Electroweak precision measurements, encoded in the oblique parameters, give strong constraints
on physics beyond the Standard Model. The oblique parameters S,T,U(V,W,X) are calculated in the next-to-minimal supersymmetric model (NMSSM). We outline the calculation of the oblique
parameters in terms of one-loop gauge-boson self-energies and find sensitive restrictions for the NMSSM parameter space.
1. Introduction
The precision measurements of the electroweak parameters give stringent constraints on physics beyond the Standard Model (SM). A very elegant method to systematically confront the electroweak precision measurements with new physics is given by the oblique parameters S, T, U [1–3]. These three parameters allow for restricting any physics beyond the SM, under the following three conditions.
The physics beyond the SM has to obey SU(2)L⊗U(1)Y gauge symmetry; that is, there are no additional electroweak gauge bosons compared to the SM.
The couplings of new particles to light fermions have to be suppressed. That is, the main contribution of couplings beyond the SM to four-fermion scattering originates from the change in the self-energies of the gauge-boson propagators. These contributions are called oblique corrections. The suppressed contributions which may for instance appear in box diagrams with four external fermions or in vertex corrections are called nonoblique corrections.
New physics enters only at a large scale compared to the electroweak scale.
From the second condition it is clear that the oblique parameters are expressed in terms of gauge-boson self-energies, as shown in detail in [3]. The main argument is that the electroweak precision measurements probe weak-interaction processes with light external fermions of mass mf (at cms energies on the electroweak scale), wherein vertex- and box-type corrections are suppressed by factors of mf2/mZ2 as compared to the self-energy loop corrections.
However, many models beyond the SM are expected to have effects at a scale not too far from the electroweak scale, which is given by the vacuum-expectation value of the neutral SM Higgs boson component v0≈174 GeV. In order to weaken the third condition for the oblique parameters S, T, U mentioned previously, allowing new physics to enter already at a scale somewhat larger than the electroweak scale, the oblique parameters were extended to the six parameters S, T, U, V, W, X [4, 5]. The explicit expressions for these oblique parameters read
(1)S=4sW2cW2α[ΠZZ(mZ2)-ΠZZ(0)mZ2-cW2-sW2sWcWΠZγ′(0)-Πγγ′(0)],T=1α[ΠWW(0)mW2-ΠZZ(0)mZ2],U=4sW2α[ΠWW(mW2)-ΠWW(0)mW2-cW2ΠZZ(mZ2)-ΠZZ(0)mZ2-2sWcWΠZγ′(0)-sW2Πγγ′(0)ΠWW(mW2)-ΠWW(0)mW2],V=1α[ΠZZ′(mZ2)-ΠZZ(mZ2)-ΠZZ(0)mZ2],W=1α[ΠWW′(mW2)-ΠWW(mW2)-ΠWW(0)mW2],X=-sWcWα[ΠZγ(mZ2)mZ2-ΠZγ′(0)].
The quantities ΠG1G2(s) with G1/2∈{γ,W,Z} denote the new contributions to the transverse part of the self-energies at a momentum-squared scale s compared to the SM,
(2)ΠG1G2(s)=ΠG1G2new(s)-ΠG1G2SM(s).
The derivatives of the self-energies ΠG1G2(s) with respect to the scale s are denoted by ΠG1G2′(s0)=dΠG1G2(s)/ds|s=s0. The fact that only relatively few parameters (besides Π(s) for s∈{0,mW2,mZ2} only Π′(s) at the same low-energy scales) enter in (2) reflects the observation that precision measurements are made only by two-particle scatterings on light fermions at those few scales, as explained in detail in [4]. Finally, sW=sin(θW) and cW=cos(θW) contain the usual weak Weinberg mixing angle θW, and α denotes the fine-structure constant.
Having defined the oblique parameters, electroweak precision observables, like for instance the W±-boson mass, may be expressed in terms of these parameters. Constraints on the oblique parameters are gained via a global fit to the electroweak precision measurements; see for example, [6]. Being exactly zero within the SM, these global fits result in error bands for the six parameters of (1), see (3), hence potentially constraining the size of effects from new physics.
