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Advances in High Energy Physics
VolumeΒ 2012Β (2012), Article IDΒ 625389, 18 pages
http://dx.doi.org/10.1155/2012/625389
Research Article

Higgs Bosons Near 125 GeV in the NMSSM with Constraints at the GUT Scale

1LPT, UMR 8627 CNRS, UniversitΓ© de Paris-Sud, 91405 Orsay, France
2LUPM, UMR 5299 CNRS, UniversitΓ© de Montpellier II, 34095 Montpellier, France

Received 21 May 2012; Accepted 3 July 2012

Academic Editor: Jun-jieΒ Cao

Copyright Β© 2012 Ulrich Ellwanger and Cyril Hugonie. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the NMSSM with universal Susy breaking terms (besides the Higgs sector) at the GUT scale. Within this constrained parameter space, it is not difficult to find a Higgs boson with a mass of about 125 GeV and an enhanced cross-section in the diphoton channel. An additional lighter Higgs boson with reduced couplings and a mass ≲ 123 GeV is potentially observable at the LHC. The NMSSM-specific Yukawa couplings πœ† and πœ… are relatively large and tan 𝛽 is small, such that πœ†, πœ…, and the top Yukawa coupling are of π’ͺ(1) at the GUT scale. The lightest stop can be as light as 105 GeV, and the fine-tuning is modest. WMAP constraints can be satisfied by a dominantly Higgsino-like LSP with substantial bino, wino, and singlino admixtures and a mass of ∼60–90 GeV, which would potentially be detectable by XENON100.

1. Introduction

Recently, the ATLAS [1–3] and CMS [4–6] collaborations have presented evidence for a Higgs boson with a mass near 126 GeV (ATLAS) and 125 GeV (CMS in [6]). Interestingly, the best fit to the signal strength πœŽπ›Ύπ›Ύβ‰‘πœŽprod(𝐻)×𝐡𝑅(𝐻→𝛾𝛾) in the 𝛾𝛾 search channel is about one standard deviation larger than expected in the Standard Model (SM), πœŽπ›Ύπ›Ύobs/πœŽπ›Ύπ›ΎSM∼2 for ATLAS [1, 2], and πœŽπ›Ύπ›Ύobs/πœŽπ›Ύπ›ΎSM∼1.6 for CMS [6].

Since then, several publications have studied the impact of a Higgs boson in the 125 GeV range on the parameter space of supersymmetric (Susy) extensions of the SM [7–38]. Whereas a Higgs boson in the 125 GeV range is possible within the parameter space of the Minimal Susy SM (MSSM) [7–13, 15, 17–21, 25, 29–37], large radiative corrections involving heavy stops are required, which aggravate the β€œlittle fine-tuning problem” of the MSSM. In addition, it would be difficult to explain a large enhancement of the diphoton signal strength in the MSSM [15, 34, 37].

Within the Next-to-Minimal Supersymmetric SM (NMSSM [39, 40]), a Higgs boson in the 125 GeV range is much more natural [7, 16, 21, 26, 27, 34]. Additional tree-level contributions and large singlet-doublet mixings in the CP-even Higgs sector can push up the mass of the mostly SM-like Higgs boson and, simultaneously, reduce its coupling to 𝑏-quarks which results in a substantial enhancement of its branching fraction into two photons [7, 16, 26, 34]. Studies of the parameter space of the general NMSSMβ€”including the dark matter relic density and dark matter nucleon cross sectionβ€”were performed in [34, 38].

An important question is whether these interesting features of the NMSSM survive universality constraints on the soft Susy breaking parameters at the GUT scale. Since this approach imposes severe restrictions on the sparticle masses and couplings, it allows to study whether these would be consistent with present constraints from direct and indirect sparticle searches, the dark matter relic density and dark matter direct detection experiments. Moreover, it allows to make predictions for future searches, both in the sparticle and the Higgs sector.

The (fully constrained) CNMSSM [41, 42] was analysed in [11] with the result that once a relic density in agreement with WMAP [43] is imposed, the Higgs boson mass can barely be above 123 GeV. We find that one should allow for deviations from full universality in the Higgs sector, both for the NMSSM-specific soft Susy breaking terms and the MSSM-like Higgs soft masses like in the MSSM studies in [8, 11, 17, 20]. In analogy to the NUHM version of the MSSM, we will refer to such a model as NUH-NMSSM for nonuniversal Higgs NMSSM. A first study of the NUH-NMSSM was made in [23] which was confined, however, to the more MSSM-like region of the parameter space of the NMSSM involving small values of the NMSSM-specific coupling πœ† and hence small NMSSM-specific effects in the CP-even Higgs sector.

In the present paper we study the NUH-NMSSM for large values of the NMSSM-specific coupling πœ† (and low tan𝛽), where the singlet-doublet mixing in the CP-even Higgs sector is large, and we find that the interesting features of the Higgs sector of the NMSSM observed in [7, 16, 21, 26, 27, 34] can remain present, including constraints from searches for squarks and gluinos from ATLAS and CMS [44–46], constraints on the dark matter relic density from WMAP [43] and on the dark matter nucleon cross section from XENON100 [47].

