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  • Isospin Mixing in the Nucleon and 4He and the Nucleon Strange Electric Form Factor

    M. Viviani,1 R. Schiavilla,2,3 B. Kubis,4 R. Lewis,5 L. Girlanda,1 A. Kievsky,1 L. E. Marcucci,1 and S. Rosati11INFN, Sezione di Pisa, and Department of Physics, University of Pisa, I-56127 Pisa, Italy

    2Jefferson Lab, Newport News, Virginia 23606, USA3Department of Physics, Old Dominion University, Norfolk, Virginia 23529, USA4HISKP (Theorie), Universitat Bonn, Nussallee 14-16, D-53115 Bonn, Germany

    5Department of Physics, University of Regina, Regina, Saskatchewan, Canada, S4S 0A2(Received 15 March 2007; published 14 September 2007)

    In order to isolate the contribution of the nucleon strange electric form factor to the parity-violatingasymmetry measured in 4He ~e; e04He experiments, it is crucial to have a reliable estimate of the mag-nitude of isospin-symmetry-breaking (ISB) corrections in both the nucleon and 4He. We examine thisissue in the present Letter. Isospin admixtures in the nucleon are determined in chiral perturbation theory,while those in 4He are derived from nuclear interactions, including explicit ISB terms. A careful analysisof the model dependence in the resulting predictions for the nucleon and nuclear ISB contributions to theasymmetry is carried out. We conclude that, at the low momentum transfers of interest in recent mea-surements reported by the HAPPEX Collaboration at Jefferson Lab, these contributions are of comparablemagnitude to those associated with strangeness components in the nucleon electric form factor.

    DOI: 10.1103/PhysRevLett.99.112002 PACS numbers: 14.20.Dh, 12.15.Ji, 25.30.Bf

    One of the challenges of modern hadronic physics is todetermine, at a quantitative level, the role that quark-antiquark pairs, and in particular ss pairs, play in thestructure of the nucleon. Parity-violating (PV) electronscattering from nucleons and nuclei offers the opportunityto investigate this issue experimentally. The PVasymmetry(APV) arises from interference between the amplitudes dueto exchange of photons and Z-bosons, which couple, re-spectively, to the electromagnetic (EM) and weak neutral(NC) currents. These currents involve different combina-tions of quark flavors, and therefore measurements of APV,in combination with electromagnetic form factor data forthe nucleon, allow one to isolate, in principle, the electricand magnetic form factors GsE and G

    sM, associated with the

    strange-quark content of the nucleon.Experimental determinations of these form factors have

    been reported recently by the Jefferson Lab HAPPEX [1]and G0 [2] Collaborations, Mainz A4 Collaboration [3],and MIT-Bates SAMPLE Collaboration [4]. These experi-ments have scattered polarized electrons from either un-polarized protons at forward angles [13] or unpolarizedprotons and deuterons at backward angles [4]. The result-ing PVasymmetries are sensitive to different linear combi-nations of GsE and G

    sM as well as the nucleon axial-vector

    form factor GZA. However, no robust evidence has emergedso far for the presence of strange-quark effects in thenucleon.

    Last year, the HAPPEX Collaboration [5,6] at JeffersonLab reported on measurements of the PV asymmetry inelastic electron scattering from 4He at four-momentumtransfers of 0:091 GeV=c2 and 0:077 GeV=c2.Because of the J 0 spin-parity assignments of thisnucleus, transitions induced by magnetic and axial-vectorcurrents are forbidden, and therefore these measurementscan lead to a direct determination of the strangeness elec-

    tric form factor GsE [7,8], provided that isospin-symmetry-breaking (ISB) effects in both the nucleon and 4He, andrelativistic and meson-exchange (collectively denoted withMEC) contributions to the nuclear EM and weak vectorcharge operators, are negligible. A realistic calculation ofthese latter contributions [8] found that they are in fact tinyat low momentum transfers. The goal of the present Letteris to provide a quantitative estimate of ISB corrections tothe PV asymmetry.

    In the following analysis, we only need to consider thetime components of the EM current and vector part of theweak NC currentthe weak vector charge referred toabove [8]. We account for isospin symmetry breaking inboth the nucleon and -particle. We first discuss it in thenucleon.

    Ignoring radiative corrections, the EM and weak vectorcharge operators can be decomposed as j0EM j0 j1and j0NC 4s2Wj0 2 4s2Wj1 js, where j0and j1 are, respectively, the isoscalar and isovector com-ponents of the EM charge operators, js is the (isoscalar)component due to strange-quark contributions, and s2W sin2W contains the Weinberg mixing angle. In terms ofquark fields u, d, s, these charge operators read: j0 u0u d0d 2s0s=6, j1 u0u d0d=2, andjs s0s. In a notation similar to that adopted by theauthors of Ref. [9], we introduce form factors correspond-ing to the following matrix elements of j0 and j1 be-tween proton (p) and neutron (n) states:

    hpjj0jpi ! G0EQ2 G;EQ2; (1)

    hnjj0jni ! G0EQ2 G;EQ2; (2)

    hpjj1jpi ! G1EQ2 G16EQ2; (3)

