Generalized polarizabilities of the nucleon studied in the linear sigma model (II)

a r X i v :n u c l -t h /9705010v 1 5 M a y 1997


Generalized polarizabilities of the nucleon studied in the linear

sigma model (II)

A.Metz and D.Drechsel

Institut f¨u r Kernphysik,Johannes Gutenberg-Universit¨a t Mainz,J.J.Becher-Weg 45,D-55099

Mainz,Germany (February 9,2008)


In a previous paper virtual Compton scattering o?the nucleon has been investigated in the one–loop approximation of the linear sigma model in or-der to determine the 3scalar generalized polarizabilities.We have now ex-tended this work and calculated the 7vector polarizabilities showing up in the spin–dependent amplitude of virtual Compton scattering.The results ful?ll 3model–independent constraints recently to the constituent quark model there exist enormous di?erences for some of the vec-tor polarizabilities.At vanishing three–momentum of the virtual photon,the analytical results of the sigma model and of chiral perturbation theory can be related.The in?uence of the π0exchange in the t channel has been discussed in some detail.Besides,the vector polarizabilities determine 2linear com-binations of the third order spin–polarizabilities appearing in real Compton scattering.12.39.Fe,13.60.Fz

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Recently,it has been proposed to study the structure of the proton by the reaction

p(e,e′p)γ.The reason is that this process contains,in addition to electron bremsstrahlung

(Bethe–Heitler scattering),the amplitude of virtual Compton scattering(VCS)o?the pro-ton,γ?+p→γ+p.In general,Compton scattering as a two step process allows to extract

information on the excitation spectrum of the target.

An overview of the various aspects of VCS can be found in Ref.[1].VCS is of particular interest below pion production threshold,where the information about the excited states of

the nucleon can be parametrized by means of the generalized electromagnetic polarizabilities. These generalized polarizabilities emerge as coe?cients if the non Born amplitude of VCS is

expanded in terms of the?nal photon energyω′=0.As has been demonstrated by Guichon

et al.[2],the leading term of such an expansion contains10generalized polarizabilities, 3of them in the spin–independent amplitude(scalar polarizabilities)and7in the spin–

?ip amplitude(vector polarizabilities).In contrast to real Compton scattering(RCS),the

generalized polarizabilities in VCS are functions depending on the four–momentum transfer Q2.

At present,three experimental programs are under way to determine the generalized polarizabilities.At MAMI,data have been taken at Q2=0.33GeV2[3].There also exist

plans to investigate VCS at even lower values of Q2at MIT–Bates[4],while the activities at

Je?erson Lab will concentrate on the region of higher Q2[5].In an unpolarized experiment only3independent linear combinations of the polarizabilities can be determined[2,10].A

separate measurement of all polarizabilities is only possible by use of double polarization observables,e.g.,the reaction p( e,e′ p)γ[7].Moreover,a careful treatment of radiative

corrections is unavoidable as their contributions are comparable to the polarizability e?ects


Regarding the expansion inω′for RCS,higher order terms can not be neglected at

photon energies larger than about80MeV.In order to guarantee the dominance of the

leading order term,the energy of the real photon should also satisfyω′?E?m N,where E is the energy of the initial nucleon in the cm system and m N the nucleon mass[6],i.e.,

ω′has to be much smaller than the three–momentum| q|of the virtual photon(see section II).Ifω′and| q|are of the same order of magnitude,the amplitude has to be expanded in both variables(see Ref.[9]for the case of a spin0target).The extension of this work to the nucleon leads to36independent low energy constants entering the non–Born VCS amplitude to?fth order[10].While it is certainly hopeless to measure these parameters,we want to emphasize the importance of a reliable estimate of the higher order terms,except for very small values ofω′.

Recently,it has been shown that the combined symmetry of charge conjugation and

nucleon crossing results in unexpected relations between the VCS multipoles,beyond the usual constraints of parity and angular momentum conservation[6,10].As a consequence, only2independent scalar and4independent vector polarizabilities exist.

The generalized polarizabilities of the proton were?rst predicted by use of the constituent quark model(CQM)[2,11].Further calculations in an e?ective Lagrangian model[12],the linear sigma model(LSM)[13],chiral perturbation theory(ChPT)[14,15],and the Skyrme model[16]focused on the scalar polarizabilities and the spin–independent VCS amplitude.

In the meantime,the vector polarizabilities have been evaluated on the basis of ChPT[17,18] and in a coupled–channel unitary model[19].

The present contribution completes our earlier work on the LSM[13]by presenting the results for the vector polarizabilities.Since the model is chirally invariant,we expect similar results as in ChPT for Q2→0,and in addition a reasonable estimate for the Q2dependence of the polarizabilities.


