Yukawa coupling unification in an SO(10) model consistent with Fermilab (g − 2)μ result

We investigate the Yukawa coupling unification for the third generation in a class of SO(10) unified models which are consistent with the 4.2 σ deviation from the standard model of the muon g − 2 seen by the Fermilab experiment E989. A recent analysis in supergravity grand unified models shows that such an effect can arise from supersymmetric loops correction. Using a neural network, we further analyze regions of the parameter space where Yukawa coupling unification consistent with the Fermilab result can appear. In the analysis we take into account the contributions to Yukawas from the cubic and the quartic interactions. We test the model at the high luminosity and high energy LHC and estimate the integrated luminosities needed to discover sparticles predicted by the model.


Introduction
Recently the Fermilab E989 experiment [1] has measured a µ = (g − 2) µ /2 with significantly greater accuracy than the previous Brookhaven experiment [2,3]. Thus the combined Fermilab experimental data and Brookhaven experimental data gives a exp µ = 116592061(41) × 10 −11 , (1.1) which is to be compared with the Standard Model (SM) prediction [4] a SM µ = 116591810(43) × 10 −11 . (1. 2) The combined Fermilab and Brookhaven result shows an excess over the SM result by an amount ∆a FB µ which is Eq. (1.3) records a 4.2σ deviation from the SM compared to 3.7σ for the Brookhaven result. hus the Fermilab experiment further strengthens the Brookhaven result on the possible existence of new physics beyond the Standard Model (see, however, ref. [5]). Subsequent to the Fermilab result, artificial neural network analysis was used to explore the parameter space of supergravity (SUGRA) unified models. It was seen that regions of the parameter space where supersymmetric loops can give the desired correction consistent with the Fermilab results are those where gluino-driven radiative breaking of the electroweak symmetry -1 - where

2)
(2. 3) The notation used above is as follows: ∆ µνρσλ and ∆ µνρσλ are fields for the 126 and 126 representations, Φ µνρσ is the field for the 210 representation and r Ω µ (r = 1, 2) are the fields for the two 10 of Higgs representations and Σ µνρ is the field for the 120-plet representation.
In the above W GUT breaks the SO(10) GUT symmetry down to the standard model gauge group SU(3) C × SU(2) L × U(1) Y by VEV formations of V 1 126 and V 1 126 and the VEVs of JHEP06(2021)002 210 . The equations that determine these VEVs are derived in [9]. Thus the 126 + 126-plet VEVs V 1 126 and V 1 126 break the SO(10) symmetry down to SU(5) × U(1) and the 210-plet VEVs V 1 210 , V 24 210 , V 75 210 further break the gauge symmetry down to SU(3) C × SU(2) L × U(1) Y . The notation for the VEVs is explicit. Thus, for example, V 1 126 stands for the VEV of the SU(5) singlet in the SU(5) × U(1) decomposition of 126 and V 24 210 stands for the VEV of the 24-plet of SU(5) field in the SU(5) × U(1) decomposition of 210. The doublet-triplet splitting is generated by W DT which contains 2 × 10 + 120-plets of light fields. Thus the heavy fields 126 + 126-plet and 210-plet contain three heavy SU (2) Higgs doublet pairs while the light fields 2 × 10 + 120-plets contain four light Higgs doublet pairs. After mixing of the light and heavy fields, three light Higgs doublets become heavy leaving one pair massless which we identify as the standard model Higgs doublet. The Yukawa couplings arise from cubic and quartic interactions. They are given by where Here B and Γ's are the SO(10) charge conjugation and gamma matrices [17] and W 4 are the higher dimensional interactions discussed below. Yukawa couplings arising from eq. (2.5) are given by where where U d r1 and V d r1 are defined by eq. (2.14) and evaluated numerically in tables 2 and 3. In addition to Yukawa couplings arising from W 3 , contributions arise from higher dimensional operators in W 4 where and where

