We present the results of a first-principles theoretical study of the inclusive semileptonic decays of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mi>s</mi></msub></math> meson. We performed a state-of-the-art lattice QCD calculation using the gauge ensembles produced by the Extended Twisted Mass Collaboration with dynamical light, strange and charm quarks with physical masses and employed the so-called Hansen-Lupo-Tantalo (HLT) method to extract the decay rate and the first two lepton-energy moments from the relevant Euclidean correlators. We have carefully taken into account all sources of systematic errors, including the ones associated with the continuum and infinite-volume extrapolations and with the HLT spectral reconstruction method. We obtained results in very good agreement with the currently available experimental determinations and with a total accuracy at the few-percent level, of the same order of magnitude of the experimental error. Our total error is dominated by the lattice QCD simulations statistical uncertainties and is certainly improvable. From the results presented and thoroughly discussed in this paper we conclude that it is nowadays possible to study heavy mesons inclusive semileptonic decays on the lattice at a phenomenologically relevant level of accuracy. The phenomenological implications of our physical results are the subject of a companion Letter [A. De Santis et al., Inclusive semileptonic decays of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mi>s</mi></msub></math> meson: Lattice QCD confronts experiments, Phys. Rev. Lett. 135, 121901 (2025).].

Germany

Alessandro De Santis

Italy

Antonio Evangelista

Roberto Frezzotti

Giuseppe Gagliardi

Paolo Gambino

Marco Garofalo

Christiane Franziska Groß

Bartosz Kostrzewa

Vittorio Lubicz

Francesca Margari

Finland

Marco Panero

Francesco Sanfilippo

Silvano Simula

United Kingdom

Antonio Smecca

Nazario Tantalo

Carsten Urbach

Inclusive semileptonic decays of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mi>s</mi></msub></math> meson: A first-principles lattice QCD calculation

We present the results of a first-principles theoretical study of the inclusive semileptonic decays of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mi>s</mi></msub></math> meson. We performed a state-of-the-art lattice QCD calculation by taking into account all sources of systematic errors. A detailed discussion of our lattice calculation, demonstrating that inclusive semileptonic decays can nowadays be studied on the lattice at a phenomenologically relevant level of accuracy, is the subject of a companion paper [A. De Santis et al., Inclusive semileptonic decays of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mi>s</mi></msub></math> meson: A first-principles lattice QCD calculation, Phys. Rev. D 112, 054503 (2025)]. Here, we focus on the phenomenological implications of our results. Using the current best estimates of the relevant Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, our theoretical predictions for the decay rate and for the first two lepton-energy moments are in very good agreement with the corresponding experimental measurements. We also argue that, while the inclusive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mi>s</mi></msub></math> channel is not yet competitive with the exclusive channels in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">|</mo><msub><mi>V</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub><mo stretchy="false">|</mo></math> determination, the situation can be significantly improved in the near future.

Inclusive Semileptonic Decays of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mi>s</mi></msub></math> Meson: Lattice QCD Confronts Experiments

We present the full decomposition of the momentum fraction carried by quarks and gluons in the pion and the kaon. We employ three gauge ensembles generated with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></math> Wilson twisted-mass clover-improved fermions at the physical quark masses. For both mesons we perform a continuum extrapolation directly at the physical pion mass, which allows us to determine for the first time the momentum decomposition at the physical point. We find that the total momentum fraction carried by quarks is 0.575(79) and 0.683(50) and by gluons 0.402(53) and 0.422(67) in the pion and in the kaon, respectively, in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>MS</mi><mo stretchy="true">¯</mo></mover></math> scheme and at the renormalization scale of 2 GeV. Having computed both the quark and gluon contributions in the continuum limit, we find that the momentum sum is 0.984(89) for the pion and 1.13(11) for the kaon, verifying the momentum sum rule.

