1 History. The origin of Nuclear Physics The emergence of particle physics: the Standard Model and Hadrons CERN LHC Highlights 7 Units: [Length, Mass and Energy] (+ Thomson) S-­‐Relativity – Relativistic kinematics (App. B ) (+ Thomson) 2 Relativity and Antiparticles (Schrödinger, Klein-­‐Gordon, Dirac) 4 Interactions and Feynman Diagrams. 3 Space-­‐Time Symmetries and Conservation Laws. Parity
Charge Conjugation
Time Reversal (Angular momentum and Spin) 5 Particle Exchange: Forces and Potentials. Range of forces, The Yukawa potential 6 Observable Quantities Amplitudes, Cross-­‐sections, Decay rates of unstable particles 05/02/14
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1
¡
Interactions §  Electron neutrino collides with neutron to produce electron and proton e− + p → e− + p ; π − + p → π − + p
§  Elastic scattering §  Transfer i-­‐f à particle-­‐antiparticle §  Inelastic scattering ν e + n → e− + p
π +π → p + p
+
−
E min ≥ ( m p + m p ) c
π− + p → n+π0
π− + p → p+π− +π+ +π−
§  Nuclear physics §  Decays a + A → a + A* ; A* → A + γ
n → p + e − + ν e ; (Z,N) →(Z −1,N +1) + e + + ν e
Free neutron
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Bound proton
p →n + e + + ν e
2
2
¡
¡
¡
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
¡
¡
Particles (à) – Antiparticles (ß) Spin-­‐1/2 fermions as solid lines; Photons as wiggly lines 05/02/14
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¡
The Standard Model interaction vertices ¡
a+b à c+d: time ordered processes From Thomson
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¡
Weak interactions
¡
Neutrino electron scattering
ν e + e− → ν e + e−
Strong interactions quark-­‐quark scattering q+q→q+q
gluon-exchange
0
Z exchange
¡
Muon decay
µ − → e− + ν e + ν µ
−
W exchange
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¡
¡
Involving hadrons à Weak nuclear force à strong nuclear force n → p + e− + ν e
±
W exchange
¡
¡
n+ p→n+ p
π exchange
Which reactions are allowed and which are forbidden? Conservation laws: §  Electric charge, Color charge, Lepton number, Spin, … §  Other properties, Parity, Charge conjugation, time reversal, … 05/02/14
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¡
Symmetries and invariance properties of underlying interactions §  play important role in physics §  often lead to universal conservation laws ▪  (Space) translational invariance à momentum conservation ▪  (Time) translational invariance à energy conservation ▪  Rotational invariance à angular momentum conservation §
§
Gauge invariance restricts form of fundamental interactions Discrete symmetries ▪  Parity, Charge conjugation, Time reversal à very useful for classification and when we want to know whether a process is allowed or not … ¡
Symmetries are so important that even broken ones are useful ¡  Electroweak Symmetry Breaking, CP-­‐violation, … 05/02/14
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¡
Same formalism as spin (and isospin – see chapter 3) #
# yp − zp &
% L̂x = ŷp̂z − ẑp̂y
z
y
%
(
! ! !
%
L = r × p = % zpx − xpz ( ⇒ L̂ = % L̂y = ẑp̂x − x̂p̂z
%
(
%% L̂ = x̂p̂ − ŷp̂
% xpy − ypx (
y
x
$
'
$ z
.
