Large-x resummation of off-diagonal deep-inelastic parton scattering from d-dimensional refactorization

The off-diagonal parton-scattering channels $g+\gamma^*$ and $q+\phi^*$ in deep-inelastic scattering are power-suppressed near threshold $x\to 1$. We address the next-to-leading power (NLP) resummation of large double logarithms of $1-x$ to all orders in the strong coupling, which are present even in the off-diagonal DGLAP splitting kernels. The appearance of divergent convolutions prevents the application of factorization methods known from leading power resummation. Employing $d$-dimensional consistency relations from requiring $1/\epsilon$ pole cancellations in dimensional regularization between momentum regions, we show that the resummation of the off-diagonal parton-scattering channels at the leading logarithmic order can be bootstrapped from the recently conjectured exponentiation of NLP soft-quark Sudakov logarithms. In particular, we derive a result for the DGLAP kernel in terms of the series of Bernoulli numbers found previously by Vogt directly from algebraic all-order expressions. We identify the off-diagonal DGLAP splitting functions and soft-quark Sudakov logarithms as inherent two-scale quantities in the large-$x$ limit. We use a refactorization of these scales and renormalization group methods inspired by soft-collinear effective theory to derive the conjectured soft-quark Sudakov exponentiation formula.


Introduction
Resummations of logarithmically enhanced loop corrections are a powerful and often essential tool to enlarge the predictivity of QCD perturbation theory. Resummation is necessary when a ratio of kinematic invariants, λ, becomes small such that α s ln k λ, where α s is the strong coupling and k = 1 or 2, is no longer a good expansion parameter. Recent interest in this subject has focused on understanding the structure of such logarithmic terms at next-to-leading power (NLP) in λ with the aim of summing them to all orders in α s . This has been accomplished at the leading-logarithmic (LL) order in various contexts, covering final-state event shapes [1,2], threshold resummation in Drell-Yan and Higgs production [3][4][5], and Higgs production or decay through light-quark loops [6][7][8].
A number of methods has been used, but it has become evident that a generalization to the next-to-leading-logarithmic (NLL) order is not straightforward. This is to be compared to the situation at leading power (LP), where resummation is often understood to any logarithmic order, even though one faces technical challenges of high-order loop calculations in practice.
The most natural framework to formulate resummation is through the factorization of scales and evolution equations. The all-order resummed expression is then obtained as the product or convolution of the factorized pieces. At NLP, one faces the new difficulty that these convolutions are divergent. While divergent convolutions are familiar from rapidity divergences, which are not regulated dimensionally and may occur already at LP, such as in transverse-momentum dependent factorization, the problem at NLP is of a different nature. The divergences can be regulated dimensionally and arise in convolutions of factors containing the physics at different virtualities. However, since factorization and resummation refer to the renormalized factors before convolution, the standard formalism fails to deal with this situation. An explicit example can be found in [9] for the case of next-to-leading logarithms near the qq → γ * Drell-Yan threshold.
In the present paper, we address these difficulties for the threshold of off-diagonal deep-inelastic parton scattering. The off-diagonal channels vanish at LP near threshold x → 1, since they do not contain 1/[1 − x] + distributions at any order in α s . However, the failure of standard resummation methods appears already at the LL order for the DGLAP splitting functions. Vogt and collaborators [10][11][12] found that the all-order quark-gluon splitting function with LL accuracy is given in moment space by where is the Borel transform of the generating function of the Bernoulli numbers B 0 = 1, B 1 = −1/2, . . . . 1 The existence of an infinite series of double logarithmic terms shows 1 Here and in the remainder of the paper α s without argument denotes the strong coupling in the that in the off-diagonal channels even the anomalous dimension is a two-scale quantity as N 1, contrary to the diagonal anomalous dimension, a distinction that has not received as much attention as it deserves, except for [10][11][12]. This remarkable result was obtained from the structure of 1/ poles of the unfactorized parton-scattering cross sections in the exactly known d = 4 − 2 dimensional low-order results, and their consistency with factorization. These structures were then extrapolated to all orders to find closed functional forms, including the reconstruction of the series of Bernoulli numbers. To our understanding (1.1) has not yet been proven by deriving it directly from algebraic all-order expressions. With this method, further results on the finite short-distance coefficients of the off-diagonal channels in deep-inelastic scattering and Drell-Yan production at large N were also obtained [10,11].
What distinguishes the off-diagonal splitting functions from the diagonal ones in the x → 1 / large-N limit is that the former describe the splitting into an energetic parton and a soft quark. We further notice that the double-logarithmic series involves the colour factor C F − C A . A connection between soft quarks and this colour factor of the large logarithms also appears for the Sudakov resummation of the qq → φ * form factor, where φ * denotes a Higgs boson, effectively coupled to two gluons, and q a light quark [13,14]. The leading logarithms here originate from soft quark exchange. Further, the authors of [15] investigated the all-order structure of the e + e − → qqg amplitude in the kinematic configuration where the quark and anti-quark are nearly collinear with small virtuality s Q 2 and momentum fractions z andz = 1 − z, respectively, recoiling against the energetic gluon. Keeping the leading double poles 1/ 2 and logarithms of z orz, as the quark or anti-quark become soft, they conjectured the exponentiation of the corresponding one-loop terms to all orders P qq (z) = P tree qq (z) exp α s π 2 T 1 · T 3 µ 2 zQ 2 + T 2 · T 3 µ 2 zQ 2 + T 1 · T 2 µ 2 s Here T 1 , T 2 , T 3 are the colour operators of the quark, antiquark and gluon, respectively, such that T 1 · T 2 = C A /2 − C F , T 1 · T 3 = T 2 · T 3 = −C A /2. If we now take z → 0, which corresponds to the quark becoming soft, and focus on the terms involving Q 2 , we see that the coefficient of Q −2 has the colour factor T 2 · T 3 + T 1 · T 2 = −C F . The coefficient of (zQ 2 ) − , which involves the new scale √ zQ Q in the soft-quark limit, however, is T 1 · T 3 − T 1 · T 2 = C F − C A . It is tempting to conjecture that in a splitting 1 → 2+3 with soft 3, the endpoint divergence, which occurs when integrating over z since P tree qq (z) ∝ 1/z, and which requires extra resummation of the logarithms of z orz not captured by the usual formalism, is related to the difference of the Casimir charge of the energetic particles 1 and 2. Along this line, it was noted in [2,15] that in supersymmetric QCD with quarks in the adjoint representation, the endpoint divergences and extra logarithms are absent. All three examples of the appearance of C F −C A in front of double logarithms have in common that the resummed result was obtained without explicit factorization of the scales involved, using either d-dimensional arguments, diagrammatic arguments, or a conjecture.
In this paper, we establish a connection between some of these results in the context of NLP LL resummation for off-diagonal deep-inelastic parton scattering as x → 1. To this end we adapt the soft-quark Sudakov exponentiation conjecture [15] from event shapes to deep-inelastic scattering (DIS). We then • prove (1.1) for the resummed off-diagonal splitting function and the finite coefficient function from the soft-quark Sudakov exponentiation conjecture via d-dimensional consistency relations that follow from the requirement of pole cancellation between momentum regions. The adapted version of (1.3) plays the role of a "boundary condition" in the purely hard contribution to the process, from which the resummation of the full process follows in closed all-order form.
• derive the previously conjectured exponentiation formula through the refactorization of certain power-suppressed operators in soft-collinear effective theory (SCET) which have endpoint-singular matching coefficients. The renormalization group equations (RGEs) then exhibit the origin of the C F − C A colour factor.
The outline of the paper is as follows. In Section 2 we consider the q + φ * → q + g amplitude, define the light-cone momentum distribution for the qg-final state, and calculate its leading poles at the one-loop order. We then apply the exponentiation conjecture analogous to (1.3) to the soft-quark limit of this amplitude. Sections 3 and 4 contain the material related to the two bullet points above, respectively. We conclude in Section 5. Appendices A and B collect SCET conventions and the field modes which appear in different parts of the paper, and some basic facts on DIS at large x. In Appendices C and D we provide alternative derivations of a) the solution of the consistency relations at LP, and b) the resummation of the refactorized SCET operator, which confirms the result of Section 4. The application of consistency relations to the thrust event shapes considered in [15] is presented in Appendix E in order to note the similarities and differences between the two processes.

