We calculate the step scaling function, the lattice analog of the renormalization group $β$-function, for an SU(3) gauge theory with ten fundamental flavors. We present a detailed analysis including the study of systematic effects of our extensive data set generated with ten dynamical flavors using the Symanzik gauge action and three times stout smeared Möbius domain wall fermions. Using up to $32^4$ volumes, we calculate renormalized couplings for different gradient flow schemes and determine the step-scaling $β$ function for a scale change $s=2$ on up to five different lattice volume pairs. In an accompanying paper we discuss that gradient flow can promote lattice dislocations to instanton-like objects, introducing nonperturbative lattice artifacts to the step scaling function. Motivated by the observation that Wilson flow sufficiently suppresses these artifacts, we choose Wilson flow with the Symanzik operator as our preferred analysis. We study systematic effects by calculating the step-scaling function based on alternative flows (Zeuthen or Symanzik), alternative operators (Wilson plaquette, clover), and also explore the effects of the perturbative tree-level improvement. Further we investigate the effects due to the finite value of $L_s$.
Scaling behavior is analyzed for the two-flavor QCD lattice gauge theory chiral transition. Leading scaling behavior and correction to leading scaling from lattice spacing effects are examined for the quark condensate. Scaling predictions under the assumption of quark mass dominance are tested for the longitudinal correlation length. Second order scaling behavior is consistent with present data.
The finite temperature transition of QCD with two degenerate light quarks is studied on the lattice with a renormalization group improved gauge action and the Wilson quark action. We have made simulations on an $8^3\times 4$ lattice near the chiral transition point. It is shown that the chiral condensate which is the order parameter of chiral symmetry satisfies remarkably a scaling relation with the exponents of the three dimensional O(4) Heisenberg model. This indicates that the chiral transition in two-flavor QCD is of second order in the continuum limit.
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…
Nonperturbative determinations of the renormalization group (RG) $β$ function are crucial to understand properties of gauge-fermion systems at strong coupling and connect lattice simulations and the perturbative ultraviolet regime. Choosing well-understood, QCD-like systems with SU(3) gauge group and either six or four fundamental flavors, we investigate their step-scaling $β$ function. In both cases we push the simulations to the boundary of chiral symmetry breaking and study the regime $g^2_{GF} \lesssim 8.2$ with six, and $g^2_{GF} \lesssim 6.6$ with four flavors. We carefully consider the …
The question of the exact nature of the phase transition in two-flavor QCD is still under discussion. Recent results for small quark masses in simulations with 2+1 flavors show scaling behavior consistent with the O(4) or O(2) universality class. For a precise determination, an assessment of deviations from the ideal scaling behavior due to finite quark masses and finite simulation volumes is necessary. We study the scaling behavior at the chiral phase transition with an effective quark-meson model. In our Renormalization Group approach, the quark masses in the model can be varied from the chi…
We show that for small values of the chemical potential the curvature of the phase transition line can be deduced from an analysis of scaling properties of the chiral condensate and its susceptibilities. We make use of a recent analysis of the magnetic equation of state in (2+1)-flavor QCD where a connection between the QCD parameters and the universal scaling fields could be established. The remaining dependence of the reduced temperature on the chemical potential can be fixed by an analysis of a mixed susceptibility, obtained from a derivative with respect to quark mass and chemical potentia…
We investigate the nature of the chiral phase transition in the massless two-flavor QCD using the renormalization group improved gauge action and the Wilson quark action on $32^3\times 16$, $24^3\times 12$, and $16^3\times 8$ lattices. Based on the renormalization group equation, we derive the scaling relation for the effective masses of mesons at the chiral phase transition point. If the chiral phase transition is second order, the effective masses as a function of the rescaled time/space do not depend on the lattice size and show the universal behavior. We find that our numerical simulations…
The chiral transition for two-flavor QCD is predicted to be in the same universality class as the 3d O(4) model. This prediction is verified in the Wilson case, but not for the staggered-fermion case. The comparison is usually done assuming infinite-volume behavior. Here we make an analysis of existing staggered-fermion data using finite-size scaling and normalizing the QCD data. We find better agreement for larger quark masses.
