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∗ Strange form fa tors of the nu leon † R. Bijker Instituto de Cien ias Nu leares, Universidad Na ional Autónoma de Méxi o, AP 70-543, 04510 Méxi o, D.F., Méxi o (Dated: Re eivedJanuary27, 2006) Abstra t. Thestrangeness ontentof nu leonformfa tors isanalyzedinatwo- omponentmodel withaquark-likeintrinsi stru turesurroundedbyameson loud. A omparisonwiththeavailable experimentaldatafromtheSAMPLE,PVA4,HAPPEXandG0 ollaborationsshowsagoodoverall agreement. Keywords: Strange formfa tors, ve tormesondominan e Resumen. Seanaliza el ontenidodeextrañezadelosfa toresdeformadelnu leónenunmodelo dedos omponentesque onsisteenunaestru turaintrínse adetres uarks onstituyentesrodeada de una nube mesóni a. Una ompara ión on los datos experimentales disponibles de las olabo- ra iones SAMPLE, PVA4, HAPPEX y G0 muestra un buen ajuste para los fa tores de forma on 7 extrañeza. 0 Des riptores: Fa tores de forma extraños, dominan ia de mesones ve toriales 0 2 PACSnumbers: 13.40.Gp,12.40.Vv,14.20.Dh,24.85.+p,13.40.Em n a J 9 I. INTRODUCTION 1 v The ontribution of the di(cid:27)erent quark (cid:29)avors to the ele tromagneti stru ture of the nu leon an be studied 2 by ombining the nu leon's response to the ele tromagneti and weak neutral ve tor urrents [1℄. Espe ially, the 2 ontribution of strange quarks to the nu leon stru ture is of interest be ause it is ex lusively part of the quark- 0 antiquark sea. 1 In re ent experiments, parity-violating elasti ele tron-proton s attering has been used to probe the ontribution 0 7 of strange quarks to the stru ture of the nu leon [2℄. The strange quark ontent of the form fa tors an be deter- 0 minedassuming hargesymmetryand ombiningparity-violatingasymmetrieswithmeasurementsoftheele tri and / magneti form fa tors of the proton and neutron. The various experiments are sensitive to di(cid:27)erent ombinations of h the strange quark ontributions to the harge distribution and the magnetization represented by the strange ele tri t - and magneti form fa tors. There are various methods to disentangle the ele tri and magneti ontributions: by l c measuringparity-violatingasymmetriesat both forwardand ba kwardangles [3℄, by using di(cid:27)erent targets [4℄, or by u ombining parity-violating asymmetries with (anti)neutrino-proton s attering data [5℄. n The (cid:28)rst experimental results from the SAMPLE [2, 3℄, PVA4 [6℄, HAPPEX [4, 7, 8℄ and G0 [9℄ ollaborations : v haveshowneviden eforanonvanishingstrangequark ontribution to the stru ture ofthe nu leon. Inparti ular,the Xi strangeness ontent of the proton magneti moment was found to be positive [8℄, suggesting that the strange quarks redu e the proton's magneti moment. This is an unexpe ted and surprising (cid:28)nding, sin e a majority of theoreti al r a studies favors a negative value [10℄. Theaimofthis ontributionistoanalyzetheavailableexperimentaldataonstrangeformfa torsinatwo- omponent model of the nu leon. II. TWO-COMPONENT MODEL OF NUCLEON FORM FACTORS Ele tromagneti and weak form fa tors ontain the information about the distribution of ele tri harge and magnetization inside the nu leon. These form fa tors arise from matrix elements of the orresponding ve tor urrent operators i N Vµ N =u¯N F1(Q2)γµ+ F2(Q2)σµνqν uN . h | | i 2M (1) (cid:20) N (cid:21) ∗ † Invited talkatXXIXSymposiumonNu learPhysi s,Ha iendaCo oyo ,Morelos,Mexi o,January 3-6,2006 Ele troni address: bijkernu leares.unam.mx F1 F2 Q2 = q2 Here and aretheDira andPauliformfa torswhi