$n$-point correlators of twist-$2$ operators in $SU(N)$ Yang-Mills theory to the lowest perturbative order

We compute, to the lowest perturbative order in $SU(N)$ Yang-Mills theory, $n$-point correlators in the coordinate and momentum representation of the gauge-invariant twist-$2$ operators with maximal spin along the $p_+$ direction, both in Minkowskian and -- by analytic continuation -- Euclidean space-time. We also construct the corresponding generating functionals. Remarkably, they have the structure of the logarithm of a functional determinant of the identity plus a term involving the effective propagators that act on the appropriate source fields.

conf (x 1 , . . . , x n ), in the coordinate representation of the gauge-invariant twist-2 operators with maximal spin along the p + direction, both in Minkowskian and -by analytic continuation -Euclidean space-time.
In fact, our computation matches and extends the previous lowest-order perturbative computation of 2-and 3-point gluonic correlators of twist-2 operators in N = 4 SUSY YM theory [1], by including the unbalanced 1 operators with collinear twist 2 in pure YM theory and, most importantly, by calculating all the n-point correlators in the balanced and unbalanced sectors separately, and the 3-point correlators in the mixed sector as well. Our physics motivation is threefold: Firstly, our lowest-order computation has an intrinsic interest in YM theory, andaccording to [1] -in theories that extends it, such as its supersymmetric versions and QCD.
Secondly, our computation is preliminary to work out the ultraviolet (UV) asymptotics [4,5] -based on the renormalization-group (RG) improvement of perturbation theory -of the above Euclidean n-point correlators.
Thirdly, our computation is an essential ingredient to test the prediction in section 3 of [6] that, by fundamental principles of the large-N 't Hooft expansion, the generating functional of the nonperturbative leading nonplanar contributions to the aforementioned Euclidean correlators must have the structure of the log of a functional determinant [6] that sums the glueball one-loop diagrams.
Indeed, according to the philosophy of the asymptotically free bootstrap outlined in [6], the RG-improved correlators mentioned above must be asymptotic in the UV [6] to the corresponding nonperturbative correlators involving glueballs. Therefore, to the leading nonplanar order, the generating functional of the former must share with the one of the latter the very same structure of the log of a functional determinant.
Hence, our computation is the first step in both the directions mentioned above. 1 In our terminology 'unbalanced' and 'balanced' refers to either the different or the equal number of dotted and undotted indices that the aforementioned operators possess in the spinorial representation respectively. Unbalanced operators are referred to as 'asymmetric' in [2] and 'anisotropic' in [3].

