Muon and electron ( g − 2 ) anomalies with non-holomorphic interactions in MSSM

The recent Fermilab muon g − 2 result and the same for electron due to ﬁne-structure constant measurement through 133 Cs matter-waveinterferometryareprobedin relation to MSSM with non-holomorphic (NH) trilinear soft SUSY breaking terms, referred to as NHSSM. Supersymmetric contributions to charged lepton ( g − 2 ) l can be enhanced via the new trilinear terms involving a wrong Higgs coupling with left and right-handed scalars. Bino-slepton loop is used to enhance the SUSY contribution to g − 2 where wino mass stays at 1.5 TeV and the left and right slepton mass parameters for the ﬁrst two generations are considered to be the same. Unlike many MSSM-based analyses completed before, the model does not require a light electroweakino, or light sleptons, or unequal left and right slepton masses, or a very large higgsino mass parameter. In absence of popular UV complete models, we treat the NH terms at par with MSSM soft terms, in a model


Introduction
The discovery of the Higgs boson [1,2] at the Large Hadron Collider (LHC) almost a decade ago gave the Standard Model (SM) of particle physics [3] a strong foundation. However, SM has its limitations both in the theoretical as well as in the observational sides. The gauge hierarchy problem, matter-antimatter asymmetry, no candidate for dark matter [4,5] are to name a few in this regard. This demands the existence of a Beyond the Standard Model (BSM) physics. Low energy Supersymmetry (SUSY) [6][7][8][9][10][11][12][13][14] is especially attractive in this context since it can address the gauge hierarchy problem associated with the Standard Model (SM) and also it is able to provide with particle dark matter candidates. Additionally, we must not forget that the Higgs boson is found to have a mass of 125 GeV, which is well below the predicted upper limit for an SM-like Higgs particle of the Minimal Supersymmetric Standard Model (MSSM) [9][10][11][12][13][14]. Thus, over the past decades, SUSY, with its strong theoretical appeal and its ability to influence a variety of observables of phenomenological interest, continues to remain as the most attractive candidate for a BSM physics.
Undoubtedly, a BSM physics demands nothing less than direct observations of new particles at the LHC. This will then lead us to the possible new symmetries and interactions present in nature.
However, even after a decade of running of the LHC, we are yet to see a cherished new particle. We may as well need to accept the fact that BSM particles could perhaps be staying quite far from our experimental reach. Keeping hope for a collider discovery, we must at the same time continue to look for possible indirect signatures of SUSY. This may come from flavor physics, electroweak physics precision tests and dark matter. Concerning the above, we remember that the anomalous magnetic moment of muon, a µ = 1 2 (g − 2) µ , stands out prominently over the past two decades showing some degree of disagreement (over 2 to 3σ) of the experimental result as obtained in Brookhaven [15] with that of SM evaluations performed at different times. The hadronic vacuum polarization part of the SM result has a large uncertainty, particularly the lowest order part of the same that requires analysis in the non-perturbative regime. The non-perturbative aspect may require input from effective field theory like chiral perturbation theory, hadronic models, dispersion relations together with experimental data like e + e − → hadrons, and Lattice Quantum Chromodynamics (LQCD). A comprehensive analysis explaining the break-up of different contributions to the SM result of a µ may be seen in Ref. [16] 2 . While the recent Fermilab a µ data [18,19] is consistent with the same from Brookhaven, the difference ∆a µ has grown larger. The combined data from Fermilab and 2 A chosen set of references among various important past contributions to (g − 2)µ evaluation may be found in Ref. [17] Brookhaven show a 4.2σ level of discrepancy [19] as given below.
Interestingly, ∆a µ can be ascribed to a SUSY µ , the SUSY contributions to a µ which in turn will help us to constrain the SUSY model parameter space.
With further results to come from the Fermilab in the near future and the data from upcoming experiment JPARC [20], muon g − 2 can shed light on various BSM physics models. In this context, we must point out the recent Lattice result [21] for the hadronic vacuum polarization. This has effectively shifted a SM µ to move toward a exp µ rather closely causing tension between the dispersive and LQCD modes of evaluations of the hadronic uncertainty amount within a SM µ . We would also like to point out that an agreement between a exp µ and a SM µ may invite issues with global electroweak fits to electroweak precision observables. This is because the existing deviation of the above two a µ values is related to precision electroweak predictions via the common dependence on hadronic vacuum polarization effects [22][23][24]. In any case, such important issues will be transparent in future, but at this point we will use Eq.1 for a SUSY µ . On the top of (g − 2) µ , we would also include the existing deviation for (g − 2) e , the anomalous magnetic moment for electron. A smaller but not insignificant discrepancy exists for the electron g −2 anomaly arising out of the measurement of the fine-structure constant that used 133 Cs matterwave interferometry. An approximately 2.5σ level of discrepancy is given below [25], In many new physics models with flavor universality, one finds 1. This is also true in SUSY 3 . In contrast to the above, the measurement values referred to in Eqs.1 and 2 lead to an appreciably larger negative value for the above quantity, namely: Thus, using the two constraints simultaneously leads to a rather difficult situation. We note that the right hand side of Eq.4 is only the central value. Appropriate error estimates of the two magnetic moments may be used for obtaining the combined uncertainty values. On the other hand, use of Eqs.1 and 3 lead to R e,µ 8. Clearly, the later case of having simultaneous positive values for the two deviations with also a smaller R e,µ is easier to accommodate in SUSY analyses. In the absence of a resolution of the ∆a e puzzle, we choose to consider the rather difficult 133 Cs-based value of Eq.2.
This will analyze (g − 2) µ,e in the framework of Non-standard soft SUSY breaking terms [27,28] contrast it with other SUSY-based analyses that also used the 133 Cs-based result of Eq.2. We will also show the result of using the 87 Rb data briefly just for the sake of completeness.
Besides (g − 2) µ,e we will also include dark matter constraints in our analysis. The plan of our work is given below. In Section 2 we will describe the Non-holomorphic MSSM (NHSSM) model and its signature on the SUSY spectra. We will also discuss the constraints arising from avoiding charge and color breaking (CCB) minima. Apart from the above, we will also mention the difference of status between the non-holomorphic soft parameters with the ones of regular MSSM soft terms in the context of ultraviolet (UV) completion. In Section 3 we will discuss the SUSY contributions to the magnetic moment of charged leptons. The above will also emphasize the role of Yukawa threshold corrections in MSSM that may be important for satisfying Eq.2. We will then discuss the effect of non-holomorphic trilinear interactions on leptonic magnetic moments (g − 2) l and how the trilinear NH terms may provide the necessary threshold effects appropriate for Eq.2. We will particularly outline the parameter zone that would be consistent with a higgsino dark matter as a multi-component dark matter element with relic density obeying only the upper limit from the PLANCK data. Then we will discuss the combined case of obeying (g − 2) µ and (g − 2) e in MSSM and see how NHSSM effects can generate appropriate threshold corrections to y e and also to some lesser extent to y µ . We will discuss the essential points of past MSSM-based analyses in contrast to the features of NHSSM. We will see that there is no need to assume a flavor unfriendly choice for slepton masses in NHSSM, neither we do we need to consider any superheavy higgsino state to generate Yukawa threshold corrections. In Section 4 we will present the results in two separate parts namely (g − 2) µ with DM and inclusion of (g − 2) e for the combined analysis. We will also use LHC constraints from slepton pair-production. Finally, we will conclude in Section 6.

