We present our study of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi><mo stretchy="false">(</mo><mn>770</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>K</mi><mo>*</mo></msup><mo stretchy="false">(</mo><mn>892</mn><mo stretchy="false">)</mo></math> resonances from lattice quantum chromodynamics (QCD) employing domain-wall fermions at physical quark masses. We determine the finite-volume energy spectrum in various momentum frames and obtain phase-shift parametrizations via the Lüscher formalism and as a final step the complex resonance poles of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>π</mi><mi>π</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>K</mi><mi>π</mi></math> elastic scattering amplitudes via an analytical continuation of the models. By sampling a large number of representative sets of underlying energy-level fits, we also assign a systematic uncertainty to our final results. This is a significant extension to data-driven analysis methods that have been used in lattice QCD to date, due to the two-step nature of the formalism. Our final pole positions, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>M</mi><mo>+</mo><mi>i</mi><mi mathvariant="normal">Γ</mi><mo>/</mo><mn>2</mn></math>, with all statistical and systematic errors exposed, are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>M</mi></mrow><mrow><msup><mrow><mi>K</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></msub><mo>=</mo><mn>893</mn><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>54</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Γ</mi><msup><mi>K</mi><mo>*</mo></msup></msub><mo>=</mo><mn>51</mn><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>11</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>K</mi><mo>*</mo></msup><mo stretchy="false">(</mo><mn>892</mn><mo stretchy="false">)</mo></math> resonance and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mo>=</mo><mn>796</mn><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>15</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>48</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Γ</mi><mi>ρ</mi></msub><mo>=</mo><mn>192</mn><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>28</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>12</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi><mo stretchy="false">(</mo><mn>770</mn><mo stretchy="false">)</mo></math> resonance. The four differently grouped sources of uncertainties are, in the order of occurrence: statistical, data-driven systematic, an estimation of systematic effects beyond our computation (dominated by the fact that we employ a single lattice spacing), and the error from the scale-setting uncertainty on our ensemble.

United States

United Kingdom

Peter Boyle

CERN

Felix Erben

Vera Gülpers

Maxwell T. Hansen

Fabian Joswig

Michael Marshall

Nelson Pitanga Lachini

Japan

Antonin Portelli

Physical-mass calculation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi><mo stretchy="false">(</mo><mn>770</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>K</mi><mo>*</mo></msup><mo stretchy="false">(</mo><mn>892</mn><mo stretchy="false">)</mo></math> resonance parameters via <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>π</mi><mi>π</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>K</mi><mi>π</mi></math> scattering amplitudes from lattice QCD

We present the first ab initio calculation at physical quark masses of scattering amplitudes describing the lightest pseudoscalar mesons interacting via the strong force in the vector channel. Using lattice quantum chromodynamics, we postdict the defining parameters for two short-lived resonances, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi><mo stretchy="false">(</mo><mn>770</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>K</mi></mrow><mrow><mo>*</mo></mrow></msup><mo stretchy="false">(</mo><mn>892</mn><mo stretchy="false">)</mo></mrow></math>, which manifest as complex energy poles in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>π</mi><mi>π</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>K</mi><mi>π</mi></math> scattering amplitudes, respectively. The calculation proceeds by first computing the finite-volume energy spectrum of the two-hadron systems and then determining the amplitudes from the energies using the Lüscher formalism. The error budget includes a data-driven systematic error, obtained by scanning possible fit ranges and fit models to extract the spectrum from Euclidean correlators, as well as the scattering amplitudes from the latter. The final results, obtained by analytically continuing multiple parametrizations into the complex energy plane, are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mo>=</mo><mn>796</mn><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>50</mn><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi mathvariant="normal">Γ</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mo>=</mo><mn>192</mn><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>31</mn><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>M</mi></mrow><mrow><msup><mrow><mi>K</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></msub><mo>=</mo><mn>893</mn><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>54</mn><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi mathvariant="normal">Γ</mi></mrow><mrow><msup><mrow><mi>K</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></msub><mo>=</mo><mn>51</mn><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>11</mn><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>, where the subscript indicates the resonance and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>M</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Γ</mi></math> stand for the mass and width, respectively, and where the first bracket indicates the statistical and the second bracket the systematic uncertainty.

