Strangeness magnetic form factor of the proton in the extended chiral quark model
aa r X i v : . [ h e p - ph ] A ug Strangeness magnetic form factor of the proton in the extendedchiral quark model
C. S. An ∗ and B. Saghai † Institute of High Energy Physics and Theoretical PhysicsCenter for Science Facilities, CAS, Beijing 100049, China Institut de Recherche sur les lois Fondamentales de l’Univers,Irfu/SPhN, CEA/Saclay, F-91191 Gif-sur-Yvette, France (Dated: September 6, 2018)
Abstract
Background:
Unravelling the role played by nonvalence flavors in baryons is crucial in deep-ening our comprehension of QCD. The strange quark, a component of the higher Fock states inbaryons, is an appropriate tool to study nonperturbative mechanisms due to the pure sea quark.
Purpose:
Study the magnitude and the sign of the strangeness magnetic moment µ s and themagnetic form factor ( G sM ) of the proton. Methods:
Within an extended chiral constituent quark model, we investigate contributionsfrom all possible five-quark components to µ s and G sM ( Q ) in the four-vector momentum range Q ≤ . The probability of the strangeness component in the proton wave function iscalculated employing the P model. Results:
Predictions are obtained by using input parameters taken from the literature. Theobservables µ s and G sM ( Q ) are found to be small and negative, consistent with the lattice-QCDfindings as well as with the latest data released by the PVA4 and HAPPEX Collaborations. Conclusions:
Due to sizeable cancelations among different configurations contributing to thestrangeness magnetic moment of the proton, it is indispensable to i) take into account all relevantfive-quark components and include both diagonal and non-diagonal terms, ii) handle with care theoscillator harmonic parameter ω and the s ¯ s component probability. PACS numbers: 12.39.-x, 13.40.Em, 14.20.-c, 14.65.Bt ∗ [email protected] † [email protected] . INTRODUCTION Parity-violating electron scattering process, extensively investigated since more than adecade, has been proven to offer a unique experimental opportunity in probing the contri-bution of the strangeness sea to the electromagnetic properties of the nucleon. During thatperiod, results from four Collaborations have been released in several publications (for recentreviews see Refs. [1, 2]) with the latest ones for each of the Collaborations being: SAMPLE(MIT-Bates) [3], PVA4 (MAMI) [4], G0 (JLab) [5], HAPPEX (JLab) [6]. Those experimentsallowed extracting linear combinations of electric ( G sE ) and magnetic ( G sM ) strangeness formfactors of the proton as a function of four-vector momentum transfer Q .A general trend of the data published before year 2009 was to produce rather small andpositive values for G sM ( Q ), especially in the range ( Q ) ≃ ; see e.g.Table I in Ref. [7]. In this latter work a global analysis of World Data of parity-violatingelectron scattering was performed for Q < ∼ . and led to µ s = 0 . ± . ± . µ N ). Another low Q global analysis [8] disfavored negative G sM , and stilla third one [9], dedicated to the range ≃ produced two sets of solutionswith opposite signs.On the theoretical side, the strangeness contributions to the magnetic moment of theproton have also been intensively investigated. Few approaches have produced results closeto the data, with positive sign, such as heavy baryon chiral perturbation theory [10, 11],quenched chiral perturbation theory [12], chiral quark-soliton model [13], Skyrme model [14],and constituent quark models [15–17]. However, a large number of theoretical results pre-dicted negative values, notably, meson cloud model [18, 19], chiral quark model [20, 21],and unquenched constituent quark model [22]. A remarkable issue is that the lattice-QCDapproaches [23–27] have kept predicting negative strangeness magnetic moment for the pro-ton. Note that in various works prior to the advent of the first data, the general trend waspredicting negative sign for the strangeness magnetic moment of the proton µ s , as reviewedin Refs. [28, 29].In 2009, the PVA4 Collaboration [4], obtained for the first time a negative sign value G sM ( Q = 0 .
22) = − . ± . ± .