Balanced and unbalanced twist-2 conformal operators
We describe our calculation and the operators that enter it. We compute, to the lowest perturbative order in SU (N ) YM theory, n-point connected correlators in Minkowskian space-time of the gauge-invariant twist-2 operators with maximal spin along the p + direction: conf (x 1 , . . . , x n ) (2.1) To the lowest order, and to the next one 2 , YM theory is conformal invariant [7], since the beta function only affects the solution of the Callan-Symanzik equation starting from the order of g 4 . Therefore, following [7,8], we employ operators that have nice transformation properties with respect to the collinear conformal subgroup involving the coordinate x + . Primary conformal operators O j (x), with collinear conformal spin j = s + τ 2 , where τ is the collinear twist and s the collinear spin, i.e., the spin projected along the p + direction, transform under the action of the generators [7] of the collinear conformal algebra SL(2, R): [L − , L + ] = −2L 0 according [7] to: where in eq. (2. 3) x = (x + , x − , x 1 , x 2 ) is restricted [7] to the line x − = x 1 = x 2 = 0. Their conformal descendants, ∂ i + O j (x), are obtained by taking derivatives with respect to x + , and have the same τ .
Their collinear twist τ does not necessarily coincide [7] with the twist T -defined by d = T + S, where S is the spin -that refers to the conformal group instead of the collinear subgroup.
An infinite family of quasi-partonic operators is constructed as follows. A composite gauge-covariant primary conformal operator, built by two elementary 3 gauge-covariant primary conformal operators Φ j 1 , Φ j 2 , with collinear conformal spins j 1 , j 2 , has the form [7]: where P (2j 1 −1,2j 2 −1) l are Jacobi polynomials (appendix F), D + is the covariant derivative along the p + direction (appendix A), and the arrows denote the action of the derivative on the right or the left. The corresponding gauge-invariant object is obtained by taking the color trace.
The collinear conformal spin, j, of the operator, O j 1 j 2 l (x), is j = j 1 + j 2 + l, where l is the power of the derivative in eq. (2.4). By working out the definition in eq (2.4), we get: thus realizing the conformal operator O j 1 j 2 l (x) as a sum of l + 1 operators, O j 1 j 2 lk (x), that are not necessarily conformal.
Hence, the composite operators depend on a choice of the elementary conformal operators Φ j 1 , Φ j 2 . We define the standard conformal basis for primary operators with collinear twist 2, where the elementary operators are f 11 , f11 (section 4.1) with conformal spin j = 3 2 . In the standard basis the gluonic operators are classified as in [2]: with C α l Gegenbauer polynomials (appendix F), which are a special case of Jacobi polynomials.
O s andÕ s are Hermitian balanced operators with τ = T = 2. They have an equal number of undotted and dotted spinor indices (appendices C and D): S s = S1111 ...11 (2.8) Besides, we also define the extended conformal basis for primary operators with collinear twist 2, where the elementary operators are D −1 + f 11 , D −1 + f11, with conformal spin j = 1 2 , which are nonlocal in general, but local (appendix E) in the light-cone gauge A + = 0. Clearly, gauge invariance ensures that all their correlators are local, as we verify explicitly.
The extended basis is natural in SUSY calculations [11], and includes (nonlocal) operators with τ = 2 and s = 0, 1. We have chosen it in YM theory because of the simplicity of the results for the correlators. In the extended basis (section 4.2) the gluonic operators are: 2.2 Minkowskian n-point correlators

Standard basis
We have normalized our operators in such a way that the 2-point correlators in the standard basis are equal for even s: and for odd s: with: where we omit the i prescription in the propagators in the coordinate representation, in such a way that (appendix A): 1 |x| 2 (2.13) should be read (appendix B): 14) The very same correlators are evaluated by a trick [1] (appendix H): Therefore, we have discovered the following -seemingly nontrivial -identity (section 5): We have not found a direct proof of the above identity, but we have verified it numerically.

Extended basis
We normalize our operators in such a way that the 2-point correlators in the extended basis are equal for even s: and for odd s: with: The very same correlators are evaluated by a trick [1] (appendix H): Therefore, we have discovered the following -seemingly nontrivial -identity (section 5): We have not found a direct proof of the above identity, but we have verified it numerically. Moreover, the only nonvanishing 3-point correlators are: and: with: A s 1 s 2 s 3 (x, y, z) = − 1 (4π 2 ) 3 (1 + (−1) s 1 +s 2 +s 3 ) We also compute the n-point correlators. In the balanced sector, we get: The very same formula holds for an even number of operatorsÃ s , otherwise the correlators vanish. The nonvanishing correlators in the balanced sector are: In the unbalanced sector, we get: (2.33)

Standard basis
After the Wick rotation (appendix A and section 8), we obtain in the standard basis: which is equivalent to: For the nonvanishing 3-point correlators, we get: Moreover, for the n-point correlators in the balanced sector, we obtain: The very same formula holds for an even number of operatorsÕ E s , otherwise the correlators vanish. The nonvanishing correlators in the balanced sector are: (2.38) In the unbalanced sector, we get:

Extended basis
We obtain in the extended basis: which is equivalent to: For the nonvanishing 3-point correlators, we get: Moreover, for the n-point correlators in the balanced sector, we obtain: The very same formula holds for an even number ofÃ E s operators, otherwise the correlators vanish. The nonvanishing correlators in the balanced sector are: In the unbalanced sector, we get:

Plan of the paper
In section 1 we outline our main results and physics motivations.
In section 2 we display our results for the correlators to the lowest perturbative order both in Minkowskian and Euclidean space-time.
In section 4 we review the classification and construction of the gluonic operators with collinear twist 2 in Minkowskian space-time both in the standard and extended basis.
In section 5 we compute the 2-point correlators in Minkowskian space-time both in the standard and extended basis.
In section 6 we compute the 3-point correlators in Minkowskian space-time both in the standard and extended basis.
In section 7 we compute the n-point correlators in Minkowskian space-time both in the standard and extended basis in the balanced and unbalanced sectors separately.
In section 8 we compute the n-point correlators in Euclidean space-time either by analytic continuation or by employing the corresponding Euclidean operators.
In the appendices we fix the notation and provide ancillary computations.
In appendix G we verify that our results for the 2-and 3-point correlators of balanced operators with even collinear spin in Minkowskian space-time coincide with [1].

