Dark Matter Induced Proton Decays

This paper proposes a novel framework where a spontaneously broken global $U(1)_{B+L}$ symmetry generates a residual $Z_4$ symmetry that stabilizes dark matter while suppressing proton decay to the one-loop level, thereby linking the proton lifetime to dark matter mass and predicting distinctive TeV-scale leptoquark signatures accessible to current and future experiments.

The Gist

Imagine the universe is a giant, complex machine built on a set of invisible rules. For decades, physicists have been trying to solve two of the biggest mysteries in this machine:

1. The "Unbreakable" Particle: Why does the proton (the building block of all matter, including you and me) never seem to fall apart? If it did, everything would eventually dissolve.
2. The "Invisible Ghost": What is Dark Matter? We know it's there because it holds galaxies together with gravity, but we can't see it, touch it, or detect it directly. It makes up 85% of the matter in the universe.

Usually, scientists treat these two problems as completely separate. But in this paper, two researchers from India (Ranjeet Kumar and Rahul Srivastava) propose a clever new idea: What if the reason protons are stable and the reason Dark Matter exists are actually the same thing?

Here is the story of their idea, explained simply.

1. The "Do Not Disturb" Sign

In the Standard Model (our current rulebook for physics), there is a hidden symmetry called $B+L$. You can think of this as a "Do Not Disturb" sign on the door of the universe.

• B stands for Baryon number (protons/neutrons).
• L stands for Lepton number (electrons/neutrinos).

As long as this sign is up, protons cannot decay (fall apart), and Dark Matter cannot disappear.

2. Breaking the Sign (But Not All of It)

The researchers imagine a scenario where this "Do Not Disturb" sign is slightly broken, but not completely.

• Imagine you have a big, round clock face representing the symmetry.
• When the universe cooled down, the clock hands moved, breaking the symmetry.
• However, the clock didn't stop at 12:00 (total chaos). It stopped at 4:00.

This "4:00" position is a $Z_4$ symmetry. It's a smaller, residual rule that remains.

• The Good News: This remaining rule acts like a bouncer at a club. It says, "Dark Matter particles are VIPs; they can't leave the club." This ensures Dark Matter stays stable forever.
• The Twist: This same bouncer also says, "Protons are VIPs too, but they are so protected that they can't decay through the front door (Tree Level)."

3. The Secret Back Door (The Loop)

If the front door is locked, how does a proton ever decay?
The researchers suggest that protons can decay, but only through a secret back door that requires a very complicated, roundabout path.

• Instead of falling apart instantly, a proton has to go through a "loop" involving invisible Dark Matter particles and other heavy, exotic particles.
• Think of it like trying to get a message across a guarded wall. You can't walk over it (Tree Level). You have to dig a tunnel, go through a maze, and come out the other side (One-Loop Level).
• Because this path is so long and difficult, the proton takes an incredibly long time to decay. This explains why we haven't seen it happen yet!

4. The Heavyweight Connection

Here is the most fascinating part of their theory: The weight of the Dark Matter controls the lifespan of the proton.

• Heavy Dark Matter = Stronger Lock: If the Dark Matter particles are very heavy, the "tunnel" they create for the proton to decay through is even harder to dig. The proton becomes super stable.
• Light Dark Matter = Weaker Lock: If the Dark Matter is lighter, the tunnel is easier to dig, and the proton might decay faster.

This creates a direct link: The heavier the Dark Matter, the longer the proton lives.

5. The "Exotic" Particles and the Collider Hunt

To make this work, the universe needs some new, heavy particles that act as the "construction workers" digging the tunnel.

• These include a Leptoquark (a particle that is half-quark, half-lepton) and some heavy electrons.
• These particles have "exotic charges" (weird electrical properties) that make them behave differently than normal particles.
• The Catch: Because the proton decay is so slow, these heavy particles don't need to be impossibly heavy (like the size of a mountain). They can be around the 1 TeV scale.
• Why this matters: 1 TeV is a weight that our current particle smashers (like the Large Hadron Collider at CERN) or future ones (like the FCC) might actually be able to find!

The Bottom Line

This paper suggests a beautiful unification:

1. Dark Matter is stable because of a leftover symmetry rule.
2. Protons are stable because that same rule locks the front door.
3. Proton decay is a rare event that happens only through a secret loop involving Dark Matter.
4. We can test this: If we find these specific "exotic" particles at a collider, or if we finally see a proton decay, it would prove that Dark Matter and the stability of our universe are two sides of the same coin.

It's like realizing that the reason your house is secure (Dark Matter) and the reason your car won't rust (Proton stability) are both due to the same specific type of lock on your front door. And the best part? We might be able to pick that lock with our current technology.

Technical Summary

1. Problem Statement

The paper addresses two fundamental unresolved issues in the Standard Model (SM):

1. Proton Decay: While Grand Unified Theories (GUTs) predict proton decay, the required new physics scale ($\Lambda \sim 10^{16}$ GeV) is far beyond the reach of current colliders. Furthermore, no proton decay has been observed, setting strict lower bounds on the proton lifetime.
2. Dark Matter (DM): The SM lacks a viable DM candidate.
3. The Conflict: Proton decay requires the violation of Baryon ($B$) and Lepton ($L$) number (specifically $B+L$), whereas DM stability typically relies on an unbroken symmetry to prevent decay. Reconciling these seemingly contradictory requirements within a single framework accessible to current experiments is a significant theoretical challenge.

2. Methodology and Model Framework

The authors propose a novel theoretical framework based on the global $U(1)_{B+L}$ symmetry, which is an accidental symmetry in the SM.

