The electroweak sector of the NMSSM at the one-loop level

We present the electroweak spectrum for the Next-to-Minimal Supersymmetric Standard Model at the one-loop level, e.g.\ the masses of Higgs bosons, sleptons, charginos and neutralinos. For the numerical evaluation we present a mSUGRA variant with non-universal Higgs mass parameters squared and we compare our results with existing ones in the literature. Moreover, we briefly discuss the implications of our results for the calculation of the relic density.


Introduction
Supersymmetric extensions of the standard model (SM) are promising candidates for new physics at the TeV scale [1,2,3] as the solve several short-comings of the Standard Model (SM). The Minimal Supersymmetric Standard Model (MSSM) solves the hierarchy problem of the SM [4], leads to a unification of the gauge couplings [5] and introduces several candidates for dark matter depending on how SUSY is broken [6,7]. On the other hand, a new problem arises in the MSSM: the superpotential contains a parameter with mass dimension, namely the so called µ parameter which gives mass to the Higgs bosons and higgsinos. From a purely theoretical point of view, the value of this parameter is expected to be either of the order of the GUT/Planck scale or exactly zero, if it is protected by a symmetry. For phenomenological aspects, however, it is necessary that it is of the order of the scale of electroweak symmetry breaking (EWSB) and it has to be non-zero to be consistent with experimental data. This discrepancy is the so called µ-problem of the MSSM [8].
The Next-to-Minimal Supersymmetric Standard Model (NMSSM) [9] provides an elegant solution to this problem. The particle content of the MSSM is extended by an additional gauge singlet, which receives a vacuum expectation value when supersymmetry is broken. The corresponding term in the superpotential gives then rise to an effective µ-term which is naturally of the order of the EWSB scale. Also in this model several regions exist in parameter space where one obtains the correct relic density to explained the observed dark matter [10,11]. It turns out that in high scale models like mSUGRA several regions exist which are rather sensitive to mass differences of the various supersymmetric particles, in particular the masses of the Higgs bosons, the neutralinos and the staus, the supersymmetric partners of the tau-lepton, and require precise calculations of these masses. The corresponding regions are the so-called Higgs funnel(s) and the co-annihilation regions. Motivated by this observation we calculate the Higgs masses, the neutralino masses and the staus at the one-loop level.
This paper is organized as follows. We first detail our calculation of the mass spectrum in sec. 2. In sec. 3 we present the constrained NMSSM, which serves as our reference scenario and perform a numerical analysis of our implementation. sec. 4 is devoted to a comparison of our results with the public program package NMSSM-Tools [12]. Finally, we give a few examples for the calculation of the dark matter relic density in sec. 5 and draw our conclusions in sec. 6. We collect the couplings and one-loop self-energies in the appendix where we include for completeness also those for the Z-boson and neutral Higgs bosons which have already been given in ref. [13].

Calculation of the One-Loop Mass Spectrum
In this section we fix our notation and discuss briefly the DR renormalization of the relevant masses, where we follow closely ref. [14].

Superpotential and soft SUSY breaking terms of the NMSSM
As already stated above, the solution to the µ-problem of the MSSM is the replacement of the bilinear µ-term by a coupling between the Higgs superfields and an additional gauge singletŜ leading to the superpotential where the last term is introduced to forbid a Peccei-Quinn symmetry which would lead to an axion in contradiction to experimental results, see e.g. ref. [15] and refs. therein. Moreover, we have only taken into account dimensionless couplings to avoid the µ-problem of the MSSM. The scalar component S ofŜ receives after SUSY breaking a vacuum expectation value (VEV), denoted v s , which leads to where we have used the decomposition Since v s and thus also µ eff are a consequence of SUSY breaking one finds that µ eff is naturally of the order of the SUSY breaking scale. All interactions are fixed by the gauge structure and the above superpotential. We have used the Mathemtica package SARAH [16] to calculate all vertices, mass matrices including the one-loop corrections and renormalization group equations of the model.
In the following, we use the standard conventions, where for a matter superfield X,X denotes its scalar component and X denotes its fermionic component. In case of the Higgs fields and the gauge singlet, H u,d /S are the scalar components, whilẽ H u,d /S are the fermionic higgsinos and the singlino.
At tree level the scalar potential receives contributions from several sources: from the the superpotential in eq. (2.1) the so-called F -terms given by The sum runs over all chiral superfieldsφ i , which are then replaced by their scalar component φ j . The D-terms are and finally the soft breaking terms The sum in eq. (2.5) runs over all gauge groups g and over the corresponding generators a, i.e. 1 2 λ a in the case of SU (3), 1 2 σ a in the case of SU (2), and 3 5 Y 2 for the U (1). Here, λ a are the Gell-Mann matrices, σ a the Pauli-matrices, and Y is the hypercharge.