Twist-2 gluonic operators in Minkowskian space-time
We review the construction of the standard and extended conformal bases for gaugeinvariant gluonic operators with τ = 2. We also work out the dictionary between the spinorial, vectorial and complex bases (appendices D and E).

Standard basis
To construct gauge-invariant gluonic operators with τ = 2 that are primary (section 2) for the collinear conformal group, we should find -according to eq. (2.4) -suitable gaugecovariant elementary conformal operators.
The local gauge-covariant operator with lowest canonical dimension, d = 2, in YM theory is the field-strength tensor, µ T a is a traceless Hermitian matrix, with T a the generators of the SU (N ) Lie algebra: normalized in the standard way: It is convenient to write F µν in the spinorial representation [12] (appendix D): It turns out [12] that: decomposes into the sum of two chiral representations, f ab ∈ (1, 0) and fȧ˙b ∈ (0, 1), of spin S = 1 (appendix D). In Minkowskian space-time fȧ˙b =f ab . f 11 and f11 have maximal collinear spin (appendix D), s = 1, along the p + direction. Therefore, they have τ = d−s = 1 and j = s+ τ 2 = 3 2 . Hence, they are well suited (section 2) to build 2-gluon twist-2 primary conformal operators. Taking the tensor product of the above representations, we get: where + and − label the parity, and the dots denote terms that do not contribute to the components with maximal collinear spin. Hence, there are four operators with maximal s that can be constructed by means of the corresponding bilinear operators: Following [7,8], we build (section 2) conformal operators with τ = 2 and higher collinear spin inserting in eq. (4.6) the Gegenbauer polynomials (appendix F), C α l (v), in the derivatives and afterwards taking the local limit x 1 = x 2 . l is the order of the polynomial, and its relation to the collinear conformal spin is j = l + j 1 + j 2 , where j 1 and j 2 are the collinear conformal spins of the elementary operators, and α = 2j 1 − 1 2 . The Gegenbauer polynomials are either symmetric or antisymmetric for the substitution v → −v for l even or odd respectively (appendix F). The corresponding primary conformal operators match precisely the ones in [2,10] up to perhaps the overall normalization: with j = s + 1, l = s − 2 and α = 5 2 . For brevity, we define as in [1]: For l = s − 2 and α = 5 2 , we get in the light-cone gauge by means eq. (F.9):

Extended basis
There exists another choice of the basis for primary conformal operators with τ = 2 involving the elementary operators D −1 + f 11 and D −1 + f11, which are nonlocal in general, but local in the light-cone gauge. Yet, gauge invariance ensures that the corresponding gauge-invariant correlators are local. Indeed, in the light-cone gauge (appendix E): where A has d = 1, s = 0, j = 1 2 and τ = 1. The corresponding operators with τ = 2 read: with j = s + 1, l = s and α = 1 2 . This basis naturally arises in SUSY calculations [11], and also includes (nonlocal) operators with s = 0, 1. We obtain in the light-cone gauge by means of eq. (F.10):

2-point correlators of twist-2 gluonic operators
We compute to the lowest perturbative order the 2-point correlators of the operators in both bases.

Standard basis
In the light-cone gauge, the 2-point correlators of the balanced operators with even s are given by: There is only one Wick contraction: By means of eq. (B.4), we get: Now we substitute eq. (4.9) into the above equation: We compute the derivatives: by induction on the index i: We obtain: that becomes: and simplifies as follows: since the correlator is zero for s 1 = s 2 (appendix H). By setting s = s 1 = s 2 , we get: Moreover, by the substitution in eq. (5.9): we obtain: that becomes: Besides, according to the trick in [1] (appendix H), we get: Hence, comparing eqs. (5.13) and (5.14), we can virtually perform the sums in eq. (5.13) to obtain the identity: Similarly, for the balanced operators with odd s, we get as well: where the definition of C s (x, y) in eq. (5.10) has been extended to odd s. Correspondingly, eq. (5.15) extends to odd s 1 , s 2 as well. Now we compute the only correlators of two unbalanced operators that are nonzero: There are two Wick contractions but an extra factor of 1 2 in the normalization of the operators, in such a way that the result is the same as for the correlators of balanced operators with even s: All the remaining 2-point correlators vanish.

