QCD-Gravity double copy in Regge asymptotics: from $2\rightarrow n$ amplitudes to radiation in shockwave collisions

Himanshu Raj; Raju Venugopalan

REVIEW 2 major objections 4 minor 2 cited by

QCD-Gravity double copy in Regge asymptotics: from $2\rightarrow n$ amplitudes to radiation in shockwave collisions

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The gravitational Lipatov vertex is exactly the QCD Lipatov vertex squared minus a QED bremsstrahlung term, and its soft limit is Weinberg's soft graviton theorem.

desk verdict Solid, careful lecture notes that re-derive known results well, with a real but self-flagged gap in the gravitational double-log ladder that should be stated more prominently. read the letter →

arxiv 2507.21252 v2 pith:XPEDANAL submitted 2025-07-28 hep-th gr-qchep-phnucl-th

Himanshu Raj , Raju Venugopalan This is my paper

classification hep-thgr-qchep-phnucl-th

keywords doublecopyReggeasymptoticsBFKLequationLipatovvertexgravitonreggeizationsofttheoremshockwavecollisionsColorGlassCondensate

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that multi-particle production in QCD and in Einstein gravity at ultrarelativistic energies are governed by the same effective-ladder structure: nonlocal emission vertices and reggeized t-channel propagators, with the gravitational building blocks literally built from the QCD ones. Its central identity is the gravitational Lipatov vertex $C^{\mu\nu}=\frac12 C^\mu C^\nu - \frac12 N^\mu N^\nu$, where $C^\mu$ is the QCD Lipatov vertex and $N^\mu$ carries the QED bremsstrahlung factor. If this identity holds, gravitational $2\to 2+n$ amplitudes can be assembled from gauge-theory data, and the soft-graviton limit of the vertex recovers Weinberg's soft theorem. The payoff would be a weak-coupling dictionary between saturation physics in QCD shockwave collisions and gravitational radiation in trans-Planckian shockwave collisions.

What carries the argument

The carrying object is the gravitational Lipatov vertex $C^{\mu\nu}(k_1,k_2)=\frac12 C^{\mu}(k_1,k_2)C^{\nu}(k_1,k_2)-\frac12 N^{\mu}(k_1,k_2)N^{\nu}(k_1,k_2)$, where $C^\mu$ is the QCD Lipatov vertex for emitting a gluon in multi-Regge kinematics and $N^\mu=\sqrt{k_1^2k_2^2}\,(p_1^\mu/(p_1\cdot\ell)-p_2^\mu/(p_2\cdot\ell))$ contains the QED bremsstrahlung factor. This vertex is the one-graviton emission building block of the effective ladder; together with the reggeized propagator it packages the many Feynman diagrams of Einstein gravity into a single bilinear object, and in the soft limit it becomes the Weinberg emission current.

What would settle it

Compute the complete tree-level $2\to 3$ graviton amplitude in multi-Regge kinematics while keeping all contact diagrams, and check whether the coefficient of $1/(k_1^2k_2^2)$ is exactly $\frac12(C^\mu C^\nu - N^\mu N^\nu)$; any extra leading-power tensor structure, or any discrepancy in the $k\to 0$ limit against the Weinberg current, would refute the double-copy identity.

Watch

Extended reading notes

Core claim

The central claim is that, in the Regge limit where $\sqrt{s}\gg |\ell_\perp|$ and logarithms of $s$ are resummed, the $2\to 2+n$ amplitude in Einstein gravity has exactly the same ladder form as in QCD: external vertices, $n$ copies of the gravitational Lipatov vertex, and reggeized graviton propagators dressed by $(\hat s_i/k^2)^{\alpha(k_i^2)}$. The new element is the precise bilinear identity for the gravitational vertex, Eq. (3.38), which uses the QCD Lipatov vertex and a QED bremsstrahlung vector. The paper further claims that the soft-graviton limit of this vertex reproduces Weinberg's soft theorem, so the Regge framework is a smooth extension of soft-graviton physics to hard emission, and that the same correspondence holds for shockwave backgrounds, propagators, and multi-particle radiation.

