Planck Scale Boundary Conditions and the Higgs Mass
PPlanck 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, thenone should reconsider scenarios where the Standard Model with three right-handed neutrinos is validup to Planck scale. We assume in this spirit that the Standard Model couplings are remnants ofquantum gravity which implies certain generic boundary conditions for the Higgs quartic coupling atPlanck scale. This leads to Higgs mass predictions at the electroweak scale via renormalization groupequations. We find that several physically well motivated conditions yield a range of Higgs massesfrom 127 −
142 GeV. We also argue that a random quartic Higgs coupling at the Planck scale favours M H >
150 GeV, which is clearly excluded. We discuss also the prospects for differentiating differentboundary conditions imposed for λ ( M pl ) at the LHC. A striking example is M H = 127 ± λ ( M pl ) = 0, which would imply that the quartic Higgs coupling at the electroweakscale is entirely radiatively generated. I. INTRODUCTION
The Standard Model (SM) of particle physics isvery successful and it has withstood all precisiontests over almost 40 years. All SM particles havebeen discovered , except for the ingredient con-nected to electroweak Symmetry Breaking (EWSB),namely the Higgs boson. The Large Hadron Col-lider (LHC) was built to test EWSB and to find orrule out the Higgs boson. In addition, the LHC aimsat detecting new physics which is suggested to ex-ist in the TeV range in order to solve the so-calledhierarchy problem. However, the ATLAS and CMSdetectors at LHC have so far not found any signof new physics and the remaining Higgs mass rangehas shrunk considerably [1]. This suggests to thinkabout scenarios where nothing but the SM is seen.The essence of the hierarchy problem [2] is thefact that quantum corrections generically destroythe separation of two scales of scalar Quantum FieldTheories (QFT). It is thus not possible to under-stand how the electroweak scale could be many or-ders of magnitude smaller than the scale of an em-bedding QFT. Conventional solutions of the hierar-chy problem stay within QFT. One solution is to ∗ [email protected] † [email protected] ‡ [email protected] In addition neutrinos were found to be massive, which re-quires in its simplest form only the addition of three righthanded fermionic singlets. postulate a new symmetry (Supersymmetry) whichcancels the problematic quadratic divergences. Al-ternatively the scalar sector may be considered ef-fective (composite) such that form factors removelarge quadratic divergences. Another idea is that theHiggs particle could be a pseudo-Goldstone-bosonsuch that its mass is naturally somewhat lower thana scale where richer new physics exists. However,none of these ideas has so far shown up in experi-ments.This prompts us to consider new ideas. An impor-tant observation is the fact that the SM can anywaynot be valid up to an arbitrary high energy scaledue to triviality [3] and since gravity must affect el-ementary particle physics latest at the Planck scale, M pl = 1 . × GeV. Note that this introducesa conceptual asymmetry, as gravity is known to benon-renormalizable, i.e. it cannot be a QFT in theusual sense and requires fundamentally new ingredi-ents. This looks bad from the perspective of renor-malizable gauge theories, also since we can so faronly guess which concepts might be at work. How-ever, this may also be good in two ways: First, renor-malizable QFTs do not allow to calculate absolutemasses and absolute couplings. Any embedding ofthe SM into some other renormalizable QFT (likeGUTs) would therefore only shift the problem to anew theory which is also unable to determine theabsolute values of these quantities. In other words:The problems of gravity may be a sign of physicsbased on new concepts which may ultimately al-low to determine absolute masses, mixings and cou-plings. However, there is no need for the SM to be a r X i v : . [ h e p - ph ] M a r (cid:45) µm H ( µ ) (cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24) m H ( v ) M pl Non-field theoreticquantum gravityregion (cid:64)(cid:64)(cid:64)(cid:82) (cid:118)
Imprint of Higgs massleft by quantum gravity
QFTregime BeyondQFT
Figure 1. The SM Higgs mass could be determined andfixed by unknown physics connected to quantum grav-ity, which should be based on new concepts other thanconventional QFT. The running of the Higgs mass fromPlanck scale down to electroweak scale is fully dictatedby the SM as a QFT. directly embedded into gravity and various layersof conventional gauge theories (LR, PS, GUT, ...)could be in-between. The second reason why an em-bedding involving new concepts beyond renormaliz-able QFTs might be good is that this asymmetrymight offer new solutions to the hierarchy problem.In other words: The conventional approach towardsinterpreting quadratic divergences to the Higgs masscorrection by simply substituting Λ → M pl mightnot be correct if Planck scale physics is based on newphysical concepts different from conventional QFT.This view is also shared by Meissner and Nicolai [4].The point is that the unknown new concepts mayallow for mechanisms which stabilize a low-lying ef-fective QFT from the perspective of the Planck scale.From the perspective of the low-lying effective QFTthis may then appear to be a hierarchy problem ifone tries to embed into a renormalizable QFT in-stead of the theory which is based on the new, ex-tended concepts.In condensed matter physics for instance, the en-ergy of a superconductor in Ginzburg-Landau theoryis described by: E ≈ α | φ | + β | φ | + . . . (1)where α and β are phenomenological parameters.These parameters have to be determined by exper-iment itself in the Ginzburg-Landau theory frame-work, but they can be calculated from the micro-scopic theory of superconductivity, namely the BCStheory. In this sense the microscopic theory fixes theparameter or the boundary condition of the low en- ergy effective theory. The parameters in low energyGinzburg-Landau theory “know” the boundary con-dition set by the underlying BCS theory, but manydynamical details of the BCS theory are lost in theGinzburg-Landau effective theory, even though BCStheory does not provide a mechanism to explain hi-erarchies.The above considerations prompt us to specu-late that the SM might be valid up to the Planckscale, where it is embedded directly into Planck scalephysics without any intermediate energy scale. Thenew concepts behind the Planck scale physics mightthen offer a solution to the hierarchy problem whichis no longer visible when one looks at the SM only. Inanalogy to the superconductivity example the onlyway how the SM would “know” about such an em-bedding could be special boundary conditions sim-ilar to compositeness conditions or auxiliary fieldconditions in theories where redundant degrees offreedom are eliminated in embeddings. This sce-nario is depicted in Fig. (I).Following the logic outlined above it would be es-sential to have only the weak and the Planck scaleand nothing in between, since otherwise it would re-quire to solve the hierarchy problem within QFT.We therefore forbid any kind of new intermediateenergy scale between weak and Planck scale in or-der to avoid the large hierarchy between Higgs andheavy intermediate particle’s mass. This view is alsoshared by the spirit of ν MSM, proposed by Shaposh-nikov [5]. Even without any intermediate scale onemight still ask why there should not be any radiativecorrections of Planck scale size to the Higgs mass pa-rameter. We do not have an argument here but wepoint the interested reader to the works of Bardeen[6] that have argued about the spurious nature ofthe quadratic correction.In the logic of the above arguments we assumetherefore in this letter simply that the SM is valid upto the Planck scale, and that quantum gravity leavescertain boundary conditions for the Higgs quarticcoupling λ . II. BOUNDARY CONDITIONS ON λ ATTHE PLANCK SCALE
It is well known that the SM cannot be extrap-olated to arbitrarily high energies. A first con-straint comes from the fact that at very high en-ergies far above the Planck scale the U (1) Y gaugecoupling diverges. A more important piece of the-oretical input comes from the SM Higgs sector:for small Higgs masses of approximately less than227 GeV, the contributions from top loops drive theHiggs self-coupling towards negative values beforethe Planck scale and thus make the Higgs potentialunstable [7, 8]. For Higgs masses larger than approx-imately 170 GeV, the contribution from the Higgsself-coupling drives itself towards a Landau pole be-fore the Planck scale, this is the so-called trivialitybound [3].Suppose that the LHC only detects the SM Higgsboson with a mass in the range between 127 −
150 GeV and nothing else. The triviality and vac-uum stability bounds then imply that the SM canbe effectively valid up to the Planck scale. The SMparameters could then be directly determined byPlanck scale physics and one might ask the questionif there is a way to gain information on this type ofphysics from measurements performed at LHC. Mo-tivated by an asymptotic safety scenario of gravity,Shaposhnikov and Wetterich [9] have, for example,proposed that both the Higgs self-interaction λ andits beta function β λ should simultaneously vanish atthe Planck scale, from which they derive the pre-diction of m H = 126 GeV. At first sight it seemsremarkable that both conditions can be fulfilled atthe same time and this prompted us to look at suchtype of boundary conditions in more detail.We discuss therefore the following boundary con-ditions which we imagine to be imposed on the SM inthe spirit of this paper by some version of quantumgravity : • Vacuum stability λ ( M pl ) = 0 [7–9, 11–14]. • vanishing of the beta function of λβ λ ( M pl ) = 0 [9, 11]. • the Veltman conditionStr M = 0 [15–17],which states that the quadratic divergent partof the one-loop radiative correction to theHiggs bare mass parameter m should vanish : δm = Λ π v Str M (2) Boundary conditions of this type have also been discussedin the context of anthropic considerations in the multiverse[10]. Note that the notation of Str M for the Veltman condi-tion includes only the correction to m by the running cou-pling which is not the direct matching of respective polemasses. Note also, that we have only included the topquark Yukawa coupling λ t and omitted the other Yukawacouplings, as they do not contribute significantly to theHiggs mass running compared to the contributions from λ , λ t , and SU (2) L × U (1) Y gauge couplings g and g . It is = 132 π (cid:18) g + 34 g + 6 λ − λ t (cid:19) Λ . (3) • vanishing anomalous dimension of the Higgsmass parameter γ m ( M pl ) = 0 , m ( M pl ) (cid:54) = 0.As we aim to determine the Higgs mass due todifferent boundary conditions for λ imposed at thePlanck scale, we use renormalization group equa-tions (RGEs) to evolve the couplings. The relevantone- and two-loop beta functions required for solv-ing the RGE are listed in App. (A). As the uncer-tainty in the top mass is the dominant source ofuncertainty for the resulting Higgs mass predictionwe treat the top mass as a free parameter (within acertain