Delta r and the W-boson mass in the Singlet Extension of the Standard Model
∆∆ r and the W–boson mass in the Singlet Extension of theStandard Model D. L´opez-Val a , T. Robens b October 15, 2018 a Center for Cosmology, Particle Physics and Phenomenology CP3Universit´e Catholique de LouvainChemin du Cyclotron 2, B-1348 Louvain–la–Neuve, Belgium b IKTP, Technische Universit¨at DresdenZellescher Weg 19, D-01069 Dresden, Germany
E-mails: [email protected], [email protected] link between the electroweak gauge boson masses and the Fermi constant viathe muon lifetime measurement is instrumental for constraining and eventually pin-ning down new physics. We consider the simplest extension of the Standard Modelwith an additional real scalar SU (2) L ⊗ U (1) Y singlet and compute the electroweakprecision parameter ∆ r , along with the corresponding theoretical prediction for the W–boson mass. When confronted with the experimental W–boson mass measurement, ourpredictions impose limits on the singlet model parameter space. We identify regions,especially in the mass range which is accessible by the LHC, where these correspondto the most stringent experimental constraints that are currently available. The relation between the Electroweak (EW) gauge boson masses, the Fermi constant [ G F ] andthe fine structure constant [ α em ] is anchored experimentally via the muon lifetime measurementand constitutes a prominent tool for testing the quantum structure of the Standard Model (SM)and its manifold conceivable extensions. This relation is conventionally expressed in the literatureby means of the ∆ r parameter [1–6] and plays a major role in placing bounds on, and eventuallyunveiling new physics coupled to the standard electroweak Lagrangian.Aside from being interesting on its own, the quantum effects traded by ∆ r are part of theelectroweak radiative corrections to production and decay processes in the SM and beyond. Inparticular, the knowledge of ∆ r is a required footstep towards a full one–loop electroweak charac-terization of the Higgs boson decay modes in the singlet extension of the SM [7].The calculation of electroweak precision observables (EWPO) and its role in constraining man-ifold extensions of the SM has been object of dedicated attention in the literature [1, 2, 5, 6, 8–19],1 a r X i v : . [ h e p - ph ] J a n ncluding in particular the singlet extension of the SM, cf. e.g. Refs. [20–31] a . Theoretical predic-tions for ∆ r and for the W–boson mass [ m th W ] were first derived in the context of the SM [33,34] andlater on extended to new physics models such as the Two-Higgs–Doublet Model (2HDM) [35–43]and the Minimal Supersymmetric Standard Model (MSSM) [17, 44–52]. These predictions haveproven to be relevant not only to impose parameter space constraints, but also to identify newphysics structures capable to in part reconcile the well–known tension between the SM predictionand the experimental value, | m SM W − m exp W | (cid:39)
20 MeV. For instance, in Ref. [43] it was shownthat the extended Higgs sector of the general Two–Higgs–Doublet Model (2HDM) could yield m W (cid:38) m SM W , thus potentially alleviating the present discrepancy.Our main endeavour in this note is to provide a one–loop evaluation of the electroweak pa-rameter ∆ r and the W–boson mass in the presence of one extra real scalar SU (2) L ⊗ U (1) Y singlet. This model, which incorporates an additional neutral, CP -even spinless state, correspondsto the simplest renormalizable extension of the SM, and can also be viewed as an effective de-scription of the low–energy Higgs sector of a more fundamental UV completion. Pioneered byRefs. [53–55], this class of models has undergone dedicated scrutiny for the past two decades,revealing rich phenomenological implications, especially in the context of collider physics, seee.g. [22, 23, 25, 27, 28, 30, 31, 56–66, 66–73].Our starting point is the current most precise theoretical prediction for the SM W–boson mass[ m SM W ], which is known exactly at two–loop accuracy, including up to leading three–loop contri-butions [49, 74–80]. We combine these pure SM effects with the genuine singlet model one–loopcontributions and analyse their dependences on the relevant model parameters. Next we correlateour results with the experimental measurement of the W–boson mass and derive constraints onthe singlet model parameter space. Finally, we compare them to complementary constraints fromdirect collider searches, as well as to the more conventional tests based on global fits to electroweakprecision observables. ∆ r and m W as Electroweak precision measurements In the so–called “ G F scheme”, electroweak precision calculations use the experimentally measuredZ–boson mass [ m Z ], the fine–structure constant at zero momentum [ α em (0)], and the Fermi con-stant [ G F ] as input values. The latter is linked to the muon lifetime via [2, 3, 5, 6] τ − µ = G F m µ π F (cid:18) m m µ (cid:19) (cid:32) m µ m (cid:33) (1 + ∆ QED ) , (1)where F ( x ) = 1 − x − x ln x + 8 x − x . Following the standard conventions in the literature,the above defining relation for G F includes the finite QED contributions ∆ QED obtained withinthe Fermi Model – which are known to two–loop accuracy [81–85]. Matching the muon lifetime inthe Fermi model to the equivalent calculation within the full–fledged SM yields the relation: m (cid:32) − m m Z (cid:33) = πα em √ G F (1 + ∆ r ) with ∆ r ≡ ˆΣ W (0) m W + ∆ r [vert , box] , (2)which is the conventional definition of ∆ r , with m W,Z being the renormalized gauge boson massesin the on–shell scheme. Accordingly, we introduce the on–shell definition of the electroweak mixingangle [33] sin θ W = 1 − m W /m Z , along with the shorthand notations s W ≡ sin θ W , c W ≡ − s W . a Cf. also [32], which appeared after the work presented here.
