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B033 $S_{\phi K_S},\ C_{\phi K_S}$

B $ \to \phi K_S$ penguin CP asymmetry

Status REVIEWED VERIFIED High Code: NO Priority Low

Why this constrains the RS scan

In anarchic warped models, KK gluons, electroweak KK modes, or flavor-changing $Z/H$-like effects can add new weak phases to $b\to s\bar s s$ $\Delta B=1$ amplitudes. $B^0\to\phi K_S^0$ is therefore a diagnostic of new penguin phases that would not be captured by a pure $\Delta m_{B_s}$ or $\Delta m_{B_d}$ magnitude constraint. A large departure of $S_{\phi K_S}$ from $S_{\psi K_S}$ would point to decay-amplitude new physics rather than just the CKM phase measured in the golden charmonium mode.

What's changed since the original paper

Since the arXiv:0804.1954 RS-flavor baseline, the old B-factory tension in penguin CP phases has been replaced by a stable world average compatible with $S_{\psi K_S}$. HFLAV Summer 2025 combines BaBar's 2012 Dalitz analysis, Belle's 2010 time-dependent Dalitz analysis, and Belle II's 2023 time-dependent $B^0\to\phi K_S^0$ measurement. Belle II is the important new post-B-factory input: the belleii\_2023\_phik0 snapshot records $S=0.54\pm0.26^{+0.06}_{-0.08}$ and, with HFLAV's $C=-A$ convention, $C=-0.31\pm0.20\pm0.05$. The present HFLAV average differs from the charmonium benchmark by only $0.03$, far below the current $\phi K^0$ uncertainty; this difference is recorded explicitly in B033.yaml.

Validity and model dependence

This is a useful but theory-limited nonleptonic penguin observable. It is cleaner as a null test than as a standalone hard bound: the SM prediction is close to $\sin(2\beta)$, but subleading amplitudes and hadronic strong phases are mode dependent. A production implementation would need a $\Delta B=1$ operator basis, RG evolution, and a hadronic amplitude or phenomenological likelihood. Treat it as charmless-penguin contextual input unless the PI chooses a specific hadronic framework.

Code coverage in this repo

NO. Focused greps over quarkConstraints/, qcd/, flavorConstraints/, neutrinos/, yukawa/, warpConfig/, solvers/, scanParams/, and tests/ found no implementation of $B\to\phi K$, $b\to s\bar s s$ penguin CP asymmetries, $\sin(2\beta_{\rm eff})$, or time-dependent charmless $B$ CP observables. The nearest adjacent code is neutral-meson mixing: quarkConstraints/deltaf2.py:209 defines the $\Delta F=2$ input registry, quarkConstraints/deltaf2.py:903 evaluates $B_d$ mixing, and quarkConstraints/deltaf2.py:922 evaluates $B_s$ mixing. Those do not compute $\Delta B=1$ penguin decay amplitudes.

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

Implementation difficulty

HIGH. The existing $\Delta F=2$ SLL/SLR/VLL/VRR/LR1/LR2 machinery is not enough. A live constraint would require $\Delta B=1$ four-quark penguin operators, matching from RS sources, QCD running to the $b$ scale, and a hadronic treatment of the $B\to\phi K$ amplitude and strong phases.

Reason: B033 needs a $\Delta B = 1$ nonleptonic $b \to s$ sbar s operator and amplitude treatment. The existing $\Delta F = 2$ SLL/SLR/VLL/VRR/LR1/LR2 neutral-meson machinery does not compute $B \to \phi$ K decay amplitudes, strong phases, or time-dependent penguin CP likelihoods.

Key references

Process-local raw reference keys before bibliography consolidation: hflav\_triangle\_summer2025\_phiK0, babar\_2012\_b0\_kkks, belle\_2010\_b0\_kkks, belleii\_2023\_phik0, and csaki\_falkowski\_weiler\_2008.

observables[0].fallback_value_uncertainty sin(2 beta_eff)(phi K0) = 0.74 +- 0.12

RESOLVED

Match snapshot line 36

L33:     S-C statistical correlation = 0.01
L34: 
L35: HFLAV averages:
L36:   sin(2 beta_eff)(phi K0) = 0.74 +/- 0.12
L37:     chi2 = 1.5 / 2 dof, CL = 0.46, equivalent pull = 0.7 sigma
L38:   C_CP(phi K0) = -0.09 +/- 0.12
L39:     chi2 = 1.7 / 2 dof, CL = 0.42, equivalent pull = 0.8 sigma

observables[1].fallback_value_uncertainty C_CP(phi K0) = -0.09 +- 0.12

RESOLVED

Match snapshot line 38

L35: HFLAV averages:
L36:   sin(2 beta_eff)(phi K0) = 0.74 +/- 0.12
L37:     chi2 = 1.5 / 2 dof, CL = 0.46, equivalent pull = 0.7 sigma
L38:   C_CP(phi K0) = -0.09 +/- 0.12
L39:     chi2 = 1.7 / 2 dof, CL = 0.42, equivalent pull = 0.8 sigma
L40: 
L41: Comparison benchmark on the same HFLAV Summer 2025 page:

observables[0].fallback_value_uncertainty sin(2 beta), all charmonium = 0.71 +- 0.011

