Jack Laiho
We present a calculation of the form factor for B -> D* l nu using a 2+1 improved staggered action for the light quarks (on the MILC configurations), and the Fermilab action for the heavy quarks. The form factor is computed at zero recoil using a new double ratio method which yields the form factor more directly than previous approaches.
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Staggered fermions with smeared links can have greatly improved chiral properties. In a recent paper we introduced a simple and effective method to simulate four flavors of staggered smeared link fermions. In this work we extend the four flavor method to two flavors. We define the two flavor action by the square root of the four flavor fermionic determinant and show that by using a polynomial approximation the two flavor action can be evaluated with the necessary accuracy. We test this method by studying the finite temperature phase structure with hypercubic smeared (HYP) link staggered action…
Motivated by simulation results for octet and decuplet baryon masses using 2+1 flavors of light staggered quarks, we are incorporating staggered lattice artifacts into heavy baryon chiral perturbation theory, calculating the masses of various staggered baryons, and studying the connection between the staggered baryons of the chiral theory and the staggered baryons of simulations. We present order (m_q)^(3/2) loop contributions to the masses of several staggered nucleons and discuss interpolating fields that create these states.
We measure the low lying eigenmodes of an overlap Dirac operator on 2--flavor staggered configurations. By comparing the eigenmode distribution to the predictions of Random Matrix Theory we test to what accuracy staggered configurations describe continuum QCD. The agreement between the numerical data and RMT implies that at our quark mass values the lattice artifacts of the staggered configurations are comparable to overlap configurations. We identify the overlap valence mass that best matches the staggered sea quarks and predict the value of the infinite volume 2-flavor chiral condensate.
At non-zero lattice spacing the flavor symmetry of staggered fermions is broken to a discrete subgroup. We show that in the chiral limit the flavor symmetry of the pion effective Lagrangian enlarges to an SO(4) subgroup of the continuum SU(4) symmetry. This provides an explanation for observed degeneracies in the pion spectrum.
Staggered fermions with 4 tastes are expected to describe 4-flavor QCD in the continuum limit, therefore at finite lattice spacing the staggered determinant should be equivalent to an SU(4) flavor-symmetric system up to lattice artifacts. This equivalence is the starting point of the 4th root trick used to reduce the number of fermion flavors and provides the only consistent description of 2 or 1 flavor systems. In this paper we argue that the quark mass of the underlying flavor symmetric theory differs from the staggered mass by an additive term due to the taste breaking of the staggered acti…
Accurately calculating the mass of flavor-singlet meson states from numerical lattice simulations is an important milestone for lattice QCD. Careful measurement of the full pseudoscalar flavor-singlet propagator is also a crucial step in understanding the dynamics of the fermion sea on the lattice, in particular for potentially non-trivial formulations such as with 2+1-flavor staggered fermions. We briefly describe details of a dynamical QCD calculation using improved staggered fermions, with 30,000 trajectories, that was run for our studies of flavour singlet mesons.
