Proton Decay in Supersymmetric $SU(4)_c \times SU(2)_L \times SU(2)_R$

We discuss proton decay in a recently proposed model of supersymmetric hybrid inflation based on the gauge symmetry $SU(4)_c \times SU(2)_L \times SU(2)_R$. A $U(1)\, R$ symmetry plays an essential role in realizing inflation as well as in eliminating some undesirable baryon number violating operators. Proton decay is primarily mediated by a variety of color triplets from chiral superfields, and it lies in the observable range for a range of intermediate scale masses for the triplets. The decay modes include $p \rightarrow e^{+}(\mu^+) + \pi^0$, $p \rightarrow \bar{\nu} + \pi^{+}$, $p \rightarrow K^0 + e^+(\mu^{+})$, and $p \rightarrow K^+ \bar{\nu}$, with a lifetime estimate of order $10^{34}-10^{36}$ yrs and accessible at Hyper-Kamiokande and future upgrades. The unification at the Grand Unified Theory (GUT) scale $M_{\rm GUT}$ ($\sim 10^{16}$ GeV) of the Minimal Supersymmetric Standard Model (MSSM) gauge couplings is briefly discussed.


I. INTRODUCTION
In a recent paper [1] we proposed a realistic supersymmetric hybrid inflation scenario [2,3] specifically tailored for the gauge symmetry SU (4) c × SU (2) L × SU (2) R (G 4-2-2 ) [4][5][6]. The model employs shifted hybrid inflation [7,8] during which the G 4-2-2 symmetry is broken and the doubly charged monopoles [9] are inflated away. The model is fully compatible with the Planck data [10] and, for a wide choice of parameters, it predicts observable gravity waves generated during the inflationary epoch. The G 4-2-2 symmetry breaking scale is estimated to be of order M GUT (∼ 10 16 GeV).
Motivated by the above development in this follow-up paper we explore the important issue of proton decay in these supersymmetric G 4-2-2 models. It is well known that such models do not contain any superheavy gauge bosons that can mediate proton decay. However, proton decay in our G 4-2-2 model can arise from the exchange of a variety of color triplets present in the various chiral superfields. With intermediate scale masses of varying range that we estimate for these states, the proton decay rate is found to be accessible in the next generation experiments such as JUNO [11], DUNE [12], and Hyper-Kamiokande [13].
The layout of the paper is as follows. In Sec. II we describe the superpotential and the field content of our model. A U (1) R symmetry, which is required to realize hybrid inflation, is also shown to play an important role in eliminating some undesirable baryon number where i = 1, 2, 3 is the generation index, and the subscripts r, g, b, l represent the four colors in the model, namely red, green, blue, and lilac. It is sufficient to consider a right- Such doublets can remain light as a result of appropriate discrete symmetries [14]. Due to an R symmetry the color triplet pair d c H and d c H remains massless. An economical choice to remedy this problem is the introduction of a sextet superfield G = (6, 1, 1) with SM components g = (3, 1, −1/3) and g c = (3, 1, 1/3). This can provide superheavy masses to the color triplets d c H and d c H by mixing them with g and g c [15]. Finally, to realize inflation within the supersymmetric hybrid framework a gauge singlet chiral superfield S = (1, 1, 1) is introduced whose scalar component plays the role of the inflaton. The various superfields with their representation, transformation under G 4-2-2 , decomposition under G SM , and respective charge q(R) are shown in Table I. It can be noted from the Table I  This reflects the fact that the matter-parity Z mp 2 , which is usually invoked to forbid rapid proton decay operators at the renormalizable level, is contained in U (1) R as a subgroup. The superpotential W is invariant under Z mp 2 and this symmetry remains unbroken. Therefore, no domain wall problem appears here and consequently the lightest supersymmetric particle (LSP) becomes a plausible candidate for dark matter.
The superpotential employed in Ref. [1] for the shifted µ-hybrid inflation with G 4-2-2 × U (1) R symmetry is given by where κ, λ, β, a, b, λ ij 1,2 , γ ij 1,2 , and γ ij 1,2 are real and positive dimensionless couplings and M is a superheavy mass parameter. The superheavy scale Λ is assumed to lie in the range 10 16 GeV Λ m P , where m P 2.4 × 10 18 GeV is the reduced Planck mass. The first-line terms in the superpotential W in Eq. (4) are relevant for the shifted µ-hybrid inflation and the resolution of the monopole problem, as discussed in Ref. [1]. In addition, the coupling λSh u h d yields the MSSM µ term once the scalar component of the superfield S acquires a nonzero vev proportional to the gravitino mass m 3/2 with µ = −λm 3/2 /κ [16].
The achievement of low reheat temperatures T r 10 5 GeV, the possibly observable gravity waves with tensor-to-scalar ratio r 10 −4 − 10 −3 , and the gravitino dark matter with inflationary predictions consistent with the latest Planck data are the attractive features of this inflationary model as discussed in detail in Ref. [1]. For earlier work on the µ-hybrid inflation model see Refs. [17] and [18].
The first two terms in the second line of Eq. (4), which include the sextet superfield G, serve to provide superheavy masses to d c H and d c H as discussed above. The Yukawa interactions of the matter superfields F, F c are represented by the λ ij -couplings. The neutrino (ν) and right-handed neutrino (ν c ) couplings from the λ ij -and γ ij 1 -terms explain the tiny neutrino masses via the see-saw mechanism. The γ ij -and γ ij -couplings in the third line of W play an important role in generating possibly observable proton decay as discussed in the next section in detail.
The fact that the gauge bosons in the G 4-2-2 model do not mediate proton decay seems to support the observed stability of proton. We therefore only discuss proton decay mediated via the color triplets present in the chiral superfields F, F c ⊃ d, d c , G = g + g c , and This mediation can effectively generate four-Fermi proton decay operators with chirality type LLLL, RRRR, or LLRR. As discussed below, the R symmetry does not allow four-Fermi operators of the type LLLL and RRRR, whereas observable proton decay is only mediated through the color triplets d c H , d c H with four-Fermi operators of LLRR chirality.

