Virtual Compton scattering off the nucleon at low energies

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MKPH-T-96-4Virtual Compton scattering o?the nucleon at low energies S.Scherer ?Institut f¨u r Kernphysik,Johannes Gutenberg-Universit¨a t,55099Mainz,Germany A.Yu.Korchin National Scienti?c Center Kharkov Institute of Physics and Technology,310108Kharkov,Ukraine J.H.Koch National Institute for Nuclear Physics and High-Energy Physics,1009DB Amsterdam,The Netherlands Abstract We investigate the low-energy behavior of the four-point Green’s function Γμνdescribing virtual Compton scattering o?the https://www.wendangku.net/doc/c53080156.html,ing Lorentz invariance,gauge invariance,and crossing symmetry,we derive the leading terms of an expansion of the operator in the four-momenta q and q ′of the ini-tial and ?nal photon,respectively.The model-independent result is expressed in terms of the electromagnetic form factors of the free nucleon,i.e.,on-shell information which one obtains from electron-nucleon scattering experiments.Model-dependent terms appear in the operator at O (q αq ′β),whereas the orders O (q αq β)and O (q ′αq ′β)are contained in the low-energy theorem for Γμν,i.e.,no new parameters appear.We discuss the leading terms of the matrix element

and comment on the use of on-shell equivalent electromagnetic vertices in the

calculation of “Born terms”for virtual Compton scattering.

13.40.Gp,13.60.Fz,14.20.Dh

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I.INTRODUCTION

Low-energy theorems(LET)play an important role in studies of properties of parti-cles.Based on a few general principles,they determine the leading terms of the low-energy amplitude for a given reaction in terms of global,model-independent properties of the par-ticles.Clearly,this provides a constraint for models or theories of hadron structure:unless they violate these general principles they must reproduce the predictions of the low-energy theorem.On the other hand,the low-energy theorems also provide useful constraints for experiments.Experimental studies designed to investigate particle properties beyond the global quantities and to distinguish between di?erent models must be carried out with suf-?cient accuracy at low energies to be sensitive to the higher-order terms not predicted by the theorems.Another option is,of course,to go to an energy regime where the low-energy theorems do not apply anymore and model-dependent terms in the theoretical predictions are important.

The best-known low-energy theorem for electromagnetic interactions is the theorem for “Compton scattering”(CS)of real photons o?a nucleon[1–3].Based on the requirement of gauge invariance,Lorentz invariance,and crossing symmetry,it speci?es the terms in the low-energy scattering amplitude up to and including terms linear in the photon momentum.The coe?cients of this expansion are expressed in terms of global properties of the nucleon:its mass,charge and magnetic moment.In experiments,one can make the photon momentum –the kinematical variable in which one expands–small to ensure the convergence of the expansion and to allow for a direct comparison with the data.By increasing the energy of the photon one will become sensitive to terms that depend on details of the structure of the nucleon beyond the global properties.Terms of second order in the frequency,which are not determined by this theorem,can be parameterized in terms of two new structure constants, the electric and magnetic polarizabilities of the nucleon(see,for example,[4]).

As in all studies with electromagnetic probes,the possibilities to investigate the structure of the target are much greater if virtual photons are used.A virtual photon allows one to vary the three-momentum and energy transfer to the target independently.Therefore it has recently been proposed to also use“virtual Compton scattering”(VCS)as a means to study the structure of the nucleon[5–7].The proposed reaction is p(e,e′p)γ,i.e.,in addition to the scattered electron also the recoiling proton is detected to completely determine the kinematics of the?nal state consisting of a real photon and a proton.It is the purpose of this note to extend the standard low-energy theorem for Compton scattering of real photons to the general case where one or both photons are virtual.The latter would be the case,e.g., in the reaction e?+p→e?+(e?+e+)+p.We will refer to both possibilities as“VCS”.

There are several di?erent approaches to derive the LET for Compton scattering of real photons.One was?rst used by Low[1].It made use of the fact that in terms of“unitarity diagrams”the scattering amplitude is dominated by the single-nucleon intermediate state. In such unitarity diagrams–not to be confused with Feynman diagrams–all intermediate states are on their mass shell[8].Lorentz invariance and gauge invariance then allow a prediction for the amplitude to?rst order in the frequency.Another approach[2,3],?rst used by Gell-Mann and Goldberger[2],relies on a completely covariant description in terms of the basic building blocks of electromagnetic vertices and nucleon propagators.They split the amplitude into two classes,A and B:class A consists of one-particle-reducible contributions

that can be built up from dressed photon-nucleon vertices and dressed nucleon propagators. Class B contains all one-particle-irreducible two-photon diagrams,where the second photon couples into the dressed vertex of the?rst one.Application of two Ward-Takahashi identities, one relating the photon-nucleon vertex operator to the nucleon propagator,the other relating the irreducible two-photon vertex to the dressed one-photon vertex,then lead to the same result for the leading powers of the low-energy amplitude.One can use yet another technique introduced by Low[9]to describe bremsstrahlung processes.This technique relies on the observation that poles in the photon momentum can only be due to photon emission from external nucleon lines of the scattering amplitude.Low’s method has,in a modi?ed form, also been widely used in the framework of PCAC[10].

So far,there have been only a few investigations of the general VCS matrix element, since most calculations were restricted to the Compton scattering of real photons.In[11] electron-proton bremsstrahlung was calculated in?rst-order Born approximation.The pho-ton scattering amplitude,for one photon virtual and the other one real,was analyzed in terms of12invariant functions of three scalar kinematical variables.In[12]it was shown that the general VCS matrix element–virtual photon to virtual photon–requires18invari-ant amplitudes depending on four scalar variables.In[13]the reactionγp→p+e+e?was investigated.Sizeable e?ects on the dilepton spectrum from the timelike electromagnetic form factors of the proton were found.Very recently the low-energy behavior of the VCS matrix element was investigated[14].Using Low’s approach[9]the leading terms in the outgoing photon momentum were derived.It was shown that the virtual Compton scatter-ing amplitude at low energies involves10“generalized polarizabilities”which depend on the absolute value of the three-momentum of the virtual photon.These new polarizabilities were estimated in a nonrelativistic quark model.In[15]the VCS amplitude was calculated in the framework of a phenomenological Lagrangian including baryon resonances in the s and u channel as well asπ0andσexchange in the t channel.A prediction for the| q|dependence of the electromagnetic polarizabilitiesαandβwas made.

