General Properties of QCD

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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

© 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

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 therefore 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 transformation 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 quantized 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 formalism, 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

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. Therefore, the functional integral over the gauge field can be defined as the last expression in (1.23) without Dθ. Moreover, since (1.23) does

1.2 Quantization of the QCD Lagrangian 5

not depend on B, one can integrate it over B with an appropriate weight factor and define the functional integral as the result of integration. The commonly used definition is − i d4x(Ba (x))2 DBe 2ξ DA det Mˆ (A) δ (GAˆ )a(x) − Ba(x) eiSQCD a,x iS − i d4x((GAˆ )a (x))2 = DA det Mˆ (A) e QCD 2ξ . (1.24) 4 The latter exponential can be written as exp(i d x(LQCD + L fix)), where the gauge fixing term is given by 2 1 ˆ a L fix =− (GA) (x) . (1.25) 2ξ The factor det Mˆ (A) can also be presented in the Lagrangian form using nonphysical ghost fields φa(x) obeying Fermi–Dirac statistic. The functional integral formalism can be extended to the case of fermion fields, considering them as anticommuting Grassmann variables. For a finite number of such variables ψi , i =1÷N, [ψi ψ j ]+ ≡ ψi ψ j +ψ j ψi =0 for all i, j, the set of monomials

1,ψi ,ψi ψ j ,...,ψ1ψ2 ...ψN (1.26) gives the basis of the linear space of the Grassmann algebra. There are left and right derivatives in this space that are deﬁned by the rules ∂ (−1)k−1ψ ···ψ ψ ···ψ , if i = j, ψ ψ ···ψ = i1 ik−1 ik+1 in k i1 i2 in ∂ψj 0, if il = j, l = 1, 2,...,n, (1.27) ←− ∂ (−1)n−kψ ···ψ ψ ···ψ , if i = j, ψ ψ ···ψ = i1 ik−1 ik+1 in k i1 i2 in ∂ψj 0, if il = j, l = 1, 2,...,n. (1.28) Note that the derivatives are anticommuting: ∂2 ∂2 =− . (1.29) ∂ψi ∂ψj ∂ψj ∂ψi The integrals over the Grassman variables are deﬁned by the rules

dψi = 0, dψi ψ j = δij, (1.30)

and

[dψi ,ψj ]+ =[dψi , dψ j ]+ = 0. (1.31) These rules give the important relation ⎛ ⎞ ¯ ¯ ¯ ⎝ ¯ ⎠ N(N+1) dψ1dψ2 ···dψN dψ1dψ2 ···dψN exp − ψi Mijψ j = (−1) 2 det M, i, j (1.32)

6 General properties of QCD ¯ where Mij are matrix elements (which can be complex numbers) of the matrix M, and ψi are the Grassmann variables that can be considered either as complex conjugate to ψi or as independent Grassman variables. In any case, they are supposed to anticommute with ψi . Recall that for ordinary complex numbers φi instead of (1.32) one has ⎛ ⎞ φ φ ··· φ φ∗ φ∗ ··· φ∗ ⎝− φ∗ φ ⎠ = ( π)N −1 . d 1d 2 d N d 1 d 2 d N exp i Mij j 2 det M (1.33) i, j

The relation (1.32) allows det Mˆ (A) in (1.24) as the functional integral over the Grassman variables φa(x) and φa†(x) [where φa†(x) can be the complex conjugate of φa(x) or can have nothing to do with φa(x)]: det Mˆ (A) = C DφDφ† exp i d4xd4 yφ†a(x) (M(A))ab (x, y)φb(y) , (1.34)

where C is an inessential numerical constant. Usually, the operator Gˆ in (1.19) is local, so that one can write M(A) ab(x, y)) = Rˆab(x)δ(x − y), (1.35) and det Mˆ (A) = C DφDφ† exp i d4xφ†a(x)Rˆab(x)φb(x) † 4 = C DφDφ exp i d xLghost , (1.36)

a where Lghost is the Lagrangian of the ﬁelds φ (x), which are called Faddeev–Popov ghosts. Therefore, one can introduce the QCD Lagrangian unitary in physical space

