Quantum Chromo dynamics QCD

Contents

Evidence for quarks R DIS

Colour Gauge Symmetry

Gluons

Freedom Asymptotic

Interquark Forces

Mesons and Baryons

Flavour SU

Learning Outcomes

Be able to describ e the evidence for quarks

Be able to write down the QCD covariant derivative and explain the origin of

gluons in terms of a gauge symmetry

Be able to compute the interaction strength b etween quarks at tree level in

p erturbation theory

Be able to show how comp osite quark states break down into dieren t SU

colour and a vour representations

Be able to describ e the p ossible colour singlet quark states

Bo oks Reference

Intro duction to High Energy Physics Perkins

yon Elementary Particle Physics Ken

QCD

We now turn our attention to the description of the strong nuclear force and

hadrons in terms of quarks and the gauge theory describing their interactions

Evidence For Quarks

Let us b egin our study of the strong nuclear force by considering the evidence for the

+

existence of quarks The simplest studies are through e e collisions at accelerators

where jets of hadrons were discovered

jet e+ γ

_

e jet

0

In the s many such hadrons were catalogued including b easts such as p n

Understanding these jets to ok a long time but the settled picture is that a quark anti

quark pair are pro duced in the collision Their strong interactions then pro duce more

quarks with which they are b ound into the hadrons

jet

e+ γ q _ q e_

jet

We will need to build up this picture slowly to understand how quarks are conned

in hadrons

The RRatio

If we neglect the jet formation which is inevitable once the quarks are formed we

can make QED predictions for the jet pro duction cross sections Rememb er

2

+ +

e e

s

so

2

2 +

e e qq Q multiplicity

q

s

where the multiplicity is the numb er of identical copies of a particular quark

Its useful to dene the Rratio as

+

X

e e hadrons

2

multiplicity Q R

f

+ +

e e

f

where Q are charges of the quarks which can b e pro duced in the exp eriment Q

f u

Q etc

d

The multiplicity factor surprisingly turns out to b e the rst evidence that

quarks come in three colours

p

Thus for example in a machine with centre of mass energy s GeV one can

only create u d s quarks so

2 2

R

Whereas in a GeV machine we can also create charm quarks so

2

R

c

Thus at GeV we observe

R

ou can see that ab ove GeV there Heres a plot of real exp erimental data for R Y

quark is an extra step corresp onding to the discovery of the b

Quark Spin

+

In e e collisions the spin nature of the photon created inuences the spin structure

of the nal state particles pro duced The preferable conguration for the incoming

electron and p ositron to create the photon is

_ e+ e J Z = −1 helicity

−1/2 +1/2

For fermionic nal states there is a preference for the particle pair to b e pro duced

back to back along the b eam line mimicing the electron p ositron initial state This

shows up in the cross section which go es as

d

2

cos

d

This is the distribution we nd for jets which tells us the quarks themselves are

spin fermions

Deep Inelastic Scattering

A high energy virtual photon can b e used to prob e structure within the proton eg

as was done at SLAC in the s

__ e e

γ

p

p

When a lot of energy is exchanged the photon has a short wavelength E h

hc The proton app ears as a large environment to such photons and the scattering

cross section is well describ ed by it hitting an isolated quark or parton as they were

initially called γ

−i 2/3 e γ µ u u

d

A very long wavelength photon on the other hand just sees the whole proton with

charge Q p

γ

µ −i e γ

p

This transition in the form of the crosssection with scattering energy is observed

Colour Multiplicity

We have seen that there are identical copies colours of each quark We can

call them red green and blue but it is imp ortant to rememb er that they are

indistinguishable Any theory we write down must b e invariant to interchanging the

lab els

In a quantum theory this is even more subtle We can imagine a state with a wave

function

p p

R G

which is in a red state of the time and a green state of the time yet we

cant distinguish them so we could equally well describ e this as one of our basic quark

colour states

In general there are an innite set of transformations on the RGB wave function

1

vector which are describ ed by matrices acting on the vector These matrices

are called elements of a group called SU these are words b orrowed from

mathematical group theory but the details of group theory are not essential to us

here

1

A red quark is represented by the state a green quark by etc

These matrices must b e unitary as we can see from the following Both b efore

and after the transformation on our three basis wave functions we require that

Z

y 3

d x

So when we transform U

Z Z

0

y 3 y y 3

d x U U d x

From which we see that

y

U U

It turns out that these innite set of matrices may b e written compactly as

a a

U exp i T

and we are using the index summation convention where the index a

The are numb ers that you can cho ose any values for dierent values give

a

the dierent U matrices

a

The T are matrices called the generators of SU Explicitly they can

1

a b ab

so that T r T T b e written as normalized

2

i

B C C C B B

1 1 1

2 1 3

i T T T

A A A

2 2 2

i

C C C B B B

1 1 1

4 5 6

i T T T

A A A

2 2 2

i i

C B C B

1 1

7 8

p

T T

A A

2

12

We should also pause to make clear what we mean by the exp onential of a matrix

in ab ove We dene it as the sum of the innite series

2

M

M

e M

w the transformation No

a a

i T

e

iq (x)

