Studies of CP Violation and New Sources of CPV

Charge-Parity (CP) violation is a key component in explaining the matter–antimatter asymmetry in the universe. STCF will produce billions of quantum-entangled pairs of neutral D mesons, tau leptons, and hyperons, offering a unique environment for precision CPV studies. In particular, hyperon CP violation searches at STCF are expected to reach world-leading sensitivity (better than 10^-4), which is sufficient for testing Standard Model (SM) predictions.

Hadron Spectroscopy and Exotic States

Just as spectroscopy is central to atomic and molecular physics, hadron spectroscopy plays a crucial role in understanding quantum chromodynamics (QCD) confinement. STCF will yield trillions of (charmonium-like) quark-antiquark states, enabling precise studies of the light hadron spectrum, charmonium-like structures, and the systematics of exotic hadrons.

Nucleon Structure and Formation

Probing the internal structure of nucleons is essential for uncovering the properties of matter and QCD confinement. STCF will perform threshold scans of baryonic final states to measure nucleon electromagnetic form factors, baryon decay constants, and strong phases, with at least an order-of-magnitude improvement over existing measurements.

Precision Measurements of Fundamental Parameters and Searches for New Physics

STCF will significantly improve the precision of fundamental quantities such as the tau lepton mass, strong phases in neutral D meson decays, and the magnetic dipole moments of baryons and leptons. The sensitivity to potential new physics will be enhanced by up to two orders of magnitude compared to current limits.

Collider ring physics design

The most challenging aspect of the collider ring design lies in the interaction region (IR). On one hand, ultra-strong focusing is required to achieve extremely small vertical beta functions (${\beta }_y^{*}\le 1$ mm) at the IP. On the other hand, this strong focusing induces large local chromaticities, which, along with sextupoles for chromatic correction, fringe fields from superconducting magnets, and crab sextupoles, significantly degrade the dynamic aperture and momentum aperture of both the electron and positron rings. These effects become particularly critical in the low emittance regime and for high beam current, where Touschek scattering severely limits beam lifetime.

This design complexity is shared across multiple third-generation e⁺e^- collider projects worldwide, each facing varying degrees of difficulty. The SuperKEKB, the only third-generation collider currently in operation, has also struggled to reach its design beam parameters. There is a growing international consensus that new-generation colliders must comprehensively and simultaneously address a wide set of interdependent physics mechanisms, such as strong nonlinearities, Touschek effects, collective instabilities, beam–beam interactions, injection dynamics, machine errors, beam collimation, and radiation damping, through an integrated design approach. Traditionally, these physical mechanisms are studied independently. Existing simulation tools must therefore be upgraded to accommodate such complex, multi-physics studies.

Injector physics design

To accommodate the wide operational energy range (1–3.5 GeV) and short beam lifetime of the collider rings—particularly under the bunch swap-out injection scheme—the injector system must deliver electron and positron beams with the following characteristics: variable energy (1–3.5 GeV), high repetition rate (≈30 Hz), low emittance (<30 nm·rad), and high bunch charge (8.5 nC). Meeting these requirements poses significant challenges in the injector design. These challenges include how to optimize the linac design to minimize emittance growth and energy spread when accelerating high-charge electron bunches from a thermionic gun, how to design a positron production and accumulation chain—including target, capture optics, damping, and acceleration—to yield high-quality positron bunches with sufficient charge and low emittance, and how to suppress emittance degradation in beam transport lines to realize high-intensity electron and positron bunches during injection.

Key technologies

While most of the technical systems of the STCF accelerator will adopt mature and proven accelerator technologies wherever possible to ensure feasible and timely construction, several key technologies—though having a certain development base or being under development—remain immature and must be developed or advanced in significant undertakings to meet the facility’s performance goals and ensure technological advancement. The project team is actively organizing vigorous research and development (R&D) efforts in these areas, aiming for critical breakthroughs in the coming years to ensure the implementation of the engineering construction. In parallel, backup solutions to those key technologies are being prepared to guarantee the timely advancement of the construction project and compliance with final performance specifications, even if the R&D of the key technologies experiences delays or setbacks.

The key enabling technologies required for the STCF accelerator include:

Twin-Aperture Superconducting Magnets for the IR: These include the final focus quadrupoles as well as integrated corrector coils, higher-order field coils, anti-solenoids, and compensation solenoids.

Collider Ring RF Systems: Room-temperature cavities designed to support high circulating currents (≈2 A), with deep higher-order mode (HOM) suppression, high-power RF couplers, and stable low-level RF (LLRF) controls.

Beam Diagnostics and Feedback Systems: Fast-response, low-noise feedback systems and high-precision 3D bunch profiling instrumentation.

Other Advanced Accelerator Technologies: Ultra-fast kicker magnets with pulse bottom widths smaller than 6 ns; ultra-high vacuum systems with low impedance, capable of operating under very intense synchrotron radiation; high bunch-charge (>8 nC) PC electron guns; high-power positron production targets; klystrons and accelerating structures operating at up to a 90 Hz repetition rate; and complex mechanical systems for the MDI (Machine–Detector Interface) region, etc.

Baseline scheme (Two-fold symmetry)

The STCF collider rings adopt a double-ring layout, consisting of an electron ring and a positron ring. Both rings lie in the same horizontal plane and intersect at the IP in the collision region and the crossing point in the region opposite the IP. The layout is symmetric along the line connecting these two points. Each collider ring features a two-fold symmetric structure. Owing to the crossing geometry of the double-ring configuration, the drift sections in the arc regions on opposite sides of the same ring differ slightly in length, forming inner and outer rings with an approximate spacing of 2 m. Space is reserved on both sides of the IR for the future installation of spin rotators to enable beam polarization upgrades.

Each ring consists of one collision region, four large arc sections (each bending by 60°), two small arc sections (each bending by 30°), one crossing region, and several straight sections. The straight sections serve different functions, including beam injection and extraction, housing DWs, beam collimation, accommodating RF cavities, and matching the working point. The total circumference of the ring is 860.321 m.

The main features of this lattice design include:

· A symmetric lattice layout that preserves the option of upgrading to a second IP;

· Flexibility in the type and length of DWs—currently adopting a well-established room-temperature DW scheme, with adjustments in the DW section not affecting the rest of the layout;

· A relatively small injection deflection angle where both electron and positron beams require a 60° bend to enter the collider rings;

· Reserved space for future upgrades, such as spin rotators required in potential beam polarization;

· A medium-scale arc section with a maximum arc bending of 60°, which improves local chromaticity correction and allows for the better optimization of the dynamic and momentum apertures.

Figure 3 shows the layout of the collider rings, and Fig. 4 presents the optical functions. This section introduces the optical design of the arcs, straight sections, and crossing region in the two-fold symmetric lattice; the design of the IR optics is presented in Sect. 2.2.3.

Fig. 3Full-screen View

(Color online) Layout of the STCF collider rings

Fig. 4Full-screen View

(Color online) Optical functions of the full collider rings

2.2.1.1

Arc optics design

The large and small arc sections adopt the same standard FODO cell design. Each FODO cell is 4.7 m in length, providing a bending angle of 6°, a phase advance of 90°, and a momentum compaction factor of 4.9 × 10^-3. Each cell includes four 0.8 m drift sections for hosting sextupole magnets and beam collimators. Figure 5 shows the optical functions of a FODO cell. The ends of each arc section contain dispersion suppressor sections and optics matching sections. The large arc section consists of 9 FODO cells plus the dispersion suppression and matching sections, totaling 57.184 m in length and providing a total bending angle of 60°; the small arc section contains 4 FODO cells plus the dispersion suppression and matching sections, with a total length of 33.684 m and a bending angle of 30°.

Fig. 5Full-screen View

(Color online) Optical functions of a standard FODO cell

Chromaticity correction and nonlinear optimization in the arc sections are achieved using sextupole magnets. These sextupoles are placed in pairs with a phase advance of 180°, satisfying the −I transformation condition, which effectively cancels first-order geometric resonance terms. The large arc section contains 8 groups of sextupoles (4 defocusing sextupole (SD) pairs + 4 focusing sextupole (SF) pairs), and the small arc section contains 4 groups (2 SD pairs + 2 SF pairs). The field strength of each group of sextupole magnets can be adjusted independently. Figures 6 show the optical functions of the large and small arc sections.

Fig. 6Full-screen View

(Color online) Optical functions of the large arc section (top) and small arc section (bottom)

2.2.1.2

Crossing region optics

The crossing region employs two groups of dipole magnets to achieve the crossing and separation of the two beamlines. To achieve achromaticity, a triplet structure with a phase advance of π is used between the dipole magnets in each set. Additionally, two doublet structures are placed in the middle of the crossing region. To reduce the collision probabilities at the crossing point, the β functions in this section are designed to be relatively large, resulting in correspondingly larger beam sizes. The total length of the crossing region is 35 m, and the separation distance between the two rings ranges is approximately 2 m. Figure 7 shows the optical functions of the crossing region.

Fig. 7Full-screen View

(Color online) Optical functions of the crossing region

2.2.1.3

General straight section optics

In addition to generally designed straight sections that are only for connecting the arc sections, most of the straight sections in the collider rings are dedicated segments, such as the injection and extraction section, the DW sections, the RF section, and the beam collimation section, depending on their specific functions. Here, it describes only the design of the general straight section.

The general straight section adopts a standard FODO structure. Each FODO cell is 5 m long with a phase advance of 30° and contains two drift spaces that are 2.2 m long. Figure 8 shows the optical functions of the general straight section consisting of eight FODO cells.

Fig. 8Full-screen View

(Color online) Optical functions of eight FODO cells in the general straight section

2.2.1.4

Damping wiggler section optics

Each of the STCF collider rings includes two DW sections, which are used to reduce the damping time and adjust the beam emittance at different beam energies. Each DW section contains eight DWs that are 4.8 m long. To ensure the relatively gentle variation of the β-function within the wigglers, the lattice design incorporates a specially optimized structure for the wiggler sections.

A triplet configuration is chosen as the primary lattice structure for this section because of its flexibility. This configuration allows for long drift spaces to accommodate the wigglers while ensuring slow variation of the β-function at the wiggler locations. Since commonly used lattice design codes such as MADX and SAD do not include built-in DW models, the typical approach is to model the wigglers using a series of bending magnets and drift spaces that achieve equivalent radiation damping effects. Figure 9 shows the lattice functions of two focusing cells and the long drift section within the DW region.

Fig. 9Full-screen View

(Color online) Lattice functions of two triplet units and long drifts in the damping wiggler section

Because DWs influence the optical functions of the lattice, it is necessary to place several quadrupole magnets at both ends of each DW section. These are used to compensate for the optical distortions introduced by the wigglers and to match the Twiss parameters at the transition to adjacent straight sections. Table 2 lists the key parameters of the lattice design for the DW section.

Table 2Full-screen View

Key parameters of the damping wiggler section (per ring)

Parameter	Value
Total wiggler section length (m)	2 × 70
Triplet unit length (m)	7.5
Wiggler unit length (m)	4.8
Number of wigglers per ring	16
Average β-functions at wiggler, βx/βy (m)	5.08/3.78

Show more

Scheme II (Single-fold symmetry)

In conjunction with developing the above scheme, we have also studied an alternative scheme called Scheme II. This scheme has a reserved space for the installation of SSs and a multifunctional long straight section.