11; units are (GeV/c) for Q and nuclear magnetonsfor G sM . More recently the HAPPEX Collaboration [6] reported a small but also negativesign at higher Q , namely, G sM ( Q = 0 . − . ± . µ N .2he present work is motivated by interpreting the recent data [4, 6] on G sM ( Q ) withinan extended chiral constituent quark model ( EχCQM ).Our starting point was the idea put forward by Zou and Riska [30] according to which thestrangeness magnetic moment of the proton could be explained by including five-quark Fockcomponents in the proton wave function. They showed that a positive strangeness magneticmoment of the proton can rise from the ¯ s being in the ground state and the four-quarksubsystem uuds in the P -state, while ¯ s in the P -state and the four-quarks in their groundstate would lead to a negative value for µ s . Then that approach was developed and extendedto the strangeness contributions to spin of the proton [31], magnetic moments of baryons [16],electromagnetic and strong decays of baryon resonances [32–35]. The main outcome of thosestudies is that the higher Fock components play important roles in describing the propertiesof baryons and their resonances.However, in Ref. [30] only contributions from the diagonal matrix elements h uuds ¯ s | ˆ µ s | uuds ¯ s i were included, while the non-diagonal transition between three-quark andstrangeness components of the proton h uud | ˆ µ s | uuds ¯ s i also contributes. In fact, the diagonalcontributions are proportional to the probability of corresponding strangeness component P s ¯ s ≡ A s ¯ s , but the non-diagonal contributions are proportional to the product of probabilityamplitudes of three- and five-quark components A q A s ¯ s . Generally, the latter is more signif-icant than the former, given that the proton is mainly composed of three-quark component.In Ref. [16], the non-diagonal contributions were taken into account, but on the one hand,only the lowest strangeness component, with the four-quark subsystem in the P-state wasconsidered, and on the other hand, the probability amplitudes for strangeness componentsin the proton were treated as free parameters in order to obtain a positive value for µ s .In the present work, the probability amplitudes, a crucial ingredient in the extendedchiral constituent quark model, are calculated within the most commonly accepted q ¯ q paircreation mechanism, namely, the P model. Then, the q ¯ q pair is created anywhere inspace with the quantum numbers of the QCD vacuum 0 ++ , corresponding to P [36]. Thismodel has been successfully applied to the decay of mesons and baryons [37, 38], and hasrecently been employed to analyze the sea flavor content of the ground states of the SU (3)octet baryons [39]. Note that in the SU (3) symmetric case, the ratio of probabilities forfive-quark components with strange and light quark-antiquark pairs is 1 / SU (3) symmetry breaking effects, we determined [39] that ratio to3e P s ¯ s / ( P u ¯ u + P d ¯ d ) = 0 . / (0 .
098 + 0 . ∼ .
18 and putting P s ¯ s ∼ II. THEORETICAL FRAMEWORK
In this section, we first briefly review the method to derive the wave function of the protonin the extended chiral constituent quark model (Sec II A), and then present the formalismfor the strangeness magnetic moment of the proton (Sec II B).
A. Wave function of the proton
In our extended chiral constituent quark model, the wave function of the proton can beexpressed as | ψ i p = 1 √N | q i + X i,n r ,l C in r l | q, i, n r , l i . (1)The first term in Eq. (1) is just the conventional wave function for the proton with threelight constituent quarks, which reads | q i = 1 √ ] C φ ( ~ξ ) φ ( ~ξ )( ϕ pλ χ λ + ϕ pρ χ ρ ) , (2)where [1 ] C denotes the SU (3) color singlet, ϕ pλ ( ρ ) the mixed symmetric flavor wave functionsof the proton, and χ λ ( ρ ) the mixed symmetric spin wave functions for configuration [21] S with spin 1 / φ ( ~ξ i ) are the orbital wave functions with the4uantum numbers n r , l, m denoted by corresponding subscripts; ~ξ i are the Jacobi coordinatesdefined by ~ξ = 1 √ ~r − ~r ); ~ξ = 1 √ ~r + ~r − ~r ) . (3)The second term in Eq. (1) is a sum over all possible five-quark Fock components with q ¯ q pairs; q ≡ u, d, s . n r and l denote the inner radial and orbital quantum numbers, respectively.As discussed in Ref. [39], here we only consider the case for n r = 0 and l = 1, sinceprobabilities of higher radial excitations in the proton should be very small, and those ofhigher orbital excitations vanish. Different possible orbital-flavor-spin-color configurationsof the four-quark subsystems in the five-quark system with n r = 0 and l = 1 are numberedby i ; i = 1 , · · · ,
