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Updated analysis of minimal supersymmetric SO(10) with a universal soft spectrum

This paper updates the analysis of minimal supersymmetric SO(10) grand unification with a universal soft spectrum, revealing that after imposing constraints from gauge coupling unification, proton decay, the Higgs mass, and flavor observables, only a narrow "mini-split" region with heavy scalars ( $m_0 \gtrsim 7.7$ TeV) and low $\tan\beta$ ( $\lesssim 9$ ) survives, offering a testable scenario for future proton decay and electroweakino searches.

Original authors: Cheng-Wei Chiang, Takeshi Fukuyama

Published 2026-07-07

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Cheng-Wei Chiang, Takeshi Fukuyama

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, intricate machine built on a set of hidden rules. For decades, physicists have been trying to figure out the blueprint of this machine, specifically how the tiny particles that make up everything (like electrons and quarks) fit together.

This paper is like a rigorous quality-control check on one specific, very elegant blueprint called Minimal Supersymmetric SO(10). The authors are asking: "If we build the universe exactly according to this specific design, does it actually work, or does it fall apart?"

Here is the story of their investigation, broken down into simple concepts and analogies.

1. The Blueprint: A "Universal" Design

The authors are testing a theory where every particle has a heavier "shadow" partner (a supersymmetric partner). In this specific version, the rules for these shadows are universal.

The Analogy: Imagine a bakery that makes thousands of different pastries. In most bakeries, the dough for a croissant is different from the dough for a muffin. But in this "Universal" bakery, there is only one single recipe for the dough ( $m_0$ ). Every single pastry (squark, slepton, etc.) is made from this exact same dough. This makes the theory very clean and predictable, but also very rigid. If the dough is wrong, everything is wrong.

2. The Three Major Tests

To see if this bakery can stay open, the authors ran the blueprint through three brutal stress tests.

Test A: The "Proton Decay" Leak (The Lower Bound)

In this theory, protons (the building blocks of atoms) are supposed to be stable forever. However, the theory predicts they might slowly leak away (decay) over billions of years.

The Problem: If the "dough" (the mass of the particles) is too light, the protons leak too fast. We would have seen them disappear by now.
The Fix: The authors found that to stop the leak, the dough must be extremely heavy. Specifically, the particles must weigh at least 7.7 trillion times the mass of a proton (7.7 TeV).
The Catch: The heavier the dough, the slower the leak. But there's a limit to how heavy you can make it before the next test fails.

Test B: The "Higgs Mass" Scale (The Upper Bound)

There is a famous particle called the Higgs boson, which gives other particles their weight. We know its exact weight: 125 GeV.

The Problem: In this theory, the Higgs particle is naturally too light. To boost it up to 125 GeV, you need "radiative corrections" (a bit of extra energy from the heavy dough).
The Conflict: If the dough is too heavy (beyond a certain point), the Higgs particle gets boosted too high, becoming heavier than 125 GeV.
The Result: The dough cannot be infinitely heavy. It has an upper limit.

Test C: The "Gauge Unification" Lock

This is a mathematical requirement where three different forces of nature (electromagnetism, weak nuclear, and strong nuclear) must meet at a single point at very high energies.

The Lock: The theory requires a specific "key" (the mass of a colored Higgs particle) to make these forces meet perfectly.
The Twist: This key is tied to the proton leak. You can't just make the key heavier to stop the leak, because then the forces won't meet anymore. The key is locked in a narrow range.

3. The Surviving "Mini-Split" Region

After running millions of simulations, the authors found that the bakery only survives in a tiny, narrow strip of possibilities.

The "Mini-Split" Analogy: Imagine a seesaw. On one side, you have the heavy dough (the scalar particles). On the other side, you have the light shadows (the electroweakinos).
- In this surviving region, the dough is incredibly heavy (tens of trillions of electron volts).
- The shadows are relatively light (around the scale of the Large Hadron Collider, or TeV scale).
- It's a "Mini-Split" because the heavy and light parts are separated, but not as wildly separated as in some other theories.

The Surviving Recipe:

Scalar Mass ( $m_0$ ): Must be between 7.7 TeV and roughly 20 TeV. (Not too light, not too heavy).
$\tan \beta$ : A ratio of two Higgs fields must be low to moderate (less than 9).
The Key ( $M_{HC}$ ): The colored Higgs mass is pinned to a very specific range ($0.58$ to $1.43 \times 10^{16}$ GeV) by the math of unification.

4. The Dark Matter Mystery

The paper also asks: "What is the Dark Matter in this universe?" Dark Matter is the invisible stuff holding galaxies together.

