Hyperon electromagnetic timelike elastic form factors at large q 2
aa r X i v : . [ h e p - ph ] J a n LFTC-19-10/48
Hyperon electromagnetic timelike elastic form factors at large q G. Ramalho , M. T. Pe˜na and K. Tsushima Laborat´orio de F´ısica Te´orica e Computacional – LFTC,Universidade Cruzeiro do Sul and Universidade Cidade de S˜ao Paulo,01506-000, S˜ao Paulo, SP, Brazil and Centro de F´ısica Te´orica e de Part´ıculas (CFTP),Instituto Superior T´ecnico (IST), Universidade de Lisboa,Avenida Rovisco Pais, 1049-001 Lisboa, Portugal (Dated: January 23, 2020)We present estimates of the hyperon elastic form factors for the baryon octet and the Ω − baryonfor large four-momentum transfer squared, q , in the timelike region ( q > e + e − → B ¯ B and p ¯ p → B ¯ B processes, where B stands fora general baryon. Our results are based on calculations of the elastic electromagnetic form factorsin the spacelike region ( Q = − q >
0) within a covariant quark model. To connect the results inthe spacelike region to those in the timelike region, we use asymptotic relations between the tworegions which are constraints derived from analyticity and unitarity. We calculate the effective formfactors | G ( q ) | and compare them with the integrated cross section data σ Born ( q ) from BaBar,BES III, and CLEO. The available data are at the moment restricted to Λ, Σ , Σ − , Ξ − , Ξ , andΩ − as well as to e + e − → Λ ¯Σ and e + e − → Σ ¯Λ reactions. Our results provide useful referencefor future experiments and seem to indicate that the present data are still in the non-perturbativeQCD region, while the onset for the asymptotic constraints from analyticity and unitarity happensmuch before the region of the perturbative QCD falloff of the form factors. I. INTRODUCTION
The understanding of internal structure of hadrons hasbeen a great challenge after the discovery that the pro-ton is not a pointlike particle. In the last decades, greatprogress has been made in the study of the nucleon elec-tromagnetic structure, particularly through the scatter-ing of electrons with nucleon targets ( γ ∗ N → N tran-sition) which probes the spacelike momentum transferkinematic region ( Q ≥
0) [1–4]. For hyperons ( B ),however, it is difficult to get information on the inter-nal structure based on the γ ∗ B → B process due totheir very short lifetimes. The available information isrestricted, at the moment, only to the magnetic momentsof a few hyperons (determined at Q = 0).The other possibility of disclosing the electromagneticstructure of baryons is e + e − scattering. It enables us toaccess the timelike region ( q = − Q >
0) and was pro-posed a long time ago by Cabibbo and Gatto [5], however,it became possible only recently. The e + e − → B ¯ B (andthe inverse) reactions open a new opportunity to studythe role of valence quark effects, clusters of two-quarkpairs (diquarks), and different quark compositions [6–12]. The timelike region form factors appear as a viabletool to determine the hyperon structure, near the thresh-old as well as in the large- q region, where in the latter,perturbative effects are expected to dominate [4, 5, 11–16]. A significant amount of data are already availablefor the proton ( e + e − → p ¯ p ) [4, 13]. In the present studywe focus on the reactions involving hyperons in the finalstates. Data associated with hyperon electromagnetic form factors in the timelike region also became availablein facilities such as BaBar [17], BES-III [18, 19], andCLEO [11, 12]. The available data cover the high- q re-gion where we can expect to probe perturbative QCD(pQCD) physics.From the theoretical side, there have been only a fewdifferent attempts in interpreting the hyperon timelikeelectromagnetic form factor data [20–27]. Although re-sults from e + e − and p ¯ p annihilation experiments are al-ready available or being planned in the near future e.g.by the PANDA experiment at FAIR-GSI [28], theoreti-cal calculations of hyperon electromagnetic timelike formfactors are scarce. The results presented here intend tofill that gap.In the large- q region one can expect the behavior pre-dicted by pQCD [29–33]. However, some of the aspectsfrom pQCD, including the q dependence of the formfactors, can be seen only at very high q . In the regioncovered by the present experiments, finite corrections forthe large- q behavior may be still relevant.One of the goals of the present work is to provide cal-culations to be compared with the recent experimentaldeterminations of the e + e − → B ¯ B cross sections fromCLEO, BaBar, and BES-III, and to use them to guidenew experiments also for larger q . The results presentedhere can be used to study the onset of the region for thevalidity of asymptotic behavior.Our estimates are based on results of a relativisticquark model for the spacelike region [34, 35]. In this workwe focus on the general properties of the integrated crosssection σ Born ( q ) and the effective form factor | G ( q ) | for large q . Based on these, we test model-independentasymptotic relations between the form factors in thespacelike and the timelike regions [4]. We use those rela-tions to calculate the magnetic and electric form factorsin the timelike region, and give estimates for the effectiveform factor G ( q ) of the Λ, Σ + , Σ , Σ − , Ξ , Ξ − and Ω − baryons. An interesting aspect that emerges from our re-sults and the comparison with the data is that the regionof q where these model-independent relations may startto hold, differ from the (even larger) q region of pQCD.This result is discussed and interpreted in terms of thephysical scales included in our model.In addition to the effective form factor G ( q ), we cal-culate also the individual form factors | G M | and | G E | ,and determine their relative weights for the effective formfactor. Most existing studies are based on the approxi-mation G M ≡ G E , equivalent to G ( q ) = G M ( q ). How-ever, it is important to notice that although by defini-tion G M = G E at the threshold of the timelike region( q = 4 M B , where M B is the mass of the baryon), thereis no proof that this relation holds for higher values of q .Therefore, in the present work we compare the result ofthe approximation G = G M with the exact result. Thedifference between the two results is a measure of the im-pact of G E in the magnitude of the effective form factor G .It is worth mentioning that, at present, calculationsof the timelike form factors based on a formulation inMinkowski space ( q = q − q ) are very important, sincethe timelike region, in practice, is still out of reach ofthe methods as lattice QCD simulations. Also most ofthe Dyson-Schwinger-equation-based approaches, formu-lated in the Euclidean space, are still restricted to massconditions compatible with singularity-free kinematic re-gions. Their extension to regions where singularities canbe crossed requires elaborate contour deformation tech-niques [36].This article is organized as follows: In the next sectionwe describe the general formalism associated with the e + e − → B ¯ B processes and their relation with the formfactors G ( q ). In Sec. III, we review in detail the relativis-tic quark model used here, which was previously testedin calculations of several baryon elastic form factors inthe spacelike region. The model-independent relationsused for the calculations in the large- q region are dis-cussed in Sec. IV. The numerical results for the timelikeform factors are presented and compared with the exper-imental data in Sec. V. The outlook and conclusions aregiven in Sec. VI. Additional details are included in theAppendixes. II. FORMALISM