Off-diagonal DIS cross section and soft-quark Sudakov exponentiation
We begin by considering deep-inelastic scattering of a quark off a Higgs boson, which couples to quarks and gluons through the gluonic interaction Figure 1: Scattering of a quark off a virtual Higgs boson at tree level.
The coupling κ is related to the effective Higgs-gluon coupling in the infinite top-quark mass limit. 2 We are interested in the kinematic situation when the final state has small invariant mass, p 2 X Q 2 ≡ −q 2 , which corresponds to the limit x → 1 for the Bjorken scaling variable x ≡ Q 2 /(2p · q). The reason for considering this exotic process is that it is related to Drell-Yan production of a Higgs boson in the quark-gluon channel near the partonic threshold. The above DIS process allows us to extract the quark-gluon splitting kernel P gq [10], which enters Higgs production. All these quantities are NLP, i.e. suppressed by one power of (1 − x) as x → 1 relative to the leading diagonal gluongluon coupling. The off-diagonal channel g + γ * → q + g for the more standard scattering on the vector current can be obtained by substitution of colour factors [10].
We consider the dimensionally regularized and unfactorized partonic DIS cross sections. Following [10], we introduce the partonic structure functions W φ,g , W φ,q by defining where i = g or q. At the lowest non-vanishing order in α s , to NLP in (1 − x), and neglecting O( 0 ) terms not multiplied by logarithms of (1 − x) one has The exact result is provided in (B.11) and (B.13).

Momentum distribution function and lowest-order result
At the lowest order the quark-scattering off the Higgs boson is realized by the process The tree-level amplitude is shown in Figure 1. We write the d = 4 − 2 dimensional two-particle phase space as which represents the distribution of the light-cone final-state momentum between the nearly collinear quark and gluon in the final state. The vector n − represents a light-like vector which projects on the large momentum components of the final state particles. 3 We can represent the matrix element squared of the process (2.5) integrated over the phase space in the form which holds to any order in perturbation theory in the strong coupling α s . The momentum distribution function is defined as where |M 0 | 2 denotes the tree-level matrix element squared, averaged (summed) over the spin and colour of the initial (final) state for the leading diagonal channel process g + φ * → g. For x → 1, we can expand P qg (s qg , z) in s qg /Q 2 or λ ∼ √ 1 − x 1. For this purpose we note that Similarly, the expansion of |M qφ * →qg | 2 , given in (B.12), which is itself a function of s qg or x and of the momentum fraction z, gives at the lowest non-vanishing order in the coupling expansion, which implies Integrating and neglecting O( ) corrections that are not multiplied by logarithms (i.e. , (2.14) in agreement with (2.4). W φ,q qφ * →qg represents the contribution to the partonic DIS structure function when only two partons are present in the final state. As such it is an infrared (IR) divergent quantity. In lowest order in α s , the IR divergence is a single 1/ pole, which arises from the z → 0 region of the integral (2.8) owing to the 1/z behaviour of the tree-level momentum distribution function. The z → 0 limit corresponds to the kinematic configuration where the initial quark transfers all of its momentum to the final-state gluon, and the final-state quark becomes soft. It is therefore essential that the integration over z in (2.8) is done in d dimensions, a fact that will be of importance later on.

One-loop momentum distribution function
In this subsection, we calculate the one-loop (virtual) correction to the 2 → 2 scattering process qφ * → qg and obtain the corresponding momentum distribution function (2.9). We are interested in the leading double poles and logarithms. We therefore need the leading pole 1/ 2 without expanding its coefficient in a series of . The relevant Feynman diagrams are shown in Figure 2. We find that only the first five diagrams give nonvanishing leading poles. Calculating the interference of these one-loop diagrams with the tree diagram, we obtain for the leading-pole terms P qg (s qg , t, u)| 1−loop = P qg (s qg , t, u)| tree α s π employing colour operator notation, T i , for particle i (i = 0 for the incoming quark). Eq. (2.15) is valid for general values of the Mandelstam variables s qg , t, u. The first line in the curly bracket is the familiar double-pole structure when all s qg , t, u are of order of the hard scale Q 2 . The last three lines represent additional contributions, which are suppressed by 2 compared to the first line. Although technically finite as → 0, these cannot be simply omitted because after integration over the phase space to obtain W φ,q , they may generate poles and logarithms of equal order as the pole terms from the first line, as will be shown below. When the final-state quark and gluon become collinear, s qg Q. With s qg + t + u = −Q 2 , we can parameterize t = −zQ 2 − s qg /2, u = −zQ 2 − s qg /2. The collinear expansion amounts to expanding in s gq /Q 2 ∼ (1 − x) ∼ λ 2 at fixed z, which yields Given that the leading order result (2.12) becomes singular only at the end point z = 0, we can safely expand thez − terms in a series of in the above equation. Therefore, keeping only terms contributing to the leading poles after integration over the phase space, we have P qg (s qg , z)| 1−loop = P qg (s qg , z)| tree α s π In the framework of SCET this result exhibits a profound problem. The tree amplitude represented in Figure 1 corresponds to a J B1 SCET operator (see Appendix A) with a quark field in the collinear direction, and a quark and a gluon field in the anti-collinear direction with light-cone momentum fractions z andz, respectively. The tree-level matching coefficient of this operator from the diagram in Figure 1 is proportional to 1/z, which gives the 1/z behaviour of P qg (s qg , z)| tree after squaring the amplitude and accounting for a factor of z from the sum of the final-state quark spin. From the general formula for the anomalous dimension of subleading power operators [16,17], we get the double pole terms with T 1 · T 0 and T 2 · T 0 from the standard cusp anomalous dimension terms. However, one cannot obtain a cusp term for the two fields within the same collinear sector, i.e. the T 1 · T 2 term. In this part, there are three terms involving three different scales. The third contains the scale zs qg . We may disregard it here because the dependence on s qg identifies it as a term related to the final-state jet function, rather than the renormalization of the J B1 operator at the hard DIS vertex. The first two terms, however, contain the hard scales Q 2 and zQ 2 , and they are supposed to be predicted by the corresponding anomalous dimension. However, the anomalous dimension given in [16,17] applies when the convolution of the coefficient function with the anomalous dimension is convergent, which is not the case here as discussed next. The difference between these two terms in the coefficient of T 1 · T 2 is O( ) and hence does not contribute to the double pole. Instead, the expansion in produces 1/ × ln z. The important point is that the 1/z singularity of the matching coefficient promotes this term to the same leading-pole order 1/ 3 as the standard double pole terms after integration over z as in (2.8). Moreover, the integral over z must itself be regularized due to the singularity at z = 0, and the correct result is obtained by not expanding (2.18) before integration. This can easily be seen by comparing (no expansion before integration) 1 to (expansion of (2.18) before integration) If only the pole part of the integrand were kept, the result would be incomplete. This explains why it was necessary to keep the exact d-dimensional coefficient of the double pole terms in the one-loop momentum distribution function.
To summarize, when we attempt to interpret (2.18) in the SCET framework, we discover two problems with the standard treatment of factorization in SCET. First, the renormalization and logarithmic terms of some SCET operators with singular matching coefficients are not obtained correctly. Second, the convolution integrals of the hard matching coefficients with the jet functions-the integral over z above-diverge and must themselves be regularized, for instance dimensionally. These obstacles appear first at NLP.

Exponentiation conjecture
We shall pursue the SCET interpretation further in Section 4. Here, we observe that crossing symmetry relates qφ * → qg to H → qqg discussed in [15], and we follow [15] by conjecturing that the leading poles are given correctly to all orders in α s by exponentiating the one-loop expression (2.18): where, for the qg case, , but the colour-operator notation is used to emphasize the generality of the conjecture. The exponentiation refers to the d-dimensional expression, since it must be integrated in d dimensions over z as discussed above.
In the subsequent section, we employ consistency relations from the cancellation of poles between all relevant momentum regions to infer the structure of the DIS structure function from the contribution with only hard loops and a single anti-hardcollinear loop, as shown in the left diagram of Figure 4 below. The term involving s qg in the exponent in (2.21) arises from the exponentiation of the one-loop anti-hardcollinear leading pole to all orders, and should therefore be dropped for this consideration. Also the tree-level momentum distribution function is non-singular as z → 1, hence for the leading poles after the z-integration, we may replacez → 1. We can therefore simplify (2.21) to where, for the qg case, T 2 ·T 0 +T 1 ·T 2 = −C A and T 1 ·T 0 −T 1 ·T 2 = C A −C F . In SCET we interpret this as the resummation of the matching coefficient of a non-standard B1 operator, squared and convoluted with the tree-level jet function. We shall come back to this in Section 4, where we provide a derivation of this result with factorization methods. Notice that the above expression has a homogeneous power counting even when z 1, since in a d-dimensional treatment we count z − as an O(1) quantity.
Integrating (2.22) over z yields the contribution to the off-diagonal quark DIS structure function from any number of hard loops and a single anti-hardcollinear loop from the two-particle phase-space, which corresponds to the integral over the tree-level final state jet function. We obtain in the leading-pole approximation. We note that the expression contains the standard Sudakov factor with colour factor C A and a non-standard factor involving the colour factor C F − C A . The latter arises from the integral 3 Resummed off-diagonal partonic cross section and qg splitting function from consistency relations The theory of deep-inelastic scattering and of resummation for x → 1 is based on factorization formulas for the scattering of hadrons in terms of partonic quantities. The latter are usually IR and ultraviolet (UV) divergent, but can be defined in terms of a renormalization prescription. Consistency relations follow from the requirement that an observable must be finite as → 0 and allow one to deduce the expansion in of unrenormalized partonic quantities based on partial information. The LL resummation of the quark-gluon splitting function (1.1) was obtained in [10] from the requirement that the DIS cross section is an observable and hence must be finite, together with additional assumptions on the all-order colour structure as well as an exponentiation ansatz for the full partonic cross section. A stronger form of consistency relations from pole cancellations can be obtained when the regions of virtuality relevant to the observable are known. The different scaling of every region with the dimensionless parameters of the problem implies a larger number of consistency relations [18]. For example, the LL resummation of the thrust event shapes at NLP was derived in [1] from the contributions with a single collinear and an arbitrary number of hard loops alone and invoking pole cancellations between all regions.
In this section we use consistency relations to derive the NLP LL resummation of the off-diagonal DGLAP kernel (1.1) and short-distance coefficient function from the exponentiation conjecture (2.23). In this way, we infer the resummation of the shortdistance functions in the DIS factorization formula from the resummation in a single momentum region. In this section we work in moment space, following [10], to avoid dealing with convolutions. Moments of functions g(x) of the Bjorken scaling variable x are taken with the standard definition g(N ) ≡ 1 0 dx x N −1 g(x). The x → 1 limit corresponds to N → ∞ in moment space.

