Kaon $B$-parameter from improved staggered fermions in $N_f=2+1$ QCD
Taegil Bae, Yong-Chull Jang, Chulwoo Jung, Hyung-Jin Kim, Jangho Kim, Jongjeong Kim, Kwangwoo Kim, Sunghee Kim, Weonjong Lee, Stephen R. Sharpe, Boram Yoon, SWME Collaboration
We present a calculation of the kaon $B$-parameter, $B_K$, using lattice QCD. We use improved staggered valence and sea fermions, the latter generated by the MILC collaboration with $N_f=2+1$ light flavors. To control discretization errors, we use four different lattice spacings ranging down to $a\approx 0.045\;$fm. The chiral and continuum extrapolations are done using SU(2) staggered chiral perturbation theory. Our final result is $\hat{B}_K = 0.727 \pm 0.004 (\text{stat}) \pm 0.038 (\text{sys})$, where the dominant systematic error is from our use of truncated (one-loop) matching factors.
We present updated results for kaon B-parameters for operators arising in models of new physics. We use HYP-smeared staggered quarks on the $N_f = 2+1$ MILC asqtad lattices. During the last year we have added new ensembles, which has necessitated chiral-continuum fitting with more elaborate fitting functions. We have also corrected an error in a two-loop anomalous dimension used to evolve results between different scales. Our results for the beyond-the-Standard-Model B-parameters have total errors of $5-10$\%. We find that the discrepancy observed last year between our results and those of the…
The determination of $\vert V_{us}\vert$ from kaon semileptonic decays requires the value of the form factor $f_+(q^2=0)$, which can be calculated precisely on the lattice. We provide the one-loop partially quenched staggered chiral perturbation theory expressions that may be employed to analyze staggered simulations of $f_+(q^2)$ with three light flavors. We consider both the case of a mixed action, where the valence and sea sectors have different staggered actions, and the standard case where these actions are the same. The momentum transfer $q^2$ of the form factor is allowed to have an arb…
The use of improved staggered actions (HYP, Asqtad) has been proved to reduce the scaling corrections that affected previous calculations of B_K with unimproved (standard) staggered fermions in the quenched approximation. This improved behaviour allows us to perform a reliable calculation of B_K including quark vacuum polarization effects, using the MILC configurations with n_f=2+1 flavours of sea fermions. We perform such a calculation for a single lattice spacing, a=0.125 fm, and with kaons made up of degenerate quarks with m_s/2. The valence strange quark mass m_s is fixed to its physical v…
We calculate the kaon B-parameter, B_K, to next-to-leading order in staggered chiral perturbation theory. We extend the usual power counting to include the effects of using perturbative (rather than non-perturbative) matching factors. Taste breaking enters through the O(a^2) terms in the effective action, through mixing with higher-dimension operators, and through the truncation of matching factors. These effects cause mixing with several additional operators, complicating the chiral and continuum extrapolations. We summarize the results here; all details can be found in hep-lat/0507012.
Lattice QCD calculations with different staggered valence and sea quarks can be used to improve determinations of quark masses, Gasser-Leutwyler couplings, and other parameters relevant to phenomenology. We calculate the masses and decay constants of flavored pions and kaons through next-to-leading order in staggered-valence, staggered-sea mixed-action chiral perturbation theory. We present the results in the valence-valence and valence-sea sectors, for all tastes. As in unmixed theories, the taste-pseudoscalar, valence-valence mesons are exact Goldstone bosons in the chiral limit, at non-zero…
We compute pseudoscalar meson masses and decay constants using staggered quarks on lattices with three flavors of sea quarks and lattice spacings $\approx 0.12$ fm and $\approx 0.09$ fm. We fit partially quenched results to ``staggered chiral perturbation theory'' formulae, thereby taking into account the effects of taste-symmetry violations. Chiral logarithms are observed. From the fits we calculate $f_π$ and $f_K$, extract Gasser-Leutwyler parameters of the chiral Lagrangian, and (modulo rather large perturbative errors) find the light and strange quark masses.
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…
We show how to compute chiral logarithms that take into account both the $\cO(a^2)$ taste-symmetry breaking of staggered fermions and the fourth-root trick that produces one taste per flavor. The calculation starts from the Lee-Sharpe Lagrangian generalized to multiple flavors. An error in a previous treatment by one of us is explained and corrected. The one loop chiral logarithm corrections to the pion and kaon masses in the full (unquenched), partially quenched, and quenched cases are computed as examples.
