Naturalness in D-brane inspired models

We examine the naturalness of the D-brane inspired model constructed in flipped SU(5) supplemented with vector-like particles at the TeV scale, dubbed flippons. We find the model can produce a mainly Higgsino-like lightest supersymmetric particle (LSP) and small light stops, as favored by naturalness. In fact, a large trilinear scalar At term at the electroweak (EW) scale creates a large mass splitting between the top squarks, driving the light stop to near degeneracy with an LSP that is almost all Higgsino, with ΔMt˜1χ˜10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Delta M\left({\tilde{t}}_1,{\tilde{\chi}}_1^0\right) $$\end{document}< 5GeV, evading the LHC constraint on t˜1→cχ˜10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\tilde{t}}_1\to c{\tilde{\chi}}_1^0 $$\end{document} thus far. Given the smallness of the light stop, generating a 125 GeV light Higgs boson mass is aided by one-loop contributions from the Yukawa couplings between the flippons and Higgs fields. The resulting parameter space satisfying naturalness is rather constrained, thus we assess its viability by means of comparison to the LHC constraint on soft charm jets and direction detection limits on spin-independent cross-sections. Finally, we compute the level of electroweak fine-tuning and uncover a region with ΔEW< 30, i.e., fine-tuning better than 3%, regarded as low electroweak fine-tuning. Given the small light stop, the electroweak fine-tuning from only the top squarks is of O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{O} $$\end{document}(1), indicating no fine-tuning from neither the light stop t˜1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\tilde{t}}_1 $$\end{document} nor the heavy stop t˜2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\tilde{t}}_2 $$\end{document}.


Introduction
The null results at the 13 TeV LHC Run 2 (LHC2) regarding the search for supersymmetry (SUSY) have now extended through 2018, as recent results find only Standard Model (SM) background events for data collected from 2016-18, inclusive of 137 fb −1 [1]. The rather strong limits derived from these observations though rely upon gluino ( g) and light stop ( t 1 ) channels producing hard jets via g → t 1 t → tt χ 0 1 and g → qq χ 0 1 , leading to limits on the gluino mass in excess of 2 TeV and on the light stop mass above 1 TeV. Such hard jets are easily accessible at the LHC2, affirming the rapid uninterrupted march to multi-TeV exclusion limits, yet approaching tension with SUSY's solution to the hierarchy problem, a prime reason motivating SUSY in the first place. On the other hand, the empty SUSY cupboard thus far prompts one to ask as to whether SUSY could be hiding in plain sight?
Natural SUSY, referred to as naturalness, strives for negligible electroweak fine-tuning in SUSY grand unified theory (GUT) models, defined by only natural cancellations amid terms in the tree-level minimization condition on the Higgs potential, plus radiative corrections. Insignificant amounts of fine-tuning require small terms in the minimization condition such that all terms on both sides of the equation are of comparable scale in order to compute the measured Z-boson mass. Such natural dynamics are very desirable, although an additional benefit can be realized that relates to the dilemma posed in the prior paragraph. The principal contributor to loop-level corrections are top squarks, so naturalness stresses small values for M ( t 1 ), but small light stops could provoke degeneracy in the form of M ( t 1 ) ∼ M ( χ 0 1 ). This raises an element of uncertainty for accessibility at the LHC, given the softness of these light stop events and hence unreliable distinction from the ubiquitous SM background. Indeed, one could expect these soft interactions to evade observation at the LHC if the light stop becomes rather compressed with the LSP. Furthermore, a complication surfaces with insufficient 1-loop and 2-loop SUSY contributions to the light Higgs boson mass m h from a small light stop, failing to generate the observed JHEP07(2019)048 m h = 125.09 ± 0.24 GeV [2,3] light Higgs boson mass. We now introduce a model that has minimal electroweak fine-tuning but can handily achieve consistency with the light Higgs boson mass constraints, as well as other key experimental measurements, and could be flying just under the SUSY radar.
