Exploring non-holomorphic soft terms in the framework of gauge mediated supersymmetry breaking

It is known that in the absence of a gauge singlet field, a specific class of supersymmetry (SUSY) breaking non-holomorphic (NH) terms can be soft breaking in nature so that they may be considered along with the Minimal Supersymmetric Standard Model (MSSM) and beyond. There have been studies related to these terms in minimal supergravity based models. Consideration of an F-type SUSY breaking scenario in the hidden sector with two chiral superfields however showed Planck scale suppression of such terms. In an unbiased point of view for the sources of SUSY breaking, the NH terms in a phenomenological MSSM (pMSSM) type of analysis showed a possibility of a large SUSY contribution to muon g − 2, a reasonable amount of corrections to the Higgs boson mass and a drastic reduction of the electroweak fine-tuning for a higgsino dominated χ˜10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\tilde{\chi}}_1^0 $$\end{document} in some regions of parameter space. We first investigate here the effects of the NH terms in a low scale SUSY breaking scenario. In our analysis with minimal gauge mediated supersymmetry breaking (mGMSB) we probe how far the results can be compared with the previous pMSSM plus NH terms based study. We particularly analyze the Higgs, stop and the electroweakino sectors focusing on a higgsino dominated χ˜10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\tilde{\chi}}_1^0 $$\end{document} and χ˜1±\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\tilde{\chi}}_1^{\pm } $$\end{document}, a feature typically different from what appears in mGMSB. The effect of a limited degree of RG evolutions and vanishing of the trilinear coupling terms at the messenger scale can be overcome by choosing a non-minimal GMSB scenario, such as one with a matter-messenger interaction.


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gauge bosons [53][54][55]. Unlike a bino dominated LSP, a higgsino type of DM would not require any fine-adjustment of uncorrelated SUSY parameters. But a higgsino satisfying the observational relic density limits is typically quite heavy [53][54][55] (∼ 1 TeV) 1 giving rise to a large electroweak fine-tuning in MSSM. It is thus desirable to have a model that has a higgsino like LSP but is able to guard fine-tuning to become large. We further note that a higgsino like LSP scenario of phenomenological MSSM (pMSSM) [57] is significantly free from the LHC bounds [58,59].
The other issue at hand in MSSM is the stringent muon g − 2 constraint [60][61][62]. Satisfying the muon g − 2 constraint [63] in the MSSM is associated with a significant degree of reduction of parameter space. As we know, SUSY contributions to muon g − 2 are enhanced when the lighter electroweakinos ( χ ± 1 and χ 0 1 ) or the lighter smuon massμ 1 become small. Concerning Br(B → X s + γ) [64], the constraint can be effective in MSSM for a high tan β [65][66][67][68], the ratio of Higgs vacuum expectation values and non-decoupled zone of squark masses.
All the above that restrict the MSSM parameter space may be addressed reasonably well when one includes non-holomorphic (NH) soft terms on the top of the usual holomorphic soft terms of MSSM [69,70]. Staying within an MSSM setup terms like φ 2 φ * or a higgsino mass term like ψψ can be soft SUSY breaking in nature if there is no presence of a gauge singlet field [71]. There have been several works over the past two decades [69][70][71][72][73][74][75][76][77][78][79][80][81][82] that analyzed the effects of introducing NH soft terms. 2 There are three distinct signatures i) the trilinear NH terms may enhance L-R mixing thus influencing various phenomenologies, ii) higgsinos get an additional soft-term source, iii) the Higgs potential remains unaffected at the tree level thus electroweak fine-tuning may hardly change. The latter effect can give rise to a valid higgsino dark matter with low fine-tuning, whereas the L-R mixing effect may contribute to (g − 2) µ , Br(B → X s + γ), or the higgs mass radiative corrections [69,70,[78][79][80][81]. The NH soft terms can also influence on the scalar potential terms involving colored/charged scalars [82]. Thus, there are associated charge and color breaking (CCB) minima constraints that one must take care of in the limiting cases of large trilinear NH couplings.
All the above effects in combination may potentially cause very distinct outcomes in global analyses of low energy SUSY models [84][85][86][87]. In a model-dependent standpoint, one finds that such NH soft terms may arise out of interactions like 1 [71]. In the above one considers a scenario where supersymmetry is broken in the hidden sector via an auxiliary field F of the chiral superfield X. Φ is another chiral superfield and M refers to the mediation scale. Both these terms have a strength [70]. Clearly, there are large suppressions (∼ 1/M ) for the NH soft terms in supergravity scenarios where M can be very large nearing the Planck scale. However, the suppression effects may be small if one considers a SUSY breaking mechanism that would allow a low mediation scale.
