Improved prediction for the mass of the W boson in the NMSSM
aa r X i v : . [ h e p - ph ] J un DESY 15-073
Improved prediction for the mass of the W boson in the NMSSM
O. St˚al , ∗ , G. Weiglein , † , L. Zeune ‡ The Oskar Klein Centre, Department of PhysicsStockholm University, SE-106 91 Stockholm, Sweden Deutsches Elektronen-Synchrotron DESYNotkestraße 85, D-22607 Hamburg, Germany ITFA, University of AmsterdamScience Park 904, 1018 XE, Amsterdam, The Netherlands
Abstract
Electroweak precision observables, being highly sensitive to loop contributions of new physics,provide a powerful tool to test the theory and to discriminate between different models of theunderlying physics. In that context, the W boson mass, M W , plays a crucial role. The accuracy ofthe M W measurement has been significantly improved over the last years, and further improvementof the experimental accuracy is expected from future LHC measurements. In order to fully exploitthe precise experimental determination, an accurate theoretical prediction for M W in the StandardModel (SM) and extensions of it is of central importance. We present the currently most accurateprediction for the W boson mass in the Next-to-Minimal Supersymmetric extension of the StandardModel (NMSSM), including the full one-loop result and all available higher-order corrections of SMand SUSY type. The evaluation of M W is performed in a flexible framework, which facilitatesthe extension to other models beyond the SM. We show numerical results for the W boson massin the NMSSM, focussing on phenomenologically interesting scenarios, in which the Higgs signalcan be interpreted as the lightest or second lightest CP -even Higgs boson of the NMSSM. We findthat, for both Higgs signal interpretations, the NMSSM M W prediction is well compatible with themeasurement. We study the SUSY contributions to M W in detail and investigate in particular thegenuine NMSSM effects from the Higgs and neutralino sectors. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] Introduction
Supersymmetry (SUSY) is regarded to be the most appealing extension of the Standard Model (SM),as it provides a natural mechanism to explain a light Higgs boson as observed by ATLAS [1] andCMS [2]. Supersymmetry realised around the TeV-scale also comes with further desirable features,such as a possible dark matter candidate and the unification of gauge couplings.The superpotential of the Minimal Supersymmetric extension of the Standard Model (MSSM)contains a term bilinear in the two Higgs doublets, W (2) ∼ µH H . In this term a dimensionfulparameter, µ , is present, which in the MSSM has no natural connection to the SUSY breaking scale.The difficulty to motivate a phenomenologically acceptable value in this context is called the µ -problemof the MSSM [3]. This problem is addressed in the Next-to-Minimal Supersymmetric extension of theStandard Model (NMSSM), where the Higgs sector of the MSSM gets enlarged by an additional singlet.The corresponding term in the superpotential is replaced by a coupling W (3) ∼ λSH H , and the µ parameter arises dynamically from the vev of the singlet, S , and may therefore be related to the SUSYbreaking scale.Besides the solution of the µ -problem, there are additional motivations to study the NMSSM. Thephysical spectrum contains seven Higgs bosons, which leads to a rich and interesting phenomenology.Compared to the MSSM, the singlet field modifies the Higgs mass relations such that the tree-levelmass of the lightest neutral Higgs boson can be increased. Consequently, the radiative correctionsneeded to shift the mass of the lightest Higgs mass up to 125 GeV can be smaller. This relaxes therequirement of heavy stops, or a large splitting in the stop sector, in NMSSM parameter regions wherethe tree-level Higgs mass is larger than the maximal MSSM value (see e.g. Ref. [4]). The NMSSMsinglet-doublet mixing could also modify the couplings of the 125 GeV boson to explain a potentiallyenhanced rate in the diphoton signal (see e.g. Refs. [5–8]).Extensive direct searches for supersymmetric particles are carried out by the LHC experiments.These searches have so far not resulted in a signal, which leads to limits on the particle masses, see e.g.Refs. [9–11] for a compilation of the results. Indirect methods are complementary to direct searchesfor physics beyond the SM at the LHC and future collider experiments. Whereas direct methodsattempt to observe traces in the detectors arising from of the direct production of particles of newphysics models, indirect methods look for the quantum effects induced by virtual exchange of newstates. Even if not yet seen directly, signs of new physics may show up as small deviations betweenprecise measurements and SM predictions. Electroweak precision observables (EWPOs), such as the W boson mass, M W , the sine of the effective leptonic weak mixing angle at the Z boson resonance,sin θ effw , or the anomalous magnetic moment of the muon, ( g − µ , (among others) are all highlysensitive to loop contributions involving in principle all the particles of the considered model. Theycan both be theoretically predicted and experimentally measured with such a high precision that theycan be utilised to test the SM, to distinguish between different extensions, and to derive indirectconstraints on the parameters of a model. Input from indirect methods can be of great interest todirect searches for new particles. This was demonstrated, for instance, by the discovery of the topquark with a measured mass in remarkable agreement with the indirect prediction [12, 13].In this paper we focus on the W boson mass. The accuracy of the measurement of M W hasbeen significantly improved in the last years with the results presented by the Tevatron experimentsCDF [14] and DØ [15]. The current world average is [16, 17] M exp W = 80 . ± .
015 GeV . (1)This precise measurement makes M W particularly suitable for electroweak precision tests, even more2ince the precision is expected to be improved further when including the full dataset from the Teva-tron and upcoming results from the LHC. Of central importance for the theoretical precision thatcan be achieved on M W is the top quark mass measurement, since the experimental error on theinput parameter m t constitutes a dominant source of (parametric) uncertainty, see e.g. Ref. [18]. TheTevatron [19] and LHC [20–26] measurements of m t have been combined [27] to yield m exp t = 173 . ± . ± .
71 GeV . (2)In contrast to the M W measurement, a considerable improvement of the precision on m t beyond Eq. (2)seems less likely at the LHC. Furthermore, it is not straightforward to relate m exp t measured at ahadron collider (using kinematic information about the top decay products) to a theoretically well-defined mass parameter. The quantity measured with high precision at the Tevatron and the LHC isexpected to be close to the top pole mass with an uncertainty at the level of about 1 GeV [28–30].For the calculation of M W presented in this paper we adopt the interpretation of the measured value m exp t as the pole mass, but the results could easily be re-expressed in terms of a properly definedshort distance mass (such as the MS or DR mass). At an e + e − linear collider, the situation wouldimprove significantly. Estimates for the ILC show an expected precision ∆ M ILC W ∼ . − m ILC t = 0 . m t accounts both for the uncertainty in thedetermination of the actually measured mass parameter and the uncertainty related to the conversioninto a suitable and theoretically well-defined parameter such as the MS mass.For exploiting the precise current and (possible) future M W measurements, theoretical predictionsfor M W with comparable accuracy are desired both in the SM and extensions of it. In order to be ableto discriminate between different models it is necessary that the precision is comparable. In the SM,the most advanced evaluation of M W includes the full one-loop [33, 34] and two-loop [35–47] result, aswell as the leading three- and four-loop corrections [48–57]. A simple parametrization for M SM W hasalso been developed [58], see also Ref. [59]. Within the SM the LHC Higgs signal at 125 .
09 GeV [60]is interpreted as the SM Higgs boson. Setting M SM H ≃ .
09 GeV, the value of M W can be predictedin the SM without any free parameters. The result (for m t = 173 .
34 GeV, M SM H = 125 .
09 GeV) is M SM W = 80 .
358 GeV , (3)which differs by ∼ . σ from the experimental value given in Eq. (1). The theoretical uncertaintyfrom missing higher-order corrections has been estimated to be around 4 MeV in the SM for a lightHiggs boson [58].For supersymmetric theories, the one-loop result for M W [62–73] and leading two-loop corrections[74–77] have been obtained for the MSSM. A precise prediction for M MSSM W , taking into accountall relevant higher-order corrections of both SM- and SUSY-type, was presented in Ref. [61]. A firstprediction for M W in the NMSSM has also been presented in Ref. [78]. For the study of other EWPOs(mainly focusing on Z decays) in the NMSSM, see Refs. [78–80].In this work we follow the procedure employed in the MSSM to present a new prediction for M W in the NMSSM with the same level of accuracy as the current best MSSM prediction [61]. Wecombine the complete NMSSM one-loop result with the state-of-the-art SM result and leading SUSYhigher-order corrections. Our framework allows to output, besides M W , also the quantity ∆ r directly(see Sect. 3), which summarises all (non-QED) quantum correction to the muon decay amplitude.Besides its importance for electroweak precision tests, ∆ r is needed whenever a theoretical prediction We updated the SM M W prediction as discussed below in Sect. 4.3. This leads to a small difference compared tothe SM value given in Ref. [61].
3s parametrized in terms of the Fermi constant G µ (instead of M W or α ( M Z )). Our NMSSM predictionfor M W provides the flexibility to analyse SUSY loop contributions analytically and to treat possiblethreshold effects or numerical instabilities. We perform a detailed numerical analysis of M W in theNMSSM with the latest experimental results taken into account. We focus on the effects induced bythe extended Higgs and neutralino sectors, and in particular on benchmark scenarios where the LHCHiggs boson is interpreted as either the lightest or the second lightest CP -even Higgs boson of theNMSSM.This paper is organised as follows: in Sect. 2 we give a short introduction to the NMSSM, focussingon the Higgs and neutralino sectors. In Sect. 3 we describe the determination of the W boson mass inthe NMSSM. We outline the calculation of the one-loop contributions and the incorporation of higher-order contributions. In Sect. 4 we give the numerical results, analysing the NMSSM contributions tothe W boson mass, before we conclude in Sect. 5. In this section we introduce the NMSSM and specify our notation. We focus on the particle sectorswhich differ from the MSSM. Since the SM fermions and their superpartners appear in the same wayin both models, the sfermion sector of the NMSSM is unchanged with respect to the MSSM. Alsothe chargino sector is identical to that in the MSSM since no new charged degrees of freedom areintroduced. For these sectors we use the same notation as employed in Ref. [61].In addition to the two Higgs doublets of the MSSM, the NMSSM also contains a Higgs singlet, S ,which couples only to the Higgs sector. Considering the Z -symmetric version of the NMSSM, thesuperpotential takes the form W NMSSM = ¯ u y u QH − ¯ d y d QH − ¯ e y l LH + λSH H + 13 κS . (4)The new contributions of the Higgs singlet to the soft breaking terms are L NMSSMsoft = L MSSM,modsoft − m S | S | − ( λA λ SH H + 13 κA κ S + h . c . ) , (5)where L MSSM,modsoft is the soft-breaking Lagrangian L MSSMsoft of the MSSM (see e.g. Eq. (6.3.1) of Ref. [81]),but without the term bH H . The singlet couplings make it possible to dynamically generate aneffective µ parameter as µ eff = λ h S i . (6)The additional contributions (and the modified effective µ term) in the superpotential and in thesoft breaking terms lead to a Higgs potential which contains the additional soft breaking parameters m S , A λ , A κ , as well as the superpotential trilinear couplings λ and κ . Like in the MSSM, there is no CP -violation at tree-level in the couplings of the Higgs doublets. The new doublet-singlet couplingsallow in principle for CP -violation at tree level, but we will not consider this possibility here. Wechoose all parameters to be real.The minimum of the NMSSM Higgs potential triggers electroweak symmetry breaking, after whichthe Higgs doublets can be expanded around their minima according to H = v + √ ( φ − i χ ) − φ − ! , H = φ +2 v + √ ( φ + i χ ) ! . (7) For the Higgs doublets and the Higgs