Growing evidence indicates that in the continuum limit the rooted staggered action is in the correct QCD universality class, the non-local terms arising from taste breaking can be viewed as lattice artifacts. In this paper we consider the 2-flavor Asqtad staggered action at a lattice spacing of approximately 0.13 fm and probe the properties of the staggered configurations by an overlap valence Dirac operator. By comparing the distribution of the overlap eigenmodes to continuum QCD predictions we investigate if/when the lattice artifacts are small as a function of the staggered quark mass. We d…
I study two-flavor staggered chiral perturbation theory in the light pseudoscalar sector. The pion mass and decay constant are calculated through NLO in the partially-quenched case. In the limit where the strange quark mass is large compared to the light quark masses and the taste splittings, I show that the SU(2) staggered chiral theory emerges from the SU(3) staggered chiral theory, as expected. Explicit relations between SU(2) and SU(3) low energy constants and taste-violating parameters are given. The results are useful for SU(2) chiral fits to asqtad data and allow one to incorporate effe…
We resolve contradictions in the literature concerning the origins and size of unphysical flavor-changing strong interactions generated by the staggered-quark discretization of QCD. We show that the leading contributions are tree-level in $\order(a^2)$ and that they can be removed by adding three correction terms to the link operator in the standard action. These corrections are part of the systematic Symanzik improvement of the staggered-quark action. We present a new improved action for staggered quarks that is accurate up to errors of $\order(a^4,a^2α_s)$ --- more accurate than most, if not…
We have studied the 3-flavor, finite temperature, QCD phase transition with staggred fermions on an $ N_t=4$ lattice. By studying a variety of quark masses we have located the critical point, $m_c$, where the first order 3-flavor transition ends as lying in the region $0.32 \le m_c \le 0.35$ in lattice units
The Asqtad improved staggered fermion formalism has been a valuable tool in successfully calculating the non-singlet parts of the hadronic spectrum. We are engaged in a project to calculate the spectrum of the pseudoscalar singlet mesons with 2+1-flavor Asqtad staggered gauge configurations. Propagators of flavor-singlet states incorporate contributions from both disconnected and connected diagrams, and hence are sensitive to any differences in the actions governing the sea and valence fermions on the lattice. As such, they also present the possibility of a probe of the validity of the ``fourt…
The construction of the first baryon operators for staggered lattice QCD exploited the taste symmetry to emulate physical quark flavor; contemporary 2+1 flavor simulations explicitly include three physical quark flavors and necessitate interpreting a valence sector with twelve quarks. After discussing expected features of the resulting baryon spectrum, I consider the spectra of operators transforming irreducibly under SU(3)xGTS, the direct product of flavor SU(3) and the geometrical time-slice group of the 1-flavor staggered theory. I then describe the construction of a set of maximally local …
We examine the taste structure of eigenvectors of the staggered-fermion Dirac operator. We derive a set of conditions on the eigenvectors of modes with small eigenvalues (near-zero modes), such that staggered fermions reproduce the 't Hooft vertex in the continuum limit. We also show that, assuming these conditions, the correlators of flavor-singlet mesons are free of contributions singular in $1/m$, where $m$ is the quark mass. This conclusion holds also when a single flavor of sea quark is represented by the fourth root of the staggered-fermion determinant. We then test numerically, using th…
Based on recent work by Adams, I construct a lattice fermion operator that fully lifts the staggered flavor degeneracy. The resulting operator is of Wilson type but smaller by a factor of 4, better conditioned and contains 3 instead of 15 doublers. It is further suggested that this operator may be used as a candidate kernel operator to an overlap construction. Prospects for practical applications and potential problems of the new discretizations are briefly discussed.
We present an update on the MILC Collaboration's light hadron spectrum calculation with two flavors of dynamical, staggered quarks. We present extrapolations of the nucleon to rho mass ratio to the continuum limit for fixed values of the pi to rho mass ratio including the physical one.
As one test of the validity of the staggered-fermion fourth-root determinant trick, we examine the suppression of the topological susceptibility of the QCD vacuum in the limit of small quark mass. The suppression is sensitive to the number of light sea quark flavors. Our study is done in the presence of 2+1 flavors of dynamical quarks in the improved staggered fermion formulation. Variance-reduction techniques provide better control of statistical errors. New results from staggered chiral perturbation theory account for taste-breaking effects in the low-quark mass behavior of the susceptibilit…
Different versions of improved staggered fermions can be used as valence quarks to reduce discretization effects in lattice QCD calculations while increasing statistics on existing staggered gauge ensembles. Such mixed-action simulations can be used to improve determinations of light quark masses, Gasser-Leutwyler couplings, decay constants, and other parameters relevant to particle phenomenology. We recall the generalization of ordinary, unmixed staggered chiral perturbation theory required to describe data from lattice calculations with a mixed action such as with HYP staggered valence quark…