With the above motivation, we like to explore the effects of having NH soft terms in gauge mediated supersymmetry breaking (GMSB) where SUSY breaking may occur at a JHEP01(2018)158 smaller scale and the scale of mediation, the messenger scale is also low. We will try to gauge, how far the electroweak scale analysis with NH soft terms of ref. [70] referred as Non-Holomorphic Supersymmetric Standard Model (NHSSM) preserves its conclusion in a minimal GMSB (mGMSB) setup [88][89][90][91][92][93][94][95]. We remind that the mGMSB has gravitino as its dark matter candidate with its mass ranging from a few eV up to O(1) GeV [94,[96][97][98][99][100][101] whereas the Next-to-lightest-SUSY Particle (NLSP) is largely a bino dominated neutralino. We will focus here on realizing an mGMSB setup with higgsino as the NLSP. This is unlike the typical case of a bino like NLSP that is characteristic of GMSB types of analyses [88][89][90][91][92][93][94]. 3 We will investigate the effects of NH soft terms on NLSP decaying to a gravitino and a Z-boson χ 0 [94,98] while assuming the other higgs bosons to be much heavier than the NLSP.
We will now briefly describe the plan of the work. In section 2 we will outline the gauge mediated breaking SUSY mechanism and introduce non-holomorphic soft terms within the above framework. We will discuss the effect of NH terms on Higgs and electroweakinos. In section 3 we will describe the results of the relevant parameter scanning on the top-squark and higgs boson masses and investigate the phenomenologies involving Br(B → X s + γ) and muon g − 2. We will compare our results with the scenario of MSSM with NH terms where one gives all the input parameters at the electroweak scale. Furthermore, focusing on a parameter space where the NLSP is a higgsino dominated lightest neutralino, a scenario typically unavailable in mGMSB, we will estimate the relevant NLSP decay widths for a higgsino producing gravitino and other particles like h and Z bosons. Finally, we will conclude in section 4.

Gauge mediated SUSY breaking and non-holomorphic soft terms
In the Gauge-Mediated-SUSY breaking (GMSB) Models [88][89][90][91][92][93][94] SUSY is broken in a hidden/secluded sector. In the absence of the knowledge of the hidden sector, one may consider a spurion S which is a chiral superfield, singlet under the SM group. SUSY breaking in this sector is realized via S acquiring a vacuum expectation value (vev) via its scalar and auxiliary components. Thus, The parameters M and F ≡ √ F are the fundamental scales related to GMSB. Apart from the hidden sector and the observable sector where MSSM fields reside, there is a messenger sector that itself experiences the SUSY breaking and mediates the SUSY breaking it is affected with, to the observable sector. The messenger sector connects to the hidden sector via the singlet spurion field S that goes into a superpotential containing superfields of the messenger and hidden sectors. This results into a SUSY breaking via the two vevs of the scalar and the auxiliary components of S namely, M and F causing a splitting of the masses of the messenger sector scalars (m 2 φ,φ = M 2 ± F ) and fermions (m ψ,ψ = M ). The fact that the scalars should not go tachyonic, so that the vacuum stability is unaffected JHEP01(2018)158 demands F < M 2 . 4 Considering a large M , we can integrate out the messenger scalars and fermions that are charged under the SM gauge group SU(3) C × SU(2) L × U(1) Y . The low energy effective theory below M would then break supersymmetry in the observable sector. The observable sector soft terms like the masses and couplings are generated via messenger particles in loops. This ensures renormalizability, a positive feature of GMSB models. The MSSM soft terms thus obtained via gauge boson and gaugino interactions are also flavor blind. It is the messenger sector particles, assumed to be heavy for phenomenological reasons characterize the low energy phenomenology. The gaugino masses arise out of one-loop diagrams such as that involving a messenger scalar and a fermion. The scalar mass squares arise out of two loop diagrams that may include messenger scalars and fermions apart from gauge bosons and gauginos.
Thus, the gaugino masses m λ and scalar mass squares m 2 f read, is the SUSY breaking scale in MSSM. The parameters m λ and mf come with similar values. The trilinear parameters are small and considered to be vanishing at the messenger scale, a result of the fact that the messengers can only interact with SM fields via gauge interactions. Vanishing of trilinear couplings at the messenger scale may also be seen in ref. [95] that used wavefunction renormalization method without a need of using Feynman diagrams in this regard. The trilinear couplings associated with the third generation of scalars are however non-vanishing at the electroweak scale via renormalization group evolutions. A soft mass value of mf ∼ 1 TeV would set F M ∼ 10 5 GeV (eq. (2.2)), though this neither specifies F nor M . We note that the scalars should not break vacuum stability, as mentioned before, demands F < M 2 . In its limiting case of an equality, one has F M = M = 10 5 GeV. This results into M = 10 5 GeV and F = 10 5 GeV. Considering the inequality itself, one obtains the lower bounds M > 10 5 GeV and F > 10 5 GeV.
