The Return of the King: No-Scale ${\cal F}$-$SU(5)$

We revisit the viable parameter space in No-Scale ${\cal F}$-$SU(5)$, examining the Grand Unified Theory within the context of the prevailing gluino mass limits established by the LHC. The satisfaction of both the No-Scale boundary condition and the experimentally measured Standard Model (SM) like Higgs boson mass requires a lower limit on the gluino mass in the model space of about 1.9 TeV, which maybe not coincidentally is the current LHC supersymmetry search bound. This offers a plausible explanation as to why a supersymmetry signal has thus far not been observed at the LHC. On the contrary, since the vector-like flippon particles are relatively heavy due to the strict condition that the supersymmetry breaking soft term $B_{\mu}$ must vanish at the unification scale, we also cannot address the recently vanished 750 GeV diphoton resonance at the 13 TeV LHC. Therefore, No-Scale ${\cal F}$-$SU(5)$ returns as a King after the spurious 750 GeV diphoton excess was gone with the wind.


I. INTRODUCTION
Supersymmetry (SUSY) is well acknowledged for the fact it provides a natural solution to the gauge hierarchy problem in the Standard Model (SM). For supersymmetric SMs (SSMs) with R-parity in particular, gauge coupling unification can be achieved, the Lightest Supersymmetric Particle (LSP) neutralino serves as a viable dark matter (DM) candidate, and electroweak (EW) gauge symmetry can be broken radiatively due to the large top quark Yukawa coupling, etc. Furthermore, gauge coupling unification strongly implies Grand Unified Theories (GUTs), and SUSY GUTs can be elegantly constructed from superstring theory. As a result, supersymmetry is not only the most promising new physics beyond the SM, but also builds a bridge between the low energy phenomenology and high-energy fundamental physics.
The great success to date at the LHC has been the discovery of a SM-like Higgs boson with an empirically mea-sured mass of m h = 125.09±0.24 GeV [1,2]. Nonetheless, in the Minimal SSM (MSSM), obtaining such a Higgs boson mass requires multi-TeV top squarks with small mixing or TeV-scale top squarks with large mixing [3]. However, strong constraints presently exist on the parameter space in the SSMs from LHC SUSY searches. For instance, the most recent search bounds on the gluino (g) mass show that it is heavier than about 1.9 TeV, whereas the light stop (t 1 ) mass is heavier than about 900 GeV [4]. Therefore, naturalness in the SSMs is challenged from both the Higgs boson mass and the LHC SUSY searches. On the other hand, the ATLAS [5] and CMS [6] Collaborations announced in December 2015 an excess of events in the diphoton channel with invariant mass of about 750 GeV at the 13 TeV LHC run II, though this dubious excess was proven to be only a statistical fluctuation in recent LHC data [7]. Hence, any natural candidate for the GUT model of our universe must also be consistent with the vanishing of this diphoton resonance.
To achieve the string-scale gauge coupling unification, we proposed the testable flipped SU (5) × U (1) X models [8][9][10] with TeV-scale vector-like particles [11], dubbed flippons. Subsequently, we constructed these flipped SU (5) models from local F-theory model building [12,13], where these models can be obtained in freefermionic string constructions as well [14]. The models were thus referred to as F -SU (5). A brief review of the "miracles" [15] of flippons in F -SU (5) is now in order. First, the lightest CP-even Higgs boson mass can be lifted to 125 GeV easily because of the one-loop contributions from the Yukawa couplings between the flippons and Higgs fields [15,16]. In the present work, this will only be relevant for those lighter regions of the model space which have already been excluded by the LHC, hence, we shall assume the minimal Yukawa couplings amongst the flippons and Higgs fields. Second, although the dimension-five proton decays mediated by colored Higgsinos are highly suppressed due to the missing partner mechanism and TeV-scale µ term, the dimensionsix proton decays via the heavy gauge boson exchanges are within the reach of the future proton decay experiments such as the Hyper-Kamiokande experiment. The key point is that the SU (3) C × SU (2) L gauge couplings are still unified at the traditional GUT scale while the unified gauge couplings become larger due to vector-like particle contributions [17,18]. Therefore, the F -SU (5) models differ from the minimal flipped SU (5) × U (1) X model, whose proton lifetime is too lengthy for the future proton decay experiments. Third, we can consider No-Scale supergravity [19] as a result of the string model building. More specifically, the lightest neutralino fulfills the role of the LSP and is lighter than the light stau due to the longer running of the Renormalization Group Equations (RGEs), providing the LSP neutralino as a dark matter candidate [20][21][22]. Fourth, given No-Scale supergravity, there exists a distinctive mass ordering M (t 1 ) < M (g) < M (q) of a light stop and gluino in No-Scale F -SU (5), with both substantially lighter than all other squarks (q) [20][21][22]. A primary consequence of this SUSY spectrum mass pattern at the LHC is the prediction of large multijets events [23]. Fifth, with a merging of both No-Scale supergravity and the Giudice-Masiero (GM) mechanism [24], the supersymmetry electroweak fine-tuning problem can be elegantly solved rather naturally [25,26]. Conversely, to satisfy the No-Scale boundary condition B µ = 0 and obtain the experimentally observed SM like Higgs boson mass, we find that the flippons are required to be relatively heavy, and as such we cannot explain the recently vanished 750 GeV diphoton resonance at the 13 TeV LHC, which seemed to prefer rather light vector-like particle masses. In conclusion, No-Scale F -SU (5) returns post disappearance of the 750 GeV diphoton excess. In this paper, we revisit and update the viable parameter space of No-Scale F -SU (5), exhibiting that consistency with both No-Scale boundary conditions and the experimentally measured SM like Higgs boson mass necessitates a lower bound on the gluino mass in the model space of around 1.9 TeV, which perhaps not coincidentally is the current LHC supersymmetry search bound, presenting a plausible explanation for the absence to date of a definitive SUSY signal at the LHC.

