Download SU(3) - Physics

P780.02 Spring 2002 L8
Quarks
Richard Kass
Over the years inquiring minds have asked:
“Can we describe the known physics with just a few building blocks ?”
 Historically the answer has been yes.
Elements of Mendeleev’s Periodic Table (chemistry)
nucleus of atom made of protons, neutrons
proton and neutron really same “particle” (different isotopic spin)
By 1950’s there was evidence for many new particles beyond g, e, p, n
It was realized that even these new particles fit certain patterns:
pions:
p+(140 MeV)
p-(140 MeV)
po(135 MeV)
kaons:
k+(496 MeV)
k-(496 MeV)
ko(498 MeV)
Some sort of pattern was emerging, but ........... lots of questions
 If mass difference between proton neutrons, pions, and kaons is due
to electromagnetism then how come:
Mn > Mp and Mko > Mk+ but Mp+ > Mpo
Lots of models concocted to try to explain why these particles exist:
 Model of Fermi and Yang (late 1940’s-early 50’s):
pion is composed of nucleons and anti-nucleons (used SU(2) symmetry)
p+ = pn, p- = np, po = pp - n n
With the discovery of new unstable particles (L, k) a new quantum
strangeness
note this model was proposed
before discovery of anti-proton
!
number
was invented:
P780.02 Spring 2002 L8
Quarks
Richard Kass
Gell-Mann, Nakano, Nishijima realized that electric charge (Q) of all particles could be
related to isospin (3rd component), Baryon number (B) and Strangeness (S):
Q = I3 +(S + B)/2= I3 +Y/2
Coin the name hypercharge (Y) for (S+B)
Interesting patterns started to emerge when I3 was plotted vs. Y:
ko Y
1
p-
k+
p+
I3
1
po
-1
k-
-1
k
o
Particle Model of Sakata (mid 50’s):
used Q = I3 +(S + B)/2
assumed that all particles could be made from a combination of p,n, L
tried to use SU(3) symmetry
+
o
In this model: p+ = pn, p- = np, po = pp - n n, k = p L, k = n L
+
Baryons are made up of 3 at a time:  = Lpn
This model obeys Fermi statistics and explains why:
Mn > Mp and Mko > Mk+ and Mp+ > Mpo
Unfortunately, the model had major problems….
P780.02 Spring 2002 L8
Quarks
Richard Kass
Problems with Sakata’s Model:
Why should the p, n, and L be the fundamental objects ?
why not pions and/or kaons
This model did not have the proper group structure for SU(3)
What do we mean by “group structure” ?
SU(n)= (nxn) Unitary matrices (MT*M=1) with determinant = 1 (=Special) and
n=simplest non-trivial matrix representation
Example: With 2 fundamental objects obeying SU(2) (e.g. n and p)
We can combine these objects using 1 quantum number (e.g. isospin)
Get 3 Isospin 1 states that are symmetric under interchange of n and p:
|11> =|1/2 1/2> |1/2 1/2>
|1-1> =|1/2 -1/2> |1/2 -1/2>
|10> = [1/2](|1/2 1/2> |1/2 -1/2> + |1/2 -1/2> |1/2 1/2>)
Get 1 Isospin state that is anti-symmetric under interchange of n and p
|00> = [1/2](|1/2 1/2> |1/2 -1/2> - |1/2 -1/2> |1/2 1/2>)
In group theory we have 2 multiplets, a 3 and a 1:
2  2 = 3 1
Back to Sakata's model:
For SU(3) there are 2 quantum numbers and the group structure is more complicated:
3  3  3 = 1  8  8  10
Expect 4 multiplets (groups of similar particles) with either 1, 8, or 10 members.
Sakata’s model said that the p, n, and L were a multiplet which does not fit into the
above scheme of known particles! (e.g. could not account for o, +)
P780.02 Spring 2002 L8
Early 1960’s Quarks
Richard Kass
“Three Quarks for Muster Mark”, J. Joyce, Finnegan’s Wake
Model was developed by: Gell-Mann, Zweig, Okubo, and Ne’eman (Salam)
Three fundamental building blocks 1960’s (p,n,l)  1970’s (u,d,s)
mesons are bound states of a of quark and anti-quark:
Can make up "wavefunctions" by combing quarks:
p+ = ud, p- = du, po = 1 (uu - d d), k + = ds, k o= ds
2
baryons are bound state of 3 quarks:
proton = (uud), neutron = (udd), L= (uds)
anti-baryons are bound states of 3 anti-quarks:
p  uu d n  udd L uds
These quark objects are:
point like
spin 1/2 fermions
parity = +1 (-1 for anti-quarks)
two quarks are in isospin doublet (u and d), s is an iso-singlet (=0)
Obey Q = I3 +1/2(S+B) = I3 +Y/2
Group Structure is SU(3)
For every quark there is an anti-quark
quarks feel all interactions (have mass, electric charge, etc)
P780.02 Spring 2002 L8
Early 1960’s Quarks
Richard Kass
The additive quark quantum numbers are given below:
Quantum #
u
d
s
c b
t
electric charge 2/3
-1/3 -1/3 2/3 -1/3 2/3
I3
1/2
-1/2 0
0 0
0
Strangeness
0
0
-1
0 0
0
Charm
0
0
0
1 0
0
bottom
0
0
0
0 -1 0
top
0
0
0
0 0
1
Baryon number 1/3
1/3 1/3 1/3 1/3 1/3
Lepton number 0
0
0
0 0
0
Successes of 1960’s Quark Model:
Classify all known (in the early 1960’s) particles in terms of 3 building blocks
predict new particles (e.g. W-)
explain why certain particles don’t exist (e.g. baryons with S = +1)
explain mass splitting between meson and baryons
explain/predict magnetic moments of mesons and baryons
explain/predict scattering cross sections (e.g. spp/spp = 2/3)
Failures of the 1960's model:
No evidence for free quarks (fixed up by QCD)
Pauli principle violated (D++= uuu wavefunction is totally symmetric) (fixed up by color)
What holds quarks together in a proton ? (gluons!)
