Chapter 4 ¡
¡
¡
¡
¡
¡
¡
4.1 The Interaction Between Electric Charges §
§
4.1.1 The EM Coupling Constan
4.1.2 The Quantum Theory of Electromagnetism §
§
§
4.2.1 The Schr¨odinger Equation
4.2.2 Klein–Gordon Equation 4.2.3 Dirac Equation
§
§
4.5.1 The Cross-­‐Section 4.5.2 Particle Decay and Lifetime §
§
§
§
§
4.7.1 Rutherford Scattering 4.7.2 The e+e-­‐àm+ m – Process 4.7.3 Elastic Scattering e+e-­‐àe+e-­‐ (Bhabha Scattering) 4.7.4 e+e-­‐àggAnnihilation 4.7.5 Some QED Checks 4.2 Some Quantum Mechanics Concepts 4.3 Transition Probabilities in Perturbation Theory 4.4 The Bosonic Propagator 4.5 Cross-­‐Sections and Lifetime: Theory and Experiment 4.6 Feynman Diagrams 4.7 A Few Examples of Electromagnetic Processes ¡
Electromagnetism §  Unification of electrostatic and magnetic force §  Coulomb’s law: !
q1q2 !
F = K 2 r̂
r
§  Magnetic field generated by electric charges in motion – a relativistic effect ¡
Relativity: E,B interrelatedà electromagnetic interaction §  Force acting on moving charge: !
! ! !
F = qE + qv × B
¡
¡
¡
Interactions are due to exchange of particles Feynman diagrams +mathematical rules and associated techniques to calculate quantum mechanical probabilities for given reactions to occur EM – photon exchange: electric charge conserved at each vertex e +e →e +e
−
−
−
−
+
forbidden vertex: e → e + γ
−
e+ + e+ → e+ + e+
Time arrow ¡
¡
06/02/15 Particles (à) – Antiparticles (ß) Spin-­‐1/2 fermions as solid lines; Photons as wiggly lines F. Ould-­‐Saada 4 ¡
EM interaction – exchange of photon §  But electron at rest cannot emit photon – energy conservation ¡
!
ΔE
! !c
ΔE ≥ ≅
Δt Δr
α !c
ΔE ≈ V = i
r
dV 1
F≈
≈ 2
dr r
Δt ≥
§  Heisenberg’s uncertainty principle – emission and absorption of photon within time Δt §  Alternatively, E/p conserved at vertex but relation E2=m2c4+p2c2 does not hold for “virtual” particle (propagating Δr during Δt) ▪  αi characterizes intensity of EM interaction mediated by massless photon ¡
QM §  Interaction is a sequence of creation and destruction of field by the electrons §  EM coupling constant αem αi !c
e2
e2
V=
= K ⇒ α i = α em = K
r
r
!c
e2
1
α em =
=
4πε 0 !c 137.1
e2
1
α em = =
= 7.294 ⋅10 −3 cgs
!c 137.1
α em = e 2
¡
SI
! = c =1
§  αem <<1 – perturbation theory applicable for EM processes §  Order of approximation given by number of vertices in Feynman diagram ¡
First “order” §  a) Photon couples to electron with amplitude ~√αem (vertex) §  a) Transition probability= amplitude2 ~ ¡
αem Higher “orders” b,c,d,e,f ¡
Lowest order in PT ¡
Higher Orders in PT α
∝α2
§  More complicated, multi-­‐
particle exchange diagrams ∝α 4
¡
Number of vertices à order n §  Amplitude proportional to factor ( α)
§  Probability proportional to factor αn
€
06/02/15 F. Ould-­‐Saada n
α ≈ 1 / 137 ⇒ α 4 << α 2
8 ¡
EM field quantised – photon=quantum E = hν ; p = hν / c ; s = 1!