We review recent progress in calculating kaon spectrum, pseudoscalar meson decay constants, $B_K$, $ε'/ε$, $K\to ππ$ matrix elements, kaon semileptonic form factors, and moments of kaon distribution amplitudes on the lattice. We also address the issue of how best to improve the staggered fermion formulation for the action and operators.
The behavior of staggered domain wall fermions in the presence of gauge fields is presented. In particular, their response to gauge fields with nontrivial topology is discussed.
We present results for the matrix elements of the additional $ΔS=2$ operators that appear in models of physics beyond the Standard Model (BSM), expressed in terms of four BSM $B$-parameters. Combined with experimental results for $ΔM_K$ and $ε_K$, these constrain the parameters of BSM models. We use improved staggered fermions, with valence HYP-smeared quarks and $N_f=2+1$ flavors of "asqtad" sea quarks. The configurations have been generated by the MILC collaboration. The matching between lattice and continuum four-fermion operators and bilinears is done perturbatively at one-loop order. We u…
Models of new physics induce K-Kbar mixing operators having Dirac structures other than the "left-left" form of the Standard Model. We calculate the functional form of the corresponding B-parameters at next-to-leading order in both SU(3) and SU(2) staggered chiral perturbation theory (SChPT). Numerical results for these matrix elements are being generated using improved staggered fermions; our results can be used to extrapolate these matrix elements to the physical light and strange quark masses. The SU(3) SChPT results turn out to be much simpler than that for the Standard Model B_K operator,…
Improved results for $B_K$ are discussed. Scaling corrections are argued to be of $O(a^2)$, leading to a reduction in the systematic error. For a kaon composed of degenerate quarks, the quenched result is ${\widehat{B}_K} = 0.825 \pm 0.027 \pm 0.023$.
The QCD light quark mass renormalized at a 1 GeV scale in the $ \overline{MS} $ scheme is obtained from the numerical results of the lattice QCD simulation with staggered fermions. The primary emphasis is given to the connection between the lattice and continuum parameters. The results are compared with those from the QCD sum rule.
In this talk, I will give an overview of the theoretical status of staggered Lattice QCD with the "fourth-root trick." In this regularization of QCD, a separate staggered quark field is used for each physical flavor, and the inherent four-fold multiplicity that comes with the use of staggered fermions is removed by taking the fourth root of the staggered determinant for each flavor. At nonzero lattice spacing, the resulting theory is nonlocal and not unitary, but there are now strong arguments that this disease is cured in the continuum limit. In addition, the approach to the continuum limit c…
We compute O(alpha_s) lattice-to-continuum perturbative matching coefficients for the Delta S=2 flavor changing four-quark operators for the Asqtad improved staggered action. In conjunction with lattice simulations with three flavors of light, dynamical quarks, our results yield an unquenched determination of B_K, the parameter that determines the amount of indirect CP violation in the neutral kaon system. Its value is an important input for the unitarity triangle analysis of weak decays.
Consistency of present-day lattice QCD simulations with dynamical (``sea'') staggered fermions requires that the determinant of the staggered-fermion Dirac operator, $det(D)$, be equal to $det^4(D_{rg}) det(T)$ where $D_{rg}$ is a local one-flavor lattice Dirac operator, and $T$ is a local operator containing only excitations with masses of the order of the cutoff. Using renormalization-group (RG) block transformations I show that, in the limit of infinitely many RG steps, the required decomposition exists for the free staggered operator in the ``flavor representation.'' The resulting one-flav…
We report updated results for $B_K$ calculated using HYP-smeared staggered fermions on the MILC asqtad 2+1 flavor lattices. We use four different lattice spacings ($a \approx$ 0.12, 0.09, 0.06 and 0.045 fm) to control the continuum extrapolation. We use SU(2) staggered chiral perturbation theory to do the data analysis. We find that $B_K(\text{NDR}, μ=2 \text{GeV}) = 0.526 \pm 0.007 \pm 0.024$ and $\hat{B}_K = B_K(\text{RGI}) = 0.720 \pm 0.010 \pm 0.033$. Here the first error is statistical and the second systematic. The dominant source of error is that due to our use of a truncated (one-loop)…
Using staggered fermions and twisted boundary conditions, we calculate the K meson semileptonic decay vector form factor at zero momentum transfer. The HISQ formulation is used for the valence quarks, while the sea quarks are simulated with the asqtad action (MILC N_f=2+1 configurations). For the chiral and continuum extrapolation we use two-loop continuum CHPT, supplemented by partially quenched staggered CHPT at one loop. Our result is f_+^{Kπ}(0) = 0.9667+-0.0023+-0.0033, where the first error is statistical and the second is the sum in quadrature of the systematic uncertainties. This resul…
We present a phase diagram for SU(3) lattice gauge theory with 695 MeV dynamical staggered fermions, in the plane of temperature and chemical potential, obtained from effective Polyakov line actions. The derivation is via the method of relative weights, and the effective theories are solved at finite chemical potential by mean field theory.
