The LPM effect in sequential bremsstrahlung: 4-gluon vertices

The splitting processes of bremsstrahlung and pair production in a medium are coherent over large distances in the very high energy limit, which leads to a suppression known as the Landau-Pomeranchuk-Migdal (LPM) effect. In this paper, we continue study of the case when the coherence lengths of two consecutive splitting processes overlap (which is important for understanding corrections to standard treatments of the LPM effect in QCD), avoiding soft-gluon approximations. In particular, this paper completes the calculation of the rate for real double gluon bremsstrahlung from an initial gluon with various simplifying assumptions (thick media; $\hat q$ approximation; and large $N_c$) by now including processes involving 4-gluon vertices.


IV. Summary of Formula
References 34

I. INTRODUCTION AND RESULT
When passing through matter, high energy particles lose energy by showering, via the splitting processes of hard bremsstrahlung and pair production. At very high energy, the quantum mechanical duration of each splitting process, known as the formation time, exceeds the mean free time for collisions with the medium, leading to a significant reduction in the splitting rate known as the Landau-Pomeranchuk-Migdal (LPM) effect [1,2]. A longstanding problem in field theory has been to understand how to implement this effect in cases where the formation times of two consecutive splittings overlap.
Let x and y be the longitudinal momentum fractions of two consecutive bremsstrahlung gauge bosons. In the limit y x 1, the problem of overlapping formation times has been analyzed at leading logarithm order in refs. [3][4][5] in the context of energy loss of high-momentum partons traversing a QCD medium (such as a quark-gluon plasma). We subsequently developed and implemented field theory formalism needed for the more general case where x and y are arbitrary [6][7][8]. In this paper, we finally complete the calculation of the effect of overlapping formation times on the differential rate dΓ/dx dy for double bremsstrahlung from an initial high-energy gluon (with various simplifying assumptions detailed below). The missing element, presented in this paper, is the inclusion of processes involving the 4-gluon vertex.

A. What we compute (and what we do not)
The preceding work [6][7][8] computed all of the interference contributions involving only 3-gluon vertices, which are presented by the diagrams of figs. 1 and 2, which we respectively refer to as "crossed" and "sequential" diagrams. The upper (blue) part of each diagram depicts a contribution to the amplitude and the lower (red) part depicts a contribution to the conjugate amplitude. Only the high energy particles are shown; their (many) interactions with the medium are implicit. (See ref. [6] for more details.) In this paper, we will evaluate the remaining contributions, which are the diagrams involving 4-point gluon vertices, shown in figs. 3 and 4. (We will see later, by a symmetry argument, that theȳ4x contribution in fig. 3 vanishes.) Once we find the correct normalization of the 4-gluon vertex in our formalism, the evaluation of these diagrams will be a relatively straightforward application of techniques developed in previous papers [6,7].
As discussed in the preceding work [6,7], it is possible to set up the formalism in a quite general way that would require both highly non-trivial numerics and a non-trivial treatment of color dynamics to implement, but one can proceed much further analytically by making a few additional approximations. Though the methods we discuss in this paper can be applied more generally, we will follow refs. [6,7] when it comes to explicit calculations, by making the following approximations.
• We will assume that the medium is static, uniform and infinite (which in physical terms means approximately uniform over the formation time and corresponding formation length).
• We take the large-N c limit of QCD to simplify the color dynamics. The subset of interference contributions to double splitting previously evaluated in ref. [6], the "crossed" diagrams, depicted as amplitudes (blue) sewn together with conjugate amplitudes (red). The dashed lines are colored according to whether they were first emitted in the amplitude or conjugate amplitude. To simplify the drawing, all particles, including bremsstrahlung gluons, are indicated by straight or curved lines. The long-dashed and short-dashed lines are the daughters with momentum fractions x and y respectively. The naming of the diagrams indicates the time order in which emissions occur in the amplitude and conjugate amplitude. For instance, xȳyx means first (i) x emission in the amplitude, then (ii) y emission in the conjugate amplitude, then (iii) y emission in the amplitude, and then (iv) x emission in the conjugate amplitude. The interference contributions evaluated in ref. [7]: the "sequential" diagrams.
• We make the multiple-scattering approximation to interactions with the medium, appropriate for very high energies and also known as the harmonic oscillator orq approximation.
In this paper, we focus on completing the calculation of the rate for producing two real bremsstrahlung gluons (g → ggg). We defer to another time the related calculation of the change in the single-bremsstrahlung rate due to virtual corrections. (In the special limiting case y x 1, the sum of these real and virtual processes has been worked out in the context of leading parton average energy loss in refs. [3][4][5] and is related to anomalous scaling of the effective medium parameterq with energy.) Finally, as discussed in ref. [7], the double bremsstrahlung rate dΓ/dx dy by itself includes processes where two single-bremsstrahlung processes are separated by times large compared to their corresponding formation times. In the idealization of an infinite, uniform medium, this causes dΓ/dx dy to be formally infinite. But what we actually want to know is the correction to double bremsstrahlung due to overlapping formation times,

