The LPM effect in sequential bremsstrahlung: dimensional regularization

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. Of recent interest is 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). In previous papers, we have developed methods for computing such corrections without making soft-gluon approximations. However, our methods require consistent treatment of canceling ultraviolet (UV) divergences associated with coincident emission times, even for processes with tree-level amplitudes. In this paper, we show how to use dimensional regularization to properly handle the UV contributions. We also present a simple diagnostic test that any consistent UV regularization method for this problem needs to pass.

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]. As we will review shortly, calculations of the LPM effect must typically deal with ultraviolet (UV) divergences in intermediate steps, associated with effectively-vacuum evolution between nearly coincident times. For the case of computing single splitting rates, these divergences are trivial to deal with (either by subtracting out the vacuum rate a priori, or by using an appropriate iǫ prescription). However, for the case of two consecutive splittings with overlapping formation times (which we will loosely characterize as "double bremsstrahlung"), the treatment of ultraviolet divergences is much more difficult. In previous work [3], an iǫ prescription was proposed for dealing with this problem. Here, we will explain why that prescription was incomplete and missed certain contributions to the result. Then we will show how to correctly regulate the ultraviolet using dimensional regularization and will use our results to correct the QCD LPM analysis of ref. [3]. In addition, we provide a simple example-QED double bremsstrahlung in an independent emission approximation-that can be used as a test of the self-consistency of UV regularization prescriptions.
For simplicity of discussion, and in order to make contact with the double bremsstrahlung calculation of refs. [3,4], we will restrict attention in this paper to the case of mediuminduced bremsstrahlung from an (approximately) on-shell particle traversing a thick, uniform medium in the multiple scattering (q) approximation. (Thick means large compared to the formation time of the bremsstrahlung radiation.) However, the same methods should be useful for double bremsstrahlung in other situations.
A. Examples of UV divergences

Single splitting
The standard result for the single splitting rate in a thick, uniform medium in multiple scattering approximation is [5] where x is the momentum fraction of one of the daughters, and P (x) is the corresponding (vacuum) Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) splitting function. ∆t represents the time between emission in the amplitude and emission in the conjugate amplitude, as depicted in fig. 1. Ω is a complex frequency that characterizes the evolution and decoherence of this interference contribution with time ∆t. In the QCD case of g → gg, for example, it is given by 1 For a discussion specifically in the notation used in this paper, see section II of Ref. [3]. whereq characterizes transverse momentum diffusion of a high-energy particle due to interactions with the medium (average Q 2 ⊥ =qt) andq A indicatesq for an adjoint-color particle, i.e. a gluon. E is the initial particle energy. Throughout this paper, we will focus on splitting rates that have been integrated over the final transverse momenta of the (nearly collinear) daughters.
The negative imaginary part of Ω accounts for decoherence of interference (such as fig.  1) at large time separation, due to random interactions with the medium. In particular, the integrand in (1.1) falls exponentially for |Ω| ∆t ≫ 1, and so the integral is infrared (∆t→∞) convergent.
The details of the above formulas are not important yet. What is important is the behavior of the integrand in (1.1) as ∆t → 0: which makes the integral UV divergent. Note that this divergence does not depend on the medium parameterq and so represents a purely vacuum contribution to the rate. One way to deal with it is to note that an on-shell particle cannot split in vacuum, and so the purely vacuum contribution must vanish. One may then sidestep the technical issue of regulating the divergence by subtracting the necessarily-vanishing vacuum (q→0) contribution from (1.1) to get the convergent integral αP (x) π Re(iΩ). (1.4) An alternative way to deal with the divergence is to use an iǫ prescription. Notice that ∆t ≡ tx −t x in fig. 1 has the form of (i) a time in the conjugate amplitude minus (ii) a time in the amplitude. The correct iǫ prescription here is that conjugate amplitude times should be thought of as being infinitesimally displaced in the negative imaginary direction compared to amplitude times, 2 and so ∆t in (1.1) should be replaced by ∆t − iǫ. The ∆t → 0 behavior (1. 5) In this case, the UV piece of integration from ∆t → 0 does not generate a real part. That is, we can separate out the contribution proportional to from the calculation (1.1) of dΓ/dx, which leaves us again with (1.4).

Double splitting
Similar purely-vacuum divergences, which may also be easily subtracted, arise in the calculation of overlapping double splitting (e.g. overlapping double bremsstrahlung). However, as discussed in ref. [3], there are also sub-leading UV divergences which are not so easily discarded. These arise from situations such as depicted in fig. 2, in the limit where three of the four emission times become arbitrarily close together. In that short-time limit, the evolution of the system between the three close times becomes essentially vacuum evolution. But the evolution of the system from there to the further-away fourth time is not vacuum evolution and depends onq, and so this sub-leading divergence will not be subtracted away by subtracting the (vanishing) vacuum result for double splitting of an on-shell particle. Ref. [3] found that the surviving divergence of each interference diagram could be written in the form of a sum of terms proportional to where ∆t characterizes the small separation of the three emission times that are approaching each other (e.g. t x , t y , and tȳ in fig. 2) and Ω is the complex frequency characterizing the medium evolution associated with the fourth time (the right-hand part of each diagram in fig. 2). 3 Refs. [3,4] found that all the divergences (1.7) naively cancel each other in the sum over all interference contributions to the double splitting rate. But one must be careful, because finite contributions may still arise from the pole at ∆t = 0. As a simple mathematical example [3], the unregulated expression naively looks to be zero, but if it were regularized as then it would instead equal iπ. Ref. [3] attempted to find iǫ prescriptions for the poles, replacing each term (1.7) by after arguing what the sign ± of the iǫ prescription should be for each interference contribution to the double splitting rate. Unfortunately, this prescription turns out to miss some additional contributions from ∆t = 0. After discussing a relatively simple diagnostic test, we will explain (in section II B) what went wrong with the substitution (1.10).
For reasons we will discuss later, attempting to fix up the iǫ method for evaluating the pole contributions in double splitting seems complicated and fraught with subtlety. Fortunately, there is a cleaner, surer way to deal with the UV regularization of individual diagrams: We will show how to use dimensional regularization to compute the pole contributions from ∆t = 0. Dimensional regularization will turn the UV-divergent integral 0 d(∆t)/∆t in (1.7) into the UV-regularized integral 0 d(∆t)/(∆t) 1−ǫ/2 . Reassuringly, we find that dimensional regularization passes our diagnostic.
The precise details of exactly how and why, for each diagram, the earlier work [3] chose the sign of ±iǫ in the denominator of (1.10) will not be very important to the current discussion. However, for interested readers, we give a very brief review in appendix B.

