Correlative study of flavor anomalies and dark matter in the light of scalar leptoquark

This paper proposes a $U(1)_{L_e-L_\mu}$ gauge extension of the Standard Model featuring a scalar leptoquark, new fermions, and inert scalars to simultaneously explain $B$-meson flavor anomalies, generate neutrino masses, and provide a dark matter candidate, while analyzing constraints from $b \to s$ transitions and predicting observables for $\Lambda_b$ decays.

The Gist

Imagine the Standard Model of particle physics as a massive, incredibly detailed instruction manual for how the universe works. For decades, this manual has explained almost everything we see in particle accelerators. However, like any old manual, it has a few pages that seem to have typos or missing instructions. Scientists call these "anomalies."

This paper is like a team of mechanics proposing a specific, clever repair kit to fix two major problems in the manual at the same time: why the universe is full of invisible "dark matter," and why certain heavy particles (B-mesons) are behaving strangely.

Here is a breakdown of their proposal using simple analogies:

1. The Two Problems

• The Dark Matter Mystery: We know about 85% of the universe is made of "Dark Matter," an invisible substance that holds galaxies together. But we don't know what it is. The Standard Model has no candidate for it.
• The Flavor Anomalies: In the world of particles, there are "flavors" (like electron, muon, and tau). Sometimes, heavy particles decay into lighter ones. Recently, experiments found that heavy particles are decaying into "muons" slightly differently than the manual predicts. It's as if a car engine is making a specific rattle that the manual says shouldn't happen.

2. The Proposed Repair Kit: The "Leptoquark" and the "New Force"

The authors suggest adding three new ingredients to the universe's recipe to fix both problems simultaneously:

• The New Force (The $Z'$ Boson): Imagine the Standard Model has a set of rules for how particles talk to each other. The authors propose adding a new "rulebook" based on a specific difference between electrons and muons. This creates a new force carrier, a particle called $Z'$, which acts like a new type of messenger.
• The Leptoquark (The Bridge): They introduce a special particle called a Scalar Leptoquark. Think of this as a "universal translator" or a bridge. In the Standard Model, quarks (which make up protons) and leptons (like electrons) usually stay in their own neighborhoods. This new particle is a bridge that allows them to cross over and interact.
• The Dark Matter Candidate: They also add three invisible, neutral particles. The lightest one of these three is proposed to be the Dark Matter. It's the "ghost" in the machine that we can't see but feel through gravity.

3. How It Works: The "Penguin" Dance

To explain the strange behavior of the B-mesons (the flavor anomalies), the authors look at how particles interact in a loop.

• The Analogy: Imagine a particle trying to change its identity. In the Standard Model, it takes a direct path. But in this new model, the particle takes a detour. It briefly turns into a "Penguin diagram" (a physics term for a specific loop shape).
• The Detour: In this loop, the particle interacts with the new Leptoquark and the new Dark Matter particle before coming back out. This detour changes the outcome of the decay, explaining why the experimental results don't match the old manual's predictions.

4. Testing the Repair Kit

The authors didn't just draw this on a napkin; they ran the numbers to see if their repair kit holds up.

• The Dark Matter Check: They calculated how much Dark Matter would be left over from the Big Bang (relic density). They found that if the Dark Matter particle interacts with the Leptoquark and the new $Z'$ force, the amount left over matches exactly what astronomers observe in the universe today.
• The "Sniffing" Check (Direct Detection): They also checked if this Dark Matter would bump into normal matter (like a detector on Earth). They found that while it wouldn't leave a huge "spin-independent" mark (like a heavy ball hitting a wall), it would create a "spin-dependent" interaction (like a spinning top wobbling). This specific type of interaction is currently allowed by the strict limits set by experiments like LZ and XENON1T, which are trying to catch Dark Matter.
• The Flavor Check: They tested their model against the strange B-meson decays. They found that by adjusting the strength of the new forces, their model could explain the anomalies without breaking other established rules of physics.

5. The Final Prediction: A New Baryon Decay

The paper's most exciting claim is a prediction for a specific, rare event that hasn't been fully studied yet.

• The Event: They look at a heavy particle called a Lambda-b baryon decaying into an excited state called Lambda-star (1520), which then breaks apart into a proton and a kaon, while also emitting a pair of muons.
• The Prediction: Using their new model, they predict that if you measure the branching ratio (how often this happens), the forward-backward asymmetry (which direction the particles fly), and the polarization (how they spin), you will see a slight but distinct difference compared to the Standard Model.
• The "Zero-Crossing": Specifically, they predict that the point where the "forward-backward asymmetry" flips from positive to negative will shift slightly. If future experiments (like LHCb) measure this shift, it could be the "smoking gun" that proves their new repair kit is real.

Summary

In short, the authors propose a unified theory where a new force and a "bridge" particle (the leptoquark) explain why the universe has Dark Matter and why certain heavy particles are misbehaving. They show that this theory fits current data, respects the limits of Dark Matter detectors, and makes a specific, testable prediction for how a rare baryon decay should behave in the future. It's a "two-for-one" solution to some of physics' biggest mysteries.

Technical Summary

Technical Summary: Correlative study of flavor anomalies and dark matter in the light of scalar leptoquark

Problem Statement
The Standard Model (SM) of particle physics, while successful, fails to explain several fundamental phenomena, including the nature of dark matter (DM), neutrino masses, and matter-antimatter asymmetry. Furthermore, recent experimental data from LHCb and Belle collaborations have revealed anomalies in $B$-meson decays, specifically in the flavor-changing neutral current (FCNC) transitions $b \to s \ell^+ \ell^-$. While recent updates on lepton flavor universality (LFU) ratios $R_{K^{(*)}}$ align with SM predictions, other observables such as the $P'_5$ angular observable and branching fractions for $B_s \to \phi \mu^+ \mu^-$ exhibit deviations of several sigma from SM expectations. Additionally, there is a need to explore these flavor transitions in the baryon sector, specifically the decay $\Lambda_b \to \Lambda^*(1520)(\to pK) \ell^+ \ell^-$, to gain deeper insights into the underlying dynamics of potential new physics (NP).