17. Finally, C in r l / √N ≡ A in r l represents the probability amplitude for thecorresponding five-quark component, which can be calculated by C in r l = h QQQ ( Q ¯ Q ) , i, n r , l | ˆ T | QQQ i M p − E in r l , (4)where N ≡ X i =1 N i = 1 + X i =1 C in r l , (5)and ˆ T is a transition coupling operator of the P modelˆ T = − γ X j F j, C j, C OF SC X m h , m ; 1 , − m | i χ ,mj, Y , − mj, ( ~p j − ~p ) b † ( ~p j ) d † ( ~p ) , (6)with M p the physical mass of the proton.Wave functions of the five-quark components can be classified into two categories byfour-quark subsystems being in their S-state | q, i, , i = X abc X s z mm ′ s ′ z C
12 12 s z ,jm C jm m ′ , s ′ z C [1 ][31] a [211] a C [31] a [ F ] b [ S ] c [ F ] b [ S ] c [211] C,a ¯ Y m ′ ¯ χ s z ′ Φ( { ~ξ i } ) , (7)and P-state | q, i, , i = X abcde X Ms ′ z ms z C
12 12
JM, s ′ z C JM m,Ss z C [1 ][31] a [211] a C [31] a [31] b [ F S ] c C [ F S ] c [ F ] d [ S ] e [31] X,m ( b )[ F ] d [ S ] s z ( e )[211] C ( a ) ¯ χ s ′ z Φ( { ~ξ i } ) , (8)5here the flavor, spin, color and orbital wave functions of the four-quark subsystem aredenoted by the Young patterns. The coefficients C
12 12 s z ,jm and C jm m ′ , s ′ z in Eq. (7), and C
12 12
JM, s ′ z and C JM m,Ss z in Eq. (8) are Clebsch-Gordan coefficients for the angular momentum, andothers are Clebsch-Gordan coefficients of S permutation group. ¯ Y m ′ and ¯ χ s z ′ represent thewave functions of the antiquark. ~ξ i denote the Jacobi coordinates for a five-quark system,analogous to the ones in Eq. (3), and ~ξ i are defined as ~ξ i = 1 √ i + i i X j =1 ~r j − i~r i , i = 1 , · · · , . (9)Finally, the energies of five-quark components with quantum numbers n r = 0 and l = 1in constituent quark model can be expressed as E i, , = E + δ im + h H hyp i i , (10)where E is a commonly shared energy of the 17 different five-quark configurations, δ im theenergy deviation caused by the s ¯ s pairs, and h H hyp i i denote matrix elements of the quarkshyperfine interactions in the five-quark configurations. In this work, we employ the hyperfineinteractions mediated by Goldstone-boson exchange [40], H h = − X i In our model, calculations of the strangeness magnetic moment of the proton can bedivided into two parts, namely, the diagonal and non-diagonal contributions. The former canbe defined as the matrix elements of the following operator in the strangeness componentsof the proton ˆ µ Ds = M p m s X i ˆ S i (cid:16) ˆ l iz + ˆ σ iz (cid:17) , (12)6here ˆ S i is an operator acting on the flavor space, with the eigenvalue +1 for a strangequark, − µ Ds is in unit of the nuclear magneton.The non-diagonal contributions of the strangeness magnetic moment, which involve s ¯ s pair annihilations and creations, are obtained as matrix elements of the operatorˆ µ NDs = 2 M p X i ˆ S i C OF SC ~r i × ˆ σ i , (13)where ˆ µ NDs is also in unit of the nuclear magneton. C OF SC is an operator to calculate theoverlap between the orbital, flavor, spin and color wave functions of the residual three-quarkin the five-quark components after s ¯ s annihilation and the three-quark component of theproton. TABLE I: Diagonal ( µ Ds ) and non-diagonal ( µ NDs )contributions of different five-quark configurations to thestrangeness magnetic moment of the proton. Notice that the full expressions are obtained by multiplyingeach term by M p m s P is ¯ s for µ Ds and by M p ω / C A q A is ¯ s for µ NDs . The last column gives the flavor-spinoverlap factors. Category Configurations µ Ds µ NDs C iF S i) [31] X [22] S : [31] X [4] F S [22] F [22] S / √ / √ / X [31] F S [211] F [22] S / 24 2 √ / √ / X [31] F S [31] F [22] S / 24 2 √ / √ / X [31] S : [31] X [4] F S [31] F [31] S − − √ / / X [31] F S [211] F [31] S − − √ / / X [31] F S [22] F [31] S − − √ / / √ X [31] F S [31] F [31] S − / −√ / X [22] S : [4] X [31] F S [211] F [22] S − / p / √ / X [31] F S [31] F [22] S − / p / √ / X [31] S : [4] X [31] F S [211] F [31] S − − p / √ / X [31] F S [22] F [31] S − − / √ / X [31] F S [31] F [31] S − / √ −√ / As reported previously [39], among the seventeen possible different five-quark configura-tions, the probability amplitudes of twelve of them with s ¯ s pairs are nonzero in the proton.Those configurations can be classified in four categories (Table I) with respect the orbital7nd spin wave functions of the four-quark subsystem, namely, configurations with: i) [31] X and [22] S ; ii) [31] X and [31] S ; iii) [4] X and [22] S ; iv) [4] X and [31] S . Contributions of thesefour different kinds of configurations are described below. i ) [ ] X and [ ] S : The total spin of the four-quark subsystem is 0, therefore the diagonalmatrix elements h µ Ds i are only from contributions due to the four-quark orbital angularmomentum and spin of the antiquark, the resulting matrix elements are h µ Ds i i = M p m s h h X j =1 ˆ l jz ˆ S j i i i P is ¯ s , (14)where P is ¯ s is the probability of the i th strangeness component in the proton. And for thenon-diagonal matrix element h µ NDs i , explicit calculations lead to h µ NDs i i = 15 / M p C ω √ C iF S A q A is ¯ s , (15)where A q and A is ¯ s denote the probability amplitudes of the three-quark and the i th strangeness components in the proton, and C iF S is the corresponding flavor-spin overlapfactor for the i th strangeness component. C , common to all different strangeness compo-nents, is the overlap between the orbital wave function of the residual three-quark in thestrangeness component after s ¯ s annihilation and that of the three-quark component, andreads C = ω ω ω + ω ! , (16)with ω and ω the harmonic oscillator parameters of three- and five-quark components.Note that the expression for C above differs by a factor of [2 ω ω / ( ω + ω )] / from thatintroduced in, e.g. Refs. [15, 16], due to the proper handling of the center-of-motion in thepresent work. ii ) [ ] X and [ ] S : The total spin of the four-quark subsystem is 1, combined to theorbital angular momentum L [31] X = 1, the total angular momentum of the four-quark sub-system can be J = 0 , , 2, and to form the proton spin 1 / 2, only the former two are possiblealternatives. In the present case, we take the lowest one J = 0. Accordingly, the four-quarksubsystem cannot contribute to µ s , and the resulting matrix elements are h µ Ds i i = − M p m s P is ¯ s , (17) h µ NDs i i = − / M p C ω √ C F S A q A is ¯ s , (18)8 ii ) [ ] X and [ ] S : Given that the total angular momentum of the four-quark subsystemis 0, it does not contribute to µ s . Consequently, once we remove the contributions of themomentum of the proton center-of-mass motion, we obtain the following matrix elements: h µ Ds i i = − M p m s (cid:16) − h X j =1 ˆ S j i i (cid:17) P is ¯ s , (19) h µ NDs i i = 15 / M p C ω s C iF S A q A is ¯ s . (20) iv ) [ ] X and [ ] S : The total spin of the four-quark subsystem should be S [31] = 1, herewe assume that the combination of S [31] with orbital angular momentum of the antiquarkleads to J = S ⊕ L ¯ q = 0, then matrix elements read h ˆ µ Ds i i = − M p m s P is ¯ s , (21) h µ NDs i i = − / M p C ω s C iF S A q A is ¯ s . (22)Accordingly, explicit calculations of the matrix elements h P j =1 ˆ l jz ˆ S j i i , h P j =1 ˆ S j i i , and C iF S lead to the results shown in Table I. III. NUMERICAL RESULTS AND DISCUSSION As already mentioned, numerical results reported here were obtained using input param-eters (Table II) taken from the literature, as commented below.For the mass of the strange quark m s and the mass difference between constituent strangeand light quarks δm = m s − m , we adopted the commonly used values [40]. The energyshared by five-configurations between quarks E , in the absence of hyperfine interaction,and the term due to the transition between three- and five-quark components ( V ) are takenfrom our previous work [39], which allowed reproducing the experimental data for the protonflavor asymmetry ¯ d − ¯ u . The matrix elements of the flavor operators, are linear combinationsof the spatial matrix elements, A i , B i and C i , i =0,1 ; the numerical values of which werefixed to those determined in Ref. [40].The last two parameters in Table II are the harmonic oscillator parameters, ω and ω , for the three- and five-quark components, respectively, in baryons. The parameter ω can be inferred from the empirical radius of the proton via ω = 1 / q h r i , which yields9 ABLE II: Input parameters (in MeV). Parameter value Ref. m s 460 [40] δm 120 [40] E V ± 46 [35] A 29 [40] B 20 [40] C 14 [40] A 45 [40] B 30 [40] C 20 [40] ω 246 & 340 [34, 35, 41] ω 225 & 600 [34, 35, 41] ω ≃ 246 MeV for q h r i = 1 fm. However, the value of ω is rather difficult to determineempirically. As discussed in Ref. [41], the ratio R = ω ω , (23)can be larger or smaller than 1. Consequently, we used two sets for R to get the numericalresults, • Set I: ω = 246 MeV and R = q / ≃ . 