B. Outline and Referencing
In the next section, we present a diagnostic that any consistent UV regularization scheme should satisfy. We then discuss what a successful iǫ prescription would have to do to pass that test and show how the simple prescription (1.10) fails. We will characterize the type of contributions that (1.10) misses (which we call "1/π 2 " pole pieces). In section III, we turn xyyx xyxy time + xyyx + + conjugates + appropriate permutations FIG. 3: Interference contributions contributing to double splitting, showing the subset referred to as "crossed" diagrams in ref. [3]. For QED double bremsstrahlung, the "appropriate permutations" are simply x ↔ y. For g → ggg in QCD, they are instead all permutations of x, y, and z ≡ 1−x−y. to dimensional regularization by warming up with the case of the single splitting rate (1.1). Sections IV and V then apply dimensional regularization to overlapping double splitting, treating, respectively, what we call the "crossed" interference diagrams of fig. 3 and the "sequential" diagrams of fig. 4. Section VI verifies that dimensional regularization passes the diagnostic test of section II. Finally, we present a summary of results in section VII. Various matters along the way are left for appendices.
In this paper, we will occasionally (in footnotes and appendices) use the author acronyms AI and ACI as shorthand for Arnold and Iqbal [3] and Arnold, Chang and Iqbal [4] so that, for example, we may write "ACI (5.2)" to refer to eq. (5.2) of ref. [4].

A. The QED independent emission test
Consider the case of double bremsstrahlung in QED, such as shown in fig. 5. In particular, consider the soft limit where the momentum fractions x and y carried by the two photons are small: x, y ≪ 1. In that case, the backreaction on the initial high-energy electron is negligible, and so we might expect that the x and y emission are independent from each other: where dI/dx dy is the differential probability for double bremsstrahlung and dI/dx is the differential probability for single bremsstrahlung. We will refer to (2.1), and similar formulas later for emission rates, as the independent emission model. Now consider an idealized Monte Carlo (IMC) description of shower development, based on the rates for single splitting (such as single bremsstrahlung and single pair production). In ref. [4], it is explained that the important quantity for characterizing corrections to such a Monte Carlo due to overlapping formation times is the difference of actual double splitting rates from what such a Monte Carlo would predict for two consecutive splittings: However, the independent emission approximation described above already assumed that there were no effects from overlapping formation times, and so in the independent emission approximation for dΓ/dx dy. This seems trivial. Nonetheless, we will obtain below an interesting test by reorganizing the individual terms that contribute to (2.3). We have chosen QED rather than QCD bremsstrahlung for our test because the independent emission approximation is not the same thing as the Monte Carlo approximation to double bremsstrahlung in QCD. QCD Monte Carlo based on single splitting rates allows for the second bremsstrahlung to be independently emitted from either daughter of the first bremsstrahlung process, as shown in fig 6. That means that the Monte Carlo probability of y emission is different depending on whether the y emission happens before or after the x emission. For QCD, Monte Carlo is therefore inconsistent with the independent emission model of (2.1), since in (2.1) the two emissions do not affect each other in any way. 4 To turn (2.3) into a test, split dΓ/dx dy for double bremsstrahlung into a sum over the different possible time orderings of the emissions, shown in figs. 3 and 4. Then, in the independent emission approximation, 2 Re(xyȳx + xȳyx + xȳxy + xyxȳ + xxyȳ + xxȳy) + (x ↔ y) − IMC = 0. (2.4) Following refs. [3,4], the notation xyȳx indicates that the emissions in the corresponding diagram of fig. 3 happen in the time order of (i) x emission in the amplitude, followed by (ii) y emission in the amplitude, followed by (iii) y emission in the conjugate amplitude, and finally (iv) x emission in the conjugate amplitude. There are two ways to make use of (2.4). One is to apply it to the full calculation (as opposed to the independent emission approximation) of the different double bremsstrahlung interference diagrams in QED, expand the results in small x and y, and then check whether the equality (2.4) holds to leading order in that expansion. 5 The other use is to directly investigate how (2.4) works in the independent emission model itself, using one's favorite UV regularization of single bremsstrahlung rates. We'll now do just that for the case of iǫ prescriptions.

B. Application to iǫ prescriptions
Consider use of an iǫ prescription to UV-regulate the single splitting rate of (1.1): Appendix C shows that the test (2.4) is indeed satisfied (as it must be) if we use (2.5) in the independent emission model. Here we want to focus on the UV contributions to that test, which we have summarized in the second column of Table I. That column shows the small ∆t limit of the d(∆t) integrands for each diagram, corresponding to the pole pieces in (1.7). In this table, we have further assumed y ≪ x just to make the formulas as simple (and so easy to compare) as possible, and we have also introduced the short-hand notation One may now see the problem with the earlier proposal (1.10) of ref. [3] for how to regulate the 1/∆t poles in double bremsstrahlung calculations using the iǫ prescription. It is true that, if you ignore ǫ prescriptions, then all of the small-∆t divergences in Table I   Above, ∆t ± ≡ ∆t ± iǫ. Each entry is to be integrated over (small) ∆t as in (1.7) and multiplied by 4α 2 EM /π 2 xy, where α EM is the fine structure constant. The small-x complex frequency Ω x above is defined by Ω QED x ≡ −ixq/2E. Following ref. [3], ∆t (no subscript) above is defined as the time separation between the middle two emissions for all but the last entry: e.g. ∆t ≡ tȳ −t y for xyȳx and ∆t ≡ tx − tȳ for xȳxy. All other times have been integrated over. (For the identification of ∆t in the last entry, see section IIA of ref. [4] or appendix C 4 here.) The problem with the earlier analysis of ref. [3] is that it correctly identified the iǫ prescription in denominators, 6 but failed to realize that 1/∆t = (∆t)/(∆t) 2 could also have different and important iǫ dependence in a numerator. The prescription proposed by ref. [3] is shown in the last column of table I and can be obtained from the second column by replacing the full ∆t dependence by just 1/∆t, with the iǫ dependence taken from the denominator in the second column. So, for instance, the 2 Re(xȳyx) contribution (2.7) is replaced by The difference between the prescriptions (2.7) and (2.8) gives an integral that is dominated by ∆t ∼ ǫ and so is a purely "pole" contribution that can be evaluated without needing to know how the integrand behaves for large t: The one other difference in Table I can be evaluated similarly. For the y ≪ x ≪ 1 limit of the QED independent emission approximation, the total difference (summing all contributions) between the correct iǫ prescription and the naive prescription of (1.10) is This is non-zero. Because the independent emission calculation must (and does) satisfy the test (2.4), we see that the naive prescription of ref. [3] does not. Rather than only check this failure of the naive prescription in the context of the independent emission calculation, we have also done a full QED calculation (not making any assumptions about the size of x and y) of the LPM effect in double bremsstrahlung, along lines similar to the QCD calculation in refs. [3,4]. We have verified that the y ≪ x ≪ 1 limit of those full results reproduce the last column of table I if we use the naive iǫ prescription of (1.10) following ref. [3]. We have also verified that the diagnostic test (2.4) fails in this limit by exactly the amount (2.10). The details of the full QED calculation are not directly relevant to our current task, which is to find a clear, correct method for determining the UV contributions which works not just in the context of the QED independent emission approximation but generalizes to QCD and to any values of x and y. So we will defer presenting the details of full QED results for double bremsstrahlung to future work.
Why do we not simply fix up the iǫ prescription in the general case, to make it work correctly like in the second column of Table I? Despite various attempts, we were unable to find a convincing generalization that worked outside of the limiting case of y ≪ x ≪ 1. We briefly discuss the issues we encountered in appendix E. Here, we will instead turn to dimensional regularization for the general case.
C. 1/π vs. 1/π 2 pole terms Before moving on, it is interesting to note a qualitative difference between what the naive iǫ prescription does account for and what it does not. As an example, consider the naive prescription small-∆t behavior (2.8). Use the identity where "P. P." indicates the principal part prescription. The real-valued 1/∆t integration terms (the "principal part" terms above) will cancel among all the diagrams, as mentioned earlier, leaving a finite total result. The pole contributions (in the naive iǫ prescription) are given by the iπ δ(∆t) term above. Note that this term is associated with an extra factor of π. So, for instance, the corresponding pole piece of (2.8) is given by Re(Ω x ) (2.12) (in which integrating over only half a δ function is understood to give 1 2 ). 7 Looking at the factors of π in (2.12), the original, naive iǫ prescription (1.10) gives all of what we will call the "1/π" terms for the pole contribution. The correct analysis of the problem, in contrast, introduces additional 1/π 2 pole contributions such as (2.10).
We should also mention that if one analyzes all the entries this way, then the 1/π terms produced by the third column of table I add up to zero. This turns out to be an artifact of the y ≪ x ≪ 1 limit, and there is no such cancellation in the more general case.

