phy357 – lecture 8william/courses/phy357/phy357_l08.pdf · uud, udd, ddd) ! one ... in the...
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PHY357 – Lecture 8
Quark composition of hadrons
Hadron magnetic moments
Hadron masses
January 31, 2018
Quark rules for building Hadrons
! Three types of stable quark configurations established ! Baryons: qqq has baryon number +1 (eg. proton, neutron,…)
! Mesons: qq has baryon number 0 (eg. pions, kaons, ...)
! Anti-Baryons: qqq has baryon number -1 (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
π0 ! γ + γ π+ ! µ+ + νµ
January 31, 2018
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)
! Procedure would suggest that a state like
! Actually exists and is isospin singlet, parity odd
January 31, 2018
π +(ud ) : 1 1 = 1212
1212
π 0 (uu ,dd ) : 1 0 =121212
12−12
+ 1212−12
1212
π −(ud ) : 1 −1 =12−12
12−12
η = (uu ,dd ) : 0 0 =121212
12−12
− 1212−12
1212
m = 135 MeV
m = 135 MeV
m = 139 MeV
m = 549 MeV
The Strange (s) quark
! First saw K mesons in accelerators along with pions (early 1950s) ! They became known as ‘strange’ mesons due to very long lifetimes
! τπ0 ~ 10-16 s, τK ~ 10-8 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, I3 = 0
! K mesons contain only one u (or d) quark and one s quark ! Isospin doublets with:
January 31, 2018
s = 0 0
K +(us ) : 1212
0,0
K 0 (ds ) : 12−12
0,0
K −(us) : 12−12
0,0
K 0 (ds) : 1212
0,0
Strangeness = +1 Strangeness = -1
! 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
! η’ meson has (uu + dd + ss) quark composition
Eightfold way (J=0)
su sd
udud
susd
,uu dd
January 31, 2018
K − 0K
π +
0π
π −
K +0K
η
S = 1
S = 0
S = -1
Q = -1 Q = 0 Q = +1
The Meson Octet (u,d,s)
isospin doublet
isospin doublet
isospin triplet
Isospin singlet
January 31, 2018
Lowest Mass Baryons (J=1/2)
January 31, 2018
dss uss
uusdds
uududd
uds
! Can do the same trick for the lowest mass (J=1/2) Baryons ! A single diagram with different numbers of u/d/s quarks
Include Charm quarks (J=3/2)
! Charm becomes the vertical height in a pyramid
! NB. These are all J=3/2 combinations of quarks
! I=3/2 Δ resonances (uuu, uud, udd, ddd)
! One additional sss state, the Ω- baryon
! Discovered shortly after quark spectroscopy was proposed – convincing evidence of quarks
January 31, 2018
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. π0 ! γ + γ (τ =10-16 s)
! Weakly: If one of the quark numbers changes by +/- 1
! Eg. K+ ! µ+ + νµ (τ =10-8 s)
! Λ ! p + π- (τ =10-10 s)
! To first order τstrong ~ 10-22-10 -24s, τEM ~ 10-16-10-21s, τweak ~ 10-7-10-13s ! 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<1MeV, τ ~ 1000s) – very slow
January 31, 2018
Hadron Magnetic Moments
! Quark model also predicts hadron magnetic moments ! Only 3 free parameters ( µN, mu=md=m, ms)
! Can then fit all low-mass baryon moments to better than 10%
! Based on the magnetic moment of a spin-1/2 obejct ! Eg. Proton has 1 nuclear magneton: µN = e h /(2 Mp)
! 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 µΛ = µs
! Very good agreement with observations (one way to fix ms)
January 31, 2018
Magentic Moments Baryon Octet
! Baryon wavefunction constrains |aa > pair in |aab > state
! Leads to very simple prediction: µB = 4/3 µa – 1/3 µb
January 31, 2018
mu,d = 344 MeV ms = 539 MeV
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
mΞ – mΣ = mΣ – mp = mΛ - mp = ms – mu,d
! Gives mass difference ms – mu.d ~ 150-200 MeV
! Namely mu,d = 300+ MeV and ms ~ 500 MeV
! Δ(uuu,uud,udd,ddd) resonances have very different mass from proton and neutron ! Must be some spin dependent effects
January 31, 2018
Meson Masses ! Spin ½ particles with magnetic moments have interaction energy:
ΔE = µ1 µ2 / r3 ! 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 ~ S1. S2 / m1m2
! Same thing with strong force leading to mass corrections ! Can’t calculate absolute strength of strong splitting but just write
M(meson) = m1 + m2 + ΔM ! Where
ΔM ~ S1. S2 / m1m2
! Two possibilities for S1. S2: -¾ h2 (J=0) or ¼ h2 (J=1)
! Leading to ΔM = -¾a/m1m2 (J=0) and ΔM = ¼ a/m1m2 (J=1)
January 31, 2018
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
January 31, 2018
mu,d = 364 MeV ms = 537 MeV
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 mu = md and ms the corrections we discussed become less important
! mc ~ 1.5 GeV
! mb ~ 4.8 GeV
! mt = 172.38 GeV
January 31, 2018