<p>The decay <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>B</mi><mo stretchy="false">→</mo><mi>ρ</mi><mo>ℓ</mo><mover accent="true"><mrow><mi>ν</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></math> is an attractive process for determining the magnitude of the smallest Cabibbo-Kobayashi-Maskawa matrix element, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">|</mo><msub><mi>V</mi><mrow><mi>u</mi><mi>b</mi></mrow></msub><mo stretchy="false">|</mo></math>, and can provide new insights into the origin of the long-standing exclusive-inclusive discrepancy in determinations of this standard-model parameter. This requires a nonperturbative QCD calculation of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><mi>ρ</mi></math> form factors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>V</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>A</mi><mn>0</mn></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>A</mi><mn>1</mn></msub></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>A</mi><mn>12</mn></msub></math>. The unstable nature of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi></math> resonance has prevented precise lattice QCD calculations of these form factors to date. Here, we present the first lattice QCD calculation of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>B</mi><mo stretchy="false">→</mo><mi>ρ</mi></mrow></math> form factors in which the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi></math> is treated properly as a resonance in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>P</mi></mrow></math>-wave <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>π</mi><mi>π</mi></math> scattering. To this end, we use the Lellouch-Lüscher finite-volume formalism to compute the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>B</mi><mo stretchy="false">→</mo><mi>π</mi><mi>π</mi></mrow></math> form factors as a function of both momentum transfer and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>π</mi><mi>π</mi></math> invariant mass, and then analytically continue to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi></math> resonance pole. This calculation is performed with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn><mo>+</mo><mn>1</mn></math> dynamical quark flavors at a pion mass of approximately 320 MeV, and demonstrates a clear path toward results at the physical point.</p>

Slovenia

Luka Leskovec

United States

Stefan Meinel

Germany

Marcus Petschlies

John Negele

Srijit Paul

Andrew Pochinsky

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>B</mi><mo stretchy="false">→</mo><mi>ρ</mi><mo>ℓ</mo><mover accent="true"><mrow><mi>ν</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></math> Resonance Form Factors from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>B</mi><mo stretchy="false">→</mo><mi>π</mi><mi>π</mi><mo>ℓ</mo><mover accent="true"><mrow><mi>ν</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></math> in Lattice QCD

<p>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.</p>

Cyprus

Constantia Alexandrou

Simone Bacchio

Martha Constantinou

Joseph Delmar

Jacob Finkenrath

Bartosz Kostrzewa

Italy

Luis Alberto Rodriguez Chacon

Gregoris Spanoudes

Fernanda Steffens

Carsten Urbach

Switzerland

Urs Wenger

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

We investigate contributions of excited states to nucleon matrix elements computed in lattice QCD by employing, in addition to the standard nucleon interpolating operator, pion-nucleon (<math><mrow><mi>π</mi><mtext>−</mtext><mi>N</mi></mrow></math>) operators. We solve a generalized eigenvalue problem (GEVP) to obtain an optimal interpolating operator that minimizes overlap with the <math><mrow><mi>π</mi><mtext>−</mtext><mi>N</mi></mrow></math> states. We derive a variant of the standard application of the GEVP method, which allows for constructing 3-point correlation functions using the optimized interpolating operator without requiring the computationally demanding combination that includes <math><mrow><mi>π</mi><mtext>−</mtext><mi>N</mi></mrow></math> operators in both sink and source. We extract nucleon matrix elements using two twisted mass fermion ensembles, one ensemble generated using pion mass of 346 MeV and one ensemble tuned to reproduce the physical value of the pion mass. Especially, we determine the isoscalar and isovector scalar, pseudoscalar, vector, axial, and tensor matrix elements. We include results obtained using a range of kinematic setups, including momentum in the sink. Our results using this variational approach are compared with previous results obtained using the same ensembles and multistate fits without GEVP improvement. We find that for the physical mass point ensemble, the improvement, in terms of suppression of excited states using this method, is most significant for the case of the matrix elements of the isovector axial and pseudoscalar currents.

