Design of a 6 TeV muon collider

A preliminary lattice design of a muon collider ring with the center-of-mass (CM) energy of 6 TeV is presented. The ring circumference is 6.3 km, and the beta function at collision point is β* = 1 cm in each plane. The ring linear optics, a local non-linear chromaticity compensation in the Interaction Region (IR), additional IR non-linear correction knobs, and the effects of non-linear fringe field are discussed. Magnet specifications are based on the maximum pole-tip field of 20 T in dipoles and 15 T in quadrupoles. Careful compensation of the non-linear chromatic and amplitude dependent effects provides a sufficiently large dynamic aperture for the momentum range of up to ± 0.5% without considering magnet errors.


A
: A preliminary lattice design of a muon collider ring with the center-of-mass (CM) energy of 6 TeV is presented. The ring circumference is 6.3 km, and the beta function at collision point is β * =1 cm in each plane. The ring linear optics, a local non-linear chromaticity compensation in the Interaction Region (IR), additional IR non-linear correction knobs, and the effects of non-linear fringe field are discussed. Magnet specifications are based on the maximum pole-tip field of 20 T in dipoles and 15 T in quadrupoles. Careful compensation of the non-linear chromatic and amplitude dependent effects provides a sufficiently large dynamic aperture for the momentum range of up to ±0.5% without considering magnet errors.

Introduction
A muon collider is one of the potential candidates for a future energy frontier colliding machine. Taking into consideration the relationship between beam energy, luminosity and wall power in operation, an optimal choice of the center-of-mass (CM) muon beam energy appears to be near 6 TeV [1]. In order to reach the desired peak luminosity of 10 34 cm −2 s −1 in the TeV energy range, a number of demanding requirements to the collider optics has to be satisfied. Among these are a small ring circumference, limited by the maximum achievable magnetic field, a low β * function ( 1 cm) at the Interaction Point (IP), a short bunch length σ z β * which calls for a small momentum compaction factor α c < 10 −4 , and sufficiently large dynamic aperture (> 5σ) for the expected normalized beam emittance of about 25 µm-rad and rms momentum spread of 0.1%. A more detail discussion of the optics requirements can be found in refs. [2][3][4].
In this paper, we present a preliminary lattice design for a 6 TeV muon collider. The focus of this design study is a compensation of the transverse non-linear chromatic and geometric aberrations with the goal of obtaining a satisfactory dynamic aperture within a sufficiently large momentum range. Optimization of longitudinal parameters (e.g. α c ) was not part of this study.

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2 Linear optics For a 6 TeV muon collider ring, achieving a reasonably short circumference and strong focusing in the IR requires a rather high magnetic field. In this study, we limit the maximum pole-tip field to 20 T in dipoles and 15 T in quadrupoles, and use the arc cell length comparable to the 1.5 TeV design in ref. [5]. With these limitations, a longer ring containing more cells with a smaller bending angle per dipole is required as compared to the 1.5 TeV machine. We expect that achieving such a high field will be possible with future advancements in superconducting (SC) magnet design. The chosen ring circumference is ≈6.3 km, which is approximately the size of the Tevatron as a potential site for this type of collider. The designed ring lattice has a two-fold symmetry and periodicity consisting of two identical IRs and two arcs. The optics has been designed using MAD8 code [6] while the tracking simulations were performed using LEGO [7].

Interaction region
Interaction region is the most challenging part of a high luminosity collider due to the extremely high beta functions in the final focus (FF) quadrupoles creating large non-linear chromaticity and high sensitivity to magnet errors. The designed optics of one half-IR of the 6 TeV machine is shown in figure 1, where IP is on the left-hand side with β * =1 cm in x and y planes. The IP is followed by a 6 m free space and a FF quadrupole doublet, where vertical beta function reaches an extremely high value of 134 km. The remaining part of the IR is made of a FODO-like lattice, where dipole magnets fill most of the space between the quadrupoles to avoid hot spots of neutrino radiation [4]. Due to the high beta functions, the FF quadrupoles generate very large non-linear chromaticity resulting in a non-linear chromatic tune shift and strong momentum dependence of beta functions. A local compensation of these effects is required in order to avoid a severe reduction of the ring momentum acceptance. The designed IR chromaticity correction scheme is based on two (x and y) non-interleaved pairs of sextupoles on each side of the IP, where the sextupoles in each pair are separated by −I transformation for cancellation of the second order sextupole geometric (amplitude dependent) aberrations. The x and y sextupoles are placed near the high beta peaks in the corresponding plane (see figure 1) to maximize their compensating effects. This scheme will be discussed in more detail later. Dispersion function is cancelled at each end of the IR. The latter is followed by ∼40 m of dispersion-free section which can be used for beam injection, placement of RF-cavities as well as geometric (harmonic) sextupoles or octupoles for additional non-linear compensation. The dispersion-free section is followed by a dispersion suppressor that matches the IR optics to the periodic arc, as can be seen on the right-hand side of figure 1.