B. Preview of Results
Numerical results for the total ∆ dΓ/dx dy are shown in fig. 5, which includes all contributions from figs. 1-4. In ref. [7], it was shown that the contribution from crossed and sequential diagrams (figs. 1 and 2) scale as 1/xy 3/2 for y x 1, and for this reason it 3 ) of the triangular region, π 2 xy 3/2 ∆ dΓ/dx dy = 1.12 has been convenient to show the result in fig. 5 in units of In comparison to the similar plot in ref. [7], not much has changed: the inclusion of the 4-gluon vertex contributions of figs. 3 and 4 in this paper have had only a small effect on the total. We show the contributions of figs. 3 and 4 individually in figs. 6 and 7. The first of these is numerically negligible compared to the total of fig. 5. (We do not know any qualitative explanation for why it should be so small. 1 ) The second ( fig. 7) is only a very modest contribution to the total.
None of the new, 4-gluon vertex contributions to ∆ dΓ/dx dy grow as quickly as (1.2) for y x 1. We find that they instead scale as 1/y 1/2 in this limit.

C. Outline and Referencing
In the next section, we show how to calculate the 4ȳx interference diagram of fig. 3, which will be our canonical example in this paper. Section III then explains how to obtain all of the other diagrams involving 4-gluon vertices. A summary of final formulas is given in section IV, and we offer our brief conclusion in section V. Along the way, some details and cross-checks are relegated to appendices. In particular, for the sake of completeness, we have collected in Appendix D the formulas for crossed and sequential diagrams from refs. [6][7][8], so that this paper contains, in one place, all the formulas necessary for implementing the complete calculation of ∆ dΓ/dx dy. Also, the integral formula we will derive for ∆ dΓ/dx dy is a complicated expression that is painstaking to implement. In Appendix E, we provide, as an alternative, a relatively simple analytic formula that has been fitted to approximate fig. 5 very well.
In this paper, we will occasionally (in footnotes and appendices) use the author acronym AI as shorthand for Arnold and Iqbal [6] so that, for example, we may write "AI (5.2)" to refer to eq. (5.2) of ref. [6].