91 from setting the confinement strength ofthree- and five-quark configurations to be the same value [41], leading to ω ≃ 225 MeVand C ≃ . • Set II: ω = 340 MeV and ω = 600 MeV, values adopted to reproduce the data forelectromagnetic and strong decays of several baryon resonances [34, 35], correspondingto R ≃ . 76 and C ≃ . P s ¯ s , which is often left as free parameter. Here, we calculated it withinthe P formalism [36–38]. Then, that probability turns out [35] to be P s ¯ s = 5 . ± . V = 570 ± 46 MeV.In the following two sections we report our results for the strangeness magnetic moment µ s and magnetic form factor G sM of the proton and compare them with the latest data andfew most recent / relevant theoretical investigations.10 . Strangeness magnetic moment of the proton Our results for diagonal and non-diagonal components of µ s are reported in Table III, forthe central value V = 570 MeV and the two Sets with respect to the [ R , ω , ω ] ensemblespresented above. TABLE III: Diagonal µ Ds and non-diagonal µ NDs contributions to the strangeness magnetic moment ofthe proton from each configuration for Sets I and II, with A is ¯ s the probability amplitude and P is ¯ s /P tots ¯ s therelative weight of the strangeness probability in the proton; P tots ¯ s = P i =1 P is ¯ s . Set I Set IICategory Configuration A is ¯ s P is ¯ s /P tots ¯ s µ Ds µ NDs µ NDs (%) ( µ N ) ( µ N ) ( µ N )i) [31] X [22] S : [31] X [4] F S [22] F [22] S − . 099 17 0 . − . − . X [31] F S [211] F [22] S − . 060 6 0 . − . − . X [31] F S [31] F [22] S − . 051 5 0 . − . − . Subtotal 1 28 0 . − . − . ii) [31] X [31] S : [31] X [4] F S [31] F [31] S − . 079 11 − . . . X [31] F S [211] F [31] S − . 057 6 − . . . X [31] F S [22] F [31] S − . 042 3 − . . . X [31] F S [31] F [31] S . 028 1 − . . . Subtotal 2 21 − . . . iii) [4] X [22] S : [4] X [31] F S [211] F [22] S . 092 15 − . . . X [31] F S [31] F [22] S . 081 11 − . . . Subtotal 3 26 − . . . iv) [4] X [31] S : [4] X [31] F S [211] F [31] S . 088 13 − . − . − . X [31] F S [22] F [31] S . 066 8 − . − . − . X [31] F S [31] F [31] S − . 044 3 − . − . − . Subtotal 4 24 − . − . − . TOTAL - 100 − . − . − . In Table III the first column shows the four categories and the second one the associatedconfigurations. Accordingly, contributions from each one of the twelve configurations arereported. Probability amplitudes, calculates within the P model are depicted in the thirdcolumn. The fourth column gives the relative weight for each configuration in P s ¯ s = 5 . ω , are identical for the two Sets.11inally, the last two columns correspond to the contributions from non-diagonal terms forSets I and II, respectively. Several features deserve comments, which will also be useful inshedding light on the results from other sources. • A is¯s : The probability amplitudes for all [31] X configurations are negative, except forthe one with flavor-spin wave function [31] F S [31] F [31] S , while those for configurationswith [4] X are positive, except for the [31] F S [31] F [31] S configuration. • P is¯s / P tots¯s : The total contribution of each category is around 24 ± • µ Ds : The diagonal terms are positive in the first category and negative in the otherthree. The absolute values from one configuration to another show variations reachingalmost one order of magnitude. • µ NDs : The difference between Sets I and II per configuration is merely due to thedifferent [ ω , ω ] ensembles used in the present work. Non-diagonal terms have oppositesigns with respect to the corresponding diagonal ones in all categories, except the lastone. Per configuration, the magnitude of non-diagonal term is larger, in some casesby two orders of magnitudes, than that of the corresponding diagonal term. • µ Ds + µ NDs : Accordingly, the sum of the diagonal and non-diagonal terms per config-uration is dominated by far by the non-diagonal term. However, it is important tounderline the following point: the last line in Table III shows that, due to significantcancelations among the non-diagonal terms from various configurations, the ratio ofthe sum of non-diagonal terms (-0.1082 and -0.0258) over that of the diagonal ones(-0.0413), is 2.6 (Set I) or 0.6 (Set II), so very significantly different from that ratioper configuration, and even per category.From the above considerations, we infer an important finding: retaining only the diagonalterms and/or using a configuration truncated scheme will lead to unreliable results, asdiscussed in sec. III C.Finally, using values in the last line of Table III our predictions for the proton strangenessmagnetic moment µ s are − . ± . µ N for Set I and − . ± . µ N for Set II, withthe reported