Proton Decay: Flipped vs Unflipped SU(5)

We analyze nucleon decay modes in a no-scale supersymmetric flipped SU(5) GUT model, and contrast them with the predictions for proton decays via dimension-6 operators in a standard unflipped supersymmetric SU(5) GUT model. We find that these GUT models make very different predictions for the ratios $\Gamma(p \to \pi^0 \mu^+)/\Gamma(p \to \pi^0 e^+)$, $\Gamma(p \to \pi^+ \bar{\nu})/\Gamma(p \to \pi^0 e^+)$, $\Gamma(p \to K^0 e^+)/\Gamma(p \to \pi^0 e^+)$ and $\Gamma(p \to K^0 \mu^+)/\Gamma(p \to \pi^0 \mu^+)$, and that predictions for the ratios $\Gamma(p \to \pi^0 \mu^+)/\Gamma(p \to \pi^0 e^+)$ and $\Gamma(p \to \pi^+ \bar{\nu})/\Gamma(p \to \pi^0 e^+)$ also differ in variants of the flipped SU(5) model with normal- or inverse-ordered light neutrino masses. Upcoming large neutrino experiments may have interesting opportunities to explore both GUT and flavour physics in proton and neutron decays.

The outline of this paper is the following. In Section 2 we review relevant features of the no-scale flipped SU(5) GUT model, and in Section 3 we study proton (and some neutron) decay modes in this model, giving expressions in terms of the relevant hadronic matrix elements and discussing their uncertainties. The corresponding expressions in unflipped SU (5) are discussed in Section 4. In Section 5 we present predictions for ratios of proton decay rates in the flipped and unflipped SU(5) GUTs, and we review our conclusions and discuss future prospects in Section 6.

The No-Scale Flipped SU(5) Model
In the no-scale flipped SU(5) × U(1) GUT model [22][23][24][25][29][30][31][32], the three generations of the minimal supersymmetric extension of the Standard Model (MSSM) matter fields are embedded, together with three right-handed singlet neutrino chiral superfields, into three sets of 10,5, and 1 representations of SU (5), which we denote by F i ,f i and c i , respectively, where i = 1, 2, 3 is the generation index. In units of 1/ √ 40, the U(1) charges of the F i ,f i and c i are +1, −3, and +5, respectively. The assignments of the quantum numbers for the right-handed leptons, up-and down-type quarks are "flipped" with respect to the standard SU (5) assignments, giving the model its flippant name.
In addition to these matter fields, the minimal flipped SU(5) model contains a pair of 10 and 10 Higgs fields, H andH, respectively, a pair of 5 and 5 Higgs fields, h and h, respectively, and four singlet fields, φ a (a = 0, . . . , 3). The vacuum expectation values (VEVs) of the H andH fields break the SU(5) × U(1) gauge group down to the SM gauge group, and subsequently the VEVs of the doublet Higgs fields H d and H u , which reside in h andh, respectively, break the SU(2) L × U(1) Y gauge symmetry down to the U(1) of electromagnetism.
The renormalizable superpotential in this model is given by We assume here that the model possesses an approximate Z 2 symmetry, under which only the H field is odd while the rest of the fields are even. This symmetry is supposed to be violated by some Planck-scale suppressed operators, which prevent the formation of domain walls when the field H acquires a VEV. This Z 2 symmetry forbids some unwanted terms, such as F i Hh andf i Hh, which would cause baryon/lepton-number violation as well as Rparity violation. The Z 2 symmetry also forbids a vector-like mass term for H andH, which is advantageous for suppressing rapid proton decay induced by colour-triplet Higgs exchange. We embed the flipped SU(5) model in minimal N = 1 supergravity, which we assume to have a Kähler potential of no-scale form [36], as is motivated by the low-energy structure of string theory [33]. In this case the potential V has an F -and D-flat direction along a linear combination of the singlet components in H andH. These fields develop VEVs in this direction, as discussed in detail in Ref. [30]. After H andH acquire VEVs in this 'flaton' direction, the coloured components in these fields form vector-like multiplets with those in h andh via the couplings λ 4 and λ 5 in (1). On the other hand, the electroweak doublets H d and H u in h andh do not acquire masses from the flaton VEV-this is an economical realization of the missing-partner mechanism [24] that solves naturally the doublet-triplet splitting problem.
