I discuss the stability of the SM scalar potential in view of the discovery
of a Higgs boson with mass around 125 GeV. The role played by the top quark
mass in the choice between the full stability and the metastability conditions
is analyzed in detail. The present experimental value of the top mass do not
support the possibility that the SM potential is stable up to the Planck scale
but favor an electroweak vacuum sufficiently long-lived to be metastable.Comment: 7 pages, 6 figures. To appear on the Proceedings of LC13 Workshop:
  exploring QCD from the infrared regime to heavy flavour scales at
  B-factories, the LHC and a Linear Collider, 16-20 September 2013, ECT*, Villa
  Tambosi, Villazzano (TN), Ital

Degrassi, Giuseppe

English

arXiv

I discuss the stability of the SM scalar potential in view of the discovery of a Higgs boson with mass around 125 GeV. The role played by the top quark mass in the choice between the full stability and the metastability conditions is analyzed in detail. The present experimental value of the top mass do not support the possibility that the SM potential is stable up to the Planck scale but favor an

electroweak vacuum sufficiently long-lived to be metastable

Degrassi, G.

Scientific Open-access Literature Archive and Repository

DOI 10.1393/ncc/i2014-11735-1Colloquia: LC13IL NUOVO CIMENTO Vol. 37 C, N. 2 Marzo-Aprile 2014The role of the top quark in the stability of the SM Higgs potentialG. DegrassiDipartimento di Matematica e Fisica, Universita` di Roma Tre e INFN, Sezione di Roma TreRoma, Italyricevuto il 17 Febbraio 2014Summary. — I discuss the stability of the SM scalar potential in view of thediscovery of a Higgs boson with mass around 125GeV. The role played by the topquark mass in the choice between the full stability and the metastability conditionsis analyzed in detail. The present experimental value of the top mass do not supportthe possibility that the SM potential is stable up to the Planck scale but favor anelectroweak vacuum sufficiently long-lived to be metastable.PACS 14.80.Bn – Standard-model Higgs bosons.PACS 14.65.jk – Other quarks (e.g., 4th generations).1. – Vacuum stability analysisWith the discovery at the LHC of a new resonance [1] with mass around 125–126GeVand properties very compatible to those of the Standard Model (SM) Higgs boson thecomplete particle spectrum of the SM is now known. The first run of the LHC hasdelivered two important messages: i) no signal of physics beyond the SM (BSM) wasdiscovered, ii) the Higgs boson was found where predicted by the SM. Figure 1 shows theprobability density function for the SM Higgs boson mass obtained combining the infor-mation from precision measurements with the results of the Higgs search experiments,the latter expressed in terms of the likelihood of the search experiment normalized to theno-signal case [2]. In the figure only the experimental results from LEP and Tevatronbefore the turning on of the LHC are used. As shown from the figure the SM had a sharpprediction: the mass of the Higgs boson had to be between 114 and 160GeV. Indeed theHiggs boson was found by ATLAS and CMS exactly in that interval.The fact that all the parameters of the SM have been now experimentally determinedconstrains tightly the model and possibly BSM physics. New Physics (NP), if exists,should be of the decoupling type, i.e. it should have a marginal effect on the SM elec-troweak fit without spoiling its very good agreement with the experimental results. Thisfact, together with the negative result of the run I of the LHC, may put some doubtson the expectation that NP has to be “around the corner”, i.e. within the reach of thec© Societa` Italiana di Fisica 4748 G. DEGRASSI100 120 140 160 180 2000.000.020.040.060.080.100.12mH GeV ]pdfmH95 % Prob . Upper Lim.68 % Prob . Upper Lim.EW fit + LEP + T evatron[()Fig. 1. – Probability density function for the Higgs mass obtained combining the indirect infor-mation coming from precision physics with the direct search results from LEP and Tevatron.Courtesy of S. Di Vita.LHC. In this situation it is natural to ask where the scale of NP is, or if it can be aslarge as the Planck scale, MPl, implying that the validity of the SM can be extended upto MPl.One approach to answer this question is to study the stability of the SM vacuum, orif the electroweak (EW) minimum we live in is the true minimum of the SM effectivepotential, i.e. the radiatively corrected scalar potential. The effective potential, in firstapproximation, has the the same form as the tree-level one but with running parameters(μ is the