The upper limit of the messenger mass M comes from the relative degree of strengths of the gauge and gravity mediations since the superfield S may also cause gravity effects, though in much smaller strength, in addition to the SUSY breaking associated with a GMSB scenario. The scalar mass squares from the gravity and gauge mediated scenarios may now be compared. A representative factor of 1/1000 for the ratio of the two soft mass squares would mean g 2 where M P is the Planck mass [94]. The above results into an approximate upper bound M < 10 15 GeV. 5 Correspondingly, for a 1 TeV scalar mass, one finds F < 10 10 GeV. Summarizing, one finds, 10 5 < F < 10 10 GeV, and, 10 5 < M < 10 15 GeV.
Considering the supergravity relation the gravitino mass is given by Depending on F (eq. (2.3)), the gravitino mass may range from ∼1 eV to ∼1 GeV. 4 Realistic scenarios rather satisfy F M 2 . 5 This is via considering a value of g corresponding to the strong coupling at the electroweak scale.

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Now, within the MSSM framework the NH soft terms are given by [69,70] As mentioned in section 1, terms like φ 2 φ * and a higgsino mass soft-term ψψ (eq. (2.4)) may originate from D-term contributions like 1 [70,71]. Since the terms are of strength M , a low mediation scale M such as that appears in a GMSB scenario may be relevant for probing the phenomenological implications. The parameters for trilinear non-holomorphic soft terms are small, similar to the same of trilinear holomorphic terms at the mediation scale M . We will keep the other D-term soft breaking NH interaction namely the higgsino mass soft term characterized by µ (to be given at the scale M ), to have an unknown SUSY breaking origin. 7 This is considering the issue of an associated re-parametrization invariance of the higgsino mass soft term [73,76,80,81]. Reparametrization comes from unrelated quantities like µ, the higgsino mixing superpotential parameter and the higgs scalar soft mass parameters m 2 H U and m 2 H D . An assumption of an independent SUSY breaking mechanism to have a higgsino mass soft term essentially avoids such concerns (See discussions in [80,81] along with the references therein).
Coming to the minimal GMSB, to preserve the gauge coupling unification one considers messengers to belong to a complete SU(5) representation or any other complete representation of a larger gauge group that includes SU(5) as a sub-group. With S as the spurion field mentioned before one has the superpotential, In the simplest case, there will be N 5 number of flavor of messenger copies Φ andΦ transforming as 5 and5 representations of SU(5). The soft SUSY breaking parameters like 6 There can be an NH soft term like ψλ involving higgsinos and gauginos. This would however take us away from MSSM, hence ignored [71]. 7 Additionally, such a term cannot originate at a one-loop level in mGMSB, thus becoming suppressed.
Here α refers to SM gauge group ( is given as, The scalar masses are given by, Here, f (x) is given by, C α is the quadratic Casimir of the representation of the gauge group factor G α under which the scalar fieldf transforms. For the gauge groups involved, the Casimirs are: Under the assumption of F M 2 one has g(x) 1 and f (x) 1. This results into, and , The following are the model parameters where the NH higgsino soft parameter µ 0 is the input value of µ at the messenger scale M at which the common NH trilinear coupling parameter A 0 vanish. Λ, M, tan β, N 5 , sign(µ), and µ 0 . (2.12) The NH trilinear couplings of eq. (2.4) modify the off-diagonal elements of the scalar mass matrix as given below [69,70].
Similar mass matrices for sleptons (or a down-type of squarks) would have the off-diagonal term −m e (A e − (µ + A e ) tan β). Clearly, the NH trilinear couplings contribute to the offdiagonal elements where µ is replaced by µ + A f (f = u, d, e etc.). The above indicates a more significant impact of L-R mixing for i) low values of tan β for the up type of squarks and ii) high values of tan β for down type of squarks or sleptons.

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Now, the discovery of the Higgs boson with mass of 125.09±0.24 GeV [1-3] is translated into a large amount of radiative corrections to the mass of the lighter neutral CP-even Higgs boson h. The above requirement causes an increase in the masses of the top-squarks in MSSM and/or one needs an appropriately large value of |A t |, so as to have a stronger t L −t R mixing. Thus, the NH soft trilinear parameter A t that affects the L-R mixing has important contributions toward the above radiative corrections for small values of tan β.
The lightest CP-even higgs boson mass up to one loop can be read as follows [20]. (2.14) Here, Clearly, A t = 0 corresponds to the MSSM result. Herē m t refers to the running top-quark mass that includes corrections from the electroweak, quantum chromodynamics (QCD) and SUSY QCD related effects [103]. The maximal mixing scenario refers to With a suitable A t , it is possible to satisfy the Higgs mass constraint with a relatively smaller value of |A t | in NHSSM, compared to MSSM.
