A U(1)(B-L) 3 model for dark matter and the B+ to K+ nu excess reported by Belle II

This paper explores a U(1)(B−L)3 gauge symmetry model, under which only third-generation fermions are charged. The cancellation of the [U(1)(B−L)3]3 anomaly requires a chiral fermion, χL, singlet under the gauge group but charged under U(1)(B−L)3. This chiral fermion can play the role of thermal dark matter with a relic abundance set via the freeze-out scenario through the annihilation to pairs of the third-generation fermions. In the quark mass basis, the new gauge boson will have off-diagonal couplings to the b and s quarks, which can lead to a new contribution to B+ → K+ ντ ν̄τ. The paper discusses the relevant bounds from the LHC searches for a new gauge boson, dark matter searches, electroweak precision data, Υ decay, the CKM matrix, and the Bs0 − B̄s0 mixing. It entertains the possibility of explaining the B+ → K+ ν ν̄ excess recently reported by Belle II within this model.

The model introduces a gauge boson Vµ for the U(1)(B−L)3 symmetry, under which the third-generation flavor eigenstates (b and t quarks) have a charge of 1/3, and the third-generation leptons (τ and ντ) have a charge of −1. The rest of the SM fermions are neutral. A new chiral fermion χ, singlet under the SM gauge group but charged under U(1)(B−L)3, plays the role of dark matter. The gauge interaction of the fermions is given by gV Vµ [(t̄γµ t + b̄γµ b)/3 − τ̄γµ τ − ν̄τ γµ Pντ − χ̄γµ Pχ], where P is the chirality projection matrix. The contribution from chiral χ helps cancel the U(1)3 anomaly. This new fermion should be much heavier than O(MeV), otherwise it will come to thermal equilibrium with ντ for T ∼ MeV and lead to ∆Neff = 1 at the time of big bang nucleosynthesis, which is ruled out. In the early universe, χ will be thermalized via its gauge interaction with the third-generation fermions of the SM, and the dark matter production mechanism can be the canonical freeze-out.

The paper discusses the kinetic mixing between Vµ and the hypercharge gauge boson Bµ, which appears at one loop level. Through this mixing, dark matter can scatter off nuclei, and direct dark matter searches set a strong bound on the coupling, requiring the dark matter to be of Majorana nature, which leads to the suppression of the scattering cross section by the square of the dark matter velocity relative to the nucleus, vχ2. The annihilation of dark matter pairs will be p-wave too, implying that the lower bounds on the mass of thermal dark matter from indirect searches can be relaxed. In the early universe, dark matter pairs can annihilate into pairs of third-generation fermions through s-channel V exchange, χχ̄ → V* → tt̄, bb̄, ττ̄, ντν̄τ. Using micrOMEGAs7, the gauge coupling versus the gauge boson mass is determined for which the observed relic DM abundance is obtained via the freeze-out scenario, taking into account the latest bounds from direct dark matter searches and from the LHC.

Since the U(1)(B−L)3 charges of the third-generation quarks are different from those of the first and second generations, reproducing the mixings between the third generation with the rest requires spontaneous symmetry breaking through the vacuum expectation value (VEV) of a new Higgs boson charged under U(1)(B−L)3. In the quark mass basis, there will be off-diagonal couplings of V to the quarks, which can give a contribution to the flavor-changing neutral current (FCNC) processes such as B+ → K+ ντ ν̄τ. The paper discusses in detail whether this model can also explain the 2.7σ excess of Br(B+ → K+ ν ν̄) relative to the SM reported by the Belle II collaboration.

