Catalog · Beauty · B018

B018 $R_K$

Lepton-flavor universality ratio in $B+ \to K+ \ell^+ \ell^-$

Status SUBTLETY-ADDED VERIFIED High Code: NO Priority Medium

Why this constrains the RS scan

$R_K$ is a clean $\Delta B=1$ semileptonic FCNC ratio. In an RS or anarchic-flavor interpretation it probes nonuniversal contributions to the $b\to s\ell^+\ell^-$ weak Hamiltonian, especially flavor-changing $Z$ or KK-gauge effects and semileptonic Wilson coefficients $C_9^{\ell}$ and $C_{10}^{\ell}$. It is therefore complementary to the repo's neutral-meson $\Delta F=2$ constraints.

What's changed since the original paper

The CFW 2008 baseline predates the modern $R_K$ anomaly program. The key post-2008 developments are experimental: LHCb's 2021 update measured $R_K=0.846^{+0.042}_{-0.039}{}^{+0.013}_{-0.012}$ in the central bin and reported a $3.1\sigma$ tension with the SM (LHCb2021:UpdatedRK). The later LHCb simultaneous $R_K$/$R_{K^*}$ analysis using $9\,{\rm fb}^{-1}$ superseded that result and found the four LFU measurements compatible with the SM (LHCb2023:RKRKstarDetailed). HFLAV Dec. 2025 therefore treats the charged $R_K$ bins as SM-consistent averages. On the theory side, Bordone--Isidori--Pattori established the ratio as a clean probe once radiative corrections and experimental cuts are handled consistently.

Validity and model dependence

This is a robust LFU null test, but not a standalone bound on the current quark-scan model. It requires a $\Delta B=1$ semileptonic matching calculation, bin definitions, QED/radiative treatment, and experimental correlations. A universal shift in electron and muon coefficients cancels strongly in the ratio, while lepton-nonuniversal new physics can be visible. In custodial RS, reduced $Z b_L b_L$ pressure can make $R_K$ relatively more discriminating as an LFU cross-check; quote RS bounds only after specifying custodial protection, fermion embeddings, and brane kinetic terms. Use $R_K/R_{K^*}$ as precision LFU null tests; do not impose the pre-2023 anomaly narrative as a prior.

Code coverage in this repo

NO. Required greps over quarkConstraints/, qcd/, flavorConstraints/, neutrinos/, yukawa/, warpConfig/, solvers/, scanParams/, and tests/ found no $R_K$, $b\to s\ell\ell$, $C_9$, $C_{10}$, or LFU implementation. The implemented flavor surface is neutral-meson mixing: quarkConstraints/modern/phenomenology.py:23 enumerates only $\epsilon_K$, $K$, $B_d$, $B_s$, and $D^0$, and quarkConstraints/deltaf2.py:209 defines the legacy $\Delta F=2$ inputs. The only live lepton-flavor routine found is flavorConstraints/muToEGamma.py:75.

Linked evidence (opens GitHub blob at flavor-catalog-website/2026q2):

Implementation difficulty

HIGH. Production use would require a new $\Delta B=1$ semileptonic operator basis, matching and running for lepton-flavor-dependent $C_9$, $C_{10}$, and primed coefficients, SM bin predictions including radiative corrections, and likelihood handling for HFLAV/LHCb/CMS inputs.

Key references

Process-local snapshots: HFLAV2025Dec:RKBplusKLowQ2, HFLAV2025Dec:RKBplusKCentralQ2, HFLAV2025Dec:RKBplusKFullQ2Belle, LHCb2023:RKRKstarDetailed, LHCb2021:UpdatedRK, BordoneIsidoriPattori2016:RKRKstarSM, and CsakiFalkowskiWeiler2008:CompositeFlavor.

fallback_value_uncertainty R_K low-q2 = 0.994 +- +/- 0.090

AMBIGUOUS

Match 1 of 2 snapshot line 15

L12: <h1>$\frac{ {\cal{B}}(B^+ \to K^+ \mu^+ \mu^-) }{ {\cal{B}}(B^+ \to K^+ e^+ e^-) },~0.1 < m^2_{\ell^+\ell^-} < 1.1~\rm{GeV^2/c^4}$</h1>
L13: <p><img src="BR_B+_K+_mu+_mu-..B+_K+_e+_e-__bin1.png"></p>
L14: <p><table border="1"><tr><th>Experiment</th><th>Measurement [10<sup>0</sup>]</th><th>$\Delta\chi^2$</th><th>Reference</th><th>Comments</th></tr>
L15: <tr><td><b>Average</b></td><td><b>$0.994 \pm0.090$</b></td><td></td><td></td><td></td></tr>
L16: <tr><td>LHCb</td><td>$0.994\,^{+0.090}_{-0.082}\,^{+0.029}_{-0.027}$</td><td>0.00</td>
L17: <td><a href="https://inspirehep.net/literature/2684465">Phys.Rev.D <b>108</b>,032002 (2023)</a></td><td></td></tr>
L18: </table>

