Topology in SU(2) Yang-Mills theory
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。
维的准确数学定义及其意义(最新)(含“杨-米尔斯(Yang-Mills)存在性和质量缺口”的数学理论依据)
维的准确数学定义及其意义（最新）(含“杨－米尔斯(Yang-Mills)存在性和质量缺口”的数学理论依据）南京市秦淮区瓮家营汤家坝14号马双焕msh2003nj@摘要：该文对数学中维的准确定义和意义做了系统介绍。
关键词：维、多维数、素数、论证。
MR（2000）主题分类：00A05一、维的准确数学定义看到维，人们的第一个联想是几维空间。
上网检索，可以搜到：维是因素，是参数，是衡量比较的尺度。
对于维的进一步的认识是很重要的。
首先不难知道，维的学问就是关于如何衡量比较的学问。
从上面的因素、参数、尺度的概念可以看出这并不是维的根本数学定义，为什么呢？因为因素、参数、尺度有各种各样，有很多种，不同种因素、参数、尺度之间如何比较呢？仅以前述的介绍是不行的，所以维的根本数学定义是很重要的。
本人在探讨哥德巴赫猜想、庞加莱猜想的论证过程中得出了维的根本数学定义：在某前提条件下可无限任意，该前提条件就是某一维。
试想一下，无限还是有限，任意还是确定是可以衡量比较的。
什么意识呢，在某维上，如果有其他限制因素，也就是如果不是无限或是确定的，必定在该维上留下印记或标志，这就可以衡量比较了。
所以在某前提条件下可无限任意，该前提条件就是某一维。
这是维的数学定义。
而且这里的无限还是有限，任意还是确定可以是各种各样的，是跨越行业之别的。
二、维的合并与分解从前面的介绍可以知道，只要在某前提条件下可无限任意，该前提条件就是某一维。
这样维的种类数量很多很多，哪么各维之间的关系是什么样的呢？首先有些维之间是可以分解合并的。
例如学生这一维与大学生这一维就是这样的关系。
学生这一维，只要是学生，都在这一维内，无论这个学生是中学生，是大学生，是男，是女，是哪里的学生，．．．，都在这一维上。
大学生这一维仅限于大学生，显然学生这一维包含了大学生维、中学生维、男学生维、女学生维。
这就是维之间的分解合并关系。
三、不同类别维的关系我们知道，维是衡量比较的尺度，如果不同类别的维之间，不同行业单位之间不能通过维的概念进行衡量比较，那么这个维的概念是不到位的。
Generalized WDVV equations for B_r and C_r pure N=2 Super-Yang-Mills theory
a r X i v :h e p -t h /0102190v 1 27 F eb 2001Generalized WDVV equations for B r and C r pure N=2Super-Yang-Mills theoryL.K.Hoevenaars,R.MartiniAbstractA proof that the prepotential for pure N=2Super-Yang-Mills theory associated with Lie algebrasB r andC r satisfies the generalized WDVV (Witten-Dijkgraaf-Verlinde-Verlinde)system was given by Marshakov,Mironov and Morozov.Among other things,they use an associative algebra of holomorphic diffter Ito and Yang used a different approach to try to accomplish the same result,but they encountered objects of which it is unclear whether they form structure constants of an associative algebra.We show by explicit calculation that these objects are none other than the structure constants of the algebra of holomorphic differentials.1IntroductionIn 1994,Seiberg and Witten [1]solved the low energy behaviour of pure N=2Super-Yang-Mills theory by giving the solution of the prepotential F .The essential ingredients in their construction are a family of Riemann surfaces Σ,a meromorphic differential λSW on it and the definition of the prepotential in terms of period integrals of λSWa i =A iλSW ∂F∂a i ∂a j ∂a k .Moreover,it was shown that the full prepotential for simple Lie algebras of type A,B,C,D [8]andtype E [9]and F [10]satisfies this generalized WDVV system 1.The approach used by Ito and Yang in [9]differs from the other two,due to the type of associative algebra that is being used:they use the Landau-Ginzburg chiral ring while the others use an algebra of holomorphic differentials.For the A,D,E cases this difference in approach is negligible since the two different types of algebras are isomorphic.For the Lie algebras of B,C type this is not the case and this leads to some problems.The present article deals with these problems and shows that the proper algebra to use is the onesuggested in[8].A survey of these matters,as well as the results of the present paper can be found in the internal publication[11].This paper is outlined as follows:in thefirst section we will review Ito and Yang’s method for the A,D,E Lie algebras.In the second section their approach to B,C Lie algebras is discussed. Finally in section three we show that Ito and Yang’s construction naturally leads to the algebra of holomorphic differentials used in[8].2A review of the simply laced caseIn this section,we will describe the proof in[9]that the prepotential of4-dimensional pure N=2 SYM theory with Lie algebra of simply laced(ADE)type satisfies the generalized WDVV system. The Seiberg-Witten data[1],[12],[13]consists of:•a family of Riemann surfacesΣof genus g given byz+µz(2.2)and has the property that∂λSW∂a i is symmetric.This implies that F j can be thought of as agradient,which leads to the followingDefinition1The prepotential is a function F(a1,...,a r)such thatF j=∂FDefinition2Let f:C r→C,then the generalized WDVV system[4],[5]for f isf i K−1f j=f j K−1f i∀i,j∈{1,...,r}(2.5) where the f i are matrices with entries∂3f(a1,...,a r)(f i)jk=The rest of the proof deals with a discussion of the conditions1-3.It is well-known[14]that the right hand side of(2.1)equals the Landau-Ginzburg superpotential associated with the cor-∂W responding Lie ing this connection,we can define the primaryfieldsφi(u):=−∂x (2.10)Instead of using the u i as coordinates on the part of the moduli space we’re interested in,we