We investigate the entangling properties of the color flux tube between a static quark-antiquark pair in pure gauge Yang-Mills theory. In earlier works, we defined a gauge-invariant flux tube entanglement entropy (FTE ), the excess entanglement entropy of a region of gluon fields that can be attributed to the color flux tube, and demonstrated that it is finite in the continuum limit. FTE was shown to have two contributions, one from the vibrations of the QCD string, and the other from its internal (color) degrees of freedom. In this work, we further explore the internal color component in (2+1)D Yang-Mills theory for $\textit{SU}$ ($\textit{N}$ ) gauge groups, varying 2 ≤ $\textit{N}$ ≤ 5. We identify a novel physical scale in the theory, the entanglement radius $\textit{ξ}$ . This radius characterizes the transverse extent of the flux tube that must be completely severed by an entangling region to capture the entanglement entropy of color degrees of freedom. The key feature underlying this phenomenon is its topological nature. This is revealed through systematic studies of multi-slab entangling regions in which FTE changes sharply when boundaries of the slabs completely cross-cut the flux tube. We find that $\textit{ξ}$ increases approximately linearly with $\textit{N}$ and is independent of both Rényi replica number and the inter-quark separation length. We also study FTE as a function of the entangling region’s transverse displacement from the static quark pair and observe behavior consistent with a previously identified intrinsic width $\textit{λ}$ of the flux tube. The extracted value is in agreement with the inverse mass of the lightest glueball for the gauge groups studied.

United States

Rocco Amorosso

Sergey Syritsyn

United Kingdom

Raju Venugopalan

Entanglement enabled tomography of flux tubes in (2+1)D Yang-Mills theory

We present a detailed numerical investigation of semi-inclusive forward di-hadron production in proton–nucleus collisions employing the Color Glass Condensate effective theory. We focus on the regime where di-hadrons are produced nearly back-to-back in the transverse plane, thereby justifying a transverse-momentum-dependent factorization approach in terms of small- <italic>x</italic> gluon distributions. Our computation integrates several key elements: i) non-linear rapidity evolution via the Balitsky&#8211;Kovchegov equation with running coupling, ii) both perturbative and non-perturbative Sudakov resummation, and iii) a phenomenologically constrained model for the initial conditions for small- <italic>x</italic> gluon distributions. We compare this phenomenological framework to experimental data from the STAR Collaboration on azimuthal correlations in forward di-pion production in both proton&#8211;proton and proton&#8211;gold collisions. We analyze the systematic theoretical uncertainties associated with the saturation scales of nuclei at the initial scale for rapidity evolution and with those associated with the hadronization process. Finally, we make predictions for the kinematics anticipated to be covered by the ALICE Forward Calorimeter (FoCal) upgrade at the Large Hadron Collider.

France

Paul Caucal

Zhong-Bo Kang

Poland

Piotr Korcyl

Farid Salazar

Björn Schenke

Tomasz Stebel

China

Wenbin Zhao

Probing gluon saturation with forward di-hadron correlations in proton-nucleus collisions

In recent work Amorosso et al. (2024), we computed a novel flux tube entanglement entropy (FTE) of the color flux tube stretched between a heavy quark-antiquark pair on a Euclidean lattice in (2+1)D Yang–Mills theory. Our numerical results suggested that FTE can be partitioned into an internal color entanglement entropy and a vibrational entropy corresponding to the transverse excitations of a QCD string, with the latter described by a thin string model. Since the color flux tube does not have transverse excitations in (1+1)D Yang–Mills theory, we use this simpler framework to perform an exact analytical computation of the contribution of the internal color degrees of freedom to FTE. For the multipartite partitioning of the color flux tube, we find the remarkable result that FTE depends only on the dimension of the color group representation and the number of times the flux tube crosses the boundary between the traced and untraced spatial regions but not on the string length. Our proof is independent of whether the replica and region boundaries on the lattice are placed on vertices or in plaquette centers.

