CP violation studies at Super Tau-Charm Facility

Andrzej Kupsc; Fu-Sheng Yu; Haiping Peng; Hai-Yang Cheng; Jian-Ping Ma; Jianyu Zhang; Liang Liu; Pablo Roig; Qin Qin; Stephen Lars Olsen

REVIEW 5 major objections 5 minor 1 cited by

CP violation studies at Super Tau-Charm Facility

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

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Pith's one-line read At the projected event yields of the Super Tau-Charm Facility, hyperon CP-violation asymmetries become measurable at the 10^-4 Standard Model level, tau decay asymmetries reach the sensitivity needed to test the existing anomaly, and…

desk verdict A comprehensive, honest STCF prospects review; the headline sensitivities are plausible design goals but rest on fast simulation and statistical-only errors. read the letter →

arxiv 2502.08907 v1 pith:FHZBRJIJ submitted 2025-02-13 hep-ex hep-ph

Hai-Yang Cheng , Zhi-Hui Guo , Xiao-Gang He , Yingrui Hou , Xian-Wei Kang , Andrzej Kupsc , Ying-Ying Li , Liang Liu

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Xiao-Rui Lyu Jian-Ping Ma Stephen Lars Olsen Haiping Peng Qin Qin Pablo Roig Zhi-Zhong Xing Fu-Sheng Yu Yu Zhang Jianyu Zhang Xiaorong Zhou

This is my paper · ORCID

classification hep-exhep-ph

keywords CPviolationSuperTau-CharmFacilityhyperondecaystauleptoncharmmixingelectricdipolemomentCPTinvarianceneutralkaon

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

This paper argues that the proposed Super Tau-Charm Facility (STCF), running at center-of-mass energies between 2 and 7 GeV with a peak luminosity of 0.5e35 $cm^{-2}$ $s^{-1}$, would collect enough quantum-correlated particle-antiparticle pairs to push several CP-violation searches to their Standard Model targets. With roughly 3.4 trillion J/psi decays per year, hyperon CP asymmetries in Lambda, Xi, and Sigma nonleptonic decays would reach statistical sensitivities near $10^{-4}$, the level of Standard Model predictions, and electric dipole moment sensitivities of $10^{-20}$ to $10^{-21}$ e cm. A one-year tau sample would give a 9.7e-4 statistical sensitivity to the CP asymmetry in tau -> K_S pi nu, enough to confront the discrepancy between the Standard Model prediction and the existing measurement, and 10 $ab^{-1}$ would reach 3.1e-4. Correlated D-Dbar pairs would determine charm mixing and indirect CP-violation parameters at the $10^{-4}$ level, and strangeness-tagged neutral kaons would improve the kaon CPT mass-difference limit to about 4e-17 MeV, an order of magnitude beyond the current bound.

What carries the argument

The argument rides on three objects. First, the quantum-correlated, CP-odd baryon-antibaryon pairs from J/psi decays, whose joint angular distributions are described by modular decay matrices, let one separate the alpha, beta, and gamma decay parameters for hyperon and antihyperon and construct the CP observables A_CP and B_CP. Second, the empirical scaling relation sigma_ACP sqrt(N_fin) = k with k = 7.82, obtained from fast-simulation samples between 0.01 and 0.1 trillion J/psi events and extrapolated to 3.4 trillion, converts sample size into projected sensitivity. Third, for the kaon CPT test, strangeness-tagged K0 and Kbar0 events in J/psi -> K- pi+ K0 decays and the Bell-Steinberger relation tie the measurable interference phases to the CPT-violating parameter delta and to the mass difference bound.

What would settle it

A concrete check is to run the same fast-simulation analysis at 0.2, 0.5, and 1.0 trillion J/psi events: if sigma_ACP times the square root of the final event count departs from 7.82 by more than the fit uncertainty, or if a detector prototype measures the Lambda-Lambda selection efficiency below 38.2% at |cos $\theta$| < 0.93 with background above 0.5%, the projected O($10^{-4}$) hyperon sensitivities and the 4e-17 MeV CPT limit do not follow.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that the STCF program turns the tau-charm energy region into a CP-violation discovery laboratory. The projected samples translate into concrete numbers: hyperon weak-phase differences at O($10^{-4}$), hyperon EDMs at O($10^{-20}$) to O($10^{-21}$) e cm, tau EDM form-factor sensitivity near $10^{-18}$ e cm, statistical precision of 9.7e-4 per $ab^{-1}$ on A_CP(tau -> K_S pi nu), charm mixing parameters x and y with uncertainties of order 0.04%, and a CPT test reaching |M_K0bar - M_K0| near 4e-17 MeV. The paper also shows that the Standard Model prediction for the tau-channel asymmetry, about 0.36% after efficiency corrections, deviates from the current experimental value by 2.8 standard deviations, making this channel a concrete target for the facility. If the projections hold, STCF would be the first experiment able to test the Standard Model's predicted hyperon CP violation rather than merely bound it.