Load-bearing premise

The load-bearing premise is that the one-loop Regge trajectory exponentiates to all orders, dressing every t-channel gluon or graviton propagator by $(\hat s_i/k^2)^{\alpha(k_i^2)}$; the paper itself notes this was conjectured for gluons and assumed for gravitons, and that reggeization breaks down beyond next-to-leading-logarithmic accuracy.

Editorial extensions

If this is right

The $2\to 2+n$ gravitational amplitude at leading logarithmic order is fully determined once the QCD Lipatov vertex and the QED bremsstrahlung factor are substituted into the bilinear identity.
Weinberg's soft graviton theorem appears as the $k\to 0$ limit of the gravitational Lipatov vertex, so soft and hard gravitational emission are described by one continuous vertex.
The gravitational BFKL equation at $q=0$ has an analytic eigenvalue solution structurally similar to QCD, but with a UV cutoff dependence that must be completed by double-log resummation.
Gluon and graviton radiation in shockwave collisions obey parallel Wilson-line and rung structures, allowing the same dilute-dense and dense-dense power counting to be used in both theories.
The exponentiation mechanism behind the QCD pomeron also produces the trans-Planckian eikonal and its inelastic corrections, tying saturation physics to black-hole formation dynamics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable extension the paper leaves implicit is to match the subleading soft-graviton theorem against the sub-eikonal terms in the classical Yang-Mills+Wong radiation field, which the paper shows must be retained before the double-copy replacement; a match would extend the vertex identity beyond leading power.
The shockwave dictionary suggests that gravitational memory and supertranslation physics could be formulated in the same Regge/light-cone coordinates as color memory, giving gravitational wave observables a role in probing the semi-classical double copy.
One could test the reggeization assumption dynamically by constructing a rapidity renormalization-group equation for gravitational Wilson-line correlators analogous to the BK equation; a non-exponentiating remainder at higher orders would show where the Lipatov ladder construction loses control.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this is a lecture-note review, not a claim of new results. Its value is pedagogical—a mostly self-contained tour from BFKL to the gravitational Lipatov vertex to CGC shockwaves and the shockwave double copy, with the algebra checked through Sec. 3.4 and the soft-limit match in Sec. 3.5 done explicitly. The paper earns credit for that: the BFKL derivation, the pole-reconstruction of the Lipatov vertex, the gravitational vertex computation, and the Weinberg-limit matching are careful and consistent with the original literature. The Wong-equation double copy in Sec. 3.6 is also clearly attributed to the authors' earlier work, so there is no overclaiming.

The real soft spot is the one the stress-test flags: the gravitational 2→n ladder (Eq. 3.48) and BFKL equation (Eq. 3.56) are built only from the one-loop Sudakov trajectory, while the paper's own elastic analysis (Eqs. 3.76–3.81) shows the complete 2→2 amplitude requires resumming the double logs through the Riccati equation. Unitarity then forces either the elastic or the inelastic side to receive double-log corrections, and the paper derives neither for the inelastic channel. The paper does flag the UV-cutoff inconsistency and says the double-log resummation requires further work, so this is not a hidden flaw. But it does mean the tagline that identical methods and the same ladder structure govern QCD and gravity is stronger than what is established: in gravity, the inelastic ladder is a single-log object, and the complete leading-log construction is incomplete. For a review, I would like to see this stated more prominently, and ideally a sketch of how the Riccati/Bessel structure modifies the 2→n vertex.

Minor points: reggeization is a conjecture in QCD and assumed in gravity; the paper says so. The copy I saw was truncated around Sec. 4.2, so I did not independently check the CGC evolution, shockwave-collision sections, or the geodesic-focusing discussion. Self-citation is heavy in places but always tied to specific prior results and not a distortion.

Who this is for: graduate students and researchers wanting a map from BFKL to CGC to gravitational double copy, with detailed derivations. It deserves a serious referee: send it out. I would recommend acceptance as a review/lecture-note article, with a revision that separates what is established at single-log accuracy from what still needs the double-log resummation in the gravity sections. If I worked on small-x or trans-Planckian scattering, I would cite it as a modern reference.