A. Starting point
We start with the 4ȳx diagram shown in fig. 8. In the notation of ref. [6], this is |i δH|Bx Bx, tx|Bȳ, tȳ Bȳ|i δH|Cȳ 34 , Cȳ 12 × Cȳ 34 , Cȳ 12 , tȳ|C (4) 34 , C 34 , C 12 , t (4) and Bx, tx|Bȳ, tȳ represent, respectively, the (i) 4-particle evolution in the initial time interval t (4) < t < tȳ in the figure, and (ii) 3-particle evolution of the system in the final interval tȳ < t < tx. Because of the symmetries of the problem, these have been reduced to effective (i) 2-particle and (ii) 1-particle problems in non-Hermitian twodimensional quantum mechanics, described by effective transverse coordinates (i) (C 34 , C 12 ) and (ii) B. δH represents the piece of the fundamental QCD Hamiltonian associated with the splitting vertices for the high-energy particles (as opposed to the interactions of those high-energy particle with the medium, or the interaction of the medium with itself). So C  and | given in ref. [6].
where the b i are the various transverse positions of the individual particles and x i are their longitudinal momentum fractions (defined as negative for particles in the conjugate amplitude). B ≡ B 12 = B 23 = B 31 is defined similarly for the case of three particles.
The appropriately normalized results for the 3-gluon vertices were found in ref. [6]: 2 FIG. 9: The diagrammatic rules for splittings linking (via either −i δH or +i δH) the state | to |B (top rule) or |B to |C 34 , C 12 or permutation thereof (bottom rule).
and may refer, in different contexts, to ± the 3-particle B, or one of the 4-particle C uv , or to some mixture. However, note that B ji = B kj = B ik in the top rule, which can be used to always write expressions in terms of 3-particle B and/or 4-particle C ij 's. The blue arrows on the particle line indicate color flow of color representation R. (In the case of R=A, appropriate to g → gg splitting, the direction of the color flow does not matter.) b l , a l , x l , and h l indicate the transverse position, color index, longitudinal momentum, and helicity of each particle. The black arrows give the convention for the flow of x l and h l in the statement of the rule, and these values should be negated if they are instead defined by flow in the opposite direction. In the bottom rule, color and helicity indices and their contractions are not explicitly shown for the spectators because they are trivially contracted. Conservation of longitudinal momentum means x i + x j + x k = 0 (top) and additionally x m = x r and x n = x s (bottom). and where T color are color generators and the P i→jk are proportional to square roots of helicitydependent, vacuum Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) splitting functions. These were translated into the more general diagrammatic rules of fig. 9, which apply to B| −i δH| , C ij , C kl | −i δH|B , | −i δH|B and B| −i δH|C ij , C kl , as well as similar matrix elements · · · | +i δH| · · · relevant to evolution in the conjugate amplitude.
[The bar over δH here and in formulas like (2.1) is just a notation for emphasizing that δH is operating on particles in the conjugate amplitude in those cases.] In appendix B, we apply the same methodology to evaluating the 4-gluon vertex we need FIG. 10: The diagrammatic rule for a 4-gluon vertex without additional spectators (e.g. as in the 4ȳx,ȳx4 and 44 diagrams of figs. 3 and 4 but not theȳ4x diagram). The rule is symmetric under permutations of the four gluon lines, though this is not obvious from the way it is written. The upper and lower signs of ∓ apply when the 4-gluon interaction is in the amplitude and conjugate amplitude, respectively.
above and find where a i and h i are the color index and helicity ± associated with particle i. The first few lines of (2.4) can be recognized as having the structure of the usual relativistic Feynman rule for a 4-gluon vertex; the last line has the normalization factors appropriate for the way we normalize the transverse position variables C ij and the state |C 34 , C 12 [6]. The two delta functions in the last line, δ (2) , enforce that the four particles all be in the same place (b 1 =b 2 =b 3 =b 4 ) at the time of the 4-point interaction.
(Generically, two δ-functions may seem insufficient to enforce this, but in our problem the positions b i are already implicitly constrained by the additional condition See section III of ref. [6].) A diagrammatic version of (2.4) is given in fig. 10. Like the top graph of fig. 9, this particular rule only applies when there are no other particle lines present at that time. So, it can be used for 4ȳx andȳx4 in fig. 3 but not forȳ4x. The 4-point vertex requires different normalization factors in the latter case, which we give in Appendix B 2, but that detail is unimportant becauseȳ4x turns out to vanish.
The sign ∓ in fig. 10 simply reflects the fact that in the amplitude the vertex corresponds to matrix elements of −i δH whereas in the conjugate amplitude it corresponds to matrix elements of +i δH (which we denote as +i δH).  [7], and following the general convention of refs. [6,7] for discussing time-ordered large-N c planar diagrams). The top edge AB of the shaded region is to be identified with the bottom edge AB. (b) explicitly shows the corresponding color flow for an example of medium background field correlations (black) that gives a planar diagram (and so leading-order in 1/N c ). In our notation, this interference contribution could be referred to as either 4ȳx 1 or 4ȳz 2 .

