Review of a Light NMSSM Pseudoscalar Higgs-State Production at the LHC

Abstract

In this paper, we briefly review the LHC discovery potential of a light pseudoscalar Higgs boson of the NMSSM, $a_1$, produced in the gluon fusion $gg\to a_1$, bottom-quark fusion $b\overline{b}\to a_1$ and bottom-gluon fusion $bg\to b{a}_1$. We also review the LHC discovery potential of the next-to-lightest CP-even Higgs boson $h_2$ being the non-SM-like Higgs, decaying either into two light CP-odd Higgs bosons $a_1$ or into a light $a_1$ and the Z gauge boson through the gluon fusion $gg\to h_2$ in the 4$\tau$ final state. We find that the light $a_1$ can be detected at the LHC in a variety of production processes including the gluon fusion, bottom-quark fusion and bottom-gluon fusion. The latter two processes require high luminosity of the LHC and large values of tan$\beta$. We also find that the LHC has the potential to discover the non-SM-like Higgs state, $h_2$, decaying into a pair of light CP-odd Higgses $a_1$’s, allowing the distinguishing of the NMSSM Higgs sector from the MSSM one as such a light $a_1$, is impossible in the latter scenario.

1. Introduction

The observation of the neutral Higgs boson with a mass around 125 GeV by the ATLAS and CMS collaborations at CERN [1,2,3,4] in 2012 strongly supports the idea of supersymmetric models. In its minimal realization, the so-called Minimal Supersymmetric Standard Model (MSSM) [5,6], the SM Higgs sector is extended by introducing two Higgs doublets instead of the one in the SM. The two Higgs doublets of the MSSM give rise to five physical Higgs states: two CP-even states, h and H ($m_h<m_H$), one CP-odd state, A, and a pair of charged states $H^{\pm }$ instead of the only one state in the SM. However, the MSSM suffers from two critical phenomenological flaws. The first one is the $\mu$-problem [7] and the second one is a Higgs boson with a mass around 125 GeV in the context of the MSSM requires a substantial degree of fine-tuning [8,9,10,11,12].

The simplest realization of supersymmetry beyond the MSSM that can solve the $\mu$-problem is the Next-to-Minimal Supersymmetric Standard Model (NMSSM) [13,14]. Furthermore, this model can easily accommodate the 125 GeV SM-like Higgs mass without much fine-tuning (see for instance Refs. [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]). In the NMSSM one Higgs singlet superfield is added to the usual Higgs doublet ones of the MSSM. The presence of this singlet superfield leads to two additional neutral Higgs mass eigenstates and one additional neutralino mass state in the NMSSM compared to the MSSM.

The interesting phenomenological aspects of the NMSSM in comparison to the MSSM is that a light pseudoscalar Higgs boson $a_1$ is possible at the NMSSM. The discovery of such a light Higgs state with a mass less than the Z boson mass at the LHC or other colliders would then unmistakably indicate the existence of a non-minimal SUSY Higgs sector. Such a light Higgs state can be produced at the LHC and other facilities either indirectly via decays of heavier Higgs states or directly via the gluon fusion, bottom-gluon fusion and in association with bottom quarks [30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58]. On the other hand, the experimental searches for the $a_1$ signal has been explored by many collaborations. No $a_1$ signal has been observed so far, and the results have been used to set some limits on its mass [59,60,61,62,63,64,65,66,67,68].

The cross sections of Higgs boson production in association with bottom-quark pair at the LHC can be calculated using two schemes, called four-flavor scheme and five-flavor scheme. In the former scheme where the bottom quarks only appear in the final states, they are not treated as partons in the protons but as massive particles. In the latter scheme where the bottom quarks are found in both initial and final states, they are treated as massless particles. Consequently, the bottom quarks are treated as partons in the protons on the same footing as other light quarks: up, down and strange, see e.g., Refs. [69,70].

The Yukawa coupling of the Higgs state to bottom-quark at the LHC can be measured by studying the interaction of bottom quarks with the Higgs field. This interaction can be probed either through the Higgs decay into bottom-antibottom quark pair or through the Higgs production in bottom-antibottom fusion. The former one is overwhelmed by very large backgrounds even though the decay of a light Higgs state to bottom quarks is dominant. Therefore, it would be helpful to study the production of the Higgs boson through bottom-quark fusion to measure the bottom-quark Yukawa coupling at the LHC. This production channel has been studied in detail in the literature [71,72,73,74,75,76,77,78,79,80,81,82,83,84].

