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Lecture 6

● Isospin● SU(2) and SU(3)● Parity

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Reminder about isospin (1) ● We’re dealing with the strong force.● The theory of the strong force works extremely well

(can’t be proven wrong) for interactions at high energies (>> 1 GeV). Forthcoming lectures

● At low energies we can’t do much. We can’t calculate all hadron masses from first principles. We can’t calculate all reaction rates from first principles.

● We use symmetry (isospin) as an experimentally established fact to guide us and help us ”feel our way” wrt strong force observables

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Reminder (2) – conservation of isospin

0

1 1 1 10 0 -

2 2 2 2

Noether's theorem - invariance conservation law.

Study isospin from perspective of "conservation of isospin".

(a) and (b) (6.01)

Deuteron - isospin .

p p d n n d

d p n

3 3

3 3

1 1 1 -1

1 1 1 11 0 0 1 1 1

2 2 2 2

1 1 1 11 0 0 1 1 1

2 2 2 2

(6.02)

(a) LHS: ; ; RHS: (6.03)

(b) LHS: - - ; ; RHS: - (6.04)

tot tot

tot tot

I I

I I

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Reminder (3): invariance to a rotation in isospin space

180 ' '

.

A rotation by around the -axis in isospin space

converts a proton neutron and

Can measure:

(a) and (b)

(a) and (b) are the 'same' reaction as seen

by the strong force

o y

p p d n n d

if isospin is a good symmetry.

Strong reaction rates are measured to be the

same/"very, very close".

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Quarks and isospin

1 1 1 1

2 2 2 2

1 1 1 1

2 2 2 2

The up and down quarks

; - (6.05)

Anti-up and down quarks:

- ; (6.06)

(- sign is a technical and (for us) unimportant detail)

The other quarks carry no isospi

u d

u d

n.

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Isospin of antiquarks (not for lecture or exam)1 01 1 1 1

0 12 2 2 2

.

1 0 0 11 1 1 1 1 1ˆ ˆ( ) - ; ( ) -0 1 1 02 2 2 2 2 2

Light quarks form an isospin doublet: -

From spin questions in lecture 5 and eqn. 523

u d

U R U R

1 1

2 2

ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

rotation of around "y"-axis (or 2-axis) in isospin space ; (6.08)

Define charge conjugation phase factors:

; ; ;

R R u d R d u

C u u C d d C u u C d d CC u u CC d d

ˆ ˆ ˆ ˆ

(6.09)

Apply charge conjugation transformation to rotation operations:

; ; (6.10)

Use this info to define antiquark isospin doublet. Desire that antiquark doubl

R u d R d u R u d R d u

ˆ ˆ

et transforms in

the same way as the quark doublet (necessary when we combine quarks and antiquarks

together in mesons and want to transform the whole thing by a rotation).

i.e. ; R upper lower R lo

1 01 1 1 1

0 12 2 2 2

1 0 0 1ˆ ˆ ˆ ˆ0 1 1 0

set: - (6.11)

Then , ok! ; , ok! (6.12)

Without the negative si

wer upper d u

R R d u R R u d

1 01 1 1 1

0 12 2 2 2

1 0 0 1ˆ ˆ ˆ ˆ0 1 1 0

gn: -

Then ,not ok! ; ,not ok! (6.13)

The minus signs are a way ensuring symmetry und

d u

R R d u R R u d

er charge conjugation and are defined after a convention.

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Isospin with quarks - continued

0

0

1 1 1 11 1 ,

2 2 2 2

1 1 1 1 1 1 1 1 1 11 0 - -

2 2 2 2 2 2 2 22 2

1 1 1 11 -1 - -

2 2 2 2

, ,

1 1 1 1 1 10 0 -

2 2 2 22 2

(6.14)

all have similar masses 140 MeV and clearly belong together.

=

ud

uu dd

du

uu dd

1 1 1 1-

2 2 2 2

11 1 1 0 1 1

2

(6.15)

A neutral particle with a different mass, (540 MeV) is a good candidate.

Just like combining nucleons.

; ; - (5.44)

None of these

pp pn np nn

10 0

2

exist.

(5.49)

Deuteron exists.

pn np d

?

I3

d

I3

Not used by nature

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Question● Write down a particle decay which does not

conserve isospin.