The electron ring of this scheme has four arc sections, three SSs, four DWs, one IR, and one multifunctional section. The geometric layout of the dual ring and distribution of different units are shown in Fig. 10. This layout is realized by employing dipole magnets with different bending angles on both sides of the IR and arc dipole magnets with a bending radius difference of 2.14 m between the inner and outer half-rings.

Fig. 10Full-screen View

(Color online) Layout of the STCF collider ring lattice for Scheme II

Figure 11 illustrates the full-ring lattice and its linear optical functions. Table 3 presents the main collider ring parameters of Scheme II considering intra-beam scattering (IBS), synchrotron radiation, and DWs. These parameters are for the optimal energy of 2 GeV.

Fig. 11Full-screen View

(Color online) Full-ring lattice and linear optics of Scheme II

2.2.2.1

IR optics–scheme II

The IR adopts the crab-waist collision scheme, and its lattice structure is similar to that in Scheme I (see Sect. 2.2.3). The linear optics system of the right half (the left half has the same functional partitioning but with different parameters) consists sequentially of the final telescope (FT), the local chromaticity correction system (LCCS), the crab sextupole (CS) section, and the Matching Transport (MT) section, covering a total bending angle of 60°. At the IP, the β functions are ${\beta }_x^{\text{*}}=40$ mm and ${\beta }_y^{\text{*}}=0.6$ mm.

In addition to sextupoles for correcting first- and third-order chromaticities [5], the IR is also equipped with weak sextupole magnets [6] to optimize nonlinear performance. The geometric layout of the IR is shown in Fig. 12, and its linear optics functions are shown in Fig. 13.

Fig. 12Full-screen View

(Color online) IR dual-ring layout

Fig. 13Full-screen View

(Color online) IR lattice functions

2.2.2.2

Arc optics–scheme II

To implement the dual-ring layout, the arc sections are divided into inner and outer arcs, each consisting of a long arc section with a 120° bend and a short arc section with a 30° bend. Both are constructed using FODO structures with horizontal and vertical phase advances of 90° and a bending angle of 5°. In the FODO structures of the inner and outer arcs, the bending magnets have a curvature radius difference of 2.14 m, while the quadrupoles, sextupoles, and drift sections have equal lengths.

Figure 14 shows the lattice functions of the long and short arc sections in the outer half-ring. In the short arc, non-interleaved sextupole pairs with a –I transfer are used for chromaticity correction. In the FODO cells of the long arc, a focusing sextupole follows each focusing quadrupole, and a defocusing sextupole follows each defocusing quadrupole. A sequence of four identical FODO structures forms one HOA (Higher-Order Achromat) unit, effectively canceling the first- and second-order geometric driving terms [7].

Fig. 14Full-screen View

(Color online) Lattice functions of the long (left) and short (right) arc sections

2.2.2.3

Straight section optics–scheme II

The straight sections include medium straight sections dedicated to the installation of SSs and DWs, as well as the multifunctional region for hosting the injection system, RF cavities, dual-ring crossing, and phase adjustment. Their corresponding linear optical functions are shown in Figs. 15 and 16.

Fig. 15Full-screen View

(Color online) Lattice functions for the SS (left) and DW (right) sections

Fig. 16Full-screen View

(Color online) Lattice functions for the multifunctional straight section

SS refers to the space reserved for Siberian Snakes, which has an azimuthal angle of 120° between each pair. To regulate the damping time, two DWs are installed in both the inner and outer half-rings. Each wiggler is positioned at the center of its respective DW section, where a non-zero dispersion function is present. The dispersion function can be tuned by adjusting the magnetic field strength of the bending magnets, which enables control over the damping time and emittance. Six quadrupole magnets are arranged to ensure that the working point remains constant with varying wiggler field strength.

2.2.2.4

Dynamic aperture and Touschek lifetime–Scheme II

Placing phase tuners and sextupoles at the IP mirror position [8] enables the independent optimization of second- and third-order chromaticity, the W-function, and momentum acceptance. This optimization results in a large off-momentum dynamic aperture and momentum aperture. At a design luminosity of 1.03×10³⁵ cm^-2s^-1 ${\text{s}}^{\text{–1}}$, the off-momentum dynamic aperture reaches 12σ_x,y×13σ_e, and the Touschek lifetime exceeds 250 s (as rigorously evaluated by SAD [9]). The target Touschek lifetime has been preliminarily achieved. However, the transverse dynamic aperture remains insufficiently large, and the machine errors and corresponding orbit corrections have not been considered. Consequently, the actual Touschek lifetime may be further reduced (Fig. 17).

Fig. 17Full-screen View

(Color online) Momentum-dependent dynamic aperture in Scheme II

Interaction region optics design

The key objective of the lattice design of the STCF IR is compressing the β function at the IP to enhance luminosity while maintaining sufficient momentum acceptance and dynamic aperture for an adequate Touschek lifetime. This requires the careful optimization of both the linear lattice and nonlinear optics corrections in the IR. The fundamental design principle is to coordinate the linear optics with the phase advances required for nonlinear cancellation and the minimization of the effects of nonlinear sextupole fields.

Following the design philosophy of other new-generation electron–positron colliders [1, 4, 10–12], the STCF IR lattice adopts the modular structure illustrated in Fig. 18. It consists of a final focusing telescope (FFT) that minimizes the β functions at the IP, a matching section (MCY) between the FFT and the vertical chromatic correction section (CCY), local vertical/horizontal chromaticity correction sections (CCY/CCX), a matching section (YMX) from the CCY to the CCX, a dispersion suppressor (XMC), a CS section, and an MS connecting the IR to the long straight section. The linear optics functions of the STCF IR are shown in Fig. 19.

Fig. 18Full-screen View

(Color online) Lattice layout of the right half of the IR

Fig. 19Full-screen View

(Color online) Optical functions of the right half of the IR

The FFT section adopts a focusing structure composed of two groups of quadrupole magnets: one superconducting and one room-temperature. The superconducting quadrupole doublet is used to achieve an extremely low ${\beta }_y^{*}$ at the IP, which is essential for enhancing luminosity in new-generation colliders. The room-temperature quadrupole doublet forms the mirror point of the IP. At this mirror point, both ${\alpha }_x$ and ${\alpha }_y$ are zero, and the phase advances to the IP in both the horizontal and vertical planes are π [13]. The key advantage of the FFT is its ability to adjust the optical functions at the IP by only tuning the optics at the mirror point without modifying the FFT components.

A bending magnet (B0) is inserted into the FFT to generate the dispersion required for local chromaticity correction. Given its proximity to the IP, the strength of this bending magnet should remain minimal to reduce the amount of synchrotron radiation background entering the detector.

The MCY section adopts a FODO-like cell starting with a defocusing quadrupole, making ${\beta }_y\gg {\beta }_x$ at the vertical chromaticity-correction sextupole S1Y. Additionally, the phase advance from S1Y to the final focusing (FF) quadrupoles (QD0 and QF1) is designed to be approximately π [14], with fine tuning required to correct second-order chromaticity. Two dipole magnets, B1 and B2, are included to increase dispersion at S1Y, which reduces the required sextupole strength.

The local vertical chromaticity correction section (CCY) must implement –I transformation between sextupole pairs to cancel their nonlinearities. This can be realized using two center-symmetric FODO cells with horizontal and vertical phase advances of π/2. Two identical dipole magnets, B3 and B4, are used to create symmetric dispersion.

The YMX section converts the β-function condition of ${\beta }_x\gg {\beta }_y$ at the vertical chromaticity-correction sextupole S1Y to that of ${\beta }_x\gg {\beta }_y$ at the horizontal chromaticity-correction sextupole S1X. Additionally, the phase advance from S1X to the FF quadrupoles (QD0 and QF1) must be approximately 3π, and fine-tuning is required to correct second-order chromaticity. At this section’s midpoint, where ${\alpha }_x=0$, a second approximate IP mirror point is formed. Two dipole magnets, B5 and B6, are included to increase dispersion at S1X, thereby reducing sextupole strength.

The local horizontal chromaticity correction section (CCX) is designed similarly to the CCY section, requiring a –I transfer transformation between sextupole pairs to cancel nonlinearities. Two identical dipole magnets, B7 and B8, are used to form symmetric dispersion. The XMC section cancels dispersion in the collision region, ensuring zero dispersion at the crab sextupoles.

The CS section consists of six quadrupole magnets satisfying the phase advance constraint between the crab sextupole and the IP and the alpha function constraint at the crab sextupole: ${\mu }_x=6\text{π},{\mu }_y=5.5\text{π},{\alpha }_x=0,\text{ and }{\alpha }_y=0$. Additionally, ${\beta }_y\gg {\beta }_x$ at the crab sextupoles, which significantly reduces their required strength and associated nonlinearities [15]. The MS consists of several quadrupole magnets that are used to match the β/α functions required at the end of the long straight section and tune the phase advance between the IR and arc.

The dipole magnets in the IR have a total bending angle of 60°, and they are 1 m long. Apart from B0, which has a fixed bending angle of 1°, the strength of all dipoles must be chosen to adjust the dispersion function and keep the dispersion invariant ${\mathcal{H}}_x$ below 0.02 m. To create a 60 mrad crossing angle at the IP between the two rings, the dipole magnets on the inner ring (beam outgoing direction) are designed to bend 30 mrad less, while those on the outer ring (beam incoming direction) bend 30 mrad more. This approach facilitates a reasonable spacing (1.5–2 m) between the two rings but produces an asymmetry in the dispersion function about the IP, as shown in Fig. 20.

Fig. 20Full-screen View

(Color online) Optical functions of the entire STCF IR

TM020 accelerating mode stability analysis

To analyze coupled-bunch instabilities due to the fundamental modes of RF cavities, the influence of low-level feedback must be considered since it modifies the impedance experienced by the beam [20]. Two common methods are used for evaluation: analytical calculations based on closed-loop cavity impedance and particle tracking simulations [21]. For the LLRF feedback, we only consider PI (proportional-integral) feedback based on I/Q signals. The closed-loop impedance of the cavity can be expressed as: $Z_{\text{clo}}=\frac{Z_{\text{c}}\left(\omega \right)}{1+\frac{k_{\text{p}}}{R_{\text{L}}}{e}^{-i\text{Δ}\omega {\tau }_{\text{d}}}{Z}_{\text{c}}\left(\omega \right)},$ (2) where $Z_{\text{c}}\left(\omega \right)$ is the open-loop cavity impedance, $R_{\text{L}}$ is the load impedance, $k_{\text{p}}$ is the proportional gain of the PI feedback, and ${\tau }_{\text{d}}$ is the feedback loop delay.

For STCF, by setting the appropriate proportional gain and delay time of the PI feedback, the –1 mode can be effectively suppressed and the –2 mode can be maintained below the threshold determined by synchrotron radiation damping, thus avoiding instabilities caused by the fundamental mode. Using the 2 GeV & 2 A beam case as an example, Fig. 25 shows the growth rates of major low-order modes obtained by both analytical and particle tracking methods for various proportional gains and delay times. Without PI feedback, the –1 mode has a growth rate of 368 s^-1, which is much higher than the damping rate of 93.5 s^-1. When $k_{\text{p}}=1$ and the delay is between 700 and 900 buckets, the –1 mode is fully suppressed and the –2 mode becomes the dominant instability; however, its growth rate remains below the damping rate. This indicates that a low-R/Q TM020 cavity does not require a special feedback system for suppressing low-order coupled-bunch instabilities. Using standard PI feedback with suitable gain and delay is sufficient to ensure stability. The same conclusion applies to the 1 GeV & 1.5 A and 3.5 GeV & 2 A beam conditions. To ensure this approach is effective, the LLRF feedback delay should preferably not exceed 1.5 μs.

