Impacts of the Higgs mass on vacuum stability, running fermion masses and two-body Higgs decays
aa r X i v : . [ h e p - ph ] M a y Impacts of the Higgs mass on vacuum stability, running fermionmasses and two-body Higgs decays
Zhi-zhong Xing a ∗ , He Zhang b † , Shun Zhou c ‡ a Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China b Max-Planck-Institut f¨ur Kernphysik, 69029 Heidelberg, Germany c Max-Planck-Institut f¨ur Physik (Werner-Heisenberg-Institut), 80805 M¨unchen, Germany
Abstract
The recent results of the ATLAS and CMS experiments indicate 116 GeV < ∼ M H < ∼
131 GeV and 115 GeV < ∼ M H < ∼
127 GeV, respectively, for themass of the Higgs boson in the standard model (SM) at the 95% confidencelevel. In particular, both experiments point to a preferred narrow mass range M H ≃ (124 · · · M H on the SM vacuum stability by using the two-loop renormalization-groupequations (RGEs), and arrive at the cutoff scale Λ VS ∼ × GeV (for M H = 125 GeV, M t = 172 . α s ( M Z ) = 0 . M H , 1 TeV andΛ VS , with the help of the two-loop RGEs. The branching ratios of some im-portant two-body Higgs decay modes, such as H → b ¯ b , H → τ + τ − , H → γγ , H → W + W − and H → ZZ , are also recalculated by inputting the values ofrelevant particle masses at M H .PACS number(s): 12.15.Ff, 12.38.Bx, 14.80.Bn Typeset using REVTEX ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] . INTRODUCTION The Higgs mechanism [1] is responsible for the spontaneous SU(2) L ⊗ U(1) Y → U(1) em gauge symmetry breaking in the standard model (SM) of electroweak interactions [2], but theHiggs boson itself left no sort of trace in all the previous high-energy collider experiments.The main goal of the Large Hadron Collider (LHC) at CERN is just to discover this elusiveparticle, which allows other particles (except the photon and gluons) to gain finite masses.Combined with the indirect bounds obtained from the electroweak precision measurements,the recent data of the ATLAS and CMS experiments lead us to a rather narrow range ofthe Higgs mass: 114 GeV < ∼ M H < ∼
141 GeV [3]. In particular, both collaborations havereported their latest results M H ≃ ( (116 · · · , (115 · · · , (1)at the 95% confidence level. The ATLAS Collaboration has also found a preliminary hintof M H ≃
126 GeV with the 3 . σ local significance in H → γγ (2 . σ ), H → ZZ ∗ → l (2 . σ ) and H → W W ∗ → l ν (1 . σ ) decay modes [4]; and the CMS Collaboration hasobserved an excess compatible with M H < ∼
124 GeV with the 2 . σ local significance [5].These interesting results point to a preferred and narrower range M H ≃ (124 · · · • The Higgs mass theoretically suffers significant radiative corrections, and hence newsymmetries and (or) new particles should be introduced to stabilize the electroweakscale Λ EW ∼ GeV [6]. A solution to this gauge hierarchy problem calls for newphysics beyond the SM, such as supersymmetries [7] or extra spatial dimensions [8]. • The Higgs boson is indispensable to the Yukawa interactions of three-family fermionswhich makes weak CP violation possible in the SM or its simple extensions [9]. To someextent, the existence of a Higgs boson may also support the Peccei-Quinn mechanismas an appealing solution to the strong CP problem [10]. • With the help of the Higgs field, one may write out the unique dimension-five oper-ator ℓℓHH in an effective field theory [11] or implement the seesaw mechanism in arenormalizable quantum field theory [12] to generate finite but tiny neutrino masses.Therefore, the highest priority of the LHC experiment is to pin down the Higgs boson andits quantum numbers. We are approaching a success in this connection.Motivated by the encouraging ATLAS and CMS results, we aim to examine the impactsof M H ≃ (124 · · · M H is likely to cause the vacuum instability unless new physics takes effectat a proper cutoff scale [13]. Given M H ≃