range) and show the dependence on the topmass explicitly. For each top mass value we deter-mine the corresponding Higgs mass due to a givenboundary condition on λ ( M pl ). For that we need toconvert the top pole mass to its corresponding MSYukawa coupling: λ t ( M t ) = √ M t v (1 + δ t ( M t )) , (4)where δ t ( M t ) is the matching correction for topmass. The list of matching conditions used for δ t is given in App. (B). The matching scale is chosento be µ = M t , which is a suitable choice for a lowHiggs mass range [18]. We consider also the thresh-old effects in the beta functions: The known gaugecouplings g i ( M Z ) run to the scale µ = M t with-out including the top loop contribution, and thenthe value of g i ( µ = M t ) will be used in subsequentRGE, where we compute all the coupled differentialequation of g i , λ and λ t with boundary condition im-posed. With suitable boundary conditions imposedfor λ at the Planck scale, we can extract the Higgsmass at µ = M t after solving the RGE. The MSHiggs quartic coupling is then matched to the polemass with: λ ( M t ) = M H v (1 + δ H ( M t )) , (5)where the Higgs matching δ H is given at App. (B).Repeating the procedure for different values of the known that the Veltman condition is scheme dependent asthe quadratic divergence from different particles is not nec-essarily the same. However if we assume a common cut-offfor all the particle contributions, which may appear ap-propriate for our scenario, we will obtain a range of Higgsmasses which is still not excluded by the experiments. λ at the Planck scale,starting with the vacuum stability bound. To ob-tain the vacuum stability bound we need to considertwo cases in solving the coupled differential equa-tions [12, 13]:1. We first impose the boundary conditions attree-level, i.e. λ ( M pl ) = 0 and apply the one-loop beta functions and anomalous dimensionequations in solving the RGEs numerically.2. Two-loop beta functions and anomalous di-mension for m are considered in our RGE andthe effective potential is considered in one-loopapproximation. The condition that we wouldneed to impose is: λ ( M pl ) = 132 π (cid:18)
38 ( g ( M pl ) + g ( M pl )) (cid:20) − log g ( M pl ) + g ( M pl )4 (cid:21) + 6 λ t ( M pl ) (cid:20) log λ t ( M pl )2 − (cid:21) + 34 g ( M pl ) (cid:20) − log g ( M pl )4 (cid:21)(cid:19) (6)Effectively we want to be consistent with the vacuumstability bound obtained from the effective potential.With this approach we can estimate the uncertaintyfrom the difference of Higgs masses obtained via dif-ferent cases above, which is effectively due to theomission of higher-order contributions to the betafunctions and correction to the effective potential.For the case of the second boundary condition, weonly impose the tree level Veltman condition givenin Eq. (2), as higher order loops always come withlog( M pl /µ ) [19], which will be cancelled if the run-ning couplings are evaluated at Planck scale [20].This is true, however, only if the complete beta func-tions for all the couplings are used to resum the com-plete order of logarithms.The gray hatched line in Fig. (2) depicts the lowerdirect Higgs mass search bound from LEP [23]. Thecoloured hatched lines give the combined exclusionlimits from ATLAS and CMS of 141 −
476 GeV at95% confidence level (CL) and 132 −
476 GeV at90% CL [1]. The experimentally favoured param-eter range in the m t – m h plane taking into accountdirect Higgs searches and and electroweak precisionmeasurements are indicated by the GFitter region [21, 22] in the plot. We show the dependence on thetop mass outside of the range 172 . − . . ± . tt cross-section favour a smaller value of 168 . +3 . − . GeV [25].We observe that most of the Higgs masses givenby different conditions tend to overlap in the vicin-ity of the best determined value of the top mass.The triviality bound, represented by the approxi-mate condition λ ( M pl ) = π , yields a range of Higgsmasses which is already excluded at 95% CL by theTevatron and LHC. The Higgs masses generated bythe other conditions however are still allowed andnot excluded yet by the experiments. The Veltmancondition is truncated at the point where its Higgsmass calculated with two-loop beta functions startsto cross the vacuum stability bound obtained by two-loop RGEs. This is done in order to show the exactcrossing point of these two conditions from two-loopRGEs.Throughout this work, we define a measure of theuncertainties involved in the calculation as the differ-ence between using one and two-loop beta functionsfor all the relevant SM couplings in the determina-tion of the Higgs pole mass. To be more precise, wedefine a “RGE error band” as the difference in deter-mining the Higgs mass for a boundary condition of λ ( M pl ) with one and two-loop beta functions whilethe matching conditions remain the same for bothcases. Possible errors due to the matching condi-tions will be discussed below. We caution the readerthat this procedure probably overestimates the er-ror stemming from neglecting higher order contri-butions to the beta functions. While there is nouniversally accepted way of estimating the theoret-ical uncertainties , there are other approaches todefine the theoretical error used in the literature.E.g. in Ref. [27], the authors define the theoret-ical error by the scale dependence of the matchingcondition λ ( M t ) and λ t ( M t ) while neglecting the ef-fect of higher order RGEs and arrive at an 3 GeVuncertainty. In Ref. [28], the authors estimate thetheoretical uncertainty by comparing the situationwhere matching has been performed at µ = M t tothe case µ = M Z and get an uncertainty of 2 . See [26] for a recent attempt at rigorously characterizing aperturbative theoretical uncertainty. EP exclusion (cid:158) (cid:37) CLLHC exclusion (cid:158) (cid:37) CLLHC exclusion (cid:158) (cid:37) CL (cid:72) (cid:76)(cid:72) (cid:76)(cid:72) (cid:76)(cid:72) (cid:76)(cid:72) (cid:76) Legend Λ (cid:73) M pl (cid:77) (cid:135) M (cid:73) M pl (cid:77) (cid:135) Β Λ (cid:73) M pl (cid:77) (cid:135) Λ (cid:73) M pl (cid:77) (cid:135) ΠΓ m (cid:73) M pl (cid:77) (cid:135) (cid:37) (cid:37) GFitter bound (cid:37) (cid:72) (cid:76)(cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76)