2n turn, ˆΣ W ( k ) denotes the on–shell renormalized W-boson self–energy. The latter accounts forthe oblique part of the electroweak radiative corrections to the muon decay. The non–universal(i.e. process–dependent) corrections rely on the vertex and box contributions and are subsumedinto ∆ r [vert , box] . The explicit expression for ∆ r after renormalization in the on–shell scheme maybe written as a combination of loop diagrams and counterterms as follows:∆ r = Π γ (0) − c W s W (cid:32) δ m Z m Z − δ m m (cid:33) + Σ W (0) − δ m m + 2 c W s W Σ γ Z (0) m Z + ∆ r [vert , box] , (3)where Π γ (0) stands for the photon vacuum polarization, while δm W,Z denote the gauge bosonmass counterterms. Additional degrees of freedom and/or modified interactions will enter the loopdiagrams describing the muon decay, making ∆ r (and so m W ) model–dependent quantities. Atpresent, the calculation of ∆ r in the SM is complete up to two loops [49,74–79,86–94] and includesalso the leading three [80,95–99] and four–loop pieces [100,101]. The dominant contribution stemsfrom QED fermion loop corrections and is absorbed into the renormalization group running of thefine structure constant.Taking m Z and G F as experimental inputs, and using Eq. (2), the evaluation of ∆ r within theSM or beyond can be translated into a theoretical prediction for the W–boson mass [ m thW ]. Forthis we need to (iteratively) solve the equation m = 12 m Z (cid:34) (cid:115) − πα em √ G F m Z [1 + ∆ r ( m )] (cid:35) . (4)To first–order accuracy, Eq. (4) implies that a shift δ (∆ r ) promotes to the W–boson mass through∆ m W (cid:39) − m W s W c W − s W δ (∆ r ) . (5)For ∆ r = 0 one retrieves the tree-level value m tree W (cid:39) .
94 GeV. But the full theoretical resultis smaller in the SM since quantum effects yield ∆ r > ∼
126 GeV resonance with the SM Higgs boson, all experimental input values in Eq. (4) arefixed and thereby the theoretical prediction for the W–boson mass is fully determined. Setting theSM Higgs boson mass to the
HiggsSignals [102–104] best–fit value m H = 125 . r (cid:39) . >
0, wherefrom m SM W = 80 .
360 GeV. The estimated theoretical uncertainty reads∆ m th W (cid:39) m exp W = 80 . ± .
015 GeV . (6)This represents an accuracy at the (cid:39) .
02% level. The corresponding discrepancy with the SMtheoretical prediction | m exp W − m SM W | (cid:39)
20 MeV falls within the 1 σ –level ballpark; however, it is aslarge as roughly 5 times the estimated theoretical error. On the other hand, these differences shouldbe accessible by the upcoming W–boson mass measurements at the LHC, which are expected topull the current uncertainty down to ∆ m expW (cid:39)
10 MeV [109, 110]. Furthermore, a high–luminositylinear collider running in a low–energy mode at the W + W − threshold should be able to reduce iteven further, namely at the level of ∆ m expW (cid:39) r and m W to probe, constrain, or even unveil,new physics structures linked to the electroweak sector of the SM.3s a byproduct, the task of computing ∆ r involves the evaluation of the so–called δρ parameter[112–115]. The latter is defined upon the static contribution to the gauge boson self–energies,Σ Z (0) m − Σ W (0) m ≡ δρ, (7)and measures the ratio of the neutral–to–charged weak current strength. Quantum effects yielding δρ (cid:54) = 0 may be traced back to the mass splitting between the partners of a given weak isospindoublet, and so to the degree of departure from the global custodial SU (2) invariance of the SMLagrangian. The δρ parameter is finite for each doublet of SM matter fermions and is dominatedby the top quark loops δρ [ t ]SM = 3 G F m t √ π . (8)In terms of δρ , the general expression for ∆ r can be recast as [2, 5, 6]:∆ r = ∆ α − c W s W δρ + ∆ r rem = ∆ α + ∆ r [ δρ ] + ∆ r rem , (9)where ∆ r [ δρ ] ≡ − ( c W /s W ) δρ denotes the individual contribution from the static part of the self–energies. The ∆ α piece accounts for the (leading) QED light–fermion corrections, while the so–called “remainder” term [∆ r rem ] condenses the remaining (though not negligible) effects. In fact,in the SM we have ∆ α (cid:39) .
06 and ∆ r rem (cid:39) .
01, while ∆ r [ δρ ] (cid:39) − .
