Resolving the $(g-2)_{\mu}$ Discrepancy with $\cal{F}$-$SU$(5) Intersecting D-branes

A discrepancy between the measured anomalous magnetic moment of the muon $(g - 2)_{\mu}$ and computed Standard Model value now stands at a combined $4.2\sigma$ following experiments at Brookhaven National Lab (BNL) and the Fermi National Accelerator Laboratory (FNAL). A solution to the disagreement is uncovered in flipped $SU(5)$ with additional TeV-Scale vector-like $\bf 10 + \overline{10}$ multiplets and charged singlet derived from local F-Theory, collectively referred to as $\cal{F}$-$SU(5)$. Here we engage general No-Scale supersymmetry (SUSY) breaking in $\cal{F}$-$SU(5)$ D-brane model building to alleviate the $(g - 2)_{\mu}$ tension between the Standard Model and observations. A robust $\Delta a_{\mu}$(SUSY) is realized via mixing of $M_5$ and $M_{1X}$ at the secondary $SU(5) \times U(1)_X$ unification scale in $\cal{F}$-$SU(5)$ emanating from $SU(5)$ breaking and $U(1)_X$ flux effects. Calculations unveil $\Delta a_{\mu}({\rm SUSY}) = 19.0 - 22.3 \times 10^{-10}$ for gluino masses of $M(\widetilde{g}) = 2.25 - 2.56$ TeV and higgsino dark matter, aptly residing within the BNL+FNAL $1\sigma$ mean. This $(g - 2)_{\mu}$ favorable region of the model space also generates the correct light Higgs boson mass and branching ratios of companion rare decay processes, and is further consistent with all LHC Run 2 constraints. Finally, we examine the heavy SUSY Higgs boson in light of recent LHC searches for an extended Higgs sector.


INTRODUCTION
The Fermi National Accelerator Lab (FNAL) launched an experiment to update the original Brookhaven National Lab (BNL) analysis of the magnetic moment of the muon (g − 2) µ . The BNL measurements showed an enticing 3.7σ deviation [1] from the theoretical Standard Model calculation , albeit burdened with a large factor of uncertainty. The independently conducted FNAL venture sought to either confirm or exclude these initial BNL findings. Concluding the anticipation, FNAL announced recently a similar disparity with the Standard Model value, leading to a combined 4.2σ discrepancy of ∆a µ = a µ (Exp) − a µ (SM) = 25.1 ± 5.9 × 10 −10 [32], in conjunction with a reduced uncertainty by a factor of four. Even more intriguing is the magnitude of the discrepancy, which just so happens to be similar in scale to the electroweak contribution [30,31] to a µ (SM), suggesting new physics obscured at the TeV-scale.
A natural explanation for the anomaly is supersymmetry (SUSY), certainly the most auspicious extension to the Standard Model. The muon's magnetic moment maintains the benefit of precision measurements and is rather sensitive to new physics, thus it has long been viewed as a gateway to probing SUSY. The triumphs of SUSY are numerous and well known with regards to stabilizing quantum corrections to the scalar Higgs field, gauge coupling unification, mechanism for radiatively breaking electroweak symmetry, and yielding a dark matter candidate in the form of the lightest supersymmetric particle (LSP) under R-parity. Our model we investigate here showcases intersecting D-branes, therefore a critical feature of SUSY is its fundamental presence in superstring theory.
The union of these with extra vector-like matter, dubbed flippons [50], is referred to as the F -SU (5) D-brane model, which in aggregate reinforces deep theoretical constructions, furnishing a compelling natural GUT candidate for our universe. Serving as a viable high-energy candidate though, it must be capable of elegantly explaining any and all empirical observations, so we shall stress test the model to affirm whether or not it can indeed explain the BNL+FNAL 4.2σ discrepancy (Spoiler Alert: It can!). But first we must review the F -SU (5) D-brane model's foundation before we then take the deep dive into the calculations of (g − 2) µ and corresponding phenomenology later in the paper. The F -SU (5) literature is amply stocked with discussions of the minimal flipped SU (5) model (for instance, see Refs. [51,55,56,58,59,[70][71][72] and references therein). We shall provide here only a condensed review of the minimal flipped SU (5) model [73][74][75], where the gauge group SU (5)×U (1) X is embedded into the SO(10) model. First, the generator U (1) Y ′ in SU (5) is defined as which provides the hypercharge given by There are three families of Standard Model fermions, and the quantum numbers under SU (5) × U (1) X respectively are with i = 1, 2, 3. Relevant for our D-brane model notation, we associate F i ,f i andl i with the particle assignments are the left-handed quark doublets, right-handed up-type quarks, downtype quarks, left-handed lepton doublets, right-handed charged leptons, and neutrinos, respectively. Three Standard Model singlets φ i can be introduced to generate heavy right-handed neutrino masses.