MSSM: Superpotential and Soft terms
The MSSM Superpotential is given by [9], Here, for two doublet chiral superfields A and B, one has A.B = αβ A α B β , where αβ is an antisymmetric (Levi-Civita) tensor in 2-dimension. Y e ij , Y u ij and Y d ij are lepton, up and down type of Yukawa matrices respectively. H D and H U with hypercharges -1 and 1 respectively refer to down and up type of doublets of Higgs chiral superfields that contain both the Higgs scalars and and their fermionic partners higgsinos. L i , Q i and E i , U i are left handed doublet and right handed singlet chiral superfields of applicable fermions and their scalar superpartners.
The MSSM soft terms including the non-holomorphic scalar mass terms and the holomorphic trilinear coupling terms are as given below [9].
GeV. The Yukawa couplings and masses are related via the vevs as y e = me 2) etc. The above y e relates to Y e ij the leptonic Yukawa matrix as Y e 11 = y e , Y e 22 = y µ , Y e 33 = y τ 4 . Similar notation holds good for the quark Yukawa matrices.

Non-holomorphic soft terms
Ref. [27] enumerated the MSSM soft terms shown as −L sof t in Eq.6. It also listed a few additional SUSY breaking interactions in a general sense that would be regarded as hard SUSY breaking terms in presence of a gauge singlet scalar field [29]. On the other hand, in absence of any such singlets as in MSSM, such terms grouped within −L sof t shown as below, are no longer of hard SUSY breaking type [28][29][30] and these are labelled as non-holomorphic soft SUSY breaking terms or the so-called "C-terms" in SUSY texts [10]. In MSSM these are regarded as soft SUSY breaking terms.
Here, instead of the Higgs scalar doublets one has their conjugates h c d and h c u . With appropriate hypercharges, h c d couples with the up-type of squarks and h c u goes with the down type of squarks and sleptons. This is why the above is often referred as a scenario with wrong Higgs coupling.
Ref. [28] analyzed such terms (along with the MSSM soft terms) in a model independent way. Instead of a model based analysis, in an agnostic point of view the authors named the framework as "Minimal Effective Supersymmetry" where the new soft parameters considered were to be treated at par with the ones of Eq.6 and these were left to be determined from low energy data only.
In the quest of a model, Ref. [30] considered a hidden sector based F-type supesymmetry breaking scenario including two chiral superfields. The author obtained terms of −L sof t terms to be of the indicating suppression by the scale of mediation M of SUSY breaking. Arising from the same analysis, there is no such supression for the MSSM soft terms of Eq.6. Clearly, according to the model of Ref. [30] these are highly suppressed terms if the scale of mediation is close to the grand unification theory (GUT) scale or the Planck scale. However, Ref. [30] also pointed out that the above does not make the terms of −L sof t irrelevant for issues starting from a possible incorrectness in using a simplistic form of spontaneous SUSY breaking to involvement of multiple high scales etc.
It is known that building models for nonstandard soft SUSY breaking terms or the "C-terms" is difficult [10]. We cite a few available analyses here. Refs. [31][32][33] may be seen for analyses with generalized supersoft SUSY breaking that have relations to non-holomorphic soft terms. Generating non-holomorphic terms based on gauge mediated SUSY breaking may be seen in Ref. [34]. Here, the authors discussed the possibility of finding gauge invariant supersymmetric direct Yukawa couplings between the Higgs and the messenger fields. One-loop corrections involving messenger fields in the loop may lead to the wrong-Higgs gaugino operators [34] that may become important in the low energy theory below the scale of SUSY breaking 5 . Ref. [36] relates to non-holomorphic terms arising out of D-brane instantons stretched between SUSY breaking and visible sectors.
It is clear that the NH terms of −L sof t are not friendly to supergravity [30,37] types of scenarios or they may be difficult to be found from a popular UV complete theory. Similar to Ref. [10,30] we are also of the opinion that the NH terms are hardly irrelevant in spite of their difficulty in model building and we may consider the approach of "Minimal Effective Supersymmetry" of Ref. [28].
Based on the inputs given at a high or a low scale we classify below the past phenomenological analyses with non-standard soft terms that considered no explicit model or in other words were consistent with the "Effective" approach of Ref. [28]. Analyses of Refs. [38][39][40][41][42][43][44][45][46][47] used renormalization group evolutions within a Constrained MSSM (CMSSM [9]) like setup. Here the NH parameters were considered to be of unknown origin and were at par with other CMSSM mass and trilinear parameters. Similarly, there are works with phenomenological MSSM (pMSSM) [48] like inputs [49][50][51][52] where the NH parameters given at a low scale were treated at par with the MSSM soft SUSY breaking parameters. Thus as with previous analyses, we consider the new parameters to be of unknown origin given at a low scale. The tree-level Higgs potential is unaffected, but this is not so for the charge and color breaking terms of the scalar potential [50]. The presence of the higgsino mass soft terms with coupling µ may cause isolation of a fine-tuning from the Higgsino mass µ since at tree level higgsino mass would have components from the superpotential (Eq.5) as well as from the soft term of (Eq.7) [45,46,49]. The mass matrices for the scalars get modified in the off-diagonal component involving L-R mixing. For example, a slepton mass matrix may be written as, We note that going from MSSM to NHSSM, µ tan β gets replaced by (µ + A l ) tan β in the offdiagonal entries. In the electroweakino sector the higgsino mass entries are altered from µ to µ + µ leading to the following neutralino and chargino mass matrices in NHSSM.
The present analysis will consider vanishing µ .

Charge and Color Breaking:
Avoiding a Charge and Color Breaking (CCB) minima in NHSSM [50] while considering both the holomorphic and non-holomorphic trilinear couplings requires a 4-vev scenario, like the vevs for H u , Heref stands for the concerned sfermion. With A f and A f both present there is no possibility of considering a 3-vev scenario unlike in MSSM. A rather straigtforward computation as shown in Ref. [50] results into the following inequalities for avoiding a CCB minima. It is seen that unlike MSSM, there is no D-flat direction so that terms with g 2 1 + g 2 2 come into the picture arising out of the D-term potential.
where m 2 1,2 = m 2 H d ,Hu + µ 2 . One finds that even with a very large A e as in our analysis, the above constraint is easily satisfied because of the g 2 1 + g 2 2 term that becomes very large due to a small y e in the denominator. This is entirely different from the MSSM case that has a D-flat direction coming out in a 3-vev based scenario 6 .