¡
Quantum ElectroDynamics – QED, Feynman, Schwinger, Tomonaga §  Interaction of charged Dirac fields (fermions) with photon ¡
QED is renormalisable – §  redefinition of mass and electric charge such that they correspond to experimentally measured values at any scale, avoiding infinities in predictions ¡
QED is a gauge theory, based on a gauge symmetry – §  invariance of EM interaction on transformations conserving electric charge – local gauge invariance §  Gauge theories are renormalisable – predictive ¡
Correspondence principle §  Energy, momentum à operators ¡
Schrödinger eq. §  Time evolution of wave function (wf) §  Steady state – time independent: E eigenvalues ¡
Probability density ρ=|ψ|2 and flux j §  continuity relation ! *
!
i! * !
j =−
ψ ∇ψ − ψ∇ψ
2m
1 ∂ρ ! !
+ ∇j = 0 ⇒ ∂υ j υ = 0
c ∂t
(
)
!
!
∂ψ
E → i!
, p → −i"∇
∂t
!
∂
µ
µ
p → i!
= i!∂ , xµ = (ct, − x)
∂xµ
p2
∂ψ ! 2 2
E=
⇒ i"
+
∇ ψ=0
2m
∂t 2m
∂ψ
H ψ = i!
∂t
H ψ = Eψ
(steady state)
ψ = Ne
!!
(ip⋅r −iEt )/"
! p! 2
ρ = N ; j = N
m
2
¡
Classical physics: §  E,p, … of particle dynamical variables represented by time-­‐dependent quantities ¡
QM (Schrödinger picture) §  Wavefunction (WF) postulated to contain all info about state, including E,p, … §  Time-­‐independent operators act on time-­‐dependent WF §  Observable quantities A correspond to action of operator on WF §  Result of measurement of A are eigenvalues of operator ψ (x, t) = Ne−i(Et−px )/!
Âψ = aψ
a : eigenvalue, real
*
 : Hermitian ⇒ ∫ ψ Âψ 2 dt = ∫ $% Âψ1 &' ψ 2 dt
*
1
06/02/15 F. Ould-­‐Saada 11 ¡
What is the wave equation for matter-­‐waves? ω = 2πν ; v = λν
§  Wave formula ψ (x, t) = Ne−iω (t−x/v) = Ne−2 π i(vt−x/λ )
E = hν = 2π !ν "
$
§  Wave – particle duality −i(Et−px )/!
h 2π ! # ⇒ ψ (x, t) = Ne
λ= =
p
p $%
2
2
§  Wave equation – (equivalent of ∂
φ 1 ∂ φ
) ∂x
2
=
v 2 ∂t 2
2
2
!
∂
ψ
∂ψ
2
2
$
−
+V ψ = i!
∂ψ
p
∂ψ
E
2
= − 2 ψ ; = −i ψ
2m ∂x
∂t
&&
2
∂x
!
∂t
!
%⇒
∂ψ
2
2
Ĥ
ψ
=
i!
= Eψ
p
pψ
&
∂t
E = H = T +V =
+V → Eψ =
+V ψ &
'
"
" −iEt/!
2m
2m
ψ ( x, t) = φ ( x)e
06/02/15 F. Ould-­‐Saada 12 !
!
* !
¡  ρ dV: probability of finding particle density:ρ ( x, t) = ψ ( x, t)ψ ( x, t) !
3!
in volume element dV probability: ρ ( x, t)d x
! ! 1 ∂ρ
∇⋅ j +
= 0 ⇒ ∂µ j µ = 0
c ∂t
!2 2
∂ψ $ ψ * ×
−
∇ ψ = i!
&&
! *
!
i! * !
2m
∂t
⇒ j =−
ψ ∇ψ − ψ∇ψ
%
2
*
2m
!
∂ψ & ψ ×
2 *
−
∇ ψ = −i!
2m
∂t &'
(
§  Plane wave describes free particle (E,p) with ρ=|N|2 and j=p|N|2 /m 06/02/15 F. Ould-­‐Saada ψ (x, t) = Ne
!!
−i(Et− px )/!
)
!
"
!