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Thus W 4 gives additional contributions to the Yukawa couplings for the third generation which we denote by δh t , δh b , δh τ which are evaluated in the appendix. The total Yukawa couplings arising from eq. (2.4) is then given by where h b , h t , h τ act as boundary conditions on Yukawas of b, t, τ which are evolved down to the electroweak scale Q where they are related to b, t, τ masses so that Here we used the relations H d = v √ 2 cos β and H u = v √ 2 sin β, and where v = 246 GeV. As noted above there are seven Higgs doublet pairs three of which are heavy and four are light, and after the mixing of the light and heavy fields three pairs of light Higgs doublets become heavy and one pair remains light. To extract the light Higgs doublets we need to diagonalize the 7 × 7 Higgs doublet mass matrix given in [9]. The Higgs doublet mass matrix is not symmetric and is diagonalized by two unitary matrices U d and V d . Thus the down Higgs and the up Higgs doublet mass matrices are diagonalized by the transformation where H T d = ( (5 10 1 ) D a , (5 10 2 ) D a , (5 120 ) D a , (5 126 ) D a , (5 210 ) D a , (45 120 ) D a , (45 126 H T u = ( (5 10 1 ) D a , (5 10 2 ) D a , (5 120 ) D a , (5 126 ) D a , (5 210 ) D a , (45 120 ) D a , (45 126 (2.18) In the above the notation is as follows: (5 10 1 ) stands for the down Higgs doublet in the SU(5) − 5-plet in the 10 1 which is one of the two 10-plets of light Higgs of SO (10). Further, D's and D 's represent the normalized kinetic energy basis and normalized kinetic and mass eigenbasis, respectively of the Higgs doublet mass matrix. Since the muon g − 2 is one of the most accurately determined quantities in physics even a small deviation from the standard model prediction would be a significant indicator of new physics. For example, it is known that supersymmetric loop corrections could be of the same size as the electroweak corrections in the SM [25][26][27][28][29][30]. Indeed the Brookhaven result in 2001 [2] resulted in several works pointing out the impact on physics expected at colliders and elsewhere [31][32][33][34][35][36][37][38][39][40]. Thus the experiment became one of the important constraints on the parameter space of SUSY models. The discovery of the Higgs boson at 125 GeV further constrained the parameter space implying that the size of weak SUSY scale could be large lying in the TeV region [41][42][43][44][45]. Since the Fermilab result has indicated more strongly than the Brookhaven experiment for the existence of new physics, it is interesting to ask how the b−t−τ unification is affected [1]. The early work of [46] pointed out that such a unification could occur in SO(10) with appropriate choice of soft parameters. Such a unification has important effects on other phenomena such as dark matter (DM) [47]. Thus it is of interest to ask if b − t − τ unification can come about consistent with Fermilab data. We investigate this question using a neural network which is found to be useful in the analysis of large parameter spaces [48,49]). The analysis is done within the framework of supergravity grand unified models [50][51][52] using non-univeralities of gaugino masses [53][54][55][56][57][58][59][60][61][62]. The scan of the SUGRA parameter space is performed using an artificial neutral network (ANN) implemented in xBIT [63]. The ANN has three layers with 25 neurons per layer. It constructs the likelihood of a point using the three constraints on the Higgs mass, DM relic density and muon g − 2, i.e., m h 0 = 125 ± 2 GeV, The ANN first generates a set of points using the SUGRA input parameters which are used to train the neutral network based on the constructed likelihood function. The input parameters are m 0 , A 0 , m 1 , m 2 , m 3 and tan β where m 0 is the universal scalar mass, A 0 is the universal trilinear coupling, m 1 , m 2 , m 3 are the U(1), SU(2), SU(3) gaugino masses all at the GUT scale and tan β = H u / H d where H u gives mass to the up quarks and H d gives mass to the down quarks and the charged leptons. We notice that the ANN predicts a particle spectrum consistent withgSUGRA where the colored sparticles are heavy and the sleptons, staus and electroweakinos are lighter. Generating the sparticle spectrum requires evolving the renormalization group equations (RGEs) and for this we use SPheno-4.0.4 [64,65] which implements two-loop MSSM RGEs and three-loop SM RGEs while taking into account SUSY threshold effects at the one-loop level. The larger SUSY scale makes it necessary to employ a two-scale matching condition at the electroweak and SUSY scales [66]  We discuss now the results of our analysis. In table 1 we give an analysis of the VEVs of the heavy fields that enter in the GUT symmetry breaking for a range of GUT parameters η, λ, M 126 and M 210 where the VEVs are in general complex. The VEVs are obtained by solving the spontaneous symmetry breaking equations using W GUT . Using the VEVs of table 1, one solves for the Higgs doublet mass matrix using a range of a, b 1 , b 2 , c,c that appear in W DT . The diagonalization of the Higgs mass matrix allows us to identify the linear combination of the Higgs doublet fields which are massless and correspond to the pair of MSSM Higgs.
The diagonalization also allows for computation of non-vanishing elements of the U and V matrices that connect to the light Higgs. These are the matrix elements U d 11 , They are listed in tables 2 and 3. In table 4 we give a list of parameters that enter in the cubic couplings W 3 and in the quartic couplings W 4 . In table 5 we give the computations of the contributions of the cubic couplings, the quartic couplings and their sum for b, t, τ for the model points of table 1. Computation of b, t, τ masses using the analysis of table 5 as boundary conditions at the GUT scale and using RG evolution down to the electroweak scale is given in table 6. An analysis of the Higgs boson mass, the light sparticle masses, the dark matter relic  for the model points (a)-(j). The masses are in GeV.

