On the relation between circular velocity and central velocity dispersion in high and low s

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Quantum Version of Gauge Invariance and Nucleon Internal Structure

Xiao-Fu L¨ u
Department of Physics, Sichuan University, Chengdu, 610064, China The conﬂict between canonical commutation relation and gauge invariance, which both the momentum and angular momentum of quark and gluon should satisfy, is clariﬁed. The quantum version of gauge invariance is studied. The gauge independence of the matrix elements of quark momentum and angular momentum operators between physical states are proved. We suggest to use the canonical quark momentum and angular momentum distributions to describe the nucleon internal structure in order to establish an internal consistent description of hadron spectroscopy and hadron structure. The same problem for the atomic spectroscopy and structure is discussed.
arXiv:hep-ph/0510285v1 21 Oct 2005
I.
CONFLICT BETWEEN CANONICAL COMMUTATION RELATION AND GAUGE INVARIANCE

Comparison of Fifth-OrderWENO Scheme and (2)

put.Phys.doi:10.4208/cicp.110212.021112a Vol.14,No.3,pp.599-620 September2013Comparison of Fifth-Order WENO Scheme andFinite Volume WENO-Gas-Kinetic Scheme forInviscid and Viscous Flow SimulationJun Luo,Lijun Xuan and Kun Xu∗Mathematics Department,Hong Kong University of Science and Technology,Clear Water Bay,Kowloon,Hong Kong.Received11February2012;Accepted(in revised version)2November2012Available online25January2013Abstract.The development of high-order schemes has been mostly concentrated onthe limiters and high-order reconstruction techniques.In this paper,the effect of theﬂux functions on the performance of high-order schemes will be studied.Based on thesame WENO reconstruction,two schemes with differentﬂux functions,i.e.,theﬁfth-order WENO method and the WENO-Gas-kinetic scheme(WENO-GKS),will be com-pared.Theﬁfth-orderﬁnite difference WENO-SW scheme is a characteristic variablereconstruction based method which uses the Steger-Warmingﬂux splitting for invis-cid terms,the sixth-order central difference for viscous terms,and three stages Runge-Kutta time stepping for the time integration.On the other hand,theﬁnite volumeWENO-GKS is a conservative variable reconstruction based method with the sameWENO reconstruction.But,it evaluates a time dependent gas distribution functionalong a cell interface,and updates theﬂow variables inside each control volume byintegrating theﬂux function along the boundary of the control volume in both spaceand time.In order to validate the robustness and accuracy of the schemes,both meth-ods are tested under a wide range ofﬂow conditions:vortex propagation,Mach3step problem,and the cavityﬂow at Reynolds number3200.Our study shows thatboth WENO-SW and WENO-GKS yield quantitatively similar results and agree witheach other very well provided a sufﬁcient grid resolution is used.With the reduc-tion of mesh points,the WENO-GKS behaves to have less numerical dissipation andpresent more accurate solutions than those from the WENO-SW in all test cases.Forthe Navier-Stokes equations,since the WENO-GKS couples inviscid and viscous termsin a singleﬂux evaluation,and the WENO-SW uses an operator splitting technique,itappears that the WENO-SW is more sensitive to the WENO reconstruction and bound-ary treatment.In terms of efﬁciency,theﬁnite volume WENO-GKS is about4timesslower than theﬁnite difference WENO-SW in two dimensional simulations.The cur-rent study clearly shows that besides high-order reconstruction,an accurate gas evolu-tion model orﬂux function in a high-order scheme is also important in the capturing of600J.Luo,L.Xuan and K.Xu/put.Phys.,14(2013),pp.599-620 physical solutions.In a physicalﬂow,the transport,stress deformation,heat conduc-tion,and viscous heating are all coupled in a single gas evolution process.Therefore,it is preferred to develop such a scheme with multi-dimensionality,and uniﬁed treat-ment of inviscid and dissipative terms.A high-order scheme does prefer a high-ordergas evolution model.Even with the rapid advances of high-order reconstruction tech-niques,theﬁrst-order dynamics of the Riemann solution becomes the bottleneck forthe further development of high-order schemes.In order to avoid the weakness of thelow orderﬂux function,the development of high-order schemes relies heavily on theweak solution of the original governing equations for the update of additional degreeof freedom,such as the non-conservative gradients ofﬂow variables,which cannot bephysically valid in discontinuous regions.PACS:02.60Cb,47.11.Df,47.45.AbKey words:WENO scheme,gas-kinetic scheme,Euler equations,Navier-Stokes equations,high-order methods.J.Luo,L.Xuan and K.Xu/put.Phys.,14(2013),pp.599-620601 smoothness-dependent weight to each stencil and results in a(2r−1)th order accuracy. The WENO reconstruction is more accurate,efﬁcient and stable than the ENO reconstruc-tion.So far,the WENO schemes have found wide applications.The general impression is that the WENO scheme is not sensitive to theﬂuxes used,such as Lax-Friedrichs or Steger-Warming.The full accuracy of the scheme mainly depends on the order of the re-construction.One of the purpose of the current paper is to test the effect ofﬂux functions on the performance of high-order schemes.It turns out that besides the high-order ini-tial reconstruction theﬂux modeling also plays an essential role to capture accurateﬂow evolution,and to reduce the sensitive dependence of the solution on the initial recon-struction,especially in a barely resolvedﬂow region.Allﬂux functions are equivalent in a well-resolvedﬂow region,because theﬂux consistency plays a dominant role here.But their performance may deviate from each other in not well-resolved cases,such as3or4 points inside a boundary layer.Currently,in order to increase the accuracy of the WENO scheme,many attempts have been tried to develop hybrid schemes,where the WENO is used in the discontinuous region and high-order compact scheme is used in the smooth region[13,18].The aim of the hybrid scheme is to basically develop a method which could make a smooth transition between the upwind and central difference method,be-cause the fundamental basis of the1st-order Riemann solver for wave decomposition and theﬂux splitting technique contradicts with theﬂow physics in the smooth region,where the high-order spatial and temporal evolution are fully coupled.In the smooth region,the traditional central difference approximation with Cauchy-Kowalevskaya technique is far more appropriate to describeﬂow evolution than upwind concept.The Riemann solu-tion is a low order dynamic model,which is needed to model discontinuousﬂow in order to introduce enough dissipation.And this amount of dissipation is closely related to the initial jump at the cell interface[26].Without modifying the1st-order Riemann dynamic model,the development of high-order schemes becomes a game of reconstruction,since the interface jump is the only freedom many high-order schemes are able to control.Un-fortunately,there is no principle to design such an optimal and universal interface jump, and this kind of research will be endless.In order to get out of this dilemma,the use of a high-order dynamic evolution model,which could make a smooth transition between the upwind and central difference scheme,is necessary.For a second order scheme for the inviscidﬂow,we have such a high-order dynamic model,which is the generalized Riemann solver and the gas-kinetic scheme[1,11].In the past decades,a gas-kinetic scheme(GKS)based on the kinetic equation has been developed for the modeling of gas evolution process starting from a discontinu-ity[12,16,26].Theoretically,the GKS does not target to solve accurately the gas kinetic BGK model[2],but uses the kinetic equation to do the modeling around a cell interface. In the GKS evolution,the whole process from particle free transport to the Navier-Stokes (NS)or Euler solution formation has been recovered[9,28].Theﬂow regime of the gas evolution depends on the ratio of time step to the particle collision time.In the smooth ﬂow region,based on the Chapman-Enskog expansion,a time dependent gas distribu-602J.Luo,L.Xuan and K.Xu/put.Phys.,14(2013),pp.599-620 tion function corresponding to the NS or Euler solutions can be obtained accurately by the GKS.This limiting formulation is basically the central difference method.In the dis-continuity region,where the physical solution cannot be well resolved by the numerical cell size,theoretically it is not necessary to know the precise”macroscopic”governing equations here,because it is not needed to incorporate a precise amount of numerical dissipation.But,in such a region the gas evolution model should follow a path which is consistent with the physical one,such as keeping a non-equilibrium gas distribution function at a cell interface to cope with the dissipative mechanism of a physical shock layer[11,29].This limiting case is theﬂux vector splitting upwind method.The advan-tage of the GKS is that theﬂux function makes a smooth transition from a upwind(kinetic scale)to a central difference(hydrodynamic scale)method.Recently,based on the WENO reconstruction and high-order gas-kinetic solution [12],aﬁnite volume high-order WENO-gas-kinetic scheme(WENO-GKS)has been de-signed and tested for multi-dimensionalﬂows[15].Different from the traditional WENO scheme,the WENO technique is only used in the data reconstruction of the conservative ﬂow variables at the beginning of each time step.Starting from the high-order recon-struction,a space and time dependent gas distribution function is obtained along the tangential direction of a cell interface,from which the numericalﬂux is evaluated and used in aﬁnite volume scheme.In GKS,there is no need to use the Runge-Kutta time stepping and Gaussian point integration along a cell interface for the update ofﬂow variables inside each control volume.Among all WENO schemes we tested,theﬁnite difference WENO method with Steger-Warmingﬂux splitting gives the best results.The detailed formulation of this WENO scheme is presented in Section3.In order to show the effect of aﬂux function on the per-formance of high-order schemes,the numerical solutions of theﬁnite difference WENO-Steger-Warming scheme(WENO-SW)will be compared with that of theﬁnite volume WENO-GKS.The test cases are carefully chosen in order to test the accuracy,the shock-capturing ability,the robustness,and the stability of the schemes.In order to eliminate the numerical error due to the complicated geometry,all tests are conducted in rectangu-lar meshes.The5th-order WENO reconstruction is used on the characteristic variables in the WENO-SW,and the same WENO reconstruction is used on the conservative vari-ables in the WENO-GKS.The reason we use conservative variables for the WENO-GKS is that the scheme is not sensitive to the variable used in the reconstruction,because the evolution of the whole reconstructed curves will be followed dynamically in theﬂux evaluation.2The5th order WENO reconstructionThe5th order WENO reconstruction on a uniform rectangular mesh is a standard recon-struction method[20].Assume that Q is the variable to be reconstructed.¯Q i is the cell averaged value inJ.Luo,L.Xuan and K.Xu/put.Phys.,14(2013),pp.599-620603 the ith cell.Q l i and Q r i are the two pointwise values reconstructed at the left and right interfaces of the ith cell.The5th WENO reconstruction is deﬁned as,Q r i=2∑s=0w s q(s)i,Q l i=2∑s=0˜w s˜q(s)i,whereq(0)i=16¯Qi+1−16¯Qi−1+53¯Qi+1,q(2)i=16¯Qi−1+116¯Qi−73¯Qi+2,˜q(1)i=16¯Qi−16¯Qi−2+53¯Qi,w s=αs(ǫ+βs)2,˜w s=˜αs(ǫ+βs)2,s=0,1,2,β0=134(3¯Q i−4¯Q i+1+¯Q i+2)2,β1=134(¯Q i−1−¯Q i+1)2,β2=134(¯Q i−2−4¯Q i−1+3¯Q i)2,d0=˜d2=35,d2=˜d0=1604J.Luo,L.Xuan and K.Xu/put.Phys.,14(2013),pp.599-620 The BGK equation in2-D isf t+ u·∇f=g−f2(u2+v2+ξ2) T,dξ=dξ1dξ2···dξK,and K is the number of degrees of internal freedom,i.e.,K=(4−2γ)/(γ−1)for2-Dﬂow andγis the speciﬁc heat ratio.Since the mass,momentum, and energy are conserved during particle collisions,f and g satisfy the conservation con-straint, (g−f)ψαd u d v dξ=0,α=1,2,3,4,(3.3)at any point in space and time.The integral solution of(3.1)isf( x,t, u,ξ)=12g l,r0((a l,r1)2+d l,r11)x2+1J.Luo,L.Xuan and K.Xu/put.Phys.,14(2013),pp.599-620605Here g l0and g r0are two Maxwellian distribution functions which correspond to the left and right macroscopic variables respectively.For example,g l0corresponding to W l= (ρl,(ρl U l),(ρl V l),(ρl E l))T has the formg l0=ρl λl2eλl((u−U l)2+(v−V l)2+ξ2),(3.7)whereλl equals to m/2kT l,m is the molecular mass,k is the Boltzmann constant,and T l is the temperature.For the modeling of the local equilibrium distribution function g,we can use the Taylor expansion of the equilibrium state and getg( x,t, u,ξ)=¯g+¯g¯a1x+¯g¯a2y+¯g¯At+12¯g(¯a22+¯d22)y2+¯g(¯a1¯a2+¯d12)xy+1τ t0g(−ut′,y−vt′,t′, u,ξ)e−(t−t′)/τdt′=C1¯g+C2¯g¯a1u+C1¯g¯a2y+C2¯g¯a2v+C3¯g¯A+12C1¯g(¯a22+¯d22)y2+C2¯g(¯a22+¯d22)vy+12C5¯g(¯A2+¯B)+C6¯g(¯A¯a1+¯b1)u+C3¯g(¯A¯a2+¯b2)y+C6¯g(¯A¯a2+¯b2)v,(3.10) ande−t/τf0(−ut,y−vt, u,ξ)= C7f l0(−ut,y−vt, u,ξ),u>0,C7f r0(−ut,y−vt, u,ξ),u<0,(3.11) whereC1=1−e−t/τn,C2=(t+τ)e−t/τn−τ,C3=t−τ+τe−t/τn,(3.12a) C4=(−t2−2τt)e−t/τn,C5=t2−2τt,C6=−τt(1+e−t/τn),C7=e−t/τn.(3.12b) In any numerical simulation,numerical dissipation is necessary in order to cope with the limited cell resolution.In order to add numerical dissipation but not change theﬂow606J.Luo,L.Xuan and K.Xu/put.Phys.,14(2013),pp.599-620 property,in(3.12),we distinguish two particle collision times.One is the physical one (τ)which corresponds to the real collision time in the Chapman-Enskog expansion for the Navier-Stokes solution,and another one is the numerical one(τn)which takes into account the effect of initial pressure jump at the cell interface.As considered in[15],the BGK-NS solver uses√τ=µ/¯p,τn=µ/¯p+β∆x√¯λ+β∆xJ.Luo,L.Xuan and K.Xu/put.Phys.,14(2013),pp.599-620607 becomesW n+1ij =W n ij+12∆x i−1∆x∆y t n+1t n12∆y j[F i−1/2(t,y)−F i+1/2(t,y)]dydt,(3.16)where F j−1/2(t,x),F j+1/2(t,x),F i−1/2(t,y),F i+1/2(t,y)are theﬂuxes along the four cell in-terfaces respectively.Because the time-dependentﬂuxes can be explicitly integrated along a cell interface in the GKS,theﬁnal scheme presents a high-orderﬂow transport through the interface within a time step without using Runge-Kutta time stepping and ﬂux construction at Gauss points.3.2WENO-Steger-Warming schemeThe WENO-SW solves the hydrodynamic equationsW t+F(W)x+G(W)y=F v(W,W x,W y)x+G v(W,W x,W y)y,(3.17) where W is the conservative variables,F and G are the inviscidﬂuxes,and F v and G v are the viscousﬂuxes.For aﬁnite-difference scheme,we need to construct both inviscid and viscousﬂuxes at the cell interface.The following WENO scheme is the5th orderﬁnite-difference WENO-Steger-Warming scheme,where the Steger-Warming splitting[23]is used to obtain the inviscidﬂuxes at the cell interface.3.2.1Inviscidﬂux reconstructionThe x-directionﬂux F can be decomposed asF=RΛLW,(3.18) whereΛ=diag[λ1,λ2,···,λn]is a diagonal matrix with eigenvalues of∂F/∂W,R is the right eigenvector matrix and L is the left one,n is the number of the equations.Then,F can be split according toF=F++F−.(3.19) In the Steger-Warming splitting,F±=RΛ±LW,Λ=Λ++Λ−.(3.20) andΛ±=diag[λ±1,λ±2,···,λ±n]withλ±i=λi± 2(i=1,···,n),(3.21)whereεis a small constant.In our simulation,ε=1.0e−3.For the construction of the numerical x-directionﬂuxˆF i+1/2,j at the cell interface (x i+1/2,y j)of the cell(i,j),we use the following steps.608J.Luo,L.Xuan and K.Xu/put.Phys.,14(2013),pp.599-6201.Split theﬂuxes F i+l,j(l=−2,···,3)in the surrounding cells to get F±i+l,j(l=−2,···,3).e the variable W i+1/2,j=(W i,j+W i+1,j)/2to calculate the left and right eigenvector matrixL i+1/2,j and R i+1/2,j at the cell interface.3.Change the conservativeﬂuxes F±i+l,j(l=−2,...,3)to the characteristicﬂuxes˜F±i+l,j(l=−2, (3)by˜F±i+l,j=L i+l/2,j F±i+l,j(l=−2,···,3).e the5th order WENO reconstruction(see Section2)on the characteristicﬂuxes˜F±i+l,j(l=−2,···,3)to get the characteristicﬂux˜F±i+1/2,j,and calculate theﬁnal characteristicﬂux˜Fi+1/2,j =˜F+i+1/2,j+˜F−i+1/2,j.5.Construct theﬂuxes for the conservative variables at the cell interface byˆFi+1/2,j=R i+1/2,j˜F i+1/2,j.In a similar way,we can get the y-directional numericalﬂuxˆG i,j+1/2at the cell inter-face(x i,y j+1/2).3.2.2Viscousﬂux reconstructionThe viscousﬂuxes F v and G v are related to the derivatives ofﬂow variables.For the NS equations,we need to calculate T x,U x,U y,V x and V y in order to get the x-direction viscousﬂux F v,where T is the temperature,U is the x-direction velocity,and V is the y-direction velocity.So,the basic step is to construct the derivatives of a variable q.Firstly,in each cell(i,j),we use a6th-order central difference to calculate the deriva-tive,i.e., ∂q60∆x.(3.22)Then,with all the derivatives,we can calculate the x-directional and y-directional viscous ﬂuxes F v i,j and G v i,j in each cell.Finally,the numerical viscousﬂuxes at the cell interfaces can be calculated by the6th order central interpolationˆF vi+1/2,j =37(F v i,j+F v i+1,j)−8(F v i−1,j+F v i+2,j)+F v i−2,j+F v i+3,j60.(3.23b)3.2.3Time evolutionTheﬁnal numericalﬂuxes at the cell interface are denoted byˆF i+1/2,j=ˆF i+1/2,j−ˆF v i+1/2,j,(3.24a)ˆG i,j+1/2=ˆG i,j+1/2−ˆG v i,j+1/2.(3.24b)With the above numericalﬂuxes,we can get the increment∆W i,j=ˆF i+1/2,j−ˆF i−1/2,j∆y.(3.25)A3rd-order TVD Runge-Kutta method is used toﬁnish the time evolution,W(1)i,j=W n i,j−∆t∆W i,j(W n),(3.26a)W(2)i,j =34[W(1)i,j−∆t∆W i,j(W(1))],(3.26b)W(n+1)i,j =13[W(2)i,j−∆t∆W i,j(W(2))].(3.26c)4Numerical examplesFor all tests in this paper,the time step is determined by the CFL condition with CFL number0.5.4.1Mach3step problemThe Mach3step problem wasﬁrst proposed by Woodward and Colella in[25].The com-putational domain is[0,3]×[0,1].A step with height0.2is located at x=0.6.The upstream velocity is(U,V)=(3,0).The adiabatic slip Euler boundary condition is implemented at all boundaries.As explained in[25],the corner of the step is the center of a rarefaction fan and itis a singular point of theﬂow.Theﬂow will be much affected by the numerical errorsgenerated just in the neighborhood of this singular point.Almost for all WENO schemes,in order to get better results,a special treatment introduced in[25]has been applied.Or,a reﬁned mesh is used around the corner.In order to avoid confusion and compare theresults of the WENO-GKS and the WENO-SW truthfully,there is no special treatmentanywhere in the current study.The parameterǫ=10−6in the5th order WENO recon-struction is used.In the WENO-GKS,α=0andβ=100for numerical collision time in (3.14).Figs.1,2and3present the numerical solutions from both WENO-SW and WENO-GKS with different number of mesh points,i.e.,120×40,60×20,and30×10.All these results are consistent with theﬂow structures in[25].Based on the theseﬁgures,the WENO-SW gives a large tilted Mach stem above the up surface of the step,and the WENO-GKS presents a much straighter and shorter one.After a few shock reﬂections, the shock front of the WENO-SW gets smeared and disappeared in the case with30×10 mesh points case.But,for the WENO-GKS,the shock reﬂection can be observed clearly in the coarse mesh case.This illustrates that WENO-SW has large numerical dissipation in comparison with WENO-GKS.In order to understand why the WENO-SW has a tiltedFigure1:Mach3the WENO-GKS5.(upper one)and the WENO-SW(lower one).In eachﬁgure,there are50contours from0.5toFigure2:Mach3the WENO-GKS (upper one)and the WENO-SW(lower one).In eachﬁgure,there are50contours from0.5to5.Mach stem,the WENO-GKS is tested for the NS solution with Reynolds number1000, but with the Euler boundary condition.Fig.4shows the comparison of WENO-SW for the inviscidﬂow and WENO-GKS for the viscousﬂow.It shows that the solution given by the WENO-SW is more close to a viscous solution.This indicates again that a high level of numerical dissipation exists in the WENO-SW.Figure3:Mach3the WENO-GKS (upper one)and the WENO-SW(lower one).In eachﬁgure,there are50contours from0.5to5.Figure4:Mach3Upper one:NS solution with Reynolds number1000by the WENO-GKS.Lower one:Euler solution by the WENO-SW.In each ﬁgure,there are50contours from0.5to5.4.2Isentropic periodic vortex propagationThis is a test for the accuracy of the Euler solutions(see[13,20]).The initial condition is given byκ(U(x,x,0),V(x,y,0))=(1,1)+e1−r2,S(x,y,0)=1,8γπ2Figure5:Isentropic periodic vortex propagation:density distribution at t=10(1period)by the WENO-GKSFigure6:Isentropic periodic vortex propagation:density distribution at t=100(10periods)by the WENO-GKS (WGKS)and the WENO-SW(WSW).where the temperature T and the entropy S are related to the densityρand the pressurep byT=pργ,and(¯x,¯y)=(x−5,y−5),r2=¯x2+¯y2,and the vortex strengthκ=5.The computational domain is[0,10]×[0,10].The periodic boundary condition is used in both directions.The numerical results with80×80and40×40cells at t=10(1period)and t=100 (10periods)are shown in Figs.5and6for both WENO-SW and WENO-GKS.In order to show the relationship between the parameterǫin the WENO reconstruction and the accuracy of the numerical solution,we test this case with two differentǫ,which are EPS1 (ǫ=10−6)and EPS2(ǫ=10−2).Theoretically,a large value ofǫpresents a reconstruction with more equally weights for different stencils.For a smoothﬂow,a smallǫmakes thereconstruction to reduce to a3rd order interpretation.A largerǫbalances the weights of different stencils and make the reconstruction to be close to the5th order interpolation. As shown in Figs.5and6,for such a smoothﬂow,in general,the reconstruction with EPS2will give more accurate and less dissipative results than that from EPS1for both WENO-SW and WENO-GKS.At the same time,for both meshes the results from WENO-GKS are more accurate than these from WENO-SW,especially at a mesh size40×40and t=100,see Fig.6.At time t=10and40×40mesh points,see Fig.5,EPS1introduces numerical dissipation and presents undershoot for both WENO-SW and WENO-GKS. However,with EPS2,the WENO reconstruction introduces overshoot for both schemes. But,in both cases due to the time accurate gas evolution model in the gas-kinetic scheme, the WENO-GKS could increase the undershoot and decrease the overshoot in compari-son with WENO-SW.In other words,the WENO-GKS has a better capacity to drive the solution in the correct direction due to the participation of the subcellﬂow distribution in the construction of the dynamicalﬂux function.More speciﬁcally,the gas-kineticﬂux function does not only depend on theﬂow variables at the cell interface,but also takes into account the whole curve evolution around the interface.Based on the above test,we can clearly realize that even though the gas-kinetic scheme is not solving the inviscid Euler equations directly,it gives accurate inviscid solution. In general,the numerical dissipation of the gas-kinetic scheme is less than the schemes based on the Riemann solvers,since the GKS can make a smooth transition from the up-wind to the central difference.This is one of the reason for the reduction of numerical dissipation in GKS in the smoothﬂow region.4.3CavityﬂowThe cavityﬂow at low Mach number is a standard test case for validating incompress-ible or low speed NSﬂow solvers.This is also a good test case for the shock capturing scheme in validating its capacity in capturing the Navier-Stokes solutions,especially with the non-linear limiters involved.For a directional splittingﬁnite volume scheme,it will be difﬁcult to present an accurate viscous solution with strong vortex structure.Fortu-nately,theﬁnite difference WENO-SW is an intrinsic multi-dimensional scheme due to its simultaneous evaluation of∂x and∂y terms of the NS equations at a grid point.At the same time,theﬁnite volume WENO-GKS has a multi-dimensionalﬂux function as well, where both x-and y-directionﬂow derivatives contributes to theﬂux in the interface normal direction.Theﬂow is bounded by a unit square and is driven by a uniform translation of the top boundary.In the simulation,the diatomic gasγ=1.4is considered.The boundary is isothermal and nonslip,and theﬂow condition is Ma=0.3and Re=3200,where Ma is the Mach number and Re is the Reynolds number.Since the benchmark solution is from incompressible NS equations,in order to avoid kinematic dissipation[6,27],most simu-lations in the past are based on the numerical methods for the incompressible equations or the artiﬁcial compressibility ones,where a continuous initial reconstruction across acell interface is assumed.However,here we are going to use the same shock capturing WENO reconstruction in the cavity simulation.This is challenge for any shock captur-ing NSﬂow solver,because the cell interface discontinuity may generate large numerical dissipation.In order to get the best results for the WENO-SW,we tested many boundary condi-tion treatments andﬁnally choose the following reconstruction.The special treatment is the following.The temperature and velocities are given directly since the boundary is isothermal and nonslip.Other data at the boundary are extrapolated from theﬂow.For the WENO-SW,the three layer interfaces close to the cavity wall but inside the compu-tational domain are specially treated.For theﬁrst layer cell interface,a3rd-order extrap-olation is used to reconstruct the conservative variableﬂuxes.For the second interface, a4th-order interpolation is used to reconstruct the conservative variableﬂuxes.For the third interface,a5th-order upwind interpolation is used to reconstruct the characteristic ﬂuxes.We use a4th-order interpolation to reconstruct the viscousﬂuxes at all of the three interfaces.For the WENO-GKS,the boundary treatment is relatively simple.A5th-order extrapolation for the conservative variables is used at the boundary.Firstly,we compare the efﬁciency in the cavity simulation for both schemes.The results are shown in Table1.As shown in the table,the WENO-GKS is about4times slower than the WENO-SW.This shows that the speed of WENO-GKS is similar to aﬁnite volume WENO scheme.As shown in[21],the computational cost of theﬁnite volume WENO scheme is indeed at least4times more expensive than theﬁnite difference one in a two-dimensional simulation.Table1:Average computational time for one time-step.WENO-GKS33×33cells 4.3276e-0037.4369e-002For the cavityﬂow simulations,we use three sets of meshes,which are101×101, 65×65,and33×33for both schemes.Fig.7shows the distributions of streamline for both schemes with a mesh of65×65points.Except around the up left corner,the streamlines from both WENO-SW and WENO-GKS are close to each other.The results of U-velocity along the vertical symmetric line at x=0.5and V-velocity along the horizontal symmetric line at y=0.5with different mesh sizes will be presented in detail.Fig.8shows U and V velocity distributions for a mesh with101×101points. The benchmark solution is from[5,24].As shown in thisﬁgure,the results from both WENO-SW and WENO-GKS are consistent with the reference solutions.The reconstruc-tion with EPS1presenting a better shock capturing property turns out to be more dissipa-tive than those with EPS2.This is clearly observed in the V-velocity for the WENO-SW. The WENO-GKS solutions are less sensitive to the values ofǫ.At such a reﬁned mesh, it is hard to distinguish the two solutions from WENO-GKS.The overall solutions from WENO-GKS are closer to the benchmark ones than the WENO-SW results.。

