Asymptotic safety and Conformal Standard Model

Frederic Grabowski1, Jan H. Kwapisz2,∗ and Krzysztof A. Meissner21 Faculty of Mathematics, Informatics and Mechanics,

University of Warsaw ul. Banacha 2, 02-093 Warsaw, Poland and2 Faculty of Physics, University of Warsaw ul. Pasteura 5, 02-093 Warsaw, Poland

(Dated: June 30, 2020)

We show that the Conformal Standard Model supplemented with asymptotically safe gravitycan be valid up to arbitrarily high energies and give a complete description of particle physicsphenomena. We restrict the mass of the second scalar particle to ∼ 300 GeV and the masses ofheavy neutrinos to ∼ 340 GeV. These predictions can be explicitly tested in the nearby future.

I. INTRODUCTION

In the recent years there were many extensions of theStandard Model (SM) proposed to deal with the SMdrawbacks like triviality, too weak CP violation forbaryogenesis, hierarchy problem and also with lack ofdark matter candidates. Examples of these extensionsinclude Grand Unification Theories (GUT) [1, 2], super-symmetric models [3–5], Higgs portal models [6, 7] or theConformal Standard Model [8–10]. Usually free parame-ters of these models, like masses of new proposed parti-cles, are known neither theoretically nor experimentally.Possibility of narrowing them down, using some theoret-ical reasoning, to a small interval would be a huge ad-vantage in the search for new particles and interactions.

One way to restrict the low energy values of cou-plings is to include the gravitational corrections to theβ functions and demand the asymptotic safety condi-tion (AS) on the running of renormalisation group equa-tions (RGE) for given initial conditions [11]. These con-ditions impose bounds on the model free parametersvalues, which are initial conditions for RGE. Then themodel which satisfies them is UV fundamental. How-ever this reasoning is valid only if gravity indeed has anon-perturbative asymptotically safe fixed point. Thishypothesis [12] is currently under investigation, howeverthere are strong indications that it is really so, see for ex-ample [13–15]. Much of the work is done using the non-perturbative Functional Renormalisation Group methods[16–18]. While the Wetterich equation for flowing ac-tion Γk is exact (from which one gets the effective actionΓ = limk→0 Γk), however it is very difficult to deal with.To solve it and hence find the effective action, one hasto restrict himself to the finite array of couplings. This,non-perturbative, approach gave the promising results onmass difference of charged quarks [19] or on explanationof the top mass [20]. On the other hand the perturbativeapproach was used to predict the Higgs mass [11] withastonishing accuracy.In this article we analyse the Conformal Standard Model(CSM) [8–10], which is of Higgs portal type, but has ad-ditional structure. This model extends SM by adding

∗ Corresponding author: Jan.Kwapisz@fuw.edu.pl

one new complex scalar field with its phase as a darkmatter candidate and right-chiral neutrinos. The funda-mental assumption which underlies this model and othersimilar models is that there is no new physics betweenthe weak scale and the Planck scale. This mean thatfor example the masses of heavy neutrinos or vacuumexpectation value of new scalars should be of order of1 TeV. Indeed the observational abbreviations from theStandard Model such as neutrino masses and oscillations,dark matter and dark energy, baryon asymmetry of theUniverse and inflation can be understood without intro-ducing an ntermediate new scale, see for example [21, 22].The introduction of the gravitational contributions to thematter beta functions makes all the matter couplings goto non-interacting fixed point at roughly Planck scale,hence at least theoretically it is unnecessary to intro-duce the new degrees of freedom at some intermediatescale, see [23], in order to make Standard Model a CFTat high energies. Moreover the Large Hadron Collider(LHC) hasn’t detected any discrepancies from the Stan-dard Model, with no signs of supersymmetry. This iswhy the Higgs portal models and Conformal StandardModel attract a lot of attention as they can deal, in prin-ciple, with the drawbacks of the Standard Model with-out changing its structure deeply and adding only a fewparticles and interactions to the SM. Such models don’tposses any higher dimension operators in the Lagrangian,since they are the negative dimensional operators. In theFunctional Renormalisation Group they should be takeninto account. Moreover they can can affect the Higgssector however none of these effects have been confirmedyet, see for example [24–26]). Hence in our article we deal(we truncate only to the renormalisable operators) onlywith those matter operators which are in the Lagrangianof the Conformal Standard Model.By taking into account the gravitational corrections tothe beta functions and using the AS conditions (and as-suming that Weinberg hypothesis holds) we are able tocalculate the allowed range of Higgs and the second scalarcoupling parameters such that the CSM can be a UVcomplete theory. By taking into account the experimen-tal LHC data and the model restrictions for the valuesof free parameters we are able to predict the allowedsecond scalar mass. We can also narrow down massesof right-chiral neutrinos. One should also mention thatthere are some studies on models with asymptotically

