Higgs boson mass and new physics

We discuss the lower Higgs boson mass bounds which come from the absolute stability of the Standard Model (SM) vacuum and from the Higgs inflation, as well as the prediction of the Higgs boson mass coming from the asymptotic safety of the SM. We account for the three-loop renormalization group evolution of the couplings of the SM and for a part of the two-loop corrections that involve the QCD coupling αs to the initial conditions for their running. This is one step beyond the current state-of-the-art procedure (“one-loop matching-two-loop running”). This results in a reduction of the theoretical uncertainties in the Higgs boson mass bounds and predictions, associated with the SM physics, to 1–2 GeV. We find that with the account of existing experimental uncertainties in the mass of the top quark and αs (taken at the 2σ level) the bound reads MH ≥ Mmin (equality corresponds to the asymptotic-safety prediction), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ {{M}_{{\min }}}=\left( {129\pm 6} \right) $\end{document} GeV. We argue that the discovery of the SM Higgs boson in this range would be in agreement with the hypothesis of the absence of new energy scales between the Fermi and Planck scales, whereas the coincidence of MH with Mmin would suggest that the electroweak scale is determined by Planck physics. In order to clarify the relation between the Fermi and Planck scales a construction of an electron-positron or muon collider with a center-of-mass energy ~ (200 + 200 GeV) (Higgs and t-quark factory) would be needed.


Introduction
The mass M H of the Higgs boson in the SM is an important indicator of the presence of new energy scales in particle physics. It is well known that if M meta min < M H < M Landau max then the SM is a consistent effective field theory all the way from the Fermi scale up to the (reduced) Planck scale M P = 2.44 × 10 18 GeV. The upper limit comes from the requirement that the Landau pole in the scalar self-coupling 1 must not appear at energies below M P [1][2][3]. The lower limit comes from the requirement of the stability of the SM vacuum against tunneling to the states with the Higgs field φ exceeding substantially the electroweak value 250 GeV [4][5][6] (see figure 1).
The estimates of M Landau max give a number around 175 GeV [1][2][3]7] which is in the M H range excluded (at least in the range 129 − 525 GeV) by the searches for the SM Higgs JHEP10(2012)140 boson at the LHC and the Tevatron [8,9]. In other words, we already know that the SM is a weakly coupled theory up to the Planck scale.
One can distinguish between two types of the stability bounds. If M H > M stability min , the electroweak vacuum is absolutely stable, whereas if M meta min < M H < M stability min , then it is metastable with a life-time exceeding that of the Universe. Numerically, M meta min ≃ 111 GeV [10]. The existence of the Higgs boson with a mass smaller than M meta min would provide an undisputed argument in favor of the existence of new physics between the Fermi and Planck scales. However, already since LEP we know that this is not the case.
The Higgs mass M stability min is not at all special from the point of view of the validity of the SM up to the Planck scale. The value of M stability min , however, plays a crucial role if the SM is embedded to a bigger picture which includes gravity. First, only if M H > M stability min , the Higgs boson of the SM can play the role of the inflaton and make the Universe flat, homogeneous and isotropic, and produce a necessary spectrum of perturbations needed for structure formation [11,12]. Second, M H = M stability min is a prediction of an asymptotically safe scenario of the SM [13], making it consistent up to an arbitrarily large scale.
Thus, we will focus on the upgrade of existing computations of M stability min and on the discussion of the significance of the relation between the Higgs boson (to be discovered yet) mass M H and M stability min for beyond-the-SM (BSM) physics. The computation of M stability min has been already done in a large number of papers [10,12,[14][15][16][17]. It is divided into two parts. The first one is the determination of the MS parameters from the physical observables and the second one is the renormalization group running of the MS constants from the electroweak to a high-energy scale. The most advanced recent works [10,12,17,18] use the so-called "one-loop-matching-two-loop-running" procedure. It can determine the Higgs boson mass bounds with a theoretical accuracy of 2 − 5 GeV (see the discussion of uncertainties in [12] and below). Meanwhile, the most important terms in the three-loop running of the gauge and Higgs coupling constants were computed in [19,20] (we thank K. Chetyrkin and M. Zoller for sharing these results with us prior to publication). The present work accounts for O(αα s ) corrections in the MS-pole matching procedure, which were not known previously. This allows us to decrease the theoretical uncertainties in the Higgs boson mass prediction/bounds associated with SM physics down to 1-2 GeV. This is a new result, based on a superior partial "two-loop-matching-threeloop-running" procedure. These findings are described in section 2.2. We will see that the experimental errors in the mass of the top quark and in the value of the strong coupling constant are too large to settle the question of the stability of the electroweak vacuum, even if the LHC will confirm the evidence for the Higgs signal presented by the ATLAS and CMS collaborations [8,9] in the region M H = 124 − 126 GeV.
In section 3 we will discuss the significance of the relationship between the true Higgs boson mass M H and M stability min for BSM physics. We will argue that if M H = M stability min then the electroweak symmetry breaking is likely to be determined by Planck physics and that this would indicate the absence of new energy scales between the Fermi and gravitational scales. We will also address here the significance of M stability min for the SM with gravity included. Of course, this can only be done under certain assumptions. Specifically, we will discuss the non-minimal coupling of the Higgs field to the Ricci scalar (relevant for Higgs-inflation [11,12,21]) and the asymptotic-safety scenario of the SM [13].  In section 4 we present our conclusions. We will argue that if only the Higgs boson with a mass around M stability min and nothing else will be found at the LHC, the next step in high-energy physics should be the construction a new electron-positron (or muon) collider -the Higgs and t-factory. It will not only be able to investigate in detail the Higgs and top physics, but also elucidate the possible connection of the Fermi and Planck scales.