want to use the a i .For the chiral ring this implies that in the new coordinates(−∂W∂a j)=∂u x∂a jC z xy (u )∂a k∂a k )mod(∂W∂x)(2.11)which again is an associative algebra,but with different structure constants C k ij (a )=C k ij(u ).This is the algebra we will use in the rest of the proof.For the relation(2.7)weturn to another aspect of Landau-Ginzburg theory:the Picard-Fuchs equations (see e.g [15]and references therein).These form a coupled set of first order partial differential equations which express how the integrals of holomorphic differentials over homology cycles of a Riemann surface in a family depend on the moduli.Definition 6Flat coordinates of the Landau-Ginzburg theory are a set of coordinates {t i }on mod-uli space such that∂2W∂x(2.12)where Q ij is given byφi (t )φj (t )=C kij (t )φk (t )+Q ij∂W∂t iΓ∂λsw∂t kΓ∂λsw∂a iΓ∂λsw∂a lΓ∂λsw∂t r(2.15)Taking Γ=B k we getF ijk =C lij (a )K kl(2.16)which is the intended relation (2.7).The only thing that is left to do,is to prove that K kl =∂a mIn conclusion,the most important ingredients in the proof are the chiral ring and the Picard-Fuchs equations.In the following sections we will show that in the case of B r ,C r Lie algebras,the Picard-Fuchs equations can still play an important role,but the chiral ring should be replaced by the algebra of holomorphic differentials considered by the authors of [8].These algebras are isomorphic to the chiral rings in the ADE cases,but not for Lie algebras B r ,C r .3Ito&Yang’s approach to B r and C rIn this section,we discuss the attempt made in[9]to generalizethe contentsof the previoussection to the Lie algebras B r,C r.We will discuss only B r since the situation for C r is completely analogous.The Riemann surfaces are given byz+µx(3.1)where W BC is the Landau-Ginzburg superpotential associated with the theory of type BC.From the superpotential we again construct the chiral ring inflat coordinates whereφi(t):=−∂W BC∂x (3.2)However,the fact that the right-hand side of(3.1)does not equal the superpotential is reflected by the Picard-Fuchs equations,which no longer relate the third order derivatives of F with the structure constants C k ij(a).Instead,they readF ijk=˜C l ij(a)K kl(3.3) where K kl=∂a m2r−1˜C knl(t).(3.4)The D l ij are defined byQ ij=xD l ijφl(3.5)and we switched from˜C k ij(a)to˜C k ij(t)in order to compare these with the structure constants C k ij(t). At this point,it is unknown2whether the˜C k ij(t)(and therefore the˜C k ij(a))are structure constants of an associative algebra.This issue will be resolved in the next section.4The identification of the structure constantsThe method of proof that is being used in[8]for the B r,C r case also involves an associative algebra. However,theirs is an algebra of holomorphic differentials which is isomorphic toφi(t)φj(t)=γk ij(t)φk(t)mod(x∂W BC2Except for rank3and4,for which explicit calculations of˜C kij(t)were made in[9]we will rewrite it in such a way that it becomes of the formφi(t)φj(t)=rk=1 C k ij(t)φk(t)+P ij[x∂x W BC−W BC](4.3)As afirst step,we use(3.4):φiφj= Ci·−→φ+D i·−→φx∂x W BC j= C i−D i·r n=12nt n2r−1 C n·−→φ+D i·−→φx∂x W BCj(4.4)The notation −→φstands for the vector with componentsφk and we used a matrix notation for thestructure constants.The proof becomes somewhat technical,so let usfirst give a general outline of it.The strategy will be to get rid of the second term of(4.4)by cancelling it with part of the third term,since we want an algebra in which thefirst term gives the structure constants.For this cancelling we’ll use equation(3.4)in combination with the following relation which expresses the fact that W BC is a graded functionx ∂W BC∂t n=2rW BC(4.5)Cancelling is possible at the expense of introducing yet another term which then has to be canceled etcetera.This recursive process does come to an end however,and by performing it we automatically calculate modulo x∂x W BC−W BC instead of x∂x W BC.We rewrite(4.4)by splitting up the third term and rewriting one part of it using(4.5):D i·−→φx∂x W BC j= −12r−1 D i·−→φx∂x W BC j= −D i2r−1·−→φx∂x W BC j(4.6) Now we use(4.2)to work out the productφkφn and the result is:φiφj= C i·−→φ−D i2r−1·r n=12nt n D n·−→φx∂x W BC j +2rD i2r−1·rn=12nt n −D n·r m=12mt m2r−1[x∂x W BC−W BC]j(4.8)Note that by cancelling the one term,we automatically calculate modulo x∂x W BC −W BC .The expression between brackets in the first line seems to spoil our achievement but it doesn’t:until now we