The purpose of this work is to identify,in an analogous form to the real CS case,those terms of VCS which are determined on the basis of only gauge invariance,Lorentz invari-ance,crossing symmetry,and the discrete symmetries.In the following,we will refer to such terms not?xed by this LET for simplicity as“model dependent”.By introducing additional constraints,such as chiral symmetry,also statements about these terms become possible.This is,however,beyond the scope of the present work.In our study of low-energy virtual Compton scattering,below the onset of pion production,we will mainly work on the operator level.This allows us to work without specifying a particular Lorentz frame or a gauge.We combine the method of Gell-Mann and Goldberger[2]with an e?ective La-grangian approach.Class A is obtained in the framework of a general e?ective Lagrangian describing the interaction of a single nucleon with the electromagnetic?eld[4].In the spe-ci?c representation we choose,this turns out to be a simple covariant and gauge invariant “modi?ed Born term”expression,involving on-shell Dirac and Pauli nucleon form factors F1 and F2.The Ward-Takahashi identities then allow us to determine the leading-order term of the unknown class B contribution in an expansion in both the initial and?nal photon momenta.Furthermore,with the help of crossing symmetry a de?nite prediction can be made concerning the order at which one expects model-dependent terms.

Our work is organized as follows.We start out in Sec.II by outlining the general structure

of the VCS Green’s function in the framework of Gell-Mann and Goldberger.We state the ingredients for the derivation of the LET,namely,crossing symmetry and gauge invariance. Section III derives the LET for virtual photon Compton scattering and we discuss the leading terms of the matrix element for the reaction e?+p→e?+p+γin the center-of-mass frame. As the notion of“Born terms”is important for the LET,we comment on this aspect in Sec. IV and point out ambiguities that arise in their de?nition,both for real and virtual photons. Our results are summarized and put into perspective in Sec.V.

II.STRUCTURE OF THE VIRTUAL COMPTON SCATTERING TENSOR AND

GAUGE INV ARIANCE

In this section we will de?ne the Green’s functions and the kinematical variables relevant for the discussion of VCS o?the proton.We will consider the constraints imposed by the fundamental requirements of gauge invariance,Lorentz invariance and crossing symmetry. We do this in the framework of a manifestly covariant description,incorporating gauge invariance in its strong version,namely,in the form of the Ward-Takahashi identities[16,17]. The approach is similar to that of[3]using,however,a somewhat more modern formulation.

The electromagnetic three-point and four-point Green’s functions are de?ned as

Gμαβ(x,y,z)=<0|T(Ψα(x)ˉΨβ(y)Jμ(z))|0>,(2.1) Gμναβ(w,x,y,z)=<0|T(Ψα(w)ˉΨβ(x)Jμ(y)Jν(z))|0>,(2.2)

where Jμis the electromagnetic current operator in units of the elementary charge,e>0, e2/4π=1/137,and whereΨdenotes a renormalized interpolating?eld of the proton; T denotes the covariant time-ordered product[18].Electromagnetic current conservation,?μJμ=0,and the equal-time commutation relation of the charge density operator with the proton?eld,

[J0(x),Ψ(y)]δ(x0?y0)=?δ4(x?y)Ψ(y),(2.3) are the basic ingredients for deriving Ward-Takahashi identities[16,17].

Using translation invariance,the momentum-space Green’s functions corresponding to Eqs.(2.1)and(2.2)are de?ned through a Fourier transformation,

(2π)4δ4(p f?p i?q)Gμαβ(p f,p i)= d4xd4yd4z e i(p f·x?p i·y?q·z)Gμαβ(x,y,z),(2.4) (2π)4δ4(p f+q′?p i?q)Gμναβ(P,q,q′)= d4wd4xd4yd4ze i(p f·w?p i·x?q·y+q′·z)Gμναβ(w,x,y,z),(2.5)

where p i and p f refer to the four-momenta of the initial and?nal proton lines,respectively, P=p i+p f,and where q and?q′denote the momentum transferred by the currents Jμand Jν,respectively.We note that Gμαβdepends on two independent four-momenta,e.g.,p i and p f.In particular,it is not assumed that these momenta obey the mass-shell condition p2i= p2f=M2.Similarly,Gμναβdepends on three four-momenta which are completely independent as long as one considers the general o?-mass-shell case.This will prove to be an important ingredient below when analyzing the general structure of the VCS tensor.

Finally,the truncated three-point and four-point Green’s functions relevant for our dis-cussion of VCS are obtained by multiplying the external proton lines by the inverse of the corresponding full(renormalized)propagators,

Γμ(p f,p i)=[iS(p f)]?1Gμ(p f,p i)[iS(p i)]?1,(2.6)

Γμν(P,q,q′)=[iS(p f)]?1Gμν(P,q,q′)[iS(p i)]?1,(2.7) where,for convenience,from now on we omit spinor https://www.wendangku.net/doc/c53080156.html,ing the de?nitions above,it is straightforward to obtain the Ward-Takahashi identities

qμΓμ(p f,p i)=S?1(p f)?S?1(p i),(2.8) qμΓμν(P,q,q′)=i S?1(p f)S(p f?q)Γν(p f?q,p i)?Γν(p f,p i+q)S(p i+q)S?1(p i) .(2.9) Following Gell-Mann and Goldberger[2],we divide the contributions toΓμνinto two classes,A and B,Γμν=ΓμνA+ΓμνB,where class A consists of the s-and u-channel pole terms; class B contains all the other contributions.We emphasize that this procedure does not restrict the generality of the approach.The separation into the two classes is such that all terms which are irregular for qμ→0(or q′μ→0)are contained in class A,whereas class B is regular in this limit.Strictly speaking,one also assumes that there are no massless particles in the theory which could make a low-energy expansion in terms of kinematical variables impossible[1];furthermore,the contribution due to t-channel exchanges,such as aπ0,has not been considered.

The contribution from class A,expressed in terms of the full renormalized propagator and the irreducible electromagnetic vertices,reads

ΓμνA=Γν(p f,p f+q′)iS(p i+q)Γμ(p i+q,p i)+Γμ(p f,p f?q)iS(p i?q′)Γν(p i?q′,p i).

(2.10) Note thatΓμνA is symmetric under crossing,q??q′andμ?ν,i.e.,

ΓμνA(P,q,q′)=ΓνμA(P,?q′,?q).(2.11) Since also the totalΓμνis crossing symmetric,this must also be true for the contribution of class B separately[2].Using the Ward-Takahashi identity,Eq.(2.8),one obtains the following constraint for class A as imposed by gauge invariance:

qμΓμνA(P,q,q′)=i(Γν(p f,p f+q′)?Γν(p i?q′,p i)

+S?1(p f)S(p i?q′)Γν(p i?q′,p i)?Γν(p f,p f+q′)S(p i+q)S?1(p i)

≡fνA(P,q,q′).(2.12) Similarly,contraction ofΓμνA with q′νresults in

q′νΓμνA(P,q,q′)=?i(Γμ(p f,p f?q)?Γμ(p i+q,p i)

+S?1(p f)S(p i+q)Γμ(p i+q,p i)?Γμ(p f,p f?q)S(p i?q′)S?1(p i) =?fμA(P,?q′,?q),(2.13)

which is,of course,the same constraint which one obtains from Eq.(2.12)using the crossing-symmetry property ofΓμνA:

q′νΓμνA(P,q,q′)=?(?q′ν)ΓνμA(P,?q′,?q)=?fμA(P,?q′,?q).(2.14) Combining Eqs.(2.9)and(2.12)generates the following constraint for the contribution of class B:

qμΓμνB=qμ(Γμν?ΓμνA)=i(Γν(p i?q′,p i)?Γν(p f,p f+q′)),(2.15) relating it to the one-photon vertex[3].Once again,the second gauge-invariance condition, obtained by contracting with q′ν,is automatically satis?ed due to crossing symmetry.