We study scaling behavior of a chiral order parameter performing a simulation of two-flavor QCD with improved Wilson quarks. It has been shown that the scaling behavior of the chiral order parameter defined by a Ward-Takahashi identity agrees with the scaling function of the three-dimensional O(4) spin model at zero chemical potential. We extend the scaling study to finite density QCD. Calculating derivatives of the chiral order parameter with respect to the chemical potential in two-flavor QCD, the scaling property of chiral phase transition is discussed in the low density region. We moreover…
Extended simulation results and their analysis are reported in a strongly coupled gauge theory with twelve fermion flavors in the fundamental SU(3) color representation. The conformality of the model is probed using mass deformed conformal finite size scaling (FSS) theory driven by the fermion mass anomalous dimension. Two independent conformal FSS fitting procedures are used in the analysis. The first one deploys physics motivated scaling functions, complemented by a second fitting procedure with spline based general B-forms for the scaling functions. The results at fixed gauge coupling show …
We report on a scaling test of several mesonic observables in the non-perturbatively O(a) improved Wilson theory with two flavors of dynamical quarks. The observables are constructed in a fixed volume of 2.4fm x (1.8fm)^3 with Schroedinger functional boundary conditions. No significant scaling violations are found. Using the kaon mass determined in \cite{cernI}, we update our estimate of the Lambda parameter to Lambda^(2)_{msbar}/m_K = 0.52(6).
I carry out a finite-size scaling study of the correlation length in SU(3) lattice gauge theory coupled to 12 fundamental flavor fermions, using recent data published by Fodor, Holland, Kuti, Nogradi and Schroeder. I make the assumption that the system is conformal in the zero-mass, infinite volume limit, that scaling is violated by both nonzero fermion mass and by finite volume, and that the scaling function in each channel is determined self-consistently by the data. From several different observables I extract a common exponent for the scaling of the correlation length xi with the fermion m…
We present state-of-the-art results for the matrix elements of flavor diagonal tensor operators within the nucleon state. The calculation of the dominant connected contribution is done using eleven ensembles of gauge configurations generated by the MILC Collaboration using the highly improved staggered quark (HISQ) action with 2+1+1 dynamical flavors. The calculation of the disconnected contributions is done using seven (six) ensembles for the strange (light) quarks. These high-statistics simulations allowed us to address various systematic uncertainties. A simultaneous fit in the lattice spac…
We investigate the discrete $β$ function of the 2-flavor SU(3) sextet model using the finite volume gradient flow scheme. Our results, using clover improved nHYP smeared Wilson fermions, follow the (non-universal) 4-loop $\overline{\textrm{MS}}$ perturbative predictions closely up to $g^2 \approx 5.5$, the strongest coupling reached in our simulation. At strong couplings the results are in tension with a recently published work using the same gradient flow renormalization scheme with staggered fermions. Since these calculations define the discrete $β$ function in the same continuum renormaliza…
We study scaling behavior of a chiral order parameter in the low density region, performing a simulation of two-flavor QCD with improved Wilson quarks. The scaling behavior of the chiral order parameter defined by a Ward-Takahashi identity agrees with the scaling function of the three-dimensional O(4) spin model at zero chemical potential. We extend the scaling study to finite density QCD. Applying the reweighting method and calculating derivatives of the chiral order parameter with respect to the chemical potential, the scaling properties of the chiral phase transition are discussed in the lo…
We explore the phase diagram spanned by the temperature and the chemical potential for 4-flavor QCD by the phase-reweighting approach. In order to determine the order of phase transition, we perform finite size scaling studies for various quantities, for example, susceptibility, kurtosis and Challa-Landau-Binder cumulant. At the parameter (beta=1.60, kappa=0.1371, c_sw=1.9655 and N_T=4), where the Kentucky group reported a first-order phase transition in their canonical simulation, we observe that the transition is consistent with being of first order.
