Flipped SU(5) GUT Phenomenology: Proton Decay and $g_\mu-2$

We consider proton decay and $g_\mu - 2$ 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 $M_{in}$ above the standard GUT scale where the QCD and electroweak SU(2) couplings unify. In this case the proton lifetime is typically $\gtrsim 10^{36}$~yrs, too long to be detected in the foreseeable future, and the supersymmetric contribution to $g_\mu - 2$ 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 $p \to e^+ \pi^0$ decay may be detectable in the Hyper-Kamiokande experiment now under construction, though the contribution to $g_\mu -2$ is still small. A substantial contribution to $g_\mu - 2$ 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 $g_\mu - 2$ discrepancy and acceptable values of $m_h$ and the relic LSP density.


Introduction
The flipped SU(5) Grand Unified Theory (GUT) was first proposed in [1], as a possible intermediate gauge group obtained from the breaking of an underlying SO(10) GUT group. Flipped SU (5) was subsequently investigated in [2] as a GUT group in its own right, independently of a possible 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 [2]. The supersymmetric version of flipped SU (5) was subsequently advocated in [3] 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 [4,5].
Following the formulation of flipped SU(5) and its derivation from string theory, there have been many phenomenological studies of the model. These have included such particlephysics topics as proton decay [6][7][8] and neutrino masses and mixing [9][10][11][12], as well as cosmological issues such as dark matter [13,14], baryogenesis, inflation and entropy generation [10][11][12]15,16]. The upshot of these studies is that flipped SU(5) can provide a complete framework for particle physics and cosmology below the Planck scale. An additional topic of current interest in flipped SU (5) is the muon anomalous magnetic moment, g µ − 2. It has been shown recently that the discrepancy between the experimental measurement [17,18] and the data-driven theoretical calculation within the SM [19] can be partially resolved within a minimal flipped SU(5) model [20] (where many relevant references can be found), and that the discrepancy with the experimental measurement is completely resolved if the lattice SM calculation [21] is adopted. The discrepancy is also completely resolved within non-minimal flipped SU(5) models [5,22] even if the data-driven SM calculation [19] is adopted.
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 nonsupersymmetric version of flipped SU(5) [1] that it predicted the same proton decay modes as conventional SU (5), but with characteristic differences in the branching fractions (see also [6]). 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 [23], 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 [14] for an earlier study), and the uncertainties associated with SM input parameters and calculations of hadronic matrix elements [24], which had been examined previously in the context of conventional SU (5) in [25].
The outline of this paper is as follows. In Section 2 we recall briefly the salient features of the minimal supersymmetric flipped SU(5) model. Then, in Section 3 we discuss the GUT-scale 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 Section 5 presents our results.
In the first version of the model that we study, in Section 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 [14,[25][26][27][28][29][30]. 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 [20]. We therefore discuss in Section 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 supersymmetry-breaking parameters. We find that the p → e + π 0 decay rate is quite insensitive to the degree of nonuniversality, whereas this can allow a much larger contribution to g µ − 2, as illustrated previously in [20]. 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.
Finally, Section 6 summarizes our conclusions.

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][10][11][12]14,20], 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 CKM ij 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 [9]. 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 [10,11,15]. 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 [10][11][12]16], and are reviewed in [32].

Matching Conditions
As a preliminary to giving the gauge coupling matching conditions, we first specify the masses of the fields that get masses from the breaking of the unified gauge symmetries. After the symmetry breaks, just as in the minimal SU(5) case, the heavy X, X gauge bosons of the SU(5) symmetry will mediate proton decay. We recall that the conventional SM hypercharge is a linear combination of the U(1) X gauge symmetry and the diagonal U(1) subgroup of SU(5): √ 14] on the right-hand side of Eq. (18), and there are similar contributions for M 1,3 . As described in detail in [10,11], 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 = 1/8πG N . We expect the soft mass for Φ 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 Φ /10 TeV), and a temperature (after decay) of about 1 MeV λ 2 i (m Φ /10 TeV) 1/2 . Lowering m Φ 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 Φ , we do not include this contribution in Eqs. (17)(18)(19).
The scalar soft masses are matched using [14,20]: 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 [34] 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 Section 5.2, we generalize the model to allow M 5 = M X1 , 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 [14]. 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 [6], 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 [24]: 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 [36,37] 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. [38] 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 [6]: 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. [6], 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 ij ≡ cos θ ij and s ij ≡ sin θ ij 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 [23].

