QED and Electroweak Corrections to Deep Inelastic Scattering

验证摩擦起电的英文作文Frictional electrification is a phenomenon in which two objects come into contact with each other and generate electric charge. This process occurs when the surface of one object, which may be a solid, liquid, or gas, is rubbed against another surface. The friction between the two surfaces causes the transfer of electrons, resulting in one object becoming positively charged and the other becoming negatively charged.The concept of frictional electrification has been known for centuries. Ancient Greek philosophers such as Thales of Miletus and Pliny the Elder observed that amber, when rubbed with fur, would attract light objects such as feathers. This early observation marked the beginning of the study of electrostatics, which is the branch of physics that deals with the behavior of electric charges at rest.In order to understand how frictional electrification works, it is important to examine the nature of electric charge. Electric charge is a fundamental property of matter, and it can exist in two forms: positive and negative. Like charges repel each other, while opposite charges attract each other. When two objects with different electrical properties come into contact with each other, the transfer of electrons occurs, leading to the generation of electric charge on the surfaces of the objects.There are several factors that influence the extent of frictional electrification. The type of materials involved, the amount of force applied during the rubbing process, and environmental conditions such as humidity and temperature can all affect the degree of charge transfer. Different materials have different tendencies to gain or lose electrons when they come into contact with each other. For example, materials such as rubber, glass, and plastic tend to gain electrons andbecome negatively charged, while materials such as silk, wool, and fur tend to lose electrons and become positively charged.Frictional electrification has practical applications in various fields. For example, it plays a crucial role in the function of everyday items such as photocopiers, laser printers, and air purifiers. In photocopiers and laser printers, the process of charging a photoconductive drum with a corona wire involves the use of frictional electrificationto create an electrostatic charge pattern that attracts toner particles to the paper. In air purifiers, the process of ionization involves creating charged particles that attract and remove airborne contaminants.In conclusion, frictional electrification is afundamental aspect of the behavior of electric charges andhas been recognized and studied for centuries. This phenomenon is important in various scientific andtechnological applications and continues to be a subject ofresearch and exploration in the field of physics. Understanding the principles of frictional electrification is essential for developing new technologies and advancing our knowledge of the natural world.。