We extend our recent unquenched ($N_f=2+1$ flavor) calculation of $B_K$ using improved staggered fermions by including in the fits the finite volume shift predicted by one-loop staggered chiral perturbation theory. The net result is to lower the result in the continuum limit by 0.6%. This shift is slightly smaller than our previous estimate of finite volume effects based on a direct comparison between different volumes. To include the finite volume effects in a reasonable time, we found it necessary to calculate them using Graphics Processing Units.
In confining lattice gauge theories in which part of the flavor group is coupled weakly to additional gauge fields, both the dynamics of the weak gauge fields as well as lattice artifacts may have non-trivial effects on the orientation of the vacuum in flavor space. Here we discuss this issue for lattice gauge theories employing staggered fermions. Staggered fermions break flavor symmetries to a much smaller group on the lattice, and orientations in flavor space that are equivalent in the continuum may be distinct on the lattice. Assuming universality, we show that in the continuum limit the w…
We calculate the axial current decay constants of taste non-Goldstone pions and kaons in staggered chiral perturbation theory through next-to-leading order. The results are a simple generalization of the results for the taste Goldstone case. New low-energy couplings are limited to analytic corrections that vanish in the continuum limit; certain coefficients of the chiral logarithms are modified, but they contain no new couplings. We report results for quenched, fully dynamical, and partially quenched cases of interest in the chiral SU(3) and SU(2) theories.
Using staggered fermions, we calculate the perturbative corrections to the bilinear and four-fermion operators that are used in the numerical study of weak matrix elements for $ε'/ε$. We present results for one-loop matching coefficients between continuum operators, calculated in the Naive Dimensional Regularization (NDR) scheme, and gauge invariant staggered fermion operators. These results, combined with existing results for penguin diagrams, provide the complete one-loop renormalization of the staggered four-fermion operators.
We calculate the kaon B-parameter, B_K, in chiral perturbation theory for a partially quenched, mixed action theory with Ginsparg-Wilson valence quarks and staggered sea quarks. We find that the resulting expression is similar to that in the continuum, and in fact has only two additional unknown parameters. At one-loop order, taste-symmetry violations in the staggered sea sector only contribute to flavor-disconnected diagrams by generating an O(a^2) shift to the masses of taste-singlet sea-sea mesons. Lattice discretization errors also give rise to an analytic term which shifts the tree-level …
We study the improvement of staggered fermions using hypercubically smeared (HYP) links. We calculate the strange quark mass and the kaon B-parameter, $B_K$, in quenched QCD on a $16^3 \times 64$ lattice at $β=6.0$. We find $m_s(\bar{\rm MS},2 {\rm GeV})=101.2\pm1.3\pm4 $MeV and $B_K(\bar{\rm MS},2 {\rm GeV}) = 0.578 \pm 0.018\pm 0.042$, where the first error is from statistics and fitting, and the second from using one-loop matching factors. The scale ($1/a=1.95$GeV) is set by $M_ρ$, and $m_s$ is determined using the kaon mass. Comparing to quenched results obtained using unimproved staggered…
The 2+1d Gross-Neveu model with finite density and finite temperature are studied by the staggered fermion discretization. The kinetic part of this staggered fermion in momentum space is used to build the relation between the staggered fermion and Wilson-like fermion. In the large Nf limit (the number Nf of staggered fermion flavors), the chiral condensate and fermion density are solved from the gap equation in momentum space, and thus the phase diagram of fermion coupling, temperature and chemical potential are obtained. Moreover, an analytic formula for the inverse of the staggered fermion m…
An explanation is proposed for the fact that Lepage--Mackenzie tadpole improvement does not work well for staggered fermions. The idea appears to work for all renormalization constants which appear in the staggered fermion self-energy. Wilson fermions are also discussed.
We discuss the behavior of free perfect staggered fermions and truncated versions thereof. The study includes flavor non-degenerate masses. We suggest a new blocking scheme, which provides excellent locality of the perfect lattice action. A truncation procedure adequate for the structure of staggered fermions is applied. We consider spectral and thermodynamic properties and compare truncated perfect actions, Symanzik improved and standard staggered fermions in two and four dimensions.