In this paper, we will give a short review of some of the production processes that could potentially lead to the discovery of a light pseudoscalar Higgs state of the NMSSM at the LHC. We notice that such a light state can be produced at the LHC in a variety of production mechanisms, including the gluon fusion $gg\to a_1$, bottom-quark fusion $b\overline{b}\to a_1$ and bottom-gluon fusion $bg\to b{a}_1$. We also show that the process $gg\to h_2\to a_1{a}_1$ is an alternative option to produce the light pseudoscalar Higgs boson, in which case our random scan reveals that the non-SM-like Higgs boson $h_2$ mainly decays into a pair of $a_1$’s with a branching ratio ≳ 0.5 as long as it is kinematically allowed. The content is based on our previous work [55,56,57,58]. The layout of the remainder of this article is as follows. In Section 2 we give a short review of the NMSSM, focusing on some important properties of the light pseudoscalar Higgs state $a_1$ inside the model and on the computation of its mass at tree-level. In Section 3, Section 4 and Section 5, we present the production rates of the light $a_1$ at the LHC for various production processes: the gluon fusion $gg\to a_1$, bottom-gluon fusion $bg\to b{a}_1$, and bottom-quark fusion $b\overline{b}\to a_1$, respectively. The LHC discovery potential of the light $a_1$ through the decays of the non-SM-like Higgs state $h_2$ is discussed in Section 6. Finally, Section 7 contains the conclusions.

2. Light Pseudoscalar Higgs State a₁ in the NMSSM

In its simplest form, the scale invariant superpotential of the NMSSM in terms of the usual two MSSM-type Higgs doublets superfields ${\widehat{H}}_u$ and ${\widehat{H}}_d$ together with the extra singlet superfield $\widehat{S}$ takes the form

$$W_{NMSSM}=\mathrm{MSSM}\mathrm{Yukawa}\mathrm{terms}+\lambda \widehat{S}{\widehat{H}}_u{\widehat{H}}_d+\frac{1}{3}\kappa {\widehat{S}}^3,$$

(1)

where both $\lambda$ and $\kappa$ are Yukawa coupling parameters. The term $\lambda \widehat{S}{\widehat{H}}_u{\widehat{H}}_d$ is introduced in the superpotential to solve the $\mu$-problem of the MSSM superpotential, which is the major motivation for the NMSSM. Expanding the singlet field around its vacuum expectation value (VEV) $⟨S⟩=\frac{1}{\sqrt{2}}{\upsilon }_s$ dynamically generates an ‘effective’ $\mu$-parameter ${\mu }_{\mathrm{eff}}=\lambda ⟨S⟩$ of the order of the scale of electroweak symmetry breaking. The last term of Equation (1) is added to break the Peccei-Quinn symmetry [85,86]. The soft breaking terms containing only the Higgs fields read

$$V_{\mathrm{NMSSM}}=m_{H_u}^2|H_u{|}^2+m_{H_d}^2|H_d{|}^2+m_S^2{|S|}^2+\left(\lambda A_{\lambda }S{H}_u{H}_d+\frac{1}{3}\kappa A_{\kappa }{S}^3+\mathrm{h}.\mathrm{c}.\right),$$

(2)

where $A_{\lambda }$ and $A_{\kappa }$ are the trilinear soft SUSY breaking parameters of the order of SUSY breaking scale $m_{\mathrm{SUSY}}$.

Following electroweak symmetry breaking (EWSB), the physical spectrum contains seven states. In the CP conserving case (as assumed here) the NMSSM contains three CP-even states $h_{1,2,3}$ ($m_{h_1}<m_{h_2}<m_{h_3}$), two CP-odd states $a_{1,2}$ ($m_{a_1}<m_{a_2}$) and a pair of charged Higgs states $h^{\pm }$. Due to introducing the Higgs singlet superfield, the NMSSM Higgs sector at the tree-level is described by six parameters: $\kappa$, $A_{\kappa }$, $\lambda$, $A_{\lambda }$, ${\mu }_{\mathrm{eff}}$ and tan$\beta =\frac{{\upsilon }_u}{{\upsilon }_d}$ (where ${\upsilon }_u$ and ${\upsilon }_d$ are the vacuum expectation values of the two Higgs doublets).

After the Higgs fields develop their VEVs, $⟨H_u⟩=\frac{1}{\sqrt{2}}{\upsilon }_u$, $⟨H_d⟩=\frac{1}{\sqrt{2}}{\upsilon }_d$ and $⟨S⟩=\frac{1}{\sqrt{2}}{\upsilon }_s$, and eliminating the massless degree of freedom, the potential has terms for the non-zero mass modes for the scalar Higgs fields $S_i(i=1,2,3)$, pseudoscalar Higgs fields $P_i(i=1,2)$ and charged Higgs fields $h^{\pm }$ as follows

$$V_{mass}=\frac{1}{2}\left(S_1{S}_2{S}_3\right){\mathcal{M}}_S\left(\begin{array}{c}{S}_1\\ S_2\\ S_3\end{array}\right)+\frac{1}{2}\left(P_1{P}_2\right){\mathcal{M}}_P\left(\begin{array}{c}{P}_1\\ P_2\end{array}\right)+m_{h^{\pm }}^2{h}^{+}{h}^{-}.$$

(3)