166 168 170 172 174 176110120130140150160170
Top mass (cid:72)
GeV (cid:76) H i gg s m a ss (cid:72) G e V (cid:76) Figure 2. Higgs and top (pole) mass determinations for different boundary conditions at the Planck scale. Thecoloured bands correspond to the conditions discussed in the text and which are also labelled in the insert. Themiddle of each band is the best value, while the width of the band is a “RGE error band” inferred from assumingthat all omitted higher orders in the beta functions beyond two loops are limited by the difference between the oneand two loop results. Note that the Veltman condition is truncated at the point where its Higgs mass predictionviolates the vacuum stability bound (both at two-loops). The gray-hatched line at the bottom is the lower directHiggs mass bound from LEP. Similarly the purple (brown) lines indicate the LHC Higgs searches at 95% (90%) CLfrom 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-tioned “RGE error band” is represented by the band-width of each curve, with its center representing theHiggs mass obtained from two-loop RGE running.The upper edges of the bandwidths consist of theHiggs masses obtained from one-loop RGEs.We consider also the uncertainty on the curvesdue to the error of strong coupling constant α s =0 . ± ± αα s correction [32] is neglected in our analysis as it onlygives a small contribution. The Higgs pole mass is5atched with λ at top pole mass scale, i.e. the renor-malization scale of λ is set to be at µ = M t . Sincethe higher order matching conditions for λ has notbeen calculated in any literature, only the expres-sion of δ H given in App. (B) will be used as ourmatching. If the matching of λ to the Higgs polemass is only performed at tree-level, the resultingdiscrepancy of the Higgs pole mass with respect tothe one-loop matching result is found to be less than1 GeV. Therefore, we can safely assume that higherorder matching condition for λ will not yield a largeruncertainty.The error estimation for the boundary condition β λ ( M pl ) = 0 is not the same as for the other con-ditions. A careful treatment of the Higgs mass ex-traction has to be implemented in this case. Theone-loop beta function of λ is a quadratic equationof λ , and for a given top mass we obtain two positivesolutions of the boundary condition β λ ( M pl ) = 0 atthe Planck scale, and thus obtain two equally validlow-energy Higgs mass determinations. In Fig.(2)these two branches of solutions generate a hook-shaped trajectory in the M H − M t plane. The hookends where λ starts to take negative value. Due tothe mismatch of the end of the trajectory when ei-ther one-loop or two-loop beta function is applied,a larger “RGE error band” has to be taken intoaccount, where we generate error bars that coverthe distance of the mismatch and plot a band tocover all the error bars. Besides the mismatch men-tioned above, there exists also another source of er-ror, namely number of loops of the beta functionsimplemented in the condition β λ ( M pl ) = 0. In prin-ciple one should apply the full beta function as theboundary condition, but in practice this is impossi-ble and therefore we have to check the possible un-certainty which arises due to the number of loops in β λ used in the boundary condition. The errors lie,however, within the band. A similar uncertainty dueto number of loops used as boundary condition alsoappears in the γ m ( M pl ) = 0 condition and its un-certainty is larger in comparison to the β λ ( M pl ) = 0condition.Another possible source of uncertainties on thedetermination of the Higgs mass from the Planckscale boundary conditions comes from the value ofthe Planck scale. The difference of the Higgs massesobtained from a certain boundary condition imposedat a different value of the Planck scale, e.g. µ = M pl / √ π , are negligible for most of the boundaryconditions. However the discrepancies are larger forthe lower branch of the β λ = 0 condition. We willnot discuss further these uncertainties.Throughout this work we assume that neutrinos do not play a significant role in our Higgs mass pre-diction. We would like to caution the reader thatneutrinos are indeed massive and could couple withthe O (1) Yukawa coupling to the Higgs, e.g. in thecanonical type I see-saw mechanism. For a neutrinomass of m ν ≈ O (1 eV) and a see-saw scale of approx-imately 10 GeV, the Yukawa coupling of neutrinowould be around O (1). This large coupling betweenHiggs boson and neutrino could alter the predictionon Higgs mass by modifying the beta function ofHiggs quartic coupling, as pointed out by Ref. [33].However the interesting case, i.e. SM plus extensionof neutrino sector with O (1) Yukawa coupling canbe valid up to Planck scale, has been excluded byrecent Tevatron and LHC searches [1]. We there-fore implicitly assume a scenario where the neutrinoYukawa couplings are not significantly larger thanfor example the bottom Yukawa coupling. III. IS A LOW HIGGS MASS FAVOURED?
