Post on 07-Aug-2020
Angular distributions of vector mesons
Johannes Bernhard
Institut fur Kernphysik Mainz
December 5th 2011
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
c d
X
where X is the exchange particle with a given JPC defining theproduction mechanism. The cross section reads1:
dσ
dΩ=
1
(2sa + 1)(2sb + 1)
p′
sp
∑λ
∣∣⟨λd λb|F (s, t)|λcλa
∣∣⟩1K. Gottfried, J.D. Jackson, Nouvo Cim. 33 (1964) 302
Cross section cntd.
Boost to rest frame of d , use ∆ = ~pa − ~pc as quantisation axis|m〉: spin states of d with amplitude
〈m;λc |G (s, t)|λbλa〉 =∑
λbλcλa
⟨mλb|F (s, t)|λcλa
⟩·d-functions for λa,b,c
d-functions take care of rotations from s to t channel. Constructspin-density matrix⟨
m|ρ|m′⟩ = N∑
λbλcλa
⟨mλb|F (s, t)|λcλa
⟩ ⟨m′λb|F (s, t)|λcλa
⟩∗
Spin density matrix2
Characteristics of ρ:
Hermeticity ρ† = ρ
Positive semi-definite: Trρ ≤ 1, Trρ2 = 1
symmetric: ρij = ρji
for spin 1 particles additionally dim(ρ)= 2J + 1 = 3 which also gives incombination Trρ = 1 = ρ00 + 2ρ11.
dσ
dΩ∝ ρ11 sin2(θ) + ρ00 cos(θ)−
√2ρ10 cos(φ) sin(2θ)
−ρ−11 cos(2φ) sin2(θ)
2K. Schilling, P. Seyboth and G. Wolf, Nucl. Phys. B 15 (1969) 397
Spin density matrix cntd.
For unpolarized beam and target3 ⇒ independent of φ
dσ
d cos(θ)∝ ρ11 sin2(θ) + ρ00 cos(θ)
Q: θ is defined in which reference frame again?A: Depends! Different definitions of frames make θ sensitive todifferent aspects of production process.
Q: And which vector defines θ?A: So-called analyser.
3K. Schonning, Dissertation, Uppsala (2009)
Analyser
1 Analysing production mechanism:Analyser is the decaying resonance b itself.
2 Analysing polaristion of resonance:for 2-body decay, use a decay particlefor 3 body decay, the story is more complicated
decay in one plane at one vertex (no isobars assumed), vertexcontribution reads ig~n · ~e, where g coupling, ~n analyser, ~epolaristion vector~e axial, product ~n · ~e cannot be 0 ⇒ ~a needs to be axial, toofor three spinless decay products π1,2,3 off a vector meson:~n = vecπi × vecπj , i 6= j
123
n
a b
a3
c
d
Reference frames
N.b.: Only cos(θ) interesting, i.e. z and analyser to be defined
Production plane: z = |~a× ~b|, overall CM system
Gottfried-Jackson frame: z = |~a|, a is beam, in CM system of b
Helicity frame: z = |~b|, overall CM system
production-relevant Gottfried-Jackson: z = |~X |, X is beam, inCM system of b
Examples4: t-channel helicity conserved when sin2(θ) distribution inproduction-relevant Gottfried-Jackson frame (JP(X ) = 0+). s-channelhelicity conserved when sin2(θ) distribution in helicity frame, deviationmeans spin-flip
4J.Barth, Dissertation, Bonn (2002)
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 z1 = |~a× ~b| andz2 = |~c × ~d | or z1 = |~a− ~b| and z2 = |~c − ~d |5.
5Symmetrisation not yet clear to me, to be clarified.
Some first results
Cf. Karin’s talk at last hadron meeting (Nov. 30th), p p → p ω p:
Results, pp pp ω
MC
Generated events Accepted
events
Events in ω region
Sidebands
Some sensitivity to the beam directions also for high x
F
Acceptance
Cos θ w.r.t. beam direction. Black: xF(p
fast) < 0.7, blue: x
F(p
fast) > 0.7
Some first results cntd.
Results, pp pp ω
MC
Generated events Accepted
events
Events in ω region
Sidebands
Polarisation changes sign when going from lowto high x
F!
Acceptance
Cos θ w.r.t. R direction. Black: xF(p
fast) < 0.7, blue: x
F(p
fast) > 0.7
Some first results cntd.
Results, pp pp ω
MC
Generated events Accepted
events
Events in ω region
Sidebands
Difficult to draw conclusions about xF dependence by eye.
Acceptance
Cos θ w.r.t. normal of production plane. Black: xF(p
fast) < 0.7, blue: x
F(p
fast) > 0.7
Some first results cntd.
Results, pp pp ω
MC
Generated events Accepted
events
Events in ω region
Sidebands
Polarisation much stronger at high xF
Acceptance
Cos θ w.r.t. Δ1 = p
beam – p
fast direction. Black: x
F(p
fast) < 0.7, blue: x
F(p
fast) > 0.7
Some first results cntd.
Results, pp pp ω
MC
Generated events Accepted
events
Events in ω region
Sidebands
Polarisation much stronger at high xF
Acceptance
Cos θ w.r.t. Δ2 = p
target – p
recoil direction. Black: x
F(p
fast) < 0.7, blue: x
F(p
fast) > 0.7
Difference betweenΔ
2 calculated from
spectrometer and RPD
Next
Check for xF dependence of ρ00 as suggested in PRD 68 (2003)034023:
Publications possible: OZI violation and angular distributions (spinalignment), both xF , t dependent
Results from summer
T (GeV)10 210
)ω
pp→
(pp
σ)/
dφ
pp→
(pp
σd
-310
-210
-110
DISTO (Phys.Rev.Lett.81 (1998) 4572)
New Series I/12 (Springer, 1998)Blobel, Baldi in Landault-Boernstein
D 25, (1982) 2241M.W. Arenton et al., Phys. Rev.
A 359, (1997) 435S.V. Golovkin et al., Z. Phys.
COMPASS 2008
New Series I/12 (Springer, 1998)Blobel, Baldi in Landault-Boernstein
OZI prediction