In this paper, we compute the oblique parameters S, T, U, V, W, X of (1) in the next-to-minimal supersymmetric extension of the SM (NMSSM); for reviews of the NMSSM, we refer to [7, 8]. Let us briefly recall that in the superpotential of the NMSSM the μ-term of the MSSM is replaced by a λ term and a κ term. In contrast to the parameter μ, the new parameters λ and κ are dimensionless; in particular a vacuum-expectation value of a Higgs singlet times λ generates an effective μ term, and the κ term avoids a Peccei-Quinn symmetry.
The NMSSM has recently received much attention; in particular, besides the scale invariant superpotential, it has a much richer Higgs sector and a fifth neutralino compared to the minimal supersymmetric extension (MSSM). Noting that in the fermion-fermion interactions there appear in principle also nonoblique corrections in the NMSSM, here we assume that the nonoblique corrections are negligible.
There are several computations of electroweak precision observables in the NMSSM, of which we would like to mention a few here. First, in a study of the Z0 boson width [9], supposing that the lightest neutralino χ~10 in the NMSSM has a mass mχ~10<mZ/2, it has been shown that the Z-boson decay width changes as compared to the SM and hence a comparison with precision measurements (in particular at LEP) gives strong constraints on the NMSSM parameters. Second, in [10], the partial decay of Z bosons into b quark pairs has been considered. Here it is found via a parameter scan that the Zb-b coupling in the SM compared to data can not be significantly improved in the MSSM/NMSSM. Third, the new contributions of the NMSSM to the transverse parts of the W±- and the Z-boson self-energies have been presented in [11]. Fourth, leptonic decays of the Z as well as W± bosons in the NMSSM were considered in [12]. In particular, in this way, constraints on the couplings respectively masses of the new particles, which appear in the loops, have been obtained.
Since the parameter space of the NMSSM is very large, there are different approaches to phenomenological studies of this model. In [13–15], for instance, the constrained version of the NMSSM is considered where it is assumed that various masses and couplings unify at the GUT scale. Another approach is to consider specific benchmarks scenarios, representing different regimes in parameter space [16, 17]. In our numerical examples mentioned later, we will adopt the former approach. Let us remark that there exist similar approaches of electroweak precision observables in the MSSM; see for instance [18–21].
2. Details of the Calculation
For the prediction of the oblique parameters of (1) we need to compute the transverse parts of the one-loop self-energies ΠG1G2(s)=ΠG1G2NMSSM(s)-ΠG1G2SM(s), where G1, G2 denote the gauge bosons γ, W±, Z0. The self-energies with exclusively leptons, quarks, and gauge bosons in the loops are exactly the same in ΠG1G2NMSSM(s) and ΠG1G2SM(s) and therefore do not need to be evaluated. As a consistency check, however, we confirmed this analytically.
Since the Higgs sector of the NMSSM is not a simple extension of the SM Higgs sector, we have to consider in ΠG1G2SM(s) all contributions which contain the SM Higgs boson HSM. In the self-energies of the NMSSM we have to consider all contributions which involve scalar neutrinos ν~, scalar leptons l~, scalar up- and down-type quarks u~, d~, neutralinos χ0, charginos χ+, the neutral Higgs bosons H1, H2, H3, A1, A2, the pair of charged Higgs bosons H±, and the Goldstone bosons G0, G+. All Feynman diagrams of the self-energy contributions to the oblique parameters are shown in the appendix. Let us note that we consider the most general NMSSM in our computation. In particular we allow for CP violation in the Higgs sector, such that the neutral Higgs bosons Hi/Aj are not necessarily CP even/odd, respectively; for details see for instance [7].