We start our discussion with the formalism associatedwith spin-1/2 baryons with positive parity (1 / + ). Inthe following we represent the mass of the baryon by M B and use the notation τ = q M B . Within the one-photon-exchange approximation(equivalent to the impulse approximation in the space-like region) one can interpret the e + e − → B ¯ B reactionas the two-step process e + e − → γ ∗ → B ¯ B , and theintegrated cross section in the e + e − center-of-massframe becomes [12, 13, 37] σ Born ( q ) = 4 πα βC q (cid:18) τ (cid:19) | G ( q ) | , (2.1)where G ( q ) is an effective form factor for the baryon B (spin 1/2 and positive parity), α ≃ /
137 is the fine-structure constant, β is a kinematic factor defined by β = q − τ , and C is a factor which depends on the chargeof the baryon. The factor C is equal to 1 for neutralbaryons. For charged baryons C , it takes into accountthe Coulomb effects near the threshold [3, 12, 13, 38],given by the Sommerfeld-Gamow factor C = y − exp( − y ) ,with y = παβ M B √ q . In the region of interest of the presentstudy, at large q ( τ ≫ C ≃ G is definedby the combination of the electric and magnetic formfactors [12, 13, 37] as | G ( q ) | = (cid:18) τ (cid:19) − (cid:20) | G M ( q ) | + 12 τ | G E ( q ) | (cid:21) , = 2 τ | G M ( q ) | + | G E ( q ) | τ + 1 . (2.2)Equations (2.1) and (2.2) are very useful, since theymean that one can describe the integrated cross section σ Born from the knowledge of a unique, effective function G ( q ) defined by the magnetic and the electric form fac-tors. Note that the form factors G M and G E are complexfunctions of q in the timelike region. The relations (2.1)and (2.2) are particularly practical to calculate σ Born ( q ),because they enable us to estimate the integrated crosssection without taking into account the relative phasesbetween the form factors G M and G E .Assuming charge invariance of the electromagnetic in-teraction, namely that the spacelike and timelike photon-nucleon vertices γpp and γp ¯ p are the same, we can esti-mate the timelike form factors in the timelike region fromthe form factors in spacelike (SL) region G SL M ( − q ) and G SL E ( − q ) by applying the large- | q | , model-independentrelations [4], G M ( q ) ≃ G SL M ( − q ) , (2.3) G E ( q ) ≃ G SL E ( − q ) , (2.4)and therefore restricting our results to the very large- q region, where the form factors are real functions tofulfill the Schwarz reflection principle. These asymptoticrelations are a consequence of general physical and math-ematical principles: unitarity as well as the Phragm´en-Lindel¨of theorem, which is valid for analytic functions(proved in Ref. [4]). They are exact in the mathematical q → ∞ limit, and they imply that the imaginary partof the form factors in the timelike region goes to zero inthat limit.In the present work we use a quark model developedin the spacelike region [34, 35] to estimate the magneticand electric form factors in the timelike region based onEqs. (2.3)–(2.4). The discussion on how these relationscan be corrected for finite q is made in Sec. IV. Devia-tions from those estimates may indicate that the imagi-nary parts of the form factors in the considered timelikeregion cannot be neglected.We will investigate, by comparing with the data, thedegree of validity of those relations for finite q . Increas-ing the value of q , we can tentatively look for the onsetof the region where they may start to be a fairly goodapproximation. It turns out that this happens much be-low the region where the pQCD falloff of the form factorsstarts to emerge, as our results will show. We also pro-vide estimates for q >
20 GeV for comparison withfuture experiments.The formalism used in the discussion of 1 / + baryonscan also be extended to 3 / + baryons based on the effec-tive form factor (2.2), re-interpreting G M as a combina-tion of the magnetic dipole and magnetic octupole formfactors, and G E as a combination of the electric chargeand electric quadrupole form factors [20]. The expres-sions associated with G M and G E for 3 / + baryons arepresented in Appendix A. Using those expressions we cal-culate our results for the Ω − baryon.Before presenting the results of the extension of ourmodel to the timelike region, we present a review of thecovariant spectator quark model in the spacelike regimethat sustains the application here. III. COVARIANT SPECTATOR QUARKMODEL
We restrict our study to baryons with one or morestrange quarks (hyperons). In our estimates we use thecovariant spectator quark model. The covariant specta-tor quark model has been applied to the studies of theelectromagnetic structure of several baryons, includingnucleon, octet baryons, and decuplet baryons (includingΩ − ) in the spacelike region [34, 35, 39–53].The model is based on three basic ingredients:1. The baryon wave function Ψ B , rearranged as anactive quark and a spectator quark pair, is repre-sented in terms of the spin-flavor structure of theindividual quarks with SU S (2) × SU F (3) symme-try [35, 39].2. By applying the impulse approximation, after in-tegrating over the quark pair degrees of freedomthe three-quark system transition matrix elementcan be reduced to that of a quark-diquark system,parametrized by a radial wave function ψ B [35, 39,40]. 3. The electromagnetic structure of the quark isparametrized by the quark form factors, j (Dirac)and j (Pauli) according to the flavor content,which encode the substructure associated withthe gluons and quark-antiquark effects, and areparametrized using the vector meson dominance(VMD) mechanism [35, 46, 49].Concerning the first two points above, the literatureemphasizes the role of diquarks in the baryons [6–11].Our model, although based on a quark-diquark configu-ration, cannot be interpreted as a quark-diquark modelin the usual sense, i.e. a diquark as a pole of the quark-quark amplitude [35, 39, 40]. In our model, the inter-nal quark-quark motion is integrated out, at the level ofimpulse approximation, but its spin structure signaturesurvives [39]. Therefore, the electromagnetic matrix el-ement involves an effective quark-diquark vertex wherethe diquark is not pointlike [40].Another difference between our model and the usualquark-diquark models is that we explicitly symmetrizein all quark pairs applying the SU (3) flavor symme-try [34, 35]. Since it is well known that the exact SU (3)flavor symmetry models are expected to fail due to themass difference between the light quarks ( u and d ) andthe strange quarks, we break SU (3) flavor symmetryin two levels. We break the symmetry at the level ofthe radial wave functions by using different forms forthose functions for systems with a different number ofstrange quarks ( N s = 0 , , N s = 0 , , , SU (3) flavor symmetry also at the level of the quarkcurrent by considering different Q dependence for thedifferent quark sectors (isoscalar, isovector and strangequark components). A. Octet baryon wave functions