Fig. 25Full-screen View

(Color online) Growth rates of low-order coupled-bunch instabilities caused by the accelerating mode. Solid lines are from analytical methods; discrete points are from tracking simulations

TM020 parasitic mode stability analysis

In the TM020 cavity, the TM020 mode is the accelerating mode, and modes with frequencies higher or lower than its nominal frequency are referred to as higher-order and lower-order modes, respectively. Both types can cause significant coupled-bunch instabilities. For simplicity, we refer to both as parasitic modes. To mitigate these instabilities, the cavity design must incorporate features that significantly suppress these parasitic modes. Assuming that beam spectral lines coincide with the resonance frequencies of parasitic modes, the associated coupled-bunch instability growth rate is typically calculated as $\frac{1}{{\tau }_{\text{HOM}}}=\frac{I_0\alpha {\omega }_{\text{r}}}{4\pi v_{\text{s}}E/e}{e}^{-{\left({\omega }_{\text{r}}{\sigma }_{\tau }\right)}^2}\text{ReZ}，,$ (3) where ${\sigma }_{\tau }$ is the rms bunch length, $v_{\text{s}}$ is the synchrotron tune, ${\omega }_{\text{r}}$ is the resonance frequency, and ReZ is the real part of the impedance.

STCF has a synchrotron oscillation period less than 0.2 ms. Assuming that the longitudinal bunch-by-bunch feedback system has a conservative damping time of 1 ms, the impedance thresholds of parasitic modes are shown in Fig. 26. The threshold is lowest for 3.5 GeV because it uses the most cavities and assumes that all HOMs are aligned. Regardless, the lowest single-cavity threshold is 1.7 kΩ at 3.75 GHz, which is achievable for TM020 cavities.

Fig. 26Full-screen View

(Color online) Parasitic mode impedance thresholds assuming 1 ms damping time from longitudinal feedback at three beam energies

Transient beam loading effects

The 5% empty buckets reserved to suppress ion trapping introduce transient beam loading effects [22], resulting in bunch-to-bunch variations in synchronous phase and bunch length. Tracking simulations of this effect show the resulting distributions of bunch centers and lengths, as illustrated in Fig. 27. The effect on bunch length is minimal (within ± 1 ps), and variations in the bunch center remain within ± 20 ps, suggesting that the impact on luminosity is negligible.

Fig. 27Full-screen View

(Color online) Bunch length and center distributions under nominal fill pattern

STCF is designed to use the TM020-type room-temperature RF cavity scheme. Owing to its relatively low R/Q, complex feedback schemes (e.g., one-turn delay feedback and direct feedback) such as those adopted by the PEP-II collider are not necessary. The stability of the fundamental mode can be maintained by choosing the appropriate PI feedback gain and delay parameters, thereby significantly simplifying the LLRF system. Radiation damping alone is insufficient for damping coupled bunch instabilities caused by parasitic modes; a longitudinal bunch-by-bunch feedback system providing at least 1 ms of damping is required. From the perspective of the RF cavity design, suppressing parasitic modes below the instability threshold under such damping conditions is entirely achievable.

Simulation results and analysis

The working point in e⁺e^- storage ring colliders is typically optimized near half-integer values to achieve maximum luminosity. The precise fractional tune values are optimized using beam–beam simulations. Figure 28 shows the luminosity versus working point calculated using the BBWS code for fractional tunes above the half-integer (beam energy is 2 GeV; other parameters refer to Table 1). Comparing the simulations with and without the crab-waist scheme reveals that the crab-waist can effectively suppress nonlinear betatron resonances induced by beam–beam effects, thus enlarging the high-luminosity region in the tune space.

Fig. 28Full-screen View

(Color online) Comparison of luminosity maps with (left) and without (right) the crab-waist scheme. The black dot marks the nominal working point; dashed lines indicate resonance lines

However, Figure 28 also shows that the crab-waist scheme cannot suppress synchro-betatron resonances induced by beam–beam interactions. Similar to SuperKEKB, STCF chooses a relatively large synchrotron tune $v_z$ and places the transverse tune between $0.5+n{v}_z$ and $0.5+\left(n+1\right)v_z$ (n = 1 or 2). Since these resonance lines are separated by $v_z$, a large $v_z$ helps prevent the tune footprint from overlapping with resonance lines (see Fig. 29), thereby avoiding emittance growth and luminosity loss. The general rule is $v_z/{\xi }_x\ge 3$. At 2 GeV, STCF chooses $v_z=0.0194$, producing $v_z/{\xi }_x=3.7$, which should be sufficient to mitigate synchro-betatron oscillations and coherent X-Z instabilities.

Fig. 29Full-screen View

(Color online) Tune footprint of STCF computed using the SAD code (including lattice). The red star indicates the nominal working point (0.543, 0.58). Solid lines in different colors represent various resonance lines

Coherent X-Z instabilities are evaluated using strong–strong beam–beam simulations. Figure 30 shows the results of scanning the horizontal tune (with vertical fractional tune fixed at 0.58) using the BBWS and BBSS codes. The results indicate that a horizontal tune near 0.543 avoids coherent X-Z instability. The strong–strong simulations suggest that the range of horizontal tune values for achieving high luminosity is narrow and can be widened by increasing the synchrotron tune ${\nu }_z$ or reducing ${\beta }_x^{*}$ (Sect. 4.11 of [27]). Longitudinal impedance primarily causes bunch lengthening and synchrotron tune shifts. Bunch lengthening directly reduces luminosity, while tune shifts can introduce Landau damping that weakens synchro-betatron resonances, though it also broadens and shifts the oscillation peaks, complicating tune optimization.

Fig. 30Full-screen View

(Color online) Luminosity (top) and horizontal bunch size (bottom) versus horizontal tune from BBWS and BBSS simulations. Brown curves show simulations including longitudinal impedance (modeled by the broadband resonator in Fig. 37)

Fixing the horizontal fractional tune at 0.543, vertical tune scans were performed using BBSS. Figure 31 shows that the vertical fractional tune should be greater than 0.56 and that luminosity and transverse beam sizes are relatively insensitive to vertical tune within this range.

Fig. 31Full-screen View

(Color online) Luminosity (top) and transverse beam sizes (bottom) versus vertical tune from BBSS simulation

Simulations considering lattice nonlinearities (for details of lattice design, see Sect. 2.2) were performed using the BBSCL code and compared with the BBSS results. As shown in Fig. 32, coupling between lattice nonlinearities and beam–beam effects affects luminosity, with the extent depending on specific lattice design details. This highlights the need for deeper lattice nonlinearity studies and optimization. Additionally, BBSCL was used to scan the single-bunch current with the lattice included. Figure 33 shows that coherent instability occurs when N_p exceeds 6.2×10¹⁰, above the nominal value of 5.2×10¹⁰.

Fig. 32Full-screen View

(Color online) Luminosity (top) and transverse beam sizes (bottom) at the working point (0.543, 0.58) computed using BBSCL (with lattice) and BBSS (without lattice)

Fig. 33Full-screen View

(Color online) Luminosity (top) and horizontal bunch size (bottom) as a function of single-bunch population at the working point (0.543, 0.58) from BBSCL simulations

Coupled effects of multiple physical mechanisms

The coupling of multiple physical mechanisms is particularly pronounced in colliders employing the crab-waist collision scheme, which significantly increases the complexity of machine design and the difficulty of achieving high-luminosity operation. In addition to the previously discussed couplings between beam–beam interactions, lattice nonlinearities, and impedance effects, several other factors warrant further investigation.

· Space charge (SC) effects: Under high-current conditions, the Coulomb field generated by the beam has a non-negligible influence on particle motion. According to the latest STCF lattice design and beam parameters, SC effects induce vertical and horizontal tune shifts of approximately −0.043 and –0.0027 (see Fig. 34), which are opposite the shift caused by beam–beam interactions. As this effect is distributed along the entire ring and influenced by the alternating-gradient focusing structure, its overall impact cannot be simply canceled by beam–beam tune shifts and thus cannot be directly utilized to enhance luminosity. Beyond simplified particle tracking models, a more comprehensive assessment requires full-ring tracking simulations incorporating the actual lattice structure. These simulations are being actively performed using the BBSCL code with full-lattice and SC effects. A preliminary luminosity result under the design beam parameters is shown in Fig. 35. Analysis of the simulation data reveals a notable luminosity degradation and is attributed to a weak coherent X-Z instability. This instability is primarily associated with the horizontal tune shift induced by SC (its absolute value is comparable with the horizontal beam–beam tune shift), rather than the vertical component. The horizontal tune shift moves the beams closer to the broad resonance peaks of the coherent X-Z instability, as illustrated in Fig. 30 under current machine conditions.

Fig. 34Full-screen View

(Color online) Simulated luminosity by the BBSCL code for STCF with and without the space charge effects

Fig. 35Full-screen View

(Color online) Space-charge tune shift as a function of the distance along the ring of STCF

· Impedance effects: In addition to the longitudinal impedance effects discussed earlier, transverse impedance can also couple with beam–beam interactions and induce coherent mode-coupling instabilities. These instabilities are closely related to the vertical tune and thereby constrain the accessible working point space. A comprehensive impedance budget for STCF is currently under development. The combined impact of impedance and beam–beam interactions will be systematically investigated in future studies.

· Imperfections in the IP optics: These include non-zero coupling and dispersion at the IP, phase deviations between the IP and the crab-waist sextupoles, and orbit errors at both the IP and sextupole locations. Such imperfections can undermine the effectiveness of the crab-waist mechanism and consequently degrade the achievable luminosity.

· Noise in the bunch-by-bunch feedback system: Under high-current operating conditions, the machine’s reliance on the feedback system for suppressing coherent instabilities increases significantly. However, because of the extremely small transverse beam sizes at the IP, noise in the feedback system may excite beam motion and contribute to luminosity degradation. To evaluate this effect, simulations using the BBSS code were performed, incorporating turn-by-turn noise modeled as vertical collision offsets at the IP with a standard deviation of $\delta y=k_y{\sigma }_y^{*}$. The results, shown in Fig. 36, indicate that even modest noise levels (e.g., $k_y=0.1$) can lead to a luminosity loss exceeding 30%, highlighting the need for the stringent control of feedback-induced offset noise.

Fig. 36Full-screen View

(Color online) Simulated specific luminosity by the BBSS code for STCF, considering turn-by-turn noises in the vertical offsets of the beams at IP

· Other collective effects: Phenomena such as electron clouds and ion trapping may also pose potential performance limitations for the machine.

Recently, GPU-accelerated codes (e.g., BBSCL at KEK, APES-T at IHEP, and Xsuite at CERN) have enabled more efficient simulations of coupled multi-physics effects. In the future, further STCF beam–beam studies will rely heavily on these advanced tools.

