Anomalous dimension of subleading-power N-jet operators

We begin a systematic investigation of the anomalous dimension of subleading power N-jet operators in view of resummation of logarithmically enhanced terms in partonic cross sections beyond leading power. We provide an explicit result at the one-loop order for fermion-number two N-jet operators at the second order in the power expansion parameter of soft-collinear effective theory.


Introduction
The scattering amplitude of N well-separated, energetic, massless particles is one of the key quantities in gauge theories. Understanding its structure is of fundamental importance, both for its own reason by revealing mathematical structure that is not at all evident from the underlying Lagrangian and its Feynman rules, and for the phenomenology of high-energy scattering in QCD.
Of particular interest are the soft and collinear divergences, which exhibit a high degree of universality. Some form of analytic calculation is usually required in order to efficiently cancel the divergences between virtual and real emission effects in infraredsafe scattering cross sections. The infrared divergences of the virtual N-parton scattering amplitude are governed by the soft-collinear anomalous dimension, which up to the twoloop order has the very simple structure in colour-operator notation [1] and for all out-going momenta p i with s ij = 2p i · p j + i0, i, j = 1 . . . N. The soft [2] and collinear [3] contributions to Γ are known up to the threeloop order. 1 The above assumes that all scalar products s ij are parametrically of the same order as some hard scale Q. If the physical observable is sensitive to a smaller scale M generated by soft or collinear radiation, the anomalous dimension is a central object in the systematic all-order resummation of large logarithms ln Q/M in the expansion in the coupling α s . When this is the case the above anomalous dimension refers to the infrared singularities at leading order in the expansion in powers of M/Q ("leading power"). Given the advances in the understanding of multi-loop corrections to the leading power anomalous dimension, it is also timely to ask about the next, subleading power term in the M/Q expansion. It has been known for a long-time that single soft emission from an N-jet amplitude is described by a universal expression, the LBK amplitude, also at next-toleading power [4,5]. This result extends the eikonal formula and has recently attracted new interest in connection with a possible relation to an asymptotic symmetry at null infinity [6]. However, little is known about the structure of divergences of loops and the anomalous dimension at the subleading powers. The exponentiation of purely soft, "next-to-eikonal" effects has been discussed in Refs. [7,8]. However, a major complication at next-to-leading power arises from the interplay of soft and collinear radiation as can be seen, for example, from the failure (or rather-depending on the point of view-generalization) of the LBK formula for jet processes beyond the tree approximation [9,10].
In this paper we begin with a systematic study of subleading power N-jet operators and their anomalous dimension with the ultimate goal of being able to sum logarithmically enhanced loop effects to all orders in perturbation theory. We base this study on soft-collinear effective theory (SCET) [11][12][13][14], which offers the advantage that the power counting required to identify all next-to-leading power terms is already built into the Lagrangian. While we will not solve the resummation problem here and do not even discuss logarithms for a physical process, our approach demonstrates a clear path how this could be done in principle and systematically. The structure of the anomalous dimension matrix of subleading-power N-jet operators will become apparent and we provide the first complete result for the class of fermion-number F = 2 operators to begin with. Previous work on anomalous dimensions at subleading power in SCET focused on specific cases, the heavy-to-light current [15,16] (related to J B1 Aχ in the operator basis defined below) in the position-space SCET formalism, and on power-suppressed tree-level currents relevant to e + e − → two jets in a different SCET framework [17,18].
Several other works have recently addressed next-to-leading power (NLP) effects from a more practical perspective. In Refs. [19][20][21] the threshold limit of the partonic Drell-Yan process has been investigated and all NLP terms of the next-to-next-to-leading order (NNLO) cross section have been successfully reproduced in a diagrammatic expansion analysis. Also a "radiative jet function" has been identified, related to collinear effects, which appear near threshold first at NLP. For colourless final states the interference of the NLP LBK amplitude with the tree process allows one to compute the NLP terms at NLO in the loop expansion [22]. Another recent development [23,24] concerns the analytic computation of the leading NLP logarithm at NNLO in the separation parameter of the N-jettiness subtraction method [25,26], making the cancellation of the dependence on the separation parameter in the full simulation of the process more efficient. All of these applications have in common that they refer at present to logarithms at fixed order in perturbation theory up to NNLO and to processes with only two collinear directions. The general approach outlined in the present paper, once developed, should allow the computation of further logarithms in these applications, and in particular their resummation to all orders. We finally take note that along a somewhat different direction a formula for fermion-mass suppressed double logarithms in the high-energy limit of certain fermion-scattering form factors has been derived [27,28].

Subleading N -jet operator basis
It was noted in Refs. [29,30] that the infrared anomalous dimension (1) must correspond to the ultraviolet divergences of soft and collinear loops in SCET, if SCET is to be the correct effective field theory for jet processes. This observation also applies to subleading powers. The following analysis is based on the position-space field representation of SCET [13,14]. The physical processes which are covered by this analysis are those for which the virtuality of collinear modes in any of the N jet directions is of the same order, and parametrically larger than the one of the soft mode. The power-counting parameter λ is set by the transverse momentum p ⊥i ∼ Qλ of collinear momenta with virtuality O(λ 2 ). 2 The components of soft momentum are all O(λ 2 ) and consequently soft virtuality scales as λ 4 . Below the term "NLP" refers to O(λ) and O(λ 2 ), since the first non-vanishing power correction to most physical processes of interest is O(λ 2 ).
