Observable proton decay and gauge coupling unification in the improved missing doublet SU(5) model

Proton decay serves as a very important discriminator in distinguishing among the various models of grand unified theories (GUTs) such as $SU(5)$, flipped $SU(5)$, Pati-Salam model, and $SO(10)$, etc. Even different versions of the same model can lead to distinct predictions. Currently, Super-K sets a lower limit on proton lifetime [1, 2, 3, 4, 5, 6], while future experiments such as JUNO [7], DUNE [8, 9], and Hyper-K [10] with enhanced sensitivities are expected to distinguish between different GUT models.

The minimal supersymmetric $SU(5)$ model has been extensively studied in relation to proton decay, particularly with regard to the mediation of color triplets through dimension-five operators [11, 12, 13, 14, 15]. However, a major challenge is the doublet-triplet splitting problem, where both electroweak doublets and color triplets in 5-plets acquire masses of similar orders. A successful realization of electroweak symmetry breaking requires the electroweak doublets to be sufficiently light. On the other hand, to prevent rapid proton decay, it is necessary for the color triplets to have a sufficiently heavy mass. To address this issue, a missing partner mechanism was proposed, which involves replacing the standard GUT Higgs with a $75$-plet and adding a pair of GUT mass $50$-plets to the minimal matter content of the $SU(5)$ model [16]. This allows the color triplets in 5-plets to mix with those in 50-plets and acquire heavy masses, while the electroweak doublets remain light. However, the presence of large representations like the $75$ and $50$-plets makes the model nonperturbative just above the GUT scale. A modified version [17, 18] overcomes this issue by assigning Planck-scale masses to the $50$-plets and introducing additional pairs of 5 and 50-plets. To delve into the modified version and its variants that have undergone a phenomenologically acceptable treatment, one can refer to the following sources: [19, 20, 21, 22, 23, 24, 25].

This paper explores the improved missing doublet model (IMDM75) [20] in the context of searching for observable proton decay. Similar to the modified version, this improved model includes additional pairs of 5 and 50-plets. In addition, it employs an anomalous $U(1)_{A}$ symmetry that solves the doublet-triplet problem without requiring any additional discrete or global symmetries. $SU(5)$ along with anomalous $U(1)_{A}$ symmetry can be embedded in a realistic string theory, as the existence of such an anomalous $U(1)_{A}$ symmetry does not imply an anomaly in the string theory [26]. The findings of this investigation show that IMDM75 predicts observable proton decay via chirality nonflipping color-triplet mediation, making it an important target for next-generation proton decay experiments. A detailed investigation of observable proton decay in IMDM75 along with a consistent two-loop gauge coupling unification is the prime focus of this paper.

The paper is structured as follows: Section II outlines the matter content and superpotential of IMDM75, along with the charge assignments under the anomalous $U(1)_{A}$ symmetry. In Sec. III, we discuss the different modes of proton decay mediated by chirality flipping and nonflipping color triplets. Section IV provides a brief comparison of IMDM75’s predictions for proton decay with those of other GUT models. Section V describes a successful realization of gauge coupling unification, along with a discussion of the matching conditions in the presence of additional intermediate-scale particles beyond the minimal content of the minimal supersymmetric standard model (MSSM). Our conclusions are summarized in Sec. VI.

The matter content of MSSM and the right-handed-neutrino superfield reside in $10_{i}+\bar{5}_{i}$ and $1_{i}$ representations of $SU(5)$ respectively, as shown in Table 1. The GUT Higgs superfield is denoted by $75_{H}$ representation whereas the electroweak doublets arise from the mixing of $5$ and $50$-plets. An anomalous $U(1)_{A}$ symmetry, broken by the nonzero vacuum expectation value (VEV) $\langle X\rangle/m_{P}\sim 10^{-1}-10^{-2}$ of a gauge singlet field $X$ via anomalous D term[27, 28, 29], is imposed on the model, as described in [20]. The charge assignment under $U(1)_{A}$ symmetry and $\mathbb{Z}_{2}$ matter parity for all superfields are provided in Table 2. Note that the $X$ field has a charge of $-q$ under $U(1)_{A}$ and $n$ takes values $0,1,2,\cdots$. The $\mathbb{Z}_{2}$ matter parity is utilized to eliminate undesired terms in the superpotential and to obtain a potential cold dark matter candidate. Interestingly, the appearance of $\mathbb{Z}_{2}$ matter parity can be an accidental by-product of the $U(1)_{A}$ symmetry breaking, as discussed in [20]. It is noteworthy that for $q=1$ and $n=2$, only the $\mathbb{Z}_{7}$ subgroup of the $U(1)_{A}$ symmetry is sufficient to restrict the model. This implies a complete breakdown of the $U(1)_{A}$ symmetry once $X$ obtains a VEV, eliminating any potential domain wall problems. An additional set of $5$ and $50$-plets ($5^{\prime},\ \bar{5}^{\prime},50^{\prime},\ \overline{50}^{\prime}$) is necessary to prevent rapid proton decay while maintaining gauge coupling unification (GCU), as explained in [20].

The superpotential for the IMDM75 model with $U(1)_{A}\times\mathbb{Z}_{2}$ symmetry, as shown in Table 2, is given by

where generation indices have been suppressed. The first two terms in $W$ break the $SU(5)$ symmetry and attain the VEV of $\langle 75_{H}\rangle\equiv\upsilon=3M/4\lambda{75}$ in the singlet direction of MSSM gauge symmetry $SU(3)_{c}\times SU(2)_{L}\times U(1)Y$. The Yukawa interactions of charged superfields are described by the terms in the second line, while the third and fourth lines provide heavy masses to the color triplets in 5- and 50-plets and leaving the electroweak doublets in 5-plets massless. This is the missing partner mechanism, which solves the doublet-triplet splitting problem without requiring parameter fine-tuning. The fifth line provides heavy masses of the order of $\sim 10^{17}-10^{18}$ GeV to 50-plets while keeping the model perturbative above the GUT scale. The first two terms in the last line explain tiny neutrino masses through the seesaw mechanism [30]. The last term, $W_{NR}$, includes all relevant nonrenormalizable terms that will be discussed later.

