Quarks, Gluons, and the Strong Interaction
8.2 The Quark-Gluon Interaction
Colour Quarks have another important property called colour which we have previously neglected. This is needed to ensure that quarks in hadrons obey the Pauli principle. Consider theCC-resonance which consists of three u-quarks. TheCC
8.2 The Quark-Gluon Interaction 105
has spinJ D 3=2and positive parity; it is the lightest baryon withJP D 3=2C. We therefore can assume that its orbital angular momentum is` D 0; so it has a symmetric spatial wave function. In order to yield total angular momentum3=2, the spins of all three quarks have to be parallel:
jCCi D ju"u"u"i:
Thus, the spin wave function is also symmetric. The wave function of this system is furthermore symmetric under the interchange of any two quarks, as only quarks of the same flavour are present. Therefore the total wave function appears to be symmetric, in violation of the Pauli principle.
Including thecolour property, a kind of quark charge, the Pauli principle may be salvaged. The quantum number colour can assume three values, which may be calledred, blueandgreen. Accordingly, antiquarks carry the anticoloursanti-red, anti-blue, andanti-green.Now the three u-quarks may be distinguished by their colour. Thus, a colour wave function antisymmetric under particle interchange can be constructed, and we so have antisymmetry for the total wave function. The quantum number colour was introduced for theoretical reasons, yet experimental clues indicate that this hypothesis is correct. This will be discussed in Sect.9.3.
Gluons The interaction binding quarks into hadrons is called thestrong interaction.
Such a fundamental interaction is, in our current understanding, always connected with a particle exchange. For the strong interaction, gluons are the exchange particles that couple to the colour charge. This is analogous to the electromagnetic interaction in which photons are exchanged between electrically charged particles.
The experimental findings of Sect.7.4led to the development of a field theory calledquantum chromodynamics(QCD). As its name implies, QCD is modelled upon quantum electrodynamics (QED). In both, the interaction is mediated by exchange of a massless field particle withJPD1(a vector boson).
The gluons carry simultaneously colour and anticolour. According to group theory, the33colour combinations form two multiplets of states: a singlet and an octet. The octet states form a basis from which all other colour states may be constructed. They correspond to an octet of gluons. The way in which these eight states are constructed from colours and anticolours is a matter of convention. One possible choice is
rg;N rb;N gb;N gNr; bNr; bNg; p
1=2 .rNrgNg/; p
1=6 .rNrCggN2bb/ :N The colour singlet
p1=3 .rNrCggNCbb/ ;N
which is symmetrically constructed from the three colours and the three anticolours is invariant with respect to a re-definition of the colour names (rotation in colour space). It, therefore, has no effect in colour space and cannot be exchanged between colour charges.
106 8 Quarks, Gluons, and the Strong Interaction
a) b) c) d)
Fig. 8.1 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)
By their exchange the eight gluons mediate the interaction between particles carrying colour charge, i.e., not only the quarks but also the gluons themselves.
This is an important difference to the electromagnetic interaction, where the photon field quanta have no charge, and therefore cannot couple with each other.
In analogy to the elementary processes of QED (emission and absorption of photons, pair production and annihilation), emission and absorption of gluons (Fig.8.1a) take place in QCD, as do production and annihilation of quark-antiquark pairs (Fig.8.1b). In addition, however, three or four gluons can couple to each other in QCD (Fig.8.1c, d).
Hadrons as colour-neutral objects With colour, quarks gain an additional 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, p,0 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 (cf. Sect.14.3) 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 by120ı (Fig.8.2).
Hence, e.g., theCmeson has three possible colour combinations:
jCi D 8<
: jurdri jubdbi jugdgi;
8.2 The Quark-Gluon Interaction 107
blue
green red antired red
red
antiblue
green antired
blue antigreen
Fig. 8.2 Graphical presentation of the colour vectors in colour space (left); colour and anticolour in a meson combine to ‘white’ (middle) as do the three colours in a baryon (right)
b
g
b
r
b
g
b
r
u d
π+
g
b g
b r
u d
u p
r g
bg bg
g b
rg bg
bg
rb
Fig. 8.3 By the exchange of coloured gluons quark and antiquark in a meson (left) and the three quarks in a baryon (right) continuously change their colour, preserving always the net colour
‘white’
where the index designates the colour or anticolour. The physical pion is a mixture of these states. By exchange of gluons, which by themselves simultaneously transfer colour and anticolour, the colour combination continuously changes; yet the net- colour “white” is preserved (Fig.8.3).
