Particle Physics
Dr Victoria Martin, Prof Steve Playfer
Spring Semester 2013
Lecture 12: Mesons and Baryons
★Mesons and baryons
★Strong isospin and
strong hypercharge
★SU(3) flavour symmetry
★Heavy quark states
1
Review from Friday
• Using high energy deep inelastic scattering, e− p → e− X, we find out
the proton consists of partons:
• three valance quarks: two up quarks, one down quark
• gluons
• pairs of quark−anti-quark sea quarks
• In today’s lecture consider the valance quark content of the mesons
and baryons
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Mesons and Baryons
• Mesons are quark-antiquark bound
states with symmetric colour
wavefunction which is colour neutral:
�
1 �
χc = √ rr̄ + bb̄ + gḡ
3
• Baryons are three quark bound states.
They have an antisymmetric colour
wavefunction which is colour neutral:
1
χc = √ [rgb − rbg + gbr − grb + brg − bgr]
6
• As hadrons are colour neutral, they do not interact with each other by single
gluon exchange.
• Instead they couple to each other by hadron exchange, typically through the
lightest qq̅ meson, pion (π+, π0, π−)
• Yukawa (1935) – the finite range of strong interactions between hadrons is due to
the pion mass of ~140 MeV
3
Constituent Quark Masses
• Because of QCD renormalisation, there is an ambiguity as how to
define an absolute mass. The most commonly used definition of a
mass is known as the “M̅S̅ scheme”.
➡mu= 2.4 MeV, md = 4.8 MeV, ms =104 MeV
• These are too small to account for the hadron masses!
• The majority of the mass of hadrons comes from QCD interactions.
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Flavour Symmetries: Isospin
• Flavour symmetries are symmetries between interchange of quark flavour.
• Flavour symmetries were proposed before quarks were hypothesised to explain observed
phenomena.
• Strong interactions are (approximately) invariant under flavour symmetry rotations.
• Assign quantum numbers to characterise these symmetries.
➡ Strong Isospin (I, I3);
➡ Quark flavour quantum numbers: Strangeness (S), Charm (C), Beauty (B), Truth (T)
➡ Strong hypercharge (Y)
• Strong isospin is a symmetry between u and d quarks
• Strong interactions are invariant under strong isospin rotations u ↔ d, or
equivalently, p ↔ n
• The u and d quarks form an isospin doublet:
• They have the same total strong isospin (I), but different third-components (I3)
I = 1/2 with I3(u)= +1/2 and I3(u)= −1/2
(by analogy to S=1/2 with spin states ↑ and ↓ )
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SU(3) Flavour Symmetry
• An SU(3) flavour symmetry is exhibited between u, d and s quarks
• The symmetry is broken by the s quark mass ms ~ 100 MeV >> mu, md
• Strong interactions are almost invariant under SU(3) flavour symmetry
• The s quark is assigned a strangeness S=−1 ( s̅ has S=+1)
• Two useful combinations:
• Strong Hypercharge Y = S + B (where B = ⅓ [N(q)−N(q̅)] is baryon number)
• Electric charge: Q = I3 + Y/2
These are the basic building blocks for constructing
meson (q q̅) and baryon (qqq) multiplets
• Baryons generally have with no orbital angular momentum (L) between the
quarks.
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The J=0 Pseudoscalar Mesons
• Total angular momentum, J=0: orbital angular momentum, L=0; one spin-up and
one spin-down ↑ ↓ quark
• The allowed flavour combinations are given by the Gell Mann λ matrices,
➡ same matrices that describe the allowed colour combinations of gluons.
