Flipped SU(5) GUT phenomenology: proton decay and gμ-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {g_\mu - 2}$$\end{document}

We consider proton decay and gμ-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_\mu - 2$$\end{document} in flipped SU(5) GUT models. We first study scenarios in which the soft supersymmetry-breaking parameters are constrained to be universal at some high scale Min\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{in}$$\end{document} above the standard GUT scale where the QCD and electroweak SU(2) couplings unify. In this case the proton lifetime is typically ≳1036\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ > rsim 10^{36}$$\end{document} years, too long to be detected in the foreseeable future, and the supersymmetric contribution to gμ-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_\mu - 2$$\end{document} is too small to contribute significantly to resolving the discrepancy between the experimental measurement and data-driven calculations within the Standard Model. However, we identify a region of the constrained flipped SU(5) parameter space with large couplings between the 10- and 5-dimensional GUT Higgs representations where p→e+π0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \rightarrow e^+ \pi ^0$$\end{document} decay may be detectable in the Hyper-Kamiokande experiment now under construction, though the contribution to gμ-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_\mu -2$$\end{document} is still small. A substantial contribution to gμ-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_\mu - 2$$\end{document} is possible, however, if the universality constraints on the soft supersymmetry-breaking masses are relaxed. We find a ‘quadrifecta’ region where observable proton decay co-exists with a (partial) supersymmetric resolution of the gμ-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_\mu - 2$$\end{document} discrepancy and acceptable values of mh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_h$$\end{document} and the relic LSP density.


Introduction
The flipped SU(5) Grand Unified Theory (GUT) was first proposed in [1,2], as a possible intermediate gauge group obtained from the breaking of an underlying SO(10) GUT group. Flipped SU (5) was subsequently investigated in [3] as a GUT group in its own right, independently of a posa e-mail: olive@umn.edu (corresponding author) sible SO(10) parent group. Gauge coupling unification and the predictions for sin 2 θ W in flipped SU(5) models with and without supersymmetry were also studied in [3]. The supersymmetric version of flipped SU(5) was subsequently advocated in [4] on several grounds. One was that breaking the initial GUT symmetry down to the Standard Model (SM) SU(3)×SU(2)×U(1) gauge group via 10 and 10 Higgs representations led to suppression of proton decay via dimension-5 operators thanks to an economical missing-partner mechanism. It was also argued that flipped SU(5) would fit naturally into string theory, as weakly-coupled string models could not accommodate the adjoint and larger Higgs representations required to break other GUT groups such as SU (5), SO (10) and E 6 , but could accommodate the 10 and 10 of flipped SU (5). Indeed, variants of flipped SU (5) were subsequently derived in the fermionic formulation of weakly-coupled heterotic string theory [5][6][7][8].
Motivated by this encouraging backdrop, in this paper we pursue further studies of proton decay in supersymmetric flipped SU (5), which we link to an investigation of the muon anomalous magnetic moment, g μ − 2. It was pointed out in the initial paper on the non-supersymmetric version of flipped SU (5) [1,2] that it predicted the same proton decay modes as conventional SU (5), but with characteristic differences in the branching fractions (see also [9]). As already mentioned, dimension-5 contributions to proton decay are suppressed in supersymmetric flipped SU (5), so the dimension-6 modes such as p → e + π 0 are expected to dominate proton decays in this model. A new generation of underground detectors with increased sensitivities to this and other proton decay modes are now under construction, led by Hyper-Kamiokande [37], so it is interesting to evaluate accurately the expected rates for p → e + π 0 and other proton decays. In this paper we address two important aspects of such calculations, namely the appropriate matching conditions at the GUT scale (see [17] for an earlier study), and the uncertainties associated with SM input parameters and calculations of hadronic matrix elements [38], which had been examined previously in the context of conventional SU (5) in [39].
The outline of this paper is as follows. In Sect. 2 we recall briefly the salient features of the minimal supersymmetric flipped SU(5) model. Then, in Sect. 3 we discuss the GUTscale matching conditions for the gauge couplings, Yukawa couplings and soft supersymmetry-breaking parameters of the model, assuming that these are initially specified at some input scale, M in , above the scale where the SU(3) and SU (2) couplings of the SM are unified. Section 4 presents the formulae for the expressions relevant to the calculations of the proton decay rates, including their uncertainties, and Sect. 5 presents our results.