harefun tionsofthesquaredmomentumtransfer − . GE GM F1 F2 GE = F1 τF2 aTnhdeGelMe t=ri Fa1n+dFm2awgnitehtiτ =forQm2/fa4 MtoN2rs., and , are obtained from and by the relations − Di(cid:27)erent models of the nu leon orrespond to di(cid:27)erent assumptions about the Dira and Pauli form fa tors. In this ontribution, I onsider the two- omponent model of [1g1(,Q122)℄ in whi h the external photon ouples boρthωto an intrinsi three-quark stru ture des ribed by the form fa tor , and to a meson loud via ve tor-meson( , and φ )dominan e(VMD). IntheoriginalVMD al ulation[11℄,theDira formfa torwasattributedtoboththeintrinsi stru ture and the meson loud, and the Pauli form fa tor entirely to the meson loud. In [12℄, it was shown that the addition of an intrinsi part to the isove tor Pauli form fa tor as suggested by studies of relativisti onstituent quark models in the light-front approa h [13℄, improves the results for the neutron ele ri and magneti form fa tors onsiderably. ω φ ρ In order to in orporate the ontribution of the iso alar ( and ) and isove tor( ) ve tor mesons, it is onvenient to introdu e the isos alar and isove tor urrent operators 1 VI=0 = u¯γ u+d¯γ d 2s¯γ s , µ 6 µ µ − µ VI=I = 1(cid:0)u¯γ u d¯γ d . (cid:1) µ 2 µ − µ (2) (cid:0) (cid:1) ω φ The orresponding isos alar Dira and Pauli form fa tors depend on the ouplings to the and mesons 1 m2 m2 FI=0(Q2) = g(Q2) 1 β β +β ω +β φ , 1 2 " − ω− φ ωm2ω+Q2 φm2φ+Q2# 1 m2 m2 FI=0(Q2) = g(Q2) α ω +α φ , 2 2 " ωm2ω+Q2 φm2φ+Q2# (3) ρ and the isove tor ones on the oupling to the meson [12℄ 1 m2 FI=1(Q2) = g(Q2) 1 β +β ρ , 1 2 " − ρ ρm2ρ+Q2# 1 µ µ 1 α m2 F2I=1(Q2) = 2g(Q2)" p−1+n−γQ2− ρ +αρm2ρ+ρQ2# . (4) Theprotonandneutronformfa tors orresgp(oQn2d)t=ot(h1e+suγmQ2a)n−d2di(cid:27)eren eoftheisos alarandisove tor ontributions. Forthe intrinsi form fa tora dipoleform isused whose asymptoti behavioris onsistentwith p-QCD [14℄ and whi h oin ides with the form used in an algebrai treatment of the intrinsi three-quark stru ture [15℄. ρ Q2 The large width of the meson plays an important role for the small behaviorof the form fa tors and is taken is taken into a ount in the same way as in [11, 12℄ by the repla ement [16℄ m2 m2+8Γ m /π ρ ρ ρ π , m2+Q2 → m2+Q2+(4m2 +Q2)α(Q2)Γ /m (5) ρ ρ π ρ π with 2 4m2 +Q2 1/2 4m2 +Q2+ Q2 α Q2 = π ln π . π (cid:20) Q2 (cid:21) p 2mπ p ! (6) (cid:0) (cid:1) Q2 The meson dynami s is importFan1t fo1r/Qsm4all vaFl2ues1o/fQ6 , whereas for large values the form fa tors satisfy the asymptoti behavior of p-QCD, ∼ and ∼ [14℄. Sin etheintrinsi partisasso iatedwiththevalen equarksofthenu leon,thestrangequark ontentofthenu leon form fa tors arises from the meson wave fun tions ω = cosǫ ω0 sinǫ φ0 , | i | i− | i φ = sinǫ ω0 +cosǫ φ0 , | i | i | i (7) 2 ω0 = uu¯+dd¯ /√2 φ0 =ss¯ where | i and | i are the ideally mixed states. Under the assumption that the strange form fa tors have t(cid:0)he same(cid:1)form as the isos alar ones, the Dira and Pauli form fa tors that orrespond to the strange urrent Vs =s¯γ s , µ µ (8) g(Q2) are expressed as the produ t of an intrinsi part and a ontribution from the meson loud as 1 m2 m2 Fs(Q2) = g(Q2) βs ω +βs φ , 1 2 ωm2 +Q2 φm2 +Q2 " ω φ # 1 m2 m2 Fs(Q2) = g(Q2) αs ω +αs φ . 