Flavor physics at the CEPC: a general perspective

The Circular Electron-Positron Collider (CEPC) [1, 2] was proposed in 2012 by the Chinese high-energy physics community to function primarily as a Higgs factory at a center-of-mass energy of 240 GeV. It is also set to operate as a Z factory at the Z pole, conduct precise WW threshold scans, and potentially be upgraded to operate at a center-of-mass energy of 360 GeV, i.e., above the $ t \bar t $ threshold. In the proposed nominal operation scenario [1, 3], the CEPC is anticipated to produce significant numbers of Higgs and Z bosons, W boson pairs and, potentially, top quarks. With respect to the accelerator design, the development of key technologies has led to a significant enhancement in the instantaneous luminosity per interaction point (IP) compared to those reported in the Conceptual Design Report (CDR), as shown in Fig. 1. Based on this progress, the CEPC study group proposes a new nominal operation scenario, shown in Table 1, which would allow for precision measurements of Higgs boson couplings, electroweak (EW) observables, and QCD differential rates. It would also provide ample opportunities to search for rare decays and new physics (NP) signals. Moreover, the large quantities of bottom quarks, charm quarks, and tau leptons from the decays of Z bosons create opportunities for numerous critical flavor physics measurements. It should be noted that the results presented here are based on the updated running scenario using a 50 MW synchrotron radiation (SR) power beam [1].

Figure 1.  (color online) Designed luminosities of the CEPC at the Z pole, Higgs, WW, and the $ t\bar t $ thresholds operation modes with the baseline and upgrade shown in solid and dashed blue curves, respectively. Luminosities for several other proposals of $ e^-e^+ $ colliders are also shown for comparison. See Ref. [1] for details.

Operation mode	Z factory	WW threshold	Higgs factory	$ t\bar{t} $
$ \sqrt{s} $/GeV	91.2	160	240	360
Run time/y	2	1	10	5
Instantaneous luminosity ($ 10^{34} {\rm{cm}}^{-2}{\rm{s}}^{-1} $, per IP)	191.7	26.7	8.3	0.83
Integrated luminosity ($ {\rm{ab}}^{-1} $, 2 IPs)	100	6.9	21.6	1
Event yields	$ 4.1 \times 10^{12} $	$ 2.1 \times 10^{8} $	$ 4.3 \times 10^{6} $	$ 0.6 \times 10^{6} $

Table 1.  Nominal CEPC operation scheme of four different modes. See [1, 3] for details.

Flavor physics, as a well-developed area within particle physics, has contributed substantially to the establishment of the Standard Model (SM) over recent decades. This was achieved through the examination of the properties of SM fermion flavors in a myriad of experiments, yielding significant findings and discoveries. The CEPC can serve as a flavor factory, and its flavor physics program enhances the CEPC's overarching physics objectives. The flavor sector provides substantial motivations for the CEPC operation, given the existing multitude of unknowns within the SM and beyond.

Understanding the flavor physics potential of the CEPC is not an isolated field of study, as it also influences other primary fields of explorations at the CEPC, including Higgs physics, EW precision observables (EWPOs), QCD, and Beyond the Standard Model (BSM) physics. For instance, within the SM the fermion mixing, specifically the Cabibbo-Kobayashi-Maskawa (CKM) matrix [4, 5] and its hierarchical structure, originates from the Yukawa couplings of the Higgs field to the fermion gauge eigenstates. While some of the diagonal Yukawa couplings will be pinned down by the direct Higgs measurements at CEPC [6], studying the origin of the off-diagonal flavor mixing terms and their CP-violating phases remains mainly within the realm of flavor physics. Conversely, while most heavy-flavored particles decay via EW transitions at the tree level, many rare processes are only induced by EW one-loop effects, such as Flavor-Changing-Neutral-Current (FCNC) transitions. Their measurements may also serve as alternative tests of the EW sector at an energy scale lower than Z-pole measurements. Meanwhile, many EWPOs necessitate precise flavor tagging and high-precision reconstruction, e.g., the forward-backward asymmetry of charm and bottom quarks. Furthermore, most flavor physics studies involve QCD since all quarks are colored and τ leptons can decay to hadronic final states. In fact, most flavor physics studies rely on the theory of QCD, both perturbatively and non-perturbatively, to provide insights into the corresponding production, spectroscopy, and decays of hadronic states. In turn, the plethora of flavor measurements could provide crucial inputs to, and calibration of, QCD theory in multiple ways. It is also noteworthy that flavor physics provides a set of probes sensitive to BSM physics. For instance, the decay of a heavy-flavored fermion is suppressed by EW scale, $ G_F^2 m_f^4 \lesssim 10^{-7} $, and consequently f becomes long-lived. Such a narrow width makes it possible to reveal even small BSM effects, which are not easily observable otherwise. Finally, the ambitious goals of flavor physics studies motivate developments on the instrumentation frontier, demanding enhanced detector performance in vertexing, tracking, particle identification (PID), and calorimetry.

The successful realization of the flavor physics program at the CEPC relies on a number of key factors:

● An abundant luminosity of the data at the CEPC Z pole, which yields substantial heavy flavor statistics. With a high integrated luminosity and the large cross section $ \sigma (e^-e^+\to Z \to b\bar{b},c\bar{c},\tau^- \tau^+) $, the Tera-Z will generate extensive statistics of heavy-flavored hadrons and τ leptons [2], rivaling other proposed flavor physics experiments. This is demonstrated by the expected yields of b-hadrons in Belle II, LHCb and a representative future Z factory, as listed in Table 2. The Tera-Z yields approximately $ 4.8\times 10^{11} $ $ B^0/\bar B^0 $ or $ B^\pm $ mesons, which is one order of magnitude larger than that expected at Belle II [7]. Even though this yield is roughly two orders of magnitude lower compared to that of LHCb, studies at the Tera-Z can benefit significantly from the clean experimental environment and the precisely known center-of-mass energy.

Particle	BESIII	STCF (1 ab−1)	Belle II (50 ab−1 on $ \Upsilon(4S) $)	LHCb (300 fb−1)	CEPC (TDR)
$ B^{0} $, $ \bar{B}^{0} $	−	−	$ 5.4 \times 10^{10} $	$ 3 \times 10^{13} $	$ 4.8 \times 10^{11} $
$ B^{\pm} $	−	−	$ 5.7 \times 10^{10} $	$ 3 \times 10^{13} $	$ 4.8 \times 10^{11} $
$ B^{0}_{s} $, $ \bar{B}^{0}_{s} $	−	−	$ 6.0 \times 10^{8} $ (5 ab−1 on $ \Upsilon(5S) $)	$ 1 \times 10^{13} $	$ 1.2 \times 10^{11} $
$ B^{\pm}_{c} $	−	−	−	$ 1 \times 10^{11} $	$ 7.2 \times 10^{8} $
$ \Lambda_{b}^{0} $, $ \bar{\Lambda}_{b}^{0} $	−	−	−	$ 2 \times 10^{13} $	$ 1 \times 10^{11} $
$ D^0 $, $ \bar{D}^0 $	$ 1.2 \times 10^{8} $	$ 7.2 \times 10^{9} $	$ 4.8 \times 10^{10} $	$ 7 \times 10^{14} $	$ 8.3 \times 10^{11} $
$ D^{\pm} $	$ 1.2 \times 10^{8} $	$ 5.6 \times 10^{9} $	$ 4.8 \times 10^{10} $	$ 3 \times 10^{14} $	$ 4.9 \times 10^{11} $
$ D^{\pm}_s $	$ 1 \times 10^{7} $	$ 1.8 \times 10^{9} $	$ 1.6 \times 10^{10} $	$ 1 \times 10^{14} $	$ 1.8 \times 10^{11} $
$ \Lambda^{\pm}_c $	$ 0.3 \times 10^{7} $	$ 1.1 \times 10^{9} $	$ 1.6 \times 10^{10} $	$ 1 \times 10^{14} $	$ 6.2 \times 10^{10} $
$ \tau^{+}\tau^{-} $	$ 3.6 \times 10^{8} $	$ 3.6 \times 10^{9} $	$ 4.5 \times 10^{10} $		$ 1.2 \times 10^{11} $

Table 2.  Expected yields of b-hadrons, c-hadrons, and τ leptons at BESIII, STCF, Belle II, LHCb Upgrade II, and CEPC (according to the TDR [1], $ 4\times 10^{12} $ Z bosons are expected). For b- and c-hadrons, their yields include both charge conjugates, while the yield of τ leptons refers to the $ \tau^- \tau^+ $ events, namely the number of τ pairs. We take the cross sections for $ b\bar b $ and $ c\bar c $ productions at center-of-mass energies corresponding to $ \Upsilon(4S) $ and $ \Upsilon(5S) $ from Ref. [7], and of the b quark productions within LHCb detector acceptance from Ref. [8]. To estimate the production fractions of $ B^0 $ and $ B^{\pm} $ at LHCb, we utilize the $ B^0_s $ and $ \Lambda^0_b $ production fractions in Ref. [9] and assume $ f_u + f_d + f_s + f_{\rm{baryon}} = 1 $, with $ f_u = f_d $, and $ f_{\Lambda^0_b} = f_{\rm{baryon}} $. For Z decays, the production fractions of $ B^0 $, $ B^{\pm} $, $ B^0_s $, and $ \Lambda^0_b $ are presented in Ref. [10]. The $ B_c $ meson production fraction at LHCb is taken from Ref. [11], while its production fraction at the Z pole (including the contribution from $ B_c^* $ decays) is taken from Ref. [12]. For inclusive charm meson productions at the Z pole, including the contribution from b-hadron decays, see Refs. [13−17]. The yields of τ leptons at the CEPC are rescaled from Ref. [2]. The particle yields at the STCF are taken from Ref. [18].

● The clean environment of $ e^-e^+ $ collisions constitutes another cornerstone, substantially diminishing the background level and systematic uncertainties associated with neutral particles. This environment is particularly beneficial to flavor physics studies involving heavy b-hadrons, especially given the significantly limited event reconstruction efficiency in the noisy data environment of the LHCb [19].

● The scale separation $ m_Z \gg m_{b,c,\tau} \gtrsim \Lambda_{\rm{QCD}} $ underpins the success of the project, as it facilitates the production of a wide array of particle species. In addition, even decay products with low momentum in the center-of-mass frame of heavy-flavored particles are expected to be boosted to higher energies and larger displacements. The significantly higher boost at the Z factory compared to the B and C factories offers substantial advantages for particle identification and measurement precision.

● Lastly, state-of-the-art detector technologies and algorithms for data analysis under development today will be crucial when deployed in the CEPC era. These technologies will enhance the investigation of extremely rare decay modes that contain neutral or invisible particles, as the cleanliness of a lepton collider enables such studies. The evolving field of advanced algorithms, especially deep learning ones, could also benefit flavor physics at the CEPC in almost all aspects by fully utilizing the large amount of data recorded from the hardware.

While the flavor physics program at the CEPC benefits from the various advantages above, it confronts new challenges. The first of these challenges is related to the significant increase in event statistics at the CEPC, which is expected to be greater by a factor of $ \gtrsim {\cal{O}}(10^5) $ than the LEP run at the Z pole. Given the improved detector systems and electronics, the volume of data to be processed will increase substantially. Meanwhile, the precision goals of flavor physics, driven by theoretical interests, will also reach an elevated level in the CEPC era. Therefore, it becomes essential to improve the understanding of backgrounds and to control systematic effects to match statistical uncertainties, thus to fully benefit from CEPC's luminosity.

