The SU(2)–gauged custodial two-Higgs-doublet model, which shares the same global-symmetry properties with the standard model, is studied non-perturbatively on the lattice. The additional Higgs doublet enlarges the scalar spectrum and opens the possibility for spontaneous breaking of the global symmetry. In this work we start by showing the occurrence of spontaneous breaking of the custodial symmetry in a region of the parameter space of the model. Following this, both the spectrum and the running of the gauge coupling of are examined at weak quartic couplings in the presence of the custodial symmetry. The calculations are performed with energy cutoffs ranging from 300 to 600 GeV on a line of constant standard model physics, obtained by tuning bare couplings to fix the ratio between the masses of the Higgs and the $\textit{W}$ bosons, as well as the value of the renormalized gauge coupling at the scale of the $\textit{W}$ boson mass. The realizable masses for the additional scalar states are explored. For the choice of bare quartic couplings in this work, the estimated lower bound of these masses is found to be well below the $\textit{W}$ boson mass, and independent of the cutoff. We also study the finite temperature electroweak transition along this line of constant standard model physics, revealing properties of a smooth crossover behavior.

Italy

Spain

Guilherme Catumba

Japan

Taiwan

Atsuki Hiraguchi

Wei-Shu Hou

Germany

Karl Jansen

Ying-Jer Kao

C.J. Lin

Alberto Ramos

Mugdha Sarkar

Lattice investigation of custodial two-Higgs-doublet model at weak quartic couplings

