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Just how different are SU(2) and SU(3) Landau-gauge propagators in the IR regime

Just how different are SU(2) and SU(3) Landau-gauge propagators in the IR regime
Just how different are SU(2) and SU(3) Landau-gauge propagators in the IR regime

a r X i v :0705.3367v 2 [h e p -l a t ] 4 N o v 2007

Just how di?erent are SU (2)and SU (3)Landau-gauge propagators in the IR regime?

A.Cucchieri and T.Mendes

Instituto de F′?sica de S?a o Carlos,Universidade de S?a o Paulo,

Caixa Postal 369,13560-970S?a o Carlos,SP,Brazil

O.Oliveira

Department of Physics,University of Coimbra,3004516Coimbra,Portugal and Instituto de F′?sica de S?a o Carlos,Universidade de S?a o Paulo,

Caixa Postal 369,13560-970S?a o Carlos,SP,Brazil

P.J.Silva

Department of Physics,University of Coimbra,3004516Coimbra,Portugal

(Dated:February 1,2008)

The infrared behavior of gluon and ghost propagators in Yang-Mills theories is of central impor-tance for understanding quark and gluon con?nement in QCD.While simulations of pure SU (3)gauge theory correspond to the physical case in the limit of in?nite quark mass,the SU (2)case (i.e.pure two-color QCD)is usually employed as a simpli?cation,in the hope that qualitative features be the same as for the SU (3)case.Here we carry out the ?rst comparative study of lattice (Landau)propagators for these two gauge groups.Our data were especially produced with equivalent lattice parameters in order to allow a careful comparison of the two cases.We ?nd very good agreement between SU (2)and SU (3)propagators,showing that in the IR limit the equivalence of the two cases is quantitative,at least down to about 1GeV.Our results suggest that the infrared behavior of these propagators is independent of the gauge group SU (N c ),as predicted by Schwinger-Dyson equations.

PACS numbers:11.15.Ha 12.38.Aw 14.70Dj

I.INTRODUCTION AND MOTIV ATION

Despite recent progress,the infrared structure of Yang-Mills theory is still not fully understood.For QCD,the study of the infrared limit is of central importance for the comprehension of the mechanisms of quark and gluon con?nement and of chiral-symmetry breaking.In what concerns con?nement,in Landau gauge,the infrared be-havior of gluon and ghost propagators is linked with the Gribov-Zwanziger [1,2]and the Kugo-Ojima [3]con?ne-ment scenarios.These con?nement mechanisms predict,at small momenta,an enhanced ghost propagator and a suppression of the gluon propagator.The strong in-frared divergence for the ghost propagator corresponds to a long-range interaction in real space,which may be related to quark con?nement.The suppression of the gluon propagator,which should vanish at zero momen-tum,implies (maximal)violation of re?ection positivity and may be viewed as an indication of gluon con?nement.Moreover,the interest in the propagators goes beyond the con?nement mechanism,as they are inputs for many phenomenological calculations in hadronic physics (see,for example,Refs.[4,5]).

Analytic studies of gluon and ghost propagators using Schwinger-Dyson equations (SDE)[6,7,8]seem to agree with the above scenarios.(The reader should however be aware that,in the literature,there are solutions of the SDE [9,10]that do not comply with the Gribov-Zwanziger or the Kugo-Ojima predictions at small mo-menta.)Moreover,when dynamic quarks are neglected