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:
jpi D 8ˆ
<
ˆ:
juburdgi jurugdbi ::: :
108 8 Quarks, Gluons, and the Strong Interaction
From this argument, it also becomes clear why no hadrons exist which arejqqi, or jqqqicombinations, or the like. These states would not be colour neutral, no matter what combination of colours were chosen.
Constituent and current quarks In (7.25) we saw that only about half of the momentum of a nucleon is carried by valence and sea quarks. In dealing with the spectroscopic properties of nucleons, sea quarks and gluons need not be explicitly dealt with. We can combine them with the valence quarks. One then acts as though there were only three valence quarks, with enlarged masses but unchanged quantum numbers. We will return to this point in Chaps.14–16. These “effective valence quarks” are calledconstituent quarks.
In interpreting deep-inelastic scattering, we neglected the rest masses of the bare u- and d-quarks. This is justified since they are small [20]:muD1:83MeV=c2, mdD4:55:5MeV=c2. These masses are commonly calledcurrent quarkmasses.
However, these are not the masses obtained from hadron spectroscopy; e.g., from calculations of magnetic moments and hadron excitation energies that we will discuss in detail in Chaps.15and16. Theconstituent-quarkmasses are much larger with values of about300MeV=c2. They are mainly due to the cloud of gluons and sea quarks. Their values for all the quark flavours are compiled in Table9.1.
The bare d-quark is heavier than the bare u-quark, which can be easily understood as follows. The proton.uud/and the neutron.ddu/are isospin symmetric as stated above; i.e., they transform into each other under interchange of the u- and d-quarks.
Since the strong interaction is independent of quark flavour, the neutron-proton mass difference can only be due to the intrinsic quark masses and to the electromagnetic interaction between them. If we assume that the spatial distribution of the u- and d-quarks in the proton corresponds to the distribution of d- and u-quarks in the neutron, then it is easily seen that the Coulomb energy must be higher in the proton.
Despite this, the neutron is heavier than the proton which implies that the mass of the d-quark is larger.
The strong coupling constant˛s In quantum field theory, the coupling “constant”
describing the interaction between two particles is an effective constant which in fact depends onQ2. In the electromagnetic interaction this dependence is very weak; in the strong interaction, however, it is rather strong. The reason for this is that gluons, the field quanta of the strong interaction, carry colour themselves, and therefore can also couple to other gluons. In Fig.8.4 the differentQ2 behaviours of the electromagnetic and the strong coupling constants are presented. The contribution of the fluctuation of the photon into an electron-positron pair as well as of the gluon into a quark-antiquark pair results in the screening of the electric and strong charge.
The higherQ2 is, the smaller are the distances between the interacting particles, and thus the effective charge of the interacting particles increases and the coupling constant increases. Gluons couple to other gluons and can fluctuate into gluons. This fluctuation causes antiscreening. The closer the interacting particles are, the smaller the charge they see. The coupling constant decreases with increasingQ2. In the case of gluons the antiscreening is far stronger than the screening.
8.2 The Quark-Gluon Interaction 109
q
QCD QED
quark confined in hadron
coupling constant
α
sα
emq
q
Q
2Fig. 8.4 TheQ2 dependence of the strong˛s and the electromagnetic˛emcoupling constants is shown. The fluctuation of the photon into an electron-positron pair leads to the screening of the electric charge. Analogously, the fluctuation of the gluon into a quark-antiquark pair leads to the screening of the strong charge. The self coupling of the gluons results in the antiscreening
A first-order perturbation calculation in QCD yields:
˛s.Q2/D 12
.332nf/ln.Q2=2/: (8.1) Here,nf 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 highQ2. Hence,nf depends onQ2, withnf 3–6. The parameteris the only free parameter of QCD. It was found to be250MeV/cby comparing the prediction with the experimental data. The application of perturbative expansion procedures in QCD is valid only if˛s 1. This is satisfied forQ220:06 .GeV=c/2.
110 8 Quarks, Gluons, and the Strong Interaction
From (7.13) we can see that the Q2-dependence of the coupling strength corresponds to a dependence on separation. For very small distances and correspondingly high values of Q2, the interquark coupling decreases, vanishing asymptotically. In the limit Q2 ! 1, quarks can be considered “free”, this is calledasymptotic freedom. By contrast, at large distances, the interquark coupling increases so strongly that it is impossible to detach individual quarks from hadrons (confinement).