The dynamics of colour .1 Colour as an SU(3) group

We now want to consider the possible dynamical role of colour – in other words, the way in which the forces between quarks depend on their colours.

We have seen that we seem to need three different quark types for each given flavour. They must all have the same mass, or else we would observe some

‘fine structure’ in the hadronic levels. Furthermore, and for the same reason,

‘colour’ must be an exact symmetry of the Hamiltonian governing the quark dynamics. What symmetry group is involved? We shall consider how some empirical facts suggest that the answer is SU(3)c.

To begin with, it is certainly clear that the interquark force must depend on colour, since we donotobserve ‘colour multiplicity’ of hadronic states: for example we do not see eight other colouredπ⁺’s (d^∗₁u2, d^∗₃u1,. . . ) degenerate with the one ‘colourless’ physical π⁺ whose wavefunction was given previ-ously. The observed hadronic states are allcolour singlets, and the force must somehow be responsible for this. More particularly, the force has to produce only those very restrictedtypesof quark configuration which are observed in the hadron spectrum. Consider again the isospin multiplets in nuclear physics discussed in section 12.1.2. There is one very striking difference in the par-ticle physics case: for mesons only T = 0,¹₂ and 1 occur, and for baryons only T = 0,¹₂, 1 and ³₂, while in nuclei there is nothing in principle to stop us findingT = ⁵₂,3, . . . states. (In fact such nuclear states are hard to iden-tify experimentally, because they occur at high excitation energy for some of the isobars – cf figure 1.8(c) – where the levels are very dense). The same restriction holds for SU(3)f also – only1’s and8’s occur for mesons; and only 1’s, 8’s and10’s for baryons. In quark terms, this of course is what is trans-lated into the recipe: ‘mesons are ¯qq, baryons are qqq’. It is as if we said, in nuclear physics, that onlyA= 2 andA = 3 nuclei exist! Thus the quark forces must have a dramatic saturation property: apparently no ¯qqq, no qqqq, qqqqq, . . . states exist. Furthermore, no qq or ¯q¯q states exist either – nor, for that matter, do single q’s or ¯q’s. All this can be summarized by saying that the quark colour degree of freedom must beconfined, a property we shall now assume and return to in chapter 16.

If we assume that only colour singlet states exist (Fritzsch and Gell-Mann 1972, Bardeen, Fritzsch and Gell-Mann 1973), and that the strong interquark force depends only on colour, the fact that ¯qq states are seen but qq and ¯q¯q are not gives us an important clue as to what group to associate with colour. One simple possibility might be that the three colours correspond to the compo-nents of an SU(2)ctriplet ‘ψ’. The antisymmetric, colour singlet, three-quark baryon wavefunction of (14.2) is then just the triple scalar productψ₁·ψ₂×ψ₃, which seems satisfactory. But what about the meson wavefunction? Mesons are formed of quarks and antiquarks, and we recall from sections 12.1.3 and

14.2. The dynamics of colour 79

12.2 that antiquarks belong to the complex conjugate of the representation (or multiplet) to which quarks belong. Thus if a quark colour triplet wavefunction ψ_α transforms under a colour transformation as

ψ_α → ψ^' _α = V_αβ ⁽¹⁾ ψ_β (14.10) where V⁽¹⁾ is a 3 × 3 unitary matrix appropriate to the T = 1 representation of SU(2) (cf (12.48) and (12.49)), then the wavefunction for the ‘anti’-triplet is ψ_α^∗ , which transforms as

ψ_α ^∗ → ψ^∗' _α = V _αβ ^(1)∗ ψ _β ^∗ . (14.11) Given this information, we can now construct colour singlet wavefunctions for mesons, built from ¯qq. Consider the quantity (cf (14.3)) E

α ψ_α^∗ ψα where ψ^∗ represents the antiquark and ψ the quark. This may be written in matrix notation as ψ^†ψ where the ψ^† as usual denotes the transpose of the complex conjugate of the column vector ψ. Then, taking the transpose of (14.11), we find that ψ^† transforms by

ψ^† → ψ^†' = ψ^†V^(1)† (14.12) so that the combination ψ^†ψ transforms as

ψ^†ψ → ψ^†'ψ^' = ψ^†V^(1)†V⁽¹⁾ψ =ψ^†ψ (14.13) where the last step follows since V⁽¹⁾ is unitary (compare (12.58)). Thus the product is invariant under (14.10) and (14.11) – that is, it is a colour singlet, as required. This is the meaning of the superposition (14.3).