singlet we use the same notation for the supermultiplets and for its scalarcomponent. S = v s + 1 √ φ s + i χ s ) , (8)where v s is the (non-zero) vacuum expectation value of the singlet.The bilinear part of the Higgs potential can be written as V H = (cid:0) φ , φ , φ S (cid:1) M φφφ φ φ φ S + (cid:0) χ , χ , χ S (cid:1) M χχχ χ χ χ S + (cid:0) φ − , φ − (cid:1) M φ ± φ ± (cid:18) φ +1 φ +2 (cid:19) + · · · , (9)with the mass matrices M φφφ , M χχχ and M φ ± φ ± .The mixing of the CP -even, CP -odd and charged Higgs fields occurring in the mass eigenstates isdescribed by three unitary matrices U H , U A , and U C , where h h h = U H φ φ φ S , a a G = U A χ χ χ S , (cid:18) H ± G ± (cid:19) = U C (cid:18) φ ± φ ± (cid:19) . (10)These transform the Higgs fields such that the resulting (diagonal) mass matrices become M diag hhh = U H M φφφ (cid:0) U H (cid:1) † , M diag aaG = U A M χχχ (cid:0) U A (cid:1) † and M diag H ± G ± = U C M φ ± φ ± (cid:0) U C (cid:1) † (11)The CP -even mass eigenstates, h , h and h , are ordered such that m h ≤ m h ≤ m h , and similarlyfor the two CP -odd Higgs bosons, a and a . Unchanged from the SM there are also the Goldstonebosons, G and G ± . Finally, there is the charged Higgs pair, H ± with mass given by M H ± = ˆ m A + M W − λ v . (12)Here ˆ m A is the effective CP -odd doublet mass given byˆ m A = λv s sin β cos β ( A λ + κv s ) . (13)The superpartner of the singlet scalar enlarging the NMSSM Higgs sector is called the singlino,˜ S . It extends the neutralino sector with a fifth mass eigenstate. In the basis ( ˜ B, ˜ W , ˜ H , ˜ H , ˜ S ) theneutralino mass matrix at tree level is given by M ˜ χ = M − M Z s W cos β M Z s W sin β M M Z c W cos β − M Z c W sin β − M Z s W cos β M Z c W cos β − µ eff − λv M Z s W sin β − M Z c W sin β − µ eff − λv − λv − λv Kµ eff , (14)where K ≡ κ/λ . This mass matrix can be diagonalised by a single unitary matrix N such thatdiag( m ˜ χ , m ˜ χ , m ˜ χ , m ˜ χ , m ˜ χ ) = N ∗ M ˜ χ N † , (15)which gives the mass eigenvalues ordered as m ˜ χ i ≤ m ˜ χ j for i < j .5 (cid:0) ✗✁❡(cid:0)✗✂ ✖(cid:0) ✗✁❡(cid:0)✗✂❲ (cid:0) Figure 1: Left: tree-level diagram with a four-fermion vertex describing muon decay in the Fermimodel. Right: W boson exchange mediating muon decay in the electroweak SM (in unitary gauge). M W The W boson mass can theoretically be predicted from the muon decay rate. Muons decay to almost100% via µ → e ¯ ν e ν µ [82]. This decay was historically described first within the Fermi model (leftdiagram in Fig. 1). Comparing the muon decay amplitude calculated in the Fermi model to the samequantity calculated in the full SM or extensions thereof (the leading-order contribution in unitarygauge is depicted in Fig. 1) yields the relation G µ √ M Z e M W (cid:0) M Z − M W (cid:1) (1 + ∆ r ( M W , M Z , m t , ... , X )) . (16)This relates the W boson mass to the Fermi constant, G µ , which by definition contains the QEDcorrections to the four-fermion contact vertex up to O ( α ) [83–87], and to the other parameters M Z and e , which are known experimentally with very high precision. The Fermi constant itself isdetermined with high accuracy from precise measurements of the muon life time [88].The factor ∆ r in Eq. (16) summarises all higher-order contributions to the muon decay amplitudeafter subtracting the Fermi-model type virtual QED corrections, which are already included in thedefinition of G µ . Working in the on-shell renormalization scheme, Eq. (16) corresponds to a relationbetween the physical masses of the W and Z bosons.Neglecting the masses and momenta of the external fermions all loop diagrams can be expressedas a term proportional to the Born matrix element [33, 43] M Loop,i = ∆ r i M Born , ∆ r = X i ∆ r i . (17)In different models, different particles can contribute as virtual particles in the loop diagrams tothe muon-decay amplitude. Therefore, the quantity ∆ r depends on the specific model parameters(indicated by the X in Eq. (16)), and Eq. (16) provides a model-dependent prediction for the W bosonmass. The quantity ∆ r itself does depend on M W as well; hence, the value of M W as the solution ofEq. (16) has to be determined numerically. In practice this is done by iteration.In order to exploit the W boson mass for electroweak precision tests a precise theoretical predictionfor ∆ r within and beyond the SM is needed. In the next two subsections we describe our calculationof ∆ r in the NMSSM. A new one-loop calculation has been performed which is combined with allavailable higher order corrections of SM- and SUSY-type.6 .2 One-loop calculation of ∆ r in the NMSSM The one-loop contributions to ∆ r consist of the W boson self-energy, vertex and box diagrams, andthe corresponding counter terms (CT). The box diagrams are themselves UV-finite in a renormalizablegauge and require no counter terms. Schematically, this can be expressed as∆ r ( α ) = W Self-energy + W Self-energy CT + Vertex + Vertex CT + Box= Σ
W WT (0) M W + (cid:18) − δZ W − δM W M W (cid:19) + Vertex+ (cid:18) δe − δs w s w + δZ W + 12 ( δZ µ + δZ e + δZ ν µ + δZ ν e ) (cid:19) + Box . (18)Here Σ W WT (0) denotes the transverse part the W boson self-energy (evaluated at vanishing momentumtransfer), δM W is the renormalization constant for the W boson mass, δe and δs w are the renormal-ization constants for the electric charge and s w ≡ sin θ W , respectively. The δZ denote different fieldrenormalization constants. Since the W boson occurs in the muon decay amplitude only as a virtualparticle, its field renormalization constant δZ W cancels in the expression for ∆ r .We employ the on-shell renormalization scheme. The one-loop renormalization constants of the W and Z boson masses are then given by M W/Z, = M W/Z + δM W/Z , δM W/Z = Re Σ
W W/ZZT ( M W/Z ) , (19)where bare quantities are denoted with a zero subscript. The renormalization constant of the electriccharge is e = (1 + δe ) e , δe = 12 Π AA (0) + s w c w Σ AZT (0) M Z , (20)with Π AA ( k ) = Σ AAT ( k ) k , Π AA (0) = ∂ Σ AAT ( k ) ∂k | k =0 . (21)Note that the sign appearing in front of s w in Eq. (20) depends on convention chosen for the SU (2)covariant derivative. The sine of the weak mixing angle is not an independent parameter in theon-shell renormalization scheme. Its renormalization constant s w , = s w + δs w , δs w s w = − c s Re (cid:18) Σ W WT ( M W ) M W − Σ ZZT ( M Z ) M Z (cid:19) (22)is fixed by the renormalization constants of the weak gauge boson masses.Finally, the renormalization constant of a (left-handed) lepton field l (neglecting the lepton mass)is l L = (cid:18) δZ l,L (cid:19) l L , δZ l,L = − Σ lL (0) , (23)where Σ lL denotes the left-handed part of the lepton self energy. We adopt the sign conventions for the SU (2) L covariant derivative used in the code FeynArts [89–94], where (forhistorical reasons) the SU (2) L covariant derivative is defined by ∂ µ − i g I a W aµ for the SM and ∂ µ + i g I a W aµ for the(N)MSSM. The expressions given here correspond to the (N)MSSM convention. r ( α ) = Σ W WT (0) − Re (cid:2) Σ W WT ( M W ) (cid:3) M W + Π AA (0) − c s Re (cid:20) Σ ZZT ( M Z ) M Z − Σ W WT ( M W ) M W (cid:21) + 2 s w c w Σ AZT (0) M Z + Vertex + Box − (cid:0) Σ eL (0) + Σ µL (0) + Σ ν e L (0) + Σ ν µ L (0) (cid:1) . (24)The quantity ∆ r is at one loop level conventionally split into three parts,∆ r ( α ) = ∆ α − c s ∆ ρ + ∆ r rem . (25)The shift of the fine structure constant ∆ α arises from the charge renormalization which contains thecontributions from light fermions. The quantity ∆ ρ contains loop corrections to the ρ parameter [95],which describes the ratio between neutral and charged weak currents, and can be written as∆ ρ = Σ ZZT (0) M Z − Σ W WT (0) M W . (26)This quantity is sensitive to the mass splitting between the isospin partners in a doublet [95], whichleads to a sizeable effect in the SM in particular from the heavy quark doublet. While ∆ α is a pure SMcontribution, ∆ ρ can get large contributions also from SUSY particles, in particular the superpartnersof the heavy quarks. All other terms, both of SM and SUSY type, are contained in the remainderterm ∆ r rem .We have performed a diagrammatic one-loop calculation of ∆ r in the NMSSM according toEq. (24), using the Mathematica -based programs
FeynArts [89–94] and
FormCalc [96]. The NMSSMmodel file for
FeynArts , first used in [6], has been adapted from output from
SARAH [97, 98].The calculation of the SM-type diagrams (being part of the NMSSM contributions) are not dis-cussed here. This calculation has been discussed in the literature already many years ago [33, 34], andwe refer to Refs. [43, 99] for details. The one-loop result for ∆ r is also known for the MSSM (withcomplex parameters), see Refs. [61, 73]. The calculation in the NMSSM follows along the same linesas for the MSSM. However, the result gets modified from differences in the Higgs and the neutralinosectors. Below we outline the NMSSM one-loop contributions to ∆ r , for completeness also includingthe MSSM-type contributions.Besides the SM-type contributions with fermions and gauge bosons in the loops (not discussedfurther here), many additional self-energy, vertex and box diagrams appear in the NMSSM withsfermions, (SUSY) Higgs bosons, charginos and neutralinos in the loop. Generic examples of gauge-boson self-energy diagrams with sfermions are depicted in Fig. 2. Their contribution to ∆ r is finiteby itself. The NMSSM Higgs bosons enter only in gauge boson self-energy diagrams, since we haveneglected the masses of the external fermions. The contributing diagrams are sketched in Fig. 3. Thesecontributions are not finite by themselves. Only if one considers all (including SM-type) gauge bosonand Higgs contributions to the gauge boson self-energy diagrams, the vertex diagrams and vertexcounterterm diagrams together, the divergences cancel, and one finds a finite result.Charginos and neutralinos enter in gauge boson self-energy diagrams (depicted in Fig. 4), fermionself-energy diagrams (depicted in Fig. 5), vertex diagrams (depicted in Fig. 6, the analogous vertexcorrections exist also for the other vertex) and box diagrams (depicted in Fig. 7). The vertex contri-butions from the chargino/neutralino sector, together with the chargino/neutralino contributions tothe vertex CT and the gauge boson self-energies are finite. Each box diagram is UV-finite by itself.8 ✶ ❱✷⑦❢✶ ❱✶ ❱✷⑦❢✶⑦❢✷ Figure 2: Generic (N)MSSM one-loop gauge boson self-energy diagrams with a sfermion loop; V , V = γ , Z , W ± and ˜ f , ˜ f = ˜ ν , ˜ l , ˜ u , ˜ d . ❱✶ ❱✷s✶ ❱✶ ❱✷s✶s✷ ❱✶ ❱✷s✶❱✸ Figure 3: Generic NMSSM one-loop gauge boson self-energy diagrams with gauge bosons, Higgs andGoldstone bosons in the loop; V , V , V = γ , Z , W ± and s , s = h , h , h , a , a , H ± , G , G ± .In order to determine the contribution to ∆ r from a particular loop diagram, the Born amplitudehas to be factored out of the one-loop muon decay amplitude, as shown in Eq. (17). While mostloop diagrams directly give a result proportional to the Born amplitude, more complicated spinorstructures that do not occur in the SM case arise from box diagrams containing neutralinos andcharginos . Performing the calculation of the box diagrams in Fig. 7 in FormCalc , the spinor chainsare returned in the form M SUSY Box( a ) = (¯ u e γ λ ω − u µ )(¯ u ν µ γ λ ω − v ν e ) b ( a ) M SUSY Box( b ) = (¯ u ν e ω − u µ )(¯ u ν µ ω + v e ) b ( b ) . (27)The expressions for the coefficients b ( a ) and b ( b ) are lengthy and not given here explicitly. In order tofactor out the Born amplitude M Born = 2 παs M W (cid:0) ¯ u ν µ γ λ ω − u µ (cid:1) (cid:16) ¯ u e γ λ ω − v ν e (cid:17) , (28)the spinor chains in Eq. (27) have to be transformed into the same structure. We modify the spinorchains following the procedure described in Ref. [73] and get M SUSY Box( a ) = − s M W πα b ( a ) M Born M SUSY Box( b ) = s M W πα b ( b ) M Born . (29)The coefficients b ( a ) and b ( b ) contain ratios of mass-squared differences of the involved particles. Thesecoefficients can give rise to numerical instabilities in cases of mass degeneracies. In the implemen-tation of our results (which has been carried out in a Mathematica ) special care has been taken of The same complication occurs in the MSSM and was discussed in Ref. [73]. ✶ ❱✷⑦✤✵✱ ⑦✤✝⑦✤✵✱ ⑦✤✝ Figure 4: Generic NMSSM one-loop gauge boson self-energy diagram with charginos/neutralinos; V , V = γ , Z , W ± , ˜ χ ± = ˜ χ ± , and ˜ χ = ˜ χ , , , , . ❧✱✗ ❧✱✗⑦✤✵✱ ⑦✤✝⑦❧✱⑦✗ Figure 5: Generic NMSSM one-loop fermion self-energy diagram with a chargino/neutralino/sfermioncontribution; ˜ χ ± = ˜ χ ± , and ˜ χ = ˜ χ , , , , .such parameter regions with mass degeneracies or possible threshold effects. By adding appropriateexpansions a numerically stable evaluation is ensured. The