M(K0, K̅0) = 498 MeV
S Y
K0 (d s ̅)
K+
(u
M(K+,K−) = 494 MeV
s ̅)
M(π+,π−) = 140 MeV
0 +1
−1 0
M(π0)=135 MeV
M(η)=550 MeV
π0,η1,η8
π− (d u̅)
π+ (u d̅)
M(η’)=960 MeV
π0 = 1/√2 [ d d̅ − u u̅ ]
η8 = 1/√6 [ d d̅ + u u̅ – 2 s s̅ ]
−2 −1
K−
−1
(s u̅)
−1/2
η1 = 1/√3 [ d d̅ + u u̅ + s s̅ ]
K̅0 (s d̅)
0
+1/2
+1
I3
Observed η,η’ mesons are
mixtures of η1 and η8
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The J=1 Vector Mesons
• Total angular momentum, J=1: L=0, both quarks with same spin ↑ ↑
M(K*+,K*−) = 892 MeV
S Y
M(K*0, K*0) = 896 MeV
0 +1
M(ρ+,ρ−) = 776 MeV
M(ρ0)=767 MeV
M(ω)=783 MeV
−1 0
M(φ)=1019 MeV
ω = 1/√2 [ d d̅ + u u̅ ]
−2 −1
φ = s s̅
ρ0 = 1/√2 [ d d̅ – u u̅ ]
−1
−1/2
0
+1/2
+1
I3
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The J=1/2 Baryon Octet
• L=0, Quark spin composition is ↑↑↓
S Y
ddu
0 +1
M(n) = 940 MeV
uud
M(p) = 938 MeV
M(Λ) = 1116 MeV
−1 0
dds
−2 −1
uus
dss
−1
−1/2
M(Σ) = 1193 MeV
M(Ξ) = 1318 MeV
uss
0
+1/2
Λ0 = [uds] isospin singlet state I=0
Σ0 = [uds] isospin triplet state I=1 +1
I3
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The J=3/2 Baryon Decuplet
• Total angular momentum J=3/2: all spins aligned, L=0
S Y
0 +1
M(Δ) = 1232 MeV
0
M(Σ∗)=1383 MeV
−2 −1
M(Ξ∗)=1532 MeV
−3 −2
M(Ω)=1672 MeV
−1
−3/2
−1
−1/2
0
+1/2
+1
+3/2 I3
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++
Δ
and Baryon Wavefunctions
• The overall wavefunction of a system of identical fermions must be
antisymmetric under exchange of any two fermions
ψ (Δ++) = u↑u↑u↑= χc χf χS χL
➡ The Δ++ wavefunction is symmetric in flavour χf and spin χS (J=3/2)
➡ There is no orbital angular momentum L=0, spatially symmetric χL
➡ Hence it must have an antisymmetric colour wavefunction χc
• Further evidence for quark colour
• Why are there no J=1/2
uuu, ddd, sss baryons?
Baryon
Colour
Δ++
p
A
A
Flavour
Spin
S
S
A or S A or S
Spatial
Total
S
S
A
A
• Full proton wave function is:
1
ψ(p) = √ [u↓ u↑ d↑ + u↑ u↓ d↑ − 2 u↑ u↑ d↓ + u↓ d↑ u↑ + u↑ d↓ u↑ − 2 u↑ d↑ u↓ + d↓ u↑ u↑ + d↑ u↓ u↑ − 2 d↑ u↑ u↓ ]
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Heavy Quark Mesons and Baryons
• Can define an SU(4) symmetry u ↔ d ↔ s ↔ c ↔
• Heavy quark mesons and baryons obtained by replacing one
(or more) of the light u,d,s quarks by a heavy c or b quark
• There are no bound state hadrons containing t quarks
• Lowest lying charm meson states with M(D) ~ 1.9 GeV:
D+ ( c d̅̅ ) , D− ( c̅̅ d ) , D0 ( c u̅ ) , D̅0 ( c̅ u ), Ds+ ( c s ) , Ds− ( c s̅ )
• Lowest lying bottom meson states with M(B) ~ 5.3 GeV:
B0 ( b̅ d ) , B̅0 ( b d̅̅ ), B− ( b u̅ ), B+ ( b̅ u ), B̅S0 ( b̅ s ) , Bs0 ( b s̅ )
• Baryons Λc (cud) , Λb (bud) …
• Charmonium ( c c̅ ) and Bottomonium ( b b̅ )
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Resonances
hadrons decay due to strong force, hadrons have very short lifetime
• Some −24
τ ~ 10
s
• Evidence for the existence of these states are resonances in the
experimental data
• Shape is Breit-Wigner distribution:
Γ2 /4
σ = σmax
(E − M )2 + Γ2 /4
• (with Γ calculated from Fermi’s Golden rule, Γ~ |M|2 ρ)
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41. Plots of cross sections and related quantities
2010 — Subatomic: Particle Physics
!"#$$%$&'()#*%+,-.