125 GeV as indicated by the latest LHC data,2t is timely to determine the energy scale at which the effective quartic Higgs coupling ˜ λ ( µ )runs to zero. We find that this cutoff scale is around Λ VS ∼ × GeV, which presumablysignifies the end of the gauge desert and the beginning of a new physics oasis. Takingaccount of the allowed range of M H and the updated values of other SM parameters, werecalculate the running fermion masses at some typical energy scales up to Λ VS by meansof the renormalization-group equations (RGEs). Such an exercise makes sense because asufficiently large value of M H (e.g., M H ≃
140 GeV) was assumed in the previous worksand hence the potential vacuum stability problem did not show up [14]. As a by-product,the branching ratios of some important two-body Higgs decay modes in the SM, such as H → b ¯ b , H → τ + τ − , H → γγ , H → W + W − and H → ZZ , are also recalculated by usingthe new values of relevant particle masses obtained at µ ∼ M H . II. THE HIGGS MASS AND VACUUM STABILITY
First of all, let us briefly review the vacuum stability issue in the SM with a relativelylight Higgs boson. In order to find out the true vacuum state and analyze its stability, oneshould calculate the effective scalar potential by taking account of radiative corrections andRGE improvements of the relevant parameters [13,15]. It has been shown that the L -loopscalar potential improved with ( L + 1)-loop RGEs actually includes all the L th-to-leadinglogarithm contributions [16]. At the one-loop level, the effective scalar potential in the ’tHooft-Landau gauge and in the MS renormalization scheme can be written as [17] V eff [ φ ( t )] = − m ( t ) φ ( t ) + 14 λ ( t ) φ ( t ) + 364 π ( m W [ φ ( t )] " ln m W [ φ ( t )] µ ( t ) ! − + m Z [ φ ( t )] " ln m Z [ φ ( t )] µ ( t ) ! − − m t [ φ ( t )] " ln m t [ φ ( t )] µ ( t ) ! − , (2)where the contributions from the Goldstone and Higgs bosons have been safely neglected,and m W [ φ ( t )] ≡ g ( t ) φ ( t ) / m Z [ φ ( t )] ≡ [ g ( t )+ g ′ ( t )] φ ( t ) / m t [ φ ( t )] ≡ y t ( t ) φ ( t ) / g ( t ) , g ′ ( t ) , λ ( t ) , y t ( t )), the mass parameter m ( t ) and the Higgs field φ ( t ) has been explicitlyindicated through the renormalization scale µ ( t ) ≡ M Z e t or equivalently the running param-eter t = ln [ µ ( t ) /M Z ]. The β functions for the dimensionless couplings ( g ( t ) , g ′ ( t ) , λ ( t ) , y t ( t ))and the γ functions for ( m ( t ) , φ ( t )) at the two-loop order can be found in Refs. [17,18].Due to the experimental observations, the scalar potential V eff must develop a realisticminimum at the electroweak scale, corresponding to the SM vacuum. Whether the SMvacuum is stable or not depends on the behavior of V eff in the large-field limit, i.e., φ ( t ) ≫ M Z . More explicitly, one can find out the extrema φ ex ( t ) of the scalar potential via ∂V eff [ φ ( t )] ∂φ ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ ( t )= φ ex ( t ) = 0 . (3)At the weak scale µ ( t Z ) = M Z , we should impose the boundary condition φ ex ( t Z ) = v ≈
246 GeV, which is the vacuum expectation value of the Higgs field. At the large-field values,the scalar potential is dominated by the quartic coupling term and the extrema φ ex ( t ) can3e evaluated at the renormalization scale µ ( t ) = φ ex ( t ) from Eqs. (2) and (3) as φ = m / ˜ λ ,where the effective quartic coupling ˜ λ is defined as˜ λ = λ − π (
