Probing unknown nonperturbative effects in $b \to s \ell\ell$ with inclusive and exclusive observables

Abstract: In this paper we revisit, from a different perspective, a long-standing question: ``Is the systematic deficit observed in all branching ratios mediated by a $b\to s μμ$ transition due to New Physics, or to a hypothetical constant unknown universal hadronic contribution that mimics New Physics?'' The key observation that allows us to distinguish between these two possibilities is that non-perturbative contributions associated with $c\bar c$ loops affect inclusive $B\to X_s \ell\ell$ and exclusive $B\to K<sup>{(*)}\ell\ell$ modes differently. In inclusive decays, factorizable contributions are exactly determined from data on $e^{+e^-\to} \mathrm{hadrons}$, while non-factorizable corrections are described by resolved-photon contributions at low $q<sup>2$ and by local power corrections at high $q<sup>2$. In exclusive decays, by contrast, hypothetical charming-penguin effects, beyond those already included in current uncertainty estimates, could appear, in a worst-case scenario, as a constant, universal contribution that it seems, in principle, indistinguishable from genuine New Physics. We identify two observables, constructed from ratios of exclusive to inclusive modes, that can discriminate between a New Physics contribution and a constant hadronic contribution. Moreover, these ratios can be measured directly by LHCb, as they do not require any normalisation to $J/ψK<sup>{(*)}$ branching fractions from B factories. A preliminary evaluation of these observables with present data shows some preference for the New Physics interpretation. In a complementary test, a comparison between inclusive measurements and the corresponding sum of exclusive modes at high $q<sup>2$ similarly disfavours an explanation based on a constant hadronic contribution. Finally, we provide projections for the new observables based on expected LHCb and Belle II measurements in the near future.

• The paper demonstrates that a universal, constant hadronic shift cannot fully explain the discrepancies observed in b→s ℓℓ decays.
• It introduces novel ratio observables derived from inclusive and exclusive branching fractions to differentiate between new physics and nonperturbative effects.
• Experimental data and theoretical comparisons, including high-q² measurements, mildly favor a new physics interpretation over hadronic explanations.

Discriminating Nonperturbative Hadronic Effects and New Physics in $b \to s \ell\ell$ Transitions

Overview and Motivation

This paper interrogates the longstanding tension in the branching ratios and angular observables of $b \to s \ell\ell$-mediated $B$-meson decays, addressing whether these discrepancies arise from genuine New Physics (NP)—typically parametrized as a universal shift in the Wilson coefficient $\mathcal{C}_9$—or from an unknown, constant, nonperturbative hadronic contribution capable of mimicking NP effects. Crucially, the study leverages the distinct sensitivity of inclusive ($B \to X_s \ell\ell$) and exclusive ($B \to K^{(*)}\ell\ell$) modes to nonlocal hadronic corrections, constructing new observables that unambiguously test the universality and constancy assumptions underlying hadronic explanations of these anomalies.

Theoretical Framework: Inclusive vs. Exclusive Modes

The analysis is rooted in the weak effective Hamiltonian formalism, focusing on semileptonic operators and predominantly on NP contributions to $\mathcal{C}_9$ in light of global fits. Exclusive decays at low and high-$q^2$ employ SCET and OPE frameworks, respectively, with nonfactorizable charm-loop effects, encoded as "charming penguin" amplitudes, contributing to soft-gluon uncertainties. The recent theoretical advancements in modeling these effects (including LCSR and lattice QCD for form factors), alongside data-driven resonance-based techniques, allow for a refined estimate of the long-distance contributions.

Inclusive modes, by contrast, rely on operator product expansions, where factorizable contributions are reconstructed via $e^+e^- \to \mathrm{hadrons}$ dispersion relations. Nonfactorizable corrections at low-$q^2$ are found to be suppressed (estimated at $b \to s \ell\ell$0), while at high-$b \to s \ell\ell$1, the OPE is normalized against the $b \to s \ell\ell$2 rate to mitigate uncertainties in the endpoint region. The distinct theoretical handling ensures that hypothetical hadronic contributions, if present, affect exclusive but not inclusive modes in the same way.

