arxiv: 2511.09283 · v2 · pith:3RKZHVFLnew · submitted 2025-11-12 · ⚛️ nucl-th · hep-ex· hep-ph· nucl-ex

Explaining higher-order correlations between elliptic and triangular flow

Mubarak Alqahtani , Jean-Yves Ollitrault This is my paper

Pith reviewed 2026-05-17 22:37 UTC · model grok-4.3

classification ⚛️ nucl-th hep-exhep-phnucl-ex

keywords elliptic flowtriangular flowcumulantsheavy-ion collisionsinitial geometryPb+Pb collisionsflow correlationsreaction plane

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The pith

In Pb+Pb collisions at fixed impact parameter, higher-order mixed cumulants of elliptic and triangular flow are determined by the mean reaction-plane elliptic flow from the initial nuclear overlap geometry.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates an unexpected simplicity in the cumulants that combine elliptic flow v2 and triangular flow v3 up to high orders in heavy-ion collisions. When the analysis is done at fixed impact parameter, increasing the power of v2 while keeping the power of v3 fixed causes the cumulants to change in ways that depend only on the average value of v2 in the reaction plane. This average arises directly from the elliptical shape of the region where the two nuclei overlap. The authors use this fact to derive simple analytic formulas connecting cumulants of different orders. These formulas agree with data from the CMS Collaboration and enable predictions for even higher orders.

Core claim

We unravel an unexpected simplicity in these complex mathematical quantities for collisions at fixed impact parameter. We show that as one increases the order in v2, for a given order in v3, the changes in the cumulants are solely determined by the mean elliptic flow in the reaction plane, which originates from the almond-shaped geometry of the overlap area between the colliding nuclei. We derive simple analytic relations between cumulants of different orders on this basis.

What carries the argument

The mean elliptic flow in the reaction plane originating from the almond-shaped geometry of the nuclear overlap area, which fixes the variations in higher-order cumulants.

If this is right

Simple analytic relations between cumulants of different orders can be derived.
These relations agree well with recent CMS Collaboration data.
Agreement improves with finer centrality binning.
Quantitative predictions are made for cumulants of order 10 which have not yet been analyzed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result implies that initial-state geometry dominates over dynamical fluctuations in determining these particular higher-order correlations.
This framework might be applied to other combinations of flow harmonics to identify similar geometric effects.
Confirmation in other experiments or collision energies would strengthen the link between observed flow and the initial almond shape.

Load-bearing premise

The analysis is performed at fixed impact parameter so that centrality binning does not mix different geometries, and higher-order cumulants receive no additional contributions from event-by-event fluctuations beyond the mean v2.

What would settle it

A measurement of the mixed cumulants in very fine centrality bins that deviates significantly from the predicted analytic relations, or a failure of the order-10 predictions when measured.

Figures

Figures reproduced from arXiv: 2511.09283 by Jean-Yves Ollitrault, Mubarak Alqahtani.

**Figure 3.** Figure 3: FIG. 3. Schematic representation of two collision events with [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. Mixed harmonic cumulants as a function of central [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Ratios in Eqs. (15) and (17). Symbols are ALICE [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Same as Fig. 4 for the ratios Eqs. (18) and (20). [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 4.** Figure 4: As in Fig. 4, the ratio is smaller than our predic [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

read the original abstract

The ALICE and CMS Collaborations have analyzed a number of cumulants mixing elliptic flow ($v_2$) and triangular flow ($v_3$), involving up to $8$ particles, in Pb+Pb collisions at the LHC. We unravel an unexpected simplicity in these complex mathematical quantities for collisions at fixed impact parameter. We show that as one increases the order in $v_2$, for a given order in $v_3$, the changes in the cumulants are solely determined by the mean elliptic flow in the reaction plane, which originates from the almond-shaped geometry of the overlap area between the colliding nuclei. We derive simple analytic relations between cumulants of different orders on this basis. These relations are in good agreement with recent data from the CMS Collaboration. We argue that agreement will be further improved if the analysis is repeated with a finer centrality binning. We make quantitative predictions for cumulants of order 10 which have not yet been analyzed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives simple analytic links between mixed v2-v3 cumulants using only the mean geometric elliptic flow at fixed impact parameter, with reasonable CMS agreement but a clear vulnerability to centrality binning.

