Preamble

Design and integration of a crab-waist interaction region for the Super Tau-Charm Facility∗ Lin-Hao Zhang§ ,1, † Tao Liu§ ,1 Ye Zou,1, ‡ Peng-Hui Yang,2 De-Min Zhou,3 Jian-Cong Bao,1 Ze Yu,1 Yu-Han Jin,1 Yi-Hao Mo,1 Sang-Ya Li,2 Tian-Long He,2 Qing Luo,1, 2 and Jing-Yu Tang1, 2

holSc of Nuclear Scien and ,gyhnolec T sityUnver of Scien and gyhnolec T of China, Hefi, 230027, China Nationl on tr hSync Raditon ,atoryLb sityerUnv of Scien and gyhnolec T of China, Hefi, 230029, China High gy Ener Acatorcel hc Resar ganizto Or (KEK), Tsukba, 305-0801, apn J The Super Tau-Charm Facility (STCF) is a next-generation electron–positron collider being developed in China, designed to achieve a peak luminosity exceeding 5 × 1034 cm−2 s−1 at the optimal beam energy of 2 GeV. Achieving this goal relies on a crab-waist collision scheme with a large crossing angle—a configuration that introduces strong nonlinearities, severely constraining the dynamic and momentum apertures. This paper presents a comprehensive physics design for the STCF interaction region (IR) that systematically addresses these challenges. The design features a modular optics framework that incorporates local chromaticity correction up to the third order, exact −I transformations between chromatic sextupole pairs for nonlinear cancellation, and minimization of the dispersion invariant to improve local momentum acceptance. Optics at the crab sextupoles are optimized to reduce their strength and associated nonlinearities. When integrated into the collider ring, the design achieves a Touschek lifetime of over 300 s at 2 GeV, meeting the STCF requirement. Furthermore, fringe fields from superconducting quadrupoles are partially compensated using octupole correctors, and detector solenoid effects are fully suppressed via local anti-solenoids. The machine-detector interface layout is also optimized to avoid synchrotron radiation background at the interaction point. This IR design represents the current optimized solution for STCF and has been incorporated into the project’s conceptual design report.

Keywords

Super Tau-Charm Facility, Interaction region design, Crab-waist, High-order chromaticity correction, Dynamic aperture, Machine-detector interface

INTRODUCTION

However, the crab-waist scheme introduces substantial challenges to the IR optics design [11]. The extremely 27 low vertical β-function at the IP leads to large β-functions 28 at the final focus (FF) quadrupole doublets, necessitating 29 ultra-strong FF quadrupoles that in turn generate high natu30 ral chromaticity, introduce significant fringe fields, and in31 crease sensitivity to field errors.

Additional complications 32 arise from high-order kinematic terms of the IP drift, strict 33 phase-advance requirements from crab sextupoles to the IP 34 (∆µx = mπ, ∆µy = (2n + 1)π/2), and the need to com35 pensate detector solenoid fields that otherwise induce hori36 zontal–vertical coupling. Collectively, these effects degrade 37 the dynamic and momentum apertures, limiting the Touschek 38 lifetime—a critical issue for high-current, low-emittance op39 eration in the STCF. Design and operational experience from 40 SuperKEKB confirms that the IR dominates the nonlinear dy41 namics of the ring [12–14]. Additionally, integrating ultra42 strong FF quadrupoles within a constrained machine–detector 43 interface (MDI) further complicates the design. Therefore, 44 coordinating these combined effects to meet the collider’s de45 sign luminosity constitutes the central challenge in the crab46 waist IR design.

Early STCF IR designs suffered from insufficient dynamic 48 aperture (DA) and a short Touschek lifetime of only 35 s 49 when integrated into the STCF ring [15, 16]. Recent global 50 optimizations of the collider ring improved performance to 51 approximately 240 s [17]. Note that both early lattice ver52 sions prioritized longitudinal beam polarization via Siberian 53 snakes—imposing strict layout constraints due to the required 54 specific bending angles between successive snakes. However, 55 the decision in 2024 to deprioritize spin polarization opened 56 a new opportunity for a fundamental redesign of the collider 57 lattice, including the arc and IR optics [18], with the goal of 58 achieving a breakthrough in nonlinear performance.

The Super Tau Charm Facility (STCF), a new-generation electron-positron collider currently under development in 4 China, is designed to operate at center-of-mass (CoM) en5 ergies from 2 to 7 GeV [1].

It targets a peak luminosity 6 of at least 5 × 10 cm−2 s−1 at the optimal CoM energy 7 of 4 GeV—approximately 50 times that of the BEPCII (cur8 rently operating Tau-Charm factory [2])—making it a unique 9 platform for precision physics studies in the tau-charm region 10 and a potential probe of physics beyond the standard model.

To achieve the desired high luminosity, the STCF adopts 12 the crab-waist collision scheme proposed by P. Raimondi [3], 13 which combines a large crossing angle with dedicated crab 14 sextupoles. This configuration enables a drastic reduction of 15 the vertical β-function at the interaction point (IP), thereby 16 enhancing the luminosity, while simultaneously suppress17 ing detrimental beam-beam resonances through sextupole18 induced vertical beam waist shifts at the IP [4]. The scheme 19 has been validated at the DAΦNE Φ-factory [5] and forms the 20 basis of all modern high-luminosity e /e colliders, includ21 ing FCC-ee [6] and CEPC [7] in the high-energy regime, Su22 perKEKB [8] and the earlier SuperB [9] proposal at medium 23 energies, and tau-charm projects such as BINP’s SCTF [10] 24 and STCF in the low-energy domain.