Minimum Conditions of the Vacuum
Once electroweak symmetry gets broken, both Higgs doublets receive a VEV and we decompose the scalars similar to eq. (2.3) At tree level, the minimum conditions for the vacuum are the so-called tadpole equations 10) Here we have chosen a phase convention where all VEVs are real. For the later calculation of the one-loop corrections to the Higgs boson masses one needs the evaluation of the tadpole equations at the one-loop level, leading to corrections δt (1) i . As renormalization condition we demand All calculations are performed in 't Hooft gauge using the diagrammatic approach. The explicit formulas for δt (1) i are given in app. D. In our subsequent analysis we will solve eqs. (2.13) for the soft SUSY breaking masses squared: m 2 H d , m 2 Hu , and m 2 S . All parameters in eqs. (2.10)-(2.12) are understood as running parameters at a given renormalization scale Q. Note that the VEVs v d and v u are obtained from the running mass m Z (Q) of the Z-boson, which is related to the pole mass m Z through 14) The transverse self-energy Π T ZZ is given in app. E.1. Details on the calculation of the running gauge couplings at Q = m Z can be found in ref. [14]. The ratio of these VEVs is denoted as in the MSSM by tan β = v u /v d .

Masses of the Higgs bosons
The tree-level mass matrices for the neutral scalar Higgs bosons and pseudo scalar Higgs bosons can be calculated from the scalar potential according to while those of the pseudo-scalar ones are Hu and m 2 S satisfy the tadpole equations. The diagonalization of the mass matrices m 2,h T and m 2,A 0 T leads in total to five physical mass eigenstates and one neutral Goldstone boson which becomes the longitudinal component of the Z-boson. The five physical degrees of freedom are: three CP-even Higgs bosons denoted h 1,2,3 and two CP-odd bosons denoted A 0 1,2 . The corresponding rotation matrices Z h und Z A 0 are defined through Moreover, we note that we order all masses such, that m i ≤ m j if i < j. The one-loop scalar Higgs masses are then calculated by taking the real part of the poles of the corresponding propagator matrices Here,m h is the tree-level mass matrix from eq. (2.15). Equation (2.29) has to be solved for each eigenvalue p 2 = m 2 i . The same procedure is also applied for the pseudo scalar Higgs bosons. The complete 1-loop expressions for the self energy of the CP-odd and even Higgs bosons are given in apps. E. 2   where the self-energy can be found in app. E.4

Chargino and neutralino masses
As for the Higgs bosons discussed in the previous section, one has to find the real parts of the poles of the propagator matrix to obtain the masses of charginos and neutralinos. At the tree-level the chargino mass matrix in the basisψ − = (W − ,H − d ) T , ψ + = (W + ,H + u ) is given by This mass matrix is diagonalized by a biunitary transformation such that U * Mχ + T V † is diagonal. The matrices U and V are obtained by diagonalizing Mχ respectively. At the one-loop level, one has to add the self-energies In case of the neutralinos one has a complex symmetric 5 × 5 mass matrix which in the basisψ 0 = (B,W 0 ,H 0 d ,H 0 u ,S) T is at the tree-level given by (2.37) One can show that for real parameters the matrix Mχ 0 T can be diagonalized directly by a 5 × 5 mixing matrix N such that N * Mχ 0 T N † is diagonal. In the complex case, one has to diagonalize Mχ The complete self-energies for neutralinos and charginos are given in apps. E.5 and E.6, respectively.

