Observable Proton Decay in Flipped SU(5)

We explore proton decay in a class of realistic supersymmetric flipped $SU(5)$ models supplemented by a $U(1)_R$ symmetry which plays an essential role in implementing hybrid inflation. Two distinct neutrino mass models, based on inverse seesaw and type I seesaw, are identified, with the latter arising from the breaking of $U(1)_R$ by nonrenormalizable superpotential terms. Depending on the neutrino mass model an appropriate set of intermediate scale color triplets from the Higgs superfields play a key role in proton decay channels that include $p^+ \rightarrow (e^{+},\mu^+)\, \pi^0$, $p^+ \rightarrow ( e^+,\mu^{+})\, K^0 $, $p^+ \rightarrow \overline{\nu}\, \pi^{+}$, and $p^+ \rightarrow \overline{\nu}\, K^+ $. We identify regions of the parameter space that yield proton lifetime estimates which are testable at Hyper-Kamiokande and other next generation experiments. We discuss how gauge coupling unification in the presence of intermediate scale particles is realized, and a $Z_4$ symmetry is utilized to show how such intermediate scales can arise in flipped $SU(5)$. Finally, we compare our predictions for proton decay with previous work based on $SU(5)$ and flipped $SU(5)$.


I. INTRODUCTION
Proton decay is rightly considered an important observable and discriminator for models of Grand Unified Theories (GUTs). The current lifetime bounds on various proton decay channels by Super-Kamiokande (Super-K) [1][2][3][4][5][6], and the anticipated experimental results from the next generation experiments such as JUNO [7], DUNE [8], and Hyper-Kamiokande (Hyper-K) [9] should provide valuable information for comparing the proton decay predictions by GUT models. In this regard proton decay induced by the dimension five operators in supersymmetric GUTs has been a subject of great interest. The expected dominant decay mode, p → K + ν, in minimal supersymmetric SU (5) has been under intense scrutiny [10,11].
In a recent paper [29] an exciting possibility of observable proton decay from a supersymmetric SU (4) c × SU (2) L × SU (2) R (4-2-2) model mediated by color triplets of intermediate mass range was identified. This 4-2-2 model nicely implements shifted hybrid inflation, as shown in [30]. These studies have prompted the present paper where we consider a supersymmetric hybrid inflation model [31][32][33] based on the flipped SU (5) gauge symmetry, supplemented by a global U (1) R symmetry and Z 2 matter parity. In order to study the contributions to proton decay from the color triplets of 5-plet and 10-plet Higgses, we employ two models of light neutrino masses. The first model utilizes an inverse seesaw mechanism with extra gauge singlets, while the second model assumes R symmetry violation at nonrenormalizable level in the superpotential and employs the type-I seesaw mechanism. These two models lead to proton decay modes in the observable range from color triplet mediation of either Higgs multiplets. The distinctive predictions of various branching fractions and comparison with SU (5) are presented. Especially, a unique prediction for p → K + ν decay is found to serve as an additional discriminator between the present model and previous models of flipped SU (5) [28,34,35] where this mode is highly suppressed. An additional Z 4 symmetry can naturally generate intermediate scale masses for the color triplets from the Higgs 5-plets. This is in contrast to another R-symmetric model recently considered in [34] where one of the color-triplets in Higgs 5-plet becomes naturally light and contributes only to the charged lepton channels. Lastly, in the present model a successful realization of gauge coupling unification is achieved in the presence of intermediate mass color triplets.
The layout of this paper is as follows: In Sec. II we briefly describe the flipped SU (5) model including its field content, the R-symmetric superpotential and some of its uniquely attractive features. Two models of neutrino masses are described in Sec. III. One is mostly based on R-symmetric interactions, and the second model assumes R-symmetry violation in the superpotential at nonrenormalizable level. In Sec. IV we discuss proton decay in Rsymmetric flipped SU (5) model mediated via both color triplets and the superheavy gauge bosons. We mainly focus on mediation by the color triplets which occurs via the chirality nonflipping operators of type LLRR from the renormalizable interactions. The estimates for the proton partial lifetimes for the various channels are presented in the observable range of Hyper-K along with the lower bounds on the color triplet masses and relevant couplings.
The role of an additional Z 4 symmetry for naturally realizing intermediate mass for the color triplets is briefly highlighted.