Load-bearing premise

The quoted sensitivities assume the statistical scaling sigma_ACP times the square root of the final event count equals 7.82, a constant fitted on Monte Carlo samples of 0.01 to 0.1 trillion J/psi events, and that this scaling continues to hold at 3.4 trillion J/psi with the same selection efficiency, background below 0.5%, vertex resolution, and particle identification performance in the final detector.

Editorial extensions

If this is right

Hyperon decay CP asymmetries A_CP and B_CP would be measured at O(10^-4), reaching the weak-phase differences the Standard Model predicts and constraining new CP phases in the strange-quark sector.
Lambda, Sigma, and Xi electric dipole moments would be probed at 10^-20 to 10^-21 e cm, five orders of magnitude beyond the current Lambda limit.
A_CP(tau -> K_S pi nu) with 1 ab^-1 would reach 9.7e-4, sharpening the 2.8-sigma tension with the Standard Model; 10 ab^-1 would reach 3.1e-4.
Charm mixing parameters and indirect CP-violating observables would be determined at the 10^-4 level from quantum-correlated D-Dbar pairs, complementing time-dependent measurements at hadron colliders.
The neutral-kaon CPT test would improve |M_K0bar - M_K0| to about 4e-17 MeV, an order of magnitude better than the current Bell-Steinberger bound.

Reading between the lines

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

If the 1/sqrt(N) scaling holds and the electron beam is 80% polarized, the roughly threefold sensitivity gain quoted for hyperons implies per-year A_CP precisions near 10^-5 for Lambda, edging into territory where Standard Model weak phases become visible in a single run.
The same quantum-correlated D-Dbar samples that yield charm mixing parameters can provide strong-phase difference inputs with ultimate uncertainties, and those inputs are also required for CP analyses at B factories, so the facility's impact may reach beyond its own CP asymmetries.
A decisive STCF measurement of the tau asymmetry that agrees with the Standard Model after efficiency and regeneration corrections would rule out the surviving tensor-interaction explanations of the existing anomaly, effectively closing it.
The fast-simulation scaling methodology could be transported to charmed-baryon pairs at higher center-of-mass energies; the paper's quoted 0.25-0.5% sensitivity for T-odd observables suggests a dedicated scaling study could show whether larger samples push charm-baryon CP violation below the Standard Model benchmark of about 0.1%.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

This is a review-plus-prospects for the Super Tau-Charm Facility, and its real contribution is a set of sensitivity projections: hyperon A_CP at O(10^-4), hyperon EDMs at 10^-20 to 10^-21 e cm, tau A_CP(KS pi nu) at 9.7e-4 per ab^-1, and a kaon CPT mass-difference limit near 4e-17 MeV. These numbers are new, and they are honestly derived. The theoretical formalism in the hyperon and tau sections is standard and correctly transmitted, the treatment of the BaBar anomaly includes the no-go theorem that limits NP explanations, and the Bell-Steinberger discussion is clear. The authors also state their assumptions: they list systematic sources for the tau measurement, admit that hyperon systematics need detailed detector simulation, and say the charm program is complementary rather than competitive with LHCb.

The soft spots are exactly where the reader and stress-test point. The headline numbers all come from fast simulation with statistical-only uncertainties. Hyperon A_CP rests on Eq. (59) with k = 7.82 fitted over 0.01 to 0.1 trillion J/psi and then extrapolated to 3.4 trillion; no full detector simulation or quantitative systematics budget is given. BESIII's current A_CP systematics are ~0.2%, so the projected 10^-4 statistical floor will only be meaningful if the detector and analysis are much better than what exists today. The tau projection assumes control samples will correct detector asymmetries, but no number is given for the K0/K0bar nuclear-interaction asymmetry in STCF material. The kaon CPT exercise is clean but assumes 99.5% purity and small regeneration without a full simulation. Section 5.1 on CPT and the Theory of Everything is speculative and not load-bearing; it could be cut without loss.

