Non-perturbative Dynamics of the Heavy-Light Quark System
On Non-Perturbative Results in Supersymmetric Gauge Theories - A Lecture
a rXiv:h ep-th/9611152v12N ov1996CERN-TH/96-268hep-th/9611152ON NON-PERTURBATIVE RESULTS IN SUPERSYMMETRIC GAUGE THEORIES –A LECTURE 1Amit Giveon 2Theory Division,CERN,CH-1211,Geneva 23,Switzerland ABSTRACT Some notions in non-perturbative dynamics of supersymmetric gauge theories are being reviewed.This is done by touring through a few examples.CERN-TH/96-268September 19961IntroductionIn this lecture,we present some notions in supersymmetric Yang-Mills(YM) theories.We do it by touring through a few examples where we face a variety of non-perturbative physics effects–infra-red(IR)dynamics of gauge theories.We shall start with a general review;some of the points we consider follow the beautiful lecture notes in[1].Phases of Gauge TheoriesThere are three known phases of gauge theories:•Coulomb Phase:there are massless vector bosons(massless photonsγ;no confinement of both electric and magnetic charges).The behavior of the potential V(R)between electric test charges,separated by a large distance R,is V(R)∼1/R;the electric charge at large distance behaves like a constant:e2(R)∼constant.The potential of magnetic test charges separated by a large distance behaves like V(R)∼1/R,and the magnetic charge behaves like m2(R)∼constant,e(R)m(R)∼1 (the Dirac condition).•Higgs Phase:there are massive vector bosons(W bosons and Z bosons), electric charges are condensed(screened)and magnetic charges are confined(the Meissner effect).The potential between magnetic test charges separated by a large distance is V(R)∼ρR(the magnetic flux is confined into a thin tube,leading to this linear potential witha string tensionρ).The potential between electric test charges isthe Yukawa potential;at large distances R it behaves like a constant: V(R)∼constant.•Confining Phase:magnetic charges are condensed(screened)and elec-tric charges are confined.The potential between electric test charges separated by a large distance is V(R)∼σR(the electricflux is confined1into a thin tube,leading to the linear potential with a string tensionσ).The potential between magnetic test charges behaves like a constant at large distance R.Remarks1.In addition to the familiar Abelian Coulomb phase,there are theorieswhich have a non-Abelian Coulomb phase[2],namely,a theory with massless interacting quarks and gluons exhibiting the Coulomb poten-tial.This phase occurs when there is a non-trivial IRfixed point of the renormalization group.Such theories are part of other possible cases of non-trivial,interacting4d superconformalfield theories(SCFTs)[3,4]. 2.When there are matterfields in the fundamental representation of thegauge group,virtual pairs can be created from the vacuum and screen the sources.In this situation,there is no invariant distinction between the Higgs and the confining phases[5].In particular,there is no phase with a potential behaving as V(R)∼R at large distance,because the flux tube can break.For large VEVs of thefields,a Higgs description is most natural,while for small VEVs it is more natural to interpret the theory as“confining.”It is possible to smoothly interpolate from one interpretation to the other.3.Electric-Magnetic Duality:Maxwell theory is invariant underE→B,B→−E,(1.1) if we introduce magnetic charge m=2π/e and also interchangee→m,m→−e.(1.2) Similarly,Mandelstam and‘t Hooft suggested that under electric-magnetic duality the Higgs phase is interchanged with a confining phase.Con-finement can then be understood as the dual Meissner effect associated with a condensate of monopoles.2Dualizing a theory in the Coulomb phase,one remains in the same phase.For an Abelian Coulomb phase with massless photons,this electric-magnetic duality follows from a standard duality transforma-tion,and is extended to SL(2,Z)S-duality,acting on the complex gauge coupling byτ→aτ+b2π+i4πBy effective,we mean in Wilson sense:[modes p>µ]e−S=e−S ef f(µ,light modes),(2.3) so,in principle,L eff depends on a scaleµ.But due to supersymmetry,the dependence on the scaleµdisappear(except for the gauge couplingτwhich has a logµdependence).When there are no interacting massless particles,the Wilsonian effec-tive action=the1PI effective action;this is often the case in the Higgs or confining phases.2.1The Effective SuperpotentialWe will focus on a particular contribution to L eff–the effective superpo-tential term:L int∼ d2θW eff(X r,g I,Λ)+c.c,(2.4) where X r=light chiral superfields,g I=various coupling constants,and Λ=dynamically generated scale(associated with the gauge dynamics): log(Λ/µ)∼−8π2/g2(µ).Integrating overθ,the superpotential gives a scalar potential and Yukawa-type interaction of scalars with fermions.The quantum,effective superpotential W eff(X r,g I,Λ)is constrained by holomorphy,global symmetries and various limits[9,1]:1.Holomorphy:supersymmetry requires that W eff is holomorphic in thechiral superfields X r(i.e.,independent of the X†r).Moreover,we will think of all the coupling constants g I in the tree-level superpotential W tree and the scaleΛas background chiral superfield sources.This implies that W eff is holomorphic in g I,Λ(i.e.,independent of g∗I,Λ∗).2.Symmetries and Selection Rules:by assigning transformation laws bothto thefields and to the coupling constants(which are regarded as back-ground chiral superfields),the theory has a large global symmetry.This implies that W eff should be invariant under such global symmetries.43.Various Limits:W eff can be analyzed approximately at weak coupling,and some other limits(like large masses).Sometimes,holomorphy,symmetries and various limits are strong enough to determine W eff!The results can be highly non-trivial,revealing interesting non-perturbative dynamics.2.2The Gauge“Kinetic Term”in a Coulomb Phase When there is a Coulomb phase,there is a term in L eff of the formL gauge∼ d2θIm τeff(X r,g I,Λ)W2α ,(2.5) where Wα=gauge supermultiplet(supersymmetricfield strength);schemat-ically,Wα∼λα+θβσµνβαFµν+....Integrating overθ,W2αgives the term F2+iF˜F and its supersymmetric extension.Therefore,τeff=θeffg2eff(2.6)is the effective,complex gauge coupling.τeff(X r,g I,Λ)is also holomorphic in X r,g I,Λand,sometimes,it can be exactly determined by using holomorphy, symmetries and various limits.2.3The“Kinetic Term”The kinetic term is determined by the K¨a hler potential K:L kin∼ d2θd2¯θK(X r,X†r).(2.7) If there is an N=2supersymmetry,τeff and K are related;for an N=2 supersymmetric YM theory with a gauge group G and in a Coulomb phase, L eff is given in terms of a single holomorphic function F(A i):L eff∼Im d4θ∂F2 d2θ∂2FA manifestly gauge invariant N =2supersymmetric action which reduces to the above at low energies is[10]Imd 4θ∂F2 d 2θ∂2F3Some of these results also appear in the proceedings [13]of the 29th International Symposium on the Theory of Elementary Particles in Buckow,Germany,August 29-September 2,1995,and of the workshop on STU-Dualities and Non-Perturbative Phe-nomena in Superstrings and Supergravity ,CERN,Geneva,November 27-December 1,1995.6N A supermultiplets in the adjoint representation,Φab α,α=1,...,N A ,and N 3/2supermultiplets in the spin 3/2representation,Ψ.Here a,b are fundamental representation indices,and Φab =Φba (we present Ψin a schematic form as we shall not use it much).The numbers N f ,N A and N 3/2are limited by the condition:b 1=6−N f−2N A −5N 3/2≥0,(3.1)where −b 1is the one-loop coefficient of the gauge coupling beta-function.The main result of this section is the following:the effective superpoten-tial of an (asymptotically free or conformal)N =1supersymmetric SU (2)gauge theory,with 2N f doublets and N A triplets (and N 3/2quartets)isW N f ,N A (M,X,Z,N 3/2)=−δN 3/2,0(4−b 1) Λ−b 1Pf 2N f X det N A (Γαβ)2 1/(4−b 1)+Tr N A ˜mM +1√4Integrating in the “glueball”field S =−W 2α,whose source is log Λb 1,gives the non-perturbative superpotential:W (S,M,X,Z )=S log Λb 1S 4−b 1Here,the a,b indices are raised and lowered with anǫab tensor.The gauge-invariant superfields X ij may be considered as a mixture of SU(2)“mesons”and“baryons,”while the gauge-invariant superfields Zαij may be considered as a mixture of SU(2)“meson-like”and“baryon-like”operators.Equation(3.2)is a universal representation of the superpotential for all infra-red non-trivial theories;all the physics we shall discuss(and beyond) is in(3.2).In particular,all the symmetries and quantum numbers of thevarious parameters are already embodied in W Nf,N A .The non-perturbativesuperpotential is derived in refs.[11,12]by an“integrating in”procedure, following refs.[14,15].The details can be found in ref.[12]and will not be presented here5.Instead,in the next sections,we list the main results concerning each of the theories,N f,N A,N3/2,case by case.Moreover,a few generalizations to other gauge groups will be discussed.4b1=6:N f=N A=N3/2=0This is a pure N=1supersymmetric SU(2)gauge theory.The non-perturbative effective superpotential is6W0,0=±2Λ3.(4.1) The superpotential in eq.(4.1)is non-zero due to gaugino(gluino)conden-sation7.Let us consider gaugino condensation for general simple groups[1].Pure N=1Supersymmetric Yang-Mills TheoriesPure N=1supersymmetric gauge theories are theories with pure superglue with no matter.We consider a theory based on a simple group G.The theorycontains vector bosons Aµand gauginosλαin the adjoint representation of G.There is a classical U(1)R symmetry,gaugino number,which is broken tosubgroup by instantons,a discrete Z2C2(λλ)C2 =const.Λ3C2,(4.2) where C2=the Casimir in the adjoint representation normalized such that, for example,C2=N c for G=SU(N c).This theory confines,gets a mass gap,and there are C2vacua associ-symmetry to Z2by gaugino ated with the spontaneous breaking of the Z2C2condensation:λλ =const.e2πin/C2Λ3,n=1,...,C2.(4.3) Each of these C2vacua contributes(−)F=1and thus the Witten index is Tr(−)F=C2.This physics is encoded in the generalization of eq.(4.1)to any G,givingW eff=e2πin/C2C2Λ3,n=1,...,C2.(4.4) For G=SU(2)we have C2=2.Indeed,the“±”in(4.1),which comes from the square-root appearing on the braces in(3.2)when b1=6,corresponds, physically,to the two quantum vacua of a pure N=1supersymmetric SU(2) gauge theory.The superpotentials(4.1),(4.3)can be derived byfirst adding fundamen-tal matter to pure N=1supersymmetric YM theory(as we will do in the next section),and then integrating it out.5b1=5:N f=1,N A=N3/2=0There is one case with b1=5,namely,SU(2)with oneflavor.The superpo-tential isΛ5W1,0=vacuum degeneracy of the classical low-energy effective theory is lifted quan-tum mechanically;from eq.(5.1)we see that,in the massless case,there is no vacuum at all.SU(N c)with N f<N cEquation(5.1)is a particular case of SU(N c)with N f<N c(N f quarks Q i and N f anti-quarks¯Q¯i,i,¯i=1,...,N f)[1].In these theories,by using holomorphy and global symmetries,U(1)Q×U(1)¯Q×U(1)RQ:100¯Q:010(5.2)Λ3N c−N f:N f N f2N c−2N fW:002onefinds thatW eff=(N c−N f) Λ3N c−N f N c−N f,(5.3) whereX i¯i≡Q i¯Q¯i,i,¯i=1,...,N f.(5.4) Classically,SU(N c)with N f<N c is broken down to SU(N c−N f).The ef-fective superpotential in(5.3)is dynamically generated by gaugino condensa-tion in SU(N c−N f)(for N f≤N c−2)8,and by instantons(for N f=N c−1).The SU(2)with N f=1ExampleFor example,let us elaborate on the derivation and physics of eq.(5.1).An SU(2)effective theory with two doublets Q a i has one light degree of freedom: four Q a i(i=1,2is aflavor index,a=1,2is a color index;2×2=4)threeout of which are eaten by SU(2),leaving4−3=1.This single light degree of freedom can be described by the gauge singletX=Q1Q2.(5.5) When X =0,SU(2)is completely broken and,classically,W eff,class=0 (when X =0there are extra masslessfields due to an unbroken SU(2)). Therefore,the classical scalar potential is identically zero.However,the one-instanton action is expected to generate a non-perturbative superpotential.The symmetries of the theory(at the classical level and with their cor-responding charges)are:U(1)Q=number of Q1fields(quarks or squarks),1=number of Q2fields(quarks or squarks),U(1)R={number of U(1)Q2gluinos}−{number of squarks}.At the quantum level these symmetries are anomalous–∂µjµ∼F˜F–and by integrating both sides of this equation one gets a charge violation when there is an instanton background I.The instanton background behaves likeI∼e−8π2/g2(µ)= Λ9For SU(N c)with N fflavors,the instanton background I has2C2=2N c gluino zero-modesλand2N f squark zero-modes q and,therefore,its R-charge is R(I)=number(λ)−number(q)=2N c−2N f.Since I∼Λb1and b1=3N c−N f,we learn thatΛ3N c−N f has an R-charge=2N c−2N f,as it appears in eq.(5.2).11and,therefore,W eff has charges:U(1)Q(5.9)1×U(1)Q2×U(1)RW eff:002Finally,because W eff is holomorphic in X,Λ,and is invariant under symme-tries,we must haveΛ5W eff(X,Λ)=c10This is reflected in eq.(3.2)by the vanishing of the coefficient(4−b1)in front of the braces,leading to W=0,and the singular power1/(4−b1)on the braces,when b1=4, which signals the existence of a constraint.12At the classical limit,Λ→0,the quantum constraint collapses into the clas-sical constraint,Pf X=0.SU(N c)with N f=N cEquations(6.1),(6.2)are a particular case of SU(N c)with N f=N c.[1]In these theories one obtains W eff=0,and the classical constraint det X−B¯B=0is modified quantum mechanically todet X−B¯B=Λ2N c,(6.3) whereX i¯i=Q i¯Q¯i(mesons),B=ǫi1...i N c Q i1···Q i N c(baryon),¯B=ǫ¯i1...¯i N c¯Q¯i1···¯Q¯iN c(anti−baryon).(6.4)6.2N f=0,N A=1,N3/2=0The massless N A=1case is a pure SU(2),N=2supersymmetric Yang-Mills theory.This model was considered in detail in ref.[17].The non-perturbative superpotential vanishesW non−per.0,1=0,(6.5) and by the integrating in procedure we also get the quantum constraint:M=±Λ2.(6.6) This result can be understood because the starting point of the integrating in procedure is a pure N=1supersymmetric Yang-Mills theory.Therefore,it leads us to the points at the verge of confinement in the moduli space.These are the two singular points in the M moduli space of the theory;they are due to massless monopoles or dyons.Such excitations are not constructed out of the elementary degrees of freedom and,therefore,there is no trace for them in W.(This situation is different if N f=0,N A=1;in this case,monopoles are different manifestations of the elementary degrees of freedom.)137b1=3There are two cases with b1=3:either N f=3,or N A=N f=1.In both cases,for vanishing bare parameters in(3.2),the semi-classical limit,Λ→0, imposes the classical constraints,given by the equations of motion:∂W=0; however,quantum corrections remove the constraints.7.1N f=3,N A=N3/2=0The superpotential isW3,0=−Pf X2Tr mX.(7.1)In the massless case,the equations∂X W=0give the classical constraints; in particular,the superpotential is proportional to a classical constraint: Pf X=0.The negative power ofΛ,in eq.(7.1)with m=0,indicates that small values ofΛimply a semi-classical limit for which the classical constraints are imposed.SU(N c)with N f=N c+1Equation(7.1)is a particular case of SU(N c)with N f=N c+1[1].In these theories one obtainsW eff=−det X−X i¯iB i¯B¯iThis is consistent with the negative power ofΛin W eff which implies that in the semi-classical limit,Λ→0,the classical constraints are imposed. 7.2N f=1,N A=1,N3/2=0In this case,the superpotential in(3.2)readsW1,1=−Pf X2Tr mX+12TrλZ.(7.5)Here m,X are antisymmetric2×2matrices,λ,Z are symmetric2×2 matrices andΓ=M+Tr(ZX−1)2.(7.6) This superpotential was foundfirst in ref.[18].Tofind the quantum vacua, we solve the equations:∂M W=∂X W=∂Z W=0.Let us discuss some properties of this theory:•The equations∂W=0can be re-organized into the singularity condi-tions of an elliptic curve:y2=x3+ax2+bx+c(7.7)(and some other equations),where the coefficients a,b,c are functions of only thefield M,the scaleΛ,the bare quark masses,m,and Yukawa couplings,λ.Explicitly,a=−M,b=Λ316,(7.8)whereα=Λ62Γ.(7.10)15•W1,1has2+N f=3vacua,namely,the three singularities of the elliptic curve in(7.7),(7.8).These are the three solutions,M(x),of the equations:y2=∂y2/∂x=0;the solutions for X,Z are given by the other equations of motion.•The3quantum vacua are the vacua of the theory in the Higgs-confinement phase.•Phase transition points to the Coulomb branch are at X=0⇔˜m= 0.Two of these singularities correspond to a massless monopole or dyon,and are the quantum splitting of the classically enhanced SU(2) point.A third singularity is due to a massless quark;it is a classical singularity:M∼m2/λ2for large m,and thus M→∞when m→∞, leaving the two quantum singularities of the N A=1,N f=0theory.•The elliptic curve defines the effective Abelian coupling,τ(M,Λ,m,λ), in the Coulomb branch:Elliptic Curves and Effective Abelian CouplingsA torus can be described by the one complex dimensional curve in C2 y2=x3+ax2+bx+c,where(x,y)∈C2and a,b,c are complex parameters.The modular parameter of the torus isτ(a,b,c)= βdx αdxIn this form,the modular parameterτis determined(modulo SL(2,Z)) by the ratio f3/g2through the relation4(24f)3j(τ)=8b1=2There are three cases with b1=2:N f=4,or N A=1,N f=2,or N A=2.In all three cases,for vanishing bare parameters in(3.2),there are extra massless degrees of freedom not included in the procedure;those are expected due toa non-Abelian conformal theory.8.1N f=4,N A=N3/2=0The superpotential isW4,0=−2(Pf X)1Λ+12N c<N f<3N c the theory is in an interacting,non-AbelianCoulomb phase(in the IR and for m=0).In this range of N f the theory is asymptotically ly,at short distance the coupling18constant g is small,and it becomes larger at larger distance.However, it is argued that for32N c<N f<3N c,the IR theory is a non-trivial4d SCFT.The elementary quarks and gluons are not confined but appear as in-teracting massless particles.The potential between external massless electric sources behaves as V∼1/R,and thus one refers to this phase of the theory as the non-Abelian Coulomb phase.•The Seiberg Duality:it is claimed[2]that in the IR an SU(N c)theory with N fflavors is dual to SU(N f−N c)with N fflavors but,in addition to dual quarks,one should also include interacting,massless scalars. This is the origin to the branch cut in W eff at X =0,because W eff does not include these light modes which must appear at X =0. The quantum numbers of the quarks and anti-quarks of the SU(N c) theory with N fflavors(=theory A)are11A.SU(N c),N f:The Electric TheorySU(N f)L×SU(N f)R×U(1)B×U(1)RQ:N f111−N cN f(8.2)The quantum numbers of the dual quarks q i and anti-quarks¯q¯i of the SU(N f−N c)theory with N fflavors theory(=theory B)and its mass-less scalars X i¯iareB.SU(N f−N c),N f:The Magnetic TheorySU(N f)L×SU(N f)R×U(1)B×U(1)R q:¯N f1N cN f ¯q:1N f−N c N fX:N f¯N f02 1−N ctheory A and theory B have the same anomalies:U(1)3B:0U(1)B U(1)2R:0U(1)2B U(1)R:−2N2cSU(N f)3:N c d(3)(N f)SU(N f)2U(1)R:−N2cN2f(8.5)Here d(3)(N f)=Tr T3f of the global SU(N f)symmetries,where T fare generators in the fundamental representation,and d(2)(N f)=Tr T2f2.Deformations:theory A and theory B have the same quantummoduli space of deformations.Remarks•Electric-magnetic duality exchanges strong coupling with weak cou-pling(this can be read offfrom the beta-functions),and it interchanges a theory in the Higgs phase with a theory in the confining phase.•Strong-weak coupling duality also relates an SU(N c)theory with N f≥3N c to an SU(N f−N c)theory.SU(N c)with N f≥3N c is in a non-Abelian free electric phase:in this range the theory is not asymptoti-cally ly,because of screening,the coupling constant becomes smaller at large distance.Therefore,the spectrum of the theory at large distance can be read offfrom the Lagrangian–it consists of the elemen-tary quarks and gluons.The long distance behavior of the potential between external electric test charges isV(R)∼1R,e(R→∞)→0.(8.6)For N f≥3N c,the theory is thus in a non-Abelian free electric phase; the massless electrically chargedfields renormalize the charge to zero21at long distance as e −2(R )∼log(R Λ).The potential of magnetic test charges behave at large distance R asV (R )∼log(R Λ)R ,⇒e (R )m (R )∼1.(8.7)SU (N c )with N f ≥3N c is dual to SU (˜N c )with ˜N c +2≤N f ≤3R ∼e 2(R )2˜N c ,the massless magnetic monopoles renormal-ize the electric coupling constant to infinity at large distance,with a conjectured behavior e 2(R )∼log(R Λ).The potential of magnetic test charges behaves at large distance R asV (R )∼1R ⇒e (R )m (R )∼1.(8.9)•The Seiberg duality can be generalized in many other cases,includ-ing a variety of matter supermultiplets (like superfields in the adjoint representation [20])and other gauge groups [21].8.2N f =2,N A =1,N 3/2=0In this case,the superpotential in (3.2)readsW 2,1=−2(Pf X )1ΛΓ+˜mM +1√•The equations∂W=0can be re-organized into the singularity condi-tions of an elliptic curve(7.7)(and some other equations),where the coefficients a,b,c are functions of only thefield M,the scaleΛ,the bare quark masses,m,and Yukawa couplings,λ.Explicitly[11,12],a=−M,b=−α4Pf m,c=α16detλ,µ=λ−1m.(8.12)•As in section7.2,the parameter x,in the elliptic curve(7.7),is given in terms of the compositefield:x≡1•As in section7.2,the negative power ofΛ,in eq.(8.10)with˜m= m=λ=0,indicates that small values ofΛimply a semi-classical limit for which the classical constraints are imposed.Indeed,for vanishing bare parameters,the equations∂W=0are equivalent to the classical constraints,and their solutions span the Higgs moduli space[22].•For special values of the bare masses and Yukawa couplings,some of the 4vacua degenerate.In some cases,it may lead to points where mutually non-local degrees of freedom are massless,similar to the situation in pure N=2supersymmetric gauge theories,considered in[3].For example,when the masses and Yukawa couplings approach zero,all the 4singularities collapse to the origin.Such points might be interpreted as in a non-Abelian Coulomb phase[1]or new non-trivial,interacting, N=1SCFTs.•The singularity at X=0(inΓ)and the branch cut at Pf X=0 (due to the1/2power in eq.(8.10))signal the appearance of extra massless degrees of freedom at these points;those are expected similar to references[2,20].Therefore,we make use of the superpotential only in the presence of bare parameters,whichfix the vacua away from such points.8.3N f=0,N A=2,N3/2=0In this case,the superpotential in eq.(3.2)readsdet MW0,2=±212The fractional power1/(4−b1)on the braces in(3.2),for any theory with b1≤2, may indicate a similar phenomenon,namely,the existence of confinement and oblique24theory has two quantum vacua;these become the phase transition points to the Coulomb branch when det˜m=0.The moduli space may also contain a non-Abelian Coulomb phase when the two singularities degenerate at the point M=0[18];this happens when˜m=0.At this point,the theory has extra massless degrees of freedom and,therefore,W0,2fails to describe the physics at˜m=0.Moreover,at˜m=0,the theory has other descriptions via an electric-magnetic triality[1].9b1=1There are four cases with b1=1:N f=5,or N A=1,N f=3,or N A=2, N f=1,or N3/2=1.9.1N f=5,N A=N3/2=0The superpotential isW5,0=−3(Pf X)1Λ12Tr mX.(9.1)This theory is a particular case of SU(N c)with N f>N c+1.The discussion in section8.1is relevant in this case too.9.2N f=3,N A=1,N3/2=0In this case,the superpotential in(3.2)readsW3,1=−3(Pf X)1Λ13+˜mM+1√confinement branches of the theory,corresponding to the4−b1phases due to the fractional power.It is plausible that,for SU(2),such branches are related by a discrete symmetry.25•The equations∂W=0can be re-organized into the singularity condi-tions of an elliptic curve(7.7)(and some other equations),where the coefficients a,b,c are[11,12]a=−M−α,b=2αM+α4Pf m,c=α64detλ,µ=λ−1m.(9.4) In eq.(9.3)we have shifted the quantumfield M toM→M−α/4.(9.5)•The parameter x,in the elliptic curve(7.7),is given in terms of thecompositefield:x≡12.(9.6)Therefore,as before,we have identified a physical meaning of the pa-rameter x.•W3,1has2+N f=5quantum vacua,corresponding to the5singularities of the elliptic curve(7.7),(9.3);these are the vacua of the theory in the Higgs-confinement phase.•From the phase transition points to the Coulomb branch,we conclude that the elliptic curve defines the effective Abelian coupling,τ(M,Λ,m,λ), for arbitrary bare masses and Yukawa couplings.As before,on the sub-space of bare parameters,where the theory has N=2supersymmetry, the result in eq.(9.3)coincides with the result in[7]for N f=3.•For special values of the bare masses and Yukawa couplings,some of the 5vacua degenerate.In some cases,it may lead to points where mutually non-local degrees of freedom are massless,and might be interpreted as in a non-Abelian Coulomb phase or another new superconformal theory in four dimensions(see the discussion in sections7.2and8.2).26•The singularity and branch cuts in W3,1signal the appearance of extra massless degrees of freedom at these points.•The discussion in the end of sections7.2and8.2is relevant here too.9.3N f=1,N A=2,N3/2=0In this case,the superpotential in(3.2)reads[12]W1,2=−3(Pf X)1/32Tr mX+12TrλαZα.(9.7)Here m and X are antisymmetric2×2matrices,λαand Zαare symmetric 2×2matrices,α=1,2,˜m,M are2×2symmetric matrices andΓαβis given in eq.(3.3).This theory has3quantum vacua in the Higgs-confinement branch.At the phase transition points to the Coulomb branch,namely, when det˜m=0⇔det M=0,the equations of motion can be re-organized into the singularity conditions of an elliptic curve(7.7).Explicitly,when ˜m22=˜m12=0,the coefficients a,b,c in(7.7)are[12]a=−M22,b=Λ˜m21132 2detλ2.(9.8)However,unlike the N A=1cases,the equations∂W=0cannot be re-organized into the singularity condition of an elliptic curve,in general.This result makes sense,physically,since an elliptic curve is expected to“show up”only at the phase transition points to the Coulomb branch.For special values of the bare parameters,there are points in the moduli space where (some of)the singularities degenerate;such points might be interpreted as in a non-Abelian Coulomb phase,or new superconformal theories.For more details,see ref.[12].9.4N f=N A=0,N3/2=1This chiral theory was shown to have W non−per.0,0(N3/2=1)=0;[24]perturb-ing it by a tree-level superpotential,W tree=gU,where U is given in(3.4), may lead to dynamical supersymmetry breaking[24].2710b1=0There arefive cases with b1=0:N f=6,or N A=1,N f=4,or N A= N f=2,or N A=3,or N3/2=N f=1.These theories have vanishing one-loop beta-functions in either conformal or infra-red free beta-functions and, therefore,will possess extra structure.10.1N f=6,N A=N3/2=0This theory is a particular case of SU(N c)with N f=3N c;the electric theory is free in the infra-red[1].1310.2N f=4,N A=1,N3/2=0In this case,the superpotential in(3.2)readsW4,1=−4(Pf X)1Λb12+˜mM+1√β2 2α+1β2α13A related fact is that(unlike the N A=1,N f=4case,considered next)in the(would be)superpotential,W6,0=−4Λ−b1/4(Pf X)1/4+1。
涉及非体积功的热力学基本方程
第49卷第8期2021年4月广州化工Guangzhou Chemical IndustryVol.49No.8Apr.2021涉及非体积功的热力学基本方程杨嫣，陈山川，谢娟，张改(西安工业大学材料与化工学院，陕西西安710021)摘要：热力学基本方程是物理化学课程中一个重要的知识点。
现行物理化学教材中重点阐述了仅含有体积功的热力学基本方程，而对涉及非体积功的热力学基本方程介绍很少。
对材料类专业的学生来说，学习和研究材料热力学，经常会涉及到非体积功。
因此，学习涉及非体积功的热力学基本方程非常必要。
本文重点讨论了涉及非体积功的表面系统、弹性杆、电、磁介质中的热力学基本方程的推导和应用，并总结了处理这类问题的方法。
关键词：物理化学；热力学；基本方程；非体积功中图分类号：G642文献标志码：A文章编号：1001-9677(2021)08-0139-04 Fundamental Equations in Thermodynamics Involving Non-expansion Work*YANG Yan,CHEN Shan-chuan,XIE Juan,ZHANG Gai(School of Material Science and Chemical Engineering,Xi9an Technological University,Shaanxi Xi'an710021,China) Abstract:Fundamental equation in thermodynamics is one of the key points of Physical Chemistry course・In current Physical Chemistry textbooks,the fundamental equations containing only expansionwork are mainly described, while the ones involving non-expansionwork are rarely introduced.For the students majoring in materials,non-expansionwork is often involved in the processes of studyand research.Therefore,it is necessary to study the fundamental equations in involving non-expansion work.The derivations and applications of the fundamental equations in surface systems,elastic rods,electric and magnetic media were discussed.Moreover,the way to deal with this kind of problem was summarized.Key words：Physical Chemistry;thermodynamics;the fundamental equation；non-expansion work热力学基本方程在处理平衡态热力学问题时非常有用，是物理化学课程内容的重点之一。
动态跟静态相关能
动态相关能:活化空间之前的电子与活化空间的电子相关能。
Although unique definitions of nondynamic, static, and dynamic electron correlation do not exist, the dynamic part is considered to be responsible forkeeping electrons apart and is attributed to a large number of configurations, that is, Slater determinants or configuration state functions, with small (absolute) coefficients in the wave function expansion, whereas the nondynamic and static contributions involve only some determinants with large (absolute) weights, which are necessary for an appropriate treatment of the quasi-degeneracy of orbitals. In particular, static electron correlation embraces a suitable combination of determinants to account for proper spin symmetries and their interactions, whereas nondynamic correlation is required to allow a molecule to separate correctly into its fragments.From /10.1021/jz301319v | J. Phys. Chem. Lett. 2012, 3,3129−3135A Complete Active Space wavefunction is used to obtain a first approximation of the so-called static correlation, which represents the contribution needed to describe bond dissociation processes correctly. This requires a wavefunction that includes a set of electronic configurations with high and very similar importance. Dynamic correlation, representing the contribution to the energy brought by the instantaneous interaction between electrons, is normally small and can be recovered with good accuracy by means of perturbative evaluations, such as CASPT2 and NEVPT./wiki/Complete_active_spaceElectron correlation is sometimes divided into dynamical andnon-dynamical (static) correlation. Dynamical correlation is the correlation of the movement of electrons and is described under electron correlation dynamics[2] and also with the configuration interaction (CI) method. Static correlation is important for molecules where the ground state is well described only with more than one (nearly-)degenerate determinant. In this case the Hartree–Fock wavefunction (only one determinant) is qualitatively wrong. The multi-configurationalself-consistent field (MCSCF) method takes account of this static correlation but not on the dynamical correlation./wiki/Electron_correlation。
A Chiral Perturbation Expansion for Gravity
a rXiv:h ep-th/0511189v212D ec25IHES/P/05/48A Chiral Perturbation Expansion for Gravity M.Abou-Zeid 1and C.M.Hull 21Theoretische Natuurkunde,Vrije Universiteit Brussel &The International Solvay Institutes,Pleinlaan 2,1050Brussels,Belgium 1Institut des Hautes Etudes Scientifiques,35route de Chartres,Le Bois-Marie,91440Bures-sur-Yvette,France 2Theoretical Physics Group,The Blackett Laboratory,Imperial College London,Prince Consort Road,London SW72BW,United Kingdom 2The Institute for Mathematical Sciences,Imperial College London,48Princes Gardens,London SW72AZ,United Kingdom Abstract A formulation of Einstein gravity,analogous to that for gauge theory arising fromthe Chalmers-Siegel action,leads to a perturbation theory about an asymmetric weak coupling limit that treats positive and negative helicities differently.We find power counting rules for amplitudes that suggest the theory could find a natural interpretation in terms of a twistor-string theory for gravity with amplitudes sup-ported on holomorphic curves in twistor space.November 2005Tree-level MHV amplitudes for(super)Yang-Mills theory[1,2]have an elegant for-mulation in twistor space[3],and Witten considered the extension of this to general amplitudes in[4],where it was conjectured that amplitudes are non-zero only if all the external particles in a scattering process are represented by points in twistor space that lie on an algebraic curve of degree d given byd=q−1+l,(1) where q is the number of negative helicity particles and l is the number of loops.This can be understood as resulting from an underlying twistor string theory[4,5,6]and twistor string theory also leads to conformal(super)gravity,where similar results apply[7].There has since been great progress in understanding general super-Yang-Mills amplitudes in twistor space;see e.g.[8]and references therein.The twistor strings of[4,5,6]have the problem that conformal supergravity is inex-tricably mixed in with the gauge theory,so that conformal supergravity modes propagate on internal lines in gauge theory loop amplitudes and there appears to be no decoupling limit giving pure super-Yang-Mills amplitudes.A twistor string that gave Einstein super-gravity coupled to super Yang-Mills would be much more useful,and might have a limit in which the gravity could be decoupled.It is known that MHV amplitudes for Einstein(super)gravity[9]also have an elegant formulation in twistor space[4,10,11,12].Our purpose here is to seek further evidence that(super)gravity amplitudes could arise from a twistor string theory.An interesting way of understanding(1)in gauge theory[4]is that it follows naturally from the perturba-tion theory of the Chalmers-Siegel chiral formulation of Yang-Mills theory[13],in which positive and negative helicities are treated very differently.Moreover,the Chalmers-Siegel formulation is precisely the form of the gauge theory that arises from the perturbative twistor string of ref.[4].We will investigate here chiral formulations of gravity,and show that perturbation theory about them again leads to the relation(1),suggesting that such a chiral formulation of gravity might arise from a twistor string.The action for Yang-Mills can be written asd4x Tr GµνFµν−ǫ2ǫ d4x Tr F(+)µνF(+)µν ,(3) where F(±)are the self-dual and anti-self dual parts of thefield strength:F(±)=1(F±i∗F)in Lorentzian signature.Here we will2present formulae for signatures(2,2)and(4,0)so that allfields are real;the generalisation to signature(3,1)is straightforward,but involves complex actions.The action(2)isTr(F∧F),(4) 14g2Y Mwhere g2Y M=ǫ/2.Thefirst term is the usual Yang-Mills action,and the second is a topological term proportional to the2nd Chern number that does not affect the equations of motion or perturbation theory.The weak coupling limit given by settingǫ=0in(2)gives Siegel’s chiral theory[14] (see also[15])with actionTr(G∧F)= d4x Tr G∧F(+) .(5) Thefield G is a Lagrange multiplierfield whose variation gives the constraintF(+)(A)=0,(6)implying F=F(−),so that thefield strength is anti-self-dual.Thefield equation obtained by varying A isDµGµν=0,(7) where Dµ=∂µ−Aµis the gauge covariant derivative.Eq.(7)is of the same form as the Yang-Mills equation DµFµν=0.The theory describes a helicity+1particle represented by the Yang-Millsfield A withfield strength satisfying(6)(so that F=F(−)satisfiesDµF(−)µν=0)and a helicity−1particle represented by the independentfield G(+)satisfying DµG(+)µν=0.The linearized spectrum is the same for(5)and(4),but the interactions are different:the action(5)has an AAG term,describing a vertex of threefields with helicities++−,but,in contrast to Yang-Mills theory,it has no−−+vertex.The theory with action(5)has the same spectrum as Yang-Mills theory,viz.particles of helicities+1(represented by A)and−1(represented by G),but differs in the interactions, and it is a non-trivial weak coupling limit of the standard theory written in the form(2). Perturbation theory inǫbased on the action(2)is an expansion about Siegel’s theory(5) and treats positive and negative helicity gluons rather differently.It is useful to attribute to the independentfields A and G the weights w[A]=0and w[G]=−1under a U(1)transformation,related to the‘anomalous’U(1)R-symmetry S in the N=4supersymmetric extension of action(4)[4],with S=4w.The Yang-Mills action(4)then has weight w=0while the Siegel action(5)has weight w=−1and the second term in(2)has weight w=−2.In[4],the w=−1term(5)was interpreted as the transform to space-time of holomorphic Chern-Simons theory on twistor space,whilethe w=−2interaction term was related to nonperturbative D-instanton contributions in twistor-string theory.Consider the perturbation theory inǫfor the action(2),which was analysed in[4]. If one attributes weights w=1toǫand w=−1to the Planck constant ,then the action rescaled by1/ has weight w=0.The generating functional of scattering matrix elements at l-loops must be a sum of terms of the forml−1f(A)ǫd G q(8) for some function f,and for this to have total weight zero it is necessary that the rela-tion(1)holds for each term in the effective action[4].The power q of G is the number of negative helicity gluons in an l-loop scattering process,while the power d ofǫis the instanton number,given by the degree of the holomorphic curve in twistor space.We now turn to gravity,formulated in terms of a vierbein e aµand spin-connection ωµbc,with corresponding one-forms e a,ωbc.The torsion and curvature2-forms are given byT a=de a+ωa b∧e b(9) andR a b(ω)=dωa b+ωa c∧w c b.(10) In a second order formalism,one imposes the constraintT a=0,(11) which determines the spin-connection in terms of the vierbein:ωµab=Ωµab(e).(12) HereΩµab(e)is the usual expression for the Lorentz connection in terms of the vierbein,Ωµab(e)≡eνa∂[µeν]b−eνb∂[µeν]a−eρa eσb∂[ρeσ]c eµc.(13)The Einstein-Hilbert action is1In Euclidean signature(4,0),the spin group factorises as Spin(4)=SU(2)×SU(2) while in split signature it factorises as Spin(2,2)=SU(1,1)×SU(1,1).The spin-connection decomposes into the self-dual pieceω(+)ab and the anti-self-dual pieceω(−)ab,ω(±) bc ≡12εbc deωde ,(15)which are the independent gaugefields for the two factors of the spin group.The curvature2-form can also be split into self-dual and anti-self-dual piecesR(±)bc ≡12εbc de R de ,(16)and it is easily seen that R(+)ab depends only onω(+)while R(−)ab depends only onω(−), withR(±)a b(ω)=dω(±)a b+ω(±)a c∧ω(±)c b.(17) In2-component spinor notation,whereα,βtransform under thefirst SU(2)or SU(1,1) factor and˙α,˙βtransform under the second,ω(+)ab becomesωαβandω(−)ab becomesω˙α˙β.An equivalent form of the Einstein-Hilbert action(14)is given using R(+)instead of R by12κ2 e a∧e b∧R ab(ω).(19) Using(9)and(10)this can be written as12ψαβγδΣαβ∧Σγδ(21)where the curvature2-form Rαβofωαβis given in eq.(17)andΣis a a self-dual2-form acting as a Lagrange multiplier.The totally symmetric Lagrange multiplierfieldψαβγδimposes the constraintΣ(αβ∧Σγδ)=0(22) which implies thatΣαβ=eα˙α∧eβ˙α(23) for some tetrad eα˙α.Solving forΣαβas in(23)and substituting in(21)yields(18).It is remarkable that one only needs the self-dual part of the spin-connection in order to formulate gravity.The torsion constructed from e,ω(+)is(settingω(−)=0in(9))˜T a=de a+ω(+)a∧e b.(24)bIf one imposes the constraint˜T a=0,one obtainsω(+)ab=Ω(+)ab(e)(25) andΩ(−)ab(e)=0.(26) Now(26)implies in turn thatR(−)ab(e)=0,(27) where R(−)ab(e)is the anti-self-dual part of the curvature of the connectionΩ(e).Then the Riemann curvature constructed from the vierbein is self-dual and hence Ricci-flat,so that the torsion constraint˜T a=0imposes thefield equations of self-dual gravity as well as solving for the spin-connection in terms of the vierbein[20].Siegel[20]gave a remarkable asymmetric action for gravity that is analogous to the asymmetric gauge theory action(5)by introducing a Lagrange multiplierfield to impose the constraint˜T=0.In the second order formalism,ω(+)ab is given in terms of e by(25) and the remaining part of˜T a=0is imposed by a Lagrange multiplierσµ(−)ab which is anti-self-dual,or in spinor notationσµ˙α˙β.This has the same index structure as the missing anti-self-dual spin-connection.Siegel’s action can be written asσ˙α˙β∧˜Tα˙α∧eα˙β.(28)Varyingσimposes the self-dual gravity equation(26)so that e represents a graviton of helicity−2.Varying e gives=0,(29) dσ˙α˙β∧eα˙βso that the Ricci tensor constructed from the linearised curvature dσfor an anti-self-dual connectionσvanishes,and the Lagrange multiplierfield represents a graviton of helicity +2.This action then represents particles of helicity±2,as in Einstein’s theory,but the interactions are different for the two helicities,and in particular the theory is linear inσ.There is also afirst-order form of this theory,in whichω(+)ab is an independentfield and a Lagrange multiplier is introduced to impose the full constraint˜T=0[20],which in turn implies thefield equations[20].Siegel also generalised(28)to give an asymmetric form of N=8supergravity,with Lagrange multipliers imposing torsion constraints of the supergravity theory[20].Siegel’s asymmetric theory of gravity can be put in a different form that arises as a weak-coupling limit of the Einstein theory,and gives a chiral perturbation theory of gravity[16,17]similar to that arising from the Yang-Mills action(2).The gravity action(18)depends on the vierbein andω(+)only,andω(−)decouples completely:we can write it in the form12 e a∧e b∧dωab,(32) which can be rewritten using eq.(13)as− e a∧e b∧ωac∧Ωc b=− e a∧e b∧ωac∧Ω(+)c b(33)whereΩ(+)=Ω(+)(e)is the self-dual part of the connection(13).This is an action for two independentfields,the vierbein eµa and the self-dual connectionωac;the latter now plays the role of a Lagrange multiplierfield.Note that the self-duality ofωac implies that only the self-dual partΩ(+)ofΩ(e)occurs in the action.Thefield equation from varying the Lagrange multiplierfieldωac sets the self-dual part ofΩ(e)to zero,Ω(+)a b(e)=0.(34)This implies that the self-dual part of the curvature constructed from the Levi-Civita connectionΩ(e)vanishesRµνab(Ω(+))=0,(35) so that the vierbein gives a metric with anti-self-dual Riemann curvature.Thefield equation for the vierbein givese b∧dωab=0.(36) Comparing with(31),this can be seen to be a version of the Einstein equation linearised around the anti-self-dual background spacetime described by the tetrad e a,where the linearised gravitonfield is the self-dual connectionωac.The fact thatωab andΩ(−)(e)are respectively self-dual and anti self-dual means that they describe particles of opposite helicity:e describes a particle of helicity+2andωdescribes a particle of helicity−2.The linearized spectrum is the same for(32)and(14), but the interactions differ as(32)has no−−+vertex.The asymmetric theory(32) is equivalent to the Siegel theory(but with opposite conventions to those used earlier; in(28)e represents a negative helicity graviton,while in(32)it is a positive helicity one).Now the form of the action(30)has the weak-coupling limit(32),and one can consider perturbation theory inκ2about this weak-coupling limit,in complete analogy with that of the gauge theory(2).As in that case,it is useful to attribute to the independent fields e a andωab the weights w[e]=0and w[ω]=−1under a U(1)transformation, corresponding to the‘anomalous’U(1)R-symmetry S in the N=8supersymmetric extension of action(28)[20],with S=8w.The chiral action(18)has weight w=−1and the second term in(30)has weight w=−2.It is tempting to conjecture that the S=−16 interaction term is related to nonperturbative contributions in a new twistor-string theory for Siegel’s truncation of N=8supergravity.A similar conjecture was made in[11]based on an analysis of maximally helicity violating scattering amplitudes for gravitons.Consider the perturbation theory inκ2for the theory defined by action(30).If one attributes weights w=1toκ2and w=−1to the Planck constant ,then the action rescaled by1/ has weight w=0.The generating functional of scattering matrix elements at l-loops must be a sum of terms of the forml−1˜f(e)κ2dωq(37) for some function˜f,and for this to have total weight zero it is again necessary that the relation(1)holds for each term in the effective action.The power q ofωis the number of negative helicity gravitons in an l-loop scattering process in the theory defined by(30).If the theory has a twistor string origin similar to that of[4],then the power d ofκ2might arise as an instanton number,and the scattering would have support on curves in twistor space characterised by the integer d.This formulation of gravity extends to one of N=8supergravity in which the Einstein term is written in the form(30)and the vectorfield kinetic terms take the form(2).In the weak coupling limit,it gives Siegel’s chiral N=8supergravity[20].We have seen that the formulation of gauge theory with action(2)and that of gravity with action(30)have many similarities:both have a non-trivial asymmetric weak-coupling limit and both have a perturbation theory that leads to the relation(1).For the gauge theory,the action(2)arises from a twistor string theory with thefirst term arising from the perturbative theory and the second term from instanton corrections.The constraint(1) then implies that amplitudes are supported on holomorphic curves of degree d in twistor space.It is natural to conjecture that the gravity action(30),or its N=8supergravity generalisation,should also have an elegant twistor theory origin,and that the formula(1) has a similar twistor space interpretation.We will discuss the twistor formulation of this theory elsewhere.AcknowledgementsWe would like to thank Lionel Mason and Prameswaran Nair for helpful discussions.We also thank the Vrije Universiteit Brussel and the Institute for Mathematical Sciences at Imperial College London for hospitality and support.The work of M.A.was supported by a PPARC Postdoctoral Research Fellowship with grant reference PPA/P/S/2000/00402, by the‘FWO-Vlaandere’through project G.0034.02,by the Belgian Federal Science Policy Office through the Interuniversity Attraction Pole P5/27,by the European Commission FP6RTN programme MRTN-CT-2004-005104,by the European Commission through its 6th Framework Programme‘Structuring the European Research Area’and by contract Nr.RITA-CT-2004-505493for the provision of Transnational Access implemented as Specific Support Action.References[1]S.Parke and T.Taylor,An amplitude for N gluon scattering,Phys.Rev.Lett.56(1986)2459.[2]F.A.Berends and W.T.Giele,Recursive calculations for processes with N gluons,Nucl.Phys.B306(1988)759.[3]V.P.Nair,A current algebra for some gauge theory amplitudes,Phys.Lett.B214(1988)215.[4]E.Witten,Perturbative gauge theory as a string theory in twistor space,Commun.Math.Phys.252(2004)189258,[arXiv:hep-th/0312171].[5]N.Berkovits,An alternative string theory in twistor space for N=4super-Yang-Mills,Phys.Rev.Lett.B93011601(2004),[arXiv:hep-th/0402045].[6]N.Berkovits and L.Motl,Cubic twistorial stringfield theory,JHEP0404(2004)056[arXiv:hep-th/0403187].[7]N.Berkovits and E.Witten,Conformal supergravity in twistor-string theory,JHEP0408009(2004),[arXiv:hep-th/0406051].[8]F.Cachazo and P.Svrcek,Lectures on twistor strings and perturbative Yang-Millstheory,Proc.Sci.RTN2005(2005)005,[arXiv:hep-th/0504194].[9]F.A.Berends,W.T.Giele and H.Kuijf,On relations between multi-gluon andmultigraviton scattering,Phys.Lett.B211(1988)91.[10]S.Giombi,R.Ricci,D.Robles-Llana and D.Trancanelli,A note on twistor gravityamplitudes,JHEP0407(2004)059[arXiv:hep-th/0405086].[11]V.P.Nair,A note on MHV amplitudes for gravitons,Phys.Rev.D71(2005)121701,[arXiv:hep-th/0501143].[12]N.E.J.Bjerrum-Bohr,D.C.Dunbar,H.Ita,W.B.Perkins and K.Risager,MHV-vertices for gravity amplitudes,[arXiv:hep-th/0509016].[13]G.Chalmers and W.Siegel,The self-dual sector of QCD amplitudes,Phys.Rev.D54(1996)7628[arXiv:hep-th/9606061].[14]W.Siegel,N=2,N=4string theory is selfdual N=4Yang-Mills theory,Phys.Rev.D46(1992)3235.[15]A.Parkes,A cubic action for self-dual Yang-Mills,Phys.Lett.B286(1992)265,[arXiv:hep-th/9203074].[16]N.Berkovits and W.Siegel,Covariantfield theory for self-dual strings,Nucl.Phys.B505(1997)139[arXiv:hep-th/9703154].[17]W.Siegel,Fields,[arXiv:hep-th/9912205].[18]R.Capovilla,J.Dell,T.Jacobson and L.Mason,Self-dual2-forms and gravity,Class.Quantum Grav.8(1991)41.[19]R.Capovilla,J.Dell and T.Jacobson,A pure spinor formulation of gravity,Class.Quantum Grav.8(1991)59.[20]W.Siegel,Selfdual N=8supergravity as closed N=2(N=4)strings,Phys.Rev.D47(1993)2504[arXiv:hep-th/9207043].10。
非牛顿流体电动力学外文文献翻译、中英文翻译、外文翻译
论文翻译原文：Electrokinetics of non-Newtonian fluids: A reviewABSTRACTThis work presents a comprehensive review of electrokinetics pertaining to non-Newtonian fluids. The topic covers a broad range of non-Newtonian effects in electrokinetics, including electroosmosis of non-Newtonian fluids, electrophoresis of particles in non-Newtonian fluids, streaming potential effect of non-Newtonian fluids and other related non-Newtonian effects in electrokinetics. Generally, the coupling between non-Newtonian hydrodynamics and electrostatics not only complicates the electrokinetics but also causes the fluid/particle velocity to be nonlinearly dependent on the strength of external electric field and/or the zeta potential. Shear-thinning nature of liquids tends to enhance electrokinetic phenomena, while shear-thickening nature of liquids leads to the reduction of electrokinetic effects. In addition, directions for the future studies are suggested and several theoretical issues in non-Newtonian electrokinetics are highlighted.1. IntroductionThe recently growing interests in electrokinetic phenomena are triggered by their diverse applications in microfluidic devices which could have the potential to revolutionize conventionalways of chemical analysis,medical diagnostics, material synthesis, drug screening and delivery aswell as environmental detection andmonitoring. The prevalent use of electrokinetic techniques in microfluidic devices is ascribed to their several distinctive advantages: (i) the devices are energized by electricitywhich is widely available and ease of control; (ii) the devices involve no moving parts and thus less mechanical failures; (iii) the induced velocity of liquid or particle is independent of geometric dimensions of devices; (iv) the devices can be readily integrated with other electronic controlling units to achieve fully-automated operation. In addition to its useful applications in microfluidics, electrokinetics is also a basis for understanding various phenomena, such as ionic transport and rectification in nanochannels [1,2], thermophoresis inaqueous solutions [3,4], electrowetting of electrolyte solutions [5,6] and so on. When a solid surface is brought into contact with an electrolyte solution, the solid surface obtains electrostatic charges. The presence of such surface charges causes redistribution of ions and then forms a charged diffuse layer in the electrolyte solution near the solid surface to naturalize the electric charges on solid surface. Such electrically nonneutral diffuse layer is usually dubbed electric double layer (EDL) which is responsible for two categories of electrokinetic phenomena, (i) electrically-driven electrokinetic phenomena and (ii) nonelectrically-driven electrokinetic phenomena. The basic physics behind the first category is as follows: when an external electric field is applied tangentially along the charged surface, the charged diffuse layer experiences an electrostatic body force which produces relativemotion between the charged surface and the liquid electrolyte solution. The liquid motion relative to the stationary charged surfaces is known as electroosmosis (Fig. 1a), and the motion of charged particles relative to the stationary liquid is known as electrophoresis (Fig. 1b). The classic electroosmosis occurs around solids with fixed surface charges (or, equivalently, zeta potential ζ) for given physiochemical properties of surface and solution, and then the effective liquid slip at the solid surface under the situation of thin EDLs is quantified by the well-known Helmholtz–Smoluchowski velocity, i.e., us = −εζE0/μ (ε is the electric permittivity of the electrolyte solution, ζis the zeta potential of the solid surface, E0 is the external electric field strength and μ is the dynamic viscosity of electrolyte solution). When a charged particle with a thin EDL is freely suspended in a stationary liquid electrolyte solution, electroosmotic slip motion of solution molecules on the particle surface induces the electrophoretic motion of particle with a velocity given by the Smoluchowski equation, U =εζE0/μ(Note that here ζ denotes the zeta potential of particle). One typical behavior of the second category is the generation of streaming potential effect in pressure-driven flows (Fig. 1c). There are surplus counterions in EDLs adjacent to the channel walls, and the pressuredriven flow convects these counterions downstream to gives rise to a streaming current. Simultaneously, the depletion (accumulation) of counterions in the upstream (downstream) sets up a streaming potential which drives a conduction current in opposite direction to the streaming current. At the steady state, the conduction current exactly counter-balances the streaming current, and the streaming potential built up across the channel under the limit of thin EDLs is givenby Es = Pεζ/(σ0μ) (P is externally applied pressure gradient and σ0 represents the bulk conductivity of electrolyte solution). More fundamental and comprehensive descriptions of electrokinetic phenomena are given in textbooks and reviews [7–13].Previous description of electrokinetics usually assumes Newtonian fluids with constant liquid viscosity, and most studies of electrokinetics in literature adopt such assumption. But in reality, microfluidic devices are more frequently involved in analyzing and/or processing biofluids (such as solutions of blood, saliva, protein and DNA), polymeric solutions and colloidal suspensions. These fluids cannot be treated as Newtonian fluids. Therefore, the characterization of hydrodynamics of such non-Newtonian fluids relies on the general Cauchy momentum equation in conjunction with proper constitutive equations which generally define the viscosity of liquid to vary with the rate of hydrodynamic shear, rather than the Navier–Stokes equation which is only applicable to Newtonian fluids. Since electrokinetics results from the coupling of hydrodynamics and electrostatics, it is straightforward to believe that non-Newtonian hydrodynamics would modify the conventional Newtonian electrokinetics. In this review, non-Newtonian effects on electrokinetics are comprehensively summarized and discussed. This review is organized as follows: Section 2 provides a review on the most widelystudied electroosmosis of non-Newtonian fluids. Section 3 presents a review for the electrophoresis of particles in non-Newtonian fluids, and Section 4 discusses the streaming potential effects of non-Newtonian fluids. Other non-Newtonian effects of particular interest on electrokinetics are given in Section 5. Lastly, Section 6 concludes the review and identifies the directions for the future studies.2. Electroosmosis of non-Newtonian fluidsThe pioneering contribution to this field is probably attributed to Bello et al.[14] who experimentally measured an electroosmotic flow of a polymer (methyl cellulose) solution in a capillary. Their investigation showed that the electroosmotic velocity of such polymer solution is much higher than that predicted with the classic Helmholtz–Smoluchowski velocity. It was then proposed that the shear-thinning induced by polymermolecules lowers the effective fluid viscosity inside the EDL. About a decade later, more interests were paid to such phenomenonboth experimentally and theoretically. Chang and Tsao [15] conducted an experiment similar to that of Bello et al. [14] to investigate an electroosmotic flowof the polyethylene glycol solution and observed that the drag aswell as the effective viscositywas greatly reduced due to the sheared polymericmolecules inside the EDL. On theoretical aspects, recent efforts have resulted in a great deal of information on electroosmotic flows of non-Newtonian fluids. Specifically, non-Newtonian effects are characterized by proper constitutive models which relate the dynamic viscosity and the rate of shear. There has been a large class of constitutive models available in the literature for analyzing the non-Newtonian behavior of fluids, such as power-law model, Carreau model, Bingham model, Oldroyd-B model, Moldflow second-order model and so on. Power-law fluid model is certainly the most popular because it is simple and able to fit awide range of non-Newtonian fluids. One important parameter in the power law fluid model is the fluid behavior index (n) which delineates the dependence of the dynamic viscosity on the rate of shear. If n is smaller (greater) than one, the fluids demonstrate the shear-thinning (shear-thickening) effect that the viscosity of fluid decreases with the increase (decrease) of the rate of shear. If n is equal to one, the fluids then exactly behave as Newtonian fluids. Das and Chakraborty [16] obtained the first approximate solution for electroosmotic velocity distributions of power-law fluids in a parallel-plate microchannels. However, their analysis did not clearly address the effect of non-Newtonian effects on electroosmotic flows. Zhao et al. [17,18] carried out theoretical analyses of electroosmosis of power-law fluids in a slit parallel-plate microchannel and fully discussed the non-Newtonian effects on electroosmotic flow. Their analyses revealed that the fluid rheology substantially modifies the electroosmotic velocity profiles and electroosmotic pumping performance. Particularly, they derived a generalized Helmholtz–Smoluchowski velocity for power-lower fluids in a similar fashion to the classic Newtonian Smoluchowski velocity and further elaborated the influencing factors of such velocity. Similar analyses were later extended to a cylindrical microcapillary by Zhao and Yang [19,20]. Recently, an experimental investigation was performed by Olivares et al. [21] who measured the electroosmotic flow rate of a non-Newtonian polymeric (Carboxymethyl cellulose) solution, and their experimental measurements agree well with the theoretical results predicted from the generalized Helmholtz–Smoluchowski velocity of power-law fluids. Paul [22] conceptuallydevised a series of fluidic devicesemploying electroosmosis of shear-thinning fluids. These devices included pumps, flow controllers, diaphragmvalves and displacement systemswhichwere all claimed to outperform their counterparts employing Newtonian fluids. Berli and Olivares [23] addressed the electrokinetic flow of non- Newtonian fluids in microchannels with the depletion layers near channel walls. Their analysis essentially considered a combined effect of electroosmosis and pressure-driven flow, and is greatly simplified due to the presence of depletion layers. Berli [24] evaluated the thermodynamic efficiency for electroosmotic pumping of power-law fluids in cylindrical and slit microchannels. It was revealed that both the output pressure and pumping efficiency for shear-thinning fluids could be several times higher than those for Newtonian fluids under the same experimental conditions. Utilizing the Lattice–Boltzmann method, Tang et al. [25] numerically investigated the electroosmotic flow of power-law fluids in microchannels. An electroosmotic body force was incorporated in the Bhatnagar–Gross–Krook collision approximation which simulates the Cauchy momentum equation. These studies of electroosmotic flow of non-Newtonian fluids however all assumed small surface zeta potentials which are much less than the so-call thermal voltage, i.e., kBT/(ze), where kB is the Boltzmann constant, T is the absolute temperature, e is the elementary charge, z denotes the valence of electrolyte ion. This assumption could be easily violated when large surface zeta potentials are present. Therefore, investigations of electroosmotic flow of power-law fluids over solid surfaces with arbitrary surface zeta potentials were reported in [26,27].However, the constitutive model for non-Newtonian fluids in abovementioned investigations is just an extreme case of the more general non-Newtonian Carreau fluid model. In comparison with the Newtonian fluid model, Carreau constitutive model includes five additional parameters and can describe the rheology of a wide range of non-Newtonian fluids. Under the limit of zero shear rates, the commonly used power-lawmodelwould predict an infinitely large viscosity for shear-thinning fluids, while the Carreau model does not have such defect but has smoothly transits to a constant viscosity. The Carreau fluid model can well characterize the rheology of various polymeric solutions, such as glycerol solutions of 0.3% hydroxyethyl-cellulose Natrosol HHX and 1% methylcellulose Tylose [28], and pure poly(ethylene oxide) [29]. These polymers arewidely used for improving selectivity and resolution in the capillary electrophoresis for separation of protein [30] andDNA [31]. Zimmerman et al. [32,33] performed finite element numerical simulations of the electroosmotic flow of a Carreau fluid in a microchannel T-junction. The analyses suggested that the flow field remarkably depends on the non-Newtonian characteristics of fluids, and therefore could guide the design of electroosmotic flow rheometers. Zhao and Yang [34] presented a general framework to address electroosmotic/electrophoretic mobility regarding non-Newtonian Carreau fluids. They concluded that electroosmotic/electrophoretic mobility can be significantly enhanced with shearthinning fluids and large surface zeta potentials.Due to the nonlinear dependence of the dynamic viscosity on the rate of shear, equations governing electroosmotic flows of non-Newtonian fluids also become highly nonlinear and then most of theoretical analyses rely on either approximate solutions or numerical simulations. Exact solutions are valuable because they not only can provide physical insight into the studied phenomena, but also can serves as benchmarks for experimental, numerical and asymptotic analyses. An exact solution for electroosmotic flow of non-Newtonian fluids was presented by Zhao and Yang[18]who considered electroosmosis of a power-law fluid in a slit parallel-plate microchannel as illustrated in Fig. 2. The channel is filledwith a non-Newtonian power-lawelectrolyte solution having a flowbehavior index n, anda flowconsistency indexm. The microchannel walls are uniformly charged with a zeta potential ζ. The application of an external electric field E0 drives the liquid into motion because of electroosmotic effect, and the velocity profile was derived for the situation of low zeta potentials as [18]()()()()1,,cosh s n G n H G n y u y u H κκκ-= （1）where the Debye parameter κ is defined as κ = 1/λ D = [2e2z 2n ∞/(εkBT)]1/2 (wherein e is the charge of an electron, z is the ionic valence, n ∞ is the bulk number concentration of ions, ε is the electric permittivity of the solution, kB is the Boltzmann constant, and T is the absolute temperature). The function G(υ,ϖ) in Eq. (1) is defined as()()()()1222121113,cosh ,;;cosh 222G F υυυυϖϖϖυυ---⎡⎤=⎢⎥⎣⎦（2） where 2F1[α1,α2;β1;z] denotes the Gauss' hypergeometric function [35]. us in Eq.