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REVIEW 3 major objections 3 minor 64 references

The paper claims that sub-eikonal corrections nearly cancel the all-path-length suppression of jet momentum broadening, so the standard GLV result for q-hat survives in small collision systems.

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2026-08-02 22:33 UTC pith:6SOH36E3

load-bearing objection Careful, honest broadening calculation whose headline sub-eikonal factor is a unitarity-motivated truncation with uncontrolled error in the region that matters. the 3 major comments →

arxiv 2602.16450 v2 pith:6SOH36E3 submitted 2026-02-18 hep-ph nucl-th

Momentum Broadening in the Opacity Expansion: All-Path-Length Corrections and Improved Regge Kinematics

Dario van den Berg , Isobel Kolb\'e This is my paper
classification hep-ph nucl-th PACS 25.75.-q12.38.Mh
keywords momentum broadeningjet quenchingopacity expansionGLV formalismall-path-length correctionsub-eikonal correctionjet transport coefficient q-hatsmall collision systems
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Jet momentum broadening is the transverse momentum a high-energy parton accumulates while crossing a quark-gluon plasma; the standard GLV formula for it assumes large separation distances and a parton energy much larger than the momentum transfer. This paper relaxes both assumptions to first order in the opacity expansion (single scattering). It finds that dropping the large-separation assumption — the all-path-length (APL) correction — suppresses broadening at low transverse momentum, with a correction scaling roughly as 1/(L p_perp) that dominates in small collision systems. Keeping the next order in the parton-energy expansion — the sub-eikonal correction — enhances broadening at high p_perp, where the momentum transfer approaches the parton's light-cone momentum. Combined, the sub-eikonal effect partially offsets the APL suppression, and for the jet transport coefficient q-hat the compensation is essentially complete: the result nearly coincides with standard GLV, which if correct would remove the large negative high-energy correction found in the authors' earlier radiative calculation.

Core claim

The paper derives first-order-in-opacity momentum broadening with four schemes — standard GLV, GLV+APL, sub-eikonal, and combined — and collects them in Table II. Two correction factors multiply the standard GLV integrand: an all-path-length factor (1 − 1/2 e^{−μΔz})², from keeping the potential pole that the large-separation approximation discards, and a sub-eikonal factor 4/(1+γ)², from replacing the light-cone propagator poles by their exact values, γ = √(1 − 2q²/P⁺²). The first factor is at most 1 and suppresses low-p⊥ broadening when μΔz is not large; the second is at least 1 and grows at high p⊥ where q approaches P⁺/√2. The paper's central claim: combined, the sub-eikonal enhancement

What carries the argument

The argument runs on the residue structure of the parton propagator in light-cone coordinates. In the eikonal limit the propagator has poles at q_z ≈ −√2 P⁺ and q_z ≈ −q²/P⁺, which is why the standard result is built from 1/P⁺ and μ; the sub-eikonal scheme replaces these with the exact quadratic poles β_± = −(√2/2)P⁺(1 ± γ), packaging all finite-energy corrections into the single all-order factor γ = √(1 − 2q²/P⁺²). Unitarity — the requirement that scattering only redistributes particles, expressed as a factorized (|J(p−q)|² − |J(p)|²) f(q) structure — forces the authors to strip the extra terms from the full single-scattering residue, keeping the P⁺ → β_+ rescaling but dropping terms such a

Load-bearing premise

The load-bearing assumption is that the non-rescaled parts of the propagator residues — the terms involving β₋² and (β₋ − β₊)² that are dropped to preserve unitarity — are negligible; near the kinematic endpoint q ~ P⁺/√2 this is not obviously true, and if those terms contribute the claimed sub-eikonal enhancement, and thus the restoration of the standard q-hat, could change in size or sign.

