Systematically Constructing the Likelihood for Boosted $H\to gg$ Decays

Andrew J. Larkoski

REVIEW 3 major objections 7 minor 60 references

Systematically Constructing the Likelihood for Boosted $H\to gg$ Decays

T0 review · 3 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A simple infrared-safe jet observable constructed from the inverse of the H→ggg matrix element becomes an essentially perfect discriminator between boosted Higgs decays and QCD jets in the high-energy limit, and yields a…

desk verdict A clean analytic benchmark for H→gg tagging, but the perfect-discriminant claim is a leading-log estimate for g→gg jets, not an inclusive-QCD proof. read the letter →

arxiv 2411.10539 v1 pith:WZUCQAP4 submitted 2024-11-15 hep-ph hep-ex

Andrew J. Larkoski This is my paper · ORCID

classification hep-phhep-ex

keywords boostedHiggsdecaysH→ggtaggingjetsubstructurelikelihoodratioinfraredandcollinearsafetyanomalydetectioncolorflowsoftgluonemission

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

This paper asks how well a boosted Higgs boson decaying to two gluons can be separated from ordinary QCD jets that fake a Higgs, and answers with a systematic expansion of the likelihood ratio in the strong coupling. At leading order, it proves that a jet mass window plus the energy fraction of the softer of the two leading subjets improves signal over background by roughly $1/\alpha_s(m_H) \sim 10$, independent of jet kinematics at high boost. At next-to-leading order, it constructs the observable $d_2/z^2$ by inverting the $H\to ggg$ matrix element with soft gluon emission; the inversion turns the matrix element's soft and collinear divergences into zeros, making the observable infrared and collinear safe. The paper argues analytically that this observable approaches the perfect discriminant as the jet transverse momentum grows, and shows in simulated data from standard event generators and parton showering that it improves the signal-to-background ratio by a factor of several hundred at $p_T > 2$ TeV. If correct, a simple three-emission measurement approaches the likelihood-ratio bound for $H\to gg$ tagging and provides a concrete physics benchmark for machine-learning jet taggers.

What carries the argument

The load-bearing object is the inverse of the next-to-leading-order $H\to ggg$ matrix element, packaged as an IRC-safe jet observable $d_2 = \frac{p_\perp^2}{m_H^2}\sum_k z_k \frac{\theta_{1k}^2\theta_{2k}^2}{\theta_{12}^2}$; the full discriminant is $L = d_2/z^2$, where $z$ is the softer-subjet momentum fraction. Its power comes from color flow: the Higgs is a color singlet, so its soft gluon emission concentrates between the two hard gluons, whereas a QCD gluon jet is a color octet and emits at wide angles. Inverting the signal matrix element turns the soft and collinear poles into zeros, which is what makes the observable infrared and collinear safe and therefore calculable order-by-order in $\alpha_s$. The $z^2$ denominator compensates for the background's strong phase-space enhancement at small $z$, and the overall normalization makes the observable boost invariant along the jet.

What would settle it

Measure the cumulative distribution of $L = d_2/z^2$ on QCD background jets at several jet transverse momenta (for example $p_\perp = 1$, $2$, and $3$ TeV) after a Higgs mass window. The paper predicts that the value of $L$ at fixed background efficiency scales as $p_\perp^4/m_H^4$; if that scaling is absent, or if the simulated signal-to-background improvement of several hundred does not appear in data, the central claim is falsified.

Watch

Extended reading notes

Core claim

The central discovery is that the likelihood ratio for boosted $H\to gg$ jets versus QCD jets can be constructed order-by-order in $\alpha_s$ and, at next-to-leading order, reduces to a simple IRC-safe observable. Working in the collinear, high-boost limit, the paper computes the soft-gluon matrix elements for a color-singlet $H\to gg$ signal and for color-octet $g\to gg$ background. The difference of these matrix elements is proportional to $\theta_{1k}^2 + \theta_{2k}^2 - \theta_{12}^2$, which is positive where a QCD gluon jet emits and negative where the color-singlet Higgs emits. Even simpler, the inverse of the signal matrix element defines $d_2 = \frac{p_\perp^2}{m_H^2}\sum_k z_k \frac{\theta_{1k}^2\theta_{2k}^2}{\theta_{12}^2}$, which is IRC safe because soft and collinear divergences of the matrix element become zeros. The paper proves that, after a fixed jet mass window, the ratio $L = d_2/z^2$ has a background cumulative distribution whose fixed-efficiency contour scales as $L \propto p_\perp^4/m_H^4$, so as $p_\perp\to\infty$ signal efficiency can grow while background efficiency stays fixed, meaning the observable becomes the perfect discriminant. In simulated events with $p_\perp > 2$ TeV, cuts on this observable improve signal over background by a factor of several hundred.

