Search

We present a lattice QCD calculation of the hadronic light-by-light scattering contribution to the anomalous magnetic moment of the muon using $N_f=2+1+1$ flavors of staggered quarks with masses tuned to their physical values. Our final result, in the continuum limit, reads $a_{\mu}^{\mathrm{hlbl}} = 125.5(11.6)_{\mathrm{stat}}(0.4)_{\mathrm{syst}} \times 10^{-11}$ where the first error is statistical and the second is systematic. Light, strange and charm-quark contributions are considered. In addition to the connected and leading disconnected contributions, we also include an estimate of the sub-leading disconnected diagrams. Our result is compatible with previous lattice QCD and data-driven dispersive determinations., Comment: 28 pages, 18 figures

The QCD cross-over line in the temperature ($T$) -- baryo-chemical potential ($\mu_B$) plane has been computed by several lattice groups by calculating the chiral order parameter and its susceptibility at finite values of $\mu_B$. In this work we focus on the deconfinement aspect of the transition between hadronic and Quark Gluon Plasma (QGP) phases. We define the deconfinement temperature as the peak position of the static quark entropy ($S_Q(T,\mu_B)$) in $T$, which is based on the renormalized Polyakov loop. We extrapolate $S_Q(T,\mu_B)$ based on high statistics finite temperature ensembles on a $16^3\times 8$ lattice to finite density by means of a Taylor expansion to eighth order in $\mu_B$ (NNNLO) along the strangeness neutral line. For the simulations the 4HEX staggered action was used with 2+1 flavors at physical quark masses. In this setup the phase diagram can be drawn up to unprecedentedly high chemical potentials. Our results for the deconfinement temperature are in rough agreement with phenomenological estimates of the freeze-out curve in relativistic heavy ion collisions. In addition, we study the width of the deconfinement crossover. We show that up to $\mu_B \approx 400$ MeV, the deconfinement transition gets broader at higher densities, disfavoring the existence of a deconfinement critical endpoint in this range. Finally, we examine the transition line without the strangeness neutrality condition and observe a hint for the narrowing of the crossover towards large $\mu_B$., Comment: 14 pages, 8 figures

We present a lattice calculation of the $\eta$ and $\eta^{\prime}$ meson masses at the physical point and in the continuum limit, based on $N_f = 2+1+1$ flavors of rooted staggered quarks. Our analysis includes gauge ensembles at the physical pion and kaon masses spread over six lattice spacings in the range [0.064-0.1315]~fm. Our main results read $m_{\eta} = 543.5(5.6)~$MeV and $m_{\eta^{\prime}} = 986(38)~$MeV, consistent with the experimental values. This is an important numerical test that supports the validity of the fourth root procedure used in the staggered quark formalism. This calculation was the first step towards extracting the pseudoscalar transition form factors of the $\eta$ and $\eta^{\prime}$ mesons that play a crucial role in the hadronic light-by-light contribution to the muon $g-2$., Comment: 19 pages, 12 figures

The precision with which hadronic vacuum polarization (HVP) is obtained determines how accurately important observables, such as the muon anomalous magnetic moment, a_\mu, or the low-energy running of the electromagnetic coupling, \alpha, are predicted. The two most precise approaches for determining HVP are: dispersive relations combined with e+e- to hadrons cross-section data, and lattice QCD. However, the results obtained in these two approaches display significant tensions, whose origins are not understood. Here we present a framework that sheds light on this issue and, if the two approaches can be reconciled, allows them to be combined. Via this framework, we test the hypothesis that the tensions can be explained by modifying the R-ratio in different intervals of center-of-mass energy sqrt(s). As ingredients, we consider observables that have been precisely determined in both approaches. These are the leading hadronic contributions to a_\mu, to the so-called intermediate window observable and to the running of \alpha between spacelike virtualities 1GeV^2 and 10GeV^2 (for which only a preliminary lattice result exists). Our tests take into account all uncertainties and correlations, as well as uncertainties on uncertainties in the lattice results. Among our findings, the most striking is that results obtained in the two approaches can be made to agree for all three observables by modifying the \rho peak in the experimental spectrum. In particular, we find that this requires a common ~5\% increase in the contributions of the peak to each of the three observables. This finding is robust against the presence or absence of one of the constraining observables. However, such an increase is much larger than the uncertainties on the measured R-ratio. We also discuss a variety of generalizations of the methods used here, as well as the limits in the information that can be extracted..., Comment: 38 pages, 8 figures

In this paper we present a first ab-initio calculation of the $\pi^0$, $\eta$ and $\eta^{\prime}$ transition form factors performed with physical light-quark masses. We provide a complete parametrization of the form factors that includes both single and double-virtual kinematics. Our results are compared with experimental measurements of the form factors in the space-like region and with the measured two-photon decay widths. In a second step, our parametrizations of the transition form factors are used to compute the dominant pseudoscalar-pole contributions to the hadronic light-by-light scattering in the muon $g-2$. Our final result reads $a_{\mu}^{\rm hlbl, ps-pole} = (85.1 \pm 5.2) \times 10^{-11}$. Although the pion-pole is dominant, we confirm that, together, the $\eta$ and $\eta^{\prime}$ provide roughly half of its contribution., Comment: 29 pages, 16 figures. Typos corrected. Add figure 6

Since the release of the 2015 Long Range Plan in Nuclear Physics, major events have occurred that reshaped our understanding of quantum chromodynamics (QCD) and nuclear matter at large densities, in and out of equilibrium. The US nuclear community has an opportunity to capitalize on advances in astrophysical observations and nuclear experiments and engage in an interdisciplinary effort in the theory of dense baryonic matter that connects low- and high-energy nuclear physics, astrophysics, gravitational waves physics, and data science, Comment: 70 pages, 3 figures, White Paper for the Long Range Plan for Nuclear Science