Apart from a global vacuum stability where a lot of MSSM parameter space can be excluded for a large value of A f corresponding to the first two generations of leptons (f ≡ e, µ), we must point out that for a long-lived universe, the CCB conditions corresponding to the light fermion cases are readily evaded. The above is quite commonly used in SO(10) based analyses (e.g Ref. [56]) to label a parameter point valid even when the absolute stability is affected. This is true as long as the CCB inequalities are satisfied for the large Yukawa coupling cases i.e. the inequalities involving the third generation trilinear couplings like A t . The rate of tunneling from the Standard Model like false vacuum to a CCB true vacuum is proportional to e −a/y 2 [55] (follows from Eq. 9 and 11 of Ref. [57]), where a is a constant and y is the associated Yukawa coupling for the colored/charged fields. If the dangerous third generation of sfermion constraints are already avoided, the rate of tunneling corresponding to a small Yukawa case may be very small. As mentioned in Ref. [57] these are the cases where D-term contributions to the potential cannot be neglected. NHSSM with large values of A e also falls into this class and it is thus additionally consistent with a long-lived universe consideration. Thus for NHSSM, the inqualities involving A e , A e and A µ , A µ are either satisfied for absolute stability or they can be ignored as in MSSM for a long-lived universe leading to cosmological stability [56,57].

Leptonic (g − 2) and Yukawa Threshold Corrections in MSSM
In MSSM, as shown in Fig.1, at the one-loop level the leading contributions to a SUSY µ come from χ 0 −μ andχ ± −ν µ loops [58,59]. The required chirality flip may be found from a SUSY Yukawa coupling of a higgsino to a lepton, an appropriate slepton (μ) or sneutrinoν µ . Otherwise, the 6 For MSSM one has the following: [11,[53][54][55]. Using Ref. [61] the one-loop contributions are given as follows.
where i and m refer to the four neutralino and two chargino states whereas k indicates the two smuon states. The referred couplings are given by, The loop functions for the neutralino and chargino loops namely F N,C 1,2 may be seen in Ref. [61]. A simplified result follows when the loops that contribute most are the ones with chargino-sneutrino and bino-smuon fields [62].
where the loop functions f are as given in Ref. [62].

Yukawa Threshold Corrections
In MSSM, the Yukawa couplings are modified because of soft interactions. We would now like to discuss Yukawa threshold corrections for fermions that affects (g − 2) l as a higher order effect. At one-loop level, the lepton Yukawa coupling in MSSM can be given as, Here ∆ l in MSSM is given by [63], Here, m 1,2 are the chargino masses and the loop function I is as given in Ref. [63]. As we will discuss for NHSSM, the last term µI(M 1 , ml L , ml R ) will be altered via µ → (µ + A l ). Including these corrections, the one-loop contributions get modified as, ∆ l , which is proportional to tan β, contains the all order re-summation of the tan β enhanced contributions [64,65]. We further note that with the assumption that all the SUSY masses are equal and much larger than the W-boson mass M W , ∆ µ = −0.0018 tan β sign(µ) [63,64]. ∆ µ may be much larger for unequal SUSY particle mass parameters [63]. A larger ∆ l that is itself proportional to tan β, can even influence a l 's tan β-dependence via y l . In the limit of radiative mass generations of fermions, a l can become almost independent of tan β [66]. Such effects of the radiative generation of mass of fermions in analyzing g − 2 were considered in Refs. [63,[66][67][68][69][70][71]. The authors pointed out the role of non-holomorphic trilinear interactions on lepton Yukawa couplings y l and (g − 2) l . A similar control of (g − 2) e via enhancement of Yukawa threshold corrections in MSSM was used in Ref. [72] where the authors considered very large values of µ (up to 500 TeV).
We will discuss some details of the work in Sec.3.2.
Following Sec.2, an off-diagonal element of a slepton matrix would look like m l (A l − (µ + A l ) tan β), indicating a generic alteration of µ of MSSM going to µ + A l in NHSSM wherever there is an L-R mixing. The same will be true in the last term (L-R) of the last line of Eq. 21. Since for an electron the aforesaid off-diagonal term is multiplied by m e which is too small, one must have a very large value of (µ + A e ) so as to get a finite L-R mixing effect in relation to the diagonal terms.
With an electroweak fine-tuning-friendly low µ, we see that A l has to be very large so that µ may be ignored in µ + A l . Thus, with an appropriately large A l , ∆ l in NHSSM may be approximately proportional to A l , and the effect on y l can be much larger than what may be possible in MSSM due to a potentially large enhancement factor 1/(1 + ∆ e ). We will demonstrate how the radiatively corrected Yukawa coupling y e may become very important while discussing the case of a e in Sec.4.
Additionally, the sign correlation of a l with respect to A l can be ascertained from the relevant bino-slepton loop result of Eq.19 when one changes µ to µ + A l . Since ∆ e is proportional to tan β we will see that the threshold corrections to y e will cause a e to be a slowly increasing function of tan β with some intercept, but unlike MSSM it would no longer be proportional to tan β (Eq.22).
There have been an appreciable number of SUSY-based analyses on (g − 2) µ after the announcement of Fermilab result . Some of the above works also involve dark matter constraints.
DM relic density could be satisfied via higgsino or wino type of LSPs provided one considers underabundant scenarios with the possibility of multiple candidates for DM. In this case, χ ± −ν µ loop may contribute dominantly to a µ . One can similarly consider bino-wino mixed LSP in this regard.
One can also consider bino type of LSP which obtains correct relic density by self-annihilation via s-channel H/A boson, with a µ constraint being satisfied via the contribution from the bino-smuon loop with relatively light smuons. Alternatively, bino-stau coannihilation may be used for DM relic density generation and a µ constraint may be similarly addressed. As we will see in our work with nonstandard trilinear soft terms, we consider higgsino to be the LSP (in an underabundant choice for DM). Regarding a µ , we use the contribution from the bino-smuon loop that is enhanced by larger L-R mixing due to the above soft terms. We will further see that the large threshold corrections (due to the nonstandard soft terms) to leptonic Yukawas, particularly y e , can be useful to accommodate the a e constraint simultaneously.

Accommodating a e constraint in addition to a µ limits
Eq.4 summarizes the requirement for a new physics to accommodate both the constraints. The ratio is required to be not only large but also negative. We would like to address the essential parts In contrast to the above analyses that had to manage the flavor issues carefully and are strongly constrained by µ → eγ or may have to live with light SUSY spectra or very large tan β, we focus on including nonstandard soft terms. We must emphasize here that as mentioned before, the MSSM soft terms and the nonstandard soft SUSY breaking terms do not have the same status. While the regular soft terms are highly supported by popular UV complete models, the NH terms are in general difficult for modelling. As mentioned earlier, in a model independent standpoint, we analyze the effect of the nonstandard terms considering an approach of "Minimal Effective Supersymmetry" of Ref. [28]. An analysis with NH terms would not demand any large µ, or light spectra and we 7 Going beyond MSSM, i) Ref. [99] is an analysis with supersymmetric gauged U (1) Lµ−Lτ model, ii) Ref. [100] pointed out the level of phenomenological stringency of the issue while citing varieties of generic models including MSSM, leptoquarks and little Higgs inspired models and showed the usefulness of abelian Lµ − Lτ to explain the anomalies. Ref. [100] also commented on the possibility of non-holomorphic trilinear terms to address the issue. iii) Ref. [101] used additional U (1) SUSY model with family dependent (non-universal) U (1) charges, iv) Ref. [102] used inverse-seesaw extended NMSSM to address the anomalies.
will use a flavor friendly scenario of having identical slepton masses for the first two generations with equal coefficients for the left and right mass parameters. However, we will use two different non-holomorphic trilinear parameters for the first two generations of leptons, both given as inputs at the low scale. In general, had these been given at a much higher scale like the grand unification scale we might run into flavor issues. Nevertheless, a detailed analysis of the exact degree of flavor violation, is beyond the scope of this work.

