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

Meson Predictions

January 31, 2018

mu,d = 308 MeV and ms = 480 MeV

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