Gluons • Color Transformations • Quark Confinement • Scaling Violations • the Ratio R • Deep Inelastic Scattering • Structure of the Proton 2 Introductionqcd

Home , Charge (physics), Color charge, Down quark, Singlet state, Up quark

Quantum Chromo Dynamics

• Evidence for Fractional Charges • Color • Gluons • Color Transformations • Quark Confinement • Scaling Violations • The Ratio R • Deep Inelastic Scattering • Structure of the Proton 2 IntroductionQCD

QUANTUM ELECTRODYNAMICS: is the quantum theory of the electromagnetic interaction. ★ mediated by massless photons ★ photon couples to electric charge, ★ Strength of interaction : .

QUANTUM CHROMO-DYNAMICS: is the quantum theory of the strong interaction. ★ mediated by massless gluons, i.e. propagator ★ gluon couples to “strong” charge ★ Only quarks have non-zero “strong” charge, therefore only quarks feel strong interaction Basic QCD interaction looks like a stronger version of QED, QED q QCD q Q √α √αS q q q

2 2 α = e /4π ~ 1/137 γ αS = gS /4π ~ 1 g

( subscript em is sometimes used to distinguish the of electromagnetism from ).

Dr M.A. Thomson Lent 2004 2 QCD

QUANTUM ELECTRODYNAMICS: is the quantum theory of the electromagnetic interaction. ★ mediated by massless photons ★ photon couples to electric charge, ★ Strength of interaction : .

QUANTUM CHROMO-DYNAMICS: is the quantum theory of the strong interaction. ★ mediated by massless gluons, i.e. propagator ★ gluon couples to “strong” charge ★ Only quarks have non-zero “strong” charge, therefore onlyIntroductionquarks feel strong interaction Basic QCD interaction looks like a stronger version of QED, QED q QCD q Q √α √αS q q q

2 2 α = e /4π ~ 1/137 γ αS = gS /4π ~ 1 g 3

( subscript em isCOLOUR sometimes used to distinguish the of electromagnetism from ).

Dr M.A.In Thomson QED: Lent 2004 ★ Charge of QED is electric charge. ★ Electric charge - conserved quantum number. In QCD: ★ Charge of QCD is called “COLOUR” ★ COLOUR is a conserved quantum number with 3 VALUES labelled “red”, “green” and “blue”

Quarks carry “COLOUR” Anti-quarks carry “ANTI-COLOUR”

Leptons, , , DO NOT carry colour, i.e. “have colour charge zero” DO NOT participate in STRONG interaction. Note: Colour is just a label for states in a non-examinable SU(3) representation

Dr M.A. Thomson Lent 2004 Evidence for Fractional Charge Production of lepton pairs in Pion-Carbon scattering: Electromagnetic annihilation of a quark and an anti-quark in pion- nucleon collisions: a virtual photon converts into a muon pair

qq annihilate → photon → μ– + μ+ The cross section σ ~ charge of q × charge of q Since the q and q have opposite charges:

σ ~ [charge of q]2 Evidence for Fractional Charge

Use an 'Isoscalar' target – same number of u and d quarks

12 6C has 6 protons (uud) and 6 neutrons (udd) → 18 u + 18 d For π– (ud) → uu annihilation to μ– μ+ For π+ (ud) → dd annihilation to μ– μ+

− + − 2 σ(π C → µ µ + anything) ∝18 Qu =18 × 4 /9 + + − 2 σ(π C → µ µ + anything) ∝18 Qd =18 ×1/9

€ € Good agreement with experiment Also results from electron and neutrino scattering from nuclei Quark Colour Why do we need coloured quarks?

P + ++ In the baryon decuplet J = 3/2 , the Δ is u↑ u↑ u↑ with angular momentum ℓ = 0 i.e. 3 fermions in the same state

→ violates Pauli Exclusion Principle

Wavefunction symmetric in space, spin and flavour not allowed for fermions

Introduce a new Degree of Freedom → Colour

3 'colours' red r, blue b, green g

Now the 3 fermions are in different states:

u↑ (red) u↑ (blue) u↑ (green)

Not real colour – just a new quantum number that distinguishes them! Colour The corner states of the decuplet (uuu), (ddd) and (sss) are symmetric under flavor- interchange (therefore, also the other states are assumed to behave the same). The decuplet members are indeed the 10 quark combinations (out of 27) symmetric under space, spin and flavor exchange

We also assume that each baryon is a properly normalized combination of permuted 1 states, e.g. : (ddu) = (ddu + udd + dud) 3 1 (dus) = (dsu + uds + sud + sdu + dus + usd) 6 Since the decuplet members are spin 3/2 baryons of the lowest mass, we assume the three quarks are in the (spatially symmetric) ground state (l=0). This implies that the three spins are parallel: is the Pauli principle violated? Need another degree of freedom (quantum number) that can be different for the three quarks: color(red, blue, green) such that the hadron net color is 0. Indeed, experimentally one only detects protons and not red or green protons…. The three colors are the source of the strong-force charge. The strong force is independent

of the quark color, i.e. the SU(3)color symmetry is exact. Quark Colour However – they work like colours on a computer red green blue Red 1 0 0 Green 0 1 0 Blue 0 0 1 Cyan 0 1 1 = anti-red r Magenta 1 0 1 = anti-green g Yellow 1 1 0 = anti-blue b White 1 1 1 Black 0 0 0 Baryons consist of r + g + b = 1 1 1 = white – colourless Mesons consist of r + r = 1 0 0 + 0 1 1 = 1 1 1 = white – colourless (or g + g or b + b) Gluons

Expect 9 combinations (r, g, b) × (r, g, b)

These are: r b Colourless and symmetric r g under r → b → g colour g r change. Doesn't interact g b b g b r mix r r g g 8 interacting gluons b b Colour Charge The 'Color Charge' is the Strong Interaction equivalent of Electric Charge in Electromagnetism Quantum Chromodynamics (QCD) is the quantum field theory of the strong 'color' interaction between quarks

The strong force is carried by 8 massless vector gluons (JP = 1– ). They are equivalent to qq pairs i.e. rb, gr, .. etc ..