The equation of state of the quark gluon plasma is a key ingredient of heavy ion phenomenology. In addition to the traditional Taylor method, several novel approximation schemes have been proposed with the aim of calculating it at finite baryon density. In order to gain a pragmatic understanding of the limits of these schemes, we compare them to direct results at $\mu_B>0$, using reweighting techniques free from an overlap problem. We use 2stout improved staggered fermions with 8 time-slices and cover the entire RHIC BES range in the baryochemical potential, up to $\mu_B/T=3$., Comment: 7 pages, 3 figures

Contribution from the USQCD Collaboration to the Proceedings of the US Community Study on the Future of Particle Physics (Snowmass 2021)., Comment: 27 pp. main text, 4 pp. appendices, 29 pp. references, 1 p. index

New tests are applied to two $\beta$-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 $\beta$-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 $\beta$-function in the strong coupling regime of the theory arXiv:2004.00754. We analyze our substantially extended set of ten-flavor lattice ensembles at strong renormalized gauge couplings and find no evidence or hint for IRFP in the finite-volume step $\beta$-function within controlled lattice reach. We also discuss new ten-flavor tests of the recently introduced lattice definition and algorithmic implementation of the $\beta$-function defined on the gradient flow of the gauge field over infinite Euclidean space-time in the continuum. Originally we introduced this new algorithm to match finite-volume step $\beta$-functions in massless near-conformal gauge theories with the infinite-volume $\beta$-function reached in the chiral limit from small fermion mass deformations of spontaneous chiral symmetry breaking. Results from the lattice analysis of the ten-flavor infinite-volume $\beta$-function are consistent with the absence of IRFP from our step $\beta$-function based analysis. We make important contact at weak coupling in infinite volume with gradient flow based three-loop perturbation theory, serving as a first pilot study toward the long-term goal of developing alternate approach to the determination of the strong coupling $\alpha_s$ at the Z-boson pole in QCD., Comment: To appear in the Proceedings of the 38th Annual International Symposium on Lattice Field Theory - LATTICE2021 26-30 July, 2021; 18 pages, 13 figures

We calculate a resummed equation of state with lattice QCD simulations at imaginary chemical potentials. This work presents a generalization of the scheme introduced in 2102.06660 to the case of non-zero $\mu_S$, focusing on the line of strangeness neutrality. We present results up to $\mu_B/T \leq 3.5$ on the strangeness neutral line $\left\langle S \right\rangle = 0$ in the temperature range $130 \rm{MeV} \leq T \leq 280 \rm{MeV}$. We also extrapolate the finite baryon density equation of state to small non-zero values of the strangeness-to-baryon ratio $R=\left\langle S \right\rangle / \left\langle B \right\rangle$. We perform a continuum extrapolation using lattice simulations of the 4stout-improved staggered action with 8, 10, 12 and 16 timeslices., Comment: 15 pages, 12 figures; v2: contains ancillary files with tabulated data

Most lattice studies of hot and dense QCD matter rely on extrapolation from zero or imaginary chemical potentials. The ill-posedness of numerical analytic continuation puts severe limitations on the reliability of such methods. We studied the QCD chiral transition at finite real baryon density with the more direct sign reweighting approach. We simulate up to a baryochemical potential-temperature ratio of $\mu_B/T=2.7$, covering the RHIC Beam Energy Scan range, and penetrating the region where methods based on analytic continuation are unpredictive.This opens up a new window to study QCD matter at finite $\mu_B$ from first principles., Comment: 10 pages, 3 figures; Contribution to the XXXIII International (ONLINE) Workshop on High Energy Physics "Hard Problems of Hadron Physics: Non-Perturbative QCD & Related Quests"; Based on 2108.09213 [hep-lat]. arXiv admin note: substantial text overlap with arXiv:2112.02134

Approaches to finite baryon density lattice QCD usually suffer from uncontrolled systematic uncertainties in addition to the well-known sign problem. We test a method - sign reweighting - that works directly at finite chemical potential and is yet free from any such uncontrolled systematics: with this approach the only problem is the sign problem itself. In practice the approach involves the generation of configurations with the positive fermionic weights given by the absolute value of the real part of the quark determinant, and a reweighting by a sign. There are only two sectors, +1 and -1 and as long as the average $\left\langle \pm \right\rangle \neq 0$ (with respect to the positive weight) this discrete reweighting has no overlap problem - unlike reweighting from $\mu=0$ - and the results are reliable. We also present results based on this algorithm on the phase diagram of lattice QCD with two different actions: as a first test, we apply the method to calculate the position of the critical endpoint with unimproved staggered fermions at $N_\tau=4$; as a second application, we study the phase diagram with 2stout improved staggered fermions at $N_\tau=6$. This second one is already a reasonably fine lattice - relevant for phenomenology. We demonstrate that the method penetrates the region of the phase diagram where the Taylor and imaginary chemical potential methods lose predictive power., Comment: 9 pages, 4 figures; Contribution to the Proceedings of The 38th International Symposium on Lattice Field Theory, LATTICE2021; Based on 2004.10800 and 2108.09213