The changes in the neutralino and chargino mass matrices are as shown below. Essentially, µ is replaced by µ + µ as in the tree level results.
We note that the collider bound on the lighter chargino mass, except the issue of radiative corrections, will apply to essentially |µ + µ | instead of |µ| in case χ ± 1 is higgsino dominated in nature. Since the Higgs potential is unaffected, the electroweak fine-tuning measure at tree level is still dependent on µ [70,80,81] rather than it has anything to do with µ . A large higgsino mass with less electroweak fine-tuning becomes a possibility.
In contrast to the NHSSM that is based on electroweak scale inputs we should keep in mind that the dependence of relevant quantities especially µ on µ and A t can be significant in NHmGMSB due to RGE effects. Besides, it is important to mention that a fine-tuning measure analyzed in a scenario with NH terms in a predictive model that uses RGEs can be less independent with respect to the higgsino mass [81]. This is unlike what is seen in electroweak fine-tuning in NHSSM as mentioned above.

Results
We have realized the non-holomorphic MSSM on a GMSB setup, that is going to be referred as the NHmGMSB model, by using the codes SARAH-4.9.1 [104][105][106][107] and SPheno-3.3.8 [108]. Two-loop RGEs of the MSSM soft parameters, plus the same at one-loop for JHEP01(2018)158 the NH soft parameters are used to generate the sparticle mass spectra [73,74,80,113]. The codes use two-loop corrections for Higgs states [114,115] and additionally compute all the relevant flavor observables [112]. As mentioned earlier, the free parameters that are to be scanned are Λ, M mess , 8 tan β, N 5 , sign(µ) and µ 0 , where the number of the messenger copies is taken to be one (N 5 = 1). At the messenger scale M mess , the soft SUSY breaking mass parameters are given via eqs. (2.6), (2.8) and the values of (non) holomorphic trilinear coupling parameters (A 0 ), A 0 are taken to be zero. Choosing sign(µ) = 1 we scan the following volume of the mGMSB parameters. Additionally, we note that a higher value of M mess than what is given in eq. (3.2) would not be so consistent with our motivation of choosing a low scale SUSY breaking model like mGMSB. This is in keeping with limiting the mediation scale suppression of the NH soft terms as mentioned in section 1. On the other hand, reducing M mess further than the lower limit of eq. (3.2) would hardly provide with a reasonable range of RG running of both A t and A t that is essential for satisfying the higgs mass data (without trying to compensate it via choosing a larger value of Λ).
The SUSY Higgs mass (m h ) limits [116] and the constraints from B-physics namely B → X s + γ, B s → µ + µ − are mentioned as below [116].

GeV
We consider a 3 GeV theoretical uncertainty in computing m h as given above. Some of the reasons behind considering the spread are uncertainty in computing loop corrections up to three loops, top quark mass, renormalization scheme and scale dependence etc. [117][118][119][120][121][122][123].
We will now discuss the effect of introducing NH parameters within the mGMSB scenario and especially study the dependence of the SUSY spectra and observables of phenomenological interest on Λ, M mess and µ 0 . We remind that in this analysis the trilinear NH couplings are vanishing at the messenger scale whereas the higgsino mass soft-parameter µ 0 is considered to arise from sources other than GMSB. This is following the discussion of section 2. of the lighter CP-even neutral Higgs boson h which is standard-model like in its couplings satisfies the range mentioned in eq. (3.3). The yellow and blue zone refer to the results of mGMSB and NHmGMSB respectively. The lower limit of Λ corresponding to the lower limit of m h (eq. (3.3)) for any given scenario either mGMSB or NHmGMSB decreases with an increase in tan β. The spread of points for m h shown in blue (NHmGMSB) is quite large compared to the yellow region (mGMSB) for tan β = 10. However, this is not so for the case of tan β = 40. We note that µ+A t appearing in the off-diagonal elements of the top-squark mass matrix is suppressed by tan β. The larger top-squark mixing causes more prominent spread for low tan β. Regarding A t , it is seen that A t turns out to be negative. However, X t is not large enough to be in the maximal mixing zone characterized by X t = √ 6M S . This is also true for X t (= A t − µ cot β) for the mGMSB case. Furthermore, it is found that A t at the EWSB scale approximately varies from −550 (−600) GeV to 550 (600) GeV for tan β = 10 (40). A t comes with either sign because of its dependence via RGE on µ 0 , while the latter is scanned for both positive and negative values. Additionally, we note that µ 0 may also contribute [73,74,80] quite significantly to the value of µ which is obtained via REWSB. With all the above effects and estimates, X t is seen to be larger in the negative direction for tan β = 10 than tan β = 40. Because of a larger top-squark mixing, that results into a larger radiative corrections to the Higgs boson mass, the lowest value of Λ satisfying the lower limit of m h for NHmGMSB is less than the same for mGMSB.