and assuming that g 2~1/N c —as suggested by analysis of the large N c limit [11]—the SDE become indepen-dent of the number of color N c .Thus,they predict that gluon and ghost propagators be independent of N c .The Landau gauge gluon propagator D (k 2)has been investigated with lattice techniques in quenched QCD [i.e.pure SU (3)Yang-Mills theory][12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],in pure SU (2)Yang-Mills theory (in 2,3and 4space-time dimensions)[31,32,33,34,35,36,37,38,39,40,41]and in full QCD [42,43,44,45].All lattice studies in 4d suggest a ?nite nonzero infrared gluon propaga-tor [20,24,26,27,42],in contradiction with the infrared Schwinger-Dyson solution.On the other hand,?nite-size e?ects are very large and not yet well-controlled,even in the 3d case [34].Only in two space-time di-mensions [41],using a lattice side L up to about 40fm,does one ?nd that D (0)extrapolates to zero as L goes to in?nity.Let us note that investigation of SDE on a 4-torus [46]suggests that the gluon propagator indeed approaches the in?nite-volume limit very slowly,espe-cially for its low-momentum components.On the other hand,even with an infrared-?nite propagator,one clearly ?nds [27,28,33,36,45]that re?ection-positivity is vio-lated when su?ciently large lattice volumes are consid-ered.Finally,in the 2d SU (2)case [41]and in the 4d SU (3)case (using asymmetric lattices)[26,30]it was found that the gluon propagator complies with the pure power-law behavior predicted analytically [6,8].The lattice-Landau-gauge SU (2)and SU (3)ghost

2

TABLE I:Lattice setup.The lattice spacing was computed

from the string tension,assuming

1640.1021.6322.44696.0

2440.0731.7522.55016.2

3240.0541.7282.64086.4

3240.1023.2642.44696.0

propagator G(k2)has been studied in[22,25,28,29,

30,31,38,39,40,41,43,47,48,49,50,51,52,53]and

in all cases an enhancement of the propagator compared

to the tree-level behavior1/k2was observed.Concern-

ing the comparison between lattice results and the SDE

solution,the two propagators seem to agree only qual-

itatively.In particular,in three and in four space-time

dimensions,the infrared exponent obtained using lattice

simulations is always smaller than the one predicted an-

alytically.On the other hand,on the2d SU(2)case[41],

the ghost propagator shows an infrared behavior1/k2.4,

in agreement with the SDE solution[8].

In summary,for the Landau gauge,the SDE gluon and

ghost propagators agree,at least qualitatively,with the

lattice propagators.However,while analytic studies us-

ing Schwinger-Dyson equations predict the same infrared

behavior for the SU(2)and SU(3)gauge groups,lattice

simulations usually assume that the two cases are dif-

ferent,although their qualitative infrared features may

be the same.In this paper,we carry out a comparative

study of lattice Landau gauge propagators for these two

gauge groups.Our data were especially produced by con-

sidering equivalent lattice parameters in order to allow a

careful comparison of the two cases.We note that we

do not assume a power-law behavior for the propagators,

but just compare the raw data in the two cases.

II.NUMERICAL SIMULATIONS

We consider four di?erent sets of lattice parameters,

with the same lattice size N4and the same physical lat-

tice spacing a for the two gauge groups(see Table I).

The?rst three cases are chosen to yield approximately

the same physical lattice volume V=(Na)4≈(1.7fm)4.

This allows a comparison of discretization e?ects.The

fourth case corresponds to a signi?cantly larger physical

volume,i.e.V≈(3.2fm)4,in order to study?nite-size

e?ects.For all four cases,50con?gurations were gen-

erated[61]using the Wilson action.The gluon and the

ghost propagators

D abμν(k2)=δab δμν?kμkν

squared magnitude k2of the four-momentum k,for on-axis momenta(k,0,0,0).The data are organized as in Fig.1. infrared behavior of the propagators[21,31],with our set of lattice volumes and for the statistics considered here these e?ects should always be smaller than the statistical error.

The propagators were computed in the minimal Lan-dau gauge,obtained by minimizing the functional

S[?]=? x,μTr U?μ(x),(3)

where U?μ(x)=?(x)Uμ(x)??(x+?eμ)is the gauge-transformed link and?eμis the unit vector along theμdirection.For SU(2)the gauge?xing was performed using a stochastic-overrelaxation algorithm(see[35]for details),while for SU(3)a Fourier-accelerated steepest-descent algorithm was used(see[26]for details).