∗ Supported

by the National Natural Science Foundation of China (No.12341501 and No. 12405174) and the National Key R&D Program of China (No. 2022YFA1602201) † Corresponding author, zhanglinhao@ustc.edu.cn ‡ Corresponding author, zouye@ustc.edu.cn § These authors contributed equally to this work.

This paper presents a comprehensive crab-waist IR design 114 βy∗ : developed under this new paradigm. The design overcomes 115 The vertical beta function at the IP is a primary factor A smaller βy∗ directly in61 previous limitations through several key features: (1) a mod- 116 for achieving high luminosity. 62 ular optics approach with minimized dispersion invariant; (2) 117 creases the luminosity but requires stronger FF quadrupoles. 63 local chromaticity correction extended to the third order; (3) 118 These stronger quadrupoles, in turn, introduce larger natuTherefore, 64 exact −I transformation between chromatic sextupole pairs 119 ral chromaticity and fringe field nonlinearities. 65 for nonlinear cancellation; and (4) integrated compensation of 120 the choice of βy involves a trade-off between luminosity 66 fringe fields and detector solenoid effects. When incorporated 121 and nonlinear effects. To reach the STCF design luminos35 cm−2 s−1 (twice the engineering goal of 67 into the latest collider ring lattice, the design achieves a Tou- 122 ity of 1 × 10 cm−2 s−1 ), βy∗ should be below 1 mm, consider68 schek lifetime exceeding 300 s, fulfilling the STCF require- 123 5 × 10 69 ment. This IR configuration represents the current optimized 124 ing the vertical beam-beam parameter below 0.1 and beam∗ 70 design for the STCF collider ring, and has been incorporated 125 intensity effects. Our current design adopts βy = 0.8 mm, in126 stead of the previously considered 0.6 mm [17], aligning with 71 into the STCF conceptual design report [19].

The paper is organized as follows: Section II details the 127 the lowest operational level achieved to date by SuperKEKB 73 physics-driven selection of IR key parameters. Section III de- 128 [22], to maintain high luminosity while mitigating nonlinear 74 scribes the modular IR lattice design and its nonlinear prop- 129 challenges. 75 erties. Section IV evaluates the performance of the IR inte- 130 βx :

The horizontal beta function at the IP influences the hor76 grated into the full collider ring. Section V presents the MDI 131 77 layout and physics-related design. Section VI concludes the 132 izontal beam-beam parameter and the IR chromaticity. 133 smaller βx helps reduce ξx , which is beneficial for suppress78 paper. 134 ing the coherent X-Z instability. However, it also increases 135 the horizontal chromaticity and nonlinearity in the IR. After 136 careful consideration, we have chosen βx = 60 mm, revised II. IR KEY PARAMETER SELECTION: A earlier designs and 40 mm [17], PHYSICS-DRIVEN TRADE-OFF 138 which strikes a balance between beam-beam effects and non81 The IR design is governed by several key parameters that 139 linear dynamics.

The optimized set of IR parameters for STCF, alongside 82 directly influence the luminosity, nonlinear dynamics, and en+ − 141 a comparison with other new-generation e /e colliders, is 83 gineering feasibility. The selection of these parameters in142 presented in Table 1 [TABLE:1]. In comparison, the selected parameters 84 volves balancing competing physical effects and constraints. 143 represent a carefully balanced set that optimizes the IR per85 We focus on four primary parameters: the full crossing angle 144 formance within the physical and engineering constraints of 86 (2θc ), the drift length from the IP to the first FF quadrupole 145 the STCF. 87 (L ), and the horizontal and vertical beta functions at the IP

(βx∗ and βy∗ ). 89 2θc : A large crossing angle (e.g., 60 mrad) allows for a reduced 91 βy without the hourglass effect. It also enables rapid beam Colliders Chrom. (mrad) (mm) (mm) 92 separation and reduces the strength of the crab sextupoles, SuperKEKB 83 0.835/1.41 0.27/0.30 32/25 3462/2352 93 thereby mitigating their nonlinear impact.

Furthermore, a (LER/HER) 94 large crossing angle lowers the horizontal beam-beam paramSuperB 0.21/0.25 32/26 2857/2400 95 eter (ξx ), which helps satisfy the condition νz ≫ ξx (νz is the (LER/HER) 96 synchrotron tune) for suppressing the coherent X-Z beamCEPC(Z) FCC-ee(Z) 97 beam instability [20, 21]. However, a larger crossing angle BINP-SCTF 60 98 leads to geometric luminosity loss owing to reduced vertical 99 beam-beam parameter (ξy ) and necessitates higher beam cur100 rents to meet the luminosity target, which poses greater chal101 lenges from collective effects. Therefore, the current choice 102 of 2θc = 60 mrad for STCF represents a trade-off between III. MODULAR LATTICE DESIGN AND NONLINEAR 103 luminosity and beam-beam stability.

