<p>The equation of state of quantum chromodynamics with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>3</mn></math> flavors is determined nonperturbatively in the range of temperatures between 3 and 165 GeV with a precision of about 0.5%–1.0%. The calculation is carried out by numerical simulations of lattice gauge theory discretized <i>à la</i> Wilson with shifted boundary conditions in the compact direction. At each given temperature the entropy density is computed at several lattice spacings in order to extrapolate the results to the continuum limit. Taken at face value, data point straight to the Stefan-Boltzmann value by following a linear behavior in the strong coupling constant squared. They are also compatible with the known perturbative formula supplemented by higher order terms in the coupling constant, a parametrization which describes well our data together with those present in the literature down to 500 MeV.</p>

Italy

Matteo Bresciani

Mattia Dalla Brida

Leonardo Giusti

Michele Pepe

QCD Equation of State with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mn>3</mn></mrow></math> Flavors up to the Electroweak Scale

We study the renormalization group (RG) running of the nonsinglet tensor operator, for <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>3</mn></math> QCD with Wilson fermions in a mixed action setup, with standard Schrödinger functional (SF) boundary conditions for sea quarks and chirally rotated Schrödinger functional (<math><mi>χ</mi><mi>SF</mi></math>) boundary conditions for valence quarks. Based on a recursive finite-size scaling technique we compute nonperturbatively the tensor step-scaling function for an energy range between a hadronic scale and an electroweak scale, above which perturbation theory may be safely applied. Our result is expressed as the RG-running factor <math><msup><mi>T</mi><mi>RGI</mi></msup><mo>/</mo><mo>[</mo><mi>T</mi><mo>(</mo><msub><mi>μ</mi><mi>had</mi></msub><mo>)</mo><msub><mo>]</mo><mi>R</mi></msub></math>, where the numerator is the scale independent (renormalization group invariant—RGI) tensor operator and the denominator is its renormalized counterpart at a hadronic scale <math><msub><mi>μ</mi><mi>had</mi></msub><mo>=</mo><mn>233</mn><mo>(</mo><mn>8</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> in a given scheme. We determine the step-scaling function in four distinct renormalization schemes. We also compute the renormalization parameters of these schemes at <math><msub><mi>μ</mi><mi>had</mi></msub></math> which, combined with the RG-running factor, gives the scheme-independent quantity <math><msubsup><mi>Z</mi><mi>T</mi><mi>RGI</mi></msubsup><mo>(</mo><msubsup><mi>g</mi><mn>0</mn><mn>2</mn></msubsup><mo>)</mo></math> in four schemes and for a range of bare gauge couplings in which large volume hadronic matrix element simulations are performed by the CLS consortium in <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> QCD. All four results are compatible and also agree with a recent determination based on a unitary setup for Wilson quarks with Schrödinger functional boundary conditions [arXiv:2309.04314]. This provides a strong universality test.

Spain

Isabel Campos Plasencia

CERN

Giulia Maria de Divitiis

United States

Andrew Lytle

Mauro Papinutto

Ludovica Pirelli

Anastassios Vladikas

Nonperturbative running of the tensor operator for <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>3</mn></math> QCD from the chirally rotated Schrödinger functional

With Wilson quarks, on-shell O(a) improvement of the lattice QCD action is achieved by including the Sheikholeslami-Wohlert term and two further operators of mass dimension 5, which amount to a mass-dependent rescaling of the bare parameters. We here focus on the rescaled bare coupling,   <math> <msubsup> <mover> <mi>g</mi> <mo>~</mo> </mover> <mn>0</mn> <mn>2</mn> </msubsup> <mo>=</mo> <msubsup> <mi>g</mi> <mn>0</mn> <mn>2</mn> </msubsup> <mfenced> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>b</mi> <mi>g</mi> </msub> <mi>a</mi> <msub> <mi>m</mi> <mi>q</mi> </msub> </mrow> </mfenced> </math>  $$ {\tilde{g}}_0^2={g}_0^2\left(1+{b}_{\textrm{g}}a{m}_{\textrm{q}}\right) $$ , and the determination of   <math> <msub> <mi>b</mi> <mi>g</mi> </msub> <mfenced> <msubsup> <mi>g</mi> <mn>0</mn> <mn>2</mn> </msubsup> </mfenced> </math>  $$ {b}_{\textrm{g}}\left({g}_0^2\right) $$  which is currently only known to 1-loop order of perturbation theory. We derive suitable improvement conditions in the chiral limit and in a finite space-time volume and evaluate these for different gluonic observables, both with and without the gradient flow. The choice of β-values and the line of constant physics are motivated by the ALPHA collaboration’s decoupling strategy to determine α  s (m  Z ) [1]. However, the improvement conditions and some insight into systematic effects may prove useful in other contexts, too.