2 2 " ωm2ω+Q2 φm2φ+Q2# (9) β α The 's and 's in Eqs. (4) and (9) are not independent of one another. The oe(cid:30) ients appearing in the isos alar and strange form fa tors depend on the same nu leon-meson and urrent-meson ouplings [17℄. In addition, they are onstrained by the ele tri harges and magneti moments of the nu leon whi h leads to two independent isos alar ouplings α = µ +µ 1 α , ω p n φ − − βω = βφtan(θ0+ǫ)/tanǫ . − (10) The strange ouplings are then given by βωs/βω =αsω/αω = −√6 sinǫ/sin(θ0+ǫ) , βφs/βφ =αsφ/αφ = −√6 cosǫ/cos(θ0+ǫ) . (11) tanθ0 =1/√2 ǫ ω φ where . The mixing angle an be determined from the de ay properties of the and mesons. Here ǫ=0.053 I take the value rad obtained in [18℄. III. RESULTS γ Inorderto al ulate the nu leonform fa torsin the two- omponentmodel the(cid:28)ve oe(cid:30) ients, from the intrinsi β α β α φ φ ρ ρ form fa tor, and from the isos alar ouplings, and and from the isove tor ouplings,are determined in a least-square(cid:28)t to the ele tri and magneti form fa tors of the proton and the neutron using the same data set as in [12℄. Theele tromagneti formfa toroftheprotonandneutronarefoundtobeingoodagreementwithexperimental data [19℄. A oβrdsin=g toβsE=q.0(.1210)2, tαhse=st0ra.6n4g8e oupαlsin=gs a0n.01b8e determined from the (cid:28)tted values of the isos alar ouplings to be φ − ω , φ and ω − [19℄. Q2 GsFi=gs.F1sandτF2sshow the strange ele tri and magneti form fa toFrss as a fun tion of . The theoreti al values of E 1 − 2 are small and negative. The Dira form fa tor 1 is small due to an eling ontributions of the ω φ and oupliGngss(0w)h=i hFas(r0o)se=asβsa+ oβnsse=qu0en e of the fa t that the strange (anti)quarks do not ontribute to the ele tri harge E Fs 1 ω φ . Moreovτer,=foQr2th/4isMra2nge of momentum transfer the ontribution from the Pauli form fa tor 2 is suppressed by the fa tor N. The theoGrseti al values are in good agreement with the re ent experimental result of the H4APPEX Collaboration in whi hGsE =was0d.e0t3e8rmin0e.0d42for t0h.0e1(cid:28)0rst time in parity-violQat2in=g0e.l0e9 t1ron s atte2ring from He [4℄. The experimental value E − ± ± ( ir le) measured at (GeV/ ) is Gosnsi=steFntsw+itFhszero. The strange magneti form fa tor M 1 2 is positive, sin e it is dominated by the ontribution from the Pauli form fa tor. The SAMPLE experiment measured tQhe2 =pa0r.i1ty-violatin2g asyGmsm=etr0y.3a7t b0a. 2k0war0d.2a6ngl0e.s0,7whi h allowed to determine the straGngsemagGnesti form fa torat (GeV/ ) as M ± ± ± . The otherexperimentalvalues of E and M in Figs.1 and2 wereobtained [5, 20℄ by ombiningthe (anti)neutrino data from E734 [21℄ with the parity-violating asymmetries from HAPPEX [07℄<anQd2G<01[.90℄. The th2eoreti al values are in good overall agreement with the experimental ones for the entire range (GeV/ ) . The strange magneti moment is al ulated to be positive 1 µ =Gs (0)= (αs +αs)=0.315µ , s M 2 ω φ N (12) 3 0.2 s GE 0.0 -0.2 0.0 0.5 1.0 2 2 Q (GeV/c) FIG. 1: Comparison between theoreti al and experimental values of the strange ele tri form fa tor. The experimental values are takenfrom [4℄ ( ir le) and [20℄ (triangle). 0.5 s M G 0.0 0.0 0.5 1.0 2 2 Q (GeV/c) FIG.2: Comparisonbetweentheoreti alandexperimentalvaluesofthestrangemagneti formfa tor. Theexperimentalvalues are takenfrom [3℄ ( ir le) and [20℄ (triangle). µ = e~/2M c N N in units of the nu lear magneton, . This value is in aGgrseem=e0n.t55with0.r2e8 entQex2p=er0im.1ental evid2en e from the SAMPLE ollaboration [3℄ and an analysis of the world data M ± at (GeV/ ) [8℄. µ s By onvention, the de(cid:28)nition of the strange magneti moment does not involve the ele tri harge of the strange quark. Theoreti al al ulations of the strange magneti moment show a large variation, although most QCD-inspired 0.6 . µ . 