to the value M H = 125.7 GeV and increase M t , we see that τ decreaes and reaches the value τ = T U for M t = 178.04 GeV. This is how the instability line is obtained.
As in the previous section, we want to perform now the stability analysis when the presence of new physics is taken into account. To this end, we consider again the potential . We should then begin by considering the tree level contribution that comes from the new bounce solution for the potential (3). Differently from the previous case (φ 4 term alone), due to the presence of the terms λ 6 φ 6 and λ 8 φ 8 , this bounce cannot be found analytically, and we have to solve the Euclidean equation of motion numerically. Then, we should compute the quantum fluctuations around the bounce. This complete analysis is presented elsewhere [40].
We have carefully checked the validity of this approximation against the numerical computation of the bounce and of the corresponding quantum fluctuations [40] and found that the two results are in good agreement. As for the case of absolute stability considered in the previous section, from the example considered above we get the strong suggestion that new physics interactions at the Planck scale are far from being negligible. We come back to this point in the next section, where the phase diagrams are considered.

New physics, new phase diagrams and top mass
In the two previous Sections, we set up the tools for our analysis. We are now in the position to draw the stability phase diagram for different cases, with and without new physics interactions taken into account.
The phase diagram for the case when new physics interactions are neglected, as they are supposed to have no impact on it, is well known [24,25], and we have reproduced this case in fig.1. From this figure we see that, according to this stability analysis, for the central values of M H and M t , the EW vacuum is metastable with a lifetime extremely larger than the age of the universe (τ = 10 613 T U ). From the same diagram, we also see that, within 3 σ, the SM point could reach and even cross the stability line.
Due to the great sensitivity of the results on the stability analysis to the value of the top mass, it is usually believed that a more precise measurement of M t would provide a definite answer to the question of whether we live in the stability region, in the metastability region, or at the edge of stability (criticality). In particular, it was stressed in [33] that the identification of the measured mass with the pole mass is not free of ambiguities (quarks do not appear as asymptotic states, and the pole top mass has to be defined with care), and that these difficulties can be overcame if we refer to the running M M S t (µ) top mass. At the same time, the authors observe that, when the translation to the pole mass is appropriately realized, the error on M t turns out to be much larger than the experimental error usually reported. As a result, the Tevatron and LHC results for M t , within two sigma, turn out to be consistent with stability, metastability, and instability at once. This analysis seems to point towards the conclusion that our knowledge of the stability condition of the EW vacuum critically depends on the precise determination of the top mass.
As we will see in a moment, however, while the remarks on the top pole mass [33] have to be seriously taken into account, the expectation that a more precise determination of the top mass will allow to discriminate between stability or metastability (or criticality) of the dashed lines are for comparison and reproduce the phase diagram of fig.1). As a result of the presence of new physics at the Planck scale, the stability and metastability lines move down. For this choice of λ 6 and λ 8 , the SM point is still in the metastability region, but its distance from the stability line is larger than before (more than 5 σ).
Comparing the stability phase diagram of fig.4 with the one of fig.1, we clearly see that even if the top mass is measured with very high precision, this is not going to give any definite indication on the stability condition of the EW vacuum. As long as we don't know the specific form of new physics, we cannot say anything on stability. Lowering the error in the determination of M t is certainly important, but definitely not discriminating for the stability condition of the EW vacuum.
In order to better appreciate the strong dependence of the stability phase diagram on new physics, let us consider now a second example, where we use for λ 6  However, the phase diagram of fig.1 is usually presented as if it was the generic result that we obtain whenever we assume that the SM is valid all the way up to the Planck scale. In particular, referring to this phase diagram, it is stated that for the present experimental central values of M H and M t , our universe lives within the metastability region, at the edge of the stability line [25], and that better measurements of M H and M t will definitely allow to discriminate between stability, metastability or criticality for the EW vacuum [34].
In the light of what we have shown in the present work, these statements appear to be unjustified and misleading. What really discriminates between different stability conditions for the EW vacuum is New Physics. If new physics provides results of the kind that we have shown in fig.4, the phase diagram of fig.1 turns out to be simply wrong, and it has definitely nothing to say on the stability condition of the EW vacuum.  fig.1). We have seen, however, that the stability condition of the EW vacuum depends on new physics, i.e. λ 6 and λ 8 in our present case (see also [43]).
The left panel of fig.6 shows the vacuum stability phase diagram in the (λ 6 , λ 8 ) -plane. The line separating the yellow and the red regions is the instability line (τ = T U ). The  fig.6 shows a zoom on this region (dark green area).
The lesson from the above example is clear. Any beyond SM (BSM) candidate theory has to be tested with the help of a "stability test". A BSM theory is acceptable only if it provides either a stable EW vacuum or a metastable one, but with lifetime larger than the age of the universe. A phase diagram of the kind shown in fig.6 allows to determine the regions of the parameter space that are permitted by the stability test.
At the same time, it is also clear that a more refined measurement of the top (as well as of the Higgs) mass provides more stringent constraints on the parameter space. GeV [26]. This is another important lesson of our analysis. While not discriminating for the stability issue, a better measurement of the top mass has an impact in the determination of the allowed regions in the parameter space of the theory.

