Cutoff effects of heavy quark vacuum polarization at one-loop order

The charm-quark mass is typically not so far from the cutoff 1/a in lattice simulations. Its determinant may then potentially introduce large cutoff effects. We choose the O(a)-improved Wilson formulation and compute the vacuum polarization effects in two rather different observables at one-loop order. One is the quark-antiquark static force and the other the Schroedinger functional coupling; in addition we investigate two more quantities resulting from the latter. In all the cases the lattice artifacts due to the charm-quark are small when compared to the gluonic effects. This indicates that the inclusion of charm-quarks in dynamical fermion simulations is typically not a problem.Comment: 7 pages, 5 figures, talk presented at the 2011 Lattice conference, Lake Tahoe, Californi

On the spectrum and string tension of U(1) lattice gauge theory in 2+1 dimensions

We calculate the low-lying spectra of glueballs and confining flux tubes in the U(1) lattice gauge theory in 2+1 dimensions. We see that up to modest lattice spacing corrections, the glueball states are consistent with being multiparticle states composed of non-interacting massive JPC=0-- particles. We observe that the ag^2 -> 0 limit is, as expected, unconventional, and follows the well-known saddle-point analysis of Polyakov to a good approximation. The spectrum of closed (winding) flux tubes exhibits the presence of a massive world-sheet excitation whose mass is consistent with that of the bulk screening mass. These U(1) calculations are intended to complement existing lattice calculations of the properties of SU(N) and SO(N) gauge theories in D=2+1.Comment: 39 pages; 15 figures. Extra discussion, calculation, figures and reference

Closed flux tubes in D=2+1 SU(N) gauge theories: dynamics and effective string description

We extend our earlier calculations of the spectrum of closed flux tubes in SU(N) gauge theories in 2+1 dimensions, with a focus on questions raised by recent theoretical progress on the effective string action of long flux tubes and the world-sheet action for flux tubes of moderate lengths. Our new calculations in SU(4) and SU(8) provide evidence that the leading O(1/l^gamma) non-universal correction to the flux tube ground state energy does indeed have a power gamma greater than or equal to 7. We perform a study in SU(2), where we can traverse the length at which the Nambu-Goto ground state becomes tachyonic, to obtain an all-N view of the spectrum. Our comparison of the k=2 flux tube excitation energies in SU(4) and SU(6) suggests that the massive world sheet excitation associated with the k=2 binding has a scale that knows about the group and hence the theory in the bulk, and we comment on the potential implications of world sheet massive modes for the bulk spectrum. We provide a quantitative analysis of the surprising (near-)orthogonality of flux tubes carrying flux in different SU(N) representations, which implies that their screening by gluons is highly suppressed even at small N.Comment: 72 pages, including 42 figure

On the mass of the world-sheet `axion' in SU(N) gauge theories in 3+1 dimensions

There is numerical evidence that the world sheet action of the confining flux tube in D=3+1 SU(N) gauge theories contains a massive excitation with 0- quantum numbers whose mass shows some decrease as one goes from SU(3) to SU(5). It has furthermore been shown that this particle is naturally described as arising from a topological interaction term in the world-sheet action, so that one can describe it as being `axion'-like. Recently it has been pointed out that if the mass of this `axion' vanishes as N -> oo then it becomes possible for the world sheet theory to be integrable in the planar limit. In this paper we perform lattice calculations of this `axion' mass from SU(2) to SU(12), which allows us to make a controlled extrapolation to N=oo and so test this interesting possibility. We find that the `axion' does not in fact become massless as N -> oo. So if the theory is to possess planar integrability then it must be some other world sheet excitation that becomes massless in the planar limit.Comment: 14 pages, 2 tables, 3 figures; some typos corrected plus minor clarification

On the weak N-dependence of SO(N) and SU(N) gauge theories in 2+1 dimensions

We consider (continuum) mass ratios of the lightest `glueballs' as a function of N for SO(N) and SU(N) lattice gauge theories in D=2+1. We observe that the leading large N correction is usually sufficient to describe the N-dependence of SO(N.geq.3) and SU(N.geq.2), within the errors of the numerical calculation. Just as interesting is the fact that the coefficient of this correction almost invariably turns out to be anomalously small, for both SO(N) and SU(N). We point out that this can follow naturally from the strong constraints that one naively expects from the Lie algebra equivalence between certain SO(N) and SU(N') theories and the equivalence of SO(infinity) and SU(infinity). The same argument for a weak N-dependence can in principle apply to SU(N) and SO(N) gauge theories in D=3+1.Comment: 17 pages, 6 figures. Clearer discussion and extra, relevant reference

Topological charge using cooling and the gradient flow

The equivalence of cooling to the gradient flow when the cooling step $n_c$ and the continuous flow step of gradient flow $\tau$ are matched is generalized to gauge actions that include rectangular terms. By expanding the link variables up to subleading terms in perturbation theory, we relate $n_c$ and $\tau$ and show that the results for the topological charge become equivalent when rescaling $\tau \simeq n_c/({3-15 c_1})$ where $c_1$ is the Symanzik coefficient multiplying the rectangular term. We, subsequently, apply cooling and the gradient flow using the Wilson, the Symanzik tree-level improved and the Iwasaki gauge actions to configurations produced with $N_f=2+1+1$ twisted mass fermions. We compute the topological charge, its distribution and the correlators between cooling and gradient flow at three values of the lattice spacing demonstrating that the perturbative rescaling $\tau \simeq n_c/({3-15 c_1})$ leads to equivalent results.Comment: 21 pages, 10 figure

The critical temperature of the 2D-Ising model through Deep Learning Autoencoders

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 $T_c$. 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 $\mathbb{Z}_2$ 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 ${\tilde z}$, 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 $T_c(L)$ at different lattice sizes and that these values suffer from only small finite scaling effects. We demonstrate that $T_c(L)$ extrapolates to the known theoretical value as $L \to \infty$ 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.Comment: 17 pages, 14 figures, accepted for publication in Eur. Phys. J.