⚛️ phenomenology

Light-cone sum rules with $B$ -meson distribution amplitudes for the $B\to p$ form factors in $B$ -mesogenesis models

This paper calculates $B^+\to p$ transition form factors using QCD light-cone sum rules with $B$ -meson distribution amplitudes to derive lower limits on the $B^+\to p \Psi$ branching fraction required for $B$ -mesogenesis, concluding that experimental sensitivity must reach the $10^{-8}$ to $10^{-7}$ level to decisively probe this invisible decay mode.

Original authors: Aritra Biswas, Alexander Khodjamirian, Ali Mohamed

Published 2026-03-03

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CC BY 4.0

Original authors: Aritra Biswas, Alexander Khodjamirian, Ali Mohamed

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, bustling city. For a long time, physicists have been puzzled by two major mysteries in this city:

The Missing People: Why is there so much more matter (us, stars, planets) than antimatter (the "anti-city")?
The Invisible Neighbors: What is "Dark Matter"? We know it's there because of its gravity, but we can't see it or touch it.

A theory called B-mesogenesis suggests a clever solution: Maybe the "missing people" and the "invisible neighbors" are actually the same thing. It proposes that a specific type of particle, the B-meson (a heavy, unstable particle that lives for a tiny fraction of a second), can decay into two things at once:

A normal proton (a building block of atoms).
A "Dark Antibaryon" (let's call it Ψ). This Ψ is the invisible dark matter particle, and it's the "anti" version of the missing matter.

If this happens, the B-meson disappears, leaving behind a visible proton and a ghostly Ψ that flies away undetected. This would explain both mysteries in one go!

The Problem: The "Black Box"

To prove this theory, scientists need to catch a B-meson decaying into a proton and "missing energy" (the invisible Ψ). Experiments at places like Belle II and BaBar are looking for exactly this.

However, there's a catch. To know if the experiments are looking in the right place, physicists need to calculate how likely this decay is to happen. This probability depends on a complex mathematical object called a Form Factor.

Think of the Form Factor as the "strength of the handshake" between the B-meson and the proton. If the handshake is weak, the decay rarely happens. If it's strong, it happens often. To calculate this, you need to understand the internal structure of these particles, which is governed by the rules of Quantum Chromodynamics (QCD)—the physics of quarks and gluons.

The Old Way vs. The New Way

Previously, scientists tried to calculate this "handshake strength" using a method that looked at the proton's internal structure (like looking at the proton's blueprint). They used something called "Nucleon Distribution Amplitudes."

But this method had a problem: the blueprint was messy. The calculations were unstable, and the results varied wildly depending on how you tweaked the numbers. It was like trying to measure the strength of a handshake by only looking at one person's hand while the other person was a blur.

This paper introduces a new, cleaner approach.
Instead of looking at the proton's blueprint, the authors decided to look at the B-meson's blueprint (its "Distribution Amplitudes").

The Analogy:
Imagine you want to know how well a specific car (the B-meson) fits into a specific parking spot (the proton).

The Old Method: You tried to measure the parking spot's dimensions while the car was moving at high speed. It was hard to get a clear picture.
The New Method (This Paper): You measure the car's dimensions very precisely while it's parked, and then use that to predict how it fits into the spot.

What Did They Do?

The authors, Aritra Biswas, Alexander Khodjamirian, and Ali Mohamed, used a sophisticated mathematical tool called Light-Cone Sum Rules (LCSRs).

They treated the B-meson as the "main character" and the proton as the "interloper."
They calculated the interaction up to a very high level of detail (called "twist-5"), which is like checking the car not just for its size, but for its suspension, engine, and tire pressure.
They found that their new method was much more stable and reliable than the old one. In fact, they discovered that in one version of the theory, the "handshake" is exactly twice as strong as in the other version.

The Results: The "Goldilocks" Zone

They calculated the probability of this decay happening for different masses of the invisible particle (Ψ).

The Good News: Their calculations show that this decay is possible and could happen frequently enough to be detected.
The Bad News: The current experiments (BaBar and Belle) haven't seen it yet, but they haven't looked hard enough. Their "sensitivity" is like trying to hear a whisper in a noisy stadium.
The Prediction: The authors say that to definitely prove or disprove this theory, experiments need to improve their sensitivity by 10 to 100 times. They need to be able to spot a branching fraction (probability) as low as 1 in 100 million (10⁻⁸).