All the generic boundary conditions which we dis-cussed prefer a low Higgs mass 127 GeV < ∼ M H < ∼
145 GeV which fits amazingly good to the existingexperimental direct lower and upper bounds fromLEP, Tevatron and LHC. These values fit further-more very well to those preferred by electroweak pre-cision measurement. One could ask therefore if thisis by chance or if it has a specific reason. Let ustherefore first look at random values of λ ( M pl ) pre-tending in this way that every value could be realizedby the new physics at the Planck scale. We assumetherefore for a moment, that all values of λ ( M pl ) inthe range of λ ∈ [0 , π ] have equal probability . Weshow therefore first in Fig.(3) the running of λ fromthe Planck scale to the weak scale for some valuesin the interval λ ∈ [0 , π ]. Note that most values of λ ( M pl ) lead to λ at the Fermi scale which is greaterthan 0.2, which corresponds to M H >
150 GeV.To analyse the effect further we randomly gen-erate 600 values for λ ( M pl ) and the top pole massranging from 170 −
175 GeV and put the resultingHiggs masses into the scatter plot Fig.(4). Note thatmost values of λ at the Planck scale lead to a Higgs In principle one could consider all values up to infinity sincethere seems to be no a priori reason to limit λ ( M pl ). Ex-tending the upper end is however only strengthening the ar-guments, since this would favour even more a higher Higgsmass. Note that this is connected to the triviality boundand the corresponding focussing of RGE trajectories to-wards low scales. See Fig.(3) for this effect. (cid:76) (cid:72) GeV (cid:76) Λ (cid:72) (cid:76) (cid:76) Figure 3. Running of λ from the Planck scale to theFermi scale. Different values of λ ( M pl ) and it can beseen that a large parameter space of λ ( M pl ) tends toinduce λ ( M t = 173 GeV) > ∼ .
186 which is equivalent toHiggs mass greater than 150 GeV.
Excluded Region byLHC (cid:158) (cid:37) CL Λ (cid:73) M pl (cid:77) H i gg s m a ss (cid:72) G e V (cid:76) Figure 4. Scatter plot of Higgs mass at the Fermi scaledetermined by random λ at the Planck scale with ran-dom top mass constrained to the interval [170 , mass between 160 GeV and 175 GeV. For the choseninterval λ ( M pl ) ∈ [0 , π ] we find ≈
90% in this rangewhich has been excluded by the Tevatron and by theCMS and ATLAS experiments. Note that only lessthan 5% of the generated Higgs masses are allowedby experiments.Looking at Fig.(4) one immediately notices that λ ( M pl ) = 0 which corresponds to M H = 127 GeV is a special condition which fits very well to the ex-perimental findings. This is also the vacuum stabil-ity bound and it corresponds therefore to the light-est Higgs mass in the SM when it is valid up tothe Planck scale. A lower Higgs mass would requiresome extra new physics at lower scales, which wouldrule out the logic of our paper. Including errors thisleads to a lower bound M H > ∼
122 GeV for our sce-nario.Fig.(4) also shows that it is non-trivial that allthe boundary conditions which we discuss in thispaper lead to viable Higgs masses. However, thereis a systematic understanding why all our condi-tions work. The point is that all our conditionssuch as β λ ( M pl ) = 0 or Str M ( M pl ) = 0 are con-ditions connected to quantum loops for the Higgsmass parameter m and the quartic coupling λ andwe can therefore always write λ as a function of thegauge couplings and the top Yukawa coupling, i.e. λ = f ( g i , λ t ). This leads generically to smaller val-ues than a random choice, since loop factors, i.e. fac-tors of 1 / π are present and since top mass loops(fermion loop) carry a minus sign, which leads tocancellations, pushing the value of λ even smaller.One can therefore understand why the cases whichwe discussed all systematically lead to viable Higgsmass predictions. In that sense none of them is spe-cial and there may exits more interesting boundaryconditions which also lead to viable Higgs masses.This also implies that a Higgs mass in the currentlymost favoured range does not clearly select anymodel or scenario which leads to a boundary condi-tion that has such loop and top mass suppression fac-tors. However, vice versa one might argue that cur-rent data point to boundary conditions which mustinvolve such suppression factors, which is an inter-esting observation. It is also intriguing to ask in thiscontext why the top mass is much heavier than otherquarks such that it compensates other loop contri-butions which would drive the Higgs boson heav-ier. This appears to be an interesting “conspiracy”in favour of the logic of this paper which works ifthe boundary condition is imposed at a high scalelike the Planck scale. All these arguments break ofcourse down if LHC or any future experiment detectsany sign of new physics that couples directly to theHiggs boson. Even an indirect coupling, i.e. radia-tive correction to λ at loop level is severe enoughto alter the running of λ drastically. So far thereis no compelling evidence for additional physics be-yond the SM, whereas the SM Higgs search seemsto indicate some excesses of Higgs like events in therange of 130 GeV to 140 GeV, albeit at only about2 σ [1].7 V. DISCUSSION AND CONCLUSIONS