The octet baryon wave functions associated with aquark-diquark system in the S -wave configuration canbe expressed in the form [34, 52]Ψ B ( P, k ) = 1 √ (cid:2) φ S | M A i + φ S | M S i (cid:3) ψ B ( P, k ) , (3.1)where P ( k ) are the baryon (diquark) momentum, φ , S are the spin wave functions associated with the com-ponents S = 0 (scalar) and S = 1 (vector) of the di-quark states, and | M A i , and | M S i are the mixed an-tisymmetric and mixed symmetric flavor states of theoctet. The explicit expressions for | M A i and | M S i andfor φ , S are included in Appendix B. For more details,see Refs. [34, 39, 52].Since the baryons are on-shell and the intermediatediquark in the covariant spectator model is taken also tobe on-shell, the radial wave functions ψ B can be writtenin a simple form using the dimensionless variable χ B : χ B = ( M B − m D ) − ( P − k ) M B m D , (3.2)where m D is the diquark mass [39]. One can now writethe radial wave functions in the Hulthen form, accordingto [34, 39]: ψ B ( P, k ) = N B m D ( β + χ B )( β i + χ B ) , (3.3)where N B is a normalization constant and β i ( i =1 , , ,
4) are momentum-range parameters (in units M B m D ). The form of our baryon wave functions (3.3)was judiciously chosen to produce at large- Q the samebehavior of the form factors as pQCD [39, 50], as dis-cussed in Sec. III D.In Eq. (3.3) β is the parameter which establishes thelong-range scale ( β < β , β , β ), common to all theoctet baryons and β i ( i = 2 , ,
4) are parameters asso-ciated with the short-range scale, varying with the dif-ferent quark flavor contents. The short-range scale isdetermined by β ( N ), β (Λ and Σ) and β (Ξ).The magnitudes of β i establish the shape of the radialwave function and determine the falloff of the baryonform factors. Heavier baryons have slower falloffs [34].According to the uncertainty principle, the values of theparameters β i ( i = 2 , ,
4) can also be interpreted interms of the compactification in space of the baryons.The relative ordering β > β > β specifies that Λ andΣ , ± are more compact than the nucleon, and that Ξ , − are more compact than Λ and Σ , ± . B. Electromagnetic current
The contribution of the valence quarks for the transi-tion current in relativistic impulse approximation is ex-pressed in terms of the quark-diquark wave functions Ψ B by [34, 39] J µB = 3 X Γ Z k Ψ B ( P + , k ) j µq Ψ B ( P − , k ) , (3.4)where j µq is the quark current operator, P + , P − and k arethe final, initial and diquark momenta, respectively, andΓ labels the diquark scalar and vector diquark polariza-tions. The factor 3 takes into account the contributionsassociated with the different diquark pairs, and the in-tegral symbol represent the covariant integration in theon-shell diquark momentum.In Eq. (3.4) the quark current has a generic structure j µq = j ( Q ) γ µ + j ( Q ) iσ µν M N , (3.5)where M N is the nucleon mass and j i ( i = 1 ,
2) are SU (3)flavor operators. The components of the quark current j i ( i = 1 ,
2) canbe decomposed as the sum of operators acting on thethird quark in the SU (3) flavor space j i ( Q ) = f i + ( Q ) λ + f i − ( Q ) λ + f i ( Q ) λ s , (3.6)where λ = , λ = − ,λ s ≡ − , (3.7)are the flavor operators. These operators act on thequark wave function in flavor space, q = ( u d s ) T . Thefunctions f i + , f i − ( i = 1 ,
2) represent the quark isoscalarand isovector form factors, respectively, based on thecombinations of the quarks u and d . The functions f i ( i = 1 ,
2) represent the structure associated with thestrange quark. The explicit form for the quark form fac-tors is included in Appendix B.For the discussion of the results of this paper it is rel-evant that the parametrization of these form factors isbased on the VMD picture. The dressed photon-quarkcoupling is tied to the vector meson spectra. Therefore,the isoscalar and the isovector form factors include con-tributions from the ρ and ω mass poles in the light quarksector. As for the strange quark form factors we include adependence on the φ mass pole. In both cases, we includealso a contribution of an effective heavy meson with mass2 M N in order to take into account shorter-range effectsin the quark current. The parametrization of the currentfor the three quark sectors includes five parameters (co-efficients of the vector meson terms) in addition to thethree quark anomalous magnetic moments. C. Model for the nucleon and decuplet baryons
The model was first applied to the study of the electro-magnetic structure of the nucleon. The free parametersof the model (in the quark current and in the radial wavefunctions) were calibrated by the electromagnetic formfactor data for the proton and the neutron [39]. The nu-cleon data are well described without an explicit inclusionof pion cloud contributions.Taking advantage of the VMD form of the quarkcurrent and of the covariant form of the radial wavefunction, the model was extended to the lattice QCDregime [35, 46, 49]. This extension was performed byreplacing the vector meson and nucleon masses in theVMD parametrization of the current and in the baryonwave functions by the nucleon and vector meson massesfrom the lattice. This extension is valid for the region ofthe large pion masses, where there is a suppression of themeson cloud effects.The extension has proved to be very successful in thedescription of the lattice QCD data for the nucleon and γ ∗ N → ∆(1232) transition for pion masses above 400MeV [46, 49]. In the case of the γ ∗ N → ∆(1232) transi-tion the lattice data enabled us to fix the valence quarkcontribution and after the extrapolation to the physicalpion mass limit, indirectly infer from the physical datathe meson cloud effects [46, 54]. The meson cloud effectswas then seen to be significant in the case of the ∆(1232)due to the vicinity of its mass to the pion-nucleon thresh-old [45–47].The formalism was later applied to all baryons of thedecuplet using an SU F (3) extension of the model for the∆(1232) [45, 46, 51], constrained by the scarce availablelattice data for the decuplet baryon electromagnetic formfactors and the experimental magnetic moment of theΩ − [35]. The strange quark component of the current andthe decuplet radial wave functions were then determinedby the fit to the data (lattice QCD and experimental Ω − magnetic moment). No meson cloud contributions wereconsidered in this description of the baryon decuplet sincethose effects are suppressed in lattice calculations. Also,the only physical information on the meson cloud comesfrom the ∆(1232) calibrated in the previous works [46],and for the Ω − . In this last case, meson cloud effectsare expected to be very small, since pion excitations aresuppressed due to the content of the valence quark core(only strange quarks) implying reduced kaon excitationsgiven the large mass of the kaon [35, 48].The model for the Ω − was later re-calibrated with thefirst lattice QCD calculation of the Ω − form factors atthe physical mass point which we used to determine theelectric quadrupole and magnetic octupole moments [48]. D. Model for the octet baryons
