AppendixA2
¡
S.I.units
§  [kg,m,s]everydayobjects,macroscopicphenomena,but
▪  Mass:me=9.1x10-31kg
▪  Distance:1fm=10-15m
▪  Cross-section(area):1barn=10-28m2,1pb=10-12b,1N=10-15b
¡
Naturalunits
¡
!=c=1
§  [h,c,GeV]
▪  UnitofactioninQM:!=h/2π=1.055x10–34J.s=6.58x10–16eV.s
▪  Speedoflight(relativity):c=2.998x108m/s=2.998x1023fm/s
▪  Protonrestmassenergy:mp~1GeV/c2=109eV/c2=1.602x10-10J
▪  !c =0.197GeV.fm
▪  E2=p2c2+m2c4àE2=p2+m2
▪  Momenta:eV/c;MeV/c;GeV/c;TeV/c
▪  Masses:eV/c2:Me=0.51MeV/c2;Mp=0.94GeV/c2
▪  !=c=1à[M]=[E]=[P]=[L–1]=[T–1]àGeV
24/01/16
F.Ould-Saada
2
Quantity
[kg, m, s] [!, c, GeV ]
! = c =1
Energy
kg m 2 s −2
GeV
GeV
Momentum kg m s −1
GeV / c
GeV
Mass
kg
GeV / c 2
GeV
Time
s
(GeV / !)
Length
m
(GeV / !c)
−1
GeV −1
Area
m2
(GeV / !c)
−2
GeV −2
¡
−1
GeV −1
1MeV=1.53×1021s–1
1MeV–1=197fm
1s=3×1023fm
1s–1=6.5×10-16eV
1ps–1=0.65meV
1m=5.07×106eV–1
1m–1=1.97×10–7eV–1
Example
§  Root-mean-squarecharge
radiusofproton
§  NUàSI(Dimensions!)
24/01/16
F.Ould-Saada
r2
1/2
= 4.1GeV −1 → ×!c [GeV fm ]
= 4.1GeV −1 ( 0.197GeV fm ) = 0.8 fm
3
§  Distances:1fm=10-15m
▪  ProtonRadius:~0.8fm
▪  RangeofStrongnuclearfore:~1-2fm
▪  Rangeofweakforce:~10-3fm
▪  Radiusofatom:~105fm
§  Crosssections–unitforarea.Barn:1b=10-28m2
▪  ππscattering(strongprocess):few10mb,asppscattering(previousfigure)
▪  νpscattering(weakprocess):few10N(1N=10-15b)
§  Energies:1eV=1.6X10-19joules
▪  1keV=103eV;1MeV=106eV;1GeV=109eV;1TeV=1012eV;
▪  Atomicionisation:feweV
▪  Bindingofnucleonsinheavynuclei:7-8MeVperparticle
§  Momenta:eV/c;MeV/c;GeV/c;TeV/c
§  Masses:eV/c2:Me=0.51MeV/c2;Mp=0.94GeV/c2;
▪  MW=80.4GeV/c2;MZ=91.2GeV/c2
▪  SIunits:1MeV/c2=1.78X10-30kg
▪  Atomicmassunit:1u=mC/12=1.661X10-27kg=931.5MeV/c2
24/01/16
F.Ould-Saada
4
¡
Strengthof
Quantum
Electrodynamics
§  finestructure
constantα
§  αdimensionless:
samenumerical
valueregardless
ofunitsystem
used
!
#
"
#
! = c = ε 0 = µ 0 = 1$
e2
α=
4πε 0 !c
e2
1
⇒α =
≈
4π 137
24/01/16
F.Ould-Saada
5
AppendixA3
"t ' = t
$
$x ' = x
Galilei #
$ y' = y
$% z' = z − vt
"v(light) = c = constant
$ 2 2 2
2 2
$x + y + z = c t
Einstein # 2
2
2
2 2
$ x ' + y' + z' = c t '
$c 2 t 2 − x 2 − y 2 − z 2 = c 2 t '2 − x '2 − y'2 − z'2
%
"ct ' = γ ( ct − β z ) = ct cosh θ − zsinh θ
$
$x ' = x
Lorentz #
$ y' = y
$ z' = γ ( z − β ct ) = −ct sinh θ + z cosh θ
%
¡
¡
v
β = = tanh θ
c
1
γ=
= cosh θ
(1− β 2 )
βγ = sinh θ
v << c → β << 1 → γ ≈ 1 cosiθ=coshθ,siniθ=isinhθ
èLorentztransformation–rotationofimaginaryangleiθ inthe(ict,z)plane
24/01/16
F.Ould-Saada
7
¡
¡
Timedilatation
Distancecontraction
24/01/16
F.Ould-Saada
è
Timeand
space
coordinates
makeupa
4-vector
è
Space-time
unification
8
¡  4-vectors,matrixnotation
§  InverseLT:Σ’àΣ(và-v)
§  (RememberccanbeomittedinNUwithh=c=1)
!
