Transcript L7-Symmetries_and_conservation_laws

Symmetries and
conservation laws
•Introduction
•Parity
•Charge conjugation
•Time reversal
•Isospin
•Hadron quantum numbers
•G parity
•Quark diagrams
1
Symmetry and Group Theory
Symmetries – linked closely to dynamics of system. Symmetries –
the most fundamental explanation for the way things behave (laws
of physics).
Symmetry – described by group theory. A group = collection of
elements with specific interrelationships defined by the group
transformations. Demand: repeated transformations between
elements of the group equivalent to another group transformation
from the initial to the final elements. Example: equilateral triangle.
R+ - clockwise rotation through 120°
R- - counterclockwise rotation through 120°
A
Ra, Rb, Rc – flipping about axis Aa, Bb, Cc, respectively
I – doing nothing
B
C
a
Rotation clockwise through 240°  rotation
counterclockwise by 120° : R+2=R- .
2
Properties of a group
• closure – if Ri and Rj in set  RiRj=Rk also in set
• identity – there is element I so that IRi=RiI=Ri
• inverse – for every Ri there is an inverse Ri-1: RiRi-1=Ri-1Ri=I
• associativity – Ri(RjRk)=(RiRj)Rk
The group elements need not commute. If all elements do commute, the
group is called Abelian. Translation in space and time – Abelian.
Rotation – non Abelian.
Groups can be finite (like in
triangle example) or infinite.
There are continuous groups
(rotation) and discrete
groups (finite groups).
3
Nöther’s theorem
If the Lagrangian governing the phenomenon does not change under
the group transformation  a conserved quantity exists.
Nöther’s theorem: to every symmetry of a Lagrangian, there corresponds
a quantity which is conserved by its dynamics, and vice versa.
Symmetry
Conservation law
Translation in time
Energy
Translation in space
Momentum
Rotation
Angular momentum
Gauge transformation
Electric charge
4
Special unitary transformations
Most of the groups of interest in physics are groups of matrices. In
particle physics the most common groups are of the type U(n): the
collection of all unitary n x n matrices.
A unitary matrix:
U
1

U
*
(inverse = transpose conjugate)
If the group is restricted to all unitary matrices
with determinant 1, the group is called :
SU ( n)
(“special unitary”)
5
Geometrical Symmetries
Translation
Reflection
(parity)
Rotation
R
continuous
R
R
RR
RR
RR
RR
RR
continuous
discrete
6
Parity operator
Parity operator P reflects the system through origin of coor system 
left-handed coor system  right-handed one. Parity operation  mirror
reflection followed by a rotation through 180°:
Pxi  xi'   xi
xi – position vector of particle i.
A system is said to be invariant under parity operation if the Hamiltonian
remains unchanged by the transformation:
H ( x1' , x2' ,...)  H ( x1 ,  x2 ,...)  H ( x1 , x2 ,...)
Consider single particle wavefunction
 ( x, t ) : P ( x, t )  Pa ( x, t )
where a identifies particle, Pa – constant phase factor. Two
successive parity operations leave system unchanged 
P  ( x, t )   ( x, t )  Pa  1
2
7
Intrinsic Parity
Eigenfunction of momentum given as
 p ( x, t )  ei ( p xEt ) 
P p ( x, t )  Pa p ( x, t )  Pa  p ( x, t )
Thus particle at rest (p = 0) is eigenstate of parity operator with
eigenvalue Pa – which is called the intrinsic parity of particle a.
Particle with definite orbital angular momentum l, also eigenstate of P:
 nlm ( x)  Rnl (r )Yl ( ,  )
m
r,, spherical polar coordinates
Yl m ( ,  )  Yl m (   ,    )  (1)l Yl m ( ,  )
P nlm ( x)  Pa nlm ( x)  Pa (1)l nlm ( x)
eigenvalue:
Pa (1)l
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Parity conservation
If Hamiltonian H is invariant under parity transformation 
 P, H   0
In strong and electromagnetic interactions, parity is conserved.
Thus parity of the final state, Pf, equals parity of initial sate Pi.
Example: atomic bound states, parity is a good quantum number.
Atomic states s,d,g,… have even parity, while p,f,h,… have odd
parity. Electric dipole transitions between states have the
selection rule l = 1  parity of atomic state changes. However,
since a  is emitted in the transition (P = -1), parity of the
system is conserved.
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Parity of Particles
• Intrinsic parity – Fermions
– Consider electrons and positrons represented by a
wave function .
( )
(
P Y x , t = Pe ± Y - x , t
)
– Dirac equation is satisfied by a wave function
representing both electrons and positrons
 intrinsic parity related:
Pe  Pe   1
– Strong and EM reactions always produce e+e- pairs.
– Arbitrarily have to set one =1 and the other = -1.
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Parity of Fermions
• Assign positive parity state to particles,
negative to antiparticles:
Pe   P   P   1
Pe   P   P   1
• Make same assumption about quarks to
be consistent:
Pd  Pu  Ps  Pc  Pb  Pt  1
Pd  Pu  Ps  Pc  Pb  Pt  1
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Example: parapositronium
The ground state of an e+e- bound system is called parapositronim.
The initial state has 0 orbital momentum (s state). The system can
decay into a state of two photons:
e e  