B. Color routings
The diagram for 4ȳx shown in fig. 8 is technically symmetric under the permutation x ↔ z, where z ≡ 1−x−y. However, in this paper we will work in the large-N c limit in order to simplify the color dynamics of 4-particle evolution. In this limit, there are two distinct color routings of the 4ȳx diagram which are not individually y ↔ z symmetric, just like the situation for the xyxȳ diagram discussed in ref. [7]. We show these two large-N c color routings in figs. 11 and 12, which we will refer to as 4ȳx 1 and 4ȳx 2 respectively. Note that the two routings are related by x ↔ z, and so we could also call them 4ȳz 2 and 4ȳz 1 respectively.
Like the situation for the xyxȳ diagram discussed in ref. [7], the distinguishing difference between the calculation of the two color routings is the assignment of the longitudinal momentum fractions x i for the 4-particle part of the evolution, which occurs here for t (4) < t < tȳ. Going around the cylinder depicted in fig. 11, the first routing 4ȳx 1 has whereas the second routing 4ȳx 2 of fig. 12 has applied to splitting of particles in the amplitude to the same rule applied to splitting of particles in the conjugate amplitude, the value of the B ji will automatically negate. Since ∇δ (2) (B ji ) is an odd function of B ji , this automatically takes care of the sign difference between −i δH and +i δH.  fig. 11 but showing the other distinct color routing of 4ȳx. In our notation, this interference contribution could be referred to as either 4ȳx 2 or 4ȳz 1 .
Note that the ordering of the x i does not matter until we take the large-N c limit and decide that the 4-particle propagator Cȳ 34 , Cȳ 12 , tȳ|C (4) 34 , C 12 , t (4) will henceforth represent only a single color routing. That is why the x i assignment of fig. 8, before we discussed large-N c , could represent the entire contribution of 4ȳx, but in our convention after we implement the large-N c limit for discussion of the 4-particle propagator, the same assignment (2.6) now represents only a single color routing ( fig. 12).
We will focus on the second routing (2.6) just because the assignment x i =x i is identical to the one used for the canonical diagram analyzed in ref. [6]. We can obtain the other routing via x ↔ z: dI dx dy The details of extracting what pieces of the color and helicity factors given by fig. 10 correspond to which of the two large-N c color routings are a bit untidy. One can either (i) figure out how to split up the factors in fig. 10 or else (ii) switch to large-N c Feynman rules. Here we'll take the first option, as we found it the least confusing way to keep track of overall normalization factors.
If we label the gluon lines as (i, x, y, z) for the initial, x, y, and z bosons, then the color and helicity factors given by fig. 10 for the 4-point vertex are The large-N c routing 4ȳx 2 of fig. 12 corresponds to the first term above plus half of the second term, while the rest of (2.8) corresponds to the routing of fig. 11. The advantage of the large-N c limit is that it then allows us to do a naive color contraction of the vertices in fig. 11a and 12a for each routing. 4 In fig. 12a, (2.9) is contracted with adjoint color factors associated with the two 3-point vertices and averaged over initial color a i , giving overall. Using the rules for 3-gluon vertices, the general expression (2.1) then becomes 34 , C 12 , t (4) for the routing 4ȳx 2 . (See appendix A for details on the overall sign.)

C. Helicity Sums
For the helicity sums, we need hx,hy,hz,h Note that we have found it convenient to include the |x 1x2x3x4 | −1/2 factor from (2.12) here.
By transverse parity invariance, we may average over the initial helicity. By transverse rotational invariance, the initial helicity average of (2.14) must be of the form for some function ζ(x, y). Taking the formulas for the splitting functions P from ref. [6], 5 we find   Now take the harmonic oscillator approximation. As reviewed in ref. [6], for 3-particle evolution this corresponds to treating B, t|B , t as evolution of a two-dimensional harmonic oscillator with a certain effective mass M and complex natural frequency Ω. In the case of the final 3-particle evolution tȳ < t < tx in figs. 8 and 12, these are [6] and Using a harmonic oscillator propagator gives 7 +∞ tȳ dtx ∇ Bx Bx, tx|Bȳ, tȳ which recasts (2.17) as where ∆t ≡ tȳ − t (4) . We now treat the 4-particle propagator Cȳ 34 , Cȳ 12 , ∆t|C (4) 34 , C (4) 12 , 0 just as in section V.C of ref. [6], except that here we have chosen to use the same basis (C 34 , C 12 ) in both the bra and the ket. The propagator is given by 34 , C where we have included on the left-hand side of (2.21) the additional factor (2.20) because that makes the definitions of the symbols X, Y , and Z more convenient for later use. Those symbols are then given by (2.23) Above, Ω ≡ Ω + Ω − . Formulas from [6] for the two 4-particle evolution frequencies Ω ± and the corresponding normal modes (C ± 34 , C ± 12 ) are collected in Appendix D 2. Using (2.21) in (2.20) gives (2.25) Our final result for the 4ȳx diagram is the above formula together with the corresponding version of (2.7), dΓ dx dy (2.26)

