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One-Point energy correlator inside jets

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The one-point energy correlator inside jets admits a TMD factorization in soft-collinear effective theory that isolates gluon transverse-momentum fragmentation, with global logarithms resummed to NNLL.

desk verdict The 'new' observable is the differential jet shape in TMD clothing; the factorization is sound, but the advertised scale independence rests on an unproven cancellation and the gluon sensitivity is asserted, not shown. read the letter →

arxiv 2507.21613 v2 pith:OZDR2CCS submitted 2025-07-29 hep-ph

Zihao Mi , Zhan Wang This is my paper
classification hep-ph
keywords energycorrelatorsjetsubstructureTMDfragmentationfunctionssoft-collineareffectivetheoryresummationnon-globallogarithmsgluonTMDFFhadronization
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 introduces a new jet substructure observable, the one-point energy correlator (EC), which measures the energy deposited inside a jet at a fixed angle $\chi$ from the jet axis, weighted by each hadron's energy fraction relative to the jet. It claims that this observable admits a transverse-momentum-dependent (TMD) factorization within soft-collinear effective theory, built from hard functions, in-jet soft functions, and TMD fragmentation functions. On top of that factorization the authors resum global logarithms to NNLL and non-global logarithms to LL, and show that the normalized EC has greatly reduced dependence on the factorization scale while remaining sensitive to the jet scale $p_T R$. Because the EC in $pp$ collisions is dominated by gluon TMDFFs, the paper argues it offers a practical way to constrain gluon TMD fragmentation, which is difficult to extract from existing data. Numerical predictions agree with PYTHIA 8 simulations in the small-$\chi$ region where the factorization applies.

What carries the argument

The load-bearing object is the semi-inclusive TMD fragmenting jet function (siTMDFJF) and its factorized decomposition, Eqs. (2.4)-(2.5): the EC jet function is obtained by integrating the siTMDFJF over the hadron transverse momentum with the weight $z_h$ and the angular delta function $\delta(2j_\perp/(z_h\omega_J)-\chi)$, and the siTMDFJF itself factorizes into a hard function $H_{c\to i}$, an in-jet soft function $S_i$, and a TMDFF $D_{h/i}$, with rapidity divergences cancelling between $S_i$ and the TMDFF. The resummation machinery is standard TMD evolution in $b$-space: each ingredient is evolved from its natural scale to the common hard scale using its anomalous dimensions, the $b_*$ prescription separates perturbative from non-perturbative $b$, and the non-global logarithms are collected into the exponentiated factor $U_{\rm NG}$ from Eq. (3.30). This structure is what lets the paper quote NNLL accuracy for global logarithms and LL accuracy for non-global logarithms in the final expression (3.37).

What would settle it

Measure the normalized EC at a fixed small angle, say $\chi = 0.01$, for two different jet radii at the same jet $p_T$: the factorized prediction fixes the ratio through the resummed jet-scale dependence, so a measured ratio that deviates from the quoted scale-uncertainty band would show that the hierarchy $j_\perp \ll p_T R$ is violated. A second check is to fit the gluon TMD model parameters in one $p_T$ bin and see whether the same parameters describe another $p_T$ bin; if they do not, the claim that the $pp$ EC is governed by gluon TMDFFs would be called into question.

Watch

Extended reading notes

Core claim

The central claim is that the normalized one-point energy correlator inside a jet can be written as a factorized, resummed expression from Eq. (3.37): a hard matching coefficient, a $b$-space Bessel transform of the energy-angle constraint, the perturbative Sudakov factor with the $b_*$ prescription, a non-perturbative TMD model function, and the non-global evolution factor. Within the TMD regime $\Lambda_{\rm QCD} \lesssim j_\perp \ll p_T R$, the EC jet function is built from the semi-inclusive TMD fragmenting jet function, whose convolution of an in-jet soft function and a TMDFF carries the transverse-momentum information. The authors find that normalization cancels the $\ln R$ terms and the dependence on the factorization scale, leaving the jet scale $p_T R$ as the main remaining scale. They further find that in $e^+e^-$ annihilation the EC is mainly sensitive to quark TMDFFs, while in $pp$ collisions it is mainly governed by gluon TMDFFs; the gluon-dominated prediction is more sensitive to the TMD model parameters, which is exactly where the observable could add new constraints.