A second challenge arises from the multitude of viable channels to be studied at the CEPC. Compared to the other proposed future flavor physics experiments (or the upgrades of the current ones), the improvement achievable at the CEPC varies significantly channel by channel. Initial studies indicate that while the CEPC could enhance the precision of measurements by orders of magnitude in many instances, the improvement could be marginal in others. Therefore, identifying the most valuable systems, or "golden channels" - those with the highest potential for significant progress or even discovery potential - for investigation in the CEPC context could substantially reduce the allocation of future resources. As it stands, some of these golden modes at the CEPC may have been overlooked as they are not suited for the existing experiments.

Besides these aforementioned experimental challenges, control of theoretical uncertainties is critical for CEPC flavor physics measurements and their interpretation. Theoretical inputs come in multiple forms, such as the non-perturbative theory of hadronization, perturbative QCD and EW corrections to fermion production, lattice extrapolations of heavy flavor form factors, the relation between the CKM matrix elements and the observed CP asymmetries, as well as the proper modeling of the electron beam and detector system. To accurately scrutinize the SM and to search for NP, the precision of these theoretical tools must align with those of the experimental outputs.

The principal objective of this document is to present a general perspective on the discovery potential of flavor physics at the CEPC, through Monte Carlo (MC) simulations and relevant phenomenological analyses. During the compilation of this white paper, simultaneous efforts were dedicated to promoting flavor physics programs at other future lepton colliders, such as the Future Circular Collider (FCC-ee) [20, 21] and the International Linear Collider (ILC) [22], both of which also include a Z factory phase and higher energy operations. In particular, the FCC-ee Z pole run has a similar integrated luminosity (180 ab⁻¹) to the current CEPC proposal, and the higher-energy runs are likewise comparable. Since both proposals share similar detector performances [2, 23], and both adopt a particle flow algorithm (PFA)-oriented detector design [2] and IDEA (Innovative Detector for Electron-positron Accelerator) detector design [24], some relevant FCC-ee studies were also incorporated into the current summaries, with only minimal rescaling applied as necessary. For the same reason, many physical discussions and conclusions in this white paper could also be applied to the FCC-ee project.

This document is structured as follows. In Sec. II, we provide an overview of the CEPC facility, delineating key features of the collider and the detector that are crucial for flavor physics. Additionally, the simulation methods utilized at the CEPC are explained. Sec. III delves into Flavor-Changing-Charged-Current (FCCC) semileptonic and leptonic b decays, discussing their theoretical framework, recent progress and future research directions. Rare b decays mediated by FCNC are explored in Sec. IV, featuring a preliminary theoretical interpretation and discussion of di-leptonic, neutrino and radiative modes. Sec. V is dedicated to the discussions on the measurements of CP asymmetries. Secs. VI and VII focus on charm/ strange and τ physics respectively. Flavor physics measurements via leptonic or hadronic Z decays are discussed in Sec. VIII. Sec. IX extends the discussions to flavor physics at higher energies, including $ |V_{cb}| $ measurements through on-shell W boson decays, Higgs exotic and FCNC decays, as well as touching upon other possibilities. Prospects of hadron spectroscopy and exotic states are covered in Sec. X. The production of light BSM particles by heavy flavor interactions forms the central theme of Sec. XI. The detector performance requirements for a successful flavor physics program at the CEPC are discussed in Sec. XII. Finally, we summarize the topics covered in this document and provide an outlook for future explorations in Sec. XIII.

As an $ e^-e^+ $ collider operating around the EW scale, flavor physics studies at the CEPC are affected by three major features. Firstly, as $ \sqrt{s}\gg m_{b,c,\tau} $, the CEPC produces highly relativistic heavy-flavored quarks or leptons. Their boosted decay products allow for precise momentum and lifetime measurements. This is in contradistinction to the situations at low energy $ e^-e^+ $ colliders such as Belle II [7], BaBar [25], BESIII [26], and other future proposals, such as the Super Tau-Charm Factory (STCF) [18]. Secondly, as an $ e^-e^+ $ collider, the CEPC provides a clean environment for flavor physics studies with low QCD backgrounds, negligible pileup events, and an almost fixed $ E_{cm} $. Compared to hadron collider experiments, such as the LHCb [27], the CEPC enables more effective identification and reconstruction of final states that include neutral or invisible particles. The above arguments show the uniqueness of CEPC flavor physics studies. Thanks to advanced accelerator design, the large instantaneous luminosity will allow us to collect $ {\cal{O}}(10^5) $ times more statistics than the LEP Z pole run [28]. As a consequence, the search and analysis strategies may differ significantly from those employed in the relevant studies at LEP. For instance, high signal statistics allows sharper cuts to reduce backgrounds. At the same time, one needs to carefully address other systematic uncertainty sources using the plethora of data. Hence, the large luminosity of the CEPC brings new challenges and existing projections based on LEP must be reconsidered. Such challenges are especially severe for precision measurements. According to the CEPC CDR [2], the beam energy spread could typically be controlled to the level of 0.1%. This, together with a detector that can reconstruct precisely hadronic events – allowing for precise determination of missing energy/momentum – thus enables relevant physics measurements with high precision; for instance, tagging semileptonic heavy quark decay and searching for dark matter candidates in hadronic events, especially in the Z factory mode.

The CEPC uses a nano-beam scheme and therefore the typical beam spot sizes are of order µm in the x direction, of order nm in the y direction, and correspondingly of order a few hundred µm in the z direction. The beam sizes at different operation modes of the CEPC are summarized in Table 3. The accelerator will provide a collision area with a typical size of order µm in the transverse direction and of order 100 µm along the beam direction. The spatial uncertainty of the interaction point can therefore be limited, enabling high precision measurements with τ final states – for example, in dark matter searches with $ Z\to\tau^- \tau^+ $ events at the Z factory.

Operation mode	Z factory	WW threshold	Higgs factory	$ t\bar{t} $
$ \sqrt{s} $/GeV	91.2	160	240	360
Beam size $ \sigma_x $/µm	6	13	14	39
Beam size $ \sigma_y $/µm	0.035	0.042	0.036	0.113
Bunch length (total, mm)	8.7	4.9	4.1	2.9
Crossing angle at IP (mrad)	33

Table 3.  Beam size, bunch length, and crossing angle at different operation modes of the CEPC [1, 3].

Flavor physics program at Tera-Z is enormously rich and extremely demanding on detector performance. In general, a Tera-Z detector would have a large acceptance with a solid angle coverage up to $ |\cos\theta| < 0.99 $. This detector would also have low energy and momentum thresholds at the 100 MeV level to record and recognize low energy objects that characterize certain hadron decays, e.g., soft photons and pions generated from excited heavy hadrons, as well as some low energy hadrons that are essential for understanding relevant QCD processes [29].

To efficiently separate signal events from background, it is essential to identify the relevant physics objects and to precisely reconstruct their properties — especially their energies and momenta. For a Tera-Z detector, a typical benchmark is to reconstruct the intermediate particles, such as $ \pi^0\to\gamma\gamma $, $ K^0_S\to\pi^+\pi^- $, $ \phi\to K^+K^- $, $ \Lambda\to p\pi^- $, etc., inside hadronic Z events. A more challenging case would be to identify the decay products of a target heavy-flavored hadron which may decay into $ {\cal{O}}(10) $ particles with a complicated and rich decay cascading order inside a jet. These decay products include not only charged final state particles (leptons and charged hadrons), but also photons, neutral hadrons, and the missing energy/momentum induced by neutrinos. A hadronic Z event could have up to 100 final state particles, as shown in Fig. 2. To successfully separate and reconstruct the decay products of the target particle is a key challenge for measurements performed in hadronic Z events, for which it is necessary to employ the particle flow method [30, 31]. Such a method emphasizes the separation of final state particles and has been proven capable of providing better reconstruction of both the hadronic system and the missing energy/momentum.

Figure 2.  (color online) Multiplicities of different types of final state particles in $ Z\to q\bar{q} $ (91.2 GeV) and $ Z(\to q\bar{q})H(\to {\rm{inclusive}}) $ (240 GeV) events.

In addition, good intrinsic resolution of subdetectors (i.e., momentum reconstruction by the tracker and energy measurement by the calorimeter), is always critical for flavor physics measurements. It not only enables the precise reconstruction of physics properties, such as particle masses, but also significantly reduces the combinatorial background, which is especially present in physics measurements involving narrow resonances. In particular, achieving excellent electromagnetic (EM) energy resolution with a particle-flow-oriented, high-granularity calorimeter is both challenging and necessary for the flavor physics program, as photons and neutral pions are common decay products in many fundamental flavor physics measurements. A benchmark analysis [32] of the measurement of the standard CKM unitarity triangle angle α via $ B^0\to\pi^0\pi^0 $ suggests an EM resolution of approximately $ {\cal{O}}( $3%$/\sqrt{E{\rm{(GeV)}}}) $ to fulfill the requirement of 3 σ separation between $ B^0 $ and $ B^0_s $ with a 30 MeV B-meson mass resolution.

Most flavor physics measurements at the CEPC will involve hadronic events, particularly di-jet events at the Z pole. It is essential to identify the origin of a jet, i.e., to determine whether it originates from a quark, an anti-quark, or even a gluon. The jet origin identification [33], to a certain extent, shall be regarded as a natural extension of jet flavor tagging, quark-gluon jet separation, and jet charge measurements, which is indispensable in flavor physics measurements such as CKM and CP violation measurements.

A successful flavor physics program also needs high efficiency/purity PID. An efficient PID not only suppresses the combinatorial background, induced by misidentified particles, but also separates decays with similar topologies in the final states, such as $ B^0_{(s)}\to\pi^+\pi^- $, $ B^0_{(s)}\to K^+K^- $, and $ B^0_{(s)}\to K^{\pm}\pi^{\pm} $ [34]. A decent PID is also critical for the jet origin identification [33, 35] and relevant physics measurements such as the Higgs rare/exotic decay measurement [33]. The benchmark analysis of $ B_s^0\to \phi\nu\bar{\nu} $ [36] shows that a relative uncertainty of BR($ B_s^0\to \phi\nu\bar{\nu} $) less than 2% at a Tera-Z collider requires a 3 σ $ K^\pm/\pi^\pm $ separation for the identification of charged hadrons, see the left panel of Fig. 3. This requirement can be addressed by multiple PID technologies. For instance, the CEPC CDR detector [2] can separate different species of hadrons using ${\rm d}E/{\rm d}x$ information measured by the time projection chamber (TPC) and time-of-flight (TOF) information provided by either a dedicated TOF device, or by combining TOF and EM calorimeter (ECAL) together. The detector optimization study in Ref. [37] suggests that $ {\rm d}E/{\rm d}x $ needs to reach 3% in combination with a TOF resolution of 50 ps to satisfy this PID requirement. In addition, the $ {\rm d}N/{\rm d}x $ cluster-counting method proposed by the IDEA drift chamber [38] is promising to further improve the PID performance.

Figure 3.  (color online) LEFT: Precision of measuring BR($ B_s^0\to\phi\nu\bar{\nu} $) as a function of PID performance, parameterized by the $ K/\pi $ separation power [36]. RIGHT: Precision variance of measuring $ {\rm{BR}}(H_b\to H_c \tau\nu_\tau ) $ as a function of detector vertex uncertainties [39], with starred reference point set by a vertex uncertainty of 10 μm.

