Polarized light-antiquark distributions in a meson-cloud model

a r X i v :h e p -p h /0110097v 1 6 O c t 2001SAGA-HE-176-01,TMU-NT-01-04Polarized light-antiquark distributions in a meson-cloud modelS.Kumano ∗Department of Physics,Saga UniversitySaga,840-8502,JapanM.Miyama †Department of Physics,Tokyo Metropolitan UniversityTokyo,192-0397,Japan(Dated:October 6,2001)Flavor asymmetry is investigated in polarized light-antiquark distributions by a meson-cloudmodel.In particular,ρmeson contributions to ∆¯u −∆¯dare calculated.We point out that the g 2part of ρcontributes to the structure function g 1of the proton in addition to the ordinary longitudinally polarized distributions in ρ.This kind of contribution becomes important at medium x (>0.2)with small Q 2(∼1GeV 2).Including N →ρN and N →ρ∆splitting processes,we obtain the polarized ρeffects on the light-antiquark flavor asymmetry in theproton.The results show ∆¯d excess over ∆¯u ,which is very different from some theoreticalpredictions.Our model could be tested by experiments in the near future.PACS numbers:13.60.Hb,13.88.+e,12.39.-xCONTENTSI Introduction1II Vector-meson contributions 2III Meson momentum distributions 5IV Results 7VConclusions10Acknowledgments 10AAnalytical expressions of meson momen-tum distributions 10References12I.INTRODUCTIONLight antiquark distributions are expected to be al-most flavor symmetric according to perturbative quan-tum chromodynamics (QCD).The next-to-leading-ordereffects contribute to the difference between ¯u and ¯d;how-ever,the contribution is tiny as long as they are esti-mated in the perturbative QCD region.Therefore,it was rather surprising to find the antiquark flavorasymmetry2 are some semi-inclusive data[11]which could be sensi-tive to the light antiquarkflavor asymmetry,they arenot accurate enough to provide strong constraint for thepolarized antiquarkflavor asymmetry[10,11].However,the Relativistic Heavy Ion Collider(RHIC)[12]and theCommon Muon and Proton Apparatus for Structure andSpectroscopy(COMPASS)[13]experiments should clar-ify the details of the polarized antiquark distributions ina few years.It is the right time to investigate the anti-quarkflavor asymmetry∆¯u/∆¯d by possible theoreticalmodels and to summarize various predictions.In this paper,we intend to shed light on the virtualmeson model which has been successful in the unpolar-ized studies[14].The purpose of this paper is to extendthe studies of the virtualρ-meson contributions by Friesand Sch¨a fer in Ref.[7].In particular,we point out thatthe g2part of the polarizedρcontributes to the polarizedflavor asymmetry in addition to the ordinary longitudi-nal part,which was calculated in Ref.[7].Because weshow new g2terms in this paper and because the situa-tion is still confusing in the sense that anotherρ-mesonpaper[8]claims major differences from Ref.[7]in sup-posedly the sameρcontributions,the detailed formalismis shown in the following sections.The meson model wasextended recently to a different direction in Ref.