The elements of the mass matrix for the pseudoscalar Higgs states at tree-level are [87]

$${\mathcal{M}}_{P11}=m_A^2,$$

(4)

$${\mathcal{M}}_{P12}=\frac{1}{2}\left(m_A^2\sin 2\beta -6\frac{\kappa {\mu }_{\mathrm{eff}}^2}{\lambda }\right)\frac{\lambda \upsilon }{\sqrt{2}{\mu }_{\mathrm{eff}}},$$

(5)

$${\mathcal{M}}_{P22}=\frac{1}{8}\left(m_A^2\sin 2\beta +6\frac{\kappa {\mu }_{\mathrm{eff}}^2}{\lambda }\right)\frac{{\lambda }^2{\upsilon }^2}{{\mu }_{\mathrm{eff}}^2}\sin 2\beta -3\frac{\kappa {\mu }_{\mathrm{eff}}{A}_{\kappa }}{\lambda },$$

(6)

where $m_A^2=\sqrt{2}\frac{{\mu }_{\mathrm{eff}}}{sin2\beta }\left(A_{\lambda }+\frac{\kappa {\mu }_{\mathrm{eff}}}{\lambda }\right)$ and ${\upsilon }^2={\upsilon }_u^2+{\upsilon }_d^2$. To a good approximation, the tree-level mass of the lightest pseudoscalar Higgs boson is given by

$$m_{a_1}^2=\frac{9A_{\lambda }{\mu }_{\mathrm{eff}}}{2sin2\beta }{cos}^2{\theta }_A-3\frac{\kappa A_{\kappa }{\mu }_{\mathrm{eff}}}{\lambda }{sin}^2{\theta }_A.$$

(7)

It is clear from the above equation that all the six tree-level parameters of the Higgs sector jointly affects $m_{a_1}$. In practice, one must include the various radiative corrections to the Higgs mass matrices, which affect Higgs observables [88,89,90,91,92,93,94,95,96,97,98,99,100].

The light pseudoscalar state $a_1$ is in fact a composition of both the CP-odd doublet scalar in the MSSM sector, $a_{\mathrm{MSSM}}$, and the new CP-odd singlet scalar of the NMSSM, $a_{\mathrm{S}}$. This can be written as [101]:

$$a_1=a_{\mathrm{MSSM}}cos{\theta }_A+a_{\mathrm{S}}sin{\theta }_A,$$

(8)

where $cos{\theta }_A$ and $sin{\theta }_A$ are the mixing angles. The $a_1$ becomes highly singlet-like at small values of $cos{\theta }_A$ as in the case of the approximate R-symmetry, in which $A_{\lambda }\to 0$ and $A_{\kappa }\to 0$ or in the case of the Peccei-Quinn symmetry, in which $\kappa \to 0$. Therefore, the NMSSM parameter space can naturally accommodate the light CP-odd Higgs $a_1$, which is of great interest. The $a_1$ could be very light, a few GeV $\lesssim m_{a_1}\lesssim 300$ GeV.

In our work to find the regions of the NMSSM parameter space that offer a light $a_1$, we used the NMSSMTools 5.5.2 package [102,103,104,105]. Please note that our aim is exploring the NMSSM parameter space that has a light $a_1$ so we assumed small values of $A_{\kappa }$. It is a numerical package calculating the masses, couplings, branching fractions and decay widths of all the Higgs bosons in addition to the SUSY spectrum of the NMSSM for each model point in terms of its input parameters. Details of our parameter space scan including theoretical and experimental constraints can be found in Refs. [55,56,57,58]. To study the LHC discovery potential of the $a_1$, the inclusive production rates were calculated by multiplying the production cross sections extracted from CalcHEP [106] by the branching ratios computed with the NMSSMTools for all surviving data points, assuming the center of mass energy $\sqrt{s}=14$ TeV.

