The hyperon spectrum from lattice QCD

We employ a chiral effective field theory-based approach to connect D D ⁎ scattering observables at the physical and variable pion masses accessible in lattice QCD simulations. We incorporate all relevant scales associated with three-body DDπ dynamics and the left-hand cut induced by the one-pion exchange for pion masses higher than the physical one, as required by analyticity and unitarity. By adjusting the contact interactions to match experimental data at the physical pion mass and lattice finite-volume energy levels at m π = 280 MeV, we predict the trajectory of the T c c pole as a function of the pion mass, finding it consistent with the hadronic-molecule scenario. In particular, we find that the explicit treatment of the one-pion exchange has a pronounced effect on the pole trajectory for m π ≳ 230 MeV by pushing it into the complex energy plane.

We present a calculation of the electromagnetic form factor of the pion in N f = 2 + 1 flavor lattice QCD. Calculations are made on the PACS-CS gauge field configurations generated using Iwasaki gauge action and Wilson-clover quark action on a 323 × 64 lattice volume with the lattice spacing estimated as a = 0.0907(13) fm at the physical point. Measurements of the form factor are made using the technique of partially twisted boundary condition to reach small momentum transfer as well as periodic boundary condition with integer momenta. Additional improvements including random wall source techniques and a judicious choice of momenta carried by the incoming and outgoing quarks are employed for error reduction. Analyzing the form factor data for the pion mass at M π ≈ 411 MeV and 296 MeV, we find that the NNLO SU(2) chiral perturbation theory fit yields 〈r 2〉 = 0.441 ± 0.046 fm2 for the pion charge radius at the physical pion mass. Albeit the error is quite large, this is consistent with the experimental value of 0.452 ± 0.011 fm2. Below M π ≈ 300 MeV, we find that statistical fluctuations in the pion two-and three-point functions become too large to extract statistically meaningful averages on a 323 spatial volume. We carry out a sample calculation on a 644 lattice with the quark masses close to the physical point, which suggests that form factor calculations at the physical point become feasible by enlarging lattice sizes to M π L ≈ 4.

We report our recent study of nucleon electromagnetic form factor and its charge radius in lattice QCD. We compute the form factor using large size of lattice volume, more than 8 fm3, at (almost) physical pion mass which is the first realistic computation by reducing the systematic errors of the finite size correction and chiral extrapolation. We present preliminary results of not only isovector form factor but also isoscalar one, and discuss a comparison with experimental result.

We calculate the double-virtual $\ensuremath{\eta}\ensuremath{\rightarrow}{\ensuremath{\gamma}}^{*}{\ensuremath{\gamma}}^{*}$ transition form factor ${\mathcal{F}}_{\ensuremath{\eta}\ensuremath{\rightarrow}{\ensuremath{\gamma}}^{*}{\ensuremath{\gamma}}^{*}}({q}_{1}^{2},{q}_{2}^{2})$ from first principles using a lattice QCD simulation with ${N}_{f}=2+1+1$ 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 $\ensuremath{\eta}$-pole contribution to the hadronic light-by-light scattering in the anomalous magnetic moment ${a}_{\ensuremath{\mu}}=(g\ensuremath{-}2)/2$ of the muon. From the form factor calculation we extract the partial decay width $\mathrm{\ensuremath{\Gamma}}(\ensuremath{\eta}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma})=338(87{)}_{\mathrm{stat}}(17{)}_{\mathrm{syst}}\text{ }\text{ }\mathrm{eV}$ and the slope parameter ${b}_{\ensuremath{\eta}}=1.34(28{)}_{\mathrm{stat}}(14{)}_{\mathrm{syst}}\text{ }\text{ }{\mathrm{GeV}}^{\ensuremath{-}2}$. For the $\ensuremath{\eta}$-pole contribution to ${a}_{\ensuremath{\mu}}$ we obtain ${a}_{\ensuremath{\mu}}^{\ensuremath{\eta}\text{-pole}}=13.8(5.2{)}_{\mathrm{stat}}(1.5{)}_{\mathrm{syst}}\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}11}$.