Consistency relations for DIS
In this section, we consider the hadronic DIS process p + φ * → X. From standard factorization theorems at leading twist in Λ/Q, where Λ denotes the QCD scale, we can write the hadronic tensor as where i sums over all partonic scattering channels, 4 and f i denotes the unfactorized (unrenormalized) parton distribution function (PDF) of i in the proton p. Thus f i contains dimensionally regulated UV divergences. The finite, MS subtracted parton distributions and short-distance coefficients (partonic cross sections) are related to W φ,i , Note that the short-distance coefficientsC φ,k have finite limits for → 0, but are still d-dimensional. The splitting kernels are given by For generic N the leading-twist DIS factorization formula involves hard and collinear physics related to the scales Q and Λ. The latter is non-perturbative and factorized into the PDFs. For large N , the small invariant mass of the final state (see also Appendix B) introduces a new scale into the problem, which is also the source of the large logarithms that we wish to sum. The four relevant virtualities are: The anti-hardcollinear virtuality arises from the requirement of a small-mass final state X. In the adopted large-momentum frame, its large momentum is in the opposite direction of the incoming proton, hence "anti-hardcollinear". We also need a softcollinear virtuality Λ/N Λ, which accounts for the anomalously small momentum of the target remnant as x → 1 [19]. Note, however, that there is no soft region in DIS.
The calculation of the DIS process is imagined to be strictly factorized into contributions from the different virtualities. A multi-loop diagram is considered as a sum of terms, in which every loop momentum has one of the above virtualities, in the spirit of the strategy of expanding by regions [20]. Each loop is then associated with a factor (µ 2 /p 2 ) times a function of , which will usually be singular. The consistency relations follow from the requirement that the sum of all terms is non-singular as → 0. We note that dimensional regularization only factorizes regions with different virtualities. It is not sensitive to the scaling of the momentum components separately. However, for the present problem, this will be sufficient to obtain non-trivial consistency constraints.

Leading power
The resummation of leading large-N logarithms at leading power is simple and wellknown. We rederive it here from consistency relations and the RGE for the hard function to illustrate the method.
At LP there is no mixing of partonic channels. Only the gluon channel contributes to the DIS cross section. We expand the diagonal gluon channel in α s according to Here k denotes the number of hard plus anti-hardcollinear loops, which determines the dependence Q −2k on Q, j is the number of anti-hardcollinear and softcollinear loops, which determines the number of times the factor N appears. n−k is then the number of collinear plus softcollinear loops related to the PDF. At O(α n s ), the leading singularity is 1/ 2n . This factor has been extracted, such that the coefficients b (n) kj ( ) can be expanded in non-negative powers of . For the LL resummation, we focus on the leading poles, and need only the constant part b kj (0). At LP, we also drop the O(1/N ) correction. The above expansion holds when expressed in terms of the dimensionless bare coupling µ −2 α s0 , since some of the poles are related to coupling renormalization. However, since the relation between the bare and renormalized coupling involves at most a single pole per loop, we can identify the expansion parameter with the renormalized MS coupling α s ≡ α s (µ) to leading-pole accuracy.
We regard f g on the left-hand side as the unrenormalized gluon PDF at the factorization scale µ. To make the dependence on the collinear and soft-collinear scale explicit, we relate it to a non-perturbative reference PDF via where U gg (µ), defined by this equation, contains the evolution from the scale Λ to µ. Another way of reading (3.5) is that it represents DIS on a gluon with IR singularities regulated non-dimensionally rather than DIS on a hadron. It is only important that the left-hand side is finite as → 0, so all poles on the right-hand side originate from factorization, and have to cancel. The requirement of pole cancellation implies not only the obvious consistency condition from the vanishing of the coefficient of the 1/ 2n pole. In addition, the coefficients of all terms of the form must vanish for s + r < 2n, r, s ≥ 0, since other subleading pole terms from the expansion of b Using the binomial expansion of (n − k) s , they are equivalent to the consistency relations At order O(α n s ) this provides 2n 2 + n equations, but only n 2 + 2n of them are linearly independent. There are (n + 1) 2 coefficients b (n) kj , hence at any order in perturbation theory we can determine all b (n) kj through consistency equations in terms of a single remaining one.
A particularly convenient choice is b n0 , which corresponds to n-loop diagrams with only hard loops. We show below that b (n) n0 can be determined from the one-loop hard coefficient b (1) 10 by solving a RGE, such that the purely hard-loop contribution is given to all orders by which captures the leading-pole part, denoted by "LL", and implies This provides the single condition required at LP to fix all of the b (n) kj . We make the ansatz which is finite for → 0 and contains only products of factors of the form (µ 2 /p 2 ) , with p 2 any of the four relevant virtualities. This corresponds to where m 0 = max(0, k+j −n). We checked (up to n = 10) that this ansatz indeed satisfies the consistency conditions (3.10). Since the system is fully constrained (equal number of free coefficients and linearly independent consistency conditions), this solution is the unique solution of the consistency relations given (3.11). Clearly, (3.13) factorizes into where the first expression is the unfactorized partonic cross section, and the second the unfactorized PDF. The latter shows that the gluon PDF in the x → 1 limit must be considered as a two-scale object already at LP, since f LP,LL g depends on the softcollinear in addition to the collinear virtuality. Recall that in the derivation of these expressions N is treated as an O(1) quantity that must not be expanded in . However, by definition of the MS scheme, to define the MS renormalization constants, the pole part is extracted by expanding in at fixed (large) N . From (3.16) and the requirement thatf g in (3.2) be finite, we obtain The anomalous dimension in the gluon channel is obtained from In the leading (double) pole approximation, the evolution of d-dimensional MS coupling is given by with no (α s ln 2 N ) k corrections. This corresponds to the well-known fact that the DGLAP kernel for x → 1 at LP is

Derivation of the resummed hard function (3.11)
In the x → 1 limit the QCD part of the Higgs-gluon interaction is closely related to the Sudakov form factor for gluon scattering. In SCET notation (see Appendix A), it matches to the operator with a collinear gauge-invariant transverse gluon field in the collinear direction of the initial-state gluon and an anti-hardcollinear one for the outgoing gluon. The square of the hard matching coefficient C A0 of this operator contains the large logarithms at LP as x → 1 in the DIS structure functions, which we associated with W LP,LL φ,g | hard loops above. 5 Here we need the resummation of the pole part of the bare coefficient rather than of the large logarithms in the renormalized coefficient.
The anomalous dimension of J A0 takes the general form we can solve (3.24) for [21] ln The bare coefficient function is given by where C A0 (µ) is free of poles. The bare coefficient does not depend on µ and is a function of the dimensionless quantities α s0 /Q 2 and . The resummation of the pole part is obtained most easily by choosing µ = Q, in which case C A0 (Q) contains no large logarithms, and by expressing α s (Q) in terms of the bare coupling α s0 . The cusp anomalous dimension is responsible for the double logarithms. To sum the leading poles, the one-loop approximation suffices. We therefore set and obtain It is sufficient to use the tree-level coefficient C A0 (Q) = 1 to obtain the leading poles. We then find which proves (3.11). The above method can be used to include running coupling and higher-order effects. However, we restrict ourselves to the leading double logarithms here.