Germany

Roman Höllwieser

Francesco Knechtli

Tomasz Korzec

Ireland

Stefan Sint

Rainer Sommer

Heavy Wilson quarks and O(a) improvement: nonperturbative results for b g

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.

Alessandro Nada

Alberto Ramos

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

We compute non-perturbatively the renormalization constant of the flavour-singlet local vector current <math><msub><mrow><mi>Z</mi></mrow><mrow><mi>V</mi></mrow></msub></math> in lattice QCD with 3 massless flavours. Gluons are discretized by the Wilson plaquette action while quarks by the O(a)-improved Wilson–Dirac operator. The constant <math><msub><mrow><mi>Z</mi></mrow><mrow><mi>V</mi></mrow></msub></math> is fixed by comparing the expectation values (1-point functions) of the conserved and local vector currents at finite temperature in the presence of shifted boundary conditions and at non-zero imaginary chemical potential. Monte Carlo simulations with a moderate computational cost allow us to obtain <math><msub><mrow><mi>Z</mi></mrow><mrow><mi>V</mi></mrow></msub></math> with an accuracy of about 8‰ for values of the inverse bare coupling constant <math><mi>β</mi><mo>=</mo><mn>6</mn><mo>/</mo><msubsup><mrow><mi>g</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup></math> in the range <math><mn>5.3</mn><mo>≤</mo><mi>β</mi><mo>≤</mo><mn>11.5</mn></math>.

Federico Rapuano

Non-perturbative renormalization of the QCD flavour-singlet local vector current

We present a strategy based on the step-scaling technique to study non-perturbatively thermal QCD up to very high temperatures. As a first concrete application, we compute the flavour non-singlet meson screening masses at 12 temperatures covering the range from T ∼ 1 GeV up to ∼ 160 GeV in the theory with three massless quarks. The calculation is carried out by Monte Carlo simulations on the lattice by considering large spatial extensions in order to have negligible finite volume effects. For each temperature we have simulated 3 or 4 values of the lattice spacing, so as to perform the continuum limit extrapolation with confidence at a few permille accuracy. Chiral symmetry restoration manifests itself in our results through the degeneracy of the vector and the axial vector channels and of the scalar and the pseudoscalar ones. In the entire range of temperatures explored, the meson screening masses deviate from the free theory result, 2πT, by at most a few percent. These deviations, however, cannot be explained by the known leading term in the QCD coupling constant g up to the highest temperature, where other contributions are still very relevant. In particular the vector-pseudoscalar mass splitting turns out to be of O(g4) in the entire range explored, and it remains clearly visible up to the highest temperature, where the two screening masses are still significantly different within our numerical precision. The pattern of different contributions that we have found explains why it has been difficult in the past to match non-perturbative lattice results at T ∼ 1 GeV with the analytic behaviour at asymptotically high temperatures.

United Kingdom

Tim Harris

Davide Laudicina

Non-perturbative thermal QCD at all temperatures: the case of mesonic screening masses

