arXiv subjects

Marc Gillioz

Publications and source records attributed to Marc Gillioz.

At least 19 recordsLinked to original sources

arXiv2026

Anomaly Detection with Machine Learning Algorithms in Large-Scale Power Grids

We apply several machine learning algorithms to the problem of anomaly detection in operational data for large-scale, high-voltage electric power grids. We observe important differences in the performance of the algorithms. Neural networks typically outperform classical algorithms such as k-nearest neighbors and support vector machines, which we explain by the strong contextual nature of the anomalies. We show that unsupervised learning algorithm work remarkably well and that their predictions are robust against simultaneous, concurring anomalies.

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arXiv2025

The momentum-space conformal bootstrap in 2d

A new bootstrap equation in 2-dimensional conformal field theory is derived starting from the momentum-space representation of the correlation functions. Since Wightman functions are not crossing-symmetric, the analyticity properties of the commutator are leveraged instead to obtain a relation between two distinct operator product expansions. The procedure requires evaluating a 4-point function with two light-like momenta. The result is an asymmetric equation valid for arbitrary theories in 1d and 2d. The new crossing equation admits two simple projections onto orthogonal bases of Jacobi polynomials, reproducing known sets of analytic functionals. One of these, with zeros at double-twist dimensions, was known only as a contour integral. The closed-form expression found in this work is new. We provide a few examples of applications of the new crossing equation.

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arXiv2024

A large synthetic dataset for machine learning applications in power transmission grids

With the ongoing energy transition, power grids are evolving fast. They operate more and more often close to their technical limit, under more and more volatile conditions. Fast, essentially real-time computational approaches to evaluate their operational safety, stability and reliability are therefore highly desirable. Machine Learning methods have been advocated to solve this challenge, however they are heavy consumers of training and testing data, while historical operational data for real-world power grids are hard if not impossible to access. This manuscript presents a large synthetic dataset of power injections in an electric transmission grid model of continental Europe, and describes the algorithm developed for its generation. The method allows one to generate arbitrarily large time series from the knowledge of the grid -- the admittance of its lines as well as the location, type and capacity of its power generators -- and aggregated power consumption data, such as the national load data given by ENTSO-E. The obtained datasets are statistically validated against real-world data.

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arXiv2023

Conformal field theory for particle physicists

This is a set of introductory lecture notes on conformal field theory. Unlike most existing reviews on the subject, CFT is presented here from the perspective of a unitary quantum field theory in Minkowski space-time. It begins with a non-perturbative formulation of quantum field theory (Wightman axioms), and then gradually focuses on the implications of scale and special conformal symmetry, all the way to the modern conformal bootstrap. This approach includes topics often left out, such as subtleties of conformal transformations in Minkowski space-time, the construction of Wightman functions and time-ordered correlators both in position- and momentum-space, unitarity bounds derived from the spectral representation, and the appearance of UV and IR divergences.

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arXiv2022

Spinors and conformal correlators

In conformal field theory, momentum eigenstates can be parameterized by a pair of real spinors, in terms of which special conformal transformations take a simpler form. This well-known fact allows to express 2-point functions of primary operators in the helicity basis, exposing the consequences of unitarity. What is less known is that the same pair of spinors can be used, together with a pair of scalar quantities, to parameterize 3-point functions. We develop this formalism in 3 dimensions and show that it provides a simple realization of the operator product expansion (OPE) for scalar primary operators acting on the vacuum.

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arXiv2021

From Schwinger to Wightman: all conformal 3-point functions in momentum space

All conformal correlation functions of 3 scalar primary operators are constructed in an axiomatic way, relying only on conformal symmetry and causality. The construction makes use of the R-product and its analyticity properties in Minkowski momentum space. The R-product is completely determined by a system of partial differential equations (the Ward identities for conformal transformations) and a boundary condition (permutation symmetry), up to an OPE coefficient. Wightman functions and time-ordered products are then derived from the R-product using operator identities, and the Schwinger function is recovered after a Wick rotation to Euclidean space. This construction does not rely on the Fourier transform of position-space correlation functions.

