Flipping SU(5) out of Trouble

Minimal supersymmetric SU(5) GUTs are being squeezed by the recent values of alpha_s, sin^2 theta_W, the lower limit on the lifetime for p to nubar K decay, and other experimental data. We show how the minimal flipped SU(5) GUT survives these perils, accommodating the experimental values of alpha_s and sin^2 theta_W and other constraints, while yielding a p to e/mu+ pi0 lifetime beyond the present experimental limit but potentially accessible to a further round of experiments. We exemplify our analysis using a set of benchmark supersymmetric scenarios proposed recently in a constrained MSSM framework.

One of the key pieces of circumstantial evidence in favour of grand unification has long been the consistency of the gauge couplings measured at low energies with a common value at some very high energy scale, once renormalization effects are taken into account. This consistency is significantly improved when light supersymmetric particles are included in the renormalization-group running, in which case the agreement improves to the per-mille level [1].
However, this circumstantial evidence is not universally accepted as convincing. For example, it has recently been suggested that the logarithmic unification of the gauge couplings is as fortuitous as the apparent similarity in the sizes of the sun and moon [2]. Alternatively, it has been argued that the unification scale could be as low as 1 TeV, either as a result of power-law running of the effective gauge couplings in theories with more than four dimensions [3], or in theories with many copies of the SU(3) × SU(2) × U(1) gauge group in four dimensions [4].
For some time now, detailed calculations have served to emphasize [5] how much fine tuning is needed in models with power-law running to reproduce the effortless success of supersymmetric grand unification with logarithmic running of the gauge couplings. Moreover, data from particle physics and cosmology provide independent hints for low-energy supersymmetry. Precision electroweak data favour quite strongly a low-mass Higgs boson [6], as required in the minimal supersymmetric extension of the Standard Model (MSSM) [7], and the lightest supersymmetric particle is a perfect candidate [8] for the cold dark matter thought by astrophysicists to infest the Universe. Many studies have shown that these and other low-energy data -such those on b → sγ decay [9] and g µ − 2 [10] -are completely consistent with low-energy supersymmetry, and a number of benchmark supersymmetric scenarios have been proposed [11].
Issues arise, however, when one considers specific supersymmetric grand unified theories. One is the exact value of sin 2 θ W , which acquires important corrections from threshold effects at the electroweak scale, associated with the spectrum of MSSM particles [12,13], and at the grand unification scale, associated with the spectrum of GUT supermultiplets [12,14]. Precision measurements indicate a small deviation of sin 2 θ W even from the value predicted in a minimal supersymmetric SU(5) GUT, assuming the range of α s (M Z ) now indicated by experiment [15].
The second issue is the lifetime of the proton. Minimal supersymmetric SU(5) avoids the catastrophically rapid p → e + π 0 decay that scuppered non-supersymmetric SU(5). However, supersymmetric SU(5) predicts p →νK + decay through d = 5 operators at a rate that may be too fast [16] to satisfy the presently available lower limit on the lifetime for this decay [17,18]. The latter requires the SU(5) colour-triplet Higgs particles to weigh > 7.6 × 10 16 GeV, whereas conventional SU(5) unification for α s (M Z ) = 0.1185 ± 0.002, sin 2 θ W = 0.23117 ± 0.00016 and α em (M Z ) = 1/(127.943 ± 0.027) [18] would impose the upper limit of 3.6 × 10 15 GeV at the 90% confidence level [16]. This problem becomes particularly acute if the sparticle spectrum is relatively light, as would be indicated if the present experimental and theoretical central values of g µ − 2 [10] remain unchanged as the errors are reduced.
The simplest way to avoid these potential pitfalls is to flip SU(5) [19,20]. As is well known, flipped SU(5) offers the possibility of decoupling somewhat the scales at which the Standard Model SU(3), SU(2) and U(1) factors are unified. This would allow the strength of the U(1) gauge to become smaller than in minimal supersymmetric SU(5), for the same value of α s (M Z ) [13]. Moreover, in addition to having a longer p → e/µ + π 0 lifetime than nonsupersymmetric SU(5), flipped SU(5) also suppresses the d = 5 operators that are dangerous in minimal supersymmetric SU(5), by virtue of its economical missing-partner mechanism [19].