Match 2 of 2 snapshot line 16

L13: <p><img src="BR_B+_K+_mu+_mu-..B+_K+_e+_e-__bin1.png"></p>
L14: <p><table border="1"><tr><th>Experiment</th><th>Measurement [10<sup>0</sup>]</th><th>$\Delta\chi^2$</th><th>Reference</th><th>Comments</th></tr>
L15: <tr><td><b>Average</b></td><td><b>$0.994 \pm0.090$</b></td><td></td><td></td><td></td></tr>
L16: <tr><td>LHCb</td><td>$0.994\,^{+0.090}_{-0.082}\,^{+0.029}_{-0.027}$</td><td>0.00</td>
L17: <td><a href="https://inspirehep.net/literature/2684465">Phys.Rev.D <b>108</b>,032002 (2023)</a></td><td></td></tr>
L18: </table>
L19: </p>

fallback_value_uncertainty R_K central-q2 = 0.947 +- +/- 0.047

RESOLVED

Match snapshot line 15

L12: <h1>$\frac{ {\cal{B}}(B^+ \to K^+ \mu^+ \mu^-) }{ {\cal{B}}(B^+ \to K^+ e^+ e^-) },~1.1 < m^2_{\ell^+\ell^-} < 6.0~\rm{GeV^2/c^4}$</h1>
L13: <p><img src="BR_B+_K+_mu+_mu-..B+_K+_e+_e-__bin2b.png"></p>
L14: <p><table border="1"><tr><th>Experiment</th><th>Measurement [10<sup>0</sup>]</th><th>$\Delta\chi^2$</th><th>Reference</th><th>Comments</th></tr>
L15: <tr><td><b>Average</b></td><td><b>$0.947 \pm0.047$</b></td><td>0.19</td><td>$p=0.66$ (ndf=1)</td><td></td></tr>
L16: <tr><td>LHCb</td><td>$0.949\,^{+0.042}_{-0.041} \pm0.022$</td><td>0.00</td>
L17: <td><a href="https://inspirehep.net/literature/2684465">Phys.Rev.D <b>108</b>,032002 (2023)</a></td><td></td></tr>
L18: <tr><td>CMS</td><td>$0.78\,^{+0.46}_{-0.23}\,^{+0.09}_{-0.05}$</td><td>0.19</td>

fallback_value_uncertainty R_K full-q2 Belle-only = 1.08 +- +/- 0.16

AMBIGUOUS

Match 1 of 2 snapshot line 15

L12: <h1>$\frac{ {\cal{B}}(B^+ \to K^+ \mu^+ \mu^-) }{ {\cal{B}}(B^+ \to K^+ e^+ e^-) },~\rm{Full}~m^2_{\ell^+\ell^-}~\rm{range}$</h1>
L13: <p><img src="BR_B+_K+_mu+_mu-..BR_B+_K+_e+_e-.png"></p>
L14: <p><table border="1"><tr><th>Experiment</th><th>Measurement [10<sup>0</sup>]</th><th>$\Delta\chi^2$</th><th>Reference</th><th>Comments</th></tr>
L15: <tr><td><b>Average</b></td><td><b>$1.08 \pm0.16$</b></td><td></td><td></td><td></td></tr>
L16: <tr><td>BELLE</td><td>$1.08\,^{+0.16}_{-0.15} \pm0.02$</td><td>0.00</td>
L17: <td><a href="https://inspirehep.net/literature/1748231">JHEP <b>03</b>,105 (2021)</a></td><td></td></tr>
L18: </table>

Match 2 of 2 snapshot line 16

L13: <p><img src="BR_B+_K+_mu+_mu-..BR_B+_K+_e+_e-.png"></p>
L14: <p><table border="1"><tr><th>Experiment</th><th>Measurement [10<sup>0</sup>]</th><th>$\Delta\chi^2$</th><th>Reference</th><th>Comments</th></tr>
L15: <tr><td><b>Average</b></td><td><b>$1.08 \pm0.16$</b></td><td></td><td></td><td></td></tr>
L16: <tr><td>BELLE</td><td>$1.08\,^{+0.16}_{-0.15} \pm0.02$</td><td>0.00</td>
L17: <td><a href="https://inspirehep.net/literature/1748231">JHEP <b>03</b>,105 (2021)</a></td><td></td></tr>
L18: </table>
L19: </p>

fallback_value_uncertainty superseded LHCb 2021 central-q2 R_K = 0.846 +- +0.042/-0.039 stat, +0.013/-0.012 syst

RESOLVED

Match snapshot line 16

L13:     <updated>2021-05-21T12:10:06Z</updated>
L14:     <link href="https://arxiv.org/abs/2105.10303v1" rel="alternate" type="text/html"/>
L15:     <link href="https://arxiv.org/pdf/2105.10303v1" rel="related" type="application/pdf" title="pdf"/>
L16:     <summary>In the Standard Model of particle physics, charged leptons of different flavour couple to the electroweak force carriers with the same interaction strength. This property, known as lepton flavour universality, was found to be consistent with experimental evidence in a wide range of particle decays. Lepton flavour universality can be tested by comparing branching fractions in ratios such as $R_K = \mathcal{B}(B^+ \rightarrow K^+ μ^+ μ^-)/\mathcal{B}(B^+ \rightarrow K^+ e^+ e^-)$. This observable is measured using proton-proton collision data recorded with the LHCb detector at CERN's Large Hadron Collider corresponding to an integrated luminosity of $9 \rm{~fb}^{-1}$. For a dilepton invariant mass range of $q^{2} \in [1.1,6.0]~\rm{Ge}\kern -0.1em \rm{V}^{2}$, the measured value of $R_{K}=0.846\,^{+\,0.042}_{-\,0.039}\,^{+\,0.013}_{-\,0.012}$, where the first uncertainty is statistic and the second systematic, is in tension with the Standard Model predicted value at the $3.1σ$ level raising evidence for lepton flavour universality violation in $B^+ \rightarrow K^+ \ell^+ \ell^-$ decays.</summary>
L17:     <category term="hep-ex" scheme="http://arxiv.org/schemas/atom"/>
L18:     <category term="hep-ph" scheme="http://arxiv.org/schemas/atom"/>
L19:     <published>2021-05-21T12:10:06Z</published>