All this may seem fine, but there is a problem. The three-dimensional representation of SU(2)_c which we are using here has a very special nature:

the matrix V⁽¹⁾ can be chosen to be real. This can be understood ‘physically’

if we make use of the great similarity between SU(2) and the group of rota-tions in three dimensions (which is the reason for the geometrical language of isospin ‘rotations’, and so on). We know very well how real three-dimensional vectors transform, namely by an orthogonal 3 × 3 matrix. It is the same in SU(2). It is always possible to choose the wavefunctions ψ to be real, and the transformation matrix V⁽¹⁾ to be real also. Since V⁽¹⁾ is, in general, unitary, this means that it must be orthogonal. But now the basic difficulty appears:

there is no distinction between ψ and ψ ^∗ ! They both transform by the real matrix V⁽¹⁾. This means that we can make SU(2) invariant (colour singlet) combinations for ¯q¯q states, and for qq states, just as well as for ¯ states – qq indeed they are formally identical. But such ‘diquark’ (or ‘antidiquark’) states are not found, and hence – by assumption – should not be colour singlets.

The next simplest possibility seems to be that the three colours corre-spond to the components of an SU(3)_c triplet. In this case the quark colour wavefunction ψ_α transforms as (cf (12.74))

ψ → ψ^' = Wψ (14.14)

14.2. The dynamics of colour 81 Then, in just the same way, we can introduce the total colour operator

F = 1

2(λ1+λ2), (14.22)

so that

F²= 1

4(λ²₁+ 2λ1·λ2+λ²₂) (14.23) and

λ1·λ2= 2F²−λ², (14.24) where λ²₁ = λ²₂ = λ², say. Here λ² ≡E8

a=1(λa)² is found (see (12.75)) to have the value 16/3 (the unit matrix being understood). The operator F² commutes with all components of λ1 and λ2 (as T² does with τ1 and τ2) and represents the quadratic Casimir operator ˆC2 of SU(3)c (see section M.5 of appendix M), in the colour space of the two quarks considered here. The eigenvalues of ˆC2play a very important role in SU(3)c, analogous to that of the total spin/angular momentum in SU(2). They depend on the SU(3)c repre-sentation: indeed, they are one of the defining labels of SU(3) representations in general (see section M.5). Two quarks, each in the representation3c, com-bine to give a 6c-dimensional representation and a 3^∗_c (see problem 14.1(b), and Jones (1990) chapter 8). The value of ˆC2for the singlet6crepresentation is 10/3, and for the 3^∗_c representation is 4/3. Thus the ‘λ1·λ2’ interaction will produce a negative (attractive) eigenvalue -8/3 in the 3^∗_c states, but a repulsive eigenvalue +4/3 in the6c states, for two quarks.

The maximum attraction will clearly be for states in which F² is zero.

This is the singlet representation 1c. Two quarks cannot combine to give a colour singlet state, but we have seen in section 12.2 that a quark and an antiquark can: they combine to give1cand8c. In this case (14.24) is replaced by

λ1·λ2= 2F²−1

2(λ²₁+λ²₂), (14.25) where ‘1’ refers to the quark and ‘2’ to the antiquark. Thus the ‘λ1 ·λ2’ interaction will give a repulsive eigenvalue +2/3 in the 8c channel, for which Cˆ2 = 3, and a ‘maximally attractive’ eigenvalue -16/3 in the1c channel, for a quark and an antiquark.

In the case of baryons, built from three quarks, we have seen that when two of them are coupled to the3^∗_c state, the eigenvalue ofλ1·λ2is -8/3, one half of the attraction in the ¯qqcolour singlet state, but still strongly attractive.

The (qq) pair in the3^∗_c state can then couple to the remaining third quark to make the overall colour singlet state (14.2), with maximum binding.

