Catalog · Beauty · B002

B002 $S_{\psi K_S},\ \sin 2\beta$

CP phase in $B_d \to J/\psi K_S$

Status REVIEWED VERIFIED Low Code: PARTIAL Priority High

PDG / equivalent values

Observable	Value	Year	Experiment / source	Provenance
sin(2 beta) == sin(2 $\phi_1$), all charmonium including earlier measurements	0.71 dimensionless	2025	HFLAV CPV & Unitarity Triangle Parameters, Summer 2025	source ↑
S coefficient in $B_d \to J/\psi K_S$	0.712 dimensionless	2025	HFLAV CPV & Unitarity Triangle Parameters, Summer 2025	source ↑
beta == $\phi_1$ from sin(2 beta), physical CKM solution	?	2025	HFLAV CPV & Unitarity Triangle Parameters, Summer 2025	source ↑
$S_{\psi K0_S}$	0.717	2024	LHCb Collaboration 2024	source ↑
S in B0 $ \to J/\psi K0_S$	0.724	2024	Belle II Collaboration 2024	source ↑
Long-distance penguin phase shift in $B_d \to J/\psi K_S$	?	2015	Frings, Nierste, and Wiebusch 2015	source ↑

Why this constrains the RS scan

In anarchic warped models, tree-level KK-gluon and electroweak exchange can generate a new complex contribution to $M_{12}^d$. $S_{\psi K_S}$ is therefore the phase counterpart of the existing $B_d$ mass-difference constraint: a scan point can satisfy $\Delta m_d$ while still shifting $\arg(M_{12}^d)$. This observable is especially useful because the decay amplitude is CKM-favored and theoretically clean compared with many hadronic beauty modes.

What's changed since the original paper

Since the arXiv:0804.1954 RS-flavor baseline, the central experimental story is precision rather than discovery. Belle and BaBar final samples established the B-factory anchor, and the post-2023 update added two important inputs: LHCb measured \[ S_{\psi K_S}=0.717\pm0.013_{\rm stat}\pm0.008_{\rm syst} \] using $6\,{\rm fb}^{-1}$ (lhcb\_2024\_spsiks), and Belle II measured \[ S=0.724\pm0.035_{\rm stat}\pm0.009_{\rm syst} \] with $362\,{\rm fb}^{-1}$ (belleii\_2024\_sin2phi1). These inputs moved the HFLAV average to the percent-level regime. On the theory side, the main development is sharper accounting of doubly Cabibbo-suppressed penguin pollution; the frings\_nierste\_wiebusch\_2015 snapshot quotes $|\Delta\phi_d|\le 0.68^\circ$.

Validity and model dependence

This is a robust neutral-meson-mixing phase constraint, but a catalog number is not a standalone exclusion without a CKM convention. A live implementation must decide whether the SM $\beta$ input is fixed from a global fit or fitted simultaneously, and how to budget the small penguin phase. New physics in the decay amplitude would make the usual $S_{\psi K_S}\simeq \sin 2\beta$ identification model-dependent, but for RS quark-flavor scans the dominant target is the new phase in $B_d$ mixing.

Code coverage in this repo

PARTIAL. No implementation named S\_psi, J/psi K\_S, sin2beta, or equivalent was found in the required greps. However, quarkConstraints/deltaf2.py:225 defines the existing $B_d$ mixing input, quarkConstraints/deltaf2.py:631 and quarkConstraints/deltaf2.py:638 define its hadronic and $\Delta m_d$ inputs, and quarkConstraints/deltaf2.py:903 evaluates $M_{12}^{d,\rm NP}$. That evaluator reduces the result to $\lvert M_{12}^{d,\rm NP}\rvert$ at quarkConstraints/deltaf2.py:910, so the phase needed for $S_{\psi K_S}$ is not exposed. The shared $\Delta F=2$ running basis is documented at quarkConstraints/qcd\_running.py:7.

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

Implementation difficulty

LOW. The existing $\Delta F=2$ Wilson and hadronic machinery covers the needed operator basis. The missing work is to retain the complex $B_d$ mixing amplitude, define the SM phase convention and allowed penguin uncertainty, and add an observable policy for the measured $S_{\psi K_S}$.