&
0
(
0
(
( with /
0
((
0
'
1
&
(
(
L̂2 = L̂x2 + L̂y2 + L̂z2 ⇒ '
(
(
)
&
(
L̂+ = L̂x + iL̂y
(
with '
L̂− = L̂x − iL̂y
(
(
)
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* L̂ , L̂ , = i"L̂
z
+ x y* L̂ , L̂ , = i"L̂
x
+ y z* L̂ , L̂ , = i"L̂
y
+ z x-
" L̂2 , L̂ $ = 0
x%
#
" L̂2 , L̂ $ = 0
y%
#
" L̂2 , L̂ $ = 0
z%
#
" L̂2 , L̂ $ = 0
±%
#
" L̂ , L̂ $ = ± L̂
±
# z ±%
L̂2 = L̂− L̂+ + L̂z + L̂z2
8
¡
Pictorial representation of the 2l+1 states of l=2 L̂z , L̂2 → common eigenstates l, m
L̂z l, m = m l, m
−l ≤ m = −l, −l +1,..., +l −1, +l
L̂2 l, m = l(l +1) l, m
L̂+ l, m = l(l +1) − m(+1) l, m +1
L̂− l, m = l(l +1) − m(−1) l, m −1
¡
Coupling of 2 am / spins §
Clebsh-­‐Gordon coefficients From Thomson
! ! !
%l = l + l
1
2
'
l1, m1 ⊕ l2 , m2 → l, m & l1 − l2 ≤ l ≤ l1 + l2 '
( m = m1 + m2
l, m =
∑ C(m , m ;l, m) l , m
1
2
1
1
l2 , m2
m1,m2
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¡
Coupling of ½X½, 1X½ §  Spin multiplicity: 2l+1 §  Symmetric, anti-­‐symmetric and mixed configurations 05/02/14
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1
2
1
× : 2 ⊗ 2 = 3⊕ 1
2
1
1× : 3 ⊗ 2 = 4 ⊕ 2
2
1 1
1
2
2
× × : 2 ⊗ 2 ⊗ 2 = 4S ⊕ 2 MS ⊕2 MA
2
10
¡
Particles ¡
§  ½ integer spin (1/2, 3/2, …), Dirac-­‐
statistics à fermions §  Integer spin (1,2,…), Bose-­‐Einstein statistics à bosons ¡
¡
§
§  I ψ(1,2)=±ψ(1,2) è ψ(2,1)=±ψ(1,2) Total WF product of 2 functions ▪  describes orbital motion of particle wrt to the other à spherical harmonics Ylm(θ,φ) à (-­‐1)L §  Spin function β §  I(1,2)à(2,1) Eigen values: I2=1 ; I=±1 §  Bosons must be symmetric §  Fermions must be anti-­‐symmetric §  Spatial function α Statistics fix symmetry properties of WF for a pair of identical particles wrt their exchange §  I ψ(1,2)=ψ(2,1) è I2 ψ(1,2)=ψ(1,2) Under exchange, WF for 2 identical ¡
▪  Symmetric if the 2 spins are parallel ▪  Anti-­‐symmetric if anti-­‐parallel §  Identical bosons must have both α and β sym or anti-­‐symmetric §  Identical fermions must have α sym and β anti-­‐symmetric or vice-­‐
versa From Braibant
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¡
!
!
r !P̂!
→ −r
Behavior of a state under a spatial reflection §  P reverses spatial coordinates r and p §  Application on wave function ¡
Parity applied twice §  Eigenvalue equation ¡
¡
! P̂
! ! P̂ !
t !!
→ t ' = t ; p !!
→ − p; J ! !
→J
!
!
P̂ ψ (r,t) ≡ P ψ (−r,t)
P̂
!
!
!
P̂ 2ψ (r,t) = PP̂ψ (−r,t) = P 2ψ (r,t)⇒ P = ±1
P̂ψ = Pψ = ±1ψ
A particle at rest (p=o) is eigenstate of parity with eigenvalue P=±1 (intrinsic parity) Examples of WF with §  Positive parity: cos x §  Negative parity: sin x §  Undefined parity: sin x + cos x 05/02/14
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¡
¡
¡
Parity – multiplicative QN §  P(ψ1ψ2)=P1.P2 SI, EM invariant under Parity WI violates Parity (maximally) 12
!
ψ nlm (r) = Rnl (r)Yl m (θ , φ )
n, l, m : principal, orbital, magnetic QNs
x = r sin θ cos φ ! r → r
#
y = r sin θ sin φ " θ → π − θ
# φ → π +φ
z = r cosθ
$
Yl m : spherical harmonics , Pl m : Legendre polynomials
Yl m (θ , φ ) =
(2l +1)(l − m)! m
Pl (cosθ )eimφ
4π (l + m)!