The stability bound
The stability bound will be found in the "canonical" SM, without any new degrees of freedom or any extra higher-dimensional operators added, see figure 2.

The benchmark mass
It will be convenient for computations to introduce yet another parameter, the "benchmark mass", which we will call M min (without any superscript). Suppose that all parameters of the SM, except for the Higgs boson mass, are exactly known. Then M min , together with the normalisation point µ 0 , are found from the solution of the two equations: where β λ is the β-function governing the renormalisation group (RG) running of λ. Here we define all the couplings of the SM in the MS renormalisation scheme, which is used de-facto in the most of the higher-loop computations. Clearly, if any other renormalization scheme is used, the equations λ = β λ = 0 will give another benchmark mass, since the definition of all the couplings are scheme dependent. The procedure of computing M min is very clean and transparent. Take the standard MS definition of all coupling constants of the SM, fix all of them at the Fermi scale given the experimentally known parameters such as the mass of the top quark, the QCD coupling, etc., and consider the running Higgs self-coupling λ(µ) depending on the standard 't Hooft-Veltman parameter µ. Then, adjust M min in such a way that eq. (2.1) are satisfied at some µ 0 .
For the stability bound one should find the effective potential V (φ) and solve the equations where φ SM corresponds to the SM Higgs vacuum, and φ 1 correspond to the extra vacuum states at large values of the scalar field. Though the effective potential and the field φ JHEP10(2012)140 are both gauge and scheme dependent, the solution for the Higgs boson mass to these equations is gauge and scheme invariant. In fact, M stability min is very close to M min . Numerically, the difference between them is much smaller, than the current theoretical and experimental precisions for M min , see below. The following well known argument explains why this is the case. The RG improved effective potential for large φ can be written as [15,16,22]  Note that in many papers the stability bound is shown as a function of the cutoff scale Λ (the energy scale up to which the SM can be considered as a valid effective field theory). It is required that V (φ) > V (φ SM ) for all φ < Λ. This can be reformulated as λ(µ) > 0 for all µ < Λ with pretty good accuracy. Interestingly, if Λ = M P , this bound is very close to the stability bound following from eq. (2.2), having nothing to do with the Planck scale (see also below). Note also that the uncertainties in experimental determinations of M t and α s together with theoretical uncertainties, described in the next section, lead to significant changes in the scale Λ. Figure 1 illustrates that for Higgs boson masses 124 − 127 GeV this scale may vary from 10 8 GeV up to infinity within currently available precisions.