Why This Matters

This paper is a crucial map for the explorers.

It gives a reliable target: Instead of guessing where to look, experimentalists now know exactly what signal to hunt for.
It rules out the "messy" math: By showing that the old method was unstable, they prevent scientists from wasting time on calculations that might be wrong.
It sets the goal: It tells the experimental community, "You need to get your detectors 10 times better to solve the mystery of Dark Matter and the Matter-Antimatter imbalance."

In summary: The authors built a better, more stable ruler to measure a tiny, invisible interaction. They found that the interaction is real and measurable, but we need much sharper eyes (better experiments) to actually see it. If we do, we might finally solve two of the biggest riddles in the universe.

1. Problem Statement and Motivation

The paper addresses the theoretical challenge of calculating the branching fraction for the rare decay $B^+ \to p \Psi$ , where $\Psi$ is a dark antibaryon. This decay is a signature of the B-mesogenesis scenario, a theoretical framework proposed to simultaneously explain the baryon asymmetry of the universe and the abundance of dark matter.

The Gap: Previous theoretical estimates for the $B \to p$ $B \to p$ transition form factors relied on Light-Cone Sum Rules (LCSRs) using nucleon distribution amplitudes (DAs). However, this approach suffered from:
- Slow convergence of the twist expansion (large higher-twist contributions).
- Significant uncertainties in the parameters of nucleon DAs.
- A lack of independent verification.
The Goal: To provide an independent, more robust estimate of the $B \to p$ form factors using a different LCSR approach based on B-meson distribution amplitudes, and to use these results to constrain the B-mesogenesis scenario against experimental data from BaBar and Belle/Belle II.

2. Methodology

The authors employ QCD Light-Cone Sum Rules (LCSRs) but invert the standard roles of the initial and final states compared to previous nucleon-DA approaches.

A. Correlation Function and Setup

Correlation Function: They define a vacuum-to- $B$ correlation function involving the time-ordered product of the proton interpolating current ( $\eta_p$ , chosen as the Ioffe current) and the effective three-quark operator ( $O^{(d)}$ or $O^{(b)}$ ) responsible for the decay.
$F(P, q) = i \int d^4x e^{iP\cdot x} \langle 0 | T \{ \eta_p(x), O(0) \} | B^+(P+q) \rangle$
OPE Expansion: The Operator Product Expansion (OPE) is performed in terms of B-meson DAs (defined in Heavy Quark Effective Theory, HQET) up to twist-5 accuracy.
Models Considered: Two specific models of the $\Psi$ $Ψ$ -triquark interaction are analyzed:
- Model (d): The $d$ -quark couples directly to $\Psi$ and a heavy mediator.
- Model (b): The $b$ -quark couples directly to $\Psi$ and a heavy mediator.

B. Key Theoretical Findings in OPE

Diquark Loop: Unlike standard $B \to$ meson transitions, the leading-order diagram contains a diquark loop, generating logarithmic terms in the OPE.
Vanishing Three-Particle Contributions: A crucial result is that the contributions from three-particle DAs (quark-antiquark-gluon) vanish for both models (d) and (b). This is attributed to the specific Dirac structure of the Ioffe current and the scalar nature of the effective operators, combined with the massless $u,d$ quark limit.
Twist Hierarchy: The twist expansion shows a reasonable hierarchy where higher-twist (twist-4, 5) contributions are suppressed relative to twist-2 and twist-3, unlike the nucleon-DA approach where higher twists were dominant.
Model Relation: The form factors for Model (b) are found to be exactly twice those of Model (d) ( $F^{(b)} = 2F^{(d)}$ ).

