Flavor structure of GUTs and uncertainties in proton lifetime estimates

We study the flavor aspects of proton lifetime estimates in simple grand-unified models, paying particular attention to their inherent fragility due to the notorious lack of control of some of the key parameters governing the relevant hard-process amplitudes. Among these, the theoretical uncertainties in the flavor structure of the baryon-and lepton-number-violating charged currents due to the potential higher-order effects afflicting the matching of the underlying Yukawa couplings to the low-energy data often play a prominent role. Focusing on the minimal variants of the most popular unified models, we study the potential instabilities of the corresponding proton lifetime estimates based on the renormalizable-level Yukawa fits with respect to the Planck-scale-induced flavor effects. In particular, we perform a detailed numerical analysis of all minimal $SO(10)$ Yukawa sector fits available in the literature and show that the proton lifetime estimates based on these inputs exhibit a high degree of robustness with respect to moderate-size perturbations, well within the expected “improvement window” of the upcoming proton decay searches.

Figure 1

The total proton lifetime for different choices of the flavor-dependent coefficients (8)–(10) is displayed for different gauge groups (with $M_{1,2}\sim 0.3\times {10}^{16}\mathrm{GeV}$ generally assumed in order for the current Super-Kamiokande (SK) bound on the partial width in the $p\to {\pi }^0{e}^{+}$ channel to be saturated in the $SO(10)$ case; see also Fig. 2). The red bars correspond to “minimal” renormalizable scenarios which, in all cases of our interest, feature extra correlations among the relevant mixings due to, e.g., symmetry of the underlying Yukawa matrices in certain sectors; see Sec. 2c. In the case of the blue bars, no such correlations are assumed.

Figure 2

The $p\to {\pi }^0{e}^{+}$ partial proton lifetime in the case of the $SO(10)$ unifications as a function of the size of the Planck-induced perturbations $|\mathrm{\Delta }Y|$ (20). The renormalizable-level setting of the Yukawa matrices $Y_f^{\prime }$ corresponds to the fit explicitly given in Ref. [22], where the $10_H\oplus 126_H$ Higgs content and the normal neutrino mass hierarchy are assumed. Similar behavior with a large spread of the partial proton lifetime values was obtained also for other individual decay channels and also when other fits in Ref. [22] served as the initial point.

Figure 3

The inverse of the sum of the partial $p\to X^{+}\overline{\nu }$ decay widths for $X=\pi$, $K$ in the case of the SO(10) unifications (the same renormalizable-level point as in Fig. 2 was used, and again similar behavior was observed also for other initial settings based on fits from Ref. [22]). Note, however, that the behavior of the individual neutrino-final-state channels is similar to the situation for the golden channel (see Fig. 2).

Figure 4

The total proton lifetime in the case of the minimal $SO(10)$ unification.

Figure 5

Total proton lifetime in the case of flipped SU(5) unifications where $D=D_C$ is assumed for the renormalizable-level ansatz for the Yukawa sector. The large spread of points with $|\mathrm{\Delta }Y|$ only slightly below ${10}^{-2}$ would suggest that the proton decay can be indeed rotated away for this class of scenarios; however, if the minimal schemes are considered, the leading Planck-scale corrections are expected at $d=6$ level only, and for the corresponding $|\mathrm{\Delta }Y|\sim {10}^{-4}$ the total proton lifetime is constrained.