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II. MODEL AND METHOD

A. Higgs portal models and Conformal StandardModel

In this paragraph we briefly introduce the ConformalStandard Model. The CSM lagrangian [10] is given by

L = Lkin + LY − V, (1)

with the following kinetic terms

Lkin = LSMkin + (DµH)†(DµH) + (∂µφ∗∂µφ)

+iNj

ασµαβ∂µN

jβ , (2)

where H is the SM scalar SU(2) doublet. The φ(x) isa gauge sterile complex scalar field carrying the leptonnumber, and couples only to gauge singlet neutrinos N i

α.The LSMkin contains all the non-scalar Standard Model de-grees of freedom and can be found in any QFT textbooks,see for example [31, 32]. The potential reads as

V (H,φ) = −m21H†H −m2

2φ?φ+ λ1(H†H)2

+λ2(φ?φ)2 + 2λ3(H†H)φ?φ. (3)

For the potential (3), if the vacuum expectation valuesare:

√2〈Hi〉 = vHδi2,

√2〈φ〉 = vφ, (4)

then the tree-level mass parameters are:

m21 = λ1v

2H + λ3v

2φ, (5)

m22 = λ3v

2H + λ2v

2φ. (6)

Hence the model possesses two particles of masses:m1,m2, where m1 is identified with the Higgs particlemass. The potential is bounded from below if:

λ1 > 0, λ2 > 0 λ3 > −√λ2λ1. (7)

If, in addition to (7) the λ3 <√λ1λ2 holds, then (4) is

the global minimum of V . For given λ1, λ2, λ3 one cancalculate vφ and m2 at tree level using (5) and at looplevel by changing the renormalisation schemes, becausevH is known experimentally and the Higgs mass resultingfrom AS can be calculated, when we take λ3 = 0. With

λ0 = 12m2

1

v2H≈ 0.13, one can parametrize the deviation

from SM as:

tanβ =λ0 − λ1λ3

vHvφ. (8)

We assume that its value is restricted by | tanβ| < 0.35what is the limit allowed by the present LHC data [10].

This constraint will be used to narrow down the possi-ble values of m2. In the Conformal Standard Model thecoupling constants YMij and Y νij are introduced, which areresponsible for interactions of right-chiral neutrinos:

LY =1

2YMji φN

jαN iα+Y νjiN

jαH>εLiα+LSMY +h.c., (9)

where LSMY is Yukawa part of the Standard Model La-grangian part and ε is the antisymmetric SU(2)L metric.Following [10] we assume the degeneracy of Yukawa cou-plings YMij = yMδij , which amplifies the CP violationand makes the resonant leptogenesis scenario possible,see [9, 10, 33] for details. The masses of right-chiral neu-trinos are given by:

MN = yMvφ/√

2, (10)

for leptogenesis to take place, one requires: MN > m2, sothat the heavy neutrinos can decay. Moreover the Con-formal Standard Model introduces a phase of the secondscalar particle, called minoron, which can be a poten-tial dark matter candidate, with mass v4/M2

P originat-ing from quantum gravity effects, where v is some newparameter with v ∼ vφ. The CSM beta functions in the

MS-scheme, where βCSM = 16π2βCSM , are given by[10]:

βg1 = 416 g

31 , βg2 = − 19

6 g32 , βg3 = −7g33 ,

βyt = yt(92y

2t − 8g23 − 9

4g22 − 17

12g21

),

βλ1 = 24λ21 + 4λ23 − 3λ1(3g22 + g21 − 4y2t

)+ 9

8g42 + 3

4g22g

21 + 3

8g41 − 6y4t ,

βλ2=(20λ22 + 8λ23 + 6λ2y

2M − 3y4M

),

βλ3 = 12λ3 [24λ1 + 16λ2 + 16λ3

−(9g22 + 3g21

)+ 6y2M + 12y2t

],

βyM = 52y

3M ,

(11)

where g1, g2, g3 are U(1), SU(2), SU(3) Standard Modelgauge couplings respectively, yt is the top Yukawa cou-pling. Following [10] we don’t take into account the run-ning of Y ν . If one takes yM = 0 then after the redefini-tions of the couplings the CSM beta functions reduce tothe Higgs portal ones with [6, 34–38].

