利用格元和Symanzik改进的规范作用量以及Stout Link Clover费米子，从离壳夸克-夸克-胶子三点函数出发，确定了单圈格点微扰理论中的改进系数$c_{SW}$.此外，我们计算了夸克场改进的最一般形式所需的系数，并给出了临界跳跃参数$κ_c$的单圈结果。我们讨论了平均场的改进$c_{SW}$和$κ_c$和选择的平均场耦合的行动，我们已经考虑。
英文摘要：
Using plaquette and Symanzik improved gauge action and stout link clover fermions we determine the improvement coefficient $c_{SW}$ in one-loop lattice perturbation theory from the off-shell quark-quark-gluon three-point function. In addition, we compute the coefficients needed for the most general form of quark field improvement and present the one-loop result for the critical hopping parameter $κ_c$. We discuss mean field improvement for $c_{SW}$ and $κ_c$ and the choice of the mean field coupling for the actions we have considered.
We discuss a 3-flavour lattice QCD action with clover improvement in which the fermion matrix has single level stout smearing for the hopping terms together with unsmeared links for the clover term. With the (tree-level) Symanzik improved gluon action this constitutes the Stout Link Non-perturbative Clover or SLiNC action. To cancel O(a) terms the clover term coefficient has to be tuned. We present here results of a non-perturbative determination of this coefficient using the Schroedinger functional and as a by-product a determination of the critical hopping parameter. Comparisons of the resul…
We report on the results of a numerical simulation concerning the low-lying spectrum of four-dimensional N=1 SU(2) Supersymmetric Yang-Mills (SYM) theory on the lattice with light dynamical gluinos. In the gauge sector the tree-level Symanzik improved gauge action is used, while we use the Wilson formulation in the fermion sector with stout smearing of the gauge links in the Wilson-Dirac operator. The ensembles of gauge configurations were produced with the Two-Step Polynomial Hybrid Monte Carlo (TS-PHMC) updating algorithm. We performed simulations on large lattices up to a size of 24^3 x 48 …
In this paper we present the perturbative computation of the renormalization functions for the quark field and for a complete set of ultra-local fermion bilinears. The computation of the relevant Green's functions was carried out at 1-loop level for the staggered action using massive fermions. The gluon links which appear both in the fermion action and in the definition of the bilinears have been improved by applying a stout smearing procedure up to 2 times, iteratively. In the gluon sector we employed the Symanzik improved gauge action for different sets of values of the Symanzik coefficients…
We calculate corrections to the fermion propagator and to the Green's functions of all fermion bilinear operators of the form $\barΨΓΨ$, to one-loop in perturbation theory. We employ the Wilson/clover action for fermions and a family of Symanzik improved actions for gluons. The novel aspect of our calculations is that they are carried out to second order in the lattice spacing, $O(a^2)$. Consequently, they have addressed a number of new issues, most notably the appearance of loop integrands with strong IR divergences (convergent only beyond 6 dimensions). Such integrands are not present in $O(…
We performed a rough estimate of the non-perturbative value of the clover term coefficient cSW for the APE stout link Wilson fermion. We varied the number of smearings from Nsmear=1 to 6 and adopted beta values roughly corresponding to the lattice spacing of 0.1 fm. We used the Schroedinger functional technique for an evaluation of cSW and found that cSW decreases monotonically as we increase Nsmear but has a 10% order of deviation from the tree level value for Nsmear=6.
Lambda-parameter of lattice QCD with Symanzik improved gluon actions
我们计算了Lambda_L/Lambda_MS的比值，其中尺度参数Lambda_L与QCD的晶格公式相关联。我们考虑了一个3参数的胶子作用量族，这是最常用的O（a）的改进` la Symanzik。将胶子作用量与费米子的标准离散化（Wilson/Clover，overlap）结合起来，给出了几种可能的费米子和胶子作用量组合的Lambda_L。我们采用背景场技术来计算背景场的1 PI两点函数，从而得到单圈水平的耦合常数重整化函数Z_g。我们的研究结果得到了广泛的范围内的Symanzik系数的值。
We discuss the problem of lattice artefacts in QCD simulations enhanced by the introduction of dynamical charmed quarks. In particular, we advocate the use of a massive renormalization scheme with a close to realistic charm mass. To maintain O(a) improvement for Wilson type fermions in this case we define a finite size scheme and carry out a nonperturbative estimation of the clover coefficient $c_\mathrm{sw}$. It is summarized in a fit formula $c_\mathrm{sw}(g_0^2)$ that defines an improved action suitable for future dynamical charm simulations.