The equation of state of Quantum Chromodynamics (QCD) at finite density is currently known only in a limited range in the baryon chemical potential $\mu_B$. This is due to fundamental shortcomings of traditional methods such as Taylor expansion around $\mu_B=0$. In this contribution, we present an alternative scheme that displays substantially improved convergence over the Taylor expansion method. We calculate the alternative expansion coefficients in the continuum, and show our results for the thermodynamic observables up to $\mu_B/T\le3.5$., Comment: 8 pages, presentation at the 38th International Symposium on Lattice Field Theory, 26th-30th July 2021, Massachusetts Institute of Technology, USA

State-of-the-art lattice QCD studies of hot and dense strongly interacting matter currently rely on extrapolation from zero or imaginary chemical potentials. The ill-posedness of numerical analytic continuation puts severe limitations on the reliability of such methods. Here we use the more direct sign reweighting method to perform lattice QCD simulation of the QCD chiral transition at finite real baryon density on phenomenologically relevant lattices. This method does not require analytic continuation and avoids the overlap problem associated with generic reweighting schemes, so has only statistical but no uncontrolled systematic uncertainties for a fixed lattice setup. This opens up a new window to study hot and dense strongly interacting matter from first principles. We perform simulations up to a baryochemical potential-temperature ratio of $\mu_B/T=2.5$ covering most of the RHIC Beam Energy Scan range in the chemical potential. We also clarify the connection of the approach to the more traditional phase reweighting method., Comment: 11 pages, 6 figures

The hadron resonance gas (HRG) model is often believed to correctly describe the confined phase of QCD. This assumption is the basis of many phenomenological works on QCD thermodynamics and of the analysis of hadron yields in relativistic heavy ion collisions. We use first-principle lattice simulations to calculate corrections to the ideal HRG. Namely, we determine the sub-leading fugacity expansion coefficients of the grand canonical free energy, receiving contributions from processes like kaon-kaon or baryon-baryon scattering. We achieve this goal by performing a two dimensional scan on the imaginary baryon number chemical potential ($\mu_B$) - strangeness chemical potential ($\mu_S$) plane, where the fugacity expansion coefficients become Fourier coefficients. We carry out a continuum limit estimation of these coefficients by performing lattice simulations with temporal extents of $N_\tau=8,10,12$ using the 4stout-improved staggered action. We then use the truncated fugacity expansion to extrapolate ratios of baryon number and strangeness fluctuations and correlations to finite chemical potentials. Evaluating the fugacity expansion along the crossover line, we reproduce the trend seen in the experimental data on net-proton fluctuations by the STAR collaboration., Comment: 13 pages, 6 figures

New results are reported from tests of a low-energy effective field theory (EFT) that includes a dilaton field to describe the emergent light scalar with ${ 0^{++} }$ quantum numbers in the strongly coupled near-conformal gauge theory with a massless fermion flavor doublet in the two-index symmetric (sextet) representation of the SU(3) color gauge group. In the parlor of walking --- based on the observed light scalar, the small $\beta$-function at strong coupling, and the large anomalous scale dimension of the chiral condensate --- the dilaton EFT hypothesis is introduced to test if it explains the slowly changing nearly scale invariant physics that connects the asymptotically free UV fixed point and the far-infrared scale of chiral symmetry breaking. The characteristic dilaton EFT signatures of scale symmetry breaking are probed in this report in the small Compton wavelength limit of Goldstone bosons relative to the size of the lattice volume (p-regime) and in the limit when the Goldstone wavelength exceeds the size of the volume ($\epsilon$-regime). Random matrix theory (RMT) analysis of the dilaton EFT is applied to the lowest part of the Dirac spectrum in the $\epsilon$-regime to directly test predictions for the fundamental EFT parameters. The predictions, sensitive to the choice of the dilaton potential, were limited before to the p-regime, using extrapolations from far above the chiral limit with untested uncertainties. The dilaton EFT analysis of the $\epsilon$-regime was first suggested in \cite{Fodor:2019vmw}, with some results presented at this conference and with our continued post-conference analysis added to stimulate discussions., Comment: To appear in the Proceedings of the 37th Annual International Symposium on Lattice Field Theory - LATTICE2019 16-22 June, 2019; 13 pages, 9 figures

We present cross-correlators of QCD conserved charges at $\mu_B=0$ from lattice simulations and perform a Hadron Resonance Gas (HRG) model analysis to break down the hadronic contributions to these correlators. We construct a suitable hadronic proxy for the ratio $-\chi_{11}^{BS}/\chi_2^S$ and discuss the dependence on the chemical potential and experimental cuts. We then perform a comparison to preliminary STAR results and comment on a possible direct comparison of lattice and experiment., Comment: 6 pages, 3 figures, contribution to the proceedings from the 18th International Conference on Strangeness in Quark Matter (SQM 2019)

Like fluctuations, non-diagonal correlators of conserved charges provide a tool for the study of chemical freeze-out in heavy ion collisions. They can be calculated in thermal equilibrium using lattice simulations, and be connected to moments of event-by-event net-particle multiplicity distributions. We calculate them from continuum extrapolated lattice simulations at $\mu_B=0$, and present a finite-$\mu_B$ extrapolation, comparing two different methods. In order to relate the grand canonical observables to the experimentally available net-particle fluctuations and correlations, we perform a Hadron Resonance Gas (HRG) model analysis, which allows us to completely break down the contributions from different hadrons. We then construct suitable hadronic proxies for fluctuations ratios, and study their behavior at finite chemical potentials. We also study the effect of introducing acceptance cuts, and argue that the small dependence of certain ratios on the latter allows for a direct comparison with lattice QCD results, provided that the same cuts are applied to all hadronic species. Finally, we perform a comparison for the constructed quantities for experimentally available measurements from the STAR Collaboration. Thus, we estimate the chemical freeze-out temperature to 165 MeV using a strangeness-related proxy. This is a rather high temperature for the use of the Hadron Resonance Gas, thus, further lattice studies are necessary to provide first principle results at intermediate $\mu_B$., Comment: 20 pages, 17 figures, 1 table, version published in Phys.Rev.D