A. Theȳx4 diagram
Theȳx4 diagram is the third diagram of fig. 3. Instead of going through an explicit calculation, we can relate the answer for this diagram to the 4ȳx diagram computed in the last section, along the lines of how the xȳyx and xyȳx diagrams of fig. 1 were related in ref. [6].
The first thing to note is that all three diagrams shown explicitly in fig. 3 have the same factors of helicity contractions and DGLAP splitting functions associated with their vertices-these factors are unaffected by the time ordering of the 4-point vertex in the amplitude relative to the two vertices in the conjugate amplitude. As to the rest of the computation, note that the diagramsȳx4 and 4xȳ in fig. 3 look like mirror images of each other except for the identification of which gluon has which momentum fraction. For each color routing, we show one way of making this change of identification in fig. 13. There, when reflecting 4ȳx intoȳx4, we change for the first color routing, and for the second. Both cases can be summarized as We also need to appropriately change the mass M used for the 3-particle part of the evolution. As for similar diagram transformations in ref. [6], this will be taken care of automatically if we write this mass in terms of the 4-particle x i as in (2.18a): which, for example, gives M = x(1−y)(1−x−y)E (2.18a) for 3-particle evolution in the 4ȳx 2 case of (x 1 , The upshot is that we can convert the result for 4ȳx into a result forȳx4 by (i) making the change (3.5) to the 4-particle x i , (ii) always using the form (3.6) for the 3-particle evolution mass, and (iii) leaving ζ(x, y) unchanged. 8 For the sake of readers wary of the glibness of the above argument, we give a more straightforward derivation ofȳx4 2 in appendix C and verify that the result is the same.  There are many topologically-equivalent ways to draw the same diagram: we've chosen to draw theȳx4 diagrams above in a way that gives a straightforward pictorial correspondence to our rule (3.5) for going from the 4ȳx diagrams on the left to theȳx4 diagrams on the right.

B. Theȳ4x diagram
Now consider theȳ4x interference contribution, depicted by the second diagram in fig. 3. The starting point, analogous to (2.1), is |i δH|Bx Bx, tx|B (4) , t (4) × B (4) |i δH|B (4) B (4) , t (4) |Bȳ, tȳ Bȳ|i δH| . (3.7) We will not need to work out the explicit normalization of the 4-gluon vertex matrix element B (4) |i δH|B (4) (though we give it in Appendix B) because we will find that (3.7) is zero. The important point is that the helicity factors and splitting factors P are the same as they were for 4ȳx in section II C, and so, using fig. 9, dI dx dy ȳ4x ∝ ζδmn tȳ<t (4) <tx pol.
∇n Bx Bx, tx|B (4) , t (4) Bx=0=B (4) × ∇m Bȳ B (4) , t (4) |Bȳ, tȳ Bȳ=0=B (4) . (3.8) The reason that B (4) and B (4) are set to zero above is because in 3-particle evolution (analogous to the earlier statement about 4-particle evolution), the transverse positions b i in our problem are implicitly constrained by the condition (See section III of ref. [6].) One may use this constraint to show that there is but one relevant transverse degree of freedom for the three transverse positions in 3-particle evolution: 9 So, in our application, when any two of the three particles are coincident, then B = 0 and all three of the particles are necessarily coincident. But now we can see the result. The factors ∇n Bx Bx, tx|B (4) , t (4) Bx=0=B (4) and ∇m Bȳ B (4) , t (4) |Bȳ, tȳ Bȳ=0=B (4) (3.10) must both be zero by parity, and so theȳ4x contribution (3.8) vanishes.

