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REVIEW 3 major objections 5 minor 64 references

Time-Dependent Precision Measurement of $B_s^0\rightarrow \phi \mu^+\mu^-$ Decay at FCC-$ee$

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A future Z factory could measure time-dependent CP violation in the rare decay $B_s^0\to\phi\mu^+\mu^-$ with percent-level precision.

desk verdict A solid FCC-ee feasibility study with plausible statistical projections, but the flat decay-time acceptance assumption is the soft spot that should be examined before the quoted three-digit sensitives on D_f, C_f, and S_f are taken at face value. read the letter →

arxiv 2506.08089 v1 pith:CAVAPFBP submitted 2025-06-09 hep-ph hep-ex

Tsz Hong Kwok , Zachary Polonsky , Valeriia Lukashenko , Jason Aebischer , Ben Kilminster This is my paper
classification hep-phhep-ex
keywords CPviolationrareBdecaysB_smesonflavor-changingneutralcurrentFCC-eeWilsoncoefficientsweakeffectivetheorytime-dependentdecay-ratemeasurement
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that a future electron-positron collider running at the Z pole (FCC-ee) can, for the first time, measure time-dependent CP violation in the rare decay $B_s^0\to\phi\mu^+\mu^-$. Because the Standard Model prediction for these CP asymmetries is essentially zero, even a small new-physics phase would stand out. Using a Monte Carlo simulation of signal and backgrounds, the authors project a relative branching-ratio precision of 0.515%, an untagged determination of $D_f$ to 0.103, and tagged determinations of $C_f$ and $S_f$ to about 0.021. Interpreted through the weak effective theory, these sensitivities constrain the imaginary parts of the Wilson coefficients $C_7$, $C_9$, and $C_{10}$ roughly an order of magnitude more tightly than pre-FCC projections. If realized, this would open a new window on CP-violating new physics at the FCC-ee.

What carries the argument

The load-bearing object is the time-dependent decay-rate parametrization of Eq. (3.1), whose coefficients reorganize into the observables of Eq. (3.6): $A_{CP}(t) = (C_f\cos\Delta m_s t - S_f\sin\Delta m_s t)/(\cosh(\Delta\Gamma_s t/2)+D_f\sinh(\Delta\Gamma_s t/2))$. The argument rests on four pieces: the nonzero $B_s$ width difference $\Delta\Gamma_s$ makes $D_f$ accessible in the untagged rate; the finite vertex resolution enters only through a mild time-dilution factor $D_{\rm time}\simeq 0.995$; flavor tagging power $P_{\rm tag}=0.3$ controls the tagged asymmetry uncertainty; and a weak-effective-theory expansion expresses the integrated observables as at-most-quadratic polynomials in the Wilson coefficients of $O_7$, $O_9$, and $O_{10}$, allowing direct constraints on their imaginary parts.

What would settle it

Generate pseudo-experiments or a full simulation with a deliberately time-dependent reconstruction efficiency, such as a 20% linear slope across the 1-8 ps window, and refit Eq. (3.3): if the fitted $D_f$ shifts by more than its quoted 0.103 statistical uncertainty, then the reported sensitivity is not robust to the flat-acceptance assumption.

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Extended reading notes

Core claim

The central claim is that the FCC-ee's combination of a large $B_s^0$ yield, a clean leptonic environment, and excellent vertex resolution makes time-dependent CP violation in $B_s^0\to\phi\mu^+\mu^-$ measurable with previously unreachable precision. After the full selection, about $3.93\times10^4$ signal events survive. Fitting the untagged decay-time distribution gives $D_f = -0.71 \pm 0.103$, while the tagged asymmetry $A_{CP}(t)$ yields $\sigma(C_f)=0.0210$ and $\sigma(S_f)=0.0214$ with a small correlation of $\rho=-0.0118$; the time-integrated asymmetry is projected to a statistical uncertainty of $9.21\times10^{-3}$ at a tagging power of 30%. The paper further shows that branching-ratio and CP observables are complementary in the weak effective theory: the branching ratio mainly constrains the real parts of the Wilson coefficients, while the CP observables mainly constrain the imaginary parts, improving pre-FCC projections by about an order of magnitude and probing new-physics scales up to roughly 1-10 TeV.

Load-bearing premise

The projections assume the efficiency for reconstructing $B_s^0 \to \phi\mu^+\mu^-$ is the same at every decay time in the fitted 1-8 ps window; if the efficiency varies with decay time inside that window, the fitted $D_f$ and the extracted $C_f$ and $S_f$ would be biased.

