Quantum Gravitational Contributions to the Standard Model Effective Potential and Vacuum Stability

# Quantum Gravitational Contributions to the Standard Model Effective Potential and Vacuum Stability

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HU-EP-15/07

Quantum Gravitational Contributions to the Standard Model Effective Potential and Vacuum Stability

Florian Loebbert and Jan Plefka

Institut für Physik and IRIS Adlershof,

Humboldt-Universität zu Berlin,

Zum Großen Windkanal 6, D-12489 Berlin. Germany

loebbert@physik.hu-berlin.de

Abstract

We compute the quantum gravitational contributions to the standard model effective potential and analyze their effects on the Higgs vacuum stability in the framework of effective field theory. Einstein gravity necessarily implies the existence of higher dimension and operators with novel couplings in the Higgs sector. The beta functions of these couplings are established and the impact of the gravity induced contributions on electroweak vacuum stability is studied. We find that the true minimum of the standard model effective potential now lies below the Planck scale for almost the entire parameter space (). In addition quantum gravity is shown to contribute to the minimal value of the standard model NLO effective potential at the percent level. The quantum gravity induced contributions yield a metastable vacuum for a large fraction of the parameter space in the flowing couplings .

A central outcome of the recent Higgs boson discovery [1, 2] and the absence of new physics signals at the LHC is that the standard model (SM) as a quantum field theory stays perturbatively self-consistent all the way up to the Planck scale [3, 4]. From this perspective the conservative scenario of having no beyond-SM-physics up to —except for gravity—is viable and has attracted attention [5, 6, 7, 8, 9, 10].111Of course neutrino masses, dark matter and baryogenesis still require some (mild) extensions of the SM. Moreover, the measured values for the Higgs pole mass and the top mass have an intriguing consequence for the question of stability of the Higgs vacuum: The SM lies close to the border of absolute electroweak vacuum stability and metastability. Vacuum stability is usually studied by determining the renormalization group improved effective Higgs potential : If develops a negative minimum below the value of the electroweak minimum, the SM becomes unstable; if the inverse decay rate for tunneling from the false electroweak minimum at to the second (true) minimum at is larger than the lifetime of our universe, then the SM is said to be metastable. The value at the minimum is very sensitive to the value of and . State of the art high precision perturbative calculations [3, 4] using the latest experimental data indicate that the SM has a negative metastable minimum. However, it lies deep within the Planck regime: The gauge invariant value of at next-to-leading order (NLO) precision is [11]. While it is fascinating that the SM may be extrapolated to such high energy scales, it is obvious that quantum gravitational effects cannot be ignored any longer in these regimes—even for the conservative no-new-physics scenario. In consequence, the celebrated statement of metastability of the SM based on the above value for is spurious.

This is the motivation for the present analysis. We adopt the conservative viewpoint of having no new physics up to and study the quantum gravity contributions to the SM effective potential. This is done by treating Einstein gravity as an effective quantum field theory [12]. For scales below , the SM coupled to quantum gravity is perturbatively well defined albeit of limited predictability due to the necessity of including higher dimensional operators as counterterms at every loop order. In fact, the main impact of the gravitational contributions to the Higgs effective potential is that non-renormalizability induces higher dimension counter terms of the form and into the effective field theory, with a priory undetermined couplings and . The addition of such higher dimension operators to the tree-level potential has been studied in [13, 14, 11] as a means to parametrize potential new physics effects on top of the SM. Here we show that these terms are necessarily present due to the existence of gravity. Even if one chooses the new couplings to be absent at low scales, they are turned on at high scales by the quantum gravity contributions to the renormalization group equations (RGE). The higher dimensional counter terms have a profound effect: For generic positive values of the true minimum of the SM effective potential is pushed to sub-Planckian scales. It is generically metastable or even stable for a large range of initial values of at the scale . See Figure 1 for generic configurations of with and without gravitational contributions. On top, quantum gravity effects contribute to the effective potential at NLO with orders of magnitude at the percent level.

## Standard model coupled to Einstein gravity.

We consider the standard model coupled to gravity with a cosmological constant :222Note that we could also include an additional non-minimally curvature-coupled scalar term into the Lagrangian. However, the freedom of Higgs-field redefinitions and Weyl rescalings may be used to set in (1) [15] at the cost of introducing an additional kinetic term. Here we set from the start.