2 p
⇒j=N
≡ nv
m
13 ¡
High Energy Physics (HEP) requires: §  Quantum theory §  Theory of special relativity §  Particle – antiparticle symmetry Quantum Field Theory See also Appendix A.4 and Thomson, Modern particle physics and 06/02/15 F. Ould-­‐Saada 14 Equations of motion ¡
Quantum mechanics §  Schrödinger equation (linear in time, not in space) è +
p2
E = T +V =
+V
-2m
,
!
#
∂ & #!
∂ &% E →ih ( ; % p →− ih∇ = −ih ! (
$
∂t ' $
∂x ' -.
! !
3! !
∂ψ ( x, t) 0 h 2 ∂2
⇒ i"
= 2−
+V 5ψ ( x, t)
2
∂t
1 2m ∂x
4
§  Plane wave describes free particle of energy-­‐momentum (E,p) with probability density ρ=|N|2 and current j=p|N|2 /m 06/02/15 F. Ould-­‐Saada 15 Equations of motion ¡
Relativistic version §  Klein-­‐Gordon equation (quadratic in time and space derivatives) §  Expressed in Lorentz Invariant form §  Spin-­‐0 fields 2
E =± p +m
¡
!
%
E 2 = m2c 4 + p2c2
' * 1 ∂2 " 2 - m 2 c 2
" & ⇒ , 2 2 − ∇ /φ + 2 φ = 0
∂ "
!
E →ih ; p →− ih∇ ' + c ∂t
.
(
∂t
2
Inconsistency §  Negative solutions lead to negative probability densities 06/02/15 F. Ould-­‐Saada *
,
,
,,
∂
p µ →ih ∂µ = ih
⇒+
∂xµ
,
,
,
,-
$ ∂2
m2c2 '
&&
+ 2 ))φ = 0
µ
! (
% ∂xµ∂x
(∂ ∂
µ
2
+
m
)φ = 0
µ
m2c2
φ+ 2 φ =0
!
! !
ρ ∝ E, j ∝ p ⇒ j µ ∝ p µ
16 ¡
Solutions §  m=0: propagation of EM wave §  φ interpreted as potential U in coordinate space, or as wave amplitude of associated photons §  Static potential U(r) ¡
!
m2c2
1 ∂ % 2 ∂U (
2
φ (r, t) → U(r) ⇒ ∇ U = 2 U = 2 ' r
*
!
r ∂r & ∂r )
g 1 −r/R
Solution: ⇒ U(r) =
e
4π r
!
Range: R =
; g: "intensity strength"
mc
Analogy EM – “Nuclear force” §  QED infinite range, mγ=0 §  NF: short range, massive EM : R = ∞, m = 0, ∇ 2U = 0 ⇒ U =
exchange particle §  Yukawa: pion … R ≈ 1.2fm ⇒ mc
06/02/15 F. Ould-­‐Saada 2
=
Q
r
!c
≈ 150MeV ≈ mπ
R
17 ¡
Dirac’s goal: §  Linear equation in both space and time derivatives §  Reproducing KG equation – satisfying E2=p2+m2 §  Relativistic Quantum Mechanics + Spin & Magnetic moment of spin ½ fermions 06/02/15 F. Ould-­‐Saada 18 ¡
Start with Schrödinger equation ¡
Choose parameters αi
and β such that
§  Equation is linear §  And solutions are also solutions to Klein-­‐
Gordon equation ∂ψ
i!
= Hψ
∂t
! !
! !
2
H = c α. p + β mc = −ihc α.∇ + β mc 2
!