B. Asymptotic safety and gravitational corrections

The Weinbergs’ notion of asymptotic safety (AS) [12]can be summarised by his quote: “A theory is said to beasymptotically safe if the essential coupling parametersapproach a fixed point as the momentum scale of theirrenormalisation point goes to infinity.” Let us assumethat g is a set of all the couplings of a theory and let g|µ0

be a given a set of initial conditions at some momentumscale µ0. In our case we take: µ0 = 173.34 GeV, see[39, 40]. These initial conditions together with the set of

3

equations for running of couplings

βi(g(µ)) = µ∂

∂µgi(µ), (12)

describe completely and uniquely the behaviour of aphysical theory. If for some gi we have βi(g

∗) = 0, thenwe call this g∗ a fixed point of the i-th equation. Thestable fixed points are called attractors, the unstable oneare called repellers. If lim

µ→+∞g∗ ≡ 0 for all the couplings

gi we call such a point a Gaussian fixed point. Theo-ries where the couplings posses a Gaussian fixed at UVscales are called asymptotically free. Otherwise, wheng∗ 6= 0 we call such fixed point non-Gaussian/interactingand such theories are called asymptotically safe. If equa-tion for running of the coupling gi possesses an unstablefixed points at UV scale, then there is only one low en-ergy initial initial condition gi|µ0

per repeller such thatthe theory is fundamental up to the UV scale.On the other hand gravity cannot be perturbativelyquantized. Nevertheless one can utilize the effective fieldtheory approach for energies below Planck scale to de-termine predictions of quantum gravity. In particularthe Standard Model (and its extensions) β functions aremodified with the gravitational corrections at high ener-gies [11, 13, 39, 41–45]:

βi(g) = βmatteri (g) + βgrav

i (g, µ). (13)

The gravitational contributions to the beta functionsacquire the general form for all the matter couplings[11, 13, 39, 41, 45]:

βgravi (g, µ) =

ai8π

µ2

M2P + ξ0µ2

gi, (14)

due to the universal character of gravitational interac-tions. The MP = (8πGN )−1/2 = 2.4 × 1018 GeV is thelow energy Planck mass. The ξ0 is some dimensionlessconstant. Based on the results Functional Renormalisa-tion Group investigation of pure gravity [11, 44, 46, 47]its value is taken as ξ0 = 0.024. The ai are dimensionlessconstants and can be calculated for a given coupling gi.According to [39, 41, 42] one have ag1 = ag2 = ag3 = −1and ayt = −0.5 at the one-loop level. Since yM is agauge coupling we assume that also ayM = −1. Thevalue aλ1

= +3 is based on [11, 39, 47, 48]. For sim-plicity we assume that all of the aλi

have the same ab-solute value, which can be supported by the calculationsof the aλ parameters done for the Higgs Portal Models[49]. However this calculation doesn’t take into accountfermions or higher order operators for gravity. For exam-ple the theory with the R2 term in the gravity lagrangiancan be identified with the scalar tensor theory of grav-ity [50, 51] which is used to describe inflation and thisterm has negative aR2 [52] and so does the scalar. Sowe investigate all the possibilities in our article. To showthat the absolute value of aλ2

, aλ3isn’t an important fac-

tor when concerning the possible masses we have scannedover several values lying in the interval |aλ2 |, |aλ3 | ∈ [1, 3].Its change affects the allowed mass very weakly, of theorder of ±2 GeV, this is because the gravitational cor-rections are heavily suppressed below the Planck scale.Indeed, the sign of ai is much more important than itsexact value because the positive sign corresponds to theunstable Gaussian fixed point while the negative to thestable one. In result we investigate the four possibilities:aλ2

, aλ3= ±3.

The asymptotic safety assumption for quantum gravityallows us treat (extensions of) the Standard Model asfundamental (UV complete) only under the conditionthat the running of coupling constants doesn’t possessany pathological behaviour up to the Planck scale. Itimposes two conditions [10, 11]. We will call them asymp-totic safety (AS) conditions. Firstly, there should be noLandau poles up to the Planck scale. Secondly, the elec-troweak vacuum should be stable for all scales:

λ1(µ) > 0, λ2(µ) > 0, λ3(µ) > −√λ2(µ)λ1(µ). (15)

The second condition comes from the assumption thatthere is essentially no new physics between EW scaleand Planck scale, despite the one described by ConformalStandard Model. Obviously at Planck scale all the mat-ter couplings goes to zero, and hence all the beta func-tions goes to zero. The next paragraph is dedicated tothe calculation of the lambda-couplings (λ1, λ2, λ3, yM )satisfying these conditions.