Arcs
The arc lattice of the 6 TeV muon collider presented here is based on the earlier lattice design for the 1.5 TeV machine [5]. To accommodate the four times higher beam energy while limiting the magnetic field, the cell length is increased by 20%, and the number of cells per arc is increased from 6 to 16 relative to the 1.5 TeV design. The bending angle per cell is reduced accordingly. As in the 1.5 TeV design, each arc cell includes three sextupole families, which can be used to adjust linear chromaticity and momentum compaction. Phase advance per arc cell is changed from the original µ x /µ y =0.833/0.833 [2π] to 0.875/0.875 [2π]. The latter provides a unit transformation per each half-arc (8 cells) leading to cancellation of the second order sextupole geometric aberrations [8]. Figure 2 shows the optics functions of one arc cell. The optics of one-half of the ring is presented in figure 3.

IR magnet parameters
The IR magnets must have sufficient physical aperture while satisfying the limitation on the pole-tip field described earlier. Specification of the minimum half-aperture in the IR magnets is based on the definition of 5σ x,y + 15 mm, where σ is an rms beam size, and the 15 mm is a contingency for hardware including cryogenic cooling channel [9]. The resulting minimum half-aperture corresponding to the normalized beam emittance of 25 µm-rad and rms energy spread of 0.1% is shown in figure 4. Due to the limitation on the pole-tip field, the large aperture in the FF quadrupole doublet leads to a relatively weak gradient, therefore long FF quadrupoles are needed to provide the necessary focusing strength. To minimize the FF length, the doublet is made of several short magnets, where the individual quadrupole apertures are adjusted according to the rapidly changing beta functions in this area. This reduces the required magnet aperture where beta functions are lower, thus increasing the corresponding quadrupole gradient and reducing the total length of the doublet. The other IR quadrupoles are divided into three groups based on their aperture. The resulting parameters of the IR magnets are presented in table 1. The main parameters of the 6 TeV ring are listed in table 2. Note that the calculated luminosity does not include the beam-beam and the hourglass effects at IP; this luminosity value should not be considered an "official" value. The momentum compaction has not been optimized in this design study; the latter was limited to optimization of beam transverse dynamics.

