# Neutral Higgs production at proton colliders in the CP-conserving NMSSM

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## Abstract

We discuss neutral Higgs boson production through gluon fusion and bottom-quark annihilation in the CP-conserving \(\mathbb {Z}_3\)-invariant Next-to-Minimal Supersymmetric Standard Model at proton colliders. For gluon fusion we adapt well-known asymptotic expansions in supersymmetric particles for the inclusion of next-to-leading order contributions of squarks and gluinos from the Minimal Supersymmetric Standard Model (MSSM) and include electroweak corrections involving light quarks. Together with the resummation of higher-order sbottom contributions in the bottom-quark Yukawa coupling for both production processes we thus present accurate cross section predictions implemented in a new release of the code SusHi. We elaborate on the new features of an additional SU\((2)_L\) singlet in the production of CP-even and -odd Higgs bosons with respect to the MSSM and include a short discussion of theoretical uncertainties.

## 1 Introduction

After the discovery of a scalar boson at the Large Hadron Collider (LHC) [1, 2] in 2012 an essential task of particle physicists is to reveal the nature of the Higgs-like state and thus the nature of electroweak symmetry breaking. Apart from deviations from the Standard Model (SM) prediction of the properties of the found Higgs-like state, further work includes the search for additional less and/or more massive scalar bosons, which can nicely be accommodated in supersymmetric models. The Next-to-Minimal Supersymmetric Standard Model (NMSSM) extends the Minimal Supersymmetric Standard Model (MSSM) by an SU\((2)_L\) singlet and allows the dynamical generation of the \(\mu \)-term through electroweak symmetry breaking [3, 4]. The latter singlet–doublet mixing term in the superpotential lifts the MSSM tree-level upper bound of the Higgs mass given by the \(Z\)-boson mass. Thus, the NMSSM can easily accommodate the SM-like Higgs boson with a mass close to \(125\) GeV. Whereas for the calculation of the NMSSM Higgs spectrum and branching ratios various spectrum generators are available, including higher orders in perturbation theory (see Sect. 3), the calculation of neutral Higgs production cross sections did not exceed the leading order (LO) in quantum chromodynamics involving squarks and gluinos (SQCD) [5] and did not include electroweak corrections—apart from private implementations in e.g. HIGLU [6].

It is therefore timely to present the missing ingredients and a code for the calculation of accurate neutral Higgs production cross sections in the NMSSM, where the five neutral Higgs bosons are predominantly generated through gluon fusion and bottom-quark annihilation at a proton collider. For this purpose we extend the code SusHi [7]. For the time being we restrict our implementation to the real NMSSM without additional CP violation, such that CP-even \(H_1,H_2\), and \(H_3\) and CP-odd Higgs bosons \(A_1\) and \(A_2\) can be distinguished in the Higgs sector. Most recent efforts related to Higgs physics at the LHC are summarized in the reports of the LHC Higgs cross section working group [8, 9, 10]. The SM Higgs is mainly produced through gluon fusion, where the Higgs–gluon coupling is mediated through virtual top and bottom quarks [11]. Higher-order QCD corrections at next-to-leading order (NLO) are of large importance [12, 13, 14]. In the effective theory of a heavy top quark the inclusive cross section is known to next-to-next-to-leading order (NNLO) in QCD [15, 16, 17], in addition finite top-quark mass effects at NNLO were calculated [18, 19, 20, 21, 22]. Beyond NNLO QCD effects are accessible through resummation [23, 24, 25, 26, 27, 28] and electroweak corrections are known [29, 30, 31]. Meanwhile next-to-NNLO (NNNLO) QCD contributions were estimated in the so-called threshold expansion [32, 33, 34, 35], but they are not further considered in this publication.

