Cambridge University Press 978-0-521-63148-8 - Quantum Chromodynamics: Perturbative and Nonperturbative Aspects B. L. Ioffe, V. S. Fadin and L. N. Lipatov Excerpt More information 1 General properties of QCD 1.1 QCD Lagrangian As in any gauge theory, the quantum chromodynamics (QCD) Lagrangian can be derived with the help of the gauge invariance principle from the free matter Lagrangian. Since quark fields enter the QCD Lagrangian additively, let us consider only one quark flavour. We will denote the quark field ψ(x), omitting spinor and colour indices [ψ(x) is a three- component column in colour space; each colour component is a four-component spinor]. The free quark Lagrangian is: Lq = ψ(x)(i ∂ − m)ψ(x), (1.1) where m is the quark mass, ∂ ∂ ∂ ∂ = ∂µγµ = γµ = γ0 + γ . (1.2) ∂xµ ∂t ∂r The Lagrangian Lq is invariant under global (x–independent) gauge transformations + ψ(x) → Uψ(x), ψ(x) → ψ(x)U , (1.3) with unitary and unimodular matrices U + − U = U 1, |U|=1, (1.4) belonging to the fundamental representation of the colour group SU(3)c. The matrices U can be represented as U ≡ U(θ) = exp(iθata), (1.5) where θa are the gauge transformation parameters; the index a runs from 1 to 8; ta are the colour group generators in the fundamental representation; and ta = λa/2,λa are the Gell-Mann matrices. Invariance under the global gauge transformations (1.3) can be extended to local (x-dependent) ones, i.e. to those where θa in the transformation matrix (1.5) is a x-dependent. This can be achieved by introducing gluon fields Aµ(x) which transform according to (θ) −1 i −1 Aµ(x) → Aµ (x) = UAµ(x)U + (∂µU)U , (1.6) g 1 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-63148-8 - Quantum Chromodynamics: Perturbative and Nonperturbative Aspects B. L. Ioffe, V. S. Fadin and L. N. Lipatov Excerpt More information 2 General properties of QCD where g is a coupling constant, a a Aµ(x) = Aµ(x)t , (1.7) and the partial derivative ∂µ in (1.1) is replaced by the covariant derivative ∇µ ∇µ ≡ ∂µ + igAµ. (1.8) The transformation law of the covariant derivative, (θ) (θ) + ∇µ = ∂µ + igAµ = U∇µU , (1.9) ensures the gauge invariance of the Lagrangian Lq + Lqg = ψ(x)(i ∇−m)ψ(x) = Lq − gψ(x) A(x)ψ(x). (1.10) The gauge field Lagrangian 1 Lg =− Tr Gµν(x)Gµν(x) (1.11) 2 is expressed in terms of the field intensities Gµν, which are built from the covariant derivatives ∇µ i Gµν(x) =− ∇µ, ∇ν = ∂µ Aν(x) − ∂ν Aµ(x) + ig[Aµ(x), Aν(x)]. (1.12) g For the components a a Gµν(x) = 2Tr t Gµν(x) (1.13) we have a ( ) = ∂ a( ) − ∂ a ( ) − a b ( ) c ( ), Gµν x µ Aν x ν Aµ x igTbc Aµ x Aν x (1.14) a =− abc abc where Tbc if are group generators of the adjoint representation; f are the group structure constants. Invariance of the Lagrangian (1.11) follows from the transformation law for the field strengths (θ) i (θ) (θ) + Gµν =− ∇µ , ∇ν = UGµνU . (1.15) g Thus, the QCD Lagrangian is LQCD = Lg + Lq + Lqg. (1.16) 1.2 Quantization of the QCD Lagrangian The Lagrangian (1.16) is invariant under the transformations (1.3)–(1.6). However, because of this invariance, a canonical quantization (i.e. exploiting operators with commutation laws [pi q j ]=−iδij) of this Lagrangian is impossible, since the momentum, canonically a( ) conjugate to coordinate A0 x , is zero. The reason is that the gauge invariance means the presence of superfluous, nonphysical fields (degrees of freedom) in the Lagrangian. In other words, massless gluons have only two possible helicities: ±1. Only a two-component © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-63148-8 - Quantum Chromodynamics: Perturbative and Nonperturbative Aspects B. L. Ioffe, V. S. Fadin and L. N. Lipatov Excerpt More information 1.2 Quantization of the QCD Lagrangian 3 a field is required to describe them, but we have introduced the four-component field Aµ(x). This problem is similar to those that arise in quantum electrodynamics (QED), but the solution in QCD is more complicated because the symmetry group SU(3)c is non-abelian, unlike the abelian U(1) in QED. The quantization problem can be solved in several ways. One is elimination of superflu- ous degrees of freedom with the help of constraint equations (the Lagrange–Euler equation a a for the A0-component does not contain the second-order time derivative of A0 and is there- fore a constraint equation) and some gauge condition (for example, the Coulomb gauge ∂ a = condition i Ai 0), which can be imposed due