None of this invalidates the paper as a physics case. The authors are clear that these are feasibility studies, and they explicitly call for more detailed simulation. I agree with the reader's conditional verdict: the strongest claims are design goals, not established results.

I would send this to a serious referee. It deserves referee time because it is a comprehensive, honest prospects review that the STCF community and flavor physicists will use as a reference. The referee should ask for a prominent caveat that all projected sensitivities are statistical-only fast-simulation estimates, and a short section spelling out what would be needed to turn them into systematics-limited projections. I would cite it in my own work on STCF physics cases.

Referee Report

5 major / 5 minor

Summary. This Physics Reports submission reviews the physics case for CP-violation and CPT studies at the proposed Super Tau-Charm Facility (STCF), covering four sectors: hyperon decays, tau decays, charm mixing and direct CP violation, and neutral-kaon CPT tests. The analytical backbone is the modular spin-density-matrix formalism for entangled baryon-antibaryon production and the standard hyperon, tau, and D-meson CP-violation observables. The paper combines a review of existing BESIII, Belle, LHCb, and PDG results with original feasibility studies based on fast Monte Carlo simulations, and it presents concrete projected sensitivities: O(10^-4) hyperon A_CP, hyperon EDM sensitivities of O(10^-20) to O(10^-21) e cm, a statistical sensitivity of 9.7e-4 for A_CP(tau -> KS pi nu) per ab^-1, and a projected CPT limit |M_Kbar0 - M_K0| near 4e-17 MeV. The central claim is that STCF would reach Standard Model levels for hyperon CPV, provide a decisive test of the BaBar tau anomaly, and improve the neutral-kaon CPT limit by an order of magnitude.

Significance. If the projected sensitivities hold after full detector validation, the physics case is strong and timely: the hyperon CPV measurements would probe the Standard Model weak phases in the strange sector, the tau asymmetry would address the BaBar anomaly, and the kaon CPT test would improve a three-decade-old limit. The paper's strengths include the use of well-established BESIII and PDG inputs, a clear modular formulation of spin observables, and explicit fast-MC studies that convert event yields into quantitative projections. The projections are falsifiable and the paper is candid about many of its limitations. However, the headline numbers rest on an internally fitted MC scaling constant, statistical-only extrapolations, and a qualitative systematics discussion, so the significance of the central claims is conditional on additional validation rather than fully established by the manuscript.

major comments (5)

[Section 2.6, Eq. (59)] The headline hyperon sensitivity O(10^-4) is obtained by fitting k = 7.82 in Eq. (59), sigma_ACP x sqrt(N_fin) = k, to MC samples between 0.01 and 0.1 trillion J/psi events and then extrapolating to 3.4 trillion J/psi, a factor of 34 beyond the fitted range. The paper does not quote an uncertainty on k, nor does it demonstrate that the Poisson scaling, the 38.2% selection efficiency, the <0.5% background fraction, or the simplified p > 500 MeV/c PID rule remain valid at the extrapolated luminosity. At minimum, a full-simulation cross-check at a statistically meaningful sample size and a propagation of the uncertainty on k are needed before the statement in Section 6 that STCF 'will reach levels of precision compatible with SM predictions' is supported.
[Section 2.6, systematic uncertainties] The text asserts that selection-related uncertainty sources for the hyperon CP test 'are statistically related, and are estimated to be in a level of 10^-4 ~ 10^-5', but no derivation, control-sample strategy, or numerical budget is provided. Since the BESIII measurements quoted in Section 1 carry ~0.2% systematic errors, and the projected statistical floor is 10^-4, the systematic discussion is a load-bearing gap: charge-asymmetric tracking and PID, material asymmetries, and fit biases must be quantified before the SM-reach claim is credible.
[Section 3.6, tau CPV projection] The 9.7e-4 statistical sensitivity for A_CP(tau -> KS pi nu) is derived from a fast simulation with simplified Breit-Wigner form factors in Eqs. (72)-(73), and the systematic discussion lists sources without numerically estimating them. In particular, the K0/K0bar nuclear-interaction asymmetry in STCF material is only addressed by analogy to K± nucleon cross sections from Ref. [192]; no material-budget calculation or expected correction uncertainty is provided. Because the BaBar anomaly is a 2.8-sigma effect, a measurement intended to settle it requires an end-to-end systematic error estimate, not only a statement that control samples will reduce the errors.
[Section 5.6, CPT projection] The claimed order-of-magnitude improvement to |M_Kbar0 - M_K0| ~ 4e-17 MeV, via Eq. (129) and the fit result in Eq. (143), is based on a MC sample of 3.9e9 tagged K -> pi+pi- decays generated with a phase-space model under the assumption of a nearly perfect detector (|cos theta| <= 0.85) and a background ratio below 0.5%, with statistical-only errors. The listed systematic sources (time resolution, background, regeneration, fixed parameters) are not assigned numerical values, even though the CPLEAR and E773 systematic errors quoted in Eq. (144) show that systematics dominate at this precision. Without a realistic detector simulation and a numerical systematic budget, the 'order of magnitude improvement' statement in Section 6 is not yet established.
[Section 4.7, charm mixing projections] The projected sensitivities for x, y, r_CP, and phi in Tables 8-10 and in the global fit (sigma(x)=0.036%, sigma(y)=0.015%, sigma(r_CP)=0.028, sigma(phi)=2.14 deg) are statistical-only and assume external strong-phase parameters with their current uncertainties; the text acknowledges strong phases as a prerequisite but does not propagate their uncertainties into the quoted numbers. Since the C-odd strong-phase measurements at STCF are themselves part of the program, the global sensitivities in Fig. 16 should either include a systematic component from the strong-phase inputs or be explicitly labeled as statistical-only.