0.0 µ s N models seem to favor a negative value in the range − [10℄. There are relatively few al ulations 0.074 0.115 µ SU(3) N that give a sizeable positive strange magneti moment, ranging from − in the hiral quark 0.16 0.03 µ SU(3) 0.37 µ N N soliton model [22℄, ± in a group theoreti al approa h with (cid:29)avor breaking [23℄ to in the SU(3) 0.05 0.06 hiral bag model [24℄. Re ent quen hed latti e-QCD al ulations give a small value, e.g. ± [25℄ and 0.046 0.019 − ± [26℄. Thestrangeformfa torsdeterminedinthePVA4[6℄,HAPPEX [7,8℄andG0[9℄experiments orrespondtoalinear ombination of ele tri and magneti form fa tors. Fig. 3 shows the results for the strange form fa tor ombination 4 Gs +ηGs E M measured re ently by the G0 Collaboration at forward angles [9℄. A omparison of the al ulated values in the two- omponent model with the PVA4 and HAPPEX data shows a similar good agreement as for the G0 data shown in Fig. 3 [19℄. 0.2 0.1 s M G + s E G 0.0 -0.1 0.0 0.5 1.0 2 2 Q (GeV/c) GsE+ηGsM FIG. 3: Comparisonbetweentheoreti al andexperimentalvaluesofstrange formfa tors . Theexperimentalvalues were measuredbytheG0 Collaboration [9℄. IV. SUMMARY AND CONCLUSIONS Insummary,inthis ontributionitwasshownthatthere entexperimentaldataonthestrangenu leonformfa tor an be explained very well in a two- omponent model of the nu leon onsisting of an intrinsi three-quark stru ture 0.49 with a spatial extent of ∼ fm surrounded by a meson loud. The present approa h is a ombination of the two- omponentmodel of [12℄ with the treatment of the strangequark ontent of the ve tormesonsa ording to [17℄. The parameters in the model are ompletely determined by the ele tri and magneti form fa tors of the proton and neutron. It is noted, that the strange ouplings do not involve any new parameters. On the ontrary, the ondition that the strange quarks do not ontribute to the ele tri harge of the nu leon, leads to an extra onstraint relating β β ω φ and , thus redu ing the number of independent oe(cid:30) ients of the two- omponent model of [12℄ by one. Thegoodoverallagreementbetweenthetheoreti alandexperimentalvaluesfortheele tromagneti formfa torsof theprotonand neutronand theirstrangequark ontentshowsthat thetwo- omponetmodelprovidesasimultaneous and onsistent des ription of the ele tromagneti and weak ve tor form fa tors of the nu leon. The(cid:28)rstresultsfromtheSAMPLE,PVA4,HAPPEXandG0 ollaborationshaveshowneviden eforanonvanishing strange quark ontribution to the harge and magnetization distributions of the nu leon. Future experiments on parity-violating ele tron s attering at ba kward angles (PVA4 and G0 [27℄) and neutrino s attering (FINeSSE [28℄) will makeit possible to disentangle the ontributions of the di(cid:27)erent quark (cid:29)avorsto the ele tri , magneti and axial form fa tors, and thus to provide new insight into the omplex internal stru ture of the nu leon. A knowledgments This work was supported in part by a grant from CONACYT, Mexi o. [1℄ D.B. Kaplan and A. Manohar, Nu l. Phys.B 310, 527 (1988); R.D. M Keown, Phys.Lett. B 219, 140 (1989); D.H. Be k, Phys. Rev. D 39, 3248 (1989). 5 [2℄ E.J. Beise, M.L. Pitt and D.T. Spayde, Prog. Part. Nu l. Phys.54, 289 (2005). [3℄ D.T. Spaydeet al.,Phys.Lett. B 583, 79 (2004). [4℄ K.A. Aniol et al.,Phys. Rev. Lett. 96, 022003 (2006). [5℄ S.F. Pate, Phys. Rev. Lett. 92, 082002 (2004). [6℄ F.E. Maas et al., Phys.Rev. Lett. 93, 022002 (2004); ibid.,94, 152001 (2005). [7℄ K.A. Aniol et al.,Phys. Rev. C 69, 065501 (2004). [8℄ K.A. 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