III. CALCULATION OF LAMBDA COUPLINGS

In this paragraph we calculate the set of allowedlambda-couplings satisfying the asymptotic safety con-ditions. If not specified otherwise, the value of a cou-pling (for example on the plots below) means its value atµ0 = 173.34. In the low energy regime the graviton loopscan be neglected [11, 53, 54] and they become importantnear the Planck scale and manifest in the form of thegravitational corrections to the β functions. The gaugeand Yukawa couplings renormalisation group equationsdynamics is not affected by the running of λi and yM .The low-energy values at µ0 = 173.34 are taken as [39]:g1(µ0) = 0.35940, g2(µ0) = 0.64754, g3(µ0) = 1.1888 andyt(µ0) = 0.95113. Then the evolution of these couplingswith energy is obtained. The running of yM is also inde-pendent from all other couplings. So far the experimentalvalue for yM is unknown, so to obtain the allowed λi(yM )one has to scan over all possible values of yM . Using theAS conditions we have calculated the allowed interval ofyM for ayM = −1 as yM ∈ [0.0, 0.925], for ayM = +1we have yM = 0.0. We plug the evolution of the gaugeand Yukawa couplings and each allowed yM into coupledequations for λ1, λ2, λ3. Moreover, the magnitudes ofai coefficients are such that the theory becomes asymp-totically free near the Planck scale, see [11] for further

4

details, which justifies the use of perturbation approach.Furthermore we expect that if the cosmological constantruns then it does not affect the matter couplings belowthe Planck scale. This reasoning is supported by thefact that both Higgs mass [11] and mass difference be-tween top and bottom quark were accurately predicted[55] without taking into account the running of cosmolog-ical constant. Furthermore in the case of the unimodulargravity [56], which is equivalent to the Einstein theoryat the classical level, the cosmological constant isn’t adynamical degree of freedom hence it doesn’t run at all.These arguments suggest that we don’t take the cosmo-logical constant running into account. As a result we arelooking for the sets of initial values yM , λi such that theywill all drop to zero near the Planck scale.

A. Coefficients: aλ3 = −3, aλ2 = −3

We start with the most general case, when aλ2 = aλ3 =−3. Since for each possible combination of λ2, λ3, yMonly one λ1 is allowed, then we consider a set M of al-lowed λ2, λ3, yM (λ2 and λ3 depend mutually on eachother and on yM ), and there is a λ1(λ2, λ3, yM ) assignedto each of the points of this set. This set looks roughlylike a sea wave, where yM is the height. On the Fig. 1 weshow the surfaces of maximal and minimal possible yMall the points in between are also in the set M.

FIG. 1: Maximal (left) and minimal (right) yM (λ3, λ2),aλ2

= −3, aλ3= −3

On the right figure there is a special line λ†2 = 0.2.

Namely, for the region where λ2 < λ†2 the minimal yM is

0.0, while when λ2 > λ†2 the minimal and maximal valuesare very close to each other: yM (max)−yM (min) ∼ 0.01.This behaviour can be explained by the observation that

λ2 > λ†2 requires yM > 0.70 to keep βλ2small enough

throughout the evolution. As we have checked in allother cases the sets of allowed couplings form a 1D or2D subsets of M.The λ1 renormalisation group equation is affected di-

rectly only by λ3, however the running of λ3 dependson λ2 and yM . On the Fig. 2 we show the λ1 dependenceon other couplings for three chosen yM as an example.

FIG. 2: Values of λ1 for yM = 0.0 (left)and yM = 0.5 (mid), yM = 0.77 (right)

B. Coefficients: aλ3 = −3, aλ2 = +3

In this case there is one allowed λ1 and λ2 per set of λ3and yM satisfying the AS conditions. On the Fig. 3 wepresent this dependence. As we can see there is much lessspread in possible values of λ1 than λ2. This is becauseall the gauge coupling initial values are fixed for λ1, whichis not the case for λ2.

FIG. 3: λ1 dependence on λ3 and yM ,aλ2 = +3, aλ3 = −3

For this case the domain of allowed λ3 and yM issmaller than for M. However for this domain we have:λ2(aλ3

= +3, λ3, yM ) = minλ2(λ2(aλ2

= −3, λ3, yM )} .

