The Yang-Mills gradient flow and the observable E(t) , defined by the square of the field strength tensor at t > 0, are calculated at finite lattice spacing and tree-level in the gauge coupling. Improvement of the flow, the gauge action and the observable are all considered. The results are relevant for two purposes. First, the discretization of the flow, gauge action and observable can be chosen in such a way that O(a 2 ), O(a 4 ) or even O(a 6 ) improvement is achieved. Second, simulation results using arbitrary discretizations can be tree-level improved by the perturbatively calculated correction factor normalized to one in the continuum limit.

We report new results on the conformal properties of an important strongly coupled gauge theory, a building block of composite Higgs models beyond the Standard Model. With twelve massless fermions in the fundamental representation of the SU(3) color gauge group, an infrared fixed point of the β-function was recently reported in the theory [1] with uncertainty in the location of the critical gauge coupling inside the narrow [6.0 < g 2 * < 6.4] interval and widely accepted since as the strongest evidence for a conformal fixed point and scale invariance in the theory with model-building implications. Using the exact same renormalization scheme as the previous study, we show that no fixed point of the β-function exists in the reported interval. Our findings eliminate the only seemingly credible evidence for conformal fixed point and scale invariance in the N f = 12 model whose infrared properties remain unresolved. The implications of the recently completed 5-loop QCD beta function for arbitrary flavor number are discussed with respect to our work. (1) a near-conformal and unexpectedly light scalar particle, perhaps dilaton-like with mass at the Electroweak scale or (2) a parametrically light pseudo NambuGoldstone boson (PNGB) combined with partial compositeness for fermion mass generation to avoid the flavor problem. Both paradigms are based on strongly coupled gauge dynamics to address important aspects of conformal and chiral symmetries and their symmetry breaking patterns in BSM theories. The precise determination of nearconformal or conformal behavior of SU(3) gauge theory with twelve flavors is relevant for both paradigms.

We compute the renormalized running coupling of SU(3) gauge theory coupled to N f = 8 flavors of massless fundamental Dirac fermions. The recently proposed finite volume gradient flow scheme is used. The calculations are performed at several lattice spacings allowing for a controlled continuum extrapolation. The results for the discrete β-function show that it is monotonic without any sign of a fixed point in the range of couplings we cover. As a cross check the continuum results are compared with the wellknown perturbative continuum β-function for small values of the renormalized coupling and perfect agreement is found.

We analyze three sets of gauge ensembles in our extended physics program of a particularly important BSM gauge theory with a fermion doublet in the two-index symmetric (sextet) representation of the SU(3) BSM color gauge group. Our investigations include chiral symmetry breaking (χSB) in the p-regime and ε-regime, the mass of the composite 0 ++ scalar, resonance spectroscopy, new physics from gauge anomaly constraints, and the role of stable sextet BSM baryons with Electroweak interactions in dark matter searches. Important new goals include studies of the 0 ++ scalar entangled with Goldstone dynamics in the p-regime and the ε-regime, the resonance spectrum with particular attention to emerging LHC signals, like recent hints for diphoton excess at 750 GeV or diboson anomalies in the 2 TeV range. All results reported here are preliminary before journal publication including some post-conference material for the discussion.