a r X i v :h e p -t h /9906141v 2 19 O c t 1999UPR-0849-T hep-th/9906141One-loop effective potential of N=1supersymmetrictheory and decoupling effectsI.L.Buchbinder ∗,M.Cvetiˇc +and A.Yu.Petrov ∗∗Department of Theoretical Physics,Tomsk State Pedagogical University634041Tomsk,Russia+Department of Physics and Astronomy,University of PennsylvaniaPhiladelphia,PA 19104–6396,USAAbstractWe study the decoupling effects in N =1(global)supersymmetric theories with chiral superfields at the one-loop level.Examples of gauge neutral chiral superfields with minimal (renormalizable)as well as non-minimal (non-renormalizable)couplings are considered,and decoupling in gauge theories with U (1)gauge superfields that cou-ple to heavy chiral matter is studied.We calculate the one-loop corrected effective Lagrangians that involve light fields and heavy fields with mass of order M .Elimina-tion of heavy fields by equations of motion leads to decoupling effects with terms that grow logarithmically with M .These corrections renormalize light fields and couplings in the theory (in accordance with the “decoupling theorem”).When the field theory is an effective theory of the underlying fundamental theory,like superstring theory,where the couplings are calculable,such decoupling effects modify the low energy predictions for the effective couplings of light fields.In particular,for the class of string vacua with an “anomalous”U (1),the vacuum restabilization triggers decoupling effects,which can significantly modify the low energy predictions for couplings of the surviving light fields.We also demonstrate that quantum corrections to the chiral potential depend-ing on massive background superfields and corresponding to supergraphs with internal massless lines and external massive lines can also arise at the two-loop level.Contents1Introduction22Effective action in the model of interacting light and heavy superfields42.1General structure of the effective action ......................52.2The effective equations of motion .........................82.3Calculation of the one-loop k¨a hlerian effective potential .............93One-loop effective action for minimal models133.1Calculation of the effective action (13)3.1.1The one-loop k¨a hlerian effective potential (13)3.1.2Corrections to the chiral potential (14)3.2The effective action for light superfields (18)3.2.1Contribution of the self-interaction of the light superfield (20)3.2.2Absence of the self-interaction of the light superfield (23)4One-loop effective action for non-minimal models244.1The model with heavy quantum superfields and external light superfields (25)4.1.1Calculation of the effective action (25)4.1.2Solving the effective equations of motion (27)4.2The model with light and heavy quantum superfields and light and heavyexternal superfields (29)4.2.1Calculation of the effective action (29)4.2.2Solution of the effective equations of motion (32)5Quantum corrections to the effective action in gauge theories335.1Gauge invariant model of massive chiral superfields (33)5.2One-loop k¨a hlerian potential in supersymmetric gauge theory (35)5.3Chiral potential corrections (40)5.4Strength depending contributions in the effective action (41)6Summary45 Appendix A47 Appendix B49 Appendix C52 1IntroductionThis paper is devoted to the calculation of the one-loop effective action for several models of the global N=1supersymmetric theory with chiral superfields and a subsequent study of some of their phenomenologically interesting aspects.In particular,we investigate in detail the decoupling effects due to the couplings of heavy and light chiral superfields in the theory and subsequent implications for the low energy effective action of light superfields.In principle the decoupling effects of heavyfields infield theory are well understood. According to the decoupling theorem[1,2](for additional references see,e.g.,[3])in thefield theory of interacting light(with masses m)and heavyfields(with masses M)the heavyfields decouple;the effective Lagrangian of the lightfields can be written in terms of the original classical Lagrangian of lightfields with loop effects of heavyfields absorbed into redefinitionsof new lightfields,masses and couplings,and the only new terms in this effective Lagrangianare non-renormalizable,proportional to inverse powers of M(both at tree-and loop-levels).In afield theory as an effective description of phenomena at certain energies,the rescaling of thefields and couplings due to heavyfields does not affect the structure of the couplings,since those are free parameters whose values are determined by experiments.On the other hand if thefield theory is describing an effective theory of an underlying fundamental theory,like superstring theory,where the couplings at the string scale are calculable,the decouplingeffects of the heavyfield can be important and can significantly affect the low energy pre-dictions for the couplings of lightfields at low energies.Therefore the quantitative study ofdecoupling effects at the loop-level in effective supersymmetric theories is important;it should improve our understanding of such effects for the effective Lagrangians from superstring the-ory and provide us with calculable corrections for the low energy predictions of the theory.We also note that as the decoupling theorem is based onfinite renormalization offields and parameters as all parameters in the effective theory(fields,masses,couplings)are determinedfrom the corresponding string theory and hence cannot be renormalized.Therefore we willuse consistence with the decoupling theorem only to check the results.Effective theories of N=1supersymmetric four-dimensional perturbative string vacuacan be obtained by employing techniques of two-dimensional conformalfield theory[4].In particular the k¨a hlerian and the chiral(super-)potential can be calculated explicitly at thetree level.While the chiral potential terms calculated at the string tree-level are protectedfrom higher genus corrections(for a representative work on the subject see,e.g.,[5,6],and references therein),the k¨a hlerian potential is not.Such higher genus corrections to thek¨a hlerian potential could be significant;however,their structure has not been studied verymuch.In this paper we shall not address these issues and assume that the string theory calculation provides us with a(reliable,calculable)form of the effective theory at M string,which would in turn serve as a starting point of our study.One of the compelling motivations for a detailed study of decoupling effects is the phe-nomenon of vacuum restabilization[7]for a class of four-dimensional(quasi-realistic)heteroticsuperstring vacua with an“anomalous”U(1).(On the open Type I string side these effects are closely related to the blowing-up procedure of Type I orientifolds and were recently stud-ied in[8].)For such string vacua of perturbative heterotic string theory,the Fayet-Iliopoulos (FI)D-term is generated at genus-one[9],thus triggering certainfields to acquire vacuumexpectation values(VEV’s)of order M String∼g gauge M P lanck∼5×1017GeV along D-and F-flat directions of the effective N=1supersymmetric theory.(Here g gauge is the gauge coupling and M P lanck the Planck scale.)Due to these large string-scale VEV’s a numberof additionalfields obtain large string-scale masses.Some of them in turn couple through(renormalizable)interactions to the remaining lightfields,and thus through decoupling ef-fects affect the effective theory of lightfields at low energies.(For the study of the effective Lagrangians and their phenomenological implications for a class of such four-dimensional string vacua see,e.g.,[10]–[12]and references therein.)The tree level decoupling effects within N=1supersymmetric theories,were studiedwithin an effective string theory in[13].In a related work[14]it was shown that the lead-ing order corrections of order1important next order effects in the effective chiral potential[15].In addition,in[15]the nonrenormalizable modifications of the k¨a hlerian potential(as was also pointed out in[16]) were systematically studied.These tree level decoupling effects(as triggered by,e.g.,vac-uum restabilization for a class of string vacua)lead to new nonrenormalizable interactions which are competitive with the nonrenormalizable terms that are calculated directly in the superstring theory.In this paper we consider one-loop decoupling effects in N=1supersymmetric theory. We study both the effects on chiral(gauge neutral)superfields and on the effects of gauge superfields.(In another context see[17].)It turns out that an essential modification of low energy predictions takes place not only for chiral superfields[18]but also for gauge superfields. As stated earlier such effective Lagrangians arise naturally due to the vacuum restabilization for a class of supersymmetric string vacua and trigger couplings between heavyfields with mass scale M∼1017GeV and the light(massless)fields[19].(Note however,that we do not include supergravity effects which could also be significant.)As a result wefind that the one-loop effective action after a redefinition offields,masses and couplings coincides with the one-loop effective action of the corresponding theory where heavy superfields are completely absent,in accordance with the decoupling theorem.How-ever,since the masses and the couplings of thefields are calculable in string theory(at the mass scale M string),the decoupling effects add additional corrections to the effective action of the light superfields.These corrections grow logarithmically with M(mass of the heavy superfields)and modify the effective couplings in an essential way,which for a class of string vacua under consideration can be significant.Another interesting result presented in this paper pertains to the chiral effective potential. When the chiral potential depends on massive superfields,quantum corrections due to these fields appear earlier than in the case when one considers lightfields only.This paper is organized as follows.In Section2the general structure of the effective action studied is given and the general approach to addressing the decoupling effects is presented. Section3is devoted to the study of the effective action for the“minimal”model with one heavy and one light(gauge neutral)chiral superfield.In Section4the leading order decou-pling corrections to the effective action for non-minimal models(with more general couplings) are considered.Section5is devoted to the investigation of the one-loop decoupling effects in N=1supersymmetric theory with U(1)gauge vector superfields and chiral superfields charged under U(1).A summary and discussion of the obtained results are given in Section 6.In Appendix A details of the calculation of the one-loop k¨a hlerian effective potential for the minimal model are presented,in Appendix B the calculation of the one-loop k¨a hlerian effective action via diagram technique for the minimal model is described,and in Appendix C details of the calculation for the effective action of non-minimal models are given.2Effective action in the model of interacting light and heavy superfields2.1General structure of the effective actionN=1supersymmetric actions with chiral supermultiplets arise as a subsector of an effective theory of N=1supersymmetric string vacua.Such calculations are carried out for per-turbative string vacua primarily by employing conformalfield theory techniques.(Though less powerful techniques,e.g.,sigma-model approach,in which the integration over massivestring modes is carried out in the the background of the ten-dimensional manifold with thestructure M4×K where M4is a four-dimensional Minkowski space and K is a suitable six-dimensional compact(Calabi-Yau)manifold,can also be employed.)The resulting effectivetheories contain as an ingredient N=1chiral superfieldsΦi with actionS[Φ,¯Φ]= d8zK(¯Φi,Φi)+( d6zW(Φi)+h.c.)(2.1) HereΦi=Φi(z),z A≡(x a,θα,¯θ˙α);a=0,1,2,3;α=1,2,˙α=˙1,˙2,d8z=d4xd2θd2¯θ. Real function K(¯Φi,Φi)is called the k¨a hlerian potential,the holomorphic function W(Φi)is called the chiral potential[20].Expression(2.1)represents the most general action of gauge neutral chiral superfields which does not contain higher derivatives at a component level[20]. We refer to this action as the chiral superfield model of a general form.In a special case K(¯Φi,Φi)=Φ¯Φ,W(Φi)∼Φ3we obtain the well-known Wess-Zumino model.For W(Φi)=0 the present theory represents itself as a N=1supersymmetric four-dimensional sigma-model (see,e.g.,[6]).The action(2.1),which originates from superstring theory,can be treated as a classical effective action of the fundamental theory,suitable for description of phenomena at energies much less than the Planck scale.Such models of chiral superfields are widely used for the study of possible phenomenological implications of superstring theories(see,e.g.,recent papers[8,10,11,12,18]and references therein).One of the most important aspects of the study of these models pertains to the investigation of the decoupling effects,which is the main subject of the present paper.The starting point in the study of the decoupling effects is the model with the classicalaction(2.1)and,for the sake of simplicity,two chiral superfields:a light one,φ,and a heavyone,Φ,i.e.