Collective effects

Beam collective effects encompass various phenomena correlated with beam current. On one hand, they can degrade beam quality, including by increasing emittance and energy spread; on the other hand, they can cause various instabilities that limit beam current and consequently reduce collision luminosity. Therefore, carefully studying the influence of these collective effects is essential. The types of beam collective effects are diverse and include intra-beam scattering (IBS) and Touschek scattering, which are both intra-scattering effects within the beam; various instabilities induced by the coupling of the beam with the impedance of the vacuum chamber environment, such as longitudinal microwave instabilities, transverse single-bunch instabilities, and coupled-bunch instabilities; and instabilities caused by residual gas molecules, ions, and electron clouds within the vacuum chamber. To meet the high-luminosity requirements of a collider, it is necessary to reduce the beta function ${\beta }^{*}$ at the IP. This reduction implies the need for short bunch lengths and high total beam current, along with the additional requirement that the bunches remain stable, that is, transverse and longitudinal oscillations must be well suppressed. Therefore, for STCF, beam current limitation due to collective beam stabilities is a critical issue. During the design phase, it is crucial to carefully evaluate collective effects and propose effective mitigation strategies to ensure the stable operation of STCF at high beam currents, thereby achieving the desired machine performance.

Impedance-driven single-bunch collective effects

Single-bunch collective effects [28] are predominantly governed by the broadband impedance of the storage ring (from flanges, BPMs, bellows, collimators, etc.) and include bunch lengthening, microwave instabilities, and transverse mode coupling instability (TMCI).

To suppress these effects, strict control of broadband impedance is required. During engineering implementation, this requires optimization of the vacuum chamber structure through streamlining geometry and minimizing discontinuities, combined with rigorous quality control in fabrication to mitigate impedance contributions—particularly from crucial vacuum components. Additionally, avoiding structures that could trap HOMs is crucial because they can lead to parasitic energy loss, local heating of vacuum chambers, and coupled-bunch instabilities.

When the single-bunch current remains below the microwave instability threshold, bunch lengthening is dominated by potential well distortion and can be described by ${\left(\frac{{\sigma }_z}{{\sigma }_{z0}}\right)}^3-\frac{{\sigma }_z}{{\sigma }_{z0}}=-\frac{c{I}_{\text{b}}}{4\sqrt{\pi }{\alpha }_{\text{p}}{\omega }_0{\sigma }_{z0}{\sigma }_{\delta 0}^2{E}_0/e}\text{Im}{\left(\frac{Z_{\parallel }}{n}\right)}_{\text{eff}},$ (11) where ${\sigma }_{z0}$ is the rms bunch length, ${\sigma }_{\delta 0}$ is the rms energy spread, ${\alpha }_{\text{p}}$ is the momentum compaction factor, $E_0$ is the beam energy, and $\text{Im}{\left(\frac{Z_{\parallel }}{n}\right)}_{\text{eff}}$ is the effective longitudinal impedance. For the STCF design, with a current target of 2 A and a filling factor of 50%, the required single-bunch current target is 2.8 mA. By optimizing impedance control strategies and referencing successful practices from existing storage rings, an effective impedance of 0.2 Ω is expected. Under these conditions, the bunch length would increase from the natural value of 6.8 mm to 8.9 mm at an RF voltage of 3 MV.

Microwave instability, a typical longitudinal single-bunch instability, arises when the current exceeds a certain threshold. While it does not cause particle loss, it degrades luminosity by amplifying the energy spread and elongating the bunch length. A conservative threshold estimate is provided by the Keil–Schnell–Boussard criterion $I_{\text{th}}{\left|\frac{Z_{\parallel }}{n}\right|}_{\text{eff}}<\frac{{\left(2\pi \right)}^{\frac{3}{2}}{E}_0{\alpha }_{\text{p}}{\sigma }_{\delta }^2{\sigma }_z}{C},$ (12) where $C$. is the ring circumference. Assuming |Z_L/n| = 0.2 Ω, the estimated threshold current is 1.06 mA. Since the Boussard formula tends to underestimate the actual threshold, a more precise analysis should use macro-particle tracking simulations after the impedance model is built. Here, a simplified broadband resonator model is assumed with $f_{\text{r}}=20\text{ GHz}$, $Q=1$, and $R_{\text{s}}=11.6\text{ kΩ}$, corresponding to an effective impedance of 0.2 Ω. The tracking simulation results for the bunch length and energy spread versus single-bunch current are shown in Fig. 37.

Fig. 37Full-screen View

(Color online) Variation of single-bunch rms length and energy spread with bunch charge

Simulations reveal that the microwave instability threshold is approximately 2 mA, which is smaller than the STCF’s target single-bunch current (2.8 mA). This proximity indicates potential operational risks. Increasing the momentum compaction factor ${\alpha }_{\text{p}}$ and the natural energy spread ${\sigma }_{\delta 0}$ while carefully managing the impedance budget during design optimization can help increase the threshold.

Coherent synchrotron radiation (CSR) and coherent wiggler radiation (CWR) [29] can also drive microwave instability. Assuming a wiggler vacuum chamber with a gap height of 35 mm, a peak field of 1.6 T, a single wiggler length of 4.8 m, and a period length of 0.6 m, the simulations employing CSR impedance models obtained using the CSRZ code and macro-particle tracking yield a single-bunch instability threshold of approximately 0.6 mA at a beam energy of 1 GeV; however, the instability is weaker at 2 GeV with a threshold of 3 mA. Strategies to enhance the current threshold at 1 GeV include modifying the wiggler period and optimizing the vacuum chamber geometry. The lattice could also be further optimized by increasing ${\alpha }_{\text{p}}$ and ${\sigma }_{\delta 0}$, as mentioned above. In addition, reducing the operating beam current to sacrifice the target luminosity at low energies could also be adopted if hardware modifications are insufficient.

Head–tail instability and TMCI are critical transverse single-bunch instabilities. Negative natural chromaticity can trigger the zero-mode head–tail instability, which is conventionally suppressed by using sextupoles to correct chromaticity to positive values. While positive chromaticity may excite higher-order head–tail modes, the mode tends to grow slowly and can be suppressed by radiation damping and transverse feedback systems. At zero chromaticity, increasing the beam current shifts the 0 mode toward the −1 mode, and after the threshold is crossed, the merging of the two modes triggers bunch instability, which is the strongest TMCI; hence, it is also called the strong head–tail instability. Once it occurs, the TMCI develops very fast and results in the rapid growth of transverse oscillations and particle loss. Thus, this instability must be strictly avoided.

Here, we only discuss the TMCI threshold under resistive wall impedance, assuming zero chromaticity, which is estimated using $I_{\text{th}}=\frac{4\pi {\nu }_{\text{s}}\left(E_0/e\right)}{T_0{{\displaystyle \sum }}_i{\beta }_{y,i}{k}_{y,i}},$ (13) where $I_{\text{th}}$ represents the threshold current, $v_{\text{s}}$ is the synchrotron tune, and ${\beta }_{y,i}$ and $k_{y,i}$ are the vertical beta function and the transverse kick factor corresponding to the i^th vacuum chamber element in the ring, respectively. Based on design experience from advanced colliders worldwide, collimators and IR are the primary sources of transverse impedance. The former is due to their narrow apertures, and the latter is due to large beta functions. To ensure beam stability, in future design iterations, the total transverse impedance ${{\displaystyle \sum }}_i{\beta }_{y,i}{k}_{y,i}$ should be maintained below 78 kV/pC. For reference, at the SuperKEKB low-energy collider ring, which has a ${\beta }_y^{*}$ of 0.27 mm, the total ${{\displaystyle \sum }}_i{\beta }_{y,i}{k}_{y,i}$ is 138 kV/pC, and the collimators contribute more than half of this value. Thus, at STCF, the collimator design, including apertures, locations, and materials, will critically influence the impedance optimization.

To suppress the TMCI, the chromaticity can be corrected to positive values to help damp the head–tail zero-mode; however, this creates the risk of exciting higher-order head–tail modes. Nonetheless, these HOMs grow slowly and can be effectively suppressed by enhancing radiation damping through damping wigglers and implementing a transverse bunch-by-bunch feedback system, thereby raising the single-bunch transverse instability threshold current. In addition, the nonlinear collimation technique being developed at SuperKEKB—designed to mitigate impedance effects—may offer further benefits. Its potential application to STCF will be explored.

Impedance-related coupled-bunch instability

This analysis focuses on long-range wakefield effects, including narrowband impedance from RF cavities or vacuum structures and low-frequency resistive wall impedance. Instabilities induced by the RF cavity modes have already been discussed in Sect. 2.4 and will not be repeated here. Geometric narrowband impedances, which can trap resonant modes, should be minimized through the optimization of the geometric structures of the devices.

Resistive wall instability is mainly due to the strong transverse impedance near zero frequency. These long-range wakefields can cause transverse coupled-bunch instability. For Gaussian beams, the instability growth rate is governed by [30] $\frac{1}{{\tau }_{\text{n,m}}}=-\frac{1}{1+m}\frac{M{I}_{\text{b}}c}{4\pi \frac{E_0}{e{\upsilon }_{\perp }}}\frac{{\displaystyle {\sum }_{p=-\infty }^{\infty }{\text{ReZ}}_{\perp }}\left({\omega }_{p,n,m}\right)H_n\left({\omega }_{p,n,m}-{\omega }_{\xi }\right)}{{\displaystyle {\sum }_{p=-\infty }^{\infty }{H}_n}\left({\omega }_{p,n,m}-{\omega }_{\xi }\right)}.$ (14)

Here, the mode frequency is given by ${\omega }_{p,n,m}=\left(Mp+n\right){\omega }_0+{\omega }_{\beta }+m{\omega }_{\text{s}}$, where M denotes the number of bunches, n is the coupled-bunch mode number, I_b is the single-bunch current, ${\omega }_{\xi }=\frac{\xi }{\alpha }{\omega }_0$ is the chromatic angular frequency, v_⊥ is the transverse betatron tune, $Z_{\perp }$ is the real part of the transverse resistive wall impedance, and the $H_n$ function is given by Eq. (15): $H_n\left(\omega \right)=\frac{2\pi }{3\sqrt{\frac{\pi }{2}}\cdot \text{Γ}\left(n+\frac{1}{2}\right)}{\left(\omega {\sigma }_{\text{t}}\right)}^{2n}\exp \left(-{\omega }^2{\sigma }_{\text{t}}^2\right).$ (15)

Assuming a uniform aluminum beam pipe with a radius of 25 mm for the full ring circumference, Eq. (14) predicts an instability growth rate of 1.6 ms^–1 at zero chromaticity. Although modern bunch-by-bunch feedback techniques can achieve damping times of approximately 100 μs or better, the SuperKEKB experience [31] shows that the noise of the feedback system has an important impact on the luminosity stability. A slower feedback system combined with other measures, such as correcting chromaticity to positive values to reduce the growth rate, is being considered. With these measures, STCF is expected to manage this instability effectively.

Beam lifetime

The Touschek scattering effect is the dominant factor limiting the beam lifetime in the collider rings. It refers to large-angle Coulomb scattering between two electrons or positrons within a bunch, where, during collisions, one particle transfers its transverse momentum to the longitudinal momentum of another particle. This interaction makes it possible for some particles to exceed the energy acceptance and be lost. From the local momentum aperture (LMA) shown in Fig. 24, the calculated beam lifetime at 2 GeV is approximately 252 s.

Residual gas scattering includes elastic scattering and inelastic scattering. Elastic scattering leads to transverse oscillations, and the particle will be lost if its amplitude exceeds the dynamic aperture. Inelastic scattering causes the deceleration and energy deviation of the particle, and it will also lead to a loss if it is beyond the momentum acceptance. Based on the SuperKEKB experience, elastic scattering is the dominant factor for vacuum-related beam losses in this collider ring type.