Value of M min
The state-of-the-art computation of M min has contained up to now the so-called one-loop MS-pole matching, relating the experimentally measured physical parameters to the parameters of the SM in the MS scheme (to be more precise, the two-loop α s corrections to the top pole mass-MS mass relation has been included [10]). Then the results of the first step are plugged into two-loop RG equations and solved numerically.
Before discussing the upgrade of the one-loop-matching-two-loop-running procedure, we will remind of the results already known and their uncertainties. We will make use of our computations of M min presented in [12]. 2 Somewhat later papers [17,18,23] contain exactly the same numbers for M stability min (note, however, that the theoretical uncertainties were not discussed in [23]). See also earlier computations in [7,10,[14][15][16]18].
and estimated the theoretical uncertainties as summarized in table 1 (see also [17]). 3 While repeating this analysis, we found some numerical errors which are given at the bottom section of this table (see a detailed discussion below). In total, they shift the value given in eq. (2.4) up by 0.7 GeV. As for the uncertainties, they were estimated as follows. The one-loop matching formulas can be used directly at µ = m t , or at some other energy scale, e.g. at µ = M Z , and then the coupling constants at µ = m t can be derived using RG running. The difference between the procedures gives an estimate of the two-loop effects in the matching procedure. This is presented by the first two lines in table 1 (in fact, we underestimated before the uncertainty from the λ matching -previously we had here 1.2 GeV and now 1.7 GeV). The next two lines are associated with three-and four-loop corrections to the top Yukawa coupling y t . The three-loop corrections were computed in [24][25][26] and the four-loop α s contribution to the top mass was estimated to be of the order δy t (m t )/y t (m t ) ≃ 0.001 in [27,28]. The non-perturbative QCD effects in the top pole mass-MS mass matching are expected to be at the same level [29][30][31]. For threeloop running we put the typical coefficients in front of the largest couplings α s and y t . If these uncertainties are not correlated and can be summed up in squares, the theoretical uncertainty is 2.5 GeV. If they are summed up linearly, then the theoretical error can be as large as ∼ 5 GeV.
Now, this computation can be considerably improved. First, in [19] the three-loop corrections to the running of all gauge couplings has been calculated. Second, in [20] the

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Contribution ∆M min , GeV Three loop beta functions -0.23 leading contributions (containing the top Yukawa coupling and α s ) to the running of the top quark Yukawa and the Higgs boson self-coupling have been determined. This removes the uncertainty related to the three-loop RG running. In addition, in the present paper, we determine the two-loop corrections of the order O(αα s ) to the matching of the pole masses and the top quark Yukawa and Higgs boson self coupling constants. Also, the known [24][25][26] three-loop QCD correction to the top quark mass relation of the order O(α 3 s ) can be included (previously it had been used for estimates of uncertainties). All this considerably decreases the theoretical uncertainties in M min .
The individual contributions of the various new corrections on top of the previous result are summarized in table 2. It is clearly seen that there are two new significant contributions -one is the three-loop pure QCD correction to the top quark mass [24][25][26], and another is the two-loop correction of O(αα s ) to the Higgs boson mass, found in the present paper. Together the new contributions add −0.89 GeV to the overall shift of the previous prediction [12], giving the result 4 The new result (2.5) is less than 0.2 GeV away from the old one (2.4) if the same central values for M t and α s are inserted. This coincidence is the result of some magic. In the old evaluation several mistakes were present, summarized in table 1. The largest one was the double counting of δ QED t in eq. (A.5) of [10], as compared to the original result [32]. Also, there were minor typos in the computer code for the matching of the Higgs coupling constant, and finally there was a small correction coming from the use of an approximate rather than exact one-loop formula for the O(α) corrections from [32]. These corrections add 0.7 GeV to the original number in [12]. By chance this almost exactly canceled the −0.89 GeV contribution from the higher loops, table 2, nearly leading to a coincidence of eqs. (2.5) and (2.4). Table 3 summarizes the uncertainties in the new computation. It contains fewer lines. Now we can ignore safely the error from higher-order (four-loop) RG corrections for the running up to the Planck scale. The first two lines were derived in the same manner as previously. For the Higgs boson self-coupling we can use the matching formulas (A. 35) to get the value of λ(µ) at scale µ = M t directly, or to get the value λ(M Z ) and then