Of course, such a simple potential model does not imply that the energy difference between the 1c states and all coloured states is infinite, as our strict ‘colour singlets only’ hypothesis would demand, and which would be one (rather crude) way of interpreting confinement. Nevertheless, we can ask:

what single particle exchange process between quark (or antiquark) colour triplets produces a λ1·λ2 type of term? The answer is the exchange of 80 14. QCD I: Introduction, Tree Graph Predictions, and Jets

where W is a special unitary 3 × 3 matrix parametrized as

W = exp(iα · λ/2), (14.15) and ψ^† transforms as

ψ^† → ψ^†' = ψ^†W^† . (14.16) The proof of the invariance of ψ^†ψ goes through as in (14.13), and it can be shown (problem 14.1(a)) that the antisymmetric 3q combination (14.2) is also an SU(3)c invariant. Thus both the proposed meson and baryon states are colour singlets. It is not possible to choose the λ’s to be pure imaginary in (14.15), and thus the 3×3 W matrices of SU(3)c cannot be real, so that there is a distinction between ψ and ψ^∗, as we learned in section 12.2. Indeed, it can be shown (see Carruthers 1966, chapter 3, Jones 1990, chapter 8, and also problem 14.1(b)) that, unlike the case of SU(2)_c triplets, it is not possible to form an SU(3)_c colour singlet combination out of two colour triplets qq or anti-triplets ¯q¯q. Thus SU(3)_c seems to be a possible and economical choice for the colour group.

14.2.2 Global SU(3)_c invariance, and ‘scalar gluons’

As stated above, we are assuming, on empirical grounds, that the only phys-ically observed hadronic states are colour singlets – and this now means sin-glets under SU(3)c. What sort of interquark force could produce this dramatic result? Consider an SU(2) analogy again, the interaction of two nucleons be-longing to the lowest (doublet) representation of SU(2). Labelling the states by an isospin T , the possible T values for two nucleons are T = 1 (triplet) and T = 0 (singlet). We know of an isospin-dependent force which can produce a splitting between these states, namely V τ 1 · τ 2, where the ‘1’ and ‘2’ refer to the two nucleons. The total isospin is T = (τ 1₂ 1 + τ 2), and we have

T ² = 1

(τ ²₁ + 2τ 1 · τ 2 + τ ²₂) = 1

(3 + 2τ 1 · τ 2 + 3) (14.17)

4 4

whence

τ 1 · τ 2 = 2T ² − 3. (14.18) In the triplet state T ² = 2, and in the singlet state T ² = 0. Thus

(τ 1 · τ 2)T =1 = 1 (14.19) (τ 1 · τ 2)_{T =0} = −3 (14.20) and if V is positive the T = 0 state is pulled down. A similar thing happens in SU(3)c. Suppose this interquark force depended on the quark colours via a term proportional to

λ₁ · λ2. (14.21)

80 14. QCD I: Introduction, Tree Graph Predictions, and Jets whereWis a special unitary 3×3 matrix parametrized as

W= exp(iα·λ/2), (14.15)

andψ^† transforms as

ψ^† →ψ^†'=ψ^†W^†. (14.16)

The proof of the invariance ofψ^†ψ goes through as in (14.13), and it can be shown (problem 14.1(a)) that the antisymmetric 3q combination (14.2) is also an SU(3)c invariant. Thus both the proposed meson and baryon states are colour singlets. It is not possible to choose the λ’s to be pure imaginary in (14.15), and thus the 3×3Wmatrices of SU(3)c cannot be real, so that there is a distinction betweenψ and ψ^∗, as we learned in section 12.2. Indeed, it can be shown (see Carruthers 1966, chapter 3, Jones 1990, chapter 8, and also problem 14.1(b)) that, unlike the case of SU(2)c triplets, it is not possible to form an SU(3)c colour singlet combination out of two colour triplets qq or anti-triplets ¯q¯q. Thus SU(3)c seems to be a possible and economical choice for the colour group.