Masses of sleptons
In the basis (ẽ L ,μ L ,τ L ,ẽ R ,μ R ,τ R ), the mass matrix of the charged sleptons at the tree-level is given by with the diagonal entries Where 1 3 is the 3 × 3 unit matrix. This matrix is diagonalized by a unitary mixing matrix Z E : The corresponding mass matrix at the one-loop level is again obtained by taking into account the self-energy according to (2.43) and the one-loop masses are obtained by calculating the real parts of the poles of the propagator matrix. The expression for Πll(p 2 i ) can be found in app. E.7. Finally, in the basis (ν e ,ν µ ,ν τ ) the tree-level sneutrino mass matrix is given by This matrix is diagonalized by a unitary mixing matrix Z E : Similarly as above the one loop mass matrix is given by The one-loop masses are obtained by calculating the real parts of the poles of the propagator matrix. The expression for Πνν(p 2 i ) can be found in app. E.8.

The model and its free parameters
In the subsequent numerical analysis, we are mainly interested in precision calculation of the SUSY masses and potential effects in the calculation of the relic density. To reduce the number of free parameters we therefore focus on a scenario motivated by minimal supergravity (mSUGRA) [17]. More precisely, we study a variant of the constrained NMSSM [18,19] where we allow for non-universal Higgs mass parameters squared at the GUT scale. In our setup, these parameters are determined with the help of the tadpole equations (2.13) at the electroweak scale. As a side remark, we note that also other recently used mSUGRA versions of the NMSSM contained non-minimal features either for the scalar mass parameter and/or for the trilinear couplings.
We apply the following boundary conditions for the gaugino masses M 1 , M 2 , M 3 and the soft breaking masses of the squarks and sleptons m 2 i at the GUT scale, which is defined as the scale where the U Y (1) and SU (2) L couplings fulfill 5 3 g 1 = g 2 : The trilinear scalar couplings T i are given by Here, A 0 is defined at the GUT scale, while λ, κ, A λ and A κ can be defined either at the GUT or at the SUSY scale. Together with the values for tan β = vu v d and v s the spectrum is fixed. To summarize, we have nine input parameters, M 1/2 , m 0 , A 0 , λ, κ, A λ , A κ , v s , and tan β. Note, that we allow for non-universalities in the trilinear parameters for an easier comparison with the existing literature but in principal we could take all Aparameters equal at the GUT scale. We choose in the following v s > 0 and λ, κ ∈ [−1, 1].

Procedure to evaluate the SUSY parameters at the electroweak scale
In order to connect the parameters at various scales, we use the renormalization group equations (RGEs), which are calculated at the two-loop level in the most general form using the generic formulas given in ref. [20]. We have compared the obtained expressions for the RGEs with those given in ref. [15] in the limit where only the third generation Yukawa couplings contributes. There has been a slight difference in the two-loop β-function of A λ = T λ /λ, but it was confirmed by the authors of ref. [15] that our result is correct. The RGEs themselves can easily be calculated by the CalcRGEs command of SARAH and a print-out can be found at [21].
In the calculation of the gauge and Yukawa couplings we follow closely the procedure described in ref. [22]: the values for the Yukawa couplings giving mass to the SM fermions and the gauge couplings are determined at the scale M Z based on the measured values for the quark, lepton and vector boson masses as well as for the gauge couplings. Here, we have included the one-loop corrections to the mass of W-and Zboson as well as the SUSY contributions to δ V B for calculating the gauge couplings. Similarly, we have included the complete one-loop corrections to the self-energies of SM fermions extending the formulas of [14] to include the additional neutralino and Higgs bosons. Moreover, we have resummed the tan β enhanced terms for the calculation of the Yukawa couplings of the b-quark and the τ -lepton as in [22]. The vacuum expectation values v d and v u are calculated with respect to the given value of tan β at M Z . Furthermore, we solve the tadpole equations to get initial values for m 2 H d , m 2 Hu and m 2 S . Afterwards the RGEs are used to obtain the values at the GUT scale and all boundary conditions including λ and κ are set as described above. Then, an RGE running to the SUSY scale is performed and the SUSY masses are calculated at the one-loop level and for the neutral and pseudo scalar Higgs bosons we include beside the one-loop contributions presented here also the known two-loop contributions [13]. For this purpose also the numerical the values for the VEVs at M SUSY are needed. These are derived using the two-loop RGEs are the anomalous dimensions for the two Higgsdoublets at the one-and two-loop level, respectively. The corresponding expressions are given in app. B. Let us recall that the input value for v s is already given at M SUSY . These steps are iterated until the masses converge with a relative precision of 10 −5 . The complete procedure has been implemented in SPheno [22] 1 .