In the first version of the model that we study, in Sect. 5.1, the values of the soft supersymmetry-breaking parameters used as inputs at the input scale M in > M GU T are constrained to be universal [17,[39][40][41][42][43][44][45][46][47][48]. Universal boundary conditions arise in minimal models of supergravity [49,50], though a more general Kähler structure could lead to nonuniversalities. However, here we wish to compare the effects derived from flipped SU(5) directly with previously studied models such as the constrained minimal supersymmetric standard model or models based on SU(5) that satisfy universality conditions. In this case we find that the proton lifetime is generally beyond the reach of the next generation of experiments. However, the decay p → e + π 0 may be accessible if the couplings λ 4,5 between the GUT Higgs fields in the 10 and 10 representations and the SM Higgs fields in the 5 and 5 representations are both relatively large, λ 4,5 ∼ 0.5. However, even in this case the supersymmetric contribution to g μ − 2 is far smaller than the discrepancy between the experimental value and that from data-driven or lattice theoretical calculations in the SM [34]. We therefore discuss in Sect. 5.2 the possibilities for the combination of detectable proton decay and a substantial contribution to g μ − 2 in flipped SU(5) with non-universal input soft supersymmetrybreaking parameters. We find that the p → e + π 0 decay rate is quite insensitive to the degree of non-universality, whereas this can allow a much larger contribution to g μ − 2, as illustrated previously in [34]. We exhibit 'quadrifecta' domains of the multi-dimensional unconstrained flipped SU(5) parameter space where observable proton decay can co-exist with a (partial) supersymmetric resolution of the g μ −2 discrepancy, while the calculated value of m h is compatible with experiment within conservative calculational uncertainties and the relic LSP density is similar to the observed value.

The model
The model we consider is the minimal supersymmetric flipped SU(5)(FSU(5)) GUT with the gauge symmetry SU(5)×U(1) X [1][2][3][4][5][6][7][8][13][14][15]17,34], where U(1) X is an 'external' Abelian gauge factor. Here, we review only the essential components of the model. The model contains three generations of minimal supersymmetric Standard Model (MSSM) matter fields, together with three right-handed neutrino chiral superfields. These are embedded into 10,5 and 1 representations, which are denoted by F i ,f i , and c i , respectively, with i = 1, 2, 3 the generation index. The SU(5) and U(1) X charges of the matter sector of the theory arē A characteristic feature of the FSU(5) GUT is that the assignments of the quantum numbers for right-handed leptons and the right-handed up-and down-type quarks are "flipped" with respect to their assignments in standard SU (5). In Eq. (1), the V C K M i j are the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, U ν c , U l , and U l c are unitary matrices, and the phase factors ϕ i satisfy the condition i ϕ i = 0 [12]. The components of the doublet fields Q i and L i are written as where U PMNS is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. 1 The FSU(5) theory must be broken to the SM gauge symmetry. This is accomplished by including a pair of 10 and 10 Higgs fields, H and H , respectively, with the decompositions We note that the phase transition associated with this symmetry breaking was discussed in detail in [13,14,[18][19][20]. We recall also that the supersymmetric SM Higgs bosons are embedded in a pair of 5 and 5 Higgs multiplets, h andh, respectively, with the decompositions In addition, the theory has three (or more) SU(5) singlets φ a that generate the masses of the right-handed neutrinos.
The superpotential for this theory is where the indices i, j run over the three fermion families, the indices a, b, c have ranges ≥ 3, and for simplicity we have suppressed gauge group indices. We note that we have imposed a Z 2 symmetry H → −H to prevent the Higgs colour triplets or elements of the Higgs decuplets from mixing with SM fields. This symmetry also suppresses the supersymmetric mass term for H and H , and thus suppresses dimension-five proton decay operators. The first three terms of the superpotential (5) provide the SM Yukawa couplings, and the fourth and fifth terms in (5) account for the splitting of the triplet and doublet masses in the Higgs 5-plets. The masses of the color triplets are where V is the common vacuum expectation value (vev) of the H and H fields that break FSU(5), with V = N c H = N c H . The sixth term in (5) accounts for neutrino masses, and the seventh term plays the role of the μ-term of the MSSM. The last two terms may play roles in cosmological inflation, along with λ 6 , and also play roles in neutrino masses. GUT symmetry breaking, inflation, leptogenesis, and the generation of neutrino masses in this model have been discussed recently in [13][14][15]21,22], and are reviewed in [52].
