ρ meson impact parameter distributions
aa r X i v : . [ h e p - ph ] M a y Chinese Physics C Vol. xx, No. x (201x) xxxxxx ρ meson impact parameter distributions *Bao-Dong Sun , Yu-Bing Dong , , Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, P. R. China School of Physics, University of Chinese Academy of Sciences, Beijing 100049, P. R. China Theoretical Physics Center for Science Facilities (TPCSF), CAS, Beijing 100049, P. R. China
Abstract:
In this paper, the ρ meson impact parameter dependent parton distributions and the impact parameterdependent form factors are introduced and discussed. By employing a Gaussian form wave packet, we calculate theimpact parameter distributions of the ρ meson based on a light-cone constituent quark model. Key words: ρ meson, impact parameter distribution, light-cone approach PACS:
It is common believed that the usual parton distribu-tions (PDFs) can only give the longitudinal informationof a hadron target in the deep inelastic scattering (DIS)processes, while the generalized parton distributions(GPDs) have the promising ability to shade light on thetransverse information, which gives rise to the idea of“quark/gluon imaging” of hadrons [1]. Moreover, theimpact parameter distributions (IPDs), obtained by theFourier transform of GPDs with respect to the transversemomentum transfer, may show some information aboutthe transverse impact space position of partons [2]. Thisimpact parameter representation is useful in processessuch as high-energy scattering and hard processes [3].It is also argued that, in position space, IPDs play asimilar role to the charge distributions, and are, thus,very promising for understanding the hadron internalstructures.As we know, G C ( Q ) is the form factor of the con-served local current, and is thus independent of therenormalization scale µ . It can be obtained through thesum rules from GPDs, which by definition are probedin hard processes [3]. In the case of Fourier transformsof GPDs, Burkardt pointed out that, when ξ = 0, theFourier transforms of GPDs have the interpretation of adensity of partons with longitudinal momentum fraction x , localized at b ⊥ relative to the transverse center in theimpact parameter space, which is allowed by the Heisen-berg uncertainty principle [4, 5]. Due to the significanceof the form factors in the impact parameter space, manytheoretical works have been devoted to study the IPDsof pions, kaons and nucleons [5–15].It should be mentioned that our recent work [16]gave a discussion of the ρ meson unpolarized GPDs inmomentum space with a Light-Cone Constituent QuarkModel (LCCQM). The form factors and some other low-energy observables of the ρ meson were calculated andour numerical results agreed with the previous publica-tions and some experimental data [17]. In the literature,the constituent quark model is also used to describethe form factors of pions, nucleons, deuterons, etc. [18–20]. Moreover, the contributions from the valence andnon-valence regimes to the form factors and generalizedparton distributions were discussed and analyzed in de-tail. In addition, the reduced matrix elements, whichare the moments of the DIS structure functions, werealso estimated and the obtained values were compat-ible with the available lattice calculation at the