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Spin entanglement in $pp$ and $pA$ collisions: high-energy TMD effects and fragmentation probes

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Michael Fucilla

High Energy Physics - Phenomenology · Submitted 8 Nov 2026 · 135 references

Pith's verdict: Solid CGC derivation of entanglement observables in pp/pA; the back-to-back truncation is the weak link but not a fatal one.

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arXiv ID2611.11568 (version 1)
Pith ID5WN4SVOL, Pith's permanent record of this paper and its review
CitationMichael Fucilla. "Spin entanglement in $pp$ and $pA$ collisions: high-energy TMD effects and fragmentation probes." arXiv:2611.11568 (2026). https://pith.science/paper/5WN4SVOL
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The paper

The paper claims

Linearly polarized gluons and saturation shape spin entanglement in quark-antiquark pairs from hadron collisions.

The reading

This paper derives the full spin-density matrix for a back-to-back quark-antiquark pair produced in inclusive and diffractive proton-proton and proton-nucleus collisions within the shockwave formalism at small x. It shows that in inclusive production, the concurrence, stabilizer Rényi entropy, and quantum discord exhibit a cos(2φ) modulation controlled by the ratio of linearly polarized to unpolarized gluon transverse-momentum-dependent distributions, a modulation that is maximal in the dilute BFKL limit but strongly suppressed by saturation effects. In diffractive production, it proves that the pair is always entangled and Bell-nonlocal across the entire phase space, except at the endpoints, because the exclusive eikonal Pomeron exchange forces the spin-density matrix to have rank at most two and maximally mixed marginals. The paper proposes using double-Collins asymmetries and dihadron fragmentation functions to access these spin correlations experimentally, and shows that for massless quarks a suitably chosen double-Collins asymmetry directly equals the concurrence.

What carries the argument

The central objects are the gluon TMD distributions F_g(z,q) (unpolarized) and H_g(z,q) (linearly polarized) defined via the twist expansion of shockwave correlators (eqs. 4.41-4.43). The key identity is the spin-density matrix decomposition (4.46): $C^{{ab}}$ = [F_g C̃^{(0)} + cos(2φ) H_g C̃^{(2)}]^{ab} / A, where the hard coefficients C̃^{(0)} and C̃^{(2)} encode the Dirac structure of the g→qar q splitting. In diffraction, the crucial property is that the density matrix (5.17) has rank ≤ 2 and maximally mixed marginals, which via eqs. (5.26)-(5.27) guarantees entanglement and Bell-nonlocality coincide.

Extended reading notes

Core claim

The paper establishes that in inclusive pp/pA collisions, the spin-density matrix of a back-to-back quark-antiquark pair has a cos(2φ)-modulated concurrence, stabilizer Rényi entropy, and quantum discord whose magnitude is governed by the ratio R_{Hg/Fg} of linearly polarized to unpolarized gluon TMDs. In the BFKL limit R_{Hg/Fg}=1, yielding maximal modulation independent of the transverse momentum imbalance q; saturation effects strongly suppress the modulation when |q| ∼ Q_s. In diffractive production, the spin-density matrix has rank ≤ 2 and maximally mixed single-quark marginals, which forces the pair to be entangled and Bell-nonlocal for all non-endpoint kinematics, a property the paper traces to the exclusive and eikonal nature of Pomeron exchange. For massless quarks, the paper finds that a double-Collins asymmetry constructed from unlike-sign minus like-sign charged-hadron pairs directly equals the concurrence, providing a practical entanglement witness.

Load-bearing premise

The entire derivation of the inclusive spin-density matrix rests on the back-to-back limit |P| >> |q|, which enables TMD factorization and a twist expansion of the shockwave correlators; the validity of this expansion at moderate |q| is not quantified.

Editorial extensions

If this is right

  • Measurement of double-Collins asymmetries in unpolarized pp collisions at small x can directly extract the concurrence of the produced light-quark pair, providing a new quantum-information observable in hadronic collisions.
  • The cos(2φ) modulation of concurrence, magic, and discord offers a sensitive probe of gluon saturation: BFKL predicts maximal modulation independent of q/Q_s, while saturation suppresses it strongly for |q| ≲ 2Q_s.
  • Diffractive production of heavy-quark pairs in pA collisions yields a state that is always entangled and Bell-nonlocal, making it a unique resource for studying quantum correlations beyond the top-quark sector.
  • The rank-2, maximally-mixed-marginal structure found in diffraction implies that entanglement and Bell nonlocality coincide for mixed states, a property that can be tested via spin correlations in exclusive photoproduction at the EIC or in UPCs.
  • For massive quarks (charm, bottom), the proposed Collins/dihadron fragmentation strategy can be combined with baryon decay chains to access spin correlations over a wider kinematic range.