The dilaton low-energy effective field theory (EFT) of an emergent light scalar is probed in the paradigm of strongly coupled near-conformal gauge theories. These studies are motivated by models which exhibit small $\beta$-functions near the conformal window (CW), perhaps with slow scale-dependent walking and a light scalar with ${ 0^{++} }$ quantum numbers. We report our results from the hypothesis of a dilaton inspired EFT analysis with two massless fermions in the two-index symmetric (sextet) representation of the SU(3) color gauge group. With important caveats in our conclusions, conformal symmetry breaking entangled with chiral symmetry breaking would drive the near-conformal infrared behavior of the theory predicting characteristic dilaton signatures of the light scalar from broken scale invariance when probed on relevant scales of fermion mass deformations. From a recently reasoned choice of the dilaton potential in the EFT description~\cite{Golterman:2016lsd} we find an unexpectedly light dilaton mass in the chiral limit at $m_d/f_\pi = 1.56(28)$, set in units of the pion decay constant $f_\pi$. Subject to further statistical and systematic tests of continued post-conference analysis, this result is significantly lower than our earlier estimates from less controlled extrapolations of the light scalar (the $\sigma$-particle) to the massless fermion limit of chiral perturbation theory. We also discuss important distinctions between the dilaton EFT analysis and the linear $\sigma$-model without dilaton signatures. For comparative reasons, we comment on dilaton tests from recent work with fermions in the fundamental representation with $n_f=8$ flavors., Comment: 14 pages, 34 figures, Proceedings of the 36th International Symposium on Lattice Field Theory (Lattice 2018), July 22-28, 2018, East Lancing, USA; elimination of some fit redundancies with minor changes in related figures

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 $\beta$-function. Our results suggest the opposite conclusion, namely we find no $\beta$-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 and other studies., Comment: 7 pages, 6 figures; Proceedings for the 36th Annual International Symposium on Lattice Field Theory, 22-28 July 2018, Michigan State University, East Lansing, Michigan, USA

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 flavors., Comment: 7 pages, 7 figures; Proceedings for the 36th Annual International Symposium on Lattice Field Theory, 22-28 July 2018, Michigan State University, East Lansing, Michigan, USA

We discuss near-conformal gauge theories beyond the standard model (BSM) where interesting results on the twelve-flavor $\beta$-function of massless fermions in the fundamental representation of the SU(3) color gauge group and dilaton tests of the light scalar with two massless fermions in the two-index symmetric tensor (sextet) representation can be viewed as parts of the same BSM paradigm under investigation. We report results from high precision analysis of the twelve-flavor $\beta$-function \cite{Fodor:2016zil} refuting its published IRFP \cite{Cheng:2014jba,Hasenfratz:2016dou}. We present our objections to recent claims \cite{Hasenfratz:2017mdh,Hasenfratz:2017qyr} for non-universal behavior of staggered fermions used in our analysis. We also report our first analysis of dilaton tests of the light $0^{++}$ scalar in the sextet model and comment on related post-conference developments. The dilaton test is the main thrust of this conference contribution including presentation #405 on the $n_f=12$ $\beta$-function and presentation #260 on dilaton tests of the sextet model. They are both selected from the near-conformal BSM paradigm., Comment: 14 pages, 10 figures; Proceedings of the 35th International Symposium on Lattice Field Theory, 18-24 June 2017, Granada, Spain

We report new results from high precision analysis of an important BSM gauge theory with twelve massless fermion flavors in the fundamental representation of the SU(3) color gauge group. The range of the renormalized gauge coupling is extended from our earlier work {Fodor:2016zil} to probe the existence of an infrared fixed point (IRFP) in the $\beta$-function reported at two different locations, originally in {Cheng:2014jba} and at a new location in {Hasenfratz:2016dou}. We find no evidence for the IRFP of the $\beta$-function in the extended range of the renormalized gauge coupling, in disagreement with {Cheng:2014jba,Hasenfratz:2016dou}. New arguments to guard the existence of the IRFP remain unconvincing {Hasenfratz:2017mdh}, including recent claims of an IRFP with ten massless fermion flavors {Chiu:2016uui,Chiu:2017kza} which we also rule out. Predictions of the recently completed 5-loop QCD $\beta$-function for general flavor number are discussed in this context., Comment: 7 pages, 9 figures

The first principle lattice QCD methods allow to calculate the thermodynamic observables at finite temperature and imaginary chemical potential. These can be compared to the predictions of various phenomenological models. We argue that Fourier coefficients with respect to imaginary baryochemical potential are sensitive to modeling of baryonic interactions. As a first application of this sensitivity, we consider the hadron resonance gas (HRG) model with repulsive baryonic interactions, which are modeled by means of the excluded volume correction. The Fourier coefficients of the imaginary part of the net-baryon density at imaginary baryochemical potential -- corresponding to the fugacity or virial expansion at real chemical potential -- are calculated within this model, and compared with the $N_t = 12$ lattice data. The lattice QCD behavior of the first four Fourier coefficients up to $T \simeq 185$ MeV is described fairly well by an interacting HRG with a single baryon-baryon eigenvolume interaction parameter $b \simeq 1$ fm$^3$, while the available lattice data on the difference $\chi_2^B - \chi_4^B$ of baryon number susceptibilities is reproduced up to $T \simeq 175$ MeV., Comment: 11 pages, 4 figures, new Fig. 2 depicting $\chi_2^B - \chi_4^B$, version accepted for publication in Physics Letters B