How many different types of quarks exist ? (6?)
P780.02 Spring 2002 L8
Dynamic Quarks
Richard Kass
Dynamic Quark Model (mid 70’s to now!)
Theory of quark-quark interaction  QCD
includes gluons
Successes of Quark Model (QCD):
“Real” Field Theory i.e.
Gluons instead of photons
Color instead of electric charge
explains why no free quarks  confinement of quarks
calculate cross sections, e.g.
e+e-  qq
calculate lifetimes of baryons, mesons
Failures/problems of the model:
Hard to do calculations in QCD (non-perturbative)
Polarization of hadrons (e.g. L’s) in high energy collisions
How many quarks are there ?
Historical note:
Original quark model assumed approximate SU(3) for the quarks.
Once charm quark was discovered SU(4) was considered.
But SU(4) is a badly “broken” symmetry.
Standard Model puts quarks in SU(2) doublet,
COLOR exact SU(3) symmetry.
P780.02 Spring 2002 L8
Richard Kass
From Quarks to Particles
How do we "construct" baryons and mesons from quarks ?
M&S P133-140
Use SU(3) as the group (1960’s model)
This group has 8 generators (n2-1, n=3)
Each generator is a 3x3 linearly independent traceless hermitian matrices
Only 2 of the generators are diagonal  2 quantum numbers
Hypercharge = Strangeness + Baryon number = Y
Isospin (I3)
In this model (1960’s) there are 3 quarks, which are the eigenvectors (3 row column vector)
of the two diagonal generators (Y and I3)
Baryons are made up of a bound state of 3 quarks
Mesons are a quark-antiquark bound state
The quarks are added together to form mesons and baryons using the rules of SU(3).
It is interesting to plot Y vs. I3 for quarks and anti-quarks:
Y 1
Y 1
2/3
d
1/3
u
I3
1
-1
s
s
-1
u
d
2/3
-1
-1
I3
1
P780.02 Spring 2002 L8
Making Mesons with Quarks
Richard Kass
Making mesons with (orbital angular momentum L=0)
The properties of SU(3) tell us how many mesons to expect:
3 3  1 8
Thus we expect an octet with 8 particles and a singlet with 1 particle.
If SU(3) were a perfect
symmetry then all particles
in a multiplet would have
the same mass.
P780.02 Spring 2002 L8
Baryon Octet
Richard Kass
Making Baryons (orbital angular momentum L=0).
Now must combine 3 quarks together:
3  3  3  1  8  8  10
Expect a singlet, 2 octets, and a decuplet (10 particles) 27 objects total.
Octet with J=1/2:
P780.02 Spring 2002 L8
Baryon Decuplet
Baryon Decuplet (J=3/2)
Expect 10 states.
Prediction of the W- (mass =1672 MeV/c2, S=-3)
Use bubble chamber to find the event.
1969 Nobel Prize to Gell-Mann!
“Observation of a hyperon with strangeness minus 3”
PRL V12, 1964.
Richard Kass
P780.02 Spring 2002 L8
Quarks and Vector Mesons
Richard Kass
Leptonic Decays of Vector Mesons
What is the experimental evidence that quarks have non-integer charge ?
 Both the mass splitting of baryons and mesons and baryon magnetic moments
depend on (e/m) not e.
Some quark models with integer charge quarks (e.g. Han-Nambu) were also successful
in explaining mass patterns of mesons and baryons.
Need a quantity that can be measured that depends only on electric charge !
Consider the vector mesons (V=r,w,f,y,U): quark-antiquark bound states with:
mass  0
electric charge = 0
orbital angular momentum (L) =0
spin = 1
charge parity (C) = -1
parity = -1
strangeness = charm = bottom=top = 0
These particles have the same quantum numbers as the photon.
The vector mesons can be produced by its coupling to a photon:
e+e- gV e.g. : e+e- gY(1S) or y
The vector mesons can decay by its coupling to a photon:
Vg e+e- e.g. : rg e+e- (BR=6x10-5) or yg e+e- (BR=6.3x10-2)
P780.02 Spring 2002 L8
Quarks and Vector Mesons
Richard Kass
The decay rate (or partial width) for a vector meson to decay to leptons is:
g 
16p2em
M
2
V
|  ai Qi | y(0)
2
2
The Van RoyenWeisskopf Formula
i
In the above MV is the mass of the vector meson, the sum is over the amplitudes that
make up the meson, Q is the charge of the quarks and y(0) is the wavefunction for the
two quarks to overlap each other.
meson
quarks
r
1 ( uu - d d)
2
1 ( 2 -(2 3
w
1 ( uu + dd)
2
1 ( 2 +(- 1 ) )
3
2 3
f
ss
y
cc
(- 1 )
1)
3
2
)
2
2
=
1
2
=
7 KeV
1 3.2
1
18
0 .8 Ke V
1 2.8
=
1
9
1 .3 Ke V
1 1.8
=
4
9
4 .7 Ke V
1 0.5
=
1
9
1 .2 Ke V
1 0.6
3
(2 )
2
3
Y
L(exp) L(exp) |aiQi|-2
|aiQi|2
bb
(- 1 )
3
2
If we assume that |y(o)|2/M2 is the same for r,w,f, (good assumption since masses are 770
MeV, 780 MeV, and 1020 MeV respectively) then:
expect:
measure:
Good agreement!
L(r) : L(w) : L(f) = 9 : 1 : 2
(8.8 ± 2.6) : 1 : (1.7 ± 0.4)