An example spectrum
In Table 1 we give as an example the masses of the Higgs bosons, chargino, neutralinos and third generation sfermions at tree-level as well as at the one-and two-loop level for the parameter set which is close to the benchmark scenario 1 of ref. [18]. As can be seen in Table  1, the corrections are sizable ranging from 0.1 % to 23.6 % in case of the lightest Higgs boson. This large correction is well known and the main reason for including the two-loop corrections. The corresponding two-loop Higgs masses as well as the relative correction with respect to the one-loop results are also displayed in Table 1. Again the largest correction with 5.2 % is in case of the lightest Higgs boson mass.
As an estimate of the remaining theoretical uncertainty we have varied the renormalization scale in SPheno. We show in fig. 1  980 neutral scalar Higgs boson at the one-and two-loop masses normalized to their values at Q = 1 TeV and vary the renormalization scale Q between 200 GeV and 2.2 TeV. As can be seen, the large variation of 8% at one-loop for the lightest Higgs, which is mainly the lighter SU (2) doublet Higgs in this case, is reduced at two-loop to less than 2%. In case of the heavier Higgs bosons the scale dependence is significantly smaller showing a significant improvement when going from the one-loop level to the two-loop level. However, we remark that the values of λ and κ are small in this scenario and we expect a stronger dependence in case of larger couplings.
The picture changes slightly in the case of the pseudo scalar bosons as can be seen in fig. 2. While the heavier pseudo scalar behaves exactly as the second scalar field since both originate to 99.5% from H d , the scale dependence for the lighter pseudo scalar is smaller compared to the lightest scalar field, but hardly improves at the two-loop level. This is because in the two-loop part contain 'only' the strong contributions of the third generation squarks whereas this state is mainly a singlet state and, thus, the contributions due to the Yukawa couplings would be needed for a further improvement. In Figure 3 the scale dependence for different neutralinos is shown. As can be seen, in case of the three lighter states the scale dependence is reduced from the level of about 1.5% to 3-5 per-mill. In case of the singlet stateχ 5 the scale dependence is already small due to the small values of λ and κ. We note that the scale dependence ofχ + 1 (χ + 2 andχ 0 4 ) is nearly the same as that ofχ 0 2 (χ 0 3 ) as these state have their main origin in the same electroweak multiplet.
Finally we show in fig. 4 the scale dependence of the staus. The scale dependence at tree level amounts to about 2-2.5% and is reduced at one-loop level to about 1% and less where theτ 1 shows still the larger dependence. The sleptons of the first two generations show a somewhat smaller scale dependence as in their cases the Yukawa couplings do not play any role.