minor comments (5)

[Abstract] The sentence 'opportunities for improved tests of CPT invariance test in K0-Kbar0 mixing' contains a duplicated and grammatically inconsistent 'test'; it should be rephrased, for example as 'improved tests of CPT invariance in K0-Kbar0 mixing'.
[Section 2.6, Eq. (57)] The normalization constant C is used in Eq. (57) before it is defined in the following sentence, and the text 'estimated the NMC events generated with the phase space model' appears to be missing a word such as 'using'; the undefined term 'mDIY MC' should also be clarified.
[Figure 1 caption] The caption first states 'figures taken from arXiv:1712.06147' and then says the figure was 'created using a modified script from Ref. [16]'; these attributions are inconsistent and should be reconciled.
[Section 5.3, Eq. (119)] The notation in Eq. (119) mixes psi and Psi and uses both lambda_k and lambda_K for the eigenvalue; the equation should be made internally consistent.
[Table 1] The column header 'Lumi Samples sigma Numbers (ab^-1) (nb) of Events' is unclear; the units should be attached to the individual columns and the sample labels separated for readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the projected sensitivities are Monte-Carlo extrapolations and PDG-input simulations, not redefinitions of fitted inputs.

full rationale

The paper's central numerical claims do not reduce to their inputs by construction. The hyperon A_CP projection is obtained by fitting Eq. (59), sigma_ACP sqrt(N_fin) = k with k = 7.82, to fast-simulation samples between 0.01 and 0.1 trillion J/psi events, and then extrapolating to 3.4 trillion J/psi. This is an internal calibration of a statistical scaling law, not a definition of the CP-violating observable; the target A_CP is extracted from a maximum-likelihood fit to the joint angular distribution. The tau-sector sensitivity of 9.7 x 10^-4 per ab^-1 follows from efficiency-corrected MC event counts and the standard 1/sqrt(N) scaling, with input values checked against generated samples rather than being used to define the asymmetry. The kaon CPT projection uses PDG values of eta_+- and phi_SW to simulate K0 and K0bar decay-time distributions, then fits phi_SW + Delta_phi_CPT; the quoted |M_K0bar - M_K0| limit follows from the independently derived relation in Eq. (129). The charm prospects use external strong-phase inputs from BESIII/CLEO and explicit formulas for coherent D-Dbar decays. Some sensitivity estimates cite prior work by overlapping authors, notably Refs. [19, 65, 69] for hyperon polarization formulas and EDM pseudo-data studies, but these are used as supporting methodological references rather than as a substitute for the derivation, and the authors' own manuscript describes the relevant fit procedure and inputs. No equation or fitted parameter is renamed as a prediction, and no uniqueness claim is imported from the authors' prior work to force a choice. The extrapolation of k to 3.4 trillion J/psi is an assumption about statistical scaling and detector performance, which is a validity risk rather than circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The projections rest on standard quantum mechanics, measured inputs from BESIII and PDG, and Monte Carlo assumptions about the future STCF detector. The main ledger items are the fitted MC scaling constant k, the data-fitted tau form-factor coefficients, the assumed luminosity and CP-odd pair production, and the fast-simulation fidelity. No new particles, mediators, forces, dimensions, or conserved quantities are introduced.

