In general NON-ABELAIN GAUGE THEORIES:

•introduce more interactions (vertices)•for SU(2) we saw both 3 and 4 particle interaction vertices

•have (still) massless gauge particles (like the photon!)•the gauge field particles posses “charge” just like the fundamental Dirac states

•not electric charge - we’re trying to think of NEW forces

The YANG-MILLS was built on the premise that

there existed •2 elementary Dirac (spin-1/2) particles of ~equal mass•serving as sources for the force fields through which they interact

NO SUCH PAIRS EXIST

proton/neutron isospin states were the inspiration, but•there is NO massless vector (spin 1) iso-triplet (isospin 1) of known particle states

• -mesons? 770 MeV/c2 •p,n, now recognized as COMPOSITE particles•isospin of up,dn quarks generalized into SU(3) SU(4)

The strong force must be independent of FLAVOR

up charm topdown strange bottom

i.e., the strong force does not couple to flavors.SO WHERE DOES THE STRONG FORCE COME FROM?

1911 Rutherford

Elastic scattering of α – particles on atoms Discovery of atomic nucleus Size of nucleus 10-5 size of atom

1968 SLAC-MIT

Deep inelastic scattering of e of p, dobservation of ~flat q2 dependence of R= σinel/σMott

R ~ const → point-like scatterers inside proton

Fermi referred to these as “partons” which eventually became identified with

the quarks!

Cro

ss se

ctio

n

Energy

We’ve argued for the simple energy dependence

possibly punctuated by resonances

Cro

ss se

ctio

n

Energy

and threshold effectsas higher energy

opens up new production of heavier particles

s = cms energy, GeV

Cro

ss-s

ectio

n, c

m2

e+e anihilation cross section

Dirac particles fundamental,charged Fermions (spin- 1/2)

e- e+ + - + u d c s t b u d c s t b

q: -1 +1 -1 +1 -1 +1 +2/3 -1/3 +2/3 -1/3 +2/3 -1/3

-2/3 +1/3 -2/3 +1/3 -2/3 +1/3

All can be described by DIRAC equations w/charge coupling to -fields

105 MeV/c2

u 7.5 MeV/c2

d 4.2 MeV/c2

c 1100 MeV/c2

s 150 MeV/c2

b 4200 MeV/c2

e-

e-

q=e

-

-

q=e

u

u??2/3 e

Consider e+e annihilation in an electron-positron collider

ee

e

Q-couplingee

qq

unclear if annihilated

easy signature to detect!

anyflavor

R =(hadrons)(+)

N e+ehadrons

Ne+e +

=

Q total probability of all possible quarks i qi

2

105 MeV/c2

u 7.5 MeV/c2

d 4.2 MeV/c2

c 1100 MeV/c2

s 150 MeV/c2

b 4200 MeV/c2

When above the u,d,s threshold

1)()()(

1

2312

312

32

2

2

e

QR i

i 23=

above the J/ mass (c-production threshold)

R = 109 = 1.11

above the upsilon mass (b-production threshold)

R = 119 = 1.22

0.66 1.111.22

3.673.33

Off by a factor of 3!!!!!

Could every known quark have a previously

un-noticed3-fold degeneracy????

Baryon States

State Quark content Mass Spin p uud 938.272 MeV 1/2 n udd 939.565 MeV 1/2 uds 1115.683 MeV 1/2 + uus 1189.37 MeV 1/2 0 uds 1192.632 MeV 1/2 - dds 1197.449 MeV 1/2 0 uss 1314.9 MeV 1/2 dss 1321.32 MeV 1/2 uuu 1230. MeV 3/2 uud 1231 MeV 3/2 0 udd 1233 MeV 3/2 - ddd 1234 MeV 3/2 *+ uus 1382.8 MeV 3/2 *0 uds 1383.7 MeV 3/2 * dds 1387.2 MeV 3/2 *0 uss 1531.80 MeV 3/2 * dss 1535.0 MeV 3/2 sss 1672.45 MeV 3/2

U(1) introduced an interaction term: (q)A

2

Remember:SU(2) includes ALL POSSIBLE traceless HERMITIAN 2x2 matricesthe most GENERAL such matrix is

a1 + a2 + a3 = a . 0 11 0

0 -ii 0

1 00 -1

· G

•the G = (G1, G2

, G3) are three independent

4-vector fields, but together form an ISO-vector•transform under SU(2) the same way does:

U( · G)U† = U U†· UGU†

a charged current coupled to a vector field

Yang-Mills extended this to SU(2) with an interaction term of: (q).G

· G

each is a “vector” on “iso-space”

Under the SU(2) Iso-spin transformations:

2cig

eU

1

2recall it was from

U( · G)U† 2

that we determined the correct transformation property of G

G / =RTG-

the 3 independent fields are isospin components which transform like under a

basis transformation

G must obey the same COMMUTATOR relationsthat does, i.e., they form an adjoint representationof (a set of functions isomorphic – mathematicallyequivalent to the matrices).