Extended basis
Similarly, employing eq. (4.12), we obtain the correlators in the extended basis: We extend the above definition of A s (x, y) to odd s. We get for even s: and for odd s: We obtain (appendix H): Similarly, it follows the identity: All the remaining 2-point correlators vanish.

3-point correlators of twist-2 gluonic operators
We compute to the lowest perturbative order the 3-point correlators of the operators in both bases.

Extended basis
The 3-point correlators in the extended basis are computed analogously.

n-point correlators of twist-2 gluonic operators
We compute to the lowest perturbative order n-point correlators of the operators in both bases.

Standard basis
Given the bilocal operators: in the light-cone gauge, the corresponding n-point correlator that is connected in the local limit, The factor of 1 n arises because, if the first index -for example i 1 = 1 -is kept fixed, there are only (n − 1)! Wick contractions that contribute to the connected correlator. A nicer -but completely equivalent -formula is written in terms of permutations. If we denote by P n the set of permutations of 1 . . . n, it follows identically: Besides, eq. (B.4) reads: Hence, for the balanced operators with even s, we get in the light-cone gauge: where: It follows from eq. (5.5) that, correspondingly, the n-point correlator contains factors of the form: Therefore, we get: where we have set x A i = x B i = x i in order to implement the local limit of the bilocal operators. The color factor comes from the contraction of the n Kronecker delta: The overall factor of (−1) n occurs because of the factor of i −2 , which is a partial factor of i s−2 in eq. (7.6). After cancelling between themselves the pairs of factors of the kind (−1) sa−ka in eq. (7.8), and moving out of the sum over the permutations the product of the binomial coefficients, since it is independent of the permutations, we obtain: Actually, if n is even, eq. (7.10) also holds for the n-point correlator of the operatorsÕ s , with the only difference that their collinear spin is odd. Otherwise, if n is odd, the correlator vanishes. To verify it, it suffices to notice that, in the sum over the permutations, for every permutation the inverse permutation also occurs with the opposite sign. For example, for 3-point correlators we get pairs of terms of the kind: Employing the substitution k i = s i − 2 − k i , we obtain for the last term above: that cancels the first term in eq. (7.11).
The same reasoning applies to the n + 2m + 1-point correlators of balanced operators: Otherwise, we get: For the correlators of the unbalanced operators, we obtain: The factor of 1 2 2n arises from the normalization of the color trace in eq. (7.1), while the factor of 1 2 n comes from the normalization of the operators. We get the very same correlator by exchanging A and B in all the couples (x A i , x B i ) and (y A k , y B k ) simultaneously: Indeed, in eq. (7.17) we may conveniently relabel the coordinates, , and vice versa for each i and k, since they coincide in the local limit. Moreover, according to eq. (6.4), G 5 2 s−2 (∂ x A + , ∂ x B + ) in eqs. (7.15) and (7.16) is symmetric for the exchange of its arguments, because the collinear spin is even.
We evaluate the Wick contractions: We exploit the symmetry above: We only perform the Wick contractions involving the pairing of x A i with y A k and of x B i with y B k for any i, k, since all the remaining contractions provide the very same result due to the symmetry, and can be taken into account by a symmetry factor that we compute momentarily.
Besides, since we only are interested in the connected correlator, once x A i has been contracted with some y A k , x B i cannot be contracted with y B k , because the corresponding contribution to the correlator is not connected.
Hence, we construct the correlator as follows: We contract all the x A i with the y A k and all the x B i with the y B k with i = i if k = k and k = k if i = i , in such a way that we build a single connected loop. This is realized by summing over two sets of independent permutations arranged in such a way that no disconnected piece may be created: Firstly, we contract x A i 1 with y A k 1 , secondly, we contract y B k 1 with x B i 2 for i 1 = i 2 , then, we contract x A i 2 with y A k 2 for k 2 = k 1 , afterwards, we contract y B k 2 with x B i 3 for i 3 = i 2 = i 1 and so on, until we arrive at x B i 1 , which we contract with the last remaining y A kn with k n = k n−1 = . . . = k 1 , in order to close the loop. We end up with a chain that looks like: However, now we are creating a redundancy, since we also are summing on the possible n choices of the starting point of the loop. Therefore, we divide the sum by a factor of n: A nicer -but completely equivalent -formula is written in terms of permutations. It follows identically: 1 n σ∈Pn ρ∈Pn . . . f All the remaining contractions are obtained from this formula by exchanging the coordinates in each couple, (x A i , x B i ) and (y A k , y B k ), for each i and k. There are 2 2n of such exchanges. However, the actual degeneration factor is 2 2n−1 . Indeed, the extra factor of 1 2 comes from the fact that the simultaneous exchange of the coordinates in each couple, (x A i , x B i ) and (y A k , y B k ), yields a contraction that has already been counted due to the symmetry of eqs. (7.15) and (7.16) with respect of the simultaneous exchange of A with B in all the coordinate pairs. Hence, by combining the degeneration factor of 2 2n−1 with the factor of 1 2 n from the normalization of the operators, the overall factor of 2 n−1 survives. It follows: . . . s n k n s n k n + 2