[8]byincludingπ−ρinterference terms;however,this paperis intended to investigate a different kinematical aspectwithin the meson model.The paper consists of the following.The formalismofρcontributions to∆¯u−∆¯d is presented in Sec.II.Meson momentum distributions are obtained in Sec.III,and numerical results of∆¯u−∆¯d are shown in Sec.IV.Our studies are summarized in Sec.V.II.VECTOR-MESON CONTRIBUTIONSThe cross section of polarized electron-nucleon scat-tering is generally written in terms of lepton and hadrontensors:dσ| p e|α22π X(2π)4δ4(p N+q−p X)×<p N,s N|Jµ(0)|X><X|Jν(0)|p N,s N>,(2.3)where the factorεµνρσis the antisymmetric tensor withthe conventionǫ0123=+1.p'NBFIG.1:Virtual vector-meson contribution.Next,we consider the process in Fig.1,where thenucleon splits into a virtual vector meson and a baryon,then the virtual photon interacts with the polarized me-son.Because scalar mesons do not contribute directly tothe polarized structure functions due to spinless nature,the lightest vector meson,namelyρ,is taken into ac-count in this paper.In future,we may extend the presentstudies by including heavier vector mesons.As thefinalstate baryon,the nucleon and∆are considered.Express-ing the V NB vertex multiplied by the meson propagatoras J V NB(k,s V,p N,s N,p B,s B)and calculating the crosssection due to the process in Fig.1,weobtain dσ| pe|α2(2π)3m B3tion to the nucleon tensor:W µν(p N ,s N ,q )=d 3p BE B×λV ,λB|J V NB |2W (V )µν(k,s V ,q ),(2.5)where m V is the meson mass,and the meson tensor isdefined byW (V )µν(k,s V,q )=1p N ·q+(p N ·q s σN −s N ·q p σN )g 22p N ·qi εµναβq αs N β,(2.8)is then applied to giveP µνW Aµν(p N ,s N ,q )=m 2NQ 2,(2.10)with Q 2=−q 2.In the same way,g 1and g 2structure functions of the vector meson are defined in the asym-metric part of the tensor.Operating the projection also on the meson tensor,we obtain PµνW (V )A µν(k,s V,q )=m Np N ·q k ·q(s N ·q s V ·q −q 2s N ·s V ),(2.12)A 2=m N m V q 2(p N ·q )2[q 2+(s N ·q )2]g 1(p N ,q )−γ2g 2(p N ,q )=d 3p BE BλV ,λB|J V NB |2×[A 1g V 1(k,q )+A 2g V2(k,q )].(2.14)Then,the above integration variables (p x B ,p y B ,p zB )are changed for the meson momentum fraction y ,the trans-verse momentum k ⊥of the meson,and the angle φbe-tween k ⊥and the transverse spin vector of the nucleon ( s T N ):y =k ·q(p N ·q )2[q 2+(s N ·q )2]g 1(x,Q 2)−γ2g 2(x,Q 2)=1xdy (2π)3E B∂y ′1+1+γ24 and the baryon B should move in the forward direction.The maximum transverse momentum is given by( k2⊥)max=m2N1+γ2+1)( yy′m2N(y m N,A2=γ2m2Vy′m N((2π)3E B∂y′yy′m2N(ym N,(2.26)fλV 2L (y)= ( k2⊥)max0d k2⊥γ22πCλV L m2Vy′y2m3N(y λN f1L(y)−τ2N f1T(y) g V1(x/y,Q2)+ −λN f2L(y)+τ2N f2T(y) g V2(x/y,Q2) .(2.29)Combining the longitudinal polarizationλN=1(τN=0)with the transverse polarizationτN=1(λN=0),wecan extract the g1part asg1(x,Q2)=1y× ∆f1L(y)+∆f1T(y) g V1(x/y,Q2)− ∆f2L(y)+∆f2T(y) g V2(x/y,Q2) ,(2.30)where the functions∆f V Ni(y)with i=1L,2L,1T,and2T are defined by∆f i(y)=fλV=+1i(y)−fλV=−1i(y).(2.31)The g2part is obtained in the same way asg2(x,Q2)=1y× −∆f1L(y)+∆f1T(y)/γ2 g V1(x/y,Q2)+ ∆f2L(y)−∆f2T(y)/γ2 g V2(x/y,Q2) .(2.32)In the limit Q2→∞,namelyγ2→0,only the momen-tum distribution∆f1L(y)remainsfinite,and Eq.(2.30)agrees with the expression in Ref.[7].In Eq.(2.30),there are additional terms associ-ated with g2of the meson.For discussing these g2type contributions to∆¯u−∆¯d,g V2is approximated bythe Wandzura-Wilczek(WW)relation[15]by neglectinghigher-twist terms:g V(W W)2(x,Q2)=−g V1(x,Q2)+ 1x dy5 Then,providing the leading-order expression for g V1,wehaveg V(W W) 2(x,Q2)=1y∆¯q V i(y,Q2),(2.35)and the same equation for∆q V(W W)i (x,Q2).From theseequations,we obtain a vector meson contribution to the polarized antiquark distribution∆¯q i in the proton as∆¯q V NBi (x,Q2)=1y× ∆f1L(y)+∆f1T(y) ∆¯q V i(x/y,Q2)− ∆f2L(y)+∆f2T(y) ∆¯q V(W W)i(x/y,Q2) .