Parity of initial state:
Parity of final state:

Pi  Pe Pe  1
Pf  P (1)
2
l
Conclusion: relative angular momentum between the two ’s has to
be odd, to conserve parity. This was confirmed experimentally.
12
Parity of Mesons
Mesons are quark antiquark pairs:
PM  Pq Pq   1   1
L
L 1
Since the  meson in the lowest energy
bound state of a q-qbar system, it has L=0,
and thus expected to have negative parity.
This was confirmed from completely
different considerations (before the quark
model was introduced).
13
Parity of Baryons
Baryons contain three quarks:
PB  Pq1 Pq2 Pq3  1 (1)   1
L12
q1
L3
L12
q2
q3
L3
L12  L3
The lowest energy
state baryons, like
proton, neutron, , C,
have L12+L3=0, and
therefore positive
parities.
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Naming convention
P a · b = a · b Þ Pscalar = +1
P a · b ´ c = - a · b ´ c Þ Ppseudoscalar = -1
P a =- a
Þ Pvector = -1
P a ´ b = a ´ b Þ Paxialvector = +1
The  has spin 0 and negative parity  called pseudoscalar meson.
The  has spin 1 and negative parity  called vector meson.
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Charge conjugation
Charge conjugation operator, C, replaces all particles by their
antiparticles in the same state, so that momenta, positions, spins,
etc., are unchanged. C changes charge, and other additive quantum
numbers like Baryon number or Lepton number.  only those
particles with additive quantum numbers = 0, like  or 0, are
eigenstates of C. Denote these states by :
C   C  ; C 2   C2     C  1
C are called C-parities, and if
C , H   0
C-parity is conserved.
This is true for strong and electromagnetic interactions.
The electromagnetic field is produced by a moving charge, which
changes sign under charge conjugation  C = -1.
C is multiplicative  C(0  ) = +1.
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C of particle-antiparticle system
Particle-antiparticle system has additive quantum numbers = 0  is
eigenstate of C. However, since the operation changes particle into
antiparticle, this means also a spatial interchange and thus will depend on
the relative orbital momentum L and total spin S of the system:
C aa  (1) L  S aa
is the lowest energy state of u-ubar
or d-dbar, which are fermionantifermion pairs with L=0 and S=0,
therefore C(0 )=+1, as before.
0
A hydrogen-like electron-positron
bound state is called positronium. The
total spin of the electron-positron
state can be 0 or 1. L is restricted to
be ≤ n-1. The following J, P, C
values are for n≤2:
n=1
state
J
P
C
1S
0 -1
1
0
3S
n=2
1S
1
1 -1 -1
0
0 -1
3S
1
1
1 -1 -1
1P
1
1
1 -1
3P
0
0
1
1
3P
1
1
1
1
3P
2
2
1
1
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C of particle-antiparticle system
p
x2, 2
p
x1, 1
C | px11 px2 2   | px11 px2 2 
Need to interchange space and spin
coordinates to make an eigenvalue equation.
Interchange of spin: (-1)s+1.
Interchange of space: (-1)l from orbital
momentum and another (-1) from the
opposite intrinsic parities of the proton and
antiproton.
C aa  (1) L  S aa
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Time reversal
Time reversal makes the transformation t  t’ = -t. Invariance of
time reversal means that the probability of finding a particle at
position x and time t is the same as finding it at position x and time –t:
T
 ( x, t ) 
  '( x, t )   ( x, t )
2
2
2
The operation of time reversal changes the sign of momentum p
and of the direction of the total angular momentum J:
T
p 
 p'  p
T
J 
 J '  J
Time reversal on a+b  c+d  c+d  a+b. However, p and J
opposite to that of the original reaction. Operation with parity
operator on the reversed reaction, all momenta will reverse sign.
Average over spin  back to original configuration.
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Principle of detailed balance
If time reversal and parity operation are both invariants of a given
reaction, the rate of the original one and of the time reversed one is
the same, provided the spin states of the initial states are averaged
and that of the final state are summed over.
This principle, called principle of detailed balance, was used to
determine that the spin of the  meson is 0, using the reactions:
p  p   d