In what concerns the evaluation of the ghost prop-agator,in the SU(2)case the Faddeev-Popov matrix was inverted using the method described in[31],while the SU(3)simulation relies on the method discussed in Ref.[47](considering more than one source).In the calculation of the gluon and of the ghost propagators,the four lattice setups considered.

the statistical errors were computed with the(single-elimination)jackknife method in the SU(3)case and with the bootstrap method(using1000bootstrap samples)in the SU(2)case.We checked that these errors are in agreement with those obtained considering one standard deviation.

In order to compare the propagators from the di?er-ent simulations,the gluon and ghost propagators were renormalized accordingly to

D(k2) k2=μ2=1μ2,(4)

usingμ=3GeV as a renormalization point.The lattice data were interpolated(using splines)to allow the use of such a renormalization point in all the simulations.We have checked that the interpolation reproduces perfectly the lattice data.Let us note that,due to breaking of rotational invariance,the renormalization factors Z(μ2) depend,in general,slightly on the type of momenta.Here we use,for all momenta k,the factor Z(μ2)obtained from the on-axis momenta(k,0,0,0).

4

the four lattice setups considered.

III.RESULTS

The renormalized SU(2)and SU(3)propagators can be seen for the various lattice setups in Fig.1(gluon) and in Fig.2(ghost)for the on-axis momenta(k,0,0,0). (Results are similar when considering the other types of momenta.)In all?gures we report,on the horizontal axis,the squared magnitude k2(in GeV2)of the four-momentum k.These?gures show that,for the set of momenta accessible in our simulations,?nite-volume and ?nite-spacing e?ects are under control.Moreover,they show that the SU(2)and SU(3)propagators are essen-tially equal,with slight di?erences in the low-momenta region.Similar results have been recently presented at Lattice2007by Anthony G.Williams[56].In Figs.3 (gluon)and4(ghost)we show the ratios of SU(3)over

(2)propagators.The statistical errors were com-

assuming Gaussian-error propagation.Note that the case of the gluon propagator there are momenta which the discrepancy from1for the ratio is about or larger.However,these deviations are not sys-and are probably due to a combination of sev-e?ects.These may include breaking of rotational small statistics and?nite-size e?ects,such as related to the global Z(N c)symmetry of the lattice

[57,58,59,60].

IV.CONCLUSIONS

In summary,considering a careful choice of the lattice we were able to carry out an unambiguous

of the lattice Landau gluon and ghost prop-for SU(2)and SU(3)gauge theories.The data

that the two cases have very similar?nite-size and

e?ects.Moreover,we?nd very good agree-between the two Yang-Mills theories(for our values momenta larger than1GeV),for all lattice parame-and for all types of momenta.Below1GeV,the for the two gauge groups show some di?erences, for the gluon propagator.Note,however,that ratios are compatible with1within two standard de-In this sense,our results suggest that the propagators the same for all SU(N c)groups in the nonpertur-region,as predicted by Schwinger-Dyson equa-

Of course,given the lattice volumes considered,

studies are required before drawing?nal conclu-

about the comparison below1GeV.In particular, it will be interesting to investigate if this agreement per-sists also in the deep-infrared region,where the gluon propagator may show a turnover and a suppression,as predicted in the Gribov-Zwanziger scenario.

Acknowledgments

The authors thank R.Alkofer,A.Maas and C.Fischer for discussions.O.O.and P.J.S.acknowledge FCT for?-nancial support under contract POCI/FP/63923/2005. P.J.S.acknowledges?nancial support from FCT via grant SFRH/BD/10740/2002.O.O.was also supported by FAPESP(grant#06/61514-8)during his stay at IFSC-USP.A.C.and T.M.were supported by FAPESP and by CNPq.Parts of our simulations have been done on the IBM supercomputer at S?a o Paulo Univer-sity(FAPESP grant#04/08928-3)and on the super-computer Milipeia at Coimbra University.

