High-energy dynamics of QCD: Theoretical and phenomenological results

Gabriele Gatto

REVIEW 3 major objections 4 minor 2 cited by

High-energy dynamics of QCD: Theoretical and phenomenological results

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The real NLO corrections to the Higgs impact factor are computed with a physical top-quark mass and shown to reduce to the infinite-top-mass result after the rapidity divergence cancels against the BFKL counter-term.

desk verdict A serious, honest thesis whose one genuinely new result is the real NLO Higgs impact factor with finite top mass; the virtual part is deferred and the rapidity-cancellation proof needs the missing Package X expansions, but the core calculation is likely right. read the letter →

arxiv 2506.03222 v1 pith:5NZOI22M submitted 2025-06-03 hep-ph hep-th

Gabriele Gatto This is my paper

classification hep-phhep-th

keywords BFKLformalismHiggsimpactfactortop-quarkmassnext-to-leadingorderrapiditydivergencetetraquarkfragmentationfunctionsvariable-flavornumberschemeColorGlassCondensate

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 central claim of this thesis is that the real next-to-leading-order corrections to the Higgs impact factor in BFKL factorization can be computed with a physical (finite) top-quark mass, and that the resulting expression reduces to the known infinite-top-mass impact factor when the top is taken heavy. The calculation is a step toward a complete NLO impact factor for forward-Higgs processes at the LHC and FCC, where the Higgs transverse momentum approaches the top mass and the old effective-operator approximation is expected to fail. The paper also supplies updated fragmentation-function parametrizations for bottomonium-like tetraquarks in a variable-flavor-number scheme and a leading-order analytical cross section for diffractive di-hadron production in the saturation regime, but the load-bearing new result is the finite-top-mass Higgs impact factor and the cancellation of its rapidity divergence against the BFKL counter-term.

What carries the argument

The central object is the Higgs impact factor $\Phi_{PP}^{(Hg)}$ defined by BFKL factorization, which describes the transition of a collinear parton plus a Reggeized gluon into a forward Higgs plus an extra parton, integrated over the unobserved part of the final state. The computation is carried by the off-shell $ggH$ and $gggH$ tensor form factors $F_T$, $F_L$ and $B_a,B_b,B_c$ built from massive top-quark loop integrals, together with the BFKL counter-term that cancels the rapidity divergence. The crucial mechanism is the expansion around $z_H=1$ of the six box-type coefficients, where all enhanced $\frac{1}{1-z_H}\ln(1-z_H)$ and $\frac{1}{1-z_H}\ln^2(1-z_H)$ terms cancel term by term, and the surviving $D_0$ box integral is evaluated through its asymptotic large-$s$ form, leaving the single soft-divergent $\frac{1}{1-z_H}$ term that combines with the counter-term to yield the finite Eq. (2.80).

What would settle it

Ask for the full expansion of the ten coefficients in Eq. (2.47) around $z_H=1$: if any of the $\frac{1}{1-z_H}\ln^2(1-z_H)$ or $\frac{1}{1-z_H}\ln(1-z_H)$ terms survive, the claimed finite term Eq. (2.80) would be incomplete. Alternatively, evaluate the box diagrams numerically at $z_H$ close to 1 and compare the integrand with the asymptotic $D_0$ expression (2.66).

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the real part of the NLO Higgs impact factor can be evaluated keeping the full top-quark mass dependence. The amplitudes are organized into quark-initiated and gluon-initiated contributions; the gluon-initiated part is further split into triangular- and box-type diagrams. In the $z_H \to 1$ (high-rapidity) limit the naive $\ln(1-z_H)$ and $\ln^2(1-z_H)$ terms in the box coefficients cancel, leaving only the single $1/(1-z_H)$ divergence that matches the BFKL prediction and is removed by the counter-term. The final finite contribution is Eq. (2.80), proportional to $|F_T(0,-\vec p_H^2,m_t^2)|^2$, which in the infinite-top limit reproduces Ref. [60]. The paper also argues that the $s \to \infty$ and $m_t \to \infty$ limits do not commute: in the finite-mass calculation the $t$- and $s$-channel triangular diagrams are suppressed by $(1-z_H)$, and only the $u$-channel diagram contributes to the rapidity limit, whereas in the constant-coupling infinite-mass case all diagrams contribute and cancel.

