arXiv subjects

Franz R. Sattler

Publications and source records attributed to Franz R. Sattler.

14 recordsLinked to original sources

arXiv2026

Multi-scattering processes and spectral properties of low-energy QCD

We compute quark and meson spectral functions in low-energy QCD, including all-order scatterings and decays of pions and the scalar $σ$-mode. For low energies the gluons decouple and the dynamics of two-flavour QCD is well-described by a Quark-Meson model. The computations are performed directly in Minkowski space, using the spectral functional renormalisation group. The full mesonic realtime dynamics is captured with the emergent composite approach which is extended here to realtime processes. The inclusion of all-order scatterings and decays is achieved through the self-consistent treatment of the scattering tails and the momentum-dependent resummation of the four-meson vertex via its Bethe-Salpeter equation. We illustrate the importance of higher-order scatterings using the example of the $π\to 3π$ scattering threshold - the lowest kinematically accessible channel for the pion.

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arXiv2026

CosmoLattice 2.0

This paper introduces $\tt {\mathcal C}osmo{\mathcal L}attice$ $\tt v2.0$, a major upgrade that substantially broadens the physical scope and computational capabilities of the code. It introduces lattice implementations of scalar fields non-minimally coupled to gravity through $ϕ^2R$, as well as axion-like fields coupled to Abelian gauge sectors as $ϕF_{μν}\widetilde F^{μν}$. It also provides new procedures for generating specialized initial conditions, including scaling networks of cosmic defects ($\it e.g.$ strings and domain walls), and fields with arbitrary power spectra. The release also incorporates low-storage Runge-Kutta integrators for non-symplectic systems (suitable $\it e.g.$ for non-minimal scalar kinetic terms as $\mathcal{G}_{ab}\partial_μϕ^a\partial^μϕ^b$), scalar-field simulations on reduced $(1+1)$- and $(2+1)$-dimensional lattices, new optimized gravitational-wave evolution, more flexible field and energy-density outputs, and GPU support that can accelerate simulations by a factor $\mathcal{O}(10)$ relative to CPU execution. Extensive documentation on the use of the code is provided on https://www.cosmolattice.com

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arXiv2026

TempLat: a versatile C++ engine for lattice field theories

We present TempLat, a C++ framework for lattice field theory simulations in arbitrary dimensions. Its symbolic language, built on expression templates, translates mathematical expressions almost verbatim into C++ while compiling to highly optimized kernels, in contexts ranging from classical-statistical to Monte Carlo simulations. TempLat is performance-portable: building on Kokkos, it supports large CPU clusters, as well as NVIDIA and AMD GPUs. Its abstraction of hardware into a device concept makes TempLat extensible to future architectures. We demonstrate excellent strong and weak scaling on both CPUs and GPUs. We also present ParaFaFT, a standalone parallel discrete Fourier transform library supporting arbitrary dimensions on all of the above hardware, and benchmark its scaling. Both are open source and available on GitHub, and power a new release of CosmoLattice, a widely used early-universe simulation library now available on GPUs.

hep-lat↗

arXiv2026

Classical equipartition dynamics between axions and non-Abelian gauge fields

Motivated by axion-like inflation and its warm embedding within the Standard Model, we study the early stages of the energy transfer between an axion condensate and an SU(2) gauge ensemble, by employing non-linear classical real-time lattice simulations. The discretized equations of motion are worked out, elaborating on Gauss constraints. A numerical solution is implemented on the CosmoLattice platform. Adopting a quadratic potential, and omitting universe expansion, we establish initial exponential growth of the low-momentum gauge modes; damping of axion oscillations after some delay; and subsequent energy equipartition between axion and gauge ensembles. A clear difference between the SU(2) and U(1) dynamics is observed, likely associated with non-Abelian self-interactions. We elaborate on what this implies for the possible thermalization of the SU(2) ensemble.