In our numerical calculations we match at a scale close to M GU T , in which case the exponent has a small effect on this relationship. Keeping the exponential correction, we use the expression in Eq. (11) and (13) We then solve this equation numerically for g 5 , which can then be used in Eq. (13) to obtain the vev: Once we have the vev and g 5 , we can obtain M X from Eq. (9). In general, the loop corrections used in the matching conditions are important when the scale M in at which universality is imposed on the soft supersymmetry-breaking parameters is > M GU T . The SM Yukawa couplings are also matched at M GU T , to λ 1,2,3 [17,34]: Unlike minimal SU(5), the neutrino Yukawa couplings are naturally fixed to be equal to the up-quark Yukawa couplings. 2 This is a consequence of the flipping that puts the right-handed neutrinos into decuplets in FSU(5), instead of being singlets as in minimal SU (5), where their Yukawa couplings would be viewed as independent parameters. The supersymmetric FSU(5) GUT model is specified by the following GUT-scale parameters. There are two independent soft supersymmetry-breaking gaugino masses, namely a common mass M 5 for the SU(5) gauginosg,W andB, and another mass M X 1 for the 'external' gauginoB X that is independent a priori. There are also three independent soft supersymmetry-breaking scalar masses that we assume to be generation-independent, namely m 10 for the sfermions in the 10 representations of SU(5), m5 for the sfermions in the 5 representations of SU(5), and m 1 for the right-handed sleptons in the SU(5)-singlet representations. We assume for simplicity that the trilinear soft supersymmetry-breaking parameters A 0 are universal.
We assume initially that the gaugino and scalar masses and trilinear parameters are universal at M in [17,21,22], i.e., we take M 5 = M X 1 = m 1/2 and m 10 = m5 = m 1 = m 0 at M in . In general, as in the NUHM2 [53,54], one may assume independent soft supersymmetry-breaking for the 5 and 5 Higgs representations, m H 1,2 . However, we begin by assuming that these are also universal so that m H 1 = m H 2 = m 10 = m5 = m 1 . We treat the ratio of SM Higgs vevs, tan β, as a free parameter. Finally, we assume that the Higgs mixing parameter μ > 0, so as to obtain a supersymmetric contribution to g μ − 2 with the same sign as the discrepancy between the experimental measurement and the data-driven theoretical value in the SM.
The matching conditions for the the soft supersymmetrybreaking gaugino mass terms at M GU T are where A 4 and A 5 are the trilinear A-terms associated with the superpotential couplings λ 4 and λ 5 , respectively. We note that there are additional 1-loop contributions to the gaugino masses that could in principle be as large as the those included in Eqs. (17)(18)(19). These are proportional to the soft mass term along the flat direction ( = N c H = N c H ) that breaks the FSU(5) gauge symmetry. For example, M 2 includes an additional term, 14] on the right-hand side of Eq. (18), and there are similar contributions for M 1,3 . As described in detail in [13,14], this flat direction is lifted by a non-renormalizable superpotential term of the form λ(HH ) n /M 2n−3 P with n ≥ 4 to obtain a sufficiently large vev, where M P is the reduced Planck mass, M 2 P = 1/8π G N . We expect the soft mass for to be of the same order as the other soft mass parameters, in which case the late decay of the flat direction releases entropy leading to a dilution factor of order = 10 4 (m /10 TeV), and a temperature (after decay) of about 1 MeV λ 2 i (m /10 TeV) 1/2 . Lowering m would require some Yukawa coupling (e.g., λ 7 ) to be increased to maintain a temperature 1 MeV, and would decrease and the contribution to the gaugino masses. However, due to the model dependence of m , we do not include this contribution in Eqs. (17)(18)(19).
The scalar soft masses are matched using [17,34]: The trilinear terms are initially set to be universal at M in with A i = A 0 , corresponding to the Yukawa couplings λ i for i = 1 − 5. Each A i is run down to the GUT scale and matched using Finally, the magnitude of the MSSM μ-term and the bilinear soft supersymmetry-breaking B-term are determined at the electroweak scale by the minimization of the Higgs potential. This also determines the pseudoscalar Higgs mass, M A , which we use as an input in FeynHiggs 2.18.10 [55][56][57][58] to determine the masses of the remaining physical Higgs degrees of freedom. 3 Our constrained FSU(5) model is therefore specified by the following set of parameters: Later, in Sect. 5.2, we generalize the model to allow M 5 = M X 1 , as well as allowing the soft masses m H 1 , m H 2 , m 10 , m5, and m 1 to differ from each other. The relevant RGEs for flipped SU(5) were given in [17]. In principle, it is also necessary to specify the mass of the heaviest left-handed neutrino, m ν 3 , which we take to be 0.05 eV. This and λ 6 fix the right-handed neutrino mass and μ φ . However, our results are quite insensitive to this choice.

