Resummation of non-global logarithms and the BFKL equation

We consider a `color density matrix' in gauge theory. We argue that it systematically resums large logarithms originating from wide-angle soft radiation, sometimes referred to as non-global logarithms, to all logarithmic orders. We calculate its anomalous dimension at leading- and next-to-leading order. Combined with a conformal transformation known to relate this problem to shockwave scattering in the Regge limit, this is used to rederive the next-to-leading order Balitsky-Fadin-Kuraev-Lipatov equation (including its nonlinear generalization, the so-called Balitsky-JIMWLK equation), finding perfect agreement with the literature. Exponentiation of divergences to all logarithmic orders is demonstrated. The possibility of obtaining the evolution equation (and BFKL) to three-loop is discussed.


Introduction
Collimated sprays of particles, or jets, figure prominently in high-energy collider physics. This has led to a growing interest in the characterization of jet shapes and event shapes, with the goal to extract as much information as possible about underlying hard scattering events. The pencil-like nature of jets implies that one often encounters disparate angular and energy scales. These lead to large logarithms in theoretical calculations, whose resummation is necessary to obtain controlled, precise predictions. Theoretically, in analytic studies these large logarithms are often the only terms which one may hope to predict in an amplitude or cross section at higher orders in perturbation theory, and thus could potentially help reveal new structures. Both of these reasons make them especially important.
Thanks to developments spanning many years, resummation for most observables of interest is now possible. In the case of so-called global observables, which involve complete ('global') integrals over final state phase spaces, a critical ingredient the exponentiation of infrared and collinear divergences [1][2][3][4][5][6]. This predicts in a quantitative way the logarithms left after the cancelation of infrared and collinear divergences, guaranteed on general grounds by the Kinoshita-Lee-Neuenberg (KLN) theorem [7,8]. There exists however non-global observables, for which phase space cuts lead to soft radiation not being integrated over all angles ('not globally'), which are considerably more difficult to resum [9,10].
The aim of this paper is to propose a comprehensive theory of non-global logarithms, to all logarithmic orders and finite N c . This theory will be closely related to the BFKL theory, which controls another class of difficult logarithms in the Regge limit (high-energy scattering at fixed momentum transfer) [11,12].
To set the stage we consider a generic weighted cross-section of the form σ = n dΠ n A Q→n (p 1 , . . . , p n ) 2 u({p i }) (1.1) where dΠ n is the phases space measure for n partons. The measurement function u({p i }) specifies the details of the measurement, including various vetoes etc. For suitable infraredand collinear-safe choices the cross-section will be finite order by order in perturbation theory. As a preliminary simplification (to avoid initial state radiation), in this paper we will assume that the initial state is a color-singlet state of mass Q, and assume massless final states. A time-tested strategy to resum large logarithms is to introduce intermediate, divergent, matrix elements, which are to be renormalized at a suitable factorization scale. The template is Wilson's operator product expansion, which expresses the short-distance limit of a correlator in terms of short-distance OPE coefficients, anomalous dimensions, and long-distance matrix elements. The factorization scale µ, whose dependence is controlled by anomalous dimensions, cancels between the OPE coefficients and matrix elements and provides a handle on large logarithms. Our main proposal is that the pertinent operator for resumming non-global logarithms is the color density matrix : (1.2) We call it a density matrix because it is linear in both the amplitude and its complex conjugate. It is a functional of a continuous field of unitary matrices U ab (θ), which depend on a twodimensional angle and live in the adjoint representation of the gauge group. Pictorially, U , shown in fig. 1, is a (local) color rotation between the matrix element and its conjugate. A closely related construction has been used to describe parton showers at finite N c [13]. The physical motivation for eq. (1.2) is that the information carried by σ[U ] is clearly necessary to fully characterize the distribution soft gluons. Since soft radiation can be triggered by any other colored parton with a higher energy, keeping track of color charges in every direction, as σ[U ] does, indeed seems unavoidable. The information in σ[U ] is also intuitively sufficient: due to coherence effects, soft gluons are affected by the color charge carried by harder partons but generally not by other details.
Contrary to the original weighted cross-section, the density matrix σ[U ] is infrared divergent. We propose, and will demonstrate, that these infrared divergences exponentiate in terms of a well-defined anomalous dimension. This supports our claim that the information in σ[U ] is sufficient. After cancelling these divergences (see eq. (2.7)), the 'renormalized' density matrix then depends on a factorization scale as The anomalous dimension or Hamiltonian K assumes the form of a functional differential operator. Its one-loop expression, computed in eq. (2.14), repdoduces earlier formulas derived in the literature to deal with non-global logarithms [10,14].