Fluctuations of conserved charges allow to study the chemical composition of hadronic matter. A comparison between lattice simulations and the Hadron Resonance Gas (HRG) model suggested the existence of missing strange resonances. To clarify this issue we calculate the partial pressures of mesons and baryons with different strangeness quantum numbers using lattice simulations in the confined phase of QCD. In order to make this calculation feasible, we perform simulations at imaginary strangeness chemical potentials. We systematically study the effect of different hadronic spectra on thermodynamic observables in the HRG model and compare to lattice QCD results. We show that, for each hadronic sector, the well established states are not enough in order to have agreement with the lattice results. Additional states, either listed in the Particle Data Group booklet (PDG) but not well established, or predicted by the Quark Model (QM), are necessary in order to reproduce the lattice data. For mesons, it appears that the PDG and the quark model do not list enough strange mesons, or that, in this sector, interactions beyond those included in the HRG model are needed to reproduce the lattice QCD results., Comment: 9 pages, 9 figures

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 $\beta$-function was recently reported in the theory (Cheng:2014jba) with uncertainty in the location of the critical gauge coupling inside the narrow $[ 6.0

We present a calculation of the leptonic decay-constant ratio $f_K/f_\pi$ in 2+1 flavor QCD. Our data set includes five lattice spacings and pion masses reaching down below the physical one. Special emphasis is placed on a careful study of all systematic uncertainties, especially the continuum extrapolation. Our result is perfectly compatible with the first-row unitarity constraint of the Standard Model., Comment: 19 pages, 6 figures, 3 tables; v2: added supplementary analysis, version published in Phys. Rev. D

The Electroweak interactions of a strongly coupled gauge theory are discussed with outlook beyond the Standard Model (BSM) under global and gauge anomaly constraints. The theory is built on a minimal massless fermion doublet of the SU(2) BSM flavor group (bsm-flavor) with a confining gauge force at the TeV scale in the two-index symmetric (sextet) representation of the BSM SU(3) color gauge group (bsm-color). The intriguing possibility of near-conformal sextet gauge dynamics could lead to the minimal realization of the composite Higgs mechanism with a light $0^{++}$ scalar, far separated from strongly coupled resonances of the confining gauge force in the 2-3 TeV range, distinct from Higgsless Technicolor. In previous publications we have presented results for the meson spectrum of the theory, including the light composite scalar, perhaps the emergent Higgs impostor. Here we discuss the critically important role of the baryon spectrum in the sextet model investigating its compatibility with what we know about thermal evolution of the early Universe including its galactic and terrestrial relics. For an important application, we report the first numerical results on the baryon spectrum of this theory from non-perturbative lattice simulations with baryon correlators in the staggered fermion implementation of the strongly coupled gauge sector. The quantum numbers of composite baryons and their spectroscopy from lattice simulations are required inputs for exploring dark matter contributions of the sextet BSM model, as outlined for future work., Comment: 23 pages, 3 figures

We compute the renormalized running coupling of SU(3) gauge theory coupled to N_f = 2 flavors of massless Dirac fermions in the 2-index-symmetric (sextet) representation. This model is of particular interest as a minimal realization of the strongly interacting composite Higgs scenario. A recently proposed finite volume gradient flow scheme is used. The calculations are performed at several lattice spacings with two different implementations of the gradient flow allowing for a controlled continuum extrapolation and particular attention is paid to estimating the systematic uncertainties. For small values of the renormalized coupling our results for the beta-function agree with perturbation theory. For moderate couplings we observe a downward deviation relative to the 2-loop beta-function but in the coupling range where the continuum extrapolation is fully under control we do not observe an infrared fixed point. The explored range includes the locations of the zero of the 3-loop and the 4-loop beta-functions in the MSbar scheme. The absence of a non-trivial zero in the beta-function in the explored range of the coupling is consistent with our earlier findings based on hadronic observables, the chiral condensate and the GMOR relation. The present work is the first to report continuum non-perturbative results for the sextet model., Comment: 21 pages, 19 figures, published version

We perform a detailed, fully-correlated study of the chiral behavior of the pion mass and decay constant, based on 2+1 flavor lattice QCD simulations. These calculations are implemented using tree-level, O(a)-improved Wilson fermions, at four values of the lattice spacing down to 0.054 fm and all the way down to below the physical value of the pion mass. They allow a sharp comparison with the predictions of SU(2) chiral perturbation theory (\chi PT) and a determination of some of its low energy constants. In particular, we systematically explore the range of applicability of NLO SU(2) \chi PT in two different expansions: the first in quark mass (x-expansion), and the second in pion mass (\xi-expansion). We find that these expansions begin showing signs of failure around M_\pi=300 MeV for the typical percent-level precision of our N_f=2+1 lattice results. We further determine the LO low energy constants (LECs), F=88.0 \pm 1.3\pm 0.3 and B^\msbar(2 GeV)=2.58 \pm 0.07 \pm 0.02 GeV, and the related quark condensate, \Sigma^\msbar(2 GeV)=(271\pm 4\pm 1 MeV)^3, as well as the NLO ones, l_3=2.5 \pm 0.5 \pm 0.4 and l_4=3.8 \pm 0.4 \pm 0.2, with fully controlled uncertainties. We also explore the NNLO expansions and the values of NNLO LECs. In addition, we show that the lattice results favor the presence of chiral logarithms. We further demonstrate how the absence of lattice results with pion masses below 200 MeV can lead to misleading results and conclusions. Our calculations allow a fully controlled, ab initio determination of the pion decay constant with a total 1% error, which is in excellent agreement with experiment.