Φi={Φ,φ}.The aim is to to calculate the low-energy effective action in the one-loop approximation and to compute the one-loop corrected effective action of light superfield, only.We refer to the model in which the k¨a hlerian potential is of the canonical(minimal)form:K(Φ,¯Φ,φ,¯φ)=Φ¯Φ+φ¯φ(2.2) as the minimal model,and the model in whichK(Φ,¯Φ,φ,¯φ)=Φ¯Φ+φ¯φ+˜K(Φ,¯Φ,φ,¯φ)(2.3) with˜K=0–as the non-minimal one(in analogy with[10]).We assume that the function ˜K(Φ,¯Φ,φ,¯φ)can be expanded into power series in superfieldsΦ,¯Φ,φ,¯φwhere the leading order term is at least of the third order in the chiral superfields(and thus proportional to at least one inverse power of M)˜K(Φ,¯Φ,φ,¯φ)=φ¯Φ2M...(2.4)The chiral potential M is taken to be of the form:W=MM+...(2.6) with M as a massive parameter.Hence the possible vertices of interaction of superfields havethe formφΦ2,Φφ2,Φ¯φ2M...The effective actionΓ[Φ,¯Φ,φ,¯φ]is defined as the Legendre transform from the generating functional of connected Green functions[21]W[J,¯J]:exp(i¯h(S[Ψ,¯Ψ,ϕ,¯ϕ]++( d6z(JΨ+jϕ)+h.c.)))(2.7)Γ[Φ,¯Φ,φ,¯φ]=W[J,¯J]−( d6z(JΦ+jφ)+h.c.)Γ[Φ,¯Φ,φ,¯φ]can be calculated using the loop-expansion method.This method employs the splitting of all the chiral superfields into a sum of the background superfieldsΦ,φand the quantum onesΦq,φq,using the ruleΦ→Φ+√¯hφqAs a result the action(2.1)after such changes can be written asS q= d8zK(Φ+√¯h¯Φq,φ+√¯h¯φq)++[ d6zW(Φ+√¯hφq)+h.c.](2.9) and the effective action takes the form:exp(i¯hS[Φ+√¯h¯Φq,φ+√¯h¯φq]−−√δΦ(z)Φq(z)+δΓ(for details see[20,21]).The effective action(2.10)can be cast in the formΓ[Φ,¯Φ,φ,¯φ]= S[Φ,¯Φ,φ,¯φ]+˜Γ[Φ,¯Φ,φ,¯φ].Here˜Γ[Φ,¯Φ,φ,¯φ]is a quantum correction in effective action which can be expanded into power series in¯h as˜Γ[Φ,¯Φ,φ,¯φ]=∞ n=1¯h nΓ(n)[Φ,¯Φ,φ,¯φ](2.11) The one-loop quantum correctionΓ(1)to the effective action is defined through the fol-lowing expression[21]:e iΓ(1)= DΦq D¯Φq Dφq D¯φq exp(iS(2)q)(2.12) Here S(2)q corresponds to the part of S q(2.9)which is quadratic in quantum superfields.It is of the formS(2)q= d8z(KΦ¯ΦΦq¯Φq+Kφ¯Φφq¯Φq+KΦ¯φΦq¯φq+Kφ¯φφq¯φq)++[ d6zWΦΦΦ2q+WφΦΦqφq+Wφφφ2q+h.c.](2.13) As a result we arrive at the one-loop effective action of the formΓ[Φ,¯Φ,φ,¯φ]=S+¯hΓ(1)= d8zK(Φ,¯Φ,φ,¯φ)+[ d6zW(Φ,φ)+h.c.]++¯h( d8zK(1)(Φ,¯Φ,φ,¯φ)+( d6zW(1)(Φ,φ)+h.c.))(2.14)Here we suppose that the one-loop correction in the effective actionΓ(1)can be represented in the formΓ(1)= d8zK(1)(Φ,¯Φ,φ,¯φ)+( d6zW(1)(Φ,φ)+h.c.)+...(2.15) Dots denote terms that depend on the supercovariant derivatives of the chiral superfields.The loop corrected effective action has the following structureΓ[Φ,¯Φ,φ,¯φ]= d8zL eff(Φ,D AΦ,D A D BΦ,¯Φ,D A¯Φ,D A D B¯Φ,φ,D Aφ,D A D Bφ,¯φ,D A¯φ,D A D B¯φ)+( d6zL(c)eff(Φ,φ)+h.c.)+...(2.16)Here D A are supercovariant derivatives,D A=(∂a,Dα,¯D˙α).L eff is the effective super-Lagrangian that we write in the formL eff=K eff(Φ,¯Φ,φ,¯φ)+...K=K(Φ,¯Φ,φ,¯φ)+∞n=1¯h n K(n)(2.17)and L(c)is the effective chiral LagrangianL(c)=W eff(Φ,φ)+...(2.18)K eff is the k¨a hlerian effective potential that depends only on the chiral superfieldsΦ,¯Φ,φ,¯φbut not on their(covariant)derivatives.W eff is the chiral effective potential that depends on on(holomorphic)chiral superfields{Φ,φ},only.Dots denote the terms that depend on the the space-time derivatives of chiral superfields only.Furthermore,one can prove that the one-loop correction to the chiral potential is zero(for the N=1supersymmetric theory which does not include gauge superfields).However,higher corrections can exist(cf.[22]–[24]),i.e.W eff(Φ,φ)=W(Φ,¯φ)+∞n=2¯h n W(n)(Φ,φ)(2.19)Here K(n)and W(n)are loop corrections to the k¨a hlerian and chiral potential,respectively.Since in this paper we concentrate on the one-loop corrected effective action only,we are mainly interested in the correction to the k¨a hlerian potential which is the leading term in the one-loop corrected low-energy effective action.(At low energies(E≪M)higher derivative terms are suppressed.)Our ultimate goal is to obtain the effective action for light superfields,only.For that purpose one must eliminate heavy superfields from the one-loop effective actionΓ[Φ,¯Φ,φ,¯φ] (2.14)by means of the effective equations of motion.These equations can be solved by an iterative method up to a certain order in the inverse mass M of heavy superfield.Substituting a solution of these equations into the effective action(2.14)we then obtain the one-loop corrected effective action of light superfields only.In the following subsection we shall describe the procedure in detail.2.2The effective equations of motionThe effective equations of motion for heavy superfields in the model with the effective action (2.14,2.15)are of the formδΓ4¯D2(∂K∂Φ)+∂Wδ¯Φ=0:−1∂¯Φ+∂K(1)∂¯Φ=0(2.20)The effective equations of motion for light superfields have an analogous form.We consider the case when the interactions with the gauge superfields are absent(see however Section5) and W(1)=0(which is absent at one-loop level(cf.,discussion above)).The equations(2.20)can be solved via an iterative method,described below.We can represent the heavy superfieldΦin the formΦ=Φ0+Φ1+...+Φn+...(2.21) whereΦ0is zeroth-order approximation,Φ1isfirst-order one,etc..We assume that|D2Φ|≪M¯Φsince the superfieldΦis heavy,and thus the assumption is valid.The zeroth-order approximationΦ0can be found from the condition∂WAfter a substitution of the expansion(2.21)into equations(2.20)we arrive at the following equation for the(n+1)-th-order solution for¯Φn+1(∂¯W∂¯Φ)|Φ=Φ0+...+Φn=(2.22)=¯D2∂Φ|Φ=Φ0+...+Φn−∂K∂Φ|Φ=Φ0+...+Φn−∂K(1)2Φ2+˜W(see eqs.(2.5-2.6),eq.(2.23))can be rewritten in the formM¯Φn+1+(∂¯˜W∂¯Φ)|Φ=Φ0+...+Φn=(2.23)=¯D2∂Φ|Φ=Φ0+...+Φn−∂K∂Φ|Φ=Φ0+...+Φn−∂K(1)which represents itself as a column q =uv.The action for q reads asS 0q =−14¯D 2q −ψand ¯χ[q ]=14D 2q −¯ψ)δ(14¯D2−1214D 2q −¯ψ)δ(12T r log ∆(2.32)Here T r is a functional supertrace,andS [q ]=116(K ψ¯ψ−1){D 2,¯D2}−14¯W′′D 2(2.34)The terms proportional to the supercovariant derivatives of K ψ¯ψ,W ′′and ¯W′′are omitted since the one-loop k¨a hlerian effective potential by definition does not depend on the deriva-tives of superfields.In order to determine T r log ∆we use the Schwinger representationT r log ∆=trds∂sΩ=Ω˜∆(2.35)Here ˜∆is a matrix operator of the form ˜∆=−14W ′′¯D2−116A (s )D 2¯D2+18B α(s )D α¯D2+14C (s )D 2+1i˙A=F +AF 2−CW ′′(2.38)1i˙C=−¯W ′′−A ¯W ′′2+CF 2and an analogous system of equations for ˜A,˜B,˜C ,which can be obtained from this one by changing W ′′into ¯W ′′and vice versa.Here F =1−K ψ¯ψ.Since the initial condition for ΩisΩ|s =0=1the initial conditions for A,˜A,B,˜B,C,˜C )are A |s =0=˜A |s =0=C |s =0=˜C |s =0=0.The solution for B α,˜B ˙αevidently has the form B α=˜B ˙α=0.The manifest form of thematrices A,˜AC,˜C ,necessary for exact calculations,is of the form A =A 11A 12A 21A 22;C =C 11C 12C 21C 22(2.39)Here index 1denotes the sector of heavy superfield Φand 2the sector of light superfield φ.Now let us solve the system for matrices A,C .The solution for ˜A,˜C can be easily obtained in an analogous way since the system with B α=˜B ˙α=0is invariant under the change A →˜A,C →˜C.Let us study the solution for A,C which should be chosen in the formA =A i +A 0(2.40)C =C i +C 0Here A i ,C i is a partial solution of the inhomogeneous system,and A 0,C 0is a general solution of the homogeneous system.It is straightforward to see that A i =−12−1,C i =0.And A 0,C 0should satisfy the system of equations1i˙C0=A 0¯G 2+C 0F 2A 0,C 0should be chosen to be of the form A 0=a 0exp(iωs ),C 0=c 0exp(iωs )where a 0,c 0,ωare some functions of the background superfields and the d’Alembertian operator,but are independent of s .As a result we arrive at the equations for a 0,c 0:a 0(ω1−F 2)+c 0W ′′=0(2.42)c 0(ω1−F 2)+a 0¯W′′2=0This system of equations has a non-trivial solution atdetω12−F 2W′′¯W ′′2ω12−F 2=0(2.43)In principle,parameters ωcan be found from this equation.Their exact form is determined by the structure of the matrix W ′′and F .It turns out that for the specific cases studied in detail the subsequent sections (minimal (Section 3)and non-minimal (Section 4)cases)these parameters are different.As a result the final solution can be cast in the form:A =ka 0k exp(iωk s )−12∞dsi∂16π2(is )2.A ,and ˜Aare functions of 2.Hence in order to calculate the one-loop k¨a hlerian effective potential it is necessary to find A and ˜Aand to expand them into a power series in 2.In this section we addressed in detail the techniques needed to calculate the one-loop corrected effective action,to eliminate the heavyfields and to obtain the effective action of lightfields only.In the subsequent sections these techniques will be applied to obtain the explicit form of the one-loop corrected actions for specific models.In Section5we shall also include interactions with the U(1)vector superfields and modify the procedure accordingly. 3One-loop effective action for minimal modelsIn this section we study decoupling effects for the model with minimal k¨a hlerian potential (2.2)K=Φ¯Φ+φ¯φ.Thefirst part consists of calculating the one-loop correction to the k¨a hlerian effective potential.In the second part we solve the effective equations of motion for the heavy superfields.As a result we arrive at the effective action of the light superfields.3.1Calculation of the effective action3.1.1The one-loop k¨a hlerian effective potentialHere we are going to calculate the one-loop contribution to k¨a hlerian effective potential by means of the effective equations of motion.We study the minimal model with the chiral potential in the formW=13!gφ3(3.1)with the corresponding functions in W′′(see eq.(2.25))Wφφ=λΦ+gφWφΦ=λφWΦΦ=M(3.2) The total classical action with the chiral potential(3.1)is of the formS= d8z(φ¯φ+Φ¯Φ)+[ d6z(13!gφ3)+h.c.](3.3)Note that the chiral potential used is of the“minimal”form:it involves the renormalizable terms only and the renormalizable coupling between the light and heavy superfields is linear in the heavyfields,which yields a dominant contribution in the study of the decoupling effects.These types of couplings are typical for a class of effective string models after the vacuum restabilization was taken into account,and thus this minimal model provides a prototype example for the study of decoupling effects in N=1supersymmetric theories. (The results for this model and the physics consequences were presented in[18].For the sake of completeness we present here the intermediate steps in the derivation.)In order tofind the one-loop k¨a hlerian effective potential we should determine the operator Ω(s)that satisfies the equation(2.35).For the case of this minimal model this equation leads to the following system of equations for matrices A,C:1i˙C=−¯W′′−A¯W′′2with the analogous equations for˜A,˜C.Initial conditions are A|s=0=˜A|s=0=C|s=0=˜C|s=0=0.Calculations described in Appendix A show that the one-loop contribution to k¨a hlerian effective potential is of the form:K(1)=−1(|λΦ+gφ|2−M2)2+4|λ2Φ¯φ+λMφ+λg|φ|2)|2)××log(|λΦ+gφ|2+2λ2φ¯φ+M2+ µ2+ +(|λΦ+gφ|2+2λ2φ¯φ+M2− (|λΦ+gφ|2−M2)2+4|λ2Φ¯φ+λMφ+λg|φ|2|2)2MΦ2+Φφ2)+h.c.)−1(|λΦ+gφ|2−M2)2+4|λ2Φ¯φ+λMφ+λg|φ|2)|2)××log(|λΦ+gφ|2+2λ2φ¯φ+M2+ µ2+ +(|λΦ+gφ|2+2λ2φ¯φ+M2− (|λΦ+gφ|2−M2)2+4|λ2Φ¯φ+λMφ+λg|φ|2|2)corrections to the chiral effective potential one must set¯Φ=¯φ=0.Possible vertices contributing to one-loop effective potential should be quadratic in quantum superfields[21]. They have the formK¯φ¯φ¯φ2,Kφφφ2,(Kφ¯φ−1)φ¯φ,12WΦφΦφ,K¯Φ¯Φ¯Φ2,KΦΦΦ2,(KΦ¯Φ−1)Φ¯Φ,142)g(Φ)(3.6)Namely,after a transformation to an integral over the chiral superspace by the ruled8zF(Φ,¯Φ)= d6z(−¯D24(2−m2))g(Φ)(3.8) A transformation to the form of an integral over the chiral superspace leads tod6zf(Φ)(2(2π)4f(Φ)(p2decreases a number of D,¯D-factors by4and the corresponding scaling dimension by2.Each propagator of a massless superfield gives no contribution(scaling dimension0)since it has the form(cf.[20])G(z1,z2)=−D21¯D2216δ12=0.Hence a contribution of such a diagram is equal to zero,and a one-loop contribution to the chiral effective potential is absent:W(1)(Φ)=0.We note that this situation is analogous to the general model of one chiral superfield studied in[27].However,higher order(loop)corrections to the chiral effective potential can arise not only for diagrams with external massless lines but also for those with heavy external lines, in spite of the fact that it was commonly believed that quantum corrections to the chiral effective potential for massive superfields are absent.For example,consider the supergraph|¯D 2||¯D 2¯D2D 2D 2D 2D 2−−−−Here a double line denotes the external superfield Φ,and a single line corresponds to thepropagator <φ¯φ>of the massless superfield φ.A contribution of such a supergraph is of the form I =d 4p 1d 4p 2(2π)8d 4θ1d 4θ2d 4θ3d 4θ4d 4θ5(gk 2l 2(k+p 1)2(l +p 2)2(l +k )2(l +k +p 1+p 2)2××δ13¯D 2316δ14δ42D 21¯D 253!)2λ3d 4p 1d 4p 2(2π)8d 2θΦ(−p 1,θ)Φ(−p 2,θ)Φ(p 1+p 2,θ)××k 2p 21+l 2p 22+2(kl )(p 1p 2)3!)2λ3d 2θd 4p 1d 4p 2(2π)8k 2p 21+l 2p 22+2(kl )(p 1p 2)3!)2λ3d 2θd 4x 1d 4x 2d 4x 3d 4p 1d 4p 2。

