Recent results on light hadron and quark masses

Recent results for the spectrum of light hadrons provide clear evidence for the failure of quenched QCD and encouraging signs that simulations with dynamical sea quarks rectify some of the discrepancies, although string breaking has not yet been observed. The use of perturbation theory to match lattice quark masses to continuum schemes remains questionable, but non-perturbative methods are poised to remove this uncertainty. The inclusion of dynamical sea quarks substantially reduces estimates of the light quark masses. New results for the lightest glueball and the lightest exotic hybrid state provide useful input to phenomenology, but still have limited or no treatment of mixing. The $O(a)$-improved Wilson quark action is well-established in quenched QCD for $\beta\geq 5.7$, with most parameters obtainable non-perturbatively, in which range scaling violations are small. Progress has also been made with high-order improvement schemes for both Wilson and staggered quarks.Comment: LATTICE98(Plenary Talk

QuickXsort: Efficient Sorting with n log n - 1.399n +o(n) Comparisons on Average

In this paper we generalize the idea of QuickHeapsort leading to the notion of QuickXsort. Given some external sorting algorithm X, QuickXsort yields an internal sorting algorithm if X satisfies certain natural conditions. With QuickWeakHeapsort and QuickMergesort we present two examples for the QuickXsort-construction. Both are efficient algorithms that incur approximately n log n - 1.26n +o(n) comparisons on the average. A worst case of n log n + O(n) comparisons can be achieved without significantly affecting the average case. Furthermore, we describe an implementation of MergeInsertion for small n. Taking MergeInsertion as a base case for QuickMergesort, we establish a worst-case efficient sorting algorithm calling for n log n - 1.3999n + o(n) comparisons on average. QuickMergesort with constant size base cases shows the best performance on practical inputs: when sorting integers it is slower by only 15% to STL-Introsort

Scaling functions for O(4) in three dimensions

Monte Carlo simulation using a cluster algorithm is used to compute the scaling part of the free energy for a three dimensional O(4) spin model. The results are relevant for analysis of lattice studies of high temperature QCD.Comment: 12 pages, 6 figures, uses epsf.st

Strong contribution to octet baryon mass splittings

We calculate the $m_d-m_u$ contribution to the mass splittings in baryonic isospin multiplets using SU(3) chiral perturbation theory and lattice QCD. Fitting isospin-averaged perturbation theory functions to PACS-CS and QCDSF-UKQCD Collaboration lattice simulations of octet baryon masses, and using the physical light quark mass ratio $m_u/m_d$ as input, allows $M_n-M_p$, $M_{\Sigma^-}-M_{\Sigma^+}$ and $M_{\Xi^-}-M_{\Xi^0}$ to be evaluated from the full SU(3) theory. The resulting values for each mass splitting are consistent with the experimental values after allowing for electromagnetic corrections. In the case of the nucleon, we find $M_n-M_p= 2.9 \pm 0.4 \textrm{MeV}$, with the dominant uncertainty arising from the error in $m_u/m_d$

The equation of state for two flavor QCD at N_t=6

We calculate the two flavor equation of state for QCD on lattices with lattice spacing a=(6T)^{-1} and find that cutoff effects are substantially reduced compared to an earlier study using a=(4T)^{-1}. However, it is likely that significant cutoff effects remain. We fit the lattice data to expected forms of the free energy density for a second order phase transition at zero-quark-mass, which allows us to extrapolate the equation of state to m_q=0 and to extract the speed of sound. We find that the equation of state depends weakly on the quark mass for small quark mass.Comment: 24 pages, latex, 11 postscipt figure

Lattice determination of the critical point of QCD at finite T and \mu

Based on universal arguments it is believed that there is a critical point (E) in QCD on the temperature (T) versus chemical potential (\mu) plane, which is of extreme importance for heavy-ion experiments. Using finite size scaling and a recently proposed lattice method to study QCD at finite \mu we determine the location of E in QCD with n_f=2+1 dynamical staggered quarks with semi-realistic masses on $L_t=4$ lattices. Our result is T_E=160 \pm 3.5 MeV and \mu_E= 725 \pm 35 MeV. For the critical temperature at \mu=0 we obtained T_c=172 \pm 3 MeV.Comment: misprints corrected, version to appear in JHE

Quenched Lattice QCD with Domain Wall Fermions and the Chiral Limit

Quenched QCD simulations on three volumes, $8^3 \times$, $12^3 \times$ and $16^3 \times 32$ and three couplings, $\beta=5.7$, 5.85 and 6.0 using domain wall fermions provide a consistent picture of quenched QCD. We demonstrate that the small induced effects of chiral symmetry breaking inherent in this formulation can be described by a residual mass (\mres) whose size decreases as the separation between the domain walls ($L_s$) is increased. However, at stronger couplings much larger values of $L_s$ are required to achieve a given physical value of \mres. For $\beta=6.0$ and $L_s=16$, we find \mres/m_s=0.033(3), while for $\beta=5.7$, and $L_s=48$, \mres/m_s=0.074(5), where $m_s$ is the strange quark mass. These values are significantly smaller than those obtained from a more naive determination in our earlier studies. Important effects of topological near zero modes which should afflict an accurate quenched calculation are easily visible in both the chiral condensate and the pion propagator. These effects can be controlled by working at an appropriately large volume. A non-linear behavior of $m_\pi^2$ in the limit of small quark mass suggests the presence of additional infrared subtlety in the quenched approximation. Good scaling is seen both in masses and in $f_\pi$ over our entire range, with inverse lattice spacing varying between 1 and 2 GeV.Comment: 91 pages, 34 figure