Giannis Koutsou

Yan Li

Ferenc Pittler

Investigation of pion-nucleon contributions to nucleon matrix elements

We present an investigation of pion-nucleon elastic scattering in the <math><mi>I</mi><mo>(</mo><msup><mi>J</mi><mi>P</mi></msup><mo>)</mo><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>(</mo><msup><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>+</mo></msup><mo>)</mo></math> channel using lattice QCD with degenerate up and down, strange and charm quarks with masses tuned to their physical values. We use an ensemble of twisted mass fermions with box size <math><mi>L</mi><mo>=</mo><mn>5.1</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and lattice spacing <math><mi>a</mi><mo>=</mo><mn>0.08</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and we consider the <math><mi>π</mi><mi>N</mi></math> system in rest and moving frames up to total momentum <math><mrow><msup><mrow><mover><mrow><mi>P</mi></mrow><mrow><mo>→</mo></mrow></mover></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>3</mn><mo>(</mo><mn>2</mn><mi>π</mi><mo>/</mo><mi>L</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>0.17</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>. We take into account the finite volume symmetries and <math><mi>S</mi></math>- and <math><mi>P</mi></math>-wave mixing, and use the Lüscher formalism to simultaneously constrain the <math><mi>J</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>,</mo><mo>ℓ</mo><mo>=</mo><mn>0</mn></math> and <math><mi>J</mi><mo>=</mo><mn>3</mn><mo>/</mo><mn>2</mn><mo>,</mo><mo>ℓ</mo><mo>=</mo><mn>1</mn></math> scattering amplitudes. We estimate the <math><mi>Δ</mi></math> resonance pole in the <math><mi>P</mi></math>-wave channel as well as the <math><mi>S</mi></math>-wave isospin-<math><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></math> scattering length.

Theodoros Leontiou

United Kingdom

Elastic nucleon-pion scattering amplitudes in the <math><mi>Δ</mi></math> channel at physical pion mass from lattice QCD

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

<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.

Alessandro De Santis

Petros Dimopoulos

Roberto Frezzotti

Giuseppe Gagliardi

Marco Garofalo

Vittorio Lubicz

Francesco Sanfilippo

Silvano Simula

Nazario Tantalo

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 determine the <math><mi>Δ</mi><mo>(</mo><mn>1232</mn><mo>)</mo></math> resonance parameters using lattice QCD and the Lüscher method. The resonance occurs in elastic pion-nucleon scattering with <math><mrow><msup><mrow><mi>J</mi></mrow><mrow><mi>P</mi></mrow></msup><mo>=</mo><mn>3</mn><mo>/</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>+</mo></mrow></msup></mrow></math> in the isospin <math><mi>I</mi><mo>=</mo><mn>3</mn><mo>/</mo><mn>2</mn></math>, <math><mi>P</mi></math>-wave channel. Our calculation is performed with <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> flavors of clover fermions on a lattice with <math><mi>L</mi><mo>≈</mo><mn>2.8</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>. The pion and nucleon masses are <math><msub><mi>m</mi><mi>π</mi></msub><mo>=</mo><mn>255.4</mn><mo>(</mo><mn>1.6</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> and <math><msub><mi>m</mi><mi>N</mi></msub><mo>=</mo><mn>1073</mn><mo>(</mo><mn>5</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, respectively, and the strong decay channel <math><mi>Δ</mi><mo>→</mo><mi>π</mi><mi>N</mi></math> is found to be above the threshold. To thoroughly map out the energy dependence of the nucleon-pion scattering amplitude, we compute the spectra in all relevant irreducible representations of the lattice symmetry groups for total momenta up to <math><mrow><mover><mrow><mi>P</mi></mrow><mrow><mo>→</mo></mrow></mover><mo>=</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mi>L</mi></mrow></mfrac><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math>, including irreps that mix <math><mi>S</mi></math> and <math><mi>P</mi></math> waves. We perform global fits of the amplitude parameters to up to 21 energy levels, using a Breit-Wigner model for the <math><mi>P</mi></math>-wave phase shift and the effective-range expansion for the <math><mi>S</mi></math>-wave phase shift. From the location of the pole in the <math><mi>P</mi></math>-wave scattering amplitude, we obtain the resonance mass <math><msub><mi>m</mi><mi>Δ</mi></msub><mo>=</mo><mn>1378</mn><mo>(</mo><mn>7</mn><mo>)</mo><mo>(</mo><mn>9</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> and the coupling <math><msub><mi>g</mi><mrow><mi>Δ</mi><mo>−</mo><mi>π</mi><mi>N</mi></mrow></msub><mo>=</mo><mn>23.8</mn><mo>(</mo><mn>2.7</mn><mo>)</mo><mo>(</mo><mn>0.9</mn><mo>)</mo></math>.