III. SINGLE SPLITTING WITH DIMENSIONAL REGULARIZATION
We now turn to dimensional regularization and will start with the single splitting formula (1.1). As reviewed in the introduction, dealing with the UV divergence for single splitting by other means is trivial, but it will provide a simple and useful warm-up example.

A. Straightforward method
Following Zakharov [6], one way to view the source of the single splitting formula (1.1) is as an effective 2-dimensional non-Hermitian non-relativistic quantum mechanics problem for the three high-energy particles shown in fig. 1b. Using symmetries of the problem, the threeparticle quantum mechanics problem can be reduced to a one-particle quantum mechanics problem. In this language, the basic formula corresponding to fig. 1 is where B is a single, convenient combination of the transverse positions of the three particles. For a more complete discussion in the notation used here, see section II of ref. [3]. In the multiple scattering (q) approximation appropriate for high energy particles traversing thick media, the problem turns out to become a harmonic oscillator problem with complex frequency Ω and mass For constant Ω (appropriate to the case of a thick, homogeneous medium), the propagator of a 2-dimensional harmonic oscillator is Using this in (3.1) reproduces the earlier formula (1.1). 7 The factors of 2 are irrelevant to the point here, but see section VII.B.1 of ref. [3] if a more convincing discussion that does not rely on δ functions is desired.
To implement dimensional regularization, we simply generalize the d=2 analysis for two transverse dimensions to an arbitrary number d of transverse dimensions. (Note that we are defining d to be the number of transverse spatial dimensions, not the total number of spacetime dimensions, and so the real world is d = 2 in this paper, not d = 4.) The propagator for a d-dimensional Harmonic oscillator is Changing integration variable to τ ≡ iΩ ∆t, this becomes We've been seemingly cavalier here about the complex phase of the upper limit of integration-see appendix A for a more careful discussion. The integral (3.6) converges for −2 < d < 0. The result for other d is defined by analytic continuation, giving (see appendix A) is the Euler beta function.
We can now take the d → 2 limit, giving Because this regularized result for the integral is finite for d=2, we do not have to worry about the generalization of the prefactor αP (x)/[x(1−x)E] 2 in the dΓ/dx formula (3.1) to general dimension: we can just use the known d=2 version. Combining (3.1), (3.2) and (3.9) correctly reproduces the usual result (1.4) for single splitting.

B. Alternative derivation
Before we launch into the complexities of the double bremsstrahlung calculation, we can introduce another formula that we will need by repeating the previous calculation in a slightly more roundabout manner: We will do the ∆t integral in (3.1) before taking the Bx derivative and so also before setting Bx to zero. The integral we need has the form (3.10) Using the d-dimensional propagator (3.4), this becomes with τ ≡ iΩ ∆t as before. The integral converges for d > −2 when B = 0. Using This is a formula we will need later for double splitting, where it will be convenient to rewrite it equivalently as Let's finish up the single splitting calculation by showing that we can use these formulas to get the same answer as before. We need to take the gradient ∇ B of (3.13) and set B to zero. The last step raises a subtlety which happily will not arise in the double splitting calculation later: the original integral (3.11) has convergence problems for B=0 unless d < 0. Eventually we want to focus on d → 2, but we should be cautious about whether we analytically continue to that limit before or after setting B to zero. To see that there is an issue, use the generic expansion of the Bessel function for small arguments: Which term dominates depends on the sign of ν and so, in our case, on the sign of d. Keeping both of the potentially leading terms above, the small B behavior of (3.13) is Now dot ∇ B into the above expression and then set B to zero. The second term gives exactly our earlier answer (3.7), which leads to the correct result (3.9) when we analytically continue to d = 2. The first term gives a (UV) singularity unless d ≤ 0. Since the point of dimensional regularization was to regulate the UV, we learn that in this application to single splitting we need to keep the dimension as d < 0 until after we take B to zero.

IV. CROSSED DIAGRAMS WITH DIMENSIONAL REGULARIZATION
We now turn to double bremsstrahlung, focusing on g → ggg and starting with the crossed diagrams of fig. 3, which were evaluated for d=2 (other than the missing 1/π 2 pole terms) in ref. [3]. As in ref. [3], we will start with the xyȳx diagram, to which the others can be related.

