_{c}

^{+}(3900), which decays into J/ψ π

^{+}, and thus cannot be a simple quark-antiquark bound state. In order to elucidate this question, the variational method has to be used with a basis of operators containing both one- and two-meson operators as well as possible tetraquark operators, and this then requires the use of all-to-all propagators (with distillation now being the most commonly used approach) as well as a Lüscher-type method to treat the multiparticle states. These added difficulties mean that studies in this area are still a bit rough at the moment, with the physical-pion, large-volume and continuum limits generally not yet taken. For the Z

_{c}

^{+}, Sasa

*et al.*find a candidate state only when including both two-meson and tetraquark operators in their basis. The more charmonium-like states, such as the X(3872), are better studied, and the X(3872) in particular appears likely to be mostly a DD

^{*}molecule. The greatest challenges in spectroscopy are the mixing between quarkonia and light hadron states, which is still mostly ignored, and the inclusion of more-than-two particle states, for which the theoretical tools aren't quite there yet.

A topical talk on new algorithms for finite-density QCD given by Denes Sexty followed. QCD at finite chemical potential μ suffers from the well-known sign problem; while there are a number of methods to evade it (in particular analytically continuing from imaginary μ and Taylor expansion methods), the newer methods attempt to address it directly. One of these is the complex Langevin method, which responds to the complex action by complexifying the fields and noise term in the Langevin equation (which for gauge links means continuing from SU(N) to SL(N,

**C**) and requires some means of restraining the links from wandering off too far into the unphysical part of the group manifold, e.g. by gauge cooling steps interspersed with the dynamical updates). In the past, this method was hampered by a lack of theoretical understanding and the presence of possibly unphysical runaway trajectories; now, it has been established that for holomorphic actions, the complex Langevin time average does converge to the ensemble average. Unfortunately, the action for QCD with a chemical potential is not holomorphic, but some studies indicate that this case may nevertheless be okay. The other new method to directly address the sign problem is the Lefschetz thimble, which relies on shifting the integration contour for the path integral into the complex plane, and for which simulation algorithms exist in the case of various toy models. For the complex Langevin method, there are now a number of results which look promising.

This was followed by another topical talk, Alberto Ramos speaking about the applications of the Wilson flow to scale setting and renormalization. It has long been known that the Wilson flow yields renormalized operators, and besides its use in setting the lattice scale, it is now widely used to define a renormalized coupling, where the renormalization scale is set by μ

^{2}=1/(8t). To avoid the need for a window where both cut-off and finite-volume effects are small, one can tie the renormalization scale to the volume as μ=1/(cL), however, this means that the boundary conditions become relevant. The errors on the Wilson flow coupling are orders of magnitude smaller than those on the Schrödinger functional coupling, but the SF coupling becomes less noisy at small coupling and thus provides information complementary to that from the WF coupling. Cut-off effects are important for Wilson flow obervables, and tree-level improvement has a big effect there. There is a small-flowtime expansion analogous to the OPE, and a fermionic version of the flow can be used to determine the chiral condensate. All in all, this is a very active field of current research.

After the coffee break, the Ken Wilson Award was announced. The award goes to Gergely Endrődy for

*significant contributions to our understanding of QCD matter in strong magnetic fields and to QCD thermodynamics*. Gergely gave his prize talk on the topic of QCD in magnetic fields, starting from Hofstadter's butterfly, which is a self-similar fractal describing the energy levels accessible to an electron in a crystal (which tries to enforce Bloch waves) in a magnetic field (which tries to enforce Landau levels). The Dirac operator for a free lattice fermion in a magnetic field has a similar structure, which however disappears in the continuum limit, since the magnetic flux through a plaquette scales as a

^{2}. The quark condensate is related to the Dirac eigenvalues, and hence contains the same self-similar structure, which is washed out by the quark mass, however. When QCD interactions are turned on, these similarly wash out the fractal structure. What is left over is a growth of the quark condensate with the magnetic field at zero temperature ("magnetic catalysis"). At finite temperature, a similar effect was expected from models, but Gergely

*et al.*have shown that in fact the opposite effect happens ("inverse magnetic catalysis").

This was followed by Tetsuya Onogi speaking about a hidden exact symmetry of graphene. Graphene, which is the most conductive material known under terrestrial conditions, has a band structure with a Dirac point resembling the dispersion relation for a massless relativistic fermion, with no gap. The symmetry preserving the vanishing of the gap against perturbations can be derived by treating the actual graphene lattice as a staggered version of a coarser hexagonal lattice, where six sites correspond to six internal degrees of freedom (three flavours, two spins), which then reveals a hidden flavour-chiral symmetry.

The afternoon saw the last set of parallel sessions. There were two more talks from members of the Mainz group (PhD student Hanno Horch and former postdoc Gregorio Herdoiza, now a Ramón y Cajal Fellow at the Universida Autónoma de Madrid) on work related to (g-2) and the Adler function.

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