Probing the Supersymmetric Grand Unified Theories at the Future Proton-Proton Colliders and Hyper-Kamiokande Experiment

Gauge coupling unification in the Supersymmetric Standard Models strongly implies the Grand Unified Theories (GUTs). With the grand desert hypothesis, we show that the supersymmetric GUTs can be probed at the future proton-proton (pp) colliders and Hyper-Kamiokande experiment. For the GUTs with the GUT scale $M_{GUT} \le 1.0\times 10^{16}$ GeV, we can probe the dimension-six proton decay via heavy gauge boson exchange at the Hyper-Kamiokande experiment. Moreover, for the GUTs with $M_{GUT} \ge 1.0\times 10^{16}$ GeV, we for the first time study the upper bounds on the gaugino and sfermion masses. We show that the GUTs with anomaly and gauge mediated supersymmetry breakings are well within the reaches of the future 100 TeV pp colliders such as the ${\rm FCC}_{\rm hh}$ and SppC, and the supersymmetric GUTs with gravity mediated supersymmetry breaking can be probed at the future 160 TeV pp collider.

Introduction.-Supersymmetry (SUSY) provides a natural solution to the gauge hierarchy problem in the Standard Model (SM). In the supersymmetric SMs (SSMs) with R-parity, gauge coupling unification can be achieved [1], the Lightest Supersymmetric Particle (LSP) such as the lightest neutralino can be a dark matter (DM) candidate [2], and the electroweak (EW) gauge symmetry can be broken radiatively due to the large top quark Yukawa coupling, etc. In particular, gauge coupling unification strongly suggests Grand Unified Theories (GUTs) [3][4][5][6][7], which may be constructed from superstring theory. Therefore, supersymmetry is a bridge between the low energy phenomenology and highenergy fundamental physics, and thus is the promising new physics beyond the SM.
Because the gauge coupling unification in the SSMs strongly suggests GUTs, the interesting and challenging question is: can we probe the supersymmetric GUTs at the future pp colliders and other experiments even if there does exist the SUSY EWFT problem? If yes, what is the center-of-mass energy of the future pp collider needed? We shall study it in this paper. In the GUTs, the well-know prediction is the dimension-six proton decay p → e + π 0 via heavy gauge boson exchange, and the proton lifetime is given by [35] where A R is the dimensionless one-loop renormalization factor associated with anomalous dimension of the relevant baryon-number violating operators, α GUT is the unified gauge coupling, and M GUT is the GUT scale. The current lower limit on the proton lifetime from the Super-Kamiokande experiment is τ p > 1.6 × 10 34 years [36]. Thus, we obtain M GUT ≥ 1.0 × 10 16 GeV. At the future Hyper-Kamiokande experiment, we can probe the proton lifetime at least above 1.0 × 10 35 years [37]. Therefore, the GUTs with M GUT ≤ 1.0 × 10 16 GeV is within the reach of the future Hyper-Kamiokande experiment.
In the following, with the grand desert hypothesis from the EW scale to the GUT scale, we shall show that the supersymmetric GUTs with M GUT ≥ 1.0 × 10 16 GeV can be probed at the future pp colliders. The supersymmetry searches at the 100 TeV pp colliders have been studied previously [33,[38][39][40][41]. For the integrated luminosity 30 ab −1 , Wino via Bino decay, gluinog via heavy flavor decay, gluino via light flavor decay, first-two generation squarksq, and stop can be discovered for their masses up to about 6.5 TeV, 11 TeV, 17 TeV, 14 TeV, and 11 TeV, respectively. Moreover, if the gluino and first-two generation squark masses are similar, they can be probed up to 20 TeV. Moreover, in the SSMs, supersymmetry is broken in the hidden sector, and then supersymmetry breaking is mediated to the SM observable sector via gravity mediation [42][43][44], gauge mediation [45][46][47], or anomaly mediation [48,49]. For the supersymmetric GUTs with M GU T ≥ 1.0 × 10 16 GeV, we for the first time study the upper bounds on the gaugino and sfermion masses. We show that the GUTs with anomaly and gauge mediated supersymmetry breakings are well within the reaches of the future 100 TeV pp colliders such as the FCC hh and SppC, and the supersymmetric GUTs with gravity mediated supersymmetry breaking can be probed at the future 160 TeV pp collider. The interesting viable parameter spaces for gravity mediation, which can be probed at the FCC hh and SppC, have been discussed as well.
The collected data points all satisfy the requirement of the Radiative Electroweak Symmetry Breaking (REWSB), has the lightest neutralino being the LSP for gravity and anomaly mediations, SM-like Higgs boson mass m h ⊂ [123, 127] GeV, and gluino mass mg ≥ 2.2 TeV. After collecting the data, we impose the constraints from rare decay processes B s → µ + µ − [53], b → sγ [54], and B u → τ ν τ [55]. To be general, we do not require the relic abundance of the LSP neutralino to satisfy the Planck bound within 5σ 0.114 ≤ Ω CDM h 2 (Planck) ≤ 0.126 [56].
