Upper Bound of Proton Lifetime in Product-Group Unification

TL;DR

This work analyzes proton decay in SUSY GUTs built from SU(5)$_{\rm GUT}$ × U($N$)$_{\rm H}$ with $N=2,3$, where an unbroken R symmetry eliminates dimension-five operators and gauge-boson exchange dominates decay. By performing a detailed gauge-couping matching and RG analysis, including 1- and 2-loop effects and threshold corrections, the authors derive upper bounds on the GUT gauge-boson mass $M_G$ and translate these into upper bounds on the proton lifetime: $\tau(p\to \pi^0 e^+) \lesssim 6.0\times 10^{33}$ yr for the U(2) model and $\lesssim 5.3\times 10^{35}$ yr for the U(3) model, with uncertainties from $\alpha_s$, SUSY spectra, vector-like multiplets, and non-renormalizable operators. They show that the conservative upper bounds are sensitive to the SUSY-threshold corrections and possible non-renormalizable operators, while vector-like multiplets at low energy can shorten the lifetime by factors up to ~0.7. The results provide testable predictions for next-generation proton-decay experiments and help discriminate among realistic GUT constructions in the SUSY framework.

Abstract

Models of supersymmetric grand unified theories based on SU(5)_GUT \times U(N)_H gauge group (N = 2,3) have a symmetry that guarantees light Higgs doublets and absence of dimension-five proton decay operators. We analysed the proton decay induced by gauge-boson exchange in the models. Upper bounds of proton lifetime are obtained; τ(p\to π^0e^+) \lsim 6.0 \times 10^{33} yrs in the SU(5)_GUT \times U(2)_H model and τ(p\to π^0e^+) \lsim 5.3 \times 10^{35} yrs in the SU(5)_GUT \times U(3)_H model. Various uncertainties in the predictions are also discussed.

Figures (5)