Referee Report

2 major / 4 minor

Summary. These lectures re-derive the QCD and gravitational 2→2+n amplitudes in multi-Regge kinematics using dispersive methods: the multi-particle phase space, the Lipatov vertex reconstructed from pole residues, the BFKL equation and its eigenvalue solution, and the gravitational analog with reggeized propagators. The paper's central claim is that the gravitational Lipatov vertex is the double copy C^{μν}(k1,k2) = (1/2) C^{μ}(k1,k2) C^{ν}(k1,k2) - (1/2) N^{μ}(k1,k2) N^{ν}(k1,k2) of Eq. (3.38), that the QCD ladder structure and BFKL equation carry over to gravity, that Weinberg's soft graviton theorem is recovered as the soft limit of the Lipatov framework, and that the Color Glass Condensate shockwave picture has a gravitational counterpart with a classical Wong-equation double copy. The manuscript also discusses saturation, color memory, trans-Planckian scattering, and the relation to the ACV eikonal program.

Significance. The paper has substantial pedagogical and reference value: it gives parameter-free re-derivations of the QCD Lipatov vertex from pole residues, the n-particle phase space in MRK, the BFKL eigenvalue (2.116), the gravitational BFKL eigenvalue (3.74), and the matching of the Lipatov vertex to Weinberg's soft graviton current in Sec. 3.5. It also provides an explicit classical double-copy construction from the Wong equations in Sec. 3.6 and a transparent map between CGC Wilson lines and color memory in Sec. 4.1.1. The manuscript is honest about its axioms: gluon reggeization is flagged as conjectural in Sec. 2.3, the breakdown of reggeization beyond NLLx is stated in Sec. 2.6, and the extension of CGC power counting to gravitational shockwaves is presented as an assumption. If the graviton double-log issue identified below is addressed, the paper would be a reliable modern synthesis of the Lipatov–ACV and CGC approaches.

major comments (2)

[Secs. 3.2–3.4, Eqs. (3.48), (3.56), (3.78)–(3.81)] The central gravitational claim is internally incomplete. The 2→2+n amplitude in Eq. (3.48) and the gravitational BFKL equation in Eq. (3.56) are built from the one-loop trajectory α(k²) of Eq. (3.46), which is UV divergent as shown in Eq. (3.60). With the natural cutoff Λ_UV² ∼ s, the single virtual insertion σ(k_i²) in Eq. (3.44) contains a leading double logarithm log²(s/k²). The paper itself shows in Eqs. (3.76)–(3.81) that the complete elastic amplitude requires resummation of these double logs through the Riccati equation, producing Regge poles (s/t)^{±√(−3κ²t/8π²)} and the amplitude (3.81), which is not of the simple (s_i/k²)^{α(k²)} form used in Eq. (3.47). Since s-channel unitarity at Eqs. (3.49)–(3.50) requires the imaginary part of the complete elastic amplitude to be reproduced by the sum over the inelastic channels of Eq. (3.48), either the elastic amplitude (3.81) or the inelastic amplitudes (3.48) must carry double-log corrections. The manuscript derives neither for the inelastic channel; the statement immediately before Sec. 3.1 that double logs 'play a similarly crucial role' in constructing the 2→2+n amplitude is therefore not matched by the derivation. This is load-bearing for the claim that the QCD ladder and BFKL structure transfer to gravity.
[Sec. 3.3, after Eq. (3.60)] The text correctly notes that replacing Λ_UV by √s inside the BFKL eigenvalue is inconsistent because the Mellin transform over s precedes the transverse-momentum integration. However, Eq. (3.56) and its eigenvalue solution (3.74) are obtained from the Mellin-transformed equation in Eqs. (3.51)–(3.53), so they already use the fixed-cutoff trajectory. A consistent leading-logarithmic derivation would either reorder the Mellin transform and the transverse integration, or explicitly restrict the domain of Eqs. (3.56)/(3.74) to fixed Λ_UV and treat the √s case as an open problem. As written, the paper moves from the correct warning to using the cutoff-dependent eigenvalue for the cross-section without supplying the promised reanalysis.

minor comments (4)