Our results in the Higgs sector originate essentially from the strong mixing between all three CP-even Higgs states in the NMSSM: first, a Higgs boson with a mass in the 125 GeV range can have an enhanced diphoton signal strength up to πœŽπ›Ύπ›Ύobs/πœŽπ›Ύπ›ΎSM∼2.8. Second, a lighter less SM-like Higgs boson 𝐻1 exists, with small couplings to electroweak gauge bosons if 𝑀𝐻1≲114 GeV (complying with LEP constraints [48]), but a possibly detectable production cross section at the LHC if 𝑀𝐻1≳114 GeV. In fact, a strongly enhanced diphoton signal strength ≳2 of the Higgs boson 𝐻2 with its mass in the 125 GeV range is possible only if 𝑀𝐻1≳90 GeV. The heaviest CP-even Higgs boson 𝐻3, like the heaviest MSSM-like CP-odd and charged Higgs bosons, has masses in the 250–650 GeV range, while the lightest mostly singlet-like CP-odd Higgs state has a mass in the 160–400 GeV range. These comply with constraints both from 𝐡-physics and direct Susy Higgs searches also for lower masses due to the low values of tan𝛽 considered here and the large singlet component of the lightest CP-odd state.

In the sparticle sector we require masses for the gluino and the first generation squarks to comply with constraints from present direct searches [44–46], but we also study the effect of a reduced sensitivity due to the more complicated decay cascades in the NMSSM [49]. The lightest stop ̃𝑑1 can be as light as ~105 GeV (still satisfying constraints from ATLAS [50] and the Tevatron [51], the latter due to its dominant decay into a chargino and a 𝑏-quark), and the required fine-tuning among the parameters at the GUT scale remains modest. In the neutralino sector, the mixings among the five states (bino, wino, two Higgsinos, and the singlino) are large. The LSP, with a dominant Higgsino component and a mass of 60–90 GeV, has a relic density complying with the WMAP constraints [43] and a direct detection cross-section possibly within the reach of XENON100 [47]. However, the supersymmetric contribution to the anomalous magnetic moment of the muon is somewhat smaller than desired to account for the deviation of the measurement [52] from the SM.

In the next section we present the analysed parameter space of the NMSSM with boundary conditions at the GUT scale and the imposed phenomenological constraints; our results are given in Section 3 and conclusions in Section 4.

2. The NMSSM with Constraints at the GUT Scale

The NMSSM differs from the MSSM due to the presence of the gauge singlet superfield 𝑆. In the simplest 𝑍3 invariant realisation of the NMSSM, the Higgs mass term πœ‡π»π‘’π»π‘‘ in the superpotential π‘ŠMSSM of the MSSM is replaced by the coupling πœ† of 𝑆 to 𝐻𝑒 and 𝐻𝑑 and a self-coupling πœ…π‘†3. Hence, in this simplest version the superpotential π‘ŠNMSSM is scale invariant and given by π‘ŠNMSSM𝑆𝐻=πœ†π‘’β‹…ξπ»π‘‘+πœ…3𝑆3+β‹―,(2.1) where hatted letters denote superfields, and the ellipsis denote the MSSM-like Yukawa couplings of 𝐻𝑒 and 𝐻𝑑 to the quark and lepton superfields. Once the real scalar component of 𝑆 develops a vev 𝑠, the first term in π‘ŠNMSSM generates an effective πœ‡-term, πœ‡eff=πœ†π‘ .(2.2)

The soft Susy breaking terms consist of mass terms for the Higgs bosons 𝐻𝑒, 𝐻𝑑, 𝑆, squarks Μƒπ‘žπ‘–β‰‘(̃𝑒𝑖𝐿,𝑑𝑖𝐿), ̃𝑒𝑖𝑐𝑅, 𝑑𝑖𝑐𝑅, and sleptons ℓ𝑖≑(Μƒπœˆπ‘–πΏ,̃𝑒𝑖𝐿) and ̃𝑒𝑖𝑐𝑅 (where 𝑖=1,…,3 is a generation index): βˆ’β„’0=π‘š2𝐻𝑒||𝐻𝑒||2+π‘š2𝐻𝑑||𝐻𝑑||2+π‘š2𝑆||𝑆||2+π‘š2Μƒπ‘žπ‘–||Μƒπ‘žπ‘–||2+π‘š2̃𝑒𝑖||̃𝑒𝑖𝑐𝑅||2+π‘š2𝑑𝑖||𝑑𝑖𝑐𝑅||2+π‘š2ℓ𝑖||ℓ𝑖||2+π‘š2̃𝑒𝑖||̃𝑒𝑖𝑐𝑅||2,(2.3) trilinear interactions involving the third generation squarks, sleptons, and the Higgs fields (neglecting the Yukawa couplings of the two first generations): βˆ’β„’3=ξ‚€β„Žπ‘‘π΄π‘‘π‘„β‹…π»π‘’Μƒπ‘’3𝑐𝑅+β„Žπ‘π΄π‘π»π‘‘ξ‚π‘‘β‹…π‘„3𝑐𝑅+β„Žπœπ΄πœπ»π‘‘β‹…πΏΜƒπ‘’3𝑐𝑅+πœ†π΄πœ†π»π‘’β‹…π»π‘‘1𝑆+3πœ…π΄πœ…π‘†3+h.c.,(2.4) and mass terms for the gauginos 𝐡 (bino), ξ‚‹π‘Šπ‘Ž (winos), and ξ‚πΊπ‘Ž (gluinos): βˆ’β„’1/2=12𝑀1𝐡𝐡+𝑀23ξ“π‘Ž=1ξ‚‹π‘Šπ‘Žξ‚‹π‘Šπ‘Ž+𝑀38ξ“π‘Ž=1ξ‚πΊπ‘Žξ‚πΊπ‘Žξƒ­+h.c.(2.5)