Under these assumptions (often referred to as SCET I ) the SCET Lagrangian including all subleading power interactions to O(λ 2 ) was already given in Ref. [14]. For N widely separated collinear directions, the Lagrangian is the sum of N copies of collinear Lagrangians with N pairs of separate light-like reference vectors n i± , i = 1, . . . , N satisfying n i− · n j− = O(1). The collinear fields ψ i all interact with the same soft field ψ s but not among each other. The SCET Lagrangian is invariant under N separate collinear gauge transformations and a soft gauge transformation, see Ref. [14]. We therefore proceed to the construction of a complete basis of subleading N-jet operators in SCET. The general structure can be described by products of operators J i associated to collinear directions n i+ , each of which is itself composed of a product of n i gauge-invariant collinear "building blocks" ψ i k [31], and a soft operator J s . In general, each of the collinear building blocks is integrated over the corresponding collinear direction in position space, where C({t i k }) is a Wilson coefficient, and dt = ik dt i k . Apart from the displacement along each of the collinear directions, the operators are evaluated at position X = 0, corresponding to the location of the hard interaction. The guiding principle for constructing building blocks is the requirement of collinear and soft gauge covariance. Because each collinear sector transforms under its own collinear gauge transformation, each collinear building block must be a collinear gauge singlet. However, the soft field may interact with different collinear sectors so we only need to assume that collinear building blocks transform covariantly under the soft gauge transformation. Note that, in general, the collinear building blocks may also contain multipole expanded soft fields. For a collinear block the transformation properties under collinear and soft gauge transformation may be summarized as follows where x µ i− = (n i+ x) n µ i− /2 and U s refers to the (not necessarily irreducible) colour representation of J i . For the matrix adjoint representation we would have The elementary collinear-gauge-invariant collinear building blocks are given by for the collinear quark and gluon field in i-th direction, respectively. W i is the pathordered exponential of n i+ A i ("i-collinear Wilson line") and the covariant derivative includes only the collinear gluon field. Both, the quark and gluon building blocks scale as O(λ) [13]. Objects containing in i+ D i or in i+ ∂ are redundant. The first can be reduced to the second with the help of in The ordinary derivatives can be removed using in i+ ∂ψ i k (t i k n i+ ) = idψ i k (t i k n i+ )/dt i k followed by an integration by parts in the t i k -integral in Eq. (3).
At leading power, only a single building block contributes for each direction, i.e. n i = 1 for all i = 1, . . . , N, and the elementary building blocks are given by The superscript in J A0 i indicates the leading-power contribution, and the reason for this nomenclature will become clear in a moment. We are interested in N-jet operators that are suppressed by one or two powers of λ relative to the leading power. This suppression can arise in three ways: (i) via higher-derivative operators, i.e. acting with either i∂ µ on the elementary building blocks ψ i k . Here it is important to note that since the elementary building blocks transform under the soft gauge transformation with U s (x i− ), the covariant soft derivative is the ordinary derivative for the transverse direction and in i− D s for the n i− projection. In other words, the soft covariant derivative on collinear building blocks is iD µ s (x) ≡ i∂ µ + g s n i− A s (x i− ) n µ i+ 2 due to the multipole expansion of the soft fields, which guarantees a homogeneous scaling in λ; (ii) by adding more building blocks in a given direction, i.e. n i > 1, since and A µ ⊥i ∼ O(λ), (iii) via new elementary building blocks that appear at subleading power, including purely soft building blocks in J s .
In the following, we label operators that consist of a single building block by J An i , where n = 1, 2 indicates the relative power suppression due to additional derivatives. Using the equation of motion derived from the leading power collinear Lagrangian, it is possible to eliminate operators with in i− D s derivatives (see below and App. B), such that the operator basis consists of Covariant derivative operators such as ( Hence all derivative basis operators are constructed from ordinary transverse derivatives acting on gauge-invariant collinear building blocks.
Operators with two collinear building blocks in the same direction i are suppressed at least by one power of λ with respect to the leading power, and we label them by J Bn i .
The first operator has fermion number one, the second two, and the last two have fermion number zero. We do not list explicitly the conjugate operators with negative fermion number.
At O(λ 2 ), the operators J B2 i are obtained by acting with a ∂ µ ⊥i derivative on J B1 i . We will use a basis where the derivative acts either on the second building block, or on both, where ψ i 1 ψ i 2 can be any combination from J B1 i . Finally, at O(λ 2 ) it is possible to have operators composed of three elementary building blocks in a single direction, which we collectively call J C2 i , This exhausts the options (i), (ii) from above at O(λ 2 ). An example for a new building block that scales as order λ 2 and hence could be used to construct O(λ) suppressed operators is where soft gauge covariance requires that in i− D i includes the collinear gluon and the multipole-expanded soft gluon field. The subtraction term −in i− D s in the second expression, which is also multipole expanded, is required to obtain a field rather than a differential operator, as is clear from the third expression, in which in i− D i acts only within the square bracket. 4 The last expression shows that in collinear light-cone gauge n i+ A c = 0 the new building block corresponds to the small component of the collinear gauge field. However, using the collinear-field equation of motion, we show in App. B that n i− A i can be expressed in terms of the elementary building blocks with only ∂ ⊥i derivatives, hence n i− A i can be removed from the basis building blocks. As noted above for the transverse derivatives other possible placements of in i− D i can always be reduced to (products of) existing objects. For example As already mentioned we show in App. B that the in i− D s soft covariant derivative, which operates on the elementary collinear building blocks in the form can be eliminated by equation-of-motion operator identities in terms of the A2, B2 and C2 structures defined in Eqs. (9), (11) and (12). This implies that in i− D s can be eliminated from any collinear operator as where the covariant derivative is understood in the colour representation of the object it operates on. Together with the above this implies that up to O (λ 2 ) we can use a basis of collinear building blocks that does not involve soft fields through covariant derivatives. It is constructed entirely from ordinary transverse derivatives and the elementary building block for the quark fields and the transverse gluon field.