The LHC has an excellent chance to find the SMHiggs boson and we emphasize in this paper thatthe left-over values lie in a range which is well moti-vated by various Planck scale boundary conditions.We argued that this Higgs mass range is special andthat it might be related to embeddings of the SMas a QFT into some form of quantum gravity, whichis based on concepts beyond
QFT. The SM wouldstill be an effective field theory which is valid upto the Planck scale, but the asymmetry in the con-cepts might allow to understand the famous hierar-chy problem from the perspective of the new con-cepts at the Planck scale, while it would only ap-pear unnatural from the low energy point of view. Inother words: The large hierarchy between the Planckand electroweak scale might only be a problem aslong as we look at it from the SM perspective. Animportant point is then that such a scenario makesonly sense if there are no intermediate scales, sincethis would require a QFT solution of the hierarchybetween the electroweak and intermediate scales.The proposed scenario requires that the Higgscoupling can be evolved to the Planck scale, whichimplies strict lower and upper bounds on the cou-pling. The upper bound is the so-called trivial-ity bound which is approximately 170 GeV and itis interesting to note that the Higgs mass is belowthis value, as otherwise the couplings could not beevolved up to the Planck scale. The lower bound isthe so-called vacuum stability bound, i.e. the con-dition λ ( M pl ) = 0. We carefully evaluated its valueand error with the latest data at two loops and pro-vided an error estimate. For M t = 173 . m H = 127 ± λ ( M pl ) = 0, is very spe-cial. It implies that the Higgs self-interaction at theelectroweak scale is entirely generated by radiativecorrections of the RGE evolution from a vanishingcoupling at the Planck scale. The Higgs mass wouldtherefore be connected to the gauge and Yukawacouplings which enter into the RGEs.Several comments should be carefully consideredin this context:1. The Higgs central value m H = 127 GeV is ob-tained via two-loop beta function running fromthe vacuum stability condition at the Planckscale to the weak scale regime. Fig. (5) showsthe uncertainties due to the omission of higherorders to be ± (cid:72) GeV (cid:76) H i gg s m a ss (cid:72) G e V (cid:76) Figure 5. A blow up of the vacuum stability bound inthe interesting Higgs and top mass region. The blueline in the center represents the vacuum stability boundobtained via two-loop beta functions, which has beenthoroughly discussed in main text. The yellow bandrepresents the uncertainties of the Higgs mass obtainedvia two-loop RGEs due to α s uncertainties. The outerblue band is identical to the blue band in Fig. (2) andit represents the full “RGE error band” estimated fromdifference between one- and two-loop RGEs. With thebest world average top pole mass 173 . λ ( M pl ) = 0 is 127 ± able way to arrive at a conservative error esti-mation for the Higgs mass due to the lack ofhigher order RGEs, but it should not be overinterpreted. This implies that the exact lowerbound for the Higgs mass is limited by thisconservative estimation.2. Precision top mass analysis is required to de-termine the exact value of the Higgs mass pre-dicted via vacuum stability. The reason whywe want to stress on this specific result is thatto date, there is no general consensus on whattype of top mass is actually measured via kine-matic reconstruction [34]. At the Tevatron,the main method used for the top mass ex-traction actually “measures” the Pythia mass,which is a Monte-Carlo simulated templatemass. Strictly speaking the top pole mass isnot a well defined quantity, as the top quarkdoes not exist as free parton. The top massthat the Tevatron has measured is based onthe final state of the decay products. On theother hand the running MS top mass can beextracted directly from the total cross sectionin the top pair production. In this sense, onecan obtain a complementary information of8he top mass from the production phase. Byconverting the MS mass to the pole mass viamatching conditions, the top pole mass value168 . +3 . − . GeV extracted with this method byLangenfeld et al. [25] is found to be lower thanthe world best average value. However, thisway of extracting the top mass suffers fromlarger numerical uncertainties. As we can beseen from Figs. (2) and (5), a change of the topmass by 2 GeV changes the Higgs mass predic-tion by 6 GeV.3. The electroweak vacuum might in principlebe metastable. However, most of the Higgsmass region for metastability has already beenruled out by LEP, although not entirely ex-cluded. The