As discussed in detail in Ref. [29], this model offers the possibility of successful Starobinskylike [44] inflation, with one of the singlet fields, φ 0 , playing the role of the inflaton [45]. For µ 00 = m s /2 and λ 000 (1) with the inflaton mass m s 3 × 10 13 GeV and M P ≡ (8πG N ) −1/2 the reduced Planck mass, the measured amplitude of the primordial power spectrum is successfully reproduced and the tensor-to-scalar ratio r 3 × 10 −3 , well within the range allowed by the Planck results and other data [46]. This prediction can be tested in future CMB experiments such as CMB-S4 [47] and LiteBIRD [48]. The predicted value of the tilt in the scalar perturbation spectrum, n s , is also within the range favoured by Planck and other data at the 68% CL [46].
As seen in Eq. (1), the inflaton φ 0 can couple to the matter sector via the couplings λ 6 and λ 7 . In Ref. [29], two distinct cases, λ i0 6 = 0 (Scenario A) or λ i0 6 = 0 (Scenario B), were studied. We focus on Scenario B in this work. In this scenario, one of the three singlet fields other than φ 0 , which we denote by φ 3 , does not have the λ 6 coupling; i.e., λ i3 6 = 0, whereas λ ia 6 = 0 for i = 1, 2, 3 and a = 0, 1, 2. We also assume λ a 7 = 0 for a = 0, 1, 2. To realize this scenario, we introduce a modified R-parity, under which the fields in this model transform as We note that this modified R-parity is slightly violated by the coupling λ 000 8 . Nevertheless, since this R-parity-violating effect is only very weakly transmitted to the matter sector, the lifetime of the lightest supersymmetric particle (LSP) is still much longer than the age of the Universe [30,49], so the LSP can be a good dark matter candidate. We also note that the singlet φ 3 can acquire a VEV without spontaneously breaking the modified R-parity. In this case, the coupling λ 3 7 , which is allowed by the modified R-parity, generates an effective µ term for h andh, µ = λ 3 7 φ 3 , just as in the next-to-minimal supersymmetric extension of the SM.
As discussed in detail in Refs. [29][30][31][32], the λ 6 coupling in this model controls i) inflaton decays and reheating; ii) the gravitino production rate and therefore the non-thermal abundance of the LSP; iii) neutrino masses; and iv) the baryon asymmetry of the Universe via leptogenesis [50]. In particular, we showed in Refs. [31,32] by scanning over possible values of λ 6 that the observed values of neutrino masses, the dark matter abundance, and baryon asymmetry can be explained simultaneously, together with a soft supersymmetrybreaking scale in the multi-TeV range. In this paper, we study nucleon decays in the scenario developed in Refs. [29][30][31][32].
Without loss of generality, we adopt the basis where λ ij 2 and µ ab are real and diagonal. In this case, the MSSM matter fields and right-handed neutrinos are embedded into the SU (5) representations as in [39] where the V 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 [39]. 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. 4 The diagonal components of λ ij 2 and µ ab (a, b = 0, 1, 2) are given by where we take m s = 3 × 10 13 GeV (see above). In what follows we express these matrices as λ ij 2 = λ i 2 δ ij and µ ab = µ a δ ab . The first equation in Eq. (5) is only an approximate expression, since in general renormalization-group effects and threshold corrections cause λ 2 to deviate from the up-type Yukawa couplings at low energies. However, since these effects are at most O(10)% and depend on the mass spectrum of the theory, we neglect them in the following analysis.
The neutrino/singlet-fermion mass matrix can be written as where i, j = 1, 2, 3 and a = 0, 1, 2, andφ 0 corresponds to the fermionic superpartner of the inflaton field φ 0 . The mass matrix of the right-handed neutrinos is then obtained from a first seesaw mechanism: where ν cH denotes the VEV of the F -and D-flat direction of the singlet components of H andH: we take ν cH = 10 16 GeV in the following analysis. We diagonalize the mass matrix in Eq. (7) using a unitary matrix U ν c : The light neutrino mass matrix is then obtained through a second seesaw mechanism [52,53]: This mass matrix is diagonalised by a unitary matrix U ν , so that We note that, given a matrix λ ia 6 , the eigenvalues of the m ν and m ν c matrices, as well as the mixing matrices U ν c and U ν , are uniquely determined as functions of µ 1 and µ 2 via Eqs. (7)(8)(9). The PMNS matrix is given by U l in Eq. (3) and U ν in Eq. (10): Using the measured values of the PMNS matrix elements, we can use this relation to obtain U l from U ν . The matrix U l plays an important role in determining the partial decay widths of proton decay modes, as we will see in the subsequent Section.

