Bottomoniumlike states in proton collisions: Fragmentation and resummation

Francesco Giovanni Celiberto; Gabriele Gatto

REVIEW 3 major objections 3 minor 3 cited by

Bottomoniumlike states in proton collisions: Fragmentation and resummation

T0 review · 3 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that unobserved bottomoniumlike tetraquarks—states containing a bottom-antibottom pair plus either light quarks or a second heavy pair—can be described at high transverse momentum by single-parton fragmentation, and it…

desk verdict A well-executed incremental update with genuinely new T4b FFs and first resummed T4b+jet predictions; the absolute normalization rests on one unvalidated, ninth-power LDME Ansatz. read the letter →

arxiv 2412.10549 v2 pith:NDP6FVID submitted 2024-12-13 hep-ph hep-exnucl-exnucl-th

Francesco Giovanni Celiberto , Gabriele Gatto This is my paper · ORCID

classification hep-phhep-exnucl-exnucl-th

keywords bottomoniumliketetraquarksfragmentationfunctionsDGLAPevolutionvariable-flavor-numberschemeNRQCDhigh-energyresummationBFKLLHCandFCCphenomenology

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

Bottomoniumlike tetraquarks—hypothetical particles made of a bottom-antibottom pair plus either two light quarks ($X_{b\bar{b}q\bar{q}}$) or a second bottom-antibottom pair ($T_{4b}$)—have not yet been observed, and this paper supplies a calculational framework for how they would form in proton collisions. It claims that at high transverse momentum the dominant production mechanism is the leading-power fragmentation of a single parton into the tetraquark, and it constructs two new families of collinear fragmentation functions, TQHL1.1 and TQ4Q1.1, that describe that splitting and evolve with energy via DGLAP equations in a variable-flavor-number scheme. The initial-scale inputs come from an improved spin-physics-inspired model for the doubly heavy states and from potential nonrelativistic QCD for the fully heavy states. If these functions are right, the paper's next-to-leading-log resummed predictions for tetraquark-plus-jet rates at the LHC and FCC are ready to be compared with future searches.

What carries the argument

The central objects are the two new fragmentation-function families, TQHL1.1 and TQ4Q1.1, each defined by initial-scale inputs at kinematic thresholds—$3m_Q+2m_q$ for the doubly heavy channel, and $4m_Q$ for the gluon and $5m_Q$ for the heavy-quark channel in the fully heavy case—and then evolved with DGLAP using a threshold-consistent heavy-flavor nonrelativistic evolution scheme. The carrying mechanism is the leading-power variable-flavor-number-scheme fragmentation picture, in which a short-distance coefficient for a single parton splitting into the tetraquark Fock state is convoluted with a nonperturbative hadronization matrix element or wave function. The gluon channel plays a dedicated role as a natural stabilizer of the high-energy resummed series, since its smooth scale dependence keeps the next-to-leading-log corrections under control.

What would settle it

A lattice QCD calculation (or an independent potential-model determination) of the ratio $\langle O_{T_{4b}}\rangle/\langle O_{T_{4c}}\rangle$ for the $0^{++}$ state would settle the normalization claim: if the ratio differs substantially from the $\simeq 400$ value used in Eq. (21), all $T_{4b}$ cross sections in Section 4 would shift by the same factor. In the charmed sector, a precise high-transverse-momentum measurement of prompt $T_{4c}$ production could already constrain the analogous matrix elements and test the fragmentation picture.

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Extended reading notes

Core claim

The central claim is that both doubly bottomed tetraquarks $X_{b\bar{b}q\bar{q}}$ and fully bottomed $T_{4b}$ (and their charmed analogues) can be treated as produced by collinear fragmentation of a single parton, with initial conditions computed from QCD-based models and then evolved by DGLAP, and that this treatment yields phenomenologically usable predictions. Concretely, the paper releases TQHL1.1 functions for $X_{c\bar{c}u\bar{u}}$, $X_{c\bar{c}s\bar{s}}$, $X_{b\bar{b}u\bar{u}}$, and $X_{b\bar{b}s\bar{s}}$, built on a spin-physics-inspired model with a fixed average transverse momentum $\langle \vec q_T^2 \rangle = 4$ GeV$^2$, and TQ4Q1.1 functions for $T_{4c/b}(0^{++})$ and $T_{4c/b}(2^{++})$, built on potential-NRQCD short-distance coefficients and long-distance matrix elements. The fully bottomed rates inherit their absolute normalization from a color-Coulomb scaling Ansatz that sets $\langle O_{T_{4b}}\rangle/\langle O_{T_{4c}}\rangle \simeq (m_b\alpha_s^{(b)}/(m_c\alpha_s^{(c)}))^9 \simeq 400$. The resulting rapidity-interval and transverse-momentum distributions for tetraquark-plus-jet production are stable under scale variation and under next-to-leading logarithmic corrections, which the paper interprets as a sign that heavy-flavor fragmentation stabilizes high-energy resummation.

