arXiv subjects

Stephen R. Sharpe

Publications and source records attributed to Stephen R. Sharpe.

At least 127 records · Page 7Linked to original sources

arXiv2006

Rooted staggered fermions: good, bad or ugly?

I give a status report on the validity of the so-called ``fourth-root trick'', i.e. the procedure of representing the determinant for a single fermion by the fourth root of the staggered fermion determinant. This has been used by the MILC collaboration to create a large ensemble of lattices using which many quantities of physical interest have been and are being calculated. It is also used extensively in studies of QCD thermodynamics. The main question is whether the theory so defined has the correct continuum limit. There has been significant recent progress towards answering this question. After recalling the issue, and putting it into a broader context of results from statistical mechanics, I critically review the new work. I also address the related issue of the impact of treating valence and sea quarks differently in rooted simulations, discuss whether rooted simulations at finite temperature and density are subject to additional concerns, and briefly update results for quark masses using the MILC configurations. An answer to the question in the title is proposed in the summary.

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arXiv2006

Pion spectrum using improved staggered fermions

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 noticeably smaller than those obtained with AsqTad-improved staggered fermions, and much smaller than those obtained using unimproved staggered fermions, while ${\cal O}(a^2 p^2)$ effects are suppressed to such a level that we cannot observe them given our statistical errors. From this numerical study, we conclude that HYP staggered fermions are significantly better that AsqTad fermions from the perspective of taste symmetry breaking.

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arXiv2006

Scaling behavior of discretization errors in renormalization and improvement constants

Non-perturbative results for improvement and renormalization constants needed for on-shell and off-shell O(a) improvement of bilinear operators composed of Wilson fermions are presented. The calculations have been done in the quenched approximation at beta=6.0, 6.2 and 6.4. To quantify residual discretization errors we compare our data with results from other non-perturbative calculations and with one-loop perturbation theory.

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arXiv2006

Discretization errors in the spectrum of the Hermitian Wilson-Dirac operator

I study the leading effects of discretization errors on the low energy part of the spectrum of the Hermitian Wilson-Dirac operator in infinite volume. The method generalizes that used to study the spectrum of the Dirac operator in the continuum, and uses partially quenched chiral perturbation theory for Wilson fermions. The leading-order corrections are proportional to a^2 (a being the lattice spacing). At this order I find that the method works only for one choice of sign of one of the three low energy constants describing discretization errors. If these constants have the relative magnitudes expected from large N_c arguments, then the method works if the theory has an Aoki phase for m of order a^2, but fails if there is a first-order transition. In the former case, the dependence of the gap and the spectral density on m and a^2 are determined. In particular, the gap is found to vanish more quickly as m_pi^2-> 0 than in the continuum. This reduces the region where simulations are safe from fluctuations in the gap.

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arXiv2006

Improved bilinears in lattice QCD with non-degenerate quarks

We describe the extension of the improvement program for bilinear operators composed of Wilson fermions to non-degenerate dynamical quarks. We consider two, three and four flavors, and both flavor non-singlet and singlet operators. We find that there are many more improvement coefficients than with degenerate quarks, but that, for three or four flavors, nearly all can be determined by enforcing vector and axial Ward identities. The situation is worse for two flavors, where many more coefficients remain undetermined.

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arXiv2005

B_K in Staggered Chiral Perturbation Theory

We calculate the kaon B-parameter, B_K, to next-to-leading order in staggered chiral perturbation theory. We find expressions for partially quenched QCD with three sea quarks, quenched QCD, and full QCD with m_u = m_d but not equal to m_s. 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 O(a^2) terms from the discretization of operators, and through the truncation of matching factors. These effects cause mixing with several additional operators, complicating the chiral and continuum extrapolations. In addition to the staggered expressions, we present B_K at next-to-leading order in continuum partially quenched chiral perturbation theory for N_f=3 sea quarks with m_u = m_d but not equal to m_s.