Effects on top-squarks and Higgs masses
We will now explore the variation of Higgs boson mass m h with top-squark mass mt For a given value of m h , the lower limit of mt 1 becomes smaller in NHmGMSB than that of mGMSB and this is more prominent for tan β = 10 because of a larger influence of (µ+A t ). With the validity of the lower Λ zone for a larger tan β via the m h lower limit, the top-squark mass mt 1 finds its smallest value to be smaller for tan β = 40 compared to the same for tan β = 10. On the other hand, between tan β = 10 and 40, the relative difference of the lowest value of mt 1 between NHmGMSB and mGMSB is more enhanced for the lower value of tan β primarily because of a lesser degree of suppression of the contribution of µ + A t in the top-squark mixing. The resulting reduction of mt 1 in NHmGMSB (compared to mGMSB) is about 500 (200) GeV for tan β = 10 (40).
We will now explore the role of the messenger scale M mess in our analysis that would shed some light on the extent of RG evolutions of relevant parameters that have influence on the scalar masses including that of the Higgs boson as well as on the masses of the electroweakinos. The effects of evolutions are not likely to be as large as that occur in minimal supergravity (mSUGRA) [69] with much higher SUSY mediation scale, but as we have seen they may assist in lowering the top squark mass mt 1 or enhancing m h . Figure 4(a) and 4(b) show color contour plots in the (Λ vs M mess ) plane for NHmGMSB corresponding to tan β = 10 and 40 respectively when µ 0 is scanned as in eq. (3.2). The side panels show the values of m h . The figures show strong dependence of Higgs mass on Λ (that sets the masses of the scalars as well as gauginos) and a weaker dependence on M mess . However, we must mention that the upper limit of M mess is chosen relatively small in our analysis. This is based on our motivation to have the NH terms of being associated with a lesser JHEP01(2018)158    degree of mass suppression as mentioned in section 2, the reason of our working in a GMSB setup. A larger M mess ∼ 10 12 GeV (not shown in this work) may increase both A t and A t significantly to higher values beyond what would be necessary to satisfy the Higgs boson mass limit for a reasonably chosen Λ.
It is important to note that there are a few beyond the mGMSB analyses [102,[124][125][126][127][128] for example those involving matter-messenger interactions in which the trilinear holomorphic soft terms may arise at one-loop level, thus becoming non-vanishing at the messenger scale. In such situations one obtains a significantly large amount of radiative corrections to m h , a friendly feature to accommodate a Higgs boson as massive as 125 GeV in models JHEP01(2018)158 away from mGMSB. The NH trilinear terms will then similarly become non-vanishing at the above scale. Depending on the sign of µ 0 , this may give rise to a larger A t at the electroweak scale. The above in turn that may result into a smaller mt 1 or positively contributing toward the radiative corrections to the Higgs boson mass m h . Thus a smaller Λ would be acceptable lowering the overall mass scale of sparticles. Additionally, as we will see below, the same consideration may significantly enhance the smuon mass mixing via a larger A µ . A large A µ may effectively contribute to the SUSY contribution to muon g − 2 significantly, because of an associated scaling by tan β [70].

Phenomenological implications through the electroweakino sector
As discussed in section 2 the electroweakino sectors are influenced by µ + µ particularly when the combination that is close to the higgsino mass is lesser than the masses of relevant gauginos M 1 and M 2 . We will discuss the phenomenological implications on Br(B → X s + γ) and muon g − 2 in this context. The diagrams that are significant in MSSM for Br(B → X s + γ) are the loops involving t − H ± andt − χ ± 1 . Agreement of the SM result of Br(B → X s + γ) with the experimental data demands delicate cancellation between the contributions of the above loops. In MSSM, Br(B → X s + γ) increases with tan β [65]. Additionally, there are Next to Leading Order (NLO) contributions from squark-gluino loops due to corrections of bottom and top Yukawa couplings, particularly for large values of tan β. Here we will discuss the effects on Br(B → X s + γ) in relation to the results of ref. [70] that used unconstrained NH soft terms given at the weak scale (NHSSM). Thẽ t − χ ± 1 loop that is relevant for our study does not have an appreciable effect from top squark L-R mixing since both A t and A t that start from vanishing values at the messenger scale have only limited degree of evolutions. This is in contrast to the larger possible values assumed by the above trilinear parameters as analyzed in ref. [70]. Figure 5 shows the variation of Br(B → X s + γ) with µ 0 in NHmGMSB for a given Λ and M mess . It shows some sharp increase in Br(B → X s + γ) for a given zone of µ 0 , particularly for a large tan β = 40. It is found that this is indeed the region when µ + µ or the lighter chargino mass becomes small. The effect was also seen earlier [69,79]. The corresponding mGMSB values of the same quantity are also shown where m χ ± 1 is not small. Apart from the cases with small m χ ± 1 , Br(B → X s + γ) does not impose any serious constraint on NHmGMSB parameter space even for a large value tan β because of the smallness of A t and A t and large values of top-squark masses so as to satisfy the Higgs mass data. This is unlike NHSSM that recovers a large amount of parameter space discarded by Br(B → X s + γ) in MSSM for a large value of tan β [70].