PERFORMANCE ANALYSIS OF THE STCF IR

104 L :

The IP drift length L∗ affects both the chromaticity and the A. Design philosophy and evolution of crab-waist IRs 106 attainable MDI space. A shorter L reduces the natural chro107 maticity of the IR, easing the requirement for chromaticity 108 correction. However, it also constrains the space available 149 In new-generation crab-waist colliders, the pursuit of ex∗ 109 for the detector and the dual-aperture FF superconducting 150 tremely low βy (millimeter scale and even sub-millimeter) 110 quadrupoles, complicating their design and installation. With 151 substantially amplifies the natural chromaticity, necessitat∗ 111 a crossing angle of 60 mrad, we have chosen L = 0.9 m to 152 ing more powerful local chromaticity correction systems. 112 provide relatively sufficient space for the quadrupoles while 153 Furthermore, a central challenge lies in integrating the 113 maintaining acceptable chromaticity levels. 154 crab sextupoles (CS), whose prescribed phase advances and

strengths—derived for on-momentum particles using a linear transfer map (see Ref. [4] or the Appendix A)—are perturbed by momentum deviation δ(= ∆p/p0 )

tupole, yielding an integrated gradient field of approximately 289 3.39 m−2 .

High-order local chromaticity correction

Matching section (MS): This section employs several 290 To achieve a Touschek lifetime exceeding 200 s, essential 291 for minimizing beam loss and ensuring high injection effi271 quadrupoles to match the β/α functions to the long straight 292 ciency, the STCF requires a momentum acceptance of ap272 section while adjusting phase advances between the IR and 293 proximately ±1.5%. Furthermore, to mitigate adverse effects 273 the adjacent arc. 294 of crab sextupoles on the momentum aperture, the IR optics The three design principles for nonlinearity mitigation have 295 must exhibit minimal chromatic dependence at the CS loca275 been fully integrated into the linear optics design. The result- 296 tions. These objectives necessitate a systematic local chro276 ing optical functions for the STCF IR are shown in Fig. 2 [FIGURE:2]. 297 maticity correction strategy extending to third order [23, 34].

The correction methodology is structured as follows: 277 A notable achievement is the maintenance of a low Hx (≤ 298 278 0.02 m) throughout the IR, which enhances the LMA by re299 • First-order vertical and horizontal chromaticity is cor279 ducing the nonlinear momentum dependence of particle tra300 rected using the main sextupole pairs S1Y and S1X, 280 jectories.

Strategic placement of dipoles—all standardized respectively, located in dispersive regions with ex281 to 1 m length for cost efficiency—enables this control while act −I transformations. This configuration simultane282 preserving the required 1.5–2 m separation between the two ously corrects linear chromaticity and cancels geomet283 rings. The total bending angle across the IR is 60 , consistent ric aberrations induced by the sextupoles. 284 with the collider ring’s geometric layout. To accommodate 285 the 60 mrad IP crossing angle, an asymmetric 30 mrad split is • Second-order chromaticity is suppressed by optimizing 286 applied: the outer-ring half-IR bends 30 mrad more, and the the phase advances from the S1Y and S1X sextupoles 287 inner-ring half-IR 30 mrad less, resulting in an asymmetric to the FF quadrupoles QD1 and QF2, respectively. 288 dispersion profile across the IR.

Fine-tuning of quadrupole strengths in the MCY and YMX matching sections enables this correction with310 out introducing additional nonlinear elements. Partic311 ularly, proper phase advance selection also minimizes leakage of the chromatic beta function amplitude in313 duced from the FF quadrupoles into the arc sections.

• Third-order chromaticity is compensated using additional sextupoles (S3Y and S3X) positioned at the first and second IP image points where the beta functions reach local minima. This optical configuration improves the effectiveness of higher-order chromatic correction by leveraging large off-momentum β-functions [35], while simultaneously suppressing geometric nonlinearities induced by the added sextupoles, thanks to their very small on-momentum β-functions.

This hierarchical correction strategy aligns with methodologies adopted in other new-generation e+ /e− collid325 ers.

Following third-order correction in the STCF IR, 326 the off-momentum optical behavior—described by the

Montague W functions [36, 37]—is well controlled at both 364 though their magnitudes remain small. This indicates that, in the IP and the CS locations, as shown in Fig. 3 [FIGURE:3]. Particularly, 365 the case of STCF, the impact of finite sextupole length is not 329 the sharp decline in Wy at S1Y and Wx at S1X in Fig. 3 366 severe. 330 indicates strong decoupling between horizontal and vertical 367 Nonetheless, in the off-momentum case, chromatic331 planes, confirming the orthogonality of the corrected optics. 368 geometric terms and chromatic ADTs remain only partially Despite these improvements, a residual second-order dis- 369 compensated due to the breakdown of the −I condition in 333 persion Dx outside the IR may still influence off-momentum 370 both CCY and CCX sections. This results in relatively large 334 dynamics.

In addition, the residual second-order vertical 371 residual driving terms at the IR exit, potentially limiting the 335 chromaticity remains non-negligible, ultimately limiting the 372 off-momentum dynamic aperture. These findings highlight 336 achievable momentum acceptance. These observations iden- 373 an important area for future local optimization of chromatic 374 RDTs within the IR, potentially via decapole pairs near chro337 tify key areas for further refinement in future optimizations. 375 matic sextupole pairs [23].

PERFORMANCE EVALUATION OF THE STCF IR INTEGRATED INTO THE COLLIDER RING

This section presents a comprehensive performance evaluation of the optimized IR design within the full STCF collider 380 ring lattice. The assessment validates the design’s robustness 381 by demonstrating: 1) excellent integrated dynamics perfor382 mance with minimal degradation from the crab sextupoles; 2) 383 conclusive evidence of the crab-waist effect; and 3) effective 384 mitigation of realistic perturbations from fringe fields and the 385 detector solenoid.