finite temperature metastabilityregion, however, with the local SM assumedto be stable against thermal fluctuations upto Planck scale temperatures, allows the en-tire region from the LEP bound to the vac-uum stability. Hence should LHC discover theSM Higgs boson with its mass lower than theone predicted by two-loop RGE vacuum sta-bility bound, there is a possibility that theSM electroweak vacuum is not the stable one(see Refs. [14, 27, 31, 35–37] for more de-tails). However, we would also like to re-mind that metastability bounds depend on thefastest process conceived for the transition tothe true vacuum. Any faster process occurringonce anywhere in the Universe would reduce oreliminate the metastability region.We discussed in Fig. (4) that only a small percent-age of randomly generated boundary conditions for λ ( M pl ) lead to Higgs masses which are still allowedby experiments. On the other hand we presentedin Fig. (2) results for a set of boundary conditionswhich all lead to Higgs masses in the allowed regionand we explained how this can be systematically un-derstood. The point is that the chosen boundaryconditions emerge from conditions which have loopsuppression factors, making them rather small com-pared to a random choice. Fig. (2) shows that thereexist different working conditions and others mightbe found which work as well. This implies that onceloop suppression factors have been included into theboundary condition it is not easily possible to distin-guish between various models or scenarios, but grow-ing precision will nevertheless reduce the number ofoptions somewhat. The conclusion from this obser-vations is in the scenario of this paper that quantumgravity should not generate a random value of λ atthe Planck scale, but it should somehow select con- ditions that lead to a small Higgs self-coupling λ atthe Planck scale. These conditions could possibly besome remnant of symmetry in the full quantum the-ory of gravity. The Yukawa and λ couplings could ormaybe even should have a common origin from thePlanck scale physics, as the top quark contributionmiraculously cancels the contribution of the Higgsboson such that the SM can be extrapolated to thePlanck scale.We have shown in this paper that the chosenboundary conditions for λ yield a Higgs mass whichis in the allowed range. The fact that differentboundary conditions have overlapping regions couldimply that more than one of them is simultane-ously realized at the Planck scale, which is an in-triguing possibility. For instance if we demand that λ ( M pl ) = 0 and Str M ( M pl ) = 0 are both satisfiedthen it is possible that the Higgs quartic coupling isonly generated radiatively and the quadratic diver-gence vanishes. This appears puzzling from a lowenergy perspective, however, from the Planck scalephysics perspective this could be natural, as thesetwo conditions could have a common origin of someunknown connection between the gauge, Yukawaand Higgs quartic couplings. Other combinations oflisted boundary conditions can also be considered, asfor instance those proposed in Ref. [9] where the au-thors obtained from the asymptotic safety of gravity λ ≈ β λ ≈ . − . Thisprecision is encouraging, but unfortunately the de-termination of the high energy boundary conditionsis plagued by the relatively large uncertainties dueto the lack of higher order RGEs. Three loop betafunctions and other improvements would be requiredin order to reduce the errors of the band in Fig. (2).Without these theoretical improvements it will behard to differentiate the different boundary condi-tions. However, such improvements would be im-portant if the SM would indeed be a theory which isvalid up to the Planck scale. Precise determinationsof the Higgs and top masses could then be used toidentify the correct boundary conditions for λ withinthe remaining uncertainties. A complete calculationof the three-loop beta functions of the SM wouldbe very important in this case. This would, how-ever, also require the determination of the matchingconditions to at least two-loop order. While the two-9oop QCD matching condition has been included for λ t ( m t ) in this work, the corresponding contributionto λ ( m t ) is not known and would be necessary toreduce the matching uncertainty alongside the othertwo-loop contributions to λ ( m t ) and λ t ( m t ).With around 5 fb − of integrated luminosity col-lected by ATLAS and CMS, it is possible to findor exclude a SM Higgs mass in the region from114 −
600 GeV by the end of next year. A discov-ery of SM Higgs boson will anyhow be significantfor advancement of particle physics. There existsan excellent chance to find all sort of TeV-scale newphysics, but if the LHC finds nothing but a SM Higgsthen this would be very much in favour of the spiritof this paper. A key question would be which newconcepts could be involved in quantum gravity suchthat the correct SM boundary conditions arise.