[Eq. (2.27)] There is a typo: 'We we combine' should read 'When we combine'.
[Sec. 3.5, displayed equation above Eq. (3.82)] The index structure in the term N^{μ}(q1,q2) N^{μ}(q1,q2) is incorrect; one index should be ν to match C^{μν}(q1,q2).
[Eqs. (2.114) and (3.71)] The angular integrals are described as 'somewhat involved' but not shown; since the eigenvalues (2.116) and (3.74) are central results, a short appendix or a few intermediate steps for these integrals would improve verifiability.
[Sec. 3 intro] The sentence that double logs 'play a similarly crucial role' in the 2→2+n amplitude should contain a forward reference to the equations where this role is realized; currently the reader is not directed to the relevant construction.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the BFKL and gravitational-Lipatov derivations are parameter-free re-derivations, and the self-citations that occur are not load-bearing.

full rationale

The paper's core derivation chain is self-contained. In QCD, the 2→2 discontinuity is computed from Cutkosky rules and the MRK phase space (Eqs. 2.2, 2.45-2.46), the Lipatov vertex is obtained both by summing the explicit gg→ggg graphs (Eqs. 2.32-2.33) and by pole reconstruction (Eqs. 2.59-2.60), and the BFKL equation (Eq. 2.99) follows from the Mellin transform of the reggeized ladder (Eqs. 2.88-2.97). None of these steps fit a parameter to the final result; the input is the set of Feynman diagrams and the MRK ordering. The gravity section similarly derives the Born amplitude (Eqs. 3.18, 3.28), reconstructs the 2→3 amplitude from t-channel poles, and obtains the gravitational Lipatov vertex C^μν = (1/2)C^μC^ν - (1/2)N^μN^ν (Eq. 3.38) by imposing unitarity to remove overlapping poles. The one-loop graviton trajectory (Eq. 3.46), the 2→n amplitude (Eq. 3.48), and the gravitational BFKL equation (Eq. 3.56) are built from these objects without any fitted input. The Weinberg-regime calculation of Sec. 3.5 is a genuine limit check: it shows the soft limit of the independently constructed Lipatov vertex matches the high-energy limit of Weinberg's current (Eqs. 3.89 and 3.103). The main self-citations occur in Sec. 3.6, where the classical double copy is attributed to the authors' prior work [113], and in Sec. 4.1.1 for the color-memory map [184,185]. These are not load-bearing for the central BFKL claims: the gravitational Lipatov vertex that the classical double copy reproduces was already derived in Sec. 3.1 from unitarity, and the color-memory discussion is an interpretive correspondence rather than an input to the BFKL or CGC derivations. The paper also explicitly flags the places where its derivation rests on assumptions rather than circularity: gluon reggeization is stated as a conjecture from [7] (Sec. 2.3), graviton reggeization is obtained by exponentiating the one-loop insertion (Sec. 3.2), and the UV/double-log limitations of the gravitational BFKL equation are acknowledged (Eqs. 3.60, 3.76-3.81, and the discussion after Eq. 3.56). Those are completeness or validity caveats, not instances of an output being equal to its input by construction. I therefore find no circular step requiring a quote-and-reduction entry.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on a small number of unproven structural assumptions inherited from the field: the dominance of multi-Regge kinematics, the all-orders reggeization conjecture, the validity of color-kinematic double copy in the Regge limit, and the extension of classical-statistical CGC methods to gravity. None of these are fitted to data; they are modeling postulates. The paper flags the reggeization conjecture and the gravitational UV issues explicitly, and labels the gravitational shockwave extension as speculative.

assumptions (5)