Comparison with the literature
To date, the program package NMSSM-Tools [12] has been the only complete spectrum calculator for the NMSSM. NMSSM-Tools uses for the constrained NMSSM the parameters m 0 , M 1/2 , A 0 and A κ at the GUT scale whereas tan β and λ are given at

Differences between the programs
Since both programs use different methods to calculate the spectrum, we have also done a comparison where we modified the codes such that both codes use equivalent methods except for small details. First, the implementation of NMSSM-Tools involves two different scales, namely the SUSY scale defined as 1) and the scale at which the masses are calculated, In SPheno, all masses are evaluated at the SUSY scale, so that we had to set Q STSB = Q SUSY in the relevant routines of NMSSM-Tools. Second, as already stated in sec. 3.2, the two-loop β function of A λ has been corrected in the public version of NMSSM-Tools. However, in general the numerical effect on the spectrum is rather small.
In the Higgs sector the loop contributions are taken into account differently in both codes. While SPheno takes the complete one-loop correction including the dependence of the external momenta, NMSSM-Tools uses the effective potential approach, e.g. setting the external momenta to zero. NMSSM-Tools calculates afterwards the momentum dependent contributions from top and bottom quarks. Also the included contributions differ: in SPheno the complete one-loop corrections to both, scalar and pseudo scalar Higgs bosons, and the two-loop contributions as given in ref. [13] are included. In NMSSM-Tools beside the dominant contributions due to third generation sfermions also electroweak corrections and some leading two-loop corrections for the scalars are calculated, while for the pseudo-scalars only the dominant one-loop corrections due to tops, stops, bottoms, and sbottoms are included. In addition, some corrections due to charginos and neutralinos are absorbed in a redefined A λ . To account for these differences we have switched off the two-loop parts in both codes. Furthermore, we have set the external momenta of the loopdiagrams of scalars in SPheno to zero. Finally, we have kept only those corrections to the pseudo-scalar masses in SPheno which are also included in NMSSM-Tools, but neglected the additional corrections absorbed in A λ . In the following, we refer to these modified versions by SPheno mod and NMSSM-Tools mod, respectively.
Also in the chargino and neutralino sector the implementations are different: in SPheno the complete one-loop corrections are implemented whereas in NMSSM-Tools the corrections to the parameters M 1 , M 2 , and µ eff are taken into account. In the slepton sector the differences are largest: SPheno contains the complete one-loop corrections whereas in NMSSM-Tools the calculation is done at tree-level. Last but not least we note that the data transfer has been done using the SLHA2 conventions [23].

Results of the comparison
As a first reference scenario, we take the benchmark point 1 proposed in ref. [18].
The corresponding input parameters for NMSSM-Tools are m 0 = 180 GeV, m 1/2 = 500 GeV, A 0 = −1500 GeV, tan β = 10, In the following we will vary m 0 and the keep the other parameters to the values shown here.
In the left graph of fig. 5, we show the mass of the lightest scalar h 1 as a function of m 0 . The largest discrepancies arise for the lighter scalar and pseudo scalar boson, where the relative differences between the complete calculation of both programs amount up to 2.5 and 35%, respectively. In case of h 0 1 this is a combination of the p 2 terms in the loop-functions and the additional two-loop contributions. The differences in case of A 0 1 can easily be understood by noting that in NMSSM-Tools only the contribution of third-generation sfermions are taken into account whereas we include the complete one-loop corrections plus the known two-loop contributions. In case of the modified program codes these differences reduce to at most 2% which is meanly due to two differences: (i) the way the top Yukawa coupling is calculated and (ii) the way the tadpole equations are solved which leads to somewhat different values between the two programs. There is no visible difference between NMSSM-Tools and NMSSM-Tools mod for the pseudo scalar and the heavy scalars. The reason is that in the case of the pseudo scalar no two-loop corrections are calculated in NMSSM-Tools and in case of the heavy scalars they are very small.
Finally, we have also cross-checked our results in the Higgs sector with ref.
[13] and we have found agreement better than one per-mill when using the set of soft SUSY parameters at the scale Q STSB . This small difference is an effect of the Yukawa and scalar-trilinear couplings of the first two generations which we take also into account. If we restrict ourself to third generation couplings there is an exact agreement between both calculations.
Concerning the chargino and neutralino masses, the agreement between the two spectrum calculators is rather good as can be seen in fig. 6. The relative differences are at most 1% and in general slightly below 0.5%. In case of the sleptons the differences are more pronounced as can be seen in fig. 7 which is due to the differences between tree-level and one-loop calculation and amounts in up to 3% and 0.6% for the light and heavy stau, respectively. Note, that although for LHC physics one expects similar experimental uncertainties, this precision necessary for a future linear collider or dark matter calculation require the inclusion of the radiative corrections to the slepton masses.