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arXiv2005

Observations on discretization errors in twisted-mass lattice QCD

I make a number of observations concerning discretization errors in twisted-mass lattice QCD that can be deduced by applying chiral perturbation theory including lattice artifacts. (1) The line along which the PCAC quark mass vanishes in the twisted mass-twisted mass plane makes an angle to the untwisted mass axis which is a direct measure of O(a) terms in the chiral Lagrangian, and is found numerically to be large; (2) Numerical results for pionic quantities in the mass plane show the qualitative properties predicted by chiral perturbation theory, in particular an asymmetry in slopes between positive and negative untwisted quark masses; (3) By extending the description of the ``Aoki regime'' (where m_q is of size a^2 Lambda_QCD^3) to next-to-leading order in chiral perturbation theory I show how the phase transition lines and lines of maximal twist (using different definitions) extend into this region, and give predictions for the functional form of pionic quantities; (4) I argue that the recent claim that lattice artifacts at maximal twist have apparent infrared singularities in the chiral limit results from expanding about the incorrect vacuum state. Shifting to the correct vacuum (as can be done using chiral perturbation theory) the apparent singularities are summed into non-singular, and furthermore predicted, forms. I further argue that there is no breakdown in the Symanzik expansion in powers of lattice spacing, and no barrier to simulating at maximal twist in the Aoki regime.

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arXiv2005

Finite-volume effects in moving frames

We determine the quantization condition for the energy levels of two interacting particles in a finite box in a ``moving frame'', i.e. one in which the total momentum of pions is non-zero. This condition is valid up to corrections which fall exponentially withe the box size, and holds only below the inelastic threshold. It is derived using field theoretic methods, using a generalization of previous summation formulae relating sums and integrals over momenta. The result agrees with that obtained earlier by Rummakainen and Gottlieb using a relativistic quantum mechanical approach. Technically, we expand the finite-volume four-point Green function in terms of the infinite-volume Bethe-Salpeter kernel, and determine the position of the poles. The final result is written in terms of the two-pion scattering phase shift. Our result can be used to facilitate the determination of the scattering phase shift, and can be used to generalize the Lellouch-Lüscher formula relating finite-volume two-particle matrix elements to those in infinite volume.

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arXiv2005

The Kaon B-Parameter in Staggered Chiral Perturbation Theory

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.

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arXiv2005

Finite-Volume Effects for Two-Hadron States in Moving Frames

We determine the finite-volume corrections to the spectrum and matrix elements of two-hadron states in a moving frame, i.e. one in which the total momentum of the two-hadrons is non-zero. The analysis is performed entirely within field theory and the results are accurate up to exponential corrections in the volume. Our results for the spectrum are equivalent to those of Rummukainen and Gottlieb which had been obtained using a relativistic quantum mechanical approach. A technical step in our analysis is a simple derivation of the summation formulae relating the loop summations over the momenta of the two hadrons in finite volume to the corresponding integrals in infinite volume.

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arXiv2005

Testing improved staggered fermions with $m_s$ and $B_K$

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 fermions and other discretizations, we argue that the size of discretization errors in $B_K$ is substantially reduced by improvement.

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arXiv2005

Twisted mass chiral perturbation theory at next-to-leading order

We study the properties of pions in twisted mass lattice QCD (with two degenerate flavors) using chiral perturbation theory (ChPT). We work to next-to-leading order (NLO) in a power counting scheme in which m_q ~ a Λ_QCD^2, with m_q the physical quark mass and a the lattice spacing. We argue that automatic O(a) improvement of physical quantities at maximal twist, which has been demonstrated in general if m_q >> a Λ_QCD^2, holds even if m_q ~ a Λ_QCD^2, as long as one uses an appropriate non-perturbative definition of the twist angle. We demonstrate this with explicit calculations, for arbitrary twist angle, of all pionic quantities that involve no more than a single pion in the initial and final states: masses, decay constants, form factors and condensates, as well as the differences between alternate definitions of twist angle. We also calculate the axial and pseudoscalar form factors of the pion, quantities which violate flavor and parity, and which vanish in the continuum limit. These are of interest because they are not automatically O(a) improved at maximal twist. They allow a determination of the unknown low energy constants introduced by discretization errors, and provide tests of the accuracy of ChPT at NLO. We extend our results into the regime where m_q ~ a^2 Λ_QCD^3, and argue in favor of a recent proposal that automatic O(a) improvement at maximal twist remains valid in this regime.