A high-precision and low-material vertex system is vital for the CEPC flavor physics program. Precise vertex measurements provide pivotal information to distinguish the species of the initial quark that fragments into a jet, namely the jet origin identification. Precise vertex information is also critical for determining the decay time or lifetime of heavy-flavored hadrons with high precision. To match the characteristic timescales such as those of $ B_s^0-\bar{B}_s^0 $ mixing ($ \sim 56 $ fs), of $ D_s $ decay ($ \sim 500 $ fs), and of τ decay ($ \sim 290 $ fs), the decay time resolution is required to reach order $ {\cal{O}}(10) $ fs. These accurate lifetime measurements will also benefit flavor tagging and time-dependent CP violation measurements. In addition, a high-performance vertex system can provide a precise reconstruction of the secondary vertices that characterize some of the heavy-flavored hadron decays, such as the example shown in Fig. 4. Such a system can also help to suppress the backgrounds, especially from the IP. One concrete application can be the measurements of FCCC $ {\rm{BR}}(H_b\to H_c \tau\nu_\tau ) $, where the reconstruction of the b hadron $ H_b $ can significantly rely on the determination of the decay vertex of the charmed hadron $ H_c $ and on the measurement of the muon track originating from the τ decay [39]. As shown in the right panel of Fig. 3, the improved resolution of the vertex system can uniformly benefit these measurements, yielding an improvement in precision of $ {\cal{O}} $(10%) level.

Figure 4.  (color online) Display of a $ Z\to b\bar{b} $ event with typical secondary vertices (SV).

The above-mentioned requirements are also highly beneficial for the physics programs at higher center-of-mass energies, i.e., the 160 GeV $ W^+W^- $ threshold scan, the 240 GeV Higgs run, and the 360 GeV top-pair operation. On top of their core physics programs, such as W mass and precise Higgs/top properties measurements, the data samples and key detector features also support an intensive flavor physics program, see Sec. IX.

To address these physics requirements, intensive efforts have been devoted to the detector conceptual design, physics performance studies, and key technology R&D. We refer to two benchmark detector concepts considered in the CDR study [2]. These concepts are used in the simulations presented in this manuscript, providing reference performance for relevant physics potential studies.

The starting point of our discussion is the particle-flow-oriented detector design in the CEPC CDR [2]. As the majority of the full simulation studies use this detector design, we will refer to it as the CDR detector for simplicity. Guided by the particle flow principle, the CDR detector features a high-precision tracking system, a high-granularity calorimeter system, and a high magnetic field. As shown in detail in Fig. 5, from inside to outside, the CDR detector consists of a silicon pixel vertex detector, a silicon tracker, a TPC, a silicon-tungsten sampling EM calorimeter (Si-W ECAL), a steel-glass Resistive Plate Chambers (RPC) sampling hadronic calorimeter (SDHCAL), a superconducting solenoid magnet providing a magnetic field of 2–3 Tesla, and a flux return yoke embedded with a muon detector. Additionally, the Si-W ECAL could also be instrumented with a few timing layers to enable TOF measurements with a precision of 50 ps or even better [2, 41].

Figure 5.  (color online) Schematic layouts of the CEPC CDR detector [2] (LEFT) and the IDEA detector [40] (RIGHT).

Alongside the CDR detector, an alternative detector concept known as IDEA [40] is also utilized in various studies covered in this white paper. In comparison to the CDR detector, the IDEA detector incorporates a dual readout calorimeter system to attain superior energy resolution for both EM and hadronic showers. Moreover, the IDEA detector operates with a reduced magnetic field of 2 Tesla while compensating for this reduction by offering a larger tracking volume. The overall structure of both the detectors can be seen in Fig. 5.

By virtue of the PFA-oriented design, the CEPC CDR detector performs well in efficient tracking, lepton identification, and precise reconstruction of hadronic systems. These excellent features of the CEPC CDR detector provide a solid basis for flavor physics studies. The expected performance of the CEPC CDR detector is summarized in Table 4. Notably, the CDR tracking system demonstrates an efficiency close to 90% and a relative momentum resolution approaching $ {\cal{O}}(10^{-3}) $ for individual tracks with momenta exceeding 1 GeV within the barrel region, as illustrated in Fig. 6. As depicted in left panel of Fig. 7, the CDR photon energy resolution is 17% / $\sqrt{E{\rm{(GeV)}}} \oplus $1%, achieved by the sampling Si-W ECAL, which features the high granularity critical for particle flow reconstruction. In terms of PID performance, the CEPC CDR design achieves a $ K/\pi $ separation better than 2 σ in the momentum range up to 20 GeV by effectively combining TOF and $ {\rm d}E/{\rm d}x $ information, as shown in Fig. 8. The inclusive $ Z\to q\bar{q} $ sample exhibits an overall $ K^\pm $ identification efficiency and purity exceeding 95% [37]. Regarding hadronic systems, the CEPC CDR detector attains a boson mass resolution (BMR) better than 4% for hadronically decaying W, Z, and Higgs bosons, as illustrated in right panel of Fig. 7. This not only enables a separation exceeding 2 σ between W and Z bosons in their hadronic decays, but also enhances the precision of missing energy/momentum measurements, which are vital for flavor physics investigations.

Item	CDR [2]	$ 4^{\rm{th}} $ concept [42]	Comments
Basic Performance
Acceptance	$ |\cos\theta|<0.99 $ [2]
Threshold	200 MeV [43, 44]	100 MeV	For tracks & photons
Beam energy spread	$ {\cal{O}} $(0.1%) [2]
Tracker momentum resolution	$ {\cal{O}} $(0.1%) [2]
ECAL energy resolution	17%$ /\sqrt{E{\rm{(GeV)}}} \oplus $1% [2]	3%$ /\sqrt{E{\rm{(GeV)}}} $ [32]
HCAL energy resolution	60%$ /\sqrt{E{\rm{(GeV)}}} \oplus $1% [2]	30%$ /\sqrt{E{\rm{(GeV)}}} $ [45]
Vertex resolution	10–200 µm [2]	5–100 µm
Jet energy resolution	3%–5% [2, 46]		For 20–100 GeV
$ \ell-\pi $ mis-ID	<1% [47]		In jet, $ |\vec{p}|> 2 $ GeV
$ \pi-K $ separation	$> 2\sigma $ [2]	$> 3\sigma $ [36]	In jet, $ |\vec{p}|> 1 $ GeV, TOF+${\rm d}E/{\rm d}x$
Flavor Physics Benchmarks (Depending on the Above)
$ \sigma(m_{H,W,Z}) $	3.7% [2]		Hadronic decays
b-jet efficiency$ \times $purity	~86% [33]		In Z hadronic decays
c-jet efficiency$ \times $purity	~64% [33]		In Z hadronic decays
b-jet charge tagging $ \epsilon_{\rm{eff}}=\epsilon(1-2\omega)^2 $	~37% [33]
c-jet charge tagging $ \epsilon_{\rm{eff}}=\epsilon(1-2\omega)^2 $	~58% [33]
$ \pi^0 $ efficiency$ \times $purity	$ \gtrsim $70% [43]	$ \gtrsim $80% [32]	In Z hadronic decays, $ |\vec{p}_{\pi^0}|>5 $ GeV
$ K_S^0 $, Λ efficiency	60%–85% [48]		In Z hadronic decays, all tracks
τ efficiency$ \times $purity	70% [49]		In $ WW\to\tau\nu q\bar{q}^\prime $, inclusive
τ mis-ID	$ {\cal{O}} $(1%) [49]		In $ WW\to\tau\nu q\bar{q}^\prime $, inclusive

Table 4.  Performance of the CEPC CDR detector and some suggested objectives.

Figure 6.  (color online) Single track reconstruction efficiency (LEFT) and momentum resolution (RIGHT) of the CEPC CDR detector [2].

Figure 7.  (color online) LEFT: Comparison of the CEPC CDR photon energy resolution achieved by the sampling Si-W ECAL [2] and expected photon energy resolution of homogeneous crystal ECAL. RIGHT: Reconstructed boson masses of cleaned $ \nu\bar{\nu}q\bar{q} $, $ l\nu q\bar{q} $, and $ \nu\bar{\nu}H,\; H\to gg $ events [46].

Figure 8.  (color online) Separation power of $ K/\pi $ (LEFT) and $ K/p $ (RIGHT) using different techniques [37].

After the release of the CEPC CDR, intensive detector R&D efforts continue to address the CEPC physics requirements. These efforts have led to the development of the $ 4^{\rm{th}} $ detector concept [42], which demonstrates significant improvements in EM energy resolution, intrinsic hadronic energy resolution, PID performance, and the vertexing. The $ 4^{\rm{th}} $ detector concept employs a PFA-compatible homogeneous crystal ECAL to enhance the EM resolution, achieving an energy resolution of 3%/ $ \sqrt{E{\rm{(GeV)}}} \oplus $1% (see the comparison in the left panel of Fig. 7). This resolution is crucial for the separation of $ B^0 $ and $ B^0_s $ that decay into EM final states [32]. It utilizes high-density glass-scintillator HCAL, which can improve the hadronic energy resolution by nearly a factor of two, consequently enhancing the BMR [45]. The $ 4^{\rm{th}} $ detector also features a pixelated TPC that provides precise $ {\rm d}E/{\rm d}x $ [37, 50] or ${\rm d}N/{\rm d}x$ [38] measurements, both of which are critical for PID. Furthermore, the $ 4^{\rm{th}} $ detector concept incorporates a vertex detector with stitching technology [51], which has significantly lower material budget.

Another significant advancement is in the jet charge measurement. The performance of jet charge measurement is typically characterized by the effective tagging efficiency (power) $ \epsilon_{\rm{eff}} \equiv \epsilon_{\rm{tag}} (1-2\omega)^2 $, where $ \epsilon_{\rm{tag}} $ is the flavor tagging efficiency and ω is the wrong tag fraction. The study [35] develops a Leading Particle Jet Charge method (LPJC) and combines it with a Weighted Jet Charge (WJC) method to form a Heavy Flavor Jet Charge method (HFJC). This study evaluates the effective tagging power for c/b jets at the CEPC Z pole and finds it to be 39%/20%, respectively. Additionally, by implementing benchmark impact parameter cuts of 0.02/0.04 mm to distinguish the origin of the leading charged particle (whether from the decay of the leading heavy hadron or QCD fragmentation), the effective tagging power for c/b jets was found to be 39%/27%. Furthermore, a dedicated b-jet charge tagging algorithm developed specifically for the study of $ B^0_s \to J/\psi\phi $ at the CEPC [52] achieved an effective tagging power of 20%.

Recently, the idea of jet origin identification has been proposed. This idea aims at simultaneously identifying jets originating from eleven different colored particle species of the SM, namely five types of quarks (u, d, s, c, b), their corresponding anti-quarks, and gluons. The jet origin identification combines the concepts of jet flavor tagging, jet charge measurement, strange jet and gluon jet identification together. The idea of jet origin identification is then realized at the full simulated data of the CEPC CDR detector and using state-of-the-art reconstruction tools, including the Arbor particle flow reconstruction and the ParticleNet algorithm [53], which simultaneously reaches jet flavor tagging efficiencies of 92%, 79%, 67%, 37%, and 41% and jet charge flip rates of 18%, 7%, 15%, 15%, and 19% for b, c, s, u, and d quarks, respectively, and meanwhile it could deliver a gluon jet identification efficiency of 66% [33], see Fig. 9. These performances infer an effective tagging power of 37%/ 58% for b/c-jets, respectively, see Table 4.

Figure 9.  (color online) Jet origin identification performance [33] of full simulated Higgs/Z to di-jet processes with CEPC conceptual detector. LEFT: The confusion matrix $ M_{11} $ with perfect identification of leptons and charged hadrons. RIGHT: Jet flavor tagging efficiency and charge flip rates for quark jets with different scenarios of particle identification: with only lepton identification, plus identification of charged hadrons, plus identification of neutral kaons.