We study the curvature of the chiral transition/crossover line between the low-temperature hadronic phase and the high-temperature quark-gluon-plasma phase at low densities, performing simulations of two-flavor QCD with improved Wilson quarks. After confirming that the chiral order parameter defined by a Ward-Takahashi identity is consistent with the scaling of the O(4) universality class at zero chemical potential, we extend the scaling analysis to finite chemical potential to determine the curvature of the chiral transition/crossover line at low densities assuming the O(4) universality. To c…
We report the current status of the on-going lattice-QCD calculations of nucleon isovector axial charge, g_A, using the RBC/UKQCD 2+1-flavor dynamical domain-wall fermion ensembles at lattice cutoff of about a^{-1}=1.4 GeV in a spatial volume (L = 4.6 fm)^3. The result from the ensemble with m_π= 250 MeV pion mass, corresponding to the finite-size scaling parameter m_πL \sim 5.8, agrees well with an earlier result at a^{-1}=1.7 GeV, L = 2.8 fm, and m_π= 420 MeV, with similar m_πL. This suggests the systematic error from excited-state contamination is small in both ensembles and about 10-% defi…
The order of the phase transition in finite-temperature QCD with two degenerate light quarks is still an open problem and corresponds to the last question mark in the zero-density phase diagram of QCD. We argue that establishing the nature of the transition in this case is also a crucial test for numerical simulations of lattice QCD, allowing precise estimates of possible systematic errors related e.g. to the choice of fermion-simulation algorithm or of discretized formulation for fermions.
Walking technicolor theory attempts to realize electroweak symmetry breaking as the spontaneous chiral symmetry breakdown caused by the gauge dynamics with slowly varying gauge coupling constant and large mass anomalous dimension. Many-flavor QCD is one of the candidates owning these features. We focus on the SU(3) gauge theory with ten flavors of massless fermions in the fundamental representation, and compute the gauge coupling constant in the Schrödinger functional scheme. Numerical simulation is performed with $O(a)$-unimproved lattice action, and the continuum limit is taken in linear in …
As a computationally less costly test case for full QCD, we investigate an SU(3) Yang-Mills theory coupled to a bosonic spinor field. This theory corresponds to QCD with minus two quark flavors and is known as the bermion model. Our central object of interest is the step scaling function which describes the scale evolution of the running coupling in the Schrodinger functional scheme. We develop a suitable algorithm for the simulation of O(a) improved bermions and compare its performance with unimproved bermions and full QCD. We study in detail the lattice artefacts and the continuum extrapolat…
We calculate the step scaling function, the lattice analog of the renormalization group $β$-function, for an SU(3) gauge theory with twelve flavors. The gauge coupling of this system runs very slowly, which is reflected in a small step scaling function, making numerical simulations particularly challenging. We present a detailed analysis including the study of systematic effects of our extensive data set generated with twelve dynamical flavors using the Symanzik gauge action and three times stout smeared Möbius domain wall fermions. Using up to $32^4$ volumes, we calculate renormalized couplin…
We present our preliminary study of the SU(2) gauge theory with 8 flavors of fermions in fundamental representation. This theory could be a candidate of the gauge theory with conformal fixed point. By using Wilson/Polyakov loop in a finite volume with twisted boundary conditions, we study the renormalization group flow of the gauge coupling constant. Our calculation gives consistent result with the perturbative prediction of the running coupling in the weak coupling region. We investigate a possible signal for conformal behavior in the strong coupling region.