a r X i v :h e p -p h /9711228v 1 4 N o v 1997hep-ph/9711228October 1997O (α)QED Corrections to Polarized Elastic µe and Deep Inelastic lN ScatteringDima Bardin a,b,c ,Johannes Bl¨u mlein a ,Penka Christova a,d ,and Lida Kalinovskaya a,caDESY–Zeuthen,Platanenallee 6,D–15735Zeuthen,GermanybINFN,Sezione di Torino,Torino,ItalycJINR,ul.Joliot-Curie 6,RU–141980Dubna,RussiadBishop Konstantin Preslavsky University of Shoumen,9700Shoumen,BulgariaAbstractTwo computer codes relevant for the description of deep inelastic scattering offpolarized targets are discussed.The code µe la deals with radiative corrections to elastic µe scattering,one method applied for muon beam polarimetry.The code HECTOR allows to calculate both the radiative corrections for unpolarized and polarized deep inelastic scattering,including higher order QED corrections.1IntroductionThe exact knowledge of QED,QCD,and electroweak (EW)radiative corrections (RC)to the deep inelastic scattering (DIS)processes is necessary for a precise determination of the nucleon structure functions.The present and forthcoming high statistics measurements of polarized structure functions in the SLAC experiments,by HERMES,and later by COMPASS require the knowledge of the RC to the DIS polarized cross-sections at the percent level.Several codes based on different approaches for the calculation of the RC to DIS experiments,mainly for non-polarized DIS,were developped and thoroughly compared in the past,cf.[1].Later on the radiative corrections for a vast amount of experimentally relevant sets of kinematic variables were calculated [2],including also semi-inclusive situations as the RC’s in the case of tagged photons [3].Furthermore the radiative corrections to elastic µ-e scattering,a process to monitor (polarized)muon beams,were calculated [4].The corresponding codes are :•HECTOR 1.00,(1994-1995)[5],by the Dubna-Zeuthen Group.It calculates QED,QCD and EW corrections for variety of measuremets for unpolarized DIS.•µe la 1.00,(March 1996)[4],calculates O (α)QED correction for polarized µe elastic scattering.•HECTOR1.11,(1996)extends HECTOR1.00including the radiative corrections for polarized DIS[6],and for DIS with tagged photons[3].The beta-version of the code is available from http://www.ifh.de/.2The Programµe laMuon beams may be monitored using the processes ofµdecay andµe scattering in case of atomic targets.Both processes were used by the SMC experiment.Similar techniques will be used by the COMPASS experiment.For the cross section measurement the radiative corrections to these processes have to be known at high precision.For this purpose a renewed calculation of the radiative corrections toσ(µe→µe)was performed[4].The differential cross-section of polarized elasticµe scattering in the Born approximation reads,cf.[7],dσBORNm e Eµ (Y−y)2(1−P e Pµ) ,(1)where y=yµ=1−E′µ/Eµ=E′e/Eµ=y e,Y=(1+mµ/2/Eµ)−1=y max,mµ,m e–muon and electron masses,Eµ,E′µ,E′e the energies of the incoming and outgoing muon,and outgoing electron respectively,in the laboratory frame.Pµand P e denote the longitudinal polarizations of muon beam and electron target.At Born level yµand y e agree.However,both quantities are different under inclusion of radiative corrections due to bremsstrahlung.The correction factors may be rather different depending on which variables(yµor y e)are used.In the SMC analysis the yµ-distribution was used to measure the electron spin-flip asymmetry A expµe.Since previous calculations,[8,9],referred to y e,and only ref.[9]took polarizations into account,a new calculation was performed,including the complete O(α)QED correction for the yµ-distribution,longitudinal polarizations for both leptons,theµ-mass effects,and neglecting m e wherever possible.Furthermore the present calculation allows for cuts on the electron re-coil energy(35GeV),the energy balance(40GeV),and angular cuts for both outgoing leptons (1mrad).The default values are given in parentheses.Up to order O(α3),14Feynman graphs contribute to the cross-section forµ-e scattering, which may be subdivided into12=2×6pieces,which are separately gauge invariantdσQEDdyµ.(2) One may express(2)also asdσQEDdyµ+P e Pµdσpol kk=1−Born cross-section,k=b;2−RC for the muonic current:vertex+bremsstrahlung,k=µµ;3−amm contribution from muonic current,k=amm;4−RC for the electronic current:vertex+bremsstrahlung,k=ee;5−µe interference:two-photon exchange+muon-electron bremsstrahlung interference,k=µe;6−vacuum polarization correction,runningα,k=vp.The FORTRAN code for the scattering cross section(2)µe la was used in a recent analysis of the SMC collaboration.The RC,δA yµ,to the asymmetry A QEDµeshown infigures1and2is defined asδA yµ=A QEDµedσunpol.(4)The results may be summarized as follows.The O(α)QED RC to polarized elasticµe scattering were calculated for thefirst time using the variable yµ.A rather general FORTRAN codeµe la for this process was created allowing for the inclusion of kinematic cuts.Since under the conditions of the SMC experiment the corrections turn out to be small our calculation justifies their neglection. 3Program HECTOR3.1Different approaches to RC for DISThe radiative corrections to deep inelastic scattering are treated using two basic approaches. One possibility consists in generating events on the basis of matrix elements including the RC’s. This approach is suited for detector simulations,but requests a very hughe number of events to obtain the corrections at a high precision.Alternatively,semi-analytic codes allow a fast and very precise evaluation,even including a series of basic cuts andflexible adjustment to specific phase space requirements,which may be caused by the way kinematic variables are experimentally measured,cf.[2,5].Recently,a third approach,the so-called deterministic approach,was followed,cf.[10].It treats the RC’s completely exclusively combining features of fast computing with the possibility to apply any cuts.Some elements of this approach were used inµe la and in the branch of HECTOR1.11,in which DIS with tagged photons is calculated.Concerning the theoretical treatment three approaches are in use to calculate the radiative corrections:1)the model-independent approach(MI);2)the leading-log approximation(LLA); and3)an approach based on the quark-parton model(QPM)in evaluating the radiative correc-tions to the scattering cross-section.In the model-independent approach the QED corrections are only evaluated for the leptonic tensor.Strictly it applies only for neutral current processes.The hadronic tensor can be dealt with in its most general form on the Lorentz-level.Both lepton-hadron corrections as well as pure hadronic corrections are neglected.This is justified in a series of cases in which these corrections turn out to be very small.The leading logarithmic approximation is one of the semi-analytic treatments in which the different collinear singularities of O((αln(Q2/m2l))n)are evaluated and other corrections are neglected.The QPM-approach deals with the full set of diagrams on the quark level.Within this method,any corrections(lepton-hadron interference, EW)can be included.However,it has limited precision too,now due to use of QPM-model itself. Details on the realization of these approaches within the code HECTOR are given in ref.[5,11].3.2O (α)QED Corrections for Polarized Deep Inelastic ScatteringTo introduce basic notation,we show the Born diagramr rr r j r r r r l ∓( k 1,m )l ∓( k 2,m )X ( p ′,M h )p ( p ,M )γ,Z ¨¨¨¨B ¨¨¨¨£¢ ¡£¢ ¡£¢ ¡£¢ ¡£¢ ¡£¢ ¡£¢ ¡£¢ ¡£¢ ¡£¢ ¡z r r r r r r r r r r r r r rr ¨¨¨¨B ¨¨¨¨r r r r j r r r r and the Born cross-section,which is presented as the product of the leptonic and hadronic tensordσBorn =2πα2p.k 1,x =Q 2q 2F 1(x,Q 2)+p µ p ν2p.qF 3(x,Q 2)+ie µνλσq λs σ(p.q )2G 2(x,Q 2)+p µ s ν+ s µ p νp.q1(p.q )2G 4(x,Q 2)+−g µν+q µq νp.qG 5(x,Q 2),(8)wherep µ=p µ−p.qq 2q µ,and s is the four vector of nucleon polarization,which is given by s =λp M (0, n )in the nucleonrest frame.The combined structure functions in eq.(8)F1,2(x,Q2)=Q2e Fγγ1,2(x,Q2)+2|Q e|(v l−p eλl a l)χ(Q2)FγZ1,2(x,Q2)+ v2l+a2l−2p eλl v l a l χ2(Q2)F ZZ1,2(x,Q2),F3(x,Q2)=2|Q e|(p e a l−λl v l)χ(Q2)FγZ3(x,Q2),+ 2p e v l a l−λl v2l+a2l χ2(Q2)F ZZ3(x,Q2),G1,2(x,Q2)=−Q2eλl gγγ1,2(x,Q2)+2|Q e|(p e a l−λl v l)χ(Q2)gγZ1,2(x,Q2),+ 2p e v l a l−λl v2l+a2l χ2(Q2)g ZZ1,2(x,Q2),G3,4,5(x,Q2)=2|Q e|(v l−p eλl a l)χ(Q2)gγZ3,4,5(x,Q2),+ v2l+a2l−2p eλl v l a l χ2(Q2)g ZZ3,4,5(x,Q2),(9) are expressed via the hadronic structure functions,the Z-boson-lepton couplings v l,a l,and the ratio of the propagators for the photon and Z-bosonχ(Q2)=Gµ2M2ZQ2+M2Z.(10)Furthermore we use the parameter p e for which p e=1for a scattered lepton and p e=−1for a scattered antilepton.The hadronic structure functions can be expressed in terms of parton densities accounting for the twist-2contributions only,see[12].Here,a series of relations between the different structure functions are used in leading order QCD.The DIS cross-section on the Born-leveld2σBorndxdy +d2σpol Borndxdy =2πα2S ,S U3(y,Q2)=x 1−(1−y)2 ,(13) and the polarized partdσpol BornQ4λp N f p S5i=1S p gi(x,y)G i(x,Q2).(14)Here,S p gi(x,y)are functions,similar to(13),and may be found in[6].Furthermore we used the abbrevationsf L=1, n L=λp N k 12πSy 1−y−M2xy2π1−yThe O(α)DIS cross-section readsd2σQED,1πδVRd2σBorndx l dy l=d2σunpolQED,1dx l dy l.(16)All partial cross-sections have a form similar to the Born cross-section and are expressed in terms of kinematic functions and combinations of structure functions.In the O(α)approximation the measured cross-section,σrad,is define asd2σraddx l dy l +d2σQED,1dx l dy l+d2σpol radd2σBorn−1.(18)The radiative corrections calculated for leptonic variables grow towards high y and smaller values of x.Thefigures compare the results obtained in LLA,accounting for initial(i)andfinal state (f)radiation,as well as the Compton contribution(c2)with the result of the complete calculation of the leptonic corrections.In most of the phase space the LLA correction provides an excellent description,except of extreme kinematic ranges.A comparison of the radiative corrections for polarized deep inelastic scattering between the codes HECTOR and POLRAD[17]was carried out.It had to be performed under simplified conditions due to the restrictions of POLRAD.Corresponding results may be found in[11,13,14].3.3ConclusionsFor the evaluation of the QED radiative corrections to deep inelastic scattering of polarized targets two codes HECTOR and POLRAD exist.The code HECTOR allows a completely general study of the radiative corrections in the model independent approach in O(α)for neutral current reac-tions including Z-boson exchange.Furthermore,the LLA corrections are available in1st and2nd order,including soft-photon resummation and for charged current reactions.POLRAD contains a branch which may be used for some semi-inclusive DIS processes.The initial state radia-tive corrections(to2nd order in LLA+soft photon exponentiation)to these(and many more processes)can be calculated in detail with the code HECTOR,if the corresponding user-supplied routine USRBRN is used together with this package.This applies both for neutral and charged current processes as well as a large variety of different measurements of kinematic variables. Aside the leptonic corrections,which were studied in detail already,further investigations may concern QED corrections to the hadronic tensor as well as the interference terms. References[1]Proceedings of the Workshop on Physics at HERA,1991Hamburg(DESY,Hamburg,1992),W.Buchm¨u ller and G.Ingelman(eds.).[2]J.Bl¨u mlein,Z.Phys.C65(1995)293.[3]D.Bardin,L.Kalinovskaya and T.Riemann,DESY96–213,Z.Phys.C in print.[4]D.Bardin and L.Kalinovskaya,µe la,version1.00,March1996.The source code is availablefrom http://www.ifh.de/~bardin.[5]A.Arbuzov,D.Bardin,J.Bl¨u mlein,L.Kalinovskaya and T.Riemann,Comput.Phys.Commun.94(1996)128,hep-ph/9510410[6]D.Bardin,J.Bl¨u mlein,P.Christova and L.Kalinovskaya,DESY96–189,hep-ph/9612435,Nucl.Phys.B in print.[7]SMC collaboration,D.Adams et al.,Phys.Lett.B396(1997)338;Phys.Rev.D56(1997)5330,and references therein.[8]A.I.Nikischov,Sov.J.Exp.Theor.Phys.Lett.9(1960)757;P.van Nieuwenhuizen,Nucl.Phys.B28(1971)429;D.Bardin and N.Shumeiko,Nucl.Phys.B127(1977)242.[9]T.V.Kukhto,N.M.Shumeiko and S.I.Timoshin,J.Phys.G13(1987)725.[10]G.Passarino,mun.97(1996)261.[11]D.Bardin,J.Bl¨u mlein,P.Christova,L.Kalinovskaya,and T.Riemann,Acta Phys.PolonicaB28(1997)511.[12]J.Bl¨u mlein and N.Kochelev,Phys.Lett.B381(1996)296;Nucl.Phys.B498(1997)285.[13]D.Bardin,J.Bl¨u mlein,P.Christova and L.Kalinovskaya,Preprint DESY96–198,hep-ph/9609399,in:Proceedings of the Workshop‘Future Physics at HERA’,G.Ingelman,A.De Roeck,R.Klanner(eds.),Vol.1,p.13;hep-ph/9609399.[14]D.Bardin,Contribution to the Proceedings of the International Conference on High EnergyPhysics,Warsaw,August1996.[15]M.Gl¨u ck,E.Reya,M.Stratmann and W.Vogelsang,Phys.Rev.D53(1996)4775.[16]S.Wandzura and F.Wilczek,Phys.Lett.B72(1977)195.[17]I.Akushevich,A.Il’ichev,N.Shumeiko,A.Soroko and A.Tolkachev,hep-ph/9706516.-20-18-16-14-12-10-8-6-4-200.10.20.30.40.50.60.70.80.91elaFigure 1:The QED radiative corrections to asymmetry without experimental cuts.-1-0.8-0.6-0.4-0.200.20.40.60.810.10.20.30.40.50.60.70.80.91elaFigure 2:The QED radiative corrections to asymmetry with experimental cuts.-50-40-30-20-100102030405000.10.20.30.40.50.60.70.80.91HectorFigure 3:A comparison of complete and LLA RC’s in the kinematic regime of HERMES for neutral current longitudinally polarized DIS in leptonic variables.The polarized parton densities [15]are used.The structure function g 2is calculated using the Wandzura–Wilczek relation.c 2stands for the Compton contribution,see [6]for details.-20-100102030405000.10.20.30.40.50.60.70.80.91HectorFigure 4:The same as in fig.3,but for energies in the range of the SMC-experiment.-20-10010203040500.10.20.30.40.50.60.70.80.91HectorFigure 5:The same as in fig.4for x =10−3.-200-150-100-5005010015020000.10.20.30.40.50.60.70.80.91HectorFigure 6:A comparison of complete and LLA RC’s at HERA collider kinematic regime for neutral current deep inelastic scattering offa longitudinally polarized target measuring the kinematic variables at the leptonic vertex.。