rewrote−D i ·r n =12nt n 2r −1C m ·−→φ+D n ·−→φx∂x W BCj(4.10)This is a recursive process.If it stops at some point,then we get a multiplication structureφi φj =r k =1C k ij φk +P ij (x∂x W BC −W BC )(4.11)for some polynomial P ij and the theorem is proven.To see that the process indeed stops,we referto the lemma below.xby φk ,we have shown that D i is nilpotent sinceit is strictly upper triangular.Sincedeg (φk )=2r −2k(4.13)we find that indeed for j ≥k the degree of φk is bigger than the degree ofQ ij5Conclusions and outlookIn this letter we have shown that the unknown quantities ˜C k ijof[9]are none other than the structure constants of the algebra of holomorphic differentials introduced in [8].Therefore this is the algebra that should be used,and not the Landau-Ginzburg chiral ring.However,the connection with Landau-Ginzburg can still be very useful since the Picard-Fuchs equations may serve as an alternative to the residue formulas considered in [8].References[1]N.Seiberg and E.Witten,Nucl.Phys.B426,19(1994),hep-th/9407087.[2]E.Witten,Two-dimensional gravity and intersection theory on moduli space,in Surveysin differential geometry(Cambridge,MA,1990),pp.243–310,Lehigh Univ.,Bethlehem,PA, 1991.[3]R.Dijkgraaf,H.Verlinde,and E.Verlinde,Nucl.Phys.B352,59(1991).[4]G.Bonelli and M.Matone,Phys.Rev.Lett.77,4712(1996),hep-th/9605090.[5]A.Marshakov,A.Mironov,and A.Morozov,Phys.Lett.B389,43(1996),hep-th/9607109.[6]R.Martini and P.K.H.Gragert,J.Nonlinear Math.Phys.6,1(1999).[7]A.P.Veselov,Phys.Lett.A261,297(1999),hep-th/9902142.[8]A.Marshakov,A.Mironov,and A.Morozov,Int.J.Mod.Phys.A15,1157(2000),hep-th/9701123.[9]K.Ito and S.-K.Yang,Phys.Lett.B433,56(1998),hep-th/9803126.[10]L.K.Hoevenaars,P.H.M.Kersten,and R.Martini,(2000),hep-th/0012133.[11]L.K.Hoevenaars and R.Martini,(2000),int.publ.1529,www.math.utwente.nl/publications.[12]A.Gorsky,I.Krichever,A.Marshakov,A.Mironov,and A.Morozov,Phys.Lett.B355,466(1995),hep-th/9505035.[13]E.Martinec and N.Warner,Nucl.Phys.B459,97(1996),hep-th/9509161.[14]A.Klemm,W.Lerche,S.Yankielowicz,and S.Theisen,Phys.Lett.B344,169(1995),hep-th/9411048.[15]W.Lerche,D.J.Smit,and N.P.Warner,Nucl.Phys.B372,87(1992),hep-th/9108013.[16]K.Ito and S.-K.Yang,Phys.Lett.B415,45(1997),hep-th/9708017.。
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
李政道与杨振宁的故事(讲稿)
The story between Tsung Dao Lee and Chen Ning Yang Good afternoon, dear classmates! I ’m Peng Jun, from Schoolof Physics Science and Engineering. I’m very happy to be here today, and I ’ll share you with the story between Chen Ning Yang and Tsung Dao Lee.As we all know, Lee and Yang are the proud of Chinese nation. 1956, they won that year’s Nobel Prize for their work about “Law of non conservation of parity ”. It was the first time that Chinese people had this honor. Their close cooperation was once enviable. Unfortunately, 5 years after winning the prize, they went their separate ways. What had happened? Now let’s see it.The whole storyYang was born in Hefei, Anhui Province in 1922. In 1945, after his postgraduate life,he went to University of Chicago,to be famous physicist,Fermi’s doctor,and graduated in 1948. Lee was born in Shanghai in 1926. At 1945, he was recommended by his teacher,Wu da You to get the scholarship for further study at America.Then, at the age of 20, Lee became Fermi’s postgraduate. Yang and Lee first met in 1946. At that time, Yang was a genius of fame at UC, while Lee was a young guy who had just completed his 2-year course in Southwest Associated University in China.Thus, Lee always turned to Yang for help, varying from course selecting to some questions. From 1946 to 1949, they had very closed relationship, and they wrote a paper together.1949, Yang went to Princeton University, and Lee went to Wisconsin. At September, 1951, Lee came to the prestigious Institute for Advanced Study of Princeton. There he met Yang again. At the guidance of Kaufman, they came to Einstein’s office.The talk was long,and Lee was very excited about that.Just from then on, these two young fellows’coorperation and their extraordinary story took leave.From 1951 to 1953,they worked together in Princeton, and wrote some papers about statistical physics. During this period, they were a little unhappy about the “ranking event” . Then Lee wanted to keep a distance with Yang. Thus, when Lee got a position in Chicago in 1953, he accepted it, due partly to this thought. Yang stayed in Princeton,and made some academic visits to Japan and Seattle.During 1953to 1956, they didn’t cooperate, and were busy for their own things.Those two years, they both worked with fruitful results. 