We present the first lattice-QCD x-dependent pion valence-quark generalized parton distribution (GPD) calculated directly at physical pion mass using the Large-Momentum Effective Theory (LaMET) with next-to-next-to-leading order perturbative matching correction. We use clover fermions for the valence action on 2+1+1 flavors of highly improved staggered quarks (HISQ), generated by MILC Collaboration, with lattice spacing a≈0.09 fm and box size L≈5.5 fm; the pion two-point measurements number up to O(106) with boost momentum 1.73 GeV. The pion valence distribution is renormalized in hybrid scheme with Wilson-line mass subtraction at large distances in coordinate space, followed by a procedure to match it to the MS‾ scheme. We focus on the zero-skewness limit, where the GPD has a probability-density interpretation in the longitudinal Bjorken x and the transverse impact-parameter distributions. We take the integral of our GPD functions to generate leading moment so that we can make comparisons with past lattice-QCD and experimental determinations of the pion form factors and found consistent agreement among them. We predict the higher GPD moments and reveal x-dependent tomography of the pion for the first time using lattice QCD.

Scattering at physical pion mass is still an exploratory field in lattice QCD. This generally involves the extraction of excited states through multi-particle correlators on systems with resonances. In that context, distillation has been demonstrated to be effective both as a smearing kernel and a computational tool. Motivated by the study of the smearing profile of the distillation operator, we compare stochastic and exact distillation cases for different numbers of Laplacian eigenvectors using a RBC-UKQCD $N_f=2+1$ domain-wall fermion lattice with a physical pion mass.

Current status of nucleon structure calculations with joint RBC and UKQCD 2+1-flavor dynamical domain-wall fermions (DWF) lattice QCD is reported: Two ensembles with pion mass of about (m_\pi=170) MeV and 250 MeV are used. The lattice cutoff is set at about 1.4 GeV, allowing a large spatial volume of about (L=4.6) fm across while maintaining a sufficiently small residual breaking of chiral symmetry with the dislocation-suppressing-determinant-ratio (DSDR) gauge action. We calculate all the isovector form factors and some low moments of isovector structure functions. We confirm the finite-size effect in isovector axialvector-current form factors, in particular the deficit in the axial charge and its scaling in terms of (m_\pi L), that we reported from our earlier calculation at heavier pion masses.

We present the status of the Mainz group's lattice QCD calculation of the pion transition form factor $\mathcal{F}_{\pi^0\gamma^\ast\gamma^\ast}$, which describes the interaction of an on-shell pion with two off-shell photons. This form factor is the main ingredient in the calculation of the pion-pole contribution to hadronic light-by-light scattering in the muon $g-2$. We use the $N_f = 2 + 1$ CLS gauge ensembles, and we update our previous work by including a physical pion mass ensemble (E250). We compute the transition form factor in the pion rest frame as well as in a moving frame in order to have access to a wider range of photon virtualities. In addition to the quark-line connected correlator we also compute the quark-line disconnected diagrams that contribute to the form factor. In this final stage of the analysis, we combine the result on E250 with the previous work published in 2019 to extrapolate the form factor to the continuum and to physical quark masses. Testing different ansätze for the fit, we explore the systematic uncertainties of the extrapolation. The contribution from the disconnected diagrams is also scrutinized.

A lattice computation of the leading-order hadronic contribution to the muon anomalous magnetic moment can potentially help reduce the error on the Standard Model prediction for this quantity, if sufficient control of all systematic errors affecting such a computation can be achieved. One of these systematic errors is that associated with the extrapolation to the physical pion mass from values on the lattice larger than the physical pion mass. We investigate this extrapolation assuming lattice pion masses in the range of 200 to 400MeV with the help of two-loop chiral perturbation theory, and find that such an extrapolation is unlikely to lead to control of this systematic error at the 1% level. This remains true even if various tricks to improve the reliability of the chiral extrapolation employed in the literature are taken into account. In addition, while chiral perturbation theory also predicts the dependence on the pion mass of the leading-order hadronic contribution to the muon anomalous magnetic moment as the chiral limit is approached, this prediction turns out to be of no practical use, because the physical pion mass is larger than the muon mass that sets the scale for the onset of this behavior.

