Branching ratios and CP violations of $B \toρ(ω) γ$ decays in the modified perturbative QCD approach and the relevant dark photonic decays of $B\to ρ(ω) γ^\prime$

Abstract: In this work we study the radiative decays of $B\to ρ(ω)γ$ processes in the modified perturbative QCD approach, where the transverse momenta of the quark and gluons envolved in the interaction process are kept, and the Sudakov factor is incorporated in the theoretical calculation, which are helpful to suppress the infrared end-point contribution. A critical infrared momentum cutoff scale is introduced, which is used to separate the soft and hard contributions in QCD. Then the form factors envolved in the radiative decays should be separated as hard and soft form factors. The hard form factors can be calculated with the perturbative QCD approach, while the soft form factors should be viewed as soft input parameters. The soft form factors can be obtained by confronting the theoretical calculation of the branching ratios of the radiative decay of $B$ meson with the experimental data. Summing the soft and hard form factors, we find that the total form factors are in fine agreement with that obtained by nonperturbative theoretical method, such as the light-cone sum rules of QCD. We also study $B\to ρ(ω) γ<sup>\prime$ decays by using the form factors obtained in the $B\to ρ(ω)γ$ decay processes, where $γ<sup>\prime$ is the dark photon introduced by the extra $U(1)$ symmetry model. The upper limit of the branching ratios of thes decays are estimated.

• The paper introduces a modified PQCD framework that systematically separates hard and soft contributions using a critical infrared cutoff.
• It computes B→ρ(ω)γ branching ratios and CP asymmetries with leading and next-to-leading order corrections, closely matching experimental data.
• The study sets upper limits on dark photon decay modes, offering essential constraints for new physics searches in flavor experiments.

Branching Ratios and CP Violations of $B \to \rho(\omega) \gamma$ Decays in Modified PQCD and Implications for Dark Photon Searches

Overview

This paper presents an in-depth analysis of the exclusive radiative decays $B \to \rho(\omega) \gamma$ within a modified perturbative QCD (PQCD) framework, incorporating both perturbative and nonperturbative contributions. The approach introduces a transverse-momentum-dependent (TMD) factorization scheme, implements Sudakov suppression, and critically separates the transition form factors into hard (perturbative) and soft (nonperturbative) components using an infrared cutoff. The study further leverages these form factor results to set upper bounds on new physics decay modes involving a hypothetical massless dark photon ($\gamma'$) as predicted by $\mathrm{U(1)}$ extensions to the Standard Model (SM).

Theoretical Framework and Methodology

Modified PQCD Factorization

The $B \to \rho(\omega)\gamma$ decays are driven by the flavor-changing neutral current (FCNC) transition $b\to d\gamma$, a loop-induced process highly sensitive to both SM and potential new physics. The authors employ a modified PQCD approach that retains the transverse momenta ($k_T$) of all partons throughout the calculation, resulting in a TMD factorization formula. Soft-gluon effects and endpoint singularities are suppressed by including the Sudakov factor and threshold resummation, as developed systematically in the literature.

A novel aspect is the explicit introduction of a critical infrared cutoff scale $\mu_c \sim 1\,\text{GeV}$, which demarcates the separation between hard and soft contributions. For scales $\mu > \mu_c$, amplitudes are calculated perturbatively. For $\mu < \mu_c$, soft transition form factors, determined phenomenologically, substitute for nonperturbative QCD effects.

Meson Wavefunctions and Input Parameters

The $B \to \rho(\omega) \gamma$0 meson wavefunction is modeled via solutions to the Bethe-Salpeter equation in a QCD-inspired relativistic potential, parameterized analytically and tested for consistency with established sum rule results. Vector meson ($B \to \rho(\omega) \gamma$1, $B \to \rho(\omega) \gamma$2) distribution amplitudes are expanded in terms of Gegenbauer polynomials, capturing leading and subleading twist contributions.