MODELS
We now briefly review the minimal flipped SU (5) model [8][9][10]. The gauge group for the flipped SU (5) model is SU (5) × U (1) X , which can be embedded into the SO(10) model. We define the generator U (1) Y ′ in SU (5) as and the hypercharge is given by There are three families of the SM fermions whose quantum numbers under SU (5) × U (1) X are respectively where i = 1, 2, 3. The SM particle assignments in F i ,f i andl i are where Q i and L i are respectively the superfields of the left-handed quark and lepton doublets, U c i , D c i , E c i and N c i are the CP conjugated superfields for the righthanded up-type quarks, down-type quarks, leptons and neutrinos, respectively. To generate the heavy righthanded neutrino masses, we can introduce three SM singlets φ i .
The breaking of the GUT and electroweak gauge symmetries results from introduction of two pairs of Higgs representations H = (10, 1), H = (10, −1), We label the states in the H multiplet by the same symbols as in the F multiplet, and for H we just add "bar" above the fields. Explicitly, the Higgs particles are As a consequence, we naturally have the doublet-triplet splitting due to the missing partner mechanism [10]. The triplets in h and h only have small mixing through the µ term, hence, the Higgsino-exchange mediated proton decay is negligible, i.e., there is no dimension-5 proton decay problem.
String-scale gauge coupling unification [11][12][13] is achieved by the introduction of the following vector-like particles (flippons) at the TeV scale XF = (10, 1) , XF = (10, −1) , The particle content from the decompositions of XF , XF , Xl, and Xl under the SM gauge symmetry are Under the SU (3) C × SU (2) L × U (1) Y gauge symmetry, the quantum numbers for the extra vector-like particles Mass degeneracy of the superpartners has not been observed, so SUSY must be broken around the TeV scale. In GUTs with gravity mediated supersymmetry breaking, called the supergravity models, we can fully characterize the supersymmetry breaking soft terms by four universal parameters (gaugino mass M 1/2 , scalar mass M 0 , trilinear soft term A, and the low energy ratio of Higgs vacuum expectation values (VEVs) tan β), plus the sign of the Higgs bilinear mass term µ.
No-Scale Supergravity was proposed [19] to solve the cosmological flatness problem, as the subset of supergravity models which satisfy the following three constraints: i) the vacuum energy vanishes automatically due to the suitable Kähler potential; ii) at the minimum of the scalar potential there exist flat directions that leave the gravitino mass M 3/2 undetermined; iii) the quantity StrM 2 is zero at the minimum. If the third condition were not true, large one-loop corrections would force M 3/2 to be either identically zero or of the Planck scale. A simple Kähler potential that satisfies the first two conditions is [19] where T is a modulus field and Φ i are matter fields, which parameterize the non-compact SU (N, 1)/SU (N ) × U (1) coset space. The third condition is model dependent and can always be satisfied in principle [27]. For the simple Kähler potential in Eq. (17) we automatically obtain the No-Scale boundary condition M 0 = A = B µ = 0 at the ultimate unification scale M F , while the sole model parameter M 1/2 is allowed, and indeed required for SUSY breaking. Because the minimum of the electroweak (EW) Higgs potential (V EW ) min depends on M 3/2 , the gravitino mass is determined by the equation d(V EW ) min /dM 3/2 = 0. Thus, the supersymmetry breaking scale is determined dynamically. No-Scale supergravity can be realized in the compactification of the weakly coupled heterotic string theory [28] and the compactification of M-theory on S 1 /Z 2 at the leading order [29].
Given that the B µ parameter is determined at the M F scale from the No-Scale boundary conditions, this in principle determines tanβ, though in the analytical procedure to follow here we use a consistency check to uncover those values of tanβ that are consistent with B µ (M F ) = 0, rather than solve for the explicit values of tanβ directly. The scale at which the vector-like flippon particles decouple is defined as M V , and as we shall show, is a function of M 1/2 via the RGE running. So in effect, all parameters are reduced to a dependence on M 1/2 , providing a genuine one-parameter model.