1 a a Lef f = LQCD + L fix + Lghost = ψ(x)(i ∇−m)ψ(x) − Gµν(x)Gµν(x) 4 1 2 − (GAˆ )a(x) + φ†a(x)Râb(x)φb(x), (1.37) 2ξ and consider it as a quantum Lagrangian, where the ghosts are fermions. We also can use this Lagrangian to describe several quark flavours; in this case ψ and ψ, besides being Dirac bispinors and colour triplets, have also the flavour indices, ∇ is multiplied by the unit matrix in flavour space and m is the mass matrix. The Lagrangian (1.37) also can be used to construct the generating functional ¯ † 4 a a Z[J,η,η¯]= DADψDψDφDφ exp i d x Lef f + Aµ(x)Jµ(x) +ψ(¯ x)η(x) +¯η(x)ψ(x) , (1.38)

a a ¯ where Jµ(x), η(x), and η(¯ x) are sources for the gluon fields Aµ(x) and quark fields ψ(x) and ψ(x), respectively. All sources and fields here are considered as classical, but whereas a a ¯ Jµ(x) and Aµ(x) are usual c–numbers, η(x), η(¯ x) and ψ(x), ψ(x) are anticommuting

1.3 The Gribov ambiguity 7

Grassman variables. This property has to be taken into account in deﬁning Green functions with participation of quarks and antiquarks. For example, the quark propagator is deﬁned as 2 2 (−i) δ Z[J,η,η¯] |T ψα(x)ψ (y) | =− , 0 β 0 [ , , ]δη¯ ( )δη ( ) (1.39) Z 0 0 0 α x β y J=η=¯η=0

where ψα(x) and ψβ (y) are Heisenberg operators, and α and β symbolize both spinor and colour indices. Green functions for any number of particles can be deﬁned in a similar way as corresponding functional derivatives of the generating functional (1.38).

1.3 The Gribov ambiguity When deriving (1.37), it was supposed that at any fixed A and B the equation (GAˆ (θ))a(x) = Ba(x) has a unique solution with respect to θ, i.e. the absence of any solution or the existence of several solutions were excluded. There are no examples for the first possibility (i.e. the absence of any solution), however, the existence of many solutions (i.e. many gauge-equivalent fields obeying the same gauge condition) is an ordinary case, as was pointed out by Gribov [6],[7]. The simplest example presented by Gribov [6] corresponds to the Coulomb gauge defined by the equation

∂i Ai (x) = 0, (1.40) for the case of the SU(2) colour group, where ta = τ a/2 and τ a are the Pauli matrices. The existence of gauge-equivalent ﬁelds means that the equality ∂ (θ)( ) = , i Ai x 0 (1.41) where

(θ) − i − A (x) = UA (x)U 1 + (∂ U)U 1, U = exp(iθa(x)ta), ∂ A (x) = 0, (1.42) i i g i i i is valid for nontrivial θa(x) and the matrix U tending to the identity matrix at |r|→∞. Taking account of the equalities −1 −1 −1 −1 −1 U ∂i U =−(∂i U )U,∂i (∂i U)U =−U ∂i (∂i U )U U , (1.43)

it follows from (1.41), (1.42), that −1 ∇i (A), (∂i U )U = 0, (1.44)

where ∇i (A) = ∂i + igAi . Let us consider ﬁrst the case A = 0. It can be shown, using (1.42), that in this case the solutions of (1.44) are transverse ﬁelds, which are gauge equivalent to A = 0. It is easy to demonstrate the existence of such solutions for the gauge group SU(2). One can look for spherically symmetric solutions of the form θ( ) τ U = ei r n = cos θ + inτ sin θ, (1.45)

8 General properties of QCD

where n = r/r. Equation (1.44) follows from the extremum condition for the functional 3 −1 −1 W = d x Tr (∂i U)(∂i U ) − 2igAi (∂i U )U . (1.46)

For A = 0 and U given by (1.45), W is equal to dθ 2 2sin2 θ W = 2 d3x + , (1.47) dr r 2

so that Equation (1.44) gives

d2 r (θr) − sin(2θ) = 0. (1.48) dr2 With t = ln r, the equation turns into the equation of motion for a particle in the potential − sin2 θ with friction: d2θ dθ + − sin(2θ) = 0. (1.49) dt2 dt (θ) The ﬁelds Ai (1.42) are expressed through the solutions of (1.49) as: 2 (θ) 2 dθ sin(2θ) sin θ A =− nani + (δai − nani ) + aibnb . (1.50) ai g dr 2r r

Fields nonsingular at r = 0 (t =−∞) correspond to motion of the particle which is at unstable equilibrium θ = 0att →−∞[points θ = nπ are equivalent because the ﬁelds (1.50) do not change at θ → θ ± π]

t θr→0 = ce = cr (1.51)

where the constant c deﬁnes the motion. Because of friction −π<θ<πat any t and √ π c1 7 θ|r→∞ =± + √ cos ln r + c2 , (1.52) 2 r 2

where the constants c1,2 are determined by c in (1.51). The corresponding fields given by (1.50) (θ) 2 1 A | →∞ =− n (1.53) ai r g aib b r decrease as 1/r. Thus, there is a family of pure gauge transverse fields, i.e. nonzero fields A(θ), which up to gauge transformations are equivalent to zero fields and satisfy the gauge condition (1.41). It is a particular case of a general statement that the gauge condition (1.41) does not fix uniquely the field from a family of gauge-equivalent fields. a The existence of a solution of (1.44) for an arbitrary field Ai means the existence of local extremes of the action (1.46), and can be easily understood for large fields. First, it is