Editorial extensions

If this is right

  • A relative branching-ratio precision of 0.515% (0.810% in the low-$q^2$ region and 1.577% in the high-$q^2$ region) would be about five times better than the current LHCb statistical reach.
  • The untagged fit determines $D_f$ to 0.103, making the $B_s$ system the first place this observable is practically measurable because $\Delta\Gamma_s$ is large enough.
  • The tagged time-dependent fits give $C_f$ and $S_f$ each to about 0.021, roughly an order of magnitude better than the time-integrated asymmetry and inaccessible to current flavor experiments.
  • Branching-ratio and CP-asymmetry measurements combine to pin down the imaginary parts of $C_7$, $C_9$, and $C_{10}$ at the $\mathcal{O}(10^{-2})$ level and to probe new physics up to scales of roughly 1-10 TeV.
  • The CP observables are largely unaffected by the 5-10% theoretical uncertainties that plague branching-ratio predictions, so a nonzero $S_f$ would be a comparatively clean signal of physics beyond the Standard Model.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same time-dependent machinery should transfer to other $B_s$ and $B^0$ modes with CP-eigenstate final states, such as $B^0\to K_S^0\mu^+\mu^-$ and $B_s\to\phi\nu\bar\nu$, albeit with additional vertex-topology or missing-energy complications; the qualitative conclusion that a Z factory opens time-dependent CPV in rare FCNC decays would likely survive in those channels.
  • The flat-acceptance assumption over the 1-8 ps window is the weakest point of the projection; a realistic acceptance that tilts with decay time would shift $D_f$ and distort $A_{CP}(t)$. This could be tested before the physics fit by extracting the acceptance from control channels with known lifetimes.
  • Splitting the CP observables into smaller $q^2$ bins would sharpen the Wilson-coefficient interpretation, since the low- and high-$q^2$ regions weight the operators differently; the paper already exploits this binning for the branching ratio but not for the CP observables.
  • If such a time-dependent asymmetry is seen, the next question would be whether it comes from new CP phases in $C_9$/$C_{10}$ or from the dipole $C_7$; combining these observables with angular analyses of the same final state would help separate those sources.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This paper presents a Monte Carlo feasibility study of time-dependent CP-violation measurements in B_s^0 -> phi mu+ mu- at the FCC-ee. Using Pythia 8 for event generation and Delphes with the IDEA detector concept, the authors apply a sequential kinematic selection and obtain a projected signal yield of 3.93e4 events with small backgrounds. They report a statistical precision of 0.515% on the branching fraction, sigma(D_f)=0.103 from an untagged fit to the decay-time distribution, sigma(<A_CP>)=9.21e-3 for the time-integrated CP asymmetry, and sigma(C_f)=0.0210, sigma(S_f)=0.0214 from tagged time-dependent fits. The results are interpreted in the Weak Effective Theory, yielding projected constraints on the real and imaginary parts of C_7, C_9, and C_10 that are claimed to improve over pre-FCC projections by about an order of magnitude.

Significance. If the quoted sensitivities hold, this is a useful and well-documented projection: it is one of the first full-simulation-based studies of time-dependent CP violation in b->s mu+ mu- at a future Z factory, and the WET interpretation connects the projected observables to new-physics parameters in a concrete way. The paper is transparent about many technical ingredients: the yield accounting is internally consistent, the uncertainty formulas are stated explicitly, the WET coefficients are tabulated in appendices, and the SM values are cross-checked with flavio and EOS. The comparison with Belle II and HL-LHC projections helps calibrate the claimed improvement. The main weakness is that the central time-dependent sensitivities rest on an unvalidated flat decay-time acceptance and on fixing D_f to its SM value without propagating its uncertainty.

