Are Subleading Effects Really Subleading? $B$-Meson Decays in Mesogenesis

Technical Summary: "Are Subleading Effects Really Subleading? B-Meson Decays in Mesogenesis"

Problem Statement
The B-Mesogenesis framework proposes a mechanism to simultaneously explain the baryon asymmetry of the universe and the nature of dark matter through new decay channels of B-mesons into visible baryons and a dark-sector antibaryon ($\psi$). While previous studies have analyzed the impact of these new channels on B-meson lifetimes using the Heavy Quark Expansion (HQE), they largely relied on the leading-order dimension-three term (free $b$-quark decay).

The central problem addressed in this work is the validity of the HQE hierarchy in this Beyond the Standard Model (BSM) scenario. In the Standard Model (SM), subleading power-suppressed corrections (dimension-five and dimension-six terms) are typically small. However, in B-Mesogenesis, the kinematics involve a dark fermion $\psi$ with a mass ($m_\psi$) that can approach the $b$-quark mass ($m_b$). The authors investigate whether, in this regime, the "subleading" corrections (specifically the chromomagnetic, Darwin, and four-quark spectator terms) remain negligible or if they become dominant, potentially invalidating the standard HQE truncation.

Methodology
The authors perform a systematic calculation of inclusive $B$-meson decay rates within the Mesogenesis framework using the HQE up to dimension-six operators.

1. Theoretical Framework:

   • The calculation utilizes the optical theorem, expressing the total decay width $\Gamma(B)$ as the imaginary part of the forward scattering amplitude of the effective Hamiltonian $H_{\text{eff}} = H_{\text{SM}} + H_{-1/3}$.
   • The non-local operator is expanded in local $\Delta B = 0$ operators ordered by dimension. The authors explicitly compute the contributions of the dimension-five chromomagnetic operator ($\mu_G^2$) and the dimension-six two-quark Darwin operator ($\rho_D^3$), as well as the dimension-six four-quark spectator operators ($\tilde{\Gamma}_6$).
   • The calculation involves expanding the quark propagator in a soft background gluon field (Fock-Schwinger gauge) to derive the necessary loop integrals.
   • Infrared divergences arising from light quark masses are regulated and shown to cancel against the renormalized matrix elements of the four-quark operators, consistent with SM cancellation mechanisms.

2. Operator Analysis:

   • The study focuses on the effective operators derived from the Lagrangian $L_{-1/3}$, specifically the $\psi$-type operators proportional to Wilson coefficients $C_{\psi}^{(ud)}$ and $\tilde{C}_{\psi}^{(ud)}$.
   • The analysis is performed for $B^+$, $B_d$, and $B_s$ mesons, with a primary focus on $B^+$ to derive constraints on exclusive decays.

3. Numerical and Phenomenological Analysis:

   • The authors compute the individual contributions of $\Gamma_3$ (leading), $\Gamma_5$, $\Gamma_6$ (Darwin), and $\tilde{\Gamma}_6$ (four-quark) as functions of the mass ratio $\rho_f = (m_f/m_b)^2$.
   • They compare the ratio of subleading to leading terms ($R_d = |\Gamma_d/\Gamma_3|$) to identify regions where the hierarchy breaks down ($R_d > 1$).
   • The study extends to lifetime ratios $\tau(B_s)/\tau(B_d)$ and $\tau(B^+)/\tau(B_d)$, incorporating $SU(3)_F$ breaking effects.
   • Finally, the updated inclusive width is used to update lower bounds on the exclusive decay $B^+ \to p^+ \psi$ via the ratio $\Gamma_{\text{excl}}/\Gamma_{\text{incl}}$, utilizing Light-Cone Sum Rules (LCSR) form factors.

Key Contributions

• Systematic Calculation of Subleading Terms: This is the first systematic computation of dimension-five and dimension-six two-quark Darwin terms specifically for the Mesogenesis operators, moving beyond the leading-order approximation used in previous works.
• Identification of Hierarchy Breakdown: The authors identify specific regions in the parameter space (specifically for $m_\psi \gtrsim 2\text{--}3$ GeV) where subleading terms exceed the leading dimension-three contribution.
• Operator-Dependent Behavior: The study reveals that the breakdown of the HQE hierarchy is highly operator-dependent. For certain topologies (e.g., Pauli Interference topologies generated by operators like $(b u^c)(\psi d^c)$), the four-quark contributions become large and negative, causing the total width to potentially turn negative at higher $m_\psi$, signaling a failure of the perturbative expansion.
• Updated Constraints: By incorporating the full HQE-corrected inclusive width, the paper provides updated lower bounds on the exclusive branching fraction $B^+ \to p^+ \psi$.

Results

• Breakdown of HQE: For operators involving specific field configurations (resembling Pauli Interference), the dimension-six four-quark contribution ($\tilde{\Gamma}_6$) dominates the Darwin term ($\Gamma_6$), which in turn dominates the dimension-five term ($\Gamma_5$). Crucially, for $m_\psi \gtrsim 2$ GeV, $\tilde{\Gamma}_6$ can exceed $\Gamma_3$. For $m_\psi \gtrsim 3$ GeV, the Darwin term also exceeds the leading term.
• Negative Widths: In the region where subleading terms dominate, the sum of contributions for certain operators yields a negative total decay width. This indicates that the HQE is no longer a reliable tool for predicting inclusive rates in this specific kinematic regime.
• Lifetime Ratios:
  • The ratio $\tau(B^+)/\tau(B_d)$ remains insensitive to the new two-quark operators due to isospin symmetry, consistent with SM expectations.
  • The ratio $\tau(B_s)/\tau(B_d)$ is sensitive to $SU(3)_F$ breaking effects from the new operators. However, the analysis finds no additional constraints on the couplings beyond existing collider bounds, consistent with analogous results for the $B^+/B_d$ ratio.
• Exclusive Bounds: The inclusion of the updated inclusive width (which can be significantly larger or even negative in the breakdown region) modifies the derived lower limits for $B^+ \to p^+ \psi$. The paper notes that in regions where the HQE prediction becomes unreliable (negative width), the derived bounds are not robust.

Significance and Claims
The paper claims that the assumption of "subleading" effects being negligible is not universally valid in B-Mesogenesis. The behavior observed is reminiscent of the SM only under artificially heavy charm masses, suggesting that the Mesogenesis scenario provides a unique laboratory to study the limits of the HQE near its breakdown.

The authors emphasize that their findings are based on Leading Order (LO) calculations. They caution that Next-to-Leading Order (NLO) QCD corrections and higher-dimensional terms could modify the relative sizes of these contributions, as seen in the SM. Therefore, the observed hierarchy breakdown should be regarded as an empirical LO result pending confirmation from more complete studies. The primary significance lies in highlighting that models with non-SM operator structures must be scrutinized for such effects before deriving phenomenological conclusions, as the standard HQE truncation may fail in regions of interest for dark matter and baryogenesis.