Improved prediction for the mass of the W boson in the NMSSM

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 the underlying physics. In that context, the $W$ boson mass, $M_W$, plays a crucial role. The accuracy of the $M_W$ measurement has been significantly improved over the last years, and further improvement of the experimental accuracy is expected from future LHC measurements. In order to fully exploit the precise experimental determination, an accurate theoretical prediction for $M_W$ in the Standard Model (SM) and extensions of it is of central importance. We present the currently most accurate prediction for the $W$ boson mass in the Next-to-Minimal Supersymmetric extension of the Standard Model (NMSSM), including the full one-loop result and all available higher-order corrections of SM and SUSY type. The evaluation of $M_W$ is performed in a flexible framework, which facilitates the extension to other models beyond the SM. We show numerical results for the $W$ boson mass in the NMSSM, focussing on phenomenologically interesting scenarios, in which the Higgs signal can be interpreted as the lightest or second lightest CP-even Higgs boson of the NMSSM. We find that, for both Higgs signal interpretations, the NMSSM $M_W$ prediction is well compatible with the measurement. We study the SUSY contributions to $M_W$ in detail and investigate in particular the genuine NMSSM effects from the Higgs and neutralino sectors.

since the precision is expected to be improved further when including the full dataset from the Tevatron and upcoming results from the LHC. Of central importance for the theoretical precision that can be achieved on M W is the top quark mass measurement, since the experimental error on the input parameter m t constitutes a dominant source of (parametric) uncertainty, see e.g. Ref. [18]. The Tevatron [19] and LHC [20][21][22][23][24][25][26] measurements of m t have been combined [27] to yield m exp t = 173.34 ± 0.27 ± 0.71 GeV. ( 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 a hadron collider (using kinematic information about the top decay products) to a theoretically welldefined mass parameter. The quantity measured with high precision at the Tevatron and the LHC is expected to be close to the top pole mass with an uncertainty at the level of about 1 GeV [28][29][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 defined short distance mass (such as the MS or DR mass). At an e + e − linear collider, the situation would improve significantly. Estimates for the ILC show an expected precision ∆M ILC W ∼ 2.5 − 5 MeV and ∆m ILC t = 0.1 GeV [31,32], where the stated precision for m t accounts both for the uncertainty in the determination of the actually measured mass parameter and the uncertainty related to the conversion into a suitable and theoretically well-defined parameter such as the MS mass.
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]. We combine the complete NMSSM one-loop result with the state-of-the-art SM result and leading SUSY higher-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 is parametrized in terms of the Fermi constant G µ (instead of M W or α(M Z )). Our NMSSM prediction for M W provides the flexibility to analyse SUSY loop contributions analytically and to treat possible threshold effects or numerical instabilities. We perform a detailed numerical analysis of M W in the NMSSM with the latest experimental results taken into account. We focus on the effects induced by the extended Higgs and neutralino sectors, and in particular on benchmark scenarios where the LHC Higgs boson is interpreted as either the lightest or the second lightest CP-even Higgs boson of the NMSSM.
This paper is organised as follows: in Sect. 2 we give a short introduction to the NMSSM, focussing on the Higgs and neutralino sectors. In Sect. 3 we describe the determination of the W boson mass in the NMSSM. We outline the calculation of the one-loop contributions and the incorporation of higherorder contributions. In Sect. 4 we give the numerical results, analysing the NMSSM contributions to the W boson mass, before we conclude in Sect. 5.

The Next-to-Minimal Supersymmetric Standard Model
In this section we introduce the NMSSM and specify our notation. We focus on the particle sectors which differ from the MSSM. Since the SM fermions and their superpartners appear in the same way in both models, the sfermion sector of the NMSSM is unchanged with respect to the MSSM. Also the chargino sector is identical to that in the MSSM since no new charged degrees of freedom are introduced. 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. 2 Considering the Z 3 -symmetric version of the NMSSM, the superpotential takes the form W NMSSM =ūy u QH 2 −dy d QH 1 −ēy l LH 1 + λSH 2 H 1 + 1 3 The new contributions of the Higgs singlet to the soft breaking terms are where L MSSM,mod soft is the soft-breaking Lagrangian L MSSM soft of the MSSM (see e.g. Eq. (6.3.1) of Ref. [81]), but without the term bH 2 H 1 . The singlet couplings make it possible to dynamically generate an effective µ parameter as µ eff = λ S .
The additional contributions (and the modified effective µ term) in the superpotential and in the soft breaking terms lead to a Higgs potential which contains the additional soft breaking parameters m 2 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 couplings allow in principle for CP-violation at tree level, but we will not consider this possibility here. We choose all parameters to be real.
The minimum of the NMSSM Higgs potential triggers electroweak symmetry breaking, after which the Higgs doublets can be expanded around their minima according to Similarly, the singlet scalar component can be expanded as where v s is the (non-zero) vacuum expectation value of the singlet. The bilinear part of the Higgs potential can be written as 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 is described by three unitary matrices U H , U A , and U C , where (10) These transform the Higgs fields such that the resulting (diagonal) mass matrices become The CP-even mass eigenstates, h 1 , h 2 and h 3 , are ordered such that m h 1 ≤ m h 2 ≤ m h 3 , and similarly for the two CP-odd Higgs bosons, a 1 and a 2 . Unchanged from the SM there are also the Goldstone bosons, G and G ± . Finally, there is the charged Higgs pair, H ± with mass given by Herem A is the effective CP-odd doublet mass given bŷ 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 0 ,H 0 1 ,H 0 2 ,S) the neutralino mass matrix at tree level is given by where K ≡ κ/λ. This mass matrix can be diagonalised by a single unitary matrix N such that which gives the mass eigenvalues ordered as mχ0 3 Predicting the W boson mass The W boson mass can theoretically be predicted from the muon decay rate. Muons decay to almost 100% via µ → eν e ν µ [82]. This decay was historically described first within the Fermi model (left diagram in Fig. 1). Comparing the muon decay amplitude calculated in the Fermi model to the same quantity calculated in the full SM or extensions thereof (the leading-order contribution in unitary gauge is depicted in Fig. 1) yields the relation This relates the W boson mass to the Fermi constant, G µ , which by definition contains the QED corrections to the four-fermion contact vertex up to O(α 2 ) [83][84][85][86][87], and to the other parameters M Z and e, which are known experimentally with very high precision. The Fermi constant itself is determined 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 amplitude after subtracting the Fermi-model type virtual QED corrections, which are already included in the definition of G µ . Working in the on-shell renormalization scheme, Eq. (16) corresponds to a relation between the physical masses of the W and Z bosons.
Neglecting the masses and momenta of the external fermions all loop diagrams can be expressed as a term proportional to the Born matrix element [33,43] In different models, different particles can contribute as virtual particles in the loop diagrams to the 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 boson mass. The quantity ∆r itself does depend on M W as well; hence, the value of M W as the solution of Eq. (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 prediction for ∆r within and beyond the SM is needed. In the next two subsections we describe our calculation of ∆r in the NMSSM. A new one-loop calculation has been performed which is combined with all available higher order corrections of SM-and SUSY-type.