3. Light Pseudoscalar Higgs Boson Production in Gluon-Gluon Fusion at the LHC

The light NMSSM pseudoscalar Higgs boson could be produced at the LHC via gluon fusion $gg\to a_1$, which is mediated by heavy quark and squark loops. This production mechanism is dominant for Higgs bosons at hadron colliders so a lot of attention has been paid to this production channel by the theory community. To study the LHC discovery potential of the $a_1$, we present in Figure 1 and Figure 2 the production cross sections for the $a_1$ in a variety of decay channels as functions of $m_{a_1}$ and of the corresponding branching ratios. As expected, the production cross sections decrease with increasing $m_{a_1}$ as shown in the left plots of the figures. The $a_1$ mass for the surviving points can reach maximum about several hundreds GeV. The production cross sections for $a_1$ decaying into $t\overline{t}$ are quite small, below 1 fb level, see the top plots of Figure 1. Therefore, for the allowed parameter space there is no point has a cross section large enough to be detectable. As for the bottom-quark pair final states, the cross sections are quite large, reaching the nanobarn (nb) level, for $m_{a_1}\lesssim 100$ GeV, see the middle plots of Figure 1. It is clear that this decay channel is dominant in large region of the parameter space as shown in the right middle plot of the figure. This region corresponds to the mass range $m_{a_1}\lesssim 140$ GeV where the branching ratio is close to 90% for most points when the decay channel is kinematically allowed. It is also noticeable from the bottom plots of the figure that the production rates into ${\tau }^{+}{\tau }^{-}$ are also sizable, reaching up to $7\times {10}^5$ fb for very light $a_1$ with masses $\lesssim 10$ GeV, and decreasing rapidly with increasing $m_{a_1}$ (left plot). Moreover, it is clear that the BR $(a_1\to {\tau }^{+}{\tau }^{-})$ reaches about 10% for most allowed points in the parameter space that has $m_{a_1}\gtrsim 10$ GeV. However, there is some points with very light $a_1$, $m_{a_1}\lesssim 10$ GeV, yielding large BR$(a_1\to {\tau }^{+}{\tau }^{-})\gtrsim 90\%$, in which case the $a_1$ decay into $b\overline{b}$ is kinematically not allowed. The mass region with $m_{a_1}\lesssim 10$ GeV is highly constrained, see, e.g., Ref. [107]. These points are the ones lying in the right of the bottom-right plot of Figure 1. In short, the production rates in some regions of the NMSSM parameter space are large enough to help extracting $a_1$ signals in the ${\tau }^{+}{\tau }^{-}$ and $b\overline{b}$ final states if the backgrounds are successfully reduced to manageable levels. However, the discovery of $a\to t\overline{t}$ is impossible because the production rates are quite small and due to the complicated final state.

Figure 1. The production rates $\sigma (gg\to a_1)\mathrm{Br}(a_1\to t\overline{t})$ (top) $\sigma (gg\to a_1)\mathrm{BR}(a_1\to b\overline{b})$ (middle) and $\sigma (gg\to a_1)\mathrm{BR}(a_1\to {\tau }^{+}{\tau }^{-})$ (bottom) as functions of $m_{a_1}$ (left) and of the corresponding branching ratios (right) [55].

Figure 2. The production rates $\sigma (gg\to a_1)\mathrm{BR}(a_1\to \gamma \gamma )$ (top) and $\sigma (gg\to a_1)\mathrm{BR}(a_1\to Z\gamma )$ (bottom) as functions of $m_{a_1}$ (left) and of the corresponding branching ratios (right) [55].

In Figure 2, we show the total production cross sections for the $gg\to a_1\to \gamma \gamma$ and $gg\to a_1\to Z\gamma$ channels as functions of $m_{a_1}$ and of the corresponding branching ratios $\mathrm{BR}(a_1\to \gamma \gamma )$ and $\mathrm{BR}(a_1\to Z\gamma )$. One notices in the figure that the production rates reach up to about several hundreds fb for $\gamma \gamma$ and several tens fb for $Z\gamma$ final states. Such high rates correspond to $m_{a_1}\lesssim 200$ GeV. This is interesting as such signal events could be detectable particularly at very large luminosities of the LHC. Interestingly, one also notices in Figure 3 that the $\mathrm{BR}(a_1\to \gamma \gamma )$ and $\mathrm{BR}(a_1\to Z\gamma )$ can reach unity in certain regions of the NMSSM parameter space. The reason for such dominant decay channels is that the $a_1$ is a singlet-like with suppressed couplings to fermions at the tree level but the $a_1$ can couple to $\gamma \gamma$ and $Z\gamma$ at one loop with charginos in the loop. In this case, the mixing angle $cos{\theta }_A$ gets close to 0, see Equation (8). Unfortunately, one cannot take advantage of this phenomenon because even if the $a_1$ couplings to $\gamma \gamma$ and $Z\gamma$ are enhanced but its couplings to gluons in the production channel are suppressed so the production cross sections are quite small when the $\mathrm{BR}(a_1\to \gamma \gamma )$ and the $\mathrm{BR}(a_1\to Z\gamma )$$\sim 100\%$, see the right plots of Figure 2. The enhancements of the partial widths of $a_1\to \gamma \gamma$ and $a_1\to Z\gamma$ are characteristic features of the NMSSM in comparison to the MSSM.

Figure 3. The correlations between the mixing angle $cos{\theta }_A$ and the branching ratios $\mathrm{BR}(a_1\to \gamma \gamma )$ (left) and $\mathrm{BR}(a_1\to Z\gamma )$ (right) [55].

4. Light Pseudoscalar Higgs Boson Production with a Single Bottom-Quark at the LHC

The $a_1$ production in association with a single bottom-quark at hadron colliders can occur at the tree-level through either the process $gb\to b{a}_1$ or the process $g\overline{b}\to \overline{b}{a}_1$. Since the production cross sections for the process $pp\to b{a}_1$ at the LHC and its charge conjugate $pp\to a_1\overline{b}$ are alike, we only consider here the production process $pp\to a_{1}b$. This process has an advantage that the existence of the bottom-quark with a high transverse momentum in the final state can be tagged, allowing the rejection of light jets. Furthermore, this process can be significantly enhanced at large values of tan$\beta$. Since the $a_1$ production through the process $pp\to a_{1}b$ followed by the $a\to b\overline{b}$ decay is plagued by large backgrounds, we only discuss in this section the LHC discovery potential of a light pseudoscalar through the process $pp\to b{a}_1$ followed by the $a_1\to {\tau }^{+}{\tau }^{-}$ decay.