The NMSSM Feynman rules are implemented in the FeynRules program package [22, 23] following the conventions of [24]. As a caveat, let us remark here that the Goldstone components of the neutral Higgs boson squared mixing matrix have to be carefully constructed to guarantee unitarity, which is violated by the parameters chosen in the model file nmssm.fr. We link this list of Feynman rules with the packages FeynArts/FormCalc [25], resulting in analytic expressions for the various one-loop self-energies in terms of basic scalar master integrals. Next, we assemble the parameters of (2) and numerically evaluate the results using the program package LoopTools [26]. We observe that all ultraviolet singularities cancel between the different self-energies in the oblique parameters. On a more technical note, the matrix γ5 is treated naively (i.e., anticommuting) with (γ5)2=14×4, while we have checked explicit gauge parameter independence.
3. Results
First let us note that we expect in general different results for the oblique parameters in the NMSSM compared to that of the MSSM: on the one hand the (complex) Higgs singlet, introduced in the NMSSM, gives two more physical Higgs bosons compared to the MSSM which show up in the gauge-boson self-energies. On the other hand the additional singlino mixes, even that it is a gauge singlet, with the other neutralinos and therefore gives changed contributions. In general only in the limit of arbitrary small parameters λ and κ keeping the ratio κ/λ as well as the product λvs fixed (with vs the vacuum-expectation value of the singlet) and keeping also the trilinear couplings Aλ and Aκ fixed, the NMSSM becomes the MSSM.
As a simple numerical example, we assume unification of scalar masses M0, fermion masses M1/2, and trilinear couplings A0. This scenario is usually called constrained NMSSM (cNMSSM). To be precise, we employ the program package NMSPEC [15], with a unification of the scalars, fermions, and trilinear couplings but with the exception of the singlet mass ms and the Peccei-Quinn parameter κ, which are given as output parameters, determined by the minimization equations of the Higgs potential. In addition Aκ is not unified with the other trilinear couplings and is varied separately. Furthermore, the ratio of the vacuum-expectation value of the two Higgs boson doublets, tan(β), the Higgs coupling parameter λ, and the sign of μ have to be fixed in addition to the parameters of the SM. The program package NMSPEC eventually computes the mass spectra and mixing angles at the electroweak scale. Let us note that our calculation of the oblique parameters is performed in the general NMSSM such that the oblique parameters for arbitrary parameter values can be easily computed. The program code for the oblique parameters is available as C-code from the URL [27].
The explicit values for the NMSSM parameters we choose in our numerical examples are given in Table 1. These parameter sets are inferred from the figures presented in [15]. The scales at which the NMSSM parameters are fixed are written as a superscript, with MSUSY and GUT the supersymmetry breaking scale, respectively, the grand unification scale, respectively; both scales are derived from the input parameters in NMSPEC. We note that the chosen parameter sets as given in Table 1 pass all the different constraints, gained from Higgs boson searches, partial decay width of the Z boson into neutralinos, and mass bounds for the charginos; for further details we refer to [15].
Parameter values for studies in case of the constrained NMSSM (Mi and Ai in GeV). These sets are inspired by the ranges given in Figures 1–3 of [15].
Set #
M0GUT
M1/2GUT
A0GUT
AκGUT
tan(β)MSUSY
sgn(μ)
λMSUSY
1
500
500
-800
-100
5
+
0.15
2
500
500
-800
-1500
1.7
+
0.5
3
100
200
-700
-75
5
+
0.2
In Figures 1 and 2 we present the results for the oblique parameters S and T for the different parameter sets given in Table 1. All other oblique parameters turn out to be rather small and are therefore not shown explicitly. In the Figures we vary successively the parameters A0, tan(β), M0, and M1/2 about the central values from Table 1 as indicated in the figures (where we suppress the superscripts MSUSY and GUT). From the lines we see how the oblique parameters S and T change under variations of the parameter values. We also draw the 1σ and 2σ error ellipses corresponding to the recent experimental fits to S and T [6]:
(3)S=0.01±0.1,T=0.03±0.11,ρ=0.87.
Here, ρ denotes the correlation coefficient. Note that in this fit a SM Higgs boson mass of mHSM=117 GeV is assumed, which we also use consistently as a parameter value in the SM self-energies.