1954, Yang wrote a very important paper with Mills, which was called “Yang-Mills theory” . This theory opened up a road for the Unified Field Theory, it was named as important as Newton ’Law and Maxwell Equations.1953,after his arriving at UC,Lee gave a so-called “Lee model” . It was very simple but had huge influence to Field Theory and renormalization.Lee had been working for the so-called “θ -τ mystery” from 1955 to 1956. One day on April in 1956, Lee was talking with Steinberger about their experiment, all of a sudden,he sparked an idea,leading to the thought of parity nonconservation.He came to aware that we must assume that other particles—not only θ -τ —may also occur parity nonconservation.He conjectured,parity violation is likely to be a basicprinciple of weak force.One morning at May, 1956, Yang came to see Lee, and Lee introduced his Lee ’s latest works. Yang was strongly rejected at first. After listening carefully to Lee, however,Yang changed his mindand expressed a willingness to work together.The following one year is amazing, with big or small arguments. In order to discuss physical problems,they kept in touch with each other.Since the two places were not far away, they made an appointment every week.Finally, in 1956, they completed the great paper. In this paper, Lee and Yang proposed “Lee -Yang hypothesis” as they worked together . That was the so-called “Law of non conservation of parity”, which means that elementary particles parity in the weak interaction may not be conserved.Nov 31,1957, 31 year-old T.D.Lee and 35 year-old C.N. Yang, won that year's Nobel Prize in Physics. Before The Nobel Prize award, Yang suddenly requested a transfer order. Lee was surprised at that moment, this had never happened before in Nobel Prize’s history. He agreed, however, without being aware of all those misfortunes afterwards.In 1962, the relationship between them bankrupt because of the name order in the paper.The once good friends then went separate ways regrettably. 20 years later, Yang published his collected papers, which gave some inaccurate descriptions of past. Then in 1986, Lee published an article against Yang. In my opinion, Lee isn ’t a liar. After all, it was a really big tragedy for Chinese people.Thank you!。
微分几何第一节
在现代物理（从量子力学到量子场论与量子统计）中。物理 体系整体大范围性质愈演愈突出，局域性质也往往与整体性 质有关，受整体性质的约束，近年来。量子场论由定域局部 微扰向整体大范围非线性深入发展，使得微分几何这一数学 武器愈来愈重要。微分几何中联络、曲率、示性类等概念渗 透到现代理论物理的各方面，而不是只限于引力理论，电磁 规范理论、及其推广Yang-Mills 场论。现代物理学家需要 利用微分几何，尤其是大范围微分几何的成就来推动物理学 的发展，大范围微分几何中一些成就，例如Atiyah-Singer 指标定理，将场论的解析性质与场位型的拓扑性质联系起来， 在现代量子场论与量子统计中得到重要应用，各方面各种物 理模型，例如超对称场论模型也验证了Atiyah-Singer指标 定理的正确性。这些都说明现代物理学与现代微分几何学的 紧密关系。
微分几何
一、教学要求
48学时，共3学分，教学周2~13周，15周考 试。 成绩认定：平时成绩50%，期末试卷50%。 期末考试，闭卷，考核基本知识和基本技能 90%~95%，其他10%~5%。
考勤占总评成绩的3%即3分：签 名考勤表，要与班上学委或信息 员记载信息一致或者测验时，收 作业时，或者点名时候都可以作 为考勤记录； 作业等占总评成绩24%，即24分:: 合计10节,少1节扣2分. 作业：抄袭、字迹不清楚、题量 没有按规定去完成酌情扣分。对 于题目做的正确还是错误属于正 常现象不扣分。
微分形式（现称为黎曼测度）是加到流形上去的一个结 构，因此在同一流形上可以有众多的黎曼测度。 Riemann意识到这件事是非凡的重要，把诱导测度与外 加的黎曼测度两者区分开来，从而开创了黎曼几何，作 出了杰出的贡献。其后，Levi-Civita等人进一步丰富 了经典的黎曼几何。 二十世纪二、三十年代E.Cartan开创并发展了 外微分形式与活动标架法，建立起李群与微分几何之间 的联系，从而为微分几何的发展奠定了重要基础且开辟 了广阔的园地，影响极为深远。 从局部微分几何到黎曼几何、微分流形与纤维 从理论的发展过程可以看到，除了微分几何本身研究中 所产生的研究问题外，其他数学学科及物理学、力学等 也推动了微分几何的发展。我们特别在这里强调一下理 论物理与微分几何的相互影响，黎曼几何与广义相对论 的
Conformal Field Theory Techniques in Large N Yang-Mills Theory
arXiv:hep-th/9311130v1 22 Nov 1993
Michael R. Douglas Dept. of Physics and Astronomy Rutgers University
1
Introduction
The first part of this article is an introduction to what might be called “large N representation theory,” Lie group representation theory with the focus on the limit N → ∞ of SU (N ) and the other classical groups. This has many applications in physics and mathematics, and good mathematical introductions exist, which tie it to its applications in group theory, soliton theory, combinatorics, and so forth. (See [1], 5.4 for a treatment very much like the one here; see also [2]) Now although it might seem that this theory would be invaluable for studying the large N limit of models with SU (N ) symmetry, and some examples in the physics literature are in [3], it does not get as much use as one might expect. Whether this isБайду номын сангаасsimply because the language is taking time to standardize,
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.).。
