Magnetic moments of octet baryons, angular momenta of quarks and sea antiquark polarizations
aa r X i v : . [ h e p - ph ] M a r Magnetic moments of octet baryons, angularmomenta of quarks and sea antiquarkpolarizations
Jan BartelskiInstitute of Theoretical Physics,Faculty of Physics, Warsaw University,Ho ˙ z a 69, 00-681 Warsaw, Poland.Stanis law TaturNicolaus Copernicus Astronomical Center,Polish Academy of Sciences,Bartycka 18, 00-716 Warsaw, Poland. Abstract
One can determine antiquark polarizations in a proton using the in-formation from deep inelastic scattering, β decays of baryons, orbitalangular momenta of quarks, as well as their integrated magnetic dis-tributions. The last quantities were determined previously by us per-forming a fit to magnetic moments of a baryon octet. However, be-cause of the SU (3) symmetry our results depend on two parameters.The quantity Γ V , measured recently in a COMPASS experiment, givesthe relation between these parameters. We can fix the last unknownparameter using the ratio of up and down quark magnetic momentswhich one can get from the fit to radiative vector meson decays. Wecalculate antiquark polarizations with the orbital momenta of valencequarks that follow from lattice calculations. The value of the differenceof up and down antiquark polarizations obtained in our calculationsis consistent with the result obtained in a HERMES experiment. PACS numbers: 12.39.-x, 13.40.Em, 13.88.+e1
Introduction and Framework
In Ref. [1] we have proposed a model for magnetic moments of SU(3) octetbaryons. We get an excellent fit using few parameters. In this model mag-netic moments of baryons are sums of products of magnetic moments ofquarks and corresponding integrated quark densities. The corrections, whichtake into account exchange phenomena, were also included. To determinethe size of such corrections we use sum rules for magnetic moments of octetbaryons (as in [1] and [2]). After subtraction of the exchange contributionswe are left with a SU (3) symmetric part of these moments, which can beexpressed as independent contributions from quarks and antiquarks in con-sidered baryon. The magnetic moment of a quark as well as its integratedmagnetic density are Q dependent, whereas its product is not. Other cor-rections coming from exchange effects of pions and gluons are incorporatedin redefinition magnetic moments of quarks and their integrated densities;hence magnetic moments of quarks are not equal to their Dirac values.So, after subtracting the pion correction to nucleon magnetic momentsand taking into account Σ − Λ mixing, we are left with independent oneparticle contributions to baryon magnetic moments (sum rules for magneticmoments are satisfied) and we use for them high energy parametrization (in-tegrated parton densities) to describe such contributions. We believe thatmost of all other pion exchange and gluon exchange corrections are takeninto account in the high energy parametrization (see, e.g., [3]). For suchparametrization, in the case of axial densities, one does not include explicitlypion and gluon corrections and we do the same for integrated magnetic den-sities. In contrary such corrections are present in models of bound quarks,e.g., in [4], [5] and [6]. One gluon correction with gluon exchanged betweendifferent quarks can also correspond to higher twist diagrams in deep inelasticscattering.In integrated magnetic quark densities, besides of spin contributions, wehave also orbital angular momentum contributions (see also [7]). Here weshall consider two models: the one in which we neglect orbital angular mo-mentum contribution and the second with such contribution included. In thefirst case we have for such integrated densities δq ≡ ∆ q val + ∆ q sea − ∆¯ q, (1)In the case with angular momentum the formulas are2 L q ≡ ∆ q val + ∆ q sea − ∆¯ q + L q , (2)where L q = < ˆ L qz > − < ˆ L ¯ qz > = < ˆ L q val z > + < ˆ L q sea z > − < ˆ L ¯ q sea z > . (3)Taking into account exchange contributions (as was explained in detailin [1]), i.e., isovector contribution connected with charged pion exchange be-tween different quarks (see also Franklin [8, 9]) and Σ − Λ mixing, the SU (3)symmetric part of baryon octet magnetic moments can be parametrized interms of four quantities: c , c , c , r . From the fit we get for these parameters[1] c = 0 . ± . n.m. ,c = 1 . ± . n.m. ,c = 0 . ± . n.m. , (4) r = 1 . ± . . Hence, six quantities: three quark magnetic moments and three quark den-sities cannot be determined using only four parameters given in Eq. (4). Soas in [1], we introduce two additional parameters, ǫ and g , and our quantitiesbecome the functions of them. The parameters ǫ and g are defined in Eqs.