The issue of gauge coupling unification in the presence of these intermediate color triplet masses is mentioned in Sec. V. In Sec. VI we allow R-symmetry breaking terms at nonrenormalizable level to generate the right handed neutrino masses and also study their impact on proton decay. We identify the dominant contribution of color triplets from the 10-plet Higgses lying within the observable range of Hyper-K. Finally we conclude in Sec. VII.

II. SUPERSYMMETRIC FLIPPED SU (5) MODEL
The Flipped SU (5) gauge group is defined as F SU (5) ≡ SU (5) × U (1) X [21][22][23][24][25][26][27]. The MSSM matter superfields including the right handed neutrino superfield (N c ) belong in the 10 1 , 5 −3 and 1 5 representations of F SU (5). Here and later, if necessary, the U (1) X charge, q(X), of F SU (5) representations are labeled with superscripts. In contrast to SU (5), the right handed neutrinos are required by the gauge symmetry in F SU (5 Table I. It is clear from the table that we can obtain the MSSM decomposition of F SU (5) mutliplets by flipping U c ↔ D c and E c ↔ N c in the corresponding multiplets of the standard SU (5) model [23].
The superpotential suitable for supersymmetric hybrid inflation in F SU (5) with the additional U (1) R × Z 2 symmetry listed in Table-I is given by [31][32][33] where λ, λ and κ are real and positive dimensionless couplings. The SU (5) gauge indices will be suppressed.  2 h ) do not acquire mass from these terms. This ultimately provides the solution of doublet-triplet splitting problem via the missing partner mechanism [25]. The significance of U (1) R symmetry is quite evident here as it forbids the 5 −2 h 5 2 h term to all orders while keeping the electroweak Higgs doublets massless, and by also avoiding dimension five proton decay mediated via the expected 5 −2 h 5 2 h mass term [31]. The MSSM µ term, however, is assumed to be generated by the Giudice-Masiero mechanism [36]. Note that the U (1) R symmetry also forbids the quadratic and cubic terms of S for successful realization of susy hybrid inflation.
The Yukawa couplings, y ij , in third line of Eq. (1) provide the Dirac masses for all fermions. The discussion of tiny neutrino masses and its possible connection with proton decay is included in the next sections. Some additional terms such as, S10 −1 and 10 1 H 5 −2 h 10 1 i , appear at the renormalizable level. Although U (1) R symmetric these terms are forbidden by Z 2 matter parity. The key feature of Z 2 matter parity lies in making the lightest supersymmetric particle a dark matter candidate while forbidding the dangerous dimension four proton decay terms. The last term, W HN , in Eq. (1) is responsible for generating the heavy Majorana neutrino masses necessary for the implementation of seesaw mechanism as described in the next section.

III. NEUTRINO MASSES
In order to accommodate the light neutrino masses responsible for solar and atmospheric neutrino oscillations [37,38], we can employ a inverse seesaw mechanism [39][40][41] with the help of extra gauge singlet superfields S a which have odd matter-parity with R(S a ) = 1. This allows us to include the following additional term in the superpotential at renormalizable level, where i, a = 1, 2, 3. Other terms at the nonrenormalizable level relevant for proton decay are S a 10 1 H 10 1 10 15−3 and S a 10 1 H5 −35−3 1 5 . However, their contribution to proton decay rates is highly suppressed. To implement a double seesaw mechanism we also need a mass term for the gauge singlet superfields S a . However, an explicit mass term, µ ab S a S b , is not allowed due to R-symmetry. We, therefore, include a spurion gauge singlet superfield Σ through the Kähler potential term, y ab where we adopt a basis in which both m (u,ν) = y (u,ν) υ u and µ are real and diagonal, m (u,ν) diag(m u , m c , m t ) and µ = diag(µ 1 , µ 2 , µ 3 ). Applying the inverse seesaw mechanism with µ a |γ ja M |, we obtain the light neutrino mass matrix, which is diagonalized by a unitary matrix U N , namely m diag ν = U * N m ν U † N . This is in contrast to the double seesaw mechanism where µ a |γ ja M | is assumed. See Refs. [42,43] for a recent analogous treatment of double seesaw mechanism in an inflation model based on  In general for a given matrix γ ia , the mixing matrix U N can be determined as a function of µ a . However, for numerical estimates we will assume normal-ordered (NO) light neutrino masses with U N equal to a unit matrix. This also allows us to write the mixing matrix This enables us to estimate all relevant proton decay rates mediated by the color triplets in the Higgs 5-plets as discussed in the next section.
