QCD Hadrons Heavy quarks ˇ
Strong Interactions
Leif Lönnblad
Institutionen för Astronomi och teoretisk fysik Lunds Universitet
2018-12-03
QCD Hadrons Heavy quarks ˇ
The potential Jets
Quantum Chromo Dynamics
The Potential in QED:
V(r ) = −αEM
r + k1~L · ~S + k2µeµp
In QCD?
V(r ) = −4 3
αS
r + k1′~L · ~S + k2′µqµ¯q
(4/3 from group theory) No! QCD is different:
V(r ) = −4αS
+ · · · + κr,
QCD Hadrons Heavy quarks ˇ
The potential Jets
V(r ) = −4 3
αS
r + κr V(r) 0
r Coulomb
linear total
◮ Comes from gluon self interactions
◮ κ ∼ 1 GeV/fm (15 metric tons per meter)
◮ Acts like a spring-like force between a q and a ¯q
◮ A quark can never be free! Confinement
QCD Hadrons Heavy quarks ˇ
The potential Jets
The Lund Model
◮ The field is compressed into a flux tube, d⊥≈ 0.7 fm
◮ a (mass less relativistic)string
◮ As q and ¯q flies apart more and more energy is stored in the field.
Virtual q ¯q pairs in the string can come on-shell and break
QCD Hadrons Heavy quarks ˇ
The potential Jets
Jets
◮ The break-ups are causally disconnected.
◮ The average distance between break-ups is independent of the string length.
◮ length and distance defined in terms ofrapidity y ≡ 1
2lnE+ pz
E− pz = lne+ pz
m⊥ ≈ ln|p| + pz p⊥ ≡ η
◮ Rapidity differences are invariants under Lorentz transformations along the string.
◮ The mesons formed along the strings will have limited
QCD Hadrons Heavy quarks ˇ
The potential Jets
Assuming e+e− → q¯q with some√
s, the maximum rapidity of a hadron is
ymax∼ ln√ s/m⊥ And the number of particles produced is
Ntot∝ ln
√s
hm⊥i + const
QCD Hadrons Heavy quarks ˇ
Light hadrons Classification
ˇMasses
Light hadrons
Hadrons are coloursinglets. That requires the q and ¯q in a meson to have opposite colour, but that’s not enough.
S = 1,
Sz = 1: | ↑↑i
= 0: √1
2(| ↑↓i + | ↓↑i)
= −1 : | ↓↓i or a spin 0 system
S= 0, Sz = 0 : 1
√2(| ↑↓i − | ↓↑i)
QCD Hadrons Heavy quarks ˇ
Light hadrons Classification ˇMasses
Mesons:
q ¯q= 1
√3(|r¯ri+|g¯gi+|b¯bi) Baryons:
qqq = 1
√6(|rgbi−|rbgi+|brgi−|bgri+|gbri−|grbi) = 1
√6εijk|qiqjqki
QCD Hadrons Heavy quarks ˇ
Light hadrons Classification ˇMasses
◮ QCD is SU(3). For a general SU(N) we would get baryons with N quarks.
◮ In principle we can in QCD also have tetraquark hadrons (q ¯qq ¯q)
◮ . . . or pentaquarks (q ¯qqqq)
◮ . . . or even glue balls (gg or ggg)
◮ . . . or maybe hermaphrodites (q ¯qg)
The first exotic state to be found was a pentaquark (uudd ¯s) which decayed into n+ K+.
QCD Hadrons Heavy quarks ˇ
Light hadrons Classification
ˇMasses
Classification of hadrons
◮ flavour contents (gives charge, decay patterns etc.)
◮ mass
◮ internal spin S. Mesons S= 0, 1. Baryons S = 12,32.
◮ internal angular momentum L= 0, 1, 2, . . .
◮ total spin J, ¯J = ¯S+ ¯L. (Often also simply called spin.)
◮ Radial excitation n.
◮ Relativistic corrections and mixing (quite messy).