We present recent results from lattice simulations of 2+1 flavors of improved staggered fermions at zero baryon number density near the high temperature crossover. Included are new results from simulations of asqtad fermions at Nt = 12 and a nearly physical Goldstone pion mass and from simulations of HISQ fermions at Nt = 6 and 8. We focus on observables sensitive to chiral symmetry and confinement. A companion HotQCD talk discusses the effects of staggered-fermion taste-symmetry breaking on thermodynamic quantities.
Many results from lattice QCD of broad importance to particle and nuclear physics are obtained with 2+1 flavors of staggered sea quarks. In the continuum limit, staggered fermions yield four species, called tastes. To reduce the number of tastes to one (per flavor), the simulation employs the fourth root of the four-taste staggered fermion determinant. This talk surveys evidence in favor of this procedure, refutes recent criticisms, and reviews recent algorithmic and technical improvements. Physics results are covered in other plenary talks.
Lattice fermion actions are constructed with path integrals which are equivalent to the free one-flavour staggered fermion determinant. The Dirac operators used are local and have an identical spectrum of states to the staggered theory. Operators obeying a generalised Ginsparg-Wilson relation are developed.
We present results for the pion multiplet spectrum calculated using both unimproved staggered fermions and improved HYP-smeared staggered fermions. In the case of unimproved staggered fermions, we observe (consistent with previous work) that ${\cal O}(a^2)$ taste symmetry breaking effects are large and comparable to the $\approx {\cal O}(p^2)$ contributions to their masses. Higher order ${\cal O}(a^2 p^2)$ effects are also substantial enough to be seen. For HYP-smeared staggered fermions, we find that taste breaking is much reduced. The ${\cal O}(a^2)$ effects are observable, but are noticeabl…
We construct a perfect lattice action for staggered fermions by blocking from the continuum. The locality, spectrum and pressure of such perfect staggered fermions are discussed. We also derive a consistent fixed point action for free gauge fields and discuss its locality as well as the resulting static quark-antiquark potential. This provides a basis for the construction of (classically) perfect lattice actions for QCD using staggered fermions.
We discuss the construction of a chiral random matrix model for staggered fermions. This model includes $O(a^2)$ corrections to the continuum limit of staggered fermions and is related to the zero momentum limit of the Lee-Sharpe Lagrangian for staggered fermions. The naive construction based on a specific expansion in lattice spacing ($a$) of the Dirac matrix produces the term which gives the dominant contribution to the observed taste splitting in the pion masses. A more careful analysis can include extra terms which are also consistent with the symmetries of staggered fermions. Lastly I wil…
We extend the construction of the Symanzik effective action to include rooted staggered fermions, starting from a generalization of the renormalization-group approach to rooted staggered fermions. The Symanzik action, together with the usual construction of a partially quenched chiral effective theory from a local, partially quenched, fundamental theory, can then be used to derive the chiral effective theory. The latter reproduces rooted staggered chiral perturbation theory.
We study chiral properties of eigenvalue spectrum for staggered quarks. We present a new method to identify would-be zero modes and nonzero modes using their symmetry and chiral properties. Here, we review the traditional method with HYP improved staggered quarks, and extend it to a completely new method which uses the chiral Ward identities and leakage patterns to achieve the goal.
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…
It is conventional wisdom that staggered fermions do not feel gauge field topology. However, the response of staggered fermion eigenmodes to the topology of the gauge field can depend quite sensitively on the way in which the staggered fermion action is improved. We study this issue using a variety of improved staggered quark actions. We observe that the separation between the ``would be'' zero modes and the non-chiral modes increases with the level of improvement. This enables the ``zero modes'' to be identified unambiguously. The distribution of the remaining non-chiral modes is compared wit…
We discuss how to formulate a staggered chiral perturbation theory. This amounts to a generalization of the Lee-Sharpe Lagrangian to include more than one flavor (i.e. multiple staggered fields), which turns out to be nontrivial. One loop corrections to pion and kaon masses and decay constants are computed as examples in three cases: the quenched, partially quenched, and full (unquenched) case. The results for the one loop mass and decay constant corrections have already been presented in Ref. [1].
