arXiv subjects

Mattia Dalla Brida

Publications and source records attributed to Mattia Dalla Brida.

At least 19 recordsLinked to original sources

arXiv2026

The QCD energy-momentum tensor on the lattice: non-perturbative renormalization with $N_f=3$

We construct the traceless components of the energy-momentum tensor on the lattice for QCD with $N_f=3$ flavours, such that their correlation functions satisfy the appropriate Ward identities in the continuum limit. To carry out this program, we define the theory on the lattice by the Wilson-plaquette and the $O(a)$-improved Wilson actions for gluons and quarks respectively. The discretization of the space-time entails that (i) the irreducible nonet representation of the SO($4$) group splits into a triplet and a sextet irreducible representations of the hypercubic group, and (ii) for each multiplet non-perturbative determinations of the the gluonic and fermionic renormalization constants are required. The bare gluonic components of the energy-momentum tensor are defined via the clover discretization of the field strength tensor, while the fermionic ones are discretized by appropriate combinations of symmetric covariant derivatives. Either for the triplet or the sextet representations, the two independent renormalization constants are then fixed non-perturbatively by imposing discretized versions of continuum Ward identities for one-point correlation functions in the presence of shifted boundary conditions and an imaginary chemical potential. The non-perturbative calculation is then carried out by Monte Carlo simulations, and the resulting renormalization constants are determined with a final accuracy of a few percent for values of the bare coupling constant squared in the range $0 \leq g_0^2\leq 0.96$.

hep-lat↗

arXiv2026

QCD Equation of State at very high temperature: computational strategy, simulations and data analysis

We present a detailed account of the theoretical progress and the computational strategy that led to the non-perturbative determination of the QCD Equation of State at temperatures up to the electroweak scale reported in [Phys. Rev. Lett. 134, 201904 (2025)]. The two key ingredients that make such a calculation feasible with controlled uncertainties are: (i) the definition of lines of constant physics through the running of a non-perturbatively defined finite-volume coupling across a wide range of energy scales, and (ii) the use of shifted boundary conditions which allow a direct determination of the entropy density thus without the need for a zero-temperature subtraction. Considering the case of QCD with $N_f =3$ massless flavours in the temperature interval between 3 GeV and 165 GeV, we describe the numerical strategy based on integrating in the bare coupling and quark mass, the perturbative improvement of lattice observables, the optimization of numerical simulations, and the continuum extrapolation. Extensive consistency checks, including finite-volume and topological-freezing effects, confirm the robustness of the method. The final results have a relative accuracy of about $1\%$ or better, and the errors are dominated by the statistical fluctuations of the Monte Carlo ensembles. We also compare our non-perturbative results with predictions from standard and hard thermal loop perturbation theory showing that at the level of $\%$-precision contributions beyond those known, including non-perturbative ones due to ultrasoft modes, are relevant up to the highest temperatures explored. The methodological framework is general and readily applicable to QCD with four and five massive quark flavours and to other thermal observables, paving the way for systematic non-perturbative studies of thermal QCD at very high temperatures.

hep-lat↗

arXiv2025

Equation of State of QCD with $N_f=3$ flavours up to the electroweak scale

The Equation of State of Quantum Chromodynamics with $N_f=3$ flavours is determined non-perturbatively with a precision of about $0.5\%-1.0\%$ in the range of temperatures between 3 GeV and 165 GeV. The computation is carried out by numerical simulations of the gauge theory discretized on the lattice. At each given temperature the entropy density is computed at several lattice spacings in order to extrapolate the results to the continuum limit. The pressure and energy density are then determined by integrating the entropy density with respect to the temperature. The numerical data show a linear behaviour in the strong coupling constant squared, which points to the Stefan-Boltzmann limit at infinite temperature. They are also compatible with the known perturbative formula supplemented by higher order terms in the coupling constant, containing non-perturbative contributions. This parametrization describes well our data together with those present in the literature down to 500 MeV.