Global dynamics performance of the collider ring with the integrated IR

To evaluate performance under realistic operating conditions, the optimized IR design was integrated into the full 390 STCF collider ring lattice. The ring features a quasi-two-fold To quantitatively evaluate the effectiveness of the nonlin- 391 symmetric structure comprising one IR, four 60° arcs, two 340 ear suppression strategy after local chromaticity correction 392 30° arcs, and multiple straight sections for injection, extrac341 in the STCF IR, we computed the distribution of resonance 393 tion, damping wigglers, RF cavities, and collimation. The 342 driving terms (RDTs) [38, 39] along the IR using E. Forest’s 394 arcs employ standard 90° FODO cells—replacing the ear343 FPP/PTC code [40], as shown in Fig. 4 [FIGURE:4]. 395 lier HMBA design [17]—to achieve a target emittance of The results indicate that first-order geometric terms are 396 5 nm · rad while improving the momentum compaction fac345 fully canceled outside the crab sextupole pairs, while trans- 397 tor, minimizing the dispersion invariant Hx , and enabling 346 verse coupling terms such as h10110 , h10200 , and h10020 per- 398 more flexible sextupole configuration. 347 sist at the IP. This is consistent with the intentional crab-waist 399 Chromaticity outside the IR is corrected using an inter2 348 effect generated by the xy Hamiltonian component of the 400 leaved −I transformation sextupole scheme, reducing indi349 crab sextupole [38]. Owing to the exact −I transformation ap- 401 vidual magnet strength while canceling first-order geometric 350 plied between the CCY and CCX sextupole pairs, amplitude- 402 aberrations. Global nonlinear optimization was performed us351 dependent tune shifts (ADTs) and second-order geometric 403 ing PAMKIT [41] code integrating multi-objective algorithms 352 terms are also largely canceled outside these sections. 404 with 48 sextupole families (8 in IR, 40 in arcs). During op353 It is worth noting that a sextupole pair separated 405 timization, crab sextupoles remained active and IR chromatic 354 by a −I transformation can, in principle, compensate 406 sextupole strengths were allowed slight variations from nom355 for geometric aberrations—both second-order and higher- 407 inal values. 357 ever, this result is strictly valid only under the thin-lens ap- 409 order dispersion along the ring. The Montague function re358 proximation and for on-momentum particles. When account- 410 mains well-controlled within the IR, while second-order dis359 ing for the finite sextupole length or off-momentum beam 411 persion exhibits relatively large global perturbations due to 360 conditions, the ideal −I transformation is perturbed, leading 412 leakage from the IR into the arcs. Figure 6 [FIGURE:6] presents the tune 361 to residual nonlinearities. For instance, certain RDTs such 413 shift with momentum for the full ring (top) and the varia362 as h00220 and h00310 are not completely canceled outside the 414 tion of phase advance for the IR only (bottom), indicating 363 CCY sextupole pair due to finite-length effects, see Fig. 4, 415 a momentum acceptance of approximately ±1.5%—meeting

Nonlinear analysis via resonance driving terms

the Touschek lifetime requirement. The similarity between 417 full-ring tune shift and IR-only phase shift confirms that the

momentum acceptance is primarily limited by residual verti- 428 impact indicates that the IR design strategies, namely the cal second-order and horizontal third-order chromaticity orig- 429 CS optics design, high-order local chromaticity correction, 430 and nonlinear cancellation via −I transformations, have suc420 inating in the IR. 431 cessfully suppressed the IR inherent nonlinearities. The re432 sulting Touschek lifetime with CS active reaches 350 s at 433 0.94×10 cm−2 s−1 and 2 GeV, satisfying the STCF design 434 goal. Performance under more realistic conditions, including 435 Maxwellian fringe fields and detector solenoid effects, is ad436 dressed in Sections IV C and IV D.

ring.

STCF ring.

RDTs across the full ring, see Fig. 8 [FIGURE:8], confirm that nonlinear contributions from the arcs remain small, with the IR remain439 ing the primary source of nonlinearities—highlighting the im440 portance of rigorous local chromaticity correction and non441 linear suppression. Additionally, crab sextupoles introduce 442 first- and second-order chromo-geometric terms and chro443 matic amplitude-dependent tune shifts for off-momentum 444 particles (see Figs. 4 and 8), further justifying global non445 linear optimization with CS activated.

for the ring (top) and the fractional phase advance variation for IR only (bottom).

Validation of the crab-waist effect for the integrated IR

The 6D DA simulations using the SAD code [42], includ- 447 The critical question of whether the crab sextupoles funcing synchrotron motion, synchrotron radiation, quantum fluc- 448 tion as intended is addressed through the following validation 423 tuations, tapering, and high-order kinematic terms (depend- 449 approaches. 424 ing on high powers of transverse momentum [11]), reveal 450 (1) Hamiltonian and RDT analysis 425 that activating the crab sextupoles causes only a slight re- 451 The xy term in the Hamiltonian of a crab sextupole induces 426 duction in both on-momentum and off-momentum DA com- 452 the desired crab-waist effects at the IP, see Appendix A. From 427 pared with the CS off case, as shown in Fig. 7 [FIGURE:7]. This small 453 the perspective of RDTs, this xy component generates the

driving terms including h10110 , h10200 , and h10020 [38]. As 456 non-zero, consistent with the RDTs analysis in Section III D, shown in Fig. 8, these terms propagate to the IP and remain 457 confirming that the crab sextupoles can effectively introduce