The H and F multiplet states are labeled similarly, including only a "bar" added above the fields for H. More precisely, the Higgs particles are such that H d and H u are a single pair of MSSM Higgs doublets. We introduce this GUT scale Higgs superpotential to break the SU (5) × U (1) X gauge symmetry down to the Standard Model gauge symmetry:  [75]. There is only a small mixing through the µ term in the triplets h and h, so colored higgsinoexchange mediated proton decay remains negligible, i.e., the dimension-5 proton decay problem is absent [76].
String-scale gauge coupling unification at about 10 17 GeV can be realized by introducing the following vector-like particles (referred to as flippons) at the TeV scale derived from local F-theory model building [77][78][79] XF = (10, 1) , XF = (10, −1) , Under Standard Model gauge symmetry, the particle content resulting from decompositions of XF , XF , Xl, and Xl are The additional vector-like particles under the SU (3) C × SU (2) L ×U (1) Y gauge symmetry have the quantum numbers The superpotential is and after the SU (5) × U (1) X gauge symmetry is broken down to the Standard Model gauge symmetry, the superpotential presented directly above gives where y D ij , y Uν 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 new particle masses. The vector-like particle flippons are of course these new particles. The masses M φ ij , M XF ,and M Xl have not been explicitly computed yet, reserving that in-depth project for the future. Regardless, a common mass decoupling scale M V for the vector-like multiplets is implemented.
Contributions from vector-like multiplets require changes to the one-loop gauge β-function coefficients b i that promote a flat SU (3) Renormalization Group Equation (RGE) running (b 3 = 0) [49], separating the secondary SU (3) C × SU (2) L unification around 10 16 GeV, which we refer to as the mass scale M 32 , and the primary SU (5)×U (1) X unification near the string scale 10 17 GeV, defined as the mass scale M F . This is significant as it elevates unification close to the Planck mass. The M 3 and M 2 gaugino mass terms and couplings α 3 and α 2 unify at the scale M 32 into a single mass parameter M 5 and coupling α 5 [48], where M 5 = M 3 = M 2 and α 5 = α 3 = α 2 between M 32 and M F [49]. The flattening of the M 3 gaugino RGE running between M 32 and M V produces a characteristic mass texture of M ( t 1 ) < M ( g) < M ( q), spawning a light stop and gluino that are lighter than all other squarks [55].
Two critical effects naturally transpire at the scale M 32 . First, U (1) X flux effects [48] cause a small increase in M 1 from the evolution of M 1X via the following where α 1 = 5α Y /3 is the U (1) Y gauge coupling. Second, in the SU (5) × U (1) X models motivated by Dbrane model building, there exist three chiral multiplets in the SU (5) adjoint representation, for example, see Ref. [80]. These chiral multiplets can obtain vacuum expectation values around the SU (3) C × SU (2) L unification scale M 32 , and then the gaugino masses for can be split due to the high-dimensional operators [81,82]. Because the bino mass is a linear combination of the U (1) Y ′ and U (1) X gaugino masses, the bino mass M 1 , wino mass M 2 , and gluino mass M 3 can be independent free parameters at M 32 . Thus, we shall consider such effects by introducing the following relationship at the scale M 32 that stimulates mixing between M 5 and M 1X : This effect drives the wino to small values at the electroweak scale and we use the resulting phenomenology to constrain the coefficients in Eq. (18). A larger contribution from M 5 decreases the wino contribution to (g − 2) µ . On the contrary, a smaller contribution from M 5 shifts the SU (3) C × SU (2) L unification scale M 32 lower, to below 10 16 GeV. The lower M 32 in turn pushes the proton decay rate τ p (p → e + π 0 ) down to unacceptably fast time periods of 10 34 yrs or less. We study the nominal mixing highlighted in Eq. (18) in this analysis, though plan a more in-depth study later regarding maximum limits on the mixing parameters of Eq. (18). Following upon the F i ,f i , andl i of Eq. (4), the general No-Scale SUSY breaking soft terms at (3). General SUSY breaking soft terms of this type are motivated by D-brane model building [80], where F i , f i , l i , and h/h arise from intersections of different stacks of D-branes. As a result, the SUSY breaking soft mass terms and trilinear A terms will be different. The Yukawa terms HHh and HHh of Eq. (8) and F i F j h, XF XF h, and XF XF h of Eq. (15) are forbidden by the anomalous global U (1) symmetry of U (5), nonetheless, these Yukawa terms can be generated from high-dimensional operators or instanton effects. Differing from SU (5) models, the Yukawa term F i F j h in the F -SU (5) model gives down-type quark masses, so these Yukawa couplings can be small and hence generated via high-dimensional operators or instanton effects.