Structure of the resummation
For concreteness, we briefly sketch how the formalism would be applied in a complete calculation of a specific non-global event shape, the (cumulative) hemisphere mass function, at finite N c . The hemisphere mass function is the distribution of the invariant masses in the left-and right-hemispheres of a decaying color-singlet state of mass Q (for example, a Z-boson). In the di-jet limit m L , m R Q, the observable simplifies to (1.4) Here θ L/R (i) = θ(∓p z i ) project onto respective hemispheres. Logarithms of X/Q and Y /Q are global and can be resummed using standard techniques (see ref. [15] and references therein). When X Y Q, R receives additional logarithms log(Y /X) due to the phase space cut between the two hemispheres. These are the non-global logarithms.
· · · U anbn (θ n ) * Figure 1. Color density matrix. For each colored final state, an independent color rotation is applied between the amplitude and its complex conjugate.
To resum them we take Y as the hard scale and X as the soft scale. At the hard scale we drop the X-dependent step function, and define a density matrix involving only Y : (1.5) This is of the form of eq. (1.2). Infrared divergences introduced by the U matrices exponentiate and renormalizing at the scale µ = Y (see eq. (2.7)) removes large infrared logarithms. Concretely, in perturbation theory, σ ren is polynomial in U 's and can be viewed as a bookkeeping device encoding the orientations of radiated particles. It starts with the dijet term (1.6) with n,n null vectors along the positive and negative z directions. To resum logarithms log(Y /X) we now use the RG equation (1.3) to run σ ren down to the scale X, where we deal with the infrared part of the measurement. In the leading-log approximation, the θ(X − . . .) factor in the observable effectively removes left-hemisphere radiation between the scales X and Y (except within a cone of small radius R ∼ X/Q centered around the left jet). The IR measurement can thus be phrased in terms of an averaging The step function θ R (i) = θ R (i) + θ −p z i /p 0 i + cos R allows real radiation inside the right hemisphere and left cone. It is important that the averaging depends only on angles, since at this stage information about energies is lost.
In principle both the UV and IR measurements (1.5), (1.8) are to be computed order by order in perturbation theory. Resummation is considered successful if theses series contain no large logarithms. Since the Hamiltonian K is only meant to resum soft logarithms between X and Y , the UV and IR measurement functions actually do contain further logarithms of Q and R, but these should be resummed using other, conventional, means. Comparison of this procedure with the leading-log prescriptions of refs. [9,10,16] is discussed in section 2. In keeping with the logic of factorization, in this paper we will concentrate on the (universal) evolution K and leave calculations of (process-dependent) measurement functions to future work.
A remarkable fact about K is that it is also (essentially) the Balitsky-Fadin-Kuraev-Lipatov (BFKL) Hamiltonian, that is, the boost operator of the theory in the high-energy limit. The same Hamiltonian K thus simultaneously governs non-global logarithms and the Regge limit. This was observed mathematically from the one-loop expressions in refs. [10,14]. A general explanation is available, however, using a conformal transformation, which extends to higher loop orders [17]. One thus anticipate the difference in QCD to be at most proportional to the β-function.
Since this correspondence will play an prominent role it is helpful to include a rough explanation here. High-energy forward scattering (for example the elastic pp → pp amplitude) amounts to taking an instantaneous snapshot of a hadron's wavefunction, so pictorially it measures the amplitude for a virtual shower to form inside the hadron and then recombine. This is illustrated in fig. 2(a). This is also roughly what the density matrix σ Q→(··· ) [U ] of a decaying virtual hadron measures. Importantly, however, one measurement is instantaneous while the other takes place at infinity. To relate them requires the nontrivial conformal transformation of ref. [17].
In this correspondence with the Regge limit, the color rotations in σ[U ] implement the shockwave of the Balitsky-JIMWLK framework [18][19][20][21]. Here the 'shockwave' is inserted at infinity between the matrix element and its complex conjugate. This was our original motivation for defining eq. (1.2). (Mathematically similar considerations were used in refs. [22,23] to exploit the conformal symmetry of the BFKL equation.) The UV and IR measurements represent, respectively, the projectile and target impact factors.
The aim of this paper is to analyze the properties of the Hamiltonian K and to calculate it explicitly to the next-to-leading order. The lessons learned from this calculation will then lead to an immediate proof of exponentiation. As a cross-check of the calculation we will compare against results obtained in the context of Regge limit scattering. This paper is organized as follows. In section 2 we review known facts regarding the exponentiation of infrared divergences and factorization of soft emissions. We illustrate the formulas by giving the leading terms in perturbation theory for the various ingredients. We also verify that the procedure around eq. (1.8) reproduces the established resummation of non-global logarithms at leading-logarithm order. In section 3 we perform the two-loop calculation. A key finding will be the possibility to express all terms in K as finite integrals over well-defined, finite and gauge-invariant building blocks. The final result is recorded in subsection 3.6. In section 4 we compare our result for K against the two-loop BFKL equation. We will find perfect agreement in conformal theories, with, as expected, a relatively compact correction term proportional to the β-function in QCD. In section 5, using the lessons learned from the two-loop calculation we derive formal expressions for K at three-loop and beyond, and demonstrate exponentiation in general. Conclusions are in section 6. A technical appendix reports complete details of our evaluation of the real-virtual contributions at twoloop.

Conventions and review
To set our conventions we now review the exponentiation of infrared divergences and give explicit formulas for the relevant objects at one loop. We also discuss the resummation of non-global logarithms at leading-log.

Soft factorization
As described in refs. [4][5][6], the exponentiation of infrared and collinear divergences is controlled by a soft anomalous dimension: For a gentle introduction we refer to ref. [24]. The infrared-renormalized amplitude H n is also called the hard function and is finite (in this paper we use only ultraviolet-renormalized amplitudes). It depends on the choice of a factorization scale µ, It is important to note that since Γ n acts as a matrix in the space of color structures, the path-ordering symbol cannot be omitted.
We work in D = 4−2 dimensions and the coupling constant depends on scale through in a theory with n F flavors of Dirac fermions and n S complex scalars (in QCD C A = 3 and T F = 1 2 ). The solution is The integral in (2.1) converges and produces the desired 1/ poles provided that is negative enough that the coupling vanishes in the infrared.
In the literature Γ n is often written as being -independent, which defines minimal subtraction schemes. We keep the more general notation since it will be useful to consider non-minimal schemes. As long as Γ n remain finite as → 0, different schemes are related by finite renormalizations of H n .
Amplitudes with m soft gluons are also known to factorize in a simple way [6,25], up to powers of k provided that all of {k 1 , . . ., k m } are softer than other scales in M . We have only indicated the color and Lorentz indices of the soft gluons (to be contracted with polarization vectors µ i i ). Since H n is the same as in eq. (2.1), this formula states that soft gluons can be 'tacked onto' an amplitude without recomputing it.
Similarly to Γ n , the soft currents S m are matrices in the space of color structures. According to eq. (2.2) they are finite and have factorization scale dependence Our main proposal is that the color density matrix admits a similar factorization,