A. First equations
Our starting point here will be the d=2 expression 8 Cȳ 34 , Cȳ 12 , tȳ|C y 41 , C y 23 , t y developed for xyȳx in section IV of ref. [3]. B y , t y |B x , t x and Cȳ 34 , Cȳ 12 , tȳ|C y 41 , C y 23 , t y and Bx, tx|Bȳ, tȳ represent, respectively, the (i) 3-particle evolution of the system in the initial time interval t x < t < t y of the figure, (ii) 4-particle evolution in the intermediate interval t y < t < tȳ, and (iii) 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) 1-particle, (ii) 2-particle, and (iii) 1-particle problems in non-Hermitian d=2 quantum mechanics. Each vertex in the diagram is associated with one of the gradients above. The dimensionless functions α(x, y), β(x, y), and γ(x, y) contain normalization factors and combinations of spin-dependent DGLAP splitting functions associated with those vertices. 9 The variablesx i refer to the momentum fractions associated with the four particles involved in the 4-particle evolution in this diagram, which are The overall factors of |x 1 +x 4 | −1 and |x 3 +x 4 | −1 are additional normalization factors associated with the vertices at the intermediate times t y and tȳ given our choice of normalization of the transverse position variables and of corresponding states such as |C 41 , C 23 and |B . 10 The expression (4.1) also assumes the large N c limit in order to simplify the color dynamics of the problem associated with the 4-particle propagation in the medium. However, we believe that the results for the pole contributions calculated in this paper do not depend on the assumption of large N c because the 4-particle propagation in pole contributions is effectively vacuum propagation (as discussed earlier with regards to fig. 2).
8 AI (4.40) 9 AI (4.30-39) 10 AI (4.6) and (4.22-25) The d-dimensional generalization of (4.1) is Cȳ 34 , Cȳ 12 , tȳ|C y 41 , C y 23 , t y There are no important changes to this formula other than the fact that transverse positions B and C are now d-dimensional, but we should comment on the other, mostly unimportant differences between (4.1) and (4.3): • The functions α, β, and γ will be different in d dimensions. For one thing, the number of "helicities" to be summed over depends on d. Except as noted below, we absorb all this dependence on d into new d-dimensional versions of α, β, and γ. However, similar to the discussion of P (x) in section III A, we will see when we take d → 2 at the end of the day that we only need explicit formulas for the original d=2 versions given in ref. [3].
• One of the exceptions to absorbing all of the differences into the d-dimensional definitions of (α, β, γ): We have changed the overall E −4 in (4.1) to E −2d in order to keep α, β, and γ dimensionless (and so make dimensional analysis of our formulas easier). This difference originates with the factors of E associated with the vertices (see appendix A).
• We have also treated separately the generalization of the normalization factors . The reason that we do not also absorb these differences is that, unlike (α, β, γ), these factors will not be the same for the three diagrams shown explicitly in fig. 3. Since we will later relate these diagrams to each other, we need to keep track of factors that change between them. These factors are nonetheless fairly uninteresting because |x 1 +x 4 | −d/2 |x 3 +x 4 | −d/2 will simply cancel some related normalization factors when we later write more explicit formulas for the 4-particle propagator.
We now want to use (3.14) to integrate over the first time t x in (4.3). The derivation of (3.14) relied on the M associated with the 3-particle evolution being positive, and the associated Ω having phase √ −i. All is well for now, but when we later relate the other crossed diagrams to xyȳx, we will encounter situations where instead both M is negative and Ω ∝ √ +i. It's therefore convenient to use an appropriate generalization of (3.14) to cover both situations: (see appendix A). The similar result for integration over the final time is Using these for the initial t x and final tx time integrations in (4.3) gives 11 Cȳ 34 , Cȳ 12 , ∆t|C y 41 , C y 23 , 0 where (M i , Ω i ) are associated with the initial 3-particle evolution (t x < t < t y ) and (M f , Ω f ) with the final 3-particle evolution (tȳ < t < tx).

B. 4-particle propagator
For evaluation of the pole pieces (which are the pieces that require UV regularization), we only need the small ∆t limit of (4.5). In that limit, medium effects on the propagator Cȳ 34 , Cȳ 12 , ∆t|C y 41 , C y 23 , 0 are small. We will therefore use the vacuum result for Cȳ 34 , Cȳ 12 , ∆t|C y 41 , C y 23 , 0 . (Readers who would prefer to see a derivation closer to ref. [3], where we first find full expressions for the propagator before taking the small ∆t limit to find the poles, may turn instead to Appendix F.) As discussed in ref. [3], 12 the effective 4-particle evolution in interference diagrams such as fig. 3 is given (in the high-energy limit) by a Lagrangian of the form b i are the transverse positions of the individual particles, and The imaginary-valued potential V implements medium effects which cause decoherence of interference over times ∆t of order the formation time. In vacuum, V = 0. 13 Were we to 11 The d=2 cases of (4.4) and (4.5) reproduce AI (5.9) and AI (5.10) respectively. 12 Specifically, see the discussion leading up to AI (5.15-18). 13 As in refs. [3,4], we have for simplicity assumed that the energy is high enough that we may ignore the effects of the physical masses of the high-energy particles. If one does not ignore them, their effects contribute a real-valued constant to V [6], even in vacuum. See the discussion surrounding AI (2.15).
express the propagator associated with (4.6) solely in terms of the variables (C 34 , C 12 ), it would then (in this limit) simply be Changing variables from (C 34 , C 12 ) to (C 41 , C 23 ) in just the ket |C ′ 34 , C ′ 12 , 0 then gives the version of this propagator that we need: In order to keep notation as close as possible to ref. [3], it will be useful to rewrite the exponential in (4.10) as Here, the unspecified +O(∆t) contributions represent the size of effects due to the medium. Using (4.10) and (4.12) in (4.5) then gives 15 (4.14) We could try to follow the d=2 analysis of ref. [3] by next doing the two B integrations in (4.14). Unfortunately, the integrals are complicated. Fortunately, we can simply the calculation because we need general-d expressions for only the pole pieces, corresponding to ∆t → 0. The exponential factors in (4.14) become highly oscillatory for B ≫ X −1/2 ∼ ∆t/E and so cause the integrals to be dominated by the scale B ∼ ∆t/E (where we are not showing the x i dependence). So, for the purpose of extracting the small ∆t behavior, we may expand the Bessel functions in (4.14) for small arguments, as in (3.15). For −4 < d < 4 (which includes both the physical point d=2 and the region −2 < d < 0 where all of our earlier integrals were convergent), the terms shown explicitly in (3.15) are the leading ones and give We will see later that the integrals we have left to do converge for d = 2 − ǫ with ǫ small, and so we may as well take d = 2−ǫ now. In that case, the first term in (4.15) dominates over the second. By itself, however, the first term is uninteresting because it does not depend on Ω and so does not depend onq. If we use just the first term for both of the (|M|Ω/B 2 ) d/4 K d/4 factors in (4.14), we will obtain a contribution that will be canceled when we subtract away 15 It is easy to get confused about cuts associated with the fractional exponents.
In the xyȳx case at hand, they are resolved by the facts that (i) M and M ′ are positive definite, so that See appendix H for a more general discussion of cuts. Additionally, the d=2 case of (4.14) reproduces AI (5.43) with X y = X y + |M i |Ω i and Xȳ = Xȳ + |M f |Ω f , except that we have expanded here in the small ∆t limit. the purely vacuum result. We should therefore focus on the next term: The (non-vacuum) small ∆t limit of the integrand in (4.14) then gives for the sake of compactness.

D. Scaling
Before diving into the details of explicitly performing the B integrals in (4.17), it is worthwhile to see the general structure of the result using a simple scaling argument. We can quickly see the ∆t dependence by rescaling B =B ∆t/E, whereB is dimensionless, and noting that (X, Y, Z) ∝ E/∆t in (4.13). This rescaling pulls out all of the ∆t (and E) dependence from the two d d B integrals, identifying the small ∆t behavior as Note that this result has the right dimension to be dΓ/dx dy and also gives the usual dependence dΓ/dx dy ∝ q/E onq and E for d = 2.
The UV behavior of the ∆t integral above is convergent and well-defined for d = 2−ǫ. On the infrared (IR) end of the integration region (large ∆t), our small-∆t expansion formulas are no longer valid. But the formulas above will be good enough to study the contribution we get, if any, from arbitrarily small ∆t. Imagine taking the full expression for dΓ/dx dy as an integral over ∆t, without having made any small ∆t approximation. Then divide the integration region up into ∆t < a and ∆t > a for some a chosen very small compared to the formation times in the problem: (4.19) The ∆t > a integration can be handled with the d=2 formulas of ref. [3]: the difference between d=2 − ǫ and d=2 in this region can be ignored as ǫ → 0. The UV contributions that required UV regularization appear only in the ∆t < a integration. In (4.18), that integration region gives (4.20) The 1/ǫ and ln a terms will cancel between diagrams for the same reason that 1/∆t terms cancel between diagrams in ref. [3]. However, there can be finite O(ǫ 0 ) contributions to diagrams that do not cancel. The UV piece of a diagram D will have the generic form For the xyȳx diagram, for example, f D (ǫ, {x j }) represents all the factors in (4.18) besides the ∆t integral. Now consider the sum of some set of diagrams, Using (4.20), we see that we can still get a non-vanishing result from (i) the O(ǫ) pieces of f D (ǫ) multiplying (ii) the 2/ǫ piece of (4.20): This finite piece, which survives in the a → 0 limit, represents the pole contribution that we are looking for: it is a contribution associated with ∆t = 0 in the limit d → 2.