domain assumption Multi-Regge kinematics dominance and strong rapidity ordering in the ladder
All 2->n derivations assume rho1 >> rho2 >> ... and |lambda_{n+1}| >> |lambda_n| >> ... with crossed ladders, fermion pairs, and contact graphs suppressed by powers of rho_i/rho_{i-1} (Sec. 2.2.3). If this ordering fails, the effective ladder is not the leading contribution.
domain assumption Gluon reggeization: exponentiation of the one-loop Regge trajectory
The paper states 'it was conjectured in [7] that the one-loop result in Eq. (2.87) can be exponentiated' (Sec. 2.3). The dressing of every t-channel propagator by (s_i/k^2)^{alpha(k_i^2)} is essential for the BFKL equation.
domain assumption Graviton reggeization in the trans-Planckian Regge limit
Sec. 3.2 assumes the same all-orders exponentiation for gravitons with trajectory alpha(k^2) from Eq. (3.46), even though the reggeization terms are t/s suppressed relative to the eikonal phase; the paper notes the construction is 'more delicate' than in QCD.
domain assumption Validity of color-kinematic double copy for the Lipatov vertex
The gravitational vertex is built as 1/2 C^mu C^nu - 1/2 N^mu N^nu, assuming the double copy survives the Regge limit. The paper states the usual BCJ replacement 'does not work fully' (Sec. 3.6) and the Wong-equation replacements in Eq. (3.107) are motivated by structural similarity.
ad hoc to paper Classical-statistical CGC power counting applies to gravitational shockwaves
The extension of the CGC EFT to overoccupied graviton states and gravitational shockwave radiation (Secs. 4-5) assumes a semi-classical, stochastic description; the paper labels the universality of this physics as speculation (Sec. 3.7).

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Cite this review

Pith. "Pith review of QCD-Gravity double copy in Regge asymptotics: from $2\rightarrow n$ amplitudes to radiation in shockwave collisions." pith.science (2026). https://pith.science/paper/XPEDANAL

@misc{pith2026250721252,
  author       = {Pith},
  title        = {Pith review of: QCD-Gravity double copy in Regge asymptotics: from $2\rightarrow n$ amplitudes to radiation in shockwave collisions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XPEDANAL}},
  note         = {Machine review of arXiv:2507.21252}
}

read the original abstract

These lectures discuss multi-particle production in QCD and in gravity at ultrarelativistic energies, their double copy relations, and strong parallels in emergent shockwave dynamics. Dispersive techniques are applied to derive the BFKL equation for multi-gluon production in Regge asymptotics. Identical methods apply in gravity and are captured by a gravitational Lipatov equation. The building blocks in both cases are Lipatov vertices and reggeized propagators satisfying double copy relations; in gravity, Weinberg's soft theorem is recovered as a limit of the Lipatov framework. BFKL evolution in QCD generates wee parton states of maximal occupancy characterized by an emergent semi-hard saturation scale. Renormalization group equations in the Color Glass Condensate (CGC) EFT describe wee parton correlations and their rapidity evolution. A shockwave picture of deeply inelastic scattering and hadron-hadron collisions follows, with multi-particle production described by Cutkosky's rules in strong time-dependent fields. Gluon radiation in the CGC EFT has a double copy in gravitational shockwave collisions, with a similar correspondence applicable between gluon and graviton shockwave propagators. Possible extensions of this semi-classical double copy are outlined for computing multi-particle production in gravitational shockwave collisions, self-force and tidal contributions, and classical and quantum noise in the focusing of geodesics.

Figures

Figures reproduced from arXiv: 2507.21252 by the authors.

**Figure 2.** 2 → 2 gluon scattering amplitude at order g 2 and g 4 . Fig. (a) represents the leading order one-gluon exchange Born diagram. Figs. (b) and (c) are the O(g 4 ) two-gluon exchanges that give the lowest order contribution to color-singlet pomeron exchange. These diagrams have an imaginary part that can be computed using Cutkosky rules. First, we need the tree-level Born amplitude in the Regge limit, shown in [PITH_F… view at source ↗

**Figure 3.** Feynman graphs for the process gg → ggg that contribute to the imaginary part of the 2-to-2 scattering at order g 6 . The momenta ℓi correspond to the momenta of the produced on-shell gluons, as shown in [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (37 more)

**Figure 4.** Figure 4: The nonlocal central gluon emission Lipatov vertex, represented by the black blob. [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: Cut Feynman graphs contributing to Im A at order g 6 . diagrams, the contribution to the imaginary part from these diagrams is Im A (1)µµ′νν′ 2 = − Ncg 6 4π η µµ′ η νν′ G (1) 0 s ln s/k 2 × Z d 2k1 (2π) 2 d 2k2 (2π) 2 1 k 2 1 (k1 − k2) 2 (k2 − q) 2 + 1 k 2 2 (k2 − …