PHENOMENOLOGICAL RESULTS
The general No-Scale soft SUSY breaking terms in the F -SU (5) D-brane model are implemented at the SU (5) × U (1) X unification scale M F , and we concurrenly float the low-energy parameters tanβ, m t , and M V . Over 1.2 billion points in the model space are sampled by computing the SUSY mass spectra, rare decay process branching ratios, spin-independent dark matter cross-sections, and relic density using a proprietary mpi codebase built on scaled down versions of Micromegas 2.1 [83] and SuSpect 2.34 [84]. The intervals scanned are:    plied around the World Average top quark mass [85]. We only consider a positive Higgs bilinear mass term µ, and the strong coupling constant is fixed at α S (M Z ) = 0.1184, within the Particle Data Group 1σ variation [85]. The SUSY contribution to ∆a µ is calculated using GM2Calc 1.7.5 [86].
The computational results are constrained by filtering the data through current limits on pertinent beyond the Standard Model (BSM) experiments. A tolerance of 2σ is allowed around the branching ratio Br(b → sγ) of the bquark decay [89] and branching ratio Br(B 0 S → µ + µ − ) of the rare B-meson decay to a dimuon [90]. The WMAP 9year [91] and 2015-18 Planck [92,93] upper limit of about Ωh 2 ≃ 0.12 is required, though no lower limit on the relic density is applied. A lower bound on the proton decay rate τ p (p → e + π 0 ) ≥ 1.7 × 10 34 yrs [94] is also enforced. In summary, the points are filtered to ensure consistency with the following requirements:  Higgs boson mass of m h = 125.09 GeV [95,96], providing a range of 123 ≤ m h ≤ 127 GeV on the computed total m h . The total light Higgs boson mass calculation includes the vector-like particle contribution, which couples through the vector-like multiplet Yukawa coupling. We assume a maximum coupling, implying the (XD, XD c ) Yukawa coupling is Y XD = 0 and the (XU, XU c ) Yukawa coupling is Y XU = 1, and furthermore, the trilinear coupling A-term is A XD = 0 while the (XU, XU c ) A-term is A XU = A U [55,97]. These numerical values ensure a maximal coupling between the vector-like particles and light Higgs boson. Given only an upper limit established on the abundance of the lightest neutralino χ 0 1 , multi-component dark matter is generally necessary. Therefore, we rescale the spin-independent cross-section σ SI on nucleonneutralino collisions as such: The SU (5) breaking effects drive the wino to near degeneracy with the lightest neutralino, generating higgsino LSPs. Compressed spectra with M ( χ ± 1 )−M ( χ 0 1 ) = 3 − 7 GeV and M ( χ 0 2 ) < 0 are typical conditions identifying higgsino spectra, and indeed these are the char- 25 TeV [98][99][100][101][102][103][104] correlates to the current chargino constraint for higgsino LSP since the BNL+FNAL 1σ and 2σ points that satisfy the ATLAS chargino constraint show a coincident lower bound of M ( g) ≃ 2.25 TeV.
To more accurately assess any tension with LHC electroweakino production constraints so that we can provide a frame of reference for our relatively small wino, we superimpose the D-brane model points of FIG. 1 onto the ATLAS electroweakino production summary plot for higgsino LSP [87], as displayed in FIG. 2, and also the CMS electroweakino production plot for higgsino LSP [88], as shown in FIG. 3. The points are likewise separated into those that reside within either the BNL+FNAL 1σ or 2σ limits on ∆a µ . Given the compressed nature of the D-brane spectra, these higgsino simplified models appear to be the only search regions capable of probing the Dbrane model space that resolves the BNL+FNAL 4.2σ discrepancy. In the CMS plot in FIG. 3, for clarity we show only those points that satisfy the ATLAS chargino constraint of FIG. 2. The CMS simplified model scenario applied in FIG. 3 is not a precise fit for the F -SU (5) Dbrane model space, but it does give a good summary as to how our model stands against the current chargino constraints. The CMS analysis of Ref. [88] also studies the higgsino model in the phenomenological MSSM (pMSSM), though constraints are given in terms of the wino mass as a function of µ and all points in our model are well beyond these exclusion limits due to the large µ term.
