Understanding real-time dynamics of interacting quantum fields in curved space-time remains a major theoretical challenge. We employ tensor network methods to study such dynamics using interacting scalar and gauge theories in 1+1 spacetime dimensions, subject to a quench modeling a homogeneously expanding gravitational background. The models considered are the scalar $\textit{λϕ}$ theory and the Schwinger model, i.e. a Dirac fermion coupled to a U(1) gauge field which is equivalent via bosonization to a scalar field with a cosine self-interaction. In the free scalar limit, both theories reproduce known analytical results, providing a nontrivial numerical validation of bosonization in curved spacetime for the Schwinger model. Our central finding is that self-interactions lead to a suppression of gravitational particle production compared to the free-field case, as evidenced by two-point functions and the spectra of produced particles. We further examine the behavior of entanglement generation and find that interactions suppress entanglement growth in the $\textit{λϕ}$ theory, while in the Schwinger model, the interplay between suppressed particle production and enhanced inter-particle correlations leads to more complex entanglement behavior. Our results pave the way for further explorations of nonperturbative quantum real-time dynamics of interacting scalar and gauge theories in arbitrary gravitational backgrounds.

United States

Evan Budd

Germany

Adrien Florio

David Frenklakh

Swagato Mukherjee

Quantum dynamics of cosmological particle production: interacting quantum field theories with matrix product states

We present lattice QCD calculations of the odd Mellin moments of pion valence-quark generalized parton distribution up to fifth order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">⟨</mo><msup><mi>x</mi><mn>4</mn></msup><mo stretchy="false">⟩</mo></math> and for the skewness range <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">[</mo><mo>−</mo><mn>0.33</mn><mo>,</mo><mn>0</mn><mo stretchy="false">]</mo></math> using operator product expansion of bilocal quark-bilinear operators. The calculations are performed on an ensemble with lattice spacing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>=</mo><mn>0.04</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and valence pion mass 300 MeV, employing boosted pion states with momenta up to 2.428 GeV and momentum transfers reaching <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2.748</mn><mtext> </mtext><mtext> </mtext><msup><mi>GeV</mi><mn>2</mn></msup></math>. We employ ratio-scheme renormalization and next-to-leading logarithmic resummed perturbative matching. At zero skewness, our results are consistent with previous lattice studies. By combining matrix elements at multiple values of skewness and momentum transfer, skewness-dependent moments are obtained through simultaneous polynomiality-constrained fits.

Xiang Gao

Qi Shi

Fei Yao

Yong Zhao

Skewness-dependent moments of the pion GPD from nonlocal quark-bilinear correlators

We investigate thermal properties of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math> lattice gauge theory in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1</mn><mo>+</mo><mn>1</mn></math> dimensions through real-time simulations. We extract the spectral functions directly coupling to the pseudoscalar and scalar mesons, demonstrating the thermal modifications of these states with increasing temperatures. Introducing the notion of energy-flow operators, we quantify the temporal buildup of correlations in the energy flows across the lattice. We demonstrate that energy-energy correlators fail to factorize to products of energy flows, both in the vacuum and at nonzero temperature, indicating the presence of nontrivial correlations in the quantum states. Our results constitute a first real-time ab initio study of bound-state thermal broadening and finite temperature energy-flow correlations in a gauge theory, providing a benchmark for future studies of hadronic matter under extreme conditions.

CERN

João Barata

Thermal modifications of mesons and energy-energy correlators from real-time simulations of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></math> lattice gauge theory

We present the first lattice QCD calculations of the isovector helicity transverse momentum-dependent parton distribution function (TMDPDF) and the flavor-dependent unpolarized TMDPDFs for up and down quarks in the proton. Our computations utilize domain-wall fermion discretization with physical quark masses. Employing Coulomb-gauge-fixed quark correlation functions within the large-momentum effective theory framework, we access nonperturbative transverse quark separations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mi>T</mi></msub></math> up to approximately 1 fm, corresponding to transverse momenta as low as 200 MeV. Based on the quasi-TMD factorization theorem, we construct renormalization-group–invariant ratios that are equal to the corresponding light-cone TMDPDF ratios. At moderate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math>, our results reveal that the isovector helicity and unpolarized TMDPDFs exhibit nearly identical transverse structure up to a normalization factor, and the unpolarized distributions display only mild flavor dependence. These findings not only support key trends observed in recent global analyses but also provide robust, nonperturbative constraints that can distinguish between different parameterizations. This Letter establishes a first-principles benchmark for TMDPDFs, offering valuable input for ongoing and future experimental efforts to map the proton’s three-dimensional structure.

Dennis Bollweg

Lattice QCD Benchmark of Proton Helicity and Flavor-Dependent Unpolarized Transverse Momentum-Dependent Parton Distribution Functions at Physical Quark Masses

We present a lattice determination of the heavy-quark diffusion coefficient in (2+1)-flavor QCD with almost physical quark masses. The momentum and spatial diffusion coefficients are extracted for a wide temperature range, from $\textit{T}$ = 163 MeV to 10 GeV. The results are in agreement with previous works from the HotQCD collaboration, and show fast thermalization of the heavy quark inside the QGP. Near the chiral crossover temperature $\textit{T}$ ≃ 150 MeV, our results are close to the AdS/CFT estimation computed at strong coupling.