C. Coefficients: aλ3 = +3, aλ2 = ±3

In this case, we found that only λ3(µ0) = 0 satisfiesthe AS conditions, hence λ3 is zero at all energy scales.Then the φ sector is decoupled from the rest of Stan-dard Model, which makes it a scalar dark matter can-didate, like in [49]. The numerical solution for λ1 givesλ1 = 0.1537, which agrees the Standard Model predic-tions from asymptotic safety, see [11, 39]. The allowedregion for λ2 is determined by the AS conditions for thiscoupling and depends heavily on yM .

5

FIG. 4: Allowed range of λ2 coupling,aλ2 = −3, aλ3 = +3

We checked that the allowed value for λ2 shown on theFig. [4] is the subset of M with the condition λ3 = 0.0.We have also calculated that λ2(aλ2 = +3, yM ) mimicsthe lower bound for λ2(aλ2

= −3, yM ). Moreover thisbound is the same as for the running of λ2 without grav-itational corrections.

IV. THE SECOND SCALAR MASS

In this paragraph we calculate the vφ,m2 and MN .For the decoupled (Standard Model) case one obtainsλ1 = 0.1537 hence the the Higgs mass is given by:

m1 =√

2λ1v2H ≈ 136GeV, (16)

where the λ1 is calculated from the AS requirementsand is in MS scheme. So one can see that the effectsregarding the renormalisation of mass and the ones con-cerning the changing of renormalisation scheme (fromMS to the physical one) have the contribution of or-der of 10 GeV to the masses calculated at the tree levelrelations (without renormalisation of mass), which wasdemonstrated in [11, 39]. Hence due to other much big-ger sources of uncertainties, like the value of yM we as-sume that the CSM tree level relations (5, 6) hold. Thisassumption restricts λ3 to be nonnegative, otherwise onewould obtain negative values for v2φ. sufficient for the cal-culations of the allowed λ couplings. By comparing theone loop [39] and two-loop calculation [11] of λ1, wherethe outcome is almost identical, we conclude that higherthan the first non-trivial order is sufficient for our pur-pose. In order to calculate vφ and m2 at tree level wetake vH = 246 GeV, then we solve the equations (5,6) with m1 = 136 GeV. Obviously we are able to pre-dict the m2 and vφ only in the case when the secondscalar sector is coupled to the SM sector. Hence, werestrict only to the case when: λ3 6= 0. For small λ3tiny changes in λ1 results in enormous changes in vφ,

so we treat λ3 < 0.01 as a decoupled case. Our claimthat λ3 should be large enough can also be justified withthe LHC condition | tanβ| < 0.35, because small λ3 re-sults in large β. In our analysis we have excluded all thesets of parameters not satisfying the LHC and the globalstability conditions (at µ0:

√λ1(µ0)λ2(µ0) > λ3(µ0)).

After doing that we have two separate cases: aλ2 = +3and aλ2 = −3. Moreover if the second scalar particle isunstable (following CSM [10]), then at the tree level wehave: 2m1 < m2. Also the yM = 0.0 situation is an inter-esting situation, when CSM reduces to the Higgs portalcase. The masses of right handed neutrinos are can alsobe calculated, MN = yMvφ/

√2 in case when yM 6= 0.

Below we illustrate these combinations of possible condi-tions.

TABLE I: Arrays of allowed masses

aλ2 2m1 < m2 yM m2 [GeV] vφ [GeV] MN [GeV]+3 yes 0.84 275 538 319+3 yes 0.85 296 574 345

−3 yes 0.77+0.07−0.06 300+28

−28 586+60−46 342+41

−41−3 no 0.00 160+103

−100 300+275−15 NA

In the second row “yes” means that we’ve taken intoaccount this condition, while “no” means the opposite.As we can see for yM = 0.0 the second scalar mass is m2

is smaller than 2m1 making this particle stable. In theConformal Standard Model case the right-chiral neutri-nos turns out to be unstable (m2 < yMvφ/

√2 holds for

each of the sets of parameters) making the leptogenesisscenario possible. Furthermore if we assume that v = vφ,then the mass of the minoron is given by: v2φ/MP ≈ 10−4

eV, which is of the same order as estimated in [10].We have also analyzed the running of the β functions

for m2 and m1, where we took ami = −1. It givesno new bounds on m2 and lambda-couplings. This re-sult is in close relation with scenario B analysed in [49],however in our case the φ particle isn’t decoupled fromSM. We have also taken into account the restrictionsfrom quadratic divergences cancellation, which can beachieved if Softly Broken Conformal Symmetry [53] re-quirements are satisfied (these requirements in CSM arenot essentially changed even after adding gravitationalcorrections [54]). Apparently it gives no new restrictionsfor the parameters additional to the asymptotic safetyconditions. Moreover the SBCS requirements are satis-fied at E = MP with all the couplings becoming asymp-totically free, which was suggested in [53].