In this paper, we re-analyze the issues of sin 2 θ W and proton decay in flipped SU(5) [13], in view of the most recent precise measurements of α s (M Z ) and sin 2 θ W , and the latest limits on supersymmetric particles. We study these issues in the MSSM, constraining the soft supersymmetry-breaking gaugino masses m 1/2 and scalar masses m 0 to be universal at the GUT scale (CMSSM), making both a general analysis in the (m 1/2 , m 0 ) plane and also more detailed specific analyses of benchmark CMSSM parameter choices that respect all the available experimental constraints [11]. We find that the p → e/µ + π 0 decay lifetime exceeds the present experimental lower limit [17], with a significant likelihood that it may be accessible to the next round of experiments [21]. We recall the ambiguities and characteristic ratios of proton decay modes in flipped SU(5).
We first recall the lowest-order expression for α s (M Z ) in conventional SU(5) GUTs, namely The present central experimental value of α s (M Z ) = 0.118 is obtained if one takes sin 2 θ W = 0.231 and α −1 = 128, indicating the supersymmetric grand unification is in the right ball-park. However, at the next order, one should include two-loop corrections δ 2loop as well as electroweak and GUT threshold corrections, that we denote by δ light and δ heavy . Their effects can be included by making the following substitution in (1) [12]: where δ 2loop ≈ 0.0030, whereas δ light and δ heavy can have either sign. If one neglects δ light and δ heavy , the conventional SU(5) prediction increases to α s (M Z ) ≈ 0.130 [15]. A value of α s (M Z ) within one standard deviation of the present central experimental value requires δ light and/or δ heavy to be non-negligible, so that the combination (δ 2loop + δ light + δ heavy ) is suppressed. However, in large regions of parameter space δ light > 0, which does not help. Moreover, in conventional SU (5), as was pointed out in [12,15], a compensatory value of δ heavy is difficult to reconcile with proton decay constraints. This problem is exacerbated by the most recent lower limit on τ (p →νK + ) [17] 1 .
As has been advertized previously [13], an alternative way to lower α s (M Z ) is to flip SU(5). In a flipped SU(5) model, there is a first unification scale M 32 at which the SU(3) and SU(2) gauge couplings become equal, which is given to lowest order by [24] 1 where α 2 = α/ sin 2 θ W , α 3 = α s (M Z ), and the one-loop beta function coefficients are b 2 = +1, b 3 = −3. The hypercharge gauge coupling α Y = 5 3 (α/ cos 2 θ W ) has, in general, a lower value α ′ 1 at the scale M 32 : where b Y = 33/5. Above the scale M 32 , the gauge group is the full SU(5) × U(1), with the U(1) gauge coupling α 1 related to α ′ 1 and the SU(5) gauge coupling α 5 as follows: The SU(5) and U(1) gauge couplings then become equal at some higher scale M 51 . The maximum possible value of M 32 , namely M max 32 , is obtained by substituting α ′ 1 = α 5 (M 32 ) into (5), and coincides with the unification scale in conventional SU (5) [18]. In general, one has and the flipped SU(5) prediction for α s (M Z ) is in general smaller than in minimal SU(5), for the same value of sin 2 θ W . The next-to-leading order corrections to (7) are also obtained by the substitution in (2). Numerically, an increase of ∼ 10% in the denominator in (1), which would compensate for the decrease due to δ 2loop , could be achieved simply by setting M 32 ≈ 1 3 M max 32 in (7).