(2.36)If this kind of vector-meson contribution is the only source for the polarizedflavor asymmetry,the∆¯u−∆¯d distribution is then calculated by taking the difference∆¯q V NBi=u−∆¯q V NBi=din the above equation.III.MESON MOMENTUM DISTRIBUTIONSIn order to estimate the meson contributions numeri-cally,it is necessary to calculate the momentum distribu-tions∆f V Ni (y)of the meson.We calculate them by con-sidering the vector-meson creation processes N→V N′and N→V∆through the interactionsV V NN= φ∗V· T F V NN(k)2m Niσµν Kν u(p N,s N),(3.1) V V N∆= φ∗V· T F V N∆(k)m Vγ5γµ× Kµεν∗− Kνεµ∗ u(p N,s N),(3.2)where u(p N,s N)is the Dirac spinor,Uµ(p∆,s∆)is the Rarita-Schwinger spinor,andεµis the polarization vec-tor of the vector meson.The V NN and V N∆coupling constants are denoted as g V,f V,and f V N∆,and form factors are denoted as F V NN(k)and F V N∆(k).Isospin dependence is taken into account by the factor φ∗V· T, and it is defined in terms of areducedmatrixelementandaClebsch-Gordan coefficient[16,17]<B| φ∗V· T|N>=(−1)M V<T B T T N>2T B+1×<T N M N:1−M V|T B M B>(3.3)with<1/2 T 1/2>=√k=p N−p B=(E N−E B, k),(3.4) where E V=4m2N g2V−s V +g V f V −2k·ε∗− K·ε∗4m2N ( K2+ K·ε K·ε∗)(s V +2(p N· K+p′N· K)(p N·ε K·ε∗+p N·ε∗ K·ε)−2p N· K( K·εk·ε) ,(3.5)where s′µ,ˆsµV,m Nεµναβε∗νεαp Nβ,(3.6)ˆsµV=−isµV=−ikβ.(3.8)6 The meson polarization vector is given byεµ= k· ελV m V(E V+m V) k ,(3.9)whereλV is the meson helicity.The spherical unit vectorελVis defined in the frame with theˆz′axis parallel to k:ε±1=∓12(ˆx′±iˆy′), ε0=ˆz′.(3.10)In the same way,the V N∆term is calculated from Eq.(3.2)asλ∆|V V N∆|2=−| φ∗V· T|2F2V N∆(k)f2V N∆s V 2(p∆· K)2+m2∆ K2−4m N m2∆m V p∆· K s N·ˆs V+2m2N m∆m V( K2s N·s1−p∆· Ks N·ˆs2) ,(3.11) where sµ1andˆsµ2are defined bysµ1=ikβ,(3.12)ˆsµ2=i2E V(E N−E V−E B)+1y′(m2N−m2V B),(3.14)where m2V B is defined bym2V B=(k+p B)2=m2V+ k2⊥1−y′.(3.15)y∆f(y)ρNNFIG.2:Meson momentum distributions∆f1L(y)and∆f2L(y)from theρNN process.The solid and dashed curvesare obtained at Q2=1and2GeV2,respectively,with x=0.2.The isospin factors are taken out from the distributions asexplained in thetext.y∆f(y)ρNNFIG.3:Meson momentum distributions∆f1T(y)and∆f2T(y)from theρNN process.The solid and dashed curvesare obtained at Q2=1and2GeV2,respectively,with x=0.2.Therefore,J V NB is expressed asJ V NB=17y∆f (y )ρN ∆FIG.4:Meson momentum distributions ∆f 1L (y )and ∆f 2L (y )from the ρN ∆process.The solid and dashed curves are obtained at Q 2=1and 2GeV 2,respectively,with x=0.2.y∆f (y )ρN ∆FIG.5:Meson momentum distributions ∆f 1T (y )and ∆f 2T (y )from the ρN ∆process.The solid and dashed curves are obtained at Q 2=1and 2GeV 2,respectively,with x =0.2.dependence,the unphysical region y <x is not shown in these figures.In Figs.2and 3,the distributions due to the ρNN process are shown.In Figs.4and 5,the dis-tributions due to the ρN ∆process are shown.In these distributions,the isospin factors are taken out from thedistributions,so that the distributions ∆f (y )/| φ∗V· T |2are actually shown,and there are differences of factors 3(in ρNN )and 2(in ρN ∆)from those in Ref.[7].The coupling constants are taken as g 2V /(4π)=0.84,f V =6.1g V ,and f 2V N ∆/(4π)=20.45[21].In spite of the positive ∆f 1L (y )and ∆f 2L (y )distri-butions from the ρNN process in Fig.2,the distribu-tions from the ρN ∆are mainly negative.In the un-polarized case,the meson momentum distributions are,of course,positive.However,because the distribution ∆f (y )is defined by the helicity difference in Eq.