read the letter

The main takeaway is that higher-order mixed cumulants of elliptic and triangular flow simplify at fixed impact parameter, with changes driven solely by the average reaction-plane v2 from the initial almond geometry. The authors derive explicit analytic relations between cumulants of different orders on that basis and report decent agreement with CMS Pb+Pb data, plus predictions for order-10 cumulants not yet measured. This is a practical simplification for people who already work with multi-particle flow observables. It turns a set of complicated quantities into something more directly tied to the mean initial geometry without needing full event-by-event modeling for every order. That is the useful part. The derivation rests on symmetry at fixed impact parameter and the claim that extra fluctuation terms beyond the mean v2 do not enter. The data comparison is presented as supportive, though the paper itself flags that finer centrality bins should tighten the match. That note is telling. Experimental centrality bins average over a range of impact parameters, so different overlap shapes are mixed inside each bin. Any geometry-dependent contributions that the fixed-b fixed derivation excludes could be leaking in and helping the current agreement. Without the full steps it is hard to judge how much post-hoc adjustment or averaging is doing the work. The math itself looks internally consistent on the symmetry argument, and the citation pattern to prior cumulant work is standard for the field. This is aimed at heavy-ion physicists who analyze or interpret flow cumulants at the LHC. A reader who needs a quick way to relate different-order measurements or to forecast unmeasured ones will get something concrete out of it. The idea is modest but grounded enough, with data contact and testable predictions, that it deserves a serious referee rather than a desk reject.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that in Pb+Pb collisions at fixed impact parameter, higher-order cumulants mixing elliptic flow v2 and triangular flow v3 exhibit simple analytic relations determined solely by the mean reaction-plane elliptic flow arising from the almond-shaped initial overlap geometry. As the order in v2 is increased for fixed order in v3, the changes in the cumulants follow from this geometric mean v2. The derived relations are stated to agree with CMS data, with the suggestion that finer centrality binning would improve the agreement, and quantitative predictions are made for order-10 cumulants.

Significance. If the central relations hold under the stated assumptions, the work provides a transparent geometric explanation for otherwise complex multi-particle flow cumulants, reducing them to dependence on the mean v2 from initial-state almond geometry. This offers analytic, low-parameter relations between different-order cumulants that could simplify data interpretation and guide future measurements, with the order-10 predictions adding direct falsifiability.

major comments (2)

[Central derivation and data comparison] The derivation assumes analysis at fixed impact parameter so that centrality binning does not mix different geometries and higher-order cumulants receive no additional contributions from event-by-event fluctuations beyond the mean v2. However, the comparison is to CMS data in centrality bins, which average over ranges of impact parameters; the manuscript notes that finer binning would improve agreement but does not quantify the size of any contamination from geometry mixing in the current bins. This assumption is load-bearing for the claim that changes are solely determined by the mean reaction-plane v2.
[Derivation of analytic relations] The relations are derived from the geometric mean v2 rather than fitted directly to the mixed cumulants, with the mean v2 ultimately taken from data or models. This introduces moderate external dependence that should be made fully explicit when claiming the relations are a direct test of the geometric picture.

minor comments (1)

[Abstract] The abstract states 'good agreement' with CMS data but does not specify the exact cumulant orders compared or any quantitative measure of agreement (e.g., relative deviation or chi-squared).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and valuable suggestions. We address each major comment below and indicate the revisions made to the manuscript.

read point-by-point responses

Referee: The derivation assumes analysis at fixed impact parameter so that centrality binning does not mix different geometries and higher-order cumulants receive no additional contributions from event-by-event fluctuations beyond the mean v2. However, the comparison is to CMS data in centrality bins, which average over ranges of impact parameters; the manuscript notes that finer binning would improve agreement but does not quantify the size of any contamination from geometry mixing in the current bins. This assumption is load-bearing for the claim that changes are solely determined by the mean reaction-plane v2.