Load-bearing premise

The whole consistency argument rests on the claim that the most divergent pieces of the box diagrams cancel exactly when expanded near the high-rapidity limit, a cancellation that is stated but not displayed in full.

Editorial extensions

If this is right

The full NLO Higgs impact factor, with virtual corrections added, will enable precision BFKL-resummed predictions for forward Higgs production and Higgs-plus-jet at the LHC and the FCC, free of the infinite-top-mass approximation.
In the kinematic region where the Higgs transverse momentum is comparable to $m_t$, the finite-mass result will differ from the effective-operator prediction in a way that can be tested by measuring transverse-momentum distributions at large rapidity separations.
The explicit cancellation of the rapidity divergence strengthens the consistency of BFKL factorization with gluon Reggeization at NLO.
The updated TQHL1.1 and TQ4Q1.1 fragmentation functions give concrete, DGLAP-evolved predictions for tetraquark-plus-jet rates at 14 and 100 TeV, so a comparison with LHC data will test both the tetraquark production model and the BFKL dynamics.
The leading-order analytical cross section for diffractive di-hadron production provides the starting point for numerical CGC predictions that can be compared with saturation and non-saturation scenarios at the EIC and in ultraperipheral LHC collisions.

Reading between the lines

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

If the finite-mass result survives the check of virtual corrections, it would suggest that high-energy resummations of forward-Higgs processes should be re-run at FCC energies, because the infinite-top approximation is precisely the region where large energy logarithms and top-threshold effects overlap.
The non-commutativity of the $s \to \infty$ and $m_t \to \infty$ limits highlighted for the triangular diagrams is a caution for effective-field-theory treatments of heavy-quark loops: the same vertex that looks safe in the effective theory at leading power can change the high-energy singularity structure when mass effects are restored.
The pattern that the gluon-initiated tetraquark fragmentation functions grow slowly with the factorization scale and help stabilize the resummed distributions suggests that similar stabilization could appear for other exotic-hadron channels, such as charmed tetraquarks or pentaquarks; this is an extrapolation beyond the thesis.
A testable extension would be to compare the back-to-back diffractive di-hadron cross section with the CGC dipole model against the same observable computed without saturation, since the analytical expression derived here is directly amenable to that numerical comparison.

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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

Short version: this is a serious, honest thesis whose one genuinely new result is the real NLO corrections to the Higgs impact factor with finite top mass. It is not a complete NLO impact factor—the virtual part is explicitly deferred—and the rapidity-cancellation proof rests on Package X expansions the text does not show. I still think the central claim is probably right and worth refereeing.

The new piece is Chapter 2. Previous NLO Higgs impact factors [58–60] used the infinite-top-mass effective operator. Here they keep the physical m_t in the real gluon-emission vertices, compute both quark- and gluon-initiated pieces, check gauge invariance at q=0, reproduce the infinite-mass limit, and confirm the on-shell amplitudes against VBFNLO. The infrared structure is clean and the subtraction scheme is spelled out. That is a real advance, even if incremental. The tetraquark chapter is also solid: TQHL1.1 and TQ4Q1.1 are updated fragmentation sets, evolved with DGLAP and released publicly; the phenomenological plots come with uncertainty bands. Chapter 4 is explicitly labelled unfinished; it is a preview, not a result.