Light and Strange Vector Resonances from Lattice QCD at Physical Quark Masses

Hadronic matrix elements involving tensor currents play an important rôle in decays that allow to probe the consistency of the Standard Model via precision lattice QCD calculations. The non-singlet tensor current is a scale-dependent (anomalous) quantity. We fully resolve its renormalisation group (RG) running in the continuum by carrying out a recursive finite-size scaling technique. In this way ambiguities due to a perturbative RG running and matching to lattice data at low energies are eliminated. We provide the total renormalisation factor at a hadronic scale of 233 MeV, which converts the bare current into its RG-invariant form.

Germany

Leonardo Chimirri

Ireland

Patrick Fritzsch

Jochen Heitger

Italy

Marco Panero

Spain

Carlos Pena

David Preti

Non-perturbative renormalisation and improvement of non-singlet tensor currents in N f = 3 QCD

We present results for the leptonic decay constants of the D and Ds mesons from N f = 2 + 1 lattice QCD. We employ a set of 49 high statistics gauge ensembles generated by the Coordinated Lattice Simulations (CLS) effort utilising non-perturbatively improved Wilson fermions and the tree-level Symanzik improved gauge action at six values of the lattice spacing in the range a = 0.098 fm down to a = 0.039 fm, with pion masses varying from around 420 MeV down to below the physical point. The ensembles lie on three trajectories in the quark mass plane, two trajectories intersecting close to the physical quark mass point and the third one approaching the SU(3) chiral limit, enabling tight control of the light and strange quark mass dependence. We obtain <math> <msub> <mi>f</mi> <msub> <mi>D</mi> <mi>s</mi> </msub> </msub> </math> $$ {f}_{{\textrm{D}}_{\textrm{s}}} $$ = 246.8(1.3) MeV, f D = 208.4(1.5) MeV and <math> <msub> <mi>f</mi> <msub> <mi>D</mi> <mi>s</mi> </msub> </msub> </math> $$ {f}_{{\textrm{D}}_{\textrm{s}}} $$ /f D = 1.1842(36), where the precision of our results is mostly limited by the determination of the scale.

Simon Kuberski

Sara Collins

Wolfgang Söldner

D and Ds decay constants in N f = 2 + 1 QCD with Wilson fermions

Pia Petrak

Anastassios Vladikas

Erratum to: Ratio of flavour non-singlet and singlet scalar density renormalisation parameters in $$N_\mathrm {f}=3$$ <math> <mrow> <msub> <mi>N</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>3</mn> </mrow> </math> QCD with Wilson quarks