Chromaticity correction
Chromaticity of a collider with low IP beta functions is dominated by the FF quadrupoles where beta functions are extremely high. These quadrupoles generate not only a significant part of ring linear chromaticity, but are also the main source of non-linear chromaticity such as a non-linear chromatic tune shift and energy dependent beta functions. These effects must be compensated locally to prevent  the large non-linear chromatic perturbations to severely limiting the ring momentum acceptance. The designed IR chromaticity correction scheme is based on two non-interleaved pairs of sextupoles on each side of IP (SIRY1-SIRY2 and SIRX1-SIRX2) for x and y correction, where the identical sextupoles in each pair are separated by −I transformation for cancellation of the second order sextupole geometric aberrations. The following optics conditions are implemented in order to obtain an efficient correction of the FF chromaticity: 1) nπ phase advance (in the correcting plane) between the −I sextupoles and the FF; 2) large beta function and dispersion at the sextupoles to provide the compensation with a limited sextupole magnetic field; 3) large β x / β y or β y / β x ratio for orthogonal x and y correction; 4) local correction, where the first sextupole is placed as close as possible to the FF. The dedicated high beta functions for the −I sextupoles can be seen in figure 1.
After correction of the IR non-linear chromaticity, the remaining linear chromaticity of the machine is canceled using two sextupole families in the periodic arc cells. The x/y phase advance per cell is chosen to be 0.875/0.875 [2π]. The resulting +I transformation per each half-arc (8 periodic cells) provides cancellation of many of the sextupole driven non-linear resonances up to fourth order [8,10]. The non-vanishing non-linear effects of the arc sextupoles are the resonance driving term 2ν x − 2ν y , two second order chromatic tune terms and three amplitude dependent tune terms.
The strength of the third sextupole in the arc cell was scaled from the 1.5 TeV design. This sextupole can be used for adjustment of the momentum compaction, however the latter was not part of optimization in this design. The scope of this study was limited to the transverse dynamics; variation of momentum compaction would have minimal impact on dynamic aperture.
The set-up of the IR and the arc sextupole strengths was performed in MAD8 using the following method. First, the IR sextupole strengths were set for local cancellation of the x and y Wfunctions [6] at the IP and outside of the IR as shown in figure 5. This way, the chromatic variation of IP beta functions and, therefore, the IP beam size, are minimized. The above procedure also compensates the second order terms of the chromatic tune shift generated by the FF quadrupoles, where linear dispersion is zero [11]. Note that the second order dispersion generated by the IR sextupoles is self-compensated within each −I pair due to the identical sextupole strengths, same linear dispersion, and 180 • phase advance.
As a second step, the remaining ring linear chromaticity is cancelled using two sextupole families in the arcs. This two-step procedure can be used iteratively, if needed, to obtain simultaneous cancellation of the IP W-functions and the ring linear chromaticity. The effect of this compensation on beam dynamic aperture was verified in LEGO tracking simulations for 1024 turns for the machine without magnet errors. Figure 6 shows the dynamic aperture at IP for on and off-momentum particles up to ∆p/p = 0.4%, while the particles with initial ∆p/p of 0.5% did not survive. The tune footprint and the amplitude dependent tunes obtained in tracking are shown in figures 7 and 8. Figure 9 presents the chromatic tune shifts which are calculated analytically based -6 -

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on tune derivatives with momentum obtained from the tracking data. This method also applies to other plots in this paper showing the chromatic tune shift.
One can see that the vertical dynamic aperture is rather poor (≈2σ y ), and the momentum range is below the desired ∆p/p = 0.5%. Figures 8 and 9 reveal that the vertical amplitude dependent tune shift and the horizontal chromatic tune shift are quite large. Therefore, further non-linear compensation is required in order to reduce these effects. The goal of this additional correction is to increase the dynamic aperture to more than 5σ (where σ = 3 µm at IP at beam energy of 3 TeV) and the momentum range to at least ±0.5%.

Further compensation of non-linear effects
Dynamic aperture simulations with the implemented IR chromaticity correction reveal that the vertical dynamic aperture is not sufficient, likely due to the large vertical amplitude dependent tune shift (see figure 8). Secondly, the momentum range is below the desired 0.5% due to the large third order term of horizontal chromatic tune shift (figure 9). In order to improve the dynamic aperture, additional orthogonal non-linear correction systems (knobs) were implemented in the IR. These knobs include (per each half-IR): 1) one octupole (OCT1) at a non-dispersive location with a large vertical beta function to correct the vertical amplitude dependent tune shift; 2) a pair of weak sextupoles placed symmetrically around each main sextupole (SIRY1, SIRY2) to compensate the effect of the finite sextupole length (the latter is amplified by the very high beta function); and 3) −I pair of opposite polarity octupoles (OCT2) next to the main SIRX1, SIRX2 sextupoles (where β x and dispersion are large) to correct the third order term of horizontal chromatic tune shift. Positions of these sextupoles and octupoles in one half-IR are shown in figure 10. These correction knobs are discussed in more detail below.

