Ultrarelativistic Heavy-Ions

Transcription

1 Kinematics November 11, 2010 / GSI

2 Outline Introduction 1 Introduction 2 3 3

3 Notation Introduction A parallel to z-axis (beam): A A = A A transverse to z-axis: A = A A A = A Transverse mass: m = m 2 + p 2 m is invariant under Lorentz transformations along z-axis p = γ(p + vε) ε = m 2 + p 2 ε = γ(ε + vp ) γ = 1/ 1 v 2 p = p

4 Four vectors Introduction p µ = (p 0, p x, p y, p z ) (p 0, p) on shell = (ε, p) contravariant p µ = (p 0, p) covariant A µ B µ = A 0 B 0 A B four-vector product p 2 = p µ p µ = p0 2 p 2 on shell = ε 2 p 2 four-vector squared p 2 > 0 p 2 < 0 time like space like

5 Lorentz invariance (ε ) 2 ( p ) 2 = (ε) 2 ( p) 2 : p 2 Lorentz invariant m 2 Lorentz invariant Exponential spectra: dn m dm e m /λ p < 1 2 GeV λ: inverse slope parameter (apparent temperature) typical value λ 200 MeV ( ) λ 1 = d dm log dn m dm useful concept if dependence on m weak

6 Rapidity Introduction [ ] ( ) y = 1 2 log E+p p E p = arctanh E = arctanhv E = m 2 + p 2 = m 2 + p2 Rapidity is additive under Lorentz transformations along z-axis v = p E = tanh y p m = sinh y E m = cosh y To determine y in an experiment, need particle ID!

7 Pseudorapidity [ ] η = 1 2 log p +p = log [cot(θ/2)] = log [tan(θ/2)] p p Need only to measure angle θ p = p cosh η p = p sinh η sin θ = 1/ cosh η For m 0 (or for p ) η = y In general: dn dηd 2 p = p E dn dyd 2 p

8 CM frame Introduction In CM frame y η 0 central rapidity region (mid rapidity region) y = y P, y = y T projectile and target rapidity regions Central rapidity region: newly created particles, or original particles that have suffered several collisions dn dηd 2 p η=0 = p dn m dyd 2 p y=0 p = 0 p m = v < 1 dn/dη dn/dy (flattening of dn/dη)

9 Invariant cross section p = γ(p + vε) dp = γ(dp + vdε) = γ(1 + v p ε )dp ε = γ(ε + vp ) = γ(1 + v p ε )ε dp ε = dp ε Lorentz invariant ε dσ d 3 p = ε dσ = dσ d 2 p dp p dεd 2 Ω Invariant cross section

10 System expanding from a point (explosion) Ballistic trajectories: z = vt t v=z/t v=c Particle at (z, t) has velocity v = z/t Particle distribution g(z, t) = density of particles at (z, t) Change variables (z, t) (v, τ), τ = t 2 z 2 (proper time) f (v, τ) = g(z, t) Lorentz transformation f (v, τ) = f (v, τ) z τ = τ Rapidity: v = tanh y, y 1 = y 1 + y boost

11 If over a range in y τ t v=z/t f (y, τ) = f (τ) f is boost invariant In p p and A A collisions at very high energy, expect boost invariant dn/dy dn dyd 2 p = F(p ) ( y < y scaling < y beam ) If y scaling >> 1 a large range of Lorentz frames where system looks identical L.T. by V = v 2 : (v 2 t, t) (0, τ 2 ), τ 2 = t 1 (v 2 ) 2 z

12 Assumptions i) object created in inelastic collision interacts with the same σ inel ii) nuclei travel on straight lines Eikonal approximation classical approximation to angular momentum Elastic scattering amplitude f (s, t) = 1 2ip l (2l + 1) [ e 2iδ l 1 ] P l (cos θ) p 1 p 2 Mandelstam variables s = (p 1 + p 2 ) 2 t = (p 1 p 1 )2 p 1 p 2

13 Identical particles, CM frame s = 4(m 2 + p 2 ) 2t s 4m 2 = cos θ 1 Semiclassical approximation Large l, small θ pb = l P l (cos θ) 0 0 dφ 2π ei(2l+1) sin θ/2 cos φ dφ ei q b 2π

14 l pb l (l ) p2 db b f (s, t) = ip [ ] 2π d 2 b e i q b 1 e iχ(s, b) χ(s, b) = 2δ(s, b) Optical theorem σ tot = 4π p Im f (s, t = 0) = 2 d 2 b [ 1 Re e iχ] Elastic scattering Integrate over angles dω d 2 q p 2 (forward peaked) σ el = d 2 q 4π d 2 b d 2 b e i q b [ 1 e iχ] [ e i q b 1 e iχ ] 2 σ el = d 2 b 1 e iχ 2 σ in = σ tot σ el = d 2 b (1 e iχ 2)