Oblique parameters S and T in the NMSSM in the constrained case with central parameters tan(β)=5, M0=M1/2=500 GeV, A0=-800 GeV, Aκ=-100 GeV, sgn(μ)=+, and λ=0.15 (first row of Table 1) with variation of the parameters successively in the ranges tan(β)=5,10,…,40 (tan(β)=5,10 lie very close together), M0=250 GeV, 350 GeV,…,1950 GeV, M1/2=150 GeV, 250 GeV,…,650 GeV as indicated in the figure. The shaded regions show the 1σ and 2σ error ellipses of the electroweak precision measurements fitted to S and T corresponding to (3) [6].
(a): Oblique parameters S and T in the NMSSM in the constrained case with central parameters tan(β)=1.7, M0=M1/2=500 GeV, A0=-800 GeV, Aκ=-1500 GeV, sgn(μ)=+, and λ=0.5 (second row in Table 1) with successive variations of the parameters M0=250 GeV, 300 GeV,…,500 GeV, M1/2=450 GeV, 500 GeV,…,650 GeV, A0=-800 GeV, −700 GeV,…,−100 GeV, as indicated in the figure. (b): Same as (a) but with central parameters tan(β)=5, M0=100 GeV, M1/2=200 GeV, A0=-700 GeV, Aκ=-75 GeV, sgn(μ)=+, and λ=0.2 with successive variations of the parameters tan(β)=2,3,…,15, M0=0 GeV, 100 GeV,…,800 GeV, M1/2=200 GeV, 225 GeV,…,400 GeV.
As expected, in our numerical examples we find suppressed contributions to the oblique parameters V, W, X, which is due to the large masses of the additional particles as compared to the electroweak scale. The sensitivity of the oblique parameters under variations of NMSSM parameters is clearly visible: for the central parameter set 1 in Table 1. We infer from Figure 1 that the 2σ error ellipse constrains tan(β)≲40, M0≳250 GeV, and M1/2≲650 GeV. For the other central values in Table 1 we can easily read off the constraints from Figure 2. In general the oblique parameters S and T are highly sensitive to new particles which are nonisosinglets. However, only under certain assumptions the origin of the contributions to the different oblique parameters is immediately evident; we refer to the discussion in [3] for further details. Since these assumptions are not fulfilled in our case we refrain from a detailed study here. It would be also interesting to discuss the impact of the NMSSM parameters at the electroweak scale on the oblique parameters. However, this is beyond the scope of the present paper, where we focus on the calculation of the oblique parameters itself.
4. Conclusions
For a large class of models beyond the Standard Model, the so-called oblique parameters give very sensitive constraints coming from electroweak precision measurements. We have computed the set of extended oblique parameters S, T, U, V, W, X for the next-to-minimal supersymmetric model (NMSSM).
We have presented numerical examples with the parameters of the NMSSM chosen in a constrained case, as explained in Section 3. We observe the oblique parameters S and T to be highly sensitive on variations of the model parameters. In fact, fairly modest changes of the NMSSM parameters easily violate the constraints from the electroweak precision measurements.
The oblique parameters have been computed for the general case, in particular with a general CP violating Higgs sector, such that they may be applied to arbitrary parameter values, in a more complete parameter scan, which we reserve for future work.
AppendixFeynman Diagrams for the Oblique Parameters
Here we present the Feynman diagrams which contribute to the oblique parameters of (1). For self-energy diagrams which exclusively have leptons, quarks, and gauge bosons in the loops, the contributions to ΠG1G2NMSSM(s) and ΠG1G2SM(s) exactly are canceled in (2) and do not have to be computed.
The contributions to ΠG1G2SM(s) consist of diagrams which contain the SM Higgs boson (HSM) in the loop. There are only contributions of this kind to the W+ and Z0 self-energies as shown in Figure 3.