Cyprus

Constantia Alexandrou

Simone Bacchio

United States

Martha Constantinou

Joseph Delmar

Jacob Finkenrath

Marcus Petschlies

Luis Alberto Rodriguez Chacon

Gregoris Spanoudes

Fernanda Steffens

Switzerland

Urs Wenger

Quark and Gluon Momentum Fractions in the Pion and in the Kaon

We present a lattice determination of the inclusive decay rate of the process <math><mrow><mi>τ</mi><mo>↦</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>u</mi><mi>s</mi></mrow></msub><msub><mrow><mi>ν</mi></mrow><mrow><mi>τ</mi></mrow></msub></mrow></math> in which the <math><mi>τ</mi></math> lepton decays into a generic hadronic state <math><msub><mi>X</mi><mrow><mi>u</mi><mi>s</mi></mrow></msub></math> with <math><mover><mi>u</mi><mo>¯</mo></mover><mi>s</mi></math> flavor quantum numbers. Our results have been obtained in <math><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math> isosymmetric QCD with full nonperturbative accuracy, without any operator product expansion approximation and, except for the presently missing long-distance isospin-breaking corrections, include a solid estimate of all sources of theoretical uncertainties. This has been possible by using the Hansen-Lupo-Tantalo method [M. Hansen et al., Phys. Rev. D 99, 094508 (2019)] that we have already successfully applied [A. Evangelista et al., Phys. Rev. D 108, 074513 (2023)] to compute the inclusive decay rate of the process <math><mi>τ</mi><mo>↦</mo><msub><mi>X</mi><mrow><mi>u</mi><mi>d</mi></mrow></msub><msub><mi>ν</mi><mi>τ</mi></msub></math> in the <math><mover><mi>u</mi><mo>¯</mo></mover><mi>d</mi></math> flavor channel. By combining our first-principles theoretical results with the presently available experimental data, we extract the Cabibbo-Kobayashi-Maskawa matrix element <math><mo>|</mo><msub><mi>V</mi><mrow><mi>u</mi><mi>s</mi></mrow></msub><mo>|</mo></math>, the Cabibbo angle, with a 0.9% accuracy, dominated by the experimental error.

Simone Romiti

Inclusive Hadronic Decay Rate of the <math><mi>τ</mi></math> Lepton from Lattice QCD: The <math><mover><mi>u</mi><mo>¯</mo></mover><mi>s</mi></math> Flavor Channel and the Cabibbo Angle

We compute the nucleon axial and pseudoscalar form factors using three <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></math> twisted-mass fermion ensembles with all quark masses tuned to approximately their physical values. The values of the lattice spacings of these three physical point ensembles are 0.080, 0.068, and 0.057 fm and spatial sizes 5.1, 5.44, and 5.47 fm, respectively, yielding <math><msub><mi>m</mi><mi>π</mi></msub><mi>L</mi><mo>&gt;</mo><mn>3.6</mn></math>. Convergence to the ground-state matrix elements is assessed using multistate fits. We study the momentum dependence of the three form factors and check the partially conserved axial-vector current (PCAC) hypothesis and the pion pole dominance (PPD). We show that in the continuum limit, the PCAC and PPD relations are satisfied. We also show that the Goldberger-Treimann relation is approximately fulfilled and determine the Goldberger-Treiman discrepancy. Our final results are <math><msub><mi>g</mi><mi>A</mi></msub><mo>=</mo><mn>1.245</mn><mo>(</mo><mn>28</mn><mo>)</mo><mo>(</mo><mn>14</mn><mo>)</mo></math> for the nucleon axial charge, <math><mo>⟨</mo><msubsup><mi>r</mi><mi>A</mi><mn>2</mn></msubsup><mo>⟩</mo><mo>=</mo><mn>0.339</mn><mo>(</mo><mn>48</mn><mo>)</mo><mo>(</mo><mn>06</mn><mo>)</mo><msup><mtext>fm</mtext><mn>2</mn></msup></math> for the axial radius, <math><msub><mi>g</mi><mrow><mi>π</mi><mi>N</mi><mi>N</mi></mrow></msub><mo>≡</mo><munder><mo>lim</mo><mrow><msup><mi>Q</mi><mn>2</mn></msup><mo>→</mo><mo>−</mo><msubsup><mi>m</mi><mi>π</mi><mn>2</mn></msubsup></mrow></munder><msub><mi>G</mi><mrow><mi>π</mi><mi>N</mi><mi>N</mi></mrow></msub><mo>(</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo><mo>=</mo><mn>13.25</mn><mo>(</mo><mn>67</mn><mo>)</mo><mo>(</mo><mn>69</mn><mo>)</mo></math> for the pion-nucleon coupling constant, and <math><msub><mi>G</mi><mi>P</mi></msub><mo>(</mo><mn>0.88</mn><msubsup><mi>m</mi><mi>μ</mi><mn>2</mn></msubsup><mo>)</mo><mo>≡</mo><msubsup><mi>g</mi><mi>P</mi><mo>*</mo></msubsup><mo>=</mo><mn>8.99</mn><mo>(</mo><mn>39</mn><mo>)</mo><mo>(</mo><mn>49</mn><mo>)</mo></math> for the induced pseudoscalar form factor at the muon capture point.