SCET intepretation
The SCET interpretation of DIS at LP as x → 1 is sketched in Figure 3. A collinear gluon from the PDF is converted into an anti-hardcollinear gluon by the A0 current (3.23), which sources the final-state jet. The figure shows the cross section W φ,g f g with hard vertices and lines corresponding to the (anti-hard) collinear fields. It does not show the Wilson lines attached to these fields. Since the softcollinear PDF modes enter at LP only through Wilson lines, they do not appear in the graph, despite the fact that the result (3.16) for the resummed PDF shows that they are necessary at LP to achieve pole cancellation. The picture is consistent with the explicit SCET computations [19]. The appearance of softcollinear modes only in Wilson lines is also the reason why they do not appear explicitly in the LP factorization theorem [19], once the standard PDFs are introduced. Their presence in (3.16) leads us to suspect that this will no longer hold at NLP. There is some similarity of this with collinear functions in the factorization of the Drell-Yan process near the kinematic threshold. These do not appear in the wellknown LP factorization theorem, because soft modes appear only through Wilson lines. However, this no longer holds beyond LP, and collinear functions do appear at NLP [3,9].

Next-to-leading power
Having introduced the method for the well-understood case of LP large-N resummation, we proceed to the main subject of this paper, the NLP suppressed off-diagonal quark-Higgs scattering channel. All partonic channels are relevant at NLP, hence we consider was considered before in (3.5). The NLP term in the hadronic cross section consists of The evolution factors that express the unrenormalized PDF at the scale µ in terms of the PDFs at the initial scale Λ must also be expanded. We generalize (3.6) to and find The LP leading-pole resummed factor can be inferred from (3.16). U LP,LL qq (µ) and U LP,LL qq (µ) are obtained by replacing C A → C F . The hadronic cross section should be finite for any choice of non-perturbative initial conditions f g (Λ), f q (Λ) and fq(Λ). For the off-diagonal quark-gluon channel we focus on the terms proportional to f q (Λ), given by

Consistency relations
Assuming that the same hard, anti-hardcollinear, collinear and softcollinear virtualities describe the physics of large-x DIS at NLP, we expand the hadronic cross section as Compared to the previous LP expansion formula (3.36), we note the overall NLP factor 1/N , the power 2n − 1 rather than 2n for the leading pole and the absence of a tree term n = 0. This follows from the fact that a quark must be radiated into the final state in the off-diagonal quark-gluon channel. This first emission brings a factor of α s but produces only a single 1/ pole. As mentioned above, the poles must cancel in all channels separately, and we can therefore disregard the fq(Λ), f g (Λ) terms. 6 In the following we will only be interested in the leading pole at any order, in which case we can replace c The similarity of (3.5) and (3.36) implies that the consistency relations from pole cancellation take the same form: However, the absence of the 1/ 2n pole now leads to the condition s + r < 2n − 1 as compared to s + r < 2n at LP, (3.10). There are still (n + 1) 2 coefficients c The number of linearly independent consistency relations is related to the rank of matrix M (n) , which is (n + 1) 2 − 3. Hence, the consistency relations allow us to determine all (n + 1) 2 coefficients c (n) kj in terms of three unknowns at every order n.

Solution
Two of the three "initial conditions" at every n for solving the consistency relations can be fixed trivially. In the absence of collinear and softcollinear loops (k = n), there must be at least one anti-hardcollinear loop, since the final state cannot be made up of hard modes for x → 1. This implies c (n) for all n. Similarly, without any hard or anti-hardcollinear loops (k = 0), the necessary off-diagonal q → qg splitting always produces a softcollinear quark. Thus there must be at least one softcollinear loop, such that The third "initial condition" is provided by the Sudakov exponentiation conjecture (2.23). Recall that this refers to the all-hard loop corrections to the square of the qφ * → qg amplitude integrated over the anti-hardcollinear two-particle phase space, which gives the series of terms c (n) n1 in present notation. In moment space, we replace This particular finite series of C F and C A terms has already been seen in [10,15]. The consistency equations can now be solved. For example, at n = 2, we can rewrite (3.38) as equations for the unknown coefficients as where we used c 21 is given by (3.42) for n = 2. All unknown coefficients are uniquely determined, since the matrix has full rank (equal to 6) and is therefore invertible. At general n we can proceed analogously, and write the n(2n − 1) consistency conditions in the form for s + r < 2n − 1, r, s ≥ 0 , where the right-hand side is fixed by (3.42) and the left-hand side can be written as a matrix-vector multiplication with a matrix of dimension ((n + 1) 2 − 3) × n(2n − 1). This matrix is quadratic only for n = 1 and n = 2, and has more rows than columns for n ≥ 3. This means the free coefficients are over-constrained. Nevertheless, as expected from the previous discussion, not all consistency conditions are linearly independent, and the rank of the matrix is such that there is a solution, which is then the unique solution.
Rather than attempting a direct solution of these linear systems, we will guess a suitable ansatz. From (3.35), (3.36), we deduce that we must match appears to be the unique finite combination of all four regions in the leading-pole approximation, suggests the ansatz where n( ) is a normalization factor yet to be determined. 7 At O(α s ), (3.37) implies c With the normalization determined, (3.47) reproduces all c (n) n1 or (2.23). Since (3.47) is finite as → 0, and since the content of consistency relations is the finiteness of the physical cross section assuming (2.23), (3.47) provides the unique solution.
Given that W N LP φ,q must not depend on (µ 2 /Λ 2 ), while U N LP gq must not depend on (µ 2 /Q 2 ), the solution (3.47) implies The first of these equations reproduces Eq. (17) in [10] for µ = Q and C A = 0 as assumed there, and therefore proves and generalizes the conjectured all-order structure of the full partonic cross section. In addition, the dependence of W N LP,LP φ,q on C F and C A is consistent with the colour structure conjectured in Eq. (13) of [10].
It is remarkable that in the leading-pole approximation, the full result follows from the exponentiation conjecture for the hard-only amplitude (2.23) by a simple substitution. Let us define Then the solution of the consistency equations in terms of the hard-only amplitude (2.23) can be summarized in moment space as N ). The appearance of the factor (N − 1) is characteristic of the leading-pole solution. The prefactor of the Sudakov factors accounts for the anti-hardcollinear O(α s ) contribution that must always be present at NLP.

DGLAP kernel and coefficient function
To determine the resummed off-diagonal splitting function and the MS-subtracted shortdistance partonic cross section, we decompose the unfactorized partonic cross section W N LP,LL φ,q into its finite and divergent parts. From (3.2) we deduce where Z LP qq and C LP φ,g are known from (3.17) (replacing C A by C F ) and (3.18), respectively, and the NLP off-diagonal factors Z N LP gq ,C N LP φ,q are to be determined. From the structure of the LP expressions (3.17), (3.18) it is apparent that the split into pole and finite part in the exponents must be done according to It will be convenient to introduce the abbreviations which allow us to write (3.50) as into its pole and finite part. Using where are the Bernoulli numbers, we obtain by expanding e a/w that where the sums over n on the right-hand side can be regarded as a generalization of Bernoulli polynomials. Inserting this decomposition into (3.58) and matching the resulting expression to (3.54), we identify the splitting kernel and short-distance coefficient as Note that by construction Z N LP,LP gq is a pure pole term, hence corresponds to the MS PDF renormalization factor, while the short-distance coefficient is finite as it should be. Indeed, for → 0 at fixed N , which implies w → 0, we find The anomalous dimension in the q → gq splitting channel is obtained from where the second equality holds to NLP accuracy and uses the vanishing of the offdiagonal terms at LP. Inserting the leading-pole resummed results for the Z-factors and the LP leading-pole anomalous dimension (3.21) in the gluon-gluon channel, we obtain after a short calculation which has no poles as it must be and proves (1.1) first given in [10]. We close this derivation with three observations: pdf c pdf c Figure 4: SCET representation of the content of (3.35) for quark-Higgs scattering at NLP as x → 1. Wilson lines are set to 1.
• Comparison of the summed large-N anomalous dimensions (3.21), (3.69) shows that there is an infinite series of (double) logarithmic terms only for the off-diagonal channel. This implies that not only the short-distance coefficient, but also the anomalous dimension is a two-scale object in the off-diagonal channel. The double logarithms are associated with the colour charge change C F −C A of the partons that carry large momentum. The absence of large logarithms in the diagonal channel seems related to the fact that the energetic particles have the same colour charge.
• The Sudakov exponentiation conjecture was originally proposed and explored in [15] for the case of e + e − or Higgs-decay event-shape distributions in the two-jet limit, when the final state includes a soft quark. There are interesting differences and similarities between the DIS and event-shape case, which we elaborate on in Appendix E. The solution of the consistency relations takes a form similar to (3.50). The Bernoulli series does not arise for event shapes. Event shapes are infrared finite, and the Bernoulli numbers arise in DIS from the need to factorize the pole part of (3.50) to obtain the renormalized short-distance coefficient and parton distribution, as seen above.
• Let us comment on similarities and differences compared to the derivation of (3.69) presented in [10]. Both derivations use finiteness and pole cancellation. In addition, [10] conjectures a specific form of the full unfactorized partonic cross section (including hard and hardcollinear contributions to all orders) as well as a particular assumption for the colour structure (as stated in Eqs. (13) and (14) in [10]). We require (2.22) as an input for the derivation, which involves a single region only (specifically the hard region), that we consider as a weaker assumption compared to those used in [10]. In addition, the exponentiation conjecture (2.22) lends itself to a derivation based on RGE methods, that we turn to in Section 4. Finally, we obtain the Bernoulli series in (3.69) by an algebraic derivation in a closed form.