pπ scattering
2. Draw a Feynman diagram for the process
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2
pπ + → ∆++ → pπ + .
2
π + p total
⇓
10
π + p elastic
Plab GeV/c
10
√s GeV
10
-1
πp
πd
1
10
1.2
2.2
2
3
3
4
4
5
2
⇓
6
10
5
7
6
2
7 8 9 10
8 9 10
20
20
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You have to think of the constituents quarks. We have uud+ d̄u → uuu → uud+ d̄u.
The down and anti-down quarks annihilate by producing a gluon. The gluon can
be absorbed by any of the quarks. Remember as long as the flavour of the quark
doesn’t change, a gluon can be emitted or absorbed by any quark. After about
30
40
10−24 s (any) one of the quarks emits a gluon which turns into a dd̄ pair. It decays
so quickly as the strong coupling constant is large, so the quarks are very likely to
40 50 emit
60 an energetic enough gluon to make a dd̄, and because the minimum energy
state is favoured. The mass of the pion and the proton is smaller than the mass of
the ∆++ .
This looks like a pretty silly Feynman diagram, but the uuu state actually lives for
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slightly longer than just three quarks would if they didn’t form a bound state.
Discovery of the Heavy Quarks
• Collider experiments discovered the charm (1974),
bottom (1977) and top quarks (1995)
+e−→cc̅ e+e−→bb̅
Produced
in
pairs
by
e
•
• At threshold (E~2mq) bound cc̅, bb̅ states are
narrow resonances
• Produced in pairs at hadron colliders through one
gluon and two gluon diagrams
• (Single c,b,t production requires an intermediate
W boson)
• Can identify heavy quark jets by tagging decays of
c and b quarks with lifetimes τc ~0.4ps, τb~1.5ps
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Charmonium & Bottomonium
}
1st observed by ATLAS in 2011
• Analogous to hydrogen spectroscopy with quark-quark potential
Vqq̅ (r) = −4/3 αS/r + kr (see lecture 9)
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tt̅
+
→W b
−
W b̅
candidate event
• Lines are project paths
of charged particles
through the detector.
• Not all particles
originate from collision
point.
• Particle produced and
travelled short distance
before decaying,
indicates production of
a b-quark!
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Summary
• Quarks are confined to colourless bound states, collectively known as hadrons:
➡ mesons: quark and anti-quark. Bosons (s=0, 1) with a symmetric colour
wavefunction.
➡ baryons: three quarks. Fermions (s=1/2, 3/2) with antisymmetric colour
wavefunction.
➡ anti-baryons: three anti-quarks.
• Lightest mesons & baryons characterised by strong isospin (I, I3), strangeness (S) and
strong hypercharge Y
➡ strong isospin I=½ for u and d quarks; (isospin combined as for spin)
➡ I3=+½ (strong isospin up) for up quarks; I3=−½ (strong isospin down) for down quarks
➡ S=+1 for strange quarks (additive quantum number)
➡ strong hypercharge Y = S + B
• Hadrons display SU(3) flavour symmetry between u d and s quarks.
The symmetry
predicts the allowed meson and baryon states.
• Strong decays of hadrons cause resonances due to very short lifetimes.
• Residual strong force interactions between colourless hadrons propagated by mesons.
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