Transcript Elementary Particle Physics

Lecture 2
Theory concepts and leptons
●
Units
●
Scattering and decays
●
Leptons and the weak force
●
Lepton universality.
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Units
We conventionally measure and calculate in MKS/SI (metre-kilogram-second) units
Quantities expressed in terms of units which are combinations of L, M , T (m,kg,s) with
various dimensions
Eg distance   L ,force   M  L T  , energy  E    M  L T 
2
2
In particle physics it is preferable to use natural units (nu) for which
 c  1 (2.1)
c  1 (pure number - dimensionless)
Implications:
c   L T 
1
(MKS)
c  1 (natural units)   L   T 
E   mc and E   pc    mc
2
2
2

2 2
(MKS)
 E  m and E 2  p 2  m 2 (nu)   E    M    P  (nu)
E  

 E T  (MKS)
;
 1  T    E 
1
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(nu); E   (nu)
2
Conversion
To convert is often trivial:
5  105
5 105
Speed= 5  10 ms 
;
c  1.67  103 c (MKS)  Speed=1.6  103 nu
8
c
3 10
In general, follow a simple rule based on dimensional arguments:
1
5
Every natural unit is based on energy, eg distance expressed in  E 
1
Take a MKS quantity and decompose into units of energy, c and
at various dimensions
Eg cross sectional area   1031m2 (MKS)
Can rewrite:  
Units of energy 
 L

1031
n
a
 3 10 8  6.582  1025 


 
 M  L  T 
2
2
 GeV   c ms-1 
b
,units of c 
 L T 
 M  L T    LT    M  L T  
2
2 1
 n  2, a  2, b  2   
1
2
1
1
a

,units of
GeVs  (MKS)
b

 M  L  T 

2
2
  L     M  L  T 

2
1031
2
 3 10 8  6.582 1025 


 
2
2
1
  L T   M  L  T  
1
1
2
1


2
 2.56 GeV 2 (nu)
Quantity p in natural units  pnu  and MKS  pMKS  :
pnu  c  a
b
pMKS (2.2) ; pMKS  c a
b
pnu (2.3)
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Dimensions and units
Quantity
MKS
Natural
Time
[T]
[E]-1
Distance
[L]
[E]-1
Momentum
[E]=[M][L][T]-1
[E]
Energy
[E]=[M][L]2[T]-2
[E]
Mass
Fermi constant GF
[M]
[M ]1[L]5[T]-2
[E]
[E]-2
[M ][L]2[T]-1
[E]01
[L][T]-1
[E]01
(this lecture)
 (angular
momentum)
c (speed)
Natural units are derived from a base unit of energy.
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Question
e2
What are the dimensions of the fine structure constant  =
? How does the value change
4 0 c
1
if natural units are used ?
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Question
If the natural energy unit is chosen to be 1GeV calculate the
MKS/SI values of the natural length, natural time and natural mass units.
c
1 natural length unit =
 0.197fm
E
1 natural time unit =
 6.58 1025 s
E
E
1 natural mass unit = 2  1.78 1027 kg
c
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Theoretical concepts – amplitude and golden rule
Non-relativistic, spinless treatment in this section unless stated.
Fermi's golden rule: Perturbation theory: transition rate of a state from initial state  to 
W  2 f | H int | i
2
  E '  (2.4)
f | H int | i    *f H int  i dV  M if matrix element/probability amplitude (2.5)
M if must be calculated (Feynman diagrams) or through other approximations.
  E '   density of states (2.6) - next slides.
Target particles
Beam particles
A
B
Scattering
A
C
Decay
D
C
Consider set of beam particles scattering off target particles in a volume.
W  reaction rate per beam and target particle. (2.7)
Unstable particles: W   
1

= decay rate (2.8)
For different decay modes eg: A  B  C  1  or A  C  D   2  ...
Lifetime  =

1
(2.9) tot  1   2  ... (2.10) and branching ratios Bi  1 (2.11)
tot
tot
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Density of states
Consider a particle is scattered by another particle into a volume V and has
momentum between p ' and p ' dp '.
Classically a particle state can be found at any position x and momentum p
QM: a state has a size in position and momentum: phase space
In "2-dimensions" size = xp  h  2 (2.12)
"Volume" of individual state in "6-dimensions" Vs   2 
px
Classical
px
.
.
.
.
.
(2.13)
Quantum
size  2
p
.
3
p
.
L
L
x
x
Particle states in 2 space and momentum dimensions
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Density of states cont.
p
p’
dp’
dp’
p’
2-D (ring)
or 3-D (spherical shell)
1-D (line)
Concentrate on the momentum part of the phase space:
the particle is scattered between p ' and p '  dp '
In 3 momentum dimensions this is a shell:
Volume of shell=4 p '2 dp ' (2.14)
V 4 p '2 dp ' V 4 p '2 dp '
 Total number of states available=

