Higgs Mass and Planck Scale

Planck Scale Boundary Conditions and the Higgs Mass

Martin Holthausen,∗ Kher Sham Lim,† and Manfred Lindner‡ Max Planck Institute for Nuclear Physics,Saupfercheckweg 1, 69117 Heidelberg, Germany (Dated: March 5, 2012) If the LHC does only find a Higgs boson in the low mass region and no other new physics, then one should reconsider scenarios where the Standard Model with three right-handed neutrinos is valid up to Planck scale. We assume in this spirit that the Standard Model couplings are remnants of quantum gravity which implies certain generic boundary conditions for the Higgs quartic coupling at Planck scale. This leads to Higgs mass predictions at the electroweak scale via renormalization group equations. We find that several physically well motivated conditions yield a range of Higgs masses from 127−142 GeV. We also argue that a random quartic Higgs coupling at the Planck scale favours MH > 150 GeV, which is clearly excluded. We discuss also the prospects for differentiating different boundary conditions imposed for λ(Mpl) at the LHC. A striking example is MH = 127 ± 5 GeV corresponding to λ(Mpl) = 0, which would imply that the quartic Higgs coupling at the electroweak scale is entirely radiatively generated.

I. INTRODUCTION postulate a new symmetry (Supersymmetry) which cancels the problematic quadratic divergences. Al- The Standard Model (SM) of particle physics is ternatively the scalar sector may be considered ef- very successful and it has withstood all precision fective (composite) such that form factors remove tests over almost 40 years. All SM particles have large quadratic divergences. Another idea is that the been discovered1, except for the ingredient con- Higgs particle could be a pseudo-Goldstone-boson nected to electroweak Symmetry Breaking (EWSB), such that its mass is naturally somewhat lower than namely the Higgs boson. The Large Hadron Col- a scale where richer new physics exists. However, lider (LHC) was built to test EWSB and to ﬁnd or none of these ideas has so far shown up in experi- rule out the Higgs boson. In addition, the LHC aims ments. at detecting new physics which is suggested to ex- This prompts us to consider new ideas. An impor- ist in the TeV range in order to solve the so-called tant observation is the fact that the SM can anyway hierarchy problem. However, the ATLAS and CMS not be valid up to an arbitrary high energy scale detectors at LHC have so far not found any sign due to triviality [3] and since gravity must aﬀect el- of new physics and the remaining Higgs mass range ementary particle physics latest at the Planck scale, 19 has shrunk considerably [1]. This suggests to think Mpl = 1.2 × 10 GeV. Note that this introduces about scenarios where nothing but the SM is seen. a conceptual asymmetry, as gravity is known to be The essence of the hierarchy problem [2] is the non-renormalizable, i.e. it cannot be a QFT in the fact that quantum corrections generically destroy usual sense and requires fundamentally new ingredi- the separation of two scales of scalar Quantum Field ents. This looks bad from the perspective of renor-

arXiv:1112.2415v2 [hep-ph] 2 Mar 2012 Theories (QFT). It is thus not possible to under- malizable gauge theories, also since we can so far stand how the electroweak scale could be many or- only guess which concepts might be at work. How- ders of magnitude smaller than the scale of an em- ever, this may also be good in two ways: First, renor- bedding QFT. Conventional solutions of the hierar- malizable QFTs do not allow to calculate absolute chy problem stay within QFT. One solution is to masses and absolute couplings. Any embedding of the SM into some other renormalizable QFT (like GUTs) would therefore only shift the problem to a