In order to understand the implications for τ (p → e/µ + π 0 ) decay, we first calculate M 32 , using (7) with sin 2 θ W replaced by sin 2 θ W − δ 2loop , leaving for later discussions of the possible effects of δ light,heavy . We now explore the possible consequences of δ light for M 32 , following [12,13]. We approximate the δ light correction by where L(x) = ln(x/M Z ). As already mentioned, we assume that the soft supersymmetrybreaking scalar masses m 0 , gaugino masses m 1/2 and trilinear coefficients A 0 are universal at the GUT scale (CMSSM). We used ISASUGRA [25] to calculate the sparticle spectra in terms of The unknown parameters in (8) were constrained by requiring that electroweak symmetry breaking be triggered by radiative corrections, so that the correct overall electroweak scale and the ratio tan β of Higgs v.e.v.'s fix |µ| and m A in terms of m 1/2 and m 0 . Before making a more general survey, we recall that a number of benchmark CMSSM scenarios have been proposed [11], which include these constraints and are consistent with all the experimental limits on sparticle masses, the LEP lower limit on m h , the world-average value of b → sγ decay, the preferred range 0.1 < Ω χ h 2 < 0.3 of the supersymmetric relic density, and g µ − 2 within 2 σ of the present experimental value. These points all have A 0 = 0, but otherwise span the possible ranges of m 1/2 , m 0 , tan β and feature both signs for µ. Fig. 1 also shows the change in M 32 induced by the values of δ light in these benchmark models, assuming a fixed value α s (M Z ) = 0.1185. In general, these benchmark models increase M 32 for any fixed value of α s (M Z ) and sin 2 θ W . As α s (M Z ) varies, the predicted value of M 32 in each model varies in the same way as indicated by the sloping lines. We recall that the estimated error in α s (M Z ) is about 0.002, corresponding to an uncertainty in M 32 of the order of 20%, and hence a 2 Heavy singlet neutrinos were not used in the renormalization-group equations.
corresponding uncertainty in the proton lifetime of a factor of about two. The error associated with the uncertainty in sin 2 θ W is somewhat smaller 3 .
We now turn to the calculation of τ (p → e/µ + π 0 ). We recall first that the form of the effective dimension-6 operator in flipped SU(5) is different [24,26] from that in conventional SU(5) [27,28]: where θ c is the Cabibbo angle 4 . Also appearing in (9) are two unknown but irrelevant CPviolating phases η 11,21 and lepton flavour eigenstates ν L and ℓ L that are related to mass eigenstates by unknown but relevant mixing matrices: Despite our ignorance of the mixing matrices (10), some characteristic flipped SU (5) predictions can be made [24]: In the light of recent experimental evidence for near-maximal neutrino mixing, it is reasonable to think that (at least some of) the e/µ entries in U ℓ are O(1). In what follows, we assume that the lepton mixing factors |U ℓ 11,12 | 2 are indeed O(1), and do not lead to large numerical suppressions of both the p → e/µ + π 0 decay rates. Note that there is no corresponding suppression of the p →νπ + and n →νπ 0 decay rates, since all the neutrino flavours are summed over. However, without further information, we are unable to predict the ratio of p → e + X and p → µ + X decay rates. Hereafter, wherever we refer to p → e + π 0 decay, this mixing-angle ambiguity should be understood.
The p → e + π 0 decay amplitude is proportional to the overall normalization of the proton wave function at the origin. The relevant matrix elements are α, β, defined by The reduced matrix elements α, β have recently been re-evaluated in a lattice approach [29], yielding values that are very similar and somewhat larger than had often been assumed previously, and therefore exacerbating the proton-stability problem for conventional supersymmetric SU (5). Here, we use here the new central value α = β = 0.015 GeV 3 for reference. The error quoted on this determination is below 10%, corresponding to an uncertainty of less than 20% in τ (p → e + π 0 ), which would be negligible compared with other uncertainties in our calculation. Thus, we have the following estimate, based on [26,16] and references therein: We see in Fig. 2 that the 'bulk' regions of the parameter space preferred by astrophysics and cosmology, which occur at relatively small values of (m 1/2 , m 0 ), generally correspond to τ (p → e + π 0 ) ∼ (1 − 2) × 10 35 y. However, these 'bulk' regions are generally disfavoured by the experimental lower limit on m h and/or by b → sγ decay. Larger values of τ (p → e + π 0 ) are found in the 'tail' regions of the cosmological parameter space, which occur at large m 1/2 where χ−l coannihilation may be important, and at larger m 1/2 and m 0 where resonant direct-channel annihilation via the heavier Higgs bosons A, H may be important.