Leading-order expressions
The tree-level emission of one soft gluon is controlled by Weinberg's well-known soft current: where R a i is the operator which inserts a color generator on leg i and β µ i = (1, v i ) µ is a null vector proportional to p i . These obey [R a j , R b k ] = if abc δ jk R c j and our normalizations are such that Tr[T a T b ] = 1 2 δ ab and Tr [1] = N c . The soft anomalous dimension at one-loop is [4] Γ (1) where the collinear anomalous dimensions are γ (1) q = −3C F for quarks. We loop-expand using the uniform notation: To find K at one loop it suffices to compute σ[U ] to that accuracy and compare the divergence with eq. (2.7). The real emission contribution to σ[U ] has an infrared divergence when an additional gluon is emitted at a wide angle, as shown in fig. 3(a). It is given as the square of the soft current (2.8): ij;0 is the (infrared) pole from the energy integral, . Here the sums run over the U -matrices present in σ[U ] (which at finite order in perturbation theory is a polynomial) and we use the abbreviation U ab k = U ab (β k ). The notation R a i here is slightly different, but compatible with, that used in eq. (2.8): R a i is the operator which replaces U i with U i T a . Similarly L a i , representing the color charge in the complex conjugate amplitude, replaces U i with T a U i . These obey: The virtual corrections ( fig. 3(b)) generate products of the type LL and RR with no extra U . An important constraint is that σ ren [U ab = δ ab ] must be evolution-invariant, since this correspond to the total cross-section which is finite by the KLN theorem. Equivalently K must vanish when U is the identity field. This unambiguously determines the LL and RR terms. Using the identities which in particular yield L i = R i when U ab i = δ ab , the (unique) solution is easily seen to be 2 (2.14) This gives the complete scale dependence of the density matrix σ ren [U ], including non-planar effects (and therefore, by the expected factorization, any non-global logarithm at leading-log).
We review a few known facts about this equation.
• Taking the 't Hooft planar limit N c → ∞ with λ = g 2 N c fixed, eq. (2.14) becomes for the dipole U ij = 1 Nc Tr U i U † j : Using simple color identities this reduces to a closed nonlinear equation: 3 This is the Banfi-Marchesini-Smye (BMS) equation governing non-global logarithms in the planar limit [10]. 4 Let us be more precise. As stated in the introduction, the functional RG equation (1.3) is to be integrated from the UV to the IR, starting from e.g. the dijet initial condition σ[U ] = U nn (1.6). In the IR one performs the average (1.8). In the planar limit, the averaging reduces to evaluating the functional at one point, σ[U ij = θ R (i)θ R (j)], so the procedure is equivalent to evolving the argument of the functional from the IR to the UV, e.g. the function U ij . This is how the relevance of eq. (2.16) arises.
would also satisfy the KLN theorem and preserve the reality of σ provided that its coefficient is imaginary. The imaginary part of the explicit expression (2.9) however shows that fij ∝ iπ is constant and thus cancels out using color conservation. 3 We have used: . 4 In addition, compared with ref. [26] which deals with the hemisphere function, one needs to set U here ; the step-functions factors are stable under evolution. At leading-log collinear divergences exponentiate independently so the R term in θ R in eq. (1.8) does not interfere with the non-global part.
• Away from the planar limit, eq. (2.14) coincides with the generalization of the BMS formula derived in ref. [14]. Again, as in the BMS case, the two forms differ only by multiplication of the U matrices by step functions, which commute with the evolution. The averaging procedure (1.8), done in the infrared, is as in refs. [14].
• The double-sum notation in eq. (2.14) is most natural in a perturbative context where σ[U ] is a polynomial in the U 's. Since the evolution increases the number of U 's, for solution it can be better to view K as a functional differential operator acting on σ[U ]. This is achieved by the following simple substitutions (done after normal-ordering all L, R's to the right of U 's) [27,28]: These L a i and R a i obey the same commutation relations as those defined previously, and in fact after substitution into eq. (2.14) one finds the same action on any polynomial σ[U ]. This reveals eq. (2.14) as a functional second-order differential equation of the Fokker-Planck type, whose solution can be importance-sampled via lattice Monte-Carlo techniques [27,28]. For studies of 1/N c effects in the context of non-global logarithms, see [16,29].
• Also well-studied is the weak-field regime where all matrices are close to the identity.
Following ref. [30] and references therein, we write U j = e igT a W a j and expand (2.17) in powers of W . This can be streamlined using the Baker-Campbell-Hausdorff formula, which gives Plugging this into the one-loop Hamiltonian yields after a small bit of algebra (only the first two terms contribute) up to nonlinear terms of the form δK ∼ g 4 W 4 δ 2 δW 2 . This is one form of the one-loop BFKL equation and its ('BJKP') multi-Reggeon generalization [11,12], valid for colorsinglet states (see ref. [30] and references therein). It acts on functionals σ[W ] where W a is identified as the Reggeized gluon field. This identification will play a useful role later in this paper. • Finally, we did not prove in this subsection that divergences do exponentiate according to eq. (2.7). We simply read off the exponent from a one-loop fixed-order calculation. Proofs to leading-logarithm accuracy are in refs. [10,14] and an all-order demonstration is given in section 5.

Evolution equation to next-to-leading order
We now present a calculation of K to the next-to-leading order, by matching two-loop infrared divergences in σ[U ] against eq. (2.7). The computation will be phrased exclusively in terms of convergent integrals over building blocks with a clear physical interpretation (renormalized soft currents), which will shed light on the exponentiation mechanism. We perform the computation in a general gauge theory, although at intermediate steps we only write formulas for color-adjoint matter. The reader not interested in the technical details can skip directly to the final result in subsection 3.6.

Building blocks: soft currents
A natural building block is the tree-level amplitude for emitting two soft gluons. It can be written naturally as a sum of disconnected and connected contributions: follows directly from the Feynman graphs shown in fig. 4(a) [25]. To optimize the notation all color generators are implicitly symmetrized: which is relevant when i = j. This notational convention (borrowed from ref. [31]) is why the connected part is proportional to f abc . Figure 5. Second building block: one-loop soft current.
To familiarize ourselves with the notation we review the transverseness check: We need to use color conservation in the form of the identity i R a i = 0. Since this holds when i R i is inserted to the right of an operator product, the implicit symmetrization in eq. (3.1) produces commutators. For example the divergence of the first sum is This is easily seen to cancel the second term in the parenthesis in (3.3), up to a β i -independent term which itself cancels due to i R c i = 0, proving transverseness. Pairs of soft fermions or soft scalars can also be emitted ( fig. 4(b)). For notational simplicity we carry out all intermediate steps in a theory with n adj Weyl color-adjoint Weyl fermions and n adj s real adjoint scalars (the final result will be trivial to generalize). Then: The second natural building block is the next-to-leading order soft gluon amplitude S 1 . We take it from the literature and quote the result in (A.1); representative graphs are shown in fig. 5.
Equation (2.7) gives the next-to-leading order kernel as the divergent part of the following combination: . (3.6) We will now see that this can be expressed in terms of soft currents.

Double-real emission
We begin with the terms in the NLO kernel which involve two wide-angle soft partons, and thus generate two additional U factors. The double-real contribution to σ (2) is by definition (suppressing color indices) (3.7) (The integrals have compact support due to the momentum-conserving δ-function in A, and we do not show a factor dΠ n u({p i }) associated with the underlying hard event.) The trick to evaluate (3.6) is to find compatible integral representations for K (1) and σ (1)ren . For K (1) we already have eq. (2.12) and subtracting it from σ (1) leaves simply The essential point here is that the matrix element factorizes in the soft region, |A n+1 | 2 (aβ 0 ) → |S n A n | 2 , so that subtracting K (1) 2 σ (0) is equivalent to removing the integration region a < µ (to all orders in ). Invoking factorization similarly, eq. (3.6) can be re-written as: Here F (a, b) denotes the integrand in (3.7). This formula is also exact in . One can see that the second integral is finite and the first integral has no subdivergences. (Except from collinear regions, which are dealt with in the next subsection.) After scaling a → ab in the first integral to extract the pole, one thus just get: This is the desired formula, which expresses the double-real contribution to K (2) as a convergent integral over tree-level soft currents. The integrand measures the extent to which two soft emissions are not independent from each other. Using the explicit expressions (3.1) the formula yields two nontrivial color structures fig. 6). These multiply angular functions We note the absence of a fully disconnected (Abelian) color structure: since its squared amplitude is proportional to 1/a 2 it disappeared before integration. The matter contribution in the last parenthesis is, in full: The a-integrals are elementary (and convergent!) and after some straightforward algebra starting from (3.2) yield For K (2) ij;00 we have used symmetry in (i ↔ j) to simplify. The expression is especially compact in N = 4 SYM (the first term). The second and third terms represent, respectively, an (adjoint) N = 1 chiral multiplet and a scalar. The rational structures are such that all potential divergences associated with the β i , β j and β k regions cancel. There remains a divergence as β 0 → β 0 , proportional to the gluon collinear anomalous dimension γ (1) g ∝ b 0 , which will be canceled shortly.
At this point we could stop: using simple and not-so-simple physical considerations one could determine the full result using only what we have so far. We find it instructive, however, to continue with the explicit computation.