∗ new theory which is also unable to determine the [email protected] absolute values of these quantities. In other words: † [email protected] ‡ [email protected] The problems of gravity may be a sign of physics 1 In addition neutrinos were found to be massive, which re- based on new concepts which may ultimately al- quires in its simplest form only the addition of three right low to determine absolute masses, mixings and cou- handed fermionic singlets. plings. However, there is no need for the SM to be 2 mH (µ) ergy effective theory. The parameters in low energy 6 QFT Beyond Ginzburg-Landau theory “know” the boundary con- regime QFT dition set by the underlying BCS theory, but many dynamical details of the BCS theory are lost in the Imprint of Higgs mass left by quantum gravity Ginzburg-Landau effective theory, even though BCS theory does not provide a mechanism to explain hi- @ Non-field theoretic quantum gravity erarchies. @@R region The above considerations prompt us to specu- late that the SM might be valid up to the Planck 2 v mH (v) scale, where it is embedded directly into Planck scale - physics without any intermediate energy scale. The new concepts behind the Planck scale physics might M µ pl then offer a solution to the hierarchy problem which is no longer visible when one looks at the SM only. In Figure 1. The SM Higgs mass could be determined and analogy to the superconductivity example the only fixed by unknown physics connected to quantum grav- way how the SM would “know” about such an em- ity, which should be based on new concepts other than bedding could be special boundary conditions sim- conventional QFT. The running of the Higgs mass from ilar to compositeness conditions or auxiliary field Planck scale down to electroweak scale is fully dictated conditions in theories where redundant degrees of by the SM as a QFT. freedom are eliminated in embeddings. This scenario is depicted in Fig. (I). Following the logic outlined above it would be es- directly embedded into gravity and various layers sential to have only the weak and the Planck scale of conventional gauge theories (LR, PS, GUT, ...) and nothing in between, since otherwise it would re- could be in-between. The second reason why an em- quire to solve the hierarchy problem within QFT. bedding involving new concepts beyond renormaliz- We therefore forbid any kind of new intermediate able QFTs might be good is that this asymmetry energy scale between weak and Planck scale in or- might offer new solutions to the hierarchy problem. der to avoid the large hierarchy between Higgs and In other words: The conventional approach towards heavy intermediate particle’s mass. This view is also interpreting quadratic divergences to the Higgs mass shared by the spirit of νMSM, proposed by Shaposh- correction by simply substituting Λ2 → M 2 might pl nikov [5]. Even without any intermediate scale one not be correct if Planck scale physics is based on new might still ask why there should not be any radiative physical concepts different from conventional QFT. corrections of Planck scale size to the Higgs mass pa- This view is also shared by Meissner and Nicolai [4]. rameter. We do not have an argument here but we The point is that the unknown new concepts may point the interested reader to the works of Bardeen allow for mechanisms which stabilize a low-lying ef- [6] that have argued about the spurious nature of fective QFT from the perspective of the Planck scale. the quadratic correction. From the perspective of the low-lying effective QFT In the logic of the above arguments we assume this may then appear to be a hierarchy problem if therefore in this letter simply that the SM is valid up one tries to embed into a renormalizable QFT in- to the Planck scale, and that quantum gravity leaves stead of the theory which is based on the new, ex- certain boundary conditions for the Higgs quartic tended concepts. coupling λ. In condensed matter physics for instance, the energy of a superconductor in Ginzburg-Landau theory is described by: II. BOUNDARY CONDITIONS ON λ AT THE PLANCK SCALE E ≈ α|φ|2 + β|φ|4 + ... (1) where α and β are phenomenological parameters. It is well known that the SM cannot be extrap- These parameters have to be determined by exper- olated to arbitrarily high energies. A first con- iment itself in the Ginzburg-Landau theory frame- straint comes from the fact that at very high en- work, but they can be calculated from the micro- ergies far above the Planck scale the U(1)Y gauge scopic theory of superconductivity, namely the BCS coupling diverges. A more important piece of the- theory. In this sense the microscopic theory fixes the oretical input comes from the SM Higgs sector: parameter or the boundary condition of the low en- for small Higgs masses of approximately less than

2 1 9 3 127 GeV, the contributions from top loops drive the = g2 + g2 + 6λ − 6λ2 Λ2. (3) Higgs self-coupling towards negative values before 32π2 4 2 4 1 t the Planck scale and thus make the Higgs potential unstable [7,8]. For Higgs masses larger than approx- • vanishing anomalous dimension of the Higgs imately 170 GeV, the contribution from the Higgs mass parameter self-coupling drives itself towards a Landau pole be- γm(Mpl) = 0, m(Mpl) 6= 0. fore the Planck scale, this is the so-called triviality As we aim to determine the Higgs mass due to bound [3]. different boundary conditions for λ imposed at the Suppose that the LHC only detects the SM Higgs Planck scale, we use renormalization group equa- boson with a mass in the range between 127 − tions (RGEs) to evolve the couplings. The relevant 150 GeV and nothing else. The triviality and vac- one- and two-loop beta functions required for solv- uum stability bounds then imply that the SM can ing the RGE are listed in App. (A). As the uncer- be effectively valid up to the Planck scale. The SM tainty in the top mass is the dominant source of parameters could then be directly determined by uncertainty for the resulting Higgs mass prediction Planck scale physics and one might ask the question we treat the top mass as a free parameter (within a if there is a way to gain information on this type of certain range) and show the dependence on the top physics from measurements performed at LHC. Mo- mass explicitly. For each top mass value we deter- tivated by an asymptotic safety scenario of gravity, mine the corresponding Higgs mass due to a given Shaposhnikov and Wetterich [9] have, for example, boundary condition on λ(M ). For that we need to proposed that both the Higgs self-interaction λ and pl convert the top pole mass to its corresponding MS its beta function βλ should simultaneously vanish at the Planck scale, from which they derive the pre- Yukawa coupling: √ diction of mH = 126 GeV. At first sight it seems 2Mt remarkable that both conditions can be fulfilled at λt(Mt) = (1 + δt(Mt)) , (4) the same time and this prompted us to look at such v type of boundary conditions in more detail. where δt(Mt) is the matching correction for top We discuss therefore the following boundary con- mass. The list of matching conditions used for δt ditions which we imagine to be imposed on the SM in is given in App. (B). The matching scale is chosen the spirit of this paper by some version of quantum to be µ = Mt, which is a suitable choice for a low gravity2: Higgs mass range [18]. We consider also the thresh- • Vacuum stability old effects in the beta functions: The known gauge couplings gi(MZ ) run to the scale µ = Mt with- λ(Mpl) = 0 [7–9, 11–14]. out including the top loop contribution, and then • vanishing of the beta function of λ the value of gi(µ = Mt) will be used in subsequent βλ(Mpl) = 0 [9, 11]. RGE, where we compute all the coupled differential • the Veltman condition equation of gi, λ and λt with boundary condition im- StrM2 = 0 [15–17], posed. With suitable boundary conditions imposed which states that the quadratic divergent part for λ at the Planck scale, we can extract the Higgs of the one-loop radiative correction to the mass at µ = Mt after solving the RGE. The MS Higgs bare mass parameter m2 should vanish3: Higgs quartic coupling is then matched to the pole mass with: Λ2 δm2 = StrM2 (2) M 2 32π2v2 λ(M ) = H (1 + δ (M )), (5) t 2v2 H t where the Higgs matching δ is given at App. (B). 2 Boundary conditions of this type have also been discussed H in the context of anthropic considerations in the multiverse Repeating the procedure for different values of the [10]. 3 Note that the notation of StrM2 for the Veltman condition includes only the correction to m by the running coupling which is not the direct matching of respective pole known that the Veltman condition is scheme dependent as masses. Note also, that we have only included the top the quadratic divergence from different particles is not nec- quark Yukawa coupling λt and omitted the other Yukawa essarily the same. However if we assume a common cut-off couplings, as they do not contribute significantly to the for all the particle contributions, which may appear ap- Higgs mass running compared to the contributions from λ, propriate for our scenario, we will obtain a range of Higgs λt, and SU(2)L × U(1)Y gauge couplings g2 and g1. It is masses which is still not excluded by the experiments.