What would settle it

Perform the same first-order opacity calculation keeping the full residue (eq. A75) instead of substituting 1/(β₊² μ⁴), and numerically evaluate the broadening distribution and q-hat for, say, P⁺ = 7 GeV and q within 20% of P⁺/√2. If the integrated result moves away from the standard GLV curve by more than the few-percent level claimed, or changes sign relative to the APL-only curve, the complete-mitigation result is falsified.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • At fixed system size L, the APL-corrected distribution falls below standard GLV at low p_perp, with the suppression growing as L shrinks; the sub-eikonal correction raises the distribution at high p_perp, with the effect appearing at lower p_perp for smaller P⁺.
  • For the integrated jet transport coefficient q-hat, the combined correction nearly coincides with standard GLV, removing the large negative APL-only deviation reported by the authors' previous radiative calculation.
  • The results extend the validity of the broadening calculation from the regime q ≪ P⁺ to the extended range μ_D ≤ q ≤ (√2/2) P⁺, i.e. to parton energies low enough that the transverse momentum transfer is no longer negligible compared with the light-cone momentum.
  • Since the APL correction becomes negligible for large L and the sub-eikonal correction vanishes for q ≪ P⁺, every corrected distribution reduces to standard GLV in the appropriate limits, so existing large-system phenomenology is unchanged.
  • In small systems such as pp, pA, OO, and NeNe, where the large-separation and eikonal assumptions are questionable, these corrections constrain how much medium-induced broadening should actually be suppressed relative to the standard estimate.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The paper itself shows (Appendix A) that the raw single-scattering residue contains terms (β₋ − β₊)² and β₋² that it drops to keep unitarity; the drop is asserted, not estimated, so the size of the sub-eikonal enhancement in the region where γ differs most from 1 remains an open question.
  • If those dropped terms are retained, the sub-eikonal factor is not simply 4/(1+γ)²; near the kinematic endpoint γ ≪ 1 the differences are order one, so the complete-mitigation result and even the sign of the sub-eikonal correction are not settled by the present calculation.
  • The same residue mechanism, applied to the radiative energy-loss calculation, would be the natural place to test whether the large negative high-energy APL correction the authors identified earlier is similarly undone by sub-eikonal terms; this paper's broadening result is evidence for that possibility but not a proof.
  • Because the two correction factors multiply the same integrand, one could experimentally probe the claimed compensation by measuring broadening (or q-hat-like) distributions in small versus large systems at moderate parton energies and checking whether the ratio returns to the standard GLV curve once P⁺ is lowered.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper extends the first-order opacity-expansion (GLV) calculation of momentum broadening to include two corrections: all-path-length (APL) terms, which relax the large separation distance assumption, and sub-eikonal terms, which retain finite-P^+ corrections in the propagator kinematics. The results are summarized in Table II and Eq. (A133): the APL correction multiplies the standard GLV kernel by (1 - (1/2)e^{-\mu \Delta z})^2, the sub-eikonal correction multiplies it by 4/(1+\gamma)^2 with \gamma = \sqrt{1-2q^2/P^{+2}}, and the combined scheme uses both factors. The paper claims that the APL suppression is partially mitigated by the sub-eikonal enhancement, and that the combined result for \hat{q} nearly coincides with the standard GLV result. The standard GLV limit is correctly recovered when the correction factors tend to 1, and the numerical ratios show the claimed qualitative trends. The APL part is essentially the authors' earlier result, while the new sub-eikonal factor is obtained by keeping only the 'rescaling' parts of the residues and dropping terms to preserve the factorized unitary form.

Significance. If the central result were established, this would be an important step: it would soften the large negative APL correction found in the authors' earlier radiative energy-loss work and would provide a concrete broadening prediction for small systems, where eikonal and large-L approximations are questionable. The paper is clearly organized, the standard GLV rederivation is useful, and the analytic expressions are explicit and easy to test numerically. However, the main new ingredient—the sub-eikonal factor 4/(1+\gamma)^2—rests on an uncontrolled truncation of the exact residues, and the abstract's \hat{q} claim is not backed by any calculation in the body. Until those points are addressed, the claimed mitigation of the APL suppression is not established.