lo oks like a generalization of the gauge transform of QED e a set of

transformations incidentally called the group U It is therefore a natural step to

promote the symmetry to a lo cal level to obtain gauge b osons for the strong nuclear

force

QCD

The Dirac equation for the three colour wave functions is

i mI

where the I is the identity matrix in colour space We will require that this

equation is invariant under the transformation

a a

(x)T ig

s

e

where g will b e the strong forces coupling and note the lo cal form of the transfor

s

mation

a

For ease lets just think ab out a transformation where the are innitesimal so

a a

ig xT

s

a

As usual the op erators will not let the x pass by unhindered as in QED

into the derivative whose shifts under an enlarged we must intro duce vector elds A

gauge transformation will undo the damage Clearly we will now need such vector

a a a

elds to cancel the degrees of freedom x Further each comes with a T so

a

so must each of the A to o

Thus we intro duce the covariant derivative

a a

ig G T D

s

a

The G are the gluon elds of QCD Their Feynman rules to quarks are _ q gluon a γµ

q ig s T

Note that the presence of the generators in the vertices means that the gluons

interchange quark colours

2 2

eg G has a coupling including T which acts as

R G R

C B C B C B B C

2

G T R G

A A A A

B B

In other words the vertex lo oks like _ _ R R

G G

In a sense the gluon here is carrying away G antiR colour the gluon is coloured

Note also this guy do es not couple to blue quarks

The mathematics of the generalized Maxwell equations is a quagmire that we

shall avoid here Not surprisingly though having learnt that the gluons are coloured

it turns out that they interact directly with each other unlike photons

2 gs

gs

Collider Evidence For Gluons

The simplest way to lo ok for gluons is to directly make them For example

_ q e γ gluon

_

e+ q

2

We exp ect to see three jet events suppressed by g rememb er to square the

s

diagram to get the probability

Such events are seen and from them we can extract the strong coupling constant

2

M GeV g

s Z

s

Asymptotic Freedom

The quark and gluon mo del of QCD is highly successful at predicting results in high

energy collider exp eriments but we need to understand why we never see free quarks

or gluons In fact we never see any colour charged ob jects in normal life Formally

this is still an op en question within QCD but there is enough understanding of the

theory to see that it will answer the question The main concrete understanding for

which the Nob el Prize was awarded in is asymptotic freedom

We saw in QED that the running coupling runs with energy scale due to lo op

corrections In QCD the same happ ens but there are additional contributions since

the gluons self interact

QED like

new

The result of the computation is again that the coupling runs logarithmically

2

M

s

2

Q

s

0

2 2 2

M ln Q M

s

4

where is the numb er one extracts from the diagrams ab ove

0

In QED just including a single electron and the running lo oks like 0

α

1/137

Q 10277 GeV

In QCD though the extra contributions enter with the opp osite sign

0

N N

f c 0

where N the numb er of colours and N counts the numb er of quark avours

c f

accessible at a particular energy scale

If N then and the coupling runs as shown

f 0 αs

ΛQCD Q

At large energies or small quark separation the theory is weakly coupled asymp

totically free

At low energies or big quark separation the coupling grows At the scale

QC D

the equations we have give an innite value of the coupling Of course our calculation

has b een done in p erturbation theory which assumes that g so to extrap olate to

s

this p oint is b eyond the validity of the metho d however that the coupling b ecomes

nonp erturbative is known

Exp erimental measurement gives M which implies M eV

s Z QC D

Asymptotic freedom explains much of our picture of QCD At short distances

quarks and gluons are weakly interacting but as they separate the force b etween

them grows This explains connement As we separate a quark and an antiquark

for example the energy of their interaction grows until it will b ecome energetically

favourable to pair create a quark antiquark pair in the middle The hadron has then

broken into two which can then b e separated at no extra energy cost Never is a

quark freed

_ _ _

q q q qq q

We can also understand jet formation The quark antiquark pair are created close

to each other and b ehave weakly As they move apart the coupling grows and the

energy of that interaction then creates more quark antiquark pairs and everything

pairs up We see hadronic jets not the original quarks

The Role of 

QC D

We have just seen that QCD creates an energy scale we can trade describing the

theory by the value of the dimensionless coupling constant for describing it by the scale

at which that coupling blows up This is called dimensional transmutation

QC D

The scale created by the running is the only scale in low energy QCD

QC D

physics The u d s quarks have masses b elow and can b e eectively set to zero

QC D

the other more massive quarks we consider not to b e part of the low energy theory