Extended basis
Similarly, in the extended basis, we get: in the light-cone gauge, where: It follows from eqs. (B.5) and (5.5) that, correspondingly, the n-point correlator contains factors of the form: Therefore: where now the overall factor of (−1) n occurs because of the extra minus sign in eq. (7.24) with respect to eq. (7.7).
The very same formula holds for an even number of operatorsÃ s , otherwise the correlators vanish. We obtain as well: Similarly, for the unbalanced operators in the extended basis, we get: 8 n-point correlators and twist-2 gluonic operators in Euclidean spacetime

Analytic continuation of n-point correlators to Euclidean space-time
The Minkowskian n-point correlators can be analytically continued to Euclidean space-time by substituting (appendix A): x + → −ix z (8.1) and: We describe the effect of the analytic continuation about various numerical factors.
In the standard basis, for the (n + 2m)-point correlators of balanced operators, the extra factor of (−i) Similarly, for the 2n-point correlators of unbalanced operators, the factor of (−i) n l=1 s l +s l , which arises from the substitution of eq. (8.1) into the numerators of eq. (2.22), cancels out the factor of i n l=1 s l +s l -already present in eq. (2.22) -that comes from the definition of the Minkowskian operators. Thus, only the factor of (−1) n l=1 (s l +1+s l +1) = (−1) n l=1 s l +s l , which comes from the substitution of eq. (8.2) into the denominators of eq. (2.22), survives after the analytic continuation. Exactly the same factors survive in the analytic continuation of the corresponding correlators in the extended basis.

Twist-2 gluonic operators in Euclidean space-time
Alternatively, the correlators may be computed by first defining the Euclidean operators and afterwards evaluating them in Euclidean space-time. Of course, the two procedures must furnish identical results, as we verify momentarily.
By performing the Wick rotation (appendix A) to Euclidean space-time, the operators get rotated as follows. The derivative along the p + direction transforms as: Correspondingly, for the elementary operators in the light-cone gauge (appendix E), we get: and: place, which combines with the factor of (−1) n already present in the Minkowskian correlators, in such a way that only the factor of (−1) n l=1 s l survives, thus providing the same result as in the preceding discussion about the analytic continuation.