V. CONCLUSIONS

Our investigation shows that the Conformal Stan-dard Model has a set of parameters compatible with theasymptotic safety conditions, so supplemented with grav-itational corrections it can be a UV fundamental the-ory and [10]: “allows for a comprehensive treatment of

6

all outstanding problems of particle physics.” Moreover,CSM can be slightly modified to incorporate an inflationscenario [38, 57], which agrees with 2013 Planck dataanalysis [58–60]. Inflation can also take place due to theasymptotically safe scenario [61].

The restrictions on the couplings and masses derived inthis article allow us to make theoretical predictions forfree parameters of the models extending the StandardModel. The lambda-couplings can be directly measured(or explicitly calculated from masses of the new scalarparticles) in LHC or indirectly measured in cosmologi-cal observations. In particular our investigation supportsthe claim [62] that the excess of events with four chargedleptons at E ∼ 325 GeV seen by the CDF [63] and CMS[64] Collaborations can be identified with a detection ofa new ‘sterile’ scalar particle proposed by the ConformalStandard Model. On the other hand the compatibilityof the LHC run 1 data with the heavy scalar hypothe-sis was investigated in [65, 66]. The hypothetical heavyboson mass is measured to be around 272 GeV (in the270−320 GeV range). Hence we would like to emphasisethat this experimental analysis agrees with the theoreti-cal range provided by our calculations and with the claim

discussed in [62] at least up to the order of several GeV.The fact that m2 > 2m1 implies the yM 6= 0.0 underliesthe role of right-handed neutrinos not only in contextof smallness of neutrino masses, but also in the case ofasymptotically safe BSM physics, which makes the Con-formal Standard Model unique among the 2HDM andHiggs portal models.

We hope that our analysis will be helpful in search fornew particles at LHC and future colliders and the newscalar particle predicted by this model can be detectedin the nearby future.

ACKNOWLEDGMENTS

We thank Piotr Chankowski for valuable discussions.J.H.K. would like to thank Yukawa Institute for The-oretical Physics for hospitality and support during thiswork. The PL-Grid Infrastructure is gratefully acknowl-edged. K.A.M. was partially supported by the Polish Na-tional Science Center grant DEC-2017/25/B/ST2/00165.J.H.K. was supported by the National Science Centre,Poland grant 2018/29/N/ST2/01743.

[1] H. Georgi and S. L. Glashow, Phys. Rev. Lett. 32, 438(1974).

[2] A. Buras, J. Ellis, M. Gaillard, and D. Nanopoulos, Nuc.Phys. B 135, 66 (1978).

[3] S. Dimopoulos, S. Raby, and F. Wilczek, Phys. Rev. D24, 1681 (1981).

[4] L. Ibanez and G. Ross, Phys. Lett. B 105, 439 (1981).[5] S. Dimopoulos and H. Georgi, Nuc. Phys. B 193, 150

(1981).[6] B. Patt and F. Wilczek, (2006), arXiv:hep-ph/0605188

[hep-ph].[7] M. Shaposhnikov and I. Tkachev, Phys. Lett. B 639, 414

(2006), arXiv:hep-ph/0604236 [hep-ph].[8] K. A. Meissner and H. Nicolai, Phys. Lett. B 648, 312

(2007), arXiv:hep-th/0612165 [hep-th].[9] A. Latosinski, A. Lewandowski, K. A. Meissner, and

H. Nicolai, JHEP 2015, 170 (2015), arXiv:1507.01755[hep-ph].

[10] A. Lewandowski, K. A. Meissner, and H. Nicolai, Phys.Rev. D 97, 035024 (2018), arXiv:1710.06149 [hep-ph].

[11] M. Shaposhnikov and C. Wetterich, Phys. Lett. B 683,196 (2010), arXiv:0912.0208 [hep-th].

[12] S. Weinberg, in General Relativity: An Einstein cente-nary survey, edited by S. W. Hawking and W. Israel(1979) pp. 790–831.