New tests are applied to two $β$-functions of the much-discussed BSM model with ten massless fermion flavors in the fundamental representation of the SU(3) color gauge group. The renormalization scheme of the two $β$-functions is defined on the gauge field gradient flow in respective finite or infinite physical volumes at zero lattice spacing. Recently published results in the ten-flavor theory led to indicators of an infrared fixed point (IRFP) in the finite-volume step $β$-function in the strong coupling regime of the theory arXiv:2004.00754. We analyze our substantially extended set of ten-…
Report is made of a systematic scaling study of the finite-temperature chiral phase transition of two-flavor QCD with the Kogut-Susskind quark action based on simulations on $L^3\times4$ ($L$=8, 12 and 16) lattices at the quark mass of $m_q=0.075, 0.0375, 0.02$ and 0.01. Our finite-size data show that a phase transition is absent for $m_q\geq 0.02$, and quite likely also at $m_q=0.01$. The scaling behavior of susceptibilities as a function of $m_q$ is consistent with a second-order transition at $m_q=0$. However, the exponents deviate from the O(2) or O(4) values theoretically expected.
We present a study of the flavor symmetry breaking in the pion spectrum for various improved staggered fermion actions. To study the effects of link fattening and tadpole improvement, we use three different variants of the p4 action - p4fat3, p4fat7, and p4fat7tad. These are compared to Asqtad and also to naive staggered. To study the pattern of symmetry breaking, we measure all 15 meson masses in the 4-flavor staggered theory. The measurements are done on a quenched gauge background, generated using a one-loop improved Symanzik action with $β=10/g^2 = 7.40, 7.75,$ and 8.00, corresponding to l…
SU(3) gauge theory with $N_f$ fermions in the fundamental representation serves as a theoretical testing ground for possible infrared conformal behavior, which could play a role in BSM composite Higgs models. We use lattice simulations to study the 10-flavor model, for which it has been claimed there is an infrared fixed point in the gauge coupling $β$-function. Our results suggest the opposite conclusion, namely we find no $β$-function fixed point in the explored range, with qualitative agreement with the 5-loop $\overline{MS}$ prediction. We comment on the inconsistency between our findings …
We apply the Grassmann tensor renormalization group (GTRG) to the one-flavor lattice Gross-Neveu model in the presence of chemical potential. We compute the fermion number density and its susceptibility and confirm the validity of GTRG for the finite density system. We introduce a method analogous to the reweighting method for Monte Carlo method and test it for some parameters.
We present results of the location of the closest singularities in the complex chemical potential plane using a novel method. These results are obtained with (2+1)-flavor of highly improved staggered quarks (HISQ) on lattices with temporal extent of Nt=4,6. We show that the scaling is consistent with the expected scaling of the Lee-Yang edge singularities in the vicinity of the Roberge-Weiss (RW) transition. We determine various non-universal parameters using 3D Ising model scaling functions that map QCD in the scaling region of the RW transition. Furthermore, as a preliminary result we discus…
We present first results for lattice simulations, on a single volume, of the low-lying spectrum of an SU(3) Yang-Mills gauge theory with ten light fermions in the fundamental representation. Fits to the fermion mass dependence of various observables are found to be globally consistent with the hypothesis that this theory is within or just outside the strongly-coupled edge of the conformal window, with mass anomalous dimension consistent with 1 over the range of scales simulated. We stress that we cannot rule out the possibility of spontaneous chiral-symmetry breaking at scales well below our i…
The mode number of the Dirac operator scales with an exponent related to the mass anomalous dimension gamma_m. This relation holds both in IR-conformal systems, as well as in confining systems for large enough eigenvalues. We investigate the Nf=4, 8 and 12 flavor SU(3) systems at several couplings near the chiral limit, and show that in general the scaling exponent varies with the eigenvalue, describing the dependence of gamma_m on the energy (or, equivalently, on the running coupling). This energy dependence can be explored even with fixed lattice parameters (bare coupling and mass). We find …
We present a new lattice study of the discrete beta function for SU(3) gauge theory with Nf=8 massless flavors of fermions in the fundamental representation. Using the gradient flow running coupling, and comparing two different nHYP-smeared staggered lattice actions, we calculate the 8-flavor step-scaling function at significantly stronger couplings than were previously accessible. Our continuum-extrapolated results for the discrete beta function show no sign of an IR fixed point up to couplings of g^2~14. At the same time, we find that the gradient flow coupling runs much more slowly than pre…