Single-real emission
We now turn to terms with one radiated parton; since these contain only one U field at a wide angle, these will combine with and cancel the collinear divergences in eq. (3.12).
A good starting point is to express the single-real contribution in terms of the hard function (2.1): In the soft region we can replace the amplitude by a soft current times H n . It is useful to run the soft current to its natural scale, µ = a, using eq. (2.6). The energy integral then becomes, formally to all orders in perturbation theory, (3.14) Comparing against the factorization formula (2.7), since Γ n K virtual we see that the second integral represents a finite correction to the finite coefficient σ ren . The a-integrand on the first line, on the other hand, is nicely identified as the following shift to the exponent: This gives the single-real contribution to K, formally to all loop orders. This generalizes in the simplest conceivable way the leading-order result: one simply evaluates the loop-corrected soft current with energy set equal to the renormalization scale µ. The 'bar' onS accounts for a collinear subtraction, to be defined shortly (as well as for an O( ) mismatch between Γ n and K (1) virtual ). Because the soft current is used at its natural scale, the series for (3.15) contains no large logarithms and the b 0 term in eq. (3.6) is automatically accounted for.
We now deal with collinear divergences. We do so by subtracting suitable splitting functions. First, in the preceding subsection, we replace eq. (3.8) by (3.16) The next-to-last term removes the a → 0 limit of everything to its left. The formula differs from eq. (3.8) only by the splitting functions, which are added in such a way as to introduce no new soft divergences. The function Split i (p 1 ; k i ) (representing the amplitude for particle i to split into two with momenta aβ 0 and k i − aβ 0 , symmetrized between the two), is required to have the same integrand-level collinear singularity as |A n+1 | 2 when β 0 k i , ensuring convergence. This is guaranteed to exist by the factorization of amplitudes in collinear limits, and an explicit expression is given in (A.3).
We stress that the subtraction is not written as an integral over the phase space of (n + 1) particles with the original total momentum, since a and β 0 do not enter A n . This would probably be very inconvenient for numerical integration, which is a limitation compared to more general methods, for example, dipole subtraction [32]. However this is simpler and will suffice for our purposes.
To avoid changing σ (1)ren one needs to make a compensating subtraction in the virtual contribution: (3.17) This differs from the naive subtraction of the virtual part of K (1) (the R a i R a j term), which is essentially Γ n up to O( ) terms, by terms involving the splitting amplitude and compensating those in eq. (3.16). The iπ term is such that, as verified in appendix,H n is finite. The barred hard functions can thus viewed as simply the hard functions in a different scheme.
What is important is that the same subtractions can be applied to all expressions in the preceding subsection. For example, in the double-real term in eq. (3.7) one simply subtracts Split g (aβ 0 , bβ 0 ) 2 + i Split i (aβ 0 , k i ) 2 A n+1 (bβ 0 ) 2 from the integrand. In the Γ (1) terms one subtracts only the soft limit. In this way all two-particle-collinear divergences are removed from the preceding subsection, at the only cost of adding a piece to eq. (3.10): (3.18) This makes it collinear-safe when added to the integrand. The argument of the splitting function is such that most energetic particle of the pair has energy µ (which we scaled out). Regarding the virtual corrections, the only effect of (3.17) is to put the bar on S in (3.15).
Both eqs. (3.15), (3.18) are explicitly finite and to be evaluated directly in four dimensions. Their evaluation is conceptually straightforward and detailed in appendix A. It turns out that the result could have been anticipated using (not so trivial) physical considerations, so here we concentrate on explaining these considerations.
The most critical consideration is gluon Reggeization, or, more broadly, the connection with BFKL. As mentioned in introduction, K is the BFKL Hamiltonian in disguise (up to βfunction terms). Interactions between Reggeized gluons are constrained by physical principles such as Hermiticity of the boost operator and signature conservation U ab → U ba , which are not self-evident from the perspective of non-global logarithms.
We consider the weak-field regime where U = e igT a W a with the 'Reggeized gluon' field W small. Linearizing the Hamiltonian yields Reggeon-number conserving terms given in eq. (2.19), as well as 2 → 4 transitions (between states with with different powers of W ) at order g 4 , and so on. Hermiticity of the boost operator (with respect to the specific inner product given by the scattering amplitude of left-and right-Wilson lines) then predicts 4 → 2 transitions at the same order, whose existence is indeed well-known [33]. These are the terms which close the so-called Pomeron loop. Now when reverting to the current power-counting which treats (U − 1) as O(1), instead of O(g), these 4 → 2 transition become a three-loop effect (see ref. [30] and references therein). Signature forbids 3 → 2 transitions. Hence the remarkable statement that K must be triangular at one-and two-loop [30]: Mathematically, this formula (just with the L = 1 case) can be seen as equivalent to gluon Reggeization, since it ensures that sectors with different powers of W 's can be diagonalized independently at one loop. One then expects the Reggeized gluon (W field) to provide a good degree of freedom upon which to organize the perturbative spectrum of K to any order (as usually happens in degenerate perturbation theory after degeneracies are lifted at one-loop). For our immediate purposes, eq. (3.19) constrains two-loop color structures. One easily sees that no double-real color structure satisfies it by itself: for example, using (2.18), the first line of eq. (3.11) linearizes to give terms which replace three Reggeons W i W j W k by a single one W 0 . Cancelling this term uniquely fixes the range-three part of the single-real contribution (to the form in eq. (A.10)). From other terms one constrains the double-virtual and range-two kernels. In this way, using in addition that double-real terms are signatureeven, we find that the two-loop Hamiltonian can be parametrized by at most three angular functions: (3.20) The first one is shown in fig. 7. Since K ijk;00 and K ij;00 have already been determined from double-real emissions, effectively eq. (3.19) predicts all virtual corrections, up to a term proportional to the leading-order structure (the last line). The physical interpretation is that gluon Reggeization entails nontrivial real-virtual connections, which was indeed the original observation [11,12].
In appendix A the prediction (3.20) is compared with the direct evaluation of singlereal terms eqs. (3.15), (3.18). It turns out that there is a subtle loophole in the above argument: in the non-global log context, U ab → U ba is not symmetry but only need to send the Hamiltonian to its complex conjugate. Thus signature is not conserved. The ansatz fails, by a single signature-odd term: (3.21) The origin of this term is simple: the imaginary part of the one-loop soft current (A.1). Its physical significance will be discussed shortly. The explicit computation in appendix also yields the yet-undetermined signature-even function: (3.22) This contains only the two-loop cusp anomalous dimension [34] (normalized to γ (1) K = 1) and the one-loop β-function. 5 Physically, the b 0 term is fully dictated by the collinear anomaly discussed subsection 3.5, while the cusp anomalous dimension term can be checked to provide the correct Sudakov double logarithms in the limit of a narrow jet cone.
How to explain the real-virtual connections (3.20) from the perspective of non-global logarithms? Perhaps one could use the Feynman tree theorem [36]. This is a way of computing loops by putting one (or more) propagator per loop on-shell. Indeed, putting a gluon on-shell in fig. 5 one can recognize diagrams of fig. 4, so at least schematically this seems to work. The tree theorem was streamlined and generalized to higher loops, with at least partial success, in refs. [37,38]; it would be interesting to see its implications in detail.