The jet origin identification has significant impact on many physics measurements at the future electron-positron Higgs factories. For instance, the rare and exotic hadronic Higgs boson decays (see Sec. IX.B), the determination of CKM matrix elements directly from W boson decays (see Sec. IX.A), the time-dependent CP measurements, the measurements of weak mixing angle, the differential measurements with multi-jet final states, etc.

To explore the flavor physics potential of the CEPC, various benchmark analyses that have been evaluated at the simulation level are covered in this manuscript. Many of them are performed in the CEPC official software framework, illustrated in Fig. 10, with full simulation and reconstruction of the CEPC CDR detector. Limited by the available computing resource, a dataset of $ {\cal{O}}(10^9) $ generator level inclusive $ Z\to q\bar{q} $ events is generated for the physics potential studies at Tera-Z. Since the full simulation of the whole dataset is computationally expensive and time-consuming, pre-selections are generally applied to refine the dataset into core subsets. The analysis of $ B_c \to \tau\nu_{\tau} $ in Sec. III, the study of $ B_s^0 \to \phi\nu\bar{\nu} $ in Sec. IV, and the $ \phi_s $ measurement via $ B_s^0 \to J/\psi\phi $ in Sec. V are three typical examples.

Figure 10.  (color online) The CEPC official software chain and analysis flow [54]. More detailed information can be found in the CEPC CDR [2].

For some studies, especially those that are oriented towards phenomenology and detector requirements, fast simulation is usually adopted. Based on the understanding of detector responses and validated by the full simulation results, key detector performance is parameterized and modeled, and its effect on final physics observables is evaluated accordingly. This evaluation is used in studies such as the measurement of the α angle via $ B^0_{(s)}\to\pi\pi $ channels discussed in Section V. In this way, we can investigate the whole parameter space as much as possible with fast convergence.

To make the physics picture complete, we also list many benchmarks that have not been fully explored via simulation, but via first principle estimation, such as τ relevant studies in Sec. VII and exclusive hadronic Z decays in Sec. VIII.B.

Historically, β decays, probably the best-known FCCC processes, have resulted in the discovery of weak interactions. While sensitivities to heavy-flavored leptonic and semileptonic FCCC decays in ongoing experiments are relatively limited, their explorations will continue to be significant for flavor physics in the CEPC era. Firstly, measuring the signal rates of these channels can be used to determine the values of the CKM matrix elements such as $ V_{cb} $ and $ V_{ub} $ [55]. Moreover, by performing these measurements, one can test lepton flavor universality (LFU), one of the most important predictions of the SM, see Refs. [56−59] for reviews. So the FCCC measurements can be an efficient way to probe NP that couples to leptons family-dependently. For instance, given a relative deviation $ \delta_{\rm{SL}} $ in the signal rate from the SM prediction, the energy scale probed can reach

for $ b\to c\ell\nu $ transitions and

for $ b\to u\ell\nu $ transitions. Notice that here the NP effective interactions have been assumed to be agnostic w.r.t. the SM flavor structure and have a strength of $ {\cal{O}}(1) $.

The operation of the CEPC at the Z pole enables the detector to access a full spectrum of b hadrons with high statistics, including multiple heavy-flavored mesons like $ B_c $ and baryons like $ \Lambda_b $, which are b-hadrons not accessible or planned to produce at B-factories. Measuring their (semi)leptonic decays would cross-validate our current understanding of FCCCs and further reveal hitherto unexplored physics. Particularly interesting among the list of expected measurements are the ones involving τ decays. These measurements are crucial for, inter alia, achieving a full test of LFU. However the multi-body decays of τ leptons complicate the event topology and kinematics. Even worse, the signature of neutrinos as missing momentum is hardly accessible at hadron colliders. The event reconstruction thus becomes a challenging task. In contrast, the reconstruction of these events including the τ leptons and other particles may greatly benefit from the excellent collider environment of the CEPC and the high-performance of its detector. These measurements thus define one of the "golden" channels for flavor physics at the CEPC.

The above discussion can also be applied to the measurement of FCNC processes. Since such processes are forbidden at tree level and suppressed at loop level in the SM, these channels are capable of probing NP (see detailed discussions in Sec. IV). The results obtained from both classes of measurements can be interpreted in various NP models. In a simplified NP model, these processes can arise from either colorless or colored mediators. The simplest colorless example might be a family non-universal $ Z' $ boson with off-diagonal couplings to both quarks and leptons, thus yielding FCNC processes, see, e.g., [60−64]. This setup can be extended to a framework with an extra $ SU(2) $ gauge triplet, where the additional W' gauge bosons will contribute to the FCCC processes [65]. Another example is provided by leptoquarks, namely scalar or vector bosons that couple to quarks and leptons simultaneously and therefore carry color. Leptoquarks are predicted by a wide range of ultraviolet (UV) theories such as grand unified theories, supersymmetry, composite Higgs models, etc. – for a review see Ref. [66]. Such interpretations are model-dependent, and hence often limited in their applicability.

Alternatively, one can interpret the results in an Effective Field Theory (EFT) framework. The EFT is usually defined to parameterize the NP effects by integrating out the short distance physics. As a manifestation of physics at a low energy scale, the EFT is insensitive to the concrete format of UV physics. Here, let us consider the low-energy EFT (LEFT) [67] with a natural cutoff at the EW-breaking scale. For $ b\to c \ell \nu $ transitions, we have the dimension-6 LEFT Hamiltonian

where $ O_i $ denote the left(right)-handed scalar, vector, and tensor operators, namely

and $ C_i $ represent the corresponding Wilson coefficients. The SM can only contribute to $ C_{V_{L}} $ via the exchange of a W boson. Any deviation from this prediction will indicate the presence of NP, and the specific pattern of such deviation will carry crucial information on the nature of the underlying NP sector.

One important case regarding the $ b\to c \ell \nu $ transitions is the leptonic decay of $ B_q \to \tau\nu $ ($ q=u,c $). As shown in Fig. 11, this decay mode is sensitive to the axial vector $ (C_{V_L} - C_{V_R} ) $ and pseudoscalar $ (C_{S_L} - C_{S_R}) $ Wilson coefficients, with the branching ratio (BR) given by

Figure 11.  (color online) Illustrative Feynman diagrams for the decay $ B_c^+\to \tau^+\nu_\tau $. LEFT: SM example. RIGHT: BSM example.

where $G_{\rm F}$ is the Fermi constant, $ m_\tau $ is the mass of the τ lepton, and $ m_{B_q^+} $, $ \tau_{B_q^+} $ and $ f_{B_q^+} $ denote the $ B_q^+ $ mass, lifetime and decay constant, respectively. The SM prediction for the BR of the decay $ B_c \to \tau \nu $ is rather large, $ \sim 2.3\times 10^{-2} $ [69], but the current constraint is relatively weak, BR$ (B_c\to \tau\nu)\lesssim $30%. Detailed studies indicate that a Tera-Z factory can measure this BR with a precision of $ {\cal{O}}(10^{-4}) $ [68−70]. Specifically, the CEPC study in Ref. [68] employs a full simulation and incorporates leptonic τ decays $ \tau^\pm\to\ell^\pm\nu\bar{\nu} $. The major features that differentiate $ B_u^+ $ from $ B_c^+ $ stem from their differing lifetimes and hadrons associated with their hadronization. As illustrated by Fig. 12, a measurement of the rate of $ B_c \to \tau\nu $ with a relative precision $ \sim {\cal{O}} $(1%) can be achieved at the Tera-Z run of the CEPC. The study in Ref. [69] instead focuses on the 3-prong τ decay, namely $ \tau^\pm \to \pi^\pm \pi^\pm \pi^\mp \nu $. Within the considered analysis scenarios, the expected precision of the measurements of the rates ranges from 1.6% to 2.3% for $ B_c^+ \to \tau^+\nu_\tau $ and from 1.8% to 3.6% for $ B^+ \to \tau^+\nu_\tau $.

Figure 12.  (color online) Relative precision of measuring the $ B_{(c)}^\pm \to \tau\nu $ rate at the CEPC Tera-Z, as a function of $R_{B_{c}/B}\equiv $$ N(B_c^\pm\to\tau\nu)/N(B^\pm\to\tau\nu)$ [68]. Here the red band denotes the SM prediction for $ R_{B_{c}/B} $.

Within the SM, Eq. (5) can be further used to extract the $ |V_{qb}| $ value by measuring the $ B_q \to \tau \nu $ decay rates [69]. Such a determination depends on precise inputs on the decay constants of the $ B_q^+ $ mesons $ f_{B_q^+} $ as well as their production fractions. Currently, the relative precision is ~0.7% for $ f_{B_u^+} $ [71] and ~4.6% for $ f_{B_c^+} $ [72], which could be improved in the coming decade. The $ B^+ $ production fraction is known with a precision ~2% [10] and could be significantly improved in the CEPC era due to the abundant $ Z\to b\bar{b} $ data. As for the $ B_c^+ $ production fraction, however, no information is available from any existing measurements or future projections.

With the high precision measurement of BR($B^+ \to \tau^+ \nu_\tau$) expected at Tera-Z factories [68, 69] and the theoretical uncertainties described above, we expect that the $ |V_{ub}| $ value can be determined with a relative precision of 1% or better. In comparison, the Belle II experiment is expected to perform a similar determination with a relative precision of 2%−3% employing the full integrated luminosity [7]. Notably, these measurements may cast new insights on the long-standing discrepancy of more than $ 3\sigma $ between the inclusive and exclusive determinations of $ |V_{cb}| $ [10, 73−76]. ¹

The semileptonic decays induced by the $ b \to c \ell\nu $ transitions are often applied for the test of LFU. The LFU is predicted in the SM, because all three lepton families possess the same gauge charges. Consequently, any differences in decays involving different leptons can only arise from the Yukawa sector, in addition to any variations due to phase space. To highlight the special role of τ flavor, we introduce

as an indicator for the LFU, where $ H_{b(c)} $ represents a $ b(c) $-hadron, and $ \ell'=e, \mu $ unless stated otherwise. Such an observable can be also defined for the decays of $ B_c \to \tau \nu_\tau $ and $ B_c \to \ell'\nu_{\ell'} $. For these observables, the systematics, such as the uncertainties from the CKM matrix elements and form factors, largely cancel. As an illustration, we show the Feynman diagrams for the SM and BSM contributions to the $ H_b\to H_c \ell^+\nu_\ell $ transitions in Fig. 13.

Figure 13.  Illustrative Feynman diagrams for the transition $ H_b\to H_c \ell^+\nu_\ell $. LEFT: SM example. RIGHT: BSM example.

For the test of LFU at the Z pole, a variety of $ R_{H_c} $ observables ($ R_{D_s} $, $ R_{D_s^\ast} $, $ R_{J/\psi} $, and $ R_{\Lambda_c} $) have been recently investigated employing the fast simulation template of the CEPC [39]. The relative precisions that can be achieved, considering statistical errors only, are summarized in Table 5. Systematics in the $ R_{H_c} $ measurements, as mentioned before, are expected to cancel largely since $ R_{H_c} $ denotes a ratio of two aligned measurements. This study indicates that at CEPC, a relative precision of $ \lesssim $3% for $ R_{J/\psi} $, as well as $ \lesssim $0.2% and ~0.05% for $ R_{D_s^{(\ast)}} $ and $ R_{\Lambda_c} $, respectively, could be reached. Due to the complex topology and dynamics, these outcomes rely heavily on a vertex-based strategy for event reconstruction. They would benefit from a higher detector performance in general. Concretely, the $ R_{J/\psi} $ measurement benefits the most from the improvement of tracker resolution, (see right panel of Fig. 3 also), in reconstructing the $ B_c^\pm $ vertex as well as in identifying the $ J/\psi $ one, while the $ R_{D_s^{(\ast)}} $ measurements gain more from the increase of soft photon identification efficiency in distinguishing the $ D_s^{\ast} $ and $ D_s $ modes via the decay $ D_s^{\ast} \to D_s \gamma $.