In addition to the collinear building blocks, the N-jet operator may also contain a pure soft building block J s . The soft fields do not transform under the collinear gauge transformation, such that J s is trivially a singlet under collinear gauge transformations. In the pure soft sector there is no need to perform the SCET multipole expansion of the soft fields and therefore the soft gauge transformation U s (x) in this case depends on x rather than on x − . The soft transformation of J s is with U s taken in the appropriate representation. In the adjoint matrix representation in the fundamental. The covariant pure soft building blocks start at O (λ 3 ), for example where on soft building blocks iD µ s (x) = i∂ µ + g s A µ s (x) and the soft field strength tensor is defined as ig s F µν s = [iD µ s , iD ν s ]. We can therefore drop J s (0) in Eq. (3) at O(λ 2 ). Therefore, soft fields enter neither via the soft nor via the collinear building blocks for our basis choice, up to O(λ 2 ). This implies that the emission of a soft gluon from the hard process, which generates the N-jet operator, is entirely accounted for by Lagrangian interactions.
The case of N-jet operators differs from that of heavy-to-light currents, which consist of one collinear direction and a soft heavy-quark field, whose decay is the source of large energy for the collinear final state. The basis of subleading SCET operators listed in Ref. [31] does contain soft covariant derivatives at O(λ 2 ) due to the presence of the soft heavy-quark building block at leading power. The absence of soft building blocks in Njet operators at O(λ 2 ) is also an important difference and simplification of the positionspace vs. the label-field SCET formalism [11,12], where soft fields must be included in the basis operators at O(λ 2 ) [10,32]. The difference arises from a different split into collinear and soft, since in the label formalism only the large and transverse component of collinear momentum are treated as labels, while the residual spatial dependence of all fields, collinear and soft, is soft. The difference in the operator basis due to this is compensated by a corresponding difference in the soft-collinear interactions in the Lagrangian in the two formulations of SCET.
It is useful to consider Fourier transformation with respect to the positions t i k in the collinear direction, for operators with one, two and three building blocks, respectively, wherex i = 1 − x i , Here we adopt the convention that n i+ p i k = x i k n i+ P i > 0 for an outgoing momentum in direction i, such that from Eq. (20) also P i > 0 and x i k ∈ [0, 1] for all momenta outgoing, which we shall assume in the following. 5 In general, the basis of N-jet operators can then be written in the form 5 Equivalently, one could assume n i+ p > 0 for ingoing momenta. It is possible to translate between both cases by flipping the signs n i+ → −n i+ and n i− → −n i− of all directions. This sign change can be compensated by substituting t i k → −t i k , such that the form of the building blocks in position space is unchanged. The only difference is then the sign in the exponents in Eq. (20), such that in collinear momentum space P i > 0 for ingoing momenta in that case. We do not consider here the situation where some momenta are ingoing and others are outgoing.
where x i k are momentum fractions of the collinear momentum in direction i, carried by the k-th building block. The operators are given by J i ∈ {J An i , J Bn i , J Cn i }, depending on the number of collinear building blocks and the order in λ. For each direction i one of the x i k can be eliminated using the constraint k x i k = 1, in accordance with the previous definitions. For brevity, we will omit the arguments P i indicating the total collinear momentum in direction i if there is no danger of confusion, because it is conserved in all processes we consider.
The total power suppression of the N-jet operator is then obtained from adding up the suppression factors in λ from each direction. For example, at O(λ 2 ), it is possible to either have a J X2 The infrared divergences of N-jet processes at NLP follow from the ultraviolet divergences of the matrix elements of the above operators computed with the SCET Lagrangian including NLP interactions. For the derivation of the anomalous dimension and renormalization group equation it is convenient to adopt the interaction picture and treat the subleading SCET Lagrangian as an interaction, such that all operator matrix elements are understood to be evaluated with the leading-power SCET Lagrangian. The basis of subleading power N-jet operators at a given order in λ then includes further "non-local" operators from the time-ordered products of the current operators J at lower order in λ with the subleading terms in the SCET Lagrangian. The "local" (in reality, light-cone) currents do not mix into the non-local time-ordered product operators, but the latter can, in principle, mix into the former. The non-local operators mix into themselves but the corresponding matrix of renormalization factors is given by the one for the local currents of lower order in λ contained in the time-ordered product. The absence of further renormalization from the subleading soft-collinear interactions in the time-ordered product follows from the non-renormalization of the SCET Lagrangian to all orders in the strong coupling constant at any order in λ [13].
At O(λ) the time-ordered product operators are of the form where L YM refers to the O(λ) suppressed terms in the SCET Lagrangian given in Ref. [14]. It is understood that the collinear fields in these terms are those of direction i. The generalization to O(λ 2 ) should be evident.
In the following, we will focus on the case in which one of the collinear directions carries fermion number F = 2. The simplification of this choice results from the absence of a leading-power operator J A0 i (and consequently all J An i ), since one needs two fermion fields in the same direction to begin with. Nevertheless, this simpler case allows us to display most of the features of the anomalous dimension at O(λ 2 ). The F = 2 operator basis at O(λ) consists of the single collinear operator We keep open the Dirac spinor indices α, β, because they will in general be contracted with components of the N-jet operator from the other collinear directions j = i. The same rule applies to Lorentz and colour indices, and we only assume that the total N-jet operator transforms as a colour singlet. At O(λ 2 ), we have We will omit the Dirac indices in the following for brevity and drop the direction index i in the notation for the operator unless ambiguities can arise. The time-ordered product operators at O (λ 2 ) are The inclusion of these operators guarantees that the anomalous dimension matrix does not mix operators with different λ scaling. Note that in contrast to the local current operators, the time-ordered products always contain the soft fields.

General structure
The operator renormalization in renormalized perturbation theory is given by where P, Q label the N-jet operators as well as time-ordered products of N-jet operators with insertions of power-suppressed interactions L SCET . The products run over all fields and couplings that enter O Q , respectively. We omit the argument in the following for brevity. At one-loop, writing Z P Q = δ P Q + δZ P Q and demanding that the left-hand side is finite, implies For the operator basis we are interested in we need to consider also the continuous operator label x = {x i k }, and generalize the anomalous dimension to include integrations as well as summation over different types of operators Note that for n i collinear building blocks in one direction we need n i − 1 integrals, because k x i k = 1. If there is only a single building block for a given direction i, then x i 1 = 1, and no integration over momentum fractions occurs. We use the convention that empty products are unity, so that the above equation covers also this case.