Figure 4 shows the distribution of the production rates $\sigma (gb\to a_1)\mathrm{BR}(a_1\to {\tau }^{+}{\tau }^{-})$ as functions of $m_{a_1}$ and of $\mathrm{BR}(a_1\to {\tau }^{+}{\tau }^{-})$. These rates decreases rapidly with increasing $m_{a_1}$ as shown in the left plot of the figure. The production rates can reach up to 10 nb for BR$(a_1\to {\tau }^{+}{\tau }^{-})\approx 0.1$ and 25 nb for BR$(a_1\to {\tau }^{+}{\tau }^{-})\approx 1$ as shown in the right plot of the figure. In short, it is remarkable to notice that these production rates are quite sizable for considerable points in the NMSSM parameter space, which can reach detectable levels.

Figure 4. The production rates $\sigma (gg\to a_1)\mathrm{BR}(a_1\to {\tau }^{+}{\tau }^{-})$ as functions of both $m_{a_1}$ (left) and $\mathrm{BR}(a_1\to {\tau }^{+}{\tau }^{-})$ (right) [56].

A partonic signal-to-background (S/B) analysis for the process $pp\to a_{1}b\to b{\tau }^{+}{\tau }^{-}$ was done in Ref. [56] where extraction of the $a_1$ signature was proven for several benchmark points given in Table 1. Figure 5 shows the signal significance $S/\sqrt{B}$ (where S and B are the signal and background rates, respectively), left plot, and the corresponding signal event rate S, right plot, as functions of the integrated luminosity. One can see that the discovery of the light $a_1$ at the LHC may occur for $a_1$ masses between ≈90 and ≈150 GeV at a luminosity of $\mathcal{O}$(300 $f{b}^{-1}$) or more where the signal significances, $S/\sqrt{B}$, are good enough to discover the $a_1$ in this mass region.

Table 1. Four benchmark points used in the signal-to-background (S/B) analysis [56].

Point	$\mathit{\lambda }$	$\mathit{\kappa }$	$tan\mathit{\beta }$
1	0.17633	0.248102	15.9481
2	0.295035	0.481850	17.2552
3	0.259318	−0.548899	13.4015
4	0.120601	0.294640	13.2896
Point	${\mu }_{\mathrm{eff}}\left[\mathrm{GeV}\right]$	$A_{\lambda }\left[\mathrm{GeV}\right]$	$A_{\kappa }\left[\mathrm{GeV}\right]$
1	165.357	34.9856	0.0221315
2	169.105	−15.7234	−2.63603
3	177.252	540.586E	16.2739
4	766.903	−1821.2	−8.76219
Point	${\mathrm{m}}_{{\mathrm{a}}_1}\left[\mathrm{GeV}\right]$	$\mathrm{BR}({\mathrm{a}}_1\to$${\tau }^{+}{\tau }^{-}$)	$\sigma$ (gb$\to b{a}_1).BR(a_1\to$ ${\tau }^{+}{\tau }^{-}$) [fb]
1	8.80412	0.767892	16,340.7
2	45.0156	0.0808669	371.75
3	92.0510	0.0929458	796.7
4	147.483	0.108216	400.9

Figure 5. The signal significance $S/\sqrt{B}$ (left plot) and the signal event rate S (right plot) of the process $bg\to a_{1}b\to b{\tau }^{+}{\tau }^{-}$ as functions of the integrated luminosity for the four benchmark points mentioned in Table 1. The lines for both $m_{a_1}$ = 8.8 GeV and 45.02 GeV may be hardly visible but they were added for clarity.

5. Light Pseudoscalar Higgs Boson Production in Bottom-Quark Annihilation at the LHC

Higgs production in association with bottom quarks in supersymmetric theories with large tan$\beta$ can be the dominant Higgs production mechanism in hadronic collisions as the bottom Yukawa coupling can be enhanced compared to that of the SM. This enhancement of the bottom Yukawa coupling in supersymmetric theories occurs only at large tan$\beta$. As mentioned in the introduction, the cross sections for this process can be calculated using either the four-flavor scheme where the leading order processes is $pp\to b\overline{b}H$ or using the five-flavor scheme where the leading order one is $b\overline{b}\to H$.

