Strange form factors of the nucleon in the chiral quark-soliton model

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  • DOI: 10.1140/epjad/s2005-04-020-4Eur Phys J A (2005) 24, s2, 9396 EPJ A direct

    electronic only

    Strange form factors of the nucleonin the chiral quark-soliton model

    A. Silva1 a, D. Urbano1,2, H.-C. Kim3, and K. Goeke4

    1 Centro de Fsica Computacional, Departamento de Fsica da Universidade de Coimbra, P-3004-516 Coimbra, Portugal2 Faculdade de Engenharia da Universidade do Porto, R. Dr. Roberto Frias s/n, P-4200-465 Porto, Portugal3 Department of Physics, Pusan National University, 609-735 Pusan, Republic of Korea4 Institut fur Theoretische Physik II, Ruhr-Universitat Bochum, D-44780 Bochum, Germany

    Received: 15 October 2004 / Published Online: 8 February 2005c Societa` Italiana di Fisica / Springer-Verlag 2005

    Abstract. The results for the strange electromagnetic and axial form factors in the chiral quark-solitonmodel are reviewed. The roles of a new quantization method and of the meson asymptotic behaviour arediscussed. Predictions for the A4 and G0 experiments are presented.

    PACS. 12.40.-y Other models for strong interactions 14.20.Dh Protons and neutrons 13.40.Gp Elec-tromagnetic form factors

    1 Introduction

    The question of the strange content of the nucleon is avery important one in the understanding of hadron struc-ture, which eventually is to be settled by experiment. In-deed, on one hand there is no obvious theoretical reasonwhy strange ss pairs should not contribute to the nucleonproperties. On the other hand the theoretical mechanismand the actual contribution of strange quarks are not yetunderstood as well.

    Early results about the strange quark bilinears in thenucleon included the analysis of the sigma term [1] andpolarized DIS. These rst results indicated strange quarknonvanishing contributions to the nucleons mass and spin,respectively. The strange vector currents associated withelectromagnetic form factors were discovered to be acces-sible by means of weak neutral current experiments [2,3,4]. For reviews see [5,6].

    Due to its importance, the question of the strange formfactors has been addressed before in the context of the chi-ral quark-soliton model (CQSM) [7,8]. This work reviewsthe CQSM results [9,10] for the strange electromagneticform factors and presents also the strange axial form fac-tor. These form factors are obtained using a new quan-tization method [11] and studied phenomenologically interms of the eects of the asymptotic behavior of the me-son elds.

    a Financial support from the organization of PAVI04 andfrom the Center for Computational Physics (CFC) for theparticipation in PAVI04 is kindly acknowledged. The work ofA.S. has been partially supported by the grant PRAXIS XXIBD/15681/98.

    2 The chiral quarksoliton model (CQSM)

    The chiral quark-soliton model (CQSM) has been appliedsuccessfully to the description of many baryon observ-ables, both in avours SU(2) and SU(3). See [12,13] forreviews. The Lagrangian of the CQSM reads,

    L = (x) (i m MU5) (x) (1)where m = diag(mu,md,ms) is the current quark massmatrix and M the constituent quark mass. M is the onlyfree parameter in the model. The main property of themodel in relation with hadron physics is chiral symmetryand its spontaneous breaking. The auxiliary chiral eldsU5 = ei

    5 in (1), may be interpreted as physical me-son elds and, thus, the Lagrangian (1) corresponds to atheory of constituent quarks interacting through the ex-change of mesons.

    However, the eld description embodied in (1) is notrenormalizable. A regularization method is therefore nec-essary to completely dene the model. This work usesthe proper-time regularization, which introduces one cut-o parameter into the model. The remaining parametersof the model are the current quark masses. In this workisospin breaking eects have been neglected and thereforethe current mass parameters are the equal current masses(m) of the u and d quarks and the current mass (ms) of thestrange quarks. In order to t these parameters, one cal-culates the pion and kaon masses as well as the pion decayconstant in terms of the parameters of the model. Equat-ing the model results for these quantities with their exper-imental values allows to x the parameters of the model:the current nonstrange quark masses, the strange quarkmass and the cut-o come out around 8 MeV, 164 MeVand 700 MeV, respectively.