(5) and (6): ǫ = − − µ d µ u , (5) g = δ L u − δ L d. (6)Now we can express magnetic quark densities as δ L u = g r [ f ( ǫ ) + 1 + 3 r ] ,δ L d = g r [ f ( ǫ ) + 1 − r ] , (7) δ L s = g r [ f ( ǫ ) − , where 3 ( ǫ ) = (3 + ǫ ) rc c − rc − ǫ ( c + rc ) . (8)One can also express magnetic moments of u, d and s quarks in terms ofour parameters ǫ and g : µ u = 8 c g (3 + ǫ ) ,µ d = − ǫ ) c g (3 + ǫ ) , (9) µ s = − rc − c + ǫ ( c + 3 rc )] g (3 + ǫ ) . The parameter g sets a scale at which we have calculated our quantities.From Eqs. (7) and (9) we have that µ q δ L q and quark magnetic momentratios, e.g., µ u /µ d , do not depend on g .The new quantity Γ V , which in our notation isΓ V ≡ δu + δd = δ L u + δ L d − L u − L d , (10)is measured in the COMPASS experiment [10] and one getsΓ V = 0 . ± . stat. ) ± . syst ) . (11)In the case when ∆ q sea = ∆¯ q this quantity is not a valence one: ∆ u val +∆ d val .Using Eqs. (7) and (10) we can express our parameter g as: g = 3 r (Γ V + L u + L d ) f ( ǫ ) + 1 . (12)Hence, the COMPASS measurement gives the relation between intro-duced parameters ǫ and g . So we will have only one unknown parameterhowever, the orbital angular momenta of quarks are present in the formulas.We know that integrated axial densities, used in deep inelastic scatteringanalysis, differ from δq by a sign in an antiquark term:∆ q ≡ ∆ q val + ∆ q sea + ∆¯ q . (13)From Eqs. (1,2) and (13) we can express ∆¯ q as4¯ q = 12 (∆ q − δq ) = 12 (∆ q − δ L q + L q ) . (14)Let us express the function f ( ǫ ) in the form f ( ǫ ) = 3 r (Γ v + L u + L d ) a − η + L u − L d − , (15)where η is defined by η ≡ ∆¯ u − ∆ ¯ d. (16)Equation (15) gives us a relation between our two basic parameters ǫ and η (which replaces parameter g ). The quark integrated axial densities ∆ u ,∆ d , ∆ s can be determined from∆ u = 13 a + 16 a + 12 a , ∆ d = 13 a + 16 a − a , (17)∆ s = 13 a − a , where the values of a , a and a are obtained from neutron and hyperon β decays [11] and deep inelastic scattering spin experiments [12]. We have a = 0 . ± . ,a = 0 . ± . , (18) a = 1 . ± . . We can get additional information (although not very precise) using thevalue of η from the HERMES experiment [13]. We will take η = 0 . ± . x : (0 . ≤ x ≤ . u , ∆ ¯ d and ∆¯ s as a function of parameter η usingEqs. (7), (12), (15), and (17), getting∆¯ u = 16 a + 112 a −
14 Γ V + 12 η , ∆ ¯ d = 16 a + 112 a −
14 Γ V − η , (19)∆¯ s = 16 a − a −
14 Γ V + a − η r + L u − L d r − L u + L d − L s . Η -0.15-0.1-0.050.050.10.15 D q (cid:143) Figure 1:
The antiquark polarizations for ¯ u (solid line), ¯ d (short-dashed line),and ¯ s (long-dashed line) versus η in the model where angular momenta ofquarks are neglected. One sees that ∆¯ u , and ∆ ¯ d do not depend directly on orbital angularmomenta. Knowing the precise value of η in the whole range (0 ≤ x ≤ u and ∆ ¯ d in our model. Let us start with an assumption that all angular momenta of quarks arenegligible (i.e., we put them equal to zero). In Fig. 1 we present antiquarkpolarizations ∆¯ q for u, d and s quarks as a functions of parameter η .The fact that ∆¯ u +∆ ¯ d ≈ V measured bythe COMPASS experiment. We can also see how the values of ∆¯ u , ∆ ¯ d , and∆¯ s change when we change η between − .
01 and 0 .
11, i.e., within 1 standarddeviation off central value. When we take the number from the HERMESexperiment ( η = 0 . u = 0 . ± . , Ε -0.2-0.10.10.2 D q (cid:143) Figure 2:
The antiquark polarizations for ¯ u (solid line), ¯ d (short-dashed line),and ¯ s (long-dashed line) versus ǫ in the model where angular momenta ofquarks are neglected. ∆ ¯ d = − . ± . , (20)∆¯ s = 0 . ± . . The values are a little bit different from the antiquark values quoted by HER-MES [13]. Using Eqs. (4), (7) and (15) we can calculate the correspondingvalue of parameter ǫ . We get ǫ = − .
17 for η = 0 .