on-shell renormalization conditions correspond to the definition of the W and Z boson massesaccording to the real part of the complex pole of the propagator, which from two-loop order on isthe only gauge-invariant way to define the masses of unstable particles (see Ref. [43] and referencestherein). The expansion around the complex pole results in a Breit-Wigner shape with a fixed width(fw). Internally we therefore use this definition (fw) of the gauge boson masses. The experimentallymeasured values of the gauge boson masses are obtained using a mass definition in terms of a Breit-Wigner shape with a running width (rw). As the last step of our calculation, we therefore transformthe W boson mass value to the running width definition, M rw W to facilitate a direct comparison to theexperimental value of M W . The difference between these two definitions is M rw W = M fw W + Γ W M rw W , (30)where M fw W corresponds to the fixed width description, see Ref. [100]. For the prediction of the W decay width we use Γ W = 3 G µ ( M rw W ) √ π (cid:18) α s π (cid:19) , (31)parametrized by G µ and including first order QCD corrections. The difference between the fixed- andrunning width definitions amounts to about 27 MeV, which is very relevant in view of the currenttheoretical and experimental precisions. For the Z boson mass, which is used as an input parameter inthe prediction for M W , the conversion from the running-width to the fixed-width definition is carried10 ✗(cid:0) ✗❡✁❲⑦✗(cid:0)✱⑦✖⑦✤✝✱ ⑦✤✵ ⑦✤✵✱ ⑦✤✝ ✖ ✗(cid:0) ✗❡✁❲⑦✤✵⑦✖ ⑦✗(cid:0) Figure 6: Generic one-loop vertex correction diagrams in the NMSSM; ˜ χ ± = ˜ χ ± , and ˜ χ = ˜ χ , , , , .Analoguous diagrams exist for the other W vertex. ✭❛✮✖ ⑦✤✝✱ ⑦✤✵⑦✗(cid:0)✱⑦✖ ⑦✗❡✱⑦✁⑦✤✵✱ ⑦✤✝✁✗❡✗(cid:0) ✭❜✮✖ ⑦✤✵✱ ⑦✤✝⑦✖✱⑦✗(cid:0) ⑦✗❡✱⑦✁⑦✤✝✱ ⑦✤✵✗❡✁✗(cid:0) Figure 7: Generic one-loop box diagrams contributing to the muon decay amplitude in the NMSSM;˜ χ ± = ˜ χ ± , and ˜ χ = ˜ χ , , , , .out in the first step of the calculation. Accordingly, keeping track of the proper definition of thegauge boson masses is obviously important in the context of electroweak precision physics. For theremainder of this paper we will not use the labels (rw, fw) explicitly; if we do not refer to an internalvariable, M W and M Z will always refer to the mass definition according to a Breit-Wigner shape witha running width (see e.g. Ref. [43] for further details; see also Refs. [101, 102]).We combine the SM one-loop result (which is part of the NMSSM calculation) with the relevantavailable higher-order corrections of the state-of-the-art prediction for M SM W . As we will describebelow in more detail, the higher-order corrections of SM-type are also incorporated in the NMSSMcalculation of ∆ r in order to achieve an accurate prediction for M NMSSM W . For a discussion of theincorporation of higher-order contributions to M W in the MSSM see Refs. [61, 73].In a first step, we write the NMSSM result for ∆ r as the sum of the full one-loop and the higher-order corrections, ∆ r NMSSM = ∆ r NMSSM( α ) + ∆ r NMSSM(h.o.) , (32)where ∆ r NMSSM( α ) denotes the NMSSM one-loop contributions from the various particle sectors∆ r NMSSM( α ) = ∆ r fermion( α ) + ∆ r gauge-boson/Higgs( α ) + ∆ r sfermion( α ) + ∆ r chargino/neutralino( α ) , (33)as discussed in the previous subsection. The term ∆ r NMSSM(h.o.) denotes the higher-oder contributions,which we split into a SM part and a SUSY part,∆ r NMSSM(h.o.) = ∆ r SM(h.o.) + ∆ r SUSY(h.o.) . (34)The terms ∆ r SM/SUSY(h.o.) describe the SM/SUSY contributions beyond one-loop order. For ∆ r SM(h.o.) we employ the most up-to-date SM result including all relevant higher-order corrections, while ∆ r SUSY(h.o.) r SM(h.o.) , we incorporate the following contributionsup to the four-loop order∆ r SM(h.o.) =∆ r ( αα s ) + ∆ r ( αα s ) + ∆ r ( α )ferm + ∆ r ( α )bos + ∆ r ( G µ α s m t ) + ∆ r ( G µ m t ) + ∆ r ( G µ m t α s ) . (35)The contributions in Eq. (35) consist of the two-loop QCD corrections ∆ r ( αα s ) [35–40], the three-loopQCD corrections ∆ r ( αα s ) [48–51], the fermionic electroweak two-loop corrections ∆ r ( α )ferm [42–44], thepurely bosonic electroweak two-loop corrections ∆ r ( α )bos [45–47], the mixed QCD and electroweak three-loop contributions ∆ r ( G µ α s m t ) [52,53], the purely electroweak three-loop contribution ∆ r ( G µ m t ) [52,53],and finally the four-loop QCD correction ∆ r ( G µ m t α s ) [55–57].The radiative corrections in the SM beyond one-loop level are numerically significant and lead toa large downward shift in M W by more than 100 MeV. The largest shift (beyond one-loop) is causedby the two-loop QCD corrections [35–41] followed by the three-loop QCD corrections ∆ r ( αα s ) [48–51].Most of the higher-order contributions are known analytically (and we include them in this form),except for the full electroweak two-loop contributions in the SM which involve numerical integrationsof the two-loop scalar integrals. These contributions are included in our calculation using a simpleparametrization formula given in [59]. This fit formula gives a good approximation to the full resultfor a light SM Higgs (the agreement is better than 0 . M W ) [59]. Using this parametrizationdirectly for the SM prediction of ∆ r ( α )ferm + ∆ r ( α )bos (rather than for the full SM prediction of M W — anapproach followed for the MSSM case in Ref. [73] and for the NMSSM case in Ref. [78]) allows us toevaluate these contributions at the particular NMSSM value for M W in each iteration step. The outputof this formula approximates the full result of ∆ r ( α )ferm + ∆ r ( α )bos using the fixed-width definition, suchthat it can directly be combined with other terms of our calculation. In our expression for ∆ r SM(h.o.) we use the result for ∆ r ( αα s ) given in Ref. [37], which contains also contributions from quarks of thefirst two generations and is numerically very close to the result from Ref. [41], and the result for The corrections beyond one-loop order are in fact crucial for the important result that the M W prediction in the SMfavours a light Higgs boson, whereas the one-loop result alone would favour a heavy SM Higgs. In [59] the electroweak two-loop contributions are expressed via ∆ r ( α ) ≡ ∆ r ( α )ferm + ∆ r ( α )bos = (∆ α ) + 2∆ α ∆˜ r ( α ) +∆ r ( α )rem , where a simple fit formula for the remainder term ∆ r ( α )rem is given. The quantity ∆˜ r ( α ) in the second termdenotes the full one-loop result without the ∆ α term. It should be noted, however, that the gauge boson masses with running width definition are needed as input for thefit formula given in [59]. This is the only part of our calculation where the running width definition is used internally. This is an improvement compared to Ref. [61], where the two-loop QCD contributions from Ref. [50] were employed,which include only the top und bottom quark contributions. r ( αα s ) from Ref. [50]. Both contributions are parametrized in terms of G µ . A comparison betweenour evaluation of M SM W and the fit formula for M SM W given in Ref. [58] can be found in Sect. 4.3 below.For the higher-order corrections of SUSY type, see Eq. (34), we take the following contributionsinto account, ∆ r SUSY(h.o.) = ∆ r SUSY( α )red − c s ∆ ρ SUSY , ( αα s ) − c s ∆ ρ SUSY , ( α t ,α t α b ,α b ) , (36)incorporating all SUSY corrections beyond one-loop order that are known to date. The first term inEq. (36) denotes the leading reducible O ( α ) two-loop corrections. Those contributions are obtainedby expanding the resummation formula [103]1 + ∆ r = 1(1 − ∆ α ) (cid:16) c s ∆ ρ (cid:17) − ∆ r rem , (37)which correctly takes terms of the type (∆ α ) , (∆ ρ ) and ∆ α ∆ ρ into account if ∆ ρ is parametrized by G µ . The pure SM terms are already included in ∆ r SM(h . o . ) , and because of numerical compensationsthose contributions are small beyond two-loop order [41]. Thus, we only need to consider the leadingtwo-loop terms with SUSY contributions,∆ r SUSY( α )red = − c s ∆ α ∆ ρ SUSY + c s (∆ ρ SUSY ) + 2 c s ∆ ρ SUSY ∆ ρ SM . (38)The other two terms in Eq. (36) denote irreducible two-loop SUSY contributions. The two-loop O ( αα s ) SUSY contributions [74, 75], ∆ ρ SUSY , ( αα s ) , contain squark loops with gluon exchange andquark/squark loops with gluino exchange (both depicted in Fig. 8). While the formula for the gluinocontributions is very lengthy, a compact result exists for the gluon contributions to ∆ ρ [74, 75]. Usingthese two-loop results for the SUSY contributions to M W requires the on-shell (physical) values forthe squark masses as input. The SU (2) relation M ˜ t L = M ˜ b L implies that one of the stop/sbottommasses is not independent, but can be expressed in terms of the other parameters. Therefore, whenincluding higher-order contributions, one cannot choose independent renormalization conditions forall four (stop and sbottom) masses. Loop corrections to the relation between the squark masses mustbe taken into account in order to be able to insert the proper on-shell values for the squark massesinto our calculation. This one-loop correction to the relation between the squark masses is relevantwhen inserted into the one-loop SUSY contributions to M W , while it is of higher order for the two-loopSUSY contributions. In our evaluation of M W this is taken into account by a “mass-shift” correctionterm. For more details see Ref. [75]. The gluon, gluino and mass-shift corrections, which are identicalin the MSSM and the NMSSM, are included in our NMSSM result for M W .The third term in Eq. (36) denotes the dominant Yukawa-enhanced electroweak two-loop correc-tions to ∆ ρ of O ( α t ), O ( α t α b ) and O ( α b ) [76, 77]. These contributions consist of heavy quark ( t/b )loops with Higgs exchange, squark (˜ t/ ˜ b ) loops with Higgs exchange, and mixed quark-squark loopswith Higgsino exchange, see Fig. 9. The corrections of this kind, which depend on the specific formof the Higgs sector, are only known for the MSSM so far [76, 77]. It is nevertheless possible to takethem into account also for the NMSSM in an approximate form. To this end, the considered NMSSMparameter point needs to be related to appropriate parameter values of the MSSM. Besides the valuesfor tan β , the sfermion trilinear couplings A f , and all the soft mass parameters, which can be directly In principle one could also include the term ∆ α ∆ r rem , which however is numerically small. ✶ ❱✷⑦t❀ ⑦❜⑦t❀ ⑦❜ ❣ ❱✶ ❱✷⑦t❀ ⑦❜t❀ ❜ ⑦❣ Figure 8: Generic O ( αα s ) two-loop self-energy diagrams in the (N)MSSM. Here g denotes a gluonand ˜ g a gluino; V , V = γ , Z , W ± . ❱✶ ❱✷t❀ ❜t❀ ❜ s ❱✶ ❱✷⑦t❀ ⑦❜⑦t❀ ⑦❜ s ❱✶ ❱✷⑦t❀ ⑦❜t❀ ❜ ⑦❍ Figure 9: Generic O ( α t + α t α b + α b ) two-loop diagrams, where ˜ H denotes a Higgsino and is s eitheran NMSSM Higgs or a Goldstone boson; V , V = γ , Z , W ± .taken over from the parameter point in the NMSSM, we determine the parameters in the followingway: we set the MSSM µ parameter equal to µ eff and we use the physical value of the charged Higgsmass as calculated in the NMSSM (see below) as input for the calculation of the MSSM Higgs masses.This prescription is motivated by the fact that in this way the value of the mass of the charged Higgsboson, which is the only Higgs boson appearing without mixing in both models, is the same in theNMSSM and the MSSM. The MSSM Higgs masses are calculated with FeynHiggs [104–108], usingthe calculated physical mass value of M H ± as on-shell input parameter. The MSSM Higgs massesand the Higgsino parameter µ determined in this way are then used as input for the calculation of ∆ ρ to O ( α t ), O ( α t α b ), O ( α b ). In order to avoid double-counting the dominant Yukawa-enhanced elec-troweak two-loop corrections in the SM [109,110] have been subtracted according to Eq. (34). We findthat the impact of the Yukawa-enhanced electroweak corrections of SUSY type on M W is relativelysmall (typically . M W in the NMSSM, we therefore leave it as an option to choose whether these contributionsshould be included or not. For the results presented in Sect. 4.4 they have not been included, unlessstated otherwise. For the evaluation of the W boson mass prediction, the masses of the NMSSM particles are needed. Weuse the NMSSM on-shell parameters as input to calculate the sfermion, chargino and neutralino masses.For the calculation of the Higgs boson masses we use NMSSMTools (version 4.6.0) [111–114]. Forother tools that are available to calculate the NMSSM Higgs masses including higher-order radiativecorrections see Refs. [115–117]. The implementation of the Higgs mass results of Ref. [117] (usingdirectly the on-shell parameters as input) is in progress. In two plots below the NMSSM Higgs boson masses at tree-level are used. They are calculated using the tree-levelrelations given in Sect. 2.