Topological susceptibility for the SU(3) Yang--Mills theory
ar X i v:h e p -l a t /0409100v 1 16 S e p 2004Topological susceptibility for the SU(3)Yang–Mills theory Luigi Del Debbio a ,Leonardo Giusti b ∗and Claudio Pica c a CERN,Department of Physics,TH Division,CH-1211Geneva 23,Switzerland b Centre de Physique Th´e orique,Case 907,CNRS Luminy,F-13288Marseille Cedex 9,France c Dipartimento di Fisica dell’Universit`a di Pisa and INFN,Via Buonarroti 2,I-56127Pisa,Italy We present the results of a computation of the topological susceptibility in the SU(3)Yang–Mills theory performed by employing the expression of the topological charge density operator suggested by Neuberger’s fermions.In the continuum limit we find r 40χ=0.059(3),which corresponds to χ=(191±5MeV)4if F K is used to set the scale.Our result supports the Witten–Veneziano explanation for the large mass of the η′.1.Introduction The topological susceptibility in the Yang–Mills (YM)gauge theory can be formally defined in Euclidean space-time as χ= d 4x q (x )q (0) ,(1)where the topological charge density q (x )is q (x )=−12N f =χ,(3)where F πis the pion decay constant.The dis-covery of a fermion operator [5]that satisfies the Ginsparg–Wilson (GW)relation [6]triggered a breakthrough in the understanding of the topo-logical properties of the YM vacuum [7,8,4,9,10],and made it possible to give a precise and unam-biguous implementation of the WV formula [4].Indeed the naive lattice definition of the topo-logical susceptibility has a finite continuum limit,V (4)with Q ≡ x q (x )=n +−n −being the topo-logical charge,V the volume,and n +(n −)the number of zero modes of D with positive (nega-tive)chirality in a given ing new simulation algorithms [11],it is now possible to investigate the WV scenario from first principles for the first time.More precisely,the aim of the work presented here,and fully described in [12],is to achieve an accurate and reliable de-termination of χin the continuum limit,which in turn allows a verification of the WV mecha-nism for the η′mass.Several exploratory com-putations have already studied the susceptibility employing the GW definition of the topological charge [13,14,15,16,17,18,19]ttice computation The ensembles of gauge configurations are gen-erated with the standard Wilson action and peri-odic boundary conditions,using a combination of heat-bath and over-relaxation updates.More de-tails on the generation of the gauge configurations can be found in Refs.[18,19].Table 1shows the 1list of simulated lattices,where the bare coupling constantβ=6/g20,the linear size L/a in each direction and the number of independent config-urations are reported for each lattice.The topo-logical charge density is defined asq(x)=−¯aL/a N conf Q2 r40χ1213492.79(12)0.0543(24)F5.951212911.955(79)0.0551(24)G6.01235861.489(37)0.0596(18)H6.1169622.45(13)0.0599(33)J6.21817212.114(76)0.0591(24)3mensional product r40χ.The s=0.0and s=0.4data sets are represented by black circles andwhite squares respectively.The dashed lines rep-resent the results of the combinedfit describedin the text.Thefilled diamond at a=0is theextrapolated value in the continuum limit.earfit of the data.Thisfit gives a very goodvalue ofχ2dof when all sets are included,and isvery stable if some points at larger values of a2/r20are removed.In particular a combinedfit of allpoints with a2/r20<0.05gives c0=0.059(3)withχ2dof≈0.73.Since r0is not directly accessibleto experiments,we express our result in phys-ical units by using the lattice determination ofr0F K=0.4146(94)in the quenched theory[21]and we obtainχ=(191±5MeV)4,(6)which has to be compared with[2]F2π。
用英语介绍杨振宁的事迹作文
用英语介绍杨振宁的事迹作文Title: The Remarkable Journey of Dr. Chen Ning YangDr. Chen Ning Yang, a towering figure in the realm of theoretical physics, stands as a beacon of intellectual brilliance and scientific perseverance. Born on September 22, 1922, in Hefei, Anhui Province, China, Yang embarked on a journey that would not only revolutionize our understanding of fundamental particles and forces in nature but also forge bridges between East and West in the realm of scientific collaboration.From a young age, Yang demonstrated a profound curiosity and aptitude for mathematics and physics. His academic prowess led him to National Southwestern Associated University, where he excelled under the guidance of renowned professors. However, the tumultuous political landscape of China during the 1940s prompted him