hep-lat↗

arXiv2025

QCD Equation of State with $N_f=3$ Flavors up to the Electroweak Scale

The equation of state of Quantum Chromodynamics with $N_f=3$ flavors is determined non-perturbatively 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 à la 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.

hep-lat↗

arXiv2025

The Equation of State of QCD up to very high temperatures

We present the non-perturbative computation of the entropy density in QCD for temperatures ranging from 3 GeV up to the electro-weak scale, using $N_f=3$ flavours of massless O$(a)$-improved Wilson fermions. We adopt a new strategy designed to be computationally efficient and based on formulating thermal QCD in a moving reference frame, where the fields satisfy shifted boundary conditions in the temporal direction and periodic boundary conditions along the spatial ones. In this setup the entropy density can be computed as the derivative of the free-energy density with respect to the shift parameter. For each physical temperature, we perform Monte Carlo simulations at four values of the lattice spacing in order to extrapolate the numerical data of the entropy density to the continuum limit. We achieve a final accuracy of approximatively $0.5$-$1.0\%$ and our results are compared with predictions from high-temperature perturbation theory.

hep-lat↗

arXiv2025

The strength of the interaction between quarks and gluons

Modern particle physics experiments, e.g. at the Large Hadron Collider (LHC) at CERN, crucially depend on the precise description of the scattering processes in terms of the known fundamental forces. This is limited by our current understanding of the strong nuclear force, as quantified by the strong coupling, $α_s$, between quarks and gluons. Relating $α_s$ to experiments poses a major challenge as the strong interactions lead to the confinement of quarks and gluons inside hadronic bound states. At high energies, however, the strong interactions become weaker ("asymptotic freedom") and thus amenable to an expansion in powers of the coupling. Attempts to relate both regimes usually rely on modeling of the bound state problem in one way or another. Using large scale numerical simulations of a first principles formulation of Quantum Chromodynamics on a space-time lattice, we have carried out a model-independent determination of $α_s$ with unprecedented precision. The uncertainty, about half that of all other results combined, originates predominantly from the statistical Monte Carlo evaluation and has a clear probabilistic interpretation. The result for $α_s$ describes a variety of physical phenomena over a wide range of energy scales. If used as input information, it will enable significantly improved analyses of many high energy experiments, by removing an important source of theoretical uncertainty. This will increase the likelihood to uncover small effects of yet unknown physics, and enable stringent precision tests of the Standard Model. In summary, this result boosts the discovery potential of the LHC and future colliders, and the methods developed in this work pave the way for even higher precision in the future.

hep-ph↗

arXiv2024

Non-perturbative thermal QCD at very high temperatures: computational strategy and hadronic screening masses

We discuss a recently introduced strategy to study non-perturbatively thermal QCD up to temperatures of the order of the electro-weak scale, combining step scaling techniques and shifted boundary conditions. The former allow to renormalize the theory for a range of scales which spans several orders of magnitude with a moderate computational cost. Shifted boundary conditions remove the need for the zero temperature subtraction in the Equation of State. As a consequence, the simulated lattices do not have to accommodate two very different scales, the pion mass and the temperature, at the very same spacing. Effective field theory arguments guarantee that finite volume effects can be kept under control safely. With this strategy the first computation of the hadronic screening spectrum has been carried out over more than two orders of magnitude in the temperature, from $T\sim 1$ GeV up to $\sim 160$ GeV. This study is complemented with the first quantitative computation of the baryonic screening mass at next-to-leading order in the three-dimensional effective theory describing QCD at high temperatures. Both for the mesonic and the baryonic screening masses, the known leading behaviour in the coupling constant is found to be not sufficient to explain the non-perturbative data over the entire range of temperatures. These findings shed further light on the limited applicability of the perturbative approach at finite temperature, even at the electro-weak scale.