α = (α 1, α 2, α 3) and β ⇒ to be fixed
&
∂ψ #
∂
∂
∂
i
= % −iα x − iα y − iα z + β m (ψ
∂t $
∂x
∂y
∂z
'
square + compare to KG ⇒
{αi, α j} =2δij ; {αi, β } = 0 ; β 2 = 1
{αi, α j} = αiα j + α jαi
⇒ α i , β ⇒ 4 × 4 matrices
06/02/15 F. Ould-­‐Saada 19 ¡
¡
γ i ≡ βα i ; γ 0 ≡ β "$
µ
⇒ (i
γ
∂µ −m)ψ =0
#
µ
0 1 2
3
γ = (γ , γ , γ , γ ) $%
Introduce Dirac γ 4x4 matrices and their properties (γ 0 )2 = I
invariance γ 5 ≡i γ 0γ 1γ 2γ 3 with {γ µ , γ 5 } = 0; γ 5γ 5 = I
#
%
§  Anti-­‐commutation i 2
(γ ) = −I
$ ⇒ {γ µ , γ ν } ≡ γ µγ ν + γ ν γ µ = 2g µν
and hermiticity %
µ ν
ν µ
relations γ γ = −γ γ for µ ≠ υ &
Ensure Lorentz γ i↑ =−γ i ; γ 0↑ = γ 0
"I 0%
" 0 σ i% 5 "0 I %
i
γ =$
' ; γ =$ i
' ; γ =$
'
#0 − I &
# −σ 0 &
# I 0&
" 0 1%
"0 − i%
"1 0 %
"1
2
1
3
σ =$ ' ;σ =$
' ;σ =$
' ; 1= $
#1 0 &
#i 0 &
# 0 −1&
#0
F. Ould-­‐Saada 0
06/02/15 0%
'
1&
20 4-­‐component wave-­‐functions – Spinors ¡
§  2 spins (up and down) of e-­‐ §  2 spins (up and down) of e+ §  (“e-­‐ with negative energy”) "
$
γ0 =$
$
$
#
"
$
2
γ =$
$
$
#
1
0
0
0
0
0
0
−i
"
0 0 0 %
'
$
1 0 0 '
; γ 1 = $
$
0 −1 0 '
$
0 0 −1 '&
#
"
0 0 −i %
'
$
0 i 0 '
3
; γ = $
$
i 0 0 '
$
0 0 0 '&
#
06/02/15 0 0 0
0 0 1
0 −1 0
−1 0 0
0
0
−1
0
0
0
0
1
1
0
0
0
1 0
0 −1
0 0
0 0
F. Ould-­‐Saada %
'
'
'
'
&
%
'
'
'
'
&
(iγ µ∂µ −m)ψ =0
"∂ / ∂t %
$
'
∂
/
∂x
'
∂µ = $
$∂ / ∂y '
$
'
#∂ / ∂z &
iγ 0
!ψ1 $
# &
ψ2 &
#
ψ
#ψ 3 &
# &
"ψ 4 %
∂ψ
∂ψ
∂ψ
∂ψ
+ iγ 1
+ iγ 2
+ iγ 3
− mψ = 0
∂t
∂x
∂y
∂z
21 ¡
Wavefunctions 4-­‐component spinors §  Complex conjugate à Hermitian conjugates
ψ † = (ψ1*, ψ 2*, ψ3*, ψ 4* )
†
2
2
2
2
ψ = ψ1 + ψ 2 + ψ3 + ψ 4 !
†!
Current: j = ψ γψ !
µ
† 0 µ
Covariant current: j = ( ρ, j ) = ψ γ γ ψ
Density: ρ = ψ
ψ ≡ ψ †γ 0 ⇒ j µ = ψγ µψ ∂ρ ! !