E. Actually doing the B integrals
We now return to the unscaled B's, just to keep the discussion as close as possible to the d=2 analysis of ref. [3]. 16 Contracting the various transverse spatial indices m, n,m, andn 16 There is a slight difference with AI as far as line-by-line comparisons go. The small-argument expansion (4.15) for the Bessel functions corresponds (for d=2) to making the expansion e − 1 2 |M|ΩB 2 ≃ 1 − 1 2 |M |ΩB 2 to the 3-particle factors in AI before doing the integration over the B's. In the end, that should get us to the same small-∆t behavior, but AI did the operations in the opposite order: AI did the B integrals first and only then extracted the small ∆t limit. of (4.17) gives where 17 .  The integrals J of (4.26) here correspondingly play the roles of the integrals I of AI (5.44).
simplifies to 18 (4.28) Using the small ∆t limit (4.13) for X, and using Eqs. (4.13) and (4.28) then give 20 which (using x 1 + x 2 + x 3 + x 4 = 0) can be rewritten as If desired, the Γ functions may be manipulated into the same form as in the single splitting result (3.7) using (4.32) 18 (4.28) plays a role similar to AI (D4). 19 We do not need J i3 and J i4 in (4.25), but their small ∆ limits may be found in eqs. (G7) of appendix G. For d=2, our result reproduces the corresponding (unregulated) small-∆t behavior in ref. [3], 21 which is, for future reference, (after subtracting the purely vacuum contribution).

F. Branch cuts
We will need to be somewhat careful about branch cuts when we expand in ǫ for d = 2−ǫ. As an example, or some other variant. The derivation of (4.31) took the cavalier attitude that, if we keep formulas as simple as possible and never isolate factors that might be fractional powers of negative real numbers, then standard choices of branch cuts would give the correct answer. That is, we assumed it was okay to write (ix 1 x 2 x 3 x 4 Ω sgn M) d/2 but not (without further branch-cut clarification) (iΩ sgn M) d/2 (x 1 x 2 x 3 x 4 ) d/2 , since x 1 x 2 x 3 x 4 < 0 for the xyȳx interference diagram. Moreover, we want expressions that will also work correctly for other diagrams when we later relate them to the results for xyȳx, in which case M, Ω, and x 1 x 2 x 3 x 4 turn out to have other signs and phases. In appendix H, we check that the combination (ix 1 x 2 x 3 x 4 Ω sgn M) d/2 used in (4.31), with the standard choice to run the branch cut along the negative real axis, produces the correct overall phase for all cases of interest. We will see in section IV G below that these complex phases generate the 1/π pole terms previously found in ref. [3] using naive ǫ prescriptions.
Despite the ambiguity of the expression (x 1 x 2 x 3 x 4 ) d/2 for x 1 x 2 x 3 x 4 < 0, it will be convenient to rewrite below. This rewriting works if we adopt the convention that negative real numbers have phase e −iπ . See appendix H for verification of (4.34a) in all relevant cases. That is, one should interpret in what follows, where θ is the step function.

G. Expansion in ǫ
Now take the ǫ → 0 limit of (4.31) for d = 2 − ǫ. To isolate the pole contribution, introduce a small cut-off a as in  With this cut-off, we can rewrite (4.31) as dΓ dx dy and c 1 = 1 + ln(2π 2 ) is an uninteresting numerical constant that will not appear in our final results. It's convenient not to explicitly expand the factors of (iΩ sgn M) d/2 above. We save some effort in what follows by having separated (4.37) from the other terms in (4.36). Specifically, note that (4.37) is almost proportional to the coefficient of 1/∆t in the integrand of (4.33), which is the 1/∆t behavior found previously in the d=2 analysis of ref. [3] and which we'll call the "usual" behavior. We say "almost" proportional because (4.37) has • d-dimensional (α, β, γ) instead of 2-dimensional (α, β, γ).
These differences inspires our designation "almost usual" in (4.37). In ref. [3], the 1/∆t pieces of the integrand canceled between the six diagrams of fig. 7 and so canceled in the sum of all crossed diagrams as well. So, we might expect the corresponding contribution (4.37) to similarly cancel here. Indeed, the replacement of iΩ sgn M by (iΩ sgn M) d/2 does not affect the cancellation, because the pieces with different values of iΩ sgn M canceled separately in ref. [3]. 22 The replacement of 2-dimensional (α, β, γ) with d-dimensional (α, β, γ) similarly does not affect the cancellation because the cancellation in ref. [3] did not depend on the specific form of the functions (α, β, γ). The upshot is that the contributions (4.37) will cancel between diagrams. Among other things, this means that the ln a terms which depended on our choice of cut-off a in (4.35) will disappear. So, we may ignore the "almost usual" contributions of (4.37) and focus exclusively on the other terms in (4.36). 23 Those other terms are finite, and so we may now take (α, β, γ) to be given by their d=2 formulas because the discrepancy will only contribute to the final result at O(ǫ).

H. Other diagrams
In ref. [3], it was shown that xȳyx could be obtained from xyȳx by 24 where we have found it convenient to use the fact that | before rewriting the x ′ i in terms ofx i . Similarly, xȳxy was shown to be obtainable from xyȳx by 24 Specifically, section VI of ref. [3].
The Ω i piece arising from adding together the three diagrams (4.36), (4.39), and (4.43) is In fig. 7, we further added in diagrams corresponding to swapping x ↔ y while also conjugating. If it weren't for the e −iπ factor inside the logarithm, this would have the effect, after including all Ω terms and evaluating all sgn M, of replacing where The last term (the π/π 2 = 1/π term) is the same as the (incomplete) pole term that was found in ref. [3]. 25 The other term (the 1/π 2 term) is new.

V. SEQUENTIAL DIAGRAMS WITH DIMENSIONAL REGULARIZATION
We now turn to the sequential diagrams of fig. 4. Ref. [4] analyzed these diagrams except for the pole pieces, for which results were quoted by reference to this paper. Here we present the analysis of those poles, using techniques similar to those of the previous section.