Expressions for the mass matrices of the physical CP-even and CP-odd Higgs statesβ€”after 𝐻𝑒, 𝐻𝑑, and 𝑆 have assumed vevs 𝑣𝑒, 𝑣𝑑, and 𝑠 and including the dominant radiative correctionsβ€”can be found in [40] and will not be repeated here. The couplings of the CP-even Higgs states depend on their decompositions into the weak eigenstates 𝐻𝑑, 𝐻𝑒, and 𝑆, which are denoted by 𝐻1=𝑆1,𝑑𝐻𝑑+𝑆1,𝑒𝐻𝑒+𝑆1,𝑠𝐻𝑆,2=𝑆2,𝑑𝐻𝑑+𝑆2,𝑒𝐻𝑒+𝑆2,𝑠𝐻𝑆,3=𝑆3,𝑑𝐻𝑑+𝑆3,𝑒𝐻𝑒+𝑆3,𝑠𝑆.(2.6)

Then the reduced tree-level couplings (relative to a SM-like Higgs boson) of 𝐻𝑖 to 𝑏 quarks, 𝜏 leptons, 𝑑 quarks, and electroweak gauge bosons 𝑉 are 𝑔𝐻𝑖𝑏𝑏𝑔𝐻SM𝑏𝑏=π‘”π»π‘–πœπœπ‘”π»SM𝜏𝜏=𝑆𝑖,𝑑,𝑔cos𝛽𝐻𝑖𝑑𝑑𝑔𝐻SM𝑑𝑑=𝑆𝑖,𝑒,sin𝛽𝑔𝑖≑𝑔𝐻𝑖𝑉𝑉𝑔𝐻SM𝑉𝑉=cos𝛽𝑆𝑖,𝑑+sin𝛽𝑆𝑖,𝑒.(2.7)

Mixings between the SU(2)-doublet and singlet sectors are always proportional to πœ†.

As compared to two independent parameters in the Higgs sector of the MSSM at tree level (often chosen as tan𝛽 and 𝑀𝐴), the Higgs sector of the NMSSM is described by the six parameters, πœ†,πœ…,π΄πœ†,π΄πœ…,tan𝛽≑𝑣𝑒/𝑣𝑑,πœ‡eff.(2.8)

Then the soft Susy breaking mass terms for the Higgs bosons π‘š2𝐻𝑒, π‘š2𝐻𝑑, and π‘š2𝑆 are determined implicitly by 𝑀𝑍, tan𝛽, and πœ‡eff.

In constrained versions of the NMSSM (as in the constrained MSSM) one assumes that the soft Susy breaking terms involving gauginos, squarks, or sleptons are identical at the GUT scale: 𝑀1=𝑀2=𝑀3≑𝑀1/2,π‘š2Μƒπ‘žπ‘–=π‘š2̃𝑒𝑖=π‘š2𝑑𝑖=π‘š2ℓ𝑖=π‘š2Μƒπ‘’π‘–β‰‘π‘š20,𝐴𝑑=𝐴𝑏=π΄πœβ‰‘π΄0.(2.9)

In the NUH-NMSSM considered here one allows the Higgs sector to play a special role: the Higgs soft mass terms π‘š2𝐻𝑒, π‘š2𝐻𝑑, and π‘š2𝑆 are allowed to differ from π‘š20 (and determined implicitly as noted above), and the trilinear couplings π΄πœ†, π΄πœ… can differ from 𝐴0. Hence, the complete parameter space is characterized by πœ†,πœ…,tan𝛽,πœ‡eff,π΄πœ†,π΄πœ…,𝐴0,𝑀1/2,π‘š0,(2.10) where the latter five parameters are taken at the GUT scale.

Subsequently we are interested in regions of the parameter space implying large doublet-singlet mixing in the Higgs sector, that is, large values of πœ† (and πœ…) and low values of tan𝛽, which lead naturally to a SM-like Higgs boson 𝐻2 in the 125 GeV range [7, 16, 21, 26, 27, 34]. Requiring 124GeV<𝑀𝐻2<127GeV and πœŽπ›Ύπ›Ύobs(𝐻2)/πœŽπ›Ύπ›ΎSM>1, we find 0.41<πœ†<0.69,0.21<πœ…<0.46,1.7<tan𝛽<6,(2.11) with many points for tan𝛽≲2.5. It is intriguing that with these choices at the weak scale, one obtains πœ†βˆΌπœ…βˆΌβ„Žπ‘‘βˆΌπ’ͺ(1) for the running couplings at the GUT scale; hence, all 3 Yukawa couplings are close to (but still below) a Landau singularity.

