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Event Shape/Energy Flow Correlations

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Carola F. Berger, Tibor Kucs, George Sterman (Stony Brook)

High Energy Physics - Phenomenology · Submitted 6 Mar 2003 · 38 references

Pith's verdict: A sound factorization and resummation paper whose numerical section sits outside the kinematic window it is meant to illustrate.

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arXiv IDhep-ph/0303051 (version 3)
Pith IDDV5LDEYI, Pith's permanent record of this paper and its review
CitationCarola F. Berger, George Sterman (Stony Brook), Tibor Kucs. "Event Shape/Energy Flow Correlations." arXiv:hep-ph/0303051 (2003). https://pith.science/paper/DV5LDEYI
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The paper

The paper claims

Correlating interjet energy flow with a thrust-like event shape replaces the logarithms that spoil fixed-order predictions with ones that can be pushed to subleading order, keeping the observable tied to radiation from the leading jets.

The reading

Predictions for how much energy lands in the angular region between two jets are contaminated by 'non-global' logarithms: soft gluons radiated at wide angles off intermediate-energy partons generate large corrections that do not exponentiate in any simple way, so the measured flow no longer reflects the short-distance color structure of the hard event. This paper proposes to fix that by measuring energy flow into the interjet region jointly with an event shape that forces the jets to be narrow, and by weighting each event with $e^{-nu f}$ so that the shape enters through its Laplace transform. The authors show that this weighting turns the offending logarithm $ln(Q/Q_Omega)$ into $ln(1/(epsilon nu))$, which for large $nu$ sits below the primary logarithm $ln(1/epsilon)$ that they do resum. If the argument holds, interjet energy flow becomes a controlled probe of how color flows at short distances, rather than a quantity dominated by soft secondary radiation at all scales.

What carries the argument

The load-bearing device is the Laplace transform in the $a$-dependent event shape. Because the shape weight suppresses final states with wide-angle radiation away from the jet axes, and because a jet of parallel-moving particles emits soft radiation coherently, the transform acts as a knob: it converts the secondary-emission logarithm into $ln(1/(epsilon nu))$ and thereby demotes it. The rest of the machinery is standard factorized resummation made to work with this knob — a hard function, two jet functions defined as matrix elements with Wilson lines, an eikonal soft function whose ratio to eikonal jets is free of collinear logarithms, and the Collins-Soper $K$-$G$ decomposition with the universal Sudakov anomalous dimension controlling the $nu$ dependence. The parameter $a$ is restricted to $a<1$; the appendix argues that for $a<1$ the recoil of the jet axis from soft radiation shifts the shape only by a power-suppressed amount, which is what licenses treating the jet axis as fixed inside each factorized function.

Extended reading notes

Core claim

The paper introduces 'shape/flow correlations': differential distributions in the energy $Q_Omega = epsilon Q$ deposited in a chosen interjet angular region $Omega$, taken together with a one-parameter family of event shapes $f(N,a) = s^{-1/2} sum_i k_{i,perp}^a, omega_i^{1-a} (1 - n_i cdot n_c)^{1-a}$ summed over the two jet hemispheres, which reduces to $1-T$ (thrust) at $a=0$ and is related to jet broadening at $a=1$. Working in $e^+e^-$ dijet events and Laplace-transforming in the shape variable with conjugate $nu$, the authors derive a factorized cross section in hard, jet and soft functions, resum it with Collins-Soper evolution equations, and obtain a closed expression at leading logarithm in $epsilon$ and next-to-leading logarithm in $nu$ (their Eq. 113). The central technical claim is that the transform replaces the non-global logarithm $ln(Q/Q_Omega)$ of the inclusive flow cross section by $ln(1/(epsilon nu))$, so that for sufficiently large $nu$ the secondary-radiation logarithm is subleading while the primary logarithm $ln(1/epsilon)$ is still resummed. As a consistency check, setting $a=0$ reproduces the known next-to-leading-logarithm thrust resummation, and the paper exhibits the soft function's explicit geometry dependence for a cone and for a rapidity ring.

Load-bearing premise

The whole construction rests on there being a real kinematic window in which the primary logarithm $alpha_s ln(1/epsilon)$ is large enough to need resummation while the secondary logarithm $alpha_s ln(1/(epsilon nu))$ is small enough to ignore, which requires the transform variable $nu$ to be large at accessible energies.