 LSMgrav=2κ2√−gR+LGB+LF+√−g(Λ+gμν∂μH∂νH†+m2|H|2−λ|H|4). (1)

Here denotes the dimensionful gravitational coupling constant related to Newton’s constant . As we are interested in the quantum gravitational contributions to the Higgs effective potential at the one-loop order, it is sufficient to study the Higgs-gravity sector of the above model since the gauge bosons in and the matter fermions in decouple at this leading perturbative order. Moreover, due to the non-renormalizability of gravity we will be forced to include higher dimension scalar operators as counterterms

 LCT=√−g(−η1κ2|H|6−η2κ4|H|8), (2)

carrying their own dimensionless bare couplings and . We seek the one-loop corrections to the SM effective potential , where we expand the Higgs doublet about the constant real background field with and . The tree-level Higgs potential is then given by

 Vtree(ϕ)=−m22ϕ2+λ4ϕ4+η18κ2ϕ6+η216κ4ϕ8. (3)

In the gravitational sector we work in de Donder gauge expanding the metric field as . This yields the following standard expansions of the Einstein–Hilbert and de Donder gauge fixing terms :

 2κ2√−g R+Lg.f.=12hαβPαβ;γδ∂2hγδ+O(h3), (4) √−g =1+κ2hαβηαβ+κ24hαβPαβ;γδhγδ+O(h3), Pαβ;γδ =12(ηαβηγδ−ηαγηδβ−ηαδηγβ).

For the quadratic fluctuations of the graviton and Higgs field in our model described by (1) we then have

with the effective masses

 m2A =κ24(−m2ϕ2+12λϕ4+14κ2η1ϕ6+18κ4η2ϕ8+2Λ), m2B =κ2(−m2ϕ+λϕ3+34κ2η1ϕ5+12κ4η2ϕ7), m2C =−m2+3λϕ2+154κ2η1ϕ4+72κ4η2ϕ6, m2D =−m2+λϕ2+34κ2η1ϕ4+12κ4η2ϕ6. (6)

We see that the graviton obtains a small mass generated by the Higgs field. While beyond the scope of this letter, it would be important to further analyze the SM coupled to gravity in the light of different Higgs mechanisms for the graviton discussed in e.g. [16, 17].

Here we proceed by writing with the collective quantum field . We perform the path integral at 1-loop order to find the gravitational contributions to the effective 1-loop Higgs potential, where333We have .

 ΔV[ϕ]=−i2¯μ4−d∫ddp(2π)d(9ln(p2+m2A)+3ln(p2−m2D)+ln[(p2+m2A)(p2−m2C)−4m4B]). (7)

The relevant dimensionally regularized integral reads

 (8)

with 444We stress the use of dimensional regularization here, which only sees logarithmic divergences. A (naive) cutoff regularization of the integral would also induce a power divergence in (8). However, this term does not contribute to the -functions in perturbatively renormalized effective field theory. Nevertheless, in the framework of the functional renormalization group approach this term contributes, c.f. [18].. The pole in yields a renormalization of the scalar field couplings and . Proceeding in the scheme gives the following gravitational contribution to the renormalized effective potential of the standard model:

 ΔV [ϕ]=964π2m4A(lnm2Aμ2−32)+3m4D64π2(lnm2Dμ2−32)+∑i=±C2i64π2(lnCiμ2−32), (9)

where for conciseness we have defined555Note that the term under the square root is always positive at on the parameter space of .

 C±=12(m2C−m2A±√(m2C+m2A)2−16m4B). (10)

## The β-functions.

Adding the counterterms necessary to absorb the -pole terms to the bare Lagrangian yields the renormalized Lagrangian. An equivalent statement is that the effective potential obeys a renormalization group equation (RGE)

 (μ∂∂μ+∑iβi∂∂λi−γϕϕ∂∂ϕ)Veff(ϕ)=0, (11)

where are the -functions of the couplings and is the anomalous dimension of the SM Higgs field. From (9) one thus establishes the one-loop -functions of the novel couplings , which take the following form in the limit:

 β(1)η1 =6η1γ(1)ϕ+116π2[108λη1−8λ2], (12) β(1)η2 =8η2γ(1)ϕ+116π2[192λη2+126η21+54λ2−24η1λ], γ(1)ϕ =116π2[3y2t−34g21−94g22].

The scaling dimension contains the top-Yukawa and electroweak coupling constants . We stress that the term in as well as the and terms in are quantum gravity induced contributions despite the fact that they are not proportional to , see Figure 2.

The remaining non-gravitational term in has been reported in [19], the non-gravitational terms in were considered in [20], however we disagree with the numerical factor for the term given there. We hence see that even in the absence of higher dimensional operators () at an initial scale, these terms will be created in the renormalization group flow.666We assume that contributions to the RG-flow from other higher dimensional operators in the effective field theory are consistently neglectable at this order of perturbation theory. It would be important to investigate this point in full detail, cf. e.g. [19, 21]. Including the Higgs mass and cosmological constant in the analysis leads to quantum gravitational contributions of order and to all the -functions including and . However, these terms are of order or less and absolutely negligible. We therefore set in the remaining analysis. This also puts the electroweak minimum to zero, i.e. .