And the time reversed one
 d  p p
A study of the ratio of the differential cross sections of both
reactions, which is a function of the spin of the pion, led to the
experimental measurement of S ≈ 0.
20
Isospin
Isospin invariance follows from the fact that strong interactions are
independent of quark type, and so do not distinguish up quarks from
down quarks. Furthermore, the masses of the up and down quarks are
small compared to their energy in the proton or neutron, and thus protons
and neutrons have close to equal masses.
As far as strong interactions are concerned, protons and neutrons
behave identically. Isospin is the invariance that relates strong
interaction processes or states that differ only by replacing some
number of protons by equal number of neutrons.
Isospin is a compound word which suggests two ideas – isotopes and spin.
Isospin is only very loosely related to the concept of an isotope and has
nothing whatever to do with spin! However, the mathematics of isospin is
similar to the mathematics of spin for spin ½ particles, and this is where
the spin part of the name arose.
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Isospin (2)
• Heisenberg (1932) suggested that the similarity of the neutron
and proton (M~938 MeV, spin ½) indicated that they were two
states of the same particle – the nucleon
• The formalism was the same as for spin-1/2, thus the name
“isospin” (self spin)
• Isospin was found to be conserved in strong interactions, not in
weak & EM
– Only the magnitude I matters, not particular Iz
• Evidence
– Equivalence of nn, np, pp interactions
– Equivalence of “mirror” nuclei
 p
 
n
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Mirror nuclei
23
Isospin in the Quark Model
• Isospin can be understood in the quark model of hadrons
p  uud
• The only difference between
n and p is interchange du
n  udd
• If d & u had the same mass, this would be a true symmetry
– No feature except charge would distinguish p/n
• But it’s not perfect
– We know that Mn-Mp = 1.3 MeV
– Coulomb effects should be stronger for p than n
• So we conclude that Mq=Md-Mu~2-3 MeV
– And Mq/MN~.2% so a small effect!
24
Isospin in other hadrons
• We now know the fundamental doublet for isospin is
1
1  u  I3   2
I
 
2  d  I3   12
• The comparable doublet for anti-particles is
Bigger charge
has larger I3
1
1  d  I3   2
I
 
2  u  I 3   12
• We can combine two isospin-1/2 states just like spin
2  2  1 3
I3  1
   ud