03 [3, 5, 6].At variance with this significant contribution, the counterpart Higgs boson–mediated effects arecomparably milder in the SM and feature a trademark logarithmic dependence on the Higgs mass[113] b , δρ [H] (cid:39) − √ G F m π s W c W (cid:26) ln m H m W − (cid:27) + ... . (10)Remarkably, this telltale screening behavior does not hold in general for extended Higgs sectors –viz. in the general 2HDM [43]. ∆ r and m W in the singlet model Our starting point is the most general form of the gauge invariant, renormalizable potential in-volving one real SU (2) L ⊗ U (1) Y singlet S and one doublet Φ, the latter carrying the quantumnumbers of the SM Higgs weak isospin doublet (see e.g. [29, 54, 55]): L s = ( D µ Φ) † D µ Φ + ∂ µ S∂ µ S − V (Φ , S ) , (11)with the potential V (Φ , S ) = − µ Φ † Φ − µ S + (cid:0) Φ † Φ S (cid:1) (cid:18) λ λ λ λ (cid:19) (cid:18) Φ † Φ S (cid:19) = − µ Φ † Φ − µ S + λ (Φ † Φ) + λ S + λ Φ † Φ S . (12) b One should bear in mind that the Higgs boson contribution in the SM [ δρ [H]SM ] is neither UV finite nor gaugeinvariant on its own, but only in combination with the remaining bosonic contributions. Z symmetry forbidding additional terms in the potential. We allow both of the scalar fields to acquirea Vacuum Expectation Value (VEV), in which case the Z symmetry is spontaneously broken by thesinglet VEV. The breaking of such a discrete symmetry during the electroweak phase transition inthe early universe may in principle lead to problematic weak–scale cosmic domain walls [116–118].However, analyses of the stability and evolution of such topological defects in multiscalar extensionsof the SM (cf. e.g. Refs. [119–121]) identify a variety of mechanisms that may sidestep these issues.These can also be evaded by extending this minimal setup with additional Z breaking terms [122],which would nevertheless have no direct impact on our analysis. In this sense, let us emphasizethat we interpret the singlet model as the low–energy effective Higgs sector of a more fundamentalUV–completion (cf. e.g. a model with an extended gauge group [123, 124]), whose specific detailsare either way not relevant for the purposes of our study.The neutral components of these fields can be expanded around their respective VEVs asfollows: Φ = G ± v d + l + iG √ S = v s + s √ . (13)The minimum of the above potential is achieved under the conditions µ = λ v d + λ v s µ = λ v s + λ v d , (14)while the quadratic terms in the fields generate the mass–squared matrix M ls = (cid:18) λ v d λ v d v s λ v d v s λ v s (cid:19) . (15)Requiring this matrix to be positively–defined leads to the stability conditions c λ , λ >
0; 4 λ λ − λ > . (16)The above mass matrix in the gauge basis M ls can be transformed into the (tree–level) massbasis through the rotation R ( α ) M ls R − ( α ) = M hH = diag( m m ), with R ( α ) = (cid:18) cos α − sin α sin α cos α (cid:19) and tan(2 α ) = λ v d v s λ v d − λ v s . (17)Its eigenvalues then read m , H = λ v d + λ v s ∓ | λ v d − λ v s | (cid:113) (2 α ) with the convention m > m , (18)and correspond to a light [h ] and a heavy [H ] CP -even mass–eigenstate. From Eq. (17), we seethat both are admixtures of the doublet [ l ] and the singlet [ s ] neutral componentsh = l cos α − s sin α and H = l sin α + s cos α. (19) c Cf. e.g. [29] for a more detailed discussion. Z h / H Z Z h / H G Z Z h / H Z W Wh / H W Wh / H G W Wh / H W Figure 1: One–loop Higgs boson–mediated contributions to the weak gauge boson self–energiesin the singlet model. The charged and neutral Goldstone boson contributions appear explicitly inthe ’tHooft-Feynman gauge. The Feynman diagrams are generated using
FeynArts.sty [126].The Higgs sector in this model is determined by five independent parameters, which can bechosen as m h , m H , sin α, v d , tan β ≡ v d v s , where the doublet VEV is fixed in terms of the Fermi constant through v d = G − F / √
2. Further-more, we fix one of the Higgs masses to the LHC value of 125 . g xxh = g SM xxh (1 + ∆ xh ) with 1 + ∆ xh = (cid:40) cos α h = h sin α h = H . (20) Let us now focus on the calculation of ∆ r and m W in the singlet extension of the SM. The pureSM contributions [∆ r SM ] and the genuine singlet model effects [ δ (∆ r sing )] can be split into twoUV-finite, gauge–invariant subsets and treated separately:∆ r sing = ∆ r SM + δ (∆ r sing ) . (21)We here include the state–of–the–art ∆ r SM evaluation, extracted from Eq. (2) and the numer-ical parametrization given in Ref. [78], which renders the central values m SM W = 80 .
360 GeV and ∆ r SM = 37 . × − . (22)We set the top-quark mass [ m t = 173 .