3 top pole mass, we obtain the Higgs mass determina- [21, 22] in the plot. We show the dependence on the tions plotted in Fig. (2). top mass outside of the range 172.3 − 174.1 GeV, Next we discuss some details regarding the bound- corresponding to the best world average value of top ary condition imposed for λ at the Planck scale, quark mass 173.2 ± 0.9 GeV [24], as methods that starting with the vacuum stability bound. To ob- determine the top mass directly from the tt cross- +3.5 tain the vacuum stability bound we need to consider section favour a smaller value of 168.9−3.4 GeV [25]. two cases in solving the coupled differential equa- We observe that most of the Higgs masses given tions [12, 13]: by different conditions tend to overlap in the vicin- ity of the best determined value of the top mass. 1. We first impose the boundary conditions at The triviality bound, represented by the approxi- tree-level, i.e. λ(M ) = 0 and apply the one- pl mate condition λ(Mpl) = π, yields a range of Higgs loop beta functions and anomalous dimension masses which is already excluded at 95% CL by the equations in solving the RGEs numerically. Tevatron and LHC. The Higgs masses generated by the other conditions however are still allowed and 2. Two-loop beta functions and anomalous di- not excluded yet by the experiments. The Veltman mension for m are considered in our RGE and condition is truncated at the point where its Higgs the effective potential is considered in one-loop mass calculated with two-loop beta functions starts approximation. The condition that we would to cross the vacuum stability bound obtained by two- need to impose is: loop RGEs. This is done in order to show the exact crossing point of these two conditions from two-loop 1 3 2 2 2 1 λ(Mpl) = (g (Mpl) + g (Mpl)) RGEs. 32π2 8 1 2 3 Throughout this work, we define a measure of the 2 2 g1(Mpl) + g2(Mpl) 4 uncertainties involved in the calculation as the differ- − log + 6λ (Mpl) 4 t ence between using one and two-loop beta functions λ2(M ) 3 1 for all the relevant SM couplings in the determina- log t pl − 1 + g4(M ) 2 4 2 pl 3 tion of the Higgs pole mass. To be more precise, we define a “RGE error band” as the difference in deter- g2(M ) − log 2 pl (6) mining the Higgs mass for a boundary condition of 4 λ(Mpl) with one and two-loop beta functions while the matching conditions remain the same for both Effectively we want to be consistent with the vacuum cases. Possible errors due to the matching condi- stability bound obtained from the effective potential. tions will be discussed below. We caution the reader With this approach we can estimate the uncertainty that this procedure probably overestimates the er- from the difference of Higgs masses obtained via dif- ror stemming from neglecting higher order contri- ferent cases above, which is effectively due to the butions to the beta functions. While there is no omission of higher-order contributions to the beta universally accepted way of estimating the theoret- functions and correction to the effective potential. ical uncertainties4 , there are other approaches to For the case of the second boundary condition, we define the theoretical error used in the literature. only impose the tree level Veltman condition given E.g. in Ref. [27], the authors define the theoret- in Eq. (2), as higher order loops always come with ical error by the scale dependence of the matching log(Mpl/µ)[19], which will be cancelled if the run- condition λ(Mt) and λt(Mt) while neglecting the ef- ning couplings are evaluated at Planck scale [20]. fect of higher order RGEs and arrive at an 3 GeV This is true, however, only if the complete beta func- uncertainty. In Ref. [28], the authors estimate the tions for all the couplings are used to resum the com- theoretical uncertainty by comparing the situation plete order of logarithms. where matching has been performed at µ = Mt to The gray hatched line in Fig. (2) depicts the lower the case µ = MZ and get an uncertainty of 2.2 GeV. direct Higgs mass search bound from LEP [23]. The We comment on possibilities to reduce the theoreti- coloured hatched lines give the combined exclusion cal uncertainty at the end of the paper. If one rather limits from ATLAS and CMS of 141 − 476 GeV at believes in this method of error estimation, the error 95% confidence level (CL) and 132 − 476 GeV at 90% CL [1]. The experimentally favoured parameter range in the mt–mh plane taking into account direct Higgs searches and and electroweak precision 4 See [26] for a recent attempt at rigorously characterizing a measurements are indicated by the GFitter region perturbative theoretical uncertainty.

