Correlated $b \to s$ and $s \to d$ Rare Semileptonic Transitions in the Standard Model Effective Field Theory

Abstract: The persistent anomalies observed in $b \to s\,(\ell<sup>+\ell<sup>-,\,ν\barν)$ transitions continue to provide strong motivation for exploring possible extensions of the Standard Model (SM). Motivated by these discrepancies, we present a comprehensive analysis of semileptonic flavor changing neutral current processes within the Standard Model Effective Field Theory (SMEFT), encompassing both $b \to s\,(μ<sup>+μ<sup>-,\,ν\barν)$ and $s \to d\,(μ<sup>+μ<sup>-,\,ν\barν)$ transitions. We perform a combined fit to $b \to s\,(μ<sup>+μ<sup>-,\,ν\barν)$ observables, allowing the relevant dimension-six Wilson coefficients to be complex. We find that the four-fermion operators involving left-handed quark and lepton doublets provide the preferred description of the current $b \to s$ data, while the electroweak operator modifying the $Z$-boson couplings also plays an important role in improving the fit. We show that flavor-universal SMEFT couplings lead to strongly enhanced rare semileptonic kaon decay branching ratios, in conflict with current experimental bounds and thus motivating the implementation of Minimal Flavor Violation. In particular, we demonstrate that flavor-symmetric frameworks based on $U(3)<sup>5$ and $U(2)<sup>5$ naturally restore the required CKM hierarchies and bring the predicted kaon observables into agreement with present data. We further analyze the differential distributions with respect to the dineutrino invariant mass squared $q<sup>2$, as well as the reconstructed variable $q<sup>2_{\mathrm{rec}}$, in $B \to K<sup>{(*)}ν\barν$ decays, demonstrating their sensitivity to different new physics operators. In addition, we investigate the impact of complex Wilson coefficients on $\mathcal{CP}$ asymmetries in $B \to K<sup>{(*)}μ<sup>+μ<sup>-$ decays and find that percent-level effects can arise in specific $q<sup>2$ regions.

• The paper presents a comprehensive SMEFT analysis demonstrating that left-handed four-fermion and Z-penguin operators best address b→s anomalies while respecting kaon decay constraints.
• The study employs a global χ² fit on 31 rare decay observables and uses MFV to align new physics effects with the CKM hierarchy, ensuring consistency with experimental data.
• The results predict CP asymmetries at the percent level and enhanced dineutrino modes, emphasizing the need for precise measurements at Belle II and LHCb.

Comprehensive SMEFT Analysis of Correlated $b \to s$ and $s \to d$ Rare Semileptonic Transitions

Introduction

Rare flavor-changing neutral current (FCNC) semileptonic transitions, specifically $b \to s$ and $s \to d$ modes, are highly suppressed in the Standard Model (SM) and therefore sensitive probes of short-distance new physics (NP). The persistent anomalies in $b \to s\ell^+\ell^-$ observables, alongside recent evidence for $B^+ \to K^+ \nu\bar\nu$, motivate an exploration of these processes within the Standard Model Effective Field Theory (SMEFT). SMEFT, constructed from dimension-six gauge-invariant operators, provides a systematic model-independent framework that incorporates both SM and possible NP contributions. Importantly, the SU(2)$_L$ gauge symmetry in SMEFT enforces correlations between charged lepton and neutrino transitions, allowing for simultaneous and correlated analyses of $b \to s$ (and $s \to d$) semileptonic decay modes.

SMEFT Framework and Operator Structure

The analysis employs the Warsaw basis for dimension-six SMEFT operators. The effective Hamiltonian for $b \to s$ and $s \to d$0 transitions includes four-fermion operators involving left-handed and right-handed quark–lepton combinations and modified $s \to d$1-boson couplings via the $s \to d$2 operators. The fit allows all relevant Wilson coefficients to be complex, permitting CP-violating effects beyond the SM. Crucially, due to the SU(2)$s \to d$3 structure, operators modifying left-handed quark doublets contribute with correlated strengths to both dileptonic and dineutrino final states.

Two classes of flavor symmetries are considered:

• Flavor-universal SMEFT: Wilson coefficients are generation-independent.
• Minimal Flavor Violation (MFV): NP sources are aligned with SM Yukawas, realizing the CKM suppression across generations. Both U(3)$s \to d$4 and U(2)$s \to d$5 scenarios are analyzed, representing full and minimal symmetry cases.

Statistical Analysis and Preferred Operator Structures

A global $s \to d$6 fit is performed using 31 rare $s \to d$7-meson and $s \to d$8-meson observables, incorporating both differential and angular distributions in the $s \to d$9 system, as well as $b \to s$0 and rare kaon decays.