As discussed below, the soft loops within a single collinear direction vanish. Therefore, we split the renormalization constant to soft and collinear contributions via where we have used the fact that the soft loops are diagonal in x. The collinear loop along direction i is diagonal in the x j k for j = i, which is reflected by . This gives the MS scheme renormalization conditions where in the collinear part x j k = y j k for j = i. In the last line we include only those fieldand coupling renormalization factors that are associated to collinear building blocks of the direction i (that is, 1 2 δZ χ = − αsC F 8πǫ for each collinear fermion, and 1 2 δZ A + δZ gs = − αsC A 4πǫ for each collinear gluon). We also use the notation

The anomalous dimension matrix is defined by
where we use matrix notation involving both discrete indices (P, Q) labelling the set of N-jet operators including open Lorentz, spinor and colour indices as well as continuous indices (x, y) for the collinear momentum fractions associated to each building block. Before we proceed to discuss the details of each contribution, let us make a technical remark about the extraction of ultraviolet (UV) divergences. To compute the anomalous dimension we need to separate the UV and infrared (IR) poles of the amplitude. In our computation of the soft and collinear contributions we assume that the external states have small off-shellness p 2 i k = 0. This choice regularizes the IR divergences of the amplitude and guarantees that all the 1/ǫ n divergences are related to UV poles of the SCET amplitude. At the end of the computation, the soft and collinear part are combined and only then the limit p 2 i k → 0 can be taken. The cancellation of the off-shell regulator dependence serves as an additional check of our computation.

Collinear part
The collinear contribution to the anomalous dimension can be extracted by computing one-loop matrix elements with a collinear loop. These loops do not contain soft fields, and therefore it is sufficient to concentrate on purely collinear interactions. In principle, there could be collinear one-loop diagrams with external soft gluons generated by the insertion of a power-suppressed Lagrangian interaction. The divergent part of any such diagram would correspond to the mixing of one of the time-ordered product operators into a current operator with a soft field. However, as shown in the previous section there are no such operators at O(λ 2 ) that cannot be removed by the field equations. It is therefore sufficient to focus on collinear loop amplitudes with external collinear lines only.
Since each collinear sector is interacting only with itself, collinear contributions factorize into individual contributions from each of the collinear directions n i+ , i = 1, . . . , N, respectively. Therefore, it is sufficient to consider only the contribution J i to the N-jet operator that contains collinear fields along the n i+ -direction, while the other contributions J j =i are irrelevant. Moreover, in the position-space SCET formulation there are no purely collinear power-suppressed interactions, so the power counting of the collinear loop is determined solely by the operator. We first consider the case of an O(λ) power suppressed operator J i , and then turn to the more involved case of O(λ 2 ), where operator mixing occurs. We will often omit the label i of the collinear quantities in this section for brevity, since only a single collinear direction is involved.

Order O(λ)
In order to extract the anomalous dimension, we consider the matrix element of J B1 χχ defined in Eq. (23) with two external fermions with external momenta p 1 and p 2 . To be specific, we take the two fermions to be distinguishable by their flavours, which we do not show explicitly. The extension to identical particles will be discussed below Eq. (59). We show the collinear one-loop diagrams in Fig. 1. The labels t i 1 and t i 2 indicate whether the corresponding line is attached to the first or second building block of J B1 χχ . For the first two diagrams, all internal lines contributing to the collinear loop are attached to a single building block. In the following, we refer to these contributions as type-(a) loops. Since effectively only a single building block is involved, type-(a) loops can be inferred from the leading-power result. In particular, collecting the sum of the two type-(a) one-loop diagrams, the tree-level diagram, and the contributions from wave-function renormalization from the right-hand side of Eq. (31) for the two external building blocks of J B1 χχ in a matrix element labelled with subscript (a), we find where is the leading-power collinear contribution from a single fermionic building block [30,33]. The third and fourth diagram in Fig. 1 appear similar to the first and second one at first sight, but differ in an important respect. Namely, the two internal lines of the collinear loop are attached to two different building blocks of J B1 χχ . As a consequence, the fractions of collinear momenta of the two lines attached to the operator will in general be different from the momentum fractions of the external lines. We consider the operator J B1 χχ (x) in Fourier space with respect to the collinear direction, where x denotes the momentum fraction associated to the first building block, and correspondinglyx = 1 − x for the second building block. For the external momenta, we label the collinear momentum fractions by y = n i+ p 1 /(n i+ p 1 + n i+ p 2 ) = n i+ p 1 /P andȳ = 1 − y. In this notation, the tree-level diagram is in collinear momentum space given by where v α (p) is the collinear spinor for the outgoing antiquark with momentum p and spinor index α. In order to compute the one-loop matrix element in collinear momentum space, we express loop integrals d d l = 1 2 dn + l dn − l d d−2 l ⊥ in light-cone coordinates, and first perform the n − l integration by closing the contour either in the upper or lower complex plane. Then the integration over l ⊥ can be performed by standard techniques, while the integration over n + l is trivial and set by the fixed value of the momentum fraction x in collinear momentum space. Finally, we express the result in terms of the tree-level matrix element by first renaming y → y ′ , inserting 1 = dyδ(y − y ′ ), and using For example, in position space we find for the contribution from diagram (b, i) where z = n + l/n + p 1 ,z = 1 − z, and l is the momentum of the gluon in the loop. Fourier transforming to collinear momentum space yields a delta function that allows to trivially evaluate the z integration. Following the steps described above we obtain where we used colour-space operator notation for the generators, Here T i 1 and T i 2 are understood to act on the fundamental colour index of the first and second building block of J B1 χχ , respectively. Diagram (b, ii) gives a similar result, that differs only by the replacement x ↔x, y ↔ȳ and p 2 1 ↔ p 2 2 outside of the matrix elements. For diagram (c) we find (39) Note that this contribution induces a spin-dependent structure, i.e. it is non-diagonal in Dirac indices. Collecting all results, we can read off the collinear contribution to the anomalous dimension using Eq. (31). It has a diagonal part ∝ δ(x − y) in collinear momentum space, and a non-diagonal part. Using (34) and T 2 i k = C F for quarks, we can write the anomalous dimension in the form with and Here we also expressed the Dirac gamma matrices in terms of σ µν Note that the contributions from wave-function renormalization in Eq. (31) were already included in Eq. (33), and are thus contained in the diagonal part, with c i 1 = c i 2 = 3/2 for quarks.