4 170 4 Legend 1 Λ Mpl 0 H L 160 2 2 Str M Mpl 0 H3L ΒIΛ MplM 0

L 4 Λ Mpl Π H L I M 3 150 H5L ΓmIMplM 0 GeV

H H L I M % LHC exclusionH L 95 CLI M H L

mass 140 LHC exclusion 90% CL

Higgs 130 5 GFitter bound 120 H L 68% 95% 99% 1

110 H L 2 LEP exclusion 95% CL 166 168 170 172 174 176 H L Top mass GeV

Figure 2. Higgs and top (pole) mass determinations for different boundary conditions at the Planck scale. The coloured bands correspond to the conditions discussed in the textH andL which are also labelled in the insert. The middle of each band is the best value, while the width of the band is a “RGE error band” inferred from assuming that all omitted higher orders in the beta functions beyond two loops are limited by the difference between the one and two loop results. Note that the Veltman condition is truncated at the point where its Higgs mass prediction violates the vacuum stability bound (both at two-loops). The gray-hatched line at the bottom is the lower direct Higgs mass bound from LEP. Similarly the purple (brown) lines indicate the LHC Higgs searches at 95% (90%) CL from the 2010 data. The black dashed lines show the electroweak precision fit from GFitter [21, 22] for 68%, 95% and 99% confidence intervals (which include limits from radiative corrections and also the direct searches).

curves shink accordingly. In our plot, the aforemen- band obtained with Casas et al. [12, 13], a discrep- tioned “RGE error band” is represented by the band- ancy of around ±7 GeV for the Higgs mass value ob- width of each curve, with its center representing the tained via two-loop RGE is observed. This mismatch Higgs mass obtained from two-loop RGE running. can be explained by the omission of two-loop QCD The upper edges of the bandwidths consist of the matching condition by the authors of Refs. [12, 13], Higgs masses obtained from one-loop RGEs. as they only considered one-loop QCD, electroweak and QED contribution in the top mass matching We consider also the uncertainty on the curves condition. Since we would like to consider only the due to the error of strong coupling constant αs = uncertainties due to the number of loops of the beta 0.1184(7) [29] and we obtain ±1 GeV uncertainty to function used but not the errors caused by omis- the Higgs mass, which is negligible when quadrati- sion of better matching precision, we include the cally added to the bandwidth in Fig.(2). Due to the QCD matching between top Yukawa MS coupling relatively large “RGE error band”, the error prop- and top pole mass up to three-loop. The resulting agation from the strong coupling constant can be Higgs mass determined by the vacuum stability with safely ignored. The theoretical error on the Higgs two-loop RGE agrees with Ellis et al. [31]. The ααs mass due to the matching uncertainty [18, 30] be- correction [32] is neglected in our analysis as it only tween top Yukawa MS coupling and top pole mass gives a small contribution. The Higgs pole mass is is also considered. Comparing our vacuum stability

5 matched with λ at top pole mass scale, i.e. the renor- do not play a significant role in our Higgs mass pre- malization scale of λ is set to be at µ = Mt. Since diction. We would like to caution the reader that the higher order matching conditions for λ has not neutrinos are indeed massive and could couple with been calculated in any literature, only the expres- the O(1) Yukawa coupling to the Higgs, e.g. in the sion of δH given in App. (B) will be used as our canonical type I see-saw mechanism. For a neutrino matching. If the matching of λ to the Higgs pole mass of mν ≈ O(1 eV) and a see-saw scale of approx- mass is only performed at tree-level, the resulting imately 1015 GeV, the Yukawa coupling of neutrino discrepancy of the Higgs pole mass with respect to would be around O(1). This large coupling between the one-loop matching result is found to be less than Higgs boson and neutrino could alter the prediction 1 GeV. Therefore, we can safely assume that higher on Higgs mass by modifying the beta function of order matching condition for λ will not yield a larger Higgs quartic coupling, as pointed out by Ref. [33]. uncertainty. However the interesting case, i.e. SM plus extension The error estimation for the boundary condition of neutrino sector with O(1) Yukawa coupling can βλ(Mpl) = 0 is not the same as for the other con- be valid up to Planck scale, has been excluded by ditions. A careful treatment of the Higgs mass ex- recent Tevatron and LHC searches [1]. We there- traction has to be implemented in this case. The fore implicitly assume a scenario where the neutrino one-loop beta function of λ is a quadratic equation Yukawa couplings are not significantly larger than of λ, and for a given top mass we obtain two positive for example the bottom Yukawa coupling. solutions of the boundary condition βλ(Mpl) = 0 at the Planck scale, and thus obtain two equally valid low-energy Higgs mass determinations. In Fig.(2) III. IS A LOW HIGGS MASS FAVOURED? these two branches of solutions generate a hook- shaped trajectory in the MH − Mt plane. The hook All the generic boundary conditions which we dis- < < ends where λ starts to take negative value. Due to cussed prefer a low Higgs mass 127 GeV ∼ MH ∼ the mismatch of the end of the trajectory when ei- 145 GeV which fits amazingly good to the existing ther one-loop or two-loop beta function is applied, experimental direct lower and upper bounds from a larger “RGE error band” has to be taken into LEP, Tevatron and LHC. These values fit further- account, where we generate error bars that cover more very well to those preferred by electroweak pre- the distance of the mismatch and plot a band to cision measurement. One could ask therefore if this cover all the error bars. Besides the mismatch men- is by chance or if it has a specific reason. Let us tioned above, there exists also another source of er- therefore first look at random values of λ(Mpl) pre- ror, namely number of loops of the beta functions tending in this way that every value could be realized implemented in the condition βλ(Mpl) = 0. In prin- by the new physics at the Planck scale. We assume ciple one should apply the full beta function as the therefore for a moment, that all values of λ(Mpl) in boundary condition, but in practice this is impossi- the range of λ ∈ [0, π] have equal probability5. We ble and therefore we have to check the possible un- show therefore first in Fig.(3) the running of λ from certainty which arises due to the number of loops in the Planck scale to the weak scale for some values βλ used in the boundary condition. The errors lie, in the interval λ ∈ [0, π]. Note that most values of however, within the band. A similar uncertainty due λ(Mpl) lead to λ at the Fermi scale which is greater to number of loops used as boundary condition also than 0.2, which corresponds to MH > 150 GeV. appears in the γm(Mpl) = 0 condition and its un- To analyse the effect further we randomly gen- certainty is larger in comparison to the βλ(Mpl) = 0 erate 600 values for λ(Mpl) and the top pole mass condition. ranging from 170 − 175 GeV and put the resulting Another possible source of uncertainties on the Higgs masses into the scatter plot Fig.(4). Note that determination of the Higgs mass from the Planck most values of λ at the Planck scale lead to a Higgs scale boundary conditions comes from the value of the Planck scale. The difference of the Higgs masses obtained from a certain boundary condition imposed 5 at a√ different value of the Planck scale, e.g. µ = In principle one could consider all values up to infinity since M / 8π, are negligible for most of the boundary there seems to be no a priori reason to limit λ(Mpl). Ex- pl tending the upper end is however only strengthening the ar- conditions. However the discrepancies are larger for guments, since this would favour even more a higher Higgs the lower branch of the βλ = 0 condition. We will mass. Note that this is connected to the triviality bound not discuss further these uncertainties. and the corresponding focussing of RGE trajectories to- Throughout this work we assume that neutrinos wards low scales. See Fig.(3) for this effect.