uncertainties corresponding to the range V = 570 ± 46 MeV [35].12t is worth to underline two features: both Sets lead to small and negative values for µ s ,though the two results differ one from another by more than 20 σ . This latter observationshows the high sensitivity of the strangeness magnetic moment to the ratio R = ω /ω . -0.20-0.16-0.12-0.08-0.040 0 1 2 3 4 5 µ s ( µ N ) R FIG. 1: (Color online) The strangeness magnetic moment of the proton µ s in units of nuclear magnetons( µ N ) as a function of the ratio R = ω /ω , for ω = 246 MeV (full red curve) and ω = 340 MeV (dottedgreen curve). In Fig. 1 µ s is depicted as a function of R , varying from 0 . . ω fixed at 246 MeV (full curve) and at 340 MeV (dotted curve). Themaximum discrepancy between the two curves is roughly 20% at the minimum values for µ s , located at R ≃ . 71. So, µ s depends mildly on the exact value of ω , but strongly onthat of ω and hence R . The proton strangeness magnetic moment turns out then to besignificantly sensitive to that ratio in the range 0 . < R < 3, where µ s varies by a factor of4. In any case, according to our study, µ s is small and negative. B. Strangeness magnetic form factor of the proton In order to extend the present approach to the Q -dependent strangeness magnetic formfactor of the proton G sM , for which experimental data are available, we need to calculatethe matrix elements of the transitions h uuds ¯ s | ~J | uuds ¯ s i and h uud | ~J | uuds ¯ s i for both diagonaland non-diagonal terms. For the former ones, explicit calculations lead to( G sM ) D = µ Ds e − q / (5 ω ) , (24)13xcept for two of the configurations with four-quark subsystem wave functions being[4] X [31] F S [211] F [22] S and [4] X [31] F S [31] F [22] S , for which the expression reads( G sM ) D = (cid:16) µ Ds − q ω (cid:17) e − q / (5 ω ) . (25)For the non-diagonal transitions between all the strangeness configurations and the three-quark component of the proton, the strangeness magnetic form factor is:( G sM ) ND = µ NDs e − q / (15 ω ) , (26)with the photon three-momentum term ( q ) related to the four-momentum transfer Q = q − k γ as q = Q (cid:16) Q M p (cid:17) . (27) TABLE IV: Diagonal and non-diagonal contributions to the strangeness magnetic form factor of the protonfrom each configuration for Sets I and II, at momentum transfer values Q =0.220 and 0.624 (GeV/c) . Set I, Q = 0 . 220 Set II, Q = 0 . 220 Set I, Q = 0 . 624 Set II, Q = 0 . Configuration ( G sM ) Di ( G sM ) NDi ( G sM ) Di ( G sM ) NDi ( G sM ) Di ( G sM ) NDi ( G sM ) Di ( G sM ) NDi i) [31] X [22] S : [31] X [4] F S [22] F [22] S . − . . − . . − . . − . X [31] F S [211] F [22] S . − . . − . . − . . − . X [31] F S [31] F [22] S . − . . − . . − . . − . Subtotal 1 .0066 –.6206 .0148 –.4299 .0009 –.0439 .0112 –.2965 ii) [31] X [31] S : [31] X [4] F S [31] F [31] S − . . − . . − . . − . . X [31] F S [211] F [31] S − . . − . . − . . − . . X [31] F S [22] F [31] S − . . − . . − . . − . . X [31] F S [31] F [31] S − . . − . . − . . − . . Subtotal 2 –.0097 .5714 –.0216 .3961 –.0014 .0405 –.0163 .2734 iii) [4] X [22] S : [4] X [31] F S [211] F [22] S − . . − . . − . . − . . X [31] F S [31] F [22] S − . . − . . − . . − . . Subtotal 3 –.0020 .4842 –.0044 .3354 –.0003 .0343 –.0034 .2314 iv) [4] X [31] S : [4] X [31] F S [211] F [31] S − . − . − . − . − . − . − . − . X [31] F S [22] F [31] S − . − . − . − . − . − . − . − . X [31] F S [31] F [31] S − . − . − . − . − . − . − . − . Subtotal 4 –.0112 –.4671 –.0250 –.3235 –.0016 –.0331 –.0189 –.2232 TOTAL − . − . − . − . − . − . − . − . Q = 0 . 22 and 0 . 624 (GeV/c) . Table IV contains the outcome of ourcalculations on the proton strangeness magnetic form factor for all 12 configurations and forboth Sets I and II, bringing in few comments: • ( G sM ) Di : Because of the ω dependence of G sM , the diagonal terms are not identicalin Sets I and II, as it was the case for µ s . The magnitude of this component, perconfiguration, decreases with Q as well as in going from Set II to Set I at a fixed Q . • ( G sM ) NDi : The magnitude of the non-diagonal terms are larger than those of diagonalones, and they decrease with Q and also in going from Set I to Set II at a fixed Q . • ( G sM ) Di / ( G sM ) NDi : the Q dependence of this ratio turns out to be quite different forSets I and II, as shown in Fig. 2. For Set I, between Q = 0 and 1 (GeV/c) the ratiodecreases by a factor of more than 3 and above Q ∼ . , the diagonal termsbecome larger than the diagonal ones, while in Set II the non-diagonal terms standfor roughly 37 ± 2% of the sum of the two terms in the whole shown Q range. • Signs: There are no sign changes in diagonal and non-diagonal terms for a givenconfiguration at different Q s, including Q = 0. ( G s M ) N D / G s M ( % ) Q (GeV /c ) FIG. 2: (Color online) The ratio of non-diagonal to diagonal + non-diagonal terms in the strangenessmagnetic form factor of the proton G sM as a function of Q for Sets I (full red curve) and II (dotted greencurve). In the next section we proceed to comparisons between our results and relevant onesreported in the literature. 