Yang-Mills instantons on 7-dimensional manifold of G_2 holonomy
Running Coupling in SU(3) Yang-Mills Theory
cal running coupling constant at small separation r. This requires control of the force from short to long distance in one simulation with a r L holding for the whole range of physical r involved. With L=a always limited to feasible lattice sizes like 32 or 48, compromises on the above conditions have to be accepted, and it is hardly possible to vary all scale ratios signi cantly to check for the stability of the results. State of the art calculations along these lines are reported in 1,2]. It has to be noted that the highest physical energies r 1 that can be reached here are below about 2 GeV, if one only stays a factor 2 : : : 3 away from the cuto energy. Cuto e ects are corrected semi-empirically using the lattice Coulomb propagator. While these are di cult and careful simulations, we nd it somewhat hard to assess the systematic errors in a completely convincing fashion. An alternative attempt to derive the coupling in QCD has been pioneered by the Fermilab group 3]. Here, in a quenched simulation, the spin averaged 1P-1S charmonium splitting is determined on a physically large lattice. Although this is a nice experimentally known scale with little sensitivity to the quark masses, also other masses could in principle be used here to set the scale. The point relevant in the present context is, that they extract from such a simulation the bare lattice coupling g0 together with the corresponding lattice spacing a in GeV. A perturbative method is then used to relate g0 to a physical coupling at a scale of the order of the cuto . The scale problem is clearly alleviated in comparison to the quark force method, as e ectively the cuto a 1 is identi ed with the high energy physical scale.
Recent results in the center vortex model for the infrared sector of Yang-Mills theory
a rXiv:h e p-la t/821v125Aug2RECENT RESULTS IN THE CENTER VORTEX MODEL FOR THE INFRARED SECTOR OF YANG-MILLS THEORY a M.ENGELHARDT,b H.REINHARDT Institut f¨u r theoretische Physik,Universit¨a t T¨u bingen,Auf der Morgenstelle 14,72076T¨u bingen,Germany M.F ABER Institut f¨u r Kernphysik,Technische Universit¨a t Wien,A-1040Vienna,Austria A model for the infrared sector of SU (2)Yang-Mills theory,based on magnetic vortices represented by (closed)random surfaces,is presented.The model quan-titatively describes both confinement and the topological aspects of Yang-Mills theory.Details (including an adequate list of references)can be found in the e-prints hep-lat/9912003and hep-lat/0004013,both to appear in Nucl.Phys.B.Diverse nonperturbative effects characterize strong interaction physics.Color charge is confined,chiral symmetry is spontaneously broken,and the axial U (1)part of the flavor symmetry exhibits an anomaly.Various model explanations for these phenomena have been advanced;to name but two widely accepted ones,the dual superconductor mechanism of confinement,and instan-ton models,which describe the U A (1)anomaly and spontaneous chiral symme-try breaking.However,no clear picture has emerged which comprehensively describes infrared strong interaction physics within one common framework.The vortex model presented here 1,2aims to bridge this gap.On the basis of a simple effective dynamics,it simultaneously reproduces the confinement properties of SU (2)Yang-Mills theory (including the finite-temperature de-confinement transition),as well as the topological susceptibility,which encodes the U A (1)anomaly.Remarks on the chiral condensate,an important point ofinvestigation which has not yet been carried out,will be made in closing.Center vortices are closed chromomagnetic flux lines in three-dimensional space;thus,they are described by closed two-dimensional world-surfaces in four-dimensional space-time.In the SU (2)case,their magnetic flux is quan-tized such that they modify any Wilson loop by a phase factor (−1)when they pierce an area spanned by the loop.To arrive at a tractable vortex model,it is useful to compose the vortex world-surfaces out of plaquettes on a hypercubic lattice.The spacing of this lattice is a fixed physical quantity (related to a00.20.40.60.81.01.21.41.600.51 1.52T T co σ σT T cMeVχ1/4Figure 1:Observables in the random vortex surface model on 163×N t lattices,with c =0.24,as a function of temperature.Left:String tension between static color sources (crosses)and spatial string tension (circles).Whereas the quantitative behavior of the static quark string tension has largely been fitted using the freedom in the choice of c (see text),the spatial string tension σs is predicted.In the deconfined regime,it begins to rise with temperature;the value σs (T =1.67T C )=1.39σ0corresponds to within 1%with the value measured in full SU (2)Yang-Mills theory.5Right:(Fourth root of)the topological susceptibility;also this result is quantitatively compatible with measurements in full Yang-Mills theory.6thickness of the vortex fluxes),and represents the ultraviolet cutoffinherent in any infrared effective framework.The model vortex surfaces are regarded as random surfaces,and an ensemble of them is generated using Monte Carlo methods.The corresponding weight function penalizes curvature by associat-ing an action increment c with every instance of two plaquettes which are part of a vortex surface,but which do not lie in the same plane,sharing a link (note that several such pairs of plaquettes can occur for any given link).Via the definition given above,Wilson loops (and,in complete analogy,Polyakov loop correlators)can be evaluated in the vortex ensemble,and string tensions extracted.For sufficiently small curvature coefficient c ,one finds a confined phase (non-zero string tension)at low temperatures,and a transition to a high-temperature deconfined phase.For c =0.24,the SU (2)Yang-Mills relation between the deconfinement temperature and the zero-temperature string tension,T C /√which the set of tangent vectors to the surface configuration spans all fourspace-time directions(a simple example are surface self-intersection points). Since a vortex surface carries afield strength characterized by a nonvanishingtensor component associated with the two space-time directions locally orthog-onal to the surface,4these singular points are precisely the points at which the topological charge densityǫµνλτTr FµνFλτis non-vanishing.In practice,imple-menting this result for the hypercubic lattice surfaces used in the present model involves resolving ambiguities2reminiscent of those contained in lattice Yang-Mills link configurations.The resulting topological susceptibilityχ= Q2 /V, where V denotes the space-time volume under consideration,is exhibited in Fig.1(right)as a function of temperature.Taken together,the measurementsin Fig.1show that the vortex model provides,within one common framework,a quantitative description not only of the confinement properties,but also of the topological properties of the SU(2)Yang-Mills ensemble.One obvious generalization of the present work is the treatment of SU(3) color.Also,the coupling of the vortex degrees of freedom to quarks must be investigated,e.g.whether the correct chiral condensate is induced in the vortex background.In this respect,the vortex picture has an important advantage to offer.It is possible to associate any arbitrary vortex surface with a continuum gaugefield,4including the surfaces generated within the present model.As a consequence,the Dirac operator,encoding quark propagation,can be con-structed directly in the continuum,and some of the difficulties associated with lattice Dirac operators,such as fermion species doubling,may be avoidable.AcknowledgmentsM.E.and H.R.acknowledge DFGfinancial support under grants En415/1-1 and Re856/4-1,respectively.M.F.is supported by Fonds zur F¨o rderung der Wissenschaftlichen Forschung under P11387-PHY.References1.M.Engelhardt and H.Reinhardt,hep-lat/9912003,to appear in Nucl.Phys.B.2.M.Engelhardt,hep-lat/0004013,to appear in Nucl.Phys.B.3.M.Engelhardt,ngfeld,H.Reinhardt and O.Tennert,Phys.Rev.D61(2000)054504.4.M.Engelhardt and H.Reinhardt,Nucl.Phys.B567(2000)249.5.G.S.Bali,J.Fingberg,U.M.Heller,F.Karsch and K.Schilling,Phys.Rev.Lett.71(1993)3059.6.B.All´e s,M.D’Elia and A.Di Giacomo,Phys.Lett.B412(1997)119.3。
英文翻译 太阳能定日镜 2012
连续追踪的定日镜亚伯拉罕.克劳布斯陈晓东译摘要:即使太阳的运动是连续的,但是在当前的定日镜跟踪运动通常为不连续的步骤。
瞄准误差离散步骤通常大约1 毫弧度或更多。
为了有效降低跟踪误差提出了光滑连续跟踪。
就是在一个现有的定日镜中使用两个利用电子速度控制单元调整转速的交流电机。
持续跟踪系统在以色列魏兹曼科学中心定日镜领域成功的实现了。
定日镜运动的测量表明目标跟踪误差间隔几乎是消除的。
一种在定日镜运动和不稳定目标的不连续跟踪和连续跟踪的对比的报道。
关键词：跟踪误差；连续跟踪系统；交流电机；定日镜运动1 简介：许多现代的太阳能转化工艺要求的温度升高，温度一般都在1000℃以上，为了高效的操作这些高温过程需要高强度的阳光照射。
经常在相当大的程度上高于1000个太阳。
目前的技术领域的定日镜通常是用来生产平均浓度比值已达到1000。
虽然从理论上讲有这种可能性更高的浓度【1】。
由于几何误差在定日镜结构面误差和操作系统跟踪误差精度中存在，导致定日镜领域的表现是有限的。
比起原来的太阳能磁盘，他们能有效分散反射辐射在更大的角范围内，使得定日镜中集中而又潜在的错误减少了。
当在定日镜领域里不能提供必要的水平时，二级集中器就常常被用于增加聚光浓度【2-4】。
这包含额外的损失和附加的功率损失。
另外，因为定日镜的误差，二级集中器不能弥补失去的潜在热能。
所以这就有利于提高定日镜的性能，减少对二级集中器的依赖。
定日镜误差其中之一的跟踪误差存在于运转的定日镜中。
在定日镜跟踪控制中一般误差在1-1.5毫弧度，在某些特别的情况下会更高点【5】。
这些误差是油很多因素造成的。
比如:驱动公差以及齿侧间隙、太阳的位置精度模型、定日镜运动之间的间隔（由于编码器分辨率）、由重力和风造成的结构变形。
这些因素的影响可能有十分不同的时间段内。
类似于太阳模拟不正确以及重心转移的偏移的误差发生在相对较长的时间段内。
一个可能的解决方法来检测和消除这些误差是一个闭环控制系统【8】。
Singularity of the density of states in the two-dimensional Hubbard model from finite size
a r X i v :c o n d -m a t /9503139v 1 27 M a r 1995Singularity of the density of states in the two-dimensional Hubbard model from finitesize scaling of Yang-Lee zerosE.Abraham 1,I.M.Barbour 2,P.H.Cullen 1,E.G.Klepfish 3,E.R.Pike 3and Sarben Sarkar 31Department of Physics,Heriot-Watt University,Edinburgh EH144AS,UK 2Department of Physics,University of Glasgow,Glasgow G128QQ,UK 3Department of Physics,King’s College London,London WC2R 2LS,UK(February 6,2008)A finite size scaling is applied to the Yang-Lee zeros of the grand canonical partition function for the 2-D Hubbard model in the complex chemical potential plane.The logarithmic scaling of the imaginary part of the zeros with the system size indicates a singular dependence of the carrier density on the chemical potential.Our analysis points to a second-order phase transition with critical exponent 12±1transition controlled by the chemical potential.As in order-disorder transitions,one would expect a symmetry breaking signalled by an order parameter.In this model,the particle-hole symmetry is broken by introducing an “external field”which causes the particle density to be-come non-zero.Furthermore,the possibility of the free energy having a singularity at some finite value of the chemical potential is not excluded:in fact it can be a transition indicated by a divergence of the correlation length.A singularity of the free energy at finite “exter-nal field”was found in finite-temperature lattice QCD by using theYang-Leeanalysisforthechiral phase tran-sition [14].A possible scenario for such a transition at finite chemical potential,is one in which the particle den-sity consists of two components derived from the regular and singular parts of the free energy.Since we are dealing with a grand canonical ensemble,the particle number can be calculated for a given chem-ical potential as opposed to constraining the chemical potential by a fixed particle number.Hence the chem-ical potential can be thought of as an external field for exploring the behaviour of the free energy.From the mi-croscopic point of view,the critical values of the chemical potential are associated with singularities of the density of states.Transitions related to the singularity of the density of states are known as Lifshitz transitions [15].In metals these transitions only take place at zero tem-perature,while at finite temperatures the singularities are rounded.However,for a small ratio of temperature to the deviation from the critical values of the chemical potential,the singularity can be traced even at finite tem-perature.Lifshitz transitions may result from topological changes of the Fermi surface,and may occur inside the Brillouin zone as well as on its boundaries [16].In the case of strongly correlated electron systems the shape of the Fermi surface is indeed affected,which in turn may lead to an extension of the Lifshitz-type singularities into the finite-temperature regime.In relating the macroscopic quantity of the carrier den-sity to the density of quasiparticle states,we assumed the validity of a single particle excitation picture.Whether strong correlations completely distort this description is beyond the scope of the current study.However,the iden-tification of the criticality using the Yang-Lee analysis,remains valid even if collective excitations prevail.The paper is organised as follows.In Section 2we out-line the essentials of the computational technique used to simulate the grand canonical partition function and present its expansion as a polynomial in the fugacity vari-able.In Section 3we present the Yang-Lee zeros of the partition function calculated on 62–102lattices and high-light their qualitative differences from the 42lattice.In Section 4we analyse the finite size scaling of the Yang-Lee zeros and compare it to the real-space renormaliza-tion group prediction for a second-order phase transition.Finally,in Section 5we present a summary of our resultsand an outlook for future work.II.SIMULATION ALGORITHM AND FUGACITY EXPANSION OF THE GRAND CANONICALPARTITION FUNCTIONThe model we are studying in this work is a two-dimensional single-band Hubbard HamiltonianˆH=−t <i,j>,σc †i,σc j,σ+U i n i +−12 −µi(n i ++n i −)(1)where the i,j denote the nearest neighbour spatial lat-tice sites,σis the spin degree of freedom and n iσis theelectron number operator c †iσc iσ.The constants t and U correspond to the hopping parameter and the on-site Coulomb repulsion respectively.The chemical potential µis introduced such that µ=0corresponds to half-filling,i.e.the actual chemical potential is shifted from µto µ−U412.