Assuming all residual gas molecules are CO, the vacuum-limited lifetime is ${\tau }_{\text{C}}\left[\text{hrs}\right]=11.1\times \frac{E_0^2\left[{\text{GeV}}^2\right]{ϵ}_A\left[\text{mm}\cdot \text{mrad}\right]}{\langle {\beta }_{\perp }\left[m\right]\rangle P\left[\text{nTorr}\right]},$ (16) ${ϵ}_A\equiv \min \frac{A^2}{{\beta }_{\perp }\left(s\right)},$ (17) where A is the local physical aperture and “min” represents taking the minimum value of the entire ring. Assuming ${ϵ}_A=0.1$ μm rad, to achieve a vacuum lifetime longer than 40 min, the vacuum system must maintain an average pressure below $1\times {10}^{-7}$ Pa.

Ion effects and electron cloud effects

Ion instabilities originate from the interaction between the electron beam and their trapped ions. The ions are generated from the ionization of residual gas in the vacuum and desorption from the vacuum wall. In modern electron storage rings, ultra-high vacuum systems can provide a pressure lower than 10^–7 Pa and can generally minimize these effects. When designing the bunch filling scheme, large gaps are typically introduced between bunch trains to help suppress ion instabilities. In addition, transverse bunch-by-bunch feedback systems can effectively suppress fast ion–induced oscillations. Transient fast ion instabilities may still occur during early machine operation under poor vacuum conditions but will diminish as the vacuum improves.

In the positron ring, synchrotron radiation photons strike the vacuum chamber wall and release photoelectrons. These photoelectrons are accelerated by the positron beam and can be multiplied by multiple reflections between the walls. The electrons captured by the positron beam form the so-called electron cloud [32]. The electron cloud interacts with the positron bunches and can couple the motions of successive positron bunches and cause both coupled-bunch instability and single-bunch head–tail instability. Using the PEI and PyECLOUD codes for simulations, at 2 GeV and with an average chamber diameter of 50 mm and a secondary electron yield (SEY) of 1.3, the cloud density near the beam is ${\rho }_{\text{e}}\text{=}{10}^{13}$ m^–3 without additional mitigation measures. The density can be reduced to 5×10¹² m^-3 with antechambers. Using the PEHTS code, the single-bunch instability threshold is predicted to be ${\rho }_{\text{e},\text{th}}\text{=}1.4$ ×10¹² m^–3. Thus, the vacuum system design should aim to keep ${\rho }_{\text{e}}$ below 10¹² m^–3. Multiple measures for suppressing the instability should be considered. These include antechambers to localize photon impact, TiN/NEG coatings to reduce SEY; chamber grooves, clearing electrodes and external magnetic fields to disrupt electron accumulation; and bunch-by-bunch feedback for dynamic stabilization. Based on experience with similar machines, these combined measures are expected to effectively suppress the electron cloud effect in STCF.

Off-axis injection

The key to off-axis injection design is to use local orbit bumps to steer the injected beam into the ring acceptance while minimizing its transverse offset from the central orbit and reducing beam losses at the septum magnet [35]. First, the optics of the injection section must be configured with large β functions to reduce the impact of septum thickness and closed-orbit errors on injection efficiency. Second, the beam clearance aperture must be optimized to balance dynamic aperture and beam losses—a clearance that is too large requires an excessively large dynamic aperture, while one that is too small causes significant beam loss. Lastly, the local bump orbit must be designed with appropriate bump height and angle so that the injected beam exits the septum magnet as close as possible to the pipe center, without excessive loss of the injected beam, while also controlling the influence of the stray magnetic field of the septum magnet to the circulating beam.

The off-axis injection system in the STCF collider rings consists of three components: quadrupoles for optics matching, bump magnets to form the local orbit, and septum magnets for large-angle bending of the injected beam. Table 7 lists the key parameters. Figure 38 shows a schematic of the off-axis injection scheme, while Fig. 39 presents the layout and optics of the injection section, located in one of the 20 m-long straight sections of the collider rings, where 10 quadrupoles are installed for optics matching. Figure 40 illustrates the local bump orbit formed by four bump magnets. Each magnet provides a maximum bending angle of less than 3.5 mrad and can be adjusted independently to vary the orbit height and angle.

Fig. 38Full-screen View

(Color online) Schematic layout of the STCF off-axis injection design

Fig. 39Full-screen View

(Color online) Injection section element layout and optics for the off-axis scheme

Fig. 40Full-screen View

(Color online) Local orbit bump for off-axis injection scheme

Figure 41 shows the evolution of the horizontal geometric emittance (phase-space area) of the injected bunch over five damping times, simulated with Elegant. Initially, the filamentation of the injected bunch dominates emittance growth. Then, synchrotron radiation exponentially damps the off-axis oscillation of the injected beam. After five damping times, the emittance decreases by more than an order of magnitude, fully merging with the stored bunch and reaching an equilibrium emittance of approximately 4.17 nm·rad. Considering the physical aperture of the septum and the dynamic aperture of the collider rings, the final injection efficiency reaches approximately 90%, meeting the design goals. Injection simulations with all realistic errors, which ensure that the injection physics design is robust, are ongoing. More simulations with all error types, including the combined beam–beam effect for merging bunches, will also be performed.

Fig. 41Full-screen View

Evolution of horizontal emittance oscillation for the injected bunch under the off-axis scheme

Bunch swap-out injection

The key to the physics design of the bunch swap-out injection scheme is arranging the components in a geometrically rational layout so that the injected bunch enters the closed orbit of the collider rings accurately, thereby minimizing disturbance to the stored beam and reducing the required deflection angle of the kicker magnets [36]. The STCF bunch swap-out injection system primarily consists of a septum magnet for large-angle deflection and a set of ultra-fast horizontal kicker magnets. The septum magnet design is identical to that used in the off-axis injection scheme. The kicker system consists of five identical modules spanning a 1.9 m drift.

Table 8 lists the relevant design parameters. Figure 42 shows a schematic of the injection system, while Fig. 43 presents the layout and optics of the injection section. This section, shared with the off-axis injection scheme, spans 20 m and incorporates ten quadrupoles. To accommodate the deflection angle and the good-field region of the kicker magnets, approximately 6 m of drift space is reserved for the injected beam to travel from the septum magnet to the kickers. Figure 44 illustrates the beam trajectory and envelope in this region.

Fig. 42Full-screen View

(Color online) Schematic layout of the STCF bunch swap-out injection design

Fig. 43Full-screen View

(Color online) Layout and optics of the injection section for the bunch swap-out scheme

Fig. 44Full-screen View

(Color online) Beam trajectory and envelope from the septum exit to the kicker exit for the bunch swap-out scheme

In the bunch swap-out injection scheme, the injected bunch has a relatively large emittance owing to the high bunch charge from the injector. A key factor for injection efficiency and final luminosity is whether the injected beam can avoid significant losses during the first few turns and then be damped by synchrotron radiation to the equilibrium emittance of the collider rings. Figure 45 shows the evolution of the horizontal beam emittance over five damping times, as simulated with the Elegant tracking code. After five damping times, the emittance converges to approximately 4.20 nm·rad, and the injection efficiency exceeds 98%, just meeting the basic physical design requirements. In future upgrades, smaller injection emittance will be adopted to further reduce the loss rate.

Fig. 45Full-screen View

(Color online) Evolution of horizontal emittance oscillation for the injected bunch under the bunch swap-out scheme

Beam dumping and extraction

Beam extraction can be considered as the reverse process of bunch swap-out injection and is realized using a set of pulsed kicker magnets and a downstream septum magnet. STCF requires two extraction systems to fulfill the following two beam extraction functions:

The first is during normal operation under the bunch swap-out injection scheme. Ultra-fast kicker magnets deflect the bunch to move it away from the equilibrium orbit, and the downstream septum magnet subsequently extracts it from the ring. One bunch is extracted every 33.3 ms to enable on-axis replacement. The hardware requirements are identical to those for swap-out injection, as shown in Table 8. The extraction section layout is also similar to that in Fig. 43, with the exception of having the septum magnet placed downstream in the beam direction. The main requirement for the extraction system in this case is minimizing circulating beam disturbances caused by the kicker magnets.

The second is for beam dumping during machine protection or experimental shutdown. This requires kicker magnets with a rise time shorter than the bunch train spacing and a flat-top width longer than the revolution period for the entire ring to be emptied in a single extraction. This process does not affect luminosity or experiments, and the design is relatively straightforward, with ample reference designs available. Therefore, this is not a technical challenge for the STCF beam injection/extraction design. However, considering the large stored beam energy in the collider rings, minimizing beam loss during extraction is critical. Additionally, kicker magnets with both short rise time and long flat-top width present certain technical challenges. A dedicated beam dump will be located outside the tunnel to receive the high-energy, high-charge (approximately 5.7 μC) dumped beam. Figure 46 shows the beam orbits in the two extraction modes.

Fig. 46Full-screen View

(Color online) Schematic of the beam extraction (Upper: extraction trajectories for swap-out and for dumping; Lower: layout of the extraction section)

The off-axis injection scheme, widely used in positron/electron storage rings over the past decades, has proven effective and reliable in numerous experiments. Hence, adopting the septum-and-bump-based off-axis injection as the baseline scheme for STCF is feasible.

The bunch swap-out injection scheme, which is considered ideal for some fourth-generation light sources and new-generation e⁺e⁻ colliders, has gained much attention in recent years. Its main challenge lies in developing kicker magnets with ultra-short pulses and ultra-fast rise times. Preliminary numerical simulations confirm the feasibility of this scheme, with injection efficiency reaching up to 98%. Therefore, the bunch swap-out injection scheme—based on fast pulsed kickers and septum magnets—will serve as an upgrading path for STCF. The bunch swap-out injection scheme is generally compatible with the off-axis scheme, especially in terms of injector requirements. Hence, adopting the off-axis method at the initial stages of development is advisable. After upgrading to the bunch swap-out injection scheme, the off-axis injection may still serve as a commissioning and operational fallback, ensuring the feasibility of maintaining peak luminosity via the top-up injection mode at STCF.

Beam collimation methods

Calculation and simulation results

2.9.2.1

Beam loss simulation from Touschek scattering

Owing to the very short Touschek lifetime, beam losses during routine STCF collider operation are primarily caused by the Touschek scattering effect. To better collimate Touschek-induced losses, the beam loss distribution in the absence of collimators was first evaluated. The vacuum pipe geometry was modeled as follows: a radius of 15 mm at the IP and the first quadrupole magnets on either side, expanding uniformly through one drift section to 25 mm at the second quadrupole, and then further increasing uniformly to 33.5 mm in the main IR. For the remainder of the ring, a constant vacuum pipe radius of 26 mm was assumed.

Under the condition of a single-bunch charge of 8 nC and a transverse coupling ratio of 0.5%, the distribution of beam losses due to Touschek scattering—assuming scattering points at every quadrupole magnet—was simulated using the Accelerator Toolbox (AT). The result is shown in Fig. 56. Analysis indicates that approximately 58.2% of the lost particles are absorbed in the IR (between 370 m and 380 m), 11.2% in the CCX section preceding the IR (300–320 m), and 13.4% near the CS section upstream of the IR (270–290 m). Most losses occur in the horizontal plane; however, vertical losses are also observed in the crab sextupole region, where β_y is large. In the IR, losses primarily arise in the transition drifts where the pipe aperture changes, within 10 m of the IP, where β-functions are large and complex and the aperture is relatively narrow.

