Overview What is P ? Some Physics motivation for P Initial P - PowerPoint PPT Presentation

Estimating P µ for the T WIST Measurement of P µξ Blair Jamieson Ph.D. Candidate University of British Columbia for the T WIST Collaboration LLWI’04 February 16-21, 2004

Overview • What is P µ ξ ? • Some Physics motivation for P µ ξ • Initial P µ and Depolarization Effects • Statement of the problem • Review of Spin • Spin propogation in Magnetic Fields • Overall T WIST Muon Depolarization estimate

What is P µ ξ ? • P µ is the polarization of the muon, ξ is asymmetry in angle of decay positrons from normal µ decay • Standard Model (V-A) predicts ξ = 1 and P µ = 1 dxd cos θ ∝ x 2 − x 3 + 2 9 ρ (4 x 3 − 3 x 2 ) + ηx 0 ( x − x 2 )+ d 2 Γ (1) 3 P µ ξ cos θ ( x 2 − x 3 + 2 3 δ (4 x 3 − 3 x 2 )) 1 x = E e /W eµ m 2 µ + m 2 Reconstructed Data Muon Decay Spectrum W eµ = e 2 m µ 7 4.5 × 10 Entries x 0 = m e Target 18000 16000 W eµ 14000 12000 10000 8000 Upstream 6000 4000 2000 0 Downstream -0.8 -0.6 -0.4 -0.2-0 0 0.2 10 0.4 20 0.6 30 cos( Total Momentum (MeV/c) 0.8 40 50 θ )

Physics and Motivation for P µ ξ • Best Measurements: – P µ ξ = 1 . 0027 ± 0 . 0079 ± 0 . 0030 (Beltrami et. al., PL B194 326) – P µ ξδ/ρ > 0 . 99682 , 90% conf. level (Jodidio et.al., PR D34 1967, PR D37 237) • ξ and δ together give limit on probability of right-handed muon decaying into any handed positron: R = 1 2(1 + 1 3 ξ − 16 Q µ 9 ξδ ) (2) • In Left-right symmetric model, P µ ξ sets limit on W R mass ( ǫ ) and left/right mixing parameter ( ζ ): P µ ξ = 1 − 2 ǫ 2 − 2 ζ 2 − 2 ǫ 2 ( V R ) 2 − ǫζV R ud ud (3) V L V L ud ud

0.1 0.1 CDF D0 0.075 0.075 PmuXi 0.05 0.05 mixing angle mixing angle TWIST PmuXi 0.025 0.025 TWIST rho 0 0 TWIST PmuXi, MLRS -0.025 -0.025 -0.05 -0.05 -0.075 -0.075 PmuXiDeltaRho 200 200 400 400 600 600 800 800 1000 1000 1200 1200 M W R , GeV M W R , GeV

Initial P µ and Depolarization Effects • Muon from π decay at rest has spin opposite direction from momentum since: – Standard Model ν is left handed – Conservation of Angular Momentum • Depolarization Effects: – Precession of Spin in Magnetic Fields ∗ Beam Divergence ∗ Radial Fringe Fields – Muonium Formation in Non-metals

Statement of the Problem • What is the average ∆ P µ as µ goes from production to stopping?

Review of Spin 1/2 Leptons • Spin “angular momentum” is a fundamental property of a particle • Magnetic dipole moment due to spin is: � � 2 m � M = − ge ¯ h S h , µ B = 5 . 788381749(43) × 10 − 11 MeV/T i S = − gµ B ¯ g ≈ 2 . due to relativistic kinematics, called Thomas Precession τ ), and Force ( � • Torque ( � F ) due to the intrinsic spin are: τ = � M × � � B F = ∇ ( � � M · � B ) • Quantization of spin • Spin must be 1/2 (ie 2s+1=2) • Spin precesses about � h B , along direction of B (z-axis): S z = ± ¯ 2 • Time average of Spin perpendicular to B is zero

Non-Relativistic Propogation of Spin in Uniform B • The equation for propogation of spin in a uniform magnetic field is: d� dt ′ = ge S S × � � B ′ (4) 2 mc • Prime means defined in rest frame of the particle, � S is the spin in that frame • For perfect alignment of � S and � B : h ¯ S x = 2 sin γ z t √ ¯ h S y = 2 cos γ z t √ (5) h S z = − ¯ 2 ge γ z = 2 mc B z • Misalignment α between � S and � B results in depolarization: ∆ P µ = 1 − | cos α |

Relativistic Propogation of Spin • Spin propogation is given by Bargmann, Michel, Telegdi (BMT) equation: d� dt = e s s × [( g 2 − 1 + 1 B − ( g γ γ ) � γ + 1( � β · � B ) � mc� 2 − 1) β ] (6) • For non-uniform field solve by stepwise integration in Monte-Carlo

Inputs to Depolarization Calculation • Field map • Beam Tune

Radial Magnetic Field Map (Gauss) Br vs r and z 800 800 600 600 400 400 200 0 5 x (cm) for y=0 200 4 3 2 0 1 -50 -300 -250 -200 -150 -100 ) 0 m c 0 ( z

Summary • Estimated ∆ P µ for current tune is ≈ 3 × 10 − 3 • Further reduction of beam size and divergence is desireable to reduce fringe field depolarization • T WIST goal is for knowledge of ∆ P µ to better than 10 − 4

Contents 1 Overview 2 2 What is P µ ξ ? 3 3 Physics and Motivation for P µ ξ 4 4 Initial P µ and Depolarization Effects 6 5 Statement of the Problem 7 6 Review of Spin 1/2 Leptons 8 7 Non-Relativistic Propogation of Spin in Uniform B 9 8 Relativistic Propogation of Spin 10

9 Inputs to Depolarization Calculation 11 10 Entrance Region Field Map 12 11 Summary 13