SCET interpretation
The SCET interpretation of DIS at NLP as x → 1 in the off-diagonal channel is sketched in Figure 4. The figure shows from (3.35), the external quark line representing the quark PDF f q (Λ). As before, Wilson lines of whatever fields are set to 1. The left diagram in Figure 4 represents the first term in this expression and describes the hard scattering of a quark off the Higgs boson. The corresponding hard vertex is of the B1 type with field content A hc⊥χhc χ c (see Appendix A), and the presence of two anti-hardcollinear fields provides the power suppression. The circled operator vertex labelled "B1" in the graph represents the hard subgraph, which corresponds to Figure 1 at tree level, and to the hard region of the diagrams shown in Figure 2 at one-loop order, respectively. At this vertex the incoming quark is converted into the anti-hardcollinear quark and gluon in the final state. The right diagram describes the second term in (3.70). Here the hard scattering occurs through the LP gluon-Higgs scattering vertex of the A0 type with field content A hc⊥ A c⊥ . In the x → 1 limit, the gluon in the q → gq splitting carries almost the entire momentum of the initial quark, leaving a remnant softcollinear q. The interaction that couples soft(collinear) quarks to collinear modes is power-suppressed, and part of the NLP L (1) SCET Lagrangian [22,23]. In (3.70) this process is part of U N LP qg , the off-diagonal evolution of the PDF in the x → 1 limit. Now there is a softcollinear mode in the final state, which does not arise from a Wilson line, which is shown explicitly in the right diagram in Figure 4.
It is evident that both diagrams in the Figure are related. If the gluon propagators labelled pdf c in the right diagram became hard, the diagram would turn into the left one. What prevents standard SCET factorization methods to be applied to this situation, is that the convolution of the short-distance coefficient of the B1 operator with the finalstate jet function is divergent after SCET renormalization of the hard and jet functions. The above treatment through consistency relations circumvents this problem, since it is d-dimensional to the end. The divergent convolutions do not appear, but they are implicit, and done in d dimensions, where they exist. In the LL NLP resummation of the diagonal quark-quark and gluon-gluon channels for Drell-Yan production near threshold with SCET methods [3,4] these complications did not appear, but they do at the next-to-leading logarithmic order [9].
The appearance of an endpoint divergence and the breakdown of standard SCET factorization points to the emergence of a new scale in the problem, which requires a refactorization of the B1-type SCET operator. In the following, we show how this idea can be implemented. The resummation of logarithms from the new scale will allow us to derive the exponentiation conjecture that was used above as a boundary condition to solve the consistency equations for the NLP LL resummation of the quark-gluon channel.

Derivation of the soft-quark Sudakov factor
In the previous sections, we have seen that the SCET interpretation of NLP off-diagonal DIS involves a B1-type current, i.e. an operator constructed from one collinear and two anti-hardcollinear SCET fields, where the latter two are light-like separated. The operator creates the anti-hardcollinear final-state quark and gluon carrying momentum fractions z and 1 − z, respectively, of the total anti-hardcollinear final-state momentum. The peculiarity of our problem manifests itself in the fact that the convolution of treelevel matching coefficient and the anomalous dimension is not well defined, because the integral exhibits an endpoint divergence as z → 0, i.e. when the quark becomes soft. This prevents the standard application of the RGE to the summation of the large logarithms. Instead, we must first consider the limit z → 0 and sum ln z terms, which become large in the singular region, to all orders, while still working in d dimensions. This goes beyond the standard paradigm of SCET, where the large component of collinear momenta are assumed to be of the order of the hard scale, hence the momentum fraction z appearing in a B-type current is treated as an order one parameter z = O(1). z is nevertheless integrated from 0 to 1. This is justified since the contribution to the integral from an interval [0, δ] can be made arbitrarily small as δ → 0, as long as the matching coefficient is less singular than 1/z.
A similar problem has recently been discussed in [24], where it was noted that the singular part of the hard matching coefficient can be included into the definition of the operator. This leads to the concept of singular and regular B1 operators. In that case, the singular B1 operator was related by reparameterization invariance (RPI) to the leading-power A0 operator. The singular B1 operator can then be combined with the time-ordered product of the leading-power A0 operator and the NLP Lagrangian to obtain a well-defined operator, whose evolution is governed by the standard RGE. The analogous construction does not work for DIS considered here, as there is no RPI relation of the NLP B1 operator to the LP one. This fact is easily understood from the observation that the LP and NLP operators relevant to the dicussion of off-diagonal DIS have different fermion numbers in the collinear and anti-collinear sectors, unlike in [24], where the NLP current was obtained by adding a gluon building block to the LP current.
From Figure 1 it is evident that the intermediate gluon propagator is proportional to 1/z as z → 0. When the momentum fraction of the outgoing quark is O(1), the intermediate gluon propagator is hard, and it has to be integrated out, which gives the matching coefficient of the B1 operator that behaves as 1/z for z → 0. In this limit, however, the virtuality of the intermediate propagator approaches zero, and the intermediate gluon should still be present as a dynamical mode in the effective theory (EFT) rather than having been integrated out. This causes a breakdown of the standard application of SCET to this problem, and its failure to reproduce the IR singularity structure of full QCD correctly. We are therefore forced to revise the structure of the modes in the presence of an endpoint-singular matching coefficient.

Scales relevant for the endpoint-singular B1 operator
To cure the lack of proper scale separation due to the endpoint singularity, we must identify the intermediate scales and modes relevant to the z → 0 limit, which goes beyond the SCET I paradigm. Only then, the soft-quark Sudakov expoentiation conjecture (2.22) can be derived with EFT methods.
To understand the scales relevant for our problem, we perform a method-of-region analysis [20] of the integrals appearing in the one-loop diagrams shown in Figure 2. We assume that the external momenta are slightly off-shell so that we can identify all modes which contribute to the loop integrals when there is a dimensionful infrared scale. As elsewhere in this paper, we focus on the terms giving rise to the leading logarithms. This means that we want to capture terms which diverge in the limit when the quark momentum fraction goes to zero, z → 0, and we focus on the double poles in or single poles multiplied by ln z. This allows us to drop many terms and perform simplifications such that the final expression for the leading one-loop correction takes the form qφ * →qg the Born amplitude for the process q(p) + φ * (q) → q(p 1 ) + g(p 2 ). The result can be expressed in terms of the three master integrals (4.4) The integral I 1 corresponds to diagram (1) of Figure 2, while the sum of diagrams (2) and (5) can be expressed in terms of the integrals I 2 and I 12 . The remaining diagrams in Figure 2 do not contribute to the terms that we consider here. The modes relevant to the endpoint problem can be identified by expanding these integrals. To perform the expansion by regions we introduce the variable z as defined in (2.7) as a new power counting parameter and take the limits 1 z λ. We work to leading order in the z and λ expansion. We assume the following scaling for the external momenta p 1 , p 2 , p, respectively: z-anti-softcollinear anti-hardcollinear with component notation l ∼ (n + l, l ⊥ , n − l). With off-shell external momenta all three integrals are IR and UV finite. When the off-shell regulator is set to zero, this computation corresponds to standard one-loop matching of the B1 operator in SCET I . Note that we choose the momentum p to have hardcollinear rather than collinear virtuality here to facilitate the interpretation of the result as a SCET I matching computation-only subsequently the hardcollinear modes shall be matched on the collinear modes corresponding to the external initial-state momentum in DIS as discussed before. In the following, we focus only on the pole parts of contributing regions.
We begin with the I 2 integral. This is a standard vertex integral exhibiting a double logarithmic enhancement, which can be decomposed into the following loop momentum regions: • hard k ∼ (n + k, k ⊥ , n − k) ∼ Q (1, 1, 1) • hardcollinear k ∼ Q (1, λ, λ 2 ) • anti-hardcollinear k ∼ Q (λ 2 , λ, 1) As expected, I 2 | h + I 2 | hc + I 2 | hc + I 2 | s = O( 0 ), and this integral is reproduced by the standard SCET I modes. The soft modes appear here because we assumed that momentum p has hardcollinear virtuality. The complete EFT description must take into account that the external momentum p has collinear virtuality, and in this case, the soft modes ought to be replaced by the softcollinear modes. The integral I 1 has a similar mode structure, but in this case the hard mode results in a scaleless expression and gives a vanishing contribution. Instead, a new z-hardcollinear mode appears. It is obtained by combining z-anti-softcollinear (4.5) and hardcollinear (4.7) momenta, in analogy with the hard mode being a sum of hardcollinear (4.7) and anti-hardcollinear (4.6) momenta. We find following decomposition of the I 1 integral into modes: • hardcollinear k ∼ Q (1, λ, λ 2 ) • z-anti-softcollinear k ∼ Q λ 2 , z 1/2 λ, z • soft k ∼ Q (λ 2 , λ 2 , λ 2 ) These results show the emergence of the new scale √ zQ related to the endpoint singularity in the z-integral (2.8). The new scale is not directly related to the scales present in the factorization of the DIS process at LP. Rather, it is generated dynamically at NLP, due to the breakdown of naïve factorization for the B1 current with a singular matching coefficient. As the momentum fraction of a collinear quark becomes parametrically small, the scalar product p 1 · p cannot be treated as being of the same order as p 2 · p.
The presence of the new scale manifests itself in a particularly subtle manner for the I 12 integral. Before we proceed to consider the relevant regions for I 12 , let us look at the on-shell result. For I 1 and I 2 , the on-shell results are equal to the hard and z-hardcollinear contributions, respectively. Thus, they are both single scale integrals, though the scale for I 1 is zQ 2 , while for I 2 it is Q 2 . On the contrary, the on-shell result for I 12 contains a large logarithm (4.16) which cannot be removed by any choice of µ. This observation is troublesome as the on-shell integral is already a two-scale object, and it needs to be factorized to achieve a proper EFT interpretation of the result. Returning to the case of off-shell external momenta, we find contributions from the following integration regions: • hard k ∼ Q (1, 1, 1) • anti-hardcollinear k ∼ Q (λ 2 , λ, 1) • z-anti-softcollinear k ∼ Q λ 2 , z 1/2 λ, z We observe that I 12 | on−shell = I 12 | h + I 12 | z−hc , and the large logarithm appears as the result of a cancellation between the hard and z-hardcollinear contributions. When considering the limit z → 0 it is thus crucial to factorize (2.22) into these two contributions.
The cusp anomalous dimension governing the LL resummation should not itself contain large logarithms, which cannot be removed by some choice of the scale µ. If that is not the case, then the anomalous dimension should itself be resummed or factorized. In the following, we perform a refactorization such that the logarithms from the hard scale may be resummed independently of the logarithms whose origin is the z-hardcollinear scale. Such refactorization is necessary to correctly sum all the logarithms of z.