X = (ct, x)
⎛ ct '⎞ ⎛ γ
⎜ ⎟ ⎜
⎜x' ⎟ = ⎜ 0
⎜ y' ⎟ ⎜ 0
⎜ ⎟ ⎜⎜
⎝ z' ⎠ ⎝ −γβ
24/01/16
X ' = ΛX
0 0 −γβ ⎞⎛ ct ⎞
⎟⎜ ⎟
1 0 0 ⎟⎜ x ⎟
0 1 0 ⎟⎜ y ⎟
⎟⎟⎜ ⎟
0 0 γ ⎠⎝ z ⎠
F.Ould-Saada
X = Λ −1 X '
(ΛΛ
⎛ ct ⎞ ⎛ γ
⎜ ⎟ ⎜
⎜x ⎟ = ⎜ 0
⎜y ⎟ ⎜ 0
⎜ ⎟ ⎜⎜
⎝ z ⎠ ⎝ +γβ
0 0 +γβ ⎞⎛ ct '⎞
⎟⎜ ⎟
1 0 0 ⎟⎜ x ' ⎟
0 1 0 ⎟⎜ y' ⎟
⎟⎜ ⎟
0 0 γ ⎟⎠⎝ z' ⎠
−1
= I)
9
¡
Goal
§  Expressphysicalpredictions–interactioncrosssections,decayrates–inexplicitly
LorentzInvariant(LI)formvalidinallinertialframes
¡
Contravariant4-vectors
§  Setofquantitiesmeasuredin2inertial
framesandrelatedbyLT
§  (magnitudeofnormal3-vectorx2
isinvariantunderrotations)
¡
x µ = (ct, x, y, z) = (x 0 , x1, x 2 , x 3 )
E
p µ = ( , px , py , pz ) = ( p 0 , p1, p 2 , p3 )
c
x 'µ = Λ µ ν xν Λ µ ν : elements of Λ
Covariant4-vector
§  LIspace-timeinterval:c2t2-r2;LIE-pinvariant:E2/c2-p2=m2
xµ = (ct, −x, −y, −z) = (x0 , x1, x2 , x3 )
x µ xµ = x 0 x0 + x1 x1 + x 2 x2 + x 3 x3 = c 2 t 2 − x 2 − y 2 − z 2
x 'µ = Λ µ ν xν
24/01/16
F.Ould-Saada
Λ µ ν : elements of Λ −1
10
x 'µ = Λ µ ν xν
¡
# ct ' & # γ
% ( %
% −x '( = % 0
% −y' ( % 0
% ( %%
$ −z' ' $ +γβ
Covariant–contravariant
§  Metrictensorgµν(alsoηµν)
¡
Scalarproduct
§  GuaranteedtobeLI
0 0 +γβ &# ct &
(% (
1 0 0 (% −x (
0 1 0 (% −y (
((% (
0 0 γ '$ −z '
xµ = gµν xν
#
%
gµν ≡ %
%
%
$
gµν : metric tensor
1 0 0 0 &
(
0 −1 0 0 (
0 0 −1 0 (
0 0 0 −1 ('
a µ bµ = aµ b µ = gµν a µ b µ
24/01/16
F.Ould-Saada
11
!
!
!
E = γ mc ,
p = γ mv = γ mcβ,
!
§  Contravariantpµ=(E/c,p)
!
" = c = 1 ⇒ E = γ m,
p = γ mβ ;
transforms(asxµ)
p µ = (E / c, px , py , pz ) = (E, px , py , pz )
accordingtoΛ
¡
Energy/momentum
¡
Scalarproduct
§  E/p/mrelation
2
! 2!
v =c p/E
! !
β = p/E
2
§  (Lorentz)-Invariantmass
! E $ !2
!2
µ
2
p pµ = # & − p = E − p
"c%
µ
particle at rest: p = (mc, 0, 0, 0) = (m, 0, 0, 0)
→ p µ pµ = m 2 c 2 = m 2 (LI!)