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Model  density and of the supersymmetric correction to the muon anomaly is given in table 7. A  comparison between table 6 and table 7 shows that one has a unification of Yukawas and a g − 2 anomaly consistent with the Fermilab result of eq. (1.3). One may note that the dark matter relic density is not fully saturated by the model points of table 7. This implies that the dark matter may likely be multicomponent which includes other forms of dark matter, such as dark fermions of the hidden sector [75][76][77] or possibly a dark photon [78] or an axion [79,80]. A scan on the parameter space using the GUT scale input of SO(10) results in a larger set of points than those presented in tables 1-5. The range of values the input parameters take are: 0.5 < η, λ < 6.0, 0.1 < a, b 1 , b 2 , c,

Sparticle hierarchies and signal region analysis
The set of data points retained after satisfying the constraints from the Higgs sector, the DM relic density, dark matter direct detection and the LHC is further processed and points consistent with Yukawa coupling unification are kept. We observe that the spectrum consisting of light electroweakinos, sleptons (selectron and smuons) and staus belong to three cases of mass hierarchy.  Case 1. The electroweakinos,χ 0 2 ,χ ± 1 are almost degenerate, with the stau being the next-to-lightest supersymmetric particle (NLSP). The mass hierarchy here is where EW = (χ 0 2 ,χ ± 1 ) and˜ represents the sleptons.
Here we distinguish two subcategories (I) and (II) where

Case 3.
The last category also includes stau as the NLSP but the electroweakino and slepton hierarchy is inverted, i.e., Benchmarks (a), (f) belong to Case 1, while (b), (e) and (i) belong to Case 2 and (c), (d), (g), (h) and (j) belong to Case 3. Figure 2 shows the obtained data set categorized according to the above three cases.
An illustration of such a complex spectrum is given in figure 3. The upper panels correspond to benchmark (a) while the lower ones are for (d). Cascade decays are common  Figure 3. A display of the particle spectrum using PySLHA [89] for benchmarks (a) (upper panels) and (d) (lower panels). The left panels represent the spectrum up to 13 TeV while the right panels give the low-lying masses of the spectrum.
in high scale models which, unlike simplified models considered by ATLAS and CMS, produce more complicated event topology. Thus, for slepton pair production, analyses by ATLAS [81,82] and CMS [83,84] consider a 100% branching ratio of˜ → χ 0 1 which can happen in spectra belonging to Case 3. However, Cases 1 and 2 do not necessarily abide by this and one can get several decay channels making the final states more complicated.
In the next section, we select a set of benchmarks belonging to the three cases discussed above. We study slepton pair production and decay at HL-LHC and HE-LHC. We design a set of signal regions to target the rich final states corresponding to the three cases of mass hierarchies. For earlier works on SUSY discovery at HL-LHC and HE-LHC, see refs. [85,86] and the CERN yellow reports [87,88].