14.2.2 Global SU(3)_c invariance, and ‘scalar gluons’

As stated above, we are assuming, on empirical grounds, that the only phys-ically observed hadronic states are colour singlets – and this now means sin-glets under SU(3)c. What sort of interquark force could produce this dramatic result? Consider an SU(2) analogy again, the interaction of two nucleons be-longing to the lowest (doublet) representation of SU(2). Labelling the states by an isospinT, the possibleT values for two nucleons areT = 1 (triplet) and T = 0 (singlet). We know of an isospin-dependent force which can produce a splitting between these states, namelyVτ1·τ2, where the ‘1’ and ‘2’ refer to the two nucleons. The total isospin isT = ¹₂(τ1+τ2), and we have

T²= 1

4(τ²₁+ 2τ1·τ2+τ²₂) =1

4(3 + 2τ1·τ2+ 3) (14.17) whence

τ1·τ2= 2T²−3. (14.18)

In the triplet stateT²= 2, and in the singlet stateT²= 0. Thus

(τ1·τ2)T=1 = 1 (14.19)

(τ1·τ2)T=0 = −3 (14.20)

and ifV is positive the T = 0 state is pulled down. A similar thing happens in SU(3)c. Suppose this interquark force depended on the quark colours via a term proportional to

λ1·λ2. (14.21)

14.2. The dynamics of colour 81

Then, in just the same way, we can introduce the total colour operator F = 1 (λ1 + λ2), (14.22)

2 so that

1(λ²

F ² = ₁ + 2λ₁ · λ2 + λ²₂) (14.23) 4

and

λ1 · λ2 = 2F ² − λ² , (14.24) where λ² ₁ = λ² ₂ = λ² , say. Here λ² ≡E8 _a=1(λa)² is found (see (12.75)) to have the value 16/3 (the unit matrix being understood). The operator F ² commutes with all components of λ₁ and λ₂ (as T ² does with τ 1 and τ 2) and represents the quadratic Casimir operator Cˆ₂ of SU(3)_c (see section M.5 of appendix M), in the colour space of the two quarks considered here. The eigenvalues of Cˆ₂ play a very important role in SU(3)c, analogous to that of the total spin/angular momentum in SU(2). They depend on the SU(3)c repre-sentation: indeed, they are one of the defining labels of SU(3) representations in general (see section M.5). Two quarks, each in the representation 3c, com-bine to give a 6c-dimensional representation and a 3^∗ _c (see problem 14.1(b), and Jones (1990) chapter 8). The value of Cˆ₂ for the singlet 6c representation is 10/3, and for the 3^∗ _c representation is 4/3. Thus the ‘λ1 · λ2 ’ interaction will produce a negative (attractive) eigenvalue -8/3 in the 3^∗ _c states, but a repulsive eigenvalue +4/3 in the 6_c states, for two quarks.

The maximum attraction will clearly be for states in which F ² is zero.

This is the singlet representation 1c. Two quarks cannot combine to give a colour singlet state, but we have seen in section 12.2 that a quark and an antiquark can: they combine to give 1_c and 8c. In this case (14.24) is replaced by

λ₁ · λ2 = 2F ² −1

(λ²₁ + λ²₂), (14.25) 2

where ‘1’ refers to the quark and ‘2’ to the antiquark. Thus the ‘λ₁ · λ2 ’ interaction will give a repulsive eigenvalue +2/3 in the 8_c channel, for which Cˆ₂ = 3, and a ‘maximally attractive’ eigenvalue -16/3 in the 1_c channel, for a quark and an antiquark.

In the case of baryons, built from three quarks, we have seen that when two of them are coupled to the 3^∗ _c state, the eigenvalue of λ1 · λ2 is -8/3, one half of the attraction in the ¯qq colour singlet state, but still strongly attractive.

The (qq) pair in the3^∗ _c state can then couple to the remaining third quark to make the overall colour singlet state (14.2), with maximum binding.

Of course, such a simple potential model does not imply that the energy difference between the 1_c states and all coloured states is infinite, as our strict ‘colour singlets only’ hypothesis would demand, and which would be one (rather crude) way of interpreting confinement. Nevertheless, we can ask:

what single particle exchange process between quark (or antiquark) colour triplets produces a λ₁ · λ2 type of term? The answer is the exchange of

14.2. The dynamics of colour 83 as in (13.58 ), and the vertex (14.27) becomes

−igs

λa

2 γ^μ (14.30)

as in (13.60). One motivation for this is the desire to make the colour dynamics as much as possible like the highly successful theory of QED, and to derive the dynamics from a gauge principle. As we have seen in the last chapter, this involves the simple but deep step of supposing that the quark wave equation is covariant underlocalSU(3)c transformations of the form

ψ→ψ^' = exp(igsα(x)·λ/2)ψ. (14.31) This is implemented by the replacement

∂μ→∂μ+ igs

λa

2 Aaμ(x) (14.32)

in the Dirac equation for the quarks, which leads immediately to (14.29) and the vertex (14.30).