18 ( g ′ + g ) " − ln g ′ + g ! + 2 y t " ln y t ! − + 14 g " − ln g ! . (4)Now it is clear that V eff ≈ ˜ λφ / λ becomes negative [19–23]. To maintain the absolutestability of the SM vacuum, new physics should come into play below or at the energy scaleΛ VS where the effective coupling ˜ λ vanishes, i.e., ˜ λ (Λ VS ) = 0. One can derive a lower massbound on the Higgs boson by requiring that the SM vacuum is absolutely stable up to apossible grand-unified-theory (GUT) scale or the Planck scale [13,19–23].In view of the allowed range of the Higgs mass, we may conversely implement the vacuumstability argument to determine the energy scale Λ VS at which new physics should take effect.Our strategy is as follows. First, we have to specify the matching conditions relating thequartic coupling λ to the Higgs mass M H , as well as the top-quark Yukawa coupling y t tothe top-quark pole mass M t . Although the complete effective potential V eff must be scale-independent, the one with one-loop approximation is not. The solution is to find an optimalscale µ ∗ = µ ( t ∗ ) for which the effective potential has the least scale-dependence, as shown inRef. [19], where one can observe that µ ∗ = M t is a reasonable choice. Therefore, we choosethe matching conditions for λ and y t at µ ∗ = M t : λ ( M t ) = M H v [1 + δ H ( M t )] ,y t ( M t ) = √ M t v [1 + δ t ( M t )] , (5)where the correction terms δ H ( M t ) and δ t ( M t ) have been given in Ref. [23–25]. The valuesof the other input parameters are taken from Ref. [26] and will be specified in Sec. III whenwe turn to the running fermion masses. Second, we run λ ( µ ) to a much higher energy scaleby solving the complete two-loop RGEs. Third, the cutoff scale Λ VS can be identified withthe solution to ˜ λ (Λ VS ) = 0, where ˜ λ and λ are related via Eq. (4). Note that the cutoffscale Λ VS determined by ˜ λ (Λ VS ) = 0 could be an order of magnitude larger than the one by λ (Λ VS ) = 0, which has not taken account of the one-loop radiative corrections to the scalarpotential [19,21].Our numerical result for the correlation between the Higgs mass and the energy scale isshown in FIG. 1. Some comments are in order.1. If M H > ∼
129 GeV holds, the vacuum stability can be guaranteed even around apossible GUT scale (e.g., 10 GeV) or the Planck scale Λ Pl ∼ GeV [27]. Thecutoff scale Λ VS increases as the Higgs mass M H increases, but this observation issensitively dependent on the value of the top-quark pole mass M t .4. Given M H ≃
125 GeV, some kind of new physics should come out around Λ VS ∼ GeV to stabilize the SM vacuum . For example, it is interesting to notice thatthe canonical seesaw mechanism for neutrino mass generation is expected to workaround this cutoff scale. In such a seesaw model the heavy Majorana neutrinos couldhave masses of O (10 ) GeV, so that the leptogenesis mechanism [28] may work wellto account for the observed matter-antimatter asymmetry of the Universe.Although the existence of a cutoff scale is robust for the SM with a relatively light Higgsboson, it remains unclear what kind of new physics could take effect over there. In anyevent, if the new physics responsible for the vacuum stability could also offer a solution tothe flavor puzzles of leptons and quarks (especially the origin of tiny neutrino masses), therunning fermion masses at the cutoff scale Λ VS will be very helpful for model building. Weshall focus on this issue in the following section. III. RUNNING LEPTON AND QUARK MASSES
A systematic analysis of the RGE running masses of leptons and quarks has been done inRef. [14], where M H ≃
140 GeV has typically been taken just for illustration. As discussedabove, such a value of the Higgs mass makes the situation simple because it does not giverise to the vacuum instability problem in the SM. Here we want to update the runningfermion masses for two good reasons: (a) the latest ATLAS and CMS data point to M H ≃ (124 · · · • Six quark masses are m u (2 GeV) = (1 . · · · .
1) MeV, m d (2 GeV) = (4 . · · · .
7) MeV, m s (2 GeV) = (80 · · · m c ( m c ) = 1 . +0 . − . GeV, m b ( m b ) = 4 . +0 . − . GeV and M t = 172 . +1 . − . GeV [26], where M t represents the pole mass of the top quark extractedfrom the direct measurements. In addition, the pole masses of three charged leptons aregiven by M e = (0 . ± . M µ = (105 . ± . M τ = (1776 . ± .