Entanglement entropy of a color flux tube in (1+1)D Yang–Mills theory

<p>We extend our prior results on the worldline computation of the axial vector-vector-vector (AVV) triangle anomaly in polarized deeply inelastic scattering (DIS) to the finite mass case by computing in addition the pseudoscalar-vector-vector (PVV) triangle graph. For the well-studied QED case, we show explicitly how the off-forward AVV pole exactly cancels an identical PVV pole. We then demonstrate the dramatic difference in QCD due to the chiral condensate, which qualitatively modifies anomalous Ward identities. As in the massless case, the anomaly pole in QCD is canceled by the dynamics of a primordial isosinglet pseudoscalar <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>η</mi><mo stretchy="false">¯</mo></mover></math>-meson, whose Wess-Zumino-Witten coupling to the topological charge density shifts the pole to the physical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>η</mi><mo>′</mo></msup></math> mass, with the finite quark mass contribution differing by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>O</mi><mo stretchy="false">(</mo><mn>10</mn><mo>%</mo><mo stretchy="false">)</mo></math> from the Witten-Veneziano formula. We obtain a compact analytic expression for the finite mass corrections to Shore and Veneziano’s result that the proton’s net quark helicity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Δ</mi><mi mathvariant="normal">Σ</mi><mo>∝</mo><msqrt><mrow><msubsup><mi>χ</mi><mi>QCD</mi><mo>′</mo></msubsup><msub><mo stretchy="false">|</mo><mrow><mi>m</mi><mo>=</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msqrt></math>, the forward slope of the topological susceptibility in the chiral limit, and show they are of the order of a few percentages. Our prior prediction that the polarized DIS structure function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>g</mi><mn>1</mn></msub></math> is quenched by sphaleronlike topological transitions at small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> is unaffected by quark mass effects. Our results illustrate how worldline computations of anomalous processes, in synergy with lattice computations and nonet chiral perturbation theory, can uncover novel nonperturbative features of QCD at the Electron-Ion Collider.</p>

Andrey Tarasov

Role of the chiral anomaly in polarized deeply inelastic scattering. III. Wess-Zumino-Witten contributions and chiral Ward identities for finite quark mass

We construct a novel flux tube entanglement entropy (FTE ), defined as the excess entanglement entropy relative to the vacuum of a region of color flux stretching between a heavy quark-anti-quark pair in pure gauge Yang-Mills theory. We show that FTE can be expressed in terms of correlators of Polyakov loops, is manifestly gauge-invariant, and therefore free of the ambiguities in computations of the entanglement entropy in gauge theories related to the choice of the center algebra. Employing the replica trick, we compute FTE for SU(2) Yang-Mills theory in (2+1)D and demonstrate that it is finite in the continuum limit. We explore the properties of FTE for a half-slab geometry, which allows us to vary the width and location of the slab, and the extent to which the slab cross-cuts the color flux tube. Following the intuition provided by computations of FTE in (1+1)D, and in a thin string model, we examine the extent to which our FTE results can be interpreted as the sum of an internal color entropy and a vibrational entropy corresponding to the transverse excitations of the string.

Entanglement entropy of a color flux tube in (2+1)D Yang-Mills theory

In a previous paper [H. Raj and R. Venugopalan, Phys. Rev. D 109, 044064 (2024)], we demonstrated a double-copy relation between inclusive gluon radiation in shock wave collisions of ultrarelativistic nuclei and inclusive graviton radiation in trans-Planckian gravitational shock wave collisions. We compute here the corresponding gravitational shock wave propagators in general relativity and demonstrate that they too obey a double-copy relation to gluon shock wave propagators computed previously. These results provide key input in a renormalization group approach toward computing the high frequency radiation spectrum in close black hole encounters.

Himanshu Raj

QCD-gravity double-copy in the Regge regime: Shock wave propagators

Self-polarizing weak decays of <math><mi>Λ</mi></math>-hyperons provide unique insight into the role of entanglement in the fragmentation of QCD strings through measurements of the spin correlations of <math><mi>Λ</mi><mover><mi>Λ</mi><mo>¯</mo></mover></math> pairs produced in collider experiments. The simplest quantum field theory representing the underlying parton dynamics is the four-flavor massive Schwinger model plus an effective spin-flip term, where the flavors are mapped to light (up or down) and heavy (strange) quarks and their spins. This construction provides a novel way to explore hyperon spin correlations in <math><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math> dimensions. We investigate the evolution of these correlations for different string configurations that are sensitive to the rich structure of the model Hamiltonian.