The SM results for Higgs production through gluon fusion can be adjusted to the MSSM and the NMSSM through a proper reweighting of the Higgs couplings to the quarks. However, the gluon fusion process can also be mediated through their superpartners, the squarks. With respect to the MSSM the only generically new ingredient, which goes beyond the projection of the physical Higgs bosons onto the neutral components of the two Higgs doublets, are couplings of the NMSSM singlet to squarks, since no couplings of the singlet to quarks or gauge bosons are present in the tree-level Lagrangian. It is therefore of importance to include squark contributions to gluon fusion at the highest order possible, even though they decrease in size with increasing squark masses. For the pseudoscalars \(A_i\) squark contributions to gluon fusion are only induced at NLO, which motivates to go beyond just LO squark contributions for all Higgs bosons. For this purpose we adapt the works of Refs. [36, 37, 38] for the MSSM to present NLO SQCD contributions for the NMSSM, which are based on an expansion in terms of heavy supersymmetric particles taking into account terms up to \(\mathcal {O}(m_{\phi }^2/M^2\)), \(\mathcal {O}(m_t^2/M^2\)), \(\mathcal {O}(m_b^2/M^2\)) and \(\mathcal {O}(m_Z^2/M^2\)), where \(m_\phi \) denotes the Higgs mass and \(M\) a generic SUSY mass. In contrast to the MSSM we are at present only working in this expansion of inverse SUSY masses and do not include an expansion in the so-called VHML, the vanishing Higgs mass limit (\(m_\phi \rightarrow 0\)) for the SQCD contributions, as implemented in evalcsusy [39, 40, 41] or discussed in Ref. [42]. In the latter limit higher-order stop-induced contributions up to NNLO level are known [43, 44, 45] and were partially included in previous discussions of precise MSSM neutral Higgs production cross sections [46]. Although for a pure CP-odd singlet component NNLO stop-induced contributions are the first non-vanishing contributions to the gluon fusion cross section, we leave an inclusion of these to future work. For completeness, we add that in the MSSM a numerical evaluation of NLO squark/quark/gluino contributions was also reported in Refs. [47, 48], whereas Refs. [49, 50, 51] presented analytic results for the pure squark-induced NLO contributions. Electroweak contributions to the gluon-fusion production process mediated through light quarks [30, 31] can be adjusted from the SM to the MSSM [7] and similarly to the NMSSM and are known to capture the dominant fraction of electroweak contributions for a light SM-like Higgs with a mass below the top-quark mass, whereas they are generically small for larger Higgs masses.

For large values of \(\tan \beta \), the ratio of the vacuum expectation values of the neutral components of the two Higgs doublets, the bottom-quark Yukawa coupling is enhanced, such that the bottom sector gets more important for gluon fusion, and the associated production with a pair of bottom quarks \(pp\rightarrow b\bar{b}\phi \) is significantly enhanced. SusHi includes bottom-quark annihilation \(b\bar{b}\rightarrow \phi \), which in the case of non-tagged final state \(b\)-quarks is a good theoretical approach, since it resums logarithms through the \(b\)-parton distribution functions. The latter process is known as five-flavor scheme (5FS) up to NNLO QCD [52, 53] and can easily be reweighted from the SM to the MSSM/NMSSM by effective couplings [54, 55]. In the NMSSM the singlet does not couple to the quarks at LO, however, taking into account the singlet-induced component into the resummation of higher-order sbottom effects is mandatory, since also the singlet to sbottom couplings are enhanced by \(\tan \beta \).

The new release of SusHi thus provides gluon-fusion cross sections at NLO QCD taking into account the third generation quarks and their superpartners, the squarks, for all the five neutral Higgs bosons of the NMSSM. The squark and squark/quark/gluino contributions are implemented in asymptotic expansions of heavy SUSY masses. Electroweak corrections induced by light quarks through the couplings of the Higgs bosons to \(Z\) and \(W^\pm \) bosons can be added consistently like in the MSSM. Similarly, the NNLO top-quark induced contributions are included. In addition, sbottom contributions can be resummed into an effective bottom-quark Yukawa coupling, also taking into account the additional singlet to sbottom couplings. The latter also applies to the calculated bottom-quark annihilation cross section at NNLO QCD. All features SusHi provides for the MSSM are available for the NMSSM as well, in particular distributions with respect to the (pseudo)rapidity and transverse momentum of the Higgs boson under consideration can be obtained. Left for future work is a link to MoRe-SusHi [56] to allow for the calculation of momentum resummed transverse momentum distributions.

We proceed as follows: We start with a discussion of the theory background in Sect. 2, where we elaborate on the NMSSM Higgs sector and the calculation of the gluon-fusion cross section. Then we present the NLO virtual amplitude for gluon fusion as well as the calculation of bottom-quark annihilation including the resummation of sbottom-induced contributions to the bottom-quark Yukawa coupling in the NMSSM. Subsequently we comment on the implementation in SusHi in Sect. 3, before we investigate the phenomenological features of the singlet-like Higgs boson in the CP-even and CP-odd sector with regard to Higgs production in Sect. 4. We also include a short discussion of theoretical uncertainties. Finally, we conclude and present the Higgs–squark–squark couplings in Appendix A.