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, and the corresponding counter terms (CT). The box diagrams are themselves UV-finite in a renormalizable gauge and require no counter terms. Schematically, this can be expressed as ∆r (α) = W Self-energy + W Self-energy CT + Vertex + Vertex CT + Box Here Σ W W T (0) denotes the transverse part the W boson self-energy (evaluated at vanishing momentum transfer), δM W is the renormalization constant for the W boson mass, δe and δs w are the renormalization constants for the electric charge and s w ≡ sin θ W , respectively. The δZ denote different field renormalization constants. Since the W boson occurs in the muon decay amplitude only as a virtual particle, 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 where bare quantities are denoted with a zero subscript. The renormalization constant of the electric charge is with Note that the sign appearing in front of s w in Eq. (20) depends on convention chosen for the SU (2) covariant derivative. 3 The sine of the weak mixing angle is not an independent parameter in the on-shell renormalization scheme. Its renormalization constant 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 where Σ l L denotes the left-handed part of the lepton self energy. 3 We adopt the sign conventions for the SU (2)L covariant derivative used in the code FeynArts [89][90][91][92][93][94], where (for historical reasons) the SU (2)L covariant derivative is defined by ∂µ − ig2I a W a µ for the SM and ∂µ + ig2I a W a µ for the (N)MSSM. The expressions given here correspond to the (N)MSSM convention.
Inserting these expressions for the renormalisation constants into Eq. (18) yields The quantity ∆r is at one loop level conventionally split into three parts, The shift of the fine structure constant ∆α arises from the charge renormalization which contains the contributions 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 This quantity is sensitive to the mass splitting between the isospin partners in a doublet [95], which leads to a sizeable effect in the SM in particular from the heavy quark doublet. While ∆α is a pure SM contribution, ∆ρ can get large contributions also from SUSY particles, in particular the superpartners of the heavy quarks. All other terms, both of SM and SUSY type, are contained in the remainder term ∆r rem . We have performed a diagrammatic one-loop calculation of ∆r in the NMSSM according to Eq. (24), using the Mathematica-based programs FeynArts [89][90][91][92][93][94] and FormCalc [96]. The NMSSM model 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 discussed here. This calculation has been discussed in the literature already many years ago [33,34], and we refer to Refs. [43,99] for details. The one-loop result for ∆r is also known for the MSSM (with complex parameters), see Refs. [61,73]. The calculation in the NMSSM follows along the same lines as for the MSSM. However, the result gets modified from differences in the Higgs and the neutralino sectors. Below we outline the NMSSM one-loop contributions to ∆r, for completeness also including the MSSM-type contributions.
Besides the SM-type contributions with fermions and gauge bosons in the loops (not discussed further here), many additional self-energy, vertex and box diagrams appear in the NMSSM with sfermions, (SUSY) Higgs bosons, charginos and neutralinos in the loop. Generic examples of gaugeboson self-energy diagrams with sfermions are depicted in Fig. 2. Their contribution to ∆r is finite by itself. The NMSSM Higgs bosons enter only in gauge boson self-energy diagrams, since we have neglected the masses of the external fermions. The contributing diagrams are sketched in Fig. 3. These contributions are not finite by themselves. Only if one considers all (including SM-type) gauge boson and Higgs contributions to the gauge boson self-energy diagrams, the vertex diagrams and vertex counterterm 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), fermion self-energy diagrams (depicted in Fig. 5), vertex diagrams (depicted in Fig. 6, the analogous vertex corrections exist also for the other vertex) and box diagrams (depicted in Fig. 7). The vertex contributions from the chargino/neutralino sector, together with the chargino/neutralino contributions to the vertex CT and the gauge boson self-energies are finite. Each box diagram is UV-finite by itself.
In order to determine the contribution to ∆r from a particular loop diagram, the Born amplitude has to be factored out of the one-loop muon decay amplitude, as shown in Eq. (17). While most loop diagrams directly give a result proportional to the Born amplitude, more complicated spinor structures that do not occur in the SM case arise from box diagrams containing neutralinos and charginos 4 . Performing the calculation of the box diagrams in Fig. 7 in FormCalc, the spinor chains are returned in the form The expressions for the coefficients b (a) and b (b) are lengthy and not given here explicitly. In order to factor out the Born amplitude the spinor chains in Eq. (27) have to be transformed into the same structure. We modify the spinor chains following the procedure described in Ref. [73] and get The coefficients b (a) and b (b) contain ratios of mass-squared differences of the involved particles. These coefficients can give rise to numerical instabilities in cases of mass degeneracies. In the implementation of our results (which has been carried out in a Mathematica) special care has been taken of Figure 4: Generic NMSSM one-loop gauge boson self-energy diagram with charginos/neutralinos; Figure 5: Generic NMSSM one-loop fermion self-energy diagram with a chargino/neutralino/sfermion contribution;χ ± =χ ± 1,2 andχ 0 =χ 0 1,2,3,4,5 .
such parameter regions with mass degeneracies or possible threshold effects. By adding appropriate expansions a numerically stable evaluation is ensured.

