particle physics chapter · ( particle physics - chapter 3 4 / 27. ... made solely of gluons?can...

1 79 7Université Hadj Lak dh ar

B A T N AEl

Particle Physicschapter 3

The simple quark model – continued

Yazid Delenda

Departement des Sciences de la matiereFaculte des Sciences - UHLB

http://theorique05.wordpress.com/f422

Batna, 07 May 2015

(http://theorique05.wordpress.com/f422) Particle Physics - chapter 3 1 / 27

The simple quark model

QCD compared to QED

In table 1 we compare the basic properties of QED and QCD.

QED QCD

Source of interaction electric charge colour charge

Force carrier photon gluon

Intrinsic strength α = 1/137 αs � α

Table: QCD and QED compared.

QCD compared to QEDGluons

gluons are the force carriers for the strong interactions, they have zeromass, spin-1, just like the photon, thus the strong force between quarks isinfinite range.However because of the confinement the force is short rangebetween hadrons. To see this consider two hadrons as shown in the figure.

Coloured quarks

Over all zero colour charge Over all zero colour charge

The constituents of hadrons are coloured quarks which interact strongly.The overall colour charge of hadrons is zero. Thus the force between twohadrons is analogous to Van-der-Waal force between atoms: The atom hasoverall zero electric charge, but atoms interact with each other throughthe same electric force (dipole interactions).

QCD compared to QEDFlavour-independence:

Gluons couple to colour charges, they do not care whether the involvedquarks are u, d, c, s, b or t quarks.Forces are independent of the quarktype (flavour). This leads to the striking consequence of isospin symmetry.

QCD compared to QEDFlavour-independence:

Gluons couple to colour charges, they do not care whether the involvedquarks are u, d, c, s, b or t quarks.Forces are independent of the quarktype (flavour). This leads to the striking consequence of isospin symmetry.

QCD compared to QEDDifferences between QED and QCD:

Photons have zero electric charge, however gluons have non-zero colourcharge.In fact there are eight different charges a gluon can have. This factleads to the concept of gluon self-interactions. Particularly, as well as thevertices shown in the following figure,

whose analogous is also present in QED,

Photons have zero electric charge, however gluons have non-zero colourcharge.In fact there are eight different charges a gluon can have. This factleads to the concept of gluon self-interactions. Particularly, as well as thevertices shown in the following figure,

whose analogous is also present in QED,

we also have the vertices of QCD (which are not allowed in QED) whichare shown in the following figure.

An important consequence of the triple and quadric gluon vertices is thatthe QCD coupling αs grows at small energy scales (high distancescales).This is in contrast to QED where the coupling grows smaller atsmall energy scales which makes the theory purely perturbative.For QCDthis means as the quarks become more and more separated the forcebetween them gets stronger. Hence this implies the concept ofconfinement.However at large energy scales (small distances) the concept asymptoticfreedom takes over, which means that at very high energies the couplingbecomes small and quarks essentially can be treated as free states andperturbation theory is valid in this domain.Also another consequence of the above vertices is the concept of“glue-balls”, since gluons interact with each other can we have hadronsmade solely of gluons?Can they form bound states? Fairly strong evidencesuggests they do.

Quark-Quark interactions and the QCD potential

Single gluon exchange Double gluon exchange

Multiple gluon exchanges are important (examples shown in above figure)and they lead to the concept of renormalisation.Using “lattice gaugetheory” it is possible to numerically calculate amplitudes of the abovediagrams approximately.

Single gluon exchange Double gluon exchange

Multiple gluon exchanges are important (examples shown in above figure)and they lead to the concept of renormalisation.Using “lattice gaugetheory” it is possible to numerically calculate amplitudes of the abovediagrams approximately.

It is also possible to simulate (with the aid of computers) force linesbetween, say a quark and anti-quark, as shown in the following figure.

At short distances

At large distances

Narrow flux tube

Thus the interaction between a quark and anti-quark gets weaker at smalldistance scales (i.e. high energy limit, which means the concept ofasymptotic freedom).We can parameterise the potential as in QED by thedominant one-gluon exchange (αs becomes small):

V (r) ∼ −αsr

At large distances field lines attract (flux tube) and V (r) becomes linear inr:

V (r) ∼ +Kr, as r →∞

This means that quarks cannot escape each other, i.e. we haveconfinement,so we have static potential (which particularly is a gooddescription of heavy quark systems, such as cc and bb).

V (r) ∼ −αsr

V (r) ∼ +Kr, as r →∞

V (r) ∼ −αsr

V (r) ∼ +Kr, as r →∞

V (r) ∼ −αsr

V (r) ∼ +Kr, as r →∞

V (r) ∼ −αsr

V (r) ∼ +Kr, as r →∞

In this figure we show the static qq potential. Here R is measured inlattice units a = 0.0544 fermi.

The potential is parameterised by:

V (r) = V0 +Kr − e/r + f/r2

The value of K from experiments (string tension) is K = (440 MeV)2.The linear behaviour is a clear indication of the confinement property.Thispotential also leads to the string model for hadronisation.Consider forexample meson decay in QCD. What happens when the separation of theq and q pair increases? For light mesons (qi = uds) we have the picturefrom the following figure.

V (r) = V0 +Kr − e/r + f/r2

E = m1 +m2 +Kr

E = (m1 +mq +Kr1) + (m2 +mq +Kr2)

So we get meson decay M1 →M2 +M3 via qq creation as shown in thefollowing figure.

This picture also leads to jets in the following way: Suppose that we give aquark in a meson high momentum transfer (e.g. through collision), thenwe get the scenario shown in the following figure.

