Tensor properties of the nucleon

INHA-NTG-05/2011 Tensor properties of the nucleon Tim Ledwig,1 Antonio Silva,2, 3 and Hyun-Chul Kim4, ∗ 1 arXiv:1103.2255v1 [hep-ph] 11 Mar 2011 Institut für Kernphysik, Universität Mainz, D -55099 Mainz, Germany 2 Faculdade de Engenharia da Universidade do Porto, R. Dr. Roberto Frias s/n, P-4200-465 Porto, Portugal 3 Centro de Fisica Computacional (CFC), P-3004-516 Coimbra, Portugal 4 Department of Physics, Inha University, Incheon 402-751, Korea (Dated: March, 2011) We report in the present talk recent results of the tensor properties of the nucleon within the framework of the chiral quark-soliton model. The tensor and anomalous tensor magnetic form factors are calculated for the momentum transfer up to Q2 ≤ 1 GeV2 and at a renormalization scale of 0.36 GeV2 . The main results are summarized as follows: the flavor tensor charges of the nucleon are yielded as δu = 1.08, δd = −0.31, δs = −0.01, while the up and down anomalous tensor magnetic moments are evaluated as κuT = 3.56 and κdT = 1.83, respectively. The strange anomalous tensor magnetic moment turns out to be κsT = 0.2 ∼ −0.2, compatible with zero. We discuss their physical implications, comparing them in particular with those from the lattice QCD. Keywords: Tensor charges, transversity, the chiral quark-soliton model I. INTRODCUTION The transversity of the nucleon has been one of the most important issues in hadronic physics well over decades. While the non-polarized parton distribution functions (PDFs) and longitudinally polarized PDFs have been extensively studied experimentally as well as theoretically, the transversity of the nucleon suffers from experimental difficulties to measure it. Only very recently, the tensor charges δq were extracted [1] from the experiments [2–4]. The QCDSF and UKQCD lattice collaborations reported also the tensor charges in the context of the spin structure of the nucleon [5]. In this talk, we present recent results of the tensor and tensor anomalous magnetic form factors within the framework of the self-consistent SU(3) chiral quark-soliton model (χQSM) [7, 8]. The χQSM was known to describe successfully the mass splittings of SU(3) baryons and static properties and form factors of the nucleon [9]. The model has certain merits: First it does not have any adjustable parameters. The cut-off parameter is fixed to the pion decay constant, the current quark mass is fitted in such a way that it reproduces kaon properties. The constituent quark mass, which is regarded as only one free parameter of the model, is also fixed to 420 MeV with which the electric properties of the nucleon are well reproduced. Secondly, it is a fully relativistic field-theoretic model. We will see why it is essential to describe the tensor properties of the nucleon. Tensor charges of the nucleon were also investigated within this framework [10, 11]. However, the old version of the SU(3) χQSM was hampered by the symmetry-nonconserving quantization, which causes the breaking of gauge invariance. Thus, it is necessary to revisit the tensor charges of the nucleon within the χQSM and we present in this talk the new results of the tensor charges and their form factors, and those of the anomalous tensor magnetic form factors, closely following Refs. [7, 8]. We will also compare the results of the present talk with the empirical data as well as those lattice calculation. II. RESULTS AND DISCUSSION The detailed formalism can be found in Refs. [7, 8]. We briefly report the results of the tensor charges and anomalous tensor magnetic moments and their form factors. The tensor charges are identical to the axial-vector charges in the 3 0 = gT3 = 5/3, respectively. In Table I, = gT0 = 1 and gA non-relativistic limit, i.e. the singlet and triplet parts are gA we summarize the results of the tensor and axial-vector charges from the SU(2) and SU(3) χQSM. The flavor tensor charges can be obtained by using the singlet, triplet, and octet ones. In Table II, the results of the up, down, and strange tensor charges are listed. Note that the tensor charges are scale-dependent. So, in order to compare the present results with those of the lattice QCD, one needs to scale down the lattice results to the scale of ∗ Electronic address: hchkim@inha.ac.kr 2 Table I: Tensor charges for the singlet, triplet, and octet components in comparison with the axial-vector charges. 