The4×4matrixΓμνB of class B is a function of the three independent four-momenta P,q,and q′.It is important to realize that the twelve components of these momenta are independent variables only for the complete o?-shell case,i.e.,if one allows for arbitrary values of p2i and p2f.This will be important when making use of the constraints imposed by gauge https://www.wendangku.net/doc/c53080156.html,ing Lorentz invariance,gauge invariance,crossing symmetry,parity and time-reversal invariance,it was shown in[12]that the generalΓμνfor VCS o?a free nucleon with both photons virtual consists of18independent operator structures.The functions associated with each operator depend on four Lorentz scalars,e.g.,q2,q′2,ν=P·q=P·q′, and t=(p i?p f)2.When allowing the external nucleon lines to be o?their mass shell,one will have an even more complicated structure[19].

However,in our derivation of the low-energy behavior of the electromagnetic four-point Green’s function we will not require the full structure as discussed in[12].At low energies we expandΓμνB in terms of the four-momenta qμand q′μ,

ΓμνB=aμν(P)+bμν,ρ(P)qρ+cμν,σ(P)q′σ+···,(2.16) where the coe?cients are4×4matrices and can be expressed in terms of the16independent Dirac matrices1,γ5,γμ,γμγ5,σμν.An expansion of the type of Eq.(2.16)is expected to work below the lowest relevant particle-production threshold,in this case the pion-production threshold;we refer the reader to,e.g.,[20]where a similar discussion for the case of pion photoproduction can be found.

So far we have considered general features of the operators entering into the description of VCS.It is clearly one of the advantages of using a covariant description of the type of Eq.

(2.16)that it neither uses a particular Lorentz system nor a speci?c gauge.When referring to powers of q or q′,we mean the ones coming from the Dirac structures and their associated functions.This is di?erent when one works on the level of nucleon matrix elements or the invariant amplitude,where also kinematical variables from the spinors or normalization factors enter into the power counting.

We conclude this section by noting that the above framework can easily be applied to VCS o?a spin-0particle,such as the pion.In that caseΓμν,of course,has no complicated spinor structure.The building blocks for class A are simply the corresponding irreducible, renormalized electromagnetic vertex for a spinless particle and the full,renormalized prop-agator?(p)[21].

III.LOW-ENERGY BEHA VIOR OF VCS

In a two-step reaction on a single nucleon,such asγ?N→γ?N,the intermediate nucleon lines in the s-and u-channel pole diagrams of class A are o?mass shell,while the external nucleons are on shell.The early,manifestly covariant derivations of the low-energy theorem for Compton scattering[2,3]took into account that the associated half-o?-shell electromag-netic vertex of the nucleon has a di?erent and more complicated structure than the free vertex.However,it was shown that the model-and representation-dependent properties of an o?-shell nucleon do not enter in the leading terms of the full Compton scattering ampli-tude when the irreducible two-photon amplitude of class B is included consistently.This was explicitly shown by expanding Eq.(2.10)to?rst order in qμand by constructing the leading-order term ofΓμνB with the help of Eq.(2.15)and crossing symmetry or,equivalently, the second gauge-invariance constraint.The?nal result for the amplitude was given in the laboratory frame and in Coulomb gauge.

In order to obtain the LET for VCS,we will proceed on the operator level and combine the method of[2,3]with ideas of an e?ective Lagrangian approach to Compton scattering [4].Let us?rst recall that the electromagnetic three-point and four-point Green’s functions of Eqs.(2.1)and(2.2)depend on the choice of the interpolating?eldΨof the proton,i.e., are“representation dependent”.Therefore,the truncated Green’s functions in momentum space,Eqs.(2.6)and(2.7),are in general not directly related to observables,except for p2i= p2f=M2.Consequently,the separation into class A and class B is necessarily representation dependent,since the total Green’s function,Eq.(2.6),as well as the individual building blocks of class A are representation dependent;this has of course no e?ect on the?nal on-shell result,which is representation independent.This was explicitly shown in[22]for the case of real Compton scattering o?the pion.On the other hand,for any given appropriate interpolating?eld satisfying the equal-time commutation relation of Eq.(2.3)the Ward-Takahashi identities,Eqs.(2.8)and(2.9),hold,providing important consistency relations between the di?erent Green’s functions.In the following we will make use of these relations for arbitrary P,q,and q′,which,in particular,includes arbitrary p2i and p2f.Only at the end,the observable on-shell case will be considered.

A.Derivation of the Low-Energy Theorem

We will derive this theorem by using a convenient representation,the“canonical form”, of the most general and gauge invariant e?ective Lagrangian.In the framework of e?ective Lagrangians,a canonical form is de?ned as a representation with the minimal number of independent structures(see,e.g.,[23]).For the class A terms,we need the electromagnetic vertex and the propagator of the nucleon.Below the pion-production threshold,the La-grangian for a single proton,interacting with an electromagnetic?eld,can be brought into the canonical form[4]

LγNN=ˉΨ(iD/?M)Ψ?e∞ n=1((?2)n?1?νFμν)F1nˉΨγμΨ

?e

where DμΨ=(?μ+ieAμ)Ψ,Fμν=?μAν??νAμ.The electromagnetic structure of the proton is accounted for through the Dirac and Pauli form factors,F1and F2,respectively, which are expanded according to

F1(q2)=1+

∞

n=1(q2)n F1n,F2(q2)=κ+∞ n=1(q2)n F2n,κ=1.79.(3.2)

To this representation of the Lagrangian belongs,of course,a particular canonical La-grangian of order e2that generates the class B terms.However,as will be seen below,we do not need to know it in detail for this derivation.Clearly,Eq.(3.1)is invariant under the gauge transformationΨ→exp(?ieα(x))Ψ,and Aμ→Aμ+?μα.In order to arrive at Eq.

(3.1),use has implicitly been made of the method of?eld transformations(see,e.g.,[23–26]). It should be stressed that all ingredients needed for Eq.(3.1)are on-shell quantities that can be determined model independently from electron-proton scattering.Any explicit o?-shell dependence of the irreducible three-point Green’s function has been transformed away and will thus not show up in the class A contribution.Such transformations,however,generate concomittant irreducible class B terms for the amplitude that must be treated consistently (see[4]for details).

Using standard Feynman rules,the irreducible vertex associated with the e?ective La-grangian of Eq.(3.1)is found to be

Γμeff(p f,p i)=Γμeff(p f?p i)=γμF1(q2)+1?F1(q2)2M F2(q2),q=p f?p i.