1.3 The Gribov ambiguity 9

clear that W = 0atU = 1 (θa = 0), and it is the absolute minimum of the contribution of the first term in (1.46). At small θa, the functional W (1.46) becomes 1 W = d3x θa(x)Râb(x)θ b(x) , (1.54) 0 2 âb( ) =−∂ ab( ) where R x i Di A is the Faddeev–Popov operator (1.35) in the case of Coulomb ab( ) = ∂ δab + cab c ˆ gauge, Di A i gf Ai . The operator R is analogous to the nonrelativistic a Hamiltonian for a particle with spin in a velocity-dependent field. Evidently, if the field Ai is small (more precisely, the product of a typical magnitude of the field on a typical length of the space occupied by the field is small), then Rˆ has only positive eigenvalues. But if the field is sufficiently large and has an appropriate configuration (with a deep potential well), the appearance of negative eigenvalues is quite natural. Negative eigenvalues of Rˆ mean negative values of the functional W0 (1.54). Since min W ≤ W0 and W = 0at U = 1, for such fields there is a nontrivial extremum of W (1.46), i.e. a solution of (1.41). Thus, for such fields the Coulomb gauge is not uniquely defined. Note that the extremum is reached for U → 1atr →∞, since the contribution of the first term in (1.46) is non-negative. The existence of solutions of (1.42) can be demonstrated explicitly in the case when

1 x j Aa(x) = f (r). (1.55) i g iaj r 2 ∂ a( ) = It is easy to see that i Ai x 0. For such fields with the parameterization (1.45), the functional (1.46) takes the form of dθ 2 2sin2 θ W = 2 d3x + (1 − f (r)) , (1.56) dr r 2 and instead of (1.49) one has d2θ dθ + − sin(2θ)(1 − f (r)) = 0. (1.57) dt2 dt Besides solutions of the same type as for f = 0, for sufficiently large f there can be other solutions corresponding to the particle in positions of unstable equilibrium θ = nπ at t →∞as well at t →−∞(with or without change of n). The corresponding fields (θ)( ) ( ) Ai x (1.42) rapidly decrease with r if f r decreases. The problem of the existence of many gauge equivalent fields (Gribov copies) satisfying the same gauge condition is inherent not only in the Coulomb gauge, but for covariant gauge conditions as well. Thus, for the gauge ∂ Aµ/∂xµ = 0, the existence of Gribov copies follows from the same line of argument as for (1.41), if the theory is formulated in the four-dimensional Euclidean space. On the contrary, axial and planar gauges were found to be free of Gribov ambiguity [8]. The existence of Gribov copies means that Eqs. (1.24) and (1.38) have to be improved. Gribov suggested [7] that the problem of copies can be solved if the integration in the functional space is restricted by the potentials for which the Faddeev–Popov determinant is positive (in Euclidean space). This restriction does not concern small fields,

10 General properties of QCD

and therefore is not significant for perturbation theory. In particular, it is not significant for hard processes, where perturbation theory is applied. But the region of sufficiently large fields, where the operator Rˆ could have negative eigenvalues, is excluded from the integral (1.38). The existence of Gribov copies is evidently important for lattice QCD. For investigation of their influence see, for instance, [9]–[11] and references therein.

1.4 Feynman rules Feynman diagrams and Feynman rules are deﬁned by the effective QCD Lagrangian (1.37). The rules for external lines are the same as in QED apart from evident colour indices. The rules for internal lines can be divided in two parts: independent and dependent of a choice of gauge. The quark propagator belongs to the ﬁrst part. Using α, β for the Lorentz indices and i, j for colour indices, we have for the propagator

β, j p α, i (p + m)αβ iδij . (1.58) p2 − m2 + i0

We will omit quark indices in the following. Then we have

a,µ

a − igt γµ (1.59)

b,β

q abc gf (q − p)γ δαβ + (r − q)αδβγ + (p − r)β δαγ =− c ( − ) (−δ ) + a ( − ) (−δ ) ig Tab q p γ αβ Tbc r q α βγ p r + b ( − ) (−δ ) Tca p r β αγ (1.60)

a,α c,γ