A. 2 Re(xxyȳ + xxȳy) minus Monte Carlo
Consider the last two diagrams in fig. 4 plus their conjugates. As discussed in ref. [4], these diagrams factorize into separate pieces associated with x emission followed by y emission, and they are almost the same thing as the corresponding idealized Monte Carlo calculation based on single splitting rates. The "almost" has to do with restrictions on the time ordering, which cause the difference with idealized Monte Carlo to be 26 ∆ dΓ dx dy xxyȳ+xxȳy where dΓ/dx d(∆t) is the integrand associated with the single-splitting result (3.1): In this calculation, it will be important to keep in mind that P (x) is, for now, the ddimensional generalization of the DGLAP splitting function. We have written the overall factor of E as E −d in order to keep P (x) dimensionless, similar to our choice of convention in section IV A. The argument [4] for (5.1) was valid in any transverse dimension d.
We've already seen from (3.7) that in dimensional regularization. For (5.1), we also need integrals of the form However, for the pole contribution, we'll really be interested in the small-∆t contribution to this integral, which is where a is a tiny cut-off on ∆t just like the one we introduced for the crossed diagrams in (4.35).
Combining (5.4) and (5.5b) as in (5.1) gives (See notes in Appendix A.) Note for future reference that We now turn to the remaining diagram of fig. 4. For d=2, ref. [4] showed that one of the large-N c color routings xyxȳ 2 of xyxȳ gives 27 which is identical to the similar formula for xyȳx [3], 28 except for the addition of the superscript "seq" on some symbols, the bars on (α, β, γ), and the purely notational change of relabelingȳ subscripts asx. (See ref. [4] for details.) From having seen earlier how the small-∆t behavior of the xyȳx diagram generalized from d=2 to 27 ACI (2.36) 28 AI (5.45) (4.25) for arbitrary d, it is easy to see how (5.8) similarly generalizes: where the J seq 's are defined the same as (4.26) but using (X seq , Y seq , Z seq ) instead of (X, Y, Z). In the small ∆t limit, ref. [4] found that 29 Correspondingly, (5.10) reduces to analogous to (4.28). The analogs of the J integrals (4.29) are which is the analog of (4.30).
The only effect of different color routings on xyxȳ is on how color is distributed during the 4-particle evolution in the diagram, which in turn affects how the 4-particle evolution interacts with the medium. Because the pole terms we are interested in here only arise from situations where ∆t is small enough that the 4-particle propagation is effectively vacuum propagation, the color routing has no effect. For this reason, adding in the other color routing xyxȳ 1 described in ref. [4] simply multiplies (5.14) by two. 31 So, summing both routings, and adding the conjugate diagram, where we have now used the fact that sgn M i and sgn M seq f are both +1.

C. Combining sequential diagrams
The sequential diagram results (5.6) and (5.15) are individually UV divergent, and it is only when we combine them that the divergences will cancel. But we need to be careful because one formula is expressed directly in terms of DGLAP splitting functions P (x) and P (y), whereas the other is expressed instead in terms of the splitting-function combinations (ᾱ,β,γ) defined in ref. [4]. For d=2, the two are related by 32 However, because the diagrams are individually UV divergent, we need the O(ǫ) corrections to this relation. To do that correctly is finicky: it requires diving into the details of how the splitting function factors are defined and normalized for d dimensions. We find that the generalization of (5.16) is (5.17) However, rather than justifying this directly, we find it easier (and less prone to error) to simply repeat the derivation of the xxyȳ diagram from the beginning directly in the language of d-dimensional (ᾱ,β,γ). We carry this out in appendix I. One may then (i) identify the conversion (5.17) by comparison with our earlier result (5.16). Alternatively, one can avoid the need for (5.17) altogether by (ii) keeping all sequential diagrams in terms of (ᾱ,β,γ), combining the diagrams to get a finite total, and only then taking d → 2. In the final, finite 31 Alternatively, ref. [4] explains that the other color routing differs only by a permutation of the 4-particle x i to (x 1 , x 2 , x 3 , x 4 ) = (−1, 1−x−y, y, x) = (x 1 ,x 3 ,x 2 ,x 4 ). One may check explicitly that this permutation leaves (5.12) invariant above. 32 ACI (E5) result, one only needs known d=2 formulas for (ᾱ,β,γ), which at that point may be related to the d=2 splitting functions using (5.16).
The final result is the same either way, but the intermediate formulas are a little simpler to present by using the conversion (5.17) to rewrite (5.15) as 2 Re dΓ dx dy where we have also used (4.2), (4.32), Using (4.20) and taking ǫ → 0, As promised, the final answer is finite, and so one may now use ordinary d=2 results for the splitting functions P .

VI. DIMENSIONAL REGULARIZATION PASSES DIAGNOSTIC
We should check our methods by checking that the diagnostic test (2.4) works when we apply dimensional regularization to computing individual time-ordered diagrams in the QED independent emission model. That calculation is carried out in appendix D. Taking the y ≪ x limit for simplicity, the results for the total contribution of crossed vs. sequential where we have split each case into the pole contribution (which requires UV regularization) and the non-pole contribution. The contributions of (6.1) indeed sum to zero, and so dimensional regularization passes this test. It had to because dimensional regularization is a fully consistent regularization scheme and the test (2.4) was ultimately a tautology, but it is reassuring to verify that we can correctly carry out the technical procedures of calculating the pole terms.
There is also a reassurance test, of sorts, for QCD results. Ref. [4], which makes use of the QCD pole terms derived in this paper, shows that 33 (i) the pole terms computed with dimensional regularization have the right behavior to effect a Gunion-Bertsch-like cancellation of logarithmic enhancements to ∆ dΓ/dx dy, which (ii) would not occur were one to use the naive iǫ prescription to determine those poles (i.e. neglect the 1/π 2 pole terms).