**Figure 6.** Figure 6: Cut diagrams that contribute to pole reconstruction of the Lipatov vertex in the 2 [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

**Figure 7.** Figure 7: The effective (half) ladder for two gluon production in the MRK regime. [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]

**Figure 8.** Figure 8: A subset of bare Feynman graphs that contribute to the effective diagram for two gluon [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]

**Figure 9.** Figure 9: Cut diagrams that contribute to pole reconstruction of two Lipatov vertices in the 2 [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]

**Figure 10.** Figure 10: Tree level multi-gluon production amplitude in multi-Regge kinematics. Black blobs [PITH_FULL_IMAGE:figures/full_fig_p023_10.png]

**Figure 11.** Figure 11: Illustration of the 2 → n + 2 amplitude A2→2+n(p1, k1, k2, · · · kn+1, p2) factorized into two sub-ampitudes Mµ (p1, k1, k2, · · · ki) and N ν (ki , ki+1, · · · kn+1, p2) separated by a cut propagator represented by the horizontal dashed line. and N µ satisfy the Ward…

**Figure 12.** Figure 12: Feynman diagrams representing the i’th segment of uncrossed and crossed ladders. the uncrossed and crossed ladder diagrams. In the uncrossed case, this segment of the ladder is proportional to the product of two Lipatov vertices C σi−1C σi . This product, when expande…

**Figure 13.** Figure 13: i’th segment of the ladder with a three-point gluon fusion and a quartic vertex. Let us now look at the left diagram in [PITH_FULL_IMAGE:figures/full_fig_p027_13.png]

**Figure 14.** Figure 14: i’th segment of the ladder with fermionic lines. 2.3 Leading virtual graphs: Reggeization In addition to multiple tree level contributions to the 2 → n amplitude that we discussed thus far (culminating in Eq. (2.78)), there are also virtual corrections that contribute…

**Figure 15.** Figure 15: The reggeized gluon, depicted as a thick gluon line, resums the leading double logarithmic [PITH_FULL_IMAGE:figures/full_fig_p029_15.png]

**Figure 16.** Figure 16: Contribution of a soft virtual gluon to the tree-level MRK amplitude. One of the Lipatov [PITH_FULL_IMAGE:figures/full_fig_p029_16.png]

**Figure 17.** Figure 17: The n-rung ladder contribution to the imaginary part of the amplitude, where the bold [PITH_FULL_IMAGE:figures/full_fig_p032_17.png]

**Figure 18.** Figure 18: Plot of the BFKL eigenvalues in Eq. (2.116) for n = 0, 1, 2, 3, 4, 5. The maximum is attained for n = 0 at ν = 0, with the maximum value ω ∗ = 4αSNc ln(2)/π. 2.6 Leading and next-to-leading log xBj DIS cross-sections and gluon saturation It is useful to connect our pr…

**Figure 19.** Figure 19: Illustration of the underlying 2 → N structure of deeply inelastic scattering (DIS) at small x (high energies). The DIS process is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p040_19.png]

**Figure 20.** Figure 20: A twist-four 1 → 2 pomeron “fan” diagram contributing to the DIS dipole cross-section. xBj → 0, the rapid growth in the dipole cross-section can lead to increasingly large contributions from higher-twist operators. An example of such a contribution is illustrated by t…

**Figure 21.** Figure 21: The eikonal scattering series comprised of horizontal ladder and crossed ladder diagrams. [PITH_FULL_IMAGE:figures/full_fig_p045_21.png]

**Figure 22.** Figure 22: The H-diagram, which gives the leading inelastic correction to the eikonal scattering [PITH_FULL_IMAGE:figures/full_fig_p047_22.png]

**Figure 23.** Figure 23: In general relativity, there are infinitely many higher point interactions that are sup [PITH_FULL_IMAGE:figures/full_fig_p050_23.png]

**Figure 24.** Figure 24: Feynman graphs for the 2 → 3 tree-level graviton scattering that sum up to the gravitational Lipatov vertex in MRK kinematics. There are two additional diagrams relative to the QCD case since the four-point vertex in gravity is not suppressed in energy. µν µ ′ ν ′ ρσ…