samescale ratio [21]. In general, our numerical results for theunpolarized GPDs [16] were reasonable and satisfying.Therefore, in this work, we extend the phenomenologicalmodel to study the IPDs of the ρ meson and to calcu-late the impact parameter dependent PDFs of q ( x, b ⊥ ) ∗ Supported by the National Natural Science Foundation of China under Grant No. 11475192, by the fund provided to the Sino-German CRC 110 ”Symmetries and the Emergence of Structure in QCD” project by the NSFC under Grant No. 11621131001, and theKey Research Program of Frontier Sciences, CAS, Grant No. Y7292610K1.1) E-mail: [email protected] c (cid:13) xxxxxx-1hinese Physics C Vol. xx, No. x (201x) xxxxxx and q ( b ⊥ ) and the form factors of q C,M,Q ( x, b ⊥ ) and q C,M,Q ( b ⊥ ).The paper is organized as follows. In Section , theframework of the impact parameter dependent PDFs ispresented. In Section , we discuss the wave packets andthe cutoff for the numerical calculation. The definitionsof the impact parameter dependent FFs are given inSection . Our numerical results for the PDFs and FFsin the impact parameter space are shown in Section ,and Section gives a short summary and conclusion. When considering the nucleon GPDs without helicityflip, Burkardt [22] identifies the Fourier transform of itsGPD H q ( x, ξ = 0 , − ∆ ⊥ ) w.r.t. − ∆ ⊥ as a distributionof partons in the transverse plane, i.e., the probabilityof finding a quark with longitudinal momentum fraction x and at transverse impact space position b ⊥ . The im-pact parameter dependent PDF for a nucleon (a spin-1/2target), given by Ref. [22], reads q N ( x, b ⊥ ) = |N | Z d p ⊥ (2 π ) Z d p ′⊥ (2 π ) × h p + , p ′⊥ , λ | (cid:20)Z dz − π ¯ q ( − z − , b ⊥ ) γ + q ( z − , b ⊥ ) e − ıxp + z − (cid:21) | p + , p ⊥ , λ i = |N | Z d p ⊥ (2 π ) Z d p ′⊥ (2 π ) H q ( x, ξ = 0 , − ( p ⊥ − p ′⊥ ) ) e i b ⊥ · ( p ⊥ − p ′⊥ ) = Z d ∆ ⊥ (2 π ) H q ( x, , − ∆ ⊥ ) e − i b ⊥ · ∆ ⊥ = Z ∞ ∆ ⊥ d ∆ ⊥ π J ( b ∆ ⊥ ) H q ( x, , − ∆ ⊥ )= q N ( x, b ) , (1)where the normalization factor N satisfies |N | R d p ⊥ (2 π ) = 1, and ∆ ⊥ = | ∆ ⊥ | = p ∆ x + ∆ y and b = | b ⊥ | = p b x + b y . Cylindrical symmetry is appliedin the last but one step and J is the Bessel functionof the first kind J ν ( z ) with ν = 0. The parton distri-bution depends on transverse impact space position b ⊥ only through its norm b being the consequence of thelongitudinal polarization. In the third step the integralturns to the total and transverse momentum transfer,i.e., d p ⊥ d p ′⊥ = d ∆ ⊥ d P ⊥ , with ∆ ⊥ = p ′⊥ − p ⊥ and P ⊥ = ( p ′⊥ + p ⊥ ) / , and using the fact that GPD H is independent of total transverse momentum P ⊥ . Ig-noring the helicity flip, the spin projection λ can bedropped. In the forward limit, namely ξ = 0, we have t = ( p ′ − p ) = − ∆ ⊥ .Note that Hoodbhoy [23] has already pointed out theDIS structure function F , F , g , and g of spin-1 tar-gets can be precisely measured in the same way as thatof spin-1/2 targets. Analogous to the fact that the struc-ture function F connects to GPD H q for spin-1/2 tar-gets, we simply assume F connects to the GPD H q forspin-1 targets as well. As shown by Eqs. (37 ∼