We assume πœ‡eff>0. π΄πœ…β‰²0 at the weak scale is required for positive CP-odd Higgs masses squared. We found that, as a consequence, the coupled RG equations imply essentially 𝐴0,π΄πœ†,π΄πœ…<0 at the GUT scale. Constraints on the soft Susy breaking parameters depend strongly on the sparticle decay cascades. Using the absence of signal at the LHC in the jets and missing transverse momentum search channels, bounds in the π‘š0, 𝑀1/2 plane have been derived in the CMSSM with tan𝛽=10 [44–46]. In the NUH-NMSSM, however, we find lighter stops (due to the lower values of tan𝛽 implying a larger value of the top Yukawa coupling, which affects the RGE running of the soft Susy breaking stop masses) and a modified neutralino sector which reduces the sensitivity in these search channels [49].

Hence, to start with, we impose only constraints from sparticle searches at LEP [53] and the Tevatron [54, 55], and from stop searches at the Tevatron [51] and the LHC [50]. These imply π‘š0≳140GeV,𝑀1/2π‘šβ‰³270GeV,Μƒπ‘žβ‰³580GeV,𝑀̃𝑔≳640GeV.(2.12)

In addition we require that the fine-tuning Ξ” defined in (3.4) satisfies Ξ”<120, which implies upper bounds which will be discussed below. However, it is possible that the stronger CMSSM-like constraints in the π‘š0,𝑀1/2 plane [46] also apply to the NUH-NMSSM considered here. These stronger constraints further reduce the allowed points in the parameter space approximately to π‘šΜƒπ‘žβ‰³1250GeV,𝑀̃𝑔≳850GeV,(2.13) but we implemented the constraints from [46] in the π‘š0, 𝑀1/2 plane exactly. These constraints are used for the points shown in the Figures 1–3 below, but the difference between the constraints (2.12) and (2.13) has practically no impact on our results in the Higgs sector. Remarkably, regardless of the constraints in the π‘š0, 𝑀1/2 plane, the lightest stop mass can be as low as ~105 GeV.

fig1
Figure 1: Reduced signal cross sections 𝑅2 for 𝐻2 with a mass in the 124–127 GeV range, as a function of 𝑀𝐻1 for a representative sample of viable points in parameter space. (a) 𝑅2𝛾𝛾(𝑔𝑔) (diphoton channel, 𝐻2 production via gluon fusion), (b) 𝑅2𝛾𝛾 (VBF) (diphoton channel, 𝐻2 production via VBF), (c) 𝑅2𝑉𝑉 (𝑍𝑍, π‘Šπ‘Š channels, 𝐻2 production via gluon fusion), and (d) 𝑅2𝜏𝜏 (VBF) (𝜏𝜏 channel, 𝐻2 production via VBF).
fig2
Figure 2: (a) 𝑅2𝑉𝑉(𝑔𝑔) as function of 𝑅2𝛾𝛾(𝑔𝑔). (b) 𝑅𝑏𝑏2 (𝑉𝐻) (associate production π‘Š/𝑍+𝐻2 with 𝐻2→𝑏𝑏) as a function of 𝑅2𝛾𝛾(𝑔𝑔).
fig3
Figure 3: Reduced signal cross sections 𝑅1 as a function of 𝑀𝐻1. (a) 𝑅1𝛾𝛾(𝑔𝑔) (diphoton channel, 𝐻1 production via gluon fusion), (b) 𝑅1𝛾𝛾 (VBF) (diphoton channel, 𝐻1 production via VBF), (c) 𝑅1𝑉𝑉(𝑔𝑔) (𝑍𝑍 and π‘Šπ‘Š channels, 𝐻1 production via gluon fusion), and (d) 𝑅1𝜏𝜏 (VBF) (𝜏𝜏 channel, 𝐻1 production via VBF).

Together with these bounds on π‘š0, 𝑀1/2, the above constraints on the Higgs sector and the LEP bound on the chargino mass lead to 105<πœ‡eff<205GeVforweakconstraints(2.12),105<πœ‡eff<160GeVforstrongconstraints(2.13).(2.14)

Appropriate combinations of the bino/wino/higgsino/singlino components of πœ’01 are required in order to satisfy the WMAP and XENON100 bounds. This leads to constraints on πœ‡eff—relevant for the higgsino componentsβ€”as function of 𝑀̃𝑔, which is proportional to the bino/wino mass terms.

We have scanned the parameter space of the NUH-NMSSM given in (2.10) using a Markov Chain Monte Carlo (MCMC) technique, which yields a very large number of points (~106) satisfying all the phenomenological constraints described below. To do so, we have modified the code NMSPEC [56] inside NMSSMTools [57, 58] in order to allow for πœ… and πœ‡eff to be used as input parameters at the weak scale and to compute the Higgs soft masses π‘š2𝐻𝑒, π‘š2𝐻𝑑 at the GUT scale; a corresponding version 3.2.0 will be made public soon. In NMSPEC, the two-loop renormalization group equations (RGEs) between the weak and GUT scales are integrated numerically for all parameters. In the presence of boundary conditions both at the weak and the GUT scales as it is the case here, these can be satisfied only through an iterative process. This iterative process is not guaranteed to converge, notably for large Yukawa couplings as in (2.11). In fact, the RGE integration algorithm within the latest public version 3.1.0 of NMSSMTools had to be modified in version 3.2.0 to achieve convergence for large Yukawa couplings.