A Notation and Wick rotation
We mostly follow the notation in [7]. We define the Minkowskian metric as: The light-cone coordinates are: The corresponding Minkowskian (squared) distance is: where: We denote the derivative with respect to x + by: We define the light-like vectors n µ andn µ : n µ n µ =n µn µ = 0 n µn µ = 1 (A.6) that can be parametrized as (n µ ) = 1 √ 2 (1, 0, 0, 1) and (n µ ) = 1 √ 2 (1, 0, 0, −1). The Minkowskian metric can be decomposed into orthogonal and longitudinal parts with respect to the light-like vectors: The Euclidean metric is: The corresponding Euclidean (squared) distance is: with: .10) and: We define the Wick rotation by: .12) and: Eq. (A.12) ensures that exp(iS M ) → exp(−S E ), where S M and S E are the Minkowskian and Euclidean actions respectively, with S E positive definite. By defining p · x = p µ x µ and px = p µ x µ in Minkowskian and Euclidean space-time respectively, eq. (A.13) ensures that, by the Wick rotation, p · x → px , in such a way that the pairings p · x and px are actually independent of the Minkowskian and Euclidean metric respectively. Therefore, by a slight abuse of notation, we also write p · x in Euclidean space-time, instead of px . Besides, |x| 2 → −x 2 and |p| 2 → −p 2 .
As a consequence, the Wick rotation of the scalar propagator of mass m in Minkowskian space-time: reads in Euclidean space-time: as it should be. Moreover, the Wick rotation of the light-cone coordinates is: and: Correspondingly, the Wick rotation of the derivative with respect to x + is:

D Relation between the spinorial and vectorial bases in Minkowskian space-time
The components of F µν , with their s, j, τ assignments, are: The component with maximal s, F +µ , is well suited (section 2) to build twist-2 operators that are primary for the collinear conformal subgroup. In the light-cone gauge: with µ = 1, 2. Similarly, twist-2 primary conformal operators can also be built by means of with s = 1, j = 3 2 and τ = 1, where:F We define: with µ, ν = 1, 2 and 12 = 1. Hence:F and:F In the spinorial representation [12]: It turns out that F µν decomposes [12] into the (1, 0) ⊕ (0, 1) representation of the Lorentz group: where 4 : that coincides with eq. (D.10) by the antisymmetry of F µν and the definition of σ µν ab in eq. (C.13). Similarly, we obtain eq. (D.11). It follows that: where: From the definition of the matrices (σ µ aȧ ) (appendix C): it follows that σ µ 11 is nonvanishing only for µ = +, and σ µ 12 is nonvanishing only for µ = 1, 2. Hence, by employing the antisymmetry of F µν , we obtain: Therefore: We can now build the dictionary from the spinorial to the vectorial basis of the twist-2 operators: and: In principle, the unbalanced operators with τ = 2 in the vectorial basis should be constructed by means of the tensor: and its Hermitian conjugate, with µ, ν = 1, 2. However, a simple computation shows that all the components of the operators above are actually proportional to f 11 f 11 and its Hermitian conjugate respectively. Indeed: and: It follows that in the standard basis: and in the extended basis:

E Complex basis
The complex basis is defined by means of: In the light-cone gauge, it follows from eq. (D.18) that: Hence, the operators in the standard basis are: in the light-cone gauge, and analogously in the extended basis:

F Jacobi and Gegenbauer polynomials
We work out the formulas for the Jacobi and Gegenbauer polynomials that are employed in the present paper. For x real, the Jacobi polynomials, P (α,β) l (x), admit the representation [14]: with α, β real and l a natural number. Moreover, they satisfy the symmetry property: The Gegenbauer polynomials, C α l (x), are a special case of the Jacobi polynomials: Therefore, they satisfy the symmetry property: From now on, we set: in such a way that: Hence, eq. (F.1) becomes: Besides, setting l = J − α + 1 2 in eq. (F.3) and α = β = α − 1 2 in eq. (F.1), we obtain: Specializing the above equation to J = s and α = 5 2 , we get: Moreover, for J = s and α = 1 2 , we obtain: From now on, we restrict α, β to the natural numbers and, correspondingly, α to the positive half-integers and J to the natural numbers. By employing the identity: it follows from eq. (F.7) that: Corrispondingly, eq. (F.3) reads: which reduces to: Specializing the above equation to α = 5 2 , we obtain: to be compared with eq. (F.9).

G Matching 2-and 3-point Minkowskian correlators with [1]
We verify that our results for the 2-and 3-point Minkowskian correlators of the balanced operators with even collinear spin in the standard basis coincide with the ones in [1] up to the different normalization of the operators. in terms of gamma functions. Besides, we rewrite the above correlator in terms of the collinear conformal spin, j = s − 1, to match the notation of eq. (2.11) in [1]: This is the very same result in [1] up to the overall factor of σ j 1 σ j 2 , which is missing as -contrary to eq. (2.2) in [1] -we have defined the operators O s in eq. (4.7) without the factor of σ j in front.