辩证法事物的主从关系英文

辩证法事物的主从关系英文Dialectics and the Master-Slave Relationship in Things.Dialectics, a fundamental principle of philosophy, explores the interactions and relationships between opposing forces within a system. At its core, dialectics posits that everything is in a state of constant flux, with opposing forces constantly interacting and transforminginto each other. This principle extends to the master-slave relationship, which exists not only in human society but also in the natural world and in abstract concepts.In the context of human society, the master-slave relationship often refers to a hierarchy of power and control. The master exercises authority over the slave, dictating their actions and decisions. This relationship is based on the unequal distribution of power and resources, with the master controlling the means of production and distribution.However, dialectics offers a different perspective on this relationship. It suggests that the master and slave are not static entities but are constantly interacting and transforming into each other. The master, in their exercise of power, is constantly shaping and reshaping the slave, while the slave, in their submission, is also influencing the master. This interaction creates a dynamic and fluid relationship that is constantly evolving.Moreover, dialectics suggests that the master-slave relationship is not absolute. The master can become the slave, and the slave can become the master, depending on the changing context and power structures. This transformation is not a linear process but a circular one, with the master and slave constantly swapping roles and positions.The master-slave relationship also exists in thenatural world. For example, in the ecosystem, certain species may exercise control over others, dictating their behavior and survival. However, dialectics suggests that this relationship is also fluid and dynamic. The dominantspecies can be overthrown by a weaker one, and the balanceof power can shift constantly. This transformation is not just a matter of survival but also of adaptation and evolution.In addition, the master-slave relationship can be observed in abstract concepts and ideas. For example, in philosophy and science, certain theories and concepts maybe considered dominant and authoritative for a period of time. However, with the advancement of knowledge and understanding, these theories and concepts can beoverturned and replaced by new ones. This transformation is not just a matter of intellectual curiosity but also of progress and development.Dialectics also highlights the role of contradiction in the master-slave relationship. Contradiction, a fundamental principle of dialectics, refers to the tension and opposition between two forces or ideas. In the master-slave relationship, the contradiction lies in the fact that the master and slave are simultaneously dependent on each other. The master needs the slave to perform tasks and maintaintheir position of power, while the slave needs the master for protection and survival. This dependence creates a tension and opposition that is constantly being resolved and reshaped.In conclusion, dialectics offers a unique perspective on the master-slave relationship. It suggests that this relationship is not static or absolute but is constantly interacting and transforming into each other. The master and slave are not separate entities but are part of a larger system that is constantly evolving and changing. This understanding of the master-slave relationship has important implications for our understanding of power, control, and hierarchy in both human society and the natural world.。