39) in Ref. [16], the isospin combination implies that Z − dx H ui ( x, ξ, t ) = Z − dx H I =1 i ( x, ξ, t ) . (2)Hereafter we omit the label of quark flavor u and isospin I = 1 for simplicity. Due to the similar roles of H q and H , we introduce the impact parameter dependent PDFfor spin-1 targets (for the u quark), q ( x, b ) = Z ∞ ∆ ⊥ d ∆ ⊥ π J ( b ∆ ⊥ ) H ( x, , − ∆ ⊥ ) , (3)One can further define the total parton distribution inthe impact parameter space as q ( b ) = Z dx q ( x, b ) . (4)Notice that R d b ⊥ q ( x, b ) = H ( x, , q ( x ) in the forward limit t = ∆ → q ( x, b ), the Fourier transform of the GPD H ( x, ξ = 0 , − ∆ ⊥ ) w.r.t. − ∆ ⊥ , can be identified, inanalogy to the nucleon case, with the probability of find-ing a quark with longitudinal momentum fraction x andtransverse impact space position b ⊥ in the ρ meson.It should be emphasized that in Ref. [2], the nucleonimpact parameter dependent PDF q N was proved to xxxxxx-2hinese Physics C Vol. xx, No. x (201x) xxxxxx satisfy the positive constraints for the so-called “good”quark field. In our model calculation, the phenomenolog-ical vertexes (see Eq. (24) in Ref. [16]) involve the loopmomentum ( k ), and the form of the vertexes is fixed ac-cording to the constraints from isospin symmetry. Oursophisticated model cannot simply reproduce the proce-dure of Ref. [2] to fold the correlation function into anorm of a quantity (see Eq. (23) of Ref. [2]). Therefore,the positive constraint for q ( x, b ) with a realistic modelcalculation needs to be proven further. The Fourier transform of a plane wave is not welldefined, thus, one can start with the wave packets in-stead of the plane wave. In the non-relativistic limit, theFourier transform of the charge form factor G C ( Q ) canbe interpreted as the charge distribution in the trans-verse direction. In other words, as long as the wavepackets peak sharply at some point in position space, bytaking the non-relativistic limit, the Fourier transformof the charge distribution equals the form factor. By theway, a Gaussian weighting factor was also adopted in arecent lattice QCD calculation [24], in order to suppressthe unphysical oscillatory behaviour. The oscillation isdue to the finite lattice size and nucleon momentum. Theresult in the small Bjorken x ( < .
3) region is changedby weighting. In Ref. [25], the Gaussian ansatz is alsoapplied to shape the hadron when calculating general-ized distribution amplitudes of the pion pair productionprocess.Moreover, as pointed out by Burkardt [2, 4], the in-terpretation of the Fourier transform of the form factoras the charge distribution may receive relativistic correc-tions in the rest frame. However, such a problem maydisappear in either Breit frame or infinite momentumframe (IMF). In the relativistic case, the transform re-ceives relativistic corrections when the wave packet islocalized with a size smaller than the Compton wave-length of the system. In the IMF, the relativistic cor-rection can be managed to be very small, and therefore,the wave packet does not change the interpretation, aslong as the wave packets are set slowly varying w.r.t. ∆ ⊥ . To be specific, the width of the wave packetsmust be much larger than a typical QCD scale Λ QCD ( ∼ .