Load-bearing premise

The load-bearing premise is that the color-Coulomb scaling Ansatz of Eq. (21), which sets the ratio of bottom- to charm-tetraquark nonperturbative matrix elements to about 400, correctly fixes the overall normalization of all fully bottomed tetraquark rates.

Editorial extensions

If this is right

Predictions are now available for $X_{b\bar{b}u\bar{u}}$, $X_{b\bar{b}s\bar{s}}$, $T_{4b}(0^{++})$, and $T_{4b}(2^{++})$ plus a jet in rapidity-interval and transverse-momentum bins at 14 TeV LHC and 100 TeV FCC, with rates from about 1 pb down to $10^{-5}$ pb.
The gluon fragmentation channel, though smaller than the heavy-quark one, controls the stability of the resummed distributions, so future data can test the natural-stability picture directly.
The released machine-readable sets let other groups compute single-inclusive and semi-inclusive bottom-tetraquark observables without redoing the model inputs.
Variants evolved without the initial heavy-quark input differ from the full sets by factors of 1.5 to 10 in the heavy-quark channel, so precision measurements could discriminate the initial condition.
Cross sections grow by roughly an order of magnitude from LHC to FCC energies, which would make a future 100 TeV collider a substantially more sensitive discovery channel.

Reading between the lines

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

The absolute normalization of every $T_{4b}$ prediction rests on the color-Coulomb Ansatz of Eq. (21); a lattice or potential-model computation of the four-body wave function at the origin could rescale all $T_{4b}$ rates by orders of magnitude without changing their shapes.
The same initial-scale inputs could be applied to photoproduction and deep-inelastic channels at an electron-ion collider, where the gluon-initiated fragmentation channel would be probed directly.
Comparing the $T_{4c}$ analogue predictions against existing LHC double-$J/\psi$ data at high transverse momentum would calibrate the long-distance matrix elements and thereby sharpen the $T_{4b}$ predictions.
The transverse-momentum parameter that fixes the doubly heavy fragmentation functions is chosen by heuristic peak scans rather than data; a data-driven determination of this parameter would convert the fragmentation functions from model-guided into measured inputs.

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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

Two things to know. First, the new content is real: TQHL1.1 and TQ4Q1.1 fragmentation functions, an NRQCD-based heavy-quark channel for fully heavy tetraquarks, and the first T4b-plus-jet NLL/NLO+ distributions at 14 and 100 TeV. The framework is inherited from the authors' own earlier TQHL1.0/TQ4Q1.0 papers and the HF-NRevo/JETHAD machinery, so this is an update rather than a new paradigm. Second, the paper is honest about its main vulnerability. Sec. 2.3.2 presents Eq. (21), a color-Coulomb dimensional-analysis estimate for the T4b/T4c LDME ratio, states that exact T4b LDMEs have not been computed, and calls the result "a reasonable Ansatz." Since Eq. (10) is linear in each LDME, any error in this ratio enters every T4b FF and all T4b cross sections linearly, and the ninth-power sensitivity means the factor ~400 is effectively unconstrained. No lattice, data, or independent potential-model check is offered. This is a load-bearing limitation, not an internal inconsistency, and the stress-test note has the right target; its factor of 2.6 for a 15% shift in the mass-coupling ratio is conservative in my own check.

What is done well: the DGLAP/HF-NRevo evolution is threshold-consistent; the TQ4Q1.1 vs TQ4Q1.1- comparisons cleanly show the effect of the charm initial-scale input; the sets are publicly released in LHAPDF; and the uncertainty bands from scale variation and integration are reported. The long symbolic SDCs are elaborate and the authors explicitly flag the unresolved z->1 behavior of NRQCD fragmentation functions rather than hiding it.