For convenience we brie?y repeat the de?nition of the kinematical variables for the VCS reaction[13,20]

γ?(qμ,εμ)+N(pμ,Sμ)→γ(q′μ,ε′μ)+N′(p′μ,S′μ),with q2=?Q2<0,q′2=0.(1) The cm energies and three–momenta of the involved particles are denoted by qμ= (ω, q),q′μ=(ω′, q′),pμ=(E,? q),p′μ=(E′,? q′).The polarization vectors of both photons can be chosen to be spacelike,i.e.,εμ=(0, ε)andε′μ=(0, ε′),noting that the time–like component of the virtual photon can be eliminated by current conservation.While the outgoing photon is purely transverse(helicityλ=±1),the incoming photon has also longitudinal polarization(λ=0),denoted byεμ(λ=0)=(0,?q).In the following we also use the de?nitions

ω0=ω|ω′=0=m N?E=m N?



q·p.In the following we will use an equivalent,non–covariant set of arguments,ω′,ˉq and cos?=?q·?q′.The orthonormal coordinate system

?e x=?q′?cos??q


,?e z=?q(7)

is?xed by the momenta of the photons.


The scattering amplitude is decomposed into the Born contribution T B and the non–Born or residual contribution T R,

T=T B+T R,(8) and the generalized polarizabilities are given by a low energy expansion of T R with respect to ω′[2].As has been pointed out in Refs.[2,23],the splitting in(8)is not unique.Contributions which are regular in the limitω′→0and separately gauge invariant can be shifted from the Born amplitude to the residual amplitude and vice versa.Therefore,when calculating the generalized polarizabilities,one has to specify which Born terms have been subtracted from the full amplitude,since di?erent Born terms lead to di?erent numerical values of the polarizabilities.In our calculation we use the Born amplitude as de?ned in[13].

To be speci?c,the generalized polarizabilities have been derived from the multipoles

H(ρ′L′,ρL)S R (ω′,ˉq)of the residual amplitude[2].In this notationρ(ρ′)represents the charge

(0),magnetic(1)or electric(2)nature of the initial(?nal)photon,while L(L′)refers to its angular momentum.The quantum number S characterizes the no spin–?ip(S=0)and the spin–?ip(S=1)transitions.

For the de?nition of the generalized polarizabilities it is necessary to know the low energy behaviour of the multipoles if(ω′,ˉq)→(0,0).While the Coulomb and the magnetic transi-tions are well behaved in that limit,the electric transitions of the virtual photon depend on the path along which the origin in theω′-ˉq-plane is approached.Since we are only interested in the leading order term in| q′|=ω′,we can express the electric transitions of the outgoing photon by the Coulomb transitions[2],



(ω′,ˉq)=? L′H(0L′,ρL)S R(ω′,ˉq)+O(ω′L′+1).(9) The corresponding equation for the incoming photon reads



(ω′,ˉq)=? Lω2L+1

The generalized polarizabilities are de?ned according to [2]as


(ρ′L ′,ρL )S

(ˉq )=


ω′L ′ˉ

q L +1?H (ρ′L ′,L )S R (ω′,ˉq )




In the following we restrict ourselves to electric and magnetic dipole transitions of the

real photon,which is equivalent to keep only the leading,linear term in ω′of the residual amplitude T R .In that case there exist,due to conservation of parity and angular momentum,3scalar multipoles (S =0)and 7vector multipoles (S =1).

To evaluate the vector polarizabilities it is su?cient to treat the following 5out of the 9spin–dependent amplitudes of eqs.(5)and (6):

b l R,1=e 2

E sin ?√2



+O (ω′2),(13)b l R,3=e 2E 3ˉ

q ?P (11,00)1


√4π m N ω′


2ω0P (01,01)1+

√3?P (01,1)1 +O (ω′2),(15)b t ′


=e 2

E sin ?332ω0ˉq P (11,02)1+√√4π m N ω′4

2ω0P (01,01)1+√3?P (01,1)1 +cos ? ˉq P (11,11)1


ˉq 3?P

(11,2)1 +O (ω′2).(17)Because of their angular dependence,b l R,1and b t ′

R,3contain two independent linear combina-tions of the polarizabilities.

Based upon gauge invariance,Lorentz invariance,parity conservation,and charge con-jugation in connection with nucleon crossing,it was shown in Ref.[10]that only four inde-pendent vector polarizabilities exist.This fact is re?ected by the equations

0=P (11,11)1(ˉq )+


ω0P (11,02)1(ˉq )+


ˉq 2?P (11,2)1(ˉq ),(18)



(ˉq )+2ˉq 2

2ˉq 2

P (01,12)1

(ˉq )+




(ˉq )?


have been established[10].Atˉq=0,further speci?c relations between the polarizabilities and their derivatives can be obtained by expanding(18)–(20)with respect to Q20orˉq2.We have used such expansions up to Q40in order to test our analytical results.Furthermore we note that the polarizabilities are actually functions ofˉq2[10].