  • 94 A. Silva et al.: Strange form factors of the nucleon in the chiral quark-soliton model

    3 The model baryon state

    In broad terms, the construction of the model baryonstates necessary to compute baryon matrix elements areobtained in a projection after variation procedure, withthe variation referring to the mean eld solution.

    Taking the large Nc limit of the correlation functionof two baryon currents and neglecting the strange quarkmass lead to an eective action from which the mean eldsolution is obtained. At this level the eective action isproportional to the energy of the mean eld, Se [U ] M [U ], which is a functional of the chiral elds U : M [U ] =Ncv [U ] + s [U ], with v the energy of the valence level,occupied by Nc quarks, and s the regularized energy ofthe Dirac sea. Both result from the solution of the Diracone-particle problem n [U ] = n|h (U) |n with

    h (U) = i0ii + 0MU5 + m0 . (2)In order to solve such problem one further restricts themesons to the hedgehog shape, = P (r) n , witha prole function P (r) vanishing at large distances. Theminimization of the action corresponds to a prole func-tion Pc(r) which represents the mean eld soliton. Theformalism in avour SU(3) is built upon the embeddingof the SU(2) hedgehog into an SU(3) matrix [14]:

    U = eiP (r)8

    a=1 naa

    (eiP (r)

    3i=1 n

    i i 00 1

    ). (3)

    3.1 Quantization

    The quantization of the mean eld is achieved in the con-text of collective coordinates. The collective coordinatesare the orientations of the soliton in conguration as wellas in avour spaces (related by the hedgehog) and the po-sition of the center of mass. These coordinates are mostsuited to describe the rotational zero modes, i.e. uncon-strained large amplitude motion, which should be treatedexactly. The model calculation neglects modes normal tothe zero modes, which are subjected to restoring forcesand hence suppressed.

    The time dependent solutions are constructed on thebasis of the restriction to the zero modes by the ansatz1

    U(x) A(t)Uc(x)A(t) (4)with the collective coordinates contained in A.

    In the laboratory system the Dirac operator becomes

    D(U) = A[D(Uc) + AA + 0AmA

    ]A , (5)

    which clearly shows the two expansion parameters usedin this work: the angular velocity AA (t) = iE

    /2 andthe strange mass related parameter m, discussed below,in Sect. 3.2, in connection with the asymptotic meson be-haviour eects.

    1 Not showing translational zero modes explicitly.

    The collective coordinate Hamiltonian is

    H = M(Uc) + Hsccoll + Hsbcoll , (6)i.e. it is composed by a avour symmetry conserving (sc)and a symmetry breaking (sb) pieces. While the rst istreated exactly, the second is treated perturbatively. Thewave functions of the symmetry conserving part are theWigner SU(3) matrices. For the octet

    |B, 8sc (8)sc (A) =8 ()Y /2+J3 D(8) (A) . (7)

    with = (Y, T, T3) and = (Y , J, J3) standing for thequantum numbers of the baryon state (Y = 1). Thesymmetry breaking part of the Hamiltonian leads to arepresentation mixing, in rst order perturbation theory,from which the baryon wave function |B, 8sc picks termsfrom higher order representations [14].

    3.2 Asymptotics of the meson eldsand symmetry conserving quantization

    The question of the asymptotics of the meson elds orig-inates in the fact that the embedding (3) forces all themeson elds to have a common asymptotic behavior. Thisis not satisfactory in the SU(3) case due to the large massdierence between pions and kaons. In order to have someinformation on how large these eects may be, one ex-ploits in this work the fact that there is no prescription tox the multiples of the unity mass matrices in (2) and inthe symmetry breaking piece m of (5):

    0D(Uc) +AmA = iii +MU5 +m+AmA (8)where m = M1 + M88, with M1 = (ms m)/3 andM8 = (m ms)/

    3. In this case one may obtain larger

    mass asymptotics of the meson elds, by, in (8), increasingm at the expense of a lower M1. The pion asymptotics willbe denoted in the following by ( 140 MeV) andthe kaon asymptotics by K( 490) MeV).

    The symmetry conserving quantization [11] used inthis work avoids the problem which is encountered in thismodel by using the ansatz (4) to compute the relation be-tween isospin (T ) and spin (S) operators. While the resultfrom (4) reads

    T = D(n) (A)J 3D(n)8 (A)

    I 2I2

    , (9)

    the correct relation implies I 2 = 0. The symmetry con-serving quantization identies the terms like I 2 in the ex-pressions for observables on the basis of the quark modellimit of the CQSM.