Small, Dense Quark Stars from Perturbative QCD

As a model for nonideal behavior in the equation of state of QCD at high density, we consider cold quark matter in perturbation theory. To second order in the strong coupling constant, $\alpha_s$, the results depend sensitively on the choice of the renormalization mass scale. Certain choices of this scale correspond to a strongly first order chiral transition, and generate quark stars with maximum masses and radii approximately half that of ordinary neutron stars. At the center of these stars, quarks are essentially massless.Comment: ReVTeX, 5 pages, 3 figure

Chirality Correlation within Dirac Eigenvectors from Domain Wall Fermions

In the dilute instanton gas model of the QCD vacuum, one expects a strong spatial correlation between chirality and the maxima of the Dirac eigenvectors with small eigenvalues. Following Horvath, {\it et al.} we examine this question using lattice gauge theory within the quenched approximation. We extend the work of those authors by using weaker coupling, $\beta=6.0$, larger lattices, $16^4$, and an improved fermion formulation, domain wall fermions. In contrast with this earlier work, we find a striking correlation between the magnitude of the chirality density, $|\psi^\dagger(x)\gamma^5\psi(x)|$, and the normal density, $\psi^\dagger(x)\psi(x)$, for the low-lying Dirac eigenvectors.Comment: latex, 25 pages including 12 eps figure

Direct CP violation and the ΔI=1/2 rule in K→ππ decay from the standard model

We present a lattice QCD calculation of the ΔI=1/2, K→ππ decay amplitude A0 and ϵ′, the measure of direct CP violation in K→ππ decay, improving our 2015 calculation [1] of these quantities. Both calculations were performed with physical kinematics on a 323×64 lattice with an inverse lattice spacing of a-1=1.3784(68)  GeV. However, the current calculation includes nearly 4 times the statistics and numerous technical improvements allowing us to more reliably isolate the ππ ground state and more accurately relate the lattice operators to those defined in the standard model. We find Re(A0)=2.99(0.32)(0.59)×10-7  GeV and Im(A0)=-6.98(0.62)(1.44)×10-11  GeV, where the errors are statistical and systematic, respectively. The former agrees well with the experimental result Re(A0)=3.3201(18)×10-7  GeV. These results for A0 can be combined with our earlier lattice calculation of A2 [2] to obtain Re(ϵ′/ϵ)=21.7(2.6)(6.2)(5.0)×10-4, where the third error represents omitted isospin breaking effects, and Re(A0)/Re(A2)=19.9(2.3)(4.4). The first agrees well with the experimental result of Re(ϵ′/ϵ)=16.6(2.3)×10-4. A comparison of the second with the observed ratio Re(A0)/Re(A2)=22.45(6), demonstrates the standard model origin of this “ΔI=1/2 rule” enhancement.We present a lattice QCD calculation of the $\Delta I=1/2$, $K\to\pi\pi$ decay amplitude $A_0$ and $\varepsilon'$, the measure of direct CP-violation in $K\to\pi\pi$ decay, improving our 2015 calculation of these quantities. Both calculations were performed with physical kinematics on a $32^3\times 64$ lattice with an inverse lattice spacing of $a^{-1}=1.3784(68)$ GeV. However, the current calculation includes nearly four times the statistics and numerous technical improvements allowing us to more reliably isolate the $\pi\pi$ ground-state and more accurately relate the lattice operators to those defined in the Standard Model. We find ${\rm Re}(A_0)=2.99(0.32)(0.59)\times 10^{-7}$ GeV and ${\rm Im}(A_0)=-6.98(0.62)(1.44)\times 10^{-11}$ GeV, where the errors are statistical and systematic, respectively. The former agrees well with the experimental result ${\rm Re}(A_0)=3.3201(18)\times 10^{-7}$ GeV. These results for $A_0$ can be combined with our earlier lattice calculation of $A_2$ to obtain ${\rm Re}(\varepsilon'/\varepsilon)=21.7(2.6)(6.2)(5.0) \times 10^{-4}$, where the third error represents omitted isospin breaking effects, and Re$(A_0)$/Re$(A_2) = 19.9(2.3)(4.4)$. The first agrees well with the experimental result of ${\rm Re}(\varepsilon'/\varepsilon)=16.6(2.3)\times 10^{-4}$. A comparison of the second with the observed ratio Re$(A_0)/$Re$(A_2) = 22.45(6)$, demonstrates the Standard Model origin of this "$\Delta I = 1/2$ rule" enhancement