Proton lifetime
Proton decay in FSU(5) was discussed in detail in [9], and we quote here only the essential results from that work. Thanks to the suppression of dimension-5 operators by the FSU(5) missing-partner mechanism, the main contribution to nucleon decay is due to the exchanges of SU(5) gauge bosons. 4 The relevant gauge interaction terms are where the X α a are the SU(5) gauge vector superfields, where the Wilson coefficient can be written as evaluated at the weak scale. The partial proton decay widths to + i π 0 can be expressed as follows in terms of these coefficients at the hadronic scale: where and we use the following determinations of the matrix elements by lattice calculations [38]: For proton decays with a final-state lepton i ( 1 = e, 2 = μ), we have where m p and m π denote the proton and pion masses, respectively, and the subscript on the hadronic matrix element indicates that it is evaluated at the corresponding lepton kinematic point. The renormalization factor between the GUT scale and the electroweak scale is [65,66] where m Z , μ SUSY , and μ GUT denote the Z -boson mass, the SUSY scale and the GUT scale, respectively, and α A ≡ g 2 A /(4π) with g A (A = 1, 2, 3) the gauge coupling constants of the SM gauge groups. Below the electroweak scale, we take into account the perturbative QCD renormalization factor, which was computed in Ref. [67] at the two-loop level to be A L = 1.247.
Using Eq. (29), we can readily compute the partial lifetime of the p → e + π 0 mode as [9]: A similar expression can be obtained for the partial lifetime of the p → μ + π 0 mode: As seen in the above expressions, the proton decay rates depend on the unitary matrix U l associated with the embedding of the left-handed lepton fields into the5 fields (see Eq. (1)). As discussed in Ref. [9], for a light neutrino mass matrix that has a hierarchical structure that is either normally ordered (NO) or inversely ordered (IO), the relevant matrix elements of U l may be approximated by where c i j ≡ cos θ i j and s i j ≡ sin θ i j are the mixing angles, δ is the Dirac phase, and α 3 is a Majorana phase in the PMNS matrix. We use these relations in the following calculation, in which case the ratio of the μ + π 0 and e + π 0 partial decay widths is predicted to be Both of these values are much larger than the prediction in conventional supersymmetric SU (5), which is 0.008. The rate for p → μ + π 0 in the IO scenario is expected to be similar to that for p → e + π 0 in the NO scenario, and the sensitivity of Hyper-Kamiokande to the μ + π 0 final state is expected to be similar to that to the e + π 0 final state [37].

Error estimates
We provide in this section a brief derivation of the estimates of dominant errors in the proton lifetime. We look at two contributions to these error estimates, namely the effect of the uncertainty in g 3 on the mass of M X and the effects of the uncertainties in the matrix elements. To determine the effect of g 3 , we look at the dependence of on g 3 , ignoring the g 3 dependence of g 5 . We have checked numerically that this can safely be ignored. Since (36) is determined at the scale M GU T , the scale at which g 3 and g 2 unify, the variation of the exponential due to the error in g 3 has no effect. This means that the leading-order dependence of M X on g 3 is due to the change in the matching scale M GU T . To approximate this effect on the lifetime, we need the one-loop expressions for the gauge couplings g 2 , g 3 : where M GU T is a function of g 3 defined by the relation The g 3 dependence of M GU T is given by the following expression The estimated error in M X is then where α s is the strong coupling constant, and α s is its uncertainty. Since the proton lifetime scales as M 4 X , we have 6 Estimating the error in the lifetime due to the uncertainties in the matrix elements is straightforward, as the lifetime scales as the inverse of the matrix element squared. This leads to an error estimate of where M i denotes the matrix elements and M i is their uncertainties.