Slepton pair production and event simulation at the LHC
The pair production cross section of sleptons (selectrons and smuons) is proportional to the electron and muon Yukawa coupling which means that those cross sections are small compared to staus and electroweak gauginos. For our LHC analysis, we select six of the ten benchmarks shown in table 6 corresponding to sleptons in the mass range of ∼ 350 GeV to ∼ 850 GeV. The production cross sections of the slepton pairs at 14 TeV and 27 TeV are calculated at the aNNLO+NNLL accuracy using Resummino-3.0 [90,91]  shown in table 8. Also shown are the different branching ratios of sleptons but for brevity we do not exhibit the branching ratios ofχ 0 2 andχ ± 1 for benchmarks (b), (f) and (i). To have an idea of the decay channels involved, one can examine the right panel of figure 3 which shows the low-lying spectrum of benchmark (a). Since (a) and (f) both belong to Case 1, one can have an idea of the different decay channels ofχ 0 2 andχ ± 1 which involve the stau. This leads to a tau-enriched final state.
The final states which make up our signal region (SR) involve two same flavor and opposite sign (SFOS) leptons with missing transverse energy (MET). We also require at least two jets (N ≥ 2) which can be used to form kinematic variables that are effective for jetty final states. We call the signal region SR-2 Nj. For such final states, the dominant SM backgrounds are from diboson production, Z/γ+jets, dilepton production from offshell vector bosons (V * → ), tt and t + W/Z. The subdominant backgrounds are Higgs production via gluon fusion (ggF H) and vector boson fusion (VBF). The simulation of the signal and background events is performed at LO with MadGraph5_aMC@NLO-3.1.0 interfaced to LHAPDF [92] using the NNPDF30LO PDF set. Up to two hard jets are added at generator level. The parton level events are passed to PYTHIA8 [93] for showering and hadronization using a five-flavor matching scheme in order to avoid double counting of jets. For the signal events, the matching/merging scale is set at one-fourth the mass of the pair produced sleptons. Additional jets from ISR and FSR are added to the signal and background events. Jets are clustered with FastJet [94] using the anti-k t algorithm [95] with jet radius R = 0.4. DELPHES-3.4.2 [96] is then employed for detector simulation and event reconstruction using the HL-LHC and HE-LHC card. The SM backgrounds are scaled to their relevant NLO cross sections while aNNLO+NNLL cross sections are used for the signal events.

Event selection
The selected SFOS leptons must have a leading and subleading transverse momenta p T > 15 GeV for electrons and p T > 10 GeV for muons with |η| < 2.5. Each event should contain at least two non-b-tagged jets with the leading p T > 20 GeV in the |η| < 2.4 region and -12 -

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a missing transverse energy E miss T > 70 GeV. Despite the specific preselection criteria, the analysis cuts used for the six benchmarks cannot be the same. This is due to the rich final states involved. To help us discriminate the signal from the background events, we use a set of kinematic variables along with a deep neural network (DNN) which is trained and tested on two independent sets of signal and background samples. We list the kinematic variables that enter in the training of the DNN: 1. E miss T : the missing transverse energy in the event. It is usually high for the signal due to the presence of neutralinos.
2. The transverse momentum of the leading non-b tagged jets, p T (j 1 ). Rejecting btagged jets reduces the tt background.
3. The transverse momentum of the leading lepton (electron or muon), p T ( 1 ).

M T2
, the stransverse mass [97][98][99] of the leading and subleading leptons where q T is an arbitrary vector chosen to find the appropriate minimum and the transverse mass m T is given by 6. The dilepton invariant mass, m , helps in rejecting the diboson background with a peak near the Z boson mass which can be done by setting m > 100 GeV.
7. The opening angle between the MET system and the dilepton system, ∆φ(p T , p miss T ), where p T = p 1 T + p 2 T .
8. The smallest opening angle between the first three leading jets in an event and the MET system, ∆φ min (p T (j i ), p miss T ), where i = 1, 2, 3.
We use the DNN implementation in the 'Toolkit for Multivariate Analysis' (TMVA) [100] framework within ROOT6 [101]. The DNN employed has three dense hidden layers with 128 neurons per layer and tanh as an activation function to define the output neurons given the input values. The DNN trains on the signal and background events using the above set of kinematic variables in three phases with a decreasing learning rate. After the 'learning' process is over, the DNN tests the predictions on another set of signal and background samples. Despite having one background set, the training and testing must be done every time a signal sample is used, i.e., six times in our case. During the testing stage, the DNN creates a new discriminator which is called the DNN response or the DNN score. Cuts on this new variable maximizes the signal (S) to background (B) ratio, S/ √ S + B.  We give in table 9 the set of analysis cuts on a select number of kinematic variables along with the new 'DNN response' variable. Variations in cuts are used for our six benchmarks depending on the hierarchy of the spectrum which allows us to put them in three categories with (b),(i) as the first, (f) as the second and (d),(g),(h) as the third. The values shown in parentheses are the modified cuts at 27 TeV which are essential to improving the S/ √ S + B ratio.