Giorgio Silvi

Stefan Krieg

Gumaro Rendon

Sergey Syritsyn

Antonino Todaro

<math><mi>P</mi></math>-wave nucleon-pion scattering amplitude in the <math><mi>Δ</mi><mo>(</mo><mn>1232</mn><mo>)</mo></math> channel from lattice QCD

We present a lattice-QCD determination of the elastic isospin-<math><mn>1</mn><mo>/</mo><mn>2</mn></math> <math><mi>S</mi></math>-wave and <math><mi>P</mi></math>-wave <math><mi>K</mi><mi>π</mi></math> scattering amplitudes as a function of the center-of-mass energy using Lüscher’s method. We perform global fits of <math><mi>K</mi></math>-matrix parametrizations to the finite-volume energy spectra for all irreducible representations with total momenta up to <math><msqrt><mn>3</mn></msqrt><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mi>L</mi></mfrac></math>; this includes irreducible representations (irreps) that mix the <math><mi>S</mi></math>- and <math><mi>P</mi></math>-waves. Several different parametrizations for the energy dependence of the <math><mi>K</mi></math>-matrix are considered. We also determine the positions of the nearest poles in the scattering amplitudes, which correspond to the broad <math><mi>κ</mi></math> resonance in the <math><mi>S</mi></math>-wave and the narrow <math><msup><mi>K</mi><mo>*</mo></msup><mo>(</mo><mn>892</mn><mo>)</mo></math> resonance in the <math><mi>P</mi></math>-wave. Our calculations are performed with <math><mn>2</mn><mo>+</mo><mn>1</mn></math> dynamical clover fermions for two different pion masses of 317.2(2.2) and 175.9(1.8) MeV. Our preferred <math><mi>S</mi></math>-wave parametrization is based on a conformal map and includes an Adler zero; for the <math><mi>P</mi></math>-wave, we use a standard pole parametrization including Blatt-Weisskopf barrier factors. The <math><mi>S</mi></math>-wave <math><mi>κ</mi></math>-resonance pole positions are found to be <math><mrow><mrow><mo>[</mo><mn>0.86</mn><mo>(</mo><mn>12</mn><mo>)</mo><mo>−</mo><mn>0.309</mn><mo>(</mo><mn>50</mn><mo>)</mo><mi>i</mi><mo>]</mo></mrow><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math> at the heavier pion mass and <math><mrow><mrow><mo>[</mo><mn>0.499</mn><mo>(</mo><mn>55</mn><mo>)</mo><mo>−</mo><mn>0.379</mn><mo>(</mo><mn>66</mn><mo>)</mo><mi>i</mi><mo>]</mo></mrow><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math> at the lighter pion mass. The <math><mi>P</mi></math>-wave <math><msup><mi>K</mi><mo>*</mo></msup></math>-resonance pole positions are found to be <math><mrow><mrow><mo>[</mo><mn>0.8951</mn><mo>(</mo><mn>64</mn><mo>)</mo><mo>−</mo><mn>0.00250</mn><mo>(</mo><mn>21</mn><mo>)</mo><mi>i</mi><mo>]</mo></mrow><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math> at the heavier pion mass and <math><mrow><mrow><mo>[</mo><mn>0.8718</mn><mo>(</mo><mn>82</mn><mo>)</mo><mo>−</mo><mn>0.0130</mn><mo>(</mo><mn>11</mn><mo>)</mo><mi>i</mi><mo>]</mo></mrow><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math> at the lighter pion mass, which corresponds to couplings of <math><msub><mi>g</mi><mrow><msup><mi>K</mi><mo>*</mo></msup><mi>K</mi><mi>π</mi></mrow></msub><mo>=</mo><mn>5.02</mn><mo>(</mo><mn>26</mn><mo>)</mo></math> and <math><mrow><msub><mrow><mi>g</mi></mrow><mrow><msup><mrow><mi>K</mi></mrow><mrow><mo>*</mo></mrow></msup><mi>K</mi><mi>π</mi></mrow></msub><mo>=</mo><mn>4.99</mn><mo>(</mo><mn>22</mn><mo>)</mo></mrow></math>, respectively.

<math><mi>I</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></math> <math><mi>S</mi></math>-wave and <math><mi>P</mi></math>-wave <math><mi>K</mi><mi>π</mi></math> scattering and the <math><mi>κ</mi></math> and <math><msup><mi>K</mi><mo>*</mo></msup></math> resonances from lattice QCD