(5)This transformation enables one to integrate out the fermionic degrees of freedom and the resulting partition function is written as an ensemble average of a product of two determinantsZ ={s i,l =±1}˜z = {s i,l =±1}det(M +)det(M −)(6)such thatM ±=I +P ± =I +n τ l =1B ±l(7)where the matrices B ±l are defined asB ±l =e −(±dtV )e −dtK e dtµ(8)with V ij =δij s i,l and K ij =1if i,j are nearestneigh-boursand Kij=0otherwise.The matrices in (7)and (8)are of size (n x n y )×(n x n y ),corresponding to the spatial size of the lattice.The expectation value of a physical observable at chemical potential µ,<O >µ,is given by<O >µ=O ˜z (µ){s i,l =±1}˜z (µ,{s i,l })(9)where the sum over the configurations of Ising fields isdenoted by an integral.Since ˜z (µ)is not positive definite for Re(µ)=0we weight the ensemble of configurations by the absolute value of ˜z (µ)at some µ=µ0.Thus<O >µ= O ˜z (µ)˜z (µ)|˜z (µ0)|µ0|˜z (µ0)|µ0(10)The partition function Z (µ)is given byZ (µ)∝˜z (µ)N c˜z (µ0)|˜z (µ0)|×e µβ+e −µβ−e µ0β−e −µ0βn (16)When the average sign is near unity,it is safe to as-sume that the lattice configurations reflect accurately thequantum degrees of freedom.Following Blankenbecler et al.[1]the diagonal matrix elements of the equal-time Green’s operator G ±=(I +P ±)−1accurately describe the fermion density on a given configuration.In this regime the adiabatic approximation,which is the basis of the finite-temperature algorithm,is valid.The situa-tion differs strongly when the average sign becomes small.We are in this case sampling positive and negative ˜z (µ0)configurations with almost equal probability since the ac-ceptance criterion depends only on the absolute value of ˜z (µ0).In the simulations of the HSfields the situation is dif-ferent from the case of fermions interacting with dynam-ical bosonfields presented in Ref.[1].The auxilary HS fields do not have a kinetic energy term in the bosonic action which would suppress their rapidfluctuations and hence recover the adiabaticity.From the previous sim-ulations on a42lattice[3]we know that avoiding the sign problem,by updating at half-filling,results in high uncontrolledfluctuations of the expansion coefficients for the statistical weight,thus severely limiting the range of validity of the expansion.It is therefore important to obtain the partition function for the widest range ofµ0 and observe the persistence of the hierarchy of the ex-pansion coefficients of Z.An error analysis is required to establish the Gaussian distribution of the simulated observables.We present in the following section results of the bootstrap analysis[17]performed on our data for several values ofµ0.III.TEMPERATURE AND LATTICE-SIZEDEPENDENCE OF THE YANG-LEE ZEROS The simulations were performed in the intermediate on-site repulsion regime U=4t forβ=5,6,7.5on lat-tices42,62,82and forβ=5,6on a102lattice.The ex-pansion coefficients given by eqn.(14)are obtained with relatively small errors and exhibit clear Gaussian distri-bution over the ensemble.This behaviour was recorded for a wide range ofµ0which makes our simulations reli-able in spite of the sign problem.In Fig.1(a-c)we present typical distributions of thefirst coefficients correspond-ing to n=1−7in eqn.(14)(normalized with respect to the zeroth power coefficient)forβ=5−7.5for differ-entµ0.The coefficients are obtained using the bootstrap method on over10000configurations forβ=5increasing to over30000forβ=7.5.In spite of different values of the average sign in these simulations,the coefficients of the expansion(16)indicate good correspondence between coefficients obtained with different values of the update chemical potentialµ0:the normalized coefficients taken from differentµ0values and equal power of the expansion variable correspond within the statistical error estimated using the bootstrap analysis.(To compare these coeffi-cients we had to shift the expansion by2coshµ0β.)We also performed a bootstrap analysis of the zeros in theµplane which shows clear Gaussian distribution of their real and imaginary parts(see Fig.2).In addition, we observe overlapping results(i.e.same zeros)obtained with different values ofµ0.The distribution of Yang-Lee zeros in the complexµ-plane is presented in Fig.3(a-c)for the zeros nearest to the real axis.We observe a gradual decrease of the imaginary part as the lattice size increases.The quantitative analysis of this behaviour is discussed in the next section.The critical domain can be identified by the behaviour of the density of Yang-Lee zeros’in the positive half-plane of the fugacity.We expect tofind that this density is tem-perature and volume dependent as the system approaches the phase transition.If the temperature is much higher than the critical temperature,the zeros stay far from the positive real axis as it happens in the high-temperature limit of the one-dimensional Ising model(T c=0)in which,forβ=0,the points of singularity of the free energy lie at fugacity value−1.As the temperature de-creases we expect the zeros to migrate to the positive half-plane with their density,in this region,increasing with the system’s volume.Figures4(a-c)show the number N(θ)of zeros in the sector(0,θ)as a function of the angleθ.The zeros shown in thesefigures are those presented in Fig.3(a-c)in the chemical potential plane with other zeros lying further from the positive real half-axis added in.We included only the zeros having absolute value less than one which we are able to do because if y i is a zero in the fugacity plane,so is1/y i.The errors are shown where they were estimated using the bootstrap analysis(see Fig.2).Forβ=5,even for the largest simulated lattice102, all the zeros are in the negative half-plane.We notice a gradual movement of the pattern of the zeros towards the smallerθvalues with an increasing density of the zeros nearθ=πIV.FINITE SIZE SCALING AND THESINGULARITY OF THE DENSITY OF STATESAs a starting point for thefinite size analysis of theYang-Lee singularities we recall the scaling hypothesis forthe partition function singularities in the critical domain[11].Following this hypothesis,for a change of scale ofthe linear dimension LLL→−1),˜µ=(1−µT cδ(23)Following the real-space renormalization group treatmentof Ref.[11]and assuming that the change of scaleλisa continuous parameter,the exponentαθis related tothe critical exponentνof the correlation length asαθ=1ξ(θλ)=ξ(θ)αθwe obtain ξ∼|θ|−1|θ|ναµ)(26)where θλhas been scaled to ±1and ˜µλexpressed in terms of ˜µand θ.Differentiating this equation with respect to ˜µyields:<n >sing =(−θ)ν(d −αµ)∂F sing (X,Y )ν(d −αµ)singinto the ar-gument Y =˜µαµ(28)which defines the critical exponent 1αµin terms of the scaling exponent αµof the Yang-Lee zeros.Fig.5presents the scaling of the imaginary part of the µzeros for different values of the temperature.The linear regression slope of the logarithm of the imaginary part of the zeros plotted against the logarithm of the inverse lin-ear dimension of the simulation volume,increases when the temperature decreases from β=5to β=6.The re-sults of β=7.5correspond to αµ=1.3within the errors of the zeros as the simulation volume increases from 62to 82.As it is seen from Fig.3,we can trace zeros with similar real part (Re (µ1)≈0.7which is also consistentwith the critical value of the chemical potential given in Ref.[22])as the lattice size increases,which allows us to examine only the scaling of the imaginary part.Table 1presents the values of αµand 1αµδ0.5±0.0560.5±0.21.3±0.3∂µ,as a function ofthe chemical potential on an 82lattice.The location of the peaks of the susceptibility,rounded by the finite size effects,is in good agreement with the distribution of the real part of the Yang-Lee zeros in the complex µ-plane (see Fig.3)which is particularly evident in the β=7.5simulations (Fig.4(c)).The contribution of each zero to the susceptibility can be singled out by expressing the free energy as:F =2n x n yi =1(y −y i )(29)where y is the fugacity variable and y i is the correspond-ing zero of the partition function.The dotted lines on these plots correspond to the contribution of the nearby zeros while the full polynomial contribution is given by the solid lines.We see that the developing singularities are indeed governed by the zeros closest to the real axis.The sharpening of the singularity as the temperature de-creases is also in accordance with the dependence of the distribution of the zeros on the temperature.The singularities of the free energy and its derivative with respect to the chemical potential,can be related to the quasiparticle density of states.To do this we assume that single particle excitations accurately represent the spectrum of the system.The relationship between the average particle density and the density of states ρ(ω)is given by<n >=∞dω1dµ=ρsing (µ)∝1δ−1(32)and hence the rate of divergence of the density of states.As in the case of Lifshitz transitions the singularity of the particle number is rounded at finite temperature.However,for sufficiently low temperatures,the singular-ity of the density of states remains manifest in the free energy,the average particle density,and particle suscep-tibility [15].The regular part of the density of states does not contribute to the criticality,so we can concentrate on the singular part only.Consider a behaviour of the typedensity of states diverging as the−1ρsing(ω)∝(ω−µc)1δ.