Higgs Inflation and new physics
Let us come back now to the Higgs inflation scenario of [32]. As noted before, this scenario is heavily based on the standard vacuum stability analysis. In particular, it requires that new physics shows up only at the Planck scale M P , and that the SM lives at the edge of the stability region, where λ(M P ) ∼ 0 and β(λ(M P )) ∼ 0. We have seen, however, that new physics interactions, even if they live at the Planck scale, can strongly change the SM phase diagram of fig.1. The realization of the conditions λ(M P ) ∼ 0 and β(λ(M P )) ∼ 0 requires such a fine tuning [44] that even a small grain of new physics at the Planck scale can totally destroy the picture. We believe that these observation make the chance for the realization of the Higgs inflation scenario quite low.
Similar considerations also apply to an alternative implementation of Higgs inflation [45]. The idea is that we could have a second minimum that is higher than the EW one and such that this metastable state could have been the source of inflation in the early universe, later decaying in the EW stable minimum. In order to illustrate this scenario, together with our comments, we now consider the following case. In the left panel of fig.8 we plot the Higgs effective potential computed with SM interactions only for M t = 171.5, 171.5005, 171.50057 GeV. We see that in the case M t = 171.5005 GeV the potential develops a new (shallow) minimum, higher than the EW one. As is clear from fig.8, there is only a narrow band of values of M t such that this minimum is higher than the EW one. In fact (see fig.8), for M t = 171.5 GeV the minimum disappears, while for M t = 171.50057 GeV the new minimum is lower that the EW one.
Needless to say, this proposal has severe intrinsic fine tuning problems. Changing the fifth decimal in the top mass, the new minimum goes from metastable to stable. But even if we accept such a fine tuning for the top mass, and stick on the M t = 171.5005 GeV value of our example, the presence of even a little seed of new physics would be suffient to screw up the whole picture. This is clearly shown in the right panel of fig.8, where the addition of very tiny values of new physics coupling constants is able to produce either the disappearance of the new minimum or the lowering of this minimum below the EW vacuum.

Conclusions
We have shown that the stability condition of the EW vacuum and the corresponding stability phase diagram in the (M H , M t ) -plane strongly depend on new physics. On the contrary, in the past it was thought that, given the values of M H and M t , the stability of the EW vacuum could be studied with no reference to the specific UV completion of the SM. This lead to the believe that the phase diagram of fig.1 is universal, that is independent on new physics. As we have shown, see figs.4 and 5, this is not the case.
This lack of universality has far reaching consequences for phenomenology, in particular for model building. As the stability condition of the EW vacuum is sensitive to new physics, it is clear that any BSM candidate has to pass a sort of "stability test". In fact, only a BSM theory that respects the requirement that the EW vacuum is stable or metastable (but with lifetime larger than the age of the universe) can be accepted as a viable UV completion of the SM.
We have also shown that it is incorrect and misleading to refer to the phase diagram of fig.1 as if it was the snapshot of the present situation for a SM valid up to the Planck scale.
As we have seen, the SM may well be valid up to the Planck scale and still we could have a completely different stability phase diagram as compared to the phase diagram of fig.1, the latter being the only one considered in the literature [24,25]. This phase diagram is not universal, it depends on the kind of new physics that we have at the Planck scale, it is just one case out of different possibilities. Therefore, we should no longer refer to this diagram as the status of art of our knowledge concerning the stability condition of the EW vacuum.
As a consequence of that, it is clear that, despite claims to the contrary, with a more precise determination of the top mass M t we will not be able to discriminate between stability, metastability, or criticality of the EW vacuum. This expectation, in fact, is related to the (erroneous) assumption that the phase diagram of fig.1 is valid whatever new physics we have at the Planck scale.
At the same time, if we consider a specific UV completion of the SM (a specific BSM theory), a more precise knowledge of M t , as well as of M H and of the other parameters, will be important to put constraints on the parameter space of the theory. In other words, as long as we do not work with a specific BSM theory, we cannot draw any conclusions on the stability condition of the EW vacuum. Even if new physics shows up only at the Planck scale, the "fate" of our universe (that is the stability condition of the EW vacuum) crucially depends on the new physics interactions.
Moreover, it is clear that the same warnings apply to the Higgs inflation scenario of [32]. The latter is heavily based on the standard analysis, and in particular on the assumptions that new physics shows up only at the Planck scale M P and that the EW vacuum is at the edge of the stability region, where λ(M P ) ∼ 0 and β(λ(M P )) ∼ 0. As we have seen, new physics interactions can strongly change the SM phase diagram of fig.1, thus changing these relations. In fact, the realization of the conditions λ(M P ) ∼ 0 and β(λ(M P )) ∼ 0 requires an enormous fine tuning [44], and new physics interactions at the Planck scale can easily screw up these relations. Other implementations of the Higgs inflation idea [45], based on the possibility for the SM Higgs potential to develop a minimum at lower energies (a minimum where inflation could have started in a metastable state), are also subject, for the same reasons, to the same warnings.
Finally, it is important to note that our analysis on the impact of new physics interactions on the stability analysis can be repeated even when the new physics scale lies below the Planck scale, as could be the case, for instance, of GUT scale.