(3.3) Since it is an important ingredient in our derivation of the LET,we emphasize that the vertex of Eq.(3.3)satis?es the Ward-Takahashi identity of Eq.(2.8);the corresponding propagator in the representation that yields Eq.(3.1)is the Feynman propagator of a point proton.As a consequence of gauge invariance,Eq.(3.1)automatically generates a term in the vertex proportional to qμ.

We note that the e?ective Lagrangian approach provides a natural explanation for the vertex of Eq.(3.3)which has previously been used by several authors as a simple means to restore gauge invariance in the form of the Ward-Takahashi identity(see,for example, [27]).However,it is not an independent building block for any amplitude,but must be used together with the corresponding irreducible class B terms for the reaction in question!

In principle,we could now proceed to construct the most general e?ective Lagrangian relevant to generate the corresponding class B terms for VCS.This would allow us,together with a calculation of the pole terms involving the vertex of Eq.(3.3),to determine the model-dependent terms ofΓμνat low energies.However,at this point it is more straightforward to apply the results of the last section.We obtain forΓμνA in the framework of Eqs.(3.1)and (3.3)

ΓμνA,eff=Γνeff(?q′)iS F(p i+q)Γμeff(q)+Γμeff(q)iS F(p i?q′)Γνeff(?q′).(3.4) Making use of Eq.(2.15)to obtain the gauge-invariance constraint for class B in our repre-

sentation,we see that it has the simple form1

qμΓμνB=0.(3.5)

This is due to the fact that the vertex of Eq.(3.3)depends on the momentum transfer,only. Note that Eq.(3.5)is still an operator equation,valid for arbitrary P,q,and q′.

Class A is de?ned to contain for any representation the pole terms ofΓμνwhich are singular in the limit qμ→0;we will come back to this point in the next section.We thus can make for class B the following ansatz for the q dependence:

ΓμνB(P,q,q′)=aμ,ν+bμρ,νqρ+cμρσ,νqρqσ+···.(3.6)

The coe?cients aμ,ν,···are4×4matrices which have to be constructed from the16Dirac matrices,the metric tensor gαβ,the completely antisymmetric Levi-Civita pseudo tensor ?αβγδ,and the remaining independent variables Pα,and q′α.The form of these coe?cients will be constrained by Lorentz invariance,gauge invariance,crossing symmetry and the discrete symmetries,C,P,and T.Contracting this ansatz with qμ,the condition for the class B operator,Eq.(3.5),becomes

qμΓμνB=aμ,νqμ+bμρ,νqρqμ+···=0.(3.7)

Multiple partial di?erentiation of Eq.(3.7)with respect to qαresults in the following condi-tions for the coe?cients,

aμ,ν=0,bμρ,ν+bρμ,ν=0, 6perm.(μ,ρ,σ)cμρσ,ν=0,···.(3.8)

Since current conservation is expected to hold for arbitrary q,the technique of partial di?er-entiation to obtain conditions for the coe?cients can easily be extended to obtain constraints for higher-order coe?cients.The simple constraints of Eq.(3.8)are based on the fact that Pα,qα,and q′αare independent variables;an implicit dependence would make matters more complicated.

From aμ,ν=0we can already conclude that the operator of class B contains no contri-butions which involve powers of q′only,without powers of q.Taking Eq.(3.8)into account, we now expand with respect to q′,

ΓμνB(P,q,q′)=Bμρ,νqρ+Bμρ,ναqρq′α+···+Cμρσ,νqρqσ+Cμρσ,ναqρqσq′α+···.(3.9)

If we apply crossing symmetry to Eq.(3.9)we?nd,as expected,that indeed all terms vanish that involve powers of q′only.Thus we?nd for the leading term of the model-dependent class B,

ΓμνB(P,q,q′)=Bμρ,να(P)qρq′α+O(qqq′,qq′q′),Bμρ,να=?Bρμ,να=Bνα,μρ,(3.10)

where the conditions for Bμρ,ναresult from gauge invariance and crossing symmetry,re-spectively.To be speci?c,after imposing the constraints of Eq.(3.10)and of the discrete symmetries one obtains three possible structures for Bμρ,ναon the operator level: Bμρ,να(P)=i(gμνgρα?gρνgμα)f1(P2)

+i(gμνPρPα+gραPμPν?gρνPμPα?gμαPρPν)f2(P2)

+?μρναγ5f3(P2),?0123=1.(3.11) In summary,we have shown that on the operator level

?the terms of order O(q?1α),O(q′?1α),O(1),O(qα)and O(q′α)are contained inΓμνA,eff,Eq.

(3.4).They are therefore determined model independently through on-shell quantities;?model-dependent terms?rst appear in the order O(qαq′β);

?all operators which contain either only powers of q or only powers of q′can entirely be obtained from thisΓμνA,eff.

In the next section we will discuss the implications of these?ndings for the on-shell VCS matrix element.

B.Application

We now want to apply the above result to the observable case where the nucleons are on mass shell,i.e.,we consider the matrix element.For that purpose we?rst de?ne the matrix element ofΓμνbetween positive-energy spinors as

(P,q,q′)=ˉu(p f,s f)Γμν(P,q,q′)u(p i,s i).(3.12) Vμνs

i s f

At this point we have to keep in mind that for the variables we use,P,q,and q′,the on-shell condition p2i=p2f=M2is equivalent to P·(q?q′)=0,and P2+(q?q′)2=4M2.In other words,the four-momenta chosen for the description of the o?-shell Green’s function will no longer be independent for the on-shell invariant amplitude.In particular,the distinction between powers of q only or,respectively,of q′only inΓμνis not valid anymore for the matrix element since q·P=q′·P.

Let us consider as an application the case where the initial photon is virtual and spacelike and the?nal photon is real,γ?(q,?)+p(p i,s i)→γ(q′,?′)+p(p f,s f).The following discussion does not include the Bethe-Heitler terms of the physical process p(e,e′p)γ,where the real photon is radiated by the initial or?nal electron,since these terms are not part ofΓμν.For the?nal photon the Lorentz condition q′·?′=0is automatically satis?ed,and we are free to choose,in addition,the Coulomb gauge?′μ=(0, ?′)which implies q′· ?′=0.We write the invariant amplitude in the convention of[28]as M=?ie2?μMμ,where?μ=eˉuγu u/q2is the polarization vector of the virtual photon.With a suitable choice for the reference frame, qμ=(q0,0,0,| q|),the Lorentz condition q·?=0and current conservation,qμMμ=0,may be used to reexpress the invariant amplitude as[29]

M=ie2 ?T· M T+q2

Note that as q0→0,both?z and M z tend to zero such that M in Eq.(3.13)remains?nite. Making use that we now know the general structure of the?rst undetermined operator,Eq.

