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MKPH-T-96-4Virtual Compton scattering offthe nucleon at low energies S.Scherer ∗Institut f¨u r Kernphysik,Johannes Gutenberg-Universit¨a t,55099Mainz,Germany A.Yu.Korchin National Scientific 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 offthe nucleon.Using 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 final photon,respectively.The model-independent result is expresd in terms of the electromagnetic form factors of the free ,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 u 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
Typet using REVT E X
I.INTRODUCTION
Low-energy theorems(LET)play an important role in studies of properties of parti-cles.Bad 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 the general principles they must reproduce the predictions of the low-energy theorem.On the other hand,the low-energy theorems also provide uful constraints for experiments.Experimental studies designed to investigate particle properties beyond the global quantities and to distinguish between different models must be carried out with suf-ficient accuracy at low energies to be nsitive to the higher-order terms not predicted by the theorems.Another option is,of cour,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 sc
attering”(CS)of real photons offa nucleon[1–3].Bad on the requirement of gauge invariance,Lorentz invariance,and crossing symmetry,it specifies the terms in the low-energy scattering amplitude up to and including terms linear in the photon momentum.The coefficients of this expansion are expresd 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 nsitive to terms that depend on details of the structure of the nucleon beyond the global properties.Terms of cond 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(e,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 ud.A virtual photon allows one to vary the three-momentum and energy transfer to the target independently.Therefore it has recently been propod to also u“virtual Compton scattering”(VCS)as a means to study the structure of the nucleon[5–7].The propod reaction is p(e,e′p)γ,i.e.,in addition to the scattered electron also the recoiling proton is det
ected to completely determine the kinematics of thefinal state consisting of a real photon and a proton.It is the purpo of this note to extend the standard low-energy theorem for Compton scattering of real photons to the general ca where one or both photons are virtual.The latter would be the , in the reaction e−+p→e−+(e−+e+)+p.We will refer to both possibilities as“VCS”.
There are veral different approaches to derive the LET for Compton scattering of real photons.One wasfirst ud by Low[1].It made u 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 confud with Feynman diagrams–all intermediate states are on their mass shell[8].Lorentz invariance and gauge invariance then allow a prediction for the amplitude tofirst order in the frequency.Another approach[2,3],first ud 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 class,A and B:class A consists of one-particle-reducible contributions
that can be built up from dresd photon-nucleon vertices and dresd nucleon propagators. Class B contains all one-particle-irreducible two-photon diagrams,where the cond photon couples into the dresd vertex of thefirst one.Application of two Ward-Takahashi identities, one relating the phot
on-nucleon vertex operator to the nucleon propagator,the other relating the irreducible two-photon vertex to the dresd one-photon vertex,then lead to the same result for the leading powers of the low-energy amplitude.One can u yet another technique introduced by Low[9]to describe bremsstrahlung process.This technique relies on the obrvation 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 modified form, also been widely ud 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 infirst-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 effects 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.The 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 purpo of this work is to identify,in an analogous form to the real CS ca,tho 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 notfixed by this LET for simplicity as“model dependent”.By introducing additional constraints,such as chiral symmetry,also statements about the terms become possible.This is,however,beyond the scope of the prent work.In our study of low-energy virtual Compton scattering,below the ont 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 effective La-grangian approach.Class A is obtained in the framework of a general effective Lagrangian describing the interaction of a single nucleon with the electromagneticfield[4].In the spe-c
ific reprentation we choo,this turns out to be a simple covariant and gauge invariant “modified 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 andfinal photon momenta.Furthermore,with the help of crossing symmetry a definite 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 ari in their definition,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 ction we will define the Green’s functions and the kinematical variables relevant for the discussion of VCS offthe proton.We will consider the constraints impod 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 defined as
Gµαβ(x,y,z)=<0|T(Ψα(x)¯Ψβ(y)Jµ(z))|0>,(2.1) Gµναβ(w,x,y,z)=<0|T(Ψα(w)¯Ψβ(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 interpolatingfield of the proton; T denotes the covariant time-ordered product[18].Electromagnetic current conrvation,∂µJµ=0,and the equal-time commutation relation of the charge density operator with the protonfield,
[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 defined 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 andfinal 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 ,p i and p f.In particular,it is not assumed that the 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 off-mass-shell ca.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 inver 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 indices.Using the definitions 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 class,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 paration into the two class 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,expresd 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 parately[2].Using the Ward-Takahashi identity,Eq.(2.8),one obtains the following constraint for class A as impod 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 cour,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 cond gauge-invariance condition, obtained by contracting with q′ν,is automatically satisfied 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 the momenta are independent variables only for the complete off-shell ,if one allows for arbitrary values of p2i and p2f.This will be important when making u of the constraints impod by gauge invariance.Using Lorentz invariance,gauge invariance,crossing symmetry,parity and time-reversal invariance,it was shown in[12]that the generalΓµνfor VCS offa free nucleon with both photons virtual consists of18independent operator structures.The functions associated with each operator depend on four Lorentz ,q2,q′2,ν=P·q=P·q′, and t=(p i−p f)2.When allowing the external nucleon lines to be offtheir 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 discusd 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 coefficients are4×4matrices and can be expresd 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 ca the pion-production threshold;we refer the reader ,[20]where a similar discussion for the ca 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 us a particular Lorentz system nor a specific gauge.When referring to powers of q or q′,we mean the ones coming from the Dirac structures and their associated functions.This is different 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 ction by noting that the above framework can easily be applied to VCS offa spin-0particle,such as the pion.In that caΓµν,of cour,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].