1
uu  d d
2
I 3  1    du
I3  0
0 

I  0 I3  0  
1
(uu  d d )
2
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Note on  wavefunction
Following the Condon-Shortly
convention:
(arrow denotes C-operation)
Isospin shift operators:
Example:
I  ( I , I 3 )  I ( I  1)  I 3 ( I 3  1)( I , I 3  1)
I  d  u , I u  d , I u  I  d  0, I u  d , I  d  u
s

u

d
26
Note on  wavefunction
 wavefunction:
I    I  (du )  uu  dd  2 0
dd  uu ud  0  0  ud

 2ud  2 
2
2
dd  uu ud  ud
I   I 

0
2
2
I  0  I 
27
Note on  wavefunction
 wavefunction:
Symmetry determined under interchanges:
ud  du   
ud   du  

dd  uu  uu  dd   0
d u ; d u
 Triplet antisymmetric
singlet - symmetric
28
Isospin in the N-N system
• Consider a system of 2 nucleons
– Nucleon is p or n
• We again combine the isospins into triplet and singlet
combinations
2  2  1 3
 1,1  p 1 p  2 
 1, 0  
1
2
 p 1 n  2   n 1 p  2  
 1, 1  n 1 n  2 
  0, 0  
1
2
 p 1 n  2   n 1 p  2  
Symmetry under
label interchange
1
2
Symmetric
Anti-symmetric
29
Some additive quantum numbers
In the simple quark model, only two types of quark bound states (and
their anti-states) are allowed:
• baryons, made up of 3-quarks
• mesons, made up of quark-antiquark
For each state we can associate several quantum numbers, which refer to
the quark content: strangeness S, charm C, beauty B, truth T:
S   N S    N ( s)  N ( s) 
C  Nc   N (c)  N (c) 
S=-1 for s-quark, +1 for anti-s-quark, 0 – rest.
B   Nb    N (b)  N (b)  T  Nt   N (t )  N (t ) 
Number of u and d quarks
have no special names:

Nu   N (u )  N (u ) 
1
 N (q )  N (q ) 
B

Baryon number B:

3

N d   N (d )  N (d ) 
baryon: B=1, antibaryon: B=-1
rest: B=0
30
Baryons and Mesons
particle
P
n
Mass quarks
(MeV)
Q
S
C
B
B

938
940
1116
uud
udd
uds
1
0
0
0
0
-1
0
0
0
0
0
0
1
1
1
c
+
KD-
2285
140
494
1869
udc
ud
su
dc
1
1
-1
-1
0
0
-1
0
1
0
0
-1
0
0
0
0
1
0
0
0
D+s
B-
1969
5278
cs
bu
1
-1
1
0
1
0
0
-1
0
0