Inclusive weak decay rates of heavy hadrons

a r X i v :h e p -p h /0004257v 1 27 A p r 2000Theoretical Physics InstituteUniversity of MinnesotaTPI-MINN-00/17UMN-TH-1851-00April 2000Inclusive weak decay rates of heavy hadrons M.B.Voloshin Theoretical Physics Institute,University of Minnesota,Minneapolis,MN 55455and Institute of Theoretical and Experimental Physics,Moscow,117259Expanded version of a contribution to the ﬁnal report book of the Fermilab Workshop on B Physics at Tevatron Abstract A compact review of the theory,including some recent developments,of inclusive weak decay rates of charmed and b hadrons with an emphasis on predictions that can be tested in the forthcoming experiments.1IntroductionThe dominant weak decays of hadrons containing a heavy quark,c or b,are caused by the decay of the heavy quark.In the limit of a very large mass m Q of a heavy quark Q the parton picture of the hadron decay should set in,where the inclusive decay rates of hadrons, containing Q,mesons(Q¯q)and baryons(Qqq),are all the same and equal to the inclusive decay rateΓparton(Q)of the heavy quark.Yet,the known inclusive decay rates[1]are conspicuously diﬀerent for diﬀerent hadrons,especially for charmed hadrons,whose lifetimes span a range of more than one order of magnitude from the shortestτ(Ωc)=0.064±0.020 ps to the longestτ(D+)=1.057±0.015ps,while the diﬀerences of lifetime among b hadrons are substantially smaller.The relation between the relative lifetime diﬀerences for charmed and b hadrons reﬂects the fact that the dependence of the inclusive decay rates on the light quark-gluon‘environment’in a particular hadron is a pre-asymptotic eﬀect in the parameter m Q,which eﬀect vanishes as an inverse power of m Q at large mass.A theoretical framework for systematic description of the leading at m Q→∞term in the inclusive decay rateΓparton(Q)∝m5Q as well as of the terms relatively suppressed by inverse powers of m Q is provided[2,3,4]by the operator product expansion(OPE)in m−1Q. Existing theoretical predictions for inclusive weak decay rates are in a reasonable agreement, within the expected range of uncertainty,with the data on lifetimes of charmed particles and with the so far available data on decays ofB mesons.The only outstanding piece of present experimental data is on the lifetime of theΛb baryon:τ(Λb)/τ(B d)≈0.8,for which ratio a theoretical prediction,given all the uncertainty involved,is unlikely to produce a number lower than0.9.The number of available predictions for inclusive decay rates of charmed and b hadrons is suﬃciently large for future experimental studies toﬁrmly establish the validity status of the OPE based theory of heavy hadron decays,and,in particular,toﬁnd out whether the present contradiction between the theory and the data onτ(Λb)/τ(B d)is a temporary diﬃculty,or an evidence of fundamentalﬂaws in theoretical understanding.It is a matter of common knowledge that application of OPE to decays of charmed and b hadrons has potentially two caveats.One is that the OPE is used in the Minkowsky kinematical domain,and therefore relies on the assumption of quark-hadron duality at the energies involved in the corresponding decays.In other words,it is assumed that suﬃciently many exclusive hadronic channels contribute to the inclusive rate,so that the accidentals of the low-energy resonance structure do not aﬀect the total rates of the inclusive processes. Theoretical attempts at understanding the onset of the quark-hadron duality are so farlimited to model estimates[5,6],not yet suitable for direct quantitative evaluation of possible deviation from duality in charm and b decays.This point presents the most fundamental uncertainty of the OPE based approach,and presently can only be clariﬁed by confronting theoretical predictions with experimental data.The second possible caveat in applying the OPE technique to inclusive charm decays is that the mass of the charm quark,m c,may be insuﬃciently large for signiﬁcant suppression of higher terms of the expansion in m−1c. The relative lightness of the charm quark,however,accounts for a qualitative,and even semi-quantitative,agreement of the OPE based predictions with the observed large spread of the lifetimes of charmed hadrons:the nonperturbative eﬀects,formally suppressed by m−2c and m−3c are comparable with the‘leading’parton term and describe the hierarchy of the lifetimes.Another uncertainty of a technical nature arises from poor knowledge of matrix elements of certain quark operators over hadron,arising as terms in OPE.These can be estimated within theoretical models,with inevitable ensuing model dependence,or,where possible, extracted from the experimental data.With these reservations spelled out,we discuss here the OPE based description of inclusive weak decays of charm and b hadrons,with emphasis on speciﬁc experimentally testable predictions,and on the measurements,which would less rely on model dependence of the estimates of the matrix elements,thus allowing to probe the OPE predictions at a fundamental level.2OPE for inclusive weak decay ratesThe optical theorem of the scattering theory relates the total decay rateΓH of a hadron H Q containing a heavy quark Q to the imaginary part of the‘forward scattering amplitude’.For the case of weak decays the latter amplitude is described by the following eﬀective operatorL eff=2Im i d4x e iqx T{L W(x),L W(0)} ,(1) in terms of which the total decay rate is given by1ΓH= H Q|L eff|H Q .(2) The correlator in equation(1)in general is a non-local operator.However at q2=m2Q the dominating space-time intervals in the integral are of order m−1Q and one can expand thecorrelator in x ,thus producing an expansion in inverse powers of m Q .The leading term in this expansion describes the parton decay rate of the quark.For instance,the term in the non-leptonic weak Lagrangian √q 1L γµQ L )(64π3ηnlq 3,due to the relation H Q |64π3 64π3 4π i c (3)i (Q Γ′i Q ),(4)where the superscripts denote the power of m −1Q in the relative suppression of the correspond-ing term in the expansion with respect to the leading one,G µνis the gluon ﬁeld tensor,q i stand for light quarks,u,d,s ,and,ﬁnally,Γi ,Γ′i denote spin and color structures of the four-quark operators.The coeﬃcients c (a )depend on the speciﬁc part of the weak interaction Lagrangian L W ,describing the relevant underlying quark process.One can notice the absence in the expansion (4)of a term suppressed by just one power of m −1Q ,due to non-existence of operators of suitable dimension.Thus the decay rates receiveno correction of relative order m −1Q in the limit of large m Q ,and the ﬁrst pre-asymptoticcorrections appear only in the order m −2Q .The mechanisms giving rise to the three discussed terms in OPE are shown in Figure 1.The ﬁrst,leading term corresponds to the parton decay,and does not depend on the light guark and gluon ‘environment’of the heavy quark in a hadron.The second term describes the eﬀect on the decay rate of the gluon ﬁeld that a heavy quark ‘sees’in a hadron.This term in fact is sensitive only to the chromomagnetic part of the gluon ﬁeld,and contains the operator of the interaction of heavy quark chromomagnetic moment with the chromomagnetic ﬁeld.Thus this term depends on the spin of the heavy quark,but does not depend on theﬂavors of the light quarks or antiquarks.Therefore this eﬀect does not split the inclusive decay rates withinﬂavor SU(3)multiplets of heavy hadrons,but generally gives diﬀerence of the rates, say,between mesons and baryons.The dependence on the light quarkﬂavor arises from the third term in the expansion(4)which explicitly contains light quarkﬁelds.Historically, this part is interpreted in terms of two mechanisms[2,8,9]:the weak scattering(WS)and the Pauli interference(PI).The WS corresponds to a cross-channel of the underlying decay, generically Q→q1q2q1,weak-scatters(annihilates)in the process q3.The Pauli interference eﬀect arises when one of theﬁnal(anti)quarks in the decay of Q is identical to the spectator(anti)quark in the hadron,so that an interference of identical particles should be taken into account.The latter interference can be either constructive or destructive,depending on the relative spin-color arrangement of the(anti)quark produced in the decay and of the spectator one,thus the sign of the PI eﬀect is found only as a result of speciﬁc dynamical calculation.In speciﬁc calculations,however,WS and PI arise from the same terms in OPE,depending on the hadron discussed,and technically there is no need to resort to the traditional terminology of WS and PI.q3q3--Q QQ QP P PPq qm5Q(Q( σ· B)Q m2Q(QΓ′Q)vvvFigure1:Graphs for threeﬁrst terms in OPE for inclusive decay rates:the parton term, the chromomagnetic interaction,and the four-quark term.In what follows we discuss separately the eﬀects of the three terms in the expansion(4) and their interpretation within the existing and future data.3The parton decay rateThe leading term in the OPE amounts to the perturbative expression for the decay rate of a heavy quark.In b hadrons the contribution of the subsequent terms in OPE is at the level of few percent,so that the perturbative part can be confronted with the data in its own right. In particular,for the B d meson the higher terms in OPE contribute only about1%of the total non-leptonic as well as of the semileptonic decay rate.Thus the data on these rates can be directly compared with the leading perturbative term in OPE.The principal theoretical topic,associated with this term is the calculation of QCD radiative corrections,i.e.of the factorηnl in eq.(3)and of a similar factor,ηnl,for semileptonic decays.It should be noted,that even at this,perturbative,level there is a known long-standing problem between the existing data and the theory in that the current world average for the semileptonic branching ratio for the B mesons,B sl(B)=10.45±0.21%,is somewhat lower than the value B sl(B)≥11.5preferred from the present knowledge of theoretical QCD radiative corrections to the ratio of non-leptonic to semileptonic decay rates(see e.g.[10]). However,this apparent discrepancy may in fact be due to insuﬃcient‘depth’of perturbative QCD calculation of the ratioηnl/ηsl.In order to brieﬂy elaborate on this point,we notice that the standard way of analyzing the perturbative radiative corrections in the nonleptonic decays is through the renormalization group(RG)summation of the leading log terms and the ﬁrst next-to-leading terms[11,12]in the parameter L≡ln(m W/m b).For the semileptonic decays the logarithmic dependence on m W/m b is absent in all orders due to the weak current conservation at momenta larger than m b,thus the correction is calculated by the standard perturbative technique,and a complete expression in theﬁrst order inαs is available both for the total rate[13,14]and for the lepton spectrum[15].In reality however the parameter L≈2.8is not large,and non-logarithmic terms may well compete with the logarithmic ones. This behavior is already seen from the known expression for the logarithmic terms:when expanded up to the orderα2s the result of Ref.[16]for the rate of decays with singleﬁnal charmed quark takes the formΓ(b→c¯u d)+Γ(b→c¯u s)π+α2s6+2c(a)is known explicitly[16]and is quite weak:c(0)=19/2,c(1)=6,and c(m2c/m2b)≈9.0for the realistic mass ratio m c/m b≈0.3.One can see that the term with the single logarithm L contributes about two thirds of that with L2in the term quadratic inαs.Under such circumstances the RG summation of the terms with powers of L does not look satisfactory for numerical estimates of the QCD eﬀects,at least at the so far considered level of theﬁrst next-to-leading order terms,and the next-to-next-to-leading terms can be equally important as the two known ones,which would eliminate the existing impasse between the theory and the data on B sl(B).One can present some arguments[17]that this is indeed the case for the b quark decay,although a complete calculation of these corrections is still unavailable.4Chromomagnetic and time dilation eﬀects in decay ratesThe corrections suppressed by two powers of m−1Q to inclusive decay rates arise from two sources[7]:the O(m−2Q)corrections to the matrix element of the leading operator,(QQ|H Q =1−µ2π(H Q)−µ2g(H Q)Q(i D)2Q|H Q ,µ2g= H Q|2σµνGµνQ|H Q ,(7) with D being the QCD covariant derivative.The correction in equation(6)in fact corre-sponds to the time dilation factor m Q/E Q,for the heavy quark decaying inside a hadron, where it has energy E Q,which energy is contributed by the kinetic part(∝µ2π)and the chromomagnetic part(∝µ2g).The second term in OPE describes the eﬀect of the chromo-magnetic interaction in the decay process,and is also expressed throughµ2g.The explicit formulas for the decay rates,including the eﬀects up to the order m−2Q are found in[7]and for decays of the b hadrons read as follows.For the semileptonic decay rate Γsl(H b)=|V cb|2G2F m5b bb|H b 1+µ2g2dand for the non-leptonic decay rateΓnl(H b)=|V cb|2G2F m5b bb|H b 1+µ2g2d m2b I2(x) .(9) These formulas take into account only the dominant CKM mixing V cb and neglect the small one,V ub.The following notation is also used:x=m c/m b,I0(x,y,z)stands for the kinemat-ical suppression factor in a three-body weak decay due to masses of theﬁnal fermions.In particular,I0(x,0,0)=(1−x4)(1−8x2+x4)−24x4ln x,(10)I0(x,x,0)=(1−14x2−2x4−12x6)√1−4x21−4x2. Furthermore,I(x)=I0(x,0,0)+I0(x,x,0),andI2(x)=(1−x2)3+ 1+11−4x2−3x2(1−2x4)ln1+√1−√αs(m W) 4/b,(11)and b is the coeﬃcient in the QCD beta function.The value of b relevant to b decays is b=23/3.Numerically,for x≈0.3,the expressions for the decay rates can be written asΓsl(H b)=Γpartonsl 1−µ2π(H b)−µ2g(H b)m2b ,Γnl(H b)=Γpartonnl 1−µ2π(H b)−µ2g(H b)m2b ,(12)whereΓparton is the perturbation theory value of the corresponding decay rate of b quark.The matrix elementsµ2πandµ2g are related to the spectroscopic formula for a heavy hadron mass M,M(H Q)=m Q+2m Q + (13)Being combined with the spin counting for pseudoscalar and vector mesons,this formula allows toﬁnd the value ofµ2g in pseudoscalar mesons from the mass splitting:µ2g(B)=3Γnl(B)=1−µ2π(Λb)−µ2π(B)m2b.(15)The diﬀerence of the kinetic terms,µ2π(Λb)−µ2π(B),can be estimated from the mass formula:µ2π(Λb)−µ2π(B)=2m b m cM(B)−M is the spin-averaged mass of the mesons,e.g.with respect to those of the charmed hyperons in a reasonable agreement with the observed pattern of the lifetimes.It should be emphasized once again that the m−2Q eﬀects do not depend on theﬂavors of the spectator quarks or antiquarks.Thus the explanation of the variety of the inclusive decay rates within theﬂavor SU(3)multiplets,observed for charmed hadrons and expected for the b ones,has to be sought among the m−3Q terms.5L(3)eff.Coeﬃcients and operatorsAlthough the third term in the expansion(4)is formally suppressed by an extra power of m−1Q,its eﬀects are comparable to,or even larger than the eﬀects of the second term.This is due to the fact that the diagrams determining the third term(see Fig.1)contain a two-body phase space,while theﬁrst two terms involve a three-body phase space.This brings in a numerical enhancement factor,typically4π2.The enhanced numerical signiﬁcance of the third term in OPE,generally,does not signal a poor convergence of the expansion in inverse heavy quark mass for decays of b,and even charmed,hadrons the numerical enhancement factor is a one time occurrence in the series,and there is no reason for similar‘anomalous’enhancement among the higher terms in the expansion.Here weﬁrst present the expressions for the relevant parts of L(3)eff for decays of b and c hadrons in the form of four-quark operators and then proceed to a discussion of hadronic matrix elements and the eﬀects in speciﬁc inclusive decay rates.The consideration of the eﬀects in decays of charmed hadrons is interesting in its own right,and leads to new predic-tions to be tested experimentally,and is also important for understanding the magnitude of the involved matrix elements using the existing data on charm decays.,induced We start with considering the term L(3)eff in b hadron non-leptonic decays,L(3,b)eff,nlby the underlying processes b→c c s,b→c c d.Unlike the case of three-body decay,the kinematical diﬀerence between the two-body states c u,involvedis of the order of m2c/m2b≈0.1and is rather small.At present level in calculation of L(3,b)eff,nlof accuracy in discussing this term in OPE,one can safely neglect the eﬀect ofﬁnite charmedreads as[4]quark mass2.In this approximation the expression for L(3,b)eff,nl=|V bc|2G2F m2b bΓµb)(bΓµu)(L(3,b)eff,nl2The full expression for aﬁnite charmed quark mass can be found in[21]˜C5(3q Γµq )+˜C 6(3q k Γµq i )+(17)1b Γµt a b )j a µ−(5˜C 2++˜C 2−−6˜C +˜C −)(3q Γq )=(s Γs )is used,the indices i,k are the color triplet ones,Γµ=γµ(1−γ5),and j a µ=dγµt a d +3(1−κ1/2)(˜C 2+−˜C 2−),˜C2=κ1/2(˜C 2+−˜C 2−),˜C 3=−13(1−κ1/2)(5˜C2++˜C 2−+6˜C +˜C −) ,˜C4=−14(˜C ++˜C −)2+14κ1/2(5˜C 2++˜C 2−−6˜C +˜C −).(18)The expression for the CKM dominant semileptonic decays of b hadrons,associated with the elementary process b →c ℓνdoes not look to be of an immediate interest.The reason is that this process is intrinsically symmetric under the ﬂavor SU(3),and one expects no signiﬁcant splitting of the semileptonic decay rates within SU(3)multiplets of the b hadrons.The only possible eﬀect of this term,arising through a penguin-like mechanism can be in a small overall shift of semileptonic decay rates between B mesons and baryons.However,these eﬀects are quite suppressed and are believed to be even smaller than the ones arisingform the discussed m −2b terms.For charm decays there is a larger,than for b hadrons,variety of eﬀects associated with L (3)eff ,that can be studied experimentally,and we present here the relevant parts of the eﬀective Lagrangian.For the CKM dominant non-leptonic decays of charm,originatingfrom the quark process c →s u4πC 1(d Γµd )+C 2(d Γµc )+C 3(3s Γµs )+C 4(3s k Γµs i )+(19)C 5(3u Γµu )+C 6(3u k Γµu i )+1c Γµt a c )j a µ−(5C 2++C 2−)(3s and c →d u4π{C 1(q Γµq )+C 2(q k Γµq i )+C 3(3q Γµq )+C 4(3q k Γµq i )+(20)2C 5(3u Γµu )+2C 6(3u k Γµu i )+2c Γµt a c )j a µ−(5C 2++C 2−)(3q Γq )=(s Γs )is used.The semileptonic decays of charm,the CKM dominant,associated with c →s ℓν,and the CKM suppressed,originating from c →s ℓν,contribute to the semileptonic decay rate,which certainly can be measured experimentally.The expression for the part of the eﬀectiveLagrangian,describing the m −3Q terms in these decays is [17,24,25]L (3)eff,sl =G 2F m 2cc Γµc +2cγµγ5c )(c i Γµc k +2c i γµγ5c k )(3Even if the inclusive rate of these decays is not to be separated experimentally,they contribute about 10%of the total decay rate,and it is worthwhile to include their contribution in the balance of the total width.sin2θc L1(3dΓµd)+L2(3d kΓµd i) −2κ1/2(κ−2/9−1)(3B s oscillations. The data on decay rates of the cascade hyperonΞ0b are not yet available,while the currently measured lifetimes of B d and B s are within less than2%from one another.Theoretically, the diﬀerence of the lifetimes,associated with possible violation of the SU(3)symmetry and with breaking of the U symmetry of the eﬀective Lagrangian(17),is expected to not exceed about1%.For the non-vanishing matrix elements of four-quark operators over pseudoscalar mesons one traditionally starts with the factorization formula and parametrizes possible deviation from factorization in terms of‘bag constants’.Within the normalization convention adopted here the relations used in this parametrization read asP Q QΓµq)(q =1P QQ ΓµQ )(q =1q stands for pseudoscalar meson made of Q and8π (˜C 2+−˜C 2−)B (m b )+1200MeV 2ps −1,(25)Γ(D ±)−Γ(D 0)=cos 4θc G 2F m 3c f 2D 3(C 2++C 2−)˜B (m c )∼−0.8 f DOn the contrary,inΩQ the two strange quarks form a J P=1+state,and a correlation between the spins of heavy and light quarks is present.The absence of spin correlation forthe heavy quark in the triplet of hyperons somewhat reduces the number of independentfour-quark operators,having nonvanishing diagonal matrix elements over these baryons. Indeed,the operators entering L(3)eff contain both vector and axial bilinear forms for the heavy quarks.However the axial part requires a correlation of the heavy quark spin withthat of a light quark,and is thus vanishing for the hyperons in the triplet.Therefore only the structures with vector currents are relevant for these hyperons.These structures are of thetype(qγµq)and(q kγµq i)with q being d,s or u.Theﬂavor SU(3)symmetrythen allows to express,for each of the two color combinations,the matrix elements of threediﬀerent operators,corresponding to threeﬂavors of q,over the baryons in the triplet interms of only two combinations:ﬂavor octet andﬂavor singlet.Thus all eﬀects of L(3)eff inthe triplet of the baryons can be expressed in terms of four independent combinations of matrix elements.These can be chosen in the following way:x= 1QγµQ)[(sγµs)] Ξ(d)Q−ΛQ= 1QγµQ) (dγµd) ΛQ−Ξ(u)Q,(27) y= 1Q iγµQ k)[(s kγµs i)] Ξ(d)Q−ΛQ= 1Q iγµQ k) (d kγµd i) ΛQ−Ξ(u)Q, with the notation for the diﬀerences of the matrix elements: O A−B= A|O|A − B|O|B ,for theﬂavor octet part and the matrix elements:x s=1QγµQ) (dγµd)+(3 H Q|(u kγµu i)+(s kγµs i)|H Q (28)for theﬂavor singlet part,where H Q stands for any heavy hyperon in the(anti)triplet.The initial,very approximate,theoretical estimates of the matrix elements[4]were es-sentially based on a non-relativistic constituent quark model,where these matrix elements are proportional to the density of a light quark at the location of the heavy one,i.e.in terms of the wave function,proportional to|ψ(0)|ing then the same picture for the matrix elements over pseudoscalar mesons,relating the quantity|ψ(0)|2to the annihilation constant f P,and assuming that|ψ(0)|2is approximately the same in baryons as in mesons, one arrived at the estimatey=−x=x s=−y s≈f2D M Dwhere the sign relation between x and y is inferred from the color antisymmetry of the constituent quark wave function for baryons.Since the constituent picture was believed to be valid at distances of the order of the hadron size,the estimate(29)was applied to the matrix elements in a low normalization point whereαs(µ)≈1.For the matrix elements of the operators,containing s quarks over theΩQ hyperon,this picture predicts an enhancement factor due to the spin correlation:ΩQ|(sΓµs)|ΩQ =− ΩQ|(s kΓµs i)|ΩQ =10[(C5−C3)x+(C6−C4)y],4πG2F m2cδnl,02≡Γnl∆S=∆C(Λc)−Γnl∆S=∆C(Ξ+c)=cos4θcdecays in the baryon triplet isδnl,1≡Γnl ∆S =0(Ξ0c )−Γnl ∆S =0(Λc )=cos 2θc sin 2θc G 2F m 2c12π[L 1x +L 2y ].(33)Finally,the Cabibbo suppressed semileptonic decay rates are equal for Λc and Ξ0c ,due to the ∆V =0property of the corresponding interaction.Thus the only diﬀerence for these isδsl,1≡Γsl ∆S =0(Λc )−Γsl ∆S =0(Ξ+c )=−sin 2θc G 2F m 2c4πcos 2θ x cos 2θ(C 5−C 3)+sin 2θ(2C 5−C 1−C 3)−23L 2,(35)and∆2=δnl,02−2δsl,0+2δsl,1=G 2F m 2c 3(cos 2θ−sin 2θ)L 1+y cos 4θ(C 4−C 2)+2m c2,(37)while the dependence of the thus extracted matrix element y on the normalization point µis shown in Fig.24.23456-0.04-0.020.02yxκFigure 2:The values of the extracted matrix elements x and y in GeV 3vs.the normalization point parameter κ=αs (µ)/αs (m c ).The thick lines correspond to the central value of the data on lifetimes of charmed baryons,and the thin lines show the error corridors.Theextracted values of x and y scale as m −2c with the assumed mass of the charmed quark,andthe plots are shown for m c =1.4GeV .Notably,the extracted values of x and y are in a drastic variance with the simplistic constituent model:the color antisymmetry relation,x =−y ,does not hold at any reasonable µ,and the absolute value of x is substantially enhanced 5Once the non-singlet matrix elements are determined,they can be used for predicting diﬀerences of other inclusive decay rates within the triplet of charmed hyperons as well as for the b baryons.Due to correlation of errors in x and y it makes more sense to express the predictions directly in terms of the total decay rates of the charmed hyperons.The thus arising relations between the rates do not depend on the normalization parameter µ.In this way one ﬁnds [28]for the diﬀerence of the Cabibbo dominant semileptonic decay rates between either of the Ξc hyperons and Λc :Γsl (Ξc )−Γsl (Λc )≈δsl,0=0.13∆1−0.065∆2≈0.59±0.32ps −1.(38)When compared with the data on the total semileptonic decay rate of Λc ,Γsl (Λc )=0.22±0.08ps −1,this prediction implies that the semileptonic decay rate of the charmed cascade hyperons can be 2–3times larger than that of Λc .The predictions found in a similar way for the inclusive Cabibbo suppressed decay ratesare[28]:for non-leptonic decaysδnl,1=0.082∆1+0.054∆2≈0.55±0.22ps−1(39) and for the semileptonic onesδsl,1=tan2θcδsl,0≈0.030±0.016ps−1.(40) For the only diﬀerence of the inclusive rates in the triplet of b baryons,Γ(Λb)−Γ(Ξ−b), oneﬁnds an expression in terms of x and y,or alternatively,in terms of the diﬀerences∆1 and∆2between the charmed hyperons,Γ(Λb)−Γ(Ξ−b)=cos2θc|V bc|2G2F m2bm2c(0.85∆1+0.91∆2)≈0.015∆1+0.016∆2≈0.11±0.03ps−1.(41) When compared with the data on the total decay rate ofΛb this result predicts about14% longer lifetime ofΞ−b than that ofΛb.The singlet matrix elements x s and y s(cf.eq.(28))are related to the shift of the average decay rate of the hyperons in the triplet:3 Γ(ΛQ)+Γ(Ξ1Q)+Γ(Ξ2Q) .(42) For the charmed baryons the shift of the dominant non-leptonic decay rate is given by[29]δ(3,0)nl8π(C 2++C2−)κ5/18(x s−3y s),(43)while for the b baryons the corresponding expression reads asδ(3)8π(˜C+−˜C−)2˜κ5/18(x s−3y s).(44) The combination x s−3y s of the SU(3)singlet matrix elements cancels in the ratio of the shifts for b hyperons and the charmed ones:δ(3)cos4m2bC2++C2− αs(m c)Γc≈0.0025δ(3,0)nl˜C)2/(C2++C2−),which parametrically is of the second order inαs,and numerically is only −about0.12.An estimate ofδ(3)Γc for charmed baryons.The latter shift can be conservatively bounded from aboveΓc=6.0±0.7ps−1,which then by the average total decay rate of those baryons:δ(3,0)nlyields,using eq.(45),an upper boundδ(3)Γc the contribution of the‘parton’term,which can be estimated from the decay rate of D0with account of the O(m−2c)eﬀects,as amounting to about3ps−1.(One should also take into account the semileptonic contribution to the total decay rates,which however is quite small at this level of accuracy).Thus a realistic evaluation ofδ(3)Γb due to the non-singlet operators is one third of the splitting(41),i.e.about5%.Adding to this the1%shift of the average width and another1%diﬀerence from the meson decays due to the suppression of the latter by the m−2b chromomagnetic eﬀects,one concludes that at the present level of theoretical understanding it looks impossible to explain a more than10%enhancement of the total decay rate ofΛb relative to B d,where an ample3%margin is added for the uncertainties of higher order terms in OPE as well as for higher order QCD radiative eﬀects in the discussed corrections. In other words,the expected pattern of the lifetimes of the b hyperons in the triplet,relative to B d,isτ(Ξ0b)≈τ(Λb)<τ(B d)<τ(Ξ−b),(46) with the“best”theoretical estimate of the diﬀerences to be about7%for each step of the inequality.For the double strange hyperonsΩc andΩb there is presently no better approach to evaluating the four-quark matrix elements,than the use of simplistic relations,like(30) based on constituent quark model.Such relations imply that the eﬀects of the strange quark,WS and PI,in theΩQ baryons are signiﬁcantly enhanced over the same eﬀects in the cascade hyperons.In charmed baryons a presence of strange spectator quark enhances the decay through positive interference with the quark emerging from the c→s transition in the decay.ForΩc this implies a signiﬁcant enhancement of the total decay rate[4],which is in perfect agreement with the data on theΩc lifetime.Also a similar enhancement is expected for the semileptonic decay rate ofΩc.In b baryons,on the contrary,the interference eﬀect。