A. R-symmetry Breaking Proton Decay Modes
The dimension-four L-and B-violating operators may appear at the nonrenormalizable level in the superpotential as The fermionic or bosonic character of the external lines in each vertex can be interchanged independently.
which can lead to fast proton decay via the effective operator by the color-triplet d c -squark mass. However, these operators are not allowed by the R symmetry defined in Table I. Similarly, the dimension-five L-and B-violating operators arising from the following nonrenormalizable gauge invariant terms in the superpotential are forbidden by the R symmetry. The gauge invariant renormalizable interactions dimension-five fast proton decay via the chirality flipping propagator with a GG ⊃ g c g mass insertion [15,19]. Again, the R symmetry does not allow these terms in the superpotential which could otherwise generate LLLL and RRRR four-Fermi operators.
The breaking of the R symmetry in the hidden sector can also assist proton decay via the soft supersymmetry breaking terms, although the corresponding decay rates are generally expected to be suppressed. As an example, consider the following R-symmetric nonrenor-malizable terms These interactions yield effective dimension-five proton decay operators via a chirality flipping propagator involving a mass insertion κ S H c H c = −m 3/2 H c H c , as shown in Fig. 1.
Here, the solid lines refer to fermions, the dashed lines to bosons, and the dotes represent the vevs. The S field acquires a nonzero vev due to the violation of the R symmetry by the soft supersymmetry breaking terms [16]. In Fig. 1 and thereafter we use the same notation for the chiral superfields and their scalar and fermionic components.
To provide an order of magnitude estimate for the proton decay rate we assume all dimensionless coupling constants in Eq. (7) to be of the same order. Note that only the γ ij 1 ≡ γ ij coupling is actually related to the right-handed neutrino Majorana mass matrix The distinguishing feature of linking proton decay to neutrino masses via the right-handed neutrino Majorana mass terms is highlighted in Refs. [20][21][22]. Connecting external squark and/or slepton lines, in each of these dimensionfive diagrams with a Higgsino or gaugino line one can generate one-loop (box) diagrams corresponding to LLLL and RRRR type four-Fermi proton decay operators. Other possible diagrams with an external ν c line are not allowed kinematically, whereas the amplitude for diagrams with internal ν c H lines is suppressed as compared to the diagrams shown in Fig. 1. Assuming all γ ij 1,2 's and γ ij 1,2 's to be of order γ i , the amplitude of the box diagrams corresponding to the dimension-five diagrams in Fig. 1 contains the suppression fac- For typical values of m 3/2 ∼ TeV, tan β ∼ 10, and v = 2 × 10 16 GeV, we obtain the largest

