Up-type FCNC in presence of Dark Matter

This paper proposes a minimal extension of the Standard Model featuring a $\mathbb{Z}_3$-stabilized complex scalar dark matter candidate coupled to up-type quarks via a heavy vector-like quark, successfully addressing up-type flavor-changing neutral currents and dark matter relic density while offering testable signatures at future muon colliders.

The Gist

Imagine the universe as a giant, complex puzzle. For decades, physicists have been trying to fit the pieces together using a set of rules called the Standard Model. It's a brilliant theory that explains how most of the visible universe works—like how atoms stick together or how light behaves. But there are two massive pieces missing from the puzzle:

1. Dark Matter: We know it's there because it holds galaxies together with its invisible gravity, but we've never seen it. It's like a ghost in the room that you can feel pushing the furniture but can't see.
2. Flavor-Changing Neutral Currents (FCNC): This is a fancy way of saying "particles changing their identity without swapping charge." In the Standard Model, this is incredibly rare, like a chameleon suddenly turning blue in a split second. But if we see it happening too often, it means there's a new rulebook we haven't discovered yet.

This paper, written by a team from the Indian Institute of Technology Guwahati, proposes a clever new theory to solve both mysteries at once. They suggest that Dark Matter and these rare particle changes are actually connected, like two sides of the same coin.

Here is the story of their theory, broken down with some everyday analogies:

The New Cast of Characters

To fix the puzzle, the authors introduce two new characters to the Standard Model's play:

1. The Invisible Ghost (Dark Matter): They propose a "complex scalar field." Think of this as a shy, invisible ghost particle. It's stable (it doesn't disappear) because of a special rule called $Z_3$ symmetry.

   • The Analogy: Imagine a game of musical chairs where the ghost is the only one who never gets eliminated because the music stops in a specific pattern that only it understands. This rule prevents it from decaying into other things, keeping it around to be Dark Matter.

2. The Heavy Hauler (Vector-Like Quark or VLQ): This is a heavy, new type of particle that acts as a bridge. It's like a massive delivery truck.

   • The Analogy: The invisible ghost (Dark Matter) is too shy to talk to the regular particles (like the up, charm, and top quarks). The Heavy Hauler is the only one who can talk to both. It carries the ghost's "charge" and can pick up regular particles, swap them, and drop them off.

The Plot: How They Connect

The story goes like this: The Heavy Hauler (VLQ) interacts with the "up-type" quarks (the top, charm, and up quarks). Because the Hauler is so heavy, it can briefly pop into existence, grab a regular quark, swap it with the invisible ghost, and then vanish.

This swapping process causes two things to happen:

• The Ghost Hunt: The Hauler helps the ghosts (Dark Matter) destroy each other in the early universe, leaving just the right amount of them to match what we see today.
• The Identity Crisis: The swapping causes quarks to change their "flavor" (e.g., a top quark turning into a charm quark) in ways that shouldn't happen so easily. This is the FCNC part.

The Detective Work: Checking the Clues

The authors didn't just dream this up; they checked it against the real world. They looked at three main crime scenes:

1. The D0 Meson Mix-up: D0 mesons are particles that can oscillate between being matter and anti-matter. The Standard Model says this happens very slowly. If the Heavy Hauler is too light or the couplings are too strong, this mixing would happen way too fast. The authors calculated exactly how heavy the Hauler needs to be to avoid breaking the rules we've already observed.
2. The Top Quark's Secret Life: Top quarks are the heaviest particles we know. Sometimes they decay into lighter particles. The authors checked if their new Hauler would make the top quark decay into a charm quark and a photon (or a Z boson) too often. Current experiments say "no," so they had to tune their model to stay under the radar.
3. The Ghost Hunters (Direct Detection): Experiments like LUX-ZEPLIN are deep underground, waiting for Dark Matter to bump into a xenon atom. If the Hauler is too "chatty" (interacts too strongly), the ghost would have been caught by now. The authors showed that their model keeps the ghost quiet enough to hide from these detectors.

The Future: The Muon Collider

So, if this theory is true, how do we find it? The authors suggest that our current particle smashers (like the LHC) might be too messy to see this specific signal. The background noise is too loud.

Instead, they propose a Future Muon Collider.