The total error estimate is then

Universal boundary conditions
We examine first a selection of (m 1/2 , m 0 ) planes when universal boundary conditions are applied at a high input scale M in > M GU T . Our baseline plane shown in Fig. 1 is similar to that considered in [21,22] with tan β = 10, and μ > 0. 7 The pink shaded region at large m 0 m 1/2 is excluded by the absence of a consistent electroweak vacuum, and the brown shaded region where m 1/2 m 0 is excluded because the lighter stau is the LSP and/or tachyonic. The red dot-dashed lines are contours of constant Higgs masses between m h = 121 and 126 GeV in intervals of 1 GeV, as calculated using FeynHiggs 2.18.10 [55][56][57][58]. We consider calculated values of m h ∈ (122, 128) GeV to be consistent with the measured value within conservative calculational uncertainties.
The solid blue contours in Fig. 1 show values of the LSP relic density, χ h 2 , as labeled, as calculated assuming that the Universe expands adiabatically. The contour for χ h 2 = 0.1, corresponding to the measured dark matter density, appears as a thick blue curve near the pale blue shaded area, and corresponds to the focus-point region [68][69][70][71]. There is also a short contour with χ h 2 = 0.1 just above the stau-LSP region with m 1/2 1 TeV [72][73][74][75] that is almost invisible. As mentioned previously, the generation of a large amount (O(10 4 ) ) of entropy in the early Universe due to the late decay of the flat direction responsible for the breaking of FSU(5) is a generic feature of FSU(5) cosmology [13][14][15]21,22], so we do not interpret this adiabatic calculation of χ h 2 as a necessary constraint. Indeed when accounting for the late entropy production, we expect that parameters yielding h 2 ∼ 100 − 1000 would correspond better to the present relic density h 2 0.1 (see [76,77] for related work).
In addition, we show in Fig. 1 as the solid brown curve the contour where τ p ( p → e + π 0 ) = 10 36 years, as calculated assuming normal ordering (NO) of the neutrino masses. This line appears at m 0 ≈ 2.5 TeV and also runs roughly parallel to the focus-point strip. The proton lifetime varies slowly across this plane, in general, and is always within the range 5-20 ×10 35 years, beyond the foreseen experimental reach [37]. The brown dashed contour corresponds to τ p − σ τ p = 5 × 10 35 years, illustrating the effect of the uncertainty in the calculation of τ p discussed in the previ- which is shaded pink, we see a focus-point strip (the thick blue line that has χ h 2 = 0.1). We also see that, near this region, the Higgs mass is between 124 and 126 GeV for this choice of m 1/2 . This is one reason why we focus on A 0 = 0 in the subsequent exploration of the effects of GUT parameters. 9 As in Fig. 1, a thick blue contour with h 2 = 0.1 is also found outside the brown shaded region when A 0 > 0, corresponding to stop coannihilation. (In principle, there is also a corresponding line at negative A 0 , but it cannot be seen with the resolution used in this figure). Between the pink and brown regions, the relic density can be as large as 10 3 , which may even be preferred in FSU(5) if entropy is injected late, as already mentioned. For the value of m 1/2 = 4 TeV chosen for this plane, the contribution to g μ − 2 is always small, and we see only one green-dotted contour with a μ = 10 −11 . Above this line, the contribution is even smaller. The proton lifetime is rather long, and we see two contours corresponding to values of 1 and 2 ×10 36 years. In the remainder of this subsection we concentrate on two fixed values of A 0 /m 0 : A 0 /m 0 = 0 corresponding to the focus-point region and 3.8 corresponding to stop coannihilation. 9 However, we note that there is no strong reason to expect a vanishing trilinear coupling and, in fact, in minimal supergravity one expects A 0 /m 0 = 3 − √ 3 = 1.27. In our (supergravity-based) sign convention, the Higgs mass is larger at large positive A 0 . In Fig. 3, we compare analogous planes with different values of tan β. In the left panel, tan β = 4, while in the right panel tan β = 35. All other fixed parameters are the same as in Fig. 1. For tan β = 4, The Higgs mass is always less than 122 GeV, and only the m h = 121 GeV contour appears. The region without consistent electroweak symmetry breaking is pushed out beyond the range of the plot, so there is no visible focus-point region, and entropy production is required throughout the displayed plane. Compared to Fig. 1, the proton lifetime is longer and the values of a μ are smaller for given values of (m 1/2 , m 0 ). In contrast, for the larger value of tan β = 35 shown in the right panel of Fig. 3, the region without electroweak symmetry breaking extends to lower values of m 1/2 and m 0 , and the Higgs mass is higher, rising beyond 127 GeV in this plane. Though the proton lifetime is somewhat lower than in Fig. 3, it is still very large. On the other hand, values of a μ are larger than in Fig. 3, and reach 10 −10 for m h > 122 GeV.