An alternative interesting possibility for generating light neutrino masses can be realized by allowing explicit U (1) R symmetry breaking terms at the nonrenormalizable level [44].
As U (1) R is a global symmetry it could be broken in the hidden sector while mediating breaking effects to the visible sector via gravitational interactions. We will assume that the R-symmetry breaking occurs in such a way that it only allows terms with R = 0 charge in the superpotential at the nonrenormalizable level. With Z 2 matter parity present only even number of matter superfields appear with the 10-plet Higgs fields. Therefore, to leading order the following terms are allowed in the superpotential, where γ k , with k = 0, 1, 2, 3, are the dimensionless matrices with family indices suppressed.
As we will see in Sec. VI, these terms play a crucial role in the estimates of proton decay mediated by the color triplets in the Higgs 10-plets.
A neutrino mass matrix in the (N, N c ) basis can now be written as where the third term in the superpotential W II HN provides the mass matrix, M ν c = γ 2 M 2 m P , for the right handed neutrinos. The light neutrino mass matrix is obtained via the standard seesaw mechanism [45], and is diagonalized by a unitary matrix U N as m diag For numerical estimates in this second model of neutrino masses we adopt the basis where (γ 2 ) ij is real and diagonal,  16 GeV, we obtain M ν c diag(5.1 × 10 10 , 1.7 × 10 11 , 6.0 × 10 14 ) GeV for γ 2 diag(6.3 × . These values of the right handed neutrino masses are significantly larger than the corresponding estimate of heavy neutrino masses obtained in the inverse seesaw mechanism described above. This scenario can be naturally incorporated in hybrid inflation models with successful reheating and nonthermal leptogenesis [46]. Proton decay in F SU (5) mediated by the superheavy gauge bosons has been extensively studied in the past [47][48][49][50][51][52] mostly in comparison with the unflipped SU (5) model. In a recent paper [28] this is discussed in a no-scale supersymmetric F SU (5) inflation model with an approximate Z 2 symmetry and modified R parity. In this section we will explore proton decay in an R-symmetric F SU (5) model suitable for susy hybrid inflation model. As emphasized earlier the U (1) R × Z 2 symmetry plays an important role in suppressing various operators that mediate rapid proton decay. For example, the dimension four rapid proton decay mediated through the color triplet, D c ⊂ 10 1 , can appear at nonrenormalizable level via the following operators, Without the S field and with no U (1) R symmetry these operators can lead to fast proton decay incompatible with the experimental observations. The presence of S is required by the U (1) R symmetry which makes these operators highly suppressed as the S field is expected to acquire a vev of order TeV scale from the soft susy breaking terms [53]. Note that these operators are also forbidden by the Z 2 matter parity even if we allow R-symmetry breaking operators at nonrenormalizable level as discussed in the previous section. The GUT scale mass terms for Higgs 5-plets, 5 h 5 h , and Higgs 10-plets, 10 H 10 H , are also not allowed due to U (1) R symmetry and which may otherwise mediate dimension five rapid proton decay.
For proton decay via dimension five and dimension six operators we mainly focus on the mediation by color triplets in the conjugate pairs of Higgs superfields, In general, these color triplets can contribute to proton decay via operators of chirality types LLLL, RRRR and LLRR, as discussed in a recent paper on 4-2-2 model [29]. In our model R symmetry with renormalizable interactions only allows the chirality nonflipping modes which reduce to the following four Fermi operators of LLRR chirality generated via color triplet exchange from Later we also discuss the proton decay mediation by the color triplets from 10 H , 10 H by allowing explicit R-symmetry breaking terms with R-charge zero at the nonrenormalizable level.