(1) denotes the so-called Helmholtz –Smoluchowski velocity for power-law fluids and can be written as110n nn s E u n m εζκ-⎛⎫=- ⎪⎝⎭ （3） which was firstly derived by Zhao et al. [17] using an approximate method. The thickness of EDL on the channel wall is usually measured by the reciprocal of the Debye parameter (κ−1), so the nondimensional electrokinetic parameter κH = H/κ−1 characterizes the relative importance of the half channel height to the EDL thickness. Then for large values of electrokinetic parameter κH (thin EDL or large channel), the Helmholtz –Smoluchowski velocity given by Eq. (3) signifies the constant bulk velocity in microchannel flows due to electroosmosis. In electrokinetically-driven microfluidics dealing with non-Newtonian fluids, the Helmholtz –Smoluchowski velocity in Eq. (3) is of both practical and fundamental importance due to two reasons: First, the volumetric flow rate can be simply calculated by multiplying the area of channel cross-section and the Helmholtz –Smoluchowski velocity. Second, numerical computations of electroosmotic flow fields in complex microfluidic structures can be immensely simplified by prescribing the Helmholtz –Smoluchowski velocity as the slip velocity on solid walls. One can find more detailed derivation and discussion of this generalized Smoluchowski velocity in Refs. [17,18,21]. Very recently, Zhao and Yang [20] reported an interesting but counterintuitive effect that the Helmholtz –Smoluchowski velocity of non-Newtonian fluids becomes dependent on the dimension and geometry of channels owing to the complex coupling between the non-Newtonian hydrodynamics and the electrostatics.Inmicrofluidic pumping applications, the flow rate or average velocity is usually an indicator of pump performance.With the above derived electroosmotic velocity in Eq. (1), the electroosmotic average velocity along the cross-section of channel can be sought as()01Hu u y dy H =⎰ （4） where 3F2[α1,α2,α3;β1,β2;z] represents one of the generalized hypergeometric functions [35]. It needs to be pointed out that all the hypergeometric functions presented in this review can be efficiently computed in commercially-available software, such as MATLab and Mathematica.翻译：非牛顿流体电动力学：回顾摘要：本文对关于非牛顿流体电动力学进行了全面的回顾，涵盖大量非牛顿流体电动力学效应，包括非牛顿流体的电渗、非牛顿流体的电泳、流动的非牛顿流体的潜在影响以及其他电动力学中的非牛顿流体影响。
湍流燃烧模型
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 2. Balance equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
194
D. Veynante, L. Vervisch / Progress in Energy and Combustion Science 28 (2002) 193±266
6. Tools for turbulent combustion modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 6.2. Scalar dissipation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 6.3. Geometrical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 6.3.1. G-®eld equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 6.3.2. Flame surface density description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 6.3.3. Flame wrinkling description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 6.4. Statistical approaches: probability density function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 6.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 6.4.2. Presumed probability density functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 6.4.3. Pdf balance equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 6.4.4. Joint velocity/concentrations pdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 6.4.5. Conditional moment closure (CMC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 6.5. Similarities and links between the tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
全新版大学英语综合教程5 U5 heat wave 全文翻译
你说你是一个全球变暖的怀疑者？也许你不住在沿着海岸的弗洛里达洲或者希什马瑞芙洲，阿拉斯加州吧：住在那些地区的人们通常都相信全球变暖。
弗洛里达洲在2004年时遭遇了四场魔鬼般的飓风，在一年后，卡特里娜袭击了新奥尔良和密西西比州的沿岸地区。
许多科学家相信，这些飓风时节的超强破坏力应该归因于全球变暖。
大风暴加强了墨西哥州沿岸的暖流，一年一年，这些纬度地区变得越来越暖。
You say you’are a skeptic on global warming?至于，希什马瑞福州的一个小镇（人口600），它是一个因纽皮亚克的爱斯基摩小村庄，坐落于一个细长的屏障岛屿，在安克雷奇北边625英里处。
当《时代》记者Margot Roosevelt 在2004年参观它时，她发现在它正在“融入海洋”。
它已经少了100至300英尺的海岸线，而这数字中的一半发生在1997以后。
在沙滩下的永久冻土开始消融，海面上的冰也正不断减少，这使得居民对于凶猛的暴风雨来说，显得越来越脆弱。
一间房子倒了，18间其他的房子就得带着镇上大量的油桶，搬到更高的地方去。
巨浪冲走了学校的操场，毁灭了价值100000美元的船只，打猎与晒鱼的装备。
“这太可怕了，”村里的官员Luci Eningowuk告诉Roosevelt。
“每一年我们都非常害怕，下一场暴风雨会把我们都冲走。
”由于海面结冰得越来越晚，希什马瑞福州通常开始于10月的冰钓季，现在开始于12月。
浆果采摘开始于7月而非8月。
对于因纽皮亚克人来说最令人绝望的就是稀薄的冰面似的猎捕髯海豹变得非常困难，一种长胡须的海报，那是他们饮食和文化的重要组成部分。
什么正在继续？全球变暖，一部分是由石油和汽油的燃烧造成的，它使墨西哥沿岸甚至极地地区都受到创伤，而在极地地区，复杂的气候进程与雪、永久冻土和冰联系在一起，放大了全球变暖的影响。
2004年，在《科学》上出版的一篇文章发现，南极洲西部的冰川正以二十世纪九十年代两倍的速度消融。
新SAT评分详解及样题
* Combined score of two raters, each scoring on a 1– 4 scale 1-4
SAT 1. Composite Score 2 2. SAT raw score 3 3. SAT Test Score Evidence-Based Reading and Writing raw score 4. SAT Studies OG 1—15 5. SAT Subscore 7 Cross-section Score 3 Section Score 400—1600
3.
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25+15min 49
:35min 44
2-12 25min
2-8 50min
History Questions 1-5 are based on the following passage.
This passage is adapted from a speech delivered by Congresswoman Barbara Jordan of Texas on July 25, 1974, as a member of the Judiciary Committee of the United States House of Representatives. In the passage, Jordan discusses how and when a United States president may be impeached, or charged with serious offenses, while in office. Jordan’s speech was delivered in the context of impeachment hearings against then president Richard M. Nixon.
Two-dimensional Quantum Field Theory, examples and applications
Abstract The main principles of two-dimensional quantum field theories, in particular two-dimensional QCD and gravity are reviewed. We study non-perturbative aspects of these theories which make them particularly valuable for testing ideas of four-dimensional quantum field theory. The dynamics of confinement and theta vacuum are explained by using the non-perturbative methods developed in two dimensions. We describe in detail how the effective action of string theory in non-critical dimensions can be represented by Liouville gravity. By comparing the helicity amplitudes in four-dimensional QCD to those of integrable self-dual Yang-Mills theory, we extract a four dimensional version of two dimensional integrability.
2 48 49 52 54 56
5 Four-dimensional analogies and consequences 6 Conclusions and Final Remarks
On the Vacuum Energy Density and Non-Perturbative Effects in Sigma Models
a rXiv:h ep-th/942123v122Fe b1994ON THE VACUUM ENERGY DENSITY AND NON-PERTURBATIVE EFFECTS IN SIGMA MODELS V.G.Ksenzov Institute of Theoretical and Experimental Physics,Moscow Abstract The vacuum energy density is obtained for the O (N )nonlinear sigma model.It is shown that non-perturbative contributions are connected with the square of the symmetry current of the group O (N ).This result is valid for σ-fields which are subject to the constraint.After the discovery of nonperturbative fluctuations the question of their role in gener-ating the physical amplitude has been of particular interest for the field theory.A lot of effort has been invested in investigating these effects in Yang-Mills theory,but the ques-tion is still far from being solved.Therefore it makes sense to study these problems in a simple theory in order to get a new insight into the subject.Such a simple analogy to the Yang-Mills theory is the two-dimensional sigma model with fields transforming according to the vector representation of the group O (N )[1].This model provides a possibility to perform an analysis,and to get the results which might be relevant to actual physics.For a long time it has been clear that the dynamics is in some way related to a complicated vacuum structure,therefore investigation of the vacuum structure turns out to be the key to the understanding of the actual dynamics.In this paper we will discuss the influence of non-perturbative fluctuations on the vacuum energy density εvac in the framework of the two-dimensional sigma model in the large N limit.In these models non-perturbative effects occur which do not depend on the existence of instantons for N >3.We first recall what is known about nonperturbative effects and εvac in sigma model.These results were first obtained in ref.[2].Lorentz invariance implies that the vacuum energy density may be defined asg µνεvac =<0|θµν|0>,where θµνstands for the energy-momentum tensor.Taking into account the conformal anomaly we arrive atεvac =1In the large N sigma model the regularized expression forθµµmay be written as asum of two termsεvac=N√N.Thefirst term is connected with the perturbativefluctu-ations in the vacuum,while the second one is due to non-perturbativefluctuations.Let us note that<α>is a fundamental parameter of the theory also in the case of phase transitions[3].To demonstrate how the non-perturbative effects enter into the value of<α>=0 one employs the LagrangianL=1√f)}.(1)We want to decomposeσa(x)andα(x)asσa(x)=c a(x)+q a(x)(2)α(x)=αc(x)+αqu(x),where c a(x)andαc(x)are classicalfields,while q a(x)andαqu(x)describe smallfluctua-tions around the classical background.The equations of motion for c a andαcfields are∂2c a=αc aN(3)c a(x)c a(x)=NN2(J abµ)2=N √2(fBy virtue of eqs.(5)and(7)we can obtain a new nonlinear equation∂µc a(∂µc a+fNJ abµc b=0.(8) The solutions of these equations are special forms of the solutions of equation(5).The solution of thefirst equation can not give non-perturbative effect because the Lagrange multiplier is zero(5).The eq.(8)is identically satisfied provided c a(x)are subject to the constraint(4).Thesefields have a special property:their transformation under translations can be compensated by isotopic transformations as it wasfirst discussed in [4].The classical energy-momentum tensor can be found from(8)and we arrive atθµν=f2(J aµ)2)(9)with the evident properties∂µθµν=0,θµµ=0.The energy-momentum tensorθµνturns out to be proportional to the product of conserved currents in this model.Similar facts are well known infield theory[5]and in quantum mechanics[6].Tofind the conformal anomalyθµµwe define it asθµµ=lim<0|M2q a(x−∆)q a(x+∆)|0>.(10) where M2is the regulator’s mass and q a is the regulator’sfield.The limit of the right-hand-side of Eq.(10)is calculated at M2→∞and∆2→0and under a special condition∆2M2=Const.In equation(10)fast varyingfluctuations q a(x)are not arbitrary but must satisfy the constraint[2]2c a q a+q a q a=0.(11) This contraint can be used in linear form in q a at large N[2].Differentiating c a q a=0 and using(8)we get∂µq a=−fOur goal is to calculate(10)and we can do this by knowing equation(12)alone. Indeed,we can obtainθµµ=lim1N 2J abµJ bkν+...)(13)Thefirst term is the perturbative contribution toθµµ.Knowing the value of this term we can conclude that<0|q a q k|0>=−δakN)2J abµJ bkν|0>==−gµνδak<0|(f√。
Heavy-light decay constant at the 1m order of HQET
2 fA (x0 ) = −ZA Zζ 2 kV (x0 ) = −ZV Zζ
a6 2
∑ (AI)0 (x) ζ¯b (y)γ5 ζl (z) ,
y,z
(1.2) (1.3)
a6 ¯b (y)γk ζl (z) , (VI )k (x) ζ 6 y∑ ,z,k
where the improved currents AI (x) and VI (x) are defined as in [1]. We also consider the boundary to boundary correlators
Benoît Blossier ∗
DESY, Platanenallee 6, 15738 Zeuthen, Germany
Michele Della Morte
CERN, Physics Department, TH Unit, CH-1211 Geneva 23, Switzerland
Nicolas Garron†
Nicolas Garron
1. Introduction
A few years ago, a non-perturbative formulation of Heavy Quark Effective Theory (HQET) has been given in [1] - see [2] for a review given at this conference. In particular the problem of power divergences is solved through a finite volume matching. Last year, this has been applied to the quenched computation of the b-quark mass at the 1/m order [3]. In the same spirit, we present here a strategy to compute a heavy-light decay constant. We start by writing the Lagrangian at the leading order (i.e. in the static approximation) and add a kinetic and a magnetic piece (we follow the conventions of [3] and set the counterterm δm to zero) LHQET = ψ h D0 ψh − ωkin ψ h D2 ψh − ωspin ψ h σ · Bψh . (1.1)