free parameters (2)

MC sensitivity constant k in sigma_ACP x sqrt(N_fin) = k = 7.82
Fitted to fast-simulation J/psi samples between 0.01 and 0.1 trillion events in Section 2.6, then extrapolated to 3.4 trillion J/psi to quote O(10^-4) hyperon CPV sensitivity. The central projection depends on this Monte Carlo calibration.
Effective tau form-factor coefficients a_K0*(800) and a_K*(1410) = Complex coefficients from Ref. [190]
Enter the MC signal model for tau -> KS pi nu in Eqs. (72)-(73). The reconstructed yield and the quoted A_CP sensitivity depend on these data-fitted parameters.

assumptions (5)

domain assumption STCF will reach a peak luminosity of 0.5 x 10^35 cm^-2 s^-1 and deliver about 1 ab^-1 per year at the relevant energies.
Table 1 and Section 1 state the event yields; all projected sensitivities scale with these samples. If luminosity is lower, the quoted reach degrades proportionally.
domain assumption Charmonium decays into particle-antiparticle pairs produce a pure CP-odd quantum-correlated state when the production process conserves CP.
Used throughout Sections 2, 4, and 5 to build angular distributions and double-tag observables, e.g., Eqs. (31)-(35), (91), and the strangeness-tagged kaon interference.
domain assumption Fast simulation and simplified selection efficiencies faithfully represent the final STCF detector response.
Efficiency 38.2%, background below 0.5%, and vertex resolution are taken from fast simulation in Sections 2.6, 3.6, and 5.6. The numerical projections inherit these assumptions.
domain assumption Known measured inputs such as BESIII alpha_psi, Delta_Phi_psi, PDG branching fractions, form factors, and CKM parameters are correct and applicable.
These inputs set the polarization, event yields, and Standard Model expectations that anchor the sensitivity studies. Errors in these inputs propagate into the projected uncertainties.
standard math The Bell-Steinberger unitarity relation and the Wigner-Weisskopf effective Hamiltonian describe neutral kaon time evolution.
Section 5.5 derives CPT constraints from unitarity. This is a standard and accepted formal result of the neutral kaon system.

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Pith. "Pith review of CP violation studies at Super Tau-Charm Facility." pith.science (2026). https://pith.science/paper/FHZBRJIJ

@misc{pith2026250208907,
  author       = {Pith},
  title        = {Pith review of: CP violation studies at Super Tau-Charm Facility},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FHZBRJIJ}},
  note         = {Machine review of arXiv:2502.08907}
}

abstract

Charge-parity ($C\!P$) violation in the tau-charm energy region is a promising area for sensitive tests of Standard Model (SM) predictions and searches for new, beyond the SM physics. A future Tau-Charm Facility that operates at center-of-mass energies between 2.0 and 7.0 GeV, with a peak luminosity of $0.5\times10^{35}$~cm$^{-2}$s$^{-1}$, would provide huge numbers of hadrons and tau ($\tau$) leptons that are produced in low-background environments and with well understood kinematic properties. In this report, prospects for unique studies of $C\!P$ violation in the decay of charmed hadrons, and in the production and decay of hyperons and $\tau$ leptons at a next-generation tau-charm facility are discussed. In addition, opportunities for improved tests of $CPT$ invariance test in $K^{0}-\bar{K}^{0}$ mixing are presented.

Figures

Figures reproduced from arXiv: 2502.08907 by the authors.