major comments (3)
  1. [Sec. 3, Eq. (3.3), Fig. 7] The decay-time fits in Section 3 (Eq. (3.3), Figure 7, and the tagged fit using Eq. (3.6)) are performed directly on selected events in the 1-8 ps window with no acceptance function. The text itself states that the detector acceptance varies with decay length and boost, and that the window is chosen to avoid reconstruction inefficiencies, but no efficiency-versus-time curve, closure test, or systematic allowance is provided. Because a monotonic efficiency slope is partially absorbed by the cosh and sinh terms in Eq. (3.3), the fitted D_f can be biased at a level that is not negligible compared with the quoted 0.103 uncertainty, and this bias propagates into the tagged C_f and S_f fits. The authors should include an acceptance function in the fit, demonstrate that the acceptance is flat at the required precision, or quote a systematic uncertainty from this source.
  2. [Sec. 3.2, Eq. (3.6)] The quoted uncertainties sigma(C_f)=0.0210 and sigma(S_f)=0.0214 are obtained with D_f fixed to its SM value, with no propagation of the 0.103 uncertainty from the untagged D_f measurement. Over the fit window, |sinh(0.5 DeltaGamma_s t)| reaches about 0.34 at t = 8 ps, so a 1-sigma change in D_f alters the denominator of A_CP(t) by a few percent at large t; the statement that a delta D_f of order 0.1 is negligible because sinh(0.5 DeltaGamma_s t) is of order 0.1 appears to underestimate this product. The authors should marginalize over D_f or otherwise quantify the resulting bias on C_f and S_f.
  3. [Sec. 3, Table 5] All precision numbers in Table 5 are statistical only (sqrt(s+b)/s, 1/sqrt(P_tag s), and fit uncertainties). The paper does not list the dominant experimental systematics (selection efficiency, mass-window description, background subtraction, tagging calibration, time-resolution calibration) or estimate their impact. For a feasibility projection this is acceptable if stated unambiguously, but the abstract's claim that these precisions are 'achievable' needs qualification, especially because the acceptance issue in the previous comment is a systematic effect at the same order as the quoted statistical sensitivities.
minor comments (5)
  1. [Sec. 3, Eq. (3.2)] The time-resolution estimate sigma_t = 5.5e-3 ps uses only the secondary-vertex resolution; the text notes that momentum resolution and primary-vertex resolution are neglected, but no estimate of their relative size is given, so the robustness of D_time = 0.995 is not fully quantified.
  2. [Sec. 3.2] The sentence 'the SM predicts C_f = S_f = 0' is an approximation; in the standard CKM phase convention these observables are expected to be tiny but not exactly zero. The wording should be softened to 'vanish or are extremely small in the SM' to avoid an exact-statement reading.
  3. [Appendix E] There is a typo: 'The coeffieients' should read 'The coefficients'.
  4. [Fig. 7] The decay-time fit is presented without a chi2, pull distribution, or goodness-of-fit measure; adding such information would help the reader assess whether Eq. (3.3) actually describes the simulated distribution, especially in the presence of the small backgrounds.
  5. [Sec. 4.2, Eq. (4.14)] The sign conventions relating h_i in Eq. (4.12) to D_phi_mu_mu in Eq. (4.17) are not spelled out; a reader checking the derivation must infer the signs from Appendix D. A one-sentence statement of the convention would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper reports Monte Carlo sensitivity projections with theory inputs used only to generate samples and evaluate projections, not as outputs.

full rationale

The paper's central claims are projected sensitivities from a Monte Carlo study, not derivations of theoretical inputs. The SM value D_f = -0.71 is injected into the generated sample, and Eq. (3.3) is fitted only to extract the statistical uncertainty on D_f; the paper explicitly notes that the fitted uncertainty is insensitive to the input value (footnote 6). C_f and S_f are likewise extracted from tagged A_CP(t) distributions generated with SM values and quoted as uncertainties. The WET interpretation in Section 4 uses independently published form factors and matrix elements (Refs. [15, 61, 80-82]) and standard open-source codes (flavio, EOS, wilson) to evaluate the same observables as polynomial functions of Wilson coefficients; no fitted parameter from the MC is used as an input to those predictions, so there is no construction whereby an output equals an input. The assumption of a flat decay-time acceptance in the [1,8] ps window is an unvalidated modeling assumption that could bias the quoted sensitivities, but it is a systematic/correctness risk, not a circularity. No self-citation chain is load-bearing; the citations are for formalism and yields previously published by independent or partially overlapping authors, and the central feasibility result stands on this paper's own simulation.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central projections rest on assumed tagging power, a time-resolution parameter, theory inputs for the SM decay-rate shape, and the WET operator basis with external form factors. No new entities are postulated. The D_f used in the MC is a theory value, and the fit returns an uncertainty, not a measurement.

