Mass of the dark antibaryon using $B_d\rightarrow \Lambda \psi_{DS}$ channel in light cone QCD

This paper utilizes Light Cone Sum Rules with twist-6 contributions to calculate the branching fraction of the $B_d\rightarrow\Lambda \psi_{DS}$ decay, thereby determining the allowed mass range for the dark antibaryon $\psi_{DS}$ consistent with $B$-mesogenesis constraints from BaBar and Belle experimental data.

The Gist

The Big Picture: Solving Two Mysteries at Once

Imagine the universe as a giant party that started with a perfect balance of "matter" (the stuff we are made of) and "antimatter" (its evil twin). According to the laws of physics, they should have destroyed each other instantly, leaving nothing but empty space. But here we are, so clearly, something went wrong with the balance. There is too much matter and not enough antimatter. This is the first mystery: Why are we here?

The second mystery is Dark Matter. We know it's there because of its gravity, but we can't see it, touch it, or detect it with normal tools. It's like a ghost in the room that pushes furniture around but never shows its face.

This paper proposes a clever theory called "B-mesogenesis." It suggests that these two mysteries might be solved by the same event. Imagine a specific type of particle called a B-meson (a heavy, unstable particle that exists for a split second) as a "magic coin." When this coin flips and decays, it doesn't just break into normal pieces. Instead, it splits into two things:

1. A normal particle we can see (a Lambda baryon, which is a type of heavy proton).
2. A "dark" particle we can't see (a dark antibaryon, which the authors call $\psi_{DS}$).

The theory says that every time this happens, it creates a tiny bit more matter than antimatter (solving the first mystery) and creates a piece of dark matter (solving the second).

The Detective Work: Weighing the Invisible Ghost

The authors of this paper are theoretical detectives. They want to know: How heavy is this invisible dark particle ($\psi_{DS}$)?

If the dark particle is too heavy, the "magic coin" (the B-meson) doesn't have enough energy to break apart into it. If it's too light, the math doesn't add up with what we see in experiments. The goal is to find the "Goldilocks zone"—the specific weight range where this dark particle could exist without breaking the laws of physics or contradicting what scientists have already measured.

The Tool: The Light-Cone Sum Rule (LCSR)

To figure this out, the authors use a mathematical tool called Light-Cone Sum Rules (LCSR).

• The Analogy: Imagine you are trying to guess the weight of a sealed box by shaking it and listening to the sound it makes. You can't open the box (because the dark particle is invisible), but you know the laws of physics (the "sound" of the shaking).
• The Method: The authors build a complex mathematical model that connects the known properties of the B-meson and the Lambda baryon to the unknown properties of the dark particle. They use something called Distribution Amplitudes, which are like a detailed "blueprint" of how the quarks (the tiny building blocks) are arranged inside the Lambda particle. They didn't just look at the basic blueprint; they looked at the fine details (up to "twist-6"), which is like checking the wiring, the insulation, and the screws, not just the outer shell.

The Two Scenarios: The "s" and "b" Models

The paper looks at two different ways this "magic coin" could flip, which they call the (s)-model and the (b)-model.

• Think of these as two different recipes for the same cake.
• The authors calculated the "branching fraction" for both. This is a fancy way of saying: "Out of every 100,000 times a B-meson decays, how many times does it turn into a Lambda and a dark particle?"

The Results: Narrowing the Search

The authors compared their calculations against real-world data from two giant particle detectors, BaBar and Belle. These detectors have been watching B-mesons for years and have set "speed limits" (upper limits) on how often this specific decay can happen. If the dark particle were a certain weight, the detectors would have seen it by now. Since they haven't, those weights are ruled out.

Here is what they found:

1. The "b" Model (Recipe B): This version predicts that the decay happens so rarely that it is far below what the detectors can see. It's like trying to hear a whisper in a hurricane. Because the signal is so weak, this model doesn't give us any useful clues about the dark particle's weight. It's essentially a "no-go" zone for finding answers right now.

2. The "s" Model (Recipe S): This is the interesting one. The math shows that if the dark particle exists, it must be in one of two specific weight ranges to avoid being detected by the current experiments:

   • Window 1 (Lighter): Between 1.0 and 2.8 GeV.
   • Window 2 (Heavier): Between 3.6 and 4.1 GeV.

   However, the data from the Belle experiment is very strict. It cuts out almost everything except the very top of the heavy range.

   • The Final Verdict: If this theory is true, the dark particle must be extremely heavy, weighing between 4.108 and 4.164 GeV.

Why This Matters

The paper concludes that the decay of a B-meson into a Lambda and a dark particle is a very sensitive "smoke detector" for this specific theory. If future experiments (like those at the LHC or future B-factories) look in this specific heavy weight range and find nothing, this whole "B-mesogenesis" idea might be wrong. If they do find a particle there, it would be a massive breakthrough, explaining why the universe is full of matter and where all the dark matter is hiding.

In short: The authors used advanced math to predict that if a specific theory about the origin of the universe is correct, a mysterious dark particle must be hiding in a very narrow, heavy weight range, waiting to be found by future experiments.

