Strong Interactions

Strong Interactions

Leif Lönnblad

Institutionen för Astronomi och teoretisk fysik Lunds Universitet

2018-12-03

Quantum Chromo Dynamics

The Potential in QED:

V(r ) = −α^EM

r + k1~L · ~S + k2µeµp

In QCD?

V(r ) = −4 3

α_S

r + k₁^′~L · ~S + k2^′µqµ¯q

(4/3 from group theory) No! QCD is different:

V(r ) = −4αS

+ · · · + κr,

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

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

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− p^z = lnE+ pz

m_⊥ ≈ ln|p| + p^z p_⊥ ≡ η

◮ Rapidity differences are invariants under Lorentz transformations along the string.

◮ The mesons formed along the strings will have limited

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

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)

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 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⁺.

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).

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.

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 –

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

m_π = 2 · 330 −3

4 · 640 = 180 MeV The actual mass is m_π^± = 140 MeV – close enough?

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^′

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.

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)

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.

◮ 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).

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.

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

Heavy quarks – the October revolution

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.

Charmed mesons

The lightest charm mesons are D⁺(c ¯d) and D⁰(c ¯u).

But m_J/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.

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 ∼ G²_Fm_c⁵

192π³(1 + 1 + 3)

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.

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.

e

e

physics

e⁺e⁻→ γ^⋆/Z⁰→ f¯f For low√

s we can ignore Z⁰exchange and get M = e^e(¯eγµe)1

sef(¯fγ^µf) and

σ = 4π 3

Q_f²α² s .

√

Rhad = σ(e⁺e⁻→ hadrons)

+ − + − = 3X

Q²_q

The gluon

The PETRA accelerator at DESY 1978