23 GeV). For a Gaussian form wave packet, onegets σ ≪ / Λ QCD ∼ /M , with M being the ρ mesonmass.On the other hand, as Diehl [5] has discussed, a real hadron is an extended object and is smeared out by anamount σ . From the experimental viewpoint, there is alargest measured value | t | max and thus there is the accu-racy of the measurement σ ∼ ( | t | max ) − / . According tothe observations and to the limit of the effect from un-measured values of t , a Gaussian form wave packet canalso be reasonably introduced. Thus we have Z d p ⊥ dp + (2 π ) p + p + δ ( p + − p +0 ) G ( p ⊥ , σ ) | p, λ i∼ Z d p ⊥ (2 π ) exp (cid:18) − p ⊥ σ (cid:19) | p + , p ⊥ , λ i , (5)where G ( p ⊥ , /σ ) = exp( − p ⊥ σ /
2) and the mixed stateis modified to be | p + , b ⊥ , λ i σ = N σ Z d p ⊥ (2 π ) e − ı b ⊥ · p G ( p ⊥ , σ ) | p + , p ⊥ , λ i σ → = | p + , b ⊥ , λ i , (6)where the normalization factor N σ satisfies |N σ | R d p ⊥ (2 π ) = 1 and lim σ → N σ = N . Note that ournormalization of states is different from that in Ref. [5].This action will add two Gaussian functions in the ex-pression, G ( p ⊥ , σ ) and G ( p ′⊥ , σ ), into the definitionof q ( x, b ) (see eq. (2)). We can still change variablesto remove the dependence of P ⊥ , which leaves only one G ( ∆ ⊥ , σ ). Consequently, the definition of the impactparameter dependent PDF is modified to be q σ ( x, b ) = Z ∞ ∆ ⊥ d ∆ ⊥ π J ( b ∆ ⊥ ) G ( ∆ ⊥ , σ ) H ( x, , − ∆ ⊥ )= Z ∞ ∆ ⊥ d ∆ ⊥ π J ( b ∆ ⊥ ) e − ∆ ⊥ σ / H ( x, , − ∆ ⊥ ) , (7)and q σ ( b ) = Z dx q σ ( x, b ) . (8)Reference [5] also argued that in order to give awell-defined (positive, or without sign flip) longitudinalmomentum p , | p ⊥ | ≪ p + is required. However, as onecan see in Eq. (5), p ⊥ and p + are separated in the wavepacket and thus this requirement actually does not affectthe result of the integrals. This can also be seen from theproperty of GPDs. In the forward limit, H ( x, , − ∆ ⊥ )is not affected by this requirement either. Moreover,Ref. [26] emphasized that since the longitudinal mo-mentum is p + in the front form, one needs not to goto infinite momentum along the moving direction, andnot to impose the constraint on the p component either.According to the above discussions, the relation σ ∼ ( | t | max ) − / inspires us to introduce a cutoff (∆ ) of the xxxxxx-3hinese Physics C Vol. xx, No. x (201x) xxxxxx momentum transfer in the integral as well q ( x, b, ∆ ) = Z ∆ ∆ ⊥ d ∆ ⊥ π J ( b ∆ ⊥ ) H ( x, , − ∆ ⊥ ) , (9)and q ( b, ∆ ) = Z dx q ( x, b, ∆ ) . (10)This assumption is supported by a comparison betweenthe results of the integrals with a wave packet, q σ ( b )(width σ ∼ / ∆ ) and the one with a cutoff q ( b, ∆ ).This will be shown in Section . We emphasize that the unpolarized impact parame-ter dependent PDFs are proposed to describe the trans-verse distribution of unpolarized partons in an unpolar-ized target. As shown in previous sections, the IPDs canbe obtained through Fourier transform of the unpolar-ized GPD H . We notice that the conventional charge,magnetic dipole and quadrupole FFs are the integrals of the linear combination of H i . This motivates us to ex-plore the possibility of obtain the IPDs with respect tothe three FFs. The sum rules relating to the GPDs andthe FFs G i are [27] Z − dxH i ( x, ξ, t ) = G i ( t ) ( i = 1 , , , Z − dxH i ( x, ξ, t ) = 0 ( i = 4 , , (11)where G qi are the FFs in the decomposition of the localcurrent. The FFs G C,M,Q can be expressed in terms of G , , as [28] G C ( t ) = G ( t ) + 23 ηG Q ( t ) ,G M ( t ) = G ( t ) ,G Q ( t ) = G ( t ) − G ( t ) + (1 + η ) G ( t ) , (12)where η = − t/ M . Together with Eq. (11), one can ob-tain G C,M,Q directly from GPDs H , , . This allows usto bypass the well-known ambiguity of the angular con-dition [29]. With the above two equations, one can getthe relations G C ( t ) = Z − dx h H ( x, ξ, t ) + 23 η [ H ( x, ξ, t ) − H ( x, ξ, t ) + (1 + η ) H ( x, ξ, t )] i ,G M ( t ) = Z − dxH ( x, ξ, t ) ,G Q ( t ) = Z − dx h H ( x, ξ, t ) − H ( x, ξ, t ) + (1 + η ) H ( x, ξ, t ) i . (13)Notice that by taking ξ = 0 and η = − t/ M =∆ ⊥ / M , one can get quantities similar to the integrands in Eq. (1). We have the impact parameter dependent FFs q Cσ ( x, b ) = Z ∞ ∆ ⊥ d ∆ ⊥ π J ( b ∆ ⊥ ) e − ∆ ⊥ σ / × " H ( x, , − ∆ ⊥ ) + 23 ∆ ⊥ M h H ( x, , − ∆ ⊥ ) − H ( x, , − ∆ ⊥ ) + (1 + ∆ ⊥ M ) H ( x, , − ∆ ⊥ ) i , (14) q Mσ ( x, b ) = 1 G M (0) Z ∞ ∆ ⊥ d ∆ ⊥ π J ( b ∆ ⊥ ) e − ∆ ⊥ σ / H ( x, , − ∆ ⊥ ) , (15) q Qσ ( x, b ) = 1 G Q (0) Z ∞ ∆ ⊥ d ∆ ⊥ π J ( b ∆ ⊥ ) e − ∆ ⊥ σ / × (cid:20) H ( x, , − ∆ ⊥ ) − H ( x, , − ∆ ⊥ ) + (1 + ∆ ⊥ M ) H ( x, , − ∆ ⊥ ) (cid:21) , (16) xxxxxx-4hinese Physics C Vol. xx, No. x (201x) xxxxxx and q C,M,Qσ ( b ) = Z dx q C,M,Qσ ( x, b ) . (17) Comparing the impact parameter dependent FFs,Eq. (14), with the impact parameter dependent PDFs,Eq. (7), we introduce the “difference” quantities q QCσ ( x, b ) = Z ∞ ∆ ⊥ d ∆ ⊥ π J ( b ∆ ⊥ ) e − ∆ ⊥ σ / × (cid:18)
23 ∆ ⊥ M (cid:19) " H ( x, , − ∆ ⊥ ) − H ( x, , − ∆ ⊥ ) + (1 + ∆ ⊥ M ) H ( x, , − ∆ ⊥ ) , (18) q QCσ ( b ) = Z dx q QCσ ( x, b ) , (19)which receive the contribution from the quadrupolemoment. The “difference” quantities satisfy q QCσ ( x, b ) = q Cσ ( x, b ) − q σ ( x, b ) ,q QCσ ( b ) = q Cσ ( b ) − q σ ( b ) . (20)It is clear that the impact parameter dependent PDFsrelate to the impact parameter dependent FFs and Z dx Z ∞−∞ d b q C,M,Qσ ( x, b ) = 1 . (21)Thus, it is possible to interpret q Cσ , q Mσ and q Qσ as thepercentage of the contributions to the charge (normal-ized to 1), magnetic dipole µ ρ and quadrupole moment Q ρ respectively, from the parton with the longitudinalmomentum fraction x and transverse impact space posi-tion b ⊥ . In our previous work [16] with a light-cone con-stituent quark model, we took the two model parametersof the constituent mass m = 0 .
403 GeV and regulator mass m R = 1 .
61 GeV, and we calculated the GPDs ofthe ρ meson. In our LCCQM, we introduced an effectiveLagrangian for the ρ − q ¯ q interaction with a phenomeno-logical vertex Γ u and a Bethe-Salpeter amplitude. By in-tegrating the minus component of the quark momentum k − analytically and rest of the components numerically,we obtained the GPDs and FFs of the ρ meson.In this work, we simply extend the calculation tothe impact parameter dependent PDFs q ( b ) and impactparameter dependent FFs q C,M,Qσ ( b ). Figure 1 gives the q ( b ) with a wave packet, q σ ( b ), and with a cutoff on themomentum transfer, q ( b, ∆ ), respectively. The compar-ison shows that the cutoff (∆ ) has a similar effect as thewave packet with width σ ∼ / ∆ . Of course, we expectthat the prediction of the constituent quark model isreasonable only in the region of | t | / ≤ ∼ σ ) cannot be smaller than theCompton wavelength. In the later content, our numeri-cal results