Reason: No new operator basis is needed: a B_d mixing phase constraint would reuse the existing $\Delta F = 2$ Wilson, QCD-running, and hadronic machinery. The missing work is observable plumbing for arg(M12_d), a CKM/SM phase convention, penguin-systematics treatment, and a likelihood or tolerance policy.

Key references

Process-local raw reference keys before bibliography consolidation: hflav\_triangle\_summer2025\_sin2beta, lhcb\_2024\_spsiks, belleii\_2024\_sin2phi1, frings\_nierste\_wiebusch\_2015, faller\_fleischer\_jung\_mannel\_2008, and csaki\_falkowski\_weiler\_2008.

fallback_value_uncertainty sin(2 beta) == sin(2 phi_1), all charmonium including earlier measurements = 0.71 +- 0.011

AMBIGUOUS

Match 1 of 2 snapshot line 12

L9: 
L10: Canonical extracted average:
L11: All charmonium, including earlier sin(2 beta) measurements:
L12:   sin(2 beta) == sin(2 phi_1) = 0.710 +/- 0.011
L13:   stat-only uncertainty = 0.010
L14:   CL = 0.34
L15:

Match 2 of 2 snapshot line 21

L18:   or beta == phi_1 = 67.37^{+0.44}_{-0.45} degrees
L19: 
L20: Mode-specific line relevant to the assigned process:
L21:   B_d -> J/psi K_S: average S = 0.712 +/- 0.011
L22:   stat-only uncertainty = 0.011
L23: 
L24: Inputs named by HFLAV in this block:

fallback_value_uncertainty S coefficient in B_d -> J/psi K_S = 0.712 +- 0.011

RESOLVED

Match snapshot line 21

L18:   or beta == phi_1 = 67.37^{+0.44}_{-0.45} degrees
L19: 
L20: Mode-specific line relevant to the assigned process:
L21:   B_d -> J/psi K_S: average S = 0.712 +/- 0.011
L22:   stat-only uncertainty = 0.011
L23: 
L24: Inputs named by HFLAV in this block:

fallback_value_uncertainty beta == phi_1 from sin(2 beta), physical CKM solution = 22.63 +- 0.45

RESOLVED

Match snapshot line 17

L14:   CL = 0.34
L15: 
L16: Corresponding beta solutions reported by HFLAV:
L17:   beta == phi_1 = 22.63^{+0.45}_{-0.44} degrees
L18:   or beta == phi_1 = 67.37^{+0.44}_{-0.45} degrees
L19: 
L20: Mode-specific line relevant to the assigned process:

fallback_value_uncertainty S_{psi K0_S} = 0.717 +- 0.013

RESOLVED

Match snapshot line 15

L12:   and J/psi(e+e-) K0_S, with K0_S -> pi+ pi-.
L13: 
L14: Reported CP-violation parameters:
L15:   S_{psi K0_S} = 0.717 +/- 0.013(stat) +/- 0.008(syst)
L16:   C_{psi K0_S} = 0.008 +/- 0.012(stat) +/- 0.003(syst)
L17: 
L18: The abstract states that this S_{psi K0_S} result was the most precise

fallback_value_uncertainty S in B0 -> J/psi K0_S = 0.724 +- 0.035

RESOLVED

Match snapshot line 14

L11:   a 362 fb^-1 sample collected at the Upsilon(4S) resonance.
L12: 
L13: Reported B0 -> J/psi K0_S CP-violation parameters:
L14:   S = 0.724 +/- 0.035(stat) +/- 0.009(syst)
L15:   C = -0.035 +/- 0.026(stat) +/- 0.029(syst)
L16: 
L17: Flavor-tagging performance quoted in the abstract:

fallback_value_uncertainty Long-distance penguin phase shift in B_d -> J/psi K_S = 0.68

RESOLVED

Match snapshot line 15

L12:   unknown long-distance penguin contribution involving up quarks.
L13: 
L14: Numerical bound quoted in the abstract:
L15:   |Delta phi_d| <= 0.68 degrees
L16: 
L17: The paper also reports polarization-dependent bounds for |Delta phi_s| in
L18: B_s -> J/psi phi and bounds for related B_d -> psi(2S) K_S and B_d -> J/psi K*