The electromagnetic form factors of Λ_c hyperon in the vector meson dominance model
aa r X i v : . [ h e p - ph ] F e b The electromagnetic form factors of Λ c hyperon in the vector meson dominance model Junyao Wan, Yongliang Yang, ∗ and Zhun Lu † School of Physics, Southeast University, Nanjing 211189, China College of Physics, Qingdao University, Qingdao 266071, China
We applied a modified vector meson dominance (VMD) model to analyze the electromagnetic form factorsof the Λ c hyperon in the time-like reaction e + e − → Λ + c ¯ Λ − c . In the model, we include the contributions from ω (782), ω (1420), ω (1650), φ (1020), φ (1680) and φ (2170), as well as charmed mesons and their resonancestates ψ (1 S ), ψ (2 S ), ψ (3770), ψ (4040), ψ (4160) and ψ (4415). We perform a combined fit to the available dataon the Born cross section in reaction e + e − → Λ + c ¯ Λ − c and the ratio of the electromagnetic form factors | G E / G M | to obtain the values of the model parameters. Our results show that the vector meson dominance model cansimultaneously describe the data of electromagnetic form factors from the Bell and BESIII Collaboration, andthe model result for | G E / G M | qualitatively agrees with the data from BESIII. With the fitted parameters, wepredict the relative phase ∆Φ between G E and G M . Moreover, we predict the single and double polarizationobservables in e + e − → Λ + c ¯ Λ − c reactions, which are experimentally accessible in the polarized process. We alsoobtain the form factors of the Λ c hyperon in the space-like region via analytic continuing the time-like formfactors. I. INTRODUCTION
The electromagnetic form factors (EMFFs) G E and G M ofhadrons are important physical quantities that encode the in-formation of the perturbative and nonperturbative quantumchromodynamics (QCD) e ff ects in hadrons [1–3]. The time-like and space-like electromagnetic form factors of the pro-ton and neutron have been extensively studied, e.g., in the ep elastic scattering, ¯ pp annihilation and e + e − annihilation pro-cesses [1, 4–19]. In the last two decades, the EMFFs of hy-perons (i.e., Λ , Σ , Ξ ) in the time-like region have also been in-vestigated. Particularly, the enhancement of the cross sectionsin reactions e + e − → Y ¯ Y near threshold were measured andanalyzed [20–25]. It is found that the vector mesons and theirresonances, as the intermediate states of the reaction, play im-portant roles in these processes. That is, they could provide anexplanation for the threshold enhancement e ff ect of scatteringcross section and large ratio between G E and G M [26–28].In recent years, the EMFFs of charmed hyperon Λ c attractsa lot of interest both theoretically [29–32] and experimen-tally [33–37], since the Λ c hyperon is the lightest baryon con-taining the charm quark. Similar to the Λ hyperon, the Λ c target is unfeasible and the EMFFs of Λ c can not be accessedfrom exclusive experiments in the space-like region [27, 38].On the other hand, the cross section of reaction e + e − → Λ + c ¯ Λ − c have been measured by the Belle and the BESIII Collabora-tions [34, 36]. The ratio between the electric form factor G E and the magnetic form factor G M near threshold region is alsoavailable [36]. These measurements provide great opportunityto study the dynamics on the production of charmed baryonpairs and the time-like EMFFs of Λ c . Furthermore, by ana-lytic continuation, the knowledge of EMFFs in the time-likeregion could be extend to study the EMFFs in the space-likeregion.The vector meson dominance (VMD) model