Spin observables for polarizing antiprotons. Donie O Brien

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1 Spin observables for polarizing antiprotons Donie O Brien Trinity College Dublin, Ireland donie@maths.tcd.ie 9th November 006 ECT* Trento Italy Donie O Brien

2 Introduction The PAX project at GSI Darmstadt plans to polarize an antiproton beam by repeated interaction with a hydrogen target in a storage ring. Many of the beam particles are required to remain within the ring after interaction with the target so small scattering angles are important. Hence we concentrate on low momentum transfer (small t). Electromagnetic effects dominate the hadronic effects in this low t region of interest. Thus we calculate all Electromagnetic Helicity amplitudes and Spin Observables for elastic p p and p e scattering, to first order in QED. A beam of polarized electrons with energy sufficient to provide scattering of antiprotons beyond ring acceptance may polarize an antiproton beam by spin filtering. The spin observables are then used to estimate the rate of buildup of polarization of an antiproton beam. ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 1

3 The GSI Facility ECT* Trento Italy Donie O Brien donie@maths.tcd.ie

4 The Future GSI Facility ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 3

5 The Future Accelerator Layout ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 4

6 1 Normalization We investigate the general two particle elastic process with spin A ( p 1, S 1 ) + B ( p, S ) A ( p 3, S 3 ) + B ( p 4, S 4 ) where it is assumed that the beam particles (A) of mass M are antiprotons and the target particles (B) of mass m are electrons or protons. The target is initially polarized and the beam is initially unpolarized, we seek to model the buildup of polarization of the antiproton beam. p e p e p p p p p d p d ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 5

7 Define electromagnetic form factors F 1 (q ) and F (q ), with normalization F 1 (0) = 1 and F (0) = µ 1, the anomalous magnetic moment, where q = t. We use the Sach s electric and magnetic form factors G M = F 1 + F and G E = F 1 + t 4 M F respectively. The differential cross section is related to the helicity amplitudes M(Λ, λ ; Λ, λ) by s dσ dω = 1 (8π) λλ ΛΛ 1 4 M(Λ, λ ; Λ, λ) where λ, Λ and λ, Λ are the helicities of the initial and final particles respectively. The electron current is j µ = e ū(p 4, λ ) γ µ u(p, λ), and the antiproton current, after Gordon decomposition is ( J µ = e p ū(p 3, Λ ) G M γ µ p µ F + p µ ) 4 u(p 1, Λ). M ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 6

8 1 The Electromagnetic Helicity Amplitudes and Spin Observables ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 7

9 1.1 The Generic Calculation The generic equation for polarization effects in elastic spin 1/ - spin 1/ scattering to first order in QED is ( q ) 4 16 M = e [( (1+ ) ( (1 ) ] Tr /p 4 + m) γ5 /S 4 (gm γ ν + fr ν ) /p + m) + γ5 /S (gm γ µ +fr µ ) Tr [( (1+ ) ( (1+ ) ] /p 1 +M) γ5 /S 1 (GM γ µ +F R µ ) /p 3 + M) γ5 /S 3 (GM γ ν +F R ν ) where the electromagnetic form factors G M = F 1 + F, g M = f 1 + f, F = F / M and f = f / m; also R µ = p µ 1 + p µ 3, r µ = p µ + p µ 4. This generic equation can thus be used to calculate all helicity amplitudes and spin observables etc. by substituting specific values for the spin (S i ) and momenta (p i ) vectors. The result has been obtained for this equation with the traces computed and contracted, using Mathematica. ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 8

10 Centre-of-Mass Momenta vectors p 1 = ( E 1, 0, 0, k ) p 3 = ( E 1, k sin θ, 0, k cos θ ) p = ( E, 0, 0, k ) p 4 = ( E, k sin θ, 0, k cos θ ) Centre-of-Mass Normal spin vectors S N 1 = ( 0, 0, 1, 0 ) S N 3 = ( 0, 0, 1, 0 ) S N = ( 0, 0, 1, 0 ) S N 4 = ( 0, 0, 1, 0 ) Centre-of-Mass Transverse spin vectors S T 1 = ( 0, 1, 0, 0 ) S T 3 = ( 0, cos θ, 0, sin θ ) S T = ( 0, 1, 0, 0 ) S T 4 = ( 0, cos θ, 0, sin θ ) S L 1 = Centre-of-Mass Longitudinal spin vectors 1 M ( k, 0, 0, E 1 ) S3 L = 1 M ( k, E 1 sin θ, 0, E 1 cos θ ) S L = 1 m ( k, 0, 0, E ) S L 4 = 1 m ( k, E sin θ, 0, E cos θ ) ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 9