15 . Discussion Table V summarizes our numerical results for the strangeness magnetic moment of theproton and its magnetic form factor at four Q values. In Fig. 3 results for G sM withinSets I and II, spanning the range 0 ≤ Q ≤ are depicted and compared to theHAPPEX [6] and PVA4 [4] data. TABLE V: Results for the proton strangeness magnetic moment and magnetic form factor (in nuclearmagneton) at four Q values (in (GeV/c) ). Reference Year Approach µ s G sM ( Q = 0 . G sM ( Q = 0 . G sM ( Q = 0 . G sM ( Q = 0 . Present work: Set I EχCQM − . ± . − . ± . − . ± . − . ± . − . ± . Present work: Set II − . ± . − . ± . − . ± . − . ± . − . ± . et al. [24] { } LQCD − . ± . et al. [25] { } LQCD − . ± . et al. [26] { } LQCD − . ± . − . ± . et al. [27] { } LQCD − . ± . − . ± . − . ± . et al. [6] { } Data [HAPPEX] − . ± . et al. [4] { } Data [PVA4] − . ± . et al. [5] { } Data [G0] +0 . ± . − . ± . et al. [3] { } Data [SAMPLE] +0 . ± . -0.30-0.24-0.18-0.12-0.060 0 0.2 0.4 0.6 0.8 1Set ISet II G s M ( µ N ) Q (GeV /c ) FIG. 3: (Color online) The strangeness magnetic moment of the proton G sM as a function of the momentumtransfer Q for Sets I (full red curve) and II (dotted green curve). Data are from Refs. [4, 6]. The general trend in our results is that the investigated observable is negative with smallmagnitude. However, Sets I and II behave differently as a function of Q . Actually, for16et I, the harmonic oscillator parameter ω ≃ 225 MeV, is smaller than ω ≃ 600 MeV inSet II. So due to the exponential Q dependence, G sM approaches zero faster in Set I than inSet II. In the following we compare our predictions with results from other sources quotedin Table V.At Q =0.22 (GeV/c) both Sets give almost identical values, compatible with PVA4data [4], while at Q =0.624 (GeV/c) Set II is favored by the HAPPEX [6] data. At thosetwo momentum transfer values, data reported by the G0 Collaboration [5] have too largeuncertainties to allow informative comparisons with our predictions. To a lesser extent, thesame consideration is also true for the SAMPLE Collaboration data [3] at Q =0.1 (GeV/c) with a positive value, large uncertainty and compatible with zero.In Table V we also show results from lattice-QCD calculations. Quenched QCD comple-mented by chiral extrapolation techniques performed by Leinweber et al. [24] and Wang etal. [25] produce for µ s and G sM ( Q = 0 . σ for µ s in Set II and G sM in both Sets. This is also thecase for Set II results with respect to the outcome of a N f = 2 + 1 clover fermion LQCDby Doi et al. [26] for µ s and G sM ( Q = 0 . et al. [27],based on the Wilson gauge and fermion actions on an anisotropic lattice, leads to smallermagnitudes than our predictions at Q = 0 . 22 (GeV/c) . While at Q = 0 . 62 (GeV/c) result of the latter work agrees with ours for Set I, at Q = 0 . 81 GeV/c) Set II producesvalue compatible with the considered LQCD data.Here, it is worth mentioning that theoretical predictions as well as recent data (Table V)show (significant) discrepancies with the extracted values from global fits to the data releasedbefore 2009: G sM ( Q = 0 . 22) = 0 . ± . µ N (Ref. [7]), G sM ( Q = 0 . 21) = 0 . ± . µ N .(Ref. [8]) and G sM ( Q = 0 . . ± . µ N (Ref. [9]), all of them in disagreement withthe latest data from PVA4 [4] and HAPPEX [6] Collaborations.To end this section, we compare our approach to results coming from similar works [15–17, 30, 31] reported in the literature.As mentioned in Introduction, in Ref. [30] the sign of the proton strangeness magneticmoment was investigated with respect to the strange antiquark states in the five-quarkcomponent of the proton. In a subsequent paper [15] the authors calculated G sM ( Q ) inthe range 0 ≤ ( Q ) ≤ , where data were giving positive values [7–9]. There,17wo scenarios were adopted i) ω ≃ ω ( R ≃ 2) and ω ≃ ω ( R ≃ s ¯ s , namely P s ¯ s = 10% and 15%. The three combinations between R and P s ¯ s studied gave results consistent with the available data in 2006. However, outof