Jorge Golán

Olaf Kaczmarek

Rasmus Larsen

Guy Moore

Peter Petreczky

China

Hai-Tao Shu

Simon Stendebach

Johannes Weber

Temperature dependence of heavy quark diffusion from (2+1)-flavor lattice QCD

We present the first lattice quantum chromodynamics (QCD) calculation of the pion valence-quark transverse-momentum-dependent parton distribution function (TMDPDF) within the framework of large-momentum effective theory (LaMET). Using correlators fixed in the Coulomb gauge (CG), we computed the quasi-TMD beam function for a pion with a mass of 300 MeV, a fine lattice spacing of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>=</mo><mn>0.06</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>, and multiple large momenta up to 3 GeV. The intrinsic soft functions in the CG approach are extracted from form factors with large momentum transfer, and as a byproduct, we also obtain the corresponding Collins-Soper (CS) kernel. Our determinations of both the soft function and the CS kernel agree with perturbation theory at small transverse separations (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mo>⊥</mo></msub></math>) between the quarks. At larger <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mo>⊥</mo></msub></math>, the CS kernel remains consistent with recent results obtained using both CG and gauge-invariant TMD correlators in the literature. By combining next-to-leading logarithmic factorization of the quasi-TMD beam function and the soft function, we obtain an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math>-dependent pion valence-quark TMDPDF for transverse separations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mo>⊥</mo></msub><mo>≳</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>. Interestingly, we find that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mo>⊥</mo></msub></math> dependence of the phenomenological parametrizations of TMDPDF for moderate values of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> are in reasonable agreement with our QCD determinations. In addition, we present results for the transverse-momentum-dependent wave function for a heavier pion with 670 MeV mass.

Jinchen He

Transverse-momentum-dependent pion structures from lattice QCD: Collins-Soper kernel, soft factor, TMDWF, and TMDPDF

We calculate the perturbative corrections in the strong coupling to the unpolarized quark transverse-momentum dependent parton distribution function (TMDPDF) operator within a background-field framework, extending the approach of Mukherjee et al. [<a href="http://dx.doi.org/10.1103/PhysRevD.109.034035">Phys. Rev. D 109, 034035 (2024)</a>]. We focus on ensuring gauge invariance, identifying two key components needed: a gauge-invariant TMDPDF operator with a transverse gauge link at spatial infinity, and accounting of the equations of motion (EoM) of the background fields. We go beyond next-to-leading order in strong coupling expansion, considering multiple interactions with the background field at all orders of strong coupling. By examining the interplay between quark and gluon contributions, we show that spurious singularities, proportional to EoM, can be misinterpreted as genuine divergences in quantum chromodynamics factorization unless properly identified and removed.

Vladimir V. Skokov

Andrey Tarasov

Shaswat Tiwari

Perturbative corrections to quark TMDPDFs in the background-field method: Gauge invariance, equations of motion, and multiple interactions

We calculate the temperature dependence of bottomonium correlators in (2+1)-flavor lattice QCD with the aim to constrain in-medium properties of bottomonia at high temperature. The lattice calculations are performed using HISQ action with physical strange quark mass and light quark masses twenty times smaller than the strange quark mass at two lattice spacings $\textit{a}$ = 0$\textit{.}$0493 fm and 0$\textit{.}$0602 fm, and temporal extents $\textit{N}$ = 16 − 30, corresponding to the temperatures $\textit{T}$ = 133 − 250 MeV. We use a tadpole-improved NRQCD action including spin-dependent $\textit{v}$ corrections for the heavy quarks and extended meson operators in order to be sensitive to in-medium properties of the bottomonium states of interest. We find that within estimated errors the bottomonium masses do not change compared to their vacuum values for all temperatures under our consideration; however, we find different nonzero widths for the various bottomonium states.

Heng-Tong Ding

Wei-Ping Huang

Stefan Meinel

Zhanduo Tang

In-medium bottomonium properties from lattice NRQCD calculations with extended meson operators

Hadronic vacuum polarization (HVP) and light-by-light scattering (HLBL) are crucial for evaluating the Standard Model predictions concerning the muon’s anomalous magnetic moment. However, direct first-principle lattice gauge theory-based calculations of these observables in the timelike region remain challenging. Discrepancies persist between lattice quantum chromodynamics (QCD) calculations in the spacelike region and dispersive approaches relying on experimental data parametrization from the timelike region. Here, we introduce a methodology employing 1+1-dimensional quantum electrodynamics (QED), i.e. the Schwinger Model, to investigate the HVP and HLBL. To that end, we use both tensor network techniques, specifically matrix product states, and classical emulators of digital quantum computers. Demonstrating feasibility in a simplified model, our approach sets the stage for future endeavors leveraging digital quantum computers.

Kazuki Ikeda

Jonathan Raghoonanan

Towards a real-time computation of timelike hadronic vacuum polarization and light-by-light scattering: Schwinger Model tests

In this work, we report a lattice calculation of $\textit{x}$-dependent valence pion generalized parton distributions (GPDs) at zero skewness with multiple values of the momentum transfer −$\textit{t}$. The calculations are based on an $\textit{N}$ = 2 + 1 gauge ensemble of highly improved staggered quarks with Wilson-Clover valence fermion. The lattice spacing is 0.04 fm, and the pion valence mass is tuned to be 300 MeV. We determine the Lorentz-invariant amplitudes of the quasi-GPD matrix elements for both symmetric and asymmetric momenta transfers with similar values and show the equivalence of both frames. Then, focusing on the asymmetric frame, we utilize a hybrid scheme to renormalize the quasi-GPD matrix elements obtained from the lattice calculations. After the Fourier transforms, the quasi-GPDs are then matched to the light-cone GPDs within the framework of large momentum effective theory with improved matching, including the next-to-next-to-leading order perturbative corrections, and leading renormalon and renormalization group resummations. We also present the 3-dimensional image of the pion in impact-parameter space through the Fourier transform of the momentum transfer −$\textit{t}$.

Sergey Syritsyn

Three-dimensional imaging of pion using lattice QCD: generalized parton distributions