A Novel Scenario in the Semi-constrained NMSSM

In this work, we develop a novel efficient scan method, combining the Heuristically Search (HS) and the Generative Adversarial Network (GAN), where the HS can shift marginal samples to perfect samples, and the GAN can generate a huge amount of recommended samples from noise in a short time. With this efficient method, we find a new scenario in the semi-constrained Next-to Minimal Supersymmetric Standard Model (scNMSSM), or NMSSM with non-universal Higgs masses. In this scenario, (i) Both muon g-2 and right relic density can be satisfied, along with the high mass bound of gluino, etc. As far as we know, that had not been realized in the scNMSSM before this work. (ii) With the right relic density, the lightest neutralinos are singlino-dominated, and can be as light as 0-12 GeV. (iii) The future direct detections XENONnT and LUX-ZEPLIN (LZ-7 2T) can give strong constraints to this scenario. (iv) The current indirect constraints to Higgs invisible decay $h_2\to \tilde{\chi}^0_1 \tilde{\chi}^0_1$ are weak, but the direct detection of Higgs invisible decay at the future HL-LHC may cover half of the samples, and that of the CEPC may cover most. (v) The branching ratio of Higgs exotic decay $h_2\to h_1h_1, a_1a_1$ can be over 20 percent, while their contributions to the invisible decay $h_2\to4\chi_1^0$ are very small.


Introduction
Higgs boson was discovered in 2012 [1,2], and its production rate in most channels coincides with the Standard Model (SM) prediction considering uncertainties [3][4][5]. While there are still chances for physics beyond the SM. For example, for the branching ratio of Higgs boson invisible decay, the current excluding limits are only 26% by ATLAS [6] and 19% by CMS [7], with all data at Run I and data of about 36 fb −1 at Run II.
Supersymmetry is a popular theory beyond the SM, which introduces a new internal symmetry between fermions and bosons. Thus the large hierarchy problem can be solved, gauge coupling can be unified, and dark matter candidates can be provided, etc. In the Minimal supersymmetric Standard Model (MSSM) with 7 free parameters at the electroweak scale, a SM-like 125 GeV Higgs can be afforded, but need large fine-tuning, and the branching ratio of Higgs boson invisible decay can be about 10% at most [8][9][10]. The Next-to Minimal Supersymmetric Standard Model (NMSSM) with Z 3 symmetry extends the MSSM by a complex singlet superfieldŜ, but introduces four more parameters. In the graceful and simple model of fully-constrained NMSSM (cNMSSM), all Higgs and sfermion masses are assumed unified at the Grand Unified theoretical (GUT) scale, thus only four parameters at GUT scale are left free [11][12][13][14][15][16][17][18][19]. These four or five parameters run according to the Renormalization Group Equations (RGEs), forming the spectrum of NMSSM at low energy scale. While it was found that when considering all the constraints including muon g-2, the SM-like Higgs mass can not reach to 125 GeV in cNMSSM [11,12], like these in the CMSSM, NUHM1 and NUHM2 [20][21][22][23].
In this work, we consider possible scenarios of Higgs invisible decay in the semiconstrained NMSSM (scNMSSM) [24][25][26][27][28][29], which relaxing the Higgs masses at GUT scale, and also called NMSSM with non-universal Higgs mass (NUHM). As a simple and graceful SUSY model, the scNMSSM had attracted much attention. In [26,27] the constraints of LHC and dark matter to scNMSSM was considered, while the muon g-2 was left aside; in [25] muon g-2 was satisfied, while dark matter relic density is not sufficient; In [24,28,29], direct searches for higgsino sector was considered. In this work, we consider all constraints including muon g-2, and also try to get sufficient relic density.