Muon magnetic Moment in NHSSM
Our NHSSM analysis on lepton g −2 in its first part involves studying the effects of the nonstandard soft terms on the enhancement of the SUSY contributions to muon g-2. We will explore at the beginning the constraints from (g − 2) µ data and analyze the parameter space that would be consistent with dark matter (DM) and other constraints. We will demonstrate the mechanism that enhances (g − 2) µ in NHSSM. For DM, we will consider the case where the lightest supersymmetric particle (LSP) is expected to obey only the upper limit of the relic density bound from PLANCK data [106]. We will identify the parameter space that would satisfy all the direct detection limits from XENON1T namely both the spin-independent [107] and spin-dependent [108] type of data.
Once the above analysis delineates the parameter space, we will explore the effect of imposing the constraint from the electron's magnetic moment (g−2) e due to fine-structure constant measurement.
We will see how the Yukawa threshold corrections due to NHSSM effects can affect leptonic g − 2, particularly (g − 2) e . Because of the above, we will see that it is indeed possible to find NHSSM parameter regions such that the scaled magnetic moment ratio of interest R e,µ as discussed before may be consistent with Eq.4. We use SARAH-4.14.4 [109,110] and SPheno 4.0.4 [111,112] to implement the model and for computing SUSY spectra and observables.
In order to generate a sufficient amount of SUSY contributions to leptonic g − 2 in NHSSM we take the help of chirality flip via the Left-Right scalar mixing due to the non-holomorphic trilinear SUSY breaking interaction parametrized by A l . We use T l as an input defined via T l = y l A l (with 2) ). This follows the input convention used in the codes SARAH-SPheno (for example see Ref. [45]) for all the trilinear soft parameters. We should point out that with this parametrization a value of a few GeV for T l may mean a large value for A l . The largeness is most prominent for electron, whereas for top quark, the associated values of T t and A t are not so different from each other. We must also remember that because of the appearance of the fermionic mass m e in the offdiagonal entry of Eq.8, any non-negligible L-R mixing of selectrons would need a significantly large A e because of the smallness of the electronic mass. In one-loop vertex correction radiative diagrams ( Figure 1), this is achieved via the NH trilinear interactions of scalars in the bino-slepton loops 8 .
Thus, both the above trilinear parameters and the mass of bino M 1 will have significant roles in our analysis apart from the masses of sleptons for which we consider equal left and right mass parameters m L , m R for the first two generations. We must emphasize that the above NH terms may potentially enhance a l significantly in comparison with the MSSM contributions involving chargino or neutralino loops.
In order to probe NHSSM effects clearly we keep the SU(2) gaugino mass to be sufficiently heavy, much above µ or M 1 . A large part of parameter space where (g − 2) e can be accommodated in our work involves i) direct effect of larger A e on the bino-seletron loop of Fig.1 (i.e. through Eq.19 with µ → µ + A e ) and ii) via the effect of the enhancement factor 1 1+∆e directly on a e (Eq.22). Of course, 1 1+∆e influences the Yukawa coupling for electron (y e ) directly via A e through the same L-R mixing. The effect of the said NH trilinear mixing may easily supersede the contribution from the higgsino mixing superpotential term characterized by µ. On the other hand, the corrections may become smaller for larger slepton mass values. Keeping our choice of scalar mass spectra in tact, we will rather limit the associated NH trilinear SUSY breaking parameters such as A e so as to limit y e . We will label the above as the "Limited Threshold Corrections of Yukawa Coupling (LTCYC)" zone. To specify explicitly, by LTCYC zone of y e we mean that the radiatively corrected value of y e can at most be twice the corresponding MSSM value, or in other words 1 1+∆e < 2. However, with the chosen SUSY mass parameters we will see that the required agreement with data for (g − 2) e range is possible even with a factor quite smaller than two. As we will point out in Section 4.3.2 the chosen SUSY parameter space that satisfies (g − 2) µ requires only moderately large A µ in contrast to A e for the case of (g − 2) e . Thus, LTCYC for y µ is automatically satisfied for all the parts of our analysis meant for (g − 2) µ . We will explore explicitly how far LTCYC for y e as a condition becomes important when we incorporate (g − 2) e as a constraint in Section 4.3.
Toward the end, we will impose the constraints from ATLAS search for slepton-pair production [113] in the LSP-slepton mass plane and explore a few benchmark points that would satisfy all the constraints. While the neutralino-slepton loop dominated by bino type of neutralino would enhance (g − 2) l , we will primarily consider higgsino as the LSP which would be suitable for satisfying the DM relic density data from PLANCK (only the upper limit, in a multi-component DM scenario) as well as the spin independent (SI) direct-detection (DD) scattering cross section data of LSP-nucleon scattering. The bino-higgsino mixed region would obviously be disfavored or discarded via the SI DD constraints. Below, we focus on the minimal requirement for enhancing SUSY contributions 8 We use vanishing coefficients A l for the trilinear SUSY breaking parameters throughout our study.
to leptonic g − 2 and the variables that are closely connected to g − 2 and dark matter. With the above in mind, we keep the squark masses decoupled to large values and we do the same for the tau-sleptons too. All the left and right handed scalars including that of the first two generations of sleptons will assume equal SUSY breaking mass parameter input values (m L = m R ). We keep the trilinear coefficient for top-squark namely T t at a high negative value that would be consistent with Higgs mass data. The Table 1  With the above parameter space been defined, some of our constraints are that from the Higgs data, Br(b → sγ), relic density upper limit from PLANCK. This is apart from the (g − 2) µ and (g − 2) e constraints mentioned earlier and the SI and SD direct detection constraints from XENON1T that depend on the mass of the LSP. With M SUSY being large, we consider a 3 GeV theoretical uncertainty in SUSY Higgs mass leading to the following as the acceptable range [114] for the SM-like Higgs in MSSM.
