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Symmetries and conservation laws

•Introduction•Parity•Charge conjugation•Time reversal•Isospin•Hadron quantum numbers•G parity•Quark diagrams

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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.

A

CBa

R+ - clockwise rotation through 120° R- - counterclockwise rotation through 120°Ra, Rb, Rc – flipping about axis Aa, Bb, Cc, respectivelyI – doing nothingRotation clockwise through 240° rotation counterclockwise by 120° : R+

2=R- .

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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)RkThe 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).

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Nöther’s theoremIf 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 EnergyTranslation in space MomentumRotation Angular momentumGauge transformation Electric charge

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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: 1 *U U

(inverse = transpose conjugate)

If the group is restricted to all unitary matrices with determinant 1, the group is called : ( )SU n (“special

unitary”)

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Geometrical SymmetriesTranslation Rotation Reflection

(parity)

RRRRRRRRRR

RR RRRR RRRRR

R

continuous continuous discrete

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Parity operatorParity 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°:

' '1 2 1 2 1 2( , ,...) ( , ,...) ( , ,...)H x x H x x H x x

A system is said to be invariant under parity operation if the Hamiltonian remains unchanged by the transformation:

'i i iPx x x xi – position vector of particle

i.

Consider single particle wavefunction

( , ) :x t ( , ) ( , )aP x t P x t

where a identifies particle, Pa – constant phase factor. Two successive parity operations leave system unchanged 2 ( , ) ( , ) 1aP x t x t P

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Intrinsic ParityEigenfunction of momentum given as

( )( , ) i p x Etp x t e

( , ) ( , ) ( , )p a p a pP x t P x t 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:( ) ( ) ( , )mnlm nl lx R r Y r,, spherical polar coordinates

( , ) ( , ) ( 1) ( , )m m l ml l lY Y Y

( ) ( ) ( 1) ( )lnlm a nlm a nlmP x P x P x

eigenvalue:

( 1)laP

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Parity conservationIf Hamiltonian H is invariant under parity transformation

, 0P H

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 .

– Dirac equation is satisfied by a wave function representing both electrons and positrons intrinsic parity related:

– Strong and EM reactions always produce e+e- pairs.

– Arbitrarily have to set one =1 and the other = -1.

1PP ee

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Parity of Fermions• Assign positive parity state to

particles, negative to antiparticles:

• Make same assumption about quarks to be consistent:

1PPP

1PPP

e

e

1PPPPPP1PPPPPP

tbcsud

tbcsud

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Example: parapositroniumThe 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: 1i e e

P P P

Parity of final state:

2 ( 1)lfP P

Conclusion: relative angular momentum between the two ’s has to be odd, to conserve parity. This was confirmed experimentally.

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Parity of Mesons

111PPP LLqqM

Mesons are quark antiquark pairs:

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).

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Parity of BaryonsBaryons contain three quarks:

12 12 33

1 2 3P P P P 1 ( 1) 1L L LLB q q q

q1

q2

q3

L12

L3The lowest energy state baryons, like proton, neutron, , C, have L12+L3=0, and therefore positive parities.

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Naming convention

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 conjugationCharge 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 : 2 2; 1C C C C C

C are called C-parities, and if

, 0C H C-parity is conserved.

The electromagnetic field is produced by a moving charge, which changes sign under charge conjugation C = -1. C is multiplicative C(0 ) = +1.

This is true for strong and electromagnetic interactions.

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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:

( 1)L SC aa aa 0 is the lowest energy state of u-ubar or d-dbar, which are fermion-antifermion pairs with L=0 and S=0, therefore C(0 )=+1, as before.

state J P Cn=1

1S0 0 -1 1

3S1 1 -1 -1n=2

1S0 0 -1 1

3S1 1 -1 -11P1 1 1 -13P0 0 1 13P1 1 1 13P2 2 1 1

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:

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C of particle-antiparticle system

( 1)L SC aa aa

x1, 1

p

px2, 2

1 1 2 2 1 1 2 2| |C px px px px

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.

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Time reversalTime 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: 2 2 2( , ) '( , ) ( , )Tx t x t x t

The operation of time reversal changes the sign of momentum p and of the direction of the total angular momentum J: ' 'T Tp p p 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 balanceIf 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.

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IsospinIsospin 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

pn

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Mirror nuclei

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Isospin in the Quark Model• Isospin can be understood in the quark model of hadrons• The only difference between

n and p is interchange du

• 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!

p uudn udd

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Isospin in other hadrons• We now know the fundamental doublet for isospin is

• The comparable doublet for anti-particles is

• We can combine two isospin-1/2 states just like spin

13 2

13 2

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u II

d I

13 2

13 2

12

IdI

Iu

2 2 1 3

Bigger chargehas larger I3

3

03

3

1102

1

I ud

I uu dd

I du

310 0 ( )2

I I uu dd

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Note on wavefunctionFollowing the Condon-Shortly convention:

(arrow denotes C-operation)Isospin shift operators: 3 3 3 3( , ) ( 1) ( 1) ( , 1)I I I I I I I I I

