arXiv subjects

Gerrit Schierholz

Publications and source records attributed to Gerrit Schierholz.

At least 19 recordsLinked to original sources

arXiv2026

Renormalization Group Approach to Confinement

While we have several complementary models of confinement, some of which are phenomenologically appealing, we do not have the ability to calculate analytically even simple aspects of confinement, let alone have a framework to eventually prove confinement. The problem we are facing is to evolve the theory from the perturbative regime to the long distance confining regime. This is generally achieved by renormalization group transformations. With the gradient flow we now have a technique to address the problem from first principles. The primary focus is on the running coupling $α_S(μ)$, from which confinement can be concluded alone. A central point is that the gluon condensate is scale invariant, which reflects its self-similar behavior across different scales. Building on that, we derive $α_S(μ) \simeq Λ_S^2/μ^2$, which evolves to the infrared fixed point $1/α_S = 0$ in accordance with infrared slavery. The only important factor appears to be the presence of the gluon condensate, which is a universal feature that QCD shares with many other models. The analytical results are supported by numerical simulations.

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arXiv2026

Form factors of the $ρ$ meson from effective field theory and the lattice

The calculation of resonance form factors in effective field theory as well as on the lattice is a highly challenging task. In a recent paper, we proposed a novel method based on the introduction of a background field and the Feynman-Hellmann theorem to address the problem, and applied it to a toy model. In the present work we use this method for the electromagnetic form factors of the $ρ$-meson. By matching the results to Chiral Perturbation Theory, we provide a first, crude estimate of all three form factors of the $ρ$-meson within the effective field theory. Contact contributions to these form factors turn out to be substantial. A procedure for lattice calculations is outlined, paving the way for an ab initio approach to the problem.

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arXiv2025

QCD Lambda Parameter from Gradient Flow

Starting from the running coupling in the gradient flow scheme, the QCD Lambda parameter $Λ_{\,\overline{\!\rm MS\!}\;}$ is determined analytically with respect to the reference scale $w_0$ in the pure gauge theory. A key element is the gluon condensate, which, being invariant under RG transformations, establishes a formal relationship between $Λ_{\,\overline{\!\rm MS\!}\;}$ and $w_0$. Good agreement with numerical estimates is found.

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arXiv2025

Gross-Llewellyn Smith sum rule from lattice QCD

We compute the Gross-Llewellyn Smith sum rule, i.e. the lowest odd moment of the parity-violating structure function, $F_3$, of the nucleon from a lattice QCD calculation of the Compton amplitude. Our calculations are performed on $48^3 \times 96$ lattices at the $SU(3)$ symmetric point for two lattice spacings. We extract the moments for several values of the current momenta in the range $0.5 \lesssim Q^2 \lesssim 10 \; {\rm GeV}^2$, covering both the nonperturbative and perturbative regimes. We compare our moments to the Gross-Llewellyn Smith sum rule and discuss the implications for higher-twist effects, a determination of $α_s(Q^2)$ from a hadronic quantity complementing the phenomenological and other lattice approaches, and electroweak box contributions crucial for Cabibbo-Kobayashi-Maskawa matrix unitarity studies.

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arXiv2025

Absence of strong CP violation

Quantum Chromodynamics admits a CP-violating contribution to the action, the $θ$ term, which is expected to give rise to a nonvanishing electric dipole moment of the neutron. Despite intensive search, no CP violations have been found in the strong interaction. This puzzle is referred to as the strong CP problem. There is evidence that CP is conserved in the confining theory, to the extent that color charges are totally screened for $θ> 0$ at large distances. It is not immediately obvious that this implies a vanishing dipole moment though. With this Letter I will close the gap. It is shown that in the infinite volume hadron correlation functions decouple from the topological charge, expressed in terms of the zero modes. The reason is that hadrons have a limited range of interaction, while the density of zero modes vanishes with the inverse root of the volume, thus reducing the probability of finding a zero mode in the vicinity to zero. This implies that CP is conserved in the strong interaction.

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arXiv2025

Absence of CP Violation in the Strong Interaction: Vacuum thwarts Axion

QCD admits a contribution to the action, the $θ$ term, which potentially gives rise to nontrivial phases and violates CP. This is essentially a question of how the vacuum reacts to the $θ$ term. In this talk I will address the problem using new developments on the lattice. The overall solution is contrasted with the axion `solution'.