We shall study the flipped SU(5) model [4][5][6] in this paper, where the gauge group SU(5) × U(1) X can be embedded into the SO(10) model. The flipped SU(5) models can be constructed from the four-dimensional free fermionic string construction [7][8][9], orbifold [10,11] and Calabi-Yau [12] compactifications of the heterotic E 8 × E 8 string theory, intersecting D-brane model building [13][14][15], as well as F-theory model building [16,17]. In addition, two of us (TL and DVN) with Jiang proposed the testable flipped SU (5) models where the TeV-scale vector-like multiplets, dubbed flippons, are introduced, and string-scale gauge coupling unification can be achieved [18]. Such kind of models can be constructed from the four-dimensional free fermionic string construction [9], intersecting D-brane model building [15], as well as F-theory model building [16,17], and was referred to as F-SU(5). This model persists in two classes: (i) the minimalistic formalism of the oneparameter version implementing vanishing No-Scale SUGRA soft SUSY breaking terms at the unification scale (For example, see refs. [19][20][21][22] and references therein), and (ii) the general formalism with non-universal SUSY soft breaking terms mirroring the flipped SU(5) GUT representation, inspired by D-brane model building, and thus informally designated the F-SU(5) D-brane inspired model [23]. This second approach endures as merely a D-brane inspired model and not a formally constructed D-brane model by reason of forbidden Yukawa coupling terms in the Higgs and Yukawa superpotentials, though we discuss in the next section possible methods to elude these hurdles. The F-SU(5) D-brane inspired model revealed a possible region of naturalness featuring small light stops and a Higgsino-like lightest supersymmetric particle (LSP) [23], which we shall more fully unpack here in this work. For a discussion of naturalness in a Pati-Salam model constructed from intersecting D6-branes in Type IIA string theory, see ref. [24].
Fine-tuning in the minimalistic formalism of F-SU(5) has been explored [25]. In ref. [25] it was shown that the contemporary measures of fine-tuning we shall employ in this analysis are essentially structurally similar to an original fine-tuning measure, ∆ EENZ [26,27], first prescribed some 30 years ago by Ellis, Enqvist, Nanopoulos, and Zwirner (EENZ). The one-parameter version of the model possesses an intrinsic proportional dependence of all model scales on the unified gaugino mass parameter M 1/2 , inclusive of the Z-boson mass expressed as a simple quadratic function of M 1/2 . The implication was electroweak fine-tuning of unity scale [25]. The minimalistic version of F-SU(5) is presently under probe at the LHC2 [28] and has thus far survived the 13 TeV LHC2 137 fb −1 results [1]. Now we turn our attention to the less internally constrained version of F-SU(5), evaluating fine-tuning in the D-brane inspired model. Our goal here is to show that this class of F-SU(5) is inflicted with a minimal amount of fine-tuning also, and even though the one-parameter version is presently experiencing a direct probe by the LHC, the naturalness sector of the D-brane inspired model has been just under the reach of the LHC2.
In this work we first supply a brief review of the flipped SU(5) class of models and the D-brane inspired model in particular. Then we delve into the comprehensive numerical JHEP07(2019)048 procedure necessary to investigate naturalness. Once the numerical approach has been wholly dissected, we expand upon the phenomenology of the naturalness sector and the attainment of small light stops, Higgsino-like LSPs, and other associated provisions essential for low fine-tuning, accompanied by light Higgs boson masses lifted to 125 GeV for many points by the vector-like flippon contributions. Integrated into this analysis will be evidence of our naturalness sector skirting under the LHC constraints up to this point, and moreover, an evaluation against dark matter direct detection experiments and application of their results as a constraint on the naturalness region. Finally, we conclude with the fine-tuning calculations and assessment of the numerical findings.

Review of F-SU(5) model
We review here only the primary principles of F-SU (5). In the minimal flipped SU(5) model [4][5][6], the gauge group SU(5) × U(1) X can be embedded within the SO(10) model. Please see refs. [20-22, 25, 29] and references therein for a more in-depth analysis of the minimal flipped SU(5) model. The generator U(1) Y ′ in SU(5) is defined as and as a result the hypercharge is given by There are three families of the SM fermions with quantum numbers under SU(5) × U(1) X given by, respectively, where i = 1, 2, 3. The SM particle assignments in F i ,f i andl i are and N c i are the left-handed quark doublets, right-handed up-type quarks, down-type quarks, left-handed lepton doublets, right-handed charged leptons, and neutrinos, respectively. The introduction of three SM singlets φ i can generate the heavy right-handed neutrino masses.