Y00 =
1
4π
Y10 =
3
cosθ
4π
Y1±1 = ∓
3
sin θ e ∓iφ 8π
"$
P̂Yl m (θ , φ ) = (−1)l Yl m (θ , φ )
#
P̂ Pl m (cosθ ) = (−1)l+m Pl m (cosθ )$%
!
!
!
⇒ P̂ ψlmn (r) = Pψlmn (−r) = P(−1)l ψlmn (r)
P̂ eimφ = (−1)m eimφ
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¡
Invariance under parity of Dirac equation à P(e e ) = −1
¡
¡
Convention: P=+1 for leptons and quarks and P=-­‐1 for anti-­‐fermions Parity of photon: -­‐1 Intrinsic parities of hadrons follow structure in terms of quarks + orbital l between constituent quarks: l
¡
+ −
P( f f ) = −1
P(−1)
¡
¡
¡
¡
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Meson = quark-­‐antiquark: P=(-­‐1)(-­‐1)l=(-­‐1)l+1 Pion (l=0): P=-­‐1 Proton (uud,l=0): P=+1 neutron (udd,l=0): P=+1 F. Ould-Saada
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¡
¡
Operation changing particle à antiparticle Multiplicative QN conserved in SI, EM – not in WI ¡
Distinguish cases where §  (a) Particle = antiparticle: γ, π0 à
▪
#
%
n
Ĉ nγ = (−1) nγ %
$ ⇒ Cπ 0 = +1
0
%
π → γγ
%
Cγ = −1
&
Ĉ γ = − γ
Ĉ a, ψ a = Ca a, ψ a
Ĉ π 0 = + π 0
Ca = ±1 : C − parity
/ γγγ C − invariance
π0 →
EM fields produced by moving electric charges, which change sign under C, so Cγ=-­‐1 §  (b) Particle diff. antiparticle: π+àπ-­‐, nà anti-n
π 0 → γγγ
−8
<
3×10
π 0 → γγ
– only linear combinations are relevant Ĉ b, ψ b = Cb b, ψ b
Ĉ π + = π −
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¡
(b) mesons (spin-­‐less): π+à π-­‐ §  Interchanging position of particles ! reverses Ĉ π +π − ; L = (−1)L π +π − ; L
relative position in WF à (-­‐1)L ¡
(b) fermions – anti-­‐fermions §  Interchanging positions à (-­‐1)L Ĉ f f ; L, S
= (−1)L+S f f ; L, S
§  Interchange fermion-­‐antifermion à (-­‐1) §  Interchanging spins à (-­‐1)S+1 ↑1↑ 2
1
↑ 1 ↓ 2+ ↓ 1 ↑ 2
2
↓1↓ 2
)
1
↑ 1 ↓ 2− ↓ 1 ↑ 2
2
)
(
¡
(b) mesons with spin: 0,1,2 S z = +1
(
Sz = 0
S z = −1
Sz = 0
⎫
⎪⎪
⎬ S = 1
⎪
⎪⎭
S =0
Ĉ π +π − ; L, S = (−1)L+S m + m − ; L, S
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¡
(Classical) Poisson’s equation: ! ! !
)
1 !
∇⋅ E ( r ,t) = ρ ( r ,t) : P − invariant +
! !
! !
ε0
* ⇒ E ( r ,t) →− E (− r ,t)
§  Electric field E ! P !+
vs charge ρ !P !
! P
!
r →− r ⇒ ρ ( r ,t) →ρ(−r ,t) ; ∇ →− ∇,
density !
)
!
!
! !
∂A
! ! i( k!r! −Et )
¡  EM field A E = (−∇φ ) − ‚
A( r ,t) = Nε ( k )e
+
∂
t
*Pγ = −1
§  Parity: Pγ=-­‐1; C-­‐
! ! P ! !