Correction of vertical amplitude dependent tune shift
A thin lens octupole (OCT1) is included on the outer side of each FF doublet for control of vertical tune shift with vertical amplitude. Due to the large ratio of β y / β x at this location, the effect of  the octupole is mostly in the vertical plane. Additionally, due to zero dispersion, the OCT1 does not create chromaticity. Hence, this octupole can be used as a knob for tuning the vertical tune shift with amplitude. The integrated octupole strength K 3 L = 0.01031 m −3 was determined in LEGO by minimizing the vertical tune shift with vertical amplitude, where K 3 = B 6Bρ , L is the effective octupole length, and Bρ the magnetic rigidity. This setting improved the on-momentum  vertical dynamic aperture, but the off-momentum aperture was still insufficient, as can be seen in figure 11. As a second step, the octupole strength was optimized in LEGO by maximizing the -9 - vertical dynamic aperture for both on and off-momentum particles, as shown in figure 12. In this case, the optimal octupole strength is 0.022 m −3 . The amplitude dependent tune shift terms with and without the OCT1 correction are presented in table 3. The latter shows that maximizing the off-momentum dynamic aperture corresponds to changing the sign of the vertical tune shifts with x and y amplitude as compared to the case without the OCT1. This results in a different orientation of the tune footprint, as can be seen in figure 13 and in the plot of the tune shift derived from tracking in figure 14. The latter shows that the vertical tune with amplitude is driven away from the integer resonance, which improves the vertical dynamic aperture as compared to figure 6 without the OCT1. The impact of OCT1 on the horizontal tune shift is small confirming that the octupole effect is primarily in the vertical plane. Although, the OCT1 correction improves the overall dynamic aperture, the stable off-momentum range is still below the goal of 0.5%. This is likely due to the large third order horizontal chromatic tune shift, which needs to be reduced.

Compensation of the effect of finite sextupole length
Cancellation of non-linear geometric effects caused by sextupoles in a −I pair is exact in the approximation of a thin lens sextupole (i.e. for zero sextupole length). For realistic sextupoles with non-zero length the residual non-linear effects can be significant if beta functions at the sextupoles are very large, such at the SIRY1 and SIRY2 sextupoles. Figure 15 shows dynamic aperture for the case where the SIRY1, SIRY2 are modelled as thin lens sextupoles (with octupoles turned off). In this case, the on-momentum aperture is very large as compared to the case with thick sextupoles in figure 6. Table 4 shows that with the thin lens sextupoles the vertical tune shift with the vertical amplitude is reduced and changes sign as compared to the case with thick sextupoles, while the other tune shift terms remain the same in the two cases.
It has been shown [12] that the non-linear effect of the finite sextupole length can be partially compensated by introducing a −I pair of weak sextupoles next to each main −I sextupole pair, where a weak sextupole is added next to the corresponding main sextupole. An improved version  of such correction [13] requires the use of two pairs of weak sextupoles (instead of one) per each −I pair of the main sextupoles. In this case, each main sextupole is accompanied by two weak correcting sextupoles placed symmetrically on either side of the main sextupole. The four weak sextupoles, therefore, form two −I pairs as can be seen in the schematic layout in figure 16, where S1 and S2 are the strengths of the main and the weak sextupoles, L is the sextupole length, and kL is the distance between the main and the weak sextupoles. The strength S2 required for compensation Table 3. Tune shift with amplitude versus OCT1 strength.

K 3 L, m −3 d(Qx)/d(Jx) d(Qy)/(dJy) d(Qy)/d(Jx)
of the effect of the finite sextupole length (up to the order of L 5 ) is determined by the equation: 2 (7 + 6k) S2 S1 2 + 12 (1 + k) S2 S1 + 1 = 0 [13]. Using k = 0.5 and choosing a solution with a weaker S2, the resulting ratio of S2/S1 is −0.0595. The corresponding dynamic aperture, where the weak sextupoles are correcting the SIRY1, SIRY2 sextupoles, is shown in figure 17 (with octupoles off). One can see that the resulting aperture is very similar to the one with thin lens sextupoles in figure 15. This, therefore, confirms that the above correction, indeed, compensates the effect of the finite length of the SIRY1, SIRY2 sextupoles.   Further improvement of dynamic aperture was achieved when the OCT1 octupoles were included in addition to the weak sextupole correction. The OCT1 strength was re-optimized to     Figure 9 shows that the large horizontal chromatic tune shift is dominated by a cubic term. The latter may be the limiting factor for the observed dynamic aperture momentum range. To correct this term, a −I pair of thin lens octupoles (OCT2) was added in each half-IR. The octupoles were placed next to the SIRX1, SIRX2 sextupoles, where dispersion is sufficiently large. The two octupoles in this pair have opposite polarities in order to cancel the octupole contributions to the horizontal amplitude dependent tune shift and to the quadratic term of the horizontal chromatic tune shift. Due to the large β x / β y ratio at these locations, these octupoles mostly affect the horizontal plane, thus providing an orthogonal correction. Figure 19 shows that the OCT2 can effectively control the third order term of the horizontal chromatic tune shift. For the OCT2 strengths of K 3 L = ±14.5 m −3 , the corresponding dynamic aperture is shown in figure 20. In this case, the weak sextupoles and OCT1 octupoles are turned off resulting in relatively small vertical aperture; however, the momentum range is increased to 0.5% due to the reduced horizontal chromatic tune shift. The optimized dynamic aperture including the weak sextupoles and OCT1, OCT2 octupole corrections is shown in figure 21. One can see that the on-momentum aperture is sufficiently large (8σ), and the off-momentum aperture is above or near 5σ for the momentum range of up to 0.5%.