to seek further education abroad, ultimately landing at the University of Chicago, where he pursued his graduate studies under the tutelage of Enrico Fermi, a Nobel laureate himself.It was during this period that Yang's groundbreaking work began to take shape. In collaboration with Robert Mills, he proposed the Yang-Mills theory in 1954, a framework that laid the foundation forunderstanding the behavior of subatomic particles and the fundamental forces that govern them. Although the immediate impact of this theory was not fully realized at the time, it eventually became instrumental in the development of quantum chromodynamics, the theory that describes the strong force binding atomic nuclei together, earning its proponents the Nobel Prize in Physics decades later.However, Yang's most celebrated achievement came in 1956, when he and his student T.D. Lee overturned a long-held belief in physics: the conservation of parity in weak interactions. Their groundbreaking paper, "Question of Parity Conservation in Weak Interactions," challenged the notion that nature's laws were mirror-symmetric, demonstrating that certain subatomic processes do not behave identically under reflections. This discovery, later confirmed experimentally, revolutionized particle physics and earned Yang and Lee the Nobel Prize in Physics in 1957, making Yang the first Chinese-born scientist to receive this prestigious honor.Beyond his scientific accomplishments, Yang is also renowned for his contributions to education and international scientific cooperation. He has held prestigious positions at institutionsworldwide, including Princeton University, where he served as professor emeritus, fostering generations of talented physicists and promoting cross-cultural exchanges in science. His dedication to nurturing young minds and fostering a global scientific community is a testament to his commitment to advancing human knowledge. Moreover, Yang's personal life embodies the spirit of cultural harmony. Married to Du Zhidao, a former Chinese scholar and diplomat, their union symbolizes the blending of Eastern and Western cultures, reflecting Yang's belief in the power of dialogue and understanding across borders.In conclusion, Dr. Chen Ning Yang's life is a testament to the limitless potential of human ingenuity and the transformative power of scientific inquiry. His groundbreaking discoveries, unwavering commitment to education, and promotion of international cooperation have left an indelible mark on the annals of science. As a scientist, educator, and cultural bridge-builder, Yang's legacy continues to inspire future generations, reminding us all of the boundless possibilities that lie at the intersection of curiosity, perseverance, and global collaboration.。
杨振宁成就英文作文
杨振宁成就英文作文Yang Zhenning is a renowned Chinese physicist who has made significant contributions to the field of particle physics. He was born in Hefei, Anhui province in 1922 and went on to study at the National Southwestern Associated University in Kunming. 。
After completing his undergraduate studies, Yang moved to the United States to pursue a PhD in physics at the University of Chicago. It was during his time there that he became interested in the study of elementary particles. 。
In 1957, Yang and his colleague Tsung-Dao Lee proposed the theory of parity non-conservation, which states that the weak nuclear force does not obey the principle of parity, or mirror symmetry. This groundbreaking theory was later confirmed by experiments and earned Yang and Lee the Nobel Prize in Physics in 1957. 。
Yang has also made significant contributions to thestudy of Yang-Mills theory, which describes the behavior of elementary particles. He has received numerous awards and honors for his work, including the National Medal of Science in 1986 and the Albert Einstein Medal in 1987. 。
关于杨振宁的英语作文