(33)with the valueδfor the particle number governed by thedivergence of the density of states(at low temperatures)in spite of thefinite-temperature rounding of the singu-larity itself.This rounding of the singularity is indeedreflected in the difference between the values ofαµatβ=5andβ=6.V.DISCUSSION AND OUTLOOKWe note that in ourfinite size scaling analysis we donot include logarithmic corrections.In particular,thesecorrections may prove significant when taking into ac-count the fact that we are dealing with a two-dimensionalsystem in which the pattern of the phase transition islikely to be of Kosterlitz-Thouless type[23].The loga-rithmic corrections to the scaling laws have been provenessential in a recent work of Kenna and Irving[24].In-clusion of these corrections would allow us to obtain thecritical exponents with higher accuracy.However,suchanalysis would require simulations on even larger lattices.The linearfits for the logarithmic scaling and the criti-cal exponents obtained,are to be viewed as approximatevalues reflecting the general behaviour of the Yang-Leezeros as the temperature and lattice size are varied.Al-though the bootstrap analysis provided us with accurateestimates of the statistical error on the values of the ex-pansion coefficients and the Yang-Lee zeros,the smallnumber of zeros obtained with sufficient accuracy doesnot allow us to claim higher precision for the critical ex-ponents on the basis of more elaboratefittings of the scal-ing behaviour.Thefinite-size effects may still be signifi-cant,especially as the simulation temperature decreases,thus affecting the scaling of the Yang-Lee zeros with thesystem rger lattice simulations will therefore berequired for an accurate evaluation of the critical expo-nent for the particle density and the density of states.Nevertheless,the onset of a singularity atfinite temper-ature,and its persistence as the lattice size increases,areevident.The estimate of the critical exponent for the diver-gence rate of the density of states of the quasiparticleexcitation spectrum is particularly relevant to the highT c superconductivity scenario based on the van Hove sin-gularities[25],[26],[27].It is emphasized in Ref.[25]thatthe logarithmic singularity of a two-dimensional electrongas can,due to electronic correlations,turn into a power-law divergence resulting in an extended saddle point atthe lattice momenta(π,0)and(0,π).In the case of the14.I.M.Barbour,A.J.Bell and E.G.Klepfish,Nucl.Phys.B389,285(1993).15.I.M.Lifshitz,JETP38,1569(1960).16.A.A.Abrikosov,Fundamentals of the Theory ofMetals North-Holland(1988).17.P.Hall,The Bootstrap and Edgeworth expansion,Springer(1992).18.S.R.White et al.,Phys.Rev.B40,506(1989).19.J.E.Hirsch,Phys.Rev.B28,4059(1983).20.M.Suzuki,Prog.Theor.Phys.56,1454(1976).21.A.Moreo, D.Scalapino and E.Dagotto,Phys.Rev.B43,11442(1991).22.N.Furukawa and M.Imada,J.Phys.Soc.Japan61,3331(1992).23.J.Kosterlitz and D.Thouless,J.Phys.C6,1181(1973);J.Kosterlitz,J.Phys.C7,1046(1974).24.R.Kenna and A.C.Irving,unpublished.25.K.Gofron et al.,Phys.Rev.Lett.73,3302(1994).26.D.M.Newns,P.C.Pattnaik and C.C.Tsuei,Phys.Rev.B43,3075(1991);D.M.Newns et al.,Phys.Rev.Lett.24,1264(1992);D.M.Newns et al.,Phys.Rev.Lett.73,1264(1994).27.E.Dagotto,A.Nazarenko and A.Moreo,Phys.Rev.Lett.74,310(1995).28.A.A.Abrikosov,J.C.Campuzano and K.Gofron,Physica(Amsterdam)214C,73(1993).29.D.S.Dessau et al.,Phys.Rev.Lett.71,2781(1993);D.M.King et al.,Phys.Rev.Lett.73,3298(1994);P.Aebi et al.,Phys.Rev.Lett.72,2757(1994).30.E.Dagotto, A.Nazarenko and M.Boninsegni,Phys.Rev.Lett.73,728(1994).31.N.Bulut,D.J.Scalapino and S.R.White,Phys.Rev.Lett.73,748(1994).32.S.R.White,Phys.Rev.B44,4670(1991);M.Veki´c and S.R.White,Phys.Rev.B47,1160 (1993).33.C.E.Creffield,E.G.Klepfish,E.R.Pike and SarbenSarkar,unpublished.Figure CaptionsFigure1Bootstrap distribution of normalized coefficients for ex-pansion(14)at different update chemical potentialµ0for an82lattice.The corresponding power of expansion is indicated in the topfigure.(a)β=5,(b)β=6,(c)β=7.5.Figure2Bootstrap distributions for the Yang-Lee zeros in the complexµplane closest to the real axis.(a)102lat-tice atβ=5,(b)102lattice atβ=6,(c)82lattice at β=7.5.Figure3Yang-Lee zeros in the complexµplane closest to the real axis.(a)β=5,(b)β=6,(c)β=7.5.The correspond-ing lattice size is shown in the top right-hand corner. Figure4Angular distribution of the Yang-Lee zeros in the com-plex fugacity plane Error bars are drawn where esti-mated.(a)β=5,(b)β=6,(c)β=7.5.Figure5Scaling of the imaginary part ofµ1(Re(µ1)≈=0.7)as a function of lattice size.αm u indicates the thefit of the logarithmic scaling.Figure6Electronic susceptibility as a function of chemical poten-tial for an82lattice.The solid line represents the con-tribution of all the2n x n y zeros and the dotted line the contribution of the six zeros nearest to the real-µaxis.(a)β=5,(b)β=6,(c)β=7.5.。
大学普通物理-光的干涉
例如, 计算图中光通过路程 r1 和 r2 在P点的相差。
S1 n
r1 r2
·P
解： r2 d nd r1
2
r2
d
nd
r1
S2 d
2
r2
r1
n
1
d
说明： 研究光干涉问题，归结为分析两束光在相遇点的
光程差。
如果 k ，
相差 2
对应相位差为 2k
相长(明)
如果 2k 1 ， 对应相位差为 2k 1 相消(暗)
1. 平行于棱的、明暗相间的直条纹。 属于等厚条纹。
2. 在棱边处, e=0， 两反射光
棱
光程差为 / 2 , 形成o级暗
条纹，越向外级次越高.
劈尖角
3. 相邻明纹或暗纹是等间距的.
对于相邻的两 条明条纹:
2n2ek 1
2
k
1
2n2ek
2
k
L
e
ek
n2
ek 1
对于相邻的两 条明条纹:
2n2ek 1
牛顿环——由球面和平面构成的轴对称状 空气薄膜所形成的干涉条纹。
S
干涉装置如图所示:
平玻璃B，平凸透镜A(曲率半径
R很大)；A, B之间形成薄的轴对
A
称状的空气层；当单色平行光垂
B
直入射于平凸透镜时，在透镜表
面下可以观察到一组干涉条纹，
称为牛顿环.
目镜
M
理论分析: 当平行光垂直入射后，在空气层的上下表面形成两束反射
第二十二章 光的干涉
光（一般指可见光）是一定频率范围内的电磁波。
频率：3.91014 ~ 8.61014 Hz 波长： 350 ~ 770nm
Renormalizability of the massive Yang-Mills theory
a rXiv:h ep-ph/53198v12Mar25Renormalizability of the massive Yang-Mills theory rin Institute for Nuclear Research of the Russian Academy of Sciences,60th October Anniversary Prospect 7a,Moscow 117312,Russia Abstract It is shown that the massive Yang-Mills theory is on mass-shell renormalizable.Thus the Standard Model of electroweak interactions can be modified by removing terms with the scalar field from the Lagrangian in the unitary gauge.The resulting electroweak theory without the Higgs particle is on mass-shell renormalizable and unitary.1The massive Yang-Mills theory[1]is considered to be non-renormalizable [2],see also[3,4]and references therein.The known way to get renormaliz-able and unitary theory with massive Yang-Mills bosons is due to the Higgs mechanism of spontaneous symmetry breaking[5].The mechanism is used in the Standard SU(2)×U(1)Model of electroweak interactions[6]which is established to be renormalizable[7],see also[8]and references therein.In this way one introduces in the Model the scalar Higgs particle which one can hope to see in experiments.The purpose of the present paper is to show that the massive Yang-Mills theory is in fact on mass-shell renormalizable.Hence the Standard Model can be modified by discarding from the Lagrangian in the unitary gauge all terms containing the scalarfield.Let us consider the massive Yang-Mills theory of gaugefields W aµ(x)de-fined by the generating functional of Green functions in the path integral formZ(J)=14F aµνF aµν+1(2π)4gµν−kµkν/m2To establish on mass-shell renormalizability of the massive Yang-Mills theory (1)oneshould show that the S-matrix elements can be made finite by means of counterterms which can be absorbed into renormalization constants of the parameters g and m although the Green functions are divergent.We will work within perturbation theory.To regularize ultraviolet divergences we will use for convenience dimen-sional regularization [9]with the space-time dimension d =4−2ǫ,ǫbeing the regularization parameter.Let us consider the known model given by the initial SU (2)-invariant Lagrangian possessing the spontaneously broken symmetryL =−12aW a µ Φis the covariant derivative,τa are the Pauli matrices,λ>0,v 2>0.To get the complete Lagrangian one makes the shift of the scalar fieldΦ(x )=12 iφ1(x )+φ2(x )√NdW dφdχd 4F a µνF a µν+m 22∂µφa ∂µφa +12χ2+g 2χW a µW a µ+g 24mχ(χ2+φa φa )−g 2M 22ξ(∂µW a µ+ξmφa )23+∂µc a c a −g c a c a +g c a c b φc+countertermsThis theory describes three physical massive vector bosons with the mass m =gv/√4F a µνF a µν+m 22∂µχ∂µχ−M 22χW a µW a µ+g 24m χ3−g 2M 2N dW dφdχexp i dx L R +J a µW a µ+Kχ ∆L (W )δ(∂µW µ)(6)where ∆L (W )is the Faddeev-Popov determinant[13]and L R is obtained from L R ξby omitting terms depending on ξand c a (and by corresponding modification of counterterms).The Lagrangian L R is invariant under the following gauge transformationsW a µ→ W ωµ a =W a µ+∂µωa +˜g f abc W b µωc +O (ω2)(7)φa →(φω)a =φa −˜mωa −˜g 2χωa +O (ω2)4χ→χω=χ−˜g z 2g ˜m =z 1NdW dφdχexp i dx L U +J µW ˜ωµ+Kχ˜ω ∆U (χ)δ(φ)(9)where ˜ωis defined from the equation ∂µ W ˜ωµ a =∂µW a µ+∂µ˜ωa +˜g f abc W b µ˜ωc +O (˜ω2)=0(10)The Lagrangian L U is given ineq.(5).The functional ∆U (χ)can be presented on the surface φa =0as ∆U (χ)=det |˜m +˜g 2m χ(x ))3dxIn dimensional regularization this functional is just a constant and can be absorbed in the normalization factor N although this simplification is not essential for the following derivation.One obtainesZ L (J,K )=1N dW dχexp idx (L U +J µW µ+Kχ) (12)5only by source terms.It is known that this difference is not essential for the S-matrix elements,see e.g.[8].Thus the physical equivalence of the L-gauge and the U-gauge is proved.From eq.(11)one sees that the counterterms of L U are given by the coun-terterms of L R atφa(x)=0.To consider renormalization for our purpose it is convenient to use the Bogoliubov-Parasiuk-Hepp subtraction scheme[14].As it is well known in this scheme a counterterm of e.g.a primitively divergent Feynman diagram is the truncated Taylor expansion of the diagram itself at somefixed values of external momenta.Hence counterterms of mass dependent diagrams are also mass dependent.Let us now analyze the dependence of the Green functions on the Higgs mass M.We will use for this purpose the expansion in large M.The algo-rithm for the large mass expansion of Feynman diagrams is given in[15],it can be rigorously derived e.g.with the technique of[16].The representation(11)ensures for the regularized Green functions of the fields W andχthat the large M-expansion of M-dependent contributions contain either terms with integer negative powers of M2or terms with non-integer powers of M2(non-integer powers containǫ).This is because each vertex with the factor M2has three or four attachedχ-lines due to the structure of L U.Corresponding counterterms(i.e.counterterms relevant for L U)have the same property within the large-M expansion.(In contrast, counterterms of e.g.the four-φvertex in L R contain polynomial in M terms because of the M2-factors in the couplings of L R.)Let us further consider Green functions with external W-bosons only.We willfirst shortly formulate the result.The eq.(11)ensures that if one removes from a renormalized Green function M-dependent terms then the remaining part isfinite.On the Lagrangian level it means that one removes from L U all terms containing thefieldχ.Thus one obtains the theory˜Z(J)=1F aµνF aµν+m2W aµW aµ+counterterms4withfinite off-shell Green functions,where W˜ωµis given by(10).Since the difference between W˜ωµand Wµin the source term is not essential for S-matrix elements the massive Yang-Mills theory is renormalizable on mass-shell.6Let us elaborate these arguments in more detail.The representation (11)ensures,seee.g.[8],that the following on mass-shell expressions for the renormalized Green functions (relevant for the S-matrix elements)should coincide1znn i =1(k 2i −m 2)G a 1...a n µ1...νn (k 1...k n )|L −gauge k 2i =m 2=(14) 1z n n i =1(k 2i −m 2)G a 1...a n µ1...νn (k 1...k n )|U −gauge where z is the residue of the propagator poleδab g µν−k µk νk 2 e ikx δ2Z (J )NdW exp i dx L Y M +J a µW a µ (16)L Y M =−1z 2gf abc W b µW c ν)2+z m m 2W a µW a µAfter renormalizability is established one can fix renormalization con-stants z 1,z 2and z m within the theory (16)(without referring to the L-gauge)by proper normalization conditions.It is known that the Higgs theories of vector mesons posses so called tree level unitarity,see e.g.[3]and references therein.Tree level cross sections of such theories grow at high energies slowly enough and do not exceed the7so called unitary limit imposed by the unitarity condition.The reversed statement is also proved:from the condition of tree level unitarity follows that a theory of vector mesons should be a Higgs theory[17].But one can see that tree level unitarity is not the necessary condition for renormalizability. Tree level unitarity is violated in the massive Yang-Mills theory.It indicates that higher order contributions become relevant at high energies.The above derivation of on mass-shell renormalizability is applicable also to other gauge groups.It can be straightforwardly applied to the Standard SU(2)×U(1)Model of electroweak interactions.The presence of the U(1) gauge boson and of fermions does not change the derivation.One can remove from the Lagrangian in the unitary gauge all terms containing the scalarfield. The resulting electroweak theory without the Higgs particle is on mass-shell renormalizable and unitary.The author is grateful to D.S.Gorbunov and S.M.Sibiryakov for helpful discussions.References[1]C.N.Yang and ls,Phys.Rev.96(1954)191.[2]D.G.Boulware,Ann.of Phys.56(1970)140.[3]J.C.Taylor,Gauge theories of weak interactions,Cambridge UniversityPress,1976.[4]C.Itzykson and J.B.Zuber,Quantumfield theory,New York,Mcgraw-hill,1980.[5]P.W.Higgs,Phys.Lett.12(1964)132.F.Englert and R.Brout,Phys.Rev.Lett.13(1964)321.T.W.B.Kibble,Phys.Rev.155(1967)1554.[6]S.L.Glashow,Nucl.Phys.22(1961)579.S.Weinberg,Phys.Rev.Lett.19(1967)264.A.Salam,in Elementary Particle Theory,ed.N.Svartholm,Stochholm,Almquist and Wiksell,1968.8[7]G.’t Hooft,Nucl.Phys.B35(1971)167.[8]L.D.Faddeev and A.A.Slavnov,Gaugefields.Introduction to quantumtheory,Front.Phys.83(1990)1.[9]K.G.Wilson and M.E.Fisher,Phys.Rev.Lett.28(1972)240.G.’t Hooft and M.Veltman,Nucl.Phys.B44(1972)189.C.G.Bollini and J.J.Giambiagi,Phys.Lett.B40(1972)566.J.F.Ashmore,Nuovo Cimento Lett.4(1972)289.G.M.Cicuta and E.Montaldi,Nuovo Cimento Lett.4(1972)329.[10]K.Fujikawa,B.W.Lee and A.I.Sanda,Phys.Rev.D6(1972)2923.[11]A.A.Slavnov,Theor.Math.Phys.10(1972)99.[12]J.C.Taylor,Nucl.Phys.B33(1971)436.[13]L.D.Faddeev and V.N.Popov,Phys.Lett.B25(1967)30.[14]N.N.Bogoliubov and O.S.Parasiuk,Acta Math.97(1957)227.K.Hepp,Comm.Math.Phys.2(1966)301.[15]rin,T.van Ritbergen and J.A.M.Vermaseren,Nucl.Phys.B438(1995)278.[16]rin,Phys.Lett.B469(1999)220.[17]J.M.Cornwall,D.N.Levin and G.Tiktopoulos,Phys.Rev.D10(1974)1145.9。