Technical Summary

Technical Summary: Mass of the dark antibaryon using $B_d \to \Lambda \psi_{DS}$ channel in light cone QCD

Problem Statement
The Standard Model (SM) fails to explain the baryon asymmetry of the universe (BAU) and the nature of dark matter (DM). The "B-mesogenesis" framework proposes a solution where CP-violating $B$-meson oscillations simultaneously generate the BAU and a dark-sector antibaryon ($\psi_{DS}$). In this scenario, $B$-mesons decay into a visible SM baryon and a dark antibaryon. While theoretical efforts have investigated inclusive decays and other exclusive channels (e.g., $B^+ \to p \psi_{DS}$), the specific exclusive decay channel $B_d \to \Lambda \psi_{DS}$ remains a critical probe. The primary challenge addressed in this work is determining the allowed mass window for the dark antibaryon $\psi_{DS}$ by calculating the branching fraction of this decay and comparing it against existing experimental upper limits from the BaBar and Belle collaborations.

Methodology
The authors employ the Light Cone Sum Rule (LCSR) approach to calculate the transition form factors for the $B_d \to \Lambda$ decay. The methodology involves the following steps:

1. Effective Interaction: The decay is mediated by a heavy colored scalar field (mass $\sim$ TeV) integrated out to form effective four-fermion operators. Two distinct models are considered based on which quark couples to the dark state: the $(s)$-model (where the $s$-quark couples to $\psi$) and the $(b)$-model (where the $b$-quark couples to $\psi$).
2. Correlation Function: A two-point correlation function is constructed involving the $B_d$-meson interpolating current and the effective three-quark operator.
3. QCD Side (OPE): The correlation function is evaluated in the deep Euclidean region using the Operator Product Expansion (OPE) near the light cone. Crucially, the authors include non-perturbative effects by employing $\Lambda$-baryon distribution amplitudes (DAs) up to twist-6. The calculation accounts for $SU(3)_f$ symmetry-breaking effects specific to the $\Lambda$ baryon and isospin-symmetry breaking corrections.
4. Physical Side: The physical representation isolates the ground state contribution using the $B_d$ decay constant and the transition form factors.
5. Sum Rules and Extrapolation: Borel transformation and continuum subtraction are applied to both sides to derive sum rules for the form factors. Since the LCSR is valid only for spacelike $q^2$, the authors use the $z$-expansion (BCL version) to extrapolate the form factors from the calculated region ($-10 \le q^2 \le 10$ GeV$^2$) to the full physical timelike region required for the decay.
6. Branching Fraction Calculation: The decay width is computed using the derived form factors and compared with experimental upper limits to constrain the dark antibaryon mass ($m_\psi$).

Key Contributions and Results

• Form Factor Calculation: The study provides the first calculation of the $B_d \to \Lambda$ transition form factors within the B-mesogenesis framework using $\Lambda$ DAs up to twist-6. It is found that for both models, only two form factors ($F_R$ and $\tilde{F}_L$) contribute non-zero values to the decay amplitude; the others vanish due to the specific operator structures.
• Model Dependence:
  • $(b)$-model: The theoretical branching fraction is found to be several orders of magnitude below the experimental upper limits across the entire kinematically allowed mass range. Consequently, this model imposes no constraints on the dark antibaryon mass.
  • $(s)$-model: This model yields branching fractions comparable to experimental sensitivity.
• Mass Constraints: By comparing the theoretical predictions for the $(s)$-model with experimental limits, the authors identify specific allowed mass windows for $\psi_{DS}$:
  • Using the BaBar limit ($B < 5.2 \times 10^{-5}$ for $1.0 < m_\psi < 4.1$ GeV), the mass region $2.817 < m_\psi < 3.624$ GeV is excluded. This leaves two allowed windows:
    • Window I (Low Mass): $1.000 \le m_\psi \le 2.817$ GeV.
    • Window II (High Mass): $3.624 \le m_\psi \le 4.164$ GeV.
  • Using the stricter BaBar limit ($B < 0.13 \times 10^{-5}$), only the kinematic endpoint $m_\psi = 4.164$ GeV remains viable.
  • Using the Belle limit ($B < 2.1 \times 10^{-5}$ for $1.0 < m_\psi < 3.9$ GeV), the allowed region is further constrained to $4.108 \le m_\psi \le 4.164$ GeV. This region lies just above the mass interval explored by Belle in their search.

Significance and Claims
The paper claims that the exclusive decay $B_d \to \Lambda \psi_{DS}$ serves as a sensitive probe for B-mesogenesis scenarios involving dark-sector antibaryons. The study demonstrates that precision measurements of this channel can effectively constrain the mass of the dark antibaryon, particularly in the $(s)$-model scenario. The authors highlight that the decay chain $B_d \to \Lambda \psi \to p \pi \psi$ offers an experimentally favorable signature due to the reconstructable $\Lambda \to p \pi$ decay and the relatively stringent upper bounds on its branching fraction compared to other channels.

The authors conclude that while their analysis restricts the kinematically allowed mass range, a more precise comparison between theory and experiment requires higher accuracy in the calculation of $\Lambda$ distribution amplitudes and more precise experimental limits. The work establishes a theoretical baseline for future experimental searches at $B$ factories.