C. The 44 diagram
The 44 diagram, shown in fig. 4, is formally given by |i δH|C (4) 34 , C 12 , t (4) |C 34 , C 12 , t (4) × C 34 , C 12 |−i δH| . The helicity and color factors associated with the 4-gluon matrix elements do not depend on the longitudinal momentum fractions (e.g. x and y) of the various gluons and so, when summed over polarizations and colors, give the exact same helicity/color factor for each of the three color routings of fig. 14. Each is therefore a third of the total helicity/color factor S we would get in a vacuum calculation, where we would not need to split the calculation into different color routings but could simply square and initial-state average the color/helicity factors (2.8) of the 4-point vertex: (where d A is the dimension of the adjoint representation). So each color routing has a corresponding factor of S/3 = 3C 2 A . We will focus on the second color routing 44 2 , which is convenient because it again corresponds to our canonical choice (2.6), (3.13) The corresponding contribution to (3.11) is 12 , t (4) |C (4) 34 , C 12 , ∆t|C (4) 34 , C 12 , 0 (3.15) and so dΓ dx We may then sum all the color routings by adding appropriate permutations:  fig. 7 verifies this is the case. 10 We also note in passing that we can evaluate (3.16) analytically in the limit that one of the final-state gluons in soft. For y x and z, the result for the total contribution of fig. 4  where [dΓ/dx dy] crossed and [∆ dΓ/dx dy] seq are given respectively in ref. [6,8] and ref. [7]. For completeness, we have summarized those formulas in Appendix D. The contributions new to this paper, involving one or more 4-gluon vertices, are summarized below.

A. Diagrams with one 4-gluon vertex
The diagrams of fig. 3 (including all permutations, large-N c color routings, and conjugates) give the following contribution to dΓ/dx dy: dΓ dx dy (4) = A (4) (x, y) + A (4) (1−x−y, y) + A (4) (x, 1−x−y) where A (4) (x, y) is the result for one color routing of 4ȳx +ȳ4x +ȳx4 plus conjugates. We will write this as where 10 One might think of checking that the total double bremsstrahlung rate dΓ/dx dy, which is also the medium average of the magnitude squared of something (the total amplitude for double bremsstrahlung), is also positive. However, as discussed in ref. [7], the total dΓ/dx dy is formally infinite in our calculation, and the physically relevant quantity is instead ∆ dΓ/dx dy defined by (1.1). The latter is a difference of two positive quantities and so can have either sign (as seen in fig. 5).
corresponds to (i) the 4ȳx 2 color routing of 4ȳx plus (ii) the related color routingȳx4 2 of yx4. ζ = ζ(x, y) is given by (2.16). Each of the terms in (4.4) is given by which is the integrand of (2.25). Here, the X, Y, Z are defined by (2.22) and (2.23), with As mentioned earlier, explicit formulas for the 4-particle evolution frequencies Ω ± in terms of (x 1 , x 2 , x 3 , x 4 ) are collected in Appendix D 2. Unlike for the crossed and sequential diagrams analyzed in refs. [6,7], it is unnecessary to explicitly subtract the vacuum contribution from D (4) . That's because the vacuum limit q → 0 (and so Ω ± → 0 and Ω f → 0) of (4.5) already vanishes.
Also unlike the crossed and sequential diagrams [6,7], there are no 1/∆t divergences associated with the individual diagrams of fig. 3, and so there are no "pole" term contributions that need to be included in A (4) above.

B. Diagrams with two 4-gluon vertices
The diagrams of fig. 4 give the following contribution: Again, there are no 1/∆t divergences associated with the diagrams here, and so there are no "pole" term contributions that need to be included in A (44) above.