We present possible indications for flavor separation during the QCD crossover transition based on continuum extrapolated lattice QCD calculations of higher order susceptibilities. We base our findings on flavor specific quantities in the light and strange quark sector. We propose a possible experimental verification of our prediction, based on the measurement of higher order moments of identified particle multiplicities. Since all our calculations are performed at zero baryochemical potential, these results are of particular relevance for the heavy ion program at the LHC., Comment: 5 pages, 3 figures, revtex

By fitting pion masses and decay constants from 2+1 flavor staggered lattice simulations to the predictions of NLO and NNLO SU(2) chiral perturbation theory we determine the low-energy constants l_3 and l_4. The lattice ensembles were generated by the Wuppertal-Budapest collaboration and cover pion masses in the range of 135 to 435 MeV and lattice scales between 0.7 and 2.0 GeV. By choosing a suitable scaling trajectory, we were able to demonstrate that precise and stable results for the LECs can be obtained from continuum ChPT to NLO. The pion masses available in this work also allow us to study the applicability of using ChPT to extrapolate from higher masses to the physical pion mass., Comment: 8 pages, 8 figures, 1 table, talk presented at Xth Quark Confinement and the Hadron Spectrum, Munich, October 2012

The frequently discussed strongly interacting gauge theory with a fermion flavor doublet in the two-index symmetric (sextet) representation of the SU(3) color gauge group is investigated \cite{Fodor:2012ty}. The chiral condensate and the mass spectrum are consistent with chiral symmetry breaking at vanishing fermion mass. In contrast, sextet fermion mass deformations of spectral properties are not consistent with leading conformal scaling behavior near the critical surface of a conformal theory. A recent paper \cite{DeGrand:2012yq} which could not resolve the conformal fixed point of the gauge coupling from the slowly walking scenario of a very small nearly vanishing \beta -function is not in conflict with chiral symmetry breaking reported here. A light Higgs impostor could emerge as the dilaton from spontaneous symmetry breaking of scale invariance or, without the dilaton mechanism, as a composite state., Comment: 7 pages, Proceedings of the 30th International Symposium on Lattice Field Theory, June 24 - 29, 2012, Cairns, Australia

We investigate two models of much recent interest in lattice Beyond Standard Model studies: N_f=2 fermions in the 2-index symmetric (sextet) representation, and N_f=12 fermions in the fundamental representation, both with SU(3) gauge symmetry. We present results at fixed lattice spacing for the static fermion potential and force as measured via lattice simulations. We show indications that both models are confining in the chiral limit and that neither theory is conformal. This is consistent with our findings for the mass spectrum, which indicate that chiral symmetry is spontaneously broken in both theories., Comment: 7 pages, Proceedings of the 30th International Symposium on Lattice Field Theory, June 24 - 29, 2012, Cairns, Australia

New results are reported from large scale lattice simulations of a frequently discussed strongly interacting gauge theory with a fermion flavor doublet in the two-index symmetric (sextet) representation of the SU(3) color gauge group. We find that the chiral condensate and the mass spectrum of the sextet model are consistent with chiral symmetry breaking in the limit of vanishing fermion mass. In contrast, sextet fermion mass deformations of spectral properties are not consistent with leading conformal scaling behavior near the critical surface of a conformal theory. A recent paper could not resolve the conformal fixed point of the gauge coupling from the slowly walking scenario of a very small nearly vanishing \beta-function (DeGrand:2012yq). It is argued that overall consistency with our new results is resolved if the sextet model is close to the conformal window, staying outside with a very small non-vanishing \beta-function. The model would exhibit then the simplest composite Higgs mechanism leaving open the possibility of a light scalar state with quantum numbers of the Higgs impostor. It would emerge as the pseudo-Goldstone dilaton state from spontaneous symmetry breaking of scale invariance. We will argue that even without association with the dilaton, the scalar Higgs-like state can be light very close to the conformal window. A new Higgs project of sextet lattice simulations is outlined to resolve these important questions., Comment: 10 pages, 8 figures

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 fermion mass dimension with $\gamma\geq 1$. Since our sextet analysis favors the chiral symmetry breaking hypothesis without conformality, the large exponent $\gamma$ could play an important role in understanding the composite Higgs mechanism. The new results discussed here include our extended data sets and exceed what was presented at the conference., Comment: Contribution to the XXIX International Symposium on Lattice Field Theory - Lattice 2011

We extract the next-to-leading-order low-energy constants \bar\ell_3 and \bar\ell_4 of SU(2) chiral perturbation theory, based on precise lattice data for the pion mass and decay constant on ensembles generated by the Wuppertal-Budapest Collaboration for QCD thermodynamics. These ensembles feature 2+1 flavors of two-fold stout-smeared dynamical staggered fermions combined with Symanzik glue, with pion masses varying from 135 to 435 MeV, lattice scales between 0.7 and 2.0 GeV, while m_s is kept fixed at its physical value. Moderate taste splittings and the scale being set through the pion decay constant allow us to restrict ourselves to the taste pseudoscalar state and to use formulas from continuum chiral perturbation theory. Finally, by dropping the data points near 135 MeV from the fits, we can explore the range of pion masses that is needed in SU(2) chiral perturbation theory to reliably extrapolate to the physical point., Comment: 40 pages, 22 figures, 3 tables; v2: expanded discussion, matches published version