The $U(1)_{A}$ symmetry in the IMDM75 model forbids the $\mu\ 5_{h}\bar{5}_{h}$ term, which leads to the electroweak doublets in $5_{h}$ and $\bar{5}_{h}$ remaining massless up to all orders. To generate the MSSM $\mu$ term for the light electroweak doublets, the Giudice-Masiero mechanism [31] can be utilized, which is a consequence of supergravity breaking in the hidden sector. The term $y^{(l,d)}_{ij}10_{i}\ \bar{5}_{j}\bar{5}_{h}$ leads to same mass matrix for down type quark and charges leptons. This can be avoided by using nonrenormalizable term

Yukawa terms for quarks and charged leptons in superpotential of IMDM75 include

where $i$ and $j$ are generation indices. After diagonalizing Yukawa couplings and calculating mass eigenstates, down-type quark and charged lepton Yukawa couplings are redefined as

The Yukawa terms in the effective superpotential can be expressed in terms of mass eigenstates as

here $a,b,c,...$ are $SU(3)_{c}$ color indices. $Y^{(u)},Y^{(d)},Y^{(l)}$ are diagonalized Yukawa couplings effectively. Furthermore, fermion mass hierarchy is explained by introducing a generation-dependent horizontal $U(1)_{A}$ symmetry and incorporating nonrenormalizable terms from $W_{NR}$ in [20].

Proton decay can occur via the color triplets through dimension-five and -six operators. On the other hand, GUT gauge bosons can mediate proton decay only through dimension-six operators at the leading order. Chirality flipping and nonflipping operators contribute to both dimension-five and -six levels. However, we can observe testable predictions only through the chirality nonflipping operators in color-triplet mediation as explained below.

To compare the proton decay predictions in IMDM75 with other GUT models, we consider the standard $SU(5)$ model with a $24$-dimensional Higgs field [35] ($SU(5)_{std}$), the flipped $SU(5)$ model ($FSU(5)$) [32, 33], the Pati-Salam model ($PS$) [36] and the Babu-Pati-Wilczek $SO(10)$ model [37]. One can find proton decay predictions from other $SO(10)$ models in references such as [38, 39, 40]. The branching fractions for various proton decay channels are plotted in Fig. 12 and 13. The left panel displays the relationship between the branching fraction and $M_{T}$ in IMDM75, while the right panel presents the corresponding predictions from the flipped $SU(5)$ model. While IMDM75 places an upper bound on the value of $M_{T}\lesssim v^{2}/M_{50}$, the flipped $SU(5)$ model allows $M_{T}$ to extend up to $10^{16}$ without any such constraint.

We find a significant difference in the proton lifetime behavior for the $p\rightarrow\bar{\nu}K^{+}$ channel compared to the predictions from the flipped $SU(5)$ model in [32]. Our plots exhibit a spread with respect to the $\tan\beta$ values, and due to the contributions from gauge boson mediation, we notice saturation at $M_{T}\sim 10^{15}$ GeV, which is in contrast to the flipped $SU(5)$ case where gauge boson mediation was absent and the spread in plots with respect to $\tan\beta$ values was minimal for this channel. We conclude that the $\bar{\nu}_{i}K^{+}$ channel is expected to play a crucial role in differentiating between the $SU(5)$ and flipped $SU(5)$ models.

The red horizontal lines in the figures represent the predictions of different GUT models. Specifically, the line corresponding to $SU(5)_{std}$ represents the prediction where proton decay is mainly mediated via gauge bosons. On the other hand, the lines corresponding to the Pati-Salam ($PS$) and Babu-Pati-Wilczek $SO(10)$ models represent the predictions where proton decay is dominantly mediated via color triplets. Thus, the IMDM75 model is found to make unique predictions for diverse branching ratios over a broad range of involved parameters, setting it apart from the three other GUT models.

In this section, we present a comprehensive two-loop analysis of gauge coupling unification in IMDM75. Following the breaking of the GUT symmetry, we end up with a pair of electroweak doublets with a mass of $M^{\prime}_{D}$ and two pairs of color triplets, in addition to the minimal matter content of the MSSM. The color-triplet mass matrix in Eq. (11) has eigenvalues denoted by $M_{Ti}$, where $i=1,2,3,4$. It is worth noting that only one pair of color triplets acquires intermediate-scale masses, specifically $M_{T_{1}}\sim M_{T}$ and $M_{T_{2}}\sim M_{T}$, while the other pair, with masses $M_{T_{3}}$ and $M_{T_{4}}$, acquire GUT-scale masses. As a result, only one pair of color triplets, with masses $M_{T_{1}}=M_{T_{2}}\sim M_{T}$, is pertinent for the renormalization group analysis below the GUT scale, while the other pair, with masses $M_{T_{3}}=M_{T_{4}}$, is relevant for the renormalization group analysis above the GUT scale. We set $M_{GUT}$ as the unification scale where $g_{1}(GUT)=g_{2}(GUT)=g_{12}$, and we match IMDM75 with MSSM at $M_{GUT}$. The $SU(5)$ symmetry breaking into the $SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$ symmetry results in the various components of $75_{H}$ acquiring masses (Table 4), as given below