Authors: We agree that our derivation is performed at fixed impact parameter, and that the CMS data are presented in finite centrality bins that average over a range of impact parameters. This averaging can introduce some mixing of geometries. In the manuscript, we already suggest that finer centrality binning would improve the agreement, which implicitly acknowledges this effect. However, we have not provided a quantitative estimate of the contamination in the current bins. To address the referee's concern, we will add a discussion in the revised version estimating the variation of the mean v2 within the centrality bins using a standard Monte Carlo Glauber model. This will show that the effect is small for the bins considered, supporting our interpretation that the dominant behavior is still captured by the geometric mean v2. We maintain that the central claim holds under the fixed-b assumption, with the data comparison being a first test. revision: partial
Referee: The relations are derived from the geometric mean v2 rather than fitted directly to the mixed cumulants, with the mean v2 ultimately taken from data or models. This introduces moderate external dependence that should be made fully explicit when claiming the relations are a direct test of the geometric picture.

Authors: The referee correctly points out that the mean v2 is an input taken from independent sources. In the manuscript, we use the reaction-plane v2 from data or hydrodynamic models to predict the higher-order cumulants. We have revised the text to explicitly state the origin of this mean v2 for each comparison presented and to clarify that the analytic relations test the geometric picture conditional on the value of the mean v2. This makes the external dependence transparent and frames the results as a consistency check of the geometric origin rather than a parameter-free prediction. revision: yes

Circularity Check

0 steps flagged

No significant circularity: relations derived from geometric mean v2 at fixed impact parameter

full rationale

The paper's core claim derives analytic relations between mixed v2-v3 cumulants by positing that, at fixed impact parameter, increases in v2 order for fixed v3 order are determined solely by the mean reaction-plane elliptic flow from the almond-shaped nuclear overlap geometry. This geometric input is external to the cumulant data and not obtained by fitting the target quantities or by self-referential definition. The derivation proceeds mathematically from that assumption without reducing any prediction to a fitted parameter or prior self-citation chain. Agreement with CMS data and predictions for order-10 cumulants serve as external tests rather than inputs. No self-definitional, fitted-input, or ansatz-smuggling patterns appear in the load-bearing steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption of fixed impact parameter and the statement that higher-order changes are determined exclusively by the mean elliptic flow from geometry; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Analysis performed at fixed impact parameter
The unexpected simplicity and analytic relations hold only when geometry is held fixed, as stated in the abstract.

pith-pipeline@v0.9.0 · 5467 in / 1251 out tokens · 42924 ms · 2026-05-17T22:37:21.834903+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We show that as one increases the order in v2, for a given order in v3, the changes in the cumulants are solely determined by the mean elliptic flow in the reaction plane... We derive simple analytic relations between cumulants of different orders on this basis.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

If the impact parameter is constant... a cumulant of order k varies with N like N^{1-k}... cmpqr ∼ O(V^{2(m+p+q+r-1)+|m-p+3/2(q-r)|})

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

101 extracted references · 101 canonical work pages · 53 internal anchors

[1]

Explaining higher-order correlations between elliptic and triangular flow

Our goal is to extend this study to higher-order cumulants, of 6 and 8 particles, which have subsequently been measured [29]. Throughout this paper, we assume that nonflow corre- lations are negligible, so that particles in each event are emitted independently according to an underlying prob- ability distribution [37, 38]. LetP(φ) denote the az- imuthal d...

work page internal anchor Pith review Pith/arXiv arXiv 2025
[2]

The lowest-order mixed cumulant corre- sponds tom=q= 1

=c 2{2m} M HC(v 0 2, v2q 3 ) =c 3{2q}.(3) The mixed cumulants are those for which bothmand qare positive. The lowest-order mixed cumulant corre- sponds tom=q= 1. Expanding the left-hand side of Eq. (2) to orderλλ ∗µµ∗, one obtains its expression in terms of moments: M HC(v 2 2, v2

work page
[3]

cumulants

=⟨v 2 2v2 3⟩ − ⟨v 2 2⟩⟨v2 3⟩.(4) It was measured by ALICE in 2016 in Ref. [34], where it was namedSC(3,2). Higher-order cumulants with (m, q) = (2,1),(3,1),(1,2),(2,2),(1,3) were subse- quently measured in Ref. [29], where their expressions in terms of moments are provided. Deriving these expres- sions is straightforward using Eq. (2). We do not repeat th...

work page 2016
[4]