Higher-order corrections
The on-shell renormalization conditions correspond to the definition of the W and Z boson masses according to the real part of the complex pole of the propagator, which from two-loop order on is the only gauge-invariant way to define the masses of unstable particles (see Ref. [43] and references therein). 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 experimentally measured 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 transform the W boson mass value to the running width definition, M rw W to facilitate a direct comparison to the experimental value of M W . The difference between these two definitions is where M fw W corresponds to the fixed width description, see Ref. [100]. For the prediction of the W decay width we use parametrized by G µ and including first order QCD corrections. The difference between the fixed-and running width definitions amounts to about 27 MeV, which is very relevant in view of the current theoretical and experimental precisions. For the Z boson mass, which is used as an input parameter in the prediction for M W , the conversion from the running-width to the fixed-width definition is carried Figure 6: Generic one-loop vertex correction diagrams in the NMSSM;χ ± =χ ± 1,2 andχ 0 =χ 0 1,2,3,4,5 . Analoguous diagrams exist for the other W vertex. Figure 7: Generic one-loop box diagrams contributing to the muon decay amplitude in the NMSSM; out in the first step of the calculation. Accordingly, keeping track of the proper definition of the gauge boson masses is obviously important in the context of electroweak precision physics. For the remainder of this paper we will not use the labels (rw, fw) explicitly; if we do not refer to an internal variable, M W and M Z will always refer to the mass definition according to a Breit-Wigner shape with a 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 relevant available higher-order corrections of the state-of-the-art prediction for M SM W . As we will describe below in more detail, the higher-order corrections of SM-type are also incorporated in the NMSSM calculation of ∆r in order to achieve an accurate prediction for M NMSSM W . For a discussion of the incorporation 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 higherorder corrections, 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) , contains the most up-to-date higher-order contributions of SUSY type (see below). The approach followed in Eq. (34) has several advantages. It ensures in particular that the best available SM prediction is recovered in the decoupling limit, where all superpartners are heavy, the singlet decouples, and the NMSSM Higgs sector becomes SM-like. Furthermore, the approach to combine the most up-to-date SM prediction with additional "new physics" contributions (here from supersymmetry) allows one to readily compare the MSSM and NMSSM predictions on an equal footing and it provides an appropriate framework also for an extension to other scenarios of physics beyond the SM. The expression for the higher-order contributions given in Eq. (34) formally introduces a dependence of the NMSSM result on the SM Higgs mass, which enters from the two-loop electroweak corrections onwards. In those contributions we identify the SM Higgs mass with the mass of the NMSSM Higgs boson with the largest coupling to gauge bosons.
For the higher-order corrections in the SM, ∆r SM(h.o.) , we incorporate the following contributions up to the four-loop order The contributions in Eq. (35) consist of the two-loop QCD corrections ∆r (ααs) [35][36][37][38][39][40], the three-loop QCD corrections ∆r (αα 2 s ) [48][49][50][51], the fermionic electroweak two-loop corrections ∆r ferm [42][43][44], the purely bosonic electroweak two-loop corrections ∆r bos [45][46][47], the mixed QCD and electroweak threeloop contributions ∆r (G 2 µ αsm 4 t ) [52,53], the purely electroweak three-loop contribution ∆r (G 3 µ m 6 t ) [52,53], and finally the four-loop QCD correction ∆r (Gµm 2 The radiative corrections in the SM beyond one-loop level are numerically significant and lead to a large downward shift in M W by more than 100 MeV. 5 The largest shift (beyond one-loop) is caused by the two-loop QCD corrections [35][36][37][38][39][40][41] followed by the three-loop QCD corrections ∆r (αα 2 s ) [48][49][50][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 integrations of the two-loop scalar integrals. These contributions are included in our calculation using a simple parametrization formula given in [59]. 6 This fit formula gives a good approximation to the full result for a light SM Higgs (the agreement is better than 0.4 MeV for M W ) [59]. Using this parametrization directly for the SM prediction of ∆r bos (rather than for the full SM prediction of M W -an approach followed for the MSSM case in Ref. [73] and for the NMSSM case in Ref. [78]) allows us to evaluate these contributions at the particular NMSSM value for M W in each iteration step. The output of this formula approximates the full result of ∆r bos using the fixed-width definition, such that it can directly be combined with other terms of our calculation. 7 In our expression for ∆r SM(h.o.) we use the result for ∆r (ααs) given in Ref. [37], which contains also contributions from quarks of the first two generations and is numerically very close to the result from Ref. [41], 8 and the result for 5 The corrections beyond one-loop order are in fact crucial for the important result that the MW prediction in the SM favours a light Higgs boson, whereas the one-loop result alone would favour a heavy SM Higgs. 6 In [59] the electroweak two-loop contributions are expressed via ∆r (α 2 ) ≡ ∆r rem , where a simple fit formula for the remainder term ∆r rem is given. The quantity ∆r (α) in the second term denotes the full one-loop result without the ∆α term. 7 It should be noted, however, that the gauge boson masses with running width definition are needed as input for the fit formula given in [59]. This is the only part of our calculation where the running width definition is used internally. ∆r (αα 2 s ) from Ref. [50]. Both contributions are parametrized in terms of G µ . A comparison between our 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 contributions into account, incorporating all SUSY corrections beyond one-loop order that are known to date. The first term in Eq. (36) denotes the leading reducible O(α 2 ) two-loop corrections. Those contributions are obtained by expanding the resummation formula [103] 1 + ∆r = 1 which correctly takes terms of the type (∆α) 2 , (∆ρ) 2 and ∆α∆ρ into account 9 if ∆ρ is parametrized by G µ . The pure SM terms are already included in ∆r SM(h.o.) , and because of numerical compensations those contributions are small beyond two-loop order [41]. Thus, we only need to consider the leading two-loop terms with SUSY contributions, 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,(ααs) , contain squark loops with gluon exchange and quark/squark loops with gluino exchange (both depicted in Fig. 8). While the formula for the gluino contributions is very lengthy, a compact result exists for the gluon contributions to ∆ρ [74,75]. Using these two-loop results for the SUSY contributions to M W requires the on-shell (physical) values for the squark masses as input. The SU (2) relation Mt L = Mb L implies that one of the stop/sbottom masses is not independent, but can be expressed in terms of the other parameters. Therefore, when including higher-order contributions, one cannot choose independent renormalization conditions for all four (stop and sbottom) masses. Loop corrections to the relation between the squark masses must be taken into account in order to be able to insert the proper on-shell values for the squark masses into our calculation. This one-loop correction to the relation between the squark masses is relevant when inserted into the one-loop SUSY contributions to M W , while it is of higher order for the two-loop SUSY contributions. In our evaluation of M W this is taken into account by a "mass-shift" correction term. For more details see Ref. [75]. The gluon, gluino and mass-shift corrections, which are identical in 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- [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 loops with Higgsino exchange, see Fig. 9. The corrections of this kind, which depend on the specific form of the Higgs sector, are only known for the MSSM so far [76,77]. It is nevertheless possible to take them into account also for the NMSSM in an approximate form. To this end, the considered NMSSM parameter point needs to be related to appropriate parameter values of the MSSM. Besides the values for tan β, the sfermion trilinear couplings A f , and all the soft mass parameters, which can be directly Here g denotes a gluon andg a gluino; whereH denotes a Higgsino and is s either an NMSSM Higgs or a Goldstone boson; taken over from the parameter point in the NMSSM, we determine the parameters in the following way: we set the MSSM µ parameter equal to µ eff and we use the physical value of the charged Higgs mass 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 Higgs boson, which is the only Higgs boson appearing without mixing in both models, is the same in the NMSSM and the MSSM. The MSSM Higgs masses are calculated with FeynHiggs [104-108], using the calculated physical mass value of M H ± as on-shell input parameter. The MSSM Higgs masses and the Higgsino parameter µ determined in this way are then used as input for the calculation of ∆ρ . In order to avoid double-counting the dominant Yukawa-enhanced electroweak two-loop corrections in the SM [109,110] have been subtracted according to Eq. (34). We find that the impact of the Yukawa-enhanced electroweak corrections of SUSY type on M W is relatively small (typically 1 MeV), and their numerical evaluation is rather time-consuming. In our numerical code for M W in the NMSSM, we therefore leave it as an option to choose whether these contributions should be included or not. For the results presented in Sect. 4.4 they have not been included, unless stated otherwise.

Framework for the numerical analysis
For the evaluation of the W boson mass prediction, the masses of the NMSSM particles are needed. We use 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][112][113][114]. 10 For other tools that are available to calculate the NMSSM Higgs masses including higher-order radiative corrections see Refs. [115][116][117]. The implementation of the Higgs mass results of Ref. [117] (using directly the on-shell parameters as input) is in progress.
In NMSSMTools the input parameters are assumed to be DR parameters at the SUSY breaking scale. In order to use the NMSSMTools Higgs masses in our result, a transformation from the onshell parameters, needed for our evaluation, to the DR parameters, needed as NMSSMTools input, is necessary. This effect is approximately taken into account by transforming the on-shell X t parameter into its DR value by the relation given in Ref. [118] (equations (60) ff in Ref. [118]). The shift in the other 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 experimental and theoretical constraints. Besides the constraints already implemented in NMSSMTools, 11 further direct constraints on the Higgs sectors are evaluated using the code HiggsBounds (version 4.2.0) [120][121][122][123]. All programs used for the numerical evaluation are linked through an interface to the NMSSM Mathematica code for the W boson mass prediction. 12

Theoretical uncertainties
Before moving on to our numerical results for the W boson mass prediction in the NMSSM, we discuss the remaining theoretical uncertainties in the M W calculation.
The dominant theoretical uncertainty of the prediction for M W arises from the parametric uncertainty induced by the experimental error of the top-quark mass. Here one needs to take into account both the experimental error of the actual measurement and the systematic uncertainty associated with relating the experimentally determined quantity to a theoretically well-defined mass parameter, see the 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 of the hadronic contribution to the shift in the fine-structure constant, ∆α had , and by the experimental error of M Z amount to about 2 MeV and 2.5 MeV, respectively. The uncertainty of the SM M W prediction caused by the experimental error of the Higgs boson mass, δM exp H = 0.24 GeV [60], is significantly smaller ( 0.2 MeV). In Ref. [32] the impact of improved accuracies of m t and ∆α had has been discussed. With a precise top mass measurement of ∆m t = 0.1 GeV (anticipated ILC precision) the associated parametric uncertainty in M W is about 0.6 MeV.
The uncertainties from unknown higher-order corrections have been estimated to be around 4 MeV in the SM for a light Higgs boson (M H SM < 300 GeV) [58]. The prediction for M W in the NMSSM is affected 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 some SUSY particles, in particular in the scalar top and bottom sectors, are relatively light. The combined theoretical uncertainty from unknown higher-order corrections of SM-and SUSY-type has been estimated (for the MSSM with real parameters) in Refs. [73,77] as δM W ∼ (4 − 9) MeV, depending on the SUSY mass scale. 13 Since we include the same SUSY higher-order corrections in our NMSSM calculation as were considered for the uncertainty estimate in the MSSM, the uncertainty from unknown 11 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]. 12 The Mathematica code is linked to a Fortran driver program, calling NMSSMTools and HiggsBounds. The calculation of the SUSY particle masses is also included in the Fortran driver. Similarly to the MSSM case [61], we plan to additionally implement the MW calculation directly into Fortran in order to increase the speed of the MW evaluation. This will be useful in particular for large scans of the parameter space. 13 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 bounds from the SUSY searches at the LHC, the (maximal) uncertainty from missing higher orders is expected to be somewhat smaller than 9 MeV.
297.17 36.28 7.03 29.14 -1.60 1.23 higher-order corrections is estimated to be of similar size.

SM higher-order corrections
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 formula incorporates the O(αα s ) from Ref. [41], whereas we use the O(αα s ) from Ref. [37]. These results are in good numerical agreement with each other if in both cases the electric charge is parametrized in terms of the fine structure constant α. The O(α 2 α s ) three-loop corrections included in Eq. (35) are parametrized in terms of G µ . We therefore choose to parametrize the O(αα s ) contributions also in terms of G µ . The difference between the G µ parametrization of the QCD two-loop corrections that we use here and the α parametrization used in Ref. [58] leads to a prediction for M SM W that is ∼ 2 MeV lower than the result given in Ref. [58].
The numerical values of the different SM-type contributions to ∆r are given in table 1 As explained above, the values for the W and Z boson masses given above, which correspond to a Breit-Wigner shape with running width, have been transformed internally to the definition of a Breit-Wigner shape with fixed width associated with the real part of the complex pole.

Results for the M W prediction in the NMSSM
We now turn to the discussion of the prediction for M W in the NMSSM. Our evaluation has been carried out for the case of real parameters, consequently for all parameters given in this section the phases 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-loop contributions, in this result only the part of the contributions to ∆ρ SUSY,(ααs) , see Eq. (36), arising from squark loops with gluon exchange is taken into account. As we will show below in the discussion of our improved result for M W in the NMSSM, the two-loop contributions that have been neglected in Ref. [78] can have a sizeable impact. A further improvement of our results for the MSSM and the NMSSM is that they are based on contributions to ∆r that can all be evaluated at the correct input value for M W (using an iterative procedure), i.e. M , while it is the latter that is actually needed for the evaluation in the (N)MSSM (see Ref. [73] for a discussion how to remedy this effect). We have compared our result with the one given in Ref. [78] 14 taking into account only those contributions in our result that are also contained in the result of Ref. [78]. We found good agreement 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 on SUSY particle masses [124], by all theoretical constraints in NMSSMTools (checking e.g. that the Higgs potential has a viable physical minimum and that no Landau pole exists below the GUT scale), and have the neutralino as LSP. Unless stated otherwise, we choose the masses of the first and second generation squarks and the gluino to be large enough to not be in conflict with the limits from the searches for these particles at the LHC. 15 We make use of the code HiggsBounds [121][122][123] to check each parameter point against the limits from the Higgs searches at LEP, the Tevatron and the LHC.