! ! P ! ! +
!P !
parity: Cγ=-­‐1 r →− r ⇒ A( r ,t) →Pγ A(−r ,t) ; E ( r ,t) →Pγ E (−r ,t) ,
¡  π0=q-­‐qbar C
! ! C ! !
)
q →− q ⇒ A( r ,t) →Cγ A( r ,t)
+
§  Pπ0=-­‐1 *Cγ = −1
C
C
! ! C ! !
§  JPC =0-­‐+ !
!
q →− q ⇒ E ( r ,t) →− E ( r ,t); φ ( r ,t) →− φ ( r ,t) +,
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¡
¡
Symmetry of SI & EM à Invariance under transformation but violated in WI No conserved quantum number associated to time reversal (neglect WI), unlike P and C !
! T ! ! T
! ! T
t !!
→ t ' = −t ; r ! !
→ r ; p !!
→ − p; J ! !
→−J
2
! 2 T
!
2
' !
If system invariant under T, probability ψ (r,t) ! !
→ ψ (r,t) = ψ (r,−t)
!
! !
!
∂ψ (r, t)
i"
= H (r, p)ψ (r, t): SE not invariant under T
∂t
!!
!!
( p⋅r −Et )
( p⋅r −Et )
i
−i
!
!
!
Ψ(r,t) = e " ⇒ T ψ (r,t) = ψ ' (r,t) = e "
T
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If we introduce operator T̂ by analogy with P̂
!
!
!
!
: ψ (r,t) !T!
→ψ ' (r,t) = ψ * (r,−t) ≡ T̂ ψ (r,t)
!
! !
!
∂ψ * (r, t)
Then SE invariant! − i"
= H (r, p)ψ * (r, t)
∂t
!
! ! * !
∂ψ * (r, −t)
t → −t ⇒ i"
= H (r, p)ψ (r, −t)
Same form as for Ψ
∂t
!
ψ (r,t) = e
!!
( p⋅r −Et )
i
"
!
* !
T̂ ψ (r,t) = ψ (r,−t) = e
¡
!!
( p⋅r +Et )
−i
"
=e
!!
(− p⋅r −Et )
i
"
Time-­‐reversed WF describes a particle with momentum -­‐p 05/02/14
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19
¡
Quantum Mechanics operators corresponding to physical observables must be Linear Oˆ (α1ψ1 + α 2ψ 2 ) = α1 Oˆ ψ1 + α 2 Oˆ ψ 2
§  to ensure superposition principle holds ¡
and Hermitian ( ) (
∫ dx (Oˆ ψ ) ψ = ∫ dxψ (Oˆ ψ )
*
§  eigenvalues (observed values ) are real 1
2
*
1
)
2
!
!
!
T
Ψ(r ,t) #
#
→ Ψ* ( r , − t) ≡ TˆΨ(r ,t)
Tˆ (α1ψ1 + α 2ψ 2 ) = α *1 Tˆψ€1 + α *2 Tˆψ 2 ≠ α1 Tˆψ1 + α 2 Tˆψ 2
( ) ( )
∫ dx€(Tˆψ ) ψ ≠ ∫ dxψ (Tˆψ )
*
1
¡
¡
€
2
*
1
( )
( )
2
Time reversal operator does not correspond to a physical observable No observable conserved as a consequence of T invariance 05/02/14
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20
¡
¡
But T-­‐invariance leads to a relation between process and its time-­‐
reversed Reactions and time-­‐reversed counter parts are related !
!
!
!
a ( pa , ma ) + b( pb , mb ) → c( pc , mc ) + d ( pd , md )
⎫
!
!
!
!
⎬
c(− pc ,−mc ) + d (− pd ,−md ) → a (− pa ,−ma ) + b(− pb ,−mb )⎭
¡
Reactions proceed with equal rates if WI neglected §  mi: magnetic quantum number 05/02/14
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Time reversal & Parity ¡
Combination of T & P §  Same rate of reactions if P and T invariance hold (neglect WI!) !