关于杨振宁的英语作文Title: Yang Chen-Ning: A Pioneering Physicist and a Beacon of Scientific Progress In the annals of modern science, few names shine as brightly as that of Yang Chen-Ning, a Chinese-American physicist whose groundbreaking contributions to theoretical physics have revolutionized our understanding of the universe. Born on September 22, 1922, in Hefei, Anhui Province, China, Yang emerged as a giant in the field of particle physics, statistical mechanics, and condensed matter physics, earning him not only international acclaim but also numerous prestigious awards, including the Nobel Prize in Physics in 1957.Early Life and EducationYang's journey towards scientific excellence began at an early age. Inspired by his father, a mathematician and educator, he developed a keen interest in science and mathematics. After completing his undergraduate studies at Southwest Associated University, he pursued advanced degrees at the University of Chicago, where he encountered the likes of Enrico Fermi and Edward Teller, mentors who significantly shaped his intellectual pursuits. It was during this period that Yang's fascination with theoretical physics blossomed, leading him to embark on a lifelong quest for knowledge and discovery.Nobel Prize-Winning WorkYang's crowning achievement came in collaboration with Robert Mills, when they proposed the Yang-Mills theory in 1954. This groundbreaking theory laid the foundation for understanding the behavior of particles within the atomic nucleus and predicting the existence of non-Abelian gauge fields, which are essential for describing the fundamental forces of nature. Although the direct experimental evidence for the Yang-Mills theory was not immediately apparent, it eventually formed the cornerstone of quantum chromodynamics (QCD), the theory describing the strong nuclear force that binds protons and neutrons together within atomic nuclei. For this seminal work, Yang was awarded the Nobel Prize in Physics, jointly with Tsung-Dao Lee, in recognition of their groundbreaking discovery of parity non-conservation in weak interactions.Contributions Beyond PhysicsBeyond his remarkable accomplishments in physics, Yang has also been a vocal advocate for science education and international cooperation in scientific research. He has served on numerous advisory committees and has played a pivotal role in promoting scientific exchange between China and the West. Furthermore, his commitment to nurturing the next generation of scientists is evident in his mentorship of countless students and young researchers, many of whom have gone on to make their own significant contributions to the field.Legacy and InfluenceYang Chen-Ning's legacy extends far beyond his groundbreaking scientific discoveries. He stands as a symbol of intellectual curiosity, perseverance, anddedication to the pursuit of truth. His life's work serves as a testament to the power of human ingenuity and the limitless potential of scientific inquiry. Even in his later years, Yang continued to engage with the scientific community, offering insights and guidance to aspiring scientists around the world.In conclusion, Yang Chen-Ning is a towering figure in the annals of modern science, whose contributions have reshaped our understanding of the universe and inspired generations of scientists to follow in his footsteps. His legacy will endure as a beacon of scientific progress, guiding us towards new frontiers of knowledge and discovery.。
世界近代三大数学难题之一-哥德巴赫猜想（2）
世界近代三大数学难题之一----哥德巴赫猜想哥德巴赫是德国一位中学教师，也是一位著名的数学家，生于1690年，1725年当选为俄国彼得堡科学院院士。
1742年，哥德巴赫在教学中发现，每个不小于6的偶数都是两个素数(只能被和它本身整除的数)之和。
如6＝3＋3，12＝5＋7等等。
1742年6月，哥德巴赫写信将这个问题告诉给意大利大数学家欧拉，并请他帮助作出证明。
欧拉在6月30日给他的回信中说，他相信这个猜想是正确的，但他不能证明。
叙述如此简单的问题，连欧拉这样首屈一指的数学家都不能证明，这个猜想便引起了许多数学家的注意。
他们对一个个偶数开始进行验算，一直算到3.3亿，都表明猜想是正确的。
但是对于更大的数目，猜想也应是对的，然而不能作出证明。
欧拉一直到死也没有对此作出证明。
从此，这道著名的数学难题引起了世界上成千上万数学家的注意。
200年过去了，没有人证明它。
哥德巴赫猜想由此成为数学皇冠上一颗可望不可及的“明珠”。
到了20世纪20年代，才有人开始向它靠近。
1920年、挪威数学家布爵用一种古老的筛选法证明，得出了一个结论：每一个比大的偶数都可以表示为(99)。
这种缩小包围圈的办法很管用，科学家们于是从(9十9)开始，逐步减少每个数里所含质数因子的个数，直到最后使每个数里都是一个质数为止，这样就证明了“哥德巴赫”。
1924年，数学家拉德马哈尔证明了(7＋7)；1932年，数学家爱斯尔曼证明了(6＋6)；1938年，数学家布赫斯塔勃证明了(5十5)，1940年，他又证明了(4＋4)；1956年，数学家维诺格拉多夫证明了(3＋3)；1958年，我国数学家王元证明了(2十3)。
随后，我国年轻的数学家陈景润也投入到对哥德巴赫猜想的研究之中，经过10年的刻苦钻研，终于在前人研究的基础上取得重大的突破，率先证明了(l十2)。
数中自有黄金屋
Clay Mathematics Institute:
多謝各位老師和同學來臨
數學系電腦室設於邵逸夫夫人樓232室 歡迎老師和同學參觀
有興趣參與研討會的老師和同學， 請到邵逸夫夫人樓222B室
數學問題一般由以下原因而提 出的。
檢驗或發展數學理論─數學自身發展所需 要的 (內部問題) 實在的應用─其他學科發展所需要的 (現象世界所提出的「外在問題」)
外在問題 經驗) (經驗) 解決辦法
數學理論
內部問題
理論提升
新外在問題
(部份或全部 部份或全部) 部份或全部
歷史：
Hilbert 23 問題 發表於一九○○年八月八日，國際數學 家大會(International Congress of Mathematicians) 的演講影響了整個二十 世紀數學的發展。
Poincaré 猜想
背景：幾何形狀的分類
理論： 代數拓撲學(Algebraic Topology) 幾何物體→群 (Group) （基本群）
問題： 幾何物體←群？
猜想 如一個三維「緊致流形」的基本群只有 一個元素，則此流形「拓撲」等價於三 維球面。
S =
3
{(w, x, y, z ) : w
2
X +Y =1
3 3
有無窮多有理數解 沒有有趣的有理數解
問題：找通用的判別的方法 (Hilbert 第十問題) 答案：否 (Matiyaserich) 答案：
猜想：在特殊情況下 （「阿貝耳簇」上的有理點）
ζ (1) = 0 ⇒ 無窮多解 ζ (1) ≠ 0 ⇒ 有限多解
ζ = 相對應的Zeta函數