Quantum Chromodynamics, the theory of quarks and gluons, whose interactions can be described by a local SU(3) gauge symmetry with charges called “color quantum numbers”, is reviewed; the goal of this review is to provide advanced Ph.D. students a comprehensive handbook, helpful for their research. When QCD was “discovered” 50 years ago, the idea that quarks could exist, but not be observed, left most physicists unconvinced. Then, with the discovery of charmonium in 1974 and the explanation of its excited states using the Cornell potential, consisting of the sum of a Coulomb-like attraction and a long range linear confining potential, the theory was suddenly widely accepted. This paradigm shift is now referred to as the November revolution. It had been anticipated by the observation of scaling in deep inelastic scattering, and was followed by the discovery of gluons in three-jet events. The parameters of QCD include the running coupling constant,  $$\alpha _s(Q^2)$$  <math> <mrow> <msub> <mi>α</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>Q</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> </mrow> </math>  , that varies with the energy scale  $$Q^2$$  <math> <msup> <mi>Q</mi> <mn>2</mn> </msup> </math>   characterising the interaction, and six quark masses. QCD cannot be solved analytically, at least not yet, and the large value of  $$\alpha _s$$  <math> <msub> <mi>α</mi> <mi>s</mi> </msub> </math>   at low momentum transfers limits perturbative calculations to the high-energy region where  $$Q^2\gg \varLambda _{{\textrm{QCD}}} ^2\simeq $$  <math> <mrow> <msup> <mi>Q</mi> <mn>2</mn> </msup> <mo>≫</mo> <msubsup> <mi>Λ</mi> <mrow> <mtext>QCD</mtext> </mrow> <mn>2</mn> </msubsup> <mo>≃</mo> </mrow> </math>   (250 MeV) $$^2$$  <math> <msup> <mrow></mrow> <mn>2</mn> </msup> </math>  . Lattice QCD (LQCD), numerical calculations on a discretized space-time lattice, is discussed in detail, the dynamics of the QCD vacuum is visualized, and the expected spectra of mesons and baryons are displayed. Progress in lattice calculations of the structure of nucleons and of quantities related to the phase diagram of dense and hot (or cold) hadronic matter are reviewed. Methods and examples of how to calculate hadronic corrections to weak matrix elements on a lattice are outlined. The wide variety of analytical approximations currently in use, and the accuracy of these approximations, are reviewed. These methods range from the Bethe–Salpeter, Dyson–Schwinger coupled relativistic equations, which are formulated in both Minkowski or Euclidean spaces, to expansions of multi-quark states in a set of basis functions using light-front coordinates, to the AdS/QCD method that imbeds 4-dimensional QCD in a 5-dimensional deSitter space, allowing confinement and spontaneous chiral symmetry breaking to be described in a novel way. Models that assume the number of colors is very large, i.e. make use of the large  $$N_c$$  <math> <msub> <mi>N</mi> <mi>c</mi> </msub> </math>  -limit, give unique insights. Many other techniques that are tailored to specific problems, such as perturbative expansions for high energy scattering or approximate calculations using the operator product expansion are discussed. The very powerful effective field theory techniques that are successful for low energy nuclear systems (chiral effective theory), or for non-relativistic systems involving heavy quarks, or the treatment of gluon exchanges between energetic, collinear partons encountered in jets, are discussed. The spectroscopy of mesons and baryons has played an important historical role in the development of QCD. The famous X,Y,Z states – and the discovery of pentaquarks – have revolutionized hadron spectroscopy; their status and interpretation are reviewed as well as recent progress in the identification of glueballs and hybrids in light-meson spectroscopy. These exotic states add to the spectrum of expected  $$q{{\bar{q}}}$$  <math> <mrow> <mi>q</mi> <mover> <mrow> <mi>q</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   mesons and qqq baryons. The progress in understanding excitations of light and heavy baryons is discussed. The nucleon as the lightest baryon is discussed extensively, its form factors, its partonic structure and the status of the attempt to determine a three-dimensional picture of the parton distribution. An experimental program to study the phase diagram of QCD at high temperature and density started with fixed target experiments in various laboratories in the second half of the 1980s, and then, in this century, with colliders. QCD thermodynamics at high temperature became accessible to LQCD, and numerical results on chiral and deconfinement transitions and properties of the deconfined and chirally restored form of strongly interacting matter, called the Quark–Gluon Plasma (QGP), have become very precise by now. These results can now be confronted with experimental data that are sensitive to the nature of the phase transition. There is clear evidence that the QGP phase is created. This phase of QCD matter can already be characterized by some properties that indicate, within a temperature range of a few times the pseudocritical temperature, the medium behaves like a near ideal liquid. Experimental observables are presented that demonstrate deconfinement. High and ultrahigh density QCD matter at moderate and low temperatures shows interesting features and new phases that are of astrophysical relevance. They are reviewed here and some of the astrophysical implications are discussed. Perturbative QCD and methods to describe the different aspects of scattering processes are discussed. The primary parton–parton scattering in a collision is calculated in perturbative QCD with increasing complexity. The radiation of soft gluons can spoil the perturbative convergence, this can be cured by resummation techniques, which are also described here. Realistic descriptions of QCD scattering events need to model the cascade of quark and gluon splittings until hadron formation sets in, which is done by parton showers. The full event simulation can be performed with Monte Carlo event generators, which simulate the full chain from the hard interaction to the hadronic final states, including the modelling of non-perturbative components. The contribution of the LEP experiments (and of earlier collider experiments) to the study of jets is reviewed. Correlations between jets and the shape of jets had allowed the collaborations to determine the “color factors” – invariants of the SU(3) color group governing the strength of quark–gluon and gluon–gluon interactions. The calculated jet production rates (using perturbative QCD) are shown to agree precisely with data, for jet energies spanning more than five orders of magnitude. The production of jets recoiling against a vector boson,  $$W^\pm $$  <math> <msup> <mi>W</mi> <mo>±</mo> </msup> </math>   or Z, is shown to be well understood. The discovery of the Higgs boson was certainly an important milestone in the development of high-energy physics. The couplings of the Higgs boson to massive vector bosons and fermions that have been measured so far support its interpretation as mass-generating boson as predicted by the Standard Model. The study of the Higgs boson recoiling against hadronic jets (without or with heavy flavors) or against vector bosons is also highlighted. Apart from the description of hard interactions taking place at high energies, the understanding of “soft QCD” is also very important. In this respect, Pomeron – and Odderon – exchange, soft and hard diffraction are discussed. Weak decays of quarks and leptons, the quark mixing matrix and the anomalous magnetic moment of the muon are processes which are governed by weak interactions. However, corrections by strong interactions are important, and these are reviewed. As the measured values are incompatible with (most of) the predictions, the question arises: are these discrepancies first hints for New Physics beyond the Standard Model? This volume concludes with a description of future facilities or important upgrades of existing facilities which improve their luminosity by orders of magnitude. The best is yet to come!

