We evaluate the spectral dimension in causal set quantum gravity by simulating random walks on causal sets. In contrast to other approaches to quantum gravity, we find an increasing spectral ...dimension at small scales. This observation can be connected to the nonlocality of causal set theory that is deeply rooted in its fundamentally Lorentzian nature. Based on its large-scale behaviour, we conjecture that the spectral dimension can serve as a tool to distinguish causal sets that approximate manifolds from those that do not. As a new tool to probe quantum spacetime in different quantum gravity approaches, we introduce a novel dimensional estimator, the causal spectral dimension, based on the meeting probability of two random walkers, which respect the causal structure of the quantum spacetime. We discuss a causal-set example, where the spectral dimension and the causal spectral dimension differ, due to the existence of a preferred foliation.
Here, we give a pedagogical review of the recently-introduced notion of a "scalar product" between Feynman integrals and how it helps us understand the analytic structure of the perturbative S-matrix.
We reformulate the analysis of singularities of Feynman integrals in a way that can be practically applied to perturbative computations in the standard model in dimensional regularization. After ...highlighting issues in the textbook treatment of Landau singularities, we develop an algorithm for classifying and computing them using techniques from computational algebraic geometry. We introduce an algebraic variety called the principal Landau determinant, which captures the singularities even in the presence of massless particles or UV/IR divergences. We illustrate this for 114 example diagrams, including a cutting-edge 2-loop 5-point nonplanar QCD process with multiple mass scales.
Principal Landau determinants Fevola, Claudia; Mizera, Sebastian; Telen, Simon
Computer physics communications,
October 2024, 2024-10-00, Letnik:
303
Journal Article
Recenzirano
We reformulate the Landau analysis of Feynman integrals with the aim of advancing the state of the art in modern particle-physics computations. We contribute new algorithms for computing Landau ...singularities, using tools from polyhedral geometry and symbolic/numerical elimination. Inspired by the work of Gelfand, Kapranov, and Zelevinsky (GKZ) on generalized Euler integrals, we define the principal Landau determinant of a Feynman diagram. We illustrate with a number of examples that this algebraic formalism allows to compute many components of the Landau singular locus. We adapt the GKZ framework by carefully specializing Euler integrals to Feynman integrals. For instance, ultraviolet and infrared singularities are detected as irreducible components of an incidence variety, which project dominantly to the kinematic space. We compute principal Landau determinants for the infinite families of one-loop and banana diagrams with different mass configurations, and for a range of cutting-edge Standard Model processes. Our algorithms build on the Julia package Landau.jl and are implemented in the new open-source package PLD.jl available at https://mathrepo.mis.mpg.de/PLD/.
Program title:PLD.jl
CPC Library link to program files:https://doi.org/10.17632/7h5644mm4n.1
Developer's repository link:https://mathrepo.mis.mpg.de/PLD/
Licensing provisions: Creative Commons by 4.0
Programming language:Julia
Supplementary material: The repository includes the source code with documentation (PLD_code.zip), a jupyter notebook tutorial providing installation and usage instructions (PLD_notebook.zip), a database containing the output of our algorithm on 114 examples of Feynman integrals (PLD_database.zip).
Nature of problem: A fundamental challenge in scattering amplitude is to determine the values of complexified kinematic invariants for which an amplitude can develop singularities. Bjorken, Landau, and Nakanishi wrote a system of polynomial constraints, nowadays known as the Landau equations. This project aims to rigorously revisit the Landau analysis of the singularity locus of Feynman integrals with a practical view towards explicit computations.
Solution method: We define the principal Landau determinant (PLD), which is a variety inspired by the work of Gelfand, Kapranov, and Zelevinsky (GKZ). We conjecture that it provides a subset of the singularity locus, and we implement effective algorithms to compute its defining equation explicitly.
References: OSCAR 1, HomotopyContinuation.jl 2, Landau.jl 3
CHY loop integrands from holomorphic forms Gomez, Humberto; Mizera, Sebastian; Zhang, Guojun
The journal of high energy physics,
03/2017, Letnik:
2017, Številka:
3
Journal Article
Recenzirano
Odprti dostop
A
bstract
Recently, the Cachazo-He-Yuan (CHY) approach for calculating scattering amplitudes has been extended beyond tree level. In this paper, we introduce a way of constructing CHY integrands for ...Φ
3
theory up to two loops from holomorphic forms on Riemann surfaces. We give simple rules for translating Feynman diagrams into the corresponding CHY integrands. As a complementary result, we extend the Λ-algorithm, originally introduced in
arXiv:1604.05373
, to two loops. Using this approach, we are able to analytically verify our prescription for the CHY integrands up to seven external particles at two loops. In addition, it gives a natural way of extending to higher-loop orders.