As mentioned above, in this work we restrict the discussion to the case of two-fermion operators. Detailed results for all possible contributions to the N-jet operator will be presented in a forthcoming paper.

Order O(λ 2 )
At O(λ 2 ) the three operators in Eq. (24) contribute, and the anomalous dimension is correspondingly given by a 3 × 3 block matrix. We find the following structure at oneloop, that we will derive below: The equation numbers point to the results for the non-zero entries. Note that the operators J T 2 i containing insertions of the power-suppressed SCET Lagrangian contain at least one soft field and therefore do not contribute in the purely collinear sector. We first discuss the first row δZ c,i χ∂χ,Q , then the second δZ c,i ∂(χχ),Q , and finally the last row δZ c,i Aχχ,Q , where Q ∈ {χ∂χ, ∂(χχ), Aχχ}. The zero entries in the second row persist at higher orders in α s (see below).
First row: The contributions δZ c,i χ∂χ,χ∂χ and δZ c,i χ∂χ,∂(χχ) can be extracted by computing the matrix element q(p 1 )q(p 2 )|J B2 χ∂χ (x)|0 at one-loop, involving diagrams as in Fig. 1. The additional ∂ ⊥ derivative leads to an extra power of the loop momentum in the numerator, which yields a more involved structure of divergences compared to O(λ). The divergent part can be expressed in terms of the two tree-level contributions q(p 1 )q(p 2 )|J B2 χ∂χ (y)|0 tree and q(p 1 )q(p 2 )|J B2 ∂(χχ) (y)|0 tree . The coefficients yield the corresponding anomalous dimensions, and we find with The last terms in each expression arise from diagram (c) and are given in App. C. Let us now turn to Z c,i χ∂χ,Aχχ , which describes the mixing of B-into C-type operators. To extract this contribution we compute the matrix element g(q)q(p 1 )q(p 2 )|J B2 χ∂ µ χ |0 at one-loop involving a gluon and two antiquarks. To determine the mixing with J C2 Aχχ it is sufficient to consider a configuration where the gluon has only ⊥ polarization, and the external momenta of all particles have vanishing ⊥ component.
For loops that consist of two internal lines that are both attached to the same building block of the operator J B2 χ∂ µ χ (called type-(a) loops above) one of the collinear building blocks, that is not contributing to the loop, acts as a 'spectator', i.e. the matrix element factorizes, where we have also used that the ⊥ derivative acting on the second building block gives a simple factor of total momentum both at the tree-and loop-level. All contributions of type-(a) are therefore zero for vanishing external ⊥ momenta. Therefore, we can focus on loops that connect the two building blocks. They are obtained from the one-loop diagrams for a quark-quark matrix element shown in Fig. 1 (specifically from diagrams (b, i), (b, ii) and (c)) with the additional emission of the gluon off either an internal fermion line (subscript F ), internal boson (i.e. gluon) line (B), vertex (V ), or directly from the operator (J). These four possibilities are illustrated in Fig. 2  for diagram (b, i). The case (J) is only possible if the gluon is attached to a Wilson line, and therefore this contribution vanishes for ⊥ polarization. Analogous arguments hold for (b, ii) and (c). Similarly, the contributions (b, i) V , (b, ii) V are zero, because the internal gluon is in this case connected to a Wilson line, and the four-point vertex connecting two collinear gluons and two collinear quarks vanishes when contracted with n µ + . Finally, there could be a contribution from one-particle reducible (1PR) diagrams for which the 1PR propagator is canceled by a corresponding momentum-squared suppression of the loop diagram. However, it turns out that there are no such contributions because the vertex for radiating off a ⊥ polarized gluon from a quark line with momentum p vanishes for p ⊥ = 0. In summary, all relevant loop diagrams are shown in Fig. 3.
The computation of the one-loop diagrams is straightforward and we find the result where we made explicit colour indices for clarity. The y k denote the collinear momentum fractions for J C2 Aχχ with y 1 + y 2 + y 3 = 1, and y 1 corresponds to the gluonic building block A. The kernels K are defined in App. C. In colour-space notation where we defined a cross product via (T i 1 × T i 2 ) a ≡ if abc T b i 1 T c i 2 , and the subscripts refer to the first and second fermionic building block, respectively. In addition, we leave implicit the open adjoint index of the colour-space operators, which generates the additional colour label required for the gluonic building block of Aχχ.
The last line follows from the equality at any loop order together with the first line of Eq. (50). Since the O(λ) matrix element on the right-hand side is rendered finite by the δZ c,i χχ,χχ counterterm, it is not necessary to introduce new counterterms to renormalize the left-hand side at O(λ 2 ). We have checked this explicitly by computing the left-hand side of Eq. (51) at one loop.
Third row: For C-type operators with three collinear building blocks the one-loop anomalous dimension can be inferred from operators involving only two collinear building blocks. The reason is that at one-loop, at most two building blocks can be connected to the loop, while the third one acts as a spectator.
In particular, type-(a) loops operate on each building block separately, and therefore give the same result as at leading power (when including also coupling and wavefunction renormalization, as above) Figure 3: Collinear loops contributing to the anomalous dimension Z c,i χ∂χ,Aχχ , that describes mixing of B-into C-type operators. Arrows show the fermion flow for two outgoing antiquarks.