The GUT and electroweak gauge symmetries are broken through the introduction of the two pairs of Higgs representations H = (10, 1), H = (10, −1), The H multiplet states are labeled by the same symbols as the F multiplet, and for H we only add a "bar" above the fields. Specifically, the Higgs particles are >. Therefore, the doublet-triplet splitting due to the missing partner mechanism [6] naturally arises. However, the triplets in h and h only have a small mixing via the µ term, so the colored Higgsino-exchange mediated proton decay is negligible, i.e., there is no dimension-5 proton decay problem.
The following vector-like particles (flippons) at the TeV scale are introduced to realize string-scale gauge coupling unification [16][17][18]  (2.9) The particle content from the decompositions of XF , XF , Xl, and Xl under the SM gauge symmetry are The quantum numbers for the extra vector-like particles under the SU(3) C ×SU(2) L ×U(1) Y gauge symmetry are 14) The superpotential is and the above superpotential after the SU(5) × U(1) X gauge symmetry is broken down to the SM gauge symmetry gives

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where y D ij , y U ν ij , y E ij , y N ij , y XF , and y XF are Yukawa couplings, µ is the bilinear Higgs mass term, and M φ ij , M XF and M Xl are masses for new particles. The new particles are the vector-like flippons, though we shall not formally compute the masses M φ ij , M XF ,and M Xl in this study, reserving this analysis for the future. Only a common mass decoupling scale M V for the flippon vector-like particles is enforced. Present LHC constraints on vector-like T and B quarks [30] fix lower limits of about 855 GeV for (XQ, XQ c ) vector-like flippons and 735 GeV for (XD, XD c ) vector-like flippons. We therefore suitably place our lower M V limit at M V ≥ 855 GeV to guarantee inclusion of all experimentally viable flippon masses in our work.
The two-stage unification of flipped SU(5) [4][5][6] allows for fundamental GUT scale Higgs representations (not adjoints), natural doublet-triplet splitting, suppression of dimension-five proton decay [31], and a two-step see-saw mechanism for neutrino masses [32,33]. More precisely, a distinct separation between the ultimate SU(5) × U(1) X unification at around 3 × 10 17 GeV and the penultimate SU(3) C × SU(2) L unification near 10 16 GeV emerges due to revisions to the one-loop gauge β-function coefficients b i to include contributions from the vector-like flippon multiplets that induce the required flattening of the SU(3) Renormalization Group Equation (RGE) running (b 3 = 0) [19]. The M 2 and M 3 gaugino mass terms are unified into a single mass term M 5 = M 2 = M 3 [34], and hence α 5 = α 2 = α 3 , at the SU(3) C × SU(2) L unification near 10 16 GeV. The M 1 gaugino mass term runs up to the SU(5) × U(1) X unification at M F , by virtue of a small shift due to U(1) X flux effects [34] between the SU(3) C × SU(2) L unification around 10 16 GeV and the SU(5) × U(1) X unification around 3 × 10 17 GeV [19]. This shift motivates that the M 1 gaugino mass term above the unification around 10 16 GeV be referred to as M 1X . The scale M F is defined by unification of the couplings α 5 = α 1X , which boosts unification to near the string scale and Planck mass. The flattening of the M 3 gaugino mass dynamic evolution down to the electroweak scale generates the mass texture of M ( t 1 ) < M ( g) < M ( q), with the light stop and gluino lighter than all other squarks [21].