Other than the small wino, the only other class of measurements the D-brane model would seem to be experiencing tension with are the direct-detection spinindependent cross-sections, which we rescale for small Ωh 2 . However, given the difficulty with which higgsino LSPs can be detected, we are not alarmed by the larger σ rescaled SI ∼ 10 −9 pb cross-section. Recently, ATLAS and CMS have completed searches for heavy resonances that would include heavy Higgs bosons found in an extended Higgs sector. The AT-LAS search [105] involves b-jet and τ -lepton final states, whereas the CMS search [106] focuses on b-quark pairs. All 261,562 points that pass the LHC chargino constraints on higgsino spectra and concurrently reside within either the BNL+FNAL 1σ or 2σ limits on ∆a µ are binned in the histogram of FIG. 4, using 10 GeV bin widths. The histogram peak resides in the (985, 995] GeV bin, or M (H 0 ) ≃ 1 TeV.

CONCLUSIONS
The recent FNAL confirmation of the BNL discrepancy between the measured value of the anomalous magnetic moment of the muon (g − 2) µ and the Standard Model (SM) prediction presents a combined deviation of 4.2σ. Given the possible confirmation of a statistically significant 5σ discovery in forthcoming years, all natural GUT model candidates should be capable of elegantly explaining these experimental anomalies. Supersymmetry (SUSY) persists as one promising extension of the SM, hence we study in this work a merging of a realistic D-brane model with the supersymmetric GUT model flipped SU (5) with extra TeV-scale string derived vectorlike multiplets, referred to collectively as the F -SU (5) Dbrane model. Flipped SU (5) has been shown to be both phenomenologically and cosmologically favorable, therefore, it is an ideal candidate to pursue whether it can indeed resolve the BNL+FNAL observed disparity with the SM calculations.
The supersymmetric GUT model F -SU (5) produces a distinctive two-stage unification process. The primary unification occurs at the SU (5) × U (1) X scale where M 1 = M 2 = M 3 , then a secondary unification at the SU (3) C × SU (2) L scale. However, the large wino mass M 2 at SU (5) × U (1) X due to the primary unification presents difficulties since a light wino at low-energy provides a large contribution to the muon (g − 2) µ . This dilemma can be resolved when the three chiral multiplets in the SU (5) adjoint representation acquire vevs around the scale SU (3) C ×SU (2) L , and then the gaugino masses for SU (3) C × SU (2) L × U (1) Y ′ of SU (5) can be split due to high-dimensional operators, leading to independent bino, wino, and gluino masses. These effects from SU (5) breaking can drive the wino M 2 to small values at the electroweak scale, which in consequence generates a large contribution to the muon (g − 2) µ .
A deep analysis of the parameter space uncovered a region in the model that can explain the BNL+FNAL muon (g − 2) µ measurements of ∆a µ = a µ (Exp) − a µ (SM) = 25.1 ± 5.9 × 10 −10 . Calculations show that the model can produce an anomalous magnetic moment as large as ∆a µ (SUSY) = 22.3 × 10 −10 for a gluino mass of M ( g) = 2281 GeV, well within the 1σ uncertainty on the observed value. Moreover, we completed computations for gluino masses as large as M ( g) = 2560 GeV, sufficient to ensure probing of the model space for several years to come at the LHC Run 2. We found that ∆a µ (SUSY) can remain tucked just above the lower 1σ bound of ∆a µ (SUSY) ≃ 19.0 × 10 −10 all the way up to the 2560 GeV gluino mass. The small M 2 does drive the chargino mass M ( χ ± 1 ) to near degeneracy with the LSP mass M ( χ 0 1 ) along with a small negative neutralino mass M ( χ 0 2 ), generating higgsino dark matter. In addition to the favorable ∆a µ (SUSY), these same points in the model space are consistent with the observed light Higgs boson mass and other rare decay branching ratios, and also all LHC constraints, particularly those on electroweakinos, given the small chargino. Lastly, in light of recent LHC searches for heavy Higgs bosons in an extended Higgs sector, for this same region that can explain the BNL+FNAL measurements, we examined the heavy SUSY Higgs by plotting all viable (g − 2) µ points in a histogram with 10 GeV bin widths and discovered that the histogram peak resides at M (H 0 ) ≃ 1 TeV.