Load-bearing premise

The analytic prediction that $d_2/z^2$ becomes a perfect discriminant assumes that background QCD jets at high transverse momentum are dominated by collinear gluon splitting with soft wide-angle gluon emission, and that hadronization does not erase the color-flow difference; the paper tests this only in simulation, not in collider data.

Editorial extensions

If this is right

After a Higgs mass window, a cut on the softer subjet energy fraction alone reduces background by about $1/\alpha_s(m_H) \sim 10$, independent of jet kinematics at high boost.
On simulated jets with $p_\perp > 2$ TeV, the observable $d_2/z^2$ improves the signal-to-background ratio by a factor of several hundred over inclusive jet selection.
The inverse-matrix-element construction is IRC safe in general, so the same anomaly-detection recipe can produce calculable discriminants for other signal processes whose next-to-leading-order matrix elements are known.
The analytic ROC curves give a ground reference that machine-learning taggers for $H\to gg$ should reproduce at minimum, which the paper frames as a step toward interpretability of jet taggers.

Reading between the lines

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

The author leaves implicit that the same color-flow asymmetry should tag any color-singlet resonance decaying to two partons at high boost, not just the Higgs decaying to gluons.
The simulation validation uses default showering and hadronization settings; varying those models or adding underlying-event and pileup conditions would test whether the several-hundred-fold improvement survives in a more detector-like environment, a check the paper does not perform.
A natural extension of the paper's moment analysis would be to construct the full two-dimensional likelihood on the $(d_2, z)$ space rather than the ratio $d_2/z^2$; the paper's moments suggest the ratio is near-optimal, but the full likelihood could be computed from the same simulated samples.

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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 solid, mostly honest analytic benchmark paper. The d2/z^2 observable is new, the construction is well explained, and the simulation comparison is genuine. The main caveat is the 'perfect discriminant' claim: it is derived only for g→gg background using leading double logarithms, and the paper is more confident in that limit than the derivation supports.

What's actually new: a first NLO likelihood construction for H→gg versus QCD jets, including the observable d2, and clean analytic ROC expressions at LO. The inverse-matrix-element/anomaly-detection idea is a nice trick and is made concrete. The paper extends the author's earlier program, but the H→gg case has its own wrinkles, and the simulation validation with MadGraph+Pythia is independent enough that the central result—that d2/z2 is a strong discriminant—is credible.

The soft spots: First, the NLO analysis restricts the background to g→gg fragmentation. That's stated in Sec. 5.1, but the intro and the 'perfect discriminant' language in Sec. 5.1 go beyond what that calculation shows. Quark-initiated jets are never analyzed at NLO. Simulation at pT>2TeV includes quark jets and still shows strong discrimination, so the empirical result is fine, but the asymptotic claim for inclusive QCD is not established analytically. Second, the pT^4 scaling is derived by keeping only leading double logarithms and discarding the x→0 region without evaluating the full integral. That is a parametric estimate, not a proof. The paper is fairly transparent about the approximation, but the phrase 'perfect discriminant' overstates it. Minor: no statistical uncertainties on the ROC/S/B curves, no public code or data, and the abstract's 'several hundred' is a bit strong—the text gives ~150 with the mass cut and ~30 from d2/z2 alone at reasonable signal efficiency.

For whom: anyone working on jet substructure or on ML tagger benchmarks. It's not going to change experimental searches, and the paper says so. The LO part is a nice pedagogical example of likelihood-ratio construction; the NLO part is a useful baseline for what a simple IRC-safe observable can do.

I'd send it to review. It deserves referee time; the gaps I named can be fixed with a more careful statement of what is proven and what is estimated, and maybe a footnote on quark jets. I would not desk reject.

Referee Report

3 major / 7 minor

Summary. The paper presents a systematic, matrix-element-based approach to discriminating boosted H→gg decays from massive QCD jets. At leading order, using a jet-mass window and the softer-subjet momentum fraction z, it proves that the signal-to-background improvement is independent of the jet kinematics at high boosts and is approximately 1/αs(m_H). At next-to-leading order, it constructs IRC-safe observables from the soft-emission matrix elements, in particular d2 (the inverse of the H→ggg matrix element) and the ratio L=d2/z^2, and argues that L becomes a perfect discriminant as p⊥→∞. The theoretical results are compared with MadGraph+Pythia simulations for p⊥ > 500, 1000, 1500, and 2000 GeV.