11 ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 10

12 1. Helicity Amplitudes The notation of the helicity amplitudes M( A, B ; A, B ) is M( ±, ± ; ±, ± ) where the arguments are + if the spin vector is as Si L above (polarized along the direction of motion) and if the spin vector is minus Si L above (polarized opposite to the direction of motion). After using T and P invariance there are 6 independent helicity amplitudes for the scattering of two non-identical spin 1/ particles. φ 1 M(+, + ; +, +) φ M(+, + ;, ) φ 3 M(+, ; +, ) φ 4 M(+, ;, +) φ 5 M(+, + ; +, ) φ 6 M(+, + ;, +) Note for p p, p p and p p scattering φ 6 = φ 5, so there are only 5 independent helicity amplitudes. ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 11

13 1.3 Helicity Amplitudes - 1st order QED results φ 1 α = s m M t φ α = 1 ( m k f 1 k m f φ 3 α = [ s m M t ( 1 + t ) 4 k f 1 F 1 f 1 F 1 f F 1 f 1 F 1 ( f F 1 t ) 4 k ) + s m M k ( 1 + t ) 4 m M 4 k f F ] ( 1 + t ) 4 k ) ( M k F 1 k M F f 1 F 1 + f F φ 4 = φ [ φ 5 s α = t (4 f1 F 1 m k + t) 4 k [ φ 6 s α = t (4 f1 F 1 M k + t) 4 k (1 m M ) f F 1 s m + t f F (1 16 m k + m M )] s ( M m ) 1 + f 1 F s M t f F (1 16 M k + M m )] s ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 1

14 Spin Observables All the electromagnetic spin observables of a reaction (polarization transfer K ij, depolarization (1 D ij ) and asymmetries A ij where i, j, k { X, Y, Z }) can now be obtained by direct computation. See D.O B. and N. H. Buttimore hep-ph/ for complete results. For electromagnetic interactions to first order the double spin asymmetries equal the polarization transfer observables (A ij = K ij ) and all the single and triple spin asymmetries are zero (A i = A ijk = 0). Spin filtering requires evaluation of the angular integration of the product of the observables A ii = K ii and (1 D ii ) with /. Azimuthal averaging indicates that the observables with single X (i.e. K XZ, K ZX, D XZ and D ZX ) do not contribute to spin filtering. The quantities (K XX + K YY ) /, (D XX + D YY ) /, K ZZ and D ZZ play the important role, we now present results for these. ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 13

15 Antiproton-proton scattering To look at the case of antiproton-proton scattering set the form factors and masses of each particle equal (f 1 F 1, f F and m M) in the generic equation. We obtain the results to leading order in small t: K XX + K YY (1 D XX ) + (1 D YY ) α M µ s t ( α k + M ) k M s t [ M k (µ 1) ] K ZZ α µ ( k + M ) s t (1 D ZZ ) ( α k + M ) [ M k M k (µ 1) ] s t ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 14

16 .1 Antiproton-electron scattering To look at the case of antiproton-electron scattering set the form factors of the second particle to be structureless (f 1 1 and f 0) in the generic equation. We obtain the results to leading order in small t: K XX + K YY (1 D XX ) + (1 D YY ) α m M µ s t m α K ZZ α µ s t (1 D ZZ ) M α ( s m + M ) 4 k s t ( s m M ) ( s + m M ) k s t ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 15

17 Antiproton-proton spin observables K XX = K YY = α G M { 8 s k M 4 M 4 F1 8 k M [ F 1 F + 4 k 4 ( + k + t ) ] s F } 4 ( ) α M G s t E G M K ZZ = α G M 8 k s t K XZ = K ZX = α G M s M t [ ( s 4 k ) + t F1 ( + 4 k ) ] F 1 t F ( t 4 k ) + t k 4 ( M F 1 k F 1 F + t F ) 8 ( ) 1 DXX ( ) 1 DYY ( ) 1 DZZ ( ) 1 DXZ = α F 1 k M s t ( k + M ) ( M F 1 k F ) α s G M 4, complete to all orders in t α F 1 k M s t ( k + M ) ( M F 1 k F ) ( ) 1 DZX 4 α F 4 ( 1 s t k + M ) ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 16