The string solution in SU(2) Yang-Mills-Higgs theory
a rXiv:h ep-th/966124v12J un1996The string solution in SU(2)Yang-Mills-Higgs theory V.D.Dzhunushaliev ∗and A.A.Fomin Theoretical physics department,the Kyrgyz State National University,720024,Bishkek,Kyrgyzstan Abstract The tube solutions in Yang -Mills -Higgs theory are received,in which the Higgs field has the negative energy density.This solutions make up the discrete spectrum numered by two integer and have the finite linear energy density.Ignoring its transverse size,such field configuration is the rest infinity straight string.PACS number:03.65.Pm;11.17.-w At the end of 50-th years W.Heisenberg has been investigate the non-linear spinor matter theory (see,for example,[1],[2]).It is supposed that on the basis one or another nonlinear spinor equation the basic parameters of the elementary particles existing at that time will be derived:masses,charges and so on.The mathematical essence of this theory lies in the fact that the nonlinear spinor Heisenberg equation (HE)(or in the simpler case the nonlinear boson equation like nonlinear Schr¨o dinger equation)has the discrete spectrum of the solutions having physical meaning (possesing,for example,the finite energy).This solutions give the mass spectrum in clas-sical region even.This gave hope that after quantization more or less likely mass spectrum and the charges of the elementary particles would be derive.Now the string can to arise in Dirac theory with the massive vector field A µby interaction 2magnetic charges with opposite sign [3].At present timethe investigations continue along this line and explore not1-dimensional ob-ject(string)stretched between quarks(see,for example,[4])but3-dimensional (tube)filled byfield(see,for example,[5],[6]).So,for example,a tube of the chromodynamicalfield and its properties in[5]is considered.But this consideration is phenomenological because a question on the reason of the field pinching isn’t affected,also a question on thefield distribution in the tube isn’t analyzed.In this article we shown that the Yang-Millsfield interacted with Higgs scalarfield is confined in tube.In this case the Higgsfield have the negative energy density.In[2]it is showed that the nonlinear Klein-Gordon and Heisenberg equations have the regular solutions.They are the spherical symmetric par-ticlelike solutions numered by integer,i.e.they form discrete spectrum with the corresponding energy value.One would expect(and this will be showed below)that we have in axial symmetric case as well as in spherical-sym-metric case the physical interesting(string)solutions withfinite energy per unith length.Finally,we present some qualitative argument in favour of the existence suchfield configurations(tube,string)according[4].In QCD vacuumfield taken external pressure on the gluon tube.Diameter of such tube will be defined from equilibrium condition between external pressure of the vacuum field and internal pressure of the gluonfield in tube.It can be evaluate by minimizing the energy density of such tube which is the difference between the positive energy density of the chromodynamicfield and negative energy density of vacuumfield in QCD.This diameter R0after corresponding cal-culations is equal:ΦR0=F aµνFµνa−14g2where a =1,2,3is SU (2)colour index;µ,ν=0,1,2,3are spacetime indexes;F aµν=∂µA aµ−∂νA aµ+ǫabc A bµA aνis the strength tensor of the SU (2)gauge field;F µν=F aµνt a ,t a are generators of the SU (2)gauge group;D µΦ=(∂µ+A µ)Φ;V (Φ)=λ(Φ+Φ−4η2)/32;g,η,λare constant;Φis an isodoublet of the Higgs scalar field;The Yang -Mills -Higgs equations system look by following form in this model:D µF µνa =(−γ)−1/2∂µ (−γ)1/2F µνa +ǫabc A bµF µνc =g 2∂Φ+,(4)where γis the metrical tensor determinant.We seek the string solution in the following form:the gauge potential A aµand the isodoublet of the scalar field Φwe chosen in cylindrical coordinate system (z,r,θ)as :A 1t =2ηf (r ),(5)A 2z =2ηv (r ),(6)A 3θ=2ηrw (r ),(7)Φ= 2ηϕ(r )0 (8)By substituting Eq’s(5-8)in Eq’s(3-4)we receive the following equations system:f ′′+f ′x =v 4 −f 2+w2 −g 2ϕ2 ,(10)w ′′+w ′x 2=w 4 −f 2+v 2 −g 2ϕ2,(11)ϕ′′+ϕ′λis introduced;(′)means thederivative with respect to x ;and the following renaming are made:g 2λ−1/2→3g2,f(x)λ−1/2→f(x),v(x)λ−1/2→v(x),w(x)λ−1/2→w(x).We will study this system by the numerical tools.In this article we investigate the easiest case v=f=0.Thus system(9-12)look as:w′′+w′x2=−g2wϕ2,(13)ϕ′′+ϕ′2+···,(15) w=w0x+w3x32ϕ0 1−ϕ20 ,(17) w3=−3w(x)≈1√x−Cg2x2,(22)where integers m and n enumerate the knot number ofϕ(x)and w(x)func-tions respectively.According to this we shall denote the boundary value ϕ(0)and parameter g in the following manner:ϕ∗mn and g∗mn.The result of numerical calculations on Fig.1,2are displayed(w1=0.1).The asymptotic behaviour of theϕmn(x)and w mn(x)functions as in(21)-(22)results in that the energy density of thisfields drop to zero as exponent on the infinity and this means that this tube has thefinite energy per unit. It is easy to show that aflux of colour”magnetic”field H z across the plane z=const isfinite.Thus we can to speak that the Yang-Mills-Higgs theory have the tube solution if the Higgsfield have the negative energy density.It is notice that this solutions are not topological nontrivial thread.Ignoring the transversal size of obtained tube we receive the rested boson string withfinite linear energy density.References5[1]Nonlinear quantumfield theory.Ed.D.D.Ivanenko,Moskow,IL,p.464,1959.[2]R.Finkelstein,R.LeLevier,M.Ruderman,Phys.Rev.,83,326(1951).[3]Nambu Y.,Phys.Rev.1974,D10,p.4262.[4]Bars I.,Hanson A.J.,Phys.Rev.,1976,D13,p.1744.[5]Nussinov S.,Phys.Rev.D.1994,v.50,N5,p.3167.[6]Olson C.,Olsson M.G.,Dan LaCourse,Phys.Rev.D.1994,v.49,N9,p.4675.[7]Barbashow B.M.,Nesterenko W.W.Relativistic string model inhadronic physics,Moskow,Energoatomizdat,p.179,1987.[8]V.D.Dzhunushaliev,Superconductivity:physics,chemistry,technique,v.7,N5,767,1994.6。
Exact solutions in the Yang-Mills-Wong theory
2
of gauge invariant operators obeys the so-called factorization relation, and quantum fluctuations disappear [8]. Thus QCD becomes a classical theory as N → ∞. We suggest that the large-N YMW theory is intimately related to the classical limit of QCD. Note, however, that the confinement problem is out of the question now. Indeed, it is conceivable that quarks constituting a hadron experience an attractive constant force originating from a term of the potential Aµ which linearly rises with distance between the quarks, and such a behavior of Aµ is to provide the area law for the Wilson loop functional [9]. Are we correct in interpreting the area law as the evidence of the constant attractive force? As will be shown, an exact classical solution Aµ with the linearly rising term actually exists. Although this term contributes to the field strength, it produces no force. The general reason for such a surprising result is the conformal invariance. The linearly rising term violates the scale invariance. While such a violation being allowable for the gauge quantities Aµ and Fµν , it cannot be tolerated for observables. One may expect a dimensional parameter, measuring a gap in the energy spectrum and violating the scale symmetry, to emerge only upon quantization leading to anomalies. Meanwhile exact classical solutions are crucial in learning the symmetry of the vacuum. One believes two phases of the strong interacting matter to exist, hot and cold, which must be distinguished by their symmetry. At high temperatures, the asymptotical freedom dominates, hence the conventional SU(3)c symmetry is inherent in the hot phase. On the ˇ cki [10] developed an exhaustive phenomenological classifiother hand, Ne’eman and Sijaˇ cation of hadrons on the basis of infinite-dimensional unitary representations of SL(4, R), which hints that SL(4, R) is the cold phase symmetry. Where does this SL(4, R) come from? Coleman [11] argued that the symmetry of the vacuum is the symmetry of world. Given the vacuum invariant under SL(4, R), excitations about it possess the same symmetry. Since the symmetry of the gluon vacuum is nothing but the symmetry of the background field, the responsibility for SL(4, R) rests with the background described by a certain solution of the QCD equations in the classical limit. It is the background generated by quarks in hadrons that provides the SL(4, R) relief for gluon excitations. We will find two classes of exact retarded solutions to the classical Yang-Mills (YM) equations. Solutions of the first class, invariant under SU(N ), appear to be related to the background in the hot phase. Solutions of the second class might be treated as the background generated by bound quarks in the cold phase. These solutions are complex valued with respect to the Lie algebra su(N ), but one can convert them to the real form to yield the invariance under SL(N, R) or its subgroups. In particular, the background generated by any three-quark cluster is invariant under SL(4, R), and that generated by any two-quark cluster is invariant under SL(3, R). ˇ cki operates in spacetime while the present Notice that SL(4, R) of Ne’eman and Sijaˇ SL(4, R) acts in the color space. However, we attempt to interweave two arenas by reference to that color degrees of freedom may be convertible into spin degrees of freedom, the fact discovered by Jackiw and Rebbi, and Hasenfratz and ’t Hooft [12]. The paper is organized as follows. Section II outlines the general formalism of the YMW theory. The next section is devoted to a justification of the Ansatz whereby we seek exact retarded solutions of the YM equations with the source composed of several arbitrarily moving quarks. Finding such solutions is traced by the simplest example of the single-quark source, Sec. IV. Properties of the background generated by two-quark sources 3
Two-dimensional Quantum Field Theory, examples and applications
Abstract The main principles of two-dimensional quantum field theories, in particular two-dimensional QCD and gravity are reviewed. We study non-perturbative aspects of these theories which make them particularly valuable for testing ideas of four-dimensional quantum field theory. The dynamics of confinement and theta vacuum are explained by using the non-perturbative methods developed in two dimensions. We describe in detail how the effective action of string theory in non-critical dimensions can be represented by Liouville gravity. By comparing the helicity amplitudes in four-dimensional QCD to those of integrable self-dual Yang-Mills theory, we extract a four dimensional version of two dimensional integrability.