Results in the MSSM limit of the NMSSM
Before turning to the discussion of the genuine NMSSM effects, we show the NMSSM M W prediction in the MSSM limit with all other parameters (including µ eff ) held fixed (such that the MSSM is recovered). In this limit one CP-even, one CP-odd Higgs boson (not necessarily the heaviest ones) and one neutralino become completely singlet and decouple. In the discussion of M NMSSM W in the MSSM limit, the setup for the numerical evaluation is introduced and the comparison to the MSSM M W prediction serves as validation of our implementation.
The left plot of Fig. 10 shows the NMSSM predictions in the MSSM limit (blue curves) as well as the MSSM predictions (red curves) for M W as a function of the stop mixing parameter X t . 16 The parameters in For the additional NMSSM parameters we choosem A = 1000 GeV, λ → 0, K = κ/λ = 0.5, A κ = −100 GeV (the impact of A κ on M W in the MSSM limit is negligible). Here, and in the following the prediction for M W includes all higher-order corrections described above (besides the Higgsino two-loop corrections).
Our approach here is the following: We start from a NMSSM parameter point. We take the effective CP-odd doublet massm A or the parameter A λ (herem A = 1000 GeV) as input to calculate the NMSSM Higgs boson spectrum. The physical value of the charged Higgs mass (calculated in the NMSSM) is used as input for the calculation of the MSSM Higgs masses. As discussed in Sect. 3.3, this procedure ensures that the mass of the charged Higgs boson used in our M W calculation is the same in the 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 parameters which occur in both models (tan β, the sfermion trilinear couplings A f , and the soft mass parameters) 14 We thank the authors of Ref. [78] for providing us with numerical results from their code. 15 The most stringent limits from SUSY searches at the LHC are set on the masses of the first and second generation squarks and the gluino, which go beyond ∼ 1 TeV. However these limits depend on the model assumptions. Relaxing these assumptions, squarks can still be significantly lighter [125]. Substantially weaker limits have been reported for the particles of the other sectors, so that third-generation squarks, stops and sbottoms, as well as the uncoloured SUSY particles, are significantly less constrained by LHC searches. 16 The Xt parameter that we plot here is the on-shell parameter. As described in Sect. 4.1 the on-shell value is transformed into a DR value, which is used as input for NMSSMTools to calculate the Higgs masses. All numerical values given for Xt in this section refer to the on-shell parameters. The corresponding predictions for the lightest CP-even Higgs mass in the (N)MSSM are displayed in the right plot of Fig. 10. For illustration, in the plots for the Higgs mass predictions the theoretical uncertainty on the SUSY Higgs mass is combined with the experimental error into an allowed region for the Higgs boson mass, rather than displaying the theoretical uncertainty in the Higgs mass prediction as a band around the theory prediction. Consequently, the blue band in the right plot shows the region M H = 125.09 ± 3.04 GeV, which was obtained by adding a theoretical uncertainty of 3 GeV quadratically to the experimental 2σ error. Here M H represents the corresponding mass parameter in the MSSM and the NMSSM (in the considered case M h in the MSSM and M h 1 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 Higgs masses) are allowed by HiggsBounds coincides (approximately) with the region in with the lightest Higgs mass is heavy enough to be interpreted as the signal at 125.09 GeV (|X t | 1000 GeV and X t < 1900 GeV). While the tree-level Higgs masses agree exactly in the MSSM and the NMSSM in the MSSM limit, we observe a small difference between the masses for the lightest CP-even Higgs calculated with FeynHiggs and with NMSSMTools. This discrepancy arises because of differences in the higher-order corrections implemented in the two codes 18 . The tree-level Higgs masses are only used in Fig. 10 for illustration. In all following plots (if nothing else is specified) the full loop-corrected results for the Higgs masses are used.
Going back to the left plot of Fig. 10, we see that the M NMSSM at about X t = 0 is in accordance with the local minimum in the Higgs-mass predictions. The fact that the local minima in the M W predictions are somewhat shifted compared to the local maxima in the Higgs-mass predictions is caused by the stop-loop contributions to ∆r, whose effect can be directly seen for the curves based on the tree-level predictions for the mass of the lightest CP-even Higgs boson in 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 splitting between 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 and moderate values of X t .

SUSY higher-oder corrections
Now we turn to the discussion of the size and parameter dependence of the SUSY two-loop corrections. 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 correspond to 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 agreement with flag 1 of the NMSSMTools result. For the sake of comparison between the NMSSM and the MSSM predictions for MW it is useful to keep those differences arising from different higher-order corrections in the MSSM limit of the Higgs sector as small as possible. We have therefore chosen flag 1 for the Higgs-mass evaluation with NMSSMTools in our numerical analyses presented in this paper. As mentioned above, an implementation of our predictions using the Higgs-mass evaluation of Ref. [117] is in progress.
1000 GeV, M 2 = 600 GeV, mg = 1500 GeV (solid curves) and mg = 300 GeV (dashed curves), A λ = 395 GeV, λ = 0.57, κ = 0.2, A κ = −80 GeV and we vary M SUSY = MQ 3 = MŨ 3 = MD 3 . We show the results for three values of X t : X t = 2 M SUSY (left), X t = 0 (middle) and X t = −2 M SUSY (right). It should be stressed here that the parameters for these plots are chosen to demonstrate the possible size and the parameter dependence of the SUSY two-loop corrections, however they are partially excluded by 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 mg = 300 GeV is clearly disfavoured by the negative LHC search results. Fig. 11 shows the contribution to the W boson mass, δM W , from the O(αα s ) two-loop corrections with gluon exchange (dark blue curves), with gluino exchange (orange curves) and from the 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 results have been subtracted from each other. Starting with the dark blue curves, we find that the gluon contributions lead to a maximal shift of ∼ 3 MeV in M W for all three choices of X t and that the size of the gluon contributions decreases with increasing M SUSY . Turning to the orange curves, we find that for mg = 1500 GeV (solid curves) the δM W shift, induced by the gluino two-loop corrections, is small (< 1 MeV) for X t = 0, while it is up to 3 − 4 MeV for X t = 2 M SUSY and X t = −2 M SUSY . Making the gluino light -choosing mg = 300 GeV (dashed curves) -the gluino corrections can get large. 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 on the stop mixing parameter. Threshold effects occur in the gluino corrections and cause kinks in the orange 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 ) contributions to ∆ρ SUSY . The pink curves show the impact of this additional correction term. Starting with the solid curves (mg = 1500 GeV), we observe that for large stop mixing, X t = ±2 M SUSY , the mass-shift corrections are positive and the maximal shift is ∼ 4 MeV. For zero mixing the mass-shift corrections lead to a large negative shift in M W (up to −12 MeV for small M SUSY ). For mg = 300 GeV, the size 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-shift corrections leads to a smooth curve and no kink is found in the full M W prediction. This can be seen in Fig. 12, where we plot the sum of the gluon, gluino and mass-shift corrections (all parameters are the same as in Fig. 11 b )) formula. The size of these contributions can be seen in Fig. 13. Here, and in some 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 Higgs signal, as starting point for our study. Here we take the following parameters: m t = 173. 34 Fig. 11) as a function of M SUSY . The parameters are the same as in Fig. 11. The solid curves correspond to mg = 1500 GeV while the dashed curves correspond to mg = 300 GeV. In the left plot we set X t = 2 M SUSY , in the middle one X t = 0 and in the right one X t = −2 M SUSY . κ = 0.18, A κ = −10 GeV, and we vary X t . These parameter points are HiggsBounds allowed in the regions 700 GeV < X t < 1000 GeV and 1100 GeV < X t < 1400 GeV.  (∼ 1 MeV). It was shown in Ref. [77] that the contributions to M W from the Higgsino corrections can be slightly larger (∼ 5 MeV) for lightert/b. The right plot shows the M W prediction plotted against M h 1 . We can clearly see here that this scenario, in which the Higgs signal can be interpreted as the lightest CP-even NMSSM Higgs, gives a W boson mass prediction in good agreement with the M W measurement indicated by the grey band.

