Strange magnetism in the nucleon

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  • 1 October 1998

    .Physics Letters B 437 1998 184190

    Strange magnetism in the nucleonThomas R. Hemmert a,1, Ulf-G. Meiner a,2, Sven Steininger a,b,3,4

    a ( )Forschungszentrum Julich, IKP Theorie , D-52425 Julich, Germany b Department of Physics, Harard Uniersity, Cambridge, MA 02138, USA

    Received 15 June 1998Editor: H. Georgi

    Abstract

    Using heavy baryon chiral perturbation theory to one loop, we derive an analytic and parameter-free expression for the s. 2.momentum dependence of the strange magnetic form factor of the nucleon G Q and its corresponding radius. ThisM

    should be considered as a lower bound. We also derive a model-independent relation between the isoscalar magnetic and the .strange magnetic form factors of the nucleon based on chiral symmetry and SU 3 only. This gives an upper bound on the

    strange magnetic form factor. We use these limites to derive bounds on the strange magnetic moment of the proton from the s. 2 2.recent measurement of G Q s0.1 GeV by the SAMPLE collaboration. We further stress the relevance of this result forM

    the on-going and future experimental programs at various electron machines. q 1998 Published by Elsevier Science B.V. Allrights reserved.

    PACS: 13.40.Cs; 12.39.Fe; 14.20.Dh

    1. There has been considerable experimental andtheoretical interest concerning the question: Howstrange is the nucleon? Despite tremendous efforts,we have not yet achieved a detailled understandingabout the strength of the various strange operators inthe proton. These are ss, as extracted from theanalysis of the pion-nucleon S-term, sg g s as mea-m 5sured e.g. in deep-inelastic lepton scattering off pro-

    1 Email: Th.Hemmert@fz-juelich.de.2 Email: Ulf-G.Meissner@fz-juelich.de.3 Email: S.Steininger@fz-juelich.de.4 Work supported in part by the Graduiertenkolleg Die Er-

    forschung subnuklearer Strukturen der Materie at Bonn Univer-sity and by the DAAD.

    tons and the vector current sg s, which is accesibleme.g. in parity-violating electron-nucleon scattering. Adedicated program at Jefferson Laboratory preceded

    . .by experiments at BATES MIT and MAMI Mainzis aimed at measuring the form factors related to thestrange vector current. In fact, the SAMPLE collabo-ration has recently reported the first measurement of

    w xthe strange magnetic moment of the proton 1 . To beprecise, they give the strange magnetic form factor at

    s. 2 2 .a small momentum transfer, G q sy0.1 GeVMs q0.23 " 0.37 " 0.15 " 0.19 nuclear magnetons .n.m. . The rather sizeable error bars document thedifficulty of such type of experiment. On the theoret-ical side, there is as much or even more uncertainty.To document this, let us pick one particular ap-

    w xproach. Jaffe 2 deduced rather sizeable strange

    0370-2693r98r$ - see front matter q 1998 Published by Elsevier Science B.V. All rights reserved. .PII: S0370-2693 98 00889-2

  • ( )T.R. Hemmert et al.rPhysics Letters B 437 1998 184190 185

    vector current matrix elements from the dispersion-theoretical analysis of the nucleon electromagneticform factors, assuming that the isoscalar spectralfunctions are dominated at low momentum transferby the v and f mesons. This analysis was updated

    w xin 3 with similar results. However, if one improvesthe isoscalar spectral function by considering also

    w x w xthe correlated pr-exchange 4 or kaon loops 5 , thecorresponding strange matrix elements can changedramatically. Also, the spread of the theoretical pre-dictions for the strange magnetic moment, y0.8Fm s.F0.5 n.m. underlines clearly the abovemadep

    w x.statement for a review, see Ref. 6 . As we willdemonstrate in the following, there is, however, onequantity of reference, namely we can make a param-eter-free prediction for the momentum dependence

    .of the nucleons strange magnetic Sachs form fac-tor based on the chiral symmetry of QCD solely. Inaddition, we derive a leading order model-indepen-dent relation between the strange and the isoscalarmagnetic form factors, which allows to give an upper

    s. 2 .bound on the momentum dependence of G Q .MThese two different results can then be combined toextract a range for the strange magnetic moment ofthe proton from the SAMPLE measurement of theform factor at low momentum transfer.

    2. The strangeness vector current of the nucleon isdefined as

    0 8 < < : < < :N sg s N s N qg l r3yl r 3 q N .m m0 8s 1r3 J y 1r 3 J , 1 . . .m m

    .with qs u,d,s denoting the triplet of the light0 a. quark fields and l s I l the unit the as8

    . .Gell-Mann SU 3 matrix. Assuming conservation ofall vector currents, the corresponding singlet andoctet vector current for a spin-1r2 baryon can then

    be written as from here on, we mostly consider the.nucleon

    X .0,8 0,8 2J su p F q g . .m N 1 m

    nis qmn .0,8 2qF q u p . 2 . . .2 N2mN

    Here, q spX yp corresponds to the four-momen-m m mtum transfer to the nucleon by the external singlet 0. 0. 8. 8. s l and the octet s l vectorm m m msource , respectively. The strangeness Dirac andmPauli form factors are defined via