6 is a special condition which ﬁts very well to the ex- 1.4 perimental ﬁndings. This is also the vacuum stability bound and it corresponds therefore to the light- 1.2 est Higgs mass in the SM when it is valid up to the Planck scale. A lower Higgs mass would require 1.0 some extra new physics at lower scales, which would

L 0.8 rule out the logic of our paper. Including errors this L

H > Λ leads to a lower bound MH ∼ 122 GeV for our sce- 0.6 nario. Fig.(4) also shows that it is non-trivial that all 0.4 the boundary conditions which we discuss in this 0.2 paper lead to viable Higgs masses. However, there is a systematic understanding why all our condi-

0.0 5 8 11 14 17 20 tions work. The point is that all our conditions 10 10 10 10 10 10 2 such as βλ(Mpl) = 0 or StrM (Mpl) = 0 are con- L GeV ditions connected to quantum loops for the Higgs mass parameter m and the quartic coupling λ and Figure 3. Running of λ fromH theL Planck scale to the we can therefore always write λ as a function of the Fermi scale. Diﬀerent values of λ(M ) and it can be pl gauge couplings and the top Yukawa coupling, i.e. seen that a large parameter space of λ(Mpl) tends to > λ = f(gi, λt). This leads generically to smaller val- induce λ(Mt = 173 GeV) ∼ 0.186 which is equivalent to Higgs mass greater than 150 GeV. ues than a random choice, since loop factors, i.e. factors of 1/16π2 are present and since top mass loops (fermion loop) carry a minus sign, which leads to cancellations, pushing the value of λ even smaller. 170 One can therefore understand why the cases which we discussed all systematically lead to viable Higgs 160 mass predictions. In that sense none of them is spe- L cial and there may exits more interesting boundary Excluded Region by

GeV conditions which also lead to viable Higgs masses. H 150 LHC 95% CL This also implies that a Higgs mass in the currently most favoured range does not clearly select any mass model or scenario which leads to a boundary condi- 140 tion that has such loop and top mass suppression fac- Higgs tors. However, vice versa one might argue that cur-

130 rent data point to boundary conditions which must involve such suppression factors, which is an interesting observation. It is also intriguing to ask in this 120 context why the top mass is much heavier than other 0.0 0.5 1.0 1.5 2.0 2.5 3.0 quarks such that it compensates other loop contri- Λ Mpl butions which would drive the Higgs boson heavier. This appears to be an interesting “conspiracy” Figure 4. Scatter plot of HiggsI massM at the Fermi scale in favour of the logic of this paper which works if determined by random λ at the Planck scale with ran- the boundary condition is imposed at a high scale dom top mass constrained to the interval [170, 175] GeV. like the Planck scale. All these arguments break of course down if LHC or any future experiment detects any sign of new physics that couples directly to the mass between 160 GeV and 175 GeV. For the chosen Higgs boson. Even an indirect coupling, i.e. radia- interval λ(Mpl) ∈ [0, π] we ﬁnd ≈ 90% in this range tive correction to λ at loop level is severe enough which has been excluded by the Tevatron and by the to alter the running of λ drastically. So far there CMS and ATLAS experiments. Note that only less is no compelling evidence for additional physics be- than 5% of the generated Higgs masses are allowed yond the SM, whereas the SM Higgs search seems by experiments. to indicate some excesses of Higgs like events in the Looking at Fig.(4) one immediately notices that range of 130 GeV to 140 GeV, albeit at only about λ(Mpl) = 0 which corresponds to MH = 127 GeV 2σ [1].