The expression for J q (p 2 ) is given in Eq. (34), and J g (p 2 ) is given by the same expression with C F → C A , 3/(2ǫ) → 0. All other loops connect two building blocks. There are three possibilities to select a pair. For each pair, the computation is analogous to the corresponding case where the third collinear building block is absent. Therefore, we can obtain the anomalous dimension by rescaling the corresponding momentum fractions. For example, for the case where the loop connects the second and third building block (indicated by the subscript 23), the contribution to the anomalous dimension is related to the O(λ) result from Eq. (40), x 1 ) and y = y 2 /(y 2 + y 3 ) = y 2 /(1 − y 1 ). The momentum fractions in the first building block are not affected by the loop, and therefore identical, leading to the δ(x 1 − y 1 ), and a similar argument applies to the Lorentz indices leading to g µρ ⊥ . The prefactor is due to the Jacobian 6 dy/dy 2 = 1/(1 − y 1 ).
To obtain the full anomalous dimension we need to sum over the three pairs of collinear building blocks, 13, 23, 12. Note that the anomalous dimension on the righthand side of Eq. (53) captures also the contributions from type-(a) loops attached to either the second or the third building block. This will also be the case for the 23 and 12 contributions, such that the type-(a) loops are counted twice. We therefore need to subtract them once to obtain the correct result. In addition each term contains the tree-level contribution, which we need to subtract twice. Altogether, The last line contains the subtractions accounting for the over-counting (see Footnote 6 for the normalization). The anomalous dimension Z c,i Aχ,Aχ is given in App. C (see also 6 This can be seen by writing the corresponding delta functions in the tree-level matrix element in the form δ(y 1 P − n + p 1 )δ(y 2 P − n + p 2 )δ(y 3 P − n + p 3 ) = δ(y 1 P − n + p 1 )δ(yP 23 − n + p 2 )δ(ȳP 23 − n + p 3 ) where P 23 ≡ (1 − y 1 )P = (1 − x 1 )P is the collinear momentum of the two building blocks that are connected by the loop. Then the product δ(yP 23 − n + p 2 )δ(ȳP 23 − n + p 3 ) has the same form as for the case with only two building blocks (except that P → P 23 ). The remaining factor δ(y 1 P − n + p 1 ) is not affected by the loop integration, and therefore the same for the one-loop and tree-level matrix elements, leading to δ(x 1 − y 1 ). Therefore the only re-scaling factor is the Jacobian obtained from the change of integration measure Z c A µ χχ,A ρ χχ (x 1 , x 2 , y 1 , y 2 ) 23 dy 2 dy 1 = δ(x 1 − y 1 )g µρ ⊥ dy 1 × Z c χχ,χχ (x, y)dy. For example, the Jacobian ensures that the 'diagonal' contributions to Z c χχ,χχ (x, y) have the correct normalization, because δ(x 1 − y 1 )δ(x − y) = (1 − y 1 )δ(x 2 − y 2 )δ(x 1 − y 1 ). Note also that the anomalous dimension does not explicitly depend on the total collinear momentum P in the direction n i+ under consideration.
Refs. [15,16]). Notice that the above equation is valid only up to one-loop. At higher loops, the three building blocks may be connected together. Eq. (54) can be brought into the form where with T 2 i 1 = C A and c i 1 = 0 for the gluonic building block and T 2 i 2 = T 2 i 3 = C F , c i 2 = c i 3 = 3/2 for the fermionic building blocks. The non-diagonal part is given by In addition, there is no mixing with operators with two building blocks, inherited from δZ c,i Aχ,∂χ = 0 at O(λ) (see App. C), that is,

General structure of the collinear anomalous dimension
The previous findings suggest a general structure for the collinear contributions to the anomalous dimension. We can write schematically for the contribution from collinear direction i with n i building blocks (n i = 1, 2, 3 for A-, B-,C-type operators, respectively), where the first term is the diagonal contribution, δ P Q is non-zero for identical operators P = Q and then stands for the product of δ αβ for Dirac and g µν ⊥ for Lorentz indices, x i k and y i k denote the collinear momentum fractions in direction i for the building blocks k = 1, . . . , n i , and γ i P Q (x, y) encapsulates the non-diagonal contribution. Here x and y denote the vectors of momentum fractions as introduced above.
The non-diagonal contributions in general encapsulate rather lengthy results that depend on the Lorentz structure and on momentum fractions in a generic way. The diagonal contribution can be summarized in a universal way, where c i k = 3/2 for fermionic building blocks, and c i k = 0 for gluonic building blocks. For clarity we added an additional label to the off-shell regulator p 2 i k for the collinear direction it corresponds to.
So far we assumed that the two fermionic building blocks considered above have different flavours. It is straightforward to generalize the result in Eq. (58) to the case of identical building blocks, which is relevant e.g. for quarks of identical flavour or when considering operators with more than one gluonic building block. For gluons (quarks), one has to symmetrize (anti-symmetrize) the anomalous dimension with respect to exchanging them (including a factor 1/N s where N s is the number of terms). 7 Moreover, if more than one ⊥ derivative acts on the same building block at O(λ 2 ), the corresponding Lorentz indices need to be symmetrized too.
The final result for the collinear contribution to the anomalous dimension is obtained by adding together the collinear contributions from all directions, which gives an additional sum over i, where we used that δ [i] (x−y) k>1 δ(x i k −y i k ) = δ(x−y) in the compact vector notation introduced above. This result is consistent with all individual results obtained above, for the fermion number two case. We checked that the diagonal contributions are in accord with Eq. (60) also for fermion number one and zero up to O(λ 2 ).

Soft part
The soft fields mediate interactions between collinear fields in different directions. Here, we need to consider two types of contributions: first, soft loops with leading-power Figure 4: The leading power diagrams with a soft-gluon exchange. The j-direction parton is either a (anti)quark or a gluon created by either A0 or A1 current. In the two-fermion sector, the current can be either B1 or B2.
interactions, for which the power suppression arises purely from the N-jet operator, giving rise to current-current mixing. Second, soft loops containing insertions of the power-suppressed contributions to the SCET Lagrangian that describe subleading softcollinear interactions. They give rise to operator mixing involving J T 2 i operators featuring time-ordered products, see Eq. (25). This approach helps to keep the power-counting manifest and ensures that the anomalous dimension does not mix operators with different powers of λ. Because the leading two-fermion operator is O (λ), in this work we need to consider only a single insertion of the subleading interaction. The leading-power interaction between soft gluons and collinear particles can be used any number of times when constructing the amplitude.

