Povh B., Rith K., Scholz Ch., Zetsche F. Particles and Nuclei: An Introduction to the Physical Concepts

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8.3 The Quark–Gluon Interaction 105

a) b) c) d)

Fig. 8.3. The fundamental interaction diagrams of the strong interaction: emission

of a gluon by a quark (a), splitting of a gluon into a quark–antiquark pair (b)and

“self-coupling” of gluons (c, d).

Hadrons as colour-neutral objects. With colour, quarks gain an addi-

tional degree of freedom. One might therefore expect each hadron to exist in

a multitude of versions which, depending upon the colours of the constituent

quarks involved, would have diﬀerent total (net) colours but would be equal

in all other respects. In practice only one type of each hadron is observed

(one π

−

,p,∆

etc.). This implies the existence of an additional condition:

only colourless particles, i. e., with no net colour, can exist as free particles.

This condition explains why quarks are not observed as free particles. A

single quark can be detached from a hadron only by producing at least two

free objects carrying colour: the quark, and the remainder of the hadron.

This phenomenon is therefore called conﬁnement. Accordingly, the potential

acting on a quark limitlessly increases with increasing separation, in sharp

contrast to the Coulomb potential. This phenomenon is due to the inter-

gluonic interactions.

The combination of a colour with the corresponding anticolour results in a

colourless (“white”) state. Putting the three diﬀerent colours together results

in a colourless (“white”) state as well. This can be graphically depicted by

three vectors in a plane symbolising the three colours, rotated with respect

to each other by 120

◦

blue

green

red

red antired

red

antiblue

green

antired

blue

antigreen

Hence, e.g., the π

meson has three possible colour combinations:

106 8 Quarks, Gluons, and the Strong Interaction

|π

 =

⎧

⎨

⎩



,

where the index designates the colour or anticolour. The physical pion is a

mixture of these states. By exchange of gluons, which by themselves simulta-

neously transfer colour and anticolour, the colour combination continuously

changes; yet the net-colour “white” is preserved.

In baryons, the colours of the three quarks also combine to yield “white”.

Hence, to obtain a colour neutral baryon, each quark must have a diﬀerent

colour. The proton is a mixture of such states:

|p =

⎧

⎪

⎨

⎪

⎩



From this argument, it also becomes clear why no hadrons exist which are

|qq,or|qq

q combinations, or the like. These states would not be colour

neutral, no matter what combination of colours were chosen.



udu

The strong coupling constant α

. In quantum ﬁeld theory, the coupling

“constant” describing the interaction between two particles is an eﬀective

constant which is in fact dependent on Q

. In the electromagnetic interaction

this dependence is only a weak one; in the strong interaction, however, it

is very strong. The reason is that gluons as the ﬁeld quanta of the strong

interaction carry colour themselves, and therefore can also couple to other

gluons. A ﬁrst-order perturbative calculation in QCD yields:

12π

(33 − 2n

) · ln(Q

/Λ

)

. (8.6)

8.4 Scaling Violations of the Structure Functions 107

Here, n

denotes the number of quark types involved. Since a heavy virtual

quark–antiquark pair has a very short lifetime and range, it can be resolved

only at very high Q

. Hence, n

depends on Q

,withn

≈ 3–6. The pa-

rameter Λ is the only free parameter of QCD. It is determined by comparing

predictions with experimental data to be Λ ≈ 250 MeV/c. The application

of perturbative expansion procedures in QCD is valid only if α

1. This is

satisﬁed for Q

Λ

≈0.06 (GeV/c)

From (7.15), the Q

-dependence of the coupling strength corresponds to a

dependence on separation. For very small distances and corresponding high

values of Q

, the interquark coupling decreases, vanishing asymptotically.

In the limit Q

→∞, quarks can be considered to be “free”, this is called

asymptotic freedom. By contrast, at large distances, the interquark coupling

increases so strongly, that it is impossible to detach individual quarks from

hadrons (conﬁnement).

8.4 Scaling Violations of the Structure Functions

In Sect. 7.2 we showed that the structure function F

depends on the quantity

x only. We thereby concluded that the nucleon is composed of point-like,

charged constituents. Yet, high precision measurements show that to a small

degree, F

does depend on Q

. Figure 8.4 shows the experimental results of

as a function of Q

at several ﬁxed values of x. The data cover a large

kinematic range of x and Q

. We see that the structure function increases

with Q

at small values of x; and decreases with increasing Q

at large values

of x. This behaviour, called scaling violation, is sketched once more in Fig. 8.5.

