Precision Spin Tracking for Electric Dipole Moment Searches

Transcription

1 Precision Spin Tracking for Electric Dipole Moment Searches

2 Outline: Spin Evolution in Storage Rings Technique for EDM Measurement and Systematic Errors Integration Methods, Current Status of Our Program Benchmarks Outlook Oct. 15, 2015 Martin Gaisser 2

3 Spin Evolution In particles' rest frame: (fields in rest frame) Transform fields from laboratory system into rest system and take into account that the rest system is rotating, use accelerator coordinates obtain T-BMT Equation: Oct. 15, 2015 Martin Gaisser 3

4 Spin Evolution All terms depend on position Fields may explicitly depend on time Spin evolution is nonlinear Have to solve equation for orbit (independent of spin) first No general analytical solution Need numerical solution! Oct. 15, 2015 Martin Gaisser 4

5 Technique for pedm Search V Use all-electric ring, no B-field Run at magic momentum V Radial E-Field Green: clockwise beam Red: counterclockwise beam V V frozen spin, no horizontal precession EDM-signal: vertical precession Expected precession rate: nrad/s Measure for ~1000s Need good alignement of spins relative to each other, i.e. long Spin Coherence Time (SCT) Use counter-rotating beams with different helicities for systematics Oct. 15, 2015 Martin Gaisser 5

6 Systematic Errors Effects that can mimic a possible EDM signal: Radial B-Field (few at are critical!) Geometric Phase (rotations don't commute) Field errors, misalignments Polarization profile of the beam Fields for feedbacks, extraction Need to understand all effects Requires simulations Need fast and precise long term tracking capability Have to proof correctness Oct. 15, 2015 Martin Gaisser net rotation around z-axis! 6

7 Which way to go? All paths lead to Rome : DA (Differential Algebra) TPSA (Truncated Power Series Algebra) RK4 (4 th order Runge-Kutta) Kick-Bend Romberg Integration... Take all of them Find their advantages/disadvantages See if we meet at the same place Oct. 15, 2015 Martin Gaisser 7

8 Our way: Numerical Integration Trade-off: Complexity Speed Choose simple numerical integration Requires few (but allows many) simplifications Easy to use, even with difficult and time-dependent fields Very precise for short and medium time range Fast for short simulations Many well tested algorithms like Runge-Kutta, Bulirsch-Stoer, Different algorithms are suitable for different situations Oct. 15, 2015 Martin Gaisser 8

9 Implementation Details Code written in C++ Modular design for flexibility Use Boost-Odeint library for algorithms Use VexCL (Vector Expression Template Library for OpenCL/CUDA) for parallelization Use Boost-Multiprecision library for arbitrary precision numbers VexCL can handle different devices of different manufacturers Solving N systems of 3 coupled ODEs (Lorenz attractors) with Odeint Oct. 15, 2015 Martin Gaisser 9

10 Choice of Algorithms Boost offers many different algorithms: Single & multistep methods Explicit/Implicit methods W/o step size control For some steppers: dense output (interpolation between large steps) Symplectic/not symplectic Many are (currently) not suitable because: They require constant step size (multistep methods) solution: Nordsieck Method Ublas vectors required, not for parallel version (implicit methods) Splitting of system + potentials of fields needed (symplectic methods) soon to come Oct. 15, 2015 Martin Gaisser 10

11 Symplecticity Currently none of our algorithms is symplectic Symplecticity crucial requirement for long term tracking otherwise Spin/Energy gets lost orbit incorrect spin incorrect Δx x n x n+1 Standard integrator (left): Geometric integrator (right): x n θ x n+1 Plan to implement high order symplectic solvers in the near future Oct. 15, 2015 Martin Gaisser 11

12 Symplecticity Currently none of our algorithms is symplectic Not a problem in the short term: Just reduce step size to get a more accurate result Loss of spin with 8 th order Runge-Kutta algorithm with dt=10-8 s for Muon g-2 ring For dt=3*10-9 s error remains below 3* th order Runge-Kutta needs a step size below s for error < 3*10-16 Step size reduction not a viable option for long term tracking! Oct. 15, 2015 Martin Gaisser 12

13 Performance Simulate 1ms of 3D lattice (right), electric bends, dt=10-8 s Every time measured in single run! Bulirsch-Stoer algorithm with dense output and error limits (absolute and relative) at Oct. 15, 2015 Martin Gaisser 13

14 Benchmarking We have to get the correct results... and we need a way to prove it! Compare with other algorithms (can only detect wrong settings, problems of algorithms) Compare with other codes (may or may not detect systematic problems) Compare with analytical results (have to match and understand what goes wrong if not) E.M. Metodiev et al., Nucl. Instrum. Methods Phys. Res., Sect. A 797, 311 (2015) Oct. 15, 2015 Martin Gaisser 14

15 Benchmarking (Pitch Effect) Magnetic ring with weak vertical focusing: Significance detected by tracking E=0 Causes spin tune for particles on flat orbits Oct. 15, 2015 Causes correction C to spin tune such that with Martin Gaisser 15

16 Benchmarking (Pitch Effect) Comparison of analytical and simulated pitch corrections for a muon with and Agreement better than ppb! Resonance behavior of pitch correction in the general formula (without approximation ) Oct. 15, 2015 Martin Gaisser 16

17 Benchmarking (Systematic Error Study) rf Wien filter (no Lorentz force, only effect on spin): Magnetic ring, B = Bv (Wien filter condition) Idea: Kick spin resonantly such that the EDM causes a build-up of a vertical spin component SV What if the Wien filter is tilted with respect to the vertical? Systematic Error! Oct. 15, 2015 Martin Gaisser 17

18 Benchmarking (Systematic Error Study) (Signal) (systematic error for tilt angle θ versus vertical) Useful fact: Values in rad/s for deuteron with assumed EDM and θ = 0.1 mrad. Oct. 15, 2015 Martin Gaisser 18

19 Planned Future Improvements Include high order symplectic solvers Implement Nordsieck method Implement algorithms with global error control (Runge-Kutta pairs, Nordsieck methods, general linear multistep algorithms) Optimize parallel version Implement special functions for parallel version Write script for automatic benchmarking tests Use the program for studies in Muon g-2 and sredm experiments Oct. 15, 2015 Martin Gaisser 19

20 Summary Simple numerical integration can be very precise for short term simulations Good method to deal with complicated fields Arbitrary precision numbers may increase precision, especially when EDMs are considered Very interesting and promising future possibilities Parallel (CPU/GPU) version will help significantly with systematic studies Program may profit from future hard- and software developments (higher CPU frequency, possible symbolic math capabilities of compilers,...) Maybe 1000s are not out of range forever... Oct. 15, 2015 Martin Gaisser 20

21 Thank you! Oct. 15, 2015 Martin Gaisser 21 October 2014