V. CONCLUSION
We have now completed the calculation of the overlapping formation time correction to double bremsstrahlung for the process g → ggg of emitting two real bremsstrahlung gluons from an initial gluon. The size of interference terms involving 4-gluon vertices had to be computed (i) for completeness and (ii) to see how big they are. However, the conclusion we can take from the numerical results of figs. 5-7 is that their effect on the result is small and one would not go far wrong in ignoring them, at least insofar as ∆ dΓ/dx dy is concerned.
An important reason for calculating the overlapping formation time correction is to test whether it is large or small for realistic value of α s . It is already known that the corrections due to soft bremsstrahlung (y 1) are large due to large logarithms but that such soft corrections can be resummed into a running value ofq that depends on energy [3][4][5][9][10][11]. But what about the contribution from overlapping hard bremsstrahlung, which cannot be absorbed intoq? In the thick-medium approximation used here, these corrections are controlled by the value of α s at scales of order 11 Q ⊥ ∼ (qE) 1/4 . An answer concerning the size of these non-absorbable corrections will need to wait longer until we are in a position to calculate an infrared-safe physical quantity characterizing shower development, which will require (i) including the effects of virtual corrections to single bremsstrahlung and (ii) consistent factorization of the effects of soft bremsstrahlung intoq.

Acknowledgments
This work was supported, in part, by the U.S. Department of Energy under Grant No. DE-SC0007984.
Appendix A: More details on some formulas Eq. (2.12): The overall sign of this formula arises as follows, similar to the discussion of AI (4.16) in Appendix A of ref. [6]. Consider first the rule associated with the t = tȳ vertex in fig. 8 (remembering that the ordering of x i used in that figure was chosen to match the ordering of the large-N c color routing 4ȳx 2 of fig. 12). According to the rules of fig. 9, this vertex comes with a factor of (T a k R ) a j a i ∇δ (2) (B ji ), with lines (i, j, k) identified as in the figure. Using the cyclic permutation identity B ji = B kj = B ik noted in the caption, and comparing fig. 9 to the t = tȳ vertex in fig. 8, we can identify these factors . Similarly, the vertex at t = tx comes with a factor of (T a k R ) a j a i ∇δ (2) (B ik ) = (T ax A )ā az ∇δ (2) (C 34 ). Since we have identified C 34 with B in (2.12), the latter is (T ax A )ā az ∇δ (2) (B). The color factors (T ay A ) a iā (T ax A )ā az from these two vertices (and the signs that arise from them) have already been accounted for in (2.10), which has already been combined with the 4-gluon vertex factor (and its signs) in (2.11). We are left with the δ-function factors ∇δ (2) (C 21 ) ∇δ (2) (B). Since C 21 = −C 12 , these may be rewritten as −∇δ (2) which is the form used in (2.12), where both C 12 and B have been integrated by parts. This minus sign combines with the minus sign in (2.11) and the ∓ = − in fig. 10 to give the overall minus sign in (2.12). Eq. (3.18): In the limit that y is small compared to both x and z ≡ 1−x−y, the formulas for the 4-particle frequencies Ω ± collected in appendix D 2 satisfy, for the case x i =x i , The factor of csc(Ω + ∆t) in (3.16) means that the integrand is negligible unless Ω + ∆t 1, in which case Ω − ∆t 1. So the integral may be approximated as This approximation is the same for all three color routings. Correspondingly multiplying by 3, and then adding in the conjugate diagram44 by taking twice the real part, ∆t Ω y csc(Ω y ∆t).
As we do with all diagrams, we now subtract out the vacuum contributionq → 0 (i.e. Ω y → 0), leaving which gives (3.18). To derive the matrix element C 34 , C 12 |δH| , we will follow the method used for deriving other matrix elements in Appendix B of ref. [6]. We start in a description of states where we individually distinguish each high-energy particle, using the conventions of fig. 15a. First, the δH matrix element in the amplitude, written conventionally in terms of the individual particles in the Hilbert space H (as opposed to the Hilbert spaceH⊗H used to simultaneously describe particles in the amplitude and conjugate amplitude), is FIG. 15: The notation used in (a) appendix B 1 and (b) appendix B 2 to label different particles' states immediately before and after a 4-gluon vertex. The dashed connection in (a) indicates the fact that in this case the initial particles in the amplitude and conjugate amplitude represent the same particle (and so, for instance, b 1 = b 2 and, given our conventions, (B1) is the usual relativistic formula except for a few small differences. The factors of (2E i ) −1/2 = (2|x i |E) −1/2 for each particle above are included because we use non-relativistic rather than relativistic normalization for the states. We have written the rule in transverse b-space instead of transverse momentum space, so there are δ-functions requiring the points to be coincident at the vertex instead of a δ-function for overall transverse momentum conservation. We have assumed that the longitudinal momenta have already been chosen to satisfy longitudinal momentum conservation, and we have (just as in ref. [6]) chosen a normalization of our states where we implicitly drop the corresponding momentum-space δ(p 2z −p 2z −p 3z −p 4z ). Finally, we have used the fact that the initial state represents a single on-shell particle to link the color and helicity of particle 2 to that of 1 and thus, via fig. 15a, to particle 1. We have accordingly chosen to label the corresponding color and helicity indices in (B2) by 1 instead of by 2 . The convention used for the flow of helicity here is that of fig. 10. The δ ··· δ ··· terms in (B2) for helicity come from contracting the usual factors g µν g αβ in the Feynman rule for the 4-point vertex with normalized helicity polarizations µ (h) for each particle. The corresponding matrix element in the spaceH ⊗ H that includes the particle in the conjugate amplitude is Next we want to use the symmetry of the problem to project each state onto a subspace with two fewer degrees of freedom, as discussed in AI section III and AI Appendix B [6].
Using the notation of that reference, where it is understood that both the initial and final positions satisfy the constraint i x i b i = 0 and where V ⊥ is a formally infinite normalization given by Using (B1) and (B3), The initial state |b 1 , b 2 satisfies the constraint x 1 b 1 + x 2 b 2 = 0 with x 1 + x 2 = 0, and therefore b 12 = 0, giving Given the presence of the other two δ-functions, the first one can be rewritten as where the last equality uses Since as well, the substitution (B8) in (B7) gives by (B5). Because of the constraints (B9), the variables b 32 and b 42 are related to C 12 ≡ b 12 /(x 1 +x 2 ) and C 34 ≡ b 34 /(x 3 +x 4 ) by 12 and the Jacobean for the change of variables is ∂(b 32 , b 42 )/∂(C 12 , C 34 ) = [x 1 (x 3 +x 4 )] 2 . So (B10) can be rewritten as Changing normalization as in ref. [6], 13 then gives the matrix element (2.4) displayed in the main text.