to gauge invariance of the Lagrangian. Clearly, this leads to the loss of an explicitly relativistic invariance of calculations. Of course, physical values obtained as a result of these calculations have correct transforma- tion properties (because of their gauge invariance); however, the loss of explicit invariance of intermediate calculations can produce a sense of aesthetic protest; moreover, it can lead to more cumbersome calculations. This method is applied as a rule in cases where a noncovariant approach has evident advantages (as the Coulomb gauge in nonrelativistic problems). In another approach, a term L fix, which allows one to perform the canonical quantiza- L =−1 (∂ a )2 tion, is added to the Lagrangian (for example, fix 2 µ Aµ ). In classical theory, this term is neutralized by the corresponding gauge condition (in our example, the Lorentz a condition ∂µ Aµ = 0). In quantum theory, such a condition cannot be imposed since com- a ponents Aµ are quantized independently. Therefore, the total space contains states with arbitrary numbers of pseudogluons, i.e. massless particles with nonphysical polarization vectors – timelike and longitudinal. In such an approach, the canonical quantization leads to an indefinite metric, i.e. a not positive definite norm of states, which clearly is incom- patible with the probabilistic interpretation. Of course, the physical subspace contains only gluons with transverse polarizations, and the indefinite metric does not show up there. But the S-matrix built using LQCD + L fix as a total Lagrangian has matrix elements between physical and nonphysical states. Recall that the same applies to QED. However, in QED, the corresponding S-matrix is unitary in the physical subspace, so that it can be success- fully used for calculation of physical observables. This is not quite so in the QCD case because of self-interaction of gluons. Here, the unitarity in the physical subspace can be achieved only by embedding ghost fields with an unusual property: They must be quan- tized as fermions with spin zero. Originally, this was pointed out by Feynman [1], who suggested the method of resolving the problem in the one-loop approximation. Later the method was developed in more detail by DeWitt [2]–[4]. Now the most common method of quantization of gauge fields is based on the functional integral formalism. In this for- malism, the impossibility of canonical quantization becomes apparent as the divergence 4 of the functional integral DA exp (iSQCD), where SQCD = d xLQCD is the classical action for the gluon fields. The reason of the divergence is that, since both the integration measure a DA = dAµ(x), (1.17) a,µ,x © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-63148-8 - Quantum Chromodynamics: Perturbative and Nonperturbative Aspects B. L. Ioffe, V. S. Fadin and L. N. Lipatov Excerpt More information 4 General properties of QCD and SQCD are invariant under the gauge transformations (1.3), (1.6), the integral contains as a factor the volume of the gauge group Dθ = dθa(x). (1.18) a,x However, it is sufficient to integrate only over a subspace of gauge nonequivalent fields a Aµ(x), i.e. fields not interconnected by the transformations (1.6). Such a subspace can be defined by the gauge condition (GAˆ )a(x) = Ba(x), (1.19) ˆ c with some operator G acting on the fields Aµ. The factor (1.18) can be extracted explicitly using the trick suggested in [5], namely by exploiting the equality (θ) det Mˆ (A) Dθ δ((GAˆ )a(x) − Ba(x)) = 1, (1.20) a,x where the operator Mˆ (A) is defined by its matrix elements δ(GAˆ (θ))a(x) (M(A))ab (x, y) = , (1.21) δθb(y) and det Mˆ (A) is called the Faddeev–Popov determinant. Here δ/δθ is the functional derivative, which is defined for any functional F(θ) as δF(θ) F(θa(x) + δabδ(x − y)) − F(θa(x)) = lim , (1.22) δθb(y) →0 and the values of θ in (1.21) are determined by the solution of the equation (GAˆ (θ))a(x)= Ba(x) with given A and B. It is supposed that this solution is unique. ˜ From the definition (1.20), it follows that det Mˆ (A) is gauge invariant, since (A(θ))θ = ˜ A(θ+θ). Therefore, we have (θ) DA eiSQCD = DA eiSQCD det Mˆ (A) Dθ δ (GAˆ )a(x) − Ba(x) a,x (θ) (θ) (θ) = Dθ DA det Mˆ (A ) δ (GAˆ )a(x) − Ba(x) eiSQCD a,x = Dθ DA det Mˆ (A) δ (GAˆ )a(x) − Ba(x) eiSQCD , (1.23) a,x ˆ where det M(A) is now given by (1.21) at θ = 0, so that the integrand is θ-independent and we can omit the factor Dθ.