major comments (3)
  1. [Appendix A, Eqs. (A75)-(A76) and (A102)-(A108)] The exact sub-eikonal single-scattering residue is proportional to [(\beta_- - \beta_+)^2(\beta_-^2 + \mu^2)^2]^{-1}, but Eq. (A76) replaces it by (\beta_+^2 \mu^4)^{-1}. The paper says this is needed to avoid unitarity violation, but no small parameter controls the replacement. With \beta_\pm = -(\sqrt{2}/2)P^+(1\pm\gamma), one has (\beta_- - \beta_+)^2/\beta_+^2 = 4\gamma^2/(1+\gamma)^2, which tends to 0 as \gamma\to 0, and \beta_-^2 \sim \tfrac12 P^{+2}(1-\gamma)^2 is not small relative to \mu^2 = q^2+\mu_D^2 near the kinematic endpoint. The same uncontrolled 'rescaling-only' truncation is applied to the double-scattering residues (A104)-(A108), whose exact forms contain (\beta_3^- - \beta_2^-)^2, \mu_1\mu_2 and e^{-(\mu_1+\mu_2)\Delta z} terms. Since the factor 4/(1+\gamma)^2 in the central result (A133) is precisely what produces the claimed high-p enhancement and mitigation of the
  2. [Abstract and Sec. VI] The abstract states that for \hat{q} the combined APL+SUB result 'coincides closely' with the standard GLV result, i.e. that the mitigation is essentially complete. However, no definition or numerical result for \hat{q} appears in Sec. VI or in the appendices; the numerical section presents only ratios of dN^{(1)}/d^3p as functions of p, L, and P^+. The \hat{q} claim is therefore unsupported by the manuscript as written. If \hat{q} is to be a central result, its definition in terms of the broadening distribution and the corresponding calculation or plot must be included.
  3. [Secs. III and B] The treatment of the crossed diagram needs clarification. The text says unitarity 'ensures' the crossed diagram does not contribute, but Appendix B computes that its contribution starts at O(1/P^{+4}), which is beyond the paper's stated O(1/P^{+3}) accuracy. If that power-counting statement is correct, the diagram's absence follows from order counting and the unitarity discussion is irrelevant. If the diagram is not beyond that order, excluding it because its inclusion would violate unitarity is circular. The manuscript should state explicitly whether the crossed diagram is excluded by power counting or by a physical selection rule.
minor comments (3)
  1. [Eq. (A62)] The factorization is written as (q_z - \beta_+)(q_z + \beta_+), but the poles used immediately afterward are \beta_- and \beta_+. It should read (q_z - \beta_-)(q_z - \beta_+).
  2. [Eq. (A111)] In the first term of the square bracket, '1/(\beta_+^2 \mu)' should evidently be '1/(\beta_+^2 \mu^4)' to match Eq. (A112) and the final result.
  3. [General] There are several typographical errors: 'distribtution', 'logitidual', and inconsistent use of 'all order' vs. 'all-orders'. Also, the APL ingredient is taken from the authors' earlier works [48,49]; the text should state more precisely which elements of Table II are new to this paper.

Circularity Check

1 steps flagged

Sub-eikonal factor 4/(1+γ)^2 is imposed by the unitarity factorization ansatz, not derived from the residues.

specific steps
  1. self definitional [Appendix A, (GLV)SUB calculation, single-scattering replacement at eqs. (A75)-(A76); double-scattering residues eqs. (A102)-(A108)]
    "The additional terms present a problem as one can clearly see the structure is not consistent from M1 to M2, resulting in not factoring like J(p−q)f(q)−J(p)f(q) = (J(p−q)−J(q))f(q). However if we focus on the rescaling part of the correction. We preserve P+ → β+ i.e. incorporating the γ-factor but dropping additional terms the residues become..."

    The exact single-scattering residue computed from the diagram is 1/[(β− − β+)²(β−² + µ²)²] (eq. A75). The paper replaces it with 1/(β+²µ⁴) (eq. A76) solely to maintain the factorized form J(p−q)f(q)−J(p)f(q), i.e., to enforce unitarity. Since β+² = ½P+²(1+γ)², the advertised sub-eikonal factor 4/(1+γ)² is just the inverse of this imposed replacement. The same 'dropping additional terms' is applied to the double-scattering residues (A102–A108). Therefore the claimed enhancement is not an independent output of the diagrammatic calculation; it is the unitarity ansatz itself. The combined result (A133) inherits this by-construction factor.

full rationale

The APL ingredient is re-derived in Appendix A (eqs. A37–A54) and is not circular merely because it agrees with the authors' earlier paper [48]; self-citation of [48,49] is therefore not load-bearing for the APL suppression itself. The sub-eikonal step, however, is constraint-selected: exact residues containing (β− − β+)² and β−² are discarded with the stated reason that keeping them would violate unitarity, and only the β+ rescaling is kept. That makes the central mitigation claim depend on the chosen ansatz rather than on a controlled evaluation of all terms. The paper is self-contained in that no data are fitted and the APL factor is genuinely computed, but the headline sub-eikonal correction reduces by construction to the factorization assumption, with the dropped non-factorizing terms unsuppressed precisely in the γ≠1 region where the correction matters. No external benchmark is used to validate the truncation. Score 6 reflects a partial circularity in the central correction factor; it is not a full 8–10 because the derivation does contain independent residue calculations and the APL suppression is independently re-derived here.