05. In general we can useEq. (15) to eliminate η and express ∆¯ q as a function of ǫ in the case whenwe neglect dependence on orbital angular momenta. In Fig. 2 we show suchdependence, i.e., ∆¯ q ( ǫ ).One can try to determine the value of parameter ǫ using the experimentaldata for radiative vector meson decays. The model which is used to determine µ u µ d is not as sophisticated as is the one for baryon magnetic moments; wehave used similar formulas as in [14]. One does not include the contributionfrom orbital momenta of quarks in such a model; however, in [15] it wasshown that such contributions may be small. Performing the fit one gets µ u µ d = − . ± .
07 which gives, with the help of Eq. (5), ǫ = 0 . ± .
04. Ifwe use this value of parameter ǫ we will get for ∆¯ q ∆¯ u = 0 . ± . , d = − . ± . , (21)∆¯ s = − . ± . . In this case the corresponding value of parameter η is 0 . ± .
08. It looksas if ∆¯ q calculated from the HERMES value of η and ∆¯ q calculated using ǫ gotten from vector meson decays are not consistent. Now we shall consider the model with nonzero orbital angular momenta ofquarks. We will use the values of such momenta calculated numerically onthe lattice. From [16] we have L u = L valu ( lattice ) = − . ± . , (22) L d = L vald ( lattice ) = 0 . ± . . These values are determined at Q = 4 GeV . Let us make some comments.From Eq. (3) angular momentum of quarks consists of angular momentumof valence quarks, sea quarks, and antiquarks. From [16] we have only infor-mation on valence quark contribution. We neglect the rest because of lack ofknowledge; it actually means that we assume that orbital angular momentaof sea quarks and antiquarks are equal [see Eq. (3)]. The existence of asmall correction to this hypothesis cannot be excluded. The orbital angularmomentum of quarks is scale dependent [17]. There are two possibilities:First one can take into account evolution equations for angular momentaand start with initial conditions at low energies taking as is suggested by A.W. Thomas values that follow from the cloudy bag model [18], [19], whichtake into account the relativistic motion of quarks, chiral pion cloud, and onegluon exchange corrections. Second, one can take initial conditions at highenergy as was done in [20]. In our case we use high energy parameters so it isnatural to use high energy initial conditions as in [20]. From our procedureit seems that we cannot go in Q scale below 1 GeV . From [20] it followsthat the angular momenta of quarks for Q > GeV are only weakly scaledependent. The angular momenta of valence quarks and Γ V determine thescale used in our equations. Using Eq. (19) and eliminating η [using Eq.(15)] we can get formulas for ∆¯ q (for u, d and s quarks) with orbital angularmomenta taken into account:∆¯ u = 16 a + 112 a + 14 a −
14 Γ V − r V + L u + L d f ( ǫ ) + 1 + 14 ( L u − L d ) , Ε -0.15-0.1-0.050.050.10.15 D q (cid:143) Figure 3:
The antiquark polarizations for ¯ u (solid line), ¯ d (short-dashed line),and ¯ s (long-dashed line) versus ǫ in the model where angular momenta ofquarks are taken into account. ∆ ¯ d = 16 a + 112 a − a −
14 Γ V + 3 r V + L u + L d f ( ǫ ) + 1 −
14 ( L u − L d ) , (23)∆¯ s = 16 a − a −
14 Γ V + 3 r V + L u + L d f ( ǫ ) + 1 −
14 ( L u + L d − L s ) . The dependence of ∆¯ q on ǫ , calculated from Eq. (23), is shown in Fig. 3.When we use the result for ǫ from the fit to radiative decays of vectormesons we get for ∆¯ u , ∆ ¯ d and ∆¯ s ∆¯ u = 0 . ± . , ∆ ¯ d = − . ± . , (24)∆¯ s = − . ± . . The errors are quite big so the determination is not very conclusive. For η = ∆¯ u − ∆ ¯ d we get the value 0 . ± .
09 which has to be compared with0 . ± .
06. The agreement is reasonable despite the fact that all errors arerelatively big. Let us stress that we have used the value of ǫ calculated from9 Η -0.06-0.04-0.020.020.040.06 D q (cid:143) Figure 4:
The antiquark polarizations for ¯ u (solid line), ¯ d (short-dashed line),and ¯ s (long-dashed line) versus η in the model where angular momenta ofquarks are taken into account. the fit to experimental data on radiative vector meson decays and have takeninto account orbital angular momenta of quarks from lattice calculations.If we do not want to use the information about ǫ from the fit to mesondecays, we can use Eq. (19) where this parameter is eliminated and ∆¯ q arefunctions of η . The dependence of ∆¯ q on η for u, d, and s antiquarks isshown in Fig. 4.If we knew precisely the value of η we could predict ∆¯ u , ∆ ¯ d , and ∆¯ s val-ues. From Eq. (19) we see that ∆¯ u and ∆ ¯ d do not depend on orbital angularmomenta of quarks; only ∆¯ s does. It means that precise determination of ∆¯ s could be the additional test of the importance of orbital angular momentaof quarks. When we use the η value obtained in the HERMES experimentwe will get ∆¯ u and ∆ ¯ d as in Eq. (20) and ∆¯ s = − . ± .