Effects on the relic density of dark matter
It is well known that the prediction of the dark matter relic density Ω CDM h 2 is very sensitive to the exact mass configuration of the scenario under consideration [24]. For a neutralino LSP, this is, e.g., the case for the annihilation through Higgs-resonances, but also in the case of neutralino-sfermion co-annihilation. For the latter, the mass difference between the two particles plays a key role in the calculation of the resulting relic density. Therefore, it is necessary to calculate the complete spectrum as precisely as possible to get viable results of allowed regions of parameter space with respect to the constraints imposed by the presence of dark matter. Let us recall that recent measurements by the WMAP satellite in combination with further cosmological data lead to the favored interval at 3σ confidence level [25]. We compute the relic density of the lightest neutralino using the public program package micrOMEGAs 2.4.O [26]. To this end, we have implemented the NMSSM particle content and corresponding interactions into a model file for CalcHEP [27], which is used by micrOMEGAs to evaluate the (co)annihilation cross-section. The relevant interactions have again been calculated and written into the model files by     the program package SARAH. Let us note, that we take into account important QCD effects, such as the running strong coupling constant and the running quark masses [28,29,30].
As an example, we illustrate the effect of the one-loop correction to the slepton masses on the dark matter relic density in a region of dominant neutralino-stau coannihilations. In fig. 8, we show the isolines corresponding to the upper and lower limit of eq. One clearly sees that the allowed parameter range gets shifted depending on the precision with which the spectrum is calculated. More, the two regions shown do not overlap as can also be clearly be seen in the figure to the right.

Conclusion
The NMSSM is an attractive extension of the MSSM, in particular as it solves the µ-problem of the MSSM and as it leads to new phenomenology at present and future collider experiments. It can also explain the observed amount of dark matter in the universe. However, in particular for comparison of the WMAP data improved theoretical predictions are necessary. We therefore have presented in this paper the complete one-loop calculation of the electroweak sector: Higgs bosons, charginos, neutralinos and sleptons. While in case of the Higgs bosons we have reproduced known results, the corrections to the other particles have not yet been discussed in the literature. We have shown that the corrections amount to the order of a few percent. While the corrections are most likely below the precision of the coming LHC data, they are clearly important for comparison with WMAP data and also with a future international linear collider, and thus crucial for precision investigations of the NMSSM parameter space.
The corresponding expressions for the up-type squarks in the basis ũ L ,c L ,t L ,ũ R ,c R ,t R are These matrix are diagonalized by unitary mixing matrices Z Q :

B. Anomalous Dimensions
In this app., we give the detailed expressions of the anomalous dimension of the Higgs-fields, which are needed for the RGE evaluation of the VEVs.

C. Couplings
We list in the following all couplings of the NMSSM which contribute to the electroweak self-energies or influence the annihilation or coannihilation of the neutralino.
These and all other couplings of the NMSSM can be derived with the command MakeVertexList[EWSB] of SARAH. A pdf version is also available at [21]. We define the following abbreviations: Furthermore, c Θ is cos(Θ W ) and s Θ is sin(Θ W ).

C.6 Four Scalar
We define With this definitions often appearing terms in the vertices involving squarks and sleptons are given by

D. One-loop tadpoles
In this and the subsequent Apps., particles that are denoted with a hat, e.g.ĥ i , are the unrotated external states. In the corresponding vertices the associated mixing matrix has to be replaced by the identity matrix. Moreover, we have summed her and in the subsequent section in all the vertices implicitly over the colour indices of quarks and squarks. At the one-loop level, the expressions for the tadpoles of eq. (2.13) are given by

E. One-loop self-energies
The definitions of the scalar one-loop functions and their explicit analytic expressions can be found in ref. [14].