This means that at increasing values of Q

, there are fewer quarks with large

momentum fractions in the nucleon; rather, quarks with small momentum

fractions predominate.

This violation of scaling is not caused by any ﬁnite size of the quarks. In

the framework of QCD, the above violation can be traced back to fundamental

processes in which the constituents of the nucleon continuously interact with

each other (Fig. 8.3). Quarks can emit or absorb gluons, gluons may split

into q

q pairs, or emit gluons themselves. Thus, the momentum distribution

between the constituents of the nucleon is continually changing.

q(x)

q(y)

g(y)

108 8 Quarks, Gluons, and the Strong Interaction

(x,Q

)

0.5

x= 0.008

(x 4.0)

x= 0.0125

(x 3.2)

x= 0.0175

(x 2.5)

x= 0.025

(x 2.0)

x= 0.035

(x 1.5)

x= 0.050

(x 1.2)

x= 0.070

(x 1.0)

110

Deuteron

NMC

SLAC

BCDMS

[(GeV/c)

]

(x,Q

)

110

x= 0.009

(x 7.5)

x=0.11

(x 5.2)

x=0.14

(x 3.7)

x=0.18

(x 2.5)

x= 0.225

(x 1.7)

x=0.50

(x 1.0)

x= 0.275

(x 1.2)

x=0.35

(x 1.0)

0.1

Deuteron

NMC

SLAC

BCDMS

[(GeV/c)

]

Fig. 8.4. Structure function F

of the deuteron as a function of Q

at diﬀerent

values of x on a logarithmic scale. The results shown are from muon scattering

at CERN (NMC and BCDMS collaboration) [Am92a, Be90b] and from electron

scattering at SLAC [Wh92]. For clarity, the data at the various values of x are

multiplied by constant factors. The solid line is a QCD ﬁt, taking into account

the theoretically predicted scaling violation. The gluon distribution and the strong

coupling constant are free parameters here.

0.5

0.4

0.3

0.2

0.1

0 0.2 0.4 0.6 0.8 1

0.5

0.4

0.3

0.2

0.1

[(GeV/c)

]

Fig. 8.5. Schematic representation of the deuteron structure function F

as a

function of x at various values of Q

(left); and as a function of Q

at constant x

(right).

Figure 8.6 is an attempt to illustrate how this alters the measurements

of structure functions at diﬀerent values of Q

. A virtual photon can resolve

dimensions of the order of /



. At small Q

= Q

, quarks and any emitted

8.4 Scaling Violations of the Structure Functions 109

> Q

q(y)

q (x,Q

)

q (x,Q

)

q(y)

Fig. 8.6. A quark with momentum fraction x can be emitted from a quark or gluon

that carries a larger momentum fraction (x<y<1), (left). The resolving power of

the virtual photon increases with Q

(right). The diagram shows the interaction of

a photon with a quark after it has emitted a gluon. At small Q

= Q

,thequark

and the gluon are seen as a unit. At larger Q

, the resolution increases and

the momentum fraction of the quark alone is measured, i.e., without that of the

gluon; hence, a smaller value is obtained.

gluons cannot be distinguished and a quark distribution q(x, Q

)ismeasured.

At larger Q

and higher resolution, emission and splitting processes must be

considered. Thus, the number of partons seen to share the momentum of

the nucleon, increases. Therefore, the quark distribution q(x, Q

)atsmall

momentum fractions x is larger than q(x, Q

); whereas the eﬀect is reversed

for large x. This is the origin of the increase of the structure function with

at small values of x and its decrease at large x. The gluon distribution

g(x, Q

) shows a Q

-dependence as well, which originates from processes of

gluon emission by a quark or by another gluon.

The change in the quark distribution and in the gluon distribution with

, at ﬁxed values of x, is proportional to the strong coupling constant α

)

and depends upon the size of the quark and gluon distributions at all larger

values of x. The mutual dependence of the quark and gluon distributions

can be described by a system of coupled integral-diﬀerential equations [Gr72,

Li75, Al77]. If α

) and the shape of q(x, Q

)andg(x, Q

)areknownat

a given value Q

,thenq(x, Q

)andg(x, Q

)canbepredictedfromQCD

for all other values of Q

. Alternatively, the coupling α

)andthegluon

distribution g(x, Q

), which cannot be directly measured, can be determined

from the observed scaling violation of the structure function F

(x, Q

The solid lines in Fig. 8.4 show a ﬁt to the scaling violation of the measured

structure functions from a QCD calculation [Ar93]. The ﬁt value of Λ ≈

250 MeV/c corresponds to a coupling constant:

=100 (GeV/c)