    4 Strange electromagnetic form factors

    The knowledge of the form factors of the octet vector cur-rents is enough to provide the form factors for each avour:

    V 0 = uu + dd + ss , (10)

    V 3 = uu dd , (11)3V 8 = uu + dd 2ss . (12)

  • A. Silva et al.: Strange form factors of the nucleon in the chiral quark-soliton model 95

    Table 1. Strange magnetic moment (in n.m.) and electricand magnetic radii (in fm2). The constituent quark mass is420 MeV and the strange quark mass 180 MeV. and K standfor the two asymptotic descriptions of the mesons as discussedin Sect. 3.2.

    Kr2sE 0.220 0.095s 0.074 0.115r2sM 0.303 0.631

    Fig. 1. Strange electric form factor of the nucleon. Conventionsand model parameters as in Table 1.

    In this work the form factors for these currents were rstobtained for the two asymptotic behaviours. The eectsof the asymptotics was then studied at the level of baryonform factors by combining avour form factors with pionasymptotics for nonstrange quarks with kaon asymptoticsfor strange form factors, i.e. by taking

    GBE,M (Q2) = GuB()E,M (Q

    2) + GdB()E,M (Q2) + GsB(K)E,M (Q

    2) .(13)

    The most signicant consequences of this ansatz for thebaryon octet electromagnetic form factors is the improve-ment in the overall description of magnetic form factorsand the neutron electric form factor. This supports theexpectation that the strange form factors are better de-scribed in terms of kaon ( K) rather than in pion( ) asymptotics. The results for ( ) are never-theless always presented.

    The strange magnetic moment and the radii are pre-sented in Table 1 and the strange electric and magneticform factors in the CQSM are shown in Fig. 1 and Fig. 2,respectively. A particular aspect of these results is the pos-itive value for the strange magnetic moment, which seemsto be at variance with many theoretical calculations basedin dierent approaches. The CQSM seems nevertheless tobe able to accommodate higher values of this quantity,up to 0.41 n.m., as is found in the model independentanalysis of [15]. This pinpoints the SU(3) structure ofthe rotational and mass corrections as the reason for such

    Fig. 2. Strange magnetic form factor in nuclear magnetons.Conventions and model parameters as in Table 1.

    positive value, similarly to what is observed in other mod-els by maintaining just the SU(3) structure [16].

    4.1 The SAMPLE, HAPPEX, A4, and G0 experiments

    The experimental value for the magnetic form factor ob-tained by the SAMPLE collaboration [17] is

    GsM (Q2 = 0.1) = +0.370.200.260.07 (n.m.) . (14)

    The CQSM results of this work underestimates this exper-imental result, still falling however within the error bar,shown in Fig. 2 at Q2 = 1. The HAPPEX result [18]

    (GsE + 0.392GsM )(Q

    2 = 0.477) = 0.014 0.030 , (15)is, on the contrary, overestimated in the CQSM, with thenearest result (kaon asymptotics) being 0.073.

    Using the necessary form factors, one may study thecombination GsE(Q

    2) + (Q2, )GsM (Q2), with (Q2, ) =

    GpM /GpE , = Q

    2/(4M2N ), 1 = 1+2(1+ ) tan2(/2),

    for the values of and Q2 used in current or near future ex-periments. Figure 3 compares the model predictions withthe recent forward data,

    (GsE + 0.225GsM )(Q

    2 = 0.23) = 0.039 0.034 (16)(GsE + 0.106G

    sM )(Q

    2 = 0.10) = 0.074 0.036 , (17)with (16) from [19] and (17) from [20], by the A4 Col-laboration and gives predictions for the backward angle.Figure 4 gives predictions for the G0 experiment.

    5 Strange axial form factor

    The axial currents of interest are expressed in term ofavour components in the same way as (12). The axialform factors for the octet axial currents have also beencomputed in the model with the same accuracy of the

  • 96 A. Silva et al.: Strange form factors of the nucleon in the chiral quark-soliton model

    0.00

    0.10

    0.20

    0.30

    0.40

    0.50

    0.0 0.2 0.4 0.6 0.8 1.0

    Q2 [GeV2]

    GsE(Q2) + (Q2, )GsM(Q

    2)

    = 35

    K

    = 145

    K

    A4

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