United States

Franz Gross

Eberhard Klempt

Stanley Brodsky

Andrzej Buras

Volker Burkert

Gudrun Heinrich

Karl Jakobs

Curtis Meyer

Kostas Orginos

Michael Strickland

Johanna Stachel

Giulia Zanderighi

Nora Brambilla

Peter Braun-Munzinger

Daniel Britzger

Simon Capstick

Tom Cohen

Volker Crede

Martha Constantinou

United Kingdom

Christine Davies

Luigi Debbio

Achim Denig

Carleton DeTar

Alexandre Deur

Russia

Latvia

Yuri Dokshitzer

Hans Dosch

Jozef Dudek

Monica Dunford

Evgeny Epelbaum

Miguel Escobedo

Harald Fritzsch

Kenji Fukushima

Paolo Gambino

Canada

Sweden

Dag Gillberg

Steven Gottlieb

CERN

Per Grafstrom

Switzerland

Massimiliano Grazzini

Boris Grube

JINR

Alexey Guskov

Toru Iijima

Xiangdong Ji

Frithjof Karsch

Stefan Kluth

John Kogut

Frank Krauss

Shunzo Kumano

Australia

Derek Leinweber

Heinrich Leutwyler

China

Hai-Bo Li

Yang Li

France

Bogdan Malaescu

Chiara Mariotti

Pieter Maris

Simone Marzani

Wally Melnitchouk

Johan Messchendorp

Harvey Meyer

Ryan Mitchell

Chandan Mondal

Frank Nerling

Sebastian Neubert

Marco Pappagallo

Saori Pastore

José Peláez

Andrew Puckett

Jianwei Qiu

Klaus Rabbertz

Patrizia Rossi

Azerbaijan

Anar Rustamov

Andreas Schäfer

Stefan Scherer

Matthias Schindler

Steven Schramm

Mikhail Shifman

Edward Shuryak

Torbjörn Sjöstrand

George Sterman

Iain Stewart

Joachim Stroth

Eric Swanson

Costa Rica

Guy Téramond

Ulrike Thoma

Antonio Vairo

Danny Dyk

James Vary

Javier Virto

Marcel Vos

Christian Weiss

Markus Wobisch

Sau Wu

Christopher Young

Feng Yuan

Xingbo Zhao

Xiaorong Zhou

50 Years of quantum chromodynamics

David Albandea

Pilar Hernández

Fernando Romero-López

Erratum to: Topological sampling through windings

We present the details and first results of a new strategy for the determination of  $$\alpha _s(m_Z)$$  <math> <mrow> <msub> <mi>α</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>Z</mi> </msub> <mo>)</mo> </mrow> </mrow> </math>   (ALPHA Collaboration et al. in Phys. Lett. B 807:135571, 2020). By simultaneously decoupling 3 fictitious heavy quarks we establish a relation between the  $$\Lambda $$  <math> <mi>Λ</mi> </math>  -parameters of three-flavor QCD and pure gauge theory. Very precise recent results in the pure gauge theory (Dalla Brida and Ramos in Eur. Phys. J. C 79(8):720, 2019; Nada and Ramos in Eur Phys J C 81(1):1, 2021) can thus be leveraged to obtain the three-flavour  $$\Lambda $$  <math> <mi>Λ</mi> </math>  -parameter in units of a common decoupling scale. Connecting this scale to hadronic physics in 3-flavour QCD leads to our result in physical units,  $$\Lambda ^{(3)}_{\overline{\textrm{MS}}} = 336(12)\, {\textrm{MeV}}$$  <math> <mrow> <msubsup> <mi>Λ</mi> <mover> <mtext>MS</mtext> <mo>¯</mo> </mover> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mn>336</mn> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> <mspace width="0.166667em"></mspace> <mtext>MeV</mtext> </mrow> </math>  , which translates to  $$\alpha _s(m_Z) = 0.11823(84)$$  <math> <mrow> <msub> <mi>α</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>Z</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.11823</mn> <mrow> <mo>(</mo> <mn>84</mn> <mo>)</mo> </mrow> </mrow> </math>  . This is compatible with both the FLAG average (Aoki et al. in FLAG review 2021.  arXiv:2111.09849   [hep-lat]) and the previous ALPHA result (ALPHA Collaboration et al., Phys. Rev. Lett. 119(10):102001, 2017), with a comparable, yet still statistics dominated, error. This constitutes a highly non-trivial check, as the decoupling strategy is conceptually very different from the 3-flavour QCD step-scaling method, and so are their systematic errors. These include the uncertainties of the combined decoupling and continuum limits, which we discuss in some detail. We also quantify the correlation between both results, due to some common elements, such as the scale determination in physical units and the definition of the energy scale where we apply decoupling.