7 IV. DISCUSSION AND CONCLUSIONS 134

132 The LHC has an excellent chance to find the SM L Higgs boson and we emphasize in this paper that 130 GeV the left-over values lie in a range which is well moti- H 128 vated by various Planck scale boundary conditions. mass We argued that this Higgs mass range is special and 126 that it might be related to embeddings of the SM Higgs as a QFT into some form of quantum gravity, which 124 is based on concepts beyond QFT. The SM would 122 still be an effective field theory which is valid up 120 to the Planck scale, but the asymmetry in the con- 172.5 173.0 173.5 174.0 cepts might allow to understand the famous hierar- Top mass GeV chy problem from the perspective of the new concepts at the Planck scale, while it would only ap- Figure 5. A blow up of the vacuumH L stability bound in pear unnatural from the low energy point of view. In the interesting Higgs and top mass region. The blue other words: The large hierarchy between the Planck line in the center represents the vacuum stability bound and electroweak scale might only be a problem as obtained via two-loop beta functions, which has been long as we look at it from the SM perspective. An thoroughly discussed in main text. The yellow band important point is then that such a scenario makes represents the uncertainties of the Higgs mass obtained only sense if there are no intermediate scales, since via two-loop RGEs due to αs uncertainties. The outer blue band is identical to the blue band in Fig. (2) and this would require a QFT solution of the hierarchy it represents the full “RGE error band” estimated from between the electroweak and intermediate scales. difference between one- and two-loop RGEs. With the The proposed scenario requires that the Higgs best world average top pole mass 173.2 GeV the in- coupling can be evolved to the Planck scale, which ferred Higgs mass from the vacuum stability condition implies strict lower and upper bounds on the cou- λ(Mpl) = 0 is 127 ± 5 GeV. pling. The upper bound is the so-called triviality bound which is approximately 170 GeV and it is interesting to note that the Higgs mass is below able way to arrive at a conservative error esti- this value, as otherwise the couplings could not be mation for the Higgs mass due to the lack of evolved up to the Planck scale. The lower bound is higher order RGEs, but it should not be over the so-called vacuum stability bound, i.e. the con- interpreted. This implies that the exact lower dition λ(Mpl) = 0. We carefully evaluated its value bound for the Higgs mass is limited by this and error with the latest data at two loops and pro- conservative estimation. vided an error estimate. For Mt = 173.2 GeV we find mH = 127±5 GeV which implies that the Higgs 2. Precision top mass analysis is required to de- mass must be heavier than 122 GeV. It is impor- termine the exact value of the Higgs mass pre- tant to note that the vacuum stability bound, or dicted via vacuum stability. The reason why equivalently the condition λ(Mpl) = 0, is very spe- we want to stress on this specific result is that cial. It implies that the Higgs self-interaction at the to date, there is no general consensus on what electroweak scale is entirely generated by radiative type of top mass is actually measured via kine- corrections of the RGE evolution from a vanishing matic reconstruction [34]. At the Tevatron, coupling at the Planck scale. The Higgs mass would the main method used for the top mass ex- therefore be connected to the gauge and Yukawa traction actually “measures” the Pythia mass, couplings which enter into the RGEs. which is a Monte-Carlo simulated template Several comments should be carefully considered mass. Strictly speaking the top pole mass is in this context: not a well defined quantity, as the top quark does not exist as free parton. The top mass 1. The Higgs central value mH = 127 GeV is ob- that the Tevatron has measured is based on tained via two-loop beta function running from the final state of the decay products. On the the vacuum stability condition at the Planck other hand the running MS top mass can be scale to the weak scale regime. Fig. (5) shows extracted directly from the total cross section the uncertainties due to the omission of higher in the top pair production. In this sense, one orders to be ±5 GeV. This appears a reason- can obtain a complementary information of