Softer spots: the TQHL1.1 = 4 GeV^2 choice is a scanned parameter selected partly to put the FF peak in a desired z-range, so shape statements built on that scan are partly imposed. The T4c LDMEs come from one of three potential models in Ref. [104], and that model uncertainty is not propagated. The citation pattern is self-heavy, but the self-citations point to the actual code and prior releases, which is appropriate for an incremental update.

Who is this for: heavy-flavor QCD phenomenologists planning LHC or FCC exotic-hadron searches, and the BFKL semihard community looking for another stable heavy-flavor tag. It deserves a serious referee. My recommendation: send it to review, and ask the authors to either anchor Eq. (21) with an independent estimate or present T4b predictions both normalized to T4c and with the LDME ratio varied by a plausible factor, plus a sensitivity scan for TQHL1.1. I would not desk-reject; the released FFs are a concrete, reusable asset.

Referee Report

3 major / 3 minor

Summary. The paper constructs two new families of DGLAP-evolving, VFNS collinear fragmentation functions for tetraquark production: TQHL1.1, describing the fragmentation of a heavy quark into a doubly heavy tetraquark X_{Q\bar Q q\bar q} via an improved Suzuki model, and TQ4Q1.1, describing the fragmentation of a gluon or heavy quark into a fully heavy tetraquark T_{4Q} via potential-NRQCD short-distance coefficients and long-distance matrix elements. These FFs are evolved with DGLAP/HF-NRevo and released in LHAPDF format. Using the NLL/NLO+ hybrid factorization implemented in (sym)JETHAD, the authors provide predictions for rapidity-interval and transverse-momentum distributions for bottomoniumlike-state-plus-jet production at 14 TeV LHC and 100 TeV FCC. The central claim is that these FF sets provide usable leading-power, factorized predictions for future searches, superseding the earlier 1.0 versions.

Significance. If the underlying model inputs are accepted, the paper makes a useful and concrete contribution: it provides public, DGLAP-evolved VFNS FFs for doubly and fully bottomed tetraquarks, with transparent symbolic computations through symJETHAD and numerical results through JETHAD. The phenomenology extends the group's established program on high-energy resummation to the exotic bottom sector, and the LHAPDF release is a genuine asset for future experimental and theoretical work. The main scientific value lies in the predictions for tetraquark-plus-jet rates at LHC/FCC energies, which are falsifiable once data on these hypothetical states become available. However, the absolute normalization of the T4b predictions rests on an unvalidated LDME scaling, and the TQHL1.1 shape relies on a heuristic transverse-momentum parameter; these inputs are acknowledged in the text but are not quantified as uncertainties, which limits the strength of the central claim.

major comments (3)

[Sec. 2.3.2, Eq. (21)] The T4b predictions inherit their absolute normalization from the assumption ⟨O_{T4b}⟩/⟨O_{T4c}⟩ ≃ (m_b α_s^{(b)}/(m_c α_s^{(c)}))^9 ≃ 400. Since Eq. (10) is linear in each color-composite LDME, the same factor multiplies every [g→T4b] and [Q→T4b] initial-scale FF and, through the linear convolution in Eq. (33), every T4b cross section shown in Figs. 13, 16, and 17. The ninth-power exponent makes this input extremely sensitive: changing the bracketed ratio from 1.8 to 2.07 changes the ratio by roughly a factor of 2.6, and the paper gives no explicit values of α_s, renormalization scales, or velocities that produce the number 400. The text itself states that exact T4b LDMEs have not been computed and labels Eq. (21) a 'reasonable Ansatz,' with no lattice, data, or independent potential-model value used to anchor it. Please provide an uncertainty band or a sensitivity scan over this ratio, and ideally a cross-check from an alternative LDME model, before presenting the T4b rates as quantitative predictions.
[Sec. 2.2.1, Eq. (9)] The TQHL1.1 ⟨q_T^2⟩ parameter is fixed at 4 GeV² through the relation sqrt(⟨q_T²⟩_{X}) ≈ sqrt(⟨q_T²⟩_{T4Q})/2, where ⟨q_T²⟩_{T4Q}=70 GeV² was itself chosen in Ref. [106] by requiring ⟨z⟩≳0.4 and that the quark channel have the same order of magnitude as the gluon one. This is a heuristic tuning of the peak position, and the statement that the [Q→X_{Q\bar Q q\bar q}] FFs peak in the window 0.65<z<0.85 is therefore partly imposed rather than predicted. Since the phenomenological rates in Figs. 12, 14, and 15 depend directly on these FFs, please show how the rapidity-interval and transverse-momentum distributions respond to a variation of ⟨q_T^2⟩ over a plausible range, or justify the chosen value with an independent observable.
[Sec. 2.3.3, Fig. 7] The paper notes that the [g→T4Q] initial-scale FFs do not vanish as z→1 and acknowledges that this behavior raises questions about compatibility with collinear factorization. Because these FFs enter the convolution in Eq. (33) over a z range that reaches unity, the endpoint region contributes to the cross section. Please estimate the numerical impact of the z→1 region, for instance by comparing with a version of the FF regulated near z=1, or state explicitly the z-range over which the predictions are meant to be trusted.