Load-bearing premise

The entire factorized prediction rests on the assumed hierarchy $\Lambda_{\rm QCD} \lesssim j_\perp \ll p_T R$ holding for the measured hadrons, so that the in-jet transverse momentum is small enough for a TMD separation and the rapidity divergences of the soft function and TMDFF cancel at a common scale; if this hierarchy breaks down at moderate $p_T$, large $R$, or larger angles $\chi$, the factorization no longer applies and only the fixed-order matching covers the large-$\chi$ region.

Editorial extensions

If this is right

  • If the factorization holds, the normalized EC provides a new jet-substructure observable whose main scale is the jet scale $p_T R$, not the hard scale.
  • In $pp$ collisions the EC is dominated by gluon TMDFFs, so measurements of the distribution can directly constrain gluon TMD model parameters that are poorly fixed by current data.
  • The small-$\chi$ behavior of the EC is predicted to scale approximately as $\sim\chi$, corresponding to a fairly uniform in-jet energy distribution, and the resummed result matches the fixed-order calculation for $\chi > 0.1$.
  • Because the rapidity divergences cancel between the in-jet soft function and the TMDFF, the observable tests the universality of TMD fragmentation inside jets: the same gluon TMDFF should appear in other TMD processes.
  • The framework can be extended to spin-dependent TMD fragmentation functions, since the EC probes the same transverse-momentum structure that underlies Collins-type asymmetries inside jets.

Reading between the lines

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

  • One consequence the paper leaves implicit is a direct test of the TMD hierarchy: measuring the normalized EC at fixed small $\chi$ while varying $p_T$ and $R$ should trace the resummed $p_T R$ dependence, and any departure would locate where the two-scale factorization fails.
  • The reduced sensitivity to the factorization scale suggests that at small $\chi$ the dominant theory uncertainty moves into the non-perturbative TMD models, so the small-$\chi$ predictions inherit the assumptions of those models.
  • If the gluon-TMDFF dominance in $pp$ collisions is confirmed, the same observable could be adapted to polarized collisions, where the in-jet transverse-momentum distribution would carry spin-dependent TMD information.
  • A practical extension would be to compare EC predictions at different jet radii with a single fitted gluon TMD model; consistency across $R$ and $p_T$ bins would be a non-trivial check of factorization that the present Monte Carlo comparison only partially covers.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The manuscript defines a one-point energy correlator (EC) inside reconstructed jets, constructs it by integrating the semi-inclusive TMD fragmenting jet function over the hadron transverse momentum with an energy-fraction weight and an angular delta function (Eq. (2.4)), and factorizes it in SCET into hard functions, in-jet soft functions, and TMDFFs (Eq. (2.5)). It then resums global logarithms to NNLL and non-global logarithms to LL, includes non-perturbative TMD and hadronization models, and presents normalized EC distributions for e+e- and pp collisions compared with PYTHIA 8 and NLOJet++. The central claims are that the normalized EC has strongly reduced factorization-scale dependence, is mainly sensitive to the jet scale pT R, and can be used to constrain gluon TMDFFs.

Significance. If the factorization and the normalization argument are correct, this is a genuinely useful and experimentally accessible jet substructure observable. The paper imports a well-established SCET/TMD framework, uses public PDF and FF sets, and makes explicit resummed predictions for both quark- and gluon-dominated channels, including comparisons with PYTHIA 8 and NLOJet++. The proposed sensitivity to gluon TMDFFs is valuable because those functions are poorly constrained by current data. However, the advertised scale-independence is exactly the step that is not proven, so the significance is conditional on completing and validating the normalization derivation.