Axiom & Free-Parameter Ledger

3 free parameters · 8 axioms · 0 invented entities

The central claims rest on the GLV kinematic/diagrammatic framework inherited from Refs. [18,45]; the numerical study adds hand-chosen input scales. No new physical entities are postulated — γ = √(1−2q²/P⁺²) is a resummation of existing kinematic poles, not a new object. The main conceptual burden is the ad-hoc unitarity selection rule in the sub-eikonal computation and the phase approximation e^{−iβ₋∆z} ≈ 1.

free parameters (3)
  • Strong coupling αs = 0.3
    Hand-chosen for the numerical illustrations; a standard pQCD value. The qualitative central claims do not depend on it.
  • Debye mass µ_D = 0.5 GeV
    Hand-chosen scale that sets the infrared cutoff p ≥ µ_D of the kinematic domain in eq. (26); not fitted to data.
  • Parton momentum P⁺ and system size L scan = P⁺ ∈ {7,8,10,40} GeV; L ∈ {1,4,10} fm
    Parameter scan for Figures 2–4; the body claims the qualitative behavior holds across these values. The abstract's (unsupported) q̂ claim would depend on these, but no q̂ calculation is given.
axioms (8)
  • domain assumption Gyulassy–Wang static Debye-screened potential V(q) = 4παs/(q² + µ²) with δ(q⁰)
    Eqs. (1)–(2); the medium potential, its screening scale, and the static-potential approximation are inherited from GLV literature [18].
  • domain assumption Exponential distribution of scattering centers ρ(∆z) = (2/L)e^{−2∆z/L}
    Eq. (10); a modeling choice inherited from GLV. The paper notes other choices exist ([48]) and are not explored.
  • domain assumption Collimated source J(p) with f(E) = 1/E⁴ and slow variation of J with p up to O(1/P⁺⁴)
    Eq. (3) and Appendix B; needed for the factorization (|J(p−q)|² − |J(p)|²) and for dropping higher-order J corrections at O(1/P⁺⁴).
  • ad hoc to paper Unitarity (particle-number conservation) enforced by requiring the factorized form, used to select which sub-eikonal terms are kept
    Section III and Appendix A: 'keeping such additional terms results in a unitarity violation.' The reported P⁺²/β₊²µ⁴ is the unitarity-compatible truncation, not the complete residue.
  • domain assumption Phase approximation e^{−iβ₋∆z} ≈ 1 (∆z q²/P⁺ ≪ 1) while e^{−µ∆z} APL terms are retained
    Appendix A; stated to be required 'to preserve unitarity' following Ref. [22]. Inconsistent in principle with keeping the µ-pole path-length dependence, justified only by that unitarity condition.
  • domain assumption Contact limit z₂ → z₁ for double scattering, with impact-parameter averaging setting µ₁ = µ₂
    Eqs. (7)–(9) and Appendix A; the standard GLV contact approximation used to obtain the (1 − ½e^{−µ∆z})² factor.
  • domain assumption Physical kinematic domain µ_D ≤ q ≤ (√2/2)P⁺ with γ real
    Eq. (26); the UV cutoff is imposed by requiring γ = √(1−2q²/P⁺²) real; above it the sub-eikonal kinematics are undefined and the paper claims perturbative control only within this window.
  • standard math Residue theorem, Fourier transforms, SU(N_c) color algebra
    Used throughout Appendices A and B; standard background, not in question.

pith-pipeline@v1.3.0-alltime-deepseek · 23379 in / 30396 out tokens · 268244 ms · 2026-08-02T22:33:01.919467+00:00 · methodology