Double-virtual terms
Let us now make sure that the ansatz (3.20) is not missing any virtual corrections. A priori these could involve two color structures: The coefficients are constrained by the KLN theorem (vanishing for U ab = δ ab ) and by Lorentz invariance, but without signature, unfortunately this does not fix them uniquely. (Below we 5 The appendix uses the so-called dimensional reduction scheme. In conventional dimensional regularization (CDR), more commonly used in QCD, a simple coupling redefinition [35] gives: 64C A 9 → 67C A 9 .
give an example of a signature-odd function satisfying both constraints.) We resort to explicit computation. The two-loop soft anomalous dimension is known to take the 'dipole' form [5,39,40] Γ (2) n = −2γ This gives the divergence of the amplitude after subtracting the square of Γ (1) n . Since we are instead subtracting K (1) , we need again to switch to the collinear-subtracted barred scheme (A.5):Γ We omit 'collinear terms' which depend on only one leg at a time, since these are trivial to fix using the KLN theorem. Concentrating on the terms which have nontrivial color structures and which are not so easily fixed, the calculation of eq. (3.25) is rather straightforward and detailed in appendix A. The outcome confirms that no additional terms besides (3.21) need to be added to the Ansatz (3.20).
We can now interpret this term. First we observe that it can be mostly removed by a finite scheme transformation. Namely, if we set where the MS density matrix is the minimally subtracted one we have been working with so far, then the two-loop Hamiltonian in MS gets shifted by a commutator with K (1) and a β-function term. It is easy to check that the commutator term precisely cancels (3.21). The β-function term then replaces it by This combination is Lorentz-invariant in an interesting way: under a rescaling of β i , the jsum become telescopic and simplifies to (C i − C i ) = 0. This also satisfies the KLN theorem, being zero when L = R. The existence of this structure is the only reason we needed to use the explicit formula (3.24) to get the virtual corrections, otherwise the KLN theorem and Lorentz invariance would have sufficed. It violates the triangular structure (3.19) but since it is proportional to the β-function this does not contradict the BFKL-based argument leading to it. It also has a simple and suggestive physical interpretation: effectively it replaces the spacelike couplings in the one-loop evolution, by timelike counterparts: With hindsight, had we used timelike couplings in the one-loop evolution, we would never have had to write down eqs. (3.21), (3.26) nor (3.27). We will nonetheless continue to use the (more conventional) spacelike coupling.

Lorentz invariance and (lack of ) collinear anomaly
We have assembled all ingredients of the kernel, but we notice that the angular functions are not Lorentz-covariant: the arguments of the logarithms (3.12) are not homogeneous in β 0 , β 0 (and thus depend on the frame choice implicit in the normalization β µ i = (1, v i )). This may be surprising given that dimensional regularization preserves Lorentz invariance.
The simple explanation is that we did not write the one-loop evolution in a D-dimensional covariant form. What would constitute a Lorentz-invariant version is instead: ij;0 + . . . , (3.29) which differs at order by the amount δ (1) The integrand is homogeneous in all of β i , β j , β 0 and one may check that under a Lorentz transformation the Jacobian factor precisely cancels the change in the parenthesis. (The factor 4 is for future convenience.) An O( ) shift to an anomalous dimension, as usual, is equivalent to a finite renormalization of σ ren , e.g. a scheme transformation. The density matrix in the 'Lorentz' scheme (3.29) is related to the MS one used so far, or better the MS scheme just defined, as: (3.30) This shifts K (2) by a commutator [K (1) , δ (1) ] as well as a β-function term.
This transformation is only well-defined because it contains both real and virtual terms: the middle integral in eq. (3.29) would otherwise be un-regulated even for = 0. This clash between Lorentz covariance and collinear divergences reflects the (now called) collinear anomaly of refs. [39,40]. Here, the anomaly cancels between real and virtual terms.
To achieve manifest Lorentz covariance we must still manipulate the expressions by using color conservation to add terms independent of some of the β i , being careful with commutators as below eq. (3.3). Collecting these commutators is tedious but fortunately the task can be easily automated on a computer. We find (as it should) that the color structures in eq. (3.20) are preserved under these operations (see also ref. [41]). Only the angular functions change: The functions E ij;0 and F ij;0 are arbitrary, with E ij;00 = −E ji;0 0 . The formula (3.33) below arises for E ij;00 = α ij α 0i α 00 α 0 j log α 0 i α 0 j α 0i α 0j and F ik;00 = α ik 2α 0i α 00 α 0 k log α ik α 00 α 0k α 0 i . (With these choices the integrand on the last line vanishes.)

Final result for the evolution equation
We record our final result for the two-loop Hamiltonian in the 'Lorentz' scheme (superscript ), which combines eqs. (3.20)- (3.22) with the finite renormalizations (3.26) and (3.30). For convenience we repeat the color structures, switching to the integro-differential notation (2.17): Here and i = d 2 Ω i , the color rotations L and R being differential operators defined in eq. (2.17). All products of L a i 's and R a i 's are implicitly symmetrized and normal-ordered to the right of U 0 , U 0 . The third term is simply the one-loop result (2.14) times the cusp anomalous dimension (3.22). The angular functions are: This is the complete result in N = 4 SYM. In a general gauge theory with n F flavors of Dirac fermions and n S complex scalars in the representation R, there additional contributions from matter loops, also obtained in eq. (3.12). Upon restoring group theory factors corresponding to representation R, in accordance with the square of fig. 4(b), these can be written:

Comparison with BFKL and conformal transformation
As mentioned in the introduction, the same Hamiltonian K governs the Regge limit. Hence the reader familiar with the literature on the Regge limit, in particular the Balitsky-JIMWLK equation, will have recognized several equations by this point. Let us now discuss the connection in detail. Physically, as sketched in the introduction, the connection originates from the existence of a conformal transformation which interchanges the x + = 0 light-sheet and future (null) infinity. This interchanges the target residing at x + = 0 with the color rotations in the definition (1.2) of σ[U ]. It is given explicitly as [17,42,43] where µ is a reference scale. This maps the Minkowski metric ds 2 = −2dx + dx − +dx 2 ⊥ to a multiple of itself, as one may verify. Points approaching the BFKL target, x + → 0, are mapped to infinity along the null direction y µ ∝ (β 0 , β ⊥ , β z ) = ( ). In this way the transverse plane of the BFKL problem is mapped stereographically onto the two-sphere at infinity of the non-global log problem. Provided that the conformal transformation (4.1) preserves the Lagrangian, this map predicts that K should go into the BFKL Hamiltonian upon substituting [17] 6 : We now verify this equivalence directly, beginning with the case of N = 4 SYM where conformal symmetry is unbroken. In the general case we will find a discrepancy proportional to the β-function, as anticipated.