Soft spots, in proportion. The load-bearing step of Chapter 2 is the claim in Section 2.4.2.5 that, as z_H→1, all ln(1-z_H) and ln^2(1-z_H) enhanced terms cancel coefficient-by-coefficient in Eq. (2.47), leaving the D0 asymptotic term (2.66). The expansions are not displayed. This is a verifiability problem, not an obvious error: the authors use Package X, verify the infinite-mass limit, and explicitly check the cancellation with the BFKL counter-term to get (2.80). Still, I would demand the expansions or a notebook before endorsing. Second, the result is only half of an NLO impact factor; without the virtual corrections it is not by itself infrared-finite and cannot feed phenomenology yet. The thesis states this honestly. Third, there are no numerical checks of the finite-top result beyond the on-shell amplitude comparison, so the size of top-mass effects is not demonstrated. I do not see circular fitting or invented machinery.

Who is this for? People working on BFKL Higgs phenomenology and on NLO impact factors, though they will want the follow-up with virtual corrections. I would not cite it as a final number for forward-Higgs production, but I would cite it for the new real-correction expressions once journaled.

Recommendation: this deserves a serious referee. Send it to review, with the referee asked to verify the z_H→1 expansions—ideally with the authors' auxiliary files—and to judge it as an explicitly partial result. Not a desk reject.

Referee Report

3 major / 4 minor

Summary. This manuscript, formatted as a PhD thesis, develops the BFKL formalism and applies it to three topics in high-energy QCD. Its central new result, in Chapter 2, is the computation of the real next-to-leading-order (NLO) corrections to the forward-Higgs impact factor in gluon-Reggeon collisions with a finite top-quark mass. The chapter derives quark- and gluon-initiated contributions, verifies gauge invariance, isolates soft, collinear, and rapidity singularities, and claims that after inclusion of the BFKL counter-term the rapidity divergence cancels, leaving the finite term in Eq. (2.80). Chapter 3 presents two new families of collinear fragmentation functions for tetraquarks (TQHL1.1 and TQ4Q1.1) and uses them in a hybrid NLLA/NLO+ factorization to produce rapidity-interval and transverse-momentum distributions at 14 and 100 TeV. Chapter 4 derives a leading-order analytic expression for diffractive di-hadron production in the CGC/saturation framework, with the GBW dipole model, and states that numerical results are still in progress.

Significance. If the Chapter 2 claim is correct, the real-emission part of the NLO Higgs impact factor with physical top mass is a meaningful advance beyond the infinite-top-mass approximation and provides a necessary ingredient for a future complete NLO impact factor. The manuscript contains several verifiable checks: agreement of the amplitudes with VBFNLO and with known helicity amplitudes, explicit gauge-invariance cancellation at q=0, and reduction to the infinite-top-mass result of Ref. [60] in the appropriate limit. The comparison with the infinite-top-mass case is a legitimate consistency check rather than a fit to the target result. Chapter 3 is strengthened by the public release of the TQHL1.1 and TQ4Q1.1 fragmentation functions and by numerical predictions obtained with the JETHAD framework. The significance of Chapter 4 is presently limited because it is explicitly an ongoing analysis with no numerical results. However, the central rapidity-divergence proof in Chapter 2 is not fully documented, and this is the main obstacle to endorsing the paper's central claim.

major comments (3)