Using the SU (3) quark current determined in the stud-ies of the nucleon and decuplet baryon systems, the co-variant spectator quark model was also extended to theoctet baryon systems. However, different from the de-cuplet case, where a fair description of the data can beobtained based exclusively on the valence quark degreesof freedom, in the case of the octet, there is evidence thatthe pion cloud effects are significant [52]. Therefore, inthe model for the baryon octet, in addition to the valencequarks we consider also explicit pion cloud contributionsbased on the SU (3) pion-baryon interaction [34, 52].The valence quark contributions, regulated by the ra-dial wave functions (3.3), were fixed by lattice QCDdata. The pion cloud contributions were calibrated bythe physical data (nucleon electromagnetic form factorsand octet magnetic moments). Compared to the previousstudies of the nucleon [39], we readjusted the values ofthe momentum-range parameters β and β of the radialwave functions (3.3), and the quark anomalous moments κ u and κ d in order to take into account the effects of thepion cloud. More details can be found in Appendixes B 3 and B 4.We discuss now the contributions from the valencequarks to the form factors. From the structure for thequark current and radial wave functions, we obtain thefollowing expressions for the valence quark contributionsto the octet baryon form factors: F B ( Q ) = B ( Q ) × (cid:20) j A + 12 3 − τ τ j S − τ τ M B M N j S (cid:21) , (3.8) F B ( Q ) = B ( Q ) × (cid:20)(cid:18) j A −
12 1 − τ τ j S (cid:19) M B M N − τ j S (cid:21) , (3.9)with τ = Q M B , and B ( Q ) = Z k ψ B ( P + , k ) ψ B ( P − , k ) , (3.10)the overlap integral between the initial and final scalarwave functions. The function B ( Q ) is independent ofthe diquark mass [39].The coefficients j A,Si ( i = 1 ,
2) are combinations ofthe quark form factors dependent on the baryon quarkcontent. The explicit expressions are included in Ap-pendix B. One concludes that the results are an interplayof both the structure of the quark form factors and of theradial wave functions.The results for the electric and magnetic form factorsare then determined by G EB = F B − τ F B , G MB = F B + F B . (3.11)The asymptotic behavior of the form factors G E and G M is determined by the asymptotic results for F B and F B from Eqs. (3.8)–(3.9). The terms between bracketsdepend only on the quark form factors and for large Q ,and contribute to F B and Q F B with constants. As aconsequence, the results for G E and G M are determinedat very large Q by the function B ( Q ), which in turn ex-clusively depends on the radial wave functions and theiroverlap. In Ref. [50] it was shown that, if we use theradial wave functions (3.3), one has B ∝ /Q plus loga-rithmic corrections. We can conclude then that the com-bination of the quark current with the radial structureinduces falloffs for the form factors consistent with thepower law of pQCD result: G E ∝ /Q and G M ∝ /Q ,in addition to logarithmic corrections [30, 31, 33, 50].The deviations of our results from the simple power law1 /Q are originated by contaminations from logarithmiccorrections, or from the difference of the quark form fac-tors from their asymptotic result ( j A,S , Q j A,S → con-stant). The latter is regulated by large momentum scalesassociated with the VMD parametrization. But the dif-ferent momentum falloff tails of the baryon form factorsalso play a role and relate to the difference in the flavorcontent of the constituent valence quarks described bythe wave functions, as well as the VMD structure of thequark form factors.As mentioned above, an accurate description of theelectromagnetic structure of the octet baryons is achievedwhen we include an explicit parametrization of the pioncloud contributions [34, 44]. The consequence of the in-troduction of the pion cloud effects is that the transi-tion form factors (3.8)–(3.9) have additional contribu-tions, which can be significant below Q < , andthat the two contributions have to be normalized by aglobal factor Z B < √ Z B is the factor associated witheach wave function).In the large- Q region, the pion cloud contributionsare suppressed and the form factors are then reduced to G EB → Z B G EB , G MB → Z B G MB , (3.12)where G EB and G MB on the r.h s. represent the valencequark estimate. From Eq. (3.12), we conclude that thepion cloud dressing affects only the normalization of theform factors at large Q . The normalization factor Z B depends only on a parameter associated with the pioncloud parametrization: the parameter which determinesthe pion cloud contribution to the proton charge ( Z N ).In Appendix B, we show that all normalization factorscan be determined by the normalization of the nucleonwave functions Z N . The values of Z B (between 0.9 and1) are also presented in Appendix B.Our calculations for the baryon octet presented in thenext sections for the Q > region depend essen-tially on seven parameters: four momentum-range pa-rameters ( β i ), two anomalous magnetic moments ( κ u and κ d ) and one pion cloud parameter associated with thenormalization of the octet baryon wave functions. Theparametrization for the quark current was determinedpreviously in the studies of the nucleon and baryon de-cuplet systems. IV. MODEL-INDEPENDENT RELATIONS INTHE LARGE- q REGIME
In the present work we test the results of extrapolatingthe parametrizations in the spacelike region ( q = − Q <
0) to the timelike region ( q > Q .Concerning the relations (2.3)–(2.4), they map the re-gion of q : ] − ∞ ,
0] into the region [0 , + ∞ [. Note, how-ever, that q = 0 is not the center point of the reflec-tion symmetry that relates timelike and spacelike regions,because of the unphysical gap region ]0 , M B [ betweenthem. The reflection symmetry center point between thetwo regions lies, in fact, inside this interval and can betentatively taken as q = 2 M B instead of q = 0. Thisconsideration leads us to correct the relations (2.3)–(2.4) by introducing finite corrections to q , G M ( q ) ≃ G SL M (2 M B − q ) , (4.1) G E ( q ) ≃ G SL E (2 M B − q ) . (4.2)While the difference between using (2.3)–(2.4) and (4.1)–(4.2) is naturally negligible for very large q , and is im-material in the mathematical q → ∞ limit, it can benon-negligible otherwise. In the next section we checkthat this is indeed the case when one gets to values inthe range q = 10–20 GeV .In the calculations presented in the next section,Eqs. (4.1)–(4.2) provide a central value for our results ofthe form factors, Eqs. (2.3)–(2.4), a lower limit, while theestimate where we replace G SL l ( − q ) by G SL l (4 M B − q )( l = M, E ) gives the upper limit.An important point that is addressed in the next sec-tion is to know how far the region of the asymptotic rela-tions (4.1)–(4.2) is from the pQCD region characterizedby the relations G M ∝ /q and G E ∝ /q [30, 31, 33]. V. RESULTS