We plot Figure 2 that describes our result of the SUSY contributions to a µ (referred hereafter as a µ itself) in relation to the relevant SUSY breaking parameter M 1 , the mass of bino. Fig. 2(a)  We continue to study the behavior of a µ concerning the relevant SUSY breaking parameters in Figure 3. Undoubtedly, a µ is enhanced most prominently via the trilinear parameter A µ via its strong effect through chirality flipping L-R scalar interaction in the bino-smuon loop. For tan β = 10, m L = µ = 600 GeV and M 2 = 1.5 TeV, Fig.3(a) shows the variation of a µ with respect to M 1 for a few different values of T µ , 100 GeV to 500 GeV in steps of 100 GeV. A substantial amount of enhancement of a µ occurs due to a change in T µ . In this analysis, we satisfy all the charge-breaking related constraints for NHSSM (Eq.11) [50] and stay within the LTCYC zone for the threshold corrections to y l due to the variation over T l . Fig.3 to dominate over all the other diagrams, at the lowest order, µ tan β gets replaced by (µ + A µ ) tan β in Eq. 19. With vanishing A µ , and A µ (= T µ /y µ ) large enough to offset µ, the sign of a µ is determined by the sign of A µ . Numerically this is seen to be valid as long as |T µ | > ∼ 20 GeV or so. Consistency with the 1σ band demands that (g − 2) µ data can be satisfied with only positive T µ . Fig.3(c) displays the variation of a µ with M 1 for µ satisfying 150 < µ < 1000 GeV. The blue and red regions refer to tan β = 10 and 40 respectively. The limited level of thickening of lines even for a large tan β demonstrates that there is only a mild degree of dependence of a µ on µ over the range mentioned above. This is consistent with the fact that a higgsino-smuon or a charged higgsino-sneutrino loop for a µ are hardly important in our analysis with a dominant effect due to a NH trilinear term.
We now use the (g − 2) µ constraint in the most important M 1 − T µ plane in Fig.4(a). The fixed parameter under the study are tan β = 10, µ = 300 GeV, m L = 600 GeV and M 2 = 1.5 TeV.
We divide the region into 1σ (blue), 2σ (green), and 3σ (gray) zones with respect to the (g − 2) µ constraint. The 1σ region extends from T µ = 175 GeV to 450 GeV while M 1 spans the entire space chosen in our analysis. The region bents toward the left and this is indeed consistent with what follows from Fig.3(a). Fig.4(b) shows similarly constrained region with the same color convention as above. A generic decrease of a µ with increase in m L is apparent since large T µ is necessary to stay in a given fixed colored zone for large m L values.   Clearly, a µ is large for small m L or small smuon mass regions. We must note that the reach of the smallness of m L to obtain an enhanced a µ effectively may lead to picking up small M 1 regions. This is particularly true for higher µ cases. The requirement ofχ 0 1 to be the LSP, which in our case is either a bino or a higgsino dominated state, to a reasonable degree of approximation demands m L to be higher than M 1 or µ whichever is the lowest among the last two. There are other reasons why the bottom white regions of each of the figures are excluded. These may be due to the requirements of avoidance of tachyonic sleptons or charge-breaking minima. On the other hand, small m L regions find stringent constraints from the LHC data which we will discuss later. Comparing the figures, we can further see that, as expected, larger T µ values enhance a µ . For a given a µ , and a fixed M 1 , a larger T µ creates a possibility to accommodate a larger m L . A large tan β like 40 enhances a µ .

Constraints from Dark Matter
We now include the constraints from dark matter in this NHSSM (g − 2) µ analysis. Of course, with our choice of an underabundant scenario of a higgsino dark matter, no NHSSM trilinear parameter would directly affect the DM analysis. We will study DM for its effects on the combined  parameter space. In our analysis, wino mass M 2 is chosen to be quite heavy (1.5 TeV), whereas the higgsino mass has a range of 150 < µ < 1000 GeV, and for bino, we have 100 < M 1 < 1000 GeV.
Our parameter space further consists of heavy tau-sleptons (2 TeV) along with decoupled squarks, whereas we choose the range of the first two generations of sleptons to vary within 200 GeV to 1 TeV.
With m A = 2.5 TeV as an input the neutral CP-even or CP-odd Higgs masses are way above to encounter the so-called funnel region that is characterized by bino-dominated LSPs undergoing selfannihilation via s-channel processes involving A or H-bosons. Coming to the higgsinos, the chosen mass range is such that the upper limit of 1 TeV is what is necessary for higgsino LSP to become a single component DM but the upper limit of M 1 is also chosen to be the same. A chance equality of M 1 and µ would produce a large bino-higgsino mixing, but this would not be friendly with the SI direct detection. On the other hand, regions with M 1 < µ would produce overabundance. Hence, in most of the available parameter space, the LSP is of higgsino type, except in a few occasions when there is a possibility of χ 0 1 -slepton coannihilations. With higgsino LSPs to have mass below a TeV the associated relic density is supposed to satisfy only the upper limit of DM relic density constraint from PLANCK data. One of the important channels for DM production would be the coannihilation channel between the higgsno LSP with the lighter chargino state (χ 0 1 −χ ± 1 ) which is also higgsino dominated in nature. Fig.6(a) shows the Spin-independent scattering cross-section of the LSP with a proton (σ SI χp ) as a function of the LSP mass for tan β = 10 for the parameter space mentioned in Table 1. We used micrOMEGAs 5.2.13 for dark matter-related computations [118][119][120]. Considering the large number of points with underabundance of DM we multiply σ SI χp with a scale factor ξ which was defined in Eq.24. Only those points satisfying the DM relic density upper bound, Higgs mass data, B-physics related limits and (g − 2) µ at 2σ level are plotted and these are shown in green. The blue line is the constraint from the spin-independent (SI) direct detection (DD) experiment of XENON1T indicating discarded regions above the line (at 90%CL).
Using interpolation we will further use the XEONON1T line to judge whether a parameter point with a given mass of the LSP would survive the SI DD experiment limit. Fig.6(b) shows the results for tan β = 40.  Fig.7(b), shows a similar plot for tan β = 40. Here m χ 0 1 ranges from 150 GeV to 760 GeV. Both for tan β = 10 and 40, the upper limit of the mass of χ 0 1 is restricted via the XENON1T SI-DD data. We now include the Spin-dependent direct detection of DM study in our analysis. Fig.8(a) shows the Spin-dependent (SD) scattering cross-section of the LSP with a neutron (σ SD χn ) as a function of the LSP mass for tan β = 10. As before, we multiply σ SD χn with a scale factor ξ (Eq.24). Only those points satisfying the DM relic density upper bound as well as (g − 2) µ at 2σ level are plotted and these are shown in green. The blue line is the constraint from the SD direct-detection experiment of XENON1T indicating discarded regions above the line at 90% CL. We further checked that the constraint from σ SD χp is weaker than σ SD χn . Fig.8(b) is a similar plot for tan β = 40. Compared to the SI DD cross-sections of Fig.6, here we do not find any new discarded region. This is true for both choices of tan β. Hence, all our conclusion for the SI DD cross-section analysis remain valid and analyzing σ SI χp itself is sufficient for a conclusion regarding the direct detection of DM. Below, we probe how large a µ can go depending on the variation of basic MSSM parameters like M 1 and m L and NHSSM parameter T µ . Thus, Fig.9(a) shows a scatter plot of a µ vs. a µ typically arise in small M 1 zones and this is likely to be case due to the associated smallness of slepton masses (see Figure 5). The figure shows that the SI direct detection cross-section is below the XENON1T limit throughout the domain of variation chosen for M 1 . The results shown in Fig.9(b) confirm larger MSSM contributions to a µ for an increase of tan β to 40, although the enhancement is much smaller than the NHSSM effects toward a µ , hence not so visible.  Fig.11(a). In combination, the resulting off-diagonal slepton mass values for large T µ and small m L are likely to generate excluded regions because of the appearance of tachyonic slepton states or vacuum instability. With T µ = y µ A µ , the effect becomes more prominent for a larger tan β as may be seen in Fig.10(b) and Fig.11(b). We further note that the large a µ regions corresponding to small m L values in each of the parts of Figure 11 refer to small M 1 and large T µ zones. The LSP that satisfies the DM constraints is of higgsino type and associated with underabundance.