2
! E $ !2
!2 2
2 2
2
⇒ # & − p = m c ⇒ E − p c = m2c 4
"c%
!
c = 1: E 2 − p 2 = m 2
24/01/16
F.Ould-Saada
12
a µ = (a 0 , a1, a 2 , a 3 )
Scalarproduct
¡
a ⋅ b ≡ a µ bµ ≡ gµν a µ bν = a 0 b0 − a1b1 − a 2 b2 − a 3b3
!
p2 = p ⋅ p = m2 = E 2 − p2
! !
p µ xµ = Et − p ⋅ x
§  a.b
N
¡
Systemofparticles
N particles:
E = ∑ Ei
i=1
! N !
p = ∑ pi
i=1
2
"N % "N !%
µ
p pµ = $ ∑ Ei ' − $ ∑ pi '
# i=1 & # i=1 &
¡
Particledecayaà1+2
a →1+ 2
( p1 + p2 )
µ
2
( p1 + p2 )µ = paµ paµ = ma2
! ! 2
2
ma2 = ( E1 + E2 ) − ( p1 + p2 )
24/01/16
F.Ould-Saada
13
¡
ΣàΣ’(LTinz-direction)
§  t’=γ(t-βz);x’=x;y’=y;z’=γ(z-βt);Inverse:t=γ(t’+βz’);z=γ(z’+βt’)
∂ " ∂z % ∂ " ∂t % ∂
∂
∂
= $ ' + $ ' = γ + βγ
∂z' # ∂z' & ∂z # ∂z' & ∂t
∂z
∂t
∂ " ∂z % ∂ " ∂t % ∂
∂
∂
= $ ' + $ ' = βγ + γ
∂t ' # ∂t ' & ∂z # ∂t ' & ∂t
∂z
∂t
"∂ / ∂t ' % " γ
0 0 +γβ %"∂ / ∂t %
'$
$
' $
'
$∂ / ∂x '' = $ 0 1 0 0 '$∂ / ∂x '
$∂ / ∂y' ' $ 0 0 1 0 '$∂ / ∂y '
''$
$
' $$
'
#∂ / ∂z' & # +γβ 0 0 γ &#∂ / ∂z &
24/01/16
F.Ould-Saada
14
¡
4-derivative
§  transformsascovariant(+signinfrontofβγ!)
"∂
∂
∂
∂ % " ∂ !%
∂
$ , + , + , + ' = $ , +∇ ' = ∂µ = µ
&
∂x
# ∂t ∂x ∂y ∂z & # ∂t
"∂
∂
∂
∂ % " ∂ !% µ
∂
,
−
,
−
,
−
=
,
−
∇
=
∂
=
'
$
' $
&
∂xµ
# ∂t ∂x ∂y ∂z & # ∂t
!2
Laplacian: ∇
covariant!
contravariant
∂2
∂2
∂2
= 2+ 2+ 2
∂x ∂y ∂y
∂2 ! 2
d'Alembertian: = ∂ ∂µ = 2 − ∇
∂t
µ
24/01/16
F.Ould-Saada
15
¡
¡
"
∂ !
E → i! ; p → −i!∇ ⇒ p µ → i!∂µ
∂t
Correspondence
principleinQM
EanBfields
transformation
§
"E ' = E
z
$$ z
itisimpossibleto
# E y' = γ ( E y + β Bx ) considertheelectric $
'
E
fieldandthemagnetic $
% x = γ ( E x − β By )
fieldasindependent
entities.
24/01/16
F.Ould-Saada
" B' = B
z
$$ z
# By' = γ ( By − β E x )
$ '
$% Bx = γ ( Bx + β E y )
16
¡
¡
Reaction1+2à3+4
4-vectorsscalarproducts
§  Lorentzinvariant,canbeevaluatedinanyframe
§  s-channel,t-channel,u-channel
▪  u-relevantwhenidenticalparticlesinfinalstate
¡
2
2
2
2
s = ( p1 + p2 ) = ( p3 + p4 )
t = ( p1 − p3 ) = ( p2 − p4 )
2
u = ( p1 − p4 ) = ( p2 − p3 )
2
Centreofmasssystem
§  s:centerofmassenergy
squared
!
!
p1 = ( E1*, p* ) ; p2 = ( E2*, − p* )
2
s = ( p1 + p2 ) = ( E + E
= ( E + E
*
1
24/01/16
* 2
2
)
*
1
* 2
2
)
!* !* 2
−(p − p )
⇒ s = E1* + E2*
F.Ould-Saada
s + t + u = m12 + m22 + m32 + m42
Show!