We explore the idea of asymptotic silence in causal set theory and find that causal sets approximated by continuum spacetimes exhibit behavior akin to asymptotic silence. We make use of an intrinsic ...definition of spatial distance between causal set elements in the discrete analogue of a spatial hypersurface. Using numerical simulations for causal sets approximated by D=2,3 and 4 dimensional Minkowski spacetime, we show that while the discrete distance rapidly converges to the continuum distance at a scale roughly an order of magnitude larger than the discreteness scale, it is significantly larger on small scales. This allows us to define an effective dimension which exhibits dimensional reduction in the ultraviolet, while monotonically increasing to the continuum dimension with increasing continuum distance. We interpret these findings as manifestations of asymptotic silence in causal set theory.
Strong gravitational lensing provides fundamental insights into the understanding of the dark matter distribution in massive galaxies, galaxy clusters, and the background cosmology. Despite their ...importance, few gravitational arcs have been discovered so far. The urge for more complete, large samples and unbiased methods of selecting candidates increases. Several methods for the automatic detection of arcs have been proposed in the literature, but large amounts of spurious detections retrieved by these methods force observers to visually inspect thousands of candidates per square degree to clean the samples. This approach is largely subjective and requires a huge amount of checking by eye, especially considering the actual and upcoming wide-field surveys, which will cover thousands of square degrees. In this paper we study the statistical properties of the colours of gravitational arcs detected in the 37 deg2 of the CFHTLS-Archive-Research Survey (CARS). Most of them lie in a relatively small region of the (g′ − r′, r′ − i′) colour–colour diagram. To explain this property, we provide a model that includes the lensing optical depth expected in a ΛCDM cosmology that, in combination with the sources’ redshift distribution of a given survey, in our case CARS, peaks for sources at redshift z ~ 1. By furthermore modelling the colours derived from the spectral energy distribution of the galaxies that dominate the population at that redshift, the model reproduces the observed colours well. By taking advantage of the colour selection suggested by both data and model, we automatically detected 24 objects out of 90 detected by eye checking. Compared with the single-band arcfinder, this multi-band filtering returns a sample complete to 83% and a contamination reduced by a factor of ~6.5. New gravitational arc candidates are also proposed.
You might've heard about various mathematical properties of scattering amplitudes such as analyticity, sheets, branch cuts, discontinuities, etc. What does it all mean? In these lectures, we'll take ...a guided tour through simple scattering problems that will allow us to directly trace such properties back to physics. We'll learn how different analytic features of the S-matrix are really consequences of causality, locality of interactions, unitary propagation, and so on. These notes are based on a series of lectures given in Spring 2023 at the Institute for Advanced Study in Princeton and the Higgs Centre School of Theoretical Physics in Edinburgh.
Recent advances in machine learning establish the ability of certain neural-network architectures called neural operators to approximate maps between function spaces. Motivated by a prospect of ...employing them in fundamental physics, we examine applications to scattering processes in quantum mechanics. We use an iterated variant of Fourier neural operators to learn the physics of Schr\"odinger operators, which map from the space of initial wave functions and potentials to the final wave functions. These deep operator learning ideas are put to test in two concrete problems: a neural operator predicting the time evolution of a wave packet scattering off a central potential in \(1+1\) dimensions, and the double-slit experiment in \(2+1\) dimensions. At inference, neural operators can become orders of magnitude more efficient compared to traditional finite-difference solvers.
Singularities, such as poles and branch points, play a crucial role in investigating the analytic properties of scattering amplitudes that inform new computational techniques. In this note, we point ...out that scattering amplitudes can also have another class of singularities called natural boundaries of analyticity. They create a barrier beyond which analytic continuation cannot be performed. More concretely, we use unitarity to show that \(2 \to 2\) scattering amplitudes in theories with a mass gap can have a natural boundary on the second sheet of the lightest threshold cut. There, an infinite number of ladder-type Landau singularities densely accumulates on the real axis in the center-of-mass energy plane. We argue that natural boundaries are generic features of higher-multiplicity scattering amplitudes in gapped theories.
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