Inclusion of Electron Magnetic Moment limits; Yukawa coupling enhancement in NHSSM
We will now probe the effect of including the 133 Cs-based fine-structure constant measurement derived electron g −2 data in our analysis on top of the muon g −2 constraint. We will only consider the necessary amount for T e that would generate the required value for a e . We will also highlight the threshold corrections to the electron Yukawa coupling y e due to T e and constrain the NHSSM parameter space accordingly. This will involve both cases namely a free enhancement of y e as well as limiting y e by requiring to stay within the LTCYC zone as mentioned earlier. Fig.12(a) shows the effect of T e on the electron Yukawa coupling y e for µ = 400 GeV and m L (= m R ) = 500 GeV.
The blue lines are for tan β = 10, and the red lines are for tan β = 40. For each tan β, the solid lines refer to M 1 = 300 GeV whereas the dotted lines are drawn for M 1 = 700 GeV. No LTCYC condition for y e is applied here in order to display the extent of the Yukawa threshold correction of y e . A large negative value for T e gives rise to a larger |y e | compared to the same for an identical positive value of T e . y e is also seen to flip its sign as T e becomes positive. A larger M 1 also enhances |y e |. We point out that no LTCYC condition is applied here in order to show the extent how y e is affected due to an increased T e . Depending on the value of m L and M 1 , the above may in turn cause appearance of tachyonic selectron states. We now try to understand how a e behaves as T e is varied. Fig.12(b) displays a e for two different values of tan β, and two different values of M 1 identical with those of Fig.12(a). The other SUSY parameters are µ = 400 GeV and m L (= m R ) = 500 GeV as used before. Unless T e is tiny, negative value of a e that is required due to the experimental data is correlated with negative value of T e . We note that the solid lines of a e for tan β = 10 and 40 for M 1 = 300 GeV are very close to each other. The same is true for the two dotted lines corresponding to M 1 = 700 GeV. The reason for the approximate independence of a e on tan β would be clear in Section 4.3. The display of y µ for a variation of T µ as shown in Fig. 12(c) indicates a larger available zone of variation of T µ compared to that of T e of Fig. 12(a). This arises because of the difference of mass values of electron and muon. No LTCYC for y µ is an issue here since the threshold correction of y µ is moderate corresponding to the given span of variation of T µ and the chosen SUSY mass parameters. A large negative value for T µ gives rise to a larger |y µ | compared to the same for a similar positive value for T µ . Unlike the case of y e of Fig.12(a), here y µ does not change sign when T µ becomes positive, albeit within its given range of interest for our study. A larger M 1 also enhances |y µ |. Fig. 12(d) shows the variation of a µ over T µ . We note that with a given value T (e,µ) , varying M 1 will alter y (e,µ) which in turn would affect a e,µ .      at 90% CL. The constraint from σ SD χp is weaker than σ SD χn . Fig.8  space (using T l = 0). None of the dark matter constraints are applied here. Throughout the analyses, we confirm that the points that satisfy σ SI χp also respect both n and p SD DD cross-sections namely, σ SD χn and σ SD χp respectively. The 1σ allowed band for a µ is shown as black horizontal lines.     Fig.12(a) for electron. Fig.12(d): A similar plot like Fig.12(b) for muon.   The non-holomorphic soft terms may be able to cause large Yukawa corrections for fermions, particularly the leptons so that the radiatively corrected values may be a few times the MSSM specified value namely y l,(ref ) ≡ y l = me 2) . This was already discussed for phenomenological analyses in Refs. [63,[66][67][68][69][70][71] where lepton masses were generated radiatively, obviously requiring a large radiative corrections. We should also point out that the present Higgs decay to e + e − from the LHC can only give an upper bound for y e which is about a few hundred times of its SM value [121][122][123] 11 ..
In our analysis, a negative a e may be accommodated by a relatively large radiative corrections to y e . Of course a choice of a very large slepton mass would be able to suppress the radiative corrections, but here we want to explore the minimal zone of values of |T e | that with the chosen SUSY mass spectra of our analysis would be consistent with limits from a e . Since a e is driven by y e , in general, both for muon and electron, as mentioned before we choose to discard any parameter point that leads to y l /y l(ref ) > 2 with l ≡ e, µ and denote it as Limited Threshold Corrections of Yukawa Coupling (LTCYC) criterion. Fig.14(a)   not produce any appreciably large amount of radiative corrections to y µ . Hence, LTCYC for y µ is not an issue for a µ , so that the condition gets automatically satisfied for the full parameter space under consideration. Indeed it may be seen that a positive T µ as required for positive a µ leads to values below unity for y µ /y µ(ref ) . 11 As described in Ref. [122] Yukawa couplings may not be proportional to fermion mass in presence of higher dimensional operators (6 or more )that respect SM gauge symmetry. Thus a very different Higgs-electron-electron coupling may be possible to arise from new physics effects from these operators. There may be a significant amount of cancellation between the contributions to the mass of electron coming from the SM Yukawa coupling and that obtained from higher dimensional operators. Similarly, the Higgs-electron-electron coupling will have two components, one from SM and the other from the higher dimensional operators. See also Ref. [69] for a SUSY context The fact that NHSSM parameter T e is able to generate a significant amount of correction to y e , while on the other hand there is an approximate sign correlation between a e,µ with T e,µ gives one a reasonable expectation for the scaled magnetic moment ratio R e,µ = (ae/m 2 e ) (aµ/m 2 µ ) to be within the desired zone, namely R e,µ −15 (see Eq.4). Figure 15 is a scatter plot of the above ratio with parameters in the x-axis, namely T e and M 1 that are important for the radiative corrections to Yukawa coupling y e . All other parameters including also T µ are varied according to the mentioned respective ranges of Table 1 considering tan β = 10 while also applying the LTCYC conditions for the two leptons as mentioned earlier. All the points (shown in green) satisfy the DM relic density constraint for its upper limit, the XENON1T data for SI direct detection, a e and a µ limits showing the desired R e,µ values well within its uncertainty limits. Fig.15 T e for tan β = 10. All the points (shown in green) satisfy the DM relic density constraint for its upper limit, the XENON1T data for SI direct detection, a e and a µ limits. LTCYC conditions are applied in addition. Fig.15(b): Similar scatter plot for a variation of M 1 .