[Sec. 2.4.2.5, Eqs. (2.47)-(2.68)] The cancellation of all ln^2(1-z_H) and ln(1-z_H) enhanced terms among the ten coefficients contributing to the box amplitude is the load-bearing step that reduces the z_H -> 1 limit to the BFKL-compatible 1/(1-z_H) form of Eq. (2.68). The manuscript states that Package X was used, but it does not display the expansions, provide an auxiliary file, or give a reproducible derivation. This cancellation is nontrivial because the coefficients in Eq. (2.47) are linear combinations of tensor integrals with s ~ 1/(1-z_H), and several terms carry explicit (1-z_H)^{-1} prefactors. If the cancellation is incomplete, the impact factor would contain non-BFKL rapidity logarithms that the counter-term in Eq. (2.1) cannot absorb, invalidating Eq. (2.80). The authors should supply the full z_H -> 1 expansions, or a machine-readable notebook that reproduces the cancellation for all ten coefficient combinations.
[Sec. 2.4.2.5, Eq. (2.66)] The asymptotic formula for the scalar box integral D0 in the s -> infinity limit is quoted without derivation. The expression contains logarithms of ratios involving sqrt(1 - 4 m_t^2 / m_H^2), which is imaginary for the physical values m_t = 172 GeV and m_H = 125 GeV, yet no branch convention or prescription for the analytic continuation is given. The finite remainder after the rapidity cancellation, Eq. (2.80), and the subtraction terms in Eqs. (2.70)-(2.75) depend on the precise values and phases of these logarithms. The authors should provide the derivation of Eq. (2.66), specify the branch choices, or validate the asymptotic form numerically against a direct evaluation of D0 in the z_H -> 1 limit.
[Sec. 2.5 and Abstract] The manuscript computes only the real-emission part of the NLO impact factor; the virtual corrections are explicitly deferred to future work in Sec. 2.5. As a consequence, the claimed cancellation of soft divergences and the organization of the collinear-remnant terms in Eqs. (2.71)-(2.80) are statements about how the real part is expected to combine with the future virtual part, not a complete demonstration for the full NLO impact factor. The abstract's wording, which refers to 'demonstrating the cancellation of divergences in the full impact factor', overstates what is established. The chapter title and abstract should either explicitly restrict the claim to the real-emission contribution or the full NLO calculation should be completed.

minor comments (4)

[Sec. 2.2.2, Eqs. (2.19)-(2.22)] The notation for the D-coefficients is introduced through the mapping in Eq. (2.20) before the tensor decomposition in Eqs. (2.21)-(2.22) is defined, which forces the reader to jump forward and back; reordering or a short defining table would improve readability.
[Sec. 3.4, Figs. 3.12-3.15] The figure captions state that the ancillary panels show the ratio of LL/LO to NLLA/NLO+ predictions, but the main text says the panels emphasize the ratio between 'LL/LO or HE-NLO+' and NLLA/NLO+ predictions; the captions should be updated to describe exactly which ratios are plotted.
[Sec. 3.2.2.2, Eq. (3.23)] The estimate <O_{T4b}>/<O_{T4c}> ~ 400 from (m_b alpha_s^{(b)} / m_c alpha_s^{(c)})^9 should be accompanied by the numerical values of the couplings used, since different choices for alpha_s at the respective scales can change the estimate by a large factor and thereby affect the T4b predictions.
[Sec. 4.2.1, Eqs. (4.25)-(4.28)] The passage from Eq. (4.25) to Eq. (4.28) states that the z1 and z2 integrations can be performed analytically, but the intermediate integral identities are not shown; for a chapter that is otherwise presented as a derivation, the analytic evaluation of the z1-z2 integrals should be given or referenced explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: central Chapter 2 derivation is an ab initio amplitude computation checked against external benchmarks; Chapter 3-4 use model parameters, not fitted circular predictions.

full rationale

The derivation chain in Chapter 2 is self-contained: the ggH and gggH amplitudes are computed from the Standard Model with FormCalc, cross-checked against VBFNLO and against known on-shell helicity amplitudes, and the infinite-top-mass limit is used only as a consistency boundary condition, not as an input that fixes the finite-m_t result. The rapidity-divergence analysis is an algebraic verification (via Package X expansions) that the ln^n(1-z_H) terms cancel among the ten coefficients and that the surviving D0 integral produces the BFKL-compatible 1/(1-z_H) term, which is then cancelled by the standard counter-term in Eq. (2.1); this is a check of BFKL factorization rather than a definitional identity. Chapter 3 contains parameter choices (e.g., <q_T^2> and LDMEs) and the use of the authors' own fragmentation functions, but these are model inputs that are not fitted to the output observables, and the hybrid-factorization formulae are standard BFKL/NLLA expressions with external PDFs/FFs. Chapter 4 presents an LO derivation from established shockwave/GBW ingredients. No equation used as an input is equivalent by construction to a claimed prediction, and no load-bearing premise rests on a self-citation chain. The absence of displayed Package X expansions is a reproducibility and verification-detail issue, not circularity.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central Higgs impact-factor result has no fitted parameters, which is a strength. The phenomenological parts inherit model parameters: FF shape inputs and GBW dipole parameters. The main unstated load-bearing element is the Package X-based high-rapidity expansion of box integrals.