major comments (3)
  1. [§3.1, Eqs. (3.18)–(3.21)] The normalization derivation is not valid as written. Eq. (3.19) repeats σ^(0)⊗J^(0) in the numerator and writes the NLO expansion of the ratio in a way that conflates the expansion of the numerator with the expansion of the ratio. Eq. (3.20) is the identity A = B A/B and therefore carries no content. The subsequent statement that the hard matching coefficients C_{c→i} reduce to δ(1−z) d^{c,alg}_J and that the NLO partonic cross sections vanish for the normalized EC is unsupported. Concretely, C_{q→q'} in Eq. (3.8a) contains P_qq(z)L plus z-dependent plus-distribution and non-log terms, while J_q^(1) in Eq. (3.17a) contains L[P_qq + P_gq] plus different finite terms; whether the z_c, v, ω, pT, and η integrations make these cancel is exactly the non-trivial step. This must be shown explicitly, because Eq. (3.37) and the factorization-scale-independence claim are built on it.
  2. [§3.1, Eqs. (3.10)–(3.11)] The key step from the convolution definition (2.4)–(2.5) to the resummed EC jet function is stated rather than derived. The text does not show how the sum over intermediate partons, the transverse-momentum convolution, the δ(2j⊥/zhωJ − χ) constraint, and the b-space transform combine to produce C_{c→i} times D_i^pert times the Sudakov factor, nor is it explained why the collinear-matching approximation of the TMDFF is legitimate under the same scale hierarchy. Since Eqs. (3.10)–(3.11) feed directly into the final resummed expression (3.37), this omission should be repaired by a derivation or by a precise reference to the corresponding derivation in the siTMDFJF literature.
  3. [§2.2, hierarchy and validation] The factorization assumes Λ_QCD ≲ j⊥ ≪ pT R, but the pp predictions in Fig. 3 integrate over pT and η, and the small-χ region includes configurations where the transverse momentum of the hadron relative to the jet axis is not parametrically small compared with pT R. The paper does not quantify the range of zh and χ for which the hierarchy is satisfied after the integrations in Eq. (2.6), nor does it test the stability of the small-χ predictions under variations of the pT bin or the jet radius R. Because matching is applied only for χ > 0.1, the small-χ claims rest entirely on the validity of the TMD factorization; a quantitative check, such as varying pT and R and monitoring the ratio of resummed to fixed-order results in the overlap region, would substantially strengthen the central claim.
minor comments (5)
  1. [§3.1, Eq. (3.21)] There is a typographical double equals sign, "EC(χ) = =", and the notation in Eq. (3.19) uses the same symbol σ⊗J for numerator and denominator, which makes the expansion hard to follow.
  2. [§3.1, Eq. (3.12)] The Sudakov factor S_i^pert(µb, µJ) is written as an integral followed by the factor (νD/νS)^{γR[αs(µb)]}, but as typeset it is missing an explicit exponential and has an unclosed parenthesis in the integrand; please rewrite this equation.
  3. [§4, Eqs. (4.4)–(4.5)] The transition function is written as t(χ)=1/(1+e^{a χ−0.1/0.1}), which is ambiguous; it should presumably be t(χ)=1/(1+exp(a(χ−0.1)/0.1)), and the parameter a should be defined.
  4. [§3.2, Eq. (3.29)] The statement that the result (3.29) is "exact for hemisphere jet masses in e+e− collisions, implies that the result obtained does not depend on the jet cone radius" is confusing, because Eq. (3.29) is written as an expansion in R with R-dependent terms while the hemisphere statement refers to a different geometry; please clarify.
  5. [Appendix B, Eqs. (B.1)–(B.3)] The function Θalg is used before it is defined, and in Eq. (B.2) the quantity "ωj" should presumably be "ωJ"; please clean up the notation and order of definitions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the EC observable is constructed from, not fitted to, TMDFFs, and all load-bearing factorization input is cited from non-overlapping prior work.

full rationale

The only definitional tautology is Eq. (2.4), where the EC jet function is defined as an integral over the siTMDFJF, so the statement that EC is sensitive to TMDFFs follows immediately from the observable's construction; this is not a derived prediction but a design choice, and the paper does not fit TMDFF parameters to EC data and then call the result a prediction. The factorization (2.5) and the in-jet soft function are taken from refs [34,37], whose authors (Kang, Liu, Ringer, Xing, Vitew, Waalewijn) do not overlap with the present authors, so there is no load-bearing self-citation chain; the NGL parameterization (3.30) is likewise imported from independent literature [65,72]. The central claim of reduced factorization-scale dependence rests on an analytic cancellation between the hard coefficients (3.8) and the siFJFs (3.17); whether that cancellation is fully demonstrated is a separate correctness concern, since Eqs. (3.19)-(3.21) are schematic and include a tautological identity, but it is not circular because the paper does not impose the result by construction or by fitting. The numerical predictions are checked against PYTHIA 8 and NLOJet++ as external benchmarks, and the non-perturbative models come from global fits to other data, so the derivation is self-contained and not circular.