In this section, we consider the direct production of a light $a_1$ through the process $b\overline{b}\to a_1$ in the context of the NMSSM rather than looking for its traditional production $pp\to b\overline{b}{a}_1$, which has been studied in the literature [42,43,44]. Figure 6 presents the production cross sections $\sigma (b\overline{b}\to a_1)\mathrm{BR}(a_1\to {\tau }^{+}{\tau }^{-})$ as functions of tan$\beta$ (top plot), $m_{a_1}$ (middle plot) and $\mathrm{BR}(a_1\to {\tau }^{+}{\tau }^{-})$ (bottom plot) for all the surviving points of the scan. We can see from the figure that the production rates are sizable, topping ${10}^9$ pb. These largest rates correspond to the region of large values of tan$\beta$ in which the $a_1$ is a light, $m_{a_1}\lesssim 200$ GeV. The bottom plot of the figure shows the dependence of the production rates on the $\mathrm{BR}(a_1\to {\tau }^{+}{\tau }^{-})$. It is clear that the population density of the points corresponds to the region of $\mathrm{BR}(a_1\to {\tau }^{+}{\tau }^{-})$ ∼10%. In summary, the $a_1$ production in bottom-quark annihilation in the tauonic decay channel at the LHC seems to be a promising production channel for the discovery of the light $a_1$ in the region of large values of tan$\beta$. Some scope could also be afforded by $\gamma \gamma$ signature in this production channel, see Ref. [108].

Figure 6. Production rates for the lightest pseudoscalar Higgs state, $\sigma (b\overline{b}\to a_1\to {\tau }^{+}{\tau }^{-})$, as functions of tan$\beta$ (top), $m_{a_1}$ (middle) and $\mathrm{BR}(a_1\to {\tau }^{+}{\tau }^{-})$ (bottom) [57].

Several analyses for signals and dominant backgrounds for the process $b\overline{b}\to a_1\to {\tau }^{+}{\tau }^{-}$ were done for the six benchmark points given in Table 2, see Ref. [57] for more details. Figure 7 shows the signal significance and the signal event rates for these benchmark points. As it can be seen from this figure, the LHC has the potential to observe the $a_1$ signal in the range $150\lesssim m_{a_1}\lesssim 200$ GeV but this requires very high luminosity of 600 fb${}^{-1}$ or more, see the left plot of the figure. Lighter or Heavier states could not be resolvable via this production channel.

Table 2. The benchmark points used in the signal-to-background analysis for the process $b\overline{b}\to a_1\to {\tau }^{+}{\tau }^{-}$ [57].

	P1	P2	P3	P4	P5	P6
$\lambda$	0.0549371	0.248102	0.217092	0.197646	0.172677	0.249246
$\kappa$	0.496695	0.618832	0.392990	−0.349694	0.481966	0.547502
$tan\beta$	13.8036	18.8314	32.9745	53.9098	20.6563	12.2270
${\mu }_{\mathrm{eff}}\left[\mathrm{GeV}\right]$	106.941	139.910	328.891	546.213	622.326	875.295
$A_{\lambda }\left[\mathrm{GeV}\right]$	490.984	826.568	339.250	1125.35	−1434.74	−1201.27
$A_{\kappa }\left[\mathrm{GeV}\right]$	−0.0759634	−0.310352	−4.45516	4.52072	−4.38217	−4.45711
${\mathrm{m}}_{{\mathrm{a}}_1}\left[\mathrm{GeV}\right]$	8	52	100	150	197	243
$\mathrm{BR}({\mathrm{a}}_1\to$${\tau }^{+}{\tau }^{-}$)	0.870856	0.0835167	0.103431	0.121666	0.116006	0.118482
$\sigma$ [fb]	$4\times {10}^9$	$5.1\times {10}^8$	$3\times {10}^7$	$4.8\times {10}^7$	$3.9\times {10}^9$	$3.6\times {10}^7$

Figure 7. The signal significance $S/\sqrt{B}$ (left plot) and the total event rate S (right plot) of the production process $b\overline{b}\to a_1\to {\tau }^{+}{\tau }^{-}$ as functions of the integrated luminosity for the six benchmark points mentioned in Table 2.

6. Light Pseudoscalar Higgs Boson Production via Decays of a Heavy Scalar Higgs Boson at the LHC

In this section, we consider the production of the light $a_1$ through the non-SM Higgs boson, $h_2$, decaying either into two light CP-odd Higgs bosons $a_1{a}_1$ or into a light $a_1$ and the gauge boson Z. One of the interesting features of the NMSSM is the fact that Higgs-to-Higgs decays can be kinematically open in large areas of the parameter space in which the existence of the light $a_1$ is quite natural, see e.g., Refs. [30,31,32,33]. Here, we only consider the gluon fusion $gg\to h_2$ followed by either $a_1{a}_1$ or and $Z{a}_1$ as a potential production channel to produce the light $a_1$.

