Conformal symmetry of QCD in $d$-dimensions

QCD in $d=4-2\epsilon$ space-time dimensions possesses a nontrivial critical point. Scale invariance usually implies conformal symmetry so that there are good reasons to expect that QCD at the critical point restricted to the gauge invariant subsector provides one with an example of a conformal field theory. The aim of this letter is to present a technical proof of this statement which is important both as a matter of principle and for applications.


1.
Coupling constants in quantum field theory (QFT) models usually depend on the renormalization scale. This dependence is described by beta-functions which enter renormalization group equations (RGEs) for correlators of the fundamental fields and/or local composite operators. If the beta-functions vanish, the theory enjoys scale invariance and the RGEs reduce to equations describing the behavior of the correlation functions under scale transformations. In four-dimensional models, the only zero of the betafunctions accessible in perturbation theory corresponds to a trivial situation when all couplings vanish, i.e. the free theory. In noninteger d = 4 − 2 dimensions, the situation is different. In this case it is common that the beta-functions vanish for some special values of the couplings g = O( ) (critical couplings). If is considered a small parameter, the critical couplings can be calculated in perturbation theory. QFT models at the critical point thus provide one with examples of scale-invariant theories.
As was first suggested by Polyakov [1], scale invariance of a quantum field theory usually implies conformal invariance. Recently, considerable effort was invested to make this statement more precise [2][3][4][5][6][7][8][9]. In non-gauge theories a clear picture is emerging, but the case of gauge theories is less studied and still subject to considerable debate.
In non-abelian gauge theories and in particular QCD there are additional complications due to the gauge-fixing and ghost terms * Corresponding author. in the Lagrangian that are not invariant under conformal transformations even in d = 4 dimensions. As a consequence, there is no hope that correlators of fundamental fields may transform in a proper way under scale and conformal transformations -good symmetry properties can only be expected for the correlators of gauge-invariant operators. The subtlety is that gauge-invariant operators mix under renormalization with gauge-variant operators of a special type (BRST variations) and Equation of Motion operators (EOMs). These counterterms -BRST and EOM operators -are believed to be artifacts of the Faddeev-Popov approach to quantization of gauge theories and all troubles caused by them are likely to be of technical character. In this letter we clarify the structure of such "unwanted" contributions in conformal Ward identities, which is important for practical applications. This analysis can be viewed as an extension of the work by Joglekar and Lee [10][11][12] on the structure of gauge-variant operators in the RGE equations.
It has been observed, see e.g. [13][14][15][16][17][18], that apparently unrelated perturbative QCD observables differ only by terms involving the beta-function, and one possibility to understand this connection [17][18][19] is to start from the theory in d = 4 − 2 dimensions at the critical point where they are related by a conformal transformation. Similar ideas have been used to derive the RGEs for leading-twist QCD operators in general off-forward kinematics [20][21][22][23]. Our intention is to put these methods on a more rigorous footing.
To be specific, we will consider QCD near four dimensions, d = 4 − 2 , in perturbation theory assuming the minimal subtraction renormalization scheme. With the above mentioned applications in mind, we are interested in the behavior of correlation functions of local operators in this theory under conformal transformations. This question can be answered, at least within perturbation theory, by the study of scale and conformal Ward identities. In this way conformal invariance of the correlators of fundamental fields in non-gauge theories can be proven along the lines of Refs. [24][25][26], see also [4] for recent developments. A detailed description of this technique and its extension to the case of local composite operators can be found in the book [27].
On a more technical level, let O q , q = 1, 2, . . . , n be a (finite) set of local composite operators with the same quantum numbers so that they mix under renormalization. In Ref. [27] it was shown that in scalar theory the scale and conformal Ward identities for these operators at the critical point imply that the symmetry transformations take the following form: where the sum over q is implied. The generators of scale and conformal transformations D and K μ are defined as where is the canonical scaling dimension of the operators O q , μν is the spin generator and O μ q are certain local operators with canonical dimension − 1. These expressions can be simplified by going over to a basis of operators that diagonalize the anoma- Here c αq is a left eigenvector of γ qq , q c αq γ qq = γ α c αq , and α = + γ α is the scaling dimension of the operator O α . In this basis the transformations in Eqs. (1) simplify to The analysis of scale and conformal Ward identities given in Ref. [27] can be extended to gauge theories. We will show that Eqs. At first sight the appearance of a gauge non-invariant operator on the r.h.s. of Eqs. (3) can be ruled out by observing that its anomalous dimension would depend on the gauge-fixing parameter. This is not always the case, however. To give an example, the gauge-invariant operator O = F F in four dimensions can be written as a divergence of the topological current K μ , Evidently, F F and K μ have the same anomalous dimensions and the current K μ can be a natural candidate for the role of the non-