The SUSY breaking soft terms at the M F scale in the F-SU(5) model are appropri- Non-universal SUSY breaking soft terms such as these are inspired partially by D-brane model building [15], where F i , f i , l i , and h/h result from intersections of different stacks of D-branes. In this event, the associated SUSY breaking soft mass terms and trilinear scalar A terms are different, while M Hu is equal to M H d . Despite the fact the Yukawa terms HHh and HHh of eq. (2.8) and F i F j h, XF XF h, and XF XF h of eq. (2.15) are forbidden by the anomalous global U(1) symmetry of U(5), these Yukawa terms could be generated from high-dimensional operators or instanton effects. In fact, the SU(5) × U(1) X models differ from SU(5) models such that in F-SU(5) the Yukawa term F i F j h gives down-type quark masses, so their Yukawa couplings can be small and be generated via high-dimensional operators or instanton effects.  . Layered illustration of the F -SU(5) D-brane inspired naturalness region (represented by the small red points) superimposed upon the ATLAS Collaboration exclusion curve plot on t 1 t 1 production in the monojet search region for the channel t 1 → c χ 0 1 , reprinted from ref. [39]. The branching fraction for g → t 1 t and t 1 → c χ 0 1 is nearly 100% for our points shown in this figure. The naturalness region in this diagram involves the 74 points that satisfy M ( g) ≥ 1.6 TeV, 124 ≤ m h ≤ 128 GeV, Ωh 2 ≤ 0.1221, σ Rescaled SI ≤ 1.5 × 10 −9 pb, and ∆M ( t 1 , χ 0 1 ) ≤ 5 GeV. All 74 points depicted here possess an LSP that is at least 92% Higgsino, but no more than 98% Higgsino. D Figure 4. Layered illustration of the F -SU(5) D-brane inspired naturalness region (represented by the small red points) superimposed upon the ATLAS exclusion curve plot in the charm jets plus zero lepton (0L) search region for the channel t 1 → c χ 0 1 , reprinted from ref. [40]. This ATLAS exclusion plot also includes the monojet search region of ref. [39]. The branching fraction for g → t 1 t and t 1 → c χ 0 1 is nearly 100% for our points shown in this figure. The naturalness region in this diagram involves the 74 points that satisfy M ( g) ≥ 1.6 TeV, 124 ≤ m h ≤ 128 GeV, Ωh 2 ≤ 0.1221, σ Rescaled SI ≤ 1.5 × 10 −9 pb, and ∆M ( t 1 , χ 0 1 ) ≤ 5 GeV. All 74 points depicted here possess an LSP that is at least 92% Higgsino, but no more than 98% Higgsino.

JHEP07(2019)048 3 Numerical approach
At the unification scale of M F ∼ 3 × 10 17 GeV, the F-SU(5) general SUSY breaking soft terms are applied, namely The F-SU(5) unification scale M F near the string and Planck scale is in contrast to the usual lower GUT scale of about 10 16 GeV in the MSSM. All SUSY breaking soft terms are allowed to float up to 5 TeV, with the A terms varying between ±5 TeV, though specifically for the A t term we establish an extended lower limit of -7 TeV. A ±1.5 GeV margin of error is permitted around the top quark world average of 173.2 GeV [35]. The ratio of the vacuum energy expectation values tanβ is allowed to span its entire range of 5 ≤ tanβ ≤ 60. The flippon vector-like decoupling scale is sampled within the range 855 ≤ M V ≤ 23, 000 GeV. We adopt µ > 0 for all points as suggested by the results of g µ − 2 for the muon.
The model is constrained to be consistent with both the WMAP 9-year [36] and Planck 2018 [37] relic density measurements, imposing an upper limit of Ωh 2 ≤ 0.1221. Given the large annihilation cross-section of a Higgsino-like LSP, no lower limit is placed on Ωh 2 . The strongest LHC gluino limits arise from the search regions g → qq → qq χ 0 1 and g → t 1 t → tt χ 0 1 , however, in our study here we are interested in the channel producing a top+charm via g → t 1 t → ct χ 0 1 , which persists with weaker limits. Accordingly, we implement a somewhat weaker lower boundary of M ( g) ≥ 1.6 TeV given that these gluinos are not easily accessible.
The theoretical calculation of the light Higgs boson mass is allowed to vary from the experimental central value of m h = 125.09 GeV, where we account for a 2σ experimental uncertainty and theoretical uncertainty of 1.5 GeV. The allocated range for the flippon Yukawa coupling spans from its minimal value (no coupling between the flippons and Higgs fields) to its maximal value (maximum coupling between flippons and Higgs fields). In the maximum case, the light Higgs boson mass calculation consists of the 1-loop and 2-loop SUSY contributions, mainly from the coupling to the light stop, plus the vector-like flippon contributions. This maximal value implies the (XD, XD c ) Yukawa coupling is fixed at Y XD = 0 and the (XU, XU c ) Yukawa coupling is set at Y XU = 1, while the (XD, XD c ) trilinear coupling A term set at A XD = 0 and the (XU, XU c ) A term is fixed at A XU = A U = A 0 [21,38]. In total, after including all contributions, the light Higgs boson mass calculation must return a value of 124 ≤ m h ≤ 128 GeV.