(2.31),it becomes either positive or negative depending on the helicity structure at the V NB vertex.For the meson angular momentum state ℓz ,the V NB vertex amplitudeis proportional to k ℓz⊥and higher order terms.Since the momentum k ⊥is in general much smaller than the nu-cleon mass,the vertex amplitudes with ℓz =0contribute dominantly to ∆f (y ).There is only one ρNN amplitude with ℓz =0and λV =0,and it has the helicity structure λV =+1and λ′N =−1/2,where the initial helicity isfixed at λN =+1/2.This fact indicates that f λV =+11L(y )and f λV =+12L (y )are certainly larger than f λV =−11L (y )and f λV =−11L(y ),respectively,which results in the positive dis-tributions ∆f 1L (y )and ∆f 1L (y )from the ρNN process.On the other hand,there are two amplitudes with ℓz =0and λV =0in the ρN ∆process,and they have helicity states λV =−1,λ∆=+3/2and λV =+1,λ∆=−1/2.Actually calculating these helicity amplitudes,we find that both amplitudes depend much on the momentum choice,namely (A)or (B),at the ρN ∆vertex.Therefore,f λV =+11L (y )and f λV =+12L(y )are either larger or smaller than f λV =−11L (y )and f λV =−12L (y ),respectively,depending on the momentum choice.In the prescription (B),thepositive helicity distributions f λV =+11L (y )and f λV =+12L(y )are mostly smaller,so that ∆f 1L (y )and ∆f 2L (y )become negative distributions in the wide x region.However,the situation is opposite in the prescription (A),where the distributions are mostly positive.As expected,the distributions ∆f 1L (y )in Figs.2and 4are the dominant ones and they are almost independent of Q 2.However,∆f 2L (y ),∆f 1T (y ),and ∆f 2T (y )are roughly proportional to 1/Q 2,so that these new contri-butions become more important as Q 2becomes smaller.Figures 2−5clearly show this tendency.The transverse distributions ∆f 1T (y )and ∆f 2T (y )are an order of mag-nitude smaller than the longitudinal ones ∆f 1L (y )and ∆f 2L (y ).Therefore,the major correction comes from the distribution ∆f 2L (y ),which is almost comparable mag-nitude with ∆f 1L (y )in Figs.2and 4.Because the cor-rection terms are proportional to γ2=4m 2N x 2/Q 2,they are small contributions in the kinematical range x <0.05with Q 2>1GeV 2.However,their effects become more pronounced as x becomes larger.IV.RESULTSFor calculating the ∆¯u −∆¯ddistribution numerically,we need the polarized antiquark distributions in ρ,the isospin factors,and the vertex form factors.The ρ-meson parton distributions are not known,so that the same prescription is used as the one in Refs.[7,8].Considering a lattice QCD estimate [22],the polarized valence-quark distribution is assumed as∆V ρ(x,Q 2)=0.6V π(x,Q 2),(4.1)at Q 2=1GeV 2.The distribution in the pion is taken from the GRS (Gl¨u ck,Reya,and Schienbein)parametrization in 1999[23].The charge symmetry suggests the relation for the valence-quark distributions:(∆¯u )val ρ−=(∆¯d )val ρ+=2(∆¯u )val ρ0=2(∆¯d )val ρ0=∆V ρ.(4.2)For the sea-quark distributions,they are assumed to beflavor symmetric.Then,we obtain the ∆¯u −∆¯ddistri-8 butions in theρmeson:(∆¯u−∆¯d)ρ+=−∆Vρ,(∆¯u−∆¯d)ρ0=0,(∆¯u−∆¯d)ρ−=+∆Vρ.(4.3)For the g2part ofρ,the Wandzura-Wilczek relation isused as discussed in Sec.II:∆V W Wρ(x,Q2)=−∆Vρ(x,Q2)+ 1x dy3∆fρN∆1L+1T ⊗∆Vρ− −2∆fρNN2L+2T+21+γ2 1x dy| φ∗ρ· T|2,i=1,2.(4.8) The expression of Eq.(4.6)may seem to be different from Refs.