[13] A. Eichhorn, in Black Holes, Gravitational Waves andSpacetime Singularities Rome, Italy, May 9-12, 2017(2017) arXiv:1709.03696 [gr-qc].

[14] O. Lauscher and M. Reuter, in Quantum gravity: Mathe-matical models and experimental bounds (2005) pp. 293–313, arXiv:hep-th/0511260 [hep-th].

[15] A. Salvio and A. Strumia, Eur. Phys. J. C78, 124 (2018),arXiv:1705.03896 [hep-th].

[16] C. Wetterich, Phys. Lett. B 301, 90 (1993),

arXiv:1710.05815 [hep-th].[17] T. R. Morris, Int. J. Mod. Phys. A 9, 2411 (1994),

arXiv:hep-ph/9308265 [hep-ph].[18] T. R. Morris, Phys. Lett. B 329, 241 (1994), arXiv:hep-

ph/9403340 [hep-ph].[19] A. Eichhorn and A. Held, Phys. Rev. Lett. 121, 151302

(2018), arXiv:1803.04027 [hep-th].[20] A. Eichhorn and A. Held, Phys. Lett. B777, 217 (2018),

arXiv:1707.01107 [hep-th].[21] A. Boyarsky, O. Ruchayskiy, and M. Shaposhnikov, Ann.

Rev. Nucl. Part. Sci. 59, 191 (2009), arXiv:0901.0011[hep-ph].

[22] M. Shaposhnikov, in Astroparticle Physics: Current Is-sues, 2007 (APCI07) Budapest, Hungary, June 21-23,2007 (2007) arXiv:0708.3550 [hep-th].

[23] G. Marques Tavares, M. Schmaltz, and W. Skiba, Phys.Rev. D89, 015009 (2014), arXiv:1308.0025 [hep-ph].

[24] B. Grzadkowski, M. Iskrzynski, M. Misiak, andJ. Rosiek, JHEP 10, 085 (2010), arXiv:1008.4884 [hep-ph].

[25] J. Borchardt, H. Gies, and R. Sondenheimer, Eur. Phys.J. C76, 472 (2016), arXiv:1603.05861 [hep-ph].

[26] R. Sondenheimer, (2017), arXiv:1711.00065 [hep-ph].[27] M. Badziak and K. Harigaya, Phys. Rev. Lett. 120,

211803 (2018), arXiv:1711.11040 [hep-ph].[28] G. M. Pelaggi, A. D. Plascencia, A. Salvio, F. Sannino,

J. Smirnov, and A. Strumia, Phys. Rev. D 97, 095013(2018), arXiv:1708.00437 [hep-ph].

[29] R. Mann, J. Meffe, F. Sannino, T. Steele, Z. Wang,and C. Zhang, Phys. Rev. Lett. 119, 261802 (2017),arXiv:1707.02942 [hep-ph].

[30] D. Barducci, M. Fabbrichesi, C. M. Nieto, R. Percacci,and V. Skrinjar, JHEP 11, 057 (2018), arXiv:1807.05584[hep-ph].

7

[31] S. Weinberg, The Quantum Theory of Fields, Vol. 1(Cambridge University Press, 1995).

[32] S. Pokorski, Gauge Field Theories, 2nd ed., CambridgeMonographs on Mathematical Physics (Cambridge Uni-versity Press, 2000).

[33] A. Pilaftsis and T. E. J. Underwood, Nuc. Phys. B 692,303 (2004), arXiv:hep-ph/0309342 [hep-ph].

[34] G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo,M. Sher, and J. P. Silva, Phys. Rept. 516, 1 (2012),arXiv:1106.0034 [hep-ph].

[35] J. F. Gunion and H. E. Haber, Phys. Rev. D67, 075019(2003), arXiv:hep-ph/0207010 [hep-ph].

[36] J. D. Wells, in 39th British Universities Summer Schoolin Theoretical Elementary Particle Physics (BUSSTEPP2009) Liverpool, United Kingdom, August 24-September4, 2009 (2009) arXiv:0909.4541 [hep-ph].

[37] J. O.Gong, H. M. Lee, and S. K. Kang, JHEP 04, 128(2012), arXiv:1202.0288 [hep-ph].

[38] O. Lebedev and H. M. Lee, Eur. Phys. J. C 71, 1821(2011), arXiv:1105.2284 [hep-ph].

[39] L. Laulumaa, Higgs mass predicted from the standardmodel with asymptotically safe gravity, Master’s thesis,Jyvaskyla U. (2016).