This is an non-Abelian theory where the force carriers have the charge and can interact with each other – unlike electromagnetism where the photons carry NO charge and don't interact with each other QCD Colour Transformations

QCD is a gauge theory. Gauge means scale or absolute potential. Consider a proton made up of one red, one green and one blue quark. Do a global colour transformation red ⇌ green everywhere

Still 'colourless' – nothing has changed QCD ColourLocal Colour TransformationsTransformations Now do a local colour transformation red ⇌ green at one quark only

Can be made colourless if A emits an rg gluon which is absorbed at C. When the gluon is absorbed at C the green and anti-green No longer colourless annihilate to make C red.

g + rg → r Colourless again

Local colour transformation → exchange of gluons → force Particle Physics

Quantum Chromodynamics

The Local Gauge The Local Gauge PrinciplePrinciple

, All the interactions between fermions and spin-1 bosons in the SM are specified by the principle of LOCAL GAUGE INVARIANCE

, To arrive at QED, require physics to be invariant under the local phase transformation of particle wave-functions

, Note that the change of phase depends on the space-time coordinate: •Under this transformation the Dirac Equation transforms as

•To make “physics”, i.e. the Dirac equation, invariant under this local phase transformation FORCED to introduce a massless gauge boson, . + The Dirac equation has to be modified to include this new field:

•The modified Dirac equation is invariant under local phase transformations if: Gauge Invariance The Local Gauge Principle

, For physics to remain unchanged – must have GAUGE INVARIANCE of the new field, i.e. physical predictions unchanged for

,Hence the principle of invariance under local phase transformations completely specifies the interaction between a fermion and the gauge boson (i.e. photon):

interaction vertex:

QED !

, The local phase transformation of QED is a unitary U(1) transformation i.e. with

Now extend this idea…

From QED to QCD , Suppose there is another fundamental symmetry of the universe, say “invariance under SU(3) local phase transformations”

• i.e. require invariance under where are the eight 3x3 Gell-Mann matrices introduced in handout 7 are 8 functions taking different values at each point in space-time 8 spin-1 gauge bosons wave function is now a vector in COLOUR SPACE QCD ! , QCD is fully specified by require invariance under SU(3) local phase transformations Corresponds to rotating states in colour space about an axis whose direction is different at every space-time point

interaction vertex: , Predicts 8 massless gauge bosons – the gluons (one for each ) , Also predicts exact form for interactions between gluons, i.e. the 3 and 4 gluon vertices – the details are beyond the level of this course , For physics to remain unchanged – must have GAUGE INVARIANCE of the new field, i.e. physical predictions unchanged for

,Hence the principle of invariance under local phase transformations completely specifies the interaction between a fermion and the gauge boson (i.e. photon):

interaction vertex:

QED !

, The local phase transformation of QED is a unitary U(1) transformation i.e. with

Now extend this idea…

From QED to QCDFrom QED to QCD , Suppose there is another fundamental symmetry of the universe, say “invariance under SU(3) local phase transformations”

interaction vertex: , Predicts 8 massless gauge bosons – the gluons (one for each ) , Also predicts exact form for interactions between gluons, i.e. the 3 and 4 gluon vertices – the details are beyond the level of this course ColourColourin QCD in QCD ,The theory of the strong interaction, Quantum Chromodynamics (QCD), is very similar to QED but with 3 conserved “colour” charges In QED: • the electron carries one unit of charge • the anti-electron carries one unit of anti-charge • the force is mediated by a massless “gauge boson”–the photon In QCD: • quarks carry colour charge: • anti-quarks carry anti-charge: • The force is mediated by massless gluons , In QCD, the strong interaction is invariant under rotations in colour space

i.e. the same for all three colours SU(3) colour symmetry

•This is an exact symmetry, unlike the approximate uds flavour symmetry discussed previously.

, Represent SU(3) colour states by:

, Colour states can be labelled by two quantum numbers: ( colour isospin ( colour hypercharge Exactly analogous to labelling u,d,s flavour states by and , Each quark (anti-quark) can have the following colour quantum numbers:

quarks anti-quarks Colour in QCD ,The theory of the strong interaction, Quantum Chromodynamics (QCD), is very similar to QED but with 3 conserved “colour” charges In QED: • the electron carries one unit of charge • the anti-electron carries one unit of anti-charge • the force is mediated by a massless “gauge boson”–the photon In QCD: • quarks carry colour charge: • anti-quarks carry anti-charge: • The force is mediated by massless gluons , In QCD, the strong interaction is invariant under rotations in colour space

i.e. the same for all three colours SU(3) colour symmetry

•This is an exact symmetry, unlike the approximate uds flavour symmetry discussed previously.