has been rec- ∗ Electronic address: [email protected] † Electronic address: [email protected] ognized as a reliable theoretical approach in the study of thespace-like electromagnetic form factors of hadrons. It can de-scribe the existing data of proton and neutron EMFFs in thespace-like region quite well. The approach was also extendedto investigate the EMFFs of the Λ hyperon [27] in the time-like region. In the VMD model, the electromagnetic form fac-tors receive contributions from two parts. One is the intrinsicstructure defined by the valence quarks, the other is the contri-bution form the meson clouds in terms of vector mesons. Dueto the isoscalar property of Λ , the contribution of ρ meson andits resonances should excluded. In order to introduce a com-plex structure of EMFFs in time-like region, the decay widthsof the vector mesons and their resonance states are taken intoaccount [27, 28, 39]. Particularly, the contributions from theresonance states below the threshold of Λ ¯ Λ pair are involved.The study shows that the inclusion of these resonance statesare essential to simultaneously describe the experiment dataof the e ff ective form factors, ratio | G E / G M | and relative phase ∆Φ for Λ hyperon in a wide range of √ s .Encouraged by the success on the nucleon and Λ hyperon,in this work, we extend the VMD model to explore the EMFFsof the Λ c hyperon. There is some di ff erence between theVMD model for the Λ hyperon and that for the Λ c hyperon.Firstly, the production of the Λ + c ¯ Λ − c pair are related to c ¯ c pair, which has the quantum numbers I G ( J PC ) = − (1 −− ).Secondly, the production threshold of Λ + c ¯ Λ − c is 2 M Λ c = . ω , φ and their resonance states, the J / Ψ and their resonance states below the threshold should be in-cluded. Thus, we take into account the contributions from allthe resonances states below the threshold: ω (782), ω (1420), ω (1650), φ (1020), φ (1680) and φ (2170), as well as ψ (1 S ), ψ (2 S ), ψ (3770), ψ (4040), ψ (4160) and ψ (4415). Thirdly, theCoulomb final-state interactions should be considered, whichis similar to the case of the proton [28, 40]. Based on the aboveconsideration, we can obtain the formula of the time-like formfactors for Λ c by analytic continuation of the space-like formfactors.The remained content of the paper is organized as follows.In Section II, we present a detailed framework on the formfactors of Λ c hyperon in the VMD model. In Section III, wefit theoretical expressions for G E and G M the to experimentaldata of reaction e + e − → Λ + c ¯ Λ − c from the Belle and the BESIIIcollaborations. We also provide our predictions for the singleand double polarization observables, the relative phase angle ∆Φ , as well as space-like form factors of Λ c . We summarizethe paper in Section IV. II. FORM FACTORS OF Λ c HYPERON IN THE VMDMODEL
The process e + e − → Λ + c ¯ Λ − c which we study in the frame-work of the VMD model is shown in Fig. 1. That is, thephotons formed in e + e − annihilation are first transformed intoneutral vector mesons, the latter ones then decay into Λ + c ¯ Λ − c pairs through some specified couplings. Since one-photon ex-change dominates the production of spin − / B , theBorn cross section of the process e + e − → B ¯ B can be parame-terized [3] in terms of EMFFs. Generally, the integrated crosssection of the Λ c hyperon pairs production can be given in thefollowing way: σ ( s ) = πα β s C Λ c (cid:20) | G M ( s ) | + τ | G E ( s ) | (cid:21) (1)Here, G E ( s ) and G M ( s ) are the electric form factor