We further assess the model against rare decay processes, to include the branching ratio of the rare b-quark decay of Br(b → sγ) = (3.43±0.21 stat ±0.24 th ±0.07 sys )×10 −4 [41], the branching ratio of the rare B-meson decay to a dimuon of Br(B 0 s → µ + µ − ) = (2.9 ± 0.7 ± 0.29 th )×10 −9 [42], and the 3σ intervals around the Standard Model result and experimental measurement of the SUSY contribution to the anomalous magnetic moment of the muon of −17.7 × 10 −10 ≤ ∆a µ ≤ 43.8 × 10 −10 [43]. We only inspect the model versus these rare decay processes, and do not explicitly constrain the model per these experimental limits.
The naturalness region is also evaluated against dark matter direct detection constraints on spin-independent cross-sections σ SI for neutralino-nucleus interactions established by the Large Underground Xenon (LUX) experiment [44], PandaX-II Experiment [45], and XENON100 Collaboration [46]. The relic density calculations involve only JHEP07(2019)048 the SUSY lightest neutralino χ 0 1 abundance, hence all points must admit alternative components to maintain compatibility with the WMAP 9-year and 2018 Planck total observed relic density, thus the spin-independent cross-section calculations are rescaled as follows: The 150 million points scanned in ref. [23] were enhanced in this effort by an additional 250 million points. The Higgs and SUSY mass spectra, relic density, dark matter direct detection cross-sections, LSP composition, and rare decay processes are calculated with MicrOMEGAs 2.1 [47] employing a proprietary mpi modification of the SuSpect 2.34 [48] codebase to run flippon and general No-Scale F-SU(5) enhanced RGEs, implementing non-universal soft supersymmetry breaking parameters at the M F scale. Supersymmetric particle decays are calculated with SUSY-HIT 1.5a [49]. The Particle Data Group [50] world average for the strong coupling constant is α S (M Z ) = 0.1181 ± 0.0011 at 1σ, and we assume a value in this work of α S = 0.1184.

Naturalness phenomenology
Naturalness demands no disproportionate cancellations amongst the terms within the minimization of the Higgs scalar potential with respect to the H u and H d directions. The tree-level minimization condition is The largest contributions from the radiative corrections Σ u u and Σ d d emanate from the top squarks t 1 and t 2 , so we will only consider those loop corrections in this study. Provided that we desire no auspicious cancellations on the right-hand side of eq. (4.1) in order to produce the correct Z-boson mass, we also require a small bilinear Higgs mixing term µ in addition to the top squarks. Moreover, the quadratic Higgs mass term M 2 Hu evolves from a large positive value at the ultimate unification scale M F to a negative value at the EW scale through RGE running due to the large top quark Yukawa coupling, provoking the need for a small negative M 2 Hu as well. In summary, the leading culprits to engender contrived results within eq. (4.1) are t 1 , t 2 , µ, and H 2 u , motivating minimal values for these quantities. We correspondingly seek regions of the D-brane inspired F-SU(5) parameter space yielding small top squarks, small µ parameter, and a small negative M 2 Hu term. A small µ parameter at the EW scale in turn produces light Higgsinos since the Higgsino mass is near µ, and more practically, a dominant Higgsino component of the LSP. Therefore, we further search for regions of the model space with a dominant Higgsino-like LSP.
The initial step involves a search for an LSP that is greater than 80% Higgsino. These All 74 points depicted here possess an LSP that is at least 92% Higgsino, but no more than 98% Higgsino. The ∆ EW stem exclusively from either M 2 Hu (EW ) or µ 2 (EW ). These plots highlight a region with ∆ EW < 30, regarded as low electroweak fine-tuning. below the gaugino mass terms M 1 and M 2 at the electroweak scale via RGE running, sending χ 0 2 to negative values. Another characteristic of spectra with a Higgsino-like LSP is the compressed nature of the χ 0 1 , χ ± 1 , and χ 0 2 . The mass deltas expected to produce a Higgsino-like LSP are ∆M ( χ ± 1 , χ 0 1 ) ∼ 5 GeV and ∆M ( χ 0 2 , χ 0 1 ) ∼ 17 GeV. Accompanying the Higgsino-like LSP, we further require the condition ∆M ( t 1 , χ 0 1 ) ≤ 10 GeV to restrict the results to only those light stops nearly degenerate with the LSP, fulfilling the requisite small light stop limitation. Out of the 400 million points scanned, the intersection of the experimentally viable constraints on M ( g), m h , and Ωh 2 in tandem with an LSP that is > 80% Higgsino and ∆M ( t 1 , χ 0 1 ) ≤ 10 GeV only surrenders ∼ 2900 points. The resulting region is illustrated in figure 1 and figure 2, where the dark matter direct detection upper limits on spin-independent neutralino-nucleus cross-sections are superimposed, along with the neutrino scattering floor. All ∼ 2900 points are discretely depicted in figure 1, whilst JHEP07(2019)048 figure 2 delineates smoothly flowing contours of this region highlighting Higgsino percentage of the LSP. It is clear in figure 2 that the more favorable SUSY spectra in terms of smaller spin-independent cross-sections are the larger Higgsino percentages, exhibiting positive accommodation with both characteristics. All points in figure 1 and figure 2 have been rescaled in accordance with eq. (3.1).
