arXiv subjects

Stephen R. Sharpe

Publications and source records attributed to Stephen R. Sharpe.

At least 91 records · Page 5Linked to original sources

arXiv2011

Determining low-energy constants in partially quenched Wilson chiral perturbation theory

In the low energy effective theory describing the partially quenched extension of two light Wilson fermions, three low energy constants (LECs) appear in terms proportional to a^2 (a being the lattice spacing). We propose methods to separately calculate these LECs, typically called W'_6, W'_7 and W'_8. While only one linear combination of these constants enters into physical quantities, different combinations enter into the description of the spectral density and eigenvalue distributions of the lattice Dirac operator and its Hermitian counterpart. Thus it is useful to be able to determine the LECs separately. Our methods require studying certain correlation functions for either two or three pion scattering, which are accessible only in the partially quenched extension of the theory.

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arXiv2011

Constraint on the Low Energy Constants of Wilson Chiral Perturbation Theory

Wilson chiral perturbation theory (WChPT) is the effective field theory describing the long- distance properties of lattice QCD with Wilson or twisted-mass fermions. We consider here WChPT for the theory with two light flavors of Wilson fermions or a single light twisted-mass fermion. Discretization errors introduce three low energy constants (LECs) into partially quenched WChPT at O(a^2), conventionally called W'_6, W'_7 and W'_8 . The phase structure of the theory at non-zero a depends on the sign of the combination 2W'_6 + W'_8, while the spectrum of the lattice Hermitian Wilson-Dirac operator depends on all three constants. It has been argued, based on the positivity of partition functions of fixed topological charge, and on the convergence of graded group integrals that arise in the epsilon-regime of ChPT, that there is a constraint on the LECs arising from the underlying lattice theory. In particular, for W'_6 = W'_7 = 0, the constraint found is W'_8 \le 0. Here we provide an alternative line of argument, based on mass inequalities for the underlying partially quenched theory. We find that W'_8 \le 0, irrespective of the values of W'_6 and W'_7. Our constraint implies that 2W'_6 > |W'_8| if the phase diagram is to be described by the first-order scenario, as recent simulations suggest is the case for some choices of action.

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arXiv2011

Large-N reduction with two adjoint Dirac fermions

We study the single site SU(N) lattice gauge theory with N_f=2 adjoint Wilson fermions for values of N up to 53. We determine the phase diagram of the theory as a function of the hopping parameter kappa and the inverse 't Hooft coupling b, searching for the region in which the Z_N^4 center symmetry is unbroken. In this region the theory is equivalent to the infinite volume theory when N goes to infinity. We find a region of values of kappa on both sides of kappa_c for which the symmetry is unbroken, including both light physical quarks and masses ~O(1/a). This is surrounded by a region with a complicated sequence of partially broken phases. We calculate Wilson loop expectation values and find that using N <= 53 it is possible to extract the heavy-quark potential at small distances (1-3 links) but not at longer distances. For this, larger values of N, or lattices with more sites, are needed.

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arXiv2011

Continuum extrapolation of $B_K$ with staggered fermions

We report on recent progress in the calculation of $B_K$ using HYP-smeared staggered fermions on the MILC asqtad lattices. Our main focus is on the continuum extrapolation, which is done using (up to) four different lattice spacings---$a\approx$ 0.12, 0.09, 0.06 and 0.045 fm. Since Lattice 2010, we have reduced the statistical errors on the $a\approx 0.09\;$fm lattices by a factor of $\sim 3$, and roughly doubled the size of the $a\approx0.045\;$fm ensemble. We find that these improvements have a very significant impact on the continuum extrapolation, with the $a\approx 0.12\;$fm data lying outside the range of applicability of simple functional forms. Hence we use only the three smallest lattice spacings to perform the extrapolation, finding $\hat{B}_K = B_K(\text{RGI}) = 0.725 \pm 0.004(\text{stat}) \pm 0.038(\text{sys}) $. This value is consistent with our published value from 2010 (based the three coarsest lattice spacings), but has smaller errors.

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arXiv2011

Finite Volume Errors in $B_K$

We discuss finite volume errors in our calculations of $B_K$ using improved staggered fermions on the MILC asqtad lattices. Using GPUs, we are now able to extrapolate using next-to-leading order (NLO) staggered SU(2) chiral perturbation theory including the finite volume corrections arising from pion loops. We find that the impact of FV fitting is very small, giving a 0.5% shift in the continuum limit.