$ R_{H_c} $	SM Value	4 Tera-Z
$ R_{J/\psi} $	0.289	$ 2.1\times 10^{-2} $
$ R_{D_s} $	0.393	$ 2.1\times 10^{-3} $
$ R_{D_s^*} $	0.303	$ 1.6\times 10^{-3} $
$ R_{\Lambda_c} $	0.334	$ 4.9\times 10^{-4} $

Table 5.  SM predictions for the $ R_{H_c} $ observables and relative precision for their measurements at 4 Tera-Z, considering statistical uncertainties only [39].

Note that these measurements cover a variety of $ b\to c \tau \nu $ transitions: such as the ones from pseudoscalar ($ B_{s,c} $) to vector ($ D_s^\ast $, $ J/\psi $) or pseudoscalar ($ D_s $); those from baryon ($ \Lambda_b $) to another baryon ($ \Lambda_c $); and the decays of a pseudoscalar ($ B_{c} $) to a pair of fermions. Consequently, they can be employed to constrain different LEFT operators that can induce $ b\to c \tau \nu $ transitions. Following the approach in Ref. [39], we present in Fig. 14 the marginalized constraints on the Wilson coefficients of $ b\to c\tau \nu $ LEFT at the CEPC, based on the results of [39, 68]. In this context, these Wilson coefficients can be universally constrained to a level of $ {\cal{O}}(10^{-3}) $. ²

Figure 14.  (color online) Marginalized constraints on the Wilson coefficients of $ b\to c\tau \nu $ LEFT at the CEPC, with $ \delta C_{V_L}^{\tau} = C_{V_L}^{\tau} - \delta C_{V_L,{\rm{SM}}}^{\tau} $. This plot is taken from Ref. [39].

Additionally, several unexplored topics of FCCC physics deserve attention. Firstly, in view of the scientific significance of testing LFU, it is necessary to establish the CEPC sensitivity for a full list of $ R_{H_c} $ measurements including the traditional $ R_D $ and $ R_{D^\ast} $, higher-resonant $ R_{D^{\ast\ast}} $ [77], remaining baryonic modes such as $ R_{\Xi_c} $, etc., and their corresponding differential measurements. Also, to provide an LFU test for all three generations, it is natural to extend studies to the measurement of $ {{\rm{BR}}(b\to c\mu\nu)}/ {{\rm{BR}}(b\to ce\nu)} $, where it is crucial to reduce the systematics to a level comparable to the statistical errors. The relevant benchmark channels that can be investigated at CEPC are listed in Table 6. Secondly, the superior precision of measuring the B meson flight distance at the CEPC creates a new opportunity for the measurement of time-dependent CP-violation in semileptonic $ b\to c \ell \nu $ decays. With this approach, the CP-violating markers in $ B^0_{(s)}-\bar{B}^0_{(s)} $ mixing, which are encoded as $ {\cal{A}}^{d}_{\rm{SL}} $ and $ {\cal{A}}^{s}_{\rm{SL}} $ [78, 79] respectively, can be extracted by measuring the $ B^0 $ and $ B_s^0 $ decays. As these measurements can contribute significantly to the global constraints on the parameters β and $ \beta_s $ [80, 81], where the current experimental precision remains far from the SM predictions, it is of high value to perform a more dedicated sensitivity analysis with either fast or full simulations.

Process	Observable
$ b\to cl\nu $	$ R_{H_{c}}(R_{J/\psi},R_{D_s^{(*)}},R_{\Lambda_c}) $
$ B_c\to \tau \nu $	$ |V_{cb}| $
$ B\to \tau \nu $	$ |V_{ub}| $

Table 6.  List of benchmark FCCC semileptonic and leptonic b-decay channels that can be investigated at CEPC.

FCNC transitions are prohibited at tree level in the SM. While being enabled by EW penguin or box diagrams (see Fig. 15), these transitions are subject to a joint suppression by off-diagonal CKM matrix elements and loop factors, and thus are rare. Because of this feature, the FCNC processes emerge uniquely sensitive to weak NP effects that may otherwise evade detection. Given a relative deviation of $ \delta_{\rm{rare}} $ in signal rate from the SM prediction, the energy scale probed can reach [82]

Figure 15.  (color online) Illustrative Feynman diagrams for the transition $ H_b\to H_s\ell^+\ell^- $. UPPER: SM examples. BOTTOM: BSM examples.

and

for the $ b\to s $ and $ b\to d $ transitions, respectively. Notably, while the FCNC processes are rarer than the FCCC ones in the SM, $ \Lambda_{\rm{NP}}^{\rm{rare}} $ can be comparable to, or even higher than, $ \Lambda_{\rm{NP}}^{\rm{SL}} $ as long as $ \delta_{\rm{rare}} \lesssim 100 \delta_{\rm{SL}} $ is achieved.

Similar to the $ b\to c\ell \nu $ transitions investigated in Section III, we have the dimension-6 LEFT Hamiltonian to parametrize the $ b\to s $ transitions:

where the operators of interest include

Among these operators, the first five encode the scalar-, vector-, and tensor-mediated $ b\to s $ transitions with a pair of charged leptons and may violate LFU. The presence and absence of a "prime" denote the $ b\to s $ currents which are subject to the left- and right-handed chiral projections respectively, while the opposite convention applies to the dipole operators $ O_{7}^{(\prime)} $. $ O_{L(R)} $ encodes the vector-mediated $ b\to s $ transitions with a pair of neutrinos. $ O_{7}^{(\prime)} $ is an EM dipole operator which can either yield decays with an on-shell photon or mediate $ b\to s \ell\ell $ transitions (see the bottom left panel in Fig. 15). Note that, when the strange-quark and lepton masses are neglected, the SM contributes to $ O_{9} $, $ O_{10} $, $ O_{L} $ and $ O_{7} $ only.

In this section, we will mostly focus on the measurements of $ b\to s \tau\tau $, $ b \to s \nu \bar\nu $ and $ b\to s \gamma $ transitions. The CEPC offers a great platform for these studies, particularly during its Z pole run. The extraordinarily high luminosity delivered by the CEPC ensures considerable signal statistics for even the most elusive decay modes with BRs typically $ \lesssim 10^{-5} $. Moreover, as compared to the LHCb detector, the planned detectors of the CEPC are better suited for the reconstruction of τ leptons and thus the measurement of $ b\to s \tau\tau $, for the measurement of missing energy and hence of $ b \to s \nu \bar\nu $, and for photon identification as needed for the measurement of $ b\to s \gamma $. A combination of these advantages yields an enhanced sensitivity for both testing the SM and probing NP effects. The CEPC thus represents an ideal facility for investigating these rare FCNC decays and the underlying physics. It is worth noting that both $ b \to s \nu \bar\nu $ and, especially, $ b \to s \tau\tau $ transitions, for which we have very poor experimental information so far, are extremely sensitive to test a wide class of motivated NP models with new dynamics coupled mainly to the third generation [83, 84]. For the convenience of the discussion below, we summarize the projected sensitivities to $ b\to s\tau\tau $ and $ b\to s\nu\bar{\nu} $ transitions, together with the $ b\to c\tau\nu $ processes discussed in Section III, in Fig. 16. At the end of this section, we will extend the discussions to the possibilities of testing the SM global symmetries with forbidden b-hadron decays.

Figure 16.  (color online) Projected sensitivities of measuring the $ b\to s\tau\tau $ [85], $ b\to s\nu\bar{\nu} $ [36, 86] and $ b\to c\tau\nu $ [39, 68] transitions at the Z pole. The sensitivities at Belle II @ 50 $ {\rm{ab}}^{-1} $ [7, 87] and LHCb Upgrade II [19, 57] have also been provided as a reference. Note that LHCb sensitivities are generated by combining the analyses of $ \tau^+ \to \pi^+\pi^-\pi^-(\pi^0)\nu $ and $ \tau \to \mu \nu \bar \nu $. This plot is taken from Ref. [39], with additional $ b\to s\nu\bar{\nu} $ modes included.

In general, the reconstruction of $ b\to s \tau\tau $ is more involved compared to the reconstruction of $ b\to s ee, s\mu\mu $. As the τ decays result in neutrino production, the $ b\to s \tau\tau $ events are not fully visible to a detector. This difficulty, however, can be well-addressed at a machine like the CEPC. In a recent study [85] (for discussions on $ B^0\to K^{\ast 0} \tau^- \tau^+ $, also see [88]), the sensitivity for measuring a set of benchmark $ b\to s\tau\tau $ transitions, including $ B^0\to K^{\ast 0} \tau^- \tau^+ $, $ B_s^0\to\phi \tau^- \tau^+ $, $ B^+ \to K^+ \tau^- \tau^+ $ and $ B_s^0 \to \tau^- \tau^+ $, at the Z pole has been systematically analyzed. To utilize the machine's capability, a tracker-based scheme to reconstruct the signal B mesons that works for these $ b\to s\tau\tau $ channels has been developed, achieved by using the decay modes of $ \tau^\pm\to \pi^\pm\pi^\pm\pi^\mp\nu $. Such a tracker-based scheme also benefits from the particle kinematics at the Z pole. Due to their boost, the signal b hadrons tend to travel further (compared to, e.g., Belle II) before their decay, which benefits the relevant tracking measurements. The predominant backgrounds for these measurements are the Cabibbo-favored $ b\to c+X $ processes. Recall that both $ D^\pm $ and $ D_s^\pm $ mesons have masses and lifetimes comparable to those of τ leptons and thus may decay to a vertex of $ \pi^\pm\pi^\pm\pi^\mp $ with extra particles. Therefore, they can fake the τ leptons in the signal. In Fig. 17 we demonstrate the mass reconstruction for the signal b-mesons in the measurements of $ B^0\to K^{\ast 0} \tau \tau $ and $ B^+ \to K^+ \tau^- \tau^+ $ at the Z pole. These two channels involve the decay of b-mesons into vector and pseudoscalar mesons respectively. They are sensitive to the LEFT in approximately orthogonal ways and thus are complementary in probing NP [85].

Figure 17.  (color online) Mass reconstruction for the signal b-mesons in the measurements of $ b\to s\tau\tau $ at the Z pole, with $ \tau^\pm \to \pi^\pm\pi^\pm\pi^\mp\nu $ [85]. LEFT: $ B^0\to K^{\ast 0} \tau^- \tau^+ $. RIGHT: $ B^+ \to K^+ \tau^- \tau^+ $. The major backgrounds arise from the $ b\to c \tau \nu $ and $ b\to c c s $ transitions and are both reconstructed.

As illustrated in Fig. 16, the Tera-Z and $ 10\times $Tera-Z machines would be able to measure the BRs of $ B^0\to K^{\ast 0} \tau^- \tau^+ $, $ B_s^0\to\phi \tau^- \tau^+ $ and $ B^+ \to K^+ \tau^- \tau^+ $ with an absolute precision of $ {\cal{O}}(10^{-7} - 10^{-6}) $, as well as BR($B_s^0 \to \tau^- \tau^+$) with an absolute precision of $ {\cal{O}}(10^{-6} - 10^{-5}) $. In comparison, Belle II and LHCb either have no sensitivity to these measurements or can only yield a sensitivity that is one to two orders of magnitude weaker. With the baseline luminosity, this indicates that the CEPC will be able to identify $ \sim {\cal{O}}(1) $ deviations from the SM predictions and further probe the $ b\to s\tau \tau $ LEFT operators. Fig. 18 shows the marginalized constraints on the corresponding Wilson coefficients in the presence of the vector-mediated operators only.