(3.11),we can now consider the matrix element up to second order in q or q′.This will enable us to explicitly show how far the amplitude is determined and where the?rst model-dependent terms enter.After a reduction from Dirac spinors to Pauli spinors the transverse and longitudinal parts of Eq.(3.13)may be expressed in terms of8and4independent structures,respectively[4,14,30]:

?T· M T= ?′?· ?T A1+i σ·( ?′?× ?T)A2+(?q′× ?′?)·(?q× ?T)A3+i σ·(?q′× ?′?)×(?q× ?T)A4 +i?q· ?′? σ·(?q× ?T)A5+i?q′· ?T σ·(?q′× ?′?)A6+i?q· ?′? σ·(?q′× ?T)A7

+i?q′· ?T σ·(?q× ?′?)A8,(3.14)?z M z=?z ?′?·?q A9+i?z ?′?·?q σ·(?q′×?q)A10+i?z σ·(?q× ?′?)A11+i?z σ·(?q′× ?′?)A12.

(3.15) The results for the functions A i in the CM system expanded up to O(2),i.e.| q′|2,| q′|| q| and| q|2,are shown in Tables I and II.The derivation of the corresponding expressions is outlined in Appendix A.

In the amplitudes A5and A9,terms of order1/| q′|appear.We have not further expanded them in| q|and kept,e.g.,the on-shell form factors G M and G E.This was done mainly to stress that according to Low’s theorem[9]these divergent terms are already entirely?xed. They are due to soft radiation o?external lines,with the intermediate line approaching the mass shell in the pole terms.They can be obtained from,e.g.,the Born terms that contain the same on-shell information(see however the caveats in Sec.IV).In order to uniquely identify these singular contributions we have expanded all relevant expressions in terms of | q|and| q′|.This is the reason why the argument of the form factors is Q2=q2|| q′|=0=?2M(E i?M),where E i=√

In Table III we also show the results for the transverse amplitudes in the CM frame for real Compton scattering.Since A5=?A6and A7=?A8the result e?ectively involves6 independent amplitudes as required by time reversal invariance[4].When comparing with other expressions in the literature[1,2,4,30],one has to keep in mind that they usually are given in the laboratory frame.This observation accounts,for example,for the di?erence of the LET for A1and A3of real Compton scattering in the lab and the CM frame.Note also that the1/| q′|singularity disappears in the real Compton scattering limit,sinceω=| q|= | q′|in this case.

In conclusion,the low-energy behavior of the VCS matrix element for e?+p→e?+p+γ, expanded up to order O(2)in| q|and| q′|,contains,in addition to the structure coe?cients that enter into real Compton scattering,also the electric mean square radius and the electric and magnetic Sachs form factors in the spacelike region which all can be obtained from electron scattering o?the proton.For the reactionγ+p→p+e+e?[13],the analogous information for timelike momentum transfers is needed.However,here one has to keep in mind that this information is not directly accessible for0

IV.THE BORN TERMS

From the above discussion in a particular representation one might be tempted to con-clude that the leading terms for VCS–on the operator level or for the amplitude–are in general simply given by“Born terms”.However,some caveats are in order.First,one has to keep in mind that the low-energy behavior was obtained from considerations involving the most general ansatz for the truncated four-point Green’s function.All terms one can think of are included into class A and class B.In,e.g.,the derivation of[3]of the LET for Compton scattering,where no special representation is chosen,it is clearly shown that both class A and class B terms are needed.Implicit in all derivations is of course the assumption that a description in terms of observable asymptotic hadronic degrees of freedom is su?cient and complete at low energies.Even though subnucleonic degrees of freedom,quarks and gluons,are ultimately the origin of the structure of the nucleon,it was shown[32,33]that an e?ective?eld-theory approach[34–37]in terms of hadrons is meaningful at low energies, thus allowing a classi?cation into class A and class B.

Secondly,there is an ambiguity concerning what exactly is meant by“Born terms”,once phenomenological form factors are introduced and the result is not obtained from a micro-scopic Lagrangian.We will illustrate this ambiguity by considering di?erent representations of the photon-nucleon vertex.All representations contain the same information concern-ing the electromagnetic structure of the on-shell nucleon,as obtained in electron-nucleon scattering,but di?er in the half-o?-shell situation encountered in the s-and u-channel pole terms of Compton scattering.This di?erence is,of course,accompanied by di?erent class B terms such that the total result is the same.

To explain the above points in more detail,we will?rst reconsider the most general expression for class A and,without going to a special representation,identify those terms which contain the irregular contribution for qμ→0or q′μ→0.We?nd that these contribu-tions can be expressed in terms of on-shell quantities,in our case the Dirac and Pauli form factors F1and F2.We then show that the use of on-shell equivalent electromagnetic vertices

gives rise to the same results for the VCS matrix element as far as the irregular terms are concerned,but not every choice will result in“Born terms”which are gauge invariant.

A.Irregular Contribution to the VCS matrix element

When calculating Vμνs

i s f ,the irregular contribution originates from the singularities of the

propagators S(p i+q)and S(p i?q′)in the s and u channel,respectively.To be speci?c, below pion-production threshold the renormalized,full propagator can be written as S(p+q)=S F(p+q)+regular terms,(p+q)2<(M+mπ)2,(4.1) where S F(p)denotes the free propagator of a nucleon with mass M.By“regular terms”we mean terms which have a well-de?ned,non-singular limit for qμ→0.Thus,as long as we are only interested in the irregular terms,we can simply replace the full,renormalized propagator by the free Feynman propagator,S F.

The most general form of the irreducible,electromagnetic vertex of the nucleon can be expressed in terms of12operators and associated form functions[38,39].These functions depend on three scalar variables,e.g.,the squared momentum transfer and the invariant masses of the initial and?nal nucleon lines.A convenient parametrization ofΓμ(p f,p i)is given by

Γμ(p f,p i)= α,β=+,?Λα(p f) γμFαβ1+iσμνqνM Fαβ3 Λβ(p i),Fαβi=Fαβi(q2,p2f,p2i),

(4.2) where q=p f?p i,andΛ±(p)=(M±p/)/2M.We have chosen a form which di?ers slightly from the convention of[39]in the de?nition of the projection operators and the normalization of the F3form functions.We only need the following on-shell properties of the form functions:

F++ 1(q2,M2,M2)=F1(q2),F++

2

(q2,M2,M2)=F2(q2),F++

3

(q2,M2,M2)=F3(q2)=0,

(4.3)

where F1and F2are the standard Dirac and Pauli form factors and F3vanishes because of time-reversal invariance.One can also show that F3(q2)vanishes due to current conservation.