In the Higgs sector we have used two-loop radiative corrections from [59], and for the top quark pole mass we use π‘štop=172.9 GeV. Our results in the next section use the reduced Higgs production rates (normalized with respect to the SM production rates) in various channels. For gluon-gluon fusion we use the reduced Higgs-gluon coupling as computed in NMSSMTools, which takes care of all colored (s)particles in the loop. For the low values of tan𝛽 considered here, the top quark loop dominates by far and leads essentially to 𝑔𝐻𝑖𝑑𝑑/𝑔𝐻SM𝑑𝑑 as given in (2.7). Since a single particle loop dominates, radiative corrections not considered in NMSSMTools tend to cancel in the ratio to the SM. Likewise, Higgs production rates via associate production with π‘Š/𝑍 (≑𝑉) or vector boson fusion (VBF) are simply proportional to the SM rates rescaled by 𝑔2𝑖 defined in (2.7). The Higgs branching fractions are computed in NMSSMTools to the same accuracy both for the NMSSM and a SM-like Higgs boson, such that radiative corrections not considered in NMSSMTools tend again to cancel in the ratio to the SM.

Next we turn to the imposed phenomenological constraints. In the Higgs sector we impose constraints from LEP [48], which still allow for a Higgs mass below 114 GeV if its coupling to the 𝑍 boson is reduced. Constraints on Higgs bosons from the LHC are those implemented in version 3.1.0 of NMSSMTools, which are based on public ATLAS and CMS results available at the end of 2011, including constraints from CMS on heavy MSSM-like Higgs bosons decaying to tau pairs [60]. In version 3.2.0, however, we have updated the important 𝛾𝛾 search channel with the results from ATLAS [2] and CMS [5]. In order to fit the evidence of both experiments in the 𝛾𝛾 channel, we impose 124GeV<𝑀𝐻<127GeV, which is satisfied exclusively by 𝐻2 for larger values of πœ†, and we require a good visibility of 𝐻2 in the 𝛾𝛾 channel, that is, πœŽπ›Ύπ›Ύobs(𝐻2)/πœŽπ›Ύπ›ΎSM>1 in both the gluon fusion and VBF production modes.

Also the constraints from 𝐡-physics are those implemented in version 3.1.0 of NMSSMTools. In spite of charged (resp., CP-odd) Higgs masses as low as ~250 (resp., 160) GeV, these are easily satisfied for low values of tan𝛽 or a large singlet component for the lightest CP-odd state (i.e., the couplings of these Higgs bosons to 𝑏-quarks are hardly enhanced with respect to the SM Higgs).

The dark matter relic density and direct detection cross section of the LSP πœ’01 (the lightest neutralino) are computed with the help of MicrOmegas [61–63] implemented in NMSSMTools. The default constraints 0.094<Ξ©β„Ž2<0.136 are slightly weaker than the most recent ones from WMAP [43], but this has no impact on the viable regions in parameter space (only on the number of points retained). We also apply the bounds from XENON100 [47] on the spin-independent πœ’01-nucleon cross section (roughly πœŽπ‘ π‘–(𝑝)≲10βˆ’8 pb for π‘€πœ’01∼60– 90 GeV).

Since we hardly find light sleptons of the second generation (and again due to the low values of tan𝛽), the Susy contribution Ξ”π‘Žπœ‡ to the anomalous magnetic moment of the muon is below Ξ”π‘Žπœ‡β‰²7β‹…10βˆ’10, violating the constraint implemented in NMSSMTools. However, it still improves the discrepancy between the SM and the measured value [52] and can reduce the discrepancy to two standard deviations depending on the employed SM value.

3. Results

Some remarks on our results have already been made above. Notably we have no difficulties to find points in the parameter space satisfying the above constraints, including the dark matter relic density and the direct detection cross section, 124GeV<𝑀𝐻2<127 GeV and 𝑅2𝛾𝛾>1, where we define 𝑅2π›Ύπ›Ύβ‰‘πœŽπ›Ύπ›Ύobs𝐻2ξ€ΈπœŽπ›Ύπ›ΎSM.(3.1)

The mechanism behind this enhancement has been discussed earlier in [7, 16, 64]: the 𝐡𝑅(𝐻2→𝛾𝛾) is strongly enhanced due to a reduced total width (dominated by Ξ“(𝐻2→𝑏𝑏)) for a small reduced coupling 𝑔𝐻2𝑏𝑏/𝑔𝐻SM𝑏𝑏 in (2.7), that is, a small value of the mixing angle 𝑆2,𝑑, in spite of a milder reduction of the partial width Ξ“(𝐻2→𝛾𝛾). This occurs for large singlet-doublet mixing (which also leads to an increase of 𝑀𝐻2) and requires that the third eigenstate 𝐻3 is not decoupled, that is, not too heavy. The enhancement of the 𝐡𝑅(𝐻2→𝛾𝛾) also overcompensates a milder reduction of the production cross section of 𝐻2 due to singlet-doublet mixing.