Giannis Koutsou

Nucleon axial and pseudoscalar form factors using twisted-mass fermion ensembles at the physical point

We calculate the double-virtual <math><mrow><mi>η</mi><mo>→</mo><msup><mrow><mi>γ</mi></mrow><mrow><mo>*</mo></mrow></msup><msup><mrow><mi>γ</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> transition form factor <math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>η</mi><mo>→</mo><msup><mrow><mi>γ</mi></mrow><mrow><mo>*</mo></mrow></msup><msup><mrow><mi>γ</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></msub><mo>(</mo><msubsup><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo></mrow></math> from first principles using a lattice QCD simulation with <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></math> quark flavors at the physical pion mass and at one lattice spacing and volume. The kinematic range covered by our calculation is complementary to the one accessible from experiment and is relevant for the <math><mi>η</mi></math>-pole contribution to the hadronic light-by-light scattering in the anomalous magnetic moment <math><msub><mi>a</mi><mi>μ</mi></msub><mo>=</mo><mo>(</mo><mi>g</mi><mo>−</mo><mn>2</mn><mo>)</mo><mo>/</mo><mn>2</mn></math> of the muon. From the form factor calculation we extract the partial decay width <math><mrow><mi>Γ</mi><mo>(</mo><mi>η</mi><mo>→</mo><mi>γ</mi><mi>γ</mi><mo>)</mo><mo>=</mo><mn>338</mn><mo>(</mo><mn>87</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>17</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>syst</mi></mrow></msub><mtext> </mtext><mtext> </mtext><mi>eV</mi></mrow></math> and the slope parameter <math><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>η</mi></mrow></msub><mo>=</mo><mn>1.34</mn><mo>(</mo><mn>28</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>14</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>syst</mi></mrow></msub><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></math>. For the <math><mi>η</mi></math>-pole contribution to <math><msub><mi>a</mi><mi>μ</mi></msub></math> we obtain <math><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mi>η</mi><mtext>-pole</mtext></mrow></msubsup><mo>=</mo><mn>13.8</mn><mo>(</mo><mn>5.2</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>1.5</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>syst</mi></mrow></msub><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>11</mn></mrow></msup></mrow></math>.

Sebastian Burri

Andrew Gasbarro

Kyriakos Hadjiyiannakou

Karl Jansen

Gurtej Kanwar

Konstantin Ottnad

Ferenc Pittler

<math><mrow><mi>η</mi><mo>→</mo><msup><mrow><mi>γ</mi></mrow><mrow><mo>*</mo></mrow></msup><msup><mrow><mi>γ</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> transition form factor and the hadronic light-by-light <math><mi>η</mi></math>-pole contribution to the muon <math><mi>g</mi><mo>−</mo><mn>2</mn></math> from lattice QCD

We present a first-principles lattice QCD investigation of the <math><mi>R</mi></math> ratio between the <math><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> cross section into hadrons and into muons. By using the method of Ref. [1], that allows one to extract smeared spectral densities from Euclidean correlators, we compute the <math><mi>R</mi></math> ratio convoluted with Gaussian smearing kernels of widths of about 600 MeV and central energies from 220 MeV up to 2.5 GeV. Our theoretical results are compared with the corresponding quantities obtained by smearing the KNT19 compilation [2] of <math><mi>R</mi></math>-ratio experimental measurements with the same kernels and, by centering the Gaussians in the region around the <math><mi>ρ</mi></math>-resonance peak, a tension of about 3 standard deviations is observed. From the phenomenological perspective, we have not included yet in our calculation QED and strong isospin-breaking corrections, and this might affect the observed tension. From the methodological perspective, our calculation demonstrates that it is possible to study the <math><mi>R</mi></math> ratio in Gaussian energy bins on the lattice at the level of accuracy required in order to perform precision tests of the standard model.