Figure 18.  (color online) Marginalized constraints on the Wilson coefficients of $ b\to s\tau \tau $ LEFT (vector current only) at the CEPC, with $ \delta C_9^\tau = C_9^\tau - C_{9,{\rm{SM}}}^\tau $ and $ \delta C_{10}^\tau = C_{10}^\tau - C_{10,{\rm{SM}}}^\tau $. This plot is adapted from Ref. [85].

In spite of this progress, the study of FCNC b rare decays at CEPC should be extended in multiple directions. Firstly, the CEPC constraints on the LEFT operators in Eq. (9) should be improved. Currently, the sensitivity to BR($ B_s\to \tau^- \tau^+ $) is too weak to probe unconstrained LEFT parameter space. BR($ B^0\to K^{\ast 0} \tau^- \tau^+ $) and BR($ B_s^0\to\phi \tau^- \tau^+ $) are both pseudoscalar to vector transitions and have a similar dependence on the NP parameters. To improve the constraints on the relevant LEFT coefficients, one can consider: (i) introducing differential observables, such as forward-backward asymmetry and τ polarimetry [88]; and (ii) incorporating $ b\to s\tau\tau $ transitions of different nature, such as the baryonic decay $ \Lambda_b \to \Lambda\tau^- \tau^+ $. Interestingly, within the context of an $S U(2)_L$-invariant EFT, sizable NP contributions to the $ b\to s\tau\tau $ transitions are often accompanied with large effects on the left-handed vector current NP operators that contribute to the LFU observables $ R_{D^{(*)}} $, which currently exhibit some tension with the SM predictions [89, 90].

A second area of improvement would be to advance the study on LFU tests at the CEPC. The CEPC analysis in Ref. [85] focuses on the di-τ mode of $ b\to s $ transitions. To paint a full picture in this context, it is of high value to extend the analysis to $ b\to s \ell \ell $. The measurements of, e.g., $ R_{K^{(\ast)}} $, $ R_{pK} $ [91], $ R_\phi $ [92], $ R_{f_2^\prime(1525)} $ [92] and even $ R_{\Lambda} $ could provide important insights regarding LFU. For some of these measurements, the systematic uncertainties induced by PID could be dominant. The superior electron- and muon-ID capabilities of future detectors are anticipated to offer an edge over LHCb. Notably, the luminosity advantage of the CEPC in measuring the $ b\to s \tau\tau $ transitions could be extended to ultra-rare channels such as $ B_s^0 \to \mu^+ \mu^- $. The measurement of $ {\rm{BR}}(B_s^0 \to \mu^+ \mu^-) $ in the SM is known to be statistically limited, due to its tiny value of around $ \sim 3.0 \times 10^{-9} $ [93]. With a yield of $ \sim 1.2 \times 10^{11} $ for $ B_s^0 $ mesons at the CEPC, about 360 $ B_s^0 \to \mu^+ \mu^- $ events are expected to be produced, which provides a good opportunity to improve the precision of its measurements.

Finally, sensitivity studies should be extended to $ b\to d \ell^+\ell^- $ transitions at the CEPC. The $ b\to d \ell^+\ell^- $ transitions represent another independent category of FCNC rare b-decays, and hence play a role complementary to the $ b\to s \ell^+\ell^- $ transitions in exploring flavor physics. The measurements of these channels including both signal rate and CP violation [94, 95] may share difficulties similar to those of $ b\to s\ell^+\ell^- $ decays, and hence would impose similar requirements for the detector performance at the CEPC. All of these issues deserve further detailed examinations.

The $ b\to s\nu\bar{\nu} $ decay is immune to non-factorizable charm-loop corrections and photonic-penguin contributions. Therefore, the theoretical calculation for its SM rate is cleaner than that for the $ b\to s\ell\ell $ transitions, which yields BR($ B_s^0 \to \phi\nu\bar{\nu})_{\rm{SM}}=(9.93\pm 0.72)\times 10^{-6} $ [36]. The $ b\to s\nu\bar{\nu} $ decay can be used to probe light dark sectors, such as dark photons, sterile neutrinos, axions/axion-like-particles (ALPs), or neutral scalars, which may significantly alter the kinematics of visible particles [96−98], (for discussions on the light dark sectors at CEPC, also see Sec. XI). Also, due to the constraints of electroweak gauge symmetry, the impacts of NP on the $ b \to s\nu\bar{\nu} $ and $ b \to s\ell^+\ell^- $ decays could be interconnected. Thus, the measurement of $ b\to s\nu\bar{\nu} $ offers a complementary probe to look into the underlying physics [83, 99].

A dedicated study of the $ B_s^0\to\phi\nu\bar{\nu} $ decay (see Fig. 19) at the Z pole has been conducted, using full simulation samples aligned with the CEPC detector profile [36]. This study, facilitated by the large $ B_s^0 $ statistics at the CEPC (see Table 2), suggests that a precise measurement of such a rare decay is possible. Explicitly, the accurate ϕ and $ B_s^0 $ reconstructions in this analysis reduce the $ Z\to q\bar{q} $ events by a factor $ \sim {\cal{O}}(10^{-8}) $, with a signal efficiency ~3%, leaving primarily the $ Z\to b\bar{b} $ events as the backgrounds. As a result, a relative precision $ \lesssim $2% can be achieved for measuring the SM $ B_s^0\to\phi\nu\bar{\nu} $ signal, as shown in the left panel of Fig. 20. Particularly, with a high signal-to-background ratio of $ \simeq $77%, the robustness of this measurement against potential systematic uncertainties is largely assured. This study has also shown that the constraints obtained from this measurement can contribute pivotally to the global determination of NP effects, e.g., the ones encoded in the LEFT, (see the right panel of Fig. 20).

Figure 19.  Illustrative Feynman diagrams for the transition $ B_s^0\to \phi\nu\overline{\nu} $ in the SM. LEFT: EW penguin diagram. RIGHT: EW box diagram.

Figure 20.  (color online) LEFT: Relative precision for measuring the signal strength of $ B_s^0 \to \phi\nu\bar{\nu} $ at Tera-Z, as a function of its BR. RIGHT: Constraints on the LEFT coefficients $ C_L^{\rm{NP}} \equiv C_{L}-C_L^{\rm{SM}} $ and $ C_R $ with the measurements of the overall $ B_s^0 \to \phi\nu\bar{\nu} $ decay rate (green band) and the ϕ polarization $ F_L $ (orange regions). These plots are taken from Ref. [36].

In addition to the $ B_s^0\to\phi\nu\bar{\nu} $ decay, there exist a set of other physics processes that can be applied to study the $ b\to s\nu\bar{\nu} $ transitions at the CEPC, for example $ B^+ \to K^+\nu\bar{\nu} $, $ B^+ \to K^{+\ast} \nu\bar{\nu} $, and $ B^0 \to K^{0\ast} \nu\bar{\nu} $. Interestingly, the Belle II collaboration has recently performed a search for the rare $ B^+ \to K^+\nu\bar{\nu} $ decay using an inclusive tagging approach, and obtained a branching fraction of $ (2.7\pm0.7)\times 10^{-5} $ [100], with a significance of $ 3.5 $ standard deviation with respect to the background-only hypothesis. This measurement also shows a 2.9 standard deviation departure from the SM expectation [101, 102]. The expected precision of the branching ratios for $ B\to K^{(*)}\nu\bar\nu $ with $ 50\; {\rm{ab}}^{-1} $ by combining the charged and neutral B decay modes are of the order of 10% [87]. Yet, by leveraging its advantages in reconstructing the missing energy and producing the b-hadrons, the CEPC may push this precision to a much higher level. Such expectations have been confirmed by a recent study at FCC-ee [86].

Furthermore, probes of other decay modes involving long-lived s-hadrons, such as $ B^0 \to K^0_S\nu \bar\nu $, $ \Lambda_b \to \Lambda \nu\bar{\nu} $ and $ \Xi_b^\pm \to \Xi^\pm \nu\bar{\nu} $ could also help pin down the $ b\to s\nu \bar\nu $ transition. The decays of the intermediate neutral particles in general give rise to vertices with a displacement of $ {\cal{O}}(10) $ cm. Therefore the precision of these channels highly depends on the reconstruction and resolution of these significantly displaced vertices. From a preliminary estimate [103], it is possible to achieve an 80% reconstruction efficiency for the $ K_S^0 $ and Λ vertices at a CEPC environment, opening up the opportunity to perform a combined constraint of $ b s\nu\bar{\nu} $ effective interactions with all the aforementioned decay modes. In particular, the baryonic processes such as $ \Lambda_b \to \Lambda \nu\bar{\nu} $ and $ \Xi_b^\pm \to \Xi^\pm \nu\bar{\nu} $ are unique opportunities at the CEPC as they are above the production threshold of the Belle II experiment. Since form factors of these baryonic modes are different from those of the mesonic modes, studies of these channels will bring independent information to understand the dynamics underlying the $ b\to s\nu\bar{\nu} $ transition in a global fit.

The third category of FCNC rare B decays consists of radiative ones, such as $ b \to s\gamma, d\gamma $. These modes are sensitive to the EM dipole operators $ O_7 $ and $ O_7^\prime $. A wealth of data, including the inclusive $ B \to X_{s,d}\gamma $ decays, as well as the direct CP violation $A_{\rm CP}$ and time-dependent CP violation $S_{\rm CP}$ in various $ b\to s\gamma $ decays, has yielded complementary insights into the corresponding Wilson coefficients $ C_7 $ and $ C_7^\prime $. At the CEPC, however, the reach for FCNC radiative modes is yet to be fully explored, despite their scientific significance [104]. One such example is the $ B_s^0 \to \phi\,(\to K^+ K^-) \,\gamma $ decay, illustrated in Fig. 21. Achieving a high accuracy in reconstructing the signal $ B_s^0 $ meson necessitates superior photon angular and energy resolution. For the LHCb Upgrade II, it was found that BR$ (B_s^0\to\phi\gamma) $ could be measured with a statistical uncertainty ~0.1%, and the CP parameters can also be well measured [19, 105]. These sensitivities are expected to be further improved at the CEPC due to the potentially high performance of its ECAL. This study can be extended to baryonic radiative decays of the $ b\to s\gamma $ type, such as $ \Lambda_b \to \Lambda\gamma $ and $ \Xi_b\to \Xi \gamma $, again with an expected sensitivity better than the LHCb [106]. The study can also be extended to $ b \to d\gamma $ decays, which can broaden our understanding of the FCNC transition amplitudes and potentially refine the CKM matrix determinations. Finally, if the ECAL of the CEPC allows an efficient reconstruction of $ \pi^0, \eta \to \gamma\gamma $ [32], the double-radiative decays of $ B_{s,d} \to \gamma\gamma $ could be measured [107]. Theoretical studies show that the $ \Lambda_{\rm{QCD}}/m_b $ power corrections in these channels are well under control, making them new benchmark probes of non-standard dynamics [108, 109]. The SM predictions for their BRs are given by [108, 109]

Figure 21.  (color online) Illustrative Feynman diagrams for the decay $ B_s^0\to \phi \gamma $. LEFT: SM example. RIGHT: BSM example.

Belle II has assessed its sensitivities to be respectively ~23% and ~10% [7] relative to the theoretical estimates in Ref. [110] that, we notice, are a factor of few larger than those provided above. Recently, an analysis combining the Belle and available Belle II data sets an upper limit of $ {\rm{BR}}(B^0\to\gamma\gamma)<6.4\times 10^{-8} $ at 90% confidence level [111].