+ ∇ ⋅ j = 0 ⇒ ∂µ j µ = 0
∂t
06/02/15 F. Ould-­‐Saada !ψ1 $
# &
ψ2 &
#
ψ
#ψ 3 &
# &
"ψ 4 %
ψ ≡ ψ †γ 0 = (ψ * )T γ 0 = (ψ1*, ψ 2*, −ψ3*, −ψ 4* )
22 ¡
Dirac interpretation negative energy solutions as holes in the vacuum that correspond to antiparticle states §  Vacuum corresponds to the state where all negative energy states are occupied ¡
In Dirac “sea” picture, Pauli exclusion principle prevents e-­‐ (E>0) from falling into states (E<0) §  Holes correspond to positive energy antiparticles (e+) §  Limitation of interpretation: bosons (Pauli principle does not hold) have also antiparticles 06/02/15 F. Ould-­‐Saada 23 ¡
E<0 solutions interpreted as negative energy particles propagating backwards in time §  Physical E>0 antiparticles propagating forwards in time §  Time dependence of WF: exp(-­‐iEt) unchanged under simultaneous transformation (Eà-­‐E & tà-­‐t) §  The 2 pictures mathematically equivalent è e
[−iEt ]
≡e
[−i(−E )(−t )]
¡
Electron-­‐positron annihilation ¡
e+ and e-­‐ can be created and destroyed just like γ’s, the quanta of the EM field 06/02/15 F. Ould-­‐Saada 24 ¡
Positron discovered in 1933 by Carl Anderson in a cloud chamber photograph of cosmic rays ¡
Antiproton discovered in 1955 by Chamberlain et al., required high energy accelerators Emilio Segrè and Owen Chamberlain
were awarded the
Nobel Prize in Physics 1959 for the
discovery of the antiproton
Show that
p+ p→ p+ p+ p+ p
.
E thr
p = 7 m p = 6.6 GeV
C.D. Anderson,
Physical Review 43,
491 (1933).
Carl Anderson won
the 1936 Nobel Prize
for Physics for this
discovery.
¡  Initial state system §  Stationary wf ψ with defined Em ψ (t < 0) = φ m e
H 0φ m = Emφ m
M nk = ∫ φk*V φn d τ , ( dτ = d 3 x = dv)
t ≥ 0 ⇒ V = g0U ⇒ (φ m → φn )
ψ (t) = ∑ cn (t) φn e
i
− En t
!
n=0
H ψ = ( H 0 +V ) ψ = i!
06/02/15 i
-/
− ( En −Ek )t
" dck % ∞
!
. ⇒ i! $
' = ∑ cn (t)M nk e
*
#
& n=0
dt
/
φk φn = 0 (n ≠ k)0
×ψ k*
M nk = ∫ φk*V φn d τ , ( dτ = d 3 x = dv)
i
− Em t
!
∞
i
i
∞
− En t
− En t
" dcn %
!
!
⇒ i!∑$
= ∑Vcn (t) φn e
' φ n e
#
&
dt
n=0
n=0
∞
∂ψ
∂t
F. Ould-­‐Saada PT ⇒ cm (t) ≅ 1, cn (t) ≈ 0 (n ≠ m) ⇒ M nk ≈ 0 (n ≠ m)
i
'
− ( Ek −Em )t *
i
t
− ( Ek −Em )t
1
1− e !
)
,
!
ck (t) = ∫ M km e
dt = M km )
i! 0
Ek − Em ,
)(
,+
' '
t **
i
) sin )( Ek − Em ) ,,
− ( Ek −Em )t
2! +,
) (
= 2iM km e !
Ek − Em
)
,
)(
,+
26 W=
1
1 +∞
2
2 dN
c
(t)
→
ck (t)
dE
∑
∫
i
−∞
t i
t
dE
2
2
+∞ & dN ) sin x
2
W = M km ∫ (
+ 2 dx
−∞ '
!
dE * x
& dN )
2 dN
2π
(
+ ≈ ctant (−π , +π )/
W
=
M
if /
' dE *
!
dE f
.⇒
2
+∞ sin x
∫ −∞ x 2 dx = π ! //0 M if = ∫ φi*V φ f dτ 06/02/15 F. Ould-­‐Saada 27 ¡
! !
ipi ⋅r /!
Free particle (E) φi = e
scattered by potential → φ
f
=e
! !
ip f ⋅r /!
g 1 −r/R
!
U(r) =
e
; R=
4π r
mc
! !
! !
−ipi ⋅r /! ip f ⋅r /!
*
i
§  Transition amplitude ¡
U
M if = ∫ φ V (r)φ f d τ = g0 ∫ U(r)e
e
dτ Bosonic propagator !!
! !
!
iq⋅r /!
M if = g0 ∫ U(r)e
d τ , q = p f − pi
§  Central potential !