Fringe field effects
Besides the effects of large non-linear chromaticity generated by the FF quadrupoles, another concern is the effect of their non-linear fringe field. The latter was not included in the dynamic aperture calculations presented in the previous sections. Since the beta functions in the FF quadrupoles and the other IR magnets are very high, the impact of their fringe field on the beam dynamics may not be negligible, and therefore requires evaluation.
The LEGO code includes the model of non-linear fringe field in quadrupoles which can be turned on or off in tracking. Recently, the code has been updated to also include the non-linear fringe field in dipoles and the effects of a large bending curvature h. The third-order kinematic terms h x (P 2 x + P 2 y ) have been added to the main body of the dipoles, where P is the canonical momentum. The dipole non-linear fringe field was implemented by including the sextupole-like terms proportional to h.  Tracking simulations have been carried out including the previously optimized IR corrections and the effects of non-linear fringe field in all quadrupoles and dipoles. Figure 22 shows that the main impact of the fringe field on momentum dependent tune is a modest increase of the quadratic term of the vertical chromaticity. The vertical tune shift, however, is still small. The amplitude dependent tune shift terms with and without the fringe field are shown in table 5. One can see that the largest term dQ y /dJ y is further increased when the fringe field is included. The resulting dynamic aperture with the previously optimized IR corrections and the fringe field in all quadrupoles and dipoles is shown in figure 23. The effects of fringe field make the horizontal dynamic aperture more symmetric and, as a result, larger than without the fringe field in figure 21. The on-momentum vertical aperture is reduced by ≈20%, possibly due to the larger dQ y /dJ y , however it is still better than 5σ. The off-momentum aperture is also above or near 5σ even at ∆p/p = 0.5%.  Table 5. Amplitude dependent tune shift terms for the optimized 6 TeV lattice with and without fringe field effects in quadrupoles and dipoles.

Summary
A preliminary lattice design of a muon collider ring with 6 TeV CM energy, β * = 1 cm, and circumference of 6.3 km is presented. The very high beta functions in the FF quadrupoles create strong non-linear chromatic aberrations severely limiting the dynamic aperture. To compensate these effects, we use a system of non-interleaved −I pairs of IR sextupoles for local chromaticity correction. In addition, we introduce the −I pairs of weak sextupoles and two families of octupoles in the IR to compensate the third order horizontal chromatic tune shift, the non-linear effects of finite sextupole length, and the vertical amplitude dependent tune shift. The arc lattice design features  a local cancellation of many non-linear resonance driving terms up to fourth order. The effect of non-linear fringe field in quadrupoles and dipoles is evaluated; its main impact is a modest reduction of vertical dynamic aperture. Optimization of the IR non-linear correction systems allows to reach and exceed the target dynamic aperture of 5σ and the momentum range of ±0.5%, including the fringe field effects.
In addition to the presented design, a possibility to lower the β * value has been preliminarily investigated indicating that it can be reduced to 2.5 mm, assuming the same IR length and a modest increase of the quadrupole pole-tip field from 15 to 20 T. Optimization of dynamic aperture of this lattice requires a separate study.