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Source of uncertainty Nature of estimate ∆ theor M min , GeV 3-loop matching λ Sensitivity to µ 1.0 3-loop matching y t Sensitivity to µ 0.2 4-loop α s to y t educated guess [27,28] 0.4 confinement, y t educated guess ∼ Λ QCD 0.5 4-loop running M W → M P educated guess < 0.2 total uncertainty sum of squares 1.2 total uncertainty linear sum 2.3 Table 3. Theoretical uncertainties in the present M min evaluation.
evolve the constants to the scale µ = M t with the RG equations. The obtained difference figure 12) corresponds to the error δm ∼ 1.0 GeV. A similar procedure of comparing the evolution between M t and M Z using RG equations and direct matching formulas to the order O(α 3 s , α, αα s ) leads for the change in the top quark Yukawa coupling δy t /y t ∼ 0.0005 (cf. figure 12), leading to δM min ∼ 0.2 GeV. Note, however, that strictly speaking this test verifies the error of the µ dependent terms in the matching formulas, while the constant ones may lead to larger contributions. We also do not estimate now the contributions of the order O(α 2 ), where numerically one order in α may correspond to y 4 t . Thus, this estimate should be better considered as a lower estimate of the error. The four-loop matching and confinement contributions are the same as before.
As an indication of the dependence on the matching point we present figure 3, where the reference Higgs boson mass M min was obtained using the matching formulas at scale µ 0 varying between the Z-boson and top quark masses. One can see that the overall change of the Higgs boson mass is about 1 GeV.
If we assume that these uncertainties are not correlated and symmetric we get a theoretical error in the determination of the critical Higgs boson mass δM min ≃ 1.2 GeV. If they are summed up linearly, we get an error of 2.4 GeV. We leave it to the reader to decide which estimate of the uncertainties is more adequate. The precision of the theoretical value of M min can be further increased by computing the O(α 2 ) two-loop corrections to the matching procedure. Numerically, the most important terms are those when α corresponds to y 2 t and λ.
The result (2.5) is visualized by figure 4. The experimentally allowed regions for the top mass M t and strong coupling α s are adopted from the PDG 2010 edition [33]. 5 On top of these allowed regions the bands corresponding to the reference values of the Higgs boson mass M min being equal to 124, 125, 126, 127 GeV are shown, with the dashed and dotted lines corresponding to quadratically or linearly added estimates of the theoretical uncertainties.
One can see that the accuracy of the theoretical computations and of the experimental measurements of the top and the Higgs boson masses does not allow yet to conclude with  Figure 3. The dependence of the reference Higgs boson mass M min on the matching scale µ 0 (the MS constants are obtained by matching formulas at the scale µ 0 and then used for the solution of eq. (2.1)). The thick solid line corresponds to the full matching formulas λ ∼ O(α, αα s ), y t ∼ O(α 3 s , α, αα s ); the thin lines correspond to using matching formulas of lower order. The thick dashed line corresponds to using additionally the two-loop electroweak contributions to the Higgs coupling constant in the gauge-less limit, eq. (48) of [119], see the discussion in the "Note added". Here M t = 172.9 GeV and α s = 0.1184.
confidence whether the discovery of the Higgs boson with a mass 124 − 127 GeV would indicate stability or metastability of the SM vacuum. All these reference values of Higgs masses are compatible within 2σ with current observations.

M min and BSM physics
Our definition of the "benchmark" Higgs boson mass consists of the solution of the two equations (2.1) and gives, in addition to M min , the value of the scale µ 0 at which the scalar self-coupling and its β-function vanish simultaneously. The central value of µ 0 is 2.9 × 10 18 GeV and is quite stable if m t and α s are varied in their confidence intervals (see figure 5). One can see that there is a remarkable coincidence between µ 0 and the (reduced) Planck scale M P = 2.44 × 10 18 GeV. The physics input in the computation of µ 0 includes the parameters of the SM only, while the result gives the gravity scale. A possible explanation may be related to the asymptotic safety of the SM, see [13] and below. 6 It remains to be seen if this is just the random play of the numbers or a profound indication that the electroweak symmetry breaking is related to Planck physics. If real, this coincidence indicates that there should be no new energy scales between the Planck and Fermi scales, as they would remove this coincidence unless some conspiracy is taking place. We will discuss below two possible minimal embeddings of the SM into the theory of gravity and discuss the significance of M min in them.