! iq⋅! r! /"
! !
f (q) = g0 ∫ U(r )e
d τ , q ⋅ r = qr cosθ ; dτ = r 2 dϕ sin θ dθ dr
∫
π
0
sin θ eiqr cosθ
e
(
2sin qr
dθ =
; sin z =
qr
iz
− e−iz )
2i
iqr
∞
∞ −r/R e
!
sin qr 2
− e−iqr
f (q) = f (q) = 4π g0 ∫ U(r)
r dr = g0 g ∫ e
dr
0
0
qr
2iq
g0 g
f (q) = 2
q + m2
06/02/15 1
¡  Propagator è 2
2
q
+
m
F. Ould-­‐Saada 28 W=
¡
a+bàc+d particles, velocity vi relative to target §  Φ=navi :flux [cm-­‐2s-­‐1] of incident particles: ▪  na =1 , Φ=vi |ψ|2 à ψ [cm-­‐3/2]
§  W=σ Φ=σ navi [s-­‐1]: probability that beam particle interacts with a target per second §  dN: #states of a particle ▪  Cartesian ▪  Spherical: in volume V and momentum range (p,p+dp) ¡
Scattering on Heavy particle M §  Scattered particle E,p,m dx dy dz dpx dpy dpz
!3
dΩ
2
dN =
p
dp
3
(2π !)
dN
dΩ
2 dp
=
g
p
f
dE0 (2π )3
dE0
dN =
§  na: density [cm-­‐3] of incident ▪  Spin factor: gf=(2sc+1) (2sd+1) 2 dN
2π
M if
!
dE f
dΩ = sin θ dθ dφ
p
dp
E
dp E 1
E = γ mc 2 ; p = γ mv, c = 1 ⇒
= =
dE0 p v
E0 = M + p 2 + m 2 ⇒ dE0 =
1
2 2 gf
2
dσ (a + b → c + d) =
f
(q
)
p
dΩ
4π 2
vi v
¡
Parameter: lifetime τ
§  ∞ (e,p), 109s y (some nuclei), 900s (n), 10-­‐23s (hadrons) ¡
Decay law: §  Decay probability: P(Δt)=Δt/τ §  Number of decaying particles: N P(Δt) dN
N
= − ⇒ N(t) = N 0 e−t/τ
dt
τ
tN(t)dt
∫
τ≡
∫ N(t)dt
§  Initial number N0 decreases by: -­‐Δt (dN/dt) ¡
NΔt
dN
= −Δt
τ
dt
N P(Δt) =
Transition probability W vs Lifetime §  Decay = transition from a state to another of smaller mass ▪  According to conservation laws W=
1
τ
¡
Decay fraction Γi/Γvs branching ratio BR §  Decay mode – characteristic τ §  Muon and pion decay modes Γi
∑i Γ = 1
Γ /Γ
Wi = i
τ
¡
Feynman diagrams §  offer a simple visualization of the interaction mechanism between particles and provide rules for calculating the transition amplitude. §  Fermions, bosons, basic vertex ¡
e-­‐e-­‐àe-­‐e-­‐ ¡
e+e-­‐àe+e-­‐ à ¡
Bremsstrahlung ¡
Higher order corrections ¡
Calculation of transition probability and of the cross-­‐
sections is based on the use of the perturbation expansion 1
2 2 gf
2
dσ (a + b → c + d) =
f
(q
)
p
dΩ
2
4π
vi v
¡
f (q) =
g0 g
q2 + m2
Bosonic propagator §  ~ exchange of a boson between two particles connected by two vertices and with a probability proportional to (q2+m2c2)-­‐1 §  Photon: m=0 à 1/q2 à cross section ~1/q4 ¡
Vertex factor §  g0, g §  Each vertex contributes with √αQED to Matrix Element ¡
Rutherford scattering ¡
e+e-­‐àµ+µ-­‐ ¡
e+e-­‐àe+e-­‐ ¡
e+e-­‐àγγ Coulomb scattering neglected ¡
Momentum & energy conservation !
!
!
mα vi = mα v f + mt vt !