16) MeV [26]. Following the same approach as the one describedin Ref. [14], we can calculate the running masses of charged leptons and quarks at sometypical energy scales in the SM, including µ = M W , M Z , M H , 1 TeV and Λ VS . • The strong and electromagnetic fine-structure constants at M Z are α s ( M Z ) = 0 . ± . α ( M Z ) − = 127 . ± . θ W ( M Z ) = Since the cutoff scale depends sensitively on the Higgs mass in the range [120 GeV ,
130 GeV],as shown in FIG. 1, one has to take care of experimental errors from the top quark mass M t andthe strong coupling α s ( M Z ), as well as the theoretical uncertainties involved in the two-loop RGEsand one-loop matching conditions [21]. For instance, the SM vacuum for M H ≃
125 GeV couldeven be stable up to the Planck scale Λ Pl ≃ GeV if the relevant uncertainties are included. . ± . g s = 4 πα s , g = 4 πα/ sin θ W and g ′ = g tan θ W at theenergy scale µ = M Z . • The four parameters of quark flavor mixing and CP violation in the modified Wolfen-stein parametrization are λ = 0 . ± . A = 0 . +0 . − . , ¯ ρ = 0 . +0 . − . and¯ η = 0 . ± .
013 [26]. These values, together with the values of quark masses, allowus to reconstruct the quark Yukawa coupling matrices Y u and Y d at the electroweakscale. The RGEs of Y u and Y d can therefore help us to run the quark masses and flavormixing parameters to a much higher energy scale. • The allowed ranges of three lepton flavor mixing angles are 30 . ◦ ≤ θ ≤ . ◦ ,35 . ◦ ≤ θ ≤ . ◦ and 1 . ◦ ≤ θ ≤ . ◦ [29], and the allowed ranges of twoneutrino mass-squared differences are 6 . × − eV ≤ δm ≤ . × − eV and2 . × − eV ≤ | ∆ m | ≤ . × − eV [29]. For simplicity, we only take the best-fit values θ = 33 . ◦ , θ = 40 . ◦ and θ = 8 . ◦ together with δm = 7 . × − eV and | ∆ m | = 2 . × − eV as the inputs at M Z in our numerical calculations. Inparticular, the value of θ taken above is essentially consistent with the latest DayaBay [30] and RENO [31] results. The unknown CP-violating phases in the lepton sectorare all assumed to be zero. In view of the fact that the absolute neutrino mass scale isalso unknown, we shall only consider the normal mass hierarchy with m = 0 .
001 eVand m < m ≪ m at M Z for illustration. For the same reason, only the one-loopRGE for neutrino masses is considered. It is then possible to reconstruct the charged-lepton Yukawa coupling matrix Y l and the effective neutrino coupling matrix κ at M Z from the given lepton masses and flavor mixing parameters [32].For a complete list of the RGEs to be used in our numerical analysis, we refer the reader toRef. [14] and references therein.TABLES I and II summarize our numerical results for the running quark and charged-lepton masses at some typical energy scales, respectively. Different from the previous works,here the scales characterized by the Higgs mass M H and the vacuum stability cutoff Λ VS aretaken into account for the first time. The values of fermion masses at M H will be used tocalculate the branching ratios of some important Higgs decay modes in Sec. IV, and thoseat Λ VS are expected to be very useful for building possible flavor models beyond the SM.In studying the running behaviors of twelve fermion masses above M Z , we have used theinputs at M Z to numerically solve the RGEs of the Yukawa coupling matrices Y u , Y d , Y l andthe effective neutrino coupling matrix κ as well as the two-loop RGEs of the quartic Higgscoupling λ ( µ ) and gauge couplings at µ ≥ M Z . After Y u , Y d , Y l and κ are diagonalized, onecan obtain the running quark masses m q ( µ ) = y q ( µ ) v/ √ q = u, c, t and d, s, b ), therunning charged-lepton masses m l ( µ ) = y l ( µ ) v/ √ l = e, µ, τ ) and the running neutrinomasses m i ( µ ) = κ i ( µ ) v / i = 1 , , R f ( µ ) ≡ m f ( µ ) /m f ( M Z ), where the subscript f runs over the mass-eigenstate indices of six quarksand six leptons. We find that R u ( µ ) ≈ R d ( µ ) ≈ R s ( µ ) ≈ R c ( µ ) ≈ R e ( µ ) ≈ R µ ( µ ) ≈ µ is below the cutoff scale Λ VS . So we only plot the6umerical results of R t ( µ ), R b ( µ ) and R τ ( µ ) in FIG. 2. The ratios R i ( µ ) for three neutrinomasses are shown in FIG. 3. Some comments are in order.1. The mass ratios R f ( µ ) are not very sensitive to the quartic Higgs coupling λ ( µ ) orequivalently the Higgs mass M H , simply because the latter enters the RGEs of fermionmasses only at the two-loop level. As observed