bound states

Originally form Brian Meadows, U. Cincinnati

Bound States

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 #


“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


“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


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+


Elect. chargeQ = Y + I3/2


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



Discovery of the - Hyperon

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


Hadron and Meson

Wavefunctions


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.


Mesons – Flavour Wave-functions


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


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


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


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


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


Flavor Wave-functions {10}

Completely symmetric wrt

interchange of any two quarks


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


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)


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


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.


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:


Magnetic Moments of Ground State Baryons


Masses of Ground State Baryons

bound states

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