ATLAS denotes this quantity bynsc 2,3{4}

in Pb+Pb collisions at 5.02 TeV per nucleon pair, as a function of the collision centrality. ATLAS denotes this quantity bynsc 2,3{4}. ALICE normalizes the cumulants as follows: nM HC(v 2m 2 , v2q 3 )≡ M HC(v 2m 2 , v2q 3 ) ⟨v2m 2 ⟩⟨v2q 3 ⟩ .(5) This normalization suppresses the sensitivity to kine- matic cuts, and also provides an intuitive, dimension- l...

work page
[5]

is displayed in Fig. 1. There are sizable differences between the two ex- periments, whose origin is unknown. One possible expla- nation is the wider centrality bins used by ALICE. How- ever, one would typically expect wider bins to increase the value ofnM HC(v 2 2, v2

work page
[6]

Interestingly, ATLAS observes a variation ofnM HC(v 2 2, v2

[49], and ALICE is below AT- LAS for most centralities. Interestingly, ATLAS observes a variation ofnM HC(v 2 2, v2

work page
[7]

going in the same direction as ALICE (down for centralities<40%, up for central- ities>40%) when only particles withp T >2 GeV/c are included. Since these high-p T particles are likely to have sizable nonflow correlations from jet production, it is tempting to postulate that the difference between AT- LAS and ALICE may be due to larger nonflow effects in ...

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[8]

3 precise than ALICE results, and we will take this as an excuse for not understanding precisely the ALICE results on higher-order cumulants in Sec

reaches 1 in peripheral collisions [29, 35], which is a natural consequence of the non-linear coupling betweenv 2 andv 4 [48]. 3 precise than ALICE results, and we will take this as an excuse for not understanding precisely the ALICE results on higher-order cumulants in Sec. V. 0 10 20 30 40 50 60 Centrality [%] 10 15 10 13 10 11 10 9 10 7 MHC(v2 2, v2 3)...

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[9]

intrinsic frame

are positive for the most cen- tral bins, and the three corresponding data points are circled. For our analysis, we will need the un-normalized cu- mulantsM HC(v 2m 2 , v2q 3 ). In order to compute them, we evaluate the moments appearing in the denominator of Eq. (5) using standard formulas which are recalled in Ap- pendix A. Results are displayed in Fig....

work page 2000
[10]

(4) is of order V 4, while the difference is of orderV 6, i.e., much smaller

=c 1111, each of the moments in the right-hand side of Eq. (4) is of order V 4, while the difference is of orderV 6, i.e., much smaller. This systematic expansion scheme will allow us to single out the dominant contributions to each of the MHCs. IV. RELATIONS BETWEEN EXPERIMENTAL CUMULANTS AND INTRINSIC CUMULANTS We now relate the two sets of cumulants de...

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[11]

(7) and used the symmetryc mpqr =c pmrq

=c 1000c0111 +c 0100c1011 +c 1111 = 2 ¯V2c0111 +c 1111 (12) where, in the last equality, we have introduced ¯V2 de- fined by Eq. (7) and used the symmetryc mpqr =c pmrq. For central collisions, ¯V2 = 0 andM HC(v 2 2, v2

work page
[12]

coincides withc 1111, which is of orderV 6 as explained at the end of Sec. III B. For non-central collisions, the first term in the right-hand side of Eq. (12) differs from zero, but is also of orderV 6 according to Eq. (10). We now list the expressions of the higher-order cumu- lants measured by ALICE [29], which we truncate by keeping only the leading t...

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[13]

=−4 ¯V 3 2 c0111 M HC(v 6 2, v2

work page
[14]

= 24 ¯V 5 2 c0111 M HC(v 2 2, v4

work page
[15]

= 2(2c 2 0111 + ¯V2 c0122) +c 1122 M HC(v 4 2, v4

work page
[16]

=−4 ¯V 2 2 (6c2 0111 + ¯V2c0122) M HC(v 2 2, v6

work page
[17]

More generally,M HC(v 2m 2 , v2q 3 ) is of orderV 2m+4q

= 2(9c0111c0122 + ¯V2c0133) +c 1133.(13) They are respectively of orderV 8,V 10,V 10,V 12,V 14. More generally,M HC(v 2m 2 , v2q 3 ) is of orderV 2m+4q. These orders of magnitude are reflected in the hierar- chy observed in Fig. 2. In particular, they explain why M HC(v 6 2, v2

work page
[18]

Sincev 2 andv 3 are both of orderV, the normalized symmetric cumulant (5) is of orderV 2q

are of comparable mag- nitude, despite being cumulants of different orders (8 and 6 respectively). Sincev 2 andv 3 are both of orderV, the normalized symmetric cumulant (5) is of orderV 2q. This explains why the magnitude ofnM HC(v2m 2 , v2q 3 ) de- creases strongly asqincreases, as pointed out in Sec. II. We finally provide leading-order expressions for ...