2 48 49 52 54 56
5 Four-dimensional analogies and consequences 6 Conclusions and Final Remarks
Lattice Gauge Fields and Discrete Noncommutative Yang-Mills Theory
Contents
1 Introduction and summary 2 Quantum field theory on noncommutative spaces 2.1 2.2 2.3 2.4 Sc . . . . . . . . . . . . . . . . . . . . . . . . . . Noncommutative Yang-Mills theory . . . . . . . . . . . . . . . . . . . . . . . Star-gauge invariant observables . . . . . . . . . . . . . . . . . . . . . . . . . The noncommutative torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6 6 9 10 13 14 15 17 23 26 27 31 33 34 38 40 41 44
J. Ambjørn1) ∗ , Y.M. Makeenko1) 2) † , J. Nishimura1) ‡ and R.J. Szabo1) §
1)
The Niels Bohr Institute Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
2)
Institute of Theoretical and Experimental Physics B. Cheremushkinskaya 25, 117218 Moscow, Russia
2
geometry provides a natural framework to describe nonperturbative aspects of string theory [2, 5]. This belief is further supported by the fact that Matrix Theory [6] and the IIB matrix model [7], which are conjectured to provide nonperturbative definitions of string theories, give rise to noncommutative Yang-Mills theory on toroidal compactifications [8]. The particular noncommutative toroidal compactification is interpreted as being the result of the presence of a background Neveu-Schwarz two-form field, and it can also be understood in the context of open string quantization in D-brane backgrounds [9, 10]. Furthermore, in Ref. [11] it has been shown that the IIB matrix model with D-brane backgrounds is described by noncommutative Yang-Mills theory. The early motivation [12] for studying quantum field theory on noncommutative spacetimes was that, because of the spacetime uncertainty relation, the introduction of noncommutativity would provide a natural ultraviolet regularization. However, more recent perturbative calculations [13]–[16] have shown that planar noncommutative Feynman diagrams contain exactly the same ultraviolet divergences that their commutative counterparts do, which implies that the noncommutativity does not serve as an ultraviolet regulator. One therefore needs to introduce some other form of regularization to study the dynamics of noncommutative field theories. On the other hand, it has been found that the ultraviolet divergences in non-planar Feynman diagrams [16, 17] exhibit an intriguing mixing of ultraviolet and infrared scales, which can also be described using string-theoretical approaches [18, 19]. Heuristically, this UV/IR mixing can be understood in terms of the induced uncertainty relations among the spacetime coordinates. If one measures a given spacetime coordinate with some high precision, then the remaining spacetime directions will generally extend because of the smearing. Furthermore, noncommutative solitons which do not have counterparts in ordinary field theory have been discovered [20] for sufficiently large values of the noncommutativity parameters, and it has also been shown [19] that noncommutative Yang-Mills theory in four dimensions naturally includes gravity. In order to investigate further the non-trivial dynamics of noncommutative field theories, it is important therefore to develop a nonperturbative regularization of these theories. Such a program has been put forward in Refs. [11, 15, 19],[21]–[24] and it is similar to earlier works [25] based on the mapping between large N matrices and spacetime fields. In particular, in Ref. [22] a unified framework was presented which naturally interpolates between the two ways that noncommutative Yang-Mills theory has appeared in the context of matrix model formulations of string theory, namely the compactification of Matrix theory and the twisted large N reduced model. The model proposed was a finite N matrix model defined by the twisted Eguchi-Kawai model [26, 27] with a quotient condition analogous to the ones considered in Refs. [8, 28]. It was interpreted as a lattice formulation of noncommutative
Quantization of Field Theories Generalizing Gravity-Yang-Mills Systems on the Cylinder
Quantization of Field Theories Generalizing Gravity-Yang-Mills Systems
on the Cylinder
P. Schaller∗ and T. Strobl† Inst. f. Theor. Physik, Techn. Univ. Vienna, Wiedner Hauptstr. 8-10, A-1040 Wien, Austria
TUW-94-02, gr-qc/9406027
Abstract Pure gravity and gauge theories in two dimensions are shown to be special cases of a much more general class of field theories each of which is characterized by a Poisson structure on a finite dimensional target space. A general scheme for the quantization of these theories is formulated. Explicit examples are studied in some detail. In particular gravity and gauge theories with equivalent actions are compared. Big gauge transformations as well as the condition of metric nondegeneracy in gravity turn out to cause significant differences in the structure of the corresponding reduced phase spaces and the quantum spectra of Dirac observables. For R2 gravity coupled to SU(2) Yang Mills the question of quantum dynamics (‘problem of time’) is addressed. ∗email: schaller@email.tuwien.ac.at, †tstrobl@email.tuwien.ac.at
N=4 Super-Yang-Mills Theory, QCD and Collider Physics
a rXiv:h ep-th/04121v12O ct24SLAC–PUB–10739,IPPP/04/59,DCPT/04/118,UCLA/04/TEP/40Saclay/SPhT–T04/116,hep-th/0410021October,2004N =4Super-Yang-Mills Theory,QCD and Collider Physics Z.Bern a L.J.Dixon b 1 D.A.Kosower c a Department of Physics &Astronomy,UCLA,Los Angeles,CA 90095-1547,USA b SLAC,Stanford University,Stanford,CA 94309,USA,and IPPP,University of Durham,Durham DH13LE,England c Service de Physique Th´e orique,CEA–Saclay,F-91191Gif-sur-Yvette cedex,France1Introduction and Collider Physics MotivationMaximally supersymmetric (N =4)Yang-Mills theory (MSYM)is unique in many ways.Its properties are uniquely specified by the gauge group,say SU(N c ),and the value of the gauge coupling g .It is conformally invariant for any value of g .Although gravity is not present in its usual formulation,MSYMis connected to gravity and string theory through the AdS/CFT correspon-dence[1].Because this correspondence is a weak-strong coupling duality,it is difficult to verify quantitatively for general observables.On the other hand, such checks are possible and have been remarkably successful for quantities protected by supersymmetry such as BPS operators[2],or when an additional expansion parameter is available,such as the number offields in sequences of composite,large R-charge operators[3,4,5,6,7,8].It is interesting to study even more observables in perturbative MSYM,in order to see how the simplicity of the strong coupling limit is reflected in the structure of the weak coupling expansion.The strong coupling limit should be even simpler when the large-N c limit is taken simultaneously,as it corresponds to a weakly-coupled supergravity theory in a background with a large radius of curvature.There are different ways to study perturbative MSYM.One approach is via computation of the anomalous dimensions of composite,gauge invariant operators[1,3,4,5,6,7,8].Another possibility[9],discussed here,is to study the scattering amplitudes for(regulated)plane-wave elementaryfield excitations such as gluons and gluinos.One of the virtues of the latter approach is that perturbative MSYM scat-tering amplitudes share many qualitative properties with QCD amplitudes in the regime probed at high-energy colliders.Yet the results and the computa-tions(when organized in the right way)are typically significantly simpler.In this way,MSYM serves as a testing ground for many aspects of perturbative QCD.MSYM loop amplitudes can be considered as components of QCD loop amplitudes.Depending on one’s point of view,they can be considered either “the simplest pieces”(in terms of the rank of the loop momentum tensors in the numerator of the amplitude)[10,11],or“the most complicated pieces”in terms of the degree of transcendentality(see section6)of the special functions entering thefinal results[12].As discussed in section6,the latter interpreta-tion links recent three-loop anomalous dimension results in QCD[13]to those in the spin-chain approach to MSYM[5].The most direct experimental probes of short-distance physics are collider experiments at the energy frontier.For the next decade,that frontier is at hadron colliders—Run II of the Fermilab Tevatron now,followed by startup of the CERN Large Hadron Collider in2007.New physics at colliders always contends with Standard Model backgrounds.At hadron colliders,all physics processes—signals and backgrounds—are inherently QCD processes.Hence it is important to be able to predict them theoretically as precisely as possi-ble.The cross section for a“hard,”or short-distance-dominated processes,can be factorized[14]into a partonic cross section,which can be computed order by order in perturbative QCD,convoluted with nonperturbative but measur-able parton distribution functions(pdfs).For example,the cross section for producing a pair of jets(plus anything else)in a p¯p collision is given byσp¯p→jjX(s)= a,b1 0dx1dx2f a(x1;µF)¯f b(x2;µF)׈σab→jjX(sx1x2;µF,µR;αs(µR)),(1)where s is the squared center-of-mass energy,x1,2are the longitudinal(light-cone)fractions of the p,¯p momentum carried by partons a,b,which may be quarks,anti-quarks or gluons.The experimental definition of a jet is an in-volved one which need not concern us here.The pdf f a(x,µF)gives the prob-ability forfinding parton a with momentum fraction x inside the proton; similarly¯f b is the probability forfinding parton b in the antiproton.The pdfs depend logarithmically on the factorization scaleµF,or transverse resolution with which the proton is examined.The Mellin moments of f a(x,µF)are for-ward matrix elements of leading-twist operators in the proton,renormalized at the scaleµF.The quark distribution function q(x,µ),for example,obeys 10dx x j q(x,µ)= p|[¯qγ+∂j+q](µ)|p .2Ingredients for a NNLO CalculationMany hadron collider measurements can benefit from predictions that are accurate to next-to-next-to-leading order(NNLO)in QCD.Three separate ingredients enter such an NNLO computation;only the third depends on the process:(1)The experimental value of the QCD couplingαs(µR)must be determinedat one value of the renormalization scaleµR(for example m Z),and its evolution inµR computed using the3-loopβ-function,which has been known since1980[15].(2)The experimental values for the pdfs f a(x,µF)must be determined,ide-ally using predictions at the NNLO level,as are available for deep-inelastic scattering[16]and more recently Drell-Yan production[17].The evolu-tion of pdfs inµF to NNLO accuracy has very recently been completed, after a multi-year effort by Moch,Vermaseren and Vogt[13](previously, approximations to the NNLO kernel were available[18]).(3)The NNLO terms in the expansion of the partonic cross sections must becomputed for the hadronic process in question.For example,the parton cross sections for jet production has the expansion,ˆσab→jjX=α2s(A+αs B+α2s C+...).(2)The quantities A and B have been known for over a decade[19],but C has not yet been computed.Figure 1.LHC Z production [22].