Colour in QCD

, Represent SU(3) colour states by:

quarks anti-quarks ColourColour ConfinementConfinement , It is believed (although not yet proven) that all observed free particles are “colourless” •i.e. never observe a free quark (which would carry colour charge) •consequently quarks are always found in bound states colourless hadrons ,Colour Confinement Hypothesis:

only colour singlet states can exist as free particles , All hadrons must be “colourless” i.e. colour singlets , To construct colour wave-functions for hadrons can apply results for SU(3) flavour symmetry to SU(3) colour with replacement g r

, just as for uds flavour symmetry can define colour ladder operators b

Colour Singlets

, It is important to understand what is meant by a singlet state , Consider spin states obtained from two spin 1/2 particles. • Four spin combinations: • Gives four eigenstates of

spin-1 spin-0 triplet singlet

, The singlet state is “spinless”: it has zero angular momentum, is invariant under SU(2) spin transformations and spin ladder operators yield zero

, In the same way COLOUR SINGLETS are “colourless” combinations: ( they have zero colour quantum numbers ( invariant under SU(3) colour transformations ( ladder operators all yield zero

, NOT sufficient to have : does not mean that state is a singlet Colour Confinement , It is believed (although not yet proven) that all observed free particles are “colourless” •i.e. never observe a free quark (which would carry colour charge) •consequently quarks are always found in bound states colourless hadrons ,Colour Confinement Hypothesis:

, just as for uds flavour symmetry can define colour ladder operators b

ColourColour SingletsSinglets , It is important to understand what is meant by a singlet state , Consider spin states obtained from two spin 1/2 particles. • Four spin combinations: • Gives four eigenstates of

spin-1 spin-0 triplet singlet

, The singlet state is “spinless”: it has zero angular momentum, is invariant under SU(2) spin transformations and spin ladder operators yield zero

, In the same way COLOUR SINGLETS are “colourless” combinations: ( they have zero colour quantum numbers ( invariant under SU(3) colour transformations ( ladder operators all yield zero

, NOT sufficient to have : does not mean that state is a singlet Meson MesonColour Colour Wave-functionWavefunction , Consider colour wave-functions for , The combination of colour with anti-colour is mathematically identical to construction of meson wave-functions with uds flavour symmetry

Coloured octet and a colourless singlet •Colour confinement implies that hadrons only exist in colour singlet states so the colour wave-function for mesons is:

, Can we have a state ? i.e. by adding a quark to the above octet can we form a state with . The answer is clear no. bound states do not exist in nature.

Baryon Colour Wave-function , Do qq bound states exist ? This is equivalent to asking whether it possible to form a colour singlet from two colour triplets ? • Following the discussion of construction of baryon wave-functions in SU(3) flavour symmetry obtain

• No qq colour singlet state • Colour confinement bound states of qq do not exist

BUT combination of three quarks (three colour triplets) gives a colour singlet state (pages 235-237) Meson Colour Wave-function , Consider colour wave-functions for , The combination of colour with anti-colour is mathematically identical to construction of meson wave-functions with uds flavour symmetry

Coloured octet and a colourless singlet •Colour confinement implies that hadrons only exist in colour singlet states so the colour wave-function for mesons is:

, Can we have a state ? i.e. by adding a quark to the above octet can we form a state with . The answer is clear no. bound states do not exist in nature.

Baryon BaryonColour Colour Wave-functionWavefunction , Do qq bound states exist ? This is equivalent to asking whether it possible to form a colour singlet from two colour triplets ? • Following the discussion of construction of baryon wave-functions in SU(3) flavour symmetry obtain

• No qq colour singlet state • Colour confinement bound states of qq do not exist

BUT combination of three quarks (three colour triplets) gives a colour singlet state (pages 235-237) Baryon Color Wavefunction ,The singlet colour wave-function is:

Check this is a colour singlet… • It has : a necessary but not sufficient condition • Apply ladder operators, e.g. (recall )

•Similarly

Colourless singlet - therefore qqq bound states exist ! Anti-symmetric colour wave-function Allowed Hadrons i.e. the possible colour singlet states Mesons and Baryons Exotic states, e.g. pentaquarks To date all confirmed hadrons are either To date all confirmed hadrons are either mesonsmesonsor baryonsor baryons. However,. However, some recent (but not entirely convincing) “evidence” for pentaquark states some recent evidence for pentaquark states (LHCb experiment).

Gluons , In QCD quarks interact by exchanging virtual massless gluons, e.g. q q q qb r qb r qb r

rb br

qr qb qr qb qr qb , Gluons carry colour and anti-colour, e.g. qb qr qr qr

br rb rr

, Gluon colour wave-functions (colour + anti-colour) are the same as those obtained for mesons (also colour + anti-colour)

OCTET + “COLOURLESS” SINGLET ,The singlet colour wave-function is:

Check this is a colour singlet… • It has : a necessary but not sufficient condition • Apply ladder operators, e.g. (recall )

•Similarly

Colourless singlet - therefore qqq bound states exist ! Anti-symmetric colour wave-function Allowed Hadrons i.e. the possible colour singlet states Mesons and Baryons Exotic states, e.g. pentaquarks

To date all confirmed hadrons are either mesons or baryons. However, some recent (but not entirely convincing) “evidence” for pentaquark states

GluonsGluons , In QCD quarks interact by exchanging virtual massless gluons, e.g. q q q qb r qb r qb r

rb br

qr qb qr qb qr qb , Gluons carry colour and anti-colour, e.g. qb qr qr qr

br rb rr

, Gluon colour wave-functions (colour + anti-colour) are the same as those obtained for mesons (also colour + anti-colour)