[7,8]even in the limit Q2→∞;however,it is just the matter of the definition of the meson momentum distributions.They included the isospin factor |<n| φ∗ρ+· T|p>|2+|<p| φ∗ρ0· T|p>|2=3,(4.9) in the distribution∆f(y)for theρNN process and the factor|<∆0| φ∗ρ+· T|p>|2+|<∆+| φ∗ρ0· T|p>|2+|<∆++| φ∗ρ−· T|p>|2=2(4.10) for theρN∆.Therefore,our expression certainly agrees on those in Refs.[7,8]at Q2→∞.The remaining quantities are the vertex form factors. They are roughly known from the studies of one-boson-exchange potentials(OBEPs);however,a slight change of the cutoffparameter could result in a large difference of antiquark distributions.Furthermore,there is an issue of the charge and momentum conservations for the splitting process[24]if a t(=(p N−p B)2)dependent form factor is used.A possible solution is to use the t dependent form factor multiplied by a u dependent one[25].For this purpose,it is more convenient to take an exponential form factor so as to become the additional form t+u within the form factor:FρNN(k)=FρN∆(k)=exp m2N−m2V B9xx (∆u –∆d )FIG.7:∆¯u −∆¯ddistributions from the ρNN process at Q 2=1GeV 2.The 1L ,2L ,1T ,and 2T type contributions and their summation areshown.xx (u –∆d )FIG.8:∆¯u −∆¯ddistributions from the ρN ∆process at Q 2=1GeV 2.The 1L ,2L ,1T ,and 2T type contributions and their summation areshown.xx (∆u –∆d )FIG.9:∆¯u −∆¯ddistributions from the ρNN and ρN ∆pro-cesses at Q 2=1GeV 2.Each term contribution has almost the same tendency in the ρN ∆process as shown in Fig.8:the 1L term is the major one and the 2L term provides some corrections depending on the x region.There are two contributing processes,p →ρ+∆0and p →ρ−∆−,and the isospin factor is three times larger in the latter one.This fact may seem to indicate that the ρN ∆processes provideaxx (∆u –d )FIG.10:∆¯u −∆¯d distributions from the ρNN and ρN ∆processes at Q 2=1GeV 2for the vertex momentum choice (A).positive contribution to ∆¯u −∆¯din the nucleon due to the valence ¯u distribution in ρ−.This kind of explanation is certainly valid in the unpolarized flavor asymmetry [5,26].However,this is not the case in Fig.8,where the 1L and 1T distributions are mostly negative.This misleading result comes from the helicity structure at the N →ρ∆vertex.Although the helicity difference ∆f (y )is positive for the ρNN ,it is negative for the ρN ∆in the case (B)as explained in Sec.III.Therefore,theρN ∆contribution becomes also negative for the ∆¯u −∆¯ddistribution.Next,the ρNN and ρN ∆contributions are compared in Fig.9.The magnitude of the ρN ∆contribution is very small compared with the ρNN one in (B).From Figs.2and 4,we find that the magnitude of ∆f ρN ∆(y )is already three times smaller than ∆f ρNN (y ),and the ρN ∆contribution is further suppressed by the isospin factor (2/3)/2=1/3.Therefore,the overall magnitude becomes much smaller.As discussed in Sec.III,we may have another vertex choice (A)instead of (B).In showing the numerical re-sults so far,the model (B)has been used.We show the choice (A)results in Fig.10.It is obvious that the dis-tributions depend much on this vertex choice.There are two major differences from Fig.9.One is that the order of magnitude is much smaller in the ρNN distribution,and the other is that the ρN ∆distribution becomes pos-itive.These are due to the difference of helicity structure at the vertices between (A)and (B).We also discuss the vertex cutoffdependence.The ver-tex cutoffhas been taken as Λe =1GeV in this section;however,it is well known that calculated antiquark dis-tributions are very sensitive to the cutoffvalue [26].In the present paper,the exponential form factor is used instead of dipole or monopole form factor,which is more popular in the studies of OBEPs.The cutoffparameters of different form factors could be related by [26]Λ1=0.62Λ2=0.78√10xx (u –∆d )FIG.11:Cutoffdependence of ∆¯u −∆¯dis shown at Q 2=1GeV 2by taking Λe =0.5,1.0,and 1.3GeV.