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Baryon Predictions. Wave functions of Baryons. Baryon Magnetic Moments Baryon masses. Need to explain Parity and Charge Conjugation. Hadrons Magnetic moments. m q related to the intrinsic spin S of the quark. m = (q/mc) S and therefore for each spin-up quark:.

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Baryon Predictions

  • Wave functions of Baryons.
  • Baryon Magnetic Moments
  • Baryon masses.
  • Need to explain Parity and Charge Conjugation
hadrons magnetic moments
Hadrons Magnetic moments
  • mq related to the intrinsic spin S of the quark.
  • m=(q/mc)S
  • and therefore for each spin-up quark:

Spin down just changes the sign

hadron magnetic moments
Hadron Magnetic moments
  • Need a particles which are long-lived and have some intrinsic spin. Proton!

Total Magnetic Moment should equal the vector sum of the

magnetic moments of the constituent quarks.

Reminder: The order of the spin arrows designates which quark has that spin.

hadron magnetic moments4
Hadron Magnetic moments

Doing the calculation for the first term:

So we expect mproton to be:

hadron masses
Hadron Masses
  • Seems Simple enough
    • Just add up the masses of the quarks
      • Mp = Mu + Md = 2*Mu = 620 MeV/c2
      • Experimentally  Mp = 139 MeV/c2
        • What????

p+ is |u, d-bar>.

This is a particle made up of two like-sign charged quarks.

Why doesn’t it fly apart?

Strong Nuclear Force!

hadron masses6
Hadron Masses

Electromagnetic Force

Hyperfine splitting in hydrogen atom:

Caused by the spin of the electron interacting with the spin of the proton

Strong Nuclear Force!

hadron masses7
Hadron Masses

Strong Nuclear Force!

Masses are more equal, Force is much more powerful.

Fit to some meson masses and find

As = 160*(4pmu/h)2 MeV/c2

S1•S2 Meson Calculated Observed

p 140 138

r 780 776

K 484 496

K* 896 892

hadron masses8
Hadron Masses

Amazingly we can take the meson mass formula as the lead for

estimating baryon masses:

Fit to some baryon masses and find

As’ = 50*(4pmu/h)2 MeV/c2

Caution: There are tricks you need in order to calculate those spin

dot products. Example: if all masses are equal (proton, neutron):

Again see Griffiths, page 182.

more conserved stuff
More Conserved Stuff
  • We need to cover some more conserved quantum numbers and explain some notation before moving on.
  • Parity and Charge Conjugation:
    • Parity Y(x,y,z)Y(-x,-y,-z) notreflection in a mirror!
    • Define the parity operator ‘P’ such that:
      • P | Y(x,y,z)> = | Y(-x,-y,-z)>
      • |> is an eigenstate of P if P|> = p|>
      • P2|> = p2|> = |> so p = 1
      • Parity is a simple group. Two elements only.
eigenstates of parity
Eigenstates of Parity
  • Suppose we have a force that only acts radially between two particles.
    • Then the wave function Y = y(r)yqyqbar
      • P | yq>  | yq> = -P| yqbar>
      • Parity is a Multiplicative quantum number, not additive.
        • Given q1 and q2
        • J = S1 + S2
        • P = P1*P2
eigenstates of parity11
Eigenstates of Parity
  • For once, Baryons are easy!
    • For Mesons with no ang. Momtenum
      • P|Yb>|Ybbar> = -1 |Yb>|Ybbar>
    • DEFINE: P |Yb> 1 P |Ybbar> -1
    • So in general, for baryons with orbital angular momentum between the quarks:
      • P |Yb> = (-1)l |Yb>
  • Unfortunately, because baryon number is conserved anyway this relation is essentially useless.
eigenstates of parity12
Eigenstates of Parity
  • y(r) can be separated into the angular part Ylm(,) and a purely radial part so:
    • y(r) = (r) Ylm(,) space-part of wave function
    • P Ylm(,) = (-1)l Ylm(,)
    • And P| Y > = (-1)l pq pqbar| Y > = (-1)l(1)(-1)| Y >
    • P| Y > =(-1)l+1 | Y >
      • For MESONS only (since pq=1, pqbar=-1)
charge conjugation
Charge Conjugation
  • Cis an operator which turns all particles into antiparticles:
    • C|q> = |q-bar>
      • changes sign of charge, baryon #, flavour quan. Num.
      • Leaves momentum, spin, position, Energy unchanged.
  • Most particles are NOT eigenstates of C
    • C|Y>  a|Y> (where a = number)
    • eg.
charge conjugation14
Charge Conjugation
  • Neutral Mesons are eigenstates of C


If we apply C to the diagram on the left we change nothing but the ‘particleness’.

This doesn’t effect |q,qbar>

but has the same effect on |Y(space)> as if we’d used the parity operator.

C |Y(space)> = (-1)l+1 |Y(space)>

charge conjugation15
Charge Conjugation
  • Neutral Mesons are eigenstates of C


If we apply C|Y(spin)> what do we get?

Lets try this on a S=1 or 0 meson |ms> = |0>

C |Y(spin)> = (-1)s+1 |Y(spin)> so

C |Y> = (-1)l+s |Y> neutral mesons only

conserved by strong force
Conserved by Strong force:
  • Isospin, Quark Flavor
    • (I, I3, U, D, S, C, B, and T)
  • Parity
  • Charge Conjugation
  • Electric Charge
  • Energy/momentum
  • Angular Momentum / Spin