) ≈ 0.16 . (8.7)

Figure 8.7 is a three dimensional representation of the Q

-dependence of the

structure function F

, here given as xq(x) ≡



x(q

(x)+¯q

(x)) ≈

110 8 Quarks, Gluons, and the Strong Interaction

and of the gluon structure function G(x, Q

)=xg(x, Q

)astheyemergefrom

this QCD analysis. The drastic change in the shape of these distributions with

2,4

0,90

0,50

0,28

0,18

0,11

0,07

0,035

0,018

0,008

11,5

7,0

4,5

2,5

1,3

xq(x, Q

)

2,0

1,6

1,2

0,8

0,4

xg(x, Q

)

11,5

7,0

4,5

2,5

1,3

0,90

0,50

0,28

0,18

0,11

0,07

0,035

0,018

0,008

[(GeV/c)

]

[(GeV/c)

]

Fig. 8.7. The distributions of quarks (top) and gluons (bottom) in the nucleon, as

functions of x and Q

[Bi92]. While q(x, Q

)andg(x, Q

) respectively describe the

probability of ﬁnding a quark or a gluon with momentum fraction x; xq(x, Q

)and

xg(x, Q

) measure the share of the momentum carried by the quarks or the gluons

in the particular x-interval.

Problem 111

is clearly seen. At small values of x, F

increases rapidly with Q

.Atsmall

values of Q

, the shape approaches that of the valence quark distribution,

since the sea quark contribution becomes less and less signiﬁcant. Similarly,

the gluon structure function xg(x) increases rapidly with Q

at small values

of x. It assumes a shape similar to that of a valence quark distribution at

small values of Q

. For large values of Q

the gluon distribution drops rapidly

with x.Thechange in the structure function with Q

can be calculated. It has

however so far proven impossible to predict the x-dependence of F

(x, Q

)in

theory; it is known only from experiment.

Summary. Scaling violation in the structure functions is a highly interesting

phenomenon. It is not unusual that particles that appear point-like turn out

to be composite when studied more closely (e.g., atomic nuclei in Rutherford

scattering with low-energy α particles or high-energy electrons). In deep in-

elastic scattering, however, a new phenomenon is observed. With increasing

resolution, quarks and gluons turn out to be composed of quarks and gluons;

which themselves, at even higher resolutions, turn out to be composite as

well (Fig. 8.6). The quantum numbers (spin, ﬂavour, colour, . . . ) of these

particles remain the same; only the mass, size, and the eﬀective coupling

change. Hence, there appears to be in some sense a self similarity in the

internal structure of strongly interacting particles.

Problem

1. Partons

Consider deep inelastic scattering of muons with energy 600 GeV oﬀ protons at

rest. The data analysis is to be carried out at Q

=4GeV

a) What is the smallest value of x which can be attained under these circum-

stances? You may assume that the minimal scattering energy is E



=0.

b) How many partons may be resolved with x>0.3, x>0.03 and in the full

measurable range of x if we parameterise the parton distribution as follows:

(x)=A(1 − x)

√

x for the valence quarks,

(x)=0.4(1 − x)

/x for the sea quarks and

g(x)=4(1− x)

/x for the gluons.

The role of the normalisation constant, A, is to take into account that there

are3valencequarks.

9 Particle Production in e

−

Collisions

So far, we have only discussed the light quarks, u and d, and those hadrons

composed of these two quarks. The easiest way to produce hadrons with

heavier quarks is in e

−

collisions. Free electrons and positrons may be

produced rather easily. They can be accelerated, stored and made to collide in

accelerators. In an electron–









positron collision process, all

particles which interact electro-

magnetically and weakly can be

produced, as long as the en-

ergy of the beam particles is

suﬃciently high. In an electron-

positron electromagnetic annihi-

lation, a virtual photon is pro-

duced, which immediately de-

cays into a pair of charged ele-

mentary particles. In a weak in-

teraction, the exchanged particle is the heavy vector boson Z

(cf. the diagram

and see Chap. 11). The symbol f denotes an elementary fermion (quark or

lepton) and

f its antiparticle. The ff system must have the quantum num-

bers of the photon or the Z

, respectively. In these reactions all fundamental,

charged particle–antiparticle pairs can be produced; lepton–antilepton and

quark–antiquark pairs. Neutrinos are electrically neutral; hence, neutrino–

antineutrino pairs can only be produced by Z

exchange.

