Improved Lattice Renormalization Group Techniques

We compute the bare step-scaling function $s_b$ for SU(3) lattice gauge theory with $N_f = 12$ massless fundamental fermions, using the non-perturbative Wilson-flow-optimized Monte Carlo Renormalization Group two-lattice matching technique. We use a short Wilson flow to approach the renormalized trajectory before beginning RG blocking steps. By optimizing the length of the Wilson flow, we are able to determine an $s_b$ corresponding to a unique discrete $\beta$ function, after a few blocking steps. We carry out this study using new ensembles of 12-flavor gauge configurations generated with exactly massless fermions, using volumes up to $32^4$. The results are consistent with the existence of an infrared fixed point (IRFP) for all investigated lattice volumes and number of blocking steps. We also compare different renormalization schemes, each of which indicates an IRFP at a slightly different value of the bare coupling, as expected for an IR-conformal theory.Comment: 31st International Symposium on Lattice Field Theory, Lattice201

Finite size scaling and the effect of the gauge coupling in 12 flavor systems

Finite size scaling is a powerful tool to study the critical properties of systems governed by one relevant operator, assuming all irrelevant operators have scaling dimensions much smaller then zero. This condition is likely not satisfied in many-fermion conformal systems where perturbation theory predicts a nearly-marginal irrelevant gauge coupling. In this work we carry out a new investigation of SU(3) lattice gauge theory with 12 fundamental flavors. Analyzing data at many different gauge couplings, our preliminary results indicate that a finite size scaling analysis that takes into account the effect of a nearly-marginal gauge coupling can resolve many of the inconsistencies observed previously in this system, leading to results consistent with conformal infrared dynamics and predicting a mass scaling anomalous around $\gamma_m=0.25$.Comment: Contribution to 31st International Symposium on Lattice Field Theory - LATTICE 201

Scale-dependent mass anomalous dimension from Dirac eigenmodes

We investigate the eigenmodes of the massless Dirac operator to extract the scale-dependent fermion mass anomalous dimension gamma_m(mu). By combining simulations on multiple lattice volumes, and when possible several gauge couplings, we are able to measure the anomalous dimension across a wide range of energy scales. The method that we present is universal and can be applied to any lattice model of interest, including both conformal or chirally broken systems. We consider SU(3) lattice gauge theories with Nf=4, 8 and 12 light or massless fermions. The 4-flavor model behaves as expected for a QCD-like system and demonstrates that systematic effects are manageable in practical lattice calculations. Our 12-flavor results are consistent with the existence of an infrared fixed point, at which we predict the scheme-independent mass anomalous dimension gamma_m^*=0.32(3). For the 8-flavor model we observe a large anomalous dimension across a wide range of energy scales. Further investigation is required to determine whether Nf=8 is chirally broken and walking, or if it possesses a strongly-coupled conformal fixed point.Comment: Version to be published in JHE

Determining the mass anomalous dimension through the eigenmodes of Dirac operator

We define a scale-dependent effective mass anomalous dimension from the scaling of the mode number of the massless Dirac operator, which connects the perturbative $\gamma_m$ of an asymptotically-free system to the universal $\gamma_m^{\star}$ at a conformal fixed point. We use a stochastic algorithm to measure the mode number up to the cutoff scale on lattices as large as $48^4$. Focusing on SU(3) lattice gauge theory with $N_f = 12$ massless fundamental fermions, we examine systematic effects due to finite volumes and non-zero fermion masses. Our results suggest the existence of an infrared fixed point with $\gamma_m^{\star} \approx 0.25$. Our method provides a unique probe to study systems from the UV to the IR. It is universal and can be applied to any lattice model of interest, including both chirally-broken and IR-conformal systems.Comment: 7 pages, 3 figure

Reaching the chiral limit in many flavor systems

We present a brief overview of our recent lattice studies of SU(3) gauge theory with N_f = 8 and 12 fundamental fermions, including some new and yet-unpublished results. To explore relatively unfamiliar systems beyond lattice QCD, we carry out a wide variety of investigations with the goal of synthesizing the results to better understand the non-perturbative dynamics of these systems. All our findings are consistent with conformal infrared dynamics in the 12-flavor system, but with 8 flavors we observe puzzling behavior that requires further investigation. Our new Monte Carlo renormalization group technique exploits the Wilson flow to obtain more direct predictions of a 12-flavor IR fixed point. Studies of N_f = 12 bulk and finite-temperature transitions also indicate IR conformality, while our current results for the 8-flavor phase diagram do not yet provide clear signs of spontaneous chiral symmetry breaking. From the Dirac eigenvalue spectrum we extract the mass anomalous dimension gamma_m, and predict gamma*_m = 0.32(3) at the 12-flavor fixed point. The N_f = 8 system again shows interesting behavior, with a large anomalous dimension across a wide range of energy scales. We use the eigenvalue density to predict the chiral condensate, and compare this approach with direct and partially-quenched measurements.Comment: 7 pages, 5 figures; Contribution to SCGT12 "KMI-GCOE Workshop on Strong Coupling Gauge Theories in the LHC Perspective", 4-7 Dec. 2012, Nagoya Universit

Improving the continuum limit of gradient flow step scaling

We introduce a non-perturbative improvement for the renormalization group step scaling function based on the gradient flow running coupling, which may be applied to any lattice gauge theory of interest. Considering first SU(3) gauge theory with $N_f = 4$ massless staggered fermions, we demonstrate that this improvement can remove $O(a^2)$ lattice artifacts, and thereby increases our control over the continuum extrapolation. Turning to the 12-flavor system, we observe an infrared fixed point in the infinite-volume continuum limit. Applying our proposed improvement reinforces this conclusion by removing all observable $O(a^2)$ effects. For the finite-volume gradient flow renormalization scheme defined by $c = \sqrt{8t} / L = 0.2$, we find the continuum conformal fixed point to be located at $g_\star^2 = 6.2(2)$Comment: 12 pages, 4 figures; Minor changes, published versio

Finite size scaling of conformal theories in the presence of a near-marginal operator

The slowly evolving gauge coupling of gauge-fermion systems near the conformal window makes numerical investigations of these models challenging. We consider finite size scaling and show that this often used technique leads to inconsistent results if the leading order scaling corrections are neglected. When the corrections are included the results become consistent not only between different operators but even when data obtained at different gauge couplings or with different lattice actions are combined. Our results indicate that the SU(3) 12-fermion system is conformal with mass anomalous dimension $\gamma_m=0.235(15)$

Bulk and finite-temperature transitions in SU(3) gauge theories with many light fermions

We investigate finite-temperature transitions in SU(3) lattice gauge theories with Nf=8 and 12 staggered fermions in the fundamental representation. For both of these systems, we have observed a strongly-coupled lattice phase in which the single-site shift symmetry of the staggered action is spontaneously broken. Here we report new results for finite-temperature transitions on 24^3x12 and 32^3x16 lattice volumes, contrasting the 8- and 12-flavor systems. While the Nf=12 finite-temperature transitions accumulate at the bulk transition bounding the strongly-coupled lattice phase, the Nf=8 finite-temperature transitions are able to pass through the bulk transition, and behave as expected for a QCD-like system. We discuss our current results and the work in progress to complete our investigation of the finite-temperature phase diagram.Comment: Contribution to the proceedings of Lattice 2012, June 24-29, Cairns, Australia. 7 page