#
"⇒
mα vi2 = mα v 2f + mt vt2 $
#
! !
mt2 2
m v = mα v +
vt + 2mt ( v f ⋅ vt )
mα
2
α i
2
f
( m +
! !
vt2 *1− t - = 2 v f ⋅ vt
) mα ,
t = e ⇒ mt = me << mα ⇒ no large angle scattering (las)
t = Au ⇒ mt >> mα ⇒ LAS
06/02/15 F. Ould-­‐Saada 36 ¡
Rutherford scattering §  Limit to elastic scattering of point-­‐like, spinless particles ▪  Kinetic energy of incident particle conserved, momentum changes ▪  Spherical symmetric potential §  Particles with an impact parameter between (b,b+db) are scattered within the angular region (θ, θ-­‐dθ) corresponding to a solid angle dΩ ▪  N0: incident pqrticles per cm2 s ; N: incident particles elastically scattered per sec in (θ, θ-­‐dθ) "
$
dσ
b db
⇒
(
θ
)
=
−
⋅
#
dσ
dσ
dσ (θ ) =
(θ )dΩ =
2π sin θ dθ $
dΩ
sin θ dθ
%
dΩ
dΩ
dN = 2π N 0 b db = N 0 dσ
Coulomb scattering: M>>m Angular momentum conservation : mvb = mr 2
dφ
dt
(5)
Scattering symmetric about y - axis →along y
p i = −mv sin(θ /2) = − p f = p ⇒ Δp = 2mv sin(θ /2)
b: impact parameter €
py due to Coulomb force: [ F=dp/dt ]
zZe 2
⇒ Δp = ∫ 2 cos φ dt
−∞ r
+∞
! 1 $ +( π −θ )/2
2mvsin(θ / 2) = zZe # & ∫ cos φ dφ
" bv % −( π −θ )/2
2
06/02/15 F. Ould-­‐Saada "θ %
zZe 2 1
Impact parameter: b =
⋅ cot $ '
#2&
2 Ec
38 "θ %
zZe 2 1
b=
⋅ cot $ '
#2&
2 Ec
d(cot x) = −(sin x)−2 dx
*
db
zZe 2 1
1
2
=−
,
2%
2
"
,
dσ
zZe
1
dθ
2 4Ec sin θ / 2
(θ ) = $
+⇒
'
# 4Ec & sin 4 "$ θ %'
dσ
b db
, dΩ
(θ ) = −
⋅
,
#2&
dΩ
sin θ dθ
¡
Rutherford scattering §  Classical Formula el
σ tot
=
§  Elastic scattering between 2 spinless particles ¡
06/02/15 F. Ould-­‐Saada ¡
∫
# zZe 2 &
dσ
(θ ) dΩ = 8π %
(
dΩ
4E
$ c'
2
d(sin (θ / 2 )
∫ 0 sin3 (θ / 2)
1
Integrate up to θ>θ0>0 to avoid Coulomb divergence due to 1/r2 for infinite range force Very large r, very small deflection 39 ¡
¡
¡
Previous (classical) formula adequate for α-­‐scattering !
g0 g
f
(q)
=
#
zZe 2
2
2
For e-­‐ (z=-­‐1) –Nucleus (Z) q +m
" ⇒ f (q) = 4π 2
q
scattering, quantum mechanics m = 0 ; g0 = ze ; g / 4π = Ze#
$
and relativity necessary ! ! 2
q 2 = ( p − p') ≅ 2p 2 (1− cosθ ) = 4p 2 sin(θ / 2)
Rutherford formula §  Scattering angle θ small §  Valid for any electric charge, provided target much heavier than projectile §  Case of ep scattering §  To be extended to include finite mass, spin and finite proton size 06/02/15 F. Ould-­‐Saada 2
dσ
1
2 2 p
=
f (q )
dΩ 4π 2
vi v
2
, m2
p
dσ )
zZe 2
2
..