Fig. 56Full-screen View

Beam loss distribution from Touschek scattering without collimators

Based on the loss distribution, global optical function map, and betatron function distortions of off-momentum particles, a layout of 11 horizontal collimators (HCs) and 6 vertical collimators (VCs) was determined through iterative simulation and optimization. The final configuration is shown in Fig. 57. After collimator placement, the Touschek-induced loss distribution is presented in Fig. 58. The simulations indicate that the current configuration achieves a collimation efficiency of approximately 97.3%. Within ±20 m of the IP, residual beam losses are approximately 1.36%, occurring primarily in the transition region upstream of QF1. As shown in Fig. 58, HCs in the CCY section (HC5, HC6) and at locations with large horizontal β-function distortions (HC1) effectively intercept horizontally scattered particles. Similarly, VCs (VC1, VC2) placed in high-β_y regions upstream of the IR efficiently intercept vertically scattered particles.

Fig. 57Full-screen View

(Color online) Full-ring collimator layout. (a) Global beam collimator layout; (b) Collimator layout with lattice; example shown for one ring. (VC = vertical collimator, HC = horizontal collimator)

Fig. 58Full-screen View

(Color online) Interception ratio distribution of lost particles due to Touschek scattering after the collimators are placed

When additional loss mechanisms such as injection losses, beam–gas scattering, and beam–beam collisions at the IP are included in the simulations, a greater number of collimators will be required. A nonlinear collimation system, as discussed above, is also foreseen to reduce the impedance contribution from collimators with very small gaps.

Requirements of the collider rings for radiation damping

The collider rings are subject to significant synchrotron radiation effects. These effects lead to the loss of beam energy, impose heavy thermal loads, and contribute adversely to experimental backgrounds. In addition, synchrotron radiation exerts a damping effect on the circulating beams, reducing transverse and longitudinal emittances. This damping effect is beneficial and crucial for beam injection, harmful collective effect suppression, and luminosity enhancement. Concurrently, the quantum excitation effect, also originating from synchrotron radiation, introduces randomness and competes with the damping. Hence, the horizontal emittance and energy spread eventually reach an equilibrium determined by the lattice design and beam energy.

The STCF collider rings are high-current, low-emittance electron and positron storage rings designed to operate over a wide energy range. Synchrotron radiation damping from the lattice alone is insufficient to achieve the required short damping times, low horizontal emittance, and relatively high energy spread, particularly at lower beam energies. To meet these requirements, additional DWs will be installed in the dedicated long straight sections to enhance the overall damping effect.

The design targets for radiation damping in the collider rings are as follows: at the optimal energy point (2 GeV), the damping time should be less than 30 ms, the horizontal emittance should be approximately 5 nm·rad, and the energy spread should not be less than 6×10^-4. At other operating energies, these parameters may vary slightly but must remain within acceptable limits.

Mechanism of synchrotron radiation damping

Synchrotron radiation is emitted as a continuous spectrum of electromagnetic waves in the tangential direction when high-speed charged particles traverse curved trajectories in magnetic fields. In the STCF collider rings, synchrotron radiation is primarily produced by the dipole bending magnets and DWs. The dipole magnets provide approximately uniform vertical magnetic fields, and electrons emit radiation as they bend in these fields. The wiggler magnets consist of alternating-polarity dipole magnets arranged periodically and perpendicular to the central plane of the beam. As electrons pass through this alternating field, their trajectories oscillate, producing synchrotron radiation.

The radiated energy $U$. per turn is expressed by Eq. (24), where $I_2$ is the second synchrotron radiation integral, $C_{\text{r}}$ is a constant (8.846×10^-15m·GeV^-3), and $E$ is the electron energy [50]. When wigglers are added, $I_2$ includes contributions from both bending and wiggler magnets. The contribution from the bending magnets $I_{20}$ is given in Eq. (25), where ${\rho }_i$ is the bending radius and $L_i$ the effective length of the i-th bending magnet. The wiggler contribution depends on the magnetic field profile. Equations (26) and (27) describe $I_{2\text{W}}$ for sinusoidal and rectangular field wigglers, respectively, where $B\rho$ is the magnetic rigidity, $B_{\text{W}}$ is the peak magnetic field, and $L_{\text{W}}$ is the total effective length of the wiggler. $U=\frac{C_{\text{r}}{E}^4}{2\pi }{I}_2$ (24) $I_{20}={\displaystyle \int \frac{1}{{\rho }^2}}\text{d}s={\displaystyle \sum }_{i=1}^N\frac{1}{{\rho }_i{}^2}\cdot L_i$ (25) $I_{2\text{W},\text{S}}={\displaystyle {\int }_0^{L_{\text{W}}}\frac{1}{{\rho }^2}}\text{d}s=\frac{B_{\text{W}}{}^2{L}_{\text{W}}}{2{\left(B\rho \right)}^2}$ (26) $I_{2\text{W},\text{R}}={\displaystyle {\int }_0^{L_{\text{W}}}\frac{1}{{\rho }^2}}\text{d}s=\frac{B_{\text{W}}{}^2{L}_{\text{W}}}{{\left(B\rho \right)}^2}$ (27)

In the collider rings, the higher energy of the electron and positron beams causes more synchrotron radiation energy loss and increases particle damping. The damping rates are different for different directions (horizontal, vertical, and longitudinal), as shown in Eq. (28), where $J_i$ is the damping factor, T is the revolution period, E is the beam energy, and U is the radiated energy per turn. The average longitudinal energy loss will be compensated by the RF cavities. ${\tau }_i=\frac{2}{J_i}T\frac{E}{U}i=x,y,s$ (28)

The radiation damping effect reduces both the horizontal and vertical emittance, while the quantum excitation during the radiation process slightly increases horizontal emittance. When quantum excitation and damping reach dynamic equilibrium in the horizontal plane, the horizontal equilibrium emittance is given by Eq. (29), where $I_5$ is the fifth synchrotron radiation integral, $C_q$ is a constant (3.832×10^-13 m), $\gamma$ is the relativistic factor, and $J_x$ is the horizontal damping partition number [51]. ${\epsilon }_{x0}=\frac{C_q{\gamma }^2{I}_5}{J_x{I}_2}$ (29)

The beam energy spread ${\sigma }_{\text{E}}$ is also affected by synchrotron radiation damping. When quantum excitation and damping reach equilibrium, the beam reaches a steady state, and the equilibrium energy spread is given by Eq. (30), where $I_3$ is the third synchrotron radiation integral and $J_{\text{s}}$ is the longitudinal damping partition number [50, 52]. The contribution from the bending magnets to $I_{30}$, denoted as $I_{30}$, is expressed by Eq. (31), where ${\rho }_i$ and $L_i$ are the bending radius and effective length of the i-th dipole magnet, respectively. Equations (32) and (33) describe the $I_{3\text{W}}$ contributions from sinusoidal-field and rectangular-field wiggler magnets, respectively, where $B\rho$ is the magnetic rigidity, and $B_{\text{W}}$ and $L_{\text{W}}$ are the peak magnetic field and total effective length of the wiggler. ${\sigma }_E{}^2=\frac{C_q{\gamma }^2{I}_3}{J_{\text{s}}{I}_2}$ (30) $I_{30}={\displaystyle \int \frac{1}{{\left|\rho \right|}^3}}\text{d}s={\displaystyle \sum }_{i=1}^N\frac{1}{\left|{\rho }_i{}^3\right|}\cdot L_i$ (31) $I_{3\text{W},\text{S}}={\displaystyle {\int }_0^{L_{\text{W}}}\frac{1}{{\rho }^3}}\text{d}s=\frac{4B_{\text{W}}{}^2{L}_{\text{W}}}{3\pi {\left(B\rho \right)}^3}$ (32) $I_{3\text{W},\text{R}}={\displaystyle {\int }_0^{L_{\text{W}}}\frac{1}{{\rho }^3}}\text{d}s=\frac{B_{\text{W}}{}^3{L}_{\text{W}}}{{\left(B\rho \right)}^3}$ (33)

Compared to the previous generation of colliders, the collider rings of the STCF have a significantly shorter beam lifetime, which necessitates frequent beam injections and a damping time of ≤ 30 ms. At lower energy (1 GeV), the radiation damping time is excessively long, necessitating the use of wiggler magnets to enhance damping and reduce the damping time. At higher energy (3.5 GeV), synchrotron radiation also affects the equilibrium emittance and energy spread of the stored beam; the introduction of wiggler magnets can also help tune these critical beam parameters.

Damping wiggler parameter design

2.10.3.1

Total effective length and peak magnetic field

For the optimized 2-GeV energy point, the transverse plane damping time of the collider rings must be shorter than 30 ms. At the beam energy of 1 GeV, the radiation damping contribution of the lattice is much smaller, and the required damping time should be less than 50 ms. Additional damping will originate from the DWs. By scanning the peak magnetic field strength $B_{\text{W}}$ from 1 T to 5 T, the theoretical total effective length of the wiggler magnets required to achieve a 50 ms damping time at 1 GeV was calculated. The results, shown in Fig. 59 for both sinusoidal and rectangular field configurations, indicate that the effective length of a practical magnet will lie between these two cases.

Fig. 59Full-screen View

(Color online) Total effective length of wiggler magnets required for a 50 ms damping time at 1 GeV versus peak magnetic field strength

Insertion magnet technologies currently used in accelerators can generally be classified into three types: room-temperature magnets, cryogenic permanent magnets, and superconducting magnets [53]. Typical magnetic field strengths are 1–1.5 T for room-temperature magnets, 1.5–2 T for cryogenic permanent magnets [54, 55], and 3–5 T or higher for superconducting magnets.

Among these, superconducting wigglers are the most complex and expensive. Moreover, for a given damping time, increasing the field strength from 3 T to 5 T provides only marginal reductions in the required total effective length. Since the STCF collider rings are designed to operate over a wide energy range (1–3.5 GeV), the wigglers must provide sufficient flexibility and tunability. Permanent magnets allow limited field adjustment through mechanical gap control, whereas room-temperature magnets offer broader and more convenient tunability via coil excitation current.

For these reasons, STCF has adopted room-temperature wigglers. Taking into account cost, technical complexity, and the overall ring layout, a peak field of $B_{\text{W}}$ of 1.6 T and a total effective length $L_{\text{W}}$ of 76.8 m have been selected.

Impact of damping wigglers on beam parameters

Based on the current design parameters—room-temperature magnets with a peak field of $B_{\text{W}}=1.6$ T, total effective length $L_{\text{W}}=76.8$ m, period length ${\lambda }_{\text{W}}=0.8$ m, and end fields arranged in an even-numbered pole configuration {+1/4, –3/4, +1, –1, ..., +1, –1, +3/4, –1/4}—each magnet has a length of 4.8 m, and a total of 16 wigglers are deployed per ring. A more realistic wiggler field was calculated using the magnet design software OPERA. The vertical magnetic field B_y along the central trajectory of a single wiggler is shown in Fig. 60, demonstrating that the actual field lies between sinusoidal and rectangular profiles and includes fringe-field effects.

Fig. 60Full-screen View

(Color online) Vertical magnetic field distribution B_y along the longitudinal axis for three types of wiggler field profiles in a single magnet

To accommodate the broad energy range of the STCF, the wigglers must support variable settings to satisfy beam dynamics requirements—including damping time, horizontal emittance, and energy spread—across different operational energies. Because sinusoidal and rectangular wiggler fields produce different damping effects, and the actual field lies between these extremes, Table 12 presents simulation results for damping time, horizontal emittance, and energy spread at four beam energies (1.0, 1.5, 2.0, and 3.5 GeV) for the cases of no wigglers, sinusoidal wigglers, rectangular wigglers, and realistic wigglers.

