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Contributions of Qqqq¯Q Components to Nucleon Spin Structure

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Contributions of Qqqq¯Q Components to Nucleon Spin Structure CPC(HEP & NP), 2009, 33(Suppl..): 18|20 Chinese Physics C Vol. 33, Suppl.., Mar., 2009 Contributions of qqqqq¯ components to nucleon spin structure * WAN Meng() TIAN Wei-Zhao(Xm) CHEN Hong(ö)1) (School of Physical Science and Technology, Southwest University, Chongqing 400715, China) Abstract The spin structure of nucleons is presented in the framework of an extended quark model which in addition to the conventional qqq structure also takes into account qqqqq¯ admixtures in the nucleon wave functions, where the qqqqq¯ components are in colored quark cluster configurations. The axial vector weak coupling constant and spin distributions for polarized nucleons as well as spin content are obtained for the lowest positive parity qqqqq¯ configurations in flavor-spin dependent interaction. In particular, the contributions of the down and strange quarks to the proton spin and the sum rule for polarized neutron are negative, in agreement with recent experiments. Key words five quark components, spin content, axial vector coupling constant, sum rule for polarized nucleon, nucleon PACS 12.39.Jh, 14.20.Dh, 13.30.Ce, 11.55.Hx 1 Introduction in the quark model which in addition to the conven- tional qqq structure also takes into account qqqqq¯ The extended quark model which takes into admixtures in the nucleon wave functions. It was [7, 8] account the explicit coupling of the lowest posi- shown that spin structure of the nucleon is deter- tive parity qqqqq¯ components to the conventional mined by the spin content of nucleon carried by all qqq structure was first proposed to study the quarks, magnetic moments of the nucleon, and the baryon spectroscopy [1] and electromagnetic transi- sum rule of polarized protons and neutrons in the [4|6, 9] tion properties[2]; in particular, it help us to under- deep inelastic scattering. Previous works have stand the long-standing puzzle of the low mass of indicated that the qqqqq¯ components in baryons seem [3] to be mainly in colored quark cluster configurations the Roper resonance P11(1440) . If this correct, one should also expect that the pentaquark components rather than in baryon-meson configurations or in the play an explicit role in the nucleon wave function. form of a sea of quark-antiquark pairs. The formal- Indeed, the implication of these quark cluster config- ism summarized in that references is elaborated and urations have been analyzed for the pentaquark com- is now applied to the weak decay constant and sum ponents with hidden strangeness in the nucleons[4, 5]. rules for polarized nucleons in the deep inelastic scat- It seem to be feasible to account for the empirical tering as well as spin contents. indications for the positive strangeness magnetic mo- 2 Wave functions of nucleons ment of proton for the configurations that are ex- pected to have the lowest energy in the flavor-spin The nucleon wave function with the qqqqq¯ com- [6] hyperfine interaction. Further, it is shown that ponents in addition to the conventional qqq structure addition of the lowest positive qqqqq¯ admixtures to may be expressed in the general form[6]: the ground state baryon can improve the overall de- N = " N;qqq +p1 "2 N;qqqqq¯ ; (1) scription of the magnetic moments of the octet and j i j i − j i decuplet baryons. where " and p1 "2 represent the probability am- − Here the spin structure of nucleons is examined plitudes of qqq and qqqqq¯ components. The wave Received 3 September 2008 * Supported by National Natural Foundation of China (10575083) 1) E-mail: [email protected] ©2009 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd Suppl.. WAN Meng et alµContributions of qqqqq¯ components to nucleon spin structure 19 function of the qqq components can be expressed in flavor and spin wave functions in the qqq component the form are listed in Tables 1 and 2. One can construct the spatial wave function in terms of the internal coor- 1 3 N;qqq = [21]f [21]s +[21]f [21]s [3]x[1 ]c; (2) j i p2 1 1 2 2 dinates ξi, which are in general defined by the con- stituent coordinates for the n-body system 3 where the partitions [3]x, [1 ]c and [21]f;s have been k used to indicate the spatial, color (or flavor) SU(3) 1 ξk = ri krk+1 ; k = 1;2; ;n 1: (3) p 2 hX − i ··· − and the spin SUs(2) symmetries. The explicit form of k +k i=1 Table 1. The flavor wave functions of the qqq components with the flavor symmetry [21]f , and of the qqqqq¯ components with the flavor symmetry [321]f in which the qqqq has the symmetry [22]f . The abbreviation notations [q1;q2] = q1q2 −q2q1 and fq1;q2g = q1q2 +q2q1 are used in table. Young partition p n 1 1 qqq [21]1 ( u;u d u;d u) ( u;d d d;d u) p6 f g −f g p6 f g −f g 1 1 [21]2 [u;d]u [u;d]d p2 p2 1 1 qqqqq¯ [321]1 u;d ; u;s u;u ; d;s )¯s u;u ; d;d u;d ; u;d )u¯ 6p2 hff g f gg−ff g f gg 6p2 hff g f gg−ff g f gg u;u ; d;d u;d ; u;d d¯ u;d ; d;s u;s ; d;d ¯s −ff g f gg−ff g f gg i −ff g f gg−ff g f gg i 1 1 [321]2 [u;d];[s;u] ¯s [u;d];[u;d] d¯ [u;d];[u;d] u¯ [u;d];[d;s] ¯s 2p6 hf g −f g i 2p6 hf g −f g i Table 2. The spin-up wave functions of the qqq may be expected to become the component of nucle- components with the symmetry [21]s, and of ons. The flavor-spin interaction generated by the one- the qqqqq¯ components with the spin symme- meson-exchange can give the correct ordering of the try [32]s in which the qqqq has the symmetry baryon spectrum in all flavor sectors, indicting that [22] . s antisymmetric flavor and spin configurations have the Young partition nucelon lowest energy[1]. Consequently for the baryon octet, 1 qqq [21]1 ; ; p6 f" "g # −{↑ #g " the lowest energy qqqq system is expected to have the 1 symmetry [31] [211] [4] [22] [22] , which also be sug- [21]2 [ ; ] x c fs f s p2 " # " 1 gested by the experimental data on the strangeness qqqqq¯ [32]1 ; ; ; 2p3 f"" ##g−f"# #"gf"# #"g " magnetic moment of the proton if one of the quarks 1 [4, 5] [32]2 [ ; ][ ; ] is strange . Thus, the wave functions of the qqqqq¯ 2 "# #" "# #" " components are given in the form[4|6] The wave function for the qqqqq¯ components with 1 1 4 1 2 2 [1 ]cx 4]fs N;qqqqq¯ = C 1 C C the total spin and the positive parity is formed in 1m; s [211]i;[31]j [22]k[22]l 2 j i X X 2 z × the similar way to the qqq component. Since the total m;sz i;j;k;l c x ;1m f ; 1 : (4) qqqqq¯ wave function is a [222]c color singlet and the i j k sl 2 sz anti-quark is in the conjugate color symmetry [11]c, where the sum over i, j, k, and l run over the config- the color part of the qqqq system has to be in a [211]c triplet. The qqqq wave function is required to be urations of [211]c, [31]x, [22]f and [22]s for which the completely antisymmetric, hence the corresponding corresponding Clebsch-Gordan coefficients are non- vanishing. The spatial wave functions x transform orbital-flavor-spin part have mixed symmetry [31]xfs. Since the quark and antiquark have opposite intrin- as three component of the permutation group S4 sym- sic parity, the positive parity demands that the qqqqq¯ metry [31]x. The qqqqq¯ flavor and spin-up wave func- components have to be in the P -state. For the qqqq tions are listed in Tables 1 and 2. system, if the qqqq are in spatially S-wave ground state with spatial symmetry [4]x, the allowed favor- 3 Spin structure of nucleons spin part is in the SUfs(6) symmetry [31]fs. In con- trast, if the qqqq are in a P -wave state [31]x, there In the constituent quark model, the explicit gluon are several allowed SUfs(6) symmetries: [4]fs, [31]fs, degree of freedom is usually neglected and its effect [22]fs and [211]fs. These flavor-spin multiplet can be is included in the quark interaction term. In fact, it split by the spin-dependent hyperfine interaction be- was found that the available data do not require the tween the quarks, only these states with lowest energy gluon to be polarized. The nucleon spin is the sum of 20 Chinese Physics C (HEP & NP) Vol. 33 1 1 quark spin and orbital angular momentum: I n = gn(x)dx = e2σi = (1 "2): (12) i Z 1 2hX i zin −18 − 1 1 i = ∆Σ +∆G(= 0)+Lq: (5) 2 2 One can see that the qqq quark model gives The contributions of all the quarks to the nucleon p n g1 (x)dx = 5=18 and g1 (x)dx = 0. Including the spin for the wave function (1) is Z Z qqqqq¯ contributions, however, gp;n(x)dx could be 1 R 1 ∆Σ = σ = "2 (1 "2): (6) different from their qqq values, and in particular, hX iiN − 3 − i gn(x)dx could be negative.
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