Currents
For the current-current mixing, the soft loops within a single collinear sector vanish to all orders in α s because the leading-power interaction contains only a single component of the soft field, n i− A s . Hence, to determine the soft part of the anomalous dimension we only need to consider soft loops connecting different collinear sectors. At the one-loop level, only two different collinear directions can be connected by a soft loop. The result is then given as a sum of all possible pairings of fields belonging to different directions. For the two-fermion operator, the relevant diagrams are presented in Fig. 4. The parton belonging to the j direction can be either a (anti)quark or a gluon.
The divergent part of the diagrams shown in Fig. 4 with soft loops and leading power interaction is where s ij = 1 2 (n i− · n j− )P i P j depends only on the total collinear momentum in the directions connected by the soft loop.
The colour-space formalism reveals the universal form of the soft factor. The result in Eq. (61) holds for gluons as well as for quarks. The soft factor depends only on the colour charge of the collinear particle but not on its spin. When there are identical partons within one collinear direction, the result should by symmetrized as in the collinear case.
The renormalization factor for the subleading currents with extra ⊥ derivatives acting on the collinear fields is also given by Eq. (61). In the soft-collinear vertices, only the n − component of the momentum is conserved. The other components are conserved only within the collinear sector as dictated by the SCET multipole expansion of the soft fields. In the ⊥ direction, the soft field wave-length is much larger than the size of typical fluctuations of the collinear field. As a result, the soft field is insensitive to the ⊥ momentum of the collinear fields. Hence, the extra momentum factor in the N-jet operator Feynman rule that comes from the ⊥ derivative does not affect the computation of the soft loop.
To summarize, the soft counterterm for the subleading local operators is universal, diagonal and given by Eq. (61). This fact is easily understood by application of the soft decoupling transformation. The collinear fields can be redefined to remove the leadingpower soft interactions from the SCET Lagrangian [12]. For example, for the fermion fields we define The fields building the N-jet operator are evaluated at n i+ t i k so the decoupling transformation commutes with the derivative ∂ ⊥i . The N-jet operator at O (λ) factorizes into a product of collinear fields χ (0) that do not interact with the soft fields and a product of soft Wilson lines. Hence, the universality of the Eq. (61) is a consequence of the standard eikonal approximation for the leading-power soft gluon coupling.

Time-ordered products
The decoupling transformation in Eq. (62) does not remove the soft fields from the nonlocal time-ordered product operators. In this case, it is necessary to compute the soft loops explicitly. To obtain non-zero mixing into local operators we compute diagrams where the soft field from the Lagrangian insertion appears as an internal line. Non-zero mixing can occur only between operators with identical quantum numbers, and as the local currents do not contain soft fields, only these diagrams can induce mixing into local operators. Nevertheless, we checked that the one-loop amplitudes with one external soft gluon are indeed finite after combining the soft and collinear loop contributions. Consider first the J T 2 χαχ β ,ξq operator. Since there is no leading-power interaction between soft quarks and collinear partons it is impossible to form a soft loop and remove the soft quark field. Therefore, no mixing into any of the local operators is allowed for this operator.
The operator J T 2 χαχ β ,YM can form a non-vanishing contraction only with the gluon fields contained in the Wilson lines that accompany the quarks. Choosing the light-cone gauge we immediately see that this operator does not mix into any of the local operators.
Finally, we investigate possible mixing of the time-ordered product containing L ξ . The O (λ) Lagrangian L (1) ξ contains interactions with the ⊥ and n − components of the and n j− , so any tensor integral can be reduced to a combination of these vectors and a metric tensor. After the tensor reduction of the loop integral, the numerator terms with k → n i− vanish by definition of the light-cone coordinates. If k → n j− then the total result is zero because of the anti-symmetric Feynman rule obtained from the soft gluon field-strength tensor.
In summary, the time-ordered product operators with O (λ) Lagrangians do not mix into local currents. The renormalization factor of the mixing of the time-ordered products containing the O (λ) Lagrangian with themselves is given by the Z-factor of its local component.

Combined result
In this section we discuss the combination of the collinear and soft contributions to the anomalous dimension. As concluded above, at fermion-number two we can focus on current-current contributions. We found that both the collinear and soft contributions can be summarized in a universal way, given by Eq. (60) and Eq. (61), respectively. In particular, the total soft contribution, summed over all pairs of collinear directions i, j with i = j, takes the form Notice that for identical building blocks, a symmetrization needs to be performed as discussed in the collinear case. We can write the logarithm as a sum of three terms involving −s ij x i l x j k /µ 2 , µ 2 /(−p 2 i l ), and µ 2 /(−p 2 j k ), respectively. The last two terms are identical after renaming i, l ↔ j, k, thus we obtain Colour-neutrality of the entire N-jet operator implies j k T j k = 0. We can use this to rewrite S as When combining this with the collinear result in Eq. (60), we find that the regulatordependent terms cancel, as expected. This is a consequence of the colour conservation and our assumption that the operator is a colour singlet. The cancellation serves as a consistency check proving that the N-jet operator matrix elements have the correct IR behaviour and no further basis operators, in particular with soft building blocks, are necessary. Therefore, all current-current contributions to the Z-factor can be summarized as From this result we obtain the anomalous dimension matrix where γ cusp (α s ) = αs π , γ i l (α s ) ≡ − αs 2π T 2 i l c i l = − 3αs 4π C F (0) for collinear quark (gluons), and the last line captures the off-diagonal contributions computed above.