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Isospin and group theoryTwo complementary ways of thinking about isospin:

(1) Hadrons (and quarks) carry a conserved quantity: isospin.

The algebra for isospin is the same as for spin angular momentum.

Conservation of isospin

0, , 3 1 .

and the rules of adding up isospin would give us,

triplet candidates: eg and a singlet candidate

(2) Rotating a co-ordinate system in "isospin space" has no effect when considerin

ˆ /2'( ) ( )

'

g

a strong force observable in the same way that rotating a co-ordinate system in real

space has no effect on nature.

Following rotation a state, eg spinor ; (5.3i

U U e

1), is a matrix.

The set of all such matrices is given by the SU(2) group.

Group theory deals with symmetries from another, complementary perspective.

For the SU(2) group, can show in group theory lang

U

2 2 3 1uage:

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Conservation of isospin

● Conserved for strong processes.● Violated for weak and electromagnetic

processes.● Can think about as: photon,W,Z, and the leptons

have zero isospin: 0 0

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Why does isospin work ?

3 5

, , .

Not a totally stupid question.

Now we're thinking in terms of quarks we can try to understand.

MeV, MeV.

Same colour charges: Small masses and mass differences.

Take the pions:

up downm m

R G B

0

0

,

1

2

, .

2

; ; (6.07)

Of course the will have similar masses 140 MeV

It doesn't matter how we arrange the quarks : they have a tiny mass

difference

mass arises fru d

ud uu dd du

m m

10 5

om potential energy in the strong field and

quark motion (forthcoming lectures).

If GeV and MeV we would see large differences in the pion masses.

(if they could even be formed!)

The protons

u dm m

( ) ( ) and neutrons would also be very different.

isospin symmetry breaks down.

(from earlier question) (a) and (b) would not have

the same rates for the same experimental c

uud udd

p p d n n d

onditions (incoming energy etc.).

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● Strange, charm and bottom quarks also form hadrons.

● Large mass differences between the quarks.

● What kind of symmetry can we obtain here.

Quark Q

(e)

Mass (GeV)

B S C B T

u- up 2/3 0.003 1/3 0 0 0 0

d- down -1/3 0.005 1/3 0 0 0 0

s- strange -1/3 0.15 1/3 -1 0 0 0

c- charm 2/3 1.2 1/3 0 1 0 0

b- bottom -1/3 4.2 1/3 0 0 -1 0

t-top 2/3 171 1/3 0 0 0 1

4 G

eV

1 G

eV 0.1

GeV

2 M

eV

The other quarks

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From SU(2) to SU(3)

2 2 3 1

(2)

,

From group theory and "adding up" isospin

combinations we expect, eg, our triplet+singlet

structure for pions: (6.16)

two quarks and two anti-quarks

for a symmetry.

Triplet: (6.14)

SU

ud

0 0

0

1

2

1

2

, =

Singlet: (6.15)

uu dd du

uu dd

?I3

( , , )

(3)

3 3 8 1

.

Only consider a three quark system :

is the relevant symmetry group:

(6.17)

three quarks and three anti-quarks

Expect mesons to lie in an octet and

a singlet and that with isospin

sub

u d s

SU

1 1, , - , , , , , 2

2 61

3

multiplets.

Octet: (6.18)

Singlet: (6.19)

ud du uu dd us ds us ds dd uu ss

dd uu ss

I3½ -½ +1-1

Meson nonet (spin 0)

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Identifying the states0 0 0

0

8

1( )

2

1( 2

6

-

Most are straightforward to assign:

(Mass 500 MeV)

(Mass 140 MeV)

Two neutral combinations with same quantum numbers:

-K ds K sd K su K ds K us

du uu dd du

uu dd ss

0

0

08 0

8 0

1) ( )

3

10

1cos sin ( 2 )

61

' sin cos ( )3

and

Nature uses a linear combination of them as observable

particles:

(Mass 550 MeV)

(Mass 950 MeV)

uu dd ss

uu dd ss

uu dd ss

SU(3) is an approximate symmetry. Particles within multiplet have large mass differences. Due to quark mass differences! Strong force is not invariant to SU(3) flavour transformations. Useful to catalogue the states and their quark composition.