In staggered QCD, many staggered baryons correspond to each physical state. Taste violations lift the continuum degeneracies of the baryons and introduce nonzero off-diagonal elements in the mass matrix. While presenting no problem of principle, these splittings and mixings complicate analyses of simulation results. However, in special cases operators with good SU(3) quantum numbers can be used to circumvent the splittings and mixings. I review what has been learned from staggered chiral perturbation theory, outline a program of attack for the amenable cases, and summarize the present status o…
We present preliminary results from exploring the phase diagram of finite temperature QCD with three degenerate flavors and with two light flavors and the mass of the third held approximately at the strange quark mass. We use an order $α_s^2 a^2, a^4$ Symanzik improved gauge action and an order $α_s a^2, a^4$ improved staggered quark action. The improved staggered action leads to a dispersion relation with diminished lattice artifacts, and hence better thermodynamic properties. It decreases the flavor symmetry breaking of staggered quarks substantially, and we estimate that at the transition t…
We present preliminary results from exploring the phase diagram of finite temperature QCD with three degenerate flavors and with two light flavors and the mass of the third held approximately at the strange quark mass. We use an order $α_s^2 a^2, a^4$ Symanzik improved gauge action and an order $α_s a^2, a^4$ improved staggered quark action. The improved staggered action leads to a dispersion relation with diminished lattice artifacts, and hence better thermodynamic properties. It decreases the flavor symmetry breaking of staggered quarks substantially, and we estimate that at the transition t…
We discuss the Schrödinger functional in lattice QCD with staggered fermions and relate it, in the classical continuum limit, to the Schrödinger functional regularized with Wilson fermions. We compute the strong coupling constant defined via the Schrödinger functional with staggered fermions at one loop and show that it agrees with the continuum running coupling constant in the Schrödinger functional formalism. We compute this running coupling in the ``weak coupling phase'' of many flavor QCD numerically at several values of the bare coupling and for several system sizes from $L/a=4$ to 12. Th…
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…
A popular approximation in lattice gauge theory is an extrapolation in the number of fermion species away from the four fold degeneracy natural with the staggered fermions formulation. I show that at finite lattice spacing and for an odd number of flavors this extrapolation misses terms which, on general principles, must be present in the continuum theory. For a correct continuum limit, this forces unphysical singularities in parameter regions where continuum physics is smooth and all physical particles are massive. These singularities are not expected with other lattice regulators. Finally, I…
We present updated results for the light-quark connected part of the leading hadronic contribution to the muon $g-2$ from configurations with 2+1+1 flavors of HISQ quarks using the time-momentum representation of the electromagnetic current correlator. We have added statistics on two ensembles as well as a fourth lattice spacing using configurations that have been generated by the MILC collaboration at the physical pion mass. Additionally we account for the leading finite-volume and taste-breaking effects using Staggered Chiral Perturbation Theory at NNLO.
In the light pseudoscalar sector, we study rooted staggered chiral perturbation theory in the two-flavor case. The pion mass and decay constant are calculated through NLO for a partially-quenched theory. In the limit where the strange quark mass is large compared to the light quark masses and the taste splittings, we show that the SU(2) staggered chiral theory emerges from the SU(3) staggered chiral theory, as expected. Explicit relations between SU(2) and SU(3) low energy constants and taste-violating parameters are given. A brief summary of updated SU(2) chiral fits to the MILC lattice data …
Staggered Domain Wall Fermions (SDWF) combine the attractive chiral properties of staggered fermions with those of domain wall fermions. SDWF describe four flavors with exact U(1)xU(1) flavor chiral symmetry. An extra lattice dimension is introduced and the full SU(4)xSU(4) flavor chiral symmetry is recovered as its size is increased. Here, the free theory of SDWF is described and a preliminary discussion of the interacting case is presented. SDWF may be well suited for numerical simulation of lattice QCD thermodynamics.
We report results for the interaction measure, pressure and energy density for nonzero temperature QCD with 2+1 flavors of improved staggered quarks. In our simulations we use a Symanzik improved gauge action and the Asqtad $O(a^2)$ improved staggered quark action for lattices with temporal extent $N_t=4$ and 6. The heavy quark mass $m_s$ is fixed at approximately the physical strange quark mass and the two degenerate light quarks have masses $m_{ud}\approx0.1 m_s$ or $0.2 m_s$. The calculation of the thermodynamic observables employs the integral method where energy density and pressure are o…
We present new results in an ongoing study of the nature of the high temperature crossover in QCD with two light fermion flavors. These results are obtained with the conventional staggered fermion action at the smallest lattice spacing to date---approximately 0.1 fm. Of particular interest are a study of the temperature of the crossover, an important indicator of continuum scaling, a determination of the induced baryon charge and baryon susceptibility, used to study the dissolution of hadrons at the crossover, the scalar susceptibility, a signal for the appearance of soft modes, and the chiral…
We present results on the sum of the masses of light and strange quark using improved staggered quarks. Our calculation uses 2+1 flavours of dynamical quarks. The effects of the dynamical quarks are clearly visible.