•real ×real:וvirtual ×real:וvirtual ×virtual:וdoubly-virtual ×real:×Figure 2.Purely gluonic contributionsto ˆσgg →jjX at NNLO.Indeed,the NNLO terms are unknown for all but a handful of collider puting a wide range of processes at NNLO is the goal of a large amount of recent effort in perturbative QCD [20].As an example of the im-proved precision that could result from this program,consider the production of a virtual photon,W or Z boson via the Drell-Yan process at the Tevatron or LHC.The total cross section for this process was first computed at NNLO in 1991[21].Last year,the rapidity distribution of the vector boson also be-came available at this order [17,22],as shown in fig.1.The rapidity is defined in terms of the energy E and longitudinal momentum p z of the vector boson in the center-of-mass frame,Y ≡1E −p z .It determines where the vector boson decays within the detector,or outside its acceptance.The rapidity is sensitive to the x values of the incoming partons.At leading order in QCD,x 1=e Y m V /√s ,where m V is the vector boson mass.The LHC will produce roughly 100million W s and 10million Z s per year in detectable (leptonic)decay modes.LHC experiments will be able to map out the curve in fig.1with exquisite precision,and use it to constrain the parton distributions —in the same detectors that are being used to search for new physics in other channels,often with similar q ¯q initial states.By taking ratios of the other processes to the “calibration”processes of single W and Z production,many experimental uncertainties,including those associated with the initial state parton distributions,drop out.Thus fig.1plays a role as a “partonic luminosity monitor”[23].To get the full benefit of the remarkable experimental precision,though,the theory uncertainty must approach the 1%level.As seen from the uncertainty bands in the figure,this precision is only achievable at NNLO.The bands are estimated by varying the arbitrary renormalization and factorization scales µR and µF (set to a common value µ)from m V /2to 2m V .A computation to all orders in αs would have no dependence on µ.Hence the µ-dependence of a fixed order computation is related to the size of the missing higher-order terms in the series.Althoughsub-1%uncertainties may be special to W and Z production at the LHC, similar qualitative improvements in precision will be achieved for many other processes,such as di-jet production,as the NNLO terms are completed.Even within the NNLO terms in the partonic cross section,there are several types of ingredients.This feature is illustrated infig.2for the purely gluonic contributions to di-jet production,ˆσgg→jjX.In thefigure,individual Feynman graphs stand for full amplitudes interfered(×)with other amplitudes,in order to produce contributions to a cross section.There may be2,3,or4partons in thefinal state.Just as in QED it is impossible to define an outgoing electron with no accompanying cloud of soft photons,also in QCD sensible observables require sums overfinal states with different numbers of partons.Jets,for example,are defined by a certain amount of energy into a certain conical region.At leading order,that energy typically comes from a single parton, but at NLO there may be two partons,and at NNLO three partons,within the jet cone.Each line infig.2results in a cross-section contribution containing severe infrared divergences,which are traditionally regulated by dimensional regula-tion with D=4−2ǫ.Note that this regulation breaks the classical conformal invariance of QCD,and the classical and quantum conformal invariance of N=4super-Yang-Mills theory.Each contribution contains poles inǫranging from1/ǫ4to1/ǫ.The poles in the real contributions come from regions ofphase-space where the emitted gluons are soft and/or collinear.The poles in the virtual contributions come from similar regions of virtual loop integra-tion.The virtual×real contribution obviously has a mixture of the two.The Kinoshita-Lee-Nauenberg theorem[24]guarantees that the poles all cancel in the sum,for properly-defined,short-distance observables,after renormal-izing the coupling constant and removing initial-state collinear singularities associated with renormalization of the pdfs.A critical ingredient in any NNLO prediction is the set of two-loop ampli-tudes,which enter the doubly-virtual×real interference infig.2.Such ampli-tudes require dimensionally-regulated all-massless two-loop integrals depend-ing on at least one dimensionless ratio,which were only computed beginning in 1999[25,26,27].They also receive contributions from many Feynman diagrams, with lots of gauge-dependent cancellations between them.It is of interest to develop more efficient,manifestly gauge-invariant methods for combining di-agrams,such as the unitarity or cut-based method successfully applied at one loop[10]and in the initial two-loop computations[28].i,ij+ i iFigure3.Illustration of soft-collinear(left)and pure-collinear(right)one-loop di-vergences.3N=4Super-Yang-Mills Theory as a Testing Ground for QCDN=4super-Yang-Mills theory serves an excellent testing ground for pertur-bative QCD methods.For n-gluon scattering at tree level,the two theories in fact give identical predictions.(The extra fermions and scalars of MSYM can only be produced in pairs;hence they only appear in an n-gluon ampli-tude at loop level.)Therefore any consequence of N=4supersymmetry,such as Ward identities among scattering amplitudes[29],automatically applies to tree-level gluonic scattering in QCD[30].Similarly,at tree level Witten’s topological string[31]produces MSYM,but implies twistor-space localization properties for QCD tree amplitudes.(Amplitudes with quarks can be related to supersymmetric amplitudes with gluinos using simple color manipulations.)3.1Pole Structure at One and Two LoopsAt the loop-level,MSYM becomes progressively more removed from QCD. However,it can still illuminate general properties of scattering amplitudes,in a calculationally simpler arena.Consider the infrared singularities of one-loop massless gauge theory amplitudes.In dimensional regularization,the leading singularity is1/ǫ2,arising from virtual gluons which are both soft and collinear with respect to a second gluon or another massless particle.It can be char-acterized by attaching a gluon to any pair of external legs of the tree-level amplitude,as in the left graph infig.3.Up to color factors,this leading diver-gence is the same for MSYM and QCD.There are also purely collinear terms associated with individual external lines,as shown in the right graph infig.3. The pure-collinear terms have a simpler form than the soft terms,because there is less tangling of color indices,but they do differ from theory to theory.The full result for one-loop divergences can be expressed as an operator I(1)(ǫ) which acts on the color indices of the tree amplitude[32].Treating the L-loop amplitude as a vector in color space,|A(L)n ,the one-loop result is|A(1)n =I(1)(ǫ)|A(0)n +|A(1),finn ,(3)where |A (1),fin nis finite as ǫ→0,and I (1)(ǫ)=1Γ(1−ǫ)n i =1n j =i T i ·T j 1T 2i 1−s ij ǫ,(4)where γis Euler’s constant and s ij =(k i +k j )2is a Mandelstam invariant.The color operator T i ·T j =T a i T a j and factor of (µ2R /(−s ij ))ǫarise from softgluons exchanged between legs i and j ,as in the left graph in fig.3.The pure 1/ǫpoles terms proportional to γi have been written in a symmetric fashion,which slightly obscures the fact that the color structure is actually simpler.We can use the equation which represents color conservation in the color-space notation, n j =1T j =0,to simplify the result.At order 1/ǫwe may neglect the (µ2R /(−s ij ))ǫfactor in the γi terms,and we have n j =i T i ·T j γi /T 2i =−γi .So the color structure of the pure 1/ǫterm is actually trivial.For an n -gluon amplitude,the factor γi is set equal to its value for gluons,which turns out to be γg =b 0,the one-loop coefficient in the β-function.Hence the pure-collinear contribution vanishes for MSYM,but not for QCD.The divergences of two-loop amplitudes can be described in the same for-malism [32].The relation to soft-collinear factorization has been made more transparent by Sterman and Tejeda-Yeomans,who also predicted the three-loop behavior [33].Decompose the two-loop amplitude |A (2)n as|A (2)n =I (2)(ǫ)|A (0)n +I (1)(ǫ)|A (1)n +|A (2),fin n,(5)where |A (2),fin n is finite as ǫ→0and I (2)(ǫ)=−1ǫ+e −ǫγΓ(1−2ǫ)ǫ+K I (1)(2ǫ)+e ǫγT 2i µ22C 2A ,(8)where C A =N c is the adjoint Casimir value.The quantity ˆH(2)has non-trivial,but purely subleading-in-N c ,color structure.It is associated with soft,rather than collinear,momenta [37,33],so it is theory-independent,up to color factors.An ansatz for it for general n has been presented recently [38].3.2Recycling Cuts in MSYMAn efficient way to compute loop amplitudes,particularly in theories with a great deal of supersymmetry,is to use unitarity and reconstruct the am-plitude from its cuts [10,38].For the four-gluon amplitude in MSYM,the two-loop structure,and much of the higher-loop structure,follows from a sim-ple property of the one-loop two-particle cut in this theory.For simplicity,we strip the color indices offof the four-point amplitude A (0)4,by decomposing it into color-ordered amplitudes A (0)4,whose coefficients are traces of SU(N c )generator matrices (Chan-Paton factors),A (0)4(k 1,a 1;k 2,a 2;k 3,a 3;k 4,a 4)=g 2 ρ∈S 4/Z 4Tr(T a ρ(1)T a ρ(2)T a ρ(3)T a ρ(4))×A (0)4(k ρ(1),k ρ(2),k ρ(3),k ρ(4)).(9)The two-particle cut can be written as a product of two four-point color-ordered amplitudes,summed over the pair of intermediate N =4states S,S ′crossing the cut,which evaluates toS,S ′∈N =4A (0)4(k 1,k 2,ℓS ,−ℓ′S ′)×A (0)4(ℓ′S ′,−ℓS ,k 3,k 4)=is 12s 23A (0)4(k 1,k 2,k 3,k 4)×1(ℓ−k 3)2,(10)where ℓ′=ℓ−k 1−k 2.This equation is also shown in fig.4.The scalar propagator factors in eq.(10)are depicted as solid vertical lines in the figure.The dashed line indicates the cut.Thus the cut reduces to the cut of a scalar box integral,defined byI D =4−2ǫ4≡ d 4−2ǫℓℓ2(ℓ−k 1)2(ℓ−k 1−k 2)2(ℓ+k 4)2.(11)One of the virtues of eq.(10)is that it is valid for arbitrary external states in the N =4multiplet,although only external gluons are shown in fig.4.Therefore it can be re-used at higher loop order,for example by attaching yet another tree to the left.N =41234=i s 12s 231234Figure 4.The one-loop two-particle cuts for the four-point amplitude in MSYM reduce to the tree amplitude multiplied by a cut scalar box integral (for any set of four external states).i 2s 12s121234+s 121234+perms Figure 5.The two-loop gg →gg amplitude in MSYM [11,39].The blob on theright represents the color-ordered tree amplitude A (0)4.(The quantity s 12s 23A (0)4transforms symmetrically under gluon interchange.)In the the brackets,black linesare kinematic 1/p 2propagators,with scalar (φ3)vertices.Green lines are color δab propagators,with structure constant (f abc )vertices.The permutation sum is over the three cyclic permutations of legs 2,3,4,and makes the amplitude Bose symmetric.At two loops,the simplicity of eq.(10)made it possible to compute the two-loop gg →gg scattering amplitude in that theory (in terms of specific loop integrals)in 1997[11],four years before the analogous computations in QCD [36,37].All of the loop momenta in the numerators of the Feynman di-agrams can be factored out,and only two independent loop integrals appear,the planar and nonplanar scalar double box integrals.The result can be writ-ten in an appealing diagrammatic form,fig.5,where the color algebra has the same form as the kinematics of the loop integrals [39].At higher loops,eq.