Amplitude Contributions

The calculation systematically includes:

• Leading Order (LO): Dominated by electromagnetic penguin ($B \to \rho(\omega) \gamma$3) and chromomagnetic ($B \to \rho(\omega) \gamma$4) operators, with hard gluon exchange computed in $B \to \rho(\omega) \gamma$5 factorization.
• Next-to-Leading Order (NLO): Incorporates diagrams with insertions of $B \to \rho(\omega) \gamma$6, $B \to \rho(\omega) \gamma$7 (tree-level four-quark operators) inside quark loops, as well as emission of photons from internal and external quark lines, and both tree-level and QCD penguin annihilation topologies.
• Soft Form Factor Contributions: Soft, nonperturbative pieces of the transition form factor (specifically $B \to \rho(\omega) \gamma$8) are introduced as free parameters and fitted to experimental results.

Numerical Results

Branching Ratios and Direct CP Asymmetries

Utilizing world-average experimental data, the paper fits the soft contributions $B \to \rho(\omega) \gamma$9 and $\gamma'$0 and rigorously computes amplitudes including LO, NLO, and soft terms. The resulting total form factors and predicted observables are:

Decay Mode	Theory ($\gamma'$1)	Experiment
$\gamma'$2	$\gamma'$3	$\gamma'$4
$\gamma'$5	$\gamma'$6	$\gamma'$7
$\gamma'$8	$\gamma'$9	$\mathrm{U(1)}$0
$\mathrm{U(1)}$1	$\mathrm{U(1)}$2	$\mathrm{U(1)}$3

The calculated form factors $\mathrm{U(1)}$4 and $\mathrm{U(1)}$5 show good consistency with contemporary LCSR results (e.g., $\mathrm{U(1)}$6).

Key findings include:

• The $\mathrm{U(1)}$7 operator contribution is overwhelmingly dominant.
• NLO and annihilation topologies yield modest, non-negligible corrections, confirming the stability of the perturbative expansion.
• The soft form factors are essential for reconciling both branching ratios and direct CP violations with experiment, especially for neutral modes.
• The modified PQCD framework with a hard-soft separation predicts observables in strong agreement with data, supporting its efficacy.

Dark Photon Searches in $\mathrm{U(1)}$8 Decays

Employing the extracted $\mathrm{U(1)}$9 form factors, the study investigates $B \to \rho(\omega)\gamma$0, where $B \to \rho(\omega)\gamma$1 denotes a hypothetical massless dark photon coupled via higher-dimension tensor operators. By inserting model-independent constraints on new physics couplings, the paper derives conservative upper limits:

Decay Mode	$B \to \rho(\omega)\gamma$2 (Upper Limit)
$B \to \rho(\omega)\gamma$3	$B \to \rho(\omega)\gamma$4
$B \to \rho(\omega)\gamma$5	$B \to \rho(\omega)\gamma$6
$B \to \rho(\omega)\gamma$7	$B \to \rho(\omega)\gamma$8

These are positioned near the current sensitivity of flavor factories such as Belle II and LHCb. The results provide both a theoretical and phenomenological baseline for experimental searches for dark sector physics in rare $B \to \rho(\omega)\gamma$9 decays.

Implications and Future Directions

The separation of hard and soft contributions via an explicit infrared scale in PQCD enables systematic control of uncertainties and more reliable confrontation with experiment, especially for exclusive hadronic processes sensitive to nonperturbative physics. The demonstrated agreement between total PQCD+soft form factors and LCSR computations underscores the utility of the method.

From a phenomenological standpoint, the results constrain possible new physics in radiative $b\to d\gamma$0 decays and establish the precision required for future dark photon searches. As flavor experiments continue to improve statistical reach and systematic control, the combined theoretical predictions here will serve as a critical reference for any potential anomalies.

Potential future developments include:

• Extension to other exclusive radiative or semi-leptonic decays involving vector mesons or potential dark sector signatures.
• Systematic evaluation of the scheme-dependence associated with the hard/soft separation and sensitivity to the infrared cutoff.
• Exploration of similar methodologies in other FCNC processes across the $b\to d\gamma$1, $b\to d\gamma$2, and $b\to d\gamma$3 meson sectors.

Conclusion

This work provides a comprehensive and quantitatively robust study of radiative $b\to d\gamma$4 decays in a modified PQCD approach that incorporates both perturbative and phenomenologically constrained soft contributions. The predictions match experimental measurements closely and supply critical input for ongoing and future searches for dark sector phenomena via rare flavor-changing decays. The methodology and numerical results represent a solid theoretical foundation for the interpretation of high-precision flavor physics experiments and the exploration of physics beyond the Standard Model.