In this work, to include constraints of muon g-2, etc., get sufficient relic density, and get as-large-as-possible branching ratio of Higgs invisible decay, we developed a novel efficient method to scan the parameter space, which consists of the Heuristically Search (HS) and the Generative Adversarial Network (GAN). Note that in [30,31], the Machine Learning Scan method has been used to explore the parameter space, and a scanning tool xBIT [32] based on Machine Learning has been developed. This Machine Learning Scan is based on several classifiers, dealing with a Classification problem that each sample gets a probability of how much it could be a perfect sample. This scan method also needs to generate samples in high-dimension space, which will cost very long time, (eg., when the dimension is 9 and each dimension has 100 grid, at least 100 9 samples need to be generated). On the contrary, we adopt a generative model, the Generative Adversarial Network (GAN) [33], which is a famous star in deep learning area and also gets much attention in high energy physics [34][35][36][37][38][39][40][41][42]. The GAN can directly generate samples with the same distribution as the training samples. So with a well-training GAN, we can get as many recommended samples as we want. And with the Heuristically Search we developed, we have a chance to shift some 'bad' samples to 'good' samples. Combined with Heuristically Search and GAN, we developed a novel method that can get a huge mount of surviving samples in a short time. Then, we used this novel method to study the parameter space of scNMSSM, under current constraints including LHC constraints, B physics, muon g-2, and dark matter, etc. We require our surviving samples to satisfy all these constraints, and part of them predict right relic density. To study Higgs invisible decay, we require the SM-like Higgs mass be 2 times larger than the LSP mass m h SM > 2mχ0 1 , and the invisible branching ratio be as large as possible. As can be seen from the following sections, this method is powerful in getting this novel scenario in the scNMSSM.
The rest of this paper is organized as follows. In section 2, we briefly introduce the Higgs and electroweakino sectors of the scNMSSM, and our search strategy consisting of Heuristically Search and GAN. In section 3, we describe the detail of our scan process and then discuss the Higgs invisible decay and light dark matter in the scNMSSM. Finally, we draw our conclusions in section 4.
2 The semi-constrained NMSSM and the search strategy NMSSM extends the MSSM particle content by adding a singlet superfieldŜ, which provides an effective µ-term. The superpotential of the Z 3 -invariant NMSSM is where the hats are used for superfields, y u,d,e stand for corresponding Yukawa couplings, and λ, κ are dimensionless coupling constants. When the singlet superfieldŜ gets a vacuum expectation value (VEV), S = v s , a effective µ-term is generated dynamically from the term λŜĤ u ·Ĥ d , with For convenience, in the following we refer to µ eff as µ. And the VEVs of two doublet Higgs The soft SUSY breaking terms in the NMSSM are only different from the MSSM in several terms:

Higgs and electroweakinos sector of scNMSSM
When the electroweak symmetry broken, the scalar component of superfieldsĤ u ,Ĥ d and S can be written as where ε = 0 1 −1 0 , and H 2 , H 1 and H 3 are the SM Higgs doublet, new doublet and singlet respectively.
In the basis (S 1 , S 2 , S 3 ), the CP-even Higgs boson mass matrix M 2 S is given by [43] In the basis (P 1 , P 2 ), the CP-odd Higgs boson mass matrix M 2 P is Three CP-even mass eigenstates h i (i = 1, 2, 3) (ordered in mass) are mixed from S i (i = 1, 2, 3), and two CP-odd mass eigenstates a i (i = 1, 2) (ordered in mass) are mixed from P i (i = 1, 2). The mixings are given by where the mixing matrix S ij and P ij can diagonalize the mass matrix M 2 S and M 2 P respectively.