Michael

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D{10117 Berlin, Germany,(dreyer@wias-berlin.de, kunik@wias-berlin.de y FB Mathematik, Universitat Kaiserslautern, Erwin-Schrodinger-Stra e, 67663 Kaiserslautern, Germany,(junk@mathematik.uni-kl.de).
1
Our main objective is to show that this impression is not justi ed. These observations also apply to more general cases: whenever the velocity space in the underlying kinetic equation Ris unbounded, when the entropy functional is essentially given by H (f ) = ? Rd f log f dc, when equilibrium states are related to Maxwellian distributions and when velocity moments of order four and higher are used (these assumptions are satis ed for the Fokker Planck equation but also for the Boltzmann equation of gas dynamics or the semiconductor Boltzmann equation with parabolic bands). It turns out that in such cases the equilibrium states are located on the boundary of the domain of de nition of the Maximum Entropy system. Moreover, the ux is singular in these states which leads to a very unexpected behavior of the systems. Note that there are kinetic equations and corresponding entropy functionals where these problems do not appear. One example is the kinetic model of phonons as carriers of heat in crystals, see 4], 5]. Both complications do not arise in the Hermite/Grad approach which will lead to two conclusions: rstly, the Grad system is a bad approximation of the Maximum Entropy system because the singular behavior is not captured. Secondly, the Grad system is in better coincidence with the Fokker Planck equation. Thus, despite of the disadvantages concerning positivity of even moments and hyperbolicity, the Hermite/Grad system seems to be favorable. We conclude the introduction with an outline of the article. In the remaining sections of the rst chapter, we introduce the Fokker Planck equation as well as the basic moment approximations. In the second chapter, we then construct explicit solutions of the Cauchy problem for the Fokker Planck equation and give analytic formulas for the rst ve moments of the solution with Riemann initial data. In chapter 3, the Maximum Entropy system is studied in more detail with respect to its domain of de nition and the singular behavior of the ux. In the last chapter, the approximation properties of both the Hermite/Grad and the Maximum Entropy approach are investigated.

The space of triangles, vanishing theorems, and combinatorics

3
These objects have been extensively explored. Combinatorists have examined the representations SD;n as part of the theory of generalized Schur modules and Young diagrams. (See 28], 32], 33], 34], 38].) The geometric theory of F3;n and its desingularizations goes back to Schubert 36] and Semple 37], and has been illuminated recently by Fulton, MacPherson, and others ( 9], 12], 19], 35], 7], 8]). In the current paper, we explore the relations between the algebraic and geometric pictures. We prove a Borel-Weil theorem realizing SD ;n (or the Schur module of any three-row diagram) as the sections of a line bundle over the triangle space, and we derive explicit character and dimension formulas generalizing those of Weyl for the irreducibles. Reversing the perspective, the Schur modules give a combinatorial construction of the space of sections of certain line bundles over the triangle spaces. The key steps relating the geometry to the combinatorics require the vanishing of certain higher cohomology groups and explicit computations on spaces of sections. For these, we make a detailed examination of the de ning equations, torus- xed points, and birational maps of the triangle spaces, in order to apply the standard techniques for dealing with (almost) homogeneous varieties. Indeed, we intend this paper partly as a primer on Frobenius splitting, natural desingularizations, Lefschetz theorems, and rational singularities, as illustrated by our simple example. Nevertheless, our character formulas for Schur modules are stated in purely elementary terms, and the interested reader can skip directly to sections 5.2 and 5.3. In previous papers 21], 23], 22] we considered the same problems for diagrams D satisfying the \northwest" or \strongly separated" conditions, for which the geometry of FD is particularly simple. D3 is the smallest diagram which does not fall into these classes, and thus needs a di erent treatment. In the case of northwest or strongly separated diagrams, we may desingularize our varieties by the usual Bott{Samelson and Zelevinsky resolutions, but for the triangle space we must use more complicated special desingularizations which appear in the literature. We hope that our methods will shed light on de ning non-singular spaces of tetrahedra and higher-dimensional simplices. Even

新教材高中英语Unit10Connections课件北师大版选择性必修第四册

主题：人与社会学科素养：思维品质难度系数：★★★★★ 【语篇导读】本文简单介绍了“小世界理论”的原理及其在社交网络中的广泛运用。
The Small-World Phenomenon and Decentralized Search The small-world phenomenon—the principle that we are all linked by short chains of acquaintances, or “six degrees of separation”—is a fundamental issue in social networks. It is a basic statement about the abundance of short paths in a graph whose nodes are people, with links joining pairs who know one another. It is also a topic on which the feedback between social, mathematical, and computational issues has been particularly fluid.
○句型精析
1．The small-world phenomenon—the principle that we are all linked by short chains of acquaintances, or “six degrees of separation”—is a fundamental issue in social networks.
Working much more recently, applied mathematicians Duncan Watts and Steve Strogatz proposed thinking about networks with this small-world property as a superposition: a highly clustered(成群的)sub-network consisting of the “local acquaintances” of nodes, together with a collection of random long-range shortcuts that help produce short paths.

intercourse

intercourseIntercourse: A Comprehensive Exploration of its Meaning and ImplicationsIntroductionIntercourse is a term often used to refer to sexual activity or the physical act of sexual intercourse between two individuals. However, the term encompasses a much broader spectrum of meanings and implications that go beyond a mere physical encounter. This document aims to delve into the concept of intercourse, exploring its various dimensions, implications, cultural significance, and the evolution of its understanding over time.Historical PerspectiveThroughout history, intercourse has been an essential aspect of human existence and societal development. In ancient civilizations, such as the Greeks and Romans, intercourse was often celebrated and associated with fertility, procreation, and pleasure. However, the perception of intercourse hasvaried across cultures and eras, influenced by religious beliefs, social norms, and individual interpretations.Cultural SignificanceIntercourse holds diverse cultural significance worldwide. It can be seen as a sacred act within religious contexts, symbolizing the union between individuals, spiritual connection, or the creation of life. Conversely, some cultures view intercourse as a purely physical and reproductive act, devoid of emotional or spiritual implications. Understanding these cultural perspectives can shed light on how societies regard and navigate the complexities of sexual relationships.Psychological and Emotional DimensionsBeyond the physical realm, intercourse carries deep psychological and emotional dimensions. It creates a unique connection between partners, fostering intimacy, trust, and emotional bonds. Studies suggest that engaging in consensual intercourse with a trusted partner can positively impact mental well-being, self-esteem, and overall relationship satisfaction. However, it is essential to consider the variations in individuals' emotional responses tointercourse, which can range from joy and fulfillment to feelings of guilt, shame, or trauma.Communication and ConsentEffective communication and consent play a crucial role in healthy and fulfilling intercourse. Consent implies that all involved parties actively agree to engage in sexual activity without coercion or manipulation. Lack of consent can have severe consequences, leading to emotional distress, trauma, and criminal repercussions. It is vital to foster a culture that prioritizes open dialogue, mutual respect, and consent education to ensure safe and consensual sexual encounters.Evolution of UnderstandingThe understanding of intercourse has evolved significantly over time. The advent of contraception, sexual education, and reproductive technologies has reshaped societal attitudes and behaviors related to intercourse. Topics such as safe sex, sexual health, and consent have gained prominence, challenging traditional norms and promoting a more informed and responsible approach to sexual activity.Intercourse in the Digital AgeThe rise of the internet and digital technologies has introduced new dimensions to the concept of intercourse. Online platforms enable individuals to engage in virtual intercourse experiences, connecting with others around the globe. However, this virtual realm also brings forth challenges such as consent issues, the objectification of bodies, and the blurring of boundaries between reality and fantasy. Reflecting on these modern-day developments can help us navigate the ever-changing landscape of intercourse in the digital age.ConclusionIntercourse encompasses a multifaceted range of meanings, implications, and cultural significances. Beyond its physical representation, intercourse links individuals emotionally, psychologically, and spiritually. It is essential to approach intercourse with open communication, mutual consent, and a deep understanding of its complexities. By fostering a culture that values respect, consent, and comprehensive sexual education, we can strive towards healthy, fulfilling, and safe sexual relationships for all individuals.。

Endomembrane+system

Figure 12-37. Free and membrane-bound ribosomes. A common pool of ribosomes is used to synthesize the proteins that stay in the cytosol and those that are transported into the ER. The ER signal sequence on a newly formed polypeptide chain directs the engaged ribosome to the ER membrane. The mRNA molecule remains permanently bound to the ER as part of a polyribosome, while the ribosomes that move along it are recycled; at the end of each round of protein synthesis, the ribosomal subunits are released and rejoin the common pool in the cytosol.
The structure and functions of The endoplasmic reticulum(ER)
A. A netlike labyrinth of branching tubules and flattened sacs extending throughout the cytosol
Signal Sequences and Signal Patches Direct Proteins to the Correct Cellular Address