An important class of observables include b-hadron decays that are forbidden because of the global symmetries of the SM. Aside from gauge symmetries, the SM respects or approximately respects a series of global symmetries, yielding, at different levels, the conservation of lepton family numbers, lepton and baryon numbers. The only-known breaking effects for these symmetries are highly suppressed in collider environments: lepton family numbers in the charged lepton sector are only violated through neutrino mixing and thus suppressed by the small neutrino mass differences; lepton and baryon numbers are only violated by the non-perturbative $S U(2)_L$ sphaleron which breaks both the lepton number and baryon number but conserves their difference exactly. The observation of Lepton Flavor Violation (LFV) in the charged lepton sector, as well as Lepton Number Violation (LNV), Baryon Number Violation (BNV) in any perturbative processes thus would be an indisputable evidence for BSM physics. Interestingly, LFV and LFU violation (LFUV) receive contributions respectively from the flavor off-diagonal and diagonal components of the same classes of EFT operators and thus are often correlated in UV-complete NP models. The modes that are forbidden in the SM often yield striking signals that are dramatically distinct from the background events. Just like the LFU tests, the CEPC with its large statistics and clean environment can play a significant role in examining these global symmetries.

Some of the FCNC studies presented in previous subsections can be extended to the null tests of SM global symmetries, in a straightforward way. For example, one can investigate the LFV effects in the b-hadron decays [112], such as $ H_b \to H_{d/s}\tau \ell $, where $ \ell $ denotes an electron or a muon. These decays are significant for testing current anomalies in semi-leptonic b-hadron decays [89] and, more in general, heavy NP coupling preferably to the third generation [83, 84]. In the past, experimental efforts have primarily focused on the modes $ B^{+} \to K^+ \tau\ell $, yielding $ {\cal{O}}(10^{-5}) $ upper limits on their branching ratios [113, 114]. Topological reconstruction techniques, employing a fast parametric simulation with momentum reconstruction resolutions and vertex detector performance, have been implemented to simulate LFV signal events for $ B^{0} \to K^{*0}\mu\tau $ as well. Initial explorations have demonstrated the detector requirements, offering guidance for future design and optimization goals for the vertex detector of the CEPC. As for LFV two-body decays, preliminary studies have shown that–while the CEPC constraints on the decays such as $ B^0_{(s)} \to \mu^{\pm} e^{\mp} $ and $ B^0_{(s)} \to \tau^{\pm} \mu^{\mp} $ can at most match the LHCb sensitivity [19] – an improvement in the sensitivity to $ B^0_{(s)} \to \tau^{\pm} e^{\mp} $ could be achieved at the CEPC due to the expected excellent electron identification.

The CEPC also provides a platform for testing LNV and BNV in b-hadron decays. For instance, LNV can be tested by measuring the same-sign di-lepton decay $ B^{+} \to \pi^{-}(K^{-}) \ell^{+} \ell^{+} $, where the sensitivities are primarily influenced by statistics and lepton charge identification. Unlike the LHCb analysis which has focused on the di-muon mode [115, 116], the CEPC may have a good sensitivity for the same-sign di-electron mode also, given its low misidentification rates for electrons. The BNV measurements may feature the signals such as forbidden baryon-antibaryon oscillations [117] and explicit BNV decays. One example in the latter case is $ \Lambda_b^{0} \to h^{-}(h^{0})\ell^{+} $, which arise from the dimension-6 BNV operators $ qq^{\prime}q^{\prime\prime}\ell $ where $ B-L $ is conserved.

Interestingly, BNV is one of the three Sakharov conditions [118] required for dynamically generating the baryon asymmetry of the Universe (BAU). Hence, the measurement of BNV modes may provide valuable clues for resolving this long-standing cosmological puzzle. For example, introducing a dark matter candidate carrying baryon number, the B-mesogenesis model [119] predicts the BNV separately in the visible sector and dark matter sector, simultaneously achieving baryogenesis and the correct dark matter relic abundance. This model can be tested by measuring invisible decays of neutral bottom baryons such as $ \Lambda_b^0 $ – for further discussions on its collider phenomenology, see [120−122]. In a recent study [123], it has been shown that the important constraints on the model parameters can be obtained at the Z pole run of the CEPC.

In the SM, the flavor properties of quarks are mainly encoded in the CKM matrix, including what concerns the phenomena involving CP violation. The independent entries include three Euler angles entangling the three generations and one CKM phase as the only source of CP violation in the SM [5]. Yet, addressing the puzzle of BAU dynamically requires additional CP violation, as one of the Sakharov conditions. This consideration has motivated extensive explorations in the last decades. b-hadron decays provide a handle particularly suitable for this study. Theoretically, it has been demonstrated in Ref. [124] that the CP violation in B meson systems can drive the BAU generation through EW baryogenesis. Experimentally, the heavy-flavor measurements represent one of the most important tasks in flavor physics. At the CEPC, such measurements are expected to greatly benefit from high statistics, low backgrounds, efficient hadron ID, and extreme displacement resolution. The observables, handled by proper analysis of amplitudes, can be fed into the global fit of the CKM matrix. Any deviation from the CKM unitarity would be a smoking-gun signature for NP including new CP violation.

Generally, there are three categories of observables for CP violation: CP violation in decay (direct CP violation), CP violation in mixing (indirect CP violation) and CP violation through the interference between mixing and decay. ³ The CP violation in decay can be measured through a process, where the initial particle does not mix with its CP conjugate and the final state is not a CP eigenstate, and its CP conjugate. The CP violation is then manifested as a time-integrated asymmetry in statistics between these two processes. The effective statistics is determined by both of the overall signal rate and the efficiency of tagging initial heavy-flavored particles. As introduced in Section II.B, the effective tagging efficiency $ \epsilon_{\rm{eff}} $ can be estimated as $ \epsilon_{\rm{tag}} (1 - 2\omega)^2 $ for some specific processes, where $ \epsilon_{\rm{tag}} $ and ω are the raw tagging efficiency and mistagging rate, respectively [126]. Regarding the application for determining the CKM parameters, one example is related to measuring the time-integrated CP asymmetry in the $ B^+\to D^{(\ast) 0} K^{(\ast) +} $ decay [10, 103]. Ref. [103] exploits high acceptance and excellent reconstruction of $ K^0_S $ from $ D^0\to K^0_S \pi^0 $ to study $ B^\pm \to D^0(\bar{D}^0) K^\pm $, assuming a crystal ECAL for FCC-ee, and finds that the $ \gamma_s $ parameter for the $ bs $ unitarity triangle could be determined with a precision $ \sim {\cal{O}}(1^\circ) $.

The observations of CP violations in mixing and interference between mixing and decay involve decaying processes of neutral particles which as flavor eigenstates are not identical with their mass eigenstates. In the former case, the decays are flavor-specific. The CP violation is often measured as a time-integrated asymmetry for semi-leptonic decays like $ M_0\to l^- X $ and $ \bar M_0\to l^+ X $. Differently, the latter case requests the decay products to form a CP eigenstate such that an interference can occur between the amplitudes with and without a mixing. $ B^0 $ and $ B_s^0 $ as neutral heavy-flavored mesons are especially relevant here. Because of the oscillations between them and their CP-conjugate before decay, the CP asymmetry generically demonstrates a time dependence which can be leveraged for detecting the CP violation. A general pattern holds for this time-dependence despite the diversity of possible decaying processes. The asymmetry is proportional to the oscillatory factors with the period determined by the mass gap ($ \Delta m $) between the mass eigenstates of initial particles and non-oscillatory factors caused by the decay-width difference ($ \Delta \Gamma $) of these mass eigenstates. Because $ \Delta m \gg \Delta \Gamma $ for the $ B^0 $ and $ B_s^0 $ mesons, the oscillatory factors are relatively more relevant for their CP violation measurements [10]. The mistagging probability ω becomes significant in this case, as the algorithm must determine the charge of initial b quarks after the $ b-\bar{b} $ oscillation happens. Another factor affecting the overall precision is the decay time determination, which is mainly limited by the vertex resolution of the tracking system.

The charge determination of initial b quarks is primarily affected by the mixing-induced oscillations. One way to address this difficulty is to utilize the information of the companion b-hadron. If the companion b quark hadronizes into non-oscillatory species such as $ B^\pm $ and is subsequently identified, then the charge of the original signal b quark can be identified. Alternatively, one can employ the products of QCD shower and hadronization, as they manifest the original b-quark charge before the oscillation occurs. For example, the $ B_s^0 $ meson is often accompanied by a collimated $ K^+ $ meson, where the strange quarks are pair-produced. Recent study in [52] suggests that an $ \epsilon_{\rm{eff}} $ value of $ \gtrsim $20% can be achieved at the CEPC, much higher than ~5% at LHCb [127]. This result is also consistent with another CEPC study which combines leading charged particle in a jet and momentum-weighted jet charge [35], yielding $ \epsilon_{\rm{tag}} \sim $39% and 20% for inclusive c or b jets respectively. Notably, utilizing the method of jet origin identification and the ParticleNet algorithm developed in Ref. [33], the jet charge flip rates could be controlled to 19% and 7% for inclusive b and c jets, corresponding to an effective tagging power of 37% and 54%, respectively. More details can be found in Sec. II.

The decay-time measurements at the CEPC are expected to benefit from its clean physics environment and well-designed tracking system. The full simulation in Ref. [52] reports a CEPC resolution of $ \lesssim 5 $ fs for measuring the 4-prong decay $ B_s^0\to J/\psi \phi \to \mu^+\mu^-K^+K^- $, which is much better than the typical LHCb level of $ \sim 20-30 $ fs. This will bring great benefits to the measurements of time-dependent CP violation and also, for the role of $ \Delta m $ and $ \Delta \Gamma $ as basic inputs, the global CKM fit. Additionally, a study in the FCC-ee context [128] suggests a relative uncertainty of $ \lesssim 3\times 10^{-5} $ for the $ \Delta m $ measurement of $ B_s^0 $ meson, which is about one order of magnitude better than the current level. We hope that dedicated studies in the future could help validate such results and reveal the full potential of the CEPC in measuring these basic flavor physics parameters.

The time-dependent CP measurements can be also applied to test the $ bs $ unitarity triangle. The decay of $ B_s^0\to J/\psi \phi \to \mu^+\mu^-K^+K^- $ has been widely used for this purpose [129, 130]. Figure 22 displays in its left panel the projected CEPC sensitivities of measuring the parameters $ \Delta\Gamma_s $ and $ \phi_s \approx -2 \beta_s $ in this channel [52]. The performed full simulation indicates that the CEPC could reduce the uncertainty for $ \beta_s $ to $ \sim 2.3 {\rm{mrad}} \sim 0.13^\circ $ [52], improving the existing precision by several times. FCC-ee also reported its study on the time-dependent CP measurements in the same decay mode, and additionally $ B_s^0\to D_s^\pm K^\mp $ [128], with fast simulation. The right panel of Fig. 22 shows the mass reconstruction of $ B_s^0 $ mesons achieved in this study. Most combinatoric and misidentification-induced backgrounds can be removed with the PID algorithm, yielding a sharp peak of signal events. In this context, the triangle parameter $ \alpha_s $ and $ \beta_s $ can be measured with a precision of $ 0.4^\circ $ and $ 0.035^\circ $, respectively [128]. The CEPC results are weaker than those of FCC-ee. However, considering the recent advancement of jet origin ID at the CEPC, comparable sensitivities could be finally achieved for both machines.