• Figure 1: Close-up view of the unification of the three gauge coupling constants of the MSSM. The fine structure constants in the $\overline{\rm DR}$ scheme of the U(1)$_{Y}$, SU(2)$_{L}$ and SU(3)$_{C}$ gauge interactions are denoted by $\alpha_{1,2,3}$, respectively. Three lines of $\alpha_3$ correspond to three different experimental inputs; the QCD coupling constants $\alpha_s^{\overline{\rm MS},(5)}(M_Z)=0.1132$ ($-2\sigma$), 0.1172 (central value) and 0.1212 ($+2\sigma$) are used PDG02. The 2-loop renormalization-group effects of the MSSM and the SUSY threshold corrections are taken into account. The latter corrections are those from the SUSY-particle spectrum determined by the mSUGRA boundary condition with $\tan \beta = 10$, $A_0 = 0$ GeV, ($m_0$,$m_{1/2}$) = (400 GeV, 300 GeV) and $\mu > 0$ (see the caption for Fig. \ref{['fig:Cntr-U2-mSUGRA']} for the convention on the sign of $\mu$).
• Figure 2: Parameter region of the SU(5)$_{\rm GUT}\times$U(2)$_{\rm H}$ model. The parameter space of the model spanned by two free parameters $M_G$ and $M_{3V}/M_{3C}$ are restricted by requiring that all the running coupling constants of the model remain finite while the renormalization point is below the heaviest particle of the model. The left panel shows the parameter region, where the 1-loop renormalization group is used for all the coupling constants. The right-hand sides of the four curves labelled "(gauge-coupling)-mass" are excluded. The region below a curve labelled "$\alpha_{\rm 2H}^\lambda$--$M_{3V,C}$" is also excluded. Thus, the parameter space of the model is restricted to the shaded triangular region. The right panel shows the parameter region (shaded), where 2-loop effects are included in the renormalization-group equations of gauge coupling constants. The four curves are those found in the left panel; we keep them just because they make it easier to compare the panel with the left one. The majority of the triangular region in the left panel is further excluded because of the 2-loop effects, and only a small region survives near the line $M_{3V}\simeq M_{3C}$. The upper bound of $M_G$ is indicated by an arrow. In the right panel, $M_{2-3}$ indicates the unification point between $1/\alpha_{2}$ and $1/\alpha_3(+2\sigma)$ (see Fig. \ref{['fig:closeup']} for details). It is easy to see that $M_G \mathop{}_{ \sim}^{ <} 10^{15.6}$ GeV $\simeq$ ($10^{-0.4} \simeq 0.40$) $\times$ ($M_{2-3} \simeq 10^{16.0}$ GeV). Both two panels use $\alpha_s^{\overline{\rm MS},(5)}(M_Z) = 0.1212$. The effects from the non-renormalizable operator (\ref{['eq:non-ren']}) are not included here.
• Figure 3: Parameter region of the SU(5)$_{\rm GUT}\times$U(3)$_{\rm H}$ model. The parameter space of the model is spanned by three independent parameters: $M_G$, $M_{8V}/M_{8C}$ and $(M_{H_c}M_{H_{\bar{c}}})/M_G^2$. The figure is the $\sqrt{(M_{H_c}M_{H_{\bar{c}}})/M_G^2} = 10^{0.3}$ cross section of the parameter space. We require that all the coupling constants remain finite under the renormalization group, while the renormalization point is below the heaviest particle of the model. This condition is satisfied in the shaded region when the 1-loop renormalization group is used. Thin curves and lines labelled "(gauge coupling)-mass" are lines where the corresponding gauge coupling constants become infinite at the corresponding mass scales. After two loop effects are included in the beta functions of the gauge coupling constants, the remaining allowed parameter region is only on the thick curve labelled 2-loop analysis. Points (A) and (B) denote the upper and the lower bound of the gauge-boson mass $M_G$, respectively, for fixed $\sqrt{(M_{H_c}M_{H_{\bar{c}}})/M_G^2} = 10^{0.3}$. The upper bound of $M_G$ in the model is obtained as the maximum value $M_G$ takes at (A) as $\sqrt{(M_{H_c}M_{H_{\bar{c}}})/M_G^2}$ changes. One also sees immediately that the lower bound at (B) is so low that it is of no physical importance. $M_{1-2}$ ($M_{2-3}$) indicates the unification point between $1/\alpha_{1}$ and $1/\alpha_2$ ( $1/\alpha_{2}$ and $1/\alpha_3(-2\sigma)$ ) (see Fig. \ref{['fig:closeup']} for more details). Note that ($M_{G}$ at (A)) $< 10^{16.13}$ GeV $\simeq$ (($0.60 \simeq 10^{-0.22}$) $\times$ ($M_{1-2} \simeq10^{16.35}$ GeV)). The QCD coupling constant $\alpha_s^{\overline{\rm MS},(5)}(M_Z)=0.1132$ is used. The effects from a non-renormalizable operator that corresponds to (\ref{['eq:non-ren']}) in this model are not included here.
• Figure 4: Contour plots of the upper bound of the proton lifetime on the mSUGRA parameter space. The left panel is the prediction of the SU(5)$_{\rm GUT}\times$U(2)$_{\rm H}$ model and the right one that of the SU(5)$_{\rm GUT}\times$U(3)$_{\rm H}$ model. The upper bound changes as the universal scalar mass $m_0$ and the universal gaugino mass $m_{1/2}$ are varied (other mSUGRA parameters are fixed at $\tan \beta = 10$, $A_0 =0.0$). The $\mu$ parameter is chosen to be positive, when the constraint from the branching ratio of the $b \rightarrow s \gamma$ process is less severe. The upper bound of the lifetimes varies as (1.4--3.2)$\times 10^{33}$ yrs in the SU(5)$_{\rm GUT}\times$U(2)$_{\rm H}$ model, where the QCD coupling constant $\alpha_s^{\overline{\rm MS},(5)}(M_Z)=0.1212$ is used. The upper bound varies as (1--5)$\times 10^{35}$ yrs in the SU(5)$_{\rm GUT}\times$U(3)$_{\rm H}$ model, where the QCD coupling constant $\alpha_s^{\overline{\rm MS},(5)}(M_Z)= 0.1132$ is used. In both panels, the effects from non-renormalizable operators are not included. The thick curves labelled $m_h$ and $m_{\chi}$ are the bounds on the mSUGRA parameter space from the LEP II experiment in search of the lightest Higgs ($m_h \geq 114$ GeV, 95% C.L. ) LHiggs and the lightest chargino ($m_\chi \geq 103.5$ GeV, 95% C.L.) chargino. These curves are obtained by using the SOFTSUSY1.7 code SOFTSUSY. The excluded region changes when other codes are used; lower bound of $m_{1/2}$ for fixed $m_0$ can be higher by about 100 GeV. The code we adopt yields the largest pole mass of the lightest Higgs scalar among various codes available Allanach:2003jw, and hence the excluded region is the smallest.
• Figure 5: The left panels show the contour plots of the energy scale $M_{2-3}$, where the SU(2)$_L$ and SU(3)$_C$ gauge coupling constants become the same. The right panels show those of the energy scale $M_{1-2}$, where the U(1)$_Y$ and SU(2)$_L$ gauge coupling constants become the same. The upper panels are contour plots on the $(m_0,m_{1/2})$ parameter space of mSUGRA SUSY breaking, the lower ones are for the $(M_{\rm mess},m_{\rm SUSY} \equiv ((1/24)/(4\pi))\Lambda)$ parameter space of the GMSB. Other parameters are fixed for both SUSY breakings; $A_0 = 0$ GeV for mSUGRA SUSY breaking, and $\tan \beta = 10.0$ and $\mu > 0$ for both SUSY breakings. $\alpha_s^{\overline{\rm MS},(5)}(M_Z)=0.1172$ is used as the QCD coupling constant in this figure. See the caption for Fig. 4 for more details about the region excluded by the LEP II experiments.