This article reviews lattice QCD results for the light hadron spectrum. We give an overview of different formulations of lattice QCD, with discussions on the fermion doubling problem and improvement programs. We summarize recent developments in algorithms and analysis techniques, that render calculations with light, dynamical quarks feasible on present day computer resources. Finally, we summarize spectrum results for ground state hadrons and resonances using various actions., Comment: 53 pages, 24 figures, one table; Rev.Mod.Phys. (published version); v2: corrected typo

We present the full results of the Wuppertal-Budapest lattice QCD collaboration on flavor diagonal and non-diagonal quark number susceptibilities with 2+1 staggered quark flavors, in a temperature range between 125 and 400 MeV. The light and strange quark masses are set to their physical values. Lattices with Nt=6, 8, 10, 12, 16 are used. We perform a continuum extrapolation of all observables under study. A Symanzik improved gauge and a stout-link improved staggered fermion action is utilized. All results are compared to the Hadron Resonance Gas model predictions: good agreement is found in the temperature region below the transition., Comment: 13 pages, 8 figures in Jhep style

We measure the pion mass and decay constant on ensembles generated by the Wuppertal-Budapest Collaboration, and extract the NLO low-energy constants l_3 and l_4 of SU(2) chiral perturbation theory. The data are generated in 2+1 flavor simulations with Symanzik glue and 2-fold stout-smeared staggered fermions, with pion masses varying from 135 MeV to 400 MeV, lattice scales between 0.7 GeV and 2.0 GeV, and m_s kept at its physical value. Furthermore, by excluding the lightest mass points, we are able to test the reliability of SU(2) chPT as a tool to extrapolate towards the physical point from higher pion masses., Comment: 7 pages, contribution to Lattice 2011, Squaw Valley, Lake Tahoe, California, July 2011

We report new results for a frequently discussed gauge theory with twelve fermion flavors in the fundamental representation of the SU(3) color gauge group. The model, controversial with respect to its conformality, is important in non-perturbative studies searching for a viable composite Higgs mechanism Beyond the Standard Model (BSM). To resolve the controversy, we subject the model to opposite hypotheses inside and outside of the conformal window. In the first hypothesis we test chiral symmetry breaking ($\chi{\rm SB}$) with its Goldstone spectrum, $F_\pi$, the $\chi{\rm SB}$ condensate, and several composite hadron states as the fermion mass is varied in a limited range with our best effort to control finite volume effects and extrapolation to the massless chiral limit. Supporting results for $\chi{\rm SB}$ from the running coupling based on the force between static sources and some preliminary evidence for the finite temperature transition are also presented. In the second test for the alternate hypothesis we probe conformal behavior driven by a single anomalous mass dimension under the assumption of unbroken chiral symmetry. Our results show a very low level of confidence in the conformal scenario. Staggered lattice fermions with stout-suppressed taste breaking are used throughout the simulations., Comment: 9 pages, 9 figures

We report new results for lattice gauge theories with twelve fermion flavors in the fundamental representation and two fermion flavors in the two-index symmetric (sextet) representation of the SU(3) color gauge group. Both models are important in searching for a viable composite Higgs mechanism in the Beyond the Standard Model (BSM) paradigm. We subject both models to opposite hypotheses inside and outside of the conformal window. In the first hypothesis we test chiral symmetry breaking ($\chi{\rm SB}$) with its Goldstone spectrum, $F_\pi$, the $\chi{\rm SB}$ condensate, and several composite hadron states as the fermion mass is varied in a limited range with our best effort to control finite volume effects. Supporting results for $\chi{\rm SB}$ from the running coupling based on the force between static sources is also presented. In the second test for the alternate hypothesis we probe conformal behavior driven by a single anomalous mass dimension under the assumption of unbroken chiral symmetry. Our results show very low level of confidence in the conformal scenario., Comment: 14 pages, 12 figures. Based on talks presented by J.Kuti and K.Holland at the XXVIII International Symposium on Lattice Field Theory, Lattice2010, June 14-19, 2010, Villasimius, Italy

The dual condensate is a new QCD phase transition order parameter, which connnects confinement and chiral symmetry breaking as different mass limits. We discuss the relation between the fermion spectrum at general boundary conditions and the dual condensate and show numerical results for the latter from unquenched SU(3) lattice configurations., Comment: Talk given by Bo Zhang on 'Quark Confinement and the Hadron Spectrum IX', Madrid, Aug.2010

The present paper concludes our investigation on the QCD equation of state with 2+1 staggered flavors and one-link stout improvement. We extend our previous study [JHEP 0601:089 (2006)] by choosing even finer lattices. Lattices with $N_t=6,8$ and 10 are used, and the continuum limit is approached by checking the results at $N_t=12$. A Symanzik improved gauge and a stout-link improved staggered fermion action is utilized. We use physical quark masses, that is, for the lightest staggered pions and kaons we fix the $m_\pi/f_K$ and $m_K/f_K$ ratios to their experimental values. The pressure, the interaction measure, the energy and entropy density and the speed of sound are presented as functions of the temperature in the range $100 ...1000 \textmd{MeV}$. We give estimates for the pion mass dependence and for the contribution of the charm quark. We compare our data to the equation of state obtained by the "hotQCD" collaboration., Comment: Minor changes: Figure 1 added; Figure 15, Figure 17 and Table 5 changed. Accepted for publication in JHEP

The present paper concludes our investigations on the QCD cross-over transition temperatures with 2+1 staggered flavours and one-link stout improvement. We extend our previous two studies [Phys. Lett. B643 (2006) 46, JHEP 0906:088 (2009)] by choosing even finer lattices ($N_t$=16) and we work again with physical quark masses. The new results on this broad cross-over are in complete agreement with our earlier ones. We compare our findings with the published results of the hotQCD collaboration. All these results are confronted with the predictions of the Hadron Resonance Gas model and Chiral Perturbation Theory for temperatures below the transition region. Our results can be reproduced by using the physical spectrum in these analytic calculations. The findings of the hotQCD collaboration can be recovered by using a distorted spectrum which takes into account lattice discretization artifacts and heavier than physical quark masses. This analysis provides a simple explanation for the observed discrepancy in the transition temperatures between our and the hotQCD collaborations., Comment: 25 pages, 10 figures and 3 tables

We present new results on chiral symmetry breaking in nearly conformal gauge theories with fermions in the fundamental representation of the SU(3) color gauge group. The number of fermion flavors is varied in an extended range below the conformal window with chiral symmetry breaking ($\chi{\rm SB}$) for all flavors between $N_f=4$ and $N_f=12$. To identify $\chi{\rm SB}$ we apply several methods which include, within the framework of chiral perturbation theory, the analysis of the Goldstone spectrum in the p-regime and the spectrum of the fermion Dirac operator with eigenvalue distributions of random matrix theory in the $\eps$-regime. Chiral condensate enhancement is observed with increasing $N_f$ when the electroweak symmetry breaking scale $F$ is held fixed in technicolor language. Important finite-volume consistency checks from the theoretical understanding of the $SU(N_f)$ rotator spectrum of the $\delta$-regime are discussed. We also consider these gauge theories at $N_f=16$ inside the conformal window. The importance of understanding finite volume, zero momentum gauge field dynamics inside the conformal window is pointed out. Staggered lattice fermions with supressed taste breaking are used throughout the simulations., Comment: 21 pages, 9 figures. Presented at the XXVII International Symposium on Lattice Field Theory, July 26-31, 2009, Peking University, Beijing

We report new results on nearly conformal gauge theories with fermions in the fundamental representation of the SU(3) color gauge group as the number of fermion flavors is varied in the Nf = 4-16 range. To unambiguously identify the chirally broken phase below the conformal window we apply a comprehensive lattice tool set in finite volumes which includes the test of Goldstone pion dynamics, the spectrum of the fermion Dirac operator, and eigenvalue distributions of random matrix theory. We also discuss the theory inside the conformal window and present our first results on the running of the renormalized gauge coupling and the renormalization group beta function. The importance of understanding finite volume zero momentum gauge field dynamics inside the conformal window is illustrated. Staggered lattice fermions are used throughout the calculations., Comment: 9 pages and 7 figures

One exciting possibility of new physics beyond the Standard Model is that the fundamental Higgs sector is replaced by a strongly-interacting gauge theory, known as technicolor. A viable theory must break chiral symmetry dynamically, like in QCD, to generate Goldstone bosons which become the longitudinal components of the W and Z. By measuring the eigenvalues of the Dirac operator, one can determine if chiral symmetry is in fact spontaneously broken. We simulate SU(3) gauge theory with n_s=2 and 3 staggered flavors in the fundamental representation, corresponding to N_f=8 and 12 flavors in the continuum limit. Although our first findings show that both theories are consistent with dynamically broken chiral symmetry and QCD-like behavior, flavor breaking effects in the spectrum may require further clarifications before final conclusions can be drawn. We also compare various improved staggered actions, to suppress this potentially large flavor breaking., Comment: 7 pages, 5 figures, talk presented at The XXVI International Symposium on Lattice Field Theory, July 14 - 19, 2008, Williamsburg, Virginia, USA

SU(3) gauge theory with dynamical overlap fermions in the 2-index symmetric (sextet) representation is considered. This model may be a viable model of the electroweak symmetry breaking sector along the lines of the walking technicolor paradigm. The number of fermion species is chosen such that the theory is expected to be below the conformal window. We will discuss how the epsilon-regime and random matrix theory can be used to test whether at any given set of parameters (N_c, N_f, representation) the theory is in the conformal phase or indeed just below it. Quenched Monte Carlo results are included in the fundamental representation and also preliminary dynamical ones in the 2-index symmetric representation., Comment: 7 pages, 2 figures, talk presented at The XXVI International Symposium on Lattice Field Theory, July 14 - 19, 2008, Williamsburg, Virginia, USA

We report the first results from our comprehensive lattice tool set to explore non-perturbative aspects of Higgs physics in the Standard Model. We demonstrate in Higgs-Yukawa models that Higgs mass lower bounds and upper bounds can be determined in lattice simulations when triviality requires the necessity of a finite cutoff to maintain non-zero interactions. The vacuum instability problem is investigated and the lattice approach is compared with the traditional renormalization group procedure which sets similar goals to correlate lower and upper Higgs mass bounds with the scale of new physics. A novel feature of our lattice simulations is the use of Ginsparg-Wilson fermions to represent the effects of Top quark loops in Higgs dynamics. The need for chiral lattice fermions is discussed and the approach is extended to full Top-Higgs-QCD dynamics. We also report results from our large $N_F$ analysis of Top-Higgs Yukawa models to gain analytic insight and to verify our new lattice tool set which is deployed in the simulations. The role of non-perturbative lattice studies to investigate heavy Higgs particle scenarios is illustrated in extensions of the Standard Model., Comment: 28 pages, based on contributions from K. Holland, J. Kuti, D. Nogradi, and C. Schroeder at The XXV International Symposium on Lattice Field Theory, July 30 - August 4 2007, Regensburg, Germany