## 2 Theory background

In this section we discuss the Higgs sector of the CP-conserving \(\mathbb {Z}_3\)-invariant NMSSM, before we proceed to the resummation of \(\tan \beta \) enhanced sbottom contributions in the bottom-quark Yukawa coupling. Subsequently we move to the discussion of the Higgs production cross section in gluon fusion, where we present the adapted formulas for the NLO SQCD virtual amplitude, and finally comment on the consequences of the additional singlet to bottom-quark annihilation.

### 2.1 The Higgs sector of the CP-conserving NMSSM

### 2.2 Resummation of higher-order sbottom contributions

### 2.3 Gluon-fusion cross section

### 2.4 NLO virtual amplitude for gluon fusion

#### 2.4.1 CP-even Higgs bosons

^{1}

#### 2.4.2 CP-odd Higgs bosons

### 2.5 Bottom-quark annihilation cross section in the 5FS

The generalization of the calculation of bottom-quark annihilation cross sections in the five-flavor scheme (5FS) from the MSSM to the NMSSM case is straightforward by using the appropriate couplings of the Higgs bosons to the bottom quarks. For this purpose the resummation of the sbottom contributions as described in Sect. 2.2 is taken into account. For the specific case of the singlet-like Higgs boson we point out that in the case that the coupling to the bottom quark vanishes (due to cancellations in the mixing with the Higgs doublets) a priori the coupling to sbottom squarks can still be present. This is not taken into account by the resummation procedure.

## 3 Implementation in SusHi

In the current implementation of neutral Higgs production in the real NMSSM within the code SusHi the Higgs mixing matrices as well as the Higgs masses have to be provided as input in SUSY Les Houches Accord (SLHA) form [69, 70] and can be obtained by spectrum generators for the NMSSM. Common codes are NMSSMTools [71, 72, 73, 74], NMSSMCALC [57, 65, 75, 76, 77], SOFTSUSY [78, 79], SPheno+Sarah [80, 81, 82] and FlexibleSUSY+Sarah [80, 83].

Two options for the pseudoscalar Higgs mixing matrix are accepted as input by SusHi, namely the full Higgs mixing matrix, which corresponds to the multiplication \(\mathcal {R}^P\mathcal {R}^G\) in the above notation, but instead also the rotation matrix \(\mathcal {R}^P\) can be used as input. Following SLHA2 [70] the full matrix \((\mathcal {R}^P\mathcal {R}^G)_{ij}\) is provided in Block NMAMIX and asks for entries \(ij\) with \(i\in \lbrace 2,3\rbrace \) and \(j\in \lbrace 1,2,3\rbrace \). The matrix \(\mathcal {R}^P_{ij}\) can be specified in Block NMAMIXR, which only asks for entries \(ij\) with \(\lbrace i,j\rbrace \in \lbrace 2,3\rbrace \). We point out that in contrast to other codes the Goldstone boson remains the first mass eigenstate, such that Block NMAMIXR does not ask for entries with \(i=1\) or \(j=1\). The elements of the CP-even Higgs boson mixing matrix are specified in Block NMHMIX [70]. The Higgs masses need to be given in Block MASS using entries \(25,35\) and \(45\) for the CP-even Higgs bosons and \(36\), \(46\) for the CP-odd Higgs bosons.

The block Block EXTPAR still contains the gluino mass as well as the soft-breaking parameters for the third generation squark sector. Entry \(23\) for the \(\mu \) parameter is, however, replaced by entry \(65\), where the effective value of \(\mu \) needs to be specified. Moreover, entry \(61\) asks for the choice of \(\lambda \). SusHi extracts the VEV \(v_s\) from \(\mu \) and \(\lambda \). Since the Higgs sectors including their mixing are provided, there is no need to provide the parameters \(\kappa \), \(A_\kappa \), \(A_\lambda \) (or \(m_{H^\pm }\)) in the SusHi input, since they do not enter the couplings relevant for Higgs production. The Block SUSHI entry \(2\) specifies the Higgs boson, for which cross sections are requested. The CP-even Higgs bosons are numbered \(11,12\), and \(13\), the CP-odd Higgs bosons \(21\) and \(22\). Similarly the options \(11,12\), and \(21\) also work in the 2-Higgs-Doublet Model (2HDM) and the MSSM and \(11\) and \(21\) in the SM. A CP-odd Higgs boson \(21\) in the SM is obtained from the 2HDM case with \(\tan \beta =1\). We note that SusHi is still compatible with input files with \(0\) (light Higgs), \(1\) (pseudoscalar) and \(2\) (heavy Higgs) as options for entry \(2\). Output files, however, stick to the new convention.