17
¡
Mandelstamvariables
http://en.wikipedia.org/wiki/Mandelstam_variables
A + B →C + D
2
s = ( pA + pB ) /c
• t+t+u=
2
t = ( pA − pC ) /c
2
2
u = ( pA − pD ) /c 2
∑m
2
j
j =A,B ,C ,D
• elasticscattering, p,θ in CMSrelative toparticle A,
¡
2
⇒ t = −2 p 2 (1 − cosθ ) /c 2
Rapidityandpseudo-rapidity
€
http://en.wikipedia.org/wiki/Pseudorapidity
1 ⎛ E + pz ⎞
Rapidity :y = ln ⎜
⎟
2 ⎝ E − pz ⎠
⎡ ⎛ θ ⎞⎤
Pseudo-rapidity:η = − ln ⎢tan ⎜ ⎟⎥
⎣ ⎝ 2 ⎠⎦
18
¡
Rapidityy
§  Rapiditydifferencesare
invariantunderboosts
alongbeamdirection
1 " E '+ p'z % 1 " γ ( E − β pz ) + γ ( pz − β E ) %
''
y' = ln $
' = ln $$
2 # E '− p'z & 2 # γ ( E − β pz ) − γ ( pz − β E ) &
1 " (1− β ) ( E + pz ) %
1 " 1− β %
'' = y + ln $
= ln $$
'
2 # (1+ β ) ( E − pz ) &
2 # 1+ β &
⇒ Δy' = Δy
¡
ParticleMassM
§  Highenergyparticles:
M<pz~Ecosθ
1 # 1+ cosθ & 1 # 2 θ &
y ≈ ln %
( = ln % cot (
§  Approximationfory
2 $ 1− cosθ ' 2 $
2'
§  Definepseudo-rapidityη
* θ-
η ≡ − ln ,tan /
+ 2.
R-measure
R=
2
(η1 − η2 ) + (φ1 − φ2 )
2
¡
¡
Protons@LHC
§  Speed=?
Cosmicmuons
§  paradox
24/01/16
F.Ould-Saada
21
AppendixA3.2
¡
Maxwell’sequations
§
¡
!
dq !
; j = ρ v
E,Bfieldsknowingchargeρandcurrentjdensities ρ =
dv
!!
!!
%∇E = 4πρ
%∇B = 0
'
'
!
!
&! !
&
!
!
!
1 ∂B
4π
1 ∂E
j+
'∇ × E = −
'∇ × B =
(
(
c ∂t
c
c ∂t
Forceactingonmovingcharge
!
! ! !
F = qE + qv × B
!
%!
1 ∂A
' E = −∇ϕ −
¡  E,Bfieldexpressedintermsofscalarandvectorpotentials &
c ∂t
!
!
!
' B = ∇ × A (
24/01/16
F.Ould-Saada
23
!2
Laplacian: ∇
∂2
∂2
∂2
= 2+ 2+ 2
∂x ∂y ∂y
∂2 ! 2
d'Alembertian: = ∂ ∂µ = 2 − ∇
∂t
µ
¡
¡
Relativisticnotation:contravariant4-vectors
!
!
!
!
µ
µ
µ
µ
x = (ct, x); p = (E / c, p); j = (cρ, j ); A = (ϕ, A) IntroduceAntisymmetrictensor
F
µυ
∂Aυ ∂A µ
=
−
= ∂µ Aυ − ∂ν A µ
∂xµ ∂xν
!
⎧ ! !
2
!!
!
1
∂
A
4
π
2
⎧∇E = 4πρ
⎪ −∇ A +
=
j
2
2
⎪
⎪
4π µ
!
c ∂t
c
J ⎨
⎨ ! ! 4π ! 1 ∂E ⇒ A µ =
c
1 ∂2ϕ
j+
⎪∇ × B =
⎪ !2
⎩
c
c ∂t
⎪ −∇ ϕ + c 2 ∂t 2 = 4πρ
⎩
èMaxwell’sEquations–compactform
∂µ F
µν
4π ν
=
j
c
F
µν
#
%
%
=%
%
%%
$
0
−E x
−E y
Ex
0
−Bz
Ey
Bz
0
Ez
−By
Bx
−Ez &
(
By (
(
−Bx (
(
0 (
'
24