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 present the first lattice QCD determination of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi><mo stretchy="false">→</mo><mi>N</mi></math> vector and axial-vector form factors, which are essential inputs for studying the semileptonic decay <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi><mo stretchy="false">→</mo><mi>p</mi><mo>ℓ</mo><msub><mover accent="true"><mi>ν</mi><mo stretchy="false">¯</mo></mover><mo>ℓ</mo></msub></math>. This channel provides a clean, theoretically controlled avenue for extracting the 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>s</mi></mrow></msub><mo stretchy="false">|</mo></math> from the baryon sector. Our analysis uses a gauge ensemble with physical light, strange, and charm quark masses and yields the most precise determination to date of the full set of transition form factors—including second-class contributions—as well as the associated couplings, radii, and the ratio of muon-to-electron decay rates, an observable sensitive to possible nonstandard scalar and tensor interactions. We compare our nonperturbative results with next-to-next-to-leading order expansions in the small parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>δ</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>m</mi></mrow><mrow><mi mathvariant="normal">Λ</mi></mrow></msub><mo>−</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>N</mi></mrow></msub><mo stretchy="false">)</mo><mo>/</mo><msub><mrow><mi>m</mi></mrow><mrow><mi mathvariant="normal">Λ</mi></mrow></msub><mo>≈</mo><mn>0.16</mn></mrow></math>. We find that the common phenomenological approximation of neglecting the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>q</mi><mn>2</mn></msup></math> dependence of the form factors leads to a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>∼</mo><mn>4</mn><mo>%</mo></math> deviation in the decay rate. This underscores the critical importance of precise, fully nonperturbative form factor inputs for achieving the subpercent precision targets of upcoming experimental programs.

Andreas Konstantinou

Study of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="normal">Λ</mi><mo stretchy="false">→</mo><mi>p</mi><mo>ℓ</mo><msub><mrow><mover accent="true"><mrow><mi>ν</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mo>ℓ</mo></mrow></msub></mrow></math> Semileptonic Decay in Lattice QCD

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 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 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.

Giannis Koutsou

Theodoros Leontiou

United Kingdom

Srijit Paul

Ferenc Pittler

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

We show that an infinitesimal step of gradient flow can be used for defining a novel approach for computing gradients of physical observables with respect to action parameters. Compared to the commonly used perturbative expansion, this approach does not require calculating any disconnected contribution or vacuum expectation value and can provide results up to 3 orders of magnitudes more precise. On the other hand, it requires a nontrivial condition to be satisfied by the flow action, the calculation of its force and its Laplacian, and the force of the observable, whose gradient needs to be measured. As a proof of concept, we measure gradients in <math><mi>β</mi></math> of Wilson loops in a four-dimensional SU(3) Yang-Mills theory simulated on a <math><msup><mn>16</mn><mn>4</mn></msup></math> lattice using the Wilson action.

Novel approach for computing gradients of physical observables

We investigate the low-lying baryon spectrum 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> ensembles simulated with physical values of the quark masses and lattice spacings of 0.080, 0.069 and 0.057 fm. The ensembles are generated using twisted mass clover-improved fermions and the Iwasaki gauge action. The spatial length is kept approximately the same at about 5.1 fm to 5.5 fm fulfilling the condition <math><msub><mi>m</mi><mi>π</mi></msub><mi>L</mi><mo>&gt;</mo><mn>3.6</mn></math>. We investigate isospin splitting within isospin multiples and verify that for most cases the isospin splitting for these lattice spacing is consistent with zero. In the couple of cases, for which there is a nonzero value, in the continuum limit, the mass splitting goes to zero. The baryon masses are extrapolated to the continuum limit using the 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> ensembles and are compared to other recent lattice QCD results. For the strange and charm quark masses we find, respectively, <math><msub><mi>m</mi><mi>s</mi></msub><mo>(</mo><mn>2</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi><mo>)</mo><mo>=</mo><mn>99.2</mn><mo>(</mo><mn>2.7</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> and <math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mn>3</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi><mo>)</mo><mo>=</mo><mn>1.015</mn><mo>(</mo><mn>39</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math>. The values predicted for the masses of the doubly charmed <math><msubsup><mi>Ξ</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>⋆</mo></msubsup></math>, <math><msub><mi>Ω</mi><mrow><mi>c</mi><mi>c</mi></mrow></msub></math> and <math><msubsup><mi>Ω</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>⋆</mo></msubsup></math> baryons are 3.676(55) GeV, 3.703(51) GeV and 3.803(50) GeV, respectively, and for the triply charmed <math><msub><mi>Ω</mi><mrow><mi>c</mi><mi>c</mi><mi>c</mi></mrow></msub></math> baryon is 4.785(71) GeV.

Georgios Christou

Low-lying baryon masses using twisted mass fermions ensembles at the physical pion mass

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

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