Issue 76

Beam Dynamics Newsletter

3.7 nuSTORM Racetrack Decay Ring

J.-B. Lagrange, STFC/RAL/ISIS, UK.
J. Pasternak, Imperial College London and STFC/RAL/ISIS, UK.
K. Long, Imperial College London, UK.
R.B. Appleby, J.M. Garland, S. Tygier, Cockcroft Institute and Manchester University, UK.
A. Bross, D. Neuffer, FNAL, IL, USA.
A. Liu, Euclid Technologies LLC, NJ, USA.

Abstract

The nuSTORM project addresses essential questions in neutrino physics by providing the means for a precise measurement of neutrino cross sections and opening a way for a search for light sterile neutrinos [1]. This is possible because of the precisely known flavour content and spectrum of a neutrino beam produced from muon decay. In the proposed nuSTORM facility pions would be directly injected into a racetrack storage ring, where a circulating muon beam would be captured. A FODO solution with large aperture quadrupoles, a racetrack FFA (Fixed Field alternating gradient Accelerator) and a hybrid version of the two previous solutions are the options discussed in this paper.

Introduction

The production of a neutrino beam with a defined spectrum and flux composition using muon decay is a well-established idea. The concept was developed in the Neutrino Factory proposal, which was then addressed in several dedicated research and development studies culminating in the International Design Study for the Neutrino Factory (IDS-NF)  [2]. The Neutrino Factory consists of a high power proton driver, the output of which is directed towards a pion production target; a decay channel, where the muon beam is formed; the muon front end, where the beam is prepared for acceleration and the muon accelerator to boost the energy to the required value. The muon beam is then injected into the decay ring, with straight sections pointing towards near and far detectors producing neutrino beams for both interaction and oscillation physics. Although it has been shown  [2] that such a facility will have superior physics potential for leptonic CP violation searches to a conventional neutrino beam based on pion decay, it requires the construction of new accelerator components and presents many technological challenges. To allow for the start of neutrino physics experiments based on muon decay using conventional accelerator technology, the “neutrinos from STORed Muons” (nuSTORM) project was proposed  [3–5]. The main goal of nuSTORM is to precisely study neutrino interactions for electron and muon neutrinos and their antiparticles, but the facility could also contribute to sterile neutrino searches, and serve as proof-of-principle R&D for the Neutrino Factory concept.

In nuSTORM high energy pions produced at the target are first focused with a magnetic horn  [6], and directly injected into the ring after passing through a short transfer line equipped with a chicane to select the charge of the beam. Once in the ring, decaying pions will form the muon beam. A fraction of the muon beam with momentum lower than the injected parent pions will be stored in the ring, and a fraction with similar or larger momentum will be extracted with a mirror system at the end of the long straight section to reduce beam loss and avoid activation in the arcs. The extracted beam may also be used for accelerator R&D studies for future muon accelerators, which may serve as another application for nuSTORM. The design and performance of the decay ring is tightly linked to the neutrino physics reach of the experiment, with intrinsic design challenges arising from the large range of beam momenta in the ring.

At the present time there are three options under study for the design of the decay ring. The first option is a FODO solution with large bore, conventional quadrupoles with alternating gradients  [7] in the long straight sections and with a lattice based on separated function magnets in the arcs. This solution provides excellent performance with respect to the transverse acceptance, but it has very limited longitudinal acceptance, resulting directly from the alternating gradient conventional magnet approach. The second solution uses recent developments in FFAs. In these machines, which come in scaling and non-scaling flavours, large aperture, non-linear magnets allow the beam to move through the aperture at varying momenta, with a constant betatron tune in the case of a scaling FFA. The advantage of such a lattice is the large momentum acceptance together with the possibility of a large transverse acceptance, thus increasing the number of stored muons in the ring. The design is realised by keeping the ring zero-chromatic over the whole momentum range, and choosing the tune point far from harmful resonances. Furthermore, to maximise the portion of the ring pointing towards the detector, an FFA racetrack shape is now possible  [8], while keeping the ring zero-chromatic for a large momentum range, thanks to the use of straight scaling FFA cells  [9]. The third option is a combination of the FODO and FFA solutions, called the hybrid option. It features a production straight section with conventional quadrupoles, as in the FODO solution, and FFA magnets in the rest of the ring, as in the FFA solution.

The constant betatron tune with momentum provides a strong constraint on the fields in a scaling FFA. In the arcs, the vertical magnetic field \( B_{\mathrm {az}} \) in the median plane produced by the combined-function magnets follows the circular scaling law  [10], proportional to a power of the orbit radius. This type of magnet has been successfully built for several machines in the past  [11–16]. In the straight sections, the vertical magnetic field \( B_{\mathrm {sz}} \) in the median plane produced by the combined-function magnets follows the (more complicated) straight scaling law  [9]. The beam orbit oscillations through scaling magnets in this type of straight produce a characteristic periodic beam angle oscillation known as a scallop angle, which ultimately limits the achievable neutrino flux. Straight scaling FFA magnets have been successfully demonstrated experimentally  [9]. This paper discusses the FFA solution and gives preliminary results for the hybrid option.