Reading between the lines

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

  • The paper's central mechanism—that exclusive eikonal exchange enforces rank ≤ 2 and maximally mixed marginals—may generalize to any color-singlet exchange in the Regge limit, suggesting that entanglement is a generic feature of diffractive dijet production.
  • The double-Collins asymmetry for massless quarks (eq. 6.13) is cleanest in ultraperipheral collisions where photon–gluon fusion dominates; the paper mentions this but could have emphasized it more strongly as a near-ideal entanglement witness.
  • The saturation suppression of quantum-information observables is modeled within the MV model; JIMWLK evolution is deferred to future work, but the qualitative pattern (strong suppression for |q| ∼ Q_s) is robust and testable at the LHC and future EIC.
  • For top quarks, the back-to-back limit is harder to satisfy at LHC energies, but future FCC-hh collisions would extend the accessible kinematic range and make the modulation observable in top-pair production.
Original abstract

Spin-spin correlations of a quark-antiquark pair produced in proton-proton or proton-Ion collisions are investigated in $k_T$-factorization within the shockwave formalism. We construct the full spin-density matrix of both inclusive and diffractive channels in the limit in which the two quarks are back-to-back. In the inclusive channel, due to linearly polarized gluons, we show that a $\cos (2 \phi)$-modulation of the concurrence, stabilizer R\'enyi entropy, and quantum discord emerges and depends on the interplay between six small-$x$ gluon TMDs. The modulation is predicted to be maximally strong and independent of the transverse momentum imbalance, $q$, from a dilute (BFKL) perspective, while it is considerably suppressed by saturation effects. In diffraction, we observe that the pair is always entangled and exhibits the same properties found in diffractive DIS. We explain how these properties follow directly from the exclusive and eikonal nature of the Pomeron exchange. We trace this behavior to a spin-density matrix of rank at most two with maximally mixed marginals. For this class of states, entanglement and Bell nonlocality coincide even when the state is mixed, while the explicit form of the light-front helicity amplitudes ensures that both persist in the whole phase space. We suggest accessing the spin correlations in unpolarized hadron collisions through the Collins and dihadron fragmentation functions. We point out that, in inclusive production, for massless quarks, suitably chosen double-Collins asymmetries provide promising entanglement witnesses.

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-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-11-10: 1 request, 7,733 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.45 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.17 · compares with prior results 0.44 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Spin-spin correlations of a quark-antiquark pair produced in proton-proton or proton-Ion collisions are investigated in $k_T$-factorization within the shockwave formalism. We construct the full spin-density matrix of both inclusive and diffractive channels in the limit in which the two quarks are back-to-back. In the inclusive channel, due to linearly polarized gluons, we show that a $\cos (2 \phi)$-modulation of the concurrence, stabilizer R\'enyi entropy, and quantum discord emerges and depends on the interplay between six small-$x$ gluon TMDs. The modulation is predicted to be maximally strong and independent of the transverse momentum imbalance, $q$, from a dilute (BFKL) perspective, while it is considerably suppressed by saturation effects. In diffraction, we observe that the pair is always entangled and exhibits the same properties found in diffractive DIS. We explain how these properties follow directly from the exclusive and eikonal nature of the Pomeron exchange. We trace this behavior to a spin-density matrix of rank at most two with maximally mixed marginals. For this class of states, entanglement and Bell nonlocality coincide even when the state is mixed, while the explicit form of the light-front helicity amplitudes ensures that both persist in the whole phase space. We suggest accessing the spin correlations in unpolarized hadron collisions through the Collins and dihadron fragmentation functions. We point out that, in inclusive production, for massless quarks, suitably chosen double-Collins asymmetries provide promising entanglement witnesses.

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 Spin-spin correlations of a quark-antiquark pair produced in proton-proton or proton-Ion collisions are investigated in $k_T$-factorization within the shockwave formalism. contribution 0.90 0.79 0.06 asserted 0.90
s1 We construct the full spin-density matrix of both inclusive and diffractive channels in the limit in which the two quarks are back-to-back. contribution 0.64 0.95 0.06 asserted 0.99
s2 In the inclusive channel, due to linearly polarized gluons, we show that a $\cos (2 \phi)$-modulation of the concurrence, stabilizer R\'enyi entropy, and quantum discord emerges and depends on the interplay between six small-$x$ gluon TMDs. result 0.99 0.92 0.33 asserted 1.00
s3 The modulation is predicted to be maximally strong and independent of the transverse momentum imbalance, $q$, from a dilute (BFKL) perspective, while it is considerably suppressed by saturation effects. result 0.88 0.64 0.18 asserted 0.95
s4 In diffraction, we observe that the pair is always entangled and exhibits the same properties found in diffractive DIS. result 1.00 0.89 0.15 asserted 1.00
s5 We explain how these properties follow directly from the exclusive and eikonal nature of the Pomeron exchange. interpretation 1.00 0.78 0.06 asserted 1.00
s6 We trace this behavior to a spin-density matrix of rank at most two with maximally mixed marginals. result 0.55 0.85 0.10 asserted 1.00
s7 For this class of states, entanglement and Bell nonlocality coincide even when the state is mixed, while the explicit form of the light-front helicity amplitudes ensures that both persist in the whole phase space. interpretation 0.62 0.79 0.13 asserted 1.00
s8 We suggest accessing the spin correlations in unpolarized hadron collisions through the Collins and dihadron fragmentation functions. future work 0.88 0.39 0.11 hedged possibility 0.95
s9 We point out that, in inclusive production, for massless quarks, suitably chosen double-Collins asymmetries provide promising entanglement witnesses. result 0.39 0.69 0.14 hedged possibility 0.83