489 tonian of a quadrupole fringe field and that of a thin octhe intended crab-waist effects. 490 tupole magnet [44, 45]. Octupole coils are installed at the (2) Beam-beam simulation with lattice 460 Beam-beam simulations incorporating the full collider ring 491 IP-opposite ends of the FF quadrupoles QD1 and QF2, with 461 lattice and realistic beam parameters (e.g., bunch charge and 492 strengths optimized as part of the global nonlinear tuning pro462 emittance) provide the most direct validation of the crab-waist 493 cess with crab sextupoles active. 463 effect, as luminosity is its ultimate performance metric. The 494 The compensation proves particularly crucial when con464 simulation results [43] demonstrate the luminosity with CS 495 sidering the interference between fringe fields and crab sex465 on is significantly higher than with CS off. This gain arises 496 tupoles. Without octupole correction, the combined nonlinear 466 because the CS suppress detrimental betatron beam-beam res- 497 effects cause a severe drop in DA compared with the case only 467 onances that would otherwise broaden the beam tune spread 498 considering FF quadrupole fringe fields but CS off. With op468 and increase the beam envelope. 499 timized octupoles, a clear recovery is achieved, though not 469 (3) Multi-particle tracking 500 complete restoration—indicating the challenging nature of 470 Multi-particle tracking offers direct visualization of the crab- 501 this compensation. The resulting configuration maintains a 471 waist effect via the beam density distribution at the IP. Fig- 502 Touschek lifetime of exceeding 300 s at 0.94×10 cm−2 s−1 472 ure 9 displays the beam density distribution at the IP after 503 and 2 GeV, meeting the STCF design target despite these sub473 single-pass tracking, with two distinct scenarios: with CS 504 stantial nonlinear perturbations. 474 off, the z-x plane shows normal density distribution; with CS 475 on, clear density modulation emerges, confirming the vertical 476 beam waist shift induced by the crab sextupole.

sextupole on and off.

IR fringe field compensation

compensation with CS on for the STCF ring. In comparison, the DA is also shown considering FF quadrupoles fringe fields with CS off.

Fringe fields in all magnets introduce additional nonlinearities that degrade the dynamic aperture. Unlike high-energy D. Detector solenoid field compensation 480 colliders such as FCC-ee, where fringe field effects are rela481 tively minor [29], the STCF—operating at lower energy—is The detector solenoid field presents another critical per482 more susceptible to such effects. Simulations using the SAD 506 483 code, incorporating Maxwellian fringe fields from dipoles, 507 turbation, potentially increasing vertical emittance through 484 quadrupoles, and sextupoles, identify the ultra-strong final fo- 508 two mechanisms: the longitudinal Bz component introduces 485 cus superconducting quadrupoles as the dominant source of 509 horizontal-vertical coupling, while the radial Br fringe field 510 generates vertical dispersion due to the crossing angle. 486 DA degradation, as illustrated in Fig. 10 [FIGURE:10].

To mitigate this effect, we implement a compensation strat- 511 To suppress these effects, STCF implements local perfect 488 egy based on the mathematical similarity between the Hamil- 512 compensation using anti-solenoids, requiring complete can478

cellation of the integrated Bz field between the IP and the pole face of QD1 and zero Bz field inside QD1 and QF2 515 quadrupoles. A step-function solenoid model similar to FCC516 ee [29] is adopted for evaluation: a detector solenoid field of 517 1 T extends from −0.5 m to +0.5 m around the IP, while an 518 anti-solenoid field of −1.25 T covers ±0.5 m to ±0.9 m.

As shown in Fig. 11 [FIGURE:11], with perfect local compensation, both the vertical dispersion and vertical closed orbit are effectively 521 confined near the IP, with no significant leakage into outside 522 regions. The apparent sharp change in the horizontal closed 523 orbit observed in Fig. 11 is not physical but arises from 524 the coordinate system transition from the beam axis to the 525 solenoid central axis in the SAD simulation when the beam 526 enters the solenoid field. The integrated Bz field cancellaFig. 12 [FIGURE:12]. The influence of the local perfect anti-solenoid compensation scheme on dynamic aperture. 527 tion eliminates horizontal–vertical coupling, while the result528 ing vertical emittance growth from the combined effect of 529 the anti-solenoid fringe field and the crossing angle is be530 low 1 pm—negligible compared to the STCF ring’s design 546 V. STCF MACHINE-DETECTOR INTERFACE: 531 vertical emittance of about 50 pm. This substantial margin 547 CO-DESIGN FOR PHYSICS PERFORMANCE 532 contrasts sharply with high-energy colliders like FCC-ee and 533 CEPC, where the sub-2 pm design vertical emittance imposes A. Integrated MDI layout and spatial optimization 534 far stricter constraints on solenoid-induced distortions [46].

The machine-detector interface design for the STCF embodies a philosophy of co-design, where accelerator compo551 nents and detector systems are optimized as an integrated sys552 tem rather than separate entities. The fundamental constraint 553 driving the MDI layout stems from the detector’s requirement 554 to maximize the solid angle for physics coverage, confining 555 all IR accelerator elements within a stringent 15 conical re556 gion centered on the IP. This spatial limitation, combined with 557 the demanding optical requirements for ultra-low βy achieve558 ment, creates one of the most compact and challenging MDI 559 environments among new-generation colliders. 561 ±3.5 m of the IP, detailing the spatial arrangement of detec562 tor boundaries, beam pipe assembly, superconducting mag563 net systems, and support structures.