Nucleon Decay in Flipped SU(5)
We are now ready to discuss nucleon decay in our model. In view of the suppression of the dimension-5 contribution mediated by coloured Higgs fields thanks to the missing-partner mechanism in the flipped SU(5) GUT [24], the main contribution to nucleon decay is due to exchanges of SU(5) gauge bosons. The relevant gauge interaction terms are where g 5 is the SU(5) gauge coupling constant, the X α a are the SU(5) gauge vector superfields, Below the GUT scale, the effects of SU(5) gauge boson exchanges are in general described by the dimension-six effective operators where O 6(1) with G and B the SU(3) C and U(1) Y gauge vector superfields, respectively, and g 3 and g the corresponding gauge couplings. In the unflipped SU(5) GUT both of the Wilson coefficients C ijkl 6(1,2) are non-zero, but in flipped SU(5) only C ijkl 6(1) is non-zero, and is given by 5 where M X is the SU(5) gauge boson mass. The Wilson coefficients are run down to low energy scales using the renormalisation group equations. The renormalisation factors for C ijkl 6(n) (n = 1, 2) between the GUT scale and the electroweak scale, A Sn , are evaluated at the one-loop level 6 as [55,56]: 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. We give the electroweak-scale matching conditions for each decay mode in what follows. Below the electroweak scale, we take into account the perturbative QCD renormalization factor, which is computed in Ref. [57] at the two-loop level: A L = 1.247. We then calculate the partial decay widths of various proton decay modes by using the corresponding hadronic matrix elements, for which we use the results obtained from the QCD lattice simulation performed in Ref. [58]. The relevant hadronic matrix elements are listed in Table 1.
In the following we summarise the partial decay widths for the proton decay modes that we discuss in this paper, as well as two relevant neutron decay modes. 7 5 However, although C ijkl 6(2) vanishes in flipped SU(5), we retain it in the following formulae so that it can also be used for the unflipped case. 6 The two-loop RGEs for these coefficients above the SUSY-breaking scale are given in Ref. [54]. 7 We note that these partial decay widths do not depend on the phases ϕ i .  (4)(14) p → π 0 e + The relevant effective operators below the electroweak scale are where Note that, since C ijkl 6(2) = 0 in flipped SU(5), the second term in Eq. (18) vanishes for this model. The partial decay width can be expressed as follows in terms of these coefficients at the hadronic scale: where Setting i = 1 in Eq. (20), we obtain where m p and m π denote the masses of the proton and pion, respectively, and here and subsequently the subscript on the hadronic matrix element indicates that it is evaluated at the corresponding lepton kinematic point. From Eq. (22), we can readily compute the partial lifetime of the p → π 0 e + mode as We note that this tends to be longer than the lifetime predicted in unflipped SU(5) by a factor (see also Eq. (45)) as found in Refs. [11,37,39,41]. 8 p → π 0 µ + By using the effective Lagrangian in Eq. (18) and the rate in Eq. (20) for i = 2, we have and the partial lifetime of the p → π 0 µ + mode is τ (p → π 0 µ + ) flipped 9.7 × 10 35 × |(U l ) 21 | −2 M X 10 16 GeV 4 0.0378 n → π − l + We note in passing that the rates of neutron decay modes that include a charged lepton can be obtained from Γ(p → π 0 l + i ) through SU(2) isospin relations: which applies to both the flipped and unflipped SU(5) models.
p → π +ν i The relevant effective Lagrangian term in this case is with the following matching condition at the electroweak scale The partial decay width is then computed as with We then have There is a relation between the partial decay widths for n → π 0ν i and those of p → π +ν i given by isospin: which applies to both the flipped and unflipped SU(5) models.
The effective interactions in this case are given by with We then obtain the partial decay width where m K is the kaon mass and In particular, for i = 1, we have The low-energy effective interactions for this decay mode is given by with We note that the unitarity of the CKM matrix leads to in the case of flipped SU (5). As a result, we have as found in Ref. [39].
In unflipped SU(5), the Wilson coefficients of the effective operators in Eq. (13) are given by The rest of the calculation is exactly the same as before, so we just summarize the resultant expression for each partial decay width.