The fit results show:

• Four-fermion operators, especially $b \to s$1 and $b \to s$2, provide the most favored descriptions of the current $b \to s$3 data, notably alleviating tensions in $b \to s$4 branching fractions and the angular observable $b \to s$5 in $b \to s$6.
• Electroweak operators affecting $b \to s$7 couplings, particularly $b \to s$8, further improve the fit when included in two-dimensional operator scenarios.
• Preferred scenarios allow for complex Wilson coefficients, resulting in sizeable new weak phases, a feature reflected in the potential for enhanced CP-violating observables.

The SM-only fit is significantly disfavored compared to NP scenarios involving left-handed operators, with goodness-of-fit statistics indicating nontrivial evidence for BSM effects within this formalism.

Predictions for Rare $b \to s$9-Meson and Kaon Observables

$s \to d$0 and $s \to d$1

• Differential branching fractions: Best-fit NP scenarios bring theoretical predictions into close agreement with experimental results across all $s \to d$2 bins for both $s \to d$3 and $s \to d$4. For $s \to d$5, angular observables ($s \to d$6, $s \to d$7, $s \to d$8) are accurately described within 1$s \to d$9 of data.
• Dineutrino modes: Branching ratios are enhanced in NP scenarios with sizable left-handed SMEFT coefficients, yet do not overshoot current experimental upper bounds—except for tension in $b \to s\ell^+\ell^-$0, where the experimental central value remains marginally higher than the SM+NP prediction.

Rare Kaon Decays and Flavor Structure

• Flavor-universal coefficients: Applying the same SMEFT Wilson coefficients to $b \to s\ell^+\ell^-$1 transitions results in $b \to s\ell^+\ell^-$2 level enhancements in $b \to s\ell^+\ell^-$3 and $b \to s\ell^+\ell^-$4, which strongly violate experimental constraints.
• MFV/U(3)$b \to s\ell^+\ell^-$5 and U(2)$b \to s\ell^+\ell^-$6 scenarios: Implementation of CKM-aligned suppressions naturally restores compatibility with all kaon data. The rare kaon decay rates are brought back to SM-like values due to the hierarchical suppression by $b \to s\ell^+\ell^-$7, thereby reconciling the SMEFT description with the stringent bounds on $b \to s\ell^+\ell^-$8 FCNCs.

CP Violation and Complex Wilson Coefficient Effects

Allowing for complex Wilson coefficients, the analysis evaluates direct CP asymmetries ($b \to s\ell^+\ell^-$9) in $B^+ \to K^+ \nu\bar\nu$0. The predicted asymmetries can reach the percent level in certain $B^+ \to K^+ \nu\bar\nu$1 regions, particularly for the favored left-handed operators. While currently consistent with experimental statistics, these values are within reach of near-future experimental sensitivity, rendering them a distinctive probe of NP weak phases in semileptonic transitions.

Implications and Future Directions

The results deliver several key theoretical and experimental implications:

• The SMEFT fit, restricted to left-handed four-fermion and $B^+ \to K^+ \nu\bar\nu$2-penguin operators, can simultaneously alleviate present $B^+ \to K^+ \nu\bar\nu$3 tensions and comply with all $B^+ \to K^+ \nu\bar\nu$4-meson rare decay constraints, only when CKM hierarchies are imposed via MFV-like flavor symmetries.
• The observed operator correlations across leptonic and neutrino channels highlight the necessity of complete SMEFT analyses over ad hoc model-specific approaches.
• Differential distributions in $B^+ \to K^+ \nu\bar\nu$5—notably with respect to the reconstructed variable $B^+ \to K^+ \nu\bar\nu$6—retain strong NP discriminatory power and should be prioritized at Belle II and future flavor factories.
• Predicted CP asymmetries at the percent level motivate dedicated experimental analyses in $B^+ \to K^+ \nu\bar\nu$7 at LHCb and Belle II.
• Violation of kaon constraints under flavor-universal NP demonstrates the exceptional sensitivity of rare $B^+ \to K^+ \nu\bar\nu$8 processes to flavor structure and supports the principle that any viable NP addressing $B^+ \to K^+ \nu\bar\nu$9 anomalies must be CKM-aligned.

Future theoretical work should focus on improved theoretical control over hadronic uncertainties in rare $_L$0 and $_L$1 decays, higher-order SMEFT matching, and dedicated studies on the interplay of tensor and scalar operators. From the experimental side, increased statistics and better differential measurements—especially in the neutrino modes and $_L$2—will provide sharper discrimination between allowed NP flavor structures.

Conclusion

This comprehensive study demonstrates that the present suite of rare semileptonic $_L$3- and $_L$4-decay data, analyzed within SMEFT with complex Wilson coefficients, favors left-handed four-fermion and electroweak operators, provided the flavor structure follows MFV. Future improvements in experimental precision and complementary observables (differential distributions, CP violation, and rare kaon rates) will provide pivotal tests for the SMEFT-encoded structure of NP in flavor transitions.