07 GeV] and the Z-boson mass [ m Z = 91 . HiggsSignals best–fitvalue of 125.7 GeV. This result for ∆ r SM includes the full set of available contributions, combiningthe full–fledged two–loop bosonic [79, 94] and fermionic [49, 78, 93] effects, alonside the leadingthree–loop corrections at O ( G F m t ) and O ( G F α s m t ) [80].The genuine singlet model contributions [ δ (∆ r sing )] originate from the Higgs–boson mediatedloops building up the weak gauge boson self–energies, which are shown in Fig. 1. This model–dependent part relies on the Higgs masses [ m h , m H ] and the mixing angle [sin α ], and we computeit analytically to one–loop order. As the Higgs self–interactions do not feature at one–loop, theresults are insensitive to tan β . 6t this point, care must be taken not to double–count the pure SM Higgs–mediated contribu-tions. To that aim we define [ δ (∆ r sing )] in Eq. (21) upon subtraction of the SM contribution: δ (∆ r sing ) ≡ ∆ r [H]sing − ∆ r [H]SM where ∆ r [H]SM = ∆ r [H]sing (cid:12)(cid:12)(cid:12) sin α =0 , (23)while the superscript [H] selects the Higgs–mediated contributions in each case. In this expressionwe explicitly identify the SM–like Higgs boson with the lighter of the two mass–eigenstates [h ],while the second eigenstate [H ] is assumed to describe a (so far unobserved) heavier Higgs compan-ion. Analogous expressions can be derived for the complementary case [ m H = 125 . > m h ],wherein the SM limit corresponds to cos α = 0. The phenomenology of both possibilities is analysedseparately in section 3.3.With this in mind, the purely singlet model contributions to the gauge boson self–energies giveΣ ZZ ( p ) = α em sin α πs W c W (cid:40) (cid:104) A ( m ) − A ( m ) (cid:105) m Z (cid:104) B ( p , m , m Z ) − B ( p , m , m Z ) (cid:105) − (cid:104) B ( p , m , m Z ) − B ( p , m , m Z ) (cid:105) (cid:41) (24)Σ WW ( p ) = α em sin α πs W (cid:40) (cid:104) A ( m ) − A ( m ) (cid:105) m W (cid:104) B ( p , m , m W ) − B ( p , m , m W ) (cid:105) − (cid:104) B ( p , m , m W ) − B ( p , m , m W ) (cid:105) (cid:41) . (25)The loop integrals in the above equations are expressed in terms of the standard Passarino–Veltmancoefficients in the conventions of [127]. The overlined notation Σ indicates that the overlap with theSM Higgs–mediated contribution has been removed according to Eq. (23). Analogous expressionswhere [ m h ↔ m H ] and [cos α ↔ sin α ] are valid if we identify the heavy scalar eigenstate [H ]with the SM–like Higgs boson.The presence of the additional singlet has a twofold impact: i) first, via the novel one–loopdiagrams mediated by the exchange of the additional Higgs boson, as displayed in Fig. 1; ii) second,via the reduced coupling strength of the SM–like Higgs to the weak gauge bosons, rescaled by themixing angle (cf. Eq. 20).At this stage, we in fact do not yet have to specify a complete renormalization scheme forthe model. It suffices to consider the weak gauge boson field and mass renormalization enteringEq. (2). The relevant counterterms therewith are fixed in the on–shell scheme [2, 33, 128, 129], i.e.by requiring the real part of the transverse renormalized self–energies to vanish at the respectivegauge boson pole masses, while setting the propagator residues to unity:Re ˆΣ WT ( m W ) = 0 , Re ˆΣ ZT ( m Z ) = 0 , Re ∂ ˆΣ WT ( p ) ∂p (cid:12)(cid:12)(cid:12) p = m W = 0 , Re ∂ ˆΣ ZT ( p ) ∂p (cid:12)(cid:12)(cid:12) p = m Z = 0 . The use of the on–shell scheme, which is customary in this context, provides an unambiguousmeaning to the free parameters of the model, allowing for a direct mapping between the bare7arameters in the classical Lagrangian and the physically measurable quantities in the quantizedrenormalizable Lagrangian. For instance, choosing on–shell renormalization conditions ensuresthat the weak gauge boson masses in Eqs. (1)-(2) correspond to their physical masses d .The complete singlet model prediction in Eq. (21) is exact to one–loop order and, as alluded toabove, it includes in addition all known higher order SM effects up to leading three–loop precision.Finally, let us also remark that the additional singlet–mediated contributions to the vertex andbox diagrams contained in ∆ r [vert,box] (cf. Eq. (3)) are suppressed by the light fermion Yukawacouplings and therefore negligible.In turn, the static contributions traded by the δρ parameter, as defined in Eq. (7), can beobtained by taking the limit p → δρ sing ) ≡ δρ [H]sing − δρ [H]SM G F sin α √ π (cid:40) m Z (cid:34) log (cid:32) m m (cid:33) + m Z m − m Z log (cid:32) m m Z (cid:33) − m Z m − m Z log (cid:32) m m Z (cid:33) + m m − m Z ) log (cid:32) m m Z (cid:33) − m m − m Z ) log (cid:32) m m Z (cid:33)(cid:35) − m W (cid:34) log (cid:32) m m (cid:33) + m W m − m W log (cid:32) m m W (cid:33) − m W m − m W log (cid:32) m m W (cid:33) + m m − m W ) log (cid:32) m m W (cid:33) − m m − m W ) log (cid:32) m m W (cid:33)(cid:35) (cid:41) (26)(cf. also the expression for the T -parameter in the M S scheme [21] e ). The logarithmic dependenceon both the light and the heavy Higgs masses follows the same screening–like pattern of the SM,as shown in Eq. (10). The model–specific new physics imprints are again to be found in i) theadditional Higgs contribution; and ii) the universally rescaled Higgs couplings to the gauge bosons.The size of ∆( δρ sing ) is controlled by the overall factor ∼ sin α , while its sign, which is fixed bythe respective Higgs and gauge boson mass ratios, is negative in all cases. Equation (26) thereforepredicts a systematic, negative yield from the new physics effects [∆( δρ sing ) < δρ sing < δρ SM . Finally, and owing to the fact that δρ is linked to ∆ r via Eq. (9), we may foresee∆ r sing ≡ ∆ r SM + δ (∆ r sing ) > ∆ r SM and hence m sing W < m SM W . Keeping in mind the current | m exp W − m SM W | (cid:39)