We give an explicit proof for the locality of staggered overlap operators. The proof covers the original two flavor construction by Adams as well as a single flavor version. As in the case of Neuberger's operator, an admissibility condition for the gauge fields is required.
We present results from an ongoing lattice study of the lowest lying charmonium and bottomonium level splittings using the Fermilab heavy quark formalism. Our objective is to test the performance of this action on MILC-collaboration ensembles of (2+1) flavors of light improved staggered (asqtad) quarks. Measurements are done on 16 ensembles with degenerate up and down quarks of various masses, thus permitting a chiral extrapolation, and over lattice spacings ranging from 0.09 fm to 0.18 fm, thus permitting study of lattice-spacing dependence. We examine combinations of the mass splittings that…
Improved staggered quark actions are designed to suppress flavour changing strong interactions. We discuss the perturbation theory for this type of actions and show the improvements to reduce the quark mass renormalisation compared to naive staggered quarks. The renormalisations are of similar size as for Wilson quarks.
We report on progress in our study of high temperature QCD with three flavors of improved staggered quarks. Simulations are being carried out with three degenerate quarks with masses less than or equal to the strange quark mass, $m_s$, and with degenerate up and down quarks with masses in the range $0.1 m_s \leq m_{u,d}\leq 0.6 m_s$, and the strange quark mass fixed near its physical value. For the quark masses studied to date we find rapid crossovers, which sharpen as the quark mass is reduced, rather than bona fide phase transitions.
We use the dynamical gluon configurations provided by the MILC collaboration in a study of the charmonium spectrum and psi leptonic width. We examine sea quark effects on mass splitting and on the leptonic decay matrix element for light masses as low as m_s/5, while keeping the strange quark mass fixed and the lattice spacing nearly constant.
The perfect action of free staggered fermions is calculated by blocking from the continuum for degenerate and non-degenerate flavor masses. The symmetry structure, connecting flavor transformations and translations, is explained directly from the blocking scheme. It is convenient to use a modified Fourier transformation, respecting this connection, to treat the spin-flavor structure of the blockspins. The perfect action remains local in the non-degenerate case; it is explicitly calculated in two dimensions. I finally comment on the relation of the blocking scheme to the transition from Dirac-K…
It is widely believed that chiral symmetry is spontaneously broken at zero temperature in the strong coupling limit of staggered fermions, for any number of colors and flavors. Using Monte Carlo simulations, we show that this conventional wisdom, based on a mean-field analysis, is wrong. For sufficiently many fundamental flavors, chiral symmetry is restored via a bulk, first-order transition. This chirally symmetric phase appears to be analytically connected with the expected conformal window of many-flavor continuum QCD. We perform simulations in the chirally symmetric phase at zero quark mas…
The cost of dynamical quark simulations with improved staggered quarks is estimated based on current and planned running by the MILC collaboration. I find that a few 10s of Teraflop years should be sufficient to calculate down to a lattice spacing of 0.045 fm.
We present a calculation of the transition temperature for 3 flavor QCD using p4-improved staggered fermions with two different variants of fattened links. We examine various susceptibilities at two different values for the temporal extent - $N_t = 4$ and $N_t = 6$ - in the vicinity of the 3 flavor transition at vanishing chemical potential. For $N_t = 4$, we study bare quark masses in the range $m_qa = 0.005$ to $m_qa = 0.1$, and for $N_t = 6$, we use $m_qa = 0.02$ to $m_qa = 0.2$. We also calculate hadron masses and the static quark potential at zero temperature in order to set the scale for…
I look at the rooting controversy from a historical point of view and review how I have come to the conclusion that these simulations involving staggered quarks must be discarded.