Resummation
Having understood the scales relevant to the one-loop result, we attempt to construct the EFT framework to derive (2.22). At this point, we restrict ourselves to LL accuracy and do not claim that the construction presented here could be used to perform resummation at NLL accuracy or beyond. As the analysis of regions showed, we must distinguish the hard scale Q 2 and the z-hardcollinear scale zQ 2 . This suggests that the matching of QCD to SCET I should be separated into two steps: first we integrate out the hard modes, then we the remove z-hardcollinear modes to obtain the EFT at the hardcollinear scale λ 2 Q 2 . The intermediate gluon propagator in the tree diagram shown in Figure 1 has zhardcollinear virtuality and thus it must be associated with the dynamical degrees of freedom at the z-hardcollinear scale. The matching equation at the hard scale reads where the LP SCET I current represents the point-like coupling of the Higgs boson to two gluons in SCET, and C A0 (Q 2 , µ 2 ) = 1 at tree level. In this theory, the diagram in Figure 1 is represented by the matrix element of the time-ordered product ξq z−sc (x) |q(p) . The Lagrangian mediating the z-anti-softcollinear quark interaction with z-hardcollinear modes is L Let us note a peculiarity which distinguishes this problem from the one discussed in [24]. The z-anti-softcollinear quark is generated by a subleading-power Lagrangian insertion in the collinear sector. This observation suggests that the endpoint-singular contribution in a collinear sector should be combined with a time-ordered product in the corresponding anti-collinear sector, while in [24], the time-ordered product and the singular current belong to the same direction. The hard matching coefficient C A0 of the operator (4.22) satisfies a standard RGE (3.24), which at LL accuracy reads After this first matching, the SCET with z-hardcollinear modes is defined at the zQ 2 scale. To describe DIS factorization in the limit x → 1, we need an EFT at the scale λ 2 Q 2 where λ 2 ∼ 1 − x. This EFT should contain only modes with hardcollinear virtuality or lower, as is the case for SCET factorization of DIS at LP. Besides, we need to include z-anti-softcollinear modes as separate entities represented in the EFT by their own set of fields. The detailed construction of this EFT is left for future work. Fortunately, it is not needed to perform LL resummation, since we already possess all the essential ingredients. We discuss the resummation in the following, and then provide some partly speculative comments on the SCET with z-anti-softcollinear modes in the following subsection.
We assume the existence of an operator J B1 , such that we can match the time-ordered product (4.23) to this J B1 , which must be composed of a hardcollinear quark, a z-anti-softcollinear quark and an anti-hardcollinear gluon field. While the matching of the z-hardcollinear quark field to the hardcollinear quark field is trivial, the non-trivial content of this equation is the reinterpretation of the anti-collinear sector, where zQ 2 now appears as the large scale on which the matching coefficient and anomalous dimension of the operator can depend. As this operator is supposed to reproduce the IR poles of the QCD amplitude, its renormalization factor can be deduced from our previous computation of the amplitude (4.1) in the on-shell limit. Using (4.8), (4.12) and (4.16) we find that the LL UV divergence of this operator can be removed by the counterterm (4.27) We implicitly assumed that there is no operator mixing at the LL level involved in the matching equation (4.26), i.e. the complete IR pole in QCD is reproduced in the EFT by Z B1,B1 . Using we find that the matching coefficient obeys at LL accuracy the RGE It is pivotal that Γ B1 depends only on the scale zQ 2 . The fact that the other logarithms dropped out serves as a consistency check of this construction. The anomalous dimension of the D B1 coefficient is proportional to C F − C A , reflecting earlier observations that double logarithmic enhancement [25] and endpoint divergence [15] vanish in N = 1 supersymmetric QCD. It is straightforward to solve (4.25) and (4.29). Recalling that we work with bare, ddimensional objects for the boundary terms for the d-dimensional consistency relations, we solve for the bare coefficients, and find The product of the square of these two coefficients gives (2.22), proving the exponentiation of soft-quark Sudakov logarithms conjectured in [15].

Tentative EFT interpretation
While it was not essential to know the exact form of the operator J B1 to achieve LL resummation, we nonetheless comment on the possible structure of SCET with z-antisoftcollinear modes. We expect that the operator J B1 has the form 8 In this theory, we must decompose the large component of the momentum in the anticollinear sector into a hardcollinear part, which is of the order of Q, and a residual momentum of the order of zQ. Above we accordingly decomposed the anti-hardcollinear derivative as in − ∂ = in − ∂ hc + in − ∂ z−sc . Unlike in SCET II , where the soft modes do not interact with the collinear modes, here the z-anti-softcollinear modes can still interact with the anti-hardcollinear modes. This interaction is responsible for the part of the anomalous dimension proportional to T 1 · T 2 . To compute the anomalous dimension of (4.31), one would need to derive the Lagrangian for this z-SCET, which we leave for future work. Instead, extending the notion of singular SCET I operators in [24], we consider the family of SCET I operators J (n) (4.32) which are defined at a single point x = 0, but contain the singular part of the matching coefficient in its definition through the inverse derivative. Computing the mixing of these operators into themselves together with the assumption that the relevant scale is zQ 2 , we find again the product of the resummed coefficient functions in (4.30). We provide this alternative derivation in Appendix D. In addition, we recover the renormalization factor (4.27) as the standard MS renormalization constant of J B1 . This gives us confidence that a SCET operator whose anomalous dimension is equal to the QCD IR poles can in principle be constructed.
The next step of the EFT construction involves integrating out the hardcollinear scale and matching hardcollinear fields onto the collinear fields which describe the modes inside the PDF. The hierarchy of scales is such that soft modes are also integrated out, and instead soft-collinear modes appear. Similar to the leading power, the anti-hardcollinear fields in the operator (4.31) give rise to the so-called jet function at the level of the amplitude squared. Besides, the cross-section contains a contribution due to the timeordered product of the LP current and subleading-power soft-collinear Lagrangian, as shown in Figure 4.

