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## Bound States

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**Bound States**Originally form Brian 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, … originally from B. Meadows, U. Cincinnati**Gell-Mann-Nishijima Relationship**Applies to all hadrons • Define hypercharge Y = B + S+ C + B’ + T • Then electric charge is Q = I3 + Y / 2 Relatively recently added Bayon # Third component of I-spin Brian Meadows, U. Cincinnati**Y**K+ K0(497) +1 p- p+ p0(135) 0 /’ (548/960) -1 K- K0(497) I3 -1 0 +1 “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 p’s and the K’s Pseudo-scalar mesons: All mesons here have Spin J = 0 and Parity P = -1 Centroid is at origin Brian Meadows, U. Cincinnati**Y**K*+ K*0(890) +1 - + 0(775) 0 0/ (783)/(1020) -1 K*0(890) K*- I3 -1 0 +1 “Eight-Fold Way” (Meson Resonances) • Also works for all the vector mesons (JP = 1-) Vector mesons: All mesons here have Spin J = 1 and Parity P = -1 Brian Meadows, U. Cincinnati**Y**p n (935) {8} +1 S- S+ S0(1197) 0 L0(1115) X0(1323) X- -1 I3 -1 0 +1 “Eight-Fold Way” (Baryons) • Also works for baryons with same JP JP = 1/2+ Elect. charge Q = Y + I3/2 Centroid is at origin Brian Meadows, U. Cincinnati**“Eight-Fold Way” (Baryons)**• Also find {10} for baryons with same JP JP = 3/2+ Y D++ D+ D0(1238) D- +1 {10} S- S0(1385) S+ 0 Centroid is at origin X0(1532) X- -1 W-(1679) ??? I3 -1 0 +1 G-M predicted This to exist Brian Meadows, U. Cincinnati**Discovery of the W- Hyperon**Brian Meadows, U. Cincinnati**SU(3) Flavor**• 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: +1 Y T+/- u d • Y • I3 • T+, T- • U+, U- • V+, V- 0 U+/- V+/- s -1 I3 -1/2 0 +1/2**+1**Y {3} u d 0 s -1 I3 -1/2 0 +1/2 SU(3) Flavor • 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 Brian 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**Hadron and Meson**Wavefunctions 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: p+ = |1,+1>= -ud p0 = |1, 0> = (uu-dd)/sqrt(2) p- = |1,-1> = +ud • An iso-singlet (e.g. h or h’) would be h = |0,0> = (uu+dd)/sqrt(2) Brian Meadows, U. Cincinnati**Mesons – Flavour Wave-functions**• 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 h1 and octet h8. Brian Meadows, U. Cincinnati**Mesons – Mixing(of I=Y=0 Members)**• In practice, neither h1 nor h8 corresponds to a physical particle. We observe ortho-linear combinations in the JP=0- (pseudo-scalar) mesons: h = h8 cosq + h1 sinq¼ ss h’ = - h8 sinq + h1 cosq ¼ (uu+dd)/sqrt 2 • Similarly, for the vector mesons: w = (uu+dd)/sqrt 2 f = ss • What is the difference between w and h’ (or f and h, or K0 and K*0(890), etc.)? • The 0- mesons are made from qq with L=0 and spins opposite J=0 • The 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=310MeV/c2 and ms=483MeV/c2. • The operator produces (S=1) or for (S=0) Determine empirically Brian Meadows, U. Cincinnati**Mesons – Masses in MeV/c2**JP = 1- S1.S2 = +1/4 h2 JP = 0 - S1.S2 = -3/4 h2 q q L=0 L=0 q q What is our best guess for the value of A? See page 180 Brian Meadows, U. Cincinnati**L**x x l Baryons • Baryons are more complicated • Two angular momenta (L,l) • Three spins • Wave-functions must be anti-symmetric (baryons are Fermions) • Wave-functions are product of spatial(r) x spin x flavor x color • For ground state baryons, L = l = 0 so that spatial(r)is symmetric • Productspin 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) S = ½ or 3/2 Brian Meadows, U. Cincinnati**JP = 3/2+**JP = 1/2+ Y p n (935) Y {8} +1 D++ D+ D0(1238) D- +1 S- S+ S0(1197) 0 {10} S- S0(1385) S+ 0 L0(1115) X0(1323) X- -1 X0(1532) X- -1 I3 W-(1679) ??? I3 -1 0 +1 -1 0 +1 Ground State Baryons • We find {8} and {10} for baryons 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**Anti-Symmetric wrt interchange of 1 and 2:**Anti-Symmetric wrt interchange of 2 and 3: Flavor Wave-functions {812} and {823} • Two possibilities: 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: {1} = [(u(ds-ds) + d(su-us) + s(ud-du)] / sqrt(6) Color Wave-functions 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 for • + 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: • Neutron, spin down: Brian Meadows, U. Cincinnati**Magnetic Moments of Ground State Baryons**Brian Meadows, U. Cincinnati**Masses of Ground State Baryons**Brian Meadows, U. Cincinnati