and themagnetic form factor in the time-like region, respectively, α isthe fine-structure constant, s the square of the center of mass(c.m.) energy, τ = s / M Λ c , and β = √ − /τ is the veloc-ity of the Λ c hyperon. The Coulomb factor C Λ c = ε R pa-rameterizes the electromagnetic interaction between the out-going baryon and antibaryon, with ε = πα/β an enhance-ment factor resulting in a nonzero cross section at thresholdand R = / (1 − e − πα/β ) the Sommerfeld resummation fac-tor [36, 41].In the space-like region, the EMFFs G E ( Q ) and G M ( Q )of Λ c hyperon can be expressed as G M = F + F , G E = F − τ F , (2)where τ = Q / M Λ c , and the F ( Q ) and F ( Q ) are the Diracform factor and Pauli form factor respectively, which can bedecomposed into F i = F Si + F Vi , (3)where F Si and F Vi denote the isoscalar and isovector compo-nents of the form factors, respectively. Since Λ c hyperon isan isospin singlet, the contribution from the isovector part F Vi should be excluded, which is similar to the case of the Λ hy-peron. We note that the kinematic constraint G E ( − M Λ c ) = G M ( − M Λ c ) is apparently satisfied in Eq. (2) [42].Previously, the VMD model have been widely used to studythe EMFFs of the nucleon and the Λ hyperon, showing thatit has the advantage to well describe the experimental datain the space-like and time-like region [15, 27, 43–46]. Veryrecently, it has also been applied to investigate the EMFFsfor Σ + and Σ − [25]. Encouraged by its success, we extendthe model to study the EMFFs of the Λ c hyperon. In the e − e + Λ + c Λ − c γ FIG. 1: The reaction e + e − → Λ + c ¯ Λ − c depicted in the VMD model. VMD model, two parts contributes to the Dirac form factor,one is the the intrinsic structure, the other is the vector me-son clouds; while the Pauli form factor only receives the con-tribution from the meson cloud [45]. Due to the isoscalarproperty of the Λ c hyperon, we consider the contributionsof vector mesons ω , φ and their resonance states. More-over, as Λ c is charmed hyperon, so we should also considerthe contribution of charmed mesons J /ψ and their resonancestates. Thus, in our modified model, the contributing mesonresonance states below the threshold of Λ + c ¯ Λ − c are ω (782), ω (1420), ω (1650), φ (1020), φ (1680), φ (2170), and ψ (1 S ), ψ (2 S ), ψ (3770), ψ (4040), ψ (4160), ψ (4415), and we assumethat the expression of the form factors from the ω and φ res-onance states have the same form as those from the vectormesons ω (782) and φ (1020) [47]. As for J /ψ and its reso-nance states, we compare them to ω (782) and φ (1020) in lightof the magnitude of the mass of the charmed states, and putthem into Ω and Φ . At Q =
0, the EMFFs of Λ c hyperon canbe normalized as follow, G E (0) = , G M (0) = µ Λ c . (4)where the magneton µ Λ c = .
48 ˆ µ N is predicted in Ref. [29].One should note that result of the magnetic moment of the Λ c hyperon is given in unit of nucleon magneton. Thus the mag-netic moments with units of the Λ c hyperon natural magnetoncan be expressed as µ Λ c = .
039 ˆ µ Λ c using ˆ µ Λ c = M N M c ˆ µ N [27].Taking into account all the above constraints, we can writethe parameterized forms of scalar parts of the Dirac and Pauliform factors in VMD model as follows: F S ( Q ) = g ( Q )6 Σ Ni = (cid:20) − β Ω i − β Φ i + β Ω i m Ω i m Ω i + Q + β Φ i m Φ i m Φ i + Q (cid:21) (5) F S ( Q ) = g ( Q )6 Σ Ni = (cid:20) ( µ Λ c − − α Φ i ) m Ω i m Ω i + Q + α Φ i m Φ i m Φ i + Q (cid:21) (6)where N = Ω i ( i = , , , , ,