In this section we present the results in the timelikeregion for the Λ, Σ − , Σ , Ξ − and Ξ of the baryon octetand also for the Ω − (baryon decuplet). The results for thebaryon octet are based on the model from Ref. [34]. Theresults for the Ω − are based on the model from Ref. [48]. A. Octet baryons
The results of our model in the timelike region are pre-sented in Figs. 1, 2 and 3 for the cases of Λ, Σ − , Σ , Ξ − and Ξ . The thick solid lines represent our best estimatebased on Eqs. (4.1)–(4.2). The dashed lines representthe upper limit G l ( q ) = G SL l (4 M B − q ), and the lowerlimit G l ( q ) = G SL l ( − q ) ( l = M, E ). The thin solidline results are those obtained with the approximation G E = G M , and will be discussed later. Naturally, allcurves get closer together as q increases. In all cases,we use the experimental masses or the averages (respec-tively for Σ and Ξ). We recall that in the present modelthe SU (3) flavor symmetry is broken by the radial wavefunctions and that the quark electromagnetic structureis parametrized based on a VMD representation.Our estimates are compared with the world data forthe hyperon electromagnetic form factors in the timelikeregion. The data for the Λ, Σ , and Λ ¯Σ (from e + e − → Λ ¯Σ and e + e − → Σ ¯Λ reactions) for values of q upto 9 GeV , are from BaBar [17]. There are also datafrom BES-III for the Λ [19] below q = 10 GeV andfor Σ , Σ + , Ξ − and Ξ for q ≃ . [ ψ (3770)decay] [18]. Finally, there are data from CESR (CLEO-c detector) [11, 12] for the baryon octet (Λ, Λ ¯Σ , Σ ,Σ + , Ξ − , and Ξ ) and Ω − for q ≃ . [ ψ (3770) and ψ (4170) decays]. In the near future, we q [GeV ] -4 -3 -2 -1 | G ( q ) | Λ FIG. 1:
Timelike form factor G for the Λ. Data are from Refs. [11,12, 17, 19]. The thick solid line is based on Eqs. (4.1)–(4.2). Thedashed lines represent the upper limit G l ( q ) = G SL l (4 M B − q )and the lower limit G l ( q ) = G SL l ( − q ) ( l = M, E ). The thin solidline is obtained with the approximation G E = G M . expect results on the proton-antiproton scattering fromPANDA ( p ¯ p → B ¯ B ) [28].Contrary to the case of the proton form factor data inthe timelike region, which is about 2 times larger thanthose in the spacelike region [6, 55–57], the hyperon formfactors have about the same magnitude (central valuelines in the figures) in both regions (spacelike and time-like). Our results suggest that the available data may al-ready be within the asymptotic region where Eqs. (4.1)–(4.2) are valid, with the deviations consistent with a vari-ation of the argument of G from q (lower limit) up to q − M B (upper limit), denoting that the reflection cen-ter point is within the unphysical region. In the modelof Ref. [58] this seems also to be the case.From the figures, we can conclude that our estimates(central values) are close to the data for q > inmost cases. For the Λ case our results underestimate thedata. For the Σ + and Ξ cases our results overestimatethe data. However, in general, our results are reasonablyclose to the data. To compare our results with the datafor larger values of q ( q ≃ . fromCLEO-c [11, 12]), we show in Table I the average ratios ofthe experimental values and our estimates. Note that forthe Ξ we have an underestimate of 40% ( ≈ .
6) and forthe Λ an overestimate of more than 100% ( ≈ . G E ∝ /q and G M ∝ /q . Calculations in the spacelike region wherewe consider the leading order term of the asymptoticquark current suggest that the first signs of the pQCD behavior G E ∝ /Q and G M ∝ /Q (with log correc-tions) appear only for q ≈
100 GeV . An example ofthe convergence for | G M | and | G E | to the perturbativeregime is presented in Fig. 4 for the case of Σ + . The lineswith the label “Model” indicate the exact result; the lineswith the label “Large Q ” indicate the calculation withthe asymptotic quark current. Similar behavior can beobserved for the other hyperons.The sharp minimum on | G E | is a consequence of thezero crossing for Q ≃
10 GeV ( G E becomes negativeabove that point). This case is similar to the case of theproton, where there is the possibility of a zero-crossingnear Q ≃ , according to recent measurements atJefferson Lab [59]. The zero crossing is also expected forother hyperons.That the leading order pQCD behavior of the formfactors only appears for higher q , can be interpretedas the interplay between the meson masses that enterthe model through the constituent quark electromagneticform factors (describing the photon-quark coupling) andthe tail of the baryon wave functions that enter the over-lap integral. On one hand, the quark electromagneticform factors carry information on the meson spectrum,being parametrized using the VMD mechanism in ourmodel. Depending on the hyperon flavor, one has dif-ferent contributions from the poles associated with lightvector mesons (0.8–1.0 GeV) and an effective heavy vec-tor meson (1.9 GeV). Those vector meson masses providea natural scale, which regulates the falloff of the hyperonelectromagnetic form factors. Note that the light vectormeson masses (0.8–1.0 GeV) correspond to a large scalecompared to the low- Q scale of QCD ( ∼ . Q [46, 49, 60].For a detailed comparison with the present and futuredata, we present in Tables II and III our estimates for G at larger values of q . Note in particular that we presentpredictions for Σ − , a baryon for which there are no dataat the moment. The results in the tables can be used tocalculate the ratios between the form factors associatedwith different baryons.From the previous analysis, we can conclude that theeffective form factor G for most of the octet baryons withstrange quarks (hyperons) is well described by our ap-proximated SU F (3) model combined with the asymptoticrelations (4.1)–(4.2), since the data lie within the upperand lower limits of the theoretical uncertainty. Discussion
In the literature, there are a few estimates of hy-peron form factors based on vector meson dominance [20,22]. The first calculation (1977) [20] was performed q [GeV ] -4 -3 -2 -1 | G ( q ) | Σ + q [GeV ] -4 -3 -2 -1 | G ( q ) | Σ FIG. 2:
Timelike form factor G for the Σ + (left) and Σ (right). Data are from Refs. [11, 12, 17, 18]. See also caption of Fig. 1. q [GeV ] -4 -3 -2 -1 | G ( q ) | Ξ q [GeV ] -4 -3 -2 -1 | G ( q ) | Ξ − FIG. 3:
Timelike form factor G for the Ξ (left) and Ξ − (right). Data are from Refs. [11, 12, 18]. See also caption of Fig. 1. B D G exp G mod E Λ 2.19Σ + − Comparison between the ratios between the experi-mental value ( G exp ) and the model estimate of G ( G mod ) for thedifferent baryons, for q ≃ . [11, 12]. The lastline indicates the average of all baryons. with no adjustable parameters, before the first mea-surements (Orsay 1990) [61]. Those estimates differfrom the recent measurements by an order of magni-tude [11, 12]. An improved VMD estimate (1993) [22]gave results closer to the Λ data under the condition G M = G E [11, 14]. There are also recent estimatesfor the Λ and Σ form factors based on phenomenolog-ical parametrizations of the baryon-antibaryon interac-tion [23], asymptotic parametrizations and vector mesondominance parametrizations of the form factors [25–27].In our model, the SU F (2) symmetry is broken at thequark level since we use different parametrizations for theisoscalar and isovector quark form factors. The depen-dence on the isovector component is more relevant forthe case of the neutron for which there are almost nodata available [3, 62, 63], and for the e + e − → Λ ¯Σ and e + e − → Σ ¯Λ reactions, which we discuss at the end ofthe present section.We now discuss the difference in magnitude betweenthe electric and magnetic form factors of the octet baryonmembers. The absolute value of the magnetic form fac-tor | G M | is represented in Figs. 1 to 3 by the thin solidline, which is, with no exception, just a bit above thecentral (thick solid line). Those results mean that themagnetic form factor is larger than the electric form fac-tor ( | G E | < | G M | ) for Λ, Σ + , Σ , Ξ and Ξ − . This Q [GeV ] -5 -4 -3 -2 -1 |G M | - Model|G M | - Large Q |G E | - Model|G E | - Large Q Σ + FIG. 4: Σ + form factors. Comparison between the results for | G M | and | G E | for a model with an exact quark current (Model)and the results where we consider only the leading order term in Q for the quark current (Large Q ). conclusion is a consequence of the definition of | G ( q ) | given by Eq. (2.2). If we express | G E | in terms of theratio α G = | G E || G M | , we obtain | G | = | G M | (cid:16) α G − τ (cid:17) .Since the thick solid line is the result for the full | G ( q ) | function, and the thin solid line is the result from as-suming | G ( q ) | = | G M ( q ) | , we conclude that although | G E | < | G M | , the two form factors have similar magni-tudes.Our model can also be applied for the Λ ¯Σ and ¯ΛΣ form factors ( e + e − → Λ ¯Σ and e + e − → ¯ΛΣ reactions).It is important to notice, however, that the analysis ofthe e + e − → Λ ¯Σ and the e + e − → ¯ΛΣ reactions is abit more intricate than the analysis for the e + e − → B ¯ B reactions associated with the elastic form factors. In thiscase there are two possible final states (Λ ¯Σ and ¯ΛΣ ).From the experimental point of view, this implies thatthe background subtraction in the cross section analysisis also more complex due to the proliferation of decaychannels, including the Λ and Σ decays and the decaysof the corresponding antistates.From the theoretical point of view the γ ∗ Λ → Σ transition form factors in the spacelike region are diffi-cult to test due to the lack of experimental data: Thereare no experimental constraints for the electric and mag-netic form factors, except for the transition magnetic mo-ment. We do not discuss here in detail our results for the γ ∗ Λ → Σ transition form factors, due to the experi-mental ambiguities and also because the main focus ofthis work is the octet baryon electromagnetic form fac-tors. Still, we mention that we predict the dominance ofthe meson cloud contributions for G E and of the valencequark contributions for G M [44]. At large Q , the mag-netic form factor dominates over the electric form factor.This dominance is then mirrored to the timelike region.Our estimate of G in the timelike region overestimates q (GeV ) Σ + Σ Σ −
10 40.5 16.8 10.715 15.1 6.09 4.1220 7.68 3.01 2.1925 4.58 1.76 1.3630 3.03 1.15 0.92335 2.14 0.803 0.66740 1.60 0.592 0.50345 1.24 0.453 0.39350 0.980 0.358 0.31555 0.799 0.290 0.26060 0.663 0.239 0.216TABLE II:
Estimates for the Σ effective form factor G in units10 − . q (GeV ) Λ Ξ Ξ −
10 13.4 41.4 24.915 4.90 13.6 7.9920 2.43 6.41 3.7525 1.43 3.65 2.1330 0.927 2.33 1.3635 0.648 1.61 0.93340 0.476 1.17 0.67945 0.365 0.893 0.51450 0.288 0.700 0.40255 0.233 0.564 0.32360 0.192 0.463 0.264TABLE III:
Estimates for the Λ and Ξ effective form factor G inunits 10 − . the data by about an order of magnitude, suggesting thatthe magnetic form factor dominance is not so strong inthe timelike region. Another interesting theoretical as-pect related to the γ ∗ Λ → Σ transition is its isovectorcharacter. This property can be studied in the near fu-ture once accurate timelike data for the neutron becomeavailable at large q . From the combination of protonand neutron data, we can determine the isovector com-ponent of the nucleon form factors. Then those can beused to study the γ ∗ Λ → Σ transition form factors. B. Ω − form factors CLOE-c provided the first measurements of the Ω − form factors for nonzero q [11, 12]. Our results for theΩ − form factors are very important, because theoreticalstudies of the Ω − are scarce due to its unstable charac-ter. Fortunately, for the Ω − , lattice QCD simulations at0 q [GeV ] -4 -3 -2 -1 | G ( q ) | Ω − q [GeV ] -4 -3 -2 -1 | G ( q ) | Ω − FIG. 5:
Timelike form factor G for the Ω − . We present the full result in the left panel (including G E and G M ). In the right panel,we drop the higher order multipoles ( G E and G M ). The model from Ref. [48] predicts a large magnitude for G M . The timelike datasupport the estimates with a much smaller G M . Data are from Ref. [11]. See also caption of Fig. 1. the physical point (i.e. physical strange quark mass) ex-ist [64]. Since those simulations are at the physical pointand the meson cloud contamination (kaon cloud) is ex-pected to be small due to the large kaon mass, the latticeQCD data may be considered to describe the physicalΩ − .We consider a model where the Ω − is described by adominant S state and two different D -state components:one with total quark spin 1 /
2, another with the totalquark spin 3 / D -statemixture coefficients and three momentum-range param-eters.Our model for the Ω − [48] was calibrated by the Ω − lattice QCD data from Ref. [64]. The free parametersof the radial wave functions and D -state mixture coeffi-cients of our model were adjusted by the lattice QCD re-sults for the form factors G E , G M , and G E for Q < . The model was then used to estimate the func-tions G E (electric qudrupole form factor) and G M ( Q )(magnetic octupole form factor).In the case of the electric quadrupole form factor( G E ), one obtains a consistent description of the latticeQCD data, which allows the determination of the elec-tric quadrupole moment from G E (0) = 0 . ± . G M , however, the lattice QCD simulations are re-stricted to the result for Q = 0 .