04. We can-not expect additional experimental information about magnetic moments ofquarks. In the future more precise measurements of antiquark polarizationscould be a real verification of our model.We also want to give a comparison of two models, i.e., without and withorbital angular momenta of quarks taken into account. The relation thatfollows from the COMPASS measurement of Γ V , Eq. (15), could be rewritten10 Ε -0.2-0.10.10.20.30.40.5 Η Figure 5:
The parameter η versus ǫ in the model where angular momentaof quarks are taken into account (solid line) and corresponding errors (long-dashed lines) compared to curves in the model where angular momenta areneglected (short-dashed line) and corresponding errors (dotted lines). The sizeof the rectangle is determined by the error of ǫ from radiative vector mesondecays and the error in the measurement of η in the HERMES experiment. in the form η = 12 a − r V + L u + L d f ( ǫ ) + 1 + 12 ( L u − L d ) . (25)In Fig. 5 we show the function η ( ǫ ) for both models with the correspondingerrors. The rectangle is given by the errors (1 standard deviation) of ǫ and η .It seems that it is an indication in favor of including nonzero orbital angularmomenta of quarks. From the COMPASS measurement of Γ V we have a relation between two pa-rameters not determined in our previous fit to magnetic moments of baryons.Hence, we have only one independent parameter and it could be either ǫ or η . We have discussed two possibilities: the first with inclusion of the orbitalangular momenta of quarks suggested by calculations on the lattice and the11econd with such angular momenta neglected. We have presented in bothcases the dependence of antiquark polarizations on a single independent pa-rameter (being ǫ or η ). The results are plotted in Fig. 1 to 4. In order tofind the antiquark polarizations we can use the result for η from the HER-MES experiment or the value of ǫ obtained from the fit to radiative vectormeson decays. Unfortunately the errors are big and the results are not veryconclusive.The relation between η and ǫ plotted with errors in Fig. 5 shows that thesolution with orbital angular momenta of valence quarks taken into accountis preferred.With orbital angular momenta taken into account and the value of ǫ takenfrom the fit to radiative vector meson decays, we obtain values of antiquarkpolarizations. Such a procedure gives the prediction η = 0 . ± .
09 thatseems to be consistent with the value 0 . ± .
06 from the HERMES ex-periment. Because it is difficult to get additional information on magneticmoments of quarks, it seems that more precise values of antiquark polariza-tions (maybe from a Jefferson Lab experiment) could be a real verificationof our model. 12 eferences [1] J.Bartelski, S.Tatur, Phys. Rev. D : 014019 (2005);[2] J.Bartelski, R.Rodenberg, Phys. Rev. D , 2800 (1990);[3] J.Bartelski, R.Rodenberg, Z. Phys. C , 263 (1990);[4] A.Buchmann, E.Hern´andes, k.Yazaki, Nucl. Phys. A , 661 (1994);[5] S.Th´eberge, A.W.Thomas, Phys. Rev. D , 284 (1982); Nucl. Phys. A , 252 (1983);[6] H. Høgasen, F.Myhrer, Phys. Rev. D , 1950 (1988);[7] Z.Ye, in Proceedings of the XIVth International Workshop on Deep In-elastic Scattering Tsukuba (Japan), April 20-24, 2006 (World Scientific,Singapore, 2007); M. Mazous et al. [Jefferson Lab Hall A Collaboration]Phys. Rev. Lett. , 242501 (2007);[8] J.Franklin, Phys. Rev. D , 033010 (2002);[9] J.Franklin, Phys. Rev. D , 1542 (1984);[10] M.Alekseev et al. [COMPASS Collaboration] Phys. Lett. B , 458(2008);[11] C.Amsler et al. (Particle Data Group), Phys. Lett. B , 1 (2008);[12] V.Y.Alexakhin et al. [COMPASS Collaboration] Phys. Lett. B , 8(2007);[13] A.Airapetian et al. (HERMES Collaboration), Phys. Rev. Lett. ,012005 (2004); A.Airapetian et al. (HERMES Collaboration), Phys.Rev. D : 012003 (2005);[14] D.A.Geffen, W.Wilson Phys. Rev. Lett. , 370 (1980);[15] L.Ya.Glozman, C.B.Lang and M.Limmer Few Body Syst. , 91 (2010);[16] Ph. H¨agler et al. [LHPC Collaboration] Phys. Rev. D :094502 (2008);[17] X.Ji, J.Tang and P.Hoodbhoy, Phys. Rev. Lett. , 740 (1996);1318] A.W.Thomas, Phys. Rev. Lett. , 102003 (2008);[19] A.W.Thomas, Int. J. Mod. Phys. E18