minor comments (3)

[Summary and Sec. 5] The second doubly bottomed state is denoted X_{b\bar b u\bar s} in the Summary and Sec. 5, while elsewhere in the paper the same state is denoted X_{b\bar b s\bar s}; please make the notation uniform.
[Fig. 15 caption] The caption contains a duplicated phrase: 'ratio between LL/LO or HE-NLO+ or HE-NLO+ and NLL/NLO+ predictions.' Please correct the caption.
[References] References [332] and [333] are identical entries for the CMS JINST 16 P02010 paper; one of the duplicates should be removed or replaced with the intended citation.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the FF inputs come from external NRQCD and Suzuki calculations, while the TQHL ⟨q²_T⟩ choice and Eq. (21) T4b rescaling are openly labeled heuristics/Ansätze, making the T4b normalization a model-dependence caveat rather than a circular derivation.

full rationale

Walking the derivation chain, the TQHL1.1 initial-scale inputs are built from the Suzuki-model fragmentation function of Ref. [94], with the normalization in Eqs. (1)-(8) evaluated analytically and one free parameter, ⟨q²_T⟩, chosen so that the FF peaks at moderate-to-large z and has a magnitude comparable to gluon channels. The paper transparently calls this a heuristic choice ('Beyond the heuristic nature of the previous relation, there exists a deeper rationale supporting our choice'), and it is a model-construction input, not a quantity fitted to the high-energy observables that are later predicted. The TQ4Q1.1 family uses the external NRQCD short-distance coefficients of Refs. [104,105], with T4c LDMEs taken from the Lü-Chen-Dong potential model, Eq. (20). For T4b, Eq. (21) rescales the T4c LDMEs by a Coulomb dimensional-analysis ratio of about 400; the paper twice states that exact T4b LDMEs have not been computed and that the rescaling is 'a reasonable Ansatz.' As a result, the absolute normalization of the T4b rates in Figs. 13, 16 and 17 inherits this Ansatz, and because the ratio enters to the ninth power, a 15% change in (m_b α_s^b)/(m_c α_s^c) changes it by roughly a factor of 2.6. That is an important model-uncertainty caveat, but it is not circularity: no parameter is fitted to the rapidity or transverse-momentum distributions, and no prediction is claimed to be derived from those observables. The DGLAP evolution, BFKL resummation, PDFs, and the NLL/NLO+ hybrid factorization are standard external frameworks, and the gluon FF is benchmarked against Ref. [104]. The self-citations to HF-NRevo [115,116], JETHAD [96,117-120], and the 'natural stability' concept [197] are methodological or interpretive rather than a load-bearing uniqueness argument, and the central initial-scale inputs come from external calculations. The paper therefore does not reduce any of its central predictions to its own inputs by construction; the appropriate finding is no significant circularity, with a minor caveat for the explicitly heuristic TQHL parameter choice and the explicitly Ansatz-based T4b normalization.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central FFs are model-based constructions: the T4b normalization is set by a Coulomb-scaling Ansatz, the TQHL1.1 shape is tuned through ⟨q_T^2⟩, and the NRQCD LDMEs come from a potential model. These are inputs from prior literature or this paper's model choices, not derived from fragmentation data.

free parameters (2)