The Heuristically Search (HS)
Usually, We divide the samples into 2 categories according to whether or not the samples passed all constraints. A sample that violated several constraints might be not good enough, but there is a chance that we can lead it to become a good sample. In our case, we first leave aside the dark matter and muon g-2 constraints, only imposing other constraints in the NMSSMTools. A sample that passes other constraints will get a score to evaluate how much it violates the dark matter and muon g-2 constraints, and we call it a 'marginal sample'. In Table.1, we classify the samples into 3 cases: the bad, marginal and perfect samples. For marginal and perfect samples, they will get a score to value how much they violate the constraints. And we try to shift these marginal samples to satisfy the dark matter and muon g-2 constraints, becoming perfect samples. The score function is given as: When the score is large, it means the marginal sample violates the experiments more; while when the score is zero, it means the marginal sample becomes a perfect sample, and satisfies all constraints very well, including dark matter and muon g-2 constraints.
In Algorithm 1, we give the Heuristically Search algorithm, which can shift a marginal sample to a perfect sample satisfying all constraints. With a marginal sample, X,

Algorithm 1 Heuristically Search with NMSSMTools
Input: A marginal sample, X; Output: Find a perfect sample X passed all constraints, or failed; 1: initial step = 0 and try = 0 2: score ← f (X) 3: while step < N max and try < T max and score = 0 do 4: get a new marginal sample X around the X within radius r 5: score ← f (X ) 6: if score < score then 7: X ← X Failed 20: end if we search around it and try to find another marginal sample with a smaller score. Then we repeat the process, until we meet a perfect sample whose score is zero, or get failed.
The search can be successful or get failed. Most of the time in our case, the Heuristically Search can lead about 80% (even over 94%) marginal samples to perfect samples. Meanwhile, to avoid the program being trapped in a local minimum, we give it a chance to give up. During the search, if the search step is larger than the maximum step N max (we set it to 20), or the number of tries in one step is larger than the maximum number, T max (we set it to 50), we stop the program and the search gets failed.
To get a new marginal sample X around the X, we can treat each component x i (i = 1...9) independently. The simplest way is choosing samples around the x i within radius r i with uniform distribution. To improve the efficiency, the Gaussian distribution is adopted, since it has some chance to search samples far away and could jump out of the local minimums. The Gaussian distribution function of x i is given as: where r i (we set it to 1/50) is an important parameter and determines the search efficient.
Actually, r i can change with the score. When the score is nearly zero, it means that a perfect sample is nearby, and then r i can change to a smaller one and vice versa.

The Generative Adversarial Network (GAN)
The Generative Adversarial Network (GAN) is a Generative model. It can generate samples with similar distribution as the real data. There are two neural networks in GAN. One is the Generator G, which can generate fake samples. While the other is the Discriminator D, which can classify the generated samples into real samples and the fake samples, so it is actually a binary classifier.
When the GAN is being trained, the Discriminator D tries to classify the generated samples into real and fake samples, meanwhile the Generator G try to fool the Discriminator D and generate almost 'real' samples. After training, the Generator G and Discriminator D arrive at a Nash equilibrium. Then we can use the Generator G to generate 'real' samples as many as we need. And these 'real' samples actually have similar distribution as the real samples coming from the training dataset.
In this work, we use the Artificial Neural Networks to build the Generator G and the Discriminator D. We adopt a simple Neural Network with 3 hidden layers and each layer with 50 neurons, and the Activation Function is Leaky ReLU. Furthermore, we train our GAN with Algorithm 2. In our case, we choose k = 3, n = 1, m = 20000, and the training iterations as 2000, while for the Gradient descent we use Adadelta [46]. updating the Generator by descending its binary cross entropy 10: end for 11: end for During the training, we require the Generator to learn the general distribution of the real data, but not try hard to find perfect hyperparameters, since we need the Generator to have more creativity. As a complement, we combine GAN with the Heuristically Search. The Generator generates lots of samples, and some of them might be marginal samples, while the Heuristically Search program will try to change these marginal samples into perfect samples.