We study possibility of improving staggered fermions using various fat links in order to reduce perturbative corrections to the gauge-invariant staggered fermion operators. We prove five theorems on SU(3) projection, triviality in renormalization, multiple SU(3) projections, uniqueness and equivalence. As a result of these theorems, we show that, at one loop level, the renormalization of staggered fermion operators is identical between SU(3) projected Fat7 links and hypercubic links, as long as the action and operators are constructed by imposing the same perturbative improvement condition. In…
Expanding upon the arguments of Sharpe, we explicitly implement the Symanzik improvement program demonstrating the absence of order $a$ terms in the staggered fermion action. We propose a general program to improve fermion operators to remove $O(a)$ corrections from their matrix elements, and demonstrate this program for the examples of matrix elements of fermion bilinears and $B_K$. We also determine the additional operators which must be added to improve the staggered fermion currents.
We investigate the validity of the square rooting procedure of the staggered determinant in the context of the Schwinger model. We find some evidence that at fixed physical quark mass the square root of the staggered determinant becomes proportional to the overlap determinant in the continuum limit. We also find that at fixed lattice spacing moderate smearing dramatically improves the chiral behavior of staggered fermions.
We have studied $O(a^2)$ improved lattice QCD with the staggered fermion by using Symanzik's program. We find that there are 5 dimension-6 fermion bilinears and gauge operators. In addition, there are 10 four-fermion operators which are absent at the tree-level and tadpole-improved tree-level.
We present preliminary results for $ε'/ε$ calculated using HYP staggered fermions in the quenched approximation. We compare different choices of quenched penguin operators.
The staggered fermion approach to build models with chiral fermions is briefly reviewed. The method is tested in a U(1) model with axial vector coupling in two and four dimensions.
The topological susceptibility of the vacuum in quantum chromodynamics has been simulated numerically using the Asqtad improved staggered fermion formalism. At nonzero lattice spacing the residual fermion doublers (fermion ``tastes'') in the staggered fermion formalism give contributions to the susceptibility that deviate from conventional continuum chiral perturbation theory. In this brief report we estimate the taste-breaking artifact and compare it with results of recent simulations, finding that it accounts for roughly half of the scaling violation.
We present a chronological review of the progress in calculating weak matrix elements using staggered fermions. We review the perturbative calculation of one-loop matching formula including both current-current diagrams and penguin diagrams using improved staggered fermions. We also present preliminary results of weak matrix elements relevant to CP violation calculated using the improved (HYP (II)) staggered fermions. Since the complete set of matching coefficients at the one-loop level became available recently, we have constructed lattice operators with all the $g^2$ corrections included. Th…
We present a new formulation of the staggered fermion on the D-dimensional lattice based on the SO(2D) Clifford algebra, which is naturally present in the action. The action of the massless staggered fermion is invariant under the discrete rotation and the SO(2D) chiral and other discrete transformations. From transformation properties of the fermion, we find two local meson operators (one scalar and one pseudoscalar) in addition to two standard meson operators.
We discuss the Schrödinger functional in lattice QCD with staggered fermions including its order $O(a)$ boundary counterterms. We relate it, in the classical continuum limit, to the Schrödinger functional as obtained in the same limit 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 introduce a lattice fermion-Higgs model with one component `reduced staggered' fermions. In order to use the fermion field as efficiently as possible we couple the two {\em staggered} flavors to the O(4) Higgs field leading to a model with only one SU(2) doublet in the scaling region. The number of fermions is doubled in a numerical investigation of the model with the hybrid Monte Carlo algorithm. We present results for the phase diagram, particle masses and renormalized couplings on lattices ranging in size from $6^3 24$ to $16^3 24$.
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…
We present a calculation of the mass of the eta-prime meson using quenched and dynamical staggered fermions. We also discuss the effects of "cooling" and suggest its use as a quantitative tool.
The legality of the "rooting trick" in dynamical staggered fermion simulations is discussed, i.e. whether the theory with the Boltzmann weight $\det^{1/4}(D_\mathrm{st})$ yields the right continuum limit. Since the problem is unsolved, pieces of evidence in favor and against are collected and examined.
The variational method is used widely for determining eigenstates of the QCD hamiltonian for actions with a conventional transfer matrix, e.g., actions with improved Wilson fermions. An alternative lattice fermion formalism, staggered fermions, does not have a conventional single-time-step transfer matrix. Nonetheless, with a simple modification, the variational method can also be applied to that formalism. In some cases the method also provides a mechanism for separating the commonly paired parity-partner states. We discuss the extension to staggered fermions and illustrate it by applying it …
We construct domain wall fermions with a staggered kernel and investigate their spectral and chiral properties numerically in the Schwinger model. In some relevant cases we see an improvement of chirality by more than an order of magnitude as compared to usual domain wall fermions. Moreover, we present first results for four-dimensional quantum chromodynamics, where we also observe significant reductions of chiral symmetry violations for staggered domain wall fermions.
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…