Of course, the assumption of local SU(3)c covariance leads to a great deal more: for example, it implies that the gluons are massless vector (spin 1) particles, and that they interact with themselves via three-gluon and four-gluonvertices, which are the SU(3)c analogues of the SU(2) vertices discussed in section 13.3.2. The most compact way of summarizing all this structure is via the Lagrangian, most of which we have already introduced in chapter 13.

Gathering together (13.71) and (13.140) (adapted to SU(3)c), we write it out here for convenience:

L^QCD = E

flavours f

¯ˆ

q_f,α(i ˆD/ −mf)αβqˆf,β−1

4FˆaμνFˆ_a^μν

− 1

2ξ(∂μAˆ^μ_a)(∂νAˆ^ν_a) +∂μηˆ^†_aDˆ^μ_abηˆb. (14.33) In (14.33), repeated indices are as usual summed over: αandβ are SU(3)c -triplet indices running from 1 to 3, and a, b are SU(3)c-octet indices running from 1 to 8. The covariant derivatives are defined by

( ˆDμ)αβ=∂μδαβ+ igs

1

2(λa)αβAˆaμ (14.34) when acting on the quark SU(3)c triplet, as in (13.53), and by

( ˆDμ)ab=∂μδab+gsfcabAˆcμ (14.35) when acting on the octet of ghost fields. For the second of these, note that the matrices representing the SU(3) generators in the octet representation are as given in (12.84), and these take the place of the ‘λ/2’ in (14.34) (compare (13.141) in the SU(2) case). We remind the reader that the last two terms 82 14. QCD I: Introduction, Tree Graph Predictions, and Jets

FIGURE 14.4

Scalar gluon exchange between two quarks.

an SU(3)_c octet (8c) of particles, which (anticipating somewhat) we shall call gluons. Since colour is an exact symmetry, the quark wave equation describing the colour interactions must be SU(3)c covariant. A simple such equation is

λa

(i /∂ − m)ψ = g_s Aaψ (14.26)

2

where g_s is a ‘strong charge’ and A_a (a = 1, 2, . . . , 8) is an octet of scalar

‘gluon potentials’. Equation (14.26) may be compared with (13.58): in the latter, /A_a appears on the right-hand side, because the gauge field quanta are vectors rather than scalars. In (14.26), we are dealing at this stage only with a global SU(3) symmetry, not a local SU(3) gauge symmetry, and so the potentials may be taken to be scalars, for simplicity. As in (13.60), the vertex corresponding to (14.26) is

−igsλa/2. (14.27)

(14.27) differs from (13.60) simply in the absence of the γ^μ factor, due to the assumed scalar, rather than vector, nature of the ‘gluon’ here. When we put two such vertices together and join them with a gluon propagator (figure 14.4), the SU(3)_c structure of the amplitude will be

λ1a λ2b λ1 λ2

δ_ab = · (14.28)

2 2 2 2

the δab arising from the fact that the freely propagating gluon does not change its colour. This interaction has exactly the required ‘λ1 · λ2 ’ character in the colour space.

14.2.3 Local SU(3)_c invariance: the QCD Lagrangian

It is tempting to suppose (Fritzsch and Gell-Mann 1972, Fritzsch, Gell-Mann and Leutwyler 1973) that the ‘scalar gluons’ introduced in (14.26) are, in fact, vector particles, like the photons of QED. Equation (14.26) then becomes

λa

(i /∂ − m)ψ = g_s A/ aψ (14.29)

2

82 14. QCD I: Introduction, Tree Graph Predictions, and Jets

FIGURE 14.4

Scalar gluon exchange between two quarks.

an SU(3)coctet (8c) of particles, which (anticipating somewhat) we shall call gluons. Since colour is an exact symmetry, the quark wave equation describing the colour interactions must be SU(3)c covariant. A simple such equation is