Results and discussions
To satisfy all the constraints including muon g-2, dark matter, Higgs data, gluino and other SUSY search results, and try to get right dark matter relic density and large Higgs invisible decay, we consider following parameter space in the scNMSSM: |A 0 | < 10 TeV, |A λ | < 10 TeV, |A κ | < 10 TeV. (3.1)

Scan with HS and GAN
We developed the Heuristically Search program based on NMSSMTools-5.5.2 [47][48][49][50]. During the scan, we first require the samples satisfying the following other basic constraints: • Theoretical constraints of vacuum stability, and without Landau pole below M GUT [47][48][49].
• The lower mass bounds of charginos and sleptons from the LEP: • To study Higgs invisible decay we require the mass ofχ 0 1 smaller than half of the SM-like Higgs, Then for the marginal samples, we consider the constraints of dark matter and muon g-2, calculating the score in Eq.(2.23) for each sample. The upper and lower bounds of these observables are given in Table.2. The detail experimental constraints we consider in this work are list as following: •  • The spin-independent WIMP-nucleon cross section is constrained by XENON1T [63], where we rescale the original values by Ω/Ω 0 with Ω 0 h 2 = 0.1187; • The spin-dependent WIMP-proton cross section is constrained by LUX [64], XENON1T [65] and PICO-60 [66], where we also rescale the original values by Ω/Ω 0 ; • The muon anomalous magnetic moment (muon g-2) is constrained at 2σ level. The muon g-2 is given by [67][68][69][70][71] where a SM µ is without the Higgs boson contributions, since there is a SM-like Higgs in scNMMSM contributing to δa µ . We set the limit at 2σ level, which is 8.8 × 10 −10 ≤ δa µ ≤ 46 × 10 −10 .
If a sample satisfies the basic constraints (not including DM and muon g-2), it will get a score as a marginal sample; otherwise, it will be discarded. Then with the Heuristically Search Program, we did our first scan. We randomly searched for marginal samples in the parameter space, and then used the Heuristically Search Program changing them into perfect samples. In the first search, we got about 10k perfect samples in 24 hours 1 . In fact if we changed the random scan into a multi-path Markov Chain Monte Carlo (MCMC) scan, the scan would be more efficient.
In Fig.1, we show the score of marginal samples in the M 0 versus M 1/2 plane. Note that if the score equal to zero, the marginal sample is also a perfect sample. We can see that the area of marginal samples (colored range) is much larger than the perfect samples (black range) which get a zero score (satisfying all above constraints, including the DM and muon g-2). Besides we also show five tries, that the Heuristically Search Program shift marginal samples to perfect samples, where four get success (solid lines) and one gets failure (dashed line). In the successful tries shown, the Heuristically Search needs only 10 steps, on average, to shift a marginal sample to a perfect sample. In fact, many marginal samples need only several steps to change into perfect samples, while the direct search for perfect samples will waste much more time. That is the reason why we developed the Heuristically Search Program.
After the first search, all of the 10k perfect samples are used as the training set for the GAN. Then we trained the GAN according to Algorithm 2. With a well-training GAN 2 , we can transform random noises to recommended samples that have similar distribution as the training data. Then we can easily get millions of recommended samples from the GAN in a few seconds.
In Fig.2, we show the training set in the upper panels, and the recommended samples from GAN in the lower panels. We can see that the GAN has already learned the general distribution of the perfect samples in the training set. While the recommended samples from GAN (in the lower panels) have some creativity, which is not totally identical to the training set (in the upper panels). The well-trained GAN can exploit the parameter space and recommend samples around the training samples, which is exactly what we need.
We used the trained GAN to generate 2000k recommended samples 3 , and passed these recommended samples to the Heuristically Search Program. Then we got 280k perfect samples within 30 hours 4 , such a way is much faster than the traditional parameter scan. 2 We used Pytorch v1.3 to develop the GAN, and training cost about 5 hours. CPU: I5 6600K, GPU: GTX 1660 super. 3 Less than 1 minute on the computer with CPU: I5 6600K, GPU: GTX 1660 super. 4 We used 40 threads parallel running on Intel(R) Xeon(R) CPU E7-4830 v3 @ 2.10GHz.  At last, we impose the following additional constraints: • The upper limit of Higgs invisible decay, 19%, given by the CMS collaboration [7].