Not a column vector in iso-space! Not a matrix!

Generalizing YANG-MILLS

L ~ Fi FiGAUGEFIELDS

FiGi - Gi

- 2g ijk GjGk

c

because [i,j ] = iijkk

FiGi - Gi

- 2g Cijk GjGk

More generally

c“structure constant”

of SU(n)

The exact formdepends on the

non-abelian natureof the generators

(their commutator rules)

SU(3)

)/( cgieU

So now, in the face of evidence that quarks carry some 3-fold degeneracy that needs to be explainedLET’S TRY:

This time must be not only a Dirac spinor u(s)(p)but a 3-column vector in some

NEW SPACE as well:

3

2

1

1, 2, … 8 span the 8-dim space(the 8-independent parameters)

of the 3x3 traceless, Hermitian generators of SU(3)

0 1 0 0 -i 0 1 0 01 = 1 0 0 2 = i 0 0 3 = 0 -1 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 -i 0 0 04 = 0 0 0 5 = 0 0 0 6 = 0 0 1 -1 0 0 i 0 0 0 1 0

0 0 0 1 0 0 7 = 0 0 -i 8 = 1/ /3 0 1 0 0 i 0 0 0 -2

SU(3)

)/( cgieU

(g ) . G

will require introducing an interaction term:

G the 8-independent VECTOR fields behave like thei operators:

[i, j] = 2iC ijkk

8×8×8 different C ijk but only = 56 distinct

combinations

8·7·63·2·1

only 9 of these are non-zero (with only 3 different values!)

C jik = C ikj = C ijk

C 123 = 1C 147 = C 246 = C 257 = C 345 = C 516 = C 637 = 1/2

C 458 = C 678 = 3/2

Since we have the explicitmatrix form of the i, its

easy to check any of these!

(since exchanges/rearrangementsare already defined)

SU(3) States If we write the 3 COLOR states as 1g = 0 0

0r = 1 0

0b = 0 1

0 1 0 0 -i 0 1 0 01 = 1 0 0 2 = i 0 0 3 = 0 1 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 -i 0 0 04 = 0 0 0 5 = 0 0 0 6 = 0 0 1 1 0 0 i 0 0 0 1 0

0 0 0 1 0 0 7 = 0 0 -i 8 = 1/ /3 0 1 0 0 i 0 0 0 2

Look at

1r = g1g = r

changes r g charge1

G

HEY!!!!! Electrons don’t change their charge when emitting/absorbing photons!

but remember our “gluons” carry “charge”

SU(3) States If we write the 3 COLOR states as 1g = 0 0

0r = 1 0

0b = 0 1

0 1 0 0 -i 0 1 0 01 = 1 0 0 2 = i 0 0 3 = 01 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 -i 0 0 04 = 0 0 0 5 = 0 0 0 6 = 0 0 1 1 0 0 i 0 0 0 1 0

0 0 0 1 0 0 7 = 0 0 -i 8 = 1/ /3 0 1 0 0 i 0 0 0 2

1r = g1g = r

2r creates the same color exchangebut with different coefficients: a change of phase

3 , 8: preserve/retain color but may change phase(still exchange/transfer energy & momentum - like s)

4 , 5:4b = g4g = b

6 , 7:6b = r6r = b

6 , 7:6b = r6r = b

6

G or 7

G

How does this work?

The gluons form a color OCTET(a multiplet of 8).

How can you get an OCTET from a basis of 3?

In SU(3) the 3 fundamental COLOR “charges” (the basis setin the lowest order representation) can be combined in pairs ofCOLOR/antiCOLOR to give either a SINGLET or an OCTET x x

= x xx x x 3 3 = 8 1

x xMust be color/anticolor states likerb rg br bg gr gb

While the fundamental process of the “strong” force is q gq

with a primitive vertex:

q b r

q

G6 or G

7

rg or

gr

G1 = (1/2 ) (rg + gr )

G2 = (-i/2 )(rg gr ) or inverting

rgiGG

griGG

)(

)(21

21

212

1

6 of the gluon fields are independent linear combinations ofthe simple gluon fields we enumerated

G1 = (rg + gr)/2 G

4 = (bg + gb)/2 G6 = (rb + br)/2

G2=-i(rg gr)/2 G

5=-i(bg gb)/2 G7=-i(rb + br)/2

The COLOR SINGLET would be 1/3 ( rr + gg + bb )

If the color singlet gluon existed it would be exchanged betweencolor singlet (color-less?) stateswhich as you’ll see momentarily

would suggest a long range strong force

between neutron and proton

…and as a color singlet it would be directly observable as a free particle.

…which simply does not exist