Mattia Dalla Brida

Roman Höllwieser

Francesco Knechtli

Tomasz Korzec

Alessandro Nada

Ireland

Stefan Sint

Rainer Sommer

Determination of  $$\alpha _s(m_Z)$$  <math> <mrow> <msub> <mi>α</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>Z</mi> </msub> <mo>)</mo> </mrow> </mrow> </math>   by the non-perturbative decoupling method

In this paper we explore a finite volume renormalization scheme that combines three main ingredients: a coupling based on the gradient flow, the use of twisted boundary conditions and a particular asymmetric geometry, that for SU(N) gauge theories consists on a hypercubic box of size  $$l^2 \times (Nl)^2$$  <math> <mrow> <msup> <mi>l</mi> <mn>2</mn> </msup> <mo>×</mo> <msup> <mrow> <mo>(</mo> <mi>N</mi> <mi>l</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math>  , a choice motivated by the study of volume independence in large N gauge theories. We argue that this scheme has several advantages that make it particularly suited for precision determinations of the strong coupling, among them translational invariance, an analytic expansion in the coupling and a reduced memory footprint with respect to standard simulations on symmetric lattices, allowing for a more efficient use of current GPU clusters. We test this scheme numerically with a determination of the  $$\Lambda $$  <math> <mi>Λ</mi> </math>   parameter in the SU(3) pure gauge theory. We show that the use of an asymmetric geometry has no significant impact in the size of scaling violations, obtaining a value  $$\Lambda _{\overline{\mathrm{MS}}}\sqrt{8 t_0} =0.603(17)$$  <math> <mrow> <msub> <mi>Λ</mi> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </msub> <msqrt> <mrow> <mn>8</mn> <msub> <mi>t</mi> <mn>0</mn> </msub> </mrow> </msqrt> <mo>=</mo> <mn>0.603</mn> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow> </math>   in good agreement with the existing literature. The role of topology freezing, that is relevant for the determination of the coupling in this particular scheme and for large N applications, is discussed in detail.

Eduardo Bribián

Jorge Dasilva Golán

Margarita García Pérez

Memory efficient finite volume schemes with twisted boundary conditions

We propose a modification of the Hybrid Monte Carlo (HMC) algorithm that overcomes the topological freezing of a two-dimensional U(1) gauge theory with and without fermion content. This algorithm includes reversible jumps between topological sectors – winding steps – combined with standard HMC steps. The full algorithm is referred to as winding HMC (wHMC), and it shows an improved behaviour of the autocorrelation time towards the continuum limit. We find excellent agreement between the wHMC estimates of the plaquette and topological susceptibility and the analytical predictions in the U(1) pure gauge theory, which are known even at finite  $$\beta $$  <math> <mi>β</mi> </math>  . We also study the expectation values in fixed topological sectors using both HMC and wHMC, with and without fermions. Even when topology is frozen in HMC – leading to significant deviations in topological as well as non-topological quantities – the two algorithms agree on the fixed-topology averages. Finally, we briefly compare the wHMC algorithm results to those obtained with master-field simulations of size  $$L\sim 8 \times 10^3$$  <math> <mrow> <mi>L</mi> <mo>∼</mo> <mn>8</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>3</mn> </msup> </mrow> </math>  .

Topological sampling through windings

We propose a new strategy for the determination of the step scaling function  $$\sigma (u)$$  <math> <mrow> <mi>σ</mi> <mo>(</mo> <mi>u</mi> <mo>)</mo> </mrow> </math>   in finite size scaling studies using the gradient flow. In this approach the determination of  $$\sigma (u)$$  <math> <mrow> <mi>σ</mi> <mo>(</mo> <mi>u</mi> <mo>)</mo> </mrow> </math>   is broken in two pieces: a change of the flow time at fixed physical size, and a change of the size of the system at fixed flow time. Using both perturbative arguments and a set of simulations in the pure gauge theory we show that this approach leads to a better control over the continuum extrapolations. Following this new proposal we determine the running coupling at high energies in the pure gauge theory and re-examine the determination of the  $$\Lambda $$  <math> <mi>Λ</mi> </math>  -parameter, with special care on the perturbative truncation uncertainties.