Comparison of Proton Decay Rates in Flipped and Unflipped SU(5)
As we now discuss, the predictions for proton decay branching fractions in the flipped SU(5) GUT model are different from those generated by dimension-6 operators in the standard unflipped SU(5) GUT, 9 which may enable future experiments to distinguish these two GUT scenarios. To this end, we focus on the following five quantities and compare the predictions for them in flipped and unflipped SU(5) GUTs: From Eqs. (22) and (25), we find that this ratio in the flipped SU(5) is given by We see that this ratio depends on the unitary matrix U l , which is determined from U ν and the PMNS matrix U PMNS via Eq. (11). We also note that by taking the ratio between the two partial decay widths Γ(p → π 0 µ + ) and Γ(p → π 0 e + ), many of the factors in these quantities such as the SU(5) gauge boson mass, M X , the SU(5) gauge coupling constant, g 5 , and the renormalization factors, A L and A S 1 , are cancelled, which makes the prediction for this ratio rather robust. In unflipped SU (5), on the other hand, we obtain (see Eqs. (45) and (46)): where We find R A 1 in a typical supersymmetric mass spectrum, and for R A = 1 we have: 10 Hence, the branching fraction of the muon mode is predicted to be smaller than that of the electron mode by approximately two orders of magnitude in the unflipped SU (5) GUT. This prediction is again rather robust: the uncertainty is O(10)%, which mainly comes from the errors in the hadronic matrix elements. We note also that the contribution of the colortriplet Higgs exchange to these decay modes in supersymmetric SU (5) is suppressed by small Yukawa couplings, and thus is negligible unless there is flavor violation in the sfermion mass matrices [13].
To determine the predicted value of the ratio in flipped SU(5) given by Eq. (51), we perform a parameter scan similar to that in Refs. [31,32]. We first write the Yukawa matrix λ 6 in the form where r 6 is a real constant, which plays a role of a scale factor, and M 6 is a generic complex 3 × 3 matrix. We then scan r 6 with a logarithmic distribution over the range (10 −4 , 1) choosing a total of 1000 values. For each value of r 6 , we generate 1000 random complex 3 × 3 matrices M 6 with each component taking a value of O(1).
As discussed in Refs. [31,32], for each 3 × 3 matrix λ 6 , the eigenvalues of the m ν and m ν c matrices and the mixing matrices U ν c and U ν are obtained as functions of µ 1 and µ 2 in Eq. (5). We then determine these two µ parameters by requiring that the observed values of the squared mass differences, ∆m 2 21 ≡ m 2 2 − m 2 1 and ∆m 2 3 ≡ m 2 3 − m 2 , are reproduced within the experimental uncertainties, where = 1 for the NO case and = 2 for the IO case. For the experimental input, we use the results from ν-fit 4.0 given in Ref. [62]. By using U ν determined in this manner, we then compute the matrix U l using the relation (11). We parametrise the PMNS matrix elements following the RPP convention [51]: where c ij ≡ cos θ ij and s ij ≡ sin θ ij with the mixing angles θ ij = [0, π/2], the Dirac CP phase δ ∈ [0, 2π], and the order m 1 < m 2 is chosen without loss of generality. Again we use the values obtained in Ref. [62] for θ 12 , θ 23 , θ 13 , and δ. As for the Majorana phases α 2 and α 3 , we set α 2 = α 3 = 0 in this analysis since, as we shall see below, the result scarcely depends 10 This result is consistent with the formula given in Ref. [61] Γ(p → µ + + X) Γ(p → e + + X) X nonstrange = sin 2 θ c cos 2 θ c (1 + cos 2 θ c ) 2 + 1 0.01 , where θ c is the Cabibbo angle: sin θ c 0.2245. on these phases. We generate the same number of λ 6 matrices for each mass ordering, and find solutions for 2399 and 180 matrix choices for the NO and IO cases, respectively, out of a total of 10 6 parameter sets sampled. This difference indicates some preference for the NO case in our model. In Fig. 1 we display histograms of the ratio Γ(p → π 0 µ + )/Γ(p → π 0 e + ) in the NO and IO scenarios in blue and green, respectively. The vertical black solid line represents the predicted value in unflipped SU(5). As we see, the flipped SU(5) Model predicts this ratio to be ∼ 0.10 and ∼ 23 for the NO and IO cases, respectively. To understand the origin of these values, we first note that, due to the hierarchical structure of m ν in Eq. (9), U ν has a simple form: for NO, where sin θ is found to be ∼ 0.38, and for IO, where the first matrix in the right-hand side arranges the order of the neutrino mass eigenvalues in accordance with the RPP convention. The relevant matrix elements of U l = U * PMNS U ν are then given by which leads to for NO, and for IO. These approximate estimates are in good agreement with the results given in Fig. 1.
We also note that these two expressions do not depend on the unknown Majorana phases, α 2 and α 3 . As a consequence, although we have fixed these phases to be zero in our analysis, we expect that the results in Fig. 1 will not be changed even if we take different values for these phases. The values of Γ(p → π 0 µ + )/Γ(p → π 0 e + ) predicted in the NO and IO flipped SU(5) scenarios are rather insensitive to the mass of the lightest neutrino, as seen in Fig. 2. On the other hand, we also see there that the spread in predicted values increases with the lightest neutrino mass. It may be challenging for the envisaged next-generation neutrino experiments to measure any deviation from the central values of the model predictions, but the NO and IO predictions remain well separated and hence distinguishable.