in Fig. 3(a) agree with this point of view.Figure 2 gives the contour plots of the impact pa-rameter dependent PDF q σ ( b ) with σ = 1 GeV − and2 GeV − . Since we choose the polarization in the z direction, the parton distribution is invariant under ro-tation around the z direction. We see that as σ becomes smaller, the wave functions of the initial and final statesget closer to a plane wave, and the parton distributionalso becomes more transversely localized in the positionspace, as shown in Fig. 1 and Fig. 2.Figures 3 and 4 give the impact parameterdependent FFs q C,M,Qσ ( b ) and q QCσ ( b ) with σ =1 / − , − , − respectively. Figure 4shows that, as the wave packet becomes more sharply lo-calized ( σ decreases), the contributions are concentrated more in the small b ⊥ region for both the magneticdipole µ ρ and quadrupole moment Q ρ . For the impactparameter charge density, Fig. 3(a), the distributionswith σ less than about 1 GeV − become obscure dueto the oscillation. As we argued before, the ρ meson xxxxxx-5hinese Physics C Vol. xx, No. x (201x) xxxxxx q σ ( b )( f m - ) σ = GeV - σ = - σ = - b ( fm ) (a) q σ ( b ) with packet width σ = 1 / − , 1 GeV − ,and 2 GeV − . q ( b , Δ )( f m - ) Δ = Δ = Δ = / b ( fm ) (b) q ( b, ∆ ) with cutoff ∆ = 1 / Fig. 1. The impact parameter dependent PDF q ( b ) with (a) a wave packet and (b) a cutoff on the momentum transfer. b y ( f m ) - - - - - - σ - b x ( fm ) (a) q σ ( b ) (fm − ) with packet width σ = 1 GeV − . b y ( f m ) - - - - - - σ G(cid:0)(cid:1) - b x ( fm ) (b) q σ ( b ) (fm − ) with packet width σ = 2 GeV − . Fig. 2. Contour plots of the impact parameter dependent PDF q ( b ) with a wave packet.xxxxxx-6hinese Physics C Vol. xx, No. x (201x) xxxxxx q σ C ( b )( f m - ) (cid:5)(cid:6)(cid:7) σ = GeV - σ = - σ = - b ( fm ) (a) q Cσ ( b ) with packet width σ = 1 / − , 1 GeV − ,and 2 GeV − . q σ Q C ( b )( f m - ) (cid:8)(cid:9)(cid:10) - - - - σ = GeV - σ = - σ = - b ( fm ) (b) q QCσ ( b ) with packet width σ = 1 / − , 1 GeV − ,and 2 GeV − . Fig. 3. The impact parameter dependent FFs q Cσ ( b ) and q QCσ ( b ) with σ = 1 / − , 1 GeV − and 2 GeV − . q σ M ( b )( f m - ) (cid:11)(cid:12)(cid:13) σ = GeV - σ = - σ = - b ( fm ) (a) q Mσ ( b ) with packet width σ = 1 / − , 1 GeV − ,and 2 GeV − . q σ Q ( b )( f m - ) (cid:14)(cid:15)(cid:16) (cid:17)(cid:18)(cid:19) σ = GeV - σ = - σ = - b ( fm ) (b) q Qσ ( b ) with packet width σ = 1 / − , 1 GeV − ,and 2 GeV − . Fig. 4. The impact parameter dependent FFs q M,Qσ ( b ) with σ = 1 / − , 1 GeV − , and 2 GeV − . is an extended object and its Compton wavelength is1 /m ρ = 1 . − . The position dispersion h ∆ x i = σ inthe case of the Gaussian wave packet. The uncertaintyprinciple ( h ∆ x ih ∆ p i ≥ / h ∆ x i shouldnot be smaller than the Compton wavelength. Other-wise, localizing a wave packet to less than its Compton wavelength in size will in general induce various relativis-tic corrections [4]. With the help of Figs. 1 and 3(b), andEq. 20, the oscillation in q Cσ ( b ) can be explained as thebehaviour of q QCσ ( b ) which is related to the quadrupolemoment. From the experimental aspect, since the ρ meson quadrupole moment is small, this phenomenon ishard to determine.Figures 5 and 6 show the numerical result of q σ ( x, b )and q C,M,Q,QCσ ( x, b ) with σ = 1 GeV − and x =1 / , /