Methodology
The authors propose a specific Beyond the Standard Model (BSM) framework based on a local $U(1)_{L_e - L_\mu}$ gauge extension of the SM. The model introduces the following particle content:

1. Gauge Sector: A new $Z'$ gauge boson associated with the $L_e - L_\mu$ symmetry.
2. Fermion Sector: Three neutral fermions ($N_e, N_\mu, N_\tau$), where the lightest state ($N_1$) serves as a Weakly Interacting Massive Particle (WIMP) dark matter candidate. A discrete $Z_2$ symmetry is imposed to ensure the stability of the DM candidate.
3. Scalar Sector: A scalar singlet $\phi_2$ to spontaneously break the $U(1)_{L_e - L_\mu}$ symmetry, and a scalar leptoquark doublet $\tilde{R}_2$ (specifically the $\tilde{R}_2$ representation) to mediate flavor transitions.

The study employs a multi-step analysis:

• Model Construction: The Lagrangian is defined, including gauge, fermion, and scalar interaction terms. Mass matrices for fermions and scalars are diagonalized to identify physical mass eigenstates.
• Dark Matter Phenomenology: The relic density of the fermionic DM ($N_1$) is calculated using thermal freeze-out mechanisms, considering annihilation channels via the scalar leptoquark ($\tilde{R}_2$) and the $Z'$ boson. Direct detection prospects are evaluated by calculating spin-dependent (SD) and spin-independent (SI) WIMP-nucleon cross-sections, constrained by limits from LZ and XENON1T experiments.
• Flavor Constraints: The model is constrained by analyzing $b \to s \mu^+ \mu^-$ transitions. The new physics contributions arise primarily from penguin diagrams involving $Z'$ exchange and leptoquark loops. The authors utilize the flavio package to constrain model parameters against experimental data for $B \to K^{(*)} \mu^+ \mu^-$ and $B_s \to \phi \mu^+ \mu^-$, including branching ratios, forward-backward asymmetries ($A_{FB}$), and longitudinal polarization fractions ($F_L$). Constraints from $B_s - \bar{B}_s$ mixing and neutrino trident production are also incorporated.
• Baryonic Decay Analysis: Using the parameter space allowed by both flavor and dark matter sectors, the authors investigate the exclusive decay $\Lambda_b \to \Lambda^*(1520)(\to pK) \mu^+ \mu^-$. They compute differential branching ratios, forward-backward asymmetries, and polarization asymmetries using light-front quark model form factors.

Key Contributions and Results

• Model Viability: The study establishes a correlative framework where the same set of parameters addresses both flavor anomalies and dark matter phenomenology. The scalar leptoquark $\tilde{R}_2$ and the $Z'$ boson act as portals for DM annihilation and flavor transitions.
• Parameter Space Constraints: The analysis reveals that the parameter space is tightly constrained. Specifically, the $B_s - \bar{B}_s$ mixing imposes stringent bounds on the Yukawa coupling ($y_{qRN}$), which significantly restricts the region where the model can simultaneously satisfy flavor observables and DM relic density requirements. The allowed region is further narrowed by neutrino trident production limits and LEP-II bounds on the $Z'$ coupling ($g_{e\mu}$).
• Dark Matter Limits: The study finds that for the chosen benchmark parameters (e.g., $M_{Z'} = 705$ GeV, $M_{N_1} = 320$ GeV), the DM relic density can match Planck satellite data. However, the spin-dependent DM-neutron cross-section is constrained by LZ and XENON1T limits, particularly ruling out low-mass DM ranges (below 100 GeV) for the considered couplings.
• Impact on $\Lambda_b$ Decays: In the allowed parameter space, the new physics contributions to the $\Lambda_b \to \Lambda^*(1520) \mu^+ \mu^-$ decay are found to be modest compared to the SM.
  • Branching Ratio: The differential branching ratio shows a reduction compared to the SM prediction, but the deviation is minimal.
  • Polarization Asymmetry ($F_L$): A slight shift to lower values is observed, but no significant deviation from the SM is noted.
  • Forward-Backward Asymmetry ($A_{FB}$): This observable exhibits the most notable deviation. The zero-crossing point of the $A_{FB}$ distribution shifts from approximately $2.5$ GeV$^2$ in the SM to around $2.8$ GeV$^2$ in the presence of the new physics couplings.

Significance and Claims
The paper claims to provide a comprehensive theoretical framework that links flavor anomalies in the $B$-meson sector with dark matter phenomenology through a specific $U(1)_{L_e - L_\mu}$ extension involving a scalar leptoquark. The authors emphasize that while the model can accommodate current flavor and dark matter constraints, the stringent bounds from $B_s - \bar{B}_s$ mixing limit the magnitude of new physics effects in baryonic decays.

The significance of the work lies in its application to the $\Lambda_b \to \Lambda^*(1520)$ channel, a decay mode that has not yet been experimentally observed but is theoretically motivated by its narrow width and distinct quantum numbers. The authors suggest that while current constraints limit the size of deviations, precise future measurements of the forward-backward asymmetry in this channel could potentially disentangle SM and NP predictions, serving as a motivation for future experimental efforts at LHCb. The paper concludes modestly, noting that while the NP contributions are present, they are not large enough to drastically alter current SM expectations without further data to refine the sensitivity of these rare baryonic decays.