The analysis from this point forward now enforces two more rather strong restrictions. We want to retain only those points possessing σ Rescaled SI ≤ 1.5 × 10 −9 pb, ensuring consistency with the LUX [44], PandaX-II [45], and XENON100 [46] upper limits illuminated in figure 1 and figure 2. In the region we are exploring here, σ Rescaled SI ∼ 1.5 × 10 −9 pb prevails as an approximate upper limit, so we shall now only consider points less than this boundary. We additionally aim to filter out those points inconsistent with LHC model-independent constraints on t 1 → c χ 0 1 . The nearly degenerate light stop and LSP induce a branching fraction of nearly 100% for g → t 1 t and t 1 → c χ 0 1 . However, given the compression between the light stop and LSP, we expect a rather hard top quark but a very soft charm jet, making extraction of this signal from the SM background challenging to say the least. To assist in comparing our naturalness region to the LHC constraints on t 1 → c χ 0 1 , post application of σ Rescaled SI ≤ 1.5×10 −9 pb we overlay the remaining points onto the ATLAS Collaboration exclusion curve plot on t 1 t 1 production in the monojet search region for the channel t 1 → c χ 0 1 , reprinted from ref. [39] and displayed in figure 3. In addition, we superimpose our points onto the ATLAS exclusion curve plot in the charm jets plus zero lepton (0L) search region for the channel t 1 → c χ 0 1 , reprinted from ref. [40] and shown in figure 4. The common element in both these ATLAS figures is the maximum delta between the light stop and LSP of about 5 GeV, with ∆M ( t 1 , χ 0 1 ) 5 GeV persisting as viable due to the soft nature of these events and difficulty in differentiation from the SM background. This theme is uniform between both ATLAS and the CMS Collaboration, as the t 1 → c χ 0 1 CMS search regions of refs. [51][52][53] paint the same picture of viability for ∆M ( t 1 , χ 0 1 ) 5 GeV. The CMS ref. [54] for pair production of third-generation squarks states that "Top squark masses below 510 GeV are excluded for the scenario in which t 1 → c χ 0 1 and the mass splitting between the top squark and the LSP is small", though ref. [54] does not explicitly enumerate the value of "small", hence we shall consider ∆M ( t 1 , χ 0 1 ) 5 GeV to remain experimentally viable. The administering of σ Rescaled SI ≤ 1.5 × 10 −9 pb and ∆M ( t 1 , χ 0 1 ) ≤ 5 GeV trims the number of residual points from ∼ 2900 down to only 74 out of 400 million scan! All 74 points have an LSP composition of at least 92% Higgsino, as figure 2 had indicated, though no point is greater than 98% Higgsino, supporting small but non-negligible bino and wino components.
We highlight nine benchmark points in tables 1-2. All nine benchmarks are amongst the remaining 74 points satisfying all the constraints applied. It should be noted that the light Higgs boson mass m h in table 2 includes all SUSY contributions and the vector-like flippon contribution, lifting the Higgs mass for most of the points to their observed value. This is rather beneficial given the smallness of the light stop and hence its diminished loop-level contribution to the Higgs mass. Notice that there is a repetitive pattern to the A t and M 2 Hu = M 2 H d terms at M F such that we need A t ∼ −2000 GeV and M Hu = M H d ∼ 3500 GeV at high scale. This propels consistency within the region for our fine-tuning calculations outlined in the next section.