Spin Excitation in d-wave Superconductors A Fermi Liquid Picture

a r X i v :c o n d -m a t /0006312v 2 [c o n d -m a t .s u p r -c o n ] 12 J a n 2001Spin Excitation in d -wave Superconductors :A Fermi Liquid PictureKhee-Kyun Voo,Hong-Yi Chen,and W.C.WuDepartment of Physics,National Taiwan Normal University,Taipei 11650,Taiwan,R.O.C.(February 1,2008)A detailed study of the Inelastic Neutron Scattering (INS)spectra of the high-T c cuprates based on the Fermi liquid (FL)picture is given.We focus on the issue of the transformation between the commensurate and incommensurate (IC)excitation driven by frequency or temperature .For La 2−x Sr x CuO 4(LSCO),the condition of small ∆(0)/v F a (where a is the lattice constant,and henceforth will be set to 1)can simultaneously reproduces the always existing IC peaks in the su-perconducting (SC)and normal state,and the always ﬁxed location at temperature or frequency change.For YBa 2Cu 3O 6+x (YBCO),a moderate ∆(0)/v F a and proximity of the van Hove singu-larity (vHS)atthe commensurate peak is easily derived either from the bare or RPA susceptibility.The Stripe ordering picture[15]is an alternative which predictsﬁxed IC peaks in spin and charge excitations that have incommensurability directly proportional to, and depending only on doping.In this picture the system is segregated into one-dimensional hole-rich stripes in which the holes can freely move,and electron-rich stripes in which electrons are ordered antiferromagnetically.The electronﬁlling determines the length scale and thus the incommensurability.It is believed that the static stripe at the1/8-doping[16],and the2-to-1-coupled charge and spin excitations[17]are observed.That has boosted alot of works on the“dynamic”stripe which also aims at the connection with the superconductivity mechanism.A plain Stripe model predicts no commensurate peak. Therefore the Stripe picture seems to work well in LSCO,while the FL picture works well in YBCO.This is intriguing if one adheres to the philosophy that they all belong to the cuprate family and therefore should lay in a single uniﬁed theory.Here we argue that the discrep-ancies could be reconciled if the problem is considered more carefully.The frequency and temperature driven peak shifting present in YBCO but not in LSCO can be accounted for by aﬂatter electronic dispersion near FS such that the excitation can be scattered farther from it,while electronic dispersion of LSCO is far more steep near FS.A reﬁned consideration of the nesting eﬀect, which we have termed the dynamic local nesting eﬀect is important when the condition∆/v F≪πis violated. In the regime of having shifting IC peak in SC state,the IC peak is necessarily vanishes in the normal state.The commensurate resonance in the SC state of YBCO and BSCCO is addressed as a consequence of the experimen-tally observed closeness of the vHS atM-point vHS.An appropriate de-scription of YBCO is then given.Sec.VII is a discussion on the anisotropy eﬀect and the one-dimensional nature in INS spectrum.Finally,Sec.VIII gives a discussion of the results and conclusions.II.FORMALISMWhat we need in a FL description of the INS spectrum is the existence of a well-deﬁned FS and the dispersion nearby.All low energy processes are constituted by the excitation at the vicinity of the Fermi surface.A INS spectrum is proportional to Imχ(q,ω)besides some Bose-Einstein distribution factor due to the bosonic character of the excitation.Hereχ(q,ω)is the spin sus-ceptibility.The BCS bare spin susceptibility is given be-low and followed by the RPA-corrected susceptibility. The BCS bare spin susceptibility in a one-layer SC system(the susceptibility in a normal system could be obtained by letting∆→0)isχo(q,ω)=−1E k E k+q× 1−f(E k)−f(E k+q)ω+E k+E k+q+iΓ − 1+ξkξk+q+∆k∆k+qω−E k+E k+q+iΓ−f(E k)−f(E k+q)chemical potential.In this paper only the d x2−y2-gap is treated(except in Sec.VII)since it is meant to describe the high-T c cuprates.It is taken as∆k=∆(T)(cos k x−cos k y)/2.The widthΓ/t in numerical integration is taken within0.002∼0.008with slicing2000×2000or 4000×4000.Equation(1)describes two kinds of excitation,the pair-breaking excitation that excites two quasiparticles from the SC condensate and costs energy E k+E k+q, and the thermal one-particle excitation that excites a quasiparticle from k+q to k which costs energy E k−E k+q.The two-particle excitation vanishes in the nor-mal state while the one-particle excitation vanishes at zero-temperature SC state.Here we want to make several important remarks on the modeling[by the simple dispersion Eq.(2)]of the dispersion gradient at FS(Fermi velocity v F)of a real system:(i)It is a fact that low energy physics is only rele-vant to the part of the dispersion near the Fermilevel.Therefore though the real dispersion near theFermi level can dramatically deviate from the sim-ple tight-binding band(sometimes even shows kinkin the dispersion[24]),for practical purposes westill can model it by choosing an eﬀective scale tthat gives similar Fermi velocity v F at the Fermilevel(since v F=|dε/d k|FS∝t).Thus in our con-text,∆/t is equivalent to∆/v F,i.e.a greater∆/tcould mean a smaller v F but does not necessarilymean a greater∆.Obviously,we do not think“t”as the bandwidth of the real dispersion,which isnot of our concern.(ii)On the relative size of∆/v F in LSCO and YBCO:Though there is a consensus that∆YBCO∼2∆LSCO,the v F’s are very uncertain[25,24].Wetherefore take v F as a phenomenological inputwhich is to be veriﬁed by future ARPES measure-ments,and v F is thought as the dominant factor inthe relative size of∆/v F.(iii)Note also that since v F(or the eﬀective t)is uncer-tain,we will compare all energies(e.g.ω,T,etc)only in terms of∆.The most important correction to the susceptibility should be the existence of Coulomb or AF correlations between the quasiparticles.Such correlations are be-lieved to exist as a residual interaction between the renor-malized particles,and they are conventionally treated by meanﬁeld decoupling as a nontrivial step beyond bare theories.Then the susceptibility is written into a RPA form asχ(q,ω)=χo(k,ω)2(cos q x+cos q y).(4)J q is diﬀerent from U essentially at being positive nearQ AF and changing its sign to negative near(0,0).TheRPA-type correction is important because it is perhapsthe only way to treat correlation eﬀect analytically.III.EXCITATION AT BARE LEVEL—–SUR VEY OF BASIC PROPERTIESWe have given in this section a survey of the essentialproperties of the bare excitation spectrum.Their discus-sion will be given in Sec.IV.A.Superconducting StateThe SC state is described by∆(T)=0,which resultsin a quasiparticle excitation gap and a SC coherence fac-tor.Figure1shows the frequency-evolution of the spec-tra for two typical cases of∆/t.We have ignored thewell-discussed quasielastic node-to-node IC peaks[26]at(π+δo,π±δo)and(π−δo,π±δo),and focused on thenesting IC peaks at(π,π±δ)and(π±δ,π).In general,asωis increased,the intensity and widths of the peaksare increased,and spectrum at Q AF isﬁlled in at somefrequency near2∆which depends only on band geom-etry.When∆/t is large,theﬁll-in is a stepwise jumpofχ′′o(Q AF,ω).This sharp increase ofχ′′o(Q AF,ω)hasan important relation to the resonance at Q AF and willbe further discussed in Sec.VI.The nesting IC peaksare seen to be sharpest at some intermediate frequenciesbefore theﬁlling in.In contrast to the sharp,well-preserved and not-shiftedpeaks in the∆/t≪1case[see Fig.1(a)],the larger∆/t calculation[see Fig.1(b)]shows broader and biasedpeaks that clump into Q AF with increasing frequency.Such converging of the IC peak cluster was observed ex-perimentally in YBCO and theoretically reproduced byus[12]and other groups[27,11,10].Here we proceed togive its full explanation in Sec.IV.If the saddle vHSat k=(0,π)is far from the Fermi level,the IC struc-ture will be destroyed after the peaks clump in at someω/∆<∼2.Some stray structures around the IC peaks always existespecially in the case of larger∆/t andω/∆.When theRPA correction is introduced,they will become relativelyweaker.The main point here is that atω∼∆,a system withmoderate∆/v F(modeled by a moderate∆/t)has itsnesting peak broad and driven byωto shift.We will dis-cuss in Sec.IV that the shift direction actually dependson the orientation of the curvature of the FS near thegap nodes.B.Normal State0.000.030.060.09χ0,,0.000.010.020.03χ0,,(0.5π,0.5π)(π,π)(1.5π,π)(a)(b)π−ππk xk yFIG.1.Frequency evolution of the SC state χ′′o (q ,ω)with diﬀerent ∆/t is shown.(a)∆/t =0.03,ω/∆=1.0,1.5,and 2.0(from bottom to top);(b)∆/t =0.30,ω/∆=0.6,1.0,and 1.4(from bottom to top).In (b),the broad peak shifts at changing frequency,and some stray structure away from the IC peak is present.Including the RPA-correction [see Fig.8(d)]will make the IC peaks and shifting eﬀect more conspicuous.We particularly note that since we mean to describe two different systems in (a)and (b),the “∆”or “t”in (a)and (b)are unrelated quantities.For both panels,T =0,the dispersion is t =1,t ′=−0.25,and µ=−0.65.The FS is shown in the inset and the plots are along q =(0.5π,0.5π)−(π,π)−(1.5π,π).The normal state is described by ∆(T )=0,which leaves the bandwidth or the Fermi velocity as the only energy scale.Eq.(1)is then reduced to the Lindhard functionχo (q ,ω)=kf (ξk )−f (ξk +q )0.000.030.060.090.12χ0,,0.00.51.0T/T c0.00.51.0∆(T )/∆(0)0.000.010.020.03χ0,,(0.5π, 0.5π)(π,π)(1.5π,π)(a)(b)FIG.3.Bare spectra χ′′o (q ,ω)with diﬀerent ∆(0)/t and ω/∆(0)at varying temperature.(a)∆(0)/t =0.03:the solid lines show the T =0spectra at ω/∆(0)=1.0and 2.0(from bottom to top).The dashed lines has likewise fre-quencies but T =T c =0.25∆(0).(b)∆(0)/t =0.30and ω/∆(0)=1.0:T /T c =0,0.80,0.85,and 1.0(from bottom to top).T c =0.25∆(0)and the assumed T -dependence of ∆(T )is shown in the inset.Heating converges the IC peak cluster in (b)to a commensurate structure.Note also that neither the “∆”or “t”in (a)is related to that in (b).The dispersion used in both panels is t =1,t ′=−0.25,and µ=−0.65.The plots are along q =(0.5π,0.5π)−(π,π)−(1.5π,π).—–THE DYNAMIC LOCAL NESTINGThe IC peaks are originated from the umklapp and inversion symmetry of the dispersion.This section dis-cusses the properties of the bare excitation mentioned in Sec.III.Both the ω-and T -driven peak shifting eﬀects are essentially due to the dynamic and local nature of the nesting eﬀect.We start to illustrate the mechanism by a zero tem-perature SC system where the bare spectrum is given byχ′′o (q ,ω)=πE k E k +q×[δ(ω−E k −E k +q )−δ(ω+E k +E k +q )].(6)The integrand consists of a coherence factor which re-ﬂects the non-time-reversal invariance nature of the mag-netic measurement,and a δ-function that imposes the energy conservation rule.−1.0−0.50.00.5 1.0k x /π−1.0−0.50.00.51.0k y /πq Ιq ΙΙq ΙΙΙq ΙVFIG.4.Energy contour E k =ω/2in the extended BZ for ω=1.0∆(0).Some arbitrary local-nesting vectors q I ,q II ,q III ,and q IV are shown.The thick vector between above-FS contour piece have better nesting due to the smaller curvature.The FS is shown as dotted line,and the dispersion is t =1,t ′=−0.25,and µ=−0.65.The gap here is chosen as ∆/t =0.50.The role of the δ-function should be ﬁrst discussed since it is the most dominant.It limits the contribut-ing region of the phase space by the energy conser-vation E k +E k +q =ω.At ω∼0,the only pos-sibility is E k and E k +q both ∼0.Therefore q can only be vectors connecting the gap nodes,which areQ δo ≡(π+δo ,π±δo )and (π−δo ,π±δo ),with δo =2sin −1t 2−t ′µ/[−2t ′].At ω>0,to satisfy E k +E k +q =ω,q can be any mo-mentum that connects points on the contours E k =ω1and E k +q =ω2,if ω1+ω2=ω.These are strips of area stretched out from the gap node along the FS.For most q ,it only constitutes a featureless and “incoher-ent”background in the q -space.To constitute a salient structure out of the background one needs to have extra factors that can stack extra weight onto the background.A situation that can further provide some weight is when ω1=ω2=ω/2and all the two frequency dependent con-tours become E k =E k +q =ω/2.It spans the whole FS when ω→2∆.Now q can “nest”two contours lo-cally by passing through points k o =(mπ,nπ)(where m and n are integers)in the BZ.It occurs because the ω-dependent contour is always locally parallel at k o +k and k o −k due to the dispersion E k has the k ↔−k inversion and umklapp symmetry in the extended BZ.This nesting is thus dynamic and local .As an example,we consider χ′′o (q ,ω)at q ∈[0∼2π,0∼2π]and the integration over k in evaluating χ′′o isFIG.5.(a)A bare spectrumχ′′o(q,ω)in the full q-space atω=1.0∆(0).(b)a single branch of“ridge”,obtained when the integration over k in Eq.(1)is done only over the quadrant(k x,k y)∈[0∼π,0∼π].Four of these90-degree rotated branches superpose to giveχ′′o(q,ω)in(a).Refer the dispersion parameters to Fig.1(b).run over(k x,k y)∈[−π∼π,−π∼π].Fig.4illustrates the local-nesting at a frequencyω<2∆.In the inte-gration over theﬁrst quadrant(k x,k y)∈[0∼π,0∼π], those local nesting vectors are shown as q I.Likewise, q II for the second,q III for the third,and q IV for the fourth quadrant.Figure5(a)shows the consequence of such local-nesting eﬀect.The formation of the structure can be illustrated by an illuminating calculation,where the in-tegration is done only over theﬁrst quadrant(k x,k y)∈[0∼π,0∼π].Then only one branch of the ridge-like structure is obtained[see Fig.5(b)]and its orientation is the same as the contour in quadrant(k x,k y)∈[π∼2π,π∼2π].The ends of the ridge are determined by ∆k=ω/2.The locus of the ridge in q-space resembles the locus of the contour(which is approximately the FS) in k-space but have twice the extension since the nesting vectors are twice the vectors describing the locus of the FS in k-space[see Fig.4].An IC peak is a result of overlapping of two ridges at the symmetrical points Qδ=(π,π±δ)and(π±δ,π),whereδ=2sin−1[−µ/(2t)],whenω>ωδ≡∆(T)[−µ/(2t)].For those FS of open topology[see Fig.4],the ridges overlap also at Qδ′=(π+δ′,π±δ′) and(π−δ′,π±δ′)whenω>ωδ′≡∆(T)µ/(4t′).The above expressions forδo,δ, andδ′are estimations since the ridges haveﬁnite thick-ness.Frequencyωδ′≡∆(T)appreciably from the FS.This weakens the suppression eﬀect before cutoﬀ,and the ridges will overlap at more sizable values and gives a higher contrast of the nest-ing IC peaks.It also lowers the thresholdω/∆for the emergence of the peaks.The normal state is the lift of the energy restriction by the gap,and the disappearance of coherence factor. The ridges now always span the whole q-space and the nesting peaks always exist in principle.But due to re-placing a stronger energy constraint(due to two-particle excitation)by a looser constraint(due to one-particle ex-citation),the transition nowﬂoods over regions far away from the FS and makes the widths of ridges dispersed and the nesting peaks less distinguishable.Especially now the transition near Q AF isﬁlled high for hole-like FS.Albeit the underlying cause of formation of the ridges and nesting peaks(umklapp and k-inversion symmetry) is irrelevant to the existence or not of the SC gap,there are still some upper limits onω/∆orω/t for the IC peaks to exist.Since increasingω/t broadens the ridges, the ridge and peak structure will be always destroyed at very highω/t.The same is true for very high T/t where the excitation is deconﬁned from the FS.It is easier for the peak to survive in the SC state because the near Q AF transition is kept away atω/∆<2[see Fig.3(b)].To summarize,at lowω/t and T/t,the nesting ICpeaks should always exist in bands with umklapp and inversion(k↔−k)symmetry.In the SC state,it can be driven byωor T to shift when∆/t is appreciable and the nodal FS is curved.V.RPA-CORRECTED EXCITATIONIn this section we will discuss the eﬀect of RPA cor-rection on the frequency-driven shifting eﬀect of the IC peaks.We start with a brief discussion of the essential features of the RPA spectrum.The RPA spectrum is given byχ′′(q,ω)=χ′′o(q,ω)M-POINT V AN HOVE SINGULARITYThis is a RPA spin excitation theory for the recently reported INS data on YBCO.For simplicity,we haveFIG.7.A typical SC state−1/χ′o(q,ω=0)in q-space.The no-instability bound of the interaction strength is V q<1/χ′o(q,ω=0).For V q=J q, |J|<1/χ′o(q∼Q AF,ω=0)≃2.2t in this case.The plot has T=0,∆/t=0.03,and band dispersion t=1,t′=−0.25, andµ=−0.65.mimic the realistic extended-saddle-vHS at the proximity of Fermi level[19,22]by simple saddle-vHS just beneath Fermi level.It is believed that in RPA theories they will be qualitatively the same.Nevertheless its extended na-ture is crucial for bare theory to have quantitative agree-ment with experiments[27].A number of comprehensive theories including both commensurate and IC excitation[10,11,27,12–14]were explored since the discovery of the behavior in YBCO.It is important to note that all of them are Fermi-liquid-like theories,and most are RPA theories[10,11,13,14].The possibility of a bare theory was investigated by Abrikosov [27,29]and us[12].The commensurate peaks in bare theories are generally weaker,broader,and occurs at ωo>∼2∆.That is in contrast to the much stronger and sharper one inχ′′that occurs atωo<2∆.How-ever,recently it has been shown that incorporating theextended-vHS[27]into bare theory can also give peak intensity and width in reasonable agreement with exper-iments.Attempting to rely on the criterionωo being less or greater than2∆for choosing a theory also encounters the subtleties in the determination of the SC gap.Es-pecially in the underdoped samples where there are still controversies on whether the SC gap and the pseudogap are the same gap[30–32].Therefore eitherωo>∼2∆(see discussion in Ref.[29])orωo<2∆(e.g.in the under-doped compounds)can be occasionally favored.As a re-sult we believe that at present there is no clear evidence here to favor either scenarios.Recently the IC peaks have received more attention because of their frequency-shifted behavior.Most of the present scenarios for it are based on the nesting eﬀect which depends mainly on the nodal FS,except one recently proposed by Abrikosov[27] depends on the band mass at the extended-vHS region. The discussion of the temperature-shifted IC peaks has been missing from the literature.0.10.20.3χ,,0.000.050.10χ,,2.03.01/χ,2.53.01/χ,(0.5π,0.5π)(π,π)(1.5π,π)(a)(b)(c)(d)(0.75π,0.75π)(π,π)(1.25π,π)FIG.8.RPA eﬀect(with no resonance)on two systems with small and moderate∆/t.All graphs except(a)are plot-ted along q=(0.5π,0.5π)−(π,π)−(1.5π,π).For the case of∆/t=0.03and|J|/t=1.0:(a)1/χ′o(q,ω)is shown[along q=(0.75π,0.75π)−(π,π)−(1.25π,π)]atω/∆=0(dashed line),1.0,2.0and1.5(solid lines at Q AF from top to bottom); Correspondingly,(b)χ′′(q,ω)is shown atω/∆=1.0,1.5and 2.0(from bottom to top).For the case of∆/t=0.30and |J|/t=1.0:(c)1/χ′o(q,ω)is shown atω/∆=0(dashed line),0.60,1.0and1.4(solid lines from top to bottom);Cor-respondingly,(d)χ′′(q,ω)is shown atω/∆=0.6,1.0and1.4 (from bottom to top).Note the sharp minima of1/χ′o in(c) and its discussion in the text.Temperature T=0and the corresponding bare spectra were shown previously in Fig.1.With a suﬃciently strong correction strength,it is al-ways possible to have a RPA resonance at Q AF.The rea-son is since any susceptibility function or interaction ver-tex on the lattice is symmetric at Q AF+δq→Q AF−δq, they always have an extremum at Q AF[e.g.see Fig.7] and thus one can always have1−Vχ′o=0locally at Q AF.The dispersion taken here to mimic the YBCO dis-persion has a hole-like FS and a simple saddle-vHS atM /∆=−0.5).The interaction is chosen as an antiferromagnetic J q with|J|/t=1.6.At T=0,a commensurate reso-nance occurs atω/∆=1.5.The momentum dependence of the RPA spectrum and1/χ′o are shown in Fig.9,at diﬀerent frequencies and zero temperature.The onset of χ′′o(Q AF,ω)and the minimum of1/χ′o(Q AF,ω)occurs atω/∆=1.77.A new feature appears atω>ωo(T=0)due to the shallow saddle-vHS.The discommensuration is seen to increase at increasing frequency[see Fig.9(d)].It is also due to a dynamic local nesting eﬀect similar to that at ω→ωo from below.Owing to the shallowﬂat band atµ/t′≃1.8).The onset is sharp and large due to the enhancement by the spin coherence factor(at ∆k∆k+Q AF<0),and the picking up of states around theshallow vHS atM =4t′−µ)by theδ-function[see Eq.(6)].This widens the window and low-ers the threshold for the interaction strength giving res-onance.This may partially account for the existence of the commensurate peak in YBCO and BSCCO,but not in LSCO.The RPA eﬀect at Q AF is eﬀectively turned oﬀwhenωis increased beyond the maximum ofχ′′o(Q AF,ω), since1/χ′o(Q AF,ω)recedes rapidly from zero then[see Fig.9(c)].Since the onset frequency scales with∆(T), the RPA commensurate resonance is necessarily softened by temperature.But the softening could be very little. The reason is that as∆(T)gets smaller,divergence in χ′o(Q AF,ω)gets smaller and will fail to fulﬁll the res-onance conditionχ′o(Q AF,ω)=1/V(Q AF)for a given V(Q AF).The peak could cease to exist before its soft-ening is really detected.The case for the softening of the commensurate peak in a bare theory is similar[12].0.01.02.0χ,,2.04.00.0χ,,1.03.05.01/χ,2.03.01.01/χ,(0.5π,0.5π)(π,π)(1.5π,π)(a)(b)(c)(d)(0.75π,0.75π)(π,π)(1.25π,π)kxkyπ−ππFIG.9.Frequency-driven peak shifting at diﬀerent fre-quency regimes:(a)1/χ′o(q,ω)atω/∆=1.0,1.2,and 1.5[solid lines from top to bottom];(b)χ′′(q,ω)at ω/∆=1.0,1.2,and1.5[at decreasing incommensurability along(π,π)−(1.25π,π)].(c)1/χ′o(q,ω)atω/∆=1.5,2.3, and2.5[solid lines at increasing“incommensurability”of the minima along(π,π)−(1.5π,π)];(d)χ′′(q,ω)at ω/∆=1.5,2.3,and2.5[at increasing incommensurability along(π,π)−(1.5π,π)].The dashed lines from top to bottom in(a)and(c)show1/χ′o atω/∆=0and1.77respectively.χ′′in(b)and(d)are calculated with|J|/t=1.6,and the com-mensurate resonance occurs atω=ωo(T=0)=1.5∆(0).We have truncated the strong commensurate peak for obvious rea-son.T=0,the quasiparticle dispersion is t=1,t′=−0.20,µ=−0.65,and∆=0.30(givesξk=−1.0−0.50.00.5 1.0k x /π−1.0−0.50.00.51.0k y /πFIG.10.Energy con-tour E k =ω/2for ω=2.5∆(0)>ωo (T =0).The thick local-nesting vector between below -FS contour has more pro-nounced nesting eﬀect (compared to the thin vector)due to the smaller curvature.We remind that the dispersion here has a saddle-vHS near to Fermi level (refer the exact parameters to Fig.9).The FS is indicated as dotted line.Figure 11plots the temperature-evolution of a low fre-quency spectrum in momentum space.The most no-table should be the softened commensurate resonance at ω<∼ωo (T =0).When the system is cooled down from above T c ,the broad and weak commensurate peak in the normal state grows into a sharp and strong peak at some temperature just below T c ,then it is split up into IC peaks and recede from Q AF ,to a ﬁxed location at low temperature.The weak IC structure at ω>ωo (T =0)in SC state is found to be simply smeared oﬀto a broad commensurate at warming up to the normal state [33].Note that the observed relative excitation intensity in the course of the evolution [see Fig.4in Ref.[3]]can only be obtained correctly in χ′′(q ,ω)via an appreciable in-teraction strength.Though the “shoot and split”behav-ior of the ω<ωo (T =0)spectrum at cooling down is already contained in a bare theory invoking vHS eﬀect for the commensurate peak [12],the normal state com-mensurate structure always incorrectly has an intensity higher than the low temperature IC structure in χ′′o (q ,ω)[see,for example,a similar case in Fig.3(b)].A weak and broad commensurate structure is always seen in the normal state at all frequencies [see Fig.12]as what is observed in experiments [3,2].It is due to the enhancement by the saddle-vHS and RPA correction.For the model system considered here,its intensity reaches a maximum at some ω/∆(0)≃0.4as a result of the competition between χ′′o and RPA enhancement.In summary,the diverging incommensurability at ω>ωo in the SC state is accounted by the approaching of2.04.00.0χ,,2.03.01/χ0,(a)(b)(0.75π,0.75π)(π,π)(1.25π,π)FIG.11.Temperature-driven peak shifting at low fre-quency regime :At ω=1.0∆(0)<ωo (T =0),(a)1/χ′o (q ,ω)is shown at T /T c =0,0.8,0.88(solid lines from top to bottom),and 1.0(dashed line);(b)χ′′(q ,ω)is shown at T /T c =0,0.8,0.88(solid lines at decreasing incommensura-bility),and 1.0(dashed line).The weak commensurate struc-ture in the normal state is seen to grow into a strong peak at T =0.88T c <∼T c ,then splits into IC peaks which recede from Q AF at T →0.The strong commensurate peak is truncated for obvious reason.T c =0.25∆(0),and the T -dependence of ∆(T )is assumed as the same as in Fig.3(b).Other parame-ters are referred to Fig.9.0.00.30.60.91.21.5χ,,(0.5π, 0.5π)(π,π)(1.5π,π)FIG.12.The broad commensurate peak of normal state χ′′(q ,ω)at ω/∆(0)=0.2,0.4(dashed lines from bottom to top),0.8,1.4,and 2.0(solid lines from top to bottom).T=T c =0.25∆(0).Refer those not stated parameters to Fig.9.bic crystals or the chained-YBCO.In those systems,anisotropy may either exist in the band dispersion or en-ter into the SC order parameter,or exist in both.It was pointed out in Ref.[34]that if only one of those two kinds of anisotropy exist in the system,it can be distinguished by the criterion that the gap anisotropy exist only in the SC state while the dispersion anisotropy persists into the normal state.Some important eﬀects of it which can be seen in INS spectrum are,if there is an admixture of s -component in the d -wave gap then there should be a rendering of the four-fold symmetry of the quasielas-tic node-to-node IC peaks to two-fold,and regardless of the anisotropy source a diﬀerence of the nesting IC peak intensities on diﬀerent crystal axis should be observed.We will discuss here a case of anisotropy in the gap,the existence of an s -component and the SC order pa-rameter has the form ∆k =∆(T )[as k +(1−a )d k ],where s k =1and d k =(cos k x −cos k y )/2.The existence of an s -component is suggested by the tunneling experiments [35–37]and its percentage is at a ∼0.2.It was discussed in Ref.[34]at low frequencies,but in order to clearly dis-tinguish the eﬀect we think that it is necessary to study its ω-dependence.Figure 13shows the contrast of the nesting IC peaks along diﬀerent crystal axis at diﬀerent ω,on a system with reasonable size of s -component in the gap.The diﬀerence in peak intensities is seen to be most promi-nent at low ω,and rapidly decreased at increasing fre-quency.Since the data by Mook et al.(on a detwinned orthorhombic YBa 2Cu 3O 6.6)[23]is at a somewhat low frequency 24meV ∼∆(0),we believe data on a wider ωrange is desirable in order to make deﬁnite statement on the origin of the one-dimensional nature.The diﬀerence in peak intensities can be easily ex-plained by the picture given in Sec.IV.The twofold sym-FIG.13.Anisotropic SC stateχ′′o (q ,ω)(a)in the full q -space at ω/∆=1.0,and (b)alongq =(π,0.5π)−(π,π)−(1.5π,π)at ω/∆=0.6,0.9,and 1.2(from bottom to top).The diﬀerence in height of the peaks on diﬀerent axis is more spectacular at low frequencies.The SC order parameter is taken as ∆k =∆[0.2s k +0.8d k ],where ∆=0.10,s k =1,and d k =(cos k x −cos k y )/2.T =0and the dispersion is t =1,t ′=−0.25,and µ=−0.65.metric spectrum has the node-to-node excitation closer to one of the two crystal axis,and excitation away from the nodal excitation is suppressed by the coherence fac-tor,therefore the peaks on one of the axis has lower in-tensity.This can only dominate at frequencies as low as ∼∆and excitation along the FS has not been fully opened up.We have presented here an anisotropy eﬀect suggested by the tunneling experiment and it can introduce one-dimensional feature into the INS spectrum (other possi-bility is also suggested in Ref.[9]).The frequency de-pendence of the contrast of IC peak intensity should act as a criterion to compare with the Stripe interpretation which has no frequency dependence.Furthermore such gap anisotropy eﬀect should vanish in the normal state.。