≅ m ⇒
= ++ 4π
2
2
vi v
dΩ * 4 p sin (θ / 2 ) - ( 2π )2
2
) dσ ,
p2
z 2 Z 2e4
1
≅ Ec ⇒ +
∝
. =
* dΩ -R ( 4Ec )2 sin 4 (θ / 2 )
2m
Ec2
40 ¡
(a) EM (γ) process §  2 vertices, 1/q2 à ¡
(c) angular distribution §  From total spin of (e+,e-­‐) ▪  J=0 – isotropy (1) ▪  J=1 – cos2θ term §  QED – symmetric distribution ¡
(b) pure weak (Z0) process §  Relevant at high energies §  Asymmetric angular distribution q 2 = p 2 = s = (2Ee )2
¡
(!c)2
σ ~α
s
2
dσ α em
=
1+ cos2 θ )
(
dΩ 4s
2
4πα em
(!c)2
86.8
σ=
=
nb
2
3s
s[GeV ]
2
em
¡
(b,a) EM (γ) process §  (b,c) S-­‐channel similar to muon pairs §  (a) elastic scattering dominates at small θ ¡
Angular distribution §  (b) From total spin of (e+,e-­‐) ▪  J=0 – isotropy (1) ▪  J=1 – cos2θ term 2
dσ 4πα em
(!c)2
2
cos
θ /2
§  (a) QED – asymmetric distribution peaked at small angle 2 =
4 2
dq
qc
§  Angular dependence 1/θ4 ~1/q4 à ¡  (c) pure weak (Z0) process §  Relevant at high energies §  Similar to muon pair production ¡
¡
(a) Pure EM process ¡
1/s dependence of cross section Data in excellent agreement with QED calculation à ¡
2
2πα em
(!c)2 ! s $
σT =
ln # 2 &
s
" me %
Charged particle with spin necessarily has a magnetic moment ¡
!
! q !
q !
s
e!
µ=
s=g
s = gµ 0 with magneton µ 0 =
mc
2mc
"
2mc
e!
Bohr magneton (electron): µ B =
= 5.7884 ×10 −15 MeV / G
2mec
Nuclear magneton (proton): µ N =
¡
¡
e!
= 3.1525 ×10 −18 MeV / G
2m p c
Information about the structure of a particle is contained in the g factor – gyromagnetic factor Dirac equation predicts for point-­‐like, spin-­‐1/2 particle of mass m (e,µ) 06/02/15 F. Ould-­‐Saada g = 2 ⇒ µD =
e!
mc
!
!
e !
e !
µ p = 2.79 s and µ n = −1.91 s
mp
mn
44 ¡
QED: relativistic quantum field theory describing the interaction of electrically charged particles via photons. §  Higher order radiative corrections à g or a=(g-­‐2)/2 2
3
4
"α %
"α %
"α %
µe
1α
= 1+
− 0.32848 $ ' +1.1765$ ' − 0.8 $ ' = 1.001159652307(11)
#π &
#π &
#π &
µB
2π
Experimental value:
= 1.001159652193(10)
⇒
α −1 = 137.035 999 084 ± 0.000 000 051
§  Anomalous magnetic moment leads to the most accurate determination of the fine structure constant α 06/02/15 F. Ould-­‐Saada 45 ¡
Limit MA large §  regard B as scattered by static potential created by A ¡
Klein-­‐Gordon equation (spin neglected and X with spin-­‐0) !
!
!
∂ 2φ ( r , t )
2 2 2
2 4
−"
=
−
"
c
∇
φ
(
r
,
t
)
+
M
c
φ
(
r
, t)
X
∂t 2
2
§  Static solution obeys: ¡
!
M X2 c 2 !
∇ φ (r , t ) =
φ (r )
2
"
2
Electrostatic potential §  Coulomb potential ¡
Massive X §  Yukawa Potential §  Range R, coupling constant g M X = 0, R = ∞ ⇒V (r) = −eφ (r) = −
α
r
!
g 2 e−r/R
e−r/R
M X ≠ 0, R ≡
⇒V (r) = −
= −α X
M Xc
4π r
r
§  MX very large, zero-­‐range interaction 06/02/15 F. Ould-­‐Saada 47