Feynman diagrams for the self-energies ΠWWSM(s) and ΠZZSM(s) which contribute to the oblique parameters. All other diagrams vanish, respectively, cancel with the corresponding diagrams in ΠWWNMSSM(s) and ΠZZNMSSM(s).
We also show all self-energy diagrams contributing to the NMSSM part of the oblique parameters. These diagrams involve scalar neutrinos ν~, scalar leptons l~, scalar up- and down-type quarks u~, d~, neutralinos χ0, and charginos χ+, as well as the neutral Higgs bosons Hi, Aj, the charged Higgs bosons H±, and the Goldstone bosons G0, G+. All other contributions, for instance, the self-energy with a lepton loop, are canceled with the corresponding SM contribution.
The W+, Z0, photon, Z0-photon self-energy diagrams are shown in Figures 4, 5, 6, and 7, respectively.
Feynman diagram contribution to the self-energy ΠWWNMSSM(s).
Feynman diagram contribution to the self-energy ΠZZNMSSM(s).
Feynman diagram contribution to the self-energy ΠγγNMSSM(s).
Feynman diagram contribution to the self-energy ΠZγNMSSM(s).
Acknowledgments
The authors are grateful to B. Fuks for quick response concerning an update of the NMSSM model file contained in FeynRules. The work of Y. Schröder is supported by the Heisenberg Program of the Deutsche Forschungsgemeinschaft (DFG), Contract no. SCHR 993/1.
PeskinM. E.TakeuchiT.New constraint on a strongly interacting Higgs sectorKennedyD. C.LangackerP.Precision electroweak experiments and heavy physics: a global analysisPeskinM. E.TakeuchiT.Estimation of oblique electroweak correctionsMaksymykI.BurgessC. P.LondonD.Beyond S, T and UBurgessC. P.GodfreyS.KönigH.LondonD.MaksymykI.A global fit to extended oblique parametersBeringerJ.ArguinJ. F.BarnettR. M.Review of Particle Physics (RPP)ManiatisM.The next-to-minimal supersymmetric extension of the standard model reviewedEllwangerU.HugonieC.TeixeiraA. M.The next-to-minimal supersymmetric standard modelTeyssierD.A flavour-independent search for a hadronically decaying neutral Higgs bosons at LEP with the L3 detectorCaoJ.YangJ. M.Anomaly of Zb antib coupling revisited in MSSM and NMSSMDegrassiG.SlavichP.On the radiative corrections to the neutral Higgs boson masses in the NMSSMDomingoF.LenzT.W mass and Leptonic Z-decays in the NMSSMGunionJ. F.JiangY.KramlS.The constrained NMSSM and Higgs near 125 GeVDjouadiA.DreesM.EllwangerU.Benchmark scenarios for the NMSSMEllwangerU.HugonieC.NMSPEC: a Fortran code for the sparticle and Higgs masses in the NMSSM with GUT scale boundary conditionsKingS. F.MühlleitnerM. M.NevzorovR.NMSSM Higgs benchmarks near 125 GeVAbdusSalamS. S.AllanachB. C.DreinerH. K.Benchmark models, planes, lines and points for future SUSY searches at the LHCDedesA.LahanasA. B.TamvakisK.Effective weak mixing angle in the MSSMHeinemeyerS.HollikW.WeigleinG.Electroweak precision observables in the minimal supersymmetric standard modelRamsey-MusolfM. J.SuS.Low energy precision test of supersymmetryChoG.-C.HagiwaraK.MatsumotoY.NomuraD.The MSSM confronts the precision electroweak data and the muon g-2ChristensenN. D.DuhrC.Feyn-rules—Feynman rules made easyDuhrC.FuksB.A superspace module for the Feyn-Rules packageAllanachB. C.BalazsC.BelangerG.SUSY Les Houches Accord 2HahnT.Generating Feynman diagrams and amplitudes with FeynArts 3HahnT.Perez-VictoriaM.Automatized one loop calculations in four-dimensions and DdimensionsC-code for the oblique parametershttp://www.physik.uni-bielefeld.de/theory/e6/BI-TP-2012-16.html