白洋淀水体氮、磷和重金属的变化特征及影响因素
第31卷第6期2020$1月水资源与水工程学报Journal of Water Resources &Water EngineeringVol.31 N o.6Dec.，2020D01：10.11705/j.issn.1672 -643X.2020. 06.16白洋淀水体氮、磷和重金属的变化特征及影响因素赵黔伟1汪敬忠123，魏浩3，李金涛3，张益淼1(1.河北地质大学地球科学学院，河北石家庄050031; 2.河北地质大学河北省高校生态环境地质应用技术研发中心，河北石家庄050031; 3.河北地质大学实验实践教学中心，河北石家庄050031)摘要：为了探讨白洋淀水体污染的变化特征和来源，利用多元统计方法对白洋淀湖水中离子、营养盐（氮、磷）和重金属（^&、〇1、阳、13及24进行研究,分析其时空变化,并揭示其影响因素和主要来源。
结果表明：白洋淀湖水水化学类型包括只(：〇3-63.厘8和3〇4.€：1-63.厘8,湖水总氮、^&、〇1、？13及2〇的平均值均高于乂类水质限值。
近20年来湖水总氮和总磷的平均值均呈现波动性下降，而两者无明显相关性,主要受气候和人类活动的共同影响。
在空间上，淀内湖水营养盐和重金属表现出西部地区高、东南地区低的特征,多元统计分析进一步证实流域府河携带的污染物质是白洋淀水体氮、磷和重金属的主要来源，也是影响其空间变化的主要因素，但淀内不同地区的人类活动对湖水中N、P、P b及&的影响也不可忽视。
关键词：氮磷；重金属；时空变化；影响因素；白洋淀中图分类号：T V213.4; X824 文献标识码：A文章编号：1672-643X(2020)06-0103-06Variation characteristics of nitrogen, phosphorus and heavy metals inwaters from Baiyangdian Lake and the influencing factorsZHAO Qianwei1,WANG Jingzhong1，，,WEI Hao3,LI Jintao3,ZHANG Yimiao1(1. College of Earth Sciences,Heibei GEO U niversity,Shijiazhuang050031 ,China； 2.H ebei Center forEcological and Environmental Geology Research,Hebei GEO University,Shijiazhuang050031, China;3.Experim ent and Practice Teaching Center,Heibei GEO U niversity,Shijiazhuang050031 ,China)Abstract：In order to discuss the variation characteristics and the sources of water poll an Lake,ions,nutrients(nitrogen and phosphorus)and heavy metals (Cd,Cr,Cu,Ni,Pb and Zn)inthe waters were studied to analyze the spatial and temporal variations,as well as to discover the influencing factors and main sources using multivariate statistical methods.The results show that the hydrochemistry types included HCO3 - Ca •Mg and SO4 •Cl- Ca •Mg,and the average values of TN and heavymetals in Baiyangdian Lake exceeded the water quality limits of class V.The average values of TN andTP in the past 20 years displayed a trend of decreasing volatility and had no obvious correlation betweeneach other,which w ere mainly affected by climate and human activities.Spatially,the nutrients andheavy metals in the waters of Baiyangdian Lake were relatively high in the west and low in Multivariate statistical analysis further confirmed that the pollutants carried by Fuhe River are the mainsource of nutrients and heavy metals in waters of Baiyangdian Lake,which are also fecting their spatial variations.However,the influence of human activities on elements in different regionswithin the lake area cannot be ignored,especially N,P,Pb and Cr.Key words：nitrogen and phosphorus;heavy metal;spatial- temporal change;influencing factor;Baiyangdian Lake收稿日期：2020- 05- 06;修回日期：2020- 09- 07基金项目：河北省自然科学基金项目（D2018403115);国家自然科学基金项目（41702165)作者简介:赵黔伟（1994 -)男，贵州习水人，硕士研究生，主要从事湖泊水文与环境研究。
Quark-gluon densities in the nuclear fragmentation region in heavy ion collisions at LHC
a r X i v :n u c l -t h /0602012v 1 4 F eb 2006Quark-gluon densities in the nuclear fragmentation region in heavy ion collisions at LHCMark Strikman104Davey lab,Penn State University,University Park,PA 16802,U.S.A.Abstract.At the LHC,the leading partons in the nuclei are expected to interact with the maximal possible strength -black disk limit -up to transverse momenta of the order of few GeV .We demonstrate that in this limit the densities of the quark -gluon systems produced in the central AA collisions in the nucleus fragmentation regions should exceed 300GeV /f m 3which is at least as high as the densities discussed for the central region.Experimental signatures of such a regime are also discussed.The focus of most studies of the quark-gluon state produced in heavy ion collisions is the central region where one expects generation of high gluon densities at sufficiently high energies.The first estimates of the hadron matter densities produced in the nuclear fragmentation region were presented in [1]in the framework of the soft hadronic dy-namics.The first estimate of the quark-gluon densities in this kinematics was presented within the framework of the onset of the black disk limit (BDL)of QCD in [2]which found densities at least a factor of ten larger.The purpose of this talk is review and update analysis the analysis of [2].The starting point of [2]is the observation [3]that the transverse momenta of partons propagating through a high gluon density medium should become much larger than the scale of soft interactions.This is due to the possibility for a quark with given large x 1to interact with partons with very small x 2=4p 2t /x 1s where p t is the resolution scale.As aresult,for the case of a gluon with x ≥10−2propagating through the center of the heavy nucleus we find an average of p t ≥4GeV /c at the LHC energies,see review in [4].Let us consider nucleus -nucleus scattering in the rest frame of one of the nuclei.First let us determine the emission angle,θ,of the parton belonging to the nucleus which was at rest.Since the energy losses are small,the light-cone fraction carried by the parton is approximately conserved:(E i −p z i )=xm N ,(1)leading to p z =(µ2+p 2t )/2xm N −xm N /2≈p 2t /2xm N .Here in the last step we have neglected µ2compared to p 2t which is legitimate in the leading order.Since µ2≥0,neglected terms would increase p z making the emission angles,θ,even smaller.Thus,in the BDL the angles θ≃p t /p z ∼2xm N /p t are small.So,the length of the produced wave package is reduced from the naive value of 2R A by the large factor S =1/(1−cos θ)≈p 2t /2x 2m 2N .However,we must also take into account that the products of the nucleon fragment as a whole move forward in the target rest frame.Since the knocked out partons carry practically the whole light cone fraction of the nucleon,the mass squared of the produced system,M2and its longitudinal momentum,p z can be determined from(M2+(M2/2m N)2/M≈M/2m N.As a result wefind the total reduction in the volume:D=(2m N/M)·<p2t/2m2N x2>.(2) Since the energy of the system in its rest frame is M,wefind for the overall enhancement as compared to the nuclear density:R E=1m2N x2i.(3)To illustrate the dependence of R A on the total number of involved partons,N,and on av-erage transverse momenta we can take all x i and all p it.In this case D=N p t/m N,R E= D2,and energy density depends quadratically on the average transverse momentum of partons.Our estimates indicate that at LHC for the gluons with x≥0.0.5, p2gt ≥16GeV2 (and growing with increase of x),and that for quarks p2qt is a factor of two smaller[4]. Taking for illustration N q=3,N g=6,x q=1/6,x g=1/12and p2gt =20GeV2wefind R E=2300.This corresponds to”energy density”∼370GeV/f m3,which is at least as large as the one expected for the central region.It is much larger than our initial estimate where a very conservative value of p2gt was taken.If we assume the proximity of BDL at RHIC for the fragmentation region for p t∼1GeV/c,wefind quark-gluon energy densities∼10GeV/f m3.These densities are at least a factor of10higher than in[1]due to much larger release of energy in the BDL and due to significantly larger longitudinal compression of the interaction volume.Our estimate neglects the conversion of the released partons into hadrons before they reach the back edge of the fragmenting nucleus.If p t generated in the collision is small enough, this effect may become important.Using the logic similar to the one we used for estimating hadron formation in the color transparency phenomenon[5]we can estimate the distance over which a parton (not interacting with a medium)converts to hadrons.Onefindsl coh=2p q/∆M2(4) where∆M2is the mass gap between two lowest hadrons with the same quantum num-bers.Numerically,l coh≈0.3÷0.4f m·p q[GeV],corresponding for p q=4GeV.This is substantially larger than the expected interaction length(see below)and hence the evolution of the imploded system should be determined mostly by partonic rather than hadronic interactions.The difference in average x’s of quarks and gluons leads to a different direction of theflow of the quarks and gluons in the center of mass frame of the produced system.For theabove numerical example,k z/k t∼0.7for quarks and∼−0.25for gluons.Obviously, this pattern will enhance the interactions of quarks and gluons at the next stage of theinteractions,making equilibration more likely.It follows from the above analysis that at LHC in thefirst stage of collisions astrongly compressed hot quark-gluon state of the ellipsoidal shape is formed with thesmall principal axis of∼0.5f m and densityρ≥50partons per f m3.At the higher rapidity end,this ellipsoid boarders essentially parton free space;on the end close tocentral rapidities,it boarders a hot q¯qg state.The scattering length for parton i can be estimated as l i=1/(∑j N jσi j),corresponding to the scattering length being smaller than0.5fm forσ≥0.5mb.To estimate the interaction cross section,we note that the average invariant energy s≈2p2t∼32GeV2.The initial stage of reinteractions certainlyis a highly non-equilibrium process.Nevertheless to do a perturbative estimate we can conservatively introduce a cutoff on the momentum transfer p∼π5.G.R.Farrar,H.Liu,L.L.Frankfurt and M.I.Strikman,Phys.Rev.Lett.61,686(1988).。
Nonlinear Systems and Dynamics
Nonlinear Systems and Dynamics Nonlinear systems and dynamics are an essential aspect of modern science and engineering. These systems are characterized by their complex behavior, which cannot be described by simple linear equations. Nonlinear systems are ubiquitous in nature, from the behavior of living organisms to the dynamics of the universe. Understanding these systems is crucial for developing new technologies, predicting and controlling complex phenomena, and advancing scientific knowledge.One of the most important features of nonlinear systems is their sensitivity to initial conditions. Small changes in the initial conditions of a nonlinear system can lead to significant differences in the system's behavior over time. This phenomenon, known as the butterfly effect, is a fundamental aspect of chaos theory. Chaos theory has broad applications in various fields, including meteorology, economics, and biology. The butterfly effect can be seen in many natural phenomena, such as weather patterns, the growth of populations, and the behavior of ecosystems.Nonlinear systems are also characterized by the presence of feedback loops. Feedback loops are essential for maintaining stability in complex systems. They allow the system to adjust its behavior in response to changes in its environment. Feedback loops can be positive or negative, depending on whether they amplify or dampen the system's response to external stimuli. Positive feedback loops can lead to runaway behavior, while negative feedback loops can stabilize the system.Another important aspect of nonlinear systems is their ability to exhibit oscillatory behavior. Oscillations are a fundamental feature of many natural phenomena, from the beating of a heart to the oscillation of a pendulum. Nonlinear systems can exhibit a wide range of oscillatory behavior, from simple periodic oscillations to chaotic oscillations. Understanding the dynamics of oscillatory systems is crucial for developing new technologies, such as electronic circuits, and for predicting and controlling complex phenomena, such as the spread of diseases.Nonlinear systems also play a crucial role in the study of complex networks. Complex networks are ubiquitous in nature and society, from the networks of neurons in the brain to the networks of social interactions between individuals.Understanding the dynamics of complex networks is crucial for developing new technologies, such as the internet, and for predicting and controlling complex phenomena, such as the spread of information or the emergence of new social behaviors. Nonlinear dynamics plays a central role in the study of complex networks, as it allows us to understand how the behavior of individual nodes inthe network affects the behavior of the network as a whole.In conclusion, nonlinear systems and dynamics are a crucial aspect of modern science and engineering. These systems are characterized by their complex behavior, sensitivity to initial conditions, presence of feedback loops, ability to exhibit oscillatory behavior, and importance in the study of complex networks. Understanding the dynamics of nonlinear systems is crucial for developing new technologies, predicting and controlling complex phenomena, and advancingscientific knowledge. Nonlinear dynamics is a rapidly growing field with broad applications in various fields, including physics, biology, engineering, andsocial science. As such, it is an essential area of study for anyone interested in understanding the world around us.。