In preparation of lattice studies of the two-color QCD phase diagram we study chiral restoration and deconfinement at finite temperature with two flavors of staggered quarks using an RHMC algorithm on GPUs. We first study unquenching effects in local Polyakov loop distributions, and the Polyakov loop potential obtained via Legendre transformation, in a fixed-scale approach for heavier quarks. We also present the chiral condensate and the corresponding susceptibility over the lattice coupling across the chiral transition for lighter quarks. Using Ferrenberg-Swendsen reweighting we extract the m…
We report on progress in measuring disconnected correlators associated with pseudoscalar flavor-singlet mesons. This will eventually allow us to compute the masses of the eta and eta' mesons. Flavor-singlet physics also presents an interesting test of the staggered fermion formulation, as disconnected correlators are sensitive to whether the same action governs both sea quarks and valence quarks. It can also help test the validity of the ``fourth-root trick'' used in unquenched lattice calculations where the number of flavors $N_f<4$.
We present results from our simulations of quantum chromodynamics (QCD) with four flavors of quarks: u, d, s, and c. These simulations are performed with a one-loop Symanzik improved gauge action, and the highly improved staggered quark (HISQ) action. We are generating gauge configurations with four values of the lattice spacing ranging from 0.06 fm to 0.15 fm, and three values of the light quark mass, including the value for which the Goldstone pion mass is equal to the physical pion mass. We discuss simulation algorithms, scale setting, taste symmetry breaking, and the autocorrelations of va…
Recent developements in QCD at finite temperature are reviewed. Particular emphasis is laid on results stemming from simulations which involve quarks.
A different lattice fermion method is introduced. Staggered domain wall fermions are defined in 2n+1 dimensions and describe 2^n flavors of light lattice fermions with exact U(1) x U(1) chiral symmetry in 2n dimensions. As the size of the extra dimension becomes large, 2^n chiral flavors with the same chiral charge are expected to be localized on each boundary and the full SU(2^n) x SU(2^n) flavor chiral symmetry is expected to be recovered. SDWF give a different perspective into the inherent flavor mixing of lattice fermions and by design present an advantage for numerical simulations of latt…
A new formulation of chiral fermions on the lattice is presented. It is a version of overlap fermions, but built from the computationally efficient staggered fermions rather than the previously used Wilson fermions. The construction reduces the four quark flavors described by the staggered fermion to two quark flavors; this pair can be taken as the up and down quarks in Lattice QCD. The exact flavored chiral symmetry of the staggered fermion gets converted into an unflavored Ginsparg-Wilson chiral symmetry of the new overlap fermion, which also has pairs of exact chiral zero-modes satisfying t…
We present the results of our perturbative calculations of the static quark potential, small Wilson loops, the static quark self energy, and the mean link in Landau gauge. These calculations are done for the one loop Symanzik improved gluon action, and the improved staggered quark action.
The Fermilab Lattice and MILC collaborations are completing a comprehensive program of heavy-light physics on MILC (2+1)-flavor asqtad ensembles with lattice spacings as small as 0.045 fm and light-to-strange-quark mass ratios as low as 1/20. We use the Fermilab interpretation of the clover action for heavy valence quarks and the asqtad action for the light valence quarks. The central goal of the program is to provide ever more exacting tests of the unitarity of the CKM matrix. We present preliminary results for one part of the program, namely the analysis of the semileptonic decay B -> D l nu…
We present exploratory results for the hadron mass spectrum and pseudoscalar meson decay constants using mixed actions. We use improved staggered sea quarks and HYP-smeared overlap valence quarks. We obtain good signals on 10 configurations at one lattice spacing and two different sets of sea quark masses.
By combining experimentally measured partial branching fractions for the semileptonic decays D to K l nu and D to pi l nu with lattice calculations of the form factors f_+^{D to K}(q^2) and f_+^{D to pi}(q^2), one can extract the CKM matrix elements |V_{cs}| and |V_{cd}|. We are calculating the form factors by using Fermilab charm and asqtad staggered light and strange quarks on 2+1 flavor asqtad staggered ensembles generated by the MILC Collaboration. We vary the light valence quark masses from 0.4m_s to 0.05m_s (m_s is the strange sea-quark mass), and the lattice spacings, from about 0.12 fm…
Calculations using staggered quarks augmented with a root of the fermion determinant to reduce doubling give a qualitatively incorrect behavior in the small quark mass region. Attempts to circumvent this problem for the continuum limit involve an unproven combination of unphysical states, a loss of unitarity, and a rather peculiar non-commutation of limits.