!
!
!
a ( pa , ma ) + b ( pb , mb ) → c ( pc , mc ) + d ( pd , md )
⇓T
!
!
!
!
c (− pc , −mc ) + d (− pd , −md ) → a (− pa , −ma ) + b (− pb , −mb )
⇓ P̂
!
!
!
!
c ( pc , −mc ) + d ( pd , −md ) → a ( pa , −ma ) + ( pb , −mb )
§  If averaged over all possible spin projections ¡
Principle of detailed balance §  Confirmed experimentally in a variety of Strong and EM processes mi = −si , −si +1,... si (i = a, b, c, d)
!
!
!
!
i ≡ a ( pa ) + b ( pb ) ↔ c ( pc ) + d ( pd ) ≡ f
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¡
If P or T is conserved, Hamiltonian of interaction must not contain terms that change sign after the operation §  Magnetic dipole moments allowed §  Not electric DM! µΕn≠0 would imply èP and T violated §  Longitudinal polarization only through WI From Braibant
05/02/14
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23
¡
Although C & P violated in weak interactions (100%) §  CP violated in some weak processes (~0.1 %) §  T violated in some weak processes (~0.1%) §  there is a general result ¡
CPT theorem: “Any Quantum Theory that (i) obeys the postulates of Special Relativity, (ii) admits a state with minimum energy and (iii) respects causality is invariant under CPT” §  Causality of physical events requires that the fields obey commutation or anti-­‐
commutation relations, implying the correct statistics according to the spin of particles: §  Fermi-­‐Dirac statistics for fermions and Bose-­‐Einstein statistics for bosons ¡
CPT invariance predicts that particles and antiparticles must have exactly same masses and lifetimes, opposite magnetic moments, … 05/02/14
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24
¡
Consequences of CPT invariance §  Particle & antiparticle have same mass, lifetime, opposite magnetic moments, … §  Particle in state |a> =|m,τ, …> [CPT, H ] = 0!#
"⇒
2
(CPT ) = 1 #$
qp
mp
qp
mp
τ µ+
τ µ−
2
a H a = a H (CPT ) a = a CPT H CPT a
a H a = a H a ⇒ ma = ma
< 0.99999999991± 00000000009 ,
< 1.00002 ± 0.00008 ,
"#m + − m − $%
e
e
< 8⋅10 −9
me
"µ + − µ − $
# e
e %
< (−0.5 ± 2.1) ⋅10 −12
µe
From Braibant
05/02/14
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25
¡  05/02/14
F. Ould-Saada
From Braibant
26
¡
Charge conserved at each vertex of a Feynman graph §  Other quantum numbers also conserved – depending on interaction (strong, weak, EM) §  During exchange, Energy E and momentum p cannot be conserved simultaneously ¡
Consider §  A+BàA+B mediated by X-­‐particle exchange ¡
In center of mass system of particle A !
!
!
A M A c , 0 → A ( E A , pA c ) + X ( E X , − pA c )
(
2
)
#E ! &
!
PA = % A , pA ( ⇒ PA2 = E A2 / c 2 − pA2 = M A2 c 2
$ c
'
!
p = pA ⇒ E A = p 2 c 2 + M A2 c 4 ; E X = p 2 c 2 + M X2 c 4
&( → 2 pc for p → ∞
2
ΔE = E X + E A − M A c 2 '
⇒
ΔE
≥
M
c
for any p
X
2
() → M X c for p → 0
¡
06/02/14
Momentum conserved è E not conserved F. Ould-Saada
27
¡
But according to Heisenberg’s uncertainty principle §  Energy can be violated during a short laps of time ΔE ≠ 0 within τ ≤
!
!