Example:

, , 0 , ,I d u I u d I u I d I u d I d u

s

u d

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Note on wavefunction wavefunction:

0

0

( ) 2

0 0 2 22 2

02 2

I I du uu dd

dd uu ud udI I ud

dd uu ud udI I

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Note on wavefunction wavefunction:

Symmetry determined under interchanges:

;d u d u

0

ud du

ud du

dd uu uu dd

Triplet antisymmetric singlet - symmetric

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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

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12

1,1 1 2

1,0 1 2 1 2

1, 1 1 2

0,0 1 2 1 2

p p

p n n p

n n

p n n p

2 2 1 3

Symmetric

Anti-symmetric

Symmetry under label interchange

1 2

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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 , charm , beauty , truth :

=-1 for s-quark, +1 for anti-s-quark, 0 – rest.

( ) ( )SN N s N s

( ) ( )cN N c N c ( ) ( )bN N b N b ( ) ( )tN N t N t Number of u and d quarks have no special names:

( ) ( ) ( ) ( )u dN N u N u N N d N d

Baryon number B:

1 ( ) ( )3

B N q N q baryon: B=1, antibaryon: B=-1 rest: B=0

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Baryons and Mesonsparticle Mass

(MeV)quarks

Q B

P 938 uud 1 0 0 0 1n 940 udd 0 0 0 0 1 1116 uds 0 -1 0 0 1c 2285 udc 1 0 1 0 1+ 140 ud 1 0 0 0 0K- 494 su -1 -1 0 0 0D- 1869 dc -1 0 -1 0 0D+

s 1969 cs 1 1 1 0 0B- 5278 bu -1 0 0 -1 0 9460 bb 0 0 0 0 0

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HyperchargeHypercharge Y is defined as follows: Y B + + + + 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:

3 2YI Q

Examples:

3

3

3

3

0( ) 1 121 1( ) 12 2

1 1( ) 02 20( ) 0 02

I

I K

I n

I

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Some quantum numbers of quarks

quark B Y Q I3 Id 1/3 1/3 -1/3 -1/2 1/2u 1/3 1/3 2/3 1/2 1/2s 1/3 -2/3 -1/3 0 0c 1/3 4/3 2/3 0 0b 1/3 -2/3 -1/3 0 0t 1/3 4/3 2/3 0 0

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Hadron quantum numbersAssuming only light quarks (u,d,s), the following states are allowed:

S Q I S Q I0 2,1,0,-

13/2,1/2 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

Baryons Mesons

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J, P, C of hadronsEach 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

L S J P state

0 0 0 - 1S0

0 1 1 - 3S1

1 0 1 + 1P1

1 1 0 + 3P0

1 1 1 + 3P1

1 1 2 + 3P2

Q = +1 I3 = +1 I = 1

Since Q ≠ 0 not C eigenstate

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Hadrons from quarks0PJ

1PJ

12

PJ

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PJ

C

I3

Y

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Penta-quarksAre there states like: , ?qqqq qqqqq

T. Nakano et al., PRL 91 (2003) 012002

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Hadron quantum numbers from decay

0

2

, 1 :1

( 1) 1

( ) ( 1) 1

1

L

L S

PC

LJ L S

P P

C C

J

Example:

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G parityOnly 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 +.

yi IG Ce All non-strange (or charm, beauty, top) mesons are eigenstates of G. For a multiplet of isospin I:

( 1)

( ) 1, ( ) ( 1)

I

n

G C

G G n

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Examples with G parity0( ) 1, ( ) 1, ( ) 1I C G

has to decay into even number of pions.( ) 0, ( ) 1, ( ) 1I C G

has to decay into odd number of pions.

0

( ) 0, ( ) 1, ( ) 1I C G

has to decay into even number of pions. But J=0 state of two pions has to have even parity, while

( ) 0PJ 4 not enough

energy:( ) 547.5 , (4 ) 558m MeV m MeV

3 violating G parity decay is electromagnetic.

( 1)IG C

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Summary of conservation rules

conserved quantity SI EMI WIEnergy/momentum yes yes yesCharge yes yes yesBaryon number yes yes yesLepton number yes yes yesI-isospin yes no noG-parity yes no noS-strangeness yes yes noC,B,T (charm, bottom, top)

yes yes no

P-parity yes yes noC-charge conjugation yes yes noCP (or T) yes yes yes*CPT yes yes yes

* 10-3 violation in K0 decay

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Quark diagrams:p uuu uud ud

0 : uu ud du

uu

u

ddu

0

uuu d

uu

u

d

p

0 0 0 C-parity not conserved (see also Clebsh coeff.)

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Quark diagrams (2)p p

u

dduu

u

dduu

p p duudu

duudu

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Quark diagrams (3)K p K p

suuud

suuud

K p K p usuud

usuud

“Exotic”

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Okubo-Zweig-Iizuka (OZI) rule

Problem:

0(83.8%) , (14.9%)K K

1019.5 , 4.2M MeV MeV Why so narrow?

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.

ss

suus

ss

u

dd

d

du

K K 0