NMSSM Higgs sector contributions
Now we turn to the discussion of effects from the NMSSM Higgs sector. In the MSSM the maximal value for the tree-level Higgs mass M h is M Z . One of the features of the NMSSM Higgs sector is that the tree-level Higgs mass M h 1 gets an additional contribution λ 2 v 2 sin 2 2β, which can shift the tree-level Higgs mass upwards compared to its MSSM value (an upward shift can also be caused by singlet-doublet mixing, if the singlet state is lighter than the doublet state), and thus reduce the size of the radiative corrections needed to 'push' the lightest Higgs mass up to the experimental value. For λ = 0.7 and tan β = 2 a tree-level value for M h 1 of 112 GeV is possible [127]. This additional tree-level contribution to the Higgs mass, as well as its impact on M W are shown in Fig. 14 term λ 2 v 2 sin 2 2β pushing the tree-level Higgs mass beyond M Z for large λ. 19   (calculated with NMSSMTools as described above) can be seen in the upper right plot. Now we turn to the 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] where ∆r x(α) denotes the one-loop contribution from particle sector x (here x=gauge-boson/Higgs), as defined for the NMSSM in Eq. (33). The reference M W value is set here to M ref W = M exp W . The overall contribution from the Higgs sector is rather large and negative. As we will discuss in more detail below, the Higgs sector contributions here are predominantly SM-type contributions (with M H SM set to the corresponding Higgs mass value). The prediction for M W in the NMSSM is shown in the lower right plot. Larger values for M h 1 correspond to a lower predicted value for M W . Thus, for small tan β, where we find a significantly higher predicted value for M h 1 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 however still compatible with the experimental M W measurement at the 2σ level for the scenario chosen here. For tan β ∼ 1 the difference between the W boson mass prediction for λ = 0.65 and λ → 0 is ∼ 25 MeV. The parameter tan β enters also in the sfermion and in the chargino/neutralino sector. We checked that for the parameters used here, the tan β dependence of the contributions from these two sectors is small compared to the Higgs sector contributions, less than ∼ 3 MeV.
We continue the study of the NMSSM Higgs sector contributions in Fig. 15. In the left plot we compare the NMSSM prediction for 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, this happens 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, this behaviour is approximately retained also in the presence of higher-order corrections in the Higgs sector.
The parameters we use here are m t = 173.34 GeV, tan β = 2, µ = 200 GeV, ML /Ẽ = 1500 GeV, MQ /Ũ /D 1,2 = 1200 GeV, MŨ 3 = MQ 3 = 540 GeV, MD 3 = 1000 GeV, A τ = A b = 1000 GeV, M 2 = 370 GeV, mg = 1500 GeV, A λ = 420 GeV, λ = 0.57, κ = 0.2, A κ = −10 GeV, and we vary X t . The NMSSM parameters are allowed by HiggsBounds for X t 780 GeV. For X t 810 GeV the mass of the lightest CP-even Higgs falls in the range of the observed Higgs signal. The MSSM prediction is plotted as a comparison to illustrate and discuss the NMSSM effects on M W . Here (and in the following) we do not check any phenomenological constraints for the MSSM parameter point (but only for the considered NMSSM scenario).
The NMSSM prediction for M W differs from the MSSM prediction by ∼ 12 MeV. The chargino/neutralino 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 2 values tend to give positive contributions to ∆r. For the chosen parameters, these two effects cancel and contributions from the chargino/neutralino sector are very small, O(0.1 MeV). Consequently, different Higgs sector contributions give rise to the difference between the MSSM and the NMSSM curves. Any differences in the CP-odd Higgs sector have a negligible impact on the M W prediction (see also Ref. [78]). Since we set the 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 2 = 150 GeV) has a large singlet component (|U H 23 | 2 ≃ 95%), consequently the singlet components of h 1 and h 3 are small. h 3 is heavy and has no impact on the M W prediction. Our procedure to calculate the Higgs masses in the MSSM and the NMSSM leads to the same charged Higgs masses, but to different predictions for the lightest CP-even Higgs masses M h 1 and M h . This difference arises from the different relations between the charged Higgs mass and the lightest CP-even Higgs mass in the MSSM and the NMSSM. Further it 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 shows in addition to the NMSSM prediction for M W (blue) and the MSSM prediction (red), a blue dashed curve (with open dots). The dashed blue curve corresponds to M NMSSM 20 . As one can see the dashed blue curve is very close to the red MSSM curve, thus here the difference between the MSSM and the NMSSM Higgs sector contributions to M W essentially arises from the SM-type Higgs sector contributions, in which different Higgs mass values are inserted. It should be noted in this context that we have made a choice here by comparing the predictions for a particular NMSSM parameter point with an associated MSSM parameter point having the same value of the mass of the charged Higgs boson. Accordingly, the predictions for the other Higgs boson masses in the two models in general differ from each other, see above, leading to the effect displayed in the left plot of Fig. 15. Instead, one could have chosen, at least in principle, the associated MSSM parameter point such that the masses of the lightest CP-even Higgs masses, M h 1 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 the charged 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 Higgs mass M h 1 . In this plot we display both the blue band indicating the region M h 1 = 125.09 ± 3.04 GeV as 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 1 . It is interesting to 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 SM, on the other hand, Fig. 15 shows the well-known result that setting the SM Higgs boson mass to the measured experimental value one finds a predicted value for M W which is somewhat low compared to the experimental value. Now we want to investigate whether singlet-doublet mixing (a genuine NMSSM feature) has a significant impact on the M W prediction. Such a scenario is analysed in Fig. 16. Our parameters are m t = 173.34 GeV, tan β = 2, µ = 140 GeV, ML /Ẽ = 130 GeV MQ /Ũ /D 1,2 = 1200 GeV, MQ 3 = 800 GeV, MŨ 3 = 600 GeV, MD 3 = 1000 GeV, A t = 1300 GeV A τ = A b = 1000 GeV, M 2 = 230 GeV, mg = 1500 GeV, A λ = 210 GeV, λ = 0.55, κ = 0.31, and we vary A κ . These parameters are allowed by HiggsBounds everywhere apart from −145 GeV A κ −105 GeV, and the Higgs signal can be interpreted as either h 1 or h 2 . The left plot shows the prediction for M h 1 (solid curve) and M h 2 (dashed). The corresponding singlet components U 2 13 (solid) and U 2 23 (dashed) are shown in the middle plot. The third CP-even Higgs is heavy and has a negligible singlet component. For A κ −120 GeV, h 2 is doublet-like and has a mass in the region of the observed Higgs signal (indicated by the blue band). In the MSSM, scenarios which allow the interpretation of the Higgs signal as the heavy CPeven Higgs involve always a (relatively) light charged Higgs (see e.g. Ref. [128]). Due to changed mass 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 the interpretation of the Higgs signal as the second lightest CP-even Higgs is much less constrained by the LHC results from charged Higgs searches [129,130]. The interpretation of the Higgs signal as h 2 in this model is always accompanied by a lighter state with reduced couplings to vector bosons. In this figure the charged Higgs mass is ∼ 280 GeV. For A κ −100 GeV, h 1 is doublet-like and has a mass in the region of the observed Higgs signal. In the "transition" region (−150 GeV A κ −50 GeV) the two 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, the parameter regions of A κ corresponding to two different interpretations of the Higgs signal within the NMSSM lead to very similar predictions for the W boson mass, which are in both cases compatible with the experimental result. Even a sizeable doublet-singlet mixing has only a minor effect on the M W prediction in this case. We have demonstrated so far that, taking Higgs search constraints and the information on the discovered Higgs signal into account, 21 the genuine NMSSM effects from the extended Higgs sector are quite small, and the Higgs sector contributions that we analysed so far were dominated by SMtype 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 couplings to vector bosons) can lead to sizeable (non SM-like) Higgs contributions to M W . This effect can also be observed in the MSSM. Although it is not a genuine NMSSM effect, we want to demonstrate the impact of such a contribution here. For Fig. 17 we choose the following parameters m t = 173.34 GeV, tan β = 9.25, µ = 200 GeV, ML /Ẽ = 300 GeV MQ /Ũ /D 1,2 = 1500 GeV, MQ 3 = MŨ 3 = MD 3 = 1100 GeV, A t = −2300 GeV A τ = A b = −1500 GeV, M 2 = 500 GeV, mg = 1500 GeV, λ = 0.2, κ = 0.6, A κ = −1370 GeV, and we varym A . The left plot in Fig. 17 shows the predictions for the masses of the lightest two CP-even Higgs bosons in the NMSSM (blue) and in the MSSM (red) as a function of the charged Higgs mass. In both models the second lightest Higgs falls in the mass range 125.09 ± 3.04 GeV for the chosen parameters. This scenario is essentially excluded by the latest charged Higgs searches [129,130]. Nevertheless, we include these plots to illustrate the possible size of the 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 the NMSSM (blue) and in the MSSM (red) while the right plot shows the full M W prediction in the NMSSM (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 varyinĝ m A (or M H ± ), the change in M W with M H ± stems purely from the Higgs sector. The Higgs sector contribution to M W comes dominantly from the light charged Higgs, while the lightest CP-even Higgs gives only a rather small contribution to M W due to its reduced vector boson couplings. In the middle plot the SM result for δM W with M H SM = M h 2 is shown in black. A significant difference between the SM Higgs contribution and the MSSM/NMSSM Higgs contributions can be observed. As one can see in the right plot, the displayed variation with the charged Higgs boson mass corresponds to about a 1σ shift in M W .