Finally, after all the scan and constraints, we got about 88k surviving samples. In Fig.3, we show these surviving samples in the nine-dimensional parameter space, where the coordinates are the same as those in Fig.2. We can see that all M 1/2 are larger than 1200 GeV. The reason is that we imposed the additional constraints, especially the high mass bound of gluino and the first-two-generation squarks at the LHC in Eq.(3.8).
Comparing Fig.3 with the lower plane in Fig.2, we can see that the recommended samples from GAN are changed to perfect samples by Heuristically Search Program. While comparing Fig.3 with the upper plane in Fig.2, we can see that the GAN has recommended many marginal samples that we need, and it does have some creativity to recommend samples around the training samples. So, the combination of Heuristically Search and GAN is very crucial.

Light dark matter and Higgs invisible decay
In Fig.4 we show the final surviving samples in the plane of κ vs λ, with colors indicate the masses of the lightest neutralinoχ 0 1 , the lightest CP-even Higgs h 1 and the light CP-odd Higgs a 1 respectively. For the surviving samples, we checked that the lightest CP-even Higgs h 1 are all highly singlet-dominated, and the next-to-lightest CP-even Higgs h 2 is the SM-like Higgs of 125 GeV. Since we need the SM-like Higgs have a chance decaying to invisibleχ 0 1 , theχ 0 1 is lighter than m h 2 /2. If the LSP is singlino-dominated, according Eq.(2.20), we should have mχ0 Since we set the parameter µ from 100 to 200 GeV, we have Thus it is and we checked that theχ 0 1 are singlino-dominated for samples between the two dash line. We can also see that for the samples between the two dash lines, h 1 and a 1 are also probably lighter than m h 2 /2. In Fig.5 we show the properties of dark matter in the scNMSSM. In the lower panel, the spin-independent dark matter and nucleon scattering cross section σ SI has been rescaled by a ratio of Ω/Ω 0 , where the Ω 0 is the right dark matter relic density with Ω 0 h 2 = 0.1187. As seen from these panels, the samples can be divided into three cases: • From the upper right panel, there is a special relationship between the mass of h 1 , a 1 andχ 0 1 . For the samples with right DM relic density in Case I and Case II, the LSP χ 0 1 is highly singlino-dominated, and with small λ, κ and a sizable tan β. Combining with Eq.(2.22), we can see the two ellipse arcs: Case II : • From the lower-left panel, most samples predicted spin-independent WIMP-nucleon cross section σ SI not far below the bound from XENON1T 2018, and can be covered by future LZ and XENONnT experiments. Thus these two future direct detections are crucial to check the parameter space of the scNMMSM. But there still are some   samples that can escape from these future detections, and also can predict right relic density. Besides, there are also some samples below the neutrino floor, although most of them do not predict sufficient DM relic density.
• From the lower right panel, samples with large Higgs invisible decay branch ratio, Br(h 2 →χ 0 1χ 0 1 ) > 10%, have a large LSP mass, mχ0 1 > 30 GeV. This is because the small LSP mass, mχ0 1 < 30 GeV, always accompanying with a small h 1 and a 1 mass, which can be seen from the upper right panel of Fig.4. Then the exotic decay channels h 2 → h 1 h 1 and h 2 → a 1 a 1 will open, which can be seen in Fig.6. The the Higgs invisible decay branch ratio Br(h 2 →χ 0 1χ 0 1 ) become smaller.
• From the lower right panel, most samples which have large Higgs invisible decay branch ratio, Br(h 2 →χ 0 1χ 0 1 ) > 10%, could be covered by future LZ and XENONnT detections. But there are still some samples that can escape from these future experiments, and also can have large Higgs invisible decay branch ratio. And there are also some samples below the neutrino floor, some of them can have large Higgs invisible decay branch ratio Br(h 2 →χ 0 1χ 0 1 ) > 10%.