Colliding beams. Which particle–antiparticle pairs can be produced only

depends upon the energy of the electrons and

positrons. In a storage ring, electrons and

positrons orbit with the same energy E, but

in opposite directions, and collide head-on. It

is conventional to use the Lorentz-invariant en-

ergy variable s, the square of the centre of mass

energy:

s =(p

c + p

= m

+ m

+2E

− 2p

(9.1)

114 9 Particle Production in e

−

Collisions

In a storage ring with colliding particles of energy E,

s =4E

. (9.2)

Hence, particle–antiparticle pairs with masses of up to 2m =

√

s/c

can be

produced. To discover new particles, the storage ring energy must be raised.

One then looks for an increase in the reaction rate, or for resonances in the

cross-section.

The great advantage of colliding beam experiments is that the total beam

energy is available in the centre of mass system. In a ﬁxed target experiment,

with m satisfying mc

 E, s is related to E by:

s ≈ 2mc

· E. (9.3)

Here, the centre of mass energy only increases proportionally to the square

root of the beam energy.

Particle detection. To detect the particles produced in e

−

annihilation

one requires a detector set up around the collision point which covers as much

as possible of the the total 4π solid angle. The detector should permit us to

trace the tracks back to the interaction point and to identify the particles

themselves. The basic form of such a detector is sketched in Fig. 9.1.

9.1 Lepton Pair Production

Before we turn to the creation of heavy quarks, we want to initially consider

the leptons. Leptons are elementary spin 1/2 particles which feel the weak

and, if they are charged, the electromagnetic interaction — but not, however,

the strong interaction.

Muons. The lightest particles which can be produced in electron-positron

collisions are muon pairs:

−

→ µ

+ µ

−

The muon µ

−

and its antiparticle

the µ

both have a mass of only 105.7

MeV/c

and they are produced in all usual e

−

storage ring experiments.

They penetrate matter very easily

, whereas electrons because of their small

mass and hadrons because of the strong interaction have much smaller ranges.

After that of the neutron, theirs is the longest lifetime (2 µs) of any unsta-

ble particle. This means that experimentally they may easily be identiﬁed.

Therefore the process of muon pair production is often used as a reference

point for other e

−

reactions.

Antiparticles are generally symbolised by a bar (e.g., ν

). This symbol is generally

skipped over for charged leptons since knowledge of the charge alone tells us

whether we have a particle or an antiparticle. We thus write e

, µ

, τ

Muons from cosmic radiation can still be detected in underground mines!

9.1 Lepton Pair Production 115

Scintillation counters

Magnetic field B

Muon detector

Lead glass counter

Wire chamber

Vertex detector

Supraconductive coil

Iron yoke

}





Fig. 9.1. Sketch of a 4π-detector, as used in e

−

collision experiments. The

detector is inside the coil of a solenoid, which typically produces a magnetic ﬁeld

of around 1T along the beam direction. Charged particles are detected in a vertex

detector, mostly composed of silicon microstrip counters, and in wire chambers. The

vertex detector is used to locate the interaction point. The curvature of the tracks

in the magnetic ﬁeld tell us the momenta. Photons and electrons are detected as

shower formations in electromagnetic calorimeters (of, e.g., lead glass). Muons pass

through the iron yoke with little energy loss. They are then seen in the exterior

scintillation counters.

Tau leptons. Ifthecentreofmassenergyinane

−

reaction suﬃces, a

further lepton pair, the τ

−

and τ

, may be produced. Their lifetime, 3 ·

−13

s, is much shorter. They may weakly decay into muons or electrons as

will be discussed in Sect 10.1f.

The tau was discovered at the SPEAR e

−

storage ring at SLAC when

oppositely charged electron-muon pairs were observed whose energy was

much smaller than the available centre of mass energy [Pe75].

These events were interpreted as the creation and subsequent decay of a

heavy lepton–antilepton pair:

−

−→ τ

+ τ

−

→ µ

−

+ ν

or e

−

+ ν

−−−→ e

+ ν

or µ

+ ν

The neutrinos which are created are not detected.

The threshold for τ

−

-pair production, and hence the mass of the τ-

lepton, may be read oﬀ from the increase of the cross-section of the e

−