Angular distributions of vector mesons

Transcription

1 Angular distributions of vector mesons Johannes Bernhard Institut für Kernphysik Mainz December 5 th 2011

2 Cross section Consider the reaction a + b c + d, (e.g. p + p pω + p) which holds for the crossed channel a + c X b + d a b X c d where X is the exchange particle with a given J PC defining the production mechanism. The cross section reads 1 : dσ dω = 1 p λ d λb F (s, t) (2s a + 1)(2s b + 1) sp λ c λ a 1 K. Gottfried, J.D. Jackson, Nouvo Cim. 33 (1964) 302 λ

3 Cross section cntd. Boost to rest frame of d, use = p a p c as quantisation axis m : spin states of d with amplitude m; λ c G(s, t) λ b λ a = λ b λ cλ a m λb F (s, t) λ c λ a d-functions for λa,b,c d-functions take care of rotations from s to t channel. Construct spin-density matrix m ρ m = N λ b λ cλ a m λb F (s, t) λ c λ a m λb F (s, t) λ c λ a

4 Spin density matrix 2 Characteristics of ρ: Hermeticity ρ = ρ Positive semi-definite: Tr{ρ} 1, Tr{ρ 2 } = 1 symmetric: ρ ij = ρ ji for spin 1 particles additionally dim(ρ)= 2J + 1 = 3 which also gives in combination Tr{ρ} = 1 = ρ ρ 11. dσ dω ρ 11 sin 2 (θ) + ρ 00 cos(θ) 2ρ 10 cos(φ) sin(2θ) ρ 11 cos(2φ) sin 2 (θ) 2 K. Schilling, P. Seyboth and G. Wolf, Nucl. Phys. B 15 (1969) 397

5 Spin density matrix cntd. For unpolarized beam and target 3 independent of φ dσ d cos(θ) ρ 11 sin 2 (θ) + ρ 00 cos(θ) Q: θ is defined in which reference frame again? A: Depends! Different definitions of frames make θ sensitive to different aspects of production process. Q: And which vector defines θ? A: So-called analyser. 3 K. Schönning, Dissertation, Uppsala (2009)

6 Analyser 1 Analysing production mechanism: Analyser is the decaying resonance b itself. 2 Analysing polaristion of resonance: for 2-body decay, use a decay particle for 3 body decay, the story is more complicated decay in one plane at one vertex (no isobars assumed), vertex contribution reads ig n e, where g coupling, n analyser, e polaristion vector e axial, product n e cannot be 0 a needs to be axial, too for three spinless decay products π 1,2,3 off a vector meson: n = vecπ i vecπ j, i j a d 1 c n b 3 2 a^

7 Reference frames N.b.: Only cos(θ) interesting, i.e. ẑ and analyser to be defined Production plane: ẑ = a b, overall CM system Gottfried-Jackson frame: ẑ = a, a is beam, in CM system of b Helicity frame: ẑ = b, overall CM system production-relevant Gottfried-Jackson: ẑ = X, X is beam, in CM system of b Examples 4 : t-channel helicity conserved when sin 2 (θ) distribution in production-relevant Gottfried-Jackson frame (J P (X ) = 0 + ). s-channel helicity conserved when sin 2 (θ) distribution in helicity frame, deviation means spin-flip 4 J.Barth, Dissertation, Bonn (2002)

8 Reference frames cntd. So far only relevant for single X exchange (photo-production, diffractive scattering, etc.) double X exchange (central production, double Reggeon exchange, etc.) must be treated separately: two production planes relevant, define with either ẑ 1 = a b and ẑ 2 = c d or ẑ 1 = a b and ẑ 2 = c d 5. 5 Symmetrisation not yet clear to me, to be clarified.

9 Some first results Cf. Karin s talk at last Results, hadron meeting 30th ), p p p ω p: pp (Nov. pp ω Cos θ w.r.t. beam direction. Black: xf(pfast) < 0.7, blue: xf(pfast) > 0.7 MC Events in ω region Generated Sidebands Acceptance Some sensitivity to the beam directions also for high xf Accepted

10 Some first results Results, cntd. pp pp ω Cos θ w.r.t. R direction. Black: xf(pfast) < 0.7, blue: xf(pfast) > 0.7 MC Events in ω region Generated Sidebands Acceptance Polarisation changes sign when going from low to high xf! Accepted

11 Some first results Results, cntd. pp pp ω Cos θ w.r.t. normal of production plane. Black: xf(pfast) < 0.7, blue: xf(pfast) > 0.7 MC Events in ω region Generated Sidebands Acceptance Difficult to draw conclusions about xf dependence by eye. Accepted

12 Some first results Results, cntd. pp pp ω Cos θ w.r.t. Δ1 = pbeam pfast direction. Black: xf(pfast) < 0.7, blue: xf(pfast) > 0.7 MC Events in ω region Generated Sidebands Acceptance Polarisation much stronger at high xf Accepted

13 Some first results Results, cntd. pp pp ω Cos θ w.r.t. Δ2 = ptarget precoil direction. Black: xf(pfast) < 0.7, blue: xf(pfast) > 0.7 MC Events in ω region Generated Sidebands Accepted Acceptance Difference between Δ2 calculated from spectrometer and RPD Polarisation much stronger at high xf

14 Next Check for x F dependence of ρ 00 as suggested in PRD 68 (2003) : Publications possible: OZI violation and angular distributions (spin alignment), both x F, t dependent

15 Results from summer dσ(pp ppφ)/dσ(pp ppω) Blobel, Baldi in Landault-Boernstein New Series I/12 (Springer, 1998) DISTO (Phys.Rev.Lett.81 (1998) 4572) M.W. Arenton et al., Phys. Rev. D 25, (1982) 2241 S.V. Golovkin et al., Z. Phys. A 359, (1997) 435 COMPASS 2008 OZI prediction T (GeV)