In Fig.6, we show the decay information of the SM-like Higgs h 2 . From this figure, we can see that all of the branching ratios of h 2 →χ 0 1χ 0 1 , h 1 h 1 , a 1 a 1 can be at most about 20%. While we checked that considering in addition that of h 2 decay to 4χ 0 1 though a 1 /h 1 →χ 0 1χ 0 1 , which acquire mχ0 1 < m h 2 /4 31 GeV, the branching ratio of Higgs invisible decay increase very little compared with only that h 2 decay to twoχ 0 1 though h 2 →χ 0 1χ 0 1 . The upper limit of Higgs invisible decay branching ratio is about 19% at Run II of the LHC, while the future detections for that can reach to 5.6%, 0.24%, 0.5% and 0.26% according to HL-LHC [86], CEPC [87], FCC [88] and ILC [89] respectively.
Considering the values of |N 15 | 2 , we can have the following observations from Fig.6: • For most samples with higgsino-dominated LSP, |N 15 | 2 < 0.5, the branching ratio Br(h 2 →χ 0 1χ 0 1 ) can be sizeable, while the branching ratio Br(h 2 → h 1 h 1 ) and Br(h 2 → a 1 a 1 ) are both zero. The reason is that the higgsino-dominated LSPs are usually accompanied by a large mass of h 1 and a 1 , as can be seen from the upper panels of Fig.5, thus these two exotic decay channels are closed.
• For samples with h 2 /Z-funnel dark matter, mχ0 1 m Z,h 2 , the branching ratio of Higgs boson invisible decay can be large or small depending on the parameter λ.
• For most samples with low-mass LSP, mχ0 1 < 20 GeV, the branching ratio of Higgs boson invisible decay is small and beyond the ability of HL-LHC, while the Br(h 2 → h 1 h 1 ) can be larger than the Br(h 2 → a 1 a 1 ).
• Though the detection of Higgs invisible decay, about half of the surviving samples can be covered at the future HL-LHC, while the future CEPC can cover most.

Conclusions
In this work, we develop a novel scan method, combining the Heuristically Search (HS) and the Generative Adversarial Network (GAN). The HS can shift marginal samples to perfect samples, and the GAN can generate recommended samples as many as we need from noise. In our specific process, we first scan the parameter space randomly with NMSSMTools under basic constraints, generating marginal samples; then the HS try to shift the marginal samples to perfect samples satisfying in addition the dark matter and muon g-2 constraints; with these randomly-generated perfect samples, the GAN is trained, and then generates a huge amount of recommended samples in a short time; again the HS try to shift the recommended samples to perfect samples; finally, we check the final perfect samples with additional constraints including these of sparticle search, Higgs search and Higgs invisible decay, getting the final surviving samples.
With this efficient method, we find a new scenario in the semi-constrained Next-to Minimal Supersymmetric Standard Model (scNMSSM), or NMSSM with non-universal Higgs masses. In this scenario, • Both muon g-2 and right relic density can be satisfied, along with the high mass bound of gluino, etc. As far as we know, that had not been realized in the scNMSSM before this work.
• With the right relic density, the lightest neutralinos are singlino-dominated, and can be as light as 0-12 GeV.
• The future direct detections XENONnT and LUX-ZEPLIN (LZ-7 2T) can give strong constraints to this scenario.
• The current indirect constraints to Higgs invisible decay h 2 →χ 0 1χ 0 1 are weak, but the direct detection of Higgs invisible decay at the future HL-LHC may cover half of the samples, and that of the CEPC may cover most.
• The branching ratio of Higgs exotic decay h 2 → h 1 h 1 , a 1 a 1 can be over 20 percent, while their contributions to the invisible decay h 2 → 4χ 0 1 are very small.