The compact cryo564 stat design incorporates two superconducting quadrupoles 565 (QD1/QF2), multiple compensating and screening solenoids, 566 beam position monitors (BPMs), corrector magnets, and bottom. 570 of QD1 cancels the integrated longitudinal magnetic field 571 of the detector solenoid between the IP and the QD1; the 572 screening solenoid ensures zero net Bz field within the The weak optics focusing effect introduced by the 573 quadrupole regions; a second compensating solenoid located 536 solenoids is corrected by fine-tuning the strengths of FF 574 QF2-upstream counteracts the long-range fringe field of the 537 quadrupoles to restore the ring tunes, with negligible impact 575 detector solenoid.

The central beam pipe employs an IP beryllium section 538 on the beta functions and only minor, acceptable perturba- 576 539 tions in the horizontal dispersion. Most significantly, the dy- 577 at the center (215 mm length) with gold coating to miti540 namic aperture is fully recovered under perfect solenoid com- 578 gate synchrotron radiation background and beam impedance. 541 pensation, as shown in Fig. 12. These results confirm that the 579 Its 30 mm inner diameter is optimized to avoid higher-order 542 fully local anti-solenoid compensation approach effectively 580 mode capture. The transition section, fabricated from tan543 eliminates the adverse solenoid effects without introducing 581 talum with copper coating, evolves from a 30 mm round 544 new nonlinearities, making it well-suited to the STCF col- 582 profile to a racetrack-shaped cross-section (60 mm wide 545 lider. 583 × 30 mm high), gracefully accommodating the transition to

dual-aperture configuration while balancing impedance control.

The Y-chamber (or remote vacuum connector, RVC) rep587 resents a key junction between the single-aperture and dual588 aperture beam pipes. An eight-button BPM installed at this 589 location provides potential for real-time beam position moni590 toring near the IP.

Electromagnetic simulations using the CST code indicate 592 total beam power losses of approximately 587 W in the cen593 tral beam pipe, with only 40 W deposited in the critical beryl594 lium section—well within thermal limits. Ongoing optimiza595 tion of the Y-chamber geometry focuses on further reducing 596 capture-mode losses and impedance contributions.

Synchrotron radiation management

within MDI for e+ beam. Control of synchrotron radiation background is achieved 599 through strategic lattice design and component placement. 600 Our analysis identifies the upstream bending magnet B0 601 as the primary radiation source entering the MDI region. 618 1 mrad at B0, confirms the robustness of this approach. Even 602 Through deliberate lattice optimization, this magnet is posi◦ 619 in these scenarios, no synchrotron radiation hits the beryl603 tioned 8.5 m from the IP with a minimal bending angle of 1 , 620 lium chamber. Additionally, radiation generated in the FF 604 yielding critical energies of 0.31 keV at 2 GeV and 1.66 keV 621 quadrupoles remains negligible, assuming closed orbit devi605 at 3.5 GeV—significantly reducing the high-energy photon 622 ations within 200 µm—well above the alignment tolerance 606 flux.

The strategic combination of beam-pipe transition design 623 of 30 µm specified for these components. Compared to the 624 high-energy collider CEPC [47, 48], synchrotron radiation in 608 and outer-ring beam incidence provides an elegant solution 625 the STCF can be more effectively managed under the consid609 to the radiation challenge. As shown in Fig. 14 [FIGURE:14], simulations 626 ered misalignment conditions. 610 confirm that synchrotron radiation from B0 does not strike 611 the central beryllium pipe when beams enter the IP from the 627 We further calculated the line power density distribution 612 outer ring, owing to effective shielding by the transition beam 628 of synchrotron radiation from B0 along the beamline un613 pipe. This configuration simultaneously addresses accelera- 629 der beam energy of 3.5 GeV and beam current of 2 A, see 614 tor operational needs (compatible with injection layout) and 630 Fig. 15 [FIGURE:15], with power loss in different regions explicitly indi615 detector performance requirements (minimized background). 631 cated. These results provide essential input for the thermal Further analysis under extreme conditions, including 632 and mechanical design of the MDI’s vacuum chambers, as 617 closed orbit deviations of 1 mm and angular deviations of 633 well as for mask shielding (if required).