Abstract This is a short introduction to the Standard Model and the underlying concepts of quantum field theory.
Lectures given at the European School of High-Energy Physics, August 2005, Kitzb¨ uhel, Austria
Contents
1 Introduction 1.1 Theoretical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Phenomenological Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Quantisation of Fields 2.1 Why Fields? . . . . . . . . . . . . . . . . . . . . 2.1.1 Quantisation in Quantum Mechanics . . 2.1.2 Special Relativity Requires Antiparticles 2.2 Multiparticle States and Fies, Creation and Annihilation . . . . 2.2.2 Charge and Momentum . . . . . . . . . 2.2.3 Field Operator . . . . . . . . . . . . . . 2.2.4 Propagator . . . . . . . . . . . . . . . . 2.3 Canonical Quantisation . . . . . . . . . . . . . . 2.4 Fermions . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Canonical Quantisation of Fermions . . . 2.5 Interactions . . . . . . . . . . . . . . . . . . . . 2.5.1 φ4 Theory . . . . . . . . . . . . . . . . . 2.5.2 Fermions . . . . . . . . . . . . . . . . . . 3 Gauge Theories 3.1 Global Symmetries v Gauge Symmetries 3.2 Abelian Gauge Theories . . . . . . . . . 3.3 Non-Abelian Gauge Theories . . . . . . . 3.4 Quantisation . . . . . . . . . . . . . . . . 4 Quantum Corrections 4.1 Anomalous Magnetic Moment . . . 4.2 Divergences . . . . . . . . . . . . . 4.2.1 Dimensional Regularisation 4.2.2 Renormalisation . . . . . . . 4.2.3 Running Coupling in QED . 4.2.4 Running Coupling in QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 5 6 6 6 8 9 9 10 11 12 12 14 15 17 18 20 22 22 24 27 29 32 32 35 36 38 40 41

高三英语材料科学单选题60题1. In the field of materials science, the "atom" is considered as the basic ______.A. unitB. elementC. particleD. molecule答案：A。

本题考查名词词义辨析。

选项A“unit”有“单位；单元”的意思，“atom”（ 原子）被视为材料科学中的基本“单位”，符合语境。

选项B“element”指化学元素；选项C“particle”侧重于粒子；选项D“molecule”指分子。

在材料科学中，原子通常被描述为基本单位，其他选项不符合。

2. The study of materials science often involves the analysis of various ______.A. substancesB. compoundsC. mixturesD. materials答案：D。

此题考查名词的含义。

选项A“substances”指物质；选项B“compounds”指化合物；选项C“mixtures”指混合物；选项D“materials”指材料。

材料科学的研究对象主要是各种“材料”，其他选项虽然也与物质相关，但不如“materials”直接和准确。

3. When testing the properties of a new material, the ______ of heat transfer is an important factor.A. processB. methodC. wayD. means答案：A。

本题考查名词的用法。

选项A“process”强调过程；选项B“method”侧重方法；选项C“way”比较常用，指方式、道路；选项D“means”意为手段、工具。

在测试新材料性能时，热传递的“过程”是重要因素，A 选项最符合题意。

传送两原子直积态的腔qed方案(英文)Quantum electrodynamics (QED) is a theory of how electromagnetic interactions occur between particles at the microscopic level. It is one of the most successful theories in modern physics and plays an important role in understanding the behavior of atomic and molecular systems. In particular, QED has been used to explain many phenomena, such as the energy levels of atoms, the Zeeman effect and the Stark effect. Recently, it has been proposed that two-atom QED could be used to generate entangled states, which could have a wide range of applications in quantum information processing.Two-atom QED is a cavity version of the traditional QED system, in which two atoms interact with each other in a common electromagnetic field. In this system, the two atoms are positioned close to each other, and the electromagnetic field is confined to the space within the cavity. This allows the two atoms to interact more strongly than in the open space, leading to new and interesting effects. One of the most intriguing effects of two-atom QED is that it can create entangled states, which can then be used for a variety of applications.For instance, entangled states can be used for quantum computing, where information is encoded in the states of two or more qubits. Entangled states are also useful for quantum cryptography, where the state of two qubits is used to securely transmit sensitive information. Finally, entangled states can be used to measure quantum effects such as spincorrelations between two particles, which could be useful for understanding the behavior of complex quantum systems.In addition to these applications, two-atom QED may also be useful for sensitizing scanning tunneling microscopes or improving atom-based magnetometers. In general, two-atom QED is a promising opportunity for exploring the fascinating world of entanglement and quantum information processing. With further development, two-atom QED could potentially revolutionize our understanding of the fundamentals of quantum physics.。