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 g 2 (M GU T ) = g 3 (M GU T ). 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 5 Results

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 [16] with tan β = 10, A 0 = 0, M in = 10 16.5 GeV, λ ≡ (λ 4 , λ 5 ) = (0.3, 0.1), λ 6 = 10 −4 , 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 [34]. 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, Ω χ h 2 , as labeled, as calculated assuming that the Universe expands adiabatically. The contour for Ω 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 [39]. There is also a short contour with Ω χ h 2 = 0.1 just above the stau-LSP region with m 1/2 1 TeV [40] 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 [10][11][12]16], so we do not interpret this adiabatic calculation of Ω χ h 2 as a necessary constraint. Indeed when accounting for the late entropy production, we expect that parameters yielding Ωh 2 ∼ 100 − 1000 would correspond better to the present relic density Ωh 2 0.1. (See [41,42] 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 yrs, 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 yrs, beyond the foreseen experimental reach [23]. The brown dashed contour corresponds to τ p − σ τp = 5 × 10 35 yrs, illustrating the effect of the uncertainty in the calculation of τ p discussed in the previous Section. 8 Clearly, the relatively long proton lifetime in this plane makes detection difficult. Finally, we show a series of curves of constant ∆a µ as indicated in the caption, with the largest values appearing at small m 1/2 and m 0 . We note that all values of ∆a µ > 2 × 10 −11 appear for values of m h < 122 GeV, outside the range that we consider compatible with the measured value of m h .
In Fig. 2, 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. 2, 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. 2, it is still very large. On the other hand, values of ∆a µ are larger than in Fig. 2, and reach 10 −10 for m h > 122 GeV. We explore the dependence on λ in Fig. 3. Keeping the other parameters fixed to the values used in Fig. 1 Fig. 1 The dependence on M in is considered in Fig. 4. 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. 5, which  [43], and can be compared with Fig. 12c of [16]. 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 It is important to note that we can find Ωh 2 = 0.1 and m h = 125 GeV simultaneously in both panels in Fig. 6, but at very different values of (m 1/2 , m 0 ). For M in = 10 16.5 GeV, simultaneity occurs around (m 1/2 , m 0 ) (7, 13) TeV where the proton lifetime τ p ≈ 5 ± 3 × 10 35 years. However, for M in = M P , these conditions are both satisfied when (m 1/2 , m 0 ) (1.1, 3.8) TeV. 10 Despite the lower sparticle masses, the proton lifetime is actually longer here (around 10 36 years), mainly due to the lower values of λ 4,5 needed 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. 7, 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. 6. 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 and small m 0 when M in = 10 16.5 GeV, whereas the proton lifetime is generally longer when M in = M P .

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 substantially larger contributions to ∆a µ are possible when some degree of universality is abandoned [20,44]. 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 X1 for the 'external' gauginoB X . This is to be contrasted with our previous assumption that M 5 [20]. These were found to optimize the value of ∆a µ .
Some results are displayed in the (M X1 , m 1 ) planes shown in Fig. 8. In the left panel, only the singlet masses M X1 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 Ωh 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 However, ∆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 Ω χ h 2 ∼ 0.1. This region resembles the best ∆a µ point found in [20]. The sensitivity to λ 4,5 for similar choices of model parameters is shown in the left panel of Fig. 9, which displays a (λ 4,5 , m 1 ) plane with M X1 = 0.8 TeV and other parameters the same as those used in the left panel of Fig. 8. In this case, we see only a single relic density contour, which has Ω 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. 9 we fix M X1 = 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 X1 , 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 Ωh 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. 11 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 Ω 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. 10. 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 .

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 the GUT scale, the proton lifetime is typically 10 36 yrs 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 [23] 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 supersymmetrybreaking 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 LSP density Ω χ h 2 0.12 if the Universe expands adiabatically passes through the region. 12 This 'quadrifecta' region was previously identified in a dedicated analysis of g µ − 2 in the flipped SU(5) GUT [20], and it is encouraging that this region appears quite stable under mild variations in the input parameters. As pointed out in [20], 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 [23].