Conclusion
Contrary to the expectation that the resummation of large logarithms in 1 − x in the expansion of the off-diagonal parton scattering channels in deep-inelastic scattering or Drell-Yan production near threshold ought to be simpler than for the diagonal channels, since the former vanish at leading power in the expansion in 1 − x, the converse is true. This can already be seen from the fact that even the DGLAP splitting functions contain an infinite series of double logarithms for quark-gluon or gluon-quark transitions, for which the formula (1.1) was found [10] a decade ago, but a method for systematic improvements beyond leading logarithms is still missing. The difficulty appears to be related to the emission of a soft quark rather than gluon in the parton splitting, or more generally to the change of colour charge of the energetic partons in the splitting.
The present work was motivated by the desire to understand (1.1) from the perspective of scale separation and effective field theory as a necessary step towards a general resummation formalism at next-to-leading power. In the first part of the paper, we showed that given that the relevant modes are hard, collinear, soft-collinear and antihardcollinear, the leading-logarithmic resummation of off-diagonal deep-inelastic parton scattering as x → 1 follows from the resummation of the purely hard virtual contribution to the process. The condition that all 1/ poles in dimensional regularization cancel between the various regions is sufficient to bootstrap the full solution. For the resummed purely hard contribution, which acts as a "boundary condition" to solve these consistency conditions, we adapted the "soft-quark Sudakov" exponentiation conjecture [15] from event shapes in e + e − collisions to DIS. In this way we derived the expression for the resummed off-diagonal DGLAP kernel in terms of the series of Bernoulli numbers found previously [10] directly from algebraic all-order expressions, that is, without ex-trapolating the structure of an iteratively generated finite series of terms.
The second part of the paper is concerned with the derivation of the "soft-quark Sudakov" exponentiation of the hard function. The hard function can be alternatively interpreted either as the light-cone momentum distribution amplitude of the final-state qg pair in the off-diagonal 2 → 2 scattering process, or the matching coefficient of a B1type collinear operator in SCET. The crucial feature is that the amplitude is singular as 1/z when the quark momentum fraction z → 0. The failure of standard Sudakov resummation or SCET factorization is caused by the divergence of the convolution of the hard amplitude with the final-state jet function and the emergence of the new scale √ zQ. Based on this observation, we derive the previously conjectured exponentiation formula through the refactorization of certain power-suppressed operators in SCET which have endpoint-singular matching coefficients. The renormalization group equations then exhibit the origin of the peculiar C F − C A colour factor through an additional exponent related to the scale √ zQ.
We cannot offer a precise effective field theory formulation for this refactorization at this point, and, furthermore must note that in this treatment, the problem of endpointdivergent convolutions is side-stepped by effectively regulating them dimensionally, since the consistency relations and the "boundary condition" for their solution are formulated in d dimensions for unrenormalized objects. A truly satisfactory solution would express the result as finite convolutions of properly renormalized functions. Nevertheless, we believe that the connection between various ideas made manifest here for the first time should provide useful insight on NLP resummations. In particular, the formalization of the refactorization of SCET operators appears to be a promising avenue for the systematic understanding of resummation at next-to-leading power.

Acknowledgments
We would like to thank A. Vogt for correspondence. This work has been supported by the Bundesministerium für Bildung and Forschung (BMBF) grant no. 05H18WOCA1, by a Fellini-Fellowship for Innovation at INFN, funded by the European Union's Horizon 2020 research programme under the Marie Sk lodowska-Curie Cofund Action, grant agreement no. 754496, and by the Taishan scholars program of Shandong province. Figures were drawn with Jaxodraw [26]. Calculations were done in part with FeynCalc [27], FIRE [28], Package-X [29], and Reduze [30].

A SCET conventions and relevant modes
• We define light-like reference vectors n ± with n 2 ± = 0, n + · n − = 2. Any fourmomentum can be decomposed as Collinear modes have large n + p, anti-collinear modes large n − p.
• SCET operators are conveniently constructed from collinear quark and gluon fields, which are invariant under collinear gauge transformations is the collinear Wilson line. Similar definitions apply to hardcollinear fields. For anticollinear fields n + and n − are interchanged.
• In SCET operators, J Xn i denotes the product of collinear fields for direction i, here collinear or anticollinear. X = A, B, . . . refers to the number 1, 2, . . . of collinear fields from (A.2), and n to the power suppression relative to the leading term consisting of a single field without derivative [16,22]. In this paper, when we refer to an A0 or B1 operator, we refer to field operators involving a product of collinear fields in directions n + and n − with field content A ν ⊥hc A µA ⊥c , for which we employ the short-hand notation J A0 , and [A ν ⊥hcχ hc ]χ c , which we refer to as J B1 . Both arise in the matching of the Higgs-gluon coupling (2.2) to SCET. The explicit forms are The B1 operator as given is the one that appears in Section 3. The one in Section 4 looks similar but its precise mode content is different and the two B1 operators must be carefully distinguished. See Table 1 below for a summary of modes and their abbreviations.
• The following scaling and power counting variables are used in this work: λ ∼ √ 1 − x 1 related to factorization DIS at large x; η ∼ Λ/Q λ related to the twist expansion. The QCD scale Λ appears in modes for the non-perturbative PDFs. We consider large-x factorization at NLP, but always work at LP in the twist expansion parameter η. For the refactorization of the B1 operator in Section 4, we also need to consider z ≡ n − p 1 /n − (p 1 + p 2 ) 1, where p 1 is the momentum of the quark, which becomes soft.
The scalings assigned to the momentum modes used throughout the paper are summarized in Table 1.

B DIS at x → 1
We briefly summarize some results and definitions for DIS off a scalar particle and factorization at large x at LP here. Name (n + l , l ⊥ , n − l) virtuality l 2

B.1 DIS off a virtual scalar
We consider DIS of a particle N off a Higgs boson, as represented in Figure 5. The large momentum transfer Q and the Bjorken scaling variable x are defined by Partons in particle N carry momentum fraction ξ, such that p = ξP , 0 < ξ < 1. DIS mediated by the exchange of a scalar particle (dubbed Higgs boson) occurs via the effective gluon-gluon-scalar coupling (2.2), where the coupling κ for an actual Higgs boson would be given by The DIS structure function W φ is defined as 9 The QCD factorization theorem relates the structure function W φ to the partonic partonic short-distance coefficientsC φ,i (x) by means of the convolution with renormalized PDFsf i of parton i in N . The notation follows (3.1), (3.2) of the main text. There we often refer to unfactorized (unrenormalized) partonic structure functions W φ,i (x) and PDFs f i , related to the above by (3.2). Eq. (B.6) also holds with The equation also applies to the case when the particle N is itself a quark or gluon, but then the left-hand side is IR divergent for massless, on-shell partons. When the IR divergences are regulated non-dimensionally with a regulator introducing the scale Λ, the consistency arguments based on pole cancellations used in the main text apply to this partonic scattering. If dimensional regularization is employed, the unfactorized partonic structure functions W φ,i (x) do not change but the unrenormalized PDFs f i (x) become trivial, because the loop integrals are scaleless, and the left-hand side of (B.6) is simply W φ,i . The hadronic structure function is related to the phase-space integrated, initial-state spin-and colour-averaged and final-state spin-and colour-summed scattering amplitude as The same relation applies to the partonic structure functions for the scattering of gluons and quarks. The respective lowest order contributions in powers of the strong coupling are obtained from (B.9) The two-particle phase space dΦ 2 has been defined in (2.6), and dΦ 1 denotes the ddimensional one-particle phase space. The tree-level contribution (see diagram (a) in The quark-scattering channel starts contributing at O(α s ) with diagram (b) in Figure  6. The spin-and colour-averaged / summed matrix element squared expressed in terms of the variable z defined in (2.7) reads Inserting this expression into (B.9), we get with s qg as defined in (2.10).