1. 在电子检测中，以下哪种设备常用于测量电压？A. 万用表B. 示波器C. 频谱分析仪D. 信号发生器2. 质量控制中的“六西格玛”是指？A. 每百万次操作中的错误率不超过3.4次B. 每百万次操作中的错误率不超过6次C. 每百万次操作中的错误率不超过60次D. 每百万次操作中的错误率不超过600次3. 以下哪种测试方法用于检测电子元件的可靠性？A. 高温测试B. 低温测试C. 湿热测试D. 以上都是4. 在质量控制中，PDCA循环的四个步骤是？A. Plan, Do, Check, ActB. Plan, Design, Check, ActC. Plan, Do, Control, ActD. Plan, Design, Control, Act5. 以下哪种工具用于分析和改进生产过程中的问题？A. 5SB. 鱼骨图C. 流程图D. 直方图6. 在电子检测中，以下哪种设备用于检测电磁干扰？A. 频谱分析仪B. 网络分析仪C. 信号发生器D. 功率计7. 质量控制中的“零缺陷”理念是由谁提出的？A. 菲利普·克罗斯比B. 沃尔特·休哈特C. 爱德华兹·戴明D. 约瑟夫·朱兰8. 以下哪种测试用于评估电子产品的环境适应性？A. 振动测试B. 冲击测试C. 温度循环测试D. 以上都是9. 在质量控制中，以下哪种图表用于显示数据随时间的变化趋势？A. 控制图B. 散点图C. 柱状图D. 饼图10. 以下哪种设备用于测量电子元件的电阻？A. 万用表B. 示波器C. 频谱分析仪D. 信号发生器11. 在电子检测中，以下哪种测试用于检测电路板的焊接质量？A. 视觉检测B. X射线检测C. 红外检测D. 超声波检测12. 质量控制中的“持续改进”理念强调的是？A. 一次性改进B. 周期性改进C. 不断改进D. 不改进13. 以下哪种工具用于识别生产过程中的关键控制点？A. 流程图B. 鱼骨图C. 控制图D. 直方图14. 在电子检测中，以下哪种设备用于测量信号的频率？A. 万用表B. 示波器C. 频谱分析仪D. 信号发生器15. 质量控制中的“质量功能展开”（QFD）主要用于？A. 产品设计B. 生产过程C. 质量检测D. 市场分析16. 以下哪种测试用于评估电子产品的耐久性？A. 老化测试B. 疲劳测试C. 振动测试D. 以上都是17. 在质量控制中，以下哪种图表用于显示数据的分布情况？A. 控制图B. 散点图C. 柱状图D. 直方图18. 以下哪种设备用于测量电子元件的电容？A. 万用表B. 示波器C. 频谱分析仪D. 信号发生器19. 在电子检测中，以下哪种测试用于检测电路板的短路和开路？A. 视觉检测B. X射线检测C. 红外检测D. 电阻测试20. 质量控制中的“全面质量管理”（TQM）强调的是？A. 局部管理B. 全面管理C. 个别管理D. 不管理21. 以下哪种工具用于分析生产过程中的因果关系？A. 流程图B. 鱼骨图C. 控制图D. 直方图22. 在电子检测中，以下哪种设备用于测量信号的幅度？A. 万用表B. 示波器C. 频谱分析仪D. 信号发生器23. 质量控制中的“质量保证”（QA）与“质量控制”（QC）的主要区别在于？A. QA侧重预防，QC侧重检测B. QA侧重检测，QC侧重预防C. QA和QC都侧重检测D. QA和QC都侧重预防24. 以下哪种测试用于评估电子产品的安全性？A. 电气安全测试B. 电磁兼容性测试C. 环境适应性测试D. 以上都是25. 在质量控制中，以下哪种图表用于显示数据的中心趋势和离散程度？A. 控制图B. 散点图C. 柱状图D. 箱线图26. 以下哪种设备用于测量电子元件的电流？A. 万用表B. 示波器C. 频谱分析仪D. 信号发生器27. 在电子检测中，以下哪种测试用于检测电路板的元件布局？A. 视觉检测B. X射线检测C. 红外检测D. 超声波检测28. 质量控制中的“质量成本”包括哪些方面？A. 预防成本B. 鉴定成本C. 内部和外部失败成本D. 以上都是29. 以下哪种工具用于识别生产过程中的潜在问题？A. 流程图B. 鱼骨图C. 控制图D. 直方图30. 在电子检测中，以下哪种设备用于测量信号的相位？A. 万用表B. 示波器C. 频谱分析仪D. 信号发生器31. 质量控制中的“质量计划”主要用于？A. 产品设计B. 生产过程C. 质量检测D. 市场分析32. 以下哪种测试用于评估电子产品的性能？A. 功能测试B. 性能测试C. 可靠性测试D. 以上都是33. 在质量控制中，以下哪种图表用于显示数据的异常值？A. 控制图B. 散点图C. 柱状图D. 箱线图34. 以下哪种设备用于测量电子元件的功率？A. 万用表B. 示波器C. 频谱分析仪D. 功率计35. 在电子检测中，以下哪种测试用于检测电路板的连接可靠性？A. 视觉检测B. X射线检测C. 红外检测D. 拉力测试36. 质量控制中的“质量改进”强调的是？A. 一次性改进B. 周期性改进C. 不断改进D. 不改进37. 以下哪种工具用于分析生产过程中的数据分布？A. 流程图B. 鱼骨图C. 控制图D. 直方图38. 在电子检测中，以下哪种设备用于测量信号的波形？A. 万用表B. 示波器C. 频谱分析仪D. 信号发生器39. 质量控制中的“质量目标”主要用于？A. 产品设计B. 生产过程C. 质量检测D. 市场分析40. 以下哪种测试用于评估电子产品的兼容性？A. 电气安全测试B. 电磁兼容性测试C. 环境适应性测试D. 以上都是41. 在质量控制中，以下哪种图表用于显示数据的变异情况？A. 控制图B. 散点图C. 柱状图D. 箱线图42. 以下哪种设备用于测量电子元件的频率响应？A. 万用表B. 示波器C. 频谱分析仪D. 网络分析仪43. 在电子检测中，以下哪种测试用于检测电路板的电磁兼容性？A. 视觉检测B. X射线检测C. 红外检测D. 电磁干扰测试44. 质量控制中的“质量文化”强调的是？A. 局部文化B. 全面文化C. 个别文化D. 不文化45. 以下哪种工具用于分析生产过程中的数据趋势？A. 流程图B. 鱼骨图C. 控制图D. 直方图46. 在电子检测中，以下哪种设备用于测量信号的失真？B. 示波器C. 频谱分析仪D. 信号发生器47. 质量控制中的“质量标准”主要用于？A. 产品设计B. 生产过程C. 质量检测D. 市场分析48. 以下哪种测试用于评估电子产品的可靠性？A. 功能测试B. 性能测试C. 可靠性测试D. 以上都是49. 在质量控制中，以下哪种图表用于显示数据的分布范围？A. 控制图B. 散点图C. 柱状图D. 箱线图50. 以下哪种设备用于测量电子元件的温度？A. 万用表B. 示波器C. 频谱分析仪D. 温度计51. 在电子检测中，以下哪种测试用于检测电路板的电磁辐射？A. 视觉检测B. X射线检测C. 红外检测D. 电磁辐射测试52. 质量控制中的“质量政策”主要用于？A. 产品设计B. 生产过程C. 质量检测D. 市场分析53. 以下哪种工具用于分析生产过程中的数据关系？A. 流程图B. 鱼骨图C. 控制图54. 在电子检测中，以下哪种设备用于测量信号的频率响应？A. 万用表B. 示波器C. 频谱分析仪D. 网络分析仪55. 质量控制中的“质量审核”主要用于？A. 产品设计B. 生产过程C. 质量检测D. 市场分析56. 以下哪种测试用于评估电子产品的环境适应性？A. 电气安全测试B. 电磁兼容性测试C. 环境适应性测试D. 以上都是57. 在质量控制中，以下哪种图表用于显示数据的分布形态？A. 控制图B. 散点图C. 柱状图D. 直方图58. 以下哪种设备用于测量电子元件的电感？A. 万用表B. 示波器C. 频谱分析仪D. 信号发生器59. 在电子检测中，以下哪种测试用于检测电路板的电磁屏蔽效果？A. 视觉检测B. X射线检测C. 红外检测D. 电磁屏蔽测试60. 质量控制中的“质量培训”主要用于？A. 产品设计B. 生产过程C. 质量检测D. 市场分析61. 以下哪种工具用于分析生产过程中的数据模式？B. 鱼骨图C. 控制图D. 散点图62. 在电子检测中，以下哪种设备用于测量信号的频率稳定度？A. 万用表B. 示波器C. 频谱分析仪D. 信号发生器63. 质量控制中的“质量反馈”主要用于？A. 产品设计B. 生产过程C. 质量检测D. 市场分析64. 以下哪种测试用于评估电子产品的用户体验？A. 功能测试B. 性能测试C. 可靠性测试D. 用户体验测试答案：1. A2. A3. D4. A5. B6. A7. A8. D9. A10. A11. B12. C13. A14. C15. A16. D17. D18. A19. D20. B21. B22. B23. A24. D25. D26. A27. A28. D29. B30. B31. A32. D33. D34. D35. D36. C37. D38. B39. A40. B41. D42. D43. D44. B45. C46. B47. A48. C49. D50. D51. D52. A53. D54. D55. C56. C57. D58. A59. D60. C61. D62. C63. D64. D。

高一英语大学实验室安全规范严格练习题20题含答案解析1.In the laboratory, you need to use a device to measure temperature. Which one is it?A.microscopeB.thermometerC.beakerD.flask答案解析：B。

A 选项microscope 是显微镜，不能测量温度；C 选项beaker 是烧杯，也不能测量温度；D 选项flask 是烧瓶，同样不能测量温度。

只有B 选项thermometer 是温度计，可以测量温度。

2.When using an electrical appliance in the laboratory, what should you pay attention to?A.Keep it wetB.Connect it to multiple power sourcesC.Check for damage before useD.Leave it unattended答案解析：C。

A 选项让电器保持潮湿是非常危险的，可能会导致触电等事故；B 选项连接到多个电源也不安全，可能会引起过载等问题；D 选项让电器无人看管也不安全，可能会发生意外。

只有 C 选项在使用电器前检查是否有损坏是正确的安全规范。

3.What is the English name for the device used to hold chemicals for heating?A.test tubeB.buretteC.crucibleD.evaporating dish答案解析：A。

B 选项burette 是滴定管；C 选项crucible 是坩埚；D 选项evaporating dish 是蒸发皿。

只有 A 选项test tube 是试管，可以用来装化学药品进行加热。

4.Which of the following is not a safety rule when using a gas burner in the laboratory?A.Open the gas valve fullyB.Light the burner with a match or lighterC.Make sure there is proper ventilationD.Keep flammable materials away答案解析：A。

The Properties and Uses ofElectrolytesElectrolytes are substances that can conduct electricity when dissolved in water or melted. They are important for many everyday uses, ranging from batteries to medical treatments. In this article, we will explore the properties and uses of electrolytes in more detail.1. Properties of ElectrolytesElectrolytes are substances that can ionize when dissolved in water or melted. This means that they can break down into charged particles, such as cations (positively charged ions) and anions (negatively charged ions). Some examples of electrolytes include salt (sodium chloride), potassium chloride, and magnesium sulfate.One key property of electrolytes is their ability to conduct electricity. When they dissociate into ions, these charged particles can move freely in the solution or melt, and can carry an electric current. This property is important for many applications, such as in batteries, electroplating, and electrochemical cells.Another important property of electrolytes is their effect on the boiling and freezing points of water. When an electrolyte is dissolved in water, it can lower the freezing point and raise the boiling point of the solution. This is due to the interactions between the ions and water molecules, which affect the properties of the solution as a whole.2. Uses of ElectrolytesElectrolytes have many practical uses in various fields, including medicine, chemistry, and industry. Here are some examples:Medicine: Electrolytes are essential for the proper functioning of our bodies. They help regulate the balance of fluids and ions in our cells, and play a key role in processes such as nerve signaling, muscle contraction, and acid-base balance. Medical treatmentsoften involve the administration of electrolytes, either orally or intravenously, to restore or maintain the body's electrolyte balance.Chemistry: Electrolysis is a process that involves the use of electrolytes to conduct electricity and produce chemical reactions. This technique is used in many industrial and laboratory applications, such as electroplating, metal extraction, and the synthesis of chemicals and materials.Industry: Many industrial processes rely on the use of electrolytes to carry out electrical and chemical reactions. For example, batteries and fuel cells use electrolytes to generate and store electrical energy. Electrolytes are also used in electroplating, where a metal is deposited onto a surface by means of an electric current.In addition, electrolytes have applications in other areas such as agriculture, food processing, and environmental remediation. For example, they can be used to enhance plant growth, preserve food quality, and remove contaminants from wastewater.3. ConclusionOverall, electrolytes are an important class of substances with a wide range of properties and uses. Their ability to conduct electricity and interact with water molecules makes them essential for many technological and biological processes, and their applications are diverse and wide-ranging. As our understanding of electrolytes and their properties continues to deepen, we can expect to see even more innovative applications in the future.。

温州2024年05版小学五年级英语第3单元综合卷[有答案]考试时间：90分钟（总分:110）A卷考试人：_________题号一二三四五总分得分一、综合题(共计100题)1、听力题：An example of a noble gas is _______.2、What is the color of an emerald?A. RedB. BlueC. GreenD. Yellow答案:C3、听力题：The __________ is the process of a liquid turning into a solid.4、填空题：The __________ is a region known for its cultural diversity. (南亚)5、填空题：My grandma loves to __________ (照顾) her pets.6、填空题：A _______ is used to observe chemical reactions closely. (显微镜)7、选择题：What do you call the sound a duck makes?A. QuackB. MooC. BaaD. Roar8、填空题：A bear's claws are perfect for digging and ________________ (抓捕).The antelope gracefully moves through the grasslands, a testament to speed and ____.10、填空题：My grandma has many beautiful ________ in her garden.11、填空题：The _____ (植物生长条件) must be monitored for success.12、听力题：The dog is ______ (wagging) its tail happily.13、填空题：The __________ (历史的动态) highlight change over time.14、听力题：My cousin is a ______. She enjoys solving problems.15、What do we call the liquid that we drink?A. WaterB. SandC. AirD. Food16、填空题：I can build a _________ (玩具飞机) that really flies.17、填空题：The invention of ________ has transformed daily life.18、What is the name of the largest volcano in the solar system?A. Olympus MonsB. Mount EverestC. Mauna KeaD. Kilauea19、填空题：We should ________ (保护) the environment.20、填空题：The ________ (文化价值观) shape societies.21、填空题：A butterfly goes through a ______ (生命周期).I want to ________ a new toy.23、听力题：The turtle is very ___ (slow).24、填空题：The __________ (传统习俗) are passed down through generations.25、听力题：The chemical symbol for carbon is ______.26、填空题：A lobster's claws are powerful tools for catching ________________ (食物).27、听力题：The ______ teaches us about scientific discoveries.28、What is the capital of the Republic of the Congo?A. BrazzavilleB. KinshasaC. Pointe-NoireD. Ouesso答案:A. Brazzaville29、填空题：The __________ (历史的感知) affects how we view the past.30、What is the color of the sun?A. BlueB. YellowC. RedD. Green答案:B31、选择题：Which of these is a cold drink?A. CoffeeB. JuiceC. TeaD. Hot chocolate32、听力题：The _______ can be very large or very small.The __________ was a time of great scientific advancement. (启蒙时代)34、ts can grow in __________ (贫瘠的土壤). 填空题：Some pla35、填空题：The ________ was a significant event in the history of the United States.36、填空题：A wildcat is a small ________________ (猫).37、What do you call the place where you go to borrow books?A. SchoolB. LibraryC. StoreD. Park答案:B38、ssance was a period of great __________ (文化) and art development in Europe. 填空题：The Rena39、听力题：She enjoys ________ (mentoring) younger students.40、填空题：I love the feeling of unboxing a new ________ (玩具名) and discovering all its features.41、听力题：I enjoy _______ (drawing) in my sketchbook.42、填空题：The _____ (小鹦鹉) imitates sounds and dances. 小鹦鹉模仿声音并跳舞。

Ar-(Ar-CO2)双气体保护GTA焊中焊接成形和电极氧化现象本文推荐利用双气体保护钨极电弧焊（氩弧焊，也叫钨极惰性气体（TIG焊）保护焊），纯惰性氩气作为内层屏蔽和AR - CO2或活性气体二氧化碳作为保护层，研究SUS304不锈钢焊接中对钨电极保护和焊接变形的影响。

实验结果表明，内惰性气体氩可以成功地防止外层活性气体接触和氧化钨电极。

外层的屏蔽活性气体二氧化碳在电弧中被分解，溶于熔池，有效的降低了活性元素氧在焊缝金属溶解量。

当焊缝金属中氧含量超过70 ×10−6时，表面张力引起的Marangoni 对流方向从向外变成向内，从焊缝成形从宽而浅变为窄而深。

内层气体流动速率对焊缝形状和焊接变形的影响进行了系统的研究。

结果表明，当保护气体内层氩气流量过低时，焊缝容易氧化，焊缝成形宽而浅。

一个厚的连续氧化层是对熔池运动障碍。

关键词：双气体保护，焊接成形，Marangoni对流，钨极惰性气体保护焊1.介绍气体钨极电弧焊（氩弧焊），也称为钨极惰性气体保护（TIG）焊，因其高质量的焊接而在工业上广泛应用，特别是对不锈钢，钛合金及有色金属合金。