    PRL 99, 112002 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending

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    http://dx.doi.org/10.1103/PhysRevLett.99.112002

  • hnjj1jni ! G1EQ2 G16EQ2; (4)where the arrow indicates that only leading contributionsare listed in the nonrelativistic limit of these matrix ele-ments. While higher-order corrections associated with theDarwin-Foldy and spin-orbit terms are not displayed ex-plicitly in the equations above, they are in fact retained inthe calculations discussed later in the present work. Theform factors G;EQ2 and G16EQ2 parameterize ISB effectsin the nucleon states. We also introduce the strange formfactor via

    hpjjsjpi hnjjsjni ! GsEQ2; (5)where here ISB terms in the p, n states are neglected.Contributions from sea quarks heavier than strange arealso ignored.

    In terms of the experimental proton and neutron electricform factors, derived from the matrix elementshpjj0EM jpi ! GpEQ2 and hnjj0EM jni ! GnEQ2, we ob-tain

    G0E GpE GnE=2G16E; (6)

    G1E GpE GnE=2G;E; (7)where the Q2 dependence in these and the following twoequations is understood. In the limit in which the p, n statesform an isospin doublet, the form factors G;E and G

    16E

    vanish, and G0E and G1E reduce to the standard isoscalar

    and isovector combinations of the proton and neutronelectric form factors. The proton and neutron vector NCform factors follow from the expression for j0NC givenearlier, i.e.,

    Gp;ZE 1 4s2WGpE GnE 2G16E G;E GsE; (8)

    Gn;ZE 1 4s2WGnE GpE 2G16E G;E GsE: (9)We now turn to the nuclear charge operator. At low

    momentum transfer, it is simply given by [8]

    EMqGpEQ2XZk1

    eiqrk GnEQ2XA

    kZ1eiqrk ; (10)

    where Z is the number of protons, A Z the number ofneutrons, and q is the three-momentum transfer. An equa-tion similar to Eq. (10) holds for the weak vector chargeoperator, but with GpE and G

    nE being replaced, respectively,

    by Gp;ZE and Gn;ZE . It is also convenient to define the charge

    operators:

    0q GpE GnE2

    XAk1

    eiqrk ; (11)

    1q GpE GnE2

    XZk1

    eiqrk XA

    kZ1eiqrk

    ; (12)

    from which

    EMq 0q 1q; (13)

    NCq 4s2WEMq 2G16E GsE

    GpE GnE=20q

    21q 2G;E

    GpE GnE=21q; (14)

    where again the Q2 dependence of the nucleon form factorshas been suppressed here and in the following for brevity.The relations above lead to the definition of the followingnuclear form factors: h4Hejaqj4Hei=Z Faq witha EM, NC, 0, 1, having the normalizations FEM0 F00 1 and F10 0. The form factor F1q isvery small because 4He is predominantly an isoscalar state.Using standard techniques, it is possible to show thatAPV FNCq=FEMq (see Ref. [7], for example).Thus, ignoring second order terms like G;F1q, weobtain for the PV asymmetry measured in ( ~e, e0) elasticscattering from 4He:

    APV GQ

    2

    42

    p4s2W 2

    F1qF0q

    2G16E GsEGpE GnE=2

    ; (15)

    where G is the Fermi constant as determined from muondecays, and here s2W is taken to incorporate radiative cor-rections. The terms G16E and F

    1q=F0q are the contri-butions to APV, associated with the violation of isospinsymmetry at the nucleon and nuclear level, respectively.

    The most accurate measurement of the PV asymmetry,recently reported in Ref. [6] at Q2 0:077 GeV=c2,gives APV 6:40 0:23stat 0:12syst ppm,from which, after inserting the values for G 1:166 37 105 GeV2, 1=137:036, and s2W 0:2286 (including its radiative corrections [7]) inEq. (15), one obtains

    2F1q

    F0q 2G16E GsE

    GpE GnE=2 0:010 0:038 (16)

    at Q20:077 GeV=c2. This result is consistent with zero.In the following, we discuss the estimates for the ISBcorrections first in the nucleon and then in 4He, respec-tively G16EQ2 and F1q, at Q2 0:077 GeV=c2 (cor-responding to q 1:4 fm1).

    For G16EQ2 we use the estimate obtained in Ref. [9]adapted to our conventions, combining a leading-ordercalculation in chiral perturbation theory with estimatesfor low-energy constants using resonance saturation.Collecting the various pieces, we find

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    112002-2

  • G16EQ2g2AmNm

    F2

    MmN

    0Q24 3Q2

    Q2

    2m2N

    Q2M

    mN 0Q25 3Q2

    1162

    12logM

    MV

    v6M2mN

    g!F!Q2

    2MVM2VQ221!M

    2V

    4m2N

    ; (17)

    where the loop functions , 0=3 are given explicitly inRef. [9], along with the precise definitions of the variouscoupling constants. The chiral loop contributions inEq. (17) scale with the neutron-proton mass differencem, while the resonance part is proportiona