QCD Hadrons Heavy quarks ˇ
ˆ Classification Masses
ˇSymmetries
Hadron masses
The atom analogy gives an approximation m=X
i
mi+ kX
i<j
h¯µiµ¯ji =X
i
mi+ kX
i<j
h¯SiS¯ji mimj
k ∝ |Ψ(0)|2≈ m2u· 640 MeV (mesons) ≈ m2u· 200 MeV (baryons).
To account for the binding energy we need to consider constituentquark masses.
QCD Hadrons Heavy quarks ˇ
ˆ Classification Masses
ˇSymmetries
Quark masses
quark current mass constituent mass
(MeV) (MeV)
u 2 330
d 5 330
s 100 500
c 1 250 1 600
b 4 200 5 000
t 173 000 –
QCD Hadrons Heavy quarks ˇ
ˆ Classification Masses ˇSymmetries
m=X
i
mi+ kX
i<j
h¯µiµ¯ji =X
i
mi+ kX
i<j
h¯SiS¯ji mimj
Let’s calculate the mass ofπ+. We have= u ¯d in a spin-0 state:
S¯1·¯S2= 1
2( ¯S2−¯S12−¯S22) = 1
2(S(S+1)−S1(S1+1)−S2(S2+1)) = −3 4 so
mπ = 2 · 330 −3
4 · 640 = 180 MeV The actual mass is mπ± = 140 MeV – close enough?
QCD Hadrons Heavy quarks ˇ
ˆ Masses Symmetries Decays
Charge and parity symmetries
The lightest mesons are vectors (S = 1, ρ0,ρ±,ω, . . . ) and pseudo-scalars (S= 0−,π0,π±,η, . . . )
Flavour-diagonal states may mix:
u¯u, d ¯d, s¯s ⇒ π0, η, η′
QCD Hadrons Heavy quarks ˇ
ˆ Masses Symmetries Decays
The parity of two particles in a specific angular momentum eigen state L:
P|ab; Li = |P(a)P(b); Li(−1)L= PaPb(−1)L|ab; Li with PaPb = −1 for a fermion f¯f pair and +1 for a boson pair.
QCD Hadrons Heavy quarks ˇ
ˆ Masses Symmetries Decays
For charge conjugation we have for flavour-diagonal mesons C|a¯a; L, Si = |¯aa; L, Si = (−1)(−1)L(−1)S+1|a¯a; L, Si but other mesons are not necessarily eigenstates under C.
C|a¯b; L, Si ∝ |b¯a; L, Si 6= ±|¯ab; L, Si This is due to quark mixing (next week)
QCD Hadrons Heavy quarks ˇ
ˆ Masses Symmetries Decays
Hadron Decays
Before quarks were invented, there was iso-spin. It was used to explain why some hadrons had much shorter lifetimes that others.
proton: |12,12i, neutron: |12, −12i, π±: |1, ±1i, π0:|1, 0i.
Leptons have iso-spin zero, so n→ pe−νe breaks iso-spin and is forbidden (i.e. the neutron has a long life-time)
∆+:|32,12i so ∆+→ pπ0is allowed and it decays immediately.
Today we identify u = |12,12i and d = |12, −12i.
QCD Hadrons Heavy quarks ˇ
ˆ Masses Symmetries Decays
◮ Strong decays: conserves individual quark numbers.
E.g.∆++→ pπ+(uuu→ uud + ¯d u).
◮ EM Decays: also conserves quark numbers (but not iso-spin).
E.g.π0→ γγ (|1, 0i → |0, 0i|0, 0i).
◮ Weak Decays: does not conserve quark numbers. E.g.
π+→ µ+ν¯µ(u ¯d → W+→ µ+ν¯µ).
QCD Hadrons Heavy quarks ˇ
ˆ Masses Symmetries Decays
Rules for hadronic decays:
◮ All decays must respect energy conservation.
◮ If a strong decay possible it will dominate: τ ∼ 10−23s (difficult to calculate)
◮ Otherwise, if an EM decay is possible it will dominate:
τ ∼ 10−20− 10−10 s (approximately calculable).