Editorial extensions

If this is right

  • Interjet energy flow becomes a short-distance color probe: the shape weighting suppresses contamination from intermediate-energy partons, so the observed flow can be read as radiation from the two leading jets.
  • The formalism covers thrust and jet broadening as two values of one parameter, so a single resummed expression interpolates between known event-shape results and provides a family of correlated observables to compare against data.
  • The soft function is the only place the geometry of the interjet region enters, so the angular dependence of the correlation factorizes from the color-dependent resummation and can be computed once per region shape.
  • Because the same jet functions appear for quark and gluon primaries, ratios of correlations for different jet flavors are predicted, giving a handle on the more complex color flow expected in hadron collisions.
  • The method extends in principle to multijet events and to initial-state hadrons, where color exchange between jets is richer.

Reading between the lines

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

  • The proposal is only as good as the numerical size of the window where $alpha_s ln(1/epsilon) gtrsim 1$ while $alpha_s ln(1/(epsilon nu)) < 1$; a plot of the two logarithms versus $nu$ at LEP energies, rather than the qualitative remark that the thrust peak sits near $nu approx 10$, would settle whether the window is wide or narrow.
  • Since the correlation is defined in transform space, comparing to experiment requires an inverse Laplace transform or a binning that mimics it; the practical usefulness of the observable depends on how much the inversion smears the shape dependence.
  • The same weighting trick plausibly applies to other non-global observables — jet masses, out-of-jet $p_T$ sums, rapidity-gap energy flow — wherever a secondary logarithm can be traded for a transform variable.
  • The paper's own note that a related analysis appeared after submission suggests the transform-based suppression of non-global logarithms is one instance of a broader structural statement about which observables admit primary-emission dominance.
Original abstract

We introduce a set of correlations between energy flow and event shapes that are sensitive to the flow of color at short distances in jet events. These correlations are formulated for a general set of event shapes, which includes jet broadening and thrust as special cases. We illustrate the method for electron-positron annihilation dijet events, and calculate the correlation at leading logarithm in the energy flow and at next-to-leading-logarithm in the event shape.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,933 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.15 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.06 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

We introduce a set of correlations between energy flow and event shapes that are sensitive to the flow of color at short distances in jet events. These correlations are formulated for a general set of event shapes, which includes jet broadening and thrust as special cases. We illustrate the method for electron-positron annihilation dijet events, and calculate the correlation at leading logarithm in the energy flow and at next-to-leading-logarithm in the event shape.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We introduce a set of correlations between energy flow and event shapes that are sensitive to the flow of color at short distances in jet events. contribution 1.00 0.93 0.31 asserted 1.00
s1 These correlations are formulated for a general set of event shapes, which includes jet broadening and thrust as special cases. contribution 0.64 0.81 0.08 asserted 1.00
s2 We illustrate the method for electron-positron annihilation dijet events, and calculate the correlation at leading logarithm in the energy flow and at next-to-leading-logarithm in the event shape. method 0.59 0.94 0.14 asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.87 · evidence from proof or derivation 0.98.

What was examined, and what was not

sources read
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not examined
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  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Sources of global and non-global logarithms in dijet event
Figure 2. A kinematic configuration that gives rise to the non-global
Figure 3. The relevant two-loop cut diagrams corresponding to the
Figure 4. Factorized cross section (27) after the application of Wa
Figure 5. Differential cross section εdσ/(dεdnˆ1) dσ0/dnˆ1 , normalized by the Born cross section, at Q = 100 GeV, as a function of ε and a at fixed ν: a) ν = 10, b) ν = 50. Ω is a ring (slice) centered around the jets, with a width of ∆η = 2. Finally, we illustrate the sensitivity of these results to the flavor of the primary partons. For this purpose we study the corresponding ratio of the shape/flow correlation to the cross section for gluon jets produced by a hypothetical color singlet source
Figure 6. Differential cross section εdσ/(dεdnˆ1) dσ0/dnˆ1 , normalized by the Born cross section, at Q = 100 GeV, as a function of a at fixed ν = 20 and ε = 0.05. Ω is chosen as in
Figure 7. Ratios of differential cross sections for quark to gluon je
Figure 8. C(∆η), as defined in (125), as a function of rapidity width ∆η of the region Ω. The dashed line is its limiting value, C(∆η → ∞) = π 2/6. determined by a set of energetic, nearly collinear particles. When we make this replacement, how￾ever, the contributions to the event shape from energetic particles near the jet axis may change. This change is neglected in going from the original factorization, Eq. (21), to the factorization in convolution form, Eq. (27), which is the starting point for the resummation techniques that we employ in this paper. The weight functions ¯f N (Ni , a) in…

Figures reproduced from arXiv:hep-ph/0303051 by the authors.