For the time being we emphasize that SusHi is not strictly suitable for very low values of Higgs masses \(m_\phi <20\) GeV, where quark threshold effects start to become relevant and also electroweak corrections are not implemented. This statement mostly applies to studies of a very light CP-odd Higgs boson, which is poorly constrained by LEP experiments in contrast to a light CP-even Higgs boson [87].

## 4 Phenomenological study

In this section we elaborate on the phenomenological consequences of the additional SU\((2)_L\) singlet in the NMSSM with respect to the MSSM for neutral Higgs production. Neglecting the squark-induced contributions to gluon fusion, the only consequence of the additional singlet component is another admixture of the three CP-even/two CP-odd Higgs bosons. However, no generically new contributions to Higgs boson production arise. This differs when taking into account squark-induced contributions to gluon fusion due to the additional singlet to squark couplings. In particular for the CP-odd Higgs bosons squark contributions are only induced at the two-loop level due to the non-diagonal structure of the CP-odd Higgs bosons to squark couplings. Subsequently we work with two scenarios, start with their definition, present the Higgs boson masses and admixtures, and then discuss the behavior of cross sections, including the squark and electroweak corrections to the gluon-fusion cross section. Our studies are performed for a proton–proton collider with a center-of-mass (cms) energy of \(\sqrt{s}=13\) TeV, as planned for the second run of the LHC. Lastly we add a short discussion of renormalization and factorization scale uncertainties as well as PDF \(+\alpha _s\) uncertainties for one of the two scenarios.

### 4.1 Scenarios \(S_1\) and \(S_2\)

To present the most relevant features of the NMSSM for what concerns neutral Higgs production we pick two scenarios. The first scenario \(S_1\) is in the vicinity of the natural NMSSM [5] with a rather large value of \(\lambda =0.62\). Other input parameters are \(M_1=150\) GeV, \(M_2=340\) GeV, \(M_3=1.5\) TeV, \(\tan \beta =2\), \(A_\kappa =-20\) GeV and \(\mu =200\) GeV. \(A_\lambda \) is determined from the charged Higgs mass \(m_{H^\pm }=400\) GeV. The size of \(\lambda \) ensures a large mixing of the singlet component with the \(H_d^0\) and \(H_u^0\) doublets. All soft-breaking masses are set to \(1.5\) TeV except for the soft-breaking masses of the third generation squark sector, which are fixed to \(750\) GeV. The soft-breaking couplings are set to \(A=1.8\) TeV. The on-shell stop masses are then given by \(m_{\tilde{t}_1}=544.7\) GeV and \(m_{\tilde{t}_2}=941.2\) GeV, whereas the sbottom masses are \(m_{\tilde{b}_1}=749.4\) GeV and \(m_{\tilde{b}_2}=757.4\) GeV. We vary \(\kappa \) between \(0.15\) and \(0.80\) and thus vary the mass of the singlet-like Higgs component in particular in the CP-even Higgs sector. We note that for illustrative reasons the perturbativity limit approximately given by \(\sqrt{\lambda ^2+\kappa ^2}<0.7\) is not always fulfilled in our study. We work out the characteristics for the singlet-like component in the following discussion. The relevant input for SusHi is obtained with NMSSMCALC 1.03, which incorporates the leading two-loop corrections \(\mathcal {O}(\alpha _s\alpha _t)\) to the Higgs boson masses calculated in the gaugeless limit with vanishing external momentum [77]. We request NMSSMCALC to work with an on-shell renormalized stop sector and add local modifications to the NMSSMCALC input routines to read in on-shell parameters rather than \(\overline{\text {DR}}\) renormalized parameters.^{2} These modifications guarantee identical on-shell stop masses in NMSSMCALC and SusHi. The renormalization of the sbottom sector on the other hand is performed SusHi-internally.