in Ref. [14], there exists a maximum forthe charged-lepton masses around µ ∼ GeV, while the quark masses monotonouslydecrease as the energy scale increases. Taking account of the vacuum instability prob-lem discussed in Sec. II, we argue that the evolution of fermion masses above thecutoff scale Λ VS might not be meaningful anymore. We expect that some kind of newphysics should take effect around Λ VS and thus modify the RGEs of the SM.2. In most cases the running behaviors of three neutrino masses are neither sensitiveto their absolute values nor sensitive to their mass hierarchies in the SM [33]. Onlywhen three neutrino masses are assumed to be nearly degenerate, the RGE runningeffects of neutrino mass and mixing parameters are possible to be significant. But thedependence of m i ( µ ) on the quartic Higgs coupling λ ( µ ) or the Higgs mass M H is quiteevident, simply because the effective neutrino coupling matrix κ receives the one-loopcorrections from the quartic Higgs interaction [14,33].For simplicity, we skip the numerical illustration of the running behaviors of quark andlepton flavor mixing parameters in this paper. IV. BRANCHING RATIOS OF THE HIGGS DECAYS
The present ATLAS and CMS experiments are mainly sensitive to the Higgs bosonvia its decay channels H → γγ , H → b ¯ b , H → τ + τ − , H → W + W − (2 l ν ) and H → ZZ (4 l, l ν, l q, l τ ), where l = e or µ and ν denotes the neutrinos of any flavors [3].Which channel is dominant depends crucially on the Higgs mass. If M H < ∼
135 GeV holds,thedecay mode H → b ¯ b is expected to have the largest branching ratio; and if the Higgs massis slightly heavier, the decay mode H → W + W − will surpass the others [15].We first consider the leptonic H → l + l − decays, where l runs over e , µ or τ . Includingthe one-loop electroweak corrections, the decay width of H → l + l − is given by [34]Γ l = G F M H √ π M l − M l M H ! / (cid:16) δ QED + δ W (cid:17) , (6)where G F is the Fermi constant, δ QED = 9 α [3 − M H /M l )] / (12 π ), and δ W = G F √ π ( M t + M W θ W ln cos θ W − ! − M Z (cid:20) (cid:16) − θ W (cid:17) − (cid:21)) . (7)Note that the large logarithmic term ln( M H /M l ) in δ QED can be absorbed in the runningmass of l at the scale of M H , which has been given in TABLE II.Now we turn to the H → q ¯ q decays, where q runs over u , d , s , c or b for the Higgs massto lie in the range 114 GeV < ∼ M H < ∼
141 GeV. Since the decay rates of H → u ¯ u , d ¯ d and s ¯ s H → c ¯ c and H → b ¯ b . Up to the three-loop QCD corrections [35],Γ q = 3 G F M H √ π m q ( M H ) (cid:16) δ QCD + δ t (cid:17) , (8)where δ QCD = 1 + 5 . α s ( M H ) π ! + 29 . α s ( M H ) π ! + 41 . α s ( M H ) π ! ; (9)and δ t = α s ( M H ) π ! " . −
23 ln M H M t ! + 19 ln m q ( M H ) M H ! . (10)Note that the running quark masses m q ( M H ) and the strong coupling constant α s ( M H ) atthe scale of M H are useful here to absorb the large logarithmic terms.A detailed discussion about the two-body decay modes H → γγ , H → W + W − , H → ZZ , H → gg and H → t ¯ t can be found in Ref. [36]. For simplicity, here we do not elaboratethe relevant analytical results but do a numerical recalculation based on the updated particlemasses at M H . In order to compute the branching ratios of the above decay channels, weimplement the latest version of the program HDECAY [37] and update the input parametersaccording to our TABLES I and II together with Ref. [26]. Some comments are in order. • The pole masses of the charged leptons (i.e., M l ), instead of the running masses m l ( M H ), have been used as the input parameters in the program HDECAY. Thistreatment is consistent with the formula of Γ l given in Eq. (6). If one chooses to usethe running masses m l ( M H ) in the numerical calculation, then the correction terms inEq. (6) should take different forms. • The one-loop pole masses of c and b quarks have been used as the input parametersin the program HDECAY, because they must be consistent with the correspondingparton distribution function when the production of the Higgs boson in a hadroncollider (e.g., the LHC) is taken into account [38]. In our calculations we start fromthe values of m c ( m c ) and m b ( m b ) [26] and then evaluate the pole masses M c and M b as precisely as possible by using the relevant four-loop RGEs and three-loop matchingconditions [14]. Hence we obtain the pole masses M c = 1 .