(2.15)
3
Vs
2

 
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Energy and momentum related by :
p '2
E'
2m
 dE '  v ' dp ' (2.16)
 Density of states in energy interval dE ' :   E ' 
dn  E ' 
dE
'

V 4 p '2
v '  2 
3
(2.17)
Density of states says how much "phase space" is available for, eg, a decay.
Decays from higher energy states to the ground state happen faster than
decays with smaller energy difference even if the same force is involved.
Fast
Slow
E
Eg – atomic energy levels
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Question
Show that the density of states  for a particle trapped inside
a 1-Dimensional potential well is the same as would be expected
from our picture of a state occupying a momentum-space volume.
  Aeipx  Be  ipx (2.18)
Boundary conditions:  (0)   ( L) (2.19)  px  
2 n
n  0,1, 2 (2.20)
L
px2 4 2 n 2 2 EmL2
n mL2
E

n 
(2.21) 

(2.22)
2m 2mL2
2 2
E 4n 2
potential well
L
px
Alternatively: the maximum number of states
which can be packed into a volume Lp
Lp
L2 p 2 2mEL2 mEL2
2
n
(2.23)  n 
=
=
this is (2.21) again.
2
2
2
2
2

 2   2 
p
L
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Yukawa potential
Start with generic particle exchange
Model the potential energy for a particle B scattering off
a static particle A:
2

r
R
g e
(2.25) Yukawa potential
4 r
R=range, g  coupling  "charge"
V (r )  -
Eg EM force considering an electron scattering off a positron
g
e
0
(2.26)
Limit R   V (r )  
e2
4 0 r
(2.27) (Coulomb energy)
g2
Define dimensionless strength parameter  X 
(2.28)
4
gW2
1
g2
1
Weak force: W 

(1.39) EM  

(1.24)
4 240
4 137
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Born approximation
Amplitude (lowest order pertubation theory/single particle exchange ) for a
particle with momentum qi to be scattered to a final state
with momentum q f .
M  q    d 3r e
 M q 
iq f  r
V (r ) e  iqi r   d 3r e  iq r V (r ) (2.29)
g 2
q M
2
2
X
q  q f  qi (2.30)
qf
(2.31)
qi
Full relativistic treatment:
g2
M  q   '2
(2.32)
q  M X2
'
q  ( E f  Ei )   q f  qi 
'2
2
2
(2.33)
 weak force is weak compared to the em force
M X  M Z , M W  80 GeV (weak) M X  0 GeV (em)
g2
gW2
q'
q '  M W2
(em) >>
2
2
( weak) unless q '  M W2
2
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Short range force
g2
M (q )  '2
q  M X2
'2
M X  q
'
g2
 M (q )  2 (2.34)
MX
'2
g2
Define M (q )  2  G (2.35)
MX
'2
Interpretation: wavelengths of interacting particles greater than the range: 

1
1 
  R 
 (2.36)
q
M
W 

Interaction can be thought of as a point interaction (zero range)
For the weak force in the zero-range limit
G  GF
Fermi constant  1.166  105 GeV 2 (nu)  1.166  105  c  GeV 2 (MKS) (2.37)
3
gW2
2 gW2
GF = 2 (no spin) (2.38) GF =
(spins taken into account) (2.39)
2
MW
MW
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Summary – putting things together
Used a non-relativistic approach with spins neglected.
(1) Fermi's golden rule: W  2 f | H int | i
2
  E '   interaction rate/decay rate
f | H int | i  M fi is the matrix element/amplitude,   E '   density of states
M fi depends on the nature of the fundamental force under investigations.
(2) Then the Born approximation was used with the Yukawa potential.
(i) Yukawa potential for exchange with massive particle
V (r )  -
2

g e
4 r
r
R
(reduces to Coulomb and weak forces in appropriate limits)
(ii) Calculation of amplitude with Born approximation simple single particle exchange.
M  q    d 3r e
iq f  r
V (r ) e  iqi r   d 3r e  iq r V (r )
q  q f  qi
qf
qi


g2
M q   2
,

relativistic
M
q
'


 
2
2 

q  M X2
q
'

M
X


Weak case M X  M W , EM M X  0  weak suppression.
g 2
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Understanding better the Feynman diagrams
2
When we talked of "probabilities" in lec. 1 we meant M fi .
M fi is represented by the diagrams eg e   e      and e   e       
(a)
Each vertex contributes a factor  to M fi
M afi  a  E ' 
2
Rate (a )
3
Suppression of (a ):
=
 2 (2.40)
b 2
b
'
Rate (b)
M fi   E  
We assumed the same phase space for each interaction implicitly.
1
Also a factor
where M X is the mass of the virtual particle.
2
2
q' MX
 Relevant when comparing, eg electromagnetic and weak interactions.
W
q' M
2

q'
2
e2
Z
 0 for low momenta (2.41)
W e-
Suppressed
MZ 80 GeV
W
e-  e

M0
We can neglect the weak contribution for this interaction.
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Particles in nature
Mass
171000