23 GeV , G M =1 . ± .
50 [48, 65] (which has a significant errorbar).From the form factors G E , G M , G E and G M , we cal-culate the function G based on the results in Appendix A.The results are presented in the left panel of Fig. 5.Our estimates for the electromagnetic form factors inthe timelike region of Ω − should be taken with caution,since the model used for the radial wave functions wasnot chosen in order to describe the large- Q region butrather fitted to the Q < data. For that reason, the falloff of G E and G M at large Q is determinedby the 1 /Q behavior, and not by the falloff of pQCD(1 /Q ).From Fig. 5 (left panel), we conclude that our resultsfor G overstimate the data. In order to understand thisresult, we examine the magnitude of the higher multipoleform factors G E and G M . If we drop these contribu-tions we obtain the results presented in the right panelof Fig. 5. In this case, we observe a close agreementwith the data. From this analysis, we can conclude thatthe deviation from the data comes from the form fac-tors G E and G M . We have confirmed that it is thefunction G M that originates a contribution that makesthe total results differ from the data. Our model gives G M (0) ≃ .
5. The result presented in the right panelof Fig. 5 is more compatible with G M (0) ≈
1. We thenconclude that the timelike data are more consistent witha small magnitude for the function G M .The value of G M (0) has been estimated based on dif-ferent frameworks. Light front QCD sum rules predict G M (0) = 64 . ± . G M (0) = 48 . SU F (3) quark model and G M (0) = 12 . G M (0), and it is more consis-tent with the estimate that breaks SU F (3). The timelikedata, however, seem to indicate that G M (0) may be evensmaller.It is worth noting that the function G M is, at themoment, poorly estimated. On the contrary, the func-tions G E , G M , and G E , are well determined by thelattice QCD data. The present result suggests the needfor a determination of G M by a combined study of moreaccurate lattice QCD data with the very recent timelikeregion data for G in the region q ≈
16 GeV [11]. Infuture studies, the expected pQCD falloff of the form fac-tors for very large q should also be taken into account.1 VI. OUTLOOK AND CONCLUSIONS
A relativistic quark model which was successful in thedescription of the baryon electromagnetic form factors inthe spacelike region was extended to the timelike region.Our SU F (3) model provides a fair description of the databoth in the spacelike and timelike regions.The extension of the model from the spacelike intothe timelike regions uses asymptotic reflection symme-try relations connecting the electromagnetic elastic formfactors in the two different regions. The theoretical un-certainty in our predictions for the timelike region waspresented. An important conclusion is that the mea-sured data are consistent with the asymptotic relationsof Eqs. (4.1)–(4.2), originated from general principles asunitarity and analyticity. Finite corrections for q stillhave a role in the strength of the form factors for q = 10–30 GeV , since within this range the differences betweenthe results obtained from G SL l ( − q ), G SL l (2 M B − q ), and G SL l (4 M B − q ) ( l = M, E ) show that the strict q → ∞ limit is not yet attained numerically within that region.On the other hand, the fact that the data are withinthe theoretical uncertainty of our model seems to indi-cate that the reflection symmetry center point is insidethe unphysical region ]0 , M B [, where M B is the baryonmass.Our model leads to the correct pQCD asymptoticpower law behavior of the electromagnetic form factors.But an important conclusion of this work is that thepQCD limit onset G ∝ /q is way above the regionwhere the reflection symmetry relations are valid. Wefound that only beyond the region of q : 30 −
50 GeV ,the pQCD power law was observed. This was interpretedas an interplay of the two scales entering the model:the meson mass scales that determine the quark electro-magnetic current, and the momentum-range scales deter-mined by the extension of the hyperons.In the present work, our main focus was on the baryonoctet since the available data are mostly on that familyof baryons, and therefore, the comparison with the dataenabled us to better probe our model in the timelike re-gion. Our framework can also be applied to all baryonsof the decuplet, and as an example, we presented ourresults for the Ω − baryon and compared them with thenew data from CLEO.Under study is the possible extension of the presentmodel to charmed baryons. By this extension, the modelcan be applied to the e + e − → Λ + c ¯Λ + c process to estimatethe Λ + c timelike electromagnetic form factors, which wererecently measured at BES-III [68]. Acknowledgments
G.R. was supported by the Funda¸c˜ao de Am-paro `a Pesquisa do Estado de S˜ao Paulo (FAPESP):Project No. 2017/02684-5, Grant No. 2017/17020-BCO-JP. M.T. Pe˜na was supported in part by Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT) Grant No. CFTP-FCT (UID/FIS/00777/2015). K.T. was supported byConselho Nacional de Desenvolvimento Cient´ıfico e Tec-nol´ogico (CNPq, Brazil), Processes No. 313063/2018-4 and No. 426150/2018-0, and FAPESP Process No.2019/00763-0, and his work was also part of the projects,Instituto Nacional de Ciˆencia e Tecnologia - NuclearPhysics and Applications (INCT-FNA), Brazil, ProcessNo. 464898/2014-5, and FAPESP Tem´atico, Brazil, Pro-cess No. 2017/05660-0. M. T. Pe˜na thanks Gernot Eich-mann and Alfred Stadler for useful discussions.
Appendix A: / + baryons As discussed in the main text, the relations (2.1) and(2.2) can be used for 3 / + baryons if the form factors G M and G E are expressed as combinations of electricform factors (electric charge G E and electric quadrupole G E ) and magnetic form factors (magnetic dipole G M and magnetic octupole G M ).According to Ref. [20], we should use the followingreplacements: | G E | → | G E | + 89 τ | G E | , (A1) | G M | → | G M | + 325 τ | G M | , (A2)where τ = q M B . Appendix B: Details of the model
Below we describe some details of the model, includ-ing the spin and flavor wave functions, the explicit formof the quark form factors, the parameters of the model,and the values of the normalization factors due to theinclusion of pion cloud contributions.