We explore the dependence on λ in Fig. 4. Keeping the other parameters fixed to the values used in Fig. 1 The dependence on M in is considered in Fig. 5. The left panel assumes the same parameter values as in Fig. 1, with the exception of M in , which is now set at the Planck scale. Electroweak symmetry breaking occurs throughout the plot, and the stau LSP region is pushed to the lower right corner of the panel. The Higgs mass, proton lifetime and relic density are all slightly larger than in Fig. 1. In the right panel, we again take M in = M P but now with λ = (0.3,0.3), which is near its upper limit for this value of M in . The relation between the proton lifetime and √ λ 4 λ 5 is seen more clearly in Fig. 6, which shows a pair of (λ 4 = λ 5 , m 0 ) planes for m 1/2 = 5 TeV, A 0 /m 0 = 0, tan β = 10, μ > 0, λ 6 = 0.0001, with M in = 10 16.5 GeV (left panel) and M in = M P (right panel). When M in = 10 16.5 GeV there is a small region in the upper left corner where electroweak symmetry breaking breaks down, which is shaded pink. Bordering this region, the focus-point strip with h 2 = 0.1 is the thick blue contour. Other blue contours correspond to larger values for the relic density, but we re-emphasize that larger values of χ h 2 would be allowed in the context of FSU(5) cosmology, in which substantial entropy is likely to have been generated in the early Universe. There is a stau LSP in the brown shaded region at low m 0 in the left panel. For M in = M P , the RGEs break down when λ 4 = λ 5 0.35, as indicated by the red shading in the right panel. The red lines are contours of m h = 125 GeV. We note that m h varies slowly across this plane, so this is the only integer mass contour displayed. Finally, the solid brown lines are contours of τ ( p → e + π 0 ) in units of 10 35 years. We see that values of the proton lifetime that are 3 × 10 35 years, and hence potentially accessible to the next generation of experiment, are found in the right portions of the planes where λ 4 = λ 5 0.3.
The parameter planes displayed above have all assumed A 0 = 0, and we present in Fig. 7 a pair of planes with nonzero A 0 , specifically A 0 = 3.8 m 0 . In the left panel, we take M in = 10 16.5 GeV and λ = (0.5,0.5) to minimize the pro-ton lifetime. In the right panel, M in = M P and λ = (0.3,0.3). These planes exhibit the possible importance of a compressed stop spectrum, which introduces the possibility of stop coannihilation [78][79][80][81][82][83][84][85], and can be compared with Fig. 12c of [21,22]. The brown shaded regions where m 0 > m 1/2 are disallowed because the stop is either the LSP or tachyonic, and that in the left panel where m 1/2 > m 0 has a stau LSP. As we have seen previously, the stau LSP region recedes to larger values of m 1/2 as M in increases, and is not visible in the right panel where M in = M P . We see very large values of χ h 2 in the bulk of the uncoloured region, 10 but there are strips close to the boundaries of the shaded regions where χ h 2 is reduced. Once again, the thick blue shaded contour running along the stop LSP region corresponds to h 2 = 0.1.
It is important to note that we can find h 2 = 0.1 and m h = 125 GeV simultaneously in both panels in Fig. 7 Fig. 1 to ensure non-divergent running between M P and M GU T . In both cases, the contribution to a μ is small (< 10 −11 ). We stress again, however, that as late-time entropy production is expected in this FSU(5) model, most of the displayed plane is viable cosmologically.