(10)leads to a “rung rule”[11]for generating a class of (L +1)-loop contributions from L -loop contributions.The rule states that one can insert into a L -loop contribution a rung,i.e.a scalar propagator,transverse to two parallel lines carrying momentum ℓ1+ℓ2,along with a factor of i (ℓ1+ℓ2)2in the numerator,as shown in fiing this rule,one can construct recursively the external and loop-momentum-containing numerators factors associated with every φ3-type diagram that can be reduced to trees by a sequence of two-particle cuts,such as the diagram in fig.7a .Such diagrams can be termed “iterated 2-particle cut-constructible,”although a more compact notation might be ‘Mondrian’diagrams,given their resemblance to Mondrian’s paintings.Not all diagrams can be computed in this way.The diagram in fig.7b is not in the ‘Mondrian’class,so it cannot be determined from two-particle cuts.Instead,evaluation of the three-particle cuts shows that it appears with a non-vanishing coefficient in the subleading-color contributions to the three-loop MSYM amplitude.ℓ1ℓ2−→i (ℓ1+ℓ2)2ℓ1ℓ2Figure 6.The rung rule for MSYM.(a)(b)Figure 7.(a)Example of a ‘Mondrian’diagram which can be determined re-cursively from the rung rule.(b)Thefirst non-vanishing,non-Mondrian dia-grams appear at three loops in nonplanar,subleading-color contributions.4Iterative Relation in N =4Super-Yang-Mills TheoryAlthough the two-loop gg →gg amplitude in MSYM was expressed in terms of scalar integrals in 1997[11],and the integrals themselves were computed as a Laurent expansion about D =4in 1999[25,26],the expansion of the N =4amplitude was not inspected until last fall [9],considerably after similar investigations for QCD and N =1super-Yang-Mills theory [36,37].It was found to have a quite interesting “iterative”relation,when expressed in terms of the one-loop amplitude and its square.At leading color,the L -loop gg →gg amplitude has the same single-trace color decomposition as the tree amplitude,eq.(9).Let M (L )4be the ratio of this leading-color,color-ordered amplitude to the corresponding tree amplitude,omitting also several conventional factors,A (L ),N =4planar 4= 2e −ǫγg 2N c2 M (1)4(ǫ) 2+f (ǫ)M (1)4(2ǫ)−12(ζ2)2is replaced by approximately sixpages of formulas (!),including a plethora of polylogarithms,logarithms and=+f(ǫ)−12(ζ2)2+O(ǫ)f(ǫ)=−(ζ2+ǫζ3+ǫ2ζ4+...)Figure8.Schematic depiction of the iterative relation(13)between two-loop and one-loop MSYM amplitudes.polynomials in ratios of invariants s/t,s/u and t/u[37].The polylogarithm is defined byLi m(x)=∞i=1x i t Li m−1(t),Li1(x)=−ln(1−x).(14)It appears with degree m up to4at thefinite,orderǫ0,level;and up to degree4−i in the O(ǫ−i)terms.In the case of MSYM,identities relating these polylogarithms are needed to establish eq.(13).Although the O(ǫ0)term in eq.(13)is miraculously simple,as noted above the behavior of the pole terms is not a miracle.It is dictated in general terms by the cancellation of infrared divergences between virtual corrections and real emission[24].Roughly speaking,for this cancellation to take place,the virtual terms must resemble lower-loop amplitudes,and the real terms must resemble lower-point amplitudes,in the soft and collinear regions of loop or phase-space integration.At the level of thefinite terms,the iterative relation(13)can be understood in the Regge/BFKL limit where s≫t,because it then corresponds to expo-nentiation of large logarithms of s/t[40].For general values of s/t,however, there is no such argument.The relation is special to D=4,where the theory is conformally invariant. That is,the O(ǫ1)remainder terms cannot be simplified significantly.For ex-ample,the two-loop amplitude M(2)4(ǫ)contains at O(ǫ1)all three independent Li5functions,Li5(−s/u),Li5(−t/u)and Li5(−s/t),yet[M(1)4(ǫ)]2has only the first two of these[9].The relation is also special to the planar,leading-color limit.The subleading color-components of thefinite remainder|A(2),finn defined by eq.(5)show no significant simplification at all.For planar amplitudes in the D→4limit,however,there is evidence that an identical relation also holds for an arbitrary number n of external legs, at least for certain“maximally helicity-violating”(MHV)helicity amplitudes. This evidence comes from studying the limits of two-loop amplitudes as two of the n gluon momenta become collinear[9,38,41].(Indeed,it was by analyzing these limits that the relation for n=4wasfirst uncovered.)The collinear limits turn out to be consistent with the same eq.(13)with M4replaced by M n everywhere[9],i.e.M(2)n(ǫ)=12(ζ2)2+O(ǫ).(15)The collinear consistency does not constitute a proof of eq.(15),but in light of the remarkable properties of MSYM,it would be surprising if it were not true in the MHV case.Because the direct computation of two-loop amplitudes for n>4seems rather difficult,it would be quite interesting to try to examine the twistor-space properties of eq.(15),along the lines of refs.[31,42].(The right-hand-side of eq.(15)is not completely specified at order1/ǫandǫ0for n>4.The reason is that the orderǫandǫ2terms in M(1)n(ǫ),which contribute to thefirst term in eq.(15)at order1/ǫandǫ0,contain the D=6−2ǫpentagon integral[43],which is not known in closed form.On the other hand, the differential equations this integral satisfies may suffice to test the twistor-space behavior.Or one may examine just thefinite remainder M(L),finn definedvia eq.(5).)It may soon be possible to test whether an iterative relation for planar MSYM amplitudes extends to three loops.An ansatz for the three-loop planar gg→gg amplitude,shown infig.9,was provided at the same time as the two-loop re-sult,in1997[11].The ansatz is based on the“rung-rule”evaluation of the iterated2-particle cuts,plus the3-particle cuts with intermediate states in D=4;the4-particle cuts have not yet been verified.Two integrals,each be-ginning at O(ǫ−6),are required to evaluate the ansatz in a Laurent expansion about D=4.(The other two integrals are related by s↔t.)The triple ladder integral on the top line offig.9was evaluated last year by Smirnov,all the way through O(ǫ0)[44].Evaluation of the remaining integral,which contains a factor of(ℓ+k4)2in the numerator,is in progress[45];all the terms through O(ǫ−2)agree with predictions[33],up to a couple of minor corrections.5Significance of Iterative Behavior?It is not yet entirely clear why the two-loop four-point amplitude,and prob-ably also the n-point amplitudes,have the iterative structure(15).However, one can speculate that it is from the need for the perturbative series to=i3s12s212+s223+2s12(ℓ+k4)+2s23(ℓ+k1)21Figure9.Graphical representation of the three-loop amplitude for MSYM in the planar limit.be summable into something which becomes“simple”in the planar strong-coupling limit,since that corresponds,via AdS/CFT,to a weakly-coupled supergravity theory.The fact that the relation is special to the conformal limit D→4,and to the planar limit,backs up this speculation.Obviously it would be nice to have some more information at three loops.There have been other hints of an iterative structure in the four-point correlation func-tions of chiral primary(BPS)composite operators[46],but here also the exact structure is not yet clear.Integrability has played a key role in recent higher-loop computations of non-BPS spin-chain anomalous dimensions[4,5,6,8].By imposing regularity of the BMN‘continuum’limit[3],a piece of the anoma-lous dimension matrix has even been summed to all orders in g2N c in terms of hypergeometric functions[7].The quantities we considered here—gauge-invariant,but dimensionally regularized,scattering amplitudes of color non-singlet states—are quite different from the composite color-singlet operators usually treated.Yet there should be some underlying connection between the different perturbative series.6Aside:Anomalous Dimensions in QCD and MSYMAs mentioned previously,the set of anomalous dimensions for leading-twist operators was recently computed at NNLO in QCD,as the culmination of a multi-year effort[13]which is central to performing precise computations of hadron collider cross sections.Shortly after the Moch,Vermaseren and Vogt computation,the anomalous dimensions in MSYM were extracted from this result by Kotikov,Lipatov,Onishchenko and Velizhanin[12].(The MSYM anomalous dimensions are universal;supersymmetry implies that there is only one independent one for each Mellin moment j.)This extraction was non-trivial,because MSYM contains scalars,interacting through both gauge and Yukawa interactions,whereas QCD does not.However,Kotikov et al.noticed, from comparing NLO computations in both leading-twist anomalous dimen-sions and BFKL evolution,that the“most complicated terms”in the QCDcomputation always coincide with the MSYM result,once the gauge group representation of the fermions is shifted from the fundamental to the adjoint representation.One can define the“most complicated terms”in the x-space representation of the anomalous dimensions—i.e.the splitting kernels—as follows:Assign a logarithm or factor ofπa transcendentality of1,and a polylogarithm Li m or factor ofζm=Li m(1)a transcendentality of m.Then the most complicated terms are those with leading transcendentality.For the NNLO anomalous dimensions,this turns out to be transcendentality4.(This rule for extracting the MSYM terms from QCD has also been found to hold directly at NNLO,for the doubly-virtual contributions[38].)Strikingly,the NNLO MSYM anomalous dimension obtained for j=4by this procedure agrees with a previous result derived by assuming an integrable structure for the planar three-loop contribution to the dilatation operator[5].7Conclusions and OutlookN=4super-Yang-Mills theory is an excellent testing ground for techniques for computing,and understanding the structure of,QCD scattering amplitudes which are needed for precise theoretical predictions at high-energy colliders. One can even learn something about the structure of N=4super-Yang-Mills theory in the process,although clearly there is much more to be understood. Some open questions include:Is there any AdS/CFT“dictionary”for color non-singlet states,like plane-wave gluons?Can one recover composite operator correlation functions from any limits of multi-point scattering amplitudes?Is there a better way to infrared regulate N=4supersymmetric scattering amplitudes,that might be more convenient for approaching the AdS/CFT correspondence,such as compactification on a three-sphere,use of twistor-space,or use of coherent external states?Further investigations of this arena will surely be fruitful.AcknowledgementsWe are grateful to the organizers of Strings04for putting together such a stim-ulating meeting.This research was supported by the US Department of En-ergy under contracts DE-FG03-91ER40662(Z.B.)and DE-AC02-76SF00515 (L.J.D.),and by the Direction des Sciences de la Mati`e re of the Commissariat `a l’Energie Atomique of France(D.A.K.).。
科技翻译(supplements) 2012
科技英语翻译补充材料2012年11月Part OneFull Translation☎Section One: Chinese-English TranslationText 1四川简介四川自古以来被称为“天府之国”，其土地肥沃，气候温和，幅员辽阔，风光秀丽，历史悠久。
四川简称“川”或“蜀”，位于中国内陆腹地。
从地势上看，它西为青藏高原扼控，东为长江三峡之险，北为秦岭巴山屏障，南为云贵高原拱卫，战略地位十分重要。
全省东西绵延1，200多公里，南北最宽为900多公里，面积达57万平方公里，约占中国面积的百分之六，仅次于新疆、西藏、青海和内蒙。
四川四面环山，中间是一盆地，滔滔的长江在四川境内由西向东，横贯而过。
其地形为西高东低。
西部为川西高原，海拔7556米的贡嘎山是横亘高原的第一峰。
东部为盆地，盆地西部是富饶美丽的成都平原。
全省气候东西各异。
西部除河谷地带外，属大陆性高寒气候，气温低，日照长，长冬无夏。
东部属亚热带湿润东南风气候，冬暖，春早，夏长，最冷的一月份为5c—8c，最热的七月份为25c---29c，全年无霜期可达300 天以上。
深受中国和世界人民喜爱的珍奇动物---大熊猫，就生长在盆地西北部的卧龙自然保护区和王郎自然保护区。
Text 2新都简介新都区位于成都市北大门，面积480平方公里，人口62万，是成都市科技文化旅游卫星城，在成都市总体规划中被确定为都市区。
被定位为成都市城北副中心，承担着成都市机械电子食品制药旅游和物流基地的功能。
新都镇、新繁镇均为四川省历史文化名城。
根据四川省优先发展“成都一个特大城市，绵阳一个大城市，德阳等10个中等城市”的平原都市群发展战略，新都区处于中国西部最具活力的成都平原经济圈的核心地带，是四川省委、省政府确定的成（都）－德（阳）－绵（阳）高新技术产业带的桥头堡。
2002年全区国内生产总值90.2亿元，全口径财政收入5.9亿元。