We therefore exp ect that all the masses of QCD b ound states will b e given by

M

QC D QC D

where the is some order one numb er Similarly the widths of any resonances will

b e

QC D QC D

or in other words the lifetime of an unstable QCD particle will b e roughly

QC D

This is broadly a correct description of hadronic decay times

Colour Neutrality in QCD

Lets see if we can understand the sorts of hadrons that might favourably form in

QCD Since gluons are so strongly coupled we exp ect any free colour charges to b e

attracted to each other very quickly Nature should only have colour neutral groupings

of quarks therefore

Neutrality in QED

A neutral state in QED such as an electron and a proton has net charge zero and

do es not couple to photons That is to say that under a U QED gauge transforma

tion the wave function of the comp osite ob ject do es not transform Since it do esnt

transform into something else it is a unique state called a singlet representation of

the transformation We need to nd the equivalent ob jects for QCD

Mesons Quark AntiQuark Pairs

The following is a What states can we have out of a quark and an antiquark

complete list

RR B B GG RG RB B R B G GR RB

In fact when we make colour transformations these states split into two sets

Firstly there is

p

RR B B GG

This state stays the same and do es not transform under the SU rotation and gluons

wont couple to it we call this a colour neutral state or singlet

rstly of course the quark colour vector transforms The pro of is a little subtle

under a gauge transform as

R R

a a

C C B B

i T

G G

e

A A

B B

The anticolours actually transform as

R R

a aT

C B C B

i T

G G

e

A A

B B

The minus sign is b ecause they are antiparticles with opp osite charges The transp ose

a

of T is the subtle bit it comes ab out b ecause when we draw the vertex for example

RG

we mean the vertex for a green quark to turn into a red quark but time reversed

the transp ose takes this into account

Now we have

a a

i T

R G B R G B e

and hence our state

R R R

a a a a

C C C B B B

i T i T

G G R G B e e G R G B R G B

A A A

B B B

Now the remaining states in mix into each other though under SU

B and youll transformations For example just interchange two lab els such as R

see these mix

p p

RR B B RR GG B B RG RB B R B G GR RB

These states are called an o ctet of colour and must have identical prop erties

These app ear charged to a gluon

Formally we write this decomp osition as follows

The presence of the colour neutral singlet matches that we see mesons made of a

quark and an antiquark in nature

Baryons Quark States

There are p ossible combinations of two quarks to o

1 1

p p

RR RB BR RB BR

2 2

1 1

p p

RG GR RG GR BB

2 2

1 1

p p

GB BG GB BG GG

2 2

These two groupings are a and a These two representations cant mix under

SU rotations b ecause one set are symmetric and the other antisymmetric in the

two colours Under eg R G the states within each of the two representations do

mix though

To see that the is an antithree lo ok at the charge of these states under the

8

diagonal generator T For the neglecting normalizations

Q Q Q

R G B

whereas the three states ab ove have charge

Q Q Q

RB RG GB

so they transform as an antithree

Since there isnt a colour singlet in we do not exp ect any two quark b ound

states it would have colour charge and interact strongly with gluons

Three Quark States

We can think of a three quark state as a b ound state of a diquark the represen

tation and a further quark The group theory is one we are familiar with

There is a singlet which at the constituent level is given by

1 1 1 1

p p p p

R GB BG B BR RB RG GR G

3 2 2 2

1

p

RGB RB G B RG B GR GRB GB R

6

Note this is a totally antisymmetric combination of the colours

We therefore conclude we exp ect to nd quark states in nature which we do

eg proton uud neutron udd

++ ++

The There is a state the with spin J which is made up of solely

z

up quarks

u u u

This provides more evidence for quark colour since this lo oks like fermions all in

In fact it is totally antisymmetric since the colour the same avour and spin state

structure is that ab ove it is an allowed fermionic state

Exotics

We can make colour singlets in more exotic fashions eg

P entaquarks

We could take four quarks and combine them into two diquarks in the These

diquarks could then b e combined to form a That could then b e made into a

singlet with an antiquark

Until recently these states were not seen in the data

++

States such as uuuuu has the same quantum numb ers as the so would quan

tum mechanically mix and also decay quickly

A state such as uuuds is more clearly distinct On the other hand this could fall

apart to

+ ++

pK

One might exp ect this to happ en rather fast

++

Recent F uss a numb er of exp eriments in claimed to see the with a

very narrow width M eV Other exp eriments have not seen it since the

verdict is that it do es not exist we exp ect it to b e very unstable to falling apart into

a baryon and a meson

Glueballs

We can also imagine states that are just made from gluons eg g g

There are some candidates for such states but this picture is naive since such

a state has the same quantum numb ers as uu etc presumably the real states are

mixtures of all these states

Flavour SU

The SU colour symmetry of QCD resulted from there b eing three identical copies

of the quarks

Accidental symmetries of this sort can also o ccur The uds quarks all have

masses Their QED interactions are much weaker than QCD so as far as the

QC D

strong force knows these are three identical particles The upshot is that we exp ect

many dierent hadrons to have the same masses upto small corrections

We can keep track of these states using SU avour symmetries For example

spin quark antiquark meson states have the avour structure

Singlet

2

p

M eV c uu dd ss

Octet

Here Ive lab elled the states according to their isospin charge I which is the charge

3

under

u

C B C B

d

A A

s

and hyp ercharge Y which is the charge under

u

C B C B

d

A A

s

Note that the o ctet roughly share the same mass at least in comparison with the

splitting of the singlet from them The whole light hadron sp ectrum can b e classied

using SU

F