free parameters (5)

transverse momentum width <q_T^2> for TQHL1.1 = 4 GeV^2
Chosen via numerical scan based on correlation with the FF peak position (Sec. 3.2.1); controls the shape of the doubly heavy tetraquark fragmentation functions.
hadron decay constant f_B = 0.25 GeV
Input in the Suzuki-model normalization of Eq. (3.1); taken from the literature.
heavy quark masses m_Q = m_c = 1.5 GeV, m_b = 4.9 GeV
Model inputs for the fragmentation function thresholds and evolution scales.
T4c LDMEs and T4b/T4c LDME ratio = 0.0347, 0.0128, 0.0211, 0.072 GeV^9; ratio ~400
Potential-model estimates; the T4b ratio follows from dimensional analysis (Eq. 3.23), not from data or lattice.
GBW dipole model parameters = sigma_0 = 23.03 mb, lambda = 0.288, x_0 = 3.04e-4, Q_0 = 1 GeV
Parameters fitted to HERA data in the cited dipole model and inherited by the diffractive di-hadron cross section (Eqs. 4.23-4.24).

assumptions (6)

domain assumption Gluon Reggeization and the factorized form of high-energy amplitudes hold in LLA and NLLA.
Used throughout Chapter 1 to set up Eq. (1.6) and the BFKL equation; standard but not proven in the thesis.
domain assumption The NLLA impact factor is defined by Eq. (2.1) with the s_Lambda MRK/QMRK separation and the BFKL counter-term.
This definition is central to the Higgs impact factor calculation and is taken from the BFKL literature [47,60].
standard math The off-shell ggH vertex is fully described by the two form factors F_T and F_L of Eq. (2.7) with QED-like Ward identities.
The decomposition and Passarino-Veltman reduction are standard; the large-top-mass limit matches the effective Lagrangian Eq. (2.16).
domain assumption NLO BFKL kernel eigenvalues and NLO emission functions retain the form of Eqs. (3.29)-(3.35) when applied to heavy-flavor tetraquarks in a VFNS.
The tetraquark-plus-jet predictions assume that the light-hadron NLO emission function of Ref. [185] transfers to tetraquark fragmentation above heavy-quark thresholds.
domain assumption The dipole amplitude of the target is represented by the GBW parametrization Eq. (4.23) with fixed parameters.
The diffractive di-hadron cross section, Eq. (4.28), depends on this phenomenological model; the paper presents no derivation or uncertainty estimate for it.
ad hoc to paper The z_H to 1 asymptotic expansions of box-type scalar integrals, including Eq. (2.66) and the cancellation of ln^2 terms, are correct as extracted with Package X.
This is a paper-specific computational step on which the claimed cancellation of rapidity divergences rests, and the full expansion is not shown.

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Cite this review

Pith. "Pith review of High-energy dynamics of QCD: Theoretical and phenomenological results." pith.science (2026). https://pith.science/paper/5NZOI22M

@misc{pith2026250603222,
  author       = {Pith},
  title        = {Pith review of: High-energy dynamics of QCD: Theoretical and phenomenological results},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5NZOI22M}},
  note         = {Machine review of arXiv:2506.03222}
}

read the original abstract

This work investigates the behavior of hadronic matter in the high-energy Regge-Gribov (semi-hard) regime of Quantum Chromodynamics (QCD), accessible through current and future colliders such as the LHC, EIC, and FCC. Central to the analysis is the Balitsky-Fadin-Kuraev-Lipatov (BFKL) formalism, with detailed treatment of the BFKL equation in both leading and next-to-leading logarithmic approximations. A major focus is placed on the computation of real next-to-leading order (NLO) corrections to the Higgs boson impact factor, incorporating finite top-quark mass effects. These corrections are essential for improving precision in Higgs production processes at large rapidity separations. The study also explores the semi-inclusive production of exotic tetraquark states, employing a hybrid framework that combines collinear and BFKL dynamics within a variable-flavor number scheme. Updated fragmentation functions for bottomonium-like states are provided, offering improved predictions at 14 TeV and 100 TeV. Additionally, the work addresses diffractive di-hadron production in the small-x saturation regime using the Color Glass Condensate (CGC) formalism, presenting preliminary leading-order analytical results.