Significance. The leading-order analysis is clean, and the comparison with simulation in Fig. 2 is convincing. The construction of an IRC-safe observable from the inverted signal matrix element is a useful and well-motivated idea that connects anomaly detection to analytic QCD, and the paper explicitly derives the functional form of the observable rather than fitting it to data. However, the headline asymptotic claim—perfect discrimination for inclusive QCD jets—rests on a restricted g→gg background analysis and on a leading-double-log estimate that contains an unjustified zero for the small-angle (x→0) contribution. If the x→0 term is included, the scaling L ~ p⊥^4/m_H^4 may be weakened, which would eliminate the perfect-discriminant conclusion. The paper's finite-p⊥ results are still valuable, but the central asymptotic claim needs substantial revision.

major comments (3)

[§5.1, Eqs. (48)–(51)] The claim in Eq. (50) that the x→0 limit of the integral in Eq. (48) vanishes is incorrect. For x→0, the integrand behaves as (1/x) log[x^2/(z^2(1-z)L)] with a theta function that sets a lower bound x > z√((1-z)L) (up to O(1) factors), while the jet-radius constraint in Eq. (47) supplies an upper bound x < √(z(1-z)) R p⊥/m_H. The resulting integral is (1/2) log^2[ R^2 p⊥^2/(m_H^2 z L) ] plus terms of order log^2 L, and is not zero; it is a leading double logarithm of the same parametric order as the x→∞ contribution kept in Eq. (49). Consequently, the approximation leading to Eq. (51) is not a controlled leading-double-log estimate, and the scaling L ~ R^4 p⊥^4/m_H^4 in Eq. (53) is not established by the given calculation. The full integral in Eq. (48) should be evaluated, or at least estimated with both end-point limits retained, before drawing the perfect-discriminant conclusion.
[§5.1 and Introduction] The perfect-discriminant conclusion is derived only for the g→gg background channel: Eq. (33) uses the g→ggg soft matrix element, and Eq. (47) uses the g→gg splitting function. Quark-initiated jets are never analyzed at NLO, despite the Introduction's statement that the high-boost problem reduces to discrimination from 'color triplet or color octet jets of QCD' and the abstract's unqualified 'perfect discriminant' claim. Since quark jets can dominate the inclusive background at large transverse momentum, the asymptotic claim for inclusive QCD jets is unsupported. The simulation in Fig. 4 includes both quark and gluon jets but stops at p⊥ > 2 TeV, so it cannot establish the p⊥→∞ behavior. The paper should either explicitly restrict the asymptotic claim to the g→gg subchannel or extend the color analysis to quark-initiated backgrounds, for example by treating the q→qg dipole.
[§5.1, Eq. (53)] The conclusion from Eq. (53) that L=d2/z^2 has 'arbitrarily good signal efficiency with arbitrarily low background efficiency' as p⊥→∞ is drawn from a fixed-order, leading-double-log approximation of the background cumulative distribution. The author notes that this expansion is only meaningful where the O(αs) correction is below unity, but the asymptotic statement requires the cumulative distribution to be valid at arbitrarily small background efficiencies, where αs log^2(...) is no longer small. A resummation or a higher-order argument is needed to justify the perfect-discriminant limit, and the finite-p⊥ simulation cannot validate it. The paper should either provide such an argument or explicitly weaken the claim to a scaling prediction for the background tail at fixed, moderate background efficiencies.

minor comments (7)

[§2] The sentence 'p⊥/mH≪ 1' should read 'm_H/p_$\perp$ ≪ 1', since the highly-boosted regime is defined by the jet mass being much smaller than its transverse momentum.
[§2] There is a typo: 'signficant' should be 'significant'.
[§4.1] The word 'simualted' should be 'simulated'.
[§5.1] In the paragraph before Eq. (42), 'an observable formed from their combination that performs between than either individually' should read 'performs better than either individually'.
[Fig. 3] The left-panel legend entry for the observable (1+O_NLO)/z is garbled in the rendered text; please ensure the math displays correctly.
[§6] The phrase 'hadronc top decay' should be 'hadronic top decay'.
[Ref. [48]] Reference [48] is listed as 'E∞ Scheme, unpublished'; if this is the intended source for Winner-Take-All recombination, please provide a full reference or remove it.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: d2/z^2 is constructed from the signal matrix element by design, but its discrimination power is computed from the independent g→gg background matrix element and validated on unfitted simulation.