VII. SUMMARY OF QCD RESULTS
Our final results for QCD are that the crossed diagrams, accounting for all permutations, have pole contribution where z ≡ 1−x−y and where A pole (x, y) is twice the real part of (4.47): 33 See ACI appendix B and ACI footnote 20.
Using the same notation as ref. [4], the sequential diagrams give ∆ dΓ dx dy pole sequential = A pole seq (x, y) + A pole seq (z, y) + A pole seq (x, z) where A pole seq (x, y) is half of the y ↔ z symmetric result (5.20): Eq. (4.3): With regard to the factors of E, AI (4.29-30) wrote p j , p k |δH|p i = gT i→jk · P jk with T i→jk = T color i→jk P i→jk /2E 3/2 and P i→jk dimensionless (which is what made the later definitions of α, β, and γ dimensionless). The fact that the power E −3/2 of E in the last formula is necessitated by dimensional analysis was explained in notes on AI (4.29-31) in AI appendix A. That same argument, applied to d dimensions, gives p j , p k |δH|p i ∝ gT color i→jk P i→jk · P jk /2E (d+1)/2 . The four factors of E −(d+1)/3 then combine with a factor of E 2 in AI (4.10), whose origin is not dimension dependent [see the notes on AI (4.16) in AI appendix A], to give the E −2d of (4.3) in this paper. Alternatively, one could just determine the power by dimensionally analyzing (4.3) or later formulas.
With regard to the factors of |x 1 + x 4 | −1 and |x 3 + x 4 | −1 , these came from the vertex formula AI (B32), which originated from the normalizations of states in AI Appendix B and AI (4.22-25). Repeating the derivations of AI Appendix B in d dimensions, one finds that AI (B11) changes to with consequence that AI (4.22) is replaced by This means we need to replace AI (4.23) by in order to get the desired normalization of AI (4.24), and make a similar replacement for AI (B31). This change propagates to replacing the vertex formula AI (4.15) by which is the source of the factor |x 1 +x 4 | −d/2 and the corresponding permutation |x 3 +x 4 | −d/2 in our (4.3).
Eq. (4.4a): If M < 0 and Ω ∝ √ +i, then Re(MΩ) < 0, which means the integral in (3.11) does not converge at large τ . To get a convergent integral, we should change variables from ∆t in (3.10) to τ ≡ −iΩ∆t instead of τ ≡ iΩ∆t. This yields instead of (3.11). Similar to this appendix's notes for (3.6) above, we may deform the upper limit of the τ contour from ∞ e −iπ/4 to +∞. Recognizing that M = −|M| in the M < 0 case discussed here, we see that both this case and the M > 0 case of (3.11) can be written in the convergent form with result (4.4a). This result agrees with AI (5.9a) in the special case d = 2.
In addition to making this change for C ′ on the right-hand side of (4.9), it is necessary to account for the change in normalization of the ket |C ′ 34 , C ′ 12 vs. |C ′ 41 , C ′ 23 . The former is normalized so that C 34 , ) and the latter so that . The relationship between these δ functions is determined by the Jacobean of the transformation (A8). Specifically, the normalization change accounts for an overall factor of |( in going from (4.9) to (4.10).
Eq. (5.6): For d=2, this agrees with ACI (2.28). Note that in dimensional regularization (unlike for iǫ prescriptions), the ∆t integral in (5.5b) is real valued. When using (5.4) and (5.5b) in (5.1), remember that E is replaced by ( Eq. (D9): The last integral in (D8) is the same one as in (3.6). One way to do the first integral is to make the same change of variables described in the notes for (3.7) above to get and then simplify from there using ψ(1−z) − ψ(z) = π cot(πz), where ψ(z) ≡ Γ ′ (z)/Γ(z) is the digamma function. See AI section VII.A.2 and AI appendix D.3 [3]. The iǫ prescription above corresponds to time ordering in the first case, anti-time ordering in the second, and Wightman correlators (no time ordering) in the remaining cases. However, there is a difficulty. Double bremsstrahlung interference diagrams such as fig. 2 involve four different times, and, in the method of ref. [3], the first and last of those times are already integrated over. The time ∆t left after these integrations represents the separation between the middle two times in the diagrams, but by then the effects of the iǫ's that should have been associated with the first and last times were lost. AI appendix D.3 [3] sorted out this issue in detail in order to figure out what the net ±iǫ should be in denominators. If one naively applies the results of that appendix to 1/∆t factors, one obtains the results listed in the last column of table I of this paper. As explained in section II B of this paper, that argument was flawed because there can also be importantly different iǫ factors in numerators of ∆t/(∆t) 2 . Additionally, the situation is confusingly complicated by the fact that both amplitude and conjugate-amplitude particles are interacting with the medium between highenergy splitting times, linking (after medium-averaging) the evolution of one to the evolution of the other. As described in appendix E of this paper, it is unclear how to fully incorporate the iǫ prescription into this evolution in the case of double bremsstrahlung.

Appendix C: Test of iǫ prescription for independent emission model
In this appendix, we will take a look at how the QED independent emission test of section II works out when we regulate each single splitting amplitude with an iǫ prescription as in (2.5), which we will write here as Remembering that the QED independent emission approximation is only relevant for small x and y, we will use the corresponding limit of the splitting function and just write In this appendix, all calculations will be in the context of the independent emission model, and so we will just use equal signs (rather than ≃) in our formulas with that understanding. For QED (and not QCD!), the relevant complex frequency Ω in the small x limit is In general, the double emission rate is given in the independent emission model by where ∆t x = |tx − t x |, ∆t y ≡ |tȳ − t y |, and the time integral is (using overall time translation invariance) over any three of the four times t x , tx, t y , tȳ. To isolate different time-ordered contributions such as xyȳx, xȳyx, etc. in the diagnostic (2.4), we just need to impose the appropriate restrictions on the time integrals and take the appropriate pieces of the real parts.

xyȳx + xȳyx
As an example, we begin by computing the xyȳx contribution to the diagnostic (2.4). This corresponds to The time ordering requires ∆t y < ∆t x . Using the fact that the integrand depends only on ∆t x and ∆t y , the time integral corresponding to where the interval (t y , tȳ) lies within the interval (t x , tx) is trivial and gives a factor of ∆t x − ∆t y , so that .

xȳxy
In the case of xȳxy, we want t x < tȳ < tx < t y . In keeping with the conventions of our full analysis of crossed diagrams, we will want to write this contribution as an integral over the time separation ∆t = tx − tȳ of the two intermediate times. The corresponding constrained time integral is where the conjugation of the last factor is because the y emissions appear in the orderȳy in xȳxy. The basic integral needed is

Total crossed diagrams
Adding together (C11), (C14) and their x ↔ y permutations gives the total contribution from QED crossed diagrams in the independent emission model: In order to make contact with the discussion in the text, take the simplifying limit y ≪ x, in which case the QED frequencies (C4) are ordered as |Ω y | ≪ |Ω x |. The integrand in (C15) is negligible unless |Ω x | ∆t 1, which therefore means we can assume |Ω y | ∆t ≪ 1: (C16) Now separate out the UV divergent pieces by writing where D + P represent the divergent pieces from the ∆t → 0 behavior of the integrand and f is everything else (so that the integral of f will be finite). Specifically, we split the divergent pieces into which represents vacuum contributions (q = 0) as well as double pole terms of the form (∆t) −2 ln Ω, and which represents the (regulated) simple pole terms of the form Ω/∆t. In order for (C17) to reproduce (C16), we then have the remaining D(∆t; ǫ) turns out to integrate to zero, and so we may drop D altogether. The regulated small-∆t behavior of (C18b) is equal to the sum of the crossed diagram entries (the entries above the line) in the second column of Table I. Reviewing the above derivation of (C18b) and isolating the separate contribution from each diagram will give the individual entries for crossed diagrams in the table.

Sequential diagrams
The sequential diagrams proceed similarly. xyxȳ is the same as the earlier xȳxy except that the y factor is not conjugated. The analogs to (C12) and (C14) are For the other sequential diagrams shown in fig. 4, the starting point is just (5.1) but taking the small x and y limit appropriate to the independent emission model: .
Adding in related diagrams and also performing those integrals which correspond to simple factors of the single splitting rate, Totaling (C21) and (C23) and adding in x ↔ y permutations, Again focusing on the y ≪ x limit, Separating UV divergences as before, with and f seq (∆t) = ∆t Re(iΩ x ) Re Ω 2 y csc 2 (Ω y ∆t) + D seq (∆t; ǫ) integrates to zero, and the remaining divergences (C27b) correspond to the sequential diagram entries (the entries below the line) in the second column of Table I.