84 GeV and M b = 4 .
92 GeV,as given in TABLE I.Our numerical results for the branching ratios of H → b ¯ b , c ¯ c and τ + τ − decays are shownin FIG. 4, where the branching ratios of H → γγ , gg , W + W − , ZZ and Zγ decays are alsoplotted for a comparison. These important two-body decay channels will help discover theHiggs boson and pin down its mass in the near future. The branching ratios of H → s ¯ s and H → µ + µ − are of O (10 − ) in the range of M H ∈ [110 GeV ,
150 GeV], and thus they havebeen neglected from FIG. 4. 8 . SUMMARY
In view of the recent results from the ATLAS and CMS experiments which hint at theexistence of the Higgs boson, we have examined the impact of the Higg mass on vacuumstability in the SM by means of the two-loop RGEs. We find that M H ≃
125 GeV leads usto an interesting cutoff scale Λ VS ∼ GeV, as required by the vacuum stability. Somekind of new physics are therefore expected to take effect around Λ VS . In other words, Λ VS characterizes the end of the gauge desert and the beginning of a new physics oasis.We have argued that possible new physics responsible for the vacuum stability of theSM might also be able to help solve the flavor puzzles. Hence we have recalculated therunning fermion masses up to the cutoff scale Λ VS by inputting the allowed range of M H and the updated values of other SM parameters into the full set of the two-loop RGEs forthe quartic Higgs coupling, the Yukawa couplings and the gauge couplings. In particular,the values of lepton and quark masses at µ = M H and Λ VS are obtained for the first time.As a by-product, the branching ratios of some important two-body Higgs decay modes inthe SM, such as H → b ¯ b , H → τ + τ − , H → γγ , H → W + W − and H → ZZ , have beenrecalculated with the help of the new values of relevant particle masses obtained at M H .Our numerical results should be very useful for model building and flavor physics.We reiterate that an unambiguous discovery of the Higgs boson at the LHC in thenear future will pave the way for us to confirm the Yukawa interactions between the Higgsfield and fermion fields. That will be a crucial step towards understanding the origin offermion masses, flavor mixing and CP violation either within or beyond the SM. This pointis especially true for testing the seesaw mechanisms, which attribute the tiny masses of threeknown neutrinos to the presence of some unknown heavy degrees of freedom via the Yukawainteractions. We believe that a new era of flavor physics is coming to the surface.The authors are indebted to Peter Zerwas for valuable comments and suggestions. Thiswork was supported in part by the National Natural Science Foundation of China undergrant No. 11135009 and by the Ministry of Science and Technology of China under grantNo. 2009CB825207 (Z.Z.X.), by the ERC under the Starting Grant MANITOP and bythe DFG in the Transregio 27 “Neutrinos and Beyond” (H.Z.) and by the Alexander vonHumboldt Foundation (S.Z.). 9 EFERENCES [1] P. Higgs, Phys. Lett. B , 132 (1964); Phys. Rev. Lett. , 508 (1964); F. Englert andR. Brout, Phys. Rev. Lett. , 321 (1964); G.S. Guralnik, C.R. Hagen, and T.W.B.Kibble, Phys. Rev. Lett. , 585 (1964); P. Higgs, Phys. Rev. , 1156 (1966).[2] S.L. Glashow, Nucl. Phys. , 579 (1961); S. Weinberg, Phys. Rev. Lett. , 1264(1967); A. Salam, in Elementary Particle Theory , edited by N. Svartholm (Almqvistand Wiksells, Stockholm, 1968), p. 367.[3] The ATLAS and CMS Collaborations,
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125 GeV and the corresponding cutoff scale Λ VS ≃ × GeV. Note that the valuesof the pole masses M q and running masses m q ( M q ) themselves, rather than the running masses m q ( µ ) at these mass scales, are given in the last two rows for comparison. But the pole masses ofthree light quarks are not listed, simply because the perturbative QCD calculation is not reliablein that energy region. µ m u ( µ ) (MeV) m d ( µ ) (MeV) m s ( µ ) (MeV) m c ( µ ) (GeV) m b ( µ ) (GeV) m t ( µ ) (GeV) m c ( m c ) 2 . +0 . − . . +0 . − . +36 − . +0 . − . . +0 . − . . +8 . − . . +0 . − . . ± . +30 − . +0 . − . . +0 . − . . +5 . − . m b ( m b ) 2 . +0 . − . . +0 . − . +26 − . +0 . − . . +0 . − . . +3 . − . M W . +0 . − . . +0 . − . +18 − . +0 . − . . +0 . − . . ± . M Z . +0 . − . . ± .