⇒r≤R≡
≡ range
ΔE
mX c
EM: mγ = 0 ⇒ Rγ = ∞
WI: mW ,Z ≈ 80, 90GeV ⇒ R W ,Z ≈ 2 ×10 −18 m
SI : mπ ≈ 100MeV ⇒ R ≈ (1− 2) ×10 −15 m
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28
¡
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 05/02/14
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29
¡
Yukawa (1935) formulated a theory of strong interactions between nucleons inside nuclei §  In analogy with QED – exchange of massless γ, potential V~1/r §  Strong forces – maximum range ~1fm, exchange boson must be heavy ~150 MeV §  so change QED potential such that it it quickly vanishes with distance due to exchange of massive particles ¡
Hunting after a meson (me<mπ<mp) opened §  3 charge states to accommodate pp, pn, nn interactions 05/02/14
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30
¡
Feynman diagrams à process probability by a set of mathematical rules (Feynman rules), derived from underlying quantum field theory N
=
Observation
Experiment
Fermi’s Golden Rule
L
Luminosity
Accelerator
. σ
Cross-section
Theory
2
M (i1, i2 → f1, f2 ,..., fn ) %
d3pf (
dσ =
'∏
*(2π )4 δ 4
3
4 ( p1 ⋅ p2 )2 − m12 ⋅ m22 '& n (2π ) 2E f *)
(∑ p −∑ p ) S
i
f
M: matrix element calculable with Feynman diagrams
05/02/14
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31
¡  More about Quantum Field Theory, Calculation of amplitudes of Feynman graphs, cross sections, … §  In follow-­‐up courses §  FYS4170 (Quantum filed theory), FYS4560 (Elementary particle physics) 06/02/14
F. Ould-Saada
32
¡
¡
Use non-­‐relativistic (NR) QM to derive propagator of X Assume small coupling g 2 << 4 π!c
¡  Interaction is a small perturbation on a “free” particle (plane wave) ¡
¡
Lowest order (LO) perturbation theory (PT) ¡  M for scattering € B+Aà B+A by potential V Potential theory – A: static source, B: scatters without loss of energy (Ei=Ef,) ¡  Transition amplitude iàf – Yukawa potential !
!
M if = ∫ φi*V (r )φi d 3r ; φi, f = e
! !
ipi, f ⋅r
"
g 2 e−r/R
V (r) = −
4π r
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F. Ould-Saada
!
! !
M (q) = ∫ d 3r V (r )e
! ! !
q ⋅ r =| q | r cosθ
!!
iq⋅r
!
" ! !
q ≡ pi − p f
!
d 3r = r 2 sin θ dθ dr dφ
!2
−g 2 " 2
⇒ M (q ) = ! 2
| q | +M X2 c 2
do the calculation problem 1.12 33
¡
Amplitude derived in rest frame of A ¡
¡
Recoil neglected ¡
!2
−g 2 " 2
M (q ) = ! 2
| q | +M X2 c 2
At high energies, ¡
A recoil non negligible ! !
q 2 ≡ (E f − Ei )2 / c 2 − (q f − qi )2
explicitly Lorentz invariant (Ei=Ef)
g2!2
M (q ) = 2
q − M X2 c 2
2
propagator ¡
This is an amplitude for exchange of 1 particle ¡
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Higher orders might be important F. Ould-Saada
34
g2!2
M (q ) = 2
q − M X2 c 2
¡
Amplitude ¡
Zero-­‐range approximation: amplitude reduces to a constant 2
"
R = ! / M X c << λ ⇒ q 2 << M X2 c 2 ⇒ M(q 2 ) = −G
2
G
1% g (
4 πα X
=
≡
dim:1/ E 2 ]
'
*
[
2
3
2
(!c)
!c & M X c ) ( M X c 2 )
Resulting point interaction between A and B characterized by single dimensional coupling constant G, not g and Mx separately €
¡
Fermi “effective” coupling constant GF
−5
−2
=
1.166
×
10
GeV
(!c) 3
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F. Ould-Saada