Checking the diagnostic
For the sake of completeness, we should verify that the independent emission model results pass the diagnostic (2.4), as they must. Using (C4), we find (for y ≪ x), that the various integrals above give 2 Re dΓ dx dy As promised, these sum to zero.

Appendix D: Test of dimensional regularization for independent emission model
Testing the diagnostic (2.4) for dimensional regularization will be very similar to appendix C except that now the regularized single splitting rate is given in terms of (3.5) and (5.3), so that instead of (C2). Here, P (x) is the d-dimensional DGLAP splitting function. Remember that the test corresponds to the case of small x. The dimension dependence of time-independent factors will not be interesting, and so we rewrite (D1) as where (for small x, for which is normalized so that N x (d=2) = 1 (in small-x limit).
Note also that N x is dimensionful when d = 2.

xyȳx + xȳyx
For xyȳx and related diagrams, we need the analog of the integral (C9): We can make the integrals we need to do a little easier, however. The UV regularization is only required for the small-∆t behavior of our later results, where, for the crossed diagrams, ∆t is the separation between the two intermediate times in the diagram. What we need to do is compute the analog for dimensional regularization of the regulated small-∆t divergences D(∆t) + P (∆t), which were given by (C18) and (C27) for the iǫ prescription. The other contributions, (C29) and (C31), will be exactly the same in the two regularization methods.
For the diagrams xyȳx + xȳyx, ∆t is specifically ∆t y . So, in order to isolate the divergences, we will only need to know (D5) when ∆t y is small. We can take advantage of this by rewriting (D5) as .
We may then make a small-∆t x approximation to the last integrand above (since ∆t x < ∆t there), .

(D7)
The other integral we need is where τ ≡ iΩ x ∆t x . This evaluates to (see appendix A). Combining the above formulas and (C8), and restricting the final ∆t integration to small ∆t < a, 2 Re dΓ dx dy .

xȳxy
For xȳxy, we need the analog of the integral (C13): Using the same trick as above of rewriting the integral as and focusing on the case of small ∆t, we find The fact that the argument of the last logarithm in (D11) is not dimensionless is an artifact of (i) choosing a normalization N x (D3) whose dimension depends on d and (ii) not having expanded N x and N y in ǫ.

Total crossed diagrams
The sum of (D11) and (D14) and their x ↔ y permutations gives 2 Re dΓ dx dy for the sum of all QED crossed diagrams in the independent emission model. Because the answer is finite as d → 2, we have now been able to set d=2 for N x N y as well.
The a → 0 divergences of (D15) are canceled by divergent contributions from the region of integration ∆t > a, which was not included above. Specifically, our goal here has been to evaluate the analog, in dimensional regularization, of the ∞ 0 d(∆t) (D + P ) of appendix C (and to then reuse that appendix's result for the remaining ∞ 0 d(∆t) f ). What (D15) gives us is a 0 d(∆t) (D + P ). So, to compute the total contribution from D + P in dimensional regularization, we need to add in ∞ a d(∆t) (D + P ). Since this integral involves ∆t > a rather than ∆t → 0, it does not require further UV regularization and there is no reason not to use the d=2 expressions for D and P . We can take the latter from (C18), ignoring the iǫ prescriptions since ∆t > a. But ignoring the iǫ prescriptions in (C18b) gives P = 0, leaving Adding this to (D15) gives, finally,

Sequential diagrams
As observed in appendix C, xyxȳ is the same as the earlier xȳxy except that the y factor is not conjugated. The corresponding analog of (D14) is For xxyȳ, we take (C22) but use (D1) for dΓ/dx d(∆t x ). The analog of (C23) is then (Ω x + Ω y ) .
(D23) 5. Summary in the limit y ≪ x In the limit y ≪ x, which was taken to simplify the discussion in both the main text and in appendix C, the pole pieces (D17) and (D23) are identical to the ones computed with the iǫ prescription given by (C30) and (C32). Since the non-pole pieces are unaffected by regularization, those will be the same too, as quoted in the main text in (6.1).
which for ∆t = 0 is real and positive definite, The prescription which makes this sensible for ∆t = 0 is ∆t → ∆t − , as before. This is equivalent to the previous method after changing the basis.
For xȳyx, the corresponding normal mode propagator is instead (see appendix E of ref. [3]) with limit As in the previous method, we need to replace one ∆t by ∆t − and the other by ∆t + to make (E11) sensible when ∆t = 0. This prescription is not equivalent to what we did before, once we change basis back to (C 34 , C 12 ) and/or (C 41 , C 23 ). It does, though, at least have the virtue of treating the two ends of the 4-particle propagation symmetrically.
The new prescription also gives the same results in the y ≪ x ≪ 1 limit of QED. Unfortunately, outside of this special case, it does not agree with the result derived in this paper using dimensional regularization (nor with the first proposal in this appendix). Which method should we trust?
There is a dirty secret shared by all of the iǫ methods just proposed: they only attempted to regulate the 4-particle propagation. In the d=2 derivation of AI [3], there was a step at AI (5.7) where an infinitely oscillatory term associated with time-integrated 3-particle evolution was discarded. The argument given there was that a finite infinitely-oscillatory function would give zero when later integrated against a smooth function. However, the pole pieces of our calculation are coming from infinitesimal ∆t ∼ ǫ when we regulate the 4-particle propagation with an iǫ prescription. So, when we use the iǫ prescription, it is important that we treat arbitrarily short-time features of our expressions correctly. That's inconsistent with requiring the infinitely oscillatory piece of AI (5.7) to be integrated only against smooth functions. So we need to keep that piece and regulate it as well. Could we regulate it with yet another iǫ prescription? We attempted this but were unable to find a method that gave a convincing, unique answer independent of details of exactly how we chose the magnitudes of the various ǫ factors.
In contrast, the advantage of dimensional regularization is that it simultaneously treats all UV problems in a consistent manner.
Appendix F: 4-particle propagator in medium In this appendix, we discuss the d-dimensional generalization of the 4-particle propagator without first taking the ∆t → 0 limit relevant to the main text.
with respect to the parameters (a, b, c). Unfortunately, the above integral is UV divergent (even in d dimensions) from the region of integration B 2 → 0. However, the divergent part must vanish when we take the derivatives necessary to get the integrals (4.26) because the latter integrals are all UV convergent. We will see that this works out. For now, let us just temporarily regulate the UV divergence by replacing (G1) by |i δH|Bȳ Bȳ, tȳ|B y , t y B y |−i δH| In d dimensions, this gives The only difference here is the normalization of | −1 according to (A3c), the use of ddimensional (ᾱ,β,γ), and the overall factor of E −2d to keep (ᾱ,β,γ) dimensionless. Next, use rotation invariance to write ∇n Bȳ ∇ n B y Bȳ, tȳ|B y , t y