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


    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: Email: Email: 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


    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. , 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.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 1998 184190 187

    0 . 8.Fig. 1. Coupling of the singlet ; l and octet ; l vector . . .currents wiggly line to the nucleon solid line . a is a one kaon

    . .dashed line loop graph and b a dimension two contact term. Thelatter only contributes to the strange magnetic moment.

    be predicted without additional experimental input asw xhas already been noted in 7 . The counterterms

    b D , F, however, can be extracted from the anomalous6 bmagnetic moments k ,k of the proton and thep nneutron. Third, there is a strong renormalization of

    s. 3.m due to the kaon cloud. To OO p we findNm s. K - loops s2.0, which is large and positie. ThisN

    w xresult is in agreement with the calculation of Ref. 7 .It is well-known that such large leading order kaonloop effects generally are diminished by higher order

    . w xcorrections unitarization , see e.g. 4 . We also notethat in some models the strange magnetic moment isassumed to be generated exclusively by the kaon

    w xcontributions 16 , which is already ruled out to .leading order chiral analysis of Eq. 15 .

    To obtain the complete strange magnetic formfactors one only has to consider the diagrams shownin Fig. 1. The important observation to make is thatto leading order in the chiral analysis, the singletcurrent only couples to the strange magnetic mo-ment, whereas the Q2-dependence is entirely gov-erned by the octet current mediated by the mesoncloud. For the loop graph 1a, in case of an incomingnucleon, the only allowed intermediate states areK L and K S, i.e. the pion and the h cloud do not

    .contribute to this order. For the proton p and the .neutron n one finds

    G s. Q2 sG s. p Q2 sG s. n Q2 . . .M M Mp m MN K qN 24p F .p

    = 2 2 2 25D y6DFq9F f Q , 16 . . .3

    with Q2 syq2. The momentum dependence is givenentirely in terms of

    2 2 2(4qQ rM QK12f Q sy q arctan . 17 . . 2 2 2 /2 M(4 Q rM KK 2 .The function f Q is shown in Fig. 2. For small

    and moderate Q2, it rises almost linearly with in-2 .creasing Q . We note from Eq. 16 that the slope of

    s. 2 .G Q is uniquley fixed in terms of well-knownMlow energy parameters,

    d G s. q2 .M22

  • ( )T.R. Hemmert et al.rPhysics Letters B 437 1998 184190188

    predictions are already quite satisfactory. However, .in the SU 2 small scale expansion framework

    w x w x11 it was found 12 that the radius of the isovectorIs1 2 .magnetic Sachs form factor G q is increasedM

    by 15-20% due to intermediate Dp cloud effects. As .similar analysis in the SU 3 small scale expan-

    sion framework is in preparation to see whetherthere are similarly sizeable corrections for the mag-netic strangeness form factor due to intermediatedecuplet-octet states. In addition, there are other

    mechanisms like e.g. contributions from vector.mesons not covered at this order.

    5. We can also give an upper limit for the strangemagnetic form factor as the following argumentsshows. For that, we consider the electromagneticcurrent

    2 1 1EM < < :J s N ug uy dg dy sg s Nm m m m3 3 31

    1 < < : < < :s N qg l q N q N qg l q Nm 8 m 322 31

    18 3s J q J , 19 .m m22 38 .where J corresponds to the octet current of Eq. 1 .m

    . 3The conserved triplet current J parameterizes themresponse of a nucleon coupled to an external tripletvector source 3.s l3. The calculation proceedsm m

    .as before. We find that while in an SU 3 calculationthe magnetic form factors of the proton and theneutron both have a pion and a kaon cloud contribu-tion, the pion cloud terms drop out to leading orderfor the isoscalar magnetic form factor of the nucleon.

    . w xAlso, in contrast to the SU 2 calculations 10,12 , . Is0 2 .the leading one loop SU 3 contribution to G QM

    picks up a momentum dependence given again en- 2 . .tirely in terms of the function f Q , see Eq. 17 ,

    .SU 3Is0 2 p 2 n 2G Q sG Q qG Q . . .M M MM m pK Nsm ys 24p F .p

    = 2 2 2 25 D y6 D Fq9 F f Q , . .320 .

    with m s0.88 n.m. the isoscalar nucleon magneticsmoment. We note that to this order in the chiral

    expansion the prediction is again free of countert-erms for the momentum dependence. Interestingly,

    3.this means that to OO p the isoscalar magneticform factor of the nucleon is completely dominatedby the kaon cloud, as all virtual pion contributionscancel exactly to this order. The result is of course

    . w xconsistent with the SU 2 analyses of 10,12 as oneIs0 2 .can check that G Q m in the limit M ,M s K

    i.e. the kaon cloud contribution shows up via higher .order counterterms in the SU 2 calculation. For the

    leading chiral contribution to the isoscalar magneticradius one finds

    6 d G Is0 q2 .2 MIs0 2

  • ( )T.R. Hemmert et al.rPhysics Letters B 437 1998 184190 189

    Fig. 3. The strange magnetic form factor derived from the isoscalarmagnetic one with m s.s0.N

    This number is roughly three times larger than the .leading chiral estimate of Eq. 18 . Given that there

    are also non-strange contributions in the isoscalarmagnetic form factor, which will start to manifest at

    4 .order q , we consider Eq. 23 as an upper boundon the strange magnetic radius. The correspondingstrange magnetic form factor is shown in Fig. 3 for avanishing strange magnetic moment and using thedipole parametrization for the isoscalar magneticform factor. Any finite value for m s. can be accom-Nmodated by simply shifting the curve up or down theabzissa. Note that a similar dipole-like behaviour

    with a much smaller slope corresponding to the.lower bound discussed before was found in the

    vector meson dominance type analysis supplementedw xby regulated kaon loops presented in Ref. 18 . It is

    also important t...