The most critical BSC limitation occurs in the FF quadrupoles, where the combination of minimal physical The definition of beam-stay-clear (BSC) regions represents 657 aperture and large beta functions creates a particle loss 658 hotspot. This region proves particularly vulnerable to losses 636 a critical intersection of beam dynamics and engineering de659 from Touschek scattering and injection processes, posing 637 sign. For STCF, we adopt a hybrid approach that synthesizes 660 risks of superconducting magnet quenches and elevated de638 the established practices of BSC criteria from previous col661 tector background. Our mitigation strategy employs strategic 639 liders [49] with advanced stability requirements of modern 662 collimator placement at distances ≥ 15 m from the IP (out640 facilities. Figure 16 [FIGURE:16] illustrates the BSC boundaries at beam 663 side the MDI core), effectively localizing particle loss away 641 energies of 2 GeV and 3.5 GeV, where the dashed magenta 664 from sensitive regions. 642 and green lines represent the positron and electron beam en643 velopes, respectively. At 2 GeV, the horizontal BSC is de644 fined as 22σx + 2 mm (or 24σx ), and the vertical BSC as D. Preliminary estimation of detector background 645 22σy + 2 mm (or 26σy ); while at 3.5 GeV, the larger nat646 ural emittance permits a more aggressive 9.5σx + 2 mm (or Experimental backgrounds in the IR include beam-induced 647 10.5σx ) horizontally and 9.5σy + 2 mm (or 11.5σy ) verti- 666 648 cally. These values assume a horizontal emittance of 5 nm at 667 sources—such as Touschek scattering, beam-gas Coulomb 649 2 GeV and 27 nm at 3.5 GeV, with vertical emittance set to 668 scattering, beam-gas bremsstrahlung, synchrotron radiation, 650 be 5% of the horizontal value. The reduced BSC at 3.5 GeV 669 and injection losses—as well as luminosity-related processes 651 can be relaxed by increasing beta functions at the IP (e.g., 670 like radiative Bhabha scattering and two photon process 652 βx : 60 → 150 mm, βy : 0.8 → 1.6 mm), with accept- 671 [50, 51]. 653 able luminosity trade-offs (reduced by 24%) for high-energy 672 Comprehensive background evaluations, conducted 654 operation. 673 through collaborative accelerator-detector simulations, iden674 tify Touschek scattering and beam-gas Coulomb scattering 675 as the dominant background sources in STCF’s low-energy 676 regime [52].

With optimized collimation, Touschek losses 677 in the IR are reduced to less than 1% of ring-wide total 678 losses—an order-of-magnitude improvement over the un679 collimated scenario. The same collimator settings effectively 680 control injection losses without compromising injection 681 efficiency.

With effective Touschek scattering suppression, beam683 gas Coulomb scattering emerges as the primary background 684 source, driving the requirement for ultra-high vacuum condi−7 685 tions (10 Pa in MDI region, 10−8 Pa elsewhere).

Synchrotron radiation and beam-gas bremsstrahlung rep687 resent secondary but non-negligible background contribu688 tions.

Synchrotron radiation primarily contributes to the 689 thermal load on beam pipes, though its high-energy com690 ponents at 3.5 GeV require careful assessment. Beam-gas 691 bremsstrahlung becomes increasingly relevant at higher enerFig. 16. The beam stay-clear regions for the STCF MDI. 692 gies, reinforcing the need for the specified ultra-high vacuum

MDI beam stay-clear region

Looking forward, several strategic directions are identiconditions.

These findings establish the foundation for ongoing opti- 749 fied for further enhancement. Correction of residual second695 mization of vacuum systems, collimator configurations, and 750 order vertical and third-order horizontal chromaticity of696 background suppression strategies, ensuring the STCF will 751 fers the most immediate gains in momentum acceptance. 697 achieve its physics goals while maintaining operational ro- 752 Compensation of chromo-geometric terms arising from the 753 breakdown of −I transformations for off-momentum parti698 bustness. 754 cles—e.g., via dedicated decapole pairs near chromatic sex755 tupoles [23]—could further extend off-momentum dynamic 756 aperture.

Tailored dipole settings may also help optimize VI. SUMMARY AND DISCUSSION 757 second-order dispersion, while higher-order correctors (oc700 This paper presents a complete physics design for the 758 tupoles or decapoles) at IP image points may open routes to 759 fourth- and fifth-order chromaticity control [23]. 701 STCF crab-waist interaction region, successfully address760 In parallel, several complementary efforts will support the 702 ing the challenge of managing strong nonlinearities in a 761 development of the STCF Technical Design Report: (1) im703 low-energy, high-luminosity collider.

The integrated ap∗ 762 plementing a reduced βx (e.g., 40 mm) to further mitigate 704 proach—encompassing parameter optimization, modular op763 coherent X-Z instability [53], beneficial for considering the 705 tics design, nonlinear dynamics control, and MDI optimiza764 interplay of the beam–beam interaction with lattice nonlinear706 tion—provides a robust foundation for achieving the project’s 765 ities [54]; (2) establishing flexible IP β-function tuning proce707 ambitious performance targets.

Key IR parameters, including the crossing angle 2θc , the 766 dures and robust IP optics correction schemes; (3) optimizing 767 octupole configurations to recover dynamic aperture reduc∗ 709 IP drift length L , and the IP beta functions βy and βx , were 768 tion induced by final-focus quadrupole fringe fields; (4) ad710 selected through a balanced consideration of luminosity per769 dressing engineering challenges in local solenoid compensa711 formance, nonlinear dynamics control, and engineering con770 tion, particularly the overlap of residual Bz fields with final712 straints. These choices fundamentally shape both the optical 771 focus quadrupole fields.

Furthermore, during global opti713 design and mechanical integration while ensuring compatibil772 mization of the collider ring, minimizing resonance driving 714 ity with the detector requirements. 773 term fluctuations could serve as a promising strategy to im715 A central achievement of this work is the development and 774 prove the dynamic aperture [55]. 716 validation of a modular crab-waist IR optics that strategically As the design evolves, continued co-optimization of IR op717 decomposes the complex nonlinearity into manageable com776 tics, final focus magnets, and machine-detector interface will 718 ponents. By implementing exact −I transformations for chro777 be essential to further refine the balance among performance, 719 matic sextupole pairs, minimizing the dispersion invariant to 778 technical feasibility, and cost-effectiveness. This work thus 720 enhance local momentum acceptance, optimizing beta func779 lays a solid foundation for realizing the full potential of the 721 tions at crab sextupole locations, and extending local chro780 STCF collider ring. 722 maticity correction to third order, we have designed a practi693