We also choose a second scenario \(S_2\), in which we vary \(\lambda \) to decouple the singlet-like Higgs from the Higgs doublets. The detailed choice of parameters is \(M_1=150\) GeV, \(M_2=300\) GeV, \(M_3=1.5\) TeV, \(\tan \beta =10\), \(A=-2.0\) TeV, \(\kappa =0.2\), \(A_\kappa =-30\) GeV, \(\mu =130\) GeV, and \(m_{H^\pm }=350\) GeV. In this scenario we set the soft-breaking masses to \(1.0\) TeV. The on-shell stop and sbottom masses are given by \(m_{\tilde{t}_1}=824.1\) GeV, \(m_{\tilde{t}_2}=1173.4\) GeV, \(m_{\tilde{b}_1}=998.0\) GeV and \(m_{\tilde{b}_2}=1008.4\) GeV. We vary \(\lambda \) between \(0.04\) and \(0.25\). For small values of \(\lambda \) \(H_1\) corresponds to the SM-like Higgs boson with mass \(m_{H_1} \sim 121\) GeV. The lower bound at \(\lambda =0.04\) is to avoid tiny cross sections for a heavy singlet-like Higgs boson and to keep its mass below the SUSY masses thresholds to justify the NLO SQCD expansion employed for the gluon-fusion cross section calculation.

Both our scenarios come along with rather light third generation squark masses at the low TeV scale. Contrary to the Higgs mass calculations the squark contributions completely decouple from Higgs production for heavy SUSY spectra. Our scenarios are chosen to flash the phenomenology of an additional singlet-like Higgs boson and thus do not always include a SM-like Higgs boson with mass \(\sim 125\) GeV and are partially under tension from LEP searches [87] (for low CP-even Higgs masses below \(110\) GeV) or LHC searches [88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102].

We add for both scenarios the relevant SM input, which includes the \(\overline{\text {MS}}\) renormalized bottom-quark mass \(m_b(m_b)=4.20\) GeV, which is translated into a bottom-quark pole mass of \(m_b=4.92\) GeV. In SusHi we choose the renormalization scheme, where the bottom-quark pole mass enters all occurrences of heavy bottom-quark masses in the loops and the bottom-quark Yukawa coupling for the gluon-fusion cross section. Bottom-quark annihilation is based on the running \(\overline{\text {MS}}\) renormalized bottom-quark Yukawa coupling. As pointed out in Ref. [46] the gluon densities are hardly dependent on the bottom-quark pole mass fit value of the PDF fitting groups, emphasizing that there is no need to adjust the bottom-quark pole mass to the PDF fit value for the calculation of the gluon fusion cross section. The top-quark pole mass equals \(m_t=173.3\) GeV. The strong coupling constant \(\alpha _s(m_Z)\) is set to \(0.1172\) for the calculation of running masses, and is obtained from the corresponding PDF set for the cross section calculation. We choose MSTW2008 [103] at the appropriate order in perturbation theory. Our central scale choices for gluon fusion are \(m_\phi /2\) for both renormalization and factorization scale, \(\mu _{\scriptscriptstyle R}^0\) and \(\mu _{\scriptscriptstyle F}^0\), respectively, and \(\mu _{\scriptscriptstyle R}^0=m_\phi \) and \(\mu _{\scriptscriptstyle F}^0=m_\phi /4\) for bottom-quark annihilation.

### 4.2 Higgs boson masses and singlet admixtures

Figure 1c and d show the behavior of the singlet admixture and the masses for the two CP-odd Higgs bosons in scenario \(S_1\) as a function of \(\kappa \). We point again to the region in the vicinity of \(\kappa \sim 0.35\), where the light CP-odd Higgs boson \(A_1\) is a pure singlet-like CP-odd Higgs boson contrary to the CP-even Higgs boson \(H_2\), for which \(H_d^R\) and \(H_u^R\) components remain. The coupling of \(A_1\) to the quarks vanishes, but the coupling to the squarks is still present due to the relatively large value of \(\lambda =0.62\), which will be apparent when calculating the gluon-fusion cross section.