We present results from a systematic perturbative investigation of taste-changing in improved staggered quarks. We show one-loop taste-changing interactions can be removed perturbatively by an effective four-quark term and calculate the necessary coefficients.
In HISQ simulations by the MILC and Fermilab Lattice collaborations, both the light quarks and the charm quark are staggered. We extend staggered chiral perturbation theory (\schpt) to include such all-staggered heavy-light mesons. We assume that the heavy quark action is sufficiently improved that we may take $a m_Q <<1$ (where $m_Q$ is the heavy quark mass), but also that $m_Q>>Λ_{QCD}$ so that a continuum heavy quark expansion is appropriate. Using this \schpt, the leptonic decay constant of the heavy-light meson is calculated at next-to-leading-order. The pattern of taste splittings in the…
In highly improved staggered quark (HISQ) simulations by the HPQCD, MILC, and Fermilab Lattice collaborations, both the light quarks and the charm quark are staggered. We extend chiral perturbation theory for staggered quarks to include such all-staggered heavy-light mesons. We assume that the heavy quark action is sufficiently improved that we may take $a m_Q <<1$ (where $m_Q$ is the heavy quark mass), but also that $m_Q>>Λ_{QCD}$ so that a continuum heavy quark expansion is appropriate. We develop this effective chiral theory through next-to-leading order, and use it to study the pattern of …
We are studying the effects of light dynamical quarks on the excitation energies of a flux tube between a static quark and antiquark. We report preliminary results of an analysis of the ground state potential and the $Σ^{\prime+}_g$ and $Π_u$ potentials. We have measured these potentials on closely matched ensembles of gauge configurations, generated in the quenched approximation and with 2+1 flavors of Asqtad improved staggered quarks.
We compare several improved actions for staggered quarks. We study the effect of improvement on the taste changing interactions by calculating the splitting in the pion spectrum. We investigate the effect of the improvement on some topological properties.
In the staggered fermion formulation of lattice QCD, we construct diquark operators which are to be embedded in singly heavy baryons. The group theoretical connections between continuum and lattice staggered diquark representations are established.
We study the leading discretization errors for staggered fermions by first constructing the continuum effective Lagrangian including terms of O(a^2), and then constructing the corresponding effective chiral Lagrangian. The terms of O(a^2) in the continuum effective Lagrangian completely break the SU(4) flavor symmetry down to the discrete subgroup respected by the lattice theory. We find, however, that the O(a^2) terms in the potential of the chiral Lagrangian maintain an SO(4) subgroup of SU(4). It follows that the leading discretization errors in the pion masses are SO(4) symmetric, implying…
We present results of an ongoing study of the nature of the high temperature crossover in QCD with two light fermion flavors. These results are obtained with the conventional staggered fermion action at the smallest lattice spacing to date---approximately 0.1 fm. Of particular interest are a study of the temperature of the crossover a determination of the induced baryon charge and baryon susceptibility, the scalar susceptibility, and the chiral order parameter, used to test models of critical behavior associated with chiral symmetry restoration. From our new data and published results for N_t …
Because the propagators of flavor-singlet states incorporate disconnected diagrams, they are uniquely sensitive to any differences in the actions governing sea and valence fermions on the lattice. As such, they present an important test of the validity of the ``fourth-root trick'' in the staggered fermion formulation. The pseudoscalar's relationship to topological charge also makes it of theoretical interest. We present preliminary results from our measurements of flavor-singlet pseudoscalar mesons on 2+1 flavor Asqtad lattices, and discuss some strategies for improving the signal-to-error rat…
We perform a nonperturbative tuning of the coefficients in the $O(a^2)$ improved action for staggered fermions. The mass splitting for the pions of different doubler flavor is used as a measure of the symmetry breaking effects introduced by $O(a^2)$ discretization errors. We find that the flavor nondegeneracy can be somewhat reduced but not eliminated by such a tuning, indicating the need for new terms in the action to reduce the nondegeneracy.
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