2003年预计国内生产总值可达106亿元，全口径财政收入可达8亿元。
yang modules公式
yang modules公式YANG -NETCONF 数据建模语言YANG 是一种被用来为NETCONF 协议建模的语言。
一个YANG module 定义了具有垂直结构的数据，这些数据可以被用做基于NETCONF 的operations ，比如configuration，state date，RPCs，以及notifications。
它使得NETCONF 的client 和server 之间能有完整的数据描述。
YANG 建模得到的数据具备树形结构。
其中每一个节点都有一个名字，都有一个值或者一些子节点。
YANG 为这些节点，以及节点之间的交互提供明确清晰的描述。
YANG 使用modules 和submodule 进行数据建模。
一个module 能够从其他外部的modules 中导入数据，也可以从submodules 中包含数据。
YANG 定义的垂直结构可以扩展，使得一个module 能够增加数据节点给另一个module。
这种扩展是有条件的，只有特定条件满足，新的数据节点才会出现。
YANG 模型还能描述数据之上增加的约束，基于垂直结构中其他节点的出现与否，值为多少等等来限制一些节点的出现与赋值。
这些约束可以被client 或者server 强制执行。
不过如果要使得内容有效，MUST 双方都要遵守特定的约定。
YANG 定义了一系列的内建数据类型，也有定义新数据类型的类型命名机制。
派生数据类型可以通过像range，pattern 这样的声明限制其原生数据类型的取值范围，这样的声明在client 或者server 端都可以执行。
它们还能为派生数据类型定义常用用法，比如定义一个string-based 类型，包含主机名。
YANG 允许对可重用的grouping 中节点的定义。
这些groupings 中的实例能够通过特定的增强以满足特殊的需求。
派生类型以及groupings 能够定义在一个module 或者submodule 中，能够被本地，其他module/submodule 导入和使用。
Tech-Savvy millennials: Shaping the Digital Future
Tech-Savvy millennials: Shaping the DigitalFutureTech-Savvy Millennials: Shaping the Digital Future。
As we bid farewell to another year, it's crucial to reflect on the significant strides and transformations witnessed in the digital landscape, particularly concerning the role of tech-savvy millennials. In an era dominated by rapid technological advancements and digital innovation, millennials have emerged as the driving force behind shaping the digital future. With their inherent understanding of technology and digital fluency, this generation has not only adapted to the evolving digital landscape but has also played a pivotal role in influencing its trajectory.The Rise of Digital Natives。
Millennials, born between the early 1980s and the mid-1990s, are often referred to as digital natives due totheir upbringing amidst the burgeoning digital revolution. Unlike previous generations, millennials grew up in an era where computers, the internet, and smartphones were becoming increasingly prevalent. Consequently, they developed an innate familiarity with technology from ayoung age, seamlessly integrating it into their daily lives.Driving Innovation and Disruption。
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a r X i v :h e p -t h /9911184v 1 23 N o v 1999Yang-Mills instantons on 7-dimensional manifoldof G 2holonomy∗Sayuri MiyagiDepartment of Physics,Osaka City University ,Sumiyoshiku,Osaka 558-8585,JapanAbstractWe investigate Yang-Mills instantons on a 7-dimensional manifold of G 2holon-omy.By proposing a spherically symmetric ansatz for the Yang-Mills connection,we have ordinary differential equations as the reduced instanton equation,and give some explicit and numerical solutions.1IntroductionRecent progress in string duality has stimulated a great deal of interest in Yang-Mills in-stantons on higher dimensional manifolds of special holonomy.1,2The higher dimensional Yang-Mills instanton equations on such manifolds have been given in several papers.3−6It is known that some Yang-Mills instantons on the manifolds can be extended to solitonic solutions of the low energy effective theory of the heterotic string 7,8preserving 116of the 10-dimensional N =1supersymmetry.The manifolds of special holonomy admit covariantly constant spinors and define the supersymmetric cycles (calibrated submanifolds).6Solitons and instantons of superstring theory or M-theory can be considered to be wrapping branes around a supersymmetric cycle.9,10,11The manifolds also play an important role in generalizations of the Donaldson-Floer-Witten theory to higher dimensions.12,13Baulieu et al 12found a topological action on a 8-dimensional manifold of Spin (7)holonomy;their covariant gauge fixing conditions lead to Yang-Mills instanton equations on the pared with the 4-dimensional case,there are a few explicit Yang-Mills instantons on 7-and 8-dimensional manifolds of G 2and Spin (7)holonomy.7,8,14Recently Yang-Mills instantons on a 8-dimensional manifold of Spin (7)holonomy have been examined in detail.15In this paper we investigate Yang-Mills instantons on a 7-dimensional manifold of G 2holonomy.Our paper is organized as follows:in section 2and section 3we briefly review the octonionic algebra and the explicit metric of a manifold of G 2holonomy given byGibbons et al;16in section4we express Yang-Mills instantons on7-dimensional manifolds of G2holonomy;in section5,under a spherically symmetric ansatz,we solve the Yang-Mills instanton equations on the manifold described in section3,and present some new explicit solutions obtained from the Riccati equation and numerical solutions.2Octonionic AlgebraIn this section we briefly review the octonionic algebra following the papers.8,14,17The exceptional Lie group G2which is a subgroup of Spin(7)is the automorphism group of the octonionic algebra.The generators of Spin(7)are given byΓab=13!ǫabcd ijk C ijk,(2.5) the following relations are satisfiedC abpq C cdpq=4(δa cδb d−δa dδb c)−2C ab cd,(2.6)C apq C bcpq=−4C a bc.(2.7) By the use of the tensor C abcd,one can decompose the21-dimensional Lie algebra of Spin(7)into the direct sum of the14-dimensional Lie algebra of G2and the7-dimensional orthogonal subspace P.Indeed,the projector P1ab cd onto the Lie algebra of G2and the projector P2ab cd onto P are given byP1ab cd=22(δa cδb d−δa dδb c)+13 12C ab cd.(2.9)Thus the generators G mn of G2can be realized as8G ab=32(Γab+13Explicit metric on 7-dimensional manifold of G 2holonomyIn this section we describe the explicit metric on a 7-dimensional manifold of G 2holonomy.It hasbeen firstgiven asa metric on the R 4bundle over S 3by solving the Einstein equation:16ds 2=α2dr 2+β2(σi −A i )2+γ2Σi Σi ,(3.1)where i =1,2,3and α,β,γare functions solely of the radial coordinate r .The symbols Σi denote the left invariant 1-forms on the S 3base manifold and the symbols σi denote the left invariant 1-forms on the fiber R 4.They satisfyd Σi =−12ǫijk σj ∧σk .(3.2)Then SU (2)connection A i is given by A i =1r3 −1,β2=1r3,γ2=r 22C ab cd ωcd .(3.4)Under the spherically symmetric ansatz (3.1),the orthonormal basis ise 7=αdr,e i =γΣi ,e ˆi =β(σi −A i )(3.5)where ˆi =ˆ1,ˆ2,ˆ3=4,5,6,and the spin connection ωab is obtained by the equations de a =ωa b ∧e b and ωab =−ωba :ω7i =−γ′αβe ˆi ,(3.7)ωi ˆj =β2γe k +β2γe k +18γ2−α4γ,(3.11)which reproduce the solution (3.3).4Yang-Mills instanton on7-dimensional manifold of G2holonomyHere we define a Yang-Mills instanton equation on the7-dimensional manifold of G2 holonomy in terms of a special closed3-formΩ,3,6Ω∧F=∗F.(4.1) Using the antisymmetric tensor C abc,the3-formΩcan be written asΩ=12C ab cd F cd.(4.3) The seven components of Eq.(4.3)are as follows:F71=F26+F53,F72=F61+F34,F73=F42+F15,F74=F23+F65,F75=F46+F31,F76=F12+F54,(4.4)F63=F25+F14.The curvatures(4.3)provide the solutions of the Yang-Mills equation on the manifold, D a F ab=0,as a consequence of the Bianchi identity.We shall call solutions of(4.3) Yang-Mills instantons.The Yang-Mills instantons can be embedded into supersymmetric theories as solutions to the equation of motion.It is known that there is a spinorηon the manifold of G2holonomy which satisfies the conditions:∇η=0,G abη=0,(4.5) where∇denotes the covariant derivative of the Levi-Civita connection on the manifold.3,19 Then,one can use the covariantly constant spinor as a global supersymmetric parameter defined on the manifold.The supersymmetric transformation of a spinorfieldχ,a super partner of Yang-Mills connection,δχ=12(P1ab cd+P2ab cd)F abΓcdη=26(F ab−13G abωab.(4.7)Using(3.4)and the symmetry between thefirst and second pair of indices of the Riemann curvature R abcd,we can see that the curvature of A satisfies the Yang-Mills instanton equation(4.3).75Ansatz for G 2Yang-Mills connectionNow we construct Yang-Mills instantons on the 7-dimensional manifold of G 2holonomy described in section 3.Referring to the spin connection (3.6)-(3.10),we propose a spher-ically symmetric ansatz for the G 2Yang-Mills connection A =12C ij kl A kl(5.6)and so does their curvature,i.e.,they are just projected onto the Lie algebra of G 2.Note that the indices denote those of the Lie algebra of G 2,not of the differential forms.This connection reproduces Eq.(4.7)by puttingA =β8γ2,C =−12γ.(5.7)The components of the curvature F (=1α(A ′+γ′γ+2D )e j ∧e k −ǫijk A (3B −C )e j ∧e ˆk ,(5.8)F 7ˆi =1β(B −C ))e 7∧e ˆi −ǫijk (B −C )(12γ+D )e ˆj∧e k+ǫijkβα(A ′+γ′γ+2D )e i ∧e j −A (3B −2C )e i ∧e ˆj−ACe ˆi ∧e j +δij A (B −C )e l ∧e ˆl ,(5.10)F ij=ǫijk 1γD )e 7∧e k +ǫijk 1βB )e 7∧e ˆk−(D 4γ2B +5A 2+D 2)e i ∧e j −B (12γ+D )e ˆi ∧e j +B (1F ˆi ˆj =ǫijk1γD )e 7∧e k +ǫijk1βC )e 7∧e ˆk−(D4γ2C +A 2+D 2)e i ∧e j−(C2γ+D )e ˆi ∧e j +C (12F abij e i ∧e j :F abij =12γ+D=0,(5.14)D ′+γ′2γ+D ),D ′+γ′2γ+D,(5.15)A ′+γ′γ+2D )=0,(5.16)(B −C )′+β′4γ2(B −C )−4A 2+2α(B −C ) 1α B ′+β′γ−βBβ−B 2,(5.18)−1βC =D 4γ2+A 2+D 2−C2γ,B =C ,(ii)D =−12γ,B =C,so we study these cases separately.In case (i),we have two solutions:(A,B,C,D )=β8γ2,−12γ , −β8γ2,−12γ.(5.20)The first one is nothing but the connection(4.7),while the second one is a new solution.Although their difference is only the signature of A ,these two solutions lead to different curvatures.In cases (ii)and (iii),A is zero from (5.17)so that the Yang-Mills instanton equation (4.3)gives two nonlinear ordinary differential equations for B and D :B ′=α−1+9α23αr −2√r B −2BD,(5.22)where we have used Eqs.(3.11)and(3.3).Let us introducenewvariables X and Y,B=X+13a).Then Eqs.(5.21)and(5.22)becomedX2(X+1)+3α−αY2+α(X+1)2,(5.24)dY2γ)and Eq.(5.24)becomes theRiccati equation:dX2(X+1)+3α+α(X+1)2.(5.26) We canfind a general solution of Eq.(5.26):X=3α−1+12+c3G ab F ab=13T ij F ij,(5.28)where T ij are SU(2)generators;the gauge group reduces to SU(2)from G2.The explicit components of the curvature becomeFij7ˆk =Fˆiˆj7ˆk=ǫijk1r)32+c(−1r2)2α2 ,(5.29)F ijij=Fˆiˆjij =−1(−1r2)α2,(5.30)Fijˆiˆj =Fˆiˆjˆiˆj=−1−1r2+12+c9vol SU(2) ∞a d3α6(−1r2)−3+32+c27vol SU(2),(5.32)where T r refers to the adjoint representation of SU (2)and vol SU (2)is the volume of SU (2).Note that the value comes from the infinite surface according to 13),M =(−1,−√ds=−∂W αds =−∂W α4(X +1)2+α(3−Y 2)X +α∂X i ∂X j(X 1=X,X 2=Y )(5.35)at the four fixed points,we can see that O is a stable point and L,M,N are saddle points (the Morse index is 1).This analysis provides a qualitative explanation for the behavior of the flows in Figure 1.AcknowledgmentsI would like to thank Y.Yasui,Y.Hashimoto,H.Kanno,T.Ootsuka and T.Takaai for helpful discussions.References[1]C.M.Hull,Adv.Theor.Math.Phys.2,619(1998)[2]L.Baulieu,H.Kanno and I.M.Singer,hep-th/9705127[3]E.Corrigan,C.Devchand and D.B.Fairlie,Nucl.Phys.B214,452(1983).[4]R.S.Ward,Nucl.Phys.B236,381(1984).[5]A.D.Popov,Mod.Phys.Lett.A232077(1992).[6]H.Kanno,hep-th/9903260.[7]B.S.Acharya and M.O’Loughlin,Phys.Rev.D55,4521(1997).[8]M.G¨u naydin and H.Nicolai,Phys.Lett.B351,169(1995).[9]K.Becker,M.Becker and A.Strominger,Nucl.Phys.B456,130(1995).[10]H.Ooguri,Y.Oz and Z.Yin,Nucl.Phys.B477,407(1996).[11]K.Becker,M.Becker,D.R.Morrison,H.Ooguri,Y.Oz and Z.Yin,Nucl.Phys.B480,225(1996).[12]L.Baulieu,H.Kanno and I.M.Singer,Comm.Math.Phys.194,149(1998)[13]B.S.Acharya and M.O’Loughlin and B.Spence,Nucl.Phys.B503,657(1997).[14]S.Fubini and H.Nicolai,Phys.Lett.B155,369(1985).[15]H.Kanno and Y.Yasui,hep-th/9910003.[16]G.W.Gibbons,D.N.Page and C.N.Pope,Commun.Math.Phys.127,529(1990).[17]R.D¨u ndarer,F.G¨u rsey and C.H.Tze,J.Math.Phys.25,1496(1984).[18]D.Brecher and M.J.Perry,hep-th/9908018.[19]I.Bakas,E.G.Foratos and A.Kehagias,Phys.Lett.B445,9(1998)。