**Figure 2.** Summary of the results for the weak phases [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

**Figure 3.** Magnitudes of the hyperon polarization as a function of the production angle for: (a) [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗

Figures from the paper (18 more)

**Figure 4.** Figure 4: Sensitivity of EDM for different hyperons. The markers for hyperons [PITH_FULL_IMAGE:figures/full_fig_p029_4.png]

**Figure 5.** Figure 5: The blue dots represent the statistical errors of [PITH_FULL_IMAGE:figures/full_fig_p032_5.png]

**Figure 6.** Figure 6: Invariant masses of K0 Sπ − with combined e-tag and µ-tag from τ + decay after selection. Normalized according to the luminosity of 1 ab−1 at √ s = 4.26 GeV. final states remain. The mass spectrum of K0 Sπ − after the above selections is shown in [PITH_FULL_IMAGE:figu…

**Figure 7.** Figure 7: The charmed CKM unitarity triangle associated with V ∗ udVcd + V ∗ usVcs + V ∗ ubVcb = 0 in the complex plane. The relevant angle determining charmed CP violation in the SM is highlighted with a red dot. 50 [PITH_FULL_IMAGE:figures/full_fig_p050_7.png]

**Figure 8.** Figure 8: An illustration of the box diagrams for D0 -D¯ 0 mixing and the penguin diagrams for the D0 → π +π − decay. 4.2. CP violation in the direct decays As given in Eq. (4), at least two amplitude components with different weak phases and different strong phases are the nece…

**Figure 9.** Figure 9: The tree and penguin Feynman diagrams for [PITH_FULL_IMAGE:figures/full_fig_p052_9.png]

**Figure 10.** Figure 10: Comparison between experimental measurements (in black) and theoretical predic [PITH_FULL_IMAGE:figures/full_fig_p054_10.png]

**Figure 11.** Figure 11: Constraints on the D0 -D¯ 0 mixing parameters x and y as obtained by the HFLAV group from current data [235]. illustrates the present experimental constraints on the argument and modulus of q/p, which are consistent with the SM expectation. NP might contribute to D0 -…

**Figure 12.** Figure 12: Constraints on the modulus and argument of [PITH_FULL_IMAGE:figures/full_fig_p059_12.png]

**Figure 13.** Figure 13: The CP-conserving strong-phase differences between D0 and D¯ 0 decays in phase-space regions are required as inputs to such studies. These parameters can be accessed with the C-odd correlated DD¯ pairs produced at the same energy point [260] [PITH_FULL_IMAGE:figures/…

**Figure 14.** Figure 14: 1, 2 and 3σ regions of the charm mixing parameters and indirect CP violation parameters by analyzing the D → K0 Sπ +π − decay. Measurements of the D → K−π +π 0 decay. Similar to the D → K−π +π +π − decay, the D → K−π +π 0 decay can be studied with the C-even correlat…

**Figure 15.** Figure 15: 1, 2 and 3σ regions of the charm mixing parameters and indirect CP violation parameters by analyzing the D → K−π +π 0 decay. Overall prospects. A global fit of the DT D → K−π +π 0 , K0 Sπ +π − and K−π +π +π − signal decays and flavor-specific ST gives the sensitiviti…

**Figure 16.** Figure 16: 1, 2 and 3σ regions of the charm mixing parameters and indirect CP violation parameters by analyzing the D → K−π +π 0 , K0 Sπ +π − and K−π +π +π − decays. Prerequisites for the charm mixing and indirect CP violation studies at charm and B factories are the strong-pha…

**Figure 17.** Figure 17: In string theories, elementary particles are tiny loops of oscillating strings with no [PITH_FULL_IMAGE:figures/full_fig_p077_17.png]

**Figure 18.** Figure 18: The box diagrams for the short-distance contributions to [PITH_FULL_IMAGE:figures/full_fig_p078_18.png]

**Figure 19.** Figure 19: a) A simulated J/ψ → K−π +K0 (τ ); K0 (τ ) → π +π − event in the BESIII detector. b) The relative arrangements in the complex plane of the complex quantities discussed in the text. Here, for display purposes, the magnitudes of δ and ε ′ relative to ε are exaggerated. …

**Figure 20.** Figure 20: a) The 68% and 95% confidence level allowed region for Im mδ and Re eε from a Bell-Steinberger analysis. b) The corresponding allowed regions in ∆Γ= ΓK¯ 0−ΓK0 and ∆M = MK¯ 0 − MK0 . From Ref. [269]. 84 [PITH_FULL_IMAGE:figures/full_fig_p084_20.png]

**Figure 21.** Figure 21: a) The solid circles show the proper time distribution for simulated strangeness-tagged K0 (τ ) → π +π − decays (the open circles are K¯ 0 (τ )→π +π − decays). b) The reduced asymmetry, A′ π+π− , for the events shown in panel a). The simulated data shown in Figs. 21 (…

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