但是，浅熔深使焊缝深宽比只有0.2左右，造成较低的焊接生产。

如何增加GTAW 焊的熔深已经是长久以来人们关心的问题，目前已经提出的一些措施包括在原始材料中加入微量元素，如S，O，Se，和Bi 进VIB组，F,Br和Cl，加进VIIB组，焊接前抹助焊剂在钢板表面（的A - TIG焊或FB - TIG焊），还有加入少量的活性保护气体以提高渗透性等。

尽管有些技术已经部分应用于工业，人们仍然没有对A - TIG焊机理达成共识。

已经有四个假象的机理来解释的A - TIG焊的现象，包括氩弧焊锁孔模型，电弧收缩理论，熔池对流理论和流量绝缘模型。

自2002年以来，一系列高效率的GTA焊试验研究已经开展，包括焊接前涂氧化性助焊剂，增添少量活性气体，氧气或二氧化碳，保护气体为氩气，加入氧气或二氧化碳气体，保护气体为氦气。

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2.电化学传感器在环境监测中有广泛的应用。请举例说明电化学传感器如何用于检测水中的重金属离子，并讨论其检测原理和优点。
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3.简述电化学阻抗谱（EIS）的基本原理，并解释它在电化学分析中的应用。
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4.请讨论电化学仪器制造中，如何通过表面修饰技术来提高电极的选择性和灵敏度。
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标准答案
一、单项选择题
8.电化学传感器中，敏感材料的选择通常取决于其对于特定分析物的______和______。（）
9.微型电极技术可以显著提高电化学分析的______和______。（）
10.电化学仪器在环境监测中，可用于检测空气中的______和______等有害气体。（）
四、判断题（本题共10小题，每题1分，共10分，正确的请在答题括号中画√，错误的画×）
7.电化学滴定法中，滴定终点可以通过电流的变化来判断。（）
8.电化学传感器在生物医学领域的应用主要是用于血糖水平的监测。（）
9.电池的充放电过程属于物理变化，不涉及电化学反应。（）
10.电化学分析中，所有的电化学方法都可以用于检测金属离子。（）
五、主观题（本题共4小题，每题10分，共40分）
1.请描述电化学分析中工作电极、辅助电极和参比电极的作用及它们在电化学电池中的配置。
A.电流型传感器
B.电势型传感器
C.电容型传感器
D.光学型传感器
2.以下哪些因素会影响电化学分析中的电极反应速度？( )
A.电解质溶液的浓度
B.电极表面积
C.溶液的温度
D.电极材料的种类
3.电化学阻抗谱（EIS）分析中，低频区的半圆通常与以下哪些因素有关？( )
A.电荷转移电阻
B.传质过程
C.双电层电容

a rXiv:h ep-ph/998439v327Se p1999BNL-HET-99/20,hep-ph/9908439Muonium DecayAndrzej CzarneckiPhysics Department,Brookhaven National Laboratory,Upton,NY 11973G.Peter Lepage Newman Laboratory of Nuclear Studies,Cornell University,Ithaca,NY 14853-5001William J.Marciano Physics Department,Brookhaven National Laboratory,Upton,NY 11973Abstract Modifications of the µ+lifetime in matter due to muonium (M =µ+e −)formation and other medium effects are examined.Muonium and free µ+decay spectra are found to differ at O (αm e /m µ)from Doppler broadening and O (α2m e /m µ)from the Coulomb bound state potential.However,both types of corrections are shown to cancel in the total decay rate due to Lorentz and gauge invariance respectively,leaving a very small time dilation lifetime difference,(τM −τµ+)/τµ+=α2m 2e192π3f m 2e 5m 2µR.C.=α4−π2+m2em e−18−8π2× 1+α3ln mµπ2 4m e−2.0ln mµ2m2W(1−m2W/m2Z)(1−∆r)where∆r≃0.0358(form top=174.3GeV and Higgs mass m H≃125GeV)represents calculable electroweak radia-tive corrections[7].In that role,Gµhelped predict the top quark’s mass before its discovery and currently constrains the Higgs mass to relatively low preferred values m H<∼220GeV. In addition,it provides a sensitive probe of“new physics”such as SUSY,Technicolor,Extra Dimensions etc.[5].Recently,there have been several proposals[8–10]to further improve the measurement ofτµ(and thereby Gµ)by as much as a factor of20,bringing its uncertainty down to an incredible±1ppm.Of course,such an improvement can only by fully utilized if the other electroweak parameters with which it is compared reach a similar level of precision and radiative corrections to their relationship(e.g.∆r)are computed at least through2loops. That confluence of advances appears unlikely in the foreseeable future.Nevertheless,given the fundamental nature and importance of Gµ,efforts to improve its determination should be strongly encouraged and pushed as far as possible.In that spirit,we examine in this paper several theoretical concerns that must be addressed in any±1ppm study ofτµ.Precision muon lifetime experiments involve stoppingµ+in material.At some level,the medium in which it comes to rest will affect the muon decay rate.The most straightforward issue to consider is the formation of muonium(M=µ+e−)and its bound state effect onτµ.During the slowing down ofµ+in matter,muonium will form and be ionized many times [11,12].Eventually it comes to(thermal)rest at which point the muon or muonium atomhas a room temperature kinetic energy∼0.04eV or thermal velocityβthermalµ≃2.7×10−5.The actual fraction of stoppedµ+that end up as muonium and decay while in that bound state is very medium dependent.It can range from a small fraction in metals to nearly100%in some materials.If a significant modification ofτµoccurred in muonium,a correction would have to be applied.For example,a published study[13]has claimed a0.999516reduction factor for the bound state decay rate.Such a large−484ppm shift would be difficult to correct for at the±1ppm level unless the muonium formation fraction was very precisely known.It would also impact the interpretation of existingτµmeasurements [14]and might indicate the possibility of other large medium effects.Given its potential importance,we have reexamined the muonium bound state effect on the muon lifetime.As we shall show,the leading correction turns out to be remarkably small,α2m2eWe begin by recalling some basic properties of muonium[15].It is aµ+e−Coulombic bound state with hydrogenic features.The reduced massm e mµm==−13.5eV/n2,n=1,2 (6)2n2For the n=1ground state the virial theorem tells us that the average(electron)kinetic energy isT =−E1=13.5eV(7) while the average bound state potential energy isV =E1− T =α2m=−27.0eV.(8) Also,the n=1momentum distribution is given by[16]4πp2|ψ(p)|2=32p2+α2m2 4p2.(9)It peaks at p2=α2m2/3which is somewhat below the average p2 =α2m2.In that configuration the e−andµ+have equal but opposite momenta and r.m.s.velocities,βr.m.s.≡ β2 1/2,βe=α≃1/137,andβµ≃αm e/mµ≃3.5×10−5.Note that the muon’s r.m.s. bound state velocity is similar(somewhat larger)in magnitude to its room temperature thermal velocity.Of course,(9)represents a non-relativistic approximation and should not be used indis-criminately for large p.However,it is adequate for the analysis presented here.Infig.1,we display the bound state momentum probability distribution corresponding to(9).Muonium bound state modifications of the muon decay spectrum and lifetime must vanish asαor m e/mµ→0.It is,therefore,useful to organize such corrections as an αn(m e/mµ)m,n,m=1,2...expansion in the small parametersα≃1/137and m e/mµ≃1/207.Terms of order n+m>4are completely negligible and can be safely ignored.The leading O(αm e/mµ)effects result from the muon’s non-zero velocity in the muonium rest frame.(For this discussion we take the muonium atom to be at rest in the lab frame.) The ground state velocity,βµ∼αm e/mµ,will modify the e+decay spectrum relative to its shape for freeµ+decay at rest(seefig.2).For a givenβµ,it gives rise to Doppler broadening which corresponds to an overall dilation along with smearing of the positron energy over the range(E e+±βµp e+)/redistribution of the spectrum.Lorentz invariance requires that such effects cancel in the total integrated decay rate up to an overall time dilation factor γ=1/2 Γ(µ+→e +νe ¯νµ),(10)τM =τµ 1+α2m 2eτµ=α2m 2eenergy by the27eV needed for it to climb out of the potential well due toρ(r).Thus the entire positron energy distribution is shifted down by V0=27eV:dΓM(E)(14)d E e+This is the effect offinal-state interactions.The two effects,phase-space contraction and final-state interactions,cancel in the total decay rate because the energy distribution and the kinematic limit are shifted by the same amount,leading to the same total integral for the decay rate:mµ/2−V0dΓM(E)d E e+d E= mµ/2dΓµ(E′)1−Z2α2≃1−1theµ−lifetime in a material and subtracting out the ordinary decay rate(µ−→e−¯νeνµ) using the correction in(16).That prescription needs some adjustment for high Z nuclei where residual low energy e−−Nuclei interactions can significantly suppress the decay rate and become comparable to time dilation[20].The analog ofµ−p capture for muonium is annihilation M→νe¯νµ.We have computed that rate for the n=1ground state andfind[21,22]Γ(M→νe¯νµ)=48π αm eand electromagnetic emission spectra.However,such changes again correspond to a redis-tribution of events and should cancel(to a very good approximation)in the total decay rate or lifetime(assuming Lorentz and gauge invariance are maintained in the analysis). The space-time scale associated with the actual decay process(not the lifetime)is1/mµ. That space-time interval is much too small to be affected by long distance environmental conditions.In summary we have found that lifetimes of muonium and a freeµ+differ in leading order only by time dilation effects which are∼10−9and completely negligible for future±1ppm experiments.The smallness of that difference follows from Lorentz and gauge invariance which guarantee cancelations among considerably larger spectrum distortions.A similar suppression of other medium effects on theµ+lifetime is also expected.ACKNOWLEDGMENTSWe thank D.Hertzog,R.Hill,P.Kammel,T.Kinoshita,and A.Sirlin for stimulat-ing discussions that rekindled our interest in this problem. A.C.thanks N.Uraltsev and A.Yelkhovsky for helpful correspondence.This work was supported by the DOE contract DE-AC02-98CH10886,and by a grant from the National Science Foundation.APPENDIX A:BETHE-SALPETER AND NRQED ANALYZES In the text,we described how spectral shifts occur in muonium decay but largely cancel in the total decay rate,leaving time dilation as the primary bound state correction.Here,a proof of those cancelations is given using a Bethe-Salpeter bound state approach.A second formalism,nonrelativistic QED(NRQED)[28],is then employed to illustrate the important role of electromagnetic gauge invariance or charge conservation in the cancelation.That method provides a powerful means of parameterizing other more general medium effects and showing that they also lead to negligible corrections to the muon lifetime in matter.1.Bethe-Salpeter AnalysisWe can compute the leading bound state corrections to muonium decay using standard Bethe-Salpeter perturbation theory for the bound state energies[16].The decay rate is obtained from the imaginary part of the energy(Γ=−2Im E).The leading effect comes from theµ+→e+ν¯ν→µ+contributionΣe+ν¯ν(p)to the muon’s self energy(seefigs.3and 4(a)).Since the decay products are typically highly relativistic,we can Taylor expand theself energy about the mass-shell momentum in the usual fashion[29]:ΓµImΣe+ν¯ν(p)=−reduction of the e+ν¯νphase space.The shift in the bound state energy due to this self energy is,infirst order perturbation theory,2ψ(p e·γ−m e)(pµ·γ(µ)−mµ)ψ,(A2) whereψis theµ+e−wave function,γ(µ)denotes Dirac matrices for the muon spinor part ofψ,p e and pµare the electron and muon momenta respectively,and integration over the wave function’s momenta distribution is implicit.The constantδZ e+ν¯νis of orderΓµ/mµ.The coefficient of−Γµ/2in this expectation value is nearly equal to unity since the wave functions are normalized such that[30]u(p)1u(p)u(p)γ0u(p)≈1−p2ψ(p e·γ−m e)ψ≈ 1−β2µ/2 (A5) whereβµis the muon velocity.This factor decreases the decay rate for muonium;it is the time dilation caused by the muon’s motion within the atom.The second term in expectation value(A2)is simplified by noting that(p e·γ−m e)(pµ·γ(µ)−mµ)ψ=Vψ,(A6) from the Bethe-Salpeter equation,where V is the binding potential(dominated by the Coulomb interaction).Thus,through orderα2,2 1−β2µ/2 +δZ e+ν¯ν V ,(A7) where thefirst correction to the free decay rate is due to time dilation,and the second correction is due to the contraction offinal-state phase space.These two corrections are of relative ordersα2m2e/m2µandα2m e/mµrespectively.A second contribution is relevant in orderα2.This is from the imaginary part of themuon-photon vertex correctionΓρe+ν¯νdue to thefluctuationµ+→e+ν¯ν→µ+—that is,the correction obtained by attaching a photon to the e+in the muon self-energy diagram forΣe+ν¯ν(seefig.4(b)).Physically this corresponds to afinal-state interaction between the outgoing positron(from the decay)and the electron(from the atom).The momentum transfer carried by atomic photons(≈αm e)is tiny compared to the typical e+ν¯νmomenta, and so we can Taylor expand the vertex function to obtain the leading contribution:ImΓρe+ν¯ν=−δZ e+ν¯νγρ+···,(A8)8where the remaining terms on the right all vanish on mass shell for zero momentum transfer. The constantδZ e+ν¯νhere is the same as that appearing in the self energy;this is required by QED gauge invariance and can be proven using Ward identities in the standard fashion. This vertex correction results in a perturbation given by−δZ e+ν¯ν2mµ+···(A11)+iΓµ2m2µ−c2D4m3µ+d2σ·ψ†eσψem2µ+f2e∇·EThe couplings c1,c2,d1...in the effective Lagrangian are all dimensionless and have perturbative expansions inα.The c i’s are determined by computing muon decay,using the effective theory,for muons in different reference frames.They are tuned until the decay rate predicted by the effective theory agrees with the relativistic resultΓµ(βµ)=Γµ(0) 1−β2µ8−··· ;(A12)at tree-level c1=c2=1.The d i’s incorporate the contribution from muon-electron annihila-tion:µe→ν¯ν(see Eq.(17)).The f i’s are computed by comparing the on-shell,renormalized vertex functionΓe+ν¯ν,expanded in powers of the external momenta,with predictions from the effective theory.Two features of the effective theory deserve comment.One is that there are no terms of the formiΓµψ†µψµ(A14)2is a conserved quantity in the nonrelativistic theory;it is the pure number operator that counts muons and so cannot be renormalized.Consequentlyφ|δL|φ(A15)2for any state|φ that contains a single muon—including muonium states as well as arbi-trarily complicated many-electron states or other medium effects that describe muons inside bulk materials.This“no-renormalization theorem”implies that the only corrections to the free muon decay rate in such systems come from the explicit nonunitary,factorizable cor-rection terms in the effective Lagrangian,the terms with couplings c i,d i,f i...that are directly related to muon decay.As discussed in the text,these corrections are typically of or-der1ppb or smaller.Other effects not of this form,due to external photons or multi-electron excitations,cannot modify the total decay rate.10REFERENCES[1]C.Caso et al.(Particle Data Group),Eur.Phys.J.C3,1(1998).[2]S.Berman and A.Sirlin,Ann.Phys.20,20(1962).T.Kinoshita and A.Sirlin,Phys.Rev.113,1652(1959).S.Berman,Phys.Rev.112,267(1958).[3]T.van Ritbergen and R.G.Stuart,Phys.Rev.Lett.82,488(1999).[4]M.Roos and A.Sirlin,Nucl.Phys.B29,296(1971).[5]W.J.Marciano,Fermi constants and“New Physics”,to be published in Phys.Rev.D,hep-ph/9903451.[6]Y.Nir,Phys.Lett.B221,184(1989).[7]A.Sirlin,Phys.Rev.D22,971(1980).[8]R.Carey and D.Hertzog et al.,A precision measurement of the positive muon lifetimeusing a pulsed muon beam andµLan detector,proposal to PSI.[9]J.Kirkby et al.,FAST Proposal to PSI(May1999).[10]S.N.Nakamura et al.,Precise measurement of theµ+lifetime and test of the exponentialdecay law,RIKEN-RAL Proposal.[11]J.Brewer,K.Crowe,F.Gygax,and A.Schenk,in C.S.Wu and V.W.Hughes,eds.,Muon Physics(Academic Press,New York,1977),vol.3,p.1.[12]A.Seeger,in G.Alefeld and J.V¨o lkl,eds.,Topics in Applied Physics(Springer Verlag,New York,1978),vol.28,p.349.[13]L.Chatterjee,A.Chakrabarty,G.Das,and S.Mondal,Phys.Rev.D46,5200(1992),E:D49,6247(1994).[14]D.Hertzog,private communication.[15]V.W.Hughes and G.zu Putlitz,in T.Kinoshita,ed.,Quantum Electrodynamics(WorldScientific,Singapore,1990),p.822.[16]H.A.Bethe and E.E.Salpeter,Quantum Mechanics of One-and Two-Electron Atoms(Plenum Publ.Corp.,New York,1977).[17]The2S state of free muonium is metastable[15],decaying to1S+γγin about1/7sec.The value of β2µ =α2m2e/4m2µin that configuration leads to a factor of4reduction in the corresponding time dilation corrections of Eqs.(10)–(12).[18]I.I.Bigi,N.G.Uraltsev,and A.I.Vainshtein,Phys.Lett.B293,430(1992).B.Blok and M.Shifman,Nucl.Phys.B399,441and459(1993).For a recent discussion and further references see N.Uraltsev,hep-ph/9804275.[19]H.¨Uberall,Phys.Rev.119,365(1960).[20]R.Huff,Ann.Phys.16,288(1961).[21]Our result in Eq.(17)is about an order of magnitude smaller than a similar calculationin Ref.[13].[22]Muonium annihilation was originally estimated in B.Pontecorvo,Zh.Eksp.Teor.Fiz.33,549(1958)[Sov.Phys.JETP6,429(1958)].[23]P.-J.Li,Z.-Q.Tan,and C.-E.Wu,J.Phys.G14,525(1988).[24]G.Bardin et al.,Nucl.Phys.A352,365(1981).[25]V.Bernard,T.R.Hemmert,and U.-G.Meissner,Muon capture and the pseudoscalarform-factor of the nucleon(1998),hep-ph/9811336.[26]T.Kinoshita,J.Math.Phys.3,650(1962).[27]T.D.Lee and M.Nauenberg,Phys.Rev.133,1549(1964).[28]W.E.Caswell and G.P.Lepage,Phys.Lett.B167,437(1986).For more recent exam-ples see e.g.A.Czarnecki,K.Melnikov,and A.Yelkhovsky,Phys.Rev.Lett.82,311 (1999)and83,1135(1999).[29]The expansion is analytic because thefinal decay state is far from any threshold.Bound state effects on the low energy e+spectrum modify the total decay rate at O α3m3e/m3µ and can,therefore,be ignored in this analysis.[30]The right-hand side of the normalization condition would be the electromagnetic formfactor of muonium at zero momentum transfer if the electron had no electric charge, and provided that the potential is instantaneous(which is true up to O(α3)corrections which we ignore).Charge conservation or gauge invariance demands that this form factor equal unity.FIGURES 024********.050.10.150.20.25p [keV]4πp 2|ψ(p )|2FIG.1.Momentum probability distribution in muonium 1S state.0102030405000.0050.010.0150.020.0250.030.035E e +[MeV]1d Ee +FIG.2.Positron energy spectrum for free µ+decay at rest.νeνµW We +µ+µ+FIG.3.Self energy of the muon.Its imaginary part,computed in the external Coulombic field,determines the lowest order muonium width.νeνµWγe +e -µ+νe νµW γe +e -µ+(a)(b)FIG.4.Coulomb interactions in muonium decay.。