We will now investigate how the conclusion of Section 4.2 gets modified when we further restrict the NHSSM parameter space toward satisfying the (g − 2) e constraint. First, since we demand y e to be within the LTCYC zone, |T e | should be small and this comes to around 14 GeV or less. Considering the fact that SUSY contributions to a e needs to be negative, the limits on T e becomes −14 < ∼ T e < 0. Fig. 16(a) shows the combined results (green points) of imposing all the constraints like Higgs mass data, B-physics limits, DM constraints for relic density and SI direct detection in addition to a µ and a e limits drawn in the plane of M 1 − T e . Here and henceforth in all the subsequent figures of this work LTCYC limits for y e and y µ are understood to have been applied by default. The valid region for M 1 that satisfy all the constraints comes out to be 230 < M 1 < 800 GeV for tan β = 10. Fig. 16(b), a similar plot for tan β = 40 has a valid M 1 zone satisfying 230 < M 1 < 660 GeV.  We are to explore now the effect of imposing (g − 2) e constraint on the M 1 −µ plane of Figure 18.
We will compare the above with the corresponding result of Figure 7 drawn for (g − 2) µ . The availability of NHSSM parameter space gets reduced significantly because the threshold corrections to y e should not become too large. We remember that the above corrections at least involve masses of bino, sleptons, as well as the trilinear NH coupling T e . Hence, a given set of M 1 and m L that is available in the muon (g − 2) analysis, may not be consistent when combined with non-vanishing T e to satisfy (g − 2) e . Thus, when one imposes the (g − 2) e constraint, that in turn is sensitive on the corrections to y e , the reduction of parameter space is rather unavoidable. The green points of Fig. 18(a) satisfy (g − 2) e,µ constraints at 2σ level in addition to obeying the dark matter relic density bound. The associated σ SI χp values also fall below the XENON1T limit. Fig. 18(b) shows a similar result for tan β = 40 where a largeness of tanβ further takes away a significant amount of parameter space so as to have y e within LTCYC zone. Fig.19(a) is a scatter plot in the plane of trilinear NH parameters for electron and muon for tan β = 10. The points shown in green satisfy the DM relic density constraint, the XENON1T σ SI χp data along with (g − 2) µ and (g − 2) e limits at 2σ level. The points also have y e in the LTCYC zone. Clearly, T e should be appreciably large so as to satisfy (g − 2) e data. We find the valid region to have −14 GeV < T e < 0.

LHC Constraints from slepton pair production and constraint from com-
With no difference in SUSY sparticle content between NHSSM and MSSM, we directly apply the SUSY constraints from LHC data on our analysis. We intend to identify the exclusion region of NHSSM parameter space from the ATLAS data for slepton pair production that considered selectron and smuon in the analysis [113]. The later gives an exclusion region in the (mχ0 1 − m L ) plane. We note here that by directly applying the ATLAS bound on our parameter space we are taking a conservative approach as the exclusion can potentially get somewhat weaker for the higgsino LSP scenario. The reason is as follows. The ATLAS limit is derived for a simplified model allows for a significant branching ratio of the sleptons to final states involving ν l and χ ± 1 . This may lead to a reduction in the number of signal leptons and thus to the weakening of the exclusion limit.
We will further analyze the constraint from a compressed scenario with higgsino as LSP that is associated with closely spaced values for the masses m χ 0 1 , m χ ± 1 and m χ 0 2 . This is based on the ATLAS result given in Fig.14a of Ref. [124]. we henceforth consider the lowest value for a higgsino type of LSP to be 185 GeV. 12 Fig.21(a) shows a scatter plot in m L − mχ0 1 plane for tan β = 10. All the points satisfy the perturbativity of y e . Considering the constraint for compressed higgsino states as mentioned above, 12 On the other hand, regarding a compressed LSP-slepton scenario there is hardly any constraint from LHC (Fig.   16a of Ref. [113]) in our analysis.  1 is allowed to have values above 185 GeV. The points shown in green satisfy the DM relic density constraint, the XENON1T σ SI χp data along with (g − 2) µ and (g − 2) e limits at 2σ level. The points also have y e in the LTCYC zone. On the top of the figure we draw the black line that indicates the current exclusion bounds in the m L − mχ0 1 plane at the LHC. The green points that stay outside the contour are the residual parameter points that would also satisfy the LHC limit. On the higher LSP mass side, irrespective of slepton masses, for tan β = 10 a value of mχ0 1 between 400 to 500 GeV would satisfy all the constraints considered in this analysis. For slepton masses above 700 GeV, valid LSP mass would lie below 350 GeV. Fig.21(b) shows a similar result for tan β = 40.
Here, the LHC contour engulfs a lot of region of the parameter space with a larger mass of the LSP as shown in green. The region with slepton masses above 700 GeV allows valid LSP mass up to 275 GeV. Apart from the above, the LSP mass values for valid parameter points outside and near the black contour in its lower side are correlated with m L in an approximately linear fashion. One finds that with tan β = 10, mχ0 1 can be as large as 400 GeV when m L = 550 GeV whereas the later numbers are 400 GeV and 500 GeV respectively for tan β = 40.

Only (g − 2) µ
Since the LHC data is seen to affect the combined analysis of (g − 2) µ and (g − 2) e with dark matter rather strongly, we feel that it is important to go back and examine the situation of considering only the (g − 2) µ constraint with dark matter (as in Section4.2) in relation to the ATLAS constraints described above. The results are seen in Figure 22(a) and 22(b). Figure 22(a) shows the ATLAS Figure 20: Allowed (cyan) and disallowed (red) regions in the plane of m χ 0 2 − ∆ 21 with respect to the 1σ-level contour (black line) as given by the ATLAS data namely Fig.14a of Ref. [124]. The corresponding lower limit of m χ 0 1 satisfying the above constraint is about 185 GeV in this higgsino data [113] satisfied appropriate mass values of sparticles in the m L − mχ0 1 plane. As before, we also impose the constraint from the compresed higgsino scenario. For tan β = 10, satisfying all the constraints, valid mχ0 1 ranges from 400 GeV to 750 GeV for any value of m L . On the other hand when m L > 700 GeV, mχ0 1 assumes values up to the same upper limit namely 750 GeV. For tan β = 40, as seen in Figure 22(a) the above numbers for mχ0 1 are in between 400 to 775 GeV, whereas for m L > 700 GeV, valid mχ0 1 zone extends to 775 GeV. Apart from the above, the LSP mass values for valid (green) parameter points outside and near the black contour in its lower side are correlated with m L in an approximately linear fashion. One finds for both the values of tan β, mχ0 1 can be as large as 400 GeV when m L = 550 GeV. We now show two representative points of our analysis in Table 2 for two values of tanβ.
Referring to Figure 21(a) that corresponds to satisfying both (g − 2) µ and (g − 2) e constraints, the two points are chosen based on the ATLAS provided contours [113] for slepton pair production shown in black. We choose tanβ = 10 and 40 for specifying two benchmark points BP1 and BP2 both of which correspond to moderate values of m χ 0 1 and slepton mass parameter m L . The points are far away from having any effect due to the constraints from compressed spectra. We remind that all the left and right slepton mass parameters are taken to be equal. The LSP is almost a higgsino in nature. The relic density is satisfied via likeχ 0 1 −χ 0 2 orχ 0 1 −χ ± 1 coannihilation mechanisms. Apart from other constraints like (g − 2) e,µ and dark matter, the constraints from B-physics are easily satisfied because of large squark masses of the third generation. With a larger stop-mixing via A t , the Higgs mass is satisfied within the limits of Eq.23. Notably, a e value of BP2 for tan β = 40 is similar to that of BP1 corresponding to tan β = 10. This is because, as discussed earlier, in NHSSM, the SUSY contributions to a e has a milder tan β dependence unlike a proportional relationship as seen in MSSM. With LSP as a higgsino it is a candidate of a subdominant component of DM.