III. NUMERICAL RESULTS
The LHC will soon increase its reach to probe for a 2 TeV gluino and beyond, so we update and compute the precise upper boundary of the No-Scale F -SU (5) parameter space, extending the analysis of Ref. [30]. This upper limit is entirely defined by the requirement of neutralino dark matter. Our first constraints imposed are the WMAP 9-year [31] and 2015 Planck [32] 1σ relic density measurements, where we constrain the model to be consistent with both data sets, imposing limits of 0.1093 ≤ Ωh 2 ≤ 0.1221, as well as a sufficient range of the top quark mass around the world average [33], implementing limits in our analysis of 172.   s → µ + µ − ) are ×10 −9 , spin-independent cross-sections σSI are ×10 −11 pb, spindependent cross-sections σSD are ×10 −9 pb, and proton decay rate p → e + π 0 are in units of 10 35 years. The ∆M represents the mass difference between the light stau and lightest neutralino, given here to sufficient precision.  tolerance of ±0.24 GeV delivers the same message that the LHC is currently probing the viable region of the model space where a SUSY discovery would be expected. It should be noted that our Higgs boson mass calculations assume a minimal coupling of the flippon vectorlike multiplets. Although this has no effect on the Higgs mass calculations for a gluino mass greater than 1.9 TeV due to the rather large flippon mass M V required to satisfy the theoretical constraint of |B µ | ≤ 1 GeV, it would though provide a larger contribution to those excluded regions for M g 1 TeV, raising the Higgs mass to about 125 GeV for these lighter regions of the model space [15].
The SUSY mass spectra, relic density, rare decay processes, and direct dark matter detection cross-sections are calculated with MicrOMEGAs 2.1 [34] utilizing a proprietary modification of the SuSpect 2.34 [35]  ton lifetime numerical results, which satisfy the experimental constraints on 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 [36], 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 [37], the 3σ intervals around the SM value 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 [38], limits on spin-independent cross-sections for neutralino-nucleus interactions derived by the LUX experiment [39], limits on the proton spin-dependent cross-sections by the COUPP Collaboration [40] and XENON100 Collaboration [41], and current limits of about 1.7 × 10 34 yrs on the proton decay rate p → e + π 0 in the context of flipped SU (5) grand unification [42]. In short, there is no prominent SUSY related experiment that No-Scale F -SU (5) is not consistent with. Results of all these detailed calculations along with the primary sparticle masses are listed in TABLE I for a set of eight viable  sample benchmark   From TABLE I it can be seen that the mass difference ∆M between the light stau and lightest neutralino for the viable region we analyze in this work spans from a degenerate light stau and lightest neutralino at the upper bound of the model space, to a mass delta equivalent to the tau mass τ ± = 1.777 GeV. The branching fraction of a light stau decay to the lightest neutralino τ ± 1 → τ ± + χ 0 1 is 100%, therefore, in this particular region we study, this decay mode consists of an off-shell tau.
The recently excluded possibility of a 750 GeV diphoton resonance seemed to prefer rather light vector-like masses in order to generate the temporarily observed cross section [43,44]. In the event the diphoton resonance would have been confirmed, this requirement of light vector-like masses would have surely excluded our one-parameter version of No-Scale F -SU (5) since the viable vector-like mass M V is larger than about 23 TeV from FIG. 3 due to mostly the B µ = 0 condition. However, as would be necessary for No-Scale F -SU (5) to remain viable as a natural GUT candidate, the diphoton resonance curiously faded into oblivion. The reasoning behind the assertion noted above is depicted in

IV. CONCLUSIONS
We revisited the viable parameter space in No-Scale F -SU (5), examining the GUT model given the updated gluino mass limit of M g 1.9 TeV established by the LHC. To satisfy both the No-Scale boundary condition and the experimentally measured Higgs boson mass, we discovered that the lower limit on the gluino mass in the model space is curiously also about 1.9 TeV, rather similar to the current LHC supersymmetry search bound. This does present a legitimate explanation as to why no supersymmetry signal has been observed at the LHC to date. Moreover, due to the fact that the vector-like flippon particles are relatively heavy, primarily resulting from the No-Scale boundary condition B µ = 0 at the unification scale, the model appropriately excludes the recently fizzled 750 GeV diphoton resonance at the 13 TeV LHC, as is required of any viable GUT candidate. The natural union of the LHC gluino mass limit and experimentally measured Higgs boson mass in No-Scale F -SU (5) serves as a prime region for SUSY probing at the LHC, given also this region's quite favorable consistency with all other essential SUSY experiments involving relic density observations, rare decay processes, direct dark matter detection, and proton lifetime measurements. While SUSY enthusiasts have endured several setbacks over the prior few years amidst the discouraging results at the LHC in the search for supersymmetry, it is axiomatic that as a matter of course, great triumph emerges from momentary defeat. As the precession of null observations at the LHC has surely dampened the spirits of SUSY proponents, the conclusion of our analysis here indicates that the quest for SUSY may just be getting interesting.