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arXiv2021

Conformal partial waves in momentum space

The decomposition of 4-point correlation functions into conformal partial waves is a central tool in the study of conformal field theory. We compute these partial waves for scalar operators in Minkowski momentum space, and find a closed-form result valid in arbitrary space-time dimension $d \geq 3$ (including non-integer $d$). Each conformal partial wave is expressed as a sum over ordinary spin partial waves, and the coefficients of this sum factorize into a product of vertex functions that only depend on the conformal data of the incoming, respectively outgoing operators. As a simple example, we apply this conformal partial wave decomposition to the scalar box integral in $d = 4$ dimensions.

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arXiv2020

Momentum-space conformal blocks on the light cone

We study the momentum-space 4-point correlation function of identical scalar operators in conformal field theory. Working specifically with null momenta, we show that its imaginary part admits an expansion in conformal blocks. The blocks are polynomials in the cosine of the scattering angle, with degree $\ell$ corresponding to the spin of the intermediate operator. The coefficients of these polynomials are obtained in a closed-form expression for arbitrary spacetime dimension $d > 2$. If the scaling dimension of the intermediate operator is large, the conformal block reduces to a Gegenbauer polynomial $\mathcal{C}_\ell^{(d-2)/2}$. If on the contrary the scaling dimension saturates the unitarity bound, the block is different Gegenbauer polynomial $\mathcal{C}_\ell^{(d-3)/2}$. These results are then used as an inversion formula to compute OPE coefficients in a free theory example.

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arXiv2020

Conformal 3-point functions and the Lorentzian OPE in momentum space

In conformal field theory in Minkowski momentum space, the 3-point correlation functions of local operators are completely fixed by symmetry. Using Ward identities together with the existence of a Lorentzian operator product expansion (OPE), we show that the Wightman function of three scalar operators is a double hypergeometric series of the Appell $F_4$ type. We extend this simple closed-form expression to the case of two scalar operators and one traceless symmetric tensor with arbitrary spin. Time-ordered and partially-time-ordered products are constructed in a similar fashion and their relation with the Wightman function is discussed.

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arXiv2020

A Scattering Amplitude in Conformal Field Theory

We define form factors and scattering amplitudes in Conformal Field Theory as the coefficient of the singularity of the Fourier transform of time-ordered correlation functions, as $p^2 \to 0$. In particular, we study a form factor $F(s,t,u)$ obtained from a four-point function of identical scalar primary operators. We show that $F$ is crossing symmetric, analytic and it has a partial wave expansion. We illustrate our findings in the 3d Ising model, perturbative fixed points and holographic CFTs.

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arXiv2019

Convergent Momentum-Space OPE and Bootstrap Equations in Conformal Field Theory

General principles of quantum field theory imply that there exists an operator product expansion (OPE) for Wightman functions in Minkowski momentum space that converges for arbitrary kinematics. This convergence is guaranteed to hold in the sense of a distribution, meaning that it holds for correlation functions smeared by smooth test functions. The conformal blocks for this OPE are conceptually extremely simple: they are products of 3-point functions. We construct the conformal blocks in 2-dimensional conformal field theory and show that the OPE in fact converges pointwise to an ordinary function in a specific kinematic region. Using microcausality, we also formulate a bootstrap equation directly in terms of momentum space Wightman functions.

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arXiv2018

Graviton Scattering and a Sum Rule for the c Anomaly in 4D CFT

4D CFTs have a scale anomaly characterized by the coefficient $c$, which appears as the coefficient of logarithmic terms in momentum space correlation functions of the energy-momentum tensor. By studying the CFT contribution to 4-point graviton scattering amplitudes in Minkowski space we derive a sum rule for $c$ in terms of $TT\mathcal{O}$ OPE coefficients. The sum rule can be thought of as a version of the optical theorem, and its validity depends on the existence of the massless and forward limits of the $\langle TTTT \rangle$ correlation functions that contribute. The finiteness of these limits is checked explicitly for free scalar, fermion, and vector CFTs. The sum rule gives $c$ as a sum of positive terms, and therefore implies a lower bound on $c$ given any lower bound on $TT\mathcal{O}$ OPE coefficients. We compute the coefficients to the sum rule for arbitrary operators of spin 0 and 2, including the energy-momentum tensor.