We now systematically isolate the irregular part of,e.g.,the s-channel pole diagram,2

VμνA,s=ˉu(p f)Γν(p f,p f+q′)iS(p i+q)Γμ(p i+q,p i)u(p i)

≈ˉu(p f)Γν(p f,p f+q′)iS F(p i+q)Γμ(p i+q,p i)u(p i),

where we made use of Eq.(4.1),and where the symbol≈denotes equality up to regular terms. We now insert Eq.(4.2)forΓμ(p i+q,p i)and useΛ?(p i)u(p i)=0andΛ+(p i)u(p i)=u(p i) to obtain

VμνA,s≈ˉu(p f)Γν(p f,p f+q′)iS F(p i+q)(Λ+(p i+q)(γμF++1(q2,s,M2)+···)

+Λ?(p i+q)(γμF?+

1

(q2,s,M2)+···))u(p i), where s=(p i+q)2.Since S F(p i+q)Λ?(p i+q)results in a regular term,we have VμνA,s≈iˉu(p f)Γν(p f,p f+q′)p i/+q/+M

2M

F2(q2))u(p i),

where we made use of Eq.(4.3).UsingΛ+(p i+q)=1?Λ?(p i+q)and then repeating the same procedure forΓν(p f,p f+q′)we?nally obtain

VμνA,s≈ˉu(p f,s f)ΓνF1,F2(?q′)iS F(p i+q)ΓμF1,F2(q)u(p i,s i),(4.4) where we introduced the abbreviation

ΓμF

1,F2(q)=γμF1(q2)+i

σμνqν

ΓμG

E,G M

(p f,p i)= 1?q22M G E(q2)+γμP/q/?q/P/γμ

2M

H2(q2),(4.9) where P=p i+p f and

G E=F1+

q2

4M2 ?1p2

f?p2i

2M

H2(q2)=S?1F(p f)?S?1F(p i).(4.12)

The“Born terms”calculated with these electromagnetic vertices and free nucleon prop-agators are

VμνX=ˉu(p f,s f)(ΓνX(p f,p f+q′)iS F(p i+q)ΓμX(p i+q,p i)

+ΓμX(p f,p f?q)iS F(p i?q′)ΓνX(p i?q′,p i))u(p i,s i),(4.13) where X denotes either the vertex of Eq.(4.8)in terms of G E,G M or the vertex of Eq.(4.9) involving H1,H2,respectively.These“Born terms”are by construction crossing symmetric but in both cases not gauge invariant,

qμVμνG

E,G M

=i 1?q2

2M +

q/

2M

+

q/

correctly reproduce the on-shell electromagnetic current of the nucleon will yield the same irregular contribution to the VCS matrix element.The key to the proof of this statement is the fact that any current operator which transforms as a Lorentz four-vector can be brought into the form of Eq.(4.2).On-shell equivalence then amounts to the constraint that all

operators have the same on-shell limit of the F++

i form functions.In general,no statement

can be made for either the other form functions or o?-shell kinematics.3However,as we have seen above,the irregular contribution of class A,and thus of the total VCS matrix element,

only involves the on-shell information contained in F++

1(q2,M2,M2)and F++

2

(q2,M2,M2).

Any information beyond this will give rise to regular terms.Thus the above statement is true.

To illustrate the above,we bring for example the current operators of Eq.(4.5)and(4.9), involving F1and F2or H1and H2,respectively,into the general form of Eq.(4.2).By using 1=Λ+(p)+Λ?(p),the result for the commonly used form of Eq.(4.5)is given by

Fαβ1(q2,p2f,p2i)=F1(q2),Fαβ2(q2,p2f,p2i)=F2(q2),Fαβ3(q2,p2f,p2i)=0,α,β=+,?.

(4.16) For the vertex given in Eq.(4.9),we use{γμ,γν}=2gμνand momentum conservation at the vertex to rewrite

(p i+p f)μ

2M?i

σμνqν

2M

H2(q2) Λ+(p i)

+Λ?(p f) γμH1(q2)+iσμνqν

2M

H2(q2) Λ?(p i)

+Λ?(p f) γμ(H1(q2)+H2(q2))+iσμνqν

3Further conditions on the form functions can be derived from the Ward-Takahashi identity and discrete symmetry requirements[38,39].

di?er among each other through regular terms.Furthermore,such“Born terms”are,in general,not gauge invariant;an exception is the commonly used form involving the Dirac and Pauli form factors F1and F2.“Generalized Born terms”which are made gauge invariant by hand through an ad hoc prescription also di?er by regular terms.

Important starting point for the derivation of the LET are the irregular terms.It is thus also possible to split the total VCS amplitude into“Born terms”plus“rest”,instead of class A and B amplitudes,to arrive at the same result for the LET,i.e.up to and including terms linear in the photon three-momenta.In general,this result will have contributions from“Born terms”and the“rest”amplitude.If one uses a“generalized Born amplitude”, all the terms appearing in the LET are due to the expansion of the Born amplitude.It is a well-known feature of soft-photon theorems that they cannot make statements about terms which are separately gauge invariant[41–43].One has to keep this in mind when discussing the structure-dependent higher-order terms of VCS,i.e.,one needs to specify which Born or class-A terms have been separated.For example,in[14]the“Born terms”involving F1and F2where separated since they provide without any further manipulation a gauge-invariant amplitude.Then the residual part with respect to these particular“Born terms”was parametrized in terms of generalized polarizabilities.A natural question to ask is what would have happened had one separated a di?erent choice of“generalized Born terms”and de?ned generalized polarizabilities in an analogous fashion with respect to the corresponding residual amplitude.Obviously one would,in general,have found di?erent numerical values for the new generalized polarizabilities in order to obtain the same total result.

V.CONCLUSIONS

In studying the structure of composite strongly interacting systems the electromagnetic interaction has been the traditional and precise tool of investigation.In scattering of elec-trons from a nucleon,our knowledge is restricted to two form factors that we can extract from experiments.Even though we have not yet been able to fully explain this information on the basis of QCD,it is important to look for other observables allowing us to test ap-proximations to the exact QCD solution and e?ective,QCD inspired models.Such e?ective models are expected to work especially at low energies.The electron accelerators now make it possible to study virtual Compton scattering,which is clearly more powerful in probing the nucleon than the scattering of real photons.In analyzing Compton scattering it is im-portant to know how much of the prediction is not a true test of a model,but?xed due to general principles.These model-independent predictions for virtual Compton scattering were the main topic of our discussion.

The interest in virtual Compton scattering has also been due to another aspect:When studying reactions on a nucleus,such as(e,e′p),the nucleon interacting with the electro-magnetic probe is necessarily o?its mass shell.We have no model-independent information for the behavior of such a nucleon and any conclusion about genuine medium modi?cations must be based on?rm theoretical ground how to deal with a single nucleon under these kine-matical circumstances.In fact,such a discussion depends very much on what one chooses as an interpolating?eld for the intermediate,not observed nucleon.This clearly makes the “o?-shell behavior”of the nucleon representation dependent and unobservable.We discussed how certain features of the o?-shell electromagnetic vertex of the nucleon can be shifted into

irreducible,reaction-speci?c terms for the reaction amplitude.Two-step reactions on a free nucleon–like(virtual)Compton scattering–allow us to test many aspects of dealing with

an intermediate,o?-shell nucleon under simpler circumstances,without complications from, e.g.,exchange currents or?nal state interactions.Understanding these aspects on the single

nucleon level would seem a prerequisite before any exotic claims can be made for nuclear reactions.