The Yukawa terms in the superpotential W (Eq. (1)) relevant for proton decay mediated by the color triplets can be expressed in terms of mass eigenstates as with the diagonal Yukawa couplings, y D , given by The F SU (5) supermultiplets are expressed in terms of the following mass eigenstates [28,48] where V is the Cabibbo-Kobayashi-Maskawa (CKM) matrix and P = diag(e iϕ 1 , e iϕ 2 , e iϕ 3 ) is the phase factor matrix with the condition i ϕ i = 0 [28]. As the amplitude of dimension five diagrams involves loop factors their contribution is generally expected to be suppressed as compared to dimension six diagrams. Therefore, we will include the contribution of color triplets only from dimension six diagrams which are generated from a combination of the Yukawa terms in the Lagrangian d 2 θW and their Hermitian conjugates. Similarly, the gauge boson exchange diagram (2a) is generated from the following part of the Kähler potential [28], where X is the SU (5) gauge vector superfield. The combined effects of the superheavy SU (5) gauge boson and color triplet mediation below their mass scales are described by the dimension six effective operators, where the Wilson coefficients C ijkl 6(1,2) are given by Here the color triplet masses are written as M λ = λ M and Mλ = λ M . The first term in C ijkl 6(1) is the contribution from the gauge boson exchange diagram (2a) which has been studied recently in [28] for an inflation based model. The contribution of the first term in C ijkl 6(2) arises from the D h color triplet exchange diagram (2b). This contribution has been studied more recently in an R-symmetric flipped SU (5) model [34] which naturally predicts M λ to be of intermediate scale. With Mλ of order M G only the charged lepton channels are predicted to lie in the observable range of future experiments at Hyper-K [9]. The contribution of the second term in C ijkl 6(1) arises from the D h color triplet exchange diagram (2c) and is crucial for making a nonvanishing prediction for the K +ν decay channel which is usually assumed to be suppressed. The present model with an additional Z 4 symmetry, as described in the next section, naturally predicts both M λ and Mλ to be of intermediate scale. This leads to distinctive proton lifetime predictions especially for the neutral lepton decay channels as described below.
The Wilson coefficients C ijkl 6(n) (n = 1, 2) in Eqs. (19) and (20) are run down to low energy scales using the Renormalization Group Equations (RGEs) given in [54]. The effect of oneloop RGE between the GUT scale M G and the electroweak scale M Z are encoded in the renormalization factors, A Sn , [55,56]: where c ) are the coefficients of one-loop RGEs for Wilson coefficients C ijkl 6(1,2) above (below) the SUSY scale, M SUSY , and are given as The one-loop beta coefficients, b i , b (2) i and b (1) i , of the gauge couplings α i = g 2 i /(4π) 2 are given by ). Therefore, the decay rates for charged-lepton channels with l + i = (e + , µ + ) become, where m p , m π , m K and m l i = (m e , m µ ) are the masses of proton, pion, kaon and charged leptons l i respectively. The MSSM parameters are υ u = υ sin β and υ d = υ cos β with electroweak vev, υ = 174 GeV. Finally, the k-and the C-factors are respectively defined as Decay channel T ml = Matrix element (GeV 2 ) Super-K bound [38] Hyper-K sensitivity [9] (10 34 years) (10 34 years) For convenience, the recently updated values of hadronic matrix elements T ml from lattice computation [58] and the corresponding Super-K bounds [1-6, 38] and the Hper-K sensitivities [9] are given in Table II.
With an additional Z 4 symmetry these relatively tiny values ofλ = λ can be boosted by a factor (m P /M ) 2 ∼ 10 4 as discussed in the subsection below. prediction for this scenario is depicted in Fig. (4). As expected from the M λ contribution in Eqs. (26) and (27), the weak dependence on tan β in the range 2 ≤ tan β ≤ 60 does not exhibit any spread in the proton lifetime predictions shown in Fig. (4). In this case the Super-K bound for the decay channel p + → e + π 0 with Eq. which is somewhat smaller than the corresponding estimate of M λ contribution quoted in Eq. (30). However, a potentially observable range of this bound, with M T 10 11 GeV, is in contradiction with Super-K bounds on neutral lepton channels described below.
The proton decay rates for neutral lepton channels, π +ν i and K +ν i , based on the neutrino model described in W I HN (Eq. 5), are expressed as Apart from the gauge boson contribution in the π +ν i channel [28] the contribution of color triplet with mass M λ has been ignored so far. The numerical results are displayed in Figs. (5a) and (5b) where we have used the recently updated values of U P M N S parameters from [37] with U N equal to the unit matrix. In the large M T limit the proton lifetime of the first channel is dominated by the gauge boson contribution whereas for the second decay channel lifetime increases without bound due to the absence of the gauge boson contribution.