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a rX iv :h e p -la t /9709074v 1 20 S e p 19971Topology in SU(2)Yang-Mills theory ∗B.All´e s a †,M.D’Elia b ,A.Di Giacomo b and R.Kirchner baDipartimento di Fisica,Sezione Teorica,Universit`a degli Studi di Milano and INFN,Via Celoria 16,20133Milano,Italy.bDipartimento di Fisica,Universit`a di Pisa and INFN,Piazza Torricelli 2,56126Pisa,Italy.New results on the topology of the SU (2)Yang-Mills theory are presented.At zero temperature we obtain the value of the topological susceptibility by using the recently introduced smeared operators as well as a properly renormalized geometric definition.Both determinations are in agreement.At non-zero temperature we study the behaviour of the topological susceptibility across the confinement transition pointing out some qualitative differences with respect to the analogous result for the SU (3)gauge theory.1.INTRODUCTIONA relevant quantity to understand the breaking of the U A (1)symmetry in QCD is the topological susceptibility χin pure Yang-Mills theory [1,2]χ≡d 4x 0|T (Q (x )Q (0))|0 quenched ,(1)where Q (x )=g 22N fm 2η+m 2η′−2m 2K ≈(180MeV)4(3)where N f is the relevant number of flavours.Eq.(3)implies a well defined prescription [1,2]to deal with the x →0singularity in eq.(1).In [3]χwas evaluated at zero and finite temper-ature for SU (3)Yang-Mills theory.The value ob-tained at zero temperature was in agreement with the prediction of Eq.(3).The value of χat finite temperature displayed a sharp drop beyond the deconfinement transition.Here we give a short re-view of a similar calculation for the SU (2)gaugeMS Q (x )by a finite renormalization [6]Q L (x )=Z (β)Q (x )a 4+O (a 6)(5)where β≡2N c /g 2in the usual notation.Since χL does not obey in general the prescrip-tion [1,2]leading to eq.(3),besides the multi-plicative renormalization of eq.(5)there is also an additive renormalization χL =Z (β)2χa 4+M (β)+O (a 6)(6)where M (β)contains mixings with operators of dimension ≤4.In order to extract the physical signal χfrom eq.(6),we need a determination of the renormal-ization constants M and Z .We will determine them using the non–perturbative method of ref.[7,8,3].23.THE MONTE CARLO SIMULATIONWe have used Wilson action and the usual heat-bath updating algorithm.The scale a(β)was fixed by using the results of ref.[9,10].3.1.Zero TemperatureThe simulation was done on a164lattice.We have used various definitions for Q L.The i-smearedfield theoretical Q(i)L(x)is defined as−1Q(i)L(x)=32468x−2.0.2.4.6.8.10 G L (x)Figure 3.Correlation function for the geometrical charge at β=2.57.The line is to guide the eye.is mainly determined by the singularity at x =0.This correlator,for x lying along a coordinate axis,is shown in Figures 2for the smeared charges and in Figure 3for the geometrical charge.The peak at x =0for the geometric charge is 4−5orders of magnitude larger than for the i -smeared charges,indicating that M g is much bigger than M i ,i =0,1,2,and is ∼80%of the observed χL .3.2.Finite TemperatureThe simulation was done on a 323×8lat-tice.At this size thedeconfining transition is lo-cated at βc =2.5115(40)[10]which means that T c =1/(N t a (βc ))with N t the temporal size of the lattice.The data show again a drop at the transition.However this is less sharp than for the SU (3)case [3,4].In Figure 4we show the behaviours for SU (2)and SU (3)of the ratio χ(T )/χ(T =0),where χ(T )indicates the physical susceptibility at temperature T .The slope for the SU (3)data is steeper.In both cases the data at T <T c show a constant value consistent with the value at T =0.We thank Prof.Gerrit Schierholz for provid-ing us with the fortran code for the geometrical charge.0.51.01.52.0T/T c0.00.51.0χ(T )/χ(T =0)Figure 4.Ratio χ(T )/χ(T =0)for SU (3)and SU (2).REFERENCES1. E.Witten,Nucl.Phys.B156(1979)269.2.G.Veneziano,Nucl.Phys.B159(1979)213.3. B.All´e s,M.D’Elia,A.Di Giacomo,Nucl.Phys.B494(1997)281.4. B.All´e s,M.D’Elia, A.Di Giacomo,hep-lat/9706016,to appear in Phys.Lett.B.5. B.All´e s,M.D’Elia,A.Di Giacomo,R.Kirch-ner,in preparation.6.M.Campostrini,A.Di Giacomo,H.Pana-gopoulos,Phys.Lett.B212(1988)206.7. A.Di Giacomo,E.Vicari,Phys.Lett.B275(1992)429.8. B.All´e s,M.Campostrini,A.Di Giacomo,Y.G¨u nd¨u ¸c ,E.Vicari,Phys.Rev.D48(1993)2284.9.J.Engels,F.Karsch,K.Redlich,Nucl.Phys.B435(1995)295.10.J.Fingberg,U.Heller,F.Karsch,Nucl.Phys.B392(1993)493.11.C.Christou,A.Di Giacomo,H.Panagopou-los,E.Vicari,Phys.Rev.D53(1996)2619.12.M.L¨u scher,Commun.Math.Phys.85(1982)39.13.A.Phillips,D.Stone,Commun.Math.Phys.103(1986)599.。