(i/∂−m)ψ=gs

λa

2 Aaψ (14.26)

where gs is a ‘strong charge’ and Aa (a= 1, 2, . . . , 8) is an octet of scalar

‘gluon potentials’. Equation (14.26) may be compared with (13.58): in the latter, /Aa appears on the right-hand side, because the gauge field quanta are vectors rather than scalars. In (14.26), we are dealing at this stage only with aglobalSU(3) symmetry, not a local SU(3) gauge symmetry, and so the potentials may be taken to be scalars, for simplicity. As in (13.60), the vertex corresponding to (14.26) is

−igsλa/2. (14.27)

(14.27) differs from (13.60) simply in the absence of the γ^μ factor, due to the assumed scalar, rather than vector, nature of the ‘gluon’ here. When we put two such vertices together and join them with a gluon propagator (figure 14.4), the SU(3)c structure of the amplitude will be

λ1a

2 δab

λ2b

2 = λ1

2 · λ2

2 (14.28)

theδabarising from the fact that the freely propagating gluon does not change its colour. This interaction has exactly the required ‘λ1·λ2’ character in the colour space.

14.2.3 Local SU(3)_c invariance: the QCD Lagrangian

It is tempting to suppose (Fritzsch and Gell-Mann 1972, Fritzsch, Gell-Mann and Leutwyler 1973) that the ‘scalar gluons’ introduced in (14.26) are, in fact, vector particles, like the photons of QED. Equation (14.26) then becomes

(i/∂−m)ψ=gs

λa

2 A/ aψ (14.29)

14.2. The dynamics of colour 83

as in (13.58 ), and the vertex (14.27) becomes λa

−ig_s γ^μ (14.30)

2

as in (13.60). One motivation for this is the desire to make the colour dynamics as much as possible like the highly successful theory of QED, and to derive the dynamics from a gauge principle. As we have seen in the last chapter, this involves the simple but deep step of supposing that the quark wave equation is covariant under local SU(3)c transformations of the form

ψ → ψ^' = exp(igsα(x)· λ/2)ψ. (14.31) This is implemented by the replacement

∂μ → ∂μ + igs λa

2 Aaμ(x) (14.32)

in the Dirac equation for the quarks, which leads immediately to (14.29) and the vertex (14.30).

Of course, the assumption of local SU(3)c covariance leads to a great deal more: for example, it implies that the gluons are massless vector (spin 1) particles, and that they interact with themselves via three-gluon and four-gluon vertices, which are the SU(3)c analogues of the SU(2) vertices discussed in section 13.3.2. The most compact way of summarizing all this structure is via the Lagrangian, most of which we have already introduced in chapter 13.

Gathering together (13.71) and (13.140) (adapted to SU(3)c), we write it out here for convenience:

¯ 1 ˆ Fˆ^μν

= E

ˆ /

LQCD q_f,α(iDˆ − mf )_αβ qˆ_f,β − F_{aμν a}

flavours f 4

1 _μ

Aˆ^μ Aˆ^ν η^† ˆ

− (∂_{μ a} )(∂_{ν a}) + ∂_μ ˆ_aD η_ab ˆb. (14.33) 2ξ

In (14.33), repeated indices are as usual summed over: α and β are SU(3)_c -triplet indices running from 1 to 3, and a, b are SU(3)c-octet indices running from 1 to 8. The covariant derivatives are defined by

( ˆDμ)_αβ = ∂μδ_αβ + ig_s 1 (λa)_αβ Aˆ_aμ (14.34) 2

when acting on the quark SU(3)c triplet, as in (13.53), and by

(Dˆ_μ)ab = ∂μδab + gsfcab Aˆ_cμ (14.35) when acting on the octet of ghost fields. For the second of these, note that the matrices representing the SU(3) generators in the octet representation are as given in (12.84), and these take the place of the ‘λ/2’ in (14.34) (compare (13.141) in the SU(2) case). We remind the reader that the last two terms

14.2. The dynamics of colour 85 there is in fact one more gauge invariant term of mass dimension 4 which can be written down, namely

Lˆ^θ= θg²_s

64π²EμνρσFˆ_a^μνFˆ_a^ρσ; (14.40) this is the ‘θ-term’ of QCD. A full discussion of this term (see for example

64π²EμνρσFˆ_a^μνFˆ_a^ρσ; (14.40) this is the ‘θ-term’ of QCD. A full discussion of this term (see for example

Im Dokument Non-Abelian Gauge Theories (Seite 94-101)