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

The numerical predictions depend on a set of non-perturbative TMD models and parameters imported from earlier fits, and on domain assumptions about the TMD hierarchy. None of these are derived in this paper, and at least one parameter (0.042 in Eq. (4.1)) is given without a supporting reference.

free parameters (6)
  • bmax = 1.5 GeV^-1
    Boundary between perturbative and non-perturbative b-region in the b*-prescription (Section 3.3, Eq. (3.32)); chosen from refs [77,78], not derived.
  • D_NP hadronization coefficient = 0.042
    Coefficient in exp(-0.042 b^2 / y^2) TMDFF model function in Section 4 Eq. (4.1); no reference is given for this specific value.
  • j_q(b) coefficients = -0.59, -0.03
    Quark TMD model function in Eq. (4.2) taken from SIDIS/DY fits in ref [78].
  • j_g(b) coefficients = -0.17, -0.09
    Gluon TMD model function in Eq. (4.2) taken from fits; the gluon part is the least constrained by data.
  • g_k(b) coefficient = -0.21 C_a/C_F
    Rapidity non-perturbative exponent in Eq. (4.3) from refs [78,87].
  • NGL parametrization constants = a=0.85 C_A, b=0.86 C_A, c=1.33
    LL non-global log parametrization of Eq. (3.30) from the Dasgupta-Salam parametrization [65], precision LL.
assumptions (5)
  • domain assumption The semi-inclusive TMD fragmenting jet function factorization of Eq. (2.5) is valid, with the in-jet soft function taken from ref [37].
    Section 2.2; the paper imports this factorization from refs [34,37] without derivation.
  • domain assumption The scale hierarchy Lambda_QCD is less than or similar to j_perp much smaller than pT R holds for the measured hadrons.
    Section 2.2; defines the TMD region and the small-chi approximation chi ~ 2 j_perp/(z_h omega_J).
  • standard math The rapidity divergences of the in-jet soft function and the TMDFF cancel at a common scale nu, making the observable independent of nu.
    Section 3.1, Eqs. (3.2)-(3.6); standard TMD rapidity renormalization, assumed from ref [34].
  • domain assumption Non-perturbative TMD models from external SIDIS/DY fits apply to jet fragmentation in e+e- and pp collisions.
    Section 3.3; b*-prescription and model functions taken from refs [77,78,87].
  • domain assumption The large-Nc non-global log parametrization of ref [65] describes the cone-jet NGLs.
    Section 3.2, Eq. (3.30); the coefficients a, b, c are taken from the hemisphere NGL parametrization and applied to the jet cone.

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Cite this review

Pith. "Pith review of One-Point energy correlator inside jets." pith.science (2026). https://pith.science/paper/OZDR2CCS

@misc{pith2026250721613,
  author       = {Pith},
  title        = {Pith review of: One-Point energy correlator inside jets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OZDR2CCS}},
  note         = {Machine review of arXiv:2507.21613}
}
read the original abstract

In this work, we introduce a new jet observable, the one-point energy correlators (EC), designed to characterize the in-jet energy flow distribution by measuring energy deposition at a specific angle relative to the jet axis. Building upon the transverse momentum dependent physics, we aim for the EC to provide novel insights into jet substructure and offer a new approach to study TMD physics, particularly gluon transverse momentum dependent fragmentation functions (TMDFFs) which are notoriously difficult to extract. We obtain the factorization of the EC jet function within Soft-Collinear Effective Theory and leverage the framework of semi-inclusive TMD fragmenting jet functions. We resum large global logarithms and non-global logarithms (NGLs) and show that the normalized EC jet function exhibits significantly reduced dependence on the factorization scale and is primarily sensitive to the jet scale. Finally, after incorporating non-perturbative effects, we present numerical calculations up to NNLL accuracy for global logarithms and LL accuracy for NGLs, and we compare these predictions with PYTHIA 8 simulations.

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Forward citations

Cited by 3 Pith papers

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