We evaluate the Collins-Soper kernel and the reduced soft function in lattice QCD, incorporating <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><msub><mi>α</mi><mi>s</mi></msub><mo stretchy="false">)</mo></math> matching corrections. The calculation relies on the evaluation of the quasitransverse momentum–dependent wave function with asymmetric staple-shaped quark bilinear operators and four-point meson form factors. These quantities are computed nonperturbatively using two <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> twisted-mass fermion ensembles with the same lattice spacing of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>=</mo><mn>0.093</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>: the first ensemble has a lattice size of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>24</mn><mn>3</mn></msup><mo>×</mo><mn>48</mn></math> and a pion mass of 346 MeV, and the second one has a lattice size of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>32</mn><mn>3</mn></msup><mo>×</mo><mn>64</mn></math> and a pion mass of 261 MeV. The Collins-Soper kernel and the soft function are needed for the determination of the transverse momentum–dependent parton distribution functions.

Cyprus

Constantia Alexandrou

Simone Bacchio

Poland

Krzysztof Cichy

United States

Martha Constantinou

Germany

Aniket Sen

Gregoris Spanoudes

Fernanda Steffens

Jacopo Tarello

Collins-Soper kernel and reduced soft function in lattice QCD

We investigate static and dynamical aspects of string breaking in a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></math> lattice gauge theory coupled to Kogut-Susskind staggered fermions. Using tensor network simulations, we demonstrate that the static potential as well as the site-resolved configuration of the matter sites and gauge links allows us to identify the regimes in which string breaking occurs. Furthermore, we develop a variational quantum eigensolver that allows for reliably preparing the ground state of the theory in both the absence and presence of static charges and to capture the static aspects of the phenomenon. Carrying out state preparation on real quantum hardware for up to 19 qubits, we demonstrate its suitability for current quantum devices. In addition, we study the real-time dynamics of a flux tube between two static charges using both tensor networks and quantum hardware. Using a trotterization for the time-evolution operator, we are able to show that the breaking process starts with the creation of charges inside the string. These eventually redistribute toward the static charges and screen them, which leads to the breaking of the flux tube.

Andreas Athenodorou

Kostas Blekos

Georgios Polykratis

Stefan Kühn

Realizing string breaking dynamics in a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></math> lattice gauge theory on quantum hardware

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.

Joseph Delmar

Jacob Finkenrath

Bartosz Kostrzewa

Marcus Petschlies

Italy

Luis Alberto Rodriguez Chacon

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 study the <math><mover><mi>b</mi><mo>¯</mo></mover><mover><mi>b</mi><mo>¯</mo></mover><mi>u</mi><mi>d</mi></math> tetraquark with quantum numbers <math><mi>I</mi><mo>(</mo><msup><mi>J</mi><mi>P</mi></msup><mo>)</mo><mo>=</mo><mn>0</mn><mo>(</mo><msup><mn>1</mn><mo>+</mo></msup><mo>)</mo></math> as well as the <math><mover><mi>b</mi><mo>¯</mo></mover><mover><mi>b</mi><mo>¯</mo></mover><mi>u</mi><mi>s</mi></math> tetraquark with quantum numbers <math><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><msup><mn>1</mn><mo>+</mo></msup></math> using lattice QCD. We improve on existing work by including both local and scattering interpolating operators on both sides of the correlation functions and use symmetric correlation matrices. This allows not only a reliable determination of the energies of QCD-stable tetraquark ground states, but also of low-lying excited states, which are meson-meson scattering states. The latter is particularly important for future finite-volume scattering analyses. Here, we perform chiral and continuum extrapolations of just the ground-state energies, for which finite-volume effects are expected to be small. Our resulting tetraquark binding energies, <math><mrow><mo>−</mo><mn>100</mn><mo>±</mo><msubsup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>51</mn></mrow><mrow><mo>+</mo><mn>36</mn></mrow></msubsup><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math> for <math><mover><mi>b</mi><mo>¯</mo></mover><mover><mi>b</mi><mo>¯</mo></mover><mi>u</mi><mi>d</mi></math> and <math><mrow><mo>−</mo><mn>30</mn><mo>±</mo><msubsup><mn>3</mn><mrow><mo>−</mo><mn>31</mn></mrow><mrow><mo>+</mo><mn>11</mn></mrow></msubsup><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math> for <math><mover><mi>b</mi><mo>¯</mo></mover><mover><mi>b</mi><mo>¯</mo></mover><mi>u</mi><mi>s</mi></math>, are consistent with other recent lattice-QCD predictions.

