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

\pdfbookmark[1]Title Pagetitle

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

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].^{1}^{1}1Of 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 :^{2}^{2}2Note 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.

(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

(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

(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 :

(4) | ||||

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

(5) | ||||

with the effective masses

(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, where^{3}^{3}3We have .

(7) |

The relevant dimensionally regularized integral reads

(8) |

with ^{4}^{4}4We 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:

(9) |

where for conciseness we have defined^{5}^{5}5Note that the term under the square root is always positive at on the parameter space of .

(10) |

## The -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)

(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:

(12) | ||||

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.^{6}^{6}6We 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

(13) |

with the effective field-dependent coupling constant

(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:

(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

(16) |

with the RG-improved effective coupling

(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 :

(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 reads^{7}^{7}7See [25] for a good review on effective potentials in the context of the SM.

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

(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):

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)^{8}^{8}8Cf. [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

- [1] ATLAS Collaboration Collaboration, G. Aad et al., “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC”, Phys. Lett. B716, 1 (2012), arxiv:1207.7214.
- [2] CMS Collaboration Collaboration, S. Chatrchyan et al., “Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC”, Phys. Lett. B716, 30 (2012), arxiv:1207.7235.
- [3] G. Degrassi, S. Di Vita, J. Elias-Miro, J. R. Espinosa, G. F. Giudice et al., “Higgs mass and vacuum stability in the Standard Model at NNLO”, JHEP 1208, 098 (2012), arxiv:1205.6497.
- [4] D. Buttazzo, G. Degrassi, P. P. Giardino, G. F. Giudice, F. Sala et al., “Investigating the near-criticality of the Higgs boson”, JHEP 1312, 089 (2013), arxiv:1307.3536.
- [5] C. Froggatt and H. B. Nielsen, “Standard model criticality prediction: Top mass 173 +- 5-GeV and Higgs mass 135 +- 9-GeV”, Phys. Lett. B368, 96 (1996), hep-ph/9511371.
- [6] K. A. Meissner and H. Nicolai, “Conformal Symmetry and the Standard Model”, Phys. Lett. B648, 312 (2007), hep-th/0612165.
- [7] M. Shaposhnikov, “Is there a new physics between electroweak and Planck scales?”, arxiv:0708.3550.
- [8] M. Shaposhnikov and C. Wetterich, “Asymptotic safety of gravity and the Higgs boson mass”, Phys. Lett. B683, 196 (2010), arxiv:0912.0208.
- [9] M. Holthausen, K. S. Lim and M. Lindner, “Planck scale Boundary Conditions and the Higgs Mass”, JHEP 1202, 037 (2012), arxiv:1112.2415.
- [10] F. Bezrukov, J. Rubio and M. Shaposhnikov, “Living beyond the edge: Higgs inflation and vacuum metastability”, arxiv:1412.3811.
- [11] A. Andreassen, W. Frost and M. D. Schwartz, “Consistent Use of the Standard Model Effective Potential”, Phys. Rev. Lett. 113, 241801 (2014), arxiv:1408.0292.
- [12] J. F. Donoghue, “General relativity as an effective field theory: The leading quantum corrections”, Phys. Rev. D50, 3874 (1994), gr-qc/9405057.
- [13] E. Greenwood, E. Halstead, R. Poltis and D. Stojkovic, “Dark energy, the electroweak vacua and collider phenomenology”, Phys. Rev. D79, 103003 (2009), arxiv:0810.5343.
- [14] V. Branchina and E. Messina, “Stability, Higgs Boson Mass and New Physics”, Phys. Rev. Lett. 111, 241801 (2013), arxiv:1307.5193.
- [15] G. Isidori, V. S. Rychkov, A. Strumia and N. Tetradis, “Gravitational corrections to standard model vacuum decay”, Phys. Rev. D77, 025034 (2008), arxiv:0712.0242.
- [16] G. ’t Hooft, “Unitarity in the Brout-Englert-Higgs Mechanism for Gravity”, arxiv:0708.3184.
- [17] A. H. Chamseddine and V. Mukhanov, “Higgs for Graviton: Simple and Elegant Solution”, JHEP 1008, 011 (2010), arxiv:1002.3877.
- [18] A. Eichhorn, H. Gies, J. Jaeckel, T. Plehn, M. M. Scherer et al., “The Higgs Mass and the Scale of New Physics”, arxiv:1501.02812.
- [19] E. E. Jenkins, A. V. Manohar and M. Trott, “Renormalization Group Evolution of the Standard Model Dimension Six Operators I: Formalism and lambda Dependence”, JHEP 1310, 087 (2013), arxiv:1308.2627.
- [20] Z. Lalak, M. Lewicki and P. Olszewski, “Higher-order scalar interactions and SM vacuum stability”, arxiv:1402.3826.
- [21] C. P. Burgess, S. P. Patil and M. Trott, “On the Predictiveness of Single-Field Inflationary Models”, JHEP 1406, 010 (2014), arxiv:1402.1476.
- [22] S. R. Coleman and E. J. Weinberg, “Radiative Corrections as the Origin of Spontaneous Symmetry Breaking”, Phys. Rev. D7, 1888 (1973).
- [23] A. Andreassen, W. Frost and M. D. Schwartz, “Consistent Use of Effective Potentials”, Phys. Rev. D91, 016009 (2015), arxiv:1408.0287.
- [24] E. J. Weinberg and A.-Q. Wu, “Understanding complex perturbative effective potentials”, Phys. Rev. D36, 2474 (1987).
- [25] A. Andreassen, “Gauge Dependence of the Quantum Field Theory Effective Potential”, Master’s thesis, NTNU-Trondheim D36, A. Andreassen (2013).
- [26] G. Isidori, G. Ridolfi and A. Strumia, “On the metastability of the standard model vacuum”, Nucl. Phys. B609, 387 (2001), hep-ph/0104016.
- [27] V. Branchina, E. Messina and A. Platania, “Top mass determination, Higgs inflation, and vacuum stability”, JHEP 1409, 182 (2014), arxiv:1407.4112.
- [28] V. Branchina, E. Messina and M. Sher, “The lifetime of the electroweak vacuum and sensitivity to Planck scale physics”, arxiv:1408.5302.
- [29] P. Burda, R. Gregory and I. Moss, “Gravity and the stability of the Higgs vacuum”, arxiv:1501.04937.
- [30] E. Elizalde and S. Odintsov, “Renormalization group improved effective potential for gauge theories in curved space-time”, Phys. Lett. B303, 240 (1993), hep-th/9302074.
- [31] A. Falkowski, F. Riva and A. Urbano, “Higgs at last”, JHEP 1311, 111 (2013), arxiv:1303.1812.
- [32] D. Y. J. Chu, K. Jansen, B. Knippschild, C. J. D. Lin and A. Nagy, “A lattice study of a chirally invariant Higgs-Yukawa model including a higher dimensional -term”, arxiv:1501.05440.

[1]Referencesreferences