We determine non-perturbatively the normalisation factor $$r_{\mathrm{m}}\equiv Z_{\mathrm{S}}/Z_{\mathrm{S}}^{0}$$ <math> <mrow> <msub> <mi>r</mi> <mi>m</mi> </msub> <mo>≡</mo> <msub> <mi>Z</mi> <mi>S</mi> </msub> <mo>/</mo> <msubsup> <mi>Z</mi> <mrow> <mi>S</mi> </mrow> <mn>0</mn> </msubsup> </mrow> </math> , where $$Z_{\mathrm{S}}$$ <math> <msub> <mi>Z</mi> <mi>S</mi> </msub> </math> and $$Z_{\mathrm{S}}^{0}$$ <math> <msubsup> <mi>Z</mi> <mrow> <mi>S</mi> </mrow> <mn>0</mn> </msubsup> </math> are the renormalisation parameters of the flavour non-singlet and singlet scalar densities, respectively. This quantity is required in the computation of quark masses with Wilson fermions and for instance the renormalisation of nucleon matrix elements of scalar densities. Our calculation involves simulations of finite-volume lattice QCD with the tree-level Symanzik-improved gauge action, $$N_{\mathrm{f}}= 3$$ <math> <mrow> <msub> <mi>N</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>3</mn> </mrow> </math> mass-degenerate $${\mathrm{O}}(a)$$ <math> <mrow> <mi>O</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </math> improved Wilson fermions and Schrödinger functional boundary conditions. The slope of the current quark mass, as a function of the subtracted Wilson quark mass is extracted both in a unitary setup (where nearly chiral valence and sea quark masses are degenerate) and in a non-unitary setup (where all valence flavours are chiral and the sea quark masses are small). These slopes are then combined with $$Z \equiv Z_{\mathrm{P}}/(Z_{\mathrm{S}}Z_{\mathrm{A}})$$ <math> <mrow> <mi>Z</mi> <mo>≡</mo> <msub> <mi>Z</mi> <mi>P</mi> </msub> <mo>/</mo> <mrow> <mo>(</mo> <msub> <mi>Z</mi> <mi>S</mi> </msub> <msub> <mi>Z</mi> <mi>A</mi> </msub> <mo>)</mo> </mrow> </mrow> </math> in order to obtain $$r_{\mathrm{m}}$$ <math> <msub> <mi>r</mi> <mi>m</mi> </msub> </math> . A novel chiral Ward identity is employed for the calculation of the normalisation factor Z. Our results cover the range of gauge couplings corresponding to lattice spacings below $$0.1\,$$ <math> <mrow> <mn>0.1</mn> <mspace width="0.166667em"></mspace> </mrow> </math> fm, for which $$N_{\mathrm{f}}= 2+1$$ <math> <mrow> <msub> <mi>N</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mn>1</mn> </mrow> </math> QCD simulations in large volumes with the same lattice action are typically performed.

Ratio of flavour non-singlet and singlet scalar density renormalisation parameters in $$N_\mathrm {f}=3$$ <math> <mrow> <msub> <mi>N</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>3</mn> </mrow> </math> QCD with Wilson quarks

We present a determination of the charm quark mass in lattice QCD with three active quark flavours. The calculation is based on PCAC masses extracted from N f = 2 + 1 flavour gauge field ensembles at five different lattice spacings in a range from 0.087 fm down to 0.039 fm. The lattice action consists of the O(a) improved Wilson-clover action and a tree-level improved Symanzik gauge action. Quark masses are non-perturbatively O(a) improved employing the Symanzik-counterterms available for this discretisation of QCD. To relate the bare mass at a specified low-energy scale with the renormalisation group invariant mass in the continuum limit, we use the non-pertubatively known factors that account for the running of the quark masses as well as for their renormalisation at hadronic scales. We obtain the renormalisation group invariant charm quark mass at the physical point of the three-flavour theory to be M c = 1486(21) MeV. Combining this result with five-loop perturbation theory and the corresponding decoupling relations in the <math> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </math> $$ \overline{\mathrm{MS}} $$ scheme, one arrives at a result for the renormalisation group invariant charm quark mass in the four-flavour theory of M c(N f = 4) = 1548(23) MeV, where effects associated with the absence of a charmed, sea quark in the non-perturbative evaluation of the QCD path integral are not accounted for. In the <math> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </math> $$ \overline{\mathrm{MS}} $$ scheme, and at finite energy scales conventional in phenomenology, we quote <math> <msubsup> <mi>m</mi> <mi>c</mi> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </msubsup> </math> $$ {m}_{\mathrm{c}}^{\overline{\mathrm{MS}}} $$ ( <math> <msubsup> <mi>m</mi> <mi>c</mi> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </msubsup> </math> $$ {m}_{\mathrm{c}}^{\overline{\mathrm{MS}}} $$ ; N f = 4) = 1296(19) MeV and <math> <msubsup> <mi>m</mi> <mi>c</mi> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </msubsup> </math> $$ {m}_{\mathrm{c}}^{\overline{\mathrm{MS}}} $$ (3 GeV; N f = 4) = 1007(16) MeV for the renormalised charm quark mass.