48 57 +18 − . +0 . − . . +0 . − . . ± . M H . +0 . − . . +0 . − . +17 − . +0 . − . . +0 . − . . +1 . − . m t ( m t ) 1 . +0 . − . . ± .
46 55 +17 − . +0 . − . . +0 . − . . ± .
11 TeV 1 . ± .
35 2 . +0 . − . +15 − . +0 . − . . +0 . − . . ± . VS . +0 . − . . ± .
22 26 +8 − . +0 . − . . +0 . − . . ± . M q − − − . +0 . − . . +0 . − . . ± . m q ( M q ) − − − . +0 . − . . +0 . − . . ± . M H ≃
125 GeV and the corresponding cutoff scale Λ VS ≃ × GeV, where theuncertainties of m l ( µ ) are determined by those of M l . Note that the pole masses M l , rather thanthe running masses m l ( M l ), are given in the last row just for comparison. µ m e ( µ ) (MeV) m µ ( µ ) (MeV) m τ ( µ ) (MeV) m c ( m c ) 0 . ± . . +0 . − . . ± . m b ( m b ) 0 . ± . . +0 . − . . ± . M W . +0 . − . . ± . . +0 . − . M Z . +0 . − . . +0 . − . . +0 . − . M H . +0 . − . . +0 . − . . ± . m t ( m t ) 0 . +0 . − . . +0 . − . . ± .
161 TeV 0 . +0 . − . . ± . . ± . VS . +0 . − . . +0 . − . . ± . M l . ± . . ± . . ± .
115 120 125 130 13510 . G e V . G e V M t = . G e V s (M Z ) = 0.1184 M H [GeV] V S [ G e V ] FIG. 1. Correlation between the energy scale Λ VS and the Higgs mass M H based on the require-ment of vacuum stability, where the solid curve corresponds to the best-fit value of the top-quarkpole mass M t = 172 . σ lower and upper limits. µ [GeV] R f ( µ ) R τ R t R b FIG. 2. The running behaviors of R t ( µ ), R b ( µ ) and R τ ( µ ) with respect to the energy scale µ inthe SM, where the vertical dashed line indicates the cutoff scale Λ VS ≃ × GeV as requiredby the vacuum stability for M H ≃
125 GeV. IG. 3. The evolution of R i ( µ ) with respect to the energy scale µ , where the red band corre-sponds to the variation of the Higgs mass in the range M H ≃ (124 · · · VS ≃ × GeV. Note that R ( µ ) ≃ R ( µ ) ≃ R ( µ )holds to an excellent degree of accuracy. -3 -2 -1
110 115 120 125 130 135 140 145 150 B r an c h i ng R a t i o s Higgs Mass M H [GeV]b b τ τ c cW WZ Z g g γ γ Z γ FIG. 4. The branching ratios of two-body Higgs decays versus the Higgs mass M H . The thicklines stand for the dominant H → f ¯ f modes: b ¯ b (solid line), τ + τ − (dashed line) and c ¯ c (dottedline); and the thin lines denote H → gg (solid line), γγ (dashed line), Zγ (dotted line), W + W − (dotted-dashed line) and ZZ (double-dotted dashed line).(double-dotted dashed line).