OCTET + “COLOURLESS” SINGLET Gluons

, So we might expect 9 physical gluons: OCTET: SINGLET: , BUT, colour confinement hypothesis:

only colour singlet states Colour singlet gluon would be unconfined. can exist as free particles It would behave like a strongly interacting photon infinite range Strong force. , Empirically, the strong force is short range and therefore know that the physical gluons are confined. The colour singlet state does not exist in nature ! NOTE: this is not entirely ad hoc. In the context of gauge field theory (see minor option) the strong interaction arises from a fundamental SU(3) symmetry. The gluons arise from the generators of the symmetry group (the Gell-Mann matrices). There are 8 such matrices 8 gluons. Had nature “chosen” a U(3) symmetry, would have 9 gluons, the additional gluon would be the colour singlet state and QCD would be an unconfined long-range force. NOTE: the “gauge symmetry” determines the exact nature of the interaction FEYNMAN RULES

Gluon-Gluon Interactions , In QED the photon does not carry the charge of the EM interaction (photons are electrically neutral) , In contrast, in QCD the gluons do carry colour charge Gluon Self-Interactions , Two new vertices (no QED analogues)

triple-gluon quartic-gluon vertex vertex

, In addition to quark-quark scattering, therefore can have gluon-gluon scattering

e.g. possible way of arranging the colour flow Self Interaction

Gluons that carry colour charges can interact with each other (Non-Abelian)

16 gluons

8 gluons

4 gluons

'Strength' of the colour charge appears to decrease as distance decreases (energy increases) , So we might expect 9 physical gluons: OCTET: SINGLET: , BUT, colour confinement hypothesis:

GluonGluon-Gluon-Gluon Interactions Interactions , In QED the photon does not carry the charge of the EM interaction (photons are electrically neutral) , In contrast, in QCD the gluons do carry colour charge Gluon Self-Interactions , Two new vertices (no QED analogues)

triple-gluon quartic-gluon vertex vertex

, In addition to quark-quark scattering, therefore can have gluon-gluon scattering

e.g. possible way of arranging the colour flow Free Quarks?

Many searches for 'free' quarks – not bound into hadrons:

Search for stable fractional charged particles → 'exotic' atoms - Moon rock, deep sea sludge, . . .

Search for fractional charges in Millikan oil drop type experiments

Look in accelerators and cosmic rays for particles with 1/9 or 4/9 ionisation of normal particles

No evidence for isolated free fractionally charged quarks

Why? Quark Confinement

The reason there are no free quarks, and the only stable configurations are qqq and qq (and possible gg 'glueball' states) is due to the fact that αS decreases with increasing energy and increases with increasing distance

At short distance, inside a Try and pull one out and αS proton, αS is small and the increases and the force tries quarks are 'free' to keep it in Gluon SelfGluon self-Interactions-Interaction and Confinement and Confinement , Gluon self-interactions are believed to give e+ q rise to colour confinement , Qualitative picture: •Compare QED with QCD •In QCD “gluon self-interactions squeeze lines of force into a flux tube” e- q , What happens when try to separate two coloured objects e.g. qq q q

•Form a flux tube of interacting gluons of approximately constant energy density

•Require infinite energy to separate coloured objects to infinity •Coloured quarks and gluons are always confined within colourless states •In this way QCD provides a plausible explanation of confinement – but not yet proven (although there has been recent progress with Lattice QCD)

Hadronisation and Jets ,Consider a quark and anti-quark produced in electron positron annihilation i) Initially Quarks separate at q q high velocity q q ii) Colour flux tube forms between quarks iii) Energy stored in the q q q q flux tube sufficient to produce qq pairs iv) Process continues until quarks pair up into jets of colourless hadrons , This process is called hadronisation. It is not (yet) calculable. , The main consequence is that at collider experiments quarks and gluons observed as jets of particles + q e !

e– q Gluon self-Interactions and Confinement , Gluon self-interactions are believed to give e+ q rise to colour confinement , Qualitative picture: •Compare QED with QCD •In QCD “gluon self-interactions squeeze lines of force into a flux tube” e- q , What happens when try to separate two coloured objects e.g. qq q q

•Form a flux tube of interacting gluons of approximately constant energy density

Hadronization and Jets Hadronisation and Jets ,Consider a quark and anti-quark produced in electron positron annihilation i) Initially Quarks separate at q q high velocity q q ii) Colour flux tube forms between quarks iii) Energy stored in the q q q q flux tube sufficient to produce qq pairs iv) Process continues until quarks pair up into jets of colourless hadrons , This process is called hadronisation. It is not (yet) calculable. , The main consequence is that at collider experiments quarks and gluons observed as jets of particles + q e !

e– q Hadronization

In high energy interactions you don't see the produced quarks but 'jets' of hadrons

Analogy: Gluons are like bits of elastic and quarks are the ends There are no free ends If you pull hard enough you get two bits of elastic but still no free ends Jet production in e+e- Collisions ,e+e– colliders are also a good place to study gluons

+ + e q e q + q !"Z !"# e !"#

– – e q e q e– q Experimentally: •Three jet rate measurement of •Angular distributions gluons are spin-1 •Four-jet rate and distributions QCD has an underlying SU(3) symmetry

Quark–Gluon Interaction The Quark – Gluon Interaction •Representing the colour part of the fermion wave-functions by:

•Particle wave-functions •The QCD qqg vertex is written: q q

•Only difference w.r.t. QED is the insertion of the 3x3 Gluon a colour i $ j SU(3) Gell-Mann matrices •Isolating the colour part:

•Hence the fundamental quark - gluon QCD interaction can be written Feynman Rules of QCD Feynman Rules for QCD External Lines incoming quark outgoing quark spin 1/2 incoming anti-quark outgoing anti-quark incoming gluon spin 1 outgoing gluon Internal Lines (propagators) spin 1 gluon

a, b = 1,2,…,8 are gluon colour indices Vertex Factors spin 1/2 quark

i, j = 1,2,3 are quark colours, a = 1,2,..8 are the Gell-Mann SU(3) matrices + 3 gluon and 4 gluon interaction vertices Matrix Element -iM = product of all factors

Matrix Element for quark-quark scattering , Consider QCD scattering of an up and a down quark u •The incoming and out-going quark colours are u labelled by • In terms of colour this scattering is

• The 8 different gluons are accounted for by the colour indices d d •NOTE: the !-function in the propagator ensures a = b, i.e. the gluon “emitted” at a is the same as that “absorbed” at b , Applying the Feynman rules:

where summation over a and b (and " and #) is implied. , Summing over a and b using the !-function gives:

Sum over all 8 gluons (repeated indices) Feynman Rules for QCD External Lines incoming quark outgoing quark spin 1/2 incoming anti-quark outgoing anti-quark incoming gluon spin 1 outgoing gluon Internal Lines (propagators) spin 1 gluon

a, b = 1,2,…,8 are gluon colour indices Vertex Factors spin 1/2 quark

i, j = 1,2,3 are quark colours, a = 1,2,..8 are the Gell-Mann SU(3) matrices + 3 gluon and 4 gluon interaction vertices Matrix Element -iM = product of all factors

MatrixQuark Element–Quark Scattering for quark-quark scattering , Consider QCD scattering of an up and a down quark u •The incoming and out-going quark colours are u labelled by • In terms of colour this scattering is

where summation over a and b (and " and #) is implied. , Summing over a and b using the !-function gives:

Sum over all 8 gluons (repeated indices) QCD vs QEDQCD vs QED

QED e– e–

!– !–

QCD u u

, QCD Matrix Element = QED Matrix Element with:

• or equivalently d d

+ QCD Matrix Element includes an additional “colour factor”

Evaluation of QCD Colour Factors

•QCD colour factors reflect the gluon states that are involved

Gluons:

Configurations involving a single colour r r •Only matrices with non-zero entries in 11 position are involved

r r

Similarly find QCD vs QED

QED e– e–

!– !–

QCD u u

, QCD Matrix Element = QED Matrix Element with:

• or equivalently d d

+ QCD Matrix Element includes an additional “colour factor”

EvaluationColour of QCDFactors Colour Factors

•QCD colour factors reflect the gluon states that are involved

Gluons:

Configurations involving a single colour r r •Only matrices with non-zero entries in 11 position are involved

r r

Similarly find Colour Factors Other configurations where quarks don’t change colour e.g. r r •Only matrices with non-zero entries in 11 and 33 position are involved

b b Similarly Configurations where quarks swap colours e.g. r g •Only matrices with non-zero entries in 12 and 21 position are involved

Gluons g r

Configurations involving 3 colours e.g. r b •Only matrices with non-zero entries in the 13 and 32 position •But none of the ! matrices have non-zero entries in the 13 and 32 positions. Hence the colour factor is zero b g , colour is conserved

Colour Factors : Quarks vs Anti-Quarks

• Recall the colour part of wave-function: • The QCD qqg vertex was written: q q ,Now consider the anti-quark vertex • The QCD qqg vertex is:

Note that the incoming anti-particle now enters on the LHS of the expression

•For which the colour part is i.e indices ij are swapped with respect to the quark case

• Hence

• c.f. the quark - gluon QCD interaction Other configurations where quarks don’t change colour e.g. r r •Only matrices with non-zero entries in 11 and 33 position are involved

b b Similarly Configurations where quarks swap colours e.g. r g •Only matrices with non-zero entries in 12 and 21 position are involved

Gluons g r

Colour FactorsColour : QuarksFactors vs Anti-Quarks

• Recall the colour part of wave-function: • The QCD qqg vertex was written: q q ,Now consider the anti-quark vertex • The QCD qqg vertex is:

Note that the incoming anti-particle now enters on the LHS of the expression

•For which the colour part is i.e indices ij are swapped with respect to the quark case

• Hence

• c.f. the quark - gluon QCD interaction ,Finally we can consider the quark – anti-quark annihilation

q QCD vertex:

with

Colour Factors ,Finally we can consider the quark – anti-quark annihilation

q QCD vertex:

with

• Consequently the colour factors for the different diagrams are: e.g. q q

q q

• Consequently the colour factors for the different diagrams are: q q e.g. q q

q q q q Colour index of adjoint spinor comes first q q

q q

Colour index of adjoint spinor comes first Quark-Quark Scattering Quark-Quark Scattering jet •Consider the process which can occur in the high energy proton-proton scattering • There are nine possible colour configurations d d of the colliding quarks which are all equally p likely. u u • Need to determine the average matrix element which p is the sum over all possible colours divided by the number of possible initial colour states jet

• The colour average matrix element contains the average colour factor

•For rr$rr,.. rb$rb,.. rb$br,..

•Previously derived the Lorentz Invariant cross section for e–!– e–!– elastic scattering in the ultra-relativistic limit.