This expression summarizes the main result of this work. We have checked that its form persists for all possible current-current contributions up to O (λ 2 ), beyond the F = 2 operators considered here. Operator mixing and non-diagonal contributions with respect to collinear momentum fractions always enter via the collinear contributions γ i P Q (x, y).
As a cross-check, Eq. (68) reduces to the leading-power result (1) when there is only a single building block in each collinear direction (i.e. l, k = 1, x i l , x j k → 1), such that in the notation used above δ(x − y) ≡ i k>1 δ(x i k − y i k ) → 1 is an empty product equal to unity. Furthermore, possibly non-diagonal contributions encapsulated in γ i P Q vanish at leading power. 8 In this work, we consider the case in which one of the collinear directions contains two fermionic building blocks (direction i, say). At O(λ), there is only a single type of operators of this kind, given by the product of J i = J B1 χχ (t i 1 , t i 2 ) defined in Eq. (23) for the direction labelled by i and leading-power building blocks for all other N − 1 directions J j =i = J A0 j . In this case, the anomalous dimension is off-diagonal in the collinear momentum fractions in direction i, where the right-hand side is given by Eq. (42), and we have used γ j P Q (x, y) = 0 for all leading-power building blocks j = i. Furthermore the product of delta functions for the At O(λ 2 ), there are two cases. Let us first consider the case that the direction i which we choose to carry fermion-number two encompasses itself the O(λ 2 ) suppression, i.e. it is represented by one of the three operators in Eq. (24), J i ∈ {J B2 χ∂χ , J B2 ∂(χχ) , J C2 Aχχ }. Then the other N −1 directions have to contain leading-power building blocks, as before. The structure of the anomalous dimension follows directly from Eq. (43), and leads to operator mixing, where the non-zero contributions are given in Sec. 3.2.2 (specifically Eqs. (46), (49) for the first and Eq. (56) for the last row, and γ i ∂ µ (χχ),∂ ν (χχ) = g µν ⊥ γ i χχ,χχ is related to the O(λ) result Eq. (42)). The anomalous dimension is diagonal with respect to the other N − 1 directions.
The second case that can occur at O(λ 2 ) is that direction i with F = 2 is described by the O(λ) contribution J i = J B1 χχ (t i 1 , t i 2 ), and one of the other N − 1 directions, say direction i ′ , contributes an additional O(λ) suppression. The remaining N − 2 directions must then be represented by a leading-power building block. Since we do not require direction i ′ to have a definite fermion number, there are more possibilities, in particular χχ } (plus hermitian conjugated operators). In this case we need in addition the corresponding anomalous dimension matrices γ i ′ P Q for these operators. They will be given in future work.
In summary, we have taken the first step in a systematic investigation of the anomalous dimension of subleading power N-jet operators in view of resummation of logarithmically enhanced terms in partonic cross sections beyond the leading power. We provide an explicit result at the one-loop order for fermion-number two N-jet operators. In a forthcoming paper we will present results at O(λ), O(λ 2 ) for general N-jet operators.
• Operators J An , J Bn , J Cn with one, two, three building blocks, respectively, and power suppression O(λ n ). Here we count J A0 as leading power (n = 0) for a collinear quark and gluon, respectively. The power suppression of all other operators is then counted relative to the leading power.
• Colour-space operator for parton labelled by i k is T i k and colour conservation • We define α s = g 2 s 4π andμ 2 = µ 2 e γ E /(4π). In this Appendix, we show that the operator , that could potentially contribute to the basis of collinear building blocks at (relative) O(λ), can be expressed in terms of the operator basis discussed in Sec. 2, and is therefore redundant (see also Ref. [34] for some closely related discussion).
The equation of motion for the collinear gauge field with respect to the i-th collinear direction derived from the leading-power collinear Lagrangian [14] reads where ig s G µν i = [iD µ i , iD ν i ] and the ellipsis stand for contributions involving n µ i+ , that will drop out below. In the remainder of this Appendix we will consistently omit the index i for the collinear direction i. The covariant derivative includes the multipole-expanded soft field in the n − projection, in − D. Contracting the equation of motion with n + µ and multiplying with collinear Wilson lines from both sides gives, Next we use a t a ij t a kl = 1 2 (δ il δ jk − 1 3 δ ij δ kl ) to rewrite the colour ordering on the right-hand side (colour indices made explicit) Writing the scalar product over ν on the left-hand side in terms of collinear basis vectors, and using W † in + DW = in + ∂ to simplify gives Next, we apply the inverse derivative operator formally given by 1/(in + ∂) 2 . Note that (in + ∂) 2 i.e. we can express the operator on the left-hand side in terms of other collinear building blocks. The previous equation receives corrections from the power-suppressed interactions in the SCET Lagrangian, which can be worked out in a similar manner. Leadingpower redundant operators can always be removed iteratively from these further terms.
One peculiar property of this relation is that the soft field appears explicitly only on the left-hand side. We checked that the relation is indeed fulfilled in the matrix element with one soft and one collinear gluon. On the left-hand side, a 1PI diagram exists, where the soft gluon is attached directly to the operator. In addition, a 1PR diagram where the soft gluon is emitted from the collinear line contributes. On the right-hand side, only a 1PR diagram exists, that agrees with the sum of the 1PI and 1PR contribution from the left-hand side. We also checked explicitly that the identity holds in the matrix element with one and two collinear gluons with ⊥ polarization.
where C F ≡ 1 6 (1 − 3(T i 1 + D i 1 ) · T i 2 ) and we introduced the additional colour operator D b |a = d abc |c related to the symmetric d abc symbol defined via {t a , t b } = 1 3 δ ab + d abc t c . We checked that our result agrees with Refs. [15,16] after subtracting the soft-loop contributions to the O(λ) heavy-to-light current from the anomalous dimension computed in these references. By computing the matrix element q(p)|J B1 A µ χ (x)|0 we furthermore find δZ c,i A µ χα,∂ ν χ β (x, y) = 0 .