We have studied the virtual Compton scattering?rst on the operator https://www.wendangku.net/doc/c53080156.html,ing the

requirement of gauge invariance,as expressed by the Ward-Takahashi identity,we derived constraints for the operator that determine terms up to and including linear in the four-momenta q and q′.Also,we showed that on the operator level terms involving terms de-

pending only on q or only on q′are determined model independently in terms of on-shell properties of the nucleon.

To obtain these results,we used the method of Gell-Mann and Goldberger,by splitting the contributions into general pole terms(class A)and the one-particle irreducible two-photon contributions(class B).We calculated class A below pion-production threshold in

the framework of a speci?c representation for the most general e?ective Lagrangian compat-ible with Lorentz invariance,gauge invariance and discrete symmetries.This approach was

introduced in[4]as a method for writing the general structure of the Compton scattering amplitude in a way that allows best to discuss its low-energy behavior.In this connection, we also showed the origin for a commonly used form of the electromagetic vertex of the

nucleon and stated the consistency conditions for its use.

After discussing the leading terms of the VCS operator,we considered the matrix element forγ?p→γp in the photon-nucleon CM frame.We found that the VCS amplitude up to

and including terms linear in the initial and?nal photon three-momentum can be expressed in terms of information one can obtain from electron-proton scattering.This is the result

analogous to the LET for the real Compton scattering amplitude.As we also showed,the next order–terms involving| q′|2,| q′|| q|,and| q|2–is also completely speci?ed but now requires in addition also the electromagnetic polarizabilitiesˉαandˉβencountered in real

Compton scattering.In other words,new structure-dependent information can only appear at order three or higher in the three-momenta.Our results concern the expansion in terms of powers of both the initial and?nal photon momentum.This allowed us to determine more terms than in[14],where only the leading terms in the?nal momentum were concerned.On the other hand,by expanding in both momenta,the range of applicability is smaller since both kinematical variables should be small.

We then considered di?erent commonly used methods to include the on-shell information contained in the electromagnetic form factors in a“Born-term”calculation of the VCS matrix element.The fact that the“Born terms”calculated with F1and F2are gauge invariant is not trivial,since the vertex and free propagator do not satisfy the Ward-Takahashi identity.We explained why di?erent on-shell equivalent forms for the electromagnetic vertex operator lead to the same irregular contribution in the VCS matrix element.We emphasized the importance of stating with respect to which pole terms the structure-dependent terms are de?ned.

Using only gauge invariance,Lorentz invariance,crossing symmetry,and the discrete symmetries,we were able to make statements about the low-energy behavior up to O(2). Further conclusions can be reached by also taking into account the constraints imposed by

chiral symmetry.This would most naturally be done in the framework of Chiral Perturbation Theory.In particular,predictions for the higher-order terms could be obtained.

VI.ACKNOWLEDGEMENTS

The work of J.H.K.is part of the research program of the Foundation for Fundamental Research of Matter(FOM)and the National Organization for Scienti?c Research(NWO). The collaboration between NIKHEF and the Institute for Nuclear Physics at Mainz is sup-ported in part by a grant from NATO.A.Yu.K.thanks the Theory Group of NIKHEF-K for the kind hospitality and the Netherlands Organization for International Cooperation in Higher Education(NUFFIC)for?nancial support.A.Yu.K.also would like to thank SFB 201of the Deutsche Forschungsgemeinschaft for its hospitality and?nancial support during his stay at Mainz.S.S.would like to thank H.W.Fearing for useful discussions on the soft-photon approximation.

APPENDIX A:

In this appendix we outline the calculation of the transverse and longitudinal functions A i of Eq.(3.14)and(3.15),respectively.For that purpose we split Mμinto its contributions from classes A and B,Mμ=MμA+MμB.If we introduce

F(?q′)=?/′?(1+

κ

s?M2+

p i·?′?

s?M2

+

Γμeff(q)?/′?q/′

M p f·?′?q/′Γμ

eff

(q)

u?M2 +κ

The following considerations will be carried out in the center-of-mass (CM)frame,where the four-momenta are given by q μ=(q 0,0,0,| q |),p μi =(E q ,? q ),q ′μ=(| q ′|, q ′),and p μf =

(E q ′,? q ′).From energy conservation,q 0+E q =| q ′|+E q ′,we infer that we may choose | q |,| q ′|,and z =?q ·?q ′as a set of independent variables.In terms of the CM variables the denominators of Eq.(A3)are proportional to | q ′|,

s ?M 2=2| q ′|(E q ′+| q ′|),u ?M 2=?2| q ′|(E q +z | q |),(A5)

and thus M μ

A can be written as

M μA =a

μ

(q,q ′)

2(E q ′+| q ′|)

?Γμ

eff (q )?/′?n /′

M Γμ

eff (q )n /′2M (?/′?Γμeff (q )+Γμ

eff (q )?/′?) u (? q ,s i ),

(A8)

where we introduced n ′μ=(1,?q ′),and

K ( q )=? q · ?′?

翻译技巧和经验第17期Virtual reality 该如何翻译

近一个时期来，Virtual Reality (VR)一词在报刊等新闻媒体上频频出现，十分活跃，从而也引出了对VR 该怎么好的问题。据《光明日报》载，VR 的译名计有：“虚拟实在”（1996 年,10 月28 日）“临境”、“虚实”、“电象”、“虚拟境象”以及“灵境”（1997 年1 月16 日）；此外，还有最常见的“虚拟现实”（如《文汇报》1997 年3 月12 日“小辞典”）。 对VR 究竟怎么理解和怎么翻译，笔者不敢妄下结论。不过，从最近一期外刊上读到的一篇文章（载于 Jane's DEFENCE'97，作者为 Ian Strachan，Editor，Jane's Simulationand Training Systems）在这两方面都可给人以不小的启发。为帮助说明问题，现将此文中有关的部分摘录于下。文章标题和引语是： VIRTUALLY REAL Virtual reality is rapidly becoming the training tool o f the 21stCentury - but what exactly is it? 将此翻译过来大致是： 可说是真的Virtual reality 正在迅速成为二十一世纪的训练器具--但它确切的含义是什么呢？ 文中在讲到 virtual 一词的定义时是这样说的：One dictionary definition of "virtual" is"something which is unreal but can be considered as being real for some purposes." 文章接着对 VR 作了几种实质性的释义： Cyber-something? To some, VR is putting on a Cyber-helmet (whatever that is) and involvestotal immersion (whatever that means) in cyberspace (where ver that is). Collimation? To some, VR is a view through a collimated display system. Tactile sensors? To some, VR implies tactile simulation as well a visual syst em. 对以上这些解释或释义，Strachan 认为它们都是对的：Which of these interpretations of VR is right? Ibelieve that they all are.（对VR 的这些诠释中哪个是对的？我相信它们都是对的。）为什么都对？Strachan 说那是因为 English is a living language and insistence on any preciseinterpretation of VR is narrow and not in accordance with what has already become commo nusage.（英语是一门活的语言，而坚持对 VR 作任何精确的诠释是狭隘观念，也不符合约定俗成。） 通过以上介绍，我们似乎能够得出这样的结论：各位专家学者所给的译名都是对的。因为，汉语也是一门活的语言，坚持对 Virtual Reality 作任何精确的翻译是狭隘观念。但是，另一方面，则仍有必要从以上译名中挑选出一两个最好的来，或者也可以另起炉灶，如果还有更好的话。那么，我们认为“虚拟现实”和“虚拟实在”与原文更加“形似”。其它几个译名虽各有千秋，但总嫌“神似”有余，而“形似”不足。词和词语的翻译同句子和篇章一样应力求“神形兼具”。在这种情况下，找到一个最理想的译名实属必要。 借鉴以上众多译名，并综合Strachan 所做的分析和解释，经过反复比较和推敲，我们认为 VirtualReality 视情可译为“拟真技术”、“拟真”或“虚拟真实”。这几个译名与所介绍的那些在含义上大致相同而表述略有不同，或许是最为得体的。理由如下： 第一，这么译十分符合英文词语的本义。Virtual 在这里意为 in effect, though not in fact; not such infact but capable of being consider