We present results for the nucleon axial charge g_A at a fixed lattice spacing of 1/a=1.73(3) GeV using 2+1 flavors of domain wall fermions on size 16^3x32 and 24^3x64lattices (L=1.8 and 2.7 fm) with length 16 in the fifth dimension. The length of the Monte Carlo trajectory at the lightest m_πis 7360 units, including 900 for thermalization. We find finite volume effects are larger than the pion mass dependence at m_π= 330 MeV. We also find that g_A exhibits a scaling with the single variable m_πL which can also be seen in previous two-flavor domain wall and Wilson fermion calculati ons. Using …
The flavor singlet axial and tensor charges of the nucleon are calculated in lattice QCD. We find $ΔΣ=Δu+Δd+Δs=+0.638(54)-0.347(46)-0.109(30) = +0.18(10)$ for the axial charge and $δΣ=δu+δd+δs=+0.839(60)-0.231(55)-0.046(34) = +0.562(88)$ for the tensor charge. The result for the axial charge shows reasonable agreement with the experiment and that for the tensor charge is the first prediction from lattice QCD before experimental measurements.
We discuss universal scaling properties of (2+1)-flavor QCD in the vicinity of the chiral phase transition at vanishing as well as non-vanishing light quark chemical potential (mu_l). We provide evidence for O(N) scaling of the chiral order parameter in (2+1)-flavor QCD and show that the scaling analysis of its derivative with respect to the light quark chemical potential provides a unique approach to the determination of the curvature of the chiral phase transition line in the vicinity of mu_l/T=0.
We have calculated the step scaling function and the running coupling of QCD in the Schroedinger functional scheme with four flavors of O(a) improved Wilson quarks. Comparisons of our non-perturbative results with 2-loop and 3-loop perturbation theory as well as with non-perturbative data for only two flavors are made.
I discuss the important role that Lattice QCD plays in testing the Flavor sector of the Standard Model (SM) and in indirect searches of New Physics. I review in particular the Unitarity Triangle Analysis performed by the UTfit collaboration within and beyond the SM, presenting the recent accurate lattice results that enter the analyses. I conclude with a tentative outlook to the further progresses that we can expect in the next years from Flavor Lattice QCD.
By using the lattice Monte-Carlo simulation, we investigate the finite temperature (T) chiral phase transition at color SU(3) gauge theories with various species of fundamental fermions, and discuss the signal of the (pre-)conformality at large Nf (num. of flavors.) via their comparisons. As Nf increases, we observe stronger fermion screening effects which result from a larger fermion multiplicity. We investigate a finite T step-scaling associated with a uniqueness of the critical temperature (Tc) at each Nf, then the vanishing step-scaling indicates the emergence of the conformality around Nf…
We use lattice simulations to study SU(3) gauge theory with 13 massless fermions in the fundamental representation. We present evidence that the theory is conformal with a non-zero infrared fixed point in the gauge coupling. We use a newly-developed technique to calculate the mass anomalous dimension at the fixed point via step-scaling of the mode number, allowing us to take the continuum limit and compare to perturbative predictions. We comment on the relevance of these findings to the extended search for the conformal window in the fundamental representation and in particular 12 massless fla…
We discuss the universal critical behavior in (2+1)-flavor QCD by analyzing lattice data from improved staggered fermions generated by the HotQCD Collaboration. We present recent results from two different lattice discretizations and various lattice spacings ($N_τ=6,8,12$) at fixed physical strange quark mass ($m_s$) but varying light quark mass ($m_l$). We find that the chiral order-parameter, i.e. the chiral condensate, shows the expected universal scaling that is associated with the critical point in the chiral limit already for light quark masses $m_l/m_s \lsim 0.05$. From an analysis of t…
We test the asymptotic scaling behavior of state-of-the-art simulations of QCD with two flavors of light Wilson fermions. This is done by matching $π$ and $ρ$ masses on lattices of size $16^3\times32$ and $8^3\times16$. We find that at $β=6/g^2=5.3$ matching is not possible over a range extending down to $β=3.5$. The large lattice data at $β=5.5$ matches the small lattice values at $β=4.9(1)$ leading to a shift $Δβ=0.6(1)$, considerably larger than the perturbative prediction of $0.45$. In both cases we conclude that the simulations are very far from the asymptotic scaling region.