⟨q_T^2⟩ for doubly heavy tetraquarks (TQHL1.1) = 4 GeV^2
Chosen by numeric scan so the FF peaks with ⟨z⟩ above about 0.4 and has magnitude comparable to the gluon channel; not derived from data (Sec. 2.2, Eq. (9)).
T4b-to-T4c LDME ratio = approximately 400
Coulomb-scaling Ansatz in Eq. (21) sets the normalization of all T4b FFs; no independent lattice, potential-model, or experimental determination.

assumptions (5)

domain assumption The Suzuki model provides the correct initial-scale input for Q→X_QbarQqbar fragmentation once normalization and ⟨q_T^2⟩ are fixed as in Sec. 2.2.
Used without external validation; the model was originally developed for TMD fragmentation of mesons, and its extension to tetraquarks relies on analogy.
domain assumption NRQCD factorization applies to fully heavy tetraquark fragmentation, with LDMEs taken from potential model Ref. [235].
Assumed in Eq. (10); the authors note unresolved z→1 endpoint behavior in Sec. 2.3.3.
ad hoc to paper The T4b LDMEs follow from T4c LDMEs through the Coulomb scaling in Eq. (21), with ratio approximately 400.
Dimensional analysis Ansatz; no lattice or data cross-check is provided.
domain assumption Initial-scale contributions from light partons and nonconstituent heavy quarks are negligible for both FF families.
Stated in Secs. 2.2.2 and 2.3.4; justified by analogy with quarkonium FFs rather than by an explicit calculation.
domain assumption The NLL/NLO+ hybrid factorization with a small-cone jet selection is valid for tetraquark-plus-jet production at high energy.
Carried over from earlier hadron-jet works in Sec. 3.2; no genuine fixed-order NLO benchmark is available for this process.

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Cite this review

Pith. "Pith review of Bottomoniumlike states in proton collisions: Fragmentation and resummation." pith.science (2026). https://pith.science/paper/NDP6FVID

@misc{pith2026241210549,
  author       = {Pith},
  title        = {Pith review of: Bottomoniumlike states in proton collisions: Fragmentation and resummation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NDP6FVID}},
  note         = {Machine review of arXiv:2412.10549}
}

abstract

We study the semi-inclusive hadroproduction of doubly bottomed tetraquarks ($X_{b\bar{b}q\bar{q}}$) as well as fully bottomed ones ($T_{4b}$), to which we collectively refer as "bottomoniumlike" states. We rely upon the variable-flavor number-scheme fragmentation at leading power, where a single parton perturbatively splits into the corresponding Fock state, which then hadronizes into the color-neutral, observed tetraquark. To this end, we build new sets of DGLAP/HF-NRevo consistent, hadron-structure oriented collinear fragmentation functions, which we name TQHL1.1 and TQ4Q1.1 parametrizations. They extend and supersede the corresponding 1.0 versions recently derived in previous works. The first family describes the fragmentation of doubly heavy tetraquarks and is based on an improved version of the Suzuki model for the heavy-quark channel. The second family depicts the fragmentation of fully heavy tetraquarks and embodies initial-scale inputs for gluon and heavy-quark channels, both of them calculated by the hands of potential nonrelativistic QCD. As a phenomenological application, we provide novel predictions for tetraquark-plus-jet high-energy distributions, computed within the NLL/NLO$^+$ hybrid factorization from (sym)JETHAD, at 14 TeV and 100 TeV FCC.

Figures

Figures reproduced from arXiv: 2412.10549 by the authors.

**Figure 1.** Factorization-scale dependence of KKSS07 [146, 147], ZCW19+ [119, 121], and ZCFW22 [122, 123] collinear FFs respectively depicting Hb, Υ, Bc( 1S0), and Bc( 3S1) particle formation at z = 0.425 ≃ ⟨z⟩. tribute to this expansion. The latter becomes essential to cancel divergences arising in NLO calculations of P-wave-quarkonium hard factors [156, 157]. NRQCD is based on the premise that quarkonium production begins wit… view at source ↗

**Figure 2.** LO representative diagram for the collinear fragmentation of a constituent heavy quark into a color-singlet [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Constituent heavy-quark to doubly charmed (upper) and bottomed (lower) tetraquark collinear fragmen [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

Figures from the paper (14 more)

**Figure 4.** Figure 4: Factorization-scale dependence of TQHL1.1 collinear FFs depicting Xccu¯ u¯ (left) and Xccs¯ s¯ (right) formation, at z = 0.425 ≃ ⟨z⟩. 10 3 × 10 2 1 4 × 101 6 × 101 2 × 102 µF [GeV] 10−12 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4 zD Xbu ¯b ¯u i (z, µ F ) symJETHAD + APF…