20 MeV (1 σ level) tension, this result anticipates tight constraints on the singletmodel parameter space – at the level of, if not stronger than, those stemming from the globalfits based on the oblique parameters [ S, T, U ] [10–12] (cf. discussion in section 3.4). Conversely,when considering m H ∼
126 GeV and a light Higgs companion [h ], similar arguments predict asystematic upward shift [∆( δρ sing ) >
0] with a global cos α rescaling. In this case, the singlet modelhas the potential to bring the theoretical value [ m sing W ] closer to the experimental measurement[ m exp W ]. In the next subsection we quantitatively justify all these statements. d On–shell mass renormalization in theories with mixing between the gauge eigenstates, as in the Higgs sector of thesinglet model, must be nonetheless addressed with care. In these cases, quantum effects generate off–diagonal termsin the loop–corrected propagators, which can be absorbed into the renormalization of the mixing angle. However,it can be shown that, regardless of the specific renormalization scheme chosen for the mixing angle, the on–shellrenormalized masses coincide with the physical (pole) masses to one–loop accuracy. A detailed discussion on thisissue as well as on the complete renormalization scheme as such for the singlet model will be presented in [7]. e It is easy to check that Eq. (26) is equivalent to Eq. (5.1) of Ref. [21], recalling that in our case we identify m h with the SM Higgs mass. α ∆ r [ x ] SM Exp. m h = 125.7 GeV200 GeV400 GeV600 GeV2 σ σ σ σ
800 GeV1000 GeV -1 -0.5 0 0.5 1sin α -60-40-2002040 ∆ m W [ M e V ] SMExp.m h = 125.7 GeV800 GeV1000 GeV600 GeV 2 σ σ σ σ
200 GeV 400 GeV -1 -0.5 0 0.5 1sin α ∆ r [ x ] SM Exp.m H = 125.7 GeV 60 GeV30 GeV90 GeV2 σ σ σ σ
120 GeV -1 -0.5 0 0.5 1sin α -40-2002040 ∆ m W [ M e V ] SMExp.m H = 125.7 GeV 60 GeV90 GeV120 GeV 2 σ σ σ σ
30 GeV
Figure 2: Upper panels: full one–loop evaluation of ∆ r ≡ ∆ r sing (left) and ∆ m W ≡ m sing W − m exp W (right) for different heavy Higgs masses [ m H ] with fixed [ m h = 125 . α ]. Lower panels: likewise, for different light Higgs masses [ m h ] and fixed[ m H = 125 . σ and 2 σ C.L. exclusion regions.Compatibility with the LHC signal strength measurements requires | sin α | (cid:46) .
42 (upper panels)and | sin α | (cid:38) .
91 (lower panels) (c.f. section 3.4).
In the following we present an upshot of our numerical analysis. Figures 2 and 3 illustrate thebehavior of ∆ r ≡ ∆ r sing and ∆ m W ≡ m sing W − m exp W under variations of the relevant singlet modelparameters. In Figure 2 we portray the evolution of both quantities with the mixing angle, forillustrative Higgs companion masses. In the upper panels the SM–like Higgs particle is identifiedwith the lightest singlet model mass–eigenstate [h ]. We fix its mass to m h = 125 . m H = 200 − m H = 125 . > m h ] is examined in the lower panels, with a variable mass for the second (light) Higgsspanning m h = 5 −
125 GeV. The results shown for ∆ r are referred to both the SM prediction[∆ r SM ] and the experimental value [∆ r exp ]. The latter follows from Eq. (2) with the experimentalinputs [125] m expW = 80 . ± .
015 GeV m Z = 91 . ± . α em (0) = 1 / . G F = 1 . − GeV − , (27)wherefrom we get 9
50 500 750 m H [GeV]34363840 ∆ r [ x ] SMExp. m h = 125.7 GeVsin α = 0.2sin α = 0.52 σ σ σ σ sin α = 0.7
250 500 750 m H [GeV]-60-40-2002040 ∆ m W [ M e V ] SMExp. m h = 125.7 GeVsin α = 0.2sin α = 0.5 2 σ σ σ σ sin α = 0.7
25 50 75 100m h [GeV]343638 ∆ r [ x ] SMExp.m H = 125.7 GeVsin α = 0.2sin α = 0.52 σ σ σ σ sin α = 0.7
25 50 75 100m h [GeV]-40-2002040 ∆ m W [ M e V ] SMExp.m H = 125.7 GeV sin α = 0.2sin α = 0.5 2 σ σ σ σ sin α = 0.7 Figure 3: Full one–loop evaluation of ∆ r ≡ ∆ r sing (left) and ∆ m W ≡ m sing W − m exp W (right) fordifferent mixing angle values, as a function of the heavy Higgs mass [ m H ] (upper panels) andthe light Higgs mass [ m h ] (lower panels). The corresponding SM predictions (the experimentalvalues) are displayed in dashed (dotted) lines. The shaded bands illustrate the 1 σ and 2 σ C.L.exclusion regions.∆ r exp = √ G F πα em m (cid:32) − m m Z (cid:33) − . ± . × − . (28)The 1 σ and 2 σ C.L. regions in ∆ r exp are derived from the m exp W uncertainty bands using standarderror propagation.Figure 3 provides a complementary view of the ∆ r and ∆ m W dependence on the additionalHiggs boson mass assuming mild (sin α = 0 . α = 0 .