The gradient flow renormalized coupling offers a simple and relatively inexpensive way to calculate the step scaling function and the lattice scale, but both applications can be hindered by large lattice artifacts. Recently we introduced an empirical non-perturbative improvement that can reduce, even remove $\mathcal{O}(a^2)$ lattice artifacts. The method is easy to implement and can be applied to any lattice gauge theory of interest both in step scaling studies and for scale setting. In this talk I will briefly review this improvement method and discuss its application for determining the dis…
We report extended simulation results and their new analysis in two important gauge theories with twelve fermion flavors in the fundamental SU(3) color representation and two fermions in the sextet representation. We probe the $N_f=12$ model with respect to the conformal window using mass deformed finite size scaling (FSS) theory driven by the fermion mass anomalous dimension. Our results at fixed gauge coupling show problems with the conformal scenario of the $N_f=12$ model. In the sextet model with two flavors, under the conformal hypothesis, we determine large values for the anomalous fermi…
We apply the Grassmann tensor renormalization group to the lattice regularized Schwinger model with one-flavor of the Wilson fermion. We study the phase diagram in the $(β,κ)$ plane performing a detailed analysis of the scaling behavior of the Lee-Yang zeros and the peak height of the chiral susceptibility. Our results strongly indicate that the whole range of the phase transition line starting from $(β,κ)=(0.0,0.380665(59))$ and ending at $(\infty,0.25)$ belongs to the two-dimensional Ising universality class similarly to the free fermion case.
We present a scaling study of the QCD spectrum using a smeared P4 staggered fermion formulation, in which three, five, and seven-link staples are added to reduce the effects of flavor symmetry breaking. These studies are performed on quenched lattices generated using the one-loop improved Symanzik gauge action, with Beta=10/g^2=7.40,7.75,8.00. The corresponding lattice spacings are a = .31 fm, .21 fm, and .14 fm. Particularly, we study the O(a^2) flavor symmetry violations in the pion spectrum as well as the scaling dependence of m_rho, measured in physical units, on lattice spacing. These res…
Tensor charge of the nucleon, which will be measured in Drell-Yan processes in polarized proton-proton collisions at RHIC, are studied in quenched lattice QCD simulation. On the 16$^3\times$ 20 lattice with $β=5.7$, connected parts of the tensor charge are determined with small statistical error, while the disconnected parts are found to be small with relatively large error bars. Flavor-singlet tensor charge ($δΣ= δu + δd + δs$) is not suppressed as opposed to the flavor singlet axial charge ($ΔΣ= Δu + Δd + Δs$).