We will now discuss the SUSY contributions to (g − 2) µ , namely a SUSY µ , 9 in the context of NHmGMSB against the results of the NHSSM analysis made in ref. [70]. At the oneloop level, a SUSY µ involves contributions fromχ ± i −ν µ andχ 0 i −μ loops [129][130][131][132][133][134]. In the NHSSM analysis of ref. [70], because of a strong L-R mixing via A µ , it is theχ 0 i −μ loop that contributes a significantly large amount to a SUSY µ . A µ can be as small as 100 GeV or even 50 GeV to show very prominent effects. The largeness of the effect in NHSSM comes  from an enhancement of A µ via tan β and the diagram containing bino-μ L,R in the loop that contributes to a SUSY µ [135,136]. In NHmGMSB, A µ that starts from a vanishing value at M mess has only a limited degree of evolution [80] even with a large µ 0 . As a result, with an inadequate level of enhancement of A µ , we do not expect any large amount of contributions from the bino-smuon loops due to L-R mixing, unlike the NHSSM results of ref. [70]. Figure 6 shows a scatter plot of µ 0 vs A µ where Λ and M mess are varied within the shown ranges of eq. (3.2). The green points are those that satisfy all the relevant constraints of eq. (3.3) while the orange points correspond to a region that satisfy the muon g − 2 constraint at 3σ level. The 3σ region spreads more toward the negative zone of µ 0 . We should remind that µ is obtained via REWSB with a chosen positive sign, whereas µ 0 assumes both the signs. Thus, a larger a SUSY µ via a lighter higgsino mass can be obtained only in the negative direction of µ 0 . The fact that the large A µ regions are not associated with larger values of a SUSY µ , rather the region with smaller µ 0 satisfy the 3σ, 10 level of the constraint predominantly for a negative region of µ indicates the domination of charged and neutral higgsinos in the loop diagrams [135]. This is indeed displayed in figure 7 that shows a plot of µ 0 vs a SUSY µ for a given Λ and M mess . The figure shows a sharp rise in a SUSY µ for a small zone of µ 0 that also increases with tan β. It turns out that with the value of µ obtained from REWSB, and µ + µ being the tree level higgsino mass, this rise corresponds to the light higgsino zone. 11 The specific contributions to a SUSY µ arise from diagrams [135] like wino/higgsinosneutrino and wino/higgsino-μ L , bino/higgsino-μ L or bino/higgsino-μ R . There may also be a significant degree of cancellations between the above diagrams. The 3σ level of a SUSY µ region for µ 0 as shown in figure 6 comes out to be around −1500 GeV to 200 GeV.

Higgsino like NLSP decays
The interaction Lagrangian of gravitino ψ µ with other sparticles and SM particles is given by L = 3 α=1 L (α) where α stands for a given gauge group out of SU(3) C × SU(2) L × U(1) Y and L (α) being given as [137], The covariant derivatives D (α) µ and F (α)a ρσ are appropriately defined depending on the gauge group denoted by α and the generator index a [137].