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arXiv2024

Repercussions of the Peccei-Quinn axion on QCD

The axion, originally postulated by Peccei and Quinn to solve the strong CP problem, has become of great interest in particle and astroparticle phenomenology. Yet it has a problem. It is widely assumed that the axion leaves the nonperturbative features of QCD unscathed. This is, however, not the case. On the contrary, the axion extension is found to be in conflict with seminal results on the low-energy properties of QCD. The key is that the anomalous coupling of the axion to the gauge bosons can be integrated out, leaving behind a path integral over topologically trivial gauge potentials.

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arXiv2022

Dynamical solution of the strong CP problem within QCD ?

The strong CP problem is inseparably connected with the topology of gauge fields and the mechanism of color confinement, which requires nonperturbative tools to solve it. In this talk I present results of a recent lattice investigation of QCD with the $θ$ term in collaboration with Yoshifumi Nakamura. The tool we are using to address the nonperturbative properties of the theory is the gradient flow, which is a particular realization of momentum space RG transformations. The novel result is that within QCD the vacuum angle $θ$ is renormalized, together with the strong coupling constant, and flows to $θ= 0$ in the infrared limit. This means that CP is conserved by the strong interactions.

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arXiv2022

Resonance form factors from finite-volume correlation functions with the external field method

A novel method for the extraction of form factors of unstable particles on the lattice is proposed. The approach is based on the study of two-particle scattering in a static, spatially periodic external field by using a generalization of the Lüscher method in the presence of such a field. It is shown that the resonance form factor is given by the derivative of the resonance pole position in the complex plane with respect to the coupling constant to the external field. Unlike the standard approach, this proposal does not suffer from problems caused by the presence of the triangle diagram.

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arXiv2022

Investigating the Compton amplitude subtraction function in lattice QCD

Theoretical predictions of the proton--neutron mass difference and measurements of the proton's charge radius require inputs from the Compton amplitude subtraction function. Model-dependent and non-relativistic calculations of this subtraction function vary significantly, and hence it contributes sizeable uncertainties to the aforementioned physical quantities. We report on the use of Feynman-Hellmann methods in lattice QCD to calculate the subtraction function from first principles. In particular, our initial results show anomalous high-energy behaviour that is at odds with the prediction from the operator product expansion (OPE). Therefore, we investigate the possibility that this unexpected behaviour is due to lattice artifacts, by varying the lattice spacing and volume, and comparing different discretisations of the vector current. Finally, we explore a Feynman-Hellmann implementation that is less sensitive to short-distance contributions and show that the subtraction function's anomalous behaviour can be attributed to these short-distance contributions. As such, this work represents the first steps in achieving a complete understanding of the Compton amplitude subtraction function.

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arXiv2022

Generalised Parton Distributions from Lattice Feynman-Hellmann Techniques

We report on the use of Feynman-Hellmann techniques to calculate the off-forward Compton amplitude (OFCA) in lattice QCD. At leading-twist, the Euclidean OFCA is parameterised by the Mellin moments of generalised parton distributions (GPDs). Hence we extract GPD moments for two values of the soft momentum transfer, $t=-1.10, -2.20\;\text{GeV}^2$ and zero-skewness kinematics at an unphysical pion mass of $m_π\approx 470\;\text{MeV}$. This includes the first determination of the $n=4$ moments.

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arXiv2022

Generalised parton distributions from the off-forward Compton amplitude in lattice QCD

We determine the properties of generalised parton distributions (GPDs) from a lattice QCD calculation of the off-forward Compton amplitude (OFCA). By extending the Feynman-Hellmann relation to second-order matrix elements at off-forward kinematics, this amplitude can be calculated from lattice propagators computed in the presence of a background field. Using an operator product expansion, we show that the deeply-virtual part of the OFCA can be parameterised in terms of the low-order Mellin moments of the GPDs. We apply this formalism to a numerical investigation for zero-skewness kinematics at two values of the soft momentum transfer, $t = -1.1, -2.2 \;\text{GeV}^2$, and a pion mass of $m_π\approx 470\;\text{MeV}$. The form factors of the lowest two moments of the nucleon GPDs are determined, including the first lattice QCD determination of the $n=4$ moments. Hence we demonstrate the viability of this method to calculate the OFCA from first principles, and thereby provide novel constraint on the $x$- and $t$-dependence of GPDs.