The paper is organized as follows. In sect. 2, the model and the bounds from electroweak precision data, Υ-decay, LHC search for extra heavy gauge boson, and direct dark matter searches are reviewed. In sect. 2.1, the gauge interaction is rewritten in the quark mass basis, incorporating the information from the CKM mixing matrix and discussing the bounds from the Bs0 − B̄s0 and Bd0 − B̄d0 mixings. In sect. 2.2, it is shown how the CKM matrix elements can be reproduced in a minimal model with the VEV of a heavy Higgs doublet charged under U(1)(B−L)3 with a relatively small VEV of ∼ 7 GeV. In sect. 2.3, the model is extended to include two such doublets, Φ and Φ′, with a symmetry under Φ ↔ Φ′, and it is shown how within this model the stringent bounds from Bs0 − B̄s0 can be relaxed. In sect. 3, the contribution from the model to the Br(B+ → K+ ν ν̄) excess reported by Belle II is discussed. In sect. 4, the abundance of dark matter and the possibility of direct and indirect detection as well as the possibility of explaining the reported B+ → K+ ν ν̄ excess are discussed. The results are summarized in sect. 5.

The paper introduces two variations of the model. In the minimal version, a single Φ doublet with a VEV of ∼ 7 GeV is responsible for the mixing between the third-generation quarks with the first and second generations. The full CKM matrix can be accommodated in this minimal version. The quark mass matrices will not be symmetric, so the mixing matrices of right-handed and left-handed quarks will not be the same. Going to the mass basis of the quarks, this means FCNC currents both of vector form (i.e., s̄γµ bVµ or d̄γµ bVµ) and of axial form (i.e., s̄γµγ5 bVµ or d̄γµγ5 bVµ) can be obtained. On the axial FCNC couplings, there is a strong bound from the Bs0 − B̄s0 mixing which prevents a contribution to B+ → K+ ντ ν̄τ large enough to explain the excess reported by Belle II. In the second variation, two doublets, Φ and Φ′, are introduced with a Lagrangian symmetric under Φ ↔ Φ′, which predicts a symmetric mass matrix for the quarks. Then, in the mass basis, while the vector coupling of form b̄γµ sVµ is obtained, the axial coupling, b̄γµγ5 sVµ, disappears. As a result, a relatively large contribution to B+ → K+ ντ ν̄τ at the tree level can be obtained, explaining the reported excess. Similarly, a contribution of 10−5 to Br(B+ → K+ τ+ τ−) is expected, which is beyond the resolution of the present measurements. Φ and Φ′ can be both heavier than ∼ 500 GeV, escaping the present LHC bound, but in principle can be pair produced in the colliders via their electroweak interactions and promptly decay to two jets of third and second (or first) generations. A mechanism is introduced for Φ and Φ′ to obtain a small and equal VEV of ∼ 7 GeV, despite their large masses.

The paper concludes that within the left-right symmetric version of the model, there is ample possibility for explaining the B+ → K+ ν ν̄ excess and still obtaining the observed relic abundance, without violating the direct dark matter detection bounds. This typically requires 800 GeV < mV < 2 TeV with 0.2 < gV < 1, which may be within the discovery reach of the upcoming run of the LHC. The dark matter in this model is of Majorana type, so the annihilation will be p-wave and suppressed with the square of the velocity of the annihilating pair relative to each other. As a result, the annihilation cross section in the freeze-out epoch with vχ ∼ 0.3 will be far larger than that in the galaxy with vχ ∼ 10−3, so a conventional indirect dark matter signal is not expected. However, an indirect signal may come from dark matter spike around supermassive black hole in the center of Milky Way where the dark matter density and velocity can be very large. Within the model, it is expected that with a slightly larger exposure of the direct dark matter search experiment, the signal of a heavy dark matter with mass larger than ∼ 500 GeV could be discovered, but null results from the indirect dark matter searches in dwarf galaxies, in DM halo, and in galaxy clusters are expected. If the future measurements confirm these predictions and high luminosity LHC discovers a new gauge boson coupled to the third-generation fermions, it will still remain unknown whether the discovered dark matter is also charged under this new gauge symmetry or not, especially for mV < 2mχ for which V does not decay to a χ pair. A positive photon (neutrino) signal from the annihilation of a DM pair into tt̄, ττ̄, and bb̄ (into ντν̄τ) in the dark matter spike around the central supermassive black hole of the Milky Way will then be a strong hint in favor of the model for the dark matter.