[40] D. Buttazzo, G. Degrassi, P. P. Giardino, G. F. Giudice,F. Sala, A. Salvio, and A. Strumia, JHEP 12, 089 (2013),arXiv:1307.3536 [hep-ph].

[41] S. P. Robinson and F. Wilczek, Phys. Rev. Lett. 96,231601 (2006), arXiv:hep-th/0509050 [hep-th].

[42] O. Zanusso, L. Zambelli, G. P. Vacca, and R. Percacci,Phys. Lett. B 689, 90 (2010), arXiv:0904.0938 [hep-th].

[43] L. Griguolo and R. Percacci, Phys. Rev. D 52, 5787(1995), arXiv:hep-th/9504092 [hep-th].

[44] R. Percacci and D. Perini, Phys. Rev. D 68, 044018(2003), arXiv:hep-th/0304222 [hep-th].

[45] A. Eichhorn, Y. Hamada, J. Lumma, and M. Yamada,Phys. Rev. D 97, 086004 (2018), arXiv:1712.00319 [hep-th].

[46] M. Reuter, Phys. Rev. D 57, 971 (1998), arXiv:hep-th/9605030 [hep-th].

[47] R. Percacci and D. Perini, Phys. Rev. D 68, 044018(2003), arXiv:hep-th/0304222 [hep-th].

[48] G. Narain and R. Percacci, Class. Quant. Grav. 27,075001 (2010), arXiv:0911.0386 [hep-th].

[49] A. Eichhorn, Y. Hamada, J. Lumma, and M. Yamada,Phys. Rev. D 97, 086004 (2018), arXiv:1712.00319 [hep-th].

[50] F. L. Bezrukov and M. Shaposhnikov, Phys. Lett. B 659,703 (2008), arXiv:0710.3755 [hep-th].

[51] A. A. Starobinsky, JETP Lett. 30, 682 (1979),[,767(1979)].

[52] A. Eichhorn, (2018), arXiv:1810.07615 [hep-th].[53] P. H. Chankowski, A. Lewandowski, K. A. Meissner,

and H. Nicolai, Mod. Phys. Lett. A 30, 1550006 (2015),arXiv:1404.0548 [hep-ph].

[54] K. A. M. H., Nicolai, and J. Plefka, (2018),arXiv:1811.05216 [hep-th].

[55] A. Eichhorn and A. Held, Phys. Rev. Lett. 121, 151302(2018), arXiv:1803.04027 [hep-th].

[56] A. Eichhorn, Class. Quant. Grav. 30, 115016 (2013),arXiv:1301.0879 [gr-qc].

[57] J. H. Kwapisz and K. A. Meissner, Acta Phys. Polon. B49, 115 (2018), arXiv:1712.03778 [gr-qc].

[58] P. A. R. Ade et al. (BICEP2, Planck), Phys. Rev. Lett.114, 101301 (2015), arXiv:1502.00612 [astro-ph.CO].

[59] A. H. Guth, D. I. Kaiser, and Y. Nomura, Phys. Lett. B733, 112 (2014), arXiv:1312.7619 [astro-ph.CO].

[60] D. I. Kaiser, Fundam. Theor. Phys. 183, 41 (2016),arXiv:1511.09148 [astro-ph.CO].

[61] S. Weinberg, Phys. Rev. D 81, 083535 (2010),arXiv:0911.3165 [hep-th].

[62] K. A. Meissner and H. Nicolai, Phys. Lett. B 718, 943(2013), arXiv:1208.5653 [hep-ph].

[63] T. Aaltonen et al. (CDF), Phys. Rev. D 85, 012008(2012), arXiv:1111.3432 [hep-ex].

[64] S. Chatrchyan et al. (CMS), Phys. Rev. Lett. 108, 111804(2012), arXiv:1202.1997 [hep-ex].

[65] S. von Buddenbrock, N. Chakrabarty, A. S. Cor-nell, D. Kar, M. Kumar, T. Mandal, B. Mellado,B. Mukhopadhyaya, and R. G. Reed, (2015),arXiv:1506.00612 [hep-ph].

[66] S. von Buddenbrock, N. Chakrabarty, A. S. Cor-nell, D. Kar, M. Kumar, T. Mandal, B. Mellado,B. Mukhopadhyaya, R. G. Reed, and X. Ruan, Eur.Phys. J. C76, 580 (2016), arXiv:1606.01674 [hep-ph].