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read the original abstract

We present a detailed study of momentum broadening and the jet transport coefficient $\hat{q}$ for high-energy partons traversing the Quark-Gluon Plasma (QGP), extending the Gyulassy-Levai-Vitev (GLV) formalism to include both all-path-length (APL) and sub-Regge kinematical corrections. Traditional GLV calculations rely on the large separation distance and large formation time approximations, which are valid for large systems but whose applicability to small systems, such as $pp$ and $p/d$A collisions, may fail. We derive analytic expressions for the momentum broadening distributions and $\hat{q}$ to first order in the opacity expansion, and perform a detailed numerical investigation to quantify their impact. The APL correction suppresses momentum broadening at low $p_{\perp}$, with a correction scaling as $\propto 1/(L\, p_{\perp})$ that dominates for small systems, while converging to the standard GLV result at large $L$. The sub-Regge kinematical correction enhances momentum broadening at high $p_{\perp}$, becoming significant when the transverse momentum transfer approaches the magnitude of the parton's large light-cone momentum component, and vanishing in the Regge limit where this ratio is small. When both corrections are combined, the sub-Regge kinematical correction partially mitigates the suppression induced by the APL term; in the case of $\hat{q}$, this mitigation is essentially complete, with $\hat{q}_{(\mathrm{APL+SUB})}$ found to coincide closely with the standard GLV result. These findings demonstrate that sub-Regge kinematical corrections can resolve the long-standing problem of large negative energy-loss contributions at high energies identified in earlier studies.

Figures

Figures reproduced from arXiv: 2602.16450 by Dario van den Berg, Isobel Kolb\'e.

Figure 1
Figure 1. Figure 1: FIG. 1: The relevant Feynman diagrams for momentum broadening at leading order in the opacity. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The broadening distribution to first order in the opacity expansion [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The ratios of the various corrected broadening distributions — the all path-length–corrected (GLV + APL), [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The ratios of the various corrected broadening distributions — the all path-length–corrected (GLV + APL), [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Double scattering diagram with crossed gluon exchanges. [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗

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Reference graph

Works this paper leans on

64 extracted references · 46 linked inside Pith

  1. [1]

    e−µ1∆z′ (A26) In the contact limitz 2 →z 1, so ∆z ′ →0 Res[I3, q(3) 2,z] = i(4παs)2 4P +2µ2 2(µ2 1 −µ 2

  2. [2]

    (A20) and eq

    (A27) Combining eq. (A20) and eq. (A21) gives I3I2 = −2πi 2π 2X j=1 Res[I3I2, q(j) z ] (A28) = −(4παs)2 2P +2 1 µ2 1µ2 2 − 1 2µ2 1(µ2 2 −µ 2

  3. [3]

    − 1 2µ2 2(µ2 1 −µ 2 2) ! (A29) = −(4παs)2 2P +2 (µ2 2 −µ 2 1) 2µ2 1µ2 2 (µ2 2 −µ 2 1) ! = −(4παs)2 4P +2µ2 1µ2 2 .(A30) Now we apply the average over the impact parameter as per our discussion in eq. (7). Which setsµ 1 =µ 2 as well as J(p−q 1 −q 2) =J(p). I3I2 = −(4παs)2 4P +2µ4 .(A31) The non-interacting matrix, fig. 1a, is given by M∗ 0 =−ie −ipz0 J ∗(p...

  4. [4]

    e−µ1∆z′ ,(A45) Res[I3, q(3) 2,z] = (4παs) 2 √ 2P +µ2 2(µ2 +µ 1) 1 µ2 −µ 1 + e−(µ1+µ2)∆z 2µ1 e−µ2∆z′ .(A46) Grouping the relevant terms together and applying the contact limit ∆z ′ →0, I3 = −i(4παs)√ 2P + " 1 µ2 1µ2 2 − 1 2µ2 1(µ2 2 −µ 2

  5. [5]

    + 1 2µ2 2(µ2 2 −µ 2 1) # − 1 2 1 µ2 1µ2 2 e−µ1∆z + 1 4µ1µ2 2(µ2 +µ 1) e−(µ1+µ2)∆z ! .(A47) The bracket without any exponential term simplifies as [· · ·] =1 2µ2 1µ2 2 .(A48) Therefore applying the contact limit (eq. (7)) one has, I3I2 = −(4παs)2 4P +2µ4 1− 1 2 e−µ∆z !2 (A49) The double scattering contribution to the broadening distribution is therefore gi...

  6. [6]

    (A80) ForI 3, I3 = ˆ dq2,z 2π −i4παs (β− 3 −β + 3 )(q2 2,z +µ 2 1) ! 1 (p−q 2)2 +iϵ 4παs q2 2,z +µ 2 2 e−iq2,z∆z′ .(A81) The poles are q(1) 2,z =−iµ 2,(A82) q(2) 2,z =−β − 2 ,(A83) q(3) 2,z =−β − 3 −iµ 1.(A84) Therefore the residues Res[I3, q(1) 2,z] = lim q2,z→q(1) 2,z −i4παs (β− 3 −β + 3 )[(β− 3 −q 2,z)2 +µ 2 1] e−iβ− 3 ∆z + 4παs 2µ1(q2,z −iµ 1 −β − 3 )...