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[19]

=−264 ¯V 7 2 c0111 M HC(v 6 2, v4

work page
[20]

= 24 ¯V 4 2 (10c 2 0111 + ¯V2 c0122) M HC(v 4 2, v6

work page
[21]

=−144 ¯V2 c3 0111 −4 ¯V 2 2 (27c0111c0122 + ¯V2c0133).(14) A first comment on the expressions (12), (13) and (14) is that they only involve the mean elliptic flow in the reac- tion plane, ¯V2, and mixed cumulants of order≥3, which quantify non-Gaussian fluctuations. A second remark is that for a given value ofq,M HC(v 2 2, v2q 3 ) contains more terms than...

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[22]

/ (v2{4}2 MHC(v2 2, v2 3)) MHC(v6 2, v2

work page
[23]

/ (v2{4}2 MHC(v4 2, v2 3)) FIG. 4. Ratios in Eqs. (15) and (17). Symbols are ALICE data, where the mixed cumulants are taken from Ref. [29] and v2{4}from Ref. [50], as a function of the collision centrality in Pb+Pb collisions at 5.02 TeV per nucleon pair. Horizontal lines are our theory predictions. Eqs. (12), (13) and (14) show that to leading order, th...

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[24]

=−6.(15) This prediction is tested against ALICE data in Fig. 4. The experimental ratio in the left-hand side is in rough agreement with the predicted value in the right-hand side, but somewhat smaller in absolute magnitude. Similarly, using Eqs. (13) and (14), we predict: M HC(v 8 2, v2 3) v2{4}2M HC(v 6 2, v2

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[25]

=−11.(16) This could easily be checked experimentally, as increasing the order inv 2 does not significantly increase errors. Eqs. (15) and (16) are rigorous mathematical results to leading order inV. They generalize the well-known iden- titiesv 2{4}=v 2{6}=v 2{8}(Appendix A) to mixed cumulants. We therefore expect that their accuracy is comparable, at the...

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[26]

superskewness

≈ −2.(17) This prediction is also in fair agreement with data, as shown in Fig. 4. The experimental ratio is again system- atically smaller than our prediction in absolute magni- tude. We now move on to the cumulants involvingv 4 3, third and fourth lines of Eq. (13). They involve the mixed skewnessc 0111, and also new, higher-order cumulants: a mixed “su...

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[27]

+ 2v2{4}2M HC(v 2 2, v4 3) M HC(v 2 2, v2 3)2 ≈ −4.(18) Comparison with ALICE data is displayed in Fig. 5. The agreement with our prediction is much worse than in Fig. 4. As in Fig. 4, the ratio is smaller than our predic- tion in absolute magnitude. It decreases monotonically 0 10 20 30 40 50 60 Centrality [%] 6 4 2 0 2 4 Eq. 18 Eq. 20 FIG. 5. Same as Fi...

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(18) is one need not neglect c1111 andc 1122, which do not enter the leading-order ex- pressions of the cumulants involved in Eq

+ 6v2{4}4M HC(v 4 2, v4 3) M HC(v 4 2, v2 3)2 = 6.(19) The advantage over Eq. (18) is one need not neglect c1111 andc 1122, which do not enter the leading-order ex- pressions of the cumulants involved in Eq. (19). Like Eqs. (15) and (16), Eq. (19) is a rigorous leading-order result, and we expect that it should be fairly accurate with a fine centrality bi...

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(20) Agreement with data, displayed in Fig

+ 6v2{4}2M HC(v 2 2, v4 3)) ≈ 18 16 . (20) Agreement with data, displayed in Fig. 5, is even worse than for Eq. (18). Unlike what was observed with the previous ratios, the left-hand side is not smaller than the right-hand side for all centralities. This is not surprising, as there is no regime where the term involvingc 0133 is negligible with respect to ...

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