    11 s. 2 0. 2 8. 2F q s F q y F q , 3 . . . .1,2 1,2 1,23 3

    s. .subject to the normalization F 0 sS , with S1 B Bthe strangeness quantum number of the baryon SN

    . s. . s. s. .s0 and F 0 sk , with k the anomalous2 B Bstrangeness moment. In the following we concentrateour analysis on the magnetic strangeness form

    s . 2 .factor G q , which in analogy to theM .electro magnetic Sachs form factor is defined as

    G s. q2 sF s. q2 qF s. q2 . 4 . . . .M 1 2It is this strangeness form factor for which chiral

    .perturbation theory CHPT gives the most interest- s. 2 .ing predictions. Furthermore, G q is also theM

    strangeness form factor that figures prominently inw xthe recent Bates measurement 1 .

    3. Heavy baryon chiral perturbation theory .HBCHPT is a precise tool to investigate the low-energy properties of the nucleon. It has, however,been argued that due to the appearance of higherorder local contact terms with undetermined coeffi-cents, CHPT can not be used to make any predictionfor the strange magnetic moment or the strangeelectric radius without additional, model-dependent

    w xassumptions 7 . However, the analysis of the nucle- .ons electromagnetic form factors in SU 2 CHPT

    shows that to one loop the slope of the isovectorPauli form factor can be predicted in a parameter-freemanner. Furthermore, one obtains a constant piece

    .for the isoscalar form factor in SU 2 CHPT and aparameter-free prediction for the magnetic Sachsform factor of the proton and the neutron, see Refs.w x810,12 . It is thus natural to extend this calculationto the three flavor case with the appropriate singlet

    .and octet currents as defined in Eq. 2 .We give here the relevant HBCHPT Lagrangians

    needed for the calculation. The baryon octet isparametrized in the matrix B, which has the usualtransformation properties of any matter field under

  • ( )T.R. Hemmert et al.rPhysics Letters B 437 1998 184190186

    chiral transformation. We utilize the chiral covariantderivative D ,m

    .0D Bs E qG y i B 5 . .m m m mi

    1 . . i iG s u ,E u y u qa u .m m m m2 2i

    . .i i y u ya u , is3,8 6 . . .m m2the chiral vielbein u ,m

    u s i u= U u 7 .m m

    . .i i= UsE Uy i qa U .m m m m . .i iq iU ya , is3,8 8 . . .m m

    x . x ..where the quantities a , xs0,3,8 correspondm m . .to external vector axial-vector sources and U x s

    2 .u x parametrizes the octet of Goldstone bosons.The three flavor HBCHPT Lagrangian then readswe only give the terms relevant to the calculations

    .presented here

    1. m m : :LL s B i D B qD B S u , B 4MB m m qm :qF B S u , B 9 .m y

    Fib6 a2. m n 3. :w xLL sy B S ,S f , BMB qmn4mN

    ib D6 a m n 3. :w xy B S ,S f , B 4qm n4mNFib6 b m n 8. :w xy B S ,S f , Bqm n4mN

    ib F6 b m n 8. :w xy B S ,S f , B 4qm n4mN2 ib6 c m n 0. :w xy B S ,S B q . . . 10 .mn4mN

    with

    f i. suF R i.uquF L i.u 11 .qm n mn mn . .L , R i. L , R i. L , R i. L , R i L , R iF sE F yE F y i F ,Fmn m n n m m n

    12 .

    F R i.s i.qa i. , F L i.s i.ya i. 13 .m m m m m m0.sE 0.yE 0. , 14 .mn m n n m

    :where . . . denotes the trace in flavor space andm the nucleon mass. Furthermore, F,1r2 andN

    .D,3r4 are the conventional SU 3 axial coupling 5.constants in the chiral limit, to be precise . The

    dimension two terms are accompanied by finite . D , F D , Flow-energy constants LECs , called b ,b ,b .6 a 6 b 6 c

    Their precise meaning will be discussed later.

    4. To be specifc, we now consider the strangemagnetic form factor to one loop order in CHPT.The strange magnetic moment of the nucleon getsrenormalized by the kaon cloud, completely analo-gous to the renormalization of the nucleon isovectormagnetic moment m by the pion cloudNw x13,10,12,14,15 . It can be written as

    m s.sm s.sm s.N p n1 10. D Fs G 0 q b yb .M 6 b 6 b3 3

    m MN K 2 2q 5D y6DFq9F , 15 . .224p Fp .with M the kaon mass and F F qF r2,K p p K

    102 MeV the average pseudoscalar decay constant.We use this value because the difference between thepion and the kaon decay constants only shows up at

    3.higher order. One finds that to OO p in the chiralcalculation the strange magnetic moments of theproton and the neutron are predicted to be equalbecause symmetry breaking only starts by insertions

    .from the dimension two meson-baryon Lagrangianand consist of three distinct contributions. First, the

    0. .singlet magnetic moment G 0 is parametrized inMterms of the unknown singlet coupling b . It cannot6 c

    5 To the order we are working, it is sufficient to identify thephysical with the chiral limit values.

  • ( )T.R. Hemmert et al.rPhysics Letters B 437