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arXiv2026

FunKit: A computer algebra toolkit for functional approaches

We introduce FunKit, a Mathematica package for the derivation and tracing of functional equations from arbitrary master equations. FunKit provides an expression vocabulary and a set of rules that allow for derivations in any given field theory and master equation. It also allows users to add extensions for more specific equation systems. Therefore, it can be used in a wide range of situations, for example Dyson--Schwinger or functional RG equations, flowing reparametrisations, nPI equations, (modified) STIs and WTIs, functional Polchinski and Wegner flows, functional master equations with sources, and many others. Besides interfacing with the \FORM language to trace large tensor expressions efficiently, FunKit also provides facilities to export arbitrary Mathematica expressions to C++, Julia or Fortran code, including the results of derivations, which can then be evaluated numerically. Both the tracing and code generation can also be used independently and in combination with other packages.

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arXiv2026

Thermal precondensation in gauge-fermion theories

Precondensation is a peculiar phenomenon in phase transitions, characterised by the occurrence of a condensate only over a finite range of length scales. It is closely connected to the emergence of domains, pseudo-gapped phases and spatial inhomogeneities in equilibrium. In this work, we show its occurrence in gauge-fermion theories in the chiral limit, close to the thermal chiral phase transition. We further show that the precondensation regime becomes increasingly pronounced and extends over a wider temperature range as the number of fermion flavours is increased. We analyse the underlying dynamics which is shared by a broad class of fermionic systems, ranging from condensed matter to high-energy physics. Specifically, we discuss the potential relevance of this phenomenon for physics beyond the Standard Model.

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arXiv2025

Inhomogeneous instabilities in high-density QCD

QCD at large densities exhibits a moat regime in the scalar-pseudoscalar sector. The resolution of its dynamics is pivotal for the access to the onset of new phases including the potential critical endpoint of QCD. In this work we present the first selfconsistent analysis of this regime with the functional renormalisation group approach to QCD. We map out the moat regime, including a first analysis of potential inhomogeneous instabilities at baryon chemical potential $μ_B\gtrsim 600$ MeV on the chiral crossover line.

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arXiv2025

Soft modes in hot QCD matter

The chiral crossover of QCD at finite temperature and vanishing baryon density turns into a second order phase transition if lighter than physical quark masses are considered. If this transition occurs sufficiently close to the physical point, its universal critical behaviour would largely control the physics of the QCD phase transition. We quantify the size of this region in QCD using functional approaches, both Dyson-Schwinger equations and the functional renormalisation group. The latter allows us to study both critical and non-critical effects on equal footing, facilitating a precise determination of the scaling regime. We find that the physical point is far away from the critical region. Importantly, we show that the physics of the chiral crossover is dominated by soft modes even far beyond the critical region. While scaling functions determine all thermodynamic properties of the system in the critical region, the order parameter potential is the relevant quantity away from it. We compute this potential in QCD using the functional renormalisation group and Dyson-Schwinger equations and provide a simple parametrisation for phenomenological applications.

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arXiv2025

Juggling with Tensor Bases in Functional Approaches

Systematic expansion schemes in functional approaches require the inclusion of higher order vertices. These vertices are expanded in independent tensor bases with a rapidly increasing number of basis elements. Amongst the related tasks are the construction of bases and projection operators, the importance ordering of their elements, and the optimisation of such tensor bases, as well as an analysis of their regularity in momentum space. We present progress in all these directions and introduce the Mathematica package TensorBases designed for the aforementioned tasks.