At rest Acquires high momentum transfer

jets of hadrons

Quarks, jets and gluons

In high energy interactions, many hadrons are produced. One often sees“jets” of hadrons (nearest thing to quark and gluon tracks there is).Wecan illustrate this by considering for example the process:

e+ + e− → hadrons

at high energy limit (E � me,mq). This process has been extensivelystudied at e+e− colliding beams (with E � mZ). Consider first two jetevents, which is the dominant event through the mechanism shown in thefollowing figure.

This can be thought of as a two-stage process:

e+ + e− → q + q

q + q → jet 1 + jet 2

where the jet momenta reflect the underlying quark momenta.

This can be thought of as a two-stage process:

e+ + e− → q + q

q + q → jet 1 + jet 2

where the jet momenta reflect the underlying quark momenta.

Quarks, jets and gluonsExperimental confirmation:

By comparing to the process:

e+ + e− → γ∗ → µ+ + µ−

which has the mechanism shown in the following figure, and cross-sectionat E � mµ:

By comparing to the process:

e+ + e− → γ∗ → µ+ + µ−

which has the mechanism shown in the following figure, and cross-sectionat E � mµ:

d cos θ=

(1 + cos2 θ)

where in the centre of mass frame we have the mechanism shown in figure.

d cos θ=

(1 + cos2 θ)

where in the centre of mass frame we have the mechanism shown in figure.

e+e−

We also have the same mechanism for jet formation in:

e+ + e− → q + q

as shown in figure, and therefore jets should be produced back-to-backwith same angular distribution being proportional to (1 + cos2 θ),that is ifjet direction reflects quark direction. This is exactly what is observedexperimentally!

e+e−

e+ + e− → q + q

e+e−

e+ + e− → q + q

e+e−

Quarks, jets and gluonsThree-jet events:

Two-jet events is the dominant process but sometimes we expect hardgluon emission through the mechanism shown in figure in analogy tophoton emission in electromagnetic interactions.

In the centre of mass frame we obtain a planar 3-jet event with themechanism shown in the following figure.

e+e−

This process is a bit complicated but has two simple features: First is thatthe rate of 3 jet events is of order αs relative to 2 jet events.The secondfeature is that the angular behaviour of the distribution depends on thespin of the gluon. This process has been extensively used to study the spinof the gluon as well as to measure the QCD coupling. Experimental resultsconfirmed the spin-1 nature of gluons and measured:

αs = 0.15± 0.03.

at energy scales of order 20 GeV.

This process is a bit complicated but has two simple features: First is thatthe rate of 3 jet events is of order αs relative to 2 jet events.The secondfeature is that the angular behaviour of the distribution depends on thespin of the gluon. This process has been extensively used to study the spinof the gluon as well as to measure the QCD coupling. Experimental resultsconfirmed the spin-1 nature of gluons and measured:

αs = 0.15± 0.03.

at energy scales of order 20 GeV.

Total cross-section

Consider the ratio:

R =σ(e+ + e− → hadrons)

σ(e+ + e− → µ+ + µ−)

Experimental results, shown in the figure below for this ratio show smoothbehaviour (as a function of centre of mass energy) except at thresholds for:

e+ + e− → c+ c, b+ b

where we get resonant states and increase in R. Focusing on the region10 < ECM < 40 GeV we obtain a value of R = 3.9.

Total cross-section

Consider the ratio:

σ(e+ + e− → µ+ + µ−)

e+ + e− → c+ c, b+ b

Total cross-section

Consider the ratio:

σ(e+ + e− → µ+ + µ−)

e+ + e− → c+ c, b+ b

Total cross-section

The obtained value follows from the dominant 2-quark production processe+ + e− → q + q and the existence of colour. To see this we compare theprocess e+ + e− → q + q with e+ + e− → µ+ + µ−.In units of the electriccharge e we know that

σ(e+ + e− → µ+ + µ−) ∝ e4

whileσ(e+ + e− → q + q) ∝ e2

where for example eu = +2/3, and the sum runs over all activeflavours.Supposing now that each of the quark flavours u, d, c, s, t, bcomes in Nc colour states, and by laws of quantum mechanics we mustsum over all final states,

Total cross-section

σ(e+ + e− → µ+ + µ−) ∝ e4

whileσ(e+ + e− → q + q) ∝ e2

Total cross-section

σ(e+ + e− → µ+ + µ−) ∝ e4

whileσ(e+ + e− → q + q) ∝ e2

Total cross-section

and assuming the high energy limit ECM � mµ,mq, thus for 2-jet eventwe have:

R = R0 = Nc

e2q = Nc(e2u + e2d + e2s + e2c + e2b) =

for centre of mass energy above 10 GeV (twice the mass of bottom). Weexclude the top quark because there is not enough energy to produce it.In addition we have a small contribution from three jet events, so we maywrite:

R = R0(1 + αs/π) =11

9Nc × 1.05

which gives excellent agreement with experiment for Nc = 3. Thisconfirms the existence of jets associated with quarks, spin-1 nature ofgluons and existence of colour with Nc = 3.

Total cross-section

R = R0 = Nc

e2q = Nc(e2u + e2d + e2s + e2c + e2b) =

R = R0(1 + αs/π) =11

9Nc × 1.05

Total cross-section

R = R0 = Nc

e2q = Nc(e2u + e2d + e2s + e2c + e2b) =

R = R0(1 + αs/π) =11

9Nc × 1.05

Total cross-section

R = R0 = Nc

e2q = Nc(e2u + e2d + e2s + e2c + e2b) =

R = R0(1 + αs/π) =11

9Nc × 1.05

particle physics chapter · ( particle physics - chapter 3 4 / 27. ... made solely of gluons?can...

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