Asymptotic safety
The asymptotic safety of the SM [13], associated with the asymptotic safety of gravity [46], is strongly related to the value of the Higgs boson mass. Though General Relativity is nonrenormalizable by perturbative methods, it may exist as a field theory non-perturbatively, exhibiting a non-trivial ultraviolet fixed point (for a review see [47]). If true, all other coupling of the SM (including the Higgs self-interaction) should exhibit asymptotically safe behaviors with the gravity contribution to the RG running included.
The prediction of the Higgs boson mass from the requirement of asymptotic safety of the SM is found as follows [13]. Consider the SM running of the coupling constants and add to the β-functions extra terms coming from gravity, deriving their structure from dimensional analysis: where a 1 , a 2 , a 3 , a y , and a λ are some constants (anomalous dimensions) corresponding to the gauge couplings of the SM g, g ′ , g s , the top Yukawa coupling y t , and the Higgs self-coupling λ. In addition, is the running Planck mass with ξ 0 ≈ 0.024 following from numerical solutions of functional RG equations [48][49][50]. Now, require that the solution for all coupling constants is finite for all µ and that λ is always positive. The SM can only be asymptotically safe if a 1 , a 2 , a 3 , a y are all negative, leading to asymptotically safe behaviors of the gauge and Yukawa couplings. For a λ < 0 we are getting the interval of admissible Higgs boson masses, M safety min < M H < M safety max . However, if a λ > 0, as follows from the computations of [49,50], only one value of the Higgs boson mass M H = M safety min lead to an asymptotically safe behavior of λ. As is explained in [13], this behavior is only possible provided λ(M P ) ≈ 0 and β λ (λ(M P )) ≈ 0. And, due to a miraculous coincidence of µ 0 and M P , the difference ∆m safety ≡ M safety min − M min is extremely small, of the order 0.1 GeV. The evolution of the Higgs self-coupling for the case of a h < 0 is shown in figure 6, and for the case a h > 0 in figure 7.
In fact, in the discussion of the asymptotic safety of the SM one can consider a more general situation, replacing the Planck mass in eq. Indeed, if the Higgs field has a non-minimal coupling with gravity (see below), the behavior of the SM coupling may start to change at energies smaller than M P by a factor 1/ξ, leading to an expectation for the range of κ as 1/ξ κ 1. Still, the difference between M min and M safety min remains small even for κ ∼ 10 −4 , where M safety min ≃ 128.4 GeV, making the prediction M H ≃ M min sufficiently stable against specific details of Planck physics within the asymptotic safety scenario.

M min and cosmology
It is important to note that if the mass of the Higgs boson is smaller than the stability bound M min , this does not invalidate the SM. Indeed, if the time-life of the metastable SM vacuum exceeds the age of the Universe (this is the case when M H > M meta , with JHEP10(2012)140 M meta ≃ 111 GeV [10]) then finding a Higgs boson in the mass interval M meta < M H < M min would simply mean that we live in a metastable state with a very long lifetime. Of course, if the Higgs boson were discovered with a mass below M meta , this would prove that there must be new physics between the Fermi and Planck scales, stabilizing the SM vacuum state. However, the latest LEP results, confirmed recently by LHC, tell us that in fact M H > M meta , and, therefore, that the presence of a new energy scale is not required, if only the metastability argument is used.
The bound M H > M meta can be strengthened if thermal cosmological evolution is considered [10]. After inflation the universe should find itself in the vicinity of the SM vacuum and stay there till present. As the probability of the vacuum decay is temperature dependent, the improved Higgs boson mass bound is controlled by the reheating temperature after inflation (or maximal temperature of the Big Bang). The latter is model dependent, leading to the impossibility to get a robust bound much better than M meta . For example, in R 2 inflation [51,52] the reheating temperature is rather low, T ∼ 10 9 GeV [52], leading to the lower bound 116 GeV [53] on the Higgs boson mass, which exceeds M meta only by 4 GeV.
However, if no new degrees of freedom besides those already present in the SM are introduced and the Higgs boson plays the role of the inflaton, the bound M H M min reappears, as is discussed below.