6) denotes the vector me-son states ω (782), ω (1420), ω (1650), ψ (1 S ), ψ (3770), and ψ (4160), Φ i ( i = , , , , ,
6) represents the vector me-son states φ (1020), φ (1680), φ (2170) and ψ (2 S ), ψ (4040), ψ (4415). The intrinsic structure factor is a characteristicof valence quark structure and is chosen in a dipole form g ( Q ) = (1 + γ Q ) − , which is consistent with pQCD andfits the EMFFs of nucleon well [43, 44, 47]. In the large Q region, the forms also satisfy the constraints of the asymptoticbehavior, F ∼ / Q and F ∼ / Q . Furthermore, the coe ffi -cients β Ω i , β Φ i , α Φ i can be naturally interpreted as the productsof a V γ coupling constant and a V BB coupling constant [47],respectively. The parameter γ in g ( Q ) and the coe ffi cients β Ω i , β Φ i , α Φ i in Eqs. (5)-(6) are free parameters the values ofwhich can be obtained be fitting to the data of EMFFs.By proper analytic continuation on the complex plane, wecan obtain the form factors in the time-like region on the basisof the form factors in space-like region [11, 15]. The analyticcontinuation in the time-like region is based on the followingrelation [11]: Q = − q = q e i π . (7)Therefore, in the time-like region, the intrinsic structure g ( q )has an analytical continuation form: g ( q ) = − γ q ) . (8)where γ is a parameter larger than zero. Thus, there is a polein g ( q ) in the position q = /γ . There are two methods toremove the pole, one is to change the relations in Eq.(7) with Q → q e i θ ( θ , π ) [15, 42], the other is to impose the con-straint γ > / (4 m Λ c ) for the Λ c form factors [27]. In this work,we will choose the latter one. For the contribution of the me-son cloud to the form factor, we take into account the widthsof the vector mesons ω , φ , J /ψ and their resonance states inorder to introduce the complex structure of the EMFFs in thetime-like region [27, 28]. This leads to the following replace-ment for Eq. (7) β Ω i m Ω i m Ω i + Q → β Ω i m Ω i m Ω i − q − im Ω i Γ Ω i ,β Φ i m Φ i m Φ i + Q → β Φ i m Φ i m Φ i − q − im Φ i Γ Φ i . (9)In this way we obtain the modified VMD model in the time-like region for Λ c hyperon. These terms are crucial for con-structing the complex structure and reproducing the relativephase angle of the time-like EMFFs of Λ c . III. NUMERICAL RESULTS AND DISCUSSIONSA. Fit the time-like form factors
We fit the expressions of the form factors in Eqs. (5)-(6)and the replacement in Eq. (9) to the experimental data ofBorn scattering cross section and EMFFs ratio measured bythe Belle [34] and BESIII [36] Collaborations. The data arein the range 4 .
59 GeV < √ s < .
39 GeV. The masses andwidths of the isoscalar vector mesons used in the fit are takenfrom Table I. The χ per degrees of freedom (d.o.f) χ / d.o.f TABLE I: The masses and widths of the involved vector mesons inthe model in unit of MeV [48].State Mass Width State Mass Width ω (782) 783 8.5 φ (1020) 1019 4.25 ω (1420) 1418 104 φ (1680) 1680 150 ω (1650) 1671 315 φ (2170) 2169 125 ψ (1 S ) 3097 0.093 ψ (2 S ) 3686 0.294 ψ (3770) 3773 27.2 ψ (4040) 4039 80 ψ (4160) 4191 70 ψ (4415) 4421 62 ( pb ) s (GeV) best fit uncertainty Belle BESIII FIG. 2: Our fit to the Born cross section σ in reaction e + e − → Λ c hyperon (solid curve). The rectangles and circles with error bars rep-resent the data from the Belle [34] and BESIII [36] Collaborations,respectively. is 1 .
18, and the best values of the model parameters obtainedfrom the fit are given in Table II.It should be noted that the value of intrinsic parameter γ in our model is fitted to be 0 . − , corresponding q = .