Comparison in N = 4 SYM
It is instructive to consider a special case: we act with K (2) on a dipole U 12 = Tr[U 1 U † 2 ]. The form (3.33) is particularly convenient for this since K (2) ijk;00 vanishes when i = k or j = k. The only terms in the first line are thus K (2) 112;00 and K (2) 221;00 . Furthermore the remaining lines simplify since K In this way all two-loop color structures in the dipole case are expressed in terms of a single angular function. To evaluate the color factors we recall that while L a 1 U 1 = T a U 1 , in the 6 Here we use a normalization β 0 = 1 which differs from that adopted elsewhere in the present paper and in ref. [17]. This has not effect in Lorentz-covariant expressions such as eqs. antifundamental one has that L a 2 U † 2 = −U † 2 T a (this ensures that (L 1 +L 2 )U 12 = 0). Writing if abc T c = [T a , T b ] and collecting terms one easily finds This formula is identical to the conformal form of the two-loop evolution obtained by Balitsky and Chirilli, eq. (6) of ref. [44], with αs 4π K (1) + α 2 s 16π 2 K (2) here = − d dη there , as expected. 7 In the planar limit eq. (4.4) can be reduced to a closed nonlinear equation for a dipole function; we refer to the literature for further details.
Rapidity evolution for general products of Wilson lines in the Balitsky-JIMWLK framework has been obtained recently [41,[45][46][47], extending earlier results for two [44,48] and three Wilson lines [49,50]. Given the mutual agreement between these works, here we only compare directly against the conformal form of ref. [41]. Since the stereographic projection identifies the SL(2, C) conformal symmetry of the transverse plane with Lorentz symmetry of the two-sphere, this should match with the Lorentz scheme here.
The comparison is in fact straightforward: the range-three kernel K 3,2 shown in eq. (5.12) of ref. [41] is literally the first four terms of our K (2) ijk;00 . The remaining two terms in K (2) ijk;00 arise from the telescopic term F in eq. (3.31) hence do not affect the range-three part. (These terms are helpful to manifest the convergence at β 0 → β 0 .) Furthermore, the integral representations for K 3,1 and K 3,0 in ref. [41] reproduce the real-virtual pattern embodied in the first line of eq. (3.32). This demonstrates the agreement of range-three interactions. Combined with the agreement in the dipole case, this establishes the complete equivalence of eq. (3.32) with ref. [41] (and thus, by extension, refs. [45,46,49]).
In principle, upon linearizing around U = 1, one also expects complete agreement with the interactions between Reggeized gluons obtained in the BFKL approach. For two reggeons the agreement was demonstrated at the level of eigenvalues [44,48,51,52]. For three reggeons, it was noted in ref. [49] that a scheme transformation appeared to be lacking in order to match with ref. [53]. This issue should be clarified further. Here we simply note that there is a natural candidate: the next-to-leading order inner product (correlator of Wilson lines) [54,55]. In the BFKL approach the inner product does not receive loop corrections (the transverse part of the Reggeon propagator remains 1/p 2 ), so only after this effect is removed by a scheme transformation, should agreement be expected. 7 Since subtraction terms are not written in exactly the same, there is an apparent discrepancy: 12;00 − K However the integral vanishes, proving the agreement. This is easily shown by noting that being absolutely convergent, the integral defines a Lorentz-covariant function with the same homogeneity in β0, β1, β2 as the integrand, hence must a constant times α 12 α 01 α 02 . The constant vanishes by antisymmetry in (β1 ↔ β2).
It is interesting to compare technical aspects of the calculations. The tree-level soft current (3.1) is reminiscent of the light-cone gauge amplitudes in eq. (43) of ref [48]. The subtraction of subdivergences in eq. (3.10) is similar to the + prescription derived in refs. [48,56]. The transformation to the 'Lorentz scheme' (3.30) is identical to that leading to the 'conformal basis' in refs. [41,44]. As a significant technical simplification, however, we saved the Fourier transform step. Also the reliance on standard building blocks made it possible to benefit from results in the literature, namely the soft currents and collinear splitting functions.

Comparison including running coupling
Having demonstrated the agreement in N = 4 SYM, let us now compare the fermion and scalar loop contributions to the Balitsky-JIMWLK and non-global logarithm Hamiltonians, e.g. the terms proportional to n F and n S in eq. (3.34). Performing the comparison with refs. [44,57] we find that the two Hamiltonians agree for the most part, except for the following discrepancy (setting z ij = z i − z j ): where as before µ is the MS renormalization scale. In particular, the difference is proportional to the first β-function coefficient, as predicted [17]. The origin of the discrepancy (4.5) is clear: the inversion y + → 1/µ 2 y + in (4.1), which relates the BFKL and non-global log Hamiltonians, is only an isometry up to the Weyl rescaling ds 2 y → (µy + ) −2 ds 2 y . This is not a symmetry in a non-conformal theory. Physically, BFKL and non-global logarithms describe infinitely fast and infinitely slow measurements of an object's wavefunction, which would not normally be expected to be connected without conformal symmetry.
For future reference, we note that a general theory deals with Weyl transformations in non-conformal theories (see for example [58]). The essential feature is that, starting from the BFKL side and performing the conformal transformation (4.1), one ends up with a coordinatedependent coupling constant: In other words, the BFKL Hamiltonian in QCD in principle controls non-global logs in QCD but in an imagined setup with a coordinate-dependent coupling. Contrary to real QCD, in this setup a narrow jet never hadronizes: the increasing coupling due to the growing size of a jet, is compensated by its falloff at large y + . Thus effectively the coupling is set by the angular size. This reflects that angles map to distances in the BFKL problem. We will not pursue eq. (4.6) further here, but in any case it is clear that to all orders in perturbation theory the difference between the BFKL and non-global Hamiltonians must be proportional to the β-function (up to finite scheme transformations).