Equivalent to the parametrization of the spin–?ip amplitude of VCS in terms of general-ized vector polarizabilities,Ragusa[24]expressed the spin–?ip amplitude in RCS,to lowest order in the photon energy,by4polarizabilities,calledγ1,γ2,γ3,γ4.Two linear combinations of theseγi can be related to the generalized polarizabilities[10],




P(01,01)0 p /P(01,01)1




p≈?7.8,demonstrate that the scalar polariz-

abilitiesαandβwill dominate the residual amplitude for the kinematic typical at MAMI. Contrary to this,the remaining scalar polarizability?P(01,1)0


is comparable to the vector po-


p .For increasing Q20both quantities tend to zero but remain of the same

order of magnitude.

Among the3polarizabilities P(01,12)1,P(11,02)1and?P(01,1)1with the common unit fm4, P(01,12)1is generally suppressed.

Since the polarizabilities have di?erent dimensions due to their de?nitions(11,12),the amplitudes of(13)–(17)are constructed by multiplying the polarizabilities with di?erent kinematical factors.Therefore,the in?uence of a particular polarizability can only be seen after evaluating the expansion coe?cients(13)–(17).

The LSM and the CQM calculation of Liu et al.[11]predict signs di?erent from ours for two of the vector polarizabilities(P(01,01)1,?P(01,1)1).Furthermore,with the exception of P(11,00)1,the variation of the vector polarizabilities at low Q20is much stronger in the LSM than in the CQM,as was the case for the scalar polarizabilities[13].However,the most remarkable di?erence between the models is in the absolute values of the vector polarizabil-ities.Except for P(11,11)1and P(01,12)1,our results are substantially larger than the CQM predictions,in the most striking case of P(11,00)1by three orders of magnitude.This result indicates that for most of the vector polarizabilities the nonresonant background,described in our calculation to one–loop,is more important than the nucleon resonances.

The most obvious resonance contribution is due to the strong M1transition to the ?(1232),which leads to the large value of P(11,11)1in the CQM.On the other hand,the CQM violates the model–independent constraint P(11,11)1(0)=0.This shortcoming of the CQM can probably be traced back to the lack of covariance.Similarly,the main contri-bution to P(01,12)1in the CQM is caused by the D13(1520),which is clearly visible in the photoabsorption spectrum.

B.Contribution of anomaly diagrams

The two anomaly diagrams in Fig.1give rise to the amplitude

T a=e2g2πN





with the Mandelstam variable t=(q?q′)2.The matrixτ0acting in isospace gives opposite signs for proton and neutron.The anomaly contributes to5of the7vector polarizabilities. In the case of the proton these contributions read

P(11,00)1 p,a (Q20)=




m N






?P(01,1)1 p,a (Q20)=?



m Nω0




4m2N Q20+Q40

4m2N Q20+Q40


and I(0)=



μ2 ?18μ+3lnμ+13


μ2 ?18μ+lnμ+3

√μ2 ?18μ+3lnμ?5

√μ2 ?18μ+lnμ?1

√μ2 ?38μ+21lnμ+85

√μ2 ?38μ+7lnμ+1


m N ?π


m N ?3π4+O(μ) ,(25)

with C=


In eq.(25)the expressions in round brackets denote the contribution of the anomaly. The results con?rm that the anomaly diagrams dominate the polarizabilities whenever they contribute.With the exception of?P(11,2)1,the scalar and vector polarizabilities show two common properties:First,the leading terms of the loop contributions do not depend on isospin,and second,the dominant chiral logarithm is three times larger for the proton than for the neutron.However,the scalar and vector polarizabilities di?er in their chiral behaviour.Apart from?P(11,2)1,the leading term of the non–vanishing vector polarizabilities is proportional to m?2π,while the scalar polarizabilities diverge like m?1πin the chiral limit [13].

By means of eq.(22)one immediately obtains the third order spin–polarizabilitiesγ3 andγ2+γ4.The leading order term of our results agrees with a calculation of Bernard et al.[25]who evaluated all4spin–polarizabilities for RCS on the basis of heavy baryon chiral perturbation theory(HBChPT)to lowest order(O(p3)).

The relationship between the LSM and ChPT holds for all vector polarizabilities.A calculation of Hemmert et HBChPT to the order O(p3)[18]completely agrees with the leading terms of the chiral expansion(25),except for?P(11,2)1,which vanishes in ChPT to order O(p3).Therefore,in ChPT a O(p4)calculation is required to obtain the terms in m?1π.