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arXiv2005

Staggered Chiral Perturbation Theory at Next-to-Leading Order

We study taste and Euclidean rotational symmetry violation for staggered fermions at nonzero lattice spacing using staggered chiral perturbation theory. We extend the staggered chiral Lagrangian to O(a^2 p^2), O(a^4) and O(a^2 m), the orders necessary for a full next-to-leading order calculation of pseudo-Goldstone boson masses and decay constants including analytic terms. We then calculate a number of SO(4) taste-breaking quantities, which involve only a small subset of these NLO operators. We predict relationships between SO(4) taste-breaking splittings in masses, pseudoscalar decay constants, and dispersion relations. We also find predictions for a few quantities that are not SO(4) breaking. All these results hold also for theories in which the fourth-root of the fermionic determinant is taken to reduce the number of quark tastes; testing them will therefore provide evidence for or against the validity of this trick.

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arXiv2004

Applying chiral perturbation to twisted mass Lattice QCD

We have explored twisted mass LQCD (tmLQCD) analytically using chiral perturbation theory, including discretization effects up to O(a^2), and working at next-to-leading (NLO) order in the chiral expansion. In particular we have studied the vacuum structure, and calculated the dependence of pion masses and decay constants on the quark mass, twisting angle and lattice spacing. We give explicit examples for quantities that both are and are not automatically improved at maximal twisting.

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arXiv2004

The phase diagram of twisted mass lattice QCD

We use the effective chiral Lagrangian to analyze the phase diagram of two-flavor twisted mass lattice QCD as a function of the normal and twisted masses, generalizing previous work for the untwisted theory. We first determine the chiral Lagrangian including discretization effects up to next-to-leading order (NLO) in a combined expansion in which m_π^2/(4πf_π)^2 ~ a Λ(a being the lattice spacing, and Λ= Λ_{QCD}). We then focus on the region where m_π^2/(4πf_π)^2 ~ (a Λ)^2, in which case competition between leading and NLO terms can lead to phase transitions. As for untwisted Wilson fermions, we find two possible phase diagrams, depending on the sign of a coefficient in the chiral Lagrangian. For one sign, there is an Aoki phase for pure Wilson fermions, with flavor and parity broken, but this is washed out into a crossover if the twisted mass is non-vanishing. For the other sign, there is a first order transition for pure Wilson fermions, and we find that this transition extends into the twisted mass plane, ending with two symmetrical second order points at which the mass of the neutral pion vanishes. We provide graphs of the condensate and pion masses for both scenarios, and note a simple mathematical relation between them. These results may be of importance to numerical simulations.

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arXiv2004

Next-to-Leading-Order Staggered Chiral Perturbation Theory

We have extended staggered chiral perturbation theory to O(a^2 p^2), O(a^4), and O(a^2 m), the orders necessary for a full next-to-leading order calculation of pseudo-Goldstone boson masses and decay constants including taste-symmetry violations. We present predictions relating SO(4) taste-breaking splittings in masses, pseudoscalar decay constants, and dispersion relations. These can be used to test the fourth-root trick.

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arXiv2004

Unphysical Operators in Partially Quenched QCD

We point out that the chiral Lagrangian describing pseudo-Goldstone bosons in partially quenched QCD has one more four-derivative operator than that for unquenched QCD with three flavors. The new operator can be chosen to vanish in the unquenched sector of the partially quenched theory. Its contributions begin at next-to-leading order in the chiral expansion. At this order it contributes only to unphysical scattering processes, and we work out some examples. Its contributions to pseudo-Goldstone properties begin at next-to-next-to-leading order, and we determine their form. We also determine all the zero and two derivative operators in the $O(p^6)$ partially quenched chiral Lagrangian, finding three more than in unquenched QCD, and use these to give the general form of the analytic next-to-next-to-leading order contributions to the pseudo-Goldstone mass and decay constant. We discuss the general implications of such additional operators for the utility of partially quenched simulations

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arXiv2003

Mass dependence of the hairpin vertex in quenched QCD

The pseudoscalar ``hairpin'' vertex (i.e. quark-disconnected vertex) plays a key role in quenched chiral perturbation theory. Direct calculations using lattice simulations find that it has a significant dependence on quark mass. I show that this mass dependence can be used to determine the quenched Gasser-Leutwyler constant L5. This complements the calculation of L5 using the mass dependence of the axial decay constant of the pion. In an appendix, I discuss power counting for quenched chiral perturbation theory and describe the particular scheme used in this paper.

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