Petros Dimopoulos

Probing the Energy-Smeared <math><mi>R</mi></math> Ratio Using Lattice QCD

We perform the first full decomposition of the pion momentum into its gluon and quark contributions. We employ an ensemble generated by the Extended Twisted Mass Collaboration with <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></math> Wilson twisted mass clover fermions at maximal twist tuned to reproduce the physical pion mass. We present our results in the <math><mover><mi>MS</mi><mo>¯</mo></mover></math> scheme at 2 GeV. We find <math><mrow><mo>⟨</mo><mi>x</mi><msub><mrow><mo>⟩</mo></mrow><mrow><mi>u</mi><mo>+</mo><mi>d</mi></mrow></msub><mo>=</mo><mn>0.601</mn><mo>(</mo><mn>28</mn><mo>)</mo></mrow></math>, <math><mo>⟨</mo><mi>x</mi><msub><mo>⟩</mo><mi>s</mi></msub><mo>=</mo><mn>0.059</mn><mo>(</mo><mn>13</mn><mo>)</mo></math>, <math><mo>⟨</mo><mi>x</mi><msub><mo>⟩</mo><mi>c</mi></msub><mo>=</mo><mn>0.019</mn><mo>(</mo><mn>05</mn><mo>)</mo></math>, and <math><mo>⟨</mo><mi>x</mi><msub><mo>⟩</mo><mi>g</mi></msub><mo>=</mo><mn>0.52</mn><mo>(</mo><mn>11</mn><mo>)</mo></math> for the separate contributions, respectively, whose sum saturates the momentum sum rule.

Georg Bergner

CERN

Quark and Gluon Momentum Fractions in the Pion from <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></math> Lattice QCD

We present a lattice computation of ππ–scattering in the <math><mi>I</mi><mo>=</mo><mn>1</mn></math> channel with <math><msub><mrow><mi>N</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mn>2</mn></math> dynamical quark flavours obtained with ensembles bracketing the physical value of the pion mass. Employing a global fit to data at three values of the pion mass, we determine the universal parameters of the ρ-resonance. We carefully investigate systematic uncertainties by determining energy eigenvalues using different methods and by comparing inverse amplitude method and Breit-Wigner type parametrizations. Overall, exploring the complex energy plane we find mass <math><msub><mrow><mi>M</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mo>=</mo><mn>786</mn><mo>(</mo><mn>22</mn><mo>)</mo><mspace width="0.25em"></mspace><mtext>MeV</mtext></math> and width <math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mo>=</mo><mn>180</mn><mo>(</mo><mn>6</mn><mo>)</mo><mspace width="0.25em"></mspace><mtext>MeV</mtext></math>, including statistical and systematic uncertainties. In contrast to previous <math><msub><mrow><mi>N</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mn>2</mn></math> extrapolations from higher than physical pion mass results, our mass value is in good agreement with experiment, while the width is too high.

Matthias Fischer

Maxim Mai

Martin Ueding

Markus Werner

The -resonance From = 2 lattice QCD including the physical pion mass

We present the first direct $$N_f=2$$ <math> <mrow> <msub> <mi>N</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>2</mn> </mrow> </math> lattice QCD computation of two- and three- $$\pi ^+$$ <math> <msup> <mi>π</mi> <mo>+</mo> </msup> </math> scattering quantities that includes an ensemble at the physical point. We study the quark mass dependence of the two-pion phase shift, and the three-particle interaction parameters. We also compare to phenomenology and chiral perturbation theory (ChPT). In the two-particle sector, we observe good agreement to the phenomenological fits in s- and d-wave, and obtain $$M_\pi a_0 = -0.0481(86)$$ <math> <mrow> <msub> <mi>M</mi> <mi>π</mi> </msub> <msub> <mi>a</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>-</mo> <mn>0.0481</mn> <mrow> <mo>(</mo> <mn>86</mn> <mo>)</mo> </mrow> </mrow> </math> at the physical point from a direct computation. In the three-particle sector, we observe reasonable agreement at threshold to the leading order chiral expansion, i.e. a mildly attractive three-particle contact term. In contrast, we observe that the energy-dependent part of the three-particle quasilocal scattering quantity is not well described by leading order ChPT.