References
How later papers used it

129 later papers cite it

Evidence: substantial. 126 of the citing papers share no author with it, and 11 of those carry a confident label from Pith's citation classifier.

Pith read 14 of the 214 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 1 of 5 papers with this label

  • Followed this paper to define the jet function for quarks at fixed jet mass.

    Top Jets at the LHC (2008) · 0.93 · paraphrased: the licence does not allow quoting

How the literature uses it

125 arXiv papers cite this in 211 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 123 sentences without a confident label

Builds on it 4 papers
  • In this work, we extend the groomed jet mass calculation of [ref] to a more general class of observables which are known as (groomed) jet angularities $τ_a$ with the following definition [ref] τ_a = 1p_T∑_i∈Jp_Ti ΔR_iJ^2-a .

    Soft drop groomed jet angularities at the LHC · Introduction · 0.97

  • Larkoski:2014uqa extended that analysis to the resummation of recoil-free angularities [ref] and energy correlation functions.

    Non-Global Correlations in Collider Physics · Global Logarithm Resummation · 0.85

  • It is clear that the isolated Drell-Yan factorization theorem discussed here can be extended to observables with other exponents, $e^-a η_k$, much like the angularity event shapes in $e^+e^-$ [ref] .

    Factorization at the LHC: From PDFs to Initial State Jets · Drell-Yan Factorization Theorems · 0.81

Uses its method or tool 5 papers
  • In particular, we use jet angularities [ref] and the primary Lund plane (PLP) [ref] as inputs to train ML algorithms.

    Identification of b-jets using QCD-inspired observables · Introduction · 0.99

  • Following Ref. [ref] we define jet function for quarks at fixed jet mass by J^q_i ( m_J^2, p_0,J_i, R) = (2 π)^3/2√(2) (p_0,J_i)^2 _μ /N_c ∑_N_J_i { ^0 | q(0) ^(q̅) †_( ∞,0 ) |N_J_iN_J_i| ^(q̅)_( ∞,0 ) q̅(0) | 0} ×( m_J^2 - m̃_J^2( N_J_i,R) ) ^̣(2) ( n̂ - ñ(N_J_i)) (̣ p_0,J_i - (N_J_c) ), where $m̃_J^2( N_J_i,R)$ is the invariant mass of all particles within the cone centered on direction $n̂$ in state $N_J_i$.

    Top Jets at the LHC · Jets at Fixed Invariant Mass · 0.93

  • For $β= 1$, we compute the two-loop anomalous dimension of $S_()$ numerically using the fixed-order code [ref] .[While we will not do it in this paper, one could use the results of Bell:2015lsf which calculates the anomalous dimension of the soft function for event-wide (recoil-free) angularities [ref] or energy correlation functions with arbitrary angular exponent.

    Factorization for groomed jet substructure beyond the next-to-leading logarithm · Introduction · 0.91

Compares with it 1 paper
Cites it as support 2 papers
Cites it as background 55 papers
  • In perturbative calculations, large logarithms, called `non–global logarithms' [ref] , of the type $(α_slnQ/ℳ)^n$ appear, where $Q$ is the center–of–mass energy and $ℳ$ is a second hard scale $Q≫ℳ≫Λ_QCD$ characterizing the observable of interest.

    Soft gluons away from jets: distribution and correlation · Introduction · 0.88

  • Angularities were first introduced almost twenty years ago in [ref] and they were defined as [equation] where $θ_i$ is the angle made by $i$-th particle with the thrust axis, $E_i$ is the energy of the particle $i$ and $a$ is a continuous parameter.

    Power corrections to event shapes using eikonal dressed gluon exponentiation · Angularities · 0.93

  • Quark/gluon discrimination has mostly been studied using IRC safe observables, such as the angularities [ref] or 2-point energy correlation functions $C_1 = 2$ [ref] , which are set by a single emission at LL accuracy.[Important exceptions are (IRC unsafe) multiplicity-based observables, which have a long history in QCD [ref] (see [ref] for a recent experimental study), and more recently, shower deconstruction [ref] .] At LL order, and ignoring nonperturbative effects, one can show that the discrimination power of such observables is set by the Casimir scaling relation [equation] where $x$ is the fraction of quarks retained by the cut and $disc(x)$ is the fraction of gluons retained.

    New Angles on Energy Correlation Functions · Improving Quark/Gluon Discrimination · 0.94

Discussion

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