We address the interplay between local and global symmetries in determining the continuum limit of two-dimensional lattice scalar theories characterized by $SO(N_c)$ gauge symmetry and non-Abelian $O(N_f)$ global invariance. We argue that, when a quartic interaction is present, the continuum limit of these model corresponds in some cases to the gauged non-linear $σ$ model field theory associated with the real Grassmannian manifold $SO(N_f)/(SO(N_c)\times SO(N_f-N_c)$), which is characterized by the invariance under the color-flavor reflection $N_c\leftrightarrow N_f-N_c$. Monte Carlo simulatio…
The observed slow running of the gauge coupling in SU(3) lattice gauge theory with two flavors of color sextet fermions naturally suggests it is a theory with one relevant coupling, the fermion mass, and that at zero mass correlation functions decay algebraically. I perform a finite-size scaling study on simulation data at two values of the bare gauge coupling with this assumption and observe a common exponent for the scaling of the correlation length with the fermion mass, y_m ~ 1.5. An analysis of the scaling of valence Dirac eigenvalues at one of these bare couplings produces a similar numb…
I review recently completed (since Lattice 2013) and ongoing lattice calculations in charm and bottom flavor physics. A comparison of the precision of lattice and experiment is made using both current experimental results and projected experimental precision in 2020. The combination of experiment and theory reveals several tensions between nature and the Standard Model. These tensions are reviewed in light of recent lattice results.
We present first results for the step scaling function sigma_P of the renormalization factor Z_P of the pseudoscalar density. The simulations are performed within the framework of the Schroedinger functional with two flavors of O(a) improved Wilson fermions. The knowledge of sigma_P is required to compute the renormalization group invariant quark masses. We also study the performance of a variant of the HMC algorithm using two pseudofermion fields.
In this paper, we discuss the role of an external magnetic field on the dynamically generated fermion mass in even-flavor QED in three space-time dimensions. Based on some reasonable approximations, we present analytic arguments on the fact that, for weak fields, the magnetically-induced mass increases quadratically with increasing field, while at strong fields one crosses over to a mass scaling logarithmically with the external field. We also confirm this type of scaling behavior through quenched lattice calculations using the non-compact version for the gauge field. Both the zero and finite …
We report on a study of two-flavor finite-temperature chiral phase transition employing the Kogut-Susskind quark action and the plaquette gluon action in lattice QCD for a lattice with $N_t=4$ temporal size. Hybrid R simulations of $10^4$ trajectories are made at quark masses of $m_q=0.075, 0.0375, 0.02, 0.01$ in lattice units for the spatial sizes $8^3, 12^3$ and $16^3$. The spatial size dependence of various susceptibilities confirm the previous conclusion of the absence of a phase transition down to $m_q=0.02$. At $m_q=0.01$ an increase of susceptibilities is observed up to the largest volu…
Nonperturbative lattice field theory simulations provide a systematic framework to investigate properties of conformal systems at strong couplings. These simulations can be performed using different lattice discretizations. Here we present numerical results for the step scaling beta function in SU(3) gauge theories with ten and twelve fundamental flavors. We calculate the renormalized $β$ function in the finite volume gradient flow renormalization scheme. Using Möbius domain wall fermions with Symanzik gauge action, Zeuthen gradient flow, and perturbative tree-level improvement, we implement a…
We study the renormalization group properties of SU(3) gauge theories with N_f=8 and 12 nearly-massless fermions, using Monte Carlo Renormalization Group (MCRG) two-lattice matching techniques to predict bare step-scaling functions s_b. Traditional MCRG two-lattice matching requires that the renormalization scheme be optimized for each bare lattice coupling, so that s_b is a composite of many different discrete beta functions. We propose an improved procedure that uses the Wilson flow to eliminate the need for this optimization of the RG blocking transformation. While our 12-flavor results ind…
It is expected that when the number of light flavors of gauge theories is increased near or beyond some critical value, new and interesting behavior occurs. We discuss the qualitative properties of the RG flows for a local $SU(3)$ theory with $N_f$ light fundamental flavors for $N_f$ near 12. We discuss the realization of the chiral symmetry and remind that the Wigner mode seems an unlikely alternative. We propose to use a linear sigma model to describe the light $σ$ masses found in recent lattice calculations for $N_f$=8 and 12. Progress made after the conference are reported in arXiv:1709.09…