5) and strong (sin α = 0 . α (cos α ) dependence reflects the global rescaling ofthe light (heavy) SM–like Higgs coupling to the weak gauge bosons. Accordingly, the values of ∆ r and m sing W converge to the SM predictions in the limit sin α = 0 (sin α = ±
1) in which the newphysics effects decouple. The growing departure from the SM as we raise (lower) the mass of theheavy (ligher) Higgs companion follows the logarithmic behavior singled out in Eq. (26), and canbe traced back to the increasing breaking of the (approximate) custodial invariance.In the case where m h = 125 . m H >
130 GeV (cf. upper panels of Figs. 2 and 3),we pin down positive (negative) deviations of ∆ r sing ( m sing W ) with respect to the corresponding SM10 r sing [ × ] m singW − m expW [MeV] ∆( δρ sing ) [ × ] h SM–like [ m h = 125 . m H [GeV] 300 500 1000 300 500 1000 300 500 1000sin α = 0 . α = 0 . α = 0 . H SM–like [ m H = 125 . m h [GeV] 30 60 90 30 60 90 30 60 90sin α = 0 . α = 0 . α = 0 . r sing , ∆ m W ≡ m sing W − m exp W and δρ ≡ ∆( δρ sing ) in the singlet model for representative Higgs masses and mixing angle choices.predictions. These increase systematically for larger mixing angles and heavier Higgs companions.The stark dependence on sin α and m H , combined with the fact that m sing W − m SM W < m SM W already lies 20 GeV below the experimental measurement, explains why the results obtainedin this case can easily lie outside of the 2 σ C.L. exclusion region. In the complementary scenario (cf.lower pannels), in which we set m H = 125 . m h ≤
125 GeV,we find analogous trends – but with interchanged dependences. Here the additional one–loop effectsfrom the light Higgs companion help to release the m sing W − m exp W tension. On the other hand, theonset of 2 σ –level constraints appears for m h (cid:46)
30 GeV. These results spotlight a significant massrange in which the singlet model contributions could in principle achieve m sing W (cid:39) m exp W f . Theviability of these scenarios is nevertheless hindered in practice, due to the direct collider massbounds and the LHC signal strength measurements. The impact of these additional constraints,which at this point we have not yet included, will be addressed in section 3.4 g Our discussion is complemented by specific numerical predictions for ∆ r and m sing W − m exp W ,which we list in Table 1 for representative parameter choices. For small mixing | sin α | (cid:46) . r sing departs from ∆ r SM at the O (0 . O (10)% for mixing angles above | sin α | (cid:38) . O (1) TeV scalarcompanions. Not surprisingly, these are the parameter space configurations that maximize thenon–standard singlet model imprints, viz. the rescaled Higgs boson interactions and the non–decoupling mass dependence of the Higgs–mediated loops. As we have seen in Figures 2-3, andaccording to Eq. (5), these shifts pull the resulting prediction [ m sing W ] down to ∼ −
70 MeVbelow the SM result. Staying within 1 σ C.L. we find | m sing W − m SM W | (cid:39)
10 MeV ( m sing W < m SM W ) forrelatively tempered mixing ( | sin α | (cid:46) .
2) and heavy Higgs masses up to 1 TeV. Larger mixingsof typically | sin α | (cid:38) . m sing W into the 2 σ –level exclusion region. These results once moreillustrate that, for a second heavy Higgs resonance, the singlet model effects tend to sharpen the m th W − m exp W tension even further, and more so as we increasingly depart from the SM–like limit.Alternatively, for m h < m H = 125 . ∼
5% in ∆ r sing (with ∆ r sing < ∆ r SM ) are attainable for 50 −
100 GeV light Higgs companion masses and mixing f Let us recall that both instances m th W − m exp W ≶ g A fully comprehensive analysis of the model combining all currently available constraints deserves a dedicatedstudy and will be presented elsewhere [130]. ∆ ( δ ρ s i ng ) [ x ] -0.5 0 0.5 sin α ∆ r [ δ ρ ] r e l [ % ] h = 125.7 GeV -9-6-3 ∆ ( δ ρ s i ng ) [ x ]
250 500 750m H [GeV]246 ∆ r [ δ ρ ] r e l [ % ] sin α = 0.2sin α = 0.5sin α = 0.7m h = 125.7 GeVsin α = 0.7sin α = 0.5sin α = 0.2 ∆ ( δ ρ s i ng ) [ x ] -0.5 0 0.5 sin α -4-20 ∆ r [ δ ρ ] r e l [ % ]
60 GeV90 GeV120 GeV 30 GeV60 GeV90 GeV30 GeV120 GeVm H = 125.7 GeV ∆ ( δ ρ s i ng ) [ x ]
25 50 75 100m h [GeV]-4-20 ∆ r [ δ ρ ] r e l [ % ] sin α = 0.2sin α = 0.5sin α = 0.7m H = 125.7 GeVm H = 125.7 GeVsin α = 0.7sin α = 0.2sin α = 0.5 Figure 4: Singlet model contribution to the δρ parameter at one loop [∆( δρ sing )] for representativemixing angles and Higgs companion masses. In the bottom subpannels we quantify the relativesize of these contributions to the overall ∆ r prediction, following Eq. (29).angles above | sin α | ∼ . r and m sing W we compute the new physics one–loop con-tributions to the δρ parameter (cf. Eq. (7)). The behavior of ∆( δρ sing ) as a function of therelevant singlet model parameters [sin α ] and [ m h , H ] is illustrated in Figure 4. Again, we sepa-rately examine the two complementary situations in which either the ligher (top–row panels) or theheavier (bottom–row panels) singlet model mass–eigenstate describes the SM–like Higgs boson. Asexpected, | ∆( δρ sing ) | enlarges as we progressively separate from the SM limit. The strong depen-dence in the additional Higgs mass displays the increasing deviation from the custodial symmetrylimit, which is enhanced by the mass splitting between the Higgs mass–eigenstates. Conversely,we recover ∆( δρ sing ) → m h → m H limit. The relative size of the static one–loop effectsencapsulated in ∆( δρ sing ) is quantified in the lower subpannels of Fig. 4 through the ratio∆ r [ δρ ]rel ≡ ∆ r [ δρ ]sing / ∆ r sing = − c W /s W ∆( δρ sing ) / ∆ r sing , (29)12 H [GeV] | sin α | max . . . . . . . . . . . m exp W at the 2 σ –level, for m h =125 . | sin α | ≤ .
