January 31, PHY357 Lecture 8. Quark composition of hadrons. Hadron magnetic moments. Hadron masses

Transcription

1 January 3, 08 PHY357 Lecture 8 Quark composition of hadrons Hadron magnetic moments Hadron masses

2 January 3, 08 Quark rules for building Hadrons! Three types of stable quark configurations established! Baryons: qqq has baryon number + (eg. proton, neutron, )! Mesons: qq has baryon number 0 (eg. pions, kaons,...)! Anti-Baryons: qqq has baryon number - (eg. anti-proton,...)! Baryon number conservation means:! Baryon and anti-baryon can annihilate into mesons! Lightest baryon (proton) will be stable! Lightest meson(s) can only decay through EM or Weak process π 0! γ + γ π +! µ + + ν µ

3 January 3, 08 Meson Multiplets (J=0)! Saw last time how we could use iso-spin lowering operator to relate proton to neutron! Can do the same thing for the pion multiplet (Parity even) π + (ud ) : = m = 35 MeV π 0 (uu,dd ) : 0 = + m = 39 MeV π (ud) : = m = 35 MeV! Procedure would suggest that a state like η = (uu,dd ) : 0 0 = m = 549 MeV! Actually exists and is isospin singlet, parity odd

4 January 3, 08 The Strange (s) quark! First saw K mesons in accelerators along with pions (early 950s)! They became known as strange mesons due to very long lifetimes! τ π0 ~ 0-6 s, τ K ~ 0-8 s! Now understood because they include an s quark that cannot decay strongly or electromagnetically! In terms of isospin it is a singlet with I=0, I 3 = 0 s = 0 0! K mesons contain only one u (or d) quark and one s quark! Isospin doublets with: K + (us ) : 0,0 K (us) : 0,0 K 0 (ds ) : 0,0 K 0 (ds) : 0,0 Strangeness = + Strangeness = -

5 January 3, 08 Eightfold way (J=0)! Define Hypercharge as number of non-u/d quarks! Y= (Baryon) + S + C + T + B! Can draw all the light mesons in a single diagram sd su ud uu, dd ud su sd! η meson has (uu + dd + ss) quark composition

6 January 3, 08 The Meson Octet (u,d,s) S = K 0 K + isospin doublet 0 π S = 0 π η π + isospin triplet S = - Isospin singlet K K 0 isospin doublet Q = - Q = 0 Q = +

7 January 3, 08 Lowest Mass Baryons (J=/)! Can do the same trick for the lowest mass (J=/) Baryons! A single diagram with different numbers of u/d/s quarks udd uud dds uds uus dss uss

8 January 3, 08 Include Charm quarks (J=3/)! Charm becomes the vertical height in a pyramid! NB. These are all J=3/ combinations of quarks! I=3/ Δ resonances (uuu, uud, udd, ddd)! One additional sss state, the Ω - baryon! Discovered shortly after quark spectroscopy was proposed convincing evidence of quarks

9 January 3, 08 Hadron Decays! Proton is absolutely stable lightest combination of 3 quarks! Most other hadrons decay either:! Electromagnetically: If change of quark number is 0! Eg. π 0! γ + γ (τ =0-6 s)! Weakly: If one of the quark numbers changes by +/-! Eg. K +! µ + + ν µ (τ =0-8 s)! Λ! p + π - (τ =0-0 s)! To first order τ strong ~ s, τ EM ~ s, τ weak ~ s! After that energy released/# of possible final states can! Decrease τ (higher Q, more states)! Increase τ (lower Q, fewer states)! Eg. Neutron decay (Q<MeV, τ ~ 000s) very slow

10 January 3, 08 Hadron Magnetic Moments! Quark model also predicts hadron magnetic moments! Only 3 free parameters ( µ N, m u =m d =m, m s )! Can then fit all low-mass baryon moments to better than 0%! Based on the magnetic moment of a spin-/ obejct! Eg. Proton has nuclear magneton: µ N = e h /( M p )! Have measured these for the eight lowest lying baryon states! In the Λ(uds) ud-quark are in spin-0 sub-state (no contribution)! So µ Λ = µ s! Very good agreement with observations (one way to fix m s )

11 January 3, 08 Magentic Moments Baryon Octet! Baryon wavefunction constrains aa > pair in aab > state! Leads to very simple prediction: µ B = 4/3 µ a /3 µ b m u,d = 344 MeV m s = 539 MeV

12 January 3, 08 Baryon Masses! To first order just depends on mass of the constituent quarks! EM effects of like-charge pairs results in small corrections! Leads to the prediction m Ξ m Σ = m Σ m p = m Λ - m p = m s m u,d! Gives mass difference m s m u.d ~ MeV! Namely m u,d = 300+ MeV and m s ~ 500 MeV! Δ(uuu,uud,udd,ddd) resonances have very different mass from proton and neutron! Must be some spin dependent effects

13 January 3, 08 Meson Masses! Spin ½ particles with magnetic moments have interaction energy: ΔE = µ µ / r 3! Magnetic moment is given by µ = e/m. S! Radial dependence averages over the relative wavefunction! Leads to hyperfine splitting in paired electron orbitals (atomic physics) ΔE ~ S. S / m m! Same thing with strong force leading to mass corrections! Can t calculate absolute strength of strong splitting but just write M(meson) = m + m + ΔM! Where ΔM ~ S. S / m m! Two possibilities for S. S : -¾ h (J=0) or ¼ h (J=)! Leading to ΔM = -¾a/m m (J=0) and ΔM = ¼ a/m m (J=)

14 January 3, 08 Meson Predictions m u,d = 308 MeV and m s = 480 MeV

15 January 3, 08 Baryon Mass Predictions! Baryons are more complicated! Include three combinations of spin pairings (among 3 quarks)! Add up the mass splitting contributions from all three m u,d = 364 MeV m s = 537 MeV

16 January 3, 08 Summary! Relatively simple set of rules for building hadrons from quarks! Can group hadrons into families! Based on isospin and number of other quarks! Can predict magnetic moments and masses! With relatively naïve assumptions about quark wavefunctions inside hadron bound states! After we get beyond m u = m d and m s the corrections we discussed become less important! m c ~.5 GeV! m b ~ 4.8 GeV! m t = 7.38 GeV