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originally from B. Meadows, U. Cincinnati
What is a Bound State?
Imagine a system of two bodies that interact. They can have relative movement. If this movement has sufficient energy, they will scatter and will eventually
move far apart where their interaction will be negligible. If their interaction is repulsive, they will also scatter and move far apart to
where their interaction is negligible. If the energy is small enough, and their interaction is attractive, they can
become bound together in a “bound state”. In a bound state, the constituents still have relative movement, in
general. If the interaction between constituents is repulsive, then they cannot
form a bound state. Examples of bound states include:
Atoms, molecules, positr-onium, prot-onium, quark-onium, mesons, baryons, …
Brian Meadows, U. Cincinnati
Gell-Mann-Nishijima Relationship
Applies to all hadrons
Define hyperchargeY = B + S + C + B’ + T
Then electric charge isQ = I3 + Y / 2
Relativelyrecently added
Third componentof I-spin
Bayon #
Brian Meadows, U. Cincinnati
“Eight-Fold Way” (Mesons) M. Gell-Mann noticed in 1961 that known particles can be arranged
in plots of Y vs. I3
Use your book to find the masses of the ’s and the K’s
K- K0(497)
- 0(135) +
’
(548/960)
K K
I3
Y
Pseudo-scalar mesons:All mesons here haveSpin J = 0 andParity P = -1
Centroid is at origin
Brian Meadows, U. Cincinnati
“Eight-Fold Way” (Meson Resonances) Also works for all the vector mesons (JP = 1-)
K*- K*0(890)
- 0(775) +
0/(783)/(1020)
K K
I3
Y
Vector mesons:All mesons here haveSpin J = 1 andParity P = -1
Brian Meadows, U. Cincinnati
Also works for baryons with same JP
“Eight-Fold Way” (Baryons)
n (935) p
- 0(1197) +
0(1115)
- 0(1323)
I3
Y{8}
JP = 1/2+
Centroid is at origin
Elect. chargeQ = Y + I3/2
Brian Meadows, U. Cincinnati
Also find {10} for baryons with same JP
“Eight-Fold Way” (Baryons)
- 0(1385) +
- 0(1532)
-(1679) ???
++ + 0(1238) -Y
I3
{10}
JP = 3/2+
G-M predicted This to exist
Centroid is at origin
3 quark flavors [uds] calls for a group of type SU(3)
SU(2): N=2 eigenvalues(J2,Jz) , N2-1=4-1=3 generators (Jx,Jy,Jz)
SU(3): N=3 eigen values (uds), N2-1=9-1=8 generators (8 Gell-Man mat.) or smarter:
SU(3) Flavor
I3
Y
d u
s
V+/-
T+/-
U+/-
Y
I3
T+, T-
U+, U-
V+, V-
At first, all we needed were three quarks in an SU(3) {3}:
SU(3) multiplets expected from quarks: Mesons{3} x {3} = {1} + {8} Baryons {3} x {3} x {3} = {1} + {8} + {8} + {10}
Later, new flavors were needed (C, B, T ) so more quarks needed too
Physics 841, U. Cincinnati, Fall, 2009Brian Meadows, U.
Cincinnati
SU(3) Flavor
I3
Y
d u
s
{3}
Physics 841, U. Cincinnati, Fall, 2009Brian Meadows, U.
Cincinnati
Add Charm (C)
SU(3) SU(4) Need to add b and t too !
Many more states to find ! Some surprises to come
Brian Meadows, U. Cincinnati
Mesons – Isospin Wave-functions
Iso-spin wave-functions for the quarks:u = | ½, ½ > d = | ½, -½ >
u = | ½, -½ > d = - | ½, +½ >
(NOTE the “-” convention ONLY for anti-”d”) So, for I=1 particles, (e.g. pions) we have:
+ = |1,+1>= -ud
0 = |1, 0> = (uu-dd)/sqrt(2)
- = |1,-1> = +ud An iso-singlet (e.g. or ’) would be
= |0,0> = (uu+dd)/sqrt(2)
They form SU(3) flavor multiplets. In group theory:{3} + {3}bar = {8} X{1}
Flavor wave-functions are (without proof!):
NOTE the form for singlet 1 and octet 8.