hep-lat↗

arXiv2024

Non-perturbative thermal QCD at very high temperatures

We present a recently introduced strategy to study non-perturbatively thermal QCD up to temperatures of the order of the electro-weak scale, combining step scaling techniques and shifted boundary conditions. The former allow to renormalize the theory for a range of scales which spans several orders of magnitude with a moderate computational cost. Shifted boundary conditions avoid the need for the zero temperature subtraction in the Equation of State. As a consequence, the simulated lattices do not have to accommodate two very different scales, the pion mass and the temperature. Effective field theory arguments guarantee that finite volume effects can be kept under control safely. As a first application of this strategy, we present the results of the computation of the hadronic screening spectrum in QCD with $N_f=3$ flavours of massless quarks for temperatures from $T\sim 1$ GeV up to $\sim 160$ GeV.

hep-lat↗

arXiv2024

Heavy Wilson Quarks and O($a$) Improvement: Nonperturbative Results for $b_{\rm g}$

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, $\tilde{g}_0^2 = g_0^2(1 + b_{\rm g} am_{\rm q})$, and the determination of $b_{\rm g}(g_0^2)$, 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 $α_s(m_Z)$. However, the improvement conditions and some insight into systematic effects may prove useful in other contexts, too.

hep-lat↗

arXiv2024

A non-perturbative determination of $b_{\rm g}$

Close to the continuum limit, lattice QCD with mass-degenerate Wilson quarks can be described by Symanzik's effective continuum action, which contains the dimension 5 operator, $m\,{\rm tr}(F_{μν}F_{μν})$. Its effect can be eliminated by an O($am_{\rm q}$) rescaling of the bare lattice coupling constant. Until recently, the corresponding improvement coefficient, $b_{\rm g}$, was only known perturbatively to 1-loop order and an estimate of the remaining uncertainty is the dominant systematic error in the ALPHA collaboration's recent determination of $α_s(m_Z)$ with the decoupling method. To remove this error we have determined $b_{\rm g}$ non-perturbatively for the corresponding parameter range. We here briefly review improvement conditions for $b_{\rm g}$, perform a perturbative test and report on our non-perturbative results for $b_{\rm g}$.

hep-lat↗

arXiv2023

Progresses on high-temperature QCD: Equation of State and energy-momentum tensor

We present first non-perturbative results for the renormalization constants of the QCD energy-momentum tensor, based on the framework of thermal QCD with shifted and twisted (for quarks only) boundary conditions in the compact direction. We also show preliminary results for the entropy density obtained with the very same numerical strategy. This opens the way to the determination of the QCD Equation of State up to very high temperatures.

hep-lat↗

arXiv2023

Nonperturbative running of the tensor operator for $N_\rm{f}=3$ QCD from the chirally rotated Schrödinger Functional

We study the Renormalisation Group (RG) running of the non-singlet tensor operator, for $N_\mathrm{\scriptstyle f}=3$ QCD with Wilson fermions in a mixed action setup, with standard Schrödinger Functional (SF) boundary conditions for sea quarks and chirally rotated Schrödinger Functional ($χ$SF) boundary conditions for valence quarks. Based on a recursive finite-size scaling technique we compute non-perturbatively 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 $T^{\mathrm{RGI}}/[ T(μ_{\mathrm{had}})]_{\scriptstyle \rm R}$, where the numerator is the scale independent (Renormalisation Group Invariant - RGI) tensor operator and the denominator is its renormalised counterpart at a hadronic scale $μ_{\mathrm{had}} = 233(8)$~MeV in a given scheme. We determine the step-scaling function in four distinct renormalisation schemes. We also compute the renormalisation parameters of these schemes at $μ_{\mathrm{had}}$ which, combined with the RG-running factor, gives the scheme-independent quantity $Z^{\mathrm{RGI}}_{\mathrm T}(g_0^2)$ 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 $N_\mathrm{\scriptstyle f}=2+1$ 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.