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Figure 1: View of the racetrack FFA lattice (bottom left). Enlarged details of the straight section are shown at the top left and of the arc section to the right. Matched, minimum and maximum momenta muon closed orbits are shown in red. Effective field boundaries with collimators are shown in black.

3.7.1 FFA Ring design

The first constraint of the FFA solution is to keep the scallop angle of the reference trajectories in the straight section as small as possible. A large scallop angle, giving the deviation of the beam from a straight line, will alter the spectrum and overall value of the neutrino flux. This goal has been addressed with triplet lattice cells  [17]. However a quadruplet cell DFFD gives an additional long straight between the two F magnets, increasing the efficiency of the neutrino production towards the detector. Furthermore, since the D and F magnets are identical, only one type of magnet needs to be designed and manufactured.

The second constraint is to keep the dispersion small in the muon production straights to give a good muon capture rate. Since the central momentum of the injected pions is different from the central momentum of the circulating beam, the horizontal position of the reference trajectories must be sufficiently close to each other that the muons obtained from pion decay are within the acceptance of the circulating beam. However, the dispersion has to be large where the beam is injected to provide the necessary beam separation, so a dispersion matching section is necessary to accommodate the two constraints. A dispersion suppressor is therefore introduced in the arcs. The concept of a dispersion suppressor in a scaling FFA is to induce a betatron oscillation of the reference trajectories around the periodic trajectory of the matching cell that is different to the arbitrarily chosen, matched trajectory. A \( \pi \)-phase advance in the dispersion suppressor section allows half a complete oscillation and the reference trajectories to be matched. In order to match the horizontal beta-functions, the horizontal phase advance in the arc is an integer multiple of \( \pi \), and the vertical beta-functions are matched by adjusting the ratio of magnetic fields in the magnets, the so-called F/D ratios. Superconducting magnets are chosen to keep the arc sections compact and ensure a large ratio of the production straight length to the circumference. In the straight sections, to drive down manufacturing costs, room temperature magnets are preferred. However, the use of super-ferric magnets in the straight sections is also being considered owing to their lower power consumption.

A systematic scan of the tune diagram has been conducted to find the optimised parameters of the full FFA solution  [18]. The choice of the tune point was based on the dynamic acceptance in both horizontal and vertical planes and the muon capture efficiency in the straight sections. The muon capture efficiency is a function of the dispersion in the straight cells, the distance between the reference orbit of the injected pion beam and the reference orbit of the muon beam, and the maximum stable horizontal amplitude the lattice can accept.

Ring parameters for the chosen lattice are summarised in Table 1. Closed orbits of matching momentum, minimum momentum and maximum momentum are shown in Fig. 1. The dispersion and beta-functions at matching momentum are shown in Fig. 2. The magnetic field for the maximum momentum muon closed orbit is presented in Fig. 3. The stability of the ring tune has been studied over a \( \pm 19\% \) momentum range, and the tune shift is presented in Fig. 4. The transverse acceptance in both planes has been explored by tracking over 100 turns a particle displaced from the closed orbit with a small deviation in the other transverse direction (1 mm). This lattice shows a horizontal maximum emittance of about 2 mm.rad, as shown in Fig. 5, a vertical maximum emittance of about 1 mm.rad, as shown in Fig. 6, and a maximum stable horizontal amplitude larger than the distance between the reference orbits of the injected pion beam and and the circulating muon beam.

Table 1: Lattice parameters

Parameter Value
Total circumference 502 m
Length of one straight section 180 m
One straight section/circumference ratio 36%
Momentum acceptance 3.7 GeV/c \( \pm 19 \% \)
Ring tunes (H, V) (7.18, 4.88)
Number of cells in the ring:
Straight cells 20
Arc matching cells 8
Regular arc cells 8
m-value in straight cells 2.2 m\( ^{-1} \)
Packing factor in straight cells 0.24
Max. scallop angle in straight cells 76 mrad
k-value in regular arc cells 6.056
\( R_0 \) in regular arc cells 16.4 m
Packing factor in regular arc cells 0.92
k-value in matching cells 26.0
\( R_0 \) in matching cells 36.15 m
Packing factor in matching cells 0.57

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Figure 2: FFA design: Horizontal (plain blue), vertical (dotted red) periodic betatron functions (left scale) and dispersion (mixed green line, right scale) over half the ring for matching momentum. The plot is centred on the arc part.

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Figure 3: FFA design: Vertical magnetic field on the median plane for the maximum circulating muon momentum over half the ring. The plot is centred on the arc part.

   

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Figure 4: FFA design: Tune diagram for momenta \( \pm 19\% \) around 3.7 GeV/c. Integer (red), half-integer (green), third integer (blue) and fourth integer (purple) normal resonances are plotted. Structural resonances are in bold.