049 GeV. Thus the poles of the intrinsic structure arerestricted in the unphysical region and satisfied the constraint γ > / (4 m Λ c ) in this scenario. Since the poles are below thethreshold and we focus on the region above the Λ + c ¯ Λ − c thresh-old, we can ignore the e ff ect of the pole in the first place.In Fig. 2, we show our fit (solid line) to the Born crosssections in reaction e + e − → Λ + c ¯ Λ − c measured by the Belle(filled square) and BESIII Collaboration (filled circle). Thevertical lines depict the error bars of the data. We also pro-vide the theoretical band corresponds corresponding to the un-certainty of parameters obtained from the errors of the data.The Belle data cover the region 4 . < √ s < . . < √ s < . ff ect nearthe threshold of the Λ + c Λ − c pair due to the contribution of theresonance states of the vector mesons is also observed.In Fig. 3, we present the model result of the ratio | G E / G M | (solid line) and compare it with the BESIII data (filled circle)which are near the threshold. Again, the band correspondsto the uncertainty of parameters. It is shown that the VMD TABLE II: The values of the parameters obtained from the combined fit.Parameter Value Parameter Value Parameter Value β ω (782) . β ω (1420) . β ω (1650) − . β ψ (1 S ) . β ψ (3770) − . β ψ (4160) . β φ (1020) . β φ (1680) . β φ (2170) . β ψ (2 S ) − . β ψ (4040) . β ψ (4415) − . α φ (1020) − . α φ (1680) − . α φ (2170) . α ψ (2 S ) − . α ψ (4040) . α ψ (4415) − . best fit uncertainty BESIII | G E / G M | s (GeV) FIG. 3: The same as Fig. 2 but for the ratio G E / G M . The experimen-tal data denote by the circles are from BESIII [36] Collaboration. model can qualitatively describe the ratio. It is worth notingthat, according to the kinematic constraint, this ratio is equalto 1 at the threshold, which is an important constraint for theform factors in the time-like region. Furthermore, the ratioincreases with increasing √ s in the near threshold region andreaches the maximum value 1.6 at around √ s = . √ s region, the ratio decreases with increasing √ s . These featuresare similar to those of the Λ hyperon [27, 49]. Due the VMDmodel, the asymptotic behaviors of form factors, the ratio ofEMFFs satisfy a fact that the result tends to be constant in thelimit of q → ∞ [27].As G E and G M in the time-like region are complex, there isa relative phase ∆Φ between the two EMFFs. The measure-ment of this phase at di ff erent √ s could provide additionalinformation of EMFFs which can not been revealed by | G E | and | G M | . Using the values of the parameter extracted fromthe Belle and BESIII data, we predict the relative phase ofthe EMFFs of the Λ c as function of √ s , as shown in Fig. 4.One should be note that ∆Φ = Λ + c ¯ Λ − c threshold due to G E = G M at s = M Λ c . our numerical result also satisfies theconstraint that the phase goes to zero as s → ∞ . ( (cid:176) ) s (GeV) FIG. 4: Prediction for the relative phase ∆Φ vs √ s in e + e − → Λ + c ¯ Λ + c . A y s (GeV) FIG. 5: Prediction for the single polarization observable A y vs √ s in e + e − → Λ + c ¯ Λ + c at the fixed angle θ = ◦ . B. Prediction of polarization observables in time-like region
Using the the model results of G E and G M , we present theprediction for the spin-independent observables in reaction e + e − → Λ + c ¯ Λ − c . In the single-photon exchange approxima-tion, the single and double spin polarization observables can A x z s (GeV) s (GeV) A xx A yy s (GeV) A zz s (GeV) FIG. 6: Predictions for the double polarization observables A xz , A xx , A yy and A zz vs √ s in e + e − → Λ + c ¯ Λ + c at the fixed angle θ = ◦ . be expressed in terms of the EMFFs: [51, 52] A y = − M Λ √ s sin(2 θ ) Im(G M G ∗ E )D c − D s sin ( θ ) , A xz = M Λ √ s sin(2 θ )Re(G M G ∗ E )D c − D s sin ( θ ) , A xx = [ D c − D s ] sin ( θ ) D c − D s sin ( θ ) , A yy = − D s sin ( θ ) D c − D s sin ( θ ) , A zz = [ D s sin ( θ ) + D c cos ( θ )] D c − D s sin ( θ ) , (10) θ is the scattering angle defined in the c.m.frame, and D c = s | G M | , D s = s | G M | − M | G E | . In Fig. 5, we present ournumerical results of the single polarization observable A y vs √ s , which is dependent on the the imaginary of product of G M G ∗ E . As a demonstration, in the calculation we fixed thescattering angle θ = ◦ The prediction shows that the shapeof A y is similar to that of the relative phase ∆Φ in Fig. 4, sinceIm(G M G ∗ E ) is proportional to ∼ sin( ∆Φ ). This indicates thatexact information of ∆Φ could be obtained from the precisemeasurement of the single spin polarization A y . In addition,we plot the double polarization observables A xz , A xx , A yy and A zz vs √ s in Fig. 6. It is found that in the near thresholdregion, the polarization observables changes drastically with √ s , while in the large √ s region, the double polarization ob-servables almost remain unchanged. The shape and the √ s -dependence of the polarization observables should be sensi-tive to the model assumptions on EMFFs. Therefore, precisemeasurements on these observables will be useful for exam-ining the validity of our model. C. Form factors in space-like region
The EMFFs of Λ c in the space-like region can be directlycalculated using Eqs. (5), (6) and the model parameters in Ta-ble II. We perform the numerical calculation on the space-like G M and G E of Λ c vs Q and present the results in the leftpanel of Fig. 7, which shows that the magnitude and the shapeof G E are similar to those of G M . A more clear picture aboutthe relative size of G E and G M can be revealed by the ratio µ Λ c G E / G M , as dipicted in the right panel of Fig. 7. It is foundthat in the region Q < Q > Q , which is similar to the case of the proton [11, 53].However, it is larger than the ratio of the proton EMFFs.Finally, using the space-like G E ( Q ) and G M ( Q ) of Λ c , we G E G M Q (GeV ) Q (GeV ) c G E / G M FIG. 7: The EMFFs of Λ c hyperon from the present estimation in space-like region. The solid black curve and the dotted red line on the leftare G E and G M respectively. The solid black curve on right is the ratio µ Λ c G E / G M . estimate the magnetic and charge radius of the Λ c defined by h r i E = − dG E ( Q ) dQ | Q = h r i M = − µ Λ c dG M ( Q ) dQ | Q = (11)and obtain the following results: h r i E = .
259 fm , h r i M = .
287 fm . (12)Our result for h r i E is less than the result h r i E ∼ . in Ref. [31] and the relativistic quark model result h r i E = . [54], but is larger than h r i E = .
117 fm from theheavy quark e ff ective theory [55]. IV. SUMMARY
In this work, we have investigated both the time-like andspace-like EMFFs of Λ c using a modified VMD model. Inthis model, the EMFFs are contributed by two parts. Oneis the intrinsic structure part, the other is meson clouds part.Similar to the Λ hyperon, the contributions from the isovec-tor components to the Dirac and Pauli form factors vanishdue to the isoscalar property of the Λ c hyperon. We havetaken into account all the isospin-singlet vector mesons be-low the Λ + c ¯ Λ − c threshold: ω (782), ω (1420), ω (1650), φ (1020), φ (1680), φ (2170), and ψ (1 S ), ψ (2 S ), ψ (3770), ψ (4040), ψ (4160), ψ (4415). The inclusion of these vector mesons and their widths can naturally produce the complex structure ofthe time-like EMFFs. Using the VMD model expressions forthe EMFFs of the Λ c , we have fit the Born cross section inreaction e + e − → Λ + c ¯ Λ − c to the data from the Belle and BE-SIII experiments. We have also included the ratio | G E / G M | measured by BESIII. We find that the modified VMD modelcan describe the Born cross section of e + e − → Λ + c Λ − c at Belleand BESIII simultaneously. The enhancement e ff ect near thethreshold of the Λ + c ¯ Λ − c pair is reproduced due to the contri-bution of the resonance states of the vector mesons. It isalso shown that the VMD model can qualitatively describethe ratio. We have presented the numerical results of the rel-ative phase ∆Φ , and predicted the single and double polar-ization observables in the process e + e − → Λ + c ¯ Λ − c . The mea-surement of these quantities could be used to verify the va-lidity of the model. Finally, we have extended the time-likeEMFFs to space-like region using the parameter values ob-tained from the fit. The numerical results show that the mag-nitudes and shapes of the G E and G M is rather similar. Fur-thermore, we obtain that the electric radius squared h r i E andmagnetic radius squared h r i M of Λ c hyperon are 0 .
259 fm and 0 .
297 fm , respectively. V. ACKNOWLEDGEMENTS
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