The reduction of the total width leads also to a potential increase of the reduced signal rate in the 𝑍𝑍/π‘Šπ‘Š channels (via gluon fusion), 𝑅2π‘‰π‘‰πœŽ(𝑔𝑔)≑𝑍𝑍obs𝑔𝑔→𝐻2ξ€ΈπœŽπ‘π‘SM=𝜎(𝑔𝑔→𝐻)π‘Šπ‘Šobs𝑔𝑔→𝐻2ξ€ΈπœŽπ‘Šπ‘ŠSM,(𝑔𝑔→𝐻)(3.2) in spite of the reduction of the partial widths Ξ“(𝐻2→𝑍𝑍/π‘Šπ‘Š) due to singlet-doublet mixing. A fourth interesting reduced signal cross section is the 𝜏𝜏 channel via VBF: 𝑅2𝜏𝜏𝜎(VBF)β‰‘πœπœobsξ€·π‘Šπ‘Šβ†’π»2ξ€ΈπœŽπœπœSM,(π‘Šπ‘Šβ†’π»)(3.3) which tends to be reduced, however, for a small mixing angle 𝑆2,𝑑.

Singlet-doublet mixing angles are typically large, if the eigenstates are close in mass. Hence, we should expect that 𝑅2𝛾𝛾 is the larger, the closer 𝑀𝐻1 is to 𝑀𝐻2, that is, the heavier is 𝐻1. Subsequently we consider separately 𝑅2𝛾𝛾(𝑔𝑔) (where 𝐻2 is produced via gluon fusion) and 𝑅2𝛾𝛾(VBF) (where 𝐻2 is produced via VBF). In Figure 1 we show 𝑅2𝛾𝛾(𝑔𝑔), 𝑅2𝛾𝛾(VBF), 𝑅2𝑉𝑉(𝑔𝑔), and 𝑅2𝜏𝜏(VBF) as a function of 𝑀𝐻1 for a representative sample of ~2000 points in the scanned parameter space of the semiconstrained NMSSM described above. All points satisfy the WMAP bound on the dark matter relic density, the XENON100 bound on πœŽπ‘ π‘–(𝑝), and the stronger lower bound on 𝑀1/2 given in (2.13). Relaxing this bound to the one given in (2.12) does not lead to additional regions in Figure 1.

We see that, as expected, the ratios 𝑅2𝛾𝛾(𝑔𝑔), 𝑅2𝛾𝛾(VBF), and 𝑅2𝑉𝑉(𝑔𝑔) can increase with 𝑀𝐻1, and 𝑅2𝛾𝛾(𝑔𝑔) can become as large as 2.8 for 𝑀𝐻1≳115 GeV. The inverse conclusion does not hold: 𝑀𝐻1≳115 GeV does not imply 𝑅2𝛾𝛾>2. 𝑅2𝜏𝜏(VBF) is below ~1, and the very small values of 𝑅2𝜏𝜏(VBF) correspond to the highest values of 𝑅2𝛾𝛾(𝑔𝑔). The jump in 𝑅2𝜏𝜏(VBF) near 𝑀𝐻1∼60 GeV is caused by a combination of LEP, 𝐡-physics, and WMAP constraints as well as the condition 𝑅2𝛾𝛾>1. For 𝑀𝐻2 in the range 124–127 GeV, none of these reduced signal rates show a significant dependency on 𝑀𝐻2; corresponding plots would only transcribe the LHC constraints on each rate as a function 𝑀𝐻2, but they would not provide additional informations and are hence omitted.

In Figure 1(c) we see that 𝑅2𝑉𝑉(𝑔𝑔) may be below 1. In fact, the β€œbest fit” to 𝑅2π‘Šπ‘Š(𝑔𝑔) by ATLAS [1] and by CMS [4] is below 1, whereas the β€œbest fit” to 𝑅2𝑍𝑍(𝑔𝑔) is below 1 by CMS [4], but above 1 by ATLAS [1]. We recall that we have 𝑅2π‘Šπ‘Š(𝑔𝑔)≑𝑅2𝑍𝑍(𝑔𝑔)≑𝑅2𝑉𝑉(𝑔𝑔). Hence, it may be interesting to see to which extend 𝑅2𝑉𝑉(𝑔𝑔) and 𝑅2𝛾𝛾(𝑔𝑔) (βˆΌπ‘…2𝛾𝛾(VBF)) are correlated. This correlation is shown in Figure 2(a), and we see that 𝑅2𝑉𝑉(𝑔𝑔)<1 is possible for 𝑅2𝛾𝛾(𝑔𝑔) up to ~1.5.

Recently, excesses compatible with a Higgs boson in the 125 GeV range have also been observed at the Tevatron [65]. Here, the dominant excess originates from associated 𝑉𝐻 production with 𝐻→𝑏𝑏. The corresponding reduced signal rate for the candidate 𝐻2, 𝑅𝑏𝑏2(𝑉𝐻) (which is equal to 𝑅2𝜏𝜏(VBF)) cannot be very small given the observations at the Tevatron. In Figure 2(b) we show 𝑅𝑏𝑏2(𝑉𝐻) against 𝑅2𝛾𝛾(𝑔𝑔). We see that 𝑅𝑏𝑏2(𝑉𝐻)≳0.7 is possible only for 𝑅2𝛾𝛾(𝑔𝑔)≲2, but 𝑅2𝛾𝛾(𝑔𝑔)≳1.6 still allows for 𝑅𝑏𝑏2(𝑉𝐻)≳0.9.

Obviously the reduced signal rates of 𝐻1 are also very important. For instance, 𝐻1 could be compatible with the excess of events observed by CMS for π‘€π»βˆΌ119.5 GeV in the 𝑍𝑍 channel [4]. On the other hand, for π‘€π»βˆΌ95–100 GeV the upper bounds from LEP on its reduced coupling to the 𝑍 boson are particularly weak, and a mostly (but not completely) singlet-like 𝐻1 could explain the mild excess of events observed there [48, 66, 67]. The corresponding reduced signal cross-sections as a function of 𝑀𝐻1 are shown in Figure 3. As explained in [16], the reduced signal cross-section in the 𝑏𝑏 channel at LEP coincides with π‘…πœπœ(VBF).

We see that the reduced signal cross-sections are mostly small for 𝑀𝐻1≲110 GeV where the singlet component of 𝐻1 is large, but 𝑅1𝜏𝜏(VBF) can be as large as ~0.25 for 𝑀𝐻1∼95–100 GeV which is interesting given the mild excess of events observed at LEP. On the other hand, for 𝑀𝐻1≳110 GeV, 𝑅1𝛾𝛾(𝑔𝑔), 𝑅1𝛾𝛾(VBF) and 𝑅1𝑉𝑉(𝑔𝑔) can become as large as ~0.5 and 𝑅1𝜏𝜏(VBF) as large as ~0.9, hence, 𝐻1 is potentially detectable.

The Higgs sector of the NMSSM contains a third CP-even state 𝐻3, two CP-odd states 𝐴1 and 𝐴2, and, as in the MSSM, a charged Higgs boson 𝐻±. We find that the lightest CP-odd state 𝐴1 is mostly singlet-like with a mass in the range 160–400 GeV, and hardly visible at the LHC due to its small production cross-sections. The states 𝐻3, 𝐴2, and 𝐻± all have similar masses in the 250–650 GeV range and would also be difficult to see at the LHC due to the small value of tan𝛽 in the region of the parameter space of interest (2.11).

The masses of the sparticles are essentially determined by 𝑀1/2, π‘š0, 𝐴0, and πœ‡eff. In Figure 4 we show the mass π‘šΜƒπ‘ž of the lightest first generation squark (𝑑𝑅 for our choice of parameters) as well as the mass π‘šΜƒπ‘‘1 of the lightest (mostly right-handed) stop as a function of the gluino mass 𝑀̃𝑔. Here it makes a difference whether we impose the weaker bounds (2.12) or the stronger bounds (2.13): points in red satisfy only the weaker bounds while points in green satisfy both constraints. For 𝑀̃𝑔≳640 GeV, a stop mass as small as 105 GeV is not excluded by present searches at the LHC [50], but could become observable in the near future.

fig4
Figure 4: π‘šΜƒπ‘‘1 (a) and π‘šΜƒπ‘ž (b) as a function of 𝑀̃𝑔. Points in red (darker points) satisfy the weaker bounds (2.12), but not the stronger bounds (2.13), while points in green (brighter) satisfy both constraints.

It is known that the stop mass has an impact on the fine-tuning with respect to the fundamental parameters of Susy extensions of the SM, due to its impact on the running soft Susy breaking Higgs mass terms. In addition, both are affected by the gluino mass. We have estimated the quantitative amount of fine-tuning with respect to the parameters at the GUT scale following the procedure outlined in [68]. There, a fine-tuning measure ΔΔ=MaxGUT𝑖,Ξ”GUT𝑖=||||ξ€·π‘€πœ•lnπ‘ξ€Έξ€·π‘πœ•lnGUT𝑖||||,(3.4) was used, where 𝑝GUT𝑖 are all parameters at the GUT scale (Yukawa couplings and soft Susy breaking terms). Note that sometimes ln(𝑀2𝑍) instead of ln(𝑀𝑍) is used in the definition of Ξ”, leading to an obvious factor of 2. We find that Ξ”, shown as a function of π‘šΜƒπ‘‘1 and 𝑀̃𝑔 in Figure 5, is dominated as usual by 𝑝GUT𝑖=𝑀1/2.

fig5
Figure 5: Fine-tuning as a function of π‘šΜƒπ‘‘1 (a) and 𝑀̃𝑔 (b). The color code is as in Figure 4.

We see that the fine-tuning 1/Ξ” can be as low as π’ͺ(5%) (with the definition in (3.4)) in the range of smaller stop and gluino masses allowed by the weaker lower bounds (2.12), and still as low as π’ͺ(2.5%) in the range of stop and gluino masses allowed by the stronger bounds (2.13). Both values are an order of magnitude better than in the MSSM [69].