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arXiv2024

DiFfRG: A Discretisation Framework for functional Renormalisation Group flows

We introduce DiFfRG (Discretisation Framework for functional Renormalisation Group flows), a comprehensive computational C++ framework for solving functional Renormalisation Group flows in very general truncation schemes. Its central features are threefold: Firstly, the use of Finite Element Methods (FEM) for efficient, easy to set up and quantitatively reliable computations of field dependences. Secondly, the (simultaneous) setup of large, fully momentum-dependent vertex expansions. Thirdly, efficient time-discretisation methods, incorporating insights from studies of solving theories which exhibit spontaneous symmetry breaking, going hand in hand with shocks and an exponential increase of the information flow velocity in field space. The framework provides a Mathematica package for automatic code generation of flow equations, finite and zero temperature integration routines with support for GPU hardware and extensive parallelisation capabilities. Detailed examples and tutorials are provided and discussed herein which showcase and introduce the framework to the user. We illustrate the capabilities of DiFfRG with four examples, with the complete codes fully included: finite temperature O(N) theory, a Quark-Meson model, SU(3) Yang-Mills theory and four-Fermi flows in the QCD phase diagram.

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arXiv2024

Towards quantitative precision in functional QCD I

Functional approaches are the only first principle QCD setup that allow for direct computations at finite density. Predictive power and quantitative reliability of the respective results can only be obtained within a systematic expansion scheme with controlled systematic error estimates. Here we set up such a scheme within the functional renormalisation group (fRG) approach to QCD, aiming for full apparent convergence. In the current work we test this setup, using correlation functions and observables in 2+1 flavour vacuum QCD as a natural benchmark case. While the current work includes many evolutionary improvements collected over the past two decades, we also report on three novel important developments: (i) A comprehensive systematic error analysis based on the modular nature of the fRG approach. (ii) The introduction of a fully automated computational framework, allowing for unprecedented access and improvement of the fRG approach to QCD. (iii) The inclusion of the full effective potential of the chiral order parameter. This also gives access to all-order scattering events of pions and to the full momentum dependence of correlation functions, which is a first application of the automated computational framework (ii). The results compare very well to other state-of-the-art results both from functional approaches and lattice simulations, and provide data on general multi-scattering events of pions and the sigma mode for the first time.

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arXiv2023

Towards quantitative precision for QCD at large densities

QCD at large density reveals a rich phase structure, ranging from a potential critical end point and inhomogeneous phases or moat regimes to color superconducting ones with competing order effects. Resolving this region in the phase diagram of QCD with functional approaches requires a great deal of quantitative reliability, already for a qualitative access. In the present work, we systematically extend the functional renormalisation group approach to low energy QCD by setting up a fully self-consistent approximation scheme in a low energy effective quark-meson theory. In this approximation, all pointlike multi-scattering events of the mesonic pion and the sigma mode are taken into account in terms of an effective potential as well as all higher quark-antiquark-mesonic scattering orders. As a first application we compute the phase structure of QCD including its low temperature - large chemical potential part. The quantitative reliability of the approximation and systematic extensions are also discussed.

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arXiv2023

Numerical RG-time integration of the effective potential: Analysis and Benchmark

We investigate the RG-time integration of the effective potential in the functional renormalization group in the presence of spontaneous symmetry breaking and its subsequent convexity restoration on the example of a scalar theory in $d=3$. The features of this setup are common to many physical models and our results are, therefore, directly applicable to a variety of situations. We provide exhaustive work-precision benchmarks and numerical stability analyses by considering the combination of different discrete formulations of the flow equation and a large collection of different algorithms. The results are explained by using the different components entering the RG-time integration process and the eigenvalue structure of the discrete system. Particularly, the combination of Rosenbrock methods, implicit multistep methods or certain (diagonally) implicit Runge-Kutta methods with exact or autodiff Jacobians proves to be very potent. Furthermore, a reformulation in a logarithmic variable circumvents issues related to the singularity bound in the flat regime of the potential.

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arXiv2022

Local Discontinuous Galerkin for the Functional Renormalisation Group

We apply the Local Discontinuous Galerkin discretisation to flow equations of the O(N)-model in the Local Potential Approximation. The improved stability is directly observed by solving the flow equation for various $N$ and space-time dimensions $d$. A particular focus of this work is the numerical discretisation and its implementation. The code is publicly available, and is explained in detail here. It is realised as a module within the high performance PDE framework DUNE.

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