Figure 8 shows the correlations between $m_{h_2}$ and the lightest CP-even and CP-odd Higgs masses $m_{h_1}$ and $m_{a_1}$ for the good points. Since the surviving points emerging from the random scan have large values of $\lambda$ and small values of both ${\mu }_{\mathrm{eff}}$ and $A_{\kappa }$, only small values of $m_{h_2}$ and $m_{a_1}$ are allowed, see Ref. [58]. It is remarkable to notice that in most regions of our parameter space the $h_1$ is the SM-like Higgs boson of the mass around 125 GeV. However, the $h_2$ can also play the role of the SM-like Higgs boson but only in small regions of the parameter space in which regions the $h_1$ is mostly singlet-like with mass $\lesssim 120$ GeV, see the left-plot of the figure, so it can escape the LEP, the Tevatron, and LHC bounds [109,110,111,112,113,114,115]. The right plot of the figure shows that the smaller the $m_{h_2}$, the smaller the $m_{a_1}$.

Figure 8. The mass distribution for the next-to-lightest CP-even Higgs, $m_{h_2}$, versus the lightest CP-even Higgs mass $m_{h_1}$ (left plot) and versus the lightest CP-odd Higgs mass $m_{a_1}$ (right plot) [58].

Due to the mixing between the Higgs singlet and doublets, Higgs-to-Higgs decays can be dominant over large regions of the parameter space if these decays are kinematically allowed. In Figure 9 we display the correlations between the $h_2\to a_1{a}_1$ decay rate and the $m_{h_2}$ (left) and the correlations between the $h_2\to Z{a}_1$ decay rate and the $m_{h_2}$ (right). It is clear from the left plot of the figure that the decay $h_2\to a_1{a}_1$ is dominant, reaching unity, whenever it is kinematically allowed. The dominance of this decay mode causes a suppression of the other decay channels such as $b\overline{b}$. It is also noticed that the $h_2\to Z{a}_1$ branching ratio is quite small, topping 0.1% as shown from the right plot of the figure.

Figure 9. The next-to-lightest CP-even Higgs mass $m_{h_2}$ plotted against both $\mathrm{BR}(h_2\to a_1{a}_1)$ (left) and $\mathrm{BR}(h_2\to Z{a}_1)$ (right) [58].

Furthermore, by Looking at the inclusive production rates ending up with $a_1{a}_1\to {\tau }^{+}{\tau }^{-}{\tau }^{+}{\tau }^{-}$ and $Z{a}_1\to {\tau }^{+}{\tau }^{-}{\tau }^{+}{\tau }^{-}$ in Figure 10, it is clear that these rates decrease by increasing $m_{h_2}$, as expected. Additionally, it is clear that the $\sigma (gg\to h_2\to a_1{a}_1\to {\tau }^{+}{\tau }^{-}{\tau }^{+}{\tau }^{-})$ are sizable, reaching a maximum level at about 100 fb for small values of $m_{h_2}$ while the production rates $\sigma (gg\to h_2\to Z{a}_1\to {\tau }^{+}{\tau }^{-}{\tau }^{+}{\tau }^{-})$ are definitely too small, reaching its maximum of roughly 0.5 fb. The latter production rates are obviously not enough to detect the $a_1$ through this channel at the LHC, assuming that leptonic tau decays are around 17.5%. We do not consider the $b\overline{b}b\overline{b}$ and $b\overline{b}{\tau }^{+}{\tau }^{-}$ final states here because they are plagued by large backgrounds.

Figure 10. The production rates $\sigma (gg\to h_2).\mathrm{BR}(h_2\to a_1{a}_1\to {\tau }^{+}{\tau }^{-}{\tau }^{+}{\tau }^{-}$) (left) and $\sigma (gg\to h_2).\mathrm{BR}(h_2\to Z{a}_1\to {\tau }^{+}{\tau }^{-}{\tau }^{+}{\tau }^{-}$) (right) as functions of $m_{h_2}$ [58].

After performing analyses for signals and dominant standard model backgrounds in Ref. [58] for the four benchmark points shown in Table 3 for the process $gg\to h_2\to a_1{a}_1\to {\tau }^{+}{\tau }^{-}{\tau }^{+}{\tau }^{-}$, we conclude that there are some regions of the NMSSM parameter space where the $h_2$ and $a_1$ could spontaneously be discovered in the mass region $140\lesssim m_{h_2}\lesssim 220$ GeV and $m_{a_1}\lesssim 100$ GeV at the LHC of 1000 fb${}^{-1}$ of luminosity. The existence of such a light $a_1$ is an exciting prospect to distinguish the NMSSM from the MSSM as such a light pseudoscalar Higgs state, about $M_Z$ or below in mass, is impossible in the MSSM.

Table 3. The four benchmark points P1, P2, P3, and P4 used in the signal-to-backgrounds analysis [58].