(完整版)初级语法总结(标准日本语初级上下册)

1.名［场所］名［物/人］力*笳◎去歹/「仮歹 名［物/人］总名［场所］笳◎去歹/「仮歹 意思：“ ~有~ ”“~在~”，此语法句型重点在于助词“心‘ 例：部屋忙机力?笳◎去歹。 机总部屋 2.疑问词+哲+动(否定) 意思：表示全面否定 例：教室忙疋沁o 3?“壁①上”意思是墙壁上方的天棚，“壁才是指墙壁上 例：壁 4. ( 1)名［时间］动 表示动作发生的时间时，要在具体的时间词语后面加上助词“V”，这个一个时间点 例：森总心比7時V 起吉去To 注意：只有在包含具体数字的时间时后续助词“V”，比如“ 3月14 日V, 2008年V”；星期后面可加V,比如“月曜日V” ,也可以不加V;但是“今年、今、昨日、明日、来年、去年”等词后面不能加V。 此外：表示一定时间内进行若干次动作时，使用句型“名［时间］V 名［次数］+动” 例：李1週間V2回7°-^^行吉去T。 (2)名［时间段］动：说明动作、状态的持续时间，不能加“ V” 例：李毎日7時間働^^L^o (PS: “V”的更多用法总结请看初级上第15课) ( 3)名V 【用途】【基准】 表示用途时，前接说明用途的名词，后面一般是使"去T等动词 表示基准时，前名词是基准，后面一般是表示评价的形容词。 例：--乙①写真总何V使"去T力、。「用途」 --申請V使"去T。「用途」 乙①本总大人V易L^^ToL力'L、子供V总難L^^To 「基准」 X —X—力*近乙買⑴物V便利^To 「基准」 (4)动(基本形) OV 【用途】【基准】：使用与上述( 3)一样 例：乙O写真求一卜总申請T^OV 使"去T。 ^OV>^3>^ 買“物T^OV 便利^To (5)小句1 (简体形)OV，小句2:名/形動+肚+OV 表示在“小句1 ”的情况下发生“小句2”的情况不符合常识常理，翻译为“尽管…还是…，虽

和声学基本知识

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新版标日初级·语法解释 第2课 1.これ/それ/あれは　[名]です 相当于汉语“这是/那是~”。 “これ”“それ”“あれ”是指代事物的词，相当于汉语“这、这个”“那、那个”。用法如下： （1）说话人与听话人有一点距离，面对面时： ·これ：距离说话人较近的事物 ·それ：距离听话人较近的事物 ·あれ：距离说话人和听话人都较远的事物 （2）说话人和听话人处于同一位置，面向同一方向时： ·これ：距离说话人、听话人较近的事物 ·それ：距离说话人、听话人较远的事物 ·あれ：距离说话人、听话人更远的事物 例：これは　本です。 それは　テレビです。 あれは　パソコンですか。 2.だれですか/何ですか 相当于汉语“~是什么？/~是谁？”。不知道是什么人是用“だれ”，不知道是什么东西时用“何”。句尾后续助词“か”，读升调。例：それは　何ですか。 あの人は　だれですか。 注意：“だれ”的礼貌说法是“どなた”。对方与自己是同辈、地位相当或地位较低时用“だれ”。对方比自己年长或地位高时用“どなた”。 例：吉田さんは　どなたですか。 3.[名]の[名]【所属】 助词“の”连接名词和名词，表示所属。 例：私のかぎ。 小野さんの傘。 4.この/その/あの[名]は　[名]です 相当于汉语“这个/那个~是~”。修饰名词时，要用“この”“その”“あの”。其表示的位置关系与“これ”“それ”“あれ”相同。例：このカメラは　私のです。 その傘は　小野さんのです。 あの車は　だれのですか。 5.どれ/どの[名] 三个以上的事物中，不能确定哪一个时用疑问词“どれ”“どの”。单独使用时用“どれ”，修饰名词时用“どの”。 例：森さんのかばんは　どれですか。 長島さんの靴は　どれですか。 私の机は　どの机ですか 扩展：100以下数字 0　れい/ぜろ 1　いち 2　に 3　さん 4　し/よん　 5　ご 6　ろく

Virtual Reality

Virtual Reality is a kind of computer simulation technique that is able to assist people experience virtual world by using special lens in head-sets. It can be applied in plenty of areas such as military training and medical science. For military training, with head-sets and some additional facilities, special situations like forests, desert and ruin can be virtually simulated. Then soldiers will feel that they are in real battlefield and complete tasks to improve themselves better. In terms of medical science, when facing dangerous operations, the data of patients’ tissues and organs can be input in the computer in advance. Then computer can create the structure of the patient in details. Surgeons wearing head-sets can practice on such simulation to get more familiar with the operation so as to handle unexpected dangerous condition well and avoid some mistakes. Virtual reality (VR) typically refers to computer technologies that use virtual reality headsets, sometimes in combination with physical spaces or multi-projected environments, to generate realistic images, sounds and other sensations that simulates a user's physical presence in a virtual or imaginary environment. A person using virtual reality equipment is able to "look around" the artificial world, and with high quality VR move about in it and interact with virtual features or items. VR headsets are head-mounted goggles with a screen in front of the eyes. Programs may include audio and sounds through speakers or headphones. VR systems that include transmission of vibrations and other sensations to the user through a game controller or other devices are known as haptic systems. This tactile information is generally known as force feedback in medical, video gaming and military training applications. Virtual reality also refers to remote communication environments which provide a virtual presence of users with through telepresence and telexistence or the use of a virtual artifact (VA). The immersive environment can be similar to the real world in order to create a lifelike experience grounded in reality or sci-fi. Augmented reality systems may also be considered a form of VR that layers virtual information over a live camera feed into a headset, or through a smart phone or tablet device.