We present an update of our determination of the isovector charges gAu−d, gSu−d and gTu−d, and the isovector twist-2 forward matrix elements ⟨x⟩u−d, ⟨x⟩Δu−Δd and ⟨x⟩δu−δd on the Nf=2+1 gauge ensembles generated by the Coordinated Lattice Simulations (CLS) effort. We have significantly extended our coverage of the parameter space by adding ensembles at the physical pion mass and fine lattice spacing, at nearly physical pion masses and very fine lattice spacings, and at very large physical lattice volumes, enabling a well-controlled extrapolation to the physical point. Another major improvement is achieved owing to the extended range of source-sink separations, which allows us to perform two-state fits to summed correlator ratios, leading to a much higher level of control over excited-state effects. Systematic uncertainties from the chiral, continuum and infinite-volume extrapolations are incorporated via model averages based on the Akaike information criterion. Our final results at the physical point are gAu−d=1.254(19)stat(15)sys[24]total, gSu−d=1.203(77)stat(81)sys[112]total, gTu−d=0.993(15)stat(05)sys[16]total, ⟨x⟩u−d=0.153(15)stat(10)sys[17]total, ⟨x⟩Δu−Δd=0.207(15)stat(06)sys[16]total, and ⟨x⟩δu−δd=0.195(17)stat(15)sys[23]total. While our results for the isovector charges are in excellent agreement with the FLAG 21 averages, we note that our error for the tensor charge gTu−d is considerably smaller. Published by the American Physical Society 2024

With the operation of the RHIC heavy ion program, the theoretical understanding of QCD at finite temperature and density has become increasingly important. Though QCD at finite temperature has been extensively studied using lattice Monte-Carlo simulations over the past twenty years, most physical questions relevant for RHIC (and future) heavy ion experiments remain open. In lattice QCD at finite temperature and density there have been at least two major advances in recent years. First, for the first time calculations of real time quantities, like meson spectral functions have become available. Second, the lattice study of the QCD phase diagram and equation of state have been extended to finite baryon density by several groups. Both issues were extensively discussed in the course of the workshop. A real highlight was the study of the QCD phase diagram in (T, {mu})-plane by Z. Fodor and S. Katz and the determination of the critical end-point for the physical value of the pion mass. This was the first time such lattice calculations at, the physical pion mass have been performed. Results by Z Fodor and S. Katz were obtained using a multi-parameter re-weighting method. Other determinations of the critical end point were also presented, in particular using a Taylor expansion around {mu} = 0 (Bielefeld group, Ejiri et al.) and using analytic continuation from imaginary chemical potential (Ph. de Forcrand and O. Philipsen). The result based on Taylor expansion agrees within errors with the new prediction of Z. Fodor and S. Katz, while methods based on analytic continuation still predict a higher value for the critical baryon density. Most of the thermodynamics studies in full QCD (including those presented at this workshop) have been performed using quite coarse lattices, a = 0.2-0.3 fm. Therefore one may worry about cutoff effects in different thermodynamic quantities, like the transition temperature T{sub tr}. At the workshop U. Heller presented a study of the transition temperature for three different lattice spacings and performed a continuum extrapolation of T{sub tr} for the first time. Lattice calculations of the meson spectral functions were presented by M. Asakawa, S. Datta, E. Laermann and H. Matsufuru. These show that charmonia ground states ({eta}{sub c} and J/{psi}) continue to exist in the plasma at least up to a temperature of 1.7 T{sub tr}. At what temperature charmonia states cease to exist is not yet clear. Calculations presented by M. Asakawa show dissolution of the J/{psi} at T = 1.7 T{sub tr}, while the analysis presented H. Matsufuru provided evidence that ground state charmonia still exist at this temperature. S. Datta argued that the ground state charmonia is likely to dissolve only for temperatures T > 2.25 T{sub tr}, while the P-states are dissociated at, 1.1 T{sub tr}. It is also very interesting that, even in the case of light quarks, meson spectral functions show a resonance-like structure in the plasma phase (talk by E. Laermann). Finally attempts to calculate transport properties in the Quark Gluon Plasma were presented by S. Gupta. The workshop devoted special attention to the finite temperature modification of inter-quark forces and color screening, another area where considerable progress has been made in recent years (talks by 0. Kaczmarek, K. Petrov, O. Philipsen and F. Zantow). Many other new theoretical developments which cannot be discussed here were also presented on the workshop. Altogether the workshop was a great success, for which we thank all the participants.