|x (∆u –∆d )|Λe (GeV)FIG.12:Cutoffdependence of |∆¯u −∆¯d|is shown at x =0.2and Q 2=1GeV 2as a function of the parameter Λe .by the form factorsF (1)V NB (k )=1−m 2N /Λ21(1−m 2V B /Λ22)2.(4.13)Even in the well investigated pion-nucleon coupling,the monopole cutoffparameter Λ1ranges from 0.6GeV to 1.4GeV in quark models and OBEPs [26].It roughly corresponds to Λe from 0.5GeV to 1.3GeV accordingto Eq.(4.12).We show the ∆¯u −∆¯ddistributions for various cutoffparameters,Λe =0.5,1.0,and 1.3GeV in Fig.11for the prescription (B).In the unpolarized dis-tribution ¯u −¯din Ref.[26],it is fortunate that the cutoffdependence is rather small because of the cancellation between the πNN and πN ∆processes.However,the ρNN is the dominant contribution,as shown in Fig.9,in the present polarized studies of (B),so that the overall magnitude is much dependent on the cutoffparameter.There are orders of magnitude differences between the three curves.Next,fixing x at 0.2,we show the cutoffdependence in Fig.12.In fact,there is four orders of magnitude variation from Λe =0.5to 1.5GeV.Therefore,an accurate determination of the cutoffparameters is a key for a reliable meson-cloud prediction.There have been studies in chiral soliton models [6].They predict very different distributions,namely ∆¯u ex-cess over ∆¯d and the order of magnitude of ∆¯u −∆¯dis large (∼0.4)at x =0.2.Although the soliton models and the meson-cloud models obtain very similar distri-butions for the unpolarized ¯u −¯d,it is interesting to find opposite prediction for the polarized distributions.The physics reason for the difference is not clear at this stage.In any case,the distribution ∆¯u −∆¯dshould be clari-fied experimentally in the near future by W production process [12]and semi-inclusive experiments [13].Further-more,there is a possibility to use the polarized proton-deuteron Drell-Yan process [27]in combination with the proton-proton reaction.Until the data will be taken,the theoretical predictions should be discussed in details for comparison.V.CONCLUSIONSThe ρmeson contributions to the polarized antiquarkdistribution ∆¯u −∆¯dhave been investigated.In particu-lar,we pointed out that the g 2part of ρcontributes to the polarized distributions in the nucleon.We obtained the extra contributions denoted as 2L ,1T ,and 2T in addi-tion to the ordinary one (1L ).Although the extra terms are small in the small x region (x <0.05),the magnitude of the 2L term becomes comparable to the ordinary onein the x region x >0.2.The g ρ2contributions are impor-tant in the kinematical region of medium x with smallQ 2.The obtained ∆¯u −∆¯dis very sensitive to the cutoffparameter.The model should be investigated further in order to compare with future experimental data.ACKNOWLEDGMENTSS.K.and M.M.were supported by the Grant-in-Aid for Scientific Research from the Japanese Ministry of Edu-cation,Culture,Sports,Science,and Technology.M.M.was also supported by the JSPS Research Fellowships for Young Scientists.They would like to thank R.J.Fries for communications about the calculations in Refs.