Calculating polynomial runtime properties

Calculating Polynomial Runtime Properties
Hugh Anderson, Siau-Cheng Khoo, Stefan Andrei and Beatrice Luca
Department of Computer Science School of Computing National University of Singapore {hugh,khoosc,andrei,lucabeat}@.sg
A third motivation for calculating polynomial runtime properties is to calculate more precise relative costs of the individual calls. For example in a ﬂow analysis of a program we may be interested in which calls are used most often, with a view to restructuring a program for eﬃciency. In this scenario, the relative costs between the individual calls is of interest. In the gcc compiler, a static branch predictor [2] uses heuristics to restructure the program code, optimizing the location of code for a branch more likely to occur. The approach described here can calculate more precise relative costs to improve these heuristics. In this paper we explore the automatic calculation of each of these costs through static analysis of the source of programs which are known to be aﬃne size-change terminating [1, 13], where the focus is on recording parameter size-changes only. The overall approach has three steps: 1. Assume a (degree k ) polynomial upper bound related to the runtime or space cost. The polynomial variables are the parameter sizes. 2. Derive from the source a set of equations constrained by this upper bound. 3. Solve the equations to derive the precise runtime. If the equations reduce to a vacuous result, then the original assumption of the degree of the polynomial must have been incorrect, and we repeat the process with a degree k + 1 assumption. This technique is surprisingly useful, and it is possible to derive precise runtime bounds on non-trivial programs. We can also calculate the time or space costs for a subclass of exponential costs, in particular those of the form φ1 · K φ2 + φ3 where φ1 , φ2 and φ3 are each a limited-degree polynomial in the parameter sizes, and K ∈ is a constant. There has been some research into run-time analysis for functional programs. For example, [16] explores a technique to evaluate a program’s execution costs through the construction of recurrences which compute the time-complexity of expressions in functional languages. It focuses on developing a calculus for costs, and does not provide automated calculations. In [7], Grobauer explores the use of recurrences to evaluate a DML program’s execution costs. Our focus is more with decidability aspects and precise time-costs than either of these approaches. An alternative approach is to limit the language in some way to ensure a certain run-time complexity. For example, in [8], Hofmann proposes a restricted type system which ensures that all deﬁnable functions may be computed in polynomial time. The system uses inductive datatypes and recursion operators. In our work, we generate time and stack costs of arbitrary functions or procedures, through analysis of the derived size-change information. A compact summary of a general technique for the calculation of time and space eﬃciency is found in the book [15] by Van Roy and Haridi, where recurrence relations are used to model the costs of the language elements of the programming language. There is unfortunately no general solution for an arbitrary set of recurrence relations, and in practice components of the costs are ignored, capturing at each stage only the most costly recurrence, and leading to big-O analysis. 2

Waiting for Clear Signals of New Physics in B and K Decays

a rXiv:h ep-ph/42191v118Fe b24TUM-HEP-544/04hep-ph/0402191February 2004Waiting for Clear Signals of New Physics in B and K Decays Andrzej J.Buras Technische Universit¨a t M¨u nchen Physik Department D-85748Garching,Germany Abstract We classify the extensions of the Standard Model (SM)according to the structure of local operators in the weak eﬀective Hamiltonian and the pres-ence or absence of new ﬂavour and CP-violating interactions beyond those represented by the CKM matrix.In particular we review characteristic prop-erties of models with minimal ﬂavour violation (MFV),models with signiﬁ-cant contributions from Higgs penguins and models with enhanced Z 0pen-guins carrying a large new CP-violating phase.Within the latter models,the anomalous behaviour of certain B →πK observables implies large departures from the SM predictions for rare and CP-violating K and B decays.Most spectacular is the enhancement of Br (K L →π0ν¯ν)by one order of magni-tude and a strong violation of the MFV relation (sin 2β)πν¯ν=(sin 2β)ψK S .On the other hand our prediction for (sin 2β)φK S ≈0.9diﬀers from the Belle result by the sign but is consistent with the BaBar value.We give a personal shopping list for the coming years.Invited Talk given atthe 9th Adriatic Meeting“Particle Physics and the Universe”Dubrovnik,September 4–14,2003Waiting for Clear Signals of New Physics in B and K DecaysAndrzej J.BurasTechnical University Munich,Physics DepartmentD-85748Garching,Germany1IntroductionThe quarkﬂavour dynamics of the Standard Model(SM)is consistent with the existing data on K and B meson decays within experimental and theoret-ical uncertainties.In spite of this,most of us expect that when the precisionof experiments and also the theoretical tools improve,some clear signals of new physics(NP)at very short distance scales will be seen.As the search fornew phenomena through K and B decays is necessarily an indirect one,it will not be easy toﬁnd out what precisely this NP is.This will be even thecase in the presence of clear deviations from the SM expectations.Yet,remembering that decays like K L→µ+µ−,K L→ππand the K0−¯K0mixing played an important role in the construction of the SM[1]and led to the GIM mechanism[2]and the CKM matrix[3],we are conﬁdent thatthese indirect signals of NP will give us deﬁnitive hints where to go and where not to go.This in turn will allow us to select few competing theories which will be tested directly through high energy collider experiments.Moreover,these signals will hopefully give us additional hints for the fundamentalﬂavour dynamics at ultra short distance scales that cannot be tested directly in a foreseeable future.It is obvious that for this project to become successful,it is essential to•make the SM predictions for K and B decay observables as accurate as possible,in order to be sure that the observed deviations from the SM expectations originate in NP contributions and are not due to our insuﬃcient understanding of hadron dynamics and/or truncation of the perturbative series.•consider simultaneously as many processes as possible.Only in this man-ner the paramaters of a given theory can be fully determined and having them at hand predictions for other observables can be made.In this en-terprise correlations between various observables play an important rˆo le, as they may exclude or pinpoint a given extension of the SM even withouta detailed knowledge of the parameters speciﬁc to this theory.2Andrzej J.Buras•consider observables which while being sensitive to the short distance structure of the theory are only marginally sensitive to the long distance dynamics that is not yet fully under control at present.These notes review selected aspects of this program.Due to severe space limitations I will follow the strategy of many of my colleagues and will con-centrate to a large extent on my own work and the work done in my group in Munich.In Section2,after giving a master formula for weak decay ampli-tudes,I will group various extentions of the SM inﬁve classes.In subsequent three sections,I will discuss theﬁrst three classes that,being more predictive than the remaining two,should be easiest to test.A shopping list in Section 6,with the hope to be able to distinguish between various NP scenarios,ends this presentation.Recent more detailed reviews can be found in[4].2Theoretical Framework2.1Master Formula for Weak DecaysThe present framework for weak decays is based on the operator product ex-pansion that allows to separate short(µSD)and long(µLD)distance contri-butions to weak amplitudes and on the renormalization group(RG)methods that allow to sum large logarithms logµSD/µLD to all orders in perturbation theory.The full exposition of these methods can be found in[5,6].An amplitude for a decay of a meson M=K,B,..into aﬁnal state F=πν¯ν,ππ,πK is then simply given byA(M→F)=G F2 i V i CKM C i(µ) F|Q i(µ)|M .(1)Here G F is the Fermi constant and Q i are the relevant local operators which govern the decays in question.They are built out of quark and lep-tonﬁelds.The Cabibbo-Kobayashi-Maskawa factors V i CKM[3],the matrix elements F|Q i(µ)|M and the Wilson coeﬃcients C i(µ),evaluated at the renormalization scaleµ,describe the strength with which a given operator enters the amplitude.Formula(1)can be cast into a master formula for weak decay amplitudes that goes beyond the SM[7].It reads(we suppress G F):A(Decay)= i B iηi QCD V i CKM[F i SM+F i New]+ k B New k[ηk QCD]New V k New[G k New].(2)The non-perturbative parameters B i represent the matrix elements of local operators present in the SM.For instance in the case of K0−¯K0mixing,the matrix element of the operator¯sγµ(1−γ5)d⊗¯sγµ(1−γ5)d is represented byWaiting for Clear Signals of New Physics in B and K Decays3 the parameterˆB K.There are other non-perturbative parameters in the SM that represent matrix elements of operators Q i with diﬀerent colour and Dirac structures.The objectsηi QCD are the QCD factors resulting from RG-analysis of the corresponding operators and F i SM stand for the so-called Inami-Lim functions[8]that result from the calculations of various box and penguin diagrams in the SM.They depend on the top-quark mass.New physics can contribute to our master formula in two ways.It can mod-ify the importance of a given operator,present already in the SM,through the new short distance functions F i New that depend on the new parameters in the extensions of the SM like the masses of charginos,squarks,charged Higgs particles and tanβ=v2/v1in the MSSM.These new particles enter the new box and penguin diagrams.In more complicated extensions of the SM new operators(Dirac structures),that are either absent or very strongly sup-pressed in the SM,can become important.Their contributions are describedby the second sum in(2)with B Newk ,[ηk QCD]New,V k New,G k New being analogsof the corresponding objects in theﬁrst sum of the master formula.The V k New show explicitly that the second sum describes generally new sources of ﬂavour and CP violation beyond the CKM matrix.This sum may,however, also include contributions governed by the CKM matrix that are strongly suppressed in the SM but become important in some extensions of the SM. In this case V k New=V k CKM.Clearly the new functions F i New and G k New as well as the factors V k New may depend on new CP violating phases complicating considerably phenomenological analyses.2.2Classiﬁcation of New PhysicsClassiﬁcation of new physics contributions can be done in various ways.Hav-ing the formula(2)at hand let us classify these contributions from the point of view of the operator structure of the eﬀective weak Hamiltonian,the com-plex phases present in the Wilson coeﬃcients of the relevant operators and the distinction whether theﬂavour changing transitions are governed by the CKM matrix or by new sources ofﬂavour violation[9].For theﬁrst four classes below we assume that there are only three generations of quarks and leptons.The last class allows for more generations.Class AThis is the simplest class to which also the SM belongs.In this class there are no new complex phases andﬂavour changing transitions are governed by the CKM matrix.Moreover,the only relevant operators are those that are relevant in the SM.Consequently NP enters only through the Wilson coeﬃcients of the SM operators that can receive new contributions through diagrams involving new internal particles.4Andrzej J.BurasThe models with these properties will be called Minimal Flavour Violation (MFV)models,as deﬁned in[10].Other deﬁnitions can be found in[11,12]. In this case our master formula simpliﬁes toA(Decay)= i B iηi QCD V i CKM F i(v)F i=F i SM+F i New(real),(3) where F i(v)are the master functions of MFV models[13]S(v),X(v),Y(v),Z(v),E(v),D′(v),E′(v)(4) with v denoting collectively the parameters of a given MFV model.A very detailed account of MFV can be found in[13].In Section3some of its main features will be recalled.Examples of models in this class are the Two Higgs Doublet Model II and the constrained MSSM if tanβis not too large.Also models with one extra universal dimension are of MFV type.Class BThis class of models diﬀers from class A through the contributions of new operators not present in the SM.It is assumed,however,that no new complex phases beyond the CKM phase are present.We have thenA(Decay)= i B iηi QCD V i CKM[F i SM+F i New]+ k B New k[ηk QCD]New V k CKM[G k New](5)with all the functions F i SM,F i New and G k New being real.Typical examples of new Dirac structures are the operators(V−A)⊗(V+A),(S−P)⊗(S±P) andσµν(S−P)⊗σµν(S−P)contributing to B0d,s−¯B0d,s mixings that become relevant in the MSSM with a large tanβ.Class CThis class of models diﬀers from class A through the presence of new complex phases in the Wilson coeﬃcients of the usual SM operators.Con-tributions of new operators can be,however,neglected.An example is the MSSM with not too a large tanβand with non-diagonal elements in the squark mass matrices.This class can be summarized byA(Decay)= i B iηi QCD V i CKM F i(v)F i(v)(complex).(6)Class DHere we group models with new complex phases,new operators and new ﬂavour changing contributions which are not governed by the CKM matrix. As now the amplitudes are given by the most general expression(2),the phenomenology in this class of models is more involved than in the classes A–C[14,15].Examples of models in class D are multi-Higgs models with complex phases in the Higgs sector,general SUSY models,models with spontaneous CP violation and left-right symmetric models.Waiting for Clear Signals of New Physics in B and K Decays5 Class EHere we group models in which the unitarity of the three generation CKM matrix does not hold.Examples are four generation models and models with tree level FCNC transitions.If this type of physics is present,the unitarity triangle does not close.The most recent discussion of the possible violation of the unitarity of the three generation CKM matrix can be found in[16]. 3Class A:MFV Models3.1Model Independent RelationsOne can derive a number of relations between various observables that do not depend on the functions F i(v)and consequently are universal within the class of the MFV models.Violation of any of these relations would be a signal of the presence of new operators and/or new weak complex phases.We list here the most interesting relations of this type.1.A universal unitarity triangle(UUT)common to all MFV models can be constructed by using only observables that are independent of F i(v)[10]. In particular its two sides R b and R t[4]can be determined from|V ub/V cb| and the ratio of B0d,s−¯B0d,s mixing mass diﬀerences∆M d,s,respectively:R b=4.4 V ub1.24 ∆M s 0.50/ps,(7)whereξ=1.24±0.08[17]is a non-perturbative parameter,∆M d=(0.503±0.006)/ps and∆M s>14.4/ps.Moreover,the angleβof this triangle can be found from measurements of the time-dependent CP asymmetry aψK(t)S with the result[18,19]=0.736±0.049.(8) (sin2β)ψKSUsing(7)and(8)oneﬁnds the apex of the UUT placed within the larger ellipse inﬁg.1[20].A similar analysis has been performed in[11].2.Two theoretically clean relations are[21]Br(B→X dν¯ν)V ts 2,Br(B s→µ¯µ)ˆB sτ(B s)∆M d.(9)and consequently They do not involve the B q–meson decay constants F Bqcontain substantially smaller hadronic uncertainties than the formulae for individual branching ratios[13].The ratioˆB s/ˆB d is known from lattice cal-culations with a respectable precision[17]:ˆBs6Andrzej J.Buras00.20.40.60.81η_Fig.1.and in the 95%With a give a 3.It is only on the K →πν¯νbranching ratios and a calculable ¯Pc (X )=0.38±0.06[22,23,24]:sin 2(β−βs )=2r s B 1−B 2−¯P c (X )B 2,(11)where βs ≈−1◦enters V ts =−|V ts |exp(−iβs ),εi =±1andB 1=Br (K +→π+ν¯ν)2.09·10−10.(12)In the MFV models ε1=ε2=sgn(X )[25],where X is the relevant master function.4.With no weak phases beyond the CKM phase,we also expect(sin 2β)πν¯ν=(sin 2β)ψK S ,(sin 2β)φK S ≈(sin 2β)ψK S ,(13)with the accuracy of the last relation at the level of a few percent [26].The conﬁrmation of these two relations would be a very important test of the MFV idea.Indeed,in K →πν¯νthe phase βoriginates in the Z 0penguin diagram,whereas in the case of a ψK S in the B 0d −¯B 0d box diagrams.In the case of the asymmetry a φK S it originates also in B 0d −¯B 0d box diagrams,but the second relation in (13)could be spoiled by NP contributions in the decay amplitude for B →φK S that is non-vanishing only at the one loop level.Waiting for Clear Signals of New Physics in B and K Decays7 Interestingly,the present data from Belle may indicate the violation of the second relation in(13),although the experimental situation is very unclear at present[27,28]:(sin2β)φKS = +0.45±0.43±0.07(BaBar)−0.96±0.50+0.11−0.09(Belle),(14)A subset of theoretical papers addressing this issue is listed in[29].5.An important consequence of(11)–(13)is the following one.For a given sin2βextracted from aψKSand Br(K+→π+ν¯ν)only two values of Br(K L→π0ν¯ν),corresponding to two signs of the master function X(v),are possible in the full class of MFV models,independent of any new parameters present in these models[25].Consequently,measuring Br(K L→π0ν¯ν)will either select one of these two possible values or rule out all MFV models.6.As pointed out in[30]in most MFV models there exists a correlation be-tween the zeroˆs0in the forward-backward asymmetry A FB in B→X sµ+µ−and Br(B→X sγ).We show this correlation inﬁg.2.ˆBqY2(v)8Andrzej J.Burasallow to predict Br(B s,d→µ¯µ)in a given MFV model,characterized by Y(v)and S(v),with substantially smaller hadronic uncertainties than found by using directly the formulae for these branching ratios that suﬀer from.In particular in the SM model weﬁnd[21] large uncertainties due to F BqBr(B s→µ¯µ)=(3.4±0.5)·10−9,Br(B d→µ¯µ)=(1.00±0.14)·10−10,(16) whereσ R t cosβ+λ4P c(X)Waiting for Clear Signals of New Physics in B and K Decays9Table1.Example of branching ratios for rare decays in the MFV and the SM.Branching Ratios MFVBr(K+→π+ν¯ν)×101119.1Br(K L→π0ν¯ν)×10119.9Br(K L→µ+µ−)SD×1093.5Br(K L→π0e+e−)CPV×10114.9Br(B→X sν¯ν)×10511.1Br(B→X dν¯ν)×1064.9Br(B s→µ+µ−)×10919.4Br(B d→µ+µ−)×10106.110Andrzej J.BurasThe presence of the enhanced FCNC Higgs couplings implies in turn important contributions of new operators in the eﬀective theory that are strongly suppressed in models of class A.In particular the operatorsO S=m b(b R s L)(¯µγ5µ)(19)fully dominate the branching ratios for B s,d→µ+µ−[45]when tanβ≥30. An approximate formula for Br(B s→µ+µ−)is given then by[46,47]Br(B s→µ+µ−)≈3.5·10−5 tanβM A 4F(εi,tanβ),(20) F(εi,tanβ)= 16π2εYBr(B s→µ+µ−)= τ(B d)F B d 2 V td M B d 5,(22)which diﬀers slightly from the usual MFV formula in that the last factorhas the powerﬁve instead of two.On the other hand the branching ratios themselves can still be enhanced by almost a factor of500.If this is indeed thecase,B s,d→µ+µ−should be observed already at Tevatron and B factories, respectively.The enhanced neutral Higgs couplings and more generally large tanβeﬀects can play also a signiﬁcant role in∆M s[47]inducing in particular four–fermion operatorsQ LR1=(b Rγµs R),Q LR2=(b L s R),(23)whose Wilson coeﬃcients are negligible in the SM.Oneﬁnds then∆M s=(∆M s)SM(1+f s)≈(∆M s)SM−|∆M s|DP,(24) where the new contributions come dominantly from double Higgs penguins (DP),as indicated above.Being proportional to m q m b tan4βfor∆M q,these contributions can be neglected in∆M d that for large tanβis very close to the SM estimate.It turns out that∆M s is suppressed for any choice of supersymmetric parameters(f s<0)with the size of suppression dependent strongly on the stop mixing,M A and tanβ.As a consequence of the mismatch between Higgs contributions to∆M d and∆M s,the MFV formula for R t in(7)is modiﬁed toR t=0.90 ξ18.4/ps∆M d1+f s.(25)Waiting for Clear Signals of New Physics in B and K Decays11 However,most interesting is the correlation of the enhanced neutral Higgs eﬀects in B s,d→µ+µ−and∆M s,that is independent of F(εi,tanβ)in(21): Br(B s→µ+µ−)≈10−6 tanβM A 2 |∆M s|DP12Andrzej J.Buras5.3Weak Phases in∆F=1TransitionsNew weak phases could enter also decay amplitudes.As now these eﬀects enter in principle diﬀerently in each decay,the situation can be very involved with many free parameters,no universal eﬀects and little predictive power.Here I will only discuss one scenario,discussedﬁrst in[31,32,52,53]and recently in the context of a simultaneous analysis of prominent non–leptonic B decays like B→ππ,B→πK,B→ψK S and B→φK S and equallyprominent rare decays like K→πν¯ν,K L→π0e+e−,B s,d→µ+µ−,B→X s,d e+e−andε′/εin[23,24].It is the scenario of enhanced Z0penguins witha large complex weak phase in which the only modiﬁcation with respect to class A is the replacement in the Z0penguin function C(v)→|C(v)|e iθC that makes the master functions X(v),Y(v)and Z(v)complex quantities: X(v)=|X(v)|e iθX,Y(v)=|Y(v)|e iθY,Z(v)=|Z(v)|e iθZ.(28) The magnitudes and phases of these three functions are correlated with each other as they depend only on|C(v)|e iθC and other smaller contributions,that can be set to their SM values.This new analysis has been motivated by the experimental situation in B→ππand B→πK decays described below.While our analysis does not rely on a particular model with the properties speciﬁed above,concrete models with enhanced CP-violating Z0-mediated FCNC couplings generated either at the one-loop level or even at the tree level have been discussed in the literature.They are reviewed in[31,32,52,53],in particular in the last of these papers;see also[54].Also models with Z′-mediated FCNCs could be put in this class,provided their contributions can eﬀectively be absorbed in the function C(v).For a recent analysis,see[55].5.4The B→ππPuzzleThe BaBar and Belle collaborations have very recently reported the obser-vation of B d→π0π0decays with CP-averaged branching ratios of(2.1±0.6±0.3)×10−6and(1.7±0.6±0.2)×10−6,respectively[56,57].These measurements represent quite some challenge for theory.For example,ina recent state-of-the-art calculation[58]within QCD factorization(QCDF)[59],a branching ratio that is about six times smaller is favoured,whereas the calculation of B d→π+π−points towards a branching ratio about two times larger than the current experimental average.On the other hand,the calculation of B+→π+π0reproduces the data rather well.This“B→ππpuzzle”is reﬂected by the following quantities:Rππ+−≡2 Br(B±→π±π0)τB+=2.12±0.37(29) Rππ00≡2 Br(B d→π0π0)Waiting for Clear Signals of New Physics in B and K Decays13 5.5The B→πK PuzzleIn the B→πK system,the CLEO,BaBar and Belle collaborations have measured the following ratios of CP-averaged branching ratios[60]:R c≡2 Br(B±→π0K±)2 Br(B d→π∓K±)τB+Br(B±→π±K)14Andrzej J.Buras=°nFig.3.The situation in the R n–R c plane[23].We show contours for values of q=0.69,q=1.22and q=1.75andφ∈[0◦,360◦].The ranges from(31)and(32) (experiment)as well as from the SM are indicated in grey.Step1:Since B→ππdecays and the usual analysis of the UT are only insigniﬁ-cantly aﬀected by EW penguins,the B→ππsystem can be described as in the SM.Employing the SU(2)isospinﬂavour symmetry of strong interactions and the information onγfrom the UTﬁts[17],we could extract the relevant hadronic B→ππparameters,andﬁnd large non-factorizable contributions, in variance with the QCD factorization approach.Having these parameters at hand,we could then also predict the direct and mixing-induced CP asym-metries of the B d→π0π0channel.A future measurement of one of these observables allows a determination ofγ.Step2:Using the SU(3)ﬂavour symmetry and plausible dynamical assumptions,we could subsequently determine the hadronic B→πK parameters through the B→ππanalysis,and calculate the B→πK observables in the SM. Interestingly,we found agreement with the pattern of the B-factory data for those observables where EW penguins play only a minor rˆo le.On the other hand,the observables receiving signiﬁcant EW penguin contributions did not agree with the experimental picture,thereby suggesting NP in the EW penguin sector.Parametrizing the EW contributions to B→πK by qe iφthat is related to the Z0-penguin function C(v)through[35]C(v)=|C(v)|e iθC=2.35¯q e iφ−0.82,¯q=q |V ub/V cb|Waiting for Clear Signals of New Physics in B and K Decays 15we have demonstrated [23,24]that one can describe all the currently available B →πK data provided q =1.75+1.27−0.99,φ=−(85+11−14)◦,(35)to be compared with q =0.69[65]and φ=0in the SM.In particular as seen in ﬁg.3the values in (35)allow to describe properly the data in (31)and(32).The requirement of a large CP-violating NP phase around −90◦is the most interesting result of this study.A crucial future test of this scenario will be provided by the CP-violating B d →π0K S observables,which we could predict.Moreover,we could obtain valuable insights into SU (3)-breaking eﬀects,which support our working assumptions,and could also determine the UT angle γ,that is in agreement with the UT ﬁts.Step 3:In turn,the sizeably enhanced EW penguins with their large CP-violating NP phase have important implications for rare K and B decays as well as ε′/ε.Interestingly,several predictions diﬀer signiﬁcantly from the SM expectations and should easily be identiﬁed once the data improve.Including the constraint from B →X s l +l −[40],that selects q ≤1.0in(35),the most interesting results of this study,presented in [23,24]are:a)For the very clean K →πν¯νdecays,we ﬁndBr (K +→π+ν¯ν)=(7.5±2.1)·10−11,Br (K L →π0ν¯ν)=(3.1±1.0)·10−10,(36)to be compared with the SM predictions,for which we ﬁnd (8.0±1.1)×10−11and (3.2±0.5)×10−11in the ballpark of other estimates [66],respectively.On the experimental side the results of the AGS E787[67]collaboration and KTeV [68]are Br (K +→π+ν¯ν)=(15.7+17.5−8.2)×10−11and Br (K L →π0ν¯ν)<5.9×10−7,respectively.The enhancement of Br (K L →π0ν¯ν)by one order of magnitude and the pattern in (36)are dominantly the consequences of βX =β−θX ≈111◦with θX deﬁned in (28).Indeed,Br (K L →π0ν¯ν)X SM 2 sin βX Br (K +→π+ν¯ν)≈4.4×(sin βX )2≈(4.2±0.2).(38)Interestingly,the above ratio turns out to be very close to its absolute upper bound in [69].Fig.4illustrates nicely these features.A spectacular implica-tion of these ﬁndings is a strong violation of (sin 2β)πν¯ν=(sin 2β)ψK S [22],which is valid in the SM and any model with minimal ﬂavour violation as discussed in Section 3.Indeed,we ﬁnd(sin 2β)πν¯ν=sin 2βX =−(0.69+0.23−0.41),(39)16Andrzej J.Burasin striking disagreement with (sin 2β)ψK S =0.74±0.05in (8).b)Another implication is the large branching ratioBr (K L →π0e +e −)=(7.8±1.6)×10−11,(40)which is governed by direct CP violation in this scenario.On the other hand,the SM result (3.2+1.2−0.8)×10−11[70]is dominated by the indirect CP violation.Next,the integrated forward–backward CP asymmetry for B d →K ∗µ+µ−[53]can be very large in view of θY ≈−100◦as it is given byA CP FB =(0.03±0.01)×tan θY .(41)c)Next,Br (B →X s,d ν¯ν)and Br (B s,d →µ+µ−)are enhanced by factors of 2and 5,respectively.The impact on K L →µ+µ−is rather moderate.d)As emphasized in [31],enhanced Z 0penguins may play an important rˆo le in ε′/ε.The enhanced value of C and its large negative phase suggested by the B →πK analysis require a signiﬁcant enhancement of the relevant hadronic matrix element of the QCD penguin operator Q 6,with respect to the one of the EW penguin operator Q 8,to be consistent with the ε′/εdata.e)We have also explored the implications for the decay B d →φK S [23].Large hadronic uncertainties preclude a precise prediction,but assuming that the sign of the cosine of a strong phase agrees with factorization,we ﬁnd that (sin 2β)φK S >(sin 2β)ψK S .This pattern is qualitatively diﬀerent from thepresent B -factory data [27],which are,however,not yet conclusive as seen in (14).On the other hand,a future conﬁrmation of this pattern could be another signal of enhanced CP-violating Z 0penguins at work.6Shopping ListWe have seen that each of the NP scenarios discussed above had some spe-ciﬁc features not shared by other scenarios and with a suﬃcient number of measurements it should be possible to distinguish between them,eventually selecting one of them or demonstrating the necessity for going to scenarios in classes D and E.There is a number of questions which I hope will be answered in the coming years:•Probably the most important at present is the clariﬁcation of the discrep-ancy between Belle [28]and BaBar [27]in the measurement of (sin 2β)φK S .The conﬁrmation of the signiﬁcant departure of (sin 2β)φK S from the al-ready accurate value of (sin 2β)ψK S ,would be a clear signal of new physicsthat cannot be accomodated within classes A and B.Also in a particular scenario of class C,just discussed,one ﬁnds (sin 2β)φK S >(sin 2β)ψK S and consequently (sin 2β)φK S is of the same magnitude as the centralvalue found by the Belle collaboration but of opposite sign!Thus it is likely that the conﬁrmation of the Belle result will require the models of class D.Waiting for Clear Signals of New Physics in B and K Decays171·102·103·104·105·102·104·106·108·101·101.2·101.4·101.6·10rang B R (K +→π+ν¯ν)BR(K L →π0ν¯ν)18Andrzej J.Burasfrom the SM and the MFV expectations could be easily identiﬁed in the very clean K→πν¯νdecays as clear signals of NP.•The improved measurements of Br(B→X s l+l−)and Br(K+→π+ν¯ν) in the coming years will eﬃciently bound the possible enhancements of Z0penguins,at least within the scenarios A–C discussed here.•Also very important is an improved measurement of Br(B→X sγ)as well as the removal of its sensitivity toµc in m c(µc)through a NNLO calculation.This would increase the precision on the MFV correlation between Br(B→X sγ)and the zeroˆs0in the forward–backward asym-metry A FB(ˆs)in B→X s l+l−.A20%suppression of Br(B→X sγ) with respect to the SM accompanied by a downward shift ofˆs0would be an interesting conﬁrmation of the correlation in question and consistent with the eﬀects of universal extra dimensions with a low compactiﬁcation scale of order few hundred GeV.On the other handﬁnding no zero inA FB(ˆs)would likely point towardsﬂavour violation beyond the MFV.•Finally,improved bounds and/or measurements of processes not existing or very strongly suppressed in the SM,like various electric dipol moments and FCNC transitions in the charm sector will be very important in the search for new physics.The same applies toµ→eγand generally lepton ﬂavour violation.We could continue this list for ever,in particular in view of the expected progress at Belle and Babar,charm physics at Cornell,experimental program at LHCb,BeTeV and the planned rare K physics experiments.But the upper bound on the maximal number of pages for my contribution has been already signiﬁcantly violated which is a clear signal that I should conclude here.The conclusion is not unexpected:in this decade,it will be very exciting to follow the development in thisﬁeld and to monitor the values of various observables provided by our experimental colleagues by using the strategies presented here and other strategies found in the rich literature.AcknowledgementsI would like to thank the organizers for inviting me to such a wonderful meeting and enjoyable atmosphere,and Felix Schwab for invaluable com-ments on the manuscript.Most importantly,the material presented here would not be possible without such a fruitful collaboration with20magniﬁ-cant physicists whose names can be found below.Many thanks to all of them. The work presented here has been supported in part by the German Bun-desministerium f¨u r Bildung und Forschung under the contract05HT1WOA3 and the DFG Project Bu.706/1-2.References1.S.Weinberg,arXiv:hep-ph/0401010.2.S.L.Glashow,J.Iliopoulos and L.Maiani Phys.Rev.D2,1285(1970).。