Figure 22.  (color online) LEFT: Projected 68% confidence level (CL) sensitivities of measuring the parameters $ \Delta\Gamma_s $ and $ \phi_s \approx -2 \beta_s $ at the CEPC [52], through the time-dependent CP violation in the decay $ B_s^0 \to J/\psi(\to \mu^+\mu^-) \phi (\to K^+ K^-) $. RIGHT: $ B_s^0 $ mass reconstruction in the decays $ B_s^0 \to D^\pm_s (\to \phi \pi^\pm \to K^+K^- \pi^\pm) K^\mp $ at the Z pole of FCC-ee [128].

Yet, the oscillating effects of neutral B mesons are not always trackable. One example is the decay of $B^0_{(s)}\to \pi^0\pi^0\to 4\gamma$, where the tracker loses its power and reconstructing $ B^0_{(s)} $ decay time becomes extremely challenging. One can perform time-integrated measurements only for such decays. A sensitivity study on this case has been taken with the CEPC fast simulation in Ref. [32]. Figure 23 displays the obtained relative uncertainties (statistical only) as a function of the B-meson mass resolution $ \sigma_{m_B} $. For BR($ B^{0} \to \pi^{0}\pi^{0} \to 4\gamma $) and BR($ B^{0}_{s} \to \pi^{0}\pi^{0} \to 4\gamma $), the Tera-Z precisions are expected to be $ \lesssim {\cal{O}} $(1%) and $ \lesssim {\cal{O}} $(10%), respectively. Here the magnitude of $ \sigma_{m_B} $ is significantly influenced by the ECAL performance. The benchmark presented reflects an ECAL resolution of ~3%$ /\sqrt{E{\rm{(GeV)}}} $, which could be achieved with a fully crystal ECAL [131]. The pair of B mesons produced at Z pole are not entangled, unlike the entangled B production by $ \Upsilon(4S)\to 2B $ decays in B-factories. Consequently, the time-integrated observables for CP violation at the CEPC are slightly different from their B-factory counterparts. Combining the future CEPC and Belle II results of measuring $ B\to \pi\pi $, the CKM angle α could be constrained [132] to a level as small as 0.4$ ^\circ $ if theoretical errors are resolved. These projected results are illustrated in Fig. 24, which indicate that the CEPC measurements can constrain α much stronger than the current data. The measurement of time-dependent CP violation in the $ B^0\to \pi^0\pi^0 $ decay, using the $ \pi^0 \to e^-e^+\gamma $ Dalitz mode, has been explored by Belle II collaboration [7]. The sensitivity relies on the quality of the $ \pi^0 \to e^-e^+\gamma $ decay vertex reconstruction, which is yet to be studied at the CEPC.

Figure 23.  (color online) Relative uncertainties (statistical only) of measuring BR($ B^{0} \to \pi^{0}\pi^{0} \to 4\gamma $) (LEFT) and BR($ B^{0}_{s} \to \pi^{0}\pi^{0} \to 4\gamma $) (RIGHT) at the CEPC as a function of the B-meson mass resolution $ \sigma_{m_B} $. The plots are taken from Ref. [32].

Figure 24.  (color online) p-value for the determination of the CKM angle α [32]. With the current $ B^0\to\pi\pi $ measurements (dotted-dashed blue) and global CKM fitting (blue error bars) as a reference, we demonstrate two different scenarios of the CEPC measurements as a Tera-Z factory, where the CEPC data is used alone (dashed red) or combined with the current world average of B-factory measurements (filled green). LEFT: Scan over the whole range of α. RIGHT: Scan around the value favored by the global CKM fit.

Despite the analyses discussed above, many studies regarding the CP violation at the Z pole and the relevant physics have yet to be explored. For example, the β angle is known to be primarily determined by the measurements of the $ b\to c\bar{c} s $ transitions such as $ B^0 \to J/\psi K^{0} $ and their time-dependent CP violation [10]. A dedicated simulation is needed to validate the projected Z-factory sensitivities in Ref. [20]. Also, the CP violation in the $ b\to u\bar{u} d $ transitions (see Fig. 25) such as $ B\to \rho\rho $ and $ B\to \rho\pi $, can be relevant for the determination of the α angle. These studies also echo the recent report of the first evidence from LHCb for direct CP violation arising from the $ b\to c\bar{c} q $ transitions [133], where $ q = s, d $. But, digging out the potential of a Z factory for the CKM global fit demands systematic sensitivity studies on these measurements of CP violation.

Figure 25.  Illustrative Feynman diagrams for the transition $ \bar{b}\to \bar{u} u \bar{d} $. LEFT: tree level. MIDDLE: EW penguin diagram. RIGHT: QCD penguin diagram.

Another recent achievement is the first definitive observation of CP violation in the decays of baryons, a class of particles that had remained experimentally elusive despite decades of study [134]. Using the full Run 1 and Run 2 dataset of the LHCb, the analysis focuses on the four-body decay $ \Lambda_b^0 \to pK^-\pi^+\pi^- $ and its CP-conjugate process $ \bar{\Lambda}_b^0 \to \overline{p}K^+\pi^-\pi^+ $, comprising over 80,000 reconstructed events. The global CP asymmetry is directly measured to be ${\cal{A}}_{\rm CP} = (2.45 \pm 0.46 \pm 0.10)$%, establishing this effect with 5.2σ statistical significance after careful control of systematic uncertainties. This discovery is particularly significant as CP violation had previously only been observed in meson systems, despite both quark-level transitions being theoretically predicted to show similar effects. In view of the rich statistics of $ \Lambda_b^0 $ and $ \bar{\Lambda}_b^0 $ and their boost kinematics at Z pole (see Table 2), it is natural to extend the studies of CP violation in baryon systems from LHCb to CEPC, as we have done for measuring the $ b\to c \tau \nu $ transitions and also testing the LFU (see Sec. III).

Finally, more opportunities for studying CP violation beyond the currently well-established observables at the CEPC are also expected due to the unique detector and kinematic conditions of this machine. However, additional theoretical inputs are needed to make specific recommendations.

The branching ratio of Z boson decays to a pair of charm quarks is BR($ Z\to c\bar{c})\simeq $12% in the SM, which suggests that the CEPC Z-pole operation mode could also serve as a charm factory. Given the CEPC's high luminosity, low background, and excellent detector performance, CEPC may significantly enhance the precisions of certain studies in charm physics. The charm quark may carry the information on NP, while the recent observation of CP violation in charm decays [135−137] further strengthens the need for its study.

One benchmark case for charm physics at the CEPC, akin to the discussions in Secs. III and IV, could be semileptonic c-hadron decays. The FCNC charm decays are rare in the SM. Different from the down-type FCNC, where the quarks in the loops are dominated by top, the up-type FCNC is dominated by the loops of b, s and d. The mass hierarchy for down-type quarks is relatively small compared to up-type quarks, yielding an even stronger GIM suppression. The sensitivity of rare charm decays to the NP is thus expected to be high [138−140]. Nevertheless, due to large resonance contributions, it is much more challenging to estimate the hadronic effects in charm decays. The heavy quark expansion method usually adopted for estimating rare b decays also becomes less reliable here. The short-distance physics in rare charm decays is thus difficult to probe through the BRs. Instead, we may consider the observables with a symmetry-protected suppression in the SM and essentially free of hadronic uncertainties. For example, we can test LFU in semileptonic $ c\to u \ell^+\ell^- $ decays [141] and search for LFV decays such as $ D \to \pi e\mu $ and $ D_s \to K e\mu $ [142]. We can also examine angular distributions in semileptonic $ c\to u \ell^+\ell^- $ decays [143, 144] as well as di-neutrino decays, e.g., $ D \to \pi \nu \bar{\nu} $ and $ D_s \to K \nu \bar{\nu} $ [145, 146]. Any observation of a non-standard effect in these measurements would be an evidence for the NP.

Moreover, it is important to examine hadronic c-hadron decays for charm physics. A preliminary qualitative estimate of the potential for studying charm physics at the CEPC can be made by estimating the charm particle yields. Table 7 shows the number of $ D^0 $'s and related fully hadronic final state decay modes collected by the LHCb experiment during its Run-2 period (approximately 6 fb⁻¹), the expected data to be collected over the entire lifetime of the LHC and LHCb (approximately 300 fb⁻¹), as well as the number of corresponding decay modes expected to be collected at the CEPC in the 50MW SR power beam Z-pole operation mode. Additionally, we compared the number of relevant decay modes reconstructed in certain physics analyses at LHCb, and estimated number of selected events at CEPC. According to Ref. [14], the efficiency for reconstructing and selecting charm meson decays at a typical electron-positron collider detector operating at Z-pole is at the level of 10%. Here, we assume for all decay channels at CEPC, the reconstruction and selection efficiencies are 10%.

Decays	LHCb (6 fb−1)	LHCb (300 fb−1)	CEPC (4 Tera Z)
$ D^{*+} $	$ 4.7\times 10^{12} $	$ 2.4\times 10^{14} $	$ 4.6 \times 10^{11} $
$ D^0 $ from $ D^{*+} $	$ 3.2\times 10^{12} $	$ 1.6\times 10^{14} $	$ 3.1 \times 10^{11} $
$ D^{*+}\rightarrow (D^0 \to K^- K^+)\pi^+ $	$ 1.6\times 10^{10} $	$ 6.5\times 10^{11} $	$ 1.3\times 10^9 $
$ D^{*+}\rightarrow (D^0 \to \pi^- \pi^+)\pi^+ $	$ 4.6\times 10^{9} $	$ 2.3\times 10^{11} $	$ 4.5 \times 10^8 $
$ D^{*+}\rightarrow (D^0 \to K^- \pi^+)\pi^+ $	$ 1.6\times 10^{11} $	$ 6.3\times 10^{12} $	$ 1.2\times 10^{10} $
$ D^{*+}\rightarrow (D^0 \to \pi^- \pi^+ \pi^0)\pi^+ $	$ 4.8\times 10^{10} $	$ 2.4\times 10^{12} $	$ 4.6 \times 10^{9} $
$ D^{*+}\rightarrow (D^0\to K^- \pi^+ \pi^0)\pi^+ $	$ 4.6\times 10^{11} $	$ 2.3\times 10^{13} $	$ 4.4 \times 10^{10} $
Reco. & Sel. $ D^0 \to K^- K^+ $	$ 5.8\times 10^{7} $ [147]	$ 2.9\times 10^{9} $	$ 1.3\times 10^8 $
Reco. & Sel. $ D^0 \to \pi^- \pi^+ $	$ 1.8\times 10^{7} $ [147]	$ 9\times 10^{8} $	$ 4.5 \times 10^7 $
Reco. & Sel. $ D^0 \to K^- \pi^+ $	$ 5.2\times 10^{8} $ [147]	$ 2.6\times 10^{10} $	$ 1.2\times 10^{9} $
Reco. & Sel. $ D^0 \to \pi^- \pi^+ \pi^0 $	$ 2.5\times 10^{6} $ [148]	$ 1.2\times 10^{8} $	$ 4.6 \times 10^{8} $
Reco. & Sel. $ D^0\to K^- \pi^+ \pi^0 $	$ 1.9\times 10^{7} $ [148]	$ 9.6\times 10^{8} $	$ 4.4 \times 10^{9} $

Table 7.  The number of ($ D^0 $) and related fully hadronic final state decay modes produced at the LHCb experiment during its Run-2 period (approximately 6 fb⁻¹) and the expected data to be produced over the entire lifetime of the LHC and LHCb (approximately 300 fb⁻¹), as well as the number of corresponding decay modes expected to be produced at the CEPC Z-pole operation mode. The total yields at LHCb is estimated using the cross-section measured by Ref. [149], the reconstructed and selected events from LHCb are obtained from Refs. [147, 148], while the reconstruction and selection efficiency at CEPC is assumed to be 10%.