Potential for a new measurement of muon g -2 factor

1. Potential for a new measurement of muon g-2 factor Alexander J. Silenko Research Institute for Nuclear Problems Belarusian State University WORKSHOP ON HIGH ENERGY SPIN PHYSICS Julich 2010

2. Theoretical and experimental data • Muon g-2 experiment • New ideas • Spin coherence in a storage ring with a noncontinuous magnetic field and magnetic focusing • Avoidance of resonance • Corrections • Conclusions OUTLINE

3. Theoretical and experimental data

4. Experimental data dominated by BNL E821: • From theory: • The discrepancy is 3.2 σ: G.W. Bennett et al., Phys. Rev. D 73, 072003 (2006) F. Jegerlehner and A. Nyffeler. The muon g-2. Phys. Rep. 477, 1-110 (2009).

5. Muon g-2 experiment

6. A very important improvement as compared with early experiments was the use of electrostatic focusing at the magic energy (γm=29.3) and momentum. 0 • Two small corrections: • Vertical betatron oscillations (pitch) • Electric field correction at γ≠γm • The final CERN precision was7.3 ppm The finalE821precisionwas0.54 ppm

7. New ideas

8. An upgraded experiment, E969, with goalsof σsyst = 0.14 ppm and σstat = 0.20 ppm received scientific approval from the Brookhaven National Laboratory. There is the 3.4 σ difference between theory and experiment. The new experiment didnot begin.

9. F. J. M. Farley. A new ring structure for muon (g-2) measurements.NIMA523, 251 (2004).

10. Main distinguishing features of the Farley’s muon (g-2) ring: • A storage ring with uniform magnetic field regions separated by free space sections • Measurement of the magnetic field with polarized proton beams • Edge focusing

11. Advantages: • No electric quadrupoles • No NMR trolley, which has to be calibrated, with corrections for the diamagnetism of water • No superconducting inflector • Injection with a simple kicker in a field-free region • Higher muon energy and magnetic field giving longer lifetime, more (g − 2) cycles per lifetime, and proportionally higher accuracy • Lower counting rates for the same number of stored muons, so less problem with overlapping signals • More time for detectors to recover from initial transients and particles created at injection

12. Problems: • Noncontinuous vertical magnetic field • leads to a longitudinal magnetic field • complicates spin dynamics J. P. Miller, E. de Rafael and B. L. Roberts, Rept. Prog. Phys. 70, 795 (2007). Longitudinal magnetic field acts at a short distance!

13. Spin coherence in a storage ring with a noncontinuous magnetic field and magnetic focusing

14. Two main propositions by F. Farley are used: • Average magnetic field is (almost) independent of the beam momentum • Magnetic field is measured with polarized proton beams

15. The g-2 experiment can be carried out in a usual storage ring with magnetic focusing Average magnetic field should be independent of the particle momentum This property can be proved There is another definition of the momentum compaction factor

16. A usual storage ring with magnetic focusing It consists from two semicircles with a magnetic field separated by a free space

17. Another possible ring lattice: R0 L p0 p>p0 2

18. Betatron tunes When the momentum increases (p>p0), the magnetic field becomes weaker, but the time of flight in the magnetic field becomes longer In any magnetic field, v=const. Spin coherence is kept when An average magnetic field does not depend on the particle momentum!

19. The solution is As a result,

20. The real value of the length of the straight sections,L, can slightly differ from L0. is proportional to ! p0is the vertex of a parabola in the momentum space. One can make measurements with proton beams. To findp0 and adjust the ring lattice, threemeasurements with different values of pare necessary.

21. Avoidance of resonance

22. Betatron tunes Average vertical force is proportional to Average radial force is proportional to

23. Radial BOs are in resonance with the cyclotron motion!

24. When and one looses the beam! The resonance effect depend on a periodical perturbation and the CBO frequency E.D. Courant and H.S. Snyder, Ann. of Phys. 3, 1-48 (1958). One can choose but Appropriate choice is To improve the spin coherence, radio frequency cavities can be used Longitudinal electric field does not affect the spin

25. Corrections

26. Correction to the average vertical magnetic field for the betatron oscillations For the perturbed motion, the average longitudinal component of the velocity is approximately equal to Bp is the average magnetic field for the perturbed motion Bu is the average magnetic field for the unperturbed motion Approximately,

27. When v0r/v0~v0z/v0~0.001, λ~0.01, the correction to the average vertical magnetic field for the betatron oscillations is rather small and may be even negligible

28. Pitch correction (correction for the vertical BO) Final formula: S. Granger and G.W. Ford, Phys. Rev. Lett. 28, 1479 (1972). F.J.M. Farley, Phys. Lett. B 42, 66 (1972). F.J.M. Farley and E. Picasso, in QuantumElectrodynamics, edited by T. Kinoshita (World Scientific, Singapore, 1990). J.H. Field and G. Fiorentini, Nuovo Cimento Soc. Ital. Fis. A 21, 297 (1974). Calculation of an additional oscillatory term: A.J. Silenko, Phys. Rev. ST Accel. Beams 9,034003 (2006).

30. A similar correction for the horizontal BO is zero Correction for the longitudinal magnetic field is the trajectory length is the cyclotron frequency into bending sections b is the length of the considered trajectory segment at the magnet edge

31. We can suppose that and with

32. Corrected local angular frequency The correction caused by the longitudinal magnetic field for one segment of the beam trajectory While the total correction is nonzero owing to a non-commutativity of rotations. The total correction can be obtained with the multiplication of by the additional factor:

33. If we substitute the parameters of the BNL E821 experiment, we obtainΔΩl(a)/Ω(a) ~ 1 ppmfor both muons and protons New experiments based on different ring lattices are very important even if they do not provide better precision as compared with the usual g-2 experiments

34. Conclusions • Distinctive features of finished and planned muon g-2 experiments are the uniform magnetic field and electric focusing an the magic momentum. The planned accuracy is σsyst = 0.14 ppmandσstat = 0.20 ppm. • Alternative possibility proposed by F. Farley is the use of a noncontinuous and locally uniform magnetic field and magnetic focusing. The magnetic field is measured with a polarized proton beam at the same momentum. The muon g-2 ring with the uniform magnetic field and edge focusing can provide a high accuracy. • Our proposition consists in a storage ring with a noncontinuous nonuniform magnetic field and the definite length of straight sections for keeping the independence of the spin rotation frequency from the particle momentum. The proposed experiment may be less expensive, can be carried out at one of existing rings, and can provide a high accuracy. • The experiment could provide an independent experimental result with different systematics.

35. Thank you for attention