Higgs inflation
The inclusion of a non-minimal interaction of the Higgs field with gravity, given by the Lagrangian ξ|φ| 2 R, where R is the Ricci scalar, changes drastically the behavior of the Higgs potential in the region of large Higgs fields φ > M inflation ≃ M P / √ ξ [11]. Basically, the potential becomes flat at φ > M inflation , keeping the value it acquired at φ ≃ M P / √ ξ. This feature leads to a possibility of Higgs-inflation: if the parameter ξ is sufficiently large, 700 < ξ < 10 5 , [12]  can lead to successful inflation, the value M inflation min is somewhat special. For the lower part of the admitted interval the value of the non-minimal coupling ξ reaches its minimal value ξ ≃ 700, extending the region of the applicability of perturbation theory [12,54,55].
The computation of the lower bound on the Higgs boson mass from inflation is more complicated. It is described in detail in [12,21]. Basically, one has to compute the Higgs potential in the chiral electroweak theory associated with large values of the Higgs field One must note that simple scale analysis leads to unitarity violation in the Higgs inflation below the Planck energy scale [56][57][58][59]. This means, that calculations in Higgs inflationary models should be done with some additional assumptions about the highenergy physics, formulated in [54], specifically the approximate scale invariance at high field backgrounds. As is explained in [54], adding to the SM higher-dimensional operators with a Higgs-field dependent cutoff modifies the lower bound on the Higgs boson mass in Higgs inflation. If these operators are coming with "natural" power counting coefficients (for an exact definition see [54]) the sensitivity of the Higgs boson mass bound to unknown details of ultraviolet physics is rather small ∆M inflation min ≃ 0.6 GeV [54]. At the same time, it is certainly not excluded that the change of M inflation min can be larger.

Conclusions
If the SM Higgs boson will be discovered at LHC in the remaining mass interval 115.5 < M H < 127 GeV not excluded at 95% [8,9], there is no necessity for a new energy scale between the Fermi and Planck scales. The electroweak theory remains in a weakly coupled JHEP10(2012)140 region all the way up to M P , whereas the SM vacuum state lives longer than the age of the Universe. If the SM Higgs boson mass will be found to coincide with M min given by eq. (2.5), this would provide a strong argument in favor of the absence of such a scale and indicate that the electroweak symmetry breaking may be associated with the physics at the Planck scale.
The experimental precision in the Higgs boson mass measurements at the LHC can eventually reach 200 MeV and thus be much smaller than the present theoretical (∼ 1-2 GeV) and experimental (∼ 5 GeV, 2σ) uncertainties in the determination of M min . The largest uncertainty comes from the measurement of the mass of the top quark. It does not look likely that the LHC will substantially reduce the error in the top quark mass determination. Therefore, to clarify the relation between the Fermi and Planck scales the construction of an electron-positron or muon collider with a center-of-mass energy of ∼ (200 + 200 )GeV (Higgs and t-quark factory) would be needed. This would be decisive for setting up the question about the necessity for a new energy scale besides the two ones already known -the Fermi and the Planck scales. In addition, this will allow to study in detail the properties of the two heaviest particles of the SM, potentially most sensitive to any types on new physics.
Surely, even if the SM is a valid effective field theory all the way up the the Planck scale, it cannot be complete as it contradicts a number of observations. We would like to use this opportunity to underline once more that the confirmed observational signals in favor of BSM physics which were not discussed in this paper (neutrino masses and oscillations, dark matter and baryon asymmetry of the Universe) can be associated with new physics below the electroweak scale, for reviews see [60,61] and references therein. 7 The minimal model, νMSM, contains, in addition to the SM particles, three relatively light singlet Majorana fermions. These fermions could be responsible for neutrino masses, dark matter and the baryon asymmetry of the Universe. The νMSM predicts that the LHC will continue to confirm the SM and see no deviations from it. At the same time, new experiments at the high-intensity frontier, discussed in [64], may be needed to uncover the new physics below the Fermi scale. In addition, new observations in astrophysics, discussed in [61], may shed light to the nature of dark matter. As the running of the couplings in the νMSM coincides with that in the SM, all results of the present paper are equally applicable to the νMSM.