  7. [7]

    One can see at this point the cross terms will be very complicated

    (A95) and Res[I3, q(3) 2,z] = lim q2,z→q(3) 2,z −i4παs (β− 3 −β + 3 )(((((((((q2,z −β − 3 +iµ 1)(q2,z −β − 3 −iµ 1) e−iβ− 3 ∆z (A96) + :0 4παs 2µ1(q2,z −iµ 1 −β − 3 )(q2,z −iµ 1 −β + 3 ) e−i(−iµ1+q2,z)∆z ! (A97) 1 (q2,z −β − 2 )(q2,z −β + 2 ) 4παs (q2,z −iµ 2)(q2,z +iµ 2) e−iq2,z∆z′ (((((((((q2,z −β − 3 +iµ 1) (A98) = (4πα2)2 2µ1(β− 3 −β + 3 )(β− 3 −iµ 1 ...

  8. [8]

    ,(A105) Res[I3, q(3) 2,z] = (4πα2)2 2µ1(−β+ 3 )(−iµ1)(−β+ 2 )(µ2 2 −µ 2

  9. [9]

    Tr⟨M2M∗ 0⟩= N A⊥ C2(R)C(R)(4πα s)2 ˆ ρ(∆z) ˆ d2q⊥ (2π)2 |J(p)| 2P +2 " − 1 2β+2µ4 # (A109) The momentum broadening distribution The broadening distribution as per eq

    ,(A106) and therefore I3 =−i 3X j=1 Res[I3, q(j) 2,z] = −(4παs)2 2µ2 1µ2 2β+ 3 β+ 2 (A107) 21 Now, if one applies the averaging over the impact parameter, one has I3 = −(4παs)2 2µ4β+2 .(A108) Crucially, one has the factor of 1 2 . Tr⟨M2M∗ 0⟩= N A⊥ C2(R)C(R)(4πα s)2 ˆ ρ(∆z) ˆ d2q⊥ (2π)2 |J(p)| 2P +2 " − 1 2β+2µ4 # (A109) The momentum broadening distributio...

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    + 1 2µ1β+ 3 (µ1 +µ 2) e−(µ1+µ2)∆z ! i(4παs)2 2β+ 2 µ2 2 ,(A119) Res[I3, q(2) 2,z] = 1 β+ 3 µ2 1 − 1 2β+ 3 µ2 1 e−µ1∆z ! −i(4παs)2 µ2 2β+ 2 ,(A120) Res[I3, q(3) 2,z] = i(4παs)2 2µ2 1β+ 2 β+ 3 (µ2 2 −µ 2

  11. [11]

    Constant X Res ! = i(4παs)2 2β+ 2 β+ 3 µ2 2(µ2 1 −µ 2

    (A121) The integral is given by I3 =−i 3X j=1 Res[I3, q(j) 2,z] (A122) (A123) Adding the constant terms (terms withoute −µ∆z factors). Constant X Res ! = i(4παs)2 2β+ 2 β+ 3 µ2 2(µ2 1 −µ 2

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    + −i(4παs)2 β+ 2 β+ 3 µ2 1µ2 2 + i(4παs)2 2β+ 2 β+ 3 µ2 1(µ2 2 −µ 2

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    (A124) = −i(4παs)2 2β+ 2 β+ 3 µ2 1µ2 2 (A125) The full residue is therefore X Res = −i(4παs)2 2β+ 2 β+ 3 µ2 1µ2 2 + i(4παs)2 2β+ 2 β+ 3 µ2 1µ2 2 e−µ1∆z + −i(4παs)2 4β+ 2 β+ 3 µ1µ2 2(µ1 +µ 2) e−(µ1+µ2)∆z (A126) Applying the average over the impact parameter. I3 =−i X Res (A127) = −(4παs)2 2β+2µ2 1− 1 2 e−µ∆z !2 (A128) Tr⟨M2M∗ 0⟩= N A⊥ 1 dA C2(R)C(R)(4πα s)...

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