AMBIGUOUS

Match 1 of 3 snapshot line 42

L39:     chi2 = 1.7 / 2 dof, CL = 0.42, equivalent pull = 0.8 sigma
L40: 
L41: Comparison benchmark on the same HFLAV Summer 2025 page:
L42:   All-charmonium average sin(2 beta) = sin(2 phi_1) = 0.710 +/- 0.011
L43:     stat-only uncertainty = 0.010, CL = 0.34
L44:   J/psi K0 average sin(2 beta) = 0.710 +/- 0.011
L45:   J/psi K_S mode average = 0.712 +/- 0.011

Match 2 of 3 snapshot line 44

L41: Comparison benchmark on the same HFLAV Summer 2025 page:
L42:   All-charmonium average sin(2 beta) = sin(2 phi_1) = 0.710 +/- 0.011
L43:     stat-only uncertainty = 0.010, CL = 0.34
L44:   J/psi K0 average sin(2 beta) = 0.710 +/- 0.011
L45:   J/psi K_S mode average = 0.712 +/- 0.011
L46: 
L47: Reference labels appearing in the HFLAV table:

Match 3 of 3 snapshot line 45

L42:   All-charmonium average sin(2 beta) = sin(2 phi_1) = 0.710 +/- 0.011
L43:     stat-only uncertainty = 0.010, CL = 0.34
L44:   J/psi K0 average sin(2 beta) = 0.710 +/- 0.011
L45:   J/psi K_S mode average = 0.712 +/- 0.011
L46: 
L47: Reference labels appearing in the HFLAV table:
L48:   BaBar: PRD 85 (2012) 112010

observables[1].fallback_value_uncertainty sin(2 beta), J/psi K_S mode = 0.712 +- 0.011

RESOLVED

Match snapshot line 45

L42:   All-charmonium average sin(2 beta) = sin(2 phi_1) = 0.710 +/- 0.011
L43:     stat-only uncertainty = 0.010, CL = 0.34
L44:   J/psi K0 average sin(2 beta) = 0.710 +/- 0.011
L45:   J/psi K_S mode average = 0.712 +/- 0.011
L46: 
L47: Reference labels appearing in the HFLAV table:
L48:   BaBar: PRD 85 (2012) 112010

observables[2].fallback_value_uncertainty difference sin(2 beta_eff)(phi K0) - sin(2 beta all charmonium) = 0.03

UNRESOLVED

Automated anchoring could not pinpoint this value inside the snapshot. Manual verification is needed: open the live source and confirm against the context surrounding the value claim.

Open source manually

inputs[0].fallback_value_uncertainty = 0.05 +- 0.17

RESOLVED

Match snapshot line 26

L23: Experiment/input:
L24:   BaBar, PRD 85 (2012) 112010, N(BB)=470M:
L25:     sin(2 beta_eff) = 0.66 +/- 0.17 +/- 0.07
L26:     C_CP = 0.05 +/- 0.18 +/- 0.05
L27:   Belle, PRD 82 (2010) 073011, N(BB)=657M:
L28:     sin(2 beta_eff) = 0.90 +0.09 -0.19
L29:     C_CP = -0.04 +/- 0.20 +/- 0.10 +/- 0.02

inputs[1].fallback_value_uncertainty = -0.04 +- 0.09

RESOLVED

Match snapshot line 29

L26:     C_CP = 0.05 +/- 0.18 +/- 0.05
L27:   Belle, PRD 82 (2010) 073011, N(BB)=657M:
L28:     sin(2 beta_eff) = 0.90 +0.09 -0.19
L29:     C_CP = -0.04 +/- 0.20 +/- 0.10 +/- 0.02
L30:   Belle II, PRD 108 (2023) 072012, N(BB)=387M:
L31:     sin(2 beta_eff) = 0.54 +/- 0.26 +0.06 -0.08
L32:     C_CP = -0.31 +/- 0.20 +/- 0.05

inputs[2].fallback_value_uncertainty = -0.31 +- 0.26

RESOLVED

Match snapshot line 32

L29:     C_CP = -0.04 +/- 0.20 +/- 0.10 +/- 0.02
L30:   Belle II, PRD 108 (2023) 072012, N(BB)=387M:
L31:     sin(2 beta_eff) = 0.54 +/- 0.26 +0.06 -0.08
L32:     C_CP = -0.31 +/- 0.20 +/- 0.05
L33:     S-C statistical correlation = 0.01
L34: 
L35: HFLAV averages:

fallback_value_uncertainty = 0.54 +- 17

RESOLVED

Match snapshot line 26

L23: mesons from Upsilon(4S) decays.  Belle II reconstructs 162 +/- 17 signal
L24: events.  The measured CP asymmetries are
L25: A = 0.31 +/- 0.20 +/- 0.05 and
L26: S = 0.54 +/- 0.26 +0.06 -0.08.
L27: 
L28: Use in B033:
L29: This is the Belle II post-2008 input included in the HFLAV Summer 2025