(6.20)

(6.14)

(6.21)

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SU(3) Flavour – mesons of spin 1 (vector)*0 *0 * *0 *

0

8 0 0 8

8 0

8 0

1( )

2

35 , ,

1sin cos ( )

2cos sin

-

(Mass 890 MeV)

(Mass 780 MeV)

in analogy with

(Mass 550 MeV)

-

o

K ds K sd K su K ds K us

du uu dd du

uu dd

ss

. (Mass 950 MeV)

(6.22)

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Question

1. What is the strangeness of the meson ?

2. Give evidence that the meson is more likely to consist of ss than uu.

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Higher mass meson multiplets

Spectroscopic notation – explored further in lectures on bound states.

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BaryonsSimilarly form baryon multiplets from SU(3) flavour.

Spin ½ Spin 3/2

Historically, these arrangements of hadrons was termed ”the eight-fold way” by Murray Gell-Mann after the octet arrangement above. This work allowed the understanding of hadron masses and properties in terms of quarks and was the first evidence for quarks.

Like all good theories it gave a prediction which experimentalists could verify.

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Baryon decuplet – something missing?

1964One particle was missing to make up the baryon decuplet.

?

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Discovery of the

0

1964 - Brookhaven.

An incoming interacts with a proton in the

liquid hydrogen in a bubble chamber.

Discovery confirmed the correctness of the quark model.

K

K p K K

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Summary of isospin, SU(2) and SU(3)

● Isospin is a good symmetry of the strong force Hadron masses, reaction/decay rates respect isospin

symmetry The strong force is invariant to SU(2) isospin transformations. This is due to the small u,d mass difference

● The other quarks are much heavier than u,d and show large mass differences. SU(3) flavour symmetry is useful for enumerating and

ordering the different hadrons and understanding their quark composition.

The strong force is not invariant to SU(3) flavour transformations since the quark masses are so different.

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Parity

● The laws of physics are invariant to, eg a shifted co-ordinate system (x’ x+a) or rotated co-ordinate system.

● What happens if we make a parity transformation: invert the co-ordinate system (x,y,z) (-x,-y,-z) ?

● Parity is a discrete symmetry offering two possible states

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● Invert the spatial co-ordinate system Parity transformation (P) Change handedness of co-ordinate system (eg right to

left)

● By 1956 parity invariance had been shown for the strong and electromagnetic forces.

● A test for the weak force was needed.

.

Parity

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ParityA reflection in the x-z plane followed

by a rotation about the y-axis is

equivalent to a parity

transformation.

To test parity invariance:

study a reaction and ask if the mirror

reflection of that process has the

same probability of occuring (no need

to consider further rotation since rotation

invariance is implied by angular

momentum conservation).

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Parity violation – Wu’s experiment

• Study the decay of Co-60• At a low temperature 0.01K, the spins

can be polarised parallel to an external magnetic field.

• The electrons are dominantly emitted in the direction opposite to the spin

• In the ”mirror” image, the electron is dominantly emitted in the opposite parallel to the spin.

• Parity is violated in weak decays !• A symmetry which is violated means

that some quantity is not being conserved

e e

e e

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Helicity• A particle possesses helicity

– Arbitary z-axis for spin angular momentum– select direction of motion

• Helicity is not a useful quantity for most particles since it isn’t a Lorentz invariant

• Can be defined for luminal particles eg a massless neutrino

12 1

12

12 1

12

Helicity (right handed)

(left-handed)

sm

s

-

(6.23)

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Muon measured and found to be right-handed

Neutrino oscillations imply that they have a tiny mass (<2 eV). Maybe its better to say that most neutrinos are left-handed and most anti-neutrinos are right-handed. Certainly, it’s a good approximation since left-handed neutrino and right-handed antineutrino are just about impossible to observe in laboratory experiments.

s

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What parity did to the neutrinos in Wu’s experiment

• Parity transformation means moving from a right-handed to a left-handed co-ordinate system

• Wu’s experiment is - decay • A ”mirror reflection” of the right-handed anti-

neutrino is a left-handed anti-neutrino.

We don’t observe the ”reflected” process.

eenp

s sChangehandedness

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Parity

● Parity is a symmetry respected by the electromagnetic and strong forces

● Parity is violated by the weak force.

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Summary

● Isospin and flavour symmetries● Parity – space inversion● Weak interactions are not invariant to parity