Hence we use a scale factor ξ (Eq.24) that further reduces the SI and SD direct detection cross sections. Certainly, the valid parameter space would be larger for both small and large tan β if we had chosen only (g − 2) µ among the two magnetic moment constraints.    Figure 12. The points shown in green satisfy the DM relic density constraint, the XENON1T σ SI χp data along with (g − 2) µ and (g − 2) e limits at 2σ level. The points also have y e in the LTCYC zone. A milder deviation corresponding to Eq.3 makes the result to have similarity with the case of using only the a µ constraint. Similarly, we plot Figure 23(b) indicating parameter points in the mχ0 1 − m L plane. Further details regarding constraint from the ATLAS data are same as those of Figure 21. Expectedly, the result is closer to that of Figure 22

Conclusion
The Standard Model of particle physics has achieved a remarkable degree of success ever since the discovery of the Higgs Boson. Apart from the above discovery itself, the measurements of Higgs couplings, various results from the ATLAS and the CMS experiments of the LHC, several results from low energy precision measurements involving flavor physics also point towards a strong validity of SM. However, theoretical issues such as the gauge hierarchy problem, and observational evidences like the dark matter, baryon asymmetry, non-zero neutrino mass etc. demands the SM to be extended beyond its current boundary. Hence, a search for a BSM physics is highly relevant in the present era. In this connection, we note that the recent Fermilab experiment on muon g − 2 measurement has confirmed the deviation from the SM value obtained in the previous measurement at BNL. Assuming that the SM result that depends on low energy e + e − data will not get drastically altered in the near future, or in other words, it would take a while for the lattice-based computations of the hadronic error to reach a definite conclusion, it is important to pursue the implications of the Fermilab announcement from the BSM physics perspective.
The combined discrepancy of the Fermilab and the BNL data is at an impressive level of 4.2σ.
The anomaly amount itself is of the order of electroweak corrections in SM. Thus, it is very crucial to probe a BSM physics scenario that may produce a large contribution to the above observable at the same energy scale. SUSY, undoubtedly one of the most important candidates for new physics, is well-known to be capable of producing a large correction to (g − 2) µ . There are regions of parameter space in MSSM with appropriate wino, bino, higgsino and left and right handed smuon masses that can explain the Fermilab result, typically for a large tan β. There are also dedicated beyond the MSSM models that can enhance the (g − 2) µ contribution by virtue of the model properties, including new symmetries. In this analysis, staying within the MSSM sparticle spectra, we augment MSSM with non-holomorphic trilinear soft breaking interactions. We also pointed out that unlike MSSM soft terms, the non-holomorphic terms which for MSSM are soft SUSY breaking in nature do not have the privilege of being supported by popular UV complete models. The appropriate NH term generates an enhancement of the SUSY contributions to (g −2) µ via a coupling effect which helps to easily accommodate the data. The model neither demands any flavor unfriendly choice of different right and left slepton masses nor does it demand a large tan β or a light chargino. One uses a large bino-smuon loop contribution with stronger mixing of the L-R smuons to produce the desired effect. Using the Fermilab data to constrain the model, we further impose constraints due to dark matter related observables. It is found that a higgsino-dominated dark matter having an underabundant contribution to the DM relic density can be consistent with direct detection cross-section as well as (g − 2) µ . For (g − 2) µ , a scenario with dominant binosmuon loop contribution can satisfy the data via a large L-R mixing caused by the wrong Higgs couplings of the non-holomorphic terms.
We further impose the relevant direct search bounds from the LHC, e.g. the constraints from the slepton pair production searches from ATLAS as well as the constraint coming from the compressed higgsino searches. The conclusion of the combined (g − 2) µ analysis with dark matter and collider limits is the following (Figure 22). For tan β = 10 and 40, satisfying all the constraints, valid mχ0 1 ranges from 400 GeV to 750 GeV for any value of m L . On the other hand when m L > 700 GeV, mχ0 1 stays in between the lower limit of ∼ 185 GeV to the same upper limit namely 750 GeV. Apart from the above, the LSP mass values for the valid parameter points outside and near the ATLAS provided contour in its lower side are correlated with m L in an approximately linear fashion. One finds for both the values of tan β, mχ0 1 at its minimum can be around 400 GeV when m L 600 GeV. We then extend the muon g − 2 analysis in NHSSM to include the electron g − 2 data as derived from the fine-structure constant measurement based on 133 Cs matter-wave interferometry. The resulting (g − 2) e shows an approximately 2.5σ level of discrepancy, but interestingly, it comes with a negative sign. It becomes very challenging to accommodate the muon and electron g-2 anomalies simultaneously in MSSM. This measurement is in contrast to the 87 Rb-based measurement which shows a smaller positive deviation. Here, we use the more unfriendly 133 Cs-based data in our work to compare our results with previous MSSM-based analyses and for probing the potential of the non-standard soft terms in relation to any related phenomenology. The present work approaches to accommodate a e via appropriately enhancing the Yukawa threshold corrections for y e by using minimum possible T e values for the chosen mass spectra. The analysis satisfies constraints from vacuum stability and charge and color breaking minima. The negative sign for the SUSY contributions to a e could be generated by negative T e . An interesting feature of a e in NHSSM is that one cannot enhance it appreciably by increasing tan β. a e at best can be a slowly increasing function of tan β, but unlike MSSM, it is no longer proportional to tan β, an effect that arises out of a large threshold correction to y e .
The conclusion of the combined (g − 2) µ and (g − 2) e analysis with dark matter and collider limits is the following (see Figure 21). For tan β = 10, satisfying all the constraints, valid mχ0 1 ranges from 400 GeV to 500 GeV for any value of m L . On the other hand when m L > 700 GeV, mχ0 1 stays in between the lower limit of ∼185 GeV to the same upper limit namely 500 GeV. For tan β = 40, the ATLAS contour engulfs a lot of parameter space with larger mass of the LSP leaving a small window of m χ 0 1 between 400 to 425 GeV for any m L . For m L > 700 GeV, the valid mχ0 1 zone is from 185 GeV to 275 GeV. Apart from the above, the LSP mass values for valid parameter points outside and near the ATLAS provided contour in its lower side are correlated with m L in an approximately linear fashion. One finds that for tan β = 10, mχ0 1 can be as large as 410 GeV for m L below 550 GeV. For tan β = 40, mχ0 1 can be as large as 425 GeV when m L is below 440 GeV. Finally, we presented two representative points that satisfy all the constraints considered in this analysis. We also discussed briefly the results of considering the limits of a e as coming from the 87 Rb-based experiment indicating a smaller deviation from the SM result.