Neutralino sector contributions
We start the discussion of the contributions from the NMSSM neutralino sector, which differs from the respective MSSM sector, with  41)) induced by the chargino/neutralino contributions in the MSSM (red) and in the NMSSM (blue). At M 2 = 160 GeV the chargino mass is 108 GeV and thus just above the LEP limit. The δM W contribution from the chargino/neutralino sector in the MSSM reaches 8.5 MeV in this case. 22 In the NMSSM the maximal δM W contribution from the chargino/neutralino sector is 16.5 MeV -significantly larger than in the MSSM. Both in the MSSM and the NMSSM, the chargino/neutralino contributions decrease when increasing M 2 and therewith the chargino and neutralino masses, showing the expected behaviour when decoupling the gaugino sector. The largest difference between the NMSSM and the MSSM chargino/neutralino contributions is ∼ 8 MeV (at M 2 = 160 GeV). The difference arises from the neutralino 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 in the NMSSM than in the MSSM. The lower right plot of Fig. 18

predictions.
We continue with the discussion of the neutralino contributions to M W in the NMSSM in Fig. 19.   Fig. 19 displays the W boson mass prediction in the NMSSM (blue) and in the MSSM (red). The difference between these two predictions is largest (7 MeV) for M 2 = 150 GeV and (almost) vanishes for large M 2 . 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 that in this scenario both M h 1 and M W lie within the preferred regions indicated by the blue and grey bands for the whole parameter range displayed in the figure.
In order to investigate the reasons for the different predictions for the chargino/neutralino contributions we plot the masses of the three lightest neutralino states in the NMSSM (blue) and the MSSM (red) in the lower left plot. The other MSSM/NMSSM neutralinos are heavier than 250 GeV and hardly affect the M W prediction. We set here the (unphysical) soft masses M 1 and M 2 equal in the MSSM and the NMSSM and identify the MSSM µ parameter with the effective µ of the NMSSM. The resulting predictions for the masses ofχ 0 1 andχ 0 2 are a few GeV lower in the NMSSM than in the MSSM. The singlino components of the NMSSM neutralinos, |N i5 | 2 , 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. The singlino components ofχ 0 1 andχ 0 2 are below 10% for M 2 = 150 GeV and increase up to 40%(20%) for χ 0 1 (χ 0 2 ) for higher M 2 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 can be approximated by taking into account only the leading term −c 2 W /s 2 W ∆ρ (defined in Eq. (26)). The ∆ρ term contains only the W and Z boson self-energies at zero momentum transfer, thus this approximation neglects in particular the contributions from box, vertex and fermion self-energy diagrams containing 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 observables. For the plot in Fig. 20 we use the same parameters as in Fig. 19. Again the blue(red) solid curve shows the δM W shift as a function of M 2 , calculated as in Eq. (41) with x=chargino/neutralino in the NMSSM(MSSM) (the two solid curves are identical to the ones in the upper right plot of Fig. 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 neutralino contributions to the ∆ρ parameter: In In this scenario the singlino-higgsino mixing leads to a positive contribution to ∆ρ, but to a negative contribution to the ∆r terms beyond ∆ρ (we checked that the contribution from the box diagrams is negligible for large M 2 values). We also checked that going to large µ values, the chargino/neutralino sector decouples and all terms vanish. In this scenario the two effects largely cancel each other and for large M 2 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 the scenario discussed in Fig. 15 the negative contributions exceed the positive ones so that the full ∆r result is negative for large M 2 . Thus, we have shown that the ∆ρ approximation for the chargino and neutralino contributions works quite well in the MSSM, whereas sizeable corrections to M W beyond the ∆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 left plot of Fig. 21 shows the W boson mass prediction in the NMSSM as a function of µ, with the parameters chosen as m t = 173.34 GeV, tan β = 20, ML /Ẽ = 250 GeV, MQ /Ũ /D 1,2 = 1500 GeV, MQ 3 = 500 GeV MŨ 3 = 1500 GeV, MD 3 = 300 GeV, A τ = 0 GeV, A t = A b = −2185 GeV, M 2 = 150 GeV, mg = 1500 GeV,m A = 1500 GeV, λ = 0.2, κ = 0.6, A κ = −1370 GeV. The parameter points are HiggsBounds allowed, and h 1 falls in the mass range 125.09 ± 3.04 GeV. When increasing µ, the M NMSSM W prediction decreases first, reaches its minimum for µ ∼ 1100 GeV and then rapidly increases. This behaviour can be explained by looking at the contributions to M W from the chargino/neutralino sector (here we take again the full ∆r contributions into account) and from the stop/sbottom sector. The shift δM W arising from charginos and neutralinos is shown in the middle plot of Fig. 21. The chargino/neutralino contribution is largest for small µ and decreases with increasing µ. Going to larger µ the masses of the (higgsino-like) chargino and neutralino states increase and the M W contribution decreases. The shift δM W arising from the stop/sbottom sector is shown in the right plot of Fig. 21  butions) get smaller when µ is increased up to µ ∼ 1000 GeV and then start to rise if µ is increased further. Increasing µ, the splitting between the two sbottoms gets larger (while the stop masses stay nearly constant), which implies also an increase of the splitting between stops and sbottoms. The counteracting terms in ∆ρ (see the discussion in Sect. 4.4.1) lead to the observed behaviour.