Turning to the neutralino sector we observe, as in the CP-even Higgs sector, large mixing angles involving all 5 neutralinos of the NMSSM. The lightest eigenstate πœ’01 (the LSP) is mostly higgsino-like, but with sizeable bino, wino, and singlino components and a mass in the 60–90 GeV range. The lower bound on π‘€πœ’01 follows from the conditions that invisible decays 𝐻2β†’πœ’01πœ’01 do not spoil the condition πœŽπ›Ύπ›Ύobs(𝐻2)/πœŽπ›Ύπ›ΎSM>1, together with the lower LEP bound on chargino masses. Its direct detection cross-section is reduced with respect to pure higgsino-like neutralinos, and can well comply with the constraints from XENON100. In Figure 6 we show the spin-independent neutralino-proton scattering cross-section πœŽπ‘ π‘–(𝑝) as a function of π‘€πœ’01. We see that the stronger bounds (2.13) imply 60GeVβ‰²π‘€πœ’01≲85 GeV, similar to the range within the general NMSSM obtained in [34]. In particular, the plateau observed in Figure 6 for 80β‰²π‘€πœ’01≲90 GeV and πœŽπ‘ π‘–(𝑝)∼10βˆ’7 pb, corresponding to small values of π‘š0,𝑀1/2 and a mostly bino-like πœ’01, is excluded by both the XENON100 and the strong LHC Susy constraints. The spin-independent neutralino-proton scattering cross-section πœŽπ‘ π‘–(𝑝) can vary over a wide range both above and below the XENON100 limit [47] (which remains to be confirmed by other experiments), but plenty of points would satisfy this constraint and become observable in the future.

625389.fig.006
Figure 6: The spin-independent neutralino-proton scattering cross section πœŽπ‘ π‘–(𝑝) as a function of π‘€πœ’01. The blue line indicates the bound from XENON100 [47], and we have added points violating this bound (but respecting all the others). The color code is as in Figure 4.

It may be interesting to know some of the properties of the Higgs and sparticle sectors beyond the ones shown in the scatter plots above; to this end we show two benchmark points in Table 1. The point (1) has 𝑀𝐻1∼100 GeV, the point (2) 𝑀𝐻1∼120 GeV, and they differ in the values for 𝑀1/2 and π‘š0. The branching fractions of ̃𝑑1 are similar for both points: ̃𝑑𝐡𝑅(1β†’πœ’Β±1+𝑏)∼0.7, ̃𝑑𝐡𝑅(1β†’πœ’01+𝑑)∼0.2, ̃𝑑𝐡𝑅(1β†’πœ’02+𝑑)∼0.1.

tab1
Table 1: Properties of two benchmark points corresponding to different values of 𝑀𝐻1. All dimensionful parameters are given in GeV. The components of πœ’01, as well as the components of 𝐻𝑖, are defined such that their squares sum up to 1. The Susy contributions Ξ”π‘Žπœ‡ to the muon anomalous magnetic moment are those given by NMSSMTools without any theoretical errors.

4. Conclusions

It has already been noted in [7, 16, 21, 26, 27, 34] that the NMSSM can naturally accomodate Higgs bosons in the 124–127 GeV mass range. In addition, the NMSSM can explain excesses in the 𝛾𝛾 channel, as well as potential excesses at different values of the Higgs mass (due to the extended Higgs sector). In the present paper we have shown that these features persist in the constrained NMSSM with nonuniversal Higgs sector, designated here as NUH-NMSSM. The dominant deviation from full universality of all soft Susy breaking terms at the GUT scale originates from the need to have π‘šπ»π‘’>π‘š0.

The following properties of the Higgs sector are peculiar:(i)the signal rate in the 𝛾𝛾 channel can be 2.8 as large as the one of a SM-like Higgs boson, provided the mass of the lighter CP-even state 𝐻1 is in the 115–123 GeV range;(ii)requiring a visible signal rate in the 𝑏𝑏 channel of 0.9 times the SM value allows for a signal rate in the 𝛾𝛾 channel about 1.6 as large as the one of a SM-like Higgs boson;(iii)the lighter CP-even state 𝐻1 could explain a mild excess of events around 95–100 GeV observed at LEP, or a second visible Higgs boson below ~123 GeV.

In the sparticle sector, the assumption of universality at the GUT scale leads to the following features:(i)the mass of the lightest stop can be as small as 105 GeV, complying with present constraints for 𝑀̃𝑔≳640 GeV;(ii)the fine-tuning with respect to parameters at the GUT scale remains modest, an order of magnitude below the one required in the MSSM;(iii)the eigenstates in the neutralino sector are strongly mixed, and the lightest neutralino can have a relic density in agreement with WMAP constraints. Its direct detection cross-section can be above or below present XENON100 bounds; most of the points below these bounds should be observable in the near future.

Given the large values of the NMSSM-specific coupling πœ†, all scenarios presented here differ strongly from the MSSM (also by the low value of tan𝛽). The fact that all 3 Yukawa couplings πœ†, πœ…, and β„Žπ‘‘ are of π’ͺ(1) at the GUT scale may hint at some strong dynamics present at that scale. It is possible that the deviation from full universality of soft Susy breaking terms at the GUT scale remains confined to π‘šπ»π‘’>π‘š0; such possibilities require further studies.

Of course, first of all the present evidence for a Higgs boson in the 124–127 GeV mass range should be confirmed by more data; then possible evidence for non-SM properties of the Higgs sector like an enhanced cross-section in the diphoton channel will show whether the scenarios presented here are realistic.

Acknowledgments

U. Ellwanger acknowledges support from the French ANR LFV-CPV-LHC. C. Hugonie acknowledges support from the French ANRJ TPADMS.

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