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O. Ruchayskiy, and M. Zoller for helpful discussions, collaboration and interest to our work. M.Yu.K. is indebted to Fred Jegerlehner for a fruitful long-time collaboration on studies of the MS scheme beyond the one-loop order in the framework of quantum field theory models with spontaneously symmetry breaking, and in particular for collaboration on [65], the results and methods of which were heavily used in the present work.
Note added. After our paper was submitted to the electronic preprint archive arXiv (on May 13) a number of events happened, which require its update. First, extra corrections to the matching procedure at low energy scale, not computed in our work, were found in [119] (May 29). 8 Ref. [119] finds agreement with our results on the αα s order in the Higgs selfcoupling constant. In addition, this paper computed a part of the O(α 2 ) corrections to the top Yukawa and Higgs coupling constants in the "gauge-less" limit of the Standard model, i.e. the two-loop terms containing the top Yukawa and scalar self-coupling were accounted for. The overall effect of these terms happened to be quite small. The corrections shift the benchmark Higgs mass up by 0.2 GeV, and reduce the sensitivity of the results to the normalisation point from 1.2 GeV to 0.8 GeV, decreasing somewhat the theoretical errorbars, see figure 3.
Second, the discovery of the Higgs-like resonance was announced by the ATLAS and CMS collaborations [66,67]  And, finally, updated results on the mass of the top quark were announced at ICHEP 2012 (July 9), see [38,68]. The combination of the Tevatron results reads: The central values of the top mass are somewhat higher (by 0.3-0.4 GeV) than those which were given by the Particle Data Group at the time we were writing our paper. In figure 9, which is an update of figure 4, we show the changes due to the experimental shift in the top mass (we take the Tevatron result as having smaller errors) and due to the additional two-loop corrections found in [119]   and α s (1 and 2σ) that can be achieved with an e + e − collider operated as a tt factory (this estimate is taken from [69][70][71][72][73][74][75][76] and [77] discussing the ILC physics).
It is also important to note, that it is difficult to determine which renormalization scheme corresponds to the numerical value of m t quoted by experiments. The m t determinations by the Tevatron and LHC collaborations [68,78] are based on Monte Carlo event generators implemented with LO hard-scattering matrix elements. Although it is plausible and likely that the experimental m t values thus extracted are close to the pole mass of the top quark, this is by no means guaranteed with the due theoretical rigor [79]. In fact, rigorous determinations of the top-quark pole mass from total production cross sections yield somewhat smaller values, albeit with larger errors [30,31,120]. For the time being,

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we adopt the working hypothesis that the experimental values of m t [68,78] correspond to the pole mass, bearing in mind that this is probably subject to change once a proper NLO treatment of the resonating top-quark propagators is implemented.
Thus, all our considerations remain in force and call for further improvement of the theoretical computations, which should account for all O(α 2 ) corrections to the mapping procedure, and for a construction of a tt factory to pin down α s and m t . A recent paper [120], where uncertainties in determination m t and α s have been analyzed, reached exactly the same conclusion.
A O(αα s ) electroweak corrections to the top Yukawa and Higgs self couplings in the SM The evaluation of radiative corrections to the relations between MS parameters (coupling constants) and masses of particles includes two steps: the evaluation of radiative corrections between the Fermi constant G F and its MS counterpart [80] (see [81][82][83] for recent reviews) and the evaluation of the radiative corrections between MS and pole masses. The one-loop electroweak correction of O(α) to the relation between the self-coupling λ(µ 2 ) and the pole mass of the Higgs boson was obtained in [84] and the one to the relation between the Yukawa coupling y t and the pole mass of top quark was found in [32]. The corresponding ingredients for the two-loop mixed electroweak-QCD corrections were evaluated in [65,[85][86][87][88][89], but have never been assembled. We performed independent (re)calculations of all O(α) and O(αα s ) contributions. In the following we will denote the on-shell masses by capital M and the MS masses by lowercase m.
The numerical results for the values of the MS constants at scale M t obtained up to different loop order is presented by the figures 10, 11.