B.2 Large x
In this paper we focus on the threshold region x → 1. This region is characterized by the fact that the scattering leaves a soft target nucleon (or parton, depending on whether we consider the hadronic or partonic threshold) and a jet-like final state X with parametrically small invariant mass squared p 2 X = Q 2 (1 − x)/x Q 2 . The existence of two scales for x → 1 is the basis of the factorization of the partonic structure functions W φ,i (x) into a hard and jet function [31][32][33]. For the structure function W φ , focusing on the leading gluon scattering channel, such factorization takes the form valid to leading power in λ ∼ (1 − x) and in η ∼ Λ/Q. We assume λ η, in which case the hard and jet functions in (B.14) can be calculated in perturbation theory. A similar equation holds for the partonic structure functions W φ,i (x) themselves, as discussed above, if we interpret f g as the distribution of gluons in parton i.
The factorization theorem (B.14) can also be derived within SCET [19]. Near threshold the parton undergoing the hard scattering has collinear momentum scaling p = ξP ∼ Q(1, 0, η 2 ) and carries away almost all momentum of the initial state N (p). The target remnant is then made out of partons which must have softcollinear momenta, i.e. p remnant ∼ Q(λ 2 , λη, η 2 ). This explains the need for softcollinear modes in the PDFs, as listed in Table 1. The energetic parton scattering off the Higgs boson (or virtual photon in standard DIS) is converted into a jet of partons with anti-hardcollinear momentum, p hc ∼ Q(λ 2 , λ, 1). The factorization theorem (B.14) is derived by constructing the sequence QCD → SCET(hc, c, sc) → SCET(c, sc) (B.15) of effective theories, where in the first matching step the hard modes are integrated out, and the QCD gluon-gluon-scalar interaction is matched onto A0, B1 etc. SCET(hc, c, sc) operators built from the collinear gauge-invariant building blocks (see Appendix A). In the second step, the jet function emerges as the matching coefficient containing the antihardcollinear final-state, leaving a parton distribution made up of collinear modes and the softcollinear target remnant modes. At LP, only an A0-type current is required, and the hard function is given by the square of its short-distance coefficient, More important for the present work is the observation [19] that the softcollinear mode appears only through a Wilson line in the definition of the PDF for x → 1. It is then possible to identify this PDF with the standard PDF. The two-step matching scheme (B.15) should be expected to hold beyond the LP. However, when (B.14) is naively generalized to NLP, the convolutions of generalized renormalized hard and jet functions diverge. As discussed in the main text, this requires, at least for the present, a partly d-dimensional treatment and a refactorization within SCET(hc, c, sc) to generate the correct large-x logarithms that would otherwise be missed.

C Alternative derivation of the LP solution (3.13)
There is a simpler way to obtain the leading-power leading-pole expression (3.13) for (W φ,g f g ) LP,LL , which bypasses the combinatorially involved solution for the coefficients b (n) kj in the consistency relation.
At any N the DIS factorization theorem implies multiplicative factorization. At LP in 1/N , only the gluon channel contributes. We can therefore write the expansion for the logarithm of the DIS cross section (3.5) as using that W φ,g cannot depend on Λ, and f g cannot depend on Q. The form of this expansion contains the non-trivial statement that for the logarithm of the cross section the highest pole at O(α n s ) is 1/ n+1 . For the leading poles, we can drop the -dependence of the coefficients t and a corresponding expression for ln f LP,LL g with Q → Λ and an overall minus sign. Comparison with the resummed hard-only expression (3.11) implies that the logarithm of W LP,LL φ,g is one-loop exact, that is t , from the requirement that γ LP gg (N ) should have at most a single power of ln N at any order in α s , and the observation that γ LP gg (N ) is related to the coefficient of the single pole in (C.5). We then recover the previous results (3.15), (3.16).

D Alternative derivation of the resummed singular B1 current
In this Appendix we present an alternative derivation of the exponentiation conjecture (2.22) for the momentum distribution P qg,hard (s qg , z) in the limit z → 0 that is complementary to the presentation in Section 4. Both rely on the observation made in the first part of Section 4 concerning the relevant regions for z → 0. However, instead of considering a "refactorization" into hard and z-hardcollinear regions as in Section 4, the derivation presented here solely relies on the SCET I description of DIS for large x involving collinear, anti-hardcollinear and softcollinear modes. We start from the SCET I description of the qφ * → qg scattering process depicted in Figure 1 assuming that the incoming q has collinear scaling, and the outgoing q and g are both anti-hardcollinear. The relevant SCET B1 current has field contentχ c A hc,⊥ χ hc , and its tree-level matching coefficient diverges as 1/z for z → 0, where z is the momentum fraction of the anti-hardcollinear quark. The momentum distribution P qg,hard (s qg , z) is related to the square of the B1 matching coefficient (note that one factor of 1/z cancels when computing the matrix element squared for qφ * → qg).
As a first attempt one may naively apply RG evolution to the B1 matching coefficient. Keeping only the cusp part of its anomalous dimension (that is diagonal in Lorentz and spinor indices as well as with respect to the momentum fraction) Γ cusp B1,non−singular = α s π T 1 · T 0 ln µ 2 zQ 2 + T 2 · T 0 ln µ 2 zQ 2 , (D.1) and performing a d-dimensional RG evolution analogously as described in Section 3.1.2 yields precisely the exponential factors involving T 1 · T 0 and T 2 · T 0 in (2.22) (for z → 0, i.e. settingz → 1), but misses the contribution involving T 1 · T 2 . The reason is that the one-loop anomalous dimension for the B1 operator with generic momentum fractions of the anti-hardcollinear quark and gluon [16] contains contributions that would diverge in four dimensions, when convoluted with a matching coefficient that goes like 1/z. To avoid the problem of divergent convolutions, we retain a d-dimensional description, and define a singular B1 current analogous to [24], that absorbs the 1/z factor This operator essentially agrees with the one considered in Section 4. However, in the present discussion, we do not consider a further refactorization. Also note that, in contrast to the singular B1 current considered in [24], the inverse derivative acts on the quark building block instead of the gluon building block. For the latter case, the matching coefficient of the singular B1 current was linked to the leading power coefficient by reparameterization invariance, such that its anomalous dimension coincides with the one of the corresponding LP current. This relation does not exist in the present case.
In order to find the anomalous dimension of J (0) B1 , we compute its off-shell regulated one-loop matrix element in an external state with momentum fraction z of the antihardcollinear quark. We are interested in the double-pole part in the limit z → 0, when counting factors of z as order one. Apart from the expected cusp part in accordance with (D.1), one finds an additional piece involving a factor α s T 1 · T 2 (z − − 1)/z. When expanding for → 0 this gives a factor 1/z × ln z that cannot be interpreted as a renormalization of the singular current J (0) B1 due to the additional factor of ln z. Instead, we may interpret this result as an operator mixing, requiring us to introduce an additional singular operator. At higher orders we expect contributions of the form 1/z × ln n z and a d-dimensional z-dependence of the form z −1−n . This prompts us to consider a tower of singular operators, defined in d dimensions as for z → 0 have virtuality Q 2 or zQ 2 . We therefore make the ansatz (D.6) Inserting this ansatz along with (D.4) into (D.5) yields c n = T 2 · T 0 + T 1 · T 2 = −C A and d n = T 1 · T 0 − T 1 · T 2 = C A − C F .
Since the building blocks in J (n) B1 are all evaluated at the same space-time position, the current operator itself cannot depend on z, and therefore the same applies to the Wilson coefficients. Nevertheless, when evaluated in the matrix element, J (n) B1 tree ∝ 1/z 1+n , where z is the momentum fraction of the external anti-hardcollinear quark. Therefore, the term involving the coefficient d n has to be interpreted as an operator mixing J B1,tree = κ. In the previous equation, we can therefore set j = n and drop the sum over j.
We are interested in the matrix element for qφ * → qg, with leading poles arising from hard loops only, which is given by the hard matching coefficients multiplied with the tree-level SCET matrix element, Inserting this result into (2.9) precisely yields the exponentiation conjecture (2.22).
E Relation between DIS at large x and event shapes in the two-jet limit In this Appendix we discuss the relation between NLP contributions to DIS for large x and the thrust distribution in e + e − → γ * (Q) → jets [15]. In particular, we consider the power expansion in the two-jet limit τ = 1 − T → 0, where T is the thrust event-shape variable, such that τ plays the role of 1 − x (or 1/N in Mellin space) in DIS. The leading logarithmic corrections to the differential cross section at NLP have the form [18]  Hard loops contribute a factor α s × (µ 2 /Q 2 ) , (anti-)hardcollinear loops α s × (µ 2 /Q 2 τ ) and soft loops α s × (µ 2 /Q 2 τ 2 ) . At NLP at least one (anti-)hardcollinear or soft loop is required, such that the expansion starts at j = 1. Compared to DIS, the virtualities are composed of only two independent scales instead of three, and therefore the coefficients c  that are all linearly independent. This means only one of the 2n coefficients c (n) j is free. As noticed in [18], the finite part of (E.2) is completely determined by a single coefficient,  Figure 7: SCET representation of the content of (E.8) for the thrust distribution in e + e − → γ * (Q) → jets at NLP as τ → 0. Wilson lines are set to 1.
the LL contributions to the NLP thrust distribution from this result, which coincides with the "soft quark Sudakov" factor given in [15].
At this point one may wonder why, despite of the similarities, the LL resummed off-diagonal DGLAP kernel (3.69) obtained from the DIS process is considerably more complex than the thrust distribution. To understand this difference, it is useful to separately consider contributions with an anti-hardcollinear qq pair (denoted by IIc) and those with a soft quark or antiquark (denoted by IIs). The tentative SCET interpretation given in [15] suggests that IIc is represented by diagrams involving a B1 current operator with anti-hardcollinear qq building blocks, and IIs by diagrams with an insertion of a time-ordered product operator involving the LP current and L (1) ξq (see Figure 7). This motivates the following ansatz for the (partial) factorization of hard, (anti-)hardcollinear and soft loop contributions to IIs and IIc, given the result for c (n)