The results indicate that adding wigglers reduces the damping time and increases the energy spread in all cases, with rectangular fields producing stronger effects than sinusoidal fields, and realistic fields falling in between. This trend is consistent across all considered beam energies.

Wiggler radiation generates substantial synchrotron light. At 2 GeV and 2 A, a single wiggler produces 64 kW of synchrotron radiation power. One straight section containing 8 wigglers emits a total of 512 kW. Figure 61 shows the power distribution of this radiation at the downstream end. To manage it, photon masks must be installed along the beamline to prevent direct strikes on the vacuum chamber. The remaining radiation can be extracted near the first bending magnet in the downstream arc and absorbed by a high-power photon absorber. In the future, synchrotron light from the outer ring could also be extracted for high-flux applications.

Fig. 61Full-screen View

(Color online) Superimposed power distribution from 8 DWs at the end of a straight section (left: X direction, right: Y direction)

Evaluation of the effects of damping wigglers on linear lattice and nonlinear beam dynamics

MDI design requirements

The Machine Detector Interface (MDI) is the region connecting the collider rings to the detector spectrometer, playing a critical role in both routine accelerator operation and physics data collection. Since both beams must pass through the detector center and collide at the IP, and a tightly focused beam spot is required to achieve high luminosity using superconducting quadrupole magnets, the inner detector components must be positioned very close to the IP. This results in extremely constrained space for accelerator components. For a third-generation e⁺/e^– collider such as STCF, MDI design is particularly complex due to ultra-high luminosity, a large crossing angle, strong focusing, and short beam lifetime—all of which must be addressed.

The accelerator IR involves multiple technical systems, including lattice and beam dynamics design (accounting for nonlinear effects, chromaticity correction, beam–beam interactions, collective effects, etc.), superconducting magnets and cryostats, beam diagnostics, vacuum systems, mechanical structures, collimation, and alignment. On the detector side, MDI-related systems include background simulation and shielding, spatial arrangement of inner detectors and electronics, luminosity monitoring, and feedback. Coordinated study between the accelerator and detector teams is therefore essential, with the design objective of ensuring spatial compatibility between accelerator and detector systems, enabling the accelerator to achieve its target luminosity and stable operation while minimizing background levels for collision experiments.

Impact of interaction region design on MDI

The lattice design of the IR determines the layout and longitudinal allocation of accelerator components within the MDI. To maximize the solid angle for physics measurements, the detector imposes stringent constraints on the transverse space available to accelerator elements. Figure 62 shows the layout of the STCF MDI, illustrating the preliminary positions and dimensions of components within ±3.5 m of the IP, including detector boundaries, central beam pipe, vacuum pipe structures, superconducting magnets and cryostats, beam diagnostics components, and mechanical supports. Spatially, the detector requires that all accelerator components within the IR be confined to a conical region with a 15° opening angle centered on the IP.

Fig. 62Full-screen View

(Color online) STCF MDI layout (right side only; the layout is symmetric about the IP)

The cryostat houses compensating solenoids, two superconducting quadrupoles, several correctors, and a shielding solenoid. The compensating solenoid upstream of the superconducting quadrupoles and closest to the IP cancels the integrated longitudinal magnetic field of the detector solenoid. The shielding solenoid, wound around the superconducting quadrupoles, ensures that $B_z=0$ within the quadrupole region. A downstream compensating solenoid after QF2 counteracts the long tail of the detector solenoid field.

The central beam pipe consists of the IP pipe and a transition pipe. The IP pipe is made of beryllium, with its inner wall coated in gold to shield against synchrotron radiation and reduce impedance, and has a diameter of 30 mm. The transition pipe expands longitudinally from 110 mm to 500 mm, transitioning from a 30 mm round pipe to a racetrack-shaped cross-section measuring 60 mm wide and 30 mm high. At the Y-chamber—remote vacuum connector (RVC)—the beam pipe splits into two 30 mm diameter channels up to the end of QD0, then transitions over 0.5 m to a 50 mm diameter at the entrance of QF1, and finally increases within 0.5 m to 67 mm. The design of the transition from the dual-aperture region in the superconducting magnets to the single-aperture central beam pipe prioritizes the minimization of impedance.

The upstream bending magnet B0 is a primary source of synchrotron radiation entering the MDI. In the lattice design, this magnet is deliberately positioned away from the IP (approximately 8.5 m) and set to a relatively weak bending angle of only 1°. Synchrotron radiation distributions were analyzed for electron/positron beams entering the IP from both the outer and inner rings, as shown in Fig. 63. The directly irradiated regions are marked in yellow, and only the case of the positron beam is illustrated for simplicity. Table 13 lists the synchrotron radiation power deposited in various MDI regions. For beams entering from the outer ring, radiation does not directly strike the central beryllium pipe, which occurs for beams entering from the inner ring. Hence, the preferred configuration has the beams entering the IP from the outer ring.

Fig. 63Full-screen View

(Color online) Synchrotron radiation distribution in the MDI for beam incident into IP from outer rings and inner rings

The definition of beam stay-clear regions in the IR is a critical factor for MDI magnet design. In second-generation e⁺/e^- colliders, such as PEP-II and BEPCII, the stay-clear region is typically defined as 15–20 times the beam envelope σₓ/σᵧ plus 1–2 mm to account for orbit deviations. For newer-generation colliders, including SuperKEKB, SuperB, and FCC-ee, the extremely small beam sizes at the IP require strict control of orbit deviations near superconducting quadrupoles (less than 100 µm); consequently, the stay-clear region is defined without including orbit deviations. CEPC adopts the same definition as BEPCII.

Here, both interpretations are provided. Figure 64 shows the beam stay-clear boundaries at 2 GeV and 3.5 GeV. The magenta and green dashed lines represent the stay-clear boundaries for the positron and electron beams, respectively. At 2 GeV, the horizontal stay-clear region is defined as 22σₓ + 2 mm (or 24σₓ), and the vertical as 22σ_y + 2 mm (or 26σ_y). At 3.5 GeV, due to the larger beam emittance, the stay-clear region is reduced to 9.5σₓ + 2 mm (or 10.5σₓ) for the horizontal region and 9.5σ_y + 2 mm (or 11.5σ_y) for the vertical region. In these calculations, the horizontal emittance εₓ is taken as 5 nm at 2 GeV and 27 nm at 3.5 GeV, with the vertical emittance assumed to be 5% of εₓ, i.e., ε_y = 0.05εₓ. The small beam stay-clear regions at 3.5 GeV can be enlarged by increasing ${\beta }_{x/y}^{*}$ at the IP. Table 14 lists the current parameters of the superconducting quadrupoles near the IP that satisfy the IR optics requirements, including position, field gradient, effective length, beam pipe radius, and good field region.

Fig. 64Full-screen View

(Color online) Beam stay-clear boundaries at 2 GeV and 3.5 GeV

Beam loss and experimental background simulation

The primary sources of experimental background in the IR are non-collisional beam losses, Bhabha scattering, synchrotron radiation effects, and beam-induced bremsstrahlung. In a low-energy e⁺e^- collider such as STCF, the latter two contribute less significantly, with beam loss and Bhabha scattering being the dominant factors. Bhabha scattering is an inherent process accompanying collisions and is essentially unavoidable. While beam loss is also inevitable, its impact on experimental backgrounds can be mitigated through appropriate measures.

As a newly constructed accelerator, STCF aims to minimize beam loss in the IR by employing collimators at strategic locations in the collider rings to intercept particles from Touschek scattering and the transverse beam halo, as described in the beam collimation section. Synchrotron radiation primarily contributes to the thermal load on the beam pipe in the IR; however, its effect on experimental backgrounds—particularly from high-energy X-ray components—must also be considered. Beam-induced bremsstrahlung can influence background levels, especially when operating at the high-energy end near 3.5 GeV, providing another rationale for maintaining ultra-high vacuum conditions in the IR.

With the exception of the central beam pipe section—particularly the beryllium IP pipe, which serves as the passage for reaction products entering the detector and therefore cannot be shielded—all other beam pipe sections and components are equipped with shielding absorbers. These are designed to reduce the likelihood that secondary particles from upstream beam collimators, vacuum-induced bremsstrahlung, or synchrotron radiation reach the detector or superconducting magnet coils, while minimizing the impedance introduced by the shielding structures.

Simulation studies of collision-induced backgrounds are ongoing. Preliminary results indicate that the background levels remain within controllable limits.

Design requirements and specifications

As an integral part of the STCF accelerator complex, the injector is a sophisticated accelerator system. Its primary role is providing high-quality, full-energy electron and positron beams for injection into the collider rings, serving as a key component for realizing high luminosity at the STCF. The collider rings operate in the top-up mode, and currently, the beam injection design considers two schemes: off-axis injection and bunch swap-out injection.

The off-axis injection scheme aims to replenish particle losses during collider operation. In this approach, the electron beam is accelerated and transported directly to the collider electron ring via a linac and beamline. For positrons, the beam produced by the high bunch-charge electron beam onto a target is accelerated to 1 GeV and damped in a DR. Next, it is further accelerated through the ML and transported into the collider positron ring. In contrast, the bunch swap-out injection scheme sequentially replaces bunches whose charge has dropped below a certain threshold with injected high-charge bunches in both rings. In this scheme, high-charge electron bunches are directly supplied by the electron gun and accelerated for on-axis injection into the collider electron ring. For positrons, bunch charge accumulation in an accumulator ring is necessary due to the relatively low conversion efficiency from electrons to positrons. Following accumulation, the bunches undergo emittance damping and acceleration before on-axis injection into the collider positron ring. Bunches scheduled for replacement must be extracted in advance to accommodate incoming injections.

After comparison and evaluation, an injector scheme compatible with both off-axis and swap-out injection is proposed. This allows initial implementation with off-axis injection to control construction costs, while remaining for a swap-out injection upgrade when needed.

Although the two injection schemes impose different design requirements on the injector, their primary objective is the same: to maintain approximately 95% of the nominal electron and positron beam currents (2 A) in the collider rings, assuming a minimum beam lifetime of 200 s. The off-axis injection scheme places higher demands on beam quality, such as emittance and energy spread, whereas the swap-out scheme requires higher bunch charge and an increased repetition rate for part of the injector linac. Key injector design parameters are summarized in Table 15. The extremely short beam lifetime at STCF presents substantial challenges for both beam injection into the collider rings and the overall design and construction of the injector.

Traditional off-axis injection offers higher beam utilization, lower bunch charge per injection, and reduced frequency requirements for the injector linac, making it relatively straightforward to implement. However, the larger beam loss associated with off-axis injection introduces significant experimental background and can induce emittance growth through coupling with beam–beam effects.

To mitigate these issues, the swap-out injection scheme is under study. This approach allows, in principle, higher injection efficiency for individual bunches and can suppress experimental background arising from injection losses. However, if the extracted bunches are not reused—as in the present design—beam utilization is reduced. In addition, the large bunch charge imposes challenges for emittance control, positron bunch charge accumulation, and achieving small vertical emittance in the injector design.

The swap-out injection scheme requires beamlines and accelerator components with larger physical apertures while maintaining strictly controlled emittance growth. It also necessitates pre-extraction of the bunches to be replaced before injection, adding complexity to the extraction system, which in traditional operation only needs to remove the entire beam for machine protection, and requires a more sophisticated beam dump. Reuse of the spent high-charge bunches for other applications could mitigate beam utilization issues. A notable advantage of this scheme is that an injector designed for swap-out injection can also support off-axis injection with relatively minor modifications.