9460
bb
0
0
0
0
0
31
Hypercharge
Hypercharge Y is defined as follows: Y  B + S + C + B + T
The hypercharge has the same value for each family member.
Gel-Mann and Nishijima showed the relation between the third
component of the isospin, I3, the electric charge, Q, and the
hypercharge, Y:
0
I 3 (  )  1 
Y
I3  Q 
2
Examples:
 1
2
1
1
I 3 ( K  )  1   
2
2
1
1
I 3 ( n)  0   
2
2
0
I 3 ()  0   0
2
32
Some quantum numbers of quarks
quark
B
Y
Q
I3
I
d
1/3
1/3 -1/3 -1/2
1/2
u
1/3
1/3
2/3
1/2
1/2
s
1/3 -2/3 -1/3
0
0
c
1/3
2/3
0
0
b
1/3 -2/3 -1/3
0
0
t
1/3
0
0
4/3
4/3
2/3
33
Hadron quantum numbers
Assuming only light quarks (u,d,s), the following states are allowed:
Baryons
S
Q
Mesons
I
S
Q
I
0 2,1,0,-1 3/2,1/2
1
1,0
1/2
-1
1,0,-1
1,0
0
1,0,-1
1,0
-2
0,-1
1/2
-1
0,-1
1/2
-3
-1
0
34
J, P, C of hadrons
Each hadron characterized by mass, spin, parity, charge-conjugation,
isospin, etc. Can obtain its quantum numbers by knowing its quark content
and the relative angular momentum between them. Notation of particle
quantum numbers: JPC .
Example:
ud
mesonic states
Q = +1  I3 = +1  I = 1
Since Q ≠ 0  not C eigenstate
L
0
0
1
1
1
1
S
0
1
0
1
1
1
J
0
1
1
0
1
2
P
+
+
+
+
state
1S
0
3S
1
1P
1
3P
0
3P
1
3P
2
35
Hadrons from quarks
J P  0
J 
P
J 
P
J P  1
1
2
3
2
C
Y
I3
36
Penta-quarks
Are there states like:
qqqq , qqqqq ?
T. Nakano et al., PRL 91 (2003) 012002
37
Hadron quantum numbers from
decay
Example:
 0     , L  1 :
J   L  S  1
P  P2 (1) L  1
C  C (   )  (1) L  S  1
J PC  1
38
G parity
Only few particles are eigenstates of C. In strong interactions, one
can use rotation in isospin state to create an operator for which
more particles will become eigenstates. A rotation of 180° in
isospin space around e.g. y-axis, Iy, will change I3 to –I3, converting
for instance a + into a -. Applying now C will change the - back to
+.
i I y
G  Ce
All non-strange (or charm, beauty, top) mesons are
eigenstates of G. For a multiplet of isospin I:
G  (1) C
I
G( )  1, G(n )  (1)
n
39
Examples with G parity
G  (1) I C
I (  )  1, C (  )  1,  G(  )  1
0
  
has to decay into even number of pions.
I ( )  0, C ( )  1,  G ( )  1
has to decay into odd number of pions.
     0
I ( )  0, C ( )  1,  G( )  1
  
P

J=0 state of two pions has to have even parity, while J ( )  0
  4 not enough energy: m( )  547.5MeV , m(4 )  558MeV
has to decay into even number of pions. But
  3
violating G parity  decay is electromagnetic.
40
Summary of conservation rules
conserved quantity
SI
EMI
WI
Energy/momentum
yes
yes
yes
Charge
yes
yes
yes
Baryon number
yes
yes
yes
Lepton number
yes
yes
yes
I-isospin
yes
no
no
G-parity
yes
no
no
S-strangeness
yes
yes
no
C,B,T (charm, bottom, top)
yes
yes
no
P-parity
yes
yes
no
C-charge conjugation
yes
yes
no
CP (or T)
yes
yes
yes*
CPT
yes
yes
yes
* 10-3 violation
in K0 decay
41
Quark diagrams


 p :uuu  uud  ud



d
u
u
u
    : uu  ud  du

0

0

 0   0 0
u
d
u
u

u
u
u
d



p



d  
u
C-parity not conserved (see also Clebsh coeff.)
42
Quark diagrams (2)
 p  p


  p   p
u
u
d
d
d
d
u
u
u
u
d
u
u
d
u
d
u
u
d
u
43
Quark diagrams (3)


K pK p


K pK p
“Exotic”
s
s
u
u
u
u
u
d
u
d
u
u
s
u
s
u
u
d
u
d
44
45
46
Okubo-Zweig-Iizuka (OZI) rule
M   1019.5MeV ,   4.2MeV
Problem:
Why so narrow?
  K  K  (83.8%) ,      0 (14.9%)
Q-value is 30 MeV for KK decay, while 605 MeV for 3 decay. Why
is BR(KK)»BR(3)? Answer: OZI rule – quark diagram of 3 decay
not continuous.
K K

s
s

    0 
s
u
d
u
s
d
u
s
d
s
d
u
47