0 3 8 gT0 gT3 gT8 gA gA gA SU(3) 0.76 1.40 0.45 0.37 1.18 0.36 SU(2) 0.69 1.45 −− 0.36 1.20 −− the χQSM, i.e. µ2 ≈ 0.36 GeV2 . The results of the χQSM are comparable with the lattice ones, as shown in Table II. The strange tensor charge turns out to be compatible with zero. Table II: Tensor charges for each flavor in comparison with the axial-vector charges. ∆u δu ∆d δd ∆s δs χQSM 0.84 1.08 −0.34 −0.31 −0.05 −0.01 4 4 NRQM − 13 − 13 0 0 3 3 Lattice [5] 1.05 ± 0.16 −0.26 ± 0.01 −− −− In Table III, the numerical results of the anomalous tensor magnetic moments are listed and compared with those of the lattice QCD. While the up anomalous tensor magnetic moment is in good agreement with the lattice but that for the down quark is smaller than the lattice one. As a result, the ratio of the up and down anomalous tensor magnetic moments turns out to be smaller than the lattice result. Table III: The results for the anomalous tensor magnetic moments κqT in comparison with the lattice calculation. κuT κdT κsT u κT /κdT Present work SU(3) Present work SU(2) Lattice [6] 3.56 3.72 3.70 1.83 1.83 2.35 0.2 ∼ −0.2 1.95 2.02 1.58 In Fig. 1, we draw the results of the scaled flavor tensor form factors in comparison with those of the lattice calculation. Figure 1 shows that the lattice results decrease in general almost linearly as Q2 increases. On the other hand, the present results fall off more rapidly. This is not a surprizing result, because a similar behavior was also found in the ∆(1232) electric quadrupole form factor as shown in Ref. [12]. These differences of the Q2 dependence may be due to the fact that the heavy pion mass was employed in the lattice calculation. Moreover, this has been shown explicitly for the nucleon isovector form factor F1p−n (Q2 ) on the lattice [13]. Figure 2 shows the comparison of the results of the flavor anomalous tensor form factors with those from the lattice QCD. Being similar to the case of the tensor form factors, the lattice results decrease more slowly than those of the present work. However, the down form factor is comparable to the lattice one. III. SUMMARY AND OUTLOOK In the present talk, we have reported the recent results of the tensor properties of the nucleon. The tensor charges and anomalous tensor magnetic moments were calculated. The results are in good agreement with those of the lattice calculation except for the down anomalous tensor magnetic moment. The tensor form factors and anomalous tensor form factors were also presented and compared with the lattice results. The present results can be used to describe the transverse spin structure of the nucleon. The corresponding investigation is under way. Acknowledgments HChK is grateful to A. Hosaka for the hospitality during the Baryon 2010. The present work is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of 3 ... . ... ... ... ... ... .. . ... ... ... .. .... .... ... 0.8 .. .... Lattice δu ... .... ... ... .... . .... .... .... .... .... .... .... .... .... . .... 0.6 .... ... ..... ..... ...... ............. χQSM δu ...... ...... ....... 0.4 ........ ......... ......... .......... ............ .. 0.00 0.25 0.50 0.75 1.0 ....... δd(Q2)/δd(0) δu(Q2)/δu(0) 1.0 ...... .. 1.00 ..... ... .... ... .... ... . .... .. ... .... ..... ... .... .. .... .... Lattice δd 0.8 .. .... ..... .... .... ..... ...... .... .... ...... .... .... ...... .... .... ....... 0.6 .... ... ....... ........ ........ ......... ... ............ χQSM δd .................. 0.4 ........ 0.00 0.25 Q2 [GeV2] 0.50 0.75 1.00 Q2 [GeV2] Figure 1: flavor tensor form factors δu(Q2 ), δd(Q2 ) and δs(Q2 ) for the proton. We compare the present results of the renormalization-independent scaled tensor form factors δu(Q2 )/δu(0), δd(Q2 )/δd(0) with those of the lattice QCD [5]. The red (solid) curves designate the χQSM form factors of this work while the blue (dashed) ones corrrespond to the factors from the lattice QCD calculation. 1.0 ...... ...... ... ... . .... .... Lattice u ... .. .. . . 0.8 .... ... .. ............. χQSM u ... . ... .... ... . .... .. ... .... .... .. 0.6 ..... .. .... ..... .... .... ...... .... .... ...... .... .... . . . . .... ... ........ 0.4 ......... .......... ............ .............. .............. 0.2 ............................................................................................................................ 0.00 0.25 0.50 0.75 1.00 κdT (Q2)/κdT (0) κuT (Q2)/κuT (0) 1.0 ..... . .... .... .... .... Lattice d ..... ..... ....... 0.8 ........ . ............. χQSM d ........ .. ....... ... .......... .......... ... 0.6 ......... .... ........ .... .... ......... .... ... ......... . .... ... .......... . .... .... 0.4 ............ .......... 0.2 ............................................................................................................................ 0.00 Q2 [GeV2] 0.25 0.50 0.75 1.00 Q2 [GeV2] Figure 2: The comparison of the up and down anomalous tensor magnetic form factors with lattice results. The solid curves draw the results of the present work with M = 420 MeV, whereas the dashed ones depict the lattice results [6]. In the left panel, the result of the up anomalous tensor magnetic form factor is compared to that of the lattice. The right panel is for the down form factors. The lattice calculation was performed with mπ = 600 MeV. Education, Science and Technology (grant number: 2010-0016265). [1] [2] [3] [4] [5] M. Anselmino et al., Phys. Rev. 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