[dim1/ E ]
2
35
α
¡
¡
Lowest order in PT ∝α2
Higher Orders in PT §  More complicated, multi-­‐
particle exchange diagrams ∝α 4
¡
Number of vertices à order n §  Amplitude proportional to factor ( α)
§  Probability proportional to factor αn
€
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F. Ould-Saada
n
α ≈ 1 / 137 ⇒ α 4 << α 2
36
¡
Next step, relate amplitude to measurable quantities – cross-­‐section for scattering experiment §  b+t à c+d §
A beam (of cross section area S) – density (cm-­‐3) of particles nb hits target (nt particles per unit volume, thickness t, density ρ, (N particles in total) §  Beam particle velocity in rest frame of target: vi §  Beam flux: J = nb vi [cm −3 ][cm ⋅ s −1 ]
§  Beam intensity: I = JS [cm −2 ⋅ s −1 ][cm 2 ]
§  Luminosity L ≡ JN
[cm
−2
s −1 ] contains dependencies on densities and geometries of beam & target §  Reaction rate Wr = Lσ r [cm −2 s −1 ][cm 2 ]
§  Cross section for reaction r : σr (dimension of area, Lorentz invariant) §  Target composition: atomic mass MA 05/02/14
F. Ould-Saada
Wr = JNσ r = N σ r I / S = I σ r nt t
nt =
ρ NA
⇒ Wr = I σ r ( ρ t)N A / M A
MA
37
Rate = L . σ
¡
Total cross section: sum over all reactions r Luminosity
σ tot ≡ ∑σ r r
dσ r (θ , ϕ )
dWr ≡ JN
dΩ
dΩ
¡
¡
Differential cross-­‐ section Measured rate for the particles to be emitted into an element of solid angle dΩ in the direction (θ,φ) 2π
1
σ r = ∫ dφ ∫ d cosθ
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F. Ould-Saada
0
−1
dσ r (θ , φ )
dΩ
38
¡
Cross sections: §  Formulae in terms of amplitudes describing scattering of non-­‐relativistic spin-­‐less particle from a potential §  Consider a single beam particle interacting with single target particle – whole confined in volume V -­‐ Incident flux J and luminosity L=JN, rate dWr à ¡
In QM, transition rate given in PT by Born approximation (Golden rule) §
§
ρ(Ef) =dN/dEf : density of states factor nber of possible states with E between Ef and Ef+d Ef J = n b v i = v i /V
N = 1 ⇒ dW r =
2π
dW r =
!
€
ψi =
1
e
V
v i dσr (θ , φ )
dΩ
V
dΩ
" * " 2
∫ d r ψ f V (r )ψ i ρ(E f )
3
& q" i . r" )
(i
+
' ! *
ψf =
1
e
V
" "
& q f .r )
(( i
++
' ! *
§  In terms of Mif 05/02/14
2
2π
dWr = € 2 M if ρ (E f )
!V
F. Ould-Saada
39
¡
!
*
3!
ψ
V
(
r
)
ψ
d
r
∫
f
i
V
depends on density of states *
3!
dn is number of accessible states in the ρ = ψ *ψ ⇒
ψ
ψ
d
r=
∫V
Transition rate ¡
¡
M if =
energy range EàE+dE ¡  Fermi’s golden rule (δ insures energy conservation) ¡
ψ = Ae
! !
$ p. r −Et '
&i
)
%
" (
a
a
0
0
∫ ∫ ∫
a
*
ψ
ψ dx dy dz = 1
0
⇒ A 2 = 1 / a3 = 1 / V
Particle in a box of volume a3 Normalisation àboundary conditions ¡  à components of momentum quantised ¡
3
ψ (x + a, y, z) = ψ (x, y, z),...!
%
(
"
2π !
2
π
!
(2π !)3
3
⇒ d p = dpx dpy dpz = '
" ⇒ ( px , py , pz ) = (nx , ny , nz )
* =
&
)
a
a
V
eipx x = eipx ( x+a),...
#
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F. Ould-Saada
40
¡
Numbers of states dn within pàp+dp Momentum space volume with thickness dp divided by average volume occupied by a single state ¡  dn is number of accessible states in the energy range EàE+dE ¡  Fermi’s golden rule (δ insures energy conservation) ¡
!