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arXiv2017

Spontaneous conformal symmetry breaking and a massless Wu-Yang monopole

A formulation of $\mathcal{N} = 2$ supersymmetric Yang-Mills theory with a spacetime-dependent gauge coupling allows to study the breaking of conformal symmetry at the quantum level. The theory has an energy-momentum tensor that is only conserved if an equation of motion for the coupling is imposed. It admits non-trivial solitons, among which the Wu-Yang monopole that can be regularized and turns out to be massless. On the other hand, the ordinary BPS monopole is only a solution in the large $N_c$ limit.

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arXiv2016

Scale Anomalies, States, and Rates in Conformal Field Theory

This paper presents two methods to compute scale anomaly coefficients in conformal field theories (CFTs), such as the c anomaly in four dimensions, in terms of the CFT data. We first use Euclidean position space to show that the anomaly coefficient of a four-point function can be computed in the form of an operator product expansion (OPE), namely a weighted sum of OPE coefficients squared. We compute the weights for scale anomalies associated with scalar operators and show that they are not positive. We then derive a different sum rule of the same form in Minkowski momentum space where the weights are positive. The positivity arises because the scale anomaly is the coefficient of a logarithm in the momentum space four-point function. This logarithm also determines the dispersive part, which is a positive sum over states by the optical theorem. The momentum space sum rule may be invalidated by UV and/or IR divergences, and we discuss the conditions under which these singularities are absent. We present a detailed discussion of the formalism required to compute the weights directly in Minkowski momentum space. A number of explicit checks are performed, including a complete example in an 8-dimensional free field theory.

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arXiv2016

The local renormalization of super-Yang-Mills theories

We show how to consistently renormalize $\mathcal{N} = 1$ and $\mathcal{N} = 2$ super-Yang-Mills theories in flat space with a local (i.e. space-time-dependent) renormalization scale in a holomorphic scheme. The action gets enhanced by a term proportional to derivatives of the holomorphic coupling. In the $\mathcal{N} = 2$ case, this new action is exact at all orders in perturbation theory.

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arXiv2014

Constraining new coloured matter from the ratio of 3- to 2-jets cross sections at the LHC

The Large Hadron Collider experiments are probing the evolution of the strong coupling $α_s$ up to the TeV scale. We show how the ratio of 3- to 2-jets cross sections is affected by the presence of new physics and argue that it can be used to place a model-independent bound on new particles carrying QCD color charge. The current data potentially constrains such states to be heavier than a few hundred GeVs.

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arXiv2014

Vector-like Bottom Quarks in Composite Higgs Models

Like many other models, Composite Higgs Models feature the existence of heavy vector-like quarks. Mixing effects between the Standard Model fields and the heavy states, which can be quite large in case of the top quark, imply deviations from the SM. In this work we investigate the possibility of heavy bottom partners. We show that they can have a significant impact on electroweak precision observables and the current Higgs results if there is a sizeable mixing with the bottom quark. We explicitly check that the constraints from the measurement of the CKM matrix element $V_{tb}$ are fulfilled, and we test the compatibility with the electroweak precision observables. In particular we evaluate the constraint from the $Z$ coupling to left-handed bottom quarks. General formulae have been derived which include the effects of new bottom partners in the loop corrections to this coupling and which can be applied to other models with similar particle content. Furthermore, the constraints from direct searches for heavy states at the LHC and from the Higgs search results have been included in our analysis. The best agreement with all the considered constraints is achieved for medium to large compositeness of the left-handed top and bottom quarks.

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arXiv2013

Standard Model Vacuum Stability and Weyl Consistency Conditions

At high energy the standard model possesses conformal symmetry at the classical level. This is reflected at the quantum level by relations between the different beta functions of the model. These relations are known as the Weyl consistency conditions. We show that it is possible to satisfy them order by order in perturbation theory, provided that a suitable coupling constant counting scheme is used. As a direct phenomenological application, we study the stability of the standard model vacuum at high energies and compare with previous computations violating the Weyl consistency conditions.

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