NMSSMTools the input parameters are assumed to be DR parameters at the SUSY breakingscale. In order to use the
NMSSMTools
Higgs masses in our result, a transformation from the on-shell parameters, needed for our evaluation, to the DR parameters, needed as
NMSSMTools input, isnecessary. This effect is approximately taken into account by transforming the on-shell X t parameterinto its DR value by the relation given in Ref. [118] (equations (60) ff in Ref. [118]). The shift in theother parameters is significantly smaller and therefore neglected here.We use a setup where the NMSSM parameter space can be tested against a broad set of exper-imental and theoretical constraints. Besides the constraints already implemented in NMSSMTools , further direct constraints on the Higgs sectors are evaluated using the code HiggsBounds (version4.2.0) [120–123]. All programs used for the numerical evaluation are linked through an interface tothe NMSSM
Mathematica code for the W boson mass prediction. Before moving on to our numerical results for the W boson mass prediction in the NMSSM, we discussthe remaining theoretical uncertainties in the M W calculation.The dominant theoretical uncertainty of the prediction for M W arises from the parametric uncer-tainty induced by the experimental error of the top-quark mass. Here one needs to take into accountboth the experimental error of the actual measurement and the systematic uncertainty associated withrelating the experimentally determined quantity to a theoretically well-defined mass parameter, seethe discussion above. A total experimental error of 1 GeV on m t causes a parametric uncertainty on M W of about 6 MeV, while the parametric uncertainties induced by the current experimental error ofthe hadronic contribution to the shift in the fine-structure constant, ∆ α had , and by the experimentalerror of M Z amount to about 2 MeV and 2 . M W prediction caused by the experimental error of the Higgs boson mass, δM exp H = 0 .
24 GeV [60], is sig-nificantly smaller ( . . m t and ∆ α had hasbeen discussed. With a precise top mass measurement of ∆ m t = 0 . M W is about 0 . M H SM <
300 GeV) [58]. The prediction for M W in the NMSSM isaffected by additional theoretical uncertainties from unknown higher-order corrections of SUSY type.While in the decoupling limit those additional uncertainties vanish, they can be important if someSUSY particles, in particular in the scalar top and bottom sectors, are relatively light. The combinedtheoretical uncertainty from unknown higher-order corrections of SM- and SUSY-type has been esti-mated (for the MSSM with real parameters) in Refs. [73, 77] as δM W ∼ (4 −
9) MeV, depending onthe SUSY mass scale. Since we include the same SUSY higher-order corrections in our NMSSM cal-culation as were considered for the uncertainty estimate in the MSSM, the uncertainty from unknown NMSSMTools contains a number of theoretical and experimental constraints, e.g. constraints from collider experiments(such as LEP mass limits on SUSY particles), B -physics and astrophysics. More details on the constraints included in NMSSMTools can be found in Refs. [111, 119]. The
Mathematica code is linked to a
Fortran driver program, calling
NMSSMTools and
HiggsBounds . The calculationof the SUSY particle masses is also included in the
Fortran driver. Similarly to the MSSM case [61], we plan toadditionally implement the M W calculation directly into Fortran in order to increase the speed of the M W evaluation.This will be useful in particular for large scans of the parameter space. The lower limit of 4 MeV corresponds to the SM uncertainty, which applies to the decoupling limit of the MSSM.For the upper limit of 9 MeV very light SUSY particles were considered. In view of the latest experimental boundsfrom the SUSY searches at the LHC, the (maximal) uncertainty from missing higher orders is expected to be somewhatsmaller than 9 MeV. r ( α ) ∆ r ( αα s ) ∆ r ( αα s ) ∆ r ( α )ferm + ∆ r ( α )bos ∆ r ( G µ α s m t ) + ∆ r ( G µ m t ) ∆ r ( G µ m t α s ) Table 1: The numerical values ( × ) of the different contributions to ∆ r specified in Eq. (35) aregiven for M W = 80 .
385 GeV and M SM H = 125 .
09 GeV.higher-order corrections is estimated to be of similar size.
We compare our evaluation of M SM W to the result from the fit formula for M SM W given in Ref. [58].In the latest version of Ref. [58] all the corrections of Eq. (35) are included. The M W fit formulaincorporates the O ( αα s ) from Ref. [41], whereas we use the O ( αα s ) from Ref. [37]. These results arein good numerical agreement with each other if in both cases the electric charge is parametrized interms of the fine structure constant α . The O ( α α s ) three-loop corrections included in Eq. (35) areparametrized in terms of G µ . We therefore choose to parametrize the O ( αα s ) contributions also interms of G µ . The difference between the G µ parametrization of the QCD two-loop corrections that weuse here and the α parametrization used in Ref. [58] leads to a prediction for M SM W that is ∼ r are given in table 1 for M W =80 .
385 GeV and M SM H = 125 .
09 GeV. The other relevant input parameters that we use are m t = 173 .
34 GeV , m b = 4 . , M Z = 91 . , Γ Z = 2 . , ∆ α lept = 0 . , ∆ α (5)had = 0 . , α − = 137 . ,α s ( M Z ) = 0 . , G µ = 1 . × − GeV − . (39)As explained above, the values for the W and Z boson masses given above, which correspond toa Breit-Wigner shape with running width, have been transformed internally to the definition of aBreit-Wigner shape with fixed width associated with the real part of the complex pole. M W prediction in the NMSSM We now turn to the discussion of the prediction for M W in the NMSSM. Our evaluation has beencarried out for the case of real parameters, consequently for all parameters given in this section thephases are set to zero and will not be listed as separate input parameters.An earlier result for M W in the NMSSM was presented in Ref. [78]. Concerning SUSY two-loopcontributions, in this result only the part of the contributions to ∆ ρ SUSY , ( αα s ) , see Eq. (36), arisingfrom squark loops with gluon exchange is taken into account. As we will show below in the discussionof our improved result for M W in the NMSSM, the two-loop contributions that have been neglectedin Ref. [78] can have a sizeable impact. A further improvement of our results for the MSSM and theNMSSM is that they are based on contributions to ∆ r that can all be evaluated at the correct inputvalue for M W (using an iterative procedure), i.e. M (N)MSSM W , while the evaluation in Ref. [78] makesuse of the fitting formula for M SM W [58]. The corresponding contribution to ∆ r extracted from thefitting formula for M SM W is determined at the input value M SM W rather than M (N)MSSM W , while it is thelatter that is actually needed for the evaluation in the (N)MSSM (see Ref. [73] for a discussion how to16emedy this effect). We have compared our result with the one given in Ref. [78] taking into accountonly those contributions in our result that are also contained in the result of Ref. [78]. We found goodagreement in this case, at the level of 1–2 MeV on M W .Throughout this section, we only display parameter points that are allowed by the LEP limits onSUSY particle masses [124], by all theoretical constraints in NMSSMTools (checking e.g. that the Higgspotential has a viable physical minimum and that no Landau pole exists below the GUT scale), andhave the neutralino as LSP. Unless stated otherwise, we choose the masses of the first and secondgeneration squarks and the gluino to be large enough to not be in conflict with the limits from thesearches for these particles at the LHC. We make use of the code
HiggsBounds [121–123] to checkeach parameter point against the limits from the Higgs searches at LEP, the Tevatron and the LHC.
Before turning to the discussion of the genuine NMSSM effects, we show the NMSSM M W predictionin the MSSM limit λ → , κ → , K ≡ κ/λ = constant , (40)with all other parameters (including µ eff ) held fixed (such that the MSSM is recovered). In this limitone CP -even, one CP -odd Higgs boson (not necessarily the heaviest ones) and one neutralino becomecompletely singlet and decouple. In the discussion of M NMSSM W in the MSSM limit, the setup forthe numerical evaluation is introduced and the comparison to the MSSM M W prediction serves asvalidation of our implementation.The left plot of Fig. 10 shows the NMSSM predictions in the MSSM limit (blue curves) as wellas the MSSM predictions (red curves) for M W as a function of the stop mixing parameter X t . The parameters in Fig. 10 are m t = 173 .
34 GeV, tan β = 20, µ (eff) = 200 GeV, M ˜ L/ ˜ E = 500 GeV, M ˜ Q/ ˜ U/ ˜ D , = 1500 GeV, M SUSY = M ˜ Q = M ˜ U = M ˜ D = 1000 GeV, A τ = A b = A t , M = 200 GeVand m ˜ g = 1500 GeV. For the additional NMSSM parameters we choose ˆ m A = 1000 GeV, λ → K = κ/λ = 0 . A κ = −
100 GeV (the impact of A κ on M W in the MSSM limit is negligible). Here, andin the following the prediction for M W includes all higher-order corrections described above (besidesthe Higgsino two-loop corrections).Our approach here is the following: We start from a NMSSM parameter point. We take theeffective CP -odd doublet mass ˆ m A or the parameter A λ (here ˆ m A = 1000 GeV) as input to calculatethe NMSSM Higgs boson spectrum. The physical value of the charged Higgs mass (calculated in theNMSSM) is used as input for the calculation of the MSSM Higgs masses. As discussed in Sect. 3.3, thisprocedure ensures that the mass of the charged Higgs boson used in our M W calculation is the same inthe NMSSM and the MSSM, since we calculate the MSSM Higgs masses in FeynHiggs (version 2.10.4)where the input parameter M H ± is interpreted as an on-shell mass parameter. The other parameterswhich occur in both models (tan β , the sfermion trilinear couplings A f , and the soft mass parameters) We thank the authors of Ref. [78] for providing us with numerical results from their code. The most stringent limits from SUSY searches at the LHC are set on the masses of the first and second generationsquarks and the gluino, which go beyond ∼ The X t parameter that we plot here is the on-shell parameter. As described in Sect. 4.1 the on-shell value istransformed into a DR value, which is used as input for NMSSMTools to calculate the Higgs masses. All numerical valuesgiven for X t in this section refer to the on-shell parameters. - X t @ GeV D M W @ G e V D M H = ± - - X t @ GeV D M h H L @ G e V D Figure 10: Comparison of the NMSSM predictions in the MSSM limit (blue curves) for the W bosonmass (left plot) and the lightest CP -even Higgs mass (right plot) with the MSSM predictions (redcurves) plotted against the stop mixing parameter X t . The parameters are given in the text. Forthe two dashed curves (small blue diamonds for the NMSSM predictions in the MSSM limit, andred triangles for the MSSM predictions) the tree-level Higgs masses are used. For the solid curves(with filled dots) loop-corrected Higgs masses are used: the NMSSM Higgs masses are calculated with NMSSMTools , and the MSSM Higgs masses calculated with
FeynHiggs .are used with the same values as input for the calculation of the physical masses in the MSSM andthe NMSSM. For the Higgs mass calculation with
NMSSMTools the parameter X t is transformed into aDR parameter, while for the M (N)MSSM W calculations its on-shell value is used. The MSSM parameter µ is identified with the NMSSM effective value µ eff . For the two dashed curves in Fig. 10 (small blue diamonds for the NMSSM predictions in theMSSM limit and red open triangles for the MSSM predictions) the tree-level Higgs masses are used.For the solid curve (with filled dots) loop-corrected Higgs masses are used: the NMSSM Higgs massesare calculated with
NMSSMTools and the MSSM Higgs masses calculated with
FeynHiggs .The corresponding predictions for the lightest CP -even Higgs mass in the (N)MSSM are displayedin the right plot of Fig. 10. For illustration, in the plots for the Higgs mass predictions the theoreticaluncertainty on the SUSY Higgs mass is combined with the experimental error into an allowed region forthe Higgs boson mass, rather than displaying the theoretical uncertainty in the Higgs mass predictionas a band around the theory prediction. Consequently, the blue band in the right plot shows theregion M H = 125 . ± .