• Symmetry Breaking Mechanism:

  • The $U(1)_{B+L}$ symmetry is spontaneously broken by the vacuum expectation values (vevs) of two new SM-singlet scalar fields, $\chi$ and $\sigma$.
  • This breaking leaves a residual discrete $Z_4$ symmetry.
  • Charge Assignment: Under this residual $Z_4$, all SM particles are even, while all new Beyond-Standard-Model (BSM) dark sector particles are odd.
  • Consequences:
    • DM Stability: The lightest $Z_4$-odd particle is stable, serving as a viable DM candidate.
    • Proton Decay Suppression: Tree-level proton decay is strictly forbidden because it would require a vertex connecting even (SM) and odd (BSM) particles, violating $Z_4$. Consequently, proton decay can only occur via one-loop diagrams.

• Particle Content (Scalar DM Scenario):

  • Scalars: A leptoquark $\tilde{S}_1$ (color triplet, electric charge $4/3$) and the DM candidate $\zeta$ (SM singlet, neutral).
  • Fermions: Vector-like heavy quarks ($U_L, U_R$) and heavy charged leptons ($E_L, E_R$).
  • Scalars for Breaking: $\chi$ and $\sigma$ (singlets) drive the symmetry breaking.

• Effective Operator:

  • Proton decay is mediated by a dimension-6 effective operator $[du][ue]$.
  • The UV completion of this operator involves the dark sector particles running in a one-loop box diagram.

3. Key Contributions

1. Unification of DM Stability and Proton Decay: The paper demonstrates that the same symmetry ($Z_4$) that ensures DM stability also forbids tree-level proton decay, forcing the decay to occur at the loop level.
2. Lowering the New Physics Scale: By suppressing proton decay to the one-loop level, the model allows the mass of the mediating particles to be as low as $\Lambda \sim \mathcal{O}(1)$ TeV. This is orders of magnitude lower than the GUT scale, making the model testable at current (LHC) and future (FCC-hh) colliders.
3. Correlation between DM Mass and Proton Lifetime: The model establishes a direct phenomenological link where the proton lifetime is inversely related to the DM mass. Heavier DM particles enhance proton stability (longer lifetime), while lighter DM particles reduce it.
4. Exotic Collider Signatures: The BSM particles carry exotic $B+L$ charges and $Z_4$ charges, leading to decay modes distinct from standard leptoquark or vector-like quark searches (e.g., decays involving missing energy from the stable DM candidate).

4. Results and Numerical Analysis

• Proton Decay Rate:

  • The dominant decay channel is $p \to e^+ \pi^0$.
  • The decay width $\Gamma$ is proportional to $|Y|^4 M_E M_U I$, where $Y$ represents Yukawa couplings and $I$ is the loop integral.
  • Constraint: To satisfy the Super-Kamiokande (SK) lower limit on the proton lifetime ($\tau > 2.4 \times 10^{34}$ years), the mediator masses must be sufficiently heavy or couplings small.
  • Finding: For $\mathcal{O}(1)$ Yukawa couplings, mediator masses in the TeV range are consistent with current proton decay limits.

• Dark Matter Phenomenology:

  • Relic Density: The scalar $\zeta$ is the DM candidate. The model includes co-annihilation channels with the heavy fermions $E$ and $U$.
  • Direct Detection (DD): The spin-independent DM-nucleon cross-section is mediated by the Higgs portal and direct couplings to quarks.
  • Viable Parameter Space: The analysis identifies a region where the DM mass ranges from $\sim 500$ GeV to a few TeV. In this region, the model satisfies:
    1. Observed relic density ($\Omega_{DM}h^2 \approx 0.12$).
    2. Current DD limits (LZ, XENONnT, PandaX-4T).
    3. Proton lifetime limits.
  • Lower Bound: The requirement of proton stability imposes a lower bound on the DM mass, excluding very light DM scenarios that might otherwise be allowed by collider constraints.

• Collider Signatures:

  • Leptoquark ($\tilde{S}_1$): Produced via gluon-gluon fusion or $q\bar{q}$ annihilation.
    • Decay Modes: Depends on mass hierarchy. If $M_{\tilde{S}_1} < M_{E,U}$, it decays via 3-body channels ($j e^+ \zeta^\dagger$). If $M_{\tilde{S}_1} > M_{E,U}$, it decays via 2-body channels ($j E/U$).
    • Constraints: Current LHC searches for leptoquarks and SUSY imply a lower bound of $M_{\tilde{S}_1} \gtrsim 1$ TeV.
  • Heavy Fermions ($E, U$):
    • $U$ (color triplet) has a large production cross-section via gluon fusion.
    • $E$ (color singlet) is produced via $q\bar{q}$ annihilation.
    • Decays involve multi-body final states with missing energy ($\zeta$), distinguishing them from standard BSM searches.

5. Significance and Conclusion

This work offers a compelling solution to the "proton decay vs. DM stability" paradox by utilizing the spontaneous breaking of $U(1)_{B+L}$ to a residual $Z_4$ symmetry.

• Testability: Unlike traditional GUT models where proton decay is unobservable at colliders, this model predicts new particles (leptoquarks, vector-like fermions) at the TeV scale.
• Distinct Signatures: The exotic charges and specific decay topologies (involving stable DM) provide unique signatures that can be targeted by current and future experiments (LHC, FCC-hh, DUNE, Hyper-Kamiokande).
• Interplay: The framework highlights a deep interconnection between the stability of the universe's matter (proton), the abundance of dark matter, and the signatures of new physics at colliders, all governed by a single symmetry breaking pattern.

The paper concludes that the parameter space allowing for a stable DM candidate and a proton lifetime consistent with observations is accessible to experimental verification, making this a highly testable extension of the Standard Model.