The predicted values of Γ(p → π 0 µ + )/Γ(p → π 0 e + ) in flipped SU(5) are much larger than the standard unflipped SU(5) prediction, which is 0.008. We may therefore be able to distinguish these two models in future proton decay experiments by measuring the partial lifetimes of these two decay modes. We can also determine the neutrino mass ordering in the case of flipped SU (5). Proton decay experiments are relatively sensitive to both of these decay modes, leading to the strongest available constraints on proton partial lifetimes: the current limit on τ (p → π 0 e + ) from Super-Kamiokande is 2.4 × 10 34 yrs and that on τ (p → π 0 µ + ) is 1.6 × 10 34 yrs [63,64] which can be compared to the predicted partial lifetimes given in Eq. (23) and (26), respectively. This makes the ratio Γ(p → π 0 µ + )/Γ(p → π 0 e + ) given in Eq. (51) interesting for testing the prediction of flipped SU(5) in future proton decay experiments such as Hyper-Kamiokande [3].

5.2
Next we consider the ratio i Γ(p → π +ν i )/Γ(p → π 0 e + ). Eqs. (32) and (22) imply that for the flipped SU(5) we have whereas for unflipped SU(5) we can use Eqs. (47) and (45) to obtain Setting R A = 1 again, we find We note, however, that in the supersymmetric standard SU(5) GUT colour-triplet Higgs exchange also induces p → π +ν (see, for instance, Refs. [8,13,14,17]), which can be much larger than the contribution in Eq. (47). Therefore, the value in Eq. (66) should be regarded as a lower limit on i Γ(p → π +ν i )/Γ(p → π 0 e + ) in standard unflipped SU(5). We show in Fig. 3 histograms of i Γ(p → π +ν i )/Γ(p → π 0 e + ) in the flipped SU(5) model for the NO and IO cases in blue and green, respectively. Unflipped SU(5) has the lower limit indicated by the vertical solid line. As in the previous subsection, we can again estimate this ratio using the approximation given in Eq. (60): for NO, and for IO, which agree with the results shown in Fig. 3.  (5) for the NO and IO cases in blue and green, respectively. The unflipped SU(5) prediction has a lower limit shown as the vertical solid line.
This ratio is, however, less powerful for distinguishing the flipped and unflipped SU(5) GUTs than Γ(p → π 0 µ + )/Γ(p → π 0 e + ). First, due to the potential contribution of the colour-triplet Higgs exchange, we have only a lower limit on the unflipped SU(5) prediction. Since the predicted values in the flipped SU(5) are larger than this lower limit, the unflipped SU(5) prediction can in principle mimic the flipped SU(5) predictions. Secondly, the sensitivities of experiments to p → π +ν and n → π 0ν tend to be much worse than that to p → π 0 µ + ; the present bound on p → π +ν from Super-Kamiokande is τ (p → π +ν ) > 3.9 × 10 32 yrs and that on τ (n → π 0ν ) > 1.1 × 10 33 yrs [65], which are much lower than the limit on p → π 0 µ + . On the other hand, the value of i Γ(p → π +ν i )/Γ(p → π 0 e + ) predicted in the flipped SU(5) model in the IO case is so large that this might be detectable.

Γ(p
The ratio Γ(p → K 0 e + )/Γ(p → π 0 e + ) in flipped SU (5) is computed from Eqs. (38) and (22) to be As we see, this ratio does not depend on the matrix U l . In unflipped SU(5), we use Eqs. (48) and (45) to find for R A = 1. The contribution of the colour-triplet Higgs exchange to p → K 0 e + is negligible unless flavour violation occurs in sfermion mass matrices [13,14], so this value can be regarded as a prediction of unflipped SU(5). As we see, this unflipped SU(5) prediction is much lower than the flipped SU(5) prediction (69), and thus we can in principle also use the ratio Γ(p → K 0 e + )/Γ(p → π 0 e + ) to distinguish between these two GUT models.

Γ(p
From Eqs. (39) and (25), we have Again, this ratio does not depend on the matrix U l . In unflipped SU(5), Eqs. (49) and (46) lead to for R A = 1. The contribution of colour-triplet Higgs exchange to p → K 0 µ + is small unless flavour violation occurs in sfermion mass matrices [13,14]. Therefore, this ratio can again be used to distinguish between the flipped and unflipped SU(5) GUTs.
This is a distinctive prediction in flipped SU(5)-if this decay mode is discovered in future proton decay experiments, flipped SU(5) is excluded.
These examples show that if the upcoming large neutrino experiments do discover nucleon decay, they will have interesting opportunities to explore both GUT and flavour physics.