the twelve configurations (Table III) only [31] X [4] F S [22] F [22] S was considered. Thatconfiguration was also used in Refs. [16, 31], where only the diagonal term was included,resulting in µ s =0.17 µ N .A more recent constituent quark model [17] considered separately only two configurations,namely, [31] X [4] F S [22] F [22] S and [31] X [31] F S [211] F [22] S , corresponding to the ¯ s being in the S − or P − state, respectively. Pure P -state gave µ s = 0 . µ N and an admixture between thetwo states µ s = 1 . µ N . In that work, both diagonal and non-diagonal terms were consideredfor the retained configurations and ω was fixed at 246 MeV, while ω and the probability P s ¯ s were fitted on the G0 Collaboration [5] data reported in Table V. The extracted valuesare ω =469 MeV and P s ¯ s =0.025%, smaller by more than two orders of magnitude comparedto the P model result employed in the present work. Using their approach, the authorsfound that putting P s ¯ s =2.5%, as reported in Ref. [42], leads to ω =108 MeV. The incrediblytiny probability reported in Ref. [17] can easily be understood. As shown in Table IV,contributions from individual configurations [31] X [4] F S [22] F [22] S or [31] X [31] F S [211] F [22] S compared to the total of contributions from all twelve of them differ by up to two orders ofmagnitude. IV. SUMMARY AND CONCLUSIONS The extended chiral constituent quark model offers an appropriate frame to study thepossible manifestations of genuine five-quark components in baryons. The present work isin line with our earlier efforts [34, 35, 39] in that realm. There are several difficulties in thisendeavor: few observables have been identified carrying information on higher Fock states,the data are scarce and often bear large uncertainties due to the smallness of the effectslooked for. Moreover, there are input parameters in the approach, which basically should betaken from literature and exceptionally fitted on the data under consideration. Accordingly,we took advantage of the data on radiative and strong decays of the Λ(1405) resonance [34],strong decay of low-lying S and D nucleon resonances [35], and sea flavor content ofoctet baryons [39] to deepen our understanding of the five-quark components and select a18oherent set of input parameters.Our main findings can be summarized in three points, as follows. • i) Five-quark Fock states: we gave detailed numerical results for both diagonaland non-diagonal terms for all of the twelve relevant configurations showing stronginterplays among different components with (very) large cancellations. • ii) Probability of the s ¯ s in the proton wave function: we determined P s ¯ s usinga P pair creation model, as in a previous work [39]. • iii) Harmonic oscillator parameters: it was shown that with respect to the pa-rameters ω and ω , the important element is the ratio R = ω /ω .Based on the above observations, it becomes then obvious that using severely truncatedconfiguration sets and/or unrealistic values for P s ¯ s or R will lead to unreliable results withrespect to the magnetic moment and/or magnetic form factor of the proton.In the present paper we showed that our predictions are in reasonable agreement withrecent measurements [4, 6] and lattice-QCD results [24–27].The uncertainties associated to the available data on the one hand, and those of LQCDapproaches on the other hand, do not allow us making a sharp choice between the resultscoming from the two Sets in terms of the ratio R . It is nevertheless clear that the strangenessmagnetic moment of the proton and its magnetic form factor are small and negative. Be-tween the two Sets, Set II appears to be slightly favored by findings from other sources.Accordingly, we get µ s = − . ± . µ N and the magnitude of the strangeness mag-netic from factor of the proton evolves smoothly with increasing transfer momentum to reach G sM ( Q ) = − . ± . µ N at Q = 1 (GeV/c) .Awaited for data at Q = 0 . expected to be released by the PVA4 Collab-oration [4] and more advanced LQCD approaches will hopefully improve the accuracy ofthe experimental and theoretical data bases. Recent convergence between theory and ex-periment on the negative sign of that observable and its smallness, might also initiate newdedicated measurements. 19 cknowledgments We wish to thank the anonymous Referee for his/her careful reading of the manuscript.This work was supported by the National Natural Science Foundation of China under grantnumber 11205164. 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