Results
We begin by discussing the benchmarks (d), (g) and (h) which belong to Case 3. Here the mass splitting between the slepton and the neutralino is large, ranging from 300 GeV to 600 GeV, which produces very energetic leptons. For those benchmarks, the sleptons decay to a light lepton and a neutralino with a 100% branching ratio (see table 8) which makes for a clean final state. The most effective kinematic variables for this case are M T 2 and p 2 T where the latter is the transverse momentum of the subleading lepton. We present two-dimensional plots in these variables in the middle panels of figure 4. The left panel depicts point (d) and the right one is the dominant diboson background. One can clearly see that the largest number of background events (color axis) are concentrated at small M T 2 and p 2 T while for the signal larger values are highly populated as well due to the energetic final states. A hard cut on M T 2 and p 2 T as well as the 'DNN response' can reject most of the background events.
Next, we discuss benchmarks (b) and (i) which belong to Case 2. Here the branching ratios to a lepton and a neutralino are smaller, at 31% and the slepton-neutralino mass gaps are at 85 GeV and 140 GeV, respectively. Such a mass gap is not enough to allow harder cuts on p 2 T and that's why it has been omitted in table 9. For this reason, we make use of the leading and subleading transverse momenta of the leptons to reconstruct the total momentum of the system, p T , to form the new variable E miss    Finally, for point (f) which belongs to Case 1, the branching fraction to a lepton and a neutralino is the smallest compared to its decay to a second neutralino and a chargino. The second neutralino and chargino decay predominantly to a stau which in turn decays to a neutralino and a tau. Hence we are faced with a case of tau-enriched final state which can hadronize forming jets. In our selection, we have rejected b-tagged jets but made no special requirements on tau-tagged jets. For this particular case, jets (tau-tagged or not) can be used to reject the SM background through the variable ∆φ min (p T (j i ), p miss T ) defined above. In the bottom panels of figure 4 we show this variable plotted against m for point (f) (left panel) and the Z/γ+jets background (right panel). Excluding the region formed by ∆φ min (p T (j i ), p miss T ) < 1 rad and m < 100 GeV is effective in reducing the SM background.
Along with cuts on the variables discussed thus far, the 'DNN response' plays an important role. We show in figure 5 distributions in this variable after the above cuts have been implemented. The top panel depicts benchmark (b) which shows clearly that -16 -JHEP06(2021)002 at 14 TeV this point cannot be discovered with 3000 fb −1 while the signal is in excess over the background near 1 for 2800 fb −1 at 27 TeV. The bottom panels show point (g) also at 14 TeV (left) and 27 TeV (right). The benchmark is discoverable at both HL-LHC and HE-LHC but requires smaller integrated luminosity for discovery at HE-LHC (700 fb −1 ) than at HL-LHC (2100 fb −1 ). The evaluated integrated luminosities for discovery at both machines are summarized in the lower part of table 9. Entries with 'NV' indicate that the benchmark is not discoverable at the corresponding machine. Note that there is a modest improvement in the integrated luminosity at HE-LHC in comparison to HL-LHC but the former is expected to gather data at the rate of ∼ 820 fb −1 per month, so most of those points will be discoverable within the first two to three months of run. Note that points (f), (g), (h) and (i) are discoverable at both machines while (b) and (d) can only be discoverable at HE-LHC.

Conclusion
In this work we have investigated if high scale models can produce Yukawa coupling unification consistent with the Fermilab muon g − 2 result. We used a neural network to investigate the parameter space of a class of SO(10) models where Yukawa couplings arise from the cubic as well as the quartic interactions. As in a recent work it is found that the preferred parameter space lies in a region where gluino-driven radiative breaking of the electroweak symmetry occurs. The model produces a split spectrum consisting of a light sector and a heavy sector. The light sector contains light sleptons and light weakinos, and the heavy sector contains the gluino, the squarks and the heavy Higgs. The masses of the light sparticles lie in the few hundred GeV range and are accessible at the LHC. With the help of a deep neural network, we carried out a dedicated search of sleptons in the two-lepton final state at HL-LHC and HE-LHC. It is found that most of the considered benchmarks are discoverable within the optimal integrated luminosity of HL-LHC while all of them are discoverable at HE-LHC with less integrated luminosities.
Thus W (1) 4 gives the following contribution to the third generation Yukawas The contribution of W (2) 4 to the third generation Yukawas is given by Finally, the contribution of W to the third generation Yukawas is given by (A.10) The total Yukawas are the sum of the contributions from the cubic and from the quartic terms at the GUT scale as given in eq. (2.12).

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.