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《SU秒图系列教程①——黄昏篇》by 余德杰

出品人:余德杰

教程教学内容:讲述从sketchup中导出图片快速PS后期的方法,解决电脑配置低、时间紧迫、模型建得太烂渲染不了渲染太慢等问题。 前言: 这次写su导图加ps,完全有违以往的超写实风格,是由于赶图赶得要死然后还因为之前欠下的人情所以要帮同学出图十万火急迫不得已只有用这种NPR【non photorealistic render】【非写实】风格出图。 这种做法,我个人是不推荐的,但是可以用来救急。何出此言?因为学习阶段的渲染,不只是为了交个图,更多的是通过渲染来理解自己的设计,非写实的风格,看上去可能酷炫叼,更有艺术价值,感觉更加暧昧,但却是脱离现实的结果,建筑不是纯艺术,除非毕生只做止于图纸的方案。虽然行业甚至是学术氛围都不推荐写实的渲染,但是我推荐有时间的有心人不妨去了解一下,毕竟能把方案做得最贴近现实而且最省钱也就只有CG了。 这个方法,“一秒完事”太夸张,事实上是直接从su导出二维图形然后叠加+调色,好处在于,导图耗费的时间极其短,而且配置低的电脑也可以办到,渲染的话要用cpu或者gpu,如果电脑先天性挫逼,那就没有办法了,所以如果想高效出图就只有自己动手。如果ps,内存不能太小,起码6个G平时才能用的舒服,但是内存条便宜,起码比换个处理器要便宜得多。

教程正文/注:一下部分黑字为理解讲解部分,红字为步骤讲解部分,不想看理解部分的童鞋可以直接跳到步骤灌输。/ 我们先看看sketchup自带的显示效果如下

颜色单调、很多地方都是单色的色块,打开阴影之后,阴影僵硬无比。最重要的是,这些东西都被整合在一个画面里,无法进行有效的自定义调整,说白了就是可后期性差。 如果要获得理想的效果,就必须将元素拆分开来,进行单独的处理。这也是现在游戏引擎的惯用手法,通过美工处理之后的画面,效果会更上一层楼,例如lumion就是典型的例子之一,但是lumion的引擎过于老旧,感觉发展潜力不大,如果突破DX9这一步,将会是飞升。 一般来说,先把所有元素合在一起,然后在分别调整,最终整体调整,获得最终结果。先看看下面的一组图,裸奔的步骤基本如下

su秒图系列教程——反射篇

上一节给大家讲了光照,让大家对光照的后期有了更深的认识。其实这种方法不一定只用于小型建筑的表现,由小到大都是可以做的,毕竟同一个画面大小,信息量是有限的,并不会因为“主体变复杂”而变得极其麻烦,因为对象变复杂的同时,细节也在相对缩小,所以总工作量并没有想象中的大,甚至有些时候“复杂”反而快些。嘛,来两个极致的例子就知道:①让你裸奔一堵干净的墙②让你裸奔一个城市鸟瞰。当然水平必须相当,不然没有讨论的意义。 这一节,便是各位已经就等的《反射篇》了,在这会给大家讲讲如何在Photoshop后期制作玻璃之类带有反射和折射的物体。 也只有该篇会以pdf的形式提供给大家,完整的版本是《su秒图进阶系列——细说全流程篇》,有兴趣的同学请访问 http请删除中文://www请删除中文https://www.sodocs.net/doc/2a17314734.html,/forum.php?mod=viewthread&tid=22964&extra=&page=1【请各位见谅,不这么干不方便上传,各大网站可能不批准上传】

反射的规律 接着讲案例之前再提提反射的简单规律,虽然第一节已经解释过。镜面的反射是如何大家都懂的,镜子中便是镜外空间的镜像,所以透视关系不难确定,看下图,这是我用vrayformax渲染的一个小场景。尤其是这种明显暗示透视关系的地砖,在ps的时候便可充分利用现有画面进行复制、变形。