We turn finally to the possible implications of the GUT threshold effect δ heavy [12,14]. A 5 The horizontal spacing between points sampled was comparable to the thickness of these lines. 6 For fuller discussions of the implementations of these constraints with and without ISASUGRA, see [11,30]. general expression for this in flipped SU(5) is given in [12]: where M H 3 = λ 4 |V | and MH 3 = λ 5 |V | are the masses of the heavy triplet Higgs supermultiplets, the X, Y gauge bosons and gauginos have common masses M V = g 5 |V | where V is the common v.e.v. of the 10 and 10 Higgs supermultiplets, λ 4,5 are (largely unconstrained) Yukawa couplings, g 5 is the SU(5) gauge coupling, and r ≡ max{g 5 , λ 4 , λ 5 }. Thanks to the economical missing-partner mechanism of flipped SU (5), the H 3 andH 3 do not mix, and hence do not contribute significantly to proton decay. Thus there is no strong constraint on M H 3 ,H 3 from proton decay in flipped SU (5), and it is possible that M H 3 ,H 3 < M V (i.e., r = g 5 ). In this case, we can see from (15) that δ heavy < 0 naturally. For instance, as pointed out in [13], if λ 4 , λ 5 ∼ 1 8 g 5 , then δ heavy ≈ −0.0030, which completely compensates the δ 2loop contribution. We also recall that, in general, including δ heavy leads to a re-scaling of the M 32 /M max 32 : We display in Fig. 3 the possible numerical effects of δ heavy on τ (p → e/µ + π 0 ) in the various benchmark scenarios, assuming the plausible ranges −0.0016 < δ heavy < 0.0005 [13]. The boundary between the different shadings for each strip corresponds to the case where δ heavy = 0. The left (red) parts of the strips show how much τ (p → e + π 0 ) could be reduced by a judicious choice of δ heavy , and the right (blue) parts of the strips show how much τ (p → e + π 0 ) could be increased. The inner bars correspond to the uncertainty in sin 2 θ W . On the optimistic side, we see that some models could yield τ (p → e + π 0 ) < 10 35 y, and all models might have τ (p → e + π 0 ) < 5 × 10 35 y. However, on the pessimistic side, in no model can we exclude the possibility that τ (p → e + π 0 ) > 10 36 y.
We recall that a new generation of massive water-Čerenkov detectors weighing up to 10 6 tonnes is being proposed [21], that may be sensitive to τ (p → e + π 0 ) < 10 35 y. According to our calculations, such an experiment has a chance of detecting proton decay in flipped SU(5), though nothing can of course be guaranteed. We recall that there is a mixing-angle ambiguity (11) in the final-state charged lepton, so any such next-generation detector should be equipped to detect e + and/or µ + equally well. We also recall [24,26] that flipped SU(5) makes predictions (11) for ratios of decay rates involving strange particles, neutrinos and charged leptons that differ characteristically from those of conventional SU (5). Comparing the rates for e + , µ + and neutrino modes would give novel insights into GUTs as well as mixing patterns.
We conclude that flipped SU(5) evades two of the pitfalls of conventional supersymmetric SU(5). As we have shown in this paper, it offers the possibility of lowering the prediction for α s (M Z ) for any given value of sin 2 θ W and choice of sparticle spectrum. As for proton decay, we first recall that flipped SU(5) suppresses p →νK + decay naturally via its economical missingpartner mechanism. As in conventional supersymmetric SU(5), the lifetime for p → e/µ + π 0 decay generally exceeds the present experimental lower limit. However, as we have shown in . 36 10 . 36 10 Figure 3: For each of the CMSSM benchmark points, this plot shows, by the lighter outer bars, the range of τ (p → e/µ + π 0 ) attained by varying δ heavy over the range -0.0016 to + 0.0005 [13]. The central boundary of the narrow inner bars (red, blue) corresponds to the effect of δ light alone, with δ heavy = 0, while the narrow bars themselves represent uncertainty in sin 2 θ W . We see that heavy threshold effects could make τ (p → e/µ + π 0 ) slightly shorter or considerably longer. this paper, the flipped SU(5) mechanism for reducing α s (M Z ) reduces the scale M 32 at which colour SU(3) and electroweak SU(2) are unified, bringing τ (p → e/µ + π 0 ) tantalizingly close to the prospective sensitivity of the next round of experiments. Proton decay has historically been an embarrassment for minimal SU(5) GUTs, first in their non-supersymmetric guise and more recently in their minimal supersymmetric version. The answer may be to flip SU(5) out of trouble.