We study the renormalization group (RG) running of the quark mass, for <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>3</mn></math> QCD with Wilson fermions in a mixed action setup, with standard Schrödinger functional (SF) boundary conditions for sea quarks and chirally rotated Schrödinger functional (<math><mrow><mi>χ</mi><mi>SF</mi></mrow></math>) boundary conditions for valence quarks. This necessitates the tuning of the boundary factor <math><msub><mi>z</mi><mi>f</mi></msub><mo>(</mo><msubsup><mi>g</mi><mn>0</mn><mn>2</mn></msubsup><mo>)</mo></math> of the <math><mrow><mi>χ</mi><mi>SF</mi></mrow></math> valence action, in order to ensure that QCD symmetries are fully recovered in the continuum. The properties of this novel setup are monitored through the ratios <math><msub><mi>Z</mi><mi>S</mi></msub><mo>/</mo><msub><mi>Z</mi><mi>P</mi></msub></math> and <math><msub><mi>Σ</mi><mi>S</mi></msub><mo>/</mo><msub><mi>Σ</mi><mi>P</mi></msub></math> of the renormalization parameters and step scaling functions of the scalar and pseudoscalar densities. Where comparison is possible, our <math><msub><mi>Z</mi><mi>S</mi></msub><mo>/</mo><msub><mi>Z</mi><mi>P</mi></msub></math> results are found to agree with previous determinations, based on a mass ratio method [G. M. de Divitiis et al. (ALPHA Collaboration), Eur. Phys. J. C 79, 797 (2019)] and Ward identities [ J. Heitger et al. (ALPHA Collaboration), Eur. Phys. J. C 80, 765 (2020); J. Heitger et al., Eur. Phys. J. C 81, 606 (2021)], with Schrödinger functional boundary conditions. The behavior of <math><msub><mi>Σ</mi><mi>S</mi></msub><mo>/</mo><msub><mi>Σ</mi><mi>P</mi></msub></math> confirms the theoretical expectations of <math><mrow><mi>χ</mi><mi>SF</mi></mrow></math> QCD, related to the restoration of the theory’s symmetries in the continuum limit. From the step-scaling function of the pseudoscalar density we obtain the quark mass RG-running function from hadronic to perturbative energy scales. This is fully compatible with the earlier result obtained in a similar setup for Wilson quarks with Schrödinger functional boundary conditions [I. Campos et al., Eur. Phys. J. C 78, 387 (2018)], and provides a strong universality test for the two lattice setups.

Nonperturbative running of the quark mass for <math><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mn>3</mn></mrow></math> QCD from the chirally rotated Schrödinger functional

The hadronic contribution to the muon anomalous magnetic moment <math><msub><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>=</mo><mo>(</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>−</mo><mn>2</mn><mo>)</mo><mo>/</mo><mn>2</mn></math> has to be determined at the per-mille level for the Standard Model prediction to match the expected final uncertainty from the ongoing E989 experiment. This is 3 times better than the current precision from the dispersive approach, and 5-15 times smaller than the uncertainty on the purely theoretical determinations from lattice QCD. So far the stumbling-block is the large statistical error in the Monte Carlo evaluation of the required correlation functions which can hardly be tamed by brute force. Here we propose to solve this problem by multi-level Monte Carlo integration, a technique which reduces the variance of correlators exponentially in the distance of the fields. We test our strategy by computing the Hadronic Vacuum Polarization on a lattice with a linear extension of 3 fm, a spacing of 0.065 fm, and a pion mass of 270 MeV. Indeed the two-level integration makes the contribution to the statistical error from long-distances de-facto negligible by accelerating its inverse scaling with the cost of the simulation. These findings establish multi-level Monte Carlo as a solid and efficient method for a precise lattice determination of the hadronic contribution to <math><msub><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow></msub></math>. As the approach is applicable to other computations affected by a signal-to-noise ratio problem, it has the potential to unlock many open problems for the nuclear and particle physics community.

Multi-level Monte Carlo computation of the hadronic vacuum polarization contribution to ( − 2)

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 a strategy to define non-perturbatively the energy-momentum tensor in Quantum Chromodynamics (QCD) which satisfies the appropriate Ward identities and has the right trace anomaly. The tensor is defined by regularizing the theory on a lattice, and by fixing its renormalization constants non-perturbatively by suitable Ward identities associated to the Poincaré invariance of the continuum theory. The latter are derived in thermal QCD with a non-zero imaginary chemical potential formulated in a moving reference frame. A renormalization group analysis leads to simple renormalization- group-invariant definitions of the gluonic and fermionic contributions to either the singlet or the non-singlet components of the tensor, and therefore of their form factors among physical states. The lattice discussion focuses on the Wilson discretization of quark fields but the strategy is general. Specific to that case, we also carry out the analysis for the on-shell O(a)-improvement of the energy-momentum tensor. The renormalization and improvement programs profit from the fact that, as shown here, the thermal theory enjoys de-facto automatic O(a)-improvement at finite temperature. The validity of the proposal is scrutinized analytically by a study to 1-loop order in lattice perturbation theory with shifted and twisted (for quarks only) boundary conditions. The latter provides also additional useful insight for a precise non-perturbative calculation of the renormalization constants. The strategy proposed here is accessible to Monte Carlo computations, and in this sense it provides a practical way to define non-perturbatively the energy-momentum tensor in QCD.