Nuclear and Particle Physics
An Introduction
Spring Semester 2012 Farid Ould-­‐Saada B.1 Lorentz transformations and 4-Vectors
B.2 Frames of references
B.3 Invariants
1/16/12
F. Ould-Saada
2
Particle of rest mass m, velocity u in coordinates (t,x,y,z) in frame S   In S’ moving with speed v=βc in z-­‐direction coordinates in S’: (t’,x’,y’,z’), u’ ⎧

x
x' = x
⎪
⎪ y' = y
⎪
⎛ v ⎞
⎨ z' = γ ⎜ z − ct ⎟
⎝ c ⎠
⎪
⎪
⎛
v ⎞
⎪ct' = γ ⎜ct − z⎟
⎝
c ⎠
⎩
x’
z
S
S’
β≡
v
y
z’
⎧
⎪
u −v
⎨ u' =
uv
⎪
1− 2
⎩
c
v
c
γ=
1
1− β
2
: Lorentz factor
⎡ uv ⎤
γ (u') =
= γ (u)γ (v) ⎢1 − 2 ⎥
2
⎣ c ⎦
⎛ u' ⎞
1 − ⎜ ⎟
⎝ c ⎠
1
y’
1/16/12
F. Ould-Saada
3


Time dilatation Distance contraction  Time and space coordinat
es make up a 4-­‐
vector 1/16/12
F. Ould-Saada
4
1/16/12
F. Ould-Saada
5
1/16/12
F. Ould-Saada
6
Lorentz transformations
The Lorentz-transformation
of both space-time and
momentum-energy
four-vectors can be
expressed in matrix form:
The space-time and the energy-momentum 4-vectors result in
1/16/12
F. Ould-Saada
8


Laboratory system (LS) and Centre of mass system (CMS) In LS moving projectile a in a beam strikes a target particle b at rest 
In CMS 
4-­‐vectors in both systems (L=laboratory, T=target, B=beam) Comparison of fixed target and colliding beam accelerators 1/16/12
F. Ould-Saada
9



2
B(E L , pL )+ T(mT , 0) →P(E, q ) + ...
Scatteringangle θ L in LS and θ C inCMS


pL = (0,0, pL ) ; q = (0,qsin θ L ,qcosθ L )
' ' 
In CMS : pB + pT = 0
1
q' sin θ C
tan θ L =
γ(v) q' cosθ + vE'
C
c2
E' = mP c 2γ(u) q' = mP u γ(u) u : velocity of P in CMS
v = pL c ( E L + mT c
2
pL
γ(v) ≈
2mT c

2 −1
)
⇒
HE:E L ≈ p L c >>m B c 2,m T c 2
⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯
⎯→ v ≈ c(1 − mT c/pL ) ≈ c
2mT c
usin θ C
tan θ L ≈
⋅
pL ucosθ C + c
Unless u~c and cosθC~-­‐1, final state particles emitted in narrow cone about beam direction in LS. Similarly with decays. 1/16/12
F. Ould-Saada
10
Laboratory system (LS) and Centre of mass system (CMS)   More efficient to work with quantities that are invariant! 
c
2mT
c
InCMS : p =
2 s
In LS : pL =
[s − (m
+ mB ) 2 ][ s − (mT − mB ) 2 ]
[s − (m
+ mB ) 2 ][ s − (mT − mB ) 2 ]
T
T
 Invariant under all permutations of its arguments €
 Minimum Laboratory energy to
produce particle M
1/16/12
F. Ould-Saada
11

Mass of decaying particle = invariant mass of decay products 
Dalitz plots - Crystal barrel at LEAR (Low Energy Antiproton Ring, CERN  Meson spectroscopy
- Plot with high degree of symmetry: 3 identical particles
- Clear enhancements due to resonances
1/16/12
F. Ould-Saada
12

Mandelstam variables http://en.wikipedia.org/wiki/Mandelstam_variables
A + B →C + D
2
s = ( pA + pB ) /c
• t+t+u=
2
2
t = ( pA − pC ) /c
2
2
u = ( pA − pD ) /c 2
∑m
2
j
j =A,B ,C ,D
• elasticscattering, p,θ in CMSrelative toparticle A,
⇒ t = −2 p 2 (1 − cosθ ) /c 2

Rapidity and pseudo-­‐rapidity €
http://en.wikipedia.org/wiki/Pseudorapidity
1 ⎛ E + pL ⎞
Rapidity :y = ln⎜
⎟
2 ⎝ E − pL ⎠
⎡ ⎛ θ ⎞ ⎤
Pseudo - rapidity :η = −ln ⎢tan⎜ ⎟ ⎥
⎣ ⎝ 2 ⎠ ⎦
1/16/12
F. Ould-Saada
13
1/16/12
F. Ould-Saada
14
  Pages 358-­‐359 (see next 2 pages)   B1-­‐B10 1/16/12
F. Ould-Saada
15
1/16/12
F. Ould-Saada
16
1/16/12
F. Ould-Saada
17