非平衡热动力学 书
非平衡热动力学书英文回答：Non-equilibrium thermodynamics is a branch of thermodynamics that deals with systems that are not in thermodynamic equilibrium. In contrast to equilibrium thermodynamics, which focuses on systems in a state of balance, non-equilibrium thermodynamics studies the behavior of systems that are undergoing processes and are far from equilibrium.In non-equilibrium thermodynamics, the laws of thermodynamics still apply, but they are modified to account for the deviations from equilibrium. These modifications are necessary because non-equilibrium systems are characterized by the presence of gradients, such as temperature, pressure, and concentration gradients, which drive the flow of energy and matter.One of the key concepts in non-equilibriumthermodynamics is entropy production. Entropy is a measure of the disorder or randomness in a system, and in equilibrium, it reaches its maximum value. However, in non-equilibrium systems, entropy can be produced or consumed. This production of entropy is related to the irreversible processes that occur in non-equilibrium systems.To illustrate this concept, let's consider a cup of hot coffee placed in a room at a lower temperature. Initially, the coffee is in a non-equilibrium state because there is a temperature gradient between the coffee and the room. As time passes, heat will flow from the hot coffee to the colder room until they reach thermal equilibrium. During this process, entropy is being produced due to the irreversible heat transfer. Once equilibrium is reached, there is no more entropy production.Another important aspect of non-equilibrium thermodynamics is the concept of steady-state. A system is said to be in a steady-state when its macroscopic properties, such as temperature and pressure, remain constant over time, even though there may be continuousflow of energy or matter through the system. A common example of a steady-state system is a heat exchanger, where hot and cold fluids flow through separate channels but maintain a constant temperature difference.中文回答：非平衡热力学是热力学的一个分支，研究的是不处于热力学平衡状态的系统。
无轴承异步电机转子不平衡振动仿真补偿
第37卷第7期计算机仿真2020年7月文章编号:1006-9348(2020)07-0243-05无轴承异步电机转子不平衡振动仿真补偿李彪、卜文绍\陈有鹏1(1.河南科技大学信息工程学院，河南洛阳471023;2.河南科技大学电气学院，河南洛阳471023)摘要:针对无轴承异步电机转子质量不平衡振动问题，分析了无轴承异步电机不平衡振动产生原理，采用‘旋转坐标+低通滤 波’的方法将产生的不平衡振动位移前馈补偿到转子系统中。
即从位移传感器中提取转子径向振动位移，将转子径向振动 位移旋转变换到与转子同步旋转的坐标系上，再使变换后的信号通过一个一阶低通b u tte rw o rth滤波器，获得与转子同步旋 转的不平衡振动位移信号，不平衡振动位移信号在经过旋转坐标变换到静止坐标上，将获得的静止坐标系下的不平衡振动 信号前馈补偿到经过反馈补偿后的转子不平衡振动位移上。
搭建仿真模型，仿真结果验证了转子振动位移峰-峰值显著减 小，且转子振动位移远小于气隙间隙，无轴承异步电机运行性能得以改善。
关键词:无轴承异步电机;低通滤波器;不平衡振动位移;坐标变换;前馈补偿中图分类号:TM933 文献标识码：BSimulation of Rotor Unbalanced Vibration Simulation forBearingless Induction MotorLI Biao1 ,BU Wen-shao2,CHEN You-peng1(1. Information Engineering College,Henan University of Sci &T e c h,Luoyang Henan471023, China；2. Electrical Engineering College,Henan University of Sci &T e c h,Luoyang Henan471023, China)A B S T R A C T：For the unbalanced vibration of rotor in bearingless induction motor,the principle of generating unbalanced vibration was analyzed.The forward compensation of unbalanced vibration displacement was changed into the rotor system by combing rotation coordinate with low-pass f ilter.In other words,the radial vibration displacement of rotor was extracted from the displacement sensor,and the radial vibration displacement of rotor was rotationally transformed into the coordinate system that rotated synchronously with the rotor.A n d then,the transformed signal was passed through a first-order low-pass Butterworth filter,so as to obtain the signal of unbalanced vibration displacement that rotated synchronously with the rotor.Moreover,the signal of unbalanced vibration displacement was transformed into the stationary coordinate by rotating coordinate.Finally,the unbalanced vibration signal in stationary coordinate system was compensated to the unbalanced vibration displacement of rotor after feedback compensation.The simulation model was built and the results prove that the peak-to-peak value of rotor vibration displacement i s significantly reduced,and the displacement i s m u c h smaller than the air gap.The performance of beinngless asynchronous motori s improved.K E Y W O R D S：Bearingless induction motor；Low-pass filter；Unbalanced vibration displacement；Coordinate transformation；Feedforward compensationi引言传统电机由于定转子的机械接触,转子转动过程中存在 机械摩擦，转速越高摩擦损耗越严重，严重制约电机向高转 速方向的发展，磁轴承电机具有无接触,无润滑，无噪音等优 点，但磁轴承电机需要额外的附加励磁磁场,轴向尺寸的长基金项目：国家自然科学基金项目（51277053)收稿日期:20丨9-01-16修回日期:2019-02-27度也限制了电机向更高转速方向的发展。
Resonant CP Violation due to Heavy Neutrinos at the LHC
School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom b Center for Theoretical Physics, School of Physics, Seoul National University, Seoul, 151-722, Korea
Abstract
The observed light neutrinos may be related to the existence of new heavy neutrinos in the spectrum of the SM. If a pair of heavy neutrinos has nearly degenerate masses, then CP violation from the interference between tree-level and self-energy graphs can be resonantly enhanced. We explore the possibility of observing CP asymmetries due to this mechanism at the LHC. We consider a pair of heavy neutrinos N1,2 with masses ranging from 100 − 500 GeV and a mass-splitting ∆mN = mN2 − mN1 comparable to their widths ΓN1,2 . We find that for ∆mN ∼ ΓN1,2 , the resulting CP asymmetries can be very large or even maximal and therefore, could potentially be observed at the LHC.
基于余量谐波平衡法的质点振动系统高阶近似与频率响应分析
基于余量谐波平衡法的质点振动系统高阶近似与频率响应分析国忠金;张伟;夏丽莉【摘要】基于谐波平衡方法,构建了用于研究旋转抛物线上的质点运动系统的余量谐波平衡解程序.获得了系统高阶解析近似振动频率及稳态响应.研究了系统稳态下的振动频率随系统非线性项系数、初始振幅、线性刚度系数的变化趋势,给出了初始振幅、非线性项系数对系统振动频率响应的影响.研究结果表明,给出的2-阶余量谐波平衡近似比已有的方法结果更加精确,其相对误差大大降低,系统稳态下的振动频率随非线性项系数、初始振幅的增大而减少,随线性刚度系数的增大而增大.【期刊名称】《振动与冲击》【年(卷),期】2018(037)020【总页数】5页(P154-158)【关键词】质点振动系统;余量谐波平衡;高阶近似;频率响应【作者】国忠金;张伟;夏丽莉【作者单位】泰山学院数学与统计学院,山东泰安271000;北京工业大学机电学院,北京100124;北京工业大学机电学院,北京100124;机械结构非线性振动与强度北京市重点实验室,北京100124;北京工业大学机电学院,北京100124【正文语种】中文【中图分类】O322;O328非线性动力学与振动分析对机械、结构动力学问题研究是非常重要的，它能够全面了解和准确预测系统运动特性。
近年来，旋转抛物线上的质点运动系统引起了广泛关注[1-2]，其控制方程为如下非线性自治微分方程(1)式中：△为线性刚性系数且△>0；q为非线性项系数且q>0；A为初始振幅且A>0。
针对方程(1)，文献[3]运用同伦与摄动耦合方法，文献[4]利用改进的迭代摄动法，文献[5]运用能量平衡法、同伦摄动法、振幅-频率公式以及微分变换法，文献[6]运用哈密尔顿法，文献[7]运用变分法，文献[8]运用优化同伦分析方法获得了其稳态下的振动频率。
一方面，以上文献给出的近似振动频率与精确结果之间存在一定的误差，其近似精度有待提高。
另一反面，摄动方法适用于弱非线性系统，对系统稳态下的振动频率随系统参数尤其是非线性项系数变化的振动特性还需进一步研究。
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a r X i v :h e p -p h /9710558v 1 31 O c t 1997Non-perturbative Dynamics of the Heavy-Light Quark SystemNora Brambilla 1and Antonio Vairo 1Institut f¨u r Theoretische Physik,Universit¨a t HeidelbergPhilosophenweg 16,D-69120Heidelberg,FRGAbstract.We obtain the non-perturbative interaction kernel of the heavy-light quark system in a gauge-invariant approach.The potential and the sum rules limit are discussed.Some insight is given in the light quark interaction.INTRODUCTIONThe dynamics of a system composed by two heavy quarks is well understood in terms of a potential interaction (static plus relativistic corrections)obtained from the semirelativistic reduction of the QCD dynamics (see e.g.[1]).If at least one of the quarks is light the system behaves relativistically and a pure relativistic treatment becomes necessary (via Dirac or Bethe–Salpeter equations).A lot of phenomenologically justified relativistic equations can be found in the literature but up to now we miss a relativistic treatment which follows directly from QCD.Our work goes in this direction.In order to simplify the problem we focus on the heavy-light mesons in the non-recoil limit (i.e.infinitely heavy antiquark).Only at the end we will briefly discuss the two-body case.Our starting point is the quark-antiquark gauge-invariant Green function taken in the infinite mass limit of one particle.The only dynamical assumption is on the behaviour of the Wilson loop (i.e.on the nature of the non-perturbative vacuum).The gauge invariance of the formalism guarantees that the relevant physical information are preserved at any step of our derivation.In this way we obtain a QCD justified fully relativistic interaction kernel for the quark in the infinite mass limit of the antiquark.This kernel reduces in some region of the physical parameters to the heavy quark mass potential,and leads in some other region to the heavy quark sum rules results,providing in this way an unified description.We discuss our result with respect to the old-standing problem of the Lorentz structure of the Dirac kernel for a confining interaction.A derivation of the main results summed up here and a complete reference list can be found in [2,3].THE RELATIVISTIC INTERACTION IN THEONE-BODY LIMITThe quark-antiquark Green function in the Feynman–Schwinger representation can be written as a quantomechanical path integral over the quark trajectories (z1(t1)and z2(t2))(quenched approximation),G inv(x,u,y,v)=12× ∞0dT2 u v D z2e−iT20dt2m2+˙z224σ(1)µνFµν(z1)×e iT20dt2g2 y x dx′µ y x dy′νDµν(x′,y′) ,Dµν(x,y)≡x k y l 10dααn(µ) 10dββn(ν) F kµ(x0,αx)F lν(y0,βy) ,(2)Assumption(2)corresponds to the so-called stochastic vacuum model.Insert-ing expression(2)in Eq.(1)and expanding the exponential taking into ac-count only thefirst planar graph,we have S D=S0+S0KS D with K(y′,x′)=γνS0(y′,x′)γµDµν(x′,y′)(see Fig. 1.Therefore,K can be interpreted as the in-teraction kernel of the Dirac equation associated with the motion of a quark in thefield generated by an infinitely heavy antiquark.Assuming that the correlator Fµλ(x)Fνρ(y) depends only on the difference between the coordinates and defining F kµ(x0,αx)F lν(y0,βy) ≡f kµlν(x0−y0,αx−βy).we haveFIGURE1.The interaction kernel K.K (q,p )=−g 2(2π)δ(p 0−q 0)+∞−∞dτ1dααn (µ)10dββn (ν)×∂∂q ld 3r ei (p −q )·r γνθ(−τ)Λ+(t )γ0e −i (p0−Et )τ−θ(τ)Λ−(t )γ0e−i (p 0+E t )τγµf kµlν(τ,(α−β)r ),(3)where t ≡(βp −αq )/(β−α),E t =√2E t.Focusingon the non-perturbative part,the Lorentz structure is f n .p .µλνρ(x )=1l cF 2(0)24N c+∞−∞dτ∞dλD (τ2−λ2)we obtain the well-known Eichten and Fein-berg result,V (r )=σr −C −σ·Lr,where C =g2 F 2(0)(2π)4d 4q12αF 2(0)∂p k∂CONCLUSIONIn the literature a Dirac equation with scalar confining kernel(i. e.K≃σr) has been used to evaluate non-recoil contributions to the heavy-light meson spec-trum.The main argument in favor is the nature of the spin-orbit potential for heavy quarks.Our kernel(3)follows simply from the assumption on the gauge fields dynamics given by Eq.(2)and by taking only one non-perturbative gluon insertion on the quark fermion line.When performing the potential reduction of this kernel in the heavy quark case(A)we obtain exactly the expected static and spin-dependent potentials.Therefore our conclusion is that there exists at least one non scalar kernel which reproduces for heavy quark not only the Eichten and Feinberg potentials in the long distances limit,but also the entire stochastic vac-uum model spin-dependent potential.Moreover when considering a,the inverse of the correlation length,small with respect to all the energy scales(case B),the kernel(3)gives back the leading heavy quark sum rules results.It is possible to extend the range of applicability of Eq.(3)to more realistic cases,like D s and B s mesons where the light quark mass is smaller than a(case C).The relevant part of the kernel is also in this case not a simply scalar one.An attempt to extend the present approach to the two-body case is given in[3]. The equivalent graphs of Fig.1seem to play a crucial role(in the two-body case such kind of graph exists for any fermion line and as exchange graph).Nevertheless these graphs are not sufficient in order to provide a complete relativistic description of the two-body system.The main difficulty is that in this case it does not exist a gauge which automatically cancels the contributions coming from the end-point strings.These contributions are necessary in order to restore gauge invariance.REFERENCES1.N.Brambilla and A.Vairo,Phys.Rev.D55,3974(1997).2.N.Brambilla and A.Vairo,Phys.Lett.B407,167(1997).3.N.Brambilla and A.Vairo,Phys.Rev.D56,1445(1997).4.H.G.Dosch and Yu.A.Simonov,Phys.Lett.B205(1988)339;Yu.A.Simonov,Nucl.Phys.B307(1988)512;B324(1989)67.。