QED

•For ud ud in QCD replace and multiply by

Never see colour, but QCD enters through colour factors. Can tell QCD is SU(3)

•Here is the centre-of-mass energy of the quark-quark collision •The calculation of hadron-hadron scattering is very involved, need to include parton structure functions and include all possible interactions e.g. two jet production in proton-antiproton collisions Quark-Quark Scattering jet •Consider the process which can occur in the high energy proton-proton scattering • There are nine possible colour configurations d d of the colliding quarks which are all equally p likely. u u • Need to determine the average matrix element which p is the sum over all possible colours divided by the number of possible initial colour states jet

• The colour average matrix element contains the average colour factor

•For rr$rr,.. rb$rb,.. rb$br,..

Quark-Quark Scattering

•Previously derived the Lorentz Invariant cross section for e–!– e–!– elastic scattering in the ultra-relativistic limit.

QED

•For ud ud in QCD replace and multiply by

Never see colour, but QCD enters through colour factors. Can tell QCD is SU(3)

•Here is the centre-of-mass energy of the quark-quark collision •The calculation of hadron-hadron scattering is very involved, need to include parton structure functions and include all possible interactions e.g. two jet production in proton-antiproton collisions , Measure cross-section in terms of pp• “transverse energy” • “pseudorapidity” …don’t worry too much about the details here, what matters is that… D0 Collaboration, Phys. Rev. Lett. 86 (2001) ,QCD predictions provide an excellent description of the data

,NOTE:

• at low ET cross-section is dominated by low x partons ! = 62-90o i.e. gluon-gluon scattering

• at high ET cross-section is dominated by high x partons ! = 5.7-15o i.e. quark-antiquark scattering

Running Coupling Constants Running Coupling Constants # # " # QED • “bare” charge of electron screened -Q " " by virtual e+e– pairs "# +Q " # • behaves like a polarizable dielectric " " , # " # In terms of Feynman diagrams: #

+++……

, Same final state so add matrix element amplitudes: , Giving an infinite series which can be summed and is equivalent to a single diagram with “running” coupling constant

Note sign Running Coupling Constants

, Might worry that coupling becomes infinite at

i.e. at • But quantum gravity effects would come in way below this energy and it is “ ” OPAL Collaboration, Eur. Phys. J. C33 (2004) highly unlikely that QED as is would be valid in this regime

, In QED, running coupling increases very slowly •Atomic physics:

•High energy physics:

Running of !s QCD Similar to QED but also have gluon loops

++ ++…

Fermion Loop Boson Loops , Remembering adding amplitudes, so can get negative interference and the sum can be smaller than the original diagram alone , Bosonic loops “interfere negatively”

= no. of colours with = no. of quark flavours

2 Nobel Prize for Physics, 2004 S decreases with Q (Gross, Politzer, Wilczek) , Might worry that coupling becomes infinite at , Might worry that coupling becomes infinite at

i.e. at • But quantumi.e. gravity at effects would come in way below this energy and it is • But quantum“ gravity” effects would come OPAL Collaboration, Eur. Phys. J. C33 (2004) highly unlikely that QED as is would be valid in thisin regime way below this energy and it is “ ” OPAL Collaboration, Eur. Phys. J. C33 (2004) highly unlikely that QED as is would , In QED, running couplingbe valid increases in this regime very slowly •Atomic physics:, In QED, running coupling increases very slowly •High energy• physics:Atomic physics:

•High energy physics:

Running of αS Running of !s QCD Similar to QED but also have gluon loops

Running of !s QCD Similar to QED++ but also have gluon loops ++…

Fermion Loop Boson Loops , Remembering adding amplitudes, so can++ get negative interference and the sum ++… can be smaller than the original diagram alone , Bosonic loops “interfere negatively”

Fermion Loop Boson Loops

, Remembering adding amplitudes, so can get= no.negative of colours interference and the sum with can be smaller than the original diagram alone= no. of quark flavours , Bosonic loops “interfere negatively”

2 Nobel Prize for Physics, 2004 S decreases with Q (Gross, Politzer, Wilczek)

= no. of colours with = no. of quark flavours

2 Nobel Prize for Physics, 2004 S decreases with Q (Gross, Politzer, Wilczek) Colour Charge Strength s for strong

αS ~ 0.2 at E = 30 GeV Compare with electromagnetic interaction αEM ~ 1/137 ~ 0.007

Low energy High energy Long distance Short distance Running of αS

0.4

, Measure !S in many ways:

• jet rates 0.3 • DIS QCD • tau decays α(µ) Prediction • bottomonium decays 0.2 • +…

0.1 As predicted by QCD, 2 !S decreases with Q 0.0 12 5102050100200 µ (GeV)

, At low : !S is large, e.g. at find !S ~ 1 •Can’t use perturbation theory ! This is the reason why QCD calculations at low energies are so difficult, e.g. properties hadrons, hadronisation of quarks to jets,…

, At high : !S is rather small, e.g. at find !S ~ 0.12 Asymptotic Freedom •Can use perturbation theory and this is the reason that in DIS at high quarks behave as if they are quasi-free (i.e. only weakly bound within hadrons)

Summary , Superficially QCD very similar to QED , But gluon self-interactions are believed to result in colour confinement , All hadrons are colour singlets which explains why only observe Mesons Baryons , A low energies Can’t use perturbation theory ! Non-Perturbative regime

, Coupling constant runs, smaller coupling at higher energy scales

Can use perturbation theory Asymptotic Freedom

, Where calculations can be performed, QCD provides a good description of relevant experimental data QCD in e+e- Annihilation

- e q

e+ q

Cleanest laboratory for studying perturbative QCD: • No hadrons in initial state • No “remnants” of initial proton(s) • Quarks and gluons produced at short distances and emerge as jets Two Jet Event