Higher loops and exponentiation
It is instructive to extend the general analysis of section 3 to higher loops. We will (mostly) ignore collinear subdivergences here, concentrating on the soft divergences. We can organize terms according to the number m of wide-angle partons (U matrices) added to an underlying n-jet event. Our starting point is the known exponentiation of virtual corrections (2.1), which gives the m = 0 case: For the next case of one wide-angle gluon, a formula was derived in eq. (3.14). We reproduce it here, in abbreviated notation, omitting U matrices, the angular integration, daa 1−2 energy measure, and absolute value squared on the matrix elements: The colons instruct us to normal-orders terms according to their renormalization scale (largest argument to the right). As in subsection 3.3, the first integral is identified as a shift K 1 (µ) = −S 1 (µ; µ) to the exponent. The remaining (finite) terms then define the hard coefficient σ ren ≤1 (µ). Moving on to two real emissions, we follow eq. (3.9) and write the cross-section as independent emissions plus an additional piece: We have introduced the 'connected' squared soft current defined as (all factors being evaluated at the same renormalization scale) (5.4) (In the present abbreviated notation we recall that each factor is a squared soft amplitude, S i ≡ |S i | 2 .) Again the first integral in eq. (5.3) is identified as a shift to the exponent, which generalizes eq. (3.10) to include virtual loop effects to all orders. The (finite) remainder then defines σ ren ≤2 . Using this method it is straightforward to extend the calculation to more radiated particles. For three radiated particles, for example, after pulling out Pe − µ 0 (K 0 +K 1 +K 2 ) we find again that particles with energy > µ decouple from divergences, no subdivergences, and the following shift to the anomalous dimension: The absence of subdivergences (finiteness of K 3 as → 0) is manifest from the fact that the expression involves only connected squared amplitudes, which vanish near the boundaries a → 0 or a, b → 0. This itself is a consequence of factorization, or more precisely eq. (2.5) in the form lim a 1 ,...,a k a k+1 ,...,an S(a 1 , . . . , a n ; µ) = S(a 1 , . . . , a k ; µ)S(a k+1 , . . . , a n ; µ) .

(5.7)
It is now clear how to generalize the pattern to higher orders. In fact a simple guess appears to give the anomalous dimension K to all orders: The exponential factor has a simple physical interpretation as an 'exclusion time' effect, and we recall that a's are the energies of real radiated particles. We have verified explicitly (with the help of a computer) that exponentiating K using eq. (2.7) reproduces all contributions where up to at least 9 real particles have energy below µ, so we believe that the formula is correct to all orders. Equation (5.8) is one of the main results of this paper. It expresses, to all loop orders, the Hamiltonian governing non-global logarithms as a convergent integral over finite, well-defined building blocks, generalizing the eqs. (3.10) and (3.15) used in the two-loop computation. The building blocks are the squares of the infrared-renormalized soft currents (which include virtual loops to all orders), defined in eq. (2.5). Only the 0 part of the renormalized currents are needed, in agreement with ref. [59].
Since the exponent K is manifestly finite as → 0 (being expressed in terms of connected squared soft currents), the formula also demonstrates to all loops that infrared divergences exponentiate according to eq. (2.7). The physical inputs were the known exponentiation (2.1) of virtual corrections, and the factorization of successive real emissions (5.7); eq. (2.7) is seen to be a purely combinatorial output.
To fully prove eq. (2.7) one should address the issue of collinear subdivergences, omitted in the present discussion. Physically we expect these to cancel, since σ[U ] is collinear-safe. In subsection 3.3 this was made manifest by defining collinear-subtracted real and virtual contributions, such that their sum was unaffected by the subtraction. We have no reason to think that this couldn't be achieved at higher orders as well, following the method of ref. [25].
We should mention that eq. (5.6) gives the evolution equation in a non-minimal scheme: the two-loop exponent K (2) 2 in eq. (5.5) depends on through a factor a −2 and thus differs from the MS result of this paper by terms proportional to . These are interpretable as a renormalization which shifts K (3) by a finite commutator, giving, instead of eq. (5.6): Discrepancies in K 1 at O( ) add other commutator terms. All these are unrelated to a further finite renormalization needed to make Lorentz covariance manifest. As in eq. (3.29) it can be fully predicted by upgrading the two-loop result (3.33) to a D-dimensional covariant form. Although these finite renormalizations become combinatorially very complicated at higher loop orders, being finite they cannot interfere with the statement (2.7) of exponentiation.
Finally, we list the ingredients which enter eq. (5.8) at three-loop: • The tree-level soft current for three soft gluons S 3 , at one-loop for two gluons S 2 , and two-loop for one gluon S • The next-to-leading order 1 → 2 and tree-level 1 → 3 collinear splitting amplitudes [60].
The two-loop soft current and three-loop soft anomalous dimension are presently known for two hard partons [61][62][63]. Unfortunately this will not suffice for non-global logarithms nor BFKL in general, since each radiated gluon counts like a hard one from the point of view of softer radiation. 8 However, for dipole evolution in the planar limit, everything needed is known.