8 the top mass from the production phase. By ditions that lead to a small Higgs self-coupling λ at converting the MS mass to the pole mass via the Planck scale. These conditions could possibly be matching conditions, the top pole mass value some remnant of symmetry in the full quantum the- +3.5 168.9−3.4 GeV extracted with this method by ory of gravity. The Yukawa and λ couplings could or Langenfeld et al. [25] is found to be lower than maybe even should have a common origin from the the world best average value. However, this Planck scale physics, as the top quark contribution way of extracting the top mass suffers from miraculously cancels the contribution of the Higgs larger numerical uncertainties. As we can be boson such that the SM can be extrapolated to the seen from Figs. (2) and (5), a change of the top Planck scale. mass by 2 GeV changes the Higgs mass predic- We have shown in this paper that the chosen tion by 6 GeV. boundary conditions for λ yield a Higgs mass which is in the allowed range. The fact that different 3. The electroweak vacuum might in principle boundary conditions have overlapping regions could be metastable. However, most of the Higgs imply that more than one of them is simultane- mass region for metastability has already been ously realized at the Planck scale, which is an in- ruled out by LEP, although not entirely ex- triguing possibility. For instance if we demand that cluded. The finite temperature metastability λ(M ) = 0 and StrM2(M ) = 0 are both satisfied region, however, with the local SM assumed pl pl then it is possible that the Higgs quartic coupling is to be stable against thermal fluctuations up only generated radiatively and the quadratic diver- to Planck scale temperatures, allows the en- gence vanishes. This appears puzzling from a low tire region from the LEP bound to the vac- energy perspective, however, from the Planck scale uum stability. Hence should LHC discover the physics perspective this could be natural, as these SM Higgs boson with its mass lower than the two conditions could have a common origin of some one predicted by two-loop RGE vacuum sta- unknown connection between the gauge, Yukawa bility bound, there is a possibility that the and Higgs quartic couplings. Other combinations of SM electroweak vacuum is not the stable one listed boundary conditions can also be considered, as (see Refs. [14, 27, 31, 35–37] for more de- for instance those proposed in Ref. [9] where the au- tails). However, we would also like to re- thors obtained from the asymptotic safety of gravity mind that metastability bounds depend on the λ ≈ 0 and β ≈ 0 near the Planck scale. Similar con- fastest process conceived for the transition to λ ditions also appeared in Ref. [11], where the authors the true vacuum. Any faster process occurring demanded the principle of multiple point criticality. once anywhere in the Universe would reduce or One might ask how much the situation can be im- eliminate the metastability region. proved with better measurements of the Higgs and We discussed in Fig. (4) that only a small percent- top masses. ATLAS and CMS [38–40] should be ca- age of randomly generated boundary conditions for pable to detect a Higgs mass with a precision of 0.1% −1 λ(Mpl) lead to Higgs masses which are still allowed to 1% with an integrated luminosity of 30 fb . This by experiments. On the other hand we presented precision is encouraging, but unfortunately the de- in Fig. (2) results for a set of boundary conditions termination of the high energy boundary conditions which all lead to Higgs masses in the allowed region is plagued by the relatively large uncertainties due and we explained how this can be systematically un- to the lack of higher order RGEs. Three loop beta derstood. The point is that the chosen boundary functions and other improvements would be required conditions emerge from conditions which have loop in order to reduce the errors of the band in Fig. (2). suppression factors, making them rather small com- Without these theoretical improvements it will be pared to a random choice. Fig. (2) shows that there hard to differentiate the different boundary condi- exist different working conditions and others might tions. However, such improvements would be im- be found which work as well. This implies that once portant if the SM would indeed be a theory which is loop suppression factors have been included into the valid up to the Planck scale. Precise determinations boundary condition it is not easily possible to distin- of the Higgs and top masses could then be used to guish between various models or scenarios, but grow- identify the correct boundary conditions for λ within ing precision will nevertheless reduce the number of the remaining uncertainties. A complete calculation options somewhat. The conclusion from this obser- of the three-loop beta functions of the SM would vations is in the scenario of this paper that quantum be very important in this case. This would, how- gravity should not generate a random value of λ at ever, also require the determination of the matching the Planck scale, but it should somehow select con- conditions to at least two-loop order. While the two-

9 loop QCD matching condition has been included for Gross for interesting discussions on the exact value λt(mt) in this work, the corresponding contribution of the vacuum stability bound. M.H. acknowledges to λ(mt) is not known and would be necessary to support by the International Max Planck Research reduce the matching uncertainty alongside the other School for Precision Tests of Fundamental Symme- two-loop contributions to λ(mt) and λt(mt). tries. With around 5 fb−1 of integrated luminosity col- lected by ATLAS and CMS, it is possible to find or exclude a SM Higgs mass in the region from Appendix A: List of beta functions and 114 − 600 GeV by the end of next year. A discov- anomalous dimension of Higgs mass ery of SM Higgs boson will anyhow be significant for advancement of particle physics. There exists We give the beta function and anomalous dimen- an excellent chance to find all sort of TeV-scale new sion of Higgs mass used in our calculation. The beta physics, but if the LHC finds nothing but a SM Higgs function for a generic coupling X is given as: then this would be very much in favour of the spirit (i) of this paper. A key question would be which new dX X β µ = β = X (A1) concepts could be involved in quantum gravity such dµ X (16π2)i that the correct SM boundary conditions arise. i and the anomalous dimension for Higgs mass is given Note added: After this work has been completed, as: ATLAS has reported the latest Higgs mass exclusion regions at 95% CL [41] ranging from 112.7 − 2 (i) dm X γm 115.5 GeV, 131 − 237 GeV and 251 − 453 GeV while µ = γ = (A2) dµ m (16π2)i CMS has excluded the Higgs masses in the range i 127 − 600 GeV [42]. The list of beta functions are given below [19, 43– Acknowledgements: We would like to thank E. 46]:

3 3 β(1) = λ(−9g2 − 3g2 + 12λ2) + 24λ2 + g4 + (g2 + g2)2 − 6λ4, (A3) λ 2 1 t 4 2 8 1 2 t (1) 9 3 17 2 9 2 2 β = λ + λt − g − g − 8g , (A4) λt 2 t 12 1 4 2 3 41 19 β(1) = g3, β(1) = − g3, β(1) = −7g3, (A5) g1 6 1 g2 6 2 g3 3 45 85 β(2) = −312λ3 − 144λ2λ2 + 36λ2(3g2 + g2) − 3λλ4 + λλ2 80g2 + g2 + g2 λ t 2 1 t t 3 2 2 6 1 73 39 629 8 9 − λg4 + λg2g2 + λg4 + 30λ6 − 32λ4g2 − λ4g2 − λ2g4 8 2 4 2 1 24 1 t t 3 3 t 1 4 t 2 21 19 305 289 559 379 + λ2g2g2 − λ2g4 + g6 − g4g2 − g2g4 − g6 (A6) 2 t 2 1 4 t 1 16 2 48 2 1 48 2 1 48 1 (2) 4 2 131 2 225 2 2 1187 4 β = λt −12λ + λ g + g + 36g − 12λ + g λt t t 16 1 16 2 3 216 1 3 19 23 − g2g2 + g2g2 − g4 + 9g2g2 − 108g4 + 6λ2 (A7) 4 2 1 9 1 3 4 2 2 3 3 199 9 44 17 β(2) = g3 g2 + g2 + g2 − λ2 (A8) g1 1 18 1 2 2 3 3 6 t 3 35 3 β(2) = g3 g2 + g2 + 12g2 − λ2 (A9) g2 2 2 1 6 2 3 2 t 11 9 β(2) = g3 g2 + g2 − 26g2 − 2λ2 , (A10) g3 3 6 1 2 2 3 t

10 9 3 γ(1) = m2 12λ + 6λ2 − g2 − g2 (A11) m t 2 2 2 1 27 γ(2) = 2m2 −30λ2 − 36λλ2 + 12λ(3g2 + g2) − λ4 + 20g2λ2 m t 2 1 4 t 3 t 45 85 145 15 157 + g2λ2 + g2λ2 − g4 + g2g2 + g4 (A12) 8 2 t 24 1 t 32 2 16 2 1 96 1

Appendix B: Matching of MS coupling constant The strong coupling constant g3(MZ ) can be ex- and pole mass tracted from αs(MZ ) = 0.1184(7) [29]. The matching of MS top Yukawa coupling to its The MS gauge couplings are used in this work, pole mass is given by Eq. (4), where δt can be split therefore no matching of the gauge couplings is nec- into three part [30]: essary. The boundary conditions for MS g1(MZ ) and g2(MZ ) couplings are taken from the value of QCD QED W δt(µ) = δt (µ) + δt (µ) + δt (µ) (B3) ﬁne structure constantα ˆ(MZ ) = 127.916(15) and weak-mixing angle sin2 θˆ (M ) = 0.23116(13) [29] W Z QCD by solving the equations: where δt denotes the contribution of QCD correction, which we will take up to three-loop order g2(M ) + g2(M ) [47]. The term δQED(µ) + δW(µ) contributes to the αˆ−1(M ) ≡ 4π 1 Z 2 Z , (B1) t t Z 2 2 matching correction from the QED and electroweak g1(MZ )g2(MZ ) 2 sector. The matching terms at µ = Mt have been 2 ˆ g1(MZ ) sin θW (MZ ) ≡ 2 2 . (B2) calculated to be: g1(MZ ) + g2(MZ )

4α (M ) α (M )2 α (M )3 δQCD(M ) = − s t − 9.1253 s t − 80.4046 s t (B4) t t 3π π π 2 " 2 4 2 3/2 QED W αˆ(Mt) Mt 11 MH MH 4Mt MH δt (Mt) + δt (Mt) = −4 + 2 2 − 2 − 4 2 − 1 arccos 9π 16π v 2 4Mt 2Mt MH 2Mt 2 2 2 MH MH MH −3 −3 MH + 2 2 − 3 log 2 − 6.9 × 10 + 1.73 × 10 log 2Mt 2Mt Mt 300 GeV M − 5.82 × 10−3 log t (B5) 175 GeV

QED W Note that the above formula for δt + δt is only given as [48]: 2 2 valid for MH < 4Mt , which is our case in the above analysis. M 2 δ (M ) = Z ξf (ξ) + f (ξ) + ξ−1f (ξ) H t 32π2v2 1 0 −1 (B6)

2 2 As for the matching of λ and the Higgs pole mass where ξ ≡ MH /MZ and each of the function fi de- given by Eq. (5), the matching correction δH (Mt) is ﬁned as:

2 2 Mt 3 1 1 cw 2 9 25 π f1(ξ) = 6 log 2 + log ξ − Z − Z − log cw + − √ , (B7) MH 2 2 ξ ξ 2 9 3

11 2 2 2 Mt 2 Mt 3cwξ ξ 1 f0(ξ) = −6 log 2 1 + 2cw − 2 2 + 2 log 2 + 2Z MZ MZ ξ − cw cw ξ 2 2 2 3cw 2 2 15 2 Mt Mt + 2 + 12cw log cw − (1 + 2cw) − 3 2 2Z 2 sw 2 MZ MZ ξ 2 2 Mt 2 cw +4 log 2 − 5 + 4cwZ , (B8) MZ ξ 2 4 2 Mt 4 Mt 1 4 cw 4 2 f−1(ξ) = 6 log 2 1 + 2cw − 4 4 − 6Z − 12cwZ − 12cw log cw MZ MZ ξ ξ 4 2 2 Mt Mt Mt 4 + 24 4 log 2 − 2 + Z 2 + 8(1 + 2cw). (B9) MZ MZ MZ ξ 2A arctan(1/A) if z > 1/4 Z(z) = A = p|1 − 4z| (B10) A log [(1 + A)/(1 − A)] if z < 1/4

2 2 2 2 and sw = sin θW , cw = cos θW and θW is the Wein- berg angle in on-shell scheme.

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