renormalization scale)(1) V eff ≈ −12m2(μ)φ2(μ) + λ(μ)φ4(μ) ∼ λ(μ)φ4(μ),then if we are looking at large values of the Higgs field, φ(μ)  v where v is the EWminimum, the dominant contribution to the potential is from the quartic term.The search for the scale where V eff becomes smaller than its value at the EW mini-mum, i.e. the instability scale ΛI , can be replaced, given the steepness of the potentialaround that point, by looking for the scale where V eff = 0 or, for large values of the field,where λ(μ) = 0. The Higgs quartic coupling is special among the SM couplings. Indeedλ is the only SM coupling that is allowed to change sign during the RenormalizationGroup (RG) evolution because it is not multiplicatively renormalized. For all other SMcoupling the β functions are proportional to their respective couplings and crossing zerois not possible. In fact, one finds for βλ ≡ dλ/d lnμ at the one loop level(2) βλ =116π2[+24λ2 + λ(4NcYt − 9g2 − 3g′2)− 2NcY 4t + 98g4 +38g′4 +34g2g′2],where Nc = 3 is the color factor of SU(3)c, Yt the top Yukawa coupling and g and g′the SU(2)L and U(1)Y gauge couplings, respectively. On the r.h.s. of eq. (2) the partTHE ROLE OF THE TOP QUARK IN THE STABILITY OF THE SM HIGGS POTENTIAL 49Fig. 2. – Left: Evolution of the SM gauge couplings g1 =p5/3gY , g2, g3, of the top, bottomcouplings (yt, yb) of the Higgs quartic coupling λ and of the Higgs mass parameter m. Allparameters are defined in the MS scheme. Right: Zoom on the evolution of the Higgs quartic,with uncertainties in Mt, αs and Mh as indicated. Plots taken from ref. [5].not proportional to λ contains the top Yukawa coupling at the fourth power and with anegative sign. Thus, for small values of λ this is the term dominating βλ and λ is goingto evolve towards smaller values eventually crossing zero.In fig. 2 (left) the evolution in the SM of the gauge, Yukawa and scalar couplings isshown. The running of the various couplings has been determined using the state-of-the-art computations, i.e. three-loop beta functions [3] and two-loop matching conditions [4,5]. The three gauge couplings and the top Yukawa coupling remain perturbative andare fairly weak at high energy, becoming roughly equal, within 10%, around a scale ofabout 1016 GeV. It is amusing to note that the ordering of the coupling constants at lowenergy is completely overturned at high energy with the (GUT normalized) hyperchargecoupling g1 =√5/3gY being the largest coupling. The evolution of λ is zoomed in theright part of fig. 2. The Higgs quartic coupling remains weak in the entire energy domainbelow MPl. It decreases with energy crossing λ = 0, for the central values of top mass,Mt, the strong coupling, αs, and the Higgs mass, Mh, at a scale of about 1010 GeV.The fact that λ becomes negative at a scale lower than MPl is a signal that theeffective potential is unstable, i.e. at high scale is either not bounded from below or itdevelops a second minimum that can be deeper than the EW one. In both cases the ideathat the SM can be considered a valid theory up to MPl is in trouble because v is nolonger the true minimum of the potential and there is a tunnelling probability betweenthe false vacuum v and the true vacuum at high field values. However, we can infer thatNP must appear below ΛI to cure the instability of the SM potential only if the lifetimeof EW vacuum is shorter than the life of the universe.The rate of quantum tunnelling out of the EW vacuum, given by the probabilityd℘/dV dt of nucleating a bubble of true vacuum within a space volume dV and timeinterval dt, was first computed in the late seventies by S. Coleman [6]. The total prob-ability ℘ for vacuum decay to have occurred during the history of the universe can be50 G. DEGRASSIFig. 3. – Left: SM phase diagram in terms of Higgs and top pole masses. The plane is divided intoregions of absolute stability, meta-stability, instability of the SM vacuum, and non-perturbativityof the Higgs quartic coupling. The dotted contour-lines show the instability scale ΛI in GeVassuming αs(MZ) = 0.1184. Right: Zoom in the region of the preferred experimental range ofMh and Mt (the grey areas denote the allowed region at 1, 2, and 3σ). Plots taken from ref. [5].computed by integrating d℘/dV dt over the space-time volume of our past light-cone, or(3) ℘ ∼ τ4UΛ4B e−S(ΛB), S(ΛB) =8π23|λ(ΛB)| ,where τU is the age of the universe and S(ΛB) is the action of the bounce of size R = Λ−1B .