• The Analogy: Imagine trying to hear a whisper in a crowded, noisy stadium (the LHC). It's nearly impossible. Now imagine a quiet, soundproof library (a Muon Collider). In this clean environment, you can hear the whisper.
• The Muon Collider would smash particles together at incredibly high energies. If the Heavy Hauler exists, it would be created and then immediately decay into a top quark and a charm quark, leaving behind a trail of "missing energy" (the ghosts running away). This specific signature—Top + Charm + Missing Energy—would be the smoking gun.

The Conclusion

The paper concludes that this "Minimal Extension" is a beautiful, simple solution. It links the invisible world of Dark Matter with the strange behavior of quarks.

• It fits the data: It explains why we haven't found Dark Matter yet (it's hiding well) and why we haven't seen too many flavor changes (the Hauler is heavy).
• It makes a prediction: It tells us exactly what to look for at the next generation of particle colliders.

In short, the authors are saying: "We think we found a new character in the universe's play. It's heavy, it's shy, and it's the reason why some particles are acting weird. If we build a better stage (a Muon Collider), we might finally see it."

Technical Summary

1. Problem Statement

The Standard Model (SM) of particle physics faces two major unresolved challenges:

1. Dark Matter (DM): The SM lacks a viable candidate to explain the observed non-luminous matter ($\sim 26.8\%$ of the universe's energy budget).
2. Flavor-Changing Neutral Currents (FCNC): While the SM suppresses FCNCs (particularly in the up-type quark sector) via the GIM mechanism and loop factors, experimental constraints on processes like $D^0-\bar{D}^0$ mixing and top-quark decays ($t \to c(u)X$) provide a sensitive window for Beyond the Standard Model (BSM) physics.

The authors investigate a potential correlation between the Dark Matter sector and the Up-type Flavor sector. They propose a minimal extension of the SM that simultaneously addresses the DM relic density and explains potential anomalies or constraints in up-type FCNC processes, while evading current direct and indirect detection bounds.

2. Model Framework

The authors propose a minimal extension involving two new fields:

• Complex Scalar Dark Matter ($\Phi$): A singlet complex scalar field stabilized by a discrete $Z_3$ symmetry. This symmetry allows for semi-annihilation processes ($\Phi \Phi \to \bar{\psi} u$), providing additional channels to deplete the DM relic density and evade direct detection limits.
• Vector-Like Quark (VLQ, $\psi$): A Dirac fermion with $SU(3)_c$ triplet, $SU(2)_L$ singlet, and hypercharge $Y=2/3$. It couples exclusively to right-handed up-type quarks ($u_R, c_R, t_R$) and the DM scalar $\Phi$.

Lagrangian Terms:
The interaction Lagrangian includes:
$$\mathcal{L} \supset - (y_u \bar{\psi} u_R \Phi + y_c \bar{\psi} c_R \Phi + y_t \bar{\psi} t_R \Phi + \text{h.c.}) - \mu_3 (\Phi^3 + \Phi^{*3}) - \lambda_{\Phi H} |\Phi|^2 |H|^2$$

• Free Parameters: $\{m_\Phi, m_\psi, y_u, y_c, y_t, \lambda_{\Phi H}, \mu_3\}$.
• Symmetry: Both $\Phi$ and $\psi$ carry the same $Z_3$ charge ($\omega$ or $\omega^2$), ensuring $\Phi$ is stable as the lightest particle in the dark sector ($m_\Phi < m_\psi$).

3. Methodology

The study employs a multi-faceted phenomenological analysis:

1. Flavor Constraints: Calculation of loop-induced contributions to:
   • $D^0-\bar{D}^0$ mixing (Box diagrams).
   • Rare decays $D^0 \to \ell^+\ell^-$ (Penguin diagrams).
   • Top-FCNC decays ($t \to c(u)\gamma, g, Z, h$).
2. Dark Matter Phenomenology:
   • Relic Density: Numerical solution of the Boltzmann Equation using micrOMEGAs, accounting for annihilation ($\Phi\Phi^* \to \text{SM}$), semi-annihilation, and co-annihilation ($\Phi\psi \to \text{SM}$).
   • Direct Detection (DD): Calculation of spin-independent (SI) DM-nucleon scattering cross-sections via Higgs portal and $t$-channel VLQ exchange.
   • Indirect Detection (ID): Constraints from Fermi-LAT on DM annihilation into $b\bar{b}$ and $u\bar{u}$.
3. Collider Searches:
   • LHC: Recasting ATLAS $t\bar{t} + \text{MET}$ searches to constrain VLQ masses.
   • Future Muon Collider: Simulation of $t\bar{c} + \text{MET}$ signals at a $\sqrt{s} = 10$ TeV muon collider with $10 \text{ ab}^{-1}$ luminosity using MadGraph5_aMC@NLO, Pythia8, and Delphes.