hep-lat↗

arXiv2022

QCD mesonic screening masses and restoration of chiral symmetry at high T

We present a strategy to study QCD non-perturbatively on the lattice at very high temperatures. This strategy exploits a non-perturbative, finite-volume, definition of the strong coupling constant to renormalize the theory. As a first application we compute the flavor non-singlet mesonic screening masses in a wide range of temperature, from $T\sim 1 $ GeV up to $\sim 160 $ GeV with three flavors in the chiral limit of QCD. Our results show very interesting features of the screening spectrum at very high temperatures. On one hand the mass splitting between the vector and the pseudoscalar screening masses is clearly visible up to the electroweak scale and cannot be explained by the known 1-loop perturbative result. On the other hand the restoration of chiral symmetry manifests itself through the degeneracy of the pseudoscalar and the scalar channels and of the vector and the axial-vector ones. This degeneracy pattern is the one expected by Ward identities associated to the presence of chiral symmetry.

hep-lat↗

arXiv2022

Thermal QCD for non-perturbative renormalization of composite operators

We present our progresses in the use of the non-perturbative renormalization framework based on considering QCD at finite temperature with shifted and twisted (for quarks only) boundary conditions in the compact direction. We report our final results in the application of this method for the non-perturbative renormalization of the flavor-singlet local vector current. We then discuss the more challenging case of the renormalization of the energy-momentum tensor, and show preliminary results on the relevant one-point functions for the computation of the renormalization constants of its non-singlet components.

hep-lat↗

arXiv2022

QCD mesonic screening masses up to high temperatures

We discuss a strategy to study non-perturbatively QCD up to very high temperatures by Monte Carlo simulations on the lattice. It allows not only the thermodynamic properties of the theory but also other interesting thermal features to be investigated. As a first concrete application, we compute the flavour non-singlet mesonic screening masses and we present the results of Monte Carlo simulations at 12 temperatures covering the range from T $\sim$ 1 GeV up to $\sim$ 160 GeV in the theory with three massless quarks. On the one side, 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, and, on the other side, we observe a clear splitting between the vector and the pseudoscalar screening masses up to the highest investigated temperature. A comparison with the high-temperature effective theory shows that the known one-loop order in the perturbative expansion does not provide a satisfactory description of the non-perturbative data up to the highest temperature considered.

hep-lat↗

arXiv2022

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

We compute non-perturbatively the renormalization constant of the flavour-singlet local vector current $Z_V$ 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 $Z_V$ 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 $Z_V$ with an accuracy of about 8\textperthousand\, for values of the inverse bare coupling constant $β=6/g_0^2$ in the range $5.3 \leq β\leq 11.5$.

hep-lat↗

arXiv2022

RG-running of the tensor currents for $N_f$ =3 QCD in a $χSF$ setup

We present the preliminary results of the non-perturbative running of the flavour non-singlet tensor operator in the high-energy range $2~\rm{GeV}\lesssim μ\lesssim 128~\rm{GeV}$ in $N_f=3$ massless QCD, comparing four different definitions of the renormalisation constant. We use the configuration ensembles of arXiv:1802.05243 and arXiv:1607.06423, subject to Schrödinger functional (SF) boundary conditions, and valence quarks with chirally rotated Schrödinger functional ($χ$SF) boundary conditions. Provided that boundary counterterms have been appropriately tuned, this results in O($a$) improvement of the tensor operator, without the need of a dimension-4 Symanzik counterterm (proportional to $c_T$).

hep-lat↗

arXiv2022

Determination of $α_s(m_Z)$ by the non-perturbative decoupling method

We present the details and first results of a new strategy for the determination of $α_s(m_Z)$. By simultaneously decoupling 3 fictitious heavy quarks we establish a relation between the $Λ$-parameters of three-flavor QCD and pure gauge theory. Very precise recent results in the pure gauge theory can thus be leveraged to obtain the three-flavour $Λ$-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, $Λ^{(3)}_{\bar{\rm MS}} = 336(12)\, {\rm MeV}$, which translates to $α_s(m_Z) = 0.11823(84)$. This is compatible with both the FLAG average and the previous ALPHA result, 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.

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