## Consistent perturbation theory.

The renormalization group improved SM effective potential is traditionally written in the form

 VSM(ϕ)=λeff(μ=ϕ)ϕ44, (13)

with the effective field-dependent coupling constant

 λeff(ϕ)=e4Γ(ϕ)[λ(μ)+λ(1)%eff(μ)+λ(2)eff(μ)]∣∣μ=ϕ, (14)

where . The explicit expressions for the corrections to the effective coupling up to three loops are given in the appendix of [4]. We follow a consistent use of perturbation theory along the lines of [22, 23, 11] assuming . This is necessary in order for the tree-level term to receive non-negligible corrections by the one-loop contribution scaling as . Perturbation theory in is applicable, however, it is not the usual loop expansion. The expansion of the tree, one- and two-loop contributions of (13) along these lines yields the expansion up to next-to-leading order:

 VNLOSM(ϕ)=VLOSM(ϕ)+V(NLO)SM(ϕ), (15)

where scales as and as .

Let us now add the gravitational contributions to this picture. In analogy to the SM case above we write

 Vgrav(ϕ)=ηeff(μ=ϕ)ϕ44, (16)

with the RG-improved effective coupling

 ηeff(ϕ)= (17)

Here can be extracted from (9). To obtain a consistent perturbative expansion we assume and . Expanding in then yields the gravitational correction to the SM effective potential up to next-to-leading order :

 VNLOgrav(ϕ)=VLOgrav(ϕ)+V(%NLO)grav(ϕ). (18)

Both, and have nonzero imaginary parts and in the following we will restrict to the real part of the potential, referring to [24] for an interpretation of the imaginary contribution.

## Minimum of the effective potential.

The full leading order (LO) potential reads777See [25] for a good review on effective potentials in the context of the SM.

 VLO(ϕ)=Vtree(ϕ)−364y4tϕ4logy2tϕ22μ2+(g1,g2 terms),

where we suppress the numerically small gauge coupling terms for brevity. The position of the true minimum is then the second nontrivial solution of with flowing couplings, guaranteeing gauge invariance of the minimum, c.f. [23, 11]. At NLO for the full effective potential we then have

 Vmin=VLO(ϕmin)+V(NLO)SM(ϕmin)+V(NLO)grav(ϕmin). (19)

In Figure 3 we plot the explicit data for as a function of the initial values and at the top mass scale. Here we use the initial conditions provided in [4, 11], in particular and , the available higher loop RG equations for the SM and our one-loop RGEs for . In large regions of the parameter space, the value at the minimum is well approximated by the three planes depicted in the double logarithmic plot in Figure 3 (bounded by the four (red) lines):

 Vabmin ≃−2η21κ4108, Vbcmin ≃−1η2κ4105, Vcdmin ≃−1η21κ4105.

Here the lines bounding the planes are parametrized by (a=d) , (b)  for , (c)  for . Similar values were reported in [11] for and non-flowing . The minimum disappears approximately on the lines and in the -plane. The value turns positive shortly before, at around and . Such astronomically high values for should be read as measures of scales when new physics arises in the sense of and . In that sense the above thresholds reflect the existence of an intriguing stability scale  GeV. Finally, for positive the minimum lies beyond the Planck scale.

Let us briefly discuss the order of magnitude of the NLO gravity contributions to the effective potential. The ratio evaluated at the minimum ranges between zero and ten percent in large regions of the parameter space. At the boundaries, for large initial or , the minimal value changes sign as indicated above and the relative gravitational contribution becomes large.

## Impact on vacuum stability.

We have seen that for a huge range of the parameter space of , the SM coupled to gravity develops a negative minimum below the electroweak one. The lifetime in units of the age of the universe of the false vacuum is usually estimated in the SM by [26] In the presence of the dimension six and eight counterterms this equation needs to be modified. Numerical studies [27, 28] indicate that is still well approximated by simply replacing in the exponent of the lifetime expression by . In Figure 4 we show the lifetime of the false electroweak vacuum as a function of the initial values for at the top mass scale using this lifetime approximation. We see that the SM coupled to gravity generically yields a highly metastable vacuum for a huge part of the parameter space in . Notably, the instability only occurs for negative , i.e. . Moreover, we find that the ultrashort lifetimes reported in [27] are confined to a very small (black) region in parameter space once the RG-flow is taken into account.

Certainly the lifetime formula employed should be taken cautiously as it does not include curvature effects [15, 29]. In regions where comes close to the assumption of a flat background metric turns inconsistent. In these regions a full analysis expanding the metric around curved backgrounds should be employed (cf. e.g. [30] in this context).

Beyond the instability region derived above the values of the novel couplings and are not restricted by present observational data, as so far only single Higgs interactions have been probed experimentally (see e.g. [31] for a recent analysis)888Cf. [32] for a lattice approach towards restrictions on .. Therefore also the exponentially large values explored here are not excluded and remain perturbative.

Finally, we note that the reported results on the values of and the lifetimes are sensitive to the values of and even at the level of a 2 GeV variation. Mapping this out is left for future work.

## Acknowledgements.

We thank P. Galler, S. Huber, K. Meissner, H. Nicolai, M. Shaposhnikov and P. Uwer for discussions.

## References

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