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Figure 5: FFA design: Stable motion in the horizontal Poincaré map for maximum initial amplitude over 100 turns for matched momentum. The ellipse shows a 2 mm.rad unnormalized emittance, centred on the closed orbit of the injected central momentum pion.

   

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Figure 6: FFA design: Stable motion in the vertical Poincaré map for maximum initial amplitude over 100 turns for matched momentum. The ellipse shows a 1 mm.rad unnormalized emittance.

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Figure 7: View of the racetrack hybrid lattice (bottom left). The enlargements show the production straight section (top left), the FFA straight (top middle), and the arc section (right). Matched, minimum and maximum momenta muon closed orbits are shown in red. Effective field boundaries with collimators are shown in black.

3.7.2 Hybrid solution

Although estimates promise a higher neutrino flux with the FFA lattice than with the FODO lattice  [17], a lattice combining the advantages of the FODO and the FFA solutions would give an improved performance. A production straight made of conventional quadrupoles would remove the problem of the scallop angle, while optimising the muon capture efficiency. The chromaticity of the ring can be greatly reduced if the rest of the ring is made zero-chromatic, as for the FFA solution. Furthermore, a large beta-function is desirable in the production straight to minimise the momentum angle for a given emittance, limiting the phase advance of the section, and thus its natural chromaticity. Such a lattice can keep the tune excursion across a large momentum range confined between half-integer resonances, which allows a large transverse acceptance for a large momentum range. Since the dispersion is different in the two straight sections, i.e. null in the quadrupole section and constant non-null in the straight FFA section, two different dispersion matching sections have to be designed around the straight sections.

Preliminary design parameters are presented in Table 2. Closed orbits of matching momentum, minimum momentum and maximum momentum are shown in Fig. 7. The dispersion and beta-functions at matching momentum are shown in Fig. 8. The magnetic field for the maximum momentum muon closed orbit is presented in Fig. 9. As for the FFA design, the stability of the ring tune has been studied, in this case over a \( \pm 15\% \) momentum range. The tune shift is presented in Fig. 10. The transverse acceptance in both planes has been studied in the same way as in the FFA solution. This lattice gives a maximum emittance of more than 1 mm in both horizontal and vertical planes, as shown in Figs. 11 and 12, respectively.

Table 2: Lattice parameters

Parameter Value
Total circumference 533 m
Length of production straight section 161 m
Production straight section/circumference ratio 30%
Momentum acceptance 5.3 GeV/c \( \pm 15 \% \)
Ring tune (H, V) at 5.2 GeV/c (6.20, 3.26)
Number of cells in the ring:
Quadrupoles straight cells 6
Straight FFA cells 10
Arc first matching FFA cells 4
Arc second matching FFA cells 4
Regular arc cells 8
Field gradient in quadrupoles -2.34 T/m, 2.31 T/m
Packing factor in quadrupole cells 0.16
m-value in straight FFA cells 2.2 m\( ^{-1} \)
Packing factor in straight FFA cells 0.24
Max. scallop angle in straight FFA cells 76 mrad
k-value in regular arc FFA cells 7.252
\( R_0 \) in regular arc FFA cells 31.2 m
Packing factor in regular arc FFA cells 0.92
k-value in first matching FFA cells 24.969
\( R_0 \) in first matching FFA cells 49.08 m
Packing factor in first matching FFA cells 0.62
k-value in second matching FFA cells 13.369
\( R_0 \) in second matching FFA cells 30.42 m
Packing factor in second matching FFA cells 0.91

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Figure 8: Hybrid design: Horizontal (plain blue), vertical (dotted red) periodic betatron functions (left scale) and dispersion (mixed green line, right scale) in the ring for matching momentum. The plot is centered on the straight FFA section.

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Figure 9: Hybrid design: Vertical magnetic field on the median plane for the maximum circulating muon momentum in the ring (6.1 GeV/c). The plot is centered on the straight FFA part.

   

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Figure 10: Hybrid design: Tune diagram for momenta \( \pm 15\% \) around 5.3 GeV/c. Integer (red), half-integer (green), third integer (blue) and fourth integer (purple) normal resonances are plotted.

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Figure 11: Hybrid design: Stable motion in the horizontal Poincaré map for maximum initial amplitude over 100 turns for matched momentum. The ellipse shows a 1 mm.rad unnormalized emittance.

   

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Figure 12: Hybrid design: Stable motion in the vertical Poincaré map for maximum initial amplitude over 100 turns for matched momentum. The ellipse shows a 1 mm.rad unnormalized emittance.

Summary

The nuSTORM project aims to address essential questions in neutrino physics, in particular by offering the best possible way to measure precisely neutrino cross sections and by providing an opportunity to search for light sterile neutrinos. It would also serve as a proof-of-principle experiment for the Neutrino Factory and can contribute to the R&D for future muon accelerators. The FFA solution for the decay ring gives good performance with large transverse and momentum acceptances, but a hybrid solution would give a better result. Work on an optimised version of the hybrid lattice and a full comparison in terms of neutrino flux for the three solutions are still to be carried out.

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