[7,8,18,19].APPENDIX A:ANALYTICAL EXPRESSIONS OF MESON MOMENTUM DISTRIBUTIONSIn the limit Q 2→∞,the following meson momentumdistributions agree on those by Fries and Sch¨a fer (FS)[7]with a minor misprint in a ρN ∆term.The situation of the momentum distributions is somewhat confusing in the sense that Cao and Signal (CS)[8]pointed out two major differences from Ref.[7]although the formalism is exactly the same except for interference terms.Accord-ing to Ref.[8],all the g ρNN f ρNN (g v f v in our notation)terms should be replaced by −g ρNN f ρNN ,and the mo-mentum (B)results for ρN ∆agree on those of (A)in Ref.[7]instead of (B).11 In spite of their claim,we believe that the FS resultsare right with the following reasons.We also checkedthe helicity amplitudes in Ref.[18],which is referredto as J¨u lich in the following.In addition to obvious ty-pos,our results differ from the J¨u lich expressions.First,complex conjugate should be taken if their expressionsare given for the process N→ρB as indicated in theirappendix.Second,f v terms have different sign.If theJ¨u lich amplitudes were written for Nρ→B or B→ρN,we would agree on their expressions.Depending on theρmomentum direction,the f v term in Eq.(3.1)becomeseither positive or negative,which could lead to the dif-ferent sign of the g v f v terms.However,it is obvious thatthe outgoing meson is considered in the formalism.Fur-thermore,taking summations of the helicity amplitudes,we reproduce the unpolarized momentum distributionsof Melnitchouk and Thomas[19,28],whereas the CS re-sults are inconsistent.The V NN vertex in Eq.(3.1)isalso consistent with the one in Ref.[29].We also testedρN∆helicity amplitudes in the vertexmomentum(B),but the results disagree on the J¨u lichexpressions.However,if the momentum(A)is used,ourresults agree on them.It seems to us that the helicityamplitudes are shown for the choice(B)inρNN andfor(A)inρN∆.Therefore,as far as we investigated,we believe that the FS calculations are right also for theρN∆process.In the following,we show the helicity dependent mesonmomentum distributions.Because the distributions withλV=0are irrelevant for calculating∆f(y),so that theyare not shown.The isospin factors are extracted out fromthe expressions.fλV1L,V NB(y)= ( k2⊥)max0dk2⊥{m2N−m2V B(y′,k2⊥)}2×∂y′y′ 1+k2⊥1+γ2−1) D L,λV V NB(y′,k2⊥),(A1)fλV1T,V NB(y)= ( k2⊥)max0dk2⊥{m2N−m2V B(y′,k2⊥)}2×∂y′y′γ2k⊥16π2F2V NB(y′,k2⊥)∂y yy2m2ND L,λVV NB(y′,k2⊥),(A3)fλV2T,V NB(y)= ( k2⊥)max0dk2⊥{m2N−m2V B(y′,k2⊥)}2×∂y′y′γ2k⊥m2V1+γ2−1)D T,λV V NB(y′,k2⊥).(A4)Here,the partial derivative is given by∂y′y′= 1+γ2y′3(1−y′)2 g2V k2⊥+y′4m2N +g V f V y′ k2⊥+y′ y′2m2N−(1−y′)m2V+f2Vy′3(1−y′)2 g2V(1−y′)2−g V f V y′(1−y′)+f2Vy′3(1−y′)2 −2g2V y′2(1−y′)m N +g V f V4m2N2y′m N k2⊥+y′2m2N−(1−y′)m2V .(A8)In the V N∆process,the distributions are calculated for the vertex momentum(A)asD L,+1V N∆(y′,k2⊥)=f2V N∆3y′3(1−y′)2m2∆m2V× k4⊥m2N+k2⊥{4y′2m2N m2∆−4y′(1−y′)m N m∆m2V +(1−y′)2m4V}+3y′4m2N m4∆−6y′2(1−y′)m N m3∆m2V +3(1−y′)2m2∆m4V ,(A10)12D T,λVV N∆(y′,k2⊥)=λV f2V N∆k⊥y′3(1−y′)2 g2V k2⊥+y′4m2N+g V f V y′ (1+y′)k2⊥+2y′3m2N+f2Vy′3(1−y′)2 g2V(1−y′)2−g V f V y′(1−y′)+f2Vy′3(1−y′)2 −2g2V y′2(1−y′)m N +g V f V4m2N4y′m N k2⊥+y′2m2N ,(A14)D L,+1V N∆(y′,k2⊥)=f2V N∆3y′3(1−y′)2m2∆m2V k6⊥+k4⊥ (3+2y′)m2∆+4y′m N m∆+(1−2y′+2y′2)m2N +k2⊥ y′(6+y′)m4∆+10y′2m N m3∆−2y′(3−4y′−y′2)m2N m2∆−4y′2(1−y′)m3N m∆+y′2(1−y′)2m4N +3y′2m6∆+6y′3m N m5∆−3y′2(2−2y′−y′2)m2N m4∆−6y′3(1−y′)m3N 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