1. Baryon wave functions
In the covariant spectator quark model the spin statesassociated with Eq. (3.1) are represented by [34, 39] φ S = u B ( P, s ) , φ S = − ( ε ∗ λ ) α ( P ) U αB ( P, s ) , (B1)where u B is the Dirac spinor of the baryon, s is thebaryon spin projection, λ represents the polarization ofthe diquark, and U αB ( P, s ) = 1 √ γ (cid:18) γ µ − P α M B (cid:19) u B ( P, s ) . (B2)The wave function described by Eqs. (3.1) and (B1)generalize the nonrelativistic wave function in a covariantform [39]. The flavor states | M S i (mixed symmetric)and | M A i (mixed antisymmetric) for all octet baryons2 B | M A i | M S i p √ ( ud − du ) u √ [( ud + du ) u − uud ] n √ ( ud − du ) d − √ [( ud + du ) d − ddu ]Λ √ [ s ( du − ud ) − ( dsu − usd ) − du − du ) s ] [( dsu − usd ) + s ( du − ud )]Σ + 1 √ ( us − su ) u √ [( us + su ) u − uus ]Σ [( dsu + usd ) − s ( ud + du )] √ [ s ( du + ud ) + ( dsu + usd ) − ud + du ) s ]Σ − √ ( ds − sd ) d √ [( sd + ds ) d − dds ]Ξ √ ( us − su ) s − √ [( ud + du ) s − ssu ]Ξ − √ ( ds − sd ) s − √ [( ds + sd ) s − ssd ]TABLE B1: Flavor wave functions of the octet baryons. are presented in Table B1. Although the results for thenucleon are not discussed in the present work, we includethe proton and neutron states for completeness.
2. Quark form factors
To parametrize the quark current (3.6), we adopt thestructure inspired by the VMD mechanism as in Refs. [35,39]: f ± = λ q + (1 − λ q ) m v m v + Q + c ± M h Q ( M h + Q ) ,f = λ q + (1 − λ q ) m φ m φ + Q + c M h Q ( M h + Q ) ,f ± = κ ± (cid:26) d ± m v m v + Q + (1 − d ± ) M h M h + Q (cid:27) ,f = κ s ( d m φ m φ + Q + (1 − d ) M h M h + Q ) , (B3)where m v , m φ and M h are the masses, respectively, cor-responding to the light vector meson m v ≃ m ρ ≃ m ω ,the φ meson (associated with an s ¯ s state), and an effec-tive heavy meson with mass M h = 2 M N to represent theshort-range phenomenology. The parameter λ q is deter-mined by the study of deep inelastic scattering [39]. Therelation between the quark anomalous magnetic moments κ u and κ d is κ + = 2 κ u − κ d and κ − = (2 κ u + κ d ).We consider the parametrization from Refs. [34, 39]in the study of the nucleon and decuplet systems, ex-cept for the quark anomalous magnetic moments κ + and κ − . Those coefficients are re-adjusted in our study ofthe octet baryon electromagnetic form factors in orderto take into account the pion cloud effects. The value of κ s is fixed by the magnetic moment of the Ω − .The parametrization from Eq. (B3) for the three sec-tors includes the quark anomalous magnetic moments(three parameters) and six extra parameters. Since the a κ a c a d a + − . − − . − . Parameters associated with the quark current. Inthis notation κ s = κ . For λ q , we use λ q = 1 .
22 [39]. β β β β Parameters of the radial wave functions (3.2). Re-call that β is the global long-range parameter. The short-rangeparameters are β (nucleon); β (Λ and Σ), and β (Ξ). quark anomalous magnetic moments can be fixed inde-pendently by the proton, the neutron and the Ω − mag-netic moments, we have six parameters to adjust. Wereduce the number to five using the isospin symmetryfor the Pauli form factor ( d + = d − ). All parameters ofthe quark current are included in Table B2. The resultsin boldface indicate the values fixed by the study of theoctet baryons with pion cloud dressing.
3. Octet baryon form factors
The valence quark contributions to the octet baryonelectromagnetic form factors are then determined bythe combination of the quark form factors and theparametrization of the radial wave functions, accordingto Eqs. (3.8) and (3.9).The parameters associated with the radial wave func-tions (3.2) are presented in Table B3. The coefficients j A,Si ( i = 1 ,
2) depend on the quark form factors accord-ing to the results from Table B4. Again, the expressionsfor the nucleon are included for completeness.3
B j Si j Ai p ( f i + − f i − ) ( f i + + 3 f i − ) n ( f i + + f i − ) ( f i + − f i − )Λ f i + 118 ( f i + − f i )Σ + 118 ( f i + + 3 f i − − f i ) ( f i + + 3 f i − )Σ (2 f i + − f i ) f i + Σ − ( f i + − f i − − f i ) ( f i + − f i − )Ξ (2 f i + + 6 f i − − f i ) − f i Ξ − (2 f i + − f i − − f i ) − f i TABLE B4:
Mixed symmetric and antisymmetric coefficients forthe octet baryons.
4. Pion cloud dressing
In the low- Q region, it is necessary to include theeffects of the pion cloud dressing of the baryons. Inthe study of the electromagnetic structure of the octetbaryons those effects are taken into account in an effec-tive way. There are two main contributions to take intoaccount: the contributions associated with the photoncoupling with the pion, and the contributions associatedwith the photon coupling with intermediate octet baryonstates. All these processes can be parametrized based onan SU (3) model for the pion-baryon interaction using fiveindependent couplings and two cutoffs (regulate falloff ofpion cloud contributions) [34].The main consequence of the inclusion of the pioncloud contributions is that the estimates of G BEB and G BMB from the valence quark contributions to the octetbaryon form factors are modified by the normalization ofthe wave function which combine valence and pion cloudcontributions ( δG EB and δG MB ): G EB = Z B [ G BEB + δG EB ] , (B4) G MB = Z B [ G BMB + δG MB ] . (B5)The explicit expressions for δG EB and δG MB can be found in Refs. [34, 42]. When we increase Q the pioncloud contributions are strongly suppressed since they areregulated by higher order multipoles with square cutoffsof the order 0.8 and 1.2 GeV [34].The parameters associated with the valence quark con-tributions are determined by fits to the lattice QCD re-sults for the octet baryon electromagnetic form factors.The parameters associated with the pion cloud contribu-tion are fixed by the physical data (nucleon electromag-netic form factors and octet baryon magnetic moments).In Eqs. (B4)–(B5) the factor Z B can be written as [34,42, 52] Z B = 11 + 3 a B B , (B6)where a B is a coefficient determined by the SU (3) sym-metry, and B is a parameter which determines the nu-cleon normalization ( Z N ), based on the normalization a N = 1.The normalization constant for the nucleon Z N =1 / (1 + 3 B ), meaning that the contribution from the va-lence quarks for the proton charge is Z N and the contri-bution from the pion cloud 3 B Z N . One concludes thenthat the relative pion cloud contribution to the protonelectric form factor is 3 B , which implies that the nor-malization Z N can be determined by the estimate of thepion cloud contribution based on the comparison betweenthe valence quark contributions and the data, and viceversa B =
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