Figures

Figures reproduced from arXiv: 2506.03222 by the authors.

**Figure 1.1.** A diagram showing the Regge exchange. poles in the l-plane and within the Regge kinematic domain |s| ≫ t, it is found that: A (s, t) s→∞ −−−→ η + e −iπα(t) 2 β(t)s α(t) , (1.1) where α(t) represents the location of the leading Regge pole in the l-plane; it is not constant but it is a function that depends on the momentum transfer t. The amplitude in Eq. (1.1) can be interpreted as the exchange of an entity (that we … view at source ↗

**Figure 1.2.** In (a) the leading order diagram contributing to the quark-quark scatter [PITH_FULL_IMAGE:figures/full_fig_p028_1_2.png] view at source ↗

Figures from the paper (26 more)

**Figure 1.3.** Figure 1.3: Production of n + 2 particles in the multi-Regge kinematics Now, it proves useful the Sudakov decomposition to express a momentum k in terms of k µ = βpµ 1 + αp µ 2 + k µ ⊥ , (1.12) where k⊥ = (0, ⃗k, 0) is a four-vector, completely transverse with respect to the pla…

**Figure 1.4.** Figure 1.4: Schematic representation of the factorized amplitude. [PITH_FULL_IMAGE:figures/full_fig_p030_1_4.png]

**Figure 1.5.** Figure 1.5: Next-to-leading corrections to Regge trajectory, PPR vertices, and cen [PITH_FULL_IMAGE:figures/full_fig_p037_1_5.png]

**Figure 1.6.** Figure 1.6: Next-to-leading corrections to the vertices PPR and RRG in the quasi [PITH_FULL_IMAGE:figures/full_fig_p039_1_6.png]

**Figure 1.7.** Figure 1.7: Schematic representation of the real part of the NLO BFKL kernel. In [PITH_FULL_IMAGE:figures/full_fig_p040_1_7.png]

**Figure 2.1.** Figure 2.1: Schematic description of an impact factor. [PITH_FULL_IMAGE:figures/full_fig_p044_2_1.png]

**Figure 2.2.** Figure 2.2: The two triangular-like diagrams contribute to the Higgs impact factor [PITH_FULL_IMAGE:figures/full_fig_p051_2_2.png]

**Figure 2.3.** Figure 2.3: The two triangular-like diagrams contribute to the quark-initiated part [PITH_FULL_IMAGE:figures/full_fig_p052_2_3.png]

**Figure 2.4.** Figure 2.4: The six triangular-like diagrams that contribute to the gluon-initiated [PITH_FULL_IMAGE:figures/full_fig_p054_2_4.png]

**Figure 2.5.** Figure 2.5: The six box-like diagrams contribute to the gluon-initiated part of the [PITH_FULL_IMAGE:figures/full_fig_p055_2_5.png]

**Figure 3.1.** Figure 3.1: A leading-order representative diagram illustrating the collinear frag [PITH_FULL_IMAGE:figures/full_fig_p074_3_1.png]

**Figure 3.2.** Figure 3.2: Collinear fragmentation of a constituent heavy quark into doubly [PITH_FULL_IMAGE:figures/full_fig_p077_3_2.png]

**Figure 3.3.** Figure 3.3: Dependence of TQHL1.1 collinear FFs on the factorization scale, illustrating the formation of Xccu¯ u¯ (left) and Xccs¯ s¯ (right) at z = 0.425 ≃ ⟨z⟩. inputs. Figs. 3.3 and 3.4 display the µF -dependence of the TQHL1.1 FFs for doubly charmed and doubly bottomed tetr…