In the conceptual design phase, injector physics designs for both off-axis and swap-out injection schemes have been developed, and a compatible injector design accommodating both approaches is currently under study.

Given the wide operating energy range and the very short beam lifetime of the collider rings—particularly under swap-out injection—the injector must deliver electron and positron beams with variable energy (1–3.5 GeV), high repetition rate (30–90 Hz), low emittance (<30 nm·rad), and high bunch charge (8.5 nC) for swap-out injection. These requirements pose substantial challenges for the injector design, including:

· Electron and positron linacs: Three-stage linacs are required for both injection schemes (Fig. 65). The low-energy sections must accommodate large bunch charges while controlling emittance growth. Positron linacs (PL and SPL) must handle larger positron emittance. The maximum linac repetition rate is expected to reach 90–100 Hz.

Fig. 65Full-screen View

(Color online) Layout of the dual-electron-gun low-energy section

· Positron generation, capture, and pre-acceleration: All schemes employ conventional target-based positron production. Key tasks include efficient generation, collection, and pre-acceleration of positrons, incorporating adiabatic matching, e⁺/e⁻ separation, and longitudinal compression, necessitating detailed tracking simulations.

· Damping and accumulator rings: The damping ring (DR) reduces positron emittance via synchrotron radiation. In the swap-out scheme, the accumulator ring (AR) simultaneously accumulates and damps positrons to satisfy the collider’s bunch charge and emittance requirements. In the hybrid scheme, both the DR and AR are used sequentially.

· ML: Accelerates 1 GeV e⁺/e⁻ bunches to 1–3.5 GeV for injection into the collider rings, operating alternately at 30 Hz for the electron and positron beams.

· Beam transport lines: Include the bypass beam line (for off-axis injection only), injection and extraction lines for the DR and AR, and transport lines from the ML to the collider rings. The physics design emphasizes beam matching, emittance preservation, and high transmission efficiency.

A detailed description of the compatible injector scheme is provided in Sect. 3.6.

Overview of the design schemes

Schematic layouts of both the collider ring injection schemes, which are based on the physics design requirements and injector specifications and reference similar international accelerator layouts, are shown in Fig. 1.

The linac system for each of the two injection schemes incorporates two electron sources—one corresponding to a low-emittance electron gun for direct injection into the collider electron ring and the other to a high-charge electron gun for positron production via target bombardment. For the off-axis injection scheme, the low-emittance electron source is a PC RF gun producing 1.5 nC, and the high-charge electron source is a TC RF gun producing 10 nC. For the swap-out injection scheme, the electron sources comprise a TC gun (8.5 nC) or an L-band PITZ PC gun (8.5 nC), along with an 11.6 nC TC gun. Different bunching schemes are used for beams from different types of electron guns: a chicane scheme is applied for beams from the RF PC guns, while for all three TC guns, a subharmonic buncher (SHB) and a fundamental harmonic buncher system are adopted.

In the off-axis injection scheme (Fig. 1a), electrons from the two sources pass through respective bunching sections (BSs) and pre-injector sections (PISs) for bunching and initial acceleration. They are then merged at 200 MeV at the entrance of the first ML section (EL1), which accelerates the beam to 1.0 GeV. From there, the beam from the RF PC gun is delivered via a bypass line to the positron/electron ML and accelerated to the injection energy required by the collider rings. The 10 nC electron beam from the thermionic gun is further accelerated to 1.5 GeV by a second linac section (EL2) before striking a tungsten target to produce positrons. The resulting positrons are collected to form a bunch charge of 1.5 nC and pre-accelerated to 200 MeV. In the main accelerator section of the positron linac (PL), the positrons are accelerated to 1.0 GeV and injected into a DR for emittance reduction. The DR uses a single-turn (on-axis) injection scheme, storing five bunches simultaneously. Each bunch undergoes damping from an initial emittance of 1400 nm·rad to below 11 nm·rad. The extraction repetition rate of the DR matches the injection of the collider rings. The damped positron bunches are then transported to the ML and accelerated to the injection energy of the collider rings.

In the swap-out injection scheme (Fig. 1b), 8.5 nC electron beams from a thermionic or PITZ L-band PC gun are bunched in Bunching Section 1 (BS1) and accelerated to 1.0 GeV in the linac section SEL1. They are then accelerated by the ML to the collider ring injection energy. The electron beam from the 11.6 nC thermionic is bunched in BS2, then accelerated to 2.5 GeV in SEL2 before striking a tungsten target to generate positrons. A positron beam with a bunch charge of 2.9 nC is collected and pre-accelerated to 200 MeV, then accelerated to 1.0 GeV in the PL. These positrons are injected into an accumulator ring (AR) for bunch charge accumulation and emittance damping. The AR uses a multi-turn off-axis injection scheme, sequentially filling all stored bunches and allowing sufficient damping time before extraction. The injection repetition rate is a multiple of the extraction repetition rate, which is determined by the accumulation ratio. Once the accumulation and damping are complete, positron bunches are extracted and accelerated by ML to the required energy for injection into the collider rings.

Swap-out injection requires high bunch charges and stringent control of beam quality for both electron and positron beams. Due to the positron production mechanism, the international standard employs a high-charge electron beam to bombard a production target. Considering the sensitivity of cost to electron beam energy and the technical limits on bunch charge from the electron gun and linac, a positron AR is adopted instead of a DR for off-axis injection. This AR, combined with a higher repetition rate and modest beam energy for the driving electron linac, allows accumulation of positron bunch charge through multiple injections while simultaneously damping transverse and longitudinal emittance.

For electrons, considering technological maturity and cost considerations, direct injection of high-quality, high-charge bunches into the collider electron ring is explored to replace degraded bunches. The main challenges remain achieving high bunch charge and low beam emittance. Two feasible approaches are considered: (1) Conventional thermionic guns: Capable of generating bunches exceeding 8.5 nC. This method is technically reliable and does not require a high-power laser system; however, the relatively large emittance from the gun may necessitate beam scraping prior to injection to meet emittance requirements. (2) L-band PC guns: Capable of generating high-charge (>8.5 nC), high-quality electron bunches. While technically promising, this approach imposes strict demands on the driver laser, the high-vacuum environment, and the lifetime of the semiconductor PC, particularly when operating at high repetition rates, and still requires engineering validation.

Despite differences in injector physics design corresponding to the two collider ring injection schemes, they share common requirements: strict control of beam emittance during acceleration, high positron capture efficiency, optimized design of the positron damping and accumulator rings, and optimized injection transport lines. The following sections present the physics design details for each scheme, demonstrating how the required bunch charge and beam quality are achieved.

Electron source and low-energy acceleration section

Based on electron charge requirements for direct collider electron ring injection and bombardment-based positron production, two electron guns are used to generate bunches with different charges. Consequently, the system is divided into a photocathode low-energy section and a thermionic cathode low-energy section. At the exit of the two low-energy sections, the beams from both electron guns are merged into the first main acceleration section through a combining segment. The layout of the dual-gun low-energy section is shown in Fig. 65.

3.2.1.1

Photocathode low-energy section

The electron beam quality (e.g., emittance, current, bunch length, etc.) of the low-energy section is crucial in determining the final beam quality injected into the collider electron ring. To maximize the injector beam quality, a well-established S-band 2998 MHz PC RF gun, which has already been tested at major international and Chinese accelerator facilities, is adopted for the STCF. This gun can deliver high beam quality that meets the stringent requirements of the STCF. The basic configuration of this section is shown in Fig. 66.

Fig. 66Full-screen View

(Color online) Layout of the photocathode low-energy section

The gun operates at a cathode gradient exceeding 100 MV/m and delivers a beam energy of approximately 4 MeV at its exit. The beam is then accelerated to 120 MeV using two S-band accelerating structures, L0 and L1. The next stage, L2, consists of two additional S-band structures powered by a shared RF source with an energy doubler, which accelerates the beam to over 200 MeV while imparting an energy chirp. This is followed by an X-band accelerating structure serving as a linearizer and a chicane acting as a magnetic bunch compressor, which reduces the bunch length to below 1 ps (equivalent to 1° RF phase at S-band frequency). A Ti:sapphire mode-locked laser system, synchronization control system, and various power supplies are also required.

Compared with a conventional low-energy section based on a thermionic cathode, a key challenge for the photocathode scheme is maintaining low emittance at high bunch charge. Typically, an external solenoid is used to focus the beam after the RF gun. Following a drift section, the linear space-charge effects can be partially compensated, mitigating emittance growth. According to the design requirements, the photocathode low-energy section must deliver bunches with a charge greater than 1.5 nC and a bunch length shorter than 1 ps to maintain low energy spread in the subsequent linac sections for direct and efficient injection into the collider electron ring. Therefore, a global optimization of the parameters for the low-energy section is necessary.

The electron beam properties are mainly influenced by the following four components:

1. Accelerating structures provide energy gain and longitudinal focusing.

2. Focusing elements (e.g., solenoids) compensate for transverse emittance and provide transverse focusing to confine the beam size.

3. Laser profile on the photocathode. The transverse and longitudinal distributions of the drive laser affect space charge forces and nonlinearities.

4. Photocathode material. Its thermal emittance directly impacts the beam brightness.

Given the complex interplay among these parameters and the spatial constraints of the injector, manual tuning or scan-based optimization is inefficient. Instead, multi-objective optimization algorithms, such as genetic algorithms, are employed and integrated with ASTRA beam dynamics simulations. Realistic 3D electromagnetic field maps of the components are used in the simulations to accurately represent their performance.

The optimization targets the bunch length and transverse emittance at the exit of the second S-band linac section. Three types of laser profiles are considered (see Table 16).

Table 16Full-screen View

Laser distribution cases used in optimization

Case	Transverse laser profile	Longitudinal laser profile
Case 1	Truncated Gaussian	Gaussian
Case 2	Truncated Gaussian	Flat-top
Case 3	Uniform	Flat-top

Show more

Charge values of 1, 1.5, and 2 nC are studied. The results show that the transverse beam quality is relatively insensitive to the transverse laser shape; therefore, a truncated Gaussian profile is used to minimize energy losses associated with laser shaping. In contrast, a flat-top longitudinal laser profile significantly improves the emittance. Consequently, a flat-top longitudinal and truncated Gaussian transverse laser profile is selected. Based on the optimization results and spatial constraints, the entrance to the first accelerating structure is positioned 1.5 m from the cathode, the second structure at 5.02 m, and the center of the second solenoid at 2.0 m.

Simulation results at the exit of the two S-band sections, including the phase space, current, and energy spectrum, are shown in Fig. 67.

Fig. 67Full-screen View

(Color online) Beam distributions at the exit of the two S-band accelerating structures in the photocathode low-energy section

In the L2 section, the RF phase is before the peak, which induces an energy chirp. The third and fourth S-band structures generate the chirp, while an X-band linearizer and a chicane (providing negative R56) compress the bunch length to ≈1 ps at ≈200 MeV. At this point, both emittance and energy spread are optimized (<0.1%). With this setup, normalized emittance at 1.5 nC remains below 2 mm·mrad for all studied laser distributions. Figure 68 shows the final transverse and longitudinal beam distributions.

Fig. 68Full-screen View

(Color online) Transverse (left) and longitudinal (right) beam distributions at the exit of the photocathode low-energy section