João Barata

Wenjie Gong

Realtime dynamics of hyperon spin correlations from string fragmentation in a deformed four-flavor Schwinger model

In [1], we showed that the gravitational wave spectrum from trans-Planckian shockwave scattering in Einstein gravity is determined by the gravitational Lipatov vertex expressed as the bilinear double copy <math><msup><mrow><mi>Γ</mi></mrow><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mi>C</mi></mrow><mrow><mi>μ</mi></mrow></msup><msup><mrow><mi>C</mi></mrow><mrow><mi>ν</mi></mrow></msup><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mi>N</mi></mrow><mrow><mi>μ</mi></mrow></msup><msup><mrow><mi>N</mi></mrow><mrow><mi>ν</mi></mrow></msup></math> where <math><msup><mrow><mi>C</mi></mrow><mrow><mi>μ</mi></mrow></msup></math> is the QCD Lipatov vertex and <math><msup><mrow><mi>N</mi></mrow><mrow><mi>μ</mi></mrow></msup></math> is the QED soft photon factor. We show here that this result can be directly obtained by careful application of the classical color-kinematic duality to the spectrum obtained in gluon shockwave collisions.

Gravitational wave double copy of radiation from gluon shockwave collisions

A remarkable double-copy relation of Einstein gravity to QCD in Regge asymptotics is <math><msup><mi>Γ</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>C</mi><mi>μ</mi></msup><msup><mi>C</mi><mi>ν</mi></msup><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>N</mi><mi>μ</mi></msup><msup><mi>N</mi><mi>ν</mi></msup></math>, where <math><msup><mi>Γ</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup></math> is the gravitational Lipatov vertex in the <math><mn>2</mn><mo>→</mo><mn>3</mn></math> graviton scattering amplitude, <math><msup><mi>C</mi><mi>μ</mi></msup></math> its Yang-Mills counterpart, and <math><msup><mi>N</mi><mi>μ</mi></msup></math> the QED bremsstrahlung vertex. In QCD, the Lipatov vertex is a fundamental building block of the BFKL equation describing the <math><mn>2</mn><mo>→</mo><mi>N</mi></math> scattering of gluons at high energies. Likewise, the gravitational Lipatov vertex is a key ingredient in a 2D effective field theory framework describing trans-Planckian <math><mn>2</mn><mo>→</mo><mi>N</mi></math> graviton scattering. We construct a quantitative correspondence between a semiclassical Yang-Mills framework for radiation in gluon shockwave collisions and its counterpart in general relativity. In particular, we demonstrate the Lipatov double copy in a dilute-dilute approximation corresponding to <math><msub><mi>R</mi><mrow><mi>S</mi><mo>,</mo><mi>L</mi></mrow></msub></math>, <math><msub><mi>R</mi><mrow><mi>S</mi><mo>,</mo><mi>H</mi></mrow></msub><mo>≪</mo><mi>b</mi></math>, where <math><msub><mi>R</mi><mrow><mi>S</mi><mo>,</mo><mi>L</mi></mrow></msub></math>, <math><msub><mi>R</mi><mrow><mi>S</mi><mo>,</mo><mi>H</mi></mrow></msub></math> are the respective emergent Schwarzchild radii generated in shockwave collisions and <math><mi>b</mi></math> is the impact parameter. We outline extensions of the correspondence developed here to the dilute-dense computation of gravitational wave radiation in the close vicinity of one of the black holes, the construction of graviton propagators in the shockwave background, and a renormalization group approach to compute <math><mn>2</mn><mo>→</mo><mi>N</mi></math> amplitudes that incorporates graviton Reggeization and coherent graviton multiple scattering.

Universal features of <math><mn>2</mn><mo>→</mo><mi>N</mi></math> scattering in QCD and gravity from shockwave collisions

We compute the back-to-back dijet cross section in deep inelastic scattering at small <math><mi>x</mi></math> to next-to-leading order (NLO) in the color glass condensate effective field theory. Our result can be factorized into a convolution of the Weizsäcker-Williams gluon transverse-momentum-dependent distribution function (WW gluon TMD) with a universal soft factor and an NLO coefficient function. The soft factor includes both double and single logarithms in the ratio of the relative transverse momentum <math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mo>⊥</mo></mrow></msub></mrow></math> of the dijet pair to the dijet momentum imbalance <math><msub><mi>q</mi><mo>⊥</mo></msub></math>; its renormalization group (RG) evolution is resummed into the Sudakov factor. Likewise, the WW TMD obeys a nonlinear RG equation in <math><mi>x</mi></math> that is kinematically constrained to satisfy both the lifetime and rapidity ordering of the projectile. Exact analytical expressions are obtained for the NLO coefficient function of transversely and longitudinally polarized photons. Our results allow for the first quantitative separation of the dynamics of Sudakov suppression from that of gluon saturation. They can be extended to other final states and provide a framework for precision tests of novel QCD many-body dynamics at the Electron-Ion Collider.