C. Sum Rule Derivation

Dispersion Relation: The correlation function is matched to a dispersion relation in the variable $P^2$ (momentum of the proton).
Hadronic Side: Includes the proton pole, the $N^*(1535)$ resonance, and a continuum approximated by quark-hadron duality. The authors adopt a simplified treatment where excited nucleon resonances (including $N(1440)$ ) are absorbed into the continuum spectral density, following methods used for nucleon electromagnetic form factors.
Borel Transform: Applied to suppress higher-state contributions and eliminate subtraction terms.
Extraction: The $B \to p$ form factors ( $F_{B\to p}^R$ and $\tilde{F}_{B\to p}^L$ ) are extracted from the sum rules at spacelike momentum transfer ( $q^2 < 0$ ).

D. Extrapolation to Physical Region

Since LCSRs are valid only for $q^2 < 0$ , the results are extrapolated to the physical timelike region ( $q^2 = m_\Psi^2$ ) using a $z$ -expansion (BCL parametrization).
The analytic structure is handled by mapping the $q^2$ plane to the $z$ -plane, removing the $\Lambda_b$ pole, and fitting the LCSR data points to a polynomial in $z$ .

3. Key Results

A. Form Factors

The authors calculated the $B \to p$ form factors at $q^2=0$ and their slopes.
Comparison with Nucleon-DA LCSR (LCSR-N):
- The new LCSR-B (B-meson DA) results show a smaller central value and smaller uncertainties compared to the previous LCSR-N results.
- LCSR-N exhibited large, dominant higher-twist contributions and sign differences between models (d) and (b), which are absent in the more stable LCSR-B results.
- The LCSR-B results satisfy a clear twist hierarchy, suggesting better convergence.

B. Branching Fractions

Using the extrapolated form factors, the partial width for $B^+ \to p \Psi$ was calculated as a function of the dark antibaryon mass $m_\Psi$ .
Benchmark Values (at $m_\Psi = 2$ GeV):
- Model (d): $\text{BR} \approx 2.9 \times 10^{-6} \times (|G^{(d)}|^2 / 10^{-13} \text{ GeV}^{-4})$
- Model (b): $\text{BR} \approx 0.54 \times 10^{-6} \times (|G^{(b)}|^2 / 4.7 \times 10^{-15} \text{ GeV}^{-4})$
- Note: The couplings $G$ are set to their upper limits derived from LHC searches.

C. Exclusive-to-Inclusive Ratio and Limits

To eliminate dependence on the unknown effective coupling $G$ , the authors calculated the ratio:
$R = \frac{\text{BR}(B^+ \to p \Psi)}{\text{BR}(B^+ \to X_N \Psi)}$
where the inclusive width $\text{BR}(B^+ \to X_N \Psi)$ was taken from Heavy Quark Expansion (HQE) calculations.
Lower Limits: By combining this ratio with the theoretical requirement that the inclusive invisible branching fraction must be at least $\mathcal{O}(10^{-4})$ $O (1 0^{- 4})$ for B-mesogenesis to work, the authors derived lower limits on the exclusive branching fraction $\text{BR}(B^+ \to p \Psi)$ $BR (B^{+} \to p Ψ)$ .
- Model (d) lower limits reach $\sim 10^{-8}$ .
- Model (b) lower limits reach $\sim 10^{-7}$ .

4. Significance and Conclusion

Independent Verification: This work provides the first independent LCSR calculation of $B \to p$ form factors using B-meson DAs, confirming that the previous nucleon-DA results were likely overestimated due to poor convergence.
Experimental Constraints: The authors compare their predicted lower limits with the current experimental upper bounds from BaBar and Belle/Belle II (which are around $10^{-6}$ $1 0^{- 6}$ ).
- Result: There is currently a large gap between the theoretical lower limits ( $10^{-8} - 10^{-7}$ ) and experimental upper limits ( $10^{-6}$ ).
- Implication: To decisively probe or rule out the B-mesogenesis scenario via the $B^+ \to p \Psi$ channel, experimental sensitivity must improve by 1 to 2 orders of magnitude (reaching $10^{-8}$ to $10^{-7}$ ).
Future Applicability: The methodology is robust and can be easily extended to other baryonic modes (e.g., $B \to \Lambda \Psi$ , $B \to \Lambda_c \Psi$ ) by changing the interpolating current, making it a valuable tool for future searches for invisible B decays.

In summary, the paper refines the theoretical prediction for a key B-mesogenesis signature, demonstrating that while the scenario is not yet ruled out, significantly more sensitive experiments are required to test it.