### 4.3 Scenario \(S_1\): inclusive cross sections for \(\sqrt{s}=13\) TeV

We show the effect of squark and electroweak contributions to gluon fusion for the three CP-even Higgs bosons in Fig. 3c and d. \(\sigma _{gg}^{q+\tilde{q}}\) in Fig. 3c includes stop- and sbottom-quark induced contributions at NLO SQCD on top of the quark-induced contributions without electroweak contributions and compares to the pure quark-induced cross section \(\sigma _{gg}^q\) without electroweak contributions. All cross sections include NLO QCD quark contributions and the NNLO QCD top-quark induced contributions in the heavy top-quark effective theory. Figure 3d accordingly shows the effect of electroweak contributions induced by light quarks following Eq. (15) in combination with Ref. [46] in comparison to the quark and squark induced cross section \(\sigma _{gg}^{{QCD}{}}=\sigma _{gg}^{q+\tilde{q}}\). Note that in all our figures \(\sigma _{gg}\) corresponds to \(\sigma _{gg}^{{QCD}+\mathrm {EW}}\). As expected for \(H_2\), the region with small quark contributions induced by the admixture with the \(H_d^R\) and \(H_u^R\) components is in particular sensitive to squark corrections. For the other Higgs bosons the squarks corrections in this scenario are incidentally all of the order of \(\mathcal {O}(-10\) %) and mostly independent of \(\kappa \). We note that the squark corrections are mainly induced by stop contributions, whereas sbottom-induced contributions only account for a small fraction. Interestingly, the squark contributions show an interference-like structure with a maximum and minimum around \(\kappa \sim 0.35\), whereas the relative electroweak corrections are always positive. This can be understood from a sign change in the real part of the quark-induced LO and NLO amplitude for \(H_2\) at \(\kappa \sim 0.35\), which is of relevance for the squark contributions, whereas the imaginary part, more relevant for the electroweak contributions, does not change its sign. The size of the electroweak corrections for \(H_2\) follows from a suppression of the couplings of the second lightest Higgs \(H_2\) to the quarks in contrast to the couplings to gauge bosons. Obtaining a pure singlet-like Higgs boson in the CP-even Higgs sector, which neither couples to quarks nor to gauge bosons, rarely happens due to the mixing between both \(S^R\) and \(H_d^R\) as well as \(S^R\) and \(H_u^R\) for large values of \(\lambda \). For the SM-like Higgs boson with a mass below the top-quark mass the electroweak corrections by light quarks are typically of the order of \(\mathcal {O}(+5\) %) and cover most of the SM-electroweak correction factor. On the other hand, for Higgs masses above the thresholds \(m_\phi \gg 2m_W\) or \(2m_Z\) the electroweak corrections by light quarks are small. The structure visible for \(H_2\) in Fig. 3c for \(\kappa <0.3\) is induced by the thresholds \(2m_W\) and \(2m_Z\), which the Higgs mass \(m_{H_2}\) crosses between \(\kappa =0.1\) and \(0.3\). We leave the distortion of distributions, in particular transverse momentum distributions, for such a scenario to future studies.

### 4.4 Scenario \(S_2\): inclusive cross sections for \(\sqrt{s}=13\) TeV

We point out that the singlet-like Higgs bosons \(H_3\) and \(A_2\) both approach the SUSY mass thresholds with decreasing \(\lambda \). Therefore, we employ the lower bound of \(\lambda =0.04\), since for lower values of \(\lambda \) and thus larger masses of \(H_3\) and \(A_2\) we cannot guarantee the validity of the NLO SQCD contributions implemented in SusHi. Reference [46] therefore assigned an additional theoretical uncertainty to the heavy SUSY masses expansion. In the decoupling regime we checked that the cross sections for \(H_1\), \(H_2\), and \(A_1\) coincide with the MSSM cross sections obtained for a mixing angle of \(\alpha =-0.12347\) with an accuracy of \(\sim \) \(10^{-4}\), which resembles the remaining singlet fraction of \(H_1\), \(H_2\), and \(A_1\).

### 4.5 Theory uncertainties

In this section we briefly focus on theoretical uncertainties in the calculation of neutral Higgs boson production cross sections. Reference [46] identified the most important theory uncertainties for the MSSM, which mostly apply to our discussion of the NMSSM as well. Apart from the well-known renormalization and factorization scale and PDF \(+\alpha _s\) uncertainties for cross sections at a proton–proton collider an additional uncertainty for gluon-fusion cross section is the choice of a renormalization scheme for the bottom-quark Yukawa coupling, which is of particular relevance if the bottom-quark loop dominantly contributes. Secondly, the fact that NLO SQCD contributions are taken into account in an expansion of heavy SUSY masses induces an uncertainty, which grows for larger Higgs masses approaching SUSY particle masses thresholds. Thirdly, also relevant for bottom-quark annihilation, there are missing contributions in the resummation \(\Delta _b\), which induce an uncertainty, in particular in the limit \(\Delta _b\rightarrow -1\). All of the above theoretical uncertainties as discussed in Ref. [46] apply to the NMSSM in a similar way. In contrast to the MSSM, however, phenomenological studies of the NMSSM focus on lower values of \(\tan \beta \), where both the uncertainty from the choice of the bottom-quark Yukawa coupling and the uncertainty induced from unknown contributions to \(\Delta _b\) are of less importance. A detailed discussion in particular for the singlet-like CP-even and CP-odd Higgs boson is left for future work.