A.1 O(αα s ) corrections to the relation between the on-shell and MS Fermi constants
The relation between the Fermi coupling constant and the bare parameters is as follows [80]: where ∆R 0 includes unrenormalized electroweak corrections and g 0 , m 2 W,o are the SU(2) coupling constant and the bare W boson mass (see for details [81][82][83]). After performing MS renormalization this relation has the following form: where on the r.h.s. all masses and coupling constants are taken in the MS renormalization scheme. The one-loop coefficient, ∆ G F ,α , is known from [80] and for N c = 3, C and m b = 0 has the following form: Here, sin 2 θ W is defined in the MS scheme as  where g ′ (µ 2 ) and g(µ 2 ) are the U(1) and SU(2) MS gauge coupling constants, respectively. The matching conditions between the MS parameter, defined by eq. (A.4), and its on-shell version [80] follows from the identification where M Z and M W are the pole masses of the gauge bosons (see detailed discussion in [90][91][92]). The evaluation of the mixed QCD-electroweak coefficient, ∆ G F ,ααs , is reduced to the evaluation of the O(αα s ) corrections to the W boson self-energy at zero momenta transfer

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and may be written in the following way [93][94][95][96][97]: where for Z g,ααs , Z g,ααs and Z W,ααs we used the results 9 of [92] and n F is the number of fermion families (n F = 3 in the SM).
Using the fact, that G F is RG invariant, i.e. µ 2 d dµ 2 G F = 0, the µ-dependent terms in eq. (A.6) can be evaluated explicitly from the one-loop correction and the explicit knowledge of the anomalous dimension γ G F . As was shown in [90-92, 98, 99], the anomalous dimension γ G F can be extracted (i) via the beta-function β λ of the scalar self-coupling and the anomalous dimension of the mass parameter m 2 (in the unbroken phase) or (ii) via the β-function of the SU(2) gauge coupling g and the anomalous dimension of the W boson (in the broken phase): Eq. (A.2) can be written as G F ,ααs +C There are typos in eq. (4.41) of [92]: in all MS renormalization constants, Z αs W and Z αs Z , "m 2 t /m 2 H " should be replaced by "m 2 t /m 2 W ", i.e. Collecting all terms in eq. (A.10) we get At the end of this section we again point out that the anomalous dimension of the vacuum expectation value v 2 (µ 2 ) = 1/( √ 2G F (µ 2 )) within the diagram technique is defined by eq. (A.7) and that it is not equal to the anomalous dimension of the scalar field as in the effective-potential approach [100]. Another important property of eq. (A.7) is the appearance of an inverse power of the coupling constant λ due to the explicit inclusion of the tadpole contribution. As consequence, the limit of zero Higgs mass, m 2 H = 0, does not exist within the perturbative approach. The importance of the inclusion of the tadpole contribution to restore gauge invariance of on-shell counterterms was recognized a long time ago [101] and was explicitly included in the one-loop electroweak corrections to the matching conditions [32,84]. The RG equations for the mass parameters were discussed in [90-92, 98, 99].

A.2 O(αα s ) corrections to the relation between the MS and pole masses of the top quark
A detailed discussion and an explicit evaluation 10 have been presented in [89] (the results of [89] were also used for the analysis of the convergence of the series of the set of Feynman diagrams evaluated in [103,104]). For our analysis is enough to write the following symbolic relation between the MS and pole masses of the top quark: where σ α and σ ααs are defined by eq. (5.54) or eq. (5.57) of [89]. The pure QCD corrections can be found in [24][25][26] (only the value of σ αs (M t ) is given there, but the expression for other µ values can be readily reconstructed from the beta functions).

A.3 O(αα s ) corrections to the relation between the MS and pole masses of the Higgs boson
At the two-loop level the relation between the pole and MS masses is defined as follows: 10 There is typo in eq. (4.46) of [89]: the common factor C f was lost. The correct result is = αs 4π However, all plots, eq. (5.57) and the Maple program [102] As result of our calculation we find: The contributions of other quarks with non-zero mass are additive. Exploring the ε expansion for the master integral J 0qq from [106], we have for the t-quark contribution (q = t): and we have introduced the two functions F (y) and G(y) (see also [87,88]) defined as 11 H,ααs , which may be calculated also from the one-loop result and the mass anomalous dimensions (see [98,99] for the general case). From the parametrization  .33) and the relation between the Higgs coupling constant λ ≡ h Sirlin used in [84] and the parametrization of [89][90][91][92] follows from the comparison of the RG functions: h Sirlin = λ Jegerlehner (µ 2 )/6. where N f = 5 is the number of flavors below the top quark. Strictly speaking, the value of α s (M t ) obtained from this equation should also be shifted to the six-quark value as but this introduces a negligible effect (< 0.1 GeV) for the Higgs mass. In all the formulas of appendix A we use the values of α and α s at the matching scale µ.