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component. In realistic scenarios with small amount of gaugino mixing in NLSP, which we will though ignore here, Γ( χ 0 1 → G γ) can be quite important. However, we will work only with an almost pure higgsino. For the above two decays we have, where N 13 and N 14 are the higgsino related components of the LSP in theB,W , H 0 d , H 0 u basis of the neutralino diagonalizing matrix N ij [15]. The gravitino mass is given by, For our choice of parameter space mG is of order of few keV. This allows us to neglect it in the expression of the phase space factor compared to other masses involved in the calculation of decay widths in eqs. (3.4), (3.5). The total decay width of a higgsino type of NLSP is given by, Γ tot would be strongly influenced by Λ and M mess since Γ tot ∝ 1 (ΛMmess) 2 . Γ tot is also influenced by µ 0 through the higgsino NLSP mass as well as the effect of higgsino mixing in N ij s. We explore the decay widths of eqs. (3.4) and (3.5) against the NLSP mass in figure 8(a) and 8(b) respectively. The fixed values of the chosen parameters are sign(µ) = 1, tan β = 35, Λ = 4.5 × 10 5 GeV and M mess = 8 × 10 7 GeV. µ 0 is scanned over a range −4 TeV to 4 TeV. Both the decay widths increase with NLSP mass as long as the NLSP is substantially a higgsino. This is so until m χ 0 1 is about 550 GeV. Once, the NLSP is found to be an admixture of a higgsino and bino or essentially a bino each of the decay widths is bound to decrease rapidly. This happens after it attains a peak similar to what is apparent in figure 8(a) or 8(b). In short of other relevant decay modes, eq. (3.6) would then no longer be valid. Regarding the composition of the NLSP and its effect on Γ( χ 0 1 → G Z) and Γ( χ 0 1 → G h) we would like to point out that both N 13 and N 14 change signs over the scanned region of parameter space. However, N 14 plays a dominant role for both types of decays unless tan β is very small. In eq. (3.4) the N 13 contribution is very subdominant since cos β is small. A similar subdominant role for N 13 is also true in eq. (3.5) since we have a decoupling Higgs boson scenario where sin α becomes small [20]. The above causes both Γ( χ 0 1 → G Z) and Γ( χ 0 1 → G h) to peak at similar values of the NLSP mass. Apart from the above, because of the possibility of both the signs for N 13 and N 14 , figures 8(a) and 8(b) show that each of the decay widths has two branches or in other words is a double-valued function of m χ 0 1 depending on the two cases µ > |µ | and µ < |µ |. The heights of the decay widths of the two branches for each of Γ( χ 0 1 → G Z) and Γ( χ 0 1 → G h) differ because of a varying degree of radiative corrections to the NLSP mass. Considering  Figure 8. Plot of decay width vs. NLSP mass for χ 0 1 → G + Z and χ 0 1 → G + h channel for µ > 0 when µ is scanned over a range (eq. (3.2)) to probe the higgsino NLSP zone. The blue and red points refer to µ > |µ | and µ < |µ | respectively. m χ 0 1 575 GeV (near the peaks) the approximate sum of the two decay widths is around 2.8×10 −16 GeV for the given parameter point. This leads to 1/Γ tot 2×10 −9 sec 70 cm. In general, the mean decay length of χ 0 1 as NLSP with energy E in the laboratory frame is given by [94,139,140], (3.7) Figure 9(a) shows a scatter plot of decay width Γ tot vs. m χ 0 1 for a higgsino dominated NLSP over the scanned parameter region of eq. (3.2). The higgsino fraction is shown in graded color with a reference color bar on the right. Only highly higgsino dominated NLSP region is considered. Figure 9(b) shows a similar scatter plot in the plane of Γ tot vs. F (= ΛM mess ) where the NLSP mass is shown in a graded color with a reference color bar on the right. The range of variation of Γ tot is from 10 −22 to 10 −12 GeV implying 1/Γ tot to be within 10 −3 sec to 10 −13 sec or 1000 km to 0.1 mm respectively. The decay lengths when computed in the laboratory frame would point out a long range of values indicating decays occurring both within and outside the detector. Collider studies of probing the higgsino NLSP decays for suitable values of ΛM mess may be performed similar to the analyses made in refs. [141][142][143][144][145]. This is however beyond the scope of the present work.
Finally, we present two representative points A and B in table 1 for the spectra of NHmGMSB to demonstrate the degree of evolution of the parameters connected with the NH terms while choosing the lighter chargino and the NLSP to be higgsino dominated in nature. This is generally unavailable in mGMSB where the NLSP as the lightest neutralino it is typically bino dominated in its composition. The spectra is generally heavy because of the requirement of satisfying the Higgs boson mass limit. We remind that the effect of the NH terms can be increased significantly and the spectra may be lighter while that would also satisfy the muon g − 2 data, if we go beyond the mGMSB setup. This may JHEP01(2018)158

Conclusion
It was seen that SUSY models may include non-holomorphic terms like φ 2 φ * and ψψ that can be characterized as soft SUSY breaking in nature in the absence of a gauge singlet field. The broad applicability of the terms in various possible SUSY models makes such inclusion very important. In particular, the above terms may relax stringency to accommodate various phenomenological data in MSSM. These would be additional interactions than the usual soft SUSY breaking terms like that for the scalar masses (non-holomorphic) and trilinear and bilinear interactions that are holomorphic in nature. There have been studies on non-holomorphic MSSM referred as NHSSM that included such terms. The specific areas of impact that NHSSM makes are in the phenomenologies involving i) L-R mixing of squarks or sleptons via trilinear non-holomorphic terms containing the conjugate higgs fields, ii) higgsino mass soft SUSY breaking non-holomorphic term that results into higgsino components of electroweakinos to have parts coming from both superpotential as well as soft breaking origins and iii) the tree level electroweak fine-tuning to have no essential correlation with the higgsino mass, unlike MSSM. The authors of refs. [70,82] performed phenomenological MSSM (pMSSM) type of analysis where all the soft parameters including the NH ones are provided at the weak scale. These were away from previous works that used mostly CMSSM inspired setup where in addition to the usual CMSSM inputs corresponding to the gauge coupling unification scale, the NH soft parameters were either given a) at the unification scale or b) at the electroweak scale. It was however shown that in the absence of a gauge singlet superfield, a hidden sector based F-type SUSY breaking scenario with two chiral superfields would lead to such NH soft terms [71].  could include such NH terms with mass scale suppression nearing the Planck scale [71]. Irrespective of the above, there have been analyses that worked out the phenomenologies of including such soft terms in a CMSSM setup or extensions like non-universal gaugino or non-universal Higgs scalar scenarios.