Conclusions
We have presented the currently most accurate prediction for the W boson mass in the NMSSM, in terms of the Z boson mass, the fine-structure constant, the Fermi constant, and model-parameters entering via higher-order contributions. This result includes the full one-loop determination and all available higher-order corrections of SM and SUSY type. These improved predictions have been compared 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 between the 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 the SM 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-order corrections of about 4 MeV). The comparison with the current experimental value of M exp W = 80.385± 0.015 GeV shows the well-known feature that the SM prediction lies somewhat below the value that is preferred by the measurements from LEP and the Tevatron (at the level of about 1.8 σ). The loop contributions from supersymmetric particles in general give rise to an upward shift in the prediction for M W as compared to the SM case, which tend to bring the prediction into better agreement with the 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 in particular of a NMSSM model file for the program FeynArts and a Fortran driver for the evaluation of the masses, mixing angles, etc. needed for the numerical evaluation. The code NMSSMTools is used for the evaluation of the loop-corrected Higgs boson masses. The implementation of another result for the 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 result, all available higher-order corrections of SM-type, stop-loop and sbottom-loop contributions with gluon and gluino exchange of O(αα s ), relevant reducible higher-order contributions, as well as an approximate treatment of the MSSM-type Yukawa-enhanced electroweak two-loop corrections of O(α 2 t ), O(α t α b ), O(α 2 b ). Analytic expressions for all those contributions are implemented, except for the electroweak 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 parameter regions which are allowed by Higgs searches (tested by HiggsBounds), SUSY searches and further theoretical 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 this study in the MSSM limit, yielding an important check of our NMSSM implementation. We have investigated the size of the SUSY two-loop corrections to M W and found that the O(αα s ) corrections beyond the pure gluon exchange contributions, which were incorporated in the previous result of Ref. [78], can give sizeable contributions. On the other hand, the effect of the Yukawa-enhanced electroweak two-loop corrections of O(α 2 t ), O(α t α b ), O(α 2 b ) stays numerically relatively small in the allowed parameter region.
Concerning the investigation of genuine NMSSM effects, we started our discussion with the Higgs sector contributions to M W . The tree-level prediction for the lightest CP-even Higgs mass is modified by an additional term in the NMSSM as compared to the MSSM, which (for small tan β) leads to an upward shift of the tree-level Higgs mass. Therefore, in that region, the radiative corrections needed to push the Higgs mass to about 125 GeV can be smaller than in the MSSM, which implies that lighter stop masses and a smaller stop mixing are possible. We investigated a scenario where this additional tree-level term gives rise to a higher M h 1 prediction than in the MSSM limit. The impact on the M W prediction is a downward shift (of ∼ 25 MeV in the considered example) as compared to the corresponding prediction in the MSSM. In the prediction for M W this contribution from the Higgs 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 good agreement between the theoretical prediction and the experimental result can be reached. We have furthermore investigated the effect of doublet-singlet mixing. While a sizeable doublet-singlet mixing can occur in the region where the two NMSSM Higgs states h 1 and h 2 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 the second-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 of the LHC signal as the second-lightest CP-even Higgs h 2 is possible together with a relatively heavy charged Higgs. This is different from the situation in the MSSM, where all Higgs states have to be light in this case, so that such a scenario can be probed by searches for charged Higgs bosons in topquark 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 Higgs boson. We have found that a light charged Higgs boson (together with a light CP-even Higgs with reduced but non-zero couplings to gauge bosons) can in principle give very significant contributions to M W (as in the MSSM). In that case large deviations from the SM Higgs sector contributions occur, but as discussed above scenarios of this kind are severely constrained by limits from charged Higgs searches at the LHC. Generally we find that taking all available constrains on the Higgs sector into account, 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 difference between the W boson mass predictions in the NMSSM and the MSSM. The chargino/neutralino contributions to M W can be larger in the NMSSM compared to the MSSM, where in the scenario which we studied the difference reaches ∼ 8 MeV. Assuming the same values for the soft mass parameters in the MSSM and the NMSSM and choosing µ = µ eff , the mixing with the singlino leads to shifts in the neutralino masses as compared to the MSSM case. In the considered scenarios the lightest NMSSM states turned out to be lighter than the corresponding MSSM states. They also have a 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 the difference between the full result and the ∆ρ approximation can be sizeable in the NMSSM, where the approximation can lead both to an over-or an underestimate of the full result. Light neutralinos with a significant higgsino-singlino mixing tend to give a positive contribution to ∆ρ, but a negative contribution to the ∆r terms beyond ∆ρ. It therefore depends on the exact patterns of the admixture with the singlino whether the neutralino sector of the NMSSM leads to an upward or downward shift in 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 with the Z boson mass, the Fermi constant and the fine structure constant in comparison with highprecision measurements of those quantities provides a high sensitivity for discriminating between the SM 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 spectra of supersymmetric particles -either via improved limits or the discovery of new states -the impact of this important tool can be expected to be even more pronounced in the future.