**Figure 3.4.** Figure 3.4: The factorization-scale dependence of the [PITH_FULL_IMAGE:figures/full_fig_p079_3_4.png]

**Figure 3.5.** Figure 3.5: LO representative diagrams for the collinear fragmentation process are [PITH_FULL_IMAGE:figures/full_fig_p079_3_5.png]

**Figure 3.6.** Figure 3.6: Collinear fragmentation of a gluon into fully charmed (upper) and fully [PITH_FULL_IMAGE:figures/full_fig_p085_3_6.png]

**Figure 3.7.** Figure 3.7: Collinear fragmentation of a constituent heavy quark into fully charmed [PITH_FULL_IMAGE:figures/full_fig_p086_3_7.png]

**Figure 3.8.** Figure 3.8: Factorization-scale dependence of TQ4Q1.1 collinear FFs for T4c(0++) (left) and T4c(2++) (right) formation at z = 0.425 ≃ ⟨z⟩. The first ancillary panels display the ratio of TQ4Q1.1 to TQ4Q1.1−. The second ancillary panels compare TQ4Q1.1 with TQ4Q1.0, where the cha…

**Figure 3.9.** Figure 3.9: Factorization-scale dependence of TQ4Q1.1 collinear FFs for T4b(0++) (left) and T4b(2++) (right) formation at z = 0.425 ≃ ⟨z⟩. The lower panels display the ratio of TQ4Q1.1 to TQ4Q1.1− functions. gluon FF remains almost unchanged. Finally, the TQ4Q1.1 gluon FFs displ…

**Figure 3.10.** Figure 3.10: Rapidity-interval distributions for Xb¯buu¯ (left) and Xb¯bss¯ (right) combined with jet hadroproduction are presented for √ s = 14 TeV (LHC, top) and 100 TeV (nominal FCC, bottom). The auxiliary panels beneath the main plots display the ratio of LL/LO to NLLA/NLO+…

**Figure 3.11.** Figure 3.11: Distributions of rapidity intervals for T4b(0++) (left) and T4b(2++) (right) combined with jet hadroproduction are shown for √ s = 14 TeV (LHC, top) and 100 TeV (nominal FCC, bottom). The supplementary panels beneath the main plots illustrate the ratio of LL/LO to N…

**Figure 3.12.** Figure 3.12: Momentum distributions for Xb¯buu¯ (left panels) and Xb¯bss¯ (right panels) combined with jet production at √ s = 14 TeV (upper plots, LHC) or 100 TeV (lower plots, nominal FCC), within the range 2 < ∆Y < 4. Secondary panels beneath the main graphs display the ratio…

**Figure 3.13.** Figure 3.13: Momentum distributions for Xb¯buu¯ (left panels) and Xb¯bss¯ (right panels) combined with jet production at √ s = 14 TeV (upper plots, LHC) or 100 TeV (lower plots, nominal FCC), within the range 4 < ∆Y < 6. Supporting panels below the primary graphs present the rat…

**Figure 3.14.** Figure 3.14: Momentum distributions for T4b(0++) (left panels) and T4b(2++) (right panels) combined with jet production at √ s = 14 TeV (upper plots, LHC) or 100 TeV (lower plots, nominal FCC), within the range 2 < ∆Y < 4. Auxiliary panels beneath the main graphs show the ratio …

**Figure 3.15.** Figure 3.15: Momentum distributions for T4b(0++) (left panels) and T4b(2++) (right panels) with jet production at √ s = 14 TeV (upper plots, LHC) or 100 TeV (lower plots, nominal FCC), within the range 4 < ∆Y < 6. Supplementary panels below the primary plots display the ratio of…

**Figure 4.1.** Figure 4.1: Amplitude of the process (4.6) at LO. The grey blob symbolizes the [PITH_FULL_IMAGE:figures/full_fig_p104_4_1.png]

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