A new method for calculating parton distribution functions from lattice QCD is implemented and studied. Lattice QCD calculable matrix elements with space-like separated fields have an analogous operator product expansion to experimental scattering cross sections and thus these matrix elements are known as “Good Lattice Cross Sections”. Using the colinear factorization approach, a Good Lattice Cross Section can be factorized to the short distance matching kernels that are computed in perturbation theory and the non-perturbative parton distribution functions. As a result, using the perturbative matching kernels and the non-perturbatively computed matrix elements, one can obtain the parton distribution functions. The nucleon and pion matrix elements are determined on a set of 2+1 flavors of clover improved quarks with heavier than physical pion mass. The determination of the parton distributions from Good Lattice Cross Sections constitutes an ill-posed inverse problem. Methods for accurate determination of parton distributions from the Good Lattice Cross Sections are studied. With the calculation of several Good Lattice Cross Sections, the determination of the parton distributions can be improved with a simultaneous analysis, similar to the global parton distribution fits to experimental cross sections.

We present results for the isovector nucleon form factors measured on a $96^4$ lattice at almost the physical pion mass with a lattice spacing of 0.085 fm in 2+1 flavor QCD. The configurations are generated with the stout-smeared $O(a)$-improved Wilson quark action and the Iwasaki gauge action at $\beta$=1.82. The pion mass at the simulation point is about 146 MeV. A large spatial volume of $(8.1~{\rm fm})^3$ allows us to investigate the form factors in the small momentum transfer region. We determine the isovector electric radius and magnetic moment from nucleon electric ($G_E$) and magnetic ($G_M$) form factors as well as the axial-vector coupling $g_A$. We also report on the results of the axial-vector ($F_A$), induced pseudoscalar ($F_P$) and pseudoscalar ($G_P$) form factors in order to verify the axial Ward-Takahashi identity in terms of the nucleon matrix elements, which may be called the generalized Goldberger-Treiman relation.

We present results of a lattice calculation of tetraquark states with quark contents $q_1q_2\bar{Q}\bar{Q}, \, q_1,q_2 \subset u,d,s,c$ and $Q \equiv b,c$ in both spin zero ($J=0$) and spin one ($J=1$) sectors. These calculations are performed on three dynamical $N_f = 2 + 1 + 1$ highly improved staggered quark ensembles at lattice spacings of about 0.12, 0.09 and 0.06 fm. We use the overlap action for light to charm quarks while a non-relativistic action with non-perturbatively improved coefficients with terms up to $\mathcal{O}(\alpha_s v^4)$ is employed for the bottom quark. While considering two heavy quarks as charm or bottom, we calculate the energy levels of various four-quark configurations with light quark masses ranging from the physical strange quark mass to that of the corresponding physical pion mass. Results for the spin one states show the presence of ground state energy levels which are below their respective thresholds for all the light flavor combinations with both doubly heavy quarks and particularly for the bottom quarks. Further, we identify a trend that the energy splittings, defined as the energy difference between the ground state energy levels and their respective thresholds, increase with decreasing the light quark masses and are maximum at the physical point for all the spin one states. The rate of increase is however dependent on the light quark configuration of the particular spin one state. We also present a study of hadron mass relations involving tetraquarks, baryons and mesons arising in the limit of infinitely heavy quark and find that these relations are more compatible with the heavy quark limit in the bottom sector but deviate substantially in the charm sector. The ground state spectra of the spin zero tetraquark states with various flavor combinations are seen to lie above their respective thresholds.

We calculate the $\pi^0\to \gamma^*\gamma^*$ transition form factor ${\cal F}_{\pi^0\gamma^*\gamma^*}(q_1^2,q_2^2)$ in lattice QCD with two flavors of quarks. Our main motivation is to provide the input to calculate the $\pi^0$-pole contribution to hadronic light-by-light scattering in the muon $(g-2)$, $a_\mu^{\rm HLbL;\pi^0}$. We therefore focus on the region where both photons are spacelike up to virtualities of about $1.5~$GeV$^2$, which has so far not been experimentally accessible. Results are obtained in the continuum at the physical pion mass by a combined extrapolation. We reproduce the prediction of the chiral anomaly for real photons with an accuracy of about $8-9\%$. We also compare to various recently proposed models and find reasonable agreement for the parameters of some of these models with their phenomenological values. Finally, we use the parametrization of our lattice data by these models to calculate $a_\mu^{\rm HLbL;\pi^0}$.