No abstract

We calculate the parameters describing elastic I ¼ 1, P-wave ππ scattering using lattice QCD with 2 þ 1 flavors of clover fermions. Our calculation is performed with a pion mass of m π ≈ 320 MeV and a lattice size of L ≈ 3.6 fm. We construct the two-point correlation matrices with both quark-antiquark and twohadron interpolating fields using a combination of smeared forward, sequential and stochastic propagators. The spectra in all relevant irreducible representations for total momenta j ⃗ Pj ≤ ffiffi ffi 3 p 2π L are extracted with two alternative methods: a variational analysis as well as multiexponential matrix fits. We perform an analysis using Lüscher's formalism for the energies below the inelastic thresholds, and investigate several phase shift models, including possible nonresonant contributions. We find that our data are well described by the minimal Breit-Wigner form, with no statistically significant nonresonant component. In determining the ρ resonance mass and coupling we compare two different approaches: fitting the individually extracted phase shifts versus fitting the t-matrix model directly to the energy spectrum. We find that both methods give consistent results, and at a pion mass of am π ¼ 0.18295ð36Þ stat obtain g ρππ ¼ 5.69ð13Þ stat ð16Þ sys , am ρ ¼ 0.4609ð16Þ stat ð14Þ sys , and am ρ =am N ¼ 0.7476ð38Þ stat ð23Þ sys , where the first uncertainty is statistical and the second is the systematic uncertainty due to the choice of fit ranges.

We report a lattice QCD determination of the πγ → ππ transition amplitude for the P -wave, I = 1 two-pion final state, as a function of the photon virtuality and ππ invariant mass. The calculation was performed with 2 + 1 flavors of clover fermions at a pion mass of approximately 320 MeV, on a 32 3 × 96 lattice with L ≈ 3.6 fm. We construct the necessary correlation functions using a combination of smeared forward, sequential and stochastic propagators, and determine the finite-volume matrix elements for all ππ momenta up to | P | = √ 3 2π L and all associated irreducible representations. In the mapping of the finite-volume to infinite-volume matrix elements using the Lellouch-Lüscher factor, we consider two different parametrizations of the ππ scattering phase shift. We fit the q 2 and s dependence of the infinite-volume transition amplitude in a model-independent way using series expansions, and compare multiple different truncations of this series. Through analytic continuation to the ρ resonance pole, we also determine the πγ → ρ resonant transition form factor and the ρ meson photocoupling, and obtain |Gρπγ| = 0.0802(32)(20). arXiv:1807.08357v3 [hep-lat]

We investigate deep learning autoencoders for the unsupervised recognition of phase transitions in physical systems formulated on a lattice. We focus our investigation on the 2-dimensional ferromagnetic Ising model and then test the application of the autoencoder on the anti-ferromagnetic Ising model. We use spin configurations produced for the 2-dimensional ferromagnetic and anti-ferromagnetic Ising model in zero external magnetic field. For the ferromagnetic Ising model, we study numerically the relation between one latent variable extracted from the autoencoder to the critical temperature Tc. The proposed autoencoder reveals the two phases, one for which the spins are ordered and the other for which spins are disordered, reflecting the restoration of the ℤ2 symmetry as the temperature increases. We provide a finite volume analysis for a sequence of increasing lattice sizes. For the largest volume studied, the transition between the two phases occurs very close to the theoretically extracted critical temperature. We define as a quasi-order parameter the absolute average latent variable z̃, which enables us to predict the critical temperature. One can define a latent susceptibility and use it to quantify the value of the critical temperature Tc(L) at different lattice sizes and that these values suffer from only small finite scaling effects. We demonstrate that Tc(L) extrapolates to the known theoretical value as L →∞ suggesting that the autoencoder can also be used to extract the critical temperature of the phase transition to an adequate precision. Subsequently, we test the application of the autoencoder on the anti-ferromagnetic Ising model, demonstrating that the proposed network can detect the phase transition successfully in a similar way. Graphical abstract