Theodoros Leontiou

Stefan Meinel

Martin Pflaumer

Marc Wagner

<math><mover><mi>b</mi><mo>¯</mo></mover><mover><mi>b</mi><mo>¯</mo></mover><mi>u</mi><mi>d</mi></math> and <math><mover><mi>b</mi><mo>¯</mo></mover><mover><mi>b</mi><mo>¯</mo></mover><mi>u</mi><mi>s</mi></math> tetraquarks from lattice QCD using symmetric correlation matrices with both local and scattering interpolating operators

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.

Alessandro De Santis

Antonio Evangelista

Roberto Frezzotti

Giuseppe Gagliardi

Marco Garofalo

Vittorio Lubicz

Simone Romiti

Francesco Sanfilippo

Silvano Simula

Nazario Tantalo

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 present the first determination of the energy dependence of the <math><mrow><mi>B</mi><mtext>−</mtext><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math> and <math><mrow><msup><mrow><mi>B</mi></mrow><mrow><mo>*</mo></mrow></msup><mtext>−</mtext><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math> isospin-0, <math><mi>S</mi></math>-wave scattering amplitudes both below and above the thresholds using lattice QCD, which allows us to investigate rigorously whether mixed bottom-charm <math><mrow><mover><mrow><mi>b</mi></mrow><mrow><mo>¯</mo></mrow></mover><mover><mrow><mi>c</mi></mrow><mrow><mo>¯</mo></mrow></mover><mi>u</mi><mi>d</mi></mrow></math> tetraquarks exist as bound states or resonances. The scattering phase shifts are obtained using Lüscher’s method from the energy spectra in two different volumes. To ensure that no relevant energy level is missed, we use large, symmetric <math><mn>7</mn><mo>×</mo><mn>7</mn></math> and <math><mn>8</mn><mo>×</mo><mn>8</mn></math> correlation matrices that include, at both source and sink, <math><mrow><msup><mrow><mi>B</mi></mrow><mrow><mo>(</mo><mo>*</mo><mo>)</mo></mrow></msup><mtext>−</mtext><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math> scattering operators with the lowest three or four possible back-to-back momenta in addition to local <math><mover><mi>b</mi><mo>¯</mo></mover><mover><mi>c</mi><mo>¯</mo></mover><mi>u</mi><mi>d</mi></math> operators. We fit the energy dependence of the extracted scattering phase shifts using effective-range expansions. We observe sharp peaks in the <math><mrow><msup><mrow><mi>B</mi></mrow><mrow><mo>(</mo><mo>*</mo><mo>)</mo></mrow></msup><mtext>−</mtext><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math> scattering rates close to the thresholds, which are associated with shallow bound states, either genuine or virtual, a few MeV or less below the <math><mrow><msup><mrow><mi>B</mi></mrow><mrow><mo>(</mo><mo>*</mo><mo>)</mo></mrow></msup><mtext>−</mtext><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math> thresholds. In addition, we find hints for resonances with masses of order 100 MeV above the thresholds and decay widths of order 200 MeV.

Shallow Bound States and Hints for Broad Resonances with Quark Content <math><mover><mi>b</mi><mo>¯</mo></mover><mover><mi>c</mi><mo>¯</mo></mover><mi>u</mi><mi>d</mi></math> in <math><mrow><mi>B</mi><mtext>−</mtext><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math> and <math><mrow><msup><mrow><mi>B</mi></mrow><mrow><mo>*</mo></mrow></msup><mtext>−</mtext><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math> Scattering from Lattice QCD

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.

United Kingdom

Srijit Paul

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

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.

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

Staple-shaped Wilson line operators are necessary for the study of transverse momentum-dependent parton distribution functions (TMDPDFs) in lattice QCD and beyond. In this work, we study the renormalization of such operators in the general case of an asymmetric staple. We analyze the mixing pattern of these operators using their symmetry properties, where we find that the possible mixing is restricted within groups of four operators. We then present numerical results using the regularization independent momentum subtraction (RI/MOM) scheme to study the importance of mixing using one operator in particular, the <math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></math> operator. Based on these results, we consider the short distance ratio (SDR) scheme, which is desirable in the absence of mixing. Finally, we investigate a variant of the RI/MOM scheme, where the renormalization factors are computed at short distances.

China

Xu Feng

Karl Jansen

Chuan Liu

Nonperturbative renormalization of asymmetric staple-shaped operators in twisted mass lattice QCD