full rationale

The paper's central results are not circular. At leading order (Secs. 3–4), the quark and gluon jet mass and subjet-energy-fraction distributions are calculated from standard collinear splitting functions, and the ROC curve (Eq. 23) follows analytically from the Neyman–Pearson likelihood ratio; it is then compared to MadGraph+Pythia simulation with no parameter fitted to the comparison. At next-to-leading order (Sec. 5), the observable d2 is explicitly defined as the inverse of the H→gg soft matrix element (Eq. 36), so the cancellation that makes the signal moment independent of z (Eq. 43) is by construction; however, the claimed discrimination power is not a restatement of that definition. The background cumulative distribution (Eqs. 47–51) is computed from the independent g→gg eikonal matrix element, and the recommendation L=d2/z^2 follows from evaluating moments (Eqs. 43–44) before checking simulation; the n=2 exponent is confirmed by the simulated ROC curves, not fitted to them. No uniqueness theorem or load-bearing claim is imported from the author's prior work: Ref. [49] is cited as a direction not taken, Refs. [56,57] as related constructions, and Ref. [58] only as an equivalent power-counting context. The main caveats are non-circular correctness and completeness questions: the analytic 'perfect discriminant' argument is restricted to g→gg background and uses an asymptotic estimate in which the x→0 contribution is asserted to vanish without explicit evaluation (Eq. 50), and the several-hundred-fold improvement is demonstrated in a parton-shower model rather than collider data. These are validity risks, not reductions of a prediction to a fit or to the defining condition of the observable.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central derivation depends on standard QCD approximations (collinear splitting, eikonal soft limit, narrow width) and on the choice of alpha_s and analysis parameters. No new physical entities are introduced. The observable d2 is a constructed object, not an entity.

free parameters (3)

alpha_s (strong coupling at m_H) = 0.11
Chosen standard value used for numerical evaluation of mass-cut probabilities and ROC curves in Sec 3.2 and Fig 1; not fitted to discrimination results.
Jet radius R = 0.5
Analysis choice for anti-kT jets, fixed for all transverse momentum bins; central results depend on R through logarithmic factors like log(R^2 pT^2/m_H^2).
Jet mass window = [110, 150] GeV
Fixed window around the Higgs mass used to define signal and background fractions; the LO improvement factor scales with the window width.

assumptions (6)

standard math Neyman-Pearson lemma: likelihood ratio is the optimal discriminant
Invoked in Secs 1 and 4.1 to justify constructing the likelihood ratio.
domain assumption Narrow-width approximation for the Higgs and no signal-background interference
Stated in Sec 2: 'We further assume the narrow-width approximation throughout this analysis... no mixing of off-shell Higgs bosons with QCD jets at the amplitude level.'
domain assumption Collinear limit and leading power in m_H/pT for LO distributions
Secs 3-4 use collinear splitting functions to leading power in m_H/pT; power corrections are assumed negligible at high pT.
domain assumption Soft-gluon eikonal approximation for NLO matrix elements
Sec 5.1 uses the eikonal amplitude for soft gluon emission from two hard partons; this is standard but a leading-soft approximation.
domain assumption Background restriction to g->gg for analytic NLO results
Sec 5.1 explicitly narrows the analytic background to massive jets from g->gg fragmentation, while simulation uses the full QCD background.
domain assumption Leading double-log approximation for background cumulative distribution
Eq (51) keeps only leading double logarithms in Sigma_g(L), giving the pT^4 scaling that yields perfect discrimination in the infinite-energy limit.

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Cite this review

Pith. "Pith review of Systematically Constructing the Likelihood for Boosted $H\to gg$ Decays." pith.science (2026). https://pith.science/paper/WZUCQAP4

@misc{pith2026241110539,
  author       = {Pith},
  title        = {Pith review of: Systematically Constructing the Likelihood for Boosted $H\to gg$ Decays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WZUCQAP4}},
  note         = {Machine review of arXiv:2411.10539}
}

abstract

We study the binary discrimination problem of identification of boosted $H\to gg$ decays from massive QCD jets in a systematic expansion in the strong coupling. Though this decay mode of the Higgs is unlikely to be discovered at the LHC, we analytically demonstrate several features of the likelihood ratio for this problem through explicit analysis of signal and background matrix elements. Through leading-order, we prove that by imposing a constraint on the jet mass and measuring the energy fraction of the softer subjet an improvement of signal to background ratio that is independent of the kinematics of the jets at high boosts can be obtained, and is approximately equal to the inverse of the strong coupling evaluated at the Higgs mass. At next-to-leading order, we construct a powerful discrimination observable through a sort of anomaly detection approach by simply inverting the next-to-leading order $H\to gg$ matrix element with soft gluon emission, which is naturally infrared and collinear safe. Our analytic conclusions are validated in simulated data from all-purpose event generators and subsequent parton showering and demonstrate that the signal-to-background ratio can be improved by a factor of several hundred at high, but accessible, jet energies at the LHC.

Figures

Figures reproduced from arXiv: 2411.10539 by the authors.

**Figure 2.** Left: Distributions of the subjet momentum fraction [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Plots of the ROC curves for optimal combinations of [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Plots of two quantifications of discrimination power with the observable [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

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