◮ otherwise if weak decay is possible: τ ∼ 10−10− 103s (approximately calculable).
◮ otherwise stable.
QCD Hadrons Heavy quarks ˇ
ˆ Masses Symmetries Decays
How does a hadron decay
◮ Determine the constituent quarks
◮ Find two or more particles with summed mass below the mass of the decaying hadron.
◮ If decay products have the same net quark contents as the original (e.g.∆+→ nπ+), this can be a strong decay and will dominate.
◮ If there are leptons or photons among the decay products (e.g.∆+→ pe+e−) This can be an EM decay.
◮ In weak decays the net quark content may change via q → q′W → q′f ¯f′. (if q and q′ are from different families
Hadronsˆ Heavy quarks e+e−annihilation
Charm Beauty (bottom) ˇBeauty (Top)
Heavy quarks – the October revolution
Hadronsˆ Heavy quarks e+e−annihilation
Charm Beauty (bottom)
ˇBeauty (Top)
Heavy quarks
In November 1974 a narrow peak was found in e+e−→ µ+µ− around√
s= 3 GeV. The J particle.
In another experiment colliding protons also found a peak in the µ+µ−and called it theΨ particle.
It is now called the J/Ψ particle.
Enter the charm quark.
Hadronsˆ Heavy quarks e+e−annihilation
Charm Beauty (bottom)
ˇBeauty (Top)
Charmed mesons
The lightest charm mesons are D+(c ¯d) and D0(c ¯u).
But mJ/Ψ < 2mDso J/Ψ cannot decay strongly, hence the narrow peak.
If the J/Ψ is a atomically bound state of c and ¯c, there should be excited charmonium states:
3S1(1s) = J/ψ,3S1(2s) = ψ′,3S1(3s) = ψ′′,3PJ = χc, . . . As soon as the mass of these becomes larger than 2mD we have strong decays and the peaks broadens.
Hadronsˆ Heavy quarks e+e−annihilation
Charm Beauty (bottom) ˇBeauty (Top)
They lightest DC-mesons can only decay weakly, so we can calculate the decay width in the same way as for the muon. The weak processes are c → sW+with W+→ µ+ν¯µ, e+ν¯e, u ¯d so
Γc ∼ G2Fmc5
192π3(1 + 1 + 3)
Hadronsˆ Heavy quarks e+e−annihilation
Charm Beauty (bottom)
ˇBeauty (Top)
The Bottom quark
In 1978, history repeated itself when the E288 experiment at Fermilab discovered the Upsilon and the bottom quark.
The lightest B-mesons can only decay weakly through b → W−c and b→ W−u. (b→ W−t is forbidden).
Very long lifetimes; cτ ∼ 0.4 mm.
Hadronsˆ Heavy quarks e+e−annihilation
ˆ Beauty (bottom) Beauty (Top)
The top quark
◮ Predicted since the b-quark was found.
◮ Finally discovered at the Tevatron at Fermilab in 1995.
u¯u→ g⋆ → t¯t → bW+bW¯ −
◮ Very short-lived and decays immediately through t → bW+.
◮ There are no top hadrons.
Hadronsˆ Heavy quarks e+e−annihilation
The hadronic ratio The gluon
e
+e
−physics
e+e−→ γ⋆/Z0→ f¯f For low√
s we can ignore Z0exchange and get M = ee(¯eγµe)1
sef(¯fγµf) and
σ = 4π 3
Qf2α2 s .
√
Hadronsˆ Heavy quarks e+e−annihilation
The hadronic ratio The gluon
Rhad = σ(e+e−→ hadrons)
σ(e+e−→ µ+µ−) = 3X
q;2mq<√ s
Q2q
Allows us to measure the number of quarks with 2mq<√ s and their charge and number of colour states.
Hadronsˆ Heavy quarks e+e−annihilation
The hadronic ratio The gluon
The gluon
The PETRA accelerator at DESY 1978
The gluon gives an extra jet: e+e−→ q¯qg