More information available from the
Review of Particle Physics:
http://pdg.lbl.gov/2008/listings/contents_listings.html
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The charged leptons
• Three types of charged
lepton
• Electron,muon,tau
• e-, e+, m, m , , 
• Charged leptons
interact via the weak
and electromagnetic
forces
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Lepton
Charge (e)
Mass
(GeV)
e-
-1
0.0005
m-
-1
0.105
-
-1
1.8
+ antiparticles
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Heavier leptons
●
●
●
●
Muon m (Stevenson and Street,
1936)
Measurements of energy loss of
cosmic-ray particles.
New particle with mass between eand p (106 MeV/c2)
Interacts like a heavy electron
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Evidence for neutrinos
n
n → p + e- + n
• Electrons produced by beta decay do not all
have the same energy.
– Pauli proposed the existence of an unseen neutral
particle to explain the observed electron
spectrum.
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The lepton family
Spin 1/2
Lepton
(antilepton)
e- (e+)
Charge
(e)
-1 (+1)
Mass
(GeV)
0.0005
ne,(ne)
0
0
m (m+)
nm nm
-1 (+1)
0
0.105
0
 (+)
n n
-1 (+1)
0
1.8
0
Charged leptons interact via the electromagnetic and weak forces.
Neutrinos interact only via the weak force.
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Lepton number conservation
Leptons carry a conserved quantum number.
Flavour specific lepton numbers:
electron lepton number Le , muon lepton number Lm , tau lepton number Lm
(Obviously) for all other particles Le  Lm  L  0
Lepton
Le
Lm
Lt
e-
1
0
0
n
1
0
0
m-
0
1
0
nm
0
1
0
-
0
0
1
n
0
0
1
Antileptons carry the opposite lepton number.
Eg n e , e  , Le  1, Lm  L  0
Except for neutrino oscillations (to come) lepton number has never
been seen to be violated.
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Evidence for more than one neutrino
species
• Neutrinos produced with
muons produce muons : n =
nm
• Neutrinos produced with
electrons produce electrons: n
= ne
• Neutrinos produced with tau
leptons produce tau leptons: n
= n
- Neutrino oscillations (next
lecture modify very slightly
this picture).

X
n
m
m
n
e
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n
e
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Limits on lepton number violation in charged lepton decays
Decay
Violates
Limit on
branching ratio
m   e  e  e
Lm , Le
m   e  
Lm , Le
 1.2 1011
   e  
L , Le
 1.1107
   m  
L , Lm
 6.8  108
   e  m   m 
L , Lm
 2 107
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 1.0  1012
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Interactions of leptons
Neutral
current
Leading order
e  m   e  m 
e  e  e  e
Charged
current
n m  e n e  m



n
  nn  m  n
Higher order
suppressed.
n m  e n e  m 
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Lepton universality
Interactions between, eg e  and n e and m  and n m should give the
same rates if the electron and muon had the same mass
 Lepton universality.
Test with muon and tau decays:(a ) m   e  n e n m , (b)    e  n e  n 
At low energies, can approximate as a zero-range/point force.
 Amplitude M
GF (2.35)
Decay rates  a  M a  a  E ' (2.42) b  M b b  E ' (2.43)
2
2
Masses mm  0.1 GeV, m  2 GeV >> me  5 10 4 GeV , mn  0
 electrons and neutrinos play same role in phase space factors for
both decays. The only difference arises from the masses of the muon and
tau.
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Other dimensional constants involved are muon and tau masses: mm , m
  a  kGF2 mmn (2.44) b  kGF2 m5 (2.45) (k =dimensionless constant)
Dimension of decay rate:  E 
n5
Dim. of mass:  E  Dim. of GF :  E 
  a  kGF2 mm5 (2.46) b  kGF2 m5 (2.47)
Muon,tau lifetimes= m ,    m 
Ba 
2
a
  a m (2.50) ;
m
1
m
(2.48),  
1

(2.49)
Bb  b  (2.51)
Ba , Bb  Branching ratio for decays a and b.
Use measured branching ratios to calculate lifetime ratio:
5
5
  Bb  a Bb kGF mm Bb mm
7




1.328

0.004

10
(2.52)


5
5
 m Ba b Ba kGF m Ba m
Experimentally measured lifetimes:

 1.3227  0.0005  107 (2.53)
m
Pretty good agreement !! Lepton universality holds!!
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Summary
●
Amplitudes, couplings and rates defined.
●
Leptons

Charged leptons and neutrinos

Lepton number conservation

Lepton universality in the weak force
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