	P1	P2	P3	P4
$\lambda$	0.615706	0.650828	0.637590	0.617789
$\kappa$	0.261287	0.264725	0.339134	0.387478
$tan\beta$	5.2247	3.78738	3.82514	3.70979
${\mu }_{\mathrm{eff}}\left[\mathrm{GeV}\right]$	153.678	198.766	198.201	199.224
$A_{\lambda }\left[\mathrm{GeV}\right]$	646.778	517.464	464.215	426.835
$A_{\kappa }\left[\mathrm{GeV}\right]$	−8.00937	5.1126	−9.72344	9.09329
${\mathrm{m}}_{{\mathrm{h}}_2}\left[\mathrm{GeV}\right]$	140	180	220	260
${\mathrm{m}}_{{\mathrm{a}}_1}\left[\mathrm{GeV}\right]$	66	64	99	67
S [fb] with 300 fb${}^{-1}$	$3.168\times {10}^4$	$8.61\times {10}^3$	$2.364\times {10}^3$	$3.21\times {10}^2$
B [fb] with 300 fb${}^{-1}$	$3.9\times {10}^4$	$3.9\times {10}^4$	$3.9\times {10}^4$	$3.9\times {10}^4$
$S/\sqrt{B}$ with 300 fb${}^{-1}$	160.4	43.6	12	1.6
$S/\sqrt{B}$ with 1000 fb${}^{-1}$	292.9	79.6	21.9	3

7. Conclusions

The NMSSM can naturally accommodate the measured value of the SM-like-Higgs boson mass of the range ∼125 GeV without a significant degree of fine-tuning as compared to the MSSM. In the CP conserving case, the Higgs sector of the NMSSM consists of seven physical Higgs mass states: three CP-even, two CP-odd and two charged states. Noticeably, the discovery of any of the light Higgs states along with the SM-like-Higgs state discovered in 2012 would be a clear signature of the non-minimal supersymmetric models. In this paper, in the framework of the NMSSM, we have reviewed the LHC discovery potential of a light pseudoscalar Higgs, $a_1$, in light of experimental and theoretical constraints implemented in the program NMSSMTools apart from the dark matter and the muon anomalous magnetic moment constraints, focusing on the following Higgs production channels: gluon fusion $gg\to a_1$, bottom-gluon fusion $bg\to b{a}_1$, bottom-quark fusion $b\overline{b}\to a_1$ in addition to the production of $a_1$ via decays of the non-SM-like Higgs boson: $gg\to h_2\to a_1{a}_1$ and $gg\to h_2\to Z{a}_1$. Under the given results, we have concluded that:

1. In the gluon fusion production channel, the overall production rates are quite sizable and could help extracting the $a_1$ signal over some parts of the parameter space. We believe that the most promising decay channels for the $a_1$ discovery at the LHC are the ${\tau }^{+}{\tau }^{-}$ and $\gamma \gamma$ channels. The other promising channels are the $b\overline{b}$ and $Z\gamma$ channels but the former one suffers from large backgrounds and one should successfully reduce them to manageable levels. As for the latter one, there is only a small but well-defined region of the NMSSM parameter space where the light $a_1$ could be discovered through this decay channel.

2. It has also been noted that at high luminosity of the LHC the ${\tau }^{+}{\tau }^{-}$ decay channel can be exploited to discover the light $a_1$ produced with a single bottom-quark $bg\to b{a}_1$ in the mass region $90\lesssim m_{a_1}\lesssim 150$ GeV.

3. The production of the light pseudoscalar Higgs boson in the bottom-quark fusion $bb\to a_1$ followed by the decay channel $a_1\to {\tau }^{+}{\tau }^{-}$ is also possible at large values of tan$\beta$ in the mass region $150\lesssim m_{a_1}\lesssim 200$ GeV, provided that the luminosity of $\mathcal{O}$(600 $f{b}^{-1}$) or more.

In the above three production channels, we have found that the $\tau$-pair decay is a promising channel to discover the light $a_1$ at the LHC. Here, we assume that one $\tau$ decays leptonically and the other decays hadronically. This is because they combine the high efficiency of the lepton trigger and the largeness of the branching fractions of the hadronic tau decays. The case of fully hadronic decays is ignored due to the huge QCD backgrounds.

4. Furthermore, the $a_1$ production through the process $gg\to h_2\to a_1{a}_1$ in the 4$\tau$ final state affords the possibility to discover the light $a_1$ in the mass regions $140\lesssim m_{h_2}\lesssim 220$ GeV and $m_{a_1}\lesssim 100$ GeV. We have found that this production channel can give sizable production rates, which could allow one to discover simultaneously two Higgs bosons: the non-SM-like Higgs boson, $h_2$, and the light pseudoscalar Higgs state, $a_1$. However, the $a_1$ production of the process $gg\to h_2\to Z{a}_1\to 4\tau$ is impossible due to the smallness of the production rates. Finally, we have noticed in our choice of the NMSSM parameter space that the BR$(h_2\to a_1{a}_1)$ is dominant in most regions of the parameter space as long as it is kinematically allowed.

Overall, searching for a light pseudoscalar Higgs state at the LHC and other colliders is of great value as many models predict the existence of such a light state. We hope that our encouraging results motivate further and more detailed studies in the context of the NMSSM since it offers a much richer accessible phenomenology of the light pseudoscalar state that deserves to be comprehensively exploited.