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arXiv2011

Non-perturbative renormalization for general improved staggered bilinears

We present results for non-perturbative renormalization (NPR) factors for staggered fermion bilinears of arbitrary spin and taste. We use "covariant" bilinears which transform irreducibly under the lattice translation and rotation group, and thus do not mix. We form ~ 30 ratios which have no anomalous dimensions, and compare the NPR results to those from 1-loop perturbation theory. We also compare the absolute renormalization factors (which, in general, do have anomalous dimensions) to 1-loop perturbation theory. We use asqtad and HYP-smeared staggered valence fermions on the coarse MILC asqtad lattices.

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arXiv2011

$B_K$ from improved staggered fermions using SU(3) chiral perturbation theory

We present recent progress in our calculation of $B_K$ with improved staggered fermions using chiral extrapolations based on SU(3) staggered chiral perturbation theory. We have accumulated significantly higher statistics on the coarse, fine, and ultrafine MILC asqtad lattices. This leads to a reduction in statistical error and an improved continuum extrapolation. Our updated result is $\hat{B}_K = B_K(\text{RGI}) = 0.737 \pm 0.003(\text{stat}) \pm 0.046 (\text{sys})$. This is consistent with the result obtained using chiral extrapolations based on SU(2) staggered chiral perturbation theory, although the total error is somewhat larger with the SU(3) analysis.

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arXiv2011

Finite volume effects in $B_K$ with improved staggered fermions

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.

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arXiv2011

One-loop matching of improved four-fermion staggered operators with an improved gluon action

We present results for one-loop matching factors of four-fermion operators composed of HYP-smeared staggered fermions. We generalize previous calculations by using the tree-level improved Symanzik gauge action. These results are needed for our companion numerical calculation of $B_K$ and related matrix elements. We find that the impact on one-loop matching factors of using the improved gluon action is much smaller than that from the use of either HYP smearing or mean-field improvement. The one-loop coefficients for mean-field improved, HYP-smeared operators with the Symanzik gauge action have a maximum magnitude of $O(1)\times α_s$, indicating that perturbation theory is reasonably convergent.

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arXiv2010

$B_K$ using HYP-smeared staggered fermions in $N_f=2+1$ unquenched QCD

We present results for kaon mixing parameter $B_K$ calculated using HYP-smeared improved staggered fermions on the MILC asqtad lattices. We use three lattice spacings ($a\approx 0.12$, $0.09$ and $0.06\;$fm), ten different valence quark masses ($m\approx m_s/10-m_s$), and several light sea-quark masses in order to control the continuum and chiral extrapolations. We derive the next-to-leading order staggered chiral perturbation theory (SChPT) results necessary to fit our data, and use these results to do extrapolations based both on SU(2) and SU(3) SChPT. The SU(2) fitting is particularly straightforward because parameters related to taste-breaking and matching errors appear only at next-to-next-to-leading order. We match to the continuum renormalization scheme (NDR) using one-loop perturbation theory. Our final result is from the SU(2) analysis, with the SU(3) result providing a (less accurate) cross check. We find $B_K(\text{NDR}, μ= 2 \text{GeV}) = 0.529 \pm 0.009 \pm 0.032$ and $\hat{B}_K =B_K(\text{RGI})= 0.724 \pm 0.012 \pm 0.043$, where the first error is statistical and the second systematic. The error is dominated by the truncation error in the matching factor. Our results are consistent with those obtained using valence domain-wall fermions on lattices generated with asqtad or domain-wall sea quarks.

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arXiv2010

$B_K$ with improved staggered fermions: analysis using SU(2) staggered chiral perturbation theory

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) matching factor.

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arXiv2010

$B_K$ with improved staggered fermions: analysis using SU(3) staggered chiral perturbation theory

We report updated results for $B_K$ using HYP-smeared staggered valence quarks on MILC asqtad lattices based on an analysis using SU(3) staggered chiral perturbation theory. The most important new feature of our data sample is the inclusion of a fourth ("ultrafine") lattice spacing. This improves the control over the continuum extrapolation and errors due our use of one-loop perturbative matching. We present a complete updated error budget, which leads to $B_K(\text{NDR}, μ= 2 \text{GeV}) = 0.5309 \pm 0.0051 \pm 0.0424$ and $\hat{B}_K = B_K(\text{RGI}) = 0.727 \pm 0.07 \pm 0.058$. The results of the SU(3) analysis are inferior to those based on SU(2) staggered chiral perturbation theory, primarily because of the dependence on the Bayesian priors we use in the SU(3) fits.