- e q

e+ q dσ 2 Distribution follows ∝ (1+ cos ϑ) Just like e+ e- -> µ+ µ- d(cosϑ)cm -> quarks have spin 1/2 Proof that Gluons Exist

e- q e- q

Gluons fragment into jets similarly to quarks so expect 3 jet events Proof that Gluons Exist

3 jet events observed at rate consistent with expectations The Ratio R

[ × some threshold kinematics factors]

Sum over all Charge of μ2 contributing quarks

As quarks come in 3 colours:

The fact that the factor of 3 is necessary is very strong evidence in support of the colour hypothesis The Ratio R

Flavour : u d s c b t Charge2 : (⅔)2 (-⅓)2 (-⅓)2 (⅔)2 (-⅓)2 (⅔)2

For ECM ~ 2 GeV have uu, dd, ss → R = 3 (4/9 + 1/9 + 1/9) = 2 Need ~3 GeV to make cc

For ECM ~ 5 GeV have uu, dd, ss, cc

→ R = 3 (4/9 + 1/9 + 1/9 + 4/9) = 3⅓ Need ~10 GeV to make bb

For ECM ~ 11 GeV have uu, dd, ss, cc, bb → R = 3 (4/9 + 1/9 + 1/9 + 4/9 + 1/9) = 3⅔ The R Ratio

- e q

e+ q PDG The Ratio R

u + d + s + c → 3.3 colour factor of 3 u + d + s + c + b → 3.7 u + d + s → 2 2 2 2 u, d, s: R0 = 3 (qu + qd + qs ) = 2 2 2 2 2 u, d, s, c: R0 = 3 (qu + qd + qs + qc ) = 10/3 = 3.3 2 2 2 2 2 u, d, s, c, b: R0 = 3 (qu + qd + qs + qc + qb ) = 11/3 = 3.7 2 2 2 2 2 2 u, d, s, c, b, t: R0 = 3 (qu + qd + qs + qc + qb + qt ) = 5 Any more Quarks?

We have 3 full generations. We only need one (u, d, e, ѵe) for everyday life

So if 3 why not more? Since it is not understood why there are 3 generations there is no theoretical reason why there shouldn't be more (Periodic Table again?)

However, experimental evidence now suggests only 3

In early 1990s LEP measured the width of the Z0 to very high precision. The Z0 can decay into any fermion pairs with a combined mass less than the Z0 mass (~90 GeV/c2)

f = u, d, s, c, b, e, μ, τ, ѵe, ѵμ, ѵτ

The more ways the Z0 can decay, the shorter its lifetime τ and the larger its width Γ (Γ × τ ~ ħ) Any more Quarks?

Generation 1 2 3 4 . . . Z0 ↛ tt since

M 0 < 2M u c t h . . . Z t d s b l . . . e μ τ χ . . .

ѵe ѵμ ѵτ νχ . . .

The Z0 can decay into all these The Z0 would decay into any new neutrinos assuming they were massless (or light

compared with MZ0) The Width of the Z0

We know how much each quark/lepton pair contributes to ΓZ0

0 ΓZ = Γuu + Γdd + Γss + Γ cc + Γ bb + Each of the Γuu etc Γe+e– + Γμ+μ– + Γ�+�– + are known as partial Nѵ Γѵѵ widths

A measurement of ΓZ0 is equivalent to measuring the

number of neutrino species Nѵ

If there is one neutrino species per generation as now → total number of generations

Result from LEP: Nѵ = 2.984 ± 0.008 i.e. only 3 generations

This is only true if the new neutrinos are light but they cannot be too heavy or they would contribute too much mass to the Universe → Astrophysical constraint on neutrino masses The Width of the Z0

Data points Deep Inelastic Scattering

Shoot high energy electrons at protons

Scattering angle θ related to momentum transfer q = (k – k')

x is the fraction of the proton's The total energy of the momentum carried by the struck hadrons from the struck quark 0 ≤ x ≤ 1 quark is q + xp

Measure hadron energy + scattering angle θ → deduce x Scaling Violations

x is a 'scaled' parameter i.e. x = pquark / pproton - so naively expect quark and antiquark distributions to depend on x and not energy

But see 'scaling violations' – at higher energies probe deeper into the proton and see more 'structure i.e. more of the proton's momentum appears to be carried by gluons and qq pairs and less by valence quarks

valence quarks high energy

valence quarks low energy

Distribution depends on x and energy Structure of the Proton • Protons are very complicated and have a lot more in them than just two up quarks and a down quark. • There are zillions of gluons, antiquarks, and quarks in a proton. The standard shorthand, “the proton is made from two up quarks and one down quark”, is really a statement that the proton has two more up quarks than up antiquarks, and one more down quark than down antiquarks. • The proton consists of partons: three valence quarks (uud), an infinite sea of light quark- antiquark pairs (sea quarks), and gluons. • The protons viewed at ever higher resolution appears more and more as field energy (soft glue). Intro to PDFs Global PDF analyses LHC Phenomenology Heavy Quark PDFs

QCD factorization

Parton Distribution Functions (PDFs) are non-perturbative quantities, yet their scale dependence is given by the DGLAP evolution equations (Dokshitzer-Gribov-Lipatov- Altarelli-Parisi 73-78)