**Figure 5.** Figure 5: Factorization-scale dependence of TQHL1.1 collinear FFs depicting Xb¯buu¯ (left) and Xb¯bss¯ (right) formation, at z = 0.425 ≃ ⟨z⟩. D (0++) g (z, [6, 6]) = π 2α 4 s (4mQ) 331776 dD g (z) × [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 6.** Figure 6: LO representative diagrams for the collinear fragmentation of a gluon (left) or a constituent heavy [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: Gluon to fully charmed (upper) and bottomed (lower) tetraquark collinear fragmentation. Left and right [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]

**Figure 8.** Figure 8: Constituent heavy-quark to fully charmed (upper) and bottomed (lower) tetraquark collinear fragmentation. [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]

**Figure 9.** Figure 9: Factorization-scale dependence of TQ4Q1.1 collinear FFs depicting T4c(0++) (left) and T4c(2++) (right) formation, at z = 0.425 ≃ ⟨z⟩. First ancillary panels below primary plots show the ratio between TQ4Q1.1 and TQ4Q1.1− functions. Second ancillary panels show the rati…

**Figure 10.** Figure 10: Factorization-scale dependence of TQ4Q1.1 collinear FFs depicting T4b(0++) (left) and T4b(2++) (right) formation, at z = 0.425 ≃ ⟨z⟩. Ancillary panels below primary plots show the ratio between TQ4Q1.1 and TQ4Q1.1− functions. ing purposes, we derive supplementary sets…

**Figure 11.** Figure 11: Hybrid factorization for the T4Q plus jet (left) and XQQq¯ q¯ plus jet (right) semi-inclusive hadroproduction. Firebrick squares (green rhombi) represent XQQq¯ q¯ (T4Q) tetraquark collinear FFs. Gray arrows depict lightflavored jets. Orange ovals stand for proton col…

**Figure 12.** Figure 12: Rapidity-interval rates for Xb¯buu¯ (left) and Xb¯bss¯ (right) plus jet hadroproduction at √ s = 14 TeV (LHC, upper) or 100 TeV (nominal FCC, lower). Ancillary panels below primary plots exhibit the ratio between LL/LO and NLL/NLO+ predictions. Uncertainty bands captu…

**Figure 13.** Figure 13: Rapidity-interval rates for T4b(0++) (left) and T4b(2++) (right) plus jet hadroproduction at √ s = 14 TeV (LHC, upper) or 100 TeV (nominal FCC, lower). Ancillary panels below primary plots exhibit the ratio between LL/LO and NLL/NLO+ predictions. Uncertainty bands cap…

**Figure 14.** Figure 14: Transverse-momentum distributions for Xb¯buu¯ (left) and Xb¯bss¯ (right) plus jet hadroproduction at √ s = 14 TeV (LHC, upper) or 100 TeV (nominal FCC, lower), and for 2 < ∆Y < 4. Ancillary panels below primary plots exhibit the ratio between LL/LO or HE-NLO+ and NLL/…

**Figure 15.** Figure 15: Transverse-momentum distributions for Xb¯buu¯ (left) and Xb¯bss¯ (right) plus jet hadroproduction at √ s = 14 TeV (LHC, upper) or 100 TeV (nominal FCC, lower), and for 4 < ∆Y < 6. Ancillary panels below primary plots exhibit the ratio between LL/LO or HE-NLO+ or HE-NL…

**Figure 16.** Figure 16: Transverse-momentum distributions for T4b(0++) (left) and T4b(2++) (right) plus jet hadroproduction at √ s = 14 TeV (LHC, upper) or 100 TeV (nominal FCC, lower), and for 2 < ∆Y < 4. Ancillary panels below primary plots exhibit the ratio between LL/LO or HE-NLO+ and NL…

**Figure 17.** Figure 17: Transverse-momentum distributions for T4b(0++) (left) and T4b(2++) (right) plus jet hadroproduction at √ s = 14 TeV (LHC, upper) or 100 TeV (nominal FCC, lower), and for 4 < ∆Y < 6. Ancillary panels below primary plots exhibit the ratio between LL/LO or HE-NLO+ and NL…

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