free parameters (4)
  • P_tag (flavor tagging power) = 0.3
    Assumed input, not measured in this study, used in Eq. (C.3) for sigma(A_CP), sigma(C_f), and sigma(S_f). If P_tag = 0.1, these uncertainties grow by about 1.7x.
  • sigma_t (decay time resolution) = 5.5e-3 ps
    From vertex resolution O(10 um) and B_s lifetime; determines D_time = 0.995 in Eq. (3.2). Not measured from data.
  • D_f (SM input value) = -0.71
    Theory value inserted into the MC and used as truth in the untagged fit. The paper notes it varies between -0.6 and -0.8 with stable uncertainty.
  • Br(B_s -> phi mu+ mu-) (world average) = 8.38e-7
    Used to scale signal yield in Table 2, from HFLAV [53].
assumptions (5)
  • domain assumption The b -> s mu+ mu- transition is described by the WET with operators O7, O9, O10 and their chirality-flipped partners, with form factors from Refs. [80-82].
    Used in Eqs. (4.19)-(4.22) to express observables as quadratic polynomials in BSM Wilson coefficients. Not independently verified in this paper.
  • domain assumption B_s mixing parameters Delta m_s = 17.765 ps^-1, Delta Gamma_s = 0.084 ps^-1, and lifetime 1.529 ps from PDG.
    Table 4; entering the time-dependent formulas (3.1) and (3.6).
  • domain assumption The angular integration over polarization does not erase CP violation because the decay is dominated by the CP-even amplitude over the full q2 range [58].
    Section 3, before Eq. (3.3); if CP-odd contributions were comparable and had opposite signs, the integrated observable D_f would be suppressed.
  • ad hoc to paper The detector acceptance in the decay-time window 1 to 8 ps is flat enough that Eq. (3.3) can be fitted without an acceptance correction.
    Section 3: 'we focus exclusively on the decay time region between 1 and 8 ps'; no efficiency function is modeled in the fit.
  • domain assumption Tagging power P_tag = 0.3 is achievable at FCC-ee, matching Belle II or BaBar.
    Table 4 and Section 3.2; sensitivity of C_f and S_f scales as 1/sqrt(P_tag) via Eq. (C.3).

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Cite this review

Pith. "Pith review of Time-Dependent Precision Measurement of $B_s^0\rightarrow \phi \mu^+\mu^-$ Decay at FCC-$ee$." pith.science (2026). https://pith.science/paper/CAVAPFBP

@misc{pith2026250608089,
  author       = {Pith},
  title        = {Pith review of: Time-Dependent Precision Measurement of $B_s^0\rightarrow \phi \mu^+\mu^-$ Decay at FCC-$ee$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CAVAPFBP}},
  note         = {Machine review of arXiv:2506.08089}
}
abstract

We study the feasibility of measuring time-dependent $C\!P$ violation in the rare flavor-changing neutral current (FCNC) decay $B_s^0 \rightarrow \phi(\rightarrow K^+K^-) \mu^+ \mu^-$ at the FCC-$ee$. In the Standard Model (SM), $C\!P$ violation in this mode arises only at higher orders and is highly suppressed. Extensions of the SM, collectively referred to as New Physics (NP), can introduce additional $C\!P$-violating phases that enhance such effects. The decay $B_s^0 \rightarrow \phi \mu^+ \mu^-$, mediated by the $b \rightarrow s \ell^+ \ell^-$ transition, is therefore a promising probe of NP. The FCC-$ee$, operating as a high-luminosity $Z$-factory, offers an optimal environment for this measurement due to its large event yield, clean conditions, efficient particle identification, and excellent vertex resolution. We perform a Monte Carlo study using Pythia and Delphes with the IDEA detector concept. A relative precision better than $\mathcal{O}(1\%)$ on the branching ratio and $\mathcal{O}(10^{-2})$ on the time-integrated $C\!P$ asymmetry is found to be achievable. We determine the projected sensitivities to the observables $D_f$, $C_f$, and $S_f$, which parameterize time-dependent $C\!P$ violation. In the untagged analysis, a precision of $\mathcal{O}(10^{-1})$ on $D_f$ can be reached. With flavor tagging, sensitivities to $C_f$ and $S_f$ improve to $\mathcal{O}(10^{-2})$. These measurements remain inaccessible to current flavor experiments. Interpreting the results within the Weak Effective Theory provides model-independent constraints on $C\!P$-violating NP. This study demonstrates that FCC-$ee$ enables first-time access to $C\!P$-sensitive observables previously beyond experimental reach.

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Reviewed August 7, 2026 · model on record in the stance chip above.