B. R-symmetric Observable Proton Decay Modes
We now discuss dimension-five and dimension-six proton decay operators of type LLRR which are generated from the interference of the interactions in d 2 θ W with their Hermitian conjugates. After integrating out the heavy color triplets, the effective operators obtained fall into the category of the following four-Fermi operators: The R symmetry is automatically respected by these operators and the proton decay rates can be predicted in the observable range without the R-symmetry breaking suppression factors.

Once again the couplings
Eq. (7) play crucial role for the realization of proton decay corresponding to the operators is expected to be dominant unless the logarithms are very significant. Therefore, we only focus on the dimension-six tree diagrams of Figs. 4 and 6 with the following decay rates: with Here, m p , m π , and m K are the proton, pion, and kaon mass respectively, and l + i = e + or µ + .  [26]. It is interesting to note that the value of a (b) or the mass of d c H (d c H ) can be made small enough to reduce the proton lifetime to a measurable level. The current limits on the proton lifetime for the various decay modes mentioned above are τ (p → π 0 (e + , µ + )) > (16, 7.7) × 10 33 yrs [25], τ (p → π + ν) > 3.9 × 10 32 yrs [27], τ (p → K 0 (e + , µ + )) > (1, 1.6) × 10 33 yrs [28,29], and τ (p → K + ν) > 6.6 × 10 33 yrs [23,24]. With all γ ij 1,2 's and γ ij 1,2 's in Eq. (7) being of order γ i , the decay mode p → e + π 0 provides the following most stringent bound on the masses of the i Γ(p → π + ν i ) Γ(p → π 0 e + ) 6.06 These predictions can be compared, for example, with the predictions of the no-scale supersymmetric standard unflipped SU (5) and flipped SU (5) models recently calculated in Ref. [30]. Also see Refs. [20,21,31] for SO(10) models. Most of the above branching fractions lie, in our case, close to unity except for i Γ(p → π + ν i )/Γ(p → π 0 e + ), which lies

IV. GAUGE COUPLING UNIFICATION
It is important to emphasize that with a or b ∼ 10 −3 the proton lifetime is predicted within the potentially observable range of Hyper-Kamiokande, τ (p → e + π 0 ) < 7.8 × 10 34 yrs [13]. The corresponding values of the color-triplet masses m (d c H , g) or m (d c H , g c ) are ∼ 10 13 GeV, and therefore lie somewhat below the GUT scale. These reduced masses ultimately ruin gauge coupling unification, an attractive feature of MSSM. As G 4-2-2 is a semi-simple group, gauge unification is not a must, but it can, in any case, be achieved with a modest adjustment of the model. As an example, let us consider the two color-triplet fields d c H , g and d c H , g c to be both of intermediate mass ∼ 10 13 GeV with a and b ∼ 10 −3 . To recover gauge coupling unification we add an arbitrary number of bi-doublets H α = H 1 , H 2 , H 2 , · · · , with R charge R(H α ) = 1. To avoid any unnecessary couplings of these bi-boublets with the MSSM matter superfields F , F c , we further assume an additional discrete Z 2 symmetry under which only the H α 's are odd, and all the other superfields are even. This symmetry remains unbroken and thus does not lead to a domain wall problem. The general form of the allowed nonrenormalizable superpotential terms involving the H α superfields is with m, n, p, q = 0, 1, 2, · · · , and with at least one of them being nonzero. Here, For a recent discussion of intermediate mass fermionic dark matter see [36].