ΛB is determined as the scale at which Λ4Be−S(ΛB) is maximized [7]. In practice thisroughly amounts to minimizing λ(ΛB), which corresponds to the condition βλ(ΛB) = 0.By numerical inspection of ℘ in eq. (3) one finds that the exponential suppression winsover the large 4-volume factor if |λ(ΛB)| is less than ∼ 0.05.Figure 2 shows that λ in its RG evolution towards MPl does become negative butnever too negative. In fact the running of λ is slowing down at high energy because its βfunction at high scale becomes very small, vanishing close to MPl. At the Planck scaleone then finds [5]λ(MPl) = −0.0113 + 0.0029(MhGeV− 125.66)− 0.0065(MtGeV− 173.10)(4)+0.0018(αs(MZ)− 0.11840.0007),that implies that our vacuum is metastable, i.e. ℘ is extremely small (less than 10−100)or the lifetime of the EW vacuum is extremely long much larger than τU .The study of the two-loop effective potential [8] allows us to identify the phases ofthe SM. They are shown in fig. 3 as a function of the Higgs and top masses. Theregions of stability, metastability, and instability of the EW vacuum are shown both fora broad range of Mh and Mt, and after zooming into the region corresponding to theTHE ROLE OF THE TOP QUARK IN THE STABILITY OF THE SM HIGGS POTENTIAL 51measured values. The uncertainty from αs and from theoretical errors is indicated by thedashed lines and the color shading along the borders. Also shown are contour lines of theinstability scale. The measured values of Mh and Mt appear to be rather special, in thesense that they place the SM vacuum at the border between stability and metastability.In the neighborhood of the measured values of Mh and Mt, the stability condition is wellapproximated by(5) Mh > 129.1GeV+ 2.0(Mt − 173.10GeV)− 0.5GeVαs(MZ)− 0.11840.0007 ± 0.3GeV.Since the experimental error on the Higgs mass is already fairly small and will be furtherreduced by future LHC analyses, it is becoming more appropriate to express the stabilitycondition in terms of the pole top mass or(6) Mt < (171.53± 0.15± 0.23αs ± 0.15Mh) GeV.2. – The role of the top quarkAs is clear from the right plot in fig. 3, it is the exact value of the top mass, ratherthan a further refined computation, the factor that can discriminate between a stableand a metastable EW vacuum. Figure 3, as well as the bound (6), are obtained usingas renormalized mass for the top quark the so-called pole mass and identifying it withthe average of the Tevatron, CMS and ATLAS measurements, Mt = 173.10 ± 0.6GeV.This identification can be disputed in two aspects. i) From a theoretical point theconcept of pole mass for a quark is not well defined as quarks are not free asymptoticstates. Furthermore the quark pole mass is plagued with an intrinsic non-perturbativeambiguity of the order of ΛQCD due to the so-called infrared (IR) renormalon effects.ii) The top mass parameter extracted by the experiments, which we call MMCt , is anobject that is obtained via the comparison between the kinematical reconstruction of thetop quark decay products and the Monte Carlo simulations of the corresponding event.The latter requires a careful modeling of the jets, missing energy, initial state radiationcontributions as well as of the hadronization part. MMCt is a parameter sensitive tothe on-shell region of the top quark but it cannot be directly identified with the polemass. We can write generically Mpolet = MMCt + Δ with the understanding that theerror quoted by the experimental collaborations refers to MMCt and not to Mpolet . Thepoint now is what is the size of Δ. An analysis of the phase-space regions in the topproduction cross-section at hadron colliders shows that the region possibly sensitive toIR effects contributes very little to the total rate. Then, even assuming an uncertaintyof 100% in the modelling of that region, the extraction of MMCt from the total ratewill be affected only at the level of ∼ 30 MeV. Thus we can conclude that MMCt canbe interpreted as Mpolet within the intrinsic ambiguity in the definition of Mpolet , thatimplies Δ ∼ O(ΛQCD) ∼ 250–500MeV [9].It is well known that short distance masses, such the one defined in the MS scheme, donot suffer from the IR renormalon problem. The MS top mass, MMSt , can be extracteddirectly from the total production cross section for top quark pairs σ(tt¯ + X). A recentanalysis reports MMSt (Mt) = 163.3 ± 2.7 [10], a value that translated in terms of polemass gives for Mpolet a central value very close to that obtained via the decay products52 G. DEGRASSI[GeV]tm160 170 180 190Probability density00.050.10.15Fig. 4. – Indirect determination of the top pole mass from EW precision observables. Courtesyof S. Mishima and L. Silvestrini.but a much larger error(7) MMSt (Mt) = 163.3± 2.7GeV→Mpolet = 173.3± 2.8GeV.The use of