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Originally form Brian Meadows, U. Cincinnati Bound States

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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, …

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Brian Meadows, U. Cincinnati Gell-Mann-Nishijima Relationship Applies to all hadrons Define hypercharge Y = B + S + C + B’ + T Then electric charge is Q = I 3 + Y / 2 Relatively recently added Third component of I-spin Bayon #

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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. I 3 Use your book to find the masses of the ’s and the K’s K-K- K 0 (497) -- 0 (135) ++ ’ (548/960) K KK I3I3 Y Pseudo-scalar mesons: All mesons here have Spin J = 0 and Parity P = -1 Centroid is at origin

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Brian Meadows, U. Cincinnati “Eight-Fold Way” (Meson Resonances) Also works for all the vector mesons (J P = 1 - ) K*-K*- K * 0 (890) -- 0 (775) ++ 0 / (783)/(1020) K K I3I3 Y Vector mesons: All mesons here have Spin J = 1 and Parity P = -1

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Brian Meadows, U. Cincinnati Also works for baryons with same J P “Eight-Fold Way” (Baryons) n (935) p -- 0 (1197) ++ 0 (1115) -- 0 (1323) I3I3 Y {8} J P = 1/2 + Centroid is at origin Elect. charge Q = Y + I 3 /2

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Brian Meadows, U. Cincinnati Also find {10} for baryons with same J P “Eight-Fold Way” (Baryons) -- 0 (1385) ++ -- 0 (1532) - (1679) ??? ++ ++ 0 (1238) -- Y I3I3 {10} J P = 3/2 + G-M predicted This to exist Centroid is at origin

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Brian Meadows, U. Cincinnati Discovery of the - Hyperon

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3 quark flavors [uds] calls for a group of type SU(3) SU(2): N=2 eigenvalues(J2,Jz), N 2 -1=4-1=3 generators (J x,J y,J z ) SU(3): N=3 eigen values (uds), N 2 -1=9-1=8 generators (8 Gell-Man mat.) or smarter: SU(3) Flavor I3I3 Y d u s V +/- T +/- U +/- Y I 3 T +, T - U +, U - V +, V -

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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 I3I3 Y d u s {3}

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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

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Brian Meadows, U. Cincinnati Hadron and Meson Wavefunctions

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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)

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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

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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 J P =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 K 0 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

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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” (m 1 and m 2 ) for the quarks are: m u =m d =310 MeV/c 2 and m s =483 MeV/c 2. The operator produces (S=1) or for (S=0) Determine empirically

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Brian Meadows, U. Cincinnati Mesons – Masses in MeV/c 2 L=0 q q q q J P = 0 - S 1. S 2 = -3/4 h 2 J P = 1 - S 1. S 2 = + 1/4 h 2 What is our best guess for the value of A?See page 180

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Brian Meadows, U. Cincinnati

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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 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) Baryons L l x x S = ½ or 3/2

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Brian Meadows, U. Cincinnati We find {8} and {10} for baryons Ground State Baryons -- 0 (1385) ++ -- 0 (1532) - (1679) ??? ++ ++ 0 (1238) -- Y I3I3 {10} J P = 3/2 + n (935) p -- 0 (1197) ++ 0 (1115) -- 0 (1323) I3I3 Y {8} J P = 1/2 + L = l = 0, S = ½ L = l = 0, S = 3/2

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Brian Meadows, U. Cincinnati Flavor Wave-functions {10} Completely symmetric wrt interchange of any two quarks

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Brian Meadows, U. Cincinnati Flavor Wave-functions {8 12 } and {8 23 } 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

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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)

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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

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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.

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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:

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Brian Meadows, U. Cincinnati Magnetic Moments of Ground State Baryons

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Brian Meadows, U. Cincinnati Masses of Ground State Baryons

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