04 GeV, which was obtained by adding a theoretical uncertainty of 3 GeVquadratically to the experimental 2 σ error. Here M H represents the corresponding mass parameterin the MSSM and the NMSSM (in the considered case M h in the MSSM and M h in the NMSSM).The position of the curves relative to the blue M H band depends strongly on the other parameters,which are fixed here. The range in which the NMSSM parameter points (with NMSSMTools
Higgsmasses) are allowed by
HiggsBounds coincides (approximately) with the region in with the lightest From here on we will leave out the subscript ’eff’ for the µ parameter in the NMSSM .
09 GeV ( | X t | & X t < CP -even Higgscalculated with FeynHiggs and with
NMSSMTools . This discrepancy arises because of differences in thehigher-order corrections implemented in the two codes . The tree-level Higgs masses are only usedin Fig. 10 for illustration. In all following plots (if nothing else is specified) the full loop-correctedresults for the Higgs masses are used.Going back to the left plot of Fig. 10, we see that the M NMSSM W predictions in the MSSM limitand the M MSSM W prediction coincide exactly if tree-level Higgs masses are used (which is an importantcheck of our implementation). However, using loop-corrected masses, the difference between the FeynHiggs and
NMSSMTools predictions for the lightest CP -even Higgs mass leads to a difference in M W of ∼ . | X t | . The effect of the difference in the M W prediction induced by thedifferent Higgs mass predictions is contained in the following plots in this section. This should be keptin mind when comparing M NMSSM W with M MSSM W .The dependence of the M W predictions in Fig. 10 on X t is influenced both by the loop contributionsto ∆ r involving stops and sbottoms, which are identical at the one-loop level in the MSSM and theNMSSM, and indirectly via the behaviour of the lightest CP -even Higgs mass. In the chosen examplethe impact of the former contributions is relatively small as a consequence of the relatively high massscale in the stop and sbottom sector. The effect of the higher-order corrections in the Higgs sector isclearly visible in Fig. 10 by comparing the full predictions with the ones based on the tree-level Higgsmasses. As expected from the behaviour of the M W prediction in the SM on the Higgs boson mass, theupward shift in the mass of the lightest CP -even Higgs boson caused by the loop corrections gives riseto a sizeable downward shift in the predictions for M W . The local maximum in the M W predictions atabout X t = 0 is in accordance with the local minimum in the Higgs-mass predictions. The fact thatthe local minima in the M W predictions are somewhat shifted compared to the local maxima in theHiggs-mass predictions is caused by the stop-loop contributions to ∆ r , whose effect can be directlyseen for the curves based on the tree-level predictions for the mass of the lightest CP -even Higgs bosonin the left plot of Fig. 10. The main contribution of the stop/sbottom sector can be associated with∆ ρ and hence depends strongly on the squark mixing. ∆ ρ contains terms sensitive to the splittingbetween the squarks of one flavour and terms sensitive to the splitting between stops and sbottoms.These two contributions enter with opposite signs, which tend to compensate each other for small andmoderate values of X t . Now we turn to the discussion of the size and parameter dependence of the SUSY two-loop corrections.Fig. 11 shows the size of the O ( αα s ) two-loop corrections. The parameters used here are m t =173 .
34 GeV, tan β = 2, µ = 200 GeV, M ˜ L/ ˜ E = 1000 GeV, M ˜ Q/ ˜ U/ ˜ D , = 1500 GeV, A τ = A b = In NMSSMTools the user can set a flag determining the precision for the Higgs masses. The result from Ref. [126]containing contributions up to the two-loop level is used if the flag is set equal to 1 or 2, where the two flags correspondto the result without (flag 1) and including (flag 2) contributions from non-zero momenta in the one-loop self-energies.While in
FeynHiggs this momentum dependence is taken into account, we nevertheless find better numerical agreementwith flag 1 of the
NMSSMTools result. For the sake of comparison between the NMSSM and the MSSM predictionsfor M W it is useful to keep those differences arising from different higher-order corrections in the MSSM limit of theHiggs sector as small as possible. We have therefore chosen flag 1 for the Higgs-mass evaluation with NMSSMTools inour numerical analyses presented in this paper. As mentioned above, an implementation of our predictions using theHiggs-mass evaluation of Ref. [117] is in progress. M = 600 GeV, m ˜ g = 1500 GeV (solid curves) and m ˜ g = 300 GeV (dashed curves), A λ =395 GeV, λ = 0 . κ = 0 . A κ = −
80 GeV and we vary M SUSY = M ˜ Q = M ˜ U = M ˜ D . We show theresults for three values of X t : X t = 2 M SUSY (left), X t = 0 (middle) and X t = − M SUSY (right). Itshould be stressed here that the parameters for these plots are chosen to demonstrate the possible sizeand the parameter dependence of the SUSY two-loop corrections, however they are partially excludedby experimental data: The parameter points in the left plots with X t = 2 M SUSY are
HiggsBounds allowed for M SUSY .
800 GeV (apart from a small excluded island around M SUSY ∼
550 GeV),whereas in the middle and the right plots, the chosen parameters are
HiggsBounds excluded for most M SUSY values. A gluino mass value of m ˜ g = 300 GeV is clearly disfavoured by the negative LHCsearch results. Fig. 11 shows the contribution to the W boson mass, δM W , from the O ( αα s ) two-loopcorrections with gluon exchange (dark blue curves), with gluino exchange (orange curves) and fromthe mass-shift correction (pink curves). The shift δM W has been obtained by calculating M NMSSM W twice, once including the corresponding two-loop corrections, and once without, and the two resultshave been subtracted from each other. Starting with the dark blue curves, we find that the gluoncontributions lead to a maximal shift of ∼ M W for all three choices of X t and that the sizeof the gluon contributions decreases with increasing M SUSY . Turning to the orange curves, we findthat for m ˜ g = 1500 GeV (solid curves) the δM W shift, induced by the gluino two-loop corrections, issmall ( < X t = 0, while it is up to 3 − X t = 2 M SUSY and X t = − M SUSY .Making the gluino light — choosing m ˜ g = 300 GeV (dashed curves) — the gluino corrections can getlarge. For large positive squark mixing, X t = 2 M SUSY , they reach up to 17 MeV for small values of M SUSY . The gluino corrections can lead to both a positive and a negative M W shift, depending onthe stop mixing parameter. Threshold effects occur in the gluino corrections and cause kinks in theorange curves, as can be seen in the middle and the right plots.The gluon and gluino two-loop contributions are directly related to the mass-shift correction,which has to be incorporated in order to arrive at the complete result for the O ( αα s ) contributionsto ∆ ρ SUSY . The pink curves show the impact of this additional correction term. Starting with thesolid curves ( m ˜ g = 1500 GeV), we observe that for large stop mixing, X t = ± M SUSY , the mass-shiftcorrections are positive and the maximal shift is ∼ M W (up to −
12 MeV for small M SUSY ). For m ˜ g = 300 GeV, thesize of the mass-shift correction is smaller. The kinks, caused by threshold effects, can be observed(for the same M SUSY values) also in the mass-shift corrections. Adding up the gluino and mass-shiftcorrections leads to a smooth curve and no kink is found in the full M W prediction. This can be seen inFig. 12, where we plot the sum of the gluon, gluino and mass-shift corrections (all parameters are thesame as in Fig. 11). Generally one can see that for large M SUSY all contributions decrease, showingthe expected decoupling behaviour. However contributions from the O ( αα s ) two-loop corrections upto a few MeV are still possible for M SUSY = 1000 GeV.The Yukawa-enhanced electroweak two-loop corrections of O ( α t ), O ( α t α b ), O ( α b ) to ∆ ρ (“Hig-gsino corrections”) in the MSSM can be included in our code, as discussed in Sect. 3.3. To do so,we calculate the MSSM Higgs masses as described in Sect. 3.3 (taking the NMSSM charged Higgsmass as input for the MSSM Higgs mass calculation) and use them as input for the ∆ ρ ( O ( α t ), O ( α t α b ), O ( α b )) formula. The size of these contributions can be seen in Fig. 13. Here, and insome of the following plots, we choose modified versions of the benchmark points given in Ref. [127],which predict one of the CP -even NMSSM Higgs bosons in the mass range of the observed Higgssignal, as starting point for our study. Here we take the following parameters: m t = 173 .
34 GeV,tan β = 2, µ = 200 GeV, M ˜ L/ ˜ E = 1000 GeV, M ˜ Q/ ˜ U/ ˜ D , = 1200 GeV, M ˜ Q = M ˜ U = 700 GeV, M ˜ D = 1000 GeV, A τ = A b = 1000 GeV, M = 200 GeV, m ˜ g = 1500 GeV, A λ = 405 GeV, λ = 0 .
00 500 600 700 800 900 1000 - M SUSY @ GeV D ∆ M W @ M e V D
400 500 600 700 800 900 1000 - - - - M SUSY @ GeV D ∆ M W @ M e V D
400 500 600 700 800 900 1000 - - M SUSY @ GeV D ∆ M W @ M e V D Figure 11: Size of the O ( αα s ) two-loop corrections with gluon and gluino exchange. The solid curvescorrespond to m ˜ g = 1500 GeV while the dashed curves correspond to m ˜ g = 300 GeV. In the left plotwe set X t = 2 M SUSY , in the middle one X t = 0 and in the right one X t = − M SUSY . The plotsshow the contribution to the W boson mass, δM W , from the O ( αα s ) two-loop corrections with gluonexchange (dark blue curves), with gluino exchange (orange curves), and the mass-shift correction (pinkcurves) as a function of M SUSY . The parameter points with X t = 2 M SUSY are
HiggsBounds allowedfor M SUSY .
800 GeV, whereas the points with X t = 0 and with X t = − M SUSY predict too lowHiggs masses and are
HiggsBounds excluded for most M SUSY values. Note the different scales at they-axis. The parameters used are given in the text.
400 500 600 700 800 900 1000 - M SUSY @ GeV D ∆ M W @ M e V D
400 500 600 700 800 900 1000 - - - - M SUSY @ GeV D ∆ M W @ M e V D
400 500 600 700 800 900 1000 - - M SUSY @ GeV D ∆ M W @ M e V D Figure 12: The plots show the full O ( αα s ) two-loop corrections to M W (sum of the corrections shownseparately in Fig. 11) as a function of M SUSY . The parameters are the same as in Fig. 11. The solidcurves correspond to m ˜ g = 1500 GeV while the dashed curves correspond to m ˜ g = 300 GeV. In theleft plot we set X t = 2 M SUSY , in the middle one X t = 0 and in the right one X t = − M SUSY . κ = 0 . A κ = −
10 GeV, and we vary X t . These parameter points are HiggsBounds allowed in the re-gions 700 GeV < X t < < X t < M W prediction without Higgsino corrections (blue) and including Higgsino corrections (green) plot-ted against X t . In the middle plot the shift δM W induced by the Higgsino corrections (obtained bysubtracting the M W predictions with and without Higgsino corrections as shown in the left plot) isplotted against X t . We see that the Higgsino corrections can enter the M W prediction with bothsigns. The numerical effect of the M W shift, induced by the Higgsino corrections, is relatively small21
00 400 600 800 1000 1200 140080.36680.36880.37080.37280.37480.37680.37880.380 X t @ GeV D M W @ G e V D
200 400 600 800 1000 1200 1400 - - X t @ GeV D ∆ M W @ M e V D
118 120 122 124 12680.35580.36080.36580.37080.37580.380 M h @ GeV D M W @ G e V D Figure 13: Size of the electroweak O ( α t ), O ( α t α b ), O ( α b ) SUSY two-loop corrections. The left plotshows the NMSSM M W prediction without Higgsino corrections (blue) and including Higgsino correc-tions (green). The middle plot shows the shift δM W induced by the Higgsino corrections (obtainedby subtracting the M W predictions with and without Higgsino corrections as shown in the left plot).The right plot shows the NMSSM M W prediction without Higgsino corrections (blue) and includingHiggsino corrections (green) plotted against the lightest CP -even Higgs mass M h . The black curvein the right plot indicates the SM M W prediction with M H SM = M h . The grey band indicates the1 σ region of the experimental W boson mass measurement. The parameters used for these plots aregiven in the text.( ∼ M W from the Higgsino corrections canbe slightly larger ( ∼ t/ ˜ b . The right plot shows the M W prediction plotted against M h . We can clearly see here that this scenario, in which the Higgs signal can be interpreted as thelightest CP -even NMSSM Higgs, gives a W boson mass prediction in good agreement with the M W measurement indicated by the grey band. Now we turn to the discussion of effects from the NMSSM Higgs sector. In the MSSM the maximalvalue for the tree-level Higgs mass M h is M Z . One of the features of the NMSSM Higgs sector isthat the tree-level Higgs mass M h gets an additional contribution λ v sin β , which can shift thetree-level Higgs mass upwards compared to its MSSM value (an upward shift can also be caused bysinglet–doublet mixing, if the singlet state is lighter than the doublet state), and thus reduce the sizeof the radiative corrections needed to ’push’ the lightest Higgs mass up to the experimental value. For λ = 0 . β = 2 a tree-level value for M h of 112 GeV is possible [127]. This additional tree-levelcontribution to the Higgs mass, as well as its impact on M W are shown in Fig. 14. The parameterschosen here are m t = 173 .