For bottom-quark annihilation as shown in Fig. 7b and d the scale uncertainty is mainly dependent on the Higgs mass, rather than the specific SUSY scenario. The large uncertainty for low Higgs masses reflects the need to move toward the four-flavor scheme (4FS) [104, 105] in the description of the process.

## 5 Conclusions

We presented accurate predictions for neutral Higgs boson production at proton colliders through gluon fusion and bottom-quark annihilation in the CP-conserving NMSSM. For gluon fusion we adapt the full NLO QCD and SQCD results from the MSSM to the NMSSM, based on an asymptotic expansion in heavy SUSY masses for squark and squark/quark/gluino two-loop contributions. Top-quark induced NNLO QCD contributions are added in the heavy top-quark effective theory. Electroweak corrections to gluon fusion mediated through light quarks are taken into account and the resummation of sbottom contributions for large values of \(\tan \beta \) can be translated from the MSSM to the NMSSM. The latter procedure also applies to bottom-quark annihilation.

Our discussion comes along with an implementation of the neutral Higgs boson production cross section calculation in the code SusHi. The Higgs sector (obtained by an NMSSM spectrum generator) needs to be supplied through the SusHi input file. We briefly focused on the new features of the additional singlet-like CP-even or CP-odd Higgs boson for what concerns neutral Higgs boson production. Due to possible cancellations of quark-induced contributions, squark and electroweak corrections to gluon fusion can be of greater relevance than known in the MSSM, in particular for not too heavy third generation squark mass spectra. For a small singlet–doublet mixing term, which can be achieved by lowering the parameter \(\lambda \), the singlet-like CP-even and -odd Higgs bosons can both be decoupled from the remaining MSSM-like Higgs sector. The renormalization and factorization scale uncertainties reflect the individual contributions to neutral Higgs boson production in the case of gluon fusion, whereas scale uncertainties for bottom-quark annihilation as well as PDF \(+\alpha _s\) uncertainties for both production processes mainly remain a function of the Higgs boson mass.

We leave a more detailed investigation of theoretical uncertainties to future work. Moreover, interesting for future studies is an expansion in a light Higgs boson mass rather than heavy SUSY masses for what concerns the inclusion of NLO and NNLO SQCD contributions, in particular since for pure singlet-like CP-odd Higgs bosons NNLO stop-induced contributions are the first non-vanishing contributions to gluon fusion. Similarly a discussion of distributions and of the necessity of resummation for transverse momentum distributions is timely for the real NMSSM, but left for future work.

## Footnotes

- 1.
In the CP-even sector we adapt the MSSM results of Refs. [36, 38] to the NMSSM by isolating the terms proportional to the \(H_d^R/H_u^R\)-squark–squark couplings and replacing them by the \(S^R\)-squark–squark couplings for the form factor \(\mathcal {S}\). Similarly we proceed in the CP-odd sector starting from the form factors of Ref. [37] taking into account the prerotation of the CP-odd Higgs mixing matrix.

- 2.
We thank Kathrin Walz for instructions on how to modify the NMSSMCALC input routines.

## Notes

### Acknowledgments

The author thanks Jonathan Gaunt, Robert Harlander, Hendrik Mantler and Pietro Slavich for very helpful comments on the manuscript. The author is, moreover, indebted to Pietro Slavich for help in the translation of the MSSM NLO SQCD corrections to the NMSSM and to Robert Harlander and Hendrik Mantler for their comments on the implementation of the NMSSM in SusHi. The author also thanks Kathrin Walz for help related to the renormalization of the stop and sbottom sectors within NMSSMCALC and Peter Drechsel for help with regard to the calculation of NMSSM Higgs boson masses. The author acknowledges support by Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Center SFB 676 “Particles, Strings and the Early Universe”.

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