4. Key Contributions and Results

A. Flavor Physics Constraints

• $D^0-\bar{D}^0$ Mixing: This provides the most stringent constraint on the product of couplings $|y_u y_c|$. The model assumes the observed mixing is dominated by New Physics (NP). The analysis requires $|y_u y_c| \lesssim \mathcal{O}(10^{-2})$.
• Rare Decays ($D^0 \to \mu^+\mu^-$): The model contributes to vector current operators. However, the constraints are weaker than those from mixing.
• Top-FCNC Decays: The model induces decays like $t \to c(u)Z, \gamma, g$.
  • The branching ratios scale as $\Gamma \propto |y_t y_{c(u)}|^2$.
  • Experimental upper limits (ATLAS/CMS) constrain the parameter space, particularly for $t \to c\gamma$ and $t \to cg$.
  • Key Finding: To satisfy both $D^0$ mixing and Top-FCNC bounds simultaneously, the couplings must be hierarchical: $y_u \sim \mathcal{O}(10^{-2})$ (to suppress $D^0$ mixing and DD) while $y_c, y_t \sim \mathcal{O}(1)$.

B. Dark Matter Analysis

• Relic Density: The $Z_3$ symmetry enables semi-annihilation ($\Phi\Phi \to \bar{\psi}u$) and co-annihilation ($\Phi\psi \to \text{SM}$), which are crucial for achieving the correct relic density ($\Omega_{DM}h^2 \approx 0.12$) without violating DD bounds.
• Direct Detection: The SI cross-section is suppressed by the small $y_u$ and small Higgs portal coupling $\lambda_{\Phi H}$. This allows the model to evade the LUX-ZEPLIN (LZ) 2024 limits.
• Indirect Detection: Fermi-LAT constraints on $\langle \sigma v \rangle$ into $b\bar{b}$ and $u\bar{u}$ are satisfied by tuning couplings, particularly in the high-mass regime ($m_\Phi > 1$ TeV) where co-annihilation dominates.

C. Collider Prospects

• LHC: Current ATLAS searches for $t\bar{t} + \text{MET}$ exclude VLQ masses $m_\psi \lesssim 1.4 - 1.5$ TeV (depending on $m_\Phi$).
• Muon Collider ($\sqrt{s}=10$ TeV):
  • The signal process is $pp \to \psi\bar{\psi} \to (t\Phi)(\bar{c}\bar{\Phi}) \to t\bar{c} + \text{MET}$.
  • Due to the high energy, boosted top quarks form "top-jets." The signature is a dijet (one heavy top-jet) + Missing Energy.
  • Result: For benchmark points satisfying all flavor and DM constraints (e.g., $m_\Phi \approx 700$ GeV, $m_\psi \approx 1.5$ TeV), the signal significance reaches $>5\sigma$ at a 10 TeV muon collider. This suggests the LHC is insufficient, but future muon colliders can probe the viable parameter space.

D. Predictions

The model predicts enhanced branching ratios for rare top decays:

• $B(t \to c\ell\ell) \sim \mathcal{O}(10^{-8})$ (approx. $10^7$ times larger than SM).
• $B(Z \to cu) \sim 1.33 \times 10^{-12}$ (SM is $\sim 10^{-18}$).
  These are currently unobserved but within reach of future high-luminosity experiments.

5. Significance

• Minimal Connection: The paper establishes a concrete, minimal link between the Dark Sector and the Up-type Flavor sector, demonstrating that constraints from one sector (Flavor) directly dictate the viability of the other (DM).
• Role of $Z_3$: It highlights the utility of $Z_3$ symmetry in DM models, specifically enabling semi-annihilation to satisfy relic density while keeping direct detection cross-sections low.
• Future Discovery: The study argues that while the LHC has pushed the mass scale of such models to the TeV range, making discovery difficult due to QCD backgrounds, future high-energy muon colliders offer a unique and clean environment to detect these heavy Vector-Like Quarks via the distinctive $t\bar{c} + \text{MET}$ signature.
• Phenomenological Viability: The authors demonstrate that a specific region of parameter space exists where the model satisfies all current experimental bounds (Flavor, DM, LHC) while remaining testable in the near future.

In conclusion, this work provides a robust theoretical framework where Dark Matter stability and flavor physics are intertwined, offering clear targets for next-generation collider experiments.