B|δH|B
We do not need to figure out the correct normalization of the matrix element B|δH|B for this paper, but we do so here just for the sake of completeness. The corresponding diagrammatic rule we will find is shown in fig. 16.
Analogous to (B1), start with the amplitude matrix element using the labeling of fig. 15b. Here H is the same as (B2) except that the indices 1 and 4 are replaced by 2 and 3 . Including the particle in the conjugate amplitude, Projecting the number of degrees of freedom in each state from 3 to 1 as in ref. [6], Using the constraint x 1 +x 2 +x 3 = 0 and the primed version of the relationships (3.9) defining B, Given the other δ-functions, the middle one can be rewritten as From the constraint x 1 b 1 +x 2 b 2 +x 3 b 3 =0 and (B5), we then have where in the last line we've used x 3 = −(x 2 +x 1 ) and have noted that x 1 = x 1 in the diagram of fig. 15b. fig. 13 is , Theq → 0 limit for the vacuum piece in (D4) corresponds to taking all Ω's to zero and so making the replacements Ω ± csc(Ω ± ∆t) → (∆t) −1 . (D16)

4-particle frequencies and normal modes
Here we collect formulas for the large-N c frequencies and normal modes associated with 4-particle propagation (section V.B of ref. [6]).

Sequential Diagrams
Here we collect the result for the sequential diagrams [7]. A brief summary of the interpretation of each piece below can be found in section III of ref. [7]. Symbols such as Ω ± or a y , which are written in the exact same notation as symbols defined above, are given by their definitions above.
where the parameters s ≡ 2(x − y) t , t ≡ 2x + y (E2) each vary independently from 0 to 1. The numerical coefficients a mn and b mn are given in tables I and II. We have made no effort to make the approximation work well for y < 10 −4 .