Determination of the charm quark mass in lattice QCD with 2 + 1 flavours on fine lattices

We present the results of a non-perturbative determination of the improvement coefficient $$c_\mathrm{V}$$ <math> <msub> <mi>c</mi> <mi>V</mi> </msub> </math> and the renormalisation factor $$Z_\mathrm{V}$$ <math> <msub> <mi>Z</mi> <mi>V</mi> </msub> </math> , which define the renormalised vector current in three-flavour $$\mathrm{O}(a)$$ <math> <mrow> <mi>O</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </math> improved lattice QCD with Wilson quarks and tree-level Symanzik-improved gauge action. In case of the improvement coefficient, we consider both lattice descriptions of the vector current, the local as well as the conserved (i.e., point-split) one. Our improvement and normalisation conditions are based on massive chiral Ward identities and numerically evaluated in the Schrödinger functional setup, which allows to eliminate finite quark mass effects in a controlled way. In order to ensure a smooth dependence of the renormalisation constant and improvement coefficients on the bare gauge coupling, our computation proceeds along a line of constant physics, covering the typical range of lattice spacings $$0.04\,\mathrm{fm}\lesssim a\lesssim 0.1\,\mathrm{fm}$$ <math> <mrow> <mn>0.04</mn> <mspace width="0.166667em"></mspace> <mi>fm</mi> <mo>≲</mo> <mi>a</mi> <mo>≲</mo> <mn>0.1</mn> <mspace width="0.166667em"></mspace> <mi>fm</mi> </mrow> </math> that is useful for phenomenological applications. Especially for the improvement coefficient of the local vector current, we report significant differences between the one-loop perturbative estimates and our non-perturbative results.

The renormalised $$\mathrm{O}(a)$$ <math> <mrow> <mi>O</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </math> improved vector current in three-flavour lattice QCD with Wilson quarks

We derive chiral Ward identities for lattice QCD with Wilson quarks and $$N_{\mathrm{f}}\ge 3$$ <math><mrow><msub><mi>N</mi><mi>f</mi></msub><mo>≥</mo><mn>3</mn></mrow></math> flavours, on small lattices with Schrödinger functional boundary conditions and vanishingly small quark masses. These identities relate the axial variation of the non-singlet pseudoscalar density to the scalar one, thus enabling the non-perturbative determination of the scale-independent ratio $$Z_{\mathrm {S}}/Z_{\mathrm {P}}$$ <math><mrow><msub><mi>Z</mi><mi>S</mi></msub><mo>/</mo><msub><mi>Z</mi><mi>P</mi></msub></mrow></math> of the renormalisation parameters of these operators. We obtain results for $$N_{\mathrm{f}}=3$$ <math><mrow><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>3</mn></mrow></math> QCD with tree-level Symanzik-improved gluons and Wilson-Clover quarks, for bare gauge couplings which cover the typical range of large-volume $$N_{\mathrm{f}}= 2+1$$ <math><mrow><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> simulations with Wilson fermions at lattice spacings below $$0.1\,$$ <math><mrow><mn>0.1</mn><mspace width="0.166667em"></mspace></mrow></math> fm. The precision of our results varies from 0.3 to 1%, except for the coarsest lattice, where it is 2%. We discuss how the $$Z_{\mathrm {S}}/Z_{\mathrm {P}}$$ <math><mrow><msub><mi>Z</mi><mi>S</mi></msub><mo>/</mo><msub><mi>Z</mi><mi>P</mi></msub></mrow></math> ratio can be used in the non-perturbative calculations of $${\mathrm {O}}(a)$$ <math><mrow><mi>O</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></math> improved renormalised quark masses.

Ward identity determination of $$Z_{\mathrm {S}}/Z_{\mathrm {P}}$$ <math><mrow><msub><mi>Z</mi><mi>S</mi></msub><mo>/</mo><msub><mi>Z</mi><mi>P</mi></msub></mrow></math> for $$N_{\mathrm {f}}=3$$ <math><mrow><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>3</mn></mrow></math> lattice QCD in a Schrödinger functional setup