但是像上图这种极其清晰的反射效果,若是出现在极其简单的场景里头,那反射部分我还是推荐大家直接依赖渲染器吧,毕竟p图的工作量已经超越使用渲染器的工作量了。 偷工减料 很多时候场景中并没有大片连续的反射区域【除了水面的反射,水面的教程都烂大街了】,所以反射精确不精确,我们也很难用肉眼去鉴定,或者说,谁会纠结某个点的反射正确与否?只要大体看上去没有问题即可。如果过于纠结细节,就违背秒图的初衷。场景越是复杂,人就越难判断其中的内容正确精准与否。 所以越是复杂,反射反而越是好做。 就本次使用案例而言,我在建模的时候有意制造相对复杂的条件,使得讲解更有代表性。而一般来说都会比这个小场景要复杂,所以反射的准确度就可以更加低了。 具体如何控制所谓的准确度,大家看看接下来的操作便心中有数。 反射绘制 ①反射的叠加方法 在讲绘制方法之前,首先讲讲反射图层是怎么叠加在其他图层上边的。对passes合成有所了解的同学,可能会以为反射应该用add或者screen模式叠加,也就是“添加“和”滤色“。但裸奔毕竟只是简单的在一张图上叠东西,制作反射的时候很难像渲染器输出passes的时候那么纯净。

PLC应用技术项目化教程(S7-200)习题答案

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2. 试用两个复合式按钮设计电动机"正一反一停”控制电路。 任务三电动机“ - -△转换”控制电路的设计与调试 思考与练习 对于一台大功率(10kW以上)的电动机,若要正反转时都能够实现' -△降压起动,

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另外,在本文底部,小编会附上该教程的PDF和word文本,教程中的PSD源文件也提供给大家下载,以方便大家学习。

作者:余德杰(人人网) 前言: 这是适合学生赶图党、工作赶图党的教程,讲的是救急的出图方法,对技术和硬件的要求都相对低【尤其硬件】,那些建模时“看不见”的错误,不会造成多少影响,对建模的逻辑性要求不高。 还是那句,如果要认真学设计搞设计,还是推荐正确、真实的渲染方法。我们搞的不是艺术,是建筑。这种SU导图+PS的方法很难给我们真实的材质、空间的理解。不要以为电脑做事就是傻瓜式,如果是,你就是想多了。 PS:以下黑字部分为理解讲解部分,蓝字为步骤灌输部分,不想看理解的童鞋可以直接跳过理解部分。/ 教程正文 首先我们明确目的,这次做的是雪景。要做雪景的图,不了解雪景的特征,是不行的。 所以先看看现成的雪景图。

这是Ronen Bekerman的雪景效果图,使用max建模、vray渲染和PS后期,这个昨天刚看,所以就直接搬过来了,因为效果非常棒,可以直接当做照片来参考了。 我们看雪景有啥特点? 首先整体上看,雪的颜色比较统一,他的颜色大都和天光差不多,由于雪本身颜色就接近纯白而且呈半透明,所以相比场景中的其他物体,没那么容易受到各种复杂光线的影响。 我们再想象一下,如果雪下的更大些,把所有东西的盖住了,这样的话。。。画面是不是很像我们所熟悉的素模?对吧!就是素模!

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③第三步:下载对应版本的ROM包。 ④第四步:在电脑上用Odinv刷ROM包。(这是三星安卓手机的线刷工具) 二:正式开始 【卡刷篇】 ①:给手机Root的方法也很多,这里我就不介绍了,具体请参考我的另一篇文章《Android(安卓)手机Root教程》,请先做好这一步再继续下面的步骤哦。 ②:什么是Recovery?表面上是恢复得意思,实际上它就是一个系统底层的软件,你可以把它理解为我们电脑上的Ghost,它就是手机上的Ghost了,能够备份/还原、分区、擦除选快、可以升级补丁包,可以刷系统包等等,功能是非常强大的。 Recovery,的全称是ClockworkMod Recovery,所以很多人也简称为CWM,或者工程模式。 我们安卓手机默认是带了Recovery的,不过官方原版的,功能少得可怜,而且不可以用来刷包或者备份还原什么的,一般如图所示:

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