a rXiv:h ep-ph/38223v121Aug23S.R.Valluri Departments of Physics,Astronomy &Applied Mathematics,University of Western Ontario,London,ON,N6A 3K7,Canada valluri@uwo.ca U.D.Jentschura National Institute of Standards and Technology,Mail Stop 8401,Gaithersburg,Maryland,20899–8401,USA ulj@ mm Electro-Optics,Environment,and Materials Laboratory,GTRI,Georgia Institute of Technology,Atlanta,GA,30332,USA mm@ Abstract1INTRODUCTIONThe effective one-loop Lagrangian density[1–11]of quantum electrodynamics(QED)describes the nonlinear interaction of electromagneticfields due to a single closed electron loop.[It is also referred to as the QED effective action,or the effective Lagrangian.]One of the most compact and elegant ways to treat the symmetry properties of the vacuum is by the method of the effective Lagrangian[12].This one-loop Lagrangian,often called the Heisenberg-Euler Lagrangian(HEL),has been used to describe a variety of electromagnetic processes.The real part of the Lagrangian can be used to delineate such dispersive phenomena as photon propagation in a magneticfield,second harmonic generation,photon splitting in a magneticfield,and light scattering in a vacuum [1,4–9,11,13–17].The development of high intensity lasers,with the consequent availability of powerful,coherent light sources,can render possible the observation of delicate nonlinear effects and also the nonlinearity of Maxwell’s equations.Strong magneticfields around pulsars and around magnetars,and other astrophysical and laboratory situations might also warrant accurate calculations based on the Lagrangian for astrophysical and other applications.The imaginary part of the Lagrangian has been applied to absorption processes such as electron-positron pair creation[4,5,8,9].While the integral representation of this Lagrangian as a function of the electric and magnetic field is well known,further analytical representations and numerical procedures for arbitraryfield strengths have recently been refined.In the special cases when either the magnetic or electricfield vanishes,the Heisenberg-Euler Lagrangian may be expressed in terms of elementary functions and integrals of the natural logarithm of the generalized gamma function with real or complex arguments[18,19].Elementary function series expansions for these gamma function integrals and precise numerical values for the Lagrangian using the expressions of Dittrich et al.were provided by Valluri et al.[18].For the more general case when both an electric and magneticfield are present,a numerically useful,analytical series expression for the real part of the Lagrangian was derived and this latter series expression involves elementary functions and the sine,cosine,and exponential integrals,all of which are easily calculated[20].Motivations for studying non-linear generalizations of Maxwell’s equations in the vacuum are now quite different.Strong interest in one-loop corrections to the classical Lagrangian in Abelian (as well as non-Abelian)gauge theories has resurged in order to learn about the structure of the vacuum when it is probed by an external electromagneticfield.The problem of the existence of a stable electron has been and continues to be an interesting issue,not only relevant to pure electrodynamics but also to an unknown fundamental theory of matter and its interactions(for an interesting view on this issue,see[12]).In this paper we discuss some analytic calculations relevant for the applications of the HEL.The paper is organized as follows.Section2provides the analytical definitions and preliminaries. Section3presents the analytical results indicating the existence of the higher harmonics for the real part of the HEL,and the discussion of these results will conclude the main body of the paper.Some aspects of the renormalization and the renormalization-group invariance of the action are briefly discussed in an Appendix.Thefinal section summarizes the conclusions.2REPRESENTATION OF THE QED EFFECTIVE ACTION BY SPECIAL FUNCTIONSThe renormalized(see also the Appendix)Heisenberg-Euler Lagrangian(HEL)∆L is expressed as a one-dimensional proper-time integral[1–4]:∆L=−e2se−(m2−iǫ)s ab coth(eas)cot(ebs)−a2−b2(es)2 (1)and is a quantum correction to the Maxwellian LagrangianL cl=L0=L M=14FµνFµν=12 a2−b2 ,(3)G=1Also,the following notation is sometimes used:F =−S,G =−P ,(9)L cl =−F =−12 E 2−B 2 =14π3ab ∞ n =1[a n +d n ],(12)a n =coth(nπb/a )ea cosnπm 2easinmπm 22n exp nπm 2eb+exp −nπm 2eb,(14)while the imaginary part is given asIm ∆L =e 2|ab |n coth nπa eb .(15)We note that the Cosine and Sine integrals have an “asymmetric”form since the generally accepted definitions for these integrals are “asymmetric”and are shown below.More details are given in[20].Ci(z )=−∞zdt cos(t )t =Si(z )−πt .(18)The imaginary part Im∆L is generated by a modification of the integration contour in the exponential integral entering into the definition of d n (normally,the exponential integral is defined via a principal-value prescription).The relevant exponential integral isEinπm 2tdt for u ∈.(20)Under an appropriate deformation of the contour,a unified representation for both the real and the imaginary parts is obtained[20],∆L =limǫ→0+−e 22nexp −i nπm 2ea +exp i nπm 2ea ,(22)c n =coth (nπa/b )eb Γ 0,nπm 2eb Γ 0,−nπm 22z log(2π)−γz 22+∞ k =2(−1)k ζ(k )z k +1dt ζ(t,z ),(27)whereζ(k)is the Riemann Zeta function,used in the representation of the HEL.Due to its close connection to theΓ(z)and the Zeta function,the Barnes function may have a variety of applications in Computer Algebra and Theoretical Physics as well as in otherfields.The Mittag–Leffler theorem(see[20]for a comprehensive discussion)is a key element in the derivation of the series representation for the HEL(some useful formulas,originally derived without the explicit use of the Mittag–Leffler theorem,still based on a partial-fraction decom-position and equivalent to the results obtained by us using the Mittag–Leffler theorem,can be found in[24,p.271]).The generalization of the Mittag-Leffler theorem for a function with non-simple poles and zeros is the Riemann-Roch theorem[25,26].For Lagrangians with poles and zeros of higher order,which can occur in string theories,we anticipate the application of the Riemann-Roch theorem.Such Lagrangians with poles and zeros of arbitrary order may form a vector space of meromorphic functions over a complexfield.3MAGNETO–OPTICAL EFFECT:SECONDAND HIGHER–HARMONIC GENERATIONMaxwell’s equations receive corrections from virtual excitations of the charged quantum fields(notably electrons and positrons).This leads to interesting effects[1]:light-by-light scatter-ing,photon splitting,modification of the speed of light in the presence of strong electromagnetic fields,and–last,but not least–pair production.The dominant effect for electromagneticfields that vary slowly with respect to the Compton wavelength(frequenciesω≪2mc2/h)is described by the Heisenberg-Euler Lagrangian,which is known to all orders in the electromagneticfield. For the case of zero electricfield,the HEL can be written as[18,19]∆L(h)=m4h2− 12H=m2πF c −S+(S2+P2)12,y=12 1in analogy to a and b of equations(5)and(6)above.Here F c=H cr.Forx=0,y=0,|P|=0,we obtainRe L eff=S−α2k2 e k y +e−k y =−y2k 2N.(31)Again,it is to be noted that Ei in Equation(31)is defined as real for real(negative and/or positive)arguments.We use the formulas:ζ(4)=π4945etc.to derive the equations above and below.˜d kcan be evaluated in the case of an external static magneticfield B0along the x-axis.For an electromagnetic wave propagating along the z axis,we have for the case[7],E x=E z=0,B y=B z=0,|E y|=|B x|,P=E·B=0,(32)S=12 B20+2B x B0 .(33)We then derive an expression for the partial derivative of the Re L eff with respect to S:∂π 4F2c+16F4c+256F6c+ (34)The electric displacementfield D can then be obtained.D=∂∂E ·∂45π −B20+2B x B07 B20+2B x B0The term proportional to cos(2φ)indicates the presence of the second harmonic.Similarly,E y B2x indicates the presence of the third harmonic and E y B3x indicates the fourth harmonic etc.In a similar way,the evaluation of H can be done:H=−∂S∂SRe L eff(P=0).(41)The above expression of H is also shown to exhibit the second and higher harmonics.The spatial anisotropy is imposed by the magneticfield.The SHG depends on propagation direction and polarization direction of the fundamental electromagnetic wave with respect to B0.The maximum effect is whenˆB0is⊥ˆk(the wave propagation direction).4CONCLUSIONSWe have investigated questions related to the representation of the quantum electrodynamic (QED)effective Lagrangian and its analytical expression.In Sect.2,we briefly recalled our previous results given for special-function representations of the effective Lagrangian,and we briefly clarify the mathematical notation used in the special-function representations(12)and (21).The representation(21)unifies the real and imaginary parts.We also introduced the Barnes function that should be of use in a variety of applications that include the HEL.We very briefly mention that the key step in the derivation of our series representations is the Mittag–Leffler theorem[20].In section3,we have also given an expression from the effective Lagrangian that will facilitate the evaluation of the higher harmonics.We also briefly discuss the magneto-optical effect of the vacuum which might provide a signature of“QED’s nonlinear light.”In the Appendix,we discuss some aspects of the renormalization and the renorm-invariance of the action.Based on the results of the current paper,we expect to carry out detailed studies related to various projected and ongoing experiments and astrophysical phenomena[1,11–17]involving strong static-field conditions(orfields with frequencies that are small as compared to the electron Compton wavelength).The study of the HEL to includefinite temperature effects[29]with the series representation that we have developed is an interesting problem that warrants further investigation.ACKNOWLEDGEMENTSThe authors wish to acknowledge insightful conversations with Dr.Holger Gies and Professors Bowick and Joe Schecter of the Physics Department,Syracuse University.They would also like to thank Kato Lo and John Drozd for helpful discussions and invaluable assistance in processing the paper.S.R.V.acknowledges a supporting grant from the Natural Sciences and Engineering Research Council of Canada(NSERC).A APPENDIX–RENORMALIZATIONAND RENORMALIZATION–GROUP INV ARIANCEThe renormalization as well as the renormalization-group invariance of the QED effective La-grangian have already been discussed at length by various authors(see[30]and references therein).However,material on this topic is somewhat scattered in the literature.In the current Appendix, we would like to give a brief overview of the physical ideas that led to the formulation of the related RG equations,as well as the connection with theβfunctions of QED,complemented by an easy-to-understand presentation of the basic notion of the renormalization itself.It should be noted that the considerations in this section have only partial significance for the main topic of the current investigation,which is a treatment of higher-harmonic generation based on series representations of the Lagrangian.The current section merely provides illustrating remarks on the derivation of the effective Lagrangian,which is again the starting point of the main endeavour pursued.The”original”Lagrangian of QED from which we start to work out the S-matrix with all its radiative corrections in fact has to be identified with the”bare”Lagrangian that is expressed in terms of the”bare”physical quantities(bare charges,masses,fields).As we develop the perturbations series(in e2),we see that some terms(it does not matter if they are infinite or not)in fact modify the physical parameters that entered into the very Lagrangian from which our work started.This means that we have to renormalize the Lagrangian. Renormalization:The renormalization can be done by adding counterterms to the Lagrangian. These terms relate the bare parameters(bare charges,masses,fields)to the physical parameters of the theory,i.e.to the renormalized charges,masses,fields.The renormalizability then requires that the Renormalized Lagrangian=Bare Lagrangian+Counterterms=Bare Lagrangian but with charges,masses,fields expressed in terms of the renormalized parameters.For QED,we are in the lucky position that the theory remainsfinite in the infrared,i.e.that the physical charge of the electron remainsfinite as we move two electrons far apart.Therefore,we may renormalize QED on mass shell,i.e.renormalize QED in such a way that the renormalized charge is by definition the charge of an electron as seen by another electron in the limit of a large separation of the two electrons(this latter situation corresponds to the limit of a very soft exchanged photon —the infrared limit in which the wave vector k of the exchange photon tends to zero).To see how this works,look at p.325of[31]:The vacuum polarization modifies the Coulomb law(or Thompson scattering)in such a way that(e0is the bare charge)e20/k2→e20/[k2(1+ω(k2))]=e20/[k2(1+ω(0)+O(k2))].(42) Now at k=0,the physical charge of the electron is obtained,which means that e2=e20/(1+ω(0)),or that the Z3renormalization that relates the bare and the renormalized charge of the electron is obtained as Z3=1/(1+ω(0))where e2=Z3e20.This relates the physical renormalized charge with the bare charge of the electron.In order to construct a renormalized Lagrangian in which the“e”is indeed the renormalized,finite electron charge,we now have fulfill the condition that the vacuum polarization as derived from our renormalized Lagrangian results in a”modified”vacuum polarization correction to the Coulomb law that fulfillsωRenormalized(0)=0. This is just Eq.(8-96e)of[31].In general,a set of renormalization conditions imposed on the renormalized Lagrangian determines the physical interpretation of the parameters in terms of which the renormalized Lagrangian is written.The renormalization conditions(8-96)in[31],on which the considerations of[20]are based,ensure that the renormalized effective Lagrangian is written in terms of thefinite,renormalized,physical electron charge.The”usual”representation of the QED Effective Lagrangian(QED Effective Action)in Eq.(1)fulfills these conditions.In other words,since we are used to defining the value of the coupling by Thompson scattering, i.e.,by scattering a long-wavelength(ω→0)photon offa(static)electron,the renormalization scale is naturally set by the mass of the electron.This amounts to the so-called on−electron−mass−shell renormalization condition e2(µ=m)/(4π)≃1/137.036.Our starting point Eq.(1) for the HEL is obtained exactly by implementing this renormalization condition,and the coupling e used throughout this work should be interpreted in this way.It is precisely the second term in square brackets in Eq.(1)that guarantees this renormalization condition.This term subtracts any contribution of∆L to Thompson scattering;therefore,the latter is fully described only by the renormalized Maxwell term together with the static-eletron interaction term involving the on-shell renormalized coupling e–as it should.This renormalization procedure has therebyfixed all free parameters,such that the theory is now fully predictive for all other processes that can be described by our one-loop HEL.QED is usually defined at some high ultraviolet(UV)scale in terms of a bare action.It is predictive for electrodynamic processes at low energy scales to a high accuracy,once all renormalization-group“relevant”parameters are specified.QED has two such parameters,the coupling e and the electron mass m.During the derivation of the HEL(1),we have to give a prescription how tofix these parameters to their physical values,since they turn out to be scale-dependent.However,since the HEL of Eq.(1)describes processes with external photon lines to one loop only,the scale dependence of m remains invisible in this calculation.Hence,to this order of accuracy,we canfix the electron mass to its value measurable at the scale of,say,atomic physics,and insert this into Eq.(1).In the HEL language,the scale dependence of the mass be-comes visible at two-loop order,see,e.g.,[30,32,33].This subtlety emerges in the following way: The unrenormalized two-loop expression contains a term which may be written asδm2(∂/∂m2) (one-loop Lagrangian),where m is the electron mass andδm the radiative modification(the physical mass is then m2ph=m2+δm2.The original Maxwell Lagrangian is independent of the electron mass.The one-loop Lagrangian that we work with depends on the electron mass.The two-loop Lagrangian has effective self-energy corrections to the dressed fermion propagators that interact with themselves through a further radiative photon.This self-energy effect is known to produce a radiative renormalizationδm of the electron mass.Therefore,the above term has to be interpreted as a correction to the electron mass that enters into the one-loop Lagrangian, expressing the fact that“one-loop Lagrangian+δm2(∂/∂m2)(one-loop Lagrangian)=one-loop Lagrangian as a function of m2Renormalized=m2+δm2.”This is the mechanism by which the anomalous mass dimension creeps into the renormalization group analysis of the action[30]. Renormalization–Group Invariance:The QED Effective action is well known to be a mod-ification of the Maxwell Lagrangian relevant to the situation of strong backgroundfields.By contrast,vacuum polarization modifies the Coulomb at small distances(large momenta).How-ever,there is a connection between the two regimes,asymptotically.Specifically,Ritus(see[32] and references therein)has shown that an interesting connection exists between the corrections to the Maxwell equations in intensefields and vacuum-polarization corrections to QED at large momenta.The connection can now be found upon considering Eqs.(25)and(26)of Ref.[32].Ob-serve that the mass renormalization in Eq.(14)of Ritus also has the right asymptotic behaviour at small proper time s0.One of the basic ideas of the renormalization group invariance is that the product offield and charge,which enters into the covariant coupling,i∂µ−e0A0µ→i∂µ−eAµin going from bare to renormalized quantities,necessarily has to be preserved under the renormal-ization because space itself(∂µ≡∂/∂xµ)does not stretch under the renormalization of chargeandfield.The charge and thefield are renormalized as e0=Z−1/23e and A0µ=Z1/23Aµ.[Here,“0”denotes the bare quantity.]Let usfinally discuss the scale dependence of the coupling e as it occurs in the HEL.Starting the computation with a bare(Maxwell)action at UV scaleΛwith a bare coupling e0≡e0(Λ),we obtain aΛ-and e0-dependent contribution to the Maxwell action∼F.Together with the bareaction,we can trade these contributions for a scale-dependent coupling e=e(µ)that describes the interaction of a photon with an electron at an energy scaleµ[5,30,32].Of course,µis an arbitrary parameter,and the requirement that no physical process should depend onµdefines how the value of the coupling parameter e(µ)has to be changed upon a variation ofµ.For quantitative predictivity,we have tofix the value of e(µ)at one particular scaleµ.Starting from this consideration,one may derive functional relationships for the asymptotic behaviour of the QED Effective Action in intensefields,which in turn enable the direct transition to renormalization-group equations of the Callan-Symanzik type[see Eq.(45)of[32]].Other types of renormalization-group equations,for example those of the Gell–Mann Low type,including a thorough discussion,can be found in[5]and[30].Of course,the dependence onµof the HEL could be formally kept such as in[10]which would correspond to not specifying a renormalization condition.However,once the input of a single experiment(e.g.,Thompson scattering)is taken into account,no freedom is left in the param-eters,theµ-dependence has disappeared and QED is fully predictive.In particular,there is no room for further“logarithmic correction terms”[10].References[1]W.Dittrich and H.Gies.Probing the quantum vacuum.Springer Tracts in Physics,Vol.166(Springer,Berlin,2000).[2]W.Heisenberg and H.Euler.Z.Phys.98,714(1936).[3]V.Weisskopf.K.Dan.Vidensk.Selsk.Mat.Fys.Medd.14,1(1936).Reprinted on 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专利名称：A DEVICE AND A METHOD FORCORRECTING THE CURRENT IN A GRADIENTCOIL OF A MAGNETIC RESONANCEIMAGING APPARATUS FOR THECOMPENSATION OF EFFECT OF EDDYCURRENTS发明人：Takakura, Shinji,Yasunaka, Shigen申请号：EP17187035.5申请日：20170821公开号：EP3318887A3公开日：20180523专利内容由知识产权出版社提供专利附图：摘要：According to one embodiment, a correction device (1) is used for an apparatus including a coil and a conductor in a vicinity of the coil. The correction device (1) corrects an influence of a magnetic field generated by the conductor when a current flows through the coil. The correction device (1) includes a first measuring device (3), a second measuring device (4), and a control device (2). The first measuring device (3) measures a first signal of the coil. The second measuring device (4) measures a second signal of the coil, which is different from the first signal. The control device (2) estimates the influence acting on the coil, based on a difference between the first signal filtered by a first filter and the second signal filtered by a second filter. Furthermore, the control device (2) controls a command signal for flowing the current to the coil, based on an estimation result of the influence.申请人：KABUSHIKI KAISHA TOSHIBA地址：1-1, Shibaura 1-chome Minato-ku Tokyo 105-8001 JP国籍：JP代理机构：Hoffmann Eitle 更多信息请下载全文后查看。

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~ These equations have to be solved for a given trajectory R(t). The square of the expansion amplitudes aij taken at large t leads to the occupation probability for the state j . In this way many experiments were described with high accuracy. Figure 2 and 3 are typical examples and one could show dozens of similar plots. However, the most interesting process in Figure 1, namely the
process c which is called `spontaneous pair production' could not be observed so far and theoretical calculations show that to observe it would require the two ions to stick together much longer than usual. It is generally agreed that for the rather broad class of nuclear reactions summed over in the experiments so far, such long sticking times cannot be expected. In spite of this fact positron experiments at GSI found pronounced positron lines 2] and even extremely sharp electron-positron coincidence lines 3]. At the time of their discovery it was tempting to interpret these lines as signal for the decay of a new particle, possibly the axion. New experiments and further theoretical investigations of many low-energy phenomena showed, however, conclusively that this is no valid possibility (for a review see 4]). We summa4