MS masses in the EW theory requires some specification. Differently fromQCD, in the EW theory masses, as well as v, are not parameters of the EW Lagrangian.The parameters are the gauge, Yukawa and the scalar couplings. This implies that thedefinition of an MS mass is not unique: it depends upon the definition of the vacuum.Indeed we can define the vacuum either as the minimum of the tree-level scalar potentialor as the minimum of the radiatively corrected potential. In the first case we get anMS mass that is gauge invariant, but there will be large EW corrections in the relationbetween the pole and the MS mass [11]. In the case of the top quark they are proportionalto M4t implying that if we want to extract directly MMSt (Mt) from σ(tt¯+X) we have toconsider the EW radiative corrected cross section. In the second case, when the vacuumis defined via the corrected potential, these large corrections are absent however theresulting MS mass is not a gauge-invariant object. Although this fact can seem quiteawkward we should remember that an MS mass is not a physical object and thereforegauge invariance is not a mandatory requirement.It is clear that, because MMSt is determined with an error much larger than that ofMpolet its use in the analysis of vacuum stability will weaken the conclusion that the EWvacuum is metastable while admitting, within 1 σ in the top mass error, the possibilityof full stability. However, we can take a different point of view: the top pole mass isthe same object that enters the EW fit and it can be predicted now that we know theHiggs mass quite accurately. Is the Mpolet value obtained from the fit compatible withthe bound (6)? The answer is in fig. 4 where the probability density function for Mpoletis shown with the dark (light) region corresponding to 1(2) σ interval. From the figureit is clear that values of Mt around 171GeV are in the tail of the distribution with aprobability of few per cent.THE ROLE OF THE TOP QUARK IN THE STABILITY OF THE SM HIGGS POTENTIAL 533. – ConclusionsThe SM is in a very good status. The value of the Higgs mass found by ATLAS andCMS is very intriguing. It causes the SM potential to be at the border of the stabilityregion. The exact value of the top mass plays the central role between the full stabilityor the metastability (preferred) options. The possibility of λ > 0 up to MPl requiresa top mass value around 171GeV, a number not preferred by the EW fit. Finally, thefact that our EW vacuum is metastable with a lifetime much longer than the age of theuniverse does not allow us to conclude that NP must appear at a scale lower than thePlanck scale.∗ ∗ ∗I would like to thank the organizers of LC13 for their kind invitation. I am in debtwith all the people I have collaborated with on this subject: D. Buttazzo, S. Di Vita,J. Elias-Miro´, J. R. Espinosa, P. P. Giardino, G. Giudice, G. Isidori, F. Sala, A. Salvio andA. Strumia. Interesting discussions with F. Maltoni and P. Slavich are also acknowledged.This work was partially supported by the Research Executive Agency (REA) of theEuropean Union under the Grant Agreement number PITN-GA-2010-264564 (LHCPhe-noNet).REFERENCES[1] Aad G. et al. (ATLAS Collaboration), Phys. Lett. B, 716 (2012) 1 [arXiv:1207.7214];Chatrchyan S. et al. (CMS Collaboration), Phys. Lett. B, 716 (2012) 30[arXiv:1207.7235].[2] D’Agostini G. and Degrassi G., Eur. Phys. J. C, 10 (1999) 663 [hep-ph/9902226].[3] Tarasov O. V., Vladimirov A. A. and Zharkov A. Y., Phys. Lett. B, 93 (1980) 429;Larin S. A. and Vermaseren J. A. M., Phys. Lett. B, 303 (1993) 334 [hep-ph/9302208];Mihaila L. N., Salomon J. and Steinhauser M., Phys. Rev. Lett., 108 (2012) 151602[arXiv:1201.5868]; Phys. Rev. D, 86 (2012) 096008 [arXiv:1208.3357]; Chetyrkin K. G.and Zoller M. 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The role of the top quark in the stability of the SM Higgs potential

I discuss the stability of the SM scalar potential in view of the discovery of a Higgs boson with mass around 125 GeV. The role played by the top quark mass in the choice between the full stability and the metastability conditions is analyzed in detail. The present experimental value of the top mass do not support the possibility that the SM potential is stable up to the Planck scale but favor an electroweak vacuum sufficiently long-lived to be metastable. PACS 14.80.Bn - Standard-model Higgs bosons. PACS 14.65.jk - Other quarks (e.g., 4th generations). © Società Italiana di Fisica

The role of the top quark in the stability of the SM Higgs potential

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