In Fig. 8, we show a pair of (λ 4,5 , m 0 ) planes with A 0 /m 0 = 3.8 and M in = 10 16.5 GeV (left panel) and M in = M P (right panel), as in Fig. 7. As previously, we see brown shaded regions where the lightest neutralino is not the LSP, and a red shaded region in the right panel where the RGEs break down. We again see stop strips. For the lower value of M in , we choose m 1/2 = 7 TeV and for M in = M P , we take m 1/2 = 2 TeV. We find that a μ is small everywhere in the left plane due to the large value of m 1/2 , whereas in the right plane we see contours of a μ = 1 and 2 × 10 −11 , also too small to make a significant contribution to resolving the discrepancy between experiment and the Standard Model calculation. As previously, we find that the proton lifetime is minimized, and potentially observable, for large λ 4 = λ 5

Non-universal models and g μ − 2
From the results in the previous subsection, it is clear that the contribution to a μ is always small when universal boundary conditions applied for scalar and gaugino masses at M in . Indeed, in all of the above planes, a significant contribution to a μ occurs only at low supersymmetric masses that are in tension with LHC constraints and where the Higgs mass is well below the experimental value, even with a conservative assessment of the theoretical uncertainty in the calculation of m h . On the other hand, previous analyses have shown that  Fig. 1 substantially larger contributions to a μ are possible when some degree of universality is abandoned [34,86].
Therefore, in this subsection, we depart from full gaugino and scalar mass universality at M in , while retaining the constraints imposed by FSU (5). Thus, we include two independent gaugino masses, a common mass M 5 for the SU (5) gauginosg,W andB, and an independent mass M X 1 for the 'external' gauginoB X . This is to be contrasted with our previous assumption that M 5 = M X 1 = m 1/2 at M in . Similarly we now include five independent soft supersymmetry-breaking scalar masses, m 10  TeV for the example in [34]. These were found to optimize the value of a μ .
Some results are displayed in the (M X 1 , m 1 ) planes shown in Fig. 9. In the left panel, only the singlet masses M X 1 and m 1 break universality, i.e., we set m5 = m 10 = 1 TeV in this case. The Higgs mass varies very little in this plane and is always slightly larger than 122 GeV (no contours are shown). Similarly the proton lifetime varies very little and is approximately (1.2 ± 0.6) × 10 36 years. In contrast, the relic density (indicated by the labeled blue contours) varies significantly, reaching values as large as h 2 = 500 in the upper left corner of the panel. Also seen as vertical light blue lines are the lower limits to the mass of the lightest gaugino m χ > 100 GeV (which is valid for generic slepton masses) and > 73 GeV (which can be reached if the mass difference between the LSP and the lightest slepton < 2 GeV). Finally, we show as green dotted lines some contours of a μ in units of 10 −11 . In general, they are significantly larger than was found in the universal case, with contours of 10 and 15 × 10 −11 appearing, the largest value of a μ being 18 × 10 −11 . While an improvement over the universal case, these are still too small to account for the discrepancy between the Standard Model and experiment.
We can increase a μ by choosing a lower value of m5 relative to m 10 . As an example, in the right panel of Fig. 9 we take m5 = m 10  a μ = 200 × 10 −11 appears outside the selectron/smuon LSP region only when the lightest gaugino mass is below its lower limit of 73 GeV. However, the 150 × 10 −11 contour extends to the right of the vertical LSP mass limit of 100 GeV, in a region where the gaugino is the LSP and has a relic density χ h 2 ∼ 0.1. This region resembles the best a μ point found in [34].
The sensitivity to λ 4,5 for similar choices of model parameters is shown in the left panel of Fig. 10, which displays a (λ 4,5 , m 1 ) plane with M X 1 = 0.8 TeV and other parameters the same as those used in the left panel of Fig. 9. In this case, we see only a single relic density contour, which has χ h 2 = 10. The Higgs mass is again slightly larger than 122 GeV across the plane, and a μ (10 − 18) × 10 −11 . However, we now see a large variation in the proton lifetime, which varies from 5 × 10 37 years at low values of λ 4 = λ 5 , to 10 35 years at large values.
In contrast, in the right panel of Fig. 10 we fix M X 1 = 200 GeV, with m 10 = 2m5 = 1 TeV. The proton lifetime decreases as λ 4 = λ 5 increase, becoming potentially observable for values 0.4 (taking into account the matrix element uncertainties). The Higgs mass is not very sensitive to the choice of m5 or the change in M X 1 , with the Higgs mass being slightly above 122 GeV across the plane displayed. The relic density is decreased at low m 1 , as the mass of the selectron is lower and there is now a long relic density selectron/smuon coannihilation strip where h 2 = 0.1 just above the brown shaded region where the LSP is a selectron or smuon. The value of a μ is now larger as well, and we see contours of 10, 20, 50, 100 × 10 −11 all lying above the selectron/smuon LSP region. 12 We note the appearance of a 'quadrifecta' strip at large λ 4 = λ 5 close to the charged-LSP boundary, where a μ ∼ 100 × 10 −11 , τ ( p → e + π 0 ) is potentially detectable, m h is compatible with experiment and χ h 2 ∼ 0.1 (though the latter is not a requirement, as mentioned previously).