Brian Meadows, U. Cincinnati
Mesons – Flavour Wave-functions
Brian Meadows, U. Cincinnati
Mesons – Mixing(of I=Y=0 Members)
In practice, neither 1 nor 8 corresponds to a physical particle. We observe ortho-linear combinations in the JP=0- (pseudo-scalar) mesons:
= 8 cos+ 1 sin ¼ ss
’ = - 8 sin+ 1 cos ¼ (uu+dd)/sqrt 2 Similarly, for the vector mesons:
= (uu+dd)/sqrt 2 = ss
What is the difference between and ’ (or and , or K0 and K*0(890), etc.)?
The 0- mesons are made from qq with L=0 and spins opposite J=0The 1- mesons are made from qq with L=0 and spins parallel J=1
Brian Meadows, U. Cincinnati
Mesons – Masses
In the hydrogen atom, the hyperfine splitting is:
For the mesons we expect a similar behavior so the masses should be given by:
“Constituent masses” (m1 and m2) for the quarks are: mu=md=310 MeV/c2 and ms=483 MeV/c2.
The operator produces
(S=1) or for (S=0)
Determine empirically
€
rS 1 •
r S 2
€
+1
4h 2
€
−3
4h 2
Brian Meadows, U. Cincinnati
Mesons – Masses in MeV/c2
L=0
q
q
L=0
q
q
JP = 0 -
S1.S2 = -3/4 h2
JP = 1-
S1.S2 = +1/4 h2
What is our best guess for the value of A?See page 180
Brian Meadows, U. Cincinnati
Baryons are more complicated Two angular momenta (L,l) Three spins Wave-functions must be anti-symmetric (baryons are Fermions)
Wave-functions are product ofspatial(r) x spin x flavor x color
For ground state baryons, L = l = 0 so that spatial(r) is symmetric Product spin x flavor x color must therefore be anti-symmetric w.r.t.
interchange of any two quarks (also Fermions) Since L = l = 0, then J = S (= ½ or 3/2)
BaryonsL
l
xx
S = ½ or 3/2
Brian Meadows, U. Cincinnati
We find {8} and {10} for baryons
Ground State Baryons
- 0(1385) +
- 0(1532)
-(1679) ???
++ + 0(1238) -Y
I3
{10}
JP = 3/2+
n (935) p
- 0(1197) +
0(1115)
- 0(1323)
I3
Y{8}
JP = 1/2+
L = l = 0, S = ½ L = l = 0, S = 3/2
Brian Meadows, U. Cincinnati
Flavor Wave-functions {10}
Completely symmetric wrt
interchange of any two quarks
Brian Meadows, U. Cincinnati
Flavor Wave-functions {812} and {823}
Two possibilities:
Anti-Symmetric wrt interchange of 1 and
2:
Anti-Symmetric wrt interchange of 2 and
3:
Another combination 13 = 12+23 is not independent of these
Brian Meadows, U. Cincinnati
Flavor Wave-functions {1}
Just ONE possibility:
All baryons (mesons too) must be color-less. SU (3)color implies that the color wave-function is, therefore,
also a singlet: color is ALWAYS anti-symmetric wrt any pair:
color = [R(GB – BG) + G(BR – RB) + B(RG – GR)] / sqrt(6)
Anti-symmetric wrt interchange of any
pair:
Color Wave-functions
{1} = [(u(ds-ds) + d(su-us) + s(ud-du)] / sqrt(6)
Brian Meadows, U. Cincinnati
Spin Wave-functions
Clearly symmetric wrt interchange of any pair of quarks
Clearly anti-symmetric wrt interchange of quarks 1 & 2
Clearly anti-symmetric wrt interchange quarks 2 & 3
Another combination 13 = 12+23 is not independent of these
Brian Meadows, U. Cincinnati
Baryons – Need for Color
The flavor wave-functions for ++ (uuu), - (ddd) and - (sss) are manifestly symmetric
(as are all decuplet flavor wave-functions) Their spatial wave-functions are also symmetric So are their spin wave-functions! Without color, their total wave-functions would be too!! This was the original motivation for introducing color in the
first place.
Brian Meadows, U. Cincinnati
Example
Write the wave-functions fora) + in the spin-state |3/2,+1/2>
For {8} we need to pair the (12) and (13) parts of the spin and flavor wave-functions:a) Neutron, spin down: