Extended simulation results and their analysis are reported in a strongly coupled gauge theory wi... more 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 unresolved problems with the conformal hypothesis.
The Yang-Mills gradient flow in finite volume is used to define a running coupling scheme. As our... more The Yang-Mills gradient flow in finite volume is used to define a running coupling scheme. As our main result the discrete beta-function, or step scaling function, is calculated for scale change s=3/2 at several lattice spacings for SU(3) gauge theory coupled to N_f = 4 fundamental massless fermions. The continuum extrapolation is performed and agreement is found with the continuum perturbative results for small renormalized coupling. The case of SU(2) gauge group is briefly commented on.
We report new results for lattice gauge theories with twelve fermion flavors in the fundamental r... more 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 (χSB) with its Goldstone spectrum, F π , the χSB 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 χSB 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.
The eigenvalues of the Dirac operator at finite volume encode whether or not chiral symmetry is s... more The eigenvalues of the Dirac operator at finite volume encode whether or not chiral symmetry is spontaneously broken in a massless theory. We apply this framework in a particular BSM context, namely SU(3) gauge theory with N f = 2 massless flavors in the 2-index symmetric (sextet) representation. Our first results are at a single lattice spacing. We find that both the density of near-zero eigenvalues and the renormalization group invariant mode number indicate spontaneous symmetry breaking. Quantitatively, there is a discrepancy between the determination of the fermion condensate in the chiral limit via the eigenvalue spectrum and the determinations from direct measurements of the chiral condensate and the GMOR relation. We comment on possible explanations of this discrepancy and further refinements of this study.
This White Paper has been prepared as a planning document for the Division of High Energy Physics... more This White Paper has been prepared as a planning document for the Division of High Energy Physics of the U. S. Department of Energy. Recent progress in lattice-based studies of physics beyond the standard model is summarized, and major current goals of USQCD research in this area are presented. Challenges and opportunities associated with the recently discovered 126 GeV Higgs-like particle are highlighted. Computational resources needed for reaching important goals are described. The document was finalized on February 11, 2013 with references that are not aimed to be complete, or account for an accurate historical record of the field.
Color Confinement and Hadrons in Quantum Chromodynamics, 2004
Three distinct scales are identified in the excitation spectrum of the gluon field around a stati... more Three distinct scales are identified in the excitation spectrum of the gluon field around a static quark-antiquark pair as the color source separation R is varied. The spectrum, with string-like excitations on the largest length scales of 2-3 fm, provides clues in its rich fine structure for developing an effective bosonic string description. New results are reported from the three-dimensional Z(2) and SU(2) gauge models, providing further insight into the mechanism of bosonic string formation. The precocious onset of string-like behavior in the Casimir energy of the static quark-antiquark ground state is observed below R=1 fm where most of the string eigenmodes do not exist and the few stable excitations above the ground state are displaced. We find no firm theoretical foundation for the widely held view of discovering string formation from high precision ground state properties below the 1 fm scale.
It is widely believed that, for a given Top mass, the Higgs mass has a lower bound: if mHiggs is ... more It is widely believed that, for a given Top mass, the Higgs mass has a lower bound: if mHiggs is too small, the Higgs vacuum is unstable due to Top dynamics. From vacuum instability, the state-of-the-art calculation of the lower bound is close to the current experimental limit. Using non-perturbative simulations and large N calculations, we show that the vacuum is in fact never unstable. Instead, we investigate the existence of a new lower bound, based on the intrinsic cut-off of this trivial theory.
Hybrid bgb molecules in which the heavy bb pair is bound together by the excited gluon field g ar... more Hybrid bgb molecules in which the heavy bb pair is bound together by the excited gluon field g are studied using the Born-Oppenheimer expansion and quenched numerical simulations. The consistency of results from the two approaches reveals a simple and compelling physical picture for heavy hybrid states.
We compute the renormalized running coupling of SU (3) gauge theory coupled to N f = 2 flavors of... more 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 β-function agree with perturbation theory. For moderate couplings we observe a downward deviation relative to the 2-loop β-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 β-functions in the MS scheme. The absence of a non-trivial zero in the β-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.
Our finite size scaling analysis of the renormalized mass and renormalized vacuum expectation val... more Our finite size scaling analysis of the renormalized mass and renormalized vacuum expectation value in the broken symmetry phase of the 0(4) model at infinite bare coupling is presented. We find that the Higgs loop contributions are important in the volume range where we did the simulations. The Goldstone wavefunction renormalization constant is measured to relate the bare vacuum expectation value of the Higgs field to the W boson mass. Our Monte Carlo data at AO = oe show that the SU(2) Higgs sector with spontaneously broken 0(4) symmetry and perturbative gauge coupling is driven to a trivial Gaussian fixed point as criticality is approached. We find an upper bound m H ~ 640 GeV on the Higgs mass at a dimensionless correlation length = 2 on the lattice. The bound will decrease logarithmically with increasing ~.
The cut-off effects of the lattice gradient flow -- often called Wilson flow -- are calculated on... more The cut-off effects of the lattice gradient flow -- often called Wilson flow -- are calculated on a periodic 4-torus at leading order in the gauge coupling. A large class of discretizations is considered which includes all frequently used cases in practice. It is shown how the results lead to a smoother continuum extrapolation for the beta-function of SU(3) gauge theory with N_f = 4 flavors of fermions.
The strongly coupled near-conformal gauge theory with two fermion flavors in the two-index symmet... more The strongly coupled near-conformal gauge theory with two fermion flavors in the two-index symmetric (sextet) representation of SU(3) is potentially a minimal realization of the composite Higgs mechanism. We discuss the staggered fermion construction of baryonic states, present our first numerical results and comment on implications for dark matter.
Work in progress is reported on a particularly interesting gauge theory with a fermion doublet in... more Work in progress is reported on a particularly interesting gauge theory with a fermion doublet in the two-index symmetric (sextet) representation of the SU(3) color gauge group. Extending previous studies we outline our strategy as we investigate Goldstone dynamics and Electroweak scale setting from chiral symmetry breaking (\chi SB), test the GMOR relation from the spectrum of the Dirac operator and the related chiral condensate, begin to develop and test mixed action based improved analysis of \chi SB with new run plans at fixed topology to cross over from the p-regime to the epsilon-regime of \chi SB, continue to pursue the light $0^{++}$ scalar and its relation to the dilaton, and probe the scale-dependent running coupling from the perturbative UV scale to the scale of chiral symmetry breaking. Our observations suggest that the model is very close to the conformal window and a light composite scalar, perhaps the Higgs impostor with or without dilaton-like interpretation, appears...
Rational Hybrid Monte Carlo (RHMC) HMC is a powerful tool to simulate dynamical fermions -global ... more Rational Hybrid Monte Carlo (RHMC) HMC is a powerful tool to simulate dynamical fermions -global updating and exact. a conjugate momentum field π(x) is introduced, and a kinetic term, exp(-π(x) 2 /2), is added to the action to form a fictitious Hamiltonian detailed balance and equivalence to the original system are straightforward the fermion field is replaced with a pseudofermion field ; the root is approximated by Zolotarev's (rational polynomial) method the φ and π fields are evolved according to classical equations of motion finally, the resulting φ field is accepted or rejected based on the fictitious ∆H. Fourier Acceleration Results For the case studied, FA reduces τ by up to 2 orders of magnitude. Decreasing m FA improves decorrelation, but instability arises for too small m FA . It will be interesting to test the FA on larger lattices and see how the related critical exponent is affected as in [7]. References [1-3] Z.
We report extended simulation results and their new analysis in two important gauge theories with... more 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.
We investigate two models of much recent interest in lattice Beyond Standard Model studies: N_f=2... more 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.
The frequently discussed strongly interacting gauge theory with a fermion flavor doublet in the t... more 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. In previous studies the chiral condensate and the mass spectrum were shown to be consistent with chiral symmetry breaking ($\chi$SB) at vanishing fermion mass. The recently reported $\beta$-function is not inconsistent with this observation, suggesting that the model is very close to the conformal window and a light "Higgs impostor" could emerge as a composite state. In this work we describe the methodology and preliminary results of studying the emergence of the light composite scalar with $0^{++}$ quantum numbers.
We compute the renormalized running coupling of SU (3) gauge theory coupled to N f = 8 flavors of... more 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.
Extended simulation results and their analysis are reported in a strongly coupled gauge theory wi... more 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 unresolved problems with the conformal hypothesis.
The Yang-Mills gradient flow in finite volume is used to define a running coupling scheme. As our... more The Yang-Mills gradient flow in finite volume is used to define a running coupling scheme. As our main result the discrete beta-function, or step scaling function, is calculated for scale change s=3/2 at several lattice spacings for SU(3) gauge theory coupled to N_f = 4 fundamental massless fermions. The continuum extrapolation is performed and agreement is found with the continuum perturbative results for small renormalized coupling. The case of SU(2) gauge group is briefly commented on.
We report new results for lattice gauge theories with twelve fermion flavors in the fundamental r... more 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 (χSB) with its Goldstone spectrum, F π , the χSB 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 χSB 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.
The eigenvalues of the Dirac operator at finite volume encode whether or not chiral symmetry is s... more The eigenvalues of the Dirac operator at finite volume encode whether or not chiral symmetry is spontaneously broken in a massless theory. We apply this framework in a particular BSM context, namely SU(3) gauge theory with N f = 2 massless flavors in the 2-index symmetric (sextet) representation. Our first results are at a single lattice spacing. We find that both the density of near-zero eigenvalues and the renormalization group invariant mode number indicate spontaneous symmetry breaking. Quantitatively, there is a discrepancy between the determination of the fermion condensate in the chiral limit via the eigenvalue spectrum and the determinations from direct measurements of the chiral condensate and the GMOR relation. We comment on possible explanations of this discrepancy and further refinements of this study.
This White Paper has been prepared as a planning document for the Division of High Energy Physics... more This White Paper has been prepared as a planning document for the Division of High Energy Physics of the U. S. Department of Energy. Recent progress in lattice-based studies of physics beyond the standard model is summarized, and major current goals of USQCD research in this area are presented. Challenges and opportunities associated with the recently discovered 126 GeV Higgs-like particle are highlighted. Computational resources needed for reaching important goals are described. The document was finalized on February 11, 2013 with references that are not aimed to be complete, or account for an accurate historical record of the field.
Color Confinement and Hadrons in Quantum Chromodynamics, 2004
Three distinct scales are identified in the excitation spectrum of the gluon field around a stati... more Three distinct scales are identified in the excitation spectrum of the gluon field around a static quark-antiquark pair as the color source separation R is varied. The spectrum, with string-like excitations on the largest length scales of 2-3 fm, provides clues in its rich fine structure for developing an effective bosonic string description. New results are reported from the three-dimensional Z(2) and SU(2) gauge models, providing further insight into the mechanism of bosonic string formation. The precocious onset of string-like behavior in the Casimir energy of the static quark-antiquark ground state is observed below R=1 fm where most of the string eigenmodes do not exist and the few stable excitations above the ground state are displaced. We find no firm theoretical foundation for the widely held view of discovering string formation from high precision ground state properties below the 1 fm scale.
It is widely believed that, for a given Top mass, the Higgs mass has a lower bound: if mHiggs is ... more It is widely believed that, for a given Top mass, the Higgs mass has a lower bound: if mHiggs is too small, the Higgs vacuum is unstable due to Top dynamics. From vacuum instability, the state-of-the-art calculation of the lower bound is close to the current experimental limit. Using non-perturbative simulations and large N calculations, we show that the vacuum is in fact never unstable. Instead, we investigate the existence of a new lower bound, based on the intrinsic cut-off of this trivial theory.
Hybrid bgb molecules in which the heavy bb pair is bound together by the excited gluon field g ar... more Hybrid bgb molecules in which the heavy bb pair is bound together by the excited gluon field g are studied using the Born-Oppenheimer expansion and quenched numerical simulations. The consistency of results from the two approaches reveals a simple and compelling physical picture for heavy hybrid states.
We compute the renormalized running coupling of SU (3) gauge theory coupled to N f = 2 flavors of... more 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 β-function agree with perturbation theory. For moderate couplings we observe a downward deviation relative to the 2-loop β-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 β-functions in the MS scheme. The absence of a non-trivial zero in the β-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.
Our finite size scaling analysis of the renormalized mass and renormalized vacuum expectation val... more Our finite size scaling analysis of the renormalized mass and renormalized vacuum expectation value in the broken symmetry phase of the 0(4) model at infinite bare coupling is presented. We find that the Higgs loop contributions are important in the volume range where we did the simulations. The Goldstone wavefunction renormalization constant is measured to relate the bare vacuum expectation value of the Higgs field to the W boson mass. Our Monte Carlo data at AO = oe show that the SU(2) Higgs sector with spontaneously broken 0(4) symmetry and perturbative gauge coupling is driven to a trivial Gaussian fixed point as criticality is approached. We find an upper bound m H ~ 640 GeV on the Higgs mass at a dimensionless correlation length = 2 on the lattice. The bound will decrease logarithmically with increasing ~.
The cut-off effects of the lattice gradient flow -- often called Wilson flow -- are calculated on... more The cut-off effects of the lattice gradient flow -- often called Wilson flow -- are calculated on a periodic 4-torus at leading order in the gauge coupling. A large class of discretizations is considered which includes all frequently used cases in practice. It is shown how the results lead to a smoother continuum extrapolation for the beta-function of SU(3) gauge theory with N_f = 4 flavors of fermions.
The strongly coupled near-conformal gauge theory with two fermion flavors in the two-index symmet... more The strongly coupled near-conformal gauge theory with two fermion flavors in the two-index symmetric (sextet) representation of SU(3) is potentially a minimal realization of the composite Higgs mechanism. We discuss the staggered fermion construction of baryonic states, present our first numerical results and comment on implications for dark matter.
Work in progress is reported on a particularly interesting gauge theory with a fermion doublet in... more Work in progress is reported on a particularly interesting gauge theory with a fermion doublet in the two-index symmetric (sextet) representation of the SU(3) color gauge group. Extending previous studies we outline our strategy as we investigate Goldstone dynamics and Electroweak scale setting from chiral symmetry breaking (\chi SB), test the GMOR relation from the spectrum of the Dirac operator and the related chiral condensate, begin to develop and test mixed action based improved analysis of \chi SB with new run plans at fixed topology to cross over from the p-regime to the epsilon-regime of \chi SB, continue to pursue the light $0^{++}$ scalar and its relation to the dilaton, and probe the scale-dependent running coupling from the perturbative UV scale to the scale of chiral symmetry breaking. Our observations suggest that the model is very close to the conformal window and a light composite scalar, perhaps the Higgs impostor with or without dilaton-like interpretation, appears...
Rational Hybrid Monte Carlo (RHMC) HMC is a powerful tool to simulate dynamical fermions -global ... more Rational Hybrid Monte Carlo (RHMC) HMC is a powerful tool to simulate dynamical fermions -global updating and exact. a conjugate momentum field π(x) is introduced, and a kinetic term, exp(-π(x) 2 /2), is added to the action to form a fictitious Hamiltonian detailed balance and equivalence to the original system are straightforward the fermion field is replaced with a pseudofermion field ; the root is approximated by Zolotarev's (rational polynomial) method the φ and π fields are evolved according to classical equations of motion finally, the resulting φ field is accepted or rejected based on the fictitious ∆H. Fourier Acceleration Results For the case studied, FA reduces τ by up to 2 orders of magnitude. Decreasing m FA improves decorrelation, but instability arises for too small m FA . It will be interesting to test the FA on larger lattices and see how the related critical exponent is affected as in [7]. References [1-3] Z.
We report extended simulation results and their new analysis in two important gauge theories with... more 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.
We investigate two models of much recent interest in lattice Beyond Standard Model studies: N_f=2... more 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.
The frequently discussed strongly interacting gauge theory with a fermion flavor doublet in the t... more 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. In previous studies the chiral condensate and the mass spectrum were shown to be consistent with chiral symmetry breaking ($\chi$SB) at vanishing fermion mass. The recently reported $\beta$-function is not inconsistent with this observation, suggesting that the model is very close to the conformal window and a light "Higgs impostor" could emerge as a composite state. In this work we describe the methodology and preliminary results of studying the emergence of the light composite scalar with $0^{++}$ quantum numbers.
We compute the renormalized running coupling of SU (3) gauge theory coupled to N f = 8 flavors of... more 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.
Uploads
Papers by Julius Kuti