2.
We start with collecting the necessary definitions. The QCD ac- in the fundamental (adjoint) representation for quarks (ghosts). The field strength tensor is defined as usual, F a is the scale parameter. The theory is assumed to be multiplicatively renormalized and the renormalized action takes the form where z jk are polynomials in ξ . Formally the theory has two charges: a and ξ . The corresponding beta-functions are defined as where the first two coefficients gauge group with N f quark flavors. In this notation −2aγ g is the usual QCD β-function in physical four dimensions. The anomalous dimensions of the fields = {q, q, A, c, c} are defined as The QCD Lagrangian (4) is invariant under BRST transformations [28,29] The BRST transformation rules for the renormalized fields are obtained by replacement → 0 , e → e 0 , δλ → δλ 0 in the above equations and writing the bare fields and couplings in terms of the renormalized ones: 0 = Z , e 0 = Z e e. The renormalized BRST transformation parameter δλ is defined as δλ 0 = Z c Z A δλ so that the last equation in Eqs. (9) has the same form for bare and renormalized quantities. The BRST operator s defined by δ = s δλ is nilpotent modulo EOM terms. Namely, s 2 = 0 for all fields except for the anti-ghost in which case one finds Thus the second BRST variation of an arbitrary local functional . (11) BRST symmetry is the key ingredient in the analysis of the RGEs for gauge invariant operators [10][11][12]. The result, see Ref. [30] for a review, is that gauge invariant operators, O, mix under renormalization with BRST operators, i.e. operators that can be written as a BRST variation of another operator, B = sB , and EOM operators, so that renormalized gauge-invariant operators take the following generic form 1 operator. In principle it should be possible to constrain the operator structure of potential BRST and EOM counterterms for a given O. However, no such relation is known.
The significance of this result is that the contributions of BRST and EOM operators to physical observables have to vanish so that such terms can be dropped, at least in principle. In practice this requires some caution. Calculations are usually done in momentum space. Within perturbation theory the radiative corrections to the matrix elements of composite operators develop ultra-violet divergences as well as infrared ones, which are regularized in d dimensions. In addition, the vanishing of physical matrix elements with BRST or EOM operators requires the on-shell limit with respect to their external momentum q to be taken and, generally, the limits q 2 → 0 and d → 4 do not commute. Therefore, theorems on the renormalization of gauge invariant operators [10][11][12] directly apply to matrix elements with the operators inserted at nonzero momentum. In practice, this requires the computation of three-point functions with off-shell legs, which poses certain difficulties at higher loops. Calculations of matrix elements based on two-point functions are technically easier, but are typically realized with operators inserted at zero momentum. In this case, physical matrix elements of gauge variant operators do not vanish, the mixing matrix of operators is not triangular and matrix elements with insertions of BRST or EOM operators need to be accounted for as well, see refs. [31][32][33][34].
Considering operators with fixed position essentially corresponds to nonzero momentum flow. In this case it is indeed easy to see that a correlation function of renormalized gauge-invariant operators localized at different space-time points x = {x 1 . . . x N } is equal to the correlation function of the gauge-invariant parts of the same operators 1 We use the standard notation [O] for the operator O renormalized in the MS scheme [30].
The equality holds because the additional terms due to BRST and EOM operators are local, e.g., where C k ( x) are some functions (not necessary finite at → 0), and similar for EOM terms, so they vanish if all x k are different but can contribute to integrals over the operator positions. Our goal in this paper is to show that at the critical point, β a (a * ) = 0, the correlators (14) behave in a proper way under scale and conformal transformations. The expression on the r.h.s. of (14) is the natural starting point for this undertaking.
3. Next, we introduce the relevant Ward identities. The correlation function in Eq. (14) can be written in the path-integral representation as follows (15) where N is the normalization factor. Making the change of variables → = + δ ω in the integral (15), where δ ω correspond to the dilatation and special conformal transformation, ω = D, K μ , see Appendix A, and taking into account that the integration measure stays invariant, one obtains where α D = (x∂)α and α K μ = 2x μ (x∂) − x 2 ∂ μ α.
The scale and conformal variations of the operators that appear on the l.h.s. of Eq. (16) are defined as

y δ ω (y) δO(x)/δ (y) .
Assuming that O j have canonical dimensions j they are given by the following expressions: where D j , K μ j are given in Eq. (2) and O μ k are certain gauge invariant operators with canonical dimension j − 1. Such inhomogeneous terms typically arise from the commutators of δ w with derivatives in the operator O j , if they are present. Note that the coefficients p jk ( ) can be and, as a rule, are singular in the → 0 limit. It is easy to check that the property (17) ensures that there are no gauge-dependent addenda to these expressions.
The variation of the QCD action δ ω S R on the r.h.s. of the Ward identity (16), see Appendix, can be written as  (20) where χ D = 1 and χ K μ = 2x μ for dilatation and conformal transformations, respectively.
To proceed further we re-expand 2 L R (x) in terms of renormalized (finite) operators. The corresponding expression takes the form [21,22,35] 2 2 can be rewritten as a combination of BRST and EOM operators, It can be shown that the coefficients z b (g, ξ) and z c (g, ξ) can be calculated explicitly in Landau gauge, ξ = 0, The remaining correlation function contains renormalized (finite) local operators at separated space points and is finite. The integral is also finite. This contribution vanishes, therefore, at the critical point since it comes with the factor β(a * ) = 0. Thus only the integral over the union of small balls around the operator insertions remains, Our next aim is to bring this expression to a form suitable for further analysis.