18 Antiproton-electron spin observables K XX = α m G ( M k M F 1 k ) F M s ( ) α K Y Y = m M G s t E G M K ZZ = α {[ G M 8 k s s (M m ) ] ( 4 k ) + t F k s t K α ( m F XZ = 1 G M t 4 k ) + t ( 4 s 3/ t k 4 s m + M ) K ZX = α G M 4 M s 3/ t ( 1 DXX ) ( 1 DYY ) ( 1 DZZ ) ( 1 DXZ ) m α F 1 k s t = t (4 k + t ) [ k 4 M ( s + m M ) F 1 k ] s F ( s m + M ) α s G M, complete to all orders in t M α F 1 k s t ( s + m M ) ( ) 1 DZX 4 α F ( 1 s t s k + m M ) (4 k F 1 t F ) } ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 17

19 3 The PAX Project There has been much recent debate as to whether electrons in the hydrogen target will transfer polarization to the antiproton beam. We re investigating if a beam of polarized electrons with sufficiently high density could be used to polarize an antiproton beam. ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 18

20 3.1 Scattering is within the ring for stationary electrons θ (mrad) Plot of scattering angle θ verses t for stationary electrons t (MeV/c) ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 19

21 Effect of electron beam momentum (I).5 Electron beam momentum 1 MeV/c θ (mrad) t (MeV/c) ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 0

22 Effect of electron beam momentum (II) 7 Electron beam momentum 3 MeV/c 6 5 θ (mrad) t (GeV/c) ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 1

23 Effect of electron beam momentum (III) 5 Electron beam momentum 10 MeV/c 0 θ (mrad) t (GeV/c) ECT* Trento Italy Donie O Brien donie@maths.tcd.ie

24 4 Polarization buildup When circulating at frequency ν through a polarized target of areal density n and polarization P e oriented normal to the ring plane, d dt N J = n ν I out P e A all P e K in P e A out I all D in N J describes the rate of change of the number of beam particles N and their total spin J. These coupled differential equations involve angular integration of the spin observables presented earlier. ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 3

25 Transverse polarization requires Longitudinal polarization requires I out = π A out =π A all =π π θ acc π θacc K in =π D in =π π θ acc (A XX +A YY ) (A XX +A YY ) θ 0 (K XX +K YY ) θ 0 θacc θ 0 (D XX +D YY ) sin θ dθ I sin θ dθ A sin θ dθ A sin θ dθ K sin θ dθ D out = π out = π all = π in = π in = π π π θ acc A LL θ acc π A LL θ 0 θacc θ 0 θacc θ 0 K LL D LL sin θ dθ sin θ dθ sin θ dθ sin θ dθ sin θ dθ ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 4

26 4.1 Solution of the system The time dependence of the polarization of the beam is given by solving the coupled system of differential equations, leading to P (t) = J(t) N(t) = P e A all K in L in + L d coth (L d n ν t) where the discriminant of the quadratic equation for the eigenvalues is L d = P e A out (A all K in ) + L in and L in = ( I in D in ) / is the loss of polarization quantity. The approximate rate of change of polarization for sufficiently short times, and the limit of the polarization for large times are respectively: dp dt n ν P e (A all K in ) A all K in lim P (t) = P e. t L in + L d ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 5

27 Further work Complete this analysis and obtain a numerical estimate for the polarization buildup rate, with a hydrogen gas target and also with an electron beam. Similarly calculate all electromagnetic helicity amplitudes and spin observables for antiproton-deuteron scattering, to first order in QED. Estimate the polarization buildup rate for antiprotons scattering off a polarized deuteron target. References Antiproton polarization has been considered recently by F. Rathmann et al. (005), A. I. Milstein et al. (005), N. N. Nikolaev et al. (006) and T. Walcher et al. (006). Results are consistent with the earlier work of B. Z. Kopeliovich and L. I. Lapidus (1974); N. H. Buttimore, E. Gotsman and E. Leader (1978), J. Bystricky, F. Lehar and P. Winternitz (1978), P. La France and P. Winternitz (1980) and E. Leader (005). ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 6

28 Conclusions All Helicity Amplitudes and Spin Observables for elastic spin 1/ - spin 1/ scattering have been presented to first order in QED. A beam of polarized electrons could be used to increase the polarization of an antiproton beam by spin filtering. Using the spin observables a numerical estimate for the rate of build up of polarization of an antiproton beam is being obtained for the PAX project. Acknowledgments Queries/Comments please donie@maths.tcd.ie I thank The Embark Initiative, for a postgraduate research scholarship, I also thank ECT* and those involved for organizing this program. ECT* Trento Italy Donie O Brien donie@maths.tcd.ie 7

Donie O Brien Nigel Buttimore

Spin Observables and Antiproton Polarisation Donie O Brien Nigel Buttimore Trinity College Dublin Email: donie@maths.tcd.ie 17 July 006 CALC 006 Dubna Donie O Brien Introduction Relativistic formulae for