Parton Distribution Functions2 2 1 dqi (x, Q ) ↵s (Q ) dy 2 x 2 = Pij y, ↵s (Q ) qj , Q d ln Q2 2⇡ y y Zx „ « ` ´ • Introduced by Feynman (1969) in the parton model, to explain 2 PDFs cannot be computed from ﬁrst principles Determination of qi (x, Q0 ) from data Bjorken scaling in deep inelastic by meansscattering data of a global analysis of all hard scattering! data In the Parton Model there are 13 independent PDFs • The number densities of the partons in the proton depend on the probing energy scale, Q2, and partonu(x), u¯momentum fraction, x,(x), d(x), d¯(x), s(x), ¯s(x), c(x)with , c¯(x), b(x), b¯(x), t(x), ¯t(x), g(x)

respect to the momentum of the proton, expressed by a In the Quark Parton Model, PDFs haveparton a probabilistic interpretation: 2 distribution function (PDF), fi(x,Q ), for each parton i. x2 2 2 Pu x , x , Q dxu (x, Q ) • Interpretation as probability distributions: V 1 2 ⌘ V Zx1 • According to the QCD factorization theorem for inclusive ` ´hard Probability of ﬁnding a valence quark in the proton with a fraction of the proton’s scattering processes, universal distributions momentum! x x x containing longat a resolution scale-distance Q2 1   2 structure of hadrons; related to Juan Rojo parton model distributions at leading INFN Milano order, but with logarithmic scaling PDFs in the LHC era violations. • PDFs are extracted from fits to the deep inelastic lepton-hadron scattering experimental data using the Dokshitzer-Gribov-Lipatov- Altarelli-Parisi (DGLAP) evolution equations. Proton Structure

The proton consists of u u d + q q pairs

Valence quarks Sea At energies ~ 10 GeV

quarks

valence quarks

antiquarks Parton Distribution Functions • Most particles in the proton carry much less than 10% (that is, x< 0.1) of the proton’s energy, and you are more likely to hit one of those, with lower energies than with higher ones. • If you do hit something that carries less than 10% of the proton’s energy, then it is most likely a gluon; and quarks and antiquarks are about equally likely once you get below about 2% of the proton’s energy. • If you hit something with much above • The curves all fall off very quickly as x 20% (that is, x>0.2) of the proton’s increases; you are very unlikelyindeed energy – which is very rare – it is most to hit anything with more than 50% of likely a quark, and about twice as likely the proton’s energy. to be an up quark as a down quark Structure of the Proton

U d d U s s d

Proton is composed of u,d valence quarks which carry quantum numbers and "sea" of u,u,d,d ,s,s, g with _ no _ net _ quantum _ number Constraints on Parton Density Functions The charm, beauty and top compositions are negligible so proton is composed of u, d, s quarks and anti quarks and glue. The observed quantum number constraints

No net strangeness Isospin I3 =1/2 (s(x) − s(x))dx = 0 1 1 1 ∫ [ (u(x) −u(x))− (d(x) − d (x))]dx = ∫ 2 2 2 Charge =1 2 1 1 (u(x) −u(x))− (d(x) − d (x))− (s(x) − s(x))dx =1 ∫ 3 3 3

Leads to: ∫(u(x) −u(x))dx = 2 ∫(d(x) − d (x))dx =1 Two u quarks and 1 d quark as expected. These are called sum rules and there is one for every quantum number. Momentum sum rule: ∫ x[u(x) +u(x) + d(x) + d (x) + s(x) + s(x) +...+ g(x)]dx =1 Early Data on Proton Structure

SLAC-MIT ~20 GeV electrons, protons at rest, Q2 = few GeV2

BCDMS 200-300 GeV muons, protons at rest, Q2 = tens of GeV2 PDFs u(x) and d(x) parton distributions for proton

µ = Q

u(x) xq(x)

d(x)

x PDFs u(x) and u(x) parton distributions for proton

u(x) valence + sea xq(x)

u(x) sea only

x Gluon Distribution

• Quark distributions can be measured by e or µ scattering – but not the gluon distribution as gluons are not electrically charged. • A nonzero gluon distribution can be inferred from the momentum sum rule: ∫ x(u(x) +u(x) + d(x) + d (x) + s(x) + s(x) + g(x) )dx =1.0 so: ∫ xg(x)dx = ∫1.0−(u(x) +u(x) + d(x) + d (x) + s(x) + s(x))dx

2 2 2 More generally, Q evolution of q(x,Q ), and hence Fi(x,Q ), involves g(x,Q2), so gluon distribution can be extracted that way. Nevertheless, it is not as well known as the quark distributions. PDFs Gluons carry 50% of proton momentum

∫ xg(x)dx = 0.5

xq(x)

u(x) +d(x)

g(x)

x PDFs

At high Q2 = µ2 see more of sea

u(x) 500 GeV xq(x) u(x) 5 GeV

x PDFs At high Q2 see less valence quarks more sea antiquarks

u(x) 500 GeV xq(x) u(x) 5 GeV u(x) 5 GeV

x PDFs More low x gluons less high x valence quarks at high Q2

g(x) 500 GeV xq(x)

g(x) 5 GeV

x Quark quantum numbers

Flavor I I3 S C B* T Q/e

u 1/2 1/2 0 0 0 0 +2/3

d 1/2 -1/2 0 0 0 0 -1/3

s 0 0 -1 0 0 0 -1/3

c 0 0 0 1 0 0 +2/3

b 0 0 0 0 -1 0 -1/3

t 0 0 0 0 0 1 +2/3

1 Q/e = I + (B + C + S + B *+T) 3 2

Baryon number :1/3 for all quarks

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