34 GeV, µ = 500 GeV, M ˜ L/ ˜ E = 500 GeV, M ˜ Q/ ˜ U/ ˜ D , = 1500 GeV, M ˜ Q = M ˜ U = M ˜ D = 1000 GeV, X t = 2000 GeV, A τ = A b = A t , M = 200 GeV, m ˜ g = 1500 GeV,ˆ m A = 450 GeV, κ = λ and A κ = −
100 GeV. We vary tan β and show the results for differentvalues of λ . The red curves correspond to the MSSM limit ( λ →
0) while for the other curves the λ value is given in the corresponding colour. The upper left plot shows the tree-level prediction for thelightest CP -even Higgs mass M h . As expected, the M h prediction in the MSSM limit approaches itsmaximal value M Z for large tan β . Increasing λ , the M h prediction decreases for large tan β , causedby doublet–singlet mixing terms. For small tan β one clearly sees the positive contribution from the22 Z Λ =
Λ =
Λ =
Λ =
Λ =
Λ =
Λ = Β M h H t r ee - l e v e l L @ G e V D M H = ± Β M h @ G e V D - - - - -
40 tan Β ∆ M W @ M e V D M W exp = Β M W @ G e V D Figure 14: Predictions for M h and M W as a function of tan β . The red curves correspond to theMSSM limit ( λ →
0) while for the other curves the λ values are given in the figure. The upper left plotshows the tree-level prediction for the lightest CP -even Higgs mass M h , the upper right plot shows M h including radiative corrections (calculated with NMSSMTools as described in the text), the lowerleft plot shows the shift δM W (calculated as in Eq. (41)) from diagrams involving Higgs and gaugebosons, and the lower right plot shows the full M W prediction. The parameters used for these plotsare given in the text.term λ v sin β pushing the tree-level Higgs mass beyond M Z for large λ . The full M h prediction The mixing of the h state with the heavier singlet leads to a negative contribution to the tree-level Higgs mass,which pulls the NMSSM Higgs mass value down (compared to the MSSM case) for intermediate and large tan β values(for details see Ref. [4]). At a specific tan β value this contributions exactly cancels the positive λ v sin β shift at
00 750 800 850 900 950 1000 105080.36080.36580.37080.37580.38080.38580.390 X t @ GeV D M W @ G e V D
700 750 800 850 900 950 1000 105080.36080.36580.37080.37580.38080.38580.390 X t @ GeV D M W @ G e V D M W exp =
120 121 122 123 124 125 126 12780.35580.36080.36580.37080.375 M h @ GeV D M W @ G e V D Figure 15: The left plot shows the M NMSSM W prediction (blue, solid curve) and the M MSSM W predic-tion (red) plotted against X t . In the middle plot, the additional dashed blue curve corresponds to M NMSSM W − M SM W ( M h ) + M SM W ( M h ) ( M h is the mass of the lightest CP -even Higgs of the NMSSM,and M h is the mass of the light CP -even Higgs of the MSSM). The right plots shows the M NMSSM W prediction plotted against the lightest CP -even Higgs mass M h . The black curve in the right plotindicates the SM M W prediction with M H SM = M h . The experimental M W measurement is indicatedby the grey band; the region M H = 125 . ± .
04 GeV is indicated by the blue band. The parametersare given in the text.(calculated with
NMSSMTools as described above) can be seen in the upper right plot. Now we turn tothe M W contributions from the NMSSM Higgs and gauge boson sector, shown in the lower left plot.The shift δM W displayed here is based on the approximate relation [73] δM W = − M ref W s W c W − s W ∆ r x ( α ) , (41)where ∆ r x ( α ) denotes the one-loop contribution from particle sector x (here x =gauge-boson/Higgs), asdefined for the NMSSM in Eq. (33). The reference M W value is set here to M ref W = M exp W . The overallcontribution from the Higgs sector is rather large and negative. As we will discuss in more detailbelow, the Higgs sector contributions here are predominantly SM-type contributions (with M H SM setto the corresponding Higgs mass value). The prediction for M W in the NMSSM is shown in the lowerright plot. Larger values for M h correspond to a lower predicted value for M W . Thus, for smalltan β , where we find a significantly higher predicted value for M h for large λ than in the MSSM limit(arising from the additional tree-level term), we get a lower predicted value for M W , which is howeverstill compatible with the experimental M W measurement at the 2 σ level for the scenario chosen here.For tan β ∼ W boson mass prediction for λ = 0 .
65 and λ → ∼
25 MeV. The parameter tan β enters also in the sfermion and in the chargino/neutralino sector. Wechecked that for the parameters used here, the tan β dependence of the contributions from these twosectors is small compared to the Higgs sector contributions, less than ∼ M W (blue curve) with the MSSM prediction (red curve). the tree level, and the NMSSM Higgs mass value coincides with the MSSM value. In the scenario considered here, thishappens for all λ at the same tan β value, since we chose κ = λ . As can be seen in the upper right plot of Fig. 14, thisbehaviour is approximately retained also in the presence of higher-order corrections in the Higgs sector. m t = 173 .
34 GeV, tan β = 2, µ = 200 GeV, M ˜ L/ ˜ E = 1500 GeV, M ˜ Q/ ˜ U/ ˜ D , = 1200 GeV, M ˜ U = M ˜ Q = 540 GeV, M ˜ D = 1000 GeV, A τ = A b = 1000 GeV, M =370 GeV, m ˜ g = 1500 GeV, A λ = 420 GeV, λ = 0 . κ = 0 . A κ = −
10 GeV, and we vary X t . TheNMSSM parameters are allowed by HiggsBounds for X t &
780 GeV. For X t &
810 GeV the massof the lightest CP -even Higgs falls in the range of the observed Higgs signal. The MSSM predictionis plotted as a comparison to illustrate and discuss the NMSSM effects on M W . Here (and in thefollowing) we do not check any phenomenological constraints for the MSSM parameter point (butonly for the considered NMSSM scenario).The NMSSM prediction for M W differs from the MSSM prediction by ∼
12 MeV. The chargino/neu-tralino contributions can enter with both signs, and we find that in this scenario the relatively small µ value causes negative corrections to ∆ r . On the other hand, small M values tend to give positivecontributions to ∆ r . For the chosen parameters, these two effects cancel and contributions from thechargino/neutralino sector are very small, O (0 . CP -odd Higgs sector have a negligible impact on the M W prediction (see also Ref. [78]). Since we setthe charged Higgs masses equal to each other in the two models, differences can only come from the CP -even Higgs sector. For this parameter point the second lightest Higgs ( M h = 150 GeV) has a largesinglet component ( | U H | ≃ h and h are small. h isheavy and has no impact on the M W prediction. Our procedure to calculate the Higgs masses in theMSSM and the NMSSM leads to the same charged Higgs masses, but to different predictions for thelightest CP -even Higgs masses M h and M h . This difference arises from the different relations betweenthe charged Higgs mass and the lightest CP -even Higgs mass in the MSSM and the NMSSM. Furtherit also incorporates the (“technical”) difference due to the different radiative corrections included in FeynHiggs and
NMSSMTools (as analysed above in the MSSM limit). The middle plot of Fig. 15 showsin addition to the NMSSM prediction for M W (blue) and the MSSM prediction (red), a blue dashedcurve (with open dots). The dashed blue curve corresponds to M NMSSM W − M SM W ( M h ) + M SM W ( M h ) .As one can see the dashed blue curve is very close to the red MSSM curve, thus here the differencebetween the MSSM and the NMSSM Higgs sector contributions to M W essentially arises from theSM-type Higgs sector contributions, in which different Higgs mass values are inserted. It should benoted in this context that we have made a choice here by comparing the predictions for a particularNMSSM parameter point with an associated MSSM parameter point having the same value of themass of the charged Higgs boson. Accordingly, the predictions for the other Higgs boson masses inthe two models in general differ from each other, see above, leading to the effect displayed in the leftplot of Fig. 15. Instead, one could have chosen, at least in principle, the associated MSSM parameterpoint such that the masses of the lightest CP -even Higgs masses, M h and M h , are equal to each other.Also in that case differences in the other parameters in the Higgs sector, including the mass of thecharged Higgs boson, would induce a shift in the predictions for M W .The right plot of Fig. 15 shows the M NMSSM W prediction plotted against the lightest CP -even Higgsmass M h . In this plot we display both the blue band indicating the region M h = 125 . ± .
04 GeVas well as the grey band showing the experimental 1 σ band from the W boson mass measurement.The black curve in the right plot indicates the SM M W prediction for M H SM = M h . It is interestingto note that in the NMSSM it is possible to find both the predictions for M W and for the lightest CP -even Higgs mass in the preferred regions indicated by the blue and grey bands in Fig. 15. For the The difference in the predictions for the lightest CP -even Higgs masses in the MSSM and the NMSSM, which wesubtract this way, includes both the difference between the different mass relations in the MSSM and the NMSSM, aswell as the “technical” difference between the FeynHiggs and the
NMSSMTools evaluation. H = ± - - - - -
50 080100120140160 A Κ @ GeV D H i gg s bo s on m a ss e s @ G e V D - - - - -
50 00.00.20.40.60.81.0 A Κ @ GeV D S i ng l e t c o m ponen t U i M W exp = - - - - -
50 080.3680.3780.3880.3980.4080.41 A Κ @ GeV D M W @ G e V D Figure 16: The left plot shows the prediction for M h (solid curve) and M h (dashed curve) as afunction of A κ . The region 125 . ± .
04 GeV is indicated as a blue band. The middle plot shows thesinglet components of h and h , U (solid) and U (dashed), respectively. The right plot shows the M NMSSM W prediction. Here the grey band shows the experimental 1 σ band from the W boson massmeasurement. The parameters used for these plots are given in the text.SM, on the other hand, Fig. 15 shows the well-known result that setting the SM Higgs boson mass tothe measured experimental value one finds a predicted value for M W which is somewhat low comparedto the experimental value.Now we want to investigate whether singlet–doublet mixing (a genuine NMSSM feature) has asignificant impact on the M W prediction. Such a scenario is analysed in Fig. 16. Our parametersare m t = 173 .
34 GeV, tan β = 2, µ = 140 GeV, M ˜ L/ ˜ E = 130 GeV M ˜ Q/ ˜ U/ ˜ D , = 1200 GeV, M ˜ Q =800 GeV, M ˜ U = 600 GeV, M ˜ D = 1000 GeV, A t = 1300 GeV A τ = A b = 1000 GeV, M = 230 GeV, m ˜ g = 1500 GeV, A λ = 210 GeV, λ = 0 . κ = 0 .
31, and we vary A κ . These parameters are allowedby HiggsBounds everywhere apart from −
145 GeV . A κ . −
105 GeV, and the Higgs signal canbe interpreted as either h or h . The left plot shows the prediction for M h (solid curve) and M h (dashed). The corresponding singlet components U (solid) and U (dashed) are shown in the middleplot. The third CP -even Higgs is heavy and has a negligible singlet component. For A κ . −
120 GeV, h is doublet-like and has a mass in the region of the observed Higgs signal (indicated by the blueband). In the MSSM, scenarios which allow the interpretation of the Higgs signal as the heavy CP -even Higgs involve always a (relatively) light charged Higgs (see e.g. Ref. [128]). Due to changedmass relations between the Higgs bosons, it is possible in the NMSSM to have the second lightest CP -even Higgs at 125 .
09 GeV together with a heavy charged Higgs. Therefore in the NMSSM theinterpretation of the Higgs signal as the second lightest CP -even Higgs is much less constrained by theLHC results from charged Higgs searches [129, 130]. The interpretation of the Higgs signal as h inthis model is always accompanied by a lighter state with reduced couplings to vector bosons. In thisfigure the charged Higgs mass is ∼
280 GeV. For A κ & −
100 GeV, h is doublet-like and has a mass inthe region of the observed Higgs signal. In the “transition” region ( −
150 GeV . A κ . −
50 GeV) thetwo light CP -even Higgs bosons are close to each other in mass and “share” the singlet component.The right plot shows the NMSSM prediction for M W , which is approximately flat. Accordingly, theparameter regions of A κ corresponding to two different interpretations of the Higgs signal within theNMSSM lead to very similar predictions for the W boson mass, which are in both cases compatiblewith the experimental result. Even a sizeable doublet–singlet mixing has only a minor effect on the M W prediction in this case. 26 M H ± @ GeV D H i gg s bo s on m a ss e s @ G e V D
95 100 105 110 115 120 125 - - - - - - - M H ± @ GeV D ∆ M W @ M e V D M W exp =
95 100 105 110 115 120 12580.3680.3880.4080.4280.44 M H ± @ GeV D M W @ G e V D Figure 17: M W contribution from a light charged Higgs boson. The left plot shows the prediction forthe CP -even Higgs boson masses in the NMSSM and in the MSSM as a function of the charged Higgsmass. The solid curves correspond to the mass of the lightest CP -even Higgs in the NMSSM (blue)and the MSSM (red). The dashed curves correspond to the mass of the second lightest CP -even Higgsin the NMSSM (blue) and the MSSM (red). The middle plot shows the shift δM W (calculated as inEq. (41)) induced by the Higgs and gauge boson sector in the NMSSM (blue), in the MSSM (red)and in the SM (black) with M H SM = M h . The right plot shows the W boson mass prediction in theNMSSM (blue) and the MSSM (red). The parameters used for these plots are given in the text.We have demonstrated so far that, taking Higgs search constraints and the information on thediscovered Higgs signal into account, the genuine NMSSM effects from the extended Higgs sectorare quite small, and the Higgs sector contributions that we analysed so far were dominated by SM-type contributions. This is true in the absence of a light charged Higgs boson, as we will discuss now.Light charged Higgs bosons (together with a light CP -even Higgs with small but non-zero couplingsto vector bosons) can lead to sizeable (non SM-like) Higgs contributions to M W . This effect can alsobe observed in the MSSM. Although it is not a genuine NMSSM effect, we want to demonstrate theimpact of such a contribution here. For Fig. 17 we choose the following parameters m t = 173 .
34 GeV,tan β = 9 . µ = 200 GeV, M ˜ L/ ˜ E = 300 GeV M ˜ Q/ ˜ U/ ˜ D , = 1500 GeV, M ˜ Q = M ˜ U = M ˜ D =1100 GeV, A t = − A τ = A b = − M = 500 GeV, m ˜ g = 1500 GeV, λ = 0 . κ = 0 . A κ = − m A . The left plot in Fig. 17 shows the predictions for themasses of the lightest two CP -even Higgs bosons in the NMSSM (blue) and in the MSSM (red) asa function of the charged Higgs mass. In both models the second lightest Higgs falls in the massrange 125 . ± .
04 GeV for the chosen parameters. This scenario is essentially excluded by the latestcharged Higgs searches [129, 130]. Nevertheless, we include these plots to illustrate the possible size ofthe contributions from a light charged Higgs.The middle plot shows the shift δM W calculated as in Eq. (41) with x =gauge-boson/Higgs in theNMSSM (blue) and in the MSSM (red) while the right plot shows the full M W prediction in theNMSSM (blue) and in the MSSM (red). As one can see the MSSM and NMSSM contributions to M W are very similar. Since the masses of charginos, neutralinos and sfermions stay constant when varyingˆ m A (or M H ± ), the change in M W with M H ± stems purely from the Higgs sector. The Higgs sectorcontribution to M W comes dominantly from the light charged Higgs, while the lightest CP -even Higgs Neglecting those experimental bounds one could have very light CP -Higgs bosons with only a small singlet component,which would give large contributions to M W . However this possibility will not be discussed here. M W due to its reduced vector boson couplings. In the middleplot the SM result for δM W with M H SM = M h is shown in black. A significant difference betweenthe SM Higgs contribution and the MSSM/NMSSM Higgs contributions can be observed. As one cansee in the right plot, the displayed variation with the charged Higgs boson mass corresponds to abouta 1 σ shift in M W . We start the discussion of the contributions from the NMSSM neutralino sector, which differs from therespective MSSM sector, with Fig. 18. We choose the parameters m t = 173 .
34 GeV, tan β = 3, µ =200 GeV, M ˜ L/ ˜ E = 1000 GeV, M ˜ Q/ ˜ U/ ˜ D , = 1500 GeV, M ˜ Q = M ˜ U = 650 GeV, M ˜ D = 1000 GeV, A t = A τ = A b = 1000 GeV, m ˜ g = 1500 GeV, A λ = 580 GeV, λ = 0 . κ = 0 . A κ = −
10 GeV, andwe vary M . In the upper left plot, the blue curve shows the M NMSSM W prediction and the red curve the M MSSM W prediction. The difference between the NMSSM prediction and the MSSM prediction is smallfor M .
200 GeV and increases for larger M values. The origin of this difference is investigatedin the other three plots of Fig. 18. As before our procedure to identify an MSSM point which canbe compared to the NMSSM point implies different predictions for the lightest CP -even Higgs mass.Here we subtract again the difference in the SM contributions, arising from the different Higgs masspredictions. The additional blue dashed curve (with open dots) in the upper right plot of Fig. 18corresponds to M NMSSM, sub W = M NMSSM W − M SM W ( M h ) + M SM W ( M h ). For large M the differencebetween the NMSSM and the MSSM prediction for M W can be fully explained by the differencein the (SM-type) Higgs mass contributions, which arise from inserting different predictions for M h and M h . However after subtracting the difference from the Higgs mass contributions we observe asizeable difference between M NMSSM, sub W and M MSSM W for small M . This difference stems from differentsizes of the chargino/neutralino sector contributions between the two SUSY models, which tend tocompensate the difference between M NMSSM W and M MSSM W arising from the Higgs sector. This can beseen in the lower left plot, where we display the shift δM W (calculated as in Eq. (41)) induced by thechargino/neutralino contributions in the MSSM (red) and in the NMSSM (blue). At M = 160 GeVthe chargino mass is 108 GeV and thus just above the LEP limit. The δM W contribution from thechargino/neutralino sector in the MSSM reaches 8.5 MeV in this case. In the NMSSM the maximal δM W contribution from the chargino/neutralino sector is 16.5 MeV — significantly larger than in theMSSM. Both in the MSSM and the NMSSM, the chargino/neutralino contributions decrease whenincreasing M and therewith the chargino and neutralino masses, showing the expected behaviourwhen decoupling the gaugino sector. The largest difference between the NMSSM and the MSSMchargino/neutralino contributions is ∼ M = 160 GeV). The difference arises from theneutralino sector, since the chargino sector is unchanged in the NMSSM with respect to the MSSM.We will discuss in more detail below why the contributions from the neutralino sector are larger inthe NMSSM than in the MSSM. The lower right plot of Fig. 18 is similar to the upper right plot, butit contains a fourth curve (blue dotted with open diamonds) which was obtained by subtracting thedifferent chargino/neutralino contributions, thus it corresponds to M NMSSM,sub W − δM NMSSM W + δM MSSM W .This curve lies very close to the MSSM prediction. We have therefore identified the contributionscausing the difference between the M NMSSM W and the M MSSM W predictions.We continue with the discussion of the neutralino contributions to M W in the NMSSM in Fig. 19. This is not the maximal effect possible for the chargino/neutralino contributions in the MSSM. Thechargino/neutralino contributions depend on the slepton masses (see diagrams in Figs. 5-7). For lighter slepton massesthe chargino/neutralino contributions in the MSSM can reach up to 20 MeV, as analysed in Ref. [61]. W exp =
200 300 400 500 600 700 800 90080.3680.3780.3880.3980.40 M @ GeV D M W @ G e V D
200 300 400 500 600 700 800 90080.3680.3780.3880.3980.40 M @ GeV D M W @ G e V D
200 300 400 500 600 700 800 9000510152025 M @ GeV D ∆ M W @ M e V D
200 300 400 500 600 700 800 90080.3680.3780.3880.3980.40 M @ GeV D M W @ G e V D Figure 18: The upper left plot shows the M NMSSM W prediction (blue) and the M MSSM W prediction(red) as a function of M . The experimental M W measurement is indicated as a grey band. Theupper right plot shows additionally a dashed blue curve (open dots) corresponding to M NMSSM, sub W = M NMSSM W − M SM W ( M h ) + M SM W ( M h ). The lower left plot shows the shift in the W boson mass δM W (calculated as in Eq. (41) with x =chargino/neutralino) in the MSSM (red) and in the NMSSM (blue).The lower right plot is similar to the upper right plot but it additionally contains the dotted blue curve(open diamonds) which corresponds to M NMSSM, sub W − δM NMSSM W + δM MSSM W where δM W is the shiftin M W induced by the chargino/neutralino contributions. The NMSSM parameter points are allowedby HiggsBounds , and M h falls in the range 125 . ± .
04 GeV for M .
725 GeV. The parametersused for these plots are given in the text.The chosen parameters are m t = 173 .
34 GeV, tan β = 5 . µ = 200 GeV, M ˜ L/ ˜ E = 245 GeV,29 H = ±
200 300 400 500 600 700110115120125130 M @ GeV D M h H L @ G e V D M W exp =
200 300 400 500 600 70080.3680.3780.3880.3980.40 M @ GeV D M W @ G e V D Χ Χ Χ Χ Χ
200 300 400 500 600 700050100150200250 M @ GeV D N eu t r a li no m a ss e s @ G e V D Χ Χ Χ Χ Χ
200 300 400 500 600 7000.00.20.40.60.81.0 M @ GeV D N eu t r a li no s i ng li no c o m ponen t Figure 19: The upper left plot shows the masses of the lightest CP -even Higgs boson in the NMSSM(blue) and the MSSM (red) as a function of M . The upper right plot shows the prediction for M NMSSM W (blue) and for M MSSM W (red). The lightest three neutralino masses and the neutralino singletcomponents are displayed in the lower row. The parameters (given in the text) are chosen such thatthe Higgs sectors of the MSSM and the NMSSM are very similar to each other. The parameter regionin both models is allowed by HiggsBounds and predicts the lightest CP -even Higgs (which is SM-like)close to 125 .
09 GeV. M ˜ Q/ ˜ U/ ˜ D , = 1500 GeV, M ˜ Q = M ˜ U = M ˜ D = 1000 GeV, A t = A τ = A b ≃ m ˜ g =1500 GeV, ˆ m A = 1200 GeV, λ = 0 . κ = 0 . A κ = −
10 GeV, and M is varied. All parameterpoints are HiggsBounds allowed. Again we get the MSSM prediction by setting the
FeynHiggs M H ± input to the value of the charged Higgs mass calculated by NMSSMTools . For this set of parameters30
00 300 400 500 600 7000510152025 M @ GeV D ∆ M W @ M e V D Figure 20: The shifts δM W in the NMSSM (blue) and in the MSSM (red), calculated taking thefull chargino/neutralino contribution to ∆ r into account (solid) and using only the ∆ ρ approximation(dashed). The parameters are chosen as in Fig. 19.this procedure leads to a scenario where the MSSM and the NMSSM Higgs boson sectors are verysimilar to each other. Both models predict the lightest CP -even Higgs close to the experimental value125 .
09 GeV, as one can see in the upper left plot of Fig. 19 showing the masses of the two states M h (MSSM, red) and M h (NMSSM, blue). The difference between M h and M h is . O (1 MeV)) difference in M W from the Higgs sector contributions. The upper right plotof Fig. 19 displays the W boson mass prediction in the NMSSM (blue) and in the MSSM (red). Thedifference between these two predictions is largest (7 MeV) for M = 150 GeV and (almost) vanishesfor large M . Since differences in the Higgs sector contributions are quite small, the difference between M NMSSM W and M MSSM W arises predominately from the differences in the neutralino sector. We note thatin this scenario both M h and M W lie within the preferred regions indicated by the blue and greybands for the whole parameter range displayed in the figure.In order to investigate the reasons for the different predictions for the chargino/neutralino con-tributions we plot the masses of the three lightest neutralino states in the NMSSM (blue) and theMSSM (red) in the lower left plot. The other MSSM/NMSSM neutralinos are heavier than 250 GeVand hardly affect the M W prediction. We set here the (unphysical) soft masses M and M equal inthe MSSM and the NMSSM and identify the MSSM µ parameter with the effective µ of the NMSSM.The resulting predictions for the masses of ˜ χ and ˜ χ are a few GeV lower in the NMSSM than in theMSSM. The singlino components of the NMSSM neutralinos, | N i | , where N was defined in Eq. (15),are shown in the lower right plot, and we can observe a strong mixing between the five states. Thesinglino components of ˜ χ and ˜ χ are below 10% for M = 150 GeV and increase up to 40%(20%) for˜ χ ( ˜ χ ) for higher M values. The lighter neutralino states (with relatively small singlino component)lead to larger contributions from the neutralino sector to M W in the NMSSM compared to the MSSM.In the next step we analyse how well the full ∆ r contribution of the chargino/neutralino sector canbe approximated by taking into account only the leading term − c W /s W ∆ ρ (defined in Eq. (26)). The31 ρ term contains only the W and Z boson self-energies at zero momentum transfer, thus this approx-imation neglects in particular the contributions from box, vertex and fermion self-energy diagramscontaining charginos and neutralinos. The ∆ ρ term corresponds to the T parameter of the S, T, U parameters [131, 132], often used to parametrize new physics contribution to electroweak precision ob-servables. For the plot in Fig. 20 we use the same parameters as in Fig. 19. Again the blue(red) solidcurve shows the δM W shift as a function of M , calculated as in Eq. (41) with x =chargino/neutralinoin the NMSSM(MSSM) (the two solid curves are identical to the ones in the upper right plot ofFig. 19). The two dashed curves show the M W contributions in the NMSSM (blue) and in the MSSM(red) obtained when the full ∆ r chargino/neutralino( α ) is approximated by the chargino and neutralinocontributions to the ∆ ρ parameter: δM W = − M ref W s W c W − s W (cid:18) − c W s W (cid:19) ∆ ρ chargino/neutralino . (42)In the MSSM the ∆ ρ term containing charginos and neutralinos provides a very good approximationof the full ∆ r term in the intermediate range 200 GeV . M .
500 GeV. In the range of small andlarge M values, ∆ ρ slightly underestimates the full ∆ r contribution, the difference here is ∼ . M = 150 GeV and ∼ . M = 750 GeV. In the NMSSM the ∆ ρ term gives a δM W contribution which is larger ( & r result for the full M range plotted here. Itshould be noted that the chargino/neutralino sector does not completely decouple for large M in thiscase, which is a consequence of the presence of a light Higgsino, µ = 200 GeV. For M = 750 GeV thelightest neutralino has a mass of M = 140 GeV, with a singlino component of ∼
40% and a Higgsinocomponent of ∼ ρ , but to a negative contribution to the ∆ r terms beyond ∆ ρ (we checked that the contributionfrom the box diagrams is negligible for large M values). We also checked that going to large µ values, the chargino/neutralino sector decouples and all terms vanish. In this scenario the two effectslargely cancel each other and for large M one finds a small positive value for the full ∆ r result.This however depends on the chosen parameters and the admixture of the light neutralino, e.g. in thescenario discussed in Fig. 15 the negative contributions exceed the positive ones so that the full ∆ r result is negative for large M . Thus, we have shown that the ∆ ρ approximation for the chargino andneutralino contributions works quite well in the MSSM, whereas sizeable corrections to M W beyondthe ∆ ρ approximation can occur in the NMSSM.As a final step we want to discuss the dependence of the M W prediction in the NMSSM on the µ parameter, which enters both in the sfermion and in the chargino/neutralino sectors. The leftplot of Fig. 21 shows the W boson mass prediction in the NMSSM as a function of µ , with theparameters chosen as m t = 173 .
34 GeV, tan β = 20, M ˜ L/ ˜ E = 250 GeV, M ˜ Q/ ˜ U/ ˜ D , = 1500 GeV, M ˜ Q = 500 GeV M ˜ U = 1500 GeV, M ˜ D = 300 GeV, A τ = 0 GeV, A t = A b = − M = 150 GeV, m ˜ g = 1500 GeV, ˆ m A = 1500 GeV, λ = 0 . κ = 0 . A κ = − HiggsBounds allowed, and h falls in the mass range 125 . ± .
04 GeV. Whenincreasing µ , the M NMSSM W prediction decreases first, reaches its minimum for µ ∼ M W from thechargino/neutralino sector (here we take again the full ∆ r contributions into account) and from thestop/sbottom sector. The shift δM W arising from charginos and neutralinos is shown in the middleplot of Fig. 21. The chargino/neutralino contribution is largest for small µ and decreases with increas-ing µ . Going to larger µ the masses of the (higgsino-like) chargino and neutralino states increase andthe M W contribution decreases. The shift δM W arising from the stop/sbottom sector is shown in theright plot of Fig. 21. The contributions from the stop/sbottom sector (dominated by the ∆ ρ contri-32 W exp =
200 400 600 800 1000 1200 140080.3680.3780.3880.3980.4080.4180.42 Μ @ GeV D M W @ G e V D
200 400 600 800 1000 1200 140005101520 Μ @ GeV D ∆ M W @ M e V D H c ha r g i no (cid:144) neu t r a li no L
200 400 600 800 1000 1200 1400051015202530 Μ @ GeV D ∆ M W @ M e V D H s t op (cid:144) s bo tt o m L Figure 21: Dependence of the W boson mass prediction in the NMSSM on the µ parame-ter. The left plot shows the M NMSSM W prediction, the middle one the δM W contribution from thechargino/neutralino sector and the right one shows the δM W contribution from the stop/sbottomsector. The parameters are given in the text.butions) get smaller when µ is increased up to µ ∼ µ is increasedfurther. Increasing µ , the splitting between the two sbottoms gets larger (while the stop masses staynearly constant), which implies also an increase of the splitting between stops and sbottoms. Thecounteracting terms in ∆ ρ (see the discussion in Sect. 4.4.1) lead to the observed behaviour. We have presented the currently most accurate prediction for the W boson mass in the NMSSM, interms of the Z boson mass, the fine-structure constant, the Fermi constant, and model-parametersentering via higher-order contributions. This result includes the full one-loop determination and allavailable higher-order corrections of SM and SUSY type. These improved predictions have beencompared to the state–of–the–art predictions in the SM and the MSSM within a coherent framework,and we have presented numerical results illustrating the similarities and the main differences betweenthe predictions of these models.Within the SM, interpreting the signal discovered at the LHC as the SM Higgs boson with M H SM =125 .
09 GeV, there is no unknown parameter in the M W prediction anymore. We have updated theSM prediction for M W making use of the most up to date higher-order contributions. For M H SM =125 .
09 GeV this yields M SM W = 80 .
358 GeV (with a theory uncertainty from unknown higher-ordercorrections of about 4 MeV). The comparison with the current experimental value of M exp W = 80 . ± .
015 GeV shows the well-known feature that the SM prediction lies somewhat below the value thatis preferred by the measurements from LEP and the Tevatron (at the level of about 1 . σ ). The loopcontributions from supersymmetric particles in general give rise to an upward shift in the predictionfor M W as compared to the SM case, which tend to bring the prediction into better agreement withthe experimental result.For the calculation of the M W prediction, we made use of the highly automated programs FeynArts and
FormCalc . Our evaluation is based on a framework which was developed in Ref. [6], consisting inparticular of a NMSSM model file for the program
FeynArts and a
Fortran driver for the evaluationof the masses, mixing angles, etc. needed for the numerical evaluation. The code
NMSSMTools is used33or the evaluation of the loop-corrected Higgs boson masses. The implementation of another result forthe NMSSM Higgs masses, obtained in Ref. [117], is in progress.Our improved prediction for the W boson mass in the NMSSM consists of the full one-loop re-sult, all available higher-order corrections of SM-type, stop-loop and sbottom-loop contributions withgluon and gluino exchange of O ( αα s ), relevant reducible higher-order contributions, as well as an ap-proximate treatment of the MSSM-type Yukawa-enhanced electroweak two-loop corrections of O ( α t ), O ( α t α b ), O ( α b ). Analytic expressions for all those contributions are implemented, except for the elec-troweak two-loop contributions of SM-type, for which we make use of the fit formula given in Ref. [59].The latter allows us to properly evaluate ∆ r SM at an NMSSM value for the W boson mass.We presented a detailed investigation of the prediction for M NMSSM W , focussing on the parameterregions which are allowed by Higgs searches (tested by HiggsBounds ), SUSY searches and furthertheoretical constraints. As a first step we analysed the size of the contributions from stops/sbottoms.Since the sfermion sector is unchanged in the NMSSM with respect to the MSSM, we have done thisstudy in the MSSM limit, yielding an important check of our NMSSM implementation. We haveinvestigated the size of the SUSY two-loop corrections to M W and found that the O ( αα s ) correctionsbeyond the pure gluon exchange contributions, which were incorporated in the previous result ofRef. [78], can give sizeable contributions. On the other hand, the effect of the Yukawa-enhancedelectroweak two-loop corrections of O ( α t ), O ( α t α b ), O ( α b ) stays numerically relatively small in theallowed parameter region.Concerning the investigation of genuine NMSSM effects, we started our discussion with the Higgssector contributions to M W . The tree-level prediction for the lightest CP -even Higgs mass is modifiedby an additional term in the NMSSM as compared to the MSSM, which (for small tan β ) leads to anupward shift of the tree-level Higgs mass. Therefore, in that region, the radiative corrections neededto push the Higgs mass to about 125 GeV can be smaller than in the MSSM, which implies thatlighter stop masses and a smaller stop mixing are possible. We investigated a scenario where thisadditional tree-level term gives rise to a higher M h prediction than in the MSSM limit. The impacton the M W prediction is a downward shift (of ∼
25 MeV in the considered example) as comparedto the corresponding prediction in the MSSM. In the prediction for M W this contribution from theHiggs sector enters together with other SUSY loop contribution to ∆ r yielding an upward shift in M W compared to the SM. The overall effect is such that also in a scenario of this kind a very goodagreement between the theoretical prediction and the experimental result can be reached. We havefurthermore investigated the effect of doublet–singlet mixing. While a sizeable doublet–singlet mixingcan occur in the region where the two NMSSM Higgs states h and h are close to each other in mass,we find that it has only a minor effect on the M W prediction.In the NMSSM the Higgs signal seen at the LHC can be interpreted both as the lightest and thesecond-lightest CP -even Higgs boson of the spectrum. Both interpretations give predictions for the W boson mass in good agreement with the M W measurement. In the NMSSM the interpretation ofthe LHC signal as the second-lightest CP -even Higgs h is possible together with a relatively heavycharged Higgs. This is different from the situation in the MSSM, where all Higgs states have to belight in this case, so that such a scenario can be probed by searches for charged Higgs bosons in top-quark decays. As a consequence, the interpretation of the observed Higgs signal as the second-lightest CP -even Higgs boson is much less constrained in the NMSSM compared to the MSSM.For completeness, we have nevertheless briefly investigated also the case of a light charged Higgsboson. We have found that a light charged Higgs boson (together with a light CP -even Higgs withreduced but non-zero couplings to gauge bosons) can in principle give very significant contributionsto M W (as in the MSSM). In that case large deviations from the SM Higgs sector contributions occur,34ut as discussed above scenarios of this kind are severely constrained by limits from charged Higgssearches at the LHC. Generally we find that taking all available constrains on the Higgs sector intoaccount, the specific NMSSM effects of the Higgs sector to M W are relatively small.On the other hand, the extended neutralino sector of the NMSSM can lead to a sizeable differencebetween the W boson mass predictions in the NMSSM and the MSSM. The chargino/neutralinocontributions to M W can be larger in the NMSSM compared to the MSSM, where in the scenariowhich we studied the difference reaches ∼ µ = µ eff , the mixing with the singlino leadsto shifts in the neutralino masses as compared to the MSSM case. In the considered scenarios thelightest NMSSM states turned out to be lighter than the corresponding MSSM states. They also havea relatively small singlino component, which causes the resulting contributions to the prediction for M W to be larger than in the MSSM. While light wino/bino states typically give positive contributions,light higgsinos can give contributions entering with both signs.As a final step of our analysis, we compared the M W prediction calculated with the full ∆ r to the one where the full result is approximated by the contribution to ∆ ρ . We found that thedifference between the full result and the ∆ ρ approximation can be sizeable in the NMSSM, wherethe approximation can lead both to an over- or an underestimate of the full result. Light neutralinoswith a significant higgsino–singlino mixing tend to give a positive contribution to ∆ ρ , but a negativecontribution to the ∆ r terms beyond ∆ ρ . It therefore depends on the exact patterns of the admixturewith the singlino whether the neutralino sector of the NMSSM leads to an upward or downward shiftin the prediction for M W in comparison with the MSSM.We have demonstrated that the prediction for the W boson mass arising from the relation withthe Z boson mass, the Fermi constant and the fine structure constant in comparison with high-precision measurements of those quantities provides a high sensitivity for discriminating between theSM and possible extensions of it. With further improvements of the experimental accuracy of M W ,possible improvements in the determination of m t and further information on possible mass spectraof supersymmetric particles – either via improved limits or the discovery of new states – the impactof this important tool can be expected to be even more pronounced in the future. Acknowledgements
We are grateful to Florian Domingo for helpful discussions and for providing numerical values tocross-check our results. We also thank Ayres Freitas, Thomas Hahn, Sven Heinemeyer, WolfgangHollik and Pietro Slavich for helpful discussions. This work has been supported by the CollaborativeResearch Center SFB676 of the DFG, “Particles, Strings and the early Universe”. O.S. is supportedby the Swedish Research Council (VR) through the Oskar Klein Centre. This work is part of theD-ITP consortium, a program of the Netherlands Organization for Scientific Research (NWO) that isfunded by the Dutch Ministry of Education, Culture and Science (OCW). This research was supportedin part by the European Commission through the “HiggsTools” Initial Training Network PITN-GA-2012-316704.
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