For neutral lepton channels the Super-K bound for the decay channel p + → ν K + with Eq. (33) gives the following lower bound, This is the largest bound among the neutral and charged lepton channels with a naturally accessible value with Z 4 symmetry. This bound also allows the charged lepton channels, shown in Fig. (3) with λ 2.2 × 10 −4 > λ 1 + tan 2 β 3.24 × 10 −6 , to lie within the observable range of Hyper-K whereas the prediction of π +ν channel lies far beyond the Hyper-K reach.
In order to make a comparison of proton partial lifetime predictions among various GUT models the estimates of branching fractions play a pivotal role. For this purpose a variation of various branching fractions with respect to color triplet mass M T = Mλ = M λ for tan β in the range 2 ≤ tan β ≤ 60 is shown in Fig. (6). We particularly include the corresponding predictions from the unflipped SU (5) model recently presented in [28] by ignoring the dimension five contribution of color triplets with large sfermion masses of order 100 TeV [12][13][14][15][16][17][18][19]. For a comparison with 4-2-2 model see [29] and for SO(10) models see Refs. [59][60][61].
As is obvious from Fig. (6) the present F SU (5) model makes a very distinctive predictions of various branching fractions within the observable range of Hyper-K. Especially the branching fraction of ν K + channel plays a key role in making distinctive comparison of the current model with the other models of flipped SU (5) [28,34] where this channel is highly suppressed.

A. Z 4 Symmetry and Color Triplet Masses
An additional Z 4 symmetry can be employed to make the color triplets naturally light for observable proton decay. This is achieved with the following Z 4 -charge assignments: with all other fields carrying zero Z 4 -charge. This modifies the superpotential in Eq. (1) as follows:  (Γ ν _ π + /Γ e + π 0 ) SU (5) (Γ e + K 0 /Γ e + π 0 ) SU (5)  This superpotential can be employed to realize smooth hybrid inflation [62]. Also see [35] for a relevant model of inflation. The Z 4 symmetry is spontaneously broken during smooth hybrid inflation and the domain wall problem is therefore avoided.
It is important to note that both color triplets are now naturally light relative to M G with M λ = λM (M/m P ) 2 and M λ = λM (M/m P ) 2 , and the couplings, γ ai , relevant for the realization of light neutrino masses via a double seesaw mechanism have also been enhanced by the factor, (m P /M ) 2 . The explicit mass term, µ ab S a S b , for the gauge singlet fields S a , generated effectively from the Kähler potential, K ⊃ y ab Σ † m P S a S b + h.c, still remains intact. Note that we do not consider this symmetry in the second model of neutrino masses based on the standard seesaw mechanism arising from the explicit R-symmetry breaking terms at nonrenormalizable level. The relevant superpotential terms for these additional multiplets are In this section we assume that the U (1) R symmetry is enforced at the renormalizabe level in the superpotential and its violation is allowed at the nonrenormalizable level with operators of zero R-charge (Eq. (5)). The effective Yukawa terms in the superpotential W II HN (Eq. (5)), relevant for proton decay mediated by the color triplets (D c H , D c H ) from (10 H , 10 H ), can be expressed in terms of mass eigenstates as The additional terms arising from the effective Yukawa interactions of W II HN modify the Wilson coefficients C ijkl 6(1,2) of Eq. (17) as In chirality nonflipping mediation via dimension six operators only the last term utilizes the coupling responsible for assigning superheavy Majorana masses to right handed neutrinos. With U E c = U † L this term becomes related to the PMNS mixing matrix. A similar connection of proton decay with the right handed neutrino masses and the CKM mixing matrix is built in the so called new dimension five proton decay via the chirality flipping mediation discussed in an SO(10) model [59,60,63]. Assuming all γ matrices to be real and diagonal with U E c = U † L , the above Wilson coefficients lead to the following dimension six proton decay rates, for the charged lepton channels and  [29] and also from the predicted estimates in the gauge boson domination limit [28]. The predicted values of proton partial lifetime for the neutral lepton channels show a trend similar to what has been already shown for the color triplets from Higgs 5-plets in Fig. (6). Here again the K + channel plays a discriminating role in differentiating the present model from the other models of F SU (5) considered in [28,34].