Conclusion
In this paper we considered a 'color density matrix' which aims to characterize soft radiation in gauge theory. We argued that it should resum large logarithms arising in the presence of wide-angle phase space cutoffs, so-called non-global logarithms, to all orders in logarithms and 1/N c . We proved the all-order exponentiation of infrared divergences for this object in terms of an anomalous dimension K (see eq. (2.7)), constructed formally in eq. (5.8), modulo one technical assumption regarding collinear subdivergences. We explicitly computed K to twoloop (eqs. (3.32)-(3.34)) and performed a number of checks on this result. We also stressed the equality between K and the BFKL Hamiltonian, which allows our results to be viewed as an independent derivation of the next-to-leading order BFKL Hamiltonian, obtained here directly in a novel, compact form.
The procedure to calculate a cross-section receiving non-global logarithms was sketched in the introduction. One distinguishes infrared and ultraviolet scales, which are to be connected by evolving using K. At both ends lie finite quantities: an 'IR measurement' which contains details of the experimental definition of a 'soft' particle, and corresponding vetoes; an 'UV measurement' which depends on the initial state and possible vetoes imposing hard jets in the final state. The logic of factorization being that their calculations are independent of each other, we focused in this paper on the (universal) evolution K. Study of the (expectedly) finite, but process-dependent, measurement functions is left to future work, as well as phenomenological studies.
Mathematically, K is an integro-differential operator acting on functionals σ[U ] of a twodimensional field of unitary matrices U (θ) (e.g. SU (3) matrices in QCD), with θ an angle in the detector. This means that K cannot be diagonalized explicitly. Although it is a quite complicated object, it is a useful starting point for further approximations. These include, as reviewed in section 2, numerical Monte-Carlo techniques at finite N c , reduction to an ordinary integro-differential equation at large N c , or linearizationà la BFKL around U = 1. We hope that one of the compact forms obtained in this paper will prove convenient for a next-to-leading order numerical implementation.
For application to hadron colliders it will be important to go beyond the limitation of an initial color-singlet object, as done in this paper, and allow for initial state radiation. This could lead to additional (super-leading? [64,65]) effects related to subtle color-dependent phases in collinear limits [66,67].
The formalism does not distinguish between global and non-global logarithms, but it is easy to see how it simplifies in the case of global observables. For example, when radiation is excluded everywhere but inside narrow cones, the IR averaging procedure sets U = 0 outside these cones which effectively shuts down the real terms in the evolution. It is then dominated by virtual effects, as is usual for global observables. It is only for observables sensitive to details of wide-angle radiation that the complications of the formalism kick in. It would be interesting to connect the present approach with that of ref. [68], which deals with recursive infrared and collinear safe event shapes ('rIRC').
There has been recent activity regarding formal aspects of measurements at infinity, in connection for example with the Bondi, van der Burg, Metzner and Sachs (BMS) symmetry [69,70]. The density matrix construction could be useful in this context. From a theoretical perspective, the Hamiltonian K connects, in a unified way, the following gauge-theory concepts: the cusp anomalous dimension (governing global logarithms); the KLN theorem (cancelation of collinear and infrared divergences); the factorization of soft radiation; the BFKL equation.
The equivalence with BFKL, verified explicitly in section 4, is a consequence of conformal symmetry [17] and is an equality up to β-function terms (fixed by comparatively simpler matter loops (4.5)). The basic physical intuition is summarized in fig. 2. Remarkably, properties manifest in one context are not necessarily so in the other.
For example, one fundamental assumption in both the BFKL and Balitsky-JIMWLK frameworks is that transverse integrals should be saturated by transverse scales ∼ t s, ensuring that rapidity logarithms (log s) arise only from longitudinal integrations [18,71]. While reasonable it is unclear how one would prove this directly beyond the current state of the art, e.g. next-to-leading log. The correspondence with non-global logarithms immediately implies it to all orders, since it amounts to the amply understood cancelation of collinear divergences. The non-global logarithm formulation may also be computationally advantageous, as discussed in sections 4.1 and 5.
In the other direction, the phenomenon of gluon Reggeization suggested a compact way to write the evolution equation (see eq. (3.20)), which manifests a connection between real and virtual effects. Intriguingly, we found that these relations could perhaps also be explained by the Feynman tree theorem. It would be very interesting to see if either of these approaches generalizes to higher loop orders.
Finally, we mention that the simplest non-global logarithms to resum in this framework (beyond the planar limit) involve situations close to the linear regime U ≈ 1, where the linearized equation has lowest eigenvalue the well-known Pomeron intercept − 4αsC A log 2 π . Naively this regime might correspond to multiplicity-type measurements, e.g. counting away jet charged tracks as a function of angle and an energy cutoff. Perhaps this or some other observables could provide an indirect experimental handle on the BFKL Pomeron.
whereH implements the subtraction in eq. (3.17) of collinear splitting functions. Since the splitting functions for all but the radiated gluon cancel in the commutator, we will only need the gluon splitting function Split g (aβ 0 , bβ 0 ) 2 = C A (b − a) −2 ab(b −2 ) 2 x(1−x)α 00 + (n adj Weyl − 4) α 00 +x(1−x)(n adj s −2n adj Weyl +2)f , (A.3) where x = a/b. The prefactor has a kinematical origin and accounts for the change in the measure b 1−2 db. The computation of such functions is standard [32]. In the x-dependence one can recognize various DGLAP kernels P g→(··· ) (x), as expected. We use the dimensional reduction scheme so the parenthesis does not depend on . (Regarding color factors we recall that we show intermediate formulas only in a theory with color-adjoint matter.) The scalar contribution to the splitting function is polarization-dependent and for us the most useful information will be its dot product against β µ i β ν j , divided by β i ·β j : this is what enters eq. (3.15). This is given by 2α ij α 2 00 α 0 i α 0 j + convergent or telescopic . (A.4) The first form is obtained directly from the Feynman rules and makes manifest that the dependence on β i , β j is consistent with factorization. We will prefer the second form, which provides a closer match with eq. (3.12) and also yields a simpler integrated expression. It differs by terms which are either convergent or vanish using color conservation. Computing the integral in (3.17) we then obtain 2 log 2 (x) − π 2 6 . The sum runs over gluons to stress that we haven't computed the other cases, and the cusp anomalous dimension is in eq. (3.22).
The commutator then easily yields We stress that only the O( 0 ) terms of S 1 were needed to obtain this. It is noteworthy that the −π 2 C A /6 from the original soft function, the −π 2 C A /3 from the scheme transformation, and the −C A log(e −iπ ) 2 /2 from the phase of the logarithm have nicely canceled to leave the cusp anomalous dimension.
Substituting into eq. (3.15) the soft factor produces two color structures ij;0 U aa 0 (L a i R a j + R a i L a j ) , (A.7) which multiply the angular functions ij;0 = −4S These can be evaluated explicitly using (A.6). The remaining linear-in-U contribution, the subtraction (3.18), is simply n adj Weyl −4 α 00 + 1 6 n adj s −2n adj Weyl +2 f .
(A.10) This agrees precisely with eq. (A.6a), up to the iπ term recorded in eq. (3.21). For the other structure G (2) ij;0 actual − ansatz = α ij α 0i α 0j 2γ (2) ij;0 + eq. (A.9) + C A d 2 Ω 0 4π K (2) ij;00 = α ij α 0i α 0j γ ij;0 , (A.11) which fixes K ij;0 as recorded in the main text. Finally we check the double-virtual terms. To get the prediction from the ansatz (3.20) we need to integrate (A.10). The L 2 (α 0k ) terms look scary, but they cancel out trivially because one needs only the total antisymmetrization of G (2) ijk;00 modulo terms which do not depend on all three labels simultaneously. The integral is still a bit nontrivial but we could simplify its antisymmetric part using integration-by-parts. We omit the details and quote only the rather simple result for the G (2) ijk;0 contribution, ijk;0 = 8if abc i,j,k R a i R b j R c k log(α ij )L 2 (α jk ) . (A.12) Finally the other term in the ansatz is (dropping terms depending on one particle at a time) K log(α ij ) − b 0 (L 2 α ij ) + log 4 log α ij .
The preceding two equations are easily verified to be in perfect agreement with the commutator (3.25), proving that the ansatz does not miss any double-virtual term. As a final comment, we note that the L 2 function and most log 2's have a simple origin: the scheme change (3.30). For example ij;0 = − 1 2 + π 2 3 +L 2 (α ij )+log 4 log(2α ij )+ O( ). With hindsight, we could have saved ourselves much algebra by switching from MS to the Lorentz-covariant scheme in the first step, which would have prevented L 2 from ever appearing.