10 and 1 / x ≤ / q Cσ ( x, b ) has negative values as b < . q QCσ ( x, b ) (see Fig. 5(c)). Inthe small x region (like x < /
10 in our case), it isbelieved that the contribution of the gluon GPDs be- comes more important, which is beyond the scope of thepresent model. The symmetry around x ∼ / q Qσ ( x, b ) when 1 / ≤ x ≤ / In this work, analogous to the definition of the pionand nucleon impact parameter dependent PDFs, we in- xxxxxx-7hinese Physics C Vol. xx, No. x (201x) xxxxxx q σ ( x , b )( f m - ) (cid:20)(cid:21)(cid:22) x = x = x = b ( fm ) (a) q σ ( x, b ) with σ = 1 GeV − and x = 1 /
10, 3 /
10, and1 / q σ C ( x , b )( f m - ) (cid:23)(cid:24)(cid:25) x = x = x = b ( fm ) (b) q Cσ ( x, b ) with σ = 1 GeV − and x = 1 /
10, 3 /
10 and1 / q σ Q C ( x , b )( f m - ) (cid:26)(cid:27)(cid:28) - - - - x = x = x = b ( fm ) (c) q QCσ ( x, b ) with σ = 1 GeV − and x = 1 /
10, 3 /
10 and1 / Fig. 5. The impact parameter dependent PDFs q σ ( x, b ) and FFs q C,QCσ ( x, b ) with σ = 1 GeV − and x = 1 /
10, 3 / / q σ M ( x , b )( f m - ) (cid:29)(cid:30)(cid:31) x = x = x = b ( fm ) (a) q Mσ ( x, b ) with σ = 1 GeV − and x = 1 /
10, 3 /
10 and1 / q σ Q ( x , b )( f m - ) !" x = x = x = b ( fm ) (b) q Qσ ( x, b ) with σ = 1 GeV − and x = 1 /
10, 3 /
10 and1 / Fig. 6. The impact parameter dependent FFs q M,Q,QCσ ( x, b ) with σ = 1 GeV − and x = 1 /
10, 3 /
10 and 1 / troduce the ρ meson impact parameter dependent PDFs( q ( x, b ) and q ( b )) and impact parameter dependent FFs( q C,M,Q ( x, b ) and q C,M,Q ( b )). By employing the LCCQ,as we have done previously, we carried out the numericalcalculation of those quantities for the first time. We be-lieve that q C,M,Q ( x, b ) may be interpreted as the percent-ages of the contributions to the charge (normalized to1), magnetic dipole µ ρ , and quadrupole moment Q ρ , re-spectively, from a parton with a longitudinal momentumfraction x and a transverse impact space position b ⊥ .Considering the facts that the ρ meson is an extendedobject and there exists a largest measured value of mo-mentum transfer in realistic measurements, a Gaussianform wave packet is employed in our numerical calcula-tion. Our numerical results show that the wave packetapproach plays a similar effect to the cutoff in the in-tegral, which is due to the validity of the constituentquark model. Our numerical results for impact parame-ter charge distributions also show that the width of the Gaussian wave packet should be larger than the Comp-ton wavelength. We expect that this approach is neededin a phenomenological model calculation in order to re-move the possible negative values of the impact parame-ter charge distributions q Cσ ( x, b ), which cannot be under-stood by the density interpretation. Acknowledgements
We would like to thank Stanley J. Brodsky, M. V.Polyakov, and Haiqing Zhou for their encouragementand constructive discussions. This work is supportedby the National Natural Science Foundation of Chinaunder Grant No. 11475192, by the fund provided tothe Sino-German CRC 110 “Symmetries and the Emer-gence of Structure in QCD” project by the NSFC underGrant No.11621131001, and the Key Research Programof Frontier Sciences, CAS, Grant No. Y7292610K1.
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