Note added:
After this work has been completed,ATLAS has reported the latest Higgs mass exclu-sion regions at 95% CL [41] ranging from 112 . − . −
237 GeV and 251 −
453 GeV whileCMS has excluded the Higgs masses in the range127 −
600 GeV [42].
Acknowledgements:
We would like to thank E. Gross for interesting discussions on the exact valueof the vacuum stability bound. M.H. acknowledgessupport by the International Max Planck ResearchSchool for Precision Tests of Fundamental Symme-tries.
Appendix A: List of beta functions andanomalous dimension of Higgs mass
We give the beta function and anomalous dimen-sion of Higgs mass used in our calculation. The betafunction for a generic coupling X is given as: µ d X d µ = β X = (cid:88) i β ( i ) X (16 π ) i (A1)and the anomalous dimension for Higgs mass is givenas: µ d m d µ = γ m = (cid:88) i γ ( i ) m (16 π ) i (A2)The list of beta functions are given below [19, 43–46]: β (1) λ = λ ( − g − g + 12 λ t ) + 24 λ + 34 g + 38 ( g + g ) − λ t , (A3) β (1) λ t = 92 λ t + λ t (cid:18) − g − g − g (cid:19) , (A4) β (1) g = 416 g , β (1) g = − g , β (1) g = − g , (A5) β (2) λ = − λ − λ λ t + 36 λ (3 g + g ) − λλ t + λλ t (cid:18) g + 452 g + 856 g (cid:19) − λg + 394 λg g + 62924 λg + 30 λ t − λ t g − λ t g − λ t g + 212 λ t g g − λ t g + 30516 g − g g − g g − g (A6) β (2) λ t = λ t (cid:18) − λ t + λ t (cid:18) g + 22516 g + 36 g − λ (cid:19) + 1187216 g − g g + 199 g g − g + 9 g g − g + 6 λ (cid:19) (A7) β (2) g = g (cid:18) g + 92 g + 443 g − λ t (cid:19) (A8) β (2) g = g (cid:18) g + 356 g + 12 g − λ t (cid:19) (A9) β (2) g = g (cid:18) g + 92 g − g − λ t (cid:19) , (A10)10 (1) m = m (cid:18) λ + 6 λ t − g − g (cid:19) (A11) γ (2) m = 2 m (cid:18) − λ − λλ t + 12 λ (3 g + g ) − λ t + 20 g λ t + 458 g λ t + 8524 g λ t − g + 1516 g g + 15796 g (cid:19) (A12) Appendix B: Matching of MS coupling constantand pole mass The MS gauge couplings are used in this work,therefore no matching of the gauge couplings is nec-essary. The boundary conditions for MS g ( M Z )and g ( M Z ) couplings are taken from the value offine structure constant ˆ α ( M Z ) = 127 . ˆ θ W ( M Z ) = 0 . α − ( M Z ) ≡ π g ( M Z ) + g ( M Z ) g ( M Z ) g ( M Z ) , (B1)sin ˆ θ W ( M Z ) ≡ g ( M Z ) g ( M Z ) + g ( M Z ) . (B2) The strong coupling constant g ( M Z ) can be ex-tracted from α s ( M Z ) = 0 . δ t can be splitinto three part [30]: δ t ( µ ) = δ QCD t ( µ ) + δ QED t ( µ ) + δ W t ( µ ) (B3)where δ QCD t denotes the contribution of QCD cor-rection, which we will take up to three-loop order[47]. The term δ QED t ( µ ) + δ W t ( µ ) contributes to thematching correction from the QED and electroweaksector. The matching terms at µ = M t have beencalculated to be: δ QCD t ( M t ) = − α s ( M t )3 π − . (cid:18) α s ( M t ) π (cid:19) − . (cid:18) α s ( M t ) π (cid:19) (B4) δ QED t ( M t ) + δ W t ( M t ) = − α ( M t )9 π + M t π v (cid:34) − M H M t − M H M t (cid:18) M t M H − (cid:19) / arccos (cid:18) M H M t (cid:19) + M H M t (cid:18) M H M t − (cid:19) log M H M t (cid:21) − . × − + 1 . × − log M H
300 GeV − . × − log M t
175 GeV (B5)Note that the above formula for δ QED t + δ W t is onlyvalid for M H < M t , which is our case in the aboveanalysis.As for the matching of λ and the Higgs pole massgiven by Eq. (5), the matching correction δ H ( M t ) is given as [48]: δ H ( M t ) = M Z π v (cid:2) ξf ( ξ ) + f ( ξ ) + ξ − f − ( ξ ) (cid:3) (B6)where ξ ≡ M H /M Z and each of the function f i de-fined as: f ( ξ ) = 6 log M t M H + 32 log ξ − Z (cid:18) ξ (cid:19) − Z (cid:18) c w ξ (cid:19) − log c w + 92 (cid:18) − π √ (cid:19) , (B7)11 ( ξ ) = − M t M Z (cid:20) c w − M t M Z (cid:21) + 3 c w ξξ − c w log ξc w + 2 Z (cid:18) ξ (cid:19) + (cid:18) c w s w + 12 c w (cid:19) log c w −
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