The sensitivities of the 'quadrifecta' region to some of the input parameters are shown in Fig. 11. In the upper pair of panels we explore the sensitivity to A 0 /m 0 , which is taken to be 1 and 3 in the left and right panels, respectively. We see that there is rather small sensitivity to A 0 /m 0 , with a μ and the proton lifetime both increasing slightly with the value of A 0 /m 0 . In the lower pair of panels we explore the sensitivity to tan β, which is taken to be 25 and 40 in the left and right panels, respectively. We see that a μ is significantly smaller for tan β = 25 and larger for tan β = 40, whereas the proton lifetime is quite insensitive to tan β. We recall that the validity of the perturbation regime is quite restricted for larger values of tan β, and recall that our results are rather insensitive to the value of λ 6 .
The point in a quadrifecta region with the largest contribution to g μ − 2 for these parameter choices is located in the lower right panel of Fig. 11, and is marked with a black star. The cold dark matter density and m h corresponding to  Fig. 1 this point are consistent with experiment within the theoretical uncertainties, and are given in Table 1 together with the mass spectrum. The spectrum and relic density are similar to those found for the best-fit point in [34], but the lifetime for p → e + π 0 is considerably shorter due to the higher values of λ 4 = λ 5 , and within reach of Hyper-Kamiokande if the neutrino masses are normal-ordered.

Conclusions
We have considered in this paper various aspects of the phenomenology of supersymmetric flipped SU(5) GUTs, focusing on predictions for proton decay and g μ − 2. We have found that, if the soft supersymmetry-breaking parameters are constrained to be universal at some high scale M in above  Table   the GUT scale, the proton lifetime is typically 10 36 years and the supersymmetric contribution to g μ − 2 is small. The proton lifetime is generally too long to be detected in the foreseeable future, and the model does not contribute significantly to reducing the tension between data-driven calculations of g μ − 2 within the Standard Model and the experimental measurement. However, we have found that there is a region of the constrained flipped SU(5) parameter space with large 10 10 5 and 10 10 5 couplings where p → e + π 0 decay may be detectable in the Hyper-Kamiokande experiment [37] now under construction. Nevertheless, the flipped SU(5) GUT contribution to g μ − 2 is still small.
However, we have found that if the universality constraints on the soft supersymmetry-breaking masses are relaxed there is a region of flipped SU(5) GUT parameter space where the model contribution to g μ − 2 can be large enough to reduce significantly the discrepancy between theory and experiment while τ ( p → e + π 0 ) may simultaneously be short enough to be detected in Hyper-Kamiokande. This region appears when A 0 /m 0 ∼ 1−3 and tan β ∼ 25−40, for suitable values of the other flipped SU(5) parameters. We call this the 'quadrifecta' region, since the theoretical calculation of the light Higgs mass is compatible with the experimental measurement, within uncertainties, and the strip where the relic Table 1 Parameters and predictions of an FSU(5) point in the 'quadrifecta' region that yields a μ = 120 × 10 −11 , values of χ h 2 and m h that are consistent with experiment within theoretical uncertainties, and a lifetime for p → e + π 0 within reach of the Hyper-Kamiokande experiment if neutrino masses are normal-ordered. This point corresponds to the star in the lower right panel of Fig. 11 Input GUT parameters (masses in units of 10 16  Inverse-ordered ν masses: τ p→e + π 0 | IO = 3.0 × 10 36 years τ p→μ + π 0 | IO = 2.2 × 10 35 years LSP density χ h 2 0.12 if the Universe expands adiabatically passes through the region. 13 This 'quadrifecta' region was previously identified in a dedicated analysis of g μ − 2 in the flipped SU(5) GUT [34], and it is encouraging that this region appears quite stable under mild variations in the input parameters. As pointed out in [34], in this region both the LSP and lighter smuon masses are very close to the LEP lower limits on their masses of ∼ 100 GeV, and detection of the LSP, smuon and selectron should be possible at the LHC. Their discovery would be a striking success for the flipped SU(5) framework described here, which could be complemented by the detection of p → e + π 0 decay in the Hyper-Kamiokande experiment [37].

Data Availability Statement
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