(2π !)3
2
2
d p = dpx dpy dpz = p dpsin θ dθ dφ = p dpdΩ =
V
p 2 dpdΩ
dn dn dp dn 1
dn =
⇒
ρ
(E)
=
=
=
3
(2π !) / V
dE dp dE dp v
3
V p2
ρ (E) =
dΩ
(2π !)3 v
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F. Ould-Saada
41
¡
Non-­‐relativistically Similar result relativistically (see below) ¡
"
dq f m 1
= =
$$
q 2f
"2 2
dE f q f v f
dσ
1
=
M(q )
#⇒
€
2 4
vi dσ r (θ , φ ) $ dΩ 4π ! vi v f
dWr =
dΩ$
%
V
dΩ
¡
dq f
V
2
ρ(E f ) =
dΩ
3 qf
(2π!) dE f
Relativistic kinematics for a+b!c+d ¡  In cms qf=|qc|=|qd|
Vf,i: relative velocities c-­‐d and a-­‐b E f = Ec + Ed = q 2f c 2 + mc2 c 4 + q 2f c 2 + md2 c 4 ⇒
!
dq
! pc 2 !
!
1
v=
; pc = − pd in cms ⇒ f =
E
dE f v f
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F. Ould-Saada
"1
dE f
1 %
= q f c2 $ + '
dq f
# Ec E d &
42
¡
Generalisation of cross-­‐
section formula ¡
¡
include spins à spin multiplicity factors In general: ¡
¡
q 2f
dσ
1
"2 2
=
M(
q
)
2 4
dΩ 4 π ! v iv f
un-­‐polarized initial particles à average over initial spins and sum over final spins gi = (2sa +1)(2sb +1) ; g f = (2sc +1)(2sd +1)
€
2
gf
q 2f
dσ
=
M fi
2 4
dΩ 4 π ! v iv f
M fi
2
"2 2
≡ M(q )
Amplitude is pin-­‐average of the squared matrix element €
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F. Ould-Saada
!2
−g 2 " 2
M (q ) = ! 2
| q | +M X2 c 2
g2!2
M (q ) = 2
q − M X2 c 2
2
43
Lifetime (at rest) τ , or natural decay width Γ= h/τ ¡
¡
¡
¡
¡
measure of rate of the decay reaction Partial width Γf for specific final state f Total decay width and Branching ratio Γ = ∑ Γf
Bf ≡ Γf / Γ
f
Breit-­‐Wigner energy distribution (no spin) à resonant state at W=M ¡
include spins à spin multiplicity factors N f (W ) ∝
¡
¡
Γf
(W − M )2 c4 +Γ2 / 4
M: mass of decaying state W: invariant mass of decay products FWHM
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44
¡
Cross-­‐section for reaction iàf via resonance ¡
E: total energy of the system σ fi (W ) ∝
ΓiΓf
2 2
( E − Mc )
+Γ2 /4
¡  €Resonant state with spin j ¡  Initial particle spin (s1, s2) ¡  In practice kinematical and angular ΓiΓ f
π !2
2 j +1
σ
=
fi
momentum effects distort formula from qi2 ( 2s1 +1) ( 2s2 +1) E − Mc 2 2 +Γ 2 / 4
its perfectly symmetric shape (
¡
)
Example of resonance formation à 05/02/14
F. Ould-Saada
45
§
π-­‐ p cross section §
§
§
§
§
§
centre of mass energy 1.2-­‐2.4 GeV 2 (+2) enhancements on top of non-­‐resonant contributions Resonance widths Γ~100MeV Interactions times τ~10-­‐23s à Strong interaction à Consistent with time taken for a relativistic pion to transit the dimension of a proton 05/02/14
F. Ould-Saada
46
¡  Pages 29-­‐30 §  1.1,1.2,1.3, 1.7, 1.8, 1.9, 1.10, 1.11, 1.12,1.13,1.14,1.15 05/02/14
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47