4.
Since the balls B n do not overlap, it is sufficient to consider one term in the sum. The operator product 2 L R (x) O n (x n ) for x → x n is not necessarily finite and the argument which we used to claim that the integral over the complement R can be dropped does not work. To simplify the notation we suppress the subscript n and use x ≡ x n . The first step is to show that the product of the renormalized Lagrangian and a gauge-invariant renormalized operator O(x ) can be written in the following form x ). (26) The first term on the r.h.s. of this expression is the fully renormalized product of two operators. LT stands for local terms that have a finite expansion of the form The next term is a BRST operator. Finally, the last term is an EOM operator which has the following property: its correlation function with a product of fundamental fields X (Y ) = p p (y p ), Y = {y 1 , . . . , y p } contains only delta functions of the type δ(x − y p ) or δ(x − y p ) but not δ(x − x ). In other words if x, x = y p for any p then In order to prove Eq. (26)  • It is straightforward to show that the EOM terms give rise to x ). (29) To this end consider the correlation function of E(x) O(x ) with a set of fundamental fields X (Y ) which we can write as The first term on the r.h.s. is a local operator while the second one is a EOM term, E(x, x ), that is easy to see integrating by parts in the path integral.
we observe that the EOM term gives rise to the structure (29) whereas the product L Y M L B O (x ) contributes to the R(x, x ) term. Finally, the product of two BRST operators B L (x) = s(B L (x)) and B O (x ) = s(B O (x )) can be rewritten as The first term on the r.h.s. contributes to R(x, x ) and the second term is the sum of local (LT) and E(x, x ) (EOM) contributions. To see this, write s B O (x ) = s 2 B O (x ) and use Eq. (11) to obtain x ). (34) Obviously, the first term on the r.h.s. of this identity is a local (LT) contribution. Collecting all of the above expressions we obtain Eq. (26).
Once Eq. (26)  The proof follows closely the analysis of the RGEs for gaugeinvariant operators in Ref. [30]. To this end we consider the BRST variation of Eq. (26). Since the l.h.s. vanishes, one obtains Using where, as above, X = p p (y p ) and x, x = y p , it is easy to see that the last two terms in Eq. (35) where all terms on the r.h.s. except for the first one are singular in 1/ (do not contain finite contributions). Taking a BRST variation of the both sides we conclude that up to EOM terms (LT(x, x )). The operator on the l.h.s. of this relation is a finite operator, while the one on the r.h.s. is singular. Therefore they both are equal to zero, up to EOM terms.
Going back to Eq. (35) we conclude that s(LT(x, x )) = 0 modulo EOM operators. As shown by Joglekar and Lee [10], see also [36] for a review, vanishing of the BRST variation implies that LT(x, x ) and therefore the operators F , F μ in Eq. (27) can be written as a sum of gauge invariant, BRST and EOM operators. The last ones can safely be neglected since they do not contribute to the correlation function in question.

5.
The subsequent derivation of the scale and conformal properties of correlation functions of gauge-invariant operators follows the lines of Ref. [27]. Starting from the dilatation Ward identity in Eq. (16) and taking into account Eqs. (18), (25), (26) one obtains Taking into account that the operators in questions satisfy the RGEs and have definite canonical dimension this identity implies that 2 Since this equation must hold for arbitrary operator insertions The same relation can alternatively be achieved by the analysis of the dilatation Ward identity for the correlation function of local operators with fundamental fields in Landau gauge. In this gauge β ξ = 0 holds identically so that the both beta-functions vanish at the critical point and scale invariance holds for any Green's function. Using Eq. (42) we can rewrite the conformal Ward identity as follows: is a gauge-invariant operator and the operator equality holds up to terms that vanish for all correlation functions with any number of gauge-invariant operators. Provided that the anomalous dimension matrix can be diagonalized 3 one can go over to the basis of operators with definite scaling dimensions and rewrite these equations in the form (3).

6.
To summarize, we have shown by the BRST analysis of the corresponding Ward identities that correlation functions of gaugeinvariant operators in QCD in d = 4 − 2 dimensions at the critical point transform properly under conformal transformations, as expected in a conformal invariant theory. This result gives further support to the methods based on using conformal invariance in higher-order perturbative QCD calculations [13][14][15][16][17][18][20][21][22][23] and can be also interesting in a broader context.  (A.4) and the covariant derivative of the ghost field D ν c transform as a vector field, Here J ρ (x) =q(x)γ ρ q(x) is the flavor-singlet vector current and is a BRST operator, B μ = s(c A μ ).