Scenario A (The Strict Universal Model):
- The lightest particle (the "Bino") is the candidate.
- The Problem: It is too heavy and too "clumpy." If it were the only Dark Matter, it would have over-packed the universe, causing it to collapse long ago.
- The Fix: The authors say this Bino can only exist if it's a "sub-dominant" guest, and the real Dark Matter comes from somewhere else (like an axion), or if the universe was "diluted" by some cosmic event. It's not a perfect fit.
Scenario B (The "Free" Model):
- If the authors relax the "Universal" rule just a tiny bit (allowing the Higgs dough to be different), a new candidate appears: a Higgsino.
- The Result: A particle with a mass of about 1.1 TeV appears. This one fits the Dark Matter requirements perfectly.
- The Catch: Current experiments (like the LZ detector) have already ruled out the "middle ground" candidates. Only this specific 1.1 TeV Higgsino survives.

5. The Verdict: "It's Alive, But Barely"

The paper concludes that this specific theory is not dead, but it is cornered.

It is not ruled out: The math still works.
It is highly constrained: It only works in that tiny "Mini-Split" strip.
It is testable: The authors list exactly how we can prove or disprove this in the next decade:
1. Hyper-Kamiokande: A massive water tank in Japan looking for proton decay. If they see it, this theory is likely right. If they don't, the theory is pushed out of existence.
2. Colliders (HL-LHC / Future 100 TeV machines): They need to find the light "shadow" particles (electroweakinos) or the heavy, slow-moving "gluinos" (which would leave strange tracks in detectors).
3. Dark Matter Detectors: They need to find that specific 1.1 TeV Higgsino.

Summary

Think of this paper as a detective narrowing down a suspect. The suspect (the Minimal Supersymmetric SO(10) theory) was once a broad suspect with many alibis. The authors have eliminated all the alibis. Now, the suspect is hiding in a very small, specific room (the Mini-Split region).

The room is defined by:

Heavy furniture (Scalar particles > 7.7 TeV).
Light shadows (Electroweakinos at the TeV scale).
A locked door (Proton decay limits).

The good news? We have the keys to check this room. The next generation of experiments (Hyper-Kamiokande, HL-LHC, and new Dark Matter detectors) will either find the suspect in that room or prove they are innocent, effectively closing the case on this specific theory.

Technical Summary: Updated Analysis of Minimal Supersymmetric SO(10) with a Universal Soft Spectrum

Problem and Motivation
The paper addresses the viability of the minimal supersymmetric (SUSY) SO(10) Grand Unified Theory (GUT) under the assumption of a universal, constrained-MSSM-like (CMSSM) soft-breaking spectrum. In this framework, the matter sector is highly predictive: a single pair of symmetric Yukawa matrices ( $Y_{10}, Y_{126}$ ) in the $10 \oplus 126$ Higgs sector simultaneously determines charged fermion masses, quark mixing (CKM), the flavor structure of baryon-number-violating dimension-five operators mediating proton decay, and the right-handed neutrino Majorana masses via the type-I seesaw mechanism.

The central challenge is that these tight theoretical constraints, combined with experimental data, severely restrict the parameter space. Specifically, the model must satisfy gauge coupling unification, vacuum stability, the observed 125 GeV Higgs boson mass, proton decay limits, and flavor/precision observables, while determining the fate of the lightest supersymmetric particle (LSP) as a dark matter candidate. The authors aim to perform a global scan to identify the surviving regions of the parameter space, moving beyond previous analyses by incorporating a constrained- $\mu$ treatment derived from radiative electroweak symmetry breaking (EWSB) and examining the implications for the LSP.

Methodology
The authors perform a global scan over the soft-breaking parameters $\{m_0, M_2, \mu, \tan\beta, X_t, M_{HC}\}$ , subject to a set of rigorous constraints:

Theoretical Consistency:
- Gauge Coupling Unification: The color-triplet (colored-higgsino) mass $M_{HC}$ is not a free parameter but is fixed by the requirement that the three SM gauge couplings unify at $M_{GUT} \approx 2 \times 10^{16}$ GeV. This restricts $M_{HC}$ to a specific band $[0.58, 1.43] \times 10^{16}$ GeV.
- Vacuum Stability: The analysis enforces stop positivity and the absence of charge- and color-breaking (CCB) minima, limiting the stop mixing parameter $X_t$ .
Experimental Limits:
- Proton Decay: The dimension-five operator rate ( $p \to K^+ \bar{\nu}$ ) is calculated using the fitted Yukawa matrices. The rate is normalized by the effective colored-Higgs scale $M_{HC}$ , which is pinned by unification. The current limit $\tau_p > 6.61 \times 10^{33}$ yr imposes a lower bound on the universal scalar mass $m_0$ .
- Higgs Mass: The requirement $m_h = 125.13 \pm 0.11$ GeV imposes an upper bound on $m_0$ . The authors use an effective field theory (EFT) resummed treatment calibrated to FeynHiggs to handle the multi-tens-of-TeV stop masses, avoiding the overestimation of bare leading-log approximations.
- LEP Chargino Limit: $m_{\chi^\pm_1} > 103.5$ GeV is applied, though it is found to be non-binding in the surviving region.
Flavor and Precision Observables:
- Constraints from $\mu \to e\gamma$ , muon $g-2$ , and electric dipole moments (EDMs) are evaluated. Under the assumption of real, universal soft terms, these are found to be non-binding due to the heavy scalar spectrum suppressing SUSY loops.
Electroweakino Schemes:
- Constrained- $\mu$ : The standard CMSSM scenario where $|\mu|$ is determined point-by-point by the EWSB conditions given the GUT-scale inputs.
- Free- $\mu$ (NUHM Extension): A minimal extension where $|\mu|$ is treated as a free low-scale input, allowing for a light higgsino.

Key Results

Surviving Parameter Space: The model survives only in a narrow "mini-split" region characterized by:
- Heavy scalars: $m_0 \gtrsim 7.7$ TeV.
- Low-to-moderate $\tan\beta$ : $\tan\beta \lesssim 9$ .
- Colored-Higgs scale: $M_{HC} \in [0.58, 1.43] \times 10^{16}$ GeV.
- The lower bound on $m_0$ is driven by the proton decay limit (which rises with $\tan\beta$ ), while the upper bound is driven by the Higgs mass requirement (which falls with $\tan\beta$ ). These constraints intersect at $\tan\beta \approx 8.6$ .
Electroweakino Spectrum:
- Constrained- $\mu$ : Radiative EWSB drives $|\mu|$ to the multi-TeV scale ($7-33$ TeV), well above the gaugino masses ( $M_1 < M_2$ ). Consequently, the lightest neutralino is a nearly pure bino, and the higgsino is heavy and decoupled.
- Free- $\mu$ : A light higgsino ( $\sim 1.1$ TeV) is possible, but the "well-tempered" bino-higgsino strip is excluded by direct detection limits.
Dark Matter Fate:
- In the universal (constrained- $\mu$ ) model, the bino LSP is thermally over-abundant ( $\Omega h^2 \sim 10^4 - 10^5 \times 0.12$ ) and invisible to direct detection. It cannot serve as the sole thermal dark matter candidate without non-thermal dilution mechanisms.
- In the free- $\mu$ (NUHM) extension, a viable thermal relic exists only as a nearly pure higgsino with a mass of $\sim 1.1$ TeV. The well-tempered region is excluded by LUX-ZEPLIN (LZ) data.
Flavor and Precision: All flavor-violating and CP-violating observables ( $\mu \to e\gamma$ , $g-2$ , EDMs) are suppressed well below experimental limits due to the heavy scalar masses and the assumption of real soft parameters.

Significance and Claims
The paper claims that the minimal supersymmetric SO(10) with a universal soft spectrum is not excluded but is pushed into a sharply bounded, testable "mini-split" corner of parameter space. The significance lies in the following points:

Predictive Power: The model's predictive nature, where Yukawa couplings fix both fermion masses and proton decay amplitudes, turns the proton decay rate into a strict lower bound on the scalar mass scale rather than an adjustable parameter.
Interplay of Constraints: The analysis demonstrates that gauge unification and proton decay are inextricably linked; one cannot evade the proton decay bound by simply raising the colored-Higgs mass without violating unification.
Testability: The surviving region is directly testable in the coming decade. The lower edge of the parameter space ( $m_0$ ) is targeted by Hyper-Kamiokande proton decay searches. The electroweakino sector (bino/wino) is accessible via the High-Luminosity LHC and future 100-TeV colliders. The specific higgsino dark matter candidate in the NUHM extension is a fixed-mass prediction ( $\sim 1.1$ TeV) that will be fully probed by next-generation multi-ton xenon experiments (e.g., XLZD/DARWIN).
Robustness: The results hold even when relaxing the universal hypothesis for the Higgs sector (free- $\mu$ ), though this alters the dark matter candidate from an over-abundant bino to a viable 1.1 TeV higgsino. The core bounds on $m_0$ and $\tan\beta$ remain driven by proton decay, the Higgs mass, and unification, independent of the specific dark matter realization.

The authors conclude that if this minimal SO(10) framework is realized in nature, it manifests as a mini-split spectrum with heavy scalars and light electroweakinos, a scenario that is on the verge of being confirmed or ruled out by upcoming experimental data.