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arXiv2022

Strong CP problem, electric dipole moment, and fate of the axion

Three hard problems! In this talk I investigate the long-distance properties of quantum chromodynamics in the presence of a topological theta term. This is done on the lattice, using the gradient flow to isolate the long-distance modes in the functional integral measure and tracing it over successive length scales. It turns out that the color fields produced by quarks and gluons are screened, and confinement is lost, for vacuum angles theta > 0, thus providing a natural solution of the strong CP problem. This solution is compatible with recent lattice calculations of the electric dipole moment of the neutron, while it excludes the axion extension of the Standard Model.

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arXiv2021

Towards high partial waves in lattice QCD with a dumbbell-like operator

An extended two-hadron operator is developed to extract the spectra of irreducible representations (irreps) in the finite volume. The irreps of the group for the finite volume system are projected using a coordinate-space operator. The correlation function of this operator is computationally efficient to extract lattice spectra of the specific irrep. In particular, this new formulation only requires propagators to be computed from two distinct source locations, at fixed spatial separation. We perform a proof-of-principle study on a $24^3 \times 48$ lattice volume with $m_π\approx 900$ MeV by isolating various spectra of the $ππ$ system with isospin-2 including a range of total momenta and irreps. By applying the Lüscher formalism, the phase shifts of $S$-, $D$- and $G$-wave $ππ$ scattering with isospin-2 are extracted from the spectra.

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arXiv2020

Parton distributions and lattice QCD calculations: toward 3D structure

The strong force which binds hadrons is described by the theory of Quantum Chromodynamics (QCD). Determining the character and manifestations of QCD is one of the most important and challenging outstanding issues necessary for a comprehensive understanding of the structure of hadrons. Within the context of the QCD parton picture, the Parton Distribution Functions (PDFs) have been remarkably successful in describing a wide variety of processes. However, these PDFs have generally been confined to the description of collinear partons within the hadron. New experiments and facilities provide the opportunity to additionally explore the transverse structure of hadrons which is described by Generalized Parton Distributions (GPDs) and Transverse Momentum Dependent Parton Distribution Functions (TMD PDFs). In our previous review, we compared and contrasted the two main approaches used to determine the collinear PDFs: the first based on perturbative QCD factorization theorems, and the second based on lattice QCD calculations. In the present report, we provide an update of recent progress on the collinear PDFs, and also expand the scope to encompass the generalized PDFs (GPDs and TMD PDFs). We review the current state of the various calculations, and consider what new data might be available in the near future. We also examine how a shared effort can foster dialog between the PDF and Lattice QCD communities, and yield improvements for these generalized PDFs.

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arXiv2020

Structure functions from the Compton amplitude

We have initiated a program to compute the Compton amplitude from lattice QCD with the Feynman-Hellman method. This amplitude is related to the structure function via a Fredholm integral equation of the first kind. It is known that these types of equations are inherently ill--posed - they are, e.g., extremely sensitive to perturbations of the system. We discuss two methods which are candidates to handle these problems: the model free inversion based on singular value decomposition and one Bayesian type approach. We apply the Bayesian method to currently available lattice data for the Compton amplitude.

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arXiv2018

Control of SU(3) symmetry breaking effects in calculations of B meson decay constant

Early B-physics experiments have left us with a number of puzzles in heavy flavour physics. New lattice calculations (with a greater understanding of QCD effects in the Standard Model) will be needed to support the increase in experimental precision to be achieved by upcoming experiments such as Belle II. We extend the CSSM/UKQCD/QCDSF studies of SU(3) flavour breaking effects by presenting new results for the decay constants $f_B$ and $f_{B_s}$.

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arXiv2018

Parton distributions and lattice QCD calculations: a community white paper

In the framework of quantum chromodynamics (QCD), parton distribution functions (PDFs) quantify how the momentum and spin of a hadron are divided among its quark and gluon constituents. Two main approaches exist to determine PDFs. The first approach, based on QCD factorization theorems, realizes a QCD analysis of a suitable set of hard-scattering measurements, often using a variety of hadronic observables. The second approach, based on first-principle operator definitions of PDFs, uses lattice QCD to compute directly some PDF-related quantities, such as their moments. Motivated by recent progress in both approaches, in this document we present an overview of lattice-QCD and global-analysis techniques used to determine unpolarized and polarized proton PDFs and their moments. We provide benchmark numbers to validate present and future lattice-QCD calculations and we illustrate how they could be used to reduce the PDF uncertainties in current unpolarized and polarized global analyses. This document represents a first step towards establishing a common language between the two communities, to foster dialogue and to further improve our knowledge of PDFs.

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