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
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not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • 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. Schematic illustration of heavy quark–antiquark pair production in proton–nucleus collisions, initiated by a gluon carrying a longitudinal momentum fraction xg of the incoming proton. The collision axis defines the z direction, while θ denotes the polar angle of the outgoing quark. The orthonormal basis (k, r, n) is defined in the qq¯ center-of-mass frame, with k aligned with the quark momentum, r lying in the scattering plane, and n normal to it. the quark momentum and write k µ = M 2 (1, 0, 0, β), k′µ = M 2 (1, 0, 0, −β), β = r 1 − 4m2 M2 , (2.1) where M2 = (k + k ′ ) 2 . The…
Figure 2. Diagram contributing to the S-matrix element for the dijet production in gluon-nucleus collisions. The transverse metric and antisymmetric tensor associated with the ordered pair (n1, n2) are g µν ⊥ ≡ g µν − n µ 1n ν 2 − n µ 2n ν 1 , ϵ µν ⊥ ≡ ϵ µνρσn1ρn2σ. (3.4) For transverse indices i, j = 1, 2, these definitions give g ij ⊥ = −δ ij , ϵ12 ⊥ = +1. (3.5) Throughout the following, boldface transverse vectors are two-dimensional Euclidean vec￾tors. In particular, v⊥ · w⊥ = −v · w = −v iw i , v 2 = −v 2 ⊥. (3.6) The incoming gluon is considered to move in the +
Figure 3. Concurrence (top left), stabilizer R´enyi entropy (top right), and quantum discord (bottom) as functions of β and cos θ, after averaging over ϕP,q. In the concurrence panel, the solid white line separates entangled from separable states, while the dashed black line marks the boundary of the Bell-nonlocal region. From Eqs. (4.44), (4.32) and (4.38), the total cross-section reads  dσ dzd2P d 2q  = α 2 s 2CF 1 (|P | 2 + m2) 2 ( " z 2 + ¯z 2  2 + 2z (1 − z) m2 |P | 2 (|P | 2 + m2) 2 # Fg(z, q) + cos(2ϕP,q) 2zz m¯ 2 |P | 2 (|P | 2 + m2) 2 Hg(z, q) ) ,
Figure 4. The ratio RHg/Fg ≡ Hg/Fg as a function of |q|/Qs for z = 0.1 (green), z = 0.25 (red), and z = 0.5 (blue). The curves for z and 1 − z coincide. so that Ckr| z=1/2 |P |=m = 2Λ cos(2ϕP,q) 1 + Λ cos(2ϕP,q) = 2Λ cos(2ϕP,q) + O(Λ2 ) . (4.56) This off-diagonal correlation thus provides an additional channel for probing the lin￾early polarized gluon distributions. In particular, it has no azimuthally independent back￾ground at z = 1/2, and its leading angular modulation is enhanced by the hard coefficient C (2) kr = 2 when |P | = m. 4.3 BFKL vs saturation The difference between collinear…
Figure 5. Concurrence for different values of ϕP,q and |q|/Qs. • The down-left panels correspond to ϕP,q = π/2 and Qs = 4|q|. In this way, we study the effect of linearly polarized gluons in a transition region between the non-linear and linear regimes. • The top-right panels correspond to ϕP,q = π/2 and Qs = |q|, i.e. a deeply saturated regime. • Finally, the down-right panels correspond to ϕP,q = π/2 within the BFKL approxi￾mation. We first comment on the general pattern of the three quantities in Figs. 5, 6, and 7, which, for the concurrence and nonstabilizerness, follows that already…
Figure 6. Magic for different values of ϕP,q and |q|/Qs. collinear factorization, completely independent of the target and the kinematic condition |q|/Qs. A saturation model, instead, in addition to removing this effect when |q|/Qs ∼ 1, predicts that it remains milder even in a region where |q| is already quite larger than Qs. Although the prediction of saturation depends on the target and is therefore influenced by our model (Appendix A.2)—which is strictly valid only for a heavy nucleus—a precise determination of the ratio shown in
Figure 7. Quantum discord for different values of ϕP,q and |q|/Qs. The concurrence shown in the top-right of

Figures reproduced from arXiv:2611.11568 by the authors.

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