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Benchmark   The entire naturalness model space handily satisfies the B-meson decay and anomalous magnetic moment of the muon boundaries highlighted in the prior section, with our 74 surviving points falling within 3.2×10 −9 ≤ Br(B 0 s → µ + µ − ) ≤ 3.5×10 −9 and 1.5×10 −10 ≤ ∆a µ ≤ 2.9×10 −10 . However, with regard to the rare b-quark decay, all remaining 74 points compute to Br(b → sγ) ≤ 2.34 × 10 −4 , less than the approximate lower 2σ experimental bound of Br(b → sγ) ∼ 2.77 × 10 −4 , with the smallest of the light stop points returning a value as low as Br(b → sγ) ∼ 10 −6 . This is not surprising, given the smallness of the light stop and chargino. The charged heavy Higgs bosons H ± additionally contribute, but not of sufficient magnitude to offset the minimal SUSY contribution from loops regarding stops and charginos. We emphasize that no points have been excluded from this analysis per the inconsistency with experimental limits on the Br(b → sγ), as we merely note that the SUSY contribution to the total branching ratio is light, thereby suggesting tension with the experimental result.

Fine-tuning
It was discussed in the prior section that low fine-tuning conforms with small values for M ( t 1 ), M ( t 2 ), µ, and M 2 Hu , thus we shall conclude this work with an analytical study of how the naturalness region we uncovered here performs in this realm. We follow the prescription offered in refs. [55,56], calculating measures for electroweak scale fine-tuning. Examining each term on the right-hand side of eq. (4.1), we have interest in the three JHEP07(2019)048  Table 3. Electroweak fine-tuning measures ∆ EW and ∆ t1,2 for the F -SU(5) D-brane inspired naturalness region. The fine-tuning percentage is found via ∆ −1 , also provided here. Those points with ∆ EW < 30 are regarded as low electroweak fine-tuning. Note that the fine-tuning for only the top squarks, represented by ∆ t1,2 , is of O(1).
∆ t 1,2 versus the light stop mass, with the majority of the points having ∆ t 1,2 around O(1). The explicit ∆ t 1,2 calculations for the nine benchmark points are provided in table 3.

Conclusion
In the search for SUSY, naturalness has been elevated in significance given its prospects for an elegant natural solution to the hierarchy problems and associated low electroweak fine-tuning. In conjunction, the smallness of the higgsinos and light stops required by naturalness introduces an element of uncertainty into observation of natural models at the LHC given the soft nature of the jets. We examined the well-studied GUT model flipped SU(5) with extra vector-like flippon multiplets, known as F-SU (5). However, in this work we allowed freedom on the No-Scale Supergravity boundary conditions at the unification scale, replicating the flipped SU(5)×U(1) X GUT representation, referred to as the D-brane inspired model ( "inspired" due to its forbidden Yukawa coupling terms). After a rather comprehensive search for a naturalness sector, we uncovered a region highlighted with points exhibiting a low amount of electroweak fine-tuning, namely ∆ EW < 30. The naturalness sector was exposed by constraining the model via M ( g) ≥ 1.6 TeV, 124 ≤ m h ≤ 128 GeV, Ωh 2 ≤ 0.1221, σ Rescaled SI ≤ 1.5 pb, and ∆M ( t 1 , χ 0 1 ) ≤ 5 GeV, providing us with points possessing ∆ EW < 30. Attainment of a light Higgs boson mass consistent with the empirically measured value was strengthened by including contributions from the vector-like flippon multiplets, a crucial maneuver given the smallness of the light stops compulsory within naturalness. The resulting region was rather narrow and uniformly supported by nearly 100% branching fractions for the decay channels g → t 1 t and t 1 → c χ 0 1 , indicating the production of a very hard top quark but also a considerably soft charm jet that will be quite difficult to extract from the SM background. Bolstered by these results, we gauged the model against the LHC constraint on t 1 → c χ 0 1 , finding that indeed our naturalness region uncovered here does skirt just under the ATLAS and CMS exclusion curves on t 1 → c χ 0 1 .

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Could natural SUSY be obscured by the dense Standard Model background in this region heretofore inaccessible at the LHC? Time will tell whether the LHC will yield an affirmative answer to this provocative question. Our imperative here was to merely present a viable physical model that thrives within this elusive space, furnishing motivation to develop enhanced methods of detection for probing concealed SUSY models such as the D-brane inspired model we explored in this work.