Analysis of Universality and New Observables

Three scenarios for hypothetical hadronic contributions are considered:

• Universal and constant in $b \to s \ell\ell$3 and channels: Compatible with current global fit patterns and masquerading as a universal NP shift.
• Universal across channels but non-constant in $b \to s \ell\ell$4: Disfavored due to observed deviation signs at both low and high-$b \to s \ell\ell$5.
• Non-universal in $b \to s \ell\ell$6 and channels: Not supported by data, predicts a pattern not observed experimentally.

Scenario A, the universal and constant case, is the focal point given its ability to replicate observed NP results in fits. To test this, two new observables are defined as ratios of exclusive to inclusive branching fractions in specific $b \to s \ell\ell$7 regions:

$b \to s \ell\ell$8

$b \to s \ell\ell$9

These ratios are engineered to exploit the differing susceptibilities of exclusive and inclusive channels to nonlocal hadronic effects. Their measurements, especially possible without external normalization at LHCb, facilitate a direct confrontation between NP and hadronic scenarios.

Figure 1: Current status of experimental and theoretical determinations of the observables $B$0 and $B$1.

Experimental Determinations and Theoretical Comparisons

The study presents detailed calculations and corrections for PDG and HFLAV averages, particularly emphasizing $B$2 branching fractions, isospin-breaking effects, and uncertainties propagated from normalization modes. Experimental determinations for the new $B$3 observables are constructed from recent LHCb, CMS, and Belle II measurements.

By comparing theoretical predictions for various combinations of NP and hadronic parameters with current experimental determinations, the paper demonstrates that the $B$4 and $B$5 ratios exhibit a mild preference for NP-driven scenarios: at $B$6 and $B$7 respectively, these are more consistent with a universal $B$8 than with a constant hadronic shift.

Figure 2: Global fit taken from Ref.~\cite{Alguero:2023jeh}, showing confidence regions for $B$9 in different mode selections.

Inclusive-Exclusive Sum Approach and Constraints on Hadronic Scenarios

A further test is performed by comparing the sum of exclusive branching ratios to the inclusive rate at high-$\mathcal{C}_9$0. Employing a quadratic parameterization for each mode, the analysis shows that a universal hadronic shift—required to explain exclusive deficits—adversely impacts the agreement between inclusive and exclusive determinations, with the data disfavouring purely hadronic explanations.

$\mathcal{C}_9$1

The theoretical uncertainty is dominated by the inclusive calculation; future measurements at Belle II and LHCb are projected to halve this uncertainty, tightening constraints against the hadronic scenario.

Figure 3: $\mathcal{C}_9$2 branching ratio for $\mathcal{C}_9$3 calculated via OPE and as a sum of exclusive modes; agreement favors NP.

Figure 4: A comparison of the $\mathcal{C}_9$4-dependence of the inclusive and sum-of-exclusive modes under various NP and hadronic hypotheses.

Implications and Future Directions

The analysis exhibits strong technical robustness: the construction of the $\mathcal{C}_9$5 observables and the inclusive-exclusive sum comparison both provide systematic ways to discriminate NP shifts from constant hadronic effects. The present data, especially high-$\mathcal{C}_9$6 ratios, disfavors a universal hadronic explanation with significance exceeding $\mathcal{C}_9$7 and points toward an NP interpretation.

From a theoretical perspective, these results challenge models attributing anomalies solely to rescattering effects or charming-penguin amplitudes. Given the robustness of inclusive predictions (anchored in $\mathcal{C}_9$8 data and power corrections), the distinction between exclusive and inclusive behavior places severe constraints on the size and universality of possible hadronic mimics.

With significant improvements forecast for both experimental measurements (inclusive branching ratios, normalization uncertainties) and theory (lattice calculations of $\mathcal{C}_9$9 form factors and vacuum matrix elements), future developments are expected to further clarify the origin of these anomalies and sharpen the distinction between NP and hadronic scenarios.

Conclusion

The paper provides a comprehensive strategy for disentangling unknown nonperturbative effects and genuine NP contributions in $B \to X_s \ell\ell$0 transitions. By exploiting the differential sensitivity of inclusive and exclusive modes, and constructing robust, normalization-independent observables, the study demonstrates that current data favor the NP scenario and places stringent bounds on hypothetical universal hadronic contributions. Future advancements in measurement and theory are projected to significantly tighten these constraints and play a decisive role in resolving the $B \to X_s \ell\ell$1-flavor anomalies.