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arXiv2010

Systematic Uncertainties in $B_K$ with Improved Staggered Fermions

We study three sources of error in our calculation of $B_K$ using HYP-smeared staggered fermions on the MILC asqtad lattices. These are (1) dependence on the light sea quark mass; (2) finite volume effects; and (3) the impact of an order of magnitude increase in the number of measurements. Our main results are (1) the dependence on the light sea-quark mass is weaker than expected by naive dimensional analysis, (2) including finite volume effects in SU(2) staggered chiral perturbation theory fits leads to a very small change in $B_K$, of size $\approx 0.1%$, and (3) increasing the statistics on one of the coarse MILC lattices resolves a potential discrepancy with other coarse results.

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arXiv2010

First results for $B_K$ on the ultrafine ($a=0.045$ fm) ensemble

We present preliminary results for $B_K$ from the MILC ultrafine lattices, based on a partial ensemble of 305 configurations. We use HYP-smeared improved staggered valence quarks. The analysis is done using fitting forms based on both SU(2) and SU(3) staggered chiral perturbation thery. For the SU(2) analysis, we find that the result using the NLO fit function is consistent with that from a partial NNLO fit. For the SU(3) analysis, where we have to use partially constrained fits due to the number of fit parameters, we find that our two preferred fits ("N-BB1" and "N-BB2") are also consistent, both with each other and with the results of the SU(2) fits. These results are used in companion proceedings to improve the control over the continuum extrapolation.

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arXiv2010

One loop matching factors for improved staggered four-fermion operators with improved glue

We present results for matching factors for staggered four-fermion operators constructed using HYP-smeared fat links both in the action and the operators. We use perturbation theory to calculate the matching factors and work to one-loop order. The new feaure of this work is the use of the Symanzik-improved gauge action, as opposed to the Wilson gauge action. Our results are needed for our ongoing calculation of weak matrix elements using HYP-smeared staggered valence quarks and operators on MILC lattices. We give explicit results for matching factors of the operator needed to calculate $B_K$. We compare the impact of the improvement of the gauge action on one-loop coefficients with that of mean-field improvement of the operators.

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arXiv2010

One-loop matching factors for staggered bilinear operators with improved gauge actions

We present results for one-loop perturbative matching factors using bilinear operators composed of improved staggered fermions, using unimproved (Wilson) and improved (Symanzik, Iwasaki, and DBW2) gluon actions. We consider two fermions actions---HYP/$\bar{\text{Fat7}}$-smeared and "asqtad". The former is being used in calculations of electroweak matrix elements, while the latter have been used extensively by the MILC collaboration. We observe that using the improved gluon action leads to small reductions in the perturbative corrections, but that these reductions are smaller than those obtained when moving from the tadpole-improved naive staggered action to either HYP-smeared or asqtad action.

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arXiv2009

Determination of $B_K$ using improved staggered fermions (II) SU(2) chiral perturbation theory fit

We present results for $B_K$ calculated using HYP-smeared improved staggered fermions on the MILC asqtad lattices. In this report, the data is analyzed using the results of SU(2) staggered chiral perturbation theory (SChPT). We outline the derivation of the NLO SU(2) SChPT result, explain our fitting procedure, and outline how we estimate systematic errors. We also show the light sea-quark mass and lattice spacing dependence for both SU(2) and SU(3)-based analyses. Our preliminary result from the SU(2) analysis is $B_K(\text{NDR}, μ= 2 \text{GeV}) = 0.512 \pm 0.014 \pm 0.034$ and $\hat{B}_K = B_K(\text{RGI})= 0.701 \pm 0.019 \pm 0.047$. This is somewhat more accurate than our result from the SU(3) analysis. It is consistent with results obtained using valence domain-wall fermions

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arXiv2009

Determination of $B_K$ using improved staggered fermions (I): SU(3) chiral perturbation theory fit

We present the results for $B_K$ calculated using HYP-smeared staggered fermions using lattices generated by the MILC collaboration using the asqtad staggered action. We have done the calculation on 8 ensembles of these lattices, including three different lattice spacings ($a=0.12, 0.09, 0.06$ fm). We fit the data to forms based on those predicted by SU(3) mixed-action partially-quenched staggered chiral perturbation theory. Our preliminary result is $ B_K(\text{NDR},μ=2\text{GeV}) = 0.528 \pm 0.011 \pm 0.048 $, where the first error is statistical and the second systematic. The error turns out to be larger than that from an analysis using SU(2) chiral perturbation theory.

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