The agreement between HBChPT and the LSM is restricted to the leading m?2πterm of the chiral expansion.The next to leading order contributions will be modi?ed by various low energy constants which enter a O(p4)calculation and describe physics beyond the scope of the sigma model(e.g.resonance contributions,kaon–loops).As an example we refer to the calculation ofαandβin RCS to the order O(p4)in HBChPT[26].

At Q20=0,all vector polarizabilities have a?nite derivative whose leading terms in the chiral expansion are given in Appendix A.The derivatives of those polarizabilities which van-ish at Q20=0diverge like m?2πin the chiral limit,the slope of the remaining polarizabilities is proportional to m?4π.All analytical results given in App.A satisfy the model–independent relations(18)–(20)between the vector polarizabilities.We stress that we have obtained a complete agreement with the HBChPT calculation to order O(p3)[18],also for the?rst and second derivatives of the polarizabilities.


The non Born amplitude of VCS o?the nucleon may be parametrized by10generalized polarizabilities,to leading order in the cm energy[2].To complete our previous calculation [13]we have evaluated the7vector polarizabilities which determine the spin–dependent amplitude of the VCS reaction.To this end,we used the LSM in the one–loop approximation and in the limit of an in?nite sigma mass.Both our numerical and analytical results are in complete agreement with3relations between the vector polarizabilities which have been derived in a model–independent way[10].

We?nd that the anomaly diagrams strongly a?ect5of the vector polarizabilities.In particular,when measuring double polarization observables like in the proposed reaction p( e,e′ p)γ[7],the anomaly contributions become quite important.In the3independent

structure functions of the unpolarized cross section,the anomaly contributions cancel ex-actly.At Q 20=0,we have performed a Taylor expansion of all polarizabilities (with respect to Q 20)keeping the ?rst two non–vanishing coe?cients.The various Taylor coe?cients have been expanded in powers of m πand compared with the results of ChPT.It turns out that both calculations totally agree in the leading,isospin–independent term of the chiral expansion [18].The same is true in the case of the third order spin–polarizabilities γ3and γ2+γ4of real Compton scattering [25].Of course,a future O (p 4)calculation in ChPT will give rise to additional low energy constants.

In comparison to the CQM [11]the LSM leads to substantially larger results for 5of the 7vector polarizabilities.In the case of P (11,00)1the ratio of the predictions reaches three orders of magnitude.We conclude that most of the vector polarizabilities are dominated by virtual excitations of nonresonant πN scattering states.Only in the case of P (11,11)1,the result of the LSM is small compared to the CQM,which is due to the strong in?uence of the ?(1232)resonance on that polarizability.The CQM,however,does not ful?ll the model–independent condition P (11,11)1(0)=0.

In conclusion,the generalized vector polarizabilities represent suitable observables to distinguish between di?erent models,and,in particular,to test the chiral structure of the nucleon.Accordingly,experimental e?ort to measure these quantities is certainly worth-while,despite the huge di?culties involved in clearly separating the e?ects of individual polarizabilities.


We would like to thank G.Kn¨o chlein and S.Scherer for several useful discussions.We are also grateful to G.Q.Liu for providing us with the results of the quark model calculation.This work has been supported by the Deutsche Forschungsgemeinschaft (SFB 201).



Below we list the chiral expansion of the ?rst derivatives of the vector polarizabilities.The constant C is de?ned in equation (25).


2m π


8+O (μ)


2m π



+O (μ)


2m π



dQ 20

P (11,11)1n (0)=C





+O (μ)


d √2m π



μ+12π+O (μ)


d √2m π 12μ+7π+O (μ)




5m 3π




dQ 20P (11,02)1

n (0)=

32Cm N μ



16+O (μ)



5m 3π



dQ 20P (01,12)1n (0)=122Cm N μ



32+O (μ)



5m 3π



dQ 20?P (01,1)1n (0)=

162Cm N μ




+O (μ)


d 210m 3π 1

32+O (μ)


210m 3π



+O (μ)



We also quote the second derivatives of those polarizabilities which vanish at Q 20=0.

d 2

5m 3π


32+O (μ)

,d 2

5m 3π



32+O (μ)



5m 3π




(dQ 20)

2P (11,11)1n (0)=





64+O (μ)

,d 2

5m 3π







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FIG.1.Anomaly diagrams for virtual Compton scattering o?the nucleon.Solid lines:nucle-ons,wavy lines:photons,dotted lines:neutral pions.

Generalized polarizabilities of the nucleon studied in the linear sigma model (II)

FIG.2.The vector polarizabilities as functions of momentum transfer.Solid line:calculation with the LSM for the proton,dashed line:LSM result for the neutron,dash–dotted line:CQM for the proton[11].Note that the CQM results have been scaled.

FIG.3.In?uence of the anomaly on the polarizability?P(01,1)1

p.Solid line:without anomaly, dashed line:anomaly included.