**Figure 25.** Figure 25: Three point graviton vertex. V ρσ µνµ′ν ′(p1, q) = iκ 2 Pµνµ′ν ′ p ρ 1 p σ 1 + (p1 − q) ρ (p1 − q) σ + q ρ q σ − 3 2 η ρσq 2 + 2qλqσ h I λσ µνI ρσ µ′ν ′ + I λσ µ′ν ′I ρσ µν − I λρ µνI σσ µ′ν ′ − I σσ µνI λσ µ′ν ′ i + h qλq ρ ηµνI λσ µ′ν ′ + ηµ′ν ′I λσ µν + qλ…

**Figure 26.** Figure 26: Plot of ˜ω(ν, n) = 2π 2 κ2k 2ω(ν, n) + log( k 2 Λ2 UV ), where ω(ν, n) are the gravitational BFKL eigenvalues in Eq. (3.74) for n = 0, 1, 2, 3, 4, 5. ˜ω(ν, n) is therefore independent of Λ2 UV . As in the gauge theory case, the maximum is attained for n = 0 at ν = 0, …

**Figure 27.** Figure 27: The scattering of a quark-antiquark dipole in DIS off a boosted heavy nucleus. The [PITH_FULL_IMAGE:figures/full_fig_p079_27.png]

**Figure 28.** Figure 28: Dressed propagator of a scalar field in the shockwave background of overoccupied gluons. [PITH_FULL_IMAGE:figures/full_fig_p088_28.png]

**Figure 29.** Figure 29: Real and virtual diagrams for a dipole interacting with a shockwave. (a) Real gluon [PITH_FULL_IMAGE:figures/full_fig_p091_29.png]

**Figure 30.** Figure 30: Dilute-dilute regime of shockwave scattering in QCD. The inclusive gluon radiative [PITH_FULL_IMAGE:figures/full_fig_p098_30.png]

**Figure 31.** Figure 31: Dilute-dense scattering (ρL/∇2 ⊥ ≪ 1 and ρH/∇2 ⊥ ∼ 1). For ρH/∇2 ⊥ ∼ 1, coherent multiple scatterings from the nucleus can be resummed into a lightlike Wilson line. is then straightforward to deduce the form of C µ in light cone gauge [11] ε ∗ µ (k)C µ (q1, q2) = −2ε …

**Figure 32.** Figure 32: Spacetime diagram of collision of two gluon shockwaves. Red and blue lines represent [PITH_FULL_IMAGE:figures/full_fig_p100_32.png]

**Figure 33.** Figure 33: Illustration of Eq. (4.58) for dilute-dilute single inclusive gluon production in the CGC EFT with classical fields/reggeized gluons (dark curly lines) and the Lipatov vertex (black blobs). This contribution is the imaginary part of a two-loop Feynman diagram, with th…

**Figure 34.** Figure 34: Illustration of Eq. (4.65) for the n-gluon inclusive multiplicity in the glasma, computed in the dilute-dilute limit of the CGC EFT, generalizing the structure shown in [PITH_FULL_IMAGE:figures/full_fig_p102_34.png]

**Figure 35.** Figure 35: Trajectories of colliding gravitational shockwaves. The trajectory in red (blue) is the [PITH_FULL_IMAGE:figures/full_fig_p113_35.png]

**Figure 36.** Figure 36: Cut vacuum-to-vacuum H diagram contributing at leading order to the observable [PITH_FULL_IMAGE:figures/full_fig_p123_36.png]

**Figure 37.** Figure 37: These cut vacuum-vacuum diagrams depict both higher order insertions of the classical [PITH_FULL_IMAGE:figures/full_fig_p124_37.png]

**Figure 38.** Figure 38: Illustration of dilute-dilute single inclusive graviton production with classical [PITH_FULL_IMAGE:figures/full_fig_p126_38.png]

**Figure 39.** Figure 39: Illustration of dilute-dilute inclusive multi-graviton production representing Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p126_39.png]

**Figure 40.** Figure 40: Illustration of the rescattering of the graviton produced in shockwave collision with a [PITH_FULL_IMAGE:figures/full_fig_p127_40.png]

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300 extracted references · 12 canonical work pages · cited by 2 Pith papers

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