42, cf. Fig. 6.which we construct from the different pieces singled out in Eq. (9), retaining the singlet modelcontributions only.Interestingly, the analysis of δρ provides a handle for estimating the size of higher–order correc-tions. The leading singlet model two–loop effects [∆( δρ [2]sing )] arise from the exchange of virtual topquarks and Higgs bosons. This type of mixed O ( G F m t ) Yukawa corrections was first computedwithin the SM in the small Higgs boson mass limit in Ref. [131] and later on extended to arbitrarymasses [132,133]. The analytical expressions therewith can be readily exported to our case. Takinginto account the rescaled top–quark interactions with the light (heavy) Higgs mass–eigenstate byan overall factor ∼ cos α ( ∼ sin α ); and removing as usual the overlap with the SM contribution(which we identify here with h in the sin α = 0 limit) we find∆( δρ [2]sing ) = (30)3 G F m t sin α π (cid:16) f ( m t /m ) − f ( m t /m ) (cid:17) ∼ G F m t sin α π (cid:34)
27 log (cid:32) m t m H (cid:33) + 4 πm h m t (cid:35) , where in the latter step we have introduced the asymptotic expansions of f ( r ) [132,133]. The aboveestimate ∆( δρ [2]sing ) stagnates around O (10 − ) for fiducial parameter choices with | sin α | (cid:46) . (cid:104) ∆( m [2] W ) (cid:105) sing ∼ − m W s W c W − s W δ (∆ r δ [ ρ ]sing ) (cid:46) O (1)MeV , (31)which we can interpreted as an estimate on the theoretical uncertainty on m sing W due to the quantumeffects beyond the one–loop order. In this section, we first confront the model constraints imposed by the [ m sing W − m exp W ] compari-son to those following from global fits to electroweak precision data. The difference [ m sing W − m exp W ]13orresponds to a (pseudo)observable which can directly be linked to a single experimental measure-ment. The electroweak precision tests are customary expressed in terms of the oblique parameters[ S, T, U ], c.f. e.g. Refs. [20, 23, 25, 26, 57, 58, 66] for analyses of the singlet extension with a Z symmetry, and [21, 24] for a slightly different model setup. In the standard conventions [125], andretaining the one–loop singlet model contributions only, these parameters are given by α em s W c W S = Σ Z ( m Z ) − Σ Z (0) m ; α em T = Σ W (0) m − Σ Z (0) m ; α em s W U = Σ W ( m W ) − Σ W (0) m W − c W Σ Z ( m Z ) − Σ Z (0) m Z . (32)Notice that genuine singlet model contributions to the photon and the mixed photon–Z vacuumpolarization are absent at one loop. The overlined notation Σ is once more tracking down theconsistent subtraction of the overlap with the SM Higgs–mediated contributions, as specified byEq. (23). The T parameter can obviously be related to the δρ parameter in Eq. (7), yielding α em T = − δρ . Likewise, we may rewrite ∆ r sing as∆ r sing = α em s W (cid:32) − S + c W T + c W − s W s W U (cid:33) . (33)In Fig. 5 we portray the functional dependence [ S, T, U ] with respect to the relevant singletmodel parameters. The best–fit point has been taken from Ref. [134], including the LHC Higgsmass measurement of 126 . ± . S = 0 . ± . T = 0 . ± . U = 0 . ± . . (34)Correlations among these parameters are revelant and must be taken into account when elec-troweak precision global fit estimates are used to constrain the parameter space of the model. Tothat aim we here use the best linear unbiased estimator (see e.g. [135]) based on the Gauss–Markovtheorem which yields χ = ( O l − ˆ O l ) ( V lk ) − ( O k − ˆ O k ) with O l = { S, T, U } , (35)where ˆ O l stand for the global best–fit values of the oblique parameters in Eq. (34). The covariancematrix V lk is extracted from Ref. [134], with correlation coefficients between the parameter pairs[( S, T ) , ( S, U ) , ( T, U )] given by [+0.89, -0.54, -0.83] respectively.We carry out our analysis by fixing the heavy (resp. light) additional scalar mass and allowingfor correlated variations of up to 2 σ in each of these parameters. For a two–parameter estimate,this translates into | ∆ χ | ≤ .
99. That way we derive upper (resp. lower) mixing angle limits, whichcorrespond to the parameter space regions compatible with these global electroweak precision tests.Albeit rendering non-negligible contraints, we find the resulting limits (c.f. e.g. the magenta lineof Fig. 6) to be superseded by other constraints throughout the entire parameter space, as wediscuss below.Next, we also consider the constraints to the maximal values of the mixing angle stemmingfrom direct collider searches and the averaged LHC Higgs signal strength measurements [¯ µ exp ].For the former, we use HiggsBounds [136] which incorporates detailed information from around300 search channels from the LEP, Tevatron, and LHC experiments, to extract upper (resp. lower)14 α -0.2-0.15-0.1-0.050 SM m h = 125.7 GeV200 GeV400 GeV600 GeV2 σ σ
800 GeV1000 GeV S -1 -0.5 0 0.5 1sin α h = 125.7 GeV 200 GeV400 GeV600 GeV 2 σ σ
800 GeV1000 GeV T -1 -0.5 0 0.5 1sin α -0.0500.050.10.15 SMm h = 125.7 GeV200 GeV 1 σ σ -1 -0.5 0 0.5 1sin α H = 125.7 GeV90 GeV60 GeV30 GeV2 σ σ
120 GeVS -1 -0.5 0 0.5 1sin α -0.2-0.15-0.1-0.050 SMm H = 125.7 GeV90 GeV60 GeV30 GeV 2 σ σ
120 GeVT -1 -0.5 0 0.5 1sin α -0.0500.050.10.15 m H = 125.7 GeV30 GeV1 σ σ
120 GeVU
Figure 5: Singlet model contributions to the oblique parameters S (left), T (center) and U (right)as a function of the mixing angle for representative heavy (upper row) and light (lower row) Higgscompanion masses. The dashed–dotted line represents the fiducial SM reference value S, T, U = 0. | s i n α | m a x m H [GeV]Interplay of different limits on mixing angleW mass measurementdirect search limitsfrom signal rate ew fit Figure 6: Upper limits on the mixing angle | sin α | max from i) the m exp W measurement; ii) di-rect collider searches; iii) compatibility with the m h = 125 . S, T, U .limits on the mixing angle | sin α | max (min) as a function of the heavy (resp. light) Higgs companion.In the mass range m H = 200 − µ ATLAS = 1 . ± . , µ CMS = 0 . ± .