An analysis of systematic effects in finite size scaling studies using the gradient flow

We propose a new strategy for the determination of the QCD coupling. It relies on a coupling computed in QCD with <math><msub><mrow><mi>N</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>≥</mo><mn>3</mn></math> degenerate heavy quarks at a low energy scale <math><msub><mrow><mi>μ</mi></mrow><mrow><mi>dec</mi></mrow></msub></math>, together with a non-perturbative determination of the ratio <math><mi>Λ</mi><mo>/</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>dec</mi></mrow></msub></math> in the pure gauge theory. We explore this idea using a finite volume renormalization scheme for the case of <math><msub><mrow><mi>N</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mn>3</mn></math> QCD, demonstrating that a precise value of the strong coupling <math><msub><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow></msub></math> can be obtained. The idea is quite general and can be applied to solve other renormalization problems, using finite or infinite volume intermediate renormalization schemes.

Non-perturbative renormalization by decoupling

We present the open-source package openQ* D-1.0 (openQ*D. GitLab: https://gitlab.com/rcstar/openQxD. CSIC: https://dx.doi.org/10.20350/digitalCSIC/8591. https://hdl.handle.net/10261/173334, 2019), which has been primarily, but not uniquely, designed to perform lattice simulations of QCD+QED and QCD, with and without $$\mathrm {C}^*$$ <math><msup><mrow><mi>C</mi></mrow><mo>∗</mo></msup></math>  boundary conditions, and O(a) improved Wilson fermions. The use of $$\mathrm {C}^*$$ <math><msup><mrow><mi>C</mi></mrow><mo>∗</mo></msup></math>  boundary conditions in the spatial direction allows for a local and gauge-invariant formulation of QCD+QED in finite volume, and provides a theoretically clean setup to calculate isospin-breaking and radiative corrections to hadronic observables from first principles. The openQ*D code is based on openQCD-1.6 (Simulation program for lattice QCD (openQCD code). https://cern.ch/luscher/openQCD, 2016) and NSPT-1.4 (Numerical Stochastic Perturbation Theory (NSPT code). https://cern.ch/luscher/NSPT, 2017). In particular it inherits from openQCD-1.6 several core features, e.g. the highly optimized Dirac operator, the locally deflated solver, the frequency splitting for the RHMC, or the 4th order OMF integrator.

Isabel Campos

Patrick Fritzsch

Martin Hansen

Marina Marinkovic

Agostino Patella

Nazario Tantalo

openQ*D code: a versatile tool for QCD+QED simulations

Using finite size scaling techniques and a renormalization scheme based on the Gradient Flow, we determine non-perturbatively the $$\beta $$ <math><mi>β</mi></math> -function of the SU(3) Yang–Mills theory for a range of renormalized couplings $${\bar{g}}^2\sim $$ <math><mrow><msup><mrow><mover><mrow><mi>g</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mn>2</mn></msup><mo>∼</mo></mrow></math>  1–12. We perform a detailed study of the matching with the asymptotic NNLO perturbative behavior at high-energy, with our non-perturbative data showing a significant deviation from the perturbative prediction down to $$\bar{g}^2\sim 1$$ <math><mrow><msup><mrow><mover><mrow><mi>g</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mn>2</mn></msup><mo>∼</mo><mn>1</mn></mrow></math> . We conclude that schemes based on the Gradient Flow are not competitive to match with the asymptotic perturbative behavior, even when the NNLO expansion of the $$\beta $$ <math><mi>β</mi></math> -function is known. On the other hand, we show that matching non-perturbatively the Gradient Flow to the Schrödinger Functional scheme allows us to make safe contact with perturbation theory with full control on truncation errors. This strategy allows us to obtain a precise determination of the $$\Lambda $$ <math><mi>Λ</mi></math> -parameter of the SU(3) Yang–Mills theory in units of a reference hadronic scale ($$\sqrt{8t_0}\, \Lambda _{\overline{\mathrm{MS}}} = 0.6227(98)$$ <math><mrow><msqrt><mrow><mn>8</mn><msub><mi>t</mi><mn>0</mn></msub></mrow></msqrt><mspace width="0.166667em"></mspace><msub><mi>Λ</mi><mover><mi>MS</mi><mo>¯</mo></mover></msub><mo>=</mo><mn>0.6227</mn><mrow><mo>(</mo><mn>98</mn><mo>)</mo></mrow></mrow></math> ), showing that a precision on the QCD coupling below 0.5% per-cent can be achieved using these techniques.

The gradient flow coupling at high-energy and the scale of SU(3) Yang–Mills theory