Gravity Mediated Supersymmetry Breaking: mSUGRA/CMSSM.-The mSUGRA/CMSSM [42][43][44] is based on the GUTs and N = 1 supergravity where supersymmetry breaking is communicated through the supergravity interaction. It is one of the most widely studied SUSY scenarios, and has three supersymmetry breaking soft terms at the GUT scale: the universal gaugino mass M 1/2 , universal scalar mass M 0 , and universal trilinear coupling A 0 . The other free parameter tan β is the ratio of vacuum expectation values (VEVs) of two Higgs-doublets, and a discrete parameter sign(µ) = ±1. We perform the random scans for the following mSUGRA/CMSSM parameter space In the left panel of Fig. 2, we show results of our scans in M 1/2 − mg plane. We first find that the upper bound on the gluino mass is 15 TeV. In addition, the red points (tan β > 7.5) and blue points (tan β < 7.5) are the subsets of green points and satisfy the Planck 2018 5σ bounds on dark matter relic density. Interestingly, glunio masses for the red points are lighter than 11 TeV, and thus the glunio for the red points is within the reach of the FCC hh and SppC [38,40].
In the right panel of Fig. 2, we present the scan results in the first-two generation squark mass mq and M 0 plane. In particular, M 0 can be very heavy up to 65 TeV. Similarly, the red points (tan β > 9) and blue points ((tan β < 9)) are also the subsets of green points and satisfy the Planck 2018 5σ bounds on dark mat-  [59] and the maximum value of M 1/2 ∼ 7 TeV, we obtain that the maximum value of the first-two generation squark masses for most of red points is about mq 20 TeV, as shown clearly in mq −M 0 plot. Thus, most of the red points can be probed at the FCC hh and SppC [38,41] Because M 0 can be very large up to 65 TeV, it will be difficult to search for the squarks and sleptons at the FCC hh and SppC in general. Thus, we can look for the gauginos at the future pp colliders. For the integrated luminosity 30 ab −1 at the FCC hh and SppC, gluino via heavy and light flavor decays can be discovered for the masses up to about 11 TeV and 17 TeV, respectively. Thus, if gluino decays via light flavor squarks, it can be discovered at the FCC hh and SppC. However, in our viable parameter space, the lightest squark is generically to be light stop, and thus we do have gluino via heavy flavor decay. To probe such gluino with mass up to 15 TeV, we find that the center-of-mass energy of the future pp collider needs to be about 160 TeV. And we can discover Wino at this energy as well.
Anomaly Mediated Supersymmetry Breaking.-Anomaly mediated supersymmetry breaking (AMSB) is a special type of gravity mediated SUSY breaking. In this case, SUSY breaking is communicated to the visible sector from the hidden sector via a super-Weyl anomaly [48,49]. In the minimal AMSB, there are three basic parameters in addition to sign(µ): tan β, the universal scalar mass M 0 at the GUT scale which is introduced to solve the tachyonic slepton mass problem, and gravitino mass M 3/2 . We have performed the random scans over the following parameter space of the minimal AMSB with µ > 0 and m t = 173.2 GeV [57]. In the left panel of Fig. 3, we present the results of our scan in the mq − mg plane. All the points, which satisfy the current experimental constraints and have M U > 1 × 10 16 GeV, are shown in green color. We obtain that the upper bounds on the masses of both the first-two generation squarks and gluino are around 5 TeV, and thus they are well within the reaches of the FCC hh and SppC [38,41]. Moreover, the neutralinos, charginos, and sleptons can be discovered at the FCC hh and SppC as well.
Gauge Mediated Supersymmetry Breaking.-Finally, we study the Gauge Mediated Supersymmetry Breaking (GMSB) [45][46][47]. The GMSB is a method of communicating SUSY breaking to the SSMs from the hidden sector through the SM gauge interactions. The basic parameters of the minimal GMSB are: tan β, sign(µ), the messenger field mass scale M mess , the number of SU (5) representations of the messenger fields N mess , and the SUSY breaking scale in the visible sector Λ. The messenger fields induce the gaugino masses at one loop and then they are transmitted on to the squark and slepton masses at two loops. To preserve the gauge coupling unification, we consider the messenger fields which form the complete GUT multiplets. For simplicity, we introduce one pair of the messenger fields in the 5 and 5 representations of SU (5), i.e., N mess = 1. Also, we take parameter c grav = 1.
We perform random scans over the following minimal GMSB parameter space  Fig. 3. We see that the upper bounds on the masses of the first-two generation squarks and gluino are 8 TeV, and 6 TeV, respectively. Therefore, the first-two generation squarks and gluino are well within the reaches of the FCC hh and SppC [38,41]. Moreover, the neutralinos, charginos, and sleptons might be discovered at the FCC hh and SppC as well.
Summary and Conclusions.-Gauge coupling unification in the SSMs strong suggests the GUTs. Considering the grand desert hypothesis from the EW scale to GUT scale, we showed that the supersymmetric GUTs can be probed at the future pp colliders and Hyper-Kamiokande experiment. For the GUTs with M GU T ≤ 1.0×10 16 GeV, the dimension-six proton decay via heavy gauge boson exchange can be probed at the Hyper-Kamiokande experiment. Moreover, for the GUTs with M GU T ≥ 1.0 × 10 16 GeV, we for the first time studied the upper bounds on the gaugino and sfermion masses. We showed that the supersymmetric GUTs with anomaly and gauge mediated supersymmetry breakings are well within the reaches of the future 100 TeV pp colliders such as the FCC hh and SppC, and the supersymmetric GUTs with gravity mediated supersymmetry breaking can be probed at the future 160 TeV pp collider. The interesting viable parameter spaces for gravity mediation, which can be probed at the FCC hh and SppC, have been discussed as well.