Improvements to AI Systems Based on This Paper:

1. Anomaly-Aware Model Generation

• Improvement: Train an AI to automatically check for gauge anomaly cancellation (e.g., [U(1)]3, mixed anomalies) when proposing new physics models.

• Capability: The AI can generate self-consistent extensions of the Standard Model, flagging invalid parameter spaces (e.g., requiring chiral fermions like χ to cancel anomalies) before detailed phenomenology.

2. Automated Flavor Physics Constraint Integration

• Improvement: Incorporate a module that computes FCNC constraints (e.g., Bs0–B̄s0 mixing, B+→K+νν̄) from off-diagonal gauge couplings in the quark mass basis, including vector vs. axial coupling distinctions.

• Capability: The AI can predict whether a given new gauge boson mass/coupling is excluded by existing B-physics data, and identify which model variants (e.g., symmetric vs. asymmetric mass matrices) survive.

3. Thermal Dark Matter Relic Density Optimizer

• Improvement: Use the paper’s freeze-out framework (with p-wave annihilation, Majorana nature, and s-channel V exchange) to train a reinforcement-learning agent that scans (gV, mV, mχ) space.

• Capability: The AI can rapidly identify viable dark matter parameter regions that match observed relic abundance while satisfying direct detection (velocity-suppressed) and LHC bounds, reducing manual scanning time.

4. Collider Phenomenology Predictor

• Improvement: Train a classifier on LHC search limits (e.g., for Z′/V bosons decaying to third-generation fermions) to predict discovery potential for new gauge bosons in the 0.8–2 TeV mass range.

• Capability: The AI can estimate signal significance for upcoming LHC runs, given the model’s couplings and branching ratios (including invisible decays to χ pairs when kinematically allowed).

5. Indirect Detection Signal Discriminator

• Improvement: Implement a physics-informed neural network that accounts for p-wave suppression in galactic halos but enhanced annihilation in dark matter spikes (e.g., near SMBH).

• Capability: The AI can predict photon/neutrino fluxes from DM annihilation into tt̄, ττ̄, bb̄, ντν̄τ in high-velocity environments, distinguishing this model from velocity-independent (s-wave) dark matter.

6. Symmetry-Based Model Variant Selector

• Improvement: Use the paper’s Φ↔Φ′ symmetry mechanism to train an AI that automatically imposes discrete symmetries to eliminate unwanted axial FCNC couplings while preserving vector ones.

• Capability: The AI can propose minimal Higgs sectors that satisfy both flavor bounds and explain anomalies like the Belle II B+→K+νν̄ excess, without manual trial-and-error.

7. Uncertainty-Aware Constraint Propagation

• Improvement: Build a Bayesian inference tool that propagates experimental uncertainties (e.g., Belle II 2.7σ excess, direct detection limits) into posterior distributions for model parameters (gV, mV, mχ, VEVs).

• Capability: The AI can output probability maps for model viability, including correlations between dark matter mass, gauge coupling, and Higgs VEVs, aiding experimental design.

8. Cross-Dataset Consistency Checker

• Improvement: Train an AI to cross-validate predictions across disparate datasets: electroweak precision (S/T), Υ decay, CKM unitarity, Bs mixing, LHC dilepton/dijet searches, and DM direct detection.

• Capability: The AI can automatically flag parameter sets that satisfy one constraint but violate another, ensuring global consistency—critical for proposing realistic new physics.

9. Automated Paper-to-Code Translation

• Improvement: Use the paper’s explicit Lagrangian and anomaly conditions to train a code-generation model that outputs ready-to-run micrOMEGAs7 or MadGraph scripts.

• Capability: The AI can instantly produce simulation files for relic density, direct detection, and collider event generation from textual model descriptions, reducing manual implementation errors.

10. Predictive Benchmark Scenario Generator

• Improvement: Train a generative model on the paper’s viable parameter space (800 GeV < mV < 2 TeV, 0.2 < gV < 1) to produce benchmark points for future experimental searches.

• Capability: The AI can output specific mass/coupling combinations that maximize discovery potential at HL-LHC or next-gen direct detection, guiding experimental prioritization.