We calculate the fermion propagator and the quark-antiquark Green's functions for a complete set of ultralocal fermion bilinears, ${{\cal O}_Γ}$ [$Γ$: scalar (S), pseudoscalar (P), vector (V), axial (A) and tensor (T)], using perturbation theory up to one-loop and to lowest order in the lattice spacing. We employ the staggered action for fermions and the Symanzik Improved action for gluons. From our calculations we determine the renormalization functions for the quark field and for all ultralocal taste-singlet bilinear operators. The novel aspect of our calculations is that the gluon links whi…
In this paper we present the perturbative evaluation of the difference between the renormalization functions of flavor singlet and nonsinglet bilinear quark operators on the lattice. The computation is performed to two loops and to lowest order in the lattice spacing, for a class of improved lattice actions, including Wilson, tree-level (TL) Symanzik and Iwasaki gluons, twisted mass and SLiNC Wilson fermions, as well as staggered fermions with twice stout-smeared links. In the staggered formalism, the stout smearing procedure is also applied to the definition of bilinear operators.
Additive and multiplicative renormalization of topological charge with improved gluon/fermion actions: A test case for 3-loop vacuum calculations, using overlap or clover fermions
We present a 1-loop perturbative calculation of the fermion propagator, up to O(a^2) (a: lattice spacing). The fermions are described by Wilson, clover and twisted-mass actions; for gluons we use Symanzik improved actions (Plaquette, Tree-level Symanzik, Iwasaki, TILW, DBW2). Our results are given in a general covariant gauge, and their dependence on the coupling constant, the external momentum, the masses and the clover parameter is shown explicitly. We also study the O(a^2) corrections to matrix elements of unpolarized/polarized fermion bilinear operators, which include up to one derivative.…
We conclude our investigation on the QCD equation of state (EoS) with 2+1 staggered flavors and one-link stout improvement. We extend our previous study [JHEP 0601:089 (2006)] by choosing even finer lattices. These new results [for details see arXiv:1007.2580] support our earlier findings. Lattices with N_t=6,8 and 10 are used, and the continuum limit is approached by checking the results at N_t=12. A Symanzik improved gauge and a stout-link improved staggered fermion action is taken; the light and strange quark masses are set to their physical values. Various observables are calculated in the…
We calculate the step scaling function, the lattice analog of the renormalization group $β$-function, for an SU(3) gauge theory with twelve flavors. The gauge coupling of this system runs very slowly, which is reflected in a small step scaling function, making numerical simulations particularly challenging. We present a detailed analysis including the study of systematic effects of our extensive data set generated with twelve dynamical flavors using the Symanzik gauge action and three times stout smeared Möbius domain wall fermions. Using up to $32^4$ volumes, we calculate renormalized couplin…
We present high-statistics estimates of the isovector charges of the nucleon from four 2+1-flavor ensembles generated using Wilson-clover fermions with stout smearing and tree-level tadpole improved Symanzik gauge action at lattice spacings $a=0.114$ and $0.080$ fm and with $M_π\approx 315$ and 200 MeV. The truncated solver method with bias correction and the coherent source sequential propagator construction are used to cost-effectively achieve $O(10^5)$ measurements on each ensemble. Using these data, the analysis of two-point correlation functions is extended to include four states in the f…
We compute the difference in the renormalization of flavor singlet and nonsinglet fermion bilinear operators, to two loops in perturbation theory. Our results are applicable to a rather wide class of lattice actions with Symanzik improved gluons, stout links and clover fermions, including the Twisted Mass and SLiNC actions. A more detailed presentation of our results, along with relevant references, will appear in a forthcoming publication.
We calculate the corrections to the amputated Green's functions of 4-fermion operators, in 1-loop Lattice Perturbation theory. The novel aspect of our calculations is that they are carried out to second order in the lattice spacing, O(a^2). We employ the Wilson/clover action for massless fermions (also applicable for the twisted mass action in the chiral limit) and the Symanzik improved action for gluons. Our calculations have been carried out in a general covariant gauge. Results have been obtained for several popular choices of values for the Symanzik coefficients (Plaquette, Tree-level Syma…
We study the behavior of the confining flux tube in $N_f=2+1$ QCD at the physical point, discretized with the stout smearing improved staggered quark action and the tree level Symanzik gauge action. We discuss how it depends on a uniform external magnetic field, showing how it displays anisotropies with respect to the magnetic field direction. Moreover, we compare the observed anisotropy pattern with that of the static quark-antiquark potential we obtained in our previous works.
In this paper we present the perturbative computation of the difference between the renormalization factors of flavor singlet ($\sum_f\barψ_fΓψ_f$, $f$: flavor index) and nonsinglet ($\barψ_{f_1} Γψ_{f_2}, f_1 \neq f_2$) bilinear quark operators (where $Γ= \mathbb{1},\,γ_5,\,γ_μ,\,γ_5\,γ_μ,\, γ_5\,σ_{μ\,ν}$) on the lattice. The computation is performed to two loops and to lowest order in the lattice spacing, using Symanzik improved gluons and staggered fermions with twice stout-smeared links. The stout smearing procedure is also applied to the definition of bilinear operators. A significant pa…
Perturbative and non-perturbative results are presented on the renormalization constants of the quark field and the vector, axial-vector, pseudoscalar, scalar and tensor currents. The perturbative computation, carried out at one-loop level and up to second order in the lattice spacing, is performed for a fermion action, which includes the clover term and the twisted mass parameter yielding results that are applicable for unimproved Wilson fermions, as well as for improved clover and twisted mass fermions. We consider ten variants of the Symanzik improved gauge action corresponding to ten diffe…
We calculate the renormalized step scaling function for twelve fundamental flavors nonperturbatively by determining the gradient flow coupling on gauge field configurations generated with dynamical stout-smeared Möbius domain wall fermions and Symanzik gauge action. Using Zeuthen, Symanzik, and Wilson flow we measure the energy density with three different operators. Our updated analysis is based on up to five volume pairs ranging from L^4=8^4 up to 32^4. Predictions for the infinite volume extrapolated step scaling function based on different flows and operators are mutually consistent. Our n…
In this work we calculate the corrections to the amputated Green's functions of 4-fermion operators, in 1-loop Lattice Perturbation theory. One of the novel aspects of our calculations is that they are carried out to O(a^2) (a: lattice spacing). We employ the Wilson/clover action for massless fermions (also applicable for the twisted mass action in the chiral limit) and a family of Symanzik improved actions for gluons. Our calculations have been carried out in a general covariant gauge. Results have been obtained for several popular choices of values for the Symanzik coefficients. While our Gr…
It is expected that SU(2) gauge theory with $N_f$ fundamental fermions has an infrared fixed point when $N_f$ is between $\sim 6$ and 10. We study the hadron spectrum and scale setting in SU(2) gauge field theory with $N_f=2,4,6$ using hypercubic stout smeared Wilson-clover (HEX) action. The case $N_f=2$ is QCD-like, whereas $N_f=6$ is close to the lower edge of the conformal window. In our study the length scales are determined by using the gradient flow approach.
We present results for the renormalization of gauge invariant nonlocal fermion operators which contain a Wilson line, to one loop level in lattice perturbation theory. Our calculations have been performed for Wilson/clover fermions and a wide class of Symanzik improved gluon actions. The extended nature of such `long-link' operators results in a nontrivial renormalization, including contributions which diverge linearly as well as logarithmically with the lattice spacing, along with additional finite factors. We present nonperturbative prescriptions to extract the linearly divergent contributio…
QCD is investigated at finite temperature using Wilson fermions in the fixed scale approach. A 2+1 flavor stout and clover improved action is used at four lattice spacings allowing for control over discretization errors. The light quark masses in this first study are fixed to heavier than physical values. The renormalized chiral condensate, quark number susceptibility and the Polyakov loop is measured and the results are compared with the staggered formulation in the fixed N_t approach. The Wilson results at the finest lattice spacing agree with the staggered results at the highest N_t.
We study the correlators of Polyakov loops, and the corresponding gauge invariant free energy of a static quark-antiquark pair in 2+1 flavor QCD at finite temperature. Our simulations were carried out on $N_t$ = 6, 8, 10, 12, 16 lattices using Symanzik improved gauge action and a stout improved staggered action with physical quark masses. The free energies calculated from the Polyakov loop correlators are extrapolated to the continuum limit. For the free energies we use a two step renormalization procedure that only uses data at finite temperature. We also measure correlators with definite Euc…
The pressure and energy density of the quark-gluon plasma at finite baryon chemical potential are calculated using the Complex Langevin equation. The stout smearing procedure is generalized for the SL(3,$\mathcal{C}$) manifold allowing the usage of an improved action in the Complex Langevin setup. Four degenerate flavors of staggered quarks with $m_π=500-700$ MeV are used with a tree-level Symanzik improved gauge action on $16^3 \times 8 $ lattices. Results are compared to the Taylor expansion and good agreement is found for small chemical potentials.
We study the density of states method as well as reweighting to explore the low temperature phase diagram of QCD at finite baryon chemical potential. We use four flavors of staggered quarks, a tree-level Symanzik improved gauge action and four stout smearing steps on lattices with $N_s=4,6,8$ and $N_t=6 - 16$. We compare our results to that of the phase quenched ensemble and also determine the pion and nucleon masses. In the density of states approach we applied pion condensate or gauge action density fixing. We found that the density of states method performs similarly to reweighting. At $T \…
The present paper concludes our investigation on the QCD equation of state with 2+1 staggered flavors and one-link stout improvement. We extend our previous study [JHEP 0601:089 (2006)] by choosing even finer lattices. Lattices with $N_t=6,8$ and 10 are used, and the continuum limit is approached by checking the results at $N_t=12$. A Symanzik improved gauge and a stout-link improved staggered fermion action is utilized. We use physical quark masses, that is, for the lightest staggered pions and kaons we fix the $m_π/f_K$ and $m_K/f_K$ ratios to their experimental values. The pressure, the int…
We report on advances in the non-perturbative determination of the ratio $Z_S/Z_P$ of the pseudoscalar to the scalar renormalization constants in three-flavour lattice QCD with Wilson-clover quarks and tree-level Symanzik improved gluons. The computations are based on the Ward identity approach, using Schrödinger functional boundary conditions. Our results for $Z_S/Z_P$ cover a range of couplings along a line of constant physics with lattice spacings of about 0.09 fm and below, relevant for phenomenological applications such as the calculation of renormalized quark masses.
We study SU(2) gauge theory with two Dirac fermions in the adjoint representation of the gauge group on the lattice. Using clover improved Wilson fermion action with hypercubic truncated stout smearing we perform simulations at larger coupling than earlier. We measure the evolution of the coupling constant using the step scaling method with Schrödinger functional and study the remaining discretization effects. At weak coupling we observe significant discretization effects, which make it difficult to obtain a fully controlled continuum limit. Nevertheless, the data remains consistent with the e…
We present our progress in the non-perturbative O(a) improvement and renormalization of tensor currents in three-flavor lattice QCD with Wilson-clover fermions and tree-level Symanzik improved gauge action. The mass-independent O(a) improvement factor of tensor currents is determined via a Ward identity approach, and their renormalization group running is calculated via recursive finite-size scaling techniques, both implemented within the Schrödinger functional framework. We also address the matching factor between bare and renormalization group invariant currents for a range of lattice spacin…
We present a lattice QCD determination of light quark masses with three sea-quark flavours ($N_f = 2+1$). Bare quark masses are known from PCAC relations in the framework of CLS lattice computations with a non-perturbatively improved Wilson-Clover action and a tree-level Symanzik improved gauge action. They are fully non-perturbatively improved, including the recently computed Symanzik counter-term $b_{\rm A} - b_{\rm P}$. The mass renormalisation at hadronic scales and the renormalisation group running over a wide range of scales are known non-perturbatively in the Schrödinger functional sche…
We study the correlators of Polyakov loops, and the corresponding gauge invariant free energy of a static quark-antiquark pair in 2+1 flavor QCD at finite temperature. Our simulations were carried out on $N_t$ = 6, 8, 10, 12, 16 lattices using a Symanzik improved gauge action and a stout improved staggered action with physical quark masses. The free energies calculated from the Polyakov loop correlators are extrapolated to the continuum limit. For the free energies we use a two step renormalization procedure that only uses data at finite temperature. We also measure correlators with definite E…