In this work, we first investigate the NH soft SUSY breaking models in a low scale SUSY breaking context. We choose to perform the analysis within a backbone provided by the minimal Gauge mediated SUSY Breaking (mGMSB) framework. We however assume that the higgsino mixing soft term parameter µ to have a SUSY breaking origin away from mGMSB. Thus, by having an entirely independent SUSY breaking origin for JHEP01(2018)158 the higgsino mass soft term we essentially overcome any issue of the reparametrization invariance involving unrelated quantities as mentioned in the text. We focus on the degree of influence of the NH terms in an mGMSB setup (NHmGMSB) in comparison with the NHSSM work mentioned earlier. We note that the NH terms may affect various SUSY parameters including the Higgs scalar mass parameters through the RGEs. Thus, µ, as obtained via REWSB may be non-negligibly modified once the NH terms are included.
We show that the level of the L-R mixing of top-squarks can potentially reduce the mt 1 to a reasonably large extent, in turn increasing the radiative corrections to Higgs boson mass. Thus, a smaller than required |A t | value can do the job of satisfying the lower bound of m h compared to the mGMSB case since a compensating contribution may come from appropriate NH couplings. Of course, the enhancement of radiative corrections to the higgs boson mass is lesser when we compare our result with that of NHSSM. This is simply due to the limited degree of RG evolution of A t and A t from M mess down to the electroweak scale starting from their vanishing values at the higher scale. Regarding the effects on the electroweakino sector, we observe that the high level of enhancement (via a factor of tan β) of the SUSY contributions to muon g −2 as that happens in NHSSM via A µ through the L-R mixing contributions of smuons is absent in NHmGMSB. This is because A µ does not become sufficiently large at the weak scale. The smallness of A µ can, however, be avoided in non-minimal GMSB cases where the NH trilinear coupling A 0 need not be vanishing at the messenger scale. Simultaneously, the higgs boson mass m h would receive a larger amount of radiative correction through enhanced A t . A larger a SUSY µ comes from the limited zone of light higgsino mass in NHmGMSB. The same light higgsino mass zone may enhance the Br(B → X s + γ) contribution via the chargino loops. However, the constraint from Br(B → X s + γ) is well satisfied at the 2σ level over the relevant parameter space of NHmGMSB simply because of generally large top-squark masses. This in turn arises from the requirement of satisfying the Higgs mass bound. We further estimate the decay of the lightest neutralino as NLSP into gravitino and Z-boson or h-boson while considering only a higgsino dominated NLSP. This is in contrast to a typical mGMSB scenario where χ 0 1 is bino-dominated in its composition. We probed the entire parameter space of NHmGMSB to find a large degree of variation of decay lifetimes that may correspond to a decay length of less than a millimeter to hundreds of kilometer in the rest frame of the NLSP. Relevant collider analyses may be made by finding the length in the laboratory frame of the detector. Depending on whether the NLSP decay is happening within or outside the detector one can probe NHmGMSB for a higgsino type of NLSP. Finally, we have presented two representative points for the spectra of NHmGMSB to demonstrate the extent of RG evolutions parameters related to the NH soft terms. The two spectra are on the heavier side so as to accommodate the Higgs mass data. However, a non-minimal GMSB scenario that allows having non-vanishing trilinear parameters A 0 and A 0 at the messenger scale would show a very significant effect on the scalar sector including the Higgs mass apart from potentially providing an enhanced a SUSY µ via a large L-R mixing of the smuons. This is beyond the focus of the present analysis. Finally, considering the generic nature of the NH soft terms, it is important to explore their effect in varying SUSY breaking scenarios and scales. It can also be significant in respect of global analyses of various SUSY models.