学术英语(理工)详解答案_Unit 2

highlighted texts such as words in bold or italic text;
graphs, tables or diagrams.
Unit 2 Searching for Information
2 Scanning and skimming
1 What does “A.I.” in the title stand for? Artificial intelligence. 2 What is the main idea of the article you may predict from the title? The article may argue for/again the idea that Artificial intelligence will replace human jobs in the future. 3 What does the story in the first paragraph imply? The story tells that computerization threatens to replace
To evaluate source materials
Is the material a primary or a secondary source? Is the source the latest one? Is the author a reliable scholar or an expert in the field? Does the author have biases or prejudices? Has the author been cited frequently in the field? Are the author’s arguments supported by evidence such as statistics, experiment, recent scientific findings? Are different opinions considered and weighed or simply ignored ? Are the author’s arguments and conclusions convincing?

A model for gauge theories with Higgs fields

6. The double scaling limits
1
A. General solitary waves solutions A.1 General analysis A.2 The solitons for general N A.2.1 Basic equations A.2.2 Description of the solutions A.3 Formulas for the case N=3 B. The functional s[f ] B.1 s[f ] for constant f B.2 s[f ] for arbitrary f C. The C.1 C.2 C.3 1/N corrections to the critical hypersurface The eﬀective potential The equation for Hq Formulas for Iϕ , Iα and Iϕ − Iα
2
a large class of theories which are simple enough to be tractable even in their strongly coupled, non-supersymmetric, regime, but complex enough so that many interesting questions about gauge theories have a counterpart in the simple models. More precisely, and as will be explained in the following, our simple models have all of the following general properties: a) The models are tractable from the non-supersymmetric versions to the supersymmetric ones. In the latter case exact results can be obtained (BPS mass formulas in particular) in strict parallel with what is known for four dimensional supersymmetric gauge theories. Formulas can actually quantitatively coincide in these cases. Asymptotically free as well as conformal ﬁeld theories can be studied. b) The models have an analogue of a moduli space, with generically both weakly coupled and strongly coupled regions. Strong quantum corrections then drastically modify the classical structure. At weak coupling we can have solitons playing the rˆ ole of magnetic monopoles or dyons, and these can become massless at strong coupling singularities. Argyres-Douglas -like CFTs [7] can appear at strong coupling. All these phenomena can occur and be studied in both supersymmetric and non-supersymmetric theories. c) The supersymmetric versions of the models admit constructions in terms of branes, and they can be solved via M theory. d) The models have a non-trivial 1/N expansion ` a la ’t Hooft [8]. The large N limit can be unconventional, as for N = 2 super Yang-Mills [9]. We will illustrate a), b) and d) in this article; c) is already known, as we will review below. Two additional properties would also be desirable, e) The supersymmetric versions of the models can be geometrically engineered as in [10]. f) The models are dual ` a la Maldacena to some kind of string theory. Though I am not aware of any explicit construction, it is very likely that e) is true, as explained later. As for f), it is plausible that it could be true in view of c) and d), but we will have unfortunately nothing to say about this fascinating possibility in this paper. A fundamental question of principle, that we would also like to address, is whether it is possible to prove, or at least to get a good general understanding, that gauge theories (supersymmetric or not) or other four dimensional ﬁeld theories can have a description in terms of string theories. The modern starting point is a conjecture [11] motivated by the relationship between supersymmetric D-branes and solitons in closed string theories. It is not clear whether this intuitive understanding of the gauge theory/string theory correspondence makes sense in the general non-supersymmetric case. Interestingly, the results of the present paper suggest a way to understand the possible proliferation of dualities between four dimensional ﬁeld theories and string

欧洲药典7.5版

EUROPEAN PHARMACOPOEIA 7.5
INDEX
To aid users the index includes a reference to the supplement in which the latest version of a text can be found. For example : Amikacin sulfate...............................................7.5-4579 means the monograph Amikacin sulfate can be found on page 4579 of Supplement 7.5. Note that where no reference to a supplement is made, the text can be found in the principal volume.
English index ........................................................................ 4707
Latin index ................................................................................. 4739
EUROPEAN PHARMACOPபைடு நூலகம்EIA 7.5
Index
Numerics 1. General notices ................................................................... 7.5-4453 2.1.1. Droppers...................

ho问题回答

HO1. IntroductionHO, short for Handover, is a fundamental concept in wireless communication systems. It refers to the process of transferring an ongoing call or data session from one cell to another without interrupting the communication. Handovers are crucial in maintaining seamless connectivity and ensuring smooth user experience in mobile networks.In this article, we will explore the significance of handovers, understand the various types of handovers used in different scenarios, and delve into the technical aspects of handover procedures.2. Significance of HandoversHandovers play a vital role in mobile communication systems for several reasons:2.1 Seamless ConnectivityOne of the primary objectives of handovers is to ensure uninterrupted connectivity for mobile users. As users move from one cell to another while making calls or using data services, it is essential to transfer their ongoing sessions seamlessly without any disruptions. Handovers enable this smooth transition between cells, allowing users to maintain their connections without experiencing call drops or data loss.2.2 Load BalancingHandovers also facilitate load balancing in cellular networks. By intelligently distributing users across multiple cells, handover mechanisms help optimize network resources and prevent congestion in specific cells. This dynamic load management ensures efficientutilization of network capacity and enhances overall network performance.2.3 Quality of Service (QoS)Handovers contribute significantly to maintaining QoS parameters such as call quality and data throughput. When a user moves towards the edge of a cell’s coverage area, signal strength may weaken, leading to degradedservice quality. By triggering a handover to a neighboring cell with better signal strength, QoS can be maintained at an acceptable level.3. Types of HandoversThere are several types of handovers used in different scenarios based on network architectures and mobility patterns:3.1 Intra-Cell HandoverIntra-cell handover, also known as soft handover, occurs when a mobile device moves within the coverage area of a single base station. In this case, the handover process involves transferring the connection between different sectors or antennas within the same cell. Intra-cell handovers are typically transparent to users and ensure seamless connectivity without noticeable disruptions.3.2 Inter-Cell HandoverInter-cell handover, also referred to as hard handover, takes place when a mobile device moves from one cell to another within the same network. This type of handover involves transferring the ongoing session from the serving cell to a target cell. Inter-cell handovers require coordination between multiple base stations and involve more complex procedures compared to intra-cell handovers.3.3 Inter-System HandoverInter-system handover occurs when a mobile device moves betweendifferent wireless technologies or networks, such as from GSM (2G) to UMTS (3G) or from LTE (4G) to Wi-Fi. These handovers involve not only changing cells but also transitioning between different network architectures and protocols.3.4 Vertical HandoverVertical handovers are specific types of inter-system handovers that involve switching between heterogeneous networks with varying characteristics. For example, a vertical handover may occur when a user transitions from cellular network coverage to Wi-Fi connectivity or vice versa. These handovers aim to provide seamless connectivity while considering factors such as signal strength, network availability, and user preferences.4. Handover ProceduresHandovers involve several steps and procedures that ensure successful transfer of ongoing sessions:4.1 Measurement and TriggeringThe first step in a handover procedure is measuring the signal quality and strength of neighboring cells or systems. The serving cell continuously monitors these measurements and triggers a handover decision when certain predefined thresholds are met.4.2 Handover DecisionOnce triggered, the handover decision is made based on various factors such as signal strength, interference levels, network load, and QoS requirements. The decision-making process aims to select the most suitable target cell or system for handover.4.3 Handover PreparationAfter the handover decision, the serving cell initiates preparations for the handover. This involves allocating resources in the target cell and coordinating with neighboring base stations or systems.4.4 Handover ExecutionDuring the handover execution phase, the mobile device switches its connection from the serving cell to the target cell. The switch should be seamless and imperceptible to users, ensuring uninterrupted communication.4.5 Handover CompletionOnce the handover is executed successfully, a confirmation message is exchanged between base stations or systems to finalize the handover process. This ensures that both ends of the communication are aware of the transition and can resume normal operation.5. ConclusionHO (Handover) is an essential mechanism in wireless communication systems that enables seamless connectivity, load balancing, and maintenance of quality of service parameters. By understanding different types of handovers and their procedures, network operators can optimizetheir networks for efficient resource utilization and improved user experience.Handovers continue to evolve with advancements in technology, such as 5G networks and network slicing. These developments aim to further enhance handover capabilities and provide even better connectivity in future wireless communication systems.Remember: HO is not just a simple abbreviation; it represents a critical aspect of mobile networks that keeps us connected wherever we go!。