cal IR structure that balances high luminosity, adequate beam lifetime, and operational robustness against realistic perturba781 VII. ACKNOWLEDGEMENTS 725 tions from fringe fields and solenoid effects. Integrated into 726 the full ring, the design can achieve a momentum acceptance We thank A. Bogomyagkov (BINP) for discussions on 727 of approximately ±1.5% and a Touschek lifetime exceeding 782 −2 −1 high-order chromaticity correction and sharing SCTF lattice 728 300 s at 0.94 × 10 cm s and 2 GeV, thereby meeting files. acknowledge contributions from the MDI team 729 the STCF performance goal. 785 (Wenbin Ma, Xunfeng Li, Mingyi Liu, Zhujun Fang, et al.) This study yields insights applicable beyond the STCF con786 and beneficial discussions with k Kazuhito Ohmi (KEK) and 731 text. First, explicit control of the dispersion invariant proves 787 Etienne Forest (KEK). This work is supported by the National 732 to be a powerful strategy for alleviating local momentum 788 Natural Science Foundation of China (Project No. 12341501, 733 acceptance bottlenecks, a strategy already implemented in 789 No. 12405174) and the National Key R&D Program of China 734 BINP-SCTF [28] and particularly relevant for other low- to 790 (Project No. 2022YFA1602201). We also thank the Hefei 735 medium-energy colliders where enhanced local momentum 791 Comprehensive National Science Center for their strong sup736 acceptance is essential for achieving adequate Touschek life792 port on the STCF key technology research project. 737 time. Second, we demonstrate that a systematic local com738 pensation approach—utilizing octupoles for fringe field cor739 rection and anti-solenoids for detector field cancellation—can Appendix A: Optics design at the crab sextupole 740 effectively restore dynamic aperture against realistic pertur741 bations.

The design methodology established here—systematic 794 Following K. Oide’s derivation for the strength of the crab 743 management of nonlinearities in crab-waist IR configura- 795 sextupole (CS) [29], we present a more detailed derivation 744 tions—provides a valuable reference for other colliders in 796 and analysis to guide the optics design at the CS in our sce745 comparable energy regimes. The demonstrated integration of 797 nario.

As shown in Fig. 17 [FIGURE:17], taking the positron beam as an exam746 advanced optics design with practical engineering constraints 798 747 marks a critical step toward the STCF technical design phase. 799 ple, the crab-waist scheme relates the twisting distance ∆s of

βx,CS cos ∆µx + px βx,CS βx sin ∆µx Hs = − βy,CS ∆s = − cos ∆µy + py βy,CS βy sin ∆µy tan(2θc ) βx,CS 803 where the negative sign indicates that crab-waist twists the cos ∆µx + px βx,CS βx∗ sin ∆µx . 804 vertical βy beam waist of particles with positive x in the 805 direction opposite to the beam’s motion. Therefore, the re(A9) 806 lationship between the particle coordinates after twisting and By appropriately setting the phase advance from the crab 807 the particle coordinates at the IP is: 832 sextupole to the IP, i.e., ∆µx = mπ, and ∆µy = nπ+π/2 (to 833 eliminate undesired nonlinear terms), the Hamiltonian simpli? ? ? ? ? ∗? (A2) 834 fies to: βx,CS βy,CS βy∗ x∗ p∗2 Hs = − sgn(cos ∆µx ) 809 and the associated transformation is: (A10) sgn(cos ∆µx ) . tan(2θc ) Comparing (A10) with the required crab-waist Hamilto∗ 811 which means that the crab-waist at the IP causes y to change 837 nian Eq. (A5), the integrated strength K2 of the crab sex838 tupole is derived as: 812 by: x∗ p∗y , (A11) ∆y = − tan(2θc )βy,CS βy βx,CS sgn(cos ∆µx ) tan(2θc ) Clearly, once the key IR parameters are fixed, increasing βx 841 and βy functions at the crab sextupole can reduce its strength To achieve this ∆y, the required Hamiltonian is: 842 K2 , with a larger βy being more effective than a larger βx 843 for lowering K2 .

On the other hand, to minimize the ad∗ ∗ 844 verse effects from the crab sextupole (e.g., the second term tan(2θc ) 2 tan(2θc ) 845 in Eq. (A10)), βx function at the crab sextupole should be de815 (A5) 846 signed as small as possible. However, an excessively small This Hamiltonian required by the crab-waist scheme can be 847 βx would undesirably increase the horizontal chromaticity. 817 provided by a sextupole (the so-called crab sextupole). The 848 Therefore, the optimized beta functions at the crab sextupole 849 should satisfy βy ≫ βx while maintaining βx as small as 818 Hamiltonian of a normal sextupole is: 850 possible but larger than 1. x − 3xy 2 , the vertical βy beam waist at the IP to the horizontal displace∗ 801 ment x via:

with K2 the integral strength of the sextupole. Using the wellknown linear transfer matrix from s1 to s2 expressed by the 822 twiss parameters [56] and considering α x/y = 0 at the IP, the 823 coordinates x/y at the crab sextupole can be expressed by the 824 coordinates at the IP as:

βx,CS cos ∆µx + p∗x βx,CS βx∗ sin ∆µx , (A7)

βy,CS βy,CS βy∗ sin ∆µy , (A8)

Substituting Eq. (A7) and Eq. (A8) into the sextupole 829 Hamiltonian Eq. (A6) gives:

s-direction by ∆s onto the center of the other beam.

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