14 wherefrom ¯ µ exp = 1 . ± . . (36)15 word of caution should be given here. Note that these best–fit estimates and C.L. limits arenot tailored to any particular model. This means for instance that, although the singlet model canonly yield a suppressed Higgs signal strength µ sing ≤
1, such restriction is not enforced beforehandwhen deriving the results in Eq. (36). Dedicated model–specific analyses should therefore includesuch model–dependent fit priors, which would eventually modify the resulting bounds h .To estimate the mixing angle range for which ¯ µ sing is compatible with the LHC observations[¯ µ exp ], we identify the light (heavy) singlet model mass–eigenstate with the SM Higgs boson andassume a global rescaling ¯ µ sing / ¯ µ SM (cid:39) cos α (sin α ). In this simple estimate, we do not entertainthe possibility that two eigenstates with almost degenerate masses m h (cid:39) m H could contribute tothe LHC Higgs signal. Allowing up to 2 σ –level deviations, we obtain upper (lower) mixing anglelimits of | sin α | ≤ .
42 ( | sin α | ≥ . m h < m H = 125 . S, T, U ] parameters. This result impliesthat the parameter space for the case of m H = 125 . > m h is severely restricted. Conse-quently, most regions in Figures 2-3 for which quantum corrections would shrink the [ m th W − m exp W ]discrepancy below the 1 σ –level are in practice precluded by the LHC signal strength measure-ments. One should also bear in mind that, for very light m h masses, additional constraints fromlow–energy observables may play a significant role, see e.g. [142–144] and references therein.On the other hand, larger regions of parameter space are still allowed when m h = 125 . 300 GeV. This can once again be attributed tothe Higgs–mediated corrections encoded within ∆ r , which increase with m H and are ultimatelylinked to the custodial symmetry breaking. In turn, the limits imposed by the correlated oblique[ S, T, U ] parameters (cf. the magenta curve in Fig. 6) are also milder than those obtained from the[ m sing W − m exp W ] comparison. This result is after all not surprising (cf. e.g. Ref [145]) and reflects thefact in a global fit (in this case parametrized by [ S, T, U ]), the effect of the individual observablesinvolved in it can balance each other in part. The resulting C.L. limits are then smeared withrespect to the situation in which we separately consider the more constraining measurements (inour case m exp W ) individually. In this regard, let us recall the very accurate precision (viz. 0 . | sin α | (cid:38) . − . We have reported on the computation of the electroweak precision parameter ∆ r , along withthe theoretical prediction of the W–boson mass, in the presence of one additional real scalar SU (2) L ⊗ U (1) Y singlet. The ∆ r parameter trades the relation between the electroweak gaugeboson masses, the Fermi constant and the muon lifetime. Its precise theoretical knowledge plays asalient role in the quest for physics beyond the SM. The reason is twofold: first, because ∆ r and m W constitute a probe of electroweak quantum effects and are therefore sensitive to, and able toconstrain, extended Higgs sectors; and second, due to the current 1 σ discrepancy | m SM W − m exp W | ∼ h We thank A. Straessner for clarifying comments regarding this point. m exp W ] to the limits imposed by i) direct collider searches;ii) Higgs signal strenght measurements; and iii) the bounds on [ S, T, U ] based on global fits toelectroweak precision data.Our conclusions may be outlined as follows: • The singlet model contributions to ∆ r and m W are characterized by: i) a global rescalingfactor which depends on the mixing between the two scalar mass–eigenstates and reflectsthe universal suppression of all Higgs boson couplings in this model; ii) the additional ex-change of the second Higgs boson, which exhibits a logarithmic screening–like non-decouplingdependence with the Higgs mass. • The singlet–induced new physics effects may typically yield up to O (10)% deviations in the∆ r parameter with respect to the SM prediction. Due to the characteristic dependence onthe Higgs masses, these departures are bound to be positive if the lightest mass eigenstate isidentified with the SM Higgs boson. Such a shift ∆ r sing > ∆ r SM implies | m sing W − m SM W | ∼ − m sing W < m SM W , which raises the tension with the current [ m exp W ] measurement. Thesetrends are reverted if we exchange the roles of the two mass–eigenstates and consider instead alight Higgs companion with m h < m H = 125 . r sing < ∆ r SM and hence m sing W > m SM W , which makes in principle possible to satisfy m sing W (cid:39) m exp W . Theviability of these scenarios is nonetheless limited in practice, as they are hardly compatiblewith the Higgs signal strength measurements. • Tight upper bounds on the mixing angle parameter | sin α | max can be derived when con-fronting [ m sing W ] to [ m exp W ]. These are particularly stringent for m h = 125 . m H (cid:38) 300 GeV, and reflect the enhanced breaking of the (approximate) custodial symmetryof the SM. 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