O(a^2) corrections to the one-loop propagator and bilinears of clover fermions with Symanzik improved gluons
M. Constantinou, V. Lubicz, H. Panagopoulos, F. Stylianou
备注：
37 pages, 3 figures, 27 tables. Results are also presented online, in a Mathematica input file. Version 2 (to appear in JHEP): Added a comment regarding Fig.3, corrected typo on Eq.A3
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 $O(a^1)$ improvement calculations; there, IR divergent terms are seen to have the same structure as in the $O(a^0)$ case, by virtue of parity under integration, and they can thus be handled by well-known techniques. We explain how to correctly extract the full $O(a^2)$ dependence; in fact, our method is generalizable to any order in $a$.
The $O(a^2)$ corrections to the quark propagator and Green's functions computed in this paper are useful to improve the nonperturbative RI-MOM determination of renormalization constants for quark bilinear operators.
Our results depend on a large number of parameters: coupling constant, number of colors, lattice spacing, external momentum, clover parameter, Symanzik coefficients, gauge parameter. To make these results most easily accessible to the reader, we have included them in the distribution package of this paper, as an ASCII file named: Oa2results.m ; the file is best perused as Mathematica input.
Perturbative determination of $c_{SW}$ for plaquette and Symanzik gauge action and stout link clover fermions
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。
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 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…
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 present the current status of our computation of quark bilinear renormalization constants for Wilson fermions and Symanzik improved gauge action. Computations are performed in Numerical Stochastic Perturbation Theory. Volumes range from 10^4 to 32^4. Renormalization conditions are those of the RI'-MOM scheme, imposed at different values of the physical scale. Having measurements available at several momenta, irrelevant effects are taken into account by means of hypercubic symmetric Taylor expansions. Finite volumes effects are assessed repeating the computations at different lattice sizes. …
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 present results of a lattice QCD application of a coordinate space renormalization scheme for the extraction of renormalization constants for flavour non-singlet bilinear quark operators. The method consists in the analysis of the small-distance behaviour of correlation functions in Euclidean space and has several theoretical and practical advantages, in particular: it is gauge invariant, easy to implement and has relatively low computational cost. The values of renormalization constants in the X-space scheme can be converted to the MSbar scheme via 4-loop continuum perturbative formulae. O…
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…
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We present results for matching factors for bilinear operators composed of HYP-smeared staggered fermions and constructed using HYP-smeared fat links. The matching factors are calculated perturbatively at one-loop order. The new feature of our calculation compared to previous work on HYP-smeared staggered fermions is the use of the Symanzik-improved gluon propagator, which allows our results to be applied to our ongoing simulations based on configurations generated by the MILC collaboration. We address the issue of the relative efficiency of various improvement schemes in reducing one-loop cor…
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.
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…
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…
We compute the renormalization functions on the lattice, in the RI' scheme, of local bilinear quark operators $\barψΓψ$, where $Γ= 1, γ_5, γ_μ, γ_5γ_μ, γ_5σ_{μν}$. This calculation is carried out to two loops for the first time. We consider both the flavor non-singlet and singlet operators. As a prerequisite for the above, we compute the quark field renormalization, $Z_ψ$, up to two loops. We also compute the 1-loop renormalization functions for the gluon field, $Z_A$, ghost field, $Z_c$, gauge parameter, $Z_α$, and coupling constant $Z_g$. We use the clover action for fermions and the Wilson …
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…
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…
We compute the two-loop renormalization functions, in the RI' scheme, of local bilinear quark operators $\barψΓψ$, where $Γ$ corresponds to the Vector, Axial-Vector and Tensor Dirac operators, in the lattice formulation of QCD. We consider both the flavor nonsinglet and singlet operators. We use the clover action for fermions and the Wilson action for gluons. Our results are given as a polynomial in $c_{SW}$, in terms of both the renormalized and bare coupling constant, in the renormalized Feynman gauge. Finally, we present our results in the MSbar scheme, for easier comparison with calculatio…
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We non-perturbatively determine the renormalization factor of the axial vector current in lattice QCD with $N_f=3$ flavors of Wilson-clover fermions and the tree-level Symanzik-improved gauge action. The (by now standard) renormalization condition is derived from the massive axial Ward identity and it is imposed among Schrödinger functional states with large overlap on the lowest lying hadronic state in the pseudoscalar channel, in order to reduce kinematically enhanced cutoff effects. We explore a range of couplings relevant for simulations at lattice spacings of $\approx 0.09$ fm and below. …
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In this paper we present one-loop results for the renormalization of nonlocal quark bilinear operators, containing a staple-shaped Wilson line, in both continuum and lattice regularizations. The continuum calculations were performed in dimensional regularization, and the lattice calculations for the Wilson/clover fermion action and for a variety of Symanzik-improved gauge actions. We extract the strength of the one-loop linear and logarithmic divergences (including cusp divergences), which appear in such nonlocal operators; we identify the mixing pairs which occur among some of these operators…
This is the third of a series of papers on three-loop computation of renormalization constants for Lattice QCD. Our main point of interest are results for the regularization defined by Iwasaki gauge action and n_f=4 Wilson fermions. Our results for quark bilinears renormalized according to the RI'-MOM scheme can be compared to non-perturbative results. The latter are available for Twisted Mass QCD: being defined in the chiral limit, renormalization constants must be the same. We also address more general problems. In particular, we discuss a few methodological issues connected to summing the p…
We present results on the renormalization functions of the quark field and fermion bilinears with up to one covariant derivative. For the fermion part of the action we employ the twisted mass formulation with $N_f{=}2$ and $N_f{=}4$ degenerate dynamical